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Bridge System with Precast Concrete Double-T Girder and External Unbonded Post-tensioning
by
Yang Eileen Li
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science
Graduate Department of Civil Engineering University of Toronto
2.7. Variables considered in parametric study for torsion 28
2.8. Live load distribution under applied eccentric load – analytical approach 29
2.9. Live load distribution – analytical approach 30
2.10. Comparison of analytical approach and grillage model results 34
2.11. Anchor set loss with varying l 36
2.12. Loss due to creep and shrinkage and related parameter 38
2.13. Stress in post-tensioning steel under SLS 40
2.14. Summary of flexural response under ULS 46
2.15. Anchorage zone reinforcing steel 49
2.16. Deviation reinforcing steel 50
3.1. Material properties 58
3.2. SLS stress limits 59
3.3. Fatigue limit states load combination 62
3.4. Level of prestress used in SLS load combinations 63
4.1. Material properties 72
4.2. SLS stress limits 73
5.1. Material properties for slab-on-CPCI-girder sample design 82
5.2. Summary of live load distribution equations 88
xii
6.1. Comparison of double-T systems 95
6.2. Live load distribution comparison 96
6.3. Maximum moment intensity due to DL, SDL, and LL 98
6.4. Concrete consumption 99
6.5. Prestressing steel consumption 100
6.6. Reinforcing steel consumption 101
6.7. CFRP reinforcing system (Data provided by Sika, Canada) 101
6.8. Unit cost of structural reinforcing systems 102
6.9. Cost comparison of double-T systems 102
6.10. Qualitative cost comparison of the double-T and CPCI systems 104
6.11. Cost of precast concrete and cast-in-place deck slab 105
6.12. Cost comparison between the CPCI system and double-T alternative concept II 106
xiii
List of Symbols A Area of concrete cross-section
As Area of reinforcing steel
Astrand Area of each prestresing strand
ANC Prestress loss due to anchor set
b Deck slab width (Figure 2.9)
b0 Width between centrelines of the two webs in a double-T cross-section (Figure 2.9)
bw Average web thickness (Figure 2.9)
Cf Factor for lane width correction factor, which is used in calculating live load distribution
based on CHBDC
D D value in AASHTO equation for calcuating live load distribution among girders
DF Distribution factor characterizing transverse live load distribution in a bridge system
e(x) Distance from centroid of the gross uncracked concrete section to the centroid of
prestressing steel
Ec Modulus of Elasticity of concrete
Ef Modulus of Elasticity of FRP
Ep Modulus of Elasticity of prestressing steel
Es Modulus of Elasticity of reinforcing steel
f'c Specified compressive strength of concrete
fo Jacking stress of post-tensioning tendon
fpy Yield strength of prestressing steel
fpu Specified tensile strength of prestressing steel
fy Yield strength of reinforcing steel
F Width dimension that characterizes the load distribution for a bridge
Fm
FR Prestress loss due to friction
G Shear modulus
k Ratio between St. Venant and warping torsion, assumed to be constant along along the
span
K Torsional constant
xiv
Kg Girder longitudinal stiffness in the AASHTO LRFD equation for calculating live load
distribution
lp Total length of the post-tensioning tendon between anchors
lp0 Tendon length when force in tendon equals effective prestress
L Span length
Mg Maximum longitudinal moment per web or per girder due to live load, including effects of
live load amplification factor
Mg,avg Average moment per web or per girder due to live load if live load is shared equally among
girders or webs
Mg,tot Total moment of the cross-section if the maximum longitudinal moment per girder (web),
Mg, is applied to every girder (web)
Mp Primary moment due to prestress; equals pretressing force times tendon eccentricity
MQ Moment due to external load
MT Maximum moment per design lane
Mtot Sum of moment due to external load and primary moment due to prestressing
n Number of design lanes according to CHBDC
N Number of girders or number of webs
P Prestressing force
P1 Post-tensioning force from tendons installed in stage 1 of the post-tensioning operation.
P1i Tendon stress equals initial jacking stress.
P1∞ Tendon stress equals effective prestress after all losses.
P1ULS Tendon stress at ULS
P2 Post-tensioning force from tendons installed in stage 2 of the post-tensioning operation.
P2i Tendon stress equals initial jacking stress.
P2∞ Tendon stress equals effective prestress after all losses.
P2ULS Tendon stress at ULS
Ptot Total prestressing force
Pmin Minimum prestressing force required for equilibrium under ULS
Py Tendon yield force
P∞ Effective prestressing force after all losses
Qf External load
RL Multi-lane reduction factor
S Section modulus of concrete cross-section; girder spacing in a slab-on-girder system
ts Deck slab thickness (Figure 2.9)
T(x) Total torsion due to applied load
xv
TSV(x) St. Venant torsion
TW(x) Warping torsion
wv Vertical girder deflection due to longitudinal flexure
W Axle load of truck load model
We Width of a design lane
α(x) Tendon angle change from jacking end to location x
αD Load factor for dead load
ΔlPD Tendon elongation due to deformation
ΔlPF Tendon elongation due to tendon force
Δset Change in length of post-tensioning tendon due to anchorage slip
Δεp Change in strain in prestressing steel due to deformation
εcp Concrete strain at the level of prestressing steel
εcs(t) Concrete shrinkage strain
εct Concrete tensile stress
εfd FRP debonding strain
εfu FRP ultimate strain
θSV(x) Twist angle due to St. Venant torsion
θW(x) Twist angle due to warping torsion
μ Friction coefficient; aging coefficient of concrete; factor for lane width correction factor,
which is used in calculating live load distribution based on CHBDC
ρ Reinforcement ratio
σc0(x) Concrete stress at the level of prestressing steel due to initial load
σc,bot Concrete stress in section's bottom fibre
σc,top Concrete stress in section's top fibre
φ Curvature
φ(t) Creep coefficient of concrete
Chapter 1 Introduction
This thesis compares the consumption of primary superstructure material (concrete,
prestressing steel and other reinforcements) in a conventional single span slab-on-girder system
with those of double-T alternatives. The slab-on-girder system addressed in this thesis consists
of a series of parallel CPCI (Canadian Precast Prestressed Concrete Institute) girders, which are
standardized I sections used in Canada, with a cast-in-place deck slab. A sample design of the
CPCI girder bridge will be developed with standard methods used in the industry. The double-T
concepts involve the use of slender cross-section, fully precast concrete, and external unbonded
post-tensioning. Three double-T concepts will be developed and validated in this thesis with
sample designs. The three concepts are:
1. Base concept: a double-T system with pure external unbonded post-tensioning;
2. Alternative concept I: a double-T system with a blend of external unbonded post-tensioning
and external carbon-fibre-reinforced polymer (CFRP) laminate reinforcements;
3. Alternative concept II: a double-T system with a blend of external unbonded post-
tensioning and internal bonded unstressed tendons.
This thesis will compare the material consumption and cost of the slab-on-CPCI-girder system
with the double-T systems based on their design examples.
1.1. Motivation
Long span bridges with their grand appearance often attract most of the public attention.
Records for the longest spans in the world are constantly being challenged or broken as a
reflection of people’s fascination with long spans and the extensive technological interest that
follows it. In the bridge industry, however, the largest section comprises short and medium
spans, ranging approximately from 20 m to 45 m. These may be single span bridges, or parts of
a longer multi-span structure. Given the large market share of this type of project, it follows
that short-to-medium spans are of great economical importance to the society and deserve as
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much if not more attention than long spans (Kulka and Lin, 1984). For instance, any reduction
in material consumption of short-to-medium span structures will be magnified by the large
number of their applications and result in substantial overall economical improvement.
In most parts of Canada, the preferred structural system for spans of up to about 45 m is
the CPCI slab-on-girder system, which consists of multiple parallel precast, pre-tensioned
concrete CPCI girders with a cast-in-place concrete deck slab (Figure 1.1). CPCI girders,
shown in Figure 1.2, are precast I sections commonly used in Canada. The CPCI slab-on-girder
system has become very much standardized, making its design and construction relatively
straightforward. Consequently, when facing this type of project, owners and designers are often
reluctant to consider alternatives that may be more efficient and economical.
Figure 1.1. CPCI slab-on-girder system Figure 1.2. Cross-section of CPCI girders (CPCI, 2009) (adapted from Pre-Con, 2004)
Although the cost of the CPCI system is often considered to be acceptable by owners,
this system actually makes relatively inefficient use of materials. One primary source of
inefficiency in this type of bridge comes from the imperfect sharing of live load among multiple
parallel girders due to transverse flexibility of the deck slab. An idealized example with stick
models is illustrated in Figure 1.3. As shown in the figure, if the deck slab is infinitely stiff, it
rotates under applied eccentric load and engages multiple girders in resisting the load. On the
other hand, if the deck slab is infinitely flexible, it bends under the applied load and only
engages the girder directly below or adjacent to the load. The CPCI slab-on-girder system is in
between the two extreme cases but close to the case with the flexible deck slab. This inefficient
load distribution requires every girder in the system to be designed for relatively high loading,
which, in combination with the relatively large number of girders in a CPCI system, results in an
unnecessarily high design load for the overall structure. The inefficient live load distribution in
the CPCI slab-on-girder system is qualitatively discussed in Chapter 5.
3
Figure 1.3. Idealized model of load distribution in a slab-on-girder system
Another source of inefficiency associated with live load distribution comes from the
actual method of calculating it. Much research has been done in modeling live load distribution
in a slab-on-girder system using grillage, semi-continuum or finite element methods (CSA
2006b.). The results from these analyses are used to formulate design equations in standards
and codes. However, these equations are often simplified from the real situation to include only
a limited number of variables. They usually determine the maximum amount of load distributed
to a girder under the most unfavourable conditions. The design equations for live load
distribution from AASHTO Standards, AASHO LRFD Specifications and CHBDC, will be
examined in Chapter 5.
In addition to the structural inefficiencies associated with live load distribution, the
construction of a CPCI girder bridge can also be problematic due to its cast-in-place deck slab.
The casting of the deck slab is an inconvenient and time-consuming process which involves
installing formwork, placing the deck slab reinforcements, casting and curing concrete, and
removing formwork if necessary. The non-prestressed deck slab is also a source of durability
problems due to slab’s tendency to crack. Once crack forms, salts and other chemical agents
penetrate the concrete and induce corrosion in the reinforcing steel.
Recognizing the large demand in short and medium span bridges and the inefficiency of
current solution – the CPCI slab-on-girder system, this thesis aims to develop a new structural
system based on a double-T cross-section. The new double-T concept will be developed with
the specific intent of maximizing the efficient use of concrete and prestressing steel, as well as
simplifying the construction process. The material consumption will then be compared to the
CPCI system to provide a qualitative measure of the greater efficiency of the double-T system.
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1.2. The Double-T Concept
1.2.1. Cross-Section
Recognizing the inherent inefficiency of live load distribution in a slab-on-girder system,
the new concept is based on a two-web double-T cross-section. Double-T girders, which are not
common in North America, have seen most of their use in Europe. Post-tensioned double or
triple-T girder bridges, as shown in Table 1.1, have traditionally incorporated thick webs to
accommodate internal post-tensioning tendons. The combination of cover requirements and
clearance requirements for construction (i.e. distance between tendon ducts to allow proper
placement and vibration of concrete) generally results in a minimum web thickness of
approximately 440 mm (Figure 1.4 (a)).
Table 1.1. Examples of existing post-tensioned double or triple-T girder bridges
Bridge Cross-section Web thickness Notes (Reference)
le viaduc d'Orbe, Switzerland
1050 mm (Departement des Travaux Publics du Canton de Vaud, 1989)
Weinlandbrucke Andelfingen 500 mm
Cross-section is variable along span. The triple-T section shown is for the region close to mid-span. For regions close to support, bottom slabs are added, creating a twin-box girder. (Stussi, 1958)
Isarbrucke Munchen 700 mm (Leonhardt, 1979)
Rheinbrucke Emmerich Vorlandbrucken
1150 mm (Leonhardt, 1979)
The new double-T concept is developed with the specific intention to minimize the
amount of concrete in the system. Hence the relatively thick webs in the traditional double-T
girder need to be modified. In the new concept, web thickness is reduced by removing the
internal tendons, and replacing them with external unbonded tendons (Figure 2.7(b)). By doing
so, the web thickness can be reduced to at least 300 mm and possibly lower because web
thickness is no longer governed by detailing requirements, but rather by stress.
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(a) (b)
Figure 1.4. Web thickness of a double-T girder: (a) with internal tendons (adapted from Menn, 1990); (b) with external tendons
The double-T cross-section that will be used in this thesis is illustrated in Figure 1.5. It
consists of a 225 mm thick top slab and two slender webs. The webs have an average thickness
of 300 mm, and are tapered to facilitate the forming process. The thickness of 300 mm is
consistent with the standard practice of box girder cross-sections with similar span and girder
depth (ASBI, 2008). The deck width is governed by the roadway cross-section, which is
presented in the section of Geometrical Requirements for Sample Designs (Section 1.3). The
overall depth of the cross-section is chosen to be 2 m based on the span length (36.6m, see
Section 1.3) and typical span-to-depth ratios, which range from 17:1 to 22:1 for constant-depth
girders (Menn, 1990). The web spacing is chosen to be 7.9 m based on an optimization process
of the girder’s transverse flexural behaviour. Details of the analysis can be found in Section 2.5.
