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30 3.1 Introduction The parameters that influence the behavior and design of CFRP prestressed beams were inves- tigated in experimental and analytical programs. The experimental program included testing of material samples, small-scale beams and prism, and full-scale beams to validate the existing models and to develop design guide specifications. The factors investigated were prestress relax- ation loss, creep and shrinkage loss, thermal fluctuation loss, harping characteristics of prestress- ing CFRP, type of prestressing materials, use of prestressing CFRP for internal post-tensioning applications, transfer and development length, and long-term deflection behavior of CFRP pre- stressed beams. The analytical investigations were conducted to evaluate the appropriateness of the proposed design and material guide specifications. It included probabilistic investigations to determine strength reduction factors and numerical simulations using the finite element method to broaden the range of parameters, and to support the results of the experimental program. 3.2 Results of Experimental Investigation 3.2.1 Full-scale Beam Testing The experimental program included design, construction, testing, and analysis of 12 full-scale, AASHTO Type I CFRP prestressed concrete beams. Two types of CFRP prestressing tendons were included: 0.5 in. diameter solid circular CFRP bars, and 0.6 in. and 0.76 in. diameter CFRP cables that consisted of seven individual wires twisted together into a single strand (similar to steel prestressing strand). Five beams were post-tensioned and seven beams were pretensioned. Concrete strength at the time of prestress transfer was between 5.5 and 6 ksi. Two flexural loading conditions were considered: monotonic and cyclic fatigue. The test matrix for the flexure tests is provided in Table 3.1 (details on the construction of the beams are provided in Appendix C). Figures 3.1 and 3.2 show schematics for pretensioned and post-tensioned test beams, respec- tively. The Beam ID presented in Table 3.1 starts with the type of prestressing tendon (C for CFRP cable and B for CFRP bar), followed by the type of prestressing (Pr for pretensioned, Po for bonded post-tensioned, and Pou for unbonded post-tensioned beams), profile of the prestressing tendon (S for straight and D for draped tendons), followed by the type of loading (M for static monotonic loading and F for cyclic fatigue loading), and then concluded by the repetition number if more than one test was conducted. For example, Beam CPoDM#01 is prestressed with CFRP cables (C) in a bonded post-tensioned configuration (Po) with draped (D) cables, subjected to static monotonic loading (M) and is the first repetition among multiple similar beams (#01). The post-tensioned beams were constructed either with straight or a combination of straight and parabolic draped prestressing CFRP cables with 0.76 in. diameter. Two beams were grouted C H A P T E R 3 Research Results
Research Results 31 Type of Prestressing CFRP Type of Prestressing Prestressing CFRP Profile Type of Loading Number of Beams Beam ID CFRP Cable (C) Pretension (Pr) Straight (S) Monotonic (M) 2 CPrSM(#) Flexural fatigue (F) 1 CPrSF Post-tension (Po) Draped (D) Monotonic (M) 2 CPoDM(#) Post-tension, unbonded (Pou) Straight (S) Flexural fatigue (F) 1 CPouSF Monotonic (M) 1 CPouSM Draped (D) Flexural fatigue (F) 1 CPouDF CFRP Bar (B) Pretension (Pr) Straight (S) Monotonic (M) 2 BPrSM(#) Flexural fatigue (F) 1 BPrSF Pretension, partially debonded (Prp) Straight (S) Monotonic (M) 1 BPrpSM Table 3.1. Test matrix for full-scale beams. (a) cross-sectional dimensions (b) elevation and reinforcement details Figure 3.1. Schematic drawing of typical full-scale pretensioned beams with straight cables and bars. (a) cross-sectional dimensions (b) elevation and reinforcement details Figure 3.2. Schematic drawing of typical full-scale post-tensioned beams with straight and draped cables.
32 Design of Concrete Bridge Beams Prestressed with CFRP Systems to investigate the fully bonded case and the other three beams were unbonded. Unbonded post- tensioning was only used for beams with CFRP cables. Eight prestressed CFRP beams were tested monotonically to failure, and four beams were tested under flexural fatigue loading up to 2.3 million cycles followed by monotonic loading to failure. All beams were tested under four-point loading using a servo-hydraulic actuator and a spreader beam as illustrated in Figure 3.3. The beams were simply supported and instrumented to monitor the behavior under different types of loading. Load cells, LVDTs, string potentiometers, and strain gages were used in different configurations to measure the applied load, deforma- tions, crack widths, and strains, respectively. Non-contact measurement systems were also used to monitor the local and/or global behavior of the beams. Concrete strength of the beams at the time of testing ranged from 9 to 12 ksi. Load-Deflection Behavior Figure 3.4 shows the load-deflection relationships for all of the CFRP pretensioned beams. As shown, the beams tested under monotonic loading failed at comparable loads. As designed, the Figure 3.3. Test setup for flexure tests. BPrpSM BPrSM#02 BPrSF CPrSF CPrSM#0 CPrSM#02BPrSM#01 Figure 3.4. Load versus deflection of pretensioned beams.
Research Results 33 combined tensile capacity of the reinforcement (the number of cables multiplied by the rupture load of one cable) was similar regardless of the type of prestressing CFRP. However, the beams prestressed with CFRP bars had slightly higher stiffness than those prestressed with CFRP cables because of the difference in their material properties. Beam CPrSM#2 had a higher cracking load than Beam CPrSM#1 because of its higher effective prestressing force and the concrete strength on the day of testing. Although both beams failed at similar ultimate loads (â¼5 kips difference), the deflection of Beam CPrSM#2 was slightly lower than that of Beam CPrSM#1. The lower effective prestressing force (by 11%) resulted in a lower cracking load (by 15%). The cracking load for Beam BPrSM#1 was 12% lower than that for Beam BPrSM#2 because of the 8% lower effective prestressing force. Both beams failed at similar ultimate loads (â¼1 kip difference) with Beam BPrSM#1 having higher deflection than BPrSM#2. The load-deflection curves obtained from the monotonic tests of the five post-tensioned beams are presented in Figure 3.5. The three unbonded post-tensioned beams had compara- ble capacities. The load capacity of Beam CPouSF was 10% less than that of the similar Beam CPouSM that was tested under static monotonic loading. The load capacity of Beam CPouDF with draped cables was higher than that for the comparable Beam CPouSF with straight cables because of its higher effective prestressing force. The three unbonded CFRP post-tensioned beams failed due to concrete crushing. Once the ultimate load was reached, the beams showed a 15% drop in capacity but the load was maintained until the rupture of the prestressing tendon. The effect of bond condition can be seen in Figure 3.5 by comparing the beams with bonded cables (CPoDM#01 and CPoDM#02) to the corresponding beam with unbonded cables (CPouDF). The beams with bonded cables had a 22% higher capacity. The influence of the effective prestressing force can be observed by comparing the two bonded post-tensioned beams (CPoDM#01 and CPoDM#02). The beam with a higher effective prestressing force (CPoDM#1) had a higher cracking load and less deflection than Beam CPoDM#02 but both beams had similar peak loads. Table 3.2 summarizes the results of the full-scale beam tests, including the ultimate load and deflection, and the cracking load and concrete strength (on the day of testing) for the girder and deck (detailed discussion of the results is provided in Appendix D). Fatigue Behavior Four beams prestressed with CFRP systems (CPrSF, BPrSF, CPouSF, and CPouDF) were subjected to constant-amplitude fatigue loading. Two beams were pretensioned; one with CFRP cables (CPrSF) and one with CFRP bars (BPrSF). The other two beams were post-tensioned Figure 3.5. Load versus deflection of post-tensioned beams.
