This paper presents an improved fragility analysis methodology to estimate structural vulnerability for probabilistic seismic risk assessment. Three main features distinguish this study from previous efforts. Firstly, the updated fragility curves generated are based on experimental measurements and possess higher accuracy than those produced using design information only. The updated fragility curves take into consideration both the geometry and material properties, as well as long-term health monitoring data, to reflect the current state of the structure appropriately. Secondly, to avoid arbitrariness when selecting ground motions, probabilistic seismic hazard analysis (PSHA) is adopted to provide suggestions for ground motion selection. By considering the uncertainty of the location and intensity of future earthquakes, the PSHA deaggregation result can help to determine the most probable earthquake scenarios for the specific site. Thus, the suggested ground motions are more realistic, and the seismic demand model is much closer to the actual results. Thirdly, this study focuses on the seismic performance evaluation of a typical self-anchored suspension bridge using the form of fragility curves, which has seldom been studied in the literature. The results show that bearing is the most vulnerable part of a self-anchored suspension bridge, while failure probabilities of concrete towers are relatively lower.
Fragility curves are an important tool for representing the vulnerability of bridge structures in probabilistic seismic risk assessment. The curves are defined as a function of ground motion intensity parameters, such as peak ground acceleration (PGA) and spectral acceleration (SA), to describe the probability of exceeding a damage state. Three main methods have been developed to build fragility functions: expert opinion, empirical judgment, and analytical techniques [
Despite previous efforts to generate fragility curves using analytical models, fragility curves based only on design information may not represent the current behavior of the in-service structure. In addition, these curves fail to consider phenomena derived from structural stiffness degradation due to aging factors. With the advent of structural health monitoring (SHM) techniques, the dynamic properties of the bridge in its current state can be identified and monitored, providing the opportunity to generate a model with greater accuracy. Torbol et al. [
It is inappropriate to apply existing conclusions derived from regular girder bridges to suspension bridges directly, especially for different types of suspension bridges. Thus, more specialized research must focus on the suspension bridge type to assess seismic fragility and suggest possible retrofitting solutions if required.
In addition, insufficient attention has been paid to the selection of ground motions, which is one of the most important factors for generation of the fragility curves. Synthetic motion is mostly used as ground motion in seismic demand analysis [
This study has two primary objectives. The first is to generate fragility curves for the self-anchored suspension bridge using properly selected ground motions based on the results of seismic hazard deaggregation analysis. The second objective is to demonstrate the practical use of structural health monitoring data in seismic vulnerability analysis, achieving an updated fragility curve. This article is organized as follows: The proposed fragility methodology combined with PSHA and SHM data is first introduced. Secondly, a typical self-anchored suspension bridge is chosen to be investigated by using the proposed method, and its SHM system is illustrated briefly. The initial finite element (FE) model is then established based on the design information and is updated according to the experimental data recorded from both static and dynamic testing to achieve a more accurate model which can reflect the current state. Finally, fragility curves for the updated model are calculated.
To obtain analytical fragility curves, three steps should be considered: the simulation of ground motions, the simulation of bridges, and the generation of fragility curves. As illustrated in Figure
Framework of the proposed method.
Adequate selection of ground motions is vital to obtain a more accurate prediction of a structure’s seismic response. Selection based on deaggregation analysis, which is an in-depth PSHA study for the site, is suggested here as it can determine the most likely future seismic scenario. In this situation, the selected ground motions are more realistic, and the seismic demand model is closer to the actual ones. The representative method of PSHA was proposed by Cornell [
Once the ground motions are selected, an accurate FE model must be constructed. Experiment measurements can be used to update the initial model to better reflect the current conditions of the structure. Dynamic testing can reflect the overall information on the structures; however, its inherent limitations hinder its development. Static tests have the advantage of high accuracy and low noise interference, but testing must be carried out within the elastic range. Therefore, an FE model updating method taking both static and dynamic tests into consideration is suggested in this study and can overcome the deficiencies of using static or dynamic test data separately.
