Designers' Guide to Eurocode 8 Design of Bridges for Earthquake Resistance (Designers' Guides to the Eurocodes)

DESIGNERS’ GUIDES TO THE EUROCODES

DESIGNERS’ GUIDE TO EUROCODE 8:DESIGN OF BRIDGES FOR EARTHQUAKERESISTANCEEN 1998-2

BASIL KOLIASDENCO S.A, Greece

MICHAEL N. FARDISUniversity of Patras, Greece

ALAIN PECKERGeodynamique et Structure, France

Series editorHaig Gulvanessian CBE

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Preface

Aim of the Designers’ GuideThis Designers’ Guide to EN 1998-2:2005 covers the rules for the seismic design of bridges,following in a loose way the contents of this EN Eurocode. It highlights its important pointswithout repeating them, providing comments and explanations for its application, as well asbackground information and worked-out examples. However, it does not elaborate everysingle clause in EN 1998-2:2005, neither does it follow strictly the sequence of its clauses.

Layout of this guideAll cross-references in this guide to sections, clauses, subclauses, paragraphs, annexes, figures,tables and expressions of EN 1998-2 and EN 1998-5 are in italic type, which is also usedwhere text from EN 1998-2 and EN 1998-5 has been directly reproduced (conversely, quotationsfrom other sources, including other Eurocodes, and cross-references to sections, etc., of thisguide, are in roman type). Numbers within square brackets after cross-references in themargin refer to Parts 1, 2 and 5 of EN 1998: EN 1998-1 [1], EN 1998-2 [2], EN 1998-5 [3]. Expres-sion numbers specific to this guide are prefixed by D (for Designers’ Guide), for example,Eq. (D3.1), to prevent confusion with expression numbers from EN 1998.

AcknowledgementsThis Designers’ Guide would not have been possible without the successful completion ofEN 1998-2:2005. Those involved in the process were:

g national delegates and national technical contacts to Subcommittee 8 of CEN/TC250g the Project Team of CEN/TC250/SC8 that worked for the conversion from the ENV to

the EN: namely PT4, convened by Alex Plakas.

v

Contents

Preface v

Aim of the Designer’s guide vLayout of this guide vAcknowledgements v

Chapter 1 Introduction and scope 1

1.1. Introduction 11.2. Scope of Eurocode 8 11.3. Scope of Eurocode 8 Part 2 21.4. Use of Eurocode 8 Part 2 with the other Eurocodes 21.5. Additional European standards to be used with EN 1998-2:2005 31.6. Assumptions 41.7. Distinction between principles and application rules 41.8. Terms and definitions – symbols 4

References 4

Chapter 2 Performance requirements and compliance criteria 5

2.1. Performance-based seismic design of bridges 52.2. Performance requirements for new bridges in Eurocode 8 72.3. Compliance criteria for the non-collapse requirement and

implementation 82.4. Exemption from the application of Eurocode 8 16

References 16

Chapter 3 Seismic actions and geotechnical aspects 19

3.1. Design seismic actions 193.2. Siting and foundation soils 293.3. Soil properties and parameters 303.4. Liquefaction, lateral spreading and related phenomena 32

References 36

Chapter 4 Conceptual design of bridges for earthquake resistance 37

4.1. Introduction 374.2. General rules for the conceptual design of earthquake-resistant bridges 384.3. The choice of connection between the piers and the deck 434.4. The piers 534.5. The abutments and their connection with the deck 594.6. The foundations 64

References 65

Chapter 5 Modelling and analysis of bridges for seismic design 67

5.1. Introduction: methods of analysis in Eurocode 8 675.2. The three components of the seismic action in the analysis 685.3. Design spectrum for elastic analysis 695.4. Behaviour factors for the analysis 695.5. Modal response spectrum analysis 735.6. Fundamental mode analysis (or ‘equivalent static’ analysis) 925.7. Torsional effects in linear analysis 985.8. Effective stiffness for the analysis 1005.9. Calculation of seismic displacement demands through linear analysis 1075.10. Nonlinear analysis 110

References 117

vii

Chapter 6 Verification and detailing of bridge components for earthquake resistance 119

6.1. Introduction 1196.2. Combination of gravity and other actions with the design seismic

action 1196.3. Verification procedure in design for ductility using linear analysis 1226.4. Capacity design of regions other than flexural plastic hinges in

bridges of ductile behaviour 1246.5. Overview of detailing and design rules for bridges with ductile or

limited ductile behaviour 1296.6. Verification and detailing of joints between ductile pier columns and

the deck or a foundation element 1296.7. Verifications in the context of design for ductility based on nonlinear

analysis 1326.8. Overlap and clearance lengths at movable joints 1356.9. Seismic links 1406.10. Dimensioning of bearings 1426.11. Verification of abutments 1556.12. Verification of the foundation 1596.13. Liquefaction and lateral spreading 164

References 167

Chapter 7 Bridges with seismic isolation 171

7.1. Introduction 1717.2. Objective, means, performance requirements and conceptual design 1717.3. Design seismic action 1747.4. Behaviour families of the most common isolators 1747.5. Analysis methods 1857.6. Lateral restoring capability 191

References 191

Chapter 8 Seismic design examples 193

8.1. Introduction 1938.2. Example of a bridge with ductile piers 1938.3. Example of a bridge with limited ductile piers 2108.4. Example of seismic isolation 221References 249

Index 251

viii

Designers’ Guide to Eurocode 8: Design of Bridges for Earthquake ResistanceISBN 978-0-7277-5735-7

ICE Publishing: All rights reserved

http://dx.doi.org/10.1680/dber.57357.001

Chapter 1

Introduction and scope

1.1. IntroductionDesign of structures for earthquake resistance penetrated engineering practice for buildingsmuch earlier than for bridges. There are several reasons for this. First, seismic design is of rel-evance mainly for piers, but is secondary for the deck. The deck, though, receives in generalfar more attention than the piers, as it is more important for the function and the overall costof the bridge, while its engineering is also more challenging. So, seismic considerations, beingrelevant mainly for the less important components of bridges, have traditionally been of lowerpriority. Second, a good number of bridges are not so sensitive to earthquakes: the long-spanones – which are also the subject of lots of attention and of major design and engineeringeffort – are very flexible, and their long periods of vibration are outside the frequency range ofusual ground motions. At the other extreme, short bridges, with one or only few spans, oftenfollow the ground motion with little distress, and normally suffer only minor damage.However, with the very rapid expansion of transportation networks, the new priorities in landuse – especially in urban areas – and the sensitivities of recent times to protection of the environ-ment, bridge engineering has spread from the traditional field of short crossings of rivers, ravinesor other natural barriers or of over- and underpasses for motorways to long viaducts consistingof a large number of spans on equally numerous piers, often crossing territories with differentground or soil conditions. The heavy damage suffered by such types of engineering works inthe earthquakes of Loma Prieta in 1989 and Kobe in 1995 demonstrated their seismic vulner-ability. More recent events have confirmed the importance of proper seismic design (or lack ofit) for bridge projects.

Owing to these developments, recent decades have seen major advances in the seismic engineeringof bridges. It may now be claimed with a certain amount of confidence that the state-of-the-art inthe seismic design of bridges is catching up with that of buildings, which is more deeply rooted incommon design practice and codes. Europe, where even the moderate-to-high seismicitycountries of the south lacked modern seismic design codes for bridges, has seen the developmentof EN 1998-2:2005 as a modern and complete seismic design standard, on par with its counter-parts in California, Japan and New Zealand. Part 2 of Eurocode 8 (CEN, 2005a) is quiteadvanced from the point of view of the state-of-the-art and of seismic protection technology,not only compared with the pre-existing status at national levels but also with respect to theother parts of the new European seismic design standard (EN Eurocode 8) that address othertypes of civil engineering works. It is up to the European community of seismic designpractice to make good use of it, to the benefit of the seismic protection of new bridges inEurope and of its own professional competiveness in other seismic parts of the world. ThisDesigners’ Guide aspires to help this community become familiar with Part 2 of Eurocode 8,get the most out of it and apply it in a cost-effective way.

1.2. Scope of Eurocode 8Eurocode 8 covers the design and construction of earthquake-resistant buildings and othercivil engineering works – including bridges, but excluding nuclear power plants, offshore struc-tures and large dams. Its stated aim is to protect human life and property in the event of anearthquake and to ensure that structures that are important for civil protection remainoperational.

Eurocode 8 has six Parts, listed in Table 1.1. Among them, only Part 2 (CEN, 2005a) is covered inthis Designers’ Guide.

Clauses 1.1.1(1),

1.1.1(2) [1]

Clause 1.1.1(1) [2]

Clauses 1.1.1(4),

1.1.3(1) [1]

1

Clauses 1.1.1(2)–

1.1.1(4), 1.1.1(6) [2]

Clauses 1.1.2(1)–

1.1.2(3) [1]

Clauses 1.1(1), 1.1(2)

[3]

Clause 1.2.1 [1,2]

Clauses 1.2.2, 1.2.4 [2]

1.3. Scope of Eurocode 8 Part 2Part 2 of Eurocode 8 (CEN, 2005a) has as its sole object the seismic design of new bridges. Itfocuses on bridges having a deck superstructure supported directly on vertical or nearlyvertical concrete or steel piers and abutments. The seismic design of cable-stayed or archedbridges is only partly covered, while that of suspension bridges, timber bridges (strictlyspeaking, bridges on timber piers), masonry bridges, moveable bridges or floating bridges isnot covered at all. Part 2 of Eurocode 8 also covers the design of bridges with seismic isolation.

Unlike existing buildings, whose seismic assessment and retrofitting is covered in Eurocode 8(CEN, 2005b), existing bridges are not addressed at all.

1.4. Use of Eurocode 8 Part 2 with the other EurocodesPart 2 of Eurocode 8 builds on the general provisions of Part 1 (CEN, 2004b) for:

g the general performance requirementsg seismic actiong analysis methods and procedures applicable to all types of structures.

All the general or specific provisions of Part 5 of Eurocode 8 (CEN, 2004a) regarding:

g siting of the worksg properties and seismic verification of the foundation soilg seismic design of the foundation or of earth-retaining structuresg seismic soil–structure interaction

apply as well.

Eurocode 8 is not a stand-alone code. It is applied alongside the other relevant Eurocodes in apackage referring to a specific type of civil engineering structure and construction material.For bridges, there are four Eurocode packages:

g 2/2: Concrete bridgesg 3/2: Steel bridgesg 4/2: Composite bridgesg 5/2: Timber bridges.

To be self-sufficient, each package includes all the Eurocode parts needed for design, asfollows:

g Several EN Eurocodes are included in every single bridge package:– EN 1990: ‘Basis of structural design’ (including Annex A2: ‘Application for bridges’)– EN 1991-1-1: ‘Actions on structures – General actions – Densities, Self weight and

Imposed loads for buildings’– EN 1991-1-3: ‘Actions on structures – General actions – Snow loads’

Designers’ Guide to Eurocode 8: Design of Bridges for Earthquake Resistance

Table 1.1. Eurocode 8 parts

Part EN Title

1 EN 1998-1:2004 Design of structures for earthquake resistance. General rules,

seismic actions, rules for buildings

2 EN 1998-2:2005 Design of structures for earthquake resistance. Bridges

3 EN 1998-3:2005 Design of structures for earthquake resistance. Assessment and

retrofitting of buildings

4 EN 1998-4:2006 Design of structures for earthquake resistance. Silos, tanks, pipelines

5 EN 1998-5:2004 Design of structures for earthquake resistance. Foundations,

retaining structures, geotechnical aspects

6 EN 1998-6:2005 Design of structures for earthquake resistance. Towers, masts, chimneys

2

– EN 1991-1-4: ‘Actions on structures – General actions – Wind actions’– EN 1991-1-5: ‘Actions on structures – General actions – Thermal actions’– EN 1991-1-6: ‘Actions on structures – General actions – Actions during execution’– EN 1991-1-7: ‘Actions on structures – General actions – Accidental actions’– EN 1991-2: ‘Actions on structures – Traffic loads on bridges’– EN 1997-1: ‘Geotechnical Design – General rules’– EN 1997-2: ‘Geotechnical Design – Ground investigation and testing’– EN 1998-1: ‘Design of structures for earthquake resistance – General rules, seismic

actions, rules for buildings’– EN 1998-2: ‘Design of structures for earthquake resistance – Bridges’– EN 1998-5: ‘Design of structures for earthquake resistance – Foundations, retaining

structures, geotechnical aspects’.g Additional EN-Eurocodes are included in the Concrete Bridges package (2/2):

– EN 1992-1-1: ‘Design of concrete structures – General – General rules and rules forbuildings’

– EN 1992-2: ‘Design of concrete structures – Concrete bridges – Design and detailingrules’.

g Additional EN-Eurocodes included in the Steel Bridges package (3/2):– EN 1993-1-1: ‘Design of steel structures – General rules and rules for buildings’– EN 1993-1-5: ‘Design of steel structures – Plated structural elements’– EN 1993-1-7: ‘Design of steel structures – Strength and stability of planar plated

structures subject to out of plane loading’– EN 1993-1-8: ‘Design of steel structures – Design of joints’– EN 1993-1-9: ‘Design of steel structures – Fatigue’– EN 1993-1-10: ‘Design of steel structures – Selection of steel for fracture toughness and

through-thickness properties’– EN 1993-1-11: ‘Design of steel structures – Design of structures with tension

components’– EN 1993-2: ‘Design of steel structures – Steel bridges’.

g EN-Eurocodes which are included in addition in the Composite Bridges package (4/2) are:– EN 1992-1-1: ‘Design of concrete structures – General – General rules and rules for

buildings’– EN 1992-2: ‘Design of concrete structures – Concrete bridges – Design and detailing

rules’– EN 1993-1-1: ‘Design of steel structures –General rules and rules for buildings’– EN 1993-1-5: ‘Design of steel structures – Plated structural elements’– EN 1993-1-7: ‘Design of steel structures – Strength and stability of planar plated

structures subject to out of plane loading’– EN 1993-1-8: ‘Design of steel structures – Design of joints’– EN 1993-1-9: ‘Design of steel structures – Fatigue’– EN 1993-1-10: ‘Design of steel structures – Selection of steel for fracture toughness and

through-thickness properties’– EN 1993-1-11: ‘Design of steel structures – Design of structures with tension

components’– EN 1993-2: ‘Design of steel structures – Steel bridges’– EN 1994-1-1: ‘Design of composite steel and concrete structures – General rules and

rules for buildings’– EN 1994-2: ‘Design of composite steel and concrete structures – General rules and rules

for bridges’.

Although package 5/2, for timber bridges, does include Parts 1, 2 and 5 of Eurocode 8, EN 1998-2:2005 itself is not meant to cover timber bridges.

1.5. Additional European standards to be used with EN 1998-2:2005Part 2 of Eurocode 8 makes specific reference to the following product standards:

g EN 15129:2009: ‘Antiseismic Devices’g EN 1337-2:2000: ‘Structural bearings – Part 2: Sliding elements’g EN 1337-3:2005: ‘Structural bearings – Part 3: Elastomeric bearings’.

Clause 1.2.4 [2]

Chapter 1. Introduction and scope

3

Clauses 1.3(1), 1.3(2)

[1,2]

Clause 1.4 [1,2]

Clauses 1.5, 1.6 [2]

Although EN 1337-5 ‘Structural bearings – Part 5: Pot bearings’ is not specifically referenced, it isalso to be used, as relevant.

1.6. AssumptionsEurocode 8 refers to EN 1990 (CEN, 2002) for general assumptions, so reference is made here toDesigners’ Guides to other Eurocodes for elaboration. Also, Eurocode 8 adds the condition thatno change to the structure (not even one that increases the force resistance of members) shouldtake place during execution or afterwards without proper justification and verification.

1.7. Distinction between principles and application rulesEurocode 8 refers to EN 1990 for the distinction between principles and application rules.Accordingly, reference is made here also to Designers’ Guides to other Eurocodes for elabor-ation. It is noted, though, that, in practice, the distinction between principles and applicationrules is immaterial, as all provisions of the normative text are mandatory: non-conformity to asingle application rule disqualifies the entire design from being considered to accord with theEN Eurocodes.

1.8. Terms and definitions – symbolsTerms and symbols are defined in the various chapters of this Designers’ Guide wherever theyfirst appear.

REFERENCES

CEN (Comite Europeen de Normalisation) (2002) EN 1990: Eurocode – Basis of structural design

(including Annex A2: Application to bridges). CEN, Brussels.

CEN (2004a) EN 1998-5:2004 Eurocode 8 – Design of structures for earthquake resistance – Part 5:

Foundations, retaining structures, geotechnical aspects. CEN, Brussels.

CEN (2004b) EN 1998-1:2004. Eurocode 8 – Design of structures for earthquake resistance – Part 1:

General rules, seismic actions and rules for buildings. CEN, Brussels.

CEN (2005a) EN 1998-2:2005 Eurocode 8 – Design of structures for earthquake resistance – Part 2:

Bridges. CEN, Brussels.

CEN (2005b) EN 1998-3:2005 Eurocode 8 – Design of structures for earthquake resistance – Part 3:

Assessment and retrofitting of buildings. CEN, Brussels.


4




Chapter 2

Performance requirements andcompliance criteria

2.1. Performance-based seismic design of bridgesParaphrasing – for the particular purpose of the seismic design of bridges – the fib 2010 ModelCode ib, 2012) – the seismic performance of a bridge refers to its behaviour under seismic action:the bridge must be designed, constructed and maintained so that it adequately and in an econ-omically reasonable way performs in earthquakes that may take place during its constructionand service. More specifically, the bridge must:

g remain fit for the use for which it has been designedg withstand extreme, occasional and frequent seismic actions likely to occur during its

anticipated use and avoid damage by an exceptional earthquake to an extentdisproportionate to the triggering event

g contribute positively to the needs of humankind with regard to nature, society, economyand wellbeing.

Accordingly, three categories of performance are addressed by the fib 2010 Model Code(fib, 2012):

g Serviceability: the ability of the bridge and its structural components to perform, withappropriate levels of reliability, adequately for normal use after or even during seismicactions expected during its service life.

g Structural safety: the ability of the bridge and its structural components to guarantee theoverall stability, adequate deformability and ultimate load-bearing resistance,corresponding to occasional, extreme or exceptional seismic actions with appropriate levelsof reliability for the specified reference periods.

g Sustainability: the ability of the bridge to contribute positively to the fulfilment of thepresent needs of humankind with respect to nature, society and people, withoutcompromising the ability of future generations to meet their needs in a similar manner.

In performance-based design, the bridge is designed to perform in a required manner during itsentire life cycle, with performance evaluated by verifying its behaviour against specified require-ments, based in turn on stakeholders’ demands for the bridge performance and required servicelife. Performance-based design of a new bridge is completed when it has been shown that theperformance requirements are satisfied for all relevant aspects of performance related to service-ability, structural safety and sustainability. If the performance of a structure or a structuralcomponent is considered to be inadequate, we say we have ‘failure’.

The Eurocodes introduce limit states to carry out performance-based design for serviceabilityand safety (CEN, 2002). Limit states mark the boundary between desired and undesirable struc-tural performance of the whole structure or a component: beyond a limit state, one or more per-formance requirements are no longer met. For the particular case of seismic design, limit statesare defined conceptually for all transient situations in the service life or the execution of thebridge during which the earthquake acts in combination with any relevant persistent or transientactions or environmental influences. They correspond to discrete representations of the structuralresponse under a specified exposure for which specific losses/damages can be associated. Inpractice, they use simplified models for the exposure and the structural response (fib, 2012).

5

The Eurocodes recognise (CEN, 2002):

g serviceability limit states (SLSs)g ultimate limit states (ULSs).

SLSs are those beyond which specified requirements for the bridge or its structural componentsrelated to its normal use are no longer met. If they entail permanent local damage or permanentunacceptable deformations, the outcome of their exceedance is irreversible. It is considered to beserviceability failure, and may require repair to reinstate fitness for use. According to the fib 2010Model Code (fib, 2012), in seismic design at least one – but sometimes two – SLSs must beexplicitly considered, each one for a different representative value of the seismic action:

g The operational (OP) limit state: the facility (bridge or any other construction work)satisfies the operational limit state criteria if it has suffered practically no damage and cancontinue serving its original intention with little disruption of use for repairs; any repair, ifneeded, can be deferred to the future without disruption of normal use.

g The immediate use (IU) limit state: the facility satisfies this if all of the followingconditions apply:– the structure itself is very lightly damaged (i.e. localised yielding of reinforcement,

cracking or local spalling of concrete, without residual drifts or other permanentstructural deformations)

– the normal use of the facility is temporarily but safely interrupted– risk to life is negligible– the structure retains fully its earlier strength and stiffness and its ability to withstand

loading– the (minor) damage of non-structural components and systems can be easily and

economically repaired at a later stage.

ULSs are limit states associated with the various modes of structural collapse or stages close to it,which for practical purposes are also considered as a ULS. Exceedance of a ULS is almost alwaysirreversible; the first time it occurs it causes inadequate structural safety, that is, failure. ULSsaddress (CEN, 2002; fib, 2012):

g life safetyg protection of the structure.

In seismic design, ULSs that may require consideration include (fib, 2012):

g reduction of residual resistance below a certain limitg permanent deformations exceeding a certain limitg loss of equilibrium of the structure or part of it, considered as a rigid body (e.g.

overturning)g sliding beyond a certain limit or overturning.

In seismic design there may be several ULSs, with different consequences of limit state failure,high or medium. According to the fib 2010 Model Code (fib, 2012), in seismic design at leastone – but normally both – of the following ULSs must be explicitly considered, each one for adifferent representative value of the seismic action:

g The life safety (LS) limit state: this is reached if any of the following conditions are met(but not surpassed):– the structure is significantly damaged, but does not collapse, not even partly, retaining

its integrity– the structure does not provide sufficient safety for normal use, although it is safe enough

for temporary use– secondary or non-structural components are seriously damaged, but do not obstruct

emergency use or cause life-threatening injuries by falling down– the structure is on the verge of losing capacity, although it retains sufficient load-bearing

capacity and sufficient residual strength and stiffness to protect life for the period untilthe repair is completed


6

– repair is economically questionable and demolition may be preferable.g The near-collapse (NC) limit state: this is reached if any of the following conditions are

met:– the structure is heavily damaged and is at the verge of collapse– although life safety is mostly ensured during the loading event, it is not fully guaranteed

as there may be life-threatening injury situations due to falling debris– the structure is unsafe even for emergency use, and would probably not survive

additional loading– the structure presents low residual strength and stiffness but is still able to support the

quasi-permanent loads.

A representative seismic action, with a prescribed probability of not being exceeded during thedesign service life, should be defined for each limit state considered. According to the fib 2010Model Code (fib, 2012), multiple representative seismic actions appropriate for ordinary facilitiesare:

g For the operational (OP) limit state: a ‘frequent’ seismic action, expected to be exceeded atleast once during the design service life (i.e. having a mean return period much shorterthan the design service life).

g For immediate use (IU): an ‘occasional’ earthquake, not expected to be exceeded duringthe design service life (e.g. with a mean return period about twice the design service life).

g For life safety (LS): a ‘rare’ seismic action, with a low probability of being exceeded (10%)during the design service life.

g For near-collapse (NC): a ‘very rare’ seismic action, with very low probability of beingexceeded (2–5%) in the design service life of the structure.

For facilities whose consequences of failure are very high, the ‘very rare’ seismic action may beappropriate for the life safety limit state. For those which are essential for the immediate post-earthquake period, a ‘rare’ seismic action may be appropriate for the immediate use or eventhe operational limit state (fib, 2012).

A fully fledged performance-based seismic design of a bridge as outlined above for the case of thefib 2010 Model Code (fib, 2012) will serve well the interests and objectives of owners, in that itallows explicit verification of performance levels related to different level of operation (includingloss) of the bridge under frequent, occasional, rare or quite exceptional earthquakes. However,the design process may become too complex and cumbersome. Therefore, even the fib 2010Model Code (fib, 2012) recognises that, depending on the use and importance of the facility,competent authorities will choose how many and which limit states should be verified at aminimum and which representative seismic action they will be paired with. The seismic designof a bridge, or at least certain of its aspects, may be conditioned by just one of these limitstates. However, this may hold on a site-specific but not on a general basis, because the seismicityof the site controls the relative magnitude of the representative seismic actions for which themultiple limit states should be verified.

In closing this discussion on the performance-based design of bridges, a comment is required onsustainability performance: it is not explicitly addressed in the first generation of Eurocodes, butwill be in the next one, as the European Union recently added ‘Sustainable use of resources’to the two essential requirements of ‘Mechanical resistance and stability’ and ‘Resistance tofire’ for construction products that must be served by the Eurocodes. The fib 2010 ModelCode (fib, 2012), which has raised sustainability performance to the same level as serviceabilityand structural safety, speaks about it still in rather general terms. At any rate, the sustainability-conscious bridge designer should cater in the conceptual design phase for aesthetics and theminimisation of environmental impact (including during execution) and during all phases,from concept to detailed design, for savings in materials.

2.2. Performance requirements for new bridges in Eurocode 8Part 2 of Eurocode 8 (CEN, 2005) requires a single-level seismic design of new bridges with thefollowing explicit performance objective:

Clauses 2.1(1),

2.2.2(1), 2.2.2(4) [2]

Chapter 2. Performance requirements and compliance criteria

7

Clause 2.1(1) [1,2]

Clause 2.2.2(5) [2]

Clauses 2.1(2)–2.1(6)

[2]

Clause 3.2.1(3) [1]

Clauses 2.2.1(1),

2.2.3(1), 2.3.1(1) [2]

Clauses 2.3.4(1),

2.3.4(2) [2]

Clauses 2.2.2(3),

2.3.2.2(4) [2]

Clause 2.2.2(5) [2]

g The bridge must retain its structural integrity and have sufficient residual resistance to beused for emergency traffic without any repair after a rare seismic event – the ‘designseismic action’ explicitly defined in Parts 1 and 2 of Eurocode 8; any damage due to thisevent must be easily repairable.

Although called a ‘non-collapse requirement’, in reality this corresponds to the life safety, ratherthan to the near-collapse, limit state of the general framework of performance-based seismicdesign outlined in the previous section, since sufficient residual resistance has to be availableafter the design seismic event for immediate use by emergency traffic.

As we will see in more detail in Section 3.12.2 of this Guide, the ‘design seismic action’ ofstructures of ordinary importance is called the ‘reference seismic action’; its mean returnperiod is the ‘reference return period’, denoted by TNCR. Eurocode 8 recommends basingthe determination of the ‘design seismic action’ on a 10% exceedance probability in 50 years,corresponding to a ‘reference return period’ of 475 years.

If the seismicity is low, the probability of exceedance of the ‘design seismic action’ during thedesign life of the bridge may be well below 10%, and at any rate difficult to quantify. Forsuch cases, Eurocode 8 allows for consideration of the seismic action as an ‘accidental action’;also, in these cases it tolerates more damage to the bridge deck and secondary components, aswell to the bridge parts intended for controlled damage under the ‘design seismic action’.

Again as detailed in Section 3.12.2 of this Guide, Eurocode 8 pursues enhanced performance forbridges that are vital for communications in the region or very important for public safety, not byupgrading the performance level, as suits the general framework of performance-based seismicdesign delineated in the previous section, but by modifying the hazard level (increasing themean return period) for the ‘design seismic action’ under which the ‘non-collapse requirement’is met. This is done by multiplying the ‘reference seismic action’ by the ‘importance factor’ gI,which by definition is gI ¼ 1.0 for bridges of ordinary importance (i.e. for the reference returnperiod of the seismic action).

Part 2 of Eurocode 8 calls also for the limitation of damage under a loosely defined seismic actionwith a high probability of exceedance; such damage must be minor and limited only to secondarycomponents and to the parts of the bridge intended for controlled damage under the ‘designseismic action’. However, this requirement is of no practical consequence for design: it ispresumed to be implicitly fulfilled if all the criteria for compliance with the ‘non-collapserequirement’ above are checked and met. This should be contrasted with new buildings, forwhich Part 1 of Eurocode 8 (CEN, 2004) provides explicit checks under a well-defined‘damage limitation’ seismic action. However, these damage checks (inter-storey drifts)normally refer to non-structural elements that are not present in bridges.

Although not explicitly stated, an additional performance requirement for bridges designed toface the ‘design seismic action’ by means of ductility and energy dissipation is the preventionof the near-collapse limit state in an extreme and very rare, as yet undefined, earthquake.This implicit performance objective is pursued through systematic and across-the-board applica-tion of the capacity design concept, which allows full control of the inelastic responsemechanism.

2.3. Compliance criteria for the non-collapse requirement andimplementation

2.3.1 Design options to meet the bridge performance requirementsFor continued use after the ‘design seismic action’ (e.g. by emergency traffic), the deck of thebridge must remain in the elastic range. Damage should be local and limited to non-structuralor secondary components, such as expansion joints, parapets or concrete slabs providing top-slab continuity between adjacent simply-supported spans, most often built of precast concretegirders. The latter may yield during bending of the deck in the transverse direction.

If the seismic action is considered in the National Annex as ‘accidental’, because the probabilityof exceedance of the ‘design seismic action’ during the design life of the bridge is well below 10%


8

or undefined, Eurocode 8 allows as an exemption some inelastic action in and damage to thebridge deck.

It is today commonplace that the earthquake represents for the structure a demand to accom-modate imposed dynamic displacements – primarily in the horizontal direction – and notforces. Seismic damage results from them. The prime aim of seismic design is to accommodatethese horizontal displacements with controlled damage. The simple structural system ofbridges lends itself to the following options:

1 To place the deck on a system of sliding or horizontally flexible bearings (or bearing-typedevices) at the top of the substructure (the abutments and all piers) and accommodate thehorizontal displacements at this interface.

2 To fix or rigidly connect horizontally the deck to the top of at least one pier but let it slideor move on flexible bearings at all other supports (including the abutments). The piers thatare rigidly connected to the deck are required to accommodate the seismic horizontaldisplacements by bending. These piers develop inelastic rotations in flexural ‘plastichinges’, if they are not tall and flexible enough to accommodate the horizontaldisplacements elastically.

3 To accommodate (most of ) the seismic horizontal displacements in the foundation and thesoil, either through sliding at the base of piers or through inelastic deformations of soil–pile systems of the foundation.

4 To rigidly connect the deck with the abutments (either monolithically or via fixed bearingsor links) into an integral system that follows the ground motion with little additionaldeformation of its own. It then makes little difference if any intermediate piers are alsointegral with the deck or support it on bearings.

Option 4 (usually termed ‘integral bridges’) is encountered only in relatively short bridges withone or very few spans. It is dealt with in Section 5.4 of this Designers’ Guide as a special case.

Part 5 of Eurocode 8 explicitly allows horizontal sliding of footings with respect to the soil (aslong as residual rotation about horizontal axes and overturning are controlled), but this is anunconventional design option adopted for major bridges, notably the 2.45 km continuous-deck Rion-Antirrion bridge with a design ground acceleration of 0.48g. For typical bridges, anon-reversible sliding of one foundation support may entail serious problems. Part 5 ofEurocode 8, as well as Part 2, also allows inelastic deformations in foundation piles. This maybe the only viable option if the deck is monolithic with strong and rigid wall-like piers placedtransverse to the bridge axis.

Most common in practice are options 1 and 2, which are therefore considered as the two funda-mental options for the seismic design of bridges. Option 1 is considered in Part 2 of Eurocode 8as full seismic isolation, with the piers designed to remain elastic during the ‘design seismicaction’.

In option 2, the piers are normally designed to respond well into the inelastic range, mobilisingductility and energy dissipation to withstand the seismic action. Design based on ductility andenergy dissipation capacity is seismic design par excellence. It is at the core of Part 2 ofEurocode 8, where it is called ‘design for ductile behaviour’, as well as of this Designers’ Guide.

Ductility and energy dissipation under the ‘design seismic action’ is entrusted by Part 2 ofEurocode 8 to the piers, and is understood to entail a certain degree of damage at the plastichinges (spalling of the unconfined concrete shell outside the confining hoops, but no bucklingor fracture of bars, nor crushing of confined concrete inside the hoops). However, as thisdamage is meant to be reparable, it should be limited to easily accessible parts of the pier.Parts above the normal water level (be it in a sleeve or casing) are ideal. Those at a shallowdepth below grade but above the normal water table are also accessible. Those embeddeddeeper in fill but above the normal water level are still accessible but with increased difficulty.Part 2 of Eurocode 8 does not distinguish in great detail between these cases. It considers,though, as accessible the base of a pier deep in backfill but as inaccessible parts of the pierwhich are deep in water, or piles under large pile-caps; to reduce damage in such regions

Clauses 2.4(3), 2.4(4),

6.6.2.3(1) [2]

Clauses 4.1.6(9),

4.1.6(10) [2]

Clauses 5.4.1.1(7),

5.4.2(7) [3]

Clause 4.1.6(7) [2]

Clauses 2.3.2.1(10),

4.1.6(11) [2]

Clauses 2.2.2(2),

2.2.2(4), 2.3.2.2(1),

2.3.2.2(2), 2.3.2.2(7),

4.1.6(6) [2]

Clauses 2.2.2(4),

2.3.2.2(3) [2]


9

Clauses 2.3.5.2(1),

2.3.5.2(2), 2.3.6.1(8)

[2]

under the ‘design seismic action’, it divides seismic design forces by 0.6 should plastic hinges formthere.

2.3.2 Design of bridges for energy dissipation and ductility2.3.2.1 IntroductionSection 2.3.2 refers to one of the two fundamental options for the seismic design of bridges,namely to option 2: that of fixing horizontally the deck to the top of at least one pier but tolet it slide at the abutments and accommodate the seismic horizontal displacements throughbending of the piers, with ductile and dissipative flexural ‘plastic hinges’ forming at theirends.

2.3.2.2 Design of the bridge as a whole for energy dissipation and ductilityIt has already been pointed out that the earthquake is a dynamic action, representing for a struc-ture a requirement to sustain certain displacements and deformations and not specific forces.Eurocode 8 allows bridges to develop significant inelastic deformations under the designseismic action, provided that the integrity of individual components and of the bridge as awhole is not jeopardised. Design of a bridge to Eurocode 8 for the non-collapse requirementunder the ‘design seismic action’ is force-based, nonetheless.

The foundation of force-based seismic design for ductility and energy dissipation is the inelasticresponse spectrum of a single-degree-of-freedom (SDoF) system having an elastic–perfectlyplastic force–displacement curve, F� d, in monotonic loading. For given period, T, of theelastic SDoF system, the inelastic spectrum relates:

g the ratio q ¼ Fel/Fy of the peak force, Fel, that would have developed if the SDoF systemwere linear elastic, to the yield force of the system, Fy

g the maximum displacement demand of the inelastic SDoF system, dmax, expressed as ratioto the yield displacement, dy (i.e. as the displacement ductility factor, md ¼ dmax/dy).

Part 2 of Eurocode 8 has adopted a modification of the inelastic spectra proposed in Vidic et al.(1994):

md ¼ q if T � 1:25TC ðD2:1aÞ

md ¼ 1þ ðq� 1Þ 1:25TC

T� 5q� 4 if T , 1:25TC ðD2:1bÞ

md ¼ 1 if T , 0:033 s ðD2:1cÞ

where TC is the ‘transition period’ of the elastic spectrum, between its constant spectral pseudo-acceleration and constant spectral pseudo-velocity ranges (see Section 3.1.3). Equation (D2.1)expresses Newmark’s well-known ‘equal displacement rule’; that is, the empirical observationthat in the constant spectral pseudo-velocity range the peak displacement response of theinelastic and of the elastic SDoF systems are about the same.

With F being the total lateral force on the structure (the base shear, if the seismic action is in thehorizontal direction), the ratio q ¼ Fel/Fy is termed in Eurocode 8 the ‘behaviour factor’ (the‘force reduction factor’ or the ‘response modification factor’, R, in North America). It isused as a universal reduction factor on the internal forces that would develop in the elasticstructure for 5% damping, or, equivalently, on the seismic inertia forces that would developin this elastic structure and cause, in turn, the seismic internal forces. In this way, the seismicinternal forces for which the members of the structure should be dimensioned can be calculatedthrough linear-elastic analysis. In return, the structure must be provided with the capacity tosustain a peak global displacement at least equal to its global yield displacement multipliedby the displacement ductility factor, md, that corresponds to the value of q used for thereduction of elastic force demands (e.g. according to Eqs (D2.1)). This is termed the ‘ductilitycapacity’, or the ‘energy-dissipation capacity’ – as it has to develop through cyclic response inwhich the members and the structure as a whole dissipate part of the seismic energy inputthrough hysteresis.


10

2.3.2.3 Design of plastic hinges for energy dissipation and ductilityIn force-based seismic design for ductility and energy dissipation, flexural plastic hinges in piersare dimensioned and detailed to achieve a combination of force resistance and ductility thatprovides a safety factor between 1.5 and 2 against substantial loss of resistance to lateral (i.e.horizontal) load. To this end, they are first dimensioned to provide a design value of momentand axial force resistance Rd, at least equal to the corresponding action effects due to theseismic design situation, Ed, from the analysis:

Ed � Rd (D2.2)

The values of Ed in Eq. (D2.2) are due to the combination of the seismic action with thequasi-permanent gravity actions (i.e. the nominal permanent loads and the quasi-permanenttraffic loads, as pointed out in Section 5.4 in connection with Eq. (D5.6a) for the calculationof the deck mass). As linear analysis is normally applied, Ed may be found from superpositionof the seismic action effects from the analysis for the seismic action alone to the action effectsfrom that for the quasi-permanent gravity actions.

After having been dimensioned to meet Eq. (D2.2), flexural plastic hinges in piers are detailed toprovide the deformation and ductility capacity necessary to meet the deformation demands onthem from the design of the structure for the chosen q-factor value. The measure used for thedeformation and ductility capacity of flexural plastic hinges is the curvature ductility factor ofthe pier end section, whose supply-value is

mf ¼ fu/fy (D2.3)

where fy is the yield curvature of that section (computed from first principles) and fu its ultimatecurvature (again from first principles and the ultimate deformation criteria adopted forthe materials). At the other end, the global displacement demands are expressed through theglobal displacement ductility factor of the bridge, md, connected to the q factor used in thedesign of the bridge through the inelastic spectra, in this case Eqs (D2.1).

The intermediary between mf and md is the ductility factor of the chord rotation at the pier endwhere the plastic hinge forms, mu. Recall that the chord rotation u at a pier end is the deflectionof the inflexion point with respect to the tangent to the pier axis at the end of interest, divided bythe distance between these two points of the pier, termed the ‘shear span’ and denoted by Ls. So,the chord rotation u is a measure of member displacement, not of the relative rotation betweensections. If the pier is fixed at its base against rotation and supports the deck without the inter-vention of horizontally flexible bearings (i.e. if it is monolithically connected or supported onthe pier through fixed – e.g. pot – bearings), the chord rotation at the hinging end of the pieris related as follows to the deck displacement right above the pier top, d, in the commoncases of:

1 Pier columns monolithically connected at the top to a very stiff deck with near-fixity thereagainst rotation for seismic response in the longitudinal direction and inflexion point at thecolumn mid-height (see Section 5.4, Eqs (D5.4) if the deck cannot be considered as rigidcompared with the piers in the longitudinal direction); the horizontal displacement of thatpoint is one-half of that of the deck above, d, and the shear span, Ls, is about equal toone-half of the pier clear height, Hp; Ls � Hp/2; therefore, at the plastic hinges forming atboth ends of the pier, u � 0.5d/Ls ¼ d/Hp.

2 Multiple-column piers monolithically connected at the top to a very stiff deck or a capbeam with near-fixity there against rotation for seismic response in the transversedirection; the situation is similar to case 1 above, so in the transverse direction Ls � Hp/2and u � 0.5d/Ls ¼ d/Hp.

3 Piers supporting the deck through fixed (e.g. pot) bearings at the top and working asvertical cantilevers with a shear span Ls about equal to the pier clear height, Ls � Hp andu ¼ d/Ls ¼ d/Hp.

4 Single-column piers monolithic with the deck and working in the transverse direction ofthe bridge as vertical cantilevers; if the rotational inertia of the deck about its longitudinalaxis and the vertical distance between the pier top and the point of application of the deck

Clause 2.3.3(1) [2]

Clauses 2.3.5.1(1),

2.3.5.3(1), 2.3.5.3(2),

2.3.6.1(8), Annex B,

Annex E [2]


11

Clauses 2.3.5.4(1),

2.3.5.4(2) [2]

Clauses 2.3.3(1),

2.3.4(1), 2.3.4(2),

2.3.6.2(2) [2]

inertia force are neglected (see Section 5.4, Eqs (D5.5) for the case they are not), thesituation is similar to case 3 above, so in the transverse direction Ls � Hp andu ¼ d/Ls ¼ d/Hp.

In a well-designed bridge, all piers will yield at the same time, turning the bridge into a fullyfledged plastic mechanism. Then, in all cases 1 to 4 above, md ¼ d/dy will be (about) equal tomu ¼ u/uy:

md � mu (D2.4)

In a plastic hinge model of the inelastic deformation of the pier, all inelastic deformations arelumped in the plastic hinge, which is considered to have a finite length Lpl and to developconstant inelastic curvature all along Lpl. Then, for a linear bending moment diagram(constant shear force) along the pier, the chord rotation at its yielding end(s) is

u ¼ fy

Ls

3þ f� fy

� �Lpl 1� Lpl

2Ls

� �ðD2:5Þ

giving, for uy ¼ fyLs/3 (purely flexural elastic behaviour),

mu ¼ 1þ mf � 1� � 3Lpl

Ls

1� Lpl

2Ls

� �ðD2:6Þ

Equation (D2.6) is inverted as

mf ¼ fu

fy

¼ 1þ md � 1

3lð1� 0:5lÞ ðD2:7Þ

where l ¼ Lpl/Ls. Then, if the pier plastic hinge length Lpl is estimated through appropriateempirical relations, Eqs (D2.1) and (D2.7) translate the q factor used in the design of thebridge into a demand value for the curvature ductility factor of the piers. Note that Part 1 ofEurocode 8 has adopted for concrete members in buildings the following conservative approxi-mation of Eqs (D2.6) and (D2.7):

mu ¼ 1þ 0.5(mf� 1) i.e. mf ¼ 2mu� 1 (D2.8)

dating from the ENV version of the concrete buildings part of Eurocode 8 (ENV 1998-1-3:1994).

If linear analysis is used alongside the design spectrum involving the q factor, the required valueof the curvature ductility factor of the piers is presumed to be provided if the detailing rules ofPart 2 of Eurocode 8 are applied, prescriptive or not. If nonlinear analysis is used instead, theinelastic chord rotation demands obtained from it are compared with appropriate designvalues of chord rotation capacities, obtained by setting f ¼ fu in Eq. (D2.5). Details are givenin Chapter 6.

2.3.2.4 Capacity design for the ductile global responseThe bridge’s seismic design determines how the (roughly) given peak global displacementdemand of the design seismic action is distributed to its various components. Eurocode 8 uses‘capacity design’ to direct and limit this demand only to those best suited to withstand it.

Capacity design imposes a hierarchy of strengths between adjacent components or regions, andbetween different mechanisms of load transfer in the same member, so that those items capable ofductile behaviour and hysteretic energy dissipation are the first ones to develop inelastic defor-mations. More importantly, they do so in a way that precludes the development of inelasticdeformations in any component, region or mechanism deemed incapable of ductile behaviourand hysteretic energy dissipation.

The components, regions thereof or mechanisms of force transfer to which the peak globaldisplacement and deformation energy demands are channelled by capacity design are selected,


12

taking into account the following aspects:

1 Their inherent ductility. Ductile components, regions thereof or mechanisms of forcetransfer are entrusted through ‘capacity design’ for inelastic deformations and energydissipation, while brittle ones are shielded from them. Flexure is a far more ductilemechanism of force transfer than shear, and can be made even more so through judiciouschoice of the level of axial force and the amount, distribution and ductility of longitudinaland transverse reinforcement.

2 The role of the component for the integrity of the whole and the fulfilment of theperformance requirements of the bridge. The foundation and connections betweencomponents (bearings, links, holding-down devices, etc.) securing structural integrity aremost important for the stability and integrity of the whole; the integrity of the deck itselfdetermines the continued operation of the bridge after the earthquake.

3 Accessibility and difficulty in inspecting and repairing any damage. Accessible regions ofthe piers (above the grade and the water level) are the easiest to repair without disruptionof traffic.

On the basis of the above aspects, a clear hierarchy of the bridge components and mechanisms offorce transfer emerges, determining the order in which they are allowed to enter the inelasticrange during the seismic response: the deck, the connections between components and thefoundation are to be shielded from inelastic action; the last is channelled to flexural plastichinges at accessible ends of the piers. Capacity design ensures that this order is indeed respected.As we will see in more detail later, it works as follows.

The required force resistance of the components, regions thereof or mechanisms of forcetransfer to be shielded from inelastic response is not determined from the analysis. Simplecalculations (normally on the basis of equilibrium alone) are used instead, assuming thatall relevant plastic hinges develop their moment resistances in a way that prevents pre-emptive attainment of the force resistance of the components, etc., to be shielded frominelastic action.

2.3.2.5 How elastic deformations in flexible bearings or the foundation groundaffect the ductility of the bridge

Assume that the deck is supported on a ductile pier that can develop a curvature ductility factormfo in the plastic hinge(s) and a chord rotation ductility factor muo, and has elastic lateralstiffness Kp if fixed at the base:

(a) For a single-column pier presenting flexural rigidity (EI )c in a vertical plane in thetransverse direction of the bridge and supporting a deck mass with a radius of gyrationrm,d about its centroidal axis (rm,d

2 ¼ ratio of tributary rotational mass moment of inertiaof the deck about the deck’s centroidal axis to the tributary deck mass):

Kp �3ðEIÞc

H9r2m;d

8Ls

þ Ls

!H þ ycg� �þ y2cg

" # ðD2:9aÞ

In Eq. (D2.9a), ycg is the distance from the soffit of the deck to the centroid of itssection, and Ls is the shear span at the pier base (see Eq. (D5.5) in Section 5.4 for thisparticular case).

(b) For a pier consisting of n � 1 columns, each one with height H and presenting the rigidity(EI )c within the plane of bending considered, all having the top fixed to the soffit of avery stiff deck:

Kp ¼ 12Sn(EI )c/H3 (D2.9b)

Single-column piers (n ¼ 1) in the transverse direction of the bridge are not addressed bythis case but by case 1 above and Eq. (D2.9a).

Annex B [2]


13

(c) For a pier as in case 2 above but with the top of its n � 1 columns pin-connected to thedeck, instead of being fixed to it:

Kp ¼ 3Sn(EI )c/H3 (D2.9c)

Assume also that one or more additional components intervene between the deck and the groundin series with the pier, all designed to remain elastic until and after the pier yields (e.g. throughcapacity design according to the previous section). The generic elastic stiffness of thesecomponents is denoted as Kel. Such components can be:

g the compliance of the ground – if the foundation itself has horizontal stiffness Kfh (baseshear divided by the horizontal displacement of the foundation) and rotational stiffnessKff (moment divided by the rotation at the pier base), giving horizontal stiffness at the topof the pier Kff/H

2 – and/org an elastic (e.g. elastomeric) bearing with horizontal stiffness Kb (Kb ¼ GA/t if the bearing

has horizontal section area A, and its material – the elastomer – has total thickness t andshear modulus G) – needless to say, this case does not combine with piers of case 2 above,which have their top fixed to the soffit of the deck.

The deck sees down below a total stiffness K such that

1

K¼ 1

Kp

þX 1

Kel

¼ 1

Kp

þ 1

Kb

þ 1

Kfh

þ H2

Kff

ðD2:10Þ

If the pier yields at a base shear Vy, the displacement of the deck at yielding is

dy ¼Vy

K¼ Vy

Kp

þX Vy

Kel

ðD2:11Þ

After the pier yields, additional horizontal displacements are due to the inelastic rotation(s) of itsplastic hinge(s) alone, giving an inelastic displacement of the deck:

mddy ¼ muo � 1� �Vy

Kp

þ dy ðD2:12Þ

where md is the global displacement ductility factor of the bridge at the level of the deck.According to Eqs (D2.4), (D2.6) and (D2.7), there is proportionality between (md� 1),(mu� 1) and (mf� 1). Therefore, Eqs (D2.9)–(D2.11) state that, to achieve the same targetvalue md of the global displacement ductility factor of the bridge at the level of the deck, thecurvature ductility demand at a plastic hinge of a pier should increase from mfo to

mf ¼ 1þ mfo � 1� �

1þX Kp

Kel

� �ðD2:13Þ

The horizontal stiffness of an elastic bearing is several times smaller than that of an ordinary pier.So, Eq. (D2.13) gives unduly large values of the curvature ductility demand that a plastic hinge inthe pier may have to bear for the bridge to achieve the q factor values normally used in theseismic design of bridges for ductility. So, if piers are intended to resist the design seismicaction through ductility and energy dissipation, elastic bearings have no place on top of them.By the same token, ductile piers should be nearly fixed to the ground: compliance of the foun-dation will penalise detailing of their plastic hinges for the target q factor of seismic design forductility.

The same conclusion can be reached through energy considerations. The pier is an assembly ofcomponents in series, only one of which (the pier shaft having stiffness Kp) possesses thecapability of hysteretic energy dissipation. The other components (elastic bearings at the piertop and foundation compliance, having a composite stiffness Kel) should remain within theelastic range. As the same shear force acts on all components in series, the strain energy


14

input in each component at any instant of the seismic response is proportional to theirflexibilities, 1/Kp and 1/Kel, respectively. When the portion of the energy input in the dissipativecomponent is small compared with the input in the series system, the dissipation capability ofthe whole assembly is also small. In other words, the behaviour of such assemblies becomespractically elastic.

2.3.3 Seismic design of bridges for strength instead of ductility: limited ductilebehaviour

Part 2 of Eurocode 8 gives the option to design a bridge to resist the seismic action throughstrength alone, without explicitly resorting to ductility and energy dissipation capacity. In thisoption, the bridge is designed:

g in accordance with Eurocodes 2, 3, 4 and 7, with the seismic action considered as a staticloading (like wind)

g without capacity design considerations, except for non-ductile connections or structuralcomponents (fixed bearings, sockets and anchorages of cables and stays), but

g observing:– some minimum requirements for the ductility of steel reinforcement or steel sections and

for confinement and bar anti-buckling restraint in potential plastic hinges of concrete piers– simplified rules for the ULS verification in shear.

The seismic lateral forces are derived from the design response spectrum using a behaviourfactor, q, not higher than the value of 1.5 attributed to material overstrength. In fact:

(a) if the bridge seismic response is dominated by upper modes (as in cable-stayed bridges) or(b) concrete piers have:

– axial force ratio hk ¼ Nd/Ac fck (axial load due to the design seismic action and theconcurrent gravity loads, Nd, normalised to product of the pier section area and thecharacteristic concrete strength, Ac fck), higher than or equal to 0.6, or

– shear-span ratio, Ls/h, in the direction of bending less than or equal to 1.0,

then the behaviour factor, q, is taken equal to 1.0.

As design seismic forces are derived with a value of the behaviour factor, q, possibly greater than1.0, structures designed for strength and little engineered ductility and energy dissipationcapacity are termed ‘limited ductile’, in lieu of ‘non-ductile’.

Part 2 of Eurocode 8 recommends (in a note) designing the bridge for ‘limited-ductile’ behaviourin cases of ‘low seismicity’ (see below), but does not discourage the designer from using thisoption in other cases as well. It specifies the option as the only possible one, no matterwhether the bridge is a ‘low-seismicity’ case or not, in two very specific but also quite commoncases:

1 when the deck is fully supported on a system of sliding or horizontally flexible bearings (orbearing-type devices) at the top of the substructure (the abutments and all piers), whichaccommodate the horizontal displacements (see option 1 in Section 2.3.1 and the influenceof non-dissipative components in Section 2.3.2.5 above), or

2 when the deck is rigidly connected to the abutments, monolithically or via fixed bearingsor links (listed as option 4 in Section 2.3.1 of this Guide).

2.3.4 The balance between strength and ductilityThe option described in Section 2.3.3 above, namely to design for strength alone withoutengineered ductility and energy dissipation capacity, is an extreme, specified by Part 2 ofEurocode 8 only for cases a and b and 1 and 2 well delineated in Section 2.3.3. Outside ofthese specific cases, the designer is normally given the option to opt for more strength and lessductility (i.e. for ‘limited-ductile’ behaviour) or vice versa (for ‘ductile’ behaviour).

Equations (D2.1) show that, except for short-period bridges, the magnitude of the design seismicforces decreases when the global displacement ductility factor, md, increases. So, there is an

Clauses 2.3.2.1(1),

2.3.2.2(1), 2.3.2.3(1),

2.3.3(2), 2.3.4(3),

2.3.5.4(3) [2]

Clauses 2.3.2.3(2),

4.1.6(3), 4.1.6(5) [2]

Clauses 2.3.7(1),

2.3.2.1(1), 2.4(2),

2.4(3), 4.1.6(3),

4.1.6(9)–4.1.6(11) [2]

Clause 2.3.2.1(1)


15

Clauses 3.2.1(4) [1]

Clause 2.3.7(1) [2]

Clauses 2.2.1(4),

3.2.1(5) [1]

apparent economic incentive to increase the available global ductility, to reduce the internalforces for which the components of the bridge are dimensioned. Moreover, if the lateral forceresistance of the bridge is reduced, by dividing the elastic lateral force demands by a high q-factor value, the verification of the foundation soil, which is done for strength rather than forductility and deformation capacity, is much easier. Last but not least, a bridge with ampleductility supply is less sensitive to the magnitude and the details of the seismic action, and, inview of the large uncertainty associated with the extreme seismic action in its lifetime, may bea better earthquake-resistant design.

On the other hand, there are strong arguments for less ductility and dissipation capacity inseismic design and more lateral force resistance instead. Ductility necessarily entails damage.So, the higher the lateral strength of the bridge, the smaller will be the structural damage, notonly during more frequent, moderate earthquakes but also due to the design seismic action.From the construction point of view, detailing piers for more strength is much easier andsimpler than detailing for higher ductility. Also, some bridge configurations may impart signifi-cant lateral force resistance. In others (notably when the deck is rigidly supported on tall andflexible piers), the dominant vibration modes may fall at the long-period tail of the spectrum,where design spectral accelerations may be small even for q � 1.5, and dimensioning the piersfor the resulting lateral force resistance may be trivial. Last but not least, if the bridge fallsoutside the framework of common structural configurations mainly addressed by Eurocode 8(e.g. as in arch bridges, or those having some inclined piers or piers of very different height,especially if the height does not increase monotonically from the abutments to mid-span), thedesigners may feel more confident if they narrow the gap between the results of the linear-elastic analysis, for which members are dimensioned and the nonlinear seismic response underthe design seismic action (i.e. if q � 1.5 is used).

2.3.5 The cases of ‘low seismicity’Eurocode 8 recommends in a note designing the bridge for ‘limited ductile’ behaviour if it falls inthe case of ‘low seismicity’. Although it leaves it to the National Annex to decide which com-bination of categories of structures, ground types and seismic zones in a country correspondto the characterisation as ‘cases of low seismicity’, it recommends in a note as a criterioneither the value of the design ground acceleration on type A ground (i.e. on rock), ag (whichincludes the importance factor gI), or the corresponding value, agS, over the ground type ofthe site (see Section 3.1.2.3 of this Guide for the soil factor, S). Moreover, it recommends avalue of 0.08g for ag, or of 0.10g for agS, as the threshold for the low-seismicity cases.

2.4. Exemption from the application of Eurocode 8Eurocode 8 itself states that its provisions need not be applied in ‘cases of very low seismicity’.As in ‘cases of low seismicity’, it leaves it to the National Annex to decide which combinationof category of structures, ground types and seismic zones in a country qualify as ‘cases of verylow seismicity’. It does recommend in a note, though, the same criterion as for the ‘cases oflow seismicity’: either the value of the design ground acceleration on type A ground (i.e. onrock), ag, or the corresponding value, agS, over the ground type of the site. It recommends avalue of 0.04g for ag, or of 0.05g for agS, as the threshold for the very low seismicity cases.Because the value of ag includes the importance factor gI, ordinary bridges in a region maybe exempted from the application of Eurocode 8, while more important ones may not be.This is consistent with the notion that the exemption from the application of Eurocode 8 isdue to the inherent lateral force resistance of any structure designed for non-seismic loadings,neglecting any contribution from ductility and energy dissipation capacity. Given thatEurocode 8 considers that, because of overstrength, any structure is permitted a behaviourfactor, q, of at least 1.5, implicit in the value of 0.05g for agS recommended as the ceiling forvery low seismicity cases is a presumed lateral force capacity of 0.05� 2.5/1.5 ¼ 0.083g.

If a National Annex states that the entire national territory is considered as a ‘case of very lowseismicity’, then Eurocode 8 (all six parts) does not apply at all in the country.

REFERENCES

CEN (Comite Europeen de Normalisation) (2002) EN 1990: Eurocode – Basis of structural design

(including Annex A2: Application to bridges). CEN, Brussels.


16

CEN (2004) EN 1998-1:2004: Eurocode 8 – Design of structures for earthquake resistance – Part 1:




fib (2012) Model Code 2010, vol. 1. fib Bulletin 65. Federation Internationale du Beton, Lausanne.

Vidic T, Fajfar P and Fischinger M (1994) Consistent inelastic design spectra: strength and

displacement. Earthquake Engineering and Structural Dynamics 23: 502–521.


17




Chapter 3

Seismic actions and geotechnicalaspects

3.1. Design seismic actions3.1.1 IntroductionAs pointed out in Section 2.2 of this Guide, Eurocode 8 entails a single-tier seismic design of newbridges, with verification of the no-(local-)collapse requirement under the ‘design seismic action’alone. So, whatever is said in Section 3.1 refers to the ‘design seismic action’ of the bridge. Note,however, that in its two-tier seismic design of buildings and other structures, Part 1 of Eurocode 8(CEN, 2004a) adopts the same spectral shape for the different seismic actions to be used fordifferent performance levels or limit states. The difference in the hazard level is reflected onlythrough the peak ground acceleration to which each spectrum is anchored.

The seismic action is considered to impart concurrent translational motion in three orthogonaldirections: the vertical and two horizontal ones at right angles to each other. The motion istaken to be applied at the interface between the structure and the ground. If springs are usedto model the soil compliance under and/or around spread footings, piles or shafts (caissons),the motion is considered to be applied at the soil end of these springs.

3.1.2 Elastic response spectra3.1.2.1 IntroductionThe reference representation of the seismic action in Eurocode 8 is through the responsespectrum of an elastic single-degree-of-freedom (SDoF) oscillator having a given viscousdamping ratio (with 5% being the reference value). Any other alternative representation of theseismic action (e.g. in the form of acceleration time histories) should conform to the elasticresponse spectrum for the specified value of the damping ratio.

Because:

g earthquake ground motions are traditionally recorded as acceleration time histories andg seismic design is still based on forces, conveniently derived from accelerations,

the pseudo-acceleration response spectrum, Sa(T ), is normally used. If spectral displacements,Sd(T ), are of interest, they can be obtained from Sa(T ), assuming simple harmonic oscillation:

Sd(T ) ¼ (T/2p)2Sa(T ) (D3.1)

Spectral pseudo-velocities can also be obtained from Sa(T ) as

Sv(T ) ¼ (T/2p)Sa(T ) (D3.2)

Note that pseudo-values do not correspond to the real peak spectral velocity or acceleration. Fora damping ratio of up to 10% and for a natural period T between 0.2 and 1.0 s, the pseudo-velocity spectrum closely approximates the actual relative velocity spectrum.

3.1.2.2 Design ground accelerations – importance classes for reliability differentiationIn Eurocode 8 the elastic response spectrum is taken as proportional (‘anchored’) to the peakacceleration of the ground:

Clause 2.1(1) [2]

Clauses 2.1(1),

2.2.1(1) [1]

Clauses 3.1.1(2),

3.1.2(1)–3.1.2(3) [2]

Clause 3.2.2.1(1) [1]

Clause 3.2.1(1) [2]

Clause 7.5.4(3),

Table 7.1 [2]

Clauses 3.2.1(2),

3.2.2.1(1),3.2.2.3(1)[1]

19

Clause 2.1(3) [2]

Clause 3.2.1(3) [1]

Clauses 2.1(4)–2.1(6)

[2]

Clause 2.1(3),

Annex A [2]

Clauses 2.1(1), 2.1(3),

2.1(4) [1]

g the horizontal peak acceleration, ag, for the horizontal component(s) of the seismic actionor

g the vertical peak acceleration, avg, for the vertical component.

The basis of the seismic design of new bridges in Eurocode 8 is the ‘design seismic action’, forwhich the no-(local-)collapse requirement should be met. It is specified through the ‘designground acceleration’ in the horizontal direction, ag, which is equal to the ‘reference peakground acceleration’ on rock from national zonation maps, multiplied by the importancefactor, gI, of the bridge:

ag ¼ gIagR (D3.3)

For bridges of ordinary importance (belonging to importance class II in Eurocode 8), bydefinition gI ¼ 1.0.

Eurocode 8 recommends classifying in importance class III any bridge that is crucial for com-munications, especially in the immediate post-earthquake period (including access to emergencyfacilities), or whose downtime may have a major economic or social impact, or which by failingmay cause large loss of life, as well as major bridges with a target design life longer than theordinary nominal life of 50 years. For importance class III, it recommends gI ¼ 1.3.

Bridges that are not critical for communications, or considered not economically justified todesign for the standard bridge design life of 50 years, are recommended by Eurocode 8 to beclassified in importance class I, with a recommended value gI ¼ 0.85.

The reference peak ground acceleration, agR, corresponds to the reference return period, TNCR,of the design seismic action for bridges of ordinary importance.

Note that, under the Poisson assumption of earthquake occurrence (i.e. that the number of earth-quakes in an interval of time depends only on the length of the interval in a time-invariant way),the return period, TR, of seismic events exceeding a certain threshold is related to the probabilitythat this threshold will be exceeded, P, in TL years as

TR ¼ –TL/ln(1� P) (D3.4)

So, for a given TL (e.g. the conventional design life of TL ¼ 50 years) the seismic action mayequivalently be specified either via its mean return period, TR, or its probability of exceedancein TL years, PR.

Values of the importance factor greater or less than 1.0 correspond to mean return periods longeror shorter, respectively, than TNCR. It is within the authority of each country to select the value ofTNCR that gives the appropriate trade-off between economy and public safety in its territory, aswell as the importance factors for bridges other than ordinary, taking into account the specificregional features of the seismic hazard. Part 1 of Eurocode 8 recommends the valueTNCR ¼ 475 years.

The mean return period, TR(ag), of a peak ground acceleration exceeding a value ag is the inverseof the annual rate, la(ag), of exceedance of this acceleration level:

TR(ag) ¼ 1/la(ag) (D3.5)

A functional form commonly used for la(ag) is

la(ag) ¼ Ko(ag)�k (D3.6)

If the exponent k (the slope of the ‘hazard curve’ la(ag) in a log-log plot) is approximatelyconstant, two peak ground acceleration levels ag1, ag2, corresponding to two different meanreturn periods, TR(ag1), TR(ag2), are related as

ag1

ag2¼ TRðag1Þ

TRðag2Þ� �1=k

ðD3:7Þ


20

The value of k characterises the seismicity of the site. Regions where the difference in the peakground acceleration of frequent and very rare seismic excitations is very large, have low kvalues (around 2). Large values of k (k . 4) are typical of regions where high ground accelerationlevels are almost as frequent as smaller ones.

Tall free-standing piers, decks built as free cantilevers or incrementally launched, etc., may bemuch more vulnerable to earthquake than after completion of the full bridge. It is up to thedesigner or the owner of the bridge to specify the seismic performance requirements before com-pletion of the project and the corresponding compliance and verification criteria. Equation(D3.4) may be used then to determine the mean return period of the seismic action that has agiven probability of being exceeded P (e.g. P ¼ 0.05) in the full duration of the bridge construc-tion, Tc, to be used in Eq. (D3.4) in lieu of TL. This mean return period may be used as TR(ag1) inEq. (D3.7), to compute the peak ground acceleration, ag1, with a probability P of been exceededduring construction. In that case, ag2 ¼ agR and TR(ag2) ¼ TNCR.

3.1.2.3 Horizontal elastic response spectrumThe Eurocode 8 spectra include ranges of:

g constant spectral pseudo-acceleration for natural periods between TB and TC

g constant spectral pseudo-velocity between periods TC and TD

g constant spectral displacement for periods longer than TD.

The elastic response spectral acceleration for any horizontal component of the seismic action isdescribed in Parts 1 and 2 of Eurocode 8 by the following expressions:

Short-period range:

0 � T � TB: Sa Tð Þ ¼ agS 1þ T

TB

2:5h� 1� ��

ðD3:8aÞ

Constant spectral pseudo-acceleration range:

TB � T � TC: Sa Tð Þ ¼ agS � 2:5h ðD3:8bÞ

Constant spectral pseudo-velocity range:

TC � T � TD: Sa Tð Þ ¼ agS � 2:5hTC

T

� �ðD3:8cÞ

Constant spectral displacement range:

TD � T � 4s: Sa Tð Þ ¼ agS � 2:5hTCTD

T2

� �ðD3:8dÞ

where ag is the design ground acceleration on rock and S is the ‘soil factor’.

h ¼ ffiffiffiffiffiffiffiffiffiffiffiffi10= 5þ jð Þp � 0:55 (Bommer and Elnashai, 1999) is a correction factor for viscous damping

ratio, j, other than the reference value of 5% (from Parts 1 and 2 of Eurocode 8); the valuej ¼ 5% is considered to be representative of cracked reinforced concrete. The viscousdamping values specified in Part 2 of Eurocode 8 for components of various structural materialsare shown in Table 3.1.

Note the uniform amplification of the entire spectrum by the soil factor, S, over the spectrum forrock. By definition, S ¼ 1 over rock. The value agS plays the role of effective ground acceleration,as the spectral acceleration at the constant spectral acceleration plateau is always equal to 2.5agS.

The values of the periods TB, TC and TD (i.e. the extent of the ranges of constant spectral pseudo-acceleration, pseudo-velocity and displacement) and of the soil factor, S, are taken to dependmainly on the ground type. In the Eurocodes the term ‘ground’ includes any type of soil and

Annex A [2]

Clauses 3.2.2.1(3),

3.2.2.2(1) [1]

Clause 3.2.2.2(3) [1]

Clauses 4.1.3(1),

7.5.4(3) [2]

Clauses 3.2.2.2(2),

3.1.2(1), [1]

Chapter 3. Seismic actions and geotechnical aspects

21

Clauses 3.1.2(1)–

3.1.2(3) [1]

Clause 3.1.2(4) [1]

Clause 3.2.2.2(2) [1]

rock. Parts 1 and 2 of Eurocode 8 recognise five standard ground types, over which it rec-ommends values for TB, TC, TD and S, and two special ones, as listed in Table 3.2.

The characterisation of the ground is based on the average value of the shear wave velocity, vs,30,at the top 30 m (CEN, 2004a):

vs;30 ¼30X

i¼ 1;N

hivi

ðD3:9Þ

where hi and vi are the thickness (in metres) and the shear wave velocity at small shear strains (lessthan 10�6) of the ith layer in N layers. If the value of vs,30 is not known, for soil types B, C or DPart 1 of Eurocode 8 allows the use of alternatives to characterise a soil: for cohesionless soilsespecially, the SPT (Standard Penetration Test) blow-count number may be used (e.g. accordingto the correspondence of SPT to vs,30 in Ohta and Goto (1976)); for cohesive soils, the undrainedcohesive resistance (cu).

The two special ground types, S1 and S2, require the carrying out of special site-specific studies todefine the seismic action (CEN, 2004a). For ground type S1, the study should take into accountthe thickness and the vs value of the soft clay or silt layer and the difference from the underlyingmaterials, and should quantify their effects on the elastic response spectrum. Note that soils oftype S1 may have low internal damping and exhibit linear behaviour over a large range ofstrains, producing peculiar amplification of the bedrock motion and unusual or abnormalsoil–structure interaction effects. The scope of the site-specific study should also address thepossibility of soil failure under the design seismic action (especially at ground type S2 depositswith liquefiable soils or sensitive clays) (CEN, 2004a).

The values of TB, TC, TD and S for the five standard ground types A to E are meant to be definedby each country in the National Annex to Eurocode 8, depending on the magnitude of earth-quakes contributing most to the hazard. The geological conditions at the site may also betaken into account in addition, to determine these values. In principle, S factors that decreasewith increasing spectral value because of the soil nonlinearity effect may be introduced.Instead of spectral amplification factors that decrease with increasing design acceleration


Table 3.1. Values of viscous damping for different structural materials

Material Damping: %

Reinforced concrete components 5

Pre-stressed concrete components 2

Welded steel components 2

Bolted steel components 4

Table 3.2. Ground types in Part 1 of Eurocode 8 for the definition of the seismic action

Description vs,30: m/s NSPT cu: kPa

A Rock outcrop, with less than 5 m cover of weaker material .800 – –

B Very dense sand or gravel, or very stiff clay, several tens of metres

deep; mechanical properties gradually increase with depth

360–800 .50 .250

C Dense to medium-dense sand or gravel, or stiff clay, several

tens to many hundreds of metres deep

180–360 15–50 70–250

D Loose-to-medium sand or gravel, or soft-to-firm clay ,180 ,15 ,70

E 5–20 m surface alluvium layer type C or D – underlain by stiffer

material (with vs . 800 m/s)

S1 .10 m thick soft clay or silt with plasticity index .40 and high

water content

,100 – 10–20

S2 Liquefiable soils; sensitive clays; any soil not of type A to E or S1

22

(spectral or ground) as in US codes, the non-binding recommendation of a note in Part 1 ofEurocode 8 is for two types of spectra:

g Type 1: for moderate- to large-magnitude earthquakes.g Type 2: for low-magnitude ones (e.g. with a surface magnitude less than 5.5) at close

distance, producing over soft soils motions rich in high frequencies.

The values of TB, TC, TD and S recommended in a non-binding note in Part 1 of Eurocode 8 forthe five standard ground types A to E are given in Table 3.3 They are based on Rey et al. (2002)and European strong motion data. There are certain regions in Europe (e.g. where the hazard iscontributed mainly by strong intermediate-depth earthquakes, as in the part of the easternBalkans affected by the Vrancea region) where the two recommended spectral shapes may notbe suitable. The lower S values of type 1 spectra are due to the larger soil nonlinearity in thestronger ground motions produced by moderate to large-magnitude earthquakes. Figure 3.1depicts the recommended type 1 spectral shape.

The values recommended in Part 1 of Eurocode 8 for the period TD at the outset of the constantspectral displacement region seem rather low. Indeed, for flexible structures, such as bridges withtall piers or supported only on movable bearings, they may not lead to safe-sided designs.Accordingly, Part 2 of Eurocode 8 calls the attention of designers and national authorities tothe fact that the safety of structures with seismic isolation depends mainly on the displacement

Clause 7.4.1(1) [2]

Clause 3.2.2.5(8) [1]


Table 3.3. Recommended parameter values in Part 1 of Eurocode 8 for standard horizontal elastic

response spectra

Ground type Spectrum type 1 Spectrum type 2

S TB: s TC: s TD: s S TB: s TC: s TD: s

A 1.00 0.15 0.4 2.0 1.0 0.05 0.25 1.2

B 1.20 0.15 0.5 2.0 1.35 0.05 0.25 1.2

C 1.15 0.20 0.6 2.0 1.50 0.10 0.25 1.2

D 1.35 0.20 0.8 2.0 1.80 0.10 0.30 1.2

E 1.40 0.15 0.5 2.0 1.60 0.05 0.25 1.2

Figure 3.1. Elastic response spectra of type 1 recommended in Eurocode 8, for a peak ground

acceleration on rock equal to 1g and for 5% damping

T: s0.5 1 1.5 2 2.5 3 3.5

S a/a

g

Soil ASoil BSoil CSoil DSoil E

4

3.5

3

2.5

2

1.5

1

0.5

0

23

Clause 3.2.2.5(4) [1]

Clauses 4.1.7(1)–

4.1.7(3) [2]

Clause 3.2.2.3(1) [1]

Clause 3.2.2.2(1) [2]

demands on the isolating system that are directly proportional to the value of TD. So, as allowedin Part 1 of Eurocode 8 specifically for design with seismic isolation or energy dissipation devices,Part 2 invites its National Annex to specify a value of TD for bridges with such a design that ismore conservative (longer) than the value given in the National Annex to Part 1. If the NationalAnnex to Part 2 does not exercise this right for national choice, the designer may do so, takinginto account the specific conditions of the seismically isolated bridge.

A safeguard against the rapid decay of the elastic spectrum for T> TD, is provided by the lowerbound of 0.2ag recommended in Part 1 of Eurocode 8 for the design spectral accelerations (seeEqs (D3.12c) and (D3.12d) below).

3.1.2.4 Elastic spectra of the vertical componentThe vertical component of the seismic action needs to be taken into account only in the seismicdesign of (CEN, 2005):

g prestressed concrete decks (acting upwards)g any bearingsg any seismic linksg piers in zones of high seismicity:

– if the pier is already taxed by bending due to permanent actions on the deck or– the bridge is located within 5 km of an active seismotectonic fault (defined as one where

the average historic slip rate is at least 1 mm/year and there is topographic evidence ofseismic activity in the past 11 000 years), in which case site-specific spectra that accountfor near-source effects should be used.

Eurocode 8 gives in Part 1 a detailed description of the vertical elastic response spectrum, asfollows:

Short-period range:

0 � T � TB: Sa;vert Tð Þ ¼ avg 1þ T

TB

3h� 1� ��

ðD3:10aÞ


TB � T � TC: Sa;vert Tð Þ ¼ avg � 3h ðD3:10bÞ


TC � T � TD: Sa;vert Tð Þ ¼ avg � 3hTC

T

� �ðD3:10cÞ


TD � T � 4 s: Sa;vert Tð Þ ¼ avg � 3hTCTD

T2

� �ðD3:10dÞ

The main differences between the horizontal and the vertical spectra lie:

g in the value of the amplification factor in the constant spectral pseudo-accelerationplateau, which is 3 instead of 2.5

g in the absence of a uniform amplification of the entire spectrum due to the type of soil.

The values of control periods TB, TC, TD are not the same as for the horizontal spectra. Part 1 ofEurocode 8 recommends in a note the following non-binding values of TB, TC, TD and the designground acceleration in the vertical direction, avg:

g TB ¼ 0.05 sg TC ¼ 0.15 s


24

g TD ¼ 1.0 sg avg ¼ 0.9ag, if the type 1 spectrum is considered as appropriate for the siteg avg ¼ 0.45ag, if the type 2 spectrum is chosen.

The vertical response spectrum recommended in Eurocode 8 is based on work and data specific toEurope (Ambraseys and Simpson, 1996; Elnashai and Papazoglou, 1997). The ratio avg/ag isknown to be higher at short distances (epicentral or to causative fault). However, as distancedoes not enter as a parameter in the definition of the seismic action in Eurocode 8, the type ofspectrum has been chosen as the parameter determining this ratio, on the basis of the findingthat avg/ag also increases with increasing magnitude (Ambraseys and Simpson, 1996; Abrahamsonand Litehiser, 1989), which in turn determines the selection of the type of spectrum.

3.1.2.5 Topographic amplification of the elastic spectrumEurocode 8 provides for topographic amplification (ridge effect, etc.) of the seismic action for alltypes of structures. According to Part 1 of Eurocode 8, topographic amplification of the fullelastic spectrum is mandatory for structures (including bridges) of importance above ordinary.An informative annex in Part 5 of Eurocode 8 (CEN, 2004b) recommends amplificationfactors equal to 1.2 over isolated cliffs or long ridges with a slope (to the horizontal) less than308, or to 1.4 at ridges steeper than 308. However, as a bridge on a ridge is fairly rare, theneed to account for such an effect would be exceptional.

3.1.2.6 Near-source effects‘Directivity’ effects of the seismic motion along the direction of rupture propagation are usuallyobserved near the seismotectonic fault rupture in a land strip parallel to the fault. This may showup at the site as a large velocity pulse of long period in the direction transverse to the fault. Thegeneral rules of Eurocode 8 in Part 1 do not provide for near-source effects. However, Part 2 ofEurocode 8 (CEN, 2005) requires elaboration of site-specific spectra that take into accountnear-source effects if the bridge is within 10 km horizontally from a known active fault thatmay produce an event of moment magnitude higher than 6.5. In this respect, it gives a defaultdefinition of an active fault as one where the average historic slip rate is at least 1 mm/yearand there is topographic evidence of seismic activity within the Holocene period (i.e. duringthe past 11 000 years).

For bridges less than 15 km from a known active fault, the Caltrans Seismic Design Criteria(Caltrans, 2006) increase spectral ordinates by 20% for all periods longer than 1 s, whileleaving them unchanged for periods shorter than 0.5 s, with linear interpolation in the periodrange in between. Although not stated in Caltrans (2006), this increase, known as the directivityeffect, should only be restricted to the fault normal component of the ground motion, leaving thefault parallel component unaffected (Sommerville et al., 1997).

It should be noted that near-source effects are quite usual in seismic-prone areas.

3.1.2.7 Design ground displacement and velocityThe value given in Part 1 of Eurocode 8 for the design ground displacement, dg, corresponding tothe design ground acceleration, ag, is based on the assumption that the spectral displacementwithin the constant spectral displacement range, derived as (T/2p)2Sa(TD) with the spectral accel-eration at T ¼ TD given from Eq. (D3.8c), entails an amplification factor of 2.5 over the grounddisplacement that corresponds to the design ground acceleration, ag. Taking (2p)2 � 40, weobtain (for ag in m/s2, not in g)

dg ¼ 0:025agSTCTD ðD3:11Þ

Equation (D3.11) gives estimates of the ratio dg/ag that are rather on the high side compared withmore detailed predictions as a function of magnitude and distance on the basis of Bommer andElnashai (1999) and Ambraseys et al. (1996).

