Đề tài Numerical analysis of externally prestressed concrete beams

0 4-D10 4-D10 4-D16 4-D10 3-D10 3-D10 3-D10 Vary External Unbonded External External External External External External External 3.0 - 5.0 3.0 3.0 1.8 3.0 1.5 Vary 2 - 2 2 2 2 2 3 3 Monolithic Monolithic Monolithic Monolithic Segmental Monolithic Monolithic Monolithic Segmental -73- is taken to analyze. Beam was simply supported with the clear span of 43.25m in length. The beam has a box cross-section of 2.4m high and prestressed by six cables of type 19K15 and Fig.4.4 Layout scheme of box girder beam prestressed with external cables (Test by Takebayashi T., et al.) Fig.4.3 Layout scheme of simply supported beams with external cables (Test by Matupayont, S., et al.) 32 5 275 25 0 100 P P 18001700 900 5200 2150 1700 2150 P P 30001100 900 5200 2150 1100 2150 P P 2x15001100 900 5200 2150 1100 2150 32 5 275 Deviator section Typical section 100 32 5 25 0 32 5 6 2 A1 345 A2 43.25 9.725 10.2 13.6 9.225 10.2 0. 22 5 0. 4 2. 0 0. 2 0. 18 2.0 0. 75 3.7 1 0.328 10.2 0. 22 5 0. 4 2. 0 0. 2 0. 18 2.0 0. 75 3.7 1/2 Section D-D1/2 Section C-C D D C C B B 1/2 Section A-A 1/2 Section B-B A A 0.3 28 9.725 10.21.7x4=6.8 9.2251.7x4=6.8 0. 22 5 0. 4 2. 0 0. 2 0. 18 0. 75 0. 22 5 0. 4 2. 0 0. 2 0. 18 0. 75 -74- 12K15 at the either side of cross-section. The cables were prestressed about 70% of the ultimate strength of cable. Three deviators were provided at the distance as shown in the layout scheme of the test beam (Fig.4.4) and the material properties are shown in Table 4.1. More details of test setup and geometrical dimensions of the beams on the experimental programs above are reported elsewhere1, 64~65). As mentioned earlier, there is friction between the cable and the deviator, and the friction can be expressed in terms of the friction coefficient μ as shown in Eq.(3.43). The real value of friction coefficient depends on many factors, and it can only be determined by the experimental investigations. These factors include the deviation angle, the kind of deviator, duct type, cement, and wax or grease grout (for unbonded cables). The friction coefficient, however, is not easy to find in any of the available literature. Since it has been impossible to obtain reliable values of the friction coefficient because of the high number of variables involved. For the analytical purposes, the friction coefficient at the deviators is assumed to have a certain value, and can be taken within the range from 0 to 0.35. In this study, the friction coefficient is assumed to be 0.2 for the beams tested by Nishikawa and Takebayashi, and 0.15 for the beams tested by Matupayont. Although these values might be not true in the tested beams, they are, however, adopted only for the analytical purpose. 4.2.1.2 Discussion of analytical results Comparison between the analytical predictions and the experimental results of the beams tested by Nishikawa in terms of the load vs. deflection responses is presented in Fig.4.5. In order to show the test variables in the experimental program by Nishikawa, beam G1 is chosen as a reference beam, and the predicted results as well as the experimental observations of beam G1 are compared with results of the other beams. It can be seen from Fig.4.5 that all the beams behave essentially the same before cracking regarding the use of different parameters such as amount of reinforcement, the distance between the deviators and the type of prestressing. However, the ultimate strength of beam G1 is comparatively much larger than that of the segmental beam GS1 (see Fig 4.5a), and much smaller than that of beam G6 with additional amount of reinforcement (see Fig.4.5b). The difference of ultimate strength of the beams is obvious and is approximately 55% for the case of the segmental beam GS1 and 30% for the case of beam G6. This is because the joint of the midspan section of the segmental beam GS1 opens rapidly as the applied load increases, resulting in the reduction of stiffness of the beam, leading to premature failure at a low strength. The load-deflection curve of the -75- precast segmental beam GS1 is also indicated by a rather plateau as shown in Fig.4.5a. This implies that the deflection of the beam increases rapidly with a little increase in the applied load. While the monolithic beam G6 with additional amount of reinforcement is attributed to the resistance of the reinforcement itself as well as to its effect in distributing and limiting the cracks in the concrete, leading to give a higher strength to the beam. Beam G1 is also compared with two other beams as shown in Figs.4.5c, d. It is apparently indicated that the distance between the deviators increases from 3.0 to 5.0 m, the ultimate strength of beam G5 reduces approximately 14% compared with beam G1 (see Fig.4.5d). Because of shortening of free length of cable, beam G1 with less free length of cable produces a greater stress variation in a cable than beam G5 with more free length of cable does (see Fig.4.6a). This also leads to a higher strength of beam G1 as compared with beam G5. Since there are no second-order effects in the case of unbonded beam G4, the ultimate strength of beam G4 increases approximately 9% as compared with beam G1, which is prestressed with external cables (see Fig.4.5c). a) b) c) d) Fig.4.5 Comparison of predicted load-deflection response with experimental results 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 40 80 120 160 200 Displacement [m] A pp lie d lo ad [k N ] Calc. Exp. Calc. Exp. Beam G1 Beam G4 A pp lie d lo ad [k N ] 0 0.05 0.1 0.15 0.2 0.25 0 40 80 120 160 200 Displacement [m] A pp lie d lo ad [k N ] Calc. Exp. Calc. Exp. Beam G1 Beam G5 A pp lie d lo ad [k N ] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 40 80 120 160 200 240 Displacement [m] A pp lie d lo ad [k N ] Calc. Exp. Calc. Exp. Beam