According to the updated model, fragility curves are generated to represent the vulnerability of a bridge with a higher accuracy. Currently, the scaling approach and the cloud approach are the two most frequently used mechanisms to develop fragility curves based on nonlinear dynamic analysis [
The conditional probability of the bridge demand exceeding the capacity for a given intensity can be calculated using the followed formula:
The main motivating factor for this study is to demonstrate the practical application of SHM data and in-depth PSHA technique to evaluate the fragility of the self-anchored suspension bridge and to generate accurate fragility curves according to actual experiment data instead of only design information. A typical self-anchored suspension bridge located in Nanjing, China, is selected as a case study to verify the suitability of the proposed fragility analysis method.
The investigated bridge (Figure
The investigated suspension bridge. (a) Side view. (b) Bottom of the girder. (c) E-type elastic-plastic damping bearings.
An SHM system was installed on the investigated bridge which included 80 sensors, and the layout plan is presented in Figure
SHM system of the bridge.
Based on the monitoring system, both the static property and dynamic characteristics of the investigated bridge are studied. The static test is conducted by loading a fleet of trucks with a weight of 30 t onto the bridge. The truck model is shown in Figure
Cases of the static test. (a) Symmetric loading. (b) Partial loading. (c) Truck model. (d) Layout plan of measurement.
In the dynamic test, the ambient vibration data recorded by the accelerator mounted on the north of the 1/4 section of the main span are illustrated in Figure
Ambient vibration test. (a) Acceleration time history at 1/4 section of the main girder. (b) CMIF. (c) Acceleration time history of the sling. (d) Acceleration power spectrum of the sling.
The acceleration time history and power spectrum of the cable DS15 are plotted in Figures
Prior to model updating, it is recommended that an initial model is obtained using the first available set of design information. The initial model of the suspension bridge is built using the ANSYS software. A detailed configuration of the bridge and modeling of some important components are shown in Figure
General configuration of the suspension bridge. (a) FE model of the bridge. (b) Modeling of the main girder. (c) Tower configuration and various sections. (d) Nonlinear model of the tower. (e) Bearing layout plan. (f) Nonlinear model of the elastic-plastic energy-dissipating bearing.
Elastic-plastic energy-dissipating bearings are installed on the bridge in both lateral and longitudinal directions. The layout plan of the bearing system is presented in Figure
Illustration of the bearings.
Bearing | #7 | #8 | #9 | #10 | #11 | #12 |
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Vertical | (i) Multidirectional bearing: CSm 4000 |
(i) Multidirectional bearing: CSm 20000 |
(i) Multidirectional bearing: CSm 25000 |
(i) Multidirectional bearing |
(i) Multidirectional bearing: CSm 2500 |
(i) Multidirectional bearing: CSm 5000 |
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Transversal | (i) Elastic-plastic energy-dissipating bearings: CKPZ-Z |
(i) Elastic-plastic energy-dissipating bearings: CKPZ-Z |
(i) Elastic-plastic energy-dissipating bearings: CKPZ-Z |
(i) CKPZ-Q, GJZF4: 3000 kN | (i) Elastic-plastic energy-dissipating bearings: CKPZ-Z |
(i) Elastic-plastic energy-dissipating bearings: CKPZ-Z |
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Longitudinal | — | — | — | (i) Yield load: 2000 kN |
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The Solid 65 element is widely used to simulate nonlinearities of reinforced concrete components. As the element number of the solid model is too high and leads to convergence difficulty, a simplified tower model is adopted to improve computational efficiency. A detailed tower model (Figure
Main tower modeling. (a) Detailed model. (b) Simplified model. (c) Pushover analysis.
The selection of updating parameters is a key step in FE model updating. The uncertainties of material parameters, boundary conditions, and geometric parameters are critical and must be considered. To avoid huge calculation amounts in the optimization process due to blind selection, sensitivity analysis is executed to find the parameters that will cause a large change in the structural characteristics when varying their assigned value in the FE model. The state function is defined as
In this study, the cross-sectional area and the mass of the bridge are assumed to remain unchanged unless changes are visually discernible. The elastic modulus of the main girder, tower, pier, main cable, and sling (
Sensitivity analysis. (a) Cable force. (b) Natural frequency. (c) Deflection in symmetric loading for case 1. (d) Deflection in symmetric loading for case 2.
