Duan, L., Chen, K., Tan, A. "Prestressed Concrete Bridges." Bridge Engineering Handbook. Ed. Wai-Fah Chen and Lian Duan Boca Raton: CRC Press, 2000

Transcription

1 Duan, L., Chen, K., Tan, A. "Prestressed Concrete Bridges." Bridge Engineering Handbook. Ed. Wai-Fah Chen and Lian Duan Boca Raton: CRC Press, 2000

2 10 Prestressed Concrete Bridges Lian Duan Caliornia Department o Transportation Kang Chen MG Engineering, Inc. Andrew Tan Everest International Consultants, Inc Introduction Materials Prestressing Systems 10.2 Section Types Void Slabs I-Girders Box Girders 10.3 Losses o Prestress Instantaneous Losses Time-Dependent Losses 10.4 Design Considerations Basic Theory Stress Limits Cable Layout Secondary Moments Flexural Strength Shear Strength Camber and Delections Anchorage Zones 10.5 Design Example 10.1 Introduction Prestressed concrete structures, using high-strength materials to improve serviceability and durability, are an attractive alternative or long-span bridges, and have been used worldwide since the 1950s. This chapter ocuses only on conventional prestressed concrete bridges. Segmental concrete bridges will be discussed in Chapter 11. For more detailed discussion on prestressed concrete, reerences are made to textbooks by Lin and Burns [1], Nawy [2], Collins and Mitchell [3] Materials Concrete A 28-day cylinder compressive strength ( c ) o concrete 28 to 56 MPa is used most commonly in the United States. A higher early strength is oten needed, however, either or the ast precast method used in the production plant or or the ast removal o ormwork in the cast-in-place method. The modulus o elasticity o concrete with density between 1440 and 2500 kg/m 3 may be taken as E w c c c (10.1) where w c is the density o concrete (kg/m 3 ). Poisson s ratio ranges rom 0.11 to 0.27, but 0.2 is oten assumed.

3 The modulus o rupture o concrete may be taken as [4] r 063. c or normal weight concrete lexural 052. c or sand - lightweight concrete lexural 044. c or all - lightweight concrete lexural 0.1 c or direct tension (10.2) Concrete shrinkage is a time-dependent material behavior and mainly depends on the mixture o concrete, moisture conditions, and the curing method. Total shrinkage strains range rom to over the lie o concrete and about 80% o this occurs in the irst year. For moist-cured concrete devoid o shrinkage-prone aggregates, the strain due to shrinkage ε sh may be estimated by [4] ε sh t kk s h t (10.3) K s t ( V/ S) e t V S ( 7 / ) t t (10.4) where t is drying time (days); k s is size actor and k h is humidity actors may be approximated by K n (140-H)/70 or H < 80%; K n 3(100-H)/70 or H 80%; and V/S is volume to surace area ratio. I the moist-cured concrete is exposed to drying beore 5 days o curing, the shrinkage determined by Eq. (10.3) should be increased by 20%. For stem-cured concrete devoid o shrinkage-prone aggregates: ε sh t kk s h t (10.5) Creep o concrete is a time-dependent inelastic deormation under sustained load and depends primarily on the maturity o the concrete at the time o loading. Total creep strain generally ranges rom about 1.5 to 4 times that o the instantaneous deormation. The creep coeicient may be estimated as [4] 06. ( ) ( ) H t t ψ tt KK t i (, 1) 35. c i 10 + t t i 06. (10.6) K c (10.7) K s t ( V/ S) e t e t t ( V/ S) (10.8)

4 FIGURE 10.1 Typical stress strain curves or prestressing steel. where H is relative humidity (%); t is maturity o concrete (days); t i is age o concrete when load is initially applied (days); K c is the eect actor o the volume-to-surace ratio; and K is the eect actor o concrete strength. Creep, shrinkage, and modulus o elasticity may also be estimated in accordance with CEB-FIP Mode Code [15] Steel or Prestressing Uncoated, seven-wire stress-relieved strands (AASHTO M203 or ASTM A416), or low-relaxation seven-wire strands and uncoated high-strength bars (AASHTO M275 or ASTM A722) are commonly used in prestresssed concrete bridges. Prestressing reinorcement, whether wires, strands, or bars, are also called tendons. The properties or prestressing steel are shown in Table TABLE 10.1 Properties o Prestressing Strand and Bars Material Grade and Type Diameter (mm) Tensile Strength pu (MPa) Yield Strength py (MPa) Modulus o Elasticity E p (MPa) Strand 1725 MPa (Grade 250) % o pu except 90% o pu 197, MPa (Grade 270) or low relaxation strand Bar Type 1, Plain 19 to % o pu Type 2, Deormed 15 to % o pu 207,000 Typical stress strain curves or prestressing steel are shown in Figure These curves can be approximated by the ollowing equations: For Grade 250 [5] : ps 197, 000 εps or εps < pu or ε ps > ε ps (10.9)

