Incremental Dynamic Analysis Method Research Articles

Impact of Component Correlation on Probabilistic Seismic Responses of Steel Frames by Modifying Non-Positive Definite Correlation Matrices

ABSTRACT This paper studies the effects of spatial correlation between the structural components on probabilistic seismic performance of steel moment resisting frames, using extended incremental dynamic analysis (IDA) method to consider the effects of aleatoric uncertainties caused by ground motion records. In this probabilistic assessment, a combination of Latin hypercube sampling (LHS) and Cholesky decomposition (CD) methods are utilized to take the epistemic uncertainties of the numerical model into account. In case the target correlation matrix is not positive definite, two powerful optimization algorithms are employed to find the closest positive definite correlation matrix.

Seismic fragility analysis of shield building considering strength ratio of mainshock and aftershocks

The shield building of the AP1000 nuclear power plant serves as a crucial protective barrier against radioactive substances. However, past research indicates that structures are susceptible to experiencing aftershocks, which may lead to unforeseeable damage and potential radioactive material leakage. To address this issue, a finite element model of the shield building was established with the damage indexes of the tensile and compressive damage selected for further model analysis. According to the fundamental theory of reliability, the traditional incremental dynamic analysis method was used to analyze the seismic fragility of the shield building by inputting mainshock and aftershock sequences with three strength ratios. The results indicate that the seismic fragility of shield building may be underestimated without considering the influence of aftershocks and the damage state presents an upward tendency as the strength ratio increases. However, the cumulative damage caused by aftershocks is unlikely to exceed the initial damage induced by the corresponding mainshock. Overall, the aggravation of the compressive damage is less pronounced than the increase of the tensile damage as the strength ratio increases.

A new performance‐based seismic design method using endurance time analysis for linked column frame system and a comparison of structural systems and seismic analysis methods

SummaryThis paper presents a new performance‐based seismic design method for the design of repairable linked column frame (LCF) and linked column system (LCS). Currently, the biggest problem of these systems is the lack of a simple and practical design method that leads to the design of optimal models with sufficient seismic capacity. The interaction of the primary and secondary systems, changing the lateral load pattern during an earthquake, and the implementation of the target performance objectives have complicated the design of these systems. The evaluations carried out in this research show that the rotation of link beams must be controlled in the design. Therefore, the ultimate plastic rotation of links is determined to be 0.01 rad for the seismic intensity of design base earthquake and 0.015 rad for maximum considered earthquake. The results show that the models designed using the presented method are optimal, have sufficient seismic capacity, and achieve the target performance objectives. In addition, although previous researches have shown that these systems have a suitable seismic behavior, their seismic behavior have not been compared with other structural systems. Comparing can show the behavioral characteristics of a new structural system; hence, the elastic and plastic behavior of the LCF and LCS models have been compared with other common steel structural systems using all analysis methods. Moreover, in the presented method for the design of LCF and LCS systems, nonlinear time history analysis using the endurance time method (ETM) is used, and due to the newness of the endurance time method, its results are compared with the median results of nonlinear time history analysis at different seismic hazard levels and incremental dynamic analysis (IDA), in 45 samples. The results show that the endurance time analysis is a reasonable and efficient method, and in this comparison, the difference between the results of ETM and IDA methods is 6% on average.

Dynamic and uncertainty-based assessment of the progressive collapse probability of prestressed concrete frame structures with infill walls

Since progressive collapse usually has serious consequences, the problem of progressive collapse resistance has gradually become a topical issue in recent years. However, most of the existing studies have been conducted on common reinforced concrete (RC) structures. The methodologies used in the studies are also mostly static and deterministic. There is a lack of research that considers the dynamic effects and uncertainties of pre-stressed structures with infill walls. In this paper, the assessment of progressive collapse probability considering dynamic effect and uncertainties is performed for prestressed concrete frame with infill walls (IW-PC frame) structures. The finite element software OpenSEES is chosen as the analysis platform. By introducing uncertainties in material, geometry and load, five limit states and their quantitative indicators are identified. Based on the nonlinear incremental dynamic analysis (IDA) method, the vulnerability analysis of the IW-PC frame is performed under different working conditions. The vulnerability analysis indicates that the five levels of limit states can visually evaluate the degree of structural damage. Additionally, the resistance relationships of the four structural forms are as follows: IW-PC frame > reinforced concrete frame with infill walls (IW-RC frame) > prestressed concrete frame (PC frame) > RC frame. The load carrying capacity of the frames is significantly increased by the coupling effect of the infill walls and the prestressed strands. Moreover, the prestressed strands can reduce the probability of light and moderate damage, while infill walls have an excellent preventive effect on severe damage. The reliability of the four structures is evaluated using the Zhao-Ono moment method. The results indicate that IW-PC is the safest, followed by IW-RC, PC, and RC.

