Parametric study

-95- Chapter 5 PARAMETRIC STUDY 5.1 INTRODUCTION In an external prestressing system, since the cables are attached to the beam at some deviator points, friction exists between the cables and the deviator points, obviously. It is numerically shown that strain increase in the external cables depends not only on the overall deformation of the beam and also on the cable friction. Even though in the elastic regime, the effect of friction on the overall behavior of the beam is extremely small, it can be neglected. However, as the applied load increases, especially near the collapse stage, the effect of friction may be considerably large. In this case, the cable slip might occur at the some deviator points, and the strain distribution in the external cables obviously takes place. As a result, it may be changed in the behavior of beam prestressed with external cables. To show the effect of friction at the deviators, in this chapter a parametric study is numerically carried out in order to understand this effect on the overall behavior of the beams in general, and on the increase of cable stress in the external cables in particular. Experiments obviously show that the increase of cable strain also depends on the free length of the cable and on the loading arrangement, especially for beams with multiple continuous spans having cables continued from the one end to the other end. Since the strain increase in the external cables depends on the overall deformation of the beam, i.e., it depends on the loading arrangement. For the case of a beam with unbalanced loading arrangement, the cable slip commonly occurs at the lower level of the applied load than that of a beam with the balanced loading arrangement. Consequently, this results in the lower load capacity of the beam as compared with the beam with the balanced loading arrangement. Since the experimental works are mostly concentrated on the beams with the balanced loading arrangement, there are extremely few experiments for the beams with unbalanced loading 96 -96- arrangement. For two span continuous beams with the external load applied only on one span, the defection of unloaded span has usually upward deflection, resulting caused the adverse effect on the strain increase in the external cables. The effect of unbalanced loading arrangement for multiple span continuous beams was also indicated by experiments, which have been recently reported elsewhere20, 60, 74). In order to better understanding this phenomenon, a parametric study on the effect of loading arrangement is also carried out in this chapter. The parametric evaluation is presented in the next section. 5.2 PARAMETRIC EVALUATION In this chapter, a parametric study is performed for beams prestressed with external cables with two purposes: 1) to investigate the friction effect at the deviator points on the behavior of simply supported beam; 2) to investigate the effect of loading arrangement on the behavior of two span continuous beam with external cables continued from one end to the other end in order to examine the stress increase in the external cables under the unbalanced loading condition. The predicted results are then discussed with emphasis on the effects of friction at the deviators and the loading arrangement on both the load-deflection and the load-increase of cable stress relationships. 5.2.1 Effect of friction at deviators The effect of friction is performed on a simply supported beam with a box section, which was tested at the Research Center for Experiments and Studies on Construction and Public Fig.5.1 Layout scheme of beam tested by CEBTP 1000 400 40 0 10 0 10 0 100100 3000 15001500 48 0 40 0 10 0 10 0 48 0 -97- Work (CEBTP) in France71, 75). The dimensions of the beam, span length and loading arrangement are shown in Fig.5.1, and material properties are shown in Table 5.1. Two deviators were provided at the distance of 3.0 m from each other, and symmetrically located from the midspan section. The beam is analyzed by considering four different cases: 1) free slip; 2) slip with friction; 3) partially fixed; and 4) perfectly fixed. For the case of cables being free slip, the friction coefficient is equal to zero, whereas for the case of cables being slip with friction as usually seen in the nature, the friction coefficient is assumed to be equal to 0.17. While for the case of cables being perfectly fixed, the friction coefficient should have a value, which is big enough to restrain any movement at the deviators. In this case the value of friction coefficient referred to is from Garcia-Vargas’s model71), which was assumed to be equal to 2.0. For the case of partially fixed, the friction coefficient is assumed to be 1.0, which has an intermediate value between