The design ground velocity vg may be obtained from the design ground acceleration ag as follows:

vg ¼ STCag/2p (D3.12)

Clause 3.2.2.1(6) [1]

Annex A [3]

Clause 3.2.2.3(1) [2]

Clause 3.2.2.4(1) [1]

Clauses 3.3(6),

6.6.4(3) [2]

Clause 6.7.4.(3) [2]


25

Clauses 2.1(2),

3.2.4(1), 4.1.6(1) [2]

Clause 3.2.2.5(4) [1]

Clause 3.2.2.5(5) [1],

Clause 4.1.6(12) [2]

Clauses 3.2.3(1),

3.2.3(2) [2]

Clauses 3.2.3.1.1(1),

3.2.3.1.1(2),

3.2.3.1.2(4) [1]

Clauses 3.2.3.1.1(3),

3.2.3.1.2(3),

3.2.3.1.3(1) [1]

Clauses 3.2.3(1),

3.2.3(2), 3.2.3(4),

3.2.3(8) [2]

3.1.3 Design spectrum for elastic analysisFor the horizontal components of the seismic action the design spectrum in Eurocode 8 is:

Short-period range:

0 � T � TB: Sa;d Tð Þ ¼ agS2

3þ T

TB

2:5

q� 2

3

� �� ðD3:13aÞ


TB � T � TC: Sa;d Tð Þ ¼ agS2:5

qðD3:13bÞ


TC � T � TD: Sa;d Tð Þ ¼ agS2:5

q

TC

T

� �� bag ðD3:13cÞ


TD � T : Sa;d Tð Þ ¼ agS2:5

q

TCTD

T2

� �� bag ðD3:13dÞ

The behaviour factor, q, in Eqs (D3.13) accounts for ductility and energy dissipation, as well asfor values of damping other than the default of 5% (see also Section 2.3.2 of this Guide).

The value 2/3 in Eq. (D3.13a) is the inverse of the overstrength factor of 1.5 considered inEurocode 8 to always be available even without any design measures for ductility and energydissipation. The factor b in Eqs (D3.13c) and (D3.13d) gives a lower bound for the horizontaldesign spectrum, acting as a safeguard against excessive reduction of the design forces due toflexibility of the system (real or presumed in the design). Its recommended value in Part 1 ofEurocode 8 is 0.2. Its practical implications may be particularly important, in view of therelatively low values recommended by Eurocode 8 for the corner period TD at the outset ofthe constant spectral displacement range.

The design spectrum in the vertical direction is obtained by substituting in Eqs (D3.13) the designground acceleration in the vertical direction, avg, for the effective ground acceleration, agS, andusing the values in Section 3.1.2.4 for the three corner periods. There is no clear, well-knownenergy dissipation mechanism for the response in the vertical direction. So, the behaviourfactor q in that direction is taken equal to 1.0.

3.1.4 Time-history representation of the seismic actionRepresentation of the seismic action merely by its 5%-damped elastic response spectrum is suffi-cient for linear or nonlinear static analysis. For a nonlinear dynamic (response-history) analysis,time histories of the ground motion are needed, conforming on average to the 5%-damped elasticresponse spectrum defining the seismic action. Eurocode 8 (Parts 1 and 2) requires as input for aresponse history analysis an ensemble of at least three records, or pairs or triplets of differentrecords, for analysis under two or three concurrent components of the action.

Part 1 of Eurocode 8 accepts for this purpose historic, ‘artificial’ or ‘simulated’ records, whilePart 2 mentions only historic, ‘modified historic’ or ‘simulated’ records.

‘Artificial’ (or ‘synthetic’) records can be mathematically produced using random vibrationtheory to match almost perfectly the response spectrum defining the seismic action (Gaspariniand Vanmarcke, 1976). It is fairly straightforward to adjust the phases of the various sinusoidalcomponents of the artificial waveform, as well as the time evolution of their amplitudes(‘envelope function’), so that the artificial record resembles a specific recorded motion. This isthe ‘modified historic’ type of record mentioned in Part 2 of Eurocode 8. Figure 3.2 shows anexample of such a record and Figure 8.47 another one. Note, however, that records that areequally rich in all frequencies are not realistic. Moreover, an excitation with a smooth


26

response spectrum without peaks or troughs introduces a conservative bias in the response, as itdoes not let the inelastic response help the structure escape from a spectral peak to a trough at alonger period. Therefore, historic records are favoured in Part 2 of Eurocode 8.

Records ‘simulated’ from mathematical source models, including rupture, propagation of themotion through the bedrock to the site and, finally, through the subsoil to the surface are alsopreferred over ‘artificial’ ones, as the final record resembles a natural one and is physicallyappealing. Obviously, an equally good average fitting of the target spectrum requires more –appropriately selected – historic or ‘simulated’ records than ‘artificial’ ones. Individualrecorded or simulated records should, according to Part 1 of Eurocode 8, be ‘adequately qualifiedwith regard to the seismogenetic features of the sources and to the soil conditions appropriate tothe site’ (Part 1 of Eurocode 8). In plainer language, they should come from events with themagnitude, fault distance and mechanism of rupture at the source consistent with those of thedesign seismic action (Part 2 of Eurocode 8). The travel path and the subsoil conditions shouldpreferably resemble those of the site. These requirements are not only hard to meet but mayalso conflict with conformity (in the mean) to the target spectrum of the design seismic action.

The requirement in Part 1 of Eurocode 8 to scale individual historic or simulated records so thattheir peak ground acceleration (PGA) matches on average the value of agS of the design seismicaction may also be considered against physical reality. It is more meaningful, instead, to use indi-vidual historic or simulated records with PGA values already conforming to the target value of agS.Note also that the PGA alone may be artificially increased or reduced, without affecting at all thestructural response. So, it is more meaningful to select the records on the basis of conformity ofspectral values alone along the lines of Part 2 of Eurocode 8 (CEN, 2005), as described below.

If pairs or triplets of different records are used as the input for analysis under two or threeconcurrent seismic action components, conformity to the target 5%-damped elastic responsespectrum may be achieved by scaling the amplitude of the individual records as follows (seeFigures 8.48 and 8.49 for an application example):

g For each earthquake consisting of a triplet of translational components, the records ofhorizontal components are checked for conformity separately from the vertical one.

g The records of the vertical component, if considered, are scaled so that the average 5%-damped elastic spectrum of their ensemble is at least 90% of the 5%-damped verticalspectrum at all periods between 0.2Tv and 1.5Tv, where Tv is the period of the lowestmode having a participation factor of the vertical component higher than those of bothhorizontal ones.

g For analysis in 3D under both horizontal components, the 5%-damped elastic spectra ofthe two horizontal components in each pair are combined by applying the SRSS rule ateach period value. The average of the ‘SRSS spectra’ of the two horizontal components of

Clauses 3.2.3(3),

3.2.3(6), 3.2.3(7) [2]


Figure 3.2. Herzegnovi X record from the 1979 Montenegro earthquake modulated to match the

Eurocode 8 type 1 spectrum for ground type C with a peak ground acceleration of 0.1g

0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3

Target spectrumModified Hercegnovi

Modified Hercegnovi

a: m

/s2

S a: m

/s2

Period, T: sTime: s

3

2.5

2

1.5

1

0.5

0

1

0.8

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

–1

27

Clause 3.2.3.1.2(4) [1]

Clause 4.2.4.3(1) [2]

Clause 4.3.3.4.3(3) [1]

Clauses 3.3(1)–3.3(8),

Annex D [2]

the individual earthquakes in the ensemble should be at least 0.9p2 � 1.3 times the target

5%-damped horizontal elastic spectrum at all periods from 0.2T1 up to 1.5T1, where T1 isthe lowest natural period of the structure (the effective period of the isolation system inseismically isolated bridges) in any horizontal direction. If this is not the case, allindividual horizontal components are scaled up, so that their final average ‘SRSSspectrum’ exceeds by a factor of 1.3 the target 5%-damped horizontal elastic spectrumeverywhere between 0.2T1 and 1.5T1.

Note in this respect that for analysis under a single horizontal component, Part 1 of Eurocode 8requires the mean 5%-damped elastic spectrum of the applied motions to not fall below 90% ofthat of the design seismic action at any period from 0.2T1 to 2T1.

If the response is obtained from at least seven nonlinear time-history analyses with (triplets orpairs of ) ground motions chosen in accordance with the previous paragraphs, the relevant veri-fications may use the average of the response quantities from all these analyses as the actioneffect. Otherwise, it should use the most unfavourable value of the response quantity amongthe (three to six) analyses.

3.1.5 Spatial variability of the seismic actionUnlike typical buildings, bridges are extended structures. Therefore, it is very likely that the foun-dation supports experience different ground motions owing to the spatial variability of theseismic motion. This phenomenon is known by the generic term ‘decorrelation’. Decorrelationof seismic motions arises from three different causes:

1 The travelling wave effect: except for vertically propagating waves, the seismic wavesexhibit an apparent velocity in the horizontal direction, causing out-of-phase motionsalong the bridge, even when the amplitude remains the same.

2 Scattering of waves, especially at high frequencies: waves travelling through aheterogeneous soil medium are scattered (diffracted and/or reflected) at everyheterogeneity, no matter whether a small lens or an abrupt change in the mechanicalcharacteristics of the media. As the frequency increases, the wavelength decreases and thewaves perceive and are affected by smaller heterogeneities. Scattering causes the motion attwo adjacent locations to be different.

3 Propagation through different soil profiles under each pier location: if the bridgefoundations are not very close to each other, the soil profile under them may be different,causing different soil amplification from the bedrock.

Effects 1 and 3 can, theoretically, be accounted for (although under simplifying assumptions), if thedirection of propagation of the seismic waves is given and the soil profiles are accurately known.Effect 2 does not lend itself to calculation, without complete knowledge of the subsoil conditionsbetween the seismic source and the bridge. Effect 1 can easily be modelled, by shifting the groundmotion time histories by a time lag equal to the distance between the piers divided by theapparent travelling wave velocity. Analytical studies and observations suggest that the apparentwave velocity is not equal to the wave velocity in the upper soil layers, but rather to the wavevelocity at a significant depth (in the rock medium where the rupture initiates); typical valuesexceed 1000 m/s. Effect 3 can be computed through one-dimensional site response analysis. Foreffect 2, only random vibration models with empirically determined parameters can be used.

Informative Annex D in Part 2 of Eurocode 8 presents the theory for the generation of incoherentground motions, including all three effects above, as well as the mathematical tools for theanalysis of the bridge response under multi-support excitations. However, because the theoryis complex, requiring specific tools for its numerical implementation and statistical site datafor the determination of model parameters, the code allows the use of a simplified approachto take into account the spatial variability of seismic motions along the bridge. This spatial varia-bility should be taken into account whenever the soil properties along the bridge vary and theground type according to Table 3.2 differs from one pier to another, or if the length of a continu-ous deck exceeds Lg/1.5, where Lg is the correlation distance (beyond which the motion may beassumed to be fully uncorrelated). Recommended values of the correlation distance are given inTable 3.4.


28

The simplified methodology consists of combining via the SRSS (square root of the sums of thesquare) rule the dynamic effects of a uniform ground motion acting at every foundation, to theeffects of differential displacements imposed statically at each foundation point. Two patternsare used for the static imposed displacements, and the results of the most unfavourable oneare retained:

g a pattern with foundation displacements all in the same direction and proportional to thedistance along the bridge and to the ground displacement, but inversely proportional tothe correlation distance, combined with a small offset at any intermediate pier

g a pattern with displacements alternating between consecutive piers.

The same patterns of imposed displacements, with an increased safety factor, are also used in thechecks of deck unseating at movable joints (see Eq. (D6.34) in Sect. 6.8.1.2).

Unlike certain truly dynamic approaches, the simplified method above cannot capture dynamicfeatures of the spatial variability of the seismic action, as it is essentially a pseudo-static ‘addition’of imposed support displacements. It accounts, however, to a certain degree of approxima-tion, for all three main effects of the spatial variability (Sextos et al., 2006; Sextos andKappos, 2009).

3.2. Siting and foundation soils3.2.1 IntroductionSignificant damage to foundations may be caused by soil-related phenomena: fault rupture, slopeinstability near the bridge, liquefaction or densification of the soil due to the ground shaking.Except in very few cases, such adverse effects cannot be accommodated by a foundationdesign. Therefore, these ground hazards should be thoroughly investigated and properly miti-gated to the largest possible extent.

3.2.2 Seismically active faultsSeismological evidence suggests that, where seismogenic activity is confined in the upper 20 km orso of the Earth’s crust, co-seismic surface rupture tends to occur only if the earthquake has amoment magnitude Mw over about 6.5. Therefore, in Europe, surface faulting is a rather rareevent, except in Turkey and maybe in Greece or Italy. As pointed out in Part 5 of Eurocode 8,assessment of the surface fault rupture hazard at a site requires special geological investigations,to show that there is no active fault nearby. Official documents published by competent nationalauthorities may, of course, map the seismically active faults. Note, though, that there are noabsolute criteria to characterise a fault as seismically active and to consider a site as close toit. It is suggested in Part 5 of Eurocode 8 that evidence of movement in the late Quaternaryperiod (10 000 years) or lack of it is used as a criterion, while Part 2 of Eurocode 8 defines anactive fault as one where the average historic slip rate is at least 1 mm/year and there istopographic evidence of seismic activity in the Holocene period (i.e. in the past 11 000 years).A distance of several tens of metres may be used as the criterion for the immediate vicinity toa fault.

3.2.3 Slope stabilityWhen a structure is to be built near a natural or man-made slope, a verification of the slope stab-ility under the seismic action shall be carried out. Although the stability checks recommended inPart 5 of Eurocode 8 aim at ensuring a prescribed safety factor, the underlying criterion is anultimate limit state (ULS) beyond which the permanent displacements it entails become unaccep-table. This is reflected in the method of analysis proposed in Part 5 of Eurocode 8 for achieving asafety factor above 1.0, notably a pseudo-static approach with seismic forces taken as equal to

Clause 4.1.1(1) [3]

Clause 4.1.2(1), (2) [3]

Clause 3.2.2.3(1) [2]

Clauses 4.1.3.1(1),

4.1.3.1(2), 4.1.3.3(1),

4.1.3.3(3)–4.1.3.3(6),

4.1.3.3(8), 4.1.3.4(4),

Annex A [3]


Table 3.4. Values recommended in Part 5 of Eurocode 8 for the distance beyond which ground motions

may be considered as uncorrelated

Ground type A B C D E

Lg: m 600 500 400 300 500

29

the product of the potential sliding mass multiplied by 50% of the design PGA at the soil surface,agS, including the topographic amplification factor of Section 3.1.2.5, if relevant. The 50%fraction has been chosen empirically and with back-analysis of observed performance ofslopes in earthquakes. It nevertheless reflects the observation, first pointed out by Newmark inhis 1965 Rankine lecture, that when the maximum seismic action, equal to the product of thepotentially sliding mass multiplied by the PGA, is slightly exceeded, only permanent displace-ments occur without a catastrophic failure. With the recommended value of the ground accelera-tion, 0.5agS, it is expected that the induced displacements will not exceed a few centimetres.

The design seismic inertia forces in the horizontal and vertical direction are given by

FH ¼ 0.5agSM (D3.14)

FV ¼+0.5FH if avg . 0.6ag (D3.15a)

FV ¼+0.33FH if avg � 0.6ag (D3.15b)

where M is the potentially sliding mass and agS includes the topographic amplification factor ofSection 3.1.2.5, if relevant. The seismic design resistance of the soil should be calculated with thesoil strength parameters divided by the appropriate partial factor, defined in the National Annexto Part 5 of Eurocode 8.

It is essential not to overlook the applicability conditions of the pseudo-static method of analysis,that is:

g The geometry of the topographic profile and the ground profile are reasonably regular.g The ground materials of the slope and the foundation, if water saturated, are not prone to

developing significant pore water pressure build-up that may lead to loss of shear strengthand stiffness degradation under seismic conditions. The same limitation applies to certainunusual soils, such as sensitive clays, although the mechanism of strength degradation isdifferent.

For high values of agS it may prove hard to verify the slope stability using the pseudo-staticmethod. If so, the designer may opt for computing the actual induced permanent displacementsand checking whether they are acceptable. A simplified way to estimate the displacements is theNewmark sliding block method. This entails the preliminary choice of the most critical slidingsurface and the associated ‘critical’ value of agS for which the safety factor drops to 1.0. Withthe selection of appropriate time histories for the ground motion, double integration of the differ-ence of the input acceleration and the critical one is carried out over the time intervals duringwhich the former exceeds the latter. The outcome may be taken as the permanent slope displace-ment along the chord of the critical circular failure surface. More refined analyses, accounting forthe seismic response of the slope in the evaluation of the rigid block acceleration, may bewarranted in some cases.

One essential requirement for the application of pseudo-static analysis or of the Newmarksliding block method is that the soil strength does not vary significantly during the earthquake.When the strength is reduced by a pore pressure build-up, it may be evaluated through theexpression

tanwr ¼ 1� Du

s 0v

� �tanw ðD3:16Þ

where wr is the reduced friction angle, Du the pore pressure increase estimated from empiricalcorrelations or preferably from experimental tests, and s 0

v is the effective vertical stress.

3.3. Soil properties and parameters3.3.1 Introduction: the meaning of soil property valuesMany geotechnical tests, particularly field tests, do not allow determining directly the value ofbasic geotechnical parameters or coefficients, notably for strength and deformations. Instead,


30

these values are to be derived via theoretical or empirical correlations. Part 2 of Eurocode 7(CEN, 2007) defines derived values as:

Derived values of geotechnical parameters and/or coefficients are obtained from test results bytheory, correlation or empiricism. Derived values of a geotechnical parameter then serve asinput for assessing the characteristic value of this parameter in the sense of Eurocode 7 – Part1 and, further, its design value, by applying the partial factor gM (‘material factor’).

The philosophy regarding the definition of characteristic values of geotechnical parameters isgiven in Part 1 of Eurocode 7 (CEN, 2003):

The characteristic value of a geotechnical parameter shall be selected as a cautious estimate ofthe value affecting the occurrence of the limit state . . . the governing parameter is often themean of a range of values covering a large surface or volume of the ground. The characteristicvalue should be a cautious estimate of this mean value.

These excerpts from Eurocode 7 reflect the concern that we should be able to keep using thevalues of the geotechnical parameters traditionally used, whose determination is not standardised(they often depend on the judgment of the geotechnical engineer). However, two remarks are duein this connection: on one hand, the concept of a ‘derived value’ of a geotechnical parameter (pre-ceding the determination of the characteristic value) has been introduced, but, on the other, thereis now a clear reference to the limit state involved and the assessment of a spatial mean value, asopposed to a local value; this might appear as a specific feature of geotechnical design which,indeed, involves ‘large’ surface areas or ‘large’ ground volumes.

Statistical methods are mentioned in Eurocode 7 only as a possibility:

If statistical methods are used, the characteristic value should be derived such that thecalculated probability of a worse value governing the occurrence of the limit state underconsideration is not greater than 5%.

The general meaning is that the characteristic value of a geotechnical parameter should not bevery different from the values traditionally used. Indeed, for the majority of projects, the geotech-nical investigation is such that no meaningful statistical treatment of the data can be performed.Statistical methods are, of course, useful for very large projects where the amount of data justifiestheir use.

3.3.2 Soil propertiesEurocode 8 considers both the strength properties and the deformation characteristics. It furtherrecognises that earthquake loading is essentially of short duration. Consequently, most soilsbehave in an undrained manner. In addition, for some of them the properties may be affectedby the rate of loading.

3.3.2.1 Strength parametersFor cohesive soils, the relevant strength property is the undrained shear strength, cu. For most ofthem this value can be taken as equal to the conventional ‘static’ shear strength. However, on theone hand some plastic clays may be subject to cyclic degradation of strength, but, on the other,some clays may exhibit a shear strength increase with the rate of loading. These phenomenashould ideally be given due consideration in the choice of the relevant undrained shearstrength. The recommended partial factor gM on cu is equal to 1.4.

For cohesionless soils, relevant strength properties are the drained friction angle, w0, and thedrained cohesion, c0. They are directly usable for dry or partially saturated soil. For saturatedsoils, they require knowledge of the pore water pressure variation during cyclic loading, u,which directly governs the shear strength through the Mohr–Coulomb failure criterion:

t ¼ (s� u) tan w0 þ c0 (D3.17)

The evaluation of u is very difficult. Therefore, Part 5 of Eurocode 8 gives an alternative approach,namely using the undrained shear strength under cyclic loading, tcy,u, which may be determined

Clauses 3.1(1)–3.1(3)

[3]


31

Clauses 3.2(1)–3.2(4),

4.2.3(1)–4.2.3(3) [3]

Clauses 4.1.4(1)–

4.1.4(3) [3]

from experimental correlations with, for instance, the soil relative density or any other indexparameter, such as the blow counts number, N, in a standard penetration test (SPT).

The recommended values for the partial factors gM are:

g gM ¼ 1.25 on tan(w0) and tcy,ug gM ¼ 1.4 on c0.

3.3.2.2 Stiffness and damping parametersThe soil stiffness is defined by the shear wave velocity, vs, or equivalently the soil shear modulusG. The main role played by this parameter is in the classification of the soil profile according tothe ground types in Table 3.2 of Section 3.1.2.3 in this Guide. Additional applications thatrequire knowledge of the shear stiffness of the soil profile include the evaluation of:

g soil–structure interactiong site response analyses to define the ground surface response for special soil categories

(profile S1).

In the applications listed above, it is essential to recognise that soils are highly nonlinear materialsand that the relevant values to use in calculations are not the elastic ones but secant valuescompatible with the average strain level induced by the earthquake, typically of the order of5 � 10�4 to 10�3. Part 5 of Eurocode 8 proposes the set of values in Table 3.5, depending onthe peak ground surface acceleration. Note that the fundamental variable governing thereduction factor is the shear strain and not the peak ground surface acceleration. However, inorder to provide useful guidance to designers, the induced strains have been correlated to PGAs.

In addition to the stiffness parameters, soil internal damping should be taken into account insoil–structure interaction analyses. The soil damping ratio also depends on the averageinduced shear strain, and is correlated to the reduction factor for the stiffness, as listed inTable 3.5.

3.4. Liquefaction, lateral spreading and related phenomena3.4.1 Nature and consequences of the phenomenaLiquefaction is a process in which cohesionless or granular sediments below the water table tem-porarily lose strength and behave as a viscous liquid rather than a solid, during strong groundshaking. Saturated, poorly graded, loose, granular deposits with low fines content are most sus-ceptible to liquefaction. Liquefaction does not occur randomly: it is restricted to certain geo-logical and hydrological environments, primarily recently deposited sands and silts in areaswith a high ground water level. Dense and more clayey soils, including well-compacted fills,have low susceptibility to liquefaction.

The liquefaction process itself may not necessarily be particularly damaging or hazardous. Forengineering purposes, it is not the occurrence of liquefaction that is of importance but the poten-tial of the process and of associated hazards to damage structures. The adverse effects of lique-faction can be summarised as follows:

g Flow failures, when completely liquefied soil or blocks of intact material ride on a layer ofliquefied soil. Flows can be large and develop on moderate to steep slopes.


Table 3.5. Average soil damping ratios and reduction factors (+1 standard deviation) for the shear wave

velocity vs and the shear modulus G within the upper 20 m of soil in Part 5 of Eurocode 8 (for soil with

vs , 360 m/s)

Ground acceleration, agS: g Damping ratio: % vs/vs,max Gs/Gs,max

0.1 3 0.9 (+0.07) 0.8 (+0.1)

0.2 6 0.7 (+0.15) 0.5 (+0.2)

0.3 10 0.6 (+0.15) 0.36 (+0.2)

32

g Lateral spreading, with lateral displacement of superficial blocks of soil as a result of theliquefaction of a subsurface layer. Spreading generally develops on gentle slopes, andmoves towards a free face, such as an incised river channel or coastline. It may also occurthrough the failure of shallow liquefied layers subjected to a high vertical load on part ofthe ground surface due to a natural or artificial embankment or cut.

g Ground oscillation: where the ground is flat or the slope too gentle to allow lateraldisplacement, liquefaction at depth may disconnect overlying soils from the underlyingground, allowing the upper soil to oscillate back and forth in the form of ground waves.These oscillations are usually accompanied by ground fissures and the fracture of rigidextended structures, such as pavements and pipelines.

g Loss or reduction in bearing capacity, when earthquake shaking increases pore waterpressures, which in turn cause the soil to lose its strength and bearing capacity.

g Soil settlement, as the pore water pressures dissipate and the soil densifies afterliquefaction. Settlement of structures may occur, owing to the reduction in the bearingcapacity or the ground displacements noted above. In piled foundations the post-earthquake settlement of the liquefied layer due to pore pressure dissipation inducesnegative skin friction along the shaft, in all layers above the liquefied layer.

g Increased lateral pressures on retaining walls, when the soil behind a wall liquefies andbehaves like a ‘heavy’ fluid with no internal friction.

g Flotation of buried structures, when buried structures, such as tanks and pipes, becomebuoyant in the liquefied soil.

Other manifestations of liquefaction, such as sand boils, can also occur, and may pose a risk tostructures, particularly through loss or reduction in the bearing capacity and settlement.

Liquefaction has been extensively studied since 1964. The state of the art is now well establishedand, more importantly, allows reliable prediction of the occurrence of liquefaction. So, thisaspect is fully covered in Part 5 of Eurocode 8, including a normative annex for the use ofSPT measurements for the evaluation of the undrained cyclic strength of cohesionless soils.However, in addition to the SPT, other techniques are allowed for the determination of thesoil strength, such as cone penetration tests (CPTs) and shear wave velocity measurements. Lab-oratory tests are not recommended, because to obtain a reliable estimate of liquefaction resist-ance, very specialised drilling and sampling techniques are needed, beyond the budget of anycommon project. It should, however, be noted that there have been numerous developmentsin liquefaction assessment methodologies in recent years (e.g. Seed et al., 2003; Idriss andBoulanger, 2008). So, the methods described in Part 5 of Eurocode 8 may be potentially uncon-servative, especially for materials with a high fines content. It is therefore recommended that anexpert is involved in the liquefaction assessment.

3.4.2 Liquefaction assessmentThe verification of the liquefaction susceptibility is carried out under free field conditions, butwith the prevailing situation during the lifetime of the structure. For instance, if a tallplatform is going to be built to prevent flooding of the site, or the water table will be loweredon a long-term basis, these developments should be reflected in the evaluation.

The recommended analysis is a total stress analysis in which the seismic demand, representedby the earthquake-induced stresses, is compared with the seismic capacity (i.e. the undrainedcyclic shear strength of the soil – also called liquefaction resistance). The shear stress ‘demand’is expressed in terms of a cyclic stress ratio (CSR), and the ‘capacity’ in terms of a cyclicresistance ratio (CRR). In both ratios the normalisation is with respect to the vertical effectivestress, s 0

v.

A soil should be considered susceptible to liquefaction whenever CRR, l CSR, where l is afactor of safety, with a recommended value of 1.25. The CSR is evaluated with a simplifiedversion of the Seed–Idriss formula, which allows a rapid calculation of the induced stressalong the depth, without resorting to a dynamic site response analysis:

CSR ¼ 0:65ðagS=gÞsv=s0v ðD3:18Þ

Clauses 4.1.4(3)–

4.1.4(6), 4.1.4(10),

4.1.4(11), Annex B [3]


33

Clause 4.1.4(7),

4.1.4(8) [3]

Clauses 4.1.4(12)–

4.1.4(14) [3]

Clauses 4.1.4(12)–

4.1.4(14) [3]

where sv and s 0v are the overburden pressure and the vertical effective stress at the depth of

interest. Equation (D3.18) may not be applied for depths larger than 20 m. The liquefactionresistance ratio, CRR, may be estimated through empirical correlations with an index parameter,such as the SPT blow count, the static CPT point resistance or the shear wave velocity. Note thatall of these methods should be implemented with several corrections of the measured indexparameter for the effects of the overburden at the depth of measurement, the fines content ofthe soil, the effective energy delivered to the rods in SPTs, etc. In normative Annex B in Part 5of Eurocode 8, the CRR is assessed based on a corrected SPT blowcount, using empiricalliquefaction charts relating CRR ¼ t/s 0

v to the corrected SPT blow count N1(60) for an earth-quake surface magnitude of 7.5. Correction factors are provided in Annex B [3] for othermagnitudes.

A soil may be prone to liquefaction if it presents certain characteristics that govern its strengthand the seismic demand is large enough. Taking the opposite view, Part 5 of Eurocode 8 intro-duces cumulative conditions under which the soil may be considered as not prone to liquefactionand liquefaction assessment is not required:

g Low ground surface acceleration, agS , 0.15g, and– soils with a clay content higher than 20% and a plasticity index above 10% or– soils with a silt content higher than 35% and a corrected SPT blow count of over 20 or– clean sands with a corrected SPT blow count higher than 30.

The assessment of liquefaction is also not required for layers located deeper than 15 m below thefoundation. This does not mean that those layers are not prone to liquefaction, although suscep-tibility to liquefaction decreases with depth; it means, instead, that because of their depth theirliquefaction will not affect the structure. Although it is not spelled out in Eurocode 8, obviouslythis condition is not sufficient by itself: it should be complemented with a condition on the foun-dation dimensions relative to the layer depth.

If soils are found to be susceptible to liquefaction, mitigation measures, such as ground improve-ment and piling (to transfer loads to layers not susceptible to liquefaction), should be consideredto ensure foundation stability. The use of pile foundations alone should be considered withcaution, in view of:

g the large forces induced in the piles by the loss of soil support in the liquefiable layersg the post-earthquake settlements of the liquefied layers causing negative skin friction in all

layers located aboveg the inevitable uncertainties in determining the location and thickness of such layers.

3.4.3 Liquefaction mitigationIf soils are found to be susceptible to liquefaction and the consequences are consideredunacceptable for the structure (excessive settlement, loss of bearing capacity, etc.), groundimprovement should be considered. Several techniques are available to improve the resistanceto liquefaction: soil densification, soil replacement, sand compaction piles, drainage (graveldrains), deep soil mixing (lime–cement mixing), stone columns, blasting, jet grouting, etc. Themost commonly used among these are stone columns and soil densification, the latter throughvibroflotation, dynamic compaction or compaction grouting. Some techniques, such as stonecolumns, offer the advantage of combining several effects, for example, densification anddrainage.

Not all techniques are appropriate for any soil condition. The most appropriate ones should bechosen taking into account the depth to be treated, the fines content of the soil and the presenceof adjacent structures. Attention should also be paid to the efficiency, durability and cost of thesolution. For instance, dynamic compaction is better suited for shallow clean sand layers; jetgrouting and stone columns may be used to improve the soil and offer a good load-bearinglayer under shallow foundations; compaction grouting, albeit more costly, is very efficient foralmost any soil, etc. Methods based on drainage should be considered with care, as drainageconditions may change in time and clogging may occur, especially in environments with largefluctuation of the water table.


34

There is significant experience with all the methods listed above: when properly implemented,they have a good track record during earthquakes.

3.4.4 SettlementsThe magnitude of the settlement induced by the earthquake should be assessed when there areextended layers or thick lenses of loose, unsaturated cohesionless materials at shallow depths.Excessive settlements may also occur in very soft clays because of cyclic degradation of theirshear strength under ground shaking of long duration. If the settlements caused by densificationor cyclic degradation appear capable of affecting the stability of the foundations, considerationshould be given to ground improvement methods.

Earthquake-induced settlement can be estimated using empirical relationships between volu-metric strain, SPT N values (corrected for overburden) and the factor of safety against liquefac-tion. For example, the approach in Tokimatsu and Seed (1987) is based on relationships betweenthe volumetric strain, the cyclic shear strain and SPT N values. The peak shear strain computedfrom the one-dimensional response analysis and the SPT-corrected N value at that point areentered into the Tokimatsu and Seed chart (Figure 3.3) to yield the volumetric strain. Thetotal settlement can then be obtained by integrating these strains over depth.

3.4.5 Lateral spreadingLateral spreading is a highly unpredictable phenomenon. Nevertheless, empirical correlationshave been developed to estimate the lateral displacement of the ground, DH, in metres due toliquefaction (Youd et al., 2002):

logDH ¼ �16:713þ 1:532M � 1:406 logR� � 0:012Rþ 0:592 logW þ 0:540 logT15

þ 3:413 log 100� F15

� �� 0:795 log D5015 þ 0:1mm� � ðD3:19Þ

where M is the moment magnitude of the earthquake; R is the nearest horizontal or mapdistance from the site to the seismic energy source (if R , 0.5 km, use R ¼ 0.5 km);R� ¼ Rþ 10(0.89M�5.64); T15 is the cumulative thickness of saturated granular layers with acorrected SPT blow count (N1)60 less than 15; F15 is the average fines content of granularmaterial (fraction passing the #200 sieve) within T15; D5015 is the average mean grain size ofgranular material in T15, and W is the free-face ratio, defined as the height, H, of the free faceas a percentage of the distance, L, from the base of the free face to the point in question.

Clauses 4.1.5(1)–

4.1.5(4) [3]


Figure 3.3. Assessment of volumetric strain

10–3 10–2 10–1 110–3

10–2

10–1

1

10

Volu

met

ric s

trai

n du

e to

com

pact

ion,

εc:

%

Cyclic shear strain, γcyc: %

15 cyclesN1 < 40

<30

<20<15

<10<5

35

Such relationships should be used with care and by experienced engineers, as there is no physicaltheory as yet to confirm them. Note also that large displacements, over 6 m, should not betaken to apply in engineering practice, because they are poorly supported in the database(Youd et al., 2002).

REFERENCES

Abrahamson NA and Litehiser JJ (1989) Attenuation of peak vertical acceleration. Bulletin of the

Seismological Society of America 79: 549–580.

Ambraseys N, Simpson K and Bommer JJ (1996) Prediction of horizontal response spectra in

Europe. Journal of Earthquake Engineering and Structural Dynamics 25(4): 371–400.

Ambraseys NN and Simpson KA (1996) Prediction of vertical response spectra in Europe.

Earthquake Engineering and Structural Dynamics 25(4): 401–412.

Bommer JJ and Elnashai AS (1999) Displacement spectra for seismic design. Journal of Earthquake

Engineering 3(1): 1–32.

Caltrans (2006) Seismic Design Criteria, version 1.4. California Department of Transportation,

Sacramento, CA.

CEN (Comite Europeen de Normalisation) (2003) EN 1997–1:2003: Eurocode 7 – Geotechnical

design – Part 1: General rules. CEN, Brussels.

CEN (2004a) EN 1998-1:2004: Eurocode 8 – Design of structures for earthquake resistance – Part 1:


CEN (2004b) EN 1998-5:2004: Eurocode 8 – Design of structures for earthquake resistance – Part 5:




CEN (2007) EN 1997–2:2007: Eurocode 7 – Geotechnical design – Part 2: Ground investigation

and testing. CEN, Brussels.

Elnashai AS and Papazoglou AJ (1997) Procedure and spectra for analysis of RC structures

subjected to strong vertical earthquake loads. Journal of Earthquake Engineering 1(1): 121–155.

Gasparini DA and Vanmarcke EH (1976) Simulated Earthquake Motions Compatible with

Prescribed Response Spectra. Department of Civil Engineering, Massachusetts Institute of

Technology, Cambridge, MA. Research Report R76-4.

Idriss IM and Boulanger RW (2008) Soil Liquefaction During Earthquakes. Earthquake

Engineering Research Institute, Oakland, CA. MNO-12.

Ohta Y and Goto N (1976) Estimation of S-wave velocity in terms of characteristic indices of soil.

Butsuri-Tanko 29(4): 34–41.

Rey J, Faccioli E and Bommer JJ (2002) Derivation of design soil coefficients (S) and response

spectral shapes for Eurocode 8 using the European Strong-Motion Database. Journal of

Seismology 6(4): 547–555.

Seed HB, Cetin KO, Moss RES et al. (2003) Recent advances in soil liquefaction engineering: a

unified and consistent framework. 26th Annual ASCE LA Geotechnical Seminar, HMS Queen

Mary, Long Beach, CA, Keynote Presentation.

Sextos AG and Kappos AJ (2009) Evaluation of seismic response of bridges under asynchronous

excitation and comparison with Eurocode 8-2 provisions. Bulletin of Earthquake Engineering 7:

519–545.

Sextos AG, Kappos AJ and Kolias B (2006) Computing a ‘reasonable’ spatially variable

earthquake input for extended bridge structures. First European Conference on Earthquake

Engineering and Seismology, Geneva, paper 1601.

Sommerville PG, Smith NF, Graves RW and Abrahamson NA (1997) Modification of empirical

strong ground motion attenuation relations to include the amplitude and duration effects of

rupture directivity. Seismological Research Letters 68: 199–222.

Tokimatsu K and Seed HB (1987) Evaluation of settlements in sand due to earthquake shaking.

Journal of Geotechnical Engineering of the ASCE 113(8): 861–878.

Youd TL, Hansen CM and Bartlett SF (2002) Revised multilinear regression equations for

prediction of lateral spread displacement. Journal of the Geotechnical and Geoenvironmental

Engineering 128(12): 1007–1017.


36




Chapter 4

Conceptual design of bridges forearthquake resistance

4.1. IntroductionDuring the conceptual design phase, the layout of the structure is established, the structuralmaterials for its various parts are selected and the construction technique and procedure aredecided. Conceptual design concludes with a preliminary sizing of all members, in order toallow the next phase of the design process to be carried out, namely the analysis for thecalculation of the effects of the design actions (including seismic) in terms of internal forcesand deformations in structural members. Analysis is followed by the detailed design phase(notably the verification of member sizes, the dimensioning of the reinforcement, etc., on thebasis of the calculated action effects) and the preparation of material specifications, constructiondrawings and any other information that is necessary or helpful for the implementation of thedesign.

Conceptual design is of utmost importance for the economy, safety and fitness for use of thestructure. In addition to technical skills and knowledge, it requires judgement, experience and acertain intuition. Although conceptual design cannot be taught, several authoritative documents(recently, fib, 2009, 2012) give general principles and guidance. Useful general sources for bridgesare fib (2000) and (2004), and, for their seismic design, Priestley et al. (1996) and fib (2007).

The conceptual design of a bridge is controlled mainly by that of the deck, which in turn isgoverned by its use, the preferred construction technique (Table 4.1), aesthetics, topographyand – of course – cost issues. The three last considerations significantly influence the conceptualdesign of the piers as well. Gravity loads and the construction technique control the design of thedeck. The deck spans and their erection, alongside the terrain, determine the number andlocation of the piers. In seismic regions, the piers themselves and their connection to the deckare governed by the seismic action. In addition, seismic considerations are taken into accountfor some aspects of the deck design, notably its continuity across spans and sometimes its connec-tion with the abutments and the piers. Before delving into these purely seismic aspects, it is worthrecalling that one of the prime objectives of the conceptual design of a bridge for non-seismicloads is to reduce the deck dead load, as this is normally the prime contributor to the actioneffects in the deck, the piers and the foundation for the combination of factored gravity loads(the persistent and transient loads combination in Eurocode terminology). This is first pursuedthrough the choice of the material(s) and the shape of the deck section; once these are chosen,an effort is made to reduce its dimensions by using higher-strength materials, external prestres-sing (if relevant), etc. Needless to say, although the conceptual and detailed seismic design ofbridges concerns mainly the piers and the way they are connected to or support the deck,reducing the deck’s self-weight is of prime importance for the bridge seismic design as well:both seismic force and displacement demands increase with the deck mass (they are normallyapproximately proportional to its square root), and there is good reason to reduce it.

As pointed out in Section 2.3.1 of this Guide, the prime decision in the conceptual seismic designof the bridge is how to accommodate the horizontal seismic displacements of the deck withrespect to the ground. Four options were highlighted there, repeated below for completeness:

1 to support the deck on all abutments and piers through bearings (or similar devices) thatcan slide or are horizontally very flexible

Clauses 2.4(1)–2.4(3)

[2]

37

Clauses 2.4(4), 2.4(9),

2.4(10) [2]

2 to fix the deck to the top of at least one pier (but not at the abutments) and let those piersaccommodate the horizontal seismic displacements through inelastic rotations in flexural‘plastic hinges’

3 to let the base of the piers slide with respect to the soil or allow inelastic deformations todevelop in foundation piles

4 to lock the bridge in the ground, by fixing the deck to the abutments as in an integralsystem that follows the ground motion with little additional deformation of its own.

Option 3 is not central in Part 2 of Eurocode 8, and is not dealt with in detail in this Designers’Guide. Option 4 is also a rather special case, applicable only to short bridges with up to threespans – but often with only one. It is dealt with separately in Sections 4.5, 5.4 and 6.11.3 ofthis Guide. Options 1 and 2 are the main ones. The first amounts to effectively isolating thedeck from ground shaking; it is treated in Eurocode 8 as seismic isolation. The second optionrelies on ductility and energy dissipation in the piers. This chapter – and most of the rest –focuses on these two options.

The features of conceptual design affecting the seismic behaviour and design of a bridge the mostare:

1 (for multi-span bridges) the continuity of the deck across spans over the piers2 (for bridges with a concrete deck) whether the deck is monolithic with the piers.

To a certain extent these features are related to the method used for the erection of the deck (seeTable 4.1). It is very unlikely – or even impossible – for continuous decks to lose support and dropfrom the piers. Even unseating from some bearings – which is also unlikely – will not have cata-strophic consequences, andmay be easily reversed. Unseating and dropping can be ruled out if thedeck is monolithically connected to the piers. In that case, the prevention of horizontal movementbetween the deck and the top of the piers profoundly affects the seismic response of the bridge,which is then dominated by the inelastic deformations and behaviour of the piers. It also affectsits seismic design, which is then based on the ductility of the piers. Monolithic or rigid connectionof the deck to the pier tops also affects the bridge performance under non-seismic actions. Theeffect may be favourable (e.g. the performance under braking or centrifugal traffic actions inrailway bridges) or negative (e.g. the restraint of thermal or shrinkage deformations in a longdeck on stiff piers, which may even be prohibitive for the bridge).

4.2. General rules for the conceptual design of earthquake-resistantbridges

4.2.1 Deck continuityThe most important goal of the seismic design of a bridge is to keep the deck in place under thestrongest conceivable seismic action. In multi-span bridges, one of the risks to be faced by the


Table 4.1. Span range, erection speed and continuity of the deck across spans and with the piers for

different erection techniques

Deck erection Normal span

range: m

Erection speed:

m/week

Full deck

continuity

Integral deck

and piers

Prefabricated girders 10–50 25–100a Normally not Normally not

On scaffolding/falsework on grade 5–50 5–10 Normally yes Yes or no

On mobile launching/casting girder

or gantry (span-by-span)

30–60 10–50 Yes Yes or no

Free/balanced cantilever

Cast-in-situ deck segments 60–300 6–15 Yes Normally yes

Prefabricated deck segments 40–160 20–60a

Incremental launching

Without temporary props 30–70 10–30 Yes Normally not

With temporary props 70–120 10–30

aSpeed depends on the capacity of the prefabrication plant

38

seismic design is local loss of support of the deck, due to unseating from a pier. The best way toprevent drop of a part of a multi-span deck from one or more piers is by providing continuity ofthe spans over all piers: a deck continuous from abutment to abutment. An exception may bemade in very long bridges – of several hundred or over a thousand metres – where intermediatemovement joints may be judiciously introduced. Such joints may be essential if it is consideredlikely that a strong earthquake may induce significantly different movement at the base ofadjacent piers, notably if the bridge straddles a potentially active tectonic fault or crossesnon-homogeneous soil formations. Intermediate movement joints may be placed within aspan as Gerber-type hinges with sufficient seat length. More often they are placed betweentwo spans whose ends are supported through separate bearings on the same pier. In such alayout the movement joint should be wide enough to prevent pounding between the ends ofthe two spans, in addition to providing sufficient support length against unseating (see alsoSection 6.8.1.3).

Apart from breaking up the full continuity of the deck and increasing the chances of a drop-off,intermediate movement joints increase the uncertainty of the seismic response: the parts of thebridge separated by the movement joints (the ‘frames’ of the bridge in US parlance) mayvibrate out of phase and experience pounding at the joints. Opening and closing of joints is anonlinear phenomenon, and capturing its effects may require a nonlinear analysis (normally inthe time domain). To take some of these effects into account without recourse to nonlineartime-history analysis, the Caltrans Seismic Design Criteria (Caltrans, 2006) requires using, inaddition to a stand-alone model of each and every bridge ‘frame’ between adjacent movementjoints, two global models that consider their interactions:

g a ‘compression’ model with all movement joints taken as closedg a ‘tension’ model where movement joints are considered to be open and connected only

through the axial stiffness, EA/L, of any cable restrainers linking the deck in thelongitudinal direction across joint(s).

For bridges with several intermediate movement joints, Caltrans (2006) further requires the useof several elastic multi-’frame’ models, each one encompassing not more than five ‘frames’ plus a‘boundary frame’ or abutment at each end; ‘frames’ beyond the ‘boundary’ ones are representedby massless springs, and analysis results for ‘boundary frames’ are ignored, while adjacent modelsoverlap by at least one ‘frame’ beyond a ‘boundary frame’. The objective of this complex series ofanalysis is to better capture the out-of-phase motion of ‘frames’ and to account for the importantnormal modes and periods of vibration of each ‘frame’ without resorting to an unduly largenumber of nodes from abutment to abutment. Despite its complexity, the above proceduremay not capture important features of the system response for the following reasons:

g pounding between ‘frames’ or the activation of cable restrainers linking them are unilateralnonlinear phenomena that cannot be approximated by ‘envelope’ linear models

g if the bridge is long enough to have several intermediate deck separation joints, the effectof the spatial distribution of the seismic ground motion may be quite important and worthaccounting for, even when such joints are provided.

As Part 2 of Eurocode 8 requires none of this analysis complexity, the sole reason for highlightinghere the provisions in Caltrans (2006) is to stress the uncertainty of the seismic response ofbridges with intermediate movement joints and the complexity of the analysis that this entails.Reducing the uncertainty of the response is an important goal of conceptual design, and inthis case it is served well by avoiding such joints.

Deck spans composed of prefabricated girders, be they of concrete, steel or composite (steel–concrete), are normally simply supported on the piers. Adjacent spans can be connected byencapsulating their ends in bulky cast-in-situ crossheads. However, this is not a commonpractice. Normally, continuity of adjacent spans is pursued through a cast-in-situ topping slabcontinuous over the joint between two girder ends (see also Section 5.5.1.4). Figure 4.1 depictsan example. The slab should have sufficient out-of-plane flexibility to allow different rotationsat the ends of the adjacent spans due to traffic, creep (and the ensuing moment redistribution)and pier head rotation. This detail provides continuity of the pavement for motorist and

Clauses 2.3.2.2(4),

4.1.3(3) [2]

Chapter 4. Conceptual design of bridges for earthquake resistance

39

passenger comfort and makes redundant intermediate roadway joints and the maintenance theyentail. It also ensures the continuity of seismic displacements of the deck and prevents impactbetween adjacent spans under seismic actions. Finally, it serves as a sacrificial seismic linkbetween the two spans against span drop-off after unseating. It is of note that the precastgirders of the twin 2.3 km-long Bolu viaduct were spared from dropping and triggering acascading collapse despite their unseating in the Duzce (TR) 1999 earthquake (Figure 4.2(a)),due to their continuous topping slab. Part 2 of Eurocode 8 (CEN, 2005) makes specific referenceto such continuity slabs and their modelling.

It is normal practice to support prefabricated girders on a transversely stepped top surface of thepier or the bent-cap or on corresponding concrete plinths, in order to achieve a transverse slope


Figure 4.1. Precast girders simply supported on pier and connected for continuity via topping slab

Pier

Cross beam

Slab cast over the webs on a

2 cm expanding polystyrene layer

Bearings

Figure 4.2. Bolu viaduct in Duzce (TR) 1999 earthquake: (a) unseating of precast girders; (b) suspension

from the continuity top slab prevented span drop-off

40

of the top surface of the deck without differentiating the depth among the girders. By contrast, astep of the pier top in the longitudinal direction is not a proper means to accommodate a differ-ence in depth between two adjacent deck spans simply supported on the pier as, during thelongitudinal seismic response, the deeper of the two decks may ram the step. Instead, thedepth of the shallower deck should be increased over the support (through a deeper cross-head) to that of the deeper, so that both can be supported at the same horizontal level.

4.2.2 Uniform seismic demands on piers – piers of different height4.2.2.1 IntroductionFor reasons of aesthetics, all piers or pier columns of a bridge usually have the same cross-sectional dimensions. If the bridge has several piers with the same type of rigid connection tothe deck (i.e. all monolithically connected, or supporting it on fixed bearings) as in option 2 ofSections 2.3.1 and 4.1, differences in pier height are translated into differences in pier flexibilityin a given horizontal direction (longitudinal or transverse), as flexibility is approximately pro-portional to the third power of the pier height. This has certain implications for the longitudinalor the transverse seismic response of the bridge. Some of these are unfavourable, and should beavoided at the conceptual design stage, as highlighted in the following sections.

4.2.2.2 Conceptual design of bridges with piers of different heights for favourablelongitudinal response

The longitudinal inertia forces on an approximately straight deck (even one along which thetangent to the axis does not change direction by more than 608) are about collinear. Owing tothe high axial rigidity of the deck, no matter where they originate, these forces are shared bythe individual piers (approximately) in proportion to their longitudinal stiffness. If the piercolumns have the same cross-sectional dimensions, shorter ones will undertake larger longitudi-nal seismic shears and develop higher seismic moments (which are approximately inversely pro-portional to the square of the pier height), requiring more vertical reinforcement than the rest.This will further increase the effective stiffness of the shorter piers (cf. Section 5.8.1), and maylead to a vicious cycle. In addition, regardless of the exact amount of their reinforcement, theshorter piers will yield earlier and develop larger ductility demands than the others, possiblyfailing sooner. Note that the shorter piers are normally towards the two ends of a long deck,and, if rigidly connected to it, they constrain its thermal, creep and shrinkage deformations,inducing in the deck high tensile forces and suffering themselves from the associated longitudinalshears. The measures proposed below for the mitigation of non-uniform longitudinal seismicdemands in piers are quite effective in reducing these longitudinal constraints and their effects.

Conceptual design offers various ways around the problems posed by different pier heights:

g If the height differences are rather small, the free height of the shorter piers may beincreased to be approximately the same as in all others. The added height may be in anopen (preferably lined) shaft under grade. The base of these piers should always be easilyaccessible for inspection and repair of any damage, and above groundwater level.Figure 4.3 shows an example.

g If the pier heights are very different, rigid connection of the deck to the piers (monolithicor through fixed bearings) may be limited to a few piers of about the same height –normally the tallest ones. The deck may be supported on all other piers via bearings thatare flexible in the longitudinal direction (elastomeric or sliding). Often, the tallest piers arearound the deck mid-length; so, this choice helps to relieve the stresses building up in thedeck and the piers due to the thermal and shrinkage movements of the deck in thelongitudinal direction. A typical example is the bridge in Figure 4.4. The deck iscontinuous from abutment to abutment, with a total length of 848 m for the east-boundcarriageway and 638 m for the west-bound one, both with a radius of curvature of 450 m,interior spans of about 55 m and end ones of about 44 m. It was cast span-by-span on amobile casting girder launched from pier to pier. Each deck is monolithically connected tothe five centre-most and tallest piers, but tangentially sliding on the rest and at theabutments (see Figure 4.5).

g The cross-section of the shorter piers and of the upper part of the taller ones may bechosen to present much smaller lateral stiffness in the longitudinal direction than the lowerpart. In this way, the longitudinal stiffness of the piers can be balanced despite substantial

Clauses 2.4(4), 2.4(6),

2.4(7) [2]


41

differences in height. A usual choice for the longitudinally flexible part of the pier is a‘twin blade’ consisting of two parallel wall-like piers in the transverse direction. The lowerand stiffer part (possibly of very different heights in various piers) may be a hollow box.Figure 4.6 shows a schematic, and Figure 4.7 a real example, of a balanced cantileverconstruction (where the ‘twin-blade’ piers offer additional advantages). Depending on therelative length of its ‘twin-blade’ and hollow box parts, plastic hinges may form either atthe very base of the pier or at the base of each of the individual columns of the upper‘twin-blade’ part, or at both, one after the other. These possibilities should be taken intoaccount in the capacity design of the pier. All these potential plastic hinge regions(including the top of the individual columns of the ‘twin blades’) should be detailed forductility.

g If the deck is supported on all piers through elastomeric bearings, the stiffness of pierswith different heights may be harmonised by tailoring the total thickness t of the elastomerso that the bearing stiffness Kb ¼ GA/t counterbalances the difference in pier stiffness, Kp,giving approximately the same composite stiffness from Eq. (D2.10) for all piers (see point2 in Section 4.3.3.5). Note, however, that the large flexibility of the elastomeric bearingscontrols the horizontal stiffness of such bridges (see also Section 2.3.2.5 of this Guide).

g If the section of pier columns is hollow, its thickness may be adjusted to balance thedifference in pier height and achieve either approximately uniform shear forces orapproximately uniform maximum moments among the piers. However, as the pier stiffnessis not very sensitive to the thickness of the hollow section, only small differences in thepier height can be the accommodated in this way.

4.2.2.3 Transverse response of bridges with piers of different heightsThe transverse inertial forces are distributed all along the deck. The seismic action effects theyinduce in piers depend not only on their relative transverse stiffness but also on their tributarydeck length and the in-plane flexural rigidity of the deck, which is normally high but does notdominate the transverse response as much as the axial deck stiffness does for the longitudinalresponse. Therefore, except in the special cases pointed out below, rigid transverse connectionof a long deck to all the piers produces a fairly uniform distribution of seismic demandsamong them and is therefore acceptable, or even preferred. Exceptions to this rule are listedbelow, alongside suggested conceptual design options.


Figure 4.3. Construction of the lower part of the left pier of Votonosi bridge (GR) in a shaft for about

equal pier heights. (Courtesy of Stathopoulos et al. (2004))

L = 490.00

230.00130.00 130.00

∅10

∅12

∅10∅10

∅10

∅1225

.00

20.0

0

25.0

0

5.50 5.50 5.

50

~45

.00

~47

.00

20.0

0

~13

.50

13.5

0

42

g Relatively short bridges (e.g. overpasses of three to five spans) with transversely flexiblepiers and stiff abutments. If the intention is to resist the seismic action through ductilebehaviour of the piers (i.e. not as in an ‘integral’ bridge), the deck should be transverselyunrestrained at the abutments.

g Longer bridges, with very high transverse stiffness of the abutments and of the nearbypiers compared with the others. Rigid connection of the deck to all these stiff supportsmay lead to a very unfavourable distribution of transverse shears among the supports, asshown in Figure 4.9. The connection should be made transversely flexible either at theabutments or at the nearby piers.

g Bridges with one or more secondary piers that serve the deck erection procedure.Figure 4.10 shows an example of a balanced cantilever deck with a side span (on the left)much longer than the central span. The side span is supported on an intermediate shortpier, sliding on it in both horizontal directions to avoid the unfavourable effects of atransversely rigid connection.

4.3. The choice of connection between the piers and the deck4.3.1 Introduction: the effect of the construction techniqueThe fundamental choice is between connecting the deck monolithically with the piers andsupporting it on them through bearings – fixed (hinged, articulated) or movable (sliding orelastomeric) – or even via special isolation devices.


Figure 4.4. Krystalopighi bridge (GR), with the deck monolithically connected to five central piers and

free to move tangentially on five or three piers near each end

43

For the type of bridges and the range of spans addressed herein, the piers are made of concrete.They can be monolithic with the deck, but only if the deck is of concrete as well. If it is of steel orcomposite (steel–concrete), it can only be supported on the piers via bearings.

The construction technique adopted for a concrete deck may dictate the choice between mono-lithic connection and support on bearings. As shown in Table 4.1, prefabricated girders arenormally supported on bearings (unless they are made integral with the top of the pierthrough a cast-in-situ crosshead). Decks that are incrementally launched from the abutment(s)are also supported on the pier tops during the launching through special temporary sliders appro-priate for that operation. After the launching is completed, the deck is jacked up to replace thesesliders with the final bearings. It is not practical to try there monolithic connections. Casting thedeck span by span on a mobile girder supported on and launched from the piers (as shown in theupper right-hand corner of Figure 4.4) is convenient both for monolithic connection and forsupport on any type of bearing. Concrete decks erected as balanced cantilevers are normallymonolithic with the pier, to stabilise the cantilever during construction. There are, however


Figure 4.5. Piers for the bridge shown in Figure 4.4: (a) shorter ones near the ends of the deck,

supporting it without tangential restraint but with fixity in the radial direction; (b) taller piers near the

centre, monolithically connected to the deck

3 × 4 = 12 piles

∅120

11 1

1

3 × 5 = 15 piles

∅120

11

Figure 4.6. Schematic of a bridge with a ‘twin blade’ section in the upper 30 m of unequal piers and a

box section over the lower part (Bardakis, 2007)

91.00

7.00 7.00

7.40

7.40

1.50 1.50

144.00 144.00 144.00 91.00

30.0

010

.00

30.0

010

.00 30

.00

30.0

0

30.0

030

.00

44

cases, where a movable connection (with displacement and rotation in a longitudinal verticalplane) is desired for the final bridge, allowing, inter alia, seismic isolation. The deck may thenbe supported on the pier during erection through bearings (usually temporary), with the decksegment right above the pier head temporarily tied down to it (see Figure 4.8 for an example).As another option, a span of a deck on bearings may be segmentally erected while cantileveringout from a previously completed span, or with the segments suspended from a temporary pylonthrough stays.

4.3.2 Monolithic connectionThe simplest and most cost-effective connection of the piers to the deck is a monolithic one. Fromthe point of view of aesthetics, it is best to directly connect the pier column(s) to a cross-beamincorporated within the depth of the deck, no matter whether it is a solid or voided slab, asingle- or multiple-box girder, or even a multiple T-beam deck. Monolithic connection ofa concrete deck to the piers is very common in Japan and the seismic prone areas of theUSA, but less common in Europe. The construction technique aside, it offers the followingadvantages:


Figure 4.7. ‘Twin blade’ construction of the upper part of all piers of Arachthos bridge (GR) to reduce

and harmonise pier stiffness (see Figure 4.8 for the outer, shortest piers of the bridge)

Figure 4.8. Temporary deck tie-down over the support on the shortest pier of Arahthos bridge (GR) via

four elastomeric bearings for stability during free cantilever erection (see Figure 4.7 for an overall

illustration of this bridge)

45

1 Wherever it can be applied, it is the cheapest and easiest connection of a concrete deck tothe piers, as the costs of special devices (bearings or isolation devices) and their inspection,maintenance and possible replacement are avoided. Generally, it is preferable to anarticulated connection with bearings fixed in both directions.

2 It is the best way to prevent unseating and drop-off or large residual displacements of thedeck that would be difficult to reverse in order to restore full operation of the bridge.

3 It lends itself best to design for ductile behaviour: energy dissipation and ductility candevelop not only at the bottom of the piers but at their tops as well (this does not apply tosingle-column piers under transverse seismic action).

4 Fixity to the deck reduces slenderness and second-order effects in tall piers (except forsingle-column piers in the transverse direction, if the deck is not laterally restrained at theabutments).

There are serious disadvantages as well:

(a) For more than two spans, the piers restrain longitudinal movements of the deck due tothermal actions and concrete shrinkage. As a result, significant axial tension may build upin the deck, alongside large bending moments and shears in the deck and in the piersthemselves. The longer the deck, the largest are the stresses due to the restraint of imposeddeformations. Ways to mitigate this problem have been discussed in Section 4.2.2.2.

(b) Monolithic connection precludes the use of seismic isolation and/or supplemental energydissipation.

(c) In the ‘joint’ between the deck and the pier column, shear stresses due to the seismicaction are large; bond demands along pier vertical bars anchored there or along deck barspassing through or terminating in the joint are high. Designing and detailing the joint forthese demands is not trivial.


Figure 4.10. Secondary short pier serving the erection of the long side span of Mesovouni bridge (GR)

Figure 4.9. Abnormal distribution of seismic shears to supports (bottom) in bridge with transversely very

stiff outer piers and abutments (top)

Elevation

Plan

46

4.3.3 Support on bearings4.3.3.1 Types and structural functions of bearingsSupporting the deck on bearings is very common in Europe and Japan, but rather rare in theseismic prone areas of the USA.

All types of bearings considered here have one common structural function: their articulationcapability. The vertical force to be transmitted is concentrated within the bearing supportarea, within which the distribution of compressive stresses is essentially uniform. This iseffected by practically eliminating the rotational resistance within any vertical plane (articu-lation). To this end, the normal force is transmitted either through two concentric sphericalsteel surfaces, one sliding over the other (in spherical bearings), or through a layer of resilientmaterial presenting a sufficiently small rotational stiffness. This material can be either a laterallyencased non-laminated elastomer (as in pot-bearings) or a non-encased laminated elastomer(in elastomeric bearings where the rotational stiffness is not negligible but acceptably low forpractical purposes).

Depending on their structural role, the following types of bearings are commonly used:

1 ‘Fixed’ bearings, which are spherical or pot bearings that do not allow relativedisplacement of the end bearing plates. Pot bearings are rather common in Europe, andoccasionally used in Japan, but rarely in the USA. Spherical bearings are moreuncommon.

2 Bearings sliding in all directions, to allow free displacement in any direction in the plane ofthe bearing (plane of sliding). These are a fairly common type of movable bearing inEurope or Japan, but rather rare in the USA. The bearing comprises one stainless steelplate that can slide on another (the sliding plate), which is in turn anchored to the concreteand has larger plan dimensions to accommodate the displacements. The sliding interfacematerial is usually lubricated PTFE, for low friction. Such bearings do not have rotationalcapability. Sliding-cum-rotational capability can be achieved by adding a sliding plate toone of the end plates of a spherical or pot bearing or to the end plate of an elastomericbearing pad (see below).

3 Bearings sliding in only one direction. This is effected by introducing a shear key guidebetween the sliding plates of a bearing as above. Owing to the high contact pressure,austenitic stainless steel and hard ‘composite material’ are commonly used to form themating surfaces of the guide.

4 Laminated elastomeric bearings, presenting a low and elastic shear reaction to substantialdisplacement in the plane of the lamination, according to their low shear stiffness,Kb ¼ GA/tq (where A denotes the horizontal section area of the bearing, and tq and G thetotal thickness and the shear modulus of the elastomer, respectively). These are the mostcommon type of bearing throughout the world (high damping rubber – HDR – is mostoften used for them in Japan, to enhance energy dissipation). If the deck displacement isgoverned by the low horizontal stiffness of elastomeric bearings (option 1 in Sections 2.3.1or 4.1 of this Guide), the fundamental period of the bridge is lengthened and seismic forcedemands are reduced. This is the simplest way of achieving seismic isolation, and isrecognised in Part 2 of Eurocode 8 as such.

5 Shear-key type of devices. Factory-produced devices termed shear keys or pins are fairlycommon in Europe. They are capable of restraining the relative displacement in one orboth directions, transferring shear forces. The materials used for the mating surfaces arethe same as for the guides in bearings of type 3 above. As these devices do not transfer anynormal force, they are not bearings in the strict sense of the word. However, as they aim atproviding the other structural functions of fixed bearings, they share their problems andlimitations highlighted in Section 4.3.3.4. They should allow free rotation about the twohorizontal axes and the vertical under various loadings or imposed deformations. Suchrotations, especially about the vertical axis, may be difficult to predict in long curvedbridges, erected and post-tensioned span by span, with different fixed points at eacherection stage.

Least expensive of the above bearing types for the same axial load capacity are the elastomericbearings, with unanchored ones being much cheaper than anchored. Bearings sliding in any


47

direction come next, with those having an elastomeric component for freedom to rotate againbeing less expensive. The cost of bearings fixed in one or both directions increases with the magni-tude of their shear force capacity. Spherical bearings are more expensive than pot bearings, butpresent less resistance against and a higher capability of rotation.

4.3.3.2 Special seismic design requirements for bearings and limitations thereofThe problems and limitations highlighted here arise primarily, but not exclusively, from seismicdesign requirements. The higher, therefore, the design seismic action, the more acute thesedemands and limitations.

4.3.3.2.1 Seating on the deckSufficient overlap between supported and supporting elements should be provided at all movablesupports and directions of movement, unless fixed bearings (or equivalent devices) are placedthere to preclude relative movement. The minimum overlap length per Part 2 of Eurocode 8 isestimated according to Section 6.8.1 of this Designers’ Guide. Seismic links (e.g. via concreteshear keys) are not required in addition. Fixed bearings or equivalent devices are designed tocarry the increased seismic action effects derived from capacity design (see case (b) in Section6.3.2 of this Guide). Action effects derived directly from the seismic analysis can be used forthe design of fixed bearings only if corresponding seismic links, capable of these capacitydesign action effects, are additionally provided.

4.3.3.2.2 Transfer of horizontal forces to the concreteThe transfer of the vertical force of the bearing to the concrete follows the relevant rules ofEurocode 2, and is normally taken into account in the generic design of the bearing by itsmanufacturer. In the horizontal direction where relative displacement is not free, the designshear force of fixed bearings or equivalent devices and of anchored elastomeric bearingsshould be safely transferred from the end bearing plate to the concrete. This is usually doneby means of anchor bolts, normally designed by the bearing manufacturer. The transfer to theconcrete of concentrated shear forces alongside the normal ones is not trivial, as reflected bythe complexity of the relevant design rules in standards and guidelines (e.g. see CEN, 2008;fib, 2011). For bearings, this transfer is effected through the dowel and fastening action ofanchors, and depends heavily on the resistance along potential failure surfaces in the concretesurrounding the anchor and extending to the free surface of the concrete element. The genericbearing design cannot cover all the possibilities. This can only be done for each individualdesign case.

The higher the design seismic shears to be transferred, the harder the design of the anchors.High shear forces are quite usual, not only owing to capacity design but also because the useof a single transversely fixed bearing at the centre of each support is preferable, to avoid theuncertainty about the distribution of transverse reactions to several fixed bearings and therestraint to imposed deck deformations. By contrast, usually two or more bearings are placedfor the transfer of vertical forces; for example, one bearing underneath each web of the decksection, especially under the outer ones to minimise the vertical reactions due to the transferof torsional moments from the deck. So, in most cases the transfer of transverse horizontalforces is separated from that of the vertical ones, using the special shear-key devices of point 5of Section 4.3.3.1. As a result, the design seismic shears to be transferred to the concrete mayeasily reach 50% or more of the corresponding vertical forces. If this transfer is not feasible,vertical sliding bearings may be used instead to transfer the transverse horizontal forces. Anexample is shown in Figure 4.5(a) for the tangentially movable supports of a bridge. Thissolution is not problem-free either, as it requires provisions for the replacement of partssubject to ageing, damage, contamination, etc.

4.3.3.2.3 UpliftingHolding-down devices are normally needed to reliably prevent uplift of all the bearings at thesame deck support (on a pier or abutment) when there is no sufficient safety margin againstthis risk in the seismic design situation. Individual bearings at one support do not have tomeet the relevant requirement of Part 2 of Eurocode 8 for increased safety. Nevertheless,uplift under the action effects from the analysis in the seismic design situation should beprevented at every single bearing.


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4.3.3.2.4 Inspection, maintenance and replacementAll bearings should be accessible for inspection after an earthquake and under normal serviceconditions, as well as for maintenance and – partial or full – replacement, if needed. To thisend, the design should provide for sufficient room around the bearing for the replacementworks, including the placement against the concrete surfaces of jacks, props and other temporarymeans of support and strength. The bearing itself should also provide the appropriate separationand connections between permanent and replaceable parts.

4.3.3.3 General advantages and disadvantages of supporting the deck on bearingsThe construction technique aside, supporting the deck on bearings has its own advantages anddisadvantages. The pros and cons depend on the type of bearings used. On the pro side, nomatter the type of bearing:

1 Construction is in certain cases simpler than for monolithic connection.2 Seismic stresses in the deck are diminished, compared with those in and around a

monolithic deck-column ‘joint’; for the longitudinal seismic action, they are almosteliminated.

3 Single-column piers have similar behaviour in the longitudinal and transverse directions; ifspectral accelerations in these two directions are similar, pier seismic moments and shearswill be similar as well, allowing optimal pier design.

There are general cons as well:

(a) Bearings require inspection, maintenance and occasional replacement.(b) If the pier comprises more than one column connected by a cap-beam (a ‘bent cap’ in the

USA), its columns are not optimally used for earthquake resistance: in the longitudinaldirection they work as vertical cantilevers, while in the transverse direction their top isnearly fixed to the cap-beam.

Further pros and cons depend on the type of bearing, as highlighted below.

4.3.3.4 Fixed bearingsThe advantages offered by fixed (hinged, articulated) bearings are parallel to some of those ofmonolithic connection:

1 To the extent they do not fail during the earthquake and are used over several piers, theycan prevent unseating and drop of the deck, as well as large residual displacements,therefore facilitating restoration of full functionality of the bridge. However, they are a lessreliable means of achieving these goals than monolithic connection with some piers.

2 They are the only means, apart from monolithic connection with the deck, to mobilise theductility and energy dissipation capacity of piers. Of course, it is at the base (and not thetop) of only those piers that support the deck through at least one fixed bearing thatductility and energy dissipation develop.

They also share with monolithic connection cons (a) and (b) below:

(a) Significant axial forces build up in the deck, alongside large bending moments and shearsin the piers, owing to the longitudinal restraint of deck thermal movements and concreteshrinkage. In general, the problem is less acute than for monolithic deck/pier connection,and can be mitigated as suggested in Section 4.2.2.2 in the second and third bullet points.However, as the deck is free to rotate with respect to the pier in a vertical plane throughthe longitudinal direction, the deck bending moments due to prestress and to a verticaltemperature difference component across the deck are lower.

(b) As with monolithic connection, it is not feasible to combine with seismic isolation and/orsupplemental energy dissipation.

In addition:

(c) Fixed bearings are in general more demanding for inspection and maintenance and morecostly than the other types of bearings, except for special isolation devices.


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Bearings that are fixed in one horizontal direction (the transverse) and movable in the other (thelongitudinal) offer advantages 1 and 2 above in the direction where they are most crucial – thetransverse – while avoiding disadvantage (a) in the direction where it is relevant, namely thelongitudinal.

4.3.3.5 Elastomeric bearingsLaminated elastomeric bearings offer the following special advantages:

1 Owing to their very low horizontal stiffness, Kb ¼ GA/tq, they do not materially obstructthe longitudinal movement of the deck due to thermal actions, concrete shrinkage or (forprestressed decks) creep. However, near the ends of long continuous decks they may needto be quite thick, in order to accommodate the deck longitudinal movements (seedisadvantage (c) below). To counter this drawback, sliding bearings (even elastomeric, witha sliding end plate at the top) may be used near the ends of the deck and anchoredelastomeric ones around mid-length.

2 They offer a simple way to harmonise the stiffness of piers with different heights, bytailoring the total thickness t of the elastomer so that the bearing stiffness Kb ¼ GA/tqoffsets the difference in pier stiffness, Kp, and give, in the end, about the same compositestiffness as Eq. (D2.10) for the different piers.

3 Unless they fail (see disadvantage (f) below), their behaviour is nearly elastic; they self-centre after an earthquake, with little residual displacements.

4 They are of low cost and can be easily replaced.5 They can serve as a simple means to achieve seismic isolation (and are recognised as such

in Part 2 of Eurocode 8): the fundamental period of the bridge is lengthened owing to thelow horizontal stiffness of the elastomeric bearings, and hence the seismic force demandsare reduced. In fact, they may be considered as the simplest and least expensive means forseismic isolation. If the elastomer is not of HDR, the ensuing increase in horizontal seismicdisplacement demands may be counterbalanced by supplemental (viscous) dampers. Such acombination may be competitive with special isolation-cum-dissipation devices.

6 Unlike sliding bearings, whose friction properties strongly depend on the rate of loading,the vertical stress and the condition of the sliding interface, the horizontal stiffness ofelastomeric bearings depends on well-defined and documented geometric and materialproperties. Hence, the designer is fairly confident about the seismic action effectstransmitted by the bearing to the underlying pier or abutment as determined by the analysis.

Some of the disadvantages (notably those under (a) and (e) below) are the converse of theadvantages of monolithic connection or of fixed bearings:

(a) They have such a high flexibility compared with the pier that they accommodate almostfully the seismic displacement demands, without letting the pier use any ductility andenergy dissipation capacity it may have. Hence, the behaviour is limited ductile (withq ¼ 1.50) and the bridge is designed according to Chapter 7 of this Designers’ Guide forbridges with seismic isolation. This, however, does not necessarily imply that the design isnot cost-effective.

(b) They are not a good choice over soft soils, because the ground motion may be rich in low-frequency components due to site effects and/or the piers may tilt or displace differentlyleading to deck unseating.

(c) To accommodate the deck longitudinal movements, they may need to be quite thick nearthe ends of long continuous decks. As noted under advantage (1) above, this drawbackfavours using them near deck mid-length and placing sliding bearings closer to the ends ofthe deck.

(d) They are not appropriate – or necessary – when the piers are tall and the fundamentalperiod of the bridge would anyway be long even for rigid connection of the piers with thedeck.

(e) They cannot prevent the deck from falling (i.e. such supports are considered movable). Toavoid such a catastrophic event, Part 2 of Eurocode 8 requires the minimum overlaplength of Section 6.8.1 of this Designers’ Guide.

(f ) If they exceed their deformation capacity – normally by debonding at the interfacebetween the elastomer and a steel plate, more rarely by toppling – the deck will develop


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large residual lateral displacements that will be hard to reverse in order to restore normaloperation of the bridge.

(g) They are subject to ageing; in addition, their behaviour during an earthquake may beaffected by past loading.

(h) They will tear under net vertical tension. To avoid this, Part 2 of Eurocode 8 includesrules for holding-down devices.

4.3.3.6 Sliding bearingsBearings with a horizontal sliding surface offer a single special advantage, in addition to thegeneral ones for bearings highlighted in Section 4.3.3.3:

1 As their friction coefficient is low under slow rates, they obstruct very little thelongitudinal movement of the deck due to thermal actions, concrete shrinkage or (forprestressed decks) creep.

However, their drawbacks are several and serious. They share with elastomeric bearings thelimitations listed under points (a), (b), (d) and (e) in Section 4.3.3.5. In addition:

(a) They have no elasticity to restore large lateral displacements, therefore leading tosignificant and essentially unpredictable residual drifts, if not used alongside other elasticdevices capable of restoring displacements. As pointed out in Section 4.3.3.5 underadvantage (1) and disadvantage (c), the single pro of simple sliders may be combined withpro (1) of elastomeric bearings, if the latter are used around mid-length of a longcontinuous deck while sliding ones are used near its ends.

(b) They may be more expensive than elastomeric bearings.(c) The value of their friction coefficient under dynamic rates is very uncertain and strongly

depends on the rate of loading, the vertical stress (increasing considerably when it is low),the condition of the sliding interface, etc. If the friction coefficient is well above zero, thebearing may induce significant seismic shears in the pier below. If the value is very low,the design of the bridge (seismic or not) cannot be confidently based on the contributionof a reliable lower bound of the force of the sliding bearings to the equilibrium of anycomponent of the structure (deck, pier or abutment). By the same token, such slidingbearings are not reliable means of energy dissipation during the seismic response (see theNote in Clause 7.5.2.3.5(3) of Part 2 of Eurocode 8).

(d) Apart from the large uncertainty in the value of its coefficient and its sensitivity to variousfactors, friction is inherently a nonlinear behaviour mode: the direction of the forcechanges with the sense of sliding although its magnitude may be constant. So, in theory, itcan be accounted for only through nonlinear analysis. However, this may not be possiblein practice, because a nonlinear analysis requires a well-defined set of external horizontalactions (braking, wind or seismic) combined with imposed deformation actions (daily andseasonal temperature variations, concrete shrinkage and creep). From the very largenumber of possible scenarios for this sequence, it is evidently unfeasible to foresee theworst one for the element considered. However, there are practical means to face thesedifficulties and achieve reliable results through simplified but safe-side estimation, basedon the following considerations:– As the friction forces of such sliding devices are relatively low, their direction should

always be selected so as to induce the worst effect for the case considered.– Although the magnitude of the friction coefficient mmax decreases with increasing

contact pressure sp (see Table 11 of CEN (2000)), the friction shear stress, mmaxsp keepsincreasing. Consequently, the maximum friction force corresponds always to themaximum vertical force on the bearing.

So, both the magnitude and the direction of the friction force can be easily estimated in asafe-side way for each case considered. In the seismic design situation in particular, thefollowing should also be borne in mind:– According to Part 2 of Eurocode 8 (Clause 7.2.5.4(7)), the value of mmax specified in

EN 1337-2 (CEN, 2000) may be used– The maximum normal and the corresponding maximum friction force of the bearing in

the seismic design situation are substantially lower than in the persistent-and-transient

Clause 7.5.2.3.5(3) [2]

Clause 7.2.5.4(7) [2]


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one, because in the former the partial and combination factors applied on permanentactions, Gk, and traffic actions, Qik, are equal to 1.0 or slightly above zero, respectively(see Eq. (D6.1) in Section 6.2 of this Guide), while in the latter they are both markedlygreater than 1.0.

– The seismic effects AEd may change the above picture. They comprise in general thefollowing components:� The effect of the vertical component of the seismic action.� The effect of the seismic action in the longitudinal direction (normally very small).� The effect of the seismic action in the transverse direction due to the overturning

moment about the longitudinal axis. As far as the total friction force acting on a pier,abutment or the deck is concerned, this effect is nil; by contrast, it may be quitesignificant (up to almost the full effect of gravity loads) for an individual bearing. Ifthe bearing is guided in the longitudinal direction, or it is a shear key device asmentioned in point 5 in Section 4.3.3.1, the friction on the guide or the shear key dueto the transverse seismic shear reaction should also be added. For this case, CEN(2000) specifies a value of mmax ¼ 0.2 for non-seismic design situations, no matter thecontact pressure. For the seismic design situation conditions, mmax should prudentlybe increased to 0.3.

4.3.3.7 Special isolation bearingsIt has been pointed out in Section 4.3.3.5 that simple elastomeric bearings are considered in Part2 of Eurocode 8 as a means for seismic isolation. If their elastomer is HDR or they are usedalongside supplemental (fluid viscous) dampers to counter the increase in displacements dueto the lengthening of the fundamental period, they may be considered to give a moreadvanced isolation system like those addressed in this section. The latter combine flexibility(to lengthen the period and reduce the spectral accelerations and hence the design seismicforces), with increased damping (to reduce the spectral displacements that accompany thelonger period). The damping may be provided by the isolation bearing itself (as in HDRbearings, in lead–rubber bearings, in sliders with a horizontal or spherical sliding surface andhigher and well-controlled friction properties, etc.), or by supplemental devices that intervenebetween the deck and the top of the pier (or abutment) and are subjected to their relativedisplacements without supporting vertically the deck. They may be fluid viscous dampers,ductile steel units dissipating energy through plastic deformations, special magneto-rheologicaldampers, etc.