5 For Grade 270 [5]: ps 197, 000 εps or εps < pu or ε ps > ε ps (10.10) For Bars: ps 207, 000 εps or εps < pu or ε ps > ε ps (10.11) Advanced Composites or Prestressing Advanced composites iber-reinorced plastics (FPR) with their high tensile strength and good corrosion resistance work well in prestressed concrete structures. Application o advanced composites to prestressing have been investigated since the 1950s [6 8]. Extensive research has also been conducted in Germany and Japan [9]. The Ulenbergstrasse bridge, a two-span (21.3 and 25.6 m) solid slab using 59 iberglass tendons, was built in 1986 in Germany. It was the irst prestressed concrete bridge to use advanced composite tendons in the world [10]. FPR cables and rods made o ararmid, glass, and carbon ibers embedded in a synthetic resin have an ultimate tensile strength o 1500 to 2000 MPa, with the modulus o elasticity ranging rom 62,055 MPa to 165,480 MPa [9]. The main advantages o FPR are (1) a high speciic strength (ratio o strength to mass density) o about 10 to 15 times greater than steel; (2) a low modulus o elasticity making the prestress loss small; (3) good perormance in atigue; tests show [11] that or CFRP, at least three times the higher stress amplitudes and higher mean stresses than steel are achieved without damage to the cable over 2 million cycles. Although much eort has been given to exploring the use o advanced composites in civil engineering structures (see Chapter 51) and the cost o advanced composites has come down signiicantly, the design and construction speciications have not yet been developed. Time is still needed or engineers and bridge owners to realize the cost-eectiveness and extended lie expectancy gained by using advanced composites in civil engineering structures Grout For post-tensioning construction, when the tendons are to be bound, grout is needed to transer loads and to protect the tendons rom corrosion. Grout is made o water, sand, and cements or epoxy resins. AASHTO-LRFD [4] requires that details o the protection method be indicated in the contract documents. Readers are reerred to the Post-Tensioning Manual [12] Prestressing Systems There are two types o prestressing systems: pretensioning and post-tensioning systems. Pretensioning systems are methods in which the strands are tensioned beore the concrete is placed. This method is generally used or mass production o pretensioned members. Post-tensioning systems are methods in which the tendons are tensioned ater concrete has reached a speciied strength. This technique is oten used in projects with very large elements (Figure 10.2). The main advantage o post-tensioning is its ability to post-tension both precast and cast-in-place members. Mechanical prestressing jacking is the most common method used in bridge structures.

6 FIGURE 10.2 A post tensioned box girder bridge under construction Section Types Void Slabs Figure 10.3a shows FHWA [13] standard precast prestressed voided slabs. Sectional properties are listed in Table Although the cast-in-place prestressed slab is more expensive than a reinorced concrete slab, the precast prestressed slab is economical when many spans are involved. Common spans range rom 6 to 15 m. Ratios o structural depth to span are 0.03 or both simple and continuous spans I-Girders Figures 10.3b and c show AASHTO standard I-beams [13]. The section properties are given in Table This bridge type competes well with steel girder bridges. The ormwork is complicated, particularly or skewed structures. These sections are applicable to spans 9 to 36 m. Structural depthto-span ratios are or simple spans and 0.05 or continuous spans Box Girders Figure 10.3d shows FHWA [13] standard precast box sections. Section properties are given in Table These sections are used requently or simple spans o over 30 m and are particularly suitable or widening bridges to control delections. The box-girder shape shown in Figure 10.3e is oten used in cast-in-place prestressed concrete bridges. The spacing o the girders can be taken as twice the depth.. This type is used mostly or spans o 30 to 180 m. Structural depth-to-span ratios are or simple spans, and 0.04 or continuous spans. The high torsional resistance o the box girder makes it particularly suitable or curved alignment (Figure 10.4) such as those needed on reeway ramps.

7 FIGURE 10.3 Typical cross sections o prestressed concrete bridge superstructures Losses o Prestress Loss o prestress reers to the reduced tensile stress in the tendons. Although this loss does aect the service perormance (such as camber, delections, and cracking), it has no eect on the ultimate strength o a lexural member unless the tendons are unbounded or the inal stress is less than 0.5 pu [5]. It should be noted, however, that an accurate estimate o prestress loss is more pertinent in some prestressed concrete members than in others. Prestress losses can be divided into two categories:

8 TABLE 10.2 Precast Prestressed Voided Slabs Section Properties (Fig. 10.3a) Span Range, t (m) Width B in. (mm) Section Dimensions Depth D in. (mm) D1 in. (mm) D2 in. (mm) A in. 2 (mm ) Section Properties I x in. 4 (mm ) S x in. 3 (mm ) ,912 1,152 (7.6) (1,219) (305) (0) (0) (0.372) (2.877) (18.878) 30~ ,897 1,720 (10.1~10.70) (1,219) (381) (203) (203) (0.362) (5.368) (28.185) 40~ ,855 2,428 (12.2~13.7) (1,219) (457) (254) (254) (0.405) (10.097) ( ) ,517 3,287 (15.2) (1,219) (533) (305) (254) (0.454) (1.437) (53.864) TABLE 10.3 Precast Prestressed I-Beam Section Properties (Figs. 10.3b and c) AASHTO Section Dimensions, in. (mm) Beam Type Depth D Bottom Width A Web Width T Top Width B C E F G II 36 (914) 18 (457) 6 (152) 12 (305) 6 (152) 6 (152) 3 (76) 6 (152) III 45 (1143) 22 (559) 7 (178) 16 (406) 7 (178) 7.5 (191) 4.5 (114) 7 (178) IV 54 (1372) 26 (660) 8 (203) 20 (508) 8 (203) 9 (229) 6 (152) 8 (203) V 65 (1651) 28 (711) 8 (203) 42 (1067) 8 (203) 10 (254) 3 (76) 5 (127) VI 72 (1829) 28 (711) 8 (203) 42 (1067) 8 (203) 10 (254) 3 (76) 5 (127) Section Properties A in. 2 (mm 2 10 b ) Y b in. (mm) I x in. 4 (mm ) S b in. 3 (mm ) S t in. 3 (mm ) Span Ranges, t (m) II , ~ 45 (0.2381) (402.1) (21.22) (52.77) (41.43) (12.2 ~ 13.7) III , ~ 65 (0.3613) (514.9) (52.19) (101.38) (83.08) (15.2 ~ 110.8) IV , ~ 80 (0.5090) (628.1) (108.52) (172.77) (145.98) (21.4 ~ 24.4) V , ~ 100 (0.6535) (811.8) (216.93) (267.22) (275.16) (27.4 ~ 30.5) VI , ~ 120 (0.7000) (924.1) (305.24) (330.33) (337.38) (33.5 ~ 36.6) Instantaneous losses including losses due to anchorage set ( pa ), riction between tendons and surrounding materials ( pf ), and elastic shortening o concrete ( pes ) during the construction stage; Time-dependent losses including losses due to shrinkage ( psr ), creep ( pcr ), and relaxation o the steel ( pr ) during the service lie. The total prestress loss ( pt ) is dependent on the prestressing methods. For pretensioned members: pt pes psr pcr pr (10.12) For post-tensioned members: pt pa pf pes psr pcr pr (10.13)