G1 Beam G6 A pp lie d lo ad [k N ] 0 0.05 0.1 0.15 0.2 0.25 0.3 0 40 80 120 160 200 Displacement [m] A pp lie d lo ad [k N ] Calc. Exp. Calc. Exp. Beam G1 Beam GS1 A pp lie d lo ad [k N ] -76- Fig.4.6a shows the predicted results in terms of load vs. increase of cable stress responses. It can be seen from this figure that the curves of increase of cable stress exhibit similarly to the load-deflection relationships. This means that the stress in the external cables increases very little before cracking. However, it more rapidly increases after that, i.e., the major part of stress increase in a cable develops as the deflection of the beam becomes large. The segmental beam GS1 shows the smallest value of stress variation in the cable, while the monolithic beam with unbonded cables G4 indicates the biggest value of stress increase in series of tested beams. A comparison between the predicted results and the experimental data could not be made because the experimental data are not available in the technical literature. While a comparison between the curves of load vs. deflection and load vs. increase of cable stress for the individual beam is made, it is interesting to note that these two curves are very similar in shape, indicating a close relationship between the deflection and the stress increase in the external cables. The close relationship is also indicated by an approximately linear relationship in terms of the stress variation in the cable vs. deflection curve as shown in Fig.4.6b. The linear relationship between the midspan deflection and the stress increase in the cables confirms the findings from the experimental observations, which have been reported elsewhere38, 69~70). Consequently, the deformation compatibility of beam as mentioned earlier is verified to be suitable for the analysis of prestressed beams with external cables. Fig.4.7 shows a comparison between the predicted results and the experimental observations of the beams tested by Matupayont, S.65) in terms of both load vs. deflection and load vs. increase of cable stress curves. It can be seen from this figure that the distance between the deviators increases from 1.8 to 3.0m, the load carrying capacity of beams M2 a) Load vs. increase of cable stress b) Increase of cable stress vs. deflection Fig.4.6 Predicted results of cable stress 0 0.05 0.1 0.15 0.2 0.25 0.3 0 300 600 900 1200 Displacement [m] In cr ea se o f c ab le s tr es s [N /m m 2 ] G1 G4 G5 G6I nc re as e of c ab le s tr es s [N /m m 2 ] 0 200 400 600 800 1000 1200 0 40 80 120 160 200 240 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] G1 G4 G5 G6 GS1 A pp lie d lo ad [k N ] -77- reduces approximately 9% as compared with beam M1 (see Figs.4.7a). This is because as the applied load increases, cables with a larger distance between the deviators produce more loss in the cable eccentricity than cables with lesser distance between the deviators do, resulting in a lower strength of the beam. The reduction of load capacity of the beam due to the loss of cable eccentricity is also agreed well with the experimental observations conducted by Nishikawa, K., et al.64) as presented above. While beam M3 has one more additional deviator placed at the midspan, the loss of cable eccentricity is significantly reduced as compared with beam M2. As a result, the load carrying capacity of beam M3 increases significantly (see Fig.4.7b), it is about 20% higher as compared with beams M2. Therefore, it should be noted that a proper arrangement of deviators could enhance the ultimate strength of the beams prestressed with external cables due to the reduction of cable eccentricity. The additional deviators, however, make higher cost of construction and difficulties during the casting. a) Load vs. deflection b) Load vs. deflection c) Load vs. increase of cable stress d) Load vs. increase of cable stress Fig.4.7 Behavior of beams tested by Matupayont -0.02 0 0.02 0.04 0.06 0.08 0.1 0 20 40 60 80 100 120 A pp lie d lo ad [k N ] Displacement [m] Beam M1 Exp. Calc. Exp. Calc. Beam M2 A pp lie d lo ad [k N ] -0.02 0 0.02 0.04 0.06 0.08 0.1 0 20 40 60 80 100 120 Displacement [m] A pp lie d lo ad [k n] Beam M2 Exp. Calc. Exp. Calc. Beam M3 A pp lie d lo ad [k n] 0 100 200 300 400 500 0 20 40 60 80 100 120 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] Beam M1 Exp. Calc. Exp. Calc. Beam M2 A pp lie d lo ad [k N ] 0 200 400 600 800 0 20 40 60 80 100 120 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] Beam M2 Exp. Calc. Exp. Calc. Beam M3 A pp lie d lo ad [k N ] -78- Figs.4.7c, d shows the increase of cable stress against the applied load for two pairs of beam M1-M2 and M2-M3. The same behavior as shown in the load-deflection responses is also found. Beam M2 had more loss of eccentricity, indicates the smallest value of stress increase in the cable, while beam M3 with the smallest loss of eccentricity shows the biggest value of stress increase in series of the tested beams by Matupayont. The increase of cables stress is predicted well and in close agreement with the experimental results. The tendency of stress increase in the external cables, as expected in the numerical analysis, is indicated well for the different cases of loss eccentricity. The predicted results of a precast segmental box girder bridge tested by Takebayashi, T., et al.1) are presented in Fig.4.8. Moment-deflection response, concrete strain at the midspan section as well strain increase in the external cables is predicted well, and in close agreement with the experimental data. For the beam tested by Takebayashi, cable slip at the deviators is also investigated. a) Moment vs. deflection b) Strain increase of cable No.1 c) Concrete strain at midspan section d) Slip of cable No.5 at deviator D2 Fig.4.8 Behavior of precast segmental box girder beam with external cables (Test By