34 Design of Concrete Bridge Beams Prestressed with CFRP Systems with unbonded CFRP cables: one with a straight cable (CPouSF) and one with both straight and draped cables (CPouDF). All beams were subjected to 2.3 million load cycles (Â±1,000 cycles) at a frequency of 1 Hz. Before the application of cyclic fatigue loading, the beams were cracked to simulate accidental overloading. For the pretensioned beams, the fatigue load cycle varied from 9.5% to 34% of the ultimate capacity. The upper limit of fatigue loading is the loading that induces the tensile limit of 6âf â²c psi in the extreme concrete bottom fiber under service loading conditions (AASHTO LRFD, 2017). The gross moment of inertia was used for the calculation. The lower limit was computed by subtracting the fatigue truck moment [calculated according to AASHTO LRFD (2017) Article 184.108.40.206 considering a girder distribution factor of one] from the upper limit. For the post-tensioned beams, this range was between 10% to 53% of the ultimate capacity. The upper limit of fatigue loading for post-tensioned beams was chosen as the cracking load. The lower limit was computed in a manner similar to that for pretensioned beams (details are presented in Appendix C). The repeated loading was paused during the test to conduct monotonic tests to monitor the changes in stiffness, prestressing force, and crack width. None of the beams exhibited any indication of failure during the fatigue cycles and were subsequently loaded monotonically to failure. Figure 3.6 shows the effect of repeated loading on the stiffness of the beams. As shown, the repeated loading had little effect on the stiffness (less than 1.5% for pretensioned beams and less than 3.5% for post-tensioned beams). After completion of each fatigue test, the beams were subjected to static monotonic loading up to failure. Table 3.2 shows that the ultimate loads determined from the tests conducted after the fatigue loading were close to those obtained for the corresponding monotonically loaded beams (within 1% and 10% for pretensioned and post-tensioned beams, respectively). This variation may be attributed to differences in concrete strengths of the decks and girders of the different beams at the time of testing. Crack Distribution, Spacing, and Crack Width The monotonic tests of the full-scale beams were conducted in increments to allow monitor- ing of crack propagation and measuring crack widths. The cracks in pretensioned and bonded post-tensioned beams were distributed over up to the quarter-span of the beam on either side Beam ID Concrete Strength (ksi) Cracking Load (kips) Ultimate Load (kips) Deflection at Ultimate Load (in.) Failure Mode Girder Deck CPrSM#01 11.2 4.2 75 206 8.0 CFRP rupture CPrSM#02 12.2 11.7 88 214 7.6 CPrSF 10.9 10.2 76 210 7.7 BPrSM#01 9.4 10.4 76 207 6.0 BPrSM#02 11.0 8.9 86 209 5.8 BPrSF 9.4 10.1 84 207 5.6 BPrpSM 10.8 8.3 85 209 5.2 CPouSM 10.4 9.7 61 135 9.9 Concrete crushing CPouSF 10.9 8.2 65 122 7.4 CPouDF 10.9 9.8 72 143 8.9 CPoDM#01 10.9 11.5 81 175 5.2 CFRP ruptureCPoDM#02 63 174 6.7 Table 3.2. Results of full-scale beam tests.
Research Results 35 of the midspan. However, the cracks in the unbonded post-tensioned beams were concentrated in a few wide cracks near the constant moment region; these forked as the load was increased. The maximum crack widths in the pretensioned and the bonded post-tensioned beams were 0.06 in. and 0.12 in, respectively, and 0.83 in. in the unbonded post-tensioned beams (detailed information is provided in Appendix D). 3.2.2 Prestress Losses Prestress Relaxation Stress relaxation of prestressing CFRP materials was examined over 1 year to assess the effects of the type of the prestressing CFRP system (CFRP cable and bar), initial stress level [0.5, 0.6, and 0.7 times the design tensile strength of prestressing CFRP ( fpu)], and the length of the speci- mens (10 ft., 15 ft., and 20 ft.). The anchorage loss was quantified using specimens with 1 in. of prestressing CFRP between two end anchors to obtain stress relaxation losses of the prestressing CFRP cables and bars. Socket type anchors (threaded steel pipes) with highly expansive materials (HEM) were used to provide uniform expansive pressure inside the anchors to enhance the bond and reduce the slippage between the prestressing CFRP and steel socket. The test specimen is illustrated schematically in Figure 3.7. First the prestressing CFRP cables or bars were cut to the desired length. Then, the pre- stressing CFRP was centered and fixed inside of the socket type anchors. The HEM was poured into the sockets in a vertical position and cured for 24 hours in the laboratory. For prestressing CFRP cables, the steel sockets and HEM were provided by the manufacturers of the CFRP cables. However, for prestressing CFRP bars, the anchorage system was produced by the research team. Further information on specimen preparation and test procedure are provided in Appendix C. Figure 3.6. Effect of repeated loading on beam stiffness. Clear length [10, 15, and 20 ft.] 0.5 f pu 0.6 f pu 0.7 f pu Socket type anchor HEMPrestressing CFRP cable or bar Figure 3.7. Schematic representations of relaxation test specimens.
36 Design of Concrete Bridge Beams Prestressed with CFRP Systems After preparation of the anchorage systems, the specimen was inserted inside a steel hollow structural section. During the test, the prestressing CFRP cable or bar was subjected to a sustained load under a constant strain condition in a self-reacting configuration as shown schematically in Figure 3.8. Figure 3.9 shows the relaxation test setup for specimens with various lengths. Figure 3.10 shows the stress ratios versus time (including long-term anchorage losses) for all 15 ft. long prestressing CFRP cables and bars for three levels of initial prestressing (0.5, 0.6, and 0.7 fpu). The figure shows similar relaxation loss for all specimens of the same type (cables or bars), length, and level of initial prestressing. A linear relationship between stress ratio and the logarithm of the time was observed for all the specimens; higher initial prestressing levels resulted in higher relaxation losses. Test results also showed that the stress relaxation loss of prestressing CFRP cables and bars with 0.6 fpu initial prestressing and different lengths was independent of the length of the speci- mens (details of the tests and results are presented in Appendices C and D). Anchorage losses (attributed to the gradual slip of the prestressing CFRP and expansive material inside the steel anchors, or socket, and creep of the expansive material) were determined for three anchorage specimens of each of the CFRP prestressing cables and bars for an initial prestressing level of 0.6 fpu. These losses were subtracted from the total prestress relaxation losses of longer specimens (10, 15, and 20 ft.) to determine losses related to prestressing CFRP. Figure 3.11 shows the average stress ratio versus time for both prestressing CFRP cables and bars for three tests (anchorage seating losses that occur during and immediately after stressing are not included). Prestressing CFRP strand Steel HSS reaction frame Load cell 10, 15, or 20 ft. Strain gageDead end Jacking end Threaded anchor with nut Figure 3.8. Schematic illustration of test setup. 10 ft. 15 ft. 20 ft. Figure 3.9. Relaxation test setup for specimens with various lengths.