Figure 1.5. Double-T girder cross-section
1.2.2. Prestressing Concept
The double-T concept involves post-tensioning in both the longitudinal and transverse
directions. Longitudinally, the system is post-tensioned with external unbonded tendons.
Transversely, the deck slab is post-tensioned with internal flat-duct tendons. Transverse post-
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tensioning serves primarily two purposes: 1) providing transverse bending capacity to the deck
slab; 2) controlling crack in the deck slab thus enhancing the system’s durability. Details of
longitudinal and transverse post-tensioning design will be presented in subsequent chapters.
1.2.3. Construction
The new double-T concept employs precast segmental construction technology which
allows bridge to be built rapidly with minimal impact to traffic. Segments will be fabricated of-
site with the method of match-casting, which produces custom-fitted joints by casting a new
segment against a dry mating segment (Figure 1.6). Segments produced by such method can be
erected on site speedily without the need of cast-in-place concrete or grout (Gauvreau, 2006).
New segment
Core form
Match-cast
Completed
mate segment
segment
Figure 1.6. Match casting of box girder segments (adapted from Interactive Design Systems, 2009)
The precast segmental method is most often used for large projects. This is because the
initial cost of the segmental method, which is associated with manufacturing the forming
equipment, is usually high and hard to be justified if a large number of segments is not needed.
The most expensive part of the forming equipment is the core form (shown in Figure 1.6). It
can slide in and out during the match-casting of box girder segments, thus allowing the entire
rebar cage to be prefabricated. The decoupling of rebar fabrication and the actual casting
process can simplify the casting procedures and improve both casting speed and quality. The
forming of double-T girder segments, however, does not require such a core form due to the
absence of a bottom slab. The rebar cages can be prefabricated and used during casting with
formwork simply made of plywood which is relatively low in cost. Without the high initial cost
of forming equipment, the segmental construction method becomes a feasible and economical
choice for short-to-medium span double-T systems.
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1.3. Geometrical Requirements for Sample Designs
The CPCI slab-on-girder system and the three double-T concepts will be evaluated based
on their sample designs. To form a consistent basis of comparison, all four sample designs will
be developed under the same geometrical requirements. These requirements are representative
of the general highway bridge design conditions in Ontario. First, the bridge needs to cross a
distance of 36.6 m with one simply-supported span. Second, the roadway cross-section needs to
accommodate three traffic lanes, each 3.6 m wide, and two shoulder lanes, each 1.2 m wide
(Figure 1.7). The travelled and the total deck width are 13.2 m and 13.8 m, respectively. The
road deck wearing surface is assumed to be 90 mm in thickness.
Figure 1.7. Roadway cross-section
1.4. Objective and Scope
The objective of this thesis is to compare the consumption of primary superstructure
materials (concrete, prestressing steel, and additional reinforcements) in a conventional single
span CPCI system with those of double-T alternative systems. A total of four systems are
investigated in this thesis:
1. Double-T base concept: a double-T system with pure external unbonded post-tensioning;
2. Double-T alternative concept I: a double-T system with a blend of external unbonded post-
tensioning and external carbon-fibre-reinforced polymer (CFRP) laminate reinforcements;
3. Double-T alternative concept II: a double-T system with a blend of external unbonded post-
tensioning and internal bonded unstressed tendons
4. CPCI slab-on-girder system
A sample design is produced for each of the four systems above under the general highway
bridge design conditions in Ontario.
Chapters 2 to 4 describe and discuss the three double-T concepts within the framework
of three sample designs. Chapter 2 presents a comprehensive review on the design of the
double-T base concept, including the system’s transverse flexure, torsion, live load distribution,
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longitudinal flexure and shear, anchorage and deviation regions, as well as construction.
Chapters 3 and 4 present the design of double-T alternative concept I and II. These two
alternative concepts are modified versions of the base concept and share a number of same traits
with the base concept, such as transverse flexure, torsion, live load distribution and local designs.
As a result, Chapters 3 and 4 only focus on the differences between the alternative concepts and
the base concept, which is primarily longitudinal flexure.
Chapter 5 is dedicated to the CPCI slab-on-girder system. A sample design is presented.
Two important aspects in this type of system, live load distribution and deck slab design, are
also discussed.
Chapter 6 compares the four systems based on the sample designs developed. First, the
three double-T concepts are evaluated on their differences in flexural behaviour. Next, the
CPCI slab-on-girder system is compared with the double-T systems in terms of structural
efficiency, such as live load distribution. Finally, a comparison is made on the material
economy between the CPCI slab-on-girder system and the double-T systems.
The final chapter concludes the thesis by summarizing the important findings from the
development of the double-T systems, and the comparison between the systems’ material
economy.
Chapter 2 Double-T Base Concept
– Double-T System with Pure External Unbonded Post-Tensioning
This chapter describes the design of the double-T base concept, which is a double-T
system with pure external unbonded post-tensioning. The concept is presented within the
framework of a sample design, which is briefly described in Section 2.1. While most of details
regarding design procedures and analyses are presented in the later sections, it is helpful to
outline some of the main aspects the design and set up the framework at the beginning of the
chapter for a more clear understanding of the later discussions. Following the description of the
design, Section 2.2 and 2.3 outline the material properties and the design criteria. Section 2.4
describes the loadings and the associated factors and load combinations. Sections 2.5 to 2.6
examine the system’s transverse behaviour, torsion, and live load distribution, while Section 2.7
investigates the structural system’s global longitudinal response, such as longitudinal flexure
and shear. Local effects in the anchorage and deviation zone are analyzed in Section 2.8, while
Section 2.9 is dedicated to construction related subjects.
2.1. Brief Description of Design
The plan, elevation and cross-section of the base concept are shown in Figure 2.1. The
concrete cross-section and its features were already explained in Chapter 1 (Section 1.2.1).
Concrete details at the ends of the span are designed to accommodate expansion joints and post-
tensioning anchorages (Section 2.8). Two deviation diaphragms are provided along the span to
accommodate the deviation of the external unbonded tendons. For a complete set of drawings,
the reader can refer to Appendix A.
9
10
Figure 2.1. Sample design of double-T base concept
11
The longitudinal prestressing of the sample design, shown in Figure 2.2, has a total of 78
strands per web, grouped into 3 external unbonded tendons. The tendons are arranged in a
harped profile with a horizontal segment between deviations. Between deviations where
flexural demand is high, the tendon eccentricity (vertical distance between the centroid of
tendons and the centroidal axis of the concrete cross-section) is kept at its maximum. Close to
girders ends, the tendon eccentricity is kept as small as possible to minimize cantilever moment
in the girder overhand created by prestressing.
Typical anchorage system fora multistrand post-tensioning tendon(adapted from DSI, 2009)
Figure 2.2. Longitudinal prestressing design of double-T concept
Transversely, the system’s deck slab is post-tensioned with internal bonded tendons.
The transverse tendons, each containing 4 strands, are spaced at 933 mm. This spacing
translates to 3 tendons per precast segment, which is 2.8 m long for the sample design. The
profile of the tendon, shown in Figure 2.3, is arranged to provide maximum negative flexural
capacity at web-slab conjunction and maximum positive flexural capacity at the transverse mid-
span of the deck. Details on transverse flexural design can be found in Section 2.5.
12
Figure 2.3. Transverse prestressing design of double-T concept
2.2. Material Properties
The material properties assumed for the sample design are summarized in Table 2.1.
The design chooses to utilize concrete with a compressive strength of 70 MPa because
preliminary design indicates that 70 MPa is approximately the minimum strength required for
satisfactory structural response under SLS and ULS. Although the current standard practice in
Ontario is to use 50 MPa concrete, concrete with a compressive strength of 70 MPa or above has
become commercially available and has seen increased application in North America as a result
of the recent advancement in concrete technology (Choi et al, 2008).
Table 2.1. Material property for double-T base concept sample design
*The load factor 1.2 is applied to one of tendons only. Notation: DL Dead load P1 Force in stage 1 post-tensioning tendons P2 SDL Super-imposed dead load P1i when tendon stress equals initial jacking stress P2i SDL - B Barrier load P1∞ when tendon stress equals effective prestress after all losses. P2∞ SDL - WS Wearing surface load P1ULS at ULS P2ULS
Force in stage 2 post-tensioning tendons
LL Live load 19
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Load Combination 1A
This load case takes place immediately following the installation and stressing of stage
one post-tensioning tendons. The stress in the prestressing tendons is taken as the jacking stress.
The post-tensioning force from stage I stressing overcomes the effect of dead load and causes
overall negative moment on the girder.
Load Combination 1B
This load case takes place after the addition of wearing surface and barriers. During
actual construction, false work or erection girder may be left in place or removed at this stage.
This analysis assumes that supporting devices have been removed, thus making the structure
self-supportive. By this time, the stage I tendons have likely lost some of the stresses initially
jacked in. The degree of long-term loss depends on the time elapsed since the stage one
stressing operation, which is assumed to be 28 days in this analysis. This period is estimated
based on the time required to complete work on wearing surface and barriers. This load case is
critical in positive flexure.
Load Combination 1C
This load case occurs immediately after the stressing of stage II post-tensioning tendons.
The stress in the stage II tendons is therefore taken as the jacking stress. The stress in stage I
tendons is again assumed to be the prestress after losses at 28 days. This load case is critical in
negative flexure.
Load Combination 1D
This load case takes place during the structure’s service life, when dead load,
superimposed dead load and live load are all acting on the bridge. For SLS analysis, the tendon
stress is taken as the effective prestress after all losses, while for ULS analysis, the tendon stress
is calculated based on girder’s ULS deformation (see Section 2.7.4).
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2.5. Transverse System Design
2.5.1. Load Effects
The design process requires an evaluation of the transverse structural response under the
previously discussed loads. Under uniform loads, such as weight of deck slab, barrier and
wearing surface, the deck slab can be treated as a one-way slab with transverse moment constant
along the longitudinal span. The situation under live load is more complicated because the
transverse moment due to concentrated wheel loads varies longitudinally, making it is no longer
appropriate to consider just an arbitrary slice of the girder for the analysis (Gauvreau, 2006).
To evaluate the effect of concentrated live load, elastic influence surfaces are used.
They are diagrams analogous to influence lines, used to calculate load effect at a specific
location on an elastic plate due to applied gravity loads under a given plate geometry and
support condition (Menn, 1990). The design of the double-T girder uses two specific influence
surfaces published by Pucher (1977) – one for the transverse mid-span moment in the deck slab
between the two web supports, the other for the cantilever moment in the deck slab overhang.
2.5.2. Design Approach
For the double-T girder, the transverse span and cantilever of the deck slab are much
longer compared to those of a multi-girder system. As a result, transverse flexure becomes
critical in deck slab design. Web spacing directly affects the transverse flexural demand in the
system. For a box girder, the web-slab junction is often chosen as the quarter points from the
edges of the deck slab so that there is no transverse bending in webs under dead load (Gauvreau,
2006). This is however unnecessary for a double-T girder as the absence of a bottom slab
makes the webs free to rotate. The web spacing instead is chosen to balance the demand and
capacity at the two critical locations – the transverse mid-span of the deck slab and the fixed end
of the deck slab cantilever. A change in web spacing produces opposite effects on the flexural
demands at these two locations. For example, a wider spacing decreases the negative moment at
the end of deck cantilever but increases the positive moment at transverse mid-span. The web
spacing, however, cannot be chosen based solely on the equalization of maximum positive and
negative flexural demand because the positive and negative flexural capacity at the two
locations are different and depends on the transverse post-tensioning design.
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The deck slab is post-tensioned transversely with flat-duct tendons each containing four
0.6” diameter strands. Recognizing the pattern of the transverse bending moment, the tendon
profile is made parabolic with the highest elevation at the web-slab conjunction and the lowest
elevation at mid-span (Figure 2.9). The sizing of the post-tensioning tendons is based on
flexural demand in the deck slab. For a segmental bridge, another important constraint is that
segments with the same length should have the same number of tendons except for special
segments such as end and deviation segments. For the double-T sample design, a typical
segment is 2.8 m long. Detailed segment layout is shown in Section 2.9.1.
Figure 2.9. Transverse tendon profile
It is recognized from the above discussion that the transverse flexural design of the
double-T system, including the sizing of the post-tensioning tendons and the optimization of
web spacing, is an integrated process. As illustrated in Figure 2.10, while web spacing affects
flexural demand and capacity, it is also determined based on optimization of the ratio between
these two quantities.
Figure 2.10. Integrated process of transverse flexural design As a starting point of the design process, transverse flexural demand from external load
is plotted as a function of web spacing for the two critical locations in Figure 2.11. Web spacing
is chosen to range from 6.5 m to 8.5 m, which is equivalent to 47% to 62% of total deck width.
Demand on the system shifts toward positive flexure as web spacing increases, and vice versa.
Next, moment capacities at the two critical locations are calculated for SLS and ULS
based on the design criteria given in Section 2.2. For SLS, the moment capacity is defined here
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as a value under which concrete remains uncracked. Concrete stress under SLS can be
calculated as follows:
, or 3.35 MPaQ Pcr
M MP fA S S
σ = − + − ≤ [2-1]
where MQ is moment due to external load and MP is primary moment due to prestress. By
rearranging the above equation, an expression for SLS moment capacity can be obtained:
max,SLS cr PPM f S MA
⎛ ⎞= + × +⎜ ⎟⎝ ⎠
[2-2]
Mmax, SLS is calculated and plotted in Figure 2.11 for cases of having 2, 3 and 4 tendons in each
segment respectively. The moment capacities under ULS are also calculated and shown in
Figure 2.11.
