Connection of Simple-Span Precast Concrete Girders for Continuity (2004)

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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

D-2 The following design examples demonstrate the design of precast girder bridges made continuous. The design con- forms to the AASHTO LRFD Bridge Design Specifications and the proposed design specifications developed as part of this research project (see Appendix C). LIST OF DESIGN EXAMPLES All design examples are for bridges with two equal spans. The different girder and bridge types considered in the design examples are listed below, with a brief description of distin- guishing features for each: • Design Example 1: AASHTO Type III Girder—This is a detailed design example for a bridge with relatively small conventional prestressed concrete girders and a composite concrete deck. To demonstrate the signifi- cant effect of girder age when continuity is established, designs are performed assuming continuity is established at girder ages of 28 days, 60 days, and 90 days. • Design Example 2: PCI BT-72 Girder—This is a brief design example for a bridge with deeper prestressed con- crete girders and a composite concrete deck. Only signif- icant differences from Design Example 1 (DE1) are pre- sented. The design assumes that continuity is established at a girder age of at least 90 days. Therefore, according to the simplified approach in the proposed specifications, restraint moments are not computed. • Design Example 3: 51-in. Deep Box Girder (Spread)— This is a brief design example for a bridge with deep box girders and a composite concrete deck. The girders are spaced apart for greater efficiency. Only significant differences from DE1 are presented. The design assumes that continuity is established at a girder age of at least 90 days. Therefore, according to the simplified approach in the proposed specifications, restraint moments are not computed. • Design Example 4: AASHTO BIII-48 Box Girder (Adjacent)—This is a detailed design example for a bridge with box girders placed adjacent to each other. No composite deck is used. An asphalt wearing surface and membrane is placed on the box girders to achieve the desired cross slope. To demonstrate the significant effect of girder age when continuity is established, designs are performed assuming continuity is established at girder ages of 7 days, 28 days, and 90 days. OTHER FEATURES OF DESIGN EXAMPLES Other basic features of the design examples are summa- rized as follows. While the bridges in all design examples 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. The precast/prestressed concrete girders are made con- tinuous by the placement of a continuity diaphragm at the interior support, which fills the gap between ends of gird- ers from adjacent spans. For all examples with a compos- ite deck, the continuity diaphragm is placed with the deck concrete. Therefore, the bridge is considered to be contin- uous for all loads applied after the continuity diaphragm is in place. Only those details of design that are affected by the use of continuity are presented in the design examples. The focus of the examples is on flexural design, which is most significantly affected by the consideration of restraint moments. Design shears, reactions, and deflections are also affected by continuity, but the procedures for design are not altered. Continuous bridges may be subject to restraint moments caused by the time-dependent effects of creep and shrink- age. Two approaches have been proposed for considering these effects in design of precast concrete bridges made con- tinuous: the general and simplified. For the complete design examples (DE1 and DE4), both approaches are considered. For the brief design examples, the simplified approach is employed, which does not require the evaluation of restraint moments. Where required, the restraint moments due to creep and shrinkage are computed using the RESTRAINT spreadsheet developed in this research project. Design moments from sources such as temperature gradient or support settlement may also be considered as required by the owner. Moments from these sources would be combined with the effects con- sidered in the design examples and compared with the same design criteria. For the detailed design examples—DE1 and DE4—a simple-span design is performed to compare with the two- spans made continuous design. Both mild reinforcement and pretensioning strands are used to provide the positive moment connection between the pre- cast girder and the continuity diaphragm. Reinforcement in the composite concrete deck is propor- tioned to resist negative design moments for design examples INTRODUCTION TO DESIGN EXAMPLES

D-3 DE1, DE2, and DE3. For the final design example—DE4— negative moments are resisted by a connection between the tops of the box girders. The design examples represent typical bridges for the cross sections considered. The bridge typical section for DE1 and DE2 are the same. The bridge typical section for DE3 is wider, while the bridge typical section for DE4 is narrower. The girder spacing and span length are fixed for each design exam- ple. The examples consider only interior girders. Each example provides reinforcement details for the con- nections at the continuity diaphragm. Constructability is con- sidered in developing the details. Typical design loads are used in the designs. Conventional materials are used for all designs.

D-4 DESIGN EXAMPLE 1: AASHTO TYPE III GIRDER 1 INTRODUCTION This design example demonstrates the design of a typical continuous two-span bridge using the specifications pro- posed as part of this research. The precast/prestressed con- crete girders are made continuous by the placement of a con- tinuity diaphragm at the interior support, which fills the gap between ends of girders from adjacent spans. For this exam- ple, the continuity diaphragm is placed with the deck, so the 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 of creep and shrinkage. Restraint moments are caused by restrained deformations in the bridge. Analysis indicates that the restraint moments vary linearly between supports. For this 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 effects and applied loads. Restraint moments also affect the moments within the spans. Therefore, girder designs must be adjusted to account for the additional positive moments caused by restraint. Variations in temperature also cause restraint moments in continuous bridges. However, this condition will not be con- sidered in this example. If moments from temperature effects were included, girder designs would have to be adjusted in the same way as they are for restraint moments in this example. Only those details of design that are affected by the use of continuity are presented in this design example. Therefore, the focus of this example will be on flexural design, which is most significantly affected by the consideration of restraint moments. While design shears, reactions, and deflections are also affected when compared with design for simple-span bridges, the procedures for design are not altered; therefore, design for these quantities will not be presented. In a two-span bridge with simple-span girders made con- 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 support are resisted by mild reinforcement or pretensioning strands that extend into the continuity diaphragm from the bottom flange of the girder. This positive moment connection is proportioned using strength design methods to resist any restraint moments that may develop or to provide a minimum quantity of reinforcement. The positive moment connection is also provided to enhance the structural integrity of the bridge. Construction details for the positive moment connec- tion are discussed in this example. Negative moments at the interior pier are caused by dead loads applied to the composite continuous structure, live loads, and restraint moments. However, negative restraint moments are neglected in the design as allowed by the proposed spec- ifications. In this example, negative moments are resisted by mild reinforcement added to the deck slab, which is the most common approach to providing a negative moment connec- tion. The reinforcement in the negative moment connection is proportioned using strength design methods. 1.1 Age of Girders at Continuity To demonstrate the significant effect of girder age when continuity is established, designs will be performed assum- ing that continuity is established at the following girder ages: • 28 days, • 60 days, and • 90 days. If contract documents specify the minimum girder age at con- tinuity, the minimum age is known. If the minimum girder age at continuity is 90 days, the proposed specifications allow the designer to neglect the effect of restraint moments. This is referred to as the “simplified approach.” If the minimum girder age at continuity is not specified, the designer must use the “general approach,” which considers the effect of restraint moments. See Section 4 for a discussion of the two approaches. Since positive restraint moments have the most significant effect on designs, assuming an early age at continuity will result in higher positive restraint moments. Two early ages for continuity (less than 90 days) are considered in this example to provide information for the designer to make decisions regarding whether to set a minimum girder age at continuity and what that age would be. 1.2 Design Programs Used Most of the design calculations were performed using a commercially available computer program. This was supple- mented by hand and by spreadsheet computations to obtain the quantities needed for this design. Restraint moments were estimated using the Restraint Program. Fatigue design loads were computed using the QConBridge Program, which is available at no cost from the Washington State DOT website (see Subappendix C). Moment-curvature relationships for use

in the nonlinear analysis portion of Restraint were obtained using the Response 2000 Program, which is available at no cost from a website at the University of Toronto (see Sub- appendix B). 2 DESCRIPTION OF BRIDGE The bridge is a typical two-span structure with AASHTO Type III girders and a composite deck slab. The span length for this bridge is approaching the maximum achievable for this girder and spacing. The geometry of the bridge is shown in Figures D-2-1 through D-2-3. The girders are made continuous by a continuity diaphragm that connects the ends of the girders at the interior support. D-5 The connection is made when the deck slab is cast. The gird- ers are therefore considered continuous for all loads applied to the composite section. 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 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- ure D-2-2. The space required between ends of girders to accommodate the positive reinforcement connection should be considered when laying out the bridge (see Section 5.3). The following design example demonstrates the design of an interior girder. Design of an exterior girder would be sim- ilar except for loads. For this bridge, the interior girder design governs. Figure D-2-1. Plan view of bridge. Figure D-2-2. Longitudinal section view of bridge.

3 DESIGN ASSUMPTIONS AND INITIAL COMPUTATIONS 3.1 Specifications AASHTO LRFD Bridge Design Specifications, 2nd Edi- tion with Interims through 2002 is the primary publication to which this appendix will refer (American Assoc. of State Highway and Transportation Officials, 1998). References to articles, equations, and tables in the AASHTO LRFD Spec- ifications will be preceded by the prefix “LRFD” to differen- tiate them from other references in this design example. Proposed revisions have been developed as part of this research project (see Subappendix C). References to articles and equations in the proposed specifications will be preceded by the prefix “proposed” to differentiate them from refer- ences to items in the AASHTO LRFD Specifications. 3.2 Loads Loads are as follows. • Live load: HL-93 with 33% dynamic allowance (IM) on the design truck. Live-load distribution factors are computed using equations in LRFD Table 4.6.2.2.2b-1 for section type (k) (see LRFD Table 4.6.2.2.1-1): D-6 See Sections 3.3 and 3.6 for values used in computing these factors. The factors are different for the indicated girder ages when continuity is established because the designs require different values for f ′c . • Girder self weight: The unit weight of girder concrete is 0.150 kcf. = 0.583 klf • Deck slab (structural): The structural thickness of the deck is 73/4 in. (see Section 3.5). = 0.097 ksf on the tributary area for girders = 0.751 klf for an interior girder (noncomposite section) • Weight of additional deck thickness: The additional deck slab thickness is 1/4 in. (see Section 3.5). = 0.003 ksf on the tributary area for girders = 0.024 klf for an interior girder (noncomposite section) • 21/2 in. build-up: The full build-up thickness of 21/2 in. is used for dead load computations (see Section 3.5). = 0.042 klf for all girders (noncomposite section) • Stay-in-place (SIP) deck forms: = 0.016 ksf on formed area between girders = 0.103 klf for an interior girder (noncomposite section) • Parapet load: = 0.371 klf per parapet, or 0.742 klf for both parapets = 0.148 klf for each girder (composite section) • Future wearing surface: = 0.025 ksf on roadway width = 0.170 klf for each girder (composite section) • Dead load: dead loads placed on the composite girder are distributed equally to all girders in the cross section (LRFD Article 4.6.2.2.1). Figure D-2-3. Typical section of bridge. Distribution of Live Load Moment in Interior Beams One Design Lane Loaded 0.465 lanes / girder 28 Days Two or More Design Lanes 0.654 lanes / girder One Design Lane Loaded 0.461 lanes / girder 60 & 90 Days Two or More Design Lanes 0.648 lanes / girder

3.3 Materials and Material Properties Material properties used for design are given below. 3.3.1 Girder Concrete 3.3.1.1 Basic Properties. Girder concrete strengths are dif- ferent for the indicated girder ages at continuity because of design requirements. Initial design was performed using prop- erties shown for a girder age at continuity of 90 days (see Table D-3.3.1.1-1). 3.3.1.2 Time-Dependent Properties. Time-dependent con- crete properties (creep and shrinkage) are needed only if restraint moments are being included in the analysis and design. Therefore, the following computations are not required if the simplified approach is being used (see Section 4). Mea- sured values of the ultimate creep coefficient and the ultimate shrinkage strain for the concrete should be used if possible. However, measured creep and shrinkage properties are rarely available; these quantities are usually estimated. For this design example, the equations in LRFD Article 5.4.2.3 are used to estimate creep and shrinkage. See the AASHTO LRFD Spec- ifications for secondary equations and complete definitions of the terms used in the calculations that follow. Restraint moments are very sensitive to variations in creep and shrink- age values, so the best possible estimates should be used. Other methods for estimating creep and shrinkage properties may be used as permitted by LRFD Article 5.4.2.3.1. 3.3.1.2.1 Volume–to–surface area ratio. Both creep and shrinkage equations are dependent upon the volume–to– surface area (V/S) ratio. Since the equations are sensitive to this quantity and the analysis for restraint moments is sensi- tive to creep and shrinkage values, it is important to carefully consider the computation of this ratio. The V/S ratio is generally computed using the equivalent ratio of the cross-sectional area to the perimeter. This quan- tity can be easily computed for most sections. For the stan- dard AASHTO Type III girder, the area and perimeter can be D-7 computed, or they may be obtained from a table of section properties: A = 559.5 in2, and p = 137.9 in. LRFD Article 5.4.2.3.2 suggests that only the surface area exposed to atmospheric drying should be included in the computation of the V/S ratio. For the girder, the only sur- 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 contact with the composite deck slab in the completed structure. However, the girder will be entirely exposed prior to placement of the deck slab concrete. Furthermore, the width of the contact area is relatively small compared with the total girder perimeter. Therefore, the suggestion of LRFD Article 5.4.2.3.2 will be disregarded for this girder. It appears appropriate to neglect the reduction for the con- tact area between girder and deck in most cases, especially where top flanges are wide and thin. The reduction may be appropriate if the deck will be cast at an early girder age or if the section is stocky, such as a box girder. Please note that for box girders, a fraction of the perimeter of the interior void is included in the perimeter calculation. See DE4 for discussion. The V/S ratio is V/S = A/p = 559.5/137.9 = 4.057 in. The Commentary (LRFD Article C5.4.2.3.2) indicates that the maximum value of the V/S ratio considered in the devel- opment of the equations for the creep and shrinkage factors in which V/S appears was 6.0 in. This value should be con- sidered a practical upper limit for the ratio when using the equations in the Specifications. 3.3.1.2.2 Ultimate creep coefficient. The creep coefficient may be taken as follows: LRFD Eq. 5.4.2.3.2-1 Significant load is placed on the girder at release. Therefore, ti , the age of concrete when load is initially applied, is taken to be the age of the girder at release, or typically at 1 day. To determine the ultimate value for the creep coefficient, ψu, where t = ∞, the final term in the equation is assumed to approach unity: ψu = ψ (∞, 1 day) = 1.80 (for continuity at 60 and 90 days) = 1.62 (for continuity at 28 days), ψ t t k k H t t t t t i c f i i i , . . . . . . . ( ) ( ) ( )( )= − −+ −−3 5 1 58 120 10 0 0 60 118 0 6 Girder Age at Continuity 28 Days 60 Days 90 Days f 'ci (ksi) 7.50 6.00 5.50 f 'c (ksi) 8.50 7.00 7.00 fr (ksi) 0.700 0.635 0.635 Eci (ksi) 5,520 4,696 4,496 Ec (ksi) 5,589 5,072 5,072 wc (kcf) 0.150 0.150 0.150 Note: Eci and Ec are computed using LRFD Eq. 5.4.2.4-1. fr (modulus of rupture) is computed using LRFD Art. 5.4.2.6. TABLE D-3.3.1.1-1 Girder concrete properties

where kc = factor for V/S ratio, LRFD Eq. C5.4.2.3.2-1 = 0.781, kf = factor for the effect of LRFD Eq. 5.4.2.3.2-2 concrete strength, = 0.691 (using f ′c = 7.00 ksi, for continuity at 60 and 90 days), = 0.619 (using f ′c = 8.50 ksi, for continuity at 28 days), H = relative humidity, = 75% (assumed), and V/S = 4.057 in. (used to determine kc). 3.3.1.2.3 Ultimate shrinkage strain. While it is not always known whether the girder will be steam cured during fabri- cation, the initial strength gain is generally accelerated when compared with “normal” concretes. Therefore, it is reason- able to use the shrinkage equation for steam-cured concrete. The shrinkage strain may therefore be taken as LRFD Eq. 5.4.2.3.3-2 To determine the ultimate shrinkage strain, εshu, where t = ∞, the term in the equation that contains it is assumed to approach unity: εshu = εsh (∞) = −395 × 10−6 in./in. where ks = size factor = 0.760, and LRFD Eq. C5.4.2.3.3-1 kh = humidity factor = 0.929. LRFD Eq. C5.4.2.3.3-2 3.3.2 Deck and Continuity Diaphragm Concrete The same concrete properties are used for the deck slab and continuity diaphragm because they are placed at the same time in this example. The subscript d is used to indicate properties related to the deck slab or diaphragm concrete. 3.3.2.1 Basic Properties. The same deck slab concrete strength is used for all designs: f ′cd = 4.00 ksi, frd = 0.480 ksi, LRFD Art. 5.4.2.6 wcd = 0.150 kcf, and Ecd = 3,834 ksi. LRFD Eq. 5.4.2.4-1 According to proposed Article 5.14.1.2.7j, design at the con- tinuity diaphragm will use the concrete strength of the pre- cast girder where noted. ε sh s hk k t t = − +( ) × −55 0 0 56 10 3. . . D-8 3.3.2.2 Time-Dependent Properties. See Section 3.3.1.2.1 for discussion. 3.3.2.2.1 V/S ratio. As discussed in Section 3.3.1.2.1, determination of the V/S ratio should be carefully considered. The V/S ratio for the composite deck is computed using the equivalent ratio of the cross-sectional area to the perimeter. For a composite deck slab, the area is computed as the product of the full depth of the deck and the width of the deck extending to the center of the bay between girders or to the exterior edge of the deck. The area of the build-up could also be included in the deck area. However, such a refinement of the computation is not generally justified, since the calcula- tion is not precise. Therefore, the area of the deck for the inte- rior girder being considered will be the product of the girder spacing, S, and the total deck thickness, hf : A = Shf = 7.75 ft × 8 in. = 7.75(12)(8) = 744 in.2. The perimeter of the deck slab used to compute the V/S ratio is subject to some refinement based on the recommendation 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 that is in contact with the girder will never be exposed to dry- ing, it may be eliminated from the computed perimeter. For simplicity and since the top flange of the Type III girder is relatively narrow, this correction will not be taken. It may be appropriate to use the correction for the contact area where the contact area with the girder is wide, such as a bulb-T or box girder. For interior girders, the deck thickness is not con- sidered in computing the perimeter because it is an imaginary boundary not exposed to drying. Therefore, for the interior girder being designed, the perimeter is taken as twice the girder spacing, S: p = 2S = 2(7.75)(12) = 186.0 in. The V/S ratio is V/S = A/p = 744/186 = 4.00 in. This calculation demonstrates that, for a uniform thickness deck slab with no deducted surface area, V/S is simply half of the thickness of the deck. Since stay-in-place deck forms may be used on this bridge, the bottom of the deck slab may not be exposed to drying. This would increase the V/S ratio to 8.00 in., which exceeds the V/S limit used to develop the equations for correction fac- tors, kc and ks The increased V/S will reduce the corrections factors, but not significantly. Therefore, the effect of the deck forms is neglected. 3.3.2.2.2 Ultimate creep coefficient. The age of the deck concrete at loading is not as well defined as it is for the girder.

An early age of 14 days is assumed to provide a conservative estimate of deck creep behavior (a larger creep coefficient). An early age at loading is also a reasonable assumption because some load will be transferred to the deck shortly after casting because it restrains the continued downward deflection of the girder under the load of the deck (due to creep): LRFD Eq. 5.4.2.3.2-1 As described previously, t = ∞ is used to obtain the ultimate value for the creep coefficient, ψu: ψu = ψ (∞, 14 days) = 1.70, where kc = factor for V/S ratio LRFD Eq. C5.4.2.3.2-1 = 0.775, kf = factor for the effect of LRFD Eq. 5.4.2.3.2-2 concrete strength = 0.897 (using f ′cd = 4.00 ksi), H = relative humidity = 75% (assumed), and V/S = 4.00 in. (used for kc; this is a simplified value, based on the full 8-in.deck thickness and neglecting any effect of SIP metal forms and the area of contact with the girder). 3.3.2.2.3 Ultimate shrinkage strain. Since deck slab con- crete is normally moist cured, the equation for shrinkage for moist-cured concrete is used: LRFD Eq. 5.4.2.3.3-1 To determine the ultimate shrinkage strain, εshu, where t = ∞, the term in the equation that contains t is assumed to approach unity: εshu = εsh (∞) = −353 × 10−6 in./in., where 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 The restraining effect of longitudinal reinforcement in the deck slab on the free shrinkage is not considered in this design example. Proposed Article C5.14.1.2.7c states that the effect may be computed by proposed Equation C5.14.1.2.7c-1. 3.3.3 Prestressing Strand The material properties of the prestressing strand are as follows: ε sh s hk k t t = − +( ) × −35 0 0 51 10 3. . . ψ t t k k H t t t t t i c f i i i , . . . . . . . ( ) ( ) ( )( )= − −+ −−3 5 1 58 120 10 0 0 60 118 0 6 D-9 0.5-in.- or 0.6-in.-diameter low-relaxation seven-wire strand; Aps = 0.153 in.2 (0.5-in.-diameter strand, for continuity at 60 and 90 days) and = 0.217 in.2 (0.6-in.-diameter strand, for continuity at 28 days); fpu = 270 ksi; fpy = 0.90 fpu = 243 ksi; fpj = 0.75 fpu = 202.5 ksi; and Ep = 28,500 ksi. 3.3.3.1 Transfer Length. At the ends of pretensioned gird- ers, the force in the pretensioning strands is transferred from the strands to the girder concrete over the transfer length. The stress in the strands is assumed to vary linearly from zero at the end of the girder to the full effective prestress, fpe, at the transfer length. The transfer length,
t, may be estimated as
t = 60db, LRFD Art. 5.11.4.1 = 60(0.5 in.) = 30 in. (for continuity at 60 and 90 days), and = 60(0.6 in.) = 36 in. (for continuity at 28 days), where db = nominal strand diameter. 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 in this example. These locations are identified in the tables with the heading “Trans” or “Transfer.” Values in tables dif- fer for continuity at 28 days and for continuity at 60 and 90 days because the different strand size results in a different transfer length. 3.3.4 Mild Reinforcement Mild reinforcement is as follows: fy = 60 ksi, and Es = 29,000 ksi. 3.4 Stress Limits The following stress limits are used for the design of the girders for the service limit state. For computation of girder

stresses, the sign convention will be compressive stress is positive (+) and tensile stress is negative (−). Signs are not shown for limits in the following, they will be applied in later stress comparisons. 3.4.1 Pretensioned Strands The stress limits for low relaxation strands are as follows: Immediately prior to transfer: LRFD Table 5.9.3-1 fpi = 0.75 fpu = 202.5 ksi. At service limit state after losses: LRFD Table 5.9.3-1 fp = 0.80 fpy = 199.4 ksi. The stress limits above are not discussed in this example because they do not govern designs. 3.4.2 Concrete 3.4.2.1 Temporary Stresses at Release. See Table D-3.4.2.1-1. Compression: LRFD Art. 5.9.4.1.1 fcR = 0.60 f ′ci. Tension: LRFD Table 5.9.4.1.2-1 ftR1 = 0.0948 √f ′ci ≤ 0.2 ksi, or ftR2 = 0.24 √f ′ci with reinforcement to resist the tensile force in the concrete. 3.4.2.2 Final Stresses at Service Limit State after Losses. The following stress limits are given for the girder concrete. Compressive stresses may be checked for the deck slab, but never govern, so they are not included here. Tensile stresses in the deck slab at interior supports should not be compared with limits for the service limit state because the deck is not D-10 prestressed. Instead, it is designed to satisfy the specified requirements at the strength limit state. Compression: LRFD Table 5.9.4.2.1-1 fc1 = 0.60 φw f ′c , for full-service loads (φw = 1 for girders); fc2 = 0.45 f ′c , for effective prestress (PS) and full dead loads (DL); and fc3 = 0.40 f ′c , for live load plus one-half of effective PS and full DL. Tension: LRFD Table 5.9.4.2.2-1 For the precompressed compression zone, ft1 = 0.19√f ′c , assuming moderate corrosion conditions. For locations other than the precompressed compression zone, such as at the end of the girder where the top of the girder may go into 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 same form as those for temporary tensile stresses at release given in LRFD Table 5.9.4.1.2-1, but with the specified concrete compressive strength, f ′c , rather than the concrete compressive strength at release, f ′ci: ft2 = 0.0948√f ′c ≤ 0.2 ksi, or ft3 = 0.24√f ′c with reinforcement to resist the tensile force in the concrete. Numerical values for the stress limits are given in Table D-3.4.2.2-1. 3.5 Other Design Assumptions Intermediate diaphragms are not used in the design exam- ple. Temporary steel or timber cross frames are generally required during erection to stabilize girders. However, the weight of these temporary components is minor and is neglected in these calculations. Girder Age at Continuity 28 Days 60 Days 90 Days f 'ci (ksi) 7.50 6.00 5.50 fcR (ksi) 4.500 3.600 3.300 ftR1 (ksi) –0.200 –0.200 –0.200 ftR2 (ksi) –0.660 –0.590 –0.560 Note: Values are rounded to two significant digits. TABLE D-3.4.2.1-1 Temporary stress limits at release Girder Age at Continuity 28 Days 60 Days 90 Days f 'c (ksi) 8.50 7.00 7.00 fc1 (ksi) 5.100 4.200 4.200 fc2 (ksi) 3.830 3.150 3.150 fc3 (ksi) 3.400 2.800 2.800 ft1 (ksi) –0.550 –0.500 –0.500 ft2 (ksi) –0.200 –0.200 –0.200 ft3 (ksi) –0.700 –0.630 –0.630 TABLE D-3.4.2.2-1 Final stress limits after losses

The top 0.25 in. of the deck is assumed to be a sacrificial wearing surface; the structural deck thickness is taken as 7.75 in. for design purposes. The weight of the remaining 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 to the full length of the girder for dead-load computations. In most design situations, a value is used that is less than the specified build-up thickness at the center of the bearings because the actual thickness will vary along the length of the girder from the maximum of 21⁄2 in. at the center of bearings. The build-up is neglected when computing composite sec- tion properties that are used to calculate stresses for service limit state design since the build-up will vary along the bridge. However, for computation of section properties and strength calculations related to the reinforcement at the continuity diaphragm, the build-up is included. This is done because the build-up is specified at the center of the bearings, so the full build-up will be provided at the continuity diaphragm location. Potential deck cracking, if considered in the analysis of this continuous bridge, could increase positive design moments. However, potential deck cracking is neglected as allowed in LRFD Article 4.5.2.2. 3.6 Section Properties 3.6.1 Noncomposite Section (Girder Only) The section properties for a standard AASHTO Type III girder are as follows (see Figure D-3.6.1-1): h = 45.00 in., A = 559.5 in.2, I = 125,390 in.4, yb = 20.27 in., yt = 24.73 in., D-11 Sb = 6,185 in3, and St = 5,071 in.3. 3.6.2 Composite Section (Girder with Deck Slab) 3.6.2.1 Effective Deck Width. The effective width for the composite deck at all limit states is determined according to LRFD Article 4.6.2.6.1. The effective deck width for an inte- rior beam is the least of the following: 1. One-quarter of the span length (260 in.); 2. Average spacing of adjacent girders (93 in.) GOV- ERNS; and 3. Twelve times the average thickness of the slab (93 in.), plus the greater of a. The web thickness (7 in.) or b. One-half of the width of the top flange of the girder (8 in.). In the case of this design example, the average spacing of adjacent girders controls, resulting in an effective deck width of 93 in. 3.6.2.2 Transformed Effective Deck Width. The com- posite deck slab is transformed using the modular ratio, n, for computing stresses at the service limit state. See Table D-3.6.2.3-1. 3.6.2.3 Section Properties. The build-up is neglected when computing section properties because the build-up height varies along the length of the girder, with the minimum height at or near midspan. See Section 3.5. Composite section prop- erties vary for different girder ages at continuity because the girder concrete strength is different. Figure D-3.6.1-1. AASHTO Type III girder. 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 * Stcd (in3) 26,203 25,493 25,493 *Note: Stcd = (Ic / ytcd) / n, so that fcd = M / Stcd. TABLE D-3.6.2.3-1 Composite section properties

D-12 girders made continuous. The steps in each approach are as follows: • General Approach: – The age of girders when continuity is established may or may not be specified; – Estimate time-dependent material properties (creep and shrinkage) that will be used to compute restraint moments; – Estimate the positive restraint moment, which is strongly dependent on girder age at continuity and time-dependent material properties; – Evaluate conditions at the continuity diaphragms to determine whether the connection is fully or partially effective under the effect of the positive restraint moment; – If the restraint moment exceeds 1.2Mcr or if the joint is not fully effective, it is recommended that the design or conditions be altered to improve the situation; – Analyze and design the girders for all design loads, including positive restraint moment (positive restraint moment should be neglected when evaluating stresses in regions of negative moment); – Design and detail a positive moment connection at continuity diaphragms; and – Design and detail reinforcement to resist negative moments from design loads, neglecting both positive and negative restraint moments. • Simplified Approach: – Specify the minimum age of the girders when continu- ity is established in the contract documents; the mini- mum girder age at continuity must be at least 90 days; 3.7 Design Moments The following sections present the computed design moments for service and strength limit states. All loads applied to the bare girder are thought to act on a simple span. Loads applied after the deck slab and continuity diaphragm are placed are thought to act on a fully continuous structure. For tables in this section—Tables D-3.7.1-1 through D-3.7.1-3—rows and columns are shaded to indicate quanti- ties that apply only to the design with a girder age at continu- ity of 28 days. The rows for the transfer length are shaded because the design with continuity at 28 days required 0.6-in.- diameter strands rather than the 0.5-in.-diameter strands used in the other designs. The columns for live load are shaded because the design with continuity at 28 days required a higher concrete strength, f ′c, which altered the live-load dis- tribution factor slightly. See the table in Section 3.2. Restraint moments are not shown in tables contained in this section. Computations are made later in the example. Moments at the ends of the transfer length are identified in the following tables by “Trans.” 3.7.1 Service Limit State The following tables provide moments for the service limit state. A separate table is given for moments caused by loads applied to the noncomposite section. 3.7.2 Strength Limit State Table D-3.7.2-1 provides moments for the Strength I limit state. 4 ANALYSIS AND DESIGN OF GIRDERS FOR CONTINUITY The age at which continuity is established for precast/ prestressed concrete girders is a critical factor in the design of bridges of this type. The earlier the age of the girder at con- tinuity, the more girder creep and shrinkage contribute to the development of positive restraint moments. Minimizing the risk of developing positive moments avoids an increase in critical moments and stresses in the girders and avoids crack- ing of the continuity diaphragm. It is highly recommended that the minimum age for con- tinuity be specified in the contract documents. Analytical studies and field experience indicate that waiting to estab- lish continuity until the girders are at least 90 days old will significantly reduce or eliminate the development of positive restraint moments. The proposed specifications recognize the benefit of delay- ing continuity by allowing two approaches for the design of Location from Bearing Self Weight Dead Load on Precast Dead Load of Deck (ft) (k-ft) (k-ft) (k-ft) Load Factor 1.0 1.0 1.0 Bearing 0.0 0.0 0.0 0.0 Trans. (60 & 90) 1.9 46.4 13.5 59.8 H/2 2.2 53.1 15.4 68.3 Trans. (28 Days) 2.42 58.2 16.9 74.9 0.10 L 8.0 180.3 52.2 232.1 0.20 L 16.7 331.9 96.2 427.2 0.30 L 25.3 440.2 127.5 566.6 0.40 L 33.9 505.2 146.4 650.2 0.50 L 42.5 526.8 152.6 678.0 0.60 L 51.1 505.2 146.4 650.2 0.70 L 59.7 440.2 127.5 566.6 0.80 L 68.4 331.9 96.2 427.2 0.90 L 77.0 180.3 52.2 232.1 Trans. (28 Days) 82.58 58.2 16.9 74.9 H/2 82.8 53.1 15.4 68.3 Trans. (60 & 90) 83.1 46.4 13.5 59.8 Bearing 85.0 0.0 0.0 0.0 TABLE D-3.7.1-1 Service design moments for loads on noncomposite section

D-13 Location from Bearing Compos. DL (DC) Compos. DL (DW) LL+IM (+) [28 Days] LL+IM (+) [60 & 90] LL+IM (–) [28 Days] LL+IM (–) [60 & 90] (ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) Load Factor 1.0 1.0 1.0 1.0 1.0 1.0 Bearing 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Trans. (60 & 90) 1.9 8.9 10.2 N/A 127.6 N/A –15.1 H/2 2.2 10.2 11.6 147.0 145.7 –17.5 –17.3 Trans. (28 Days) 2.4 11.1 12.8 161.0 N/A –19.2 N/A 0.10 L 8.0 33.7 38.6 486.8 482.6 –63.9 –63.4 0.20 L 16.7 59.1 67.7 854.9 847.5 –132.5 –131.4 0.30 L 25.3 73.6 84.3 1,073.8 1,064.6 –201.1 –199.4 0.40 L 33.9 77.0 88.2 1,173.7 1,163.6 –269.7 –267.4 0.50 L 42.5 69.4 79.5 1,163.2 1,153.3 –338.3 –335.4 0.60 L 51.1 50.8 58.1 1,044.7 1,035.8 –406.9 –403.4 0.70 L 59.7 21.1 24.2 812.7 805.7 –475.5 –471.4 0.80 L 68.4 –19.5 –22.4 489.5 485.3 –632.2 –626.8 0.90 L 77.0 –71.2 –81.6 187.2 185.6 –738.5 –732.2 Trans. (28 Days) 82.6 –110.8 –126.9 74.8 N/A –998.1 N/A H/2 82.8 –112.4 –128.8 71.8 71.2 –1,011.3 –1,002.6 Trans. (60 & 90) 83.1 –114.6 –131.2 N/A 67.5 N/A –1,019.7 Bearing 85.0 –129.3 –148.1 47.1 46.7 –1,156.1 –1,146.2 CL Pier 86.0 –137.0 –156.9 35.7 35.6 –1221.93 –1,211.6 TABLE D-3.7.1-2 Service I design moments for loads on composite section Location from Bearing Compos. DL (DC) Compos. DL (DW) LL+IM (+) [28 days] Days] LL+IM (+) [60 & 90] LL+IM (–) [28 Days] LL+IM (–) [60 & 90] (ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) Load Factor 1.0 1.0 1.0 1.0 1.0 1.0 Bearing 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Trans. (60 & 90) 1.9 8.9 10.2 N/A 102.1 N/A –12.1 H/2 2.2 10.2 11.6 117.6 116.6 –14.0 –13.8 Trans. (28 Days) 2.4 11.1 12.8 128.8 N/A –15.4 N/A 0.10 L 8.0 33.7 38.6 389.4 386.1 –51.2 –50.7 0.20 L 16.7 59.1 67.7 683.9 678.0 –106.0 –105.1 0.30 L 25.3 73.6 84.3 589.0 851.7 –160.9 –159.5 0.40 L 33.9 77.0 88.2 938.9 930.9 –215.8 –213.9 0.50 L 42.5 69.4 79.5 930.6 922.6 –270.6 –268.3 0.60 L 51.1 50.8 58.1 835.8 828.6 –325.5 –322.7 0.70 L 59.7 21.1 24.2 650.2 644.6 –380.4 –377.1 0.80 L 68.4 –19.5 –22.4 391.6 388.2 –505.8 –501.4 0.90 L 77.0 –71.2 –81.6 149.7 148.5 –590.8 –585.8 Trans. (28 Days) 82.6 –110.8 –126.9 59.9 N/A –798.5 N/A H/2 82.8 –112.4 –128.8 57.5 57.0 –809.0 –802.1 Trans. (60 & 90) 83.1 –114.6 –131.2 N/A 54.0 N/A –815.8 Bearing 85.0 –129.3 –148.1 37.7 37.4 –924.8 –917.0 CL Pier 86.0 –137.0 –156.9 28.6 28.6 –975.7 –969.4 TABLE D-3.7.1-3 Service III design moments for loads on composite section