Although the specific advantages and disadvantages depend on the isolation system chosen, somegeneral observations may be made. On the advantage side:

1 Like simple elastomeric bearings or sliders, there is very little obstruction to longitudinalmovement of the deck due to thermal actions, concrete shrinkage or (for prestressed decks)creep.

2 If the necessary skills are available and the necessary design effort is made, seismicisolation may be the most cost-effective seismic design option, especially in high-seismicityregions.

3 The devices themselves are subject to much stricter quality control than simple bearings,and their properties are better known and documented.

4 Isolation systems (at least if designed to Part 2 of Eurocode 8) have self-restoring featuresthat allow the deck to sufficiently recentre even after very strong shocks.

Special isolation systems share with elastomeric or sliding bearings the limitations listed under(a) and (b) in Section 4.3.3.3 and (b), (d), (e) in Section 4.3.3.5. A list of additional ones follows:

(a) They require special expertise and experience in design and analysis (which is normally ofthe nonlinear response-history type) and suitable analysis software.

(b) The cost of the devices themselves is relatively high; however, with the necessary designeffort and a judicious choice of the type of isolation system and its devices, the total costof the bridge may well be less.

(c) There is some uncertainty regarding the behaviour of the devices and of the bridge as awhole in case the displacement capacity of the device is exceeded. To counter this, Part 2


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of the Eurocode 8 requires increased reliability for the displacement capacity of theisolation devices, by multiplying the seismic displacement demand from the analysis by afactor of 1.50.

4.3.3.8 Concluding remarksAs repeatedly pointed out and exemplified in Section 4.3.3, the designer may choose to have onetype of connection or bearing at certain locations under the deck and to combine it with one ormore other types elsewhere, in order to maximise the benefit from their advantages and countertheir drawbacks to the maximum possible extent. One example is the use of monolithic connec-tion or elastomeric bearings around mid-length of long continuous decks and of sliding ones nearthe ends. As also noted (certain), bearings may be chosen to be fixed in one horizontal directionand movable in the other. In choosing among the various options available, the designer shouldkeep in mind that the connection of the deck as a whole to the piers may be considered as aparallel system, where the stiffer and stronger component(s) control the behaviour and thedisplacements of the bridge.

4.4. The piers4.4.1 Sections of pier columns for efficient detailing4.4.1.1 Solid circular columnsSolid circular pier columns are quite common around the world. A circular section has the samestrength and rigidity in every horizontal direction. So, it seems ideal for pier columns that workas vertical cantilevers in both directions (e.g. when the deck is supported on the pier through fixedor horizontally flexible bearings), or belong to multi-column piers monolithically connected to thedeck. In addition, it lends itself better than any other section to the efficient confinement of theconcrete and antibuckling restraint of vertical bars – in this case through circular hoops or a continu-ous spiral, with little need for cross-ties or link legs at right angles to the perimeter of the section.

A circular perimeter hoop or spiral contributes to the shear resistance of the section in diagonaltension with only p/2 ¼ 1.57 tie legs, not two. However, this is not a serious drawback of circularpier columns, because they are commonly quite slender and hence not critical in shear. Also, ifshear resistance in diagonal tension is critical, interior rectangular links or straight cross-tiesmay be added, engaging vertical bars across the full section. Such tie legs contribute to theshear resistance in diagonal tension with their full cross-sectional area, to be contrasted withcircular ties contributing with a fraction of p/4 ¼ 0.785. This is less material consuming than areduction in the spacing of hoops or spirals or the arrangement of the vertical bars in twolayers with a second hoop – in addition to the perimeter one – in-between.

Solid circular sections with a large diameter (e.g. over 3–4 m) are normally avoided, not only asthey are not cost-effective compared with hollow circular ones but also because of concerns abouttheir large volume of unreinforced concrete.

It is sometimes considered harder to mesh the two-way top reinforcement of a pile cap or spreadfooting or the bottom one of the deck or a column cap-beam with the dense vertical reinforce-ment of a circular column, compared with that of a column with sides parallel to those of thereinforcing mesh in the foundation element.

4.4.1.2 Solid rectangular columnsCompact solid rectangular pier columns are common in Japan, but less so in Europe, and rare inthe USA. The detailing of large sections for plastic hinging is not cost-effective compared withcircular sections, because it requires many long cross-ties or link legs at right angles to thesides in order to mobilise the straight perimeter stirrup for confinement of the concrete and torestrain against buckling the large number of vertical bars arranged along the side. This isfeasible only if the section is relatively small and does not require unduly long cross-ties or inter-mediate link legs. If a large square section is chosen over a circular one (e.g. for convenience ofthe formwork), its vertical bars may be arranged in a ring and engaged by circular hoops or acircular spiral. The four corners of the section are unreinforced and would rather be chamfered.This idea may be extended to solid rectangular sections with a ratio of the two sides of around1.5:1, employing interlocking circular hoops or spirals. Again, the four unreinforced cornersof the section are chamfered.

Clauses 6.2.1.2,

6.2.1.4(4) [2]


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Clause 6.2.4(2) [2]

Clauses 6.2.4(2),

6.2.4(3) [2]

4.4.1.3 Wall-like piersRectangular wall-like piers, with the long dimension in the transverse direction of the bridge, canprovide continuous support almost all along the width of the deck. So, they are quite convenientunder thin concrete decks. They are popular in Japan and becoming increasingly so in Europe,but not so much in the USA. Owing to the large cross-sectional area, the level of vertical stress isquite low, and hence confinement is not essential, especially in the weak direction of the pier,where the shear-span-to-depth ratio is high and favourable for ductility. Nevertheless, it canbe conveniently provided by engaging pairs of vertical bars along the two long sides of thesection by fairly short cross-ties or hoop legs at right angles to the long side of the section.The intermediate vertical bars along the two short sides are normally few; they can be laterallyrestrained through short oblique cross-ties or link legs also engaging nearby bars on the longsides.

In the longitudinal direction of the bridge, such a wall is quite ductile owing to its slenderness.This may not be the case in the transverse direction. Indeed, if in that direction the shear span(moment-to-shear ratio) is less than three times the depth of the section, and the pier top isrigidly connected to the deck, the q factor for ductile behaviour is decreased to reflect thereduced ductility (cf. Table 5.1 in Section 5.4 of this Guide). However, the moment and shearresistances in the strong direction of such a pier are large enough to resist the design seismicaction effects even for a reduced q factor.

The main drawback of wall-like piers is the very high design seismic action effects they maydeliver to the foundation. If derived from capacity design, these action effects reflect the largeoverstrength of the plastic hinge in the pier, leading to almost ‘elastic’ seismic action effects inthe foundation (i.e. with q ¼ 1.5). An option that may prove more cost-effective is to use pilesfor the foundation and allow plastic hinges in them for seismic design in the strong directionof the pier (using the value q ¼ 2.1 in this case).

Pairs of parallel pier columns with a wall-like section (‘twin blades’) are often used to supportdecks built as balanced cantilevers (see Figures 4.6 and 4.7). The pair of pier columns is quiteeffective in stabilising the deck during free cantilever construction, but – owing to the slendernessof the individual columns – have low longitudinal stiffness and reduce the longitudinal stressesthat develop in the deck due to restraint of thermal or shrinkage strains. As pointed out in thelast bullet point of Section 4.2.2, if the various piers of the bridge have very different heights(as in Figures 4.6 and 4.7), all the piers may be chosen to have this type of ‘twin-blade’ sectionover a certain length of their upper part, while the lower part – of very different height amongthe piers – may have a hollow box section (see Figure 4.6).

4.4.1.4 Hollow rectangular piersTall piers have by necessity a large section. Large sections are normally constructed as hollow, toincrease the strength and the rigidity (against second-order effects as well) for a given volume ofconcrete. The reduction in pier weight and mass – for a given strength and rigidity – decreasesinertia forces and the vertical loads on the foundation. Hollow rectangular sections are quitecommon, especially in Europe and Japan.

The vertical bars are arranged all along the outer and inner perimeters, with particular concen-tration at the corners. Pairs of vertical bars across the thickness of the box section are engaged byfairly short cross-ties or at the corners of closed links placed within the thickness of the section(see Figure 4.11 for examples). Part 2 of Eurocode 8 requires a web thickness in the plastic hingeregion of at least one-eighth of the clear distance of the webs framing in the other direction, toprevent local buckling when the web acts as compression flange. When the pier is slender inone horizontal direction, the thickness of the pair of webs parallel to that direction may becontrolled by the provisions of Eurocode 2 for slenderness and second-order effects. If it isnot slender, ultimate limit state (ULS) resistance in shear (against diagonal compression) maygovern the web thickness. Note also that, at least for the most common layouts of transversereinforcement shown in Figure 4.11, only four link legs run the full depth of the section ineach horizontal direction, and count towards the shear resistance of the section in diagonaltension.


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It is possible to vary the thickness of the section along the pier or between piers to suit better thestrength and stiffness needs of the pier and the bridge, without changing the outer dimensionsand appearance of the pier.

What has been said for hollow rectangular piers can be readily extended to polygonal hollowpiers.

4.4.1.5 Hollow circular piersHollow circular (annular) sections are less common for tall piers than hollow rectangular or poly-gonal ones (especially in Europe). The main problem for the use of hollow circular sections inductile piers is that circular hoops cannot confine the inner face of the section: the circumferentialtensile force in such hoops produces radial deviation stresses pointing away from the confinedconcrete core, and may cause loss of the cover concrete and implosion of the inner face of thesection. So, the hoops around the inner face can serve only as shear reinforcement. To confinethe inner face, radial cross-ties or link legs should be placed across the thickness of theannular section, as in a hollow rectangular one. Alternatively, the thickness should be largeenough to keep the strain at the inner face below the ultimate strain of unconfined concrete(1cu ¼ 0.35%) even when the ultimate curvature is attained and after spalling of the concretecover at the outer face. The lower limit of one-eighth of the inner diameter imposed by Part 2of Eurocode 8 on the thickness of ductile annular piers against the occurrence of local wallbuckling unintentionally helps in protecting the inner perimeter of the section from largestrains at ultimate curvature. Note, though, that radial cross-ties or link legs across the thicknessare useful to supplement the outer face hoops, if the large diameter of the section reduces theireffectiveness for confinement and against buckling of the vertical bars.

As in hollow rectangular piers, the thickness of the section may vary up the pier or between piersto adapt the stiffness and the resistance of the pier to the seismic demands without affecting theaesthetics of the bridge.

4.4.2 Single- versus multi-column piersExcept for the ‘twin-blade’ piers often used in balanced cantilever bridges for the reasonsdiscussed in previous sections, tall piers are normally made of a single column, to increaserigidity and strength and reduce slenderness for given concrete volume. When the pier is rela-tively short and the deck wide or shallow, the designer has the option of using either a wall-like pier (with the limitations pointed out in Section 4.4.1.3) or a pier with more than onecolumn across the deck.

If bearings of any type – fixed, hinged, sliding, elastomeric or even isolation bearings – supportthe deck at the top of the pier, a single-column pier is an efficient choice, as it is effectively a

Clause 6.2.4(3) [2]


Figure 4.11. Examples of reinforcement arrangement in box piers. (Courtesy of Mechaniki SA)

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vertical cantilever in any horizontal direction. If there is more than one bearing across the deck, acapital or a hammerhead on top of a single-column pier can accommodate them. If the bearingspresent the same stiffness in the transverse and the longitudinal directions, a single circular pierwill be subjected to very similar seismic shears and moments in these two directions, and may bevery cost-effective.

The single-column option is simple and the behaviour is clear, but the redundancy is low.

If the pier is monolithically connected to the deck, then in the longitudinal direction all of itscolumns have their top nearly fixed and a shear span slightly longer than half their free height(see Eq. (D5.4b) in Section 5.4 of this Guide). If the pier has a single column, in the transversedirection it works as a vertical cantilever – in fact, its shear span is longer than its free heightowing to the rotational inertia of the deck (see Eq. (D5.5) in Section 5.4). Such a single-column pier should have a significantly larger resistance and rigidity in the transverse than inthe longitudinal direction, and is not a very cost-effective option. If two or more columns areused across the deck, they will all have the top essentially fixed against rotation. Their shearspans will be similar in the transverse and longitudinal directions, and their design seismicmoments for transverse seismic action will be minimal (equal to the seismic shear force multipliedby one-half the column clear height, in lieu of the full clear height in a single-column pier).Additionally, energy dissipation and ductility demands will be shared by plastic hinges at thetop and the bottom of each column, for both transverse and longitudinal seismic action.Needless to say, a circular cross-section is the optimal choice for multi-column piers.

As pointed out in Section 4.3.2, if the pier (column) is monolithically connected with the deck,aesthetics suggest a connection with a cross-beam that is fully incorporated within the depthof the deck. If the deck is shallow without deeper and protruding cross-beams, it is not easy toverify and detail its joints with pier columns for ductile behaviour of the bridge with plastichinges forming at column tops, unless the columns are also small and their number larger.This will reduce the proportion between the moment transferred by each connection and theeffective volume of the connection (which extends into the deck beyond the perimeter of thecolumn).

Connecting the head of a multicolumn pier to the deck through fixed bearings is not as efficient asa monolithic connection. Elastomeric bearings are preferred. A continuous slab or box-girderdeck may be directly supported on the column tops through elastomeric or seismic isolationbearings. If the deck consists of precast girders, the bearings are usually supported by a transversecap-beam framing at the tops of the columns (forming a ‘bent’ in US terminology), with frameaction in the transverse direction and free-cantilever action in the longitudinal. The cap-beam isnormally deep, to allow straight anchorage of the column bars within its depth, and hence stiff.So, the top of closely spaced columns is nearly fixed against rotation in the plane of the multi-column pier. When the cap-beam supports only horizontally movable bearings, the multi-column pier should be designed for limited ductile behaviour with q ¼ 1.5. Should even one ofthe bearings be fixed, the multi-column pier should be designed and detailed for ductile behaviourwith a q value larger than 1.5. At any rate, a multi-column pier supporting the deck through a lineof bearings may be a simple choice, but is not so cost-effective. A layout that should be avoidedfor the piers is a portal frame of two columns spaced apart by more than the deck width,supporting a cap-beam that is integral with the deck (i.e. fully or partly within the depth ofthe deck). Although such a layout is common in the USA and occasionally used in Japan,Part 2 of Eurocode 8 does not address it. A major source of uncertainty and a weak link insuch a portal frame is the outriggering length of the cap-beam between the deck and the piercolumns.

Overpasses with three to four spans are most often supported on piers with several slendercolumns each, which may be hard to design for resistance against impact of heavy traffic vehicles.

A multi-column pier (be it monolithic with the deck or supporting it through a string of bearingson a cap-beam) at a skew to the bridge provides a more complex support condition to thedeck than a single-column pier. A skew bridge may be avoided over a skewed crossing, if theabutments are placed at right angles to the deck axis and single-column piers are used.


56

4.4.3 Sizing of pier columns4.4.3.1 General sizing criteriaTo complete the conceptual design phase, once a section shape is chosen for the pier columns,their size should be selected. This section covers general criteria for the selection of thecolumn size, pertaining to general design situations, including seismic ones for bridges oflimited ductile behaviour. Section 4.4.3.2, by contrast, refers specifically to limitations imposedby seismic design for ductile behaviour.

As repeatedly said in this chapter, for aesthetic reasons normally all the pier columns of a bridgehave the same outer dimensions. Choosing the same shape for different piers or pier columns butdifferent cross-section sizes may be aesthetically worse than having a different shape altogether.

General criteria for the determination of the dimensions of the pier columns include:

g practical considerations, taking into account the construction technique and procedure forthe piers themselves and the deck, as well as the connection to the deck

g the slenderness of the column, in order to limit second-order effects under factored gravityloads (in the persistent and transient design situation of the completed bridge), or,preferably, to be able to ignore them as small.

Concerning the construction procedure, often intermediate stages before the deck completionmay be more critical for the stability and design of the piers. In other cases, the geometry ofthe piers is conditioned by the need to accommodate and support special and heavy equipmentfor the erection of the deck. That said, for the bridge to be cost-effective, the choice of its struc-tural system should strike a rational balance between the requirements of design situationspertaining to intermediate construction stages and to those of the completed bridge; the lattershould normally govern the design.

A column section depth greater than that of the pile cap or spread footing underneath may makeit harder to verify their joint region for the capacity-design action effects of Section 6.4.4 of thisGuide. By the same token, a column monolithically connected to the deck should have a sectiondepth in the plane of fixity to the underdeck (i.e. in the longitudinal direction, and for multi-column piers in the transverse as well) less than the deck itself. If the deck is supported onbearings, the pier top should be sufficiently wide to accommodate the bearings, the requiredseat lengths (see Section 6.8), any shear keys and the clearance to them. To this end, a hammer-head or a capital may be added to the pier top, or the pier may be flared upwards.

Slenderness considerations are crucial for tall piers. In a bridge designed for earthquakeresistance it is natural to size the pier columns so that second-order effects in the persistentand transient design situation may be ignored altogether as small. Clause 5.8.3.1(1) inEN 1992-1-1:2004 (Part 1-1 of Eurocode 2) gives the upper limit for the column slendernessabove which second-order effects should be taken into account in the fairly detailed andcomplex way prescribed in Eurocode 2. The effective length of the pier column (i.e. the numeratorin the column slenderness) is a prime factor in these sizing calculations. In the usual case of a decksimply supported on the abutments and free to translate there in the longitudinal direction, theeffective length of pier columns may be approximated as follows:

1 For monolithic connection with the deck:(a) in the longitudinal direction: the full clear height of the pier column (i.e. column top

free to translate horizontally without rotation)(b) in the transverse direction:

– if the deck is laterally restrained at the abutments and has high in-plane rigidity:one-half the clear height of the column (i.e. top fixed against rotation and transversetranslation)

– if the deck is free to translate laterally (i.e. it is unrestrained at the abutments, or isvery long or has low in-plane rigidity):� for single-column piers: twice the clear height of the pier (i.e. free cantilever)� for multi-column piers: the full clear height of the pier column (i.e. column top

free to translate horizontally without rotation).


57

2 For a deck supported on bearings (fixed or movable):(a) in the longitudinal direction: twice the clear height of the pier column (i.e. free

cantilever)(b) in the transverse direction:

– if the deck is laterally restrained at the abutments and has high in-plane rigidity:� for columns directly supporting the deck: 70% of the clear height of the pier (i.e.

top free to rotate but restrained against horizontal translation)� for multi-column piers framing into a stiff cap-beam: one-half of the clear

height of the column (i.e. column top fixed against rotation and transversetranslation)

– if the deck is free to translate laterally (i.e. it is unrestrained at the abutments, or isvery long or its in-plane rigidity is low):� for single-column piers: twice the full clear height of the pier (i.e. free

cantilever)� for multi-column piers framing into a stiff cap-beam: the clear height of the pier

column (i.e. column top free to translate horizontally, without rotation).

If the deck is integral with at least one abutment (see Section 4.5.3), the first bullet point undercases 1 (b) and 2 (b) applies not only to the transverse but also to the longitudinal direction of thebridge.

The above refers to the column effective length in the completed bridge. In a free cantileverbridge, the effective length of single-column piers during deck erection is twice the clear heightof the pier (i.e. a free vertical cantilever) both in the longitudinal and in the transverse directions;if the upper part of the pier consists of two parallel wall-like columns (‘twin blades’, seeFigures 4.6 and 4.7), the clear height of the column applies as its effective length in the longitudi-nal direction. Critical for pier slenderness may then be the stage after completion of the hammerand before connection at mid-span or at an abutment with other completed parts of the bridge.The permanent loads applied at that time and the value of the creep coefficient at that age of thepier concrete – and not at the end of the life of the bridge – should be used in the upper limit forcolumn slenderness above which second-order effects should be taken into account as per clause5.8.3.1(1) of Part 1-1 of Eurocode 2; this differences may offset the longer effective length of thepier at that stage.

A lower limit to the cross-section of non-slender pier columns monolithically connected to thedeck may be imposed by the verification of concrete for the fatigue ULS according toclause 6.8.7(101) of Part 2 of Eurocode 2 (equivalent damage stress) for railway bridges,or clause 6.8.7(2) of Eurocode 2-Part 1-1 (simplified verification based on the frequentcombination) for roadway bridges.

4.4.3.2 Specific criteria for bridges with ductile behaviourThe choice between ductile or limited ductile behaviour (including seismic isolation) for thebridge has a decisive impact on the sizing of its piers and vice versa, as plastic hinges forenergy dissipation zones are allowed only in the piers. If design for ductile behaviour ischosen, only the pier plastic hinges are dimensioned to resist at the ULS the bending momentand axial force from the analysis for the seismic design situation. All other sections of the pierand of all other components of the bridge structure are dimensioned to resist at the ULS the‘capacity design effects’, computed according to Section 6.4 of this Guide from the designmoment resistance of the pier plastic hinges. Should that moment resistance substantiallyexceed what is required according to the analysis results for the seismic design situation, allother parts of the bridge (including the pier itself in shear) are penalised. The moment resistanceof a plastic hinge section may be unduly large if:

g the dimensions of the section are unnecessarily largeg the vertical reinforcement ratio is governed by non-seismic design situations (including at

intermediate stages of the construction) or by detailing and minimum reinforcementrequirements.

Therefore, for cost-effective seismic design for ductile behaviour:


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g The cross-section of the piers should not be larger than rationally needed, and the verticalreinforcement of their plastic hinges should not be beyond what allows reliableconstruction.

g The final vertical reinforcement of the pier plastic hinges should be as close as feasible towhat is required for the biaxial bending moments and the axial force from the analysis forthe seismic design situation.

g The q factors to be used in the longitudinal and transverse directions should be as close asfeasible to the maximum values allowed in Part 2 of Eurocode 8, listed in Table 5.1 ofSection 5.4. To this end:– the location of the expected plastic hinges should be accessible– the shear span ratio of the pier, Ls/h, should not be less than the value of 3.0, below

which the q factor is penalised (see Section 5.4 of this Guide)– the pier maximum axial load ratio, hk ¼ NEd/Ac fck, in the seismic design situation

should not exceed the value of 0.3, above which the q factor is reduced (see Section 5.4).

The axial load ratio, hk, has a number of important influences on seismic design, especially forductile behaviour, because it is important for the ductility of the pier column. If it is high, theconfinement requirements in the plastic hinge are larger, especially in piers designed for ductilebehaviour (see rows 18, 19 and 22, 23 in Table 6.1 of Chapter 6). In addition, the overstrengthfactor applied to the moment resistance of ductile piers for capacity design calculations increaseswith increasing hk (see Eq. (D6.6b) in Section 6.4.1 of this Guide). More importantly, as alreadypointed out, the q factor of bridges designed for ductile behaviour is controlled by the maximumvalue of the axial load ratio in any pier column of the bridge (see Section 5.4). In all of these cal-culations the value of hk is that determined from the maximum axial load in the pier column,NEd, in the seismic design situation. In multi-column piers and for the transverse seismicaction, the overturning moment may significantly increase NEd over the value due to thegravity loads alone. That said, very seldom does the level of hk control the area of the piersection: the size of pier columns selected on the basis of practical or slenderness considerationsnormally gives quite low values of hk in the seismic design situation.

4.5. The abutments and their connection with the deck4.5.1 The role of abutmentsThe abutments have a dual role:

1 to provide vertical support to the deck, like the piers, but at its very end2 to act as a retaining wall for the backfill beyond the end of the bridge.

The second role normally governs the form, dimensions and cost of the abutment. In fact, anabutment has a substantially higher cost than a typical pier, despite its much lower verticalreaction from the deck and shorter height. Cost is governed by the high, nearly permanent,earth pressures of the backfill, the resultant of which and the impact of the retaining role are pro-portional to the square of the height of the retained fill. The main consequences of the retainingrole are:

g The horizontal stiffness of the abutment is much larger than that of a pier:– in the longitudinal direction owing to the contribution of the backfill and/or the

required resistance to earth pressures– in the transverse owing to the wall-like shape of the abutment.

g It is beneficial and cost-effective to brace longitudinally the abutment at the level of thedeck against the action of earth pressures, especially if the depth of the backfill is large. Amonolithic connection of both abutments to the deck is very effective to this end, as itmobilises in the bracing action the opposite abutment and the reaction of the backfillbehind it.

To serve its second role, an abutment is usually a deep retaining wall on a footing on competentground or on the cap of piles going down to competent and non-liquefiable soil. Sometimes theretaining wall consists of a single curtain of dense piles supporting a shallow beam on which thedeck is seated. A deep continuous retaining wall or a curtain of dense piles may be pushedinwards by a laterally spreading or liquefied backfill. If the nature of the backfill does not rule


59

Clause 6.7.3(1) [2]

out this possibility (e.g. at a river or lake bank), the piles supporting the shallow seat beam wouldbe better sparse and deep, for uninhibited lateral flow of the ground between them.

4.5.2 The connection optionsAs far as the seismic response and design are concerned, the abutments may play a significant or aminor role, depending on how they are connected to the deck in the horizontal direction. Themain connection options are:

1 to have a deck integral with the abutments2 to seat the deck on the abutments through bearings movable only in the longitudinal

direction or in both horizontal directions.

Option 1 can be adopted only if both the deck and the abutments are of concrete. They then movein unison in any horizontal direction. For option 2, the designer may choose:

2a to let the deck free to move horizontally at the abutment2b to restrain horizontally the deck at the abutment2c to let the deck move until it comes in hard contact with the abutment and then move with

it (or not move at all).

Option 2b is not very different from option 1. Unlike the integral connection, which refers to boththe longitudinal and the transverse directions and has similar implications for the design of thebridge in both, one option among 2a to 2c may be adopted in the transverse direction andanother in the longitudinal. More details are given in the sequel.

4.5.3 Deck integral with the abutmentsThe connection of the abutment to the deck is considered as rigid if it is monolithic (‘integral’), orif in both horizontal directions (longitudinal and transverse) it is effected via fixed bearings orlinks designed to carry the seismic action. Unlike in bridges with a movable connection of thedeck to the abutments, the abutments of integral bridges play a major role for the seismicresistance in any horizontal direction.

As pointed out in Section 4.5.1, the top of abutments integral with the deck is braced horizontallyin a very reliable way, with considerable benefits against seismic actions and earth pressures fromthe backfill. The absence of movement joints improves motorist comfort – especially over bridgesof short length – and reduces initial and maintenance costs through savings on bearings androadway joints. These cost savings are a prime advantage of integral bridges. In railwaybridges, transverse movement of the deck with respect to the abutment is avoided and tracksare protected.

Integral connection is the best way to prevent the deck from dropping from the abutment. This isespecially important if the deck is very skewed to the abutments or sits on non-parallel ones (i.e. ifthe geometry is kinematically favourable for unseating of simply supported decks – see the lasttwo paragraphs of Section 6.8.1.4 of this Guide). Note that, if the end supports are very skewed,integral connection offers the advantage of eliminating the high – and uncertain – concentrationof vertical reactions to gravity loads that takes place under the obtuse angle of bridge decks onbearings. Last, but not least, if headroom below the deck is at a premium, a deck working withthe abutments as a portal frame may be made much thinner than a simply supported one, owingto its reduced span moments.

The above advantages may be offset by a serious drawback: integral abutments restrain longi-tudinal deformation of the deck due to thermal actions and concrete shrinkage. As a result,large tensile forces may build up in the deck, alongside longitudinal forces in the abutmentsand their foundation. Additionally, the deck cannot be longitudinally post-tensioned, withoutlosing the (large) part of the prestressing force that goes not to the deck itself but to theabutments (and from there to the ground, with undesirable results): the deck may profit onlyfrom the load-balancing effect of the deviation forces of curved tendons, but very little from alongitudinal compressive force. Note also that the high bending stiffness of the systemabutment-cum-backfill prevents efficient use even of these deviation forces. For these reasons,


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only short decks are built integral with the abutments. They mostly have one span, or sometimestwo or three short continuous ones, but not more than four. Their total length is normallymuch less than 100 m. Also, normally such a deck is not post-tensioned. For such bridges,accounting for a large fraction of the bridgework along motorways (at almost every shortunder- or overpass), an integral connection is the solution of choice, and a very robust oneindeed.

As the seismic behaviour and design of integral bridges is normally dominated by the abutments,any intermediate piers have almost no contribution to the longitudinal earthquake resistance.Transversely, they have a certain contribution if they have a wall-like section and the deck isrelatively narrow and long (a rare combination for this type of bridge). A row of slender(e.g. circular) columns, monolithically connected to the deck, may then be optimal for theintermediate piers.

It makes no sense to support the deck on both abutments through fixed bearings (option 2b inSection 4.5.2) instead of building it as integral: the main drawback of integral connectionremains, while the span moments are not reduced, and deck drop-off in case the fixed bearingsfail cannot be precluded. Additionally, the fixed bearings should be designed for high horizontalforces due to the restraint. There is some scope, though, for a deck integral with one abutmentand supported on the other through horizontally flexible bearings. This option relieves thebridge from stresses due to restraint of imposed deformations, allows longitudinal prestressingand decreases significantly the chances of deck drop-off compared with a deck simply supportedat both ends. However, the seismic response and behaviour is more uncertain and the analysismore complex.

As explained in more detail in Section 5.4 of this Guide, bridges often having the deck integralwith the abutments may be considered to be ‘locked-in’ the ground, and follow its horizontalmotion without amplifying it much. If their fundamental period T is less than 0.03 s, Part 2of Eurocode 8 allows for them to be designed as elastic (i.e. with q ¼ 1) but rigid (i.e. withforces equal to the masses multiplied by the design peak ground acceleration). If, however, Tis longer than 0.03 s, Eurocode 8 requires the analysis to account for the interaction betweenthe soil and the abutments, using realistic soil stiffness parameters. In that case, the bridgemay be designed for limited ductile behaviour with q ¼ 1.50. According to Part 2 ofEurocode 8, the bridge may still be considered as ‘locked-in’ (and designed with q ¼ 1.0 andT ¼ 0 s) without the need to estimate the period T, if the abutments are embedded in stiffnatural soil over at least 80% of their total surface area that is in contact with the soil/fillover the backfill face of the abutments (Figure 4.12).

Although the seismic integrity and stability of bridges having a deck integral with the abutmentsseems assured, their modelling and analysis are quite demanding, both for gravity loads and forthe design seismic action. It normally entails a fairly detailed finite-element discretisation of thedeck and the abutments and modelling of the soil behind the abutments with springs. Eventhough these springs may be taken as linear for simplicity, the actual soil–abutment interactionis quite nonlinear and very different when the abutment moves towards the soil or pulls away.

Clauses 4.1.6(9),

4.1.6(10), 6.7.3(1)–(4),

6.7.3(9) [2]


Figure 4.12. A bridge considered as ‘locked-in’: the abutments are embedded in stiff natural soil over at

least 80% of their total surface area that is in contact with the soil over their backfill face

Wid

th b

1

Wid

th b

2

Original

ground level Original

ground level

ht1 hs1 ht2hs2

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Clauses 2.3.6.3(2),

2.3.6.3(5), 6.6.1(2),

6.6.3.1(1), 6.6.3.1(2),

6.6.3.1(4) [2]

Part 2 of Eurocode 8 gives guidance for a conservative analysis, based on superposition of theinertial response of the structure and of a simplified combination of limit equilibrium and anelastic approach for the backfill or the soil. For uncertain soil behaviour, it recommends usingupper- and lower-bound estimates of the soil stiffness. For single-span box-type culverts, it speci-fies as more realistic the use of an analysis based on kinematic compatibility of the structure andthe surrounding soil.

The Caltrans Seismic Design Criteria (Caltrans, 2006) give simple rules for the estimation of thestiffness and the ultimate resistance of an abutment in the longitudinal direction of a bridge, dueto the passive resistance of a well-compacted backfill. On the basis of static tests reported inMaroney (1995) for a 1.7 m-tall abutment, the stiffness is about 7 MN/m per square metreof the projection of the vertical face of the abutment onto a plane normal to the longitudinaldirection of the bridge. The ultimate passive resistance is reached when the top of theabutment is pushed against the backfill by about 2% of the abutment height (in units of MNand m, this gives an ultimate resistance of about 0.14 times the projection of the vertical faceof the abutment onto a plane normal to the longitudinal direction, multiplied by the heightof the abutment). The results of a seismic analysis of the bridge using this longitudinal stiffnessfor the pushed abutment are considered to apply, if the longitudinal displacement of the deck isless than about 4% of the abutment height; if it exceeds that value by a factor of 2, the abut-ments are considered to contribute little against the longitudinal seismic action. In this lattercase, the analysis should be repeated with the abutment stiffness reduced to 10% of the valuequoted above. For longitudinal displacements between 2% and 4% of the abutment height,linear interpolation between these extreme stiffness values may be used, and the analysisrepeated until convergence. Apart from a simple design tool to account for the abutment–backfill interaction without recourse to the elaborate analysis alluded to in the previousparagraph, the Caltrans (2006) simple rules may be used at the conceptual design phase toestimate the contribution of relatively flexible diaphragm abutments against the longitudinalseismic action.

All things considered, the designer’s increased effort is a heavy price to pay for choosing thisrobust bridge layout.

4.5.4 Deck on bearingsTo avoid the main shortcoming of the integral connection, notably the restraint of longitudinaldeck movements and its consequences, a deck seated on the abutments through bearings canmove freely in the longitudinal direction (option 2a in Section 4.5.2). To this end, clearanceshould be left between the main body of the deck and the abutment or its back-wall toaccommodate the sum of (see Section 6.8.2.1 of this Guide):

1 the full longitudinal displacement due to the design seismic action, plus2 the displacement of the extreme fibres of the deck end section towards the abutment (i.e.

the longitudinal displacement at the centroidal axis of the deck plus the rotation of the endsection multiplied by the distance of the extreme fibres from the centroid) due to the quasi-permanent gravity actions and prestressing, plus

3 50% of the end section displacement due to lengthening of the deck by its design thermalaction (the extreme one expected to take place during the bridge lifetime).

According to Part 2 of Eurocode 8, adequate clearance should be provided to protect importantcomponents of the bridge (in this case the abutments). If such clearance is too large to beprovided at the roadway level, the movement joint there may be made wide enough to takethe sum of 2 and 3 above plus a fraction of 1; that is, of the deck longitudinal displacementdue to the design seismic action, implying that it will close when this fraction (recommendedto be 40% in Part 2 of Eurocode 8, see Section 6.8.2.2 of this Guide) is attained. Provisionsshould then be made to limit the damage to the top of the abutment backwall, when themovement joint closes at the roadway level. Figures 8.23 and 8.24 in Section 8.2.11.3 of thisGuide show an example of a practical means to this end: the deck slab protrudes from themain body of the deck towards the backwall in order to reduce the gap at the roadway level,while the top of the backwall is sacrificial, to be knocked off without transmitting the forcesto the abutment and its foundation (this is option 2c of Section 4.5.2 turned into option 2a).


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Seismic links are sometimes used to connect the deck to the abutments in the transverse direction.They are arranged between the deck and an element (abutment or pier) supporting it at amovable support joint, and link the two in a specific horizontal direction against the seismicaction. The link (usually in the form of shear keys or cables) is normally activated after aspecific horizontal displacement is attained and exhausts the gap of a shear key or the slack ofthe cable (termed ‘slack’ of the link). The design behaviour of seismic links according to Part 2of Eurocode 8 is best understood by considering the following two limiting cases, depending onthe slack:

g At the lower limit of zero slack, the link behaves as a fixed bearing. The link is modelled inthe analysis as a fixed constraint, and is capacity designed.

g At the upper limit the slack is equal to the design seismic displacement (at the relevantpoint and direction) of the bridge without the link. Then, it is meaningless to include thelink in the analysis model. The link plays the role of a second line of defence, againstseismic actions exceeding the design action. Part 2 of Eurocode 8 neither requires norrecommends such a second line of defence. If, however, it is provided, the seismic design ofthe bridge is not affected; the design of the link should be based on rational considerationsrelated by necessity to the reasons for its provision.

Part 2 of Eurocode 8 gives rules for the use of seismic links with slack between the above twolimiting cases:

(i) The link should be included in the analysis model, at least using the composite secant-to-yield-point stiffness of the link and the supporting element (which is close to that of thesupporting element if the slack is small). Besides, measures to mitigate the impact arerequired.

(ii) The design of the link and of the supporting element should be based on capacityconsiderations.

If the abutment is extended upwards in the form of shear keys in contact with the sides of thedeck, the deck may be transversely restrained there (option 2b in Section 4.5.2). Normally,shear keys are provided only at the sides of the deck, because internal ones (against the under-deck) are harder to inspect or maintain. An elastomeric bearing may be placed between eachshear key and the side of the deck to ensure full and soft contact under any seismic action andto prevent local damage to the contact surfaces due to rotation of the ends of the deck in ahorizontal plane. In that case, under transverse seismic action the deck flexes as a beam withlateral elastic supports at the locations of the piers and the abutments. Note that the lateralforce–deformation response of the system of the shear keys, the abutment with any wing-wallsand piles, and the soil is quite complex, and a composite elastic stiffness cannot be easilyestimated for it: its overestimation (let alone the assumption that it is laterally rigid) may beunsafe for piers between the abutments. In addition, although shear keys are capacitydesigned, if they do fail they will do so in a brittle fashion. Note also that, if the bridge isdesigned for ductile behaviour in the transverse direction owing to a rigid transverse connectionwith the piers, one may question whether the elastic–perfectly plastic idealisation of the bridge(the foundation of design for ductility) applies after the piers yield but the deck and its lateralsupports at the abutments are still elastic: in reality, the system will harden, at a hardeningratio approximately equal to the elastic transverse stiffness of the deck on rigid or elastic endsupports divided by the elastic transverse stiffness of the pier system. For all these reasons, itmay be prudent to also analyse the bridge under transverse seismic action, disregarding theshear keys and any restraint they offer, and to design the bridge for the envelope of designaction effects, with or without the shear keys.

In view of the questions raised in the previous paragraph, the designer may choose to let the decktransversely free (option 2a in Section 4.5.2) and provide ample seating at the abutment to avoiddrop-off under the largest conceivable seismic action. To play it safe, shear keys may be providedthere as well, but with a clearance from the deck more than the transverse displacement due to thedesign seismic action (unless the bridge is strongly curved, there is no transverse displacement dueto thermal or quasi-permanent actions). If, instead, the gap is narrower than this transversedisplacement (option 2c in Section 4.5.2), it will close during the design seismic action. Under


63

Clause 5.4.2(1) [3]

Clause 2.2.4.2 [1]

Clause 5.2(1) [3]

Clauses 5.1(1), 5.3.2

[3]

such circumstances, Part 2 of Eurocode 8 requires taking the deck as elastically supported on theabutment, with a spring stiffness equal to the secant stiffness at yielding of the shear key afterclosure of the gap (i.e. a stiffness equal to the yield force of the shear key divided by the clearanceplus the elastic deformation of the shear key until yielding). This is not very convenient, as theshear key has not been dimensioned at this stage of the design. So, option 2c is not a very prac-tical design alternative for the transverse direction.

4.6. The foundationsBridges are nowadays built on spread footings or on pile foundations, less commonly on deepcaissons.

Bridges on spread foundations are supported close to the ground surface on firm soil layers orrock, and have performed well in earthquakes. Spread foundations directly transmit loadsfrom the superstructure to competent ground. They are defined as such foundations if theyhave a ratio of the embedment depth to the foundation width of not more than 0.5. Deeper foun-dations are referred to as caisson foundations. Deep caisson foundations of bridge piers aretoday normally used wherever the available plan area for the foundation is severely limited,not allowing the use of a spread or pile group foundation.

On a site with weak upper soil layers, the bridge is supported on deep foundations transferringthe vertical and lateral forces to stronger soil layers beneath the soft material. Bridges on softclay, silt or loose saturated sand have been damaged by the amplification of the groundmotion or soil failure in earthquakes. Unless massive soil failure has occurred, pile foundationshave performed well in past earthquakes, even when other bridge elements have suffered con-siderable damage. By contrast, bridges on liquefiable soil deposits or soft sensitive clays havebeen particularly vulnerable to earthquakes: soil liquefaction can cause a loss of bearingcapacity and, sometimes, lateral movement of the substructure.

The seismic response of pile foundations to strong earthquake shaking is very complex. It iscontrolled by inertial interaction between the superstructure and the pile foundation, kinematicinteraction between foundation soils and piles, and the nonlinear stress–strain behaviour of soilsand of the soil–pile interface. In addition, at some sites the build-up of high pore water pressuresduring the earthquake or liquefaction add to the complexity. Many different materials andgeometries have been used for pile designs. Although numerous examples of long-span bridgesin seismic areas founded on timber piles still exist in California, the current trend is to useconcrete piles (reinforced or prestressed) or steel piles (H sections, shells or concrete-filledshells). A special case is an integral pile–shaft column arrangement, where the pile is notconnected to a pile cap but extended in the superstructure as a column.

Unlike other types of structures for which only one foundation type may in general be used,different foundation types may be encountered in the same bridge.

The basic principles of foundation design require the foundation be able to safely transfer to theground the applied loads. Accordingly, it should be mechanically stable and prevent detrimentaldisplacements. Soil–structure interaction should be assessed where necessary, also taking intoaccount the relevant provisions in Part 5 of Eurocode 8 (CEN, 2004). To ensure stability, thefoundation must possess the required factors of safety against bearing, sliding and overturningfailure mechanisms. The relevance of these failure modes to each foundation type is shown inTable 4.2. In addition, the structural elements of the foundation are designed to resist theaction effects.


Table 4.2. Foundation failure mechanisms and stability verifications

Foundation type Bearing capacity Overturning Sliding Horizontal displacement

Spread foundation Yes Yes Yes –

Caissons Yes – – Yes

Piles Yes – – Yes

64

Because it is difficult to inspect or repair foundations after an earthquake, it is a common practice torestrict damage to the foundation to a minimum, so that operation of the bridge can easily restartwithout repair of the foundations. In general, bridge foundations should not be intentionally usedas sources of hysteretic energy dissipation, and therefore should, as far as practicable, be designedto remain elastic under the design seismic action. To this end, if the bridge is designed for limitedductile behaviour, its foundation is designed to remain elastic. If the bridge is designed for ductilebehaviour, the foundation is designed to resist the overstrength capacity of the pier column(s) orwall for plastic hinges presumed to form at their base. However, in some instances such a design isnot possible. A prime example is a bridge structure with a large force capacity due to factors otherthan the earthquake. Under such circumstances, piles are allowed to develop a plastic hinge nextto their connection to the pile cap, where large bending moments develop. The head of the pile upto a distance to the underside of the pile cap of three times the pile cross-sectional dimension, d, isdetailed as a potential plastic hinge region. To this end, it should be provided with transverse andconfinement reinforcement following the rules for pier columns designed for ductile behaviour.The same applies to regions of the pile up to a distance of 2d on each side of an interface betweentwo soil layers with a large shear stiffness contrast (ratio of shear moduli greater than 6.0).

Inclined piles are usually not recommended for transmitting lateral loads to the soil. If used, theyshould be designed to safely carry the axial loads as well as bending moments arising from soilsettlements. One additional reason for not favouring inclined piles is their less ductile behaviouras opposed to vertical piles subjected to pure bending.

Piles required to resist tensile forces or assumed as rotationally fixed at the top should beprovided with anchorage in the pile cap to enable the development of the pile design uplift resist-ance in the soil, or of the design tensile strength of the pile reinforcement, whichever is lower. Ifthe part of such piles embedded in the pile cap is cast before the pile cap, dowels should beprovided at the interface of the connection.

REFERENCES

Bardakis V (2007) Displacement-based seismic design of concrete bridges. Doctoral thesis,

Department of Civil Engineering, University of Patras.

Caltrans (2006) Seismic Design Criteria, version 1.4. California Department of Transportation,

Sacramento, CA.

CEN (Comite Europeen de Normalisation) (2000) EN 1337-2:2000: Structural bearings – Part 2:

Sliding elements. CEN, Brussels.





CEN (2008) CEN/TC 1992-4-1:2009: Design of fastenings for use in concrete. CEN, Brussels.

fib (2000) Guidance for Good Bridge Design. fib Bulletin 9. Federation Internationale du Beton,

Lausanne.

fib (2004) Precast Concrete Bridges. fib Bulletin 29. Federation Internationale du Beton, Lausanne.

fib (2007) Seismic Bridge Design and Retrofit – Structural Solutions. fib Bulletin 39. Federation

Internationale du Beton, Lausanne.

fib (2009) Structural Concrete – Textbook on Behaviour, Design and Performance, vol. 1, 2nd edn.

fib Bulletin 51. Federation Internationale du Beton, Lausanne.

fib (2011) Design of Anchorages in Concrete. fib Bulletin 58. Federation Internationale du Beton,

Lausanne.

fib (2012) Model Code 2010 – Final Draft, vol. 2. fib Bulletin 66. Federation Internationale du

Beton, Lausanne.

Maroney BH (1995) Large scale bridge abutment tests to determine stiffness and ultimate strength

under seismic loading. PhD thesis, University of California, Davis, CA.

Priestley MJ, Seible F, Calvi GM (1996) Seismic Design and Retrofit of Bridges. Wiley-Interscience,

New York.

Stathopoulos S, Kotsanopoulos P, Stathopoulos E et al. (2004) Votonosi bridge in Greece.

Proceedings of the fib Symposium: Segmental Construction, Delhi.

Clauses 5.8.1(1),

6.4.1(1), 6.4.2(1)–

6.4.2(4) [2]

Clause 5.4.2(7) [3]

Clause 5.4.2(1) [3]

Clause 5.8.4(3) [1]


65




Chapter 5

Modelling and analysis of bridges forseismic design

5.1. Introduction: methods of analysis in Eurocode 8An analysis carried out in the framework of seismic design determines by calculation the effectsof the design seismic actions in terms of internal forces and deformations, to be used for thedimensioning of its members; that is, to verify their cross-sectional size and – in the case ofconcrete members – to determine the amount and location of their reinforcement.

This chapter is limited to the essentials for the application of well-established analysis methods tothe design of bridges for earthquake resistance according to Eurocode 8. The reader is assumed tobe conversant with the fundamentals of structural dynamics and their application for seismicanalysis.

The seismic design of bridges according to Eurocode 8 is primarily force-based. Its main work-horse is linear-elastic analysis based on the ‘design response spectrum’ of Section 5.3; that is, onthe 5%-damped elastic spectrum divided by the ‘behaviour factor’ q, which accounts on one handfor ductility and energy dissipation capacity and on the other for overstrength. Its counterpart inthe USA is the ‘force reduction factor’ or ‘response modification factor’, R.

Eurocode 8 provides two alternative methods for linear-elastic seismic analysis:

(a) Linear static analysis (termed the ‘fundamental mode method’ in Part 2 of Eurocode 8(CEN, 2005a) and the ‘lateral force’ method in Part 1 (CEN, 2004a), also known inpractice as ‘equivalent static’ analysis).

(b) Modal response spectrum analysis, as known in practice and called in Part 1 of Eurocode8, termed in Part 2 just the ‘response spectrum’ or ‘linear dynamic’ analysis or analysiswith a ‘full dynamic model’.

Eurocode 8 adopts analysis method (b) as the reference method for the design of bridges, andfully respects its rules and results. It allows its application to any bridge, except to those witha strongly nonlinear seismic isolation system. If both methods of linear-elastic seismic analysisare applicable for the design of a given bridge, an analysis of type (b) gives on average a moreeven distribution of peak internal forces over the bridge, translated to material savings. If itsresults are used for member dimensioning, the overall inelastic performance of the bridge maybe expected to be better, because peak inelastic deformations are normally closer to its predic-tions than to those of an analysis of type (a). So, as reliable and efficient computer programsfor modal response spectrum analysis in 3D are widely available nowadays, it can be adoptedas the single analysis tool for the seismic design of bridges. It is worth noting, however, thatquite a few bridges are close to a single-degree-of-freedom (SDoF) system; hence, linear staticanalysis based on the fundamental mode provides considerable insight into their seismicresponse and still has a role as a practical seismic design tool for bridges.

Eurocode 8 recognises an important role for:

(i) nonlinear static analysis (commonly known as ‘pushover’ analysis)(ii) nonlinear dynamic (time-history or response-history) analysis

Clauses 4.2.1, 4.2.2 [2]

Clauses 4.1.6(1),

4.2.1.1(1) [2]

Clauses 4.1.9(2), 4.2.4,

4.2.5 [2]

67

Clause 4.2.3.1 [2]

Clauses 2.3.1.1(5)–

2.3.1.1(8), 2.3.3, 2.3.4,

4.2.4.4(2) [2]

Clause 3.2.3.1.1(2) [1]

Clauses 3.1.2(1),

3.1.2(2), 3.2.3(5) [2]

Clause 4.3.3.5.1(2) [1]

Clauses 4.2.1.4(1),

4.2.2.1(3) [2]

Clause 4.3.3.5.2(4) [1]

Clause 4.2.1.4(2) [2]

Clauses 3.1.2(1),

4.1.1(2) [2]

in the seismic design of bridges. However, their stand-alone application is limited to the design of‘irregular’ ductile bridges (see Section 5.10.2 of this Guide) with deformation-based verificationof their ductile members and to bridges with seismic isolation. The limitation is due to concernsabout the correct and prudent application by practitioners of these more rational methods, whichhave a very different safety format and entail analysis of structures with known member dimen-sions and reinforcement in the same phase of the design process as their verification.

Unlike Part 1 of Eurocode 8, which does not explicitly mention linear time-history analysis, Part2 of Eurocode 8 does mention the method. However, it is not a very practical alternative to linearmodal response spectrum analysis.

When linear analysis with the design response spectrum is applied, internal forces for the dimen-sioning are taken as equal to those estimated from it, except for regions or members of ductilebridges intended to remain elastic, which are dimensioned on the basis of ‘capacity design’actions. Displacements due to the seismic action are obtained from those derived from thelinear analysis after corrections for (a) any difference of the effective stiffness values fromthose initially assumed, (b) damping other than the default value of 5% and (c) deviations ofshort-period bridges from the equal displacement rule (see Sections 2.3.2.2, 5.8.4 and 5.9.1).By contrast, when nonlinear analysis is applied, all seismic action effects (internal forces,displacements and deformations) are taken as equal to those derived from it. Only safetyfactors intervene between these demands and the capacities against which they are checked.

5.2. The three components of the seismic action in the analysisThe three components of the seismic action are taken to act concurrently on the bridge.Time-history analysis, be it linear elastic or nonlinear, is indeed carried out applyingsimultaneously all seismic action components of interest (the two horizontal ones and occasion-ally the vertical).

Linear static or modal response spectrum analyses give just the peak values of each seismic actioneffect of interest due to the individual seismic action components. These peak values are statisti-cally combined as outlined below. They are denoted here by EX and EY for the two horizontalcomponents and EZ for the vertical. As they do not occur simultaneously, a combination ruleof the type: E ¼ EXþ EYþ EZ is too conservative. The reference rule in Part 2 of Eurocode 8is the square root of the sum of squares (SRSS) combination of EX, EY, EZ in Smebby andKiureghian (1985):

E ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2X þ E2

Y þ E2Z

qðD5:1Þ

Part 2 of Eurocode 8 also accepts as an alternative a linear combination rule of the type:

E ¼ |EX|þ l|EY|þ l|EZ| (D5.2a)

E ¼ l|EX|þ |EY|þ l|EZ| (D5.2b)

E ¼ l|EX|þ l|EY|þ |EZ| (D5.2c)

where the meaning of ‘þ’ is superposition. A value l � 0.275 provides the best average agree-ment with the result of Eq. (D5.1) in the entire range of possible values of EX, EY, EZ. InEurocode 8 this optimal l value has been rounded up to l ¼ 0.3, which may underestimatethe outcome of Eq. (D5.1) by at most 9% (when EX, EY and EZ are about equal) and may over-estimate it by not more than 8% (when two of these three seismic action effects are an order ofmagnitude less than the third).

To combine the effects of the seismic action components as outlined above, it is computationallyconvenient to do the analysis on a 3D model of the bridge, with all seismic action components ofinterest applied to it separately but their effects combined within the same computationalenvironment and analysis run. So, it is not convenient to use a separate model for theresponse to the horizontal component in the longitudinal direction of the bridge, another onein the transverse and a third for the vertical component (if of interest), as allowed by


68

Eurocode 8. However, as the seismic response to each one of these components is distinctlydifferent, it is intuitively appealing and very didactic to consider them separately using a differentmodel for each one. This is often done in this chapter.

The horizontal components of the seismic action are most often taken parallel to the ‘longi-tudinal’ and the ‘transverse’ direction of the bridge. If the deck is straight and the abutmentsare at right angles to its axis, the definition of these directions is clear. If the deck is curvedbut fairly long, the ‘longitudinal’ direction may be taken as that of the chord connecting thetwo points where the deck axis intersects the line of support at each abutment; the ‘transverse’direction is at right angles to it. If the bridge is skewed and it has a fairly wide deck, it maymake more sense to define the ‘longitudinal’ and ‘transverse’ directions as normal and parallelto the abutments, because these are the directions in which the bridge, the abutments and thepiers (normally aligned parallel to the abutments) primarily work. If bearings or other devices(e.g. shear keys) restrain the deck in one direction but not in the other (where it may besupported with a certain horizontal flexibility or be free to move), these two directions alsolend themselves as ‘longitudinal’ and ‘transverse’. A very important point is that the behaviourfactor, q, may be taken different in these two directions, depending on the way the piers areconnected to the deck and the ratio of their shear span (moment-to-shear ratio) to their cross-sectional depth in that direction. If these features are radically different along two orthogonaldirections, these directions are prime candidates for the horizontal seismic action componentsas well.

5.3. Design spectrum for elastic analysisFor the horizontal components of the seismic action the design response spectrum to be used inlinear elastic analysis is given by different expression in four different period ranges (repeatedbelow from Section 3.1.3 for reasons of their importance and of convenience):

Short-period range:

0 � T � TB: Sa;d Tð Þ ¼ agS2

3þ T

TB

2:5

q� 2

3

� �� ðD5:3aÞ


TB � T � TC: Sa;d Tð Þ ¼ agS2:5

qðD5:3bÞ


TC � T � TD: Sa;d Tð Þ ¼ agS2:5

q

TC

T

� �� bag ðD5:3cÞ


TD � T : Sa;d Tð Þ ¼ agS2:5

q

TCTD

T2

� �� bag ðD5:3dÞ

The design spectrum in the vertical direction has also been described in Section 3.1.3 of thisGuide.

5.4. Behaviour factors for the analysisThe concept and role of the behaviour factor q has been outlined in Section 2.3.2.2 of this Guideas far as the design of ductile bridges is concerned and in Section 2.3.3 for design for limitedductile behaviour. Section 5.3 illustrated the use of the behaviour factor q in the designresponse spectrum. Before venturing into the rules and details of analysis and modelling forthe purposes thereof, the values of the behaviour factor q specified in Part 2 of Eurocode 8 forthe design of these two different types of bridges for the horizontal components of the seismicaction are given. Note that the value of q enters in the design response spectrum and thereforeshould be known before any analysis can be carried out.

Clauses 2.1(2),

3.2.4(1), 4.1.6(1) [2]

Clause 3.2.2.5(4) [1]

Chapter 5. Modelling and analysis of bridges for seismic design

69

Clauses 4.1.6(3),

4.1.6(12) [2]

Clauses 4.1.6(3),

4.1.6(12) [2]

Clauses 4.1.6(3),

4.1.6(7) [2]

Clauses 4.1.6(9),

4.1.6(10), 6.7.3(4),

6.7.3(9) [2]

Clause 4.1.6(6) [2]

Clause 4.1.6(5) [2]

As the design response spectrum is specified separately for each component of the seismic action,the q factor can be different for each of them. and normally is. Starting with the vertical direction,bridge piers and bearings cannot develop ductile behaviour in pumping concentric axial com-pression and tension, at least at the short natural periods of the vertical eigenmodes. Addition-ally, vertical modes of vibration excite mainly bending of the deck in a vertical plane; but the deckis meant to remain elastic under the design seismic action and beyond (except in flexible andductile continuity top slabs between adjacent simply supported spans). So, we use q ¼ 1 in thevertical direction.

The value of q in the two horizontal directions depends on how the bridge is configured toaccommodate the horizontal seismic displacements of the deck in each of them. Sections 2.3.1and 4.1 of this Guide highlighted three options to accommodate these horizontal displacements:

1 in flexible bearing-type devices arranged over an effectively horizontal interface betweenthe deck and the abutments and piers

2 in flexural ‘plastic hinges’ at the base – and possibly at the top – of piers rigidly connectedwith the deck

3 at the interface between the foundation element of the piers and the ground or in ‘plastichinges’ in foundation piles

as well as the option of:

4 locking the bridge in the ground, by connecting the deck and the abutments into anintegral system that follows the ground motion with little additional deformation of itsown.

As explained in Section 2.3.2.5, neither option 1 nor option 3 with the base of the piers slidingwith respect to the soil lend themselves to the development of significant inelastic deformationsin ductile members, such as the piers. Seismic design using these options should therefore beelastic, with q ¼ 1. For seismic design with a q factor that is greater than 1.0 owing to ductilityand the energy dissipation capacity we are left with options 2 and 4 (and with option 3 forplastic hinging in foundation piles).

The maximum values of q specified in Eurocode 8 for these options are given in Table 5.1.

‘Locked-in’ bridges, or dynamically independent parts thereof, are considered to follow the hori-zontal motion of the ground without appreciably amplifying it. Their design spectral accelerationmay be taken as equal to the design peak ground acceleration (i.e. as for T ¼ 0 and q ¼ 1). Thiscategory includes abutments connected to the deck through movable bearings, or integral bridgeshaving rigid horizontal connection of the deck to the abutments and a fundamental period T inthe horizontal direction of interest less than 0.03 s. This case corresponds to the last row ofTable 5.1. The row above it refers to bridges with an essentially horizontal deck rigidly connectedto both abutments (either monolithically or through fixed bearings). For such bridges, Eurocode8 requires accounting for the interaction between the soil and the abutments, using realistic soilstiffness parameters. If T is longer than 0.03 s, the design response spectrum should be enteredwith that value of T and with q ¼ 1.5. Estimation of Tmay be avoided and the bridge consideredas ‘locked in’ if its abutments are embedded in stiff natural soil over at least 80% of their surfacearea that is in contact with the soil/fill over the backfill face of the abutments (see Figure 4.12).

As already noted in Section 2.3.1, if plastic hinges can form in an inaccessible part of any pier,the design of the bridge should be based on just 60% of the value of q in Table 5.1 (but not lessthan q ¼ 1). Part 2 of Eurocode 8 considers as accessible the base of piers deep in earth fill, butas inaccessible those in deep water or groundwater, or piles under large pile caps. At any rate,locations of the pier below the normal water table at a depth that can be accessed and drainedwith reasonable retaining and pumping effort and cost may be considered as accessible forrepairs.

The values of q given in Table 5.1 for reinforced concrete piers apply as long as the maximumvalue of the axial force ratio hk (axial load due to the design seismic action in the horizontal


70

direction of interest and the concurrent gravity loads, Nd, normalised to the product of the piersection area and the characteristic concrete strength, Ac fck) among all piers does not exceed 0.3.As already noted in Section 2.3.3, if this largest value of hk ¼ Nd/Ac fck exceeds 0.6, the q factor ofthe bridge is taken as equal to 1.0. For a value of hk between 0.3 and 0.6, linear interpolationbetween 1.0 and the value in Table 5.1 is allowed. Again, it is the largest value of hk amongall reinforced concrete piers that controls the q factor of the entire bridge. It should be noted,though, that in normal practice the values of hk are fairly low. If nothing else, the upper limitsset by Eurocode 2 on the slenderness of pier columns to avoid lengthy and cumbersome calcu-lations of second-order effects under factored gravity loads (the ‘persistent and transientdesign situation’ of EN 1990) lead to fairly large-sized pier columns and drive down their hk

values (see Section 4.4.3.1 of this Guide).

The shear span of the pier, Ls ¼M/V, in the longitudinal and the transverse directions of thebridge needs to be estimated before any analysis for the seismic action yields the values of themoment, M, and shear, V, at the sections where plastic hinges may form. The values of Ls

quoted in Section 2.3.2.3 in cases 1 to 4 may be conveniently used for that purpose. Case 1therein (namely that of seismic response in the longitudinal direction with the top of the piercolumns monolithically connected to the deck) may be refined as follows.

Let us denote by EId the elastic rigidity of the deck for bending in a vertical plane throughits longitudinal axis. For bending within that same plane, EIp is the total effective rigidity ofthe pier, which may consist of n � 1 columns, each with an effective rigidity EIn; then, EIp ¼Sn(EI )c (see Section 5.8 for the effective rigidities). If Ld is the average span length on eitherside of the pier (with an exterior span freely supported at the abutment taken with twice itsspan value in this averaging) and Hp the clear pier height, the relative stiffness of the deck tothe pier is defined as

k ¼ EIdEIp

Hp

Ld

ðD5:4aÞ


Table 5.1. Behaviour factor q for use with the horizontal components of the seismic action

Type of ductile member Seismic behaviour

Limited ductile Ductile

Reinforced concrete piers

Vertical piers in bending 1.5 3.5a

Inclined struts in bending 1.2 2.1a

Piles under pile caps

Vertical piles in bending 1.0 2.1

Inclined piles 1.0 1.5

Steel piers

Vertical piers in bending 1.5 3.5

Inclined struts in bending 1.2 2.0

Piers with normal bracing 1.5 2.5

Piers with eccentric bracing – 3.5

Abutments rigidly connected to the deck

In general 1.5 1.5

Locked-in bridges 1.0 1.0

Arches 1.2 2.0

aIf the minimum among all piers of the pier shear span ratio, Ls/h (where Ls ¼M/V is the distance from the base ofthe pier to the inflexion point for the seismic action direction considered and h is the cross-section depth in thatdirection), is less than 3.0, these value of the behaviour factor are multiplied by

ffiffiffiffiffiffiffiffiffiLs=ð3hÞ

p. For piers with sides skewed

to the seismic action component considered, use the minimum value of Ls/h along the two sides

71

Then, the shear span at the base of the pier is

Ls ¼kþ 1=6

kþ 1=12

Hp

2ðD5:4bÞ

Case 4 in Section 2.3.2.3 (namely that of single-column piers as vertical cantilevers for theresponse in the transverse direction) may be refined as follows, to include the effects of (a) thetributary rotational mass moment of inertia, Iu,d, of the deck in a vertical plane through the trans-verse direction and (b) the vertical distance between the pier top and the point of application ofthe deck inertia force, taken for convenience at the centroid of the deck section, at a distance ycgfrom the bottom of the deck:

Ls ¼ Hp þ ycg þ1:5Iu;d

Md Hp þ ycg� � ðD5:5Þ

In Eq. (D5.5) rm,d is the radius of gyration of the deck mass (square root of the ratio of Iu,d tothe tributary deck mass, Md). Both Md and Iu,d refer to the full average span length Ld. Forcompleteness and future reference, they are given here as

Md ¼ Ld rcAc;d þc2qkg

þ rsurftsurf

� �bsurf þ

gsideg

� �ðD5:6aÞ

Iu;d ¼ Ld rcJp;d þc2qkg

þ rsurftsurf

� �bsurf

b2surf12

þ ðhd � ycgÞ2 !

þ gsideg

b2side4

þ ðhd � ycgÞ2 !" #

ðD5:6bÞ

In Eqs (D5.6) rc and rsurf are the mass densities of concrete and of the surfacing material (or ofthe ballast, in railway bridges), respectively; Ac,d and Jp,d are the surface area and the polarmoment of inertia of the deck section, respectively; c2qk is the quasi-permanent value of theuniform traffic load; bsurf is the width of the deck over which it is applied; tsurf is the thicknessof the surfacing (or ballast) considered to be applied over bsurf as well; gside is the total weightof hand rails, parapets and kerbs at the two sides of the deck per linear metre of deck andbside their distance across the deck; hd is the deck depth; and g ¼ 9.81 m/s2, the acceleration ofgravity.

Part 2 of Eurocode 8 recommends taking c2 ¼ 0 for bridges with normal traffic and footbridges.For road bridges with ‘severe traffic conditions’ (defined in a note as motorways and other roadsof national importance) it recommends c2 ¼ 0.2, and for railway bridges with ‘severe trafficconditions’ (defined in a note as intercity rail links and high-speed railways) c2 ¼ 0.3. In bothcases, only the characteristic value, qk, of the uniform traffic load (UDL) of Load Model 1(LM1) – with all of its nationally applying adjustments – is recommended to contribute to thequasi-permanent part of the ‘severe traffic’ on the bridge. The quasi-permanent value of thetraffic loads is not only considered to give – together with the permanent actions – the massesthat are subjected to the seismic action and produce the inertia forces but is also combinedwith them and the design seismic action in the ‘seismic design situation’ for which the perform-ance requirements are verified (see Section 6.2 of this Guide).

The discussion above tacitly implies that the bridge deck is (effectively) straight and that the prin-cipal directions of bending of the pier sections are parallel and orthogonal to the deck and, hence,to the longitudinal and the transverse direction of the bridge. The footnote in Table 5.1 mentionspiers with sides skewed to these directions; for example, wall-like or hollow-rectangular pierswith sides parallel and orthogonal to abutments that are skewed to the axis of the deck. Suchpiers present to the longitudinal and transverse directions of the bridge their effective rigiditiesin these two directions, as well as a coupling (cross-)rigidity. In terms of moments of inertia,they work with moments of inertia IL, IT and ILT (cross-moment of inertia) resulting fromrotation of the principal ones, I1 and I2, to the directions parallel and orthogonal to the longitudi-nal direction. For bending in a vertical plane in the transverse direction of the bridge, the value ofIT applies. Such bending involves both depths of the section parallel to its principal directions (i.e.both sides). The larger of these two depths vis-a-vis the corresponding shear span controls the


72

ductility of the pier (or lack of it). So, it is the smallest of the two shear span ratios that determinesthe value of q for both directions of the seismic action.

5.5. Modal response spectrum analysis5.5.1 Modelling5.5.1.1 IntroductionGuidance for modelling given below while presenting the modal response spectrum method ofanalysis also applies to the other analysis methods highlighted in Sections 5.6 and 5.10. Bytheir nature as simplifications of modal response spectrum analysis, the linear methods inSection 5.6 are also open to simplified modelling. By contrast, the nonlinear methods ofSection 5.10 normally require extension of certain modelling aspects into the nonlinear regime.

Linear analysis methods, such as those in Sections 5.5 and 5.6, are not meant to account fornonlinear behaviour other than in locations intended for energy dissipation through ductility,notably in flexural plastic hinges. Even this type of nonlinearity is dealt with in a conventional,approximate way, via the behaviour factor, q. The effects of other types, even when they areof minor importance for the global response, cannot be properly captured by linear analysis.This includes friction in sliding bearings, abrupt engagement of shear keys when the gap withthe deck closes, the interaction between the backfill and the abutment and its wing walls,closing and re-opening of deck movement joints at the abutments or in between (owing notonly to the longitudinal response but to the transverse response as well, which may close ajoint just on one side of the deck), etc. Such phenomena should be dealt with by nonlinearanalysis. As pointed out in Chapter 4, if the designer is not prepared to resort to nonlinearanalysis, they should avoid altogether sliders, intermediate movement joints, gaps between thedeck and shear keys or the abutments that close in the seismic design situation, or engineertheir way around the nonlinearity. For example, the information from Caltrans (2006)highlighted in the last paragraph of Section 4.5.3 can be used to design a sacrificial backwallthat will be knocked off when the movement joint at roadway level closes in the seismic designsituation.

5.5.1.2 Modelling of the deck and the piersThe model of a bridge for the purposes of a modal response spectrum analysis (or a ‘fulldynamic model’ in the language of Part 2 of Eurocode 8) should account for the distributionof mass all over the bridge; that is, all over the deck and in the piers, down to the top of compe-tent ground. Regarding the mass of the piers, the universal 10% rule of thumb of structuraldesign implies that the mass of a pier should be accounted for if it is more than 10% of thetributary mass of the deck. At any rate, it costs nothing to place a string of approximatelyequidistant intermediate nodes along the centroidal axis of a pier and lump there its continu-ously distributed mass. Piers are normally 1D elements, with a straight axis and a constantor varying (tapered or flared) section. Prismatic 3D beam/column (sub)elements, withconstant cross-sectional properties along their length, connect adjacent intermediate nodes ofthe pier. If the pier section varies, the spacing of intermediate nodes should be sufficient fora good stepwise approximation of this variation. If it does not, at least three intermediate

Clauses 4.1.1(1),

4.1.2(2), 4.2.1.1(2) [2]


Figure 5.1. Discretisation of a concrete bridge with a box girder deck built with the balanced cantilever

method, showing the sections at all nodes of the piers and the deck

73

nodes are placed. Similar is the model used for decks consisting of a single girder (commonly abox girder, of concrete, steel or composite – steel and concrete). Figure 5.1 depicts an exampleof such a model for a concrete bridge with monolithic connection of the piers with the boxgirder deck.

In a discretisation with prismatic 3D beam/column elements, each node possesses all six degrees offreedom (DoFs): three translations and three rotations (certainDoFs of support nodes are of courserestrained). Masses are assigned to all three translational DoFs of each deck node. Their values aregiven by Eq. (D5.6a), with the span length, Ld, replaced by the average length of the two prismaticdeck elements on either side of the node. A rotational massmoment of inertia, Iux, should always beassigned to the rotational DoF about the centroidal axis, x, of the deck. Its value is given by Eq.(D5.6b), with the span length, Ld, again replaced by the average length of the two prismatic deckelements on each side of the node. Similar is the lumping of the mass and of the rotational massmoment of inertia, Iux, of the piers along their longitudinal axis, x. As the nodes are often closelyspaced along the longitudinal axis of the deck or the pier, there is no need to assign rotationalmass moments of inertia to the two other rotational DoFs, namely those about the centroidalaxes y and z of the section. Closely spaced nodes along the deck and the pier give amodel resemblingthe continuous spread of the mass all along the deck and the piers, and allow determining modalshapes that reflect this distribution. For example, highermodes may inducemore than one inflexionpoint between the top and bottom of a pier or between adjacent joints of the deck and piers. Thesemodes cannot be captured, unless a good number of nodes are used between adjacent physical jointsof the pier and the deck. Figures 5.2(d) and 5.3(d)–5.3(n), as well as Figure 8.6 in Section 8.2.6 of thisGuide, show examples of such modes for bridges with monolithic connection of the piers and thebox girder deck. Figure 5.3 is for a bridge having very dissimilar pier heights, with the upper30 m of all piers in the form of twin blades, and the lower part – if any – with a hollow rectangularsection. Figures 8.32 to 8.35 in Section 8.3.3.4 depict also the important modes of the bridge modelof Figure 8.27.

Masses should be assigned to all nodes down to the top of ground that is competent enough to beincluded in the model with its stiffness (be it infinite). Liquefiable and very weak cohesive or siltysoils are excluded from being considered as competent. Nodes on piles (or other foundationelements) below the top of competent ground may be considered as massless. By the sametoken, earth or water pressures should not be considered to act on these nodes.

A fairly refined model of the deck is normally necessary for the analysis for gravity loads(permanent and due to traffic, often of quite non-uniform distribution owing to wheel or laneloads), in order to dimension the deck for the ultimate limit state (ULS) and the serviceabilitylimit state (SLS) in out-of-plane bending due to the relevant combinations of these actions(including intermediate stages of construction and taking into account the redistribution ofaction effects due to creep, in case the deck consists partly or fully of concrete). If the samediscretisation is used for the modal response spectrum analysis, its results are convenientlycombined with those of the analysis for the permanent actions plus the quasi-permanent valueof the traffic loads (after redistribution due to creep, etc.), to give the design action effects inthe seismic design situation. Although convenient, the choice of the same deck model for theseismic analysis is not necessary, as inertial seismic loads, being proportional to massmultiplied by response acceleration, are fairly uniformly distributed over the deck surface.Therefore, for the purposes of the global seismic analysis, the deck may be modelled as aspine of beam elements. Such a model cannot capture the intricate distribution of seismicaction effects at and around monolithic connections of the deck with the piers – especiallymulti-column ones. If the detailed distribution of seismic action effects in these regions is ofinterest, it may be derived from those from the global seismic analysis by subjecting thedetailed model of the deck used in the gravity load analyses to unit static forces or momentsin the relevant directions coming from the pier column(s). Note also that in bridges designedfor ductile behaviour the deck and its monolithic connections to pier columns are verified forcapacity design effects.

If the deck consists of a concrete slab, its analysis for gravity loads is normally based on avery refined mesh of shell-type finite elements (i.e. a combination of plate finite elementsfor out-of-plane bending, and plane-stress ones for the in-plane loads and response).


74


Figure 5.2. Shapes, periods and participation factors of four modes of a bridge of the type depicted in

Figure 5.1, which capture approximately 90% of the total mass in each of two horizontal directions: (a, c,

d) transverse; (b) longitudinal direction (Bardakis, 2007)

(d)

Mode 16 Period: 0.1532 s

Axis X Y Z

Participation factors 0.00 0.00 24.22

Effective modal mass: % 0.00 0.00 3.37

(c)


Axis X Y Z



(b)


Axis X Y Z



(a)


Axis X Y Z



Y

X

Z

75


Figure 5.3. Shapes, periods and participation factors of 14 modes capturing approximately 90% of the

total mass of the bridge depicted in Figure 4.6 in (a, d, g, i) the longitudinal direction and (b, c, e, f, h,

j–n) the transverse direction (Bardakis, 2007)


Axis X Y Z




Axis X Y Z




Axis X Y Z




Axis X Y Z



(a)

(b)

(c)

(d)

76


Figure 5.3. Continued


Axis X Y Z



(e)


Axis X Y Z



(f)


Axis X Y Z



(g)


Axis X Y Z



(h)

77




Axis X Y Z



(i)


Axis X Y Z



( j)


Axis X Y Z



(k)


Axis X Y Z



(l)

78

An example is depicted in Figure 5.4 for a three-span bridge with a prestressed concrete deckpoint-supported on four columns. For the reasons of convenience mentioned in the previousparagraph, the modal response spectrum analysis is normally done with the same finiteelement model. The nodes of the deck are located at its mid-surface and normally have fiveDoFs each: three translations and two rotations (the rotation about an axis normal to themid-surface is normally not an independent DoF). The tributary mass of each node isassigned to all its translational DoFs. Owing to the density of the nodes, no rotational massmoment of inertia needs to be taken into account for the rotational DoFs.




Axis X Y Z



(n)


Axis X Y Z



(m)

Figure 5.4. Discretisation of a concrete slab deck in a refined mesh of shell finite elements

79

Clauses 2.3.6.1(1)–

2.3.6.1(3), 2.3.2.2(4)

[2]

When the deck consists of a number of discrete parallel girders (whether of precast concrete orsteel, or a composite steel–concrete girder), a grillage-type of model is normally used forgravity load analysis (including traffic loads). In such a model, intermediate nodes are intro-duced along each girder, and the girder is modelled as a spine of prismatic 3D beam/column(sub)elements, having a T or I section with a flange width extending up to mid-distance to theadjacent girder. Cross-(sub)elements, again of the prismatic 3D beam/column type, connecteach intermediate node along each spine to its counterparts on the spine(s) used for theadjacent girder(s). They represent cross-beams wherever the deck has such elements, includingdiaphragms over the supports on piers or abutments. In this case, the cross-(sub)elementhas a T, I or inverted L section, similar to the real one, including any effective flange widthfrom the deck slab(s) at the top and sometimes at the bottom (see Figure 8.27 in Section8.3.3.2 for an example of a deck consisting of two parallel composite girders with steel cross-beams). At all other intermediate nodes, any connecting cross-(sub)elements just represent thetransverse coupling of adjacent girder(s) by the deck slab at the top and at the bottom (ifthere is one). Then, the cross-(sub)element has a rectangular section with an area andmoment of inertia about a horizontal axis equal to the sum of those of the top and bottomdeck slabs.

For concrete slab decks with parallel voids, a similar model may be used instead of a continuous2D mesh of shell finite elements (in this case, with orthotropic properties reflecting the effect ofthe voids).

The grillage model of the deck highlighted above for gravity-load analysis may be used for modalresponse spectrum analysis as well. To this end, tributary nodal masses are assigned to all threetranslational DoFs of each node, but rotational DoFs have no rotational mass moment ofinertia, exactly as in the slab deck model with shell-type finite elements outlined two paragraphsabove. In this model, the extensional and flexural rigidity of the deck as a whole was realisticallycaptured by the finite elements. In a grillage model it is not captured, unless special provisions aremade. The main concern is the in-plane stiffness of the deck. The extensional one is sufficientlyrepresented, if each of the families of (sub)elements used in the longitudinal and the transversedirections preserve on aggregate the cross-sectional area of the deck in these two directions.The distribution across the width of the axial rigidities, EA, of the family of longitudinal(sub)elements can approximate the in-plane flexural rigidity of the deck, provided that aneffectively infinite in-plane flexural and shear rigidity is assigned to the cross-(sub)elements ofthe transverse direction, so that these cross-(sub)elements remain normal to the longitudinalaxis of the deck (as in a fibre model of a section, where the plane-sections hypothesis convertsan in-plane distribution of fibre axial rigidities, E dA, into a flexural rigidity,

ÐEy2 dA, of the

section). Finally, the in-plane shear rigidity of the deck in the transverse direction, GA, can beeffectively reproduced if the girder (sub)elements are assigned infinite shear rigidity (zero sheararea) but their length (spacing of intermediate nodes along the girder) DL is about equal toDL ¼ b

p[2(1þ n)], where b is the spacing of the girders across the deck and n is the Poisson

ratio. In this way, the in-plane flexural stiffness of a girder (sub)element (with its double fixityto the in-plane infinitely stiff cross-beams), which equals 12E(tb3/12)/DL3, becomes equal toG(tb)/DL.

Reflecting the intended and expected elastic behaviour of concrete decks under the designseismic action, Eurocode 8 specifies using in the linear analysis the properties of their fulluncracked gross section for the flexural, shear and axial stiffness, except as prescribed inSection 5.5.1.4 for continuity slabs over the joint between simply supported prefabricatedgirders of adjacent spans. The same properties may also be used for reinforced concrete piersin bridges designed for limited ductile behaviour. For those of bridges designed for ductilebehaviour, the effective flexural stiffness is the secant-to-yield point (see Section 5.8.1).Special provisions apply for the torsional rigidity of concrete decks, as this is reduced bydiagonal cracking (see Section 5.8.3).