(a) SLS (b) ULS
Figure 2.11. Flexural demand and capacity of transverse system as a function of web spacing
2.5.3. Final Design
Based on the information shown in Figure 2.11, the final transverse design is set to a
web spacing of 7.9 m and a post-tensioning design of 3 tendons (12 strands) per typical segment.
The actual number of strands required for adequate SLS and ULS behaviour is approximately 10.
However, because a whole number of tendons is used, the total number strands per segment is
increased from 10 to 12. The transverse tendon layout in a typical segment is shown in Figure
2.12. To confirm the adequacy of the design, the transverse structural responses under SLS and
-400.0
-300.0
-200.0
-100.0
0.0
100.0
200.0
6 6.5 7 7.5 8 8.5 9
Web spacing [m]
M[k
N-m
/m]
Positive moment at transverse mid-spanNegative moment at fixed end of deck slab cantilever
3
Maximum moment allowableto ensure σbot≤fcr attransverse mid-span
4 tendons per seg
2
2 tendons per seg34
Maximum moment allowableto ensure σtop≤fcr at fixed
end of cantilever-400.0
-300.0
-200.0
-100.0
0.0
100.0
200.0
6 6.5 7 7.5 8 8.5 9
Web spacing [m]
M[k
N-m
/m]
Positive moment at transverse mid-spanNegative moment at fixed end of deck slab cantilever
Positive moment capacityat transverse mid-span
4 tendons per seg32
2 tendons per seg
3
4Negative momentcapacity at fixed endof cantilever
24
ULS are calculated and summarized in Table 2.6. The SLS stresses are within the limit of 0.6f’c
≤ σ ≤ fcr, or -42 MPa ≤ σ ≤ 3.35 MPa. The ULS capacities at the two critical locations are
greater than the respective demands.
Figure 2.12. Transverse tendon layout in a typical segment Table 2.6. Transverse structural response – deck slab
Based on the above method, the TSV/TW ratio for the sample design is calculated to be
0.305, indicating that approximately 77% of total torsion is resisted by warping and only 23% is
by St. Venant action. A sample calculation can be found in Appendix A. The result agrees with
the previous discussion on that warping is the dominant action for resisting torsion in an open
cross-section. However, the actual amount of warping torsion is likely smaller than the value
predicted above since this procedure neglects the presence of concrete diaphragms, which
27
provide rigidity in refraining the differential web bending associated with warping action. This
effect will be investigated in the grillage model analysis in Section 2.6.2.
Warping torsion, as illustrated in Figure 2.16, is the phenomenon of differential bending
between two webs. The applied torque due to warping torsion causes the two webs to deflect in
opposite directions and induces additional longitudinal bending in the webs (Menn, 1990). If
the total factored load results in positive flexural demand on the girder, the warping torsion
shown in Figure 2.16 will cause increased bending in Web1 and reduced bending in Web2.
Therefore, the warping component of torsion can be dealt as additional flexural demand in
design.
Figure 2.16. Differential web bending due to warping torsion (adapted from Menn, 1990)
2.6.1.2. Parametric Study on Torsion
A parametric study is carried out to evaluate the effect of change in span length and web
thickness on the distribution of torsion. It is based on Menn’s method of calculating torsion. In
addition to the cross-section of the sample design, three other cross-sections with varying web
thicknesses are proposed to form the basis of the study. As shown in Table 2.7, their
dimensions are identical to those of the sample design cross-section, except for their web
thicknesses which range from 0.4 m to 0.6 m. Three span lengths are investigated – 30 m, 36.6
m, and 45 m. The proposed variation in span and cross-section produces 12 combinations to be
investigated and compared in the study.
The TW/TTOT and TSV/TTOT ratios for the 12 combinations are evaluated and summarized
graphically in Figure 2.17. It is shown that, as the web thickness increases, the percentage of
total torsion attributed to warping is reduced. For the span of 36.6 m, by increasing the web
thickness from 0.3 m to 0.6 m, the amount of warping torsion can be reduced by 9 percentage
points. Wider webs result in less warping torsion and more St. Venant torsion because they
provide a larger area for closed shear flow. This relationship is also confirmed by equation 2-8,
28
which suggests that the torsional stiffness K of a cross-section is a function of the web thickness
cubed. As a result, as the web thickness increases, the value of K increases, and so does the
amount of torsion attributed to St. Venant action. Another trend observed from Figure 2.17 is
that, as span increases, the amount of warping torsion decreases. For example, for a web
thickness of 0.3 m, the percentage of torsion attributed to warping is reduced from 83% to 68%.
Table 2.7. Variables considered in parametric study for torsion Average web thickness [m] Cross-section Span [m]
0.3025 - cross-section of sample design
30 m, 36.6 m, 45 m
0.4
30 m, 36.6 m, 45 m
0.5
30 m, 36.6 m, 45 m
0.6 30 m, 36.6 m, 45 m
In summary, this study indicates that increase in web thickness and span length
produces less warping torsion. As discussed earlier, warping torsion creates differential web
bending and imposes additional flexural demand on one of the two webs in a double-T girder.
This can be seen as a form of unequal load distribution. Since increase in web thickness and
span reduces warping torsion, it would also result in a more equalized load distribution between
the girder webs given that other factors remain constant.
83% 82% 80% 76% 77% 76% 72% 68% 68% 67% 63% 58%
17% 18% 20% 24% 23% 24% 28% 32% 32% 33% 37% 42%
0%
20%
40%
60%
80%
100%
tw= 0.3
03m
0.4m
0.5m
0.6m
tw= 0.3
03m
0.4m
0.5m
0.6m
tw= 0.3
03m
0.4m
0.5m
0.6m
Tsv/Ttot
Tw/Ttot
Span = 30m Span = 36.6m Span = 45m
Tsv - St. Venant torsionTw - warping torsionTtot - total torsion
Figure 2.17. Torsion distribution with varying web thickness and span length
29
2.6.1.3. Live Load Distribution based on Analytical Approach
In a double-T system, the two webs always collectively carry 100% of all live load
applied to the system. If the load is concentric, for a bridge with a straight and unskewed
alignment, it distributes equally between two webs. However, if the applied load is eccentric,
the load becomes unevenly shared. As described in Section 2.6.1.1, an applied eccentric load
can be decomposed into a symmetrical and an antisymmetrical component. The equivalent load
and resisting forces for the two components are summarized in Table 2.8. While the
symmetrical component produces a pair of equal forces in the webs, the antisymmtrical
component is resisted by a force couple, of which the magnitude is proportional to the amount
of warping torsion. The overall force in each web, which is the sum of resisting force from the
symmetrical and the antisymmetrical cases, is different in magnitude. This uneven distribution
of load can be seen as a result of warping torsion.
Table 2.8. Live load distribution under applied eccentric load – analytical approach
Live load distribution for the sample design is examined based on three load cases,
which are shown in Figure 2.18. These three load cases are associated with live load models
with discreet wheel loads, such as the CL-W Truck Load model or the point load component of
the CL-W Lane Load model. Load case 1 is a concentrically loaded with three lanes of traffic;
loads are symmetric about the centreline of the cross-section. Load case 2 and 3 are eccentric
load cases, where loads are positioned in each lane to maximize the load eccentricity and torsion
30
created in the section. Load case 2 bears 2 lanes of traffic whereas Load case 3 bears 3 lanes of
traffic.
Note: W represents CL-W truck axle load.
Figure 2.18. Load cases for evaluating live load distribution
Live load distribution in the sample design is evaluated under the three load cases using
the approach illustrated in Table 2.8, and the results are summarized in Table 2.9. The
calculation procedure (included in Appendix A) assumes that 77% of total torsion is resisted by
warping as calculated in Section 2.6.1.1. Based on this assumption, the concentric load case
Load Case 1 results in an equal sharing of live load, while Load Case 3 results in a 58%-42%
distribution. Among the three cases, Load Case 2 produces the most uneven distribution of live
load, with 80% of total live load being taken by the web on the severe loading side. In addition
to the relative percentage distribution of live load, it is also important to note the absolute
amount of live load carried per web. As shown in Table 2.9, the highest moment per web at
mid-span is 7520 kN-m produced by Load Case 2.
Table 2.9. Live load distribution – analytical approach
Live load distribution: moment at mid-span
Percentage distribution of total live load
Web 1 Web 2 Web 1 Web 2 [kN-m] [kN-m] Load Case 1 6300 6300 50% 50% Load Case 2 7520 1930 80% 20% Load Case 3 7280 5320 58% 42%
Note: (1) W represents CL-W truck axle load; (2) Calculation already accounts for multi-lane reduction factor and dynamic load allowance where applicable.
As shown in Table 2.8, warping torsion is an important quantity that affects the result of
live load distribution. Figure 2.19, which is generated based on the analytical method described
above, illustrates the relationship between the maximum moment per web and the amount of
warping torsion. In general, larger warping torsion produces a higher maximum moment per
31
web thus a more uneven distribution of live load. For Load Case 1, the concentrically applied
load does not induce any torsion in the system, thus warping torsion does not play a role in load
distribution. For both Load Case 2 and 3, the maximum moment per web is reduced as the ratio
of Tw/Ttot decreases. The rate of reduction for Load Case 2 is higher because its loading
arrangement creates higher total torsion in the system. Figure 2.19 indicates that Load Case 3
governs for lower values of lower levels of warping torsion, while Load Case 2 governs for
higher levels of warping torsion.
6000
6500
7000
7500
8000
0.40 0.50 0.60 0.70 0.80 0.90
Tw/Ttot
Max
imum
M p
er w
eb [k
N-m
] Load Case 1Load Case 2Load Case 3
Figure 2.19. Maximum moment per web as a function of k
The calculation of live load distribution in this section is based on the warping torsion
analysis in Section 2.6.1. However, the analysis in Section 2.6.1 is likely a conservative
evaluation that overestimates the amount of warping torsion because it neglects the presence of
transverse diaphragms in the double-T system. Therefore, the live load distribution calculated
in this section using the analytical approach likely overestimates the maximum live load
moment per web.
2.6.2. Grillage Model Analysis
A grillage model is developed to evaluate live load distribution in the double-T system.
It is a more refined approach than the previous analytical method because it accounts for the
stiffness of the transverse diaphragms and the two way action of the deck slab. As shown in
Figure 2.20, the model consists of a number of longitudinal and transverse beam elements that
represent the longitudinal and transverse strips of the structure. The flexural, shear and torsional
stiffness of each beam are based on the properties of their corresponding strip. The model is
simply-supported at the ends of the two webs. The detailed text input file of the grillage model
in included in Appendix C.
32
Figure 2.20. Grillage model of double-T system
The two eccentric load cases from the previous section (Load Case 2 and Load Case 3)
are considered in the grillage model analysis. Transversely, the CL-W trucks are placed to
maximize the load eccentricity; longitudinally, the loads are positioned to produce the maximum
bending moment at mid-span. The footprints of the truck wheel loads on the bridge deck slab
are illustrated in Figure 2.21. As shown in Figure 2.22, the wheel loads are applied as an
equivalent pair of gravity load and torsional moment on the centroid of the longitudinal strip
they act on.
Load Case 2 Load Case 3
Figure 2.21. Position of truck wheel load for Load Cases 2 and 3
Figure 2.22. Example of equivalent load used in applying wheel load
Figure 2.23 summarizes the results from the grillage model analysis. Member forces
including moment and torsion in webs and transverse diaphragms are plotted in the diagram. A
comparison of the results from the analytical approach and the grillage model is shown in Table
33
2.10. According to the grillage model, Load Case 3 produces the maximum moment per web at
mid-span, which is 6730 kN-m as indicated on the diagram. This value is approximately 15%
less than the maximum live load moment predicted by the analytical method in the previous
section. The fact that Load Case 3 governs over Load Case 2 indicates that the grillage model
predicts less warping torsion in the system than the analytical method. This is as expected
because the grillage model accounts for the stiffness of diaphragms which restrain differential
web bending. The moment and torsion present in the diaphragms as shown in the figure are also
evidence that they are helping reduce the warping torsion in the system. In conclusion, the live
load distribution calculated using the grillage model is used in the design of double-T system
because it is a more refined method that accounts for the stiffness of the diaphragms in the
system.
Load Case 2 Load Case 3
Moment [kN-m]
3046
6404
Web1
ED ED ED EDDD DD DD DD
Web26733
5733 150Scale
2800
Torsion [kN-m]
Scale 150
100
Web1
Web2
Deformed shape
Figure 2.23. Member forces and deformation from grillage model
34
Table 2.10. Comparison of analytical approach and grillage model results [Unit: kN-m]
Analytical approach Grillage model Web 1 Web 2 Web 1 Web 2
Load Case 1 6300 6300 - - Load Case 2 7520 1930 6400 3050 Load Case 3 7280 5320 6730 5730
2.7. Longitudinal Flexure
2.7.1. Unbonded Tendons
Unbonded tendons are commonly used today in bridge and building construction. One
type of application for unbonded tendons is external post-tensioning, where tendons are placed
outside of concrete and enclosed in plastic ducts injected with grout. The underlying principle
of unbonded tendons has a fundamental difference with internal bonded tendons (Menn, 1990).
