Introduction
Recently, many highways and railways have been built in western China because of the rapid development of the local economy. Because most of the area is mountainous terrain, a high percentage of bridges have to be used; in some area, the total length of bridges even exceeds 30 percent of the length of the entire road. Most of these bridges are constructed as continuous girder bridges or as frame bridges. However, these bridges are quite different from those built on plain-type terrain; to span some deep valleys, nearly 40 percent of bridges have a pier height exceeding 40 m. Some of them are even higher than 100 m. For instance, the highest pier on the Luohe Highway Bridge in China is approximately 143 m high, and the highest pier on the Huatupo Railway Bridge is approximately 110 m high. These high piers usually have a very high slenderness ratio, and this changes the dynamic characteristics of the whole structure and subsequently results in a different structural performance under earthquake action. Moreover, most of the mountainous areas in western China are high-activity seismic zones. Therefore, the seismic response analysis and rational design measures for these high pier bridges are a major concern.
During the past three decades, considerable experimental and theoretical research on the seismic behavior of girder bridges has been carried out [
1,
2]. Some useful seismic design strategies, such as the ductility or displacement design method, the capacity design method and so on, have been presented, widely used, and even accepted by some design codes. However, most of these strategies apply only to conventional bridges. In the newly revised Guidelines for Seismic Design of Highway Bridges in China, it is required that the maximum pier height not exceed 40 m or that specified research work needs to be conducted [
3]. Although no pier height limit is clearly specified in the AASHTO Guide Specifications for LRFD Seismic Bridge Design, this guideline should only be assumed to apply to normal bridges [
4]. Ceravolo et al. [
5] investigated a bridge with tall piers and pointed out that pier dynamics would govern the seismic behavior and that the current design strategy was inadequate. Li et al. [
6] compared the seismic behavior of shorter and taller piers and concluded that both the seismic demand and the capacity of taller piers were quite different from shorter piers because of the effects of the higher modes on the seismic response.
In this paper, the incremental dynamic analysis (IDA) method is used to evaluate the seismic performance of tall bridge piers, from the initial elastic state to the ultimate state. The effects of higher-order modes are specifically discussed. Moreover, the validity of pushover analysis in determining the local displacement capacity recommended in the code is also discussed.
Analysis models
Two multispan continuous-deck highway bridges were investigated and compared in this study, one with a pier height of 30 m, and the other with a pier height of 60 m. As is well known, bridges have a nonuniform layout of structural mass or stiffness, such as equivalent pier heights, which leads to serious higher-mode effects. However, this is due to the irregularity of the whole structure rather than to an element with a high slenderness ratio. To avoid this kind of influence, all of the piers for each bridge were assumed to be equivalent in height. Therefore, both of the two bridges can be modeled as a cantilever column with a distributed mass along the element and an equivalent superstructure concentrated mass at the top [
7]. Figure 1 shows the detailed dimensions of the two columns. The longitudinal reinforcement ratio of the two columns is equal to 1.27%, and the transverse reinforcement ratio is equal to 0.6%. To capture the higher-mode vibrations and the nonlinear behavior, the two piers were sufficiently subdivided, into 30 elements for the 30 m height pier and 60 elements for the 60 m height pier. Furthermore, for each element, proper plastic hinge properties were calculated based on a moment-curvature analysis.
An IDA was used to investigate the performance of the two bridges from the initial elastic state to the ultimate limit state. This approach is a parametric analysis method that has been widely used recently to estimate structural performance under seismic loads more thoroughly. It involves subjecting a structural model to one or more ground motion records, with each record scaled to multiple levels of intensity, thus producing one or more curves of response parameterized versus intensity level [
8]. Most of all, it is a dynamic analysis procedure, thus the effect of higher modes can be taken into account in this method.
Considering the randomness of earthquake ground motions, six earthquake records were selected with different magnitudes, peak ground acceleration (PGA) values and predominant periods, as shown in Table 1. It can be seen that the predominant period of the six selected earthquake records vary from 0.1 s to 0.92 s, representing different ground conditions. Each record was scaled into 25 grades in PGA, from 0.1 g to 2.5 g.
Seismic response
Figures 2 and 3 show the elastic seismic moment of the 30 m and 60 m high pier, respectively, in which the results based on the first mode and on multimodes are compared. It can be seen that the two results for the 30 m high pier are quite similar, and this indicates that the elastic seismic response of the 30 m high pier can be well predicted by the first mode. However, the responses of the 60 m high pier are quite different: an additional S-shaped distribution of the seismic moment occurred in the multimode response. This clearly reveals the significant higher-order mode effect result from the tall pier.
