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27Â Â Analytical Research Approaches and Findings 2.1 Introduction Design case studies were conducted to compare maximum achievable span lengths when using 0.6-in. and 0.7-in. strands and to examine the influence of girder shape and size on the potential benefits of using 0.7-in. strands. The impacts of using 0.7-in. strands on girder end- region detailing requirements and prestress transfer were examined through mechanistic models. Finite element analyses were conducted to investigate the potential impacts of strand spacing and to validate the design parametric study. Finally, the handling and erection stability of long- span girders were examined. 2.2 Case Studies A total of 584 prestressed girder design cases were generated to compare girders using 0.6-in. and 0.7-in. strands. The objective of each design case was to maximize the girder span respecting all requirements of AASHTO LRFD Bridge Design Specifications (2020). Issues of handling stability were not considered in this part of the study but are addressed in Section 2.8 for girders having the longest resulting span lengths. 2.2.1 Design Parameters The design case studies are summarized in TableÂ 2.1. The design parameters were (1) con- crete unit weight: normal weight (NWC) and lightweight (LWC); (2) concrete compressive strength, f â²c ; (3) strand diameter: 0.6Â in. and 0.7Â in.; (4) girder spacing, S; (5) girder type: single web and double web; and (6) girder shape. The slab concrete compressive strength was taken as 4.5 ksi for all cases. Using data from a previous project (Shahrooz etÂ al., 2017), the concrete strength at strand release ( f â²ci) or the cases with f â²c exceeding 10 ksi was taken as 0.6f â²c while that for f â²c less than or equal to 10 ksi was taken as 0.8f â²c. Cross sections and key properties of the selected shapes are provided in Appendix A. The thickness of deck slabs was either 8Â in. or 9Â in., depending on the girder spacing as summarized in TableÂ 2.2. The designs were performed by a bespoke spreadsheet developed by the research team that was benchmarked and validated against commercial design software: LEAP CONSPAN (Bentley Systems, 2013). The design steps and procedure are provided in Appendix A. C H A P T E R Â 2
28 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders 2.2.2 Assumed Design Loads All reported designs are for interior girders; the following additional assumptions were made. 1. The concrete unit weights for deck and girders (DC load) were selected according to AASHTO LRFD Bridge Design Specifications (2020): a. Lightweight concrete (LWC) = 0.125 kcf, which is consistent with the unit weight pre- scribed in the 9th edition of AASHTO LRFD Bridge Design Specifications (AASHTO, 2020) and the desired compressive strength of 10 ksi. b. Normal weight concrete (NWC) = 0.145 kcf if f â²c â¤ 5.0 ksi and 0.140 + 0.001fcâ if 5.0 < f â²c â¤ ksi (AASHTO specifies this equation for f â²c up to 15 ksi, but it was extended in this study to f â²c = 18 ksi). c. For both concrete types, 0.005 kcf was added to account for the weight of reinforcement. 2. NWC deck with a compressive strength of 4.5 ksi was used for all designs. 3. A 2-in.-thick wearing surface having a density of 0.150 kcf was applied (DW load). 4. Regardless of bridge width, a total allowance for rail/barrier walls equal to 1.2 klf was assumed. This load was distributed equally across all girders (DC load). 5. A 2-in.-thick haunch over the entire top flange was assumed for load (DC) calculations for all single-web girders but not included in the section or capacity calculations. 6. All live load distribution factors were determined per AASHTO LRFD Bridge Design Specifi- cations (AASHTO, 2020). 2.2.3 Results and Discussions The maximum achievable spans for bridges were computed for various girder spacings, concrete strengths, and 0.6-in. and 0.7-in. strands. The resulting main design details (number of Parameter Range NWC design 10, 15, and 18 ksi ( = 8, 9, and 10.8 ksi, respectively) LWC design 10 ksi ( = 8 ksi) Strand diameter 0.6 in. and 0.7 in. Girder spacing, Single-web girders: 6, 8, 10, and 12 ftDouble-web girders: 12, 14, and 16 ft Number of spans One simple span; no skew Single-web girder shapes AASHTO-PCI Bulb-Tee: BT-54, BT-63 and BT-72 AASHTO I Girder: Type VI Florida I Girder: FIB-96 Ohio Wide Flange: WF-36, WF-54, and WF-72 Nebraska I Girder: NU-900, NU-1100, NU-1600, and NU-2000 Washington I Girder: WF74G and WF100G Double-web girder shapes AASHTO Box Girder: BIV-48 (adjacent) Northeast Extreme Tee: 40D Texas U Beam: U-40 and U-54 Washington U Beam: U54G5, UF60G5, and UF72G5 Single web Double web (ft) (in.) (ft) (in.) 6 8 + 2-in. haunch* 12 8 8 14 10 9 + 2-in. haunch 16 9 12 : Girder spacing, : Deck slab thickness. * Haunch is provided over the entire top flange. TableÂ 2.1. Variables for design case studies. TableÂ 2.2. Deck slab thickness.
Analytical Research Approaches and Findings 29Â Â straight strands, number of harped strands, number and length of debonded strands, and web reinforcement) are provided in Appendix B. The maximum achievable span length for different girder spacing and concrete compressive strength/density is plotted separately for each section in Appendix D. The results for a representative case (AASHTO-PCI BT-72) are shown in FigureÂ 2.1. The percentage increase in the achievable span length resulting from the use of 0.7-in. strands is also indicated in the figure. In the cases shown in FigureÂ 2.1, a larger increase in span length is associated with increased concrete strength and increased girder spacing. Using the calculated maximum span lengths, some observations are made as follows. 18.104.22.168 Change in Span Length FigureÂ 2.2 compares the maximum span lengths achieved using 0.6-in. strands (L0.6â³) against the values when 0.7-in. strands (L0.7â³) are used. When 0.7-in. strands are used, the span lengths FigureÂ 2.1. Span length charts for AASHTO-PCI BT-72.
30 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders increase within a band of 1.0 to 1.22 times their corresponding cases having 0.6-in. strands. Therefore, by using a 0.7-in. strand, it was possible to increase the span length by a maximum of 22% in comparison to using a 0.6-in. strand. The largest value of 1.22 is less than the ratio of the area of a 0.7-in. strand to that of a 0.6-in. strand (1.35), which indicates a one-to-one replace- ment of strands is not possible due to the many other design constraints such as stress limits at prestress release. The average, maximum, and minimum values of L0.7â³/L0.6â³ are summarized in TableÂ 2.3. Strands in Texas U girders are not permitted to be harped. Hence, debonding strands up to the current limits was the only method for keeping release stresses below the AASHTO limits. The number of 0.7-in. strands was, accordingly, limited, and the achievable span length increases were the lowest for Texas U girders. The maximum achievable span did not change for the following cases when using 0.7-in. strands (i.e., L0.7â³/L0.6â³ = 1.0): (1) NU-900, S = 12Â ft, 10 ksi LWC; (2) NU-900, S = 8Â ft, 15 ksi NWC; and (3) NU-1100, S = 8Â ft, 18 ksi NWC. The greatest increase, L0.7â³/L0.6â³ = 1.22, occurred for WF100G, S = 10Â ft, 15 ksi. FigureÂ 2.2. Comparison of span lengths (0.6-in. strands versus 0.7-in. strands). Girder Avg. Max. Min. Girder Avg. Max. Min. AASHTO VI 1.06 1.13 1.01 WF74G 1.14 1.21 1.06 FIB-96 1.10 1.13 1.06 WF100G 1.18 1.22 1.13 BT-54 1.10 1.15 1.03 All Washington WF 1.16 1.22 1.06 BT-63 1.13 1.17 1.09 BIV-48 1.08 1.09 1.06 BT-72 1.15 1.19 1.12 NEXT40D 1.15 1.16 1.15 All AASHTO-PCI BT 1.13 1.19 1.03 U-40 1.02 1.04 1.01 NU-900 1.02 1.05 1.00 U-54 1.05 1.11 1.02 NU-1100 1.03 1.05 1.00 All Texas U 1.03 1.11 1.01 NU-1600 1.06 1.12 1.01 U54G5 1.11 1.18 1.02 NU-2000 1.09 1.12 1.05 UF60G5 1.18 1.19 1.18 All Nebraska NU 1.05 1.12 1.00 UF72G5 1.18 1.19 1.17 WF-36 1.02 1.02 1.01 All Washington U 1.16 1.19 1.02 WF-54 1.07 1.08 1.06 WF-72 1.12 1.13 1.11 All Ohio WF 1.07 1.13 1.01 TableÂ 2.3. Ratio of span with 0.7-in. strands to span with 0.6-in. strands.
Analytical Research Approaches and Findings 31Â Â 22.214.171.124 Influence of Type of Girder The influence of the type of girder on the increase in span length that may be achieved using 0.7-in. strands was examined by comparing the span length increases for girders of equal or nearly equal depths. Separate comparisons are made for single-web girders and double-web girders in FigureÂ 2.3 and FigureÂ 2.4, respectively. The âefficiencyâ of the selected single-web girders in terms of increasing the span length depends on girder spacing and concrete compressive strength. For instance, WF-72 and BT-72 girders offer the greatest benefit for 10 ksi LWC or NWC regardless of the girder spacing. The largest increases are achieved with WF74G and BT-72 girders using 15-ksi or 18-ksi concrete. The use of 0.7-in. strands in BT-72 results in considerable increases of achievable span length for all of the concrete strengths considered herein. As discussed above, the lack of harped strands in Texas U girders limits the number of 0.7-in. strands, and the achievable increases in the span length for U-40 are appreciably lower than equal-depth U54G5 girders (see FigureÂ 2.4). FigureÂ 2.5 indicates that NEXT 40D girders benefit more from the use of 0.7-in. strands than do BIV-48 girders. FigureÂ 2.3. Increase in span length of single-web girders with similar or equal depths.
32 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders FigureÂ 2.4. Increase in span length of double-web girders with similar or equal depths. FigureÂ 2.5. Increase in span length of adjacent girders with similar or equal depths.
Analytical Research Approaches and Findings 33Â Â As described by Collins and Mitchell (1997), one measure of the efficiency of a prestressed concrete section is the lever arm of the internal couple. This value is represented as e + kt (see FigureÂ 2.6), in which: e = distance between the centroid of cross section and centroid of prestressing steel kt = distance between the centroid of cross section and top kern point. The top kern point is defined as the uppermost location in the cross section at which the compression resultant may be placed such that the condition of zero tension is maintained at the bottom face of the girder. The greater the distance e + kt, the more efficient the section and the less prestress force is required to carry a given load over a given span. The value of e + kt was calculated for all the sections considered herein and was normalized for girder depth (h). The average increase of span length for all the concrete strengths and girder spacings was computed for each girder shape. The relationship between the span increase and (e + kt)/h for all the sections is plotted in FigureÂ 2.7, and the girders with the lowest and highest percentage span increases are indicated. Less âefficientâ girders [i.e., those with a small (e + kt)/h, such as UF72G5 and BT-72] tend to benefit more from using 0.7-in. strands. yt yb h e kt kb FigureÂ 2.6. Top (kt) and bottom (kb) kern points. FigureÂ 2.7. Span length versus normalized girder efficiency.
34 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders In addition to the girder shape, which affects (e + kt)/h, the achievable increase in span length using 0.7-in. strands varies as a function of the girder depth. As FigureÂ 2.8 shows, the deeper the girder, the greater the proportional span length increase. 126.96.36.199 Splitting Reinforcement at Prestress Transfer AASHTO LRFD Bridge Design Specifications Article 188.8.131.52.1 (AASHTO, 2020) requires splitting reinforcement to resist ânot less than four percent of the total prestressing force at transfer.â Bars contributing to splitting resistance are defined as those placed within a distance equal to one-quarter of the height of the girder (at prestress transfer; therefore, not including the subsequent slab), h/4. Furthermore, since this reinforcement is intended to control cracking, the stress in the reinforcement is limited to 20 ksi. The splitting force, Fsp, that must be resisted at prestress transfer is calculated as follows in this study: ( )=F f A Nsp pu ps Eq. 2.10.04 0.75 0.75fpu = assumed initial prestress force, where fpu = 270 ksi Aps = area of a single strand N = number of bonded straight and harped strands at end of the girder Based on a practical limit for splitting reinforcement of pairs of No. 5 bars spaced at 2Â in., to achieve the longest spans, the length of distribution of splitting reinforcement had to be extended to h/3 in nearly 30% (86 out of 292) and 43% (126 out of 292) of the cases using 0.6-in. and 0.7-in. strands, respectively. Extension of splitting reinforcement beyond h/4 was considered acceptable in this study and is consistent with Washington state practice (Khaleghi, 2006). The results for the larger-diameter strands do not suggest a significant deviation from those for 0.6-in. strands. This issue is discussed further in Section 184.108.40.206. 220.127.116.11.1 Delaying Prestress Transfer To Mitigate Transfer Stresses. As part of the design procedure (Appendix A), a check was made to examine whether specifying a higher concrete strength at prestress transfer ( f â²ci = Î· f â²c in which Î· â¤ 1) is sufficient to mitigate not meeting stress limits at transfer. This approach worked only for 4% (21 of 584) of cases. That is, the value of Î· required would exceed 1 for the remaining 563 cases. As a result, the span length had FigureÂ 2.8. Influence of girder depth on span length increase.
Analytical Research Approaches and Findings 35Â Â to be reduced to satisfy the tensile and compressive stress limits at prestress transfer and the SERVICE I and SERVICE II limits near the end of the girder. It is noted that regardless of the outcome of this âside check,â all designs progressed using the values of f â²ci = 0.85f â²c or 0.6f â²c. All 21 cases that could have achieved a marginally longer span if a higher release strength were permitted involved 0.6-in. strands used in AASHTO-PCI BT, AASHTO BIV, Nebraska NU, and Ohio WF (refer to Appendix C). Delaying prestress transfer until the concrete is more mature (increasing the required value of fciâ) is not a practical means of addressing stress limits at prestress transfer when 0.7-in. strands are used to achieve very long spans. 18.104.22.168 Comparison with Salazar etÂ al. (2017) Salazar etÂ al. (2017) published a similar parametric study on the use of 0.7-in. strands. This study examined AASHTO Type IV, V, and VI girders; Texas bulb Ts (TX 28, 34, 40, 46, 54, 63, and 70); Texas spread box girders (4xB20, 28, 34, and 40; 5xB20, 28, 34, and 40); and Texas U-40 and U-54 girders. The study methodology was similar to that used herein. Salazar etÂ al. concluded that I girders and bulb T girder spans could be extended up to 10Â ft through the use of 0.7-in. strands. However, this increase generally required release strengths of 10 ksi or greater and the use of harping or other methods to control end-region stresses. The authors concluded that the use of 0.7-in. strands in the U girders and the box girders did not result in greater spans than could be achieved with 0.6-in. or 0.5-in. strands. For I-shaped and bulb T-shaped girders, Salazar etÂ al. concluded that, for a given span length, the required depth of some of the girder shapes could be decreased if 0.7-in. strands were used. Again, this observation was dependent on release strength, with higher release strengths being needed to increase the reduced slenderness. While many I- or bulb T-shaped girders reported in this study achieved greater slenderness (span/depth) when 0.7-in. strands were used, not all did. Shallower cross sections seemed to benefit more from the use of larger strand diameters. Salazar etÂ al. also considered whether the use of 0.7-in. strands would allow larger girder spacing for I and bulb T shapes. The authors concluded that there was little advantage to using a 0.7-in. strand and that any advantage appeared at a girder spacing so large as to be impractical. In general, the conclusions of Salazar etÂ al. are consistent with the findings of the present study, with one exception. Salazar etÂ al. found no advantage to using 0.7-in. strands in box girders whereas the present study does indicate a small advantage. This difference is easily explained. Salazar etÂ al. only considered Texas box shapes used in a spread configuration, whereas the present study considered AASHTO boxes in an adjacent configuration. The differ- ences in the girder shapes and configuration explain the discrepancy in the conclusions regard- ing the benefits of 0.7-in. strands in box girders. 2.3 Design Verification/Validation Study: Finite Element Modeling of Full-Length Girders To validate the approach used in the design parametric study reported in Section 2.2, selected girders have been modeled using ATENA. All full-girder models were developed using the ATENA finite element modeling (FEM) package. The analyses presented are based on the extensive modeling and validation studies presented in NCHRP Research Report 849 (Shahrooz etÂ al., 2017). Appendix D of NCHRP Research Report 849 provides extensive details of model parameters and validation.