Simplified state‐dependent seismic fragility assessment

AbstractEarthquakes are clustered in time and space; therefore, structures may be subjected to multiple consecutive instances of potentially damaging shaking, with insufficient in‐between time for repair operations to take place. Methodologies to evaluate the risk dynamics in this situation require vulnerability models that are able to capture the transitions between damage states, from the intact conditions to failure, due to multiple damaging earthquakes, that is, state‐dependent fragility curves. One of the state‐of‐the‐art methods for the assessment of structure‐specific state‐dependent fragility curves relies on a variant of incremental dynamic analysis (IDA), which is often termed back‐to‐back or B2B‐IDA. The computational costs typically involved in B2B‐IDA motivate attempts to simplify the evaluation of state‐dependent fragility curves. This paper presents a simplified method for multi‐story moment‐resisting frame structures, based on pushover analysis in conjunction with a predictive model for the main features of a damaged structural system, such as residual deformations and loss of stiffness and/or strength. The predictive model enables the probabilistic definition of the post‐earthquake pushover curve of a damaged structural system, given the displacement demand imposed by a preceding damaging shock. The state‐dependent fragility curves are then evaluated via IDA of single‐degree‐of‐freedom oscillators based on these pushover curves. Illustrative applications validate the ability of the proposed methodology to provide state‐dependent fragilities with reduced computational costs compared to the back‐to‐back IDA method.

Incremental dynamic analysis method application in the seismic vulnerability of infilled wall frame structures

To reasonably evaluate the overall seismic performance of infilled wall frames, a corresponding nonlinear analysis model is constructed with infilled wall frames as the research object. It quantifies the failure levels of structural and non-structural components and proposes a method for extracting structural overall performance indicators. The vulnerability of the case frame structure is analyzed using the Incremental Dynamic Analysis (IDA) method. The results showed that the seismic performance of the infilled wall frame and the ordinary frame met the seismic fortification requirements, and the seismic capacity of the infilled wall frame was better. After implementing a 7-degree seismic fortification, the cumulative probability of basic integrity, minor damage, and moderate damage for infill wall frames reached 99.15 %, surpassing the fortification target of 1.5 g. However, the seismic capacity of ordinary frames was overestimated, as their cumulative probability of basic intact, minor damage, and moderate damage under a 7-degree seismic fortification was 99.7 %. Neglecting its impact, ordinary frames exhibited lower seismic performance compared to other structures, with a basic intact probability of only 48.49 % under frequent earthquake actions at 7 degrees. The research utilizes effective methods for evaluating the seismic vulnerability of infilled wall frames.

Application of a Deep Learning Method to the Seismic Vulnerability Analysis of Cross-Fault Hydraulic Tunnels Based on MLE-IDA

Rapidly developed deep learning methods, widely used in various fields of civil engineering, have provided an efficient option to reduce the computational costs and improve the predictive capabilities. However, it should be acknowledged that the application of deep learning methods to develop prediction models that efficiently assess the nonlinear dynamic responses of cross-fault hydraulic tunnels (CFHTs) is lacking. Thus, the objective of this study is to construct a rational artificial neural network (ANN) prediction model to generate the mass data and fragility curves of CFHTs. Firstly, an analysis of 1080 complete nonlinear dynamic time histories via incremental dynamic analysis (IDA) is conducted to obtain the mass data of the drift ratio of the CFHT. Then, the hyper-parameters of the ANN model are discussed to determine the optimal parameters based on four examined approaches to improve the prediction capacity and accuracy. Meanwhile, the traditional probabilistic seismic demand models of the predicted values obtained by the ANN model and the numerical results are compared with the statistical parameters. Eventually, the maximum likelihood estimation couping IDA method is applied to assess the seismic safety of CFHTs under different damage states. The results show that two hidden layers, ten neurons, and the ReLU activation function for the ANN model with Bayesian optimization can improve the reliability and decrease the uncertainty in evaluating the structural performance. Moreover, the amplitude of the seismology features can be used as the neurons to build the input layers of the ANN model. It is found through vulnerability analysis that the traditional seismic fragility analysis method may overestimate the earthquake resistance capacity of CFHTs compared with maximum likelihood estimation. In practical engineering, ANN methods can be regarded as an alternative approach for the seismic design and performance improvement of CFHTs.

