- Email: [email protected]
Experimental and numerical study on isolated simply-supported bridges subjected to a fault rupture
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Experimental and numerical study on isolated simply-supported bridges subjected to a fault rupture
T
Jiang Yia,b, Huaiyu Yangb,c, Jianzhong Lib,∗ a
College of Civil Engineering, Guangzhou University, Guangzhou, 510006, Guangdong, China State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China c China Railway 15 Bureaus Group Co., Ltd, Shanghai, 200070, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Fault rupture Isolated simply-supported bridge Lead rubber bearing Shake table test Seismic performance
To investigate the effects of fault crossing on the seismically isolated bridges, shake table testing was conducted on a 1/10 scaled two-span simply-supported bridge model isolated by lead rubber bearings (LRBs). A synthetic fault rupture, consisting of low- and high-frequency simulations, was used to excite the test model from low to high amplitude. Test results revealed that lead rubber bearings are effective in protecting the girders and the piers of the bridge subject to fault rupture, but at the cost of large peak and residual bearing deformation or even the failure of LRBs. The bearings at near fault (NF) span are more susceptible to fault rupture than the crossing fault (CF) span because the participation of longitudinal response compensates the transverse seismic demand of the bearings at CF span. Two numerical models were constructed with differing modeling consideration of LRBs: a sophisticated one using Bouc-wen model and a simplified one using Bilinear model. Both numerical models were able to predict the behavior of test model equally well before the failure of the bearings, validating that the existing nonlinear analytical techniques are adequate to estimate the seismic response of bridges subjected to a fault rupture.
1. Introduction Despite many seismic design codes recommended to prevent bridge construction across faults, it was not feasible to avoid the entire fault zone by revising the roadway alignments. For example, it is estimated that more than 5% of all bridges in California either cross faults or lie in very close proximity to fault-rupture zones [1]. Bridges crossing faultrupture zones will experience ground offset across the fault [2,3] and can be quite vulnerable during an earthquake. As shown by the bridge damages from the recent earthquakes, including the 1999 Chi-Chi earthquake [4], 1999 Kocaeli earthquake [5], 1999 Duzce earthquake [6], and 2008 Wenchuan earthquake [7], bridges crossing earthquake faults might be severely damaged or even collapse under a rupture of causative faults. The seismic damage called for appropriate measures to protect the bridges subjected to a fault rupture. Generally, the bridges subjected to a fault rupture can be divided into two categories: ordinary bridges and seismically isolated bridges. Several studies have been conducted to evaluate the seismic response of ordinary bridges near or across a fault rupture. Goel and Chopra [1,8] proposed a simple design approach for ordinary bridges crossing fault rupture zones by considering two specific load cases in addition to the
∗
standard loading defined in design codes of California. Both linear and non-linear analysis procedures can be used to estimate seismic demands for bridges crossing fault-rupture zones. They [9] further examined the role of shear keys at bridge abutments in the seismic behavior of ordinary bridges crossing fault-rupture zones and revealed that two shearkey conditions, no shear keys and elastic shear keys, were required to estimate the upper bounds of seismic demands. Osmar Rodriguez [10] used both response spectrum analysis and linear static analysis to evaluate seismic demand of a three-span curved bridge crossing faultrupture. Besides numerical analysis, shake table testing was conducted on a large-scale two-span bridge model subjected to ground motions that simulated fault rupture. Based on the experimental results, the adequacy of existing linear and nonlinear analytical techniques was evaluated and verified in estimating the seismic response of reinforced concrete bridges subjected to a fault rupture [11]. However, as indicated by numerical and shake test results, fault rupture could cause a substantial amount of ductile inelastic deformation in bridge piers of an ordinary bridge, and even led to the failure of the bridge. To reduce the damage concentration at the piers of an ordinary bridge, an effective way is to adopt an isolated bridge, which utilizes a seismic dissipation device that limits forces transferred from the
Corresponding author. E-mail addresses: [email protected] (J. Yi), [email protected] (H. Yang), [email protected] (J. Li).
https://doi.org/10.1016/j.soildyn.2019.105819 Received 29 October 2018; Received in revised form 23 May 2019; Accepted 13 August 2019 0267-7261/ © 2019 Elsevier Ltd. All rights reserved.
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
2. Bridge specimen
bridge model, in which the girder points were named G-Ã G-D, and the bearings were numbered B-1~B-4, while the piers were numbered P1~P-3, for later illustration. The test model is designed using Buckingham π theorem of dimensional analysis [14]. A hollow regular box section with 10 mm thick steel plate was used for the specimen. The dimensions of the box were designed so that the bending moments of inertia about both the strong and weak axes matched the inertias of the prototype at a scaled level. Besides, four mass blocks, each with a mass of 2t, was added to the superstructure in consideration of scaling rules (the volume of the concrete was 1/1000th of the prototype while the area was 1/100th the prototype area). These mass blocks were placed at the top of the girder near bearings. The longitudinal gap between two girders was 1 cm, exactly scaling down from the prototype bridge. The height of each pier measured from the centerline of cap beam to the foundation top is 311 cm. The rectangle cross section at the bottom of P1 had a width of 65 cm and a depth of 29 cm while that of P2 and P3 was 65 cm and 38 cm, respectively. To simplify the construction, longitudinal reinforcement with a diameter of 10 mm was distributed along the outside and inside of the hollow section. The number and the arrangement of longitudinal reinforcement were properly designed so that the pier of the test model would have the same bending capacity exactly scaling down from the prototype bridge. As a result, the longitudinal steel ratio of P1 was 2.06% and that of P2 and P3 was 1.89%. Transverse steel was well arranged to increase plastic-hinge rotation capacity and to prevent shear failure. The width of pier section was enlarged to 147 cm at the top of the pier, where two bearings were placed with a transverse distance of 50 cm. Since the prototype bridge had a rather firm foundation, the bases of the specimen piers were rigidly fixed to each of the three shake tables. A strong beam at the base was provided for the fixed condition and was also used to move the models onto and off the table, through a hook-up system.
2.1. Prototype bridge
2.3. Lead rubber bearing design (LRB)
The prototype bridge is an approach bridge of Puqian Bridge, which is located at Puqian sea area in the northeast of Hainan province, China. It's a multi-span simply-supported with prefabricated steel box girders, as shown in Fig. 1. The span length was 60 m and the substructure was a hollow-section single-column on group-pile foundations. Based on Bridge Site Seismic Evaluation Report from the local authority, the bridge transversely spans an active fault. To investigate fault-rupture effect on the seismic performance of the bridge, two spans, with one across the fault and the other one adjacent to the fault, were extracted as the specimen for shake table tests (see Fig. 1).
Lead rubber bearing (LRB), since its invention by Robinson [15], has been extensively applied to bridges to mitigate the magnitude of the seismic force transmitted to the substructure. It carries the weight of the structure and supplies a horizontal restoring force while the plastic deformation of the lead produces hysteretic damping, thus giving a relatively cheap choice for seismic isolation. This study uses LRBs to connect the steel girder and the piers, as shown in Fig. 3(a). Note that two concrete shear keys at the top of each pier are provided, which provided extra protection of the girder in case of bearing failure during the experiments, thus guaranteeing the safety. The gap between the girder and shear keys are large enough so that shear keys would not interact during the tests. The LRBs of the model were designed according to the bearing configuration of the prototype bridge. The configuration detail of LRBs is shown in Fig. 3 (b). The bearing has a plan dimension of 75 mm by 75 mm and an effective height of 29 mm. The bearings consisted of several layers of steel shim and elastomer, two steel plates distributed at the top and the bottom, and one lead core with a diameter of 20 mm. The bearing dimensions, elastomer layer thickness and steel shim thickness all satisfy the tolerance requirements specified in the Chinese bearing guidelines [16]. Before shake table tests, monotonic and cyclic tests were conducted on LRBs so as to capture the lateral force-deformation behavior of the bearings. The testing setup and protocols were established following the procedure specified in the code [16]. Vertical static loading, 26 kN, equal to bearing force at dead load state, was first imposed to the bearing specimen. Then, cyclical horizontal force was applied to the specimen for three times, with each time the maximum deformation of the bearings equal to the height of bearings. The force-deformation behavior of the bearings is reached using the data of the third cyclical test, as shown in Fig. 4. In Fig. 4, a bilinear model with strain hardening
superstructure to the substructure while accommodating the concomitant displacement. Limited work has been conducted regarding seismic response of seismically isolated bridges near or across a fault rupture. Ucak [12] investigated the seismic response of Bolu Viaduct, a seismically isolated bridge with a fault crossed and revealed that isolation system displacement was doubled when fault crossing is considered as compared to the case when fault crossing is ignored. They also evaluate the effect of key parameters, including fault crossing permanent tectonic displacement, fault crossing location and fault crossing angle on the response of the bridge. Recently, Shuo Yang [13] conducted an investigation on this bridge utilizing a near-fault ground motion record processed with and without a displacement offset. They addressed the influence of permanent ground displacement on the dynamic response of spatially extended engineering structures crossing fault rupture zones. However, a rational seismic design philosophy for bridges crossing active faults has not yet been established and no shake table tests have been reported on the effects of fault rupture on seismically isolated bridges. This study consisted of shake table testing of a large-scale two-span simply-supported bridge model isolated by lead rubber bearings (LRBs) subjected to a simulated fault rupture. The design of the shake table model was introduced and the generation of ground motions that simulated fault rupture was presented. The test model was excited by the simulated fault rupture from low to high amplitude subsequently until the failure of bearings. Test results were presented and seismic response of isolated bridge spans near and cross a fault rupture was compared. Furthermore, numerical analysis was conducted to evaluate the adequacy of existing analytical techniques in estimating the rupture-induced responses of isolated bridges. The main objectives of this study are to investigate the effects of fault crossing on the seismically isolated bridges.