the cases of slip with friction and perfectly fixed in order to examine the extent of fixity at the deviators. Fig.5.2 plots the predicted characteristics of the load-deflection response for four cases and also the results obtained from the experimental observations. It can be seen from this figure that the deflection responses behave essentially in the same manner as in the experimental observations until the decompression stage regardless of friction. This is because the beam a) Entire responses b) Responses after the decompression Fig.5.2 Effect of friction at the deviators on the load-deflection responses Table 5.1 Material properties (MPa) Concrete Prestressing cable f’c Ec fpy fpu Eps 41.0 3.8x104 1570 1860 1.95x105 0 0.02 0.04 0.06 0.08 400 450 500 550 600 650 A pp lie d lo ad [k N ] Displacement [m] Exp. results Free slip Slip with friction Partially fixed Perfectly fixed A pp lie d lo ad [k N ] 0 0.02 0.04 0.06 0.08 0 100 200 300 400 500 600 700 A pp lie d lo ad [k N ] Displacement [m] Exp. results Free slip Slip with friction Partially fixed Perfectly fixed A pp lie d lo ad [k N ] 98 -98- deflection is very small, which induces a small tensile force in each cable segment, leading to an extremely small unbalanced force at a deviator. As a result, the cable slip cannot occur at this stage, generally. That is the friction at the deviators does have an insignificant effect on the deflection response until the decompression stage. After the decompression, the deflection responses of beam with consideration of free slip and slip with friction are more or less identical to the experimental results, whereas for the case of perfectly fixed, the prediction overestimates the strength of the beam at ultimate. The reason for this can be explained that since the cables are assumed to be a perfectly fixed at the deviators, the stress increase in each segment is independent from that of the others. As the applied load increases, the deflection of midspan and the accompanying concrete strain at the cable level between the deviator points becomes large, resulting in a great increase of cable stress of middle segment (see Fig.5.3). A greater stress variation in the middle segment of a cable induces a higher load carrying capacity, resulting in the overestimating prediction of ultimate strength of the beam. a) Entire responses b) Responses after the decompression Fig 5.3 Effect of friction at the deviators on the load-increase of cable stress Fig.5.4 Increase of cable stress vs. deflection 0 300 600 900 1200 1500 0 100 200 300 400 500 600 700 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] Exp. results Free slip Slip with friction Partially fixed Perfectly fixed A pp lie d lo ad [k N ] 0 300 600 900 1200 1500 400 450 500 550 600 650 Increase of cable stress [N/mm2] A pp lie d lo ad [k N ] Exp. results Free slip Slip with friction Partially fixed Perfectly fixed A pp lie d lo ad [k N ] 0 0.02 0.04 0.06 0.08 0 300 600 900 1200 1500 Displacement [m] In cr ea se o f c ab le s tr es s [N /m m 2 ] Slip with friction Partially fixed Perfectly fixed Exp. results Free slip In cr ea se o f c ab le s tr es s [N /m m 2 ] -99- Fig.5.3 presents the results of stress increase in the external cables. It is apparently seen that the increase of cable stress exceeds the yielding strength for the cases of partially fixed and perfectly fixed, and remains in the elastic range for the cases of free slip and slip with friction. Although a small discrepancy has been observed in the predicted results for the cases with free slip and slip with friction, the same rate of stress increase, however, is approximately found until the ultimate state, and very similar to the experimental observations. A fairly linear relationship between the increase of cable stress and the beam deflection is also observed as shown in Fig.5.4. This indicates that the stress increase in a cable is almost proportional to the midspan deflection until the crushing strain reaches in the concrete. However, the rate of stress increase in the case of cable being perfectly fixed is quite different from the other cases. It is also seen from this figure that the rate of stress increase is reduced from the deflection of 40.0 mm as observed in the experiment. This is because the rate of stress increase in the external cables is smaller than the rate of increase in the beam deflection as the applied load increases from this point. However, the rate of stress increase observed by the predictions does not change except the case of cable being slip with friction. This may be indicated in the calculated results for the ultimate load capacity, which are a little higher than that of the experimental observations (see Table 5.2). It is also found from the results