9 FIGURE 10.4 Prestressed box girder bridge (I-280/110 Interchange, CA). TABLE 10.4 Precast Prestressed Box Section Properties (Fig. 10.3d) Span t (m) Section Dimensions Width B in. (mm) Depth D in. (mm) A in. 2 (mm ) Y b in. (mm) Section Properties I x in 4 (mm ) S b in. 3 (mm ) S t in. 3 (mm ) ,941 4,932 4,838 (15.2) (1,219) (686) (0.4471) (3310.6) (27.447) (80.821) ( ) ,499 6,767 6,629 (18.3) (1,219) (838) (0.4858) (414.8) (45.993) ( ) ( ) ,367 8,728 8,524 (21.4) (1,219) (991) (0.5245) (490.0) (70.080) ( ) ( ) ,088 9,773 9,571 (24.4) (1,219) (1,067) (0.5439) (527.8) (84.532) ( ) ( ) FIGURE 10.5 Anchorage set loss model.

10 TABLE 10.5 Friction Coeicients or Post-Tensioning Tendons Type o Tendons and Sheathing Wobble Coeicient K (1/mm) (10 6 ) Curvature Coeicient µ (1/rad) Tendons in rigid and semirigid galvanized ducts, seven-wire strands ~ 0.15 Pregreased tendons, wires and seven-wire strands 0.98 ~ ~ 0.15 Mastic-coated tendons, wires and seven-wire strands 3.3 ~ ~ 0.15 Rigid steel pipe deviations , lubrication required Source: AASHTO LRFD Bridge Design Speciications, 1st Ed., American Association o State Highway and Transportation Oicials. Washington, D.C With permission Instantaneous Losses Anchorage Set Loss As shown in Figure 10.5, assuming that the anchorage set loss changes linearly within the length (L pa ), the eect o anchorage set on the cable stress can be estimated by the ollowing ormula: pa 1 x L pa (10.14) L pa E ( L) L pf pf (10.15) 2 L pf pf L pa (10.16) where L is the thickness o anchorage set; E is the modulus o elasticity o anchorage set; is the change in stress due to anchor set; L pa is the length inluenced by anchor set; L pf is the length to a point where loss is known; and x is the horizontal distance rom the jacking end to the point considered Friction Loss For a post-tensioned member, riction losses are caused by the tendon proile curvature eect and the local deviation in tendon proile wobble eects. AASHTO-LRFD [4] speciies the ollowing ormula: ( Kx + µα) 1 e (10.17) pf where K is the wobble riction coeicient and µ is the curvature riction coeicient (see Table 10.5); x is the length o a prestressing tendon rom the jacking end to the point considered; and α is the sum o the absolute values o angle change in the prestressing steel path rom the jacking end Elastic Shortening Loss pes The loss due to elastic shortening can be calculated using the ollowing ormula [4]: pj ( ) pes Ep E cgp ci N 1 E 2 N E p ci cgp or pretensioned members or post-tensioned members (10.18)

11 TABLE 10.6 Lump Sum Estimation o Time-Dependent Prestress Losses Type o Beam Section Level For Wires and Strands with pu 1620, 1725, or 1860 MPa For Bars with pu 1000 or 1100 MPa Rectangular beams and solid slab Upper bound PPR PPR Average PPR Box girder Upper bound PPR 100 Average PPR I-girder Average PPR 41 + PPR Single T, double T hollow core and voided slab Upper bound Average Note: 1. PPR is partial prestress ratio (A ps py )/(A ps py + A s y ). 2. For low-relaxation strands, the above values may be reduced by 28 MPa or box girders 41 MPa or rectangular beams, solid slab and I-girders, and 55 MPa or single T, double T, hollow core and voided slabs. Source: AASHTO LRFD Bridge Design Speciications, 1st Ed., American Association o State Highway and Transportation Oicials. Washington, D.C With permission. c c PPR c PPR c PPR where E ci is modulus o elasticity o concrete at transer (or pretensioned members) or ater jacking (or post-tensioned members); N is the number o identical prestressing tendons; and cgp is sum o the concrete stress at the center o gravity o the prestressing tendons due to the prestressing orce at transer (or pretensioned members) or ater jacking (or post-tensioned members) and the selweight o members at the section with the maximum moment. For post-tensioned structures with bonded tendons, cgp may be calculated at the center section o the span or simply supported structures, at the section with the maximum moment or continuous structures Time-Dependent Losses Lump Sum Estimation AASHTO-LRFD [4] provides the approximate lump sum estimation (Table 10.6) o time-dependent loses ptm resulting rom shrinkage and creep o concrete, and relaxation o the prestressing steel. While the use o lump sum losses is acceptable or average exposure conditions, or unusual conditions, more-reined estimates are required Reined Estimation a. Shrinkage Loss: Shrinkage loss can be determined by ormulas [4]: H psr H or pretensioned members or post-tensioned members (10.19) where H is average annual ambient relative humidity (%). b. Creep Loss: Creep loss can be predicted by [4]: pcr cgp cdp (10.20)