Takebayashi, T. et al.) 0 0.0005 0.001 0.0015 0.002 0.0025 0 20000 40000 60000 80000 Concrete strain M om en t [ kN .m ] Exp. Cal.c M om en t [ kN .m ] 0 0.1 0.2 0.3 0.4 0.5 0 20000 40000 60000 80000 Displacement [m] M om en t [ kN .m ] Exp. Cal.c M om en t [ kN .m ] 0 3 6 9 12 15 20000 40000 60000 80000 M om en t [ kN .m ] Cable slip [mm] Exp. Cal.c D2 D1 D2 M om en t [ kN .m ] 0 0.0005 0.001 0.0015 0.002 0 20000 40000 60000 80000 Cable strain M om en t [ kN .m ] Exp. Calc. M om en t [ kN .m ] -79- Fig.4.8d shows the slip of cable No.5 at deviator D2 against the superimposed moment. The cable slip occurs once the driving force exceeds the frictional resistance at the deviator. Since cable No.5 has an extremely small angle, the friction between the cable and the concrete artificially reduces to a minimal value, resulting in a small friction force. As consequently, it induces a small frictional resistance resulting in slip at each loading step. The cable slip is also indicated by almost the same amount of strain increase in each cable segment i.e., the strain redistribution apparently took place through slippage. At the ultimate state, the cable slip at deviator D2 is about 11.6mm, and in reasonably close agreement with the experimental data. Table 4.2 shows, in summary, some of the main results measured in the experimental observations and comparison with the results obtained from the numerical analysis. It can be seen from the analytical results that the predicted results are in good agreement with the experimental observations for the both of load vs. deflection and load vs. increase of cable stress responses. The ultimate load capacity and the increase of cable stress are predicted within 10% of difference with the measured results. 4.2.2 Analysis of multiple span continuous beams with external cables 4.2.2.1 Introduction of experimental program In the experimental program by Aravinthan, T. et al.20, 67), a total of five externally prestressed concrete beams with a flanged cross section are considered for the analysis as numerical examples. The tested beams were two span continuous beams, and were prestressed Table 4.2 Summary and comparison between the predicted results and experimental data Ultimate load kN Increase of cable stress MPa Ultimate deflection mm Beam No Exp. Calc. Ratio Exp. Calc. Ratio Exp. Calc. Ratio G1 G4 G5 G6 GS1 M1 M2 M3 T1 143.9 161.5 130.0 197.5 95.5 91.25 83.40 101.25 58500* 151.0 164.0 132.6 195.0 97.8 91.2 84.5 101.6 58430* 0.953 0.985 0.980 1.013 0.976 1.001 0.987 0.996 1.001 - - - - - 350.0 313.6 465.0 0.0014** 975.0 1166.1 798.0 1072.7 446.7 342.5 297.9 563.8 0.0014** - - - - - 1.022 1.053 0.825 1.000 159.0 165.0 135.3 162.0 170.0 73.0 60.0 88.5 350.0 173.3 192.6 147.9 184.7 159.4 67.7 57.5 82.2 359.4 0.917 0.857 0.915 0.877 1.066 1.078 1.043 1.077 0.974 Exp.= Experimental results; Calc.= Calculated results; Ratio=Exp./Calc. * Moment; ** Cable strain -80- by two high strength cables with the effective prestressing of 50~55% of the ultimate strength of cable at the prestressing stage. Beam AT1 was monolithically cast and the others were precast segmental beams with epoxy joints. The difference between the precast segmental beams was the provision of confined reinforcement at the critical sections. All the beams were tested under the balanced loading arrangement except beam AT5 with the unbalanced loading arrangement. The layout scheme of the tested beams is shown in Fig.4.9, and the material properties of the beams are presented in Table 4.3. In the experimental program by Macgergor, R.G.J., et al.68), a three continuous span post- Fig.4.9. Layout scheme of two span continuous beams (Test by Aravinthan, T. et al) Fig.4.10 Layout scheme of three span continuous beams with external cables (Test by Macgregor, R.F.G. et al.) PP PP Axis of symmetry 1800 12001050 4050 4050 End section External cables 32 5 275 275 275 275 15 0 22 5 100 Typical section Deviator section Support section 1550 750 1750 75 D1 D2 D2 D1 S2 32 5 1 50 22 5 75 A B C Section A Section B Section C 2134 7620 3810 40 6 Axis of symmetry 31 1 100 11 4 14 0 5176 76 51 76 1219 343 343533 1219 343343 533 1219 343343 533 76 1409 2x686 2x686 2x68614092057 1409 1029 1B 31A 2 3 5 4A 4B 5 Area of cables NO1A,1B,2,4A & 4B -2.742 cm2 Area of cables NO 3 & 5 - 1.097 cm2 40 6 31 111 4 14 076 -81- tensioned segmental beam with external cables is considered to analyze. The beam has a box cross-section and 7.62m in length of each span. The beam was prestressed by three cables at the either side of cross section. Material properties of concrete and prestressing cable are presented in Table 4.3. Two points of the external load were applied only on the left span as shown in Fig.4.10. The main aim of the test was to examine the effect of dry joint on the strength and the ductility of box girder segmental bridge. For the analysis of multiple span continuous beams, the friction coefficient at the deviators is assumed to be 0.15 for all the beams. For the case of cables being perfectly fixed at the deviators or at the intermediately supported sections of the continuous beams, the friction coefficients referred to is from Garcia-Vargas’s model71), which is assumed to be equal to 2.0. 