of the case of slip with friction that the concrete strain at the critical section suddenly jumps as the applied load reaches the peak load. As the crushing strain reaches in the concrete at the compression region, the applied load is sharply reduced, accompanying the beam deflection increases significantly as shown in Fig.5.2. This causes the change in the rate of stress increase as shown in the curve of the increase of cable stress vs. deflection. Because the Fig.5.5 Comparison between the cases of partially fixed and perfectly fixed 0 0.02 0.04 0.06 0.08 0 300 600 900 1200 1500 Displacement [m] In cr ea se o f c ab le s tr es s [N /m m 2 ] Midspan segment End segment Exp. results Partially fixed Perfectly fixed In cr ea se o f c ab le s tr es s [N /m m 2 ] 100 -100- deflection of the beam increases noticeably after the crushing of concrete, the linear relationship, therefore, is terminated as shown obviously for the case of slip with friction. Fig.5.5 shows a comparison between the cases of perfectly fixed and partially fixed in terms of the increase of cable stress vs. deflection responses. It can be seen from this figure that since the external cables are being perfectly fixed at the deviators as in the case of perfectly fixed, the stress increase in the midspan segment and the end segment is totally different. While for the case of the cables being partially fixed at the deviators, the difference of the stress increase in the midspan segment and the end segment is lesser as compared to the case of perfectly fixed. This indicates that some cable slip might occur at the deviator points, resulting in transfer of cable stress from the midspan segment to the end segment. This phenomenon is agreed well with the experimental observations, which have been conducted by Fujioka, A., et al.76) It is also found from the predicted results that the ultimate load of the beam with consideration of partially fixed at the deviators does not increase much as compared to the cases of free slip and slip with friction (see Fig.5.2 and Table 5.2). However, the stress increase in the external cables is much higher as the comparison has been made. This is because the strain variation in the external cables depends not only on the overall deformation of the beam, but also on the free length of a cable between two successive deviators, i.e., it depends on a ratio of Ld/L (the distance between the deviators per the total span length). For the beam tested by CEBTP, this ratio of Ld/L is equal to 0.5, which seems to be considerably large. In this case the extent of fixity of cable at the deviators has significant effects on the stress increase in the external cables rather than on the load-deflection response of the beam. It is believed that when the ratio of Ld/L is rather small, both the ultimate strength and the stress increase in the cables are significantly increased due to the extent of fixity of cable at Table 5.2. Comparison between the experimental observations and the calculated results Case of study Ultimate load kN Ultimate deflection mm Increase of cable stress MPa Free slip Slip with friction Partially fixed Perfectly fixed Exp. observations 586.2 580.6 594.0 589.9 570.0 58.1 54.0 58.0 45.4 53.0 741.7 679.4 995.5 1455.0 745.0 -101- the deviators. The improvement due to the fixity of cable was also verified by the experimental observations for two pairs of beams with the different ratio of Ld/L, which have been reported elsewhere76). The results at the ultimate stage for the beams under the different bondage of cable at the deviators are presented in Table 5.2. It should be, generally, noted that friction at the deviators reduces the ultimate deflection and increases the stress in the prestressing cables. However, it is found from the analysis that the results of the case of slip with friction show somewhat contrary to the other cases. The reason for that might be the strain-jump, which is happened in the concrete at the critical section as explained early. Note that the predicted results in terms of load vs. deflection and load vs. increase of cable stress curves have been observed somehow similar for the both cases of free slip and slip with friction. It is also found from the predicted results that beam with partially fixed condition shows a higher ultimate load but a lower increase of cable stress as compared with beam having perfectly fixed condition. This is rather contrary to the previous findings that beam having a higher cable stress should also have a higher ultimate load capacity in general. The reasons for this can be explained that since the cables are perfectly fixed at the deviators as in the case of perfectly fixed, the cable stress usually reaches the yielding strength at the lower