For bonded tendons, the change in strain in prestressing steel due to deformation (Δεp) equals
the concrete strain at tendon level (εcp) for any given plane section (Figure 2.24). This
compatibility relationship however cannot be applied to unbonded tendons. Due to the lack of
bonding, the tendon strain is not directly related to concrete strain at any one plane. Instead, the
strain depends on the global deformation of the girder and can be determined by integrating the
concrete strain at the level of tendon over the span of the structure (Menn, 1990). The
prestressing steel strain for unbonded tendons can be assumed constant along the span.
Figure 2.24. Compatibility relationship for bonded and unbonded tendons under ultimate limit states Under SLS, as concrete remains uncracked, the girder deformation and the change of
strain in prestressing steel is negligible (Menn, 1990). Thus, the stress in the unbonded tendons
can be taken as the effective prestress (Menn, 1990), which is the tendon stress after all losses.
35
The stress in unbonded tendons under ULS can also be conservatively estimated as the
effective prestress (Menn, 1990). Although this approximation can be sufficient for some of the
preliminary design calculations, it is desirable to determine the actual tendon stress under ULS,
so that better material economy can be achieved. A procedure to calculate unbonded tendon
stress under ULS is described in Section 2.7.4.
For both the SLS and ULS analysis, a reliable value of the effective prestress after all
losses is needed. A procedure for calculating this for girders with unbonded tendons is given in
Section 2.7.2.
2.7.2. Prestress Losses
The effective prestress in tendons under service load is always less than the jacking force
due to prestress losses (Menn, 1990). The prestress losses considered for the sample design
include: (a) friction loss, (b) anchor set loss, (c) loss due to concrete creep and shrinkage, and (d)
relaxation of prestressing steel after transfer. While (a) and (b) are losses at transfer, (c) and (d)
are time-dependent losses taking place over long-term. The losses for the sample design are
calculated with the following procedures and the results are summarized graphically in Figure
2.26.
(a) Friction Loss
Friction loss is due to the friction force between tendon and duct incurred during
stressing of tendons. It is related to the change of tendon alignment in both the vertical and
horizontal plane. The friction loss FR at a distance x away from the jacking end can be
calculated using the following formula (Menn, 1990): ( )(1 )x
oFR f e μα−= − [2-9]
where fo is the jacking stress, α(x) is the angle change between the jacking end and location x,
and μ is the friction coefficient, which varies for different types of post-tensioning systems. For
external unbonded tendons, AASHTO (1998) suggests that μ can be taken as 0.25. Unlike
internal bonded tendons with continuous angle change in their alignment, external unbonded
tendons only incur angle changes at a couple of discreet locations along the span. As a result,
friction loss in external unbonded tendons tends to be lower in comparison with internal bonded
tendons. For the sample design, the total friction loss is calculated to be 14 MPa based on a
value of 0.25 for μ.
36
(b) Anchor Set Loss
Anchor set loss refers to the anchorage slip due to seating of wedges during the
anchoring process. The tensile stress in the tendons decreases as the tendon shortens due to slip.
The anchor set loss ANC can be calculated as follows:
setpANC E
lΔ⎛ ⎞= ⎜ ⎟
⎝ ⎠ [2-10]
where Δset is the change in tendon length due to anchorage slip, which can be assumed to be 7
mm based on specifications of conventional post-tensioning systems. The parameter l is the
tendon length over which anchor set has an effect, and is dependent on friction in the post-
tensioning system. A tendon with lower friction has a longer l and vice versa. Because friction
in the unbonded system occurs primarily at deviation, the possible termination for l would be at
the deviations or the opposite end of the beam. The three possible cases for l are evaluated and
shown in Table 2.11. With an l of 12 m under case 1, the anchor set loss is calculated to be 112
MPa. This means that the total stress differential at the first deviation becomes 126 MPa
(FR+ANC), which equals 9 times of the friction loss. This likely cannot be sustained by the
deviation in the form of friction, thus the anchor set loss propagates beyond the first deviation.
A similar case can be made for Case 2. Consequently, it is assumed that the influence of anchor
set extends to the entire span. With an l of 38 m, the anchor set loss is calculated to be 37 MPa.
Table 2.11. Anchor set loss with varying l
Anchor set loss
Case 1 112 MPa Case 2 54 MPa Case 3 37 MPa
(c) Loss due to Concrete Creep and Shrinkage
Both creep and shrinkage are time-dependent phenomenons that cause plastic shortening
in concrete. As tendons shorten with concrete, they lose some of the prestressing force initially
jacked in. Trost (1967) has developed the following expression for the change of concrete strain
due to creep and shrinkage:
( )0 1c cc CS
c cE Eσ σε φ μφ εΔ
Δ = + + + [2-11]
37
where σc0 is the initial concrete stress at the level of prestressing steel due to dead load and
prestressing force; φ and μ are the creep coefficient and the aging coefficient respectively; and
εcs(t) is the concrete shrinkage strain. Based on the compatibility relationship of Δεp = εcp
(Figure 2.21), Menn (1990) has derived from equation 2-11 the following formula to calculate
the loss of prestress due to creep and shrinkage for bonded tendons:
( ) [ ]( )
0
1 1c c cs c cn A E A
P tn
ρ σ φ ερ μφ
+Δ =
+ + [2-12]
where n = Ep / Ec and ρ = Ap / Ac. This formula, however, cannot be applied to unbonded
tendons, because the derivation assumes that Δεp = εcp, which does not hold true for unbonded
tendons.
From the same expression by Trost [2-11], Gauvreau (1993) has derived a formula to
calculate creep and shrinkage loss based on the compatibility relationship for unbonded tendons,
which requires that the total change in length in prestressing steel equals the total concrete
deformation at the level of unbonded tendons. This relationship, which assumes that the strain in
prestressing steel is constant along the length of tendon, can be expressed as follows:
( )1p cp
p
x dxl
ε εΔ = ∫ [2-13]
where Δεp is the change in strain in prestressing steel; εcp is the concrete strain at elevation of
tendon; and lp is tendon length. The integration of εcp is over the projected length of tendon.
The formula for creep and shrinkage loss developed by Gauvreau (1993) for unbonded tendons
is as follows:
( )
[ ] ( )
0
2
1
1 1 1
c c cs c cp
c
p c
n A x dx E Al
PAn e x dx
l I
ρ φ σ ε
ρ μφ
⎡ ⎤+⎢ ⎥
⎢ ⎥⎣ ⎦Δ =⎡ ⎤
+ + +⎢ ⎥⎢ ⎥⎣ ⎦
∫
∫ [2-14]
where e(x) is the distance from centroid of the gross uncracked concrete section to the centroid
of unbonded prestressing steel.
38
(a) (b) (c) (a) Nominal creep coefficient versus relative humidity for water-to-cement ratio (W/C) of 0.4 and 0.5 (b) Correction factor k versus time of loading (days / years after casting) (c) Time-varying function f(t-τ) versus time (days / years after casting) for two typical values of hef, where hef = 2Ac/U. Ac is the cross-sectional area and U denotes the exposed perimeter.
Figure 2.25. Parameters in determining creep coefficient φ (Menn, 1990)
The value of μ in equation 2-14 can normally be taken as 0.8 (Menn, 1990). Both the
shrinkage strain and the creep coefficient are time-dependent variables. While εcs can be
determined from charts in Menn (1990), creep coefficient φ can be calculated using the
following equation (Menn, 1990):
( ) ( )( , ) nt k f tφ τ φ τ τ= − [2-15]
where φn is the nominal creep coefficient; k(τ) is a correction factor for the age of concrete at
time of loading; and f(t-τ) is the function accounting for the time-varying property of creep. All
three parameters can be determined graphically from diagrams in Figure 2.25.
Based on the above information, long-term prestress loss due to creep and shrinkage for
the sample design is evaluated for a projected service life of 50 years. Also included in the
calculation is the loss in stage 1 tendons at the time of stage 2 post-tensioning, which was
assumed to be 28 days after the stage 1 operation. The creep and shrinkage losses calculated are
summarized in Table 2.12.
Table 2.12. Loss due to creep and shrinkage and related parameters
Time Shrinkage strain, εcs (Menn, 1990)
Creep coefficient, φ (Menn, 1990)
Loss due to creep and shrinkage
All tendons 50 years -0.298 mm/m 2.00 136 MPa
Stage 1 tendons 28 days -0.105 mm/m 0.50 25 MPa
39
(d) Relaxation of Prestressing Steel after Transfer
The loss of prestress due to relaxation is a function of the prestressing steel property as
well as the ratio between the initial stress (σp0) and the ultimate stress (fpu) of the prestressing
steel (Menn, 1990). This relationship is expressed by the following equation in CHBDC (CSA,
2006b):
REL = C [ A – B (CR + SH) ] [2-16]
where A and B are variables related to the property of prestressing steel, and C is a variable
depending on the ratio of σp0 / fpu . According to CHBDC, A and B can be taken as 42 and 0.053
respectively for Grade 1860 low-relaxation strand, while C can be taken as 1.00 if σp0 / fpu
equals 0.75. From equation 2-16, it is calculated that the long-term prestress loss due to
relaxation for the sample design approximately 35 MPa.
Relaxation develops faster in comparison with creep and shrinkage. Approximately 50%
of the final relaxation loss can be reached at 28 days (Menn, 1990). The relaxation loss at 56
days is conservatively taken as 100% of its final value since the stress in stage 1 tendons at 56
days will be used as a lower bound in later analysis. In summary, the relaxation loss of stage 1
tendons at 28 days and 56 days are assumed to be 17 MPa and 35 MPa, respectively.
Note: * Stress at 28 and 56 days are calculated for stage 1 post-tensioning tendons. t = time elapsed since loading
Figure 2.26. Summary of prestress losses in sample design
40
2.7.3. Flexural Response under SLS
The structural response under SLS needs to be evaluated for the load combinations
presented in Section 2.4.4. The magnitude and load factors for dead, superimposed dead and
live load were given in Section 2.4.1 and 2.4.2. The prestressing force used in each load
combination is based on the loss calculation from section 2.7.2. As shown in Figure 2.26,
prestress losses vary along the span. This variation, however, only changes the total
prestressing force in each case by approximately 1%. Due to the relatively small effect of the
variations, the stress in the prestressing steel is assumed to be constant along the span for the
SLS analysis. The prestress values used in the SLS calculation are summarized in Table 2.13.
For load case 1A which represents the loading condition during stage I post-tensioning, the
stress in stage I tendons (fP1) is assumed to be the jacking stress, 0.80 fpu. For load case 2, which
describes the loading just prior to stage II post-tensioning, fP1 is taken as the prestress after 28
days, which is 0.74 fpu, because stage II post-tensioning is assumed to commence 28 days after
stage I. Load case 1C takes place shortly after load case 1B during stage II post-tensioning
operation. Under this load case, fP1 is assumed to still be at 0.74 fpu, while fP2 is taken as the
jacking stress, 0.80 fpu. Finally, load case 1D accounts for the load combination during the
bridge’s service life. fP1 and fP2 are taken as the prestress after all losses for a period of 50 years.
Table 2.13. Stress in post-tensioning steel under SLS
Figure 4.5. Moment diagrams under ULS load combinations
Under the negative-flexure-critical load cases 1A and 1C, the system behaves in the
same manner as alternative concept I. As shown in Figure 4.6, concrete stresses remain
approximately within the linear elastic region between 0.6 f’c and fcr. This indicates that the
structural capacity is adequate for the loading under combination ULS 1A and 1C.
Concrete top fibre stress
-4.000-2.0000.0002.0004.000
[MPa
].
Concrete bottom fibre stress
-50.000-40.000-30.000-20.000-10.000
0.000
[MPa
].
ULS 1A ULS 1C
fcr
0.6 f ’c
Figure 4.6. Concrete stress under negative-flexure-critical ULS load combinations
For positive-flexure-critical load combinations, the moments due to gravity loads are
higher than the moments provided by prestressing. Under these load cases, the external
unbonded tendons elongate and develop additional stress due to girder deformation. At the
same time, the internal bonded unstressed tendons also contribute to the system’s overall
capacity. The flexural response of the girder is evaluated and summarized in Figure 4.7. The
analysis is only shown for ULS 1D because loading under ULS 1D is more severe than ULS 1B.
77
0.00500.00
1000.001500.00
[MPa]
0100002000030000
Moment[kN-m]
MQ - Moment dueto external loadMcr - crackingmomentM capacity
-1.00-0.500.00
[mm/m]
0.00
2.50
5.00
Curvature[rad/km]
0.002.004.006.00
[mm/m]
topε
σ
cpε
30000 kN-m
27500 kN-m
4.31 rad/km
-0.90 mm/m
6.52 mm/m
1460 MPa
S
Cracked
Figure 4.7. System behaviour under ULS 1D
The above moment diagram shows that the moment capacity of the section is 30000 kN-
m, which is approximately 9% above the maximum demand at mid-span. This capacity is
calculated using the same method as in Chapter 2 for the base concept, where the applied load
on the bridge girder is magnified until the stress in the external unbonded tendons reaches 90%
fpu.
The compressive strain in concrete’s top fibre, represented by εtop, has a maximum of -
0.90 mm/m. This is approximately 30% of the concrete crushing strain, indicating that the
concrete is not close to failure.