4.2.2.2 Discussion of analytical results Fig.4.11 shows the predicted characteristics of the beams tested by Aravinthan in terms of load vs. deflection responses. In this figure are also plotted the results obtained from the experimental observations for the comparison with the predicted results. It can be seen from this figure that the predicted results of load-displacement responses agree well with the experimental data. All the curves are essentially identical to the experimental observations before the peak loading range. The maximum value of applied load is defined whenever the concrete is initially crushed at the compression zone. The peak loads are approximately the same as in the experimental observations (see also in Table 4.4). The post-peak behavior of all the beams, however, does not match the observed one except beam AT5, for which there is close agreement to the experimental observation as shown in Fig.4.13c. Although a small discrepancy between the experimental data and the predicted results is observed, the precision Table 4.3 Test variables and materials for multiple span continuous beams Cable strength MPa Confined reinforcement Beam No Description of beam Concrete strength MPa Area /cable mm2 fpy fpu Midspan section Center support Loading arrangement AT1 AT2 AT3 AT4 AT5 MG1 Monolithic Segmental Segmental Segmental Segmental Segmental 38.8 41.3 39.5 36.5 40.1 41.3 93.0 93.0 93.0 93.0 93.0 vary 1500 1500 1500 1500 1500 1670 1750 1750 1750 1750 1750 1860 Yes No No Yes Yes No Yes No Yes Yes Yes No Balanced Balanced Balanced Balanced Unbalanced Unbalanced -82- of the analytical results is excellent, and the analytical method can accurately show the general behavior of prestressed concrete beams with external cables. For the sake of brevity, Fig.4.12 plots the curves of load vs. increase of cable stress for the representative beams. It can be seen from this figure that the predicted responses are quite a) Beam AT1 b) Beam AT2 a) Beam AT3 b) Beam AT4 Fig.4.11. Load-displacement relationships of beams Table 4.4. Comparison between the predicted results with experimental data Ultimate load, kN Cable stress, Mpa Beam No Exp. Calc. Ratio Exp. Calc. Ratio AT1 AT2 AT3 AT4 AT5 MG1 131.8 73.7 75.9 79.4 73.3 375.0 132.7 74.9 76.9 80.2 72.1 377.0 0.993 0.984 0.987 0.990 1.017 0.995 437.5 239.2 326.7 362.6 230.0 319.0 425.8 274.4 303.2 319.9 185.0 307.0 1.027 0.872 1.078 1.133 1.243 1.039 Exp.=Experimental data; Calc.=Predicted results; Ratio=Exp./Calc. 0 0.02 0.04 0.06 0.08 0 20 40 60 80 100 Displacement [m] A pp lie d lo ad [k N ] Exp. Calc. A pp lie d lo ad [k N ] 0 0.02 0.04 0.06 0.08 0 20 40 60 80 100 Displacement [m] A pp lie d lo ad [k N ] Exp. Calc. A pp lie d lo ad [k N ] 0 0.02 0.04 0.06 0.08 0 20 40 60 80 100 Displacement [m] A pp lie d lo ad [k N ] Exp. Calc. A pp lie d lo ad [k N ] 0 0.02 0.04 0.06 0.08 0.1 0 40 80 120 160 Displacement [m] A pp lie d lo ad [k N ] Exp. Calc. A pp lie d lo ad [k N ] -83- good in comparison with the experimental data. The monolithic beam shows the highest value of stress increase, while the segmental beam AT5 with the unbalanced loading arrangement indicates the lowest value of stress increase (see also in Table 4.4). All the beams show a small rate of stress increase in a cable before the decompression. The rate of stress increase, however, rapidly develops after that, i.e., the major part of stress increase occurs after the decompression of beam. It is also found that the cable stress increases more pronouncedly as the deflection of beam becomes large. This means that the stress increase in a cable is closely related to the beam deformation. It should be noted that the prestressing cable undergoes a small stress variation for all the beams and remains in the elastic range up to the failure of beam. For the two spans continuous beam AT5, the prestressing cables continue from one end to the other end. Also beam AT5 is applied by the unbalanced loading condition, i.e., the applied load on the right span was equal to 30% of the applied load of the left span. Therefore, the prestressing cable tends to move from the right span to the left span through the center- a) Beam AT1 b) Beam AT2 a) Beam AT3 b) Beam AT4 Fig.4.12. Increase of cable stress of beams 0 100 200 300 400 500 0 20 40 60 80 100 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] Exp. Calc. A pp lie d lo ad [k N ] 0 100 200 300 400 500 0 20 40 60 80 100 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] A pp lie d lo ad [k N ] 0 100 200 300 400 500 0 20 40 60 80 100 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] A pp lie d lo ad [k N ] 0 100 200 300 400 500 600 0 40 80 120 160 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] Exp. Calc. A pp lie d lo ad [k N ] -84- supported section. This means that the redistribution of cable strain obviously takes place through the slippage. Hence, the stress in a cable increases only slowly so that when the crushing strain has been reached in the concrete, the stress in the cable is so far below its ultimate strength. When the concrete is suddenly crushed at the extreme compression zone, the cable stress sharply reduces as shown in the predicted results (Fig.4.13d). The same phenomenon was also found in the experimental observations. However, the observed value still increases after the crushing of concrete, while the predicted results do not. Although the value of cable stress could not be properly predicted after the crushing of concrete, the analytical model, however, can predict the proper trend of stress increase in the cable up to the ultimate state. 