Research on seismic fragility of subway station based on incremental dynamic analysis method

Seismic intensity measure is an significant parameter that affects the accuracy of structural seismic risk assessment, and plays an important role in the performance-based earthquake engineering research framework. Taking a two story, three span, and shallow buried subway station as the research object, a two-dimensional finite element analysis model of soil structure underground in a typical engineering site was established using ANSYS. Ten earthquake records were selected from PEER as input, and the structural seismic fragility analysis method based on incremental dynamic analysis (IDA) method was introduced into the subway station structure. The maximum story drift angle was used as the structural damage index, and three typical seismic intensity measures were selected, the exploration is conducted on the seismic intensity measures applicable to shallow buried subway stations and the variation pattern of seismic response of subway stations with increasing seismic amplitude. The seismic fragility curve of shallow buried subway station structures is established to obtain the failure probability of the structure at different seismic intensity levels. The analysis results indicate that for shallow buried subway stations, PGA is the most suitable seismic intensity measure, and the comparison with existing empirical fragility curves shows that the vulnerability analysis method is feasible and can quantitatively provide the failure probability of structures at different performance levels, providing reference for seismic design of underground structures.

Seismic resilience analysis of self-centering prestressed concrete frames with generalized flag-shaped hysteretic behavior

To increase the stiffness and energy dissipation of conventional self-centering concrete joints, this paper proposes a self-centering prestressed concrete frame with a generalized flag-shaped hysteretic behavior (GFS-SCPC). First, the working mechanism of the GFS-SCPC joint is clarified, and the key parameters affecting the generalized flag-shaped hysteresis are proposed: the second stiffness ksθ and the energy dissipation ratio βE. Subsequently, the three-dimensional self-centering prestressed concrete frame with flag-shaped hysteretic behavior (FS-SCPC) and GFS-SCPC frames are established, and the time history analysis and incremental dynamic analysis (IDA) method are carried out to compare their seismic performances. Finally, according to the FEMA P-58 specification, the resilience indexes of the frames with different hysteretic parameters are evaluated to clarify the influence of ksθ and βE. The results show that the energy dissipation capacity of the GFS-SCPC frame increases with larger second stiffness ksθ and the energy dissipation ratio βE. The increase of ksθ effectively controls the maximum drift angle and acceleration response of the GFS-SCPC frame, which would reduce the vulnerability of the structure. The main structure of the GFS-SCPC frame remains undamaged after the earthquake, and the loss mainly comes from the damage of the acceleration-sensitive non-structural components of the upper floors. The increase of ksθ can effectively reduce the damage degree of non-structural components and further reduce the resilience indexes, including casualties, repair time and repair cost, while the increase of βE alone does not significantly improve the resilience indexes. Therefore, the proposed GFS-SCPC frame with larger second stiffness has better seismic resilience performance.

Uncertainty and bias in fragility estimates by intensifying artificial accelerations

To comply with the objectives of Performance-Based Earthquake Engineering and gain a comprehensive understanding of a structure’s seismic response, the utilization of multiple ground motion records selected or scaled at various hazard levels is often necessary. Incremental Dynamic Analysis (IDA) and cloud analysis methods represent wide-range probabilistic approaches employed for this purpose. However, due to their computational complexity, researchers and engineers have sought alternative solutions. One such solution is the application of intensifying artificial accelerations (IAAs). By employing IAAs, the structural response can be recorded from linear elastic to nonlinear, and ultimately up to the collapse capacity, all within a single simulation.This paper presents a comprehensive summary of various generations of IAAs and their unique characteristics. To assess their performance on nonlinear single degree-of-freedom systems, the IAAs are compared against IDA results. As IAAs are typically generated to be independent of specific hazard levels, a series of generic (i.e., standardized) ground motion sets are used for IDA. Our objective is to quantitatively evaluate the bias in the mean response estimation by employing different generations of IAAs, developed based on various base response spectra and optimization objectives. Additionally, this paper provides a simplified estimation of fragility curves based on IAAs, and compares the resulting uncertainty with those generated through IDA. The findings aim to enhance the understanding of IAAs’ capabilities and limitations in capturing the seismic response characteristics of structures.

