Dissertation2003-Lupoi

8/13/2019 Dissertation2003-Lupoi

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Istituto Universitario

di Studi Superiori di PaviaUniversit degli Studi

di Pavia

EUROPEAN SCHOOL OF ADVANCED STUDIES IN

REDUCTION OF SEISMIC RISK

ROSE SCHOOL

LIMITATIONS AND PERFORMANCES

OF DIFFERENT APPROACHES FOR SEISMIC

ASSESSMENT

OF EXISTING BUILDINGS

A Dissertation Submitted in Partial

Fulfilment of the Requirements for the Master Degree in

EARTHQUAKE ENGINEERING

By

GIORGIO LUPOI

Supervisors: Prof. PAOLO EMILIO PINTO

Prof. GIAN MICHELE CALVI

June, 2003


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The dissertation entitled Limitations and performances of different approaches for seismic

assessment of existing buildings, by Giorgio Lupoi, has been approved in partial fulfilment of

the requirements for the Master Degree in Earthquake Engineering.

Paolo Emilio Pinto ________________________________

Gian Michele Calvi ________________________________


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ACKNOWLEDGEMENTS

The work has been carried out under partial funding from the EU project SPEAR (Contract No.

G6RD-CT-2001-00525).


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ABSTRACT

The present study consists of a commented application of the three major guidance documents

on the assessment of existing buildings currently available: the New Zealand

Recommendations, the U.S. ASCE-FEMA356, and the Japanese Standard, to three structures

(two 2D and one 3D frames) which have been constructed at a large scale and tested. The main

purpose of the study is that of checking the practical applicability of the methods, the relative

ease of use, and of course the degree of agreement on the results.

The theoretical framework on which each document is based as well as the proposed methodsare outlined and commented. Differences of conceptual nature existing between the various

approaches are noted.

From the small number of cases examined is not possible to systematically trace the differences

in the results produced by the different approaches. The large difference in the way the shear

capacities of members and joints are evaluated has been a decisive factor in some cases for the

determination of the ultimate capacity of the entire building. However, even if this source of

discrepancy of the results from the various approaches was eliminated, the present exploration

indicates that significant differences would remain, linked to the criteria used to relate the

capacity curve to the response spectrum, or to the use of elastic analysis combined with local

ductility factors, as in the U.S. FEMA356, instead of the global mechanism analysis of New

Zealand.


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v

TABLE OF CONTENTS

Aknowledgement ... i

Abstract ... iii

Table of contents ...... v

List of figures . vii

List of tables ix

1 INTRODUCTION.................................................................................................................. 1

2 REFERENCE DOCUMENTS .............................................................................................. 3

2.1 U.S. ASCE (FEMA 356) Prestandard ................................................................................. 3

2.1.1 Linear Static Procedure (LSP) ..................................................................... ............................5

2.1.2 Linear Dynamic Procedure (LDP) ................................................................... ........................6

2.1.3 Nonlinear Static Procedure (NSP) ........................................................... ................................6

2.1.4 Non-linear dynamic procedure (NDP) .............................................................. .......................7

2.1.5 Specific rules for concrete structures .................................................................... ...................8

2.2 New Zealand guidelines ...................................................................................................... 8

2.2.1 Force-based procedure ............................................................... ............................................ 10

2.2.2 Displacement-based procedure ....................................................... ....................................... 14

2.2.3 Comments ...................................................... ........................................................... .............15

2.3 Japanese Guidelines .......................................................................................................... 16

3 COMPARATIVE STUDY................................................................................................... 21

3.1 Selected procedures........................................................................................................... 21

3.2 The test structures ............................................................................................................. 21

3.2.1 Pavia Frame .............................................................. ............................................................. 21

3.2.2 Icons Frame............................................................................................................................23

3.2.3 Spear Frame ......................................................... ........................................................... .......24

3.3 Strengths of materials........................................................................................................ 25

3.4 Terms of comparison (capacity measure and earthquake input model)............................ 26

3.5 Non-linear numerical analysis........................................................................................... 27


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3.6 Risk reduction factor ......................................................................................................... 28

4 EXAMPLE APPLICATIONS............................................................................................. 29

4.1 FEMA 356: linear static .................................................................................................... 29

4.2 FEMA 356: Nonlinear static ............................................................................................. 32

4.3 NZ: Simple lateral Mechanism Analysis........................................................................... 34

4.4 NZ: Nonlinear Static.......................................................................................................... 38

4.5 BDPA: 3rdlevel ................................................................................................................. 41

5 COMPARISONS .................................................................................................................. 43

6 CONCLUSIONS...................................................................................................................51

7 REFERENCES ..................................................................................................................... 53

APPENDIX A: EC8 SPECTRA..................................................................................................57

APPENDIX B: MATERIALS MODELS ................................................................................. 59

A.1 Concrete............................................................................................................................. 59

A.2 Steel ...................................................................................................................................61

APPENDIX C: EARTHQUAKE INDUCED AXIAL FORCES.............................................. 63


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vii

LIST OF FIGURES

Figure 1. Mechanisms of post-elastic deformation of moment resisting frames [NZSEE 2002]11

Figure 2. Plastic displacement profiles for hinging at the base of the building [NZSEE 2002] . 13

Figure 3. Degradation of nominal shear stress of concrete for columns [Priestley 1996] .......... 14

Figure 4. Equivalent stiffness and equivalent damping for the Shibata-Sozen approach [NZSEE

2002] ................................................................................................................................... 15

Figure 5. View of Pavia frame ................................................................................................... 22

Figure 6. Gravity loads for Pavia frame...................................................................................... 22Figure 7. Reinforcement of columns and beams for Pavia frame............................................... 23

Figure 8. Four-storey frame, Icons.............................................................................................. 24

Figure 9. Gravity loads for Icons frame...................................................................................... 24

Figure 10. Geometric characteristics of the three test-structures ................................................ 25

Figure 11. Plant of Ispra frame and details of the columns reinforcement ................................. 25

Figure 12. EC8 elastic and design spectra .................................................................................. 26

Figure 13. Fiber model................................................................................................................ 27

Figure 14. Numbering of joints and sections and values of the coefficients of the vertical

distribution of the lateral load ............................................................................................. 30

Figure 15. Uniform and triangular vertical distributions of lateral loads.................................... 32

Figure 16. Resultant SDOF from mechanism analysis ............................................................... 37

Figure 17. Force approach of SLM for the Pavia Frame ............................................................ 37

Figure 18. Push-over curves for two types of distributions ........................................................ 38

Figure 19. Shear and flexural failures for the triangular distribution.......................................... 39

Figure 20. Design and elastic spectra for PGA at failure ............................................................ 40

Figure 21. Failure PGA for the Pavia frame................................................................................ 43

Figure 22. Failure PGA for the Icons frame................................................................................ 44

Figure 23. Collapse points of push-over procedures for Pavia frame......................................... 44

Figure 24. Collapse points of push-over procedures for Icons frame ......................................... 45

Figure 25. Comparison of shear capacities of joints for the Pavia and Icons frames.................. 45

Figure 26. Comparison of shear capacities of beams and columns for the Pavia and Icons frames

............................................................................................................................................. 46

Figure 27. Equivalent SDOF....................................................................................................... 46


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Figure 28. PGA s for different ductility levels ............................................................................ 48

Figure 29. Resisting frames of SPEAR structure ........................................................................ 49

Figure 30: Failure PGA for the 3D Spear building......................................................................49

Figure 31: EC8 acceleration spectrum Type 1, soil C.................................................................58

Figure 32: EC8 displacement spectrum Type 1, soil C...............................................................58

Figure 33: Stress strain relationship for different level of lateral confinement........................... 60

Figure 34: Stress strain relationship of the concrete ................................................................... 61

Figure 35: Stress strain relationship of the steel..........................................................................61


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ix

LIST OF TABLES

Table 1. Classification of vertical elements ................................................................................ 18

Table 2. Value of displacement compatibility factor.................................................................. 18

Table 3. Ductility index .............................................................................................................. 18

Table 4. Materials strength of the test structures (MPa) ............................................................. 26

Table 5. Values of minimum multiplier for each limit states ................................................. 31

Table 6. Collapse point for the four push-over analyses............................................................. 33

Table 7. Sway Indexes ................................................................................................................ 35Table 8. Base Shear at yielding and collapse.............................................................................. 35

Table 9. Curvature ductility and shear demand for the four base-columns ................................ 36

Table 10. Collapse point for the four analyses............................................................................ 40

Table 11. Results of the 3rdlevel procedure for the three different floors .................................. 42

Table 12. Failure mechanism according to the different procedures ......................................... 48


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1 INTRODUCTION

Earthquake engineering experts, public Authorities and general public alike concur on the idea

that the assessment of the seismic safety of the built environment is a matter of high priority.

