Analytical seismic fragility functions for highway and railway embankments and cuts

A fundamental tool in seismic risk assessment of transportation systems is the fragility curve, which describes the probability that a structure will reach or exceed a certain damage state for a given ground motion intensity. Fragility curves are usually represented by two‐parameter (median and log‐standard deviation) cumulative lognormal distributions. In this paper, a numerical approach, in the spirit of the IDA, is applied for the development of fragility curves for highways and railways on embankments and in cuts due to seismic shaking. The response of the geo‐construction to increasing levels of seismic intensity is evaluated using a 2D nonlinear finite element model, with an elasto‐plastic criterion to simulate the soil behavior. A calibration procedure is followed in order to account for the dependency of both the stiffness and the damping to the soil strain level. The effect of soil conditions and ground motion characteristics on the response of the embankment and cut is taken into account considering different typical soil profiles and seismic input motions. This study will provide input for the assessment of the vulnerability of the road/railway network regarding the performance of the embankments and cuts; therefore, the level of damage is described in terms of the permanent ground displacement in these structures. The fragility curves are estimated based on the evolution of damage with increasing earthquake intensity, which is described by PGA. The proposed approach allows the evaluation of new fragility curves considering the distinctive features of the element's geometry, the input motion, and the soil properties as well as the associated uncertainties. A relationship between the computed permanent ground displacement on the surface of the embankment and the PGA in the free field is also suggested based on the results of the numerical analyses. Finally, the proposed fragility curves are compared with existing empirical data and the limitations of their applicability are outlined. Copyright © 2015 John Wiley & Sons, Ltd.     

An earthquake-source-based metric for seismic fragility analysis

The seismic fragility of a system is the probability that the system enters a damage state under seismic ground motions with specified characteristics. Plots of the seismic fragilities with respect to scalar ground motion intensity measures are called fragility curves. Recent studies show that fragility curves may not be satisfactory measures for structural seismic performance, since scalar intensity measures cannot comprehensively characterize site seismicity. The limitations of traditional seismic intensity measures, e.g., peak ground acceleration or pseudo-spectral acceleration, are shown and discussed in detail. A bivariate vector with coordinates moment magnitude m and source-to-site distance r is proposed as an alternative seismic intensity measure. Implicitly, fragility surfaces in the (m, r)-space could be used as graphical representations of seismic fragility. Unlike fragility curves, which are functions of scalar intensity measures, fragility surfaces are characterized by two earthquake-hazard parameters, (m, r). The calculation of fragility surfaces may be computationally expensive for complex systems. Thus, as solutions to this issue, a bi-variate log-normal parametric model and an efficient calculation method, based on stochastic-reduced-order models, for fragility surfaces are proposed.     

Seismic risk assessment of liquid storage tanks via a nonlinear surrogate model

A performance‐based earthquake engineering approach is developed for the seismic risk assessment of fixed‐roof atmospheric steel liquid storage tanks. The proposed method is based on a surrogate single‐mass model that consists of elastic beam‐column elements and nonlinear springs. Appropriate component and system‐level damage states are defined, following the identification of commonly observed modes of failure that may occur during an earthquake. Incremental dynamic analysis and simplified cloud are offered as potential approaches to derive the distribution of response parameters given the seismic intensity. A parametric investigation that engages the aforementioned analysis methods is conducted on 3 tanks of varying geometry, considering both anchored and unanchored support conditions. Special attention is paid to the elephant's foot buckling formation, by offering extensive information on its capacity and demand representation within the seismic risk assessment process. Seismic fragility curves are initially extracted for the component‐level damage states, to compare the effect of each analysis approach on the estimated performance. The subsequent generation of system‐level fragility curves reveals the issue of nonsequential damage states, whereby significant damage may abruptly appear without precursory lighter damage states.     