The stress in the external unbonded tendons is calculated using the method presented in
Section 2.8.4. By integrating the strain at the level of tendons εcp along the span, it is found that
the prestressing steel gains an additional 286 MPa and reach 1530 MPa, equivalent of 0.82 fpu,
under ULS 1D.
78
The flexural response of alternative concept I has a number of differences from the
double-T base concept and alternative concept I. A comprehensive comparison is presented in
Section 6.1.
4.4.3. Fatigue Limit States
It was shown in Section 4.4.1 that the concrete girder remains uncracked under SLS.
Thus under FLS with less loading, the concrete would also remain uncracked. As a result, the
unstressed prestressing strands would experience negligible tensile stress under FLS, thus is not
critical in fatigue.
4.5. Final Remarks
The alternative double-T concept presented in this Chapter is a modification of the base
concept presented in Chapter 2. The changes include reduced amount of post-tensioning and the
addition of unstressed internal tendons. Due to these modifications, alternative concept II’s
post-tensioning design is not as restricted as the base concept. The reduction in post-tensioning
decreases the negative moment imposed on the structure during construction thus making the
structure less critical in negative flexure. The addition of internal unstressed tendon, which is
continuous along the span, expands the allowable window for concrete stress from 0.6 f’c – 0 to
0.6 f’c – fcr under SLS, thus giving the designer more flexibility. When compared to alternative
concept I, alternative concept II is no longer limited by the CFRP debonding strain under ULS.
This allows the system to achieve larger deformation and develop higher stress in the unbonded
tendons under ULS. This is indicative of a more efficient utilization of the unbonded tendons.
Chapter 5 Slab-on-Girder Bridge System with CPCI Girders
This chapter describes some important aspects in the design of a conventional slab-on-
girder bridge system with CPCI girders. CPCI girders are standardized I sections commonly
used in Canada. A sample design is developed using a standard design spreadsheet provided by
Hatch Mott MacDonald Mississauga office. The spreadsheet calculation is included in
Appendix B and not discussed in detail in this chapter. Section 5.1 gives a brief introduction to
the CPCI slab-on-girder type of bridge system. Section 5.2 briefly describes the key features of
the sample design, while Section 5.3 gives a summary of the materials used. Section 5.4 and 5.5
examine live load distribution and deck slab design, which are two important aspects in the
design of slab-on-girder bridges. Finally, Section 5.6 is dedicated to construction related
subjects.
5.1. Introduction
Prestressed concrete I-girder bridges were introduced in the 1950s (Kulka and Lin, 1984).
This type of structural system includes a series of precast and prestressed concrete I-girders with
a cast-in-place concrete top slab. One of the first prominent bridges of this type is the Walnut
Lane Memorial Bridge built in Philadelphia, Pennsylvania in 1950 (Dunker and Rabbat, 1992).
Since then, this type of slab-on-girder bridge gained wide popularity in North America and
became the most used bridge system for short-to-medium span bridges (Kulka and Lin, 1984).
A survey done by Dunker and Rabbat in 1992 indicates that they comprise about 40% of all
prestressed concrete bridges constructed in the United States during the period from 1950 to
1989. Precast pre-tensioned I-girders have traditionally been used for spans up to approximately
50 m (Kulka and Lin, 1984). Nowadays, the range of span can be extended to 75 m by using I-
girders that are spliced together with longitudinal post-tensioning (Lounis et al, 1997).
I-girders are highly standardised in today’s bridge industry (FHWA, 2009). A number
of standard I sections have been developed, such as the Florida Bulb-Tee, AASHTO-PCI Bulb-
79
80
Tee and the CPCI standard I sections (Figure 5.1). The first two types of girders are most often
used in the US, while CPCI girders are commonly used in Canada.
Figure 5.1. Standardized I sections (adapted from Pre-Con, 2004 and FHWA, 2009) Despite the wide application of the slab-on-girder system, this type of bridge has its
inherent shortcomings. The first major shortcoming is its inefficient sharing of live load
between girders. Due to the deck slab’s flexibility, the effect of an applied load on the deck slab
can only propagate to a limited number of girders close to the location of the load. The second
shortcoming of the system comes from the inconvenience and cost associated with the cast-in-
place deck slab. Both subjects will be discussed in more detail in later sections.
5.2. Brief Description of Design
A slab-on-CPCI-girder system is developed based on the geometrical requirements
presented in Chapter 1. The design is produced with the aid of a standard design spreadsheet
provided by Hatch Mott MacDonald, Missisauga. Sample pages of the spreadsheet as well as a
more complete set of drawings are included in Appendix B for reference.
As shown in Figure 5.2, the cross-section of the bridge consists of a cast-in-place deck
slab 225 mm in thickness and six prestressed precast CPCI 1900 girders spaced at 2350 mm.
The depth of the girder is 1900 mm and the total depth of the structure is 2175 mm. The
concrete girders are pre-tensioned with internal prestressing strands. The strands are installed in
standardized locations within the girder. For the sample design, each girder is equipped with 50
S13 strands, among which 32 are straight and 18 are deflected. The strand layout is shown in
Figure 5.3.
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Figure 5.2. Sample design of the Slab-on-CPCI-girder system
Figure 5.3. Pre-tension strand layout (adapted from MTO, 2002)
5.3. Material Properties
The properties of the materials used in the sample design are summarized in Table 5.1.
There are two major differences between the materials used in the CPCI system and the double-
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T systems. First, the CPCI system uses a more conventional 50 MPa concrete while the double-
T systems use 70 MPa concrete. Second, because the current pretensioning industry usually
prefers size 13 prestressing strands, the CPCI system does not employ the size 15 strands that
were used in the double-T systems.
Table 5.1. Material properties for slab-on-CPCI-girder sample design
When vehicular live load is applied on the deck slab, its influence on each girder is
different. It is important for designers to know how load is distributed among girders, as it is a
governing factor in determining the design load for each girder.
The simplified example in Figure 5.4 (Hassanain, 1998) and the following discussion
describe two extreme cases of live load distribution. The example has a slab-on-girder system
that is idealized as three simply-supported girders connected by a transverse beam at mid-span.
A load P is applied at mid-span of the transverse beam, thus directly on the central girder. On
one extreme, if the transverse beam has no stiffness, the three girders become virtually
disconnected, and all load will be taken by the central girder. In this case, Girder 2 carries the
entire P while Girder 1 and 3 carries no load. On the other extreme, if the transverse beam is
infinitely stiff, all three girders will deflect and share the load equally, and the load distributed to
each girder is P/3. The second case illustrates a more efficient load distribution among girders.
83
Figure 5.4. Load distribution in an idealized beam-on-girder system (adapted from Hassanain, 1998)
In a more realistic case where girders are connected by a deck slab, live load distribution
becomes much more complicated. It depends on many factors, such as span length, the
transverse position of live load, the dimension and location of the diaphragms, and the
transverse and longitudinal bending stiffness of the girder-and-slab composite (CSA, 2006b).
Much research has been done in modeling live load distribution in a slab-on-girder
system using grillage, semi-continuum or finite element methods (CSA, 2006b). The results
from these analysis are used to formulate design equations in standards and codes. These
equations are often simplified from the real situation to include only a limited number of
variables. They are usually used to determine the maximum amount of load distributed to a
girder under the most unfavourable conditions. This load level then becomes the design load for
each girder in the system. The following sections describe some of the load distribution
equations used in North America. The equations are also summarized in Table 5.2.
5.4.1. AASHTO Standard
The AASHTO Standard equations have been used in the United States since the 1930s
(Yousif, 2007). They were only recently replaced by the new AASHTO LRFD specification,
which was first published in 1998.
In the AASHTO Standard, a distribution factor (DF) is defined and applied to one line of
wheel load. The equation for DF takes a simple form of S/D, where S is the girder spacing and
D is a constant that depends on the type of bridge and the number lanes loaded (AASHTO,
1996). For concrete slab on girder bridges, D is 2.13 if only one lane is loaded, and 1.68 if two
or more lanes are loaded.
The S/D formula is relatively easy to apply. However, it disregards the effect of span
length, deck slab, and girder stiffness (Yousif, 2007). It was found to produce overly
84
conservative results in some cases and unconservative results in others (Cai, 2005). Also, this
formula is developed only for bridges with typical geometry, thus not applicable to more
complicated bridges (Yousif, 2007). All these shortcomings of the S/D formula led to the
development of the new load distribution formula, which was introduced in AASHTO LRFD
Bridge Design Specifications in 1998.
5.4.2. AASHTO LRFD Specifications
The AASHTO LRFD Specifications provide a more refined method to calculate live
load distribution than the previous S/D equation. In addition to girder spacing, this formula also
takes into consideration the bridge span, slab thickness, and the longitudinal stiffness of the
cross-section (Yousif, 2007).
The distribution factor, which is also applied to one line of wheel load, is defined as
follows (AASHTO 1998):
0.10.4 0.3
30.064300
g
s
KS SDFL Lt
⎛ ⎞⎛ ⎞ ⎛ ⎞= + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
if one lane is loaded [5-1]
0.10.6 0.2
30.0752900
g
s
KS SDFL Lt
⎛ ⎞⎛ ⎞ ⎛ ⎞= + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
if two or more lanes are loaded [5-2]
where S is girder spacing, L is span length, ts is deck slab thickness and Kg is the longitudinal
stiffness. Kg mainly depends on the girder type, and can be calculated as follows (Yousif, 2007):
( )2g g g gK n I A e= + [5-3]
where n is the ratio between the girder and deck’s Elastic Modulus; Ig and Ag are the moment of
inertia and the cross-sectional area of the girder; and eg is the eccentricity between the girder and
slab’s center of gravity.
The AASHTO LRFD method is more elaborate and is shown to produce more accurate
results than the previous S/D formula (Suksawang, 2007). However it complicates the design
procedure significantly because the new equations require the knowledge of the girder cross-
sectional properties, which are generally unknown prior to the determination of design load. As
a result, the design becomes an iterative process. Research has been done in recent years trying
to simplify the new AASHTO LRFD formulations while not compromising its accuracy (Cai,
2005, Suksawang, 2007, and Yousif, 2007).
85
5.4.3. Canadian Highway Bridge Design Code
The Canadian Highway Bridge Design Code prescribes simplified methods for the
analysis of live load distribution. These methods are based on analysis results from modeling
many structures with the grillage, semi-continuum, and finite element methods (CSA 2006b).
To be analyzed using the simplified methods, the bridge must satisfy certain conditions (CSA
2006a). For a straight slab-on-girder bridge, the conditions include:
a. The deck width is constant;
b. The support conditions can be represented by line support;
c. There should be at least three longitudinal girders with approximately equal flexural
rigidity and spacing;
d. The deck slab overhang should be less than 1.80 m and 60% of the average girder
spacing.
CHBDC prescribes several methods to analyze live load distribution. Each method is
applicable to a specific type of load effect under specific limit states, such as SLS, FLS or ULS.
The following discussion focuses on the method for evaluating the distribution of longitudinal
bending moments in slab-on-girder bridges under ULS and SLS.
The method presented in CHBDC centres at the concept of an amplification factor (Fm),
which is defined as:
,
gm
g avg
MF
M= [5-4]
where Mg,avg is the average moment per girder due to live load if live load were shared equally
among girders, and Mg is the maximum longitudinal moment per girder accounting for unequal
live load distribution (CSA, 2006a). Mg,avg can be calculated from the following equation:
,T L
g avgnM RM
N= [5-5]
where n is the number of design lanes, RL is the multi-lane loading reduction factor, N is the
number of girders, and MT is the maximum moment per design lane due to the two-line axel
load (CSA 2006a).
86
It is shown in equation 5-4 that Fm is a measure of how much the extreme load
distribution deviates from the average distribution. Lower values of Fm indicate less deviation,
thus greater ability of the bridge to transfer load across its width (CSA, 2006b). Figure 5.5
shows an Fm versus span graph developed for slab-on-girder bridges with a lane width of 3.33 m.
The graph indicates that bridges with longer spans and narrower decks have a lower value of Fm,
which is a sign of more even live load distribution.
Figure 5.5. Fm for internal girders in a slab-on-girder bridge system under ULS and SLS (CSA, 2006b)
In design, Fm for slab-on-girder type of bridge can be calculated using the following
formula:
1.051
100
mf
SNFC
Fμ
= ≥⎡ ⎤+⎢ ⎥
⎣ ⎦
[5-6]
where S is the girder spacing. There are two key expressions in the above equation for Fm:
F
F is a “width dimension that characterizes the load distribution for a bridge” (CSA,
2006a). It depends on many factors, including bridge type, highway class, span length,
number of design lanes, and girder position. The concept of F is related to the constant D in
AASHTO Standard’s S/D formulation by the following expression (CSA 2006b):
2 LF nR D= [5-7]
The multiplier of 2 accounts for the fact that the AASHTO formulas are based on one line of
wheel load while CHBDC formulas are associated with the full truck load. The factor n and
RL adjusts the constant to account for the multi-lane loading effect. The values of F are
tabulated in CHBDC.
87
1100
fCμ⎡ ⎤+⎢ ⎥
⎣ ⎦
This expression represents the “lane width correction factor” (CSA, 2006a). Cf is a
“percentage correction factor”, which can be obtained from tables in CHBDC (CSA, 2006a).
The factor μ can be calculated as follows:
3.3 1.00.6
eWμ −= ≤ [5-8]
where We is the design lane width.