36 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders TableÂ 2.4 summarizes the material properties used in the models. Initial prestress force is taken as fpi = 202.5 ksi and transfer length is consistent with AASHTO LRFD Bridge Design Specifications (AASHTO, 2020), Lt = 60db: 36Â in. for 0.6-in. strand and 42Â in. for 0.7-in. strand. The modeling steps are given in TableÂ 2.5. Only the critical flexural load case is considered in Steps 4, 5, and 6. Î© (overstrength factor) is found by increasing the axle loads, LLtruck (only), until failure of the girder. Applied loads on a single girder are determined as follows: DC = dead weight of girder + 8Â in. NWC slab (100 psf) + 300 plf allowance for barrier walls DW = wearing surface allowance = 35 psf LL = LLlane + (LLtruck + IM) for HL93 loading; (Ltruck + IM) = 1.33Ltruck LLlane = g Ã 640 plf LLtruck = g Ã [HL93 axle loads] g = distribution factor for moment on interior girder Two I shapes were considered. Comparative designs with both 0.6-in. and 0.7-in. strands were modeled. BT-72 girders were shown in the design parametric study to benefit most (in terms of potential span length increase) by replacing 0.6-in. strands with 0.7-in. strands. NU-2000 girders, on the other hand, did not show such a great increase in span length. ATENA material type Modelingstepsa Modulus Strength Density Poisson ratioksi ksi kcf Prestressing strand CCReinforcement All = 29,000 = 243 = 270 na na Nonprestressed reinforcement CCReinforcement All = 29,000 = 60 na na Girder concrete CC3DNonlinCementitious2 2 = 5,950 = 9 0.155 0.2 Girder concrete CC3DNonlinCementitious2 3â6 = 7,110 = 15 0.155 0.2 Slab concrete CC3DNonlinCementitious 3 0 0 0.145 na Slab concrete CC3DNonlinCementitious 4â6 4,291 5 0.145 0.2 Steel plate CC3DElastIsotropic All 29,000 na na 0.3 Neoprene bearing CC3DElastIsotropic All 9 na na 0.3 aSee Table 2.5. na = Not applicable. Material Step Description Applied loads Girder strength Slab strength External prestress Internal prestress ksi ksi ksi ksi 1 âCastâ concrete None na na 0.75 = 202.5 na 2 Release tendons Girder self-weight only = 0.6 = 9 na 0 â 0. 9(0.75 ) = 182 3 Place deck slab Girder and slab self-weight = 15 na 0 0.56fpu = 151 4 SERVICE ISERVICE III DC + DW + (LL + IM) DC + DW + 0.8(LL + IM) = 15 5 0 0.56fpu = 151 5 STRENGTH 1.25DC + 1.50DW +1.75(LL + IM) = 15 5 0 0.56fpu = 151 6 Failure 1.25DC + 1.50DW + 1.75(LLlane) + Î©(Ltruck + IM) = 15 5 0 0.56fpu = 151 aLosses upon transfer are determined within the finite element model based on the provided bond-slip model; in general, these are approximately 10% of the initial prestress force. na = Not applicable. TableÂ 2.4. Material properties used in full-girder ATENA models. TableÂ 2.5. FEM steps.
Analytical Research Approaches and Findings 37Â Â For each of the four cases (each girder shape using 0.6-in. or 0.7-in. strands), a girder spacing S = 8Â ft was assumed; deck tributary area and live load distribution factors were calculated on this basis. Each girder is âcastâ 18Â in. longer than its span, L, and is supported on 18-in.-long, full-width bearings. The distribution factor for flexure of interior girders, gM,int, was deter- mined from AASHTO LRFD Bridge Design Specifications (AASHTO, 2020) TableÂ 22.214.171.124.2b-1 for cross section type (k) (TableÂ 126.96.36.199.1-1) and includes a calculated value of the Kg/12Lts3 term (rather than taking this value equal to 1). TableÂ 2.6 summarizes the details of the BT-72 and NU-2000 models. A summary of analysis results is shown in TableÂ 2.7 to TableÂ 2.10 for the four girder models. The FEM results accurately mirror the design requirements, and each girder mostly meets the concrete stress requirements of AASHTO LRFD Bridge Design Specifications (AASHTO, 2020). At prestress transfer, the concrete tension stresses for the BT-72 girders fall between 0.0948 â²f ci and 0.24 â²f ci , indicating a need for nonprestressed reinforcement satisfying the requirements of TableÂ 188.8.131.52.1b-1 in AASHTO, 2020 in the region of tensile stress. The tensile stresses in the NU-2000 girders at prestress transfer fell below 0.0948 â²f ci . In all but BT72-6, the maximum compression stresses at prestress transfer exceeded the AASHTO limit BT-72-6 BT-72-7 NU-2000-6 NU-2000-7 Girder BT-72 BT-72 NU-2000 NU-2000 Girder length 132.5 ft 156.5 ft 181.5 ft 202.5 ft Girder span, L 131 ft 155 ft 180 ft 201 ft DF, gM.int 0.66 0.63 0.63 0.61 Strand dia. 0.6 in. 0.7 in. 0.6 in. 0.7 in. Straight 32 32 56 46 Debonded 0 6 0 0 Harped 2 2 4 14 Harped strand pairs, yi (symmetric about midspan) 70 in. @ 0 ft 8 in. @ 51.25 ft 70 in. @ 0 ft 8 in. @ 63.25 ft 76.7 in. @ 0 ft 14 in. @ 75.75 ft 74.7 in. @ 0 ft 12 in. @ 75.75 ft 76.7 in. @ 0 ft 14 in. @ 86.25 ft 74.7 in. @ 0 ft 12 in. @ 86.25 ft 72.7 in. @ 0 ft 10 in. @ 86.25 ft 70.7 in. @ 0 ft 8 in. @ 82.25 ft 68.7 in. @ 0 ft 6 in. @ 82.25 ft 66.7 in. @ 0 ft 4 in. @ 82.25 ft 64.7 in. @ 0 ft 2 in. @ 82.25 ft Straight strand arrangement in bulb Strands at midspan (harped strands shown in box) Top straight strands 4 strands stressed to 15 kips each located at y = 70 in. 4 strands stressed to 15 kips each located at y = 76.7 in. 0.04Apsfpi 60 kips 67 kips 105 kips 143 kips Web reinforcement from end 1.5 in. cover 5 pairs #5 @ 4 in. pairs #4 @ 14 in. remainder of span 1.5 in. cover 6 pairs #5 @ 3 in. pairs #4 @ 18 in. remainder of span 1.5 in. cover 10 pairs #5 @ 2 in. pairs #4 @ 14 in. remainder of span 1.5 in. cover 13 pairs #5 @ 2 in. pairs #4 @ 14 in. remainder of span Bulb tie force 115 kips 88 kips 195 kips 351 kips Bulb reinforcement from end 1.5 in. cover 9-#3 hoops @ 4.5 in. #3 hoops @ 6 in. remainder of span 1.5 in. cover #3 hoops @ 6 in. entire span 1.5 in. cover 9-#4 hoops @ 4.75 in. #3 hoops @ 6 in. remainder of span 1.5 in. cover 15-#4 hoops @ 2.5 in. #3 hoops @ 6 in. remainder of span debond 5 ft 10 ft Model Details TableÂ 2.6. Details of BT-72 and NU-2000 finite element models.
38 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders BT-72-6 Ã support half span (66.25 ft) shown midspan Ã Release of tendons Longitudinal stress: , = 0.29 ksi = 0.10 , = 5.14 ksi = 0.57 Crack pattern , â¤ 0.012 in. SERVICE I Longitudinal stress: , = 0.22 ksi = 0.06 , = 4.70 ksi = 0.31 Crack pattern , â¤ 0.012 in. SERVICE III Longitudinal stress: , = 0.10 ksi = 0.03 , = 4.73 ksi = 0.32 Crack pattern , â¤ 0.012 in. STRENGTH I Longitudinal stress: , = 0.68 ksi = 0.17 , = 4.45 ksi = 0.30 Crack pattern , â¤ 0.08 in. Failure at TableÂ 2.7. Summary of FEM results for BT-72-6. of 0.60f â²ci. The high compressive stresses were localized near the girder ends. The discrepancy between the finite element model and section-based design procedure is expected (also identi- fied in NCHRP Research Report 849). The design procedure uses gross section properties to assess stress conditions (i.e., P/A Â± Pe/I). Near the girder end, where the prestress force is introduced primarily in the flange, the entire section area (A) is not engaged for some distance into the beam. The prestress force near the girder end is, in effect, resisted over a smaller area, resulting in higher stress. The âspreadingâ of the compression force over the depth of the beam can be visualized at the left end of the stress plots shown in TableÂ 2.7 to TableÂ 2.10. This effect is more pronounced for girders having large prestress forces and in deeper girders having uniform bottom flanges (as both BT and NU do). The highly loaded bottom flange represents a smaller portion of the gross section area for a deeper girder, also resulting in a greater discrepancy between actual local stresses and those calculated based on the gross area. This effect can be mitigated by additional debonding, thereby introducing the prestress force more gradually along the span. At SERVICE I, tensile stress is not observed to exceed 0.19 â²f c and at SERVICE III, compres- sive stress does not exceed 0.60f â²c in any case. All designs were âcontrolledâ by the STRENGTH I limit state. Consistent with the design goal of maximizing girder length, all girders meet but demonstrate relatively little reserve capacity above the STRENGTH I limit state. As may be
Analytical Research Approaches and Findings 39Â Â BT-72-7 Ã support half span (78.25 ft) shown midspan Ã Release of tendons Longitudinal stress: , = 0.43 ksi = 0.14 , = 7.90 ksi = 0.88 Crack pattern , â¤ 0.023 in. SERVICE I Longitudinal stress: , = 0.29 ksi = 0.07 , = 7.84 ksi = 0.52 Crack pattern , â¤ 0.020 in. SERVICE III Longitudinal stress: , = 0.09 ksi = 0.02 , = 7.84 ksi = 0.52 Crack pattern , â¤ 0.020 in. STRENGTH I Longitudinal stress: , = 0.84 ksi = 0.22 , = 7.81 ksi = 0.52 Crack pattern , â¤ 0.080 in. Failure at TableÂ 2.8. Summary of FEM results for BT-72-7. expected for very long girders, relatively significant decompression and cracking were observed in the midspan region, particularly for the BT-72 girders. In these comparisons, the number of strands for each girder type was the same, and, thus, the total prestress force was approximately 35% greater for the models having 0.7-in. strands. The spans, on the other hand, only increased 18% and 12% for the BT-72 and NU-2000 girders, respectively. This combination is manifest as greater precompression near midspan resulting in a higher decompression load and less cracking. Because the span increase for the NU girder is less, the counteracting effects of the applied moment are less significant, and the improved cracking behavior of NU-2000-7 over NU-2000-6 at STRENGTH I is evident in FigureÂ 2.9c. âFailureâ of the finite element models occurs because of an inability of the models to [mathematically] converge at loads greater than the Î© values given in the final rows of TableÂ 2.7 to TableÂ 2.10. This lack of convergence is associated with the relatively conservative AASHTO- compliant bond-slip model used; the bond-slip model was calibrated for a transfer length, Lt = 60db. As expected, the finite element model is predicting extensive cracking near the girder midspan (bottom images in TableÂ 2.7 to TableÂ 2.10). Although the stresses in the strands are approaching rupture, in none of the models is the strand predicted to rupture at the failure load attained; rather, the degree of cracking is leading to a bond-slip failure. It can also be seen that the 5-ksi slab experienced considerable damage in the BT-72 models. A stronger and, therefore, stiffer slab would result in a relatively minor improvement in behavior.
40 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders NU-2000-6 ï support half span (90.75 ft) shown midspan ï Release of tendons Longitudinal stress: , = 0.15 ksi = 0.05 , = 6.25 ksi = 0.69 Crack pattern , â¤ 0.012 in. SERVICE I Longitudinal stress: , = 0.25 ksi = 0.06 , = 5.89 ksi = 0.39 Crack pattern , â¤ 0.008 in. SERVICE III Longitudinal stress: , = 0.07 ksi = 0.02 , = 5.91 ksi = 0.39 Crack pattern , â¤ 0.008 in. STRENGTH I Longitudinal stress: , = 0.87 ksi = 0.22 , = 6.94 ksi = 0.46 Crack pattern , â¤ 0.080 in. Failure at TableÂ 2.9. Summary of FEM results for NU-2000-6. Based on the finite element model behavior reflecting the design objectives, the design approach for the parametric study has been validated. 2.3.1 Effects at Girder Ends Finally, the finite element models provide some insight into the control of web splitting cracks (Article 184.108.40.206.1 in AASHTO, 2020) near the girder ends. Such cracking is expected to occur due to the large prestress forces, and the models include the vertical web reinforcement required to resist this arranged over the initial h/4 length of the girder (TableÂ 2.6). The associated cracking is evident in FigureÂ 2.9a. As seen in FigureÂ 2.9b, such cracking is expected to become more significant, although not propagated along the girder, at STRENGTH I. What is evident in FigureÂ 2.9 is that the cracking extends beyond the h/4 distance over which the concentrated reinforcement is provided. This result supports providing the required splitting reinforcement over a longer length as is permitted in Washington state (Khaleghi, 2006) or as proposed by Tuan etÂ al. (2004). Washington State Department of Transportation (WSDOT, 2019) limits the splitting reinforcement to pairs of No. 5 bars at 2.25Â in. spacing but permits this detail to extend beyond h/4 to accommodate all required bars. Tuan etÂ al. proposed that one-half of the required splitting reinforcement be located in h/8, with the remainder being extended to h/2.
Analytical Research Approaches and Findings 41Â Â NU-2000-7 ï support half span (101.25 ft) shown midspan ï Release of tendons Longitudinal stress: , = 0.05 ksi = 0.02 , = 6.80 ksi = 0.76 Crack pattern , â¤ 0.016 in. SERVICE I Longitudinal stress: , = 0.10 ksi = 0.03 , = 6.20 ksi = 0.41 Crack pattern , â¤ 0.016 in. SERVICE III Longitudinal stress: , = 0.09 ksi = 0.02 , = 6.24 ksi = 0.42 Crack pattern , â¤ 0.016 in. STRENGTH I Longitudinal stress: , = 0.93 ksi = 0.24 , = 7.20 ksi = 0.48 Crack pattern , â¤ 0.020 in. Failure at TableÂ 2.10. Summary of FEM results for NU-2000-7. 2.4 Analytical Modeling of End Regions 2.4.1 Strut-and-Tie Modeling of Transverse Tie Reinforcement Requirement In NCHRP Research Report 849, Shahrooz etÂ al. (2017) identified the development of tension oriented transversely across the bulb of single-web sections as a potential failure mode requiring tie reinforcement through the bulb to control associated longitudinal cracking at the STRENGTH I limit state. By further development of a proposal by Ross etÂ al. (2013), a strut-and-tie modeling (STM) approach may be used as an alternative to prescriptive detail- ing requirements used to design bulb confinement reinforcement intended to mitigate lateral splitting failures at the ultimate limit state (Harries etÂ al., 2019). Using typical assumptions of the STM approach (Shahrooz etÂ al., 2017), the three-dimensional STM at the end of a girder (FigureÂ 2.10a) can be modeled as a two-dimensional STM, as shown in FigureÂ 2.10c, which may be used to determine the force to be resisted by tie reinforcement in single-web flanged sections. This force is written as a fraction, Î±, of the girder reaction force (Vu/Ï): [ ]( )( ) ( )( ) ( )Î± Ï = Ï â + âV V n N x h y x c yu u f w p b p p b p Eq. 2.2
42 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders Nw = total number of bonded straight strands at section (harped strands do not affect end- region STM) nf = the number of bonded strands in one side of the outer portion of the bulb (FigureÂ 2.10b). The outer portion of the bulb is defined as that extending beyond the projection of web width. Strands aligned with the edge of the web are assumed to fall in the outer portion of the bulb. xp = horizontal distance to girder centerline of the centroid of nf strands in the outer portion of the bulb yp = vertical distance to girder soffit of the centroid of nf strands in the outer portion of the bulb hb = height of the bulb, taken as dimension D5 + D6 using PCI (2014) notation cb is calculated to ensure uniform bearing pressure across the width of the bearing pad, bb, i.e., ( )( )= âc b n Nb b f w Eq. 2.32 1 The reinforcement determined by Eq. 2.2 is to be uniformly distributed over the length of the bearing plus a distance equal to one-quarter of the overall height of the girder (including the BT-72-6 Ã girder end 5ft Ã Ã girder end 5ft Ã Ã 15 ft girder midspanÃ BT-72-7 Ã girder end 5ft Ã Ã girder end 5ft Ã Ã 15 ft girder midspanÃ NU-2000 -6 Ã girder end 5ft Ã Ã girder end 5ft Ã Ã 15 ft girder midspanÃ NU-2000 -7 Ã girder end 5ft Ã Ã girder end 5ft Ã Ã 15 ft girder midspanÃ (a) Girder end at prestress transfer. (b) Girder end at STRENGTH I. (c) Girder midspan region at STRENGTH I. FigureÂ 2.9. Comparison of crack patterns in finite element models.
