Numerical analysis of externally prestressed concrete beams part 2

Numerical analysis of externally prestressed concrete beams

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

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

1500

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

41.0 3.8x104 1570 1860 1.95x105

0100200300400500600700

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

Increase of cable stress [N/mm2]

Increase of cable stress [N/mm2]

Increase of cable stress [N/mm2]

Increase of cable stress [N/mm2]

Exp resultsFree slip

Exp resultsFree slip

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 deflectionas 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

End segment

Exp resultsPartially fixedPerfectly fixed

End segment

Exp resultsPartially fixedPerfectly fixed

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

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

04080120160200

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

42.4 2.58x104 1600 1900 1.97x105

Table 5.4 Loading cases

1 2 3 4 5

Ο - - - -

Ο Ο Ο Ο Ο

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 do Through the case 1 to the case 5, the first crack occurs when the applied load reaches about 133.7 kN, 128.5 kN, 120.7 kN, 114.8 kN, 102.8 kN, respectively It is apparently shown that the load carrying capacity of a beam will be higher when the first crack occurs at the higher applied load, and it will be lower when the first crack occurs at the lower applied load It is also seen

Fig.5.7 Layout scheme of two span continuous beams with external cables

AB

from Fig.5.8a that the deflection of the beam increases with decreasing the loading ratio after cracking A lesser ultimate deflection is found in the case of balanced loading arrangement as compared to the other cases The analytical results reproduce the experimental data with remarkably good accuracy for the case of balanced loading arrangement

Fig.5.8b shows the increase of cable stress against the applied loads It can be seen that the stress in the external cable increases very little so that it still remains in the elastic range at the ultimate state The rate of stress increase in a cable develops very slowly before the decompression for all the cases 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 increase of cable stress is the greatest in the case of beam with the balanced loading arrangement as compared to the other cases This is because the increase of cable stress is a function of the overall deformation of the beam as shown in Eq.(3.42) Hence, a bigger deflection at the both spans could induce a greater stress increase in a cable Although beams with the unbalanced loading arrangement have a bigger deflection on the left span (heavily

a) Load-deflection relationship b) Load-increase of cable stress

c) Increase of cable stress-deflection d) Distribution displacement along the beam

Fig.5.8 Effect of loading arrangement on behavior of beam prestressed with external cables

Increase of cable stress [N/mm2]

Increase of cable stress [N/mm2]

Exp.

loaded span), the deflection on the lightly loaded span, however, has usually the upward deflection as shown in Fig.5.8d This reason may be caused an adverse effect on the increase of stress in the external cables The adverse effect of the stress increase in the external cables was also confirmed by experiments conducted by Aparicio, A.C., et al.60), which reported in the technical literature, recently Since the cable continues from one end to the other end of the beam, when the beam is subject to the unbalanced loading arrangement, the cable tends to move from the lightly loaded span to the heavily loaded span through the center-supported section, i.e., the redistribution of cable strain in a cable obviously takes place Consequently, this will generally result in a small change of cable stress The stress increase in the prestressing cable does not reach the yielding point at the ultimate state even in the case of beam with the balanced loading arrangement This phenomenon is agreed well with the previous findings27) that the stress variation in an external cable will never reach its yielding strength except in the case when the beam deflection can become extremely large

A fairly linear relationship between the increase of cable stress and the beam deflection is observed for all the cases as shown in Fig.5.8c 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 with balanced loading arrangement is quite different from the other cases A similar rate of stress increase is observed for all the cases with the unbalanced loading arrangement Because the deflection of beam increases noticeably after the crushing of concrete, the linear relationship, therefore, is terminated, as shown obviously in Fig.5.8c for the case with the balanced loading arrangement The analytical results are also represented the experimental data for the beam with the balanced loading arrangement with a remarkable accuracy

Fig.5.9 Distribution of moment along the beam

0.0=α

Distributions of displacement and moment along the beams for two span continuous beams with external cables are presented in Fig.5.8d and Fig.5.9, respectively Whether or not the predicted responses for the beams with unbalanced loading arrangement are true, because the experimental data are not available to compare with However, they show the proper trend for the two span continuous beams subjected to the unbalanced loading arrangement Finally, it is more important to note that the predicted responses of the beam with the balanced loading arrangement show very good agreement with the experimental data