Awareness of the problem has been accelerated by the disastrous effects observed in recent

seismic events, in terms of loss of lives as well as of immediate and long-term economic losses.

The evolution of the attitude has been, one might say, more rapid that the capacity of the

technical community to cope adequately with it. Not because the seriousness of the problem had

escaped to it, but because the core activity had to be directed towards the improvement and the

harmonisation of the codes for the design of the new structures, on one hand, and also because

of the intrinsic difficulty of dealing with the problem of existing structures with procedures atthe same time rigorous, general, and practically applicable. This latter difficulty is compounded

with the lack of experimental data and models for the behaviour and the capacity of non-

seismically detailed members. The situation is now improving, with the appearance of recent

guidance documents on assessment from New Zealand [NZSEE, 2002], U.S.A. [ASCE, 2002]

and Japan [JBDPA, 1977-90], while Europe is about to complete its new Eurocode (Eurocode 8,

Part 3) on the subject.

Though each of these documents has been thoroughly checked for internal consistency,

comparative applications to selected structures have not been performed to date. Yet these test

applications are deemed to be interesting for a number of reasons: the procedures depart in

different ways from the direct design approaches, they make use of different models for

assessing the capacity of existing members, they provide more or less explicit and stringent

guidance to the user, and of course they offer different levels of complexity. Furthermore, trying

to explain results obtained from different procedures provides an excellent opportunity for

looking into the inner mechanisms of the procedures, and from this to pinpoint the more

relevant or critical aspects.


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2 Chapter 1. Introduction

The present study consists of a commented application of the three documents mentioned above

to the three structures (two 2D and one 3D frames) which have been constructed at a large scale

and tested (except the 3D one at the time of this writing) in the experimental facilities of the

University of Pavia and of the JRC in Ispra. Complete knowledge of the material properties,

reinforcement details and of the results from the actual tests are the reasons behind this

particular choice.


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2 REFERENCE DOCUMENTS

This section provides a brief general introduction to the three regulatory documents used in this

study, in order to highlight their main features as well as the principal differences among them.

2.1 U.S. ASCE (FEMA 356) Prestandard

The ASCE Prestandard for the Seismic Rehabilitation of Buildings, commonly known as

FEMA 356, is a comprehensive State-of-the-Art document on rational quantitative seismic

assessment and rehabilitation of existing buildings of concrete, steel or cast iron, masonry and

timber construction, and represents the most recent US development in seismic assessment. In

broad terms, it can be said to belong to the category of the displacement based approaches.

A brief summary is provided hereafter, focusing on the aspects relevant to the assessment of

concrete structures.

The assessment process starts with the definition of one or more Rehabilitation Objectives, each

one consisting on the selection of both a Target Building Performance Level and of the

corresponding Seismic Hazard Level. Several options are available. The Building

Performance Level is given by a combination of a Structural Performance Level, varying

from Immediate Occupancy to Collapse Prevention, and a Non Structural PerformanceLevel, varying from Operational to Not-Considered.

The seismic hazard can be represented either by an acceleration response spectrum or by

acceleration time histories. Two basic Earthquake Hazard Levels are defined: Basic Safety

Earthquake 1, corresponding to an event with 10% probability of exceedance in 50 years (return

period equal to 475 years), and Basic Safety Earthquake 2, corresponding to an event with 2%

probability of exceedance in 50 years (return period equal to 2475 years). A response spectrum

for different soil types and damping coefficients is provided as function of two parameters: the


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4 Chapter 2. Reference Documents

short-period response acceleration, XSS , and the long-period response acceleration, 1XS , which

have to be determined using the values taken from approved maps developed by the United

States Geological Survey and modified for the soil class of the site.

The selection of the so-called knowledge factor comes next. It accounts for the

uncertainties in the collection of as-built data and depends on the selected Rehabilitation

Objective, the accuracy of the data collected and the selected analysis procedure. The values for

are tabled in the Prestandard, varying from 0.75 to 1.

The actual assessment of the building can now be undertaken, according to one of the two

possible Rehabilitation Methods: the Simplified Rehabilitation Method and the Systematic

Rehabilitation Method. The former may be applied to certain buildings of regular configuration

that do not require advanced analytical procedures; the latter may be applied to any building

and involves thorough checking of each existing structural element or component, . , and the

verification of acceptable overall performance represented by expected displacements and

internal forces.

In this study, attention is concentrated on the more accurate method, i.e. the Systematic

Rehabilitation. Four different analysis procedures are allowed for the evaluation of the response

of the building: the linear static (LSP), the linear dynamic (LDP), the non-linear static (NSP)

and the non-linear dynamic (NDP).

The two linear procedures (LSP and LDP) are permitted only for buildings with a regular

structural configuration. A regular configuration is defined by means of geometrical

requirements and of a parameter called demand-capacity ratio (DCR), which is defined as the

ratio between the force acting on a member due to gravity and earthquake loads and its expected

strength. This parameter is intended to be a rough evaluation of magnitude and distribution of

the inelastic demands in the building; it is used however only to determine the structural

regularity.

The static procedures (LSP and NSP) are permitted only for those structures in which higher

mode effects are not significant. No limitations are set to the use of the non-linear dynamicprocedure.

The members are classified as primary or secondary according to their function: primary

elements are those that provide the capacity of the structure to resist collapse under seismic

force, secondary elements are all the others. Only primary elements are included in the

mathematical model of the structure. Detailed instructions are provided to include in the

analysis torsional as well as P effects.


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U.S. ASCE (Fema 356) Prestandard 5

The internal actions on members are classified as either deformation-controlled or force

controlled: the first ones are characterised by a ductile behaviour, the second ones by a brittle

behaviour. The classification is not up to the user, but it is explicitly indicated in the document.

In RC frame-type buildings, for example, the bending moment in beams and columns is

considered as a deformation-controlled action, while the shear is considered as a force-

controlled action.

This distinction governs the way in which verification of the elements is carried out. In the first

place, it affects the evaluation of the elements capacity: the expected value for the materials

strength is used for deformation-controlled actions, a lower-bound estimate for force-controlled

actions. More importantly, the limit state equation that governs the capacity/demand comparison

changes in accordance with the type of action: this latter point will be illustrated in detail

separately for each of the analysis procedures.

2.1.1 Linear Static Procedure (LSP)

The building response is evaluated by a linear elastic analysis. The seismic action is modelled as

a horizontal lateral load V, whose intensity is determined by the following expression

WSCCCCV am321= , (1)

where: 1C is a modification factor relating the expected maximum inelastic displacements to

the calculated elastic response; 2C represents the effect of stiffness degradation and strength

deterioration on maximum displacement response; 3C accounts for the increase of the

displacements due to P effects; mC is the effective weight factor to account for higher

modal mass participation effects, depending on the structural typology and on the number of

storeys; aS is the Response Spectral Acceleration at the fundamental period and damping ratio

of the building in the direction under consideration and W is the total weight of the building.

The lateral load Vis vertically distributed at floor levels proportionally to the floor mass and to

the floor height.

The fundamental period of the building can be estimated using either an empirical expression

provided in the document, or by an eigenvalue analysis, or by the Rayleigh-Ritz method.

A capacity/demand approach is applied to carry out the assessment for each individual element.