Location from Bearing Self Weight (Max) Self Weight (Min) DL on Precast (Max) DL on Precast (Min) DL of Deck (Max) DL of Deck (Min) Compos. DL (DC) (Max) Compos. DL (DC) (Min) Compos. DL (DW) (Max) Compos. DL (DW) (Min) LL+IM (+) [28 Days] LL+IM (+) [60 & 90] LL+IM (-) [28 Days] LL+IM (-) [60 & 90] (ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) (k-ft) Load Factor 1.25 0.9 1.25 0.9 1.25 0.9 1.25 0.9 1.50 0.65 1.75 1.75 1.75 1.75 Bearing 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Trans. (60 & 90) 1.9 58.1 41.8 16.8 12.1 74.7 53.8 11.1 8.0 15.3 6.6 N/A 223.3 N/A –26.5 H/2 2.2 66.4 47.8 19.2 13.8 85.4 61.5 12.7 9.1 17.5 7.6 257.2 255.0 –30.6 –30.4 Trans. (28 Days) 2.4 72.8 52.4 21.1 15.2 93.6 67.4 13.9 10.0 19.1 8.3 281.8 N/A –33.7 N/A 0.10 L 8.0 225.4 162.3 65.3 47.0 290.1 208.9 42.1 30.3 57.8 25.1 851.9 844.6 –111.9 –110.9 0.20 L 16.7 414.9 298.7 120.2 86.5 534.0 384.5 73.9 53.2 101.6 44.0 1,496.0 1,483.2 –231.9 –229.9 0.30 L 25.3 550.3 396.2 159.4 114.8 708.2 509.9 91.9 66.2 126.4 54.8 1,879.1 1,863.0 –351.9 –348.9 0.40 L 33.9 631.5 454.7 182.9 131.7 812.7 585.2 96.2 69.3 132.3 57.3 2,053.9 2,036.3 –472.0 –467.9 0.50 L 42.5 658.5 474.1 190.8 137.4 847.6 610.2 86.7 62.4 119.2 51.7 2,035.6 2,018.2 –592.0 –586.9 0.60 L 51.1 631.5 454.7 182.9 131.7 812.7 585.2 63.5 45.7 87.2 37.8 1,828.2 1,812.6 –712.0 –705.9 0.70 L 59.7 550.3 396.2 159.4 114.8 708.2 509.9 26.4 19.0 36.3 15.7 1,422.2 1,410.0 –832.0 –824.9 0.80 L 68.4 414.9 298.7 120.2 86.5 534.0 384.5 –24.4 –17.6 –33.6 –14.5 856.6 849.2 –1,106.3 –1,096.9 0.90 L 77.0 225.4 162.3 65.3 47.0 290.1 208.9 –89.0 –64.1 –122.3 –53.0 327.5 324.7 –1,292.4 –1,281.3 Trans. (28 Days) 82.6 72.8 52.4 21.1 15.2 93.6 67.4 –138.5 –99.7 –190.4 –82.5 131.0 N/A –1,746.7 N/A H/2 82.8 66.4 47.8 19.2 13.8 85.4 61.5 –140.6 –101.2 –193.2 –83.7 125.7 124.6 –1,769.7 –1,754.6 Trans. (60 & 90) 83.1 58.1 41.8 16.8 12.1 74.7 53.8 –143.2 –103.1 –196.9 –85.3 N/A 118.1 N/A –1,784.5 Bearing 85.0 0.0 0.0 0.0 0.0 0.0 0.0 –161.6 –116.4 –222.2 –96.3 82.4 81.7 –2,023.1 –2,005.8 CL Pier 86.0 N/A N/A N/A N/A N/A N/A –171.2 –123.3 –235.4 –102.0 62.4 62.3 –2,137.6 –2,120.2 TABLE D-3.7.2-1 Strength I design moments

– The connection at continuity diaphragms may be taken to be fully continuous; – Analyze and design the girders for all design loads (neglect restraint moments); – Design and detail a positive moment connection at continuity diaphragms; and – Design reinforcement to resist negative moments from design loads. This section of the design example is divided into two sub- sections corresponding to the two approaches listed above. Both approaches are presented for completeness; however, it is anticipated that the simplified approach, with its required specification of a minimum girder age of 90 days when conti- nuity is established, will be the approach used most often by bridge designers. The main benefits of the simplified approach include the simplicity of design and the ability to use standard design aids or software to complete a design. The general approach requires a method for estimating restraint moments. The Restraint software program has been developed for this purpose, and it is used for this example. Few other programs are available, and some may have significant limitations or disadvantages. The general approach is presented first. In this section, com- putations and results will be presented for the bridge with con- tinuity established when the age of the girders is 28 and 60 days. Positive restraint moments are estimated, and their effect is considered in the design of the girders. The design for continuity at a girder age of 90 days is also discussed with the initial calculation of restraint moments. The required time- dependent material properties were computed in the previous section of this design example. The simplified approach will then be presented for the bridge with girders that are at least 90 days old when conti- nuity is established. The calculations only address concrete stresses in the girders at the service limit state. Flexural design at the strength limit state was checked and does not govern the designs, so calculations are not shown. A summary compares the results of the designs using the general and simplified approaches. The reinforcement for positive and negative continuity connections, which is deter- mined using the same methods for both approaches, is com- puted in subsequent sections. 4.1 General Approach The steps in the general approach were summarized in the previous section. The items will be addressed as they appear in the list, except that the determination of the effectiveness of the joint is considered prior to computation of the restrain- ing moment, as discussed in the following section. D-15 4.1.1 Effectiveness of Joint According to proposed Article 5.14.1.2.7e, the connection between spans at the continuity diaphragm may be consid- ered fully effective if one or both of the following conditions are satisfied: 1. The contract documents specify that the girders will be at least 90 days old when continuity is established. 2. The stress in the joint is compressive for the combina- tion of superimposed permanent loads, settlement, creep, shrinkage, 50% live load (with impact), and tempera- ture gradient, if applicable. The first criterion is addressed by the contract documents. To demonstrate the general approach, girder ages at continuity are selected that do not satisfy the first criterion. The second criterion is stated in terms of a stress, but since the continuity diaphragm is not prestressed, it can also be expressed in terms of moments. Using moments will simplify manual computations since all of the moments are known, but the stress does not have to be computed. Therefore, the sum of the positive restraint moment, composite dead-load moments (negative), and 50% of the maximum negative live-load (with impact) moment must not be positive since a positive moment would cause tension at the bottom of the diaphragm. (Please note that effects other than permanent and live loads, such as temperature effects, are not considered in this exam- ple, but could be included in the calculation.) This summation can also be backsolved to determine the maximum positive restraint moment that can develop before the net moment becomes positive or the stress at the bottom of the diaphragm becomes tensile. This computed maximum positive restraint moment can then be used to facilitate the comparison of dif- ferent designs and to eliminate the separate computation of joint stress since the restraint moment is computed as part of each design iteration. The maximum positive restraint moment at the interior sup- port is computed in Table D-4.1.1-1. The live load used in this computation is 50% of the maximum negative service moment. See Table D-3.7.1-2. The Service I load combination is used rather than the Service III load combination because the TABLE D-4.1.1-1 Calculation of moment limit for joint effectiveness Composite Dead Load (DC) (k-ft) –137.0 Composite Dead Load (DW) (k-ft) –156.9 50 % of Live Load + Impact (k-ft) –605.9 TOTAL (k-ft) –900.3 Maximum Positive Restraint Moment for Fully Effective Joint (k-ft) 900.3

connection at the continuity diaphragm is not prestressed con- crete, which is a requirement for use of Service III. When the positive restraint moment computed in the various design iter- ations remains less than or equal to the computed maximum positive restraint moment (900.3 k-ft), the joint may be con- sidered to be fully effective, and the bridge may be designed using continuity for all loads applied after continuity is estab- lished. If the positive restraint moment exceeds the maximum computed moment, the connection must be considered par- tially effective. In this case, a fraction of the loads applied to the continuous structure are considered to be carried by the girders as simple spans, and the remainder of the loads resisted by the continuous structure. The computation shown above represents the initial design moments only. Due to various adjustments that are made to the designs throughout the required iterations, the live-load moment changes slightly. Therefore, in the following, the stress at the bottom of the joint is computed as a check. 4.1.2 Initial Design Without Restraint Moments An initial design of an interior girder was performed without including any restraint moment. The strand pattern required for the specified geometry and loads is shown in Figure D-4.1.2-1. 4.1.3 Compute Restraint Moments Numerical analysis shows that both positive and negative restraint moments may occur. Positive restraint moments can have a significant effect on the design of precast/prestressed concrete girders made continuous. Negative restraint moments are temporary and are usually ignored in design. Positive restraint moments are generally larger for two- span bridges than for bridges with a greater number of spans with similar span lengths, given the same span lengths and other conditions. However, two-span bridges have limited positive live-load moments at the interior support, while D-16 bridges with a greater number of spans can develop signifi- cant positive moments at interior supports from live load. The development of restraint moments with time is computed using the Restraint Program for the initial girder design. The strand pattern shown in Figure D-4.1.2-1 is input to define the creep behavior of the girder. Material properties com- puted earlier are also used for input, including the ultimate creep coefficient for girder and deck concrete and the ulti- mate shrinkage strain for the girder concrete. Restraint uses the Dischinger effect to account for the restraining effect of reinforcement on deck shrinkage. A complete listing of input values used in Restraint is presented in Subappendix A. Results for the restraint moment analysis for continuity established at the girder ages of 28, 60, and 90 days are shown in Figure D-4.1.3-1. The restraint moment analysis for this initial design indi- cates that, for continuity established at a girder age of 28 days, a positive restraint moment of about 830 k-ft will eventually develop from the combined effect of creep and shrinkage in the girder and deck concrete. For continuity established at a girder age of 60 days, the analysis indicates that a positive restraint moment of approximately 300 k-ft will eventually develop. The analysis shows that a positive restraint moment of only 3 k-ft develops when continuity is delayed until 90 days after the girder is cast. This very small (negligible) positive moment supports the proposed Article 5.14.1.2.7d, which allows positive restraint moments to be neglected in the design of bridges where contract documents require that continuity cannot be established until the girders are at least 90 days old. This initial design will be modified for the pos- itive restraint moments in the next section. The analysis also indicates that a negative restraint moment develops rapidly after the composite deck slab has been placed. The maximum negative moment varies with the age of the girders at continuity from a negative moment of nearly −120 k-ft for the bridge with continuity at a girder age of 28 days to a moment of about −540 k-ft for continuity established at a girder age of 90 days. However, the negative restraint moments dissipate with time. Because of the tempo- rary nature of this negative restraint moment peak and the lack of observed distress in deck slabs that negative restraint moments would cause, the negative moment is neglected in this example. 4.1.4 Subsequent Design Iterations Including Positive Restraint Moments Since the analysis indicates that a significant positive restraint moment develops with time for the girders that are 28 and 60 days old when continuity is established, the design of these girders must be revised. The modifications to the ini- tial design are necessary to counteract the increased stresses caused by the additional positive restraint moments, so sev- eral iterations are required. End Midspan Figure D-4.1.2-1. Strand pattern for initial design.

For each iteration, strands are added or repositioned and other quantities are adjusted as needed to provide a design that includes the positive restraint moment and satisfies the service limit state design criteria. A new positive restraint moment is then computed using the Restraint Program for the new strand pattern, and the process is repeated. The iter- ations continue until the revised positive restraint moment does not require a change in the strand pattern. Tables of strand requirements, positive restraint moments, and stresses at the bottom of the continuity diaphragm are given for the designs with girder ages at continuity of 28 and 60 days, respectively. Initial and final concrete strengths were adjusted during the iterations. For the design with con- tinuity at 28 days, the strand size and location of the hold- down are listed because modifications were made during the design iterations. Strand patterns and other data for the final designs for girder ages of 28 and 60 days at continuity are shown in the summary, Section 4.3. Computations for the strength limit state are not shown since they do not govern. The effect of the positive restraint moment is included when computing the stresses shown. 4.1.4.1 Girder Age at Continuity of 28 Days. In the final iteration, the restraint moment exceeds the maximum positive restraint moment of 900.3 k-ft computed in Table D-4.1.1-1. The computed stress at the bottom of the continuity dia- phragm has also gone into tension for the specified loading (see darker-shaded cells), which supports the use of the pos- itive restraint moment limit. Therefore, the joint cannot be considered fully effective for analysis for all loads applied to the continuous structure. If this design were to be carried fur- D-17 ther, the stresses in the girders must be recomputed by apply- ing a portion of the live load to the continuous structure and applying the remaining live load on the structure as a simple span. This accounts for the change in behavior that would occur if a crack opens in the continuity diaphragm due to the positive restraint moment. This crack must close before con- tinuous behavior can be restored. The fraction of the live load applied to the simple span would be equal to the ratio of the sum of the net positive service load moment at the center of the interior support caused by dead load and restraint moments to the full live- load moment (including any dynamic allowance) at the cen- ter of the interior support. The remainder of the live load would then be applied to the continuous structure. This com- bination of moments from simple and continuous spans can be accomplished by applying the fractions computed as dis- cussed above to the live-load moment envelopes from the simple and continuous span analyses. The additional stress that would result from applying some of the live load on the structure acting as a simple span would likely increase stresses to a level where a solution could not be reached using normal design parameters for the bridge in this example. The analysis would also be more complex. Therefore, it is recommended that the design with continuity established at a girder age of 28 days should be abandoned as unreasonable. Further discussion of the design is contin- ued for illustration purposes only. Table D-4.1.4.1-1 also shows that the positive restraint moments for all iterations following the initial design (see lighter-shaded cells) exceed the quantity 1.2Mcr = 550.8 k-ft as shown in Table D-5.2-1. It is recommended that designs should be reconsidered where the restraint moment exceeds -600 -400 -200 0 200 400 600 800 0 1000 2000 3000 4000 5000 6000 7000 8000 Girder Age (Days) R es tr ai nt M om en t ( k- ft) Continuity at Girder Age of 28 Days Continuity at Girder Age of 60 Days Continuity at Girder Age of 90 Days INITIAL DESIGNS Moments increase for final designs Figure D-4.1.3-1. Development of restraint moment with girder age—initial design.

this limit because analytical studies have shown that increas- ing the reinforcement beyond levels required to resist a moment of 1.2Mcr is not effective. In summary, the design of this girder, with continuity established at a girder age of 28 days, is not recommended because of the positive restraint moment exceeds two signif- icant limits. For further discussion, see the summary and comparison of designs in Section 4.3. The maximum restraint moment as shown in Table D-4.1.4.1-1 occurs at the center of the interior support and decreases linearly to zero at the center of bearing at the expansion joint. A load factor of 1.0 is applied to the restraint moment for the service limit state, and a load factor of 0.5 is applied to the restraint moment for the strength limit state (LRFD Article 3.4.1). Since the moments vary linearly, a table of restraint moments along the girder is not given. Tables D-4.1.4.1-2 and D-4.1.4.1-3 present the service limit state stresses for the final design for a girder age at con- tinuity of 28 days. These stresses are compared with stress limits for release and final conditions after losses that are given in Tables D-3.4.2.1-1 and D-3.4.2.2-1. D-18 4.1.4.2 Girder Age at Continuity of 60 Days. The positive restraint moments shown in the table remain below the max- imum positive restraint moment of 900.3 k-ft computed in Table D-4.1.1-1 for all iterations. As expected, the stress at the bottom of the diaphragm also remains in compression for all iterations. Therefore, design may be performed consider- ing the joint fully effective for all loads applied to the con- tinuous structure. Additionally, the moments all remain below the quantity 1.2Mcr = 550.8 k-ft as shown in Table D-5.2-1. Designs are generally considered unreasonable if this limit is exceeded. Therefore, the design of this girder, with continuity estab- lished at a girder age of 60 days, is acceptable. For further discussion, see the summary and comparison of designs in Section 4.3. The maximum restraint moment as shown in Table D-4.1.4.2-1 occurs at the center of the interior pier and decreases linearly to zero at the center of bearing at the expansion joint. A load factor of 1.0 is applied to the restraint moment for the service limit state and a load factor of 0.5 is applied to the restraint moment for the strength limit state. Iteration f 'ci f 'c Number of Strands Strand Diameter Positive Restraint Moment Stress at Bottom of Continuity Diaphragm Hold-down Location from End (ksi) (ksi) (in.) (k-ft) (ksi) 1 5.50 7.00 38 0.5 0.0 1.049 0.40 L 2 7.00 8.50 34 0.6 826.8 0.091 0.40 L 3 7.50 9.00 34 0.6 899.6 0.008 0.40 L 4 7.50 8.50 34 0.6 884.3 0.024 0.40 L 5 7.50 8.50 34 0.6 945.1 –0.047 0.40 L TABLE D-4.1.4.1-1 Summary of design iterations—continuity at 28 days Brg. H/2 Trans. 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 2.20 2.42 8.03 16.65 25.27 33.88 42.50 Top Girder (ksi) N/A N/A 1.269 0.819 0.127 –0.564 –0.910 –0.910 Prestress at Release Bottom Girder (ksi) N/A N/A 3.305 3.675 4.241 4.808 5.092 5.092 Top Girder (ksi) N/A N/A 0.172 0.461 0.820 1.076 1.230 1.281 Self Weight Bottom Girder (ksi) N/A N/A –0.141 –0.378 –0.672 –0.882 –1.008 –1.050 Top Girder (ksi) N/A N/A 1.442 1.280 0.947 0.512 0.320 0.371 Total at Release Bottom Girder (ksi) N/A N/A 3.164 3.297 3.569 3.926 4.083 4.041 Notes: 1. Critical stresses are shaded. 2. Values for limiting stresses are given in Table D-3.4.2.1-1. 3. Compressive stresses at release are compared with the limit fcR = 4.500 ksi. The maximum compressive stress is 4.083 ksi at 0.40L. 4. Tensile stresses in regions other than the precompressed tensile zone at release are compared with the limiting tensile stress ftR1 = –0.200 ksi or ftR2 = –0.660 ksi. The latter value requires an area of reinforcement to resist the tensile force. There are no tensile stresses in the concrete at release. 5. In all cases, this design satisfies the specified stress limits at release. TABLE D-4.1.4.1-2 Summary of design stresses for final design at release—continuity at 28 days

Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L 0.60L 0.70L 0.80L 0.90L H/2 Trans. Brg. From Brg. (ft) 0.00 2.42 2.20 8.03 16.65 25.27 33.88 42.50 51.12 59.77 68.35 76.97 82.80 82.58 85.00 SERVICE STRESSES (ksi) Top Girder 0.221 0.986 0.927 0.636 0.098 -0.440 -0.708 -0.708 -0.708 -0.440 0.098 0.636 0.927 0.986 0.221 Bottom Girder 0.475 2.568 2.370 2.855 3.296 3.736 3.956 3.956 3.956 3.736 3.296 2.855 2.370 2.568 0.475 Top Girder 0.000 0.138 0.126 0.427 0.786 1.042 1.196 1.247 1.196 1.042 0.786 0.427 0.126 0.138 0.000 Bottom Girder 0.000 -0.113 -0.103 -0.350 -0.644 -0.854 -0.980 -1.022 -0.980 -0.854 -0.644 -0.350 -0.103 -0.113 0.000 Top Girder 0.000 0.217 0.198 0.673 1.239 1.643 1.885 1.966 1.885 1.643 1.239 0.673 0.198 0.217 0.000 Bottom Girder 0.000 -0.178 -0.162 -0.552 -1.015 -1.346 -1.545 -1.611 -1.545 -1.346 -1.015 -0.552 -0.162 -0.178 0.000 Top Girder 0.000 0.009 0.009 0.026 0.050 0.063 0.065 0.059 0.043 0.018 -0.017 -0.061 -0.096 -0.094 -0.110 Bottom Girder 0.000 -0.028 -0.026 -0.085 -0.150 -0.186 -0.195 -0.176 -0.128 -0.053 0.049 0.180 0.284 0.280 0.327 Top Girder 0.000 0.011 0.010 0.035 0.073 0.111 0.149 0.187 0.225 0.263 0.301 0.339 0.365 0.364 0.375 Bottom Girder 0.000 -0.032 -0.029 -0.105 -0.218 -0.331 -0.444 -0.557 -0.670 -0.783 -0.896 -1.009 -1.086 -1.083 -1.115 Top Girder 0.000 0.064 0.058 0.193 0.339 0.426 0.465 0.461 0.414 0.322 0.194 0.074 0.028 0.030 0.019 Bottom Girder 0.000 -0.190 -0.173 -0.574 -1.008 -1.266 -1.384 -1.372 -1.232 -0.958 -0.577 -0.221 -0.085 -0.088 -0.056 Top Girder 0.000 -0.008 -0.007 -0.025 -0.053 -0.080 -0.107 -0.134 -0.161 -0.188 -0.251 -0.293 -0.401 -0.396 -0.458 Bottom Girder 0.000 0.023 0.021 0.075 0.156 0.237 0.318 0.399 0.480 0.561 0.746 0.871 1.193 1.177 1.363 TOTAL SERVICE STRESSES (ksi) Top Girder 0.221 1.350 1.260 1.762 2.173 2.308 2.438 2.564 2.416 2.263 2.106 1.675 1.155 1.247 0.111 Bottom Girder 0.475 2.249 2.079 1.868 1.487 1.350 1.236 1.147 1.303 1.483 1.686 2.133 2.389 2.557 0.802 Top Girder 0.221 1.361 1.270 1.797 2.246 2.419 2.587 2.751 2.641 2.526 2.407 2.014 1.520 1.611 0.486 Bottom Girder 0.475 2.217 2.050 1.763 1.269 1.019 0.792 0.590 0.633 0.700 0.790 1.124 1.303 1.474 -0.313 Top Girder 0.221 1.425 1.328 1.990 2.585 2.845 3.052 3.212 3.055 2.848 2.601 2.088 1.548 1.641 0.505 Bottom Girder 0.475 2.027 1.877 1.189 0.261 -0.247 -0.592 -0.782 -0.599 -0.258 0.213 0.903 1.218 1.386 -0.369 Top Girder 0.221 1.342 1.253 1.737 2.120 2.228 2.331 2.430 2.255 2.075 1.855 1.382 0.754 0.851 -0.347 Bottom Girder 0.475 2.272 2.100 1.943 1.643 1.587 1.554 1.546 1.783 2.044 2.432 3.004 3.582 3.734 2.165 Top Girder 0.221 1.412 1.316 1.951 2.517 2.760 2.959 3.120 2.972 2.784 2.562 2.073 1.542 1.635 0.501 Bottom Girder 0.475 2.065 1.912 1.304 0.463 0.006 -0.315 -0.508 -0.353 -0.066 0.328 0.947 1.235 1.404 -0.358 Top Girder 0.221 1.344 1.254 1.742 2.131 2.244 2.352 2.457 2.287 2.113 1.905 1.441 0.834 0.930 -0.255 Bottom Girder 0.475 2.267 2.096 1.928 1.612 1.540 1.490 1.466 1.687 1.932 2.283 2.830 3.343 3.499 1.892 Top Girder 0.111 0.745 0.693 1.092 1.462 1.636 1.759 1.837 1.735 1.585 1.398 1.081 0.788 0.836 0.262 Bottom Girder 0.238 0.919 0.852 0.308 -0.374 -0.757 -0.988 -1.077 -0.916 -0.608 -0.182 0.341 0.567 0.649 -0.213 Top Girder 0.111 0.667 0.623 0.856 1.034 1.074 1.112 1.148 1.047 0.944 0.802 0.545 0.177 0.228 -0.403 Bottom Girder 0.238 1.148 1.061 1.009 0.900 0.912 0.936 0.973 1.132 1.303 1.589 1.938 2.388 2.456 1.764 Non-Comp. DL 5 A = 1-4 Location Full PS + DL Composite DL LL + IM (+) LL + IM (-) Restraint Mom't (RM) (+) Prestress After Losses Self Weight Special Service+RM (+) Special Service (-) 1 2 3 Service I LL+IM (+) w/RM 7 6 Full PS + DL + RM (+) 4 7 + 0.5*A Service I LL+IM (-) B = 1-5 B + 1.0*6 A + 1.0*7 Service III LL+IM (+) w/RM Service III LL+IM (-) 6 + 0.5*B B + 0.8*6 A + 0.8*7 Notes for Table D-4.1.4.1-3: 1. Maximum stresses for each condition are shaded, with the governing stresses also boxed and bolded. 2. Values for limiting stresses are shown in Table D-3.4.2.2-1. 3. Compressive stresses for both dead-load combinations (A and B, see first column) are compared with the limiting compressive stress for full dead load fc2 = 3.830 ksi. The maximum stress is 2.751 ksi at 0.50L for the combination with restraint moment (B). 4. Compressive stresses for both Service I LL+IM conditions are compared with the limiting compressive stress for full service conditions fc1 = 5.100 ksi. The maximum stress is 3.734 ksi at the interior transfer length location (Trans.). 5. Tensile stresses in the precompressed tensile zone for Service III with RM are compared with the limiting tensile stress ft1 = –0.550 ksi. The maximum stress is–0.508 ksi at 0.50L. 6. Tensile stresses in regions other than the precompressed tensile zone for Service III without RM are compared with the limiting tensile stress ft2 = –0.200 ksi or ft3 = –0.700 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum stress is –0.255 ksi at the interior bearing, so reinforcement is required to resist the tensile force in the concrete. 7. Compressive stresses for the special service cases are compared with the limiting compressive stress for that case fc3 = 3.400 ksi. The maximum stress is 2.456 ksi at the interior transfer length location. 8. In all cases, this design satisfies the specified stress limits. TABLE D-4.1.4.1-3 Summary of design stresses for final design at service limit state after losses—continuity at 28 days

Since the moments vary linearly, a table of restraint moments along the girder is not given. Tables D-4.1.4.2-2 and D-4.1.4.2-3 present the service limit state stresses for the final design for a girder age at con- tinuity of 60 days. These stresses are compared with stress limits for release and final conditions after losses that are given in Tables D-3.4.2.1-1 and D-3.4.2.2-1. 4.2 Simplified Approach If the contract documents require that the girders be at least 90 days old when continuity is established, positive restraint moments may be neglected. This greatly simplifies the design of precast concrete girder bridges made continu- ous. Once the designer has made this decision, design pro- ceeds assuming that the bridge is fully continuous for loads applied to the continuous structure as specified in proposed Article 5.14.1.2.7d; therefore, no iterations are required. The resulting design is the same as the initial design performed without restraint moments in the preceding section. D-20 4.2.1 Girder Age at Continuity of at Least 90 Days Tables D-4.2.1-1 and D-4.2.1-2 present the service limit state stresses for the final design for a girder age at continu- ity of at least 90 days. These stresses are compared with stress limits for release and final conditions after losses that are given in Tables D-3.4.2.1-1 and D-3.4.2.2-1. 4.3 Summary of Results for General and Simplified Approaches Designs for precast/prestressed concrete girders made con- tinuous have been presented in the preceding sections for girder ages at continuity of 28, 60, and 90 days. The designs using earlier girder ages were performed using the general approach, which required the consideration of positive restraint moments. The design for a girder age of 90 days at continuity was performed using the simplified approach, in which positive restraint moments are neglected. The inclusion of positive restraint moments for the designs with earlier Iteration f 'ci f 'c Number of Strands Strand Diameter Positive Restraint Moment Stress at Bottom of Continuity Diaphragm Hold-down Location from End (ksi) (ksi) (in.) (k-ft) (ksi) 1 5.50 7.00 38 0.5 0 1.049 0.40 L 2 6.00 7.00 40 0.5 299.4 0.700 0.40 L 3 6.00 7.00 42 0.5 372.7 0.615 0.40 L 4 6.00 7.00 42 0.5 455.4 0.542 0.40 L TABLE D-4.1.4.2-1 Summary of design iterations—continuity at 60 days Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 1.92 2.20 8.03 16.65 25.27 33.88 42.50 Top Girder (ksi) N/A 0.297 N/A 0.049 –0.301 –0.651 –1.001 –1.001 Prestress at Release Bottom Girder (ksi) N/A 3.540 N/A 3.743 4.030 4.317 4.604 4.604 Top Girder (ksi) N/A 0.144 N/A 0.461 0.820 1.076 1.230 1.281 Self Weight Bottom Girder (ksi) N/A –0.118 N/A –0.378 –0.672 –0.882 –1.008 –1.050 Top Girder (ksi) N/A 0.442 N/A 0.510 0.519 0.425 0.229 0.280 Total at Release Bottom Girder (ksi) N/A 3.421 N/A 3.365 3.358 3.435 3.595 3.553 Notes: 1. Critical stresses are shaded. 2. Values for limiting stresses are given in Table D-3.4.2.1-1. 3. Compressive stresses at release are compared with the limit fcR = 3.600 ksi. The maximum compressive stress is 3.595 ksi at 0.40L. 4. Tensile stresses in regions other than the precompressed tensile zone at release are compared with the limiting tensile stress ftR1 = -0.200 ksi or ftR2 = -0.590 ksi. The latter value requires an area of reinforcement to resist the tensile force. There are no tensile stresses in the concrete at release. 5. In all cases, this design satisfies the specified stress limits at release. TABLE D-4.1.4.2-2 Summary of design stresses for final design at release—continuity at 60 days

Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L 0.60L 0.70L 0.80L 0.90L H/2 Trans. Brg. From Brg. (ft) 0.00 1.83 2.20 7.97 16.60 25.23 33.87 42.50 51.13 59.77 68.40 77.03 82.80 83.17 85.00 SERVICE STRESSES (ksi) Top Girder 0.070 0.238 0.229 0.039 -0.241 -0.522 -0.802 -0.802 -0.802 -0.522 -0.241 0.039 0.229 0.238 0.070 Bottom Girder 0.651 2.840 2.847 3.003 3.233 3.463 3.692 3.693 3.692 3.463 3.233 3.003 2.847 2.840 0.651 Top Girder 0.000 0.110 0.126 0.427 0.786 1.042 1.196 1.247 1.196 1.042 0.786 0.427 0.126 0.110 0.000 Bottom Girder 0.000 -0.090 -0.103 -0.350 -0.644 -0.854 -0.980 -1.022 -0.980 -0.854 -0.644 -0.350 -0.103 -0.090 0.000 Top Girder 0.000 0.173 0.198 0.673 1.239 1.643 1.885 1.966 1.885 1.643 1.239 0.673 0.198 0.173 0.000 Bottom Girder 0.000 -0.142 -0.162 -0.552 -1.015 -1.346 -1.545 -1.611 -1.545 -1.346 -1.015 -0.552 -0.162 -0.142 0.000 Top Girder 0.000 0.007 0.008 0.026 0.046 0.057 0.059 0.054 0.039 0.016 -0.015 -0.055 -0.087 -0.088 -0.100 Bottom Girder 0.000 -0.022 -0.025 -0.084 -0.148 -0.184 -0.192 -0.173 -0.127 -0.053 0.049 0.178 0.281 0.286 0.323 Top Girder 0.000 0.004 0.004 0.015 0.032 0.049 0.065 0.082 0.099 0.115 0.132 0.148 0.160 0.160 0.164 Bottom Girder 0.000 -0.012 -0.014 -0.050 -0.104 -0.158 -0.211 -0.265 -0.319 -0.373 -0.427 -0.481 -0.517 -0.519 -0.531 Top Girder 0.000 0.046 0.052 0.174 0.305 0.383 0.419 0.415 0.373 0.290 0.175 0.067 0.026 0.024 0.017 Bottom Girder 0.000 -0.149 -0.170 -0.562 -0.987 -1.240 -1.356 -1.344 -1.207 -0.939 -0.565 -0.216 -0.083 -0.079 -0.054 Top Girder 0.000 -0.005 -0.006 -0.023 -0.047 -0.072 -0.096 -0.121 -0.145 -0.170 -0.226 -0.264 -0.361 -0.367 -0.413 Bottom Girder 0.000 0.018 0.020 0.074 0.153 0.232 0.312 0.391 0.470 0.549 0.730 0.853 1.168 1.188 1.336 TOTAL SERVICE STRESSES (ksi) Top Girder 0.070 0.528 0.561 1.165 1.830 2.220 2.338 2.465 2.318 2.179 1.769 1.084 0.466 0.433 -0.030 Bottom Girder 0.651 2.586 2.557 2.017 1.426 1.079 0.975 0.887 1.040 1.210 1.623 2.279 2.863 2.894 0.974 Top Girder 0.070 0.532 0.565 1.180 1.862 2.269 2.403 2.547 2.417 2.294 1.901 1.232 0.626 0.593 0.134 Bottom Girder 0.651 2.574 2.543 1.967 1.322 0.921 0.764 0.622 0.721 0.837 1.196 1.798 2.346 2.375 0.443 Top Girder 0.070 0.578 0.617 1.354 2.167 2.652 2.822 2.962 2.790 2.584 2.076 1.299 0.652 0.617 0.151 Bottom Girder 0.651 2.425 2.373 1.405 0.335 -0.319 -0.592 -0.722 -0.486 -0.102 0.631 1.582 2.263 2.296 0.389 Top Girder 0.070 0.523 0.555 1.142 1.783 2.148 2.242 2.344 2.173 2.009 1.543 0.820 0.105 0.066 -0.443 Bottom Girder 0.651 2.604 2.577 2.091 1.579 1.311 1.287 1.278 1.510 1.759 2.353 3.132 4.031 4.082 2.310 Top Girder 0.070 0.569 0.607 1.319 2.106 2.575 2.738 2.879 2.715 2.526 2.041 1.286 0.647 0.612 0.148 Bottom Girder 0.651 2.455 2.407 1.517 0.532 -0.071 -0.321 -0.453 -0.245 0.086 0.744 1.625 2.280 2.312 0.400 Top Girder 0.070 0.524 0.556 1.147 1.792 2.162 2.261 2.368 2.202 2.043 1.588 0.873 0.177 0.139 -0.360 Bottom Girder 0.651 2.600 2.573 2.076 1.548 1.265 1.225 1.200 1.416 1.649 2.207 2.961 3.797 3.844 2.043 Top Girder 0.035 0.312 0.335 0.764 1.236 1.518 1.621 1.689 1.582 1.437 1.126 0.683 0.339 0.321 0.084 Bottom Girder 0.326 1.138 1.102 0.422 -0.326 -0.780 -0.974 -1.033 -0.847 -0.521 0.033 0.683 1.090 1.109 0.168 Top Girder 0.035 0.259 0.275 0.560 0.868 1.038 1.073 1.112 1.014 0.920 0.659 0.278 -0.128 -0.151 -0.428 Bottom Girder 0.326 1.311 1.299 1.083 0.866 0.772 0.800 0.835 0.990 1.154 1.542 1.993 2.600 2.635 1.823 Non-Comp. DL 5 A = 1-4 Location Full PS + DL Composite DL LL + IM (+) LL + IM (-) Restraint Moment (RM) (+) Prestress After Losses Self Weight Special (+) Service + RM Special (-) Service 1 2 3 Service I LL+IM (+) w/RM 7 6 Full PS + DL + RM (+) 4 7 + 0.5*A Service I LL+IM (-) B = 1-5 B + 1.0*6 A + 1.0*7 Service III LL+IM (+) w/RM Service III LL+IM (-) 6 + 0.5*B B + 0.8*6 A + 0.8*7 Notes for Table D-4.1.4.2-3: 1. Maximum stresses for each condition are shaded, with the governing stresses also boxed and bolded. 2. Values for limiting stresses are given in Table D-3.4.2.2-1. 3. Compressive stresses for both dead-load conditions (A and B) are compared with the limiting compressive stress for full dead load fc2 = 3.150 ksi. The maximum stress is 2.894 ksi at the interior transfer length location. 4. Compressive stresses for both service I LL+IM conditions are compared with the limiting compressive stress for full-service conditions fc1 = 4.200 ksi. The maximum stress is 4.082 ksi at the interior transfer length location. 5. Tensile stresses in the precompressed tensile zone for Service III with RM are compared with the limiting tensile stress ft1 = –0.500 ksi. The maximum stress is –0.453 ksi at midspan. 6. Tensile stresses in regions other than the precompressed tensile zone for Service III without RM are compared with the limiting tensile stress ft2 = –0.200 ksi or ft3 = –0.630 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum stress is –0.360 ksi at the interior bearing, so reinforcement is required to resist the tensile force in the concrete. 7. Compressive stresses for the special cases are compared with the limiting compressive stress for that case fc3 = 2.800 ksi. The maximum stress is 2.635 ksi at the interior transfer length location. 8. In all cases, this design satisfies the specified stress limits. TABLE D-4.1.4.2-3 Summary of design stresses for final design at service limit state after losses—continuity at 60 days

continuity resulted in larger positive design moments within the spans, which required an increase in the number of pre- stressing strands and changes to other design parameters. A simple-span design has also been developed using the same girder dimensions used for the spans made continuous. This design was prepared as a benchmark for comparison to the continuous girder designs. Design moments and stresses for this design as given in Section 4.3.1. Figure D-4.3-1 summarizes the significant characteristics of the different designs. The figure clearly shows the benefit of continuity for precast/prestressed concrete girders. The girder design for the simple span required over 20% more strands than did the continuous girder design developed using the simplified approach. The design using girders that were 60 days old when conti- nuity was established required 10.5% more strands than the girders designed using the simplified approach, but less strands than the simple-span design. This design required more effort than the simplified approach, since positive restraint moments were included in the design. However, the design that estab- lished continuity when the girders were 28 days old required nearly 27% more strands than did the girder designed using the simplified approach. This design had more strands than did the simple-span design, so designing for continuity pro- vided no structural advantage. The design also required sig- nificantly more effort in design because, in addition to includ- ing positive restraint moments in the design, the joint was not considered effective in the final design. In fact, as discussed in Section 4.1.4.1, the strand pattern shown will probably not be adequate to satisfy stress limits when the joint is consid- ered partially effective. The positive restraint moments for this design also exceeded the 1.2Mcr limit, as discussed in Section 4.1.4.1. Based on the significant disadvantages men- tioned, this design is not considered to be a viable design. It is recommended to abandon the design for this bridge. D-22 It appears that the simplified approach, with its requirement for girders to be at least 90 days old when continuity is estab- lished, would be the best solution, providing economy in both the structure and in the design process. Figure D-4.3-2 com- pares the development of restraint moments with time for the final designs for the three girder ages at continuity. The increase in restraint moments for the final designs with con- tinuity at earlier ages can be compared with development of restraint moments for the initial designs in Figure D-4.1.3-1. The increase in restraint moment was caused by the increased creep resulting from the increased number of strands. For the design with continuity established at a girder age of 28 days, a portion of the increase was also caused by the change in the ultimate creep coefficient that resulted from in increased concrete strength. 4.3.1 Simple-Span Design for Comparison A simple-span design using the same girder lengths and bearing locations was performed for comparison to the con- tinuous girder designs. The concrete compressive strength at release was increased to equal the specified concrete com- pressive strength at 28 days in order to satisfy design require- ments—that is, f ′c = f ′ci = 7.00 ksi. Otherwise, the properties of the concretes used for the simple-span design are the same as for the continuous girder design with a girder age of at least 90 days when continuity is established, as given in Section 3.3. The following tables of moments and stresses are pro- vided for the simple-span bridge design used for compari- son to the continuous girder designs (see Tables D-4.3.1-1 through D-4.3.1-3). Because of symmetry, moments and stresses are only shown for half of the girder. Moments for noncomposite loads on the simple-span design are the same as shown in Table D-3.7.1-1. Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 1.92 2.20 8.03 16.65 25.27 33.88 42.50 Top Girder (ksi) N/A 0.393 N/A 0.142 –0.211 –0.563 –0.916 –0.916 Prestress at Release Bottom Girder (ksi) N/A 3.129 N/A 3.334 3.623 3.913 4.202 4.202 Top Girder (ksi) N/A 0.144 N/A 0.461 0.820 1.076 1.230 1.281 Self Weight Bottom Girder (ksi) N/A –0.118 N/A –0.378 –0.672 –0.882 –1.008 –1.050 Top Girder (ksi) N/A 0.537 N/A 0.603 0.609 0.513 0.314 0.365 Total at Release Bottom Girder (ksi) N/A 3.011 N/A 2.956 2.951 3.030 3.193 3.151 Notes: 1. Critical stresses are shaded. 2. Values for limiting stresses are given in Table D-3.4.2.1-1. 3. Compressive stresses at release are compared with the limit fcR = 3.300 ksi. The maximum compressive stress is 3.193 ksi at 0.40L. 4. Tensile stresses in regions other than the precompressed tensile zone at release are compared with the limiting tensile stress ftR1 = –0.200 ksi or ftR2 = –0.560 ksi. The latter value requires an area of reinforcement to resist the tensile force. There are no tensile stresses in the concrete at release. 5. In all cases, this design satisfies the specified stress limits at release. TABLE D-4.2.1-1 Summary of design stresses for final design at release—continuity at 90 days

Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L 0.60L 0.70L 0.80L 0.90L H/2 Trans. Brg. From Brg. (ft) 0.00 1.92 2.20 8.03 16.65 25.27 33.88 42.50 51.12 59.73 68.35 76.97 82.80 83.08 85.00 SERVICE STRESSES (ksi) Top Girder 0.091 0.324 0.315 0.117 -0.174 -0.465 -0.756 -0.756 -0.756 -0.465 -0.174 0.117 0.315 0.324 0.091 Bottom Girder 0.590 2.584 2.591 2.753 2.992 3.231 3.469 3.469 3.469 3.231 2.992 2.753 2.591 2.584 0.590 Top Girder 0.000 0.110 0.126 0.427 0.786 1.042 1.196 1.247 1.196 1.042 0.786 0.427 0.126 0.110 0.000 Bottom Girder 0.000 -0.090 -0.103 -0.350 -0.644 -0.854 -0.980 -1.022 -0.980 -0.854 -0.644 -0.350 -0.103 -0.090 0.000 Top Girder 0.000 0.173 0.198 0.673 1.239 1.643 1.885 1.966 1.885 1.643 1.239 0.673 0.198 0.173 0.000 Bottom Girder 0.000 -0.142 -0.163 -0.551 -1.016 -1.346 -1.545 -1.611 -1.545 -1.346 -1.016 -0.551 -0.163 -0.142 0.000 Top Girder 0.000 0.007 0.008 0.026 0.046 0.057 0.059 0.054 0.039 0.016 -0.015 -0.055 -0.087 -0.088 -0.100 Bottom Girder 0.000 -0.022 -0.025 -0.084 -0.148 -0.184 -0.192 -0.173 -0.127 -0.053 0.049 0.178 0.281 0.286 0.323 Top Girder 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Bottom Girder 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Top Girder 0.000 0.046 0.052 0.174 0.305 0.383 0.419 0.415 0.373 0.290 0.175 0.067 0.026 0.024 0.017 Bottom Girder 0.000 -0.149 -0.170 -0.562 -0.987 -1.240 -1.356 -1.344 -1.207 -0.939 -0.565 -0.216 -0.083 -0.079 -0.054 Top Girder 0.000 -0.005 -0.006 -0.023 -0.047 -0.072 -0.096 -0.121 -0.145 -0.170 -0.226 -0.264 -0.361 -0.367 -0.413 Bottom Girder 0.000 0.002 0.020 0.074 0.153 0.232 0.312 0.392 0.470 0.549 0.730 0.853 1.168 1.188 1.336 TOTAL SERVICE STRESSES (ksi) Top Girder 0.091 0.614 0.647 1.243 1.897 2.277 2.384 2.511 2.364 2.236 1.836 1.162 0.552 0.519 -0.009 Bottom Girder 0.590 2.330 2.300 1.768 1.184 0.847 0.752 0.663 0.817 0.978 1.381 2.030 2.606 2.638 0.913 Top Girder 0.091 0.614 0.647 1.243 1.897 2.277 2.384 2.511 2.364 2.236 1.836 1.162 0.552 0.519 -0.009 Bottom Girder 0.590 2.330 2.300 1.768 1.184 0.847 0.752 0.663 0.817 0.978 1.381 2.030 2.606 2.638 0.913 Top Girder 0.091 0.660 0.699 1.417 2.202 2.660 2.803 2.926 2.737 2.526 2.011 1.229 0.578 0.543 0.008 Bottom Girder 0.590 2.181 2.130 1.206 0.197 -0.393 -0.604 -0.681 -0.390 0.039 0.816 1.814 2.523 2.559 0.859 Top Girder 0.091 0.609 0.641 1.220 1.850 2.205 2.288 2.390 2.219 2.066 1.610 0.898 0.191 0.152 -0.422 Bottom Girder 0.590 2.332 2.320 1.842 1.337 1.079 1.064 1.055 1.287 1.527 2.111 2.883 3.774 3.826 2.249 Top Girder 0.091 0.651 0.689 1.382 2.141 2.583 2.719 2.843 2.662 2.468 1.976 1.216 0.573 0.538 0.005 Bottom Girder 0.590 2.211 2.164 1.318 0.394 -0.145 -0.333 -0.412 -0.149 0.227 0.929 1.857 2.540 2.575 0.870 Top Girder 0.091 0.610 0.642 1.225 1.859 2.219 2.307 2.414 2.248 2.100 1.655 0.951 0.263 0.225 -0.339 Bottom Girder 0.590 2.331 2.316 1.827 1.306 1.033 1.002 0.977 1.193 1.417 1.965 2.712 3.540 3.588 1.982 Top Girder 0.046 0.353 0.376 0.796 1.254 1.522 1.611 1.671 1.555 1.408 1.093 0.648 0.302 0.284 0.013 Bottom Girder 0.295 1.016 0.980 0.322 -0.395 -0.817 -0.980 -1.013 -0.799 -0.450 0.126 0.799 1.220 1.240 0.403 Top Girder 0.046 0.302 0.318 0.599 0.902 1.067 1.096 1.135 1.037 0.948 0.692 0.317 -0.085 -0.108 -0.417 Bottom Girder 0.295 1.167 1.170 0.958 0.745 0.656 0.688 0.724 0.879 1.038 1.421 1.868 2.471 2.507 1.793 Service III LL+IM (+) w/RM Service III LL+IM (-) 6 + 0.5*B B + 0.8*6 A + 0.8*7 4 7 + 0.5*A Service I LL+IM (-) B = 1-5 B + 1.0*6 A + 1.0*7 Self Weight Special (+) Service + RM Special (-) Service 1 2 3 Service I LL+IM (+) w/RM 7 6 Full PS + DL + RM (+) Non-Comp. DL 5 A = 1-4 Location Full PS + DL Composite DL LL + IM (+) LL + IM (-) Restraint Moment (RM) (+) Prestress After Losses Notes for Table D-4.2.1-2: 1. Maximum stresses for each condition are shaded, with the governing stresses also boxed and bolded. 2. Values for limiting stresses are given in Table D-3.4.2.2-1. 3. Compressive stresses for both dead-load conditions (A and B) are compared with the limiting compressive stress for full dead load fc2 = 3.150 ksi. The maximum stress is 2.638 ksi at the interior transfer length location. 4. Compressive stresses for both Service I LL+IM conditions are compared with the limiting compressive stress for full-service conditions fc1 = 4.200 ksi. The maximum stress is 3.826 ksi at the interior transfer length location. 5. Tensile stresses in the precompressed tensile zone for Service III with RM are compared with the limiting tensile stress ft1 = –0.500 ksi. The maximum stress is –0.412 ksi at mids pan. 6. Tensile stresses in regions other than the precompressed tensile zone for Service III without RM are compared with the limiting tensile stress ft2 = –0.200 ksi or ft3 = –0.630 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum stress is –0.339 ksi at the interior bearing, so reinforcement is required to resist the tensile force in the concrete. 7. Compressive stresses for the special cases are compared with the limiting compressive stress for that case fc3 = 2.800 ksi. The maximum stress is 2.507 ksi at the interior transfer length location. 8. In all cases, this design satisfies the specified stress limits. TABLE D4.2.1-2 Summary of design stresses for final design at service limit state after losses—continuity at 90 days

Tables D-4.3.1-2 and D-4.3.1-3 present the service limit state stresses for the simple-span design. The stresses are compared with stress limits in the notes following each table. 5 REINFORCEMENT FOR POSITIVE MOMENTS AT INTERIOR SUPPORTS The connections between girders at interior supports of bridges made continuous are subject to positive design moments. However, the moments are caused by minor live- load effects (for more than two-span bridges) and restraint of time-dependent effects, including creep, shrinkage, and tem- D-24 perature. Therefore, the positive design moments are not well- defined. Past research has questioned the benefit of provid- ing a positive moment connection. 5.1 Positive Moment Connection The proposed specifications recommend that positive moment connections be provided between all prestressed con- crete girders made continuous (proposed Article C5.14.1.2.7a). This recommendation is based on providing the connection to enhance the structural integrity of the structure so that it may be more robust and better able to resist unforeseen or Design End of Girder Girder at Midspan f 'ci f 'c Design Moments** Number of Strands Area of Strands Draped: 8 1.224 in 2 5.5 ksi 0.0 k-ft Straight: 30 4.590 in 2 Initial Design 90 Day Final Design Simplified Approach 7.0 ksi 2,429.0 k-ft Total: 38 5.814 in2 -- Draped: 12 * 2.604 in 2 7.5 ksi 945.1 k-ft Straight: 22 * 4.774 in 2 28 Day Final Design General Approach 8.5 ksi 2,909.5 k-ft Total: 34 * 7.378 in2 +26.9% Draped: 8 1.224 in. 2 6.0 ksi 455.4 k-ft Straight: 34 5.202 in. 2 60 Day Final Design General Approach 7.0 ksi 2,656.7 k-ft Total: 42 6.426 in.2 +10.5% Draped: 14 2.142 in. 2 7.0 ksi 0.0 k-ft Simple-Span Design 7.0 2,810.1 k-ft Straight: 32 4.896 in. 2 ksi Total: 46 7.038 in.2 +21.1% * 0.6-in.-diameter strands were required for 28 day final design; other designs used 0.5-in.- diameter strands. **Top number in cells is positive restraint moment at interior support. Bottom number in cells is Service III maximum positive design moment at midspan. Figure D-4.3-1. Summary of designs.

extreme loads. However, if analysis for restraint moments is required and the analysis indicates that a positive moment will develop, the proposed specifications require that a posi- tive moment connection be provided. While not required, a positive moment connection is provided for the design with continuity at 90 days. Reinforcement to resist the positive design moment at the interior support may be provided using either mild reinforce- ment or pretensioning strand, as demonstrated in Sections 5.4 and 5.5. 5.2 Positive Design Moments For the strength limit state, the reinforcement in the positive moment connection is proportioned to provide a factored resis- D-25 tance, φMn, greater than the larger of the factored moment, Mu, or 0.6Mcr, but not to exceed 1.2Mcr. A design moment of 1.2Mcr is typically provided because testing and field experi- ence have shown that this quantity of reinforcement, which is also the minimum quantity of reinforcement required by the AASHTO LRFD Specifications in many cases, has per- formed well. Reinforcement provided in excess of the quan- tity needed to resist 1.2Mcr has been shown to be less effec- tive. Therefore, it is not recommended to use a quantity of reinforcement greater than that required for 1.2Mcr. For the service limit state, the reinforcement is propor- tioned to resist the larger of the service load moment or Mcr. The Service I load combination is used to compute the design moment because the connection is not prestressed. Since this bridge has two spans, the only positive moment that can be developed at the interior support is caused by restraint (considering the effects included in this design exam- ple). Bridges with more spans will develop positive moments at the interior supports from live loads. Because there is no restraint moment for the design with continuity at a girder age of 90 days, there is no positive design moment. 5.2.1 Computation of Positive Cracking Moment at Continuity Diaphragm 5.2.1.1 Section Properties for Continuity Diaphragm. The cracking moment is computed using a cross section com- posed of the girder shape, the build-up, and the effective width of deck. The remainder of the continuity diaphragm could be considered, but the assumed section provides a min- imum area that may be effective. -600 -400 -200 0 200 400 600 800 1000 0 1000 2000 3000 4000 5000 6000 7000 8000 Girder Age (Days) R es tr ai nt M om en t ( k-f t) Continuity at Girder Age of 28 Days Continuity at Girder Age of 60 days Continuity at Girder Age of 90 days Figure D-4.3-2. Development of restraint moment with girder age— final design. Location from Bearing Compos. DL (DC) Compos. DL (DW) LL + IM Service I LL + IM Service III (ft) (k-ft) (k-ft) (k-ft) (k-ft) Bearing 0.0 0.0 0.0 0.0 0.0 Trans. 1.9 11.8 13.5 135.9 108.7 H/2 2.2 13.5 15.5 155.2 124.2 0.10 L 8.0 45.9 52.6 523.7 419.0 0.20 L 16.7 84.4 96.7 952.0 761.6 0.30 L 25.3 112.0 128.3 1,242.1 993.6 0.40 L 33.9 128.5 147.2 1,413.8 1,131.0 0.50 L 42.5 134.0 153.5 1,456.2 1,165.0 TABLE D-4.3.1-1 Service design moments for loads on composite section for simple-span bridge

Since the continuity diaphragm (between the ends of oppos- ing girders) is composed entirely of deck concrete, transfor- mation of the deck is not required. The build-up is included in the computation of these section properties because the full depth of the build-up is specified at the center of bearings, immediately adjacent to the continuity diaphragm. Including the build-up will increase the cracking moment. The composite section properties computed earlier in Sec- tion 3.6.2 are similar, but are not used here because the deck has been transformed (deck and girder concrete compressive strengths are different) and the build-up was neglected. The section properties for the continuity diaphragm are as follows: h′c = total depth of diaphragm section; build-up is included since full build-up must be provided at CL bearings; = 45 + 2.5 + 7.75 = 55.25 in.; A′c = 1,320 in.2; I′c = 436,486 in.4; y′bc = 38.04 in.; y′tcd = 17.21 in.; S′bc = 11,475 in.3; and S′tcd = 25,362 in.3 5.2.1.2 Cracking Moment. The positive cracking moment, Mcr, is computed using the section modulus for the bottom of the continuity diaphragm and the modulus of rupture for the deck concrete, frd , which is given in Section 3.3.2.1: Mcr = frd S′bc ; = 0.480 (11,475); and = 5,508 k-in = 459.0 k-ft. D-26 The concrete strength of the deck, f ′cd, is used for this calcula- tion because the confining effect of the precast girders in the continuity diaphragm is not significant for positive moments, where the deck slab is in compression. For reinforcement limits, the quantity 1.2Mcr is also computed: 1.2Mcr = 550.8 k-ft. 5.3 Minimum Distance Between the Ends of Girders The bridge must be laid out to provide adequate distance between the ends of girders in the continuity diaphragm to develop the reinforcement. This distance depends on the type and size of reinforcement that is used. Computations for determining required development lengths into the continu- ity diaphragm are presented for mild reinforcement and for strands in the next two sections. For proper meshing of the positive moment connection reinforcement, the distance between the ends of girders must also be adequate to allow movement of the bars or strands during erection. The greater the distance, the less bending would be required to provide clearance, especially for strands that cannot be offset to avoid conflicts. The minimum dis- tance between the ends of girders should be determined early in the design process because design spans and continuity diaphragm dimensions depend on this distance. 5.4 Mild Reinforcement Mild reinforcement is often used to provide the positive moment connection. The reinforcement for the connection must extend from the end of the girder and must be anchored into the continuity diaphragm. This reinforcement is usually placed among the pretensioned strands near the bottom of the Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 1.92 2.20 8.03 16.65 25.27 33.88 42.50 Top Girder (ksi) N/A 0.884 N/A 0.531 0.033 –0.464 –0.962 –0.962 Prestress at Release Bottom Girder (ksi) N/A 3.420 N/A 3.709 4.117 4.525 4.933 4.933 Top Girder (ksi) N/A 0.144 N/A 0.461 0.820 1.076 1.230 1.281 Self Weight Bottom Girder (ksi) N/A –0.118 N/A –0.378 –0.672 –0.882 –1.008 –1.050 Top Girder (ksi) N/A 1.028 N/A 0.992 0.853 0.612 0.268 0.319 Total at Release Bottom Girder (ksi) N/A 3.301 N/A 3.331 3.445 3.643 3.925 3.883 Notes: 1. Maximum stresses for each condition are shaded, with the governing stresses also boxed and bolded. 2. Values for limiting stresses are given below because they do not appear in any table. 3. Compressive stresses at release are compared with the limit fcR = 4.200 ksi. The maximum compressive stress is 3.925 ksi at 0.40L. 4. Tensile stresses in regions other than the precompressed tensile zone at release are compared with the limiting tensile stress ftR1 = –0.200 ksi or ftR2 = –0.635 ksi. The latter value requires an area of reinforcement to resist the tensile force. There are no tensile stresses in the concrete at release. 5. In all cases, this design satisfies the specified stress limits at release. TABLE D-4.3.1-2 Summary of design stresses for simple-span design at release

girder in order to maximize the effective depth to the rein- forcement. This additional reinforcement inserted between strands that are usually placed on the standard 2-in. × 2-in. grid increases the congestion of reinforcement at the ends of gird- ers. Therefore, extra attention must be given to this area dur- ing the placement of concrete to avoid lack of consolidation. Hooks are generally provided on the projecting reinforce- ment to improve the development of the reinforcement into the continuity diaphragm and also to shorten the required diaphragm width. The bars may be bent prior to or after fab- rication of the girder, depending on fabricator preferences and clearances within the girder forms. Hairpin bars (180° hooks) are sometimes used to address reinforcement issues. D-27 5.4.1 Development and Detailing of Reinforcement into Continuity Diaphragm It is important that the distance between the ends of oppos- ing girders be great enough within a continuity diaphragm to allow development of the positive reinforcement into the diaphragm. This should be considered during the initial stages of laying out a bridge. The mild reinforcement is developed into the continuity diaphragm using a standard 90° hook. No. 5 bars will be used for the connection. The required length of embedment,
dh, into the diaphragm to develop the No. 5 hooked bar is com- puted according to LRFD Article 5.11.2.4: Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50LLocation from Bearing (ft) 0.00 1.92 2.20 8.03 16.65 25.27 33.88 42.50 SERVICE STRESSES (ksi) Top Girder (ksi) 0.183 0.698 0.685 0.419 0.026 –0.367 –0.760 –0.760Prestress after Losses Bottom Girder (ksi) 0.613 2.700 2.711 2.929 3.251 3.573 3.895 3.895 Top Girder (ksi) 0.000 0.110 0.126 0.427 0.786 1.042 1.196 1.247 Self Weight Bottom Girder (ksi) 0.000 –0.090 –0.103 –0.350 –0.644 –0.854 –0.980 –1.022 Top Girder (ksi) 0.000 0.173 0.198 0.673 1.239 1.643 1.885 1.966 Non-Comp. DL Bottom Girder (ksi) 0.000 –0.142 –0.163 –0.551 –1.016 –1.346 –1.545 –1.611 Top Girder (ksi) 0.000 0.009 0.011 0.036 0.065 0.086 0.099 0.103 Composite DL Bottom Girder (ksi) 0.000 –0.030 –0.034 –0.114 –0.211 –0.279 –0.322 –0.335 Top Girder (ksi) 0.000 0.049 0.056 0.189 0.343 0.447 0.509 0.524 LL + IM Bottom Girder (ksi) 0.000 –0.158 –0.181 –0.610 –1.109 –1.447 –1.647 –1.697 TOTAL SERVICE STRESSES (ksi) Top Girder (ksi) 0.183 0.990 1.019 1.554 2.116 2.404 2.420 2.557 Full PS + DL Bottom Girder (ksi) 0.613 2.439 2.412 1.913 1.381 1.093 1.049 0.927 Top Girder (ksi) 0.183 1.039 1.075 1.743 2.458 2.852 2.929 3.081 Service I Bottom Girder (ksi) 0.613 2.280 2.231 1.303 0.272 –0.354 –0.598 –0.770 Top Girder (ksi) 0.183 1.030 1.064 1.705 2.390 2.762 2.828 2.976 Service III Bottom Girder (ksi) 0.613 2.312 2.267 1.425 0.493 –0.065 –0.269 –0.431 Top Girder (ksi) 0.092 0.544 0.566 0.966 1.401 1.649 1.719 1.803 Special Service Bottom Girder (ksi) 0.307 1.061 1.025 0.346 –0.419 –0.901 –1.123 –1.233 Notes: 1. Maximum stresses for each condition are shaded, with the governing stresses also boxed and bolded. 2. Values for limiting stresses are the same as for the design with continuity at a girder age of 90 days given in Table D-3.4.2.2-1. 3. Compressive stresses for full prestress and dead load are compared with the limiting compressive stress for full deadloadfc2 = 3.150 ksi. The maximum stress is 2.557 ksi at midspan. 4. Compressive stresses for Service I conditions are compared with the limiting compressive stress for full-service conditions fc1 = 4.200 ksi. The maximum stress is 3.081 ksi at the interior transfer length location. 5. Tensile stresses in the precompressed tensile zone for Service III are compared with the limiting tensile stress ft1 = –0.500 ksi. The maximum stress is–0.431 ksi at midspan. 6. Tensile stresses in regions other thanthe precompressed tensile zone for Service III are compared with the limiting tensile stress ft2 = –0.200 ksi orft3 = –0.630 ksi. The latter value requires an area of reinforcement to resist the tensileforce. There are no tensile stresses in the concrete at release. 7. Compressive stresses for the special service combination are compared with the limiting compressive stress for thatcase fc3 = 2.800 ksi. The maximum stress is 1.061 ksi at the interior transfer length location. 8. In all cases, this design satisfies the specified stress limits. TABLE D-4.3.1-3 Summary of design stresses for simple-span design at service limit state after losses

dh = 8.3 in.; USE 81/2 in. LRFD Eq. 5.11.2.4.1-1 A reduction factor of 0.7 was used because conditions pro- vide the required side and end cover. While the clear distance between the hook and the edge of the diaphragm is 1.5 in. (less than the 2 in. required), the face of the diaphragm is confined by the girder concrete. Therefore, this surface is not an exte- rior surface, and it appears appropriate to use the reduction. An embedment of the hook into the continuity diaphragm of 8.5 in. will be used. The distance between ends of girders across the continuity diaphragm will be taken as 10 in., so the 8.5-in. projection to the hook can be provided and still allow for construction tolerances. A small additional reduction in the required hook embed- ment could be taken since the provided area of reinforcement is slightly more than is required by analysis. Using this reduc- tion, the embedment could be reduced to 8 in., but the design of the connection must be complete before the magnitude of this reduction is known. Therefore, 8.5 in. will be used in this example. A cross bar should be placed in the corner of the hooks to enhance the development of the hooked reinforcement into the diaphragm. The cross bar should be at least the same size as the hooked bar. In most cases, bars with 90° hooks cannot be installed in the forms with the hooks prebent. This is because the hooks cannot fit within the girder forms; thus, girders are usually fabricated with straight bars projecting from the ends, and the hooks are bent in the precast plant after fabrication. In some situations, hooked bars may be installed in the forms with the hook tilted to clear the forms. Then, after girder fabrication, the hooked bars can be twisted to make the hooks vertical. A 180° hooked bar (hairpin) may also be used to provide two legs developed into the continuity diaphragm with only one hook. Also, the hooks are prebent, which simplifies fab- rication and eliminates hooks that may not fit within the forms during manufacturing. The use of hairpin bars is especially helpful when a large number of bars is required to satisfy pos- itive moment requirements. The development length for 90° hooked reinforcement should be used to compute the required embedment of 180° hooks into the continuity diaphragm. The placement of anchor bars through the hairpins, as shown in Figure D-5.4.1-1 for 90° hooked bars, is recommended. D-28 5.4.2 Required Area of Reinforcement Using typical strength design procedures and the design moments given in Table D-5.2-1, the required area of rein- forcement is computed to be As = 2.35 in.2 (continuity at 60 and 90 days) and As = 4.92 in.2 (continuity at 28 days), where f ′cd = 4.00 ksi; fy = 60 ksi ; Mu = 1.2Mcr = 550.8 k-ft (continuity at 60 and 90 days) = 1,147.1 k-ft (continuity at 28 days); φ = 0.9; b = width of compression face = effective width of deck slab = 93 in.; hc = total section height including build-up = 55.25 in.; g = distance from bottom of girder to centroid of rein- forcement = 3.00 in. (one row of reinforcement—continuity at 60 and 90 days) = 4.00 in. (two rows of reinforcement—continuity at 28 days); and d = effective depth from top of deck including build-up = hc − g = 52.25 in.(continuity at 60 and 90 days) = 51.25 in. (continuity at 28 days). Girder Age at Continuity 28 Days 60 Days 90 Days Msl (k-ft) 698.1 202.5 N/A Service I Limit State Mcr (k-ft) 459.0 459.0 459.0 Mu (k-ft) 1,147.1 458.7 N/A Strength I Limit State 1.2Mcr (k-ft) 550.8 550.8 550.8 TABLE D-5.2-1 Summary of positive design moments and limits at center of interior support (critical moments are shaded) Figure D-5.4.1-1. Detail of reinforcement placement at positive moment connection (section view).

5.4.2.1 Girder Age at Continuity of 60 or 90 Days. For girders that are 60 and 90 days old when continuity is estab- lished, the required area of reinforcement, As = 2.35 in.2, can be provided using eight No. 5 reinforcing bars. This satisfies the assumption that all of the reinforcement can be placed in a single layer: As prov = 8 (0.31 in.2) = 2.48 in.2 > As = 2.35 in.2 OK. A layout for the reinforcement is shown in Figure D-5.4.2-1 with opposing bars offset to allow meshing. 5.4.2.2 Girder Age at Continuity of 28 Days. For girders that are 28 days old when continuity is established, the required area of reinforcement is 4.92 in.2—nearly twice the area required for continuity established at the later ages. This area can be provided using 15 No. 5 reinforcing bars. To provide a symmetrical connection, the number of reinforcing bars is rounded up to an even number, 16. As prov = 4.96 in.2 > As = 4.92 in.2 OK. Using No. 5 bars, this area requires two layers of reinforce- ment, which greatly complicates detailing, fabrication, and erection. A larger bar size could also be used to reduce the number of bars, but this would require a greater distance between the ends of girders to properly develop the hooked bars. This large required area of reinforcement and the accompanying complications in detailing are other reasons to require that continuity to be established at girder ages greater than 28 days. 5.4.3 Development and Detailing of Reinforcement into the Girder The required development length into the girder for the positive moment reinforcement is computed as follows: D-29
dh = 15.0 in. LRFD Art. 5.11.2.1.1 This length applies for designs with continuity at all ages considered. The total length of embedment of the positive moment reinforcement into the girder should be carefully considered. It is recommended that the bar be developed beyond where an assumed crack radiating at a 2
1 slope from the inner edge of the bearing or where an embedded plate intersects the reinforcement. The general concept is illustrated in Fig- ure D-5.4.3-1, with the details of the connection for this bar shown in Figure D-5.4.2-1. Where multiple bars are required for positive moment rein- forcement, at least two different embedment lengths should be used to avoid stress concentrations and potential cracking where the bars terminate in the girder. One bar type should provide the minimum embedment length, with the second bar type providing at least an additional 1.5 ft of embedment. Where hairpin bars are used, the staggering of bar termina- tions can be accomplished by using a single bar type that is detailed with unequal legs. 5.4.4 Constructability Issues Reinforcement projecting from the end of a girder is detailed to mesh with the reinforcement projecting from the opposing girder. This is intended to provide an essentially continuous load path for any tension that may develop at the bottom of the connection, requiring that the reinforce- ment be detailed to mesh. To accomplish this, the bars must be offset or bent so that conflicts between bars are mini- mized or eliminated. The use of offset bars is illustrated for the design with con- tinuity at 60 and 90 days in Figure D-5.4.4-1. The bars are placed in a slightly eccentric pattern. This pattern simplifies fabrication and erection by allowing use of a single detail for the placement of reinforcement in all girders, avoiding fabri- cation errors, and providing a positive offset between bars. Figure D-5.4.2-1. Details of reinforcement placement at end of girder.

As mentioned in Section 5.4.2.2, the use of two layers of reinforcement greatly complicates fabrication and erection of the girders, as well as placement of reinforcement in the con- tinuity diaphragm. If possible, a single layer of reinforcement should be detailed. However, the available locations for place- ment of reinforcement are limited. Furthermore, the use of a larger number of smaller bars allows for a smaller gap between ends of opposing girders than is possible with fewer larger bars. The smaller bars also require a shorter embedment into the girder. These factors tend to lead designers to use a larger number of smaller bars. The placement of the positive moment connection rein- forcement between pretensioning strands increases conges- tion; this is significant because the congestion can inhibit the proper consolidation of concrete in the critical bearing area. Reinforcement should be positioned to facilitate placement and consolidation of concrete around the strands and bars. D-30 Reinforcement projections must be detailed to allow toler- ances in bar projections, girder lengths, and girder placement at erection. In this example, the distance from end of the hook to the face of the opposing girder is detailed as 11/2 in., which should provide an adequate tolerance. When developing a layout for the positive moment reinforcement, the locations of all other reinforcement and embedments must be carefully considered to avoid conflicts and provide adequate tolerances for fabrication. One example is that the positive moment rein- forcement must avoid locations of headed studs attached to embedded plates. 5.4.5 Control of Cracking by Distribution of Reinforcement 5.4.5.1 Girder Age at Continuity of 60 or 90 Days. Accord- ing to LRFD Article 5.7.3.4, the reinforcement will be pro-  dh Figure D-5.4.3-1. Detail for embedment of reinforcement into girder. Figure D-5.4.4-1. Detail of reinforcement placement at positive moment connection (plan view).

portioned so that the tensile stress in the mild steel rein- forcement at the service limit state does not exceed the stress limit given by LRFD Equation 5.7.3.4-1. The tensile stress in the mild reinforcement at service limit state is computed to be as follows: where Msl = 202.5 k-ft (from Table D-5.2-1); As = 2.48 in.2; fy = 60 ksi; Es = 29,000 ksi; Ecd = 3,834 ksi; n = Es /Ecd = 7.564; b = width of compression face = effective width of deck slab = 93 in.; d = effective depth to positive moment reinforcement from top of deck, including build-up = 52.25 in. (Note: the height of the build-up is included in this calculation because it is specified at the centerline of the bearings); The allowable tensile stress in the mild reinforcement is LRFD Eq. 5.7.3.4-1 where dc = 2.00 + db /2 = 2.00 + 0.625/2 = 2.31 in.; Z = 170 k/in. (moderate conditions); b′ = width of bottom flange; = 22 in.; and A d bc= ′ = =2 2 2 31 228 12 71No. of Bars in. 2( . )( ) . . f Z d A f f sa c y sa = ≤ = ≤ × = > = ( ) ( ) 1 3 1 3 0 6 170 2 31 12 71 0 6 60 55 1 36 0 36 0 / / . . ( . ) . . . . ksi USE ksi, j k= − = − =1 3 1 0 084 3 0 972 . . . k n n n = + − = + ( ) − = 2 2 0 00051 7 564 0 00051 7 564 0 00051 7 564 0 084 2 2 ρ ρ ρ ( ) ( . )( . ) ( . )( . ) ( . )( . ) . ; and ρ = = =Abd s 2 48 93 52 25 0 00051 . ( . ) . ; f MA jds sl s = = = 202 5 12 2 48 0 972 52 25 19 3 . ( ) . ( . )( . ) . ksi, D-31 For girders that are 60 and 90 days old when continuity is established, the tensile stress in the mild reinforcement is less than the allowable. Therefore, the distribution of reinforce- ment for control of cracking is adequate: fsa = 36.0 ksi > fs = 19.3 ksi. OK. 5.4.5.2 Girder Age at Continuity of 28 Days. For girders that are 28 days old when continuity is established, the same procedure as shown in Section 5.4.5.1 is used to check distri- bution of reinforcement. Equations and common values are not repeated here: Msl = 698.1 k-ft (from Table D-5.2-1); As = 4.96 in.2; d = 51.25 in. (two layers of reinforcement are required); ρ = 0.00104; k = 0.118; j = 0.961; fs = 34.3 ksi; dc = 3.31 in.; A = 9.10 in.2; and fsa = 54.6 > 36.0 ksi; USE fsa = 36.0 ksi. (maximum allowed) For girders that are 28 days old when continuity is estab- lished, the tensile stress in the mild reinforcement is less the allowable stress by 1.7 ksi. Therefore, the distribution of reinforcement for control of cracking is adequate: fsa = 36.0 ksi > fs = 34.3 ksi. OK. 5.4.6 Fatigue of Positive Moment Connection Reinforcement The reinforcement in the positive moment connection is not subjected to tension from the live load for this two-span bridge, so fatigue does not need to be investigated. The pos- itive moments caused from restraint moments or temperature effects do not occur frequently enough to be considered as loadings that cause fatigue. However, for other bridge con- figurations, where the connection is subjected to tension from live loads, fatigue should be considered. See Section 6.2.5 for the approach used for negative moment connection.

