Introduction
As is well known, China has had the most prosperous period of bridge engineering construction in the world during the past few decades. Cable-stayed bridges are the most widely built type. As their spans longer and longer, the towers become taller and taller. Meanwhile, it is preferred that hollow sections be used in the tower design in order to reduce the weight of the bridge. For most engineering practices, towers are usually required to remain elastic, even under occasional happend earthquakes. However, strong earthquakes, i.e., near-fault ground motions with large pulses lasting for 2 or more seconds [
1] and approximating to the fundamental period of the tower [
2], can drive the tower into strong vibration, introducing the possibility that the tower may yield and develop plasticity and eventually collapse.
China has many bridges built or under construction in locations where strong earthquakes are likely to occur, such as in the Sichuan Province, where the well-known Wenchuan earthquake occurred (May 12, 2008) [
3]. Therefore, the investigation of seismic performance, especially the seismic design of such bridges to ensure their performance under strong earthquakes, has attracted more and more attention from the bridge engineering and research communities [
2,
4].
This paper presents the first of a series of studies on this subject. One practical cable-stayed bridge with a 730-m long main span and two high-rise towers over 200 m (see Fig. 1) was selected for the initial case study. First, the seismic responses of the bridge under site-specific ground motions were obtained with the finite element method according to the current design principles. Then, the same analytical model was used to conduct additional time history analyses, including a study of the p-delta effect, with a focus on towers under strong pulse-type ground motions. Finally, the unscaled record was input into the model to observe the differences in the seismic behaivier of this bridge.
Description of the bridge
Bridge configuration
This bridge is a very typical Chinese cable-stayed bridge with two towers and a continuous steel box girder. It consists of one 730-m main navigation span and two 240-m auxiliary spans, plus two 110-m transitional spans. The general layout is shown in Fig. 2.
The towers are an inverted Y shape with a hollow section and are 203.17 m high in total and 148.5 m high above the bridge deck. The elevation and cross sections of the tower are shown in Fig. 3. The towers are made of C50 concrete, but with a steel skeleton in the upper part to provide a cable anchorage area. The auxiliary pier is 43.50 m high, and the side pier is 46.68 m high. Both are made of a thin-walled C40 concrete box section with dimensions of 7 × 5.5 m and a thickness of 70 cm.
The foundation underneath the tower consists of 60 D2.5~3.2 drilled C30 concrete piles with a length of 104 m.
The foundation of each pier is also made of 18 (12 for the side pier) D2.5~3.2 drilled C30 concrete piles with a length of 80 m.
Bridge FEM model
Based on the aforementioned introduction and additional design details, the concrete towers, steel girder and concrete piers are modeled by 3D beam elements, the cables are modeled by tension-only elements with strengthened pretension forces. The bridge has very strong pile-foundations that are also modeled by 3D beam elements above the scour line, assuming a certain degree of scour depth in the design life cycle. Figure 4 shows the FEM model for the seismic response analysis in this paper. Table 1 lists the boundary conditions.
Dynamic characteristics
A model analysis is first run to obtain the basic dynamic characteristics of the bridge. The result shows that the fundamental period of this bridge reaches to 14 s by moving of the main girder along with the bending of the towers longitudinally, and then 4.7 s in transverse. The vibration mode of the towers appears at 4.3 s in transverse direction, followed by the third mode of the main girder involving vertical bending at 4.6 s.
This information indicates that the bridge is a super-long-period structure and that it may be very vulnerable to ground motions with very low frequency contents, which large displacements are mainly responsible for generating. Bridge structures are more vulnerable to transverse ground shaking than to ground motion in the longitudinal direction. In this case study, because of the single-column type (in both directions) of high-rise towers, as can be expected, the bridge will have considerable seismic responses, especially in terms of displacements, if transversely excited.
Seismic responses
Herein, an overview of the seismic performance of this bridge under site-specific ground motions is presented [
5] for the purpose of comparison.