Development of fragility functions for geotechnical constructions: Application to cantilever retaining walls

Fragility curves constitute an emerging tool for the seismic risk assessment of all constructions at risk. They describe the probability of a structure being damaged beyond a specific damage state for various levels of ground shaking. They are usually represented as two-parameter (median and log-standard deviation) cumulative lognormal distributions. In this paper a numerical approach is proposed for the construction of fragility curves for geotechnical constructions. The methodology is applied to cantilever bridge abutments on surface foundation often used in road and railway networks. The response of the abutment to increasing levels of seismic intensity is evaluated using a 2D nonlinear FE model, with an elasto-plastic criterion to simulate the soil behavior. A calibration procedure is followed in order to account for the dependency of both the stiffness and the damping on the soil strain level. The effect of soil conditions and ground motion characteristics on the global soil and structural response is taken into account considering different typical soil profiles and seismic input motions. The objective is to assess the vulnerability of the road network as regards the performance of the bridge abutments; therefore, the level of damage, is described in terms of the range of settlement that is observed on the backfill. The effect of backfill material to the overall response of the abutment wall is also examined. The fragility curves are estimated based on the evolution of damage with increasing earthquake intensity. The proposed approach allows the evaluation of new fragility curves considering the distinctive features of the structure geometry, the input motion and the soil properties as well as the associated uncertainties. The proposed fragility curves are verified based on observed damage during the 2007 Niigata-Chuetsu Oki earthquake.     

Vulnerability of floating tunnel shafts for increasing earthquake loading

Fragility curves constitute the cornerstone in seismic risk evaluations and performance-based earthquake engineering. They describe the probability of a structure to experience a certain damage level for a given earthquake intensity measure, providing a relationship between seismic hazard and vulnerability. In this paper a numerical approach is applied to derive fragility curves for tunnel shafts built in clays, a component that is found in several critical infrastructure such as urban metro networks, airport facilities or water and waste water projects. The seismic response of a representative tunnel shaft is assessed using tridimensional finite difference non-linear analyses carried out with the program FLAC^3D, under increasing levels of seismic intensity. A hysteretic model is used to simulate the soil non-linear behavior during the seismic event. The effect of soil conditions and ground motion characteristics on the soil-structure system response is accounted for in the analyses. The damage is defined based on the exceedance of the concrete wall shaft capacity due to the developed seismic forces. The fragility curves are estimated in terms of peak ground acceleration at a rock or stiff soil outcrop, based on the evolution of damage with increasing earthquake intensity. The proposed fragility models allows the characterization of the seismic risk of a representative tunnel shaft typology and soil conditions considering the associated uncertainties, and partially fill the gap of data required in performing a risk analysis assessment of tunnels shafts.     

Seismic reliability‐based ductility demand evaluation for inelastic base‐isolated structures with friction pendulum devices

The aim of this work is to propose seismic reliability‐based relationships between the strength reduction factors and the displacement ductility demand of nonlinear structural systems equipped with friction pendulum isolators (FPS) depending on the structural properties. The isolated structures are described by employing an equivalent 2dof model characterized by a perfectly elastoplastic rule to account for the inelastic response of the superstructure, whereas, the FPS behavior is described by a velocity‐dependent model. An extensive parametric study is carried out encompassing a wide range of elastic and inelastic building properties, different seismic intensity levels and considering the friction coefficient as a random variable. Defined a set of natural seismic records and scaled to the seismic intensity corresponding to life safety limit state for L'Aquila site (Italy) according to NTC08, the inelastic characteristics of the superstructures are designed as the ratio between the average elastic responses and increasing strength reduction factors. Incremental dynamic analyses (IDAs) are developed to evaluate the seismic fragility curves of both the inelastic superstructure and the isolation level assuming different values of the corresponding limit states. Integrating the fragility curves with the seismic hazard curves related to L'Aquila site (Italy), the reliability curves of the equivalent inelastic base‐isolated structural systems, with a design life of 50 years, are derived proposing seismic reliability‐based regression expressions between the displacement ductility demand and the strength reduction factors for the superstructure as well as seismic reliability‐based design (SRBD) abacuses useful to define the FPS properties. Copyright © 2016 John Wiley & Sons, Ltd.     