12 FIGURE 10.6 Prestressed concrete member section at Service Limit State. where cgp is concrete stress at center o gravity o prestressing steel at transer, and cdp is concrete stress change at center o gravity o prestressing steel due to permanent loads, except the load acting at the time the prestressing orce is applied. c. Relaxation Loss: The total relaxation loss ( pr ) includes two parts: relaxation at time o transer pr1 and ater transer pr2. For a pretensioned member initially stressed beyond 0.5 pu, AASHTO-LRFD [4] speciies pr1 log24t 10 log24t 40 pi py pi py pi pi orstress-relieved strand or low relaxation strand (10.21) For stress-relieved strands pr pes 02. ( psr + pcr) pf 04. pes 02. ( psr + pcr) or pretensioning or post-tensioning (10.22) where t is time estimated in days rom testing to transer. For low-relaxation strands, pr2 is 30% o those values obtained rom Eq. (10.22) Design Considerations Basic Theory Compared with reinorced concrete, the main distinguishing characteristics o prestressed concrete are that The stresses or concrete and prestressing steel and deormation o structures at each stage, i.e., during prestressing, handling, transportation, erection, and the service lie, as well as stress concentrations, need to be investigated on the basis o elastic theory. The prestressing orce is determined by concrete stress limits under service load. Flexure and shear capacities are determined based on the ultimate strength theory. For the prestressed concrete member section shown in Figure 10.6, the stress at various load stages can be expressed by the ollowing ormula: Pj Pey j My ± ± A I I (10.23)

13 TABLE 10.7 Stress Limits or Prestressing Tendons Stress Type Prestressing Method Stress Relieved Strand and Plain High-Strength Bars Prestressing Tendon Type Low Relaxation Strand Deormed High-Strength Bars At jacking, pj Pretensioning 0.72 pu 0.78 pu Post-tensioning 0.76 pu 0.80 pu 0.75 pu Ater transer, pt Pretensioning 0.70 pu 0.74 pu Post-tensioning at anchorages 0.70 pu 0.70 pu 0.66 pu and couplers immediately ater anchor set Post-tensioning general 0.70 pu 0.74 pu 0.66 pu At Service Limit State, pc Ater all losses 0.80 py 0.80 py 0.80 py Source: AASHTO LRFD Bridge Design Speciications, 1st Ed., American Association o State Highway and Transportation Oicials. Washington, D.C With permission. TABLE 10.8 Temporary Concrete Stress Limits at Jacking State beore Losses due to Creep and Shrinkage Fully Prestressed Components Stress Type Area and Condition Stress (MPa) Compressive Pretensioned Post-tensioned 060. ci 055. ci Tensile Precompressed tensile zone without bonded reinorcement N/A Area other than the precompressed tensile zones and without bonded auxiliary reinorcement Area with bonded reinorcement which is suicient to resist 120% o the tension orce in the cracked concrete computed on the basis o uncracked section ci 138. ci Handling stresses in prestressed piles ci Note: Tensile stress limits are or nonsegmental bridges only. Source: AASHTO LRFD Bridge Design Speciications, 1st Ed., American Association o State Highway and Transportation Oicials. Washington, D.C With permission. where P j is the prestress orce; A is the cross-sectional area; I is the moment o inertia; e is the distance rom the center o gravity to the centroid o the prestressing cable; y is the distance rom the centroidal axis; and M is the externally applied moment. Section properties are dependent on the prestressing method and the load stage. In the analysis, the ollowing guidelines may be useul: Beore bounding o the tendons, or a post-tensioned member, the net section should be used theoretically, but the gross section properties can be used with a negligible tolerance. Ater bounding o tendons, the transormed section should be used, but gross section properties may be used approximately. At the service load stage, transormed section properties should be used Stress Limits The stress limits are the basic requirements or designing a prestressed concrete member. The purpose or stress limits on the prestressing tendons is to mitigate tendon racture, to avoid inelastic tendon deormation, and to allow or prestress losses. Tables 10.7 lists the AASHTO-LRFD [4] stress limits or prestressing tendons.

14 TABLE 10.9 Concrete Stress Limits at Service Limit State ater All Losses Fully Prestressed Components Stress Type Compressive Tensile Area and Condition Nonsegmental bridge at service stage Nonsegmental bridge during shipping and handling Segmental bridge during shipping and handling Precompressed tensile zone assuming uncracked section With bonded prestressing tendons other than piles Subjected to severe corrosive conditions With unbonded prestressing tendon Stress (MPa) 045. c 060. c 045. c 050. c 025. c No tension Note: Tensile stress limits are or nonsegmental bridges only. Source: AASHTO LRFD Bridge Design Speciications, 1st Ed., American Association o State Highway and Transportation Oicials. Washington, D.C With permission. The purpose or stress limits on the concrete is to ensure no overstressing at jacking and ater transer stages and to avoid cracking (ully prestressed) or to control cracking (partially prestressed) at the service load stage. Tables 10.8 and 10.9 list the AASHTO-LRFD [4] stress limits or concrete. A prestressed member that does not allow cracking at service loads is called a ully prestressed member, whereas one that does is called a partially prestressed member. Compared with ull prestress, partial prestress can minimize camber, especially when the dead load is relatively small, as well as provide savings in prestressing steel, in the work required to tension, and in the size o end anchorages and utilizing cheaper mild steel. On the other hand, engineers must be aware that partial prestress may cause earlier cracks and greater delection under overloads and higher principal tensile stresses under service loads. Nonprestressed reinorcement is oten needed to provide higher lexural strength and to control cracking in a partially prestressed member Cable Layout A cable is a group o prestressing tendons and the center o gravity o all prestressing reinorcement. It is a general design principle that the maximum eccentricity o prestressing tendons should occur at locations o maximum moments. Although straight tendons (Figure 10.7a) and harped multistraight tendons (Figure 10.7b and c) are common in the precast members, curved tendons are more popular or cast-in-place post-tensioned members. Typical cable layouts or bridge superstructures are shown in Figure To ensure that the tensile stress in extreme concrete ibers under service does not exceed code stress limits [4, 14], cable layout envelopes are delimited. Figure 10.8 shows limiting envelopes or simply supported members. From Eq. (10.23), the stress at extreme iber can be obtained Pj PeC j ± ± A I MC I (10.24) where C is the distance o the top or bottom extreme ibers rom the center gravity o the section (y b or y t as shown in Figure 10.6). When no tensile stress is allowed, the limiting eccentricity envelope can be solved rom Eq. (10.24) with e limit I M ± AC IP j (10.25)