5.5.1.3 Modelling of the connections between the deck and the piersWhat has been said in Section 5.5.1.2 applies equally well when:

1 the deck is monolithically connected with the piers


80

2 the deck is horizontally fixed to the top of the piers through rigid bearings, but is free torotate there

3 movable bearings are placed between the top of the pier and the deck.

In case 1, the top of the pier and the deck immediately above it may share the same node, locatedat the theoretical centre of the physical joint between the deck and the pier (or the pier column).The part of the deck or the pier between that node and the faces of the physical joint (the endsections of the deck or the pier) are modelled as rigid. Note that, unlike US codes (Caltrans,2006), Eurocode 8 includes in the model the full width of the deck at and near monolithicsupports on pier columns as working in bending. In case 2, separate nodes are normally intro-duced: one at the top of the pier and another on the deck centroidal axis (or its mid-surface, ifthe deck is a concrete slab as in the previous paragraph). These two nodes are constrained tohave the same translations but different nodal rotations. In case 3, separate nodes are introducedat the top of the pier and at the deck immediately above it and are connected through a specialelement modelling the bearing. This may be a truss element with extensional stiffness (kN/m) ineach direction in which the bearing presents flexibility. Special bearings for seismic isolation havenonlinear force-deformation behaviour, which should be modelled as such in the proper analysisframework (i.e. with a nonlinear method). Although common laminated elastomeric bearingsalso play a role as isolation devices, they are considered as elastic, and have a place in modalresponse spectrum analysis. Their elastic stiffness is equal to EvAb/tq in the vertical directionand to GAb/tq in the horizontal direction, where Ev is the effective elastic modulus of the elas-tomer at right angles to the lamination, G its shear modulus, Ab the horizontal surface area ofthe bearing and tq the total thickness of the layers of elastomer in the bearing. The shearmodulus G can be considered as a material constant, with its value selected as outlined inSection 6.10.4.1 of this Guide. By contrast, the effective elastic modulus Ev at right angles tothe lamination depends very much on its dimensions (its shape factor S and bulk modulus K,etc, see Sections 6.10.4.1 and 6.10.4.2). The commonly used elastomeric bearings have relativelyhigh stiffness at right angles to the lamination, and may normally be assumed to be rigid in thatdirection, at least to a first approximation. If the bearing is fixed (rigid) in one horizontal direc-tion, the pier and deck nodes it connects are constrained to have the same displacement in thatdirection.

Note that a deck monolithically or rigidly connected to certain piers or the abutment(s) may havean out-of-phase response with respect to the piers or abutment(s) supporting it through movablebearings. Such a response will be reflected in higher modes; the increase in bearing deformationsand the pier forces it entails will be appropriately captured by the mode combination rules inSection 5.5.4.

Shear keys are often provided on either side of the deck over a pier or abutment where thedeck is supported through movable bearings (e.g. sliding or elastomeric ones), in order toprevent it from lateral unseating or fall-off. Closure of the gap between the deck and the shearkey under the design seismic action is a nonlinear phenomenon and cannot be realisticallyreflected in linear analysis: the secant stiffness allowed for seismic links in Part 2 of Eurocode8, equal to the lateral resistance of a shear key divided by the travel of the deck until thislateral resistance is attained, is hard to determine before dimensioning the shear key. Tobypass the problem, the gap may be filled by vertical elastomeric bearings placed between theside of the deck and the shear key. A transverse stiffness of the spring equal to the aggregateelastic stiffness, SEvAb/tq, of the vertical bearings of one side, may then be used at that pier orabutment. If there is no gap or slack, this connection is equivalent to a fixed bearing, and thesupport transverse stiffness is that of the pier or abutment. A typical wall-type abutment hasquite high transverse stiffness and resistance, even discounting the uncertain lateral contri-bution of the soil/backfill. Such a transverse connection of the deck to the abutment may betaken as a rigid support.

5.5.1.4 Modelling and analysis of link slabs between deck spans of prefabricatedgirders

A prefabricated deck (be it of a number of parallel girders or of a single box girder) is most oftensimply supported on the piers. Elastomeric bearings are often used at both supports of the span,although a fixed bearing at one support and a movable one at the other have been used as an

Clause 6.6.1(2) [2]

Clause 4.1.3(3) [2]


81

option. This option, however, is not recommended, as it gives rise to several problems regardingthe seismic isolation of the deck, especially when used in conjunction with link slabs.

As pointed out in Section 4.2.1 of this Guide, a link slab (or continuity slab) is most often cast insitu over a joint between the girders of adjacent spans (see Figure 4.1 for an example of thedetail). The link slab provides continuity to the top slab, to avoid a roadway joint and toenhance motorist comfort, and creates a rigid diaphragm link between the adjacent deckspans, so that the deck responds to the transverse seismic action as a single extended elasticbody over its entire length. Additionally, it serves as a sacrificial seismic link between the twospans against unseating and span drop-off. The link slab above the joint should be included inthe model. Its stretch between the end of a deck span consisting of simply supported girdersand that of the adjacent span may be modelled as a 3D beam element connecting the endnodes of the beam elements modelling the deck (the ones above the support nodes at the piertop). It is preferable to use for the link slab a single beam element along the bridge axis,having a rectangular section with the vertical dimension the same as the true depth of the linkslab and the horizontal dimension as the full width of the deck slab. This applies even whenthe girders of the deck span are not lumped into a single beam element along the bridge axisbut modelled individually, as in the grillage deck model highlighted in the penultimate paragraphof Section 5.5.1.2. In this case, the single link slab element will connect the mid-point nodes alongthe spine of beam elements modelling the diaphragm beams over the support nodes of each deckspan. The model should account for the vertical eccentricity between the centroidal axis of thegirders and the mid-plane of the link slab. To reflect the serious damage and other implicationsof the unavoidable concentration of inelastic deformations in the short stretch of the shallow linkslab above the joint, Part 2 of Eurocode 8 specifies for it an effective stiffness of 25% ofthe uncracked concrete stiffness. The reduction factor of 0.25 should be incorporated bothin the out-of-plane flexural rigidity, (EI )s, of the link slab element (for its flexural behaviourin the longitudinal direction of the bridge) and in its in-plane flexural and extensional ones(for the transverse response).

The following actions and action effects should be considered for the link-slabs:

1 For gravity loads and for imposed deformations:

– The local bending due to wheel traffic loads on the slab, considered to be supportedbeyond its clear span Lc, on the top slab of the rest of the deck (which is, in turn,transversely supported on the main prefrabicated girders, see Figure 4.1).

– The effects of the relative rotation, wd, of the end sections of the deck girders, one eitherside of the link slab (Figure 5.5(a)). It induces a bending moment Mc,d (approximately)constant over the length Lc, equal to

Mc;d ¼EIð Þswd

Lc

ðD5:7Þ

where (EI )s is the out-of-plane flexural rigidity of the link slab. Owing to redistributiondue to creep, the rotation wd is due not only to the permanent loads applied afterhardening of the link slab (due to the weight of the pavement, the sidewalks, etc.) butalso to a major part of the loads (about 75%) that were already acting before (theweight of the deck girders and of the in-situ slab, if cast before the link slab).A secondary effect is a tensile axial force of

Nc;d ¼ GAb

tqwd

hd2

ðD5:8Þ

where G, Ab and tq are the shear modulus, the total horizontal area of the elastomericbearings under that span end and the total elastomer thickness of each bearing,respectively, and hd is the depth of the deck girder (see Figure 5.5(a)). This axial forceshould be added to the tensile axial forces that develop as a reaction of the elastomericbearings to the shortening of the deck due to concrete shrinkage and temperaturevariation. Note that all of these effects of the gravity actions do arise from the analysis


82

for them, if the link slab is included in the model as recommended above. If, in contrast,it is excluded, they should be determined via Eqs (D5.7) and (5.8) from the value of wd

resulting from the analysis of the girders as simply supported under gravity actions.2 For seismic loads:

– For seismic action in the longitudinal direction, the main effect is induced by therotation, wp, of the pier head. As the link slab length, Lc, is much shorter than the spansof the deck on either side, the latter may be approximately taken to remain horizontal.So, the pier head rotation induces an equal chord rotation in the link slab; that is, skew-symmetric moments at its two ends (see Figure 5.5(b)) equal to

Mc;p ¼ 6 EIð Þswp

Lc

ðD5:9Þ

Skew-symmetric end moments equal to Mc,p are applied in turn to the girder deck. If Lg

is the girder length, the girder shear of 2Mc,p/Lg accompanying these moments, plus theshear of 2Mc,p/Lc in the link slab, produce a vertical reaction in the bearings. If the tworows of bearings under the girders of adjacent spans are at a centreline distance of Lb

along the deck axis, the couple of these reactions is translated into a total top momentin the pier equal to

Mp ¼ 2Mc;pLb

1

Lg

þ 1

Lc

� �� 2Mc;p

Lb

Lc

ðD5:10Þ

This moment, acting on the pier top with a sign opposite to that of the cantilevermoment at the pier bottom, suggests that the link slab induces a secondary ‘framing’


Figure 5.5. Action effects in a link slab between precast girders due to imposed rotations: (a) from the

deck under gravity actions; (b) from the pier under seismic action

Pier axis

Pier axis

Beam axis

Elastomeric bearing

(a)

(b)

Elastomeric bearing

Vb

Lc

Lc

Vb

hd

ϕd/2

ϕd

∆d = ϕpLc

Mc,d

Mc,p

Mc,phd

Beam axis

Beam axis

Beam axis

ϕp

83

Clause 4.1.2(5),

Annex F [2]

effect between the pier top and the deck. Note that all of these seismic action effectsarise from the seismic analysis if the model includes the continuity slab as recommendedabove and separate nodes at the top of the pier for the two rows of bearings at adistance Lb apart. If it excludes these, they may be estimated via Eqs (D5.9) and (5.10)from the value of wp resulting from the seismic analysis with the pier considered as avertical cantilever supporting the mass of the girders.A secondary effect is an axial force due to the different stiffnesses of the piers and theabutments in the longitudinal direction, which affects the (otherwise equal) horizontalreactions of the elastomeric bearings to the inertia forces. This effect is minor, owing tothe large shear flexibility of these bearings.

– For seismic action in the transverse direction, the effects are usually minor: they are dueto the differences in stiffness of the individual piers and abutments, whose importance isreduced by the dominant flexibility of the bearings. However, higher mode effects arepossible in this case, due to the distribution of mass and transverse stiffness of the deckover the full length of the deck. Nevertheless, the resulting seismic action effects usuallyare not a problem for the link slab, owing to the large available section depth for thedeck in-plane bending (the full width of the deck). The seismic action effects in the linkslab deserve attention only if the bearings on either side of it (elastomeric, sliding orother) have a very different stiffness in the transverse direction.

5.5.1.5 Modelling hydrodynamic effects on immersed piers5.5.1.5.1 IntroductionWhen bridge piers are immersed in water of considerable depth, hydrodynamic effects on theseismic response to the horizontal seismic action are important. They are negligible as far asthe seismic response to the vertical component is concerned. For vertical piers, the effect forthe horizontal components may be estimated by means of an ‘added mass’ of entrained water.This approach, highlighted in the present section, is based on the following assumptions,which are generally met when the seismic response of immersed piers is concerned:

g water is incompressibleg surface wave effects due to the seismic motion are negligible.

Hydrodynamic effects during the seismic response of an immersed structure are due to variationsin the water pressures on the water–structure interface, and are, therefore, a function of themotion of the structure relative to the water. The resultant forces of these pressures may be con-veniently expressed as the product of an ‘added’ mass matrix, Ma, considered as attached to thestructure, multiplied by the minus acceleration vector of the structure relative to the water in theconsidered direction, €Usw (i.e. �Ma

€Usw). The equation of motion of the structure is

MUAþC _UþKU ¼ –Ma€Usw (D5.11)

whereM, C and K are the mass, damping and stiffness matrices of the structure, respectively; forhollow piers the mass of water enclosed within the pier is included in M; UA and U are thedisplacement vectors of the nodes of the structure, absolute and relative to the ground, respect-ively; and _U and €U are the pseudo-velocity and pseudo-acceleration vectors of the structuralnodes, respectively.

The absolute, €UA, relative, €U and ground €ug accelerations are related as €UA ¼ €Uþ €ug I, where theunit vector I has all entries in the direction of the ground motion equal to 1, and 0 in all otherdirections. So, the equations of motion become

MUþC _UþKU ¼ –MIug�Ma€Usw (D5.12)

5.5.1.5.2 Added mass for the horizontal seismic componentsHaving zero shear stiffness, the water cannot follow a horizontal ground motion. Therefore, theacceleration of the structure relative to the water is equal to its absolute acceleration: €Usw ¼ €UA.Therefore, Eq. (D5.12) becomes

(MþMa) €UþC _UþKU ¼ �(MþMa)I €ug (D5.13)


84

This is equivalent to increasing the mass matrix to MþMa. As it lengthens the periods, theadded mass may reduce spectral accelerations; however, it increases the forces and displacementsof the seismic response. Its magnitude may be quite considerable, especially in hollow piers. Apier of elliptical section and axes with length 2ax and 2ay subjected to a horizontal seismicaction component at an angle u to the x axis has an added mass per unit of immersed lengthequal to

ma ¼ rp(ay2 cos2uþ ax

2 sin2u) (D5.14)

where r is the water density. Piers of circular cross-section have ax ¼ ay ¼ R. InformativeAnnex F in Part 2 of Eurocode 8 gives the added mass of immersed rectangular piers in thedirections of its sides as a function of the aspect ratio of the pier section. Note that the addedmass of elliptical or rectangular piers is different in the two orthogonal principal directions ofthe section.

The approach above gives the force resultant in the horizontal direction of the seismiccomponent and the related moments due to the variation in the water pressures on the lateralfaces of the pier. If those faces are not vertical, the horizontal component of the forcesresulting from these water pressures is well approximated by the simplified added massapproach, but the vertical components of the pressures and their effects are not captured.These vertical components produce a moment in the vertical plane of the horizontal seismiccomponent, which is destabilising (i.e. it increases the overturning seismic moment at thebase) if the outer faces of the pier taper downwards as in Figure 5.6(a); if they taper upwardsas in Figure 5.6(b), they are stabilising. If the non-vertical lateral surface of the pier issymmetric with respect to a vertical plane at right angles to the horizontal seismic com-ponent, the vertical pressure components have a zero vertical force resultant. For immersedpiers with inclined lateral faces, it is recommended to use special finite element approachescapable of dealing with the general ‘coupled’ problem of the seismic response of an immersedstructure.

5.5.1.5.3 Vertical seismic componentBeing incompressible, the water moves vertically with the ground. So, the acceleration of thestructure relative to the water is equal to the one relative to the ground, €U. The added massmatrix associated with the vertical relative movement is denoted as Mav. An additional effectof the water pressure variation on the lateral surface of the immersed pier is not due to themotion of the structure relative to the water, and therefore not associated with an ‘added’mass. It is due to the vertical acceleration of the water, €ugv, which causes its apparent unitweight to vary. The ensuing variation of the buoyancy force on the structure is convenientlyexpressed as MbI €ugv, where Mb is the mass of the water displaced by the pier.

Before writing the equations of motion, the following should be pointed out:

g Obviously the ‘added’ mass of a pier in the vertical direction, Mav, has nothing to do withthat in the horizontal; indeed, if the lateral faces of the pier are vertical, Mav is zero.


Figure 5.6. Effect of the vertical component of water pressure on an immersed pier with non-vertical

outer surfaces: (a) destabilising on pier tapering downwards, M ¼ zPHþ bPV/2; (b) stabilising on pier

tapering upwards, M ¼ zPH – bPV/2

PH/2 PH/2

PH/2 PH/2PV/2 PV/2

PV/2 PV/2b

b

z z

85

Clauses 4.1.4(1),

4.1.4(3), 6.7.3(2) [2]

Clauses 4.2.3(1),

4.2.3(3), 5.4.2(1),

5.4.2(3), 6(1), 6(2),

Annex D [3]

g A significant added mass in the vertical direction derives from a boundary of the immersedbody in contact with water with a substantial projection on a horizontal plane (e.g. asubmerged pipe or tunnel supported on the piers).

g Even for piers with inclined lateral faces (where Mav is non-zero), the relative accelerationvector, €Usw ¼ €U, is due to the axial deformation of the pier; its magnitude is thereforemuch smaller than that of €Usw corresponding to the horizontal directions where bendingdeformation is involved.

g The hydrodynamic effect due to the ‘added’ mass in the vertical direction is very small forpiers of common bridges.

With the effects of ‘added’ and ‘buoyant’ masses included, Eq. (D5.12) is written as

(MþMav) €UþC _UþKU ¼ �(M�Mb)I €ugv (D5.15)

which shows that only the part M�Mb of the total vibrating mass MþMav is excited by €ugv.Therefore, for the analysis the mass matrix is increased to MþMav, but the modal excitationfactors must be modified to correspond to the excited mass M�Mb instead of the total.Additionally, the sum of effective modal masses in the vertical direction is less than the totalstructural mass.

In conclusion, it is justified for typical bridges to neglect the hydrodynamic effects of the verticalseismic component, as allowed in Part 2 of Eurocode 8. In exceptional cases where this effectshould be included, either Eq. (D5.15) is solved, or recourse is made to special finite elementsoftware.

5.5.1.6 Linear modelling of the foundation and the soilBridge foundations are either massive shallow footings, deep seated (caissons) or piled. For thesetypes of foundations, Part 5 of Eurocode 8 (CEN, 2004b) requires that soil–structure interactionbe taken into account in the seismic analysis. Bridge foundations are specifically quoted as beingcases where considering soil–structure interaction is mandatory.

A rigorous treatment of linear soil–structure interaction is based on elasto-dynamic theories. Adirect (or complete) interaction analysis, in which both the soil and the structure are modelledwith finite elements is very demanding computationally and not well suited for design, especiallyin 3D. A substructuring approach reduces the problem to more amenable stages, and does notnecessarily require repeating the entire analysis should the superstructure be modified. Thisapproach is of great mathematical convenience and rigour, stemming, in linear systems, fromthe superposition theorem (Kausel and Roesset, 1974), according to which the seismicresponse of the complete system may be computed in two stages (Figure 5.7):

1 Determination of the kinematic interaction motion, involving the response to baseacceleration of a system that differs from the actual one in that the mass of thesuperstructure is equal to zero.

2 Calculation of the inertial interaction effects, referring to the response of the completesoil–structure system to forces associated with base accelerations equal to the accelerationsarising from the kinematic interaction.

The second step is further divided into two subtasks:

g computation of the dynamic impedances at the foundation level; the dynamic impedanceof a foundation represents the reaction forces acting under the foundation when it isdirectly loaded by harmonic forces

g analysis of the dynamic response of the superstructure supported on the dynamicimpedances and subjected to the kinematic motion (also called the effective foundationinput motion).

For rigid foundations, such as those of bridges, the dynamic impedances can be viewed as sets offrequency-dependent springs and dashpots lumped at the underside of the footing or of the pilecap. For rigid foundations, the complex-valued impedance matrix in the most general situation is


86

a full 6 � 6 matrix. However, for regular geometries with two axes of symmetry and horizontallylayered soil profiles, some off-diagonal terms are equal to zero, and the impedance matrix takesthe simplified form

K ¼

k11 0 0

0 k22 0

0 0 k33

0 k15 0

k24 0 0

0 0 0

0 k42 0

k51 0 0

0 0 0

k44 0 0

0 k55 0

0 0 k66

26666666664

37777777775

ðD5:16Þ

DoFs 1 to 3 refer to translations, and DoFs 4 and 5 to rocking about orthogonal horizontal axes(Figure 5.8). DoF 3 is the vertical translation. Diagonal terms k33, referring to this DoF, and k66to the torsional DoF, are uncoupled with respect to the other DoFs. The two translational DoFsare always coupled with the two rocking ones for piled or embedded foundations; this couplingmay be neglected only for surface foundations or when the embedment of the footing is shallow.

It may not be possible to introduce coupling (i.e. off-diagonal) terms to the stiffness matrix inmost commercial software codes. It is, however, feasible to account for coupling terms by trans-lating the stiffness matrix to a point located at a depth z (.0) below the foundation where thestiffness matrix becomes diagonal, and modelled with simple springs in all directions. Forinstance, referring to directions 1 and 5, a diagonal stiffness matrix with a translational springstiffness of k11 and a rotational one of k55� z2k11 connected to the foundation at a depthz ¼ �k15/k11 via a rigid beam element produces the correct stiffness matrix at the foundation(Figure 5.8).


Figure 5.7. Superposition theorem in soil–structure interaction

ÿ(t)

ÿ(t)

Equivalent

Kinematic interaction

MV

θü

θü

Impedance function

Kxx KxψK = [ ]Kxψ Kψψ

Figure 5.8. Modelling of rotational-translational coupling in piles (a); DoFs and pile stiffness (b);

modelling of pile with uncoupled translational and rotational springs

Pile

Rotational: K55 – z2K11

z = K15/K11

3 δ11 = 1

I = ∞

δ55 = 1

K55

K33

K51

K11

K11

K15

6 2

51

44

Pile

(a) (b)

87

Analytical expressions for the different terms of the impedance matrix, Eq. (D5.16), for shallowfootings of various geometries, embedded foundations or pile foundations can be found in thetechnical literature (Novak, 1974; Gazetas, 1983, 1991). Informative Annex C of Part 5 ofEurocode 8 provides expressions for the stiffness of the head of individual piles for variouscases of dependence of the soil modulus on depth. Note that the pile head stiffness may bestrongly affected by pile group effects, reflecting pile–soil–pile interaction. This effect dependson the cross-sectional size and spacing of piles, the soil properties and the frequency of excitation.Unlike in static loading, it may be significant even at large pile spacing. Simplified methodologieshave been developed to compute the group efficiency under seismic loading (Dobry and Gazetas,1988). For common pile sizes and spacing, the group effect on the stiffness typically varies from2.5 to 6.

Each term in Eq. (D5.16) is a complex number that can be written as

kij ¼ ksij kdij þ i a0 cdij

� ¼ ksij kdij þ i

vB

vscdij

� �ðD5:17Þ

where kijs is the static component of the impedance; the terms in parentheses, kij

d and cijd,

correspond to the dynamic contribution to the impedance, and are frequency dependent;a0 ¼ vB/vs is a dimensionless frequency; v is the circular frequency of the excitation; B is acharacteristic dimension of the foundation (radius, width, etc.); vs is the soil shear wavevelocity; and i2 ¼ �1.

The terms in Eq. (D5.17) have the following physical interpretation: kijs kij

d represents a spring andkijs cij

dB/vs a dashpot. The equivalent damping ratio of the system is

jij ¼vB

vs

cdij

2kdij

ðD5:18Þ

It is common – and recommended – practice in the context of modal response spectrum analysisto bound upwards the value of the equivalent damping ratio from Eq. (D5.18) to 30%.

Although the substructure approach above is rigorous for the treatment of linear soil–structureinteraction, its practical implementation is subject to several simplifications:

g Fully linear behaviour of the system is assumed. However, it is well recognised that this isa strong assumption, since nonlinearities are present in the soil itself and at the soil–foundation interface (sliding, uplift, gaping near pile heads, etc.). Soil nonlinearities maybe partly accounted for, as recommended in Part 5 of Eurocode 8, by choosing for thecalculation of the impedance matrix reduced values of the soil properties that reflect thesoil nonlinear behaviour in the free field (see Section 3.3.2.2 of this Guide). This implicitlyassumes that additional nonlinearities taking place at the soil–foundation interface (pileshaft, edges of footing) do not affect significantly the overall seismic response. For piles itis recommended to further reduce the soil moduli at the top, starting from an almost zerovalue at the pile head and increasing it to the ‘full reduced’ value at a depth of 4–6 timesthe pile mean cross-sectional dimension. Recommended reduction factors for the soilmoduli are given in Table 3.5 in Section 3.3.2.2.

g Kinematic interaction is usually ignored. This means that the input motion used for thedynamic response of the structure is simply the free-field motion. Although thatassumption is exact for shallow foundations subject to vertically propagating body waves,and partly exact for shallow foundations in a more complex seismic environment or forflexible piles, it becomes very inaccurate for embedded or very stiff piled foundations. Inparticular, the true input motion to embedded foundations consists not only of atranslational component but of a rotational one as well.

g The frequency-dependent terms of the stiffness matrix are approximated as constant.Except in the very rare case of a homogeneous soil profile, this condition is far from beingfulfilled. A fair approximation for the constant value of the stiffness term is onecorresponding to the frequency of the soil–structure interaction mode. To find such a


88

value, iterations on the spring constant should be carried out for each vibration mode,until the chosen value corresponds to the computed frequency. More refined models,involving additional masses and springs, are also possible to represent the frequencydependence of the impedance term. One such example is shown in Figure 5.9. The modelparameters may be fitted to the exact variation of the impedance term (as exemplified inFigure 5.10), or assessed from simplified assumptions based on cone models (Wolf andMeek, 2004). Alternatively, frequency domain solutions may be used: they offer severaladvantages, such as a rigorous treatment of the frequency-dependent impedance matrixand full account of the radiation damping represented by the imaginary part of theimpedance matrix. Special software tools have been developed to solve soil–structureinteraction problems in the frequency domain (Lysmer et al., 2000), widely known in theindustry.

A wide class of – so-called – Winkler models, based on the concept of springs and dashpots tomodel the effect of the soil on the foundation have been used extensively. They represent the


Figure 5.9. Example of a simple rheological model for foundation impedance

Figure 5.10. Example of the accuracy of the rheological model of Figure 5.9 used to model the exact

dynamic rocking impedance of the Rion Antirrion bridge foundation (Pecker, 2006)

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

1.E+07

–1.E+08

–5.E+07

0.E+00

5.E+07

1.E+08

0 1 2 3 4 5Frequency: Hz

0 1 2 3 4 5Frequency: Hz

Stiff

ness

: MN

/mD

ashp

ot: M

N s

/m

ModelFinite element analysis

ModelFinite element analysis

89

interaction with the soil using springs and dashpots distributed across the footing or along thepile shaft. Although conceptually the soil reaction forces are still represented by the action ofsprings and dashpots, unlike the impedance matrix approach, there is no rational or scientificallysound method for the definition of these springs and dashpots. Their values, but more impor-tantly their distribution over the foundation, vary with frequency; there is no unique distributionreproducing the global foundation stiffness for all degrees of freedom. For instance, althoughhighly questionable, a uniform distribution may be chosen for the vertical stiffness, but cannotaccurately match the rocking stiffness. Furthermore, two additional difficulties arise for pilefoundations:

g the choice of the springs and dashpots should reflect the pile group effectg as the seismic motion varies with depth, different input motions should be defined at all

nodes shared between the piles and the soil; one should resort to a separate analysis forthe determination of these input motions.

Therefore, in view of all the uncertainties underlying the choice of their parameters, Winkler-typemodels, although attractive, should not be favoured. Figure 5.11 depicts both types of modelling:

g that based on the concept of the impedance matrix, which is rigorous in the framework oflinear systems

g the approximate Winkler-type model.

5.5.2 Modal analysis resultsIn modal response spectrum analysis, the modal shapes (eigenmodes) in 3D and the naturalfrequencies (eigenvalues) are first computed. In a full 3D model, each mode shape in generalhas displacement and rotations in all three directions, X, Y and Z, at all nodes i of the model.The eigenmode–eigenvalue analysis gives for each normal mode, n:

1 The natural period, Tn, and the corresponding circular frequency, vn ¼ 2p/Tn.2 The mode shape vector Fn.3 Factors of the participation of the mode to the response to the seismic action component

in direction X, Y or Z, denoted as GXn, GYn or GZn, respectively. GXn is computed as

GXn ¼ FnTMIX/Fn

TMFn ¼ SiwXi,nmXi/Si(w

2Xi,nmXiþ w2

Yi,nmYiþ w2Zi,nmZi) (D5.19)

where i stands for nodes associated with DoFs, M is the mass matrix, IX is a vector withelements of 1 for the translational DoFs parallel to direction X and all other elements


Figure 5.11. Modelling techniques for linear soil–structure interaction: (a) impedance matrix; (b) Winkler-

type model

Horizontalmotion 1

Horizontalmotion 2

Horizontalmotion 3

Horizontalmotion 4

Horizontalmotion n

Verticalmotion 1

Verticalmotion 2

Verticalmotion 3

Verticalmotion 4

Verticalmotion n

(b)(a)

khn

kh4

kh3

kh2

kh1

kvn

kv4

kv3

kv2

kv1

Superstructure

Pilefoundation

Condensed 6 × 6stiffness matrix [K]

6 × 1 kinematicmotion time series

Condensed 6 × 6mass matrix [M]

Pile cap

Superstructure

Depth varyingfree fieldmotions Pile cap

90

equal to 0, wXi,n is the element of Fn corresponding to the translational DoF of node iparallel to X and mXi is the associated element of the mass matrix. Similarly for wYi,n,wZi,n, mYi and mZi. If M contains rotational mass moments of inertia, IuXi,n, IuYi,n andIuZi,n, the associated terms are included in the sum at the denominator of GXn. Thedefinitions of GYn, GZn are similar.

4 The effective modal masses in directions X, Y and Z are MXn, MYn, and MZn, respectively:MXn ¼ (Fn

TMIX)

2/FnTMFn ¼ (SiwXi,nmXi)

2/Si(w2Xi,nmXiþ w2

Yi,nmYiþ w2Zi,nmZi); similarly for

MYn and MZn. They give the peak force resultant in mode n along direction X, Y or Z:VbX,n ¼ Sa(Tn)MXn, VbY,n ¼ Sa(Tn)MYn or VbZ,n ¼ Sa(Tn)MZn, respectively. The sum ofeffective modal masses in X, Y or Z over all modes is equal to the total mass.

Peak modal seismic action effects in the response to the seismic action component in direction X,Y or Z may be calculated as follows:

1 For each mode n the spectral displacement, SdX(Tn), is calculated from the designacceleration spectrum of the seismic action component; for example, for direction X asSdX(Tn) ¼ (Tn/2p)

2SaX(Tn).2 The nodal displacement vector of the structure in mode n due to the seismic action

component of interest, let us say direction X, UXn, is computed as UXn= SdX(Tn)GXnFn.3 Peak modal values of the effects of the seismic action component of interest are computed

from the modal displacement vector of step 2; member modal deformations (e.g.curvatures or chord rotations) directly from the nodal displacement vector of mode n;modal internal forces at member ends by multiplying the member modal deformations bythe member stiffness matrix; etc.

The so-computed peak modal responses are exact but occur at different instances in the response,and can be combined only approximately. Section 5.5.4 presents statistical combination rules forthe peak modal responses.

5.5.3 Minimum number of modesModal response spectrum analysis should take into account all modes contributing significantlyto any response quantity of interest. This is hard to achieve in practice. As the number of modesto be considered should be specified as input to the eigenvalue analysis, either a very large numbermay have to be specified from the outset or the eigenvalue analysis may have to be repeated with ahigher number of modes requested this time.

Eurocode 8 considers the total force resultant in the direction of each seismic action componentas the prime response quantity of interest, and sets as a goal of the eigenvalue analysis the captureof at least 90% of its full value. This is translated into a criterion for the participating modalmass: Eurocode 8 requires the N modes that are taken into account to provide together a totalparticipating modal mass along any individual direction of the seismic action components con-sidered in the design, at least 90% of the total mass. In the example of Figure 5.3, three modessuffice for the longitudinal direction, but nine are needed for the transverse; this requirescomputing up to the 49th eigenmode of the bridge.

If the above criterion is hard to meet, Part 2 of Eurocode 8 allows as an alternative the compu-tation of all modes with periods above 0.033 s, provided that they collectively account for atleast 70% of the total mass in the direction of each component of interest. Then, we shouldmake up for the missing mass by amplifying all computed modal seismic action effects by theratio of the total mass to that accounted for so far. If the computed modes with periodsabove 0.033 s do not collectively capture at least 70% of the total mass, more modes shouldbe computed, until the minimum of 70% is achieved. The amplification of computed modalresults follows. Clearly, this option is so conservative and uneconomic for the design that thedesigner is motivated to go after as many modes as necessary to capture at least 90% of thetotal mass.

5.5.4 Combination of modal resultsIn modal response spectrum analysis, it is convenient to take the elastic response in two differentmodes as independent of each other. The magnitude of the correlation between modes i and j

Clauses 4.2.1.2(1),

4.2.1.2(2) [2]

Clause 4.2.1.2(3) [2]

Clause 4.2.1.3 [2]


91

Next Page




Chapter 6

Verification and detailing of bridgecomponents for earthquake resistance

6.1. IntroductionThe rules for the verification of the strength and global ductility of bridge components in Section 5of Part 2 of Eurocode 8 (CEN, 2005a), alongside those for detailing the components for localductility in Section 6, elaborate and implement the compliance criteria for the no-collapserequirement set out in Section 2 and in Chapter 2 of this Designers’ Guide. These rules aredifferent for bridges with ductile or with limited ductile behaviour. Different rules apply forbridges equipped with seismic isolation: they are given in Section 7 of Part 2 of Eurocode 8,and elaborated and exemplified in Chapter 7 of this Designers’ Guide.

For better illustration and comparison, the rules for the two design alternatives of ductile orlimited ductile behaviour are presented here in tabular form (Table 6.1). Some general aspectsand concepts are highlighted first, and the rationale behind certain verification procedures andrules is presented.

6.2. Combination of gravity and other actions with the designseismic action

Both at the local level (for the verification of members and sections) and at the global level (forthe calculation of masses), the design seismic action is combined with other actions as specified inEN 1990:2002 (‘Basis of design’ – CEN, 2002) for the seismic design situation and elaborated forbridges in Part 2 of Eurocode 8. Symbolically this combination is

Ed ¼ Gk ‘þ’ Pk ‘þ’ AEd ‘þ’ c2,1Q1,k ‘þ’ Q2 (D6.1)

where Gk is the characteristic (nominal) value of the permanent actions (normally the self weightand all other dead loads), Pk is the characteristic value of prestress after losses, AEd is the designseismic action (i.e. that for the ‘reference return period’ times the importance factor of thebridge), Q1,k is the nominal value of traffic loads, c2,1Q1,k is the combination value of trafficloads (normally the quasi-permanent, i.e. arbitrary-point-in-time, traffic loads) and Q2 is thequasi-permanent value of variable actions with long duration (earth or water pressure,buoyancy, currents, etc.).

Coefficient c2,1 is defined in normative Annex A2 of EN 1990:2002 as a Nationally DeterminedParameter (NDP). As already pointed out in Section 5.4 of this Guide in connection withEqs (D5.6), Part 2 of Eurocode 8 recommends the following values:

g c2,1 ¼ 0 for bridges with normal traffic and footbridgesg c2,1 ¼ 0.2 for road bridges with ‘severe traffic conditions’ (defined in a note as motorways

and other roads of national importance)g c2,1 ¼ 0.3 for railway bridges with ‘severe traffic conditions’ (defined in a note as intercity

rail links and high-speed railways).

Part 2 of Eurocode 8 recommends taking only the characteristic value, qk, of the uniform trafficload (UDL) of Load Model 1 (LM1) – with all its nationally applying adjustments – as contri-buting to the quasi-permanent part of ‘severe traffic’ loads. Note that, at least as far as thecalculation of masses is concerned, the small values of c2,1 even for bridges with heavy traffic

Clauses 3.2.4(1) [1]

Clauses 5.5(1), 5.5(4),

4.1.2(3) [2]

119


Table 6.1. Eurocode 8 rules for dimensioning and detailing of bridge piers and deck

Limited ductile Ductile

q factor q � 1.5 (see Table 5.1) q � 1.5 (see Table 5.1)

Limitations for reinforcing bars

In plastic hinges Eurocode 2 Class B or C Eurocode 2 Class C

Outside plastic hinges Eurocode 2 Class B or C

Ultimate limit state (ULS) strength verification and dimensioning of the deck

ULS in flexure MGdþMAd(1) �M yd

(2) MGdþ aCDMAd(1),(3) �M yd

(2)

ULS in shear VGdþ qVAd(1) � VRd/gBd1

(4) VGdþ aCDV Ad(1),(3) � VRd/gBd

(5)

ULS strength verification and dimensioning of the piers in flexure

(i) In plastic hinges oMGdþMAdþ DM2nd-order

(1),(6) �MRd(7)

MGdþMAdþ DM2nd-order(1),(6) �MRd

(7)

(ii) Outside plastic hinges MC �MRd(7),(8)

ULS strength verification and dimensioning of the piers in shear

(i) In plastic hinges oVGdþ qVAd

(1) � VRd /gBd1(4)

VGdþ aCDV Ad(1),(3) � VRd,o /gBd

(9)

(ii) Outside plastic hinges VGdþ aCDV Ad(1),(3) � VRd/gBd

(5)

Detailing of pier columns for ductility within length Lh from end section of plastic hinge

Length Lh from the end section where full confinement and antibuckling reinforcement is required:

(i) If hk ¼ NEd/Acfck � 0.3 oLh ¼ 0.3LsMRd/(1.3MEd)

(10),(11)Lh ¼ max[hc; Ls/5]

(10),(11),(12),(13)

(ii) If 0.3 , hk � 0.6 Lh ¼ 1.5max[hc; Ls/5](10),(11),(12),(13)

Transverse reinforcement(14) for confinement:(15),(16)

Maximum hoop/tie spacing along pier – �min(hc; bc)/5 or Dc/5(13)

Rectangular hoops and cross-ties:

mechanical ratio in each direction

Asw/(shbc)(fyd/fcd)(17)

.0.28hkAc/Accþ 0.13(rL� 0.01)fyd/f cd(19) .0.37hkAc/Accþ 0.13(rL� 0.01)fyd/fcd

(19)

.0.08 .0.12

Circular hoops or spirals: volumetric

mechanical ratio, 4Asp/(shDsp)(fyd/fcd)(18)

�0.39hkAc/Accþ 0.18(rL� 0.01)fyd/fcd(19) �0.52hkAc/Accþ 0.18(rL� 0.01)fyd/fcd

(19)

.0.12 .0.18

Hoop legs or cross-ties engaging bars:

spacing along straight sides of section

– �200 mm

�min(hc; bc)/3(13)

Confinement reinforcement at one-half

its amount from 0 to Lh

– Between Lh and 2Lh from end section

Transverse reinforcement(14) for restraint of vertical bars against buckling:

Maximum hoop/tie spacing along pier �ddbL, where d ¼ 2.5ftk/fykþ 2.25, with d � 5, d � 6(20)

Area of restraining tie-legs per linear

metre of pier column length

PAsLfyL/(1.6fyw)

(21)

Vertical bars (index L) throughout the height of pier columns(22)

Minimum steel ratio, rmin(23) 0.1Nd/Acfyd, 0.2%

Maximum steel ratio, rmax 4%

Diameter, dbL � 8 mm

Distance of unrestrained bar from

nearest restrained along perimeter

�150 mm

Transverse bars (index w) over the part of pier columns where special detailing for ductility is not required (22)

Diameter, dbw � dbL/4

Spacing, sw � 400 mm, 20dbL, min(h; b) or D

At lap splices, if dbL . 14 mm: sw � 240 mm, 12dbL, 0.6 min(h; b) or 0.6D

120

around the clock are consistent with the action of vehicles on the deck as ‘tuned mass dampers’that reduce the global response of the bridge, as has been shown in shaking table tests.

In the special but common case where only elastomeric bearings resist the seismic action, theaction effects of imposed deformations (thermal actions, shrinkage, support settlement,ground movement due to seismic faulting, etc.) should also be included in the combination ofEq. (D6.1). Note, however, that these effects are only relevant for the design of the bearingsthemselves, which is normally based on the deformations – not the forces – imposed on thebearings by actions. Additionally, in this case the behaviour under the design seismic action islinear elastic; so is the analysis with q ¼ 1.0. Therefore, the action effects of imposed defor-mations do not vanish, as normally taken in design for the ultimate limit state (ULS) after thepresumed redistribution of self-equilibrating action effects.

Clauses 5.5(2),

5.5(3) [2]

Chapter 6. Verification and detailing of bridge components for earthquake resistance

(1) MAd, VAd: moment and shear from linear analysis for the design seismic action, Ad. MGd, VGd: moment and shear from linear analysis for thequasi-permanent gravity loads concurrent with Ad

(2) Myd: design yield moment of the deck section at yielding of the tension chord, taking into account any prestressing. For bending about thevertical axis, Myd may be taken to occur when the longitudinal reinforcement yields at a distance from the edge of the deck section of 10%its depth for bending in a horizontal plane (i.e. 10% of the width of the top slab) but not beyond the outer web of the section

(3) aCD: capacity design magnification factor. It may be approximated as the ratio of the sum of pier seismic shears when overstrength momentresistances develop at all potential plastic hinges in each pier to the sum of pier shears from the linear analysis for the design seismic action(Eq. (D6.9)). The seismic shear at overstrength moment resistance at the plastic hinges of a pier is computed assuming the simultaneouspresence of the effects of the quasi-permanent loads concurrent with the design seismic action

(4) gBd1: Nationally Determined Parameter safety factor with the recommended value gBd1 ¼ 1.25. VRd: design shear resistance per Eurocode 2,taking into account any prestressing

(5) It is allowed to take gBd equal to the value of gBd1 in note (4) above or to subtract from it qVEd/VC,o� 1 (but not to a final value belowgBd ¼ 1), where VEd is the maximum shear from the analysis for the seismic design situation and VC,o is the capacity design shear without theupper limit VC,o � qVEd

(6) DM2nd-order is the additional moment due to second-order effects. It may be taken equal to (1þ q)dEdNEd/2, where NEd is the axial force anddEd is the relative horizontal displacement of the two pier ends in the seismic design situation (see Eq. (D6.3))

(7) MRd: design moment resistance of the pier section, for the concurrent value of the pier axial force and the orthogonal moment componentfrom the analysis for the seismic design situation

(8) MC: linear bending moment diagram between the overstrength moments at the ends of the pier column where plastic hinges may form (seeFigure 6.1)

(9) VRd,o: design shear resistance per Eurocode 2 computed for a truss inclination of 458 and using as dimensions of the section those of theconfined concrete core to the centreline of the perimeter hoop

(10) Ls ¼M/V: shear span at the end section according to the analysis for the seismic design situation(11) The maximum length for the two directions of the pier section applies(12) MEd: moment at the pier end section from the analysis for the seismic design situation. MRd: as in note (7) above, for the pier end section(13) hc, bc, Dc: depth, width and diameter of the confined core to the centreline of the perimeter hoop(14) Anchored with 1358-hooks. When hk � 0.3, a 908 hook with a 10-diameter extension is allowed at one end of a cross-tie, at alternating

ends in adjacent cross-ties horizontally and vertically. Long cross-ties with 1358 hooks at both ends may be lap-spliced between the ends(15) Confinement is not required in those pier columns where:

g hk ¼ NEd/Acfck � 0.08 org a curvature ductility factor value of

– 13 for ductile behaviour or– 7 for limited ductile

can be achieved in the plastic hinge without confinement. This is possible in piers with wide compression flanges, e.g. in hollow rectangularpiers, in the weak direction of wall-like piers, etc. On this basis, confinement is not required in hollow rectangular piers withhk ¼ NEd/Acfck � 0.2

(16) If confinement all along the long dimension of a wall-like pier is not necessary per the second bullet point in (15) above, but is required inthe pier strong direction, it should be provided up to a distance from the edges where the strain drops below 50% of the ultimate strain ofunconfined concrete: 1cu2 ¼ 0.0035

(17) Asw/sh: total cross-sectional area of hoop and tie legs at right angles to the dimension bc of the concrete core (measured to the outside ofthe perimeter hoop) per unit length of the pier column

(18) Asp: cross-sectional area of the spiral or hoop bar. sh: pitch of the spiral or spacing of the hoops. Dsp: diameter of the spiral or hoop bar(19) Ac: area of gross concrete section. Acc: area of confined core to the perimeter hoop centreline. hk ¼ NEd/Acfck. rL: ratio of vertical

reinforcement(20) dbL: vertical bar diameter. ftk/fyk: ratio of the characteristic tensile strength to the characteristic yield stress of vertical bars. Eurocode 2 limits

the lower 10% fractile of ft/fy to (ft/fy)k � 1.15 for steel Class C and (ft/fy)k � 1.08 for steel Class B; these values may be used in lieu of ftk/fyk,in the absence of steel-specific information

(21) fyw: yield stress of tie. fyL: yield stress of vertical bars.P

AsL: total cross-sectional area of vertical bars restrained at. 908 hooks or corners ofthe ties (tie-legs restraining the bar at a 458 corner contribute too with their cross-sectional area divided by

p2).

(22) These rules are per Eurocode 2 and apply both to ductile and limited ductile design(23) Strictly, the minimum steel ratio in pier columns is that of Eurocode 2, because Part 2 of Eurocode 8 does not specify a minimum steel ratio

for pier columns. According to Part 1 of Eurocode 8, the minimum steel ratio of 1% applies only to concrete columns in buildings, and iscertainly high for pier columns. Such a high minimum ratio may be the source of irregular seismic behaviour if the piers have the samesection but very different seismic demands (see Sections 5.10.2.1 and 5.10.2.2). Nonetheless, it is often applied in practice by designers whoconsider the minimum steel ratio specified in Eurocode 2 for any type of concrete column as too low for earthquake resistance. Note thatthe fundamental requirement of Part 2 of Eurocode 8 for a minimum of local ductility in all ductile elements is of essence. In this respect, theamount of steel reinforcement should be sufficient to provide a design moment resistance of the pier section, MRd, not less than the crackingmoment: Mcr ¼Wc(fctmþ NEd/Ac), where Wc is the elastic section modulus, fctm is the mean tensile strength of concrete, NEd is the axialforce from the analysis for the seismic design situation (positive if compressive) and Ac is the area of the concrete section. Normally, a steelratio of 0.5% provides this minimum of local ductility, without causing significant irregularity

121

Clauses 5.4(1),

5.6.2(1), 5.6.3.1(1) [2]

Clauses 5.2.4(1)–

5.2.4(3), 6.1.3(1),

7.1.3(1) [1]

6.3. Verification procedure in design for ductility using linearanalysis

In force-based seismic design with linear analysis employing a q factor greater than 1.0, theverifications outlined in the following sections are carried out.

6.3.1 Verification of the flexural resistance of plastic hinges at the ULSFlexural plastic hinges are dimensioned so that their design moment resistance,MRd, exceeds themoment from the analysis for the seismic design situation,MEd, including second-order effects, ifapplicable:

MEd � MRd (D6.2)

Linear analysis is carried out for the combination of the actions of Eq. (D6.1). Superpositiondoes apply, so the seismic action effects from an analysis for the design seismic action alonemay be superimposed on those from the analysis for the other actions in Eq. (D6.1). Approxi-mations may be used for the second-order effects in the seismic design situation. Part 2 ofEurocode 8 invites the National Annex to introduce sufficiently accurate methods to includethem. Such methods should account for the less ominous character of second-order effects ofseismic displacements compared with those due to persistent displacements, owing to theirreversed nature. In a non-binding note, Part 2 of Eurocode 8 considers it sufficient to neglectthese effects in linear analysis and simply to increase the moment calculated from the analysisat pier ends where plastic hinges are expected to form by the following additional moment(Paulay and Priestley, 1992):

DM2nd-order ¼ (1þ q)dEeNEd/2 (D6.3a)

where NEd is the axial force and dEe is the horizontal displacement of the top relative to the baseof the pier from the analysis for the design seismic action. According to Paulay and Priestley(1992), an increase in the moment resistance by the amount of Eq. (D6.3a) compensates forthe absorbed energy loss due to the second-order effects of the peak seismic displacements,mddEe (with md taken equal to q, see Eq. (2.1a)). Note that, by mistake, Part 2 of Eurocode 8uses in Eq. (D6.3a) the total design displacement in the seismic design situation, dEd, insteadof the correct dEe; dEd is the sum of (a) the displacements dE, due to the design seismic actionalone (see Eq. (D5.48)), (b) dG, due to the quasi-permanent actions on the bridge, and(c) c2dT, due to the thermal actions present in the seismic design situation. Equation (D6.3a)accounts for second-order effects due to the seismic displacements; to account for those due todisplacement components (b) and (c), Eq. (D6.3a) may be generalised as follows:

DM2nd-order ¼ [(1þ q)dEe/2þ dGþ c2dT]NEd (D6.3b)

In tall and slender piers, dG should include:

g a possible displacement (eccentricity) d0 due to the permanent loadsg the displacement (eccentricity) dimp due to geometric imperfections, according to clause 5.2

in EN 1992-2 (CEN, 2005b)g the effect of concrete creep (through the creep coefficient w).

A convenient approximation that captures all three effects above on the basis of the nominalstiffness method per clause 5.8 of EN 1992-1-1 (CEN, 2004b) is

dG ¼ ðd0 þ dimpÞ 1þ 1þ w

n� 1

� �ðD6:4Þ

where

n ¼ NB/NEd (D6.5a)

with

NB ¼ p2(EI )eff/Lo2 (D6.5b)


122

denoting the buckling load estimated according to the nominal stiffness method of clause 5.8 ofEN 1992-1-1, using the effective stiffness (EI )eff of the pier section in the seismic design situation(see Section 5.8.1), and Lo is the effective length of the pier column in the direction considered.Guidance for the value of Lo is given in Section 4.4.3 of this Guide.

The value of MRd in Eq. (D6.2) should be calculated according to the relevant rules of thepertinent material Eurocode. It should be based on the design values of material strengths;that is, the characteristic values, fk, divided by the partial factor gM of the material. Being keysafety elements, the partial factors, gM, are Nationally Determined Parameters, with valuesdefined in the National Annexes to Eurocode 8. Eurocode 8 itself does not recommend thevalues of gM to be used in the seismic design situation: it just notes the options of choosingeither the values gM ¼ 1 appropriate for the accidental design situations, or the same values asfor the persistent and transient design situation. This latter option is very convenient, becausethe plastic hinge may then be dimensioned for the largest design value of the action effect dueto the persistent and transient or the seismic design situation. With gM ¼ 1, the plastic hingewill have to be dimensioned once for the action effect due to the persistent and transientdesign situation and then for that due to the seismic design situation, each time using differentvalues of gM for the resistance side of Eq. (D6.2).

In piers, Eq. (D6.2) is checked using as MEd the algebraically maximum and minimum value ofeach uniaxial moment component from the analysis and as MRd the design resistance for theconcurrent value of axial force, NEd, and the acting moment in the orthogonal direction of thepier in the seismic design situation. If a separate analysis is carried out for each horizontalcomponent of the seismic action, X and Y, and the outcomes are combined via Eqs (D5.2),the verification is also done separately for the moment components from Eq. (D5.2a) and forthose from Eq. (D5.2b). By contrast, Eq. (D5.1) gives only the peak absolute values of thetwo moment components for the seismic design situation. These values are not concurrent,and it is too conservative to take them as such. Statistical approaches have been proposed inGupta and Singh (1977) – and elaborated and extended in Fardis (2009) – for the estimationof the axial force and orthogonal moment components likely to take place concurrently withthe peak value of each moment component under the simultaneous action of the two – orthree – independent components of the seismic action.

6.3.2 Capacity design of regions, components or mechanisms for elastic responseRegions outside the flexural plastic hinges, and non-ductile mechanisms of force transfer withinor outside the plastic hinges, as well as components for which absolute protection from damage(e.g. the deck, normally the foundation, seismic links, etc.) is desired under the design seismicaction are dimensioned to remain elastic until and after all potential flexural plastic hingesform in the bridge. This is pursued by overdesigning:

(a) all regions outside flexural plastic hinges(b) the non-ductile structural components or mechanisms of force transfer(c) all parts that are meant to remain elastic (including the deck and the foundation)

relative to the corresponding action effects, Ed, from the linear analysis for the seismic designsituation. Non-ductile structural components in the context of (b) include seismic links, fixedbearings, sockets and anchorages for cables and stays, etc.; for concrete, the non-ductilemechanism of force transfer is by shear. In bridges of limited ductile behaviour, the overdesignis limited to case (b) and to the foundation; it is normally accomplished by multiplying the seismicaction effects from the linear analysis by q and carrying out the ULS strength verifications withthe increased action effects. In bridges of ductile behaviour, by contrast, ‘capacity design’ isemployed to prevent the components listed above under (a) to (c) from becoming inelastic. Incapacity design, all potential plastic hinges are presumed to develop overstrength moments,goMRd. Then, simple analysis is employed to estimate the action effects that develop in theregions to be protected from inelasticity when this complete plastic mechanism forms.Normally, equilibrium suffices for this.

6.3.3 Detailing of plastic hinges for ductilityFlexural plastic hinges are detailed to provide the deformation and ductility capacity that is con-sistent with the demands placed on them by the design of the structure for the chosen q factor.

Clause 5.6.1(1) [2]

Clause 4.3.3.5.1(2)c

[1]

Clauses 2.3.4(1)–

2.3.4(3), 2.3.6.2(2),

5.3(1), 5.6.2(2)a,

5.6.3.6(1), 5.7.2(1),

5.8.2(3), 6.5.2(1),

6.5.2(2), 6.6.2.1(1),

6.6.3.1(3) [2]


123

Clauses 5.3(3) 5.3(4),

5.3(6) [2]

Clauses 5.3(1), 5.3(6),

G.1(1)–G.1(5) [2]

6.4. Capacity design of regions other than flexural plastic hinges inbridges of ductile behaviour

6.4.1 Overstrength moments of flexural plastic hingesIn general, capacity design effects are established separately for seismic action in each one of thelongitudinal and the transverse directions of the bridge, each time in the positive or negativesense. The design moment resistance, MRd, entering in the overstrength moment,Mo ¼ goMRd, refers to the end section of the plastic hinge, and is calculated assuming that theconcurrent values of axial force, NEd, and of the moment in the orthogonal direction of thepier column (as these are obtained from the linear analysis for the seismic design situation andfor the considered direction – longitudinal or transverse – and sense of the seismic action –positive or negative) are acting together with MRd. The overstrength factor, go, is meant totake into account the uncertainty in material strengths and the hardening of the sectionbetween yielding and ultimate strength. It is a Nationally Determined Parameter, with thefollowing recommended values:

g for steel members: go ¼ 1.25g for concrete members:

go ¼ 1.35 if hk ¼ NEd/(Ac fck) � 0.1 (D6.6a)

go ¼ 1.35[1þ 2(hk� 0.1)2] if hk ¼ NEd/(Ac fck) . 0.1 (D6.6b)

with NEd from the linear analysis for the seismic design situation and the considereddirection and sense of the seismic action.

6.4.2 Estimation of capacity design effects in the plastic mechanism from theoverstrength moments of the flexural plastic hinges

Normative Annex G of Part 2 of Eurocode 8 outlines a multistep procedure for the estimation ofthe capacity design effects assuming the complete plastic mechanism forms. First, the over-strength moments, Mo ¼ goMRd, of the flexural plastic hinges are established. Next is thecalculation at each plastic hinge of the difference between the overstrength moment and themoment, MG, produced at the end section of the plastic hinge by the non-seismic actions inthe seismic design situation:

DMh ¼ goMRd�MG (D6.7)

Bending moments enter this calculation with the signs they have in the seismic design situationfor the considered direction and sense of the seismic action. If, by contrast, goMRd and DMh

are always taken as positive, MG is also positive if it acts on the section in the same senseas goMRd does. The physical sense of the action of goMRd and DMh should be the same asthat of the rotation of the plastic hinge: for example, at the two ends of a pier column thatis fixed against rotation at both the top and bottom, they are such that the pier column is incounterflexure.

For the general case, Part 2 of Eurocode 8 requires the capacity design effects to be determinedseparately for each horizontal direction of the seismic action, and each sense of the inertial forcesin each direction (positive or negative). This is to take into account the effect on the momentresistance of the plastic hinge sections of (a) the magnitude of the axial force and (b) anyasymmetry in the shape or reinforcement of the section (a rarity in pier columns). The sense ofthe seismic action component materially affects the magnitude of the axial force if the pier hasmore than one column in the direction of this component (e.g. in twin-blade piers – seeSection 4.2.2.2 of this Guide – under the longitudinal seismic action).

The next step is the estimation of the change in action effects in the plastic mechanism, DAC,when the moments of flexural plastic hinges increase from MG to goMRd (i.e. by DMh). Often,this can be done on the basis of equilibrium alone. In the simple example of Figure 6.1 (a piercolumn in counterflexure with overstrength moments at its two ends), DAC amounts to anincrease from a linear moment diagram up the column to another linear diagram and fromone constant value of shear force to another. In the final step, the action effect increments,


124

DAC, are superimposed on the effects of the non-seismic actions in the seismic design situation,AG. The outcome of the superposition is the capacity design effects:

AC ¼ AGþ DAC � AGþ qAE (D6.8)

where AE denotes the generic effect of the design seismic action from the linear analysis.

Section 8.2.9 of this Guide exemplifies the application of the general procedure above to estimatethe capacity design effects in a bridge deck monolithically connected to the piers, for longitudinalseismic action.

Normally, simplifications of the general procedure are employed. Simplification is indeedencouraged by normative Annex G of Part 2 of Eurocode 8, as long as it respects equilibrium.A very common simplification entails using equilibrium to estimate the seismic shearaccompanying the increase in the base moment of pier column i from MG,i

b to goMRd,ib (by

DMh,ib ) and of its top moment from MG,i

t to goMRd,it (by DMh,i

t ) – where the index t is used forthe top, and b for the bottom. If the top is supporting the deck through a hinge, or if we aredealing with the transverse direction, then DMh,i

t ¼ 0. Capacity design gives for the seismicshear of pier column i

DVC;i ¼DMb

h;i þ DMth;i

Hi

� qVE;i ðD6:9Þ

which is superimposed on the shear due to gravity and other permanent actions in the seismicdesign situation, VG, to give the total capacity design shear in pier column i:

VC,i ¼ VGþ DVC,i (D6.10)

VE,i in Eq. (D6.9) is the shear of pier column i from the linear analysis under the design seismicaction alone.

Another simplification is to neglect the moments and shears in pier column i due to gravity andother permanent actions: MG ¼ 0, VG ¼ 0. With this – usually good – approximation, thecapacity design shears are the same, no matter the sense of the seismic action – and this distinc-tion between positive or negative is no longer necessary.

By far the most important simplification is to assume that the capacity design effects areproportional to the total shear force in all pier columns of the bridge, despite the departure

Clauses G.2(1),

G.2(2) [2]


Figure 6.1. Capacity design moments in a pier column forming plastic hinges at the top (index t) and

bottom (index b). Dashed line: moments ME from the analysis for the seismic design situation. Solid line:

capacity design moments, MC, with a cut-off near each end at the design moment resistance, MRd, as

controlled by the reinforcement of the end section and the axial load from the analysis for the seismic

design situation

Plastic hinge

Plastic hinge

Lh

MC

Mo,b = γ0MRd,b MRd,b ME,b

Mo,t = γ0MRd,tMRd,t

ME,t

Pier

Deck

125

Clauses 5.6.3.1(2),

5.6.3.2(1) [2]

Clauses 5.6.3.5.1(1),

5.6.3.5.2(1),

5.6.3.5.2(2) [2]

from linearity that marks the development of the plastic mechanism with the gradual formationof flexural plastic hinges. This assumption facilitates greatly the estimation of the capacity designeffects in the deck, in any seismic links and on foundations and abutments, etc., as

DAC ffiP

VC;iPVE;i

AE � qAE ðD6:11Þ

where the summations are over all pier columns, AE and VE.i denote the generic effect and theshear in pier column i, respectively, due to the design seismic action from the linear analysis,and DAC is the generic capacity design effect, to be superimposed on the generic effect of thenon-seismic actions in the seismic design situation, AG, according to Eq. (D6.8).

The above, as well as the overall framework for the capacity design of bridges, conforms with theposition of Eurocode 8 (both Parts 1 and 2) that capacity design takes place separately in thelongitudinal and transverse directions. In case, however, a single modal response spectrumanalysis is carried out for both simultaneous horizontal seismic action components and (themore rigorous and convenient) Eq. (D5.1) is employed to estimate the peak values of theseismic action effect, one cannot associate AE and VE,i in Eq. (D6.11) to a single component:they encompass the effects of both. In that case, the ratio SVC,i/SVE,i may be computed separ-ately from the shears in the two directions of the bridge; the minimum of the two ratios is usedthen in Eq. (D6.11) to amplify all the seismic action effects, AE. Indeed, it is the minimum of thetwo ratios that is associated with the earliest development of a full plastic mechanism in thebridge.

6.4.3 Capacity design moments outside the flexural plastic hinges of ductile piersAs depicted schematically in Figure 6.1, the vertical reinforcement of ductile pier columns staysthe same all along the nominal length, Lh, of the plastic hinge. In case the section of the piercolumn decreases from the pier base upwards, the moment resistance of the pier column,MRd, may also decrease a little as we go up within the nominal length of the plastic hinge (anadditional reduction may be due to a reduction in the axial force, NEd, from the pier baseupwards). Outside that length, Part 2 of Eurocode 8 allows the termination of some verticalbars according to the linear diagram of capacity design moments, MC, which connects the over-strength moments, goMRd,i

b and goMRd,it at the two end sections of the pier column. This diagram

provides a significant margin for any increase in the pier moments due to higher modes thatreflect the distributed mass of the piers, even when the analysis ignores this mass and suchmodes. Note, however, that, in practice, part of the vertical reinforcement is terminatedbetween the two ends of only fairly tall piers.

6.4.4 Capacity design shear in a joint between a ductile pier column and the deckor a foundation element next to a column plastic hinge

High shear stresses in the core of the joint between a pier column and its foundation element(spread footing or pile cap) accompany the transfer of the bending moment at the base of thecolumn to the foundation. High shear develops also in the joint region between the deck andthe top of the column, if the pier column is monolithically connected there to the deck. Thejoint may fail under these stresses, and its failure will be brittle. So, if the bridge is designedfor ductile behaviour, the shear stresses in such joints are determined via capacity design, forthe worst possible condition: when a plastic hinge has formed at the end section of the piercolumn and has developed its overstrength moment, Mo ¼ goMRd, established according toSection 6.4.1.

In the following, the pier column is indexed by ‘c’. The horizontal component monolithicallyconnected to it – whether a deck or a foundation element – is called, for simplicity, ‘beam’and indexed by ‘b’.

Let us denote by xz the vertical plane within which bending of the pier column takes place (seeFigure 6.2). A nominal shear stress is calculated as the average within the joint core; it has thesame value no matter whether it is parallel to the horizontal axis x and acts on a horizontalplane normal to the vertical axis z (tzx, denoted for simplicity vx) or is parallel to the verticalaxis z and acts on a vertical plane normal to the horizontal axis x (txz, denoted for simplicity


126

vz). vx is obtained by smearing a joint horizontal shear force Vjx over the horizontal ‘cross-sectional area’ of the joint, while vz results from smearing a vertical shear force Vjz over itsvertical ‘cross-sectional area’ bjzb, normal to the plane of bending of the column:

vx ¼Vjx

bjzcvz ¼

Vjz

bjzbvx ¼ vz ; vj ðD6:12Þ

If the pier column has a depth hc in the plane of bending and width bc orthogonal to it (for acircular column of diameter Dc, conventionally bc ¼ hc ¼ 0.9Dc), the effective width of thejoint at right angles to the plane of bending of the pier column is

bj ¼ bcþ 0.5hc bj � bw (D6.13)

where bw is the physically available width of the deck or the foundation element (the ‘beam’)parallel to bc. The ‘depth’ dimensions of the horizontal or vertical ‘cross-sectional area’ of thejoint in Eq. (D6.12) are the internal lever arms of the column and the ‘beam’ end sections, zcand zb, respectively (see Figure 6.2). For convenience, zc and zb may be taken as 90% ofcorresponding effective section depths:

zc � 0.9dc zb � 0.9db (D6.14)

The nominal shear stress vj of Eq. (D6.12) is normally determined from that of the shear forces,Vjx, Vjz, which is in the direction of the yielding component. Unlike buildings, where the yieldingcomponent is normally the beam(s), in bridges it is the column. Then, capacity design gives forthe design vertical shear of the joint, Vjz,

Vjz ¼ goFtd,c�Vb1C � goMRd,c/zc�Vb1C (D6.15)

where Ftd,c is the tension force resultant in the column section corresponding to the designflexural resistance MRd,c, go is the overstrength factor from Section 6.4.1 and Vb1C is the shearforce of the ‘beam’ adjacent to the tensile face of the column (taken as positive if acting in thesense shown in Figure 6.2). The design horizontal shear of the joint Vjx may then be calculatedfrom Eq. (D6.12) as Vjx ¼ Vjzzc/zb.

The value of Vb1C should correspond to the capacity design effects of the plastic hinge, accordingto the general procedure in Section 6.4.2. Its magnitude is small compared with the first term inEq. (D6.15), and is normally estimated on a case-by-case basis. For example:

1 In the longitudinal direction, at a monolithic connection of one out of nc columns of a pierto the deck: Vb1C � goMRd,c/Ld�Vb1G/nc (normally negative), where Ld is the averagespan length on either side of the pier and Vb1G is the total deck shear due to quasi-permanent loads at the face of the pier.


Figure 6.2. Forces on a joint between a pier column (‘c’) and the deck (‘b’) above a column plastic

hinge. (Turn the figure upside down for a joint with a foundation element.)

x

z

y

Plastic hinge

hc

Vb2c Vb1c

Vjz

Njz

Njx

γ0MRdc

γ0FtRc

zc

zb hb

127

Clauses 5.3(7),

5.3(8) [2]

2 At the connection of a spread footing with– a single-column pier in the transverse direction– any pier column in the longitudinal direction:Vb1C � 0, assuming that the weight of the uplifting part of the footing beyond the columncompensates for any soil pressures on its underside.

3 At the connection of a pile cap with– a single-column pier in the transverse direction– any pier column in the longitudinal direction:Vb1C � goMRd,c/zpc� 0.5NEd, with NEd denoting the axial force of the pier column in theseismic design situation and– zpc ¼ (Bp/3)np/(np� 1) if np is odd or– zpc ¼ (Bp/3)(npþ 1)/np if it is even,where Bp is the distance between the extreme piles of the group in the vertical plane ofbending of the column and np is the number of piles in each row of piles parallel to thatvertical plane.

4 In the transverse direction, at a monolithic connection of a multi-column pier to the deck:Vb1C � 0, as the most adverse joint is that of the column with the largest value of MRd,c

(i.e. of the outer one where the seismic action induces a compressive axial force in thepier); then, Vb1C arises from the width of the deck beyond the outer column, and is verysmall.

5 In the transverse direction, at the connection of a multi-column pier to a spread footing:the joint below every single column may have to be considered, from the one nearest theuplifting edge of the footing, where the value of MRd,c is the lowest (because the seismicaction induces a tensile axial force) but Vb1C � 0 with the same reasoning as in case 2above, to the column closest to the down-going edge, where MRd,c is the largest (owing tothe compressive seismic axial force) but Vb1C may be large; the values of Vb1C next to thatcolumn or any intermediate one may be determined from force equilibrium under the axialforces in the columns in the seismic design situation, NEd, and the soil pressures on theunderside of the footing when the overstrength moments, goMRd,c, develop at the base ofthe columns (the soil pressures taken with a linear distribution across the footing).

6 In the transverse direction, at the connection of a multi-column pier to a pile cap: the jointbelow every column should be considered, from the column nearest to the uplifting edge,where the tensile seismic axial force reduces the value of MRd,c but Vb1C is low or aboutzero, to the opposite one where the compressive seismic axial force increases MRd,c butVb1C may be large; the values of Vb1C next to every column may be estimated from forceequilibrium under the axial forces in the columns in the seismic design situation, NEd, andthe pile reactions when the overstrength moments, goMRd,c, develop at the base of thecolumns.

6.4.5 Capacity design effects in piers or abutments of ductile bridges supportingthe deck on sliding or elastomeric bearings

In a ductile bridge, where certain piers are rigidly connected to the deck (with monolithicconnection, fixed bearings or seismic links without slack or clearance), some other piers or theabutments may support the deck on sliding or elastomeric bearings. The shears and momentsin these piers or abutments are calculated assuming that the bearings develop the following‘capacities’:

g For sliding bearings: a horizontal force of gofRdf, where Rdf is the maximum designfriction force, equal to the maximum friction coefficient multiplied by the maximumvertical reaction in the seismic design situation and gof ¼ 1.30.

g For elastomeric bearings: a horizontal force equal to gof ¼ 1.30 multiplied by thehorizontal stiffness of the bearing, GbAb/tq, multiplied by the maximum bearingdeformation corresponding to the total design displacement of the deck, dEd, at thehorizontal level where it is seated on the bearing. The value of dEd is estimated fromEq. (D6.36), according to point 3 in Section 6.8.1.2.

Coefficient gof with a value of 1.30 is meant to account for potential hardening of the bearing dueto ageing, etc., from its installation (possibly as a replacement) until the seismic event.


128

6.5. Overview of detailing and design rules for bridges with ductileor limited ductile behaviour

As mentioned in Section 6.1, Table 6.1 lists the rules for the verification of strength and thecapacity design of the deck and the piers, as well as the rules for the detailing of plastic hingesfor ductility. In addition, the detailing rules per Eurocode 2, applying outside the plastic hingeregion, are given. Supplemented with a long list of footnotes, the table is almost a standalonedesign aid.

Most of the detailing rules for Eurocode 8 listed in Table 6.1 are prescriptive and – unlike theimportant concept of capacity design – are not given much attention in the text of the presentchapter.

6.6. Verification and detailing of joints between ductile piercolumns and the deck or a foundation element

6.6.1 Stress conditions in the jointThe joint core is considered to be in a triaxial stress state, with shear stresses tzx ¼ txz ; vj givenby Eq. (D6.12)–(D6.15) and normal stresses sx ; nx, sy ; ny, sz ; nz equal to

nx ¼Njx

bjhbny ¼

Njy

hbhcnz ¼

Njz

bjhcðD6:16Þ

Njx andNjy are normal forces in the plane of the horizontal element (deck or foundation element)into which the pier column frames; they are within and at right angles, respectively, to the planeof bending of the pier column (see Figure 6.2 for the convention of the axes). They derive fromany prestressing of the horizontal element that may be effective in the joint core (after losses),plus any other in-plane normal forces in the seismic design situation (the average of analysisresults at opposite vertical faces of the column). The vertical normal force at the centre of thejoint is

Njz ¼bc2bj

NGc ðD6:17Þ

where NGc is the axial force in the column due to the quasi-permanent actions in the seismicdesign situation; the factor 2 in the denominator reflects the reduction in the normal stressesdue to NGc from NGc/(hcbc) at the end section of the column to zero at the opposite face ofthe horizontal element. All stresses and forces in Eqs (D6.16) and (D6.17) are taken aspositive for compression.

6.6.2 Verification of the integrity of the jointThe major threat to the joint comes from crushing of its core by diagonal compression. Part 2 ofEurocode 8 considers that the diagonal strut running through the joint core within the verticalplane xz will fail in compression upon exceedance of the compressive strength of concrete, asthis is reduced due to tensile strains at right angles to the strut and may be enhanced by theconfining effect of any compressive stress ny normal to the xz plane and any closed reinforcementtransverse to this plane. In the classical variable strut inclination model of Eurocode 2 forconcrete members with shear reinforcement, diagonal compression failure should be checkedagainst a shear resistance – normalised to the cross-sectional dimensions of the web – of

VRd;max

bwz¼ 0:5� 0:6 1� fck½MPa�

250

� �fcd sin 2u ðD6:18Þ

where u is the strut inclination, and the brackets and the factor 0.6 are for the reduction of theuniaxial compressive strength of concrete due to tensile strains at right angles to the strut. Themaximum resistance is for u ¼ 458. So, Part 2 of Eurocode 8 adopts the following simple verifica-tion criterion against diagonal compression failure of the joint:

vj � 0:3 1� fck½MPa�250

� �ac fcd ðD6:19Þ

Clauses 5.6.3.5.2(3),

5.6.3.5.2(4) [2]

Clauses 5.6.3.5.3(2),

5.6.3.5.3(3) [2]


129

Clause 5.6.3.5.3(3) [2]

Clause 5.6.3.5.3(4) [2]

Clauses 5.6.3.5.3(1),

5.6.3.5.3(6) [2]

where vj is the average shear stress in the joint from Eqs (D6.12)–(D6.15), while

ac ¼ 1þ 2(njyþ ry fsd)/fcd � 1.5 (D6.20)

accounts – conservatively – for the increase in the compressive strength of the diagonal strutresulting from any confining pressure (ny, from Eq. (D6.16)) and/or reinforcement (ry) in thetransverse direction y; ry ¼ Asy/(hchb) is the reinforcement ratio of any closed stirrups at rightangles to the vertical plane xz of the joint, taken to work with a reduced stress,fsd ¼ 300 MPa, in order to limit the width of cracks parallel to the xz plane. The outcome ofEq. (D6.19) is on the safe side compared with the upper bound of joint shear strength demon-strated by the available cyclic tests on unconfined beam–column joints (see Fardis, 2009).

6.6.3 Dimensioning of the joint reinforcementFor the purposes of calculating the joint reinforcement, Part 2 of Eurocode 8 adopts for the jointcore the variable strut inclination model of Eurocode 2, which also underlies Eqs (D6.18) and(D6.19). It again uses the value u ¼ 458 for the strut inclination. This gives for the joint corethe following total amounts of vertical (z) and horizontal (x) reinforcement in the verticalplane of column bending:

Asx ¼ (Vjx�Njx)/fyd, Asz ¼ (Vjz�Njz)/fyd (D6.21a)

or, in terms of steel ratios, normalised to the same joint core areas as the corresponding forces,

rx ; Asx/(bjhb) ¼ (vj� nx)/fyd rz ; Asz/(bjhc) ¼ (vj� nz)/fyd (D6.21b)

6.6.4 Maximum and minimum reinforcement in the jointYielding of the joint reinforcement is a ductile failure mode. By contrast, diagonal concretecrushing, if vj exceeds the limit value of Eq. (D6.19), is brittle. If one places more reinforcementthan required by Eqs (D6.21), the shear stress, vj, that may develop in the joint core may increaseto the limit of Eq. (D6.19), inviting diagonal concrete crushing. For this reason, Part 2 ofEurocode 8 controls the amount of horizontal joint reinforcement, so that vj from Eq. (D6.19)is not reached, even when the beneficial effects of transverse compression and confinement arediscounted:

rx ; Asx=ðbjhbÞ � rmax ry ; Asy=ðhbhcÞ � rmax where rmax ¼ 0:3 1� fck½MPa�250

� �fcdfyd

ðD6:22Þ

The joint will first crack when the principal tensile stress, sI, under the system of normal stresses,sx ; nx, sy ; ny, sz ; nz from Eqs (D6.16) and shear stresses tzx ¼ txz ; vj from Eqs (D6.12)–(D6.15), exceeds the (design value of ) tensile strength, fctd. Setting sI equal to fctd gives thejoint shear stress at diagonal cracking of the joint according to Part 2 of Eurocode 8:

vj;cr ¼ fctd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ nx

fctd

� �1þ nz

fctd

� �s� 1:5fctd ðD6:23Þ

where fctd ¼ fctk0,05/gc ¼ 0.7fctm/gc. Theoretically, the joint may rely on the tensile strength ofconcrete if vj from Eqs (D6.12)–(D6.15) is less than the joint cracking stress, vj,cr. However, inULS verifications we normally do not rely on the tensile strength of concrete, placing insteadminimum reinforcement that can provide alone the tensile stresses released in case crackingoccurs for any reason. Part 2 of Eurocode 8 derives on the basis of Eqs (D6.21b) and (D6.23)– discounting the beneficial effects of any transverse compression – the minimum joint reinforce-ment ratio as

rx ; Asx/(bjhb) � rmin ry ; Asy/(hbhc) � rmin where rmin ¼ fctd/fyd (D6.24)

The minimum and maximum reinforcement ratios apply in both horizontal directions, even whenframing action with the column takes place only in one of them.


130

6.6.5 Detailing of the vertical reinforcement at jointsThe vertical bars of the pier column should be fully anchored into the horizontal element,extending over its full depth even when it is not necessary for their anchorage. In addition,excepting those not serving as extreme tension reinforcement in any of the two horizontaldirections of bending, they should have a standard 908 hook inwards or a head at the end (seeFigure 6.3(a)).

The vertical reinforcement of the joint should be in the form of stirrups clamping inwards thebars of the horizontal element at its outer (far) face. At least 50% of them should be fullyclosed stirrups, engaging both the top and the bottom bars of the horizontal element; the restmay be U bars, engaging the bars only at the outer (far) face, but not at the face where thepier column frames into (see Figure 6.3(a)).

To reduce the congestion of reinforcement in the central part of the joint in plan, the density ofvertical stirrups may be reduced there, but not below the minimum ratio of Eqs (D6.24). Thebalance, DAsz, may be placed outside the central area in plan but next to it. If framing actionof the column with the ‘beam’ takes place in the vertical plane xz, DAsz may be placed withinthe applicable joint width bj in the y direction and not beyond 0.5hb from each column facealong the x direction (Figure 6.3(b)). If there is framing action within the vertical plane yz aswell, this operation takes place separately in that direction, with the values of DAsz and bj nowapplicable (Figure 6.3(c)). Vertical stirrup legs in the overlapping areas count for both directionsof framing action.

Clauses 5.6.3.5.4(1),

5.6.3.5.4(2),

5.6.3.5.4(4)–

5.6.3.5.4(6) [2]


Figure 6.3. Alternative arrangement of joint reinforcement: (a) vertical section within the plane xz;

(b) joint plan view, if plastic hinges form only in the x direction; (c) joint plan view if plastic hinges form in

the x and y directions

Beam-columninterface

Stirrups in common areascount in both directions

y

x

Aszb

Areas forplacing Aszb

∆Asx

hb

bj

hb/2 hb/2

hb/2 hb/2

hb/2

hb/2

hc

(a)

(c)(b)

#50%

131

Clauses 5.6.3.5.4(1),

5.6.3.5.4(3),

5.6.3.5.4(5),

5.6.3.5.4(7) [2]

Clause 4.2.4.4(2) [2]

Annex E [2]

6.6.6 Detailing of joint horizontal reinforcementUp to 50% of the top and bottom bars in the horizontal element may count towards the requiredhorizontal joint reinforcement area, Asx, provided they are continuous through the joint or fullyanchored beyond it. The rest of Asx should consist of stirrups or hoops (preferably of the sameshape and diameter as those in the column plastic hinge area) enclosing the column verticalreinforcement as well as the ends of ‘beam’ horizontal bars anchored into the joint.