5.5 Pretensioning Strand An alternate positive moment connection uses pretension- ing strands extended into the continuity diaphragm. A posi- tive moment connection that uses the pretensioning strands provides the connection without increasing congestion in the end of the girder. However, girder fabrication may be affected if the required length of extended strand is large. The service load stress limit for strands used in positive moment connec- tions serves to limit the length of strand that can be effectively used for the connection. Extended strands used for the positive moment connection must be bonded at the end of the girder— that is, they cannot be shielded or debonded at the end of the girder. 5.5.1 Development and Detailing of Extended Strands into Continuity Diaphragm The strands are developed into the continuity diaphragm using a 90° hook. The strand must be bent so the hook proj- ects at least 8 in. from the end of the girder. This distance is required by the equations used in Section 5.5.2. The distance between ends of girders in the continuity diaphragm is detailed as 10 in., so the 8-in. projection to the bent strand can be pro- vided and still allow for construction tolerances. A cross bar should be placed in the corner of the hooks to enhance the development of the hooked reinforcement into the diaphragm. The cross bar should have an area not less than the area of the strand. A No. 5 bar is shown in Figure D-5.5.1-1. 5.5.2 Required Area of Strands The area of strands required to resist the positive design moments (see Table D-5.5.2-1) is computed using both the strength design provisions given in proposed Article 5.14.1.2.7i and service load (working stress) design procedures. The working stress design procedures may be found in concrete D-32 design textbooks. Working stress design is used in addition to strength design for this calculation because the procedure is based on results of research (Salmons, 1974, 1980; Salmons and May, 1974; Salmons and McCrate, 1973). The researchers proposed a design methodology that included both approaches. The design moment for the working stress check will be Mcr, while the design moment for the strength check is 1.2Mcr. Initially, it was assumed that the total length of extended strand,
sh , was 44 in. This provides a length of strand beyond the bend (vertical leg) of 44 − 8 = 36 in. The vertical leg extends for much of the depth of the girder to maximize the development of the strand, while not using so much strand that fabrication of the girders is adversely affected. The resulting stresses in the 0.5-in.-diameter strands at ser- vice and strength conditions are fpsl = [(44 − 8)/0.228] proposed Eq. 5.14.1.2.7i-1 = 158 ksi > 150 ksi, and fpul = [(44 − 8)/0.163] proposed Eq. 5.14.1.2.7i-2 = 221 ksi. Since fpsl exceeds the maximum service load stress of 150 ksi that applies to proposed Equation 5.14.1.2.7i-1, the length of strand was reduced to provide a computed fpsl just below the limit. The resulting total length of strand extension was 42 in., with a vertical leg of 34 in. The revised stress lim- its were computed as fpsl = [(42 − 8)/0.228] proposed Eq. 5.14.1.2.7i-1 = 149 ksi < 150 ksi, and fpul = [(42 − 8)/0.163] proposed Eq. 5.14.1.2.7i-2 = 209 ksi. For the design with continuity at 28 days, 0.6-in.-diameter strands are used. Since the conventional development length for strands increases linearly with increasing strand diameter (LRFD Equation 5.11.4.2-1), the above equations have been modified by introducing the factor (0.5/db) to reflect an increased development length beyond the hook for larger strands. These equations are shown below for the 0.6-in.- diameter strands at service and strength conditions: fpsl = [{(0.5/0.6)44 − 8/0.228] proposed Eq. 5.14.1.2.7i-1 = 126 ksi < 150 ksi, and fpul = [{(0.5/0.6) 44 − 8/0.163] proposed Eq. 5.14.1.2.7i-2 = 176 ksi. The computed service load stress for the initial strand exten- sion length (44 in.) does not exceed 150 ksi for the larger strand; thus, it is used in the initial design computations. Based on these stress limits for service and strength limit states, the number of strands required to resist the positive Figure D5.5.1-1. Detail of strand placement at positive moment connection (section view).

design moments are computed for both girders with continu- ity at 28 days and for 60 and 90 days. Design moments and other parameters are found in Table D-5.2-1. Results are given in Table D-5.5.2-1. The connection for the design with girder age of 28 days at continuity is governed by strength design; the connection for the design with girders made continuous at later ages is governed by working stress design, as indicated by the shaded cells in the table. A second design iteration is used to deter- mine the least length of extended strand required for the design moments for both girder ages at continuity. The first iterations are performed using the initial length of extended strand. The results indicate an odd number of strands is required for both girder ages at continuity. An even number of extended strands is desirable to provide symmetry in the connection. The number of strands was then rounded up to an even number. Using the increased number of strands, the required stress in the strands is computed, using the stress equation for which- ever design approach governs (working stress or strength). The required stress is then used to compute the required length of strand extension. This new length of strand extension is then used for the second iteration. These calculations are illus- D-33 trated below for the design of the girders with continuity established at an age of 60 and 90 days, where working stress design governed: F = Aps fpsl = (0.705)(149.1) = 105.1 kips; Aps = 6 (0.153) = 0.918 in.2; and fpsl2 = F/Aps = 105.1/0.918 = 114.5 ksi. Solving for the length of extended strand required to develop this stress is as follows: fpsl = [
dsh – 8]/0.228, then proposed Eq. 5.14.1.2.7i-1
dsh = 0.228 fpsl + 8 = 0.228(114.5) + 8 = 34.1 in. USE
dsh = 35 in. The stress can then be recomputed using the 35-in.-strand extension: fpsl = [35 − 8]/0.228 = 118.4 ksi. Iteration: 1 2 1 2 ldsh (in.) 44 39 42 35 b (in.) 93 93 93 93 d (in.) 53.25 53.25 53.25 53.25 f 'cd (ksi) 4.00 4.00 4.00 4.00 Strand Diameter (in.) 0.6 0.6 0.5 0.5 Design Moment (k-ft) 1147.1 1147.1 550.8 550.8 fpul (ksi) 175.9 150.1 208.6 165.6 a (in.) 0.82 0.82 0.39 0.39 Aps (in2) 1.481 1.736 0.597 0.752 No. of Strands Req'd 6.8 8.0 3.9 4.9 Design Moment (k-ft) 698.1 698.1 459.0 459.0 Ec (ksi) 3605 3605 3605 3605 Es (ksi) 29000 29000 29000 29000 n 8 .04 8.04 8.04 8.04 fs (ksi) 125.7 107.5 149.1 118.4 jd (in.) 52.14 52.05 52.42 52.32 k 0 .062 0.067 0.047 0.052 rho 0.000258 0.000302 0.000142 0.000180 As (in2) 1.278 1.498 0.705 0.889 No. of Strands Req'd 5.9 6.9 4.6 5.8 60 & 90 Days Girder Age at Continuity WORKING STRESS DESIGN STRENGTH DESIGN 28 Days BASIC DESIGN INFORMATION TABLE D-5.5.2-1 Design information for positive moment connection using strand (critical values are shaded)

This stress is then used in the working stress computations to compute a required area of strand. The required area of strand is slightly less than the 6 strands determined in the computa- tions above, so no further iterations are required (see Figure D-5.5.2-1). 5.5.3 Constructability Issues Pretensioning strands are on a fixed layout within the girder. If the number of strands required for the positive moment con- nection is relatively small, a pattern of strands may be devel- oped that allows strands to mesh when the girders are erected. However, in this example, nearly all of the strands in the bot- tom row are required to make the connection. The strands must therefore be pushed over during erection to allow proper girder placement, as shown in Figure D-5.5.3-1. The D-34 length of strand extension must be adequate to allow this bending during erection of the girder, considering the size of the strand. 5.5.4 Crack Control The allowable tensile stress in the untensioned preten- sioning strands in the connection would be computed using the same procedure as is used in Section 5.4.5. 5.5.5 Fatigue of Positive Moment Connection Reinforcement For information on fatigue of positive moment connection reinforcement, see Section 5.4.6. Figure D-5.5.2-1. Details of strand placement at end of girder. Figure D-5.5.3-1. Detail of strand placement at positive moment connection (plan view).

6 REINFORCEMENT FOR NEGATIVE MOMENTS AT INTERIOR SUPPORTS The proper design of reinforcement at the negative moment connection is essential for the bridge to behave as a contin- uous structure and to provide the desired service life with little or no maintenance. This section presents the necessary steps for the design of the reinforcement for the negative moment connection. 6.1 Negative Design Moments Negative moments at interior supports of precast/prestressed concrete girders made continuous result from dead loads, live loads, and restraint moments. Negative restraint moments, however, are ignored in design, as allowed by the proposed Article C5.14.1.2.7b. Therefore, the negative design moments shown below result from dead loads and live loads only (see Table D-6-1). The negative moment connection is propor- tioned for the strength limit state, so design moments are factored. 6.1.1 Computation of Negative Cracking Moment at Continuity Diaphragm Section properties for the continuity diaphragm used to compute the cracking moment are discussed and given in Section 5.2.1.1. The negative cracking moment, Mcr, is com- puted using the section modulus for the top of the continuity diaphragm and the modulus of rupture for the deck concrete, frd, which is given in Section 3.3.2.1: Mcr = frd S′tc = 0.480(25,362) = 12,174 k-in = 1,014.5 k-ft. For reinforcement limits, the quantity 1.2Mcr is also computed: 1.2Mcr = 1,217.4 k-ft. 6.2 Negative Moment Connection The connections between girders at interior piers of a bridge with precast girders made continuous are subject to significant negative design moments. The portion of the bridge subjected D-35 to negative moments must therefore be properly reinforced to resist these negative moments if it is to perform as a continu- ous member and if the structure is to have good serviceability. The negative moment connection for precast/prestressed concrete girders made continuous is typically made using rein- forcement in the deck. Longitudinal reinforcement in the deck, which includes distribution reinforcement and other mini- mum reinforcement, can be used for this connection. The total area of reinforcement required to resist the negative design moments is computed using strength design. Fatigue of the reinforcement must also be checked. Concepts have been proposed in which the negative moment connection may be made prior to deck placement in order to make the girders continuous for the dead load of the deck. However, these concepts are not considered here. Detailing of the deck reinforcement, which is important for economy and to avoid congestion, is also addressed. 6.2.1 Required Area of Reinforcement Longitudinal reinforcement is required in all deck slabs to satisfy distribution or detail reinforcement requirements. For this bridge, the typical longitudinal deck reinforcement satisfying the distribution requirements is shown in Figure D-6.2.1-1. The area of the longitudinal reinforcement within the effective tension flange width (see Section 6.2.2.1) is shown in Table D-6.2.1-1. Based on the factored negative design moment at the interior support and neglecting restraint moments, a total required area of reinforcement is computed as shown in Table D-6.2.1-1. The build-up is included in the effective depth as discussed in Section 5.2.1. The effective depth is taken as the distance from the bottom of the dia- phragm to the center of the deck slab, assuming that the area of deck reinforcement is equal on the top and bottom of the slab. The typical longitudinal deck reinforcement is included in this area of reinforcement available to resist the factored negative design moment (LRFD Article 5.14.1.2.7b and pro- posed Article 5.14.1.2.7h). For girders with continuity established at 60 and 90 days, the total area of deck reinforcement required to resist the fac- tored negative design moment is computed using strength design as follows: As = 11.52 in.2 (total area within the effective width of 93 in.; see Section 6.2.2.1), where f ′c = 7.00 ksi (use girder concrete strength, proposed Arti- cle 5.14.1.2.7j); fy = 60 ksi; Mu = 2,526.8 k-ft; φ = 0.90 (for reinforced concrete since continuity dia- phragm is not prestressed); Girder Age at Continuity 28 Days 60 Days 90 Days Msl (k-ft) 1,515.8 1,505.5 1,505.5 Service I Limit State Mcr (k-ft) –1,014.5 –1,014.5 –1,014.5 Mu (k-ft) –2,544.1 –2,526.8 –2,526.8 Strength I Limit State 1.2Mcr (k-ft) –1,217.4 –1,217.4 –1,217.4 TABLE D-6-1 Summary of negative design moments and limits at center of interior support (critical moments are shaded)

b′ = width of compression face = width of girder bottom flange = 22.0 in.; d = effective depth to negative moment reinforcement from the bottom of girder and continuity diaphragm to center of deck including build-up (see discussion in Section 5.4.5.1); the center of the deck is an approximate location that could be refined consider- ing details of deck reinforcement = 51.375 in.; a = 2.82 in.; and c = 4.03 in. Checking the maximum reinforcement limit, c/d = 4.03/51.375 = 0.079 < 0.42 OK Use 15 No. 5 and 12 No. 7 = 11.85 in.2 > 11.52 in.2 OK Results for girders with conditions established at 28 days are shown in Table D-6.2.1-1, which gives the area of additional reinforcement required to resist the negative design moment. D-36 This area will be provided by placing bars between the typical longitudinal reinforcement, as shown in Figure D-6.2.1-2. 6.2.2 Details of Deck Reinforcement Detailing of reinforcement for the negative moment con- nection is generally simple since the reinforcement consists of straight bars placed in the deck. However, several issues must be considered in detailing this reinforcement. 6.2.2.1 Effective Tension Flange Width. When the deck slab in a typical girder is subjected to tension from the effects of negative moments at the service limit state, tension rein- forcement effective in resisting the tension must be placed within the width of deck equal to the lesser of (LRFD Arti- cle 5.7.3.4): 1. The effective flange width (LRFD Article 4.6.2.6; see Section 3.6.2.1). For this design example, the average Figure D-6.2.1-1. Typical longitudinal deck reinforcement (partial section view). Girder Age at Continuity 28 Days 60 & 90 Days Typical Longitudinal Deck Reinforcement (Fig. D6.2.1-1) No. 5 @ 18 in. Top No. 5 @ 9 in. Bottom No. 5 @ 18 in. Top No. 5 @ 9 in. Bottom Total Area of Longitudinal Reinforcement Provided (in. 2) 4.65 4.65 Factored Negative Design Moment (k-ft) 2,544.1 2,526.8 Total Area Required to Resist Negative Moment (in. 2) 11.61 11.52 Additional Area of Deck Reinforcement Required (in. 2) 6.96 6.87 Additional Reinforcement Provided (Fig. D-6.2.1-2) 12 No. 7 Bars 12 No. 7 Bars Additional Area of Deck Reinforcement Provided (in. 2) 7.20 7.20 Total As Provided (in.2) 11.85 > 11.61 OK 11.85 > 11.52 OK c/de 0.079 0.079 TABLE D-6.2.1-1 Design of deck reinforcement for negative design moments

girder spacing controls resulting in an effective deck width of 93 in. GOVERNS. 2. One-tenth of the average of adjacent spans between bearings = 0.10(86)(12) = 103.2 in. For this design example, the effective flange width computed earlier governs. Therefore, the full width of deck is effective. Where a portion of the flange width is not within the effec- tive width defined above, an area of steel must be provided that is equal to at least 0.4% of the deck outside of the effec- tive flange width. 6.2.2.2 Anchorage of Deck Reinforcement. The LRFD Specifications indicate that the longitudinal reinforcement resisting the negative design moments must be anchored in concrete that is in compression at the strength limit state (LRFD Article 5.14.1.2.7b and proposed Article 5.14.1.2.7h). Therefore, the inflection points for the negative moment envelope at the strength limit state must be located. Figure D-6.2.2.2-1 indicates the extent of the negative moment region at strength limit state. Additional deck rein- D-37 forcement for the factored negative design moment must be extended past the indicated locations for at least a develop- ment length. Therefore, the minimum length of the No. 7 reinforcing bars added at the interior support would be
min = 2(22.8 ft) + 2
d = 2(22.8 ft) + 2(22.5 in./12) = 49.35 ft, where
d = 22.5 in. (for No. 7 reinforcing bars). LRFD Art. 5.11.2.1.1 6.2.2.3 Termination of Deck Reinforcement. Although all of the additional deck reinforcement theoretically can be ter- minated at the locations shown in Figure D-6.2.2.2-1, this would probably not provide good serviceability. It is recom- mended that the same reinforcing bar mark be used, but that alternate reinforcing bars be shifted several feet to stagger the terminations. This would require that the minimum length of reinforcing bar computed above would be increased by the amount of the shift. Figure D-6.2.1-2. Longitudinal deck reinforcement for negative moment at interior support (partial section view). -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 0 20 40 60 80 100 120 140 160 180 Distance along Girders from CL Bearing (ft) M om en t E nv el op e (k- ft) 22.8 ft 22.8 ft Figure 6.2.2.2-1. Negative moment envelope and location of inflection points—continuity at 60 days.

6.2.3 Constructability Issues In this example, there are no significant constructability issues to address regarding the negative moment reinforce- ment. In some cases, a large quantity of reinforcement must be added to the typical longitudinal deck reinforcement. This can lead to difficulties in the placement of reinforcement or in placement of splices for the reinforcement. The distance between layers of reinforcement must be considered when the larger bar sizes are needed to provide a large area of reinforcement. 6.2.4 Control of Cracking by Distribution of Reinforcement 6.2.4.1 Girder Age at Continuity of 60 or 90 Days. Accord- ing to LRFD Article 5.7.3.4, the reinforcement should be proportioned so that the tensile stress in the mild steel rein- forcement at the service limit state does not exceed the stress limit given by LRFD Equation 5.7.3.4-1. The tensile stress in the mild reinforcement is computed to be where Msl = 1,505.5 k-ft (from Table D-6.1-1); As = 11.85 in.2; fy = 60 ksi; Es = 29,000 ksi; Ec = 5,072 ksi (use girder concrete strength, proposed Article 5.14.1.2.7j); n = Es /Ec = 5.718; b′ = width of compression face = width of bottom flange of girder = 22 in.; d = effective depth from bottom of girder and continu- ity diaphragm to center of deck including build-up = 51.375 in.; The allowable tensile stress in the mild reinforcement is j k= − = − =1 3 1 0 291 3 0 903 . . . k n n n = + − = + ( ) − = 2 2 0 01048 5 718 0 01048 5 718 0 01048 5 718 0 291 2 2 ρ ρ ρ ( ) ( . )( . ) ( . )( . ) ( . )( . ) . ; and ρ = = =Abd s 11 85 22 51 375 0 01048 . ( . ) . ; f MA jds sl s = = = 1 505 5 12 11 85 0 903 51 375 32 9 , . ( ) . ( . )( . ) . ksi, D-38 LRFD Eq. 5.7.3.4-1 where dc = 2.00 + (7/8)/2 (neglecting bottom row of deck rein- forcement for simplicity) = 2.44 in.; Z = 170 k/in. (moderate conditions); b = 93 in.; and For girders that are 60 and 90 days old when continuity is established, the tensile stress in the mild reinforcement is less than is allowable. Thus, the distribution of reinforcement for control of cracking is adequate: fsa = 36.0 ksi > fs = 32.9 ksi OK. 6.2.5 Fatigue of Negative Moment Connection Reinforcement LRFD Article 5.5.3.1 states that “fatigue need not be inves- tigated for concrete deck slabs in multigirder applications.” However, the commentary for the article indicates that this provision is based on the observation of low measured stresses in deck slabs, which is “most probably due to inter- nal arching action.” The longitudinal reinforcement, which is acting as main reinforcement for the negative design moments at the interior support, is not subject to arching action. Therefore, fatigue of the deck slab reinforcement at the interior support is checked in this example. Furthermore, the deck slab is not in compression from the design loads (see the second paragraph of LRFD Article 5.5.3.1), so it is not exempt from fatigue considerations. To determine whether cracked section properties must be used to evaluate the stress range, the stress in the concrete is com- puted under a specified loading combination: M = sum of unfactored permanent loads + prestress + 1.5(fatigue load) = −294 + 0 + 1.5(−193) (fatigue moment is computed below) = −584 k-ft; and fcd top = M/S′tcd = M/(I′c /[h′c − y′bc]) = −584(12)/(436,713/(55.25 − 38.03) = −0.276 ksi. A d bc= ′ = =2 2 2 44 9316 28 37No. of Bars in. 2( . )( ) . . f Z d A f f sa c y sa = ≤ = ≤ × = > = ( ) ( ) 1 3 1 3 0 6 170 2 44 28 37 0 6 60 41 4 36 0 36 0 / / . . ( . ) . . . . ksi USE ksi,

The stress limit is then computed for comparison to com- puted stress: fcd max = 0.095√ f ′cd = 0.095√4.0 = 0.190 ksi. Comparing the absolute values of the computed stress and stress limit, fcd top = 0.276 ksi > fcd max = 0.190 ksi. Since the computed tensile stress exceeds the stress limit, the effect of fatigue must be evaluated using cracked section analysis. The stress in the deck slab reinforcement is computed using the specified fatigue loading and cracked section analysis as required. The following computations are based on the designs for girders made continuous at 60 and 90 days. The results for continuity established at 28 days will differ slightly, due to the difference in live-load distribution factors. The basic negative fatigue moment is computed to be −582 k-ft, using the QConBridge program (see Subappen- dix C). The analysis assumes the bridge is fully continuous for live loads. The moment is then factored by the dynamic load allowance for fatigue (1 + 0.15, LRFD Article 3.6.2.1); the live-load distribution factor for one lane loaded (0.461, LRFD Article 3.6.1.4.3b; and LRFD Table 4.6.2.2.2b-1, cross- section type (k), which is divided by the multiple presence fac- tor for one lane loaded (1.2, LRFD Article C3.6.1.1.2) and the load factor for the fatigue limit state, which is 0.75 (LRFD Table 3.4.1-1). Therefore, the design fatigue moment is as follows: Mf = 1.15(0.461/1.2)(0.75)(−582) = −193 k-ft. The area of reinforcement provided in the deck for the inte- rior girder, as shown in Table D-6.2.1-1, is As = 11.85 in.2. The stress range in the reinforcement caused by this design fatigue moment is computed using the principles of working stress design for a cracked section (see Section 6.2.4.1). The compression zone extends through the bottom flange and into the web, so the average of the web and bottom flange width, 14.5 in., is used as a conservative estimate for the width of the compression zone: ff = 4.299 ksi. The limiting stress range, ff max, is computed as ff max = 21 − 0.33 fmin + 8 (r/h) LRFD Eq. 5.5.3.2-1 = 21 − 0.33(4.299) + 8(0.3) = 22.0 ksi, D-39 where fmin = fLLf + fDL = 0.000 + 4.299 = 4.299 ksi; fLLf = minimum live-load stress from fatigue loading = 0.000 ksi; fDL = stress in reinforcement from permanent loads from working stress analysis = 4.299 ksi; MDL = total composite dead-load moment (load factor = 1.0) at the center of the pier = −294 k-ft (see Table D-3.7.1-2); and r/h = ratio of base radius to height of rolled on transverse deformations in the reinforcement; 0.3 may be used if value not known. Comparing the computed stress range with the limiting stress range, ff = 4.299 ksi < ff max = 22.0 ksi OK. 7 DESIGN WITH NONLINEAR ANALYSIS This section provides a brief overview of the results of a nonlinear analysis for this example, DE1. Nonlinear analysis takes into account the changing stiffness of the different sec- tions in the girder as loading is increased. This allows a more refined estimate of the behavior of the bridge with time as time-dependent effects develop or as live load is applied. The age of the girders when continuity is established is consid- ered in determining the behavior of the bridge with time and with increasing load. Several limitations to the nonlinear analysis are discussed. Results from this analysis should be considered as an indica- tion of behavior, but should not be used for design without further development and verification. 7.1 Analysis The Restraint Program is used to perform the nonlinear analysis of this example. The analyses performed using Restraint in Section 4.1.3 have used linear analysis. The soft- ware is only capable of performing nonlinear analysis for two-span bridges. The program first performs an analysis to estimate the development of restraint moments in the girders with time. This is similar to the analysis performed earlier in this example. Then, a live-load analysis is performed, with the live load applied at the final girder age. The live load used in this analysis is a pair of concentrated loads, one at the midspan of each span. Therefore, the loading does not consider moving loads, nor does it consider only one span loaded, which would produce the maximum positive moment in the loaded span.

To perform the nonlinear analysis, the user must define the moment-curvature relationship for the following cross sec- tions and locations: • The continuity diaphragm, which is not prestressed and is reinforced with untensioned strands or mild reinforcement; • The precast prestressed concrete girder with composite deck slab at midspan; and • The precast prestressed concrete girder with composite deck slab at the end of the girder. The program allows the input of different moment-curvature relationships at the midspan and end of girder locations because the prestressing strand patterns will be different at these locations if draped strands are used. The Response 2000 Program was used to generate the moment-curvature relationships for use in Restraint. Infor- mation regarding the website from which the program can be obtained and results from the various analyses, including plots of the required moment-curvature relationships, are presented in Subappendix B. 7.2 Results The results of the linear and nonlinear analyses are shown in Figure D-7.2-1 for girders with continuity established at D-40 the ages of 28, 60, and 90 days. The lines representing linear behavior shown in this figure are the same lines shown in Figure D-4.3-2 for the final designs discussed in the section summarizing the general and simplified approaches. The final restraint moments corresponding to the restraint moments at 7,500 days (see Figure D-7.2-1) are summarized in Table D-7.2-1. The results for linear analysis shown in this table dif- fer slightly from the results reported earlier because these are for girder ages of 7,500 days, while the previous results were for 10,000 days. The nonlinear analysis does not extend to 10,000 days. As shown in Figure D-7.2-1 and Table D-7.2-1, results of the nonlinear analysis are not significantly differ- ent from the results of the linear analysis used for design ear- lier in this example. In general, the results of the nonlinear analysis are less than the results from the linear analysis. This is true for all situations for both positive and negative moments, except for the final positive restraint moment for continuity at 90 days. The final moments at 90 days for the two analysis methods are on both sides of the origin, making a relative comparison of values meaningless. Using nonlinear analysis, the positive moments were reduced by 11.1%, while the negative moments were reduced by a maximum of 9.1%. However, the reduc- tion of the final positive moment for the design with conti- nuity established when the girder is 28 days old is significant. It reduces the moment below the moment limit used as an indi- cation of whether the joint (continuity diaphragm) is fully effective. While this is an important change, it would not make -1000 -800 -600 -400 -200 0 200 400 600 800 1000 0 1000 2000 3000 4000 5000 6000 7000 Girder Age (Days) R es tr ai nt M om en t ( k-f t) Continuity at Girder Age of 28 Days (Linear) Continuity at Girder Age of 28 Days (Nonlinear) Continuity at Girder Age of 60 Days (Linear) Continuity at Girder Age of 60 Days (Nonlinear) Continuity at Girder Age of 90 Days (Linear) Continuity at Girder Age of 90 Days (Nonlinear) Figure D-7.2-1. Development of restraint moment with girder age— comparison of linear and nonlinear analysis results.

the design viable because the positive restraint moment is still well above the 1.2Mcr limit of 550.8 k-ft (see Section 5.2.1.2). It appears that the nonlinear analysis reveals that the struc- ture is not as stiff as would be assumed using the conventional elastic analysis. This may explain the reduction in both the maximum positive and negative moments because restraint D-41 moments are developed in response to deformations in the structure and because a less stiff structure will not develop as large a moment. Further work needs to be done to better understand the results of the nonlinear analysis for restraint moments, as well as the live-load analysis that is not dis- cussed here. Girder Age at Continuity 28 Days 60 Days 90 Days Maximum Positive Restraint Moment— Final Design Linear Analysis (k-ft) 933.7 443.4 –7.3 Nonlinear Analysis (k-ft) 829.7 394.2 10.1 % Change from Linear to Nonlinear –11.1% –11.1% — Peak Negative Restraint Moment—Final Design Linear Analysis (k-ft) –104.8 –351.5 –533.4 Nonlinear Analysis (k-ft) –95.3 –325.9 –503.5 % Change from Linear to Nonlinear –9.1% –7.3% –5.6% Note: Positive restraint moments are shown at a girder age of 7,500 days. Moments shown earlier were for a girder age of 10,000 days. TABLE D-7.2-1 Comparison of restraint moments at interior support for linear and nonlinear analyses

D-42 DESIGN EXAMPLE 2: PCI BT-72 GIRDER 1 INTRODUCTION This design example demonstrates the design and detailing of the positive moment connection for a typical continuous two-span bridge. The contract documents specify that the min- imum girder age at continuity is 90 days. Therefore, this bridge is designed for continuity using the simplified approach (see DE1). Details of the girder design are not given here (again, see DE1, which is for an AASHTO Type III girder, for a description of the simplified approach and guidance in all other details of the design of this girder). This example is limited to a discussion of the design and detailing of the pos- itive moment connection because some issues for this topic differ from those found in DE1. The design is based on the AASHTO LRFD Specifications, 2nd Edition with Interims through 2002, and the specifications proposed as part of this research. 2 DESCRIPTION OF BRIDGE The bridge is a typical two-span structure with PCI BT-72 (bulb-T) girders and a composite deck slab. The span length for this bridge is approaching the maximum achievable for this girder and spacing. The geometry of the bridge is shown in Figures D-2-1 through D-2-3. The girders are made continuous by a continuity diaphragm that connects the ends of the girders at the interior support. The connection is made when the deck slab is cast. Therefore, the girders are considered continuous for all loads applied to the composite section. A typical interior girder is considered. The distance between centers of bearings (124.75 ft) is used for computing effects of loads placed on the simple-span girders before continuity is established. After continuity, the design span for the continuous girders is assumed to be from the center of the bearing at the expansion end of the girder to the center of the interior pier, or 126.00 ft (see Figure D-2-2). The space required between the ends of girders to accommo- date the positive reinforcement connection should be consid- ered when laying out the bridge (see Section 5.3 in DE1). 3 DESIGN PARAMETERS 3.1 Loads For simplified design, loads are required for the design of the girder, but the positive moment connection is designed for 1.2Mcr. Therefore, the loads are not required for this design example. 3.2 Materials and Material Properties Material properties used for design are given below. 3.2.1 Girder Concrete Material properties required for this design example are f ′c = 7.00 ksi, and wc = 0.150 kcf. 3.2.2 Deck and Continuity Diaphragm Concrete The same concrete properties are used for the deck and continuity diaphragm because they will be cast at the same time. Material properties required for this design example are given below. The subscript d is used to indicate properties related to the deck or diaphragm concrete: fcd = 4.00 ksi, frd = 0.480 ksi, and LRFD Art. 5.4.2.6 wcd = 0.150 kcf. 3.2.3 Prestressing Strand For this example, the properties of the 0.6-in.-diameter low-relaxation seven-wire strand are: Aps = 0.217 in.2; fpu = 270 ksi; fpy = 0.90 fpu = 243 ksi; fpj = 0.75 fpu = 202.5 ksi; and Ep = 28,500 ksi. 3.2.4 Mild Reinforcement Mild reinforcement is as follows: fy = 60 ksi, and Es = 29,000 ksi.

Figure D-2.2. Longitudinal section view of bridge. D-43 3.3 Section Properties 3.3.1 Girder The section properties for a standard PCI BT-72 are h = 72.00 in., A = 767.0 in.2, I = 545,894 in.4, yb = 36.60 in., yt = 35.40 in., Figure D-2.1. Plan view of bridge.

D-44 Sb = 14,915 in.3, and St = 15,421 in.3. 3.3.2 Continuity Diaphragm The section properties of the continuity diaphragm are required to compute the cracking moment. The cross section of the continuity diaphragm is assumed to be composed of the girder shape, the build-up, and the effective width of deck for an interior girder. The remainder of the continuity diaphragm could be considered, but the assumed section provides a min- imum area that may be effective. Since the continuity diaphragm (between the ends of oppos- ing girders) is composed entirely of deck concrete, transfor- mation of the deck is not required. The build-up is included in the computation of these section properties because the full depth of the build-up is specified at the center of bearings, immediately adjacent to the continuity diaphragm. Including the build-up will increase the cracking moment. The composite section properties used for girder design are similar, but are not used here because the deck has been transformed (deck and girder concrete compressive strengths are different) and the build-up was neglected. The section properties for the continuity diaphragm are as follows: h′c = total depth of diaphragm section; the full 31/2-in. build- up is included because the full build-up must be pro- vided at CL bearings = 72 + 3.5 + 7.75 = 83.25 in.; A′c = 1,635 in.2; I′c = 1,265,630 in.4; y′bc = 58.80 in.; y′tcd = 24.45 in.; S′bc = 21,524 in.3; and S′tcd = 51,764 in.3. 4 REINFORCEMENT FOR POSITIVE MOMENTS AT INTERIOR SUPPORTS The connections between girders at interior supports of bridges made continuous are subject to positive design moments. However, the positive moments are caused by minor live-load effects (for more than two-span bridges) and the restraint of time-dependent effects including creep, shrinkage, and temperature. Therefore, the positive design moments are not well-defined. Past research has questioned the benefit of providing a positive moment connection. 4.1 Positive Moment Connection The proposed specifications recommend that positive moment connections be provided for all prestressed con- crete bridges that are made continuous. This recommenda- tion is based on providing the connection to enhance the Half Section in Span Half Section at Continuity Diaphragm D Figure D-2-3. Typical section of bridge.

D-45 structural integrity of the structure so that it may be more robust and may be better able to resist unforeseen or extreme loads. Since the simplified approach is being used, where restraint moments are not computed, a positive moment connection is not required. However, a connection will be provided for this bridge as recommended by the proposed specifications. The positive moment connection can be made using either an extended strand detail or a mild reinforcement detail. The design method for the extended strand detail is identical to that presented in DE-1, so it is not repeated here. Due to the small size of the bottom flange of the bulb-T girder, special detailing is needed for the mild reinforce- ment connection. Therefore, the design of the mild rein- forcement detail is presented here. 4.2 Positive Design Moments Since this bridge has two spans, the only positive moment that can be developed at the interior support is caused by restraint (considering the effects included in this design example). Bridges with more spans will develop positive moments at the interior supports from live loads. Because the simplified approach is being used for this design example, restraint moment is not considered, so there is no positive design moment. In general, the reinforcement in the positive moment con- nection is proportioned using strength design to provide a factored resistance, φMn, greater than the larger of the fac- tored moment, Mu or 0.6Mc, but not to exceed 1.2Mcr. A design moment of 1.2Mcr is typically provided because test- ing and field experience have shown that this quantity of reinforcement, which is also the minimum quantity of rein- forcement required by the AASHTO LRFD Specifications in many cases, has performed well. Reinforcement provided in excess of the quantity needed to resist 1.2Mcr has been shown to be less effective. Therefore, the positive moment connec- tion for this design example will be taken as 1.2Mcr. 4.2.1 Computation of Positive Cracking Moment at the Continuity Diaphragm The positive cracking moment, Mcr, is computed using the section modulus for the bottom of the continuity diaphragm and the modulus of rupture for the deck concrete, frd, which is given in Section 3.2.2: Mcr = frd S′bc = 0.480(21,524) = 10,332 k-in = 861 k-ft. The concrete strength of the deck, f ′cd , is used for this calcu- lation because the confining effect of the precast girders in the continuity diaphragm is not significant for positive moments, where the deck slab is in compression. For reinforcement limits, the quantity 1.2Mcr is also computed: 1.2Mcr = 1,033 k-ft. 4.3 Mild Reinforcement Mild reinforcement will be used for this example to pro- vide the positive moment connection. See DE1 for a more complete discussion of the issues to be considered in design of the positive moment connection using mild reinforcement, including development of the reinforcement into the girder and constructability issues. 4.3.1 Development and Detailing of Reinforcement into the Continuity Diaphragm It is important that the distance between ends of opposing girders be great enough within a continuity diaphragm to allow the development of the positive reinforcement into the diaphragm. This should be considered during the initial stages of laying out a bridge. The mild reinforcement is developed into the continuity diaphragm using a standard 90° hook. A No. 6 bar will be used for the connection. The required length of embedment,
dh, into the diaphragm to develop the hooked bar is com- puted according to LRFD Article 5.11.2.4:
dh = 9.98 in.; USE 10 in. LRFD Eq. 5.11.2.4.1-1 A reduction factor of 0.7 was used because conditions pro- vide the required side and end cover. An embedment of the hook into the continuity diaphragm of 10 in. will be used. The distance between ends of girders across the continuity diaphragm will be taken as 12 in., so the 10-in. projection to the hook can be provided and still allow for construction tol- erances. A standard hook length of 12 in. is used for the ver- tical leg (see Figure D-4.3.3-1). A small additional reduction in the required hook embed- ment could be taken if the provided area of reinforcement is more than that required by analysis, but the design of the con- nection must be complete before the magnitude of this reduc- tion is known. Therefore, 10 in. will be used in this example. A cross bar of at least the same size as the hooked bar will be placed in the corner of the hooks, as shown in Figure D-4.3.1-1, to enhance the development of the hooked rein- forcement into the diaphragm. A 180° hooked bar (hairpin) may also be used to provide two legs developed into the con- tinuity diaphragm with only one hook, as shown in Figure D-4.3.3-2. The hooks are prebent, which simplifies fabrication and eliminates the tails of 90° hooks that may not fit within the forms during manufacturing. The use of hairpin bars is espe- cially helpful when a large number of bars is required to satisfy positive moment requirements. The development length for 90° hooked reinforcement should be used to compute the required embedment of 180° hooks into the continuity diaphragm.

D-46 4.3.2 Required Area of Reinforcement Using typical strength design procedures and the assumed design moment of 1.2 Mcr, the required area of reinforcement is computed to be As = 2.90 in.2 where f ′cd = 4.00 ksi; fy = 60 ksi; Mu = 1.2Mcr = 1,033 k-ft; φ = 0.9; b′ = width of compression face = width of bottom flange of girder = 26 in.; h′c = 83.25 in.; g = distance from bottom of girder to centroid of rein- forcement = 3.00 in. (one row of reinforcement cen- tered between bottom rows of strands); and d = effective depth from top of deck including build-up = h′c − g = 80.25 in. 4.3.3 Reinforcement Layout The required area of reinforcement, As = 2.90 in.2, can be provided using seven No. 6 reinforcing bars. An odd number of bars can be used since the bars are already placed asym- metrically in order to facilitate meshing of the reinforcement from opposing girders. The seven bars can be placed in a single layer, satisfying the assumption that all of the rein- forcement can be placed in a single layer. The actual area of reinforcement provided is as follows: As prov = 7(0.44 in.2) = 3.08 in.2 > As = 2.90 in.2 OK A layout for the reinforcement is shown in Figure D-4.3.3-1. If hairpin bars are used, the dimension g would have to be adjusted. For a No. 6 hairpin, the outside dimension of a stan- dard 180° bend is 6 in. Therefore, if the bottom leg of the hairpin remained centered at 3 in. from the bottom of the girder, the new value for g would be located at the center of the hairpin, which would yield g = 3 − (5/8)/2 + (6/2) = 5.69 in.; d = 77.56 in.; and As = 3.00 in.2. The required area of reinforcement for the No. 6 hairpins require four hairpins, with As prov = 8(0.44 in.2) = 3.52 in.2 > As = 3.00 in.2 OK However, it would be difficult to provide four hairpin bars in the bottom flange of a bulb-T girder and to place them so that bars from the opposing girder would mesh. The height of the flange is insufficient to place the hairpins far from the center of the girder. Figure D-4.3.1-1. Detail of reinforcement placement at positive moment connection (section view). End View Elevation at End Figure D-4.3.3-1. Details of hooked reinforcement placement at end of girder.