Seismic design criteria and requirements
According to the report provided by the owner, this bridge is located in a moderately active earthquake region, with magnitude 7 on the Richter scale. Because no design specifications are available for long-span bridges in China (also true in other counties worldwide) [
6-
8], the design follows a two-level seismic criteria and a two-stage design requirement [
9,
10], as specified by the owner. The first seismic criteria level is a 10% possibility of exceedance in 100 years (P1), i.e., a 950-year return period; the second level is 3% possibility of exceedance in 100 years (P2), i.e., 3283-year return period. Correspondingly, the first design stage is to ensure the strength of the bridge’s main components so that they do not enter the inelastic range under the first seismic criteria level; the second design stage is to ensure the structure’s ductility, avoiding unexpected collapse of the deck from the side spans under the second seismic design criteria level.
Evaluation of seismic performance under site-specific ground motions
Linear response spectrum analysis
In the seismic response analysis of this cable-stayed bridge, the response spectrum with a 5% damping ratio is shown in Fig. 5, corresponding to 10% probability of exceedance in 100 years (P1 level) and a 3% probability of exceedance in 100 years (P2 level), respectively.
Seismic response is analyzed using the multimode spectral analysis method with the lowest 800 principal natural modes of vibration considered and the CQC combination method adopted. The results show that the designed bridge can meet the required seismic performance under two design levels; however, large displacements occur both at the end of the girder (expressed as Db) and at the top of the towers (expressed as Dt). If excited in both the longitudinal and vertical directions, Db reaches 32 cm under the P1 level and 69 cm under the P2 level, while Dt reaches 36 cm under the P1 level and 75 cm under the P2 level. If excited in both the transverse and vertical directions, Db reaches 12 cm under the P1 level and 23 cm under the P2 level, while Dt reaches 41cm under the P1 level and 83 cm under the P2 level.
Based on this observation, it was suggested that two dampers be installed between the towers and the girder to help reduce the longitudinal movement of the deck. In addition, transverse excitation-induced responses in terms of axial forces, shear forces and bending moments at the end of tower columns and piers are comparably larger, which indicates that a further nonlinear time history analysis including plastic hinges may be needed.
Nonlinear time history analysis
By revising the linear beam elements of the tower columns and piers at the abovementioned areas into plastic-elastic beam elements and including nonlinear link elements to simulate the dampers, the FEM model was again run for nonlinear time history analysis to produce more “real” seismic responses. The results show that under the same longitudinal excitations, with damping coefficient C = 10000 kN/(m·s) and ξ = 0.3, the Dt is only 20 cm (P1) and 30 cm (P2), and the Db is 16 cm (P1) and 25 cm (P2). Meanwhile, the tower columns remain elastic under both longitudinal and transverse excitations; however, under P2 design level transverse excitations, the side pier and auxiliary pier columns yielded to form plastic hinges, but the ratio of capacity/requirement is more than 7, which indicates the safety margin is sufficient.
However, dampers will not change the transverse seismic responses by a large amount. The nonlinear time history analysis also gives the result of a 90-cm displacement at the top of two tower columns, which is almost the same result obtained from the response spectrum analysis. Therefore, compared with longitudinal input, the tower columns have a smaller safety margin; for instance, the smallest capacity/requirement ratio of the tower column is 1.6 in the transverse direction compared with 4.7 in the longitudinal direction.
According to all of the above seismic analysis results, this bridge can meet all the seismic requirements corresponding to two different design levels and hence was assessed as safe under the excitation of site-specific ground motions.
Seismic responses under strong earthquakes
Although the bridge was assessed to be safe under site-specific ground motions, as reported earlier [
2], the tower columns of cable-stayed bridges may yield if strong ground motions occur; plastic hinges will develop, and in that scenario, the towers will became the most vulnerable elements of the bridge and will increase the risk of losing stability during the earthquake shaking, especially under transverse and vertical excitations.
Based on the destructive earthquakes in 1994 (Northridge), 1995 (Kobe), and 1999 (Chi Chi), it is known that near-field/fault ground motions may cause severe damage to most civil engineering structures because of their special characteristics compared with far-field ground motions [
1,
4,
11]. One of these feature is the “pulse” effect, which has been studied by many researchers. The other is the comparably larger vertical ground motions. In typical engineering practice, when the vertical ground motions are not available, 1/2 or 2/3 of the horizontal motions are used in some seismic design specifications or guidelines, however, this may not be appropriate when the bridge site is close to the fault [
4].