Analytical Research Approaches and Findings 43Â Â composite slab if provided) toward the midspan of the girder; i.e., a length of H/4 + Lbearing = (h + tslab)/4 + Lbearing where h = depth of the girder, tslab = thickness of slab, and Lbearing = length of the bearing. Regardless of the tie reinforcement requirement, AASHTO LRFD Bridge Design Specifica- tions Article 220.127.116.11.2 (AASHTO, 2020) requires minimum confinement of the flange consisting of at least No. 3 hoops spaced at 6Â in. over a length 1.5d from the end of the beam. For heavily loaded or long-span members (as considered here), this requirement will typically not control the detailing requirements at the ends of the flange. 2.4.2 Flange Transverse Reinforcement Requirements for Case Studies STM was used to assess transverse reinforcement requirements for all single-web flanged section case studies reported in Section 2.2. Requirements of AASHTO LRFD Bridge Design Specifications Article 18.104.22.168.1 (AASHTO, 2020) were also assessed (Section 22.214.171.124). Strand patterns, debonding patterns, and harped strand locations are identical to those used in the case studies. The following additional assumptions were made in the STM analyses: 1. In all cases, the bearing length, Lbearing = 18Â in. and the width, bb, was taken as 4Â in. less than the width of the bottom flange. (a) Strut-and-tie mechanism (b) Strands and dimensions of bulb section. (c) Strut-and-tie model (end elevation). (d) Strut-and-tie model (plan). hb bb V/Ï(1-2n /N)f V/Ï(n /N)f V/Ï(n /N)f yp cb cb xp xp V /Ïu z stru t A strut B Î¸ tie xp V /Ïu z strut A strut B bearing Lbearing FigureÂ 2.10. Strut-and-tie model for confinement showing BT girder bulb.
44 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders 2. In all analyses, Ï = 1.0, corresponding to âtension in steel in anchorage zonesâ [AASHTO LRFD Bridge Design Specifications Article 126.96.36.199 (AASHTO, 2020)]. TableÂ 2.11 summarizes the results from the STM in terms of the following: 1. Tie force parameter, Î±. This value is used to permit normalization for girder depth. 2. Tie force per unit length of region resisting the tie force (kip/in.), that is, Î±(Vu/Ï)/(H/4 + Lbearing). This value is the force that must be resisted by the ties. 3. Splitting force, Fsp = 4% of the prestress force at transfer given by Eq. 2.1. 4. Splitting force per unit length of region resisting the splitting force (kip/in.), that is, Fsp/(h/4). This value is the force that must be resisted by the splitting reinforcement. TableÂ 2.12 provides a summary of the available tie force provided by a range of practical reinforcing details. The tie force data shown in TableÂ 2.12 assume fy = 60 ksi. AASHTO LRFD Bridge Design Specifications (AASHTO, 2020) permits the use of higher yield strengths up to 100 ksi or welded wire reinforcement (WWR) up to D31 having fy = 80 ksi. The capacities shown in TableÂ 2.12 may be factored to account for higher yield force: multiplying reported values by 1.33 or 1.66 for fy = 80 ksi or fy = 100 ksi, respectively. Splitting resistance for the same range of reinforcing details is also provided in TableÂ 2.12. The splitting requirement does not benefit from the use of higher grades of reinforcing since fs is limited to 20 ksi. Example of Use of Table 2.12 The average reinforcing requirements for a BT-72 having 0.7-in. strands are 3 kip/in. for tie reinforcement and 4 kip/in. for splitting reinforcement (Table 2.11c). Using Table 2.12, the following details will provide these capacities: Required = 60 ksi = 100 ksi WWR= 80 ksi NCHRP Research Report 849 tie reinforcement 3 kip/in. #3 bar @ 2 in. #3 hoop @ 4 in. #4 bar @ 4 in. #4 hoop @ 6 in. #3 bar @ 3.5 in. #3 hoop @ 6 in.1 D11 Ã 2.5 Two D11 Ã 5 D31 Ã 6 AASHTO LRFD Bridge Design Specifications Article 188.8.131.52.1 splitting reinforcement 4 kip/in. Two #3 hoops @ 2 in. #4 hoop @ 2 in. Two #4 hoops @ 4 in. #5 hoop @ 3 in. Two #5 hoops @ 5.5 in. Two #3 hoops @ 3.5 in. #4 hoop @ 3 in. Two #4 hoops @ 6 in. #5 bar @ 2.5 in. #5 hoop @ 5 in. D31 Ã 2 Two D31 Ã 4 Note: Minimum reinforcement required by AASHTO LRFD Bridge Design Specifications Article 184.108.40.206.2 over length Reinforcement Type (a) AASHTO Type VI, Florida I, and Washington WF girders Girder AASHTO Type VI FIB-96 WF-74G WF-100G Strand dia. (in.) 0.7 0.6 0.7 0.6 0.7 0.6 0.7 ââ 1.33 2.34 2.74 Number of cases 16 16 16 16 16 16 16 NCHRP Research Report 849 transverse tie reinforcement Tie force coefficient., Î± average 0.12 0.25 0.16 0.42 0.39 0.42 0.42 std. dev. 0.06 0.005 0.06 0 0.04 0 0 min. 0.05 0.24 0.09 0.42 0.32 0.42 0.42 max. 0.25 0.25 0.25 0.42 0.42 0.42 0.42 Required tie force over + /4 (kip/in.)a average 1.6 3.2 2.3 5.0 5.0 4.8 5.5 min. 0.6 2.8 1.2 4.2 3.6 3.6 4.6 max. 3.3 3.7 3.7 5.6 6.3 6.3 6.2 AASHTO LRFD Bridge Design Specifications Article 220.127.116.11.1 splitting reinforcement 4% Prestress force at transfer (kips) average 125 125 140 97 121 98 131 min. 114 125 121 93 93 95 126 max. 138 125 169 100 136 100 136 Required splitting resistance over â/4 (kip/in.)a average 7.0 5.2 5.8 5.2 6.5 3.9 5.3 min. 6.4 5.2 5.1 5.0 5.0 3.8 5.0 max. 0.6 16 0.20 0.02 0.17 0.23 2.5 1.9 3.3 129 127 130 7.2 7.0 7.2 7.7 5.2 7.0 5.4 7.3 4.0 5.4 TableÂ 2.11. Summary of bulb transverse tie and splitting reinforcement requirements.
Analytical Research Approaches and Findings 45Â Â (c) AASHTO-PCI BT girders Girder BT-54 BT-63 BT-72 Strand dia. (in.) 0.6 0.7 0.6 0.7 0.6 0.7 ââ 2.1 Number of cases 16 16 16 16 16 16 NCHRP Research Report 849 transverse tie reinforcement Tie force coefficient., Î± average 0.23 0.31 0.29 0.29 0.28 0.27 std. dev. 0.06 0.10 0.09 0.06 0.07 0.06 min. 0.16 0.20 0.16 0.21 0.16 0.22 max. 0.28 0.49 0.37 0.38 0.32 0.37 Required tie force over + /4 (kip/in.)a average 2.3 3.3 2.8 3.2 2.7 3.0 min. 1.3 1.7 1.4 2.0 1.4 2.0 max. 3.2 5.0 4.0 4.7 3.6 4.6 AASHTO LRFD Bridge Design Specifications Article 18.104.22.168.1 splitting reinforcement 4% Prestress force at transfer (kips) average 54 71 55 73 55 72 min. 46 62 46 62 46 67 max. 60 81 60 81 60 81 Required splitting resistance over â/4 (kip/in.)a average 4.0 5.3 3.5 4.6 3.0 4.0 min. 3.4 4.6 2.9 3.9 2.5 3.7 max. 4.4 6.0 3.8 5.1 3.3 4.5 (d) Ohio WF girders Girder WF-36 WF-56 WF-72 Strand dia. (in.) 0.6 0.7 0.6 0.7 0.6 0.7 ââ 2.88 Number of cases 16 16 16 16 16 16 NCHRP Research Report 849 transverse tie reinforcement Tie force coefficient., Î± average 0.40 0.28 0.40 0.41 0.40 std. dev. 0.02 0.03 0.02 0.03 0.02 min. 0.38 0.25 0.36 0.35 0.38 max. 0.41 0.34 0.45 0.44 0.44 Required tie force over + /4 (kip/in.)a average 4.6 3.2 5.0 5.3 5.5 min. 3.8 2.5 3.9 4.3 4.3 max. 5.4 4.2 5.9 6.6 6.8 AASHTO LRFD Bridge Design Specifications Article 22.214.171.124.1 splitting reinforcement 4% Prestress force at transfer (kips) average 73 73 84 98 110 min. 72 72 79 93 107 max. 74 74 90 102 117 Required splitting resistance over â/4 (kip/in.)a average 8.1 8.1 6.2 7.2 6.1 min. 8.0 7.9 5.9 6.9 6.0 max. 8.2 8.2 6.6 7.6 0.54 0.09 0.36 0.61 6.8 4.4 8.7 98 83 104 5.5 4.5 5.8 6.5 aSee Table 2.12 for steel details satisfying this requirement. (b) Nebraska NU girders Girder NU-900 NU-1100 NU-1600 NU-2000 Strand dia. (in.) 0.6 0.7 0.6 0.7 0.6 0.7 0.6 0.7 ââ 3.15 Number of cases 16 16 16 16 16 16 16 16 NCHRP Research Report 849 transverse tie reinforcement Tie force coefficient., average 0.37 0.37 0.26 0.44 0.53 0.36 0.47 0.52 std. dev. 0.07 0.13 0.12 0.13 0.10 0.17 0.05 0.13 min. 0.28 0.15 0.16 0.26 0.42 0.16 0.42 0.31 max. 0.47 0.60 0.46 0.73 0.63 0.65 0.51 0.76 Required tie force over + /4 (kip/in.)a average 4.2 4.1 3.0 5.1 6.4 4.5 5.6 6.8 min. 2.6 1.8 1.6 2.8 4.4 1.8 4.4 3.4 max. 5.6 6.9 5.7 9.2 8.5 8.9 6.9 9.8 AASHTO LRFD Bridge Design Specifications Article 126.96.36.199.1 splitting reinforcement 4% Prestress force at transfer (kips) average 70 71 79 82 105 108 105 129 min. 63 71 70 71 105 95 105 114 max. 74 71 88 86 105 119 105 143 Required splitting resistance over (kip/in.)a average 7.9 8.1 7.3 7.6 6.7 6.9 5.4 6.5 min. 7.2 8.1 6.5 6.6 6.7 6.0 5.4 5.8 max. 8.3 8.1 8.1 7.9 6.7 7.6 5.4 7.3 aSee Table 2.12 for steel details satisfying this requirement. TableÂ 2.11. (Continued).
46 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders 188.8.131.52 Transverse Tie Forces and Flange Reinforcement Near Girder Ends All 448 single-web design cases exhibit transverse tie forces that may be resisted without unrealistic reinforcement requirements. The greatest tie force requirement observed, 9.8 kips/in., was for an NU-2000 girder having 0.7-in. strands and a span of 185Â ft ( f â²c = 15 ksi, S = 10Â ft). This force can be provided by No. 4 hoops at 2.25Â in., No. 5 hoops at 3.75Â in., or pairs of No. 4 hoops at 4.5Â in. Despite the greater total prestress force present when 0.7-in. strands are used, the tie force is only marginally affected, decreasing with the use of 0.7-in. strand in all cases except AASHTO- PCI BT girders (FigureÂ 2.11). The decreased tie forces result primarily from fewer strands being required and the increased debonding required when 0.7-in. strands are used. Both effects allow more favorable strand patterns at the girder ends, thereby reducing tie forces. The increased tie forces in the AASHTO-PCI BT girders reflect the proportionally greater amount of prestress force that may be used in these sections (recall that BT sections generally exhibited greater potential increases in span length when 0.7-in. strands are used). Results for all six single-web shapes considered are also consistent with the findings of NCHRP Research Report 849 in which wide, flat-bottom flanges typically result in larger tie forces. As shown in FigureÂ 2.11, the anticipated flange tie force coefficient, Î±, is proportional to the ratio of bearing width to flange height, bb/hb. As a rule-of-thumb, the relationship Î± = 0.13bb/hb provides a reasonable estimate of tie force when the girder is optimized to achieve the greatest span. Introducing greater prestress force to a section will typically result in greater end-region crack control requirements, i.e., debonding or harping. Increased debonding will typically result Bar detail at spacing, s Bars #3 bar #3 hoop Two bundled #3 hoops #4 bar #4 hoop Two bundled #4 hoops #5 bar #5 hoop Two bundled #5 hoops Equivalent WWR D11 Ã s Two D11 Ã s na na na na D31 Ã s Two D11 Ã s na legs 1 2 4 1 4 1 2 4 (in.2) 0.11 0.22 0.44 0.20 0.40 0.80 0.31 0.62 1.04 a na na 0.75 na na 1.01 na na 1.26 (in.) Tie capacity, = 60 ksi (NCHRP Research Report 849) 2.0 3.3 6.6 6.0 12.0 npb 18.6 npb 2.5 2.6 5.3 4.8 14.9 npb 3.0 2.2 4.4 4.0 12.4 24.8 3.5 1.9 3.8 3.4 10.6 21.3 4.0 1.7 3.3 3.0 9.3 18.6 4.5 1.5 2.9 2.7 5.3 8.3 16.5 5.0 1.3 2.6 2.4 7.4 14.9 5.5 1.2 2.4 2.2 6.8 13.5 6.0 1.1 2.2c 2.0 9.3 7.4 6.2 5.3 4.7 4.1 3.7 3.4 3.1 6.2 12.4 (in.) Splitting reinforcement capacity, = 20 ksi (AASHTO LRFD Bridge Design Specifications Article 184.108.40.206.1) 2.0 1.1 2.2 2.0 npb 6.2 npb 2.5 0.88 1.8 1.6 5.0 npb 3.0 0.73 1.5 1.3 4.1 8.3 3.5 0.63 1.3 1.1 3.5 7.1 4.0 0.55 1.1 1.0 3.1 6.2 4.5 0.49 1.0 0.89 2.8 5.5 5.0 0.44 0.88 0.80 2.5 5.0 5.5 0.40 0.80 0.73 2.3 4.5 6.0 0.37 0.73 13.2 10.6 8.8 7.5 6.6 5.9 5.3 4.8 4.4 4.4 3.5 2.9 2.5 2.2 2.0 1.8 1.6 1.5 0.67 2 9.6 8.0 6.9 6.0 4.8 4.4 4.0 4.0 3.2 2.7 2.3 2.0 1.8 1.6 1.5 1.3 19.2 16.0 13.7 12.0 10.7 9.6 8.7 8.0 6.4 5.3 4.6 4.0 3.6 3.2 2.9 2.7 3.1 2.5 2.1 1.8 1.6 1.4 1.2 1.1 1.0 2.1 4.1 a = equivalent diameter of bundled bars (AASHTO LRFD Bridge Design Specifications Article 220.127.116.11.5). b np = Not permitted, inadequate clear spacing between reinforcement (AASHTO LRFD Bridge Design Specifications Article 18.104.22.168.2); i.e., clear spacing < ; na = not applicable c Minimum reinforcement required by AASHTO LRFD Bridge Design Specifications Article 22.214.171.124.2 over length 1.5d. TableÂ 2.12. Capacity of tie and splitting reinforcement (kip/in.).
Analytical Research Approaches and Findings 47Â Â in lower tie force demand, provided good practice debonding patterns (aimed at minimizing nf /Nw and xp) are adopted as described above. In cases in which debonding is not sufficient to control cracking, harped strands are necessary. The presence of harped strands increases tie forces since fewer straight strands are located in the web (decreasing Nw â 2nf), and strands displaced from the web result in an increase in nf and xp. From this result, allowing debonding ratios greater than 0.25 (NCHRP Research Report 849) will mitigate the transverse tie forces and resulting reinforcing congestion. No clear trends related to concrete strength or girder spacing were evident in the STM analyses. 126.96.36.199 Splitting Forces and Reinforcement Requirement Splitting forces (and resulting reinforcement) are proportional only to prestress force. Thus, the observed increase in splitting force accompanying an increase in strand diameter and total prestress force shown in TableÂ 2.11 is expected. Only debonding (not harping) can mitigate the splitting force [since only bonded strands are included in the calculation of FSP (Eq. 2.1)]. Once again, allowing debonding ratios greater than 0.25 (NCHRP Research Report 849) will mitigate the splitting tie forces and resulting reinforcing congestion. The greatest splitting force requirement, 8.3 kips/in., was observed for several NU-900 girders having 0.6-in. strands (harping, rather than debonding, was more effective at increasing span length for this section). These cases require pairs of No. 5 hoops at 3Â in. Congestion can also be mitigated by extending the region over which this reinforcement is placed. By simply permitting splitting reinforcement to be extended over h/3, rather than h/4, the splitting reinforcement requirement for the NU-900 girders may be met using single No. 5 hoops at 2Â in., pairs of No. 5 hoops at 4Â in., or pairs of No. 4 hoops at 2Â in. 188.8.131.52 Effects of Adopting 0.7-In. Strand To Maximize Girder Length In terms of required tension tie reinforcement, there is little difference between cases with 0.6-in. and 0.7-in. strands. For the cases considered, the resulting reinforcement require- ment is generally manageable. The use of 0.7-in. strands tends to require greater debonding, which, assuming a favorable strand pattern is adopted, has the effect of reducing the tie force developed. FigureÂ 2.11. Average transverse tie forces as a function of flange geometry.