4.2.2.3 Effect of casting method and loading arrangement Figs.4.13a, b show the comparison of a pair of beams AT1-AT4 in terms of load vs. deflection and load vs. increase of cable stress curves, which were cast by different method. It a) Load vs. deflection b) Load vs. increase of cable stress c) Load vs. deflection d) Load vs. increase of cable stress Fig.4.13 Comparison of behavioral responses of two pairs of beams AT1-AT4, AT4-AT5 0 0.02 0.04 0.06 0.08 0.1 0 40 80 120 160 Displacement [m] Ap pl ie d lo ad [k N ] Beam AT1 Beam AT4 Exp. Calc. Exp. Calc. Ap pl ie d lo ad [k N ] -0.02 0 0.02 0.04 0.06 0.08 0.1 0 20 40 60 80 100 Displacement [m] A pp lie d lo ad [k N ] Beam AT4 Exp.Calc. Exp. Calc.Beam AT5 Deflection of the right span of AT5 A pp lie d lo ad [k N ] 0 100 200 300 400 500 600 0 40 80 120 160 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] Beam AT1 Beam AT4 Exp. Calc. Exp. Calc. A pp lie d lo ad [k N ] 0 100 200 300 400 500 0 20 40 60 80 100 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] Beam AT4 Exp.Calc. Exp. Calc.Beam AT5 A pp lie d lo ad [k N ] -85- can be seen from the Fig.4.13a that two beams behave the same up to about 60.0 kN of the applied load. The monolithic beam AT1, however, achieves a much higher load carrying capacity after that, and the ultimate load is about 65% higher as compared to the segmental beam AT4. This is attributed to that when the joints of the segmental beam are opened with rapid rate as the applied load increases, the stiffness of the beam sharply reduces, resulting in the lower strength of the beam. The same behavior in terms of load vs. increase of cable stress response is also found as shown in Fig.4.13b. Fig.4.13c, d present a comparison of the pair of beams AT4-AT5, in which the two beams had essentially the same parameters, but they were subjected to different loading arrangement. It can be seen from this figure that beam AT4 achieves a higher load carrying capacity and has less deflection than that of beam AT5 (80.2 kN and 46.3 mm compared to 72.1 kN and 49.0 mm for the analytical model). The reduction in the load capacity of beam AT5 could be attributed to the joint opening, which begins to open at the lower level of the applied load as compared with beam AT4 (about 34.4 kN compared to 38.3 kN). Moreover, the increase of cable stress of beam AT5 also indicates a smaller value than that of beam AT4 (see Table 4.4) because of the strain redistribution of cable through the slippage at the center-supported section. Furthermore, the right span of beam AT5 (lightly loaded span) has an upward deflection as shown in Fig.4.13c, causing an adverse effect on the increase of cable stress. Consequently, this results in a lower strength of beam AT5. The effects of unbalanced loading arrangement on the ultimate strength of two span continuous beams with external cables was also found in the experimental program, which was conducted by Aparicio, A.C., et al.60) A comparison between the predicted results and experimental data such as the ultimate load and the increase of cable stress is shown in Table 4.4. It can be seen from this table that the predicted results agree reasonably well with the observed ones. Providing confined reinforcement at the critical sections enhances both the ultimate strength and the cable stress of beams. While beam AT5 subjected to the unbalanced loading arrangement shows the smallest value of ultimate load and the increase of cable stress, the monolithic beam AT1 indicates the biggest value in the five tested beams by Aravinthan. Fig.4.14 shows the predicted results of the beam tested by Macgregor, R.G.J.68). The load- deflection curve is predicted well and in close agreement with the experimental observations. The distribution of displacement along the beam is well performed for the unbalanced loading condition. The results from the analytical model such as moment, reactions for three span continuous beams under the unbalanced loading condition are obtained well. The strength of -86- the beam as well as the increase of cable stress at ultimate is predicted with a difference of 5% as compared with the experimental results (see Table 4.4). It is well known that the deformation of the beam can be divided into two parts (flexural deformation and shear deformation) as shown in Eq.(3.2). Fig.4.14a is also plotted the shear deformation in comparison with the total deformation of the beam. It can be seen from this figure that the shear deformation has extremely small effect on the total deformation of the beam in non-cracked elastic region. However, as the applied load increases, especially near the stage of collapse of the beam, the shear deformation increases very fast and it is about 4% of the total deformation of the beam at the ultimate state. Since the tested beam is a reduced scale beam of real structure with the span-to-depth ratio of l/h=18.75, the small effect of the shear deformation on the total deformation of the beam is obviously observed. a) Load-displacement relationship b) Distribution of displacement c) Moment Vs. the applied load d) Reactions at supports Fig.4.14. Behavior of beam tested by Macgregor 0 0.01 0.02 0.03 0.04 0.05 0.06 0 100 200 300 400 500 A pp lie d lo ad [k N ] Displacement [m] Exp. Calc. Crushing of concreteShear displacement A pp lie d lo ad [k N ] 0 5 10 15 20 -0.06 -0.04 -0.02 0 0.02 Beam length [m] D is pl ac em en t [ m ] Exp. Calc. D is pl ac em en t [ m ] 0 100 200 300 400 500 -600 -400 -200 0 200 400 600 Applied load [kN] M om en t [ kN .m ] Exp. Calc. Moment at the midspan section Moment at the support NI M om en t [ kN .m ] 0 100 200 300 400 500 -200 0 200 400 600 Applied load [kN] R ea ct io ns [k N ] NE NI SI SE Exp. Calc. NI NE SE SI R ea ct io ns [k N ] -87- 4.3. EFFECT OF NON-PRESTRESSED REINFORCEMENT 4.3.1 General introduction As quoted in Chapter 1 that prestressed beams may be classified as either fully or partially prestressed. Fully prestressed beams contain only prestressing cables, and under the applied load, tensile strain is taken by prestressing cables only. While partially prestressed beams contain bonded non-prestressed reinforcement in addition to prestressing cables, and tensile strain is taken by prestressing cable together with the non-prestressed reinforcement at the tension zone. Under the applied load a fully prestressed beam with unbonded cables behaves primarily as a tied arch, with the unbonded prestressing cables acting as the tension tie and the concrete as the compressive chord of the arch. This is because several cracks have been developed near the critical sections. However, as the applied load increases, only one crack or occasionally two cracks out of several