The sensitivity analysis reveals that the elastic modulus of the cable is highly sensitive to static deformation, and the variation of the main girder’s elastic modulus causes the largest change in the natural frequency. The parameters of the tower, sling, and pier can be disregarded as they are insensitive to the structural response in both the static and dynamic tests. The stiffness of the damper affects response with different degrees and so should also be considered.
Finite element model updating is described as the optimization of the difference between actual experimental data and analytical response. To determine the parameter that best reproduces the properties extracted from measurements, a global objective function combined with static property and dynamic characteristics is given as follows:
The subobjective functions concerned with static property are obtained by using the deformation residuals between the measurements, and analytical response from the FE model. For case 1,
Variation of parameters.
Component | Main girder | Cable | Damper | |||
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Initial model | 1 | 1 | 1 | 1 | 1 | 1 |
Updated model | 1.021 | 1.076 | 1.002 | 1.039 | 0.96 | 0.918 |
Variation | 2.1% | 7.6% | 0.2% | 3.9% | −4.0% | −8.2% |
Model analysis and static analysis are also performed on the updated model. Figures
Comparison of the static deformation. (a) Case 1: symmetric loading. (b) Case 1: north defection with partial loading. (c) Case 1: south deflection with partial loading. (d) Case 2: symmetric loading. (e) Case 2: north defection with partial loading. (f) Case 2: south deflection with partial loading.
Comparison of dynamic test results. (a) First model. (b) Second model. (c) Third model. (d) Frequency. (e) Cable force.
The designed value, test value, and calculated value of the cable force are illustrated in Figure
Fragility curves are used to represent the seismic vulnerability of the investigated bridge. The bridge mainly consists of the towers, cables, slings, girders, and piers. The main cables and slings are flexible structures, with vibration periods in a long range, while the seismic wave rarely contains long-period components, so the fragility can be neglected. Moreover, the main girders and piers are assumed to maintain elasticity in an earthquake situation; thus, the seismic damage can be ignored for these components. Accordingly, the seismic fragility analysis of the investigated suspension bridge focuses on the tower and the elastic-plastic energy-dissipating bearing in this study.
The investigated suspension bridge sits within the Nanjing potential seismic source. It is mainly affected by two seismic zones: the Tanlu seismic zone and the Yangtze River-South Yellow Sea seismic zone, which have the capacity to produce a devastating earthquake. There are 32 potential seismic sources around the engineering site contributing to the seismic hazard of the site which are identified on the Chinese seismic ground motion parameter zoning map [
Probabilistic seismic hazard analysis of the site. (a) Potential seismic source. (b) Hazard deaggregation at the level of 10% in 50 years.
According to equation (
As illustrated in Figure
Property of the selected ground motions. (a) Response spectra. (b) Distribution of PGA.
Once the ground motion bin is determined, the selection of an appropriate intensity measure (IM) is the next step in fragility analysis. By means of nonlinear dynamic time-history analysis, the seismic demand of the component is estimated using the ground motions applied to the updated FE model. All of the ground motions consist of two horizontal components, which are applied to the bridge model in the longitudinal and transversal directions separately. Spectral acceleration (SA) has been widely utilized as a sufficient and practical IM in recent studies. However, taking the curvature of the slope section of the tower as an example, logistic regression between the response and PGA is compared with that of SA, and the results plotted in Figure
Probabilistic seismic demand models for the curvature of the slope section. (a) PSDM with PGA. (b) PSDM with SA.
In fragility analysis, the structural capacity is described in terms of damage index (DI). Different DIs for various components require specific limit state (LS) values, which are commonly obtained from engineering judgments or experiments [
Quantified limit states of various components.