2.2. Test model The test model modeled the specimen bridge at 1/10 scale to maximize the size of the prototype bridge, while remaining within the limits of the shake tables. Fig. 2 presents the design details of the tested
Fig. 1. Prototype bridge. 2
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 2. Geometry of two-span simple-supported bridge model (unit: cm).
is used to approximately fit the behavior of LRBs, which showed a good agreement. Table 1 lists the mechanical parameters used to fit lateral force-deformation properties of LRBs, including yield force, elastic stiffness, post-elastic stiffness and post-elastic ratio. These parameters are used for numerical analysis as will be illustrated later.
3. Shake table tests 3.1. Synthetic ground motions To date, ground motions have merely been recorded in close
Fig. 3. Design of Lead-rubber bearings. 3
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
fling step. Using the Next Generation Attenuation (NGA) database [20], the record in the fault parallel direction at Pump Station #09 in Nenana Mountain Earthquake (Mw = 6.7, 2002) was selected. Fig. 5(b) shows the time histories of the high-frequency ground motion. The broad band wave is obtained by combining the results of the low- and high-frequency ground motion using matched filtering at across over frequency of 1HZ. Fig. 5(c) shows the acceleration, velocity and displacement histories of broad band wave. Note that the permanent displacement in each record is only slightly lower than the peak displacement. 3.2. Testing protocol In the experiments, ground displacement histories are scaled to apply to the shake tables. Based on scale effect of the test model, the time axis of the prototype motion was compressed by 0.3162 (1/ 10 ). The fault rupture was simulated between P-2 and P-3, as shown in Fig. 6. The seismic input at P-1 and P-2 of near fault span (NF span) was in the same direction while P-3 at crossing fault span (CF span) had inverse excitations. The simulated fault rupture was applied to the test model from low to high amplitude in the displacement form, as listed in Table 2. Three cases were included for the bridge system by multiplying an amplification factor and the synthetic ground motion. Only the fault parallel components of the simulated ground motions were applied in shake table tests, which meant that only the transverse seismic response of the bridge model was considered in this study. Note that Table 2 also lists measured rupture displacements and the ratio between the measured and target values. With the difference less than 7%, it is concluded that the measured and target values are closely matched.
Fig. 4. Force-Displacement response of the bearing specimen at third cycle. Table 1 Lateral force-deformation properties of LRBs. Yield force Fy/ kN
Elastic stiffness K1/kN·m−1
Post-elastic stiffness Kd/kN·m−1
Kd/K1 α
3.0
1500
180
0.12
proximity to the causative fault. To investigate fault rupture effect on the seismic response of the bridges, a synthetic broad band wave was generated to simulate fault parallel component of ground motion. The broad band wave consists of low- and high-frequency simulations, which are illustrated respectively as follows: The low-frequency ground motion is simulated to represent fault offsets during the earthquake. A fourth degree polynomial function proposed by Vaez [17] is utilized in this study, as:
⎧V ⎪ p v (t ) =
4
2 λTp 2
( ) ⎡⎣ (t − t ) − ( ) ⎤⎦ cos ⎡⎣2π 4 λTp
0
2
4
⎨ < t ≤ t0 + λ Tp and λ ≥ 1 4 ⎪ ⎩ 0,else
t Tp
3.3. Instrumentations During the test, the recorded response of the test model included the absolute displacement responses of the superstructure and piers and force response of the bearings. The displacement responses were recorded with displacement transducers in both longitudinal and transverse directions. These displacement transducers were distributed at deck middle, deck ends and at different elevations of the piers. In total, 12 displacement transducers were used for deck displacement and 16 for piers. The force response of the bearings was recorded with tri-axial load transducers. These load transducers were installed in the vertical align with each bearing as shown in Fig. 3 (a) and had the relatively large stiffness so as not to affect the force-displacement relationship of the bearings.
λ
+ ν ⎤, t0 − 4 Tp ⎦ (1)
where parameter Vp is the amplitude of the signal, Tp is the prevailing period of the signal, λ is oscillatory characteristics of the signal, t0 specifies the epoch of the envelope's peak and ν is the phase of the amplitude-modulated harmonic. Eq. (1) is adopted in this study because it is not only simple in form but also simulates the long-period portion of actual velocity near fault records with a high level of precision. In Eq. (1), there are five variables to be determined. In general, Vp and Tp are two controlling factors to determine the overall variability of the spectral amplitude whereas the rest three parameters are of less importance. To simplify the simulation, following design parameters are set first, t0 = 20s, λ = 1 and ν = 0. Meanwhile, Vp is set to be 100 cm/s, which is a characteristic value that represents the peak amplitudes of the majority of the available near-fault velocity records [18]. However, Tp varies significantly for different earthquakes. It is revealed by Somerville [19] that both Tp and the permanent slip of the fault rupture (D) are strongly correlated with the moment magnitude (Mw), as:
Tp = 2 × 100.5(Mw − 6.69)
(2)
D = 100.5(Mw − 2.91)
(3)
4. Observations During the test, it was observed that NF span had obvious translation under the uniform excitation while CF span had in-plan rotation along the vertical direction under the non-uniform excitation. For all the cases, the amplitude of girder movement of the NF span was larger than that of CF span. There was no pounding between two adjacent girders or the girder and shear keys, as expected. For all runs, the girders and the piers remained in good shape without observed damage while there was some residual deformation after each run, indicating damage of the bearings during the tests. When the amplitude of ground motions was small (from Run 1 to Run 2), the residual deformation of LRBs was small and the bearings remained in good shape after the excitation. When the fault rupture increased to 200 mm (Run 3), however, the bearing residual deformation increased extensively and some obvious damage occurred. It was observed that B-1 and B-2 located at NF span suffered severe damage while B-3 and B-4 at CF span remained in the good configuration during the test. Fig. 7 shows the sample bearing profiles (B-2 at NF span and B4 at CF span) at the end of Run 3 and after detaching the test mode. Fig. 7 (a) shows that at the end of Run 3, the residual bearing
In this study, Mw was set 6.91 so that the permanent slip of the fault rupture (D) was 100 cm. As a result, Tp is calculated as 1.288s. The simulated low-frequency wave is shown in Fig. 5(a). The high-frequency ground motion is substituted by a real far-field ground motion record. A far-field ground motion is used here because it has minimum directivity effect in their signal before the addition of the 4
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 5. Simulated broad band wave for earthquake inputs.
5.1. Modeling of the bearings When modeling LRBs, generally there are two models that are widely used: Bouc-Wen model [22] and Bilinear model. In Bouc-Wen model, for LRBs with a symmetry shape, the restoring forces in two orthogonal directions are expressed as [23,24]:
fx = αkp u x + (1 − α ) kp zx Fig. 6. Fault locations for ground motion simulation.
(4a)
and
f y = αkp u y + (1 − α ) kp zy
Table 2 Loading protocol.
(4b)
Run number
Amplification factor
Fault rupture/ mm
Measured rupture/ mm
Ratio
where zx and zy = internal variables related to the restoring forces along the x and y directions, respectively; kp = initial stiffness and α = post-to-pre-yielding stiffness ratio. The orthotropic system can be transformed into an equivalent isotropic system, as:
Run 1 Run 2 Run 3
AF = 1.0 AF = 1.5 AF = 2.0
100 150 200
93.3 142.3 191.3
0.933 0.949 0.957
f = αkp u + (1 − α ) kp z
(5)
where [25].
zx = z cos θ, u x = u cos θ and zy = z sin θ and u y = u sin θ deformation of B-2 was quite large in the transverse direction, almost equal to the height of the bearing. At the top of the bearing, nearly half of the elastomer layer loss contact to the top steel plate. Contrary to B-2, B-4 at CF span had dominating residual deformation in the longitudinal direction after the test. The residual deformation was around half of the height of the bearing. After detaching the bridge model, B-2 was severely damaged with the top elastomer layer torn apart, as seen from Fig. 7 (b). while B-4 was undamaged. Due to failure of B-1 and B-2 at Run 3, the shake table tests ceased.