cracks formed are found to increase significantly in width and to propagate upward to the compression zone of the beam leading to a response commonly known in the technical literature as tied arch behavior. When the beam is loaded to failure, a plastic hinge forms at a section of the maximum moment region where cracking is concentrated, and all rotation is confined primary to the location of this hinge. However, when non-prestressed bonded reinforcement is added to the unbonded prestressing cables, the beams may behave more like a beam rather than a tied arch. In this case the stress in the bonded reinforcement varies over the length of the beam, and consequently cracking and rotation are not confined to a plastic hinge region but are distributed along the span. Thus determination of the cable stress at failure of an unbonded beam is complicated by the fact that such as member may behave as on arch or a beam. The presence of bonded non- prestressed reinforcement enables an unbonded prestressed beam to act as a flexural member after cracking rather as a shallow tied arch, which exemplifies the behavior exhibited by fully prestressed beam with unbonded cables. It is experimentally agreed that the ultimate strength of prestressed concrete beams with unbonded cables or external cables is comparatively smaller than that of the similar prestressed concrete beams with internally bonded cables, the difference being placed at 10~30%14, 63). The reasons for the lower strength of beams with unbonded cables can be explained that since the cable is generally free to slip, the strain in the cable is more or less equalized along its length, and the strain at the critical section is lessened, leading to the lower -88- strength of beams. Moreover, the beams with unbonded cables tend to develop a few large cracks in the vicinity of the critical sections instead of many small ones well distributed. Such these cracks tend to concentrate the strain in the concrete at these sections, and rapidly increase in width and depth as the load increases, thus resulting in premature failure. To avoid the undesirability of such behavior in the beams prestressed with unbonded cables, non-prestressed reinforcement is commonly added to help overcoming the formation of sparsely spaced wide cracks and the concentration of compressive strain above these cracks. Such addition is attributed to the resistance of the non-prestressed reinforcement itself as well as to its effect in distributing and limiting the cracks in the concrete, help carrying the tensile stresses in the concrete, leading to give higher strength to the beam. Actually, the addition of non-prestressed reinforcement serves two main purposes: 1) to well distribute the crack along the beam length; 2) to contribute to the ultimate load capacity of the beam. In this section, a nonlinear analysis of the flanged beams prestressed by means of external cables with various amount of non-prestressed reinforcement is performed. The predicted results in terms of moment-displacement relationship are then discussed with emphasis on the effect of non-prestressed reinforcement. 4.3.2 Numerical examples In the experimental program conducted by Zhang, Z., et al.66), among ten tested beams, nine of them are considered for the analysis. A layout scheme of beam, cross section, cable Fig.4.15 Layout scheme of simply supported beams with external cables (Test by Zhang, Z et al.) 2000 20001000 1150 11502700 5000 P P 10 0 37 5 35 0 Beams with series B Beams with series A A A B B 45 0 500 37 5 100 45 0 38 0 23 4 7 B - B 500 35 0 100 38 0 A - A 10 0 37 5 35 0 45 0 37 5 45 0 38 0 23 4 7 35 0 38 0 -89- configuration, and loading arrangement are shown in Fig.4.15. A summary of the test variables is provided in Table 4.5. All the beams were simply supported with a flanged section, and were divided into two groups with different configuration of cable. In each group, the beams were designed with different amount of non-prestressed reinforcement in order to examine its effect on the behavior of the prestressed beams with external cables at ultimate. Six beams in the first group were prestressed by two cables with the straight profile at the depth of 0.35 m from the top surface of the beam (beams with series Z). Also the beams of the first group were subdivided into three pairs. In each pair, one beam was subjected to a single concentrated load at the midspan section, while the other was subjected to two loading points, symmetrically located at the distance of 2.0 m from the each end of the beam. In the second group, three beams were prestressed by two cables with the polygonal profile at the depth of 0.375 m, and two deviator points were provided at the distance of 2.7 m (beams with series S). The main test variables included the area of non-prestressed reinforcement, cable profile (straight or polygonal profile), and loading type (one or two loading points). 4.3.3 Effect of non-prestressed reinforcement Fig.4.16 and Fig.4.17 show a comparison between the analytical predictions and the experimental results of all the beams tested by Zhang in terms of moment vs. deflection responses. It can be seen from these figures that since the stiffness of the beams prior to cracking remains the same, all the beams exhibit essentially identical before cracking, indicating no significant effect resulted in using the different area of non-prestressed reinforcement. Since the first crack of the beam with a smaller amount of non-prestressed Table 4.5 Test variables and materials of beams tested by Zhang Beam No Area of reinforce ment mm2 fy Mpa Effective depth of reinc. ds mm Area of cable mm2 Effective prestress fpe MPa Effective depth of cable dp mm Concrete strength MPa Loading type Z1-1 Z1-2 Z2-1 Z2-2 Z3-1 Z3-2 S1-2 S2-2 S3-2 157.1 157.0 235.6 235.6 358.1 358.2 157.0 201.1 402.1 