48 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders In terms of splitting reinforcement [AASHTO LRFD Bridge Design Specifications Article 184.108.40.206.1 (AASHTO, 2020)], the increased prestress force introduced into the girder is reflected in greater splitting forces and more congested reinforcing steel requirements. For all cases considered, transverse tie requirements are relatively easily met. 2.4.3 Lateral Splitting Behavior Lateral splitting failure was described by Ross (2012) as shown in FigureÂ 2.12a. This behavior was observed only in girders in which the bonded prestressing strands were placed as far from the web as possible and no strands in the web region were bonded at the girder end (as shown in FigureÂ 2.12b). Such details are contrary to recommendations 1 and 2, proposed by NCHRP Research Report 849, and the amount of debonding used exceeds that permitted by AASHTO LRFD Bridge Design Specifications Article 220.127.116.11.3 (AASHTO, 2020). The propensity for lateral splitting can be investigated using the same STM described in FigureÂ 2.10. FigureÂ 2.10d shows the plan view of the previously described STM shown in FigureÂ 2.10c. If one assumes a conventional 45-degree shear strut, the dimension z = hb â yp, making the tension force in the tie equal to that calculated by Eq. 2.2. Thus, reinforcement provided pursuant to controlling transverse splitting of the bulb is also sufficient to address lateral splitting. Typically, the shear angle (Î¸ in FigureÂ 2.10a) will be less than 45Â degrees, resulting in a larger value of z and lower calculated tie force as the value of hb â yp in Eq. 2.2 is replaced with z. As described in the previous sections, for all cases considered, transverse tie requirements are relatively easily met. 2.5 Mechanistic Modeling of Effects of Prestress Transfer Several local effects are associated with prestress transfer. Typically, stresses and strains will remain in the elastic, uncracked range. When the stresses exceed those expected to cause crack- ing, the concerns are primarily those of serviceability and long-term durability. Ross (2012) identified several issues associated with prestress transfer and conducted a parametric study using the finite element method to investigate them in the context of Florida Department of Transportation practice of using FIB-shape girders. Ross drew the following conclusions (in italics) to which the research team of the present study have provided commentary: (a) Schematic representation of lateral splitting behavior. (b) Debonding pattern in girder exhibiting splitting behavior. FigureÂ 2.12. Lateral splitting behavior (Ross, 2012).
Analytical Research Approaches and Findings 49Â Â A. For the given release sequence [reported as outside-in], the largest transverse tensile stresses during prestress transfer occur at the [vertical] centerline of a section at the girder end. Centerline tension stresses are greatest when only the strands in the outer portion of the flange have been cut. Cutting of inner strands reduces this transverse tension. Related to this finding, but considering a section away from the girder centerline: B. During prestress transfer the maximum transverse tensile stress on an arbitrary vertical line through the bottom flange occurs when only the strands outboard (closer to edge) of the line have been cut. Cutting of strands along or inboard (closer to centerline) of a line relieves tensile stresses on that line. [For the FIB shapes considered,] transverse stresses at the end of the bottom flange are compressive after all strands have been cut. C. Transverse stress and forces are inversely proportional to strand transfer length. Thus, the greatest transverse effects occur in girders with the shortest transfer lengths. Finding C illustrates the beneficial effect of debonding, which shifts some of the total prestress force transfer away from the girder ends. D. [Girder] self-weight reaction produces transverse tension forces in the bottom flange above the bearing. The effect described in finding D was shown to be relatively minor, and Ross (2012) concluded that it can be neglected for FIB shapes. The self-weight effect will be primarily affected by the bearing width, with smaller actual widths and smaller ratios of bearing width to girder width, both resulting in lower transverse tension forces. Additionally, in Rossâs study, it appears that the self-weight stress was calculated assuming a uniform reaction across the flange width. While this assumption may be a theoretically critical case (indeed, it is adopted by the research team in the STM analysis), a more realistic distribution reduces the flange bending and affects these transverse tension forces. Ross (2012) also notes that embedded steel bearing plates carry approximately 10% of the transverse tension during and after prestress transfer regardless of the stage of strand cutting. In the subsequent discussion, it will be assumed that embedded steel plates are not present. On the basis of these findings, Ross (2012) proposed a simple mechanistic approach to inves- tigate serviceability considerations associated with the âpeelingâ behavior of the bottom flange associated with prestress transfer (FigureÂ 2.13). The transverse tension stress occurs across the (a) Flange behavior due to releasing outer strands. (b) Maximum peeling condition. (c) Combined condition. FigureÂ 2.13. Peeling behavior at prestress transfer (Ross, 2012).
50 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders critical sections shown in FigureÂ 2.13(b) and FigureÂ 2.13(c); the critical sections are a function of the Hoyer effect (see Section 2.5.1) and eccentric prestress forces during prestress transfer associated with the cutting sequence. Ross further identified two critical conditions, shown in FigureÂ 2.13b and FigureÂ 2.13c. Maximum peeling stress along a given section occurs when only the outboard strands are cut (FigureÂ 2.13b). The calculation of peeling stress is discussed in Sections 2.5.2 and 2.5.4. Combined condition occurs when strands along a given section are cut, and Hoyer stresses are superimposed with peeling stress (FigureÂ 2.13c). The calculation of Hoyer and peeling stresses are discussed in Sections 2.5.1 and 2.5.2, respectively. It must be noted that these cases assume a worst-case strand release pattern in which all strands are cut (released) from outside-in. Any other release sequence lowers peeling stresses. Simultaneous (gang) release of all strands would theoretically result in no peeling stress (FigureÂ 2.13a, bottom image). 2.5.1 Application of Hoyer Effect in Prestress Transfer Peeling Model The primary interest in this study is stress at the girder end where Hoyer effects (see Section 1.6) are at their greatest. Thus, the axial stress in the strand and concrete are zero: fpz = fpc = 0. To consider Hoyer-induced transverse stresses, equilibrium of the strand embedded in the concrete may be considered as shown in FigureÂ 1.5c (Ross, 2012). From equilibrium, the maxi- mum average stress due to the Hoyer effect of a single strand at any section through the strand and concrete, fH1, is: ( )= âf pd h dH b f b Eq. 2.41 where p = is obtained from Eq. 1.5 with fpz = fpc = 0 db = diameter of the prestressing strand hf = height of the critical section through the girder flange being considered Accounting for multiple bonded and debonded strands along hf, average stress due to the Hoyer effect, fH, may be written: ( )= âf n pd h n dH bh b f h b Eq. 2.5 where nh = the total number of strands along the vertical section defined by hf (FigureÂ 2.13c) nbh = the number of bonded strands along the vertical section defined by hf 2.5.2 Peeling Due to Eccentric Prestress Force Peeling stresses along any vertical plane through the bottom flange are described by the free-body diagram shown in FigureÂ 2.14 (Ross, 2012). The maximum transverse tension stress due to peeling, fP, is obtained from the conditions of moment equilibrium about the Y-Z plane. ( )= âf n A f x L L h n dP o ps ps po y t f h b Eq. 2.62 ânoApsfps noApsfps Section A-A section being considered A A hf XpoX Ly Lt fP â0.5fpLthf 0.5fpLthf FigureÂ 2.14. Free- body diagram for determining peeling stress.
Analytical Research Approaches and Findings 51Â Â where no = the total number of released bonded strands outboard of the vertical section defined by hf Aps fps = the force in a single released strand (assumed to be 0.9 Ã 0.75Aps fpu) xpo = distance from the centroid of outboard strands to the vertical section defined by hf Ly = internal moment arm in Y-direction Lt = length of assumed tensile stress distribution Ly and Lt vary according to the number of cut strands, the shape of the cross section, and the location within the cross section at which the calculation is made. Ross (2012) determined empirical values of Ly and Lt for Florida I-beam (FIB) girders as follows: at the maximum peeling condition (FigureÂ 2.13b): 53 and 10 in.= =L h x Ly f t at the combined condition (Figure 2.13c): = =L h x Ly f t Eq. 2.736 and 10 in. 2.5.3 Peeling Stress Calculations There are two conditions for peeling as illustrated in FigureÂ 2.13. For the maximum condition, fp is calculated by Eq. 2.6 and the strands at the section being considered are not released. The number of released strands is, therefore, all strands outboard of the section being considered, no. For the combined condition, the same section is considered, but the strands at that section are released, thereby creating the Hoyer-related stress, fH. In this condition, the number of released strands is no + nh. Ly is different for the different conditions (Ross, 2012); thus, fp is not the same for each condition. In the following analyses of peeling stress, transverse forces are calculated at vertical sections corresponding to the columns of strands of each girder; these are labeled beginning with A at the outermost strands. The analyses are worst-case scenarios as they assume an outside-in strand cutting sequence and that there are no debonded strands in the rows intersected by the sections, i.e., (nh = nbh). Furthermore, the following assumptions are made: Ec = 4,700 ksi Ep = 28,500 ksi Ï c = 0.20 c = 3Â in. (FIB) and 2Â in. otherwise Ï p = 0.34 for 0.6-in. strand and 0.32 for 0.7-in. strand (see Section 1.6) fps = 0.9(0.75fpu) = 182 ksi Ly and Lt are as given in TableÂ 2.13 TableÂ 2.13 summarizes the resulting maximum stresses obtained for all girder shapes con- sidered. FIB girders are presented separately (Section 18.104.22.168) since recommended values of parameters Ly and Lt are available for these beams (Ross, 2012). Analyses of the other I sections considered are presented in Section 22.214.171.124. 126.96.36.199 FIB Girders Ross (2012) provides estimates of the parameters Ly and Lt based on experimental observa- tions (Eq. 2.6). TableÂ 2.14 summarizes the predicted transverse forces at six vertical sections,
52 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders corresponding to the six outermost columns of strands of an FIB girder, using the recommen- dations of Ross (2012). Based on the results presented in TableÂ 2.14, as expected, increasing strand size results in increased Hoyer and peeling stresses. At the maximum condition, the maximum transverse stress due to peeling is 0.19 â²f ci for 0.7-in. strand and 0.14 â²f ci for 0.6-in. strand. These values are at or below the concrete tension stress limit at prestress transfer where reinforcing steel is present, 0.19 â²f ci (based on f â²ci = 8 ksi). However, for the combined condition, the stresses increase as more strands are cut, and the maximum stress is observed at Section F, 0.52 â²f ci for 0.7-in. strand and 0.35 â²f ci for 0.6-in. strand (see TableÂ 2.14). 188.8.131.52 Other I-Shaped Girders No data or summaries are available for girders other than FIB shapes. However, the values of Ly and Lt are necessary to conduct a peeling analysis. Based on the mechanism shown in FigureÂ 2.14, peeling is analogous to the transverse flexural behavior of the portion of the flange outboard of the section being considered. The stiffness of this portion of the flange is a function Girder Strand ( â ) ( â ) (in.) Maximum Condition (Figure 2.13b) Combined Condition (Figure 2.13c) max fp (ksi) Ly max (ksi) max + (ksi) FIB 0.6 1.00 10.0 53â 0.39 36â 0.37 1.00 0.7 0.54 0.59 1.48 AASHTO Type VI 0.6 1.38 13.8 73â 0.10 50â 0.60 0.74 0.7 0.14 0.93 1.14 AASHTO BT 0.6 0.82 8.2 43â 0.36 30â 0.54 1.03 0.7 0.51 0.83 1.53 Ohio WF 0.6 0.65 6.5 34â 1.77 23â 0.57 3.19 0.7 2.53 0.89 4.64 NU 0.6 0.79 7.9 42â 1.33 28â 0.52 2.40 0.7 1.87 0.80 3.47 Washington WF 0.6 0.77 7.7 41â 1.35 28â 0.50 2.44 0.7 1.91 0.77 3.50 Note: Shaded entries exceed concrete tension stress limit for ( . TableÂ 2.13. Summary of peeling stress analysis for I-shaped girders. Section (in.) (in.) (in.) â (in.) (ksi) Maximum condition Combined condition 1 + 1 (ksi) (ksi) A 0.6 16 0.00 8.5 0.30 0 0 0.30 0.110.7 0.46 0 0 0.46 0.16 B 0.6 14 2.00 9.5 0.27 0.11 0.04 0.42 0.150.7 0.41 0.15 0.05 0.62 0.22 C 0.6 12 3.00 10.5 0.38 0.24 0.08 0.73 0.260.7 0.59 0.33 0.12 1.07 0.38 D 0.6 10 3.71 11.5 0.34 0.35 0.12 0.86 0.300.7 0.52 0.49 0.17 1.24 0.44 E 0.6 8 4.60 13.2 0.29 0.39 0.14 0.98 0.350.7 0.44 0.54 0.19 1.43 0.51 F 0.6 6 5.14 14.9 0.25 0.36 0.13 1.00 0.350.7 0.38 0.51 0.18 1.48 0.52 TableÂ 2.14. Predicted peeling stresses for FIB girders (based on f âci = 8 ksi).
Analytical Research Approaches and Findings 53Â Â of x3po and hf. The values of Ly and Lt will be proportional to this stiffness, that is, a stiffer outstand will reduce peeling stresses along the critical section. Accordingly, the calculation of Ly and Lt shown in Eq. 2.7 for BT girders is in proportion to the ratio (x3pohf)BT/(x3pohf)FIB, and similarly for the other I-shapes considered. These ratios and the resulting estimates of Ly and Lt are given in TableÂ 2.13, which also summarizes the maximum stresses obtained for all girder shapes considered. TableÂ 2.15 through TableÂ 2.19 summarize the calculated stresses due to peeling for AASHTO Type VI, AASHTO-PCI BT, Ohio WF, Nebraska NU, and Washington WF girders, respectively. 184.108.40.206 Summary of Peeling Behavior of Single-Web Girders As shown in TableÂ 2.13, behavior can be divided into two groups based on flange geometry. Shapes with wide flat-bottom flanges (Ohio WF, Nebraska NU, and Washington WF) exhibit large, predicted peeling stresses since the resultant force causing peeling (1) has a larger lever arm (xpo) and (2) the resisting section is shorter (hf). Sections with stockier flanges (FIB, AASHTO Type VI, and AASHTO-PCI BT) exhibit lower stresses overall. In addition, for FIB, AASHTO Type VI, and AASHTO-PCI BT shapes, stresses at the maximum condition remain below 0.19 â²f ci (based on f â²ci = 8 ksi) for all cases. It must be noted that the data presented in TableÂ 2.13 represent the theoretical worst-case (and likely unrealistic) scenario of (1) all strands in flange stressed; (2) no strands debonded; Section (in.) (in.) (in.) â (in.) (ksi) Maximum condition Combined condition 1 + 1 (ksi) (ksi) A 0.6 11 0 11 0.51 0 0 0.51 0.180.7 0.78 0 0 0.78 0.28 B 0.6 9 2.00 13 0.55 0.04 0.02 0.61 0.220.7 0.84 0.06 0.02 0.93 0.33 C 0.6 7 2.89 15 0.57 0.08 0.03 0.70 0.250.7 0.89 0.12 0.04 1.06 0.38 D 0.6 5 3.73 17 0.60 0.10 0.04 0.74 0.26 0.7 0.93 0.14 0.05 1.14 0.40 Section (in.) (in.) (in.) â (in.) (ksi) Maximum condition Combined condition 1 + 1 (ksi) (ksi) A 0.6 11 0 6.9 0.38 0 0 0.38 0.140.7 0.58 0 0 0.58 0.21 B 0.6 9 2.00 7.8 0.33 0.14 0.05 0.54 0.190.7 0.50 0.20 0.07 0.79 0.28 C 0.6 7 3.00 8.7 0.47 0.29 0.10 0.89 0.320.7 0.73 0.41 0.14 1.32 0.47 D 0.6 5 3.71 9.6 0.42 0.36 0.12 0.94 0.330.7 0.64 0.51 0.18 1.37 0.49 0.6 0.54 0.34 0.12 1.03 0.36E 3 4.80 10.50.7 0.83 0.48 0.17 1.53 0.54 TableÂ 2.15. Predicted peeling stresses for AASHTO VI (based on f âci = 8 ksi). TableÂ 2.16. Predicted peeling stresses for AASHTO-PCI BT girders (based on f âci = 8 ksi).