For deformation-controlled actions, the limit-state equation reads:

EGCE QQQm . (2)

In the left hand side of Eq. (2), the member capacity is given by the product of the expected

strength CEQ times the knowledge factor times the local ductility factor m . The latter is


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introduced to account for the expected ductility associated with this action. Values of m are

given as functions of section geometry, amount of reinforcement, amount of the shear force and

selected performance level. For the Collapse Prevention performance level the values of m

for beams range between 7 and 2.

On the right hand side of Eq. (2) the demand on the members is the sum of the gravity load

effects GQ and of the earthquake load effects EQ .

For force-controlled actions, the limit-state equation reads:

JCCC

QQQ EGCL

321

. (3)

The member capacity is the product of the lower-bound strength CLQ times the knowledge

factor . The effects of the earthquake loads are reduced by the coefficients1

C ,2

C ,3

C

already introduced in Eq. (1) and by the force-reduction factor J , which is greater than or equal

to 1.0. Its value may be taken as the smallest DCR of the components or, alternatively, equal to

2.0, 1.5 and 1.0 in Zones of High, Moderate and Low Seismicity, respectively.

The limit-state equations reflect the failure mechanisms associated with the two types of

actions: in the case of a ductile behaviour, Eq. (2), the capacity is increased by the ductility

factor m ; in the case of a brittle behaviour, Eq. (3), the external elastic action is reduced by the

coefficient J , to account for the inelastic behaviour of the elements delivering load to the

brittle element.

Although the verifications for both force-controlled and deformation-controlled actions are

expressed in terms of forces, the procedure belongs to the category of the displacement-based

approaches, since its qualifying feature is the implicit adoption of the equal displacement rule

(adjusted by the factor 1C ).

2.1.2 Linear Dynamic Procedure (LDP)

The only difference with respect to the LSP is the method of analysis: a modal spectral analysis

is carried out using un-reduced linear elastic response spectra. The members verifications are

analogous to those of the LSP.

2.1.3 Nonlinear Static Procedure (NSP)

The demand on the building is calculated by means of a push-over analysis. The inelastic

behaviour of the structure is explicitly accounted for in the numerical model by use of either

concentrated or diffused plasticity element formulations. A monotonically increasing lateral

load is applied to the building until a target displacement t is reached at a control node, taken


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U.S. ASCE (Fema 356) Prestandard 7

to be the centre of mass at the roof of the building. The target displacement is intended to

represent the maximum displacement likely to be experienced during the design earthquake

and it is determined by the expression:

g4

TSCCCC

2

2e

a3210

=t .(4)

where 0C is a modification factor relating the spectral displacement of an equivalent single

degree of freedom (SDOF) system to the roof displacement of the building, and 1C , 2C , 3C and

aS have been defined in section 2.1.1.

The push-over analysis yields a relation between the base shear and the displacement at the

control node. This force/displacement curve is bi-linearised and the effective fundamental

period of the building calculated according to the expression:

e

iie

K

KTT = . (5)

where iT is the elastic fundamental period, iK is the elastic lateral stiffness, eK is the effective

lateral stiffness.

The format of the limit state equation is common to both displacement-controlled and force-

controlled actions:

( )tDC > . (6)

The Eq. (6) says that the member capacity Chas to be greater than the demand D evaluated at

the target displacement t . It is worth noting that for deformation-controlled action, the check is

made in terms of ultimate rotation capacity and demand, not in terms of bending moments, as in

the two linear procedures (LSP and LDP). The expected rotational capacity is tabled in the

document for several typical section and reinforcement configurations.

A further condition is put on the base shear at the target displacement, which cannot be less than

80% of the effective yield strength of the structure.

2.1.4 Non-linear dynamic procedure (NDP)

The numerical model of the building and the limit-state equations are similar to those of the

NSP. The only difference is that the structural response is calculated using a Time History

Analysis.


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2.1.5 Specific rules for concrete structures

The flexural strength and the deformation capacity of members are calculated in accordance

with either the procedures of ACI 318 or other approved methods. Reductions in the

deformation capacity due to the shear have to be taken into consideration.

The shear strength has to be calculated in accordance with the procedures of ACI 318. A

reduction of 50% of the contribution of the transverse reinforcement is required if the

longitudinal spacing of transverse reinforcements exceeds half of the component effective

depth.

In setting up the numerical model, the members stiffness has to be calculated taking into

account the effective state of stress as well as the bar-slippage phenomenon. The reduction

coefficients for the members initial stiffness, to be applied in the case of linear analysis, are

explicitly indicated in the document. For example, the effective stiffness of beams is taken as

equal to half of the initial, un-cracked, stiffness. In columns, the reduction depends on the stress

state under gravity load; it varies form 0.5 to 0.7 of the initial stiffness.

2.2 New Zealand guidelines

The document produced by the New Zealand Society for Earthquake Engineering provides a

means of assessing the capability of existing buildings to reach an adequate level of seismic

performance and, where found necessary, improving the seismic performance up to a certain

level. A first draft was circulated for comment in 1996 and a second general draft was

completed in 2002.

Two levels of assessment are considered in the document: an Initial Evaluation of the state of

the building and a Detailed Assessment. The Initial Evaluation is a coarse screening, similar

to the simplified procedure of FEMA 356, which is intended to provide an approximate

assessment of the likely performance of a building in an earthquake. The Detailed Assessment is

intended to evaluate with greater accuracy the capability of an existing building to perform

satisfactorily under an earthquake of selected level of intensity through application of more

advanced procedures. In this study the attention is concentrated on the Detailed Assessment.

The focus of the detailed procedure is the determination of demand on structural elements,

resulting form the response of the building, and the assessment of the capacity of such elements

to meet the demand without causing loss of structural integrity. The NZ document concentrates

only on matters relating to life safety, i.e. collapse which leads to loss of life. The ultimate limit

state (ULS) is selected as boundary between the acceptable and the unacceptable performance.


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New Zealand guidelines 9

The ULS is conventionally reached when the building or any part of it: (a) loses stability, (b)

exceeds prescribed displacement limits or (c) is strained to the accepted limits of the (structural)

materials involved.

Three possible approaches for performing the assessment are indicated in the document: time

history analysis, force analysis and displacement analysis. Leaving out the first, which is by far

the most accurate but the most complex as well, the attention is set on the other two approaches.

In the document it is stated that the displacement-based approach is generally considered to

produce more rational and less conservative assessment outcome, the force-based one is more

familiar to designers. This is as far as the document goes in assessing the relative merits of the

two approaches.

Four analysis methods of different level of sophistication are proposed for the evaluation of the

structural response. Two of them rely on linear elastic theory: a traditional equivalent static

analysis and a modal response spectrum analysis. Limitations to the use of the elastic methods

similar to those of FEMA are set in the NZ document. The other two consider inelastic

behaviour; these are: Simple Lateral Mechanism analysis (SLM) and lateral push-over

analysis. It is observed that the proposed procedures are the same as those in the FEMA

document, with the exception of the SLM that is a peculiarity of the NZ document. This

approach involves a hand analysis to determine the probable collapse mechanism, its lateral

strength and displacement capacity, with the help of simplified considerations of capacity issues

(relative strengths in flexure and shear, etc.). The behaviour of the structure is reduced to that of

an equivalent SDOF system.

The main weakness of SLM approach is the inability to identify the sequence of development of

inelastic action between different members of the structure, a result achievable by means of a

push-over simulation.

The model of the earthquake action depends on the analysis method applied. The acceleration

response spectra and the displacement response spectra to be used in the assessment according

to force-based or displacement-based approaches, respectively, are those defined in the NZ

loading code [SNZ, 1976].

A brief outline of both force-based and displacement-based procedures is given in the

following, in the same way as they are presented in the original document. As it will be seen,

the route followed to determine the likely lateral inelastic mechanism and its

strength/deformation capacity is common to both procedures, i.e. the SLM analysis method and

the push-over, the essential difference intervening in the way the structural response is

evaluated.