(6)
and
z˙ = Au˙ − γ|u˙ |z|z˙|n − 1 − βu˙ |z|n
(7)
where A, β, γ, n are dimensionless quantities controlling the scale and shape of the hysteresis loop. The commonly selected parameters for lead rubber bearings are A = 1, β = γ = 0.5, and n = 2 [26], which reduces Eq. (7) into:
(1 − z 2) u˙ if u˙ ⋅z > 0 z˙ = ⎧ ⎨ otherwise u˙ ⎩
5. Numerical model
(8)
For Bilinear model, the seismic behavior of the bearing is modeled by an elastic model with strain hardening, as:
The response of the bridge model under fault-rupture excitation was calculated using OpenSees [21]. Two numerical models were established in this study based on the differing sophistications of modeling of the LRBs, as follows.
kp u if u < Δy f=⎧ αkp u + (1 − α ) kp Δy if u ≥ Δy ⎨ ⎩ 5
(9)
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 7. Bearing profiles during the test.
5.3. Establishment of the models
where Δy = the yield displacement of the LRBs. Eq. (9) is applied to two orthogonal directions of the bearings. Parameters from Table 1 are used to define Bouc-wen model and Bilinear model. Fig. 8 shows the comparison of two models. Seen from Fig. 8(a), comparing to Bilinear model, Bouc-wen model produces a smooth transition from the elastic to plastic state. Except that, both models share the same hysteretic curve in the elastic state and plastic state. Fig. 8 (b) implies the yield surface of Bouc-wen model is a circle, which represents equal yield capacity along any direction. However, for Bilinear model, the yield surface is a square, indicating no interaction along two orthotropic directions.
Considering different constitute models used for modeling the bearing element, two parallel numerical models were established: Model I using Bouc-model to model the bearing element and Model II using Bilinear model. Two numerical models had the same dimensions, linear girder properties, nonlinear pier properties, and mass distribution, except for modeling aspects of the bearings. In this way, Model I represents a sophisticated modeling of the LRB, which is used for prediction and evaluation of the response of the test model in this study whereas Model II uses a simple modeling of the bearings and is more likely to be used in a design setting. The completed finite element model is shown in Fig. 9. To determine the adequacy of above numerical in estimating the seismic response of isolated bridges subjected to a fault rupture, numerical and experimental results were compared. Achieved shake table motions were used to excite the numerical model.
5.2. Modeling of other components Since the girders remained elastic during the test, elastic frame elements are utilized to model the behavior of the girders. The added mass is treated as lumped mass placed at the central of the mass block, which is 0.5 m away from the nearest bearing. To account for the material nonlinearities inherent in bridge piers, the fiber based approach is implemented to capture the axial-flexural response. This approach is established by discretizing the section into unidirectional confined concrete, unconfined concrete and steel fibers. Both unconfined and confined concrete is modeled by using the uniaxial Kent-Scott-Park concrete material [27], while the uniaxial Giuffre-Menegotto-Pinto steel material is used to simulate the stress-strain response of steel reinforcement. Since the test model was fixed on the shake tables, the restraints of the supports in the numerical models were fixed.
6. Shake table test and numerical results 6.1. Test results A summary of the maximum response for all of the shake table tests is shown in Table 3. The position of G-Ã G-D, B-1~B-4 and P-1~P4 were depicted in Fig. 2, in which G-A, G-B, B-1 and B-2 were located at NF span whereas G-C, G-D, B-3 and B-4 at CF span. In Table 3, the absolute displacement of the girder was measured to depict a global view of the response of the bridge while the relative displacement at
Fig. 8. Comparison of Bouc-wen model and Bilinear model. 6
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 9. Finite element model constructed in OpenSees.
the span length (6 m), is 2.3° at Run 3. Such rotational response of the girder leads to pronounced biaxial response of the bearings (B-3 and B-4) in both longitudinal and transverse direction. For example, the longitudinal deformation of B-3 is 18 mm at Run 3, which was comparable to transverse deformation (21 mm). On the other hand, subjected to uniform seismic excitations, the NF span had dominated translational response in the transverse direction while the bearing longitudinal deformation is far smaller than transverse deformation to be negligible. (2) Along with the low-frequency simulations (around time = 12.5–13s), there is a sudden increase of girder absolute displacement, bearing longitudinal and transverse deformation and
pier top (equal to the absolute displacement minus the shake table displacement) was recorded since the it was directly related to the damage state of the pier. Table 3 also lists numerical results of Model I and Model II, which, however, are not discussed in this section but will be illustrated later. Sample time history displacement and deformation responses from test results are shown in Fig. 10. Some conclusions can be drawn from measured results: (1) For CF span, due to inverse seismic inputs, G-C and G-D had just opposite peak displacement, revealing horizontal rotation around the vertical axis of the CF span. The maximum rotational angle, computing as the ratio of relative displacement of G-C and G-D and Table 3 Peak displacement (mm)- Measured result (Model I) [Model II]. Girder transverse displacement (absolute value) Run number
G-A
Run 1 Run 2 Run 3
83 116 148
G-B (73) (111) (151)
[75] [113] [155]
Bearing transverse deformation Run number B-1 Run 1 Run 2 Run 3
22 35 44
2 4 16
(74) (101) (140)
[74] [104] [144]
B-2 (19) (31) (43)
[21] [35] [48]
Bearing longitudinal deformation Run number B-1 Run 1 Run 2 Run 3
88 111 138
G-C
(2) (4) (6)
35 38 40
6 7 8
(28) (42) (56)
[31] [45] [61]
1.77 2.16 2.62
(1.56) (2.47) (3.53)
[1.27] [2.20] [3.16]
2.18 1.66 2.63
[73] [97] [127]
19 20 21
(2) (4) (7)
[2] [3] [4]
9 14 18
(19) (19) (28)
[20] [22] [31]
[1.42] [2.38] [3.43]
Note: (·) and [·] are the numerical result of Model I and Model II, respectively. 7
1.27 1.42 1.74
(-73) (-98) (-123)
[-73] [-97] [−127]
22 23 23
(21) (22) (33)
[21] [26] [39]
(8) (12) (16)
[10] [13] [19]
B-4 (9) (12) (16)
[10] [13] [19]
(0.83) (1.26) (1.69)
[0.88] [1.40] [1.93]
P-3 (1.67) (2.60) (3.67)
−76 −101 −127
B-4
B-3
Pier transverse displacement (value relative to shake tables) Run number P-1 P-2 Run 1 Run 2 Run 3
(73) (98) (124)
B-3
B-2 [1] [2] [4]
68 92 115
G-D
8 11 13
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 10. Time history responses of shake table test results (Run 2).
Fig. 11. Measured force-deformation hysteretic curves of B-2 at Run 3.
pier relative displacement, clearly indicating the influence of fault offsets. (3) As comparing two spans, NF span had larger response than CF span for each run of the girder transverse displacement, bearing transverse deformation and pier displacement, while the longitudinal bearing deformation of CF span is more pronounced than NF span. Meanwhile, the bearings at CF span had comparable deformation in both transverse and longitudinal direction during the low-frequency inputs. The sound participation of longitudinal response, e.g. force resistance and energy dissipation, compensated the large demand in the transverse direction. (4) For NF span, the largest bearing transverse deformation (B-1 and B2) occurred in coincidence with the low-frequency inputs. The sudden increase of bearing deformation due to fault offsets led to a severe situation of the bearings at NF span, which might be the
Fig. 12. Comparison of test results with both computer models.
cause of the failure of B-2 at Run 3 (as shown in Fig. 7). In particular, the maximum shear strain, equal to the ratio of maximum displacement and total thickness of the elastomer layers of the bearing (20 mm, as shown in Fig. 3) [28], reached 200% during the test. Such results highlight that the bearings at NF span are prone to damage under fault-ruptures. In practical engineering, on one hand these bearings should be designed with large deformation capacity, possibly larger than 300% of bearing height (based on the numerical results), to accommodate the large deformation demands from fault-ruptures. On the other hand, appropriate overlap lengths should be provided to avoid unseating in case of bearing failure. Fig. 11 further shows the bearing force-deformation hysteretic 8
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 13. Sample recorded responses (Measured result vs. Model II results) at Run 1 (Rupture = 100 mm).