Research Results 37 The test results were used to develop expressions to calculate the stress relaxation of prestress- ing CFRP cable and bar systems (DfpR). Equations 3.1 and 3.2 were developed for post-tensioning applications in which socket type anchors were permanently installed as part of the system. For CFRP Cables (Ã = 0.6 in.): 0.020 0.0066 log 24 (Eq. 3.1)( )D = ï£« ï£ ï£¬ ï£¶ ï£¸ ï£· â ï£« ï£ ï£¬ ï£¶ ï£¸ ï£·f f f t fpR pt pu pu For CFRP Bars (Ã = 0.5 in.): 0.016 0.0057 log 24 (Eq. 3.2)( )D = ï£« ï£ ï£¬ ï£¶ ï£¸ ï£· â ï£« ï£ ï£¬ ï£¶ ï£¸ ï£·f f f t fpR pt pu pu where fpt is the stress in prestressing CFRP immediately after transfer (ksi), fpu is the design tensile strength of prestressing CFRP (ksi), and t is the time after prestress transfer (days). (a) Prestressing CFRP Cable Initial prestressing ~0.7 fpu Initial prestressing ~0.6 fpu Initial prestressing ~0.5 fpu (b) Prestressing CFRP bar Three test repetitions Figure 3.10. Stress ratio versus time for 15 ft. long prestressing CFRP at different initial prestressing level. Figure 3.11. Average stress ratios of the anchorage systems versus time for prestressing CFRP cables and bars.
38 Design of Concrete Bridge Beams Prestressed with CFRP Systems The anchorage losses (presented in Figure 3.11) were subtracted from the total stress relaxation of prestressing CFRP systems to obtain the stress relaxation of the prestressing CFRP cables and bars. Equations 3.3 and 3.4 were developed for pretentioning applications in which the anchors were not permanently installed as part of the system (e.g., for precast, prestressed concrete beams). For CFRP cables (Ã = 0.6 in.): 0.019 0.0066 log 24 (Eq. 3.3)( )D = ï£« ï£ ï£¬ ï£¶ ï£¸ ï£· â ï£« ï£ ï£¬ ï£¶ ï£¸ ï£·f f f t fpR pt pu pu For CFRP bars (Ã = 0.5 in.): 0.013 0.0057 log 24 (Eq. 3.4)( )D = ï£« ï£ ï£¬ ï£¶ ï£¸ ï£· â ï£« ï£ ï£¬ ï£¶ ï£¸ ï£·f f f t fpR pt pu pu The stress relaxation losses versus time for prestressing CFRP cables and bars stressed initially to 0.60 fpu computed according to these equations are presented in Figure 3.12. The figure also shows the relaxation losses predicted according to the models proposed by Saadatmanesh and Tannous (1999) and Enomoto et al. (1990) for prestressing CFRP cables with initial prestressing of 0.6 and 0.7 fpu as well as those predicted by the AASHTO LRFD (2014) formulation for stress relieved and low relaxation steel strands. As shown, the stress relaxation loss of prestressing CFRP bars is less than that of FRP cables due to the differences in fiber content, fiber alignment, and fiber-matrix interface characteristics. The unwinding of the twisted cables may have also contributed to this difference. Concrete Creep, Shrinkage, and Thermal Fluctuation Concrete creep and shrinkage measurements were made on 6 in. Ã 6 in., 10 ft. long concrete prisms to investigate the effect of prestressing CFRP types (cables and bars) and jacking stress Figure 3.12. Stress relaxation loss versus time for CFRP cables and bars and prestressing steel.
Research Results 39 levels (0.5, 0.6, and 0.7 times the design tensile strength of the prestressing CFRP, fpu). The influence of confinement due to transverse reinforcement on transfer length and the thermally induced prestressing losses was also studied. Twenty-seven concrete beams pretensioned with each type of prestressing CFRP tendon (cables and bars) were fabricated using a self-consolidating concrete mixture. Each prism was pretensioned with a single prestressing CFRP tendon placed at the center of the cross section. The prisms were instrumented with strain gages and thermocouples bonded to the prestressing tendon to monitor the change in strain and temperature. Simultaneously, twenty 4 Ã 4 Ã 12 in. plain concrete prisms were cast using the same batch of concrete to measure the concrete shrink- age strain; these shrinkage strains were subtracted from the combined creep and shrinkage losses measured for the CFRP prestressed prisms to quantify the creep losses. The average concrete compressive strength at transfer and at 28 days were 5.4 and 12 ksi, respectively. The strain profile of the prestressed concrete was obtained from 16 demountable mechanical strain gage points (DEMEC target points) attached to the concrete surface of each creep prism at 8 in. spacing, as shown in Figure 3.13. The concrete strain was calculated from the difference between the measurements before and after the transfer of prestressing force. Also, two sets DEMEC target points were also attached to each shrinkage specimen at 8 in. spacing to measure the shrinkage strains. After prestress transfer, the specimens were stored for 1 year under laboratory conditions, and concrete strain measurements were taken periodically to determine concrete creep and shrinkage. The concrete longitudinal compressive strains increased with time due to prestress- ing losses caused by concrete creep and shrinkage. Details of specimen fabrication and test pro- cedures are provided in Appendix C. Figure 3.14 shows the average concrete total creep and shrinkage strains (obtained from the constant strain zone of the CFRP prestressed prisms) versus time for CFRP prestressed prisms DEMEC target point Transfer zone Constant strain zone Figure 3.13. Prism specimen instrumented with DEMEC target points. Concrete total strain Concrete shrinkage strain 200 (a) Prestressing CFRP cable (b) Prestressing CFRP bar St ra in (i n. /in .) St ra in (i n. /in .) Figure 3.14. Concrete total strain (creep and shrinkage) and shrinkage strain versus time in CFRP prestressed prisms.