267.0 267.0 267.0 267.0 267.0 267.0 267.0 267.0 267.0 410.0 410.0 370.0 370.0 383.7 383.7 390.0 410.0 390.0 981.7 981.7 981.7 981.7 981.7 981.7 392.7 392.7 392.7 322.9 325.9 332.3 335.4 331.3 326.8 805.7 843.4 822.2 350.0 350.0 350.0 350.0 350.0 350.0 375.0 375.0 375.0 52.3 52.3 49.8 49.8 52.6 52.6 52.7 52.7 49.3 One point Two points One point Two points One point Two points Two points Two points Two points -90- reinforcement occurs a little earlier than the companion beams with a larger amount of reinforcement do, beam with a smaller amount of non-prestressed reinforcement, therefore, produces a lesser moment at ultimate. It is apparently shown that the ultimate moment capacity of a beam will be higher when the first crack occurs at a higher applied moment, and it will be lower when the first crack occurs at a lower applied moment. After cracking, beam with a smaller amount of non-prestressed reinforcement exhibits rather ductile, and fails by initial yielding of non-prestressed reinforcement, sequentially, collapses totally by crushing of concrete in the compression region. On the other hand, beam with a higher value of non- prestressed reinforcement exhibits rather stiff, resulting in a higher strength at ultimate. It is well known from theory that a section of the beam begins to crack whenever the applied moment on this section exceeds the cracking moment. From the analytical results, it is interesting to note that the cracks mostly concentrate within the constant moment region a) Beams with one loading points b) Beams with two loading points Fig.4.16 Moment vs. deflection of beams with series Z Fig.4.17 Moment vs. deflection of beams with series S 0 0.02 0.04 0.06 0.08 0.1 0 50 100 150 200 250 300 Displacement [m] M om en t [ kN .m ] Z1-1 Z2-1Z3-1 Exp. Calc. M om en t [ kN .m ] 0 0.02 0.04 0.06 0.08 0.1 0 50 100 150 200 250 300 Displacement [m] M om en t [ kN .m ] Exp. No S2-2 S3-2S1-2 Calc. S2-2 S1-2S3-2 M om en t [ kN .m ] 0 0.02 0.04 0.06 0.08 0.1 0 50 100 150 200 250 300 Displacement [m] M om en t [ kN .m ] Exp. No Z2-2 Z3-2Z1-2 Calc. Z1-2 Z3-2 Z2-2 M om en t [ kN .m ] -91- between the loading points. The location of cracks spread in a wider area in the beam with a higher value of non-prestressed reinforcement. And the cracks tend to appear beyond the constant moment at the ultimate state as the amount of non-prestressed reinforcement increases. The reduction in the crack spacing and the extension of cracking in the shear span of an unbonded beam is also observed as the amount of non-prestressed reinforcement is increased. Although the experiment did not show the crack pattern of the tested beams, it is, however, apparently believed that the predicted results are basically identical to the observed ones. The same observations were obtained from the tests of partially prestressed concrete beams with unbonded cables, which have been reported elsewhere38, 69). It is also seen from these figures that the ultimate moment of the beams increases with increasing the amount of non-prestressed reinforcement. The analytical results reproduce the experimental data with remarkably good agreement for the beams with different layout of external cables and amount of non-prestressed reinforcement. It is also proved by experiments that the ultimate strength of unbonded beams depends not only on the amount of non-prestressed reinforcement, but also depends on the amount of prestressing cable. A combination between the non-prestressed reinforcement and the prestressing cable is characterized by a reinforcement index qo, and is defined as: peso qqq += (4.3) '' cp pep cs ys o fbd fA fbd fA q += (4.4) where qs, qpe are the index of non-prestressed reinforcement and prestressing cable, respectively; As is the area of non-prestressed reinforcement; Ap is the area of prestressing cable; fy is the yield strength of non-prestressed reinforcement; fpe is the effective prestress of prestressing cable; f’c is the compressive strength of concrete; b is the width of the compressive face; ds is effective depth of beam to centroid of non-prestressed reinforcement; dp is the effective depth of beam to centroid of prestressing cable. The values of qs, qpe and qo given in Table 4.6 reflect the actual material properties of each beam. Since the value of qpe is essentially identical to all beams of each group, the main difference in the ultimate strength of the beams is attributed to the index of non-prestressed reinforcement. From this point of view, it could be said that the adequate addition of non- prestresed reinforcement can be substantially enhanced the ultimate strength of the beams prestressed with external cables. The beneficial effect of non-prestressed reinforcement was -92- also found by test of the partially prestressed beams with unbonded cables, which have been reported elsewhere29, 38, 69). It should be noted that in practice, it is commonly referred to the lightly reinforced beam as under-reinforced and it exhibits a ductile failure. As the amount of non-prestressed reinforcement is increased, the beam approaches the over-reinforced case. The over-reinforced beam would exhibit crushing of the concrete with the reinforcement in the elastic range resulting in a brittle failure. This type of the beam, however, should be avoided in order to prevent the sudden collapse of the beams in the design practice. 