To reduce the congestion of reinforcement in the central part of the joint in elevation, the totalarea of horizontal bars may be reduced there by DAsx � DAsz, but not below the minimum ratioof Eqs (D6.24). The balance, DAsx, should then be placed at the top and bottom faces of the hori-zontal element over and beyond its reinforcement for the verification in flexure under capacitydesign effects. The additional bars should be placed within the joint width bj and be adequatelyanchored in order to be fully effective at a distance hb from the column face (Figure 6.3(a)).

6.7. Verifications in the context of design for ductility based onnonlinear analysis

6.7.1 Format of verificationsIf the analysis for the seismic design situation is nonlinear, regions of the bridge intended to haveelastic behaviour (notably the deck, parts of the piers outside the flexural plastic hinges, shearkeys and other critical connections, etc.) are verified to remain in the elastic range. In principle,such verification may take place in terms of either forces or deformations. It is consistent with theverifications for linear analysis, if it is in terms of forces. Then, flexure with or without axial loadmay be verified as for linear analysis, with material partial factors applied on the resistance side.

Components intended to stay elastic (including the foundation and seismic links) and brittlemodes of force transfer and behaviour in ductile components (notably shear in piers rigidlyconnected to the deck), are also verified in terms of forces. Their design resistances arecomputed as for linear analysis; that is, with partial factors and, in addition, divided by theNationally Determined Parameter safety factor, gBd1, which divides the shear resistance inbridges with limited ductile behaviour and has a recommended value gBd1 ¼ 1.25 (seeTable 6.1). Alternatively, however, the value gBd ¼ 1.0 may be applied if capacity designeffects are used, computed with the minimum value of the overstrength factor recommended inPart 2 of Eurocode 8 (go ¼ 1.35 or 1.25 for concrete or steel piers, respectively, see Section 6.4.1).

Flexural plastic hinges in the piers are verified in terms of deformations, namely plastic hingerotations, upl. As pointed out in informative Annex E of Part 2 of Eurocode 8, the plastic partof a hinge rotation at a pier end may be taken as equal to the plastic part of the chordrotation at that end. Similar to all verifications for nonlinear analysis (including those of thetwo paragraphs above), the demand is the value from the nonlinear analysis for the seismicdesign situation (in this case the value of the plastic hinge rotation computed, uplE ). Thequestion is, then, how much is the corresponding design capacity, uplRd?

6.7.2 Plastic hinge rotation capacityPart 2 of Eurocode 8 is not very specific regarding uplRd. As a design value, uplRd should be obtainedby dividing the expected ultimate value, uplum, by a safety factor, gR,pl, that reflects the naturalvariability of materials and components, model uncertainty and/or experimental scatter. Part2 of Eurocode 8 mentions as possible sources for uplum relevant test results or calculation fromultimate curvatures. In a non-binding note it mentions also informative Annex E as a sourceof information for uplum and gR,pl.

In the absence of specific justification based on actual data, informative Annex E is content withthe value gR,pl ¼ 1.40. However, gR,pl is meant to also reflect model uncertainty; so its valueshould depend on the model used for the calculation of uplum.

The model presented in informative Annex E of Part 2 of Eurocode 8 is the classic point hingemodel, based on curvatures, f, and on the plastic hinge length, Lpl:

uu ¼ fy

Ls

3þ fu � fy

� �Lpl 1� Lpl

2Ls

� �ðD6:25Þ


132

where uu is the ultimate chord rotation and Ls the shear span (moment-to-shear ratio) at the endof the member. The first term on the right-hand-side of Eq. (D6.25) is the elastic part of uu; thesecond is the plastic part, uplu .

Section analysis is the basis for the calculation of the yield curvature, fy, and of the ultimatecurvature, fu. Both should be determined for the value of the section axial load from thenonlinear analysis under the seismic design situation. Informative Annex E of Part 2 ofEurocode 8 refers to the construction of a full moment–curvature curve for this value of theaxial load until the value of fu is reached and to an elastic–perfectly plastic approximation ofthis curve that maintains the same plastic deformation energy (i.e. the same area under thecurve) from first yielding of the reinforcement up to fu (see Figure 5.13(a)). The curvature at thecorner of this elastic–perfectly plastic curve may be taken as fy. The value of fu is defined onthe basis of the failure criteria of the section, and is independent of the construction of the fullmoment–curvature curve (in fact, the value of fu defines a priori the end point of the curve),while the values of uu and uplu are not very sensitive to the exact value of fy. Therefore, there islittle point in constructing the full moment–curvature curve – a sensitive and often onerous under-taking – just for the sake of the value of fy. This is underlined further by Part 2 of Eurocode 8,which itself suggests (in informative Annex C) the values for fy given by Eqs (D5.57) and (5.58)in Section 5.8.2. The same expressions may well be used for the purposes of Eq. (D6.25).

Informative Annex E of Part 2 of Eurocode 8 proposes using in the section analysis for thecalculation of fu an elastic–linearly strain-hardening stress–strain (s–1) law for the reinforcingsteel: a mean yield stress higher by 15% than fyk; a mean tensile strength, ftm, higher by 20%than the characteristic value, ftk; and the strain at ftm equal to the characteristic strain value atmaximum stress, 1uk. If the values of ftk and 1uk of the specific reinforcing steel used areunknown, they may be taken as equal to the minimum values specified in Annex C ofEurocode 2 for the steel class used.

As far as the s–1 law of concrete is concerned, informative Annex E of Part 2 of Eurocode 8proposes the one in Mander et al. (1988), with an ultimate strain of 0.35% for unconfinedconcrete. The ultimate strength of confined concrete in Mander et al. (1988) is adopted, with avalue according to Elwi and Murray (1979) of

f �c ¼ fcð1þ KÞ ðD6:26Þ

where

K ¼ 2:254

ffiffiffiffiffiffiffiffiffiffiffiffi1þ 7:94

p

fc

r� 1

� �� 2p

fcðD6:27Þ

and is taken to occur at a strain of (Richart et al., 1928)

1�co � 1co(1þ 5K ) (D6.28)

where 1co ¼ 0.002 is the strain at the ultimate strength of unconfined concrete, fc. The confiningpressure in Eq. (D6.27) is

p ¼ 0:5arw fyw ðD6:29Þ

where a is the confinement effectiveness factor (Mander et al., 1988):

g for a spiral with pitch s and diameter Dc,

a ¼ 1� s

2Dc

ðD6:30aÞ

g for circular hoops with diameter Dc and spacing s,

a ¼ 1� s

2Dc

� �2

ðD6:30bÞ


133

g for vertical bars laterally restrained at distances bi (i indexes the bars) along a rectangularperimeter tie with sides bcx, bcy and spacing s,

a ¼ 1� s

2bcx

� �1� s

2bcy

� �1�

Pb2i =6

bcxbcy

!ðD6:30cÞ

fyw is the yield stress and rw is the volumetric ratio of confining reinforcement with respect to theconfined core to the centreline of the confining hoop, spiral or perimeter tie:

g for a spiral or circular hoops of cross-sectional area Asp, pitch s and diameter Dc,

rw ¼ 4Asp

sDc

ðD6:31aÞ

g for rectangular ties with total cross-sectional areas Aswx, Aswy at right angles to sides bcx,bcy of the concrete core, respectively, and spacing s,

rw ¼ 2

ffiffiffiffiffiffiffiffiffiffiAsx

sbcx

Asy

sbcy

sðD6:31bÞ

Informative Annex E of Part 2 of Eurocode 8 adopts the ultimate strain of confined concrete inPaulay and Priestley (1992):

1�cu ¼ 0:004þ 2:81su;wp

f �cðD6:32Þ

with p from Eqs (D6.29)–(D6.31), f �c from Eqs (D6.26) and (D6.27) and 1su,w the strain at thetensile strength of the confining reinforcement.

For values of fy and fu established as above, informative Annex E of Part 2 of Eurocode 8proposes the following empirical expression for Lpl, in terms of the shear span, Ls, thecharacteristic yield stress of the vertical bars in MPa, fyk (not the mean, fym ¼ 1.15fyk, used inthe calculation of fy and fu) and their diameter, dbL:

Lpl ¼ 0.1Lsþ 0.015fykdbL (D6.33)

If Ls/d , 3 (where d is the effective depth of the pier section), informative Annex E of Part 2 ofEurocode 8 multiplies the second term of the right-hand-side of Eq. (D6.25) by

p(Ls/3d). The

outcome is the final expected value of uplu , which is further divided by gR,pl ¼ 1.40, to yield adesign plastic rotation capacity, upld .

Figure 6.4 compares the predictions of the above procedure (but without dividing uplu bygR,pl ¼ 1.40) with the experimental values of the ultimate total chord rotation, uu, in a largenumber of cyclic tests to flexural failure. The comparison is shown separately for (a) compact rec-tangular sections (constituting the vast majority of the experimental data), (b) long rectangular(wall-like) sections, (c) hollow rectangular or other flanged sections and (d) circular piers. Withthe exception of the very low (safe side) predictions for circular piers, the average agreement isgood (overall median experimental-to-predicted ratio of 1.045, including the circular piers),but the scatter of test results about the predictions is high (the overall coefficient of variationof the experimental-to-predicted ratio is 66.5%).

Figures 6.5(a) to 6.5(c) compare the same data of Figures 6.4(a) to 6.4(c) with the predictions ofthe expressions given in Part 3 of Eurocode 8 (CEN, 2005c) and Biskinis and Fardis (2010) for(a) beam/column rectangular sections, (b) rectangular walls sections and (c) non-rectangularwalls or other flanged sections. Figure 6.5(d) compares the same data as those in Figure 6.4(d)with the predictions of the expressions proposed in Biskinis and Fardis (2012) specificallyfor circular piers. There are small differences between the expressions used in each ofFigures 6.5(a) to 6.5(c); by contrast, the difference between the expression underlyingFigure 6.5(d) and those used for Figures 6.5(a) to 6.5(c) is fundamental. Owing to these


134

differences, the agreement is more balanced and good for circular piers as well. The overallmedian of the experimental-to-predicted ratio is 1.00. The scatter, although significant, ismuch less than in Figures 6.4(a) to 6.4(d) (the overall coefficient of variation of the experimen-tal-to-predicted ratio is 36.5%).

6.8. Overlap and clearance lengths at movable joints6.8.1 Minimum overlap length6.8.1.1 GeneralA deck supported on a pier or abutment via a horizontally movable device (slider, elastomericbearing or special isolation device) should be prevented from dropping off by means of aminimum horizontal overlapping of the underdeck and the top of the supporting abutment orpier in any direction along which relative displacement between the two is physically possible.The same applies at a movement joint separating the deck between adjacent piers or a pierand an abutment, where one part of the span is vertically supported on the other (normallythe shorter on the longer of the two).

6.8.1.2 Minimum overlap length at an abutmentAccording to Part 2 of Eurocode 8, the minimum overlapping length of the end of the deck(Figure 6.6) seated on an abutment is

min lov � lmþ degþ des (D6.34)

Clauses 2.3.6.2(1),

2.3.6.2(3), 6.6.4(1),

6.6.4(2) [2]

Clauses 6.6.4(1)–

6.6.4(3), 2.3.6.3(2),

3.3(6) [2]


Figure 6.4. Experimental cyclic ultimate chord rotation versus predictions of approach in informative

Annex E of Part 2 of Eurocode 8: (a) compact rectangular sections, (b) rectangular walls; (c) hollow or

flanged rectangular members; (d) circular piers

0

5

10

15

20

25

0

1

2

3

4

5

Median:θu,exp = 1.35θu,pred




0

2

4

6

8

0 2 4 6 8 0 2 4 6 8 10 12 14 16θu,pred: %

(c)

θu,pred: %

(d)

0 5 10 15 20 25 0 1 2 3 4 5θu,pred: %

(a)

θu,pred: %

(b)

θ u,e

xp: %

θ u,e

xp: %

θ u,e

xp: %

θ u,e

xp: %

0

2

4

6

8

10

12

14

16

135

where:

1 lm is the maximum of 0.4 m and the size of the bearing needed to support the verticalreaction.

2 deg ¼ 2dg min 1;Leff

Lg

� �(D6.35)

is the (static) relative displacement of the abutment and the (other support) of the deckdue to the spatially varying seismic ground displacement. dg in Eq. (D6.35) is the designdisplacement of the ground from Eq. (D3.11) in Section 3.1.2.7 of this Guide. Leff is the


Figure 6.5. Experimental cyclic ultimate chord rotation versus predictions of expressions in Biskinis and

Fardis (2012) for circular piers or in informative Annex A of Part 3 of Eurocode 8-Part 3 or Biskinis and

Fardis (2010) for other section types: (a) compact rectangular sections; (b) rectangular walls; (c) hollow or

flanged rectangular members; (d) circular piers

0

2.5

5

7.5

10

12.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0

1

2

3

4

5

6

7

0

2

4

6

8

10

12

14

16

0 2.5 5 7.5 10 12.5 0 1 2 3 4θu,pred: %

(a)

θu,pred: %

(b)

0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 16θu,pred: %

(c)

θu,pred: %

(d)

θ u,e

xp: %

θ u,e

xp: %

θ u,e

xp: %

θ u,e

xp: %

Median: θu,exp = 1.04θu,pred

Median: θu,exp = θu,pred



Figure 6.6. Minimum overlap length for seating of the deck on an abutment

lov

136

distance from the joint to the nearest point where the deck is considered not to movehorizontally with respect to the abutment in question: if a continuous deck is connectedwith fixity (including via seismic links or shock transmission units without force limitingfunction) to more than one supports along its length, Leff is the distance to the centre ofthis group of supports (including the special case of just one fixed point all along thislength of deck); if it is supported everywhere on horizontally flexible bearings, Leff ismeasured to the centre of this group of bearings. Lg is the horizontal distance beyondwhich the ground motion may be considered as fully uncorrelated (see Table 3.4 in thisGuide). If the bridge is less than 5 km from a known seismically active fault that canproduce an earthquake of magnitude M of at least 6.5, and unless a site-specificseismological investigation is available, deg is taken as double that from Eq. (D6.35).

3 des is equal to the slack of any seismic links through which the deck may be connected topiers or to an abutment, plus the total design displacement of the deck at the horizontallevel where it is seated on the abutment, due to the deformation of the structure in theseismic design situation dEd:

dEd ¼ dEþ dGþ c2dT (D6.36)

where dE is the design seismic displacement from Eq. (D5.48) in Section 5.9 (including anyeffects of a torsional response about a vertical axis according to Section 5.7 of this Guide);dG is the long-term displacement due to the permanent actions (including, for concretedecks, prestressing after losses, shrinkage and creep); dT is the displacement due to thedesign thermal actions; and c2 ¼ 0.5 is the combination factor of thermal actions in theseismic design situation in Table A2.1, A2.2 or A2.3 of EN 1990:2002.The value of dEd at the horizontal level where the deck is seated on the abutment (i.e. atthe underdeck) is equal to the displacement at the level of the deck centroidal axis plus therotation of the deck end section multiplied by its centroidal distance from the underdeck.For dE and dT this displacement and rotation should be taken with signs such that theunderdeck pulls away from the abutment (there is no choice for dG, which should becalculated on the basis of the long-term value of any concrete creep, shrinkage andprestressing). The design thermal actions considered for dT are the combination of (CEN,2003a):(a) The maximum contraction due to the difference, DTN,con, of the minimum uniform

bridge temperature component, Te,min, from the initial temperature, T0, at the time thedeck is seated on the support. If T0 is not predictable, it is taken as the mean daytimetemperature during the foreseen construction period. Te,min is derived from thecharacteristic value of the minimum annual shade air temperature, Tmin, at the siteaccording to the National Annex to EN 1991-1-5:2003 or as recommended by EN 1991-1-5:2003 itself in Figure 6.1 of EN 1991-1-5:2003. Tmin in turn is obtained from thenational isotherm maps at sea level in the National Annex to EN 1991-1-5:2003 andadjusted for altitude and local conditions (e.g. frost pockets), for instance asrecommended in Annex A of EN 1991-1-5:2003.

(b) The vertical temperature difference component, DTM,heat, when the top surface of thedeck is hotter than the bottom one. Two alternative options are given in clauses 6.1.4.1and 6.1.4.2 of EN 1991-1-5:2003 for its National Annex to choose one for thedetermination of DTM,heat. Tables or figures with recommended values are also giventhere.

The effects of the uniform and the temperature difference components are superimposedlinearly, but not in full. In symbolic terms, the combination recommended in EN 1991-1-5:2003 is the most adverse of

DTM,heat ‘þ’ 0.35DTN,con, or 0.75DTM,heat ‘þ’ DTN,con (D6.37)

Albeit indirectly, Part 2 of Eurocode 8 makes reference for the above to the longitudinal directionof the bridge, which is indeed the most critical for unseating. Regarding the transverse direction,on the supply side the underdeck is usually quite wide; as far as the demand is concerned (i.e. theright-hand-side of Eq. (D6.36)), dG and dT are normally zero, while the value of dE at the level ofthe underdeck does not include a contribution from the rotation of the end section of the deck.With these qualifications, overlapping in the transverse direction may need to be checked only


137

Clause 6.6.4(4) [2]

under the narrow bottom flange of a prefabricated girder. In such a case, the torsional seismicresponse about a vertical axis may have a non-negligible effect on the transverse displacementof the end of the outer girder.

6.8.1.3 Overlap length at piers or at intermediate movement joints in the deckImplicit in Section 6.8.1.2 is the assumption that the top of the abutment does not move. Thisclaim cannot be made for the top of an intermediate pier that supports the end of a deckgirder on bearings. If dEp denotes the design seismic displacement given by Eq. (D5.48) for thetop of the pier, Part 2 of Eurocode 8 makes the conservative assumption that dEp takes placeat the same time as the design seismic displacement of the end of the girder, dE, and out ofphase; so, it finds the minimum required overlapping of the end of the girder and the pier topby adding dEp to the right-hand side of Eq. (D6.34). This applies, of course, both in the longitudi-nal and in the transverse directions.

At an intermediate movement joint in the deck within the span between adjacent piers or betweena pier and an abutment, the upper (supported) end of the deck on one side of the joint shouldoverlap the lower and supporting one by the square root of the sum of the squares of thevalues obtained from Eq. (D6.34) for each one of these two deck ends. Note that such jointsare not very common in Europe. Part 2 of Eurocode 8 requires connection between the twoparts of the deck across the joint through seismic links (see Section 6.9).

6.8.1.4 Overlap lengths in skewed or curved bridgesParticular caution is needed if the axis of the deck is not at right angles to the line of support atthe abutments (skewed bridges), or if the two abutments are not parallel but at an angle to eachother.

If the bridge has a skew angle w . 0 (see Figure 5.12), the overlap length should be checked andprovided in the direction where the seat is narrowest; that is, at right angles to the edge of the deckrather than in the longitudinal direction of the bridge. To this end, there are a few differences inthe way Eq. (D6.34) is elaborated:

1 lm � 0.4 m still, but if there are bearings that are rectangular in plan, with sides bL and bTin the longitudinal and the transverse directions, respectively, then alsolm � bL cos wþ bT sin w.

2 If the deck is connected to the abutment or the pier via seismic links having slack sL andsT in the longitudinal and the transverse directions, respectively, the contribution of theslack to des is sL cos wþ sT sin w.

3 If the displacements of the end of the deck due to permanent and quasi-permanent actionsare dG,L and dG,T in the longitudinal and the transverse directions, respectively, and thecorresponding values due to thermal actions are dT,L and dT,T (dG,T � 0, dT,T � 0), thesedisplacements enter into dEd in Eq. (D6.36) as dG ¼ dG,L cos wþ dG,T sin w � dG,L cos w,and dT ¼ dT,L cos wþ dT,T sin w � dT,L cos w.

4 The design seismic displacement dE in Eq. (D6.36) is the peak value from Eq. (D5.48)along a local axis at right angles to the edge of the deck; if no such local axis has beenintroduced, the design seismic displacements dE,L and dE,T in the longitudinal and thetransverse directions, respectively, are projected onto the normal to the edge of the deck asdE,L cos w and dE,T sin w, and combined to a single value through Eqs (D5.1) or (5.2).

Recall from Section 5.7.1 of this Guide the effects of twisting in skew bridges and the specialprovisions in Part 2 of Eurocode 8 for their accidental eccentricities. If B and L are the widthand the length parallel to the sides in plan (see Figure 5.12), a twisting angle u produces at theacute angle a displacement of u(B cos2 wþ L sin w)/2 normal to the edge of the deck. Moreimportant, as pointed out in Section 5.7.1, if sin w . B/L, any twisting of the deck willincrease the joint width all along the width B of the support. To prevent long and narrowdecks with skew sin w . B/L from dropping by uncontrolled twisting, their seating at theabutments may have to be much wider than what the calculations suggest.

If the axis of the bridge is at right angles to the abutments but is curved and the two abutmentsare at an angle w to each other, the deck is again susceptible to dropping due to uncontrolled


138

twisting: if the projection of the outer corner of the deck (on the convex side) onto the line ofsupport at the other abutment falls outside the width B of the support (i.e. if the radius ofcurvature at the axis is R. 0.5B(1þ cos w)/(1� cos w), twisting of the deck will increase thejoint width all along B. Regarding the calculation of the overlap length with the help ofEq. (D6.34), one should note that the abutments in the global longitudinal and transversedirections are at an angle w/2 to the deck axis and the edge of the deck, respectively. So, itwould be convenient to apply Eqs (D6.34) and (D6.36) with displacement results from theanalysis in a local system of horizontal axes parallel and at right angles to the edge of the deck.

6.8.2 Clearances between the deck and critical or non-critical components6.8.2.1 Clearance with critical componentsCritical or major structural components should be protected from damage due to hard contact orimpact with the deck, by providing a clearance with it that can accommodate the total designvalue of their relative displacement in the seismic design situation. The designer chooses whichcomponents are sufficiently critical or major to be fully protected from any damage throughthis clearance. Apart from that, this clearance is also a means to avoid restraining the deck inthe seismic design situation in a way that introduces uncertainties in the response or compli-cations of the analysis or design. For example, the clearance may be provided longitudinallybetween the end of the deck and an abutment backwall that is not sacrificial and not meant tobe engaged during the seismic response of the deck. It may also be provided transverselybetween the side of the deck and a shear key that is meant to act as a second line of defenceagainst unseating in a seismic event beyond the design seismic action that may exhaust theavailable overlap length or the displacement capacity of the bearings. The clearance does notapply between the deck and a seismic link (e.g. a shear key) that is meant to be activated inthe seismic design situation. Any slack to be provided there is a design parameter.

The minimum value of the clearance is given by Eq. (D6.36), discounting the effects of the (static)relative displacement of the component to be protected and the deck due to the spatially varyingseismic ground displacement (deg in Eqs (D6.34) and (D6.35)); dE, dG, c2 and dT are defined as inSection 6.8.1.2, but only c2 has the same value, c2 ¼ 0.5, because the way Eq. (D6.36) is appliedin Section 6.8.1.2 does not serve the purposes of the present section. In Section 6.8.1.2, alldisplacement components refer to the horizontal level of the bearing; here, they are examinedat the two extreme horizontal levels where the deck and the component to be protectedoverlap in the vertical direction – the lowest and the highest (and computed as the deckdisplacement at its centroidal axis plus the rotation of the deck end section multiplied by thevertical distance of the centroid from the horizontal level considered). Second, apart from dG,which has a single sign and sense, the sense (sign) of all displacement components in Section6.8.1.2 is taken such that the horizontal gap between the end of the deck and the supportingcomponent increases. Here, by contrast, the sense is the one that decreases the gap. Mostnotable in this respect is the difference in the design thermal actions considered for dT. Theyare the combination of:

(a) The maximum extension due to the difference, DTN,ext, of the maximum uniform bridgetemperature component, Te,max, from the initial temperature, T0, at the time the deck iserected, with Te,max derived from the characteristic value of the maximum annual shade airtemperature, Tmax, at the site according to the National Annex to EN 1991-1-5:2003 or asrecommended by EN 1991-1-5:2003 itself in Figure 6.1. Tmax is, in turn, read from thenational isotherm maps at sea level in the National Annex to EN 1991-1-5:2003, adjustedfor altitude and local conditions (e.g. as recommended in Annex A of EN 1991-1-5:2003).

(b) The vertical temperature difference component across the deck. For the displacement atthe highest horizontal level where the deck and the component to be protected overlapvertically, this component is DTM,heat, when the top surface of the deck is hotter than thebottom one; for the displacement at the lowest horizontal level of vertical overlapping, it isDTM,cool when the deck top is cooler than its bottom. Tools or guidance for thedetermination of DTM,cool and DTM,heat are given in clauses 6.1.4.1 and 6.1.4.2 of EN 1991-1-5:2003 and/or its National Annex.

The method recommended in EN 1991-1-5:2003 for the combination of the effects of the uniformand the temperature difference components is similar to Eq. (D6.37); again in symbolic terms,

Clauses 2.3.6.3(1)–

2.3.6.3(4) [2]


139

Clause 2.3.6.3(5) [2]

Clauses 6.6.1(2),

6.6.3.1(1), 6.6.3.1(2),

6.6.3.1(4), 6.6.3.1(5)

[2]

the most adverse of

DTM,heat ‘þ’ 0.35DTN,ext or 0.75DTM,heat ‘þ’ DTN,ext (D6.38a)

DTM,cool ‘þ’ 0.35DTN,ext or 0.75DTM,cool ‘þ’ DTN,ext (D6.38b)

If the component to be protected moves owing to the seismic action, its design seismic displace-ment relative to the deck, dE, may be taken as the square root of the sum of squares of the valuesof the design seismic displacement calculated for each one of them at the horizontal level ofinterest via Eq. (D5.48) in Section 5.9 (including any effects of torsional seismic responseabout a vertical axis according to Section 5.7 of this Guide).

Needless to say, the clearance should be calculated and provided in the horizontal directionwhere the deck and the component to be protected may come the closest.

6.8.2.2 Clearance with non-critical componentsRepairable damage of non-critical components, such as replaceable roadway movement joints,sacrificial abutment backwalls, etc., due to the design seismic action is acceptable. For suchcases, Part 2 of Eurocode 8 reduces the required clearance between the deck and the non-critical component to a value that may be exhausted under a more frequent combination ofthe seismic and the thermal actions (to be specified in the National Annex to Part 2 ofEurocode 8). Its recommendation is for a clearance that will accommodate without damagejust 40% of the design seismic displacement, dE, plus all other displacement components inEq. (D6.36) in full; that is

dEd,oc ¼ 0.4dEþ dGþ c2dT (D6.39)

where the suffix oc denotes ‘occasional’. Note that this level of protection is consistent with theseismic action and performance requirements recommended for damage limitation in buildingsaccording to Part 1 of Eurocode 8 (CEN, 2004a).

6.9. Seismic links6.9.1 Definition and roles of seismic linksPart 2 of Eurocode 8 uses the general term ‘seismic link’ for any special-purpose componentdevised to transmit horizontal seismic forces from one part of the structure to another –usually from the deck to the piers or the abutments or across an intermediate movement jointin a deck span – without partaking in the vertical transfer of gravity loads. This differentiatesit from fixed or elastomeric bearings, whose prime role is to transfer gravity loads from thedeck to the substructure, while – sometimes – doubling as means to transfer horizontal seismicforces as well. Another feature of seismic links is that, although they are activated by thehorizontal seismic displacements of the bridge, they generally allow the non-seismic displace-ments to develop almost unrestrained.

Shear keys are usually employed as seismic links in the transverse direction, transferring horizon-tal forces across an intermediate separation joint in a deck span or from the deck to an abutmentor – less often – a pier, through compressive contact stress. Shear keys commonly have the formof a short concrete corbel (sometimes prestressed), and are dimensioned and detailed as such.Steel brackets are also used, if a proper connection detail is devised. Normally, non-seismicdisplacements are zero in the transverse direction, and there is no need for a clearancebetween the shear key and the part of the bridge from where it receives the horizontal force. Avertical elastomeric bearing is often placed, though, between them, to avoid shock effects andlocal damage upon hard contact, and to allow unrestrained relative rotation across the interface.

In the longitudinal direction, seismic links are normally steel linkage bars, bolts or cables,connecting by tension the two parts of the deck across an intermediate movement joint in thespan, or the end of the deck to an abutment or pier, or the ends of two girders simply supportedover the same pier, etc. They normally have slack to allow unrestrained non-seismic displacements;they are mobilised once the seismic displacements exhaust the slack. If the link is a rigid steel rod,the slack may be provided between a steel plate at the end of the bar and the surface of the deckcomponent or abutment; a rubber ring or another device is provided along that gap for shock


140

absorption. Links connecting the parts of the deck across an intermediate movement joint in thespan serve only as second line of defence against unseating; they are not meant to be activatedin the seismic design situation. So, their slack should accommodate the relative displacement ofthe two ends according to the following extension of Eq. (D6.38):

DdEd;12 ¼ffiffiffiffiffiffiffiffiffiffiffiffid2E;1 þ d2

E;2

qþ dG;1 þ dG;2 þ c2 dT;1 þ dT;2

� � ðD6:40Þ

where the indices 1 and 2 refer to the ends of the two parts of the deck connected by the link. Thedisplacements due to the thermal action, dT,1 and dT,2, are calculated for a uniform temperaturedifference component causing contraction of the two linked parts of the deck and whatevertemperature difference component increases their distance at the horizontal level bridged bythe link. In symbolic terms, the combination of their effects is the most adverse of

DTM,heat ‘þ’ 0.35DTN,con or 0.75DTM,heat ‘þ’ DTN,con (D6.41a)

DTM,cool ‘þ’ 0.35DTN,con or 0.75DTM,cool ‘þ’ DTN,con (D6.41b)

As pointed out in Sections 4.5.4 and 5.5.1.1 of this Guide, activation of the link only after the gapbetween the deck and the shear key closes or the slack of a tension link is exhausted is a nonlinearfeature of the behaviour, and should be taken into account in the modelling. As a minimum in thecontext of linear analysis, Part 2 of Eurocode 8 requires the use of a linear spring with a stiffnessequal to the secant stiffness at yielding of the link; that is, equal to the yield force of the linkdivided by the clearance or slack plus the elastic deformation of the link until it yields. It wasalso pointed out that this is not very convenient for the analysis, because the link has not beendimensioned yet at this stage of the design, and its yield force is not known. Of course, linksthat are not meant to be activated in the seismic design situation do not need to be included inthe model.

6.9.2 Dimensioning of seismic linksAs mentioned out in Section 6.3.2 (in point (b)), a seismic link should be dimensioned forcapacity design effects. In bridges of limited ductile behaviour, these may be taken as theseismic action effects in the link from the linear analysis multiplied by q. In bridges of ductile be-haviour, the capacity design effects should be determined through Eq. (D6.8) and the associatedprocedure highlighted in Section 6.4.2; the simplification of Eq. (D6.11) may be employed to thisend.

As an exception to the general rule above, in seismic links connecting in tension the two parts ofthe deck across an intermediate movement joint in the span, or the end of the deck to anabutment or pier, or the ends of two girders simply supported over the same pier, etc., thelinkage element may be dimensioned for (the results of ) a longitudinal force equal to1.5agSMd, where ag is the design ground acceleration on type A ground, S is the soil factorand Md is the mass of the (part of the) deck linked to the pier or abutment, or the least of themasses of the two parts of the deck on either side of the intermediate separation joint. Alterna-tively, the longitudinal force for the design may be obtained from a more accurate analysis,taking into account the dynamic interaction of the parts of the deck connected by the link.

Once the capacity design internal forces are established, the seismic link is dimensioned at theULS according to the relevant material Eurocode: a concrete shear key as a corbel accordingto EN1992-1-1:2004, a steel bracket or a linkage rod, bolt or cable according to the relevantpart of Eurocode 3. The design shear resistance of concrete shear keys is divided by a NationallyDetermined Parameter reduction factor gBd. Part 2 of Eurocode 8 recommends a value ofgBd ¼ 1.25 for bridges of limited ductile behaviour. For those of ductile behaviour, it isallowed either:

g to take the same value as for bridges of limited ductile behaviour org to subtract from it qVEd/VC,o� 1 (but not to a final value below gBd ¼ 1), where:

– VEd is the maximum shear force of the seismic link from the analysis for the seismicdesign situation

– VC,o the capacity design shear of the link, without the upper limit VC,o � qVEd.

Clauses 2.3.6.2(2),

5.3(2), 6.6.3.1(3) [2]

Clauses 5.6.2(2)b,

5.6.3.3(1)b [2]


141

Clause 7.5.2.3.5(5) [2]

Clauses 6.6.2.1(1),

6.6.2.1(2), 6.6.3.2(2)b,

6.6.3.2(3) [2]

Clauses 5.3(7),

7.6.2(1) [2]

Clauses 6.6.2.3(1), 7.2,

7.5.2.3.3(1),

7.5.2.3.3(2),

7.5.2.3.3(4) [2]

(see also notes (4) and (5) in Table 6.1). Note that, as the behaviour of any seismic link is normallynon-ductile, it is prudent to reduce its design resistance by the factor gBd specified in Part 2 ofEurocode 8 for concrete shear keys, although Part 2 of Eurocode 8 does not require it explicitly.

6.10. Dimensioning of bearings6.10.1 IntroductionSections 6.10.2 to 6.10.4 refer to bearings supporting the deck over certain piers and/or at theabutments, while other supports (usually piers) are connected to the deck monolithically orvia fixed bearings. Movable bearings and other special devices (isolators) arranged along acontinuous interface between the underdeck and all its supports (isolation interface) are dealtwith in Chapter 7.

Part 2 of Eurocode 8 also allows the use of common flat sliding bearings (without a controlledlower limit of friction coefficient) and/or common low-damping elastomeric bearings at theisolation interface of bridges with seismic isolation. Bearings for this use are a special case,covered in this section. Conformity with EN 1337-2 (CEN, 2000) and EN 1337-3 (CEN,2005d), respectively, and with certain additional design requirements specified in Part 2 ofEurocode 8 is sufficient for them.

6.10.2 Fixed bearingsAs mentioned in Section 6.3.2 (in point (b)), fixed bearings should be dimensioned in the seismicdesign situation for capacity design effects. In bridges of limited ductile behaviour this just entailsmultiplying by q the seismic action effects from a linear analysis. By contrast, in bridges of ductilebehaviour, the general procedure of Section 6.4.2 and Eq. (D6.8) should be used, possibly withthe simplification of Eq. (D6.11). As an exception, Part 2 of Eurocode 8 allows fixed bearings tobe dimensioned using only the analysis results for the seismic design situation, provided they canbe easily replaced and are supplemented with seismic links that are capacity-designed to provideinstead the required horizontal resistance (i.e. dimensioned for the horizontal resistance of thefixed bearings with capacity design amplification).

Fixed bearings are normally non-ductile. Although Part 2 of Eurocode 8 does not require it expli-citly, it is prudent to reduce their design horizontal resistance by the factor gBd mentioned at theend of Section 6.9.2 for the design shear resistance of concrete shear keys.

6.10.3 Flat sliding bearingsSection 6.4.5 pointed out that the elements supporting the deck of ductile bridges should bedesigned for a horizontal friction force in sliding bearings equal to gofRdf, where gof ¼ 1.30and Rdf is the maximum design force in the seismic design situation. The dimensions of thesliding plate of the bearing should be sufficient to accommodate, with adequate safety margin,the extreme design displacement in the design seismic situation, dEd. The value of dEd is givenby Eq. (D6.36), and computed at the location and the horizontal level of the bearing accordingto Section 6.8.1.2, except that, if the sliding bearing belongs to a seismic isolation system, thevalue dE,a ¼ gISdE, is used in lieu of the seismic displacement from the analysis, dE, with avalue of 1.50 recommended in Part 2 of Eurocode 8 for gIS. This value may be considered toprovide the ‘reasonable’ safety margin for such bearings. For those not belonging to such asystem, and in the absence of more specific guidance from Part 2 of Eurocode 8, this marginmay be taken to be between 20% and 30% of dE, depending also on the sensitivity of theestimation of dE on the value of (EI )eff used in the analysis.

6.10.4 Simple low-damping elastomeric bearings6.10.4.1 Scope and definitionsAn elastomeric bearing is a block of vulcanised elastomer – with natural or chloroprene rubber asthe rawpolymer – reinforced internally with steel plates, chemically bonded to the elastomer duringvulcanisation. Being nearly incompressible, the rubber exhibits significant lateral expansion whenin vertical compression. The steel plates restrain this expansion through shear stresses at theinterface with the rubber. These stresses are largest at the perimeter of the steel plate, and maycause debonding there. The magnitude of these stresses is usually checked through the associatedshear strains, g. The large magnitude of these shear strains is manifested by the bulging of eachelastomer layer, which is largest at the perimeter and right next to the steel plate.


142

Part 2 of Eurocode 8 distinguishes two types of elastomeric bearing, depending on the dampingthey offer under cyclic shear. The low-damping ones exhibit narrow hysteresis loops with anequivalent viscous damping ratio, j, of less than 6%. Accordingly, their shear behaviour maybe approximated as linear elastic and characterised by the shear modulus of the elastomeralone, G, which applies throughout the total thickness of the elastomer in the bearing (tq inEq. (D6.43b)). The damping they offer is consistent with the default value of 5% used inlinear analysis, as well as the baseline for the q factors adopted in Part 2 of Eurocode 8.

If the deck is not fixed to any pier or abutment (not even through seismic links) and is seated onelastomeric bearings at all of its supports (or possibly on flat sliding bearings over certainsupports), therefore forming an isolation interface between the deck and all of its supports,the full horizontal seismic force of the deck is transmitted through these elastomeric bearings.Such bearings have a decisive role in the global response of the bridge to the horizontalseismic components. Part 2 of Eurocode 8 acknowledges them as seismic isolation devices, andconsiders the deck as seismically isolated. It calls them ‘simple low-damping elastomericbearings’, defined as ‘laminated low-damping elastomeric bearings in accordance with EN 1337-3:2005, not subject to EN 15129:2009 (Antiseismic Devices)’ (CEN, 2009) nor to any specialtests for seismic performance. Accordingly, they are mainly covered in Section 7 of Part 2 ofEurocode 8, dedicated to bridges with seismic isolation. The increased reliability requiredfrom the isolation system is implemented in that case by increasing the design seismic displace-ments to dE,a ¼ gISdE, with a value of 1.50 recommended in Part 2 of Eurocode 8 for gIS. Anadditional requirement is a detailed explicit investigation of the influence of the variability ofthe design properties of the isolators on the seismic response.

If the deck is fixed to the top of one or more piers (or an abutment), directly or via seismic links,the seismic action is transferred to the substructure through this type of connection. Any elasto-meric bearings used over other supports of the deck have a local role in the overall seismicresponse of the bridge. Such bearings are designed, according to EN 1337-3:2005 (CEN,2005d), to resist all non-seismic horizontal and vertical actions. They are also designed accordingto Part 2 of Eurocode 8 to accommodate the imposed deformations due to the design displace-ment in the seismic design situation, as given by Eq. (D6.36) with the value dE as the designseismic displacement, without multiplication by gIS. A detailed investigation of the variabilityof the design properties of such bearings is not required, in contrast to bearings used asisolator units. Summing up, with these differentiations, low-damping elastomeric bearings sub-jected to seismic shear deformations are verified in the seismic design situation with the rulesgiven in EN 1337-3:2005 and complemented in Part 2 of Eurocode 8.

The following section covers the design and verification of simple low-damping elastomericbearings according to EN 1337-3:2005 and Part 2 of Eurocode 8, differentiating – whereverpertinent – between seismic and non-seismic design situations. Design with/of high-dampingelastomeric bearings is addressed in Chapter 7 of this Guide. In the remainder of this section,simple low-damping elastomeric bearings in the sense of Part 2 of Eurocode 8 are termed, forbrevity, ‘elastomeric bearings’ or even just ‘bearings’.

Elastomeric bearings are usually circular, with diameter D, or rectangular, their sides denotedhere as bx and by; the longer side is normally in the transverse direction of the bridge, tominimise the rotational restraint in the longitudinal direction and – if used under a prefabri-cated girder – to stabilise it laterally during erection. The effective plan dimensions of thebearing are considered to be those of its steel reinforcing plates, denoted here as D0, b0x, b

0y.

According to EN 1337-3:2005, at least 4 mm of elastomer should cover laterally the edge ofthe steel plate:

D0 � D� 8 b0x � bx� 8 b0y � by� 8 (dimensions in mm) (D6.42)

The effective plan area of the bearing, A0, is determined from its effective plan dimensionsabove.

If the bearing has n internal layers of elastomer with thickness ti (25 mm � ti � 5 mm according toEN 1337-3:2005) and nþ 1 steel plates of thickness ts (ts � 2 mm according to EN 1337-3:2005),

Clauses 6.6.2.3(1)–

6.6.2.3(4),

7.5.2.3.3(5),

7.5.2.3.3(6),

7.5.2.4(1), 7.5.2.4(5),

7.5.2.4(6), 7.6.2(1),

7.6.2(2), 7.6.2(5) [2]


143

and the top and bottom steel plates have an outer cover of elastomer, at least 2.5 mm thick perEN 1337-3:2005, the total nominal thickness of the bearing is

tb ¼ ntiþ (nþ 1)tsþ 5 (dimensions in mm) (D6.43a)

and that of the elastomeric part is

tq ¼ ntiþ 5 (dimensions in mm) (D6.43b)

Only the elastomer is deformable in shear. The nominal value of the shear modulus of elastomericbearings, as well as its value determined by testing, denoted in EN 1337-3:2005 as G and Gg,respectively, refer to the total elastomer thickness, tq. EN 1337-3:2005 specifies a target meanvalue of 0.9 MPa for Gg (and allows the designer to specify alternative ones of 0.7 MPa or1.15 MPa), with an acceptable test value tolerance of +0.15 MPa (if one of the two alternativesis specified, +0.1 MPa or+0.2 MPa, respectively). Accelerated ageing (3 days at 708C) may notincrease the value of Gg by more than 0.15 MPa. Note that, according to Annex F of EN 1337-3:2005, the value of Gg is measured under normal temperatures between shear strains gq,d ¼ 0.27and 0.58. Such values are representative of the normal-temperature stiffness of bearings in thestrain range allowed under non-seismic design situations (gq,d � 1.0). In the significantly widerstrain range allowed in the seismic design situation (gq,sd � 2.0), substantially higher values ofthe secant shear modulus at peak strain are expected. For this reason, Part 2 of Eurocode 8gives a higher value Gb ¼ aGg for the nominal design value of the shear modulus of elastomericbearings with the value of a (normally in the range 1.1–1.4) determined from tests. This value isused as lower-bound design property (LBDP), if the bearing is used as isolator unit, with avalue of 1.2Gb as the upper-bound design property (UBDP) for the minimum bearing temperaturefor seismic design, Tmin,b, not less than 08C, or obtained from the appropriate l factor fromAnnex JJ of Part 2 of Eurocode 8 if Tmin,b is less than 08C.

Among the various types of elastomeric bearings specified in EN 1337-3:2005, two deserve specialmention: type C, with top and bottom steel plates (profiled or allowing fixing to the parts of thebridge on either side of the bearing), and type B, without steel or any other type of plate attachedto the top or bottom surface of the elastomer. If 8 mm � ti, steel plates of type C bearingsshould be at least 15 mm thick; if ti . 8 mm, their minimum thickness is 18 mm (CEN,2005d). EN 1337-3:2005 lists recommended standard sizes for type B bearings (which are alsothe basis for type C ones). As an indication of the range:

g bearings from 100 � 150 mm (or 1200 mm) to 250 � 400 mm (or 1350 mm) have n ¼ 2–7elastomer layers with ti ¼ 8 mm and steel plates with ts ¼ 3 mm

g for bearings of 300 � 400 mm (or 1400 mm) to 500 � 600 mm (or 1650 mm), n ¼ 3–10,ti ¼ 12 mm, ts ¼ 4 mm

g for bearings of 600 � 600 mm (or 1700 mm) to 700 � 800 mm (or 1850 mm), n ¼ 4–10,ti ¼ 16 mm, ts ¼ 5 mm

g for bearings of 800 � 800 mm to 900 � 900 mm (or 1900 mm), n ¼ 4–11, ti ¼ 20 mm,ts ¼ 5 mm.

In these recommended sizes, the ratio of the mean horizontal dimension to the total thickness, tb,ranges from about 3 to about 9.

The shape factor, S, is the effective plan area of a bearing, A0, divided by the product of theperimeter defined by its effective plan dimensions and the thickness of an inner elastomerlayer:

g for rectangular bearings

S ¼ b0xb0y/[2(b

0xþ b0y)ti] (D6.44a)

g for circular bearings

S ¼ D0/4ti (D6.44b)


144

If the top of the bearing is displaced with respect to the bottom by ddx � 0 in the horizontaldirection x and by ddy � 0 along the orthogonal direction, y (for rectangular bearings, alongthe sides), the vertical load passes through the area in plan where the displaced top andbottom overlap. This is termed the reduced effective plan area, and denoted as Ar:


Ar ¼ (b0x� ddx)(b0y� ddy) � (1� ddx/b

0x� ddy/b

0y)A

0 (D6.45a)

g for circular ones (Constantinou et al., 2011)

Ar ¼ A0(d� sin d)/p (D6.45b)

where

d ¼ 2 cos�1(dd/D0) (in rad) (D6.46)

with

dd ¼ffiffiffiffiffiffiffiffiffiffiffid2dx þ d2

dy

qðD6:47Þ

6.10.4.2 Dimensioning of elastomeric bearings according to EN 1337-3:2005 andEurocode 8

6.10.4.2.1 Verification of shear strainsEN 1337-3:2005 defines the shear strain in the elastomer due to the total design displacement dddefined via Eq. (D6.47) as

gq,d ¼ dd/tq (D6.48)

where tq is the total thickness of the elastomer from Eq. (D6.43b). It further limits the value ofgq,d at the ULS of the bearing under the total design displacement induced by the fundamentalcombination of actions in EN 1990:2002, to a maximum value of 1.0 (see Section 6.10.4.3 fordetails about the fundamental combination and the horizontal displacements it induces).

For low-damping elastomeric bearings verified in the seismic design situation, whether they areused as isolators or not, Part 2 of Eurocode 8 imposes the following additional requirement onthe maximum total design strain in the seismic design situation, gb,Ed:

gb,Ed � 2.0 (D6.49)

where gb,Ed is determined according to Eqs (D6.47) and (D6.48) from the design horizontaldisplacements in the seismic design situation, as given by Eq. (D6.36). If the bearing is part ofthe isolation system, the value dE,a ¼ gISdE is used as design seismic displacement, with a valueof 1.50 recommended in Part 2 of Eurocode 8 for gIS. This definition of the maximum totaldesign strain in the seismic design situation, gb,Ed, applies also in all other verifications specifiedin EN 1337-3:2005 for elastomeric bearings, wherever bearing shear strains in the seismic designsituation are involved.

Equation (D6.49) is the only additional verification specified by Part 2 of Eurocode 8 for theseismic design situation in addition to those of EN 1337-3:2005. Equations (D6.43), (D6.47) and(D6.48) or (D6.49) readily yield a first estimate of the number of internal elastomer layers, n.If the elastomeric bearings constitute the restoring element of an isolation system, Eq. (D6.49)usually controls the elastomer thickness in the bearing, and often its plan dimensions.

Note that Section 7 of Part 2 of Eurocode 8 requires all verifications of elements of the isolationsystem, the superstructure and the substructure be carried out for the most adverse results of twoanalyses: one based on the UBDP and another on the LBDP of the isolating units.

Clauses 6.6.2.3(1),

6.6.2.3(2), 6.6.2.3(4),

7.5.2.3.3(5),

7.5.2.3.3(6), 7.6.2(1),

7.6.2(2), 7.6.2(5)–

7.6.2(7) [2]


145

6.10.4.2.2 Verification of bucklingThe buckling load of a bearing without lateral displacement dd is (Constantinou et al., 2011)

Ncr ¼ lA0r0pGS

ntiðD6:50Þ

where in a rectangular bearing l � 1.5 and in a circular one l � p2 (Constantinou et al., 2011), A0

is the effective plan area of the bearing, and r0 ¼ p(I0/A0) its minimum radius of gyration, G is the

nominal shear modulus of the elastomer and S is the shape factor from Eqs (D6.44). Therefore:

g for a rectangular bearing with b0x � b0y

Ncr ¼ 0:68A0 b02x b0yG

nt2i ðb0x þ b0yÞðD6:51aÞ

g for a circular one

Ncr ¼ A0 D02G

3:6nt2iðD6:51bÞ

If the bearing is bolted at its top and bottom to mounting plates, Eqs (D6.51) apply in goodapproximation in the laterally deformed configuration, using there the reduced effective planarea Ar from Eqs (D6.45) in lieu of A0. If it is dowelled to them or kept in recessed plates, thelateral displacement may roll the bearing over, in which case Eqs (D6.51) are not critical. Asin the fundamental combination of actions in EN 1990:2002, to which EN 1337-3:2005 refersfor the buckling stability of the bearing at the ULS, the lateral displacements are rarely large,EN 1337-3:2005 does not consider the laterally deformed configuration of not bolted bearingsas a separate case, and checks the buckling stability of the bearing on the basis of Eqs (D6.51)with a safety factor of about 2 for rectangular bearings or of 1/0.6 for circular bearings:

g for a rectangular bearing with b0x � b0y

Nd

Ar

� 2b0xGS3nti

¼ b02x b0yG

3nt2i ðb0x þ b0yÞðD6:52aÞ

g for a circular bearing

Nd

Ar

� 2D0GS3nti

¼ D02G6nt2i

ðD6:52bÞ

If both ddx and ddy are significant in magnitude, Eq. (D6.52a) reduces to a quadric equation in b0xfor given b0x/b

0y. In approximation:

b0x �1

2

ddx þ ddy

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiddx þ ddy

2

� �2

þ4tib0xb0y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3nNd

G1þ b0x

b0y

� �svuut0@

1A ðD6:53aÞ

If dd is non-negligible, Eq. (D6.52b) is strongly nonlinear in D0; the required bearing diametermay be found iteratively from Eq. (D6.46) and

D0 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffi24nt2iNd

G d� sin dð Þ4

sðD6:53bÞ

The above verification of the buckling load of the bearing should be carried out for the funda-mental combination of actions in EN 1990:2002, as well as in the seismic design situation withthe relative displacements ddx, ddy (or dd from Eq. (D6.47)) corresponding to the total designdisplacement according to Eq. (D6.36), with the value dE,a ¼ gISdE used as the design seismicdisplacement and a value of 1.50 recommended in Part 2 of Eurocode 8 for gIS, if the bearing


146

is part of the isolation system. However, as this verification is controlled by the design verticalforce on the bearing, Nd, whose value in the seismic design situation is significantly smallerthan in the fundamental combination of actions in EN 1990:2002, the seismic design situationis rarely critical.

6.10.4.2.3 Verification of the total shear strain, gt,dAs pointed out in the opening paragraph of Section 6.10.4.1, the elastomer may debond from thesteel plate, owing to large shear strains, g, at their interface. The maximum value g in theelastomer is

gt,d ¼ gq,dþ gc,dþ ga,d (D6.54)

where:

g gq,d is the shear strain in the elastomer due to the total design displacement, defined inEq. (D6.48), or in Eq. (D6.49) as gb,Ed for the seismic design situation.

g gc,d is the maximum shear strain in the elastomer due to the design value of the verticalcompression, Nd; it takes place at the interface with the steel plate at the mid-side ofrectangular bearings or all around the perimeter in circular ones, and is equal to

gc;d ¼ f1Nd

ArGSðD6:55Þ

with G, Ar and S as for Eq. (D6.50). Values of f1 are tabulated in Constantinou et al.(2011) as a function of the geometry of the bearing and of the ratio of the bulk modulus ofthe elastomer, K (K � 2000 MPa) to its shear modulus G, with an upper bound for circularbearings of

f1 � 1þ 2S2G

KðD6:56Þ

Approximate expressions are also proposed for f1 in Stanton et al. (2008), including avalue f1 ¼ 1.0 for circular bearings. EN 1337-3:2005 fixes f1 to

f1 ¼ 1.5 (D6.57)

which is on the safe side for the G values specified and the bearing sizes recommended inEN 1337-3:2005, especially for circular and square bearings.

g ga,d is the maximum shear strain in the elastomer due to the total design angular rotationsin the vertical planes through axes x and y, adx � 0, ady � 0, respectively; it takes place atthe extreme compression fibres of the bearing in the vertical plane of the rotation and isequal to (Constantinou et al., 2011)– for rectangular bearings

ga;d ¼ f2adxb’

2x þ adyb’

2y

nt2iðD6:58aÞ

– for circular bearings

ga;d ¼ f2adD

02

nt2iðD6:58bÞ

where

ad ¼ffiffiffiffiffiffiffiffiffiffiffia2dx þ a2dy

qðD6:59Þ

Constantinou et al. (2011) tabulate values of f2 as a function of the geometry of thebearing and the K/G ratio of the elastomer, while Stanton et al. (2008) propose


147

approximate expressions for it, including a value f2 ¼ 3/8 for circular bearings. EN 1337-3:2005 fixes f2 to

f2 ¼ 0.5 (D6.60)

which is again on the safe side for the G values specified and the bearing sizesrecommended in EN 1337-3:2005, especially so for circular and square bearings.

According to EN 1337-3:2005, the value of gt,d from Eqs (D6.48), (D6.54), (D6.55) and (D6.57)–(D6.60) for the fundamental combination of actions in EN 1990:2002 should be limited upwardsto 7/gm, with gm ¼ 1.0. Part 2 of Eurocode 8 also recommends the same upper limit (i.e. withgm ¼ 1.0) to the value of gt,d computed from Eqs (D6.49), (D6.54), (D6.55) and (D6.57)–(D6.60) for the seismic design situation. The upper limit on gt,d is translated to additionalconstraints on the plan dimensions or the number of elastomer layers, n. The latter is easier toimplement:


n �ddtiþ adxb

02x þ adyb

02y

2t2i7

gm� 3tiNd b0x þ b0y

� �G b0xb0y� �2

1� ddx=b0x � ddy=b

0y

� �ðD6:61aÞ

g for circular ones

n �ddtiþ adD

02

2t2i7

gm� 24tiNd

GD03 d� sin dð ÞðD6:61bÞ

with dd from Eq. (D6.47). For circular bearings, ad is given by Eq. (D6.59) and d fromEq. (D6.46).

Note that the number of layers, n, comes out of Eqs (D6.48) and (D6.49) as independent of thesize of the bearing in plan. According to Eqs (D6.52) and (D6.53), the taller the bearing, thelarger its plan dimensions should be against buckling under vertical compression. By contrast,Eqs (D6.61) give fewer layers if the size in plan increases. If the second estimate of n from Eqs(D6.61) is larger than that from Eqs (D6.48) and (D6.49), Eqs (D6.53) may have to be revisited;if in that case they suggest a larger size in plan, it is the turn of Eqs (D6.60) to be revisited, until astable combination of elastomer thickness and plan dimensions is reached. Note that, if thenumber of bearings sharing the vertical reaction increases, the required plan area per bearingdrops due to the reduction in Nd, but the total area increases (see Eqs (D6.53): as the horizontaldisplacements of the bearing are independent of its size in plan, the reduction in the effective planarea according to Eqs (D6.45) is proportionally larger in smaller bearings). Unless it is controlledby Eqs (D6.48) or (D6.49), the number of elastomer layers may decrease as well.

6.10.4.2.4 Fixing of elastomeric bearingsIf the bearing is not positively fixed to both the underdeck and the top of the pier or theabutment, (e.g. for type B bearings), EN 1337-3:2005 requires checking of the transfer of thedesign shear force VEd at the ULS through friction:

Nd;min

Ar

½MPa� � 3 ðD6:62aÞ

Vd

Nd;min

� 0:1þ bNd;min

Ar

½MPa�ðD6:62bÞ


148

where

g b ¼ 0.9 for bearings on concreteg b ¼ 0.3 for bearings on all other types of surface (metallic, bedding resin mortar, etc.).

Vd and Nd,min are the design shear force and the corresponding minimum axial force, respect-ively, on the bearing, and Ar is given by Eqs (D6.45). Equations (D6.62) are to be met:

g under the fundamental combination of actions in EN 1990:2002, with the partial factorsapplied on span permanent actions so that they produce the minimum verticalcompression on the bearing and with the factored traffic loads arranged along the deck sothat they give the maximum vertical tension

g in the seismic design situation, with the seismic action causing the maximum possiblebearing tension and the maximum horizontal displacements, dsd,x, dsd,y, that are consistentwith the design shear force, Vd.

The ratio Vd/Nd,min in Eq. (D6.62b) is defined in EN 1337-3:2005 as the friction coefficient mf.Although not explicit, such friction refers to the interface between the external elastomericlayer of a type B bearing and the material of the underdeck, the pier or the abutment incontact with it. Therefore, Eq. (D6.62b) does not apply if the external steel plates of othertypes of elastomeric bearings are appropriately bonded to the bridge elements supported onthe bearing or supporting it.

If Eqs (D6.62) are not met, the bearing should be fixed at its top and bottom. The end plates(those of type C bearings of EN 1337-3:2005) should be bolted or dowelled to the top andbottom mounting plates or kept in place in recesses of the mounting plates. Note that thisway of fixing the bearing also prevents it from slipping from its intended position duringerection and – if it is fixed by bolting – precludes partial uplift due to the combination oflateral displacements and rotation, which will increase the shear strains on the compressedside over the predictions of Eqs (D6.54)–(D6.60), and may even cause the bearing to rollover.

6.10.4.3 Action effects for the verification of simple low-damping elastomericbearings

Elastomeric bearings should meet the requirements in EN 1337-3:2005 for non-seismic actions,notably for the fundamental combination in EN 1990:2002, Eq. (6.10),

Xj� 1

gG; jGk; j ‘þ’ gPP þ gQ;1Qk;1 ‘þ’Xi> 1

gQ;ic0;iQk;i

or Eq. (6.10a),Xj� 1

gG; jGk; j ‘þ’ gPP þ gQ;1c0;1Qk;1 ‘þ’Xi> 1

gQ;ic0;iQk;i

or Eq. (6.10b),

Xj� 1

jjgG; jGk; j ‘þ’ gPP þ gQ;1Qk;1 ‘þ’Xi> 1

gQ;ic0;iQk;i

where in this case Gk are the permanent actions, Pk is prestressing after losses,Qk,1 is traffic loadswith their characteristic value and Qk,i (i . 1) is the characteristic value of thermal actions, Tk,i.The choice between Eq. (6.10), (6.10a) or (6.10b) in EN 1990:2002 as well as the partial or com-bination factors gG, j, gQ,1, gP, j or c0,1 are National Determined Parameters. The reader isreferred to EN 1990:2002 for the recommended choices, those of gQ,1 and c0,1 depending onthe type of traffic load and model. The recommended value of the product gQ,ic0,i for thethermal actions is gQ,ic0,i ¼ 0.9. In what follows, the fundamental combination is symbolicallywritten as gGP ‘þ’ gQ ‘þ’ 0.9Tk, with gGP representing the part of the fundamental combinationdue to the permanent actions Gk and Pk with the appropriate values of gG,j, gP and j applied, gQ


149

that due to the traffic actionsQk,1 with the associated values of gQ,1, c0,1 and Tk the characteristicthermal actions. The total effects of the fundamental combination are indexed by FC, and theindividual ones due to gGP, gQ and Tk as defined above by gGP, gQ and T, respectively.Note that gQ,1 ¼ 0 over the plan area of the deck whose loading by traffic is favourable. Bythe same token, gG, j ¼ 1.0 and j ¼ 1.0 over the length of the deck whose loading by permanentactions is relieving; however, if the action effects are not very sensitive to variations in thepermanent actions along the length of the deck (and in the present case they are not), constantvalues of gG, j and j may be used throughout the deck.

Equation (D6.48) should be checked for the maximum horizontal displacements ddx, ddy of thetop of the bearing relative to the bottom due to the fundamental combination of actions. Bycontrast, Eqs (D6.52), (D6.53) and (D6.62b) involve both Nd and ddx, ddy,, making it necessaryto search for an arrangement of gravity loads along the bridge that is most adverse for the veri-fication of interest. The relative horizontal displacement of the bearing due to the fundamentalcombination of actions is

dFC ¼ dgGPþ dgQþ 0.9dT (D6.63)

where dgGP is the long-term relative displacement due to the factored permanent actionsabbreviated above as gGP, including prestressing after losses, shrinkage and creep, if relevant,(no partial factor with value greater than 1.0 is applied on deformations due to concreteshrinkage); dgQ is the short-term relative displacement due to the factored traffic actionsdenoted above as gQ; and dT is the relative displacement due to the characteristic thermalactions.

These displacement components are normally very small (almost zero) in the transverse direction.Hence, in the following they are assumed to be in the longitudinal direction. They all refer to thehorizontal level where the deck is seated on the bearing (i.e. the underdeck), and are equal to thedisplacement at the level of the centroidal axis of the deck plus the rotation of the deck sectionabove the bearing multiplied by the centroidal distance to the underdeck. As it is the absolutevalue of horizontal displacements that enters into the verifications, in order for dFC to havethe largest possible absolute value, dgQ and dT should have the (fixed) sign of dgGP:

g Owing to concrete creep, shrinkage and prestressing, in concrete decks dgGP is towards themid-length of the deck and away from its nearest end. Then, dgQ and dT should also betowards the deck mid-length. For dgQ, this means that the arrangement of traffic loadsalong the bridge should be that giving the maximum hogging moment (negative) at thespan next to the support in question on the side towards the deck mid-length. (If thevalues of gG, j and j are differentiated along the deck, they are taken as gG, j ¼ 1.0 andj ¼ 1.0 in the span next to the support of interest on the side towards the deck mid-lengthand with their non-unity values in the spans to its left and right, alternating with the unityvalues thereafter.) The thermal displacement, dT, is due to the same thermal actionshighlighted in Section 6.8.1.2 in points (a) and (b). Indeed, if a value of dT at a bearing iscomputed according to Section 6.8.1.2 for the purposes of overlap length, it applies for thepurposes of Eq. (D6.57) as well.

g Under a non-concrete deck, dgGP may be in the same direction as above, or in the reversedirection (i.e. away from the deck mid-length and towards its nearest end). In this lattercase, the reverse also applies regarding the most unfavourable longitudinal arrangement oftraffic loads and of the spans where gG, j and j assume unity or non-unity values; as far asdT is concerned, it is the same as the value computed in Section 6.8.2.1 in points (a) and(b) for the purposes of clearance at the lowest horizontal level (in which case Eq. (D6.38b)applies).

The values of ddx and ddy are quite sensitive to the thermal actions but insensitive to those due totraffic that determine dgQ. The reverse holds for Nd. So, when the maximum value of Nd is ofinterest, the position of traffic loads (and the spans where gG, j and j may assume their non-unity values) are those producing the maximum vertical reaction on the bearing. By contrast,the thermal actions are chosen to maximise the concurrent absolute values of ddx and ddy (asthey always affect unfavourably the verification, see Eqs (D6.53), (D6.61) and (D6.62)).


150

When Eqs (D6.61) are verified in the seismic design situation, the value of Nd is derived from thecondition when the seismic actions induce compression in the bearing. The displacements ddx andddy are those associated with Eq. (D6.49) in Section 6.10.4.2.1 and specified there. If the deck is ofconcrete, the longitudinal displacements are computed as in Section 6.8.1.2, with movementtowards the deck mid-length and away from its nearest end. For other deck materials, thereverse direction of longitudinal movement may need to be examined in addition, with displace-ments computed as in Section 6.8.2.1 in points (a) and (b) for the clearance at the lowest horizon-tal level (with Eq. (D6.38b) applying). It is meaningful to check Eqs (D6.49), (D6.53), (D6.61) or(D6.62) in the transverse direction only when the design seismic displacement of the bearing inthat direction is markedly larger than in the longitudinal direction, offsetting the almost zerovalues of dG and dT.

6.10.4.4 Sizing of simple low-damping elastomeric bearings for seismic isolation6.10.4.4.1 IntroductionIf the elastomeric bearings resist alone the seismic action (considered in Part 2 of Eurocode 8 asseismic isolators), the elastomer thickness and the bearing dimensions in plan determine thelateral stiffness and the seismic displacement and force demands. So, the thickness cannot beobtained from the displacement through simple application of Eq. (D6.49). In the following, aprocedure is proposed for the preliminary estimation of the thickness and the dimensions inplan of the bearing. It assumes that the flexibility of the piers and the abutments may beignored compared with that of the bearings, and that all elastomeric bearings have the sametotal elastomer thickness, tq. For such a system the rigid deck model may be applied. It isfurther assumed that the centre of the horizontal bearing stiffnesses coincides with the centreof mass of the deck, so that no twisting response occurs. The total plan area of the bearings isdenoted by

PAb. The total design vertical load of the deck, corresponding to the total deck

tributary mass in the seismic design situation, is denoted asP

Nd. Further, we introduce thenotation:

a ¼ Sag

g(design ground acceleration at the top of the soil in gÞ ðD6:64aÞ

sb ¼P

NdPAb

ðD6:64bÞ

r ¼ sb

Gb

ðD6:64cÞ

Witness that, sinceP

Nd is considered as given, sb and its dimensionless counterpart r are inver-sely proportional to the sum of the plan areas,

PAb, of the elastomeric bearings that undertake

the entire horizontal seismic force. Note also that, even thoughP

Nd is carried by the same elas-tomeric bearings as this horizontal force, sb is not necessarily equal to the normal stress of eachbearing, because the distribution of

PNd to the individual bearings depends on various factors,

unlike that of the horizontal force, which is distributed in proportion to the plan area of eachbearing. Note also that sb and r can be substantially varied by varying 1/

PAb, not only by

changing the areas of individual bearings: sb and r may increase by assigning part of thegravity load to a number of flat sliding bearings, reducing thereby the elastic restoring force ofthe system.

6.10.4.4.2 Seismic demand for the bearing shear strainTotal horizontal stiffness:

Kb ¼ Gb

PAb

tqðD6:65Þ

Period:

T ¼ 2p

ffiffiffiffiffiffiffiPNd

gKb

s¼ 2p

ffiffiffiffiffiffiffiffiffiffiffiffiPNd tq

gGb

PAb

s¼ 2p

ffiffiffiffiffiffiffisb

Gb

tq

g

r¼ 2p

ffiffiffiffirtqg

rðD6:66Þ


151

Seismic shear strain of bearing:

gbE ¼ ddEtq

ðD6:67Þ

Seismic displacement demand:

dbE ¼ T2

4p2SaðTÞ ¼ rtg

SaðTÞg

ðD6:68Þ

Equations (D6.67) and (D6.68) and the elastic response spectrum of Eqs (D5.3) yield:

g if T , TC

gbE ¼ 2.5ar (D6.69a)

g if TC � T � TD

gbE ¼ 2:5arTC

TðD6:69bÞ

g if TD , T

gbE ¼ 2:5arTCTD

T2ðD6:69cÞ

The average seismic shear stress in the bearings:

tbE ¼P

VbEPAb

ðD6:70Þ

is obtained as tbE ¼ gbEGb:

g if T , TC

tbE ¼ 2.5asb (D6.71a)

g if TC � T � TD

tbE ¼ 2:5asb

TC

TðD6:71bÞ

g if TD , T

tbE ¼ 2:5asb

TCTD

T2ðD6:71cÞ

Figures 6.7–6.9 show the influence on the seismic response of the bridge of:

g the spectrum corner period TC

g the thickness tq in the range 40–300 mmg the value of r in the range 2–12.

Note that, in practice tq seldom exceeds 250 mm and r rarely exceeds 7 or 8, as, around thesevalues, buckling of the bearings in the seismic design situation may become critical.

Figure 6.7 and Eq. (D6.66) show that the period T is proportional top(rtq). In the period range

TC � T � TD, which is of high practical interest, the following are noted:


152

g Figure 6.8 shows that the same value of the seismic shear strain gb can be obtained withseveral pairs of values r and tq below a certain maximum value of r. In fact, gb fromEqs (D6.69b) and (D6.66) is proportional to

p(r/tq). Note also from Eqs (D6.64b) and

(6.64c) thatp(r/tq) – and hence gb� is inversely proportional to

p(P

Abtq) (i.e. to thesquare root of the total elastomer volume of the bearings).

g Figure 6.9 shows that, among the above pairs of values of r and tq leading to a certainshear strain demand gb (i.e. pairs giving the same total volume of elastomer), that givingthe most efficient seismic isolation (i.e. the lowest force reduction ratio) is the pair with themaximum values of r and tq (i.e. with the maximum value of tq and the minimum

PAb).

The parts of Figure 6.8 for different values of TC show that the maximum value of r requiredfor a seismic shear strain gb meeting Eq. (D6.48) drops substantially as TC increases to 0.8 s.So, for this value of TC and for large design ground accelerations, seismic isolation with low-damping elastomeric bearings seems unfeasible in practice without supplementary damping.Even if such a solution proves feasible, its efficiency will be very low, as evidenced byFigure 6.9(c). Possibilities for supplementary damping are discussed in Chapter 7 of thisDesigners’ Guide.

6.10.4.4.3 Practical selection of tq andP

Ab

Equation (D6.49) may be expanded as

gb,sd ¼ gISgbEþ gbGþ 0.5gbT � 2.0 (D6.72)

where gbG is the shear strain in the bearing due to shrinkage and creep of the deck and gbT is thatdue to temperature variations of and/or through the deck. These strains may be determined asgbG ¼ dG/tq and gbT ¼ dT/tq from the relevant displacements dG and dT, which are function ofthe maximum distance of a bearing from the centre of stiffness of all the bearings. ForgIS ¼ 1.50, Eq. (D6.72) gives

gbE � 4/3 – (dGþ 0.5dT)/1.5tq (D6.73)

On the diagram for the shear strain in Figure 6.8 at the applicable value of TC and for the selectedvalues of tq, one can mark the points corresponding to tq and the value of gbE required byEq. (D6.73). They are on a line approximately parallel to the tq axis, defining pairs of r and tqsatisfying Eq. (D6.72).

Note in the corresponding shear strain diagram that the minimum force (i.e. the optimumisolation effect) is achieved when r is maximum. This selection maximises the period and theshear strain in the bearings. The selected pair should, of course, be checked that it meets


Figure 6.7. Period T as a function of parameters tq and r of low-damping elastomeric bearings

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

50 100 150 200 250 300Elastomer thickness tq: mm

Perio

dT:

s

ρ = 12

ρ = 10

ρ = 8

ρ = 6

ρ = 4

ρ = 3

ρ = 2

153

the verifications set out in EN 1337-3:2005, notably that of buckling under the fundamentalcombination of actions and in the seismic design situation (see Section 6.10.4.2.2), which isusually more critical than that for the total shear strain in Section 6.10.4.2.3.

The above verifications of elastomeric bearings for the seismic design situation need to be carriedout only for the LBDP of the bearings (i.e. for the nominal value of Gb, as it is the critical one forthe deformation-controlled verifications of the bearings). However, the maximum forces are thecritical ones for the design of the substructure and for anchoring the bearings, which are bothforce-controlled. The corresponding design forces are determined from the UBDP of Gb,which is the nominal one increased by 20% if the minimum bearing temperature for seismic


Figure 6.8. Seismic shear strain gbE normalised to a ¼ Sag/g, as a function of the parameters tq and r of

low-damping elastomeric bearings and of the corner period of the elastic spectrum: (a) TC ¼ 0.4 s;

(b) TC ¼ 0.6 s; (c) TC ¼ 0.8 s

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

Nor

mal

ised

sei

smic

she

ar s

trai

n γ b

E/α

Nor

mal

ised

sei

smic

she

ar s

trai

n γ b

E/α

Nor

mal

ised

sei

smic

she

ar s

trai

n γ b

E/α


(a)


(b)


(c)

ρ = 12ρ = 10ρ = 8ρ = 6

ρ = 2 ρ = 3 ρ = 4

ρ = 12

ρ = 6ρ = 8ρ = 10

ρ = 2 ρ = 3 ρ = 4

ρ = 2 ρ = 3 ρ = 4

ρ = 12ρ = 10

ρ = 8ρ = 6

154

design Tmin,b is not less than 08C, or may be obtained from the appropriate l factor fromAnnex JJ of Part 2 of Eurocode 8 if it is.

6.11. Verification of abutments6.11.1 GeneralThe abutments and their foundation are designed and verified to stay elastic in the seismic designsituation. Because in these verifications the earth pressure action (including seismic effects) isvery important, action effects are normally expressed in the direction of the earth pressure(i.e. at right angles to the lateral surfaces of the abutment on which the earth pressure acts),

Clause 6.7.1(1) [2]


Figure 6.9. Force reduction ratio of the isolation tbE/tbE,max (where tbE,max ¼ 2.5asb) as a function of the

parameters tq and r of low-damping elastomeric bearings and of the corner period of the elastic

spectrum: (a) TC ¼ 0.4 s; (b) TC ¼ 0.6 s; (c) TC ¼ 0.8 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Forc

e re

duct

ion

ratio

τbE

/τbE

,max

Forc

e re

duct

ion

ratio

τbE

/τbE

,max

Forc

e re

duct

ion

ratio

τbE

/τbE

,max


(a)


(b)


(c)

ρ = 2

ρ = 3ρ = 4ρ = 6

ρ = 2

ρ = 3ρ = 4ρ = 6

ρ = 12 ρ = 10 ρ = 8

ρ = 12 ρ = 10 ρ = 8

ρ = 12 ρ = 10 ρ = 8

ρ = 2

ρ = 4ρ = 6ρ = 3

155

Clauses 6.7.2(2),

6.7.2(3), 5.8.2(2),

5.8.2(3) [2]

Clauses 7.3.2.1(1)–

7.3.2.1(3), 7.3.2.2(1)–

7.3.2.2(7), 7.3.2.3(1)–

7.3.2.3(12), 7.3.2.4(1),

7.3.2.4((2), E.3–E.9 [3]

which may deviate from the longitudinal or the transverse direction of the bridge. To this end,peak values of seismic action effects should be available along local axes parallel and at rightangles to the earth pressures. If such local axes have not been introduced at the outset, thepeak values of the effects of the longitudinal and the transverse seismic action are projectedonto the direction of the earth pressures and combined into a single value through Eqs (D5.1)or (D5.2). The same holds for the seismic action effects at right angles to the earth pressures,if they are of interest.

6.11.2 Abutments supporting the deck through movable bearingsIf the abutment supports the deck on movable bearings, it is not normally included in the globalmodel for the analysis of the bridge system, even though that model might include piles under thepiers and their interaction with the soil. Such abutments are considered in the global model asrigid ground, supporting the bearings with the appropriate kinematic restraints or springs.The analysis gives, then, reactions on/from the bearings, from which the effects of the seismicaction and of the other ones in the seismic design situation on the abutment are derived andused as loads in independent static analyses and verifications of each abutment. The loads/actions applied to the abutment in this analysis/verification are the following:

1 The non-seismic actions in the seismic design situation:– (vertical) reactions from the bearings on the abutment, derived from the analysis of the

bridge system– gravity loads of the abutment itself and of any earthfill over the foundation of the

abutment.2 The seismic action effects from the deck on the abutment, as reactions from the bearings

due to the design seismic action. In bridges designed for ductile behaviour, these arecapacity design effects according to Section 6.4.5; in those designed for limited ductilebehaviour, they are the reactions on the bearings from the seismic analysis back-multipliedby q, to obtain in the end action effects for an effective value of q ¼ 1.0.

3 Pseudo-static inertia forces arising from the mass of the abutment and of the earthfilloverlying its foundation. As pointed out in Section 5.4 of this Guide, abutmentssupporting the deck on movable bearings are considered as ‘locked in’, and their inertiaforces are computed as the pertinent mass multiplied by the design ground acceleration atthe top of the ground at the site, agS (i.e. with q ¼ 1.0 and T ¼ 0). Note that this is howseismic action effects due to the mass of the abutment and the earthfill over its foundationshould be accounted for, even when the mass and stiffness of these components have beenincluded in a global analysis model of the bridge.

4 Any hydrostatic or hydrodynamic pressures, including buoyancy.5 The earth pressures (including seismic effects). These pressures are the controlling actions

on the type of abutments considered herein, both under static conditions and in the seismicdesign situation. Therefore, the methods for determining earth pressures for free retainingwalls are applicable. According to Annex C of Eurocode 7 (CEN, 2003b), static earthpressures may be determined on the basis of two limit equilibrium states of the soil:– the passive state, corresponding to the soil resisting movement of the wall against the

earthfill– the active one, for the wall giving way to the soil pressure (and possibly to other actions)

and moving away from the earthfill.The pseudo-static Mononobe–Okabe approach adopted in Part 5 of Eurocode 8 (CEN,2004c) for the estimation of the seismic effects on earth pressures conveniently uses thesame limit equilibrium states. It is worth bearing in mind for the seismic design situationthe distinction between, on the one hand, movement of the wall relative to the earthfillthat defines the passive or active state of earth pressures and the seismic motion of the wallbase and the earthfill on the other, which both move with the ground. In fact, the activeearth pressure state of the pseudo-static Mononobe–Okabe approach develops when theseismic motion occurs towards the earthfill, with the earthfill and the wall base by andlarge following this motion. In this case the wall body moves away from the earthfill underthe inertia forces on the earthfill and the wall mass, due to the deformability of the walland mainly the rotation of its foundation. Seismic forces from the deck (under point 2above) are assumed to act in phase with the aforementioned forces, for the most adverseeffect for the abutment, which may fail only when it moves away from the earthfill.