Live load distribution in the CPCI sample design is analyzed based on CHBDC’s
simplified method. From the CHBDC tables, the live load amplification factor Fm is calculated
to be 1.501. Thus, from equation 5-4 and 5-5,
, ,1.501g m g avg g avgM F M M= = ×
0.601T Lg m T
nM RM F MN
= = ×
Live load analysis for the 36.6 m span indicates that the CL truck load model produces the
governing live load effect. Hence, the dynamic load allowance needs to be applied to the above
relationships. After factoring in a dynamic load allowance of 1.25, the above two equations
become:
, ,1.25 1.501 1.876g g avg g avgM M M= × × = ×
1.25 0.601 0.751g T TM M M= × × = ×
The first relationship indicates that the maximum longitudinal moment per girder due to unequal
live load sharing is approximately 1.9 times of the average moment per girder if load were
shared equally. The second equation, which is rearranged from the first relationship, suggests
that the live load demand for each girder is approximately 75% of the maximum moment per
design lane (MT). Based on the above calculation, the total live load demand on the six-girder
system is 4.5 MT – 50% higher than the actual maximum live load possible for a structure with
three design lanes.
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Table 5.2. Summary of live load distribution equations
Standard Equation Application
AASHTO Standards
(AASHTO, 1996)
DF = S/D where D = 2.13 for one lane D = 1.68 for two or more lanes
Mg = DF x MT where MT is the moment due to one line of wheel load
AASHTO LRFD
(AASHTO, 1998)
For one lane: 0.10.4 0.3
30.06
4300g
s
KS SDF
L Lt= +
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
For two or more lanes: 0.10.6 0.2
30.075
2900g
s
KS SDF
L Lt= +
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Mg = DF x MT where MT is the moment due to one line of wheel load
CHBDC (CSA, 2006a)
,
1.051
100
g
mfg avg
M SNF
CMF
μ= ≥
+
=⎡ ⎤⎢ ⎥⎣ ⎦
equivalent LmDF F
nRN
=
Mg = DF x MT where MT is the moment from one lane (a full truck load).
5.5. Deck Slab Design
5.5.1. Arching Action
Prior to 1970s, bridge deck slabs were primarily designed for pure bending in North
America (Batchelor, 1987). This approach was shown to be overly conservative, as deck slabs
designed with this method had much higher strength than what was required by safety
(Batchelor, 1987). Researchers found that the traditional design method neglected an important
element in predicting the deck slab response – the compressive in-plane forces, which induce
arching action in laterally restrained slabs after cracks are developed (Figure 5.6) (Batchelor,
1987). Arching action can greatly enhance the deck slab strength, and is the dominant action for
deck slabs in resisting concentrated wheel loads (Batchelor, 1987). Due to arching action, deck
slabs tend to fail in punching shear instead of flexure. In 1960, Kinnunen and Nylander
proposed a model to describe the punching shear failure mode (Figure 5.7). This model
assumes the critical portion of slab bounded by the shear cracks is loaded by a conical shell at
89
the base perimeter of the loaded area (Batchelor, 1987). A failure of the system will occur if the
conical shell becomes over stressed (Batchelor, 1987).
Figure 5.6. Arching action in deck slab Figure 5.7. Punching shear failure mode (adapted from Batchelor, 1987) (adapted from Batchelor, 1987)
The Kinnunen and Nylander model provided a good foundation for predicting the slab
failure load under arching action. However, many parameters are involved in the actual analysis.
Some parameters, such as the influence of boundary forces, are not easily quantifiable (Hewitt
and Batchelor, 1975). In Ontario, extensive theoretical, laboratory and field studies were carried
out to establish an empirical method for the design of deck slabs that accounts for the effect of
arching action (Batchelor, 1987). This method was first introduced in OHBDC in 1979 and
later refined in the following editions of the code. The new method substantially reduces the
amount of reinforcements in the deck slab in comparison with the former flexural design method
(Batchelor, 1987).
For the design of the double-T deck slab, arching action is not considered. The design is
developed based on transverse flexure of the deck slab.
5.5.2. Empirical Design Method from CHBDC
The deck slab design of the CPCI system adopts the empirical method from CHBDC.
According to CHBDC, slabs designed using to this method need not to be analyzed except for a
few special locations.
The minimum deck slab thickness required by CHBDC is 225 mm if 15M bars are used
as deck slab reinforcements. This thickness accommodates a 90 mm top cover, a 50 mm sofit
cover, and a 55 mm clear distance between top and bottom transverse reinforcements (CSA,
2006b). A typical deck slab cross-section and its reinforcement layout are shown in Figure 5.8.
90
Figure 5.8. Typical deck slab design based on CHBDC’s empirical method – cross-section view For a full-depth cast-in-space deck slab, there should be a total of four layers of
reinforcing bars arranged in two orthogonal planes – one plane near the top of the slab and the
other near the bottom. The reinforcement ratio ρ for each layer should be at least 0.003, unless
the deck slab can be proved to behave satisfactorily with less reinforcements. In the latter case,
ρ can be reduced to 0.002. The ratio ρ is defined as:
sAbd
ρ = [5-9]
where b is an arbitrary width, As is the area of reinforcement within the given width b, and d is
the effective depth of concrete. In practice, it is customary to design deck slabs of slab-on-
girder bridges with 15M bars spaced at 300 mm. This translate to a reinforcement ratio slightly
higher than 0.003.
5.6. Construction
5.6.1. CPCI Girder Fabrication
The fabrication of CPCI girders involves first placing and stressing the pretensioning
strands, then laying the reinforcing bars, and finally casting concrete. This process requires
specialized formwork and equipments. The formwork is usually made of steel to maximize its
durability for extended use. Bulkheads as shown in Figure 5.9 are used at ends of the precast
segment to anchor the pre-tensioning strands. Also shown in the figure is a precasting bed made
of steel beams. It serves the purpose of both supporting the cast segment and holding down the
pre-tensioning strands’ deviations. During the casting process, steel side forms are used to form
the side geometry of the CPCI girders. The specialized forms and equipments require high
capital investment. As a result, only a very limited number of suppliers of CPCI girders are
available in Ontario.
91
Figure 5.9. CPCI girder fabrication (photos by P. Gauvreau)
5.6.2. Erection
The superstructure of a single-span CPCI slab-on-girder girder bridge is typically
constructed in the following sequence (Figure 5.10):
1. Erect precast girders;
2. Cast end diaphragms;
3. Place deck slab reinforcements;
4. Cast deck slab once the concrete compressive strength in the diaphragms has reached
the specified value.
The erection of girder is usually done by a crane. The overall construction is relatively standard.
However the speed of construction is often redistricted by the time required to place deck slab
reinforcements and cure the cast-in-place concrete.
Figure 5.10. Typical construction sequence for the superstructure of a slab-on-girder bridge (adapted from WSDOT, 2008)
Bulkhead
Precasting bed
Reinforcing bars
Pretensioning strands
Chapter 6 Evaluation of the Double-T and CPCI Systems
This chapter evaluates the structural efficiency and the material consumption and cost of
the double-T and the CPCI slab-on-girder systems based on the sample designs developed in
Chapter 2 to 5. Section 6.1 examines three alternative double-T systems. Section 6.2 compares
the double-T and the CPCI systems in terms of live load distribution, amount of design load
incurred in the system, and the approaches for deck slab design. Section 6.3 compares the
systems’ material consumption, including the use of concrete, prestressing steel and additional
reinforcements. Finally, Section 6.4 provides a preliminary cost comparison for the systems
developed in this thesis.
6.1. Comparison of Double-T Concepts
The three double-T concepts were developed based on the same concrete cross-section
and transverse design. Differences of the systems lie in the design of their primary longitudinal
reinforcements. While the base concept is reinforced with pure post-tensioning, alternative
concept I and II have a blend of prestressed and non-prestressed primary reinforcements.
The double-T base concept is longitudinally reinforced with pure external unbonded
post-tensioning. The sample design of the double-T base concept is governed by longitudinal
flexure under SLS. It is found in the SLS analysis that the structure is somewhat sensitive to the
level of prestressing losses and the amount of post-tensioning in the system. While the sample
design developed in Chapter 2 is a feasible system responding to the specific geometrical
requirements described in Chapter 1, the double-T base concept’s applicability to different
geometrical conditions may be limited due to the limitation associated with post-tensioning
design.
To improve the double-T system’s adaptability to different geometrical conditions,
alternative concept I is developed with a blend of prestressed and non-prestressed primary
reinforcements. The pretressed reinforcements are again external unbonded post-tensioning
92
93
tendons, while the non-prestressed reinforcements are external CFRP laminates. This
alternative design has two major advantages. First, due to the addition of continuous CFRP
laminate, the concrete stress limit under SLS is increased up to fcr in the bottom fibre, giving the
designer more flexibility. Second, because the system’s positive flexural capacity is
complemented by CFPR, the amount of external post-tensioning can now be reduced, thus
creating less negative flexural demand on the system. For alternative concept I, the concrete
stresses under SLS become less critical and less sensitive to level of prestressing in comparison
with the base concept. The disadvantage of alternative concept I relates to its behaviour under
ULS, where the girder’s overall deformation is restricted by the debonding strain of CFRP
laminate. This restriction lowers the additional stress that can be developed in the unbonded
tendons under ULS and increases the amount of CFRP needed.
A second alternative double-T system is created by replacing the CFRP in alternative
concept I with unstressed internal bonded tendon, which is continuous along the span. The
number of strands provided in the unstressed tendon is governed by a criterion that requires the
stress in the strand to be less than 240 MPa under SLS. This requirement is a mean of
controlling the size of crack and segmental joint opening under SLS. Like alternative concept I,
concrete stresses under SLS is less critical and less sensitive to level of prestressing in
comparison with the base concept. The unbonded tendons under ULS develop slightly higher
stresses than the previous two concepts due to the increase in girder deformation.
Both alternative concept I and II are developed with SLS design criteria that limit the
tensile stress in concrete under fcr. It is possible to increase their SLS concrete tensile stress
limit beyond fcr. However, a thorough examination of cracking needs to be carried out.
Table 6.1 summarizes some of the SLS and ULS characteristics of the double-T systems.
The major difference among the systems under SLS is that the base concept does not allow any
tensile stress in concrete while alternative concept I and II allow up to fcr in concrete’s bottom
fibre where continuous reinforcements are present. Under ULS, although the final stress in the
external unbonded tendons is close among the three concepts, the strain distribution is quite
different. For the base concept, the strain peaks at mid-span at a value of 12.3 mm/m. For the
alternative concepts, the strain diagram displays a more parabolic distribution with a lower
maximum. This indicates that the addition of internal bonded reinforcements help better
distribute strain in the system. The governing design criteria and ULS capacity of the three
systems are also shown in the table. The base concept is governed by concrete tensile stress
94
limit under SLS and display some reserve in ULS capacity. Alternative concept I is governed
by ULS criteria of CFRP debonding. The ULS capacity is calculated based on a CFRP
debonding strain of 6 mm/m, which is a conservative limit suggested by CHBDC (2006b). If
bonding can be improved and made more reliable, the ULS capacity can improve significantly.
Alternative concept II is governed by the SLS stress limit for the internal continuous
prestressing steel. The system’s ULS demand is approximately 92% of its capacity.
Overall, alternative concept II appears to be the most versatile system among the three
concepts. Unlike the base concept which is somewhat constrained by the large amount of post-
tensioning force in the system, alternative concept II can be adapted to other design conditions
by adjusting the amount of post-tensioning based on SLS design criteria and complementing the
ULS capacity with internal unstressed prestressing strands. While there is a certain range of
geometrical conditions that alternative concept II is most suitable for, this system appears to
have a reasonable range of applicability. Alternative concept I with CFRP reinforcements also
has the potential for wider range of application if bonding between CFRP and the double-T
girder can be improved.
Table 6.1. Comparison of double-T systems
Base concept Alternative concept I Alternative concept II
University of Toronto COMPUTATIONS made by EL date 12/14/09
chc'k by date
Torsion 1 Double-T Concrete Bridge Design
Torsion in Double-T Girders(Ref. Menn, Sec. 5.1.3) - Torsional moments in open sections (i.e. double-t girder) are resisted by a combnation of St. Venant torsion (Tsv) and warping torsion (Tw).
Calculate k
ho = 1.8632609 mbo = 7.90 m
h = 2.00 mb = 13.80 m
bw = 0.2975 mts = 0.2734783 m
as: distance from middle surface of top slab to centroidal axisas = 0.216102 m I (half) = 0.53254 m4
an: distance from middle surface of top slab to neutral axis.an = 0.208767 m In bar (half) = 0.59369 m4
K: torsional constant (equivalent of I in bending)K = 0.126793 G = 0.4 E
Compatibility at mid-spanspan = 36.60 m
sv = 9.15 k*Qw*bo/(G*K) = 1425.25 k*(Qw/E) [m-2]
wv: delfection at mid-spanwv = Qw*l^3/(48*E*I n bar) = 1720.47 Qw/E [m-1]
w = 2*wv(x)/bo = 435.561 Qw/E [m-2]
Set w = sv
k = 0.305603Qw = 0.76593 Q and Qsv = 0.23407 Q
)()()( xTxTxT wsv kxTxT
w
sv
)()( k
QQ
w
sv
118
page 5 of 14
University of Toronto COMPUTATIONS made by EL date 12/14/09
Other FactorsMulti-lane reduction, RL: 0.8 for 3 lanes
0.9 for 2 lanes cl. 3.8.4.2 & Table 3.5DLA : 1.25 cl. 3.8.4.5 Applied to truck load, NOT to lane load.