54 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders Section (in.) (in.) (in.) â (in.) (ksi) Maximum condition Combined condition 1 + 1 (ksi) (ksi) A 0.6 18 0.00 6.5 0.19 0 0 0.19 0.070.7 0.28 0 0 0.28 0.10 B 0.6 16 2.00 7.5 0.35 0.24 0.09 0.70 0.250.7 0.52 0.34 0.12 1.03 0.36 C 0.6 14 2.67 8.5 0.49 0.70 0.25 1.53 0.540.7 0.75 1.00 0.35 2.23 0.79 D 0.6 12 3.33 9.5 0.43 1.17 0.42 2.16 0.760.7 0.65 1.66 0.59 3.10 1.09 E 0.6 10 4.22 10.5 0.54 1.60 0.57 2.90 1.030.7 0.83 2.28 0.81 4.20 1.49 F 0.6 8 4.92 11.5 0.48 1.75 0.62 3.07 1.080.7 0.74 2.48 0.88 4.40 1.56 G 0.6 6 5.76 12.5 0.57 1.77 0.63 3.19 1.130.7 0.89 2.53 0.90 4.64 1.64 H 0.6 4 6.45 14.5 0.47 1.22 0.43 2.28 0.800.7 0.73 1.73 0.61 3.28 1.16 Section (in.) (in.) (in.) â (in.) (ksi) Maximum condition Combined condition 1 + 1 (ksi) (ksi) A 0.6 17 0.00 6.1 0.45 0 0 0.45 0.160.7 0.68 0 0 0.68 0.24 B 0.6 15 2.00 6.9 0.38 0.37 0.13 0.93 0.330.7 0.58 0.51 0.18 1.35 0.48 C 0.6 13 3.00 7.7 0.34 0.75 0.26 1.46 0.520.7 0.51 1.05 0.37 2.08 0.74 D 0.6 11 4.00 8.5 0.49 1.12 0.39 2.17 0.770.7 0.75 1.58 0.56 3.13 1.11 E 0.6 9 4.67 9.3 0.44 1.31 0.46 2.40 0.850.7 0.67 1.85 0.65 3.44 1.22 F 0.6 7 5.50 10.0 0.40 1.33 0.47 2.39 0.850.7 0.60 1.87 0.66 3.41 1.21 G 0.6 5 6.40 10.8 0.52 1.25 0.44 2.40 0.850.7 0.80 1.78 0.63 3.47 1.23 Section (in.) (in.) (in.) â (in.) (ksi) Maximum condition Combined condition 1 + 1 (ksi) (ksi) A 0.6 16.25 0.00 6.2 0.44 0 0 0.44 0.15 0.7 0.67 0 0 0.67 0.24 B 0.6 14.25 2.00 6.9 0.38 0.36 0.13 0.92 0.32 0.7 0.58 0.51 0.18 1.33 0.47 C 0.6 12.25 3.00 7.6 0.34 0.76 0.27 1.46 0.52 0.7 0.52 1.07 0.38 2.08 0.73 D 0.6 10.25 4.00 8.3 0.51 1.15 0.41 2.19 0.78 0.7 0.78 1.64 0.58 3.17 1.12 E 0.6 8.25 4.67 9.0 0.46 1.35 0.48 2.44 0.860.7 0.70 1.91 0.68 3.50 1.24 F 0.6 6.25 5.50 9.7 0.42 1.36 0.48 2.41 0.850.7 0.64 1.92 0.68 3.45 1.22 G 0.6 4.25 6.40 11.2 0.50 1.05 0.37 2.03 0.720.7 0.77 1.49 0.53 2.94 1.04 TableÂ 2.17. Predicted peeling stresses for Ohio WF beams (based on f âci = 8 ksi). TableÂ 2.18. Predicted peeling stresses for Nebraska NU girders (based on f âci = 8 ksi). TableÂ 2.19. Predicted peeling stresses for Washington WF girders (based on f âci = 8 ksi).
Analytical Research Approaches and Findings 55Â Â and (3) an âoutside-inâ release sequence. Deviation from this scenario reduces predicted stresses considerably: 1. For all cases, it is clear that the combined Hoyer effect proposed by Ross (2012) drives the peeling behavior. Strand debonding, particularly in the recommended pattern of debonding âfrom the outside-inâ will mitigate Hoyer stresses ( fH) in proportion to the number of debonded strands at a vertical section (nh â nbh). 2. Releasing/cutting strands more uniformly, such as top-down and outside-in, will significantly mitigate peeling stresses by reducing no. Furthermore, the unreleased/uncut lower layer strands in a âtop-downâ release sequence provide restraint to the peeling moment. 3. The worst cases assumed the same number of 0.6-in. and 0.7-in. strands; thus, stresses for 0.7-in. strand cases are markedly larger than for 0.6-in. strand cases. Both Hoyer and peeling stresses are on the order of 150% greater for 0.7-in. strand cases; this value accounts for a 135% increase in strand force and decrease in resisting concrete area: (hf â nhdb) term in Eq. 2.5 and Eq. 2.6. For fully loaded bottom flanges, it is unlikely that 0.6-in. strands could be replaced strand-for-strand with 0.7-in. strands (see Section 220.127.116.11). Therefore, the realistic increase in stress due to the use of 0.7-in. strands would be considerably smaller. Additionally, it is also noted that the stresses calculated are at the end face of the girder. Both Hoyer ( fH) and peeling stresses ( fp) decrease linearly with depth into the girder. fp decreases to zero over the length Lt while fH decreases to zero at the real transfer length of the strand, typically about 30db. 2.5.4 FEM of Strand Release Sequence In the previous sections, the potential for peeling due to an improper strand release sequence was investigated. Ohio WF girders were seen to be critical in this respect (TableÂ 2.17). A sub- sequent series of FEM analyses were undertaken to validate the numerical study and extend the analyses to release sequences that cannot be modeled using the analytic approach. All full-girder models were developed using the ATENA FEM package. The analyses presented are based on the extensive modeling and validation studies presented in NCHRP Research Report 849 (Shahrooz etÂ al., 2017). A limitation of this FEM analysis is that only peeling due to eccentric prestress force, fp, that is, the âmaximum peeling conditionâ shown in FigureÂ 2.13 (Ross, 2012) can be considered. The ATENA finite element models use one-dimensional truss elements to model all reinforcing strands and bars; therefore, the Hoyer effect cannot be captured. The strand designations in the flange are as shown in FigureÂ 2.15. Both 0.6-in. and 0.7-in. strands are considered in 28 strand release sequences given in TableÂ 2.20. All strands are stressed 5 4 3 2 1 AJ I CL DM ENO FP GQ H BK FigureÂ 2.15. Strand designation for Ohio WF (strand spacing is 2Â in.; width of bottom flange is 40Â in.).
56 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders to 0.75APS fpu, and each case represents only the strands indicated being released. The remain- ing strands are assumed to be stressed in the stressing bed. This condition potentially provides significant restraint at the beam end from the unreleased strands. In this model, this restraint is simplified by restraining uplift at the beam support and enforcing a horizontal restraint. In practice, this restraint would be complex and would vary based on at least (1) the length of exposed strand between the girder end and stressing bulkhead; and (2) the release sequence itself since the girder will bend both horizontally and vertically, thereby affecting stress in unreleased strands. Thus, the FEM peeling analyses conducted, like the numeric study, are artificial but should represent trends well. To assess the theoretical maximum possible peeling stress, all strands are assumed to be present, and all (for the maximum peeling condition) are bonded. Strand release sequences BâH (see TableÂ 2.20) are the same as those given in TableÂ 2.17 and are worst- case scenariosâmodeling outside-in strand release from one side of the girder. Strand release sequences IâN represent possible poor-case release sequences that combine top-down and outside-in sequences. The number 6 or 7 following the case designation indicates the strand size, 0.6-in. or 0.7-in., respectively. Only transverse flange stresses are of interest in this study. FEM results Analytic results(Table 2.17) (1) (2) (3) (4) (5) (6) (7) (8) Case Strands released Maximum transverse tension stress, (ksi) â¦occurs at the vertical plane Maximum transverse tension stress, (ksi) â¦at the vertical plane Maximum transverse tension stress, (ksi) â¦occurs at the vertical plane 0.6-in. strand A-6 All strands â0.03 H B-6 A2 0.35 F 0.20 B 0.24 B C-6 All A and B 0.46 H 0.08 C 0.70 C D-6 All AâC 0.14 F 0.10 D 1.17 D E-6 All AâD 0.17 F 0.17 E 1.60 E F-6 All AâE 0.11 G/H â0.47 F 1.75 F G-6 All AâF 0.11 H â0.02 G 1.77 G H-6 All AâG â0.14 G/H â0.14 H 1.22 H I-6 All 4 and 5; C3; D3 â0.12 G/H J-6 All 4 and 5; C3; D3, E3; F3 â0.09 G/H K-6 All 3â5; A2; B2 0.05 G/H L-6 All 3â5; A2; B2; C2; D2 0.20 G/H M-6 All 2â5; B1; C1 0.18 G/H N-6 All 2â5; B1; C1; D1; E1 0.19 H 0.7-in. strand A-7 All strands 0.08 H B-7 A2 0.35 F 0.18 B 0.34 B C-7 All A and B 0.43 H 0.06 C 1.00 C D-7 All AâC 0.05 H â0.33 D 1.66 D E-7 All AâD 0.06 E 0.06 E 2.28 E F-7 All AâE 0.05 H 0.59 F 2.48 F G-7 All AâF 0.09 H 0.03 G 2.53 G H-7 All AâG â0.25 G/H â0.25 H 1.73 H I-7 All 4 and 5; C3; D3 â0.16 G/H J-7 All 4 and 5; C3; D3, E3; F3 â0.05 G/H K-7 All 3â5; A2; B2 â0.08 A L-7 All 3â5; A2; B2; C2; D2 0.15 G/H M-7 All 2â5; B1; C1 0.27 H N-7 All 2â5; B1; C1; D1; E1 â0.01 G/H TableÂ 2.20. Strand release sequences.
Analytical Research Approaches and Findings 57Â Â To model peeling stresses, which are a local eect, a half-span model of a WF-72 was used. e half span was made statically determinate with the following restraints intended to represent the restraint of a prestressed girder at prestress transfer: 1. Vertical deection is restrained across the entire width of the girder end; this restraint mimics the simple support of a prestress girder at release where camber is expected. Upli is not permitted at this support in this model. 2. Horizontal deection is restrained at only the centerline of the girder at the support; this boundary condition permits transverse bending of the girder due to eccentric strand release sequences. 3. Longitudinal deection is restrained at the midspan of the beam, which is the typical restraint to represent a full-span model by its half span. Cases A-6 and A-7 have all 60 strands released simultaneously (Case A-7 is shown in Figure 2.16). As expected, ange stresses are symmetric, and peak tensile stresses are at the web/ange interface, related to splitting (i.e., AASHTO LRFD Bridge Design Specications Article 18.104.22.168.1) (AASHTO, 2020). Cases B through H are the same as those shown in TableÂ 2.17 in which strand release progresses inward from the outside of one edge of the section. e stresses determined using FEM at the vertical sections reported in TableÂ 2.17 [TableÂ 2.20, columns (5) and (6)] are con- siderably lower than the analytical peeling analysis predicted at the same locations [TableÂ 2.20, columns (7) and (8)]. Indeed, as the outside-in strand release moves from B to H, the stresses at the critical planes dened by FigureÂ 2.15 become compressive. In examining the sequence from Cases B to H (top row of FigureÂ 2.16), it is clear that as the prestress force increases in the ange, the girder behaves more like a Bernoulli beam bending about both principal axes. Ross B-7 C-7 D-7 E-7 F-7 G-7 H-7 Transverse stresses at end of the girder Bottom flange stresses over first 60 in. of girder (end of girder at top) A-7 I-7 J-7 K-7 L-7 M-7 N-7 Transverse stresses at end of the girder Bottom flange stresses over first 60 in. of girder (end of girder at top) Color maps are qualitative only. Red is tension, blue is compression. Color maps vary from minimum to maximum stress in each analysis and do not represent the same stress levels in subsequent figures. FigureÂ 2.16. Transverse stress results from Cases A-7 to N-7.
58 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders (2012) assumes a shear-like transfer of stress along the plane located at x (FigureÂ 2.17). This assumption appears to capture the behavior reasonably well for Cases B and C. However, as more prestress force is introduced, weak axis flexural behavior of the girder becomes dominant, resulting in more complex behavior that is simply not captured by the simple model proposed by Ross. In particular, it appears that the restraint provided by the unreleased strands has a significant effect on the stress transfer from the released strands. This effect, as noted above, will vary and is not possible to model in any but a very general sense. Additionally, the maximum stress does not occur at the vertical planes suggested by Ross; thus, the âcombinedâ case, which superimposes peeling and Hoyer-induced stresses, is also likely a significant overestimation of the actual stress state. Based on these analyses and considering the discussion presented in Section 2.5.3, the peel- ing behavior hypothesized by Ross (2012) is not considered significant and is easily mitigated by conventional âgood practiceâ strand release sequences. It is also noted that such âpeelingâ has not been observed (to the knowledge of the authors of this report) outside of extreme strand patterns and release sequences such as those reported by Ross. 2.6 Strand Spacing Adopting 2-in. spacing (2.86db) for 0.7-in. strands remains a concern that has not been addressed (see Section 1.5.1). Nonetheless, no adverse or unexpected behavior has been observed based on testing of several NU girders with 0.7-in. strands spaced at 2Â in. (Morcous, 2013; Morcous etÂ al., 2014; Shahrooz etÂ al., 2017). It is noted, however, that the 0.7-in. strand NU girders reported by Morcous (2013) and Morcous etÂ al. (2014) were only stressed to an initial prestress of 0.64fpu. If splitting resistance is assumed to be a function of clear spacing, the ratios presented in TableÂ 2.21 are obtained. The Hoyer dilation, Î½p, represents a measure of the applied radial strain. The normalized splitting stress is a simple approximate approach that indicates the expected radial splitting stress of a 0.7-in. strand at 2Â in. is 22% greater than a 0.5-in. strand at 2Â in. Increasing the spacing to 2.25Â in. mitigates this increase. However, 0.7-in. strands at 2Â in. result = 6.5 in. = 34â / = 16 in. (see Table 2.15) â 9 in. â 17 in. x xpo hf -n A fo ps ps n A fo ps ps Lt Ly fP 0.5f L hP t f -0.5f L hP t f Section A-A AA section being considered FigureÂ 2.17. Visualization of peeling model from Case B-7.
Analytical Research Approaches and Findings 59Â Â in essentially the same stress as 0.5-in. strands at 1.75Â in. Much of this comparison hinges on the Hoyer dilation, which decreases with increasing diameter. The finite element analyses presented in Section 2.7 suggest the same trends. 2.7 FEM of Strand Transfer Lengths 2.7.1 ABAQUS Finite Element Models Two prismatic models were developed and analyzed using ABAQUS (version 6.10EF). Square sections having a dimension equal to 6Â in. for the single-strand models and three times the strand spacing (3s) for the four-strand models were used. All prisms are 50Â in. long, exceeding the longest transfer length (42Â in.) assumed in this study. Concrete strength is assumed to be f â²c = 15 ksi; therefore, the strength at transfer used in this study is f â²ci = 0.6f â²c = 9 ksi. The Poisson ratio of concrete was taken as Î½c = 0.2. Concrete modulus is determined based on AASHTO LRFD Bridge Design Specifications Article 22.214.171.124 (AASHTO, 2020): = â²E K w fc c c Eq. 2.8120,000 1 2 0.33 where wc = 0.145 + 0.001f â²c â 0.005 = 0.155 kcf for 15 ksi concrete per AASHTO LRFD Article 3.5.1. For 15-ksi concrete, Ec = 7,050 ksi. At prestress transfer, Eci = 5,950 ksi. Strand modulus (Ep) was taken as 29,000 ksi, and dilation (Poisson ratio) is varied in this analysis. Strand dilation is affected by applying a temperature field along the transfer length of the strand, Lt, varying from its maximum at the free end (z = 0) linearly to zero at the transfer length (z = Lt). The dilation strain of a prestressing strand due to initial prestress force, fpi = 202.5 ksi, is: [ ]( )( )Îµ = â = â â = =r r r r r v f E r v f E vpH s p pi p p pi p p Eq. 2.91 0.0070 0 0 0 0 where the notation is given in the discussion of the Hoyer effect (Section 1.7). ABAQUS C3D8R elements were used to model the concrete and strand. The strand was modeled as a cylinder having a diameter equal to the strand diameter. A temperature field was applied to the strand to model the Hoyer effect. The temperature, ÎT, required to produce the desired dilation strain is: â = Î± T f E vpi p s Eq. 2.10 The coefficient of thermal expansion of the strand is taken as Î±s = 12 Ã 10â5 m/mÂ°K. Strand (in.) Spacing (in.) Clear spacing â (in.) Hoyer dilation Normalized splitting stress ( Ã )/( â ) 0.5 1.75 1.25 0.40 0.160 +20% 0.5 2.0 1.50 0.40 0.133 baseline 0.6 1.75 1.15 0.34 0.186 +39% 0.6 2.0 1.40 0.34 0.146 +10% 0.6 2.25 1.65 0.34 0.124 â7% 0.7 2.0 1.30 0.30 0.162 +22% 0.7 2.25 1.55 0.30 0.134 +1% 0.7 1.75 1.05 0.30 0.200 +50% TableÂ 2.21. Effect of spacing on splitting stress.