Computation of bridge seismic fragility by large‐scale simulation for probabilistic resilience analysis

Seismic resilience of structures and infrastructure systems is a fast developing concept in the field of disaster management, promoting communities that are resistant and quickly recoverable in case of an extreme event. In this contest, probabilistic seismic demand and fragility analyses are two key elements of the seismic resilience assessment in the majority of the proposed methodologies. Several techniques are available to calculate fragility curves for different types of structures. In particular, to assess the seismic performance of the regional transportation infrastructure, methods for the fragility curve estimation for entire classes of bridges are required. These methods usually rely on a set of assumptions, partially because of the limited information. Other assumptions were introduced at the time when computational resources were inadequate for a purely numerical approach and closed‐form solutions were a convenient alternative. For instance, some of these popular assumptions are aimed at simplifying the model of the engineering demand. In this paper, a simulation‐based methodology is proposed, to take advantage of the computational resources widely available today and avoid such assumptions on the demand. The resulting increase in accuracy is estimated on a typical class of bridges (multi‐span simply supported). Most importantly, the quantitative impact of the assumptions is assessed in the context of a life‐cycle loss estimation analysis and resilience analysis. The results show that some assumptions preserve an acceptable level of accuracy, but others introduce a considerable error in the fragility curves and, in turn, in the expected resilience and life‐cycle losses of the structure. Copyright © 2015 John Wiley & Sons, Ltd.     

A performance-based adaptive methodology for the seismic evaluation of multi-span simply supported deck bridges

A performance-based adaptive methodology for the seismic assessment of highway bridges is proposed. The proposed methodology is based on an Inverse (I), Adaptive (A) application of the Capacity Spectrum Method (CSM), with the capacity curve of the bridge derived through a Displacement-based Adaptive Pushover (DAP) analysis. For this reason, the acronym IACSM is used to identify the proposed methodology. A number of Performance Levels (PLs), for which the seismic vulnerability and seismic risk of the bridge shall be evaluated, are identified. Each PL is associated to a number of Damage States (DSs) of the critical members of the bridge (piers, abutments, joints and bearing devices). The IACSM provides the earthquake intensity level (PGA) corresponding to the attainment of the selected DSs, using over-damped elastic response spectra as demand curves. The seismic vulnerability of the bridge is described by means of fragility curves, derived based on the PGA values associated to each DS. The seismic risk of the bridge is evaluated as convolution integral of the product between the fragility curves and the seismic hazard curve of the bridge site. In this paper, the key aspects and basic assumptions of the proposed methodology are presented first. The IACSM is then applied to nine existing simply supported deck bridges, characterized by different types of piers and bearing devices. Finally, the IACSM predictions are compared with the results of nonlinear response time-history analysis, carried out using a set of seven ground motions scaled to the expected PGA values.     

Modeling level selection for seismic analysis of concrete‐filled steel tube/moment‐resisting frames by using fragility curves

The seismic behavior of plane moment‐resisting frames (MRFs) consisting of I steel beams and concrete‐filled steel tube (CFT) columns is investigated in this study. More specifically, the effect of modeling details of each individual component of CFT‐MRFs, such as the composite CFT columns, the beam‐column connections, the panel zones, and the steel I beams on their seismic behavior, is studied through comparisons against available experimental results. Then, fragility curves are constructed for three typical CFT‐MRFs, designed according to European codes, for various levels of modeling sophistication through nonlinear time‐history analyses. On the basis of these fragility curves, one can select the appropriate modeling level of sophistication that can lead to the desired seismic behavior for a given seismic intensity. Copyright © 2014 John Wiley & Sons, Ltd.     