40 Design of Concrete Bridge Beams Prestressed with CFRP Systems at three prestressing levels. As shown, high concrete creep and shrinkage rates occurred during the first 100 days after prestress transfer; creep and shrinkage strains became constant as time passed. Details of the experimental results are provided in Appendix D. The creep and shrinkage strains predicted according to AASHTO LRFD (2017) and those measured at different times for beams prestressed at three levels of initial prestressing with CFRP cables and bars are shown in Figure 3.15. As shown, the measured concrete creep and shrinkage strains were within 5% of the predicted values. However, the total measured con- crete creep and shrinkage strains at the initial stage (after 3 days of prestress transfer and before 3 days of prestress removal) were higher than the predicted values because of the high rate of drying shrinkage due to diffusion of the concrete moisture immediately after removing the forms. The R2 values for the regression analysis lines were 0.845 and 0.918 for the prestress- ing CFRP cables and bars, respectively, indicating a good agreement. Thus, the method pre- sented in AASHTO LRFD (2017) can be used to estimate the prestress losses due to concrete creep and shrinkage. To study thermal effects on the CFRP prestressed prisms, the prisms were subjected to thermal cycles after completion of the creep and shrinkage tests. The prisms were placed in an environment-controlled chamber and subjected to 30 temperature cycles. Each cycle consisted of 3 hours of heating to 140Â°F, 3 hours at 140Â°F, 3 hours of cooling down to 0Â°F, and 3 hours at 0Â°F. All prisms were instrumented with thermocouples to monitor the internal temperature of concrete during the thermal cycles; the temperature and strain of the prestressed CFRP cables and bars were monitored to determine the thermally induced prestress loss (or gain). Figure 3.16 shows the average strain profile for beams prestressed with CFRP cables with 0.6 fpu initial prestressing, with and without spiral reinforcement. More details are provided in Appendix D. The transfer length for the prisms prestressed with CFRP cables and bars increased after the thermal fluctuation cycles by 8 to 16 in. and 16 to 24 in., respectively. This increase is attributed to the bond deterioration and formation of tensile cracks at the interface between prestressing CFRP and concrete due to the higher transverse CTE of the prestressing CFRP relative to that of the concrete. The concrete strain within the constant strain zones did not change due to cyclic thermal loading indicating that the effective prestressing force was not affected by the 30 thermal cycles. However, the concrete compressive strains for the prisms prestressed with CFRP bars at initial (a) Prestressing CFRP Cable (b) Prestressing CFRP Bar Figure 3.15. Average measured creep and shrinkage strains versus total predicted strain for CFRP prestressed prisms.
Research Results 41 prestressing levels of 0.6 fpu and 0.7 fpu decreased by 0.0002 and 0.0003 in./in., respectively. There was also a reduction in the longitudinal strain of the prestressing CFRP bars inside the concrete prisms with initial prestressing of 0.6 fpu and 0.7 fpu indicating an average loss of the prestressing force of 30% to 40% of the jacking stress in prisms prestressed with CFRP bars. Also, the observed change in the length of the prestressing CFRP bar protruding beyond the end of the prism indi- cated slippage of the CFRP bars inside the concrete beams at the end zone. More information is provided in Appendix D. By assuming total bond between the prestressing CFRP and concrete, an increase of the tem- perature will result in compressive stresses in concrete and tensile stresses in the prestressing CFRP (prestressing gain) because of the lower longitudinal CTE of prestressing CFRP com- pared to that of the concrete, and a reduction in temperature will result in tensile stresses in the concrete and compressive stresses in the prestressing CFRP that would lead to a prestress- ing loss. Concrete longitudinal strain was measured every 10 cycles at 14Â°F, 68Â°F, 104Â°F, and 140Â°F for both CFRP prestressed prisms and plain concrete prisms to determine the longitudi- nal elongation/contraction due to temperature change. Figure 3.17 shows the average change of strain versus temperature change, DT, for the plain concrete prisms and those prestressed with CFRP cables. The longitudinal coefficient of thermal expansion for the CFRP prestressed (a) With spiral reinforcement (b) Without spiral reinforcement 16 in.16 in. 8 in.8 in. Transfer length increase Figure 3.16. Average strain profiles of test prisms (CFRP cables with 0.6 fpu initial prestressing). (a) Plain concrete prisms (b) CFRP prestressed beams â = . Ã Ã â =0.983 â = . Ã Ã â =0.982 Figure 3.17. Change in concrete strain due to temperature change.
42 Design of Concrete Bridge Beams Prestressed with CFRP Systems prisms and plain concrete specimens were 6.64 Ã 10-6 and 6.80 Ã 10-6 (/Â°F), respectively. The lower CTE of the prestressing CFRP compared to that of plain concrete resulted in less thermal expansion for the prestressed prisms. The forces induced in the prestressing CFRP cables inside the concrete prisms (Fcfrp) and in the prestressed concrete (Fc) due to a temperature change (DT) are as follows: F T E Acfrp cm cfrp cfrp cfrp( )= Î± âÎ± D (Eq. 3.5) F T E Ac cm c c c( )= Î± âÎ± D (Eq. 3.6) where Î±cm, Î±cfrp, and Î±c are the longitudinal coefficients of thermal expansion of the prism with prestressing CFRP cables (composite), CFRP cables, and plain concrete, respectively. For equilibrium: F Fc cfrp+ = 0 (Eq. 3.7) From Equations 3.5 through 3.7, the longitudinal coefficients of thermal expansion of the composite prisms can be expressed as follows: E A E A E A E A E A E A cm cfrp cfrp cfrp c c cfrp cfrp c c c c c cfrp cfrp Î± = Î± Ã + +Î± Ã + (Eq. 3.8) where Ecfrp and Ec are the modulus of elasticity of the prestressing CFRP tendon and concrete (ksi), respectively, and Acfrp and Ac are the cross-sectional areas of the prestressing CFRP tendon and concrete (in2), respectively. Friction Losses According to AASHTO LRFD (2017), the prestress loss due to friction (DfpF) can be obtained from the following equation: f f epF pj x( )D = â ( )â ÂµÎ±+k1 (Eq. 3.9) where Âµ is the coefficient of friction, k is the wobble friction coefficient per unit length of tendon (1/ft.), fpj is the jacking stress (ksi), Î± is the total angular change between the jacking point and dead end (rad.), and x is the total length of prestressing CFRP from the jacking end to dead end (ft.). Friction tests were conducted on two full-scale post-tensioned beams to quantify the wobble coefficient, k, and the friction coefficient, Âµ, of prestressing CFRP cables in polypropylene ducts. Polypropylene ducts with an inner diameter of 2.0 in. were used in the fabrication of the post- tensioned beams with 0.76 in. diameter prestressing CFRP cables. Oversized ducts were used to accommodate the socket anchors that were pre-installed onto the CFRP cables. The friction tests were conducted on straight prestressing CFRP (Î± = 0). Two load cells were used to monitor the prestressing force at both jacking and dead ends of the prestressing CFRP during jacking. After determining the wobble coefficient of friction, the friction tests were con- ducted on three draped prestressing CFRP cables. The total angular change, Î±, of the draped prestressing CFRP cable between the jacking side and dead end was 0.17 rad. Test results showed average values of the wobble friction coefficient and coefficient of friction of 0.00022/ft. and 0.19, respectively.