15 FIGURE 10.7 Cable layout or bridge superstructures. FIGURE 10.8 Cable layout envelopes. For limited tension stress t, additional eccentricities can be obtained: e' I t PC j (10.26) Secondary Moments The primary moment (M 1 P j e) is deined as the moment in the concrete section caused by the eccentricity o the prestress or a statically determinate member. The secondary moment M s (Figure 10.9d) is deined as moment induced by prestress and structural continuity in an indeterminate member. Secondary moments can be obtained by various methods. The resulting moment is simply the sum o the primary and secondary moments.

16 FIGURE 10.9 Secondary moments Flexural Strength Flexural strength is based on the ollowing assumptions [4]: For members with bonded tendons, strain is linearly distributed across a section; or members with unbonded tendons, the total change in tendon length is equal to the total change in member length over the distance between two anchorage points. The maximum usable strain at extreme compressive iber is The tensile strength o concrete is neglected. A concrete stress o 0.85 c is uniormly distributed over an equivalent compression zone. Nonprestressed reinorcement reaches the yield strength, and the corresponding stresses in the prestressing tendons are compatible based on plane section assumptions. For a member with a langed section (Figure 10.10) subjected to uniaxial bending, the equations o equilibrium are used to give a nominal moment resistance o a a Mn A ps ps dp A s y d + s 2 2 a a h As y ds c b bw h ( )β (10.27)

17 FIGURE A langed section at nominal moment capacity state. a c β 1 (10.28) For bonded tendons: c A + A A 085. β ( b b ) h ps pu s y s y 1 c w 085. β b ka 1 c w+ ps d pu p h (10.29) k c ps pu 1 d p (10.30) k py pu (10.31) 085. β ( ) c 28 ( 0. 05) (10.32) where A represents area; is stress; b is the width o the compression ace o member; b w is the web width o a section; h is the compression lange depth o the cross section; d p and d s are distances rom extreme compression iber to the centroid o prestressing tendons and to centroid o tension reinorcement, respectively; subscripts c and y indicate speciied strength or concrete and steel, respectively; subscripts p and s mean prestressing steel and reinorcement steel, respectively; subscripts ps, py, and pu correspond to states o nominal moment capacity, yield, and speciied tensile strength o prestressing steel, respectively; superscript represents compression. The above equations also can be used or rectangular section in which b w b is taken. For unbound tendons: c A + A A 085. β ( b b ) h ps pu s y s y 1 c w 085. β b 1 c w h (10.33) dp L ps pe + Ωu Epεcu c L 2 py (10.34)

18 where L 1 is length o loaded span or spans aected by the same tendons; L 2 is total length o tendon between anchorage; Ω u is the bond reduction coeicient given by Ω u 3 L/ dp 15. L/ dp or uniorm and near third point loading or near midspan loading (10.35) in which L is span length. Maximum reinorcement limit: c 042. d e (10.36) d e A d A + A d ps ps p s y s + A ps ps s y (10.37) Minimum reinorcement limit: φ M 12 n M. cr (10.38) in which φ is lexural resistance actor 1.0 or prestressed concrete and 0.9 or reinorced concrete; M cr is the cracking moment strength given by the elastic stress distribution and the modulus o rupture o concrete. ( r pe d ) t I Mcr + y (10.39) where pe is compressive stress in concrete due to eective prestresses; and d is stress due to unactored sel-weight; both pe and d are stresses at extreme iber where tensile stresses are produced by externally applied loads Shear Strength The shear resistance is contributed by the concrete, the transverse reinorcement and vertical component o prestressing orce. The modiied compression ield theory-based shear design strength [3] was adopted by the AASHTO-LRFD [4] and has the ormula: V n Vc + Vs + Vp the lesser o 025. bd + V c v v p (10.40) where V β b d c c v v (10.41) V s A d v y v (cosθ + cot α)sinα s (10.42)

19 FIGURE Illustration o A c or shear strength calculation. (Source: AASHTO LRFD Bridge Design Speciications, 1st Ed., American Association o State Highway and Transportation Oicials. Washington, D.C With permission.) TABLE Values o θ and β or Sections with Transverse Reinorcement v c Angle (degree) ε x θ β θ β θ β θ β θ β θ β θ β θ β θ β (Source: AASHTO LRFD Bridge Design Speciications, 1st Ed., American Association o State Highway and Transportation Oicials. Washington, D.C With permission.) where b v is the eective web width determined by subtracting the diameters o ungrouted ducts or one hal the diameters o grouted ducts; d v is the eective depth between the resultants o the tensile and compressive orces due to lexure, but not to be taken less than the greater o 0.9d e or 0.72h; A v is the area o transverse reinorcement within distance s; s is the spacing o stirrups; α is the angle o inclination o transverse reinorcement to longitudinal axis; β is a actor indicating ability o diagonally cracked concrete to transmit tension; θ is the angle o inclination o diagonal compressive stresses (Figure 10.11). The values o β and θ or sections with transverse reinorcement are given in Table In using this table, the shear stress v and strain ε x in the reinorcement on the lexural tension side o the member are determined by v V u φv φbd v v p (10.43)