A simplified method of constructing fragility curves for highway bridges

Fragility curves are found to be useful tools for predicting the extent of probable damage. They show the probability of highway structure damage as a function of strong motion parameters, and they allow the estimation of a level of damage probability for a known ground motion index. In this study, an analytical approach was adopted to develop the fragility curves for highway bridges based on numerical simulation. Four typical RC bridge piers and two RC bridge structures were considered, of which one was a non‐isolated system and the other was an isolated system, and they were designed according to the seismic design code in Japan. From a total of 250 strong motion records, selected from Japan, the United States, and Taiwan, non‐linear time history analyses were performed, and the damage indices for the bridge structures were obtained. Using the damage indices and ground motion parameters, fragility curves for the four bridge piers and the two bridge structures were constructed assuming a lognormal distribution. It was found that there was a significant effect on the fragility curves due to the variation of structural parameters. The relationship between the fragility curve parameters and the over‐strength ratio of the structures was also obtained by performing a linear regression analysis. It was observed that the fragility curve parameters showed a strong correlation with the over‐strength ratio of the structures. Based on the observed correlation between the fragility curve parameters and the over‐strength ratio of the structures, a simplified method was developed to construct the fragility curves for highway bridges using 30 non‐isolated bridge models. The simplified method may be a very useful tool to construct the fragility curves for non‐isolated highway bridges in Japan, which fall within the same group and have similar characteristics. Copyright © 2003 John Wiley & Sons, Ltd.     

Analytical fragility assessment using unscaled ground motion records

It is desirable that nonlinear dynamic analyses for structural fragility assessment are performed using unscaled ground motions. The widespread use of a simple dynamic analysis procedure known as Cloud Analysis, which uses unscaled records and linear regression, has been impeded by its alleged inaccuracies. This paper investigates fragility assessment based on Cloud Analysis by adopting, as the performance variable, a scalar demand to capacity ratio that is equal to unity at the onset of limit state. It is shown that the Cloud Analysis, performed based on a careful choice of records, leads to reasonable and efficient fragility estimates. There are 2 main rules to keep in mind for record selection: to make sure that a good portion of the records leads to a demand to capacity ratio greater than unity and that the dispersion in records' seismic intensity is considerable. An inevitable consequence of implementing these rules is that one often needs to deal with the so‐called collapse cases. To formally consider the collapse cases, a 5‐parameter fragility model is proposed that mixes the simple regression in the logarithmic scale with logistic regression. The joint distribution of fragility parameters can be obtained by adopting a Markov Chain Monte Carlo simulation scheme leading directly to the fragility and its confidence intervals. The resulting fragility curves compare reasonably with those obtained from the Incremental Dynamic Analysis and Multiple Stripe Analysis with (variable) conditional spectrum–compatible suites of records at different intensity levels for 3 older reinforced concrete frames with shear‐, shear‐flexure‐, and flexure‐dominant behavior.     

增量动力弹塑性分析方法的改进及其应用

本文主要对增量动力弹塑性分析(IDA)方法的两个方面作了改进:(1)以增量动力弹塑性分析方法的基本原理为基础,以快速非线性时程分析(FNA)方法为计算工具,形成快速增量动力弹塑性分析(FIDA)方法。(2)仅以IDA分析结果为基础,建立地震易损性曲线及地震破坏概率计算的离散型式。为了校验本文方法的可行性与有效性,应用增量动力弹塑性分析、模态增量动力弹塑性分析与快速增量动力弹塑性分析计算一个14层筒中筒结构试验模型,比较了用上述三种方法的计算误差与计算效率,比较了传统方法与本文方法计算该模型结构的地震易损性曲线的差别。     

Seismic design of RC structures: A critical assessment in the framework of multi‐objective optimization

The assessment of seismic design codes has been the subject of intensive research work in an effort to reveal weak points that originated from the limitations in predicting with acceptable precision the response of the structures under moderate or severe earthquakes. The objective of this work is to evaluate the European seismic design code, i.e. the Eurocode 8 (EC8), when used for the design of 3D reinforced concrete buildings, versus a performance‐based design (PBD) procedure, in the framework of a multi‐objective optimization concept. The initial construction cost and the maximum interstorey drift for the 10/50 hazard level are the two objectives considered for the formulation of the multi‐objective optimization problem. The solution of such optimization problems is represented by the Pareto front curve which is the geometric locus of all Pareto optimum solutions. Limit‐state fragility curves for selected designs, taken from the Pareto front curves of the EC8 and PBD formulations, are developed for assessing the two seismic design procedures. Through this comparison it was found that a linear analysis in conjunction with the behaviour factor q of EC8 cannot capture the nonlinear behaviour of an RC structure. Consequently the corrected EC8 Pareto front curve, using the nonlinear static procedure, differs significantly with regard to the corresponding Pareto front obtained according to EC8. Furthermore, similar designs, with respect to the initial construction cost, obtained through the EC8 and PBD formulations were found to exhibit different maximum interstorey drift and limit‐state fragility curves. Copyright © 2007 John Wiley & Sons, Ltd.     