Research Results 43 3.2.3 Harping Characteristics of the Prestressing CFRP The hold-down device geometry, harping angle, and type of prestressing CFRP were con- sidered in this study. The radius of curvature of the prestressing CFRP was varied by changing the diameter of the hold-down point (deviators) and the harping angle. Three harping angles that represent the range of commonly used harping angles were considered. Because the ten- sile capacity of prestressing CFRP bars drastically decreased for harping angles higher than 15 degrees (Quayle, 2005; Soudki and Noel, 2010), the harping angles for the prestressing CFRP bars were limited to 15 degrees. Details of the test setup and specimen configuration are shown schematically in Figure 3.18. The test frame was designed for testing different types of prestress- ing CFRP tendons with different harping angles and different types of hold-down devices. The load was applied using a hydraulic jack; the harping angle of the prestressing CFRP was varied by raising or lowering the dead-end anchor of the specimen until the desired harping angle was achieved. A wedge and sleeve type anchorage system was used. The feasibility of using 1 in. diameter steel and 2 in. diameter plastic deviators, as commonly used for steel strands, was examined. The initial test results showed higher retention of tensile capacity in prestressing CFRP cables than in the CFRP bars for all harping configurations, but both were considered unsatisfactory for CFRP prestressing applications. Premature failure of prestressing CFRP tendons occurred due to the highly concentrated contact pressure at the location of the 1 and 2 in. diameter deviators and resulted in large reductions (up to 93%) of the tensile capacity, with more reduction in the prestressing CFRP bars than in the cables. A 20 in. diameter deviator was then used to reduce the contact pressure and increase the tensile capac- ity, but premature failure of the prestressing CFRP bars occurred at stresses less than 60% of the design tensile capacity of the material (see Figure 3.19), suggesting that harped prestressing CFRP bars should not be used in prestressed bridge girders. Further testing was carried out on CFRP cables. A summary of the results is presented in Table 3.3. Two harping devices were fabricated (see Figure 3.20) to increase the contact surface between the CFRP cables and the deviator and decrease the curvature of the depressed strand to avoid local failure of the CFRP at the deviator location. Test results showed that harping devices using 20 in. and 40 in. diameter deviators caused no local failure of the prestressing CFRP cable and increased the harping capacity to 93.6% and 113.0% of the design tensile capacity of the CFRP cables, respectively (compared to 29.8% and 55.7% for 1 and 2 in. diameter deviators, respectively, as shown in Table 3.3). Test results are presented in Table 3.4. Test results showed higher retention of tensile capacity in harped prestressing CFRP cables than bars for all test configurations. In addition to the lower retention capacity, CFRP bars exhibited premature failure in the form of splitting cracks at the location of deviators suggesting Dead EndJacking End Anchor Wedge and sleeve grips Load cell Strain gage location Alignment wedge Harping angle Deviator Hydraulic jack Back to back steel channels Figure 3.18. Schematic representation of test specimen and test setup.
44 Design of Concrete Bridge Beams Prestressed with CFRP Systems Premature failure initiated by splitting 20 in. diameter Harping device Figure 3.19. Application of 20 in. diameter deviator on CFRP bars with 5 degree harping angle. Prestressing Tendon Deviator Diameter (in.) Harping Angle (Degree) Breaking Load (kips) % of Design Tensile Strength* CFRP Bar Ã = 0.50 in. 1.0 15 3.3 6.7 2.0 5 15.6 31.2 20.0 5 22.1 44.2 CFRP Cable Ã = 0.60 in. 1.0 15 18.2 29.8 2.0 5 33.9 55.7 20.0 5 42.1 69.2 *Based on manufacturer provided values. Table 3.3. Summary of harping test results. (b) 40 in. diameter deviator(a) 20 in. diameter deviator 2.5 in. Figure 3.20. Newly designed harping devices.
Research Results 45 a need to limit their application in prestressed concrete beams. However, the harping devices used in the tests provided a tensile capacity retention of more than 92% of the design tensile strength of prestressing CFRP cables for harping angles between 10Â° and 20Â°. 3.2.4 Transfer Length To evaluate the transfer lengths of CFRP cables and bars under various jacking stresses, con- crete strains at both ends (live and dead) were measured for five full-scale pretensioned bridge girders and 54 pretensioned prisms. The transfer length was estimated from the concrete strain profiles using the 95% Average Maximum Strain Method (AMS). In this method, the strain profiles along the length of the beams and prisms were plotted; AMS was determined by com- puting the average of all the strains within the constant strain zone. The transfer length was then determined as the intersection of the 95% AMS line with the strain profile at each end of the test specimens, as shown in Figure 3.21 for the full-scale beams pretensioned with CFRP cables and bars. More detailed results are provided in Appendix D. The transfer length was calculated using the equation provided by Mahmoud et al. (1999) and compared to the experimental results (see Figure 3.22). The transfer length coefficient (Î±t) was Deviator Diameter Harping Angle (degrees) Breaking Load (kips) % of Design Tensile Strength* % of Ultimate Tensile Strength 20 in. (steel) 10 56.8 93.6 73.1 15 56.1 92.4 72.2 20 55.8 91.9 71.9 40 in. (steel) 10 68.6 113.0 88.3 15 64.1 105.6 82.6 20 61.3 101.0 79.0 *Based on manufacturer provided value. Table 3.4. Harping test results on prestressing CFRP cables. (a) Prestressing CFRP cable (b) Prestressing CFRP bar St ra in ( in ./i n. ) St ra in ( in ./i n. ) L (in.) L (in.) Figure 3.21. Strain profile and transfer length determinations of full-scale CFRP prestressed beams.