level of the applied load as compared with the case of partially fixed. As a results, the ultimate load capacity of the beam in the case of perfectly fixed is a little smaller than that obtained from the case of partially fixed. Moreover, the value of friction coefficient adopted for the case of perfectly fixed in this study is not exactly known for the real condition. This reason might also lead to overestimate the stress increase in the external cables. For the others cases of this study, the predicted results are agreed well with the findings from the previous studies. a) Beam G1 tested by Nishikawa b) Beam B1-2 tested by Zhang Fig.5.6 Evaluation of the friction effect on behavior of beams prestressed with external cables 0.10 0.02 0.04 0.06 0.08 0 50 100 150 200 250 300 Displacement [m] M om en t [ kN .m ] Exp. results Free slip Slip with friction Partially fixed Perfectly fixed M om en t [ kN .m ] 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 ] Exp. results Free slip Slip with friction Partially fixed Perfectly fixed A pp lie d lo ad [k N ] 102 -102- The effect of friction is also investigated on the beams tested by Nishikawa, K., et al.64) and Zhang, Z., et al.66). The predicted results are plotted in Fig.5.6. It is apparently shown that the friction at the deviators have some influences on the load-deflection curves of a prestressed concrete beam with external cables. Although a small difference between the cases of free slip and slip with friction has been observed, the experimental results, however, fit more closely with the assumption of slip with friction. The same effect of friction at the deviators is also found as in the case of the beams presented in Fig.5.2. Similar predictions of the friction effect on the behavior of the beams with external cables have been reported elsewhere3, 54, 71). It should be noted that since no any means to prevent the movement of a cable at the deviator points are generally provided, the assumption of either free slip or slip with friction seems to be more realistic rather than the assumption of perfectly fixed in the numerical analysis. However, it is also useful when two extreme cases of free slip and perfectly fixed at deviators are considered as many researchers do in the numerical analysis. Because the whole range of behavior of beams prestressed with external cables at ultimate is to be well understood. 5.2.2 Effect of loading arrangement on behavior of two span continuous beam The effect of loading arrangement is performed on two span continuous beams prestressed with external cables, which was tested by Umezu, K., et al.22). The beam has a rectangular section, and was prestressed by the two cables type of 1T17.8 (2.084 cm2/a cable). At the initial prestressing stage, the cables were stressed approximately 50% of the ultimate strength of cable. Two points of the applied load was provided on each span as shown in the layout of Table 5.3 Material properties (Mpa) Concrete Prestressing cable f’c Ec σpy σpu Eps 42.4 2.58x104 1600 1900 1.97x105 Table 5.4 Loading cases Case Loading ratio Exp. Calc. 1 2 3 4 5 00.1=α 75.0=α 50.0=α 25.0=α 00.0=α Ο - - - - Ο Ο Ο Ο Ο -103- analytical scheme (see Fig.5.7). The applied load on each span is arranged so that the effect of loading arrangement on the behavior of two span continuous beams with external cables can be investigated. That is the left span is heavily loaded with the applied load P, while the external load αΡ is applied on the right span. The loading ratio α will change from 0 to 1.0 in order to obtain the different loading arrangement on the both spans. The beam is analyzed in the five cases with different loading ratio as shown in Table 5.4, the material properties are presented in Table 5.3. In the analysis friction coefficient at the deviators is assumed to be equal to 0.12 for all cases. Fig.5.8a presents the predicted results in terms of load vs. deflection response at the critical section on the left span. In Fig.5.8a is also plotted the results from the experimental observation for the case α = 1.0, i.e., beam with the balanced loading arrangement. It can be seen from this figure that the load capacity of the beam reduces with decreasing the loading ratio. The maximum load carrying capacity of the beam is observed when the equalized load is applied on the both spans, i.e., beam with the balanced loading arrangement. On the other hand, the minimum load carrying capacity of the beam is found when the zero-load is applied on the right span. The reason for the reduction in the load carrying capacity of the beam can be explained that the first crack at the critical section on the left span of the beams with a smaller loading ratio occurs earlier than the beams with a larger loading ratio