acceptable for shake table tests [11]. But at Run 3, there is some increased discrepancy between numerical models and test model, especially for the bearing transverse and longitudinal deformation. The maximum difference exceeds 50%. The reason is that B-2 was severely damaged at Run 3, which lose proper function and significantly altered the seismic response of the bridge. However, numerical models do not consider the failure of the bearings, resulting in the difference between the calculated and measured results at this high intensity level. Also note in Fig. 12 that, despite different levels of sophistication for bearing modeling, Model I and Model II result in similar good quality predictions for the behavior of test model at Run 1 and Run2. It's concluded that Model II, which used simplified bilinear bearing model in two orthotropic directions, would be useful to practicing engineers for designing the isolated bridges subjected to a fault rupture. To strengthen above conclusion, a more comprehensive comparison of the measured and numerical response histories at Run 1 was shown in Fig. 13. Note in Fig. 13 that, only numerical results of Model II were presented in the interest of clarity while Model I yielded similar results. Fig. 13 reveals that, although there is a minor difference in history
curves at Run 3. It can be seen that B-2 suffered sudden force loss during the excitation, capturing the damage of the bearings. (5) For all the piers, the maximum pier drift ratio, defined as the percentage of pier-top displacement to the pier length, was within 0.1%. Such drift ratio is far less than yield drift ratio, 0.7% (from pushover analysis result), indicating the piers were within elastic range during the test. 6.2. Comparison of numerical and test results As can be seen in Table 3, Mode I and Model II, despite their differing levels of sophistication, result in similar good quality predictions for the behavior of bridges at Run 1 and Run 2. Fig. 12 shows a comparison of some of the detailed results of two different models with respect to shake table test results. As this figure shows, both models provide a good representation of the response of the test model in girder transverse displacement, bearing transverse and longitudinal deformation, and pier transverse displacement response at Run 1 and Run 2, with the maximum difference less than 20%. Such difference is 9
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 13. (continued)
curves, numerical histories of Model II generally catch the main features (including the peak response, the shape of history curves) of the measured results including the girder displacement, bearing longitudinal and transverse deformation, and pier transverse displacement. One exception is the pier transverse displacement at P-3, where numerical result fails to capture the measured histories. It is difficult to figure out the actual cause to such discrepancy. The possible reasons might be that the bottom of P-3 and the shake table were not well mounted or there was the disturbance of related displacement transducers during the test. Nevertheless, the overall well-fitted curves of other histories indicate a close agreement between numerical and measured results. Fig. 14 further compares the bearing force-deformation hysteretic response of measured and numerical results at Run 1 (Rupture = 100 mm). In this figure, both hysteretic curves of Model I and Model II are presented for comparison. It is revealed that Model I and Model II provided similar predications for bearing hysteretic curves of test results, except that Model I captured the smooth transition of hysteretic curve at peak response. 7. Conclusion This study presents both experimental and numerical analysis on an isolated, two-span, simply-supported bridge model subjected to a fault rupture. Following conclusions are drawn:
Fig. 14. Bearing force-deformation hysteretic curves of measured and numerical results at Run 1 (Rupture = 100 mm).
(1) When subjected to a fault rupture, the lead rubber bearings of the 10
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
bridge model experienced large residual deformation or even failed at high amplitude while the girders and the piers were well protected and remained within elastic range. (2) To accommodate significant dislocation of the fault rupture, the CF span had pronounced rotational response, resulting in coupled response of the bearings in the longitudinal and transverse directions. (3) For NF span, fault offsets caused a sudden increase of bearing response, leading to the failure of the LRBs under a fault rupture with a large amplitude. Meanwhile, for CF span, the participation of longitudinal response compensates the transverse seismic demand of the bearings and the bearings were less damaged than NF span. (4) Modeling the bearings with either Bouc-wen model or Bilinear model yields similar reasonable prediction for the response of the test model before the failure of the bearings. It's concluded that Bilinear model, which neglects the biaxial interaction in two orthotropic directions, would be useful to practicing engineers for designing the isolated bridges subjected to a fault rupture.
Civ Inf Eng 2001:149–86. [6] Ghasemi H, Cooper JD, Imbsen R, et al. The November 1999 Duzce earthquake: post-earthquake investigation of the structures in the TEM. Washington, D.C.: U.S. Dept. of Transportation, Federal Highway Administration; 2000. Publication No. FHwA-RD00-146. [7] Kawashima K, Takahashi Y, Ge H, et al. Reconnaissance report on damage of bridges in 2008 Wenchuan, China, earthquake. J Earthq Eng 2009;13(7):965–96. [8] Goel RK, Chopra AK. Nonlinear analysis of ordinary bridges crossing fault-rupture zones. J Bridge Eng 2009;14(3):216–24. [9] Goel RK, Chopra AK. Role of shear keys in seismic behavior of bridges crossing faultrupture zones. J Bridge Eng 2008;13(4):398–408. [10] Rodriguez O, Qu B, Goel RK. Seismic demand evaluation for a three-span curved bridge crossing fault-rupture zones. Struct Congr 2012:516–26. [11] Saiidi MS, Vosooghi A, Choi H, et al. Shake table studies and analysis of a two-span RC bridge model subjected to a fault rupture. J Bridge Eng 2013;19(8):A4014003. [12] Ucak A, Mavroeidis GP, Tsopelas P. Behavior of a seismically isolated bridge crossing a fault rupture zone. Soil Dyn Earthq Eng 2014;57:164–78. [13] Yang S, Mavroeidis GP, Ucak A, et al. Effect of ground motion filtering on the dynamic response of a seismically isolated bridge with and without fault crossing considerations. Soil Dyn Earthq Eng 2017;92:183–91. [14] Moncarz PD, Krawinkler H. Theory and application of experimental model analysis in earthquake engineering vol. 50. California: Stanford University; 1981. [15] Robinson WH, Tucker AG. A lead-rubber shear damper. Bull N Z Natl Soc Earthq Eng 1976;4:251–9. [16] Ministry of Transport of P. R. China. Lead rubber bearing isolator for highway bridges (JTT822-2011). Beijing: People’s Communication Press; 2011. [17] Vaez SH, Sharbatdar MK, Amiri GG, et al. Dominant pulse simulation of near fault ground motions. Earthq Eng Eng Vib 2013;12(2):267–78. [18] Mavroeidis GP, Dong G, Papageorgiou AS. Near-fault ground motions, and the response of elastic and inelastic single-degree-of-freedom (SDOF) systems. Earthq Eng Struct Dyn 2004;33(9):1023–49. [19] Somerville P, Irikura K, Graves R, et al. Characterizing crustal earthquake slip models for the prediction of strong ground motion. Seismol Res Lett 1999;70(1):59–80. [20] PEER. (n. d.). Available at:. https://ngawest2.berkeley.edu/ Accessed on 09 Oct 2018. [21] Mazzoni S, McKenna F, Scott MH, et al. The open system for earthquake engineering simulation (OpenSEES) user command-language manual. 2006. [22] Wen YK. Method for random vibration of hysteretic systems. J Eng Mech Div 1976;102(2):249–63. [23] Park YJ, Wen YK, Ang HS. Random vibration of hysteretic systems under bi-directional ground motions. Earthq Eng Struct Dyn 1986;14(4):543–57. [24] Narasimhan S, Nagarajaiah S, Johnson EA, et al. “Smart” base isolated benchmark building. Part I: problem definition. Struct Control Health Monit 2006;13(2–3):573–88. [25] Lee CS, Hong HP. Statistics of inelastic responses of hysteretic systems under bidirectional seismic excitations. Eng Struct 2010;32(8):2074–86. [26] Mavronicola E, Komodromos P. The effect of non-linear parameters on the modeling of multi-storey seismically isolated buildings. Department of Civil and Environmental Engineering, University of Cyprus, 15th World Conference on Earthquake Engineering; 2012. [27] Kent DC, Park R. Flexural members with confined concrete. J Struct Div 1971;97:1969–90. [28] European Committee for Standardization. EN-1337-3: structural bearings-Part 3: elastomeric bearings, Brussels, Belgium. 2005.