Seismic Vulnerability Analysis of Steel Clarification Tank

As a physical analysis method, susceptibility not only predicts the probability of failure of a structure under different hazard levels but also prevents disasters induced by structural damage due to vulnerable members. To study the susceptibility of steel clarification tank under earthquake action, this study analyzes the dynamic response of steel clarification tank under near-field impulse, near-field no impulse, and far-field ground shaking, derives IDA curve clusters by incremental dynamic analysis method, and conducts related studies to analyze the susceptibility of clarification tank structures through susceptibility curves. The results of the study show that the probability of failure of the clarification tank under different ground shaking intensities is different. The probability of liquid wave height transcendence of the clarification tank under seismic action is the largest, the probability of liquid wave height transcendence of the clarification tank under near-field pulsed seismic action is the second, the probability of liquid wave height transcendence of the clarification tank under near-field nonpulsed seismic action is the smallest, and the probability of transcendence occurs similarly in near-field pulsed and far-field seismic actions.

Wind-Induced Responses of Corroded Angle-Steel Transmission Tower

Corroded transmission towers, whose load-bearing capacities are lowered, are suffering from wind-induced damage. By simulating the member corrosion through section thinning, the structural dynamic finite element model of an angle-steel transmission tower with a corrosion depth of 0 mm to 1.4 mm was established using ANSYS/LS-DYNA software. The incremental dynamic analysis method was used to study the collapse mode of the tower under different degrees of corrosion, as well as the effects of corrosion on structural self-vibration characteristics, wind-induced member stress, and tower-top displacement. The results show that with the corrosion deepening, the first three vibration frequencies decrease, and the torsional mode becomes the first mode. When the corrosion depth is smaller, the overall structure failure occurs under strong wind. The failure starts at the most unfavorable oblique rod between the upper and middle cross arm, then expands to the rods under the lower cross arm. When the corrosion depth is larger, the overall structure failure occurs restrictedly between the upper and middle cross arm. Internal axial force and bending moment redistribution happen when corrosion deepens. Under the critical failure state, the oblique rods are always at a high stress level, while the stresses of the main rods decrease. Corrosion has a stronger impact on component strength and stability than on structural stiffness.

Seismic failure probability of adjacent frame structures retrofitted by flexible control measure

Due to the randomness of earthquake and the difference of dynamic characteristics of adjacent structures, the mechanical behavior of the adjacent structures under earthquake is very complex. A flexible control method using steel strands is proposed, which has little impact on the dynamic characteristics of the adjacent structures. A three-dimensional numerical calculation model of the adjacent frame structures is established by using LS-DYNA. The interstory drift ratios of the single structure, the adjacent uncontrolled structure and the adjacent controlled structure are analyzed by using incremental dynamic analysis (IDA) method. The failure probabilities of different limit states (LS) are studied based on the vulnerability, and the annual average and the design life failure probabilities are further discussed by the probabilistic seismic failure risk analysis. The results show that the maximum interstory drift ratio of the structure is reduced by the flexible control, and the failure probability of the adjacent controlled structure is less than that of the single structure and the adjacent uncontrolled structures. It should be noted that the influence of the flexible control on the failure probability is reduced when the PGA is close to 1.0 g. The four limit states LS1, LS2, LS3 and LS4 represent the slight damage, the moderate damage, the severe damage and the complete damage, respectively. The annual average failure probability from LS1 to LS4 decreases in turn, and the difference is about 10 times. Compared with the adjacent uncontrolled structures, the flexible control reduces the design life failure probabilities of the 10-storey structure and the 8-storey structure by 13.82% and 11.79%, respectively. The proposed control measure is helpful to reduce the failure probability of the adjacent frame structures under earthquake action.