Consideration for Eccentric Live Load- When girder is loaded eccentrically, torsion will be created.- Torsion will be carried by the means of 1) Warping torsion, and 2) St. Venant torsion. 1) Warping torsion causes differential bending in webs 2) St. Venant torsion will be carried by closed shear flow within each individual cross-section element- Warping torsion will be dealt as additional flextural demand; St. Venant torsion will be dealt later as additional shear stress.- 3 load cases will be considered, the most severe will be used as demand. Load case 1: 3 lanes loaded concentrically. Load case 2: 2 lanes loaded eccentrically. Load case 3: 3 lanes loaded eccentrically. - Note: RL = 0.8 for 3 lanes and = 0.9 for 2 lanes, this significantly affects the result of Load Case 2 and Load Case 3.
- Let W be the load of 1 truck (a pair of axle); Q1 be the reaction in the web on the severe side.
Load Case 1Load Case 2Load Case 3 MaxQ1 [*W] 1.5 1.525 1.612 1.612 Note: Values account for RL and DLA.
max. MLL (one web, unfac) 6301 6404 6773 6773 Live load distribution taken fromSAP grillage model analyiss results
Stresses (ONE web)**Note: The stresses for ULS Load Combinations in the following charts only serve as general indication of the load condition severity. Detailed ULS check will be done in other spread sheets.
University of Toronto COMPUTATIONS made by EL date 12/14/09chc'k by date
Shear Double-T base concept sample design
Shear Design according to CHBDC cl.8.9
At dv from supportmax Vf (ONE web) = 2896 kN At 1.6m from support (spread of anchor force)
Mf/dv = 2612 kN Use 2 legs of 20M at 180mm over 1.5m band.Mf/dv +Vf -Apfp = -8100 kN (per web)
x = 0 = 29 degrees = 0.40
Vc = 1045 kNVp = 913 kN
Vs = Vf-Vp-Vc = 938 kNAv/s required = 1.003 mm2/mm
Vf <= 0.125 cf'cbwdv+Vp = 3772 kNMinimum spacing from code
If use 15M bars, Av = 400 mm2 .:. s <= 398 mm 600 mmIf use 20M bars, Av = 600 mm2 .:. s <= 598 mm 600 mm
At 0.1L from supportmax Vf (ONE web) = 2552 kN
Mf/dv = 7196 kNMf/dv +Vf -Apfp = -3861 kN
x = 0 = 29 degrees = 0.40
Vc = 1045 kNVp = 913 kN
Vs = Vf-Vp-Vc = 593 kNAv/s required = 0.634 mm2/mm
Vf <= 0.125 cf'cbwdv+Vp = 3772 kNMinimum spacing from code
If use 15M bars, Av = 400 mm2 .:. s <= 630 mm 600 mmIf use 20M bars, Av = 600 mm2 .:. s <= 946 mm 600 mm
At 0.2L from supportmax Vf (ONE web) = 2070 kN
Mf/dv = 12418 kNMf/dv +Vf -Apfp = 880 kN
x = 0.000202 = 30.41059 degrees = 0.31
Vc = 803 kNVp = 913 kN
Vs = Vf-Vp-Vc = 354 kNAv/s required = 0.401 mm2/mm
Vf <= 0.125 cf'cbwdv+Vp = 3772 kNMinimum spacing from code
If use 15M bars, Av = 400 mm2 .:. s <= 997 mm 600 mmIf use 20M bars, Av = 600 mm2 .:. s <= 1496 mm 600 mm
127page 14 of 14
University of Toronto COMPUTATIONS made by EL date 12/14/09chc'k by date
Shear 0
At 0.3L (11.26m) from support (just left of deviation)max Vf (ONE web) = 1518 kN Note: At Dev, there is an additional vertical force of 900kN (per web).
Mf/dv = 16579 kN .:. For the left half of a 1.5m band centred at dev,Mf/dv +Vf -Apfp = 4488 kN Vs >= Vf-Vc-Vp = 1093 kN (1 webs)
x = 0.001027 Vs = ( s fy Av dv cot ) / s = 36.19199 degrees .:. Av / s >= 1.54 mm2/mm = 0.16 s<= 388 mm 20M
Vc = 411 kNVp = 913 kN
Vs = Vf-Vp-Vc = 193 kNAv/s required = 0.273 mm2/mm
Vf <= 0.125 cf'cbwdv+Vp = 3772 kNMinimum spacing from code
If use 15M bars, Av = 400 mm2 .:. s <= 1467 mm 600 mmIf use 20M bars, Av = 600 mm2 .:. s <= 2201 mm 600 mm
At 0.3L (11.61m) from support (just right of deviation)max Vf (ONE web) = 1469 kN Note: At Dev, there is an additional vertical force of 900kN (per web).
Mf/dv = 16823 kN .:. For the right half of a 1.5m band centred at dev,Mf/dv +Vf -Apfp = 4684 kN Vs >= Vf-Vc-Vp = 2869 kN (1 webs)
x = 0.001072 Vs = ( s fy Av dv cot ) / s = 36.50567 degrees .:. Av / s >= 4.10 mm2/mm = 0.15 s<= 146 mm 20M
Vc = 401 kNVp = 0 kN
Vs = Vf-Vp-Vc = 1069 kNAv/s required = 1.526 mm2/mm
Vf <= 0.125 cf'cbwdv+Vp = 2859 kNMinimum spacing from code
If use 15M bars, Av = 400 mm2 .:. s <= 262 mm 600 mmIf use 20M bars, Av = 600 mm2 .:. s <= 393 mm 600 mm
At 0.4L from supportmax Vf (ONE web) = 1070 kN
Mf/dv = 18455 kNMf/dv +Vf -Apfp = 5917 kN
x = 0.001355 = 38.48173 degrees = 0.13
Vc = 345 kNVp = 0 kN
Vs = Vf-Vp-Vc = 725 kNAv/s required = 1.111 mm2/mm
Vf <= 0.125 cf'cbwdv+Vp = 2859 kNMinimum spacing from code
If use 15M bars, Av = 400 mm2 .:. s <= 359 mm 600 mmIf use 20M bars, Av = 600 mm2 .:. s <= 539 mm 600 mm
Appendix B:
CPCI Slab-on-Girder System Sample Calculations and Design
- Canadian Highway Bridge Design Code 2006 (CAN/CSA-S6-06) - refer to specific references listed
General Bridge Information
- Span No. 1 of 1 spans.
- Span length: 36600 mm Bridge Skew (rad.) = 0.00000- Adjacent span length 0 mm---> adjustments for continuity, L +: 36600 mm (Appendix A5.1) L -: 0 mm (Appendix A5.1)
- Deck width (travelled), Wc: 13.20 m
- No. of design lanes: 3 (ref. Table 3.8.2)
- Barrier wall area: 0.73 m2 2 barrier walls
- Total deck width: 13.80 m
- Deck thickness: 225 mm- Haunch thickness: 75 mm- Sidewalk width ; thickness: 0 mm 0 mm
- Total asphalt + w/p thickness: 90 mm- Spacing of girders: 2.35 m- Total no. of girders: 6- Top flange width 910 mm- Half deck span, BINT: 720 mm INTERIOR GIRDER- Half deck span, BEXT: 570 mm EXTERIOR GIRDER- Bearing Width = 300 mm- Deck overhang = 1.025 m < 0.5S
Effective Flange Width (cl. 5-8.2)
For Positive Moment Region
Interior (L+)/B ratio = 50.83 ===> Be = 720 mm .:. complete slab width is effectiveExterior (L+)/B ratio = 64.21 ===> Be = 570 mm .:. complete slab width is effective
For Negative Moment Region
Interior (L-)/B ratio = 0.00 ===> Be = 0 mmExterior (L-)/B ratio = 0.00 ===> Be = 0 mm
flange width, top = 910 mm hf (equiv.) = 150 mmA = 563375 mm2 bweb = 160 mmh = 1900 mm bflange top = 910 mm
y top = 960.1 mm bflange bott.= 660 mm
y bott. = 939.9 mmS top = 285522418 mm3
S bott. = 291672976 mm3
I = 2.74E+11 mm4
Composite Girder: * consider n = E(slab)/E(gird)
1. Interior Girder at Positive Moment Region = | yc - y |
Area (mm2) y (mm) Ay (x103) d (mm) Ad^2 (x106) Io (x106)Slab x n 512967 2012.5 1032346 561.4 161681.2 2164.1Girder 563375 939.9 529516 511.2 147214.8 274136.7
Total 1076342 2952.4 1561862 308896.0 276300.8yc = Ay / A = 1451.1 mm
.:. I comp. = 585196.8 x106 mm4
y top slab = 673.9 mm ; S top slab = 8.95E+08 mm^3y top girder = 448.9 mm ; S top gird. = 1.30E+09 mm^3
y bott. girder = 1451.1 mm ; S bott. gird. = 4.03E+08 mm^3
2. Interior Girder at Negative Moment Region = | yc - y |
Area (mm2) y (mm) Ay (x103) d (mm) Ad^2 (x106) Io (x106)Slab x n 198638 2012.5 399760 561.4 62608.5 838.0Girder 563375 939.9 529516 511.2 147214.8 274136.7
Total 762013 2952.4 929276 209823.3 274974.7yc = Ay / A = 1219.5 mm
.:. I comp. = 484798.0 x106 mm4
y top slab = 905.5 mm ; S top slab = 5.52E+08 mm^3y top girder = 680.5 mm ; S top gird. = 7.12E+08 mm^3
y bott. girder = 1219.5 mm ; S bott. gird. = 3.98E+08 mm^3
To use the Simplified Method for live load distribution as given in Cl. 5.7.1, the following requirements must be met: (Cl. 5.7.1.1)a) constant width.b) supports are equivalent to line support.c) skew parameter = 0.0000 < 1 / 18 = 0.05556 as per Appendix A5.1(b)(I)d) curved bridges built with shored construction must meet requirements of Appendix A5.1(b)(ii).e) N/Af) at least 3 longit. girders are of equal flexural rigidity and equally spaced, or with variations not more than 10% from the mean.g) deck cantilever does not exceed 60% of mean girder spacing nor 1.80 m.h) assumed points of inflection as per Appendix A5.1(a) apply.I) for multispine bridges, each spine has only two webs S NLongitudinal bending moments for ULS & SLS (cl. 5.7.1.2.1) Mg = Fm n MT RL / N & Fm = F (1+ Cf/100) > 1.05
MT = maximum moment per design lane at the point of interest ---- see table belown = 3 = number of design lanes as per 3.8.2
RL = 0.8 = modification factor for multi-lane loading as per 3.8.4.2 & 14.8.4.2N = 6 = number of girders within bridge deck width B
LIVE LOADS - per girderInput the results from a moving load analysis with CHBDC ONT truck (no DLA) and max. values from pattern loadingof 9 kN/m udl for lane load. The Amplification Factors from above & DLA are used to compute the Truck Load and Lane Load.
Stress (Mpa x A ps (mm^2) = Force (kN)f pu = 1861.0 x 98.7 183.7
f sj = 0.78 f pu = 1451.6 x 98.7 = 143.3
f st max.=0.74 f pu = 1377.1 x 98.7 = 135.9f st min.=0.45 f pu = 837.5 x 98.7 = 82.7
Assume /\ fs1 = 166.9 (Revise until "assumed" = calculated) Calculated = 132.7 Mpathen, f st = f sj - /\ fs1= 1284.7 x 98.7 = 126.8 < f st max .:. O.K.!
Assume /\fs=/\fs1+/\fs2= 410.3 (Revise until "assumed" = calculated) Calculated = 316.2 Mpathen, f se = f sj - /\ fs = 1041.3 x 98.7 = 102.8 > f st min. .:. O.K.!
Girder Design - try: 50 strands ====> *** From strand arrangement on previous page ***50 @ L.END and 50 @ R. END after debonding considered
i) between hold down points yp = 147.4 mme = 792.5 mm
ii) at girder ends: LEFT END: yp = 475.8 mm RIGHT END: yp = 475.8e = 464.1 mm e = 464.1
Check hold down(HD) forces for each deflected strand group (max. 6 strands/group
HD point 1 - distance from girder end = 13000 mm- y @ gird. end for group 1= 1860 mm- y @ gird. hold down for group 1 247.5 mm- # strands at hold down point = 4
.:. Hold-down force 71.1 kN ==> Within Recommended limit of 80 kN .:. O.K.!
HD point 2 - distance from girder end = 12500 mm- y @ gird. end for group 2= 1660 mm- y @ gird. hold down for group 2 197.5 mm- # strands at hold down point = 4
.:. Hold-down force 67.1 kN ==> Within Recommended limit of 80 kN .:. O.K.!
HD point 3 - distance from girder end = 12000 mm- y @ gird. end for group 3= 1410 mm- y @ gird. hold down for group 3 147.5 mm- # strands at hold down point = 4
.:. Hold-down force 60.3 kN ==> Within Recommended limit of 80 kN .:. O.K.!