20 ε x Mu N V cot θ A dv EA + EA u u ps po s s p ps (10.44) where M u and N u are actored moment and axial orce (taken as positive i compressive) associated with V u and po is stress in prestressing steel when the stress in the surrounding concrete is zero and can be conservatively taken as the eective stress ater losses pe. When the value o ε x calculated rom the above equation is negative, its absolute value shall be reduced by multiplying by the actor F ε, taken as EA EA F s s + p ps ε EA + EA + EA c c s s p ps (10.45) where E s, E p, and E c are modulus o elasticity or reinorcement, prestressing steel, and concrete, respectively; A c is area o concrete on the lexural tension side o the member as shown in Figure Minimum transverse reinorcement: Maximum spacing o transverse reinorcement: A v min. c bs v y (10.46) Camber and Delections 08. d v For V u < 01. bd (10.47) c v v smax the smaller o 600 mm 04. For V u 01. bd (10.48) c v v smax the smaller o 300 mm As opposed to load delection, camber is usually reerred to as reversed delection and is caused by prestressing. A careul evaluation o camber and delection or a prestressed concrete member is necessary to meet serviceability requirements. The ollowing ormulas developed by the moment area method can be used to estimate midspan immediate camber or simply supported members as shown in Figure For straight tendon (Figure 10.7a): d v 2 L 8EI M c e (10.49) For one-point harping tendon (Figure 10.7b): 2 L M + EI c 8 c 2 3 M e (10.50)

21 For two-point harping tendon (Figure 10.7c): L + M M M a e 2 c e 8EI 3 L 2 2 c (10.51) For parabola tendon (Figure 10.7d): 2 L 5 M + M EI e c 8 6 c (10.52) where M e is the primary moment at end, P j e end, and M c is the primary moment at midspan P j e c. Uncracked gross section properties are oten used in calculating camber. For delection at service loads, cracked section properties, i.e., moment o inertia I cr, should be used at the post-cracking service load stage. It should be noted that long term eect o creep and shrinkage shall be considered in the inal camber calculations. In general, inal camber may be assumed 3 times as great as immediate camber Anchorage Zones In a pretensioned member, prestressing tendons transer the compression load to the surrounding concrete over a length L t gradually. In a post-tensioned member, prestressing tendons transer the compression directly to the end o the member through bearing plates and anchors. The anchorage zone, based on the principle o St. Venant, is geometrically deined as the volume o concrete through which the prestressing orce at the anchorage device spreads transversely to a more linear stress distribution across the entire cross section at some distance rom the anchorage device [4]. For design purposes, the anchorage zone can be divided into general and local zones [4]. The region o tensile stresses is the general zone. The region o high compressive stresses (immediately ahead o the anchorage device) is the local zone. For the design o the general zone, a strut-andtie model, a reined elastic stress analysis or approximate methods may be used to determine the stresses, while the resistance to bursting orces is provided by reinorcing spirals, closed hoops, or anchoraged transverse ties. For the design o the local zone, bearing pressure is a major concern. For detailed requirements, see AASHTO-LRFD [4] Design Example Two-Span Continuous Cast-in-Place Box-Girder Bridge Given A two-span continuous cast-in-place prestressed concrete box-girder bridge has two equal spans o length 48 m with a single-column bent. The superstructure is 10.4 m wide. The elevation view o the bridge is shown in Figure 10.12a. Material: Initial concrete: ci 24 MPa, E ci 24,768 MPa Final concrete: c 28 MPa, E c 26,752 MPa Prestressing steel: pu 1860 MPa low relaxation strand, E p 197,000 MPa Mild steel: y 400 MPa, E s 200,000 MPa Prestressing: Anchorage set thickness 10 mm Prestressing stress at jacking pj 0.8 pu 1488 MPa The secondary moments due to prestressing at the bent are M DA P j, M DG P j

22 FIGURE A two span continuous prestressed concrete box girder bridge. Loads: Dead Load sel-weight + barrier rail + uture wearing 75 mm AC overlay Live Load AASHTO HL-93 Live Load + dynamic load allowance Speciication: AASHTO-LRFD [4] (reerred as AASHTO in this example) Requirements 1. Determine cross section geometry 2. Determine longitudinal section and cable path 3. Calculate loads 4. Calculate live load distribution actors or interior girder 5. Calculate unactored moment and shear demands or interior girder 6. Determine load actors or Strength Limit State I and Service Limit State I 7. Calculate section properties or interior girder 8. Calculate prestress losses 9. Determine prestressing orce P j or interior girder