Seismic vulnerability assessment on porcelain electrical equipment based on Kriging model

To improve the efficiency in assessing the seismic vulnerability of porcelain electrical equipment when considering different sorts of uncertainty, this paper proposes a seismic vulnerability analysis method based on Kriging model. The one-dimension and multi-dimension Kriging models were considered in the method. Afterward, case studies were conducted on an ultra-high voltage (UHV) gas insulated switchgear (GIS) porcelain bushing. The seismic vulnerability of the UHV GIS bushing obtained by incremental dynamic analysis (IDA) method and the Kriging method were compared. The validity and applicability of the method were verified. Moreover, the effects of the dimensions of the Kriging model were analyzed. The results indicated that the Kriging method reduced the calculation quantity by 40% compared with IDA method, and the analysis error was within 5%, which significantly improved the efficiency of seismic vulnerability analysis of porcelain electrical equipment. The multi-dimension Kriging model-based method is suitable for the seismic vulnerability analysis considering different sorts of uncertainty, while the one-dimensional Kriging model-based method can obtain more stable seismic vulnerability assessment results.

Seismic damage characteristics of high arch dams under oblique incidence of SV waves

This study investigates the seismic damage characteristics of high arch dams considering the oblique incidence of seismic waves. Based on the dynamic implicit finite element method, a viscous-spring artificial boundary is applied to consider radiation damping of the infinite foundation, the seismic wave input is converted into an equivalent nodal force of the artificial boundary, and the equivalent nodal loads of the wave input under oblique incidence of the SV wave is deduced. Taking JP Ⅰ arch dam as an example, combined with the concrete damaged plasticity (CDP) model, the nonlinear dynamic response of the dam under 25° oblique incidence of the SV wave is conducted considering the geometrical nonlinearity of contraction joints and the dynamic mechanical properties of concrete materials. Based on the incremental dynamic analysis (IDA) method, the relative displacement and transverse joint opening are selected as the damage measures for fragility analysis of the arch dam. According to the damage distribution from the analysis of a large number of seismic time-history responses, the damage levels of the arch dam are classified. Three limit states of slight damage, moderate damage and severe damage are proposed for the arch dam. The average responses of the 10 earthquakes are considered as the limit values to quantitatively describe the damage of the arch dam. By fitting the results of incremental dynamic analysis, the seismic probabilistic demand model and seismic fragility curves for the two damage measures to predict the probability of the arch dam reaching damage levels under different seismic actions have been established. The fragility curve shows that the probability of severe damage is almost zero under the maximum design earthquake (PGA = 0.1971 g) and less than 5% under the maximum credible earthquake (PGA = 0.2299 g). It is concluded that JP Ⅰ arch dam has a high degree of safety redundancy and fully satisfies the “two levels” design requirements.

Seismic fragility assessment of aqueduct bent structures subjected to mainshock-aftershock sequences

This study aims to assess the seismic fragility of the aqueduct bent structure subjected to mainshock-aftershock (MS-AS) ground motion sequences by double incremental dynamic analysis method. The finite element model of the aqueduct bent structure is developed in OpenSees platform and numerical simulation technique is validated through physical test results. A series of nonlinear dynamic time-history analysis is carried out on the aqueduct bent structure to obtain its seismic response under different ground motion intensities. Furthermore, the incremental dynamic analysis (IDA) data cluster and median lines of the damage measure of the aqueduct bent structure are constructed. The advanced probabilistic seismic demand model (PSDM) of the aqueduct bent structure subjected to MS-AS ground motion sequences is proposed. Based on the proposed PSDM and the predefined damage states, the fragility surfaces of the aqueduct bent structure are also developed by adopting the vector-valued intensity measures considering the mainshock intensities (IMm) and aftershock intensities (IMa). Meanwhile, for the sake of comparation, the aftershock fragility curves of the aqueduct bent structure under different mainshock intensities are drawn, which emphasizes the importance of aftershocks in the seismic risk assessment. The analysis results indicate that the proposed piecewise probabilistic seismic demand model can better fit the logarithmic relationship between the aftershock intensities and the median of the reinforcement longitudinal tensile strain. The structural fragility probability increase with the increase of mainshock and aftershock intensities. It turns out that the mainshock intensities play a leading role in the damage probability of the aqueduct structure subjected to MS-AS ground motion sequences, the aftershock can cause additional damage to the aqueduct bent structure and reduce its seismic capacity. The meaningful results can provide supports for seismic optimization design, resilience improvement and post-earthquake repair-demand of the aqueduct structure in Southwest China.