D-47 A combination of bar sizes may prove more practical. Using two No. 5 and two No. 6 bars (the development of the No. 5 bars will be adequate since they are smaller than the No. 6 bars) provides As prov = 4(0.44 in.2) + 4(0.31 in.2) = 3.00 in.2 = As = 3.00 in.2 OK The alternate reinforcement layout using hairpin bars is shown in Figure D-4.3.3-2. Two cross bars should be passed through the projecting loops of the hairpins to enhance the develop- ment, similar to the cross bars shown in Figure D-4.3.1-1. When developing a layout for the positive moment rein- forcement, the locations of all other reinforcement and embed- ments must be carefully considered to avoid conflicts and to provide adequate tolerances for fabrication. One example is that the positive moment reinforcement must avoid locations of headed studs attached to embedded plates. 4.3.4 Development of Reinforcement into Girder The positive moment reinforcement must be developed into the girder (see discussion in DE1). The development lengths for the No. 5 and No. 6 bars are 15 in. and 18 in., respectively. 4.3.5 Termination of Positive Moment Reinforcement The termination of positive moment reinforcement in the girder should be staggered. For hooked bars, two bar marks should be used with different horizontal leg lengths. For hairpin bars, the two legs of the hairpin should be detailed with different lengths, as shown in Figure D-4.3.3-2. End View Elevation at End Figure D-4.3.3-2. Details of hairpin reinforcement placement at end of girder.

D-48 DESIGN EXAMPLE 3: 51-IN. DEEP BOX GIRDER SPREAD 1 INTRODUCTION This design example demonstrates the design and detail- ing of the positive moment connection for a typical contin- uous two-span bridge. The contract documents specify that the minimum girder age at continuity is 90 days. Therefore, this bridge is designed for continuity using the simplified approach (see DE1). Details of the girder design are not given here. See DE1, which is for an AASHTO Type III girder, for a description of the simplified approach and guidance in all other details of the design of this girder. This example is limited to a discus- sion of the design and detailing of the positive moment con- nection because some issues for this topic differ from those found in DE1. The design is based on the AASHTO LRFD Specifications, 2nd Edition with Interims through 2002, and the specifications proposed as part of this research. 2 DESCRIPTION OF BRIDGE The bridge is a typical two-span structure with spread 51-in.-deep box girders and a composite deck slab. The geom- etry of the bridge is shown in Figures D-2-1 through D-2-3. The girders are made continuous by a continuity diaphragm that connects the ends of the girders at the interior support. The connection is made when the deck slab is cast. There- fore, the girders are considered continuous for all loads applied to the composite section. A typical interior girder is considered. The distance between centers of bearings (84.83 ft) is used for computing the effects of loads placed on the simple-span girders before continuity is established. After continuity, the design span for the continuous girders is assumed to be from the center of bearing at the expansion end of the girder to the center of the interior pier, or 86 ft. (see Figure D-2-2). The space required between ends of girders to accommodate the positive reinforcement connection should be considered when laying out the bridge (see Section 5.3 in DE1). 3 DESIGN PARAMETERS 3.1 Loads For simplified design, loads are required for the design of the girder, but the positive moment connection is designed for 1.2Mcr. Therefore, the loads are not required for this design example. 3.2 Materials and Material Properties Material properties used for design are given below. 3.2.1 Girder Concrete Material properties required for this design example are as follows: f ′c = 6.00 ksi, and wc = 0.150 kcf. 3.2.2 Deck and Continuity Diaphragm Concrete The same concrete properties are used for the deck and continuity diaphragm because they will be cast at the same time. Material properties required for this design example are given below. The subscript d is used to indicate properties related to the deck or diaphragm concrete: f ′cd = 4.50 ksi, frd = 0.509 ksi, and LRFD Art. 5.4.2.6 wcd = 0.150 kcf. 3.2.3 Prestressing Strand For this example, the properties of the 0.5-in.-diameter low-relaxation seven-wire strand are: Aps = 0.153 in.2; fpu = 270 ksi; fpy = 0.90 fpu = 243 ksi; fpj = 0.75 fpu = 202.5 ksi; and Ep = 28,500 ksi. 3.2.4 Mild Reinforcement Mild reinforcement is as follows: fy = 60 ksi, and Es = 29,000 ksi.

D-49 3.3 Section Properties 3.3.1 Girder The section properties for the 51-in. box girder are as fol- lows (see Figure D-3.3.1-1): h = 51.0 in., A = 908.0 in.2, I = 309,865 in.4, yb = 22.95 in., yt = 28.05 in., Sb = 13,502 in.3, and St = 11,047 in.3. 3.3.2 Continuity Diaphragm The section properties of the continuity diaphragm are required to compute the cracking moment. The cross section Figure D-2-1. Plan view of bridge. Figure D-2-2. Longitudinal section view of bridge.

D-50 of the continuity diaphragm is assumed to be composed of the outer dimensions of the box girder shape, the build-up, and the effective width of deck for an interior girder. The sec- tion properties for the box girder given above in Section 3.3.1 are for the box girder including the void. However, the end of the girder at the continuity diaphragm is a solid rectangle, so the properties of the solid section must be used in com- puting the properties of the diaphragm. The remainder of the continuity diaphragm outside the limits of the box girder could be considered, but the assumed section provides a min- imum area that may be effective. Since the continuity diaphragm (between the ends of oppos- ing girders) is composed entirely of deck concrete, transfor- mation of the deck is not required. The build-up is included in the computation of these section properties because the full depth of the build-up is specified at the center of the bearings, immediately adjacent to the continuity diaphragm. Including the build-up will increase the cracking moment. The composite section properties used for girder design are similar, but are not used here because the deck has been transformed (deck and girder concrete compressive strengths are different) and the build-up was neglected. The section properties for the continuity diaphragm are h′c = total depth of diaphragm section; the full 3-in. build-up is included because the full build-up must be provided at CL bearings; = 51 + 3 + 8.25 = 62.25 in.; A′c = 3,681 in.2; I′c = 1,517,929 in.4; y′bc = 34.51 in.; y′tcd = 27.74 in.; S′bc = 43,982 in.3; and S′tcd = 54,724 in.3. 4 REINFORCEMENT FOR POSITIVE MOMENTS AT INTERIOR SUPPORTS The connections between girders at interior supports of bridges made continuous are subject to positive design Half Section in Span Half Section at Continuity Diaphragm Figure D-2-3. Typical section of bridge. T Figure D-3.3.1-1. 51-in. box girder dimensions.

D-51 moments. However, the positive moments are caused by minor live-load effects (for more than two-span bridges) and the restraint of time-dependent effects including creep, shrinkage, and temperature. Therefore, the positive design moments are not well defined. Past research has questioned the benefit of providing a positive moment connection. 4.1 Positive Moment Connection The proposed specifications recommend that positive moment connections be provided between all prestressed concrete girders made continuous. This recommendation is based on providing the connection to enhance the structural integrity of the structure so that it may be more robust and better able to resist unforeseen or extreme loads. Since the simplified approach is being used, where restraint moments are not computed, a positive moment connection is not required. However, a connection will be provided for this bridge as recommended by the proposed specifications. As with DE-1 and DE-2, either an extended strand or mild reinforcement detail could be used for the positive moment connection. A mild reinforcement detail is shown here. The extended strand connection would be designed and detailed exactly as in DE-1, so it is not repeated here. 4.2 Positive Design Moments Since this bridge has two spans, the only positive moment that can be developed at the interior support is caused by restraint (considering the effects included in this design example). Bridges with more spans will develop positive moments at the interior supports from live loads. Because the simplified approach is being used for this design example, restraint moment is not considered, so there is no positive design moment. In general, the reinforcement in the positive moment con- nection is proportioned using strength design to provide a factored resistance, φMn, greater than the larger of the fac- tored moment, Mu, or 0.6Mcr, but not to exceed 1.2Mcr. A design moment of 1.2Mcr is typically provided because test- ing and field experience have shown that this quantity of reinforcement, which is also the minimum quantity of rein- forcement required by the AASHTO LRFD Specifications in many cases, has performed well. Reinforcement provided in excess of the quantity needed to resist 1.2Mcr has been shown to be less effective. Therefore, the positive moment connec- tion for this design example will be taken as 1.2Mcr. 4.2.1 Computation of Positive Cracking Moment at the Continuity Diaphragm The positive cracking moment, Mcr, is computed using the section modulus for the bottom of the continuity diaphragm and the modulus of rupture for the deck concrete, frd, which is given in Section 3.2.2: Mcr = frd S′bc = 0.509(43,982) = 22,387 k-in = 1,866 k-ft. The concrete strength of the deck, f ′cd, is used for this calcu- lation because the confining effect of the precast girders in the continuity diaphragm is not significant for positive moments, where the deck slab is in compression. For reinforcement limits, the quantity 1.2Mcr is also computed: 1.2Mcr = 2,239 k-ft. 4.3 Mild Reinforcement Mild reinforcement will be used for this example to pro- vide the positive moment connection. See DE1 for a more complete discussion of the issues to be considered in design of the positive moment connection using mild reinforcement, including development of the reinforcement into the girder and constructability issues. 4.3.1 Development and Detailing of Reinforcement into the Continuity Diaphragm It is important that the distance between ends of opposing girders be great enough within a continuity diaphragm to allow development of the positive reinforcement into the diaphragm. This should be considered during the initial stages of laying out a bridge. The mild reinforcement is developed into the continuity diaphragm using a standard 90° hook. A No. 6 bar will be used for the connection. The required length of embedment,
dh, into the diaphragm to develop the hooked bar is com- puted according to LRFD Article. 5.11.2.4:
dh = 9.98 in.; USE 10 in. LRFD Eq. 5.11.2.4.1-1 A reduction factor of 0.7 was used because conditions pro- vide the required side and end cover. An embedment of the hook into the continuity diaphragm of 10 in. will be used. The distance between ends of girders across the continuity diaphragm will be taken as 12 in., so the 10-in. projection to the hook can be provided and still allow for construction tol- erances. A standard hook length of 12 in. is used for the ver- tical leg (see Figure D-4.3.3-1). A small additional reduction in the required hook embed- ment could be taken if the provided area of reinforcement is more than required by analysis, but the design of the con- nection must be complete before the magnitude of this reduction is known. Therefore, 10 in. will be used in this example.

D-52 A cross bar of at least the same size as the hooked bar will be placed in the corner of the hooks, as shown in Figure D-4.3.1-1, to enhance the development of the hooked rein- forcement into the diaphragm. A 180° hooked bar (hairpin) may also be used to provide two legs developed into the continuity diaphragm with only one hook. However, hairpin bars are not generally practical for box girder sections because the top leg of the hairpin would typically extend beyond the end of the solid portion of the section and into the void. Therefore, ade- quate development may not be available for the top leg without altering the dimensions of the end block. 4.3.2 Required Area of Reinforcement Using typical strength design procedures and the assumed design moment of 1.2 Mcr, the required area of reinforcement is computed to be As = 8.60 in.2 where f ′cd = 4.50 ksi; fy = 60 ksi; Mu = 1.2Mcr = 2,239 k-ft; φ = 0.9; b′ = width of compression face = width of bottom flange of box girder = 48 in.; h′c = 62.25 in.; g = distance from bottom of girder to centroid of reinforcement = 3.00 in. (one row of reinforcement centered between bottom rows of strands); and d = effective depth from top of deck including build-up = h′c − g = 59.25 in. The much larger section of the continuity diaphragm results in a larger design moment than for the other sec- tions considered in these design examples. However, the area in which the reinforcement can be placed is also much greater. 4.3.3 Reinforcement Layout The required area of reinforcement, As = 8.60 in.2, can be provided using twenty No. 6 reinforcing bars. Due to the width of the girder, the twenty bars can be placed in a single layer, satisfying the assumption that all of the reinforcement can be placed in a single layer. The actual area of reinforce- ment provided is as follows: As prov = 20(0.44 in.2) = 8.80 in.2 > As = 8.60 in.2 OK A layout for the reinforcement is shown in Figure D-4.3.3-1. Alternate positive reinforcement layouts could be devel- oped using two layers of reinforcement or using hairpin bars. When developing a layout for the positive moment reinforcement, the locations of all other reinforcement and embedments must be carefully considered to avoid con- flicts and to provide adequate tolerances for fabrication. One example is that the positive moment reinforcement must avoid locations of headed studs attached to embedded plates. Figure D-4.3.1-1. Detail of reinforcement placement at positive moment connection (section view). Figure D-4.3.3-1. Details of reinforcement placement at end of girder.

D-53 4.3.4 Development of Reinforcement into Girder The positive moment reinforcement must be developed into the girder (see discussion in DE1). The development length for the No. 6 bars is 18 in. 4.3.5 Termination of Positive Moment Reinforcement The termination of positive moment reinforcement in the girder should be staggered. For hooked bars, two bar marks should be used with different horizontal leg lengths. For hair- pin bars, the two legs of the hairpin should be detailed with dif- ferent lengths, as shown in Figure D-4.3.3-2 of DE2.

D-54 DESIGN EXAMPLE 4: AASHTO BIII-48 BOX GIRDER (ADJACENT) 1 INTRODUCTION This example demonstrates the design of a continuous two-span bridge using the specifications proposed as part of this research. The precast/prestressed concrete box girders are made continuous by the placement of a continuity dia- phragm at the interior support, which fills the gap between ends of girders from adjacent spans. This example illustrates the design of an adjacent box girder bridge that does not have a composite deck. Therefore, continuity is established when the continuity diaphragm is placed. The bridge then becomes continuous for loads placed on the structure after the continuity diaphragm is in place. This is an experimental concept that has not been built or tested, so this design example is intended to explore the fea- sibility of its use. Once made continuous, the bridge is subject to restraint moments that may develop from the time-dependent effects of creep and shrinkage. Restraint moments are caused by restrained deformations in the bridge. Analysis indicates that the restraint moments vary linearly between supports. For this 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 effects and applied loads. Restraint moments also affect the moments within the spans. Therefore, girder designs must be adjusted to account for the additional positive moments caused by restraint. Because this bridge does not have a composite deck, a neg- ative restraint moment due to creep and shrinkage effects does not develop since the differential shrinkage between the deck and precast girder is the source of the negative moment. This also means that the initial negative moment spike that is experienced by conventional bridges with a composite deck will not occur, and negative restraint moment will not be available to reduce the effect of the creep in the girder that produces the positive restraint moment. Variations in temperature also cause restraint moments in continuous bridges. This would be the only typical source of negative restraint moments for this bridge. However, this condition will not be considered in this example. The appli- cation of any moment from temperature effects would be combined with the effects considered in this example, and the same design criteria must be satisfied. Only those details of design that are affected by the use of continuity are presented in this design example. The focus of the example will therefore be on flexural design, which is most significantly affected by the consideration of restraint moments. While the design shears, reactions, and deflections are also affected when compared with design for simple-span bridges, the procedures for design are not altered. Therefore, design for these quantities will not be presented. In a two-span bridge made continuous, positive moments at the interior pier develop only from restraint moments. These positive design moments are resisted by mild reinforcement or strands that are extended into the continuity diaphragm from the bottom flange of the girder. This positive moment connec- tion is proportioned using strength design methods to resist any developed restraint moments or to provide a minimum quantity of reinforcement. The positive moment connection is also provided to enhance the structural integrity of the bridge. Construction details for the positive moment connection will be discussed in this example. Negative moments at the interior pier of this bridge are caused by dead loads applied to the composite continuous structure and live loads. Negative restraint moments cannot form due to creep and shrinkage. However, this design is sim- ilar to the design of other precast/prestressed concrete girder bridges made continuous because negative restraint moments are neglected in conventional designs as allowed by the pro- posed specifications. In this example, negative moments are resisted by mild reinforcement added to the top of the pre- stressed concrete girder rather than being placed in a com- posite deck, which is the most common approach to provid- ing a negative moment connection. The reinforcement in the negative moment connection is proportioned using strength design methods. 1.1 Age of Girders at Continuity To demonstrate the significance of girder age when conti- nuity is established, designs will be performed assuming that continuity is established at the following girder ages: • 7 days, • 28 days, and • 90 days. If contract documents specify the minimum girder age at continuity, the minimum age is known. If the minimum girder age at continuity is 90 days, the proposed specifica- tions allow the designer to neglect the effect of restraint moments. This is referred to as the “simplified approach.” If the minimum girder age at continuity is not specified, the designer must use the “general approach,” which considers the effect of restraint moments. See Section 4 for a discus- sion of the two approaches. Since positive restraint moments have the most significant effect on designs, assuming an early age at continuity will result in higher positive restraint moments.

D-55 Two early ages for continuity (less than 90 days) are con- sidered in this example to provide information for the designer to decide whether to set a minimum girder age at continuity and what that age would be. A girder age of 7 days at con- tinuity is used in this example because continuity could be established very early as there is no deck or continuity dia- phragm forming or reinforcement to place prior to casting the continuity diaphragm. 1.2 Design Programs Used Most of the design calculations were performed using a commercially available computer program. This was supple- mented by hand and spreadsheet computations to obtain the quantities needed for this design. Restraint moments were estimated using the RESTRAINT Program that was devel- oped as part of this research project. Fatigue design loads were computed using the QConBridge Program, which is available free of charge from the Washington State DOT website (see Subappendix C). 2 DESCRIPTION OF BRIDGE The bridge is a two-span structure in which AASHTO Type BIII-48 box girders (see Figure D-3.6.1-1) are placed side-by-side (adjacent). An asphalt wearing surface is applied after continuity is established to create the riding surface and cross slope on the bridge. The span length for this bridge is in the middle of the usual span range for this girder in a side- by-side configuration. The geometry of the bridge is shown in Figures D-2-1 through D-2-3. The girders are made continuous by a continuity diaphragm that connects the ends of the girders at the interior support. For this bridge, the continuity diaphragm will require no forming except at the end of the diaphragm, since the girders are placed side by side. The continuity connection is made when the continuity diaphragm is cast; therefore, the girders are considered continuous for all loads applied after the dia- phragm is placed. During or after the girders are erected, grout keys between girders are filled and transverse ties are installed. Details and sequencing of installation for the transverse ties and grout keys are not discussed here because they are not a feature affected by continuity. Details of such construction can be found in the PCI Bridge Design Manual. The AASHTO LRFD Specifications mention two types of connection between adjacent box girders. The type of con- nection affects the live-load distribution (see AASHTO LRFD Specifications Table 4.6.2.2.2b-1). The first type of connec- tion assumes that the transverse connection between girders is only capable of preventing relative vertical displacement at the interface. A second, more-effective type of connection between boxes causes the entire bridge to function as a unit. However, the use of the more-effective (and sophisticated) connection details has not yet become widespread. There- fore, the first type of connection, which is the more typical and conservative type of connection between adjacent box members, will be used for this design example. The distance between the centers of bearings (84.83 ft.) is used for computing effects of loads placed on the simple- span girders before continuity is established. After continu- ity, the design span for the continuous girders is assumed 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- ure D-2-2. The space required between ends of girders to accommodate the positive reinforcement connection should be considered when laying out the bridge. See Section 5.3. This design example demonstrates the design of an inte- rior girder. Design of an exterior girder would be similar except for loads. For this bridge, the interior girder design governs. Figure D-2-1. Plan view of bridge.

D-56 3 DESIGN ASSUMPTIONS AND INITIAL COMPUTATIONS 3.1 Specifications AASHTO LRFD Bridge Design Specifications, 2nd Edition with Interims through 2002, is used for this design example. References to articles, equations and tables in the AASHTO LRFD Specifications will be preceded by the prefix “LRFD” to differentiate them from other references in this design example. Proposed revisions have been developed as part of this research project (see Subappendix C). References to articles and equations in the proposed specifications will be preceded by the prefix “proposed” to differentiate them from refer- ences to items in the AASHTO LRFD Specifications. 3.2 Loads The loads are as follows. • Live load: HL-93 with 33% dynamic allowance (IM) on the design truck. Live-load distribution factors are com- puted using equations in LRFD Table 4.6.2.2.2b-1 for Figure D-2-2. Longitudinal section view of bridge. Figure D-2-3. Typical section of bridge.

D-57 section type (g) (see LRFD Table 4.6.2.2.1-1), assuming that the girders are connected only enough to prevent relative vertical displacement at the interface. = 0.333 lanes per girder, regardless of number of lanes loaded • Girder self weight: The unit weight of girder concrete is 0.150 kcf. The weight of the grout in the shear keys is neglected. = 0.847 klf • Internal diaphragms: Internal diaphragms, 9 in. thick, are placed at quarter points. The area of each diaphragm is 7.26 sf. = 0.817 kips each at 3 locations within the span • Asphalt wearing surface: An average asphalt thickness of 4 in. is used for dead load of the wearing surface, with a unit weight of 0.140 kcf (LRFD Table 3.5.1-1). This weight is considered as a wearing surface for strength design (use a load factor of 1.5). = 0.047 ksf on each 4 ft wide girder = 0.187 klf for an interior girder (on continuous span) • Parapet load: = 0.371 klf per parapet, or 0.742 klf for both parapets Assume the load of each parapet is distributed to three girders on each side of bridge. = 0.124 klf for an interior girder (on continuous span) • Future wearing surface: = 0.025 ksf on each 4-ft-wide girder = 0.100 klf for each girder (on continuous span) 3.3 Materials and Material Properties Material properties used for design are given below. 3.3.1 Girder Concrete 3.3.1.1 Basic Properties. Girder concrete strengths and properties remain the same for all designs considered in this design example. f ′ci = 4.50 ksi, f ′c = 6.00 ksi, fr = 0.587 ksi, LRFD Art. 5.4.2.6 Eci = 4,067 ksi, LRFD Eq. 5.4.2.4-1 EC = 4,696 ksi, and LRFD Eq. 5.4.2.4-1 wc = 0.150 kcf. 3.3.1.2 Time-Dependent Properties. Time-dependent con- crete properties (creep and shrinkage) are needed only if restraint moments are being included in the analysis and design. Therefore, the following computations are not required if the simplified approach (see Section 4) is being used. Measured values of the ultimate creep coefficient and the ultimate shrinkage strain for the concrete should be used if possible. However, measured creep and shrinkage properties are rarely available. Therefore, these quantities are usually estimated. For this design example, the equations in LRFD Article 5.4.2.3 are used to estimate creep and shrinkage. See the AASHTO LRFD Specifications for secondary equations and complete definitions of the terms used in the calculations that follow. Restraint moments are very sensitive to variations in creep and shrinkage values, so possible estimates should be used. Other methods for estimating creep and shrinkage properties may be used as permitted by LRFD Article 5.4.2.3.1. 3.3.1.2.1 Volume–to–surface area ratio. Both creep and shrinkage equations are dependent upon the volume–to– surface area ratio (V/S). Since the equations are sensitive to this quantity and the analysis for restraint moments is sensi- tive to creep and shrinkage values, it is important to carefully consider the computation of this ratio. The V/S is generally computed using the equivalent ratio of the cross-sectional area to the perimeter. This quantity can be easily computed for most sections. For the standard AASHTO Type BIII-48 girder, the area may be obtained from a table of section properties: A = 812.5 in.2. The area above includes deductions for the shear keys, but these could be neglected. LRFD Article 5.4.2.3.2 indicates that only the surface area exposed to atmospheric drying should be included in the computation of V/S. Prior to installation in the bridge, the entire exterior surface of the girder is exposed to drying. Once the bridge is completed with the wearing surface in place, it could be reasoned that the top of the girder would not be exposed to drying. However, this refinement will be neglected for this example. The computation of the perimeter exposed to drying requires consideration of the enclosed void. LRFD Article 5.4.2.3.2 indicates that only 50% of the perimeter of a poorly ventilated interior void should be included in the perimeter. The void in the box girder will have one or more small vents, so it can be considered as poorly ventilated. Therefore, the reduction of the perimeter for the interior void will be included in computing the perimeter. The side faces of the boxes, since they are placed very close to one another, could also be con- sidered as poorly ventilated. However, this adjustment will not be considered here since the side faces will be exposed to dry- ing prior to completion of the bridge. The slight modification

D-58 of the exterior perimeter for shear keys will be neglected. Therefore, the perimeter will be computed as follows: p = Exterior perimeter + 50% of interior perimeter = 2(48 + 39) + 50%[2(38 + 28) − 4(2(3) − 4.24)] = 236.5 in. Therefore, V/S will be V/S = A/p = 812.5/236.5 = 3.44 in. It should be noted that the commentary to the specifications indicates that the maximum value of V/S considered in the development of the equations for the creep and shrinkage factors in which V/S appears was 6.0 in. Therefore, it would appear that this value may be considered a practical upper limit for the ratio when using the equations in the specifications. 3.3.1.2.2 Ultimate creep coefficient. The creep coefficient may be taken as follows: LRFD Eq. 5.4.2.3.2-1 Significant load is placed on the girder at release. Therefore, ti, the age of concrete when load is initially applied, is taken to be the age of the girder at release, or typically 1 day. To determine the ultimate value for the creep coefficient, ψu, where t = ∞, the final term in the equation is assumed to approach unity: ψu = ψ (∞, 1 day) = 2.00 where kc = factor for V/S = 0.800; LRFD Eq. C5.4.2.3.2-1 kf = factor for the effect of LRFD Eq. 5.4.2.3.2-2 concrete strength = 0.748 (using f ′c = 6.00 ksi); H = relative humidity = 75% (assumed); and V/S = 3.44 in. (used to determine kc; ratio is computed for nominal box dimensions, using 50% of interior void area as directed by LRFD Article 5.4.2.3.2; top sur- face of box in contact with the wearing surface is neglected for simplicity). 3.3.1.2.3 Ultimate shrinkage strain. While it is not always known whether the girder will be steam cured during fabri- cation, the initial strength gain is generally accelerated when compared with “normal” concretes. It is reasonable to use the shrinkage equation for steam-cured concrete. The shrinkage strain may therefore be taken as LRFD Eq. 5.4.2.3.3-2ε sh s hk k t t = − +( ) × −55 0 0 56 10 3. . . ψ t t k k H t t t t t i c f i i i , . . . . . . . ( ) ( ) ( )( )= − −+ −−3 5 1 58 120 10 0 0 60 118 0 6 To determine the ultimate shrinkage strain, εsh u, where t = ∞, the term in the equation that contains it is assumed to approach unity. εsh u = εsh (∞) = −415 × 10−6 in./in. where ks = size factor = 0.797 and LRFD Eq. C5.4.2.3.3-1 kh = humidity factor = 0.929. LRFD Eq. C5.4.2.3.3-2 3.3.2 Continuity Diaphragm Concrete The subscript “d” will be used to indicate properties related to the concrete in the continuity diaphragm. 3.3.2.1 Basic Properties. The same diaphragm concrete strength is used for all designs: f ′cd = 4.00 ksi, frd = 0.480 ksi, LRFD Art. 5.4.2.6 wcd = 0.150 kcf, and Ecd = 3,834 ksi.LRFD Eq. 5.4.2.4-1 3.3.2.2 Time-Dependent Properties. Since the continuity diaphragm is such a limited segment when compared with the remainder of the span, the time-dependent properties of the diaphragm concrete are neglected. 3.3.3 Prestressing Strand The properties of the prestressing strand are as follows: 0.5-in.-diameter low-relaxation seven-wire strand, Aps = 0.153 in.2, fpu = 270 ksi, fpy = 0.90 fpu = 243 ksi, fpj = 0.75 fpu = 202.5 ksi, and Ep = 28,500 ksi. 3.3.3.1 Transfer Length. The stress in the pretensioning strands is transferred from the strands to the girder concrete over the transfer length. The stress in the strands is assumed to vary linearly from zero at the end of the girder to the full effective prestress, fpe, at the transfer length. The transfer length,
t, may be estimated as

D-59
t = 60db LRFD Art. 5.11.4.1 = 60(0.5 in.) = 30 in. The location at a transfer length from the end of the girder is a critical stress location at release. Therefore, moments and stresses are computed for this location and are shown in var- ious tables in this example. These locations are identified in the tables with the heading “Trans.” or “Transfer.” 3.3.4 Mild Reinforcement The properties of the mild reinforcing bars are as follows: fy = 60 ksi and Es = 29,000 ksi. 3.4 Stress Limits The following stress limits are used for the design of the girders for the service limit state. For computation of girder stresses, the sign convention will be compressive stress is positive (+) and tensile stress is negative (−). Signs are not shown for stress limits computed below, but will be applied in later stress comparisons. 3.4.1 Pretensioned Strands The stress limits for low relaxation strands are Immediately prior to transfer: LRFD Table 5.9.3-1 fpi = 0.75 fpu = 202.5 ksi. At service limit state after losses: LRFD Table 5.9.3-1 fp = 0.80 fpy = 199.4 ksi. The stress limits above are not discussed in this example because they do not govern designs. 3.4.2 Concrete 3.4.2.1 Temporary Stresses at Release. Compression: LRFD Art. 5.9.4.1.1 fcR = 0.60 f ′ci = 0.60(4.50) = 2.700 ksi. Tension: LRFD Table 5.9.4.1.2-1 ftR1 = 0.0948 √f ′ci ≤ 0.2 ksi = 0.0948 √4.50 = 0.201 ≤ 0.2 ksi; USE ftR1 = 0.200 ksi (minimum allowed) or ftR2 = 0.24 √f ′ci with reinforcement to resist the tensile force in the concrete = 0.24 (4.50 = 0.509 ksi. 3.4.2.2 Final Stresses at Service Limit State after Losses. The following stress limits are given for the girder concrete. Compression: LRFD Table 5.9.4.2.1-1 fc1 = 0.60 φw f ′c, for full service loads (φw = 1 for girders) = 0.60(1)(6.00) = 3.600 ksi fc2 = 0.45 f ′c , for effective prestress (PS) and full dead loads (DL) = 0.45(6.00) = 2.700 ksi fc3 = 0.40 f ′c , for live load plus one-half of effective PS and full DL = 0.40(6.00) = 2.400 ksi. Tension: LRFD Table 5.9.4.2.2-1 For the precompressed compression zone, ft1 = 0.19 √ f ′c , assuming moderate corrosion conditions = 0.19 (6.00 = 0.465 ksi. For locations other than the precompressed compression zone, such as at the end of the girder where the top of the girder may go into tension under the effect of the negative live-load moment, the specifications give no stress limits. The follow- ing limits have been proposed, which take the same form as those for temporary tensile stresses at release given in LRFD Table 5.9.4.1.2-1, but with the specified concrete compressive strength, f ′c, substituted for the concrete compressive strength at release, f ′ci : ft2 = 0.0948 √ f ′c ≤ 0.2 ksi = 0.0948 √6.00 = 0.232 ≤ 0.2 ksi; USE ft2 = 0.200 ksi (minimum allowed) or

D-60 ft3 = 0.24 √ f ′c with reinforcement to resist the tensile force in the concrete = 0.24 √6.00 = 0.588 ksi. 3.5 Other Design Assumptions Internal intermediate diaphragms are placed at quarter points in the bridge. No other bracing is required due to the stability of the section. The weight of the grout in the shear keys is neglected. 3.6 Section Properties With no composite concrete deck, the structural properties of the bridge are defined by the girder alone. 3.6.1 Girder The section properties for a standard AASHTO BIII-48 box girder are as follows (see Figure D-3.6.1.1): h = 39.00 in., A = 812.5 in.2, I = 168,367 in.4, yb = 19.29 in., yt = 19.71 in., Sb = 8,728 in.3, and St = 8,542 in3. 3.7 Design Moments The following sections present the computed design moments for service and strength limit states. All loads applied to the bare girder are considered to act on a simple span. Loads applied after the continuity diaphragm is placed are considered to act on a fully continuous structure. Restraint moments are not shown in tables contained in this section. Computations are made later in the example. 3.7.1 Service Limit State Tables D-3.7.1-1 through D-3.7.1-3 provide moments for the service limit state. Table D-3.7.1-1 gives moments caused by loads applied to the noncomposite section (simple span). Tables D-3.7.1-2 and D-3.7.1-3 give moments caused by loads applied to the composite section (continuous span). 3.7.2 Strength Limit State Table D-3.7.2-1 provides moments for the Strength I limit state. 4 ANALYSIS AND DESIGN OF GIRDERS FOR CONTINUITY The age of precast/prestressed concrete girders when conti- nuity is established is a critical factor in the design of bridges of this type. The earlier the age of the girder at continuity, the more girder creep and shrinkage contribute to the development Figure D-3.6.1-1. AASHTO BIII-48 box girder dimensions. Location from Bearing Self Weight Internal Diaphragms (ft) (k-ft) (k-ft) Load Factor 1.0 1.0 Bearing 0.0 0.0 0.0 H/2 1.6 57.2 2.0 Transfer 1.8 64.4 2.3 0.10 L 8.0 258.8 9.8 0.20 L 16.6 478.8 20.4 0.30 L 25.2 636.0 27.7 0.40 L 33.8 730.3 31.2 0.50 L 42.42 761.7 34.8 0.60 L 51.0 730.3 31.2 0.70 L 59.7 636.0 27.7 0.80 L 68.3 478.8 20.4 0.90 L 76.9 258.8 9.8 Transfer 83.0 64.4 2.2 H/2 83.2 57.2 2.0 Bearing 84.8 0.0 0.0 TABLE D-3.7.1-1 Service design moments for loads on simple span

D-61 of positive restraint moments. Minimizing the risk of devel- oping positive moments avoids an increase in critical moments and stresses near midspan and avoids cracking of the conti- nuity diaphragm. It is highly recommended that the minimum age for con- tinuity be specified in the contract documents. Analytical studies and field experience indicate that waiting to estab- lish continuity until the girders are at least 90 days old will significantly reduce or eliminate the development of positive restraint moments. The proposed specifications recognize the benefit of delay- ing continuity by allowing two approaches for the design of Location from Bearing Parapet DL (DC) Wrg. Surf. DL (DW) LL+IM (+) LL+IM (–) (ft) (k–ft) (k–ft) (k–ft) (k–ft) Load Factor 1.0 1.0 1.0 1.0 Bearing 0.0 0.0 0.0 0.0 0.0 H/2 1.6 6.3 15.3 55.8 –6.6 Transfer 1.8 7.1 17.2 62.8 –7.4 0.10 L 8.0 27.8 67.4 245.5 –32.2 0.20 L 16.6 49.1 119.1 433.5 –67.2 0.30 L 25.2 61.2 148.5 545.7 –102.1 0.40 L 33.8 64.1 155.6 597.0 –137.1 42.42 57.9 140.4 592.7 –172.0 0.60 L 51.0 42.5 103.0 532.8 –207.0 0.70 L 59.7 17.9 43.4 415.1 –241.9 0.80 L 68.3 –15.9 –38.6 250.8 –321.7 0.90 L 76.9 –58.8 –142.8 96.4 –374.5 Transfer 83.0 –94.9 –230.2 36.8 –521.3 H/2 83.2 –96.2 –233.4 35.5 –527.9 Bearing 84.8 –106.6 –258.6 27.0 –583.0 CL Pier 86.0 –114.1 –276.8 20.8 –622.5 Location from Bearing Parapet DL (DC) Wrg. Surf. DL (DW) LL+IM (+) LL+IM (–) (ft) (k–ft) (k–ft) (k–ft) (k–ft) Load Factor 1.0 1.0 1.0 1.0 Bearing 0.0 0.0 0.0 0.0 0.0 H/2 1.6 6.3 15.3 44.6 –5.3 Transfer 1.8 7.1 17.2 50.2 –5.9 0.10 L 8.0 27.8 67.4 196.4 –25.8 0.20 L 16.6 49.1 119.1 346.8 –53.8 0.30 L 25.2 61.2 148.5 436.5 –81.7 0.40 L 33.8 64.1 155.6 477.6 –109.7 0.50 L 42.42 57.9 140.4 474.2 –137.6 0.60 L 51.0 42.5 103.0 426.2 –165.6 0.70 L 59.7 17.9 43.4 332.1 –193.59 0.80 L 68.3 –15.9 –38.6 200.7 –257.4 0.90 L 76.9 –58.8 –142.8 77.1 –299.6 Transfer 83.0 –94.9 –230.2 29.5 –417.0 H/2 83.2 –96.2 –233.4 28.4 –422.3 Bearing 84.8 –106.6 –258.6 21.6 –466.4 CL Pier 86.0 –114.1 –276.8 16.6 –498.1 TABLE D-3.7.1-2 Service I design moments for loads on continuous span TABLE D-3.7.1-3 Service III design moments for loads on continuous span