Figures 4 and 5 show the IDA results of the 30 m and 60 m high pier under the ground motion of El Centro, respectively. With the increase of the PGA, the moment distributions of both the 30 m and 60 m high piers basically retain their original shapes, even when the structures enter into an inelastic state. Moreover, Figs. 6 and 7 show the moment distribution of the two piers under the selected six records corresponding to a scaled PGA equal to 1.0 g. It is clear from the figures that the moment distributions of the 30 m pier are quite similar, while those of the 60 m pier are quite different. Because both of the piers have entered into an inelastic state, this indicates that the 30 m pier experiences almost no higher-mode effects in the inelastic stage, while the 60 m pier experiences a serious higher-mode effect. Figure 7 also highlights the high sensitivity of the higher-mode effect to the ground motion.
Furthermore, in the case of the 30 m pier, only one plastic hinge occurs at the pier bottom until the ultimate failure. However, for the 60 m pier, after the plastic hinge at the bottom formed initially, an additional plastic hinge then occurred halfway up the pier. This is quite different from the response seen for conventional bridge piers, and the difference can be attributed to the higher-mode effects. Figure 8 shows the deformation development of the two hinges in the case of the 60 m pier with the increase of PGA of the six selected ground motion records. It can be observed that in the cases of E1, E3, E4 and E6, the plastic rotation of the bottom hinge is always larger than that of the additional hinge at the middle height, while in the cases of E2 and E5, even though the bottom hinge occurs first, the rotation of the addition hinge exceeds that of the bottom hinge at 1.1g and 1.3 g, respectively. This result demonstrates that the additional hinge at the middle height of the pier may be more critical than the hinge at the bottom. Therefore, care should be taken in designing the transverse reinforcement of tall piers at the middle height, or brittle fracture may occur.
Discussion of ultimate deformation capacity
To avoid collapse of the structure, it is necessary to check the deformation state of the hinge and to ensure that it is not exceeding the ultimate state. In a conventional bridge, this is usually accomplished by equivalently checking the displacement demand versus its capacity at the pier top. The displacement capacity of a pier is usually determined by a static pushover analysis. This method is based on an assumption of the one-to-one relationship between the hinge rotation at the bottom and the displacement at the pier top. However, whether it is suitable for tall piers is not yet clear because of the significant higher-mode effects.
Figures 9 and 10 show the relationship between the maximum rotation of the critical plastic and the displacement at the pier top of the 30 m and 60 m piers, respectively, in which the curves based on the IDA are compared with those calculated by a static pushover analysis based on the first mode. The results show that in the case of the 30 m pier, the dynamic and static methods yield similar results, while for the 60 m pier, they are quite different, especially for the E1, E2, E5 and E6 cases in which the displacement predicted by the static method is nearly twice as high as those based on the dynamic analysis at the same hinge rotation. This illustrates that the displacement capacity would be extremely overestimated if the design procedure used for conventional bridge piers were applied to tall piers. Because the dynamic analysis can take into account all mode effects, while only the first mode is considered in the static pushover analysis, this difference can also be attributed to the higher-mode effects.
A further investigation of the time-history responses, the rotation of the bottom hinge and the displacement at the top (E2 earthquake, PGA equal to 1.5 g), are shown in Fig. 11 for the 60 m pier. It can be seen that the two deformation variables do not vary simultaneously. The top displacement reached its maximum at 9.09 s, while the maximum bottom hinge rotation was reached at 11.06 s. The deformation of the whole pier at the two instants, i.e., 9.09 s and 11.06 s, is shown in Fig. 12. Furthermore, the deformation based on the static pushover analysis is also shown in Fig. 12; the rotation of the bottom hinge reaches the same value equal to the maximum value in the dynamic analysis. It is clear from the figure that the three deformation curves are completely different, especially the dynamic analysis curve at 11.06 s and the static pushover analysis curve. Even for the same bottom hinge rotation, the difference in the top displacement can be two times larger. In the figure, a contraflexure tendency can be seen in the dynamic analysis curves. This reflects the fact that the higher-mode effects are important.
Therefore, it can be concluded that the displacement-based design procedure used in conventional bridge piers cannot be applied to tall piers directly, or serious unsafe results may occur. It is recommended that a nonlinear time-history analysis be conducted and that the ultimate deformation of the critical hinge be checked directly.
Conclusions
This work primarily focused on the seismic performance of tall piers. A conventional-height pier and a tall pier model were developed and investigated through an IDA. The results showed that the higher modes had a significant effect on tall piers and that these modes changed the seismic behavior of tall piers in a fundamental way. An additional plastic hinge was formed midway up the pier, and in some cases, the rotation deformation of this hinge was found to be even higher than that of the bottom hinge. Moreover, the higher-mode effects also lead to an out-of-phase response between the rotation of the critical hinge and the displacement at the pier top, subsequently resulting in a large error if the conventional displacement-based design method is used. For the seismic design of tall bridge piers, it is recommended that a nonlinear time-history analysis be conducted and that the ultimate deformation of the critical hinge be checked directly.
Higher Education Press and Springer-Verlag Berlin Heidelberg
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