4.3.4 Effects of cable configuration and loading pattern The effect of cable configuration is examined by comparing the obtained results of beam Z1-2 with beam S1-2 as shown in Fig.4.18a. Beam Z1-2 was identical to beam S1-2 except the cable area and the cable configuration. It can be seen from Fig.4.18a that since the beam deflection and the accompanying reduction in cable eccentricity are small in the elastic range, the moment-deflection responses behave similarly in this stage, indicating insignificant effect of cable configuration on the moment-deflection response. As the applied load increases, the beam deflection becomes large. As a result, the reduction in cable eccentricity of the beam without deviator becomes more pronouncedly, leading to a lower strength of the beam as compared to the beam with deviator. Moreover, cables with deviator produce a greater stress variation than cables without deviator do because of shortening of free length of cable. This also leads to a higher strength of the beams with deviators as compared with the beams without deviators. The ultimate strength of beam S1-2 with two deviators is approximately 15% higher than that of beam Z1-2 without deviator. Table 4.6 Reinforcement indexes Beams No qs qpe qo Z1-1 Z1-2 Z2-1 Z2-2 Z3-1 Z3-2 S1-2 S2-2 S3-2 0.0039 0.0039 0.0068 0.0068 0.0095 0.0095 0.0041 0.0050 0.0112 0.0350 0.0350 0.0374 0.0378 0.0353 0.0350 0.0320 0.0335 0.0350 0.0389 0.0389 0.0446 0.0446 0.0445 0.0445 0.0361 0.0385 0.0462 -93- Fig.4.18b shows the effect of loading pattern for a pair of beams Z2-1 vs. Z2-2. Beam Z2-1 was subjected only to a single loading point at the midspan section, while beam Z2-2 was subjected to two loading points arranged symmetrically at the midspan section at the distance of 0.5 m. It can be seen from this figure that the beam subjected to a single concentrated load at the midspan section exhibits a higher strength as compared with the beam subjected to two loading points. This is because the moment distribution along the beam, which is uniformly distributed over the length between two loading points, and it is concentrated only at the midspan section as in the case of single loading point. This phenomenon agrees well with the experimental observations for the beams prestressed with external cables, which have been reported elsewhere72~73). 4.4 CONCLUDING REMARKS In this chapter, a large number of beams prestressed with external cables are considered for the analysis as numerical examples. The selected beams have various parameters such as rectangular, flanged or box girder bridges, beams with different cable configuration (straight or polygonal), beams with different constrain (simply supported or multiple span continuous beams), beams with or without deviators, beams with different loading condition (symmetrical or unsymmetrical, one or more loading points), monolithic or precast segmental beams, and beams with different amount of non-prestressed reinforcement. The following conclusions are made from the results obtained by the numerical analysis: A non-linear analysis using a finite element algorithm together with the deformation compatibility of beam is performed to predict the entire response of the beams prestressed a) Effect of cable configuration b) Effect of loading pattern Fig.4.18 Effect of cable configuration and loading pattern 0 0.02 0.04 0.06 0.08 0.1 0 50 100 150 200 250 300 Displacement [m] M om en t [ kN .m ] Beam Z1-2 Calc. Exp. Beam S1-2 Calc. Exp. M om en t [ kN .m ] 0 0.02 0.04 0.06 0.08 0.1 0 50 100 150 200 250 300 Displacement [m] M om en t [ kN .m ] Exp. Calc. Exp. Calc. Z2-1 Z2-2 M om en t [ kN .m ] -94- with external cables up to the ultimate loading stage. The accuracy of the proposed method is verified by comparing the predicted results with the experimental observations. The predicted results in terms of load vs. deflection and load vs. increase of cable stress responses are in reasonably close agreement with the experimental data. The close agreement between the experimental data and the predicted results apparently indicates a validity and potential of the proposed method for the analysis of beams prestressed with external cables. The proposed method is generally suitable for the investigation of all kinds of beam prestressed with external cables such as simply supported or multiple spans continuous beams with or without deviators. It should be noted that the proposed method might be best for the purpose of research rather than for the design practice. It should be noted that the proposed method of analysis is member-analysis, which is the consideration of the deformation in the concrete at the level of the cable along the entire length of the beam. Considering the member-analysis as a whole is a more theoretically sound approach than using a section-analysis, because the increase of cable stress is a function of the change in the concrete strain at the level of the cable along the entire length of the beam rather than the change in the concrete strain at a particular section. There is a close relationship between the two curves of load vs. deflection and load vs. increase of cable stress. This relationship is also verified by a fairly linear response between the midspan deflection and the increase of cable stress for the individual beam. The appropriate amount of non-prestressed reinforcement should be provided for externally prestressed concrete beams to improve the ultimate strength of the beams. The externally prestressed concrete beams with the adequate addition of bonded non-prestressed reinforcement exhibit like the flexural members after cracking rather than the shallow tied arch members. The ultimate strength of externally prestressed concrete beams can enhance by the adequate addition of bonded non-prestressed reinforcement. The effect of non-prestressed reinforcement on the behavior of the beam prestressed with external cables should be properly taken into consideration in the design practice. The stress variation in the external cables also depends on the magnitude of combining reinforcing index. Everything else being constant except the distance between the deviators, beams with less free length of cable produce a greater ultimate strength and stress variation in the cable than beams with more free length of cable do.