D-62 girders made continuous. The steps in each approach are sum- marized below. • General Approach: – The age of girders when continuity is established may or may not be specified; – Estimate time-dependent material properties (creep and shrinkage) that will be used to compute restraint moments; – Estimate the positive restraint moment, which is strongly dependent on girder age at continuity and time-dependent material properties; – Evaluate conditions at the continuity diaphragms to determine whether the connection is fully or partially effective under the effect of the positive restraint moment; – If the restraint moment exceeds 1.2Mcr or if the joint is not fully effective, it is recommended that the design or conditions be altered to improve the situation; – Analyze and design the girders for all design loads, including positive restraint moment (positive restraint moment should be neglected when evaluating stresses in regions of negative moment); – Design and detail a positive moment connection at continuity diaphragms; and – Design and detail reinforcement to resist negative moments from design loads, neglecting both positive and negative restraint moments. • Simplified Approach: – Specify the minimum age of the girders when continu- ity is established in the contract documents; the mini- mum girder age at continuity must be at least 90 days; – The connection at continuity diaphragms may be taken to be fully continuous; – Analyze and design the girders for all design loads (neglect restraint moments); – Design and detail a positive moment connection at continuity diaphragms; and – Design reinforcement to resist negative moments from design loads. This section of the design example is divided into two sub- sections corresponding to the two approaches listed above. Both approaches are presented for completeness; however, it is anticipated that the simplified approach, with its required specification of a minimum girder age of 90 days when con- tinuity is established, will be the approach used most often by bridge designers. The main benefits of the simplified approach include the simplicity of design and the ability to use standard design aids or software to complete a design. The general approach requires a method for estimating restraint moments. The RESTRAINT Program has been developed for this pur- pose and will be used for this example. Few other programs are available, and some may have significant limitations or disadvantages. TABLE D-3.7.2-1 Strength I design moments Loc. from Brg. Self Weight (Max) Self Weight (Min) Intern. Diaphr. (Max) Intern. Diaphr. (Min) Parapet DL (DC) (Max) Parapet DL (DC) (Min) Wrg. Surf. DL (DW) (Max) Wrg. Surf. DL (DW) (Min) LL+IM (+) LL+IM (–) (ft) (k–ft) (k–ft) (k–ft) (k–ft) (k–ft) (k–ft) (k–ft) (k–ft) (k–ft) (k–ft) Load Factor 1.25 0.9 1.25 0.9 1.25 0.9 1.50 0.65 1.75 1.75 Bearing 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 H/2 1.6 71.6 51.5 2.5 1.8 7.9 5.7 23.0 10.0 55.8 –6.6 Transfer 1.8 80.5 58.0 2.8 2.0 8.9 6.4 25.8 11.2 62.8 –7.4 0.10 L 8.0 323.5 232.9 12.2 8.8 34.7 25.0 101.1 43.8 245.5 –32.2 0.20 L 16.6 598.5 430.9 25.5 18.3 61.4 44.2 178.6 77.4 433.5 –67.2 0.30 L 25.2 795.0 572.4 34.6 24.9 76.5 55.1 222.7 96.5 545.7 –102.1 0.40 L 33.8 912.9 657.3 39.1 28.1 80.2 57.7 233.4 101.1 597.0 –137.1 0.50 L 42.42 952.2 685.61 43.5 31.3 72.4 52.1 210.7 91.3 592.7 –172.0 0.60 L 51.0 912.9 657.3 39.1 28.1 53.1 38.2 154.6 67.0 532.8 –207.0 0.70 L 59.7 795.0 572.4 34.6 24.9 22.4 16.1 65.1 28.2 415.1 –241.9 0.80 L 68.3 598.5 430.9 25.5 18.3 –19.9 –14.3 –57.8 –25.1 250.8 –321.7 0.90 L 76.9 323.5 232.9 12.2 8.8 –73.6 –53.0 –214.1 –92.8 96.4 –374.5 Transfer 83.0 80.5 58.0 2.8 2.0 –118.6 –85.4 –345.3 –149.7 36.8 521.3 H/2 83.2 71.6 51.5 2.5 1.8 –120.3 –86.6 –350.1 –151.7 35.5 –527.9 Bearing 84.8 0.0 0.0 0.0 0.0 –133.3 –96.0 –388.0 –168.1 27 –583.0 CL Pier 86.0 N/A N/A N/A N/A –142.7 –102.8 –415.3 –179.9 20.8 –622.5

D-63 The general approach will be presented first. In this section, computations and results will be presented for the bridge with continuity established when the age of the girders is 7 and 28 days. Positive restraint moments are estimated, and their effect is considered in the design of the girders. The design for continuity at a girder age of 90 days is also discussed with the initial calculation of restraint moments. The required time- dependent material properties were computed in the previous section of this design example. The simplified approach will then be presented for the bridge with girders that are at least 90 days old when continuity is established. The calculations only address concrete stresses in the gird- ers at the service limit state. Flexural design at the strength limit state was checked and does not govern these designs, so calculations are not shown. A summary compares the results of the designs using the general and simplified approaches. The reinforcement for positive and negative continuity con- nections, which is determined using the same methods for both approaches, is computed in subsequent sections. 4.1 General Approach The steps in the general approach were summarized in the previous section. The items will be addressed as they appear in the list, except that the determination of the effectiveness of the joint is considered prior to computation of the restrain- ing moment, as discussed in the following section. 4.1.1 Effectiveness of Joint According to proposed Article 5.14.1.2.7e, the joint (con- tinuity diaphragm) may be considered fully effective if one or both of the following conditions are satisfied: • The contract documents specify that the girders will be at least 90 days old when continuity is established. • The stress in the joint is compressive for the combina- tion of superimposed permanent loads, settlement, creep, shrinkage, 50% live load (with impact), and temperature gradient, if applicable. The first criterion is addressed by the contract documents. To demonstrate the general approach, girder ages at continuity are selected that do not satisfy the first criterion. The second criterion is stated in terms of a stress, but since the continuity diaphragm is not prestressed, it can also be expressed in terms of moments. Using moments will sim- plify manual computations since all of the moments are known, but the stress does not have to be computed. There- fore, the sum of the positive restraint moment, composite dead-load moments (negative), and 50% of the maximum neg- ative live-load (with impact) moment must not be positive since a positive moment would cause tension at the bottom of the diaphragm. (Effects other than permanent and live loads, such as temperature effects, are not considered in this example, but could be included in the calculation.) This sum- mation can also be backsolved to determine the maximum positive restraint moment that can develop before the net moment becomes positive or the stress at the bottom of the diaphragm becomes tensile. This computed maximum posi- tive restraint moment can then be used to facilitate the com- parison of different designs and eliminate the separate com- putation of joint stress since the restraint moment is computed as part of each design iteration. The maximum positive restraint moment at the interior support is computed in Table D-4.1.1-1. The live load used in this computation is 50% of the maximum negative service moment (see Service I, Table D-3.7.1-2). The Service I load combination is used rather than Service III because the joint (continuity diaphragm) is not prestressed concrete, which is a requirement for use of Service III. When the positive restraint moment computed in the var- ious design iterations remains less than or equal to the com- puted maximum positive restraint moment (702.2 k-ft), the joint may be considered to be fully effective and the bridge may be designed using continuity for all loads applied after continuity is established. If the positive restraint moment exceeds the maximum computed moment, the connection must be considered partially effective. In this case, a fraction of the loads applied to the continuous structure are considered to be carried by the girders as simple spans, and the remainder of the loads resisted by the continuous structure. The computation shown above represents the initial design moments only. Due to various adjustments that are made to the designs throughout the required iterations, the live-load moment changes slightly. Therefore, in the following, the stress at the bottom of the joint is computed as a check. 4.1.2 Initial Design Without Restraint Moments An initial design of an interior girder was performed without including any restraint moment. The strand pattern required for the specified geometry and loads is shown in Figure D-4.1.2-1. Continuous Dead Load (DC) (k-ft) –114.1 Continuous Dead Load (DW) (k-ft) –276.8 50 % of Live Load + Impact (k-ft) –311.3 TOTAL (k-ft) –702.2 Maximum Positive Restraint Moment for Fully Effective Joint (k-ft) 702.2 TABLE D-4.1.1-1 Calculation of moment limit for joint effectiveness

D-64 4.1.3 Compute Restraint Moments For this bridge without a composite deck, numerical analy- sis shows that only positive restraint moments develop. Posi- tive restraint moments can have a significant effect on the design of precast/prestressed concrete girder bridges made continuous. Positive restraint moments are generally larger for two span bridges than for bridges with a greater number of spans with similar span lengths, given the same span lengths and other conditions. However, two-span bridges have limited positive live-load moments at the interior support, while bridges with a greater number of spans can develop significant positive moments at interior supports from live load. The development of restraint moments with time is com- puted using the RESTRAINT Program for the initial girder design. The strand pattern shown in Figure D-4.1.2-1 is input to define the creep behavior of the girder. Material properties computed earlier are also used for input, including the ulti- mate creep coefficient and the ultimate shrinkage strain for the girder concrete. A complete listing of input values used in RESTRAINT is presented in Subappendix A. Results for the restraint moment analysis for continuity established at girder ages of 7, 28, and 90 days are shown in Figure D-4.1.3-1. The restraint moment analysis for this initial design indi- cates that, for continuity established at a girder age of 7 days, a positive restraint moment of about 89 k-ft will eventually develop from the combined effect of creep and shrinkage in the girder concrete. For continuity established at a girder age of 28 days, the analysis indicates that a positive restraint moment of approximately 54 k-ft will eventually develop. The analysis shows that a positive restraint moment of 29 k-ft develops when continuity is delayed until 90 days after the girder is cast. This small positive moment supports proposed Article 5.14.1.2.7d, which allows positive restraint moments to be neglected in the design of bridges for which continuity cannot be established until the girders are 90 or more days old. This initial design will be modified for the positive restraint moments in the next section. Since there is no composite deck slab, the analysis indicates that no negative restraint moment develops. 4.1.4 Subsequent Design Iterations Including Positive Restraint Moments Since the analysis indicates that a significant positive restraint moment develops with time for the girders that are 7 and 28 days old when continuity is established, the design of these girders must be revised. The modifications to the Figure D-4.1.2-1. Strand pattern for initial design. 0 20 40 60 80 100 0 1000 2000 3000 4000 5000 6000 7000 8000 Girder Age (Days) R es tr ai nt M om en t ( k-f t) Continuity at Girder Age of 7 Days Continuity at Girder Age of 28 Days Continuity at Girder Age of 90 Days INITIAL DESIGNS Moments will increase for Final Designs Figure D-4.1.3-1. Development of restraint moment with girder age—initial design.

D-65 initial design are necessary to counteract the increased stresses caused by the additional positive restraint moments, so several iterations are required. For each iteration, strands are added or repositioned to provide a design that includes the positive restraint moment and satisfies the service limit state design criteria. A new pos- itive restraint moment is then computed using RESTRAINT for the new strand pattern, and the process is repeated. The iterations continue until the revised positive restraint moment does not require a change in the strand pattern. While other quantities, such as strand size or concrete strengths, had to be adjusted in DE1 during the iterations, only the strand pattern was revised for this problem. Tables of strand requirements, positive restraint moments, and stresses at the bottom of the continuity diaphragm are given below for the designs with girder ages at continuity of 7 and 28 days. Strand patterns and other data for the final designs for girder ages of 7 and 28 days at continuity are shown in the summary Section 4.3. Computations for the strength limit state are not shown since they do not govern. The positive restraint moment is included. 4.1.4.1 Girder Age at Continuity of 7 Days. In the final iteration, the restraint moment remains below the maximum positive restraint moment of 702.2 k-ft computed in Table D-4.1.1-1. The computed stress at the bottom of the continu- ity diaphragm also remains in compression for the specified loading, in agreement with the positive restraint moment limit. Therefore, the joint can be considered fully effective for analysis for all loads applied to the continuous structure. Table D-4.1.4.1-1 also shows that the positive restraint moments for all iterations remain below the quantity 1.2Mcr = 584.1 k-ft (see Table D-5.2-1). This indicates that the rein- forcement used in the positive moment connection will be effective and that the design is reasonable. In summary, the design of this girder, even with continu- ity established at the very early girder age of 7 days, meets all of the requirements of the proposed specification and is a viable design. For further discussion, see the summary and comparison of designs in Section 4.3, which also contains a table displaying the strand pattern and other characteristics of the final design. The maximum restraint moment shown in Table D-4.1.4.1-1 occurs at the center of the interior support and decreases lin- early to zero at the center of bearing at the expansion joint. A load factor of 1.0 is applied to the restraint moment for the service limit state, and a load factor of 0.5 is applied to the restraint moment for the strength limit state (LRFD Article 3.4.1.) Since the moments vary linearly, a table of restraint moments along the girder is not given. Tables D-4.1.4.1-2 and D-4.1.4.1-3 present the service limit state stresses for the final design for a girder age at con- tinuity of 7 days. These stresses are compared with stress limits for release, and final conditions after losses that are given in Sections 3.4.2.1 and 3.4.2.2. 4.1.4.2 Girder Age at Continuity of 28 Days. The positive restraint moments shown in Table D-4.1.4.2-1 remain below Iteration No. of Strands Positive Restraint Moment Stress at Bottom of Continuity Diaphragm (k-ft) (ksi) 1 20 0.0 0.818 2 21 89.1 0.714 3 22 151.2 0.642 4 22 213.2 0.570 TABLE D-4.1.4.1-1 Summary of design iterations— continuity at 7 days Brg. H/2 Trans. 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 1.63 1.83 7.95 16.57 25.18 33.80 42.42 Top Girder (ksi) N/A N/A –0.284 –0.284 –0.284 –0.284 –0.284 –0.284 Prestress at Release Bottom Girder (ksi) N/A N/A 1.880 1.880 1.880 1.880 1.880 1.880 Top Girder (ksi) N/A N/A 0.127 0.411 0.736 0.966 1.104 1.153 Self Weight Bottom Girder (ksi) N/A N/A –0.125 –0.402 –0.719 –0.946 –1.080 –1.128 Top Girder (ksi) N/A N/A –0.157 0.127 0.452 0.682 0.820 0.869 Total at Release Bottom Girder (ksi) N/A N/A 1.755 1.478 1.161 0.934 0.800 0.752 Notes: 1. Critical stresses are shaded. 2. Values for limiting stresses are given in Section 3.4.2.1. 3. Compressive stresses at release are compared with the limit fcR = 2.700 ksi. The maximum compressive stress is 1.755 ksi at the transfer location. 4. Tensile stresses in regions other than the precompressed tensile zone at release are compared with the limiting tensile stress ftR1 = –0.200 ksi or ftR2 = –0.509 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum tensile stress is –0.157 at the transfer location. 5. In all cases, this design satisfies the specified stress limits at release. TABLE D-4.1.4.1-2 Summary of design stresses for final design at release— continuity at 7 days

Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L 0.60L 0.70L 0.80L 0.90L H/2 Trans. Brg. From Brg. (ft) 0.00 2.42 2.20 8.03 16.65 25.27 33.88 42.50 51.12 59.77 68.35 76.97 82.80 82.58 85.00 SERVICE STRESSES (ksi) Top Girder -0.070 -0.261 -0.239 -0.261 -0.261 -0.261 -0.261 -0.261 -0.261 -0.261 -0.261 -0.261 -0.239 -0.261 -0.070 Bottom Girder 0.460 1.724 1.581 1.724 1.724 1.724 1.724 1.724 1.724 1.724 1.724 1.724 1.581 1.724 0.460 Top Girder 0.000 0.090 0.080 0.364 0.673 0.893 1.026 1.070 1.026 0.893 0.673 0.364 0.080 0.090 0.000 Bottom Girder 0.000 -0.089 -0.079 -0.356 -0.658 -0.874 -1.004 -1.047 -1.004 -0.874 -0.658 -0.356 -0.079 -0.089 0.000 Top Girder 0.000 0.003 0.003 0.014 0.029 0.039 0.044 0.049 0.044 0.039 0.029 0.014 0.003 0.003 0.000 Bottom Girder 0.000 -0.003 -0.003 -0.013 -0.028 -0.038 -0.043 -0.048 -0.043 -0.038 -0.028 -0.013 -0.003 -0.003 0.000 Top Girder 0.000 0.010 0.009 0.039 0.069 0.086 0.090 0.081 0.060 0.025 -0.022 -0.083 -0.135 -0.133 -0.150 Bottom Girder 0.000 -0.010 -0.009 -0.038 -0.068 -0.084 -0.088 -0.080 -0.058 -0.025 0.022 0.081 0.132 0.130 0.147 Top Girder 0.000 0.006 0.006 0.028 0.058 0.089 0.119 0.150 0.180 0.211 0.241 0.271 0.294 0.293 0.300 Bottom Girder 0.000 -0.006 -0.006 -0.027 -0.057 -0.087 -0.117 -0.147 -0.176 -0.206 -0.236 -0.266 -0.287 -0.287 -0.293 Top Girder 0.000 0.071 0.063 0.276 0.487 0.613 0.671 0.666 0.599 0.467 0.282 0.108 0.040 0.041 0.030 Bottom Girder 0.000 -0.069 -0.061 -0.270 -0.477 -0.600 -0.657 -0.652 -0.586 -0.457 -0.276 -0.106 -0.039 -0.041 -0.030 Top Girder 0.000 -0.008 -0.007 -0.036 -0.076 -0.115 -0.154 -0.193 -0.233 -0.272 -0.362 -0.421 -0.593 -0.586 -0.655 Bottom Girder 0.000 0.008 0.007 0.035 0.074 0.112 0.151 0.189 0.228 0.266 0.354 0.412 0.581 0.573 0.641 TOTAL SERVICE STRESSES (ksi) Top Girder -0.070 -0.157 -0.147 0.155 0.509 0.758 0.899 0.939 0.869 0.697 0.418 0.034 -0.291 -0.301 -0.220 Bottom Girder 0.460 1.623 1.491 1.316 0.970 0.728 0.589 0.550 0.619 0.787 1.060 1.436 1.632 1.763 0.607 Top Girder -0.070 -0.151 -0.141 0.183 0.568 0.846 1.018 1.089 1.049 0.907 0.659 0.305 0.003 -0.008 0.080 Bottom Girder 0.460 1.616 1.485 1.289 0.913 0.641 0.473 0.403 0.443 0.581 0.824 1.170 1.344 1.476 0.313 Top Girder -0.070 -0.080 -0.079 0.459 1.055 1.460 1.689 1.755 1.648 1.374 0.941 0.413 0.043 0.034 0.110 Bottom Girder 0.460 1.547 1.424 1.019 0.436 0.041 -0.184 -0.249 -0.143 0.125 0.548 1.064 1.305 1.435 0.284 Top Girder -0.070 -0.166 -0.154 0.119 0.434 0.643 0.745 0.746 0.636 0.425 0.057 -0.387 -0.884 -0.886 -0.875 Bottom Girder 0.460 1.631 1.498 1.352 1.044 0.840 0.740 0.739 0.847 1.053 1.414 1.847 2.212 2.336 1.248 Top Girder -0.070 -0.094 -0.091 0.404 0.958 1.337 1.555 1.622 1.528 1.280 0.885 0.392 0.035 0.026 0.104 Bottom Girder 0.460 1.561 1.436 1.073 0.532 0.161 -0.053 -0.118 -0.026 0.216 0.603 1.085 1.313 1.444 0.290 Top Girder -0.070 -0.164 -0.153 0.126 0.449 0.666 0.776 0.785 0.683 0.479 0.129 -0.303 -0.766 -0.769 -0.744 Bottom Girder 0.460 1.629 1.497 1.345 1.030 0.818 0.710 0.701 0.801 1.000 1.343 1.765 2.096 2.221 1.120 Top Girder -0.035 -0.005 -0.008 0.368 0.771 1.036 1.180 1.211 1.123 0.920 0.612 0.261 0.041 0.038 0.070 Bottom Girder 0.230 0.739 0.681 0.374 -0.020 -0.280 -0.420 -0.450 -0.365 -0.166 0.136 0.479 0.633 0.697 0.127 Top Girder -0.035 -0.087 -0.081 0.041 0.179 0.264 0.295 0.276 0.202 0.077 -0.153 -0.404 -0.739 -0.736 -0.765 Bottom Girder 0.230 0.819 0.753 0.694 0.559 0.476 0.445 0.464 0.537 0.660 0.884 1.130 1.397 1.455 0.945 Service III LL+IM (+) w/RM Service III LL+IM (-) 6 + 0 . 5 * B B + 0 . 8 * 6 A + 0 . 8 * 7 4 7 + 0 . 5 * A Service I LL+IM (-) B = 1 - 5 B + 1 . 0 * 6 A + 1 . 0 * 7 Self Weight Special Service+RM (+) Special Service (-) 1 2 3 Service I LL+IM (+) w/RM 7 6 Full PS + DL + RM (+) Non-Comp. DL 5 A = 1 - 4 Location Full PS + DL Composite DL LL + IM (+) LL + IM (-) Restraint Mom't (RM) (+) Prestress After Losses Notes: 1. Maximum stresses are shaded, with the governing stresses boxed and bolded. 2. Values for limiting stresses are given in Section 3.4.2.2. 3. Compressive stresses for both dead-load combinations (A and B) are compared with the limiting compressive stress for full dead load fc2 = 2.700 ksi. The maximum stress is 1.763 ksi at the transfer length location for the combination without restraint moment (A). 4. Compressive stresses for both Service I LL+IM conditions are compared with the limiting compressive stress for full-service conditions fc1 = 3.600 ksi. The maximum stress is 2.336 ksi at the interior transfer length location for the combination without restraint moment. 5. Tensile stresses in the precompressed tensile zone for Service III with RM are compared with the limiting tensile stress ft1 = –0.465 ksi. The maximum stress is –0.118 ksi at 0.50L. 6. Tensile stresses in regions other than the precompressed tensile zone for Service III without RM are compared with the limiting tensile stress ft2 = –0.200 ksi or ft3 = –0.588 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum stress is –0.769 ksi at the interior bearing. This stress exceeds both stress limits. This region of the girder will be designed for the strength limit state. 7. Compressive stresses for the Special Service cases are compared with the limiting compressive stress for that case fc3 = 2.400 ksi. The maximum stress is 1.455 ksi at the interior transfer location for the combination without restraint moment. 8. In all cases, this design satisfies the specified stress limits, with the exception of tension at the interior support, which will be dealt with as noted. TABLE D-4.1.4.1-3 Summary of design stresses for final design at service limit state after losses—continuity at 7 days

D-67 the maximum positive restraint moment of 702.2 k-ft com- puted in Table D-4.1.1-1 for all iterations. As expected, the stress at the bottom of the diaphragm also remains in com- pression for all iterations. Therefore, design may be performed considering the joint fully effective for all loads applied to the continuous structure. Additionally, the moments all remain below the quantity 1.2Mcr = 584.1 k-ft as shown in Section 5.2.1.2. Therefore, the design of this girder, with continuity established at the fairly early girder age of 28 days, is acceptable. This is expected, since the design with continuity at 7 days was also viable. For further discussion, see the summary and compar- ison of designs in Section 4.3. The maximum restraint moment as shown in Table D-4.1.4.2-1 occurs at the center of the interior pier and decreases linearly to zero at the center of bearing at the expansion joint. A load factor of 1.0 is applied to the restraint moment for the service limit state, and a load factor of 0.5 is applied to the restraint moment for the strength limit state. Since the moments vary linearly, a table of restraint moments along the girder is not given. Tables D-4.1.4.2-2 and D-4.1.4.2-3 present the service limit state stresses for the final design for a girder age at con- tinuity of 28 days. These stresses are compared with stress limits for release, and final conditions after losses that are given in Sections 3.4.2.1 and 3.4.2.2. 4.2 Simplified Approach If the contract documents require that the girders be at least 90 days old when continuity is established, positive restraint moments may be neglected. This greatly simplifies the design of bridges with precast concrete girders made continuous. Once the designer has made this decision, design proceeds assuming that the bridge is fully continuous for loads applied to the continuous structure (proposed Article 5.14.1.2.7d); therefore, no iterations are required. The resulting design is the same as the initial design performed without restraint moments in the preceding section. Although it was not required, a restraint moment analysis was also performed for the bridge with continuity established at a girder age of 90 days (see Figure D-4.1.3-1). The maxi- mum positive restraint moment was nearly 30 k-ft. However, a design iteration was run for the girder with this additional positive moment, and it was found that the initial strand pat- tern was adequate to resist the additional moment without any revisions. This result appears to support the provisions in the proposed specifications allowing positive restraint moments to be neglected for a design with continuity estab- lished at a girder age of at least 90 days, even when a com- posite deck slab is not used. Iteration No. of Strands Positive Restraint Moment Stress at Bottom of Continuity Diaphragm (k-ft) (ksi) 1 20 0 0.818 2 21 54.0 0.755 3 21 96.5 0.706 TABLE D-4.1.4.2-1 Summary of design iterations—continuity at 28 days Brg. H/2 Trans. 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 1.63 1.83 7.95 16.57 25.18 33.80 42.42 Top Girder (ksi) N/A N/A –0.190 –0.190 –0.261 –0.261 –0.261 –0.261 Prestress at Release Bottom Girder (ksi) N/A N/A 1.500 1.500 1.789 1.789 1.789 1.789 Top Girder (ksi) N/A N/A 0.127 0.411 0.736 0.966 1.104 1.153 Self Weight Bottom Girder (ksi) N/A N/A –0.125 –0.402 –0.719 –0.946 –1.080 –1.128 Top Girder (ksi) N/A N/A –0.063 0.221 0.475 0.705 0.843 0.892 Total at Release Bottom Girder (ksi) N/A N/A 1.375 1.098 1.070 0.843 0.709 0.661 Notes: 1. Critical stresses are shaded. 2. Values for limiting stresses are given in Section 3.4.2.1. 3. Compressive stresses at release are compared with the limit fcR = 2.700 ksi. The maximum compressive stress is 1.375 ksi at the transfer length location. 4. Tensile stresses in regions other than the precompressed tensile zone at release are compared with the limiting tensile stress ftR1 = –0.200 ksi or ftR2 = –0.509 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum tensile stress is –0.063 ksi at the transfer length location. 5. In all cases, this design satisfies the specified stress limits at release. TABLE D-4.1.4.2-2 Summary of design stresses for final design at release— continuity at 28 days

Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L 0.60L 0.70L 0.80L 0.90L H/2 Trans. Brg. From Brg. (ft) 0.00 1.83 2.20 7.97 16.60 25.23 33.87 42.50 51.13 59.77 68.40 77.03 82.80 83.17 85.00 SERVICE STRESSES (ksi) Top Girder -0.047 -0.175 -0.160 -0.175 -0.241 -0.241 -0.241 -0.241 -0.241 -0.241 -0.241 -0.175 -0.160 -0.175 -0.047 Bottom Girder 0.369 1.382 1.267 1.382 1.648 1.648 1.648 1.648 1.648 1.648 1.648 1.382 1.267 1.382 0.369 Top Girder 0.000 0.090 0.080 0.364 0.673 0.893 1.026 1.070 1.026 0.893 0.673 0.364 0.080 0.090 0.000 Bottom Girder 0.000 -0.089 -0.079 -0.356 -0.658 -0.874 -1.004 -1.047 -1.004 -0.874 -0.658 -0.356 -0.079 -0.089 0.000 Top Girder 0.000 0.003 0.003 0.014 0.029 0.039 0.044 0.049 0.044 0.039 0.029 0.014 0.003 0.003 0.000 Bottom Girder 0.000 -0.003 -0.003 -0.013 -0.028 -0.038 -0.043 -0.048 -0.043 -0.038 -0.028 -0.013 -0.003 -0.003 0.000 Top Girder 0.000 0.010 0.009 0.039 0.069 0.086 0.090 0.081 0.060 0.025 -0.022 -0.083 -0.135 -0.133 -0.150 Bottom Girder 0.000 -0.010 -0.009 -0.038 -0.068 -0.084 -0.088 -0.080 -0.058 -0.025 0.022 0.081 0.132 0.130 0.147 Top Girder 0.000 0.003 0.003 0.013 0.026 0.040 0.054 0.068 0.082 0.095 0.109 0.123 0.133 0.133 0.136 Bottom Girder 0.000 -0.003 -0.002 -0.012 -0.026 -0.039 -0.053 -0.066 -0.080 -0.093 -0.107 -0.120 -0.130 -0.130 -0.133 Top Girder 0.000 0.071 0.063 0.276 0.487 0.613 0.671 0.666 0.599 0.467 0.282 0.108 0.040 0.041 0.030 Bottom Girder 0.000 -0.069 -0.061 -0.270 -0.477 -0.600 -0.657 -0.652 -0.586 -0.457 -0.276 -0.106 -0.039 -0.041 -0.030 Top Girder 0.000 -0.008 -0.007 -0.036 -0.076 -0.115 -0.154 -0.193 -0.233 -0.272 -0.362 -0.421 -0.593 -0.586 -0.655 Bottom Girder 0.000 0.008 0.007 0.035 0.074 0.112 0.151 0.189 0.228 0.266 0.354 0.412 0.581 0.573 0.641 TOTAL SERVICE STRESSES (ksi) Top Girder -0.047 -0.071 -0.068 0.241 0.530 0.778 0.919 0.960 0.889 0.717 0.438 0.120 -0.212 -0.215 -0.197 Bottom Girder 0.369 1.281 1.177 0.974 0.894 0.652 0.513 0.473 0.543 0.711 0.984 1.094 1.318 1.421 0.516 Top Girder -0.047 -0.068 -0.065 0.254 0.556 0.818 0.973 1.027 0.970 0.812 0.547 0.243 -0.079 -0.082 -0.061 Bottom Girder 0.369 1.278 1.174 0.962 0.868 0.612 0.460 0.407 0.463 0.618 0.877 0.973 1.188 1.291 0.383 Top Girder -0.047 0.002 -0.003 0.530 1.043 1.431 1.644 1.693 1.569 1.279 0.829 0.351 -0.039 -0.041 -0.031 Bottom Girder 0.369 1.209 1.113 0.692 0.392 0.012 -0.196 -0.245 -0.123 0.161 0.601 0.867 1.149 1.250 0.353 Top Girder -0.047 -0.080 -0.075 0.205 0.454 0.663 0.765 0.766 0.656 0.445 0.077 -0.301 -0.805 -0.800 -0.852 Bottom Girder 0.369 1.289 1.184 1.010 0.968 0.764 0.664 0.663 0.770 0.977 1.338 1.505 1.898 1.994 1.157 Top Girder -0.047 -0.012 -0.015 0.475 0.946 1.308 1.510 1.560 1.449 1.185 0.773 0.329 -0.047 -0.049 -0.037 Bottom Girder 0.369 1.222 1.125 0.746 0.487 0.132 -0.065 -0.114 -0.006 0.252 0.656 0.888 1.156 1.259 0.359 Top Girder -0.047 -0.078 -0.074 0.212 0.469 0.686 0.796 0.805 0.703 0.499 0.149 -0.217 -0.687 -0.683 -0.721 Bottom Girder 0.369 1.287 1.183 1.003 0.954 0.741 0.634 0.625 0.725 0.924 1.267 1.423 1.782 1.879 1.029 Top Girder -0.024 0.036 0.030 0.403 0.765 1.022 1.157 1.180 1.084 0.873 0.556 0.230 0.000 0.000 0.000 Bottom Girder 0.185 0.570 0.526 0.211 -0.043 -0.294 -0.426 -0.448 -0.355 -0.148 0.163 0.381 0.555 0.605 0.162 Top Girder -0.024 -0.044 -0.041 0.084 0.189 0.274 0.305 0.287 0.212 0.087 -0.142 -0.361 -0.699 -0.693 -0.754 Bottom Girder 0.185 0.648 0.596 0.523 0.521 0.438 0.407 0.426 0.499 0.622 0.846 0.959 1.240 1.284 0.899 Non-Comp. DL 5 A = 1 - 4 Location Full PS + DL Composite DL LL + IM (+) LL + IM (-) Restraint Moment (RM) (+) Prestress After Losses Self Weight Special (+) Service + RM Special (-) Service 1 2 3 Service I LL+IM (+) w/RM 7 6 Full PS + DL + RM (+) 4 7 + 0 . 5 * A Service I LL+IM (-) B = 1 - 5 B + 1 . 0 * 6 A + 1 . 0 * 7 Service III LL+IM (+) w/RM Service III LL+IM (-) 6 + 0 . 5 * B B + 0 . 8 * 6 A + 0 . 8 * 7 Notes: 1. Maximum stresses are shaded, with the governing stresses boxed and bolded. 2. Values for limiting stresses are given in Section 3.4.2.2. 3. Compressive stresses for both dead-load combinations (A and B) are compared with the limiting compressive stress for full dead load fc2 = 2.700 ksi. The maximum stress is 1.421 ksi at the transfer length location for the combination without restraint moment (A). 4. Compressive stresses for both Service I LL+IM conditions are compared with the limiting compressive stress for full-service conditions fc1 = 3.600 ksi. The maximum stress is 1.994 ksi at the interior transfer length location for the combination without restraint moment. 5. Tensile stresses in the precompressed tensile zone for Service III with RM are compared with the limiting tensile stress ft1 = –0.465 ksi. The maximum stress is –0.114 ksi at midspan. 6. Tensile stresses in regions other than the precompressed tensile zone for Service III without RM are compared with the limiting tensile stress ft2 = –0.200 ksi or ft3 = –0.588 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum stress is –0.721 ksi at the interior bearing. This stress exceeds both stress limits. This region of the girder will be designed for the strength limit state. 7. Compressive stresses for the special service cases are compared with the limiting compressive stress for that case fc3 = 2.400 ksi. The maximum stress is 1.284 ksi at the interior transfer length location for the combination without restraint moment. 8. In all cases, this design satisfies the specified stress limits, with the exception of tension at the interior support, which will be dealt with as noted. TABLE D-4.1.4.2-3 Summary of design stresses for final design at service limit state after losses—continuity at 28 days

D-69 4.2.1 Girder Age at Continuity of at Least 90 Days Tables D-4.2.1-1 and D-4.2.1-2 present the service limit state stresses for the final design for a girder age at continu- ity of at least 90 days. These stresses are compared with stress limits for release and final conditions after losses that are given in Sections 3.4.2.1 and 3.4.2.2. 4.3 Summary of Results for General and Simplified Approaches Designs for precast/prestressed concrete girders made con- tinuous have been presented in the preceding sections for girder ages at continuity of 7, 28, and 90 days. The designs using earlier girder ages were performed using the general approach, which required the consideration of positive restraint moments. The design for a girder age of 90 days at continuity was performed using the simplified approach, in which positive restraint moments are neglected. The inclu- sion of positive restraint moments for the designs with ear- lier continuity resulted in larger positive design moments within the spans, which required an increase in the number of prestressing strands. A simple-span design has also been developed using the same girder dimensions that are used for the spans made con- tinuous. This design was prepared as a benchmark for com- parison to the continuous girder designs. Design moments and stresses for this design are as given in Section 4.3.1. Figure D-4.3-1 summarizes the significant characteristics of the different designs. The figure clearly shows the benefit of continuity for precast/prestressed concrete girders. The girder design for the simple span required 30% more strands than did the continuous girder design developed using the simplified approach. For this bridge, all of the designs, even the design with continuity established at a girder age of 7 days, required fewer strands than did the simple-span design. The design using girders that were 28 days old when con- tinuity was established required 5% (1) more strands than did the girders designed using the simplified approach. This design required more effort than did the simplified approach since positive restraint moments were included in the design, although the increase was minor. This is a much less signif- icant increase than was noticed for the Type III design in DE1 for a design using girders of the same age. Even the design that established continuity when the gird- ers were only 7 days old required only two more strands (a 10% increase) than did the girder designed using the simpli- fied approach. Since this design still had fewer strands than did the simple-span design, designing for continuity provided a significant structural advantage. The simplified approach, with its requirement for girders to be at least 90 days old when continuity is established, still appeared to be the best solution, with the least strands and the least effort in design. However, if speed were needed, it would still be possible to design a bridge of this type with continu- ity established at a very early age. Figure D-4.3-2 compares the development of restraint moments with time for the final designs for the three girder ages at continuity. The increase in restraint moments for the designs with continuity at earlier ages can be seen if the fig- ure is compared with development of restraint moments for the initial designs in Figure D-4.1.3-1. The increase in restraint moment was caused by the increased creep resulting from the increased number of strands. While the increase from the initial design was relatively large, the absolute value of the increase was still small enough that all designs were still viable. Brg. H/2 Trans. 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 1.63 1.83 7.95 16.57 25.18 33.80 42.42 Top Girder (ksi) N/A N/A –0.143 –0.143 –0.238 –0.238 –0.238 –0.238 Prestress at Release Bottom Girder (ksi) N/A N/A 1.311 1.311 1.697 1.697 1.697 1.697 Top Girder (ksi) N/A N/A 0.127 0.411 0.736 0.966 1.104 1.153 Self Weight Bottom Girder (ksi) N/A N/A –0.125 –0.402 –0.719 –0.946 –1.080 –1.128 Total at Top Girder (ksi) N/A N/A –0.016 0.268 0.498 0.728 0.866 0.915 Release Bottom Girder (ksi) N/A N/A 1.186 0.909 0.978 0.751 0.617 0.569 Notes: 1. Critical stresses are shaded. 2. Values for limiting stresses are given in Section 3.4.2.1. 3. Compressive stresses at release are compared with the limit fcR = 2.700 ksi. The maximum compressive stress is 1.186 ksi at the transfer length location. 4. Tensile stresses in regions other than the precompressed tensile zone at release are compared with the limiting tensile stress ftR1 = –0.200 ksi or ftR2 = –0.509 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum tensile stress is –0.016 ksi at the transfer length location. 5. In all cases, this design satisfies the specified stress limits at release. TABLE D-4.2.1-1 Summary of design stresses for final design at release—continuity at 90 days