Simplified seismic fragility analysis of structures with two types of friction devices

This paper presents a method and results of seismic fragility estimation of frame structures with friction devices and with friction devices and restrictors. The seismic intensity parameter, defined as the mean value of the pseudovelocity spectrum in a specified periods band, is proved to allow the use of linear regression analysis of the response parameters of the considered non‐linear structures on seismic intensity. A simplified method of fragility estimation is proposed, based on the concept of ‘mean seismic excitation’ and linear regression of the seismic response parameters on seismic intensity parameter. The key risk contributors for the system with friction devices and for the system with friction devices and restrictors are identified on the basis of the fragility analysis, and recommendations for improvement of the seismic response of the respective systems are derived. The results of the fragility study show that when the initial ‘bare’ frame is retrofitted by rigidly connected bracings the effect is much lower than in the case of connecting the bracings by friction devices and especially by friction devices and restrictors. Copyright © 2000 John Wiley & Sons, Ltd.     

Generation of fragility curves for Turkish masonry buildings considering in‐plane failure modes

This study focuses on the seismic safety evaluation of masonry buildings in Turkey for in‐plane failure modes using fragility curves. Masonry buildings are classified and a set of fragility curves are generated for each class. The major structural parameters in the classification of masonry buildings are considered as the number of stories, load‐bearing wall material, regularity in plan and the arrangement of walls (required length, openings in walls, etc.), in accordance with the observations from previous earthquakes and field databases. The fragility curves are generated by using time history (for demand) and pushover (for capacity) analyses. From the generated sets of fragility curves, it is observed that the damage state probabilities are significantly influenced from the number of stories and wall material strength. In the second stage of the study, the generated fragility curves are employed to estimate the damage of masonry buildings in Dinar after the 1995 earthquake. The estimated damage by fragility information is compared with the inspected visual damage as assessed from the Damage Evaluation Form. For the quantification of fragility‐based damage, a single‐valued index, named as ‘vulnerability score’ (VS), is proposed. There seems to be a fair agreement between the two damage measures. In addition to this, decisions regarding the repair or demolition of masonry buildings in Dinar due to visual damage inspection are on comparable grounds with the relative measure obtained from VS of the same buildings. Hence, the fragility‐based procedure can provide an alternative for the seismic safety evaluation of masonry buildings in Turkey. Copyright © 2007 John Wiley & Sons, Ltd.     

Seismic fragility curves of shallow tunnels in alluvial deposits

In this paper a numerical approach is proposed for the construction of fragility curves for shallow metro tunnels in alluvial deposits, when subjected to transversal seismic loading. The response of the tunnel is calculated under quasi static conditions applying the induced seismic ground deformations which are calculated through 1D equivalent linear analysis for an increasing level of seismic intensity. The results of the present numerical analyses are compared with selected closed form solutions, highlighting the limitations of the latter, while indicative full dynamic analysis are performed in order to validate the results of the quasi-static method. The proposed approach allows the evaluation of new fragility curves considering the distinctive features of the tunnel geometries and strength characteristics, the input motion and the soil properties as well as the associated uncertainties. The comparison between the new fragility curves and the existing empirical ones highlights the important role of the local soil conditions, which is not adequately taken into account in the empirical curves.     

