Fragility curves for reinforced concrete structures in Skopje (Macedonia) region

A method for the development of earthquake intensity–damage relations, given as fragility curves and damage probability matrices is proposed in this paper. The proposed method is applied on reinforced-concrete frame-wall structures. Two sets of fragility curves and damage probability matrices are developed. The first one is for reinforced-concrete frame structures lower than 10 stories. For this purpose, a six-story frame structure is used. The other set is defined for reinforced-concrete frame-wall structures higher than 10 stories. A 16-story frame-wall structure was chosen as a sample. The sample structures were designed according to Macedonian design code. The conditions of the local seismic hazard were the subject of special concern for the development of earthquake intensity–damage relations. Because of the limited number of real time histories from the Skopje region, a set of 240 synthetic time histories were generated. Geological dates from the Skopje region were used. Response of the sample structures under earthquake excitation was defined performing nonlinear dynamic analysis. Modeling of the nonlinear behavior of the structural elements was completed according to state-of-the-art methods in this field. A modified Park and Ang damage model was chosen as a measure of the structure's response to earthquake excitation. Five damage states were defined to express the condition of damage. As a result of the analytical research, the values of the global damage index corresponding to each damage state were determined. Using the dates from the nonlinear dynamic analysis of the sample structures under all 240 synthetic time histories, the two sets of fragility curves and damage probability matrices were defined.     

基于能量方法设计的RC框架结构易损性分析

基于“强柱弱梁”的屈服机制,依据能量平衡方法设计了某6层RC框架结构,采用震级-震中距条带地震动记录选取方法,选取12条随机地震动,利用Perform-3D有限元分析软件对结构进行增量动力（IDA）分析,得到了结构的地震易损性曲线、破坏状态概率曲线以及结构破坏概率矩阵。分析结果表明:该方法设计的结构能够形成预设的“强柱弱梁”屈服机制,可以保证结构中梁充分参与耗能,同时结构具有较强的抗倒塌能力,可以满足“小震不坏,中震可修,大震不倒”的性能要求。     

Comparative response and performance of base‐isolated and fixed‐base structures

A building with a seismic isolation system, in an earthquake, is recognized as producing substantially smaller accelerations and deformations compared with a building that use other systems. This type of system is therefore expected to better protect the building's nonstructural components, equipment, and other contents that are essential for the activities conducted in the building. Unlike many available studies on building responses, only a small number of studies on a buildings' nonstructural component responses are available, and no study has directly addressed building performance with regard to nonstructural component protection. This paper therefore measures the performance of various seismically isolated buildings. Specifically, the effects of important structural parameters, namely, isolation stiffness, isolation damping ratio, and number of stories on the response of base‐isolated structures are investigated parametrically. Ground motions with 2% exceedence in 50years Maximum Considered Earthquake (MCE) are used. Performance is compared with that of fixed‐base structures in order to present data that will be useful in justifying the more costly technology. The buildings are 3, 9, and 20 stories, represented by MDOF shear‐beam models. As examples of displacement‐sensitive and acceleration‐sensitive components, partition walls and ceilings are considered, respectively. The Pacific Earthquake Engineering Research Center performance‐based earthquake engineering methodology is adopted to evaluate the failure return periods of the examples based on their available fragility curves. In addition, the curves are varied hypothetically to understand the sensitivity of the return period to the curve features. Then, the median and dispersion of fragility curves required to satisfy the components' desired failure return period are obtained. Copyright © 2015 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.     

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.     