Brg. Trans. H/2 0.10L 0.20L 0.30L 0.40L 0.50L 0.60L 0.70L 0.80L 0.90L H/2 Trans. Brg. From Brg. (ft) 0.00 1.92 2.20 8.03 16.65 25.27 33.88 42.50 51.12 59.73 68.35 76.97 82.80 83.08 85.00 SERVICE STRESSES (ksi) Top Girder -0.035 -0.132 -0.121 -0.132 -0.220 -0.220 -0.220 -0.220 -0.220 -0.220 -0.220 -0.132 -0.121 -0.132 -0.035 Bottom Girder 0.324 1.213 1.112 1.213 1.571 1.571 1.571 1.571 1.571 1.571 1.571 1.213 1.112 1.213 0.324 Top Girder 0.000 0.090 0.080 0.364 0.673 0.893 1.026 1.070 1.026 0.893 0.673 0.364 0.080 0.090 0.000 Bottom Girder 0.000 -0.089 -0.079 -0.356 -0.658 -0.874 -1.004 -1.047 -1.004 -0.874 -0.658 -0.356 -0.079 -0.089 0.000 Top Girder 0.000 0.003 0.003 0.014 0.029 0.039 0.044 0.049 0.044 0.039 0.029 0.014 0.003 0.003 0.000 Bottom Girder 0.000 -0.003 -0.003 -0.013 -0.028 -0.038 -0.043 -0.048 -0.043 -0.038 -0.028 -0.013 -0.003 -0.003 0.000 Top Girder 0.000 0.010 0.009 0.039 0.069 0.086 0.090 0.081 0.060 0.025 -0.022 -0.083 -0.135 -0.133 -0.150 Bottom Girder 0.000 -0.010 -0.009 -0.038 -0.068 -0.084 -0.088 -0.080 -0.058 -0.025 0.022 0.081 0.132 0.130 0.147 Top Girder 0.000 0.001 0.001 0.004 0.008 0.012 0.016 0.021 0.025 0.029 0.033 0.037 0.040 0.040 0.041 Bottom Girder 0.000 -0.001 -0.001 -0.004 -0.008 -0.012 -0.016 -0.020 -0.024 -0.028 -0.032 -0.036 -0.039 -0.039 -0.040 Top Girder 0.000 0.071 0.063 0.276 0.487 0.613 0.671 0.666 0.599 0.467 0.282 0.108 0.040 0.041 0.030 Bottom Girder 0.000 -0.069 -0.061 -0.270 -0.477 -0.600 -0.657 -0.652 -0.586 -0.457 -0.276 -0.106 -0.039 -0.041 -0.030 Top Girder 0.000 -0.008 -0.007 -0.036 -0.076 -0.115 -0.154 -0.193 -0.233 -0.272 -0.362 -0.421 -0.593 -0.586 -0.655 Bottom Girder 0.000 0.008 0.007 0.035 0.074 0.112 0.151 0.189 0.228 0.266 0.354 0.412 0.581 0.573 0.641 TOTAL SERVICE STRESSES (ksi) Top Girder -0.035 -0.028 -0.029 0.284 0.550 0.798 0.939 0.980 0.909 0.737 0.459 0.163 -0.173 -0.172 -0.185 Bottom Girder 0.324 1.112 1.022 0.805 0.817 0.574 0.436 0.396 0.465 0.634 0.906 0.925 1.163 1.252 0.471 Top Girder -0.035 -0.027 -0.028 0.288 0.558 0.810 0.956 1.000 0.934 0.766 0.491 0.200 -0.133 -0.132 -0.144 Bottom Girder 0.324 1.111 1.021 0.802 0.809 0.562 0.420 0.376 0.441 0.606 0.874 0.888 1.124 1.213 0.431 Top Girder -0.035 0.043 0.035 0.564 1.045 1.423 1.627 1.667 1.532 1.232 0.773 0.308 -0.093 -0.090 -0.114 Bottom Girder 0.324 1.042 0.960 0.532 0.332 -0.038 -0.237 -0.276 -0.145 0.149 0.598 0.782 1.085 1.172 0.401 Top Girder -0.035 -0.037 -0.036 0.248 0.474 0.683 0.785 0.787 0.676 0.465 0.097 -0.258 -0.766 -0.757 -0.840 Bottom Girder 0.324 1.120 1.029 0.841 0.891 0.687 0.587 0.585 0.693 0.900 1.260 1.336 1.743 1.825 1.112 Top Girder -0.035 0.029 0.022 0.509 0.948 1.301 1.492 1.533 1.413 1.139 0.717 0.286 -0.101 -0.098 -0.120 Bottom Girder 0.324 1.055 0.972 0.586 0.428 0.082 -0.105 -0.145 -0.027 0.240 0.653 0.803 1.092 1.180 0.407 Top Girder -0.035 -0.035 -0.035 0.255 0.489 0.706 0.816 0.825 0.723 0.520 0.169 -0.174 -0.648 -0.640 -0.709 Bottom Girder 0.324 1.118 1.028 0.834 0.876 0.664 0.556 0.548 0.648 0.847 1.190 1.254 1.627 1.710 0.984 Top Girder -0.018 0.057 0.049 0.420 0.766 1.018 1.149 1.166 1.066 0.849 0.528 0.208 -0.027 -0.024 -0.042 Bottom Girder 0.162 0.486 0.449 0.131 -0.072 -0.319 -0.447 -0.464 -0.365 -0.154 0.161 0.338 0.523 0.566 0.186 Top Girder -0.018 -0.022 -0.022 0.106 0.199 0.284 0.316 0.297 0.222 0.097 -0.132 -0.340 -0.680 -0.672 -0.748 Bottom Girder 0.162 0.564 0.518 0.438 0.482 0.399 0.369 0.387 0.460 0.583 0.807 0.874 1.162 1.199 0.877 Non-Comp. DL 5 A = 1 - 4 Location Full PS + DL Composite DL LL + IM (+) LL + IM (-) Restraint Moment (RM) (+) Prestress After Losses Self Weight Special (+) Service + RM Special (-) Service 1 2 3 Service I LL+IM (+) w/RM 7 6 Full PS + DL + RM (+) 4 7 + 0 . 5 * A Service I LL+IM (-) B = 1 - 5 B + 1 . 0 * 6 A + 1 . 0 * 7 Service III LL+IM (+) w/RM Service III LL+IM (-) 6 + 0 . 5 * B B + 0 . 8 * 6 A + 0 . 8 * 7 Notes: 1. Maximum stresses are shaded, with the governing stresses boxed and bolded. 2. Values for limiting stresses are given in Section 3.4.2.2. 3. Compressive stresses for both dead-load combinations (A and B) are compared with the limiting compressive stress for full dead load fc2 = 2.700 ksi. The maximum stress is 1.252 ksi at the transfer location for the combination without restraint moment (A). 4. Compressive stresses for both Service I LL+IM conditions are compared with the limiting compressive stress for full-service conditions fc1 = 3.600 ksi. The maximum stress is 1.825 ksi at the interior transfer length location for the combination without restraint moment. 5. Tensile stresses in the precompressed tensile zone for Service III with RM are compared with the limiting tensile stress ft1 = –0.465 ksi. The maximum stress is –0.145 ksi at midspan. 6. Tensile stresses in regions other than the precompressed tensile zone for Service III without RM are compared with the limiting tensile stress ft2 = –0.200 ksi or ft3 = –0.588 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum stress is –0.709 ksi at the interior bearing. This stress exceeds both stress limits. This region of the girder will be designed for the strength limit state. 7. Compressive stresses for the special service cases are compared with the limiting compressive stress for that case fc3 = 2.400 ksi. The maximum stress is 1.199 ksi at the interior transfer length location for the combination without restraint moment. 8. In all cases, this design satisfies the specified stress limits, with the exception of tension at the interior support, which will be dealt with as noted. TABLE D-4.2.1-2 Summary of design stresses for final design at service limit state after losses—continuity at 90 days

D-71 4.3.1 Simple-Span Design for Comparison A simple-span design using the same girder lengths and bearing locations was performed for comparison to the con- tinuous girder designs. The properties of the concretes used for the simple-span design are the same as for the continuous design with a girder age of at least 90 days when continuity is established. See Table D-4.3.1-1. The following tables of moments and stresses are provided for the simple-span bridge design used for comparison to the continuous girder designs. Because of symmetry, moments and stresses are only shown for half of the girder. Moments for noncomposite loads on the simple-span design are the same as shown in Table D-3.7.1-1. Tables D-4.3.1-2 and D-4.3.1-3 present the service limit state stresses for the simple-span design. These stresses are compared with stress limits for final conditions after losses that are given in Section 3.4.2.2. Stress limits for release are given in the footnotes of Table D-4.3.1-2. 5 REINFORCEMENT FOR POSITIVE MOMENTS AT INTERIOR SUPPORTS The connections between girders at interior supports of bridges made continuous are subject to positive design moments. The positive moments are caused by minor live-load effects (for bridges with more than two spans) and by restraint Design Strand Pattern Design Moments* Number of Strands Area of Strands 0 k-ft** Initial Design 90 Day Final Design Simplified Approach 1,483.6 k-ft 20 3.060 in 2 -- 213.2 k-ft 7 Day Final Design General Approach 1,575.6 k-ft 22 3.366 in 2 +10.0% 96.5 k-ft 28 Day Final Design General Approach 1,517.3 k-ft 21 3.213 in 2 +5.0% 0.0 k-ft Simple-span Design 1,772.7 k-ft 26 3.978 in 2 +30.0% *Top number in cells is positive restraint moment at interior support. Bottom number in cells is Service III maximum positive design moment at midspan. **The initial design (same as continuity at 90 days) developed a minor positive restraint moment of 29.1 k-ft, which was neglected for the simplified approach. Figure D-4.3.1. Summary of designs.

D-72 of time-dependent effects including creep, shrinkage, and temperature. Therefore, the positive design moments are not well-defined. Past research has questioned the benefit of pro- viding a positive moment connection. 5.1 Positive Moment Connection The proposed specifications recommend that positive moment connections be provided between all prestressed concrete girders made continuous. This recommendation is based on providing the connection to enhance the structural integrity of the structure so that it may be more robust and better able to resist unforeseen or extreme loads. However, if analysis for restraint moments is required and the analysis indicates that a positive moment will develop, the proposed specifications require that a positive moment connection be provided. While not required, a positive moment connection is provided for the design with continuity at 90 days. Reinforcement to resist the positive design moment at the interior support may be provided using either mild reinforce- ment or pretensioning strand, as demonstrated in Sections 5.4 and 5.5. A combination of reinforcement types could also be used to provide the positive moment connection. The use of combined reinforcement types can be accomplished by combining the procedures for using each reinforcement type separately. 5.2 Positive Design Moments For the strength limit state, the reinforcement in the positive moment connection is proportioned to provide a factored resis- tance, (Mn, greater than the larger of the factored moment, Mu, or 0.6Mcr, but not to exceed 1.2Mcr. A design moment of 1.2Mcr is typically provided because testing and field experi- ence have shown that this quantity of reinforcement, which is also the minimum quantity of reinforcement required by the AASHTO LRFD Specifications in many cases, has per- formed well. Reinforcement provided in excess of the quan- tity needed to resist 1.2Mcr has been shown to be less effec- tive. It is therefore not recommended to use a quantity of reinforcement greater than that required for 1.2Mcr. For the service limit state, the reinforcement is propor- tioned to resist the larger of the service load moment or Mcr. The Service I load combination is used to compute the design moment because the connection is not prestressed. Since this bridge has two spans, the only positive moment that can be developed at the interior support is caused by 0 20 40 60 80 100 120 140 160 180 200 220 0 1000 2000 3000 4000 5000 6000 7000 8000 Girder Age (Days) R es tr ai nt M om en t ( k-f t) Continuity at Girder Age of 7 Days Continuity at Girder Age of 28 Days Continuity at Girder Age of 90 Days Figure D-4.3-2. Development of restraint moment with girder age— final design. Location from Bearing Parapet DL (DC) Wrg. Surf. DL (DW) LL + IM Service I LL + IM Service III (ft) (k-ft) (k-ft) (k-ft) (k-ft) Bearing 0.0 0.0 0.0 0.0 0.0 H/2 1.63 8.4 20.3 59.1 47.3 Transfer 1.83 9.4 22.8 66.5 53.2 0.10 L 8.0 37.8 91.6 265.4 212.3 0.20 L 16.6 69.9 169.6 485.0 388.0 0.30 L 25.2 92.8 225.2 633.9 507.1 0.40 L 33.8 106.6 258.6 722.1 577.7 0.50 L 42.4 111.2 269.8 744.0 595.2 TABLE D-4.3.1-1 Service design moments for loads on composite section for simple-span bridge

D-73 restraint (considering the effects included in this design exam- ple). Bridges with more spans will develop positive moments at the interior supports from live loads. Because there is no restraint moment for the design with continuity at a girder age of 90 days, there is no positive design moment. 5.2.1 Computation of Positive Cracking Moment at Continuity Diaphragm 5.2.1.1 Section Properties for Continuity Diaphragm. The cracking moment is computed using a solid rectangular cross section with dimensions equal to the nominal outside dimensions of the box girder. The section properties for the continuity diaphragm are as follows: h = 39.00 in.; A′ = h × b = 39(48) = 1,872 in.2; I ′ = bh3/12 = 48(39)3/12 = 237,276 in.4; y ′b = 19.50 in.; y′t = 19.50 in.; S ′b = 12,168 in3; and S ′t = 12,168 in3. 5.2.1.2 Cracking Moment. The positive cracking moment, Mcr, is computed using the section modulus for the bottom of the continuity diaphragm and the modulus of rupture for the diaphragm concrete, frd, which is given in Section 3.3.2.1: Mcr = frd S ′b = 0.480(12,168) = 5,841 k-in. = 486.7 k-ft. For reinforcement limits, the quantity 1.2Mcr is also computed: 1.2Mcr = 584.1 k-ft. 5.3 Minimum Distance Between Ends of Girders The bridge must be laid out to provide adequate distance between the ends of girders in the continuity diaphragm to develop the reinforcement. This distance depends on the type and size reinforcement that is used. Computations for deter- mining the required development lengths into the continuity are presented for mild reinforcement and strands in the next two sections. The distance between ends of girders must also be ade- quate to allow movement of bars or strands during erection for proper meshing of the positive moment connection rein- forcement. The greater the distance, the less bending would be required to provide clearance, especially for strands that cannot be offset to avoid conflicts. This minimum distance between ends of girders should be determined early in the design process because design spans and continuity diaphragm dimensions depend on this distance. 5.4 Mild Reinforcement Mild reinforcement is often used to provide the positive moment connection. The reinforcement for the connection must extend from the end of the girder and be anchored into the continuity diaphragm. This reinforcement is usually placed among the pretensioned strands near the bottom of the girder Brg. H/2 Trans. 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 1.63 1.83 7.95 16.57 25.18 33.80 42.42 Top Girder (ksi) N/A N/A –0.361 –0.361 –0.361 –0.361 –0.361 –0.361 Prestress at Release Bottom Girder (ksi) N/A N/A 2.226 2.226 2.226 2.226 2.226 2.226 Top Girder (ksi) N/A N/A 0.127 0.411 0.736 0.966 1.104 1.153 Girder DL Bottom Girder (ksi) N/A N/A –0.125 –0.402 –0.719 –0.946 –1.080 –1.128 Top Girder (ksi) N/A N/A –0.234 0.050 0.375 0.605 0.743 0.792 Total at Release Bottom Girder (ksi) N/A N/A 2.101 1.824 1.507 1.280 1.146 1.098 Notes: 1. Critical stresses are shaded. 2. Values for limiting stresses are given in Section 3.4.2.1. 3. Compressive stresses at release are compared with the limit fcR = 2.700 ksi. The maximum compressive stress is 2.101 ksi at the transfer length location. 4. Tensile stresses in regions other than the precompressed tensile zone at release are compared with the limiting tensile stress ftR1 = –0.200 ksi or ftR2 = –0.509 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum tensile stress is –0.234 ksi at the transfer length location. This stress exceeds the lower limit, so reinforcement is required to resist the tensile force in the concrete. 5. In all cases, this design satisfies the specified stress limits at release. TABLE D-4.3.1-2 Summary of design stresses for simple-span design at release

D-74 in order to maximize the effective depth to the reinforcement. This additional reinforcement inserted between strands that are usually placed on the standard 2 in. × 2 in. grid increases the congestion of reinforcement at the ends of girders. Extra atten- tion must be given to this area during placement of concrete to avoid a lack of consolidation. However, the fact that the girder in this design example is a box section greatly improves the ability to provide positive moment reinforcement without a significant increase in congestion. Hooks are generally provided on the projecting reinforce- ment to improve the development of the reinforcement into TABLE D-4.3.1-3 Summary of design stresses for simple-span design at service limit state after losses Brg. Trans H/2 0.10L 0.20L 0.30L 0.40L 0.50L Location from Bearing (ft) 0.00 1.63 1.83 7.95 16.57 25.18 33.80 42.42 SERVICE STRESSES (ksi) Top Girder (ksi) –0.327 –0.300 –0.327 –0.327 –0.327 –0.327 –0.327 –0.327 Prestress after Losses Bottom Girder (ksi) 0.538 1.850 2.018 2.018 2.018 2.018 2.018 2.018 Top Girder (ksi) 0.000 0.080 0.091 0.364 0.673 0.893 1.026 1.070 Self Weight Bottom Girder (ksi) 0.000 –0.079 –0.089 –0.356 –0.658 –0.874 –1.004 –1.047 Top Girder (ksi) 0.000 0.003 0.003 0.014 0.029 0.039 0.044 0.049 Noncomp. DL Bottom Girder (ksi) 0.000 –0.003 –0.003 –0.013 –0.028 –0.038 –0.043 –0.048 Top Girder (ksi) 0.000 0.040 0.045 0.182 0.336 0.446 0.513 0.535 Composite DL Bottom Girder (ksi) 0.000 –0.039 –0.044 –0.178 –0.329 –0.438 –0.503 –0.524 Top Girder (ksi) 0.000 0.083 0.093 0.373 0.681 0.890 1.014 1.045 LL + IM Bottom Girder (ksi) 0.000 –0.081 –0.091 –0.365 –0.667 –0.871 –0.993 –1.023 TOTAL SERVICE STRESSES (ksi) Top Girder (ksi) –0.087 –0.177 –0.189 0.232 0.71 1.052 1.255 1.327 Full PS + DL Bottom Girder (ksi) 0.538 1.729 1.882 1.470 1.002 0.668 0.468 0.399 Top Girder (ksi) –0.087 –0.094 –0.095 0.604 1.392 1.942 2.270 2.372 Service I Bottom Girder (ksi) 0.538 1.647 1.790 1.106 0.335 –0.204 –0.524 –0.624 Top Girder (ksi) –0.087 –0.111 –0.114 0.531 1.256 1.763 2.067 2.163 Service III Bottom Girder (ksi) 0.538 1.664 1.809 1.179 0.469 –0.029 –0.326 –0.419 Top Girder (ksi) –0.044 –0.005 –0.001 0.489 1.036 1.416 1.642 1.708 Special Service Bottom Girder (ksi) 0.269 0.783 0.849 0.370 –0.166 –0.538 –0.759 –0.824 Notes: 1. Maximum stresses are shaded, with the governing stresses boxed and bolded. 2. Values for limiting stresses are given in Section 3.4.2.2. 3. Compressive stresses for full prestress and dead load are compared with the limiting compressive stress for full dead load fc2 = 2.700 ksi. The maximum stress is 1.882 ksi at 0.50L. 4. Compressive stresses for Service I conditions are compared with the limiting compressive stress for full-service conditions fc1 = 3.600 ksi. The maximum stress is 2.372 ksi at 0.50L. 5. Tensile stresses in the precompressed tensile zone for Service III are compared with the limiting tensile stress ft1 = –0.465 ksi. The maximum stress is –0.419 ksi at 0.50L. 6. Tensile stresses in regions other than the precompressed tensile zone for Service III are compared with the limiting tensile stress ft2 = –0.200 ksi or ft3 = –0.588 ksi. The latter value requires an area of reinforcement to resist the tensile force. The maximum stress is –0.087 ksi at the bearing. 7. Compressive stresses for the special service combination are compared with the limiting compressive stress for that case fc3 = 2.400 ksi. The maximum stress is 1.708 ksi at 0.50L. 8. In all cases, this design satisfies the specified stress limits. Girder Age at Continuity 7 Days 28 Days 90 Days Msl (k-ft) N/A N/A N/A Service I Limit State Mcr (k-ft) 486.7 486.7 486.7 Mu (k-ft) * * N/A Strength I Limit State 1.2Mcr (k-ft) 584.1 584.1 584.1 *Load combinations at strength limit state do not result in positive design moments. TABLE D-5.2-1 Summary of positive design moments and limits at center of interior support (critical moments are shaded)

D-75 the continuity diaphragm and also to shorten the required dia- phragm width. The bars may be bent prior to or after fabri- cation of the girder, depending on fabricator preferences and clearances within the girder forms. Hairpin bars (180° hooks) are sometimes used to address reinforcement issues, although with box girders, this should not be required. 5.4.1 Development and Detailing of Reinforcement into the Continuity Diaphragm The mild reinforcement is developed into the continuity diaphragm using a standard 90° hook. No. 5 bars will be used for the connection. The required length of embedment,
dh, into the diaphragm to develop the No. 5 hooked bar is com- puted according to LRFD Article 5.11.2.4:
dh = 8.3 in.; USE 81/2 in. LRFD Eq. 5.11.2.4.1-1 A reduction factor of 0.7 was used because conditions pro- vide the required side and end cover. While the clear distance between the hook and the edge of the diaphragm is 1.5 in. (less than the 2 in. required), the face of the diaphragm is con- fined by the girder concrete. Therefore, this surface is not an exterior surface, and it appears appropriate to use the reduc- tion. An embedment of the hook into the continuity diaphragm of 8.5 in. will be used. The distance between ends of girders in the continuity diaphragm will be taken as 10 in., so the 8.5-in. projection to the hook can be provided and still allow for con- struction tolerances. A standard hook of 10 in. is used for the vertical leg (see Figure D-5.4.2-1). A small additional reduction in the required hook embed- ment could be taken if the provided area of reinforcement is more than required by analysis, but the design of the connec- tion must be complete before the magnitude of this reduction is known. Therefore, 8.5 in. will be used in this example. A cross bar of at least the same size as the hooked bar should be placed in the corner of the hooks to enhance the develop- ment of the hooked reinforcement into the diaphragm. See Figure D-5.4.1-1. A 180° hooked bar (hairpin) may also be used for posi- tive moment reinforcement, but many of the conditions that encourage use of hairpins are eliminated for box girders. 5.4.2 Required Area of Reinforcement Using typical strength design procedures and the design moments given in the table above, the required area of rein- forcement is computed to be as follows: As = 3.67 in.2 where f ′cd = 4.00 ksi; fy = 60 ksi; Mu = 1.2Mcr = 584.1 k-ft; φ = 0.9; b = 48 in.; h = total depth of diaphragm, which is the same as h for the girder = 39.00 in.; g = distance from bottom of girder to centroid of rein- forcement = 3.00 in. (one row of reinforcement); and d = effective depth from top of diaphragm = h − g = 36.00 in. For all girder designs, the required area of reinforcement, As = 3.67 in.2, can be provided using twelve No. 5 reinforcing bars. This satisfies the assumption that all of the reinforcement can be placed in a single layer. As prov = 12(0.31 in.2) = 3.72 in.2 > As = 3.67 in.2 OK. A layout for the reinforcement is shown in Figure D-5.4.2-1. 5.4.3 Control of Cracking by Distribution of Reinforcement According to LRFD Article 5.7.3.4, the reinforcement will be proportioned so that the tensile stress in the mild steel reinforcement at the service limit state does not exceed the stress limit given by LRFD Equation 5.7.3.4-1. Since there are no positive service limit state moments, this calculation is not necessary. 5.4.4 Development and Detailing of Reinforcement into Girder The required development length into the girder for the No. 5 bars used for the positive moment reinforcement is computed as follows: Figure D-5.4.1-1. Detail of reinforcement placement at positive moment connection (section view).

D-76
dh = 15.0 in. LRFD Art. 5.11.2.1.1 This length applies for designs with continuity at all ages considered. The total length of embedment of the positive moment reinforcement into the girder should be carefully con- sidered. It is recommended that the bar be developed beyond where an assumed crack radiating at a 2
1 slope from the inner edge of the bearing or embedded plate intersects the reinforcement. The general concept is illustrated in Figure D-5.4.4-1. The details of the connection for this bar are shown in Figure D-5.4.2-1. Where multiple bars are required for positive moment reinforcement, at least two different embedment lengths should be used to avoid stress concen- trations and potential cracking where the bars terminate in the girder. One bar type should provide the minimum embed- ment length, with the second bar type providing at least an additional 1.5 ft of embedment. Where hairpin bars are used, the staggering of bar terminations can be accomplished by using a single bar type that is detailed with unequal legs. 5.4.5 Constructability Issues Reinforcement projecting from the end of a girder is detailed to mesh with the reinforcement projecting from the opposing girder. This is intended to provide an essentially continuous load path for any tension that may develop at the bottom of the connection. This requires that the reinforce- ment be detailed to mesh. To accomplish this, the bars must be offset or bent so that conflicts between bars are minimized or eliminated. The use of offset bars is illustrated for this design in Fig- ure D-5.4.5-1. The bars are placed in a slightly eccentric pat- tern. This pattern simplifies fabrication and erection by allow- ing use of a single detail for the placement of reinforcement in all girders, which avoids fabrication errors and provides a positive offset between bars. As mentioned above, the use of two layers of reinforcement greatly complicates fabrication and erection of the girder, as well as placement of reinforce- ment in the continuity diaphragm. If possible, a single layer of reinforcement should be detailed; however, the available locations for placement of reinforcement are limited. Fur- thermore, the use of a larger number of smaller bars allows for a smaller gap between ends of opposing girders than is possible with fewer larger bars. The smaller bars also require a shorter embedment into the girder. These factors tend to lead designers to use a larger number of smaller bars. The placement of the positive moment connection rein- forcement between pretensioning strands increases conges- tion. This is significant because the congestion can inhibit the proper consolidation of concrete in the critical bearing area. Reinforcement should be positioned to facilitate placement and consolidation of concrete around the strands and bars. Figure D-5.4.2-1. Details of reinforcement placement at end of girder. dh Figure D-5.4.4-1. Detail for embedment of reinforcement into girder.

D-77 Reinforcement projections must be detailed to allow toler- ances in bar projections, girder lengths, and girder placement at erection. In this example, the distance from the end of the hook to the face of opposing girder was detailed as 31/2 in., which should provide an adequate tolerance. 5.5 Pretensioning Strand An alternate positive moment connection uses pretension- ing strands extended into the continuity diaphragm. Because a positive moment connection with strands uses existing reinforcement (strands), the connection is provided without increasing congestion. However, girder fabrication may be complicated if the required length of extended strand is large. The limit on the service load stress of strands serves to limit the length of strand that can be effectively used for the con- nection. Extended strands used for the positive moment con- nection must be bonded at the end of the girder—that is, they cannot be shielded or debonded at the end of the girder. 5.5.1 Development and Detailing of Extended Strands into Continuity Diaphragm The strands are developed into the continuity diaphragm using a 90° hook. The strand must be bent so that after bend- ing, the hook projects at least 8 in. from the end of the girder. This distance is required by the equation used in Section 5.5.2. The distance between ends of girders in the continuity diaphragm is detailed as 10 in., so the 8-in. projection to the bent strand can be provided and still allow for construction tolerances. A cross bar should be placed in the corner of the hooks to enhance the development of the hooked reinforcement into the diaphragm. The cross bar should have an area not less than the area of a strand. See Figure D-5.5.1-1. 5.5.2 Required Area of Strands The area of strands required to resist the positive design moments (see Table D-5.5.2-1) is computed using both the strength design provisions given in LRFD Article 5.14.1.2.7h and service load (working stress) design procedures. The working stress design procedures may be found in concrete design textbooks. Working stress design is used in addition to strength design for this calculation because the procedure is based on results Figure D-5.4.5-1. Detail of reinforcement placement at positive moment connection (plan view). Figure D-5.5.1-1. Detail of strand placement at positive moment connection (section view).

D-78 of research. The researchers proposed a design methodology that included both approaches. The design moment for the working stress check will be Mcr, while the design moment for the strength check is 1.2 Mcr. Initially, it was assumed that the total length of extended strand,
sh, was 42 in. This provides a length of strand beyond the bend (vertical leg) of 42 − 8 = 34 in. The vertical leg extends for much of the depth of the girder to maximize the development of the strand, while not using so much strand that fabrication of the girders is adversely affected. This length of strand is effectively the maximum that can be used for this depth of box girder, allowing approximately 3 in. cover over the tip of the strand. Using
sh = 42 in., stresses in strands are computed for the available length of strand. The resulting stresses in the 0.5-in.- diameter strands at service and strength conditions are fpsl = [(42 − 8)/0.228] Proposed Eq. 5.14.1.2.7i-1 = 149 ksi < 150 ksi and fpul = [(42 − 8)/0.163] Proposed Eq. 5.14.1.2.7i-2 = 209 ksi. Since fpsl does not exceed the maximum service load stress of 150 ksi that applies to proposed Equation 5.14.1.2.7i-1, this length of strand and the corresponding stresses may be used. Based on these stress limits for service and strength limit states, the number of strands required to resist the positive design moments is computed. The design moments are gov- erned by the minimum limits of Mcr = 486.7 k-ft and 1.2Mcr = 584.1 k-ft for service and strength design, respectively. Design moments are found in Table D-5.2-1. Results are given in Table D-5.5.2-1. Eight strands will be provided for the connection. Since there are many strands that can be extended, these strands could be offset as shown in Figure D-5.5.2-1 in order to facil- itate erection, using details similar to those for the mild rein- forcement connection. Since an excess area of strands is being provided, the length of the strand extension could be reduced. See DE1 for the iterative process that can be used. Strands with debonding cannot be used for the positive moment connection. 5.5.3 Constructability Issues Offsetting the pattern of extended strands enables meshing of the bent-up strands as shown in Figure D-5.5.3-1. This simplifies erection of the girders. ldsh (in.) 42 b (in.) 48 d (in.) 37 f'cd (ksi) 4.00 Strand Diameter (in.) 0.5 Design Moment (k-ft) 584.1 fpul (ksi) 208.6 a (in.) 1.18 Aps (in2) 0.923 No. of Strands Req'd 6.03 Design Moment (k-ft) 486.7 Ec (ksi) 3605 Es (ksi) 29000 n 8 .04 fs (ksi) 149.1 jd (in.) 35.83 k 0 .095 rho 0.000615 As (in2) 1.093 No. of Strands Req'd 7.1 WORKING STRESS DESIGN STRENGTH DESIGN BASIC DESIGN INFORMATION TABLE D-5.5.2-1 Design information for positive moment connection using strand (critical values are shaded) Figure D-5.5.2-1. Details of strand placement at end of girder.

D-79 5.5.4 Fatigue of Positive Moment Connection Reinforcement Because the reinforcement in the positive moment con- nection is not subjected to tension from the live load for this two-span bridge, fatigue does not need to be investigated. The positive moments caused from restraint moments or temper- ature effects do not occur frequently enough to be considered as loadings that cause fatigue. However, for other bridge con- figurations, where the connection is subjected to tension from live loads, fatigue should be considered. See Section 5.3.2.2.3 for the approach. 6 REINFORCEMENT FOR NEGATIVE MOMENTS AT INTERIOR SUPPORTS The proper design of reinforcement at the negative moment connection is essential for the bridge to behave as a continu- ous structure and to provide the desired service life with lit- tle or no maintenance. This section presents the necessary steps for the design of the reinforcement for the negative moment connection. 6.1 Negative Design Moments Negative moments at interior supports of precast/prestressed concrete girders made continuous result from dead loads, live loads, and restraint moments. Negative restraint moments, however, are ignored in design, as allowed by proposed Arti- cle 5.14.1.2.7b. Therefore, the negative design moments shown below in Table D-6.1-1 result from dead loads and live loads only. The negative moment connection is designed using strength methods, so design moments are factored. 6.1.1 Computation of Negative Cracking Moment at Continuity Diaphragm Section properties for the continuity diaphragm used to compute the cracking moment are discussed and given in Section 5.2.1.1. The negative cracking moment, Mcr, is com- puted using the section modulus for the top of the continuity diaphragm and the modulus of rupture for the diaphragm concrete, frd, which is given in Section 3.3.2.1: Mcr = frd S′t = 0.480(12,168) = 5,841 k-in = 486.7 k-ft. For reinforcement limits, the quantity 1.2Mcr is also computed: 1.2Mcr = 584.1 k-ft. 6.2 Negative Moment Connection The connections between girders at interior piers of a bridge with precast girders made continuous are subject to significant negative design moments. The portion of the bridge subjected to negative moments must be properly reinforced to resist Figure D-5.5.3-1. Detail of strand placement at positive moment connection (plan view). Girder Age at Continuity 7 Days 28 Days 90 Days Msl (k-ft) –1,013.1 –1,013.1 –1,013.1 Service I Limit State Mcr (k-ft) –486.7 –486.7 –486.7 Mu (k-ft) –1,180.6 –1,180.6 –1,180.6 Strength I Limit State 1.2Mcr (k-ft) –584.1 –584.1 –584.1 TABLE D-6.1-1 Summary of negative design moments and limits at center of interior support (critical moments are shaded)

D-80 these negative moments if it is to perform as a continuous member and if the structure is to have good serviceability. Since this bridge does not have a composite deck, the neg- ative moment connection cannot be made using reinforce- ment in the deck, which is the typical approach. Instead, the reinforcement must be provided in the top of the box girder and developed into the continuity diaphragm in the same way as the positive moment reinforcement. Fatigue of the rein- forcement must also be checked. Detailing of the negative moment reinforcement, which is important for economy and to avoid congestion, is also addressed. 6.2.1 Required Area of Reinforcement Reinforcement for the negative moment connection could be provided by mild reinforcement, strands, or a combination of both. For this example, only mild reinforcement will be considered. Computations for a connection using strand would employ the same procedures used for the positive moment connection. Based on the factored negative design moment at the interior support, a total required area of reinforcement is computed: b = 48.0 in. (width at bottom of diaphragm = width of girder bottom flange); d = 36.50 in. (providing 2 in. minimum clear to top rein- forcement); f ′cd = 4.00 ksi; fy = 60 ksi; and Mu = −1,180.6 k-ft. Using this information, the area of reinforcement required to resist the factored negative design moment is computed using strength design to give the following results: As = 7.47 in.2 where a = 2.75 in. and c = 3.23 in. Checking the maximum reinforcement limit, c/d = 3.23/36.50 = 0.088 < 0.42. OK Using No. 6 bars, which can be developed into the diaphragm with a 10-in. extension (see DE2), 17 bars are required. Therefore, As prov = 17(0.44) = 7.48 in.2. The layout of the negative moment reinforcement in the top of the girder is shown in Figure D-6.2.1-1. 6.2.2 Details of Negative Moment Reinforcement The negative moment connection reinforcement can be detailed using the same details that are used for the positive moment connection with mild reinforcement except that the hooks are turned down into the continuity diaphragm as shown in Figure D-6.2.1-1. An alternate negative moment connection may be made by splicing straight bars across the continuity diaphragm. A blockout at the top of the beam may be required to provide the length of exposed reinforcement required to make the connection. Negative moment connections of this type have been made successfully using commercially available grouted splice connections. Requirements for mechanical splices are given in LRFD Article 5.11.5.2.2. Another option is to use high-strength threaded rods that are “coupled” with offset bars from the opposing beam by bolting at a plate (Ma et al., 1998 and Hennessey and Bexten, 2003). Welded splices con- forming to the requirements of LRFD Article 5.11.5.2.3 may also be used. Figure D-6.2.1-1. Details of strand placement at end of girder—negative moment connection.