A potential limitation of this study is that only the transverse fault rupture is considered for seismic excitation whereas the structure might suffer from both longitudinal and transverse earthquake excitations. Future experimental and analytical studies on this issue can be performed. Acknowledgements This research is supported by the Natural Science Foundation of China (No.51838010). The support is gratefully acknowledged. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.soildyn.2019.105819. References [1] Goel RK, Chopra AK. Linear analysis of ordinary bridges crossing fault-rupture zones. J Bridge Eng 2009;14(3):203–15. [2] Mavroeidis GP, Papageorgiou AS. A mathematical representation of near-fault ground motions. Bull Seismol Soc Am 2003;93(3):1099–131. [3] Oettle NK, Bray JD, Dreger DS. Dynamic effects of surface fault rupture interaction with structures. Soil Dyn Earthq Eng 2015;72:37–47. [4] Yen WH. Lessons learned about bridges from earthquake in Taiwan. Public Roads 2002;65(4). [5] Erdik M. Report on 1999 Kocaeli and düzce (Turkey) earthquakes. Struct Control
11
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Experimental and numerical study on isolated simply-supported bridges subjected to a fault rupture
T
Jiang Yia,b, Huaiyu Yangb,c, Jianzhong Lib,∗ a
College of Civil Engineering, Guangzhou University, Guangzhou, 510006, Guangdong, China State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China c China Railway 15 Bureaus Group Co., Ltd, Shanghai, 200070, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Fault rupture Isolated simply-supported bridge Lead rubber bearing Shake table test Seismic performance
To investigate the effects of fault crossing on the seismically isolated bridges, shake table testing was conducted on a 1/10 scaled two-span simply-supported bridge model isolated by lead rubber bearings (LRBs). A synthetic fault rupture, consisting of low- and high-frequency simulations, was used to excite the test model from low to high amplitude. Test results revealed that lead rubber bearings are effective in protecting the girders and the piers of the bridge subject to fault rupture, but at the cost of large peak and residual bearing deformation or even the failure of LRBs. The bearings at near fault (NF) span are more susceptible to fault rupture than the crossing fault (CF) span because the participation of longitudinal response compensates the transverse seismic demand of the bearings at CF span. Two numerical models were constructed with differing modeling consideration of LRBs: a sophisticated one using Bouc-wen model and a simplified one using Bilinear model. Both numerical models were able to predict the behavior of test model equally well before the failure of the bearings, validating that the existing nonlinear analytical techniques are adequate to estimate the seismic response of bridges subjected to a fault rupture.
1. Introduction Despite many seismic design codes recommended to prevent bridge construction across faults, it was not feasible to avoid the entire fault zone by revising the roadway alignments. For example, it is estimated that more than 5% of all bridges in California either cross faults or lie in very close proximity to fault-rupture zones [1]. Bridges crossing faultrupture zones will experience ground offset across the fault [2,3] and can be quite vulnerable during an earthquake. As shown by the bridge damages from the recent earthquakes, including the 1999 Chi-Chi earthquake [4], 1999 Kocaeli earthquake [5], 1999 Duzce earthquake [6], and 2008 Wenchuan earthquake [7], bridges crossing earthquake faults might be severely damaged or even collapse under a rupture of causative faults. The seismic damage called for appropriate measures to protect the bridges subjected to a fault rupture. Generally, the bridges subjected to a fault rupture can be divided into two categories: ordinary bridges and seismically isolated bridges. Several studies have been conducted to evaluate the seismic response of ordinary bridges near or across a fault rupture. Goel and Chopra [1,8] proposed a simple design approach for ordinary bridges crossing fault rupture zones by considering two specific load cases in addition to the
∗
standard loading defined in design codes of California. Both linear and non-linear analysis procedures can be used to estimate seismic demands for bridges crossing fault-rupture zones. They [9] further examined the role of shear keys at bridge abutments in the seismic behavior of ordinary bridges crossing fault-rupture zones and revealed that two shearkey conditions, no shear keys and elastic shear keys, were required to estimate the upper bounds of seismic demands. Osmar Rodriguez [10] used both response spectrum analysis and linear static analysis to evaluate seismic demand of a three-span curved bridge crossing faultrupture. Besides numerical analysis, shake table testing was conducted on a large-scale two-span bridge model subjected to ground motions that simulated fault rupture. Based on the experimental results, the adequacy of existing linear and nonlinear analytical techniques was evaluated and verified in estimating the seismic response of reinforced concrete bridges subjected to a fault rupture [11]. However, as indicated by numerical and shake test results, fault rupture could cause a substantial amount of ductile inelastic deformation in bridge piers of an ordinary bridge, and even led to the failure of the bridge. To reduce the damage concentration at the piers of an ordinary bridge, an effective way is to adopt an isolated bridge, which utilizes a seismic dissipation device that limits forces transferred from the
Corresponding author. E-mail addresses: [email protected] (J. Yi), [email protected] (H. Yang), [email protected] (J. Li).
https://doi.org/10.1016/j.soildyn.2019.105819 Received 29 October 2018; Received in revised form 23 May 2019; Accepted 13 August 2019 0267-7261/ © 2019 Elsevier Ltd. All rights reserved.
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
2. Bridge specimen
bridge model, in which the girder points were named G-Ã G-D, and the bearings were numbered B-1~B-4, while the piers were numbered P1~P-3, for later illustration. The test model is designed using Buckingham π theorem of dimensional analysis [14]. A hollow regular box section with 10 mm thick steel plate was used for the specimen. The dimensions of the box were designed so that the bending moments of inertia about both the strong and weak axes matched the inertias of the prototype at a scaled level. Besides, four mass blocks, each with a mass of 2t, was added to the superstructure in consideration of scaling rules (the volume of the concrete was 1/1000th of the prototype while the area was 1/100th the prototype area). These mass blocks were placed at the top of the girder near bearings. The longitudinal gap between two girders was 1 cm, exactly scaling down from the prototype bridge. The height of each pier measured from the centerline of cap beam to the foundation top is 311 cm. The rectangle cross section at the bottom of P1 had a width of 65 cm and a depth of 29 cm while that of P2 and P3 was 65 cm and 38 cm, respectively. To simplify the construction, longitudinal reinforcement with a diameter of 10 mm was distributed along the outside and inside of the hollow section. The number and the arrangement of longitudinal reinforcement were properly designed so that the pier of the test model would have the same bending capacity exactly scaling down from the prototype bridge. As a result, the longitudinal steel ratio of P1 was 2.06% and that of P2 and P3 was 1.89%. Transverse steel was well arranged to increase plastic-hinge rotation capacity and to prevent shear failure. The width of pier section was enlarged to 147 cm at the top of the pier, where two bearings were placed with a transverse distance of 50 cm. Since the prototype bridge had a rather firm foundation, the bases of the specimen piers were rigidly fixed to each of the three shake tables. A strong beam at the base was provided for the fixed condition and was also used to move the models onto and off the table, through a hook-up system.
2.1. Prototype bridge
2.3. Lead rubber bearing design (LRB)
The prototype bridge is an approach bridge of Puqian Bridge, which is located at Puqian sea area in the northeast of Hainan province, China. It's a multi-span simply-supported with prefabricated steel box girders, as shown in Fig. 1. The span length was 60 m and the substructure was a hollow-section single-column on group-pile foundations. Based on Bridge Site Seismic Evaluation Report from the local authority, the bridge transversely spans an active fault. To investigate fault-rupture effect on the seismic performance of the bridge, two spans, with one across the fault and the other one adjacent to the fault, were extracted as the specimen for shake table tests (see Fig. 1).
Lead rubber bearing (LRB), since its invention by Robinson [15], has been extensively applied to bridges to mitigate the magnitude of the seismic force transmitted to the substructure. It carries the weight of the structure and supplies a horizontal restoring force while the plastic deformation of the lead produces hysteretic damping, thus giving a relatively cheap choice for seismic isolation. This study uses LRBs to connect the steel girder and the piers, as shown in Fig. 3(a). Note that two concrete shear keys at the top of each pier are provided, which provided extra protection of the girder in case of bearing failure during the experiments, thus guaranteeing the safety. The gap between the girder and shear keys are large enough so that shear keys would not interact during the tests. The LRBs of the model were designed according to the bearing configuration of the prototype bridge. The configuration detail of LRBs is shown in Fig. 3 (b). The bearing has a plan dimension of 75 mm by 75 mm and an effective height of 29 mm. The bearings consisted of several layers of steel shim and elastomer, two steel plates distributed at the top and the bottom, and one lead core with a diameter of 20 mm. The bearing dimensions, elastomer layer thickness and steel shim thickness all satisfy the tolerance requirements specified in the Chinese bearing guidelines [16]. Before shake table tests, monotonic and cyclic tests were conducted on LRBs so as to capture the lateral force-deformation behavior of the bearings. The testing setup and protocols were established following the procedure specified in the code [16]. Vertical static loading, 26 kN, equal to bearing force at dead load state, was first imposed to the bearing specimen. Then, cyclical horizontal force was applied to the specimen for three times, with each time the maximum deformation of the bearings equal to the height of bearings. The force-deformation behavior of the bearings is reached using the data of the third cyclical test, as shown in Fig. 4. In Fig. 4, a bilinear model with strain hardening
superstructure to the substructure while accommodating the concomitant displacement. Limited work has been conducted regarding seismic response of seismically isolated bridges near or across a fault rupture. Ucak [12] investigated the seismic response of Bolu Viaduct, a seismically isolated bridge with a fault crossed and revealed that isolation system displacement was doubled when fault crossing is considered as compared to the case when fault crossing is ignored. They also evaluate the effect of key parameters, including fault crossing permanent tectonic displacement, fault crossing location and fault crossing angle on the response of the bridge. Recently, Shuo Yang [13] conducted an investigation on this bridge utilizing a near-fault ground motion record processed with and without a displacement offset. They addressed the influence of permanent ground displacement on the dynamic response of spatially extended engineering structures crossing fault rupture zones. However, a rational seismic design philosophy for bridges crossing active faults has not yet been established and no shake table tests have been reported on the effects of fault rupture on seismically isolated bridges. This study consisted of shake table testing of a large-scale two-span simply-supported bridge model isolated by lead rubber bearings (LRBs) subjected to a simulated fault rupture. The design of the shake table model was introduced and the generation of ground motions that simulated fault rupture was presented. The test model was excited by the simulated fault rupture from low to high amplitude subsequently until the failure of bearings. Test results were presented and seismic response of isolated bridge spans near and cross a fault rupture was compared. Furthermore, numerical analysis was conducted to evaluate the adequacy of existing analytical techniques in estimating the rupture-induced responses of isolated bridges. The main objectives of this study are to investigate the effects of fault crossing on the seismically isolated bridges.