Intensity measure conversion of fragility curves

In seismic risk assessment of structures, fragility functions are the probabilistic characterization of vulnerability at the component and/or structural level, expressing the probability of failure as a function of a ground motion intensity measure (IM). Fragility curves, in general, are structure- and site-specific, thus a comparison of fragility curves, then of vulnerability, is not straightforward across multiple structures. Also, it could be the case that hazard at a site of interest is not available for the IM originally considered in the fragility assessment. These situations require to convert fragility curves from an original IM to a target one. The present study addresses a hazard-consistent probabilistic framework for converting spectral acceleration-based IMs from an original IM to a target IM at a given site. In particular, three conversion cases, under different assumptions on the explanatory power of the involved IMs with respect to structural failure, are discussed: (a) a vector-valued IM consisting of the original and target IMs, magnitude, and source-to-site distance; (b) a vector-valued IM consisting of the original and target IMs; and (c) the original (scalar) IM only, assuming that structural response, given the IM, is statistically independent of the other ground motion variables. In this framework, the original fragility functions are characterized using the state-of-the-art methods in performance-based earthquake engineering, then the fragility curves as a function of the target IM are evaluated through applications of the probability calculus rules, ensuring consistency with the seismic hazard at the site of interest. The conversion strategy is illustrated through the applications to three-, six-, and nine-story Italian code-conforming reinforced concrete buildings designed for a high-hazard site in Italy. The study shows that, in most of the cases, the converted fragility curves have agreement with the reference curves directly developed in terms of the target IM. Cases in which least agreement was found are likely due to the models used to obtain the terms required by the conversion equations.     

Multiple failure function based fragility curves for structures equipped with TMD

This paper presents a procedure to develop fragility curves of structures equipped with TMD considering multiple failure functions.The failure criteria considered are maximum inter-story drift ratio as a safety criterion,maximum absolute acceleration as a convenience criterion and TMD stroke length.The relationship between intensity measure and responses of the structure was assumed to follow the power-law model,and a regression analysis was used to estimate its properties.A nonlinear eight-story shear building subjected to near-fault earthquakes was used for the numerical studies.Fragility curves using multiple and single failure functions for an uncontrolled structure and a structure equipped with optimal TMDs were developed.Numerical analysis showed that using multiple failure functions led to increasing the fragility when compared with using the single failure function for both the uncontrolled and controlled structures.However,TMDs slightly reduced the seismic fragility and have the capability to improve the reliability of the structure.Also,it was found that the fragility was significantly influenced by the values of the capacity thresholds of both the acceleration of the structure and TMD stroke length,which should be selected by considering the target performance and application of the structure and control device.     

Seismic reliability of steel moment resisting framed buildings retrofitted with buckling restrained braces

The present paper investigates the seismic reliability of the application of buckling restrained braces (BRBs) for seismic retrofitting of steel moment resisting framed buildings through fragility analysis. Samples of regular three‐storey and eight‐storey steel moment resisting frames were designed with lateral stiffness insufficient to comply with the code drift limitations imposed for steel moment resisting frame systems in earthquake‐prone regions. The frames were then retrofitted with concentrically chevron conventional braces and BRBs. To obtain robust estimators of the seismic reliability, a database including a wide range of natural earthquake ground motion records with markedly different characteristics was used in the fragility analysis. Nonlinear time history analyses were utilized to analyze the structures subjected to these earthquake records. The improvement of seismic reliability achieved through the use of conventional braces and BRBs was evaluated by comparing the fragility curves of the three‐storey and eight‐storey model frames before and after retrofits, considering the probabilities of four distinct damage states. Moreover, the feasibility of mitigating the seismic response of moment resisting steel structures by using conventional braces and BRBs was determined through seismic risk analysis. The results obtained indicate that both conventional braces and especially BRBs improve significantly the seismic behavior of the original building by increasing the median values of the structural fragility curves and reducing the probabilities of exceedance of each damage state. Copyright © 2011 John Wiley & Sons, Ltd.     

Prediction of seismic collapse risk of steel moment frame mid-rise structures by meta-heuristic algorithms

Different performance levels may be obtained for sideway collapse evaluation of steel moment frames depending on the evaluation procedure used to handle uncertainties. In this article, the process of representing modelling uncertainties, record to record (RTR) variations and cognitive uncertainties for moment resisting steel frames of various heights is discussed in detail. RTR uncertainty is used by incremental dynamic analysis (IDA), modelling uncertainties are considered through backbone curves and hysteresis loops of component, and cognitive uncertainty is presented in three levels of material quality. IDA is used to evaluate RTR uncertainty based on strong ground motion records selected by the k-means algorithm, which is favoured over Monte Carlo selection due to its time saving appeal. Analytical equations of the Response Surface Method are obtained through IDA results by the Cuckoo algorithm, which predicts the mean and standard deviation of the collapse fragility curve. The Takagi-Sugeno-Kang model is used to represent material quality based on the response surface coefficients. Finally, collapse fragility curves with the various sources of uncertainties mentioned are derived through a large number of material quality values and meta variables inferred by the Takagi-Sugeno-Kang fuzzy model based on response surface method coefficients. It is concluded that a better risk management strategy in countries where material quality control is weak, is to account for cognitive uncertainties in fragility curves and the mean annual frequency.     