Component | DI | Slight | Moderate | Extensive | Complete |
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Section of the tower | Curvature |
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Tower displacement | Drift ratio |
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Bearing (#7, 12) | Disp. (mm) |
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Bearing (#8, 9, 11) | Disp. (mm) |
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Bearing (#10) | Disp. (mm) |
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As little knowledge about the DI for the tower of a suspension bridge is available in the literature, a pushover analysis is conducted on the detailed tower model to define the failure mode and locate the most vulnerable part of the tower. In elastic range, the maximum sectional curvature occurs on the slope section shown in Figure
Determination of the LS for the key section. (a) Curvature of the main tower. (b) Steel stress in the key section. (c)
The seismic behavior of these major components governs the damage and failure modes of the suspension bridge. In conclusion, the DI for various components is defined in terms of bearing displacement, displacement of the top of the tower, and curvatures of the key section of the tower. Table
To explore the seismic performance of the suspension bridge accurately, fragility analysis is conducted from the longitudinal and transversal directions separately. In longitudinal seismic excitation, the lateral displacement of the bearing plotted in Figure
Longitudinal earthquake response of the bearing. (a) Bearings #7∼12 (L: longitudinal; T: transverse). (b) Force-displacement hysteretic curves of the damper.
Combined with the component damage index and PSDMs, the fragility curves for different damage states are calculated according to equation (
Longitudinal fragility curves of components. (a) Slight damage. (b) Moderate damage. (c) Extensive damage. (d) Complete damage.
In transversal analysis, only the lateral response of the bearings is considered. It should be noted that the lateral displacement of the #10 bearing is zero, as two isolating bearings were set between the tower and the girder to limit transversal displacement. Consequently, transversal fragility focuses on the tower and #7, #8, #9, #11, and #12 bearings, and the results are provided in Figure
Transversal fragility curves of components. (a) Slight damage. (b) Moderate damage. (c) Extensive damage. (d) Complete damage.
To evaluate the impact that variation of the updating parameter has on the seismic vulnerability of the bridge, a comparison of fragility curves between the initial and updated models is conducted. In the longitudinal direction, it can be seen from the above analysis that the fragility curves of bearings are almost the same, so it only focuses on the comparison of the #10 bearing here. In the transversal direction, the most vulnerable parts are #12, #7, and #11 bearings, so only comparison of these bearings is considered here. The results are plotted as follows.
Comparison results provided in Figure
Fragility comparison between the initial model and the updated model. (a) Longitudinal fragility curves of bearing #10. (b) Transversal fragility curves of bearing #7. (c) Transversal fragility curves of bearing #11. (d) Transversal fragility curves of bearing #12.
A typical self-anchored suspension bridge was used as an example to illustrate the practical application of structural health monitoring data for the evaluation of structural fragility in this study. Using design information and service life experimental data, FE model updating was undertaken according to the optimization of an objective function. This is described in terms of the difference between the measured bridge response and the FE model response. Prior to optimization, a sensitivity analysis was conducted to determine the updating parameters that cause a larger change in structural properties when varying their assigned value. Based on the updated model, the fragility curves are researched. The ground motions used in the seismic demand analysis were selected according to the PSHA deaggregation result and can help to determine the most likely earthquake at the site according to the PEER database. In this manner, the ground motions are more realistic and the seismic demand models are closer to the actual ones.
Conclusions about the seismic performance of the self-anchored suspension bridge are obtained as follows: The longitudinal response of the bridge is mainly related to the constraints between the tower and the girder. For the floating-type bridge, large displacement will be produced that is harmful to bearings. In the transverse direction, fixed constraints between piers and girders will induce a large seismic response because of the nonuniform distribution of girder mass, which is bad for the transverse shearing resistance of the bearing. Consequently, the bearing is the most vulnerable part of the self-anchored suspension bridge. The adoption of elastic-plastic energy-dissipating bearings can decrease the seismic reaction of the bearing and improve its seismic performance. From the fragility results, it can be seen that the effect of the damper is minimal for slight to moderate earthquakes and high for strong earthquakes in the case of complete damage. For slight and moderate damage states, the dampers do not display any visible impact on the fragility curves of the bridges. Compared with the bearing, no serious damage or complete destruction of the tower will occur under a predictable earthquake.
The response data of the FE model used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was financially supported by the National Key R&D Program of China (no. 2018YFC0705601), Jiangsu Distinguished Young Scholars Fund (no. BK20160002), and National Science Foundation of China (grant nos. 51578139 and 51608110).