23 10. Check concrete strength or interior girder Service Limit State I 11. Flexural strength design or interior girder Strength Limit State I 12. Shear strength design or interior girder Strength Limit State I Solution 1. Determine Cross Section Geometry a. Structural depth d: For prestressed continuous spans, the structural depth d can be determined using a depthto-span ratio (d/l) o 0.04 (AASHTO LRFD Table ). d 0.04L 0.04(48) 1.92 m b. Girder spacing S: The spacing o girders is generally taken no more than twice their depth. S max < 2 d 2 (1.92) 3.84 m By using an overhang o 1.2 m, the center-to-center distance between two exterior girders is 10.4 m (2)(1.2 m) 8 m. Try three girders and two bays, S 8 m/2 4 m > 3.84 m NG Try our girders and three bays, S 8 m/ m < 3.84 m OK Use a girder spacing S 2.6 m c. Typical section: From past experience and design practice, we select that a thickness o 180 mm at the edge and 300 mm at the ace o exterior girder or the overhang. The web thickness is chosen to be 300 mm at normal section and 450 mm at the anchorage end. The length o the lare is usually taken as ¹ ₁₀ o the span length, say 4.8 m. The deck and soit thickness depends on the clear distance between adjacent girders; 200 and 150 mm are chosen or the deck and soit thickness, respectively. The selected box-girder section conigurations or this example are shown in Figure 10.12b. The section properties o the box girder are as ollows: Properties Midspan Bent (ace o support) A (m 2 ) I (m 4 ) y b (m) Determine Longitudinal Section and Cable Path To lower the center o gravity o the superstructure at the ace o the bent cap in the CIP post-tensioned box girder, the thickness o soit is lared to 300 mm as shown in Figure 10.12c. A cable path is generally controlled by the maximum dead-load moment and the position o the jack at the end section. Maximum eccentricities should occur at points o maximum dead load moments and almost no eccentricity should be present at the jacked end section. For this example, the maximum dead-load moments occur at three locations: at the bent cap, at the locations close to 0.4L or Span 1 and 0.6L or Span 2. A parabolic cable path is chosen as shown in Figure 10.12c. 3. Calculate Loads a. Component dead load DC: The component dead load DC includes all structural dead loads with the exception o the uture wearing surace and speciied utility loads. For design purposes, two parts o the DC are deined as:

24 DC1 girder sel-weight (density 2400 kg/m 3 ) acting at the prestressing stage DC2 barrier rail weight (11.5 kn/m) acting at service stage ater all losses. b. Wearing surace load DW: The uture wearing surace o 75 mm with a density 2250 kg/m 3 DW (deck width barrier width) (thickness o wearing surace) (density) [10.4 m 2(0.54 m)](0.075 m)(2250 kg/m 3 )( m/s 2 ) 15,423 N/m kn/m c. Live-Load LL and Dynamic Load Allowance IM: The design live load LL is the AASHTO HL-93 vehicular live loading. To consider the wheel-load impact rom moving vehicles, the dynamic load allowance IM 33% [AASHTO LRFD Table ] is applied to the design truck. 4. Calculate Live Load Distribution Factors AASHTO [1994] recommends that approximate methods be used to distribute live load to individual girders (AASHTO-LRFD ). The dimensions relevant to this prestressed box girder are: depth d 1920 mm, number o cells N c 3, spacing o girders S 2600 mm, span length L 48,000 mm, hal o the girder spacing plus the total overhang W e 2600 mm, and the distance between the center o an exterior girder and the interior edge o a barrier d e m. This box girder is within the range o applicability o the AASHTO approximate ormulas. The live-load distribution actors are calculated as ollows: a. Live-load distribution actor or bending moments: i. Interior girder (AASHTO Table b-1): One design lane loaded: g M 035. S L N c , lanes Two or more design lanes loaded: g M 13 N c 03. S L lanes (controls) 48, 000 ii. Exterior girder (AASHTO Table d-1): g M We lanes

25 FIGURE Live load distribution or exterior girder lever rule. b. Live-load distribution actor or shear: i. Interior girder (AASHTO Table a-1): One design lane loaded: S g V d L lanes 48, 000 Two or more design lanes loaded: S g V d L lanes (controls) 48, 000 ii. Exterior girder (AASHTO Table b-1): One design lane loaded Lever rule: The lever rule assumes that the deck in its transverse direction is simply supported by the girders and uses statics to determine the live-load distribution to the girders. AASHTO-LRFD [4] also requires that when the lever rule is used, the multiple presence actor m should apply. For a one design lane loaded, m 1.2. The lever rule model or the exterior girder is shown in Figure From static equilibrium: R g v mr 1.2(0.717) (controls)

26 FIGURE Moment envelopes or Span 1. Two or more design lanes loaded Modiy interior girder actor by e: g V de egv g interior girder 3800 ( ) V( interior girder) ( ) lanes The live load distribution actors at the strength limit state: Strength Limit State I Interior Girder Exterior Girder Bending moment lanes lanes Shear lanes lanes 5. Calculate Unactored Moments and Shear Demands or Interior Girder It is practically assumed that all dead loads are carried by the box girder and equally distributed to each girder. The live loads take orces to the girders according to live load distribution actors (AASHTO Article ). Unactored moment and shear demands or an interior girder are shown in Figures and Details are listed in Tables and Only the results or Span 1 are shown in these tables and igures since the bridge is symmetrical about the bent. 6. Determine Load Factors or Strength Limit State I and Service Limit State I a. General design equation (ASHTO Article 1.3.2): η γ Q φ R i i n (10.53)

27 FIGURE Shear envelopes or Span 1 TABLE Moment and Shear due to Unactored Dead Load or the Interior Girder Unactored Dead Load DC1 DC2 DW Span Location (x/l) M DC1 (kn-m) V DC1 (kn) M DC2 (kn-m) V DC2 (kn) M DW (kn-m) V DW (kn) Face o column Note: 1. DC1 interior girder sel-weight. 2. DC2 barrier sel-weight. 3. DW wearing surace load. 4. Moments in Span 2 are symmetrical about the bent. 5. Shears in span are anti-symmetrical about the bent. where γ i are load actors and φ a resistance actor; Q i represents orce eects; R n is the nominal resistance; η is a actor related to the ductility, redundancy, and operational importance o that being designed and is deined as: η ηdηrηi 095. (10.54)