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.     

按现行规范设计的Ⅷ度区RC框架结构的倒塌性能评估

选取按照现行规范设计的既有建筑进行有限元建模,考虑地震动的不确定性对其进行大量增量动力分析(IDA),得到模型的IDA曲线簇。在此基础上对其进行地震需求概率分析和概率抗震能力分析,拟合得到结构的易损性曲线,据此对结构的倒塌概率进行定量评估,并比较基于非线性分析与性能评估软件PERFORM-3D的纤维模型和塑性铰模型的分析结果。结果表明:按照我国现行规范设计的钢筋混凝土(RC)框架结构,在预期的罕遇地震作用下倒塌概率较小,可满足"大震不倒"的要求;基于PERFORM-3D的截面纤维模型所得的RC框架结构,经非线性分析所得的倒塌概率相对保守,安全储备更高。     

Evaluation of the fishbone model in simulating the seismic response of multistory reinforced concrete moment-resisting frames

The fishbone model is a simplified numerical model for moment-resisting frames that is capable of modelling the effects of column-beam strength and stiffness ratios. The applicability of the fishbone model in simulating the seismic responses of reinforced concrete moment-resisting frames of different sets of column-beam strength and stiffness ratios are evaluated through nonlinear static, dynamic and incremental dynamic analysis on six prototype buildings of 4-, 8-and 12-stories. The results show that the fishbone model is practically accurate enough for reinforced concrete frames, although the assumption of equal joint rotation does not hold in all cases. In addition to the ground motion characteristics and the number of stories in the structures, the accuracy of the model also varies with the column-beam stiffness and strength ratios. The model performs better for strong column-weak beam frames, in which the lateral drift patterns are better controlled by the continuous stiffness provided by the strong columns. When the inelastic deformation is large, the accuracy of the model may be subjected to large record-to-record variability. This is especially the case for frames of weak columns.     

Effect of isolation on fragility curves of highway bridges based on simplified approach

The trend of isolating highway bridges is on the rise after the recent large earthquakes in Japan, the United States, and other countries. Recent investigation shows that isolated systems perform well against seismic forces as the substructures of such systems experience less lateral forces due to energy dissipation of the isolation device. Hence, it is anticipated that there might be an effect on fragility curves of highway bridges due to isolation. In this study, 30 isolated bridge models were considered (and they were designed according to the seismic design code of highway bridges in Japan) to have a wider range of the variation of structural parameters, e.g. pier heights, weights, and over-strength ratio of structures. Then, fragility curves were developed by following a simplified procedure using 250 strong motion records, which were selected from 5 earthquake events that occurred in Japan, the USA, and Taiwan. It is observed that the level of damage probability for the isolated system is less than that of the non-isolated one for a lower level of pier height. However, having the same over-strength ratio of the structures, the level of damage probability for the isolated system is found to be higher for a higher level of pier height compared to the one of the non-isolated system. The proposed simple approach may conveniently be used in constructing fragility curves for a class of isolated bridge structures in Japan that have similar characteristics.     

Seismic fragility assessment of existing sub-standard low strength reinforced concrete structures

An analytical seismic fragility assessment framework is presented for the existing low strength reinforced concrete structures more common in the building stock of the developing countries.For realistic modelling of such substandard structures,low strength concrete stress-strain and bond-slip capacity models are included in calibrating material models.Key capacity parameters are generated stochastically to produce building population and cyclic pushover analysis is carried out to capture inelastic behaviour.Secant period values are evaluated corresponding to each displacement step on the capacity curves and used as seismic demand.A modified capacity demand diagram method is adopted for the degrading structures,which is further used to evaluate peak ground acceleration from back analysis considering each point on the capacity curve as performance point.For developing fragility curves,the mean values of peak ground acceleration are evaluated corresponding to each performance point on the series of capacity curves.A suitable probability distribution function is adopted for the secant period scatter at different mean peak ground acceleration values and probability of exceedance of limit states is evaluated.A suitable regression function is used for developing fragility curves and regression coefficients are proposed for different confidence levels.Fragility curves are presented for a low rise pre-seismic code reinforced concrete structure typical of developing countries.     