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The earth pressures of type 5 acting on a lateral face of the abutment or its wing walls movingeither

(a) against the earthfill or(b) away from it

may be estimated as follows. Those under (a) may be taken as equal to the static earth pressure atrest, as very large displacements are needed for the passive earth pressure to develop. If theabutment or the wing wall is free to develop the lateral displacement necessary for an activestate of stress in the backfill (see following paragraph), those under (b) are the active earth press-ures, including the seismic effects, and are computed with the pseudo-static Mononobe–Okabeapproach in Part 5 of Eurocode 8. In that calculation, abutments may be considered as non-gravity walls, and the effect of the vertical acceleration on earth pressures may be neglected;the horizontal coefficient kh is taken as equal to the design ground acceleration at the top ofthe ground at the site in g: kh ¼ agS/g (i.e. the reduction factor r is taken equal to r ¼ 1.0).If, by contrast, the lateral displacement of the abutment or the wing wall is restrained, it ismore realistic and on the safe side to estimate the seismic earth pressures (to be added to thestatic ones for the wall at rest) as given in Clause E.9 of Part 5 of Eurocode 8 for rigid structures.Wing walls monolithic with a large abutment are normally considered to belong in this case. Asignificant lateral displacement, da, is necessary at the top of an abutment for the active state ofstresses to develop in the soil. According to informative Annex C of Eurocode 7, for rotationabout the abutment foundation it takes between 0.4% and 0.5% of the height of theabutment for loose non-cohesive backfill and between 0.1% and 0.2% for dense backfill.About half of these values apply for parallel translation of the abutment. The correspondingvalues for the full passive earth pressures in dry soil with an abutment rotating about itsfoundation are 7–25% for loose non-cohesive backfill and 5–10% for dense backfill. Thelateral displacement needed for half the full value is 1.5–4% and 1.1–2% of the height of theabutment, respectively (all these values increase by 50–100% under the water table); this iswhy the static earth pressures at rest are taken to act on the face of an abutment movingagainst the earthfill. As Part 2 of Eurocode 8 allows consideration of the active earth pressureson the abutment face moving away from the backfill, it is a requirement to check that theabutment will not fail before the lateral displacement at its top reaches the value, da, wherethe Mononobe–Okabe active earth pressure develops. As the active earth pressure is higherfor lateral displacements less than da, Part 2 of Eurocode 8 requires dimensioning of theconcrete structure of the abutment body and foundation for earth pressures higher thanthe active earth pressure; namely, for the value at rest plus 1.3 times the seismic part of theMononobe–Okabe active earth pressure (which is equal to the Mononobe–Okabe active earthpressure minus the earth pressure at rest). This increase is not required for the verifications ofthe foundation soil. The designer should provide, in addition, a clearance between the deckand the abutment backwall that accommodates a lateral displacement of the top of theabutment of at least da. Taking into account that the end of the deck is moving away fromthe backwall, the required clearance is

dad ¼ da� dEþ dGþ c2dT (D6.74)

where dE, dG, c2 and dT assume the same values as in Section 6.8.1.2, and da is the indicative valuequoted above from Part 1 of Eurocode 7 (CEN, 2003b).

Regarding loads of type 5 in the list above, if the face of the abutment towards the bridge is incontact with water, the hydrostatic pressure minus the hydrodynamic pressure – given by norma-tive Annex E in Part 5 of Eurocode 8 under case E.8 – should be considered to act on that face,but with the abutment also moving away from the earthfill, which approximately follows theground motion as explained above. The minus sign is because the seismic motion is assumedto be away from the water, which does not move. The case of hydrostatic plus hydrodynamicpressure on the front side of the abutment, when the abutment and the seismic motion areagainst the water, may be ignored alongside the passive earth pressures on the backside of theabutment. Note that, if the backfill itself is saturated and permeable in the time-scale of theseismic motion (case E.7 in normative Annex E of Part 5 of Eurocode 8), its pore water canmove independently from the fill material. So, the hydrodynamic pressure should be added to


157

Clauses 6.7.3(1)–

6.7.3(6), 6.7.3(8) [2]

the earth pressures of the soil. As a result, the hydrostatic plus hydrodynamic water pressureshould be taken to act on the backside of the abutment, while the active earth pressure on it isproportional to the buoyant unit weight of the soil. Note, however, that only a very coarse-grain fill material may be considered to allow the water to move independently of the fillduring an earthquake.

The static analysis and the verifications are carried out considering the components of the seismicaction in two (ideally orthogonal) horizontal directions. Load types 2 and 3 arise from the seismicaction alone, and are taken in these two horizontal directions. Load types 4 and 5 include staticcomponents in fixed directions, plus seismic ones which are again taken in the afore-mentionedhorizontal directions. The horizontal forces in load types 2 to 4 due to the seismic action are alltaken with the same sense (positive or negative) along the pertinent horizontal direction. Only ifthe abutment and any wing walls are parallel to the transverse and the longitudinal directions ofthe bridge, respectively, do the two horizontal directions of the abutment seismic forces coincidewith them. In any other case, the two horizontal directions of the static seismic forces are chosenad hoc. For example:

g If the abutment is skewed to the longitudinal axis and does not have wing walls, the staticanalyses and the verifications are carried out with the components of the seismic action atright angles and parallel to the abutment.

g If the abutment is either at right angle or skewed to the longitudinal axis, possiblymonolithically connected wing walls are usually parallel to the longitudinal direction of thebridge, along the edges of the roadway. Wing walls of up to 8 m horizontal length areusually triangular in elevation and do not have their own foundation but cantilever fromthe vertical edge of the abutment body. This is quite common for cost reasons, but cannotbe applied to tall abutments. A tall abutment and its wing walls usually form the threesides of a caisson, with a common foundation behind the abutment body. The mainearthfill actions on triangular cantilevering wing walls act at right angles to them, andamount to the earth pressure at rest combined with that due to a traffic surcharge or thecompaction earth pressure. These actions induce mainly out-of-plane bending of the wingwall, alongside out-of-plane bending and horizontal tensile forces in the body of theabutment. Additionally, the wing walls should be verified for the accidental action ofvehicle impact on parapets. The seismic dynamic earth pressure increment, normallyaccording to case E.9 of Part 5 of Eurocode 8, should be taken as a separate load casewhen it acts on each wing wall. Earth pressures on the abutment body are, of course, atright angles to it. The influence on the abutment of the seismic earth pressure incrementacting on the skew inside face of the wing wall and obliquely to the abutment may beestimated approximately. For example, the earth pressure increments estimated for theabutment may be multiplied by the area of the wall projection on it.

g Similar approximations may be used for the ‘caisson’-type abutments mentioned above,for an earthquake at right angles to the abutment. Seismic forces at right angles to thewing-walls of high and narrow ‘caissons’ (e.g. with a width-to-height ratio less than 2) mayhave to be estimated with the relevant rules for silos in Part 4 of Eurocode 8 (CEN, 2006).

6.11.3 Abutments rigidly connected to the deckAn important case here is that of a deck monolithic with the abutments (‘integral’). The othercase is that of an abutment connected to the deck via fixed bearings or seismic links (includingshear keys) designed to carry the seismic action, in lieu of the bearings. For the connection tobe considered as rigid (and for clause 6.7.3 of Part 2 of Eurocode 8 to apply), it should be soin both horizontal directions. The length and span limitations set out in Section 4.5.3 of thisDesigners’ Guide for integral bridges also hold for decks connected to the abutments throughfixed bearings or seismic links designed to carry the seismic action.

Most of the rules in clause 6.7.3 of Part 2 of Eurocode 8 (namely the q value of 1.50, the earthfillreaction to be considered according to the next paragraph and the upper limit to the totaldesign seismic displacement in the last paragraph of this section) refer to short integralbridges. Where the connection of the deck to the abutment is horizontally movable orflexible, at least in the longitudinal direction (as is always the case in longer bridges), noneof these rules apply.


158

Regarding the activation of earthfill reaction against the backside of abutments rigidly connectedto the deck, note the following:

g In the longitudinal direction, reaction is activated on the face of the backwall towards thebackfill. Part 2 of Eurocode 8 recommends using over it equivalent springs with a stiffnesscorresponding to the actual geotechnical conditions. To cover uncertainty in thedistribution of the seismic force among any piers and the reacting abutment, it alsorecommends using upper- and lower-bound soil compliance properties. Lower-boundearthfill stiffness leads to the maximum seismic forces for the piers and their foundationsand to maximum seismic displacements of the abutment – which are subject to the damagelimitation check at the end of the present section. Upper-bound stiffness may be criticalonly for the abutment in bending and the adjacent region of the deck.

g In the transverse direction, the area where earthfill reaction may be activated is the internalface of wing walls that are monolithically connected to the abutment and move towardsthe earthfill. Again, upper- and lower-bound soil compliance estimates should be used. Thelower-bound stiffness of the soil may be close to zero if the ratio width/height of theearthfill is low in the transverse direction. ‘Active’ dynamic earth pressures should anywaybe considered on the opposite wing wall (see Section 6.11.2).

A global model is normally used for the analysis, with a fairly detailed discretisation of the deckand the abutments. Springs model the soil against which the abutments and any wing walls maypush, with use of upper- and lower-bound estimates of the soil stiffness according to the para-graph above. Bridges with the deck flexibly supported on the abutments in the longitudinal direc-tion, but restrained there transversely by seismic links (shear keys), should be analysed fortransverse seismic action with a global model involving soil springs at the interior surface ofthe wing walls. In view of the need to consider upper- and lower-bound estimates of soil stiffness,this is quite taxing. However, there is no need for such a model if the abutments do not have(significant) wing walls pushing against the backfill.

Bridges with the deck monolithic with the abutments often meet the criteria for the application ofthe linear static analysis with the fundamental mode method. With inertia forces on the masses ofthe structure estimated this way, the soil–structure system is then analysed for the simultaneousaction of gravity loads on these masses and of earth pressures (static at rest, plus seismic wherepertinent) and hydrostatic and hydrodynamic ones, where applicable. The seismic earth pressuresand the hydrodynamic pressures are computed and applied where pertinent according to Section6.11.2, but always in the same horizontal direction and with the same sense of action as the inertiaforces on the masses of the structure.

To limit damage to the backfill behind an abutment rigidly connected to the deck, Part 2 ofEurocode 8 requires checking that the displacement of the abutment towards the backfill,computed from the analysis results according to Eq. (D5.68), does not exceed a certain limit.This limit is a Nationally Determined Parameter, with a recommended value of 60 mm inbridges of importance class II or just 30 mm for those of importance class III. No limit is setfor bridges of importance class I.

6.12. Verification of the foundation6.12.1 Design action effectsFor the purpose of the verification of resistance, the design action effects on the foundationshould be determined as follows (see Sections 6.3.2 and 6.7.2):

g For bridges designed for limited ductile behaviour (q � 1.5) or for those with seismicisolation, the design action effects are those from the analysis for the seismic designsituation, Eq. (D6.1), with the seismic action effects computed for elastic response (i.e.with a behaviour factor of 1.0).

g For bridges designed for ductile behaviour (q . 1.5):– if the linear analysis with the design response spectrum is applied, the capacity design

procedure in Section 6.4.2 should be applied, for flexural plastic hinges developing in thepiers

Clause 6.7.3(7) [2]

Clauses 4.2.4.4(2)e,

5.8.2(2)–5.8.2(4) [2]


159

Clause 5.4.1 [3]

Clause 5.4.1.1,

Annex F [3]

– if nonlinear analysis is used, the design action effects are those from the analysis, but thedesign resistance (computed with material partial factors, gM) is divided by theNationally Determined Parameter safety factor, gBd1, with a recommended value ofgBd1 ¼ 1.25 (see Sections 6.7.2 and 6.9.2).

6.12.2 Shallow foundations6.12.2.1 Stability verificationsThe verifications of shallow foundations concern sliding and the bearing capacity of the soil inthe seismic design situation.

For the verification against sliding, the total design horizontal force should satisfy the following:

VEd � FH1 þ FH2 þ 0:3FB ðD6:75Þ

where FH1 is the friction over the base of the footing, equal to NEd tan(d)/gM; FH2 is the frictionover the lateral sides of an embedded foundation; FB is the ultimate passive resistance of anembedded foundation; NEd is the vertical design force acting on the foundation; d is thefriction angle between the foundation and the soil; and gM is the partial factor, taken equal togw, with a recommended value of 1.25.

Note that, although full friction on the base and the lateral sides of the foundation may beconsidered to be mobilised, relying on more than 30% of the total passive resistance is notallowed. The rationale for this limitation is that, for mobilisation of the full passive resistance,a displacement of significant magnitude is necessary to take place that does not comply withthe usual performance goals set forth for bridges. The values of FH2 and FB should correspondto the lowest top level of the ground surface expected in the seismic design situation.

Note that, under certain circumstances, sliding may be acceptable, as an effective means fordissipating energy and shielding the superstructure by limiting the forces that enter it (as inbase isolation systems). Furthermore, a numerical simulation generally shows that the amountof sliding is limited. For this situation to be acceptable, the ground characteristics shouldremain unaltered during the seismic excitation. Additionally, sliding may not affect the function-ality of the bridge. Since soils under the water table may be prone to pore pressure build-up,which affects their shear strength, sliding is only tolerated when the foundation is locatedabove the water table. It should be further pointed out that the predicted foundation displace-ment when sliding is allowed strongly depends on the friction coefficient between the footingsurface and the soil, which in turn depends on the surface material, its drainage conditionsand the construction method. If reliable estimates are necessary, in-situ tests are warranted.

Although not required by Eurocode 8, it is common practice to keep the resultant force at thelevel of the soil–footing interface within one-sixth to one-third of the footing width from thecentre; if a linear distribution of soil stresses is assumed under the footing, these two limitsare equivalent to not allowing uplift, or restricting it to one-half of the foundation, respectively.Nowadays, uplift is more commonly tolerated, because it is recognised that rocking of thefoundation reduces the seismic forces that enter the structure and therefore protects it.However, to avoid yielding of the soil under the loaded edge and the permanent settlementand tilt of the foundation it entails, rocking must be restricted to very good soil conditions.To evaluate this behaviour for a spread footing, it is recommended that nonlinear static(pushover) analysis is conducted to establish its moment–rotation characteristics, including theeffects of uplift of the footing and soil yielding. Analytical results from such analyses not onlyprovide rotational stiffness parameters but also depict the geotechnical mode of ultimatemoment capacity. For spread footings, uplift is the most severe form of nonlinearity, albeittolerated by Eurocode 8. The foundation cannot develop overturning moments higher thanthe ultimate moment capacity and its moment–rotation curve becomes distinctly nonlinearwell before the ultimate moment (see Figure 5.20 for examples).

In addition to the sliding verifications, the bearing capacity of a foundation under seismicconditions should be checked, considering the inclination and eccentricity of the force actingon the foundation, as well as the effect of the inertia forces developed in the soil medium by


160

the passage of the seismic waves. A general expression, Eq. (D6.76), is provided in Annex F ofPart 5 of Eurocode 8, derived from theoretical limit analyses of a strip footing (Pecker, 1997).The condition for the foundation to be safe against bearing capacity failure simply expressesthat the design forces NEd (vertical), VEd (horizontal), MEd (overturning moment) and the soilseismic forces should lie within the surface depicted in Figure 6.10 and expressed by

1� eF� �cT bV

� �cTN� �a

1�mFk

� k0�N

� �b þ 1� f F� �c0M gM

� �cMN� �c

1�mFk

� k0�N

� �d � 1 ðD6:76Þ

with the following definitions:

N ¼ gRdNEd

Nmax

V ¼ gRdVEd

Nmax

M ¼ gRdMEd

BNmax

ðD6:77Þ

g for purely cohesive or saturated cohesionless soils

F ¼ gRdragSB

Cu

ðD6:78aÞ

g for purely dry or saturated cohesionless soils without significant pore pressure build-up

F ¼ gRdagS

g tanf0d

ðD6:78bÞ

whereNmax is the ultimate bearing capacity of the foundation under a vertical concentric force, asestimated by any reliable method (strength parameters, empirical correlations from field tests,etc.); B is the foundation width; �FF is the dimensionless soil inertia force; gRD is the modelfactor; r is the unit mass of the soil; ag is the design ground acceleration on type A ground,Eq. (D3.3) in Section 3.1.2.2; S is the soil factor, see Table 3.3 in Section 3.1.2.3; Cu is the soilundrained shear strength, cu, for a cohesive soil or the cyclic undrained shear strength, tcy,u,for a saturated cohesionless (including the material partial factor, gM); and f0

d is the designangle of the shearing resistance of cohesionless soil (including the partial factor, gM).

The coefficients in lower case in Eq. (D6.76) (a, b, k, cM and cT) are numerical values that dependon the soil type according to Table 6.2 (from Part 5 of Eurocode 8). The model factor gRD


Figure 6.10. Ultimate load surface for the foundation bearing capacity

161

reflects the uncertainties in the theoretical model for the seismic verification of the bearingcapacity, and as such it should be larger than 1.0; it is also meant to recognise that certainpermanent foundation displacements may be tolerated (in which case Eq. (D6.76) is violated),and then it may be less than 1.0. Tentative values that intend to combine both effects areproposed in Annex F in Part 5 of Eurocode 8 and repeated in Table 6.3, reflecting that for themost sensitive soils (loose saturated soils) the model factor should be higher than for stableones (medium-dense sand).

More recent studies have shown that Eqs (D6.76)–(D6.78), as well as these tabulated values, holdalso for circular footings, provided that the ultimate vertical force under vertical concentricforce, Nmax entering Eq. (D6.77), is computed for a circular footing and that the footing widthis replaced by the footing diameter (Chatzigogos et al., 2007). Although Eq. (D6.76) does notlook familiar to geotechnical engineers who are more accustomed to the ‘classical’ bearingcapacity formula with correction factors for load inclination and eccentricity, it reflects thesame aspect of foundation behaviour. This verification is similar to the use of interactiondiagrams in structural design for cross-sections under combined axial force and bendingmoment.

6.12.2.2 Structural design of the footingA footing is usually designed by treating it as a cantilever supported at the verificationsections, or as a simple beam, or as a continuous beam between the columns of rigid framepiers. However, as footings may behave as a slab with an internal redistribution of stresses,they may be considered as two-way beams for design purposes. The footing should havesufficient thickness to be regarded as rigid relative to the underlying soil. Then, a lineardistribution of soil stresses or a distribution corresponding to an appropriate no-tensionWinkler spring model may be assumed for the determination of the internal forces of thefooting at its structural ULS.

As the footing is designed at the ULS as a beam, unless a finite element slab model is used forthe analysis, it is necessary to define an effective width of the footing for the ULS in bending;


Table 6.2. Values of numerical parameters in Eq. (D6.76) (Part 5 of Eurocode 8)

Purely cohesive soil Purely cohesionless soil

a 0.70 0.92

b 1.29 1.25

c 2.14 0.92

d 1.81 1.25

e 0.21 0.41

f 0.44 0.32

m 0.21 0.96

k 1.22 1.00

k0 1.00 0.39

cT 2.00 1.14

cM 2.00 1.01

c0M 1.00 1.01

b 2.57 2.90

g 1.85 2.80

Table 6.3. Model factors for Eqs (D6.77) and (D6.78) (Part 5 of Eurocode 8)

Medium-dense to dense

sand

Loose dry

sand

Loose saturated

sand

Non-sensitive

clay

Sensitive

clay

1.00 1.15 1.50 1.00 1.15

162

(JRA, 2002) provides the estimates of effective width in Table 6.4 as a function of the pier width,hc, and the effective footing depth, d.

6.12.3 Pile foundations6.12.3.1 Pile internal forcesPiles and piers should be verified for the effects of the inertia forces transmitted from the super-structure to the pile heads and also for the effects of kinematic forces due to earthquake-inducedsoil deformations. However, according to Part 2 of Eurocode 8, kinematic interaction needs to beconsidered only when the pile crosses consecutive layers of sharply contrasting stiffness.According to Part 5 of Eurocode 8, consideration of kinematic forces in piles is required whenthe weak layer consists of soft deposits (as in ground types D, S1 or S2) with layers of sharpstiffness contrast, the design acceleration exceeds 0.10g and the bridge is of importance aboveordinary.

For long flexible piles, the effects of kinematic interaction may be evaluated assuming that thepiles follow the ground displacement, which may be calculated either from a 1D site responseanalysis with a computer code such as SHAKE (Idriss and Sun, 1992) or, in a homogeneoussoil layer overlying bedrock, from a simplified analytical formulation with the soil layerassumed to respond in its fundamental mode of vibration and the pile modelled with theappropriate boundary conditions at its tip and head (hinged, pinned). The displacement of thefundamental mode of vibration of the soil layer is given by

d zð Þ ¼ d sinpz

2H

� ðD6:79Þ

where H is the layer thickness and d is the relative ground displacement between the top and thebase of the soil layer. This relative ground displacement can be read from the rock responsespectrum (see Section 3.1.2.3 of this Guide) at the fundamental frequency of the soil layer( f ¼ vs/4H) for the applicable shear wave velocity, vs, and damping ratio (see Table 3.5).

The effects of the inertia forces are obtained from the dynamic analysis. If the piles are modelledas beams on a Winkler foundation, these results are directly obtained from the analysis. If thesoil–structure interaction is modelled as linear, it should, however, be checked that thepressure applied by the piles to the soil does not exceed the pertinent ultimate capacity. If itdoes, the analysis should be redone with weaker springs wherever the ultimate soil bearingcapacity is exceeded, until compatibility between computed displacements and the pressure isachieved. If the pile foundation is modelled using the concept of the impedance matrixintroduced in Section 5.5.1.6 of this Guide, the internal forces in the piles can be obtainedonly through a separate model for the pile, subjected to the same limitations as above.

The values of the action effects arising from inertial and kinematic loading may be combinedusing the SRSS (square root of the sum of the squares) rule, unless the fundamental periodsof the soil layer and the bridge are close to each other. If they are, the sum of their absolutevalues is more appropriate.

6.12.3.2 Structural designPiles are generally designed to remain elastic: a q factor of 1.0 in bridges designed for limitedductile behaviour; and through capacity design in those designed for ductility. However, withthe large bending moment that develops at the connection of a pile to its cap, designing thepile to remain elastic there may not be feasible. It is therefore more economical and oftensafer to design that location to develop a plastic hinge. This is indeed allowed in Part 2 of

Clauses 5.4.2(1)–

5.4.2(6) [3]

Clause 6.4.2.2(2)c [2]

Clauses 4.1.6(7),

6.4.2(1)–6.4.2(4) [2]

Clause 5.4.2(7) [3]


Table 6.4. Effective width of footings according to JRA (2002)

Effective footing width

Bottom reinforcement of footing b ¼ B

Top reinforcement of footing b ¼ hcþ 1.5d � B

163

Clause 4.1.4(12) [3]

Eurocode 8, with the q factors of rows 3 and 4 in Table 5.1. In this case, locations of potentialplastic hinges should be detailed:

(a) the top of the pile to a distance of three pile diameters, D, from the underside of a pile capthat is restrained against rotation in a vertical plane

(b) a length 2D of the pile on each side of the location of maximum bending moment (asestimated taking into account the effective pile flexural stiffness, the lateral soil stiffnessand the rotational stiffness of the pile group at the pile cap) and of an interface betweensoil layers with a marked stiffness contrast.

Such detailing consists of confining reinforcement with a mechanical reinforcement ratio asspecified in Table 6.1 for piers of ductile behaviour. At locations of type (b) above, Eurocode 8does not reduce the required confinement steel to account for the experimentally found (Budeket al., 1997) beneficial effect of the soil pressures exerted on the compression side of the pile(Figure 6.11). In such locations it requires a pile vertical reinforcement of not less than thatplaced at the pile head.

Piles where plastic hinges are allowed should be verified for capacity design shear forces. If thebridge is designed using nonlinear analysis, the head of the pile should also be verified as aplastic hinge according to Section 6.7.2. In such a verification, the shear span, Ls, of the pilemay be taken as the distance from the point of contraflexure to the pile cap.

6.13. Liquefaction and lateral spreading6.13.1 IntroductionMost seismic codes and current practice are against placing foundations on liquefiable deposits.According to Part 5 of Eurocode 8:

if soils are found to be susceptible to liquefaction and the ensuing effects are deemed capable ofaffecting the load bearing capacity or the stability of the foundations, measures . . . shall betaken to ensure foundation stability.

When designing a foundation on a potentially liquefiable deposit three situations need to beanalysed:

g Stage 1: during which limited pore pressures have developed and the soil retains itsoriginal stiffness and strength; at this stage the forces applied to the foundation are mostlikely less than the maximum forces that would develop during the earthquake ifliquefaction has not occurred; the response of the bridge–foundation system is purelydynamic.


Figure 6.11. Confinement of the compression zone by the soil pressure for in-ground hinges

Soilpressure

Cracking and soil pressureMoment

164

g Stage 2: here, considerable excess pore pressures have developed (typically Du/s0v . 0.5,

where s0v is the vertical effective overburden) but no widespread liquefaction has yet

occurred; at this stage the soil stiffness is drastically reduced but the soil still preserves anon-zero shear stiffness, allowing the seismic waves to propagate through the soil andaffect the foundation; the response of the bridge–foundation system is still predominantlydynamic, but significant nonlinearities take place that reduce the input motion andtherefore the dynamic forces.

g Stage 3: this takes place towards the end or after the earthquake shaking has ended; fullliquefaction has developed, and lateral spreading of the liquefied soil may take place undercertain conditions, such as an inclination of the soil layers or of the ground surface,resulting in a quasi-static, gravity-induced loading to the foundation. The response isessentially static.

6.13.2 Shallow foundationsFor shallow (footing or mat) foundations, the upward flow of water towards the ground surfaceinduces a significant, if not a total, loss of soil resistance. Because the bearing layers are at theground surface, the strength reduction occurring in stage 2 or 3 causes loss of bearingcapacity, accompanied not only by vertical settlement but also, in some instances, significanttilt. The foundation movements that accompany full development of pore pressures areunpredictable. Therefore, countermeasures need to be implemented. There is no alternative toimproving the ground conditions.

6.13.3 Pile foundations6.13.3.1 Development of design approachesFor pile foundations the situation is somewhat different, as the load-resisting elements can beembedded below the potentially liquefiable strata. Therefore, it is feasible to design pile foun-dations to accommodate soil liquefaction. This has been done in practice at least to accommo-date stages 1 and 2. However, until the Kobe earthquake of 1995, it was not foreseen todesign pile foundations to accommodate lateral spreading (stage 3), and significant soil improve-ment needed to be implemented to protect the foundations of bridge piers. Soil improvement maybe expensive, especially when a wide area is affected by lateral spreading. For instance, the northapproach viaduct of the Rion–Antirrion Bridge in Greece is located in a zone where extensivelateral spreading was predicted under the design earthquake; a cost–benefit analysis showedthat designing the piles to resist the forces induced by the soil displacement was more cost-effective than improving the ground conditions. It is only recently that this type of analysis


Figure 6.12. Pile configuration due to liquefaction

Deformedshape of pile

Non-liquefiedlayer

Non-liquefiedlayer

Liquefiedlayer

Free field soildeflection

165

has been possible. Extensive back analyses of damage to pile foundations during the Kobe earth-quake resulted in rational design methodologies (Finn, 2005). The situation to be analysed isdepicted in Figure 6.12. After liquefaction, if the residual strength of the soil is less than thestatic shear stress caused by a sloping site or a free surface, such as a river bank, significantdown-slope movement may occur. The moving soil exerts damaging pressures against thepiles, especially if a non-liquefied layer rides on top of the liquefied layer. Basically, there aretwo different approaches to evaluate the forces: a force-based approach and a displacement-based approach.

6.13.3.2 Force-based approachA force-based analysis is recommended in JRA (2002) for pile foundations in liquefied soils:

g the non-liquefied surface layer is assumed to apply a passive pressure, qNL, on thefoundation


Figure 6.13. Idealisation of ground flow for the seismic design of bridge foundations

Non-liquefiablelayer

Non-liquefiable layer

Liquefiablelayer HL

HNL qNL

qL

Figure 6.14. Winkler spring model for lateral spreading analysis

Grounddisplacement

Liquefiedlayer

166

g the liquefied layer is taken to apply a pressure, qL, less than the equivalent hydrostaticpressure.

Figure 6.13 illustrates the forces acting on the foundation. It has been found that the pressureexerted by the liquefied layer may be taken as equal to 30% of the overburden pressure. Thesefindings have been confirmed by centrifuge tests (Dobry and Abdoun, 2001), which, inaddition, showed that the moments in the pile are dominated by the lateral pressure from thenon-liquefied layer. Therefore, to minimise the bending moments in the piles arising from thepressures exerted by the top non-liquefied soils, it is highly desirable to locate the pile capabove the ground surface, without contact with the in-place soils.

6.13.3.3 Displacement-based approachFollowing this approach, forces are not applied to the piles. Instead, free field displacements areimposed at the free ends of the springs in the Winkler model in Figure 6.14. The method requiresknowledge of the free field displacements, which may be estimated via predictor equationsdescribed in Section 3.4.5 of this Guide.

The displacement-based approach relies on two input data: the predictor equations giving theamplitude of lateral spreading and the values of the spring stiffness to be used in the pilemodel. In Japanese practice, the reduction in the spring stiffness for use in liquefiable soilsdepends on the factor of safety against liquefaction, FL. The recommended reduction factorsare given in Table 6.5 as a function of the product of the resistance to liquefaction, RL, and ofparameter cw, which for the reference seismic action defined in Eurocode 8 can be taken as

cw ¼ 1.0 for RL � 0.1, (D6.80a)

cw ¼ 3.3RLþ 0.67 for 0.1 , RL � 0.4 (D6.80b)

cw ¼ 2.0 for 0.4 , RL (D6.80c)

There is no commonly accepted practice in North America for the appropriate modelling ofdegraded spring stiffness. The basis of most analysis is a degraded form of the AmericanPetroleum Institute p–y curves (API, 1993). The practice is to multiply the p–y curves by auniform degradation coefficient b, which ranges from 0.3 to 0.1 (Finn, 2005).

There is a great uncertainty in the definition of the free field displacements used as input data tothe analysis. The predictor equations (Youd et al., 2002) are strongly empirical, and based on fewobservations. Consequently, the force-based approach is often preferred and recommended(JRA, 2002).

REFERENCES

API (1993) Recommended Practice for Planning, Designing and Constructing Fixed Offshore

Platforms. American Petroleum Institute, Washington, DC.

Biskinis DE and Fardis MN (2010) Flexure-controlled ultimate deformations of members with

continuous or lap-spliced bars. Structural Concrete 11(2): 93–108.


Table 6.5. Reduction coefficients for soil springs due to liquefaction according to JRA (2002)

Safety factor, FL Depth from ground

surface, x: m

Dynamic shear strength ratio, R ¼ cwRL

R � 0.3 R . 0.3

FL � 1/3 0 � x � 10 0 1/3

10, x � 20 1/3 1/3

1/3 , FL � 2/3 0 � x � 10 1/3 2/3

10, x � 20 2/3 2/3

2/3 , FL � 1 0 � x � 10 1/3 1

10, x � 20 1 1

167

Biskinis DE and Fardis MN (2012) Effective stiffness and cyclic ultimate deformation of circular

RC columns including effects of lap-splicing and FRP wrapping. 15th World Conference on

Earthquake Engineering, Lisbon.

Budek AM, Benzoni G and Priestley MJN (1997) Experimental Investigation of Ductility of In-

Ground Hinges in Solid and Hollow Precast Prestressed Piles. Division of Structural Engineering,

University of California, San Diego, CA. Report SSRP-97.

CEN (Comite Europeen de Normalisation) (2000) EN 1337-2:2000: Structural bearings – Part 2:


CEN (2002) EN 1990:2002: Eurocode: Basis of structural design (including Annex A2: Application

to bridges). CEN, Brussels.

CEN (2003a) EN 1991-1-5:2003: Eurocode 1: Actions on structures – Part 1–5: General actions –

Thermal actions. CEN, Brussels.

CEN (2003b) EN 1997-1:2003: Eurocode 7: Geotechnical design – Part 1: General rules. CEN,

Brussels.

CEN (2004a) EN 1998-1:2004: Eurocode 8 – Design of structures for earthquake resistance – Part

1: General rules, seismic actions and rules for buildings. CEN, Brussels.

CEN (2004b) EN 1992-1-1:2004: Eurocode 2: Design of concrete structure. Part 1: General rules

and rules for buildings. CEN, Brussels.

CEN (2004c) EN 1998-5:2004: Eurocode 8 – Design of structures for earthquake resistance – Part

5: Foundations, retaining structures, geotechnical aspects. CEN, Brussels.

CEN (2005a) EN 1998-2:2005: Eurocode 8 – Design of structures for earthquake resistance – Part

2: Bridges. CEN, Brussels.

CEN (2005b) EN 1992-2:2005: Eurocode 2: Design of concrete structure. Part 2: Bridges. CEN,

Brussels.

CEN (2005c) EN 1998-3:2005: Eurocode 8 – Design of structures for earthquake resistance – Part

3: Assessment and retrofitting of buildings. CEN, Brussels.

CEN (2005d) EN 1337-3:2005: Structural bearings – Part 3: Elastomeric bearings. CEN, Brussels.

CEN (2006) EN 1998-4:2006: Eurocode 8: Design of structures for earthquake resistance – Part 4:

Silos, tanks and pipelines. CEN, Brussels.

CEN (2009) EN 15129:2009: Antiseismic devices. CEN, Brussels.

Chatzigogos CT, Pecker A and Salencon J (2007) Seismic bearing capacity of circular footing on an

heterogeneous cohesive soil. Soils and Foundations 47(4): 783–797.

Constantinou MC, Kalpakidis I, Filiatrault A and Ecker Lay RA (2011) LRFD-based Analysis and

Design Procedures for Bridge Bearings and Seismic Isolators. Department of Civil, Structural and

Environmental Engineering, State University of New York, Buffalo, NY. Technical Report

MCEER-11-004:2011.

Dobry R and Abdoun T (2001) Recent studies of centrifuge modeling of liquefaction and its effect on

deep foundations. Proceedings of the 4th International Conference on Recent Advances in Geotechnical

Earthquake Engineering and Soil Dynamics (Prakash S (ed.)), San Diego, CA, pp. 26–31.

Elwi AA and Murray DW (1979) A 3D hypoelastic concrete constitutive relationship. Journal of

Engineering Mechanics Division of the ASCE 105(EM4): 623–641.

Fardis MN (2009) Seismic Design, Assessment and Retrofitting of Concrete Buildings (Based on EN-

Eurocode 8). Springer-Verlag, Dordrecht.

Finn WDL (2005) A study of piles during earthquakes: issues of design and analysis. The tenth

Mallet Milne Lecture. Bulletin of Earthquake Engineering 3(2): 141–234.

Gupta AK and Singh MP (1977) Design of column sections subjected to three components of

earthquake. Nuclear Engineering and Design 41: 129–133.

Idriss IM and Sun JI (1992) SHAKE 91: A Computer Program for Conducting Equivalent Linear

Seismic Response Analyses of Horizontally Layered soil Deposits. Program Modified Based

on the Original SHAKE Program Published in December 1972 by Schnabel, Lysmer and Seed.

Center of Geotechnical Modeling, Department of Civil Engineering, University of California,

Davis, CA.

JRA (2002) Design Specifications for Highway Bridges, Part V. Seismic Design. Japanese Road

Association, Tokyo.

Katsaras CP, Panagiotakos TB and Kolias B (2009) Effect of torsional stiffness of prestressed

concrete box girders and uplift of abutment bearings on seismic performance of bridges. Bulletin

of Earthquake Engineering 7(2): 363–376.

Mander JB, Priestley MJN and Park R (1988) Theoretical stress–strain model for confined

concrete. ASCE Journal of Structural Engineering 114(8): 1804–1826.


168

Paulay T and Priestley MJN (1992) Seismic Design of Reinforced Concrete and Masonry Buildings.

Wiley, New York.

Pecker A (1997) Analytical formulae for the seismic bearing capacity of shallow strip foundations.

In Seismic Behavior of Ground and Geotechnical Structures (Seco e Pinto PS (ed.)). Balkema,

Rotterdam.

Richart FE, Brandtzaeg A and Brown RL (1928) A Study of the Failure of Concrete Under

Combined Compressive Stresses. University of Illinois Engineering Experimental Station,

Champaign, IL. Bulletin 185.

Stanton JF, Roeder CW, Mackenzie-Helnwein P et al. (2008) Rotation Limits for Elastomeric

Bearings. Transportation Research Board, National Research Council, Washington, DC.

NCHRP Report 596.

Youd TL, Hansen CM and Bartlett SF (2002) Revised multilinear regression equations for

prediction of lateral spread displacement. Journal of Geotechnical and Geoenvironmental

Engineering 128(12): 1007–1017.


169




Chapter 7

Bridges with seismic isolation

7.1. IntroductionThe concept of seismic isolation is more familiar to designers of bridges than of buildings or otherstructures. Indeed, to avoid restraining thermal elongation or shortening and concrete shrinkage,bridge designers have since long adopted the practice of separating the deck from the abutmentsand some piers, by introducing longitudinally movable bearings (rollers, sliders or even elasto-meric bearings). From that point it takes only a short leap to a continuous isolating interfacebetween the deck and all of its supporting elements, in order to reduce the transmission ofseismic motion from the ground to the deck.

7.2. Objective, means, performance requirements and conceptualdesign

7.2.1 Objective and meansAs pointed out above, the aim of seismic isolation is to reduce the seismic inertia forces on themain mass of the bridge. To this end, an interface of isolating devices, termed the ‘isolation inter-face’, is introduced between the superstructure (normally the deck) and the substructure (i.e. theabutments and piers). Part 2 of Eurocode 8 (CEN, 2005a) uses the generic term ‘isolators’ forthe units of the isolation system. Isolators may offer one or more of the following capabilities:

1 Vertical supporting function with high stiffness, to ensure the safe transfer of the verticalreactions without deck deformation.

2 Transfer of horizontal forces with the effective displacement stiffness much less than thatof the corresponding element of the substructure (pier or abutment). These horizontalforces act usually – but not always – as elastic restoring forces for the isolation system.

3 Enhanced dissipation of seismic input energy through increased damping.

The reduction of the horizontal stiffness of the total system Keff (point 2 above) increases itsfundamental period Teff ¼ 2p

p(M/Keff) relative to that of the initial structure with horizontally

fixed connections, Tf. The acceleration spectrum in Figure 7.1(a) shows that this period shiftbrings about a substantial reduction in the spectral acceleration, especially when Teff � TC,where TC is the corner period between the constant pseudo-acceleration and the constantpseudo-velocity ranges of the elastic spectrum of Eqs (D3.8). However, the displacementspectrum in the same figure (derived from Eq. (D3.1)) clearly shows that the displacement isalso increased substantially.

An increase in damping to a value jeff above the default value of 5% of the elastic spectrum,owing to the additional damping offered by the isolation system, reduces both the displacements,as shown in Figure 7.1(b), and the forces, by the damping modification factor h ¼ p

[10/(5þ jeff)](used also in Eqs (D3.8) and (D5.48), but allowed to be as low as 0.40 for seismic isolation). Thecombination of increased damping and a period shift is a very efficient means to a substantialreduction in seismic forces without an inconvenient increase in displacements.

7.2.2 Performance requirements7.2.2.1 General requirementsBridges with seismic isolation must meet the two general performance requirements set by Part 2of Eurocode 8 for all bridges: the non-collapse and the limitation of damage requirements (seeSection 2.2 of this Designers’ Guide). However, specifically for bridges with seismic isolation,additional requirements are set, as detailed in the following sections.

Clause 7.1(1) [2]

Clauses 2.2, 7.3(1) [2]

171

Clauses 7.3(4),

7.6.2(1), 7.6.2(2) [2]

Clauses 7.3(2),

7.6.3 [2]

7.2.2.2 Increased reliability of the isolation systemAll components of the isolation system must have increased reliability relative to the other com-ponents of a bridge, both in terms of displacement capability and of force resistance. The enhancedreliability is deemed to be available if the displacement capacity of the isolators can accommodatea total design displacement in the seismic design situation, dEd, fromEq. (D6.36) in Section 6.8.1.2,computed with the design seismic displacement, dE, from Eq. (D5.48) in Section 5.9, multiplied bya safety factor gIS having a recommended value of 1.50. The resistance of the isolators and theiranchorage should accommodate the force developed at this increased displacement level.

The rationale behind the increased reliability demanded from the components of the isolationsystem can be seen by comparing the design parameters defining the corresponding ultimatelimit states (ULS):

g The ULS of ductile or limited ductile components is defined by resistances equal to thecharacteristic design values (i.e. lower 5% fractiles) divided by the material safety factors(with recommended values of 1.50 for concrete and 1.15 for steel). Therefore, thesecomponents possess substantial safety margins. Should they momentarily and locallyexceed their force limits, the effects will normally be reversible and not catastrophic, owingto their inherent ductility margins.

g The components of the isolation system do not enjoy the above favourable situation. TheirULS is practically defined by the geometric condition of the displacement capacity,without margins. Exceeding this capacity is irreversible and usually catastrophic for theisolator and probably for the bridge as well.

7.2.2.3 Behaviour of the substructureAs pointed out in Section 7.2.1, the main benefit of the isolation system is the high horizontalflexibility at the connection between the deck (which accounts for most of the system mass)and its supporting elements (piers and abutments). This produces a fundamental natural modeof vibration in the isolated horizontal direction with a substantially longer period than highermodes. The response in the isolated mode(s) dominates the seismic response of the bridge. Inaddition, the damping introduced by the isolation system acts directly on the displacementsand forces of the dominant mode(s) and is, therefore, quite efficient for dissipating energy andreducing the global seismic response. Consequently, the seismic response of the substructure issubstantially reduced, and may be undertaken via elastic or limited ductile behaviour.

One might consider attempting a further reduction in the force response by providing for aductile behaviour of the substructure, allowing plastic hinges to form there and exploitingtheir inelastic behaviour (‘partial isolation’, instead of ‘full’). However, this, in general, willnot be beneficial to the bridge for the following reasons, due to which Eurocode 8 does notallow partial isolation neither for buildings (Part 1 (CEN, 2004)) nor for bridges (Part 2):

g Partial isolation simply transfers part of the displacement demand from the isolationsystem to the substructure, to be accommodated there by inelastic deformation (i.e.


Figure 7.1. Means for achieving seismic isolation: (a) period shift; (b) increased energy dissipation

(damping)

(a) (b)

Td = a ( )

2

2π

d

dDisplacement spectrum

a

ag

a

To Tc Teff TeffTD

Period shift

d ξo to ξeff

172

damage) of certain elements. In addition to the damage, inelastic deformation entailslarger uncertainty than displacements within the (increased if necessary) capacity of theisolation system.

g The inelastic force–displacement relation of the entire system no longer has the simplelinear shape of a fully isolated structure, where the dominant linear branch dependsmainly on the post-elastic stiffness of the isolation system and to a minor degree on theelastic stiffness of the piers. This branch starts from the early yielding of the isolationsystem and runs straight to its displacement capacity. By contrast, in partial isolation anew linear branch starts at the yield displacement of each pier, with the last onecontinuing until the minimum ultimate displacement capacity among all piers or the firstdisplacement capacity is reached among the isolators (assumed here without restrainingdevices). The determination of such a complex force–displacement law requires fairlyaccurate evaluation of many parameters, and would in any case be subject to largeuncertainties.

g Transferring significant displacement demands from the isolation system to plastic hingesin the substructure reduces significantly the beneficial effect of the energy dissipated by theisolators, which is approximately proportional to the square of their displacement.

g An important reason for choosing to seismically isolate a structure is to drasticallyminimise or practically eliminate damage in the seismic design situation, by keeping thestructure within its elastic limits. This advantage is lost for partial isolation.

7.2.2.4 Consideration of the variability of the design properties of the isolationsystem

The response of an isolated bridge depends heavily on the properties of its isolation system,which in turn derive from the corresponding mechanical properties of the isolators. Part 2 ofEurocode 8 recognises the following categories of design properties:

g Nominal values of design properties, to be defined by the designer usually in the form of arange of values characterising an industrial product, such as an isolator. These propertiesshould be validated in general via special ‘prototype’ tests. Part 2 of Eurocode 8 gives inits informative Annex K a description of such tests and the relevant requirements,applicable if the provisions of EN 15129 (CEN, 2009) are not sufficient and there is norelevant European Technical Approval (ETA). ‘Prototype’ tests are not required fromnormal low-damping elastomeric bearings (LDEBs) per EN 1337-3 (CEN, 2005b) andordinary flat sliding bearings per EN 1337-2 (CEN, 2000), whose contributions to anincrease in the damping of the isolation system are ignored.

g The variation of these properties due to external factors, such as temperature,contamination, aging (including corrosion) or wear (expressed as cumulative travel),should be taken into account in the design, according to normative Annex J of Part 2 ofEurocode 8. The influence of these factors is either determined through special tests or, forcommon isolator types, estimated on the basis of modification factors (l factors) given ininformative Annex JJ of Part 2 of Eurocode 8. The following two groups of designproperties should be determined and used in separate analysis for each one:– lower-bound design properties (LBDP), usually giving the maximum displacements– upper-bound design properties (UBDP), usually resulting in the maximum forces.Part 2 of Eurocode 8 gives explicitly, usually in a simplified way, the UBDP values fornormal LDEBs and flat sliding bearings.

7.2.2.5 Restoring capability of isolation systemLarge residual displacements of an isolation system after an earthquake should be avoided,because an offset from the accumulation of such displacements will reduce the displacementcapacity of the system. So, an isolation system should have sufficient automatic restoringcapability (see Section 7.6).

7.2.2.6 Need for sufficient stiffness for non-seismic actions and service conditionsThe increased flexibility of the horizontal support of the deck due to the isolation system shouldnot impair the functions of the bridge under service, nor lead to violation of any horizontaldisplacement limits set by other Eurocodes for service or ULSs.

Clause 7.5.2.4 [2]

Annex J,

Annex JJ [2]

Clause 7.7.1 [2]

Clause 7.7.2 [2]

Chapter 7. Bridges with seismic isolation

173

Clauses 3.2.2.2,

3.2.5(8) [1]

Clauses 3.2.2.3(1),

7.4 [2]

Clause 7.5.2.3.2 [2]

7.2.3 Conceptual design remarksThe efficiency of a seismic isolation solution is measured by the seismic force reduction it entailsin the substructure of the bridge, namely the ratio VI,i/Vf,i, where Vf,i is the seismic shear force ofelement i of the ‘fixed’ system and VI,i is the corresponding force of the isolated system, whilekeeping the displacements within manageable limits.

As far as the local soil conditions are concerned, the period shift effect referred to in Section 7.2.1is most efficient if the corner period TC between the constant pseudo-acceleration and theconstant pseudo-velocity ranges of the spectrum is short. Seismic isolation of a bridgebecomes less efficient as the ground type changes from A to B, D, C or E (see Table 3.3 inChapter 3). In addition, a longer corner period TD between the constant pseudo-velocity andthe constant displacement ranges of the spectrum adversely influences the efficiency of the iso-lation. Recall that Part 1 of Eurocode 8 recommends TD ¼ 2 s for all ground types; but see,however, Section 7.3 below.

The effect of the period shift depends also on the value of the fundamental period of the bridgeconsidered fixed, Tf. Bridges with Tf longer than TC and closer to TD do not need seismic iso-lation, since the forces induced by the seismic action are already low. Examples are bridges ontall piers, cable-stayed bridges with the deck suspended from the pylon head, etc. Such bridgesmay need supplemental damping to reduce seismic displacements and sacrificial horizontal tiesystems to control displacements under non-seismic horizontal actions, mainly wind.

A very important favourable side-effect of seismic isolation in bridges is that it very efficientlymakes uniform the horizontal seismic forces among piers and abutments of quite different stiff-ness. Similarly, it reduces the restraint of the deck due to its imposed deformations. This may bequite important for long continuous post-tensioned concrete decks, which are sensitive to alter-nating thermal actions. Similar advantages are offered against imposed horizontal displacementsof the foundation, as long as these are small enough to be neglected as far as the displacementcapacity of the isolators is concerned.

7.3. Design seismic actionAccording to Part 2 of Eurocode 8, the design seismic action of isolated bridges should not betaken as less than specified for non-isolated bridges. Consequently, whatever has been said inChapter 3 regarding elastic spectra, the time history representation of the seismic action, near-source effects and spatial distribution applies to isolated bridges as well.

Seismically isolated bridges have by design a long fundamental period, and are therefore sensitiveto ground motions rich in low frequencies. Specifically, elastic spectra with long values of TC andTD are very demanding for bridges with seismic isolation. In this respect, a note in Part 2 ofEurocode 8 allows a value of TD longer than the recommended value of 2 s to be adopted(intended for earthquakes of magnitude up to 6.5).

Spectral ordinates may increase in the long-period range of the spectrum, and TD may shift upowing to near-source effects in the vicinity of the seismotectonic fault rupture. The rules inPart 2 of Eurocode 8 highlighted in Section 3.1.2.6 of this Guide cover the special needs ofbridges with seismic isolation in this respect.

7.4. Behaviour families of the most common isolators7.4.1 Bilinear hysteretic behaviour7.4.1.1 General featuresTo this family belong many common types of isolators. The most common of them are dealt within Sections 7.4.1.2 to 7.4.1.5. Isolators of this family react with a force that depends on therelative displacement between the isolator interface to the deck and its interface to the supportingelement (pier or abutment). The shape of the cyclic force–displacement diagram is very close tobilinear: unloading branches are parallel to the initial elastic branch at a stiffness (slope) Ke; thepost-yield branches have much lower stiffness (slope) Kp. Figure 7.2 shows a cycle from anextreme point (Fmax, dbd) to its opposite (�Fmax, �dbd) and back to (Fmax, dbd). The ratio

Keff ¼ Fmax/dbd (D7.1)


174

termed the ‘effective’ stiffness, is the stiffness of an elastic system reacting with the same forceFmax to the peak displacement dbd. To provide the period shift mentioned in Section 7.2, the stiff-ness of the isolation system, which is equal to the sum

PKeff of all isolators, needs be substan-

tially lower than the stiffness of the ‘fixed’ superstructure. The area enclosed by the hysteresisloop, Ed, is the dissipated energy by the isolator at the peak displacement dbd.

The basic parameters of the bilinear loop are:

g dy and Fy: displacement and force at yield, respectivelyg F0: force at zero displacement (often termed ‘characteristic strength’ in the USA),

F0 ¼ Fy – Kpdy (D7.2a)

g Fmax: force at maximum displacement dbd.

As Fmax and dbd both depend also on the seismic demand, the behaviour of a bilinear system isintrinsically determined by three parameters: F0, the elastic stiffness Ke ¼ Fy/dy and the post-elastic stiffness Kp (see Figure 7.2). Equation (D7.2a) can be used to express Fy and dy asfunctions of these parameters:

Fy ¼ F0/(1�Kp/Ke) (D7.2b)

dy ¼ (F0/Ke)/(1�Kp/Ke) (D7.2c)

Other useful relations are

Kp ¼ (Fmax� Fy)/(dbd� dy) (D7.3)

Ed ¼ 4F0(dbd� dy) ¼ 4(Fydbd� Fmaxdy) (D7.4)

7.4.1.2 Low-damping elastomeric bearingsLDEBs (depicted in Figure 7.3) offer the possibility of lengthening substantially the fundamentalperiod of the system, alongside a practical decoupling of the horizontal response from the influ-ence of a large stiffness contrast between piers or between them and the abutments. Because in theearly years of application of the pre-standard (ENV) version of Eurocode 8 to bridges, elasto-meric bearings were the only devices of European production, widely used, relatively inexpensiveand covered by a European standard that could be used as isolator bearings, Part 2 of Eurocode 8allows the use of normal LDEBsmanufactured per EN 1337-3 (CEN, 2005b) directly as isolators.Moreover, such bearings have very narrow hysteresis loops and do not lend themselves toincreasing the damping of the bridge above the default value of 5%. Therefore, the usualelastic response spectrum analysis may be applied with the low shear stiffness of the bearings.Note, however, that when an elastomer with very low values of the shear modulus is used(G, 0.6 MPa), ‘scragging’ may have to be accounted for in determining the design value of G(see Section 7.4.1.3).

The conceptual rules and verifications of these bearings are dealt with in detail in Chapter 6 ofthis Guide and their modelling and analysis in Chapter 5. In addition, all the basic principles

Clause 7.5.2.3.3(2) [2]


Figure 7.2. Force–displacement loop of a bilinear hysteretic system

KeKeff

Kp

Fmax

Fy

F0

F

ddd

dr = F0/Kp

ED

dy

175

Clause 7.5.2.3.3(3) [2]

Clause 7.5.2.3.3(9) [2]

and general rules for seismic isolation relating to the present chapter are consistently applicableto them as well. The following points are noted specifically for this type of isolator:

g The inability of these bearings to offer additional damping increases the systemdisplacements to the point that they may prove prohibitive for the shear strains of thebearings and/or the response of the bridge (see Section 6.10.4.4), unless the bearings arecombined with viscous dampers (see Section 7.4.2). This drawback, alongside the fact thatnormal LDEBs per EN 1337 that are included in manufacturers’ lists of standard bearingsare in essence not specifically intended for seismic use, makes them inadequate for use incases of truly high seismicity or adverse soil conditions.

g Although isolation with these LDEBs is feasible only in not very demanding seismicconditions and cannot be considered as an optimal solution, it is certainly an inexpensiveand simple one. In this respect, damage to some of these bearings in a strong earthquakemay even be considered as acceptable, as long as it is feasible to replace them.

7.4.1.3 High-damping elastomeric bearingsThe damping capability of elastomeric bearings can be increased by replacing the pure elasto-meric material by special mixtures of an elastomer with special aggregates. The width of thehysteresis loops increases substantially, and a damping ratio appreciably higher than 5% canbe reached. An isolation system consisting exclusively of such high-damping elastomericbearings (HDEBs) may be analysed with the multimodal response spectrum method, if the effec-tive damping ratio of the isolation modes, jeff, is determined from the dissipated energy per cycle,Ed, as measured by the appropriate tests (see EN 15129 (CEN, 2009)).

Note that, after even a single cycle of shear deformation to the peak shear strain, this isolator typepresents substantial softening (reduction of shear modulus, G) in subsequent cycles (Figure 7.4).In fact, the virgin value of G is recovered within a few months. This effect, usually termed‘scragging’, may lead to erroneous determination of the design value of G from tests onscragged bearings. To avoid this, Part 2 of Eurocode 8 requires that the representative valueof G is determined from tests of unscragged specimens as the average in the first three cyclesat peak shear strain.

7.4.1.4 Lead–rubber bearingsThe most efficient way to achieve substantial energy dissipation capacity of the elastomericbearing is by replacing its core by lead (Figure 7.5), which yields early, offering the desiredenergy dissipation. The shear stiffness of the lead core and the rubber annulus are

KL ¼ GLAL/h (D7.5a)

KR ¼ GRAR/hR (D7.5b)


Figure 7.3. Elastomeric bearings: (a) low-damping elastomeric bearing (LDEB); (b) high-damping

elastomeric bearing (HDEB)

176

where GL, AL and h are the shear modulus, the plan area and the thickness of the lead core,respectively, and GR, AR and hR are those of the rubber annulus, respectively. If FLy is theyield force of the lead core, the elastic stiffness of the lead–rubber bearing (LRB) is

Ke ¼ KLþKR (D7.6)

and its yield force is

Fy ¼ FLy(1þKR/KL) (D7.7)

A wide variety of basic parameters of a bilinear system can be achieved by simply varying thegeometry of the lead core and the rubber annulus of the LRB. Many issues concerning thedetailed design of such bearings are given in Constantinou et al. (2011). The values to beassumed for the material parameters should be in accordance to tests per EN 15129 or to anapplicable European Technical Approval (ETA). UBDP and LBDP bounds should be con-sidered in the design according to Annex J in Part 2 of Eurocode 8. Circular bearings have thesame properties in any horizontal direction, and as such are preferred to square bearings.

7.4.1.5 Steel elastoplastic energy dissipatorsThese devices are not bearings, as they are not intended to bear gravity loads. Their objective is todissipate energy via plastic deformation of ductile steel elements yielding under the horizontalseismic forces. Such a device is depicted in Figure 7.6(a), while Figure 7.6(b) shows typicalforce–displacement loops. The restoring capability of systems with such devices may need tobe increased by combining them with other components (e.g. normal LDEBs); alternatively,the displacement capacity of the system may need to be increased.

7.4.2 Velocity-dependent devicesA velocity-dependent device is connected to the deck and a supporting element (pier orabutment) at two points, defining its direction of excitation and reaction. It reacts to therelative motion of the connection points with a force F, which is co-linear to the relativevelocity of these points, v, but in the opposite direction, and has the magnitude

F ¼ Cvab (D7.8)

Exponent ab assumes values in the range 1.00–0.01, andC is a constant of the device measured bytesting, in units kN/(m/s)ab (Figures 7.7 and 7.8).

Whenever the displacement d becomes a maximum (or absolutely minimum) in a cycle of motion,and starts decreasing (or increasing), the velocity v passes through zero, as the direction ofmotion is reversed. Consequently, the force F also becomes zero. Therefore, for these points

Clause 7.5.2.3.4 [2]


Figure 7.4. Force–displacement loops of (virgin) HDEBs

–300 –200 –100 0 100 200 300

Shear strain: %

Shea

r fo

rce:

kN

50

40

30

20

10

0

–10

–20

–30

–40

–50

177

of all cycles, including the maximum, Keff from Eq. (D7.1) is zero. In other words, velocity-dependent devices do not contribute to the effective stiffness of the isolation system.

The force–displacement relation of velocity-dependent devices can be determined, if the displace-ment–time relation in a motion cycle is assumed. It is convenient to assume that the displacementis a sinusoidal function of time t within a cycle of motion with period Teff or angular velocityv ¼ 2p/Teff. Then (see Figure 7.7),

db ¼ dbd sin(vt) (D7.9)

where dbd is the maximum displacement. The velocity v is the time derivative of d:

v ¼ vdbd cos(vt) (D7.10)


Figure 7.5. (a) Lead–rubber bearings and (b) their force–displacement loops at different temperatures

178

and the force from Eq. (D7.8) becomes

F ¼ Cvab ¼ Fmax[cos(vt)]ab (D7.11a)

with

Fmax ¼ C(dbdv)ab (D7.11b)

The dissipated energy per cycle at maximum displacement dbd is

ED ¼ l(ab)C(dbdv)abdbd (D7.12)

where the coefficient l(ab) is determined, using the Gamma function G, as

lðabÞ ¼ 22þabG2ð1þ 0:5abÞGð2þ abÞ

ðD7:13Þ

The values of l(ab) are given in Table 7.1 for a wide range of ab values.


Figure 7.6. (a) Steel elastoplastic energy dissipators and (b) typical force–displacement loops

Figure 7.7. Force–displacement loop of velocity dependent devices

Fmax

F

F

ddED

d d

αb = 1

αb < 1

179

Note that the maximum damping force Fmax, as well as the dissipated energy per cycle, Ed,depend on v (i.e. on the period of the motion, Teff). The dependence on v is linear and strongif ab ¼ 1.0 (linear damping) or 1.0 and weaker for small values of ab (e.g. 0.01, seeFigure 7.8(b)). Note also that, for any value of ab, the damper force is zero at the point ofmaximum displacement, and becomes maximum at zero displacement.

7.4.3 Frictional devices7.4.3.1 Flat sliding bearingsFriction is a feature common to all sliding bearings. The Coulomb friction law gives the simplerelation between friction force, Ffr, caused by the vertical load, NSd, sliding on a horizontal flatsliding surface (Figure 7.9) with velocity _db (where _db is the time derivative of the relative displa-cement db, i.e. the velocity):

Ffr ¼ mfrNSd sign( _db) (D7.14)

where mfr is the friction coefficient and sign( _db) is the sign of the velocity vector, independent ofthe magnitude of velocity, depending only on the direction of the motion. However, the valueof mfr depends on the kind and history of the motion. Thus, one may distinguish between:

g the dynamic coefficient of friction, md, applicable for seismic motions (i.e. at relatively highvelocities)

g the static coefficient of friction ms, representative of very slow quasi-static motiong breakaway friction, for the estimation of the friction force necessary to start sliding after a

longer period of immobility and of load dwell.


Table 7.1. Values of l(a)

a 0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00

l(a) 3.988 3.882 3.774 3.675 3.582 3.496 3.416 3.341 3.270 3.204 p 2.876 2.667

Figure 7.8. Velocity-dependent device: (a) a hydraulic viscous damper and (b) its force–velocity relation

(manufacturer: Maurer-Soehne)

Piston

3000

2500

2000

1500

1000

500

0

Damper MHD of 3000 kN

F = C × V α

Velocity: m/s

(b)(a)

Forc

e: k

N

Steel cylinder

Silicon oil

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Figure 7.9. (a) A flat sliding bearing and (b) its force-displacement loop

dbd

(b)(a)

ED

F

F0 = Fmax = µdNSdNSd

Fmax

180

During seismic motion cycles the dynamic value of the friction coefficient mfr ¼ md is applicableand depends on several factors, such as the composition of the sliding surfaces, the use or not oflubrication, the bearing pressure on the sliding surface, the velocity of sliding, etc.

The friction force remains constant (i.e. Ffr ¼ Fmax), therefore Eq. (D7.14) is written as (see alsoFigure 7.9)

Fmax ¼ mdNSd sign( _db) (D7.15)

The dissipated energy per cycle of motion at maximum displacement dbd is

ED ¼ 4mdNSddbd (D7.16)

The above relations apply to an isolator once relative sliding at the sliding surface has started;that is, when its friction capacity has been reached. This is practically always the case in theseismic design situation, and can be simply verified by comparing the peak displacement of thedeck with that of the top of the supporting element. For other horizontal actions (wind orbraking load) or imposed deck deformations, certain sliding isolators, especially those supportedon flexible piers, may remain inactive if the corresponding connection force demand (with theconnection assumed to be fixed) does not exceed the isolator friction capacity.

To use flat sliding bearings as energy dissipators in the bridge seismic isolation system, they shouldpresent a reliable lower bound of energy dissipated per cycle, Ed, and, hence, a reliable lowerbound of the friction coefficient md. Normal flat sliding bearings conforming to EN 1337-2,with lubricated PTFE sliding surfaces, may be considered to have a controlled upper bound ofmd, but no reliable lower bound under conditions of seismic motion. Therefore, according toPart 2 of Eurocode 8, these bearings may be used as isolators, but not as energy dissipators.

Clearly, flat sliding bearings have no self-restoring capability. So, they may be used as com-ponents of an isolation system only in combination with other devices providing the requiredrestoring capability (e.g. elastomeric bearings).

7.4.3.2 Bearings with one spherical sliding surfaceDevices with one spherical sliding surface consist of an articulated slider coated with a specialPTFE material having controlled low friction (Figure 7.10). Sliding occurs on a concave stainlesssteel surface with a radius of curvature of the order of 2 m. The coefficient of friction at the slidinginterface is very low, of the order of 0.05–0.10; it can be reduced even more through lubrication.The combination of low friction and a restoring force due to the concave surface gives anapproximately bilinear hysteretic behaviour of the bearing, with inherent re-centring capability(see Figure 7.10). The behaviour consists of the combined effect of:

g A hysteretic frictional component, which provides the force at zero displacement:

F0 ¼ mdNsd (D7.17)

where Nsd is the normal force through the device.g A linear-elastic component that provides a restoring force corresponding to post-yield

stiffness:

Kp ¼Nsd

Rb

ðD7:18Þ

where Rb is the radius of the spherical sliding surface.g The dissipated energy per cycle at the cyclic displacement dbd is as in Eq. (D7.16)

(i.e. ED ¼ 4mdNsddbd).

The maximum force, Fmax, and the effective stiffness, Keff, at displacement dbd are

Fmax ¼Nsd

Rb

dbd þ mdNsd sign _ddbd

� � ðD7:19Þ


181

Keff ¼Nsd

Rb

þ mdNsd

dbdðD7:20Þ

Note that Rb is in fact the radius of the spherical surface on which the pivot point is moving (seeFigure 7.10). If this point lies below the sliding surface at a distance h and that surface is concavewith radius Rs, then Rb ¼ Rsþ h; if it is above (at distance �h), then Rb ¼ Rs� h.

The ‘elastic’ stiffness of the equivalent bilinear system Ke is theoretically infinite. Numericalinstabilities may be avoided by using Ke ¼ F0/dy, with a small value of dy ¼ 0.1 mm.

The following special features of these isolators are worth noting:

g The reaction, Fmax,i, of each isolator i to mass proportional forces acting on the deck is ahorizontal force parallel to the direction of motion and proportional to the correspondingvertical reaction, Nsd,i. As the forces Nsd,i are in equilibrium with the total weight of thedeck (including overturning action effects in the direction of the motion), the resultant ofthe horizontal reactions Fmax,i of all isolators passes through the horizontal projection ofthe centre of mass of the deck. Consequently, the seismic motion causes no rotation abouta vertical axis, as long as the deck deformation and the flexibility of the piers remain small,as is the usual case.

g Since the resultant horizontal reaction force of these devices is proportional to the mass ofthe deck, as is also its inertia force, the dynamic equations of motion become independentof mass, as long as the mass of the piers may be neglected. Such a system indeed behavesas a pendulum.

7.4.3.3 Bearings with multiple spherical sliding surfacesThese isolators consist of two external (main) concave plates, within which slide:

g either a spherical bearing with convex external plate surfaces (double spherical slider, seeFigure 7.11 for a schematic) or


Figure 7.10. (a) An isolator with one spherical sliding surface and (b) its force–displacement loop

F0 = µdNsd Kp = Nsd/Rb

Force F

Displacement d

Force-displacement loopfor seismic analysis

Idealised monotonic responsefor static analysis

Area enclosed inloop = dissipatedenergy per cycle Ed

dbd

dbd

Rb Nsd

Fmax

Fmax

Pivot point(a)

(b)

Ed

182

g two intermediate convex–concave sliding plates with a central solid convex slider (triplespherical isolator, shown schematically in Figure 7.12).

The behaviour of the typical double spherical slider has no significant differences from the simpleone, except that

Rb ¼ 2(R� h) (D7.21)

The displacement capacity, dbd,max, increases to

dlim ¼ 2d(1� h/R) (D7.22)

The behaviour of the triple spherical isolator is more complex. The following concise descriptioncorresponds to the ‘triple friction pendulum’ bearing, manufactured by EPS. Detailed design pro-cedures for such bearings are given in Constantinou et al. (2011). The cross-section of the generictype is depicted in Figure 7.12, alongside the usual assumptions for typical bearings. Owing to thedistances hi between the sliding surfaces and the corresponding pivot points, the followingeffective value of curvature radii should be used:

Reff,i ¼ Ri� hi (for i ¼ 1–4) (D7.23a)

di� ¼ di(Reff,i/Ri) (for i ¼ 1–4) (D7.23b)

The following parameters are determined:

d �y ¼ 2(m1� m2)Reff,2 (D7.24)

m ¼ m1� (m1� m2)(Reff,2/Reff,1) (D7.25)

dlim ¼ d �y þ 2d �

1 (D7.26)


Figure 7.11. Isolator with a double spherical sliding surface

Typically:R1 = R2 = R; h1 = h2 = h; d1 = d2 = d; µ1 = µ2 = µd

R2, µ2

R1, µ1

d2

d1

h2

h1

Figure 7.12. Triple friction pendulum bearing: geometric and frictional properties

Typically:R1 = R4, R2 = R3; h1 = h4, h2 = h3; d1 = d4, d2 = d3; µ1 = µ4 = µd, µ2 = µ3 < µd

R4, µ4 R3, µ3

R1, µ1 R2, µ2

d3

d2

h3

h2

h4

h1

d4

d1

Rigid slider

Slide plates

183

Assuming that the maximum displacement demand, dbd, remains less than dlim (dbd � dlim), theforce–displacement loops have the trilinear form depicted in Figure 7.13. The energy dissipated is

Ed ¼ 4[mdbd� 4d �y (m� m2)]Nsd (D7.27)

As a conservative approximation, one may consider the behaviour as bilinear with

dy ¼ d �y (D7.28a)

F0 ¼ mNsd (D7.28b)

Kp ¼ Nsd/2Reff,1 (D7.28c)

Ke ¼ m1Nsd/d�y (D7.28d)

The behaviour regimes of the device are illustrated in Figure 7.14:

g Regimes I and II for �d �y � d � d �

y (see Figure 7.13) correspond to a force below theminimum friction m2 or to motion within part of regions d2 and d3 (see Figure 7.12); theresponse lies in them during minor seismic events.

g Regimes III and IV for d �y � d � dlim (see Figure 7.13) correspond to motion within the

main sliding regions d1 and d4 (see Figure 7.12); the response to the design seismic events


Figure 7.13. Design force–displacement loop of a typical triple pendulum bearing

Force F

Displacement d

µNsd

dbd

d*y = 2(µ1 – µ2)Reff,2

2dy

Nsd/2Reff,2

Nsd/2Reff,1

2µ2Nsd

µ2Nsd

µ1Nsd

Figure 7.14. Triple friction pendulum bearing: behaviour regimes

Total displacement

Regime V

Regime III

Regime II

Regime I

Regime IV

Hor

izon

tal f

orce

dlim

F

u

184

lies in these regimes. (The second slightly steeper post-yield line reflects the general casewhen R4 , R1.)

g Regime V reflects an additional displacement margin beyond the design displacementcapacity dlim of the device; it corresponds to a motion within the available remaining partin regions d2 and d3.

7.5. Analysis methods7.5.1 Analysis methods and their fields of applicationPart 2 of Eurocode 8 provides for the following analysis methods of bridges with seismicisolation:

g nonlinear time-history analysis (NLTH)g the fundamental mode method (FMM)g multimode spectrum analysis (MMS).

The first one is the most general analysis method, and may be applied to all cases. The con-ditions under which the two spectral methods may be applied in a stand-alone way are givenin Table 7.2.

7.5.2 Nonlinear time-history analysisAs the superstructure and the substructure remain essentially elastic, their stiffness is in generalestablished considering them as uncracked; the nonlinearity of the model should reflect thenonlinear properties of the isolation system, including:

g the interaction of the simultaneous response in at least the two horizontal directionsg the effect of overturning forces.

Regarding specifically the contribution of the vertical seismic component, alternatively to using asimultaneous vertical acceleration time history, Part 2 of Eurocode 8 allows its estimation via thelinear response spectrum method and combination of the action effects with those of the hori-zontal components via Eq. (D5.2c).

Note that the energy dissipation offered by the components of the isolation system, either ashysteretic damping (through bilinear hysteretic or friction devices) or as viscous (via hydraulicdampers), is directly accounted for in a nonlinear analysis. It is therefore important to ensurethat the input data given for the damping matrix of the system, intended to represent thestandard structural damping of the substructure and superstructure elements, are not interpretedby the computational algorithm as applicable to the isolated modes as well (i.e. for the modeswith the longest period). It should also be taken into account that the main part of the seismicdeformation energy in the structure refers to these modes. On the other hand, as always innonlinear analysis, it is desirable to use increased damping to filter out possibly inaccurateresults due to very short period modes (i.e. those with a period less than the time-step of thedirect time integration – which is usually of the order of 0.01 s). Usually, Raleigh damping isapplied to meet such limitations, with the damping matrix C derived as a linear combinationof the stiffness matrix K and the mass matrix M:

C ¼ aKþ bM (D7.29)

Then, for any natural period T,

j ¼ ap/Tþ bT/4p (D7.30)

Clause 7.5.3 [2]

Clauses 7.5.6(1),

7.5.7(1) [2]


Table 7.2. Conditions for the application of spectral methods

Spectrum method Conditions for applicability

Distance from active fault Ground type Effective damping, jeff

Fundamental mode method �10 km A, B or C �0.30Multimodal analysis No limit A, B or C �0.30

185

Clause 7.5.4 [2]

The above mentioned requirements can be satisfied by the selection

b ¼ 0 a ¼ jT/p ¼ 0.10 � 0.05/p ¼ 0.00159 (D7.31)

which gives (see also Figure 8.52 in Chapter 8)

g for T � 1.5 s: j � 0.0032g for T � 0.05 s: j � 0.10.

7.5.3 Fundamental mode method of analysisIn the fundamental mode method (Figure 7.15) the entire structure is taken as a single-degree-of-freedom (SDoF) system according to the rigid deck model of Sections 5.6.3 and 5.6.4. In its basicform it considers only the stiffness and dissipation energy properties of the isolation system inone horizontal direction at a time. Despite its simplified form, the method can capture themost significant aspects of the structural response. Certain corrections and adaptations arediscussed in Section 7.5.5.

The method estimates the maximum seismic displacement dcd of the system iteratively, using twobasic approximation tools:

g The effective stiffness of the system, Keff. The nonlinear force–displacement relation of theisolation (the system is shown schematically in Figure 7.15(a)) is obtained by summing upthe force contributions of all its components, i, corresponding to an assumed displacementvalue dcd,a. The effective stiffness is the secant stiffness at the point of maximumdisplacement, dcd,a:

Keff ¼ Fmax/dcd,a ¼P

Fmax,i/dcd,a ¼P

Keff,i (D7.32)

Any velocity-dependent devices do not contribute to Keff (see Section 7.4.2).g The effective damping of the system, jeff: This is the equivalent viscous damping

corresponding to the sum of dissipated energies, ED,i, of all components of the isolationsystem at the cycle of peak displacement dcd ¼ dcd,a. It is calculated as

jeff ¼1

2p

PED;i

Keffd2cd

� �ðD7:33Þ

The values of ED,i are known isolator properties in terms of the isolator displacement.

The deck seismic displacement is taken as that of a linear SDoF system having a mass equal tothat of the deck, Md, stiffness Keff and an equivalent viscous damping ratio jeff. Its period is

Teff ¼ 2p

ffiffiffiffiffiMd

Keff

sðD7:34Þ


Figure 7.15. Fundamental mode method: (a) force–displacement; (b) displacement spectra

(a) (b)

dcd.a TC Teff

Keff

TD

dcd.r

ξ = 0.05

ξ = ξeffFmax

186

The peak displacement demand, dcd,r, can be derived as shown schematically in Figure 7.15(b),from the displacement response spectrum that results from multiplication of the elastic accelera-tion spectrum for j ¼ 0.05 by the damping modification factor, heff, corresponding to theestimated value of jeff:

heff ¼ffiffiffiffiffiffiffiffiffiffiffiffi

0:10

0:05þ jeff

sðD7:35Þ

After a few iterations, the assumed value dcd,a converges to the seismic demand dcd,r.

This simple approach gives good estimates of the maximum seismic forces, only if they occursimultaneously with the estimated maximum displacement. This holds in general if thereacting force of all isolators of the system is either constant (friction) or depends only on thedisplacement (as in elastic or hysteretic isolators). Then,

Fmax,i ¼ Keff,idcd (D7.36)

When the isolation system also contains devices with a reacting force proportional to velocity(hydraulic viscous dampers), their maximum force does not take place at the instant of peakdisplacement. In such cases, a more complex procedure is required to estimate the peak forces(see Section 7.5.5.5).

7.5.4 Multimode spectrum analysisThis is an approximate method, combining via the usual combination rules of Section 5.5.4:

1 the fundamental mode method described above for the isolation modes,2 the multimodal response spectrum analysis with the default damping ratio of j ¼ 0.05 for

the higher modes.

As this is an approximation, it is preferable to apply it separately in the two horizontal directions(longitudinal and transverse) and to combine the results via Eqs (D5.2). The same applies for thecombination with the computed effects of the vertical component.

Step 2 (a multimodal analysis for the upper modes) is applied as follows in each horizontaldirection:

g All isolators i are modelled with their effective stiffness, Keff,i, as derived from thefundamental mode method in the direction considered.

g The substructure and superstructure elements are modelled with uncracked stiffness and asufficient number of intermediate nodes to capture the effects of important higher modes.

g The modal damping ratio is taken as equal to the default value j ¼ 0.05 for all modes witha period T , 0.8Teff, where Teff is the effective period of the isolated mode from thefundamental mode method; the effective damping of that mode, jeff, is used for all modeswith periods longer than 0.8Teff.

Application of this method is meaningful only to capture possibly significant contributions ofhigher modes. The limitation pointed out in Section 7.5.3 regarding the inability to estimatewell the maximum forces in combinations of viscous dampers and hysteretic isolators still holds.

7.5.5 Effects of isolation properties on the approximation of the fundamentalmode method results

7.5.5.1 IntroductionPart 2 of Eurocode 8 and its counterparts (e.g. AASHTO, 2010) use the results of the funda-mental mode method as lower limits for the results of the multimode spectral method or eventhe nonlinear time-history approach, as a sort of self-checking by the designer. It is thereforeimportant to control the approximation it offers and to improve it wherever possible.

The following deals briefly with the impact of Keff, jeff and bidirectional excitation on theapproximation of the fundamental mode method and offers suggestions for its improvement

Clause 7.5.6 [2]


187

Clause 7.5.4(3) [2]

where possible. It also describes methodologies for estimating the maximum forces in isolationsystems using viscous dampers

7.5.5.2 Effective stiffness Keff

The basic form of the fundamental mode method accounts only for the stiffness of the isolators,neglecting the flexibility of the piers. Although this assumption is acceptable in most usual cases,it is not for high and flexible piers.

The flexibility of an abutment and its foundation is usually negligible. The composite stiffness ofpier i may be obtained from Eq. (D2.10), rewritten here as (Figure 7.16)

1

Keff;i

¼ 1

Kbi

þ 1

Kti

þ 1

Ksi

þH2i

Kfi

ðD7:37Þ

where Kbi, Kti and Ksi are the translational stiffness of the bearing, the foundation and the piershaft, respectively, Kfi is the foundation rotational stiffness and Hi the height of the pier shaft.The last three summands are lumped into the pier flexibility 1/KPi. Then,

Keff,i ¼ KbiKPi/(KbiþKPi) (D7.38)

If Kbi/KPi is small, Eq. (D7.38) may be written as

Keff,i ¼ Kbi/(1þKbi/KPi) � Kbi(1�Kbi/KPi) (D7.39)

If the isolators over all piers and abutments have the same stiffness, Kbi ¼ Kb, then,

Keff ¼P

Keff,i ¼P

Kb�Kb2 P (1/KPi) (D7.40)

This correction of Eq. (D7.32) – separately in each horizontal direction – lends itself to handcalculation.

7.5.5.3 Damping ratio jeffOnly few of the force–displacement loops from an analysis of either an isolated or a non-isolatedductile bridge reach a displacement near the peak displacement demand among these loops, inthe response to an accelerogram used for the design. Most other loops have substantiallysmaller peak displacements. However, the value of jeff corresponding to the loop with themaximum displacement is used in the fundamental mode method for the estimation of thepeak displacement demand. It is therefore informative to study the variation of jeff as afunction of the displacement in a given loop.

Figure 7.17 shows the variation of jeff in a bilinear hysteretic system as a function of thedisplacement normalised by the yield displacement (the displacement ductility ratio m ¼ d/dy)and the hardening ratio, l ¼ Kp/Ke. Figures 7.18 and 7.19 depict the evolution of the ratio


Figure 7.16. Composite pier stiffness (Part 2 of Eurocode 8)

Deck

Isolator i

Pier i

did

dbi,d

Hi

Fi

Fi /KbiFi /KsiFi /Kti

FiHi2/Kri

188

jeff/jeff,c of effective damping with the displacement amplitude d of the loop normalised to thepeak displacement during the response, dc. Figure 7.18 concerns bilinear hysteretic systemswith the maximum ductility ratio mc ¼ dc/dy as the parameter, whereas Figure 7.19 refers to asystem of elastic isolators (e.g. LDEB) combined with viscous dampers, and is parametrisedby the damper exponent a.