28 TABLE Live Load Moment and Shear Envelopes and Associated Forces or the Interior Girder due to AASHTO HL-93 Location Positive Moment and Associated Shear Negative Moment and Associated Shear Shear and Associated Moment Span (x/l) M LL+IM (kn-m) V LL+IM (kn) V LL+IM (kn) M LL+IM (kn-m) V LL+IM (kn) M LL+IM (kn-m) Face o column Note: 1. LL + IM AASHTO HL-93 live load plus dynamic load allowance. 2. Moments in Span 2 are symmetrical about the bent. 3. Shears in Span 2 are antisymmetrical about the bent. 4. Live load distribution actors are considered. FIGURE Eective lange width o interior girder. For this bridge, the ollowing values are assumed: Limit States Ductility η D Redundancy η R Importance η I η Strength limit state Service limit state b. Load actors and load combinations: The load actors and combinations are speciied as (AASHTO Table ): Strength Limit State I: 1.25(DC1 + DC2) + 1.5(DW) (LL + IM) Service Limit State I: DC1 + DC2 + DW + (LL + IM) 7. Calculate Section Properties or Interior Girder For an interior girder as shown in Figure 10.16, the eective lange width b e is determined (AASHTO Article ) by:

29 TABLE Eective Flange Width and Section Properties or Interior Girder Location Dimension Mid span Bent (ace o support) h (mm) L e /4 (mm) 9,000 11,813 Top lange 12h + b w (mm) 2,700 2,700 S (mm) 2,600 2,600 b e (mm) 2,600 2,600 h (mm) L e /4 (mm) 9,000 11,813 Bottom lange 12h + b w (mm) 2,100 3,900 S (mm) 2,600 2,600 b e (mm) 2,100 2,600 Area A (m 2 ) Moment o inertia I (m 4 ) Center o gravity y b (m) Note: L e 36.0 m or midspan; L e m or the bent; b w 300 mm. b e the lesser o L 4 12 h S e + b w (10.55) where L e is the eective span length and may be taken as the actual span length or simply supported spans and the distance between points o permanent load inlection or continuous spans; h is the compression lange thickness and b w is the web width; and S is the average spacing o adjacent girders. The calculated eective lange width and the section properties are shown in Table or the interior girder. 8. Calculate Prestress Losses For a CIP post-tensioned box girder, two types o losses, instantaneous losses (riction, anchorage set, and elastic shortening) and time-dependent losses (creep and shrinkage o concrete, and relaxation o prestressing steel), are signiicant. Since the prestress losses are not symmetrical about the bent or this bridge, the calculation is perormed or both spans. a. Frictional loss pf: 1 e pf ( Kx + µα) pj ( ) (10.56) where K is the wobble riction coeicient /mm and µ is the coeicient o riction 0.25 (AASHTO Article b); x is the length o a prestressing tendon rom the jacking end to the point considered; α is the sum o the absolute values o angle change in the prestressing steel path rom the jacking end. For a parabolic cable path (Figure 10.17), the angle change is α 2e p /L p, where e p is the vertical distance between two control points and L p is the horizontal distance between two control points. The details are given in Table b. Anchorage set loss pa: For an anchor set thickness o L 10 mm and E 200,000 MPa, consider the point D where L pf 48 m and pf MPa:

30 FIGURE Parabolic cable path. TABLE Prestress Frictional Loss Segment e p (mm) L p (m) α (rad) Σα (rad) ΣL b (m) Point pf (Mpa) A A 0.00 AB B BC C CD D DE E EF F FG G L pa E ( L) LpF 200, 000( 10)( 48, 000) mm 31.6 m < 48 m pf OK 2 pf LpA 2( )( 316. ) L 48 pf MPa pa x x 1 L pa c. Elastic shortening loss pes: The loss due to elastic shortening in post-tensioned members is calculated using the ollowing ormula (AASHTO Article b): pes N 1 2 N E E p ci cgp (10.57) To calculate the elastic shortening loss, we assume that the prestressing jack orce or an interior girder P j 8800 kn and the total number o prestressing tendons N 4. cgp is calculated or ace o support section:

31 cgp 2 pj Pe j + + A I x M I DC1 x e ( ) ( )( ) kn / m MPa pes N 1 2 N E E p ci cgp ( ) 197, ( ) MPa d. Time-dependent losses ptm: AASHTO provides a table to estimate the accumulated eect o time-dependent losses resulting rom the creep and shrinkage o concrete and the relaxation o the steel tendons. From AASHTO Table : ptm 145 MPa (upper bound) e. Total losses pt: Details are given in Table pt pf pa pes ptm 9. Determine Prestressing Force P j or Interior Girder Since the live load is not in general equally distributed to girders, the prestressing orce P j required or each girder may be dierent. To calculate prestress jacking orce P j, the initial prestress orce coeicient F pci and inal prestress orce coeicient F pcf are deined as: F pci pf pa pes pj (10.58) F pcf 1 pt pj (10.59) The secondary moment coeicients are deined as: M sc x MDA L Pj x 1 L M P DG j or Span 1 or Span 2 (10.60) where x is the distance rom the let end or each span. The combined prestressing moment coeicients are deined as: M F () e + M psci pci sc (10.61)

32 TABLE Cable Path and Prestress Losses Location Prestress Losses (MPa) Force Coeicient Span (x/l) pf pa pes ptm pt F pci F pcf Note: F pci pf pa pes pj F pcf 1 pt pj M F () e + M pscf pcf sc (10.62) where e is the distance between the cable and the center o gravity o a cross section; positive values o e indicate that the cable is above the center o gravity, and negative ones indicate the cable is below the center o gravity o the section. The prestress orce coeicients and the combined moment coeicients are calculated and tabled in Table According to AASHTO, the prestressing orce P j can be determined using the concrete tensile stress limit in the precompression tensile zone (see Table 10.5): DC1 DC2 DW LL+ IM psf 05. c (10.63) in which DC1 M I DC1 x C (10.64) DC2 M I DC2 x C (10.65)