5.3 CONCLUDING REMARKS

In this chapter, a parametric study is performed with emphasis on the effects of friction at the deviators and loading arrangement for the beams prestressed with external cables Both the effects of friction at the deviators and loading arrangement on the ultimate strength and increase of cable stress are clarified The following conclusions are made from the study discussed in this chapter

In consideration of friction at the deviators, cables with free slip and slip with friction produce more or less equalized stress increase at all the loading stage, and very similar to the experimental observations While cables with consideration of either partially fixed or perfectly fixed at the deviators overestimate the stress increase as well as ultimate strength of the beam The ultimate strength of the beam is influenced greatly by free length of cable between two successive deviators, i.e., depending on the distance between the deviator points The fixity of cable at the deviators does affect mainly on the stress increase in the external

cables rather than on the ultimate strength of the beam when the ratio of Ld/L is considerably

large However, both the ultimate strength of the beam and increase of cable stress are

improvably increased due to the fixity of cable at the deviators when the ratio of Ld/L is

reduced It should be noted that the assumption of either free slip or slip with friction seems to be more realistic rather than the assumption of perfectly fixed at the deviators in the numerical analysis

In consideration of the effect of loading arrangement for two span continuous beams having cables continued from one end to the other end, the load carrying capacity of the beams and negative moment at the center-supported section reduces with decreasing the loading ratio A high load carrying capacity and a less deflection can be found in the case of the beam with the balanced loading arrangement A smaller increase of stress cable is found

in the cases two span continuous beams with unbalanced loading arrangement as compared with the beam with the balanced loading arrangement It should be noted that for two span continuous beams with unbalanced loading arrangement, the stress increase in the external cables never reaches the yielding strength of cable even at the collapsed stage The predicted results reproduce the experimental data for the beam with the balanced loading arrangement with remarkably good accuracy

It is experimentally shown that in the typical beams, the cable stress does not reach the yielding strength at the ultimate limit state Such as the cable’s material properties are not effectively utilized One of the possible methods of enhancing the strength of beams prestressed with external cables is to place the cable with large eccentricities The possibility can only do when external prestressing is used, since the cables need not be arranged within the depth of the beam section By this methodology, either improvement in the strength or reduction in the amount of prestressing can be achieved, leading to economical structures

It is also shown in the previous studies72, 77~79) that the eccentricity of the cable is one of the most important factors that influenced on the ultimate strength of prestressed beams with external cables By increasing the cable eccentricity, the substantial increase in strength can be obtained with the same effective prestressing force On the other hand, targeting the same

ultimate strength, the amount of prestressing force can be significantly reduced by increasing the eccentricity of cable In the case of typical beams, the eccentricity of cable is limited to the depth of the beam However, in the case of beams with large eccentricities, since the deviators are located outside the depth of the beam, there is no limitation in the eccentricity of cable, theoretically When the cable is provided at a large eccentricity, there may be restrictions in the amount of prestressing force because of large camber or cracking in the top fibers may be induced during prestressing

For the typical beams, the analytical methodology has been presented in Chapter 3, and numerous analyses of examples are also demonstrated in Chapter 4 and Chapter 5 In this chapter, the first attempt is made to propose an equation for the cable strain in the analysis of beam with large eccentricities The applications of the proposed equation in the analysis of several examples are then performed to verify its accuracy Finally, the analytical results are discussed at the end of this chapter with emphasis on the influence of cable eccentricity on the ultimate strength of beams prestressed with external cables

6.2 PROBLEM DEFINITION

Like the typical beams, the external cables are unbonded to the concrete because they are located outside the beam section Therefore, the same parameters that are known to influence on the behavior of the typical beam are expected to influence on the beam with large eccentricities A method for the non-linear analysis of the beam with large eccentricities is similar in principle to that of the analytical methodology for the typical beam, which has been presented in Chapter 3 Although the behavior of beams with large eccentricities is somewhat similar to the typical beams, the only additional point is to be considered in the case of beams with large eccentricities, namely, eccentricity of the cable due to the location of the external cables