It is clear from Figures 7.17 and 7.18 that at ductility levels around 10 and higher, jeff eitherremains approximately constant with displacement or even reduces slightly at larger displace-ment amplitudes. The reduction with the displacement amplitude is more pronounced inFigure 7.19 for viscous dampers with small a values. For a ¼ 1.0 (linear damping), jeffremains constant over the whole range of d/dc up to 1.0. For this reason, the use of effectivedamping, jeff, gives satisfactory results in cases of seismic isolation, because the ductility ratiodemands reached are high. By contrast, Figures 7.17 and 7.18 show that for bilinear hystereticsystems reaching ductility ratios mc less than 5–7 (as typical of reinforced concrete or steelnon-isolated ductile structures), jeff is smaller at low displacement amplitudes. More refinedhysteretic models for reinforced concrete structures exhibit similar trends (Fardis and Panagio-takos, 1996). The large scatter of experimental results shown also in Fardis and Panagiotakos(1996) is an additional reason why effective damping, jeff, cannot be reliably applied forestimating nonlinear displacement demands in ductile reinforced concrete structures.

7.5.5.4 Bidirectional excitationThe fundamental mode method refers to SDoF systems and motions in a certain direction.However, real isolation systems have, in general, two horizontal degrees of freedom, with


Figure 7.17. Variation of the effective damping ratio jeff of a bilinear hysteretic system as a function of

the ductility ratio m ¼ d/dy for various values of l ¼ Kp/Ke

0.60

0.50

0.40

0.30

0.20

0.10

0.00

ξ eff λ

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.001

0.01

0.05

0.100.15

µ = d/dy

Figure 7.18. Evolution of the ratio jeff/jeff,c of effective damping to its value at the peak displacement dcwith the ratio d/dc of the peak displacement of a loop d to dc, in a bilinear hysteretic system, as a

function of the final ductility ratio mc ¼ dc/dy, for a hardening ratio l ¼ Kp/Ke equal to 0.05

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0d/dc

µc = 3

λ = 0.05

ξ eff/ξ

eff,

c

µc = 5

µc = 15

µc = 12.5 µc = 7.5µc = 10

189

Clauses 3.2.3(3),

3.2.3(6), 3.2.3(7) [2]

Clause 7.6.3(9) [2]

usually (but not always) the same or similar properties in these two directions. Part 2 ofEurocode 8, as well as AASHTO (2010) and more recently Constantinou et al. (2011), do notexplicitly provide for the estimation of the effect of the transverse seismic action in the contextof the fundamental mode method. In fact, Eqs (D5.2) normally underestimate this effect.

The key to the modification of the fundamental mode method to make its results compatible withthose of nonlinear time-history analysis lies in the rules governing the compatibility of the pairsof horizontal acceleration histories with the elastic response spectrum, highlighted in Section3.1.4 of this Guide. These rules lead to the following options for achieving compatibilitybetween the fundamental mode method and nonlinear time-history analysis:

1 The fundamental mode method is conveniently carried out using an elastic spectrummultiplied by a factor of 1.25–1.30. This amplification slightly exceeds the requirement ofEurocode 8.

2 The results of the fundamental mode method carried out for each horizontal componentstrictly according to Part 2 of Eurocode 8 are modified by multiplying the displacementsby a factor between 1.15 and 1.25. This amounts to assuming a concurrent transversedisplacement between 57% and 75% of the estimated peak displacement.

7.5.5.5 Estimation of maximum forces in systems with viscous dampersFor isolation systems consisting of elastic isolators (e.g. elastomeric bearings) and exclusivelyviscous dampers, Part 2 of Eurocode 8 gives a methodology for the estimation of maximumforces occurring simultaneously on elastic isolators and dampers. However, it does not giveany guidance for systems of hysteretic isolators and viscous dampers. A general theoreticalsolution for free vibrations of such a system is given in Ribeiro et al. (2007). In this section,the more practical methodology in Constantinou et al. (2011) is highlighted.

Both the hysteretic isolators with dissipated energy per cycle ED,h from Eq. (D7.3) and theviscous dampers with dissipated energy ED,v from Eqs (D7.12) and (D7.13) contribute to theeffective damping as

jeff ¼ (P

ED,hþP

ED,v)/(2pP

Keffd cd2 ) (D7.41)

Designating by jv the part of jeff contributed by viscous damping,

jv ¼ (P

ED,v)/(2pP

Keffdcd2 ) (D7.42)


Figure 7.19. Evolution of the ratio jeff/jeff,c of effective damping to its value at the peak displacement dcwith the ratio d/dc of the peak displacement in a loop d to dc, in an elastic system with only viscous

damping as a function of the damper exponent a

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0d/dc

α = 1.00

α = 0.80

α = 0.60

α = 0.40 α = 0.00

α = 0.05α = 0.10α = 0.20

ξ eff/ξ

eff,

c

0.0

0.5

1.0

1.5

2.0

2.5

3.0

190

the maximum velocity vm in sinusoidal motion can then be estimated from the pseudo-velocityvdcd as

vm ¼ rvvdcd (D7.43)

The correction factors rv are given in Table 7.3 (see Constantinou et al., 2011). The maximumforces Fm on the system can be estimated as

Fm ¼ Keffdcd cos dþ 2pjvl

ðrvÞab ðsin dÞab� �

� Keffdcd ðD7:44Þ

where

d ¼ 2pajvl

� �1=ð2�abÞðD7:45Þ

and the other variables are defined in Eqs (D7.13) and (D7.42).

7.6. Lateral restoring capabilityPart 2 of Eurocode 8 requires the isolation system present in both horizontal directions sufficientlateral restoring capability to automatically prevent accumulation of large residual displace-ments. Note that the absence of residual displacements in the results of nonlinear time-historyanalyses using few (usually 3–10) horizontal pairs of spectrum-compatible ground accelerationhistories is by no means conclusive proof of sufficient restoring capability. The criteria in Part2 of Eurocode 8 have a solid basis: several hundred thousand nonlinear analyses (Katsaraset al., 2008) of bilinear hysteretic systems subjected to an exhaustive range of strictly naturalfar- and near-source records.

REFERENCES

AASHTO (2010) Guide Specifications for Seismic Isolation Design. American Association of State

Highway and Transportation Officials, Washington, DC.

CEN (Comite Europe de Normalisation) (2000) EN 1337-2:2000: Structural bearings – Part 2:




CEN (2005a) EN 1998-2:2005: Eurocode 8 – Design of structures for earthquake resistance – Part 2:


CEN (2005b) EN 1337-3:2005: Structural bearings – Part 3: Elastomeric bearings. CEN, Brussels.

CEN (2009) EN 15129:2009: Antiseismic devices. CEN, Brussels.

Constantinou MC, Kalpakidis I, Filiatrault A and Ecker Lay RA (2011) LRFD-based Analysis and

Design Procedures for Bridge Bearings and Seismic Isolators. Department of Civil, Structural and

Environmental Engineering, State University of New York, Buffalo, NY. Technical Report

MCEER-11-004:2011.

Clause 7.7.1 [2]


Table 7.3. Correction factor rv

Effective

period: s

Effective damping: %

10 20 30 40 50 60 70 80 90 100

0.3 0.72 0.70 0.69 0.67 0.63 0.60 0.58 0.58 0.54 0.49

0.5 0.75 0.73 0.73 0.70 0.69 0.67 0.65 0.64 0.62 0.61

1.0 0.82 0.83 0.86 0.86 0.88 0.89 0.90 0.92 0.93 0.95

1.5 0.95 0.98 1.00 1.04 1.05 1.09 1.12 1.14 1.17 1.20

2.0 1.08 1.12 1.16 1.19 1.23 1.27 1.30 1.34 1.38 1.41

2.5 1.05 1.11 1.17 1.24 1.30 1.36 1.42 1.48 1.54 1.59

3.0 1.00 1.08 1.17 1.25 1.33 1.42 1.50 1.58 1.67 1.75

3.5 1.09 1.15 1.22 1.30 1.37 1.45 1.52 1.60 1.67 1.75

4.0 0.95 1.05 1.15 1.24 1.38 1.49 1.60 1.70 1.81 1.81

191

Fardis MN and Panagiotakos TB (1996) Hysteretic damping of reinforced concrete elements.

11th World Conference on Earthquake Engineering, Acapulco, Paper 464.

Katsaras CP, Panagiotakos TB and Kolias B (2008) Restoring capability of bilinear hysteretic

seismic isolation systems. Earthquake Engineering and Structural Dynamics 37(4): 557–575.

Ribeiro AMR, Maia NMM, Fontul M and Silva JMM (2007) Complete Response of a SDoF

System with a Mixed Damping Model. Departamento de Engenharia Mecanica, Instituto

Superior Tecnico, Lisbon.


192




Chapter 8

Seismic design examples

8.1. IntroductionThis chapter presents examples of seismic design of bridges according to Part 2 of Eurocode 8.The generic example is a bridge having a continuous deck with one central span and twoshorter side spans. The chapter comprises the following examples:

g Section 8.2 – example of ductile piers: a bridge with a concrete deck monolithic with piersdesigned for ductile behaviour. This earthquake-resisting system is usually cost-effectivefor bridges of medium-short spans and medium total length.

g Section 8.3 – example of limited ductile piers: a longer bridge on tall piers designed forlimited ductile behaviour.

g Section 8.4 – example of seismic isolation: a longer bridge on short piers with seismicisolation.

All three examples have developed from the contribution of the senior author of this Designers’Guide (Bouassida et al., 2012).

8.2. Example of a bridge with ductile piers8.2.1 Bridge layout – design conceptThe bridge is a three-span overpass, with spans 23.50þ 35.50þ 23.5 m and a total length of82.5 m. The deck is a post-tensioned cast-in-situ concrete voided slab. The piers consist ofsingle cylindrical columns with diameter D ¼ 1.2 m, monolithic with the deck. The pier heightsare 8 m for M1 and 8.5 m for M2. The bridge is simply supported on the abutments through apair of bearings allowing free sliding and rotation in every horizontal direction. The piers andthe abutments are supported on piles. The concrete grade is C30/37. The configuration of thebridge and cross-sections of the deck and the piers are shown in Figures 8.1–8.3.

The selection of single cylindrical-column piers allows arranging the supports at right angles tothe longitudinal axis, despite the slightly skew crossing in plan. For the given geometry of thebridge, the monolithic connection between the piers and the deck minimises the use of expensivebearings or isolators (and their maintenance), without subjecting the bridge elements to excessive

193

Figure 8.1. Longitudinal section of the bridge with ductile piers

A1 A2M1 M2

restraint by imposed deck deformations. Some comments on the cost-effectiveness of the seismicresistant system are given, as conclusions, in Section 8.2.12.

8.2.2 The earthquake-resisting system8.2.2.1 Structural system and ductilityThe main elements resisting seismic forces are the piers. A ductile seismic behaviour is selected forthese elements. The value of the behaviour factor q, as given in Section 5.4, Table 5.1, depends onthe shear span ratio Ls/h of the piers:

g For the longitudinal direction (taken as the horizontal direction X), assuming the piersto be fully fixed to the foundation and to the deck and for the shortest pier M1:Ls ¼ 8.0/2 ¼ 4.0 m and Ls/h ¼ 4.0/1.2 ¼ 3.33 . 3.0; therefore:, qX ¼ 3.50.

g For the transverse direction (taken as the horizontal direction Y), assuming the piers to befully fixed to the foundation and free to move and rotate at the connection to the deck andfor the shortest pier M1: Ls ¼ 8.0 m and Ls/h ¼ 8.0/1.2 ¼ 6.67 . 3.0, allowing qY ¼ 3.50.

8.2.2.2 Stiffness of elements8.2.2.2.1 PiersThe value of the effective stiffness of the piers for the seismic analysis is estimated initially andchecked after dimensioning the vertical reinforcement of the piers. For both piers the stiffnessis assumed to be 40% of the uncracked flexural stiffness of the gross section.

8.2.2.2.2 DeckThe full uncracked flexural stiffness of the gross section of the prestressed concrete deck is taken.Considering the voided section as closed, the torsional stiffness is 50% of the uncracked grosssection stiffness.


Figure 8.2. Plan view of the bridge with ductile piers

A1 A2M1 M2

Figure 8.3. Cross-sections of (a) the piers and (b) the deck

(a) (b)

194

8.2.3 Design seismic actionA response spectrum of type 1 applies for the design seismic action. The ground type is C, withthe recommended values of the soil factor S ¼ 1.15 and the corner periods TB ¼ 0.2 s andTC ¼ 0.6 s in Table 3.3, whereas the corner period is taken as longer: TD ¼ 2.5 s. The bridge islocated at a seismic zone with a reference peak ground acceleration agR ¼ 0.16g. The importancefactor is gI ¼ 1.0, and the design peak ground acceleration in the horizontal directions is

ag ¼ gIagR ¼ 1.0 � 0.16g ¼ 0.16g

The lower bound factor for design spectral accelerations is b ¼ 0.20. For the behaviour factorsqX ¼ 3.5 in the longitudinal direction and qY ¼ 3.5 in the transverse direction, the designresponse spectrum (normalised to the design peak ground acceleration, ag) from Eqs (D5.3) inSection 5.3 is shown in Figure 8.4.

8.2.4 Quasi-permanent actions for the seismic design situationThe loads applied to the bridge deck (Figure 8.5) in the seismic design situation are as follows.

8.2.4.1 Self-weight (G)The area of the voided section is 6.89 m2, the area of the solid section is 9.97 m2, the total lengthof the voided section is 73.5 m and the total length of the solid section is 9.0 m:

qG ¼ (6.89 m2 � 73.5 mþ 9.97 m2 � 9.0 m) � 25 kN/m3 ¼ 14 903 kN

8.2.4.2 Additional dead load (G2)The area of the sidewalks is 0.50 m2/m, the weight of the safety barriers is 0.70 kN/m and thewidth and thickness of the pavement are 7.5 m and 0.10 m, respectively:

qG2 ¼ 2 � 25 kN/m3 � 0.50 m2 (sidewalks)þ 2 � 0.70 kN/m (safety barriers)þ 7.5 m

� 23 kN/m3 � 0.10 m (road pavement) ¼ 43.65 kN/m

Chapter 8. Seismic design examples

Figure 8.4. Design response spectrum (normalised to the design peak ground acceleration, ag)

Period, T: s

S d(T

)/ag

Regions corresponding to constant:

TB

β

TC TD

0.821

0.767

Acceleration Velocity Displacement

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Figure 8.5. Dead, additional dead and quasi-permanent uniform traffic load application

195

8.2.4.3 Quasi-permanent value of traffic load (LE)The quasi-permanent value of the traffic load is taken as 20% of the uniformly distributed trafficload (UDL) of load model 1 (LM1), qUDL ¼ 45.2 kN/m:

qQP ¼ 0.20qL ¼ 0.2 � 45.2 kN/m ¼ 9.04 kN/m

8.2.4.4 Thermal actions (T)The thermal actions consist of:

g a uniform temperature difference DTN,ext ¼ þ52.58C of the maximum uniform bridgetemperature component, Te,max, from the initial temperature T0 ¼ 108C at the time thedeck is erected;

g a uniform temperature difference DTN,con ¼ –458C of the minimum uniform bridgetemperature component, Te,min, from the initial temperature, T0, at the time the deck iserected.

No vertical temperature difference component, DTM, between the top and bottom surfaces of thedeck is considered.

8.2.4.5 Creep and shrinkage (CS)A total strain of –32.0 � 10�5 is considered. It is of relevance only for the bearing displacements.

The total load applied on the bridge deck in the seismic design situation is then

WE ¼ 14 903 kNþ (43.65þ 9.04) kN/m � 82.5 m ¼ 19 250 kN

8.2.5 Fundamental mode analysis in the longitudinal directionThe fundamental mode period is estimated based on a simplified single-degree-of-freedom(SDoF) cantilever model of the bridge. The mode corresponds to the oscillation of the bridgealong its longitudinal axis, assuming both ends of the piers as fixed.

For cylindrical columns of diameter 1.2 m, the uncracked moment of inertia is Iun ¼ p1.24/64 ¼0.1018 m4. The assumed effective moment of inertia of piers is Ieff/Iun ¼ 0.40 (to be checkedlater). Assuming both ends of the piers fixed, for concrete grade C30/37 with Ecm ¼ 33 GPa,the stiffness of each pier in the longitudinal direction is

K1 ¼ 12EIeff/H3 ¼ 12 � 33 000 MPa � (0.40 � 0.1018 m4)/(8.0 m)3 ¼ 31.5 MN/m

K2 ¼ 12EIeff/H3 ¼ 12 � 33 000 MPa � (0.40 � 0.1018 m4)/(8.5 m)3 ¼ 26.3 MN/m

The total longitudinal stiffness is:

K ¼ 31.5þ 26.3 ¼ 57.8 MN/m

The total seismic weight is: WE ¼ 19 250 kN, so the fundamental period is

T ¼ 2p

ffiffiffiM

K

r¼ 2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi19 250=9:81

57 800

r¼ 1:16 s

The spectral acceleration in the longitudinal direction is

Se ¼ agS(2.5/q)(TC/T) ¼ 0.16g � 1.15 � (2.5/3.5) � (0.60/1.16) ¼ 0.068g

The total seismic shear force in the piers is

VE ¼ SeWE/g ¼ 0.068g � 19 250 kN/g ¼ 1309 kN

The shear force is distributed to piers M1 and M2 in proportion to their stiffness:

V1 ¼ (31.5/57.8) � 1309 kN ¼ 713 kN

V2 ¼ 1309� 713 ¼ 596 kN


196

The longitudinal seismic moments My (assuming full fixity of pier columns at top and bottom)are

My1 � V1H1/2 ¼ 713 kN � 8.0 m/2 ¼ 2852 kN m

My2 � V2H2/2 ¼ 596 kN � 8.5 m/2 ¼ 2533 kN m

8.2.6 Modal response spectrum analysisThe finite element discretisation of the deck is very fine, the same as used for the analysis forgravity and traffic loads. The piers are also discretised vertically in a fairly large number ofelements.

The characteristics of the 15 most important modes of the structure out of a total of 50 computedin the analysis are shown in Table 8.1. Some of these modes, as well as those not listed inTable 8.1, have negligible contribution to the total response. The shapes of the most importantamong the first eight modes are depicted in Figure 8.6.

Response spectrum analysis considering the first 50 modes was carried out. The sum of the modalmasses considered amounts to 99.6%, 99.7% and 92% in the X, Y and Z directions, respectively.The combination of modal responses was carried out using the complete quadratic combination(CQC) rule, Eqs (D5.20) and (D5.22b) in Section 5.5.4. The results of the fundamental modeanalysis in the longitudinal direction and of the modal response spectrum analysis are presentedand compared in Table 8.2.

8.2.7 Design action effects and verifications8.2.7.1 Design action effects for flexure and axial force verification of plastic hingesThe combination of the components of the seismic action is carried out according to Eqs (D5.2)in Section 5.2. Table 8.3 gives the design action effects (bending moment and axial force) atthe bottom section of pier M1, together with the required reinforcement, for each designcombination.

The pier is of circular section with diameter D ¼ 1.2 m, of concrete C30/37 with Class C steel ofS500 grade. The nominal cover is c ¼ 50 mm, and the estimated distance of the bar centre fromthe surface is 82 mm. The required reinforcement at the bottom section of pier M1, which iscritical, is 19 870 mm2. The reinforcement selected is 25 132 (20 100 mm2), as shown inFigure 8.7. Figure 8.8 displays the Moment–Axial force interaction diagram for the bottomsection of Pier M1 for all design combinations.


Table 8.1. Normal modes

Mode No. Period: s Modal mass contribution: %

X direction Y direction Z direction

1 1.77 0 3.4 0

2 1.43 0 94.8 0

3 1.20 99.2 0 0

4 0.32 0 0 8.9

5 0.32 0 0.3 0

6 0.19 0 0.7 0

8 0.15 0 0 63.1

15 0.054 0 0 10.8

16 0.053 0 0 0.2

17 0.052 0 0 1.8

18 0.051 0 0 0.4

26 0.030 0.2 0 0.2

27 0.029 0 0.1 0

28 0.028 0 0 5.4

30 0.027 0.1 0 0.1

197

Table 8.4 shows the design action effects of bending moment and axial force at the bottomsection of pier M2 together with the required reinforcement, for each design combination.

The required reinforcement at the bottom section of Pier M2, which is critical, is 16 800 mm2.The reinforcement selected is 21 132 (16 880 mm2) as shown in Figure 8.9. Figure 8.10 showsthe moment–axial force interaction diagram for the bottom section of pier M2 for all designcombinations.


Figure 8.6. Modes 1, 2, 3, 4 and 8

Table 8.2. Comparison of analyses in the longitudinal direction

Pier Fundamental mode analysis: Multimode response spectrum analysis:

Fundamental period, T1 1.16 s 1.20 s (third mode)

Seismic shear, Vz M1

M2

713 kN

596 kN

662 kN

556 kN

Seismic moment, My M1

M2

2852 kN m

2533 kN m

At top: 2605 kN m. At bottom: 2672 kN m

At top: 2327 kN m. At bottom: 2381 kN m

198

8.2.7.2 Confirmation of stiffness of ductile piersThe effective stiffness of the piers for the seismic design situation is estimated according to Eqs(D5.33)–(D5.35) in Section 5.8.1.

8.2.7.2.1 Pier M1For one layer of 25 132 (20 100 mm2) and an axial force of N ¼ –7200 kN, using the designvalues of material properties, the design value of yield moment is My ¼ 4407 kN m; the steelyield strain is

1sy ¼ fyd/Es ¼ 500/(1.15 � 200 000) ¼ 0.00217


Table 8.3. Design action effects and required reinforcement at the bottom section of pier M1

Combination N: kN My: kN m Mz: kN m As: mm2

max MyþMz �7159 4576 �1270 19 870

min MyþMz �7500 �3720 1296 13 490

max MzþMy �7238 713 4355 17 240

min MzþMy �7082 456 �4355 17 000

Figure 8.7. Pier M1 cross-section with reinforcement

Figure 8.8. Moment–axial force interaction diagram of the bottom section of pier M1

–30 000

–25 000

–20 000

–15 000

–10 000

–5000

0

5000

10 000

15 000

–6000 –4000 –2000 0 2000 4000 6000

M: kN m

N: k

N

Section with 25Ø32

Designcombinations

199

while the corresponding strain at the extreme concrete fibres is 1cy ¼ 0.00272. The design value ofthe moment resistance (again for design values of material properties) isMRd ¼ 4779 kN m. Theyield curvature is

fy ¼ (0.00217þ 0.00272)/(1.2� 0.082) ¼ 4.37 � 10�3 m�1

while the approximation of Eq. (D5.35b) for circular sections yields

fy ¼ 2.41sy/d ¼ 2.4 � 0.00217/(1.2� 0.082) ¼ 4.66 � 10�3 m�1

g Equation (D5.33a) in Section 5.8.1 yields

Ic ¼ p � 1.24/64 ¼ 0.1018 m4

My/(Ecfy) ¼ 4407/(33 000 � 4.37 � 10�3) ¼ 0.0306 m4

Ieff ¼ 0.08 Icþ Icr ¼ 0.0387 m4

(EI )eff/(EI )c ¼ 0.38

g Equation (D5.33b) in Section 5.8.1 gives

(EI )eff ¼ 1.2MRd/fy ¼ 1.2 � 4779/4.37 � 10�3 ¼ 1 312 000 kN m2

Ieff ¼ 1 312 000/33 000 ¼ 0.0398 m4

(EI )eff/(EI )c ¼ 0.39

The assumed value (EI )eff/(EI )c ¼ 0.40 was a good estimate for the analysis.

8.2.7.2.2 Pier M2For pier M2 the final reinforcement is one layer of 21 132 (16 880 mm2). The axial force isN ¼ �7200 kN. For the design values of material properties the yield moment isMy ¼ 4048 kN m, the steel yield strain is 1sy ¼ 0.00217 and the corresponding strain at the


Table 8.4. Design action effects and required reinforcement at the bottom section of pier M2

Combination N: kN My: kN m Mz: kN m As: mm2

max MyþMz �7528 3370 �1072 10 320

min MyþMz �7145 �4227 1042 16 800

max Mz þMy �7317 �465 3324 8980

min MzþMy �7320 �674 �3324 9250

Figure 8.9. Pier M2 cross-section with reinforcement

200

extreme concrete fibres 1cy ¼ 0.00273. The design value of the moment resistance isMRd ¼ 4366 kN m.

g From Eq. (D5.33a) in Section 5.8.1: (EI )eff/(EI )c ¼ 0.35.g From Eq. (D5.33b) in Section 5.8.1: (EI )eff/(EI )c ¼ 0.36.

The assumed value (EI )eff/(EI )c ¼ 0.40 was a fairly good estimate for the analysis.

8.2.7.3 Dimensioning of the piers in shear8.2.7.3.1 Overstrength momentsThe overstrength moment is calculated as Mo ¼ goMRd, where go is the overstrength factor andMRd is the design value of moment resistance (see Section 6.4.1, Eqs (D6.6)). The normalisedaxial force is

hk ¼ NEd/Ac fck ¼ 7600/(1.13 � 30) ¼ 0.22

Since hk ¼ 0.22 . 0.1, the minimum value go ¼ 1.35 is multiplied by

1þ 2(hk� 0.1)2 ¼ 1þ 2(0.22� 0.1)2 ¼ 1.029

So,

go ¼ 1.35 � 1.029 ¼ 1.39

and the overstrength moments for the piers are:

g Mo1 ¼ 1.39 � 4779 ¼ 6643 kN mg Mo2 ¼ 1.39 � 4366 ¼ 6069 kN m

8.2.7.3.2 Pier capacity design shearsIn the longitudinal direction the shear forces from the analysis in piers M1 and M2 areV1 ¼ 713 kN and V2 ¼ 596 kN, while the capacity shears can be calculated directly from theoverstrength moments:

g VC1 ¼ 2Mo1/H1 ¼ 2 � 6643/8.0 ¼ 1661 kNg VC2 ¼ 2Mo2/H2 ¼ 2 � 6069/8.5 ¼ 1428 kN

In the transverse direction, the shear in each pier is calculated applying the simplifications ofAnnex G of Part 2 of Eurocode 8 (see Section 6.4.2):

VCi ¼ (Mo/MEi)VEi


Figure 8.10. Moment–axial force interaction diagram of the bottom section of pier M2

–6000 –4000 –2000 0 2000 4000 6000

M: kN m

N: k

N

–30 000

–25 000

–20 000

–15 000

–10 000

–5000

0

5000

10 000

Section with 21Ø32

Designcombinations

201

The seismic moments and shear forces are:

g ME1 ¼ 3061 kN m and VE1 ¼ 680.3 kNg ME2 ¼ 2184 kN m and VE2 ¼ 450.2 kN

So, the capacity design shear forces are

g Vc1 ¼ (6643/3061) � 680.3 ¼ 1476 kNg Vc2 ¼ (6069/2184) � 450.2 ¼ 1251 kN

Figure 8.11 shows the capacity design action effects in the transverse direction at the base of thepiers.

8.2.7.3.3 Dimensioning in shearFor a circular section

d ¼ rþ rs ¼ 0.60þ 0.52 ¼ 1.12 m

while the ‘effective depth’ is

de ¼ rþ 2rs/p ¼ 0.60þ 2 � 0.52/p ¼ 0.93 m

The shear strength of the section is calculated as

VRd,s ¼ (Asw/s)(0.9de) fywd cot u/gBd or as VRd,s ¼ (p/4)(Asw/s)(0.9d ) fywd cot u/gBd

where Asw is the total cross-sectional area of the shear hoops, s is their spacing, fywd is their designyield strength, u is the angle between the concrete compression strut and the pier axis, withcot u ¼ 1 as per Part 2 of Eurocode 2 and gBd is the safety factor in note 5 of Table 6.1, withthe value gBd ¼ 1.25� (qVEd/VC,o� 1) � 1.

For pier M1 the design shear force is VC1 ¼ 1661 kN, and that from the analysis is V1 ¼ 713 kN.Then,

1.25� (qVEd/VC,o� 1) ¼ 1.25� (3.5 � 713/1661� 1) ¼ 0.75

so gBd ¼ 1 and the required shear reinforcement is

Asw/s ¼ 1.0 � 1661/(0.84 � 0.5/1.15 � 1.0) ¼ 4550 mm2/m

or

Asw/s ¼ 1.0 � 1661/(0.7854 � 0.9 � 1.12 � 0.5/1.15 � 1.0) ¼ 4827 mm2/m

For pier M2 the design shear force is VC1 ¼ 1428 kN and that from the analysis is V1 ¼ 596 kN.Then,

1.25� (qVEd/VC,o� 1) ¼ 1.25� (3.5 � 596/1428� 1) ¼ 0.79


Figure 8.11. Capacity design action effects in the transverse direction

Mo1 = 6643 kN m

Mo2 = 6069 kN m

VC1 = 1476 kN

VC2 = 1251 kN

202

so gBd ¼ 1 and the required shear reinforcement is

Asw/s ¼ 1.0 � 1428/(0.84 � 0.5/1.15 � 1.0) ¼ 3910 mm2/m

or

Asw/s ¼ 1.0 � 1428/(0.7854 � 0.9 � 1.12 � 0.5/1.15 � 1.0) ¼ 4150 mm2/m

8.2.8 Ductility requirements for the piers8.2.8.1 Confinement reinforcementThe normalised axial force is

hk ¼ NEd/Ac fck ¼ 7600/(1.13 � 30 000) ¼ 0.22 . 0.08

so confinement of the compression zone is required (see note 15 of Table 6.1).

The longitudinal reinforcement ratio is:

g for pier M1:

rL ¼ 20 100/1 130 000 ¼ 0.0178

for pier M2:

rL ¼ 16 880/1 130 000 ¼ 0.0149

The distance from the surface to the spiral centreline is c ¼ 58 mm (Dsp ¼ 1.084 m), and the coreconcrete area is Acc ¼ 0.923 m2.

For circular spirals and ductile behaviour, the required mechanical reinforcement ratio, vw,req, ofconfinement reinforcement is (see Table 6.1) vw,req ¼ 0.52(Ac/Acc)hkþ 0.18( fyd/fcd)(rL� 0.01),and vw,min ¼ 0.18. For

g pier M1:

vw,req ¼ 0.52 � (1.13/0.923) � 0.22þ 0.18 � (500/1.15)/(0.85 � 30/1.5)

� (0.0178� 0.01) ¼ 0.176

g pier M2:

vw,req ¼ 0.52 � (1.13/0.923) � 0.22þ 0.18 � (500/1.15)/(0.85 � 30/1.5)

� (0.0149� 0.01) ¼ 0.162

For the worst case of pier M1:

vwd,c ¼ max(vw,req; vw,min) ¼ 0.18

and the required volumetric ratio of confining reinforcement is

rw ¼ vwd,c( fcd/fyd) ¼ 0.18 � (0.85 � 30/1.5)/(500/1.15) ¼ 0.0070

while the required area of confining reinforcement (one leg) is

Asp/sL ¼ rwDsp/4 ¼ 0.007 � 1.084/4 ¼ 0.0019 m2/m ¼ 1900 mm2/m

The maximum allowed hoop spacing is (see Table 6.1):

max sL ¼ 1084 mm/5 ¼ 217 mm


203

8.2.8.2 Prevention of buckling of longitudinal barsFor the transverse reinforcement required to prevent buckling of the longitudinal bars, themaximum hoop spacing sL should not exceed ddbL, where

d ¼ 2.5( ftk/fyk)þ 2.25 � 5

(see Table 6.1). For S500 steel, ftk/fyk 1.15:

d ¼ 2.5 � 1.15þ 2.25 ¼ 5.125

Therefore,

max sL ¼ ddL ¼ 5.125 � 32 mm ¼ 164 mm

8.2.8.3 Transverse reinforcement of piers – comparison of requirementsThe pier transverse reinforcement requirements for each design check are compared in Table 8.5.The transverse reinforcement is governed by shear design. The reinforcement selected for bothpiers is one spiral of 116/85 (4730 mm2/m).

8.2.9 Capacity design verifications of the deck8.2.9.1 Estimation of capacity design effects – an alternative procedureThe general procedure in Part 2 of Eurocode 8 for calculating the capacity design effects consistsof adding to the action effects of the quasi-permanent loads ‘G’, the effects of the loadingDAC ¼ ‘Mo�G’, both acting in the deck-cum-piers frame system of the bridge (see Section6.4.2 of this Guide).

An alternative procedure for the longitudinal (X) direction is to work on a continuous beamsystem of the deck, simply supported on the piers and the abutments. On this system theeffects of the quasi-permanent loads ‘G’ and those of the overstrength moments ‘Mo’ areadded. Figure 8.12 demonstrates the equivalence of the two procedures.

The effects of the quasi-permanent loads ‘G’ are displayed in Figure 8.13. Figure 8.14 shows theeffects of the overstrength loading ‘Mo’ for seismic action in theþX direction. For the effects due


Table 8.5. Comparison of transverse reinforcement requirements of piers

Requirement Confinement Buckling of bars Shear design

At/sL: mm2/m 2 � 1900 ¼ 3800 – M1: 4825, M2: 4150

max sL: mm 217 164 –

Figure 8.12. Equivalence of general and simpler capacity design procedures for the deck

MG1 MG2

MG1 MG2

MG1’ MG2’

MG1 MG2MO1 MO2

MO1 MO2

MO1 MO2

MO1 – MG1 MO2 – MG2

MO1 – MG1 MO2 – MG2

MO1 – MG1’ MO2 – MG2’

MG1 MG2

MG1’ MG2’

+;

–

G G

Permanent load ‘G’

∆Ac: Over strength – ‘G’

General procedure ; alternative procedure

204

to seismic action in �X direction, the signs of the effects are reversed. Figure 8.15 displays theresult of the superposition of these two loadings to get the capacity effects for seismic actionin the X direction.

8.2.9.2 Flexural verification of the deckThe deck section at each side of the connection of the deck with a pier is checked against thecapacity design effects, taking into account the existing reinforcement and tendons (shown inFigure 8.16). Table 8.6 lists the moment and axial force combinations for which the decksections are checked. Figure 8.17 compares the moment–axial force ultimate limit state (ULS)interaction diagram with the capacity design effects.

8.2.9.3 Other deck verificationsThe ULS verification of the deck in shear is, in general, not critical. So, it is not presentedhere.

The verification of the connection between the pier and the deck as a ‘beam/column’ jointper Section 6.4.4 is far from critical, and therefore not presented here. It is usually criticalfor the shear reinforcement of joints with slender pier columns monolithically connected to thedeck.


Figure 8.13. Quasi-permanent loads (‘G’ loading) and resulting moment and shear force diagrams

2430 22852340

2195

3211

4052

4047

3207

M: kN m

V: kN

(+)(–)

G

Figure 8.14. Overstrength for seismic action in the þX direction (‘Mo’ loading): resulting moment and

shear force diagrams

M: kN m

V: kN

3692

4491 3104

4296

256.7187.4 263.8 161.9

Mo1 = 6643 kN m Mo2 = 6069 kN m

(–)

6643

6643

1661

6069

6069

(+) (+) 1428

205

8.2.10 Design action effects for the foundation designFigure 8.18(a) displays the capacity design effects acting on the foundation of pier M1 for thelongitudinal direction, for seismic actions in the negative direction (–X). Figure 8.18(b) showsthe capacity design effects for the transverse direction. The sign of the effects is reversed forthe opposite direction of the seismic action in the transverse direction.


Figure 8.15. Capacity effects for seismic action in the þX direction (‘G’þ ‘Mo’): resulting moment and

shear force diagrams

M : kN m

V : kN

6643

6643

1661

6069

6069

(+) (+) 1428

6122

2206 764

6491

3398

3795

4311

3045(+)

(–)

Figure 8.16. Deck section, reinforcement and tendons

Top layer: 46 Ø20 + 33 Ø16 (210.8 cm2)

Bottom layer: 2 × 33 Ø16 (182.9 cm2)

4 groups of 3 tendonsof type DYWIDAG 6815(area of 2250 mm2 each)

Table 8.6. Loading combinations for the deck section

Connection to Direction of seismic action My: kN m N: kN

Pier M1 – left side þX �6122 �29 900

Pier M1 – right side þX 2206 �28 300

Pier M2 – left side þX �6491 �29 500

Pier M2 – right side þX 764 �28 100

Pier M1 – left side �X 1262 �28 100

Pier M1 – right side �X �6776 �29 500

Pier M2 – left side �X 2101 �28 300

Pier M2 – right side �X �5444 �29 900

206

8.2.11 Bearings and roadway joints8.2.11.1 BearingsThe design displacement of the bearing, dEd, is given by Eq. (D6.36).

The displacements in the longitudinal direction are shown in Figure 8.19(a), and those in thetransverse direction in Figure 8.19(b). The maximum displacement at the bearings is 93.9 mmand 110 mm, respectively.

The bridge is simply supported on the abutments through a pair of bearings allowing freesliding and rotation for both horizontal axes. Plan and side views of a bearing are depicted inFigure 8.20.


Figure 8.17. Moment–axial force interaction diagram of the deck section

M: kN m

N: k

N

Design combinations

–200 000

–150 000

–100 000

–50 000

0

50 000

–60 000 –40 000 –20 000 0 20 000 40 000

Figure 8.18. Capacity design effects on the foundation of pier M1: (a) seismic action in the longitudinal

(�X) direction; (b) seismic action in the transverse direction

Vz = 1661 kN

My = 6643 kN m

N = 7371 kN

Vy = 1476 kN

Mz = 6643 kN m

Vz = 730 kN

My = 124 kN m

N = 7371 kN

X

Y

(a) (b)

Figure 8.19. Bridge displacements (mm): (a) in the longitudinal direction; (b) in the transverse direction

207

Uplifting of the bearings is checked. The minimum vertical reaction forces in the bearings arepresented in Figure 8.21, with a minimum resultant value of 17.8 kN (compressive, so there isno uplifting). The maximum vertical reactions are shown in Figure 8.22, with a maximumvalue of 2447 kN.

8.2.11.2 Overlapping lengthThe minimum overlapping (seating) length at moveable joints is given by Eq. (D6.34), where:

g the support length is in the present case equal to lm ¼ 0.5 m . 0.4 mg the effective seismic displacement of the support from the analysis for the seismic design

situation is in the present case equal to des ¼ 0.101 mg for the effective ground displacement deg ¼ (2dg/Lg)Leff:

– the design ground displacement is in the present case

dg ¼ 0.025agSTCTD ¼ 0.025 � 0.16 � 9.81 � 1.15 � 0.6 � 2.5 ¼ 0.068 m

– the distance parameter for ground type C is Lg ¼ 400 m– the effective length of the deck is in the present case Leff ¼ 82.5/2 ¼ 41.25 m.

So, for no proximity to a known seismically active fault:

deg ¼ (2dg/Lg)Leff ¼ (2 � 0.068/400) � 41.25 ¼ 0.014 m , 2dg ¼ 0.136 m


Figure 8.20. Plan and side views of sliding bearings

Long

itudi

nal

axis

600

500

140

140

95.5

Plan

Ø40

Ø40

Ø440540

460

1/2 view; 1/2 section

360

Figure 8.21. Minimum reaction forces in the bearings in the seismic design situation (kN)

208

Therefore,

min lov ¼ 0.50þ 0.014þ 0.101 ¼ 0.615 m

The available seating length is 1.25 m . min lov.

8.2.11.3 Roadway jointsThe width of a roadway joint between the top deck slab and the top of the backwall of theabutment should be designed to accommodate the displacement given by Eq. (D6.39). The clear-ance between the deck structure and the abutment or its backwall should accommodate the largerdisplacement given by Eq. (D6.36).

Table 8.7 gives the displacements for the roadway joint and those for the clearance of the struc-ture. Due to the differences between the two clearances, Part 2 of Eurocode 8 requires detailing ofthe backwall to cater for predictable (controlled) damage. An example of such a detail is shown inFigure 8.23, where impact along the roadway joint is foreseen to occur on the approach slab.Figure 8.24 displays the selected roadway joint type and the displacement capacities for eachdirection.

8.2.12 Conclusions regarding the design conceptOptimal cost-effectiveness of a ductile bridge system is achieved when all its ductile elements(notably the piers) have dimensions that lead to a seismic demand that is critical for the mainreinforcement of all critical sections and exceeds the minimum reinforcement requirements.This is difficult to achieve when the piers resisting the earthquake have:

g substantial height differences org sections larger than those required for the purposes of seismic design.

In such cases, it may be more economical to use:

g limited ductile behaviour if the design peak ground acceleration, ag, is low org flexible connection to the deck (seismic isolation).


Figure 8.22. Maximum reaction forces in the bearings in the seismic design situation (kN)

Table 8.7. Displacement for the roadway joint and clearance at the joint region

Displacement: mm dG dT dE dEd,J ( joint) dEd (structure)

Longitudinal

Opening þ18.5 10.5 þ76 þ54.5 þ100.5

Closure 0 �8.5 �76 �34.5 �80.5

Transverse 0 0 +110 +44.0 +110

209

It is worth noting that Part 2 of Eurocode 8 does not specify a minimum reinforcement require-ment (see note 23 in Table 6.1). For the bridge in this example, the owner specified rmin ¼ 1%(a fairly high value), as required by Part 1 of Eurocode 8, but only for the columns of buildings.The longitudinal reinforcement of the piers is derived from the seismic demands, and was over theminimum requirement (rL ¼ 1.78% for pier M1 and rL ¼ 1.49% for pier M2).

8.3. Example of a bridge with limited ductile piers8.3.1 Bridge layout – design conceptA schematic of the bridge is depicted in Figure 8.25. The deck is straight, continuous, with spansof 60þ 80þ 60 m. It has a composite (steel, concrete) section, consisting of two built-up steelgirders – connected at regular intervals via built-up cross bracings – and of a concrete slab.The piers are of reinforced concrete, 40 m tall. Their section is constant throughout the pierheight, hollow circular, with a thickness of 0.4 m and an external diameter of 4 m. The twogirders are supported on the pier through bearings; each pair of bearings is supported on apier head that is 4 m wide and 1.5 m deep. Concrete class in the piers is C35/45, and the steelis of grade S500, Class B.


Figure 8.23. Clearances and detailing of the roadway joint region

Clearance ofroadway joint

Structureclearance

Approach slab

Figure 8.24. Selected roadway joint type

Roadway joint type: T120Capacity in longitudinal direction: ±60 mmCapacity in transverse direction: ±50 mm

Figure 8.25. Bridge elevation and arrangement of the bearings

Pinnedconnection

Pinnedconnection

Sliding longitudinalpinned transverse

C0R

C0L

P1R

P1L

P2R

P2L

C3R

C3L

X

Y

210

The large flexibility of the 40 m tall piers has the following structural consequences:

g The connection of the deck to both pier heads can be pinned (hinged) about the transverseaxis, without causing excessive restraints due to imposed deck deformations.

g The large flexibility of the seismic force-resisting system gives long fundamental periods inboth horizontal directions and quite low response spectral accelerations. For such lowseismic response levels it is neither expedient nor cost-effective to design the piers forductility. Therefore, a limited ductile behaviour is selected with behaviour factor q ¼ 1.5(see Table 5.1).

8.3.2 Design seismic actionA response spectrum of type 1 applies. The ground type is B, with the recommended values of thesoil factor S ¼ 1.2 and of the corner periods TB ¼ 0.15 s, TC ¼ 0.50 s and TD ¼ 2 s in Table 3.3.The bridge is in a seismic zone with reference peak ground acceleration agR ¼ 0.3g. Theimportance factor is gI ¼ 1, and the design peak ground acceleration in the horizontal directionsag ¼ gIagR ¼ 1.0 � 0.3g ¼ 0.3g. The lower bound factor for design spectral accelerations isb ¼ 0.2. The behaviour factor is q ¼ 1.5. The design response spectrum from Eqs (D5.3) inSection 5.3 is shown in Figure 8.26.

8.3.3 Seismic analysis8.3.3.1 Quasi permanent traffic loadsThe quasi permanent value c2.1Qk,1 of the UDL system of LM1 is applied in the seismic designsituation. For bridges with severe traffic, the value of c2,1 is 0.2.

The load of the UDL system of LM1, with adjustment factors aq ¼ 1.0 for the UDL, are:

g lane 1:

aqq1,k ¼ 3 m � 9 kN/m2 ¼ 27.0 kN/m

g lane 2:

aqq2,k ¼ 3 m � 2.5 kN/m2 ¼ 7.5 kN/m

g lane 3:

aqq3,k ¼ 3 m � 2.5 kN/m2 ¼ 7.5 kN/m


Figure 8.26. Design spectrum for horizontal components with q ¼ 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Acc

eler

atio

n: g

Period: s

Ag = 0.3, ground type = B, soil factor = 1.2Tb = 0.15 s, Tc = 0.5 s, Td = 2.0 s, β = 0.2

211

g residual area:

aqqr,k ¼ 2 m � 2.5 kN/m2 ¼ 5.0 kN/m

Total load ¼ 47.0 kN/m.

In the seismic design situation the traffic load applied per unit of length of the bridge is

c2,1Qk,1 ¼ 0.2 � 47.0 kN/m ¼ 9.4 kN/m

8.3.3.2 Structural modelThe structural model employs prismatic 3D beam elements for cross-bracings and the two girdersof the deck. Prismatic elements are also used for the piers and in the pier heads (Figure 8.27). Thepiers were considered fixed at the top of the foundation. The model of each bearing takes intoaccount the pertinent constraints between the DoFs of the deck and pier head nodes connectedby the bearing.


Figure 8.27. Structural model

Figure 8.28. Moment–curvature curve of a pier section with r ¼ 1%

0.00 1 × 10–3 2 × 10–3 3 × 10–3 4 × 10–3 5 × 10–3 6 × 10–3

5000

10 000

15 000

20 000

25 000

30 000

35 000

40 000

45 000

50 000

55 000

60 000

65 000

70 000

75 000

Mom

ent:

kN

m

Curvature

ρ = 1%

N = –15 000 kNN = –20 000 kN

Bottom section

Top section

212

8.3.3.3 Effective pier stiffnessThe effective pier stiffness was initially assumed as 50% of the uncracked gross section stiffness,(EI)c, giving fundamental periods of 3.88 s in the longitudinal (X) direction and 3.27 s in thetransverse direction (Y). The lower bound of the design spectrum in Eqs (D5.3c) and (D5.3d)of Section 5.3, equal to Sd ¼ bag ¼ 0.2 � 0.3g ¼ 0.06g, applies for T � 3.3 s. So, the exactvalue of (EI )eff is immaterial for the design seismic action effects, provided that(EI )eff , 0.5(EI)c. However, the analysis was carried out in the end with (EI )eff ¼ 0.3(EI)c, toavoid underestimating the displacements. As shown in Figures 8.28–8.31, in the range of axialforces (15–20 MN) and bending moments (about 60 MN m) of interest here, this value cor-responds better to the finally required reinforcement ratio of r ¼ 1.5%. With this (EI )eff value,the fundamental period in the longitudinal direction is 5.02 s, and in the transverse direction itis 3.84 s.


Figure 8.29. Moment–(EI)eff /(EI )c ratio curve of a pier section with r ¼ 1%

Mom

ent:

kN

m

N = –15 000 kNN = –20 000 kN

ρ = 1%

0.0

5000

10 000

15 000

20 000

25 000

30 000

35 000

40 000

45 000

50 000

55 000

60 000

65 000

70 000

75 000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00J/Jgross

Bottom section

Top section

Figure 8.30. Moment–curvature curve of a pier section with r ¼ 1.5%

N = –15 000 kNN = –20 000 kN

ρ = 1.5%

0 1 × 10–3 2 × 10–3 3 × 10–3 4 × 10–3 5 × 10–3 6 × 10–3

Curvature

0.05000

10 00015 00020 00025 00030 00035 00040 00045 00050 00055 00060 00065 00070 00075 00080 00085 00090 00095 000

Bottom section

Top section

Mom

ent:

kN

m

213

8.3.3.4 Response spectrum analysisThe characteristics of the 11 most important modes out of the total of 30 computed in the modalanalysis are shown in Table 8.8. Some of them, as well as those not listed in Table 8.8, havenegligible contribution to the total response.

The shapes of four modes are presented in Figures 8.32–8.35.

A response spectrum analysis considering the first 30 modes was performed. The sum of modalmasses considered amounts to 97.1% and 97.2% in the X and Y directions, respectively. Thecombination of modal responses was carried out using the CQC rule.

Figure 8.36 shows the distribution of peak bending moments along pier P1.


Figure 8.31. Moment–(EI)eff /(EI )c ratio curve of a pier section with r ¼ 1.5%

N = –15 000 kNN = –20 000 kN

ρ = 1.5%

0.05000

10 00015 00020 00025 00030 00035 00040 00045 00050 00055 00060 00065 00070 00075 00080 00085 00090 00095 000

Bottom section

Top section

Mom

ent:

kN

m

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00J/Jgross

Table 8.8. Normal modes

No. Period: s Modal mass: %

X direction Y direction Z direction

1 5.03 92.5 0.0 0.0

2 3.84 0.0 76.8 0.0

3 1.49 0.0 0.0 0.0

4 0.79 0.0 0.0 1.2

6 0.66 0.0 8.4 0.0

9 0.48 0.0 2.1 0.0

11 0.42 0.0 0.0 63.2

17 0.26 0.0 6.2 0.0

18 0.26 4.4 0.0 0.0

23 0.16 0.0 0.0 5.0

27 0.15 0.0 3.0 0.0

28 0.13 0.0 0.0 8.8

214

8.3.4 Second-order effects8.3.4.1 Geometric imperfections of the piers and second-order effectsThe inclination of the pier according to clause 5.2 of EN 1992-2:2005 (CEN, 2005a) isuı ¼ uoah ¼ 1/200 � 2/

pl, where l is the length or height of the pier (l ¼ 40 m). Therefore,

uı ¼ 1.58 � 10�3 rad. According to clause 5.2(7) of EN 1992-1-1:2004 (CEN, 2004a), thisinclination creates an eccentricity, ei ¼ uilo/2, where lo is the effective length:

g longitudinal direction, �X: (lo ¼ 80 m) ex ¼ 0.063 mg transverse direction, �Y: (lo ¼ 40 m) ey ¼ 0.032 m.


Figure 8.32. First mode – longitudinal

Figure 8.33. Second mode – transverse

Figure 8.34. Third mode – rotation

215

The eccentricities under permanent loads (G), including the creep effect (for a creep coefficientw ¼ 2.0), are amplified for second-order effects according to Eq. (D6.4) in Section 6.3.1 withn ¼ NB/NEd (NB is the buckling load and NEd the axial force). The results are shown inTable 8.9.

8.3.4.2 Second-order effects of the seismic actionThese effects are estimated using two approaches.

(i) According to clause 5.8 of EN 1992-1-1:2004The nominal stiffness method (clause 5.8.7 in Part 1-1 of Eurocode 2) is applied using(EI )eff ¼ 0.3(EI), compatible with the seismic design situation. The moment magnification


Figure 8.35. Eleventh mode – vertical

Figure 8.36. Peak bending moment distribution in the two directions of the section of pier P1

53 237.13

–53 670.89

46 208.34–46 587.88

39 354.93–39 680.25

32 719.57–32 990.67

26 256.58–26 473.46

19 836.44–19 999.10

13 324.92–13 433.36

6680.61–6734.83

27 881.86

–27 868.23

24 158.64–24 146.62

20 622.12–20 611.70

17 322.35–17 313.55

14 245.07–14 237.88

11 345.81–11 340.23

8637.09–8633.13

6282.21–6279.87

216

factor is evaluated at the bottom section as 1þ b/[(NB/NEd)� 1], where b ¼ 1, NEd is the designvalue of axial load (19 538 kN) and NB is the buckling load based on the nominal stiffnessNB ¼ p2(EI )eff/(b1L0)

2, with b1 ¼ 1. This gives the following moment magnification factors:

g 1.154 in the longitudinal direction �Xg 1.034 in the transverse direction �Y.

(ii) According to EN 1998-2:2005The increase in the bending moments at the plastic hinge section (self-weight of the pier included)is given by the first term in Eq. (D6.3b) in Section 6.3.1: DM ¼ (1þ q)dEdNEd/2, where dEd is theseismic displacement of the pier top andNEd is the axial force from the seismic analysis – given inTable 8.10. This approach from Part 2 of Eurocode 8 (CEN, 2005b) gives approximately thesame moments in the longitudinal direction as that of Eurocode 2, but much higher values inthe transverse direction. It is used in the further design calculations in Table 8.11.

8.3.5 Action effects for the design of piers and abutmentsTable 8.10 lists the action effects of the individual actions and of their combinations for theseismic design situations. The effects are given:

g for piers P1 and P2, at the base sectiong for abutments C0 and C3 at the mid-distance between the bearings.

The designation of the individual actions is as follows:

g G Permanentþ quasi-permanent traffic loadsg EX Earthquake in the X direction (for the design spectrum with q ¼ 1.5)g EY Earthquake in the Y direction (for the design spectrum with q ¼ 1.5)g P-D EC2 Additional second-order effects according to clause 5.8 of EN 1992-1-1:2004g P-D EC8 Additional second-order effects according to EN 1998-2:2005g Imperf First- and second-order effects (including creep) of geometric pier imperfections.

8.3.6 Action effects for the design of foundationTable 8.12 gives the action effects corresponding to the loading combinations of the seismicdesign situation for the design of the foundations. The seismic action effects correspond toq ¼ 1.0. The action effects are given:

g for piers P1 and P2, at the base sectiong for abutments C0 and C3 at the mid-distance between the bearings

with the designation depicted in Figure 8.37. The signs of shear forces and bending momentsgiven are mutually compatible. However, as these effects (with the exception of the verticalaxial force Fz) are due predominantly to the seismic action, their signs and senses may bereversed.


Table 8.9. First- and second-order eccentricities due to geometric imperfections of the piers

Direction ei: m n ¼ NB/NEd ei,II/ei ei,II: m

X 0.063 19.65 1.161 0.073

Y 0.032 78.62 1.039 0.033

Table 8.10. Pier top displacements dEd

Pier top displacement dx: mm dy: m

EXþ 0.3EY 0.373 0.065

EYþ 0.3EX 0.110 0.197

217


Table 8.11. Action effects for the design of piers and abutments: q ¼ 1.50 ((EI )eff ¼ 0.3(EI)c)

Actions or combination thereof Fx ¼ Vx:

kN

Fy ¼ Vy:

kN

Fz ¼ N:

kN

Mx:

kN m

My:

kN m

Mz:

kN m

P1 G 5.4 �0.2 19539.3 6.8 216.9 �0.2

P2 G �5.4 �0.2 19539.3 6.8 �216.9 0.2

C0 G 0.0 0.2 3505.2 �0.4 0.0 0.0

C3 G 0.0 0.2 3505.2 �0.4 0.0 0.0

P1 EXþ 0.3EY 1254.4 187.4 28.8 7885.5 50803.5 342.8

P2 EXþ 0.3EY 1254.4 187.4 28.8 7885.5 50803.5 342.8

C0 EXþ 0.3EY 0.0 322.2 21.5 1134.3 0.0 0.0

C3 EXþ 0.3EY 0.0 322.2 21.5 1134.3 0.0 0.0

P1 EYþ 0.3EX 376.3 624.6 8.6 26285.1 15241.0 1142.5

P2 EYþ 0.3EX 376.3 624.6 8.6 26285.1 15241.0 1142.5

C0 EYþ 0.3EX 0.0 1073.9 6.4 3781.1 0.0 0.0

C3 EYþ 0.3EX 0.0 1073.9 6.4 3781.1 0.0 0.0

P1 EXþ 0.3EYþ P-D EC2 1254.4 187.4 28.8 8153.6 58576.4 342.8

P2 EXþ 0.3EYþ P-D EC2 1254.4 187.4 28.8 8153.6 58576.4 342.8

P1 EYþ 0.3EXþ P-D EC2 376.3 624.6 8.6 27178.8 17572.9 1142.5

P2 EYþ 0.3EXþ P-D EC2 376.3 624.6 8.6 27178.8 17572.9 1142.5

P1 EXþ 0.3EYþ P-D EC8 1254.4 187.4 28.8 9279.0 58298.5 342.8

P2 EXþ 0.3EYþ P-D EC8 1254.4 187.4 28.8 9279.0 58298.5 342.8

P1 EYþ 0.3EXþ P-D EC8 376.3 624.6 8.6 30508.3 17451.4 1142.5

P2 EYþ 0.3EXþ P-D EC8 376.3 624.6 8.6 30508.3 17451.4 1142.5

P1 EXþ 0.3EYþ P-D EC8þ Imperf 1254.4 187.4 28.8 9826.3 59393.1 342.8

P2 EXþ 0.3EYþ P-D EC8þ imperf 1254.4 187.4 28.8 9826.3 59393.1 342.8

P1 EYþ 0.3EXþ P-D EC8þ Imperf 376.3 624.6 8.6 31055.6 18546.0 1142.5

P2 EYþ 0.3EXþ P-D EC8þ Imperf 376.3 624.6 8.6 31055.6 18546.0 1142.5

P1 Gþ EXþ 0.3EYþ P-D EC8þ Imperf 1259.8 187.2 19568.0 9833.1 59610.0 342.5

P2 Gþ EXþ 0.3EYþ P-D EC8þ Imperf 1249.1 187.2 19568.0 9833.1 59176.2 343.0

C0 Gþ EXþ 0.3EY 0.0 322.3 3526.6 1134.0 0.0 0.0

C3 Gþ EXþ 0.3EY 0.0 322.3 3526.6 1134.0 0.0 0.0

P1 Gþ EYþ 0.3EXþ P-D EC8þ Imperf 381.7 624.4 19547.9 31062.4 18762.9 1142.3


C0 Gþ EYþ 0.3EX 0.0 1074.1 3511.6 3780.8 0.0 0.0

C3 Gþ EYþ 0.3EX 0.0 1074.1 3511.6 3780.8 0.0 0.0

Table 8.12. Action effects for the design of the foundation: q ¼ 1.0 ((EI )eff ¼ 0.3(EI)c)

Actions or combination thereof Fx:

kN

Fy:

kN

Fz:

kN

Mx:

kN m

My:

kN m

Mz:

kN m

P1, Gþ EXþ 0.3EYþ P-D EC8þ Imperf 1887.0 280.9 19582.4 13775.8 85011.7 513.9


C0, Gþ EXþ 0.3EY 0.0 483.4 3537.4 1701.1 0.0 0.0

C3 Gþ EYþ 0.3EX 0.0 1611.1 3514.8 5671.3 0.0 0.0

218

8.3.7 Verification of the piers8.3.7.1 ULS in flexure and axial forceDimensioning of the reinforcement at the base section is as follows: the nominal cover isc ¼ 50 mm and the estimated distance of the bar centre from the surface is 80 mm. Fordesign action effects, NEd ¼ 19 568 kN, My ¼ 59 610 kN m and Mx ¼ 9833 kN m ! As,req ¼67 800 mm2. The selected longitudinal reinforcement is:

g at the exterior perimeter: 62 128 (38 100 mm2)g at the interior perimeter: 49 128 (30 100 mm2).

Figure 8.38 depicts the interaction diagram for the design of the base section.


Figure 8.37. Positive sense and direction of forces and moments for the foundation design

Longitudinaldirection X

Transversedirection Y

Verticaldirection Z

Fx

Fy

Fz

Mz

MyMx

Figure 8.38. Interaction diagram for the design of the pier base section

–140 000

–130 000

–120 000

–110 000

–100 000

–90 000

–80 000

–70 000

–60 000

–50 000

–40 000

–30 000

–20 000

–10 000

0

10 000

20 000

30 000

40 000

0 10 000 20 000 30 000 40 000 50 000 60 000 70 000 80 000

Axi

al f

orce

: kN

Moment: kN m

ρ = 1.5%ρ = 1.0%

219

8.3.7.2 ULS in shearFor limited ductile behaviour the pier seismic shear force from the analysis is multiplied byq ¼ 1.5 (see Section 6.3.2 and Table 6.1), giving

VEd ¼ 1:5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1259:22 þ 187:22

p¼ 1910 kN

while the resistance, VRd, is divided by gBd1 ¼ 1.25 (see note 4 in Table 6.1).

The shear resistance at diagonal compression failure of a member with annular section is

VR;max ¼p

4twðD� 2cÞðnfcdÞ sin 2d

with 0.4 � tan d � 1 (228 � d � 458)

and

n ¼ 0:6 1� fck½MPa�250

� �¼ 0:6 1� 35

250

� �¼ 0:516

VR;max ¼1

1:25

p

4� 0:4� ð4:0� 2� 0:08Þ � 0:516� 0:85� 35 000

1:5sin 2d

¼ 9877 sin 2d

¼ 6811 kN > VEd

after division by gBd1 ¼ 1.25 and for the lower limit of tan d ¼ 0.4.

Neglecting the contribution of the axial load to shear resistance as small in this slender and lightlyloaded pier, that of the transverse steel, VRs, is

VRs ¼p

2

Asw

shfywdðD� 2cÞ cot d

after division by gBd1 ¼ 1.25, the lower limit value of 0.4 for tan d gives

VRs ¼1

1:25

p

2

Asw

sh

0:5

1:15ð4� 0:16Þ � 2:5 � VEd ! Asw

sh� 182mm2=m

8.3.8 Ductility requirements8.3.8.1 Confining reinforcementThe normalised axial force is

hk ¼ 19 560/(35 000 � 4.52) ¼ 0.1236 . 0.08

so, confinement is required (see note 15 in Table 6.1).

For circular spirals and limited ductile behaviour the required mechanical reinforcement ratio,vw,req, of confinement reinforcement is (see Table 6.1)

vw,req ¼ 0.39(Ac/Acc)hkþ 0.18( fyd/fcd)(rL� 0.01)

and

vw,min ¼ 0.12

vw,req ¼ (4.52/3.39) � 0.39 � 0.1236þ 0.18(500/1.15)/(0.85 � 35/1.5)

� (0.015� 0.01) ¼ 0.085.


220

So, the mechanical reinforcement ratio is

vwd,c ¼ max(0.085; 0.12) ¼ 0.12

and the required volumetric ratio of confining reinforcement is

rw ¼ vwd,c( fcd/fyd) ¼ 0.12 � (0.85 � 35/1.5)/(500/1.15) ¼ 0.005474

The required area of confining reinforcement (one leg) is

Asp/sL ¼ rwAcc/(pDsp) ¼ 0.005474 � 3.39/(p � 3.384) ¼ 0.001746 m2/m ¼ 1746 mm2/m

8.3.8.2 Prevention of buckling of longitudinal barsThe maximum hoop spacing, sL, of transverse reinforcement to prevent buckling of vertical barsis ddbL, where d ¼ 2.5( ftk/fyk)þ 2.25 � 5 (see Table 6.1). For Class B steel, ftk/fyk � 1.08, and

d ¼ 2.5 � 1.08þ 2.25 ¼ 4.95

So d ¼ 5, and

sLreq ¼ ddL ¼ 5 � 28 mm ¼ 140 mm

As pointed out in Section 4.4.1.5 of this Guide, at the inside face of annular piers, hoops are notefficient in preventing buckling of vertical bars or in confining the concrete, as they cannot offertensile hoop action. If vertical bars yield in compression or the crushing strain of unconfinedconcrete is exceeded at the inside face under the design seismic action (which is not the case inthis example), cross-ties as for straight boundaries are necessary. 116/110 (1828 mm2/m) isfinally chosen.

8.3.9 Bearings and jointsTable 8.13 gives the seismic deformation and force demands on the bearings. An example of thedesign for overlapping length at the movable supports and for roadway joints has been given inSections 8.2.11.2 and 8.2.11.3, respectively.

8.4. Example of seismic isolation8.4.1 IntroductionThis section covers the design of the bridge in the example in Section 8.3 but with a special seismicisolation system capable of resisting high seismic loads. The seismic isolation system selected in


Table 8.13. Bearing deformations and force demands

Bearing Direction u1: m u2: m u3: m u1: rad u2: rad u3: rad N: kN V2: kN V3: kN

M1a X Max �0.007 0.001 0.003 0.001 0.000 0.015 �6659 674 0

M1a X Min �0.007 �0.001 �0.003 �0.001 �0.001 �0.014 �7271 �669 0

M1a Y Max �0.006 0.000 0.009 0.003 0.000 0.005 �6273 375 0

M1a Y Min �0.008 0.000 �0.010 �0.003 �0.001 �0.004 �7657 �370 0

M1b X Max �0.007 0.001 0.000 0.001 0.001 0.015 �6659 674 159

M1b X Min �0.007 �0.001 0.000 �0.001 0.000 �0.014 �7271 �669 �159

M1b Y Max �0.006 0.000 0.001 0.003 0.003 0.005 �6274 375 530

M1b Y Min �0.008 0.000 �0.001 �0.003 �0.002 �0.004 �7657 �370 �529

A1a X Max �0.002 0.396 0.000 0.001 0.001 0.003 �1532 0 174

A1a X Min �0.002 �0.380 0.000 �0.001 �0.001 0.002 �1974 0 �187

A1a Y Max �0.001 0.126 0.001 0.005 0.002 0.003 �1096 0 595

A1a Y Min �0.002 �0.111 �0.001 �0.005 �0.002 0.002 �2409 0 �609

A1b X Max �0.002 0.396 0.000 0.001 0.001 0.003 �1531 0 187

A1b X Min �0.002 �0.380 0.000 �0.001 �0.001 0.002 �1974 0 �174

A1b Y Max �0.001 0.126 0.001 0.005 0.002 0.003 �1096 0 609

A1b Y Min �0.002 �0.111 �0.001 �0.005 �0.002 0.002 �2409 0 �596

221

this case consists of triple friction pendulum bearings (see Section 7.4.3.3). The analysis of theseismic isolation system is carried out with both the fundamental mode method and nonlineartime-history analysis. The results of the two analysis methods are compared.

8.4.2 Bridge configuration – Design concept8.4.2.1 Bridge layoutThe bridge has a composite steel–concrete continuous deck, with spans of 60þ 80þ 60 m andtwo solid rectangular 10.0 m tall piers. The lower 8.0 m of the pier has rectangular cross-section 5.0 m by 2.5 m. The seismic isolation bearings are supported on a widened pier headwith rectangular plan 9.0 m � 2.5 m and 2.0 m depth. The pier concrete class is C35/45.

Figure 8.39 shows the elevation, and Figure 8.40 the typical deck cross-section of the bridge.Figure 8.41 presents the layout of the seismic isolation bearings, and Figure 8.42 that of the piers.

The large stiffness of the squat piers, in combination with the high design ground acceleration onrock (agR ¼ 0.40g) leads to the selection of seismic isolation. This selection offers the followingadvantages:

g a large reduction of constraints due to imposed deck deformationg practically equal – and therefore minimised – seismic action effects on the two piers (this

would have been achieved even if the piers had unequal heights)g drastic reduction in the seismic forces.

The additional damping offered by the isolators keeps the displacements to a cost-effective level.

8.4.2.2 Seismic isolation systemThe seismic isolation system consists of eight sliding bearings with a spherical sliding surface, ofthe triple friction pendulum system (triple FPS) type. Two triple FPS bearings support the deckat each abutment, C0 and C3, or pier, P1 and P2. The triple FPS bearings allow displacements in


Figure 8.39. Bridge elevation

C0

60.00 m 60.00 m

2.5 m2.5 m

Triple FPS Triple FPS Triple FPS Triple FPS

10 m

10 m

80.00 m

P1 P2 C3

Figure 8.40. Deck section

Girder No. 1 Girder No. 2

12 000

7000

Axl

e of

the

brid

ge

2800

600

1100

1100

222

both the longitudinal and transverse directions with a nonlinear frictional force displacementrelation. The approximate bearing dimensions are:

g at the piers: 1.20 m � 1.20 m in plan and 0.40 m in heightg at the abutments: 0.90 m � 0.90 m in plan and 0.40 m in height.


Figure 8.41. Layout of seismic isolation bearings

C0_L

C0_R

P1_L

P1_R

P2_L

P2_R

C3_L

C3_R

X

Y

Figure 8.42. Layout of the piers

Longitudinal section at piers: Transverse section at piers:

9.00B

C

B

C

AA

B

C

B

C

A

Pier cross-section A–A: Pier cross-section B–B:

Pier head plan view C–C:

5.00

2.50

2.50

2.50

9.00

7.00

9.00

A

5.00

10.0

0

10.0

0

8.00

2.00

8.00

2.00

223

The layout and the labels for the seismic isolation bearings are depicted in Figure 8.41 (X is thelongitudinal direction and Y is the transverse direction). Figure 7.12 shows a typical triple FPSbearing, and Figure 7.14 its typical force–displacement relationship.

The nominal properties of the selected triple FPS bearings for seismic analysis are:

g the effective dynamic friction coefficient: md ¼ 0.061 (+16% variability of the nominalvalue)

g the effective radius of the sliding surface: Rb ¼ 1.83 mg the effective yield displacement: Dy ¼ 0.005 m.

8.4.3 Design for horizontal non-seismic actions8.4.3.1 Imposed horizontal loads – braking forceTable 8.14 gives the distribution of the reactions on the supports due to permanent loadsaccording to the gravity load analysis of the bridge. The time variation of permanent reactionsdue to creep and shrinkage is very small. So, the reactions due to permanent loads are consideredconstant in time.

The minimum value of the longitudinal force at sliding of the whole deck on the bearings iscalculated from the minimum deck weight and the minimum coefficient of friction at thebearings as

Fy,min ¼ 25 500 � 0.051 � 1300 kN

This force is not exceeded by the braking load of Fbr ¼ 900 kN. Therefore, the pier bearings donot slide due to braking. As the horizontal stiffness of the abutments is very high, sliding willoccur at the abutments, associated with the development of friction reactions mWa, where Wa

is the corresponding reaction due to permanent loads. The appropriate static system for thisloading therefore has a pinned connection between the pier tops and the deck and a slidingconnection over the abutments with the above friction reactions (Figure 8.43). The total forcesat the abutments may be calculated from the corresponding displacement of the deck and theforce–displacement relation of the bearings (additional elastic reaction Wa/R (Figure 8.43)). Asimilar situation occurs for the transverse wind loading.


Table 8.14. Reactions at supports due to permanent loads (in MN, both girders)

Self-weight

after

construction

Minimum

equipment

load

Maximum

equipment

load

Total with

minimum

equipment

Total with

maximum

equipment

Time variation

due to creep

and shrinkage

C0 2.328 0.664 1.020 2.993 3.348 �0.172

P1 10.380 2.440 3.744 12.819 14.123 0.206

P2 10.258 2.441 3.745 12.699 14.003 0.091

C3 2.377 0.664 1.019 3.041 3.396 �0.126

Sum of

reactions

25.343 6.209 9.528 31.552 34.871 0.000

Figure 8.43. Structural system for imposed horizontal loads

Sliding

Pinnedconnection

Pinnedconnection

Sliding

µ*Wa

µ*Wa

224

8.4.3.2 Imposed deformations that can cause sliding of the pier bearingsAssuming the structural system in the longitudinal direction to be the same as above, the imposeddeformation that can cause sliding at the pier bearings is calculated from the minimum slidingload of the bearings and the stiffness of the piers:

g minimum sliding load:

Fy,min ¼ 0.051 � 12 699 ¼ 648 kN

g pier stiffness:

Kpier ¼ 3EI/h3 ¼ 3 � 34 000 000 � [9 � (2.5 m)3/12]/(10)3 ¼ 1 195 313 kN/m

g minimum displacement of deck at the pier top to cause sliding:

dmin ¼ Fy,min/K ¼ 648/1 195 313 ¼ 0.5 mm

This displacement is very small and is practically exceeded even by small temperature-imposeddeformations. Consequently, sliding occurs at the bearings of at least one of the piers, underdeformations induced by temperature.

8.4.4 Imposed deformation due to temperature variationA conservative approach for estimating forces and displacements for this case is the following:because of the inevitable difference between the sliding friction coefficients of the bearings ofthe two piers, albeit small, one of the two pier supports is assumed not to slide under non-seismic conditions. The calculation of horizontal support reactions and displacements shouldtherefore be based on two systems, with the deck pinned on one of the two piers alternatively.On the other moving supports, an elastic connection is introduced between the deck and thesupport, with stiffness equal to Kpb ¼ Wp/R (see Figure 7.10, R ¼ Rb ¼ 1.83 m), calculated onthe basis of a value of Wp equal to the corresponding permanent load. At these supports,friction forces equal to mWp, should also be introduced, where m is either the minimum or themaximum value of friction, with opposite signs on the deck and the supporting element, anddirections compatible with the corresponding sliding deformation at the support, as shown inFigure 8.44. Both displacements and forces can be derived from these systems.

8.4.5 Superposition of the effects of the braking load and imposed deckdeformations

The superposition of the effects of the braking load and imposed deformations should be donewith caution, as the two cases correspond in fact to a nonlinear response of the system due to the


Figure 8.44. Structural system for imposed deformations

Pinnedconnection Sliding

SlidingSliding

Friction forces µ*WpElastic connection Kpb = Wp/R

225

involvement of the friction forces. The application of the braking force on the system on whichimposed deformations are already acting causes, in general, a redistribution of the friction forcesestimated above: namely that the original friction forces, acting on one of the piers and the corre-sponding abutment, which had the same direction with the braking force, are reversed, startingfrom the abutment, where full reversal, amounting to a force of 2mWa, will take place. Theremaining part of the braking force,

Fbr� 2mWa ¼ 900� 2 � 0.051 � 2993 ¼ 595 kN

should be equilibrated mainly by a decrease in the reaction of the relevant pier. This decrease isassociated with a displacement of the deck in the direction of the breaking force, an upper boundof which can be estimated as

Dd ¼ (Fbr� 2mWa)/Kpier ¼ 595/1 195 313 ¼ 0.0005 m ¼ 0.5 mm

The corresponding upper bounds of the force increase on the reactions of the opposite pier andabutment amounts to

DdWp/R ¼ 0.0005 � 12 699/1.83 ¼ 3.5 kN

and

DdWa/R ¼ 0.0005 � 2993/1.83 ¼ 0.8 kN

respectively. Consequently, for this example both the displacement Dd and the force increases canbe neglected.

A comparison with the forces and displacements resulting from the seismic design situationshows that the latter are always governing in a bridge with seismic isolation.

8.4.6 Design seismic action8.4.6.1 Design spectraThe project-dependent parameters defining the horizontal elastic spectrum (see Section 3.1.2.3 ofthis Guide) are:

g type 1 horizontal elastic response spectrumg no near source effectsg importance factor gI ¼ 1.0g reference peak ground acceleration for type A ground agR ¼ 0.4gg design ground acceleration for type A ground ag ¼ gIagR ¼ 0.40gg ground type B with soil factor S ¼ 1.20, periods TB ¼ 0.15 s, TC ¼ 0.5 s and TD ¼ 2.5 s.

The value of the period TD is particularly important for the safety of bridges with seismic iso-lation because it affects proportionally the estimated displacement demands. For this reason,the National Annex to Part 2 of Eurocode 8 may specify a value of TD specifically for thedesign of bridges with seismic isolation that is more conservative (longer) than the valueascribed to TD in the National Annex to Part 1 of Eurocode 8 (CEN, 2004b). For this particularexample, the selected value is TD ¼ 2.5 s, which is longer than the value TD ¼ 2.0 s recommendedin Part 1 of Eurocode 8.

The project-dependent parameters that define the vertical response spectrum (Section 3.1.2.4)are:

g type 1 vertical elastic response spectrumg ratio of the design ground acceleration in the vertical direction to the design ground

acceleration in the horizontal direction avg/ag ¼ 0.9g periods TB ¼ 0.05 s, TC ¼ 0.15 s and TD ¼ 1 s.

The horizontal and vertical design spectra are shown in Figures 8.45 and 8.46, respectively.


226

8.4.6.2 Accelerograms for nonlinear time-history analysisFor the time-history representation of the seismic action, seven ground motion time histories areused (EQ1 to EQ7), each consisting of a pair of horizontal ground motion time-history com-ponents and a vertical ground motion time-history component. Each component is producedby modifying natural, recorded accelerograms to match the Eurocode 8 elastic spectrum(semi-artificial accelerograms). The modification procedure consists of applying unit impulsefunctions that iteratively correct the accelerogram in order to better match the targetspectrum. No scaling of the individual components is required to ensure compatibility withthe Eurocode 8 spectrum, as each component is already compatible with the correspondingspectrum owing to the applied modification procedure.

Figure 8.47 shows an example. The original record is from the Loma Prieta (CA) earthquake,Corralitos 000 record (magnitude Ms ¼ 7.1, distance to fault ¼ 5.1 km, USGS ground type B).For the produced accelerogram the acceleration, velocity and displacement time histories areshown. Comparing the initial recorded accelerogram with the final semi-artificial one, it isconcluded that the modification method does not alter significantly the natural waveform.The 5%-damped pseudo-acceleration and displacement response spectra of the semi-artificial


Figure 8.45. Horizontal elastic response spectrum

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Period: s

Spec

tral

acc

eler

atio

n: g

Damping 5%

Figure 8.46. Vertical elastic response spectrum

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

Damping 5%

0.0 0.5 1.0 1.5 2.0 2.5 3.0Period: s

Spec

tral

acc

eler

atio

n: g

227

accelerogram are also compared with the corresponding Eurocode 8 spectra, matching the targetspectrum for the full range of periods shown.