60 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders For simplicity, these analyses neglect the transfer of longitudinal prestress to the concrete; this assumption is believed to be acceptable since the focus of these analyses is on the high stresses at the free end of the prism. 126.96.36.199 Mesh Size and Material Interface Because the focus of this study was on the relatively small strains and stresses in the imme- diate vicinity of the strand-concrete interface, the following relatively fine-mesh size was adopted: The strand was divided circumferentially into 100 equal radial segments; each segment was divided into multiple elements through the radius, resulting in 1,280 elements in a strand section. Thus, the circumferential mesh dimension at the strand-concrete interface is 0.019Â in. for the 0.6-in. strand and 0.022Â in. for the 0.7-in. strand. Near the strand, the concrete was similarly modeled with a fine, radially generated mesh having 128 segments around the strand circumference and a radial mesh size of 0.05Â in. out to a distance of 1Â in. from the strand-concrete interface. Farther from the strand, the concrete mesh size is increased to 0.3Â in. by 0.3Â in. for 6-in. prisms and marginally smaller or larger for smaller or larger prisms, i.e., the number of elements was kept constant. Along the length of the prism, the mesh dimension for both strand and concrete was 0.5Â in. The resulting meshes are seen in FigureÂ 2.17. The interface between the thermally stressed strand and surrounding concrete was modeled as a tie-type constraint, where the strand is the master surface and concrete is the slave surface. 188.8.131.52 Smeared Crack Concrete Model Following an initial model validation using an isotropic elastic model (Alabdulkarim, 2021), which was shown to compare well with the theoretical predictions obtained using the theoretical approach described in Section 1.7, a more realistic plastic modeling of the concrete was adopted. Particularly due to the high local stresses predicted, the concrete is anticipated to have cracked locally (this is described in relation to the Hoyer effect in Section 1.7). Such cracking will permit stress redistribution resulting in significantly different (and more realistic) behavior than assumed using elastic analysis inherent in the approach described in Section 1.7. The ABAQUS âsmeared crackâ concrete model is adopted (ABAQUS, 2011): âThe smeared crack concrete model in ABAQUS provides a general capability for modeling concrete in all types of structures. As a âsmearedâ model, it does not track individual âmacroâ cracks. Constitutive calcu- lations are performed independently at each integration point of the finite element model. The presence of cracks enters into these calculations by the way in which the cracks affect the stress and material stiffness associated with the integration point. Cracking is assumed to occur when the stress of the element reaches the âcrack detection surfaceâ which is a linear relationship between the equivalent pressure stress and the Mises equivalent deviatoric stress. As soon as the crack detection surface has been activated, the crack direction is taken to be the direction of that part of the maximum principal plastic strain. Following the crack detection, the crack affects the response of the model because a damage elasticity model is used.â To effect a smeared crack model, nonlinear compression and tension constitutive models and a failure surface interaction are defined (Wahalathantri etÂ al., 2011) as described in the following sections. 184.108.40.206.1 Concrete Compression. The complete stress-strain curve for concrete under compression is derived using the experimentally verified numerical model proposed by Hsu and Hsu (1994). As shown in FigureÂ 2.18a, this model can be used to develop the stress-strain relationship under uniaxial compression through 0.3Ïcu in the descending portion using only the maximum compressive strength (Ïcu).
Analytical Research Approaches and Findings 61Â Â The model assumes linear behavior having stiffness Ec through 0.5Ïcu. Beyond 0.5Ïcu, the stress-strain relationship through 0.3Ïcu (at Îµd) is defined as: Ï = Î² Îµ Îµ ï£« ï£ï£¬ ï£¶ ï£¸ï£· Î² â + Îµ Îµ ï£« ï£ï£¬ ï£¶ ï£¸ï£· ï£« ï£ ï£¬ ï£¬ ï£¬ï£¬ ï£¶ ï£¸ ï£· ï£· ï£·ï£· Ï Î²c c c cu Eq. 2.11 1 0 0 where the parameter Î² â , which depends on the shape of the stress-strain diagram, is calculated from the strain at peak stress, Îµ0, by: Î² = â Ï Îµ ï£® ï£°ï£¯ ï£¹ ï£»ï£ºE cu Eq. 2.121 1 0 0 [ ]Îµ = Ã Ï + Ãâ âcu Eq. 2.138.9 10 2.114 10 ksi units0 5 3 Concrete strength and modulus are taken as those at transfer: f â²ci = 0.6f â²c = 9 ksi and Eci = 5,950 ksi. The resulting compression stress-strain relationship used in this study is shown in FigureÂ 2.18b. 220.127.116.11.2 Concrete Tension. Tension stiffening is the ability of concrete to carry tension between cracks in reinforced concrete members and is known to control deformation calcu- lations particularly at serviceability stress levels (Bischoff, 2003). The concrete tensile stress- strain model proposed by Nayal and Rasheed (2006) is integrated into ABAQUS (FigureÂ 2.19). Like compression, this is essentially a two-parameter model requiring cracking stress, Ïto, and concrete elastic modulus. AASHTO LRFD Bridge Design Specifications Commentary C18.104.22.168 (AASHTO, 2020) recommends a concrete cracking stress of Ït0 = 0.23 â²f ci = 0.69 ksi. Using Eci = 5,950 ksi, the corresponding cracking strain is Îµt0 = 0.69/5950 = 0.000116. All other control parameters for the tension-stiffening stress-strain model are shown in FigureÂ 2.19a, and the relationship used in this study is shown in FigureÂ 2.19b. (a) Generic constitutive curve. (b) Relationship adopted in present study. FigureÂ 2.18. Compressive stress-strain relationship for ABAQUS (after Hsu and Hsu, 1994).
62 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders 22.214.171.124.3 Failure Surface. The plane stress smeared crack concrete failure surface adopted in ABAQUS is that described by Kupfer and Gerstle (1973) and is shown in FigureÂ 2.20. Four failure ratios are required: 1. The ratio of the ultimate biaxial compressive stress to the ultimate uniaxial compressive stress, f2/f â²ci = 1.16 [ABAQUS default value]. 2. The absolute value of the ratio of the uniaxial tensile stress at failure to the ultimate uniaxial compressive stress, ft0/f â²ci = 0.69/9 = 0.077 [calculations shown above]. 3. The ratio of the magnitude of a principal component of plastic strain at ultimate stress in biaxial compression to the plastic strain at ultimate stress in uniaxial compression; the ABAQUS default value is 1.28. 4. The ratio of the tensile principal stress at cracking, in plane stress, when the other principal stress is at the ultimate compressive value, to the tensile cracking stress under uniaxial tension; the ABAQUS default value is 0.33. (a) Generic constitutive curve. (b) Relationship adopted in present study. FigureÂ 2.19. Modified tension-stiffening model for ABAQUS (after Nayal and Rasheed, 2006). FigureÂ 2.20. ABAQUS smeared crack concrete failure surface (after Kupfer and Gerstle 1973).
Analytical Research Approaches and Findings 63Â Â 126.96.36.199.4 Shear Retention. The ABAQUS smeared crack model also permits shear retention, that is, the degree of shear capacity retained in the cracked concrete model. In this study, full shear retention (ABAQUS default) is assumed. This assumption should not impact the results of this study in any way. 188.8.131.52.5 Application of Elastic and Smeared Crack Models. FigureÂ 2.21 shows represen- tative examples of the implementation of the smeared crack model. Model geometry and the effects of applying the temperature-induced transverse strand strains on elastic isotropic models are shown in FigureÂ 2.21b, which shows the stresses at the interface of the strand and concrete, clearly showing the correct linear application of the (temperature-induced) dilation. The local effects of this stress validate the use of the prism size adopted. The nonlinearity in the smeared crack model dramatically illustrates the effects of softening and cracking behavior. 184.108.40.206 Single-Strand Models The single-strand models, comprising a single 0.6-in. or 0.7-in. strand in a 6-in. square prism, are used to validate and better illustrate the Hoyer effect described previously. Cases are labeled as follows: X-YY-ZZ ZZ = 20 â vp = 0.20 X = 6 â db = 0.6-in. strand YY = 30 â 30db transfer length ZZ = 25 â vp = 0.25 X = 7 â db = 0.7-in. strand YY = 60 â 60db transfer length ZZ = 30 â vp = 0.30 ZZ = 35 â vp = 0.35 ZZ = 40 â vp = 0.40 Results of all single-strand smeared crack models are given in TableÂ 2.22. Due to what is interpreted as extensive predicted cracking at the free end of the prism, the peak stresses occur in the vicinity of z = 1 to 4.5Â in. As expected, the smeared crack FEM results show the same trends as the analytical approach described in Section 1.7. The larger strand results in lower circumferential stresses at the interface between the strand and concrete, and the dilation ratio affects the interface stresses proportionally. 220.127.116.11.1 Stress and Cracking Distribution along the Transfer Length. FigureÂ 2.22 shows the circumferential stress in Case 7-60-30. This is the same data shown in FigureÂ 2.21b (a) Cross section at = 0 showing distribution of horizontally oriented stress ( ); circumferential stress, , is therefore at 12 and 6 oâclock around the strand. The stress range shown is â4.21 to 1.58 ksi. ( ). (b) Smeared crack longitudinal section along the interface of strand and concrete (free end on right) showing the distribution of horizontally oriented The stress range shown is â4.21 to 1.58 ksi. stress FigureÂ 2.21. Representative FEM results of single-strand model 7-60-30 having elastic and smeared crack concrete models.
64 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders Case (in.) (FEM) (ksi) (oK) = 0 (in. ) at , = = 1 in.= = 1 in. 0.6-in. strand ( = . .) 6-30-20 0.20 18 (30 ) 0.0014 117 1.04 0.66 1 1.45 0.57 6-30-25 0.25 0.0018 150 1.02 0.73 1 1.56 0.68 6-30-30 0.30 0.0021 175 0.96 0.73 1 1.61 0.69 6-30-35 0.35 0.0025 208 0.90 0.73 1 1.68 0.69 6-30-40 0.40 0.0028 233 0.81 0.73 1 1.73 0.69 6-60-20 0.20 36 (60 ) 0.0014 117 1.04 0.66 1 1.48 059 6-60-25 0.25 0.0018 150 1.02 0.73 1 1.57 0.68 6-60-30 0.30 0.0021 175 0.97 0.73 4 1.63 0.74 6-60-35 0.35 0.0025 208 0.91 0.73 4 1.72 0.77 6-60-40 0.40 0.0028 233 0.82 0.73 4 1.78 0.79 0.7-in. strand ( = . .) 7-30-20 0.20 21 (30 ) 0.0014 117 0.94 0.70 1 1.38 0.63 7-30-25 0.25 0.0018 150 0.89 0.73 2 1.50 0.72 7-30-30 0.30 0.0021 175 0.81 0.73 1 1.56 0.70 7-30-35 0.35 0.0025 208 0.72 0.73 1 1.62 0.71 7-30-40 0.40 0.0028 233 0.61 0.73 3 1.66 0.79 7-60-20 0.20 42 (60 ) 0.0014 117 0.94 0.71 1 1.40 0.65 7-60-25 0.25 0.0018 150 0.89 0.73 4.5 1.54 0.74 7-60-30 0.30 0.0021 175 0.82 0.73 4.5 1.58 0.77 7-60-35 0.35 0.0025 208 0.73 0.73 4.5 1.66 0.81 7-60-40 0.40 0.0028 233 0.62 0.73 4.5 1.71 0.83 TableÂ 2.22. Predicted circumferential tensile stresses resulting from Hoyer effect for single-strand smeared crack concrete model. FigureÂ 2.22. Longitudinal section along the interface of strand and concrete of 7-60-30 (free end on right) showing the distribution of horizontally oriented stress (rx).
Analytical Research Approaches and Findings 65Â Â except that the gray region in FigureÂ 2.22 indicates the region in which predicted tensile strains exceed those expected to cause cracking (Ït0 > 0.69 ksi). This is the region over which smeared cracking is calculated to occur. Unsurprisingly, cracking is predicted at the region of highest dilation, at the free end of the prism. FigureÂ 2.22, however, shows a thin region of smeared cracking extending along the strand almost to the full transfer length (60db = 42Â in. in the case shown). To investigate this effect, the model was rerun with lower values of initial prestress force as shown in TableÂ 2.23. The resulting longitudinal distributions of circumferential stress are shown in FigureÂ 2.25. The behavior shown in FigureÂ 2.23a reveals a distinctly three-part response. From right to left, the end of the transfer length remains uncracked; this uncracked region is represented as the initial âstraightâ section of the stress distribution. A softened, cracked but still well- confined region follows. Finally, at the free end (at the left), there is evidence of significant stress redistribution due to local cracking. This behavior is consistent for all initial stress conditions except 0.10fpi (Case 6 in TableÂ 2.22), which does not exceed the cracking tensile stress of 0.69 ksi. The sensitivity of this behavior to the assumed dilation of the strand is seen in FigureÂ 2.23b in which three representative cases, 7-60-40, 7-60-30, and 7-60-20, each stressed to fpi, are shown. As described previously, lower dilation results in lower circumferential stress and therefore a reduced region of softened behavior. 18.104.22.168 Four-Strand Models The four-strand models are intended to investigate the effects of strand spacing. The models (FigureÂ 2.24) are the same as the single-strand models except that the square prism dimension is three times the strand spacing, s, in all cases. The results of four-strand smeared crack models in TableÂ 2.24 confirm the findings of the single-strand models. FigureÂ 2.25 shows the circumferential stresses through the section shown in FigureÂ 2.24 at z = 1Â in. and near the end of the development length at z = 25db (15Â in. for 0.6-in. strand and 17.5Â in. for 0.7-in. strand). As reported previously, the larger strand results in lower circumferential stresses at the interface between the strand and concrete. Between adjacent strands, stresses decrease with increased strand spacing although the decrease is not proportional to the increase in spacing (FigureÂ 2.25). At z = 1Â in., the tensile stresses exceed the cracking threshold of 0.69 ksi, and the effect is apparent in the relatively nonuniform circumferential stress adjacent to and moving away from the strands. At z = 1Â in. concrete damage is apparent between adjacent strands. Damage between strands is more pro- nounced at (1) smaller spacing; and (2) for the larger 0.7-in. strands. Near the end of the transfer length, at 25db, the stress results show the same trends although at lower stresses. Less damage in the concrete is apparent as evidenced by smooth stress transi- tions between and around strands. Case Initial force, (ksi) (oK) (FEM) (ksi) @ ( = 0, = 0 ) 1 1.00 = 202.5 175 1.58 2 0.75 = 151.9 131 1.47 3 0.50 = 101.2 87 1.22 4 0.25 = 50.6 44 0.91 5 0.15 = 30.4 26 0.78 6 18 0.61 TableÂ 2.23. Effect of different prestress force on circumferential stress on case 7-60-30.
66 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders (a) Effect of different initial prestress force, fpi. (b) Effect of different dilation, Ï p. FigureÂ 2.23. Distribution of rp for case 7-60.
Analytical Research Approaches and Findings 67Â Â Ã square prism dimension = 3 Ã strand spacing Ã r = s/2 The section at which concrete stresses reported in Figure 2.25 is determined Case (in.) s (in.) Prism dimension (in.) (ksi) at = 0 (ksi)at = 1 in. = 0 = /2 = 1.0 = 0 = /2 = 1.0 1 0.5 1.75 5.25 0.75 0.79 1.01 0.81 2 0.5 2.00 6.00 0.78 0.77 1.00 0.73 3 0.6 2.00 6.00 0.71 0.78 0.94 0.78 4 0.6 2.25 6.75 0.91 0.76 0.93 0.71 5 0.7 2.00 6.00 0.66 0.78 0.91 0.76 6 0.7 2.25 6.75 0.66 0.73 0.91 0.66 FigureÂ 2.24. Four-strand model geometry (elastic stresses are shown). TableÂ 2.24. The effect of applying smeared crack on circumferential tensile stresses for four strands. FigureÂ 2.25. Distribution of rp in comparable four-strand models showing the effect of strand size and spacing.