46 Design of Concrete Bridge Beams Prestressed with CFRP Systems then calculated using the experimental transfer length for all test specimens according to Equa- tion 3.10 (the average values were 1.0 and 1.3 for prestressing CFRP bars and cables, respectively): f l f dt pi ci bÎ± = â² (Eq. 3.10) t 0.67 where fpi is the initial prestressing level in the CFRP tendons (ksi), db is the diameter of the tendon (in.), f â²ci is the concrete strength at transfer (ksi), and lt is the experimental transfer length value (in.). The experimentally determined transfer length values were within 5% of the predicted values. The transfer lengths for the prestressing CFRP cables and bars were 40 to 50 times the diameter of the prestressing tendon, which is less than the transfer length prediction for steel tendons by AASHTO LRFD (2017) of lt = 60db. The effect of thermal fluctuation cycles on the transfer length of the prestressing CFRP cables and bars was also studied in CFRP prestressed concrete prisms. After 30 thermal cycles with tem- peratures ranging between 0Â°F to 140Â°F (see Section 3.2.2), the transfer length was longer than that for un-weathered specimens by 70% and 100% of the initial transfer length for prestressing CFRP cables and bars, respectively. This difference is attributed to the degradation of the bond between prestressing CFRP and concrete caused by the difference in the CTE for prestressing CFRP tendons and concrete. 3.3 Results of the Finite Element Analysis 3.3.1 Verification of Finite Element Model The finite element analyses were conducted to study a wider range of parameters than were considered in the tests. A three-dimensional finite element model (FEM) was built using the commercial finite element program ATENA-GID 5.3.3 (Cervenka Consulting, 2013). The results obtained from the full-scale beam tests were used to calibrate the FEM. Load- deflection relationship, crack distribution, and crack width were determined from the finite element analysis (FEA) for each full-scale beam and compared to those obtained from the tests. Figures 3.23 and 3.24 show the comparison for two pretensioned and one post-tensioned beams, lt=40-50 db Figure 3.22. Experimental versus calculated transfer length for CFRP prestressed beams.
Research Results 47 and Table 3.5 shows a comparison of the ultimate loads and deflections; these indicate that the FEA estimates are in good agreement with the experimental results. More details are provided in Appendix E. 3.3.2 Parametric Study Prestressed Beams with Bonded CFRP A parametric study was conducted using the FEMs calibrated models to investigate the effect of the parameters identified in studies by Kakizawa et al. (1993), Fam et al. (1997), Abdelrahman and Rizkalla (1997), and Zou (2003a) as influencing the behavior of the pretensioned beams (i.e., level of prestressing, composite action, area of prestressing CFRP, and shear span-to-depth ratio). The study predicted the behavior of the prestressed beams having the same number of prestressing CFRP tendons as the test beams but different levels of prestressing, modulus of elasticity, and composite deck; and different locations of the load application. The results, listed in Table 3.6, indicate the following: â¢ Increasing the effective prestressing ratio increases the cracking load and decreases the mid- span deflection but does not affect the ultimate load; â¢ Removing the composite deck changes the mode of failure from rupture of the prestressing CFRP tendons to concrete crushing, and decreases the cracking load and the ultimate load by nearly 40%; (a) CPrSM#01 (b) BPrSM#01 Figure 3.23. Experimental and FEA predicted load versus deflection for beams pretensioned with CFRP cables and bars. (a) CPouSF (b) CPouDM#01 Figure 3.24. Experimental and FEA predicted load versus deflection for unbonded and bonded post-tensioned beams.
Experimental FEA â âBeam ID Load, (kips) Deflection, â (in.) Load, (kips) Deflection, â (in.) CPrSM#01 206 8.0 202 8.5 1.02 0.94 CPrSM#02 214 7.6 210 7.5 1.02 1.01 CPrSF 210 7.7 209 7.6 1.00 1.01 BPrSM#01 207 6.0 205 5.7 1.00 1.05 BPrSM#02 209 5.8 204 5.4 1.02 1.07 BPrpSM 209 5.2 208 5.2 1.01 1.01 BPrSF 207 5.6 208 5.4 0.99 1.03 CPouSM 135 9.9 129 9.4 1.05 1.05 CPouSF 122 7.4 122 8.5 1.00 0.87 CPouDF 143 8.9 145 10.1 0.99 0.88 CPoDM#01 175 5.2 173 5.3 1.01 0.98 CPoDM#02 174 6.7 173 6.6 1.01 1.02 Table 3.5. Experimental results and FEA estimates for ultimate load and deflection. Beam ID Parameter Peak Failure ModeLoad (kips) Deflection (in.) S1-PR-50 Effective prestressing ratio 50% 201.3 8.3 RP S1-PR-60 60% 200.5 7.1 RP S1-PR-70 70% 199.0 5.9 RP S1-PR-78 78% 197.6 5.0 RP S2-PR-50 Composite deck With deck 201.3 8.3 RP S2-PR50-NC Without deck 114.6 8.4 CC S2-PR60 With deck 200.5 7.1 RP S2-PR60-NC Without deck 119.3 7.5 CC S3-PR60 a/d ratio 6 200.5 7.1 RP S3-PR60-a/d-4 4 297.7 9.2 RP S4-PR-78-E1 Modulus of elasticity of CFRP 17,650 ksi 197.6 5.0 RP S4-PR-78-E2 22,000 ksi 195.4 4.2 RP S5-PR-78 Reinforcement index 0.43Ïb 197.6 5.0 RP S5-PR-78-RR1 0.54Ïb 238.3 5.2 RP S5-PR-78-RR2 0.76Ïb 315.6 5.7 RP S5-PR-78-RR3 0.87Ïb 350 5.8 RP S5-PR-78-RR4 Ïb 380.3 6 RP S5-PR-78-RR5 1.1 Ïb 392.5 5.9 CC S6-PR-78 Concrete strength 6 ksi 197.6 5.0 RP S6-PR-78-D4 4 ksi 191.5 5.1 RP S6-PR-78-D9 9 ksi 202.0 4.9 RP S6-PR-78-D9- G15 9 ksi 206.2 4.9 RP Ïb = balanced reinforcement ratio NC = non-composite RP = rupture of prestressing CFRP CC = concrete crushing Table 3.6. Results of parametric study of beams prestressed with bonded CFRP tendons.