The consistency of the ensemble of ground motions is verified in accordance with Section 3.1.4, asdepicted in Figures 8.48 and 8.49 for horizontal and vertical components, respectively. It isverified that the selected accelerograms are consistent with the Eurocode 8 spectrum for all


Figure 8.47. Example of horizontal semi-artificial accelerogram produced by modifying natural record

Modified record:

–0.60–0.40–0.200.000.200.400.600.80

–1.00

–0.80

–0.60

–0.40

–0.20

0.00

0.20

0.40

–0.300

–0.200

–0.100

0.000

0.100

0.200

0.300

0.400

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0Period: s

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0Period: s

Damping: 5%

Damping: 5%

0.000

0.100

0.200

0.300

0.400

0.500

0.600

Original record:

–0.60–0.40–0.200.000.200.400.600.80

0 5 10 15 20 25 30 35 40 45

Time: s

0 5 10 15 20 25 30 35 40 45

Time: s

0 5 10 15 20 25 30 35 40 45

Time: s

0 5 10 15 20 25 30 35 40 45

Time: s

Acc

eler

atio

n:g

Acc

eler

atio

n:g

Vel

ocity

: m/s

Dis

plac

emen

t: m

Dis

plac

emen

t: m

Pseu

do a

ccel

erat

ion:

g

Maximum acceleration: 0.560 gat time t = 2.580 s

Maximum velocity: 83.734 cm/sat time t = 2.545 s

Maximum displacement: 35.575 cmat time t = 2.305 s

Vmax/Amax: 0.153 s

Acceleration RMS: 0.060 gVelocity RMS: 9.017 cm/sDisplacement RMS: 16.542 cm

Arias intensity: 2.186 m/sCharacteristic intensity (Ic): 0.092Specific energy density: 3243.860 cm2/sCumulative absolute velocity (CAV): 1035.219 cm/s

Acceleration spectrum intensity (ASI): 0.482 g sVelocity spectrum intensity (VSI): 213.207 cm

Sustained maximum acceleration (SMA): 0.257 gSustained maximum velocity (SMV): 27.014 cm/s

Effective design acceleration (EDA): 0.568 g

A95 parameter: 0.553 g

Predominant period (Tp): 0.380 sMean period (Tm): 0.630 s

228

periods between 0 and 5 s for the horizontal components or between 0 and 3 s for the verticalcomponent. Therefore, consistency is established for isolation systems with an effective periodTeff , 5/1.5 ¼ 3.33 s and a prevailing vertical period TV , 3/1.5 ¼ 2 s, which are fulfilled forthe isolation system of this example.

8.4.7 Modelling of the structural system for seismic analysis8.4.7.1 Structural model8.4.7.1.1 Bridge modelFor the purpose of nonlinear time-history analysis, the bridge is modelled in 3D with computercode SAP 2000, fully accounting for the geometry and spatial distribution of the stiffness andmass of the bridge. The superstructure and the substructure of the bridge are modelled withlinear prismatic beam elements with properties in accordance with the actual cross-section ofthe element. Masses are lumped at the nodes of the model. Where necessary, kinematicconstraints were applied to establish proper connection of the elements. The effect of thefoundation flexibility is ignored, and piers are taken as fixed at their base. The model of thebridge for the time-history analysis is shown in Figure 8.50.


Figure 8.48. Verification of consistency in the mean of accelerograms used for the horizontal

components with the Eurocode 8 spectrum

Spec

tral

acc

eler

atio

n: g

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

Damping 5%

Average SRSS spectrum ofensemble of earthquakes1.3 × elastic spectrum

Period: s0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Figure 8.49. Verification of consistency between target spectrum and mean spectrum of accelerograms

used for the vertical component

Period: s

Spec

tral

acc

eler

atio

n: g

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

Damping 5%

Average ensemble spectrum0.9 × elastic spectrum

0.0 0.5 1.0 1.5 2.0 2.5 3.0

229

8.4.7.1.2 Isolator modelThe triple FPS bearings are modelled with nonlinear hysteretic friction elements, connecting deckand pier nodes at the locations of the corresponding bearing. In SAP 2000 the behaviour of theisolator elements in the horizontal direction follows a coupled frictional law based on the Bouc–Wen model. In the vertical direction the behaviour of the isolators corresponds to stiff supportacting only in compression. The actual vertical load of the bearings at each time instant is takeninto account to establish the force–displacement relation of the bearing. The effects of bridgedeformation and vertical seismic action are taken into account in the estimation of verticalbearing loads.

8.4.7.1.3 Effective pier stiffnessThe effective pier stiffness is taken as equal to the uncracked gross section stiffness. Because thestiffness of the piers is much larger than the effective stiffness of the isolation system, piers maybe considered as rigid without significant loss of accuracy. This approach is followed in thefundamental mode analysis, presented later with detailed manual calculations. In the nonlineartime-history analysis, the effective pier stiffness is included.

8.4.8 Bridge loads for the seismic analysis8.4.8.1 Permanent loadsPermanent action effects vary little with time due to creep and shrinkage (see Table 8.14).Because of this small variation, only the action effects after fully developed creep and shrinkageare considered. According to the results of the analysis, the longitudinal displacements due topermanent actions are approximately 8 mm at the abutments and 3 mm at the piers, bothtowards the centre.

8.4.8.2 Quasi-permanent traffic loadsAccording to Part 2 of Eurocode 8, for road bridges with severe traffic (i.e. bridges of motorwaysand other roads of national importance) the quasi-permanent value c2,1Qk,1 of the traffic actionto be considered in the seismic design situation is calculated from the UDL system of trafficmodel LM1, with the value of the combination factor c2,1 ¼ 0.2. The division of the carriagewayin three notional lanes according to clause 4.2.3 of EN 1991-2:2003 (CEN, 2003a) is shown inFigure 8.51.

The values of the UDL system of LM1 are calculated with the adjustment factor for UDLaq ¼ 1.0:

g lane 1:

aqq1,k ¼ 3 m � 9 kN/m2 ¼ 27.0 kN/m


Figure 8.50. Bridge model for time–history analysis

230

g lane 2:

aqq2,k ¼ 3 m � 2.5 kN/m2 ¼ 7.5 kN/m

g lane 3:

aqq3,k ¼ 3 m � 2.5 kN/m2 ¼ 7.5 kN/m

g residual area:

aqqr,k ¼ 2 m � 2.5 kN/m2 ¼ 5.0 kN/m

Total load ¼ 47.0 kN/m.

The quasi-permanent traffic load per unit of length of the bridge in the seismic design situation is:

c2,1Qk,1 ¼ 0.2 � 47 kN/m ¼ 9.4 kN/m

The reactions of the deck supports for the quasi-permanent traffic load are presented inTable 8.15, according to the analysis of the bridge for the UDL system of LM1.

8.4.8.3 Total weight on the deck in the seismic design situationThe weight Wd of the deck in the seismic design situation includes the permanent loads and thequasi-permanent value of the traffic loads:

Wd ¼ dead loadþ quasi-permanent traffic load ¼ 34 871þ 1880 ¼ 36 751 kN


Table 8.15. Total reactions due to the quasi-permanent traffic loads

Reactions due to the quasi-permanent

traffic load (for both girders): MN

C0 0.201

P1 0.739

P2 0.739

C3 0.201

Sum of reactions 1.880

Figure 8.51. Division of carriageway into notional lanes

3.503.50

Girder No. 2Girder No. 1

Lane No. 1 Lane No. 2 Lane No. 3 Residual area

3.00

1.00 0.50

3.00 3.00 2.00

Axl

e of

the

brid

ge

Modelled girder

231

8.4.8.4 Thermal actionThe minimum ambient air temperature – with a mean return period of 50 years – for the structureis assumed equal to Tmin ¼ �208C. The maximum ambient air temperature – again with a meanreturn period of 50 years – is assumed equal to Tmax ¼ þ408C. The initial temperature is takenas T0 ¼ þ108C. The uniform bridge temperature components Te,min and Te,max are calculatedfrom Tmin and Tmax using Figure 6.1 in EN 1991-1-5:2003 (CEN, 2003b), for a type 2 deck(i.e. composite). The ranges of the uniform bridge temperature component are calculated as:

g maximum contraction range:

DTN,con ¼ T0� Te,min ¼ 108C� (�208Cþ 58C) ¼ 258C

g maximum expansion range:

DTN,exp ¼ Te,max� T0 ¼ (þ408Cþ 58C)� 108C ¼ 358C.

According to note 2 in clause 6.1.3.3(3) of EN 1991-1-5:2003, for the design of bearings andexpansion joints the temperature ranges are increased as follows:

g maximum contraction range for bearings:

DTN,conþ 208C ¼ 258Cþ 208C ¼ 458C

g maximum expansion range for bearings:

DTN,expþ 208C ¼ 358Cþ 208C ¼ 558C

8.4.9 Design properties of the isolators8.4.9.1 Upper- and lower-bound design propertiesThe nominal values of the design properties of the isolators have been presented in Section8.4.2.2. They are:

g the effective dynamic friction coefficient md ¼ 0.061 (+16% variability with respect to thenominal value)

g the effective radius of sliding surface Rb ¼ 1.83 mg the effective yield displacement Dy ¼ 0.005 m.

As pointed out in Section 7.2.2.4, two sets of design properties of the isolating system areconsidered:

g the upper-bound design properties (UBDP)g the lower-bound design properties (LBDP).

A separate analysis is performed for each one. For the selected isolation system, only the effectivedynamic friction coefficient md is subject to variability of its design value. The effective radius ofthe sliding surfaceRb is a geometric property not subject to variability. The UBDP and the LBDPfor md are calculated according to Annexes J and JJ of Part 2 of Eurocode 8:

g Nominal value: md ¼ 0.061+ 16% ¼ 0.051 to 0.071g LBDP: md,min ¼ min DPnom ¼ 0.051g UBDP: according to Annexes J and JJ of Part 2 of Eurocode 8.

8.4.9.2 Minimum isolator temperature for seismic design

Tmin,b ¼ c2Tminþ DT1 ¼ 0.5 � (�208C)þ 5.08C ¼ �5.08C

where c2 ¼ 0.5 is the combination factor for thermal actions for the seismic design situation;Tmin ¼ �208C is the minimum shade air temperature at the bridge location having an annualprobability of exceedance of 0.98, per clause 6.1.3.2 of EN 1991-1-5:2003; and DT1 ¼ þ5.08C


232

is the correction temperature for composite bridge decks according to Table J.1N of Part 2 ofEurocode 8.

8.4.9.3 lmax factors per Annex JJ of Part 2 of Eurocode 8f1 – ageing: lmax,f1 ¼ 1.1 (Table JJ.5, for normal environment, unlubricated PTFE andprotective seal).f2 – temperature: lmax,f2 ¼ 1.15 (Table JJ.6 for Tmin,b ¼ �5.08C, unlubricated PTFE).f3 – contamination lmax,f3 ¼ 1.1 (Table JJ.7 for unlubricated PTFE and sliding surface facingboth upwards and downwards).f4 – cumulative travel lmax,f4 ¼ 1.0 (Table JJ.8 for unlubricated PTFE and cumulativetravel � 1.0 km).Combination factor cfi ¼ 0.70 for importance class II (Table J.2).Combination value of lmax factors: lU,fi ¼ 1þ (lmax,fi� 1)cfi (Eq. (J.5)).f1 – ageing: lU,f1 ¼ 1þ (1.1� 1) � 0.7 ¼ 1.07.f2 – temperature: lU,f2 ¼ 1þ (1.15� 1) � 0.7 ¼ 1.105.f3 – contamination lU,f3 ¼ 1þ (1.1� 1) � 0.7 ¼ 1.07.f4 – cumulative travel lU,f4 ¼ 1þ (1.0� 1) � 0.7 ¼ 1.0.

8.4.9.4 Effective UBDP

UBDP ¼ max DPnomlU,f1lU,f2lU,f3lU,f4 (Eq. (J.4))

md,max ¼ 0.071 � 1.07 � 1.105 � 1.07 � 1.0 ¼ 0.071 � 1.265 ¼ 0.09

Therefore, the range of variation of the effective friction coefficient md is from 0.051 to 0.09.

8.4.10 Analysis with the fundamental mode method8.4.10.1 GeneralThe fundamental mode method of analysis is described in Section 7.5.3 of this Guide. In each ofthe two horizontal directions of the seismic action the response of the isolated bridge is deter-mined considering the superstructure as a linear SDoF system using:

g the effective stiffness of the isolation system Keff

g the effective damping of the isolation system jeffg the mass of the superstructure Md

g the spectral acceleration Se(Teff, jeff) corresponding to the effective period Teff and theeffective damping jeff.

The effective stiffness at each support location is the composite stiffness of the isolator unit andthe corresponding substructure, per Eqs (D2.10) or (D7.37). In this particular example, the stiff-ness of the piers is much larger than the stiffness of the isolators, and the contribution of the pierstiffness may be ignored (see Eq. (D7.32)). The effective damping is derived from Eq. (D7.33) atthe design displacement dcd. The value of dcd is calculated from the effective period Teff and theeffective damping jeff, both of which depend on the value of the still unknown design displace-ment, dcd. Therefore, the fundamental mode method is in general an iterative procedure, where avalue is assumed for the design displacement in order to calculate Teff and jeff, and a betterapproximation of dcd is then calculated from the design spectrum using Teff and jeff. The newvalue of dcd is used as the initial value for the new iteration. The procedure converges rapidly.

In this example, hand calculations are presented for the fundamental mode analysis for bothLDBP and UBDP. Only the first and the last iteration are presented.

8.4.10.2 Fundamental mode analysis for LBDPThe analysis below corresponds to the LBDP of isolators (i.e. md ¼ 0.051). The iteration steps arepresented in detail. It is recalled that the weight is Wd ¼ 36 751 kN (see Section 8.4.8.3).

Iteration 1Assume a value for the design displacement dcd: assume dcd ¼ 0.15 m.Effective stiffness of the isolation system Keff (ignoring the piers):

Keff ¼ F/dcd ¼ Wd[mdþ dcd/Rb]/dcd ¼ 36 751 � [0.051þ 0.15/1.83]/0.15 ¼ 32 578 kN/m


233

Effective period of the isolation system Teff (Eq. (D7.34)):

Teff ¼ 2p

ffiffiffiffiffim

Keff

r¼ 2p


32 578

r¼ 2:13 s

Dissipated energy per cycle ED:

ED ¼ 4Wdmd(dcd�Dy) ¼ 4 � 36 751 � 0.051 � (0.15� 0.005) ¼ 1087.09 kN m

Effective damping jeff:

jeff ¼P

ED,i/(2pKeffdcd2 ) ¼ 1087.09/2p � 32 578(0.15)2] ¼ 0.36

heff ¼p[0.1/(0.05þ jeff)] ¼ 0.591

Design displacement dcd (Table 8.1 in Part 2 of Eurocode 8):

dcd ¼ (0.625/p2)agSheffTeffTC ¼ (0.625/p2) (0.4 � 9.81) � 1.2 � 0.591

� 2.13 � 0.5 ¼ 0.188 m

Assumed displacement: 0.15 m. Calculated displacement: 0.188 m ) second iteration.

Iteration 2Assume a new value for the design displacement dcd: assume dcd ¼ 0.22 m.Effective stiffness of the isolation system Keff:

Keff ¼ 36 751 � (0.051þ 0.22/1.83)/0.22 m ¼ 28 602 kN/m

Effective period of the isolation system Teff:

Teff ¼ 2p

ffiffiffiffiffim

Keff

r¼ 2p


28 602

r¼ 2:27 s


ED ¼ 4 � 36 751 � 0.051 � (0.22� 0.005) ¼ 1611.90 kN m


jeff ¼ 1611.9/[2p 28 602 � (0.22)2] ¼ 0.1853

heff ¼p[0.1/(0.05þ jeff)] ¼ 0.652

Design displacement dcd:

dcd ¼ (0.625/p2) � 0.4 � 9.81 � 1.2 � 0.652 � 2.27 � 0.5 ¼ 0.22 m

Assumed displacement ¼ calculated displacemen ) convergence.Spectral acceleration Sa:

Sa ¼ 2.5(TC/Teff)heff

agS ¼ 2.5 � (0.5/2.27) � 0.652 � 0.4 � 1.2 ¼ 0.172g

Isolation system shear force Vd:

Vd ¼ Keffdcd ¼ 28 602 � 0.22 ¼ 6292 kN


234

8.4.10.3 Fundamental mode analysis for UBDPFor the UBDP of isolators: md ¼ 0.09.

Iteration 1Assume a value for the design displacement dcd: assume dcd ¼ 0.15 m.Effective stiffness of the isolation system Keff:

Keff ¼ 36 751 � (0.09þ 0.15/1.83)/0.15 ¼ 42 133 kN/m


Teff ¼ 2p

ffiffiffiffiffim

Keff

r¼ 2p


42 133

r¼ 1:87 s


ED ¼ 4 � 36 751 � 0.09 � (0.15� 0.005) ¼ 1984.55 kN m


jeff ¼ 1984.55/[2p � 42 133 � (0.15)2] ¼ 0.333

heff ¼p[0.1/(0.05þ jeff)] ¼ 0.511

Calculate design displacement dcd:

dcd ¼ (0.625/p2) � 0.4 � 9.81 � 1.2 � 0.511 � 1.87 � 0.5 ¼ 0.142 m

Assumed displacement: 0.15 m. Calculated displacement: 0.142 m ) do another iteration.

Iteration 2Assume a new value for the design displacement dcd: assume dcd ¼ 0.14 m.Effective stiffness of the isolation system Keff:

Keff ¼ 36 751 � (0.09þ 0.14 m/1.83 m)/0.14 ¼ 43 541 kN/m


Teff ¼ 2p

ffiffiffiffiffim

Keff

r¼ 2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi36 751½kN�=9:81½m=s2�

43 541 ½kN=m�

s¼ 1:84 s


ED ¼ 4 � 36 751 � 0.09 � (0.14� 0.005) ¼ 1799.32 kN m


jeff ¼ 1799.32/[2p � 43 541 � (0.14 m)2] ¼ 0.331

heff ¼p[0.1/(0.05þ jeff)] ¼ 0.512

Calculate design displacement dcd:

dcd ¼ (0.625/p2) � 0.4 � 9.81 � 1.2 � 0.512 � 1.84 � 0.5 ¼ 0.14 m

Assumed displacement: 0.14 m. Calculated displacement: 0.14 m ) convergence achieved.Spectral acceleration Sa:

Sa ¼ 2.5 � (0.5/1.84) � 0.512 � 0.4 � 1.2 ¼ 0.166g


235

Isolation system shear force Vd:

Vd ¼ 43 541 � 0.14 ¼ 6096 kN

Typically, LBDP analysis leads to maximum displacements of the isolating system, and UBDPanalysis leads to maximum forces in the substructure and the deck. However, the latter is notalways true, as demonstrated by this example. In this particular example, the LBDP analysisleads to larger shear force (Vd ¼ 6292 kN) in the substructure than the UBDP analysis(Vd ¼ 6096 kN). This is because the increase in forces due to the reduced effective damping inthe LBDP analysis (jeff ¼ 0.1853 for LBDP versus jeff ¼ 0.331 for UBDP) is more dominantthan the reduction in forces due to the increased effective period in the LBDP analysis(Teff ¼ 2.27 s in LBDP versus Teff ¼ 1.84 s in UBDP).

8.4.11 Nonlinear time-history analysis8.4.11.1 Analysis methodThe nonlinear time-history analysis for the ground motions of the design seismic action isperformed with direct time integration of the equation of motion using the Newmark constantacceleration integration scheme with parameters g ¼ 0.5 and b ¼ 0.25. The integration timestep is taken equal to 0.01 s, which is then divided to the half value if convergence is notachieved. At each iteration, convergence is considered to be achieved when the unbalancednonlinear force is less than 10�4 of the total force.

The damping matrix C is determined from Eq. (D7.29) (Rayleigh damping) with the coefficientvalues in Eq. (D7.31). Figure 8.52 shows the damping ratio corresponding to the applieddamping matrix C as a function of mode period. The damping for periods T . 1.5 s, whereseismic isolation dominates, is very small (j, 0.3%). For that period range, energy dissipationoccurs primarily from the nonlinear response of the isolators. For very short periods (T , 0.05 s),damping increases significantly (j. 10%). This is desirable, because modes with periods of thesame order of magnitude as the time step cannot be integrated with good accuracy, and it ispreferable to filter them out via increased damping.

8.4.11.2 Action effects on the seismic isolation systemFigures 8.53–8.56 depict the hysteresis loops for an abutment bearing (C0_L) and a pier bearing(P1_L) for both the LBDP and UBDP analyses. In Tables 8.16 and 8.17 the time-history analysisresults are presented for the left and right bearings at each pier (P1_L, P1_R, P2_L and P2_R)and abutment location (C0_L, C0_R, C3_L and C3_R). As the analysis is carried out for


Figure 8.52. Damping as a function of the period of the modes

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

Period, T : s

Dam

ping

,ξ: %

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

236


Figure 8.53. Hysteresis loops for abutment bearing C0_L from the analysis with LBDP

EQ6

EQ7

EQ2

EQ1

EQ3

EQ4

Displacement: m Displacement: m







Direction X Direction Y

EQ5

–400

–200

0

200

400

600

–0.300 –0.200 –0.100 0 0.100 0.200 0.300

–400

–200

0

200

400

600

–0.300 –0.200 –0.100 0 0.100 0.200 0.300

–0.300 –0.200 –0.100 0 0.100 0.200

–400–300–200–100

0100200300400

–0.200 –0.100 0 0.100 0.200 0.300Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

N

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

N

–400

–200

0

200

400

600

–0.300 –0.200 –0.100 0 0.100 0.200 0.300

–0.300 –0.200 –0.100 0 0.100 0.200 0.300

–400

–200

0

200

400

–400

–200

0

200

400

600

–400

–300

–200

–100

0

100

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300

–0.200 –0.150 –0.100 –0.050 0 0.050 0.100 0.150 0.200

–600

–400

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0

200

400

600

–0.300 –0.200 –0.100 0 0.100 0.200

–300

–200

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0

100

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300

–0.150 –0.100 –0.050 0 0.050 0.100 0.150

–400

–300

–200

–100

0

100

200

300

–0.200 –0.100 0 0.100 0.200

–400–300–200–100

0100200300400

–0.300 –0.200 –0.100 0 0.100 0.200

–0.200 –0.100 0 0.100 0.200

–0.200 –0.100 0 0.100 0.200

–300

–200

–100

0

100

200

300

400

–400

–200

0

200

400

600

–600

–400

–200

0

200

400

–0.300 –0.200 –0.100 0 0.100 0.200 0.300

237


Figure 8.54. Hysteresis loops for abutment bearing C0_L from the analysis with UBDP

–400–300–200

–1000

100200

300400

–0.200 –0.150 –0.100 –0.050 0 0.050 0.100 0.150

–300–200–100

0100200300

400500

–0.200 –0.100 0 0.100 0.200 0.300

–400

–200

0

200

400

–0.200 –0.150 –0.100 –0.050 0 0.050 0.100 0.150

–400

–200

0

200

400

600

–0.150 –0.100 –0.050 0 0.050 0.100 0.150 0.200 0.250

–600

–400

–200

0

200

400

–0.200 –0.150 –0.100 –0.050 0 0.050 0.100 0.150

–400

–200

0

200

400

600

–0.150 –0.100 –0.050 0 0.050 0.100 0.150 0.200

–300

–200

–100

0

100

200

300

400

–0.100 –0.050 0 0.050 0.100 0.150

–600

–400

–200

0

200

400

600

–0.250 –0.200 –0.150 –0.100 –0.050 0 0.050 0.100

–400–300–200

–1000

100200

300400

–0.100 –0.050 0 0.050 0.100

–300

–200

–100

0

100

200

300

–0.150 –0.100 –0.050 0 0.050 0.100

–400–300–200

–1000

100200

300400

–0.150 –0.100 –0.050 0 0.050 0.100

–300

–200

–100

0

100

200

300

400

–0.100 –0.050 0 0.050 0.100

–400–300–200

–1000

100200

300400

–0.150 –0.100 –0.050 0 0.050 0.100 0.150

–400

–200

0

200

400

600

–0.200 –0.150 –0.100 –0.050 0 0.050 0.100 0.150


EQ6

EQ7

EQ2

EQ1

EQ3

EQ4








EQ5

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

N

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

N

238


Figure 8.55. Hysteresis loops for pier bearing P1_L from the analysis with LBDP

–1500

–1000

–500

0

500

1000

1500

2000

–0.300 –0.200 –0.100 0.000 0.100 0.200 0.300

–1500

–1000

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0

500

1000

1500

2000

–0.300 –0.200 –0.100 0.000 0.100 0.200 0.300

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0

1000

1500

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0

500

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–2000

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0

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–1500

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0

500

1000

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–2000–1500–1000–500

0500

100015002000

–0.300 –0.200 –0.100 0.000 0.100 0.200

–1500

–1000

–500

0

500

1000

–0.150 –0.100 –0.050 0.000 0.050 0.100

–1500

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0

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1000

1500

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0

500

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0

500

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–1500

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500

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–1500

–1000

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0

500

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1500

–0.300 –0.200 –0.100 0.000 0.100 0.200


EQ6

EQ7

EQ2

EQ1

EQ3

EQ4








EQ5

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

N

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

N

239


Figure 8.56. Hysteresis loops for pier bearing P1_L from the analysis with UBDP

–1000

–500

0

500

1000

1500

–0.150 –0.100 –0.050 0.000 0.050 0.100 0.150

–1500

–1000

–500

0

500

1000

1500

2000

–0.200 –0.100 0.000 0.100 0.200 0.300

–2000

–1500

–1000

–500

0

500

1000

1500

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–1000

–500

0

500

1000

1500

2000

–0.150 –0.100 –0.050 0.000 0.050 0.100 0.150 0.200

–2000

–1500

–1000

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0

500

1000

1500

–0.200 –0.150 –0.100 –0.050 0.000 0.050 0.100 0.150

–1000

–500

0

500

1000

1500

2000

–0.150 –0.100 –0.050 0.000 0.050 0.100 0.150 0.200

–1500

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0

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–2000

–1500

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1000

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1500

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500

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1500

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0

500

1000

1500

–0.150 –0.100 –0.050 0.000 0.050 0.100

–1500

–1000

–500

0

500

1000

1500

–0.200 –0.150 –0.100 –0.050 0.000 0.050 0.100 0.150

EQ6

EQ7

EQ2

EQ1

EQ3

EQ4









EQ5

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

N

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

NFo

rce:

KN

Forc

e: K

N

240

seven seismic motions EQ1 to EQ7, the average of the individual responses may be assumed asthe design value.

The results are the combined effect of the seismic action and the quasi-permanent loads. They donot include the effects of temperature and creep or shrinkage. dEd,x denotes the displacement inthe longitudinal direction, dEd,y that in the transverse, dEd is the magnitude of the displacementvector in the horizontal plane and aEd is the magnitude of the rotation vector in the horizontalplane.NEd is the vertical force on the bearing (positive when compressive), VEd,x is the horizontalforce of the bearing in the longitudinal direction, VEd,y is that in the transverse direction and VEd

is the magnitude of the horizontal force vector.

8.4.11.3 Check of the lower bound on action effectsAccording to clauses 7.5.6(1) and 7.5.5(6) of Part 2 of Eurocode 8, the resulting displacement ofthe stiffness centre of the isolating system (dcd) and the resulting total shear force transferredthrough the isolation interface (Vd) in each of the two-horizontal directions are subject tolower bounds equal to 80% of the design displacement and the shear force through the isolationinterface from the fundamental mode analysis, dcf and Vf, respectively. The lower bounds applyfor both the modal response spectrum analysis and the time-history analysis. The verification ofthese bounds is presented below:

g displacement in the X direction: rd ¼ dcd/df ¼ 0.193/0.22 ¼ 0.88 . 0.80) bound metg displacement in the Y direction: rd ¼ dcd/df ¼ 0.207/0.22 ¼ 0.94 . 0.80) bound metg total shear in the X direction: rv ¼ Vd/Vf ¼ 6929.3/6292 ¼ 1.10 . 0.80 ) bound metg total shear in the Y direction: rv ¼ Vd/Vf ¼ 6652.1/6292 ¼ 1.06 . 0.80 ) bound met.

Witness that the time-history analysis results are 12% smaller for displacements and 10% largerfor the total shear force compared with those of the fundamental mode analysis. This discrepancy


Table 8.16. Bearings: results of the analysis for LBDP

Bearing dEd,x: m dEd,y: m dEd: m aEd: rad NEd,min: kN NEd,max: kN VEd,x: kN VEd,y: kN VEd: kN

C0_L 0.193 0.207 0.255 0.00498 848.7 3310.3 346.0 375.7 469.0

C0_R 0.193 0.207 0.254 0.00509 860.4 3359.4 363.2 389.8 482.4

C3_L 0.199 0.207 0.258 0.00486 855.3 3323.9 402.5 372.0 501.4

C3_R 0.199 0.207 0.257 0.00494 858.5 3309.3 418.4 368.4 496.0

P1_L 0.188 0.193 0.244 0.00367 4541.1 12086.0 1328.5 1295.0 1654.2

P1_R 0.188 0.192 0.243 0.00381 4435.4 11994.8 1369.8 1284.5 1690.0

P2_L 0.189 0.193 0.245 0.00369 4560.3 12084.6 1336.1 1283.5 1654.3

P2_R 0.189 0.192 0.243 0.00380 4498.0 11912.9 1365.0 1283.2 1688.5

Total 6929.3 6652.1

Table 8.17. Bearings: results of the analysis for UBDP

Bearing dEd,x: m dEd,y: m dEd: m aEd: rad NEd,min: kN NEd,max: kN VEd,x: kN VEd,y: kN VEd: kN

C0_L 0.149 0.139 0.182 0.00469 655.0 3157.9 352.6 380.4 449.8

C0_R 0.149 0.139 0.181 0.00475 624.1 3110.3 363.4 366.8 452.3

C3_L 0.157 0.139 0.185 0.00466 677.2 3112.5 400.6 368.6 489.6

C3_R 0.157 0.138 0.185 0.00461 684.8 3096.8 390.6 360.1 473.0

P1_L 0.149 0.128 0.173 0.00361 3912.7 11246.7 1361.8 1273.8 1630.8

P1_R 0.149 0.128 0.172 0.00355 3781.8 11408.5 1352.6 1185.7 1587.1

P2_L 0.150 0.128 0.173 0.00359 3793.6 11246.2 1379.7 1255.4 1605.7

P2_R 0.149 0.127 0.173 0.00354 3886.4 11378.4 1370.1 1187.1 1603.4

Total 6971.3 6377.8

241

between the comparison of displacements and forces is attributed to the effect of the verticalearthquake component on bearing forces, which is not taken into account in the fundamentalmode method. For spherical sliding bearings, the horizontal bearing shear forces are alwaysproportional to the vertical bearing loads. The variation in the vertical bearing loads due tothe vertical ground motion also affects the horizontal shear forces. This effect is evident in thewavy nature of the force–displacement hysteresis loops of the isolators presented inFigures 8.53–8.56.

8.4.12 Verification of the isolation system8.4.12.1 Displacement demands on the isolation systemThe displacement demand in each direction dm,i is determined as the sum of:

g the seismic design displacement, dbi,d, multiplied by an amplification factor gIS with therecommended value gIS ¼ 1.50

g the offset displacement dG,i due to quasi-permanent actions, long-term deformations and50% of the thermal action (cf. Eq. (D6.36) in Section 6.8.1.2).

The offset displacement due to 50% of the thermal action is determined as follows. The designvalue of the uniform component of the thermal action is in the range �258C to þ358C.Assuming that the fixed point of thermal expansion/contraction is located at one of the twopiers, this leads to an effective expansion/contraction length LT of 140 m for the abutmentbearings and 80 m for the pier bearings. With the sign ‘þ’ corresponding to deck movementtowards the abutment and ‘�’ to movement towards the bridge centre, the offset displacementdue to 50% of thermal action is:

At abutments: 0.5LT aDT ¼ 0.5 � 140 000 � 1.0 � 10�5 � (�45) ¼ �31.5 mm0.5LT aDT ¼ 0.5 � 140 000 � 1.0 � 10�5 � (þ55) ¼ þ38.5 mm

At the piers: 0.5LT aDT ¼ 0.5 � 80 000 � 1.0 � 10�5 � (�45) ¼ �18 mm0.5LT aDT ¼ 0.5 � 80 000 � 1.0 � 10�5 � (þ55) ¼ þ22.0 mm

The total offset displacement, including the effects of quasi-permanent actions, long-termdeformations and 50% of the thermal action, is calculated as follows:

At abutments: Towards the bridge centre: �8� 31.5 ¼ �39.5 mmTowards the abutment: þ38.5 mm

At the piers: Towards the bridge centre: �3� 18 ¼ �21 mmTowards the abutments: þ22 mm

According to Part 2 of Eurocode 8, the displacement demand is estimated and checked in theprincipal directions and not in the most critical direction. However, this is not adequate forbearings with the same displacement capacity in all horizontal directions, such as the FPSbearings. The maximum displacement of the isolator occurs in a direction that does notcoincide in general with one of the two principal directions. The maximum required displacementdemand in the most critical direction may be estimated by examining the time history of themagnitude of the resultant displacement vector in the horizontal plane XY, including theeffect of offset displacements due to quasi-permanent actions, long-term displacements and50% of the thermal action.

In Table 8.18 the displacement demand at the abutment and pier bearings is estimated in bothprincipal directions, alongside the critical displacement demand in the horizontal XY plane,which in the present case is larger by about 25%. Therefore, the displacement demand of theisolators is 407 mm for abutment bearings and 382 mm for pier bearings.

8.4.12.2. Restoring capability of the isolation systemThe lateral restoring capability of the isolation system is verified per clause 7.7.1 of Part 2 ofEurocode 8. The equivalent bilinear model of the isolation system is shown in Figure 8.57,where F0 ¼ mdNEd is the force at zero displacement; Kp ¼ NEd/Rb is the post-elastic stiffness;and d0 is the maximum residual displacement for which the isolation system can be in static


242

equilibrium in the considered direction. For an isolation system consisting of spherical slidingisolators the displacement d0 is:

d0 ¼ F0/Kp ¼ mdNEd/(NEd/Rb) ¼ mdRb

According to clause 7.7.1(2) in Part 2 of Eurocode 8, an isolation system has adequate self-restoring capability if dcd/d0 . d in both principal directions, where d is a coefficient with therecommended value d ¼ 0.5. This criterion is verified for both UBDP and LBDP of the isolators.Lower values of the design displacement dcd give results that are more on the safe side:

g longitudinal direction, LBDP:

dcd/d0 ¼ 0.193/(0.051 � 1.83) ¼ 2.07 . 0.5

g transverse direction, LBDP:

dcd/d0 ¼ 0.207/(0.051 � 1.83) ¼ 2.22 . 0.5

g longitudinal direction, UBDP:

dcd/d0 ¼ 0.149/(0.09 � 1.83) ¼ 0.90 . 0.5

g transverse direction, UBDP:

dcd/d0 ¼ 0.138/(0.09 � 1.83) ¼ 0.84 . 0.5

Therefore, the restoring capability of the isolation system is adequate without additional increasein the displacement capacity dm. It is noted that UBDP give more unfavourable results becausedcd is larger and d0 smaller than for LBDP.

8.4.13 Verification of the substructure8.4.13.1 Action effect envelopes for the piersIn Tables 8.19 and 8.20, action effect envelopes from the time-history analysis (average for theseven earthquake ground motions EQ1 to EQ7) are given for the substructure. For piers P1


Table 8.18. Displacement demand on isolators

Displacement demand For abutments:

C0_L, C0_R, C3_L, C3_R

For piers:

P1_L, P1_R, P2_L, P2_R

In longitudinal direction X 329 305

In transverse direction Y 311 290

In horizontal plane XY 407 382

Maximum 407 382

Figure 8.57. Properties of bilinear model for verification of restoring capability of isolator

F0

Force

Displacement

d0 d0

Kp

dcd

243

and P2 they refer to their base, and for the abutments C0 and C3 to the mid-point between thebearings (at the bearing level). The envelopes include the effect of permanent actions and thequasi-permanent value of the traffic loads and the design seismic action. They do not includethe effects of temperature and shrinkage. According to clause 7.6.3(2) of Part 2 of Eurocode 8,


Table 8.19. Substructure: envelopes of analysis for LBDP

Envelope N: kN VX: kN VY: kN T: kN m MX: kN m MY: kN m

C0 max N �1754.3 �18.3 158.3 �14.6 �1.8 824.8

C0 min N �6535.1 �347.5 123.9 23.1 �34.7 380.1

C0 max VX �4930.5 616.5 �163.2 85.1 61.6 �475.4

C0 min VX �3688.2 �660.5 �115.7 �82.2 �66.0 �482.2

C0 max VY �5623.1 617.2 684.1 �192.6 61.7 1933.0

C0 min VY �4124.3 �469.5 �694.6 �190.5 �47.0 �2002.9

C0 max T �2759.9 358.3 �393.2 183.1 35.8 �1388.7

C0 min T �2989.8 �341.2 �505.2 �216.0 �34.1 �1867.7

C0 max MX �4930.5 616.5 �163.2 85.1 61.6 �475.4

C0 min MX �3688.2 �660.5 �115.7 �82.2 �66.0 �482.2

C0 max MY �3789.3 �383.9 608.9 272.1 �38.4 2575.8

C0 min MY � 4324.0 �493.4 �730.7 �312.4 �49.3 �2701.2

C3 max N �1787.9 �105.4 113.9 31.5 �10.5 654.5

C3 min N �6439.8 379.4 134.5 �32.2 37.9 446.2

C3 max VX �4241.8 783.1 �110.8 56.9 78.3 �328.9

C3 min VX �3389.9 �562.1 �106.0 �66.8 �56.2 �429.4

C3 max VY �5460.4 666.9 680.5 �238.2 66.7 2046.7

C3 min VY �4149.3 �401.9 �660.4 �172.9 �40.2 �1867.8

C3 max T �1975.2 257.9 �301.0 172.4 25.8 �1131.7

C3 min T �2760.7 312.5 435.8 �215.7 31.2 1809.1

C3 max MX �4241.8 783.1 �110.8 56.9 78.3 �328.9

C3 min MX �3389.9 �562.1 �106.0 �66.8 �56.2 �429.4

C3 max MY �4001.7 453.0 631.7 �312.7 45.3 2622.4

C3 min MY �4533.2 591.8 �690.8 395.7 59.2 �2597.2

P1 max N �12756.8 50.1 �236.8 60.2 254.0 �3971.0

P1 min N �27232.5 228.2 640.6 451.2 2143.8 7982.2

P1 max VX �16241.5 3339.4 �500.1 105.8 29347.6 �4786.2

P1 min VX �17636.3 �2906.9 86.6 �77.7 �22629.6 �1838.1

P1 max VY �16658.7 1112.7 2666.1 �758.9 11127.1 33869.5

P1 min VY �15829.2 �909.9 �2698.2 �450.8 �9661.0 �27964.5

P1 max T �8022.6 961.5 �813.0 575.0 9403.4 �12731.0

P1 min T �13056.5 2514.3 919.9 �768.1 22613.0 17367.8

P1 max MX �16142.4 3319.0 �497.1 105.2 29168.5 �4756.9

P1 min MX �18598.0 �2499.2 �1830.2 �240.9 �26831.4 �15284.0

P1 max MY �16393.7 1095.0 2623.7 �746.9 10950.1 33330.7

P1 min MY �18669.2 �1073.1 �3182.3 �531.7 �11394.4 �32981.8

P2 max N �12560.1 �792.5 �174.2 161.5 �6724.3 4432.2

P2 min N �27066.2 �230.7 715.6 �339.1 �2180.8 8957.0

P2 max VX �16266.7 3383.2 �506.5 156.6 29890.9 �4842.4

P2 min VX �17867.1 �2879.8 84.6 �83.3 �22406.6 �1807.9

P2 max VY �16650.4 1099.1 2678.2 �777.1 11054.4 34062.3

P2 min VY �15988.2 �956.8 �2711.5 �429.9 �10018.0 �28164.0

P2 max T �7732.6 960.9 �781.8 575.5 9395.1 �12189.7

P2 min T �12784.1 2478.8 860.4 �766.8 22343.7 16575.8

P2 max MX �16195.0 3368.3 �504.3 155.9 29759.0 �4821.0

P2 min MX �18734.3 �2470.9 �1809.2 �255.8 �26514.7 �15186.2

P2 max MY �16276.3 1074.4 2618.0 �759.6 10806.1 33297.0

P2 min MY �18798.5 �1125.0 �3188.1 �505.5 �11778.9 �33114.5

244

the design seismic forces due to the design seismic action alone may be derived from thetime-history analysis forces after division by the q factor for limited ductile/essentially elasticbehaviour, q ¼ 1.50. This is not included in the present results, but will be applied at thedesign stage of the pier cross-sections.


Table 8.20. Substructure: envelopes of analysis for UBDP

Envelope N: kN VX: kN VY: kN T: kN m MX: kN m MY: kN m

C0 max N �1326.0 116.0 �80.6 �12.6 11.6 62.0

C0 min N �6076.0 �594.2 �94.3 �38.4 �59.4 �365.0

C0 max VX �3620.5 627.6 �93.9 53.1 62.8 �347.0

C0 min VX �3503.1 �693.8 �158.1 �133.9 �69.4 �687.2

C0 max VY �3737.8 149.9 686.9 �105.3 15.0 2696.7

C0 min VY �3996.2 �375.4 �640.2 �176.4 �37.5 �2085.3

C0 max T �2699.6 �22.2 �197.0 300.3 �2.2 149.2

C0 min T �3260.5 479.0 471.3 �241.3 47.9 1937.6

C0 max MX �3620.5 627.6 �93.9 53.1 62.8 �347.0

C0 min MX �3503.1 �693.8 �158.1 �133.9 �69.4 �687.2

C0 max MY �3222.0 97.5 597.8 �89.8 9.7 2655.5

C0 min MY �4111.4 �219.5 �555.6 �199.5 �21.9 �2575.7

C3 max N �1417.6 �76.4 45.9 61.4 �7.6 384.7

C3 min N �6053.3 614.1 �86.6 37.5 61.4 �339.1

C3 max VX �4215.2 768.4 �147.3 39.3 76.8 �381.3

C3 min VX �3079.4 �586.3 �151.7 �96.5 �58.6 �525.6

C3 max VY �4496.6 636.9 669.0 �347.4 63.7 2340.9

C3 min VY �3930.0 �296.3 �635.4 �149.4 �29.6 �2069.0

C3 max T �2417.7 325.0 �359.0 233.9 32.5 �1283.0

C3 min T �2709.4 390.4 425.5 �285.9 39.0 1840.4

C3 max MX �4215.2 768.4 �147.3 39.3 76.8 �381.3

C3 min MX �3079.4 �586.3 �151.7 �96.5 �58.6 �525.6

C3 max MY �3961.3 570.8 622.0 �418.1 57.1 2690.9

C3 min MY �4233.5 �117.5 �558.0 �8.8 �11.8 �2615.1

P1 max N �11444.7 �125.6 566.0 7.6 �1131.0 7410.2

P1 min N �25719.8 320.3 1735.7 382.5 3219.5 22258.8

P1 max VX �15188.6 3632.5 �168.5 83.2 32565.5 �2160.6

P1 min VX �17329.4 �3190.1 �282.4 �106.6 �27647.3 �3773.0

P1 max VY �16196.9 1183.0 2666.2 �949.5 12871.0 33694.2

P1 min VY �14597.6 1.4 �2828.3 �175.7 �580.6 �29913.1

P1 max T �12907.1 1473.0 �804.9 693.2 14798.4 �13503.0

P1 min T �11198.9 2406.9 983.3 �1016.0 21664.8 18338.9

P1 max MX �14829.4 3546.6 �164.5 81.2 31795.4 �2109.5

P1 min MX �15090.4 �3183.3 127.2 �99.0 �28584.3 �246.6

P1 max MY �15952.1 1165.1 2626.0 �935.1 12676.5 33185.2

P1 min MY �15337.1 1.4 �2971.5 �184.6 �610.0 �31428.6

P2 max N �11479.8 216.1 583.7 �1.8 2007.6 7643.4

P2 min N �25746.2 �28.7 1764.3 �409.5 �372.2 22556.9

P2 max VX �15433.8 3702.6 �165.4 75.4 33190.2 �2114.8

P2 min VX �15216.5 �3197.1 106.6 �115.6 �28697.8 �609.4

P2 max VY �20549.5 280.4 2618.8 �304.4 3324.9 29039.5

P2 min VY �14855.2 �49.9 �2856.8 �190.7 �930.3 �30281.5

P2 max T �12267.8 1464.3 �764.4 741.6 14684.7 �12727.1

P2 min T �11612.0 2520.1 953.8 �1006.9 22796.5 17940.3

P2 max MX �15110.2 3625.0 �161.9 73.8 32494.1 �2070.5

P2 min MX �15128.2 �3178.6 106.0 �114.9 �28531.2 �605.9

P2 max MY �16623.8 �340.0 2495.7 �120.7 �2377.2 32509.5

P2 min MY �15508.7 �52.1 �2982.5 �199.1 �971.2 �31613.6

245

The following notation is used:

g N is the vertical force (i.e. axial force) (positive when acting upwards)g VX is the shear force along the X axis, VY is the shear along the Y axisg T is the torsional momentg MX is the moment in the vertical plane through the X axis (i.e. that produced by an

earthquake in the longitudinal, X, direction), and MY the moment in the vertical planethrough the Y axis (produced by an earthquake acting in the transverse direction, Y).

g The signs of VX/MX are the same when their directions are compatible with earthquakeforces acting in the X direction; the signs of VY/MY are the same when their directions arecompatible with earthquake forces acting in the Y direction.

Envelopes of maximum/minimum and concurrent internal forces are presented for each pier/abutment location. For instance, envelope max N corresponds to the design situation where thevalue of the vertical force N is algebraically the maximum. The values of other forces VX, VY,T, MX and MY at max N envelope are the ‘concurrent’ forces when N becomes a maximum.The maximum/minimum and the ‘concurrent forces’ for each envelope are derived as follows:

1 The maxima/minima of each force (say maxMX, j ¼ 1–7) over all time-steps of theresponse history for each motion j ¼ 1–7 are assessed. The design value of the maximum/minimum of the examined force (say MX,d) is the average of these maxima/minima for theseven motions.

2 The results of the seismic motion producing the extreme value (say maxmax MX) of thesemaxima/minima for all motions and the corresponding time-step are used as the basis forthe assessment of the ‘concurrent’ values of the other forces. A scaling factor is applied tothese results, equal to the ratio of the design value of the examined force (MX,d) divided bythe extreme value (maxmax MX) (i.e. ¼MX,d/maxmax MX).

8.4.13.2 Section verification of the piers8.4.13.2.1 GeneralThe maximum normalised axial force of the piers is

hk ¼ NEd/(Ac fck) ¼ 27 232.5/(5 � 2.5 � 35 000) ¼ 0.062 , 0.08

Therefore, according to clause 6.2.1.1(2) of Part 2 of Eurocode 8, no confinement reinforcementis necessary. However, due to the low axial force the pier should be designed taking into accountthe minimum reinforcement requirements for beams and for columns.

8.4.13.2.2 Verification for flexure and axial forceAccording to clause 7.6.3(2) of Part 2 of Eurocode 8, for the design of the substructures the seismicforcesEE due to the design seismic action alonemay be obtained by dividing the analysis forces bythe q factor corresponding to limited-ductile/essentially-elastic behaviour, q � 1.50.

Clause 6.5.1 in Part 2 of Eurocode 8 prescribes certain reduced ductility measures (for the con-finement and restraint of reinforcement buckling). It also offers the option of avoiding thesemeasures if the piers are designed so that MRd/MEd , 1.3. This option is chosen in thisexample, for reasons to become apparent soon. Therefore, for the design of longitudinalreinforcement the design seismic forces, FEd, are derived from forces derived from the time-history analysis, FEA, as FEd ¼ 1.3FE,A/1.5. For the most adverse combination of so-computeddesign seismic action effects, NEd, MX,ed and MY,ed, the required longitudinal reinforcementamounts to As ¼ 21 370 mm2, uniformly distributed all around the section.

8.4.13.2.3 Minimum longitudinal reinforcementPart 2 of Eurocode 8 does not have a specific requirement for a minimum value of the longitudi-nal reinforcement ratio. The minimum reinforcement specified in Eurocode 2 for columns(including those of bridges) is equal to

As,min ¼ max(0.1NEd/fyd, 0.002Ac) ¼ max[0.1 � 27 232.5/(500 000/1.15), 0.002 � 5 � 2.5]

¼ 0.025 m2 ¼ 25 000 mm2

(i.e. min r ¼ 0.2%).


246

Eurocode 2 specifies for beams a minimum amount of tensile reinforcement to prevent brittlefailure right after exceedance of the cracking moment of the section (i.e. of the tensile concretestrength). For uniaxial bending, the minimum tensile reinforcement (i.e. of the side of thesection that is in tension) is r1,min ¼ max (0.26fctm/fyk, 0.0013), normalised to bd. For concreteC35/45 with fctm ¼ 3.2 MPa and steel grade 500 with fyk ¼ 500 MPa, r1,min ¼ 0.001664, whichgives for each one of the two long sides of the section

As1,min ¼ 0.001664 � 5000 � (2500� 80) ¼ 20 134 mm2

(i.e. 20 134/5.0 ¼ 4027 mm2/m of the perimeter). If the same reinforcement density is adoptedalso for the two short sides of the section (which profit from the tension reinforcement of theend of the long sides, too), we obtain 60 400 mm2 for the total minimum reinforcement of therectangular section, and rmin ¼ 0.00483 ¼ 0.483%.

In summary:

g required longitudinal reinforcement for the ULS design of the section: 21 373 mm2

(r ¼ 0.17%)g required minimum longitudinal reinforcement: 60 400 cm2 (rmin ¼ 0.483%)g provided longitudinal reinforcement: one layer 128/135 ¼ 4560 mm2/m or 64 000 mm2 in

total (r ¼ 0.51%).

Note that the cross-section of the piers may be substantially reduced.

8.4.13.2.4 ULS verification in shearAccording to clause 5.6.2(2) of Part 2 of Eurocode 8, for the design of shear reinforcement,Eurocode 2 applies with the following additional rules:

g The design action effects are multiplied by the behaviour factor q used in the linearanalysis.

g The resistance values, VRd,c, VRd,s and VRd,max derived in accordance with Eurocode 2 aredivided by an additional safety factor gBd1 against brittle failure, with the recommendedvalue gBd1 ¼ 1.25. Therefore, the design seismic forces for the ULS design in shear, FEd,may be derived from the time-history analysis forces, FEA, as FEd ¼ 1.25FE,A.

The shear reinforcement calculated as above is presented below:

g required shear reinforcement in the longitudinal direction: 5903 mm2/m.g required shear reinforcement in the transverse direction: 2366 mm2/m.g provided shear reinforcement in the longitudinal direction (116/150 on the perimeter plus

4 112/150 cross-ties with two legs each): 4 � 2 � 754 mm2/mþ 2 � 1340 mm2/m ¼8710 mm2/m (rw ¼ 0.174%).

g provided shear reinforcement in the transverse direction (116/150, only on the perimeter):2 � 1340 mm2/m ¼ 2680 mm2/m (rw ¼ 0.107%).

The provided shear reinforcement satisfies the minimum requirements of Eurocode 2 forcolumns:

g maximum spacing ¼ 0.6 � min(20 � 28 mm, 2500 mm, 400 mm) ¼ 240 mm; providedspacing ¼ 150 mm

g minimum bar diameter ¼ max (6 mm, 28 mm/4) ¼ 7 mm; provided diameter ¼ 12 mm.

The shear reinforcement satisfies even the minimum with Eurocode 2 requirements forbeams:

g maximum longitudinal spacing sl,max ¼ 0.75d ¼ 0.75 � 2420 ¼ 1815 mm; providedlongitudinal spacing ¼ 150 mm

g maximum transverse spacing st,max ¼ min(0.75d, 600 mm) ¼ min(0.75 � 2420 mm,600 mm) ¼ 600 mm; provided transverse spacing ¼ 530 mm


247

g minimum shear reinforcement ratio rw,min ¼ 0.08pfck/fyk ¼ 0.08

p35/500 ¼ 0.095%;

provided shear reinforcement ratio rw ¼ 0.174% in the longitudinal direction andrw ¼ 0.107% in the transverse direction.

The reinforcement at the pier base is shown in Figure 8.58.

8.4.14 Design action effects for the foundation8.4.14.1 Design actions effects from time-history analysisTable 8.21 lists action effect envelopes for the design of the foundation, on the basis of the time-history analysis results. The action effects for the foundation design are derived according toclauses 7.6.3(4) and 5.8.2(2) of Part 2 of Eurocode 8 for bridges with seismic isolation. Theseismic action effects for the foundation design correspond to the analysis results multipliedby the q value (q ¼ 1.5) used for the design of the substructure (i.e. effectively correspondingto q ¼ 1).

The set of forces that are critical for the foundation design are the maximum/minimum shearforce envelopes for the design of the foundation of the abutment and the maximum/minimumbending moment at the base of the pier, for the design of the pier foundation. The analysisresults for the seismic design situation are given for piers P1 and P2 at their base and forabutments C0 and C3 at the midpoint between the bearings (i.e. at the bearing level). Theseaction effects include those of permanent actions, of the combination value of traffic loadsand of the design seismic action. The signs of the forces for foundation design are as inFigure 8.37.

8.4.14.2 Comparison with the fundamental mode methodThe force and displacement results at the abutments and at the base of the piers from the time-history analysis are compared here with those of the fundamental mode method. LBDP give themost unfavourable results with respect to the forces in the substructure.


Table 8.21. Seismic design situation action effects for foundation design

Location Envelope Fx: kN Fy: kN Fz: kN Mx: kN m My: kN m Mz: kN m

C0, C3 max Fx envelope 783 111 4242 329 78 57

max Fy envelope 470 695 4124 2003 47 191

P1, P2 max My envelope 3625 162 15110 2070 32494 74

max Mx envelope 1095 2624 16394 33331 10950 747

Figure 8.58. Layout of pier reinforcement

Longitudinal reinforcement: 1 layer Ø28/13.5 = 45.6 cm2/m

Stirrups: 4 two-legged Ø12/15 = 4 × 2 × 7.54 cm2/m = 60.3 cm2/m

Perimetric hoop: 1 two-legged Ø16/15 = 2 × 13.40 cm2/m = 26.8 cm2/m

5.0 m

0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53

2.5 m

248

Before attempting this comparison, the conclusions of Section 7.5.5.4 on bidirectional excitationare applied:

g Applying the second proposal, the effective value dcd,e of the maximum displacement(length of the vector resultant of the simultaneous orthogonal displacements) is assumed asequal to dcd,e � 1.15dcd.

g The increased value dcd,e should be used also for the estimation of the maximum forcestransferred through the isolator in any direction, as the isolator has no preferred direction.

g The vertical seismic motion component also has an effect on the variation of the frictionforces of the isolator. The effect oscillates, with approximately equal positive and negativevalues and approximately zero mean value. This effect can be observed in the hysteresisloops of Figures 8.53–8.56, for some of the seismic motions used (e.g. EQ7 and EQ3). Theoscillations occur at much shorter periods than those of the horizontal motion,corresponding to the much higher frequency content of the vertical component (cf. theelastic spectra of the two components). Consequently, this influence may be ignored, atleast as far as the maximum displacements are concerned. Regarding the forces, theapplication to the friction forces of the 1.15 multiplier estimated above is a convenientapproximation.

The displacement demand of abutment bearings and the total abutment shear are presented inTable 8.22. The table compares the results of the time-history analysis with those of the funda-mental mode method multiplied by 1.15, as explained above. The estimated displacementdemand using the fundamental mode method is 3% larger than that from time-historyanalysis. The total shear estimated by the fundamental mode method is 13% less in the longitudi-nal direction and 3% less in the transverse direction than those from time-history analysis.

REFERENCES

CEN (Comite Europe de Normalisation) (2003a) EN 1991-2:2003: Eurocode 1: Actions on

structures – Part 2: Traffic loads on bridges. CEN, Brussels.

CEN (2003b) EN 1991-1-5:2003: Eurocode 1: Actions on structures – Part 1-5: General actions –

Thermal actions. CEN, Brussels.

CEN (2004a) EN 1992-1-1:2004: Eurocode 2: Design of concrete structures – Part 1-1: General

rules and rules for buildings. CEN, Brussels.



CEN (2005a) BS EN 1992-2:2005: Eurocode 2. Design of concrete structures – Part 2: Concrete

bridges – Design and detailing rules. CEN, Brussels.



Bouassida Y, Bouchon E, Crespo P et al. (2012) Bridge Design to Eurocodes Worked examples.

Worked examples presented at the Workshop ‘Bridge Design to Eurocodes’, Vienna, 4–6 October

2010. JRC European Commission (Athanasopoulou A et al. (eds)).


Table 8.22. Displacements and total shears at abutment bearings in the longitudinal direction for the

two analysis methods

Displacement

demand: mm

Total shear in longitudinal

direction: kN

Total shear in the transverse

direction: kN

Time-history analysis 407 783 695

Fundamental mode method 419 683 683

249

This page has been reformatted by Knovel to provide easier navigation.

INDEX Page locators in italics denote figures separate from the corresponding text, locators in bold denote tables.

Index Terms Links

A

abutments

conceptual design 46 58 59–64

detailing 155–159

minimum overlap lengths 135–136

modal response spectrum analyses 81

nonlinear time-history analysis 236 237–238

role 59–60

seismic design examples 217

verification 155–159

accelerograms 227–229

accidental action 8–9

accidental torsion 99

action effects

detailing 149–151

foundations 159–160 206–207 248–249

isolation 236 237–240 241

lower-bound design properties 241–242

pier envelopes 243–246

seismic design examples 197–200 217–219 236–246 248–249


active faults 29

added mass effects 84–86

additional uniform traffic load applications 195

analyses 67–118

action components 68–69

behaviour factors 69–73

design spectra 26

effective stiffness 100–107

fundamental mode 67 92–97 185–191 196–197

233–234 248–249

isolation 185–191

limited ductile piers 211–214 215 216

modal response spectrum analyses 67 68 73–92 108

112 197

nonlinear analyses 67–68 108–109 110–117 132–135

torsional effects 98–99

Index Terms Links


annular sections 55

application rules 4

Arahthos bridge (GR) 42 45

articulated connections 43 49–50

artificial time-history records 26 27

assumptions 4

axial force

seismic design examples 197–200 219 246

B

backfills 9–10 59–61 62 159

balanced cantilever methods 73–74

bars

buckling prevention 204 221

rotation capacity 134

bearings

analyses 70

capacity design effects 128

conceptual design 42–43 45 47–53 55–56

58 62–64

design examples 207–208 221

detailing 145–155 156–158

dimensioning 142–155

elastic deformations 13–14

imposed deformations 225

isolation 176–177 180–181

liquefaction 33


nonlinear time-history analysis 236 241

spherical sliding surfaces 181–185

verification 145–155 156–158

behaviour factors 10 69–73

bidirectional excitation 189–190

bilinear hysteretic behaviour 174–177

blow-count number 22

boundary frames 39

box girder decks 73–74

braking force 224

braking loads 225–226

bridge models 229 230

Index Terms Links


buckling

longitudinal bars 204 221


buoyant mass effects 86

buried structure flotation 33

C

cables 63

Caltrans Seismic Design Criteria 25 39 62 109–110

cantilevers 21 42 45 58

72–74

capability restorations 173 191 242–243

capacity curves 113

capacity design

conceptual design 58

detailing 123 124–128

performance requirements 12–13

seismic design examples 201–202 204–206 206 206

207

verification 123 124–128

cascading collapse 40

cases of low seismicity 16

chord rotations 108–109 134–135 136

circular

bearings 144–148

columns 53

hoops 133 134

piers 55 103

sections 101

clearance lengths 135–140

cohesionless soils

elastic response spectra 22

stability verifications 161

strength parameters 31–32

cohesive soils 31 161

collapse 40

columns

capacity design shear 126–128

conceptual design 53–59

effective torsional rigidity 106

Index Terms Links


complete quadratic combination (CQC) rule


nonlinear dynamic analyses 111

nonlinear static analyses 114

complex-valued impedance matrices 86–87

compliance criteria 5–17

composite pier stiffness 188

composite (steel–concrete) decks 44

compression models 39 109

compression zone confinements 164


abutments 59–64

connection choice 43–53

decks 38–41 59–64

ductile piers 193–194

earthquake resistance 38–43

foundations 64–65

general rules 38–43

isolation 171 174 222–224

piers 53–59

concrete

analyses 72

conceptual design 38 44–45 48

detailing 150

discretisation 73–74 79


modal response spectrum analyses 73–74 76–79 80

verification 150

cone penetration tests (CPTs) 33

confinement reinforcement 203 220–221

connections


modal response spectrum analyses 80–81

constant shear force 12

continuity slabs see link slabs

continuous decks 9 38–41 94–96

correction factors 191

cost–benefit analyses 165–166

coupling 87

CPTs see cone penetration tests

CQC see complete quadratic combination rule

cracking 80

Index Terms Links


creep 196

critical components 139–140

cross-elements 80

curved bridges 138–139

cyclic resistance ratios (CRR) 33–34

cyclic stress ratios (CSR) 33–34

cyclic ultimate chord rotation 134–135 136

D

damage limitation seismic action 8

damped elastic response

nonlinear dynamic analyses 111–112

time-history records 27–28

damping 10–15


detailing 142–143 145 149–155

elastic response spectra 21 22 23

nonlinear time-history analysis 236

seismic isolation 171 172 175–176 188–191

190

soil 32

verification 142–143 145 149–155

dead uniform traffic load applications 195

decks


conceptual design 38–41 43–53 59–64

detailing 120 150 156–159

dimensioning 120




flexible and nearly straight decks 95–96

inflexible and nearly straight decks 94–95

linear analyses 98–99

modal response spectrum analyses 73–84 75–79

overlap lengths 135–137 138

performance requirements 9

relative displacements 109–110

seismic design examples 205 206 206 207

225–226 231–232

stiffness of elements 194

verification 150 156–159

Index Terms Links


decorrelation 28

definitions 4

degree-of-freedom (DoF) systems

fundamental mode analyses 93–94 96

modal response spectrum analyses 74 79 80 87

see also single-degree-of-freedom systems

design examples see seismic design examples

design spectra 26 69 226–227

detailing 119–169

bearings 142–155

limited ductile behaviour 129


nonlinear analyses 132–135

seismic links 140–142

diagonal cracking 80

dimensioning

design examples 201–203

joint reinforcement 130



displacement 167

detailing 152


isolation systems 242



seismic isolation 186–187

verification 152

DoF see degree-of-freedom systems

double spherical sliding surfaces 182–183

dry cohesionless soils 161

ductility

analyses methods 67–68

balance with strength 15–16


design for 9 10–15

detailing 122–123 124–128

effective flexural stiffness 100

global response 12–13


nonlinear analyses 110–111 114 115

seismic design examples 193–221


Index Terms Links


Duzce (TR) earthquake 40

dynamic amplification 99

dynamic analyses 108–109 111–114

E

effective stiffness

analyses 100–107


linear analyses 107

seismic isolation 174–175 188


elastic analyses

design spectra 26 69

elastic behaviour 12 182

elastic deformations

flexible bearings 13–14

foundation ground 13–14

elastic multi-frame models 39

elastic nodal inertia loads 96

elastic predictions

displacement calculations 108

elastic response

design ground displacement/velocity 25

design seismic actions 19–25

detailing 123

horizontal 21–24

near-source effects 25

topographic amplification 25

verification 123

vertical component 24–25

elastomeric connections


conceptual design 42 43 45 47

50–53 55–56

detailing 145–155




elastoplastic energy dissipators 177 179

encased non-laminated elastomeric bearings 47

Index Terms Links


energy dissipation

design 10–15

seismic isolation 171 172

see also damping

envelope action effects 243–246

envelope linear models


equal displacement rule 10

equivalent static analyses 92–97

exemptions from Eurocode application 16

F

failure 5 6 31–32

fixed connections

conceptual design 43 47 49–50 55–56

58

dimensioning 142

fixings

elastomeric bearings 148–149

flat sliding bearings

dimensioning 142


flexible bearings 13–14 70

flexible deck models 99

flexible and nearly straight decks 95–96

flexural force 197–198 219 246

flexural plastic hinges

analyses 70

detailing/verification 124–126

flexural resistance

verification procedures 122–123

flexural rigidity


flexural stiffness

analyses 100–101

flexural verification

seismic design examples 205 206 206 207

flotation

liquefaction 33

flow failures 32

FMM see fundamental mode method of analysis

footing structural design 162–163

Index Terms Links


force reduction factors 10 152 153 155

force-based approaches 166–167

force–displacement loops

lead-rubber bearings 176–177 178

triple pendulum bearings 184

velocity-dependent devices 177 178–179

force–displacement spectra

seismic isolation 186

force–velocity relations 177 180

foundations 29–30



design action effects 206–207 248–249

detailing 163–164 165–167

elastic deformations 13–14

linear modelling 86–90

modal response spectrum analyses 89 90

nonlinear static analyses 116–117

seismic design examples 217 218 219


free cantilevers

conceptual design 42 45 58


free-standing piers 21

frequency-dependent terms 88–89

frictional devices 180–185

full dynamic model 67

fully linear behaviour 88

fundamental mode analyses 92–97 196–197

fundamental mode method (FMM) of analysis 67 185

design action effects 248–249



G

geometric imperfections 215–216

geotechnical aspects 19–36

Gerber-type hinges 39

girders 39–40 73–74 81–84

global displacement ductility factors 14

global models

detailing/verification 159

Index Terms Links


global response

ductility 12–13

gravity 80 82–83 119–120

Greece 41 42 43 45

46

grillage models 80

ground

accelerations 19–21 23 105

displacement 25

flow 166 167

oscillation 33

types 21–22

velocity 25

see also soil

H

high design ground acceleration 105

high-damping elastomeric bearings (HDEBs) 176

higher-mode effects 113–114

hinges

conceptual design 39 43 49–50 55–56


hollow circular piers 55

hollow rectangular piers 54–55 103

horizontal elastic response spectra 21–24

horizontal force transfers 48

horizontal non-seismic actions 224–225

horizontal reinforcement of joints 132

horizontal seismic components 84–85

horizontally flexible and nearly straight decks 95–96

horizontally inflexible and nearly straight decks 94–95

hydraulic viscous dampers 177 180

hydrodynamic effects 84–86

hysteresis

nonlinear static analyses 114 115

nonlinear time-history analysis 236 237–240


I

immediate use (IU) limit states 6 7

immersed piers 84–86

impedance 86–90 115–116

Index Terms Links


importance factors 8 20

imposed deformations 82–83

imposed horizontal loads 224

in shear dimensioning 202–203

in shear ultimate limit states 220

in-ground hinges 164

in-plane stiffness 80

inclined pile foundations 65

incrementally launched decks 21

individual pier model 93–94

inelastic deformations 108

inertia forces 30

inertia loads 96

inflexible and nearly straight decks 94–95

inspections 49

integral bridges 9

integral connections 60–62

integrity verifications of joints 129–130

intermediate design ground acceleration 105

intermediate movement joints 109–110 138

internal forces 163

irregular ductile bridges 67–68 110–111

isolation systems 171–192 230

analyses 185–191

behaviour families 174–185

conceptual design 44–45 52–53 55–56 171

174

design seismic action 174

examples 221–226

lateral restoring capability 191

low-damping elastomeric bearings 151–155

means 171

nonlinear time-history analysis 236 237–240 241

objectives 171

performance requirements 171–173



IU see immediate use limit states

J

joints


Index Terms Links


joints (Cont.)


dimensioning 130

horizontal reinforcement 132

integrity verifications 129–130

maximum reinforcement 130

minimum reinforcement 130

overlap lengths 135–140


seismic design examples 207 209 221

stress conditions 129

vertical reinforcement 131

K

kinematic interaction 88

Krystalopighi bridge (GR) 41 43

L

laminated elastomeric bearings 47 50 81

lateral force 96 112–113

lateral restoring capability 191

lateral spreading 32 33 35–36 164–167

LBDP see lower-bound design properties

LDEBs see low-damping elastomeric bearings

lead-rubber bearings (LRB) 176–177 178

life safety (LS) 6–7

limit states 5–7

conceptual design 54 58


detailing 121 146 162–163

effective flexural stiffness 103–104



shear verifications 247

slope stability 29–30

verification 121 146 162–163

see also ultimate limit states




verification 129

Index Terms Links


linear analyses 67 68 73

effective stiffness confirmation 107

seismic displacement calculations 107–110


verification procedures 122–123

linear modelling


foundations 86–90

soil 86–90

link slabs 81–84

links



liquefaction 32–36

adverse effects 32–33

assessment 33–34

detailing 164–167

mitigation 34–35

settlements 33 35


liquefied backfill 59–60

locked-in bridges 61 70

longitudinal bars

buckling prevention 204 221

longitudinal fundamental mode analyses 196–197

longitudinal reinforcement 246–247

longitudinal response


loose saturated soils 162

low design ground acceleration 105

low seismicity 16

low-damping bearings

detailing/verification 143 145 149–155

low-damping elastomeric bearings (LDEBs)


seismic isolation 173 175–176

lower-bound design properties (LBDP) 232–234

detailing/verification 144 145 154

fundamental mode methods 233–234

nonlinear time-history analysis 236 237 239 241–242

241


Index Terms Links


LRB see lead-rubber bearings

LS see life safety

M

maintenance


mass 72 84–86

maximum force estimations 190–191

maximum reinforcement in joints 130

medium-dense sands 162

Mesovouni bridge (GR) 46

minimum longitudinal reinforcement 246–247

minimum reinforcement in joints 130

MMS see multimode spectrum analyses

modal pushover analyses 114

modal response spectrum analyses 67 68 73–92

displacement calculations 108


seismic design examples 197

modelling 67–118

behaviour factors 69–73

effective stiffness 100–107

fundamental mode analyses 92–97

modal response spectrum analyses 67 68 73–92


seismic action components 68–69

modified historic time-history records 26

Mohr–Coulomb failure criteria 31–32

moment–axial force interaction diagrams 197 199

monolithic connections 11–12

conceptual design 41 43–46 49–50 56

57–58



movable connections


conceptual design 43 44–45 58



see also elastomeric connections; sliding connections

movement joints 109–110

multi-column piers 11 55–56

Index Terms Links


multi-frame models 39

multi-sliding surface bearings 182–185

multi-span bridges 38–39

multimode spectrum analyses (MMS) 185 187

N

natural eccentricities 95

near-collapse (NC) limit states 7

near-source effects 25

nearly straight decks 94–96

Newmark’s equal displacement rule 10

NLTH see nonlinear time-history analysis

nodal inertia loads 96

non-collapse requirements 8–16

non-concrete decks 150

non-critical components 139 140

non-encased laminated elastomeric bearings 47

non-laminated elastomeric bearings 47

non-liquefied surface layers 166

non-seismic actions 173 224–225

non-slender pier columns 58

nonlinear analyses 67–68 108–117 132–135 160

185–186 236–242

nonlinear dynamic analyses 67–68 108–109 111–114

nonlinear static analyses 67–68 112–114 160

nonlinear time-history analysis (NLTH) 185–186 236–242

O

operational (OP) limit states 6 7

out-of-plane flexibility 39–40

overlap lengths

movable joints 135–140


overpasses 56

overstrength moments 124–126 201

P

p multipliers 117

parallel voids 80

parametric analyses 106

partial seismic isolation 172–173

Index Terms Links


participation factors 74 75–79

peak ground acceleration (PGA) 23 27

pendulum bearings 182–183 184–185

performance requirements 5–17 171–173

period shifts 171 172

permanent loads 230

PGA see peak ground acceleration

piers

action effects 243–246

analyses 70–72


conceptual design 39–40 41–59

detailing/dimensioning 120




imposed deformations 225

modal response spectrum analyses 80–81 84–86


overlap lengths 138

second-order effects 215–216

section verifications 246–248

seismic design examples 193–221 230 246–248

stiffness



of elements 194


piles 9–10


detailing 163–164 165–167


nonlinear static analyses 114 116–117


plastic hinges

analyses 70

design 10 11–12

detailing 122–128 132–134

pile internal forces 163–164

seismic design examples 197–198 199 199 200

200


Index Terms Links


Poisson assumptions 20

post-elastic strength degradation 115

pot-bearings 47

precast girders 39–40

prefabricated girders 81–84

prestressed concrete decks 74 76–79

principles 4

prismatic 3D beam/column elements 73 74 80

protection, structural 6

purely cohesive soils 161

purely dry cohesionless soils 161

purely flexural elastic behaviour 12

pushover analyses 67–68 112–114 160

p–y criteria 116–117

p–y curves 167

Q

quasi-permanent traffic loads 119 195–196 211–212 230–231

R

Rayleigh quotients 96

records

artificial time-history 26 27

rectangular

bearings 144–148

columns 53

piers 54–55 103

sections 101

ties 134

reduction coefficients 167

reduction factors

soil 32

reference return periods 8 119

reference seismic action 8

reinforcement

detailing 131 132

dimensioning 130


seismic design examples 203–204 220–221


reliability differentiation 19–21

Index Terms Links


replacements


response modification factors 10

response spectra 67–68 70 195 214

215 216

response-history analyses 67–68

retaining walls 33

rheological models 89

rigid deck model 93 94–96

rigid foundations 86–87

rigidity

concrete decks 106

rigidly connected abutments 158–159

Rion-Antirrion Bridge 9 89 165–166

roadway joints 207 209

rocking impedance 89

rotational-translational coupling 87

S

safety

life 6–7

structural 5

sands 162

saturated soils 161 162

scattering of waves 28

scragging 176

SDoF see single-degree-of-freedom systems

seating 48

secant-to-yield-points

effective flexural stiffness 100–101 102 103 104


second-order effects 215–217


seismic actions 8–9 16 19–36

ductile piers 211

elastic response spectra 19–25

fundamental mode analyses 97

gravity 119–121


spatial variability 28–29

time-history representations 26–28

see also action effects

Index Terms Links



abutments 217

action effects 197–200 217–219 236–246 248–249

axial force 197–200 219 246

bearings 207–208 221

capacity design 201–202 204–206 206 206

207

decks 205 206 206 207

225–226 231–232


ductility 193–221

effective stiffness 213–214 230

flexural verification 205 206 206 207

foundations 217 218 219

fundamental mode method of analysis 233–234

isolation systems 221–226 232–233

joints 207 209 221

limit states 219–220

limited ductile behaviour 210–221



piers 193–221 246–248

plastic hinges 197–198 199 199 200

200

reinforcement 203–204 220–221

stiffness 194 199–201 213–214 230

ultimate limit states 219–220

verification 197–200 204–206 207

self-weight 195

service conditions 173

serviceability 5

serviceability limit states (SLS) 6 74

settlements 33 35

shallow foundations 160–163 165

shear force 12 46 126–128

shear keys




shear rigidity 80

shear span 11 71 72 132–133

shear strain 147–148 151–153

Index Terms Links


shear ultimate limit states 220

shear verifications 247–248

shear wave velocity measurements 33

shrinkage 196

simulated time-history records 26 27

single-column piers 11–12

analyses 72



single-degree-of-freedom (SDoF) systems

analyses methods 67 93–94 95

design 10


seismic isolation 186 189–190

single-sliding surface bearings 181–182

siting of soils 29–30

sizing

elastomeric bearings 151–155

pier columns 57–59

skewed bridges 138–139

skewed decks 98–99

slab decks 73–74 79 80

slack of the link 63

slenderness

pier columns 57

sliding connections


conceptual design 43 47–48 51–53 55–56

dimensioning 142


sliding surfaces 181–183


SLS see serviceability limit states

soils


damping 32

elastic response spectra 21–22

foundation 29–30

linear modelling 86–90

liquefaction 33 166–167

parameters 30–32

pile foundations 164

Index Terms Links


soils (Cont.)

properties 30–32

reduction factors 32

saturated 161 162

seismically active faults 29

siting of 29–30

spatial variability 28

stability 29–30 162

stiffness 32

strength parameters 31–32

solid circular columns 53

solid rectangular columns 53

spatial variability 28–29

special isolation bearings 52–53

spectral analyses 185

spherical sliding surfaces 181–185

spirals 133 134

springs

lateral spreading 166 167


SPT see Standard Penetration Tests

square root of the sum of the squares (SRSS) rule 29 92 98–99 111

114 163

stability verifications 64 160–162

Standard Penetration Tests (SPT) 22 32 33 34

static analyses 67–68 92–97 112–117 160

steel decks 44

steel elastoplastic energy dissipators 177 179

stiffness

analyses 100–107


design examples 194 199–201 213–214 230

linear analyses 107


seismic isolation 173 174–175 188

soil properties 32

strain hardening 115

strength

balance with ductility 15–16

design 15


soils 31–32

Index Terms Links


stress conditions of joints 129

structural protection 6

structural safety 5

structural seismic analysis models 229–230

substructure verifications 243–246

superposition 86 87 225–226

surfacing materials 72

sustainability 5

symbols 4

symmetric supports 94–96

T

target displacement 111

temperature variations 225

tension models 39 109

terms 4

theoretical eccentricities 95

thermal actions 196

three-span bridges 74 76–79

tie-downs 42 45

time-history analyses 67–68 108–109 185–186 227–229

236–242 248

time-history representations 26–28

topographic amplification 25

topping slabs 39–40


torsional rigidity 106

total shear strain 147–148

traffic loads 119 195 196 211–212

230–231

transition periods 10

transverse reinforcement of piers 204

transverse responses 42–43

travelling wave effect 28

triple pendulum bearings 182–183 184–185

truss elements 109

tuned mass dampers 119 121

Turkey 40

twin-blade parts 41–42 44 45 45

54 58

twisting phenomena 98

Index Terms Links


U

UBDP see upper-bound design properties

UDL see uniform traffic loads

ULS see ultimate limit states

ultimate limit states (ULS) 6



detailing 121–123 130 146 148–149

162–163

effective flexural stiffness 103–104



shear verifications 247


verification 121–123 130 146 148–149

162–163

uncracked flexural stiffness 100

uniform seismic demands 41–43

uniform traffic loads (UDL) 119 195

uplifting 48

upper-bound design properties (UBDP) 232–233

detailing 144 145 154–155

fundamental mode methods 235–236

nonlinear time-history analysis 236 238 240 241


verification 144 145 154–155

V

velocity 25 33 177–180

verification

abutments 155–159

bearings 142–155 156–158

of components 119–169

design examples 197–200 204–206 207

foundations 159–164 165–167

isolation system substructure 243–246

isolation systems 243–246

lateral spreading 164–167



liquefaction 164–167

Index Terms Links


verification (Cont.)



vertical bars 134

vertical components

elastic spectra 24–25

fundamental mode analyses 97

vertical loads 116

vertical reinforcement of joints 131

vertical seismic components 85–86

very wide bridge decks 98–99

viscous damping 21 22 190–191

volumetric strain assessments 35

Votonosi bridge (GR) 41 42

W

wall-like piers 54

walls 33

water 84–86

waves 28 33

Winkler foundations 163

Winkler models

lateral spreading 166 167



wish-bone shaped elements 109

Designers' Guide to Eurocode 8 Design of Bridges for Earthquake Resistance (Designers' Guides to the Eurocodes)

Documents

en eurocode

uk construction industry

thomas telford limited

division of thomas telford

thomas telford initiative

statements madeand opinions

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