68 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders 22.214.171.124 Summary of Strand Transfer Analytical Study The Hoyer effect is expected to result in circumferential stresses at the strand-concrete inter- face that exceed the concrete stress tensile strength. The following observations are made in relation to the circumferential stresses: Dilation ratio: A lower dilation ratio results in lower circumferential stresses at the interface between the strand and concrete. Strand diameter: The larger strand results in lower circumferential stresses. Spacing: The greater spacing between adjacent strands results in lower circumferential stresses proportionally. Adopting a smeared crack model of the concrete illustrates the extent of the region of con- crete whose stress exceeds the concrete cracking stress. Especially at the free end of the strand, localized damage associated with only partially restrained Hoyer effect is evident. In light of the discussion in Section 1.5.1, the effect of strand spacing is less clear. Although the concrete stress is not significantly affected, the physical extent of âcrackedâ concrete is greater for smaller spacings and larger strand sizes. 2.8 Long-Span Girder Stability By using a 0.7-in. strand, span lengths of current girder shapes may, theoretically, be increased up to 20% (Section 2.2). As girders become longer, stability considerations during lifting and handling can begin to control aspects of design. As discussed in Section 1.8, the stability of prestressed concrete girders is considered in terms of the potential for rollover and the sus- ceptibility to excessive deformations causing concrete stress limits to be exceeded. 2.8.1 Stability Case Studyâ223-Ft-Long WF100G To illustrate and validate the PCI (2015 and 2019) stability analysis procedure (Section 1.8.1), an analysis was conducted of the 223-ft-long WF100G described by West (2019). This girder design permits the investigation of the PCI approach using an extreme, although documented, case. The impact of redesigning this girder, built with 0.6-in. strand, using 0.7-in. strand is also explored. To achieve the record 223-ft span, West reports the need to modify the WF100G girder section by widening the top flange to improve stability during handling: âSpecial consideration was given to girder stability and stresses during plant handling, hauling, and erection. This analysis resulted in the selection of a modified WSDOT WF100G girder cross section, where the top flange was widened from 4 ft-1Â in. to 5 ft-1Â in. for this project to increase the weak-axis stiffness.â Applying this knowledge, stability analyses were conducted on four variations of the WF100G reported by West. Each variation considered the 223-ft-span girder reported. Case 1: WF100G with 0.6-in. strandâthis, as reported by West, apparently had issues with stability, ultimately requiring Case 2. Case 2: As-built WF100G-MOD with 0.6-in. strandâthis is the modified section reported by West (2019) having a 12-in. wider compression flange. Case 3: WF100G with 0.7-in. strandâsame as Case 1 but with 0.7-in. strands providing the same moment resistance at midspan (i.e., prestress force multiplied by lever arm to the center of gravity of prestressing [cgs]). Case 4: WF100G-MOD with 0.7-in. strandâsame as Case 3 (0.7-in. strands) but with the 12-in. wider compression flange of Case 2.
Analytical Research Approaches and Findings 69Â Â The analyses conducted considered the following conditions for each girder (PCI, 2015): 1. Initial crane lift from prestressing bed 2. Girder supported on dunnage 3. Transportation of girder to site 4. Crane lift in field 5. Girder in place in final position with the top (temporary strands still active) Although the PCI (2019) spreadsheet calculates the following additional cases, due to the length of the girders, it is expected that (1) bracing would be installed at each girder end immediately upon setting in place and (2) required bracing would be installed soon after multiple girders are in place and absolutely before the start of deck construction. 6. Girder in place following cutting top strands typically required for transportation 7. Multiple seated girders at inactive construction 8. Active deck construction 126.96.36.199 Cross-Section Geometry FigureÂ 2.26 shows the geometry of the WF100G section. The only difference between WF100G and WF100G-MOD is that the top flange width of the latter is 12Â in. wider. TableÂ 2.25 summarizes the section geometries of WF100G and WF100G-MOD. Importantly, while the 12-in.-wider flange represents only a 3% increase in section area (weight), it results in a 40% increase in the weak axis moment of inertia, Iy, affecting the stability performance of the girder. As reported by West (2019), the LWC used had a unit weight, wc = 0.125 kip/ft3, making the unit weight of the girder with reinforcement, wgirder = 0.138 kip/ft3. The modulus of elasticity of concrete was taken as Ec = 120,000wc2f c0.33 (AASHTO, 2020). 188.8.131.52.1 Strand Arrangement in WF100G and WF100G-MOD with 0.6-In. Strand. The following strand arrangement is reported by West (2019) and confirmed from available drawings of WF girders (WSDOT, 2019). 46 straight 0.6-in. strands: cgs1 = 4.08 in. E in Figure 2.26b 35 harped 0.6-in. strands: cgs at midspan = 6.7 in. F in Figure 2.26c cgs at the end of the girder = 79.5 in. F0 in Figure 2.26b harp point, b = 87.5 ft Figure 2.26d 10 temporary top 0.6-in. strands: cgs = 98.5 in. T in Figure 2.26c 184.108.40.206.2 Strand Arrangement in WF100G and WF100G-MOD with 0.7-In. Strand. The WF100G girder was redesigned using 0.7-in. strands. The criteria for the substitution of 0.6-in. with 0.7-in. strands were that total prestress force in both straight and harped strands remained essentially2 the same and that the girder moment capacityâas calculated by the pre- stress force multiplied by the depth to the cgsâremained the same. The resulting design uses fewer strands and, as a result, has lower cgs values at midspan: 32 straight 0.7-in. strands: cgs = 3.00 in. 28 harped 0.7-in. strands: cgs at midspan = 5.4 in. cgs at the end of the girder = 83.5 in. harp point, b = 87.5 ft 10 temporary top 0.6-in. strands: cgs = 98.5 in. (unchanged from West, 2019) 1 Location of center of gravity of steel in cross section measured relative to the girder soffit. 2 Keeping in mind the need to provide an even number of discrete strands, moment capacity cannot be matched exactly.
(d) Longitudinal strand layout and dimensions required for stability analysis (a) WF100G (WSDOT 2019) (b) 0.6-in. strand layout atgirder end (c) 0.6-in. strand layout at midspan FigureÂ 2.26. Geometry of WF100G. WF100G WF100G-MOD WF100G-MODWF100G 49 in. 61 in. 1.24 1,084 in.2 1,120 in.2 1.03 266 in.2 302 in.2 1.14 338 in.2 338 in.2 1.00 1,526,584 in.4 1,614,640 in.4 1.06 68,602 in.4 95,935 in.4 1.40 8,552 in.4 8,660 in.4 1.01 / 0.045 0.059 1.32 36,739 in.4 64,072 in.4 1.74 30,362 in.4 30,362 in.4 1.00 1,501 in.4 = 0.022Iy 1,501 in.4 = 0.016Iy 1.00 / 1.21 2.11 1.74 37.5 in. 38.0 in. 1.01 7.96 in. 9.26 in. 1.16 11.75 in. 14.57 in. 1.24 48.3 in. 50.6 in. 1.05 1.039 (kip/ft) 1.070 (kip/ft) 1.03 TableÂ 2.25. Geometric properties of WF100G and WF100G-MOD.
Analytical Research Approaches and Findings 71Â Â 220.127.116.11 Calculation of Camber Camber (positive value is upward deflection) is calculated at each stage as: â = â + â + âself ps ohang Eq. 2.14 Îself is the (downward) deflection due to girder self-weight calculated using the distance between supports: ( )â = â âw L a E I self g c x Eq. 2.155 2 384 4 where a = distance from the end of the girder to support (FigureÂ 2.26d); Lg = length of the girder. Îps is the (net upward) deflection due to prestress, comprising three components: ( ) â = p e L E I ps st eff st c st g c x Eq. 2.16straight temporary top strands 8 , , , 2 â = p e L E I ps sb eff sb c sb g c x Eq. 2.17straight bottom strands 8 , , , 2 â = â â²ï£« ï£ï£¬ ï£¶ ï£¸ï£· p E I e L e b ps h eff h c x c h h Eq. 2.18harped strands 8 6 , , , 2 2 Îohang is the (upward) deflection resulting from the girder overhang beyond locations of support: ( )â = âwa L a E I ohang g c x Eq. 2.192 16 2 2 Note that in Eq. 2.16 to Eq. 2.18, eccentricity is measured from the centroid of the cross section (FigureÂ 2.26d) with the strand below the centroid having positive eccentricity. Thus, Eq. 2.16 should have a negative (downward) result while Eq. 2.17 and Eq. 2.18 are positive (upward). 18.104.22.168 Sweep Tolerance As recommended by PCI (2015), sweep eccentricity is taken as 1â8Â in. per 10Â ft length of girder: = Ã = 10 1 8 in. 2.8 in.e L ft i g For the transportation stability check, 1Â in. is added to sweep (PCI, 2019): ei,trans = 3.8Â in. For the initial lift from the prestressing bed, experience from practice indicates that sweep is about one-half that recommended: ei,bed = 1.4Â in. (PCI, 2019). 22.214.171.124 Bearing Rotational Stiffness, Kq Bearing stiffness was calculated based on the recommendations of NCHRP Report 596 (Stanton etÂ al., 2008): ( )= +K EI t A B Sq r r Eq. 2.202 where E and t are Youngâs modulus and thickness of the bearing material; and =I L Wbrg brg Eq. 2.21moment of inertia 123
72 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders ( ) =S L W t L W brg brg brg brg Eq. 2.22shape factor 2 in which Lbrg = plan dimension of the bearing parallel to the axis of rotation (length along beam); Wbrg = plan dimension of the bearing perpendicular to the axis of the rotation (width across beam section); Ar is a dimensionless constant taken as 1.0 for rectangular bearing pads; and Br is a dimensionless constant calculated as follows: ( ) ( )â â Î» + â Î» â âï£« ï£ï£¬ ï£¶ ï£¸ï£· ï£« ï£ï£¬ ï£¶ ï£¸ï£· B L W r brg brg brg brg Eq. 2.230.24 0.024 1.15 0.89 1 exp 0.64 The compressibility index is: Î» = S G K brg Eq. 2.24 3 where G = shear modulus of bearing; and K = bulk modulus of bearing. For the WF100G bearings, elastomeric bearings having E = 1.04 ksi, G = 0.1275 ksi, K = 450 ksi, t = 2Â in., Lbrg = 12Â in., and Wbrg = width of bottom flange â 2Â in. = 36Â in. (accounting for assumed 1-in. chamfer). With these parameters, Kq = 79,303 kip â in./rad. 126.96.36.199 Hauling Rig Stiffness, Kq,trans Upon consultation with practitioners, an initial value of Kq,tran Ã² 82,000 kip - in./rad was selected. As described previously, this value may be revised to address transportation stability issues. 188.8.131.52 Stability Analysis Input Parameters TableÂ 2.26 summarizes girder geometry- and prestressing-related input parameters for the stability analysis. Material properties vary with the presumed age of the girder at each step. TableÂ 2.27 summarizes the remainder of assumed parameters that do not vary from case to case. Except for the transportation and âin-placeâ stages, in each analysis, an initial value of a = 0.1Lg = 23Â ft was assumed. This value was revised to those shown in TableÂ 2.26 to maximize the factors of safety calculated. A maximum value of a = 20Â ft was used for the transportation stage. A photograph in West (2019) shows the girder on a transportation vehicle; the rear overhang appears to be marginally less than 20Â ft. The value of a = 0.5Â ft is used for all analyses of girders on their bearings and cannot be revised. 184.108.40.206 Stability Analysis Results Results of the stability analyses are provided in TableÂ 2.28 in terms of the three factors of Safety prescribed by PCI (2015) and described in Section 1.8.1. The analyses conducted in this study suggest that the girder is susceptible to cracking during transportation (FScr < 1). The low FS values for the transportation stage result from the imposed practical limit of a < 20Â ft. With a = 37Â ft, for instance, all four analyses presented result in FScr > 1.0. The susceptibility to cracking is dramatically reduced (FS increases) by simply increasing the compression flange width 12Â in., which increases Iy 40% (TableÂ 2.25). Given the extreme dimensions of the case study girder, this approach is deemed reasonable and efficient for improving stability. Nonetheless, in the analysis presented, FScr remains less than unity. By further revising analysis parameters (TableÂ 2.27), the modified WF100G-MOD girder can be made stable for all
Analytical Research Approaches and Findings 73Â Â Condition Lift from bed Dunnage Transport Lift in field In place In place (top strands cut) Assumed age Release Release >28 days >28 days >28 days >28 days Strand (in.) 0.6 0.7 0.6 0.7 0.6 0.7 0.6 0.7 0.6 0.7 0.6 0.7 (ksi) 8.4 8.4 10.0 10.0 10.0 10.0 Prestress losses 10% 10% 50% 50% 50% 50% (ksi) 182 182 167 167 167 167 (kips) 3,594 3,605 3,594 3,605 3,298 3,308 3,298 3,308 3,298 3,308 2,935 2,946 , (in.) 15.45 14.46 15.45 14.46 15.45 14.46 15.45 14.46 15.45 14.46 5.31 4.12 . (in.) 1.4 2.8 3.8 2.8 2.8 2.8 (ft) 29 26 20(max. permitted) 29 0.5 0.5 WF100G (in) ( /â) 14.03 (191) 14.79 (181) 13.60 (197) 14.36 (186) 10.66 (251) 11.32 (236) 11.99 (223) 12.65 (212) 6.36 (421) 7.02 (381) 8.54 (313) 9.20 (291) â (in.) â2.46 â2.46 â2.84 â2.84 â3.51 â3.51 â2.32 â2.32 â7.61 â7.61 â7.61 â7.61 â (in.) 16.13 16.89 16.13 16.89 13.97 14.63 13.97 14.63 13.97 14.63 16.15 16.81 â (in.) 0.36 0.36 0.31 0.31 0.20 0.20 0.34 0.34 0.00 0.00 0.00 0.00 WF100G-MOD (in.) ( /â) 14.12 (190) 14.83 (180) 13.70 (195) 14.41 (186) 10.76 (249) 11.38 (235) 12.07 (222) 12.69 (211) 6.57 (408) 7.18 (372) 8.54 (314) 9.15 (292) â (in.) â2.40 â2.40 â2.77 â2.77 â3.43 â3.43 â2.27 â2.27 â7.44 â7.44 â7.44 â7.44 â (in.) 16.16 16.88 16.16 16.88 14.00 14.62 14.00 14.62 14.00 14.62 15.97 16.59 (in.) 0.36 0.36 0.31 0.31 0.45 0.20 0.20 0.34 0.00 0.00 0.00 0.00 Lifting Rigid extension of lift device above top of girder, 0.000 in. Lateral tolerance of lift device from centerline of girder, econn 0.250 in. Lateral wind force at lifting from bed, . = . 0.015 klf Lateral wind force at lifting in field, . = . 0.015 klf Seating on dunnage Plan dimension of the bearing parallel to the axis of the rotation, . width of bottom flange â 2 in. = 36 in. Height from roll center to bottom of girder, . 2.000 in. Height of roll center from bearing seat, â . 2.000 in. Bearing tolerance from CL girder to CL support, . 0.250 in. Bearing rotational stiffness, Eq. 0.38 Transverse seating tolerance from level, 0.005 ft/ft Lateral wind force, . 0.055 klf Transportation Bunking tolerance from CL girder to CL support, . 1.000 in. Hauling rig stiffness, . 82,000 kip-in./rad Superelevation, 0.020 ft/ft Turn radius for adverse cross slope, 120.00 ft Hauling rig velocity in turn, 10.000 mph Height from roll center to bottom of girder, . 12.000 in. Horizontal distance from roll axis to center of tire group, . 36.000 in. Height of roll center above roadway, â . 48.000 in. Lateral wind force, . 0.055 klf Single girder on bearings Plan dimension of the bearing parallel to the axis of the rotation, . width of bottom flange â 2 in. = 36 in. Height of bearing, â . 2.000 in. Height from roll center to bottom of girder, . = â . /2 1.000 in. Height of roll center from bearing seat, â . = . 1.000 in. Bearing tolerance from CL girder to CL support, . 0.250 in. Bearing rotational stiffness, . Eq. 0.38 Transverse seating tolerance from level, 0.005 ft/ft Lateral wind force, 0.015 klf TableÂ 2.26. WF100G geometry- and prestressing-related parameters for stability analysis. TableÂ 2.27. Other input parameters for stability analysis (PCI, 2015).