Research Results 49 â¢ Reducing the span-to-depth (a/d) ratio from 6.05 to 4.0 (1/3 reduction) did not affect the ulti- mate moment capacity of the section but increased the deflection of the beams by 30 percent; â¢ Changing the modulus of elasticity of the prestressing CFRP tendons did not affect the initial elastic behavior or the ultimate load of the pretensioned beams, but increasing the modulus of elasticity of the prestressing CFRP reduced the net midspan deflection after cracking; â¢ Reinforcement ratio and level of prestressing had similar effect on the cracking behavior of the beams (i.e., increasing the reinforcement ratio above the balanced ratio changed the mode of failure from rupture of the CFRP tendons to concrete crushing); and â¢ Concrete strengths of the deck and girder did not significantly influence the strength and deflection of the prestressed beams. Details of the parametric study results are presented in Appendix E; these results were used in evaluating the proposed equations for the design of bonded CFRP prestressed beams. Prestressed Beams with Unbonded CFRP The parametric study of unbonded post-tensioned beams considered the parameters that influence the force in the unbonded prestressing CFRP. The parameters identified in studies by Naaman and Alkhairi (1991a), Lee et al. (1999), Harajli (2006), Ozkul et al. (2008), Au et al. (2009), and Lee et al. (2017) as affecting the behavior of the post-tensioned beams with pre- stressing steel were considered and their effect on post-tensioned beams with prestressing CFRP systems was investigated. These parameters were loading type, cable profile, effective prestressing ratio, unstressed bonded reinforcement, span-to-depth ratio, reinforcement ratio, and concrete strength. The results, shown in Table 3.7, indicated the following: â¢ Increasing the distance between the loading points increased capacity and deformability of the beams and increases the cable stress with a larger increase for straight cables, â¢ Increasing the level of prestressing increased the cracking load of the beams but did not affect the ultimate capacity or deformation, â¢ Adding auxiliary bonded reinforcement (steel or CFRP) increased the capacity significantly with higher ultimate capacity obtained with CFRP but the presence of unstressed reinforce- ment had no effect on the ultimate deflection, â¢ Post-cracking stiffness of the beams with auxiliary bonded reinforcement was higher than that for the fully unbonded CFRP (5% to 10% difference), â¢ Crack width at the ultimate was lower for bonded reinforcement, â¢ Increasing the reinforcement index increased the cracking and peak loads of the beams but decreased the net deflection at the peak load and the change in stress of unbonded cables, â¢ Increasing the span-to-depth ratio did not affect the moment capacity but increased the deflection and cable stress significantly, and â¢ Increasing the compressive strength of the deck increased the capacity and deflection of the beams. More details are presented in Appendix E. The results of this study were used in evaluating the performance of the proposed design equations for bonded CFRP prestressed beams. 3.4 Evaluation of Proposed Design Methods The method for the design of beams prestressed with CFRP tendons described in Section 2.5.6 follows the approach used in AASHTO LRFD (2017) but recognizes the failure mode related to CFRP. For beams prestressed with bonded CFRP tendons, strain compatibility and equilibrium equations are used as in the case of steel prestressed sections, except that CFRP rupture is gov- erned by the extreme layer of CFRP. In addition, the concrete strain at the extreme compres- sion fiber when the CFRP ruptures is less than 0.003 in./in.âthe ultimate value stipulated in
50 Design of Concrete Bridge Beams Prestressed with CFRP Systems ID Parameters Peak Load (kips) Deflection (in.) Moment Capacity (kip-ft.) S-P Loading type Single Load 105 7.0 1011 S-2P-60 2P-60" 138 9.6 1159 S-2P-90 2P-90" 160 11.2 1237 S-2P-115.5 2P-115.5" 193 14.8 1396 S-2P-154 2P-154'' 199 12.4 1279 D-P Cable profile Single Load 120 10.3 1158 D-2P-90 2P-60" 145 10.1 1214 D-2P-60 2P-90" 168 11.9 1302 S-PR-40 Effective prestressing ratio 0.4 fpu 140 10.6 1170 S-PR-50 0.5 fpu 155 10.3 1298 S-PR-60 0.6 fpu 171 10.3 1432 S-PR-70 0.7 fpu 185 9.7 1545 S-B-St Auxiliary bonded reinforcement 0.004 Act* Steel 224 10.8 1876 S-B-CFRP 0.004 Act * CFRP 270 10.7 2261 S-B-Non Non 171 10.3 1432 S-RI-08 Reinforcement index 0.083 70 11.7 585 S-RI-11 0.110 870 11.5 7290 S- RI-13 0.138 98 11.0 818 S-RI-19 0.193 127 10.0 1062 S-RI-27 0.275 171 10.3 1432 S-RI-55 0.550 284 8.0 2379 S-L/d-10 Span-to-depth ratio 10 277 3.5 1437 S-L/d-12.5 12.5 213 5.2 1397 S-L/d-15.8 15.8 171 10.3 1427 S-L/d-17.5 17.5 148 11.3 1368 S- L/d-20 20 125 14.9 1326 S-C-6.5 Concrete strength (ksi) 6.5 150 7.1 1256 S-C-7.5 7.5 162 8.7 1358 S-C-8.5 8.5 166 9.3 1386 S-C-10.5 10.5 168 9.5 1410 S-C-11.5 11.5 170 10.0 1422 *Act is the area of concrete cross section between the flexural face and centroid of gross section. Table 3.7. Results of parametric study of post-tensioned beams with unbonded CFRP tendons.
Research Results 51 AASHTO LRFD (2017). Using Whitneyâs stress block, the two factors, Î±1 and Î²1, provided in AASHTO LRFD (2017) may not be applicable; another approach to calculate the stress-block factors is proposed; details are provided in Appendix E. The results from the FEA (included in Appendix E), the tests performed in this project, and those reported in the literature were used to evaluate the proposed design methods. The FEM was first calibrated using the results of the large-scale tests conducted in this study, and then used to generate data points for different material properties, prestressing force, and dimensional properties. Figure 3.25 shows the moment capacities of the test beams versus those predicted using the proposed design methods. As shown in the figure, the design equations provide close but slightly conservative values when compared to those obtained from the tests and FEA. For prestressed beams with unbonded CFRP tendons, two design models [ACI 440.4R-04 (2011) and AASHTO LRFD (2017)] were selected to predict the capacities of all beams included in the parametric study. ACI 440.4R-04 (2011) uses a strain reduction approach to account for the incompatibility between the prestressing CFRP and concrete. However, the approach used in AASHTO LRFD (2017) assumes a failure mechanism to calculate the increase in unbonded tendon strain based on the total deformation of the tendon between the anchorage ends. For both models, the concrete compressive strain at failure was taken as 0.003. Figure 3.26 shows the moment capacities of the beams included in the database and the FEA versus those pre- dicted using the design models. The ratio of the mean of the experimental or FEA values to the predicted values is 1.2 with a COV of 0.06 for the ACI 440.4R-04 (2011) model and 1.4 with a COV of 0.21 for the AASHTO LRFD (2017) model. The figure shows no correlation of either ACI 440.4R-04 (2011) or AASHTO LRFD (2017) predictions with the experimental and FEA results. Figure 3.27 shows the increase in stress for the unbonded cables reported in the literature and those obtained from FEA versus those predicted by ACI 440.4R-04 (2011) and AASHTO LRFD (2017). The results show some correlation of ACI 440.4R-04 (2011) predictions with the experimental and FEA data but no correlation of AASHTO LRFD (2017) predictions with the experimental and FEA data. M om en t C ap ac ity fr om th e D es ig n Eq ua tio ns (k ip -ft .) Moment Capacity from Testing or FEA (kip-ft.) Figure 3.25. Moment capacities calculated according to proposed design methods versus those obtained from FEA and tests for bonded pretensioned beams.