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2.2.1 Force-based procedure

The procedure starts with the evaluation of members capacities. The probable flexural strengths

are calculated according to standard theory. The shear strengths in beams, columns and joints

are derived by means of the expressions provided in the document. A strength reduction factor

equal to 0.85 is applied to account for the approximation in the shear theory. The calculations

are to be based on the expected values for the materials strengths. If the data available are not

reliable, a variation of either one standard deviation or +/- 20% of the mean is suggested. It is

required to consider the earthquake-induced axial force in the calculation of the flexural and

shear capacities of the columns.

The post-elastic critical mechanism is investigated next. To this end, thesway potential index is

introduced. It is defined as the ratio between the sums, for all the joints at that horizontal level,

of the probable flexural strengths of beams over the probable flexural strengths of columns:

( )

( )+

+=

cbca

brbl

iMM

MMS , (7)

where blM and brM are the flexural capacities of the beams at left and right of the joint and

caM and cbM refer to the columns above and below the joint.

If 85.0>iS , the NZ document suggests that plastic hinges would develop in the columns;

otherwise they would develop in the beams. The most likely post-elastic mechanism may be

established from the calculation of the iS s at all levels. The possible outcomes are: a column

sidesway mechanism, a beam sidesway mechanism or a mixed mechanism: see Figure 1. In the

latter case, column plastic hinges would occur at some levels and beam plastic hinges at others.

The possibility of members shear failures has not been yet investigated at this point of the

procedure.

If the mechanism of post-elastic deformation is obvious from the onset, the SLM analysis

method can be applied for the calculation of the probable lateral seismic force capacity, bV . For

example, if a column sidesway mechanism is detected at the bottom floor, the probable lateralforce capacity of the frame is given by the sum of the shear forces in the columns of that storey,

found from the sum of the probable flexural strengths of the plastic hinges at the top and the

bottom of the columns of that storey divided by the storey height. This estimate is an upper

bound of the lateral force capacity.


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Figure 1. Mechanisms of post-elastic deformation of moment resisting frames [NZSEE 2002]

If the sway-index approach does not provide a clear indication on the failure mechanism, either

a standard linear elastic analysis or a non-linear push-over analysis have to be carried out. In the

first case, a linear-elastic model of the frame is set up; the equivalent static horizontal forces are

increased until the first plastic hinge forms. This method provides a lower bound to the probable

lateral force capacity of the frame, not accounting for moment redistributions. In the second

case, the inelastic deformation capacity of members is explicitly taken into account in the

structural model and the applied lateral loads are increased until the flat branch of the

force/displacement curve is reached. The NZ document suggests to apply both an inverted

triangular and a uniform horizontal force distribution.

The push-over analysis is more accurate than the SLM or the LSP, but it is also more complex.

The benefit of using the SLM approach is obvious, since it does not require numerical

modelling of the structure.

The probable base shear coefficient, i.e. the spectral acceleration at failure, is calculated by

means of the expression:

( )

RZSW

TC

pt

bsdh

V=, , (8)

where tW is the total weight of the structure, pS is the structural performance factor, R is the

return period factor based on the life span of the structure, Zis the zone factor and corresponds

to the response acceleration at 0.5 sec on a rock site with a 500-year return period. The values of

Zand R are specified in the document; their product defines the anchoring of the spectrum.

The force-based approach comes here into play: using the appropriate seismic hazard

acceleration spectra of NZS 4203:1192, the required structural ductility factor sd is derived

on the basis of hC and of the elastic period of the frame T. The NZ document does not indicate


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explicitly how to compute the fundamental period of the structure; it is however required to

include the effects of cracking on the section properties. The use of the acceleration spectrum

implicitly assumes that the structure behaviour can be represented as that of a SDOF oscillator.

The assessment whether the required demand sd can be satisfied by the available structural

ductility sc comes next. Three methods for the evaluation of sc are outlined in the NZ

document. The first one is a simple and qualitative evaluation of sc , based on the most likely

mechanism and on the detailing of the lateral reinforcement. A second possibility is to derive

the global displacement ductility sc directly from the (local) inelastic capacity of members

sections. To this end, empirical expressions of the plastic deformation are provided in the

document for the cases of beam sidesway and column sidesway mechanisms (see Figure 2,

which refers to plastic hinging occurring at the base of the building):

mmhLh ePycepp 391 === , (9)

where p is the plastic rotation of the column, eh is the equivalent height of the building, c

is the curvature ductility, y is the yielding curvature and pL is the plastic-hinge length. The

latter is function of the diameter of the longitudinal reinforcement bd and the yield strength of

the steel yf :

ybP fdLL += 022.008.0 , (10)

with L being the length of the member.

In addition lateral displacement of the frame at first yield y has to be evaluated. In the

document it is generically suggested to carry out a linear elastic push-over analysis for the

determination of y (it is believed that the analysis has to stop when the yielding moment in

any of the members sections is reached). This solution, however, represents a limitation to the

use of the SLM method, whose key feature is to avoid numerical modelling. Alternatively, an

empirical expression for y is provided in the context of a development of the SLM approach

proposed by Priestley into an appendix of the document:

= e

b

byy h

h

l5.0 , (11)

where y is the steel yielding strain, bl is the length of the shortest span, eh is the equivalent

height of the building and bh is the beam section height.


27/78


Figure 2. Plastic displacement profiles for hinging at the base of the building [NZSEE 2002]

The third method for the evaluation of sc is the non-linear static push-over: the lateral forces

are increased until the available ultimate curvature is reached at the critical plastic hinge. The

ultimate displacement is then obtained from numerical simulation. This option represents the

most advanced and accurate method available thus far.

If sdsc < , the procedure is terminated, with the conclusion that structure needs to be

retrofitted. If sdsc > , a check whether shear failures occur before the ultimate flexural plastic

rotations are reached needs to be carried out.

For each member, the reduction of the shear strength due to the curvature ductility demand

associated with the development of plastic hinges is estimated using the degradation laws

proposed by Priestley and co-workers (see for example Figure 3). The reduced shear strength,

crV , is compared with the shear demand on member generated by the plastic hinges (flexural)

mechanism, flV , the latter obtained from the equilibrium condition considering the ultimate

values of bending moments. If flcr VV < , strengthening is necessary.

A final check on the inter-storey drift is required to ensure against significant P effects and

damage to non-structural elements.


28/78


Figure 3. Degradation of nominal shear stress of concrete for columns [Priestley 1996]

2.2.2 Displacement-based procedure

The first part of the procedure dealing with the calculation of the members strengths, theidentification of most-likely failure mechanism and the evaluation of the frame probable lateral

force capacity, is the same as the force-based procedure.

The next step consists in checking the actual available shear strength, accounting for its possible

reduction due to curvature ductility demand, following the same procedure as described in the

previous section. Following this check the available local rotation capacities are reduced, if

necessary, to the values pertaining to actual shear capacity. It is noted that this check on shear is

anticipated with respect to the force-based procedure.

The global ductility capacity of the structure, sc , is found as in the force-based procedure. The

available displacement capacity at failure is obtained simply as:

( )scysc += 1 . (12)

For the evaluation of the yielding displacement, the indications given in the previous section

apply.

The demand due to the earthquake action is now evaluated applying the substitute-structure

approach [Shibata and Sozen 1976]. The frame behaviour is again reduced to that of a SDOF

oscillator, but the effective stiffness at maximum displacement is used:

sc

beff

VK= , (13)

from which the effective period T for the structure is calculated as:

effK

MT 2= , (14)

where is the mass of the structure.


29/78


The maximum displacement demand sd is found from a set of displacement response spectra,

relative to different levels of the equivalent viscous damping, v . The level of damping of the

frame depends on the ductility capacity sc and on the predominant form of plastic hinging

developed. Curves are provided relating v to sc for the most common failure mechanisms

(see Figure 4).

Thus, in the displacement procedure the seismic response is characterised by an effective

stiffness and an equivalent damping, rather than the elastic stiffness and the 5% damping as

used in the force-based design or assessment.