2.2. Test model The test model modeled the specimen bridge at 1/10 scale to maximize the size of the prototype bridge, while remaining within the limits of the shake tables. Fig. 2 presents the design details of the tested
Fig. 1. Prototype bridge. 2
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 2. Geometry of two-span simple-supported bridge model (unit: cm).
is used to approximately fit the behavior of LRBs, which showed a good agreement. Table 1 lists the mechanical parameters used to fit lateral force-deformation properties of LRBs, including yield force, elastic stiffness, post-elastic stiffness and post-elastic ratio. These parameters are used for numerical analysis as will be illustrated later.
3. Shake table tests 3.1. Synthetic ground motions To date, ground motions have merely been recorded in close
Fig. 3. Design of Lead-rubber bearings. 3
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
fling step. Using the Next Generation Attenuation (NGA) database [20], the record in the fault parallel direction at Pump Station #09 in Nenana Mountain Earthquake (Mw = 6.7, 2002) was selected. Fig. 5(b) shows the time histories of the high-frequency ground motion. The broad band wave is obtained by combining the results of the low- and high-frequency ground motion using matched filtering at across over frequency of 1HZ. Fig. 5(c) shows the acceleration, velocity and displacement histories of broad band wave. Note that the permanent displacement in each record is only slightly lower than the peak displacement. 3.2. Testing protocol In the experiments, ground displacement histories are scaled to apply to the shake tables. Based on scale effect of the test model, the time axis of the prototype motion was compressed by 0.3162 (1/ 10 ). The fault rupture was simulated between P-2 and P-3, as shown in Fig. 6. The seismic input at P-1 and P-2 of near fault span (NF span) was in the same direction while P-3 at crossing fault span (CF span) had inverse excitations. The simulated fault rupture was applied to the test model from low to high amplitude in the displacement form, as listed in Table 2. Three cases were included for the bridge system by multiplying an amplification factor and the synthetic ground motion. Only the fault parallel components of the simulated ground motions were applied in shake table tests, which meant that only the transverse seismic response of the bridge model was considered in this study. Note that Table 2 also lists measured rupture displacements and the ratio between the measured and target values. With the difference less than 7%, it is concluded that the measured and target values are closely matched.
Fig. 4. Force-Displacement response of the bearing specimen at third cycle. Table 1 Lateral force-deformation properties of LRBs. Yield force Fy/ kN
Elastic stiffness K1/kN·m−1
Post-elastic stiffness Kd/kN·m−1
Kd/K1 α
3.0
1500
180
0.12
proximity to the causative fault. To investigate fault rupture effect on the seismic response of the bridges, a synthetic broad band wave was generated to simulate fault parallel component of ground motion. The broad band wave consists of low- and high-frequency simulations, which are illustrated respectively as follows: The low-frequency ground motion is simulated to represent fault offsets during the earthquake. A fourth degree polynomial function proposed by Vaez [17] is utilized in this study, as:
⎧V ⎪ p v (t ) =
4
2 λTp 2
( ) ⎡⎣ (t − t ) − ( ) ⎤⎦ cos ⎡⎣2π 4 λTp
0
2
4
⎨ < t ≤ t0 + λ Tp and λ ≥ 1 4 ⎪ ⎩ 0,else
t Tp
3.3. Instrumentations During the test, the recorded response of the test model included the absolute displacement responses of the superstructure and piers and force response of the bearings. The displacement responses were recorded with displacement transducers in both longitudinal and transverse directions. These displacement transducers were distributed at deck middle, deck ends and at different elevations of the piers. In total, 12 displacement transducers were used for deck displacement and 16 for piers. The force response of the bearings was recorded with tri-axial load transducers. These load transducers were installed in the vertical align with each bearing as shown in Fig. 3 (a) and had the relatively large stiffness so as not to affect the force-displacement relationship of the bearings.
λ
+ ν ⎤, t0 − 4 Tp ⎦ (1)
where parameter Vp is the amplitude of the signal, Tp is the prevailing period of the signal, λ is oscillatory characteristics of the signal, t0 specifies the epoch of the envelope's peak and ν is the phase of the amplitude-modulated harmonic. Eq. (1) is adopted in this study because it is not only simple in form but also simulates the long-period portion of actual velocity near fault records with a high level of precision. In Eq. (1), there are five variables to be determined. In general, Vp and Tp are two controlling factors to determine the overall variability of the spectral amplitude whereas the rest three parameters are of less importance. To simplify the simulation, following design parameters are set first, t0 = 20s, λ = 1 and ν = 0. Meanwhile, Vp is set to be 100 cm/s, which is a characteristic value that represents the peak amplitudes of the majority of the available near-fault velocity records [18]. However, Tp varies significantly for different earthquakes. It is revealed by Somerville [19] that both Tp and the permanent slip of the fault rupture (D) are strongly correlated with the moment magnitude (Mw), as:
Tp = 2 × 100.5(Mw − 6.69)
(2)
D = 100.5(Mw − 2.91)
(3)
4. Observations During the test, it was observed that NF span had obvious translation under the uniform excitation while CF span had in-plan rotation along the vertical direction under the non-uniform excitation. For all the cases, the amplitude of girder movement of the NF span was larger than that of CF span. There was no pounding between two adjacent girders or the girder and shear keys, as expected. For all runs, the girders and the piers remained in good shape without observed damage while there was some residual deformation after each run, indicating damage of the bearings during the tests. When the amplitude of ground motions was small (from Run 1 to Run 2), the residual deformation of LRBs was small and the bearings remained in good shape after the excitation. When the fault rupture increased to 200 mm (Run 3), however, the bearing residual deformation increased extensively and some obvious damage occurred. It was observed that B-1 and B-2 located at NF span suffered severe damage while B-3 and B-4 at CF span remained in the good configuration during the test. Fig. 7 shows the sample bearing profiles (B-2 at NF span and B4 at CF span) at the end of Run 3 and after detaching the test mode. Fig. 7 (a) shows that at the end of Run 3, the residual bearing
In this study, Mw was set 6.91 so that the permanent slip of the fault rupture (D) was 100 cm. As a result, Tp is calculated as 1.288s. The simulated low-frequency wave is shown in Fig. 5(a). The high-frequency ground motion is substituted by a real far-field ground motion record. A far-field ground motion is used here because it has minimum directivity effect in their signal before the addition of the 4
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 5. Simulated broad band wave for earthquake inputs.
5.1. Modeling of the bearings When modeling LRBs, generally there are two models that are widely used: Bouc-Wen model [22] and Bilinear model. In Bouc-Wen model, for LRBs with a symmetry shape, the restoring forces in two orthogonal directions are expressed as [23,24]:
fx = αkp u x + (1 − α ) kp zx Fig. 6. Fault locations for ground motion simulation.
(4a)
and
f y = αkp u y + (1 − α ) kp zy
Table 2 Loading protocol.
(4b)
Run number
Amplification factor
Fault rupture/ mm
Measured rupture/ mm
Ratio
where zx and zy = internal variables related to the restoring forces along the x and y directions, respectively; kp = initial stiffness and α = post-to-pre-yielding stiffness ratio. The orthotropic system can be transformed into an equivalent isotropic system, as:
Run 1 Run 2 Run 3
AF = 1.0 AF = 1.5 AF = 2.0
100 150 200
93.3 142.3 191.3
0.933 0.949 0.957
f = αkp u + (1 − α ) kp z
(5)
where [25].
zx = z cos θ, u x = u cos θ and zy = z sin θ and u y = u sin θ deformation of B-2 was quite large in the transverse direction, almost equal to the height of the bearing. At the top of the bearing, nearly half of the elastomer layer loss contact to the top steel plate. Contrary to B-2, B-4 at CF span had dominating residual deformation in the longitudinal direction after the test. The residual deformation was around half of the height of the bearing. After detaching the bridge model, B-2 was severely damaged with the top elastomer layer torn apart, as seen from Fig. 7 (b). while B-4 was undamaged. Due to failure of B-1 and B-2 at Run 3, the shake table tests ceased.