HD point 4 - distance from girder end = 11500 mm- y @ gird. end for group 4= 1110 mm- y @ gird. hold down for group 4 85 mm- # strands at hold down point = 6
.:. Hold-down force 76.6 kN ==> Within Recommended limit of 80 kN .:. O.K.!
f cir = conc. stress @ c.g. prestress due to prestress at transfer + self-wt of member @ loc. of max. moment= [(Fst / Ag) + (Fst * e2) / I - (Md * e) / I]
e = e between hold downs = 0.793 mMd = M girder at midspan = 2311.2 kN-m
.:. f cir = 19.10 Mpa
f cds = conc. stress @ c.g. prestress due to all loads except dead load at transfer @ loc. of max. moment ("+" tensile)=Msl*e / Ig + Msdl*(yb - yp) / Icomp.
.:. f cds = 9.76 Mpa
2. Elastic Shortening, ES: (cl. 8.7.4.2.5)
ES = (Ep / E ci) * f cir= 119.9 MPa
.:. Total Losses At Transfer = fs1 = REL1 + ES = 132.7 Mpa
Stress in strand at transfer,fst = fsj - fs1
= 1319 Mpa < 0.74fPU OK (cl. C8.7.4.2.4)
Force per strand at transfer,Fst = 130 kN per strand
Total prestressing force at transfer,Fst = 6509 kN
Confirm that As/Aps < 1.0. If not, a more detailed analysis is needed to calculate losses after transfer.3. Creep, CR: Mean relative humidity (H) = 70 % (Figure A3.1.3) (cl. 8.7.4.3.2)
Case Type InitialCond ModalCase RunCase Text Text Text Text Yes/No
DEAD LinStatic Zero No MODAL LinModal Zero No
LC2 LinStatic Zero Yes LC3 LinStatic Zero Yes
Table: Case - Static 1 - Load Assignments
Case LoadType LoadName LoadSFText Text Text Unitless
DEAD Load case DEAD 1.000000LC2 Load case LC2-gravity 1.125000LC2 Load case LC2-torsion 1.125000LC3 Load case LC3-gravity 1.000000LC3 Load case LC3-torsion 1.000000
Table: Coordinate Systems
Name Type X Y Z AboutZ AboutY AboutXText Text m m m Degrees Degrees Degrees
GLOBAL Cartesian 0.00000 0.00000 0.00000 0.000 0.000 0.000 Table: Frame Loads - Point, Part 1 of 2
Frame LoadCase CoordSys Type Dir DistType RelDistText Text Text Text Text Text Unitless
2 LC3-gravity GLOBAL Force Gravity RelDist 0.22792 LC3-gravity GLOBAL Force Gravity RelDist 0.32622 LC3-gravity GLOBAL Force Gravity RelDist 0.35902 LC3-gravity GLOBAL Force Gravity RelDist 0.53932 LC3-gravity GLOBAL Force Gravity RelDist 0.71972 LC3-torsion GLOBAL Moment X RelDist 0.71972 LC3-torsion GLOBAL Moment X RelDist 0.22792 LC3-torsion GLOBAL Moment X RelDist 0.32622 LC3-torsion GLOBAL Moment X RelDist 0.35902 LC3-torsion GLOBAL Moment X RelDist 0.53935 LC2-gravity GLOBAL Force Gravity RelDist 0.22795 LC2-gravity GLOBAL Force Gravity RelDist 0.32625 LC2-gravity GLOBAL Force Gravity RelDist 0.35905 LC2-gravity GLOBAL Force Gravity RelDist 0.53935 LC3-gravity GLOBAL Force Gravity RelDist 0.22795 LC3-gravity GLOBAL Force Gravity RelDist 0.32625 LC3-gravity GLOBAL Force Gravity RelDist 0.35905 LC3-gravity GLOBAL Force Gravity RelDist 0.53935 LC3-gravity GLOBAL Force Gravity RelDist 0.71975 LC2-gravity GLOBAL Force Gravity RelDist 0.71975 LC2-torsion GLOBAL Moment X RelDist 0.71975 LC3-torsion GLOBAL Moment X RelDist 0.71975 LC2-torsion GLOBAL Moment X RelDist 0.22795 LC2-torsion GLOBAL Moment X RelDist 0.32625 LC2-torsion GLOBAL Moment X RelDist 0.35905 LC2-torsion GLOBAL Moment X RelDist 0.53935 LC3-torsion GLOBAL Moment X RelDist 0.22795 LC3-torsion GLOBAL Moment X RelDist 0.32625 LC3-torsion GLOBAL Moment X RelDist 0.35905 LC3-torsion GLOBAL Moment X RelDist 0.53931 LC2-gravity GLOBAL Force Gravity RelDist 0.22791 LC2-gravity GLOBAL Force Gravity RelDist 0.3262
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Frame LoadCase CoordSys Type Dir DistType RelDistText Text Text Text Text Text Unitless
1 LC2-gravity GLOBAL Force Gravity RelDist 0.35901 LC2-gravity GLOBAL Force Gravity RelDist 0.53931 LC3-gravity GLOBAL Force Gravity RelDist 0.22791 LC3-gravity GLOBAL Force Gravity RelDist 0.32621 LC3-gravity GLOBAL Force Gravity RelDist 0.35901 LC3-gravity GLOBAL Force Gravity RelDist 0.53931 LC3-gravity GLOBAL Force Gravity RelDist 0.71971 LC2-gravity GLOBAL Force Gravity RelDist 0.71971 LC2-torsion GLOBAL Moment X RelDist 0.71971 LC3-torsion GLOBAL Moment X RelDist 0.71971 LC3-torsion GLOBAL Moment X RelDist 0.22791 LC3-torsion GLOBAL Moment X RelDist 0.32621 LC3-torsion GLOBAL Moment X RelDist 0.35901 LC3-torsion GLOBAL Moment X RelDist 0.53931 LC2-torsion GLOBAL Moment X RelDist 0.22791 LC2-torsion GLOBAL Moment X RelDist 0.32621 LC2-torsion GLOBAL Moment X RelDist 0.35901 LC2-torsion GLOBAL Moment X RelDist 0.53934 LC2-gravity GLOBAL Force Gravity RelDist 0.22794 LC2-gravity GLOBAL Force Gravity RelDist 0.32624 LC2-gravity GLOBAL Force Gravity RelDist 0.35904 LC2-gravity GLOBAL Force Gravity RelDist 0.53934 LC3-gravity GLOBAL Force Gravity RelDist 0.22794 LC3-gravity GLOBAL Force Gravity RelDist 0.32624 LC3-gravity GLOBAL Force Gravity RelDist 0.35904 LC3-gravity GLOBAL Force Gravity RelDist 0.53934 LC3-gravity GLOBAL Force Gravity RelDist 0.71974 LC2-gravity GLOBAL Force Gravity RelDist 0.71974 LC2-torsion GLOBAL Moment X RelDist 0.71974 LC3-torsion GLOBAL Moment X RelDist 0.71974 LC2-torsion GLOBAL Moment X RelDist 0.22794 LC2-torsion GLOBAL Moment X RelDist 0.32624 LC2-torsion GLOBAL Moment X RelDist 0.35904 LC2-torsion GLOBAL Moment X RelDist 0.53934 LC3-torsion GLOBAL Moment X RelDist 0.22794 LC3-torsion GLOBAL Moment X RelDist 0.32624 LC3-torsion GLOBAL Moment X RelDist 0.35904 LC3-torsion GLOBAL Moment X RelDist 0.53939 LC3-gravity GLOBAL Force Gravity RelDist 0.22799 LC3-gravity GLOBAL Force Gravity RelDist 0.32629 LC3-gravity GLOBAL Force Gravity RelDist 0.35909 LC3-gravity GLOBAL Force Gravity RelDist 0.53939 LC3-gravity GLOBAL Force Gravity RelDist 0.71979 LC3-torsion GLOBAL Moment X RelDist 0.71979 LC3-torsion GLOBAL Moment X RelDist 0.22799 LC3-torsion GLOBAL Moment X RelDist 0.32629 LC3-torsion GLOBAL Moment X RelDist 0.35909 LC3-torsion GLOBAL Moment X RelDist 0.53936 LC2-gravity GLOBAL Force Gravity RelDist 0.22796 LC2-gravity GLOBAL Force Gravity RelDist 0.32626 LC2-gravity GLOBAL Force Gravity RelDist 0.35906 LC2-gravity GLOBAL Force Gravity RelDist 0.53936 LC3-gravity GLOBAL Force Gravity RelDist 0.22796 LC3-gravity GLOBAL Force Gravity RelDist 0.32626 LC3-gravity GLOBAL Force Gravity RelDist 0.35906 LC3-gravity GLOBAL Force Gravity RelDist 0.53936 LC3-gravity GLOBAL Force Gravity RelDist 0.71976 LC2-gravity GLOBAL Force Gravity RelDist 0.71976 LC2-torsion GLOBAL Moment X RelDist 0.71976 LC3-torsion GLOBAL Moment X RelDist 0.71976 LC3-torsion GLOBAL Moment X RelDist 0.22796 LC3-torsion GLOBAL Moment X RelDist 0.3262
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Frame LoadCase CoordSys Type Dir DistType RelDistText Text Text Text Text Text Unitless
6 LC3-torsion GLOBAL Moment X RelDist 0.35906 LC3-torsion GLOBAL Moment X RelDist 0.53936 LC2-torsion GLOBAL Moment X RelDist 0.22796 LC2-torsion GLOBAL Moment X RelDist 0.32626 LC2-torsion GLOBAL Moment X RelDist 0.35906 LC2-torsion GLOBAL Moment X RelDist 0.5393
Table: Frame Loads - Point, Part 2 of 2
Frame LoadCase AbsDist Force MomentText Text m KN KN-m
Table: Frame Section Properties 01 - General, Part 4 of 6
SectionName ConcCol ConcBeam Color TotalWt TotalMass FromFile AModText Yes/No Yes/No Text KN KN-s2/m Yes/No Unitless
NLONGR No No Blue 1916.045 195.22 No 1.000000NLONGT No Yes Blue 2020.146 206.00 No 1.000000
NTRANSD No No Magenta 746.203 76.09 No 1.000000NTRANSE No Yes White 810.301 82.63 No 1.000000NTRANSR No No Blue 2438.467 248.45 No 1.000000
Table: Grid Lines, Part 1 of 2
CoordSys AxisDir GridID XRYZCoord LineType LineColor Visible BubbleLocText Text Text m Text Text Yes/No Text
GLOBAL X x1 0.00000 Primary Gray8Dark Yes End GLOBAL X x2 1.52500 Primary Gray8Dark Yes End GLOBAL X x3 3.05000 Primary Gray8Dark Yes End GLOBAL X x4 4.57500 Primary Gray8Dark Yes End GLOBAL X x5 6.10000 Primary Gray8Dark Yes End GLOBAL X x6 7.62500 Primary Gray8Dark Yes End GLOBAL X x7 9.15000 Primary Gray8Dark Yes End GLOBAL X x8 10.67500 Primary Gray8Dark Yes End GLOBAL X x9 12.20000 Primary Gray8Dark Yes End GLOBAL X x10 13.72500 Primary Gray8Dark Yes End GLOBAL X x11 15.25000 Primary Gray8Dark Yes End GLOBAL X x12 16.77500 Primary Gray8Dark Yes End GLOBAL X x13 18.30000 Primary Gray8Dark Yes End GLOBAL X x14 19.82500 Primary Gray8Dark Yes End GLOBAL X x15 21.35000 Primary Gray8Dark Yes End GLOBAL X x16 22.87500 Primary Gray8Dark Yes End GLOBAL X x17 24.40000 Primary Gray8Dark Yes End GLOBAL X x18 25.92500 Primary Gray8Dark Yes End GLOBAL X x19 27.45000 Primary Gray8Dark Yes End GLOBAL X x20 28.97500 Primary Gray8Dark Yes End GLOBAL X x21 30.50000 Primary Gray8Dark Yes End GLOBAL X x22 32.02500 Primary Gray8Dark Yes End GLOBAL X x23 33.55000 Primary Gray8Dark Yes End GLOBAL X x24 35.07500 Primary Gray8Dark Yes End GLOBAL X x25 36.60000 Primary Gray8Dark Yes End GLOBAL Y y1 -1.96875 Primary Gray8Dark Yes End GLOBAL Y y2 0.00000 Primary Gray8Dark Yes End GLOBAL Y y3 1.97500 Primary Gray8Dark Yes End GLOBAL Y y4 3.95000 Primary Gray8Dark Yes End GLOBAL Y y5 5.92500 Primary Gray8Dark Yes End GLOBAL Y y6 7.90000 Primary Gray8Dark Yes End GLOBAL Y y7 9.86875 Primary Gray8Dark Yes End GLOBAL Z z1 0.00000 Primary Gray8Dark Yes End
Table: Load Case Definitions
LoadCase DesignType SelfWtMult AutoLoad Text Text Unitless Text
LC2-gravity LIVE 0.000000 LC2-torsion LIVE 0.000000 LC3-gravity LIVE 0.000000 LC3-torsion LIVE 0.000000
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Table: Program Control, Part 1 of 2
ProgramName
Version ProgLevel LicenseOS LicenseSC LicenseBR LicenseHT CurrUnits
Text Text Text Yes/No Yes/No Yes/No Yes/No Text SAP2000 10.0.1 Advanced Yes Yes Yes No KN, m, C