The required structure displacement demand is compared to the ultimate displacement capacity.

Acceptable performance is indicated by the ratio sdsc resulting greater than one.

Figure 4. Equivalent stiffness and equivalent damping for the Shibata-Sozen approach [NZSEE 2002]

2.2.3 Comments

The above summaries show that the objective common to both approaches is the determination

of the structure lateral capacity and of the structure yielding and ultimate displacements. For this

purpose, either a simplified hand procedure, i.e. the SLM approach, or a more accurate push-

over analysis may be resorted to. Furthermore, both procedures reduce the structure behaviour

to that of a SDOF oscillator, characterised by a force-displacement relation. Thus, their only

difference lies in the way the earthquake demand on the frame is established, which is however

not a secondary point. The force-based approach uses an acceleration spectrum in conjunction

with the initial elastic period of the structure to estimate the required ductility demand; the

displacement-based approach uses a displacement spectrum and an equivalent period to estimate

the maximum displacement demand.


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Both criteria are admittedly approximate, and a judgement on their relative merits is not within

the scope of the present study. A comment that is relevant to the present study, however, is that

the magnitude of the difference between the results obtainable with the two criteria can be easily

absorbed in the overall approximation of the SLM method, which varies from case to case and

rests to a considerable extent on the skill of the analyst. The push-over analysis, on the other

hand, which incidentally is a displacement-based procedure, gives a more uniform level of

approximation and, in spite of its own limitations, covers a much more wide range of practical

applications than the SLM method.

2.3 Japanese Guidelines

The first publication of the Standard for Seismic Vulnerability Assessment of Existing

Reinforced Concrete Buildings, edited by a committee organised by the Ministry of

Construction, goes back to 1977. The Standard was revised in December 1990 and its

effectiveness has been tested at the occasion of several earthquakes, for school buildings in

particular.

The standard may be applied to evaluate the seismic performance of an existing reinforced

concrete building whose structural system is made of moment resisting frames, with or without

shear walls. Its application is limited to low-rise buildings, since the standard assumes constant

acceleration response (flat response acceleration spectrum with periods) [Otani 2000].

Three procedures of different accuracy and reliability are available for the assessment. Thefirst-

level procedure is a simple screening procedure aimed at classifying earthquake resistant

buildings based on their storey shear strengths, provided by either columns or/and structural

walls. The buildings classified as questionable by the first-level procedure, must be analysed

by the more sophisticated second-level procedure, in which the deformation capacities of

vertical members are also considered. Finally, the third-levelprocedure is a refined version of

the second level, in which also the weak beam-strong column mechanism is included [FIB

2003]. The attention of the present study is concentrated on the latter procedure.

The Japanese assessment consists in the comparison of the floor shear demand due to the

earthquake action versus the floor shear capacity, for every storey and every frame of the

building:

ic

id VV .

(15)

The floor shear capacity is defined by the following product:

iiic ESTV 00= .

(16)


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Japanese guidelines 17

In Eq. (16), the factor T accounts for the time-dependent deterioration of the members strength

and the factori

S0 accounts for geometry or stiffness irregularity and/or mass concentration at

floor i . Both Tand iS0 are functions of several factors, whose number depends on the level of

the assessment. Their evaluation is based on the results of field-surveys.

The most relevant factor in Eq. (16) is the basic structural performance index iE0 , which is

defined by the following product:

iiU

iqVE =0 ,

(17)

wherei

UV is the shear resistance of all vertical elements at floor i andi

q is the ductility index,

a measure of the deformation capacity of floor i .

The shear resistancei

UV depends on the type of failure mechanism. The latter is established

through simple equilibrium considerations. The lesser between the shear force due to a plastic

hinges (flexural) mechanism and the members shear strengths gives the floor shear resistance

iUV .

The possibility of a weak-beam/strong-column failure mechanism is also investigated. To this

end, a joint index similar to that of New Zealand document may be resorted to:

UcaUcb

UbrUblj

MM

MMS

+=

++ or

UcaUcb

UbrUblj

MM

MMS

+=

+ . (18)

where +UblM is the positive ultimate bending moment of the beam on the left side of the node,

UbrM is the negative ultimate bending moment of the beam on the right side of the node, UcbM

and UcaM are the ultimate bending moment of the column below and above the node,

respectively. If 1jS , a premature beam failure is expected and the shear force on the columns

is limited by the resistance of the beams.

An interesting feature of the Japanese third-level method is about the mixed-type structures, in

which different structural types for the vertical elements are present. It is observed that thepresence of a stiffer and less ductile vertical element may significantly reduce the resistance of

the whole structure. To account for this fact, the expression of the basic performance index in

Eq. (16) is modified as:

( ) 1332210 qVVVE iUiUiUi ++= , (19)

where the shear capacity of the brittle member 1is fully accounted for, while the capacity of the

others are reduced by a coefficient that accounts for the limited deformation capability


32/78


imposed by member 1. The deformation capacity of the whole structure is thus governed by the

less ductile, i.e. 1qq= .

The s and 1q depend on the type of vertical elements. A classification of them on the basis

of their type of failure is provided in the standard: the eight classes considered are reported in

the table below. All the vertical elements of the structure have to be classified in a maximum of

three groups.

The values for and q are given as functions of the class combination. The corresponding

tables from the Japanese standard are provided below.

Table 1. Classification of vertical elements

Element Definition

Col. m Column whose failure mechanism is in flexureCol. s Column whose failure mechanism is in shear

Col. ss Short column whose failure mechanism is in shear failure and its ratio between

height and depth is smaller than two

Col. mb Column whose ultimate lateral force is caused by flexural yielding of beams

Col. sb Column whose ultimate lateral force is caused by shear failure of beams

Wall m Wall whose failure mechanism is in flexure

Wall s Wall whose failure mechanism is shear

Wall upl Wall whose ultimate lateral capacity is caused by uplift of foundation

Table 2. Value of displacement compatibility factor

Element of the first group

Col. ss Col. s or Wall s

Col. m 0.5 0.7

Col. s 0.7 1.0

Wall m or Wall 0.7 -

Table 3. Ductility index

Mechanism q

Beam Shear 1.5Beam Flexure 3

Column Shear 1

Column Flexure ( )( ) 05.0175.012 + Short Column Shea 0.8

Wall Shear 1

Wall Flexure 1 2

Wall Uplift 3.0

In Table 3, the coefficient is evaluated by:


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36/78

22 Chapter 3. Comparative Study

Figure 5. View of Pavia frame

Figure 6. Gravity loads for Pavia frame

The column have square cross-sections with dimension equal to 20cm while the beam have

dimensions of 20x33cm. The geometry and the reinforcement of the frame are shown in the

Figure 7.


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38/78


Figure 8. Four-storey frame, Icons

Figure 9. Gravity loads for Icons frame

3.2.3 Spear Frame

The last case study refers to a structure built in the framework of the European program SPEAR

[Fardis, 2002] and is a simplification of an actual three-storey building representative of older

construction in Southern Europe. It has been designed for gravity loads alone following the

prescriptions of the Greek design code in use between 1954 and 1995. The storey height is

3.0m.


39/78

Terms of comparison 25

Figure 10. Geometric characteristics of the three test-structures

COLUMNS C1-C5 & C7-C9

STIRRUPS8/25414

1014

STIRRUPS8/25

COLUMN C6

Figure 11. Plant of Ispra frame and details of the columns reinforcement

3.3 Strengths of materials

A reliable estimation of the materials strengths is fundamental in assessment. All codes require

both to use the best estimates of the actual strength instead of nominal values and to account for

the uncertainty of the available information.

The mean values of the tests performed on concrete and steel are assumed as the most probable

strengths; the values used in the applications of the procedures are given in Table 4.


40/78


Note that in both the Pavia and the Icons frame two concrete qualities were used. In the Pavia frame the

type 2 is referred only to the columns of the first floor, while in Icons frame type 1 is for all the beams

and type 2 for the columns.