Estimation of earthquake safety escape time and casualties in Chinese masonry buildings

To predict the number of casualties in dwellings when a strong earthquake occurs, this paper investigated the seismic performances of masonry buildings located in a rural town where earthquake occurs frequently with high earthquakes. The incremental dynamic analysis method was employed to estimate the available escape time and the collapse probability of building under those selected ground motions with a series of peak accelerations. Meanwhile, the required safety escape time of the building was obtained through evacuation simulation analysis. According to the safe evacuation criteria, the probability of casualties in terms of earthquake intensity measure was predicted. The results show that available escape time decreases with the increase of peak ground acceleration level. While the required safety escape time increases with the number of building layers subjoining; residents on the upper floors have a lower probability of escape. The present work implied that the residents on the higher floors must seek temporary security zones in their rooms to reduce casualties.

Vulnerability Analysis of Main Aftershock Sequence of Aqueduct Based on Incremental Dynamic Analysis Method

At present, traditional seismic design methods often ignore the structural damage caused by aftershocks in the evaluation of structural stability. In this paper, seven main aftershock sequences were constructed by using the attenuation method. The incremental dynamic analysis method (IDA) was used to analyze the nonlinear dynamic time history of the aqueduct structure. The main aftershock vulnerability curve of the aqueduct structure was obtained by taking the seismic intensity IM and the maximum ratio of the plastic strain energy to the total strain energy Dp as the structural performance parameter. The analysis results show that the residual displacement of the aqueduct increases by 33%, 66%, 44%, 37%, 0.01%, 60%, and 59%, respectively, under the seven main aftershock sequences. The incremental damage percentages of the aftershock at the end of the period were 9.85%, 15.00%, 26.53%, 2.10%, 0.9%, 35.97%, and 9.85%, respectively. The main aftershock made the damage at the bottom of the arch and the aqueduct more extensive. When the earthquake intensity is 0.3 g, the exceedance probabilities of moderate damage and severe damage are 62.68% and 14.39%, respectively, under the action of the main aftershock sequence. The exceedance probabilities under the action of the main aftershock sequence are 38.52% and 12.08% higher than that of the single main earthquake, respectively.

Seismic fragility analysis of suspended cable trays in civil buildings

The cable tray is a kind of non-structural component used to distribute the electric cable, which plays a vital role in maintaining the function of the building. Post-earthquake investigations proved that the collapse of the cable tray led to the loss of human life and business continuity. This study aims to understand the seismic fragility of typical suspended cable trays in civil buildings through full-scale shaking table tests and numerical simulation. Based on the shaking table test results, the damage states of the cable tray are defined. The finite element model for cable tray is established and verified by test results. The seismic fragility analysis of the cable tray is carried out by the incremental dynamic analysis method. The peak floor acceleration and the spectral acceleration are taken as intensity measures of seismic motions. The calculation results show that the peak floor acceleration is more suitable to be an intensity measure for the seismic fragility analysis of the cable tray. Seismic fragility curves of two typical types of cable trays are generated.

Seismic Risk Analysis of Offshore Bridges Considering Seismic Correlation between Vulnerable Components

To comprehensively evaluate the seismic performance of offshore bridges, a seismic risk analysis of an example bridge was developed based on improved two-dimensional (2D) seismic fragility analysis. Taking a simply-supported beam bridge in an offshore tidal environment as an example, the adverse effects of chloride ion erosion are considered and the seismic response process of the example bridge is simulated using the Incremental Dynamic Analysis (IDA) method. The appropriate damage indexes are chosen for the plate rubber bearing and the pier, and the one-dimensional (1D) seismic fragility curves of single components and the entire bridge are obtained. The correlation coefficients of vulnerable components are quantitatively proposed based on the correlation analysis method, and the 2D seismic fragility curves of the entire bridge are achieved while accounting for seismic correlation between vulnerable components. The seismic risk probability of the entire bridge is finally determined after combining hazard analysis at the bridge site and seismic loss analysis. The results show that there is a significant correlation between vulnerable components and that 2D seismic fragility analysis based on the reliable correlation coefficients of vulnerable components can more comprehensively evaluate the seismic performance of the entire bridge. In the context of seismic disaster reduction research focused on slight and moderate damage caused by moderate-to-small ground motions, this research can provide scientific and technical support for seismic design and seismic risk assessment of offshore bridges.