In the typical beams, the external cables are located outside the beam but within the depth of cross section Therefore, the strain variation in a cable can be determined on the basis of deformation compatibility between the concrete and the cable, which is expressed in Eq.(3.42), i.e., the strain variation in a cable depends on the overall deformation of beam On the other hand, in the case of beams with large eccentricities, since the cable portions are almost located outside the depth of cross section, the concrete, therefore, does not exist at the cable level Therefore, it is difficult to say, however, that the global compatibility requirement between the concrete and the cable, which has been proposed for the analysis of the typical beams, is

appropriately satisfied in the case of beam with large eccentricities Hence, the overall deformation of beam in terms of the concrete strain, which is usually used in the calculation of cable strain for the typical beams, cannot be used in the analysis of the beams with large eccentricities This makes a main difference in the analytical methodology for the both kinds of beams with external cables

Therefore, there is a need to have a computation method for the cable strain that should take account of the cable eccentricity; friction at the deviators and continuity of the structure, and the method can be used in the analysis of both kinds of the beams with external cables To satisfy these conditions, the elongation of the cable must be consistent with its deflection and to that effect, geometrical deformation of the cable must be correctly evaluated regardless of the deformed shape of the beam One of the possibilities is to be assumed that the cable strain depends only on the deformation of points, to which the cable is attached It turns out that the cable strain depends on the total length variation of cable between the extreme ends Therefore, the previous equation for the cable strain, which is to be used in the analysis of the typical beams, will be modified and presented hereinafter

6.3 STRAIN COMPATIBILITY OF A CABLE

6.3.1 Review of computing method for cable strain

A large number of analytical models for prestressed concrete beams with external cables were carried out in the past There were, however, extremely few analytical models for the beams with large eccentricities Here, only the most relevant models to the current research are briefly reviewed in order to have a general image of the computing method for the cable strain

In many studies20~24, 79), when the behavior of typical beams was investigated, most analytical approaches are usually based on the deformation compatibility of beam, i.e., the total elongation of cable elements must be equal to the total elongation of concrete elements at the cable level between the extreme ends, and its mathematical expression is shown in Eq.(3.42) of Chapter 3 This assumption is considered to be the effective tools for the evaluation of cable strain in the analysis of the typical beams and beams with unbonded cables, and good agreement with the experimental data has been reported, previously It also points out from the analytical results that the strain variation in a cable depends mainly on the overall deformation of beams because of the lack of bond between the concrete and the cable

Aravinthan, T., et al.72, 77~79) extended the approach of deformation compatibility of beam for the analysis of the beams with large eccentricities by the additional assumption of an imaginary concrete strain at the portion of cable, at which the concrete does not exist (see Fig.6.1) This extension, however, seems to be limited because of difficulties in defining the value of imaginary concrete strain at the cable level In the case the beam having external cables placed at very large eccentricity as compared to the depth of the beam, and the application of this assumption seems to be not appropriated to apply, evidently Because the imaginary strain of concrete at the cable level is too big as the concrete strain of the extreme fibers changes a little

Virlogeux, M.2~3) proposed another approach based on the geometrical change of external cable Due to the rectilinear shape of external cable between the points, at which the cable attaches to the concrete beam, the strain variation of cable can be defined on the basis of deformation of attachment points Therefore, the cable strain can be evaluated regardless of the overall deformation of the beam, and it depends only on the deformation of deviator points The cable length variation in each segment between two successive deviators can be defined as:

This approach seems to be more realistic than the assumption of an imaginary concrete strain at the cable level, which was proposed by Aravinthan Because it is based on the change of geometry of external cable as the applied load increases, and it is rather easy to visualize how the stress in the external cables can be increased at any stage of loading However, no movement of cable at the deviators should be assumed while calculating the cable strain

Fig.6.1 Imaginary concrete strain at the cable level

External CableConcrete beam