Table 4. Materials strength of the test structures (MPa)

Pavia Icons Spear

Concrete 1, fc 14.01 14.28 18.00

Concrete 2, fc 18.53 18.34

Steel, fy 365.7 331.0 255.0

3.4 Terms of comparison (capacity measure and earthquake input model)

The comparison between the selected procedures is made in terms of the PGA value that causes

the collapse of the structure. The PGA has been arbitrarily related to the Type 1 spectrum of

EC8 for a site of soil category C; it is believed that the results of the comparisons would not

change to any significant extent if a different reference spectrum were selected.

The fundamental parameters of the spectrum, whose analytical expression are reported in

Appendix A, are the corner periods BT , CT and DT (see Figure 12), which are respectively

equal to 0.2, 0.6 and 2.0 seconds, and the ratio between the spectral acceleration at the plateau

and the peak ground acceleration, which is equal to 2.875 assuming a site coefficient Sequal to

1.15. In Figure 12 are shown the elastic spectrum for gPGA 1= , %5= and the design

spectrum for 5=q .

Soil C, Type 1

Tb Tc Td0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Period [sec]

agS[g]

Elastic

Design

Figure 12. EC8 elastic and design spectra

The displacement spectrum is related to the elastic spectrum by: ( ) ( )

2

,,

TST ee = .


41/78

Terms of comparison 27

Once the actual spectral acceleration at failure faS , is evaluated, the corresponding PGA at

failure is obtained as:

( ) ( )[ ]scscelfaelelaf TSTSPGA = ,

1,

,, (24)

where ( )elela TS , is the elastic spectral acceleration corresponding to the normalized spectrum,

i.e. for gPGA 1= .

Since the purpose here is not to check whether a structure is able to withstand an earthquake of

given intensity, but to assess its capacity, i.e. the PGA at failure, some minor adjustments are

needed in the practical application of the procedures: they are indicated in the appropriate

places.

3.5 Non-linear numerical analysis

Two-dimensional models of the frames and a three-dimensional model of the SPEAR building

have been set up, employing fibre elements to represent columns and beams. The Manders

constitutive model [Mander et al. 1988] has been adopted for concrete, a bilinear material

relationship for the steel, with the hardening factor for the plastic branch equal to 0.002. The

analyses have been carried out by means of the finite element code OPENSEES [McKenna

1997].

Figure 13. Fiber model


42/78


3.6 Risk reduction factor

The purpose of the codes for assessment and upgrading of existing buildings is to reduce

earthquake risk to the community. However, since it is generally recognised the impracticability

of bringing all existing buildings up to the standard of the new ones, risk factors are often

adopted, allowing for a reduced protection. This aspect has not been introduced in the present

study, whose focus is on the relative efficiency of different approaches in providing estimates of

the actual risk.


43/78

4 EXAMPLE APPLICATIONS

The application of each of the selected procedures to the Pavia frame is described in detail in

this section. For the other two structures only the results are given, in the following chapter 5.

4.1 FEMA 356: linear static

The effects due to the gravity loads and to the earthquake loads are evaluated separately and

then combined.

The evaluation of the seismic action that causes failure of the frame is carried out in two steps: a

lateral load of unit value, 1V , is applied to the frame and the corresponding demandsi

D1 on all

considered members/mechanisms are evaluated. Then, for each mechanism, the lateral load

multiplier which satisfies the corresponding limit state equation is evaluated from:

iiiG

i DDC 1+= , (for member/mechanism i)(25)

wherei

C is the member capacity andiGD is the demand on the member due to gravity loads.

The collapse of the frame is identified by the minimum of over all mechanisms:

{ }ii

f = min .

The spectral acceleration at failure is simply evaluated from Eq. (1), where the elastic lateral

force is given by:

1VV f = . (26)

All the members of the frame are classified as primary elements, i.e. essential to the resistance

of the frame against the seismic lateral load. The members flexural capacities are evaluated

using the probable material strength (see section 3.3) in accordance to the procedures of ACI

318-99; the maximum tensile deformation of the longitudinal bars is set equal to 5%. No


44/78

30 Chapter 4. Example applications

reduction on member flexural strength, which is required if the shear demand exceeds

grossc Af '

6 , is necessary for the frame under examination.

The shear capacities of beams, columns and joints are calculated using the expressions provided

in the Prestandard, which take into account the contributions of both concrete and lateral

reinforcement. A lower-bound of the material strength is used in the calculation of the ultimate

shear capacity, since this action is classified as force-controlled.

The demands on members, iD s, are calculated by means of a linear elastic finite element

analysis. In setting up the model, the stiffness of beams and columns is reduced by factor equal

to 0.5 with respect to the un-cracked value, as explicitly required by the FEMA.

The seismic action 1V is vertically distributed at the floor levels in accordance with the

coefficients VxC :

11 VCF Vxx = , (27)

=

=n

i

kii

kxxVx hwhwC

1

, (28)

where iw is the portion of the total weight lumped at floor x, xh is the height of floorxfrom the

base. The exponent kis equal to 0.1 for sec5.0T , to 0.2 for sec5.2T and varies linearly

in between. The values of the coefficients VxC are shown in Figure 14: the load distribution

along the frame height is almost triangular. The horizontal actions are applied in both +X

direction and X direction.

0.43

0.38

0.19 1 2 3 4 5 6

1 2 3 4 5 6

1 2 3 4 5 6

7

8

9

10

11

12

13

14

15

16

17

18

1 2 3 4

5 6 7 8

9 10 11 12

Figure 14. Numbering of joints and sections and values of the coefficientsof the vertical distribution of the lateral load


45/78

FEMA 356: linear static 31

The failure mechanisms investigated are: flexure and shear failures at the end sections of all

beams and all columns; shear failure in all joints.

The flexural failures are governed by the following limit state equation:

1MMMm GC = . (29)

The values of the local ductility factor m are tabled in the Prestandard as function of the

Performance Level, of the Component Type, of the amount of transverse reinforcement and of

the levels of axial and shear forces acting on the members. The selected values are about 4 for

columns and about 7 for beams.

The shear failures are governed by the following limit state equation:

JCCC

VVV GC

=

321

1 , (30)

where the coefficients 1C , 2C , 3C , J , defined in section 2.1, are equal to 1.0, 1.0, 0.9 and 2.0,

respectively.

The minimum value of the multiplier is reported in Table 5, for each of the failure

mechanism investigated.

It is assumed that the collapse of the structure occurs for the smallest among those listed in

Table 5; which corresponds to the shear failure of joint 1 (see Figure 14). Thus, the elastic base

shear at failure is equal to 124.4 kN.

Table 5. Values of minimum multiplier for each limit states

Shear MomentMechanism

Beam Column Joint Beam Column

172.7 204.8 124.4 372.2 202.9

The (elastic) spectral acceleration at failure is calculated as:

gWCCCC

VS

m

Ef

fa 508.0321

1

, =

=

=

where the values of the coefficient 1C , 2C and 3C , are the same as above and the coefficient

mC is equal to 1.

The period of the structure from an eigenvalue analysis is equal to sec57.0=elT . Finally, the

failure peak ground acceleration is evaluated by:

( )gS

TSPGA fa

elela

f 177.0875.2

508.01,

,

=== .


46/78

4.2 FEMA 356: Nonlinear static

The seismic capacity of the structure is assessed by carrying out a push-over analysis; the lateral

loads are increased until failure occurs in one of the monitored sections or joints. In this case the

format of the limit state equation is the same for force-controlled actions and deformation-

controlled actions, and it consists on a direct comparison between capacity and demand, this

latter being correctly provided by the analysis, in terms either of forces or of deformations. In

this application, a fibre element formulation is used to model both beams and columns, hence

use of local ductility factors or reduction coefficients, as in Eq. (2) and in Eq. (29), is not

necessary.

The flexural capacity is expressed in terms of plastic rotation, c . These are tabled in the

Prestandard as function of the ratio between the amount of transverse reinforcement and the

cross section area, of the level of the shear forces acting on the member, of the amount of the

transverse reinforcement and, for columns, of the level of the axial force acting on member.