Aftershock collapse fragility curves for non‐ductile RC buildings: a scenario‐based assessment

Recent studies have addressed the computation of fragility curves for mainshock (MS)‐damaged buildings. However, aftershock (AS) fragilities are generally conditioned on a range of potential post‐MS damage states that are simulated via static or dynamic analyses performed on an intact building. Moreover, there are very few cases where the behavior of non‐ductile reinforced concrete buildings is analyzed. This paper presents an evaluation of AS collapse fragility conditioned on various return periods of MSs, allowing for the rapid assessment of post‐earthquake safety variations based solely on the intensity of the damaging earthquake event. A refined multi‐degree‐of‐freedom model of a seven‐storey non‐ductile building, which includes brittle failure simulations and the evaluation of a system level collapse, is adopted. Aftershock fragilities are obtained by performing an incremental dynamic analysis for a number of MS–AS ground motion sequences and a variety of MS intensities. The AS fragilities show that the probability of collapse significantly increases for higher return periods for the MS. However, this result is mainly ascribable to collapses occurred during MSs. When collapse cases that occur during a MS are not considered in the assessment of AS collapse probability, a smaller shift in the fragility curves is observed as the MS intensity increases. This result is justified considering the type of model and collapse modes introduced, which strongly depend on the brittle behavior of columns failing in shear or due to axial loads. The analysis of damage that is due to MSs when varying the return period confirms this observation. Copyright © 2017 John Wiley & Sons, Ltd.     

Practice‐oriented probabilistic seismic performance assessment of infilled frames with consideration of shear failure of columns

The objective of the study presented in this paper is to investigate the effects of masonry infills on the shear demand and failure of columns for the case when reinforced concrete frames with such infills are modeled by means of simplified nonlinear models that are not capable of the direct simulation of these effects. It is shown that an approximate simulation of the shear failure of columns can be achieved through an iterative procedure that involves pushover analysis, post‐processing of the analysis results using limit‐state checks of the components, and model adaptation if shear failure of columns is detected. The fragility parameters and the mean annual frequency of limit‐state exceedance are computed on the basis of nonlinear dynamic analysis by using an equivalent SDOF model. The proposed methodology is demonstrated by means of two examples. It was shown that the strength of the four‐story and seven‐story buildings and their deformation capacity are significantly overestimated if column shear failure due to the effects of masonry infills is neglected, whereas the mean annual frequency of limit‐state exceedance for the analyzed limit states is significantly larger than that estimated for the case if the shear failure of columns is neglected. Copyright © 2012 John Wiley & Sons, Ltd.     

Seismic fragility assessment of highway bridges using support vector machines

Seismic fragility curves provide a powerful tool to assess the reliability of structures. However, conventional fragility analysis of structures comprising a large number of components requires enormous computational efforts. In this paper, the application of probabilistic support vector machines (PSVM) for the system fragility analysis of existing structures is proposed. It is demonstrated that support vector machine based fragility curves provide accurate predictions compared to rigorous methodologies such as component based fragilities developed by Monte Carlo simulations. The proposed method is applied to an existing bridge structure in order to develop fragility curves for serviceability and collapse limit states. In addition, the efficiency of using the PSVM method in the application of vector-valued ground motion intensity measures (IM) as well as traditional single-valued IM are investigated. The results obtained from an incremental dynamic analysis of the structure are used to train PSVMs. The application of PSVM in binary and multi-class classifications is used for the fragility analysis and reliability assessment of the bridge structure.     

A simplified fragility analysis of fan type cable stayed bridges

A simplified fragility analysis of fan type cable stayed bridges using Probabilistic Risk Analysis (PRA) procedure is presented for determining their failure probability under random ground motion. Seismic input to the bridge support is considered to be a risk consistent response spectrum which is obtained from a separate analysis. For the response analysis, the bridge deck is modeled as a beam supported on springs at different points. The stiffnesses of the springs are determined by a separate 2D static analysis of cable-tower-deck system. The analysis provides a coupled stiffness matrix for the spring system. A continuum method of analysis using dynamic stiffness is used to determine the dynamic properties of the bridges .The response of the bridge deck is obtained by the response spectrum method of analysis as applied to multidegