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D-1
APPENDIX D
DESIGN EXAMPLES
CONTENTS
D-2 INTRODUCTION TO DESIGN EXAMPLES
D-4 DESIGN EXAMPLE 1: AASHTO TYPE III GIRDER
1. Introduction, D-4
2. Description of Bridge, D-5
3. Design Assumptions and Initial Computations, D-6
4. Analysis and Design of Girders for Continuity, D-12
5. Reinforcement for Positive Moments at Interior Supports, D-24
6. Reinforcement for Negative Moments at Interior Supports, D-35
7. Design with Nonlinear Analysis, D-39
D-42 DESIGN EXAMPLE 2: PCI BT-72 GIRDER
1. Introduction, D-42
2. Description of Bridge, D-42
3. Design Parameters, D-42
4. Reinforcement for Positive Moments at Interior Supports, D-44
D-48 DESIGN EXAMPLE 3: 51-IN. DEEP BOX GIRDER (SPREAD)
1. Introduction, D-48
2. Description of Bridge, D-48
3. Design Parameters, D-48
4. Reinforcement for Positive Moments at Interior Supports, D-50
D-54 DESIGN EXAMPLE 4: AASHTO BIII-48 BOX GIRDER (ADJACENT)
1. Introduction, D-54
2. Description of Bridge, D-55
3. Design Assumptions and Initial Computations, D-56
4. Analysis and Design of Girders for Continuity, D-60
5. Reinforcement for Positive Moments at Interior Supports, D-71
6. Reinforcement for Negative Moments at Interior Supports, D-79
D-84 REFERENCES FOR APPENDIX D
D-85 SUBAPPENDIX A: INPUT DATA FOR RESTRAINT
D-95 SUBAPPENDIX B: INPUT AND OUTPUT FROM RESPONSE 2000
D-105 SUBAPPENDIX C: INPUT AND OUTPUT FROM QCONBRIDGE
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D-2
INTRODUCTION TO DESIGN EXAMPLES
The following design examples demonstrate the design OTHER FEATURES OF DESIGN EXAMPLES
of precast girder bridges made continuous. The design con-
forms to the AASHTO LRFD Bridge Design Specifications Other basic features of the design examples are summa-
and the proposed design specifications developed as part of rized as follows. While the bridges in all design examples
this research project (see Appendix C). have two equal spans, the design is similar with additional
spans, unequal spans, or both. For bridges with more than
two spans, live load will cause positive moments at the inte-
rior supports, which do not develop in these examples.
LIST OF DESIGN EXAMPLES The precast/prestressed concrete girders are made con-
tinuous by the placement of a continuity diaphragm at the
All design examples are for bridges with two equal spans. interior support, which fills the gap between ends of gird-
The different girder and bridge types considered in the design ers from adjacent spans. For all examples with a compos-
examples are listed below, with a brief description of distin- ite deck, the continuity diaphragm is placed with the deck
guishing features for each: concrete. Therefore, the bridge is considered to be contin-
uous for all loads applied after the continuity diaphragm is
· Design Example 1: AASHTO Type III Girder--This in place.
is a detailed design example for a bridge with relatively Only those details of design that are affected by the use
small conventional prestressed concrete girders and a of continuity are presented in the design examples. The
composite concrete deck. To demonstrate the signifi- focus of the examples is on flexural design, which is most
cant effect of girder age when continuity is established, significantly affected by the consideration of restraint
designs are performed assuming continuity is established moments. Design shears, reactions, and deflections are also
at girder ages of 28 days, 60 days, and 90 days. affected by continuity, but the procedures for design are not
· Design Example 2: PCI BT-72 Girder--This is a brief altered.
design example for a bridge with deeper prestressed con- Continuous bridges may be subject to restraint moments
crete girders and a composite concrete deck. Only signif- caused by the time-dependent effects of creep and shrink-
icant differences from Design Example 1 (DE1) are pre- age. Two approaches have been proposed for considering
sented. The design assumes that continuity is established these effects in design of precast concrete bridges made con-
at a girder age of at least 90 days. Therefore, according to tinuous: the general and simplified. For the complete design
the simplified approach in the proposed specifications, examples (DE1 and DE4), both approaches are considered.
restraint moments are not computed. For the brief design examples, the simplified approach is
· Design Example 3: 51-in. Deep Box Girder (Spread)-- employed, which does not require the evaluation of restraint
This is a brief design example for a bridge with deep moments.
box girders and a composite concrete deck. The girders Where required, the restraint moments due to creep and
are spaced apart for greater efficiency. Only significant shrinkage are computed using the RESTRAINT spreadsheet
differences from DE1 are presented. The design assumes developed in this research project. Design moments from
that continuity is established at a girder age of at least sources such as temperature gradient or support settlement
90 days. Therefore, according to the simplified approach may also be considered as required by the owner. Moments
in the proposed specifications, restraint moments are not from these sources would be combined with the effects con-
computed. sidered in the design examples and compared with the same
· Design Example 4: AASHTO BIII-48 Box Girder design criteria.
(Adjacent)--This is a detailed design example for a For the detailed design examples--DE1 and DE4--a
bridge with box girders placed adjacent to each other. No simple-span design is performed to compare with the two-
composite deck is used. An asphalt wearing surface and spans made continuous design.
membrane is placed on the box girders to achieve the Both mild reinforcement and pretensioning strands are used
desired cross slope. To demonstrate the significant effect to provide the positive moment connection between the pre-
of girder age when continuity is established, designs are cast girder and the continuity diaphragm.
performed assuming continuity is established at girder Reinforcement in the composite concrete deck is propor-
ages of 7 days, 28 days, and 90 days. tioned to resist negative design moments for design examples
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D-3
DE1, DE2, and DE3. For the final design example--DE4-- girder spacing and span length are fixed for each design exam-
negative moments are resisted by a connection between the ple. The examples consider only interior girders.
tops of the box girders. Each example provides reinforcement details for the con-
The design examples represent typical bridges for the cross nections at the continuity diaphragm. Constructability is con-
sections considered. The bridge typical section for DE1 and sidered in developing the details.
DE2 are the same. The bridge typical section for DE3 is wider, Typical design loads are used in the designs. Conventional
while the bridge typical section for DE4 is narrower. The materials are used for all designs.
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D-4
DESIGN EXAMPLE 1: AASHTO TYPE III GIRDER
1 INTRODUCTION Negative moments at the interior pier are caused by dead
loads applied to the composite continuous structure, live loads,
This design example demonstrates the design of a typical and restraint moments. However, negative restraint moments
continuous two-span bridge using the specifications pro- are neglected in the design as allowed by the proposed spec-
posed as part of this research. The precast/prestressed con- ifications. In this example, negative moments are resisted by
crete girders are made continuous by the placement of a con- mild reinforcement added to the deck slab, which is the most
tinuity diaphragm at the interior support, which fills the gap common approach to providing a negative moment connec-
between ends of girders from adjacent spans. For this exam- tion. The reinforcement in the negative moment connection
ple, the continuity diaphragm is placed with the deck, so the is proportioned using strength design methods.
bridge becomes continuous for loads placed on the structure
after the deck and continuity diaphragm are in place.
Once made continuous, the bridge is subject to restraint
moments that may develop from the time-dependent effects 1.1 Age of Girders at Continuity
of creep and shrinkage. Restraint moments are caused by
To demonstrate the significant effect of girder age when
restrained deformations in the bridge. Analysis indicates that
continuity is established, designs will be performed assum-
the restraint moments vary linearly between supports. For this
ing that continuity is established at the following girder ages:
two-span bridge, the restraint moments reach maximum val-
ues at the center of the interior pier. Reinforcement is provided
at the interior pier to resist moments caused by time-dependent · 28 days,
effects and applied loads. Restraint moments also affect the · 60 days, and
moments within the spans. Therefore, girder designs must · 90 days.
be adjusted to account for the additional positive moments
caused by restraint.
If contract documents specify the minimum girder age at con-
Variations in temperature also cause restraint moments in
tinuity, the minimum age is known. If the minimum girder age
continuous bridges. However, this condition will not be con-
at continuity is 90 days, the proposed specifications allow the
sidered in this example. If moments from temperature effects
designer to neglect the effect of restraint moments. This is
were included, girder designs would have to be adjusted in the
referred to as the "simplified approach." If the minimum girder
same way as they are for restraint moments in this example.
age at continuity is not specified, the designer must use the
Only those details of design that are affected by the use of
"general approach," which considers the effect of restraint
continuity are presented in this design example. Therefore,
moments. See Section 4 for a discussion of the two approaches.
the focus of this example will be on flexural design, which is
Since positive restraint moments have the most significant
most significantly affected by the consideration of restraint
effect on designs, assuming an early age at continuity will
moments. While design shears, reactions, and deflections are
result in higher positive restraint moments. Two early ages for
also affected when compared with design for simple-span
continuity (less than 90 days) are considered in this example
bridges, the procedures for design are not altered; therefore,
to provide information for the designer to make decisions
design for these quantities will not be presented.
regarding whether to set a minimum girder age at continuity
In a two-span bridge with simple-span girders made con-
and what that age would be.
tinuous, positive restraint moments may develop at the inte-
rior support. Positive moments do not develop from live loads
for a two-span bridge, and the effect of support settlement is
not considered. The positive design moments at the interior 1.2 Design Programs Used
support are resisted by mild reinforcement or pretensioning
strands that extend into the continuity diaphragm from the Most of the design calculations were performed using a
bottom flange of the girder. This positive moment connection commercially available computer program. This was supple-
is proportioned using strength design methods to resist any mented by hand and by spreadsheet computations to obtain
restraint moments that may develop or to provide a minimum the quantities needed for this design. Restraint moments were
quantity of reinforcement. The positive moment connection estimated using the Restraint Program. Fatigue design loads
is also provided to enhance the structural integrity of the were computed using the QConBridge Program, which is
bridge. Construction details for the positive moment connec- available at no cost from the Washington State DOT website
tion are discussed in this example. (see Subappendix C). Moment-curvature relationships for use
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D-5
in the nonlinear analysis portion of Restraint were obtained The connection is made when the deck slab is cast. The gird-
using the Response 2000 Program, which is available at no ers are therefore considered continuous for all loads applied
cost from a website at the University of Toronto (see Sub- to the composite section.
appendix B). The distance between centers of bearings (85.00 ft) is
used for computing effects of loads placed on the simple-
span girders before continuity is established. After conti-
nuity, the design span for the continuous girders is assumed
2 DESCRIPTION OF BRIDGE to be from the center of bearing at the expansion end of the
girder to the center of the interior pier, or 86.00 ft. See Fig-
The bridge is a typical two-span structure with AASHTO ure D-2-2. The space required between ends of girders to
Type III girders and a composite deck slab. The span length accommodate the positive reinforcement connection should
for this bridge is approaching the maximum achievable for be considered when laying out the bridge (see Section 5.3).
this girder and spacing. The geometry of the bridge is shown The following design example demonstrates the design of
in Figures D-2-1 through D-2-3. an interior girder. Design of an exterior girder would be sim-
The girders are made continuous by a continuity diaphragm ilar except for loads. For this bridge, the interior girder design
that connects the ends of the girders at the interior support. governs.
Figure D-2-1. Plan view of bridge.
Figure D-2-2. Longitudinal section view of bridge.
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D-6
Figure D-2-3. Typical section of bridge.
3 DESIGN ASSUMPTIONS AND INITIAL See Sections 3.3 and 3.6 for values used in computing
COMPUTATIONS these factors. The factors are different for the indicated
girder ages when continuity is established because the
3.1 Specifications designs require different values for fc.
· Girder self weight: The unit weight of girder concrete
AASHTO LRFD Bridge Design Specifications, 2nd Edi-
is 0.150 kcf.
tion with Interims through 2002 is the primary publication to
= 0.583 klf
which this appendix will refer (American Assoc. of State
· Deck slab (structural): The structural thickness of the
Highway and Transportation Officials, 1998). References to
deck is 7 3/4 in. (see Section 3.5).
articles, equations, and tables in the AASHTO LRFD Spec-
= 0.097 ksf on the tributary area for girders
ifications will be preceded by the prefix "LRFD" to differen-
= 0.751 klf for an interior girder (noncomposite
tiate them from other references in this design example.
section)
Proposed revisions have been developed as part of this
· Weight of additional deck thickness: The additional
research project (see Subappendix C). References to articles
deck slab thickness is 1/4 in. (see Section 3.5).
and equations in the proposed specifications will be preceded
= 0.003 ksf on the tributary area for girders
by the prefix "proposed" to differentiate them from refer-
= 0.024 klf for an interior girder (noncomposite
ences to items in the AASHTO LRFD Specifications.
section)
· 2 1/2 in. build-up: The full build-up thickness of 21/2 in.
3.2 Loads is used for dead load computations (see Section 3.5).
= 0.042 klf for all girders (noncomposite section)
Loads are as follows. · Stay-in-place (SIP) deck forms:
= 0.016 ksf on formed area between girders
· Live load: HL-93 with 33% dynamic allowance (IM) = 0.103 klf for an interior girder (noncomposite
on the design truck. Live-load distribution factors are section)
computed using equations in LRFD Table 4.6.2.2.2b-1 · Parapet load:
for section type (k) (see LRFD Table 4.6.2.2.1-1): = 0.371 klf per parapet, or 0.742 klf for both parapets
= 0.148 klf for each girder (composite section)
· Future wearing surface:
Distribution of Live Load Moment in Interior Beams
One Design Lane Loaded 0.465 lanes / girder = 0.025 ksf on roadway width
28 Days
Two or More Design Lanes 0.654 lanes / girder
= 0.170 klf for each girder (composite section)
· Dead load: dead loads placed on the composite girder
60 & 90 One Design Lane Loaded 0.461 lanes / girder
Days are distributed equally to all girders in the cross section
Two or More Design Lanes 0.648 lanes / girder (LRFD Article 4.6.2.2.1).
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D-7
3.3 Materials and Material Properties computed, or they may be obtained from a table of section
properties:
Material properties used for design are given below.
A = 559.5 in2, and
3.3.1 Girder Concrete p = 137.9 in.
3.3.1.1 Basic Properties. Girder concrete strengths are dif-
LRFD Article 5.4.2.3.2 suggests that only the surface area
ferent for the indicated girder ages at continuity because of
design requirements. Initial design was performed using prop- exposed to atmospheric drying should be included in the
erties shown for a girder age at continuity of 90 days (see computation of the V/S ratio. For the girder, the only sur-
Table D-3.3.1.1-1). face that is not exposed to drying, for the life of the mem-
ber, is the top surface of the top flange, which will be in
3.3.1.2 Time-Dependent Properties. Time-dependent con- contact with the composite deck slab in the completed
crete properties (creep and shrinkage) are needed only if structure. However, the girder will be entirely exposed
restraint moments are being included in the analysis and prior to placement of the deck slab concrete. Furthermore,
design. Therefore, the following computations are not required the width of the contact area is relatively small compared
if the simplified approach is being used (see Section 4). Mea- with the total girder perimeter. Therefore, the suggestion of
sured values of the ultimate creep coefficient and the ultimate LRFD Article 5.4.2.3.2 will be disregarded for this girder.
shrinkage strain for the concrete should be used if possible. It appears appropriate to neglect the reduction for the con-
However, measured creep and shrinkage properties are rarely tact area between girder and deck in most cases, especially
available; these quantities are usually estimated. For this design where top flanges are wide and thin. The reduction may be
example, the equations in LRFD Article 5.4.2.3 are used to appropriate if the deck will be cast at an early girder age or
estimate creep and shrinkage. See the AASHTO LRFD Spec- if the section is stocky, such as a box girder. Please note that
ifications for secondary equations and complete definitions for box girders, a fraction of the perimeter of the interior
of the terms used in the calculations that follow. Restraint void is included in the perimeter calculation. See DE4 for
moments are very sensitive to variations in creep and shrink- discussion.
age values, so the best possible estimates should be used. The V/S ratio is
Other methods for estimating creep and shrinkage properties
may be used as permitted by LRFD Article 5.4.2.3.1. V/S = A/p = 559.5/137.9 = 4.057 in.
3.3.1.2.1 Volumetosurface area ratio. Both creep and The Commentary (LRFD Article C5.4.2.3.2) indicates that
shrinkage equations are dependent upon the volumeto the maximum value of the V/S ratio considered in the devel-
surface area (V/S) ratio. Since the equations are sensitive to opment of the equations for the creep and shrinkage factors
this quantity and the analysis for restraint moments is sensi- in which V/S appears was 6.0 in. This value should be con-
tive to creep and shrinkage values, it is important to carefully sidered a practical upper limit for the ratio when using the
consider the computation of this ratio. equations in the Specifications.
The V/S ratio is generally computed using the equivalent
ratio of the cross-sectional area to the perimeter. This quan- 3.3.1.2.2 Ultimate creep coefficient. The creep coefficient
tity can be easily computed for most sections. For the stan- may be taken as follows:
dard AASHTO Type III girder, the area and perimeter can be
(t , t i )
( )
H -0.118 (t - t i ) LRFD Eq. 5.4.2.3.2-1
0.6
TABLE D-3.3.1.1-1 Girder concrete properties = 3.5k c k f 1.58 - ti 0.6.
120 10.0 + (t - t i )
Girder Age at Continuity
28 Days 60 Days 90 Days Significant load is placed on the girder at release. Therefore,
'
f ci (ksi) 7.50 6.00 5.50 ti , the age of concrete when load is initially applied, is taken
f 'c (ksi) 8.50 7.00 7.00 to be the age of the girder at release, or typically at 1 day.
fr (ksi) 0.700 0.635 0.635 To determine the ultimate value for the creep coefficient,
Eci (ksi) 5,520 4,696 4,496 u , where t = , the final term in the equation is assumed to
Ec (ksi) 5,589 5,072 5,072 approach unity:
wc (kcf) 0.150 0.150 0.150
Note: Eci and Ec are computed using LRFD Eq. 5.4.2.4-1. u = (, 1 day) = 1.80 (for continuity at 60 and 90 days)
fr (modulus of rupture) is computed using LRFD Art. 5.4.2.6. = 1.62 (for continuity at 28 days),
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D-8
where 3.3.2.2 Time-Dependent Properties. See Section 3.3.1.2.1
for discussion.
kc = factor for V/S ratio, LRFD Eq. C5.4.2.3.2-1
= 0.781,
3.3.2.2.1 V/S ratio. As discussed in Section 3.3.1.2.1,
kf = factor for the effect of LRFD Eq. 5.4.2.3.2-2
concrete strength, determination of the V/S ratio should be carefully considered.
= 0.691 (using f c = 7.00 ksi, The V/S ratio for the composite deck is computed using the
for continuity at 60 and equivalent ratio of the cross-sectional area to the perimeter.
90 days), For a composite deck slab, the area is computed as the
= 0.619 (using f c
= 8.50 ksi, product of the full depth of the deck and the width of the deck
for continuity at 28 days), extending to the center of the bay between girders or to the
H = relative humidity, exterior edge of the deck. The area of the build-up could also
= 75% (assumed), and be included in the deck area. However, such a refinement of
V/S = 4.057 in. (used to determine kc). the computation is not generally justified, since the calcula-
tion is not precise. Therefore, the area of the deck for the inte-
3.3.1.2.3 Ultimate shrinkage strain. While it is not always rior girder being considered will be the product of the girder
known whether the girder will be steam cured during fabri- spacing, S, and the total deck thickness, hf :
cation, the initial strength gain is generally accelerated when
compared with "normal" concretes. Therefore, it is reason- A = Shf = 7.75 ft × 8 in. = 7.75(12)(8) = 744 in.2.
able to use the shrinkage equation for steam-cured concrete.
The shrinkage strain may therefore be taken as The perimeter of the deck slab used to compute the V/S ratio
is subject to some refinement based on the recommendation
sh = - ks kh (55.0t + t)0.56 × 10 -3
. LRFD Eq. 5.4.2.3.3-2
of LRFD Article 5.4.2.3.2, which indicates that only the sur-
face area exposed to atmospheric drying should be included
in the computation of the V/S ratio. Since the area of the deck
To determine the ultimate shrinkage strain, shu , where t = , that is in contact with the girder will never be exposed to dry-
the term in the equation that contains it is assumed to ing, it may be eliminated from the computed perimeter. For
approach unity: simplicity and since the top flange of the Type III girder is
relatively narrow, this correction will not be taken. It may be
shu = sh () = -395 × 10-6 in./in. appropriate to use the correction for the contact area where
the contact area with the girder is wide, such as a bulb-T or
where box girder. For interior girders, the deck thickness is not con-
sidered in computing the perimeter because it is an imaginary
ks = size factor = 0.760, and LRFD Eq. C5.4.2.3.3-1
boundary not exposed to drying. Therefore, for the interior
kh = humidity factor = 0.929. LRFD Eq. C5.4.2.3.3-2
girder being designed, the perimeter is taken as twice the
girder spacing, S:
3.3.2 Deck and Continuity Diaphragm Concrete
p = 2S = 2(7.75)(12) = 186.0 in.
The same concrete properties are used for the deck slab
and continuity diaphragm because they are placed at the The V/S ratio is
same time in this example. The subscript d is used to indicate
properties related to the deck slab or diaphragm concrete. V/S = A/p = 744/186 = 4.00 in.
3.3.2.1 Basic Properties. The same deck slab concrete This calculation demonstrates that, for a uniform thickness
strength is used for all designs: deck slab with no deducted surface area, V/S is simply half
of the thickness of the deck.
d = 4.00 ksi,
fc Since stay-in-place deck forms may be used on this bridge,
the bottom of the deck slab may not be exposed to drying.
frd = 0.480 ksi, LRFD Art. 5.4.2.6
This would increase the V/S ratio to 8.00 in., which exceeds
wcd = 0.150 kcf, and the V/S limit used to develop the equations for correction fac-
tors, kc and ks The increased V/S will reduce the corrections
Ecd = 3,834 ksi. LRFD Eq. 5.4.2.4-1 factors, but not significantly. Therefore, the effect of the deck
forms is neglected.
According to proposed Article 5.14.1.2.7j, design at the con-
tinuity diaphragm will use the concrete strength of the pre- 3.3.2.2.2 Ultimate creep coefficient. The age of the deck
cast girder where noted. concrete at loading is not as well defined as it is for the girder.
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D-9
An early age of 14 days is assumed to provide a conservative 0.5-in.- or 0.6-in.-diameter low-relaxation seven-wire strand;
estimate of deck creep behavior (a larger creep coefficient). An
early age at loading is also a reasonable assumption because Aps = 0.153 in.2 (0.5-in.-diameter strand, for continuity at
some load will be transferred to the deck shortly after casting 60 and 90 days) and
because it restrains the continued downward deflection of the = 0.217 in.2 (0.6-in.-diameter strand, for continuity at
girder under the load of the deck (due to creep): 28 days);
(t , t i ) fpu = 270 ksi;
( )
H -0.118 (t - t i )
0.6
LRFD Eq. 5.4.2.3.2-1
= 3.5k c k f 1.58 - ti 0.6.
120 10.0 + (t - t i ) fpy = 0.90 fpu = 243 ksi;
As described previously, t = is used to obtain the ultimate fpj = 0.75 fpu = 202.5 ksi; and
value for the creep coefficient, u :
Ep = 28,500 ksi.
u = (, 14 days) = 1.70,
3.3.3.1 Transfer Length. At the ends of pretensioned gird-
where ers, the force in the pretensioning strands is transferred from
the strands to the girder concrete over the transfer length. The
kc = factor for V/S ratio LRFD Eq. C5.4.2.3.2-1
stress in the strands is assumed to vary linearly from zero at
= 0.775,
the end of the girder to the full effective prestress, fpe , at the
kf = factor for the effect of LRFD Eq. 5.4.2.3.2-2
transfer length. The transfer length, t , may be estimated as
concrete strength
= 0.897 (using f c
d = 4.00 ksi),
t = 60 db, LRFD Art. 5.11.4.1
H = relative humidity = 75% (assumed), and
= 60(0.5 in.)
V/S = 4.00 in. (used for kc; this is a simplified value, based
= 30 in. (for continuity at 60
on the full 8-in.deck thickness and neglecting any
effect of SIP metal forms and the area of contact and 90 days), and
with the girder). = 60(0.6 in.) = 36 in.
(for continuity at 28 days),
3.3.2.2.3 Ultimate shrinkage strain. Since deck slab con-
crete is normally moist cured, the equation for shrinkage for where
moist-cured concrete is used: db = nominal strand diameter.
sh = - ks kh (35.0t + t)0.51 × 10 -3
. LRFD Eq. 5.4.2.3.3-1 The location at a transfer length from the end of the girder is
a critical stress location at release. Therefore, moments and
stresses computed for this location are shown in various tables
To determine the ultimate shrinkage strain, shu , where t = , in this example. These locations are identified in the tables
the term in the equation that contains t is assumed to with the heading "Trans" or "Transfer." Values in tables dif-
approach unity: fer for continuity at 28 days and for continuity at 60 and 90
days because the different strand size results in a different
shu = sh () = -353 × 10-6 in./in., transfer length.
where
3.3.4 Mild Reinforcement
ks = size factor = 0.745, and LRFD Eq. C5.4.2.3.3-1
kh = humidity factor = 0.929. LRFD Eq. C5.4.2.3.3-2 Mild reinforcement is as follows:
The restraining effect of longitudinal reinforcement in the fy = 60 ksi, and
deck slab on the free shrinkage is not considered in this design
example. Proposed Article C5.14.1.2.7c states that the effect
Es = 29,000 ksi.
may be computed by proposed Equation C5.14.1.2.7c-1.
3.3.3 Prestressing Strand 3.4 Stress Limits
The material properties of the prestressing strand are as The following stress limits are used for the design of the
follows: girders for the service limit state. For computation of girder
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stresses, the sign convention will be compressive stress is prestressed. Instead, it is designed to satisfy the specified
positive (+) and tensile stress is negative (-). Signs are not requirements at the strength limit state.
shown for limits in the following, they will be applied in later
stress comparisons. Compression: LRFD Table 5.9.4.2.1-1
fc1 = 0.60 w f c
, for full-service loads (w = 1 for girders);
3.4.1 Pretensioned Strands
fc2 = 0.45 f c
, for effective prestress (PS ) and full dead loads
The stress limits for low relaxation strands are as follows: (DL); and
Immediately prior to transfer: LRFD Table 5.9.3-1 , for live load plus one-half of effective PS and
fc3 = 0.40 f c
full DL.
fpi = 0.75 fpu = 202.5 ksi.
Tension: LRFD Table 5.9.4.2.2-1
At service limit state after losses: LRFD Table 5.9.3-1
For the precompressed compression zone, ft1 = 0.19 f c ,
fp = 0.80 fpy = 199.4 ksi. assuming moderate corrosion conditions. For locations
other than the precompressed compression zone, such as at
The stress limits above are not discussed in this example the end of the girder where the top of the girder may go into
because they do not govern designs. tension under the effect of the negative live-load moment,
the LRFD Specifications give no stress limits. Therefore,
the following limits have been proposed, which take the
3.4.2 Concrete
same form as those for temporary tensile stresses at release
3.4.2.1 Temporary Stresses at Release. See Table given in LRFD Table 5.9.4.1.2-1, but with the specified
D-3.4.2.1-1. concrete compressive strength, f c, rather than the concrete
:
compressive strength at release, f ci
Compression: LRFD Art. 5.9.4.1.1
ft2 = 0.0948 f c
0.2 ksi, or
fcR = 0.60 f c
i .
ft3 = 0.24 f c with reinforcement to resist the tensile force
Tension: LRFD Table 5.9.4.1.2-1 in the concrete.
ftR1 = 0.0948 f c
i 0.2 ksi, or Numerical values for the stress limits are given in Table
D-3.4.2.2-1.
ftR2 = 0.24 f ci with reinforcement to resist the tensile force
in the concrete. 3.5 Other Design Assumptions
3.4.2.2 Final Stresses at Service Limit State after Losses. Intermediate diaphragms are not used in the design exam-
The following stress limits are given for the girder concrete. ple. Temporary steel or timber cross frames are generally
Compressive stresses may be checked for the deck slab, but required during erection to stabilize girders. However, the
never govern, so they are not included here. Tensile stresses weight of these temporary components is minor and is
in the deck slab at interior supports should not be compared neglected in these calculations.
with limits for the service limit state because the deck is not
TABLE D-3.4.2.2-1 Final stress limits after losses
TABLE D-3.4.2.1-1 Temporary stress limits at Girder Age at Continuity
release 28 Days 60 Days 90 Days
'
Girder Age at Continuity f c (ksi) 8.50 7.00 7.00
28 Days 60 Days 90 Days fc1 (ksi) 5.100 4.200 4.200
f '
(ksi) 7.50 6.00 5.50 fc2 (ksi) 3.830 3.150 3.150
ci
fcR (ksi) 4.500 3.600 3.300 fc3 (ksi) 3.400 2.800 2.800
ftR1 (ksi) 0.200 0.200 0.200 ft1 (ksi) 0.550 0.500 0.500
ftR2 (ksi) 0.660 0.590 0.560 ft2 (ksi) 0.200 0.200 0.200
Note: Values are rounded to two significant digits. ft3 (ksi) 0.700 0.630 0.630
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The top 0.25 in. of the deck is assumed to be a sacrificial Sb = 6,185 in3, and
wearing surface; the structural deck thickness is taken as
7.75 in. for design purposes. The weight of the remaining St = 5,071 in.3.
0.25 in. of the deck is included as additional load on the non-
composite girder.
For simplicity, the full thickness of the build-up is applied 3.6.2 Composite Section (Girder with Deck Slab)
to the full length of the girder for dead-load computations. In
most design situations, a value is used that is less than the 3.6.2.1 Effective Deck Width. The effective width for the
specified build-up thickness at the center of the bearings composite deck at all limit states is determined according to
because the actual thickness will vary along the length of the LRFD Article 4.6.2.6.1. The effective deck width for an inte-
girder from the maximum of 2 1/2 in. at the center of bearings. rior beam is the least of the following:
The build-up is neglected when computing composite sec-
tion properties that are used to calculate stresses for service 1. One-quarter of the span length (260 in.);
limit state design since the build-up will vary along the bridge. 2. Average spacing of adjacent girders (93 in.) GOV-
However, for computation of section properties and strength ERNS; and
calculations related to the reinforcement at the continuity 3. Twelve times the average thickness of the slab (93 in.),
diaphragm, the build-up is included. This is done because the plus the greater of
build-up is specified at the center of the bearings, so the full a. The web thickness (7 in.) or
build-up will be provided at the continuity diaphragm location. b. One-half of the width of the top flange of the girder
Potential deck cracking, if considered in the analysis of this (8 in.).
continuous bridge, could increase positive design moments.
However, potential deck cracking is neglected as allowed in In the case of this design example, the average spacing of
LRFD Article 4.5.2.2. adjacent girders controls, resulting in an effective deck width
of 93 in.
3.6 Section Properties
3.6.2.2 Transformed Effective Deck Width. The com-
3.6.1 Noncomposite Section (Girder Only) posite deck slab is transformed using the modular ratio, n,
for computing stresses at the service limit state. See Table
The section properties for a standard AASHTO Type III D-3.6.2.3-1.
girder are as follows (see Figure D-3.6.1-1):
3.6.2.3 Section Properties. The build-up is neglected when
h = 45.00 in., computing section properties because the build-up height
varies along the length of the girder, with the minimum height
A = 559.5 in.2, at or near midspan. See Section 3.5. Composite section prop-
erties vary for different girder ages at continuity because the
I = 125,390 in.4, girder concrete strength is different.
yb = 20.27 in.,
TABLE D-3.6.2.3-1 Composite section properties
yt = 24.73 in.,
Girder Age at Continuity
28 Days 60 Days 90 Days
hc (in.) 52.75 52.75 52.75
n = Ecd/Ec 0.686 0.756 0.756
beff (in.) 93.00 93.00 93.00
beff tr = n beff (in.) 63.80 70.31 70.31
Ac (in2) 1,053.9 1,104.3 1,104.3
Ic (in4) 342,585 353,928 353,928
ybc (in.) 33.69 34.38 34.38
ytc (in.) 11.31 10.62 10.62
ytcd (in.) 19.06 18.37 18.37
Sbc (in3) 10,168 10,293 10,293
Stc (in3) 30,294 33,340 33,340
Figure D-3.6.1-1. AASHTO * Stcd (in3) 26,203 25,493 25,493
*Note: Stcd = (Ic / ytcd) / n, so that fcd = M / Stcd.
Type III girder.
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Geometric Properties Strain Discontinuity in Concrete
Gross Conc. Trans (n=6.56)
Area (in2 ) 1169.1 1295.5
Concrete 93.0
9 - #5
Inertia (in4 ) 405953.4 465041.1 Types
2 layers of
y t (in)
4000 9 - #6
18.7 18.9
9 - #5
y b (in) 36.5 36.4 4 - S.5
p = 7.10 ms
55.3
St (in3 ) 7000
21693.1 24641.0 base 7.0 38 - S.5
type p = 7.10 ms
Sb (in3 ) 11110.9 12783.8
8 - #5
Crack Spacing
22.0
2 x dist + 0.1 db /
Loading (N,M,V + dN,dM,dV)
0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0
Concrete Rebar P-Steel
fc' = 7000 psi fu = 90 ksi fpu = 266 ksi All dimensions in inches
Clear cover to reinforcement = 1.75 in
a = 0.75 in fy = 60 Low Relax
End of Girder - 60 Day
ft = 308 psi (auto)
c' = 2.22 ms s = 100.0 ms p = 43.0 ms TJT 8/20/2002
Figure DB.2.2.2-1. Cross section and material information.
6000.0
5000.0
4000.0
Moment (ft-kips)
3000.0
2000.0
1000.0
0.0
-90.0 0.0 90.0 180.0 270.0 360.0 450.0 540.0
-1000.0
-2000.0
-3000.0
Curvature (rad/106 in)
Figure DB.2.2.2-2. Moment curvature analysis plot.
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D-101
Geometric Properties
Gross Conc. Trans (n=8.22)
Area (in2 ) 1312.0 1427.4
93.0
Inertia (in4 ) 433853.5 473331.8
9 - #6
9 - #5
y t (in) 17.1 16.7 9 - #5
9 - #6
y b (in) 38.1 38.5
55.3
St (in3 ) 25332.4 28342.6 7.0
Sb (in3 ) 11380.2 12278.5
8 - #5
Crack Spacing
22.0
2 x dist + 0.1 db /
Loading (N,M,V + dN,dM,dV)
0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0
Concrete Rebar
fc' = 4000 psi fu = 90 ksi All dimensions in inches
Clear cover to reinforcement = 1.69 in
a = 0.75 in fy = 60 Diaphragm - 60 Day
ft = 246 psi (auto)
c' = 1.93 ms s = 100.0 ms TJT 8/20/2002
Figure DB.2.2.3-1. Cross section and material information.
1000.0
500.0
-200.0 0.0 200.0 400.0 600.0 800.0
0.0
Moment (ft-kips)
-500.0
-1000.0
-1500.0
-2000.0
-2500.0
-3000.0
Curvature (rad/106 in)
Figure DB.2.2.3-2. Moment curvature analysis plot.
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Geometric Properties Strain Discontinuity in Concrete
Gross Conc. Trans (n=6.08)
Area (in2 ) 1094.0 1160.2
Concrete 93.0
Inertia (in4 ) 351550.9 386683.8 Types 2 layers of
y t (in)
4000 9 - #5
18.3 18.8
2 - S.6
y b (in) 34.4 33.9
p = 7.10 ms
52.8
8500
St (in3 ) 19189.6 20541.9 base 7.0
type
Sb (in3 ) 10210.6 11397.9 32 - S.6
p = 7.10 ms
Crack Spacing
22.0
2 x dist + 0.1 db /
Loading (N,M,V + dN,dM,dV)
0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0
Concrete Rebar P-Steel
fc' = 8500 psi fu = 90 ksi fpu = 266 ksi All dimensions in inches
Clear cover to reinforcement = 1.70 in
a = 0.75 in fy = 60 Low Relax
Girder Midspan - 28 Days
ft = 333 psi (auto)
c' = 2.37 ms s = 100.0 ms p = 43.0 ms TJT 8/20/2002
Figure DB.2.3.1-1. Cross section and material information.
6000.0
5000.0
4000.0
Moment (ft-kips)
3000.0
2000.0
1000.0
0.0
-800.0 -600.0 -400.0 -200.0 0.0 200.0 400.0 600.0
-1000.0
-2000.0
Curvature (rad/106 in)
Figure DB.2.3.1-2. Moment curvature analysis plot.
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Geometric Properties Strain Discontinuity in Concrete
Gross Conc. Trans (n=6.08)
Area (in2 ) 1134.0 1225.9
Concrete 93.0
Inertia (in4 ) 396800.1 452140.6 Types
2 layers of
4000
y t (in) 9 - #5
19.2 20.1
4 - S.6
y b (in) 36.0 35.1 p = 7.10 ms
55.3
8500
St (in3 ) 20659.8 22488.1 base 7.0 30 - S.6
type p = 7.10 ms
Sb (in3 ) 11008.9 12865.3
8 - #5
8 - #5
Crack Spacing
22.0
2 x dist + 0.1 db /
Loading (N,M,V + dN,dM,dV)
0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0
Concrete Rebar P-Steel
fc' = 8500 psi fu = 90 ksi fpu = 266 ksi All dimensions in inches
Clear cover to reinforcement = 1.70 in
a = 0.75 in fy = 60 Low Relax
End of Girder - 28 Days
ft = 333 psi (auto)
c' = 2.37 ms s = 100.0 ms p = 43.0 ms TJT 8/20/2002
Figure DB.2.3.2-1. Cross section and material information.
6000.0
5000.0
4000.0
Moment (ft-kips)
3000.0
2000.0
1000.0
0.0
-160.0 -80.0 0.0 80.0 160.0 240.0 320.0 400.0 480.0
-1000.0
-2000.0
-3000.0
Curvature (rad/106 in)
Figure DB.2.3.2-2. Moment curvature analysis plot.
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D-104
Geometric Properties
Gross Conc. Trans (n=8.22)
Area (in2 ) 1312.0 1388.1
93.0
Inertia (in4 ) 433853.5 481993.3
2 layers of
y t (in) 17.1 17.6 9 - #5
y b (in) 38.1 37.6
55.3
St (in3 ) 25332.4 27334.8 7.0
Sb (in3 ) 11380.2 12813.2
2 layers of
8 - #5
Crack Spacing
22.0
2 x dist + 0.1 db /
Loading (N,M,V + dN,dM,dV)
0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0
Concrete Rebar
fc' = 4000 psi fu = 90 ksi All dimensions in inches
Clear cover to reinforcement = 2.44 in
a = 0.75 in fy = 60 Diaphragm - 28 Days
ft = 246 psi (auto)
c' = 1.93 ms s = 100.0 ms TJT 8/20/2002
Figure DB.2.3.3-1. Cross section and material information.
1800.0
1500.0
1200.0
900.0
Moment (ft-kips)
600.0
300.0
-800.0 -600.0 -400.0 -200.0 0.0 200.0 400.0 600.0 800.0
0.0
-300.0
-600.0
-900.0
-1200.0
-1500.0
-1800.0
Curvature (rad/106 in)
Figure DB.2.3.3-2. Moment curvature analysis plot
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D-105
SUBAPPENDIX C: INPUT AND OUTPUT FROM QCONBRIDGE
The examples discussed within this subappendix are as DC.1 PROGRAM INFORMATION
follows:
The program QConBridge was used to compute the fatigue
load effects. While QConBridge also reports load effects for
· Design Example 1: AASHTO Type III Girder Bridge. other types and combinations of loadings, these were not used
· Design Example 2: PCI BT-72 Girder Bridge. in this study because the design for these load effects was per-
· Design Example 3: 51-IN.-Deep Spread Box Girder formed using another computer program. Therefore, these
Bridge--The design spans for the bridge in this example other results have been deleted from the output that follows.
are the same as Design Example 1; therefore, the output The program QConBridge was developed by the Washing-
from Design Example 1 (see Section DC.1) was used for ton State DOT and is available free of charge on the depart-
this example. ment website:
· Design Example 4: AASHTO BIII-48 Adjacent Box
Girder Bridge--The design spans for the bridge in this www.wsdot.wa.gov/eesc/bridge/software/index.cfm.
example are the same as Design Example 1; therefore,
the output from Design Example 1 (see Section DC.1) The version of the program used for this study is shown in
was used for this example. the figure below taken from the program.
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D-106
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D-107
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D-108
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D-109
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