Their values vary from 0.002 to 0.02 for the columns and from 0.025 to 0.05 for the beams; the

average values adopted are around 0.020 and 0.026 for columns and beams, respectively.

The shear capacities are evaluated using the same expressions as for the linear procedure.

The horizontal load is vertically distributed as shown in Figure 15: (1) a uniform distribution

and (2) a triangular distribution. They are applied in the two opposite directions, +X and X. A

total of four push-over analyses are performed.

Figure 15. Uniform and triangular vertical distributions of lateral loads


47/78

FEMA 356: Non linear static 33

The capacity/demand check on all sections and all joints is carried out at each step of the

analysis.

The results of the analysis are summarised in Table 6, where U is the displacement at failure

of the control node, which is by definition the node located at the centre of mass of the top

floor, and UV is the base shear at failure. The element and the type of failure are also indicated.

The worst condition among those investigated are cases number 3 and number 4, to which the

smaller ultimate displacement corresponds; the base shear capacity is about 50 kN.

The push-over curve corresponding to case number 3 is bi-linearised to evaluate the yielding

displacement and the effective stiffness of the frame whose values are:

mY 021.0= , mkNKeff 2908= ;

where, as indicated by the Prestandard, the effective stiffness has been taken equal to the secantstiffness calculated at a base shear force equal to 60% of the effective yield strength of the

structure.

Table 6. Collapse point for the four push-over analyses

Case

number

Lateral

load typeDir. U [m] UV [kN] Mechanism

1 (1) X 0.070 51.6 Flexural failure of column 3

2 (1) -X 0.070 51.4 Flexural failure of column 2

3 (2) X 0.063 51.7 Flexural failure of column 2

4 (2) -X 0.063 51.5 Flexural failure of column 2

The period is evaluated taking into account the elongation effect through the expression:

sec68.0==eff

ineleff

K

KTT ,

where the initial stiffness, mkNKin 4132= , is the elastic lateral stiffness of the building in the

direction under consideration.

The spectral acceleration which causes the collapse of frame is:

ggTCCCC

Seff

ufa 380.0

42

2

3210

, =

=

,

with 0C , 1C , 2C and 3C taken equal to 1.2, 1.0, 1.2 and 1.0 respectively. The corresponding

failure PGA is:

( )g

TS

SPGA

effela

fa

f 150.0543.2

380.0

,

,=== .


48/78


It is noted that the displacement ductility capacity of the frame estimated from the push-over

analysis, i.e.:

0.3=

=

y

usc ,

is not explicitly used in the evaluation of faS , and hence of PGAf. This is a feature common to

all displacement-based approaches which adopt the equal displacement rule.

4.3 NZ: Simple lateral Mechanism Analysis

As already pointed out in section 2.2.3, the differences between the force-based and the

displacement-based procedures consist essentially in the way the demand on the structure is

calculated, i.e. either by using an acceleration spectrum or a displacement spectrum. In fact, the

evaluation of the structural capacity, which represents the main task of the assessment, is

common to both procedures. Accordingly, in the application instead of following the sequence

of steps outlined in the NZ document, for each of the two procedures, the evaluation of the

frame capacity has been carried out first, using both the SLM method and the push-over

analysis. The two approaches (i.e. force and displacement based) have then been separately

applied for the assessment of the PGA at failure.

The ultimate flexural capacitiesU

M of members are evaluated by the standard theory [Park and

Paulay, 1975] assuming the maximum tensile strain of the reinforcement bars equal to 0.10, at a

difference with the FEMA value of 0.05. The columns capacity UM is calculated taking into

account the earthquake-induced axial forces, which are estimated through simplified

considerations based on global equilibrium between external and internal actions. Furthermore,

a reduction of UM due to the P effects is applied. The ultimate shear strengths, CUV , of

beams, columns and joints are evaluated by means of the expressions provided in the document.

The degradation of shear strength due to inelastic deformation associated to plastic-hinges is not

accounted for at this stage.

The identification of the most-likely failure mechanism is carried out as follows. First, the type

of failure at the end sections of beams and columns is assessed by comparing the shear demands

on members generated by the plastic hinge (flexural) mechanism, flV , with the shear strengths

CUV . The condition flCU VV > is found to be satisfied at all members sections, therefore

flexural failures are expected.


49/78

NZ: Simple lateral Mechanism Analysis 35

The sway indexes are then calculated according to the Eq. (7). They are given in Table 7 for

both directions of application of the ground motion. Since the values are all larger than 0.85, a

failure mechanism of column-sway type at first floor is predicted.

Table 7. Sway Indexes

Si 1stfloor 2

ndfloor

X dir. 2.03 1.73

-X dir. 2.21 1.95

In this case the probable lateral load capacity can be simply evaluated by means of equilibrium

conditions: the maximum base shear UV is taken equal to the sum of the ultimate flexural

capacities at top and bottom of the columns of the first floor divided by the storey height. The

values of UV are given in Table 8 together with the probable base shear at yielding, calculated

using the yielding flexural capacities of members. The lower values of UV with respect to YV

are due to the P effect.

Table 8. Base Shear at yielding and collapse

Base Shear YV UV

X dir. 45.9 37.9

-X dir. 50.9 43.9

A simplified hand approach is also followed for the determination of the available (global)

displacement ductility of the frame. The (local) ductility capacities in curvature c are

evaluated first by means of sectional analysis. The expressions provided in the NZ document

might have been used alternatively. The values of c of columns at the first floor are given in

Table 9: the expected available curvature ductility is in the order of 10.

On the basis of the estimated c , the reduced shear strengths are evaluated by means of the

degradation law proposed by Priestley et al. included in the document. The values of the

reduced shear strengths crV for the columns at the first floor are also given in Table 9. They are

compared with flV : the condition flcr VV > is satisfied. Thus, the reduction of the (local)

curvature ductility capacities is not necessary.


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Table 9. Curvature ductility and shear demand for the four base-columns

Section 7 10 13 16

c 11.4 12.0 9.8 11.6

crV 17.2 18.4 18.1 16.9flV 11.1 12.2 11.9 10.7

The (global) inelastic displacement capacity is evaluated as the sum of the yield and the plastic

components.

The frame yielding displacement is evaluated by the simplified relation Eq. (11) proposed by

Priestley:

mmhh

le

b

byy 164371

330

13400018.05.05.0 =

=

The estimated displacement capacity of the frame is calculated using Eq. (9) and Eq. (10) as:

mmpyu 553916 =+=+= .

The global displacement ductility of the frame is equal to:

48.3=

=

y

usc

The evaluation of the capacity of structure is thus completed; the corresponding bilinear

approximation of the force/displacement curve is shown in Figure 16.

The seismic performance is now determined by either a force or a displacement based approach.

In the force approach, the period is evaluated by an eigenvalue analysis, sec57.0=elT , and one

obtains for the peak ground acceleration at failure:

( )g

M

V

TSPGA Usc

elela

f 204.075.27

9.3748.3

875.2

11

,

=== .


51/78

NZ: Simple lateral Mechanism Analysis 37

0 10 20 30 40 50 600

10

20

30

40

50

60

displacement [mm]

Force,

V[kN]

Edx

Esx

Figure 16. Resultant SDOF from mechanism analysis

0.1690.204

0.586

0.00

0.25

0.50

0.75

0.0 0.5 1.0 1.5 2.0 2.5

Period [sec]

Sa

[g]

elastic

design q=3.48

Figure 17. Force approach of SLM for the Pavia Frame

In the displacement approach, the characteristics of the substitute structure are:

sec26.12 ==

eff

eff

k

MT

( ) %20, == mechfailuref sceq .

The PGA at failure is:

( )g

TSPGA effu

eqeffela

f 161.0,

1 2

,

==

In the above expression, elaS , is the normalized elastic spectrum evaluated for the damping

coefficient eq using the reduction factor 63.05

10=

+=

eq .


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4.4 NZ: Nonlinear Static

The NSP procedure is intended as a refinement of the SLM analysis: in particular, the lateral

force capacity and the available displacement ductility of the structure are evaluated through a

push-over analysis instead of using the hand approach illustrated in the previous section.