(6)
and
z˙ = Au˙ − γ|u˙ |z|z˙|n − 1 − βu˙ |z|n
(7)
where A, β, γ, n are dimensionless quantities controlling the scale and shape of the hysteresis loop. The commonly selected parameters for lead rubber bearings are A = 1, β = γ = 0.5, and n = 2 [26], which reduces Eq. (7) into:
(1 − z 2) u˙ if u˙ ⋅z > 0 z˙ = ⎧ ⎨ otherwise u˙ ⎩
5. Numerical model
(8)
For Bilinear model, the seismic behavior of the bearing is modeled by an elastic model with strain hardening, as:
The response of the bridge model under fault-rupture excitation was calculated using OpenSees [21]. Two numerical models were established in this study based on the differing sophistications of modeling of the LRBs, as follows.
kp u if u < Δy f=⎧ αkp u + (1 − α ) kp Δy if u ≥ Δy ⎨ ⎩ 5
(9)
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 7. Bearing profiles during the test.
5.3. Establishment of the models
where Δy = the yield displacement of the LRBs. Eq. (9) is applied to two orthogonal directions of the bearings. Parameters from Table 1 are used to define Bouc-wen model and Bilinear model. Fig. 8 shows the comparison of two models. Seen from Fig. 8(a), comparing to Bilinear model, Bouc-wen model produces a smooth transition from the elastic to plastic state. Except that, both models share the same hysteretic curve in the elastic state and plastic state. Fig. 8 (b) implies the yield surface of Bouc-wen model is a circle, which represents equal yield capacity along any direction. However, for Bilinear model, the yield surface is a square, indicating no interaction along two orthotropic directions.
Considering different constitute models used for modeling the bearing element, two parallel numerical models were established: Model I using Bouc-model to model the bearing element and Model II using Bilinear model. Two numerical models had the same dimensions, linear girder properties, nonlinear pier properties, and mass distribution, except for modeling aspects of the bearings. In this way, Model I represents a sophisticated modeling of the LRB, which is used for prediction and evaluation of the response of the test model in this study whereas Model II uses a simple modeling of the bearings and is more likely to be used in a design setting. The completed finite element model is shown in Fig. 9. To determine the adequacy of above numerical in estimating the seismic response of isolated bridges subjected to a fault rupture, numerical and experimental results were compared. Achieved shake table motions were used to excite the numerical model.
5.2. Modeling of other components Since the girders remained elastic during the test, elastic frame elements are utilized to model the behavior of the girders. The added mass is treated as lumped mass placed at the central of the mass block, which is 0.5 m away from the nearest bearing. To account for the material nonlinearities inherent in bridge piers, the fiber based approach is implemented to capture the axial-flexural response. This approach is established by discretizing the section into unidirectional confined concrete, unconfined concrete and steel fibers. Both unconfined and confined concrete is modeled by using the uniaxial Kent-Scott-Park concrete material [27], while the uniaxial Giuffre-Menegotto-Pinto steel material is used to simulate the stress-strain response of steel reinforcement. Since the test model was fixed on the shake tables, the restraints of the supports in the numerical models were fixed.
6. Shake table test and numerical results 6.1. Test results A summary of the maximum response for all of the shake table tests is shown in Table 3. The position of G-Ã G-D, B-1~B-4 and P-1~P4 were depicted in Fig. 2, in which G-A, G-B, B-1 and B-2 were located at NF span whereas G-C, G-D, B-3 and B-4 at CF span. In Table 3, the absolute displacement of the girder was measured to depict a global view of the response of the bridge while the relative displacement at
Fig. 8. Comparison of Bouc-wen model and Bilinear model. 6
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 9. Finite element model constructed in OpenSees.
the span length (6 m), is 2.3° at Run 3. Such rotational response of the girder leads to pronounced biaxial response of the bearings (B-3 and B-4) in both longitudinal and transverse direction. For example, the longitudinal deformation of B-3 is 18 mm at Run 3, which was comparable to transverse deformation (21 mm). On the other hand, subjected to uniform seismic excitations, the NF span had dominated translational response in the transverse direction while the bearing longitudinal deformation is far smaller than transverse deformation to be negligible. (2) Along with the low-frequency simulations (around time = 12.5–13s), there is a sudden increase of girder absolute displacement, bearing longitudinal and transverse deformation and
pier top (equal to the absolute displacement minus the shake table displacement) was recorded since the it was directly related to the damage state of the pier. Table 3 also lists numerical results of Model I and Model II, which, however, are not discussed in this section but will be illustrated later. Sample time history displacement and deformation responses from test results are shown in Fig. 10. Some conclusions can be drawn from measured results: (1) For CF span, due to inverse seismic inputs, G-C and G-D had just opposite peak displacement, revealing horizontal rotation around the vertical axis of the CF span. The maximum rotational angle, computing as the ratio of relative displacement of G-C and G-D and Table 3 Peak displacement (mm)- Measured result (Model I) [Model II]. Girder transverse displacement (absolute value) Run number
G-A
Run 1 Run 2 Run 3
83 116 148
G-B (73) (111) (151)
[75] [113] [155]
Bearing transverse deformation Run number B-1 Run 1 Run 2 Run 3
22 35 44
2 4 16
(74) (101) (140)
[74] [104] [144]
B-2 (19) (31) (43)
[21] [35] [48]
Bearing longitudinal deformation Run number B-1 Run 1 Run 2 Run 3
88 111 138
G-C
(2) (4) (6)
35 38 40
6 7 8
(28) (42) (56)
[31] [45] [61]
1.77 2.16 2.62
(1.56) (2.47) (3.53)
[1.27] [2.20] [3.16]
2.18 1.66 2.63
[73] [97] [127]
19 20 21
(2) (4) (7)
[2] [3] [4]
9 14 18
(19) (19) (28)
[20] [22] [31]
[1.42] [2.38] [3.43]
Note: (·) and [·] are the numerical result of Model I and Model II, respectively. 7
1.27 1.42 1.74
(-73) (-98) (-123)
[-73] [-97] [−127]
22 23 23
(21) (22) (33)
[21] [26] [39]
(8) (12) (16)
[10] [13] [19]
B-4 (9) (12) (16)
[10] [13] [19]
(0.83) (1.26) (1.69)
[0.88] [1.40] [1.93]
P-3 (1.67) (2.60) (3.67)
−76 −101 −127
B-4
B-3
Pier transverse displacement (value relative to shake tables) Run number P-1 P-2 Run 1 Run 2 Run 3
(73) (98) (124)
B-3
B-2 [1] [2] [4]
68 92 115
G-D
8 11 13
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 10. Time history responses of shake table test results (Run 2).
Fig. 11. Measured force-deformation hysteretic curves of B-2 at Run 3.
pier relative displacement, clearly indicating the influence of fault offsets. (3) As comparing two spans, NF span had larger response than CF span for each run of the girder transverse displacement, bearing transverse deformation and pier displacement, while the longitudinal bearing deformation of CF span is more pronounced than NF span. Meanwhile, the bearings at CF span had comparable deformation in both transverse and longitudinal direction during the low-frequency inputs. The sound participation of longitudinal response, e.g. force resistance and energy dissipation, compensated the large demand in the transverse direction. (4) For NF span, the largest bearing transverse deformation (B-1 and B2) occurred in coincidence with the low-frequency inputs. The sudden increase of bearing deformation due to fault offsets led to a severe situation of the bearings at NF span, which might be the
Fig. 12. Comparison of test results with both computer models.
cause of the failure of B-2 at Run 3 (as shown in Fig. 7). In particular, the maximum shear strain, equal to the ratio of maximum displacement and total thickness of the elastomer layers of the bearing (20 mm, as shown in Fig. 3) [28], reached 200% during the test. Such results highlight that the bearings at NF span are prone to damage under fault-ruptures. In practical engineering, on one hand these bearings should be designed with large deformation capacity, possibly larger than 300% of bearing height (based on the numerical results), to accommodate the large deformation demands from fault-ruptures. On the other hand, appropriate overlap lengths should be provided to avoid unseating in case of bearing failure. Fig. 11 further shows the bearing force-deformation hysteretic 8
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 13. Sample recorded responses (Measured result vs. Model II results) at Run 1 (Rupture = 100 mm).