D-81 Using any of these mechanical connection options instead of splicing hooked bars in the continuity diaphragm would permit the use of larger bars, which would reduce the num- ber of bars required to extend from the end of the beam. This would reduce the congestion at the end of the beam and in the continuity diaphragm. 6.2.2.1 Anchorage of Negative Moment Reinforcement. The specifications indicate that the longitudinal reinforcement resisting the negative design moments must be anchored in concrete that is in compression at the strength limit state (LRFD Article 5.14.1.2.7b and proposed Art. 5.14.1.2.7h). Therefore, the inflection points for the negative moment envelope at the strength limit state must be located. Figure D-6.2.2.1-1 indicates the extent of the negative moment region at strength limit state. The reinforcement for the factored negative design moment must be extended past the indicated locations for at least a development length. Therefore, the minimum length of the No. 6 bars placed in the top of the girder would be
min = 17.3 ft +
d = 17.3 ft + (25.2 in./12) = 19.4 ft where
d = 25.2 in. (for No. 6 bar, LRFD Art. 5.11.2.1.1 “top bar” designation). 6.2.2.2 Terminations of Negative Moment Reinforcement. Although all negative moment reinforcement can theoretically be terminated at one location, as shown in Figure D-6.2.2.1-1, this would probably not provide good serviceability. It is rec- ommended that the same bar mark be used, but that alternate reinforcing bars be shifted several feet to stagger the termina- tions. This would require that the minimum length of bar com- puted above would be increased by the amount of the shift. 6.2.3 Constructability Issues In this example, constructability issues for negative moment reinforcement are the same as for the positive moment rein- forcement, although congestion issues are not as significant since the reinforcement is in the top of the girder where con- solidation is easier. The connection shown uses bars that are hooked into the continuity diaphragm. However, the top of the girder could also be blocked out, and a bar could be placed to lap with the negative moment reinforcement. 6.2.4 Control of Cracking by Distribution of Reinforcement According to LRFD Article 5.7.3.4, the reinforcement will be proportioned such that the tensile stress in the mild steel reinforcement at the service limit does not exceed the allow- able given by LRFD Equation 5.7.3.4-1. The tensile stress in the negative moment reinforcement is computed to be where Msl = 1013.1 k-ft; As = 7.48 in.2; fy = 60 ksi; Es = 29,000 ksi; f MA jds sl s = = = 1 013 1 12 7 48 0 931 36 0 48 5 , . ( ) . ( . )( . ) . ksi -1500 -1000 -500 0 500 1000 1500 0 20 40 60 80 100 120 140 160 180 Distance along Girders from CL Bearing (ft) M om en t E nv el op e (k- ft) 17.3 ft 17.3 ft Figure D-6.2.2.1-1. Negative moment envelope and location of inflection points.

D-82 Ec = 4,696 ksi; n = Es /Ecd = 6.175; b′ = width of compression face = 48 in.; d = effective depth from bottom of girder including build- up = 36.0 in.; The allowable tensile stress in the mild reinforcement is LRFD Eq. 5.7.3.4-1 (maximum allowed) where dc = 2.22 in.; Z = 170 k/in.; and For the girders, the tensile stress in the mild reinforcement is more than the allowable. Thus, the distribution of reinforce- ment for control of cracking is inadequate and more rein- forcement is required: fsa = 36.0 ksi < fs = 48.5 ksi. NG (no good) Using No. 7 bars instead of No. 6 bars, the tensile stress in the mild reinforcement would be 35.4 ksi, which is less then the allowable stress of 36.0 ksi. 6.2.5 Fatigue of Negative Moment Reinforcement Fatigue of the negative moment reinforcement is evalu- ated according to LRFD Article 5.5.3.1. The conditions in this design do not meet the requirements to be exempt from evaluation. To determine whether cracked section properties must be used to evaluate the stress range, the stress in the concrete is computed under a specified loading combination: A d bc= = =2 2 2 22 4817 12 54No. of Bars in. 2( . )( ) . . f Z d A f f sa c y sa = ( ) = ( ) = = ≤ ≤ × > 1 3 1 3 0 6 170 2 22 12 54 0 6 60 56 1 36 0 36 0 / / . . ( . ) . . . . ksi USE ksi j k= − = − =1 3 1 0 206 3 0 931 . . . k n n n = + − = + ( ) − = 2 2 0 0043 6 175 0 0043 6 175 0 0043 6 175 0 206 2 2 ρ ρ ρ ( ) ( . )( . ) ( . )( . ) ( . )( . ) . ; and ρ = = =Abd s 7 48 48 36 0 0 0043 . ( . ) . ; M = sum of unfactored permanent loads + prestress + 1.5(fatigue load) = −391 + 0 + 1.5(−139.3) (fatigue moment is computed below) = −600 k-ft; and fcd top = M/S′t = M/(I′/[h′ − y′b]) = −600(12)/(237,276/(39.00 − 19.50) = −0.592 ksi. Stress limit is then computed for comparison to computed stress: fcd max = −0.095√ f ′cd = −0.095√4.0 = −0.190 ksi. Comparing the absolute values of the computed stress and stress limit, fcd top = 0.592 ksi > fcd max = 0.190 ksi. Since the computed stress exceeds the stress limit, the effect of fatigue must be evaluated using cracked section analysis. The stress in the reinforcement is computed using the specified fatigue loading and cracked section analysis as required. The basic negative fatigue moment is −582 k-ft, using the QConBridge Program (see Subappendix C). The analysis assumes the bridge is fully continuous for live loads. This moment is factored by the dynamic load allowance for fatigue (1 + 0.15) (LRFD Article 3.6.2.1); by the live-load dis- tribution factor for one lane loaded (0.333) (LRFD Article 3.6.1.4.3b; and by LRFD Table 4.6.2.2.2b-1 (cross-section type g), which is divided by the multiple presence factor for one lane loaded (1.2) (LRFD Article C3.6.1.1.2) and the load factor for the fatigue limit state, which is 0.75 (LRFD Table 3.4.1-1). Therefore, the design fatigue moment is Mf = 1.15(0.333/1.2)(0.75)(−582) = −139.3 k-ft. The area of reinforcement provided in the top of the girder is Asd = 7.48 in.2. The stress range in the reinforcement caused by this design fatigue moment is computed using the principles of working stress design for a cracked section. The width of the com- pression zone is the width of the box girder = 48 in.: ff = 6.63 ksi. The limiting stress range, ff max, is computed as ff max = 21 − 0.33 fmin + 8(r/h) LRFD Eq. 5.5.3.2-1 = 21 − 0.33(18.61) + 8(0.3) = 17.3 ksi where

D-83 fmin = fLL f + fDL = 0.00 + 18.61 = 18.61 ksi; fLL f = minimum live-load stress from fatigue loading = 0.00 ksi; fDL = stress in reinforcement from permanent loads from working stress analysis = 18.61 ksi; MDL = total composite dead-load moment (load factor = 1.0) at the center of the pier = −391 k-ft (see Table D-3.7.1-2); and r/h = ratio of base radius to height of rolled on transverse deformations in the reinforcement = 0.3 may be used if value not known. Comparing the computed stress range to the limiting stress range, ff = 6.63 ksi < ff max = 17.3 ksi. OK

D-84 REFERENCES AND BIBLIOGRAPHY FOR APPENDIX D AASHTO LRFD Bridge Design Specifications, 2nd Edition. Amer- ican Assoc. of State Highway and Transportation Officials; Washington, DC; 1998. Hennessey, S.A. and Bexten, K.A. “Value Engineering Results In Successful Precast Bridge Solution,” Proceedings of the 2002 Concrete Bridge Conference, Precast/Prestressed Concrete Insti- tute, Chicago, 2003. Ma, Z., Huo, X., Tadros, M.K., and Baishya, M. “Restraint Moments in Precast/Prestressed Concrete Continuous Bridges,” PCI Journal, Vol. 43, No. 6, Nov/Dec 1998; pp. 40–56. Salmons, J.R. Behavior of Untensioned-Bonded Prestressing Strand, Final Report 77-1, Missouri Cooperative Highway Research Pro- gram, Missouri State Highway Department, June 1980. Salmons, J.R. End Connections of Pretensioned I-Beam Bridges, Final Report 73-5C, Missouri Cooperative Highway Research Program, Missouri State Highway Department, Nov. 1974. Salmons, J.R. and May, G.W. Strand Reinforcing for End Connec- tion of Pretensioned I-Beam Bridges, Interim Report 73-5B, Mis- souri Cooperative Highway Research Program, Missouri State Highway Department, May 1974. Salmons, J.R. and McCrate, T.E. Bond of Untensioned Prestress Strand, Interim Report 73-5A, Missouri Cooperative Highway Research Program, Missouri State Highway Department, Aug. 1973.

D-85 SUBAPPENDIX A: INPUT DATA FOR RESTRAINT DA.1 DESIGN EXAMPLE 1: AASHTO TYPE III GIRDER BRIDGE DA.1.1 Initial Design III 7.3333 2 37 85 5 100 30 0 8 0.4 0.153 7.75 202500 7.75 LL 50 Eps 28500 5500 Strand Age at Prestress Transfer (days) 1 7000 Girder Age at Continuity (days) 4000 Girder Age at Time Deck is in Place (days) 150 Y 150 30 1.8 395.4 353 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 spans) Strand Draping Length / Span Length Girder Spacing (ft) Centroid of Straight Strands (in) Second Interior Span Length (5 spans) Initial Strand Tension (psi) Girder Concrete Unit Weight (pcf) Number of Straight Strands Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Strands Girder Concrete Compressive Stress at 28 Days (psi) Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep Coefficient Cross Sectional Area of Strand (in 2 ) Deck Concrete Compressive Stress at 28 Days (psi) Centroid of Draped Strands at Girder End (in) Centroid of Draped Strands at Midspan (in) Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? (Y/N) If Yes, Deck Age for Dischinger Modification (days) Relative Humidity (%) Deck Concrete Ultimate Shrinkage (millionths) Girder Concrete Ultimate Shrinkage (millionths) Deck Concrete Unit Weight (pcf) DA.1.2 Girder Age at Continuity of 90 Days – Simplified Approach TABLE DA.1.1-1 Initial design III 7.3333 2 37 85 5 100 30 0 8 0.4 0.153 7.75 202500 7.75 LL 50 Eps 28500 5500 Strand Age at Prestress Transfer (days) 1 7000 Girder Age at Continuity (days) 90 4000 Girder Age at Time Deck is in Place (days) 90 150 Y 150 30 1.8 395.4 353 75 Include Dischinger Effect? (Y/N) If Yes, Deck Age for Dischinger Modification (days) Relative Humidity (%) Deck Concrete Ultimate Shrinkage (millionths) Girder Concrete Ultimate Shrinkage (millionths) Deck Concrete Unit Weight (pcf) Input Girder Concrete Utlimate Creep Coefficient Cross Sectional Area of Strand (in 2 ) Deck Concrete Compressive Stress at 28 Days (psi) Centroid of Draped Strands at Girder End (in) Centroid of Draped Strands at Midspan (in) Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Strands Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Strands Girder Concrete Compressive Stress at 28 Days (psi) Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 spans) Strand Draping Length / Span Length Girder Spacing (ft) Centroid of Straight Strands (in) Second Interior Span Length (5 spans) Initial Strand Tension (psi) TABLE DA.1.2-1 Final design for girder age at continuity of 90 days

DA.1.3 Girder Age at Continuity of 60 Days D-86 TABLE DA.1.3-1 Run 2—Design for girder age at continuity of 60 days III 7.5 2 37 85 5 100 32 0 8 0.4 0.153 7.75 202500 7.75 LL 50 Eps 28500 6000 Strand Age at Prestress Transfer (days) 1 7000 Girder Age at Continuity (days) 60 4000 Girder Age at Time Deck is in Place (days) 60 150 Y 150 30 1.8 395.4 353 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 spans) Strand Draping Length / Span Length Girder Spacing (ft) Centroid of Straight Strands (in) Second Interior Span Length (5 spans) Initial Strand Tension (psi) Girder Concrete Unit Weight (pcf) Number of Straight Strands Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Strands Girder Concrete Compressive Stress at 28 Days (psi) Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep Coefficient Cross Sectional Area of Strand (in 2 ) Deck Concrete Compressive Stress at 28 Days (psi) Centroid of Draped Strands at Girder End (in) Centroid of Draped Strands at Midspan (in) Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? (Y/N) If Yes, Deck Age for Dischinger Modification (days) Relative Humidity (%) Deck Concrete Ultimate Shrinkage (millionths) Girder Concrete Ultimate Shrinkage (millionths) Deck Concrete Unit Weight (pcf) DA.1.4 Girder Age at Continuity of 28 Days TABLE DA.1.3-2 Run 3 (final)—Design for girder at age continuity of 60 days III 7.294 2 37 85 5 100 34 0 8 0.4 0.153 7.75 202500 7.75 LL 50 Eps 28500 6000 Strand Age at Prestress Transfer (days) 1 7000 Girder Age at Continuity (days) 60 4000 Girder Age at Time Deck is in Place 60 150 Y 150 30 1.8 395.4 353 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Second Interior Span Length (5 spans) Initial Strand Tension Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in 2 ) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Girder End Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight

DA.1.4 Girder Age at Continuity of 28 Days D-87 TABLE DA.1.4-1 Run 2—Design for girder age at continuity of 28 days III 7.4 2 37.0 85 5.0 100 30 0 8 0.4 0.153 7.75 202500 7.75 LL 50 Eps 28500 5500 Strand Age at Prestress Transfer (days) 1 7000 Girder Age at Continuity (days) 28 4000 Girder Age at Time Deck is in Place (days) 28 150 y 150 30 1.8 395.4 353 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 spans) Strand Draping Length / Span Length Girder Spacing (ft) Centroid of Straight Strands (in) Second Interior Span Length (5 spans) Initial Strand Tension (psi) Girder Concrete Unit Weight (pcf) Number of Straight Strands Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Strands Girder Concrete Compressive Stress at 28 Days (psi) Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep Coefficient Cross Sectional Area of Strand (in 2 ) Deck Concrete Compressive Stress at 28 Days (psi) Centroid of Draped Strands at Girder End (in) Centroid of Draped Strands at Midspan (in) Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? (Y/N) If Yes, Deck Age for Dischinger Modification (days) Relative Humidity (%) Deck Concrete Ultimate Shrinkage (millionths) Girder Concrete Ultimate Shrinkage (millionths) Deck Concrete Unit Weight (pcf) TABLE DA.1.4-2 Run 3—Design for girder age at continuity of 28 days III 7.2 2 34.0 85 8.0 100 20 0 14 0.4 0.217 7.75 202500 7.75 LL 50 Eps 28500 7000 Strand Age at Prestress Transfer (days) 1 8500 Girder Age at Continuity (days) 28 4000 Girder Age at Time Deck is in Place (days) 28 150 y 150 30 1.62 395.4 353 75 Include Dischinger Effect? (Y/N) If Yes, Deck Age for Dischinger Modification (days) Relative Humidity (%) Deck Concrete Ultimate Shrinkage (millionths) Girder Concrete Ultimate Shrinkage (millionths) Deck Concrete Unit Weight (pcf) Input Girder Concrete Utlimate Creep Coefficient Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days (psi) Centroid of Draped Strands at Girder End (in) Centroid of Draped Strands at Midspan (in) Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Strands Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Strands Girder Concrete Compressive Stress at 28 Days (psi) Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 spans) Strand Draping Length / Span Length Girder Spacing (ft) Centroid of Straight Strands (in) Second Interior Span Length (5 spans) Initial Strand Tension (psi)

D-88 TABLE DA.1.4-3 Run 4—Design for girder age at continuity of 28 days III 7.09 2 35.0 85 9.0 100 22 0 12 0.4 0.217 7.75 202500 7.75 LL 50 Eps 28500 7500 Strand Age at Prestress Transfer (days) 1 9000 Girder Age at Continuity (days) 28 4000 Girder Age at Time Deck is in Place (days) 28 150 y 150 30 1.56 395.4 353 75 Include Dischinger Effect? (Y/N) If Yes, Deck Age for Dischinger Modification (days) Relative Humidity (%) Deck Concrete Ultimate Shrinkage (millionths) Girder Concrete Ultimate Shrinkage (millionths) Deck Concrete Unit Weight (pcf) Input Girder Concrete Utlimate Creep Coefficient Cross Sectional Area of Strand (in 2 ) Deck Concrete Compressive Stress at 28 Days (psi) Centroid of Draped Strands at Girder End (in) Centroid of Draped Strands at Midspan (in) Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Strands Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Strands Girder Concrete Compressive Stress at 28 Days (psi) Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 spans) Strand Draping Length / Span Length Girder Spacing (ft) Centroid of Straight Strands (in) Second Interior Span Length (5 spans) Initial Strand Tension (psi) III 7.09 2 35.0 85 9.0 100 22 0 12 0.4 0.217 7.75 202500 7.75 LL 50 Eps 28500 7500 Strand Age at Prestress Transfer (days) 1 8500 Girder Age at Continuity (days) 28 4000 Girder Age at Time Deck is in Place (days) 28 150 y 150 30 1.62 395.4 353 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 spans) Strand Draping Length / Span Length Girder Spacing (ft) Centroid of Straight Strands (in) Second Interior Span Length (5 spans) Initial Strand Tension (psi) Girder Concrete Unit Weight (pcf) Number of Straight Strands Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Strands Girder Concrete Compressive Stress at 28 Days (psi) Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep Coefficient Cross Sectional Area of Strand (in 2 ) Deck Concrete Compressive Stress at 28 Days (psi) Centroid of Draped Strands at Girder End (in) Centroid of Draped Strands at Midspan (in) Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? (Y/N) If Yes, Deck Age for Dischinger Modification (days) Relative Humidity (%) Deck Concrete Ultimate Shrinkage (millionths) Girder Concrete Ultimate Shrinkage (millionths) Deck Concrete Unit Weight (pcf) TABLE DA.1.4-4 Run 5 (final)—Design for girder age at continuity of 28 days

DA.2 DESIGN EXAMPLE 2 – PCI BT-72 GIRDER BRIDGE DA.2.1 Initial Design D-89 TABLE DA.2.1-1 Initial design BT 72 8.615 2 66 124.75 4 100 26 0 6 0.4 0.217 7.75 202500 7.75 LL 50 Eps 28500 5500 Strand Age at Prestress Transfer (days) 1 7000 Girder Age at Continuity (days) 4000 Girder Age at Time Deck is in Place 150 y 150 30 1.889 430.7 353 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Second Interior Span Length (5 spans) Initial Strand Tension Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Deck Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep CrossSectional Area of Strand (in2) Centroid of Draped Strands at Girder End Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Girder Concrete Ultimate Shrinkage Deck Concrete Ultimate Shrinkage Deck Concrete Unit Weight DA.2.2 Girder Age at Continuity of 90 Days – Simplified Approach BT 72 8.615 2 66 124.75 4 100 26 0 6 0.4 0.217 7.75 202500 7.75 LL 50 Eps 28500 5500 Strand Age at Prestress Transfer (days) 1 7000 Girder Age at Continuity (days) 90 4000 Girder Age at Time Deck is in Place 90 150 y 150 30 1.889 430.7 353 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Second Interior Span Length (5 spans) Initial Strand Tension Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Girder End Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight TABLE DA.2.2-1 Final design for girder age at continuity of 90 days

DA.2.3 Girder Age at Continuity of 60 Days D-90 TABLE DA.2.3-1 Final design for girder age at continuity of 60 days BT 72 8.62 2 64 124.75 6 100 26 0 10 0.4 0.217 7.75 202500 7.75 LL 50 Eps 28500 5600 Strand Age at Prestress Transfer (days) 1 7000 Girder Age at Continuity (days) 60 4000 Girder Age at Time Deck is in Place 60 150 y 150 30 1.889 430.7 353 75 Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Centroid of Draped Strands at Girder End Second Interior Span Length (5 spans) Initial Strand Tension Type of Strand (SR or LL) DA.2.4 Girder Age at Continuity of 28 Days TABLE DA.2.4-1 Final design for girder age at continuity of 28 days BT 72 8.4 2 61 124.75 9 100 30 0 16 0.4 0.217 7.75 202500 7.75 LL 50 Eps 28500 7500 Strand Age at Prestress Transfer (days) 1 8000 Girder Age at Continuity 28 4000 Girder Age at Time Deck is in Place 28 150 y 150 30 1.754 430.7 353 75 Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Centroid of Draped Strands at Girder End Second Interior Span Length (5 spans) Initial Strand Tension

DA.3 DESIGN EXAMPLE 3 – 51-IN. DEEP SPREAD BOX GIRDER BRIDGE DA.3.1 Initial Design D-91 TABLE DA.3.1-1 Initial design 7.33 2 0 84.83333 0 100 39 0 0 0.4 0.153 11 202500 8.25 LL 50 Eps 28500 5000 Strand Age at Prestress Transfer (days) 1 6000 Girder Age at Continuity 4500 Girder Age at Time Deck is in Place 150 y 150 30 2.025 423.6 359.6 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Second Interior Span Length (5 spans) Initial Strand Tension Girder Concrete Unit Weight (pcf) Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Girder End Centroid of Draped Strands at Midspan Number of Straight Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight DA.3.2 Girder Age at Continuity of 90 Days – Simplified Approach TABLE DA.3.2-1 Final design for girder age at continuity of 90 days 7.33 2 0 84.83333 0 100 39 0 0 0.4 0.153 11 202500 8.25 LL 50 Eps 28500 5000 Strand Age at Prestress Transfer (days) 1 6000 Girder Age at Continuity 90 4500 Girder Age at Time Deck is in Place 90 150 y 150 30 2.025 423.6 359.6 75 Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Second Interior Span Length (5 spans) Initial Strand Tension Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Girder End Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight

DA.3.3 Girder Age at Continuity of 60 Days D-92 TABLE DA.3.3-1 Final design for girder age at continuity of 60 days 7.25 2 0 84.83333 0 100 40 0 0 0.4 0.153 11 202500 8.25 LL 50 Eps 28500 5000 Strand Age at Prestress Transfer (days) 1 6000 Girder Age at Continuity 60 4500 Girder Age at Time Deck is in Place 60 150 y 150 30 2.025 423.6 359.6 75 Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Centroid of Draped Strands at Girder End Second Interior Span Length (5 spans) Initial Strand Tension DA.3.4 Girder Age at Continuity of 28 Days TABLE DA.3.4-1 Final design for girder age at continuity of 28 days 6.68 2 0 84.83333 0 100 56 0 0 0.4 0.153 11 202500 8.25 LL 50 Eps 28500 5250 Strand Age at Prestress Transfer (days) 1 6000 Girder Age at Continuity 28 4500 Girder Age at Time Deck is in Place 28 150 y 150 30 2.025 423.6 359.6 75 Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Centroid of Draped Strands at Girder End Second Interior Span Length (5 spans) Initial Strand Tension

DA.4 DESIGN EXAMPLE 4 – AASHTO BIII-48 ADJACENT BOX GIRDER BRIDGE DA.4.1 Initial Design D-93 TABLE DA.4.1-1 Final design for girder age at continuity of 28 days BIII-48 5.4 2 0.0 85 0.0 100 20 0 0 0.4 0.153 4 202500 0 LL 50 Eps 28500 4500 Strand Age at Prestress Transfer (days) 1 6000 Girder Age at Continuity 4500 Girder Age at Time Deck is in Place 150 y 150 30 1.999 414.5 100 75 Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Centroid of Draped Strands at Girder End Second Interior Span Length (5 spans) Initial Strand Tension DA.4.2 Girder Age at Continuity of 90 Days – Simplified Approach TABLE DA.4.2-1 Final design for girder age at continuity of 90 days BIII-48 5.4 2 0.0 85 0.0 100 20 0 0 0.4 0.153 4 202500 0 LL 50 Eps 28500 4500 Strand Age at Prestress Transfer (days) 1 6000 Girder Age at Continuity 90 4500 Girder Age at Time Deck is in Place 90 150 y 150 30 1.999 414.5 100 75 Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 ) Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Centroid of Draped Strands at Girder End Second Interior Span Length (5 spans) Initial Strand Tension

DA.4.3 Girder Age at Continuity of 28 Days D-94 TABLE DA.4.3-1 Final design for girder age at continuity of 28 days BIII-48 5.4 2 0.0 85 0.0 100 20 0 0 0.4 0.153 4 202500 0 LL 50 Eps 28500 4500 Strand Age at Prestress Transfer (days) 1 6000 Girder Age at Continuity 28 4500 Girder Age at Time Deck is in Place 28 150 y 150 30 1.999 414.5 100 75 Include Dischinger Effect? If Yes, Deck Age for Dischinger Modification Relative Humidity Deck Concrete Ultimate Shrinkage Girder Concrete Ultimate Shrinkage Deck Concrete Unit Weight Input Girder Concrete Utlimate Creep Cross Sectional Area of Strand (in2) Deck Concrete Compressive Stress at 28 Days Centroid of Draped Strands at Midspan Span Data Select Girder Type Exterior Span Length (ft) Number of spans Girder Concrete Unit Weight (pcf) Number of Straight Girder Concrete Compressive Stress at Transfer (psi) Number of Draped Girder Concrete Compressive Stress at 28 Days Concrete Data Type of Strand (SR or LL) Time Data Strand Data Deck Thickness (in) Additional Dead Load (psf) First Interior Span Length (ft) ( for 3,4,5 Strand Draping Length / Span Girder Spacing (ft) Centroid of Straight Strands Centroid of Draped Strands at Girder End Second Interior Span Length (5 spans) Initial Strand Tension

D-95 SUBAPPENDIX B: INPUT AND OUTPUT FROM RESPONSE 2000 DB.1 PROGRAM INFORMATION The Response 2000 Program was developed at the University of Toronto and is available free of charge at www.ecf. utoronto.ca/~bentz/r2k.htm. The version of the program used for this study is shown in the figure below taken from the program.

D-96 Girder Midspan - 90 Days TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 1.75 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1136.6 360317.7 17.8 35.0 20266.9 10303.2 1200.6 393955.8 18.2 34.6 21671.2 11395.5 Gross Conc. Trans (n=6.56) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 Strain Discontinuity in Concrete 22.0 93.0 7.052 .8 Types Concrete 4000 7000 base type 2 layers of 9 - #5 2 - S.5 ∆ε p = 7.10 ms 36 - S.5 ∆ε p = 7.10 ms Concrete ε c' = 2.22 ms fc' = 7000 psi a = 0.75 in ft = 308 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 P-Steel ε p = 43.0 ms fpu = 266 ksi Low Relax Figure DB.2.1.1-1. Cross section and material information. M om en t (f t-k ip s) Curvature (rad/106 in) -800.0 -1600.0 0.0 800.0 1600.0 2400.0 3200.0 4000.0 4800.0 -200.0-400.0-600.0-800.0 0.0 200.0 400.0 600.0 Figure DB.2.1.1-2. Moment curvature analysis plot.

D-97 End of Girder - 90 Day TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 1.75 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1169.1 405953.4 18.7 36.5 21693.1 11110.9 1292.1 462094.8 18.8 36.5 24585.7 12675.9 Gross Conc. Trans (n=6.56) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 Strain Discontinuity in Concrete 22.0 93.0 7.055 .3 Types Concrete 4000 7000 base type 9 - #6 9 - #5 9 - #6 9 - #5 2 - S.5 ∆ε p = 7.10 ms 28 - S.5 ∆ε p = 7.10 ms 8 - #5 8 - S.5 ∆ε p = 7.10 ms Concrete ε c' = 2.22 ms fc' = 7000 psi a = 0.75 in ft = 308 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 P-Steel ε p = 43.0 ms fpu = 266 ksi Low Relax Figure DB.2.1.2-1. Cross section and material information. M om en t (f t-k ip s) Curvature (rad/106 in) -900.0 -1800.0 -2700.0 -3600.0 0.0 900.0 1800.0 2700.0 3600.0 4500.0 5400.0 -90.0 0.0 90.0 180.0 270.0 360.0 450.0 540.0 Figure DB.2.1.2-2. Moment curvature analysis plot.

D-98 Diaphragm - 90 Day TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 1.69 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1312.0 433853.5 17.1 38.1 25332.4 11380.2 1427.4 473331.8 16.7 38.5 28342.6 12278.5 Gross Conc. Trans (n=8.22) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 22.0 93.0 7.055 .3 9 - #6 9 - #5 9 - #5 9 - #6 8 - #5 Concrete ε c' = 1.93 ms fc' = 4000 psi a = 0.75 in ft = 246 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 M om en t (f t-k ip s) Curvature (rad/106 in) -500.0 -1000.0 -1500.0 -2000.0 -2500.0 -3000.0 0.0 500.0 1000.0 -200.0 0.0 200.0 400.0 600.0 800.0 Figure DB.2.1.3-1. Cross section and material information. Figure DB.2.1.3-2. Moment curvature analysis plot.

D-99 Girder Midspan - 60 Day TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 1.75 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1136.6 360317.7 17.8 35.0 20266.9 10303.2 1204.0 396537.3 18.3 34.5 21720.1 11496.1 Gross Conc. Trans (n=6.56) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 Strain Discontinuity in Concrete 22.0 93.0 7.052 .8 Types Concrete 4000 7000 base type 2 layers of 9 - #5 2 - S.5 ∆ε p = 7.10 ms 40 - S.5 ∆ε p = 7.10 ms Concrete ε c' = 2.22 ms fc' = 7000 psi a = 0.75 in ft = 308 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 P-Steel ε p = 43.0 ms fpu = 266 ksi Low Relax M om en t (f t-k ip s) Curvature (rad/106 in) -900.0 -1800.0 0.0 900.0 1800.0 2700.0 3600.0 4500.0 5400.0 -200.0-400.0-600.0-800.0 0.0 200.0 400.0 Figure DB.2.2.1-1. Cross section and material information. Figure DB.2.2.1-2. Moment curvature analysis plot.

D-100 End of Girder - 60 Day TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 1.75 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1169.1 405953.4 18.7 36.5 21693.1 11110.9 1295.5 465041.1 18.9 36.4 24641.0 12783.8 Gross Conc. Trans (n=6.56) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 Strain Discontinuity in Concrete 22.0 93.0 7.055 .3 Types Concrete 4000 7000 base type 9 - #5 2 layers of 9 - #6 9 - #5 4 - S.5 ∆ε p = 7.10 ms 38 - S.5 ∆ε p = 7.10 ms 8 - #5 Concrete ε c' = 2.22 ms fc' = 7000 psi a = 0.75 in ft = 308 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 P-Steel ε p = 43.0 ms fpu = 266 ksi Low Relax M om en t (f t-k ip s) Curvature (rad/106 in) -1000.0 -2000.0 -3000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 6000.0 -90.0 0.0 90.0 180.0 270.0 360.0 450.0 540.0 Figure DB.2.2.2-1. Cross section and material information. Figure DB.2.2.2-2. Moment curvature analysis plot.

D-101 Diaphragm - 60 Day TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 1.69 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1312.0 433853.5 17.1 38.1 25332.4 11380.2 1427.4 473331.8 16.7 38.5 28342.6 12278.5 Gross Conc. Trans (n=8.22) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 22.0 93.0 7.055 .3 9 - #6 9 - #5 9 - #5 9 - #6 8 - #5 Concrete ε c' = 1.93 ms fc' = 4000 psi a = 0.75 in ft = 246 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 Figure DB.2.2.3-1. Cross section and material information. M om en t (f t-k ip s) Curvature (rad/106 in) -500.0 -1000.0 -1500.0 -2000.0 -2500.0 -3000.0 0.0 500.0 1000.0 -200.0 0.0 200.0 400.0 600.0 800.0 Figure DB.2.2.3-2. Moment curvature analysis plot.

D-102 Girder Midspan - 28 Days TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 1.70 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1094.0 351550.9 18.3 34.4 19189.6 10210.6 1160.2 386683.8 18.8 33.9 20541.9 11397.9 Gross Conc. Trans (n=6.08) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 Strain Discontinuity in Concrete 22.0 93.0 7.052 .8 Types Concrete 4000 8500 base type 2 layers of 9 - #5 2 - S.6 ∆ε p = 7.10 ms 32 - S.6 ∆ε p = 7.10 ms Concrete ε c' = 2.37 ms fc' = 8500 psi a = 0.75 in ft = 333 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 P-Steel ε p = 43.0 ms fpu = 266 ksi Low Relax M om en t (f t-k ip s) Curvature (rad/106 in) -1000.0 -2000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 6000.0 -200.0-400.0-600.0-800.0 0.0 200.0 400.0 600.0 Figure DB.2.3.1-1. Cross section and material information. Figure DB.2.3.1-2. Moment curvature analysis plot.

D-103 End of Girder - 28 Days TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 1.70 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1134.0 396800.1 19.2 36.0 20659.8 11008.9 1225.9 452140.6 20.1 35.1 22488.1 12865.3 Gross Conc. Trans (n=6.08) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 Strain Discontinuity in Concrete 22.0 93.0 7.055 .3 Types Concrete 4000 8500 base type 2 layers of 9 - #5 4 - S.6 ∆ε p = 7.10 ms 30 - S.6 ∆ε p = 7.10 ms 8 - #5 8 - #5 Concrete ε c' = 2.37 ms fc' = 8500 psi a = 0.75 in ft = 333 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 P-Steel ε p = 43.0 ms fpu = 266 ksi Low Relax Figure DB.2.3.2-1. Cross section and material information. M om en t (f t-k ip s) Curvature (rad/106 in) -1000.0 -2000.0 -3000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 6000.0 -80.0-160.0 0.0 80.0 160.0 240.0 320.0 400.0 480.0 Figure DB.2.3.2-2. Moment curvature analysis plot.

D-104 Diaphragm - 28 Days TJT 8/20/2002 All dimensions in inches Clear cover to reinforcement = 2.44 in Inertia (in4) Area (in2) yt (in) yb (in) St (in3) Sb (in3) 1312.0 433853.5 17.1 38.1 25332.4 11380.2 1388.1 481993.3 17.6 37.6 27334.8 12813.2 Gross Conc. Trans (n=8.22) Geometric Properties Crack Spacing Loading (N,M,V + dN,dM,dV) 2 x dist + 0.1 db / ρ 0.0 , -100.0 , 0.0 + 0.0 , 1.5 , 0.0 22.0 93.0 7.055 .3 2 layers of 9 - #5 2 layers of 8 - #5 Concrete ε c' = 1.93 ms fc' = 4000 psi a = 0.75 in ft = 246 psi (auto) Rebar ε s = 100.0 ms fu = 90 ksi fy= 60 M om en t (f t-k ip s) Curvature (rad/106 in) -300.0 -600.0 -900.0 -1200.0 -1500.0 -1800.0 0.0 300.0 600.0 900.0 1200.0 1500.0 1800.0 -200.0-400.0-600.0-800.0 0.0 200.0 400.0 600.0 800.0 Figure DB.2.3.3-1. Cross section and material information. Figure DB.2.3.3-2. Moment curvature analysis plot

D-105 SUBAPPENDIX C: INPUT AND OUTPUT FROM QCONBRIDGE The examples discussed within this subappendix are as follows: • Design Example 1: AASHTO Type III Girder Bridge. • Design Example 2: PCI BT-72 Girder Bridge. • Design Example 3: 51-IN.-Deep Spread Box Girder Bridge—The design spans for the bridge in this example are the same as Design Example 1; therefore, the output from Design Example 1 (see Section DC.1) was used for this example. • Design Example 4: AASHTO BIII-48 Adjacent Box Girder Bridge—The design spans for the bridge in this example are the same as Design Example 1; therefore, the output from Design Example 1 (see Section DC.1) was used for this example. DC.1 PROGRAM INFORMATION The program QConBridge was used to compute the fatigue load effects. While QConBridge also reports load effects for other types and combinations of loadings, these were not used in this study because the design for these load effects was per- formed using another computer program. Therefore, these other results have been deleted from the output that follows. The program QConBridge was developed by the Washing- ton State DOT and is available free of charge on the depart- ment website: www.wsdot.wa.gov/eesc/bridge/software/index.cfm. The version of the program used for this study is shown in the figure below taken from the program.

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