The calculation of members capacities both in flexure and in shear is carried out as for the SLM

case. Therefore, no further comments are necessary here, except that the flexural failure is

evaluated in term of curvature deformations instead of bending moments. The ultimate

curvature capacities of the sections are derived from sectional analysis as before: for the

columns of the Pavia frame, the corresponding values vary between 0.2 and 0.3 mm-1.

The finite element model of the frame has been described in section 3.5. The two horizontal

loads distributions specified by FEMA have been applied also for this case, since no explicit

indication on this subject is provided in the NZ-document.

The four curves obtained by combining the two directions of application and the two vertical

distributions of horizontal loads are shown in Figure 18. The probable lateral load capacity is in

the order of 50 kN.

PAVIA Frame

0

10

20

30

40

50

60

0 0,02 0,04 0,06 0,08 0,1 0,12

Displacement [m]

BaseShear[kN]

Triangular distribution

Uniform Distribution

Figure 18. Push-over curves for two types of distributions

To establish the available displacement ductility of the frame, in the NZ document it is

suggested to push the frame further, deforming as a mechanism, until the available ultimate

curvature is reached at the critical plastic hinge. The corresponding ultimate displacement is

read directly from curve.


53/78

NZ: Nonlinear static 39

In the case under consideration, the flexural capacity is exhausted at the bottom of the external

column 4, for the +X direction of application, and at the bottom of the opposite external column

2, for the -X direction of application.

The reductions of shear strengths due to local inelastic deformations are evaluated next, in the

same way as described in the previous section. It is noted, however, that the NZ document

requires to perform this check at failure, i.e. to use the values of the local inelastic deformation

and of the flexural shear demand flV corresponding to the frame configuration at failure. If

the check is negative, the ultimate deformation is reduced to that pertaining to shear failure.

In two out of the four cases investigated, shear failures are observed well before the flexural

ultimate capacity is reached. The corresponding points in the force/displacement curve are

shown in Figure 19: it is noted, however, that the two shear failures occur when the lateral force

has practically reached the ultimate capacity, implying that this result is quite sensitive to

accuracy of the shear strength model used.

PAVIA Frame

0

10

20

30

40

50

60

0 0,02 0,04 0,06 0,08 0,1 0,12

Displacement [m]

BaseShear[kN]


NZ shear failures

NZ flexural failures

Figure 19. Shear and flexural failures for the triangular distribution

The significant quantities derived from the push-over analyses are summarized in

Table 10: while the lateral load capacity is almost equivalent for all the cases, the reduction in

the available ductility due to shear failure in the two cases discussed above is considerable.

The values of the yielding displacement given in the table have been derived after the

bilinearisation of the push-over curves.


54/78


Table 10. Collapse point for the four analyses

Shape Dir. UV Y U sc mechanism

(1) X 50.0 0.023 0.034 1.52 Shear failure of joint 4

(1) -X 50.6 0.023 0.036 1.55 Shear failure of joint 1

(2) X 51.1 0.021 0.106 5.03 Flexural failure of section 10

(2) -X 50.9 0.021 0.101 4.91 Flexural failure of section 10

As for the SLM analysis, the failure PGA is calculated following both a force approach and a

displacement approach.

In the force approach, the peak ground acceleration at failure is:

( )g

M

V

TSPGA Usc

elela

f 091.075.27

0.5052.1

875.2

11

,

=== .

0.261

0.091

0.171

0.00

0.10

0.20

0.30

0.0 0.5 1.0 1.5 2.0 2.5

Period [sec]

Sa

[g]

elastic

design q=1.52

Figure 20. Design and elastic spectra for PGA at failure

In the displacement approach, the characteristics of the substitute structure are:

sec866.02 ==eff

effk

MT

where mkNkeff /1460034.0

0.50== and %8.502.0

112.0 =+

=sc

eq

.

The PGA at failure is:

( )g

TSPGA effu

eqeffela

f 096.0,

1 2

,

==

In the above expression, elaS , is the normalized elastic spectrum evaluated for the damping

coefficient eq using the reduction factor 96.0= .


55/78

4.5 BDPA: 3rd

level

The first step in the application of the Japanese procedure consists in the classification of the

vertical elements into the eight classes available. In the Pavia frame, all the vertical elements are

columns having the same section and reinforcement layout. To establish the type of failure of

each column, their flexure and shear strengths are calculated, at all floors, by means of the

expressions provided in the guidelines. From the equilibrium of the ultimate bending moment,

the shear forces on members associated with a plastic-hinges (flexural) mechanism are derived

and compared with the ultimate shear. It results that all the columns at all floors fail in flexure,

i.e. flCU VV > .

The possibility of beams failure is investigated by means of the joint-indexes, which are

calculated with the expressions in Eq. (18). Values greater than 1 are found at all nodes. Thus, a

strong-beam/weak-column type of failure mechanism is expected for the Pavia frame. It is

concluded that all the vertical elements belong to class m: columns whose failure mechanism is

flexural yielding.

In this case, the expression of the basic structural performance index defined in Eq. (17) reduces

to:

( )==nc

j

iUjj

i VqE1

0 min , (31)

where iUjV is the shear capacity of column jat floor i , nc is the number of columns at that floor

and jq is the ductility index of column j . The latter depends on the coefficient defined in

Eq. (20), which for the case under examination is the same for all columns at all floors.

The floor shear resistance is evaluated by the expression:

( )=

=nc

j

iUjj

iic VqTSV

1

0 min , (32)

For the Pavia frame the two indexesi

S0 and Tare both equal to 1 because the frame has a

regular layout and it is a new construction, i.e. the materials are not deteriorated. The values

ofi

cV are given in Table 11.

According to Eq. (23), the failure spectral acceleration at floor iis calculated as:

i

ii

akm

ES= 0 , (33)


56/78


The values of iaS are given in Table 11. The collapse of the frame occurs for the smallestiaS ,

which is equal to 0.477g.

In Eq. (33) the coefficients ik are equal to 0.86, 1.07 and 1.28 respectively for the first, the

second and the third floor.

Table 11. Results of the 3rdlevel procedure for the three different floors

Storey Mechanism q Vu[kN] E0[kN] Sa [g]

1 Col. Flexural 3.2 50.5 151.4 0.649

2 Col. Flexural 3.2 43.8 140.2 0.775

3 Col. Flexural 3.2 37.0 118.6 1.402

The PGA is given by the expression in Eq. (24). The elastic period, evaluated from the

eigenvalue analysis is equal to: sec57.0=elT . The factor ( )ela TS1 is equal to 2.875. Finally,

the peak ground acceleration at failure is equal to 0.226g.


57/78


58/78

44 Chapter 5. Comparisons

Icons Frame - Basic-Procedures

0.2290.209

0.253

0.298

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

NZSEE MA

Force

NZSEE MA

Disp

FEMA LSP BDPA

PGA[g]

Pushover Procedures

0.113

0.191

0.140

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

NZSEE

NPA Force

NZSEE

NPA Disp

FEMA

NSP

Figure 22. Failure PGA for the Icons frame

The variation of failure PGA s which results from the two push-over procedures is, on the

contrary, unexpected. Since the model of the frames and the external action applied are the

same, the difference in the results has to be related either to the capacities of members or to the

way the push-over curve is related to the PGA at failure. The points on the push-over curves

which correspond to the exhaustion of capacity of the indicated member are shown in Figure 23

and Figure 24.

PAVIA Frame

shear joint 4

shear joint 1

flexure col 4

flexure col 4

flex col 3

flex col 2flex col 2

flex col 2

0

10

20

30

40

50

60

0 0.02 0.04 0.06 0.08 0.1 0.12

Displacement [m]

BaseShear[kN]


Uniform Distribution

NZ failures

FEMA failures

Figure 23. Collapse points of push-over procedures for Pavia frame


59/78

45

ICONS Frame

flexure col 2

flexure col 5

shear col 2

shear joint 8