acceptable for shake table tests [11]. But at Run 3, there is some increased discrepancy between numerical models and test model, especially for the bearing transverse and longitudinal deformation. The maximum difference exceeds 50%. The reason is that B-2 was severely damaged at Run 3, which lose proper function and significantly altered the seismic response of the bridge. However, numerical models do not consider the failure of the bearings, resulting in the difference between the calculated and measured results at this high intensity level. Also note in Fig. 12 that, despite different levels of sophistication for bearing modeling, Model I and Model II result in similar good quality predictions for the behavior of test model at Run 1 and Run2. It's concluded that Model II, which used simplified bilinear bearing model in two orthotropic directions, would be useful to practicing engineers for designing the isolated bridges subjected to a fault rupture. To strengthen above conclusion, a more comprehensive comparison of the measured and numerical response histories at Run 1 was shown in Fig. 13. Note in Fig. 13 that, only numerical results of Model II were presented in the interest of clarity while Model I yielded similar results. Fig. 13 reveals that, although there is a minor difference in history
curves at Run 3. It can be seen that B-2 suffered sudden force loss during the excitation, capturing the damage of the bearings. (5) For all the piers, the maximum pier drift ratio, defined as the percentage of pier-top displacement to the pier length, was within 0.1%. Such drift ratio is far less than yield drift ratio, 0.7% (from pushover analysis result), indicating the piers were within elastic range during the test. 6.2. Comparison of numerical and test results As can be seen in Table 3, Mode I and Model II, despite their differing levels of sophistication, result in similar good quality predictions for the behavior of bridges at Run 1 and Run 2. Fig. 12 shows a comparison of some of the detailed results of two different models with respect to shake table test results. As this figure shows, both models provide a good representation of the response of the test model in girder transverse displacement, bearing transverse and longitudinal deformation, and pier transverse displacement response at Run 1 and Run 2, with the maximum difference less than 20%. Such difference is 9
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
Fig. 13. (continued)
curves, numerical histories of Model II generally catch the main features (including the peak response, the shape of history curves) of the measured results including the girder displacement, bearing longitudinal and transverse deformation, and pier transverse displacement. One exception is the pier transverse displacement at P-3, where numerical result fails to capture the measured histories. It is difficult to figure out the actual cause to such discrepancy. The possible reasons might be that the bottom of P-3 and the shake table were not well mounted or there was the disturbance of related displacement transducers during the test. Nevertheless, the overall well-fitted curves of other histories indicate a close agreement between numerical and measured results. Fig. 14 further compares the bearing force-deformation hysteretic response of measured and numerical results at Run 1 (Rupture = 100 mm). In this figure, both hysteretic curves of Model I and Model II are presented for comparison. It is revealed that Model I and Model II provided similar predications for bearing hysteretic curves of test results, except that Model I captured the smooth transition of hysteretic curve at peak response. 7. Conclusion This study presents both experimental and numerical analysis on an isolated, two-span, simply-supported bridge model subjected to a fault rupture. Following conclusions are drawn:
Fig. 14. Bearing force-deformation hysteretic curves of measured and numerical results at Run 1 (Rupture = 100 mm).
(1) When subjected to a fault rupture, the lead rubber bearings of the 10
Soil Dynamics and Earthquake Engineering 127 (2019) 105819
J. Yi, et al.
bridge model experienced large residual deformation or even failed at high amplitude while the girders and the piers were well protected and remained within elastic range. (2) To accommodate significant dislocation of the fault rupture, the CF span had pronounced rotational response, resulting in coupled response of the bearings in the longitudinal and transverse directions. (3) For NF span, fault offsets caused a sudden increase of bearing response, leading to the failure of the LRBs under a fault rupture with a large amplitude. Meanwhile, for CF span, the participation of longitudinal response compensates the transverse seismic demand of the bearings and the bearings were less damaged than NF span. (4) Modeling the bearings with either Bouc-wen model or Bilinear model yields similar reasonable prediction for the response of the test model before the failure of the bearings. It's concluded that Bilinear model, which neglects the biaxial interaction in two orthotropic directions, would be useful to practicing engineers for designing the isolated bridges subjected to a fault rupture.
Civ Inf Eng 2001:149–86. [6] Ghasemi H, Cooper JD, Imbsen R, et al. The November 1999 Duzce earthquake: post-earthquake investigation of the structures in the TEM. Washington, D.C.: U.S. Dept. of Transportation, Federal Highway Administration; 2000. Publication No. FHwA-RD00-146. [7] Kawashima K, Takahashi Y, Ge H, et al. Reconnaissance report on damage of bridges in 2008 Wenchuan, China, earthquake. J Earthq Eng 2009;13(7):965–96. [8] Goel RK, Chopra AK. Nonlinear analysis of ordinary bridges crossing fault-rupture zones. J Bridge Eng 2009;14(3):216–24. [9] Goel RK, Chopra AK. Role of shear keys in seismic behavior of bridges crossing faultrupture zones. J Bridge Eng 2008;13(4):398–408. [10] Rodriguez O, Qu B, Goel RK. Seismic demand evaluation for a three-span curved bridge crossing fault-rupture zones. Struct Congr 2012:516–26. [11] Saiidi MS, Vosooghi A, Choi H, et al. Shake table studies and analysis of a two-span RC bridge model subjected to a fault rupture. J Bridge Eng 2013;19(8):A4014003. [12] Ucak A, Mavroeidis GP, Tsopelas P. Behavior of a seismically isolated bridge crossing a fault rupture zone. Soil Dyn Earthq Eng 2014;57:164–78. [13] Yang S, Mavroeidis GP, Ucak A, et al. Effect of ground motion filtering on the dynamic response of a seismically isolated bridge with and without fault crossing considerations. Soil Dyn Earthq Eng 2017;92:183–91. [14] Moncarz PD, Krawinkler H. Theory and application of experimental model analysis in earthquake engineering vol. 50. California: Stanford University; 1981. [15] Robinson WH, Tucker AG. A lead-rubber shear damper. Bull N Z Natl Soc Earthq Eng 1976;4:251–9. [16] Ministry of Transport of P. R. China. Lead rubber bearing isolator for highway bridges (JTT822-2011). Beijing: People’s Communication Press; 2011. [17] Vaez SH, Sharbatdar MK, Amiri GG, et al. Dominant pulse simulation of near fault ground motions. Earthq Eng Eng Vib 2013;12(2):267–78. [18] Mavroeidis GP, Dong G, Papageorgiou AS. Near-fault ground motions, and the response of elastic and inelastic single-degree-of-freedom (SDOF) systems. Earthq Eng Struct Dyn 2004;33(9):1023–49. [19] Somerville P, Irikura K, Graves R, et al. Characterizing crustal earthquake slip models for the prediction of strong ground motion. Seismol Res Lett 1999;70(1):59–80. [20] PEER. (n. d.). Available at:. https://ngawest2.berkeley.edu/ Accessed on 09 Oct 2018. [21] Mazzoni S, McKenna F, Scott MH, et al. The open system for earthquake engineering simulation (OpenSEES) user command-language manual. 2006. [22] Wen YK. Method for random vibration of hysteretic systems. J Eng Mech Div 1976;102(2):249–63. [23] Park YJ, Wen YK, Ang HS. Random vibration of hysteretic systems under bi-directional ground motions. Earthq Eng Struct Dyn 1986;14(4):543–57. [24] Narasimhan S, Nagarajaiah S, Johnson EA, et al. “Smart” base isolated benchmark building. Part I: problem definition. Struct Control Health Monit 2006;13(2–3):573–88. [25] Lee CS, Hong HP. Statistics of inelastic responses of hysteretic systems under bidirectional seismic excitations. Eng Struct 2010;32(8):2074–86. [26] Mavronicola E, Komodromos P. The effect of non-linear parameters on the modeling of multi-storey seismically isolated buildings. Department of Civil and Environmental Engineering, University of Cyprus, 15th World Conference on Earthquake Engineering; 2012. [27] Kent DC, Park R. Flexural members with confined concrete. J Struct Div 1971;97:1969–90. [28] European Committee for Standardization. EN-1337-3: structural bearings-Part 3: elastomeric bearings, Brussels, Belgium. 2005.
A potential limitation of this study is that only the transverse fault rupture is considered for seismic excitation whereas the structure might suffer from both longitudinal and transverse earthquake excitations. Future experimental and analytical studies on this issue can be performed. Acknowledgements This research is supported by the Natural Science Foundation of China (No.51838010). The support is gratefully acknowledged. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.soildyn.2019.105819. References [1] Goel RK, Chopra AK. Linear analysis of ordinary bridges crossing fault-rupture zones. J Bridge Eng 2009;14(3):203–15. [2] Mavroeidis GP, Papageorgiou AS. A mathematical representation of near-fault ground motions. Bull Seismol Soc Am 2003;93(3):1099–131. [3] Oettle NK, Bray JD, Dreger DS. Dynamic effects of surface fault rupture interaction with structures. Soil Dyn Earthq Eng 2015;72:37–47. [4] Yen WH. Lessons learned about bridges from earthquake in Taiwan. Public Roads 2002;65(4). [5] Erdik M. Report on 1999 Kocaeli and düzce (Turkey) earthquakes. Struct Control
11