74 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders stages up to placing the girder on its bearings. It must be understood that many assumptions go into these analyses, and results are quite sensitive, in some cases, to these parameters. Regardless of the analysis, however, it is clear that this extra-long girder requires immediate shoring to resist rollover when placed on its bearings. When using larger strands, the stability factors of safety decrease marginally in some cases (TableÂ 2.28). This reduction is due primarily to the greater camber resulting from the lower center of gravity of the strand in each case (TableÂ 2.25). The stability can be improved by adjusting some of the analysis parameters such as crane hook locations (a, as was done in this analysis; see TableÂ 2.26), increasing bearing rotational stiffness (Kq), increasing hauling rig stiffness (Kq,trans), decreasing the hauling rig velocity (Vel,trans), or increasing the dimension of bearing (Wbrg, Lbrg) where possible. Other parameters listed in TableÂ 2.27 can also be varied to improve stability although it is not clear how practical each revision may be in the field. If revising these parameters does not improve stability sufficiently, the girder requires intermediate bracing, which is only practical once erected. 2.8.2 Evaluation of Cases Reported in Section 2.2 The objective of the study presented in Section 2.2 was to maximize girder spans for different cross-section shapes using 0.6-in. and 0.7-in. strands. Typically, the larger strand allowed longer lengths to be achieved due to the greater prestress force available using the same, already maxi- mized, strand pattern. Ball (2019) considered all AASHTO design limits to maximize the span lengths but did not verify girder stability limitations. The present study focused on the stability of those cross-section and span combinations with the greatest achievable increases in span length when replacing 0.6-in. with 0.7-in. strands. These cases potentially represent the most efficient use of 0.7-in. strands but also introduce the greatest potential impacts on stability. TableÂ 2.29 summarizes the cases selected. Only the longer 0.7-in. strand-reinforced girders were analyzed. Similar to the case study presented in the previous section, girder and analysis parameters were assembled or are assumed as previously described. Strand arrangement, geometric, and prestressing-related input parameters are shown in TableÂ 2.30 through TableÂ 2.32. Girder unit Acceptance criteria WF100G WF100G-MOD WF100G WF100G-MOD 0.6-in. strands 0.6-in. strands 0.7-in. strands 0.7-in. strands FScr FS FSroll FScr FS FSroll FScr FS FSroll FScr FS FSroll 1.0 1.5 1.5 1.0 1.5 1.5 1.0 1.5 1.5 1.0 1.5 1.5 Lift from bed 1.57 1.57 na 1.83 1.83 na 1.53 1.53 na 1.76 1.76 na On dunnage 1.43 2.63 1.79 1.65 2.87 1.93 1.18 2.63 1.78 1.42 2.86 1.93 Transportation 0.78 1.90 1.55 0.95 2.11 1.70 0.64 1.89 1.55 0.81 2.11 1.70 Lift in field 1.55 1.55 na 1.78 1.78 na 1.51 1.51 na 1.72 1.72 na Place on bearings 1.32 1.21 0.65 1.53 1.43 0.76 1.29 1.21 0.65 1.50 1.43 0.76 na = Not applicable. TableÂ 2.28. Summary of stability analysis of 223-ft-long WF100G girders. Case (ksi) Girder spacing (ft) (ft) with 0.6-in. strands (ft) with 0.7-in. strands Potential increase in span length using 0.7-in. strand WF100G 15 10 170 207 21.8% WF74G 18 10 150 181 20.7% BT-72 18 12 113 135 19.5% OHWF-72 10 8 164 185 12.8% FIB-96 18 8 207 223 12.6% NU-2000 18 6 196 220 12.2% TableÂ 2.29. Critical cases selected from Section 2.2 study.
Analytical Research Approaches and Findings 75Â Â weight is assumed to be 0.150 kips/ft3 in all cases. Other analysis parameters not indicated are the same as those given in TableÂ 2.27. The girder support location, a, was varied to maximize the calculated factors of safety. In all analyses, an initial assumption of a = 0.1Lg was made and the analyses revised until adequate (or maximum) factors of safety were achieved; the resulting values of a used are reported in TableÂ 2.32. Results of stability analyses are given in TableÂ 2.33 in terms of the three factors of safety pre- scribed by PCI (2015) and described in Section 1.8.1. In a few analyses (as noted in TableÂ 2.23), additional revisions to assumptions were necessary to achieve adequate factors of safety. A com- plete set of sample calculations for the NU-2000 case are provided in Appendix E. As seen in TableÂ 2.33, despite the long spans, adequate stability could be achieved with all cross sections. Sections having the lowest ratio Iy/IxâWF100G and FIB-96ârequired additional flange width, similar to that described in Section 220.127.116.11.1, to achieve stability at all stages. These girders also tended to require stiffer supports. For example, the WF100G with an 18-in.-wider top flange described in TableÂ 2.33 fails the cracking check for the transportation stage. The factor of safety against cracking (FScr) for this case becomes 1 by increasing the hauling rig stiffness from Kq,trans Ã² 82,000 kip â in./rad. to Kq,trans = 117,000 kip â in./rad. The question becomes, Is this required rig stiffness achievable? The BT-72 section has a thinner bottom flange width, bbot flange = 26Â in., than the other sections (all others are 38 or 40Â in. as shown in TableÂ 2.30). The smaller flange width results in a signifi- cantly lower bearing rotational stiffness, Kq.seat2 = 22,700 kip-in./rad (Eq. 2.20). When placed on dunnage, this stiffness is inadequate. Increasing the stiffness to 34,000 kip â in./rad is sufficient to mitigate this instability. It should not be surprising that the required factor of safety against rollover cannot be achieved when the long girders are placed on bearings. Such long girders require immediate installation of braces at their ends. The effect of increasing top flange width on girder stability is shown in FigureÂ 2.27 and summarized in TableÂ 2.34 in which the top flange width is increased 12Â in., 18Â in., and 24Â in. WF100G and WF74G 46 straight 0.7-in. strands 11 harped 0.7-in. strands cgs = 4.08 in. cgs at midspan = 8.5 in.; at the end = 91.5 in. BT-72 32 straight 0.7-in. strands 2 harped 0.7-in. strands 4 debonded 0.7-in. strands cgs = 3.875 in. cgs at midspan = 6 in.; at the end = 70 in. 2@5 ft, 2@10 ft OHWF-72 57 straight 0.7-in. strands 2 harped 0.7-in. strands 14 debonded 0.7-in. strands cgs = 7.6 in. cgs at midspan = 10 in.; at the end = 69.5 in. 4@5 ft, 2@10 ft, 2@15 ft, 4@25 ft, 2@30 ft FIB-96 66 straight 0.7-in. strands 5 harped 0.7-in. strands cgs = 5.91 in. cgs at midspan = 14.2 in.; at the end = 91.5 in. NU-2000 52 straight 0.7-in. strands 8 harped 0.7-in. strands cgs = 4.08 in. cgs at midspan = 11.00 in.; at the end = 70.7 in. WF100G WF74G BT-72 OHWF-72 FIB-96 NU-2000 (ft) 207 181 135 185 223 220 (in.) 49 49 42 49 48 48.25 (in.) 38.375 38 26 40 38 38.375 (in.2) 1,084 825 767 1,163 1,176 904 (in.4) 1,524,912 734,356 545,894 844,069 1,464,296 790,592 (in.4) 68,602 72,018 41,083 104,334 77,066 60,817 / 0.045 0.098 0.075 0.124 0.055 0.077 (in.4) 8552 6560 6178 11,414 11,043 7224 (in.) 48.3 35.6 36.6 35.8 42.8 35.7 (kip/ft) 1.128 0.859 0.799 1.212 1.278 0.942 Input Parameter TableÂ 2.30. Geometric properties used in stability analysis. TableÂ 2.31. Strand arrangement.
76 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders Girder Condition Lift from bed Dunnage Transport Lift in field In place Age Release Release >28 days >28 days >28 days Prestress losses 10% 10% 50% 50% 50% (ksi) 182 182 167 167 167 WF100G (207 ft) (ksi) 12.5 12.5 15 15 15 (kips) 3,050 3,050 2,799 2,799 2,799 , (in.) 4.06 4.06 4.06 4.06 4.06 . (in.) 1.3 2.6 3.6 2.6 2.6 (ft) 22 0.5 0.5 22 0.5 â (in.) 9.70 6.6 5.32 8.24 5.32 ( /â) 256 377 467 301 467 WF100G (207 ft) (+12 in. to top flange width) (ft) 19 6 4 19 0.5 â (in.) 8.78 7.02 5.47 7.43 4.89 ( /â) 283 354 454 334 508 WF100G (207 ft) (+18 in. to top flange width) (ft) 16 4 4 16 0.5 â (in.) 8.21 6.48 5.28 6.91 4.71 ( /â) 303 383 470 360 528 WF74G (ksi) 15 15 18 18 18 (kips) 3,050 3,050 2,799 2,799 2,799 , (in.) 4.06 4.06 4.06 4.06 4.06 . (in.) 1.1 2.3 3.3 2.3 2.3 (ft) 12 1 1 12 0.5 â (in.) 9.81 8.07 6.66 8.29 6.57 ( /â) 221 269 326 262 331 BT-72 (ksi) 15 15 18 18 18 (kips) 1,605 1,605 1,472 1,472 1,472 , (in.) 4.00 4.00 4.00 4.00 4.00 . (in.) 0.8 1.7 2.7 1.7 1.7 (ft) 4 1 1 4 0.5 â (in.) 3.54 3.27 2.71 2.96 2.66 ( /â) 458 495 599 548 609 OHWF-72 (ksi) 8.4 8.4 10 10 10 (kips) 2,408 2,408 2,209 2,209 2,209 , (in.) 7.68 7.68 7.68 7.68 7.68 . (in.) 1.2 2.3 3.3 2.3 2.3 (ft) 15 1 1 15 0.5 â (in.) 5.89 2.52 1.66 4.85 1.52 ( /â) 377 882 1,335 457 1,457 FIB-96 (ksi) 15 15 18 18 18 (kips) 3,799 3,799 3,486 3,486 3,486 , (in.) 6.49 6.49 6.49 6.49 6.49 . (in.) 1.4 2.8 3.8 2.8 2.8 (ft) 25 15 15 25 0.5 â (in.) 10.38 8.89 7.39 8.80 4.59 ( /â) 258 301 362 304 584 FIB-96 (+12 in. to top flange width) a (ft) 21 11 17 21 0.5 â (in.) 9.42 7.75 7.34 7.92 4.21 ( /â) 284 345 364 338 635 NU-2000 (ksi) 15 15 18 18 18 (kips) 3,210 3,210 2,946 2,946 2,946 , (in.) 5.00 5.00 5.00 5.00 5.00 . (in.) 1.4 2.8 3.8 2.8 2.8 (ft) 26 10 13 26 0.5 â (in.) 14.13 10.57 9.38 11.99 5.89 ( ) 187 250 281 220 448 TableÂ 2.32. Girder geometry- and prestressing-related parameters for stability analysis.
Analytical Research Approaches and Findings 77Â Â Girder Factor of safety Acceptance criteria Lift from beda Dunnage Transport Lift in field a Place on bearings WF100G (207 ft) 1 1.20 0.25 0.02 1.23 1.60 â 1.5 1.58 1.55 1.44 1.54 1.86 1.5 na 1.15 1.23 na 0.95 WF100G (207 ft) (+12 in. to top flange width) 1 1.44 0.58 0.31 1.38 1.90 â 1.5 1.63 2.15 1.85 1.59 2.15 1.5 na 1.53 1.53 na 1.06 WF100G (207 ft) (+18 in. to top flange width) 1 1.51 1.01 1.00 1.42 2.13 â 1.5 1.58 2.41 2.89 1.52 2.34 1.5 na 1.64 2.05 na 1.12 Changes to parameters to achieve â¥ 1.0 (shaded cells): = 87,000 kip-in./rad (+6%) for dunnage support . = 117,000 kip-in./rad (+43%) and . = 0.050 klf (â10%) for transportation WF74G 1 1.61 2.72 2.08 1.52 5.06 â 1.5 1.61 4.24 3.80 1.52 4.95 1.5 na 2.09 2.33 na 1.52 BT-72 1 1.59 1.00 1.82 1.48 2.37 â 1.5 1.59 3.66 7.35 1.51 2.76 1.5 na 1.80 3.26 na 1.14 Changes to parameters to achieve â¥ 1.0 (shaded cells): . = 34,000 kip-in./rad (+50%) for dunnage support OHWF-72 1 1.84 2.74 2.12 1.58 3.63 â 1.5 1.84 2.81 2.49 1.58 3.20 1.5 na 1.91 1.98 na 1.37 FIB-96 1 1.53 0.90 0.60 1.50 1.50 â 1.5 1.53 1.97 1.79 1.50 1.45 1.5 na 1.52 1.56 na 0.83 FIB-96 (+12 in. to top flange width) 1 1.63 1.25 1.01 1.55 1.77 â 1.5 1.63 1.97 2.28 1.55 1.69 1.5 na 1.52 1.89 na 0.93 Changes to parameters to achieve â¥ 1.0 (shaded cells): . = 88,000 kip-in./rad (+7%) for transportation NU-2000 1 1.54 1.34 1.00 1.51 2.27 â 1.5 1.54 2.34 2.37 1.51 2.08 1.5 na 1.52 1.82 na 0.93 aWind speed during lifts may need to be limited in some cases. na = Not applicable. TableÂ 2.33. Summary of stability analysis. (a) Effect of top flange width on FScr. (b) Effect of top flange width on FSroll. FigureÂ 2.27. Effect of Iy/Ix on stability factors of safety.
78 Use of 0.7-in. Diameter Strands in Precast Pretensioned Girders Added flange width WF100G BT-72 OHWF-72 / / / As built 0.045 1.60 0.95 0.075 2.37 1.14 0.124 3.73 1.37 +12 in. 0.059 1.90 1.06 0.105 2.63 1.19 0.165 4.12 1.47 +18 in. 0.072 2.13 1.12 0.129 2.79 1.21 0.192 4.32 1.52 +24 in. 0.083 2.29 1.17 0.158 2.93 1.23 0.224 4.51 1.56 FIB-96 NU-2000 WF100G (case study) / / / As built 0.053 1.50 0.83 0.076 2.27 0.93 0.045 1.29 0.65 +12 in. 0.069 1.77 0.93 0.098 2.61 1.04 0.059 1.50 0.76 +18 in. 0.080 1.93 0.99 0.112 2.82 1.10 0.072 1.71 0.83 +24 in. 0.092 2.09 1.04 0.129 3.06 1.16 0.083 1.87 0.89 TableÂ 2.34. Effects of Iy/Ix on stability. As Iy/Ix increases, the factors of safety against cracking (FScr; FigureÂ 2.27a) and rollover (FSroll; FigureÂ 2.27b) improve when the girder is placed on its final supports. In all cases, the effect of increasing top flange width is greater on FScr than on FSroll. The limited beneficial effect on the BT-72 girder reflects the more dominant contribution of this shapeâs smaller bottom flange on stability. Increasing the ratio Iy/Ix has considerable effect although typically not enough to achieve an adequate safety against rollover. The rollover must be mitigated by the immediate installation of braces in any event. 2.9 Summary of Analytical Studies From the results presented in this chapter, the following general conclusions and observa- tions are made: 1. The design approach for the parametric design case study was validated by the finite element model. Reinforcement requirements in the girder end regions were shown to be significant but constructible in every case. 2. Design case studies show that a one-to-one replacement of 0.6-in. strands by 0.7-in. strands is not possible due to the many other design constraints such as stress limits at release. 3. When not permitted to harp strands (e.g., Texas U girders) or when the current AASHTO LRFD limits on debonding are applied to 0.7-in. strands, the full benefits of using the larger-diameter strands are not fully realized. 4. The span length of girders shapes optimized for 0.6-in. strands is not appreciably increased by using 0.7-in. strands. Less efficient shapes benefit more from using larger-diameter strands. 5. The percentage of increase in achievable span length changes as a function of the girder depth; the deeper the girder, the larger the proportional span length increase that can be realized. 6. For a similar number of 0.6-in. and 0.7-in. strands, the use of 0.7-in. strands tends to require greater debonding, which, assuming a favorable strand pattern is adopted, has the effect of reducing the tie force. Therefore, in terms of required tension tie reinforcement, there is little difference between cases with 0.6-in. and 0.7-in. strands. 7. The increased prestress force from the use of 0.7-in. strands results in greater splitting forces, leading to potentially more congested reinforcing steel requirements at the beam ends. Nevertheless, the required reinforcement was easily met for all the cases considered with no anticipated constructability issues.
Analytical Research Approaches and Findings 79Â Â 8. Peeling stresses are not unique to 0.7-in. strands. Shapes having wide flat flanges exhibit large, predicted peeling stresses, and sections having stockier flanges exhibit lower stresses overall. Peeling stresses can be mitigated by debonding strands in the recommended pattern of âfrom the outside-inâ or by releasing/cutting strands in a uniform manner, such as top-down or gang release. It should be noted that it is unlikely that 0.6-in. strands could be replaced strand-for-strand with 0.7-in. strands; hence, the realistic peeling stresses generated due to the use of 0.7-in. strands would be smaller or comparable to those for 0.6-in. strands. 9. In comparison to 0.6-in. strands, the circumferential stresses around strands are marginally lower for 0.7-in. strands. Effects of interaction between adjacent strands were shown to be little different between 0.6-in. and 0.7-in. strands. 10. The finite element models suggest that cracking extends beyond the h/4 distance over which the concentrated reinforcement is provided. This result supports providing the required splitting reinforcement over a longer length as is permitted by some states. 11. The use of 0.7-in. strands, which may result in longer spans, will increase the susceptibility of girders to instabilities. As is required for much shorter girders than those considered here, end braces must be installed immediately upon placement on bearings to provide safety against rollover. For other conditions, the following typically practical measures improve safety against stability effects: a. Refining hanging (lift points) and dunnage support locations (parameter a) can optimize resistance to stability effects. The value of a is practically limited during the transportation based on vehicle geometry and routes chosen. b. Increasing the width of the top flange of a girder, thereby increasing Iy/Ix, has a pro- nounced effect on improving stability. c. Providing stiffer transportation or dunnage support (parameter Kq)âassuming this is possibleâimproves stability. d. Girders having relatively thin bottom flanges (BT sections in this study) are more susceptible to rollover while supported on dunnage or in transportation; such girders are not well suited for long spans.