Input ground motions
The strong earthquake selected in Fig. 6 is the Rinaldi record from the 1994 Northridge earthquake, which is one of the most important strong motion records in earthquake engineering (it is the largest recorded horizontal peak ground velocity in the western United States, ~170 cm/s). It has an “apparent” pulse and nearly the same peak vertical acceleration (0.852 g) as horizontal acceleration (0.838 g).
For comparison purposes, the acceleration time history is scaled to the same peak horizontal acceleration value corresponding to a P2 level for site-specific ground motions to investigate the “pulse” effect.
Seismic responses
Transverse input
Four cases (cases 1 and 2 correspond to site-specific input with and without the p-delta effect, respectively; cases 3 and 4 correspond to Rinaldi input with and without the p-delta effect, respectively) of nonlinear time history analysis were run to obtain the seismic responses of the bridge, with a special focus on tower columns under the selected strong earthquake.
Table 2 shows the axial forces and bending moments at the end of piers and tower columns in each case. It can be seen that the p-delta effect does not seem to influence the structure’s seismic responses (including displacements); however, the pulse-type Rinaldi input does have a considerable effect on the seismic responses of the bridge at the focused elements. If comparing the values in the table under case 3 to those of case 1, one finds that the “pulse” effect is different. For the tower column, this effect is positive. Even the acceleration input is scaled to the same value. The increase of the bending moment is over 100%. For piers, the effect is negative, which indicates the tower structure is more vulnerable to “pulse” effects because of its own dynamic characteristics. This, however, is not a common observation. Further investigations including additional strong ground motions are necessary.
It should be noted that although a large increase of the bending moment was induced by the Rinaldi input, a transverse displacement decrease of nearly 70% of the tower top was observed, as shown in Fig. 7. The reason for this occurrence might be due to the different frequency domains of the input, especially the low-frequency contents that contribute to the displacement.
Longitudinal input
As shown in Table 3, under the same PGA excited longitudinally, the pulse-type strong motion decreases the seismic responses to approximately 30%.
Meanwhile, the maximum longitudinal displacement between the tower and deck induced by the Rinaldi input is also half that of site-specific ground motions. If we look at the displacement time history between the tower and the deck under the P2 level Rinaldi input with dampers, the maximum is 17 cm, which indicates that there might be no need for dampers in this case.
If no dampers were installed, the corresponding displacement would be 65 cm for site-specific input and only 22 cm for the Rinaldi input. The seismic responses are shown in Table 3 in italics. From Table 3, one can find that for the Rinaldi input, the current consideration of adopting dampers does not seem to improve the bridge’s seismic performance.
Unscaled strong-motion input
Recalling that the smallest capacity for the seismic requirement ratio of the tower column is 1.6 transversely under site-specific input, if using the unscaled record as the transverse input, the bending moment at the tower bottom will reach nearly 4 times that value, which means they may already have yielded and undergone considerable plasticity development. Meanwhile, in the longitudinal direction, even though dampers were adopted to control the relative tower-deck displacement, a large one-sided offset of approximately 1 m is still observed. Comparing the damping hysteretic loops (see Fig. 8) with those of site-specific ground input, the Rinaldi record causes unsymmetrical energy dissipation because of the “pulse” effect. More importantly, dampers work well with site ground motion but do not work well with “pulse”-like strong motion, which indicates that reoptimization of the damper system or even reconsideration of alternative energy dissipation devices will be needed.
Conclusions
In an attempt to study the seismic behavior of cable-stayed bridges with high-rise towers under strong ground motions, this paper describes the modeling of an existing bridge excited by pulse-type ground motion. The preliminary results show that the tower is vulnerable to transverse earthquake shaking and that it may yield and develop plastic hinges at certain locations under strong ground motions. In addition, because the damper behaves quite differently under pulse-like ground motion, reoptimization of damper parameters or reconsideration of other energy dissipation devices will be needed. However, it should be mentioned that this conclusion might be immature. Further investigations that include plasticity development of tower columns in the analytical model excited by additional strong ground motions are necessary.
Higher Education Press and Springer-Verlag Berlin Heidelberg
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