52 Design of Concrete Bridge Beams Prestressed with CFRP Systems 3.5 Reliability Analysis A Monte Carlo Simulation approach was used to calibrate the strength reduction factor for CFRP prestressed beams failing due to the rupture of the CFRP tendons. A total of five bridges with different span lengths, roadway widths, girder positions, and number of girders were con- sidered; these are listed in Table 3.8. Concrete strength (6 ksi for deck and 9 ksi for girder) and the thickness of the deck were the same for all bridges. An earlier study (Forouzannia et al., 2016) suggested that changing the concrete strength of the deck from 4 to 8 ksi resulted in a minimal change (1.5%) in the nominal moment capacity and the deck thickness has a minor effect on moment capacity (6.3%). Therefore, a uniform deck thickness of 8 in. was selected for all design cases. The ratio of dead load to live load for the beams ranged from 0.6 to 1.77. The girders were designed according to the CFRP proposed in the design and material guide (a) ACI 440.4R-04 (2011) (b) AASHTO LRFD (2017) M om en t C ap ac ity fr om th e D es ig n Eq ua tio ns (k ip -ft .) M om en t C ap ac ity fr om th e D es ig n Eq ua tio ns (k ip -ft .) Moment Capacity from Testing or FEA (kip-ft.) Moment Capacity from Testing or FEA (kip-ft.) Figure 3.26. Moment capacities calculated according to ACI 440.4R-04 and AASHTO LRFD versus FEA and experimentally obtained results for unbonded post-tensioned beams. (a) ACI 440.4R-04 (2011) (b) AASHTO LRFD (2017) Figure 3.27. Increase in stress for unbonded tendons obtained from reported tests and FEA versus predicted values.
Research Results 53 Section Type Span Length (ft.) Girder Spacing (ft.) No. of Girders Roadway Width (ft.) Type I 40 6 5 30 Type I 60 6 5 30 Type III 80 9 7 60 Type IV 100 8 5 38 BT72 140 6 12 72 Table 3.8. Bridge data for calibration of resistance factors. specifications (developed in this project) using the load combinations from the AASHTO LRFD (2017) specifications. Because the loads are independent of reinforcement type, the data available in the literature for random load effects was used (Nowak, 1993; Nowak 1994; Nowak 1999; Moses, 2001). The dead loads used in the design were categorized as the weight of factory-made elements (girder), cast-in-place elements (diaphragm), wearing surface (asphalt), and mis- cellaneous elements (railings and luminaries). All dead loads (see Table 3.9) were assumed to be normally distributed random variables with the bias and COV values adopted from Nowak (1999). The live load model specified in AASHTO LRFD (2017), HL-93 loading (Design Truck/Tandem + Design Lane Load) was used. The descriptors for the random variables in the model are listed in Table 3.9. The descriptors for the random variables for the resistance model are listed in Table 3.10. The resistance models involve properties of concrete (e.g., compressive strength, failure strain), CFRP (e.g., rupture strength, Youngâs modulus), and related statistical parameters. Different distributions and statistical parameters for the rupture strength of the CFRP are found in the lit- erature; there are inconsistencies in reporting the strength of the materials by the manufacturers. The guide specification prepared in this project recommends the use of a two-parameter Weibull distribution to characterize the strength of CFRP by the value computed according to ASTM D7290 (2017). Statistical parameters for concrete have been derived in previous studies using large data-sets and were combined with the data obtained from the material tests performed in Variable Distribution Type Bias COV Reference Factory-Made Elements Normal 1.03 0.08 Nowak (1999) Cast-in-Place Elements Normal 1.05 0.10 Nowak (1999) Wearing Surface Normal 1.00 0.25 Nowak (1999) Live Load plus Impact Normal 1.28* 0.18 Nowak (1999) Girder Distribution Factor (Interior) Normal 1.11 0.10 BridgeTech, Inc., et al. (2007) *Average value of bias factors (1.2 to 1.35 for span lengths of 50 to 170 ft). In the Monte Carlo Simulation, this value is taken according to the span length. Table 3.9. Descriptors of random variable for load model.
54 Design of Concrete Bridge Beams Prestressed with CFRP Systems Variable Distribution Bias COV Reference Model error Normal 1.15 0.14 Current Study Height of deck Normal 1.00 0.03 Okeil et al. (2012) Height of girder Normal 1.00 0.03 Okeil et al. (2012) Web thickness Normal 1.01 0.04 Okeil et al. (2012) Prestressing CFRP area Normal 1.00 0.03 Shield et al. (2011) CFRP tensile strength Weibull 1.17* 0.04 Current study (cable) 1.12* 0.03 Current study (bar) CFRP modulus of elasticity Normal 0.97 0.14 Forouzannia et al. (2016) Concrete strength Normal Eq. 3.11 0.10 Nowak and Szersen (2003) *Mean/Design Value calculated for CFRP. Table 3.10. Descriptors of random variable for resistance model. this project and used to validate the distribution models and parameters selected for the reli- ability analysis. Other parameters such as the CFRP prestressing level, width and effective depth of the beam, and CFRP bar/strand cross-sectional area were taken as random variables with biases, COVs, and distribution models based on available literature. The bias and COVs for these parameters are well-documented in previous studies (Nowak, 1999; Shield et al., 2011; Okeil et al., 2012). Nowak and Szersen (2003) proposed the following equation for the bias factor of concrete compressive strength: 0.0081 0.15091 0.9338 3.0649 (Eq. 3.11)3 2Î» = â â² + â² â â²+ â²â² f f f ffc c c c c where f â²c is the concrete compressive strength in ksi. The professional factor accounts for the epistemic uncertainty in the prediction model, which depends on the failure mode and design objective. A database of CFRP prestressed concrete beam tests reported in the literature was compiled and supplemented with tests performed in this project and grouped by objective and failure mode. The database of 44 beams from 11 research studies (including this project) was used to calculate the professional factor. The dominant failure mode of all the beams was the rupture of the CFRP (41 beams failed by CFRP rupture and three beams failed by concrete crushing). The bias and COV for the professional factor were calculated as 1.15 and 0.14, respectively. The reliability index was calculated by changing the resistance factor for each of the bridges included in the design space. Figure 3.28 shows an example of the reliability index versus the resistance factor for one of the beams. The resistance factor for the target reliability index was obtained by linear interpolation between data points. Resistance factors of 0.85 and 0.80 for reliability indices of 3.8 and 4.0, respectively, were found to be appropriate for interior girders failing due to CFRP rupture. However, because of
Research Results 55 = 4.0 = 0.8 Figure 3.28. Reliability index versus resistance factors (Type BT72). the nature of failure of CFRP prestressed beams (with a drop of all applied load after failure), a lower resistance factor of 0.75 seems appropriate. For compression-controlled failures (con- crete crushing), the resistance factor of 0.75 stipulated in AASHTO LRFD (2017) provisions seems appropriate for beams prestressed with CFRP tendons because of their similar behavior. A resistance factor of 0.75 for tension-controlled beams (CFRP rupture) seems also appropriate although somewhat conservative based on the analysis but it eliminates the need for the transi- tion region between the modes of failure.