既有钢筋混凝土框架结构的多指标云模型地震损伤评估

目前的既有钢混结构地震损伤研究没有同时考虑不同抗震设计规范差异和耐久性两个因素对结构抗震性能的影响,且损伤指标较简单,在动力损伤分析中也存在局限。基于云模型的特点,提出了包括弹塑性耗能差率、刚度损伤指数、层间位移角和顶点位移角的多元结构损伤状态综合评估方法,能够同时考虑结构各损伤指数的随机性和模糊性。考虑不同版本抗震设计规范造成的结构性能差异和耐久性下降对结构性能的影响,设计3个典型五层钢混框架结构,进行增量动力分析,验证损伤评估方法的准确性。结果表明：随着抗震规范版本的更新,结构的损伤程度有适当减轻;同一结构的损伤程度因混凝土碳化作用先减轻后加重;采用弹塑性耗能差率表征既有结构的地震损伤效果优于刚度损伤指数;基于多指标云模型损伤评估方法获得的云模型综合隶属度和综合损伤值能够更加细化和精确地描述结构损伤状态。     

Sliding fragility of block‐type non‐structural components. Part 1: Unrestrained components

Motivated by the development of performance‐based design guidelines with emphasis on both structural and non‐structural systems, this paper focuses on seismic vulnerability assessment of block‐type unrestrained non‐structural components under sliding response on the basis of seismic inputs specified by current seismic codes. Two sliding‐related failure modes are considered: excessive relative displacement and excessive absolute acceleration. It is shown that an upper bound for the absolute acceleration response can be assessed deterministically, for which a simple yet completely general equation is proposed. In contrast, fragility curves are proposed as an appropriate tool to evaluate the excessive relative displacement failure mode. Sample fragility curves developed through Monte‐Carlo simulations show that fragility estimates obtained without taking into account vertical base accelerations can be significantly unconservative, especially for relatively large values of the coefficient of friction. It is also found that reasonable estimates of relative displacement response at stories other than the ground in multistorey buildings cannot in general be obtained by simply scaling the ground acceleration to the peak acceleration at the corresponding storey. Failure modes considered in this study are found to be essentially independent of each other, a property that greatly simplifies assessment of conditional limit states. Copyright © 2002 John Wiley & Sons, Ltd.     

Seismic fragility assessment of low-rise stone masonry buildings

Many historic buildings in old urban centers in Eastern Canada are made of stone masonry reputed to be highly vulnerable to seismic loads.Seismic risk assessment of stone masonry buildings is therefore the first step in the risk mitigation process to provide adequate planning for retrofit and preservation of historical urban centers.This paper focuses on development of analytical displacement-based fragility curves reflecting the characteristics of existing stone masonry buildings in Eastern Canada.The old historic center of Quebec City has been selected as a typical study area.The standard fragility analysis combines the inelastic spectral displacement,a structure-dependent earthquake intensity measure,and the building damage state correlated to the induced building displacement.The proposed procedure consists of a three-step development process:(1) mechanics-based capacity model,(2) displacement-based damage model and(3) seismic demand model.The damage estimation for a uniform hazard scenario of 2% in 50 years probability of exceedance indicates that slight to moderate damage is the most probable damage experienced by these stone masonry buildings.Comparison is also made with fragility curves implicit in the seismic risk assessment tools Hazus and ELER.Hazus shows the highest probability of the occurrence of no to slight damage,whereas the highest probability of extensive and complete damage is predicted with ELER.This comparison shows the importance of the development of fragility curves specific to the generic construction characteristics in the study area and emphasizes the need for critical use of regional risk assessment tools and generated results.     

地震地面运动的模糊随机模型

未来地震地面运动不仅具有随机性而且具有强烈的模糊性。这种模糊性是由于地震烈度和场地土类別的定义和评定标准的模糊性所引起的,在其模型中应该予以考虑。 本文从不同特征和条件的工程实际出发,在充分考虑上述模糊性和随机性的基础上,提出了地震地面运动的两种模糊随机模型。第一种模型是在预测地震烈度和应用地震反应谱的基础上建立起来的;第二种模型是在现有的随机过程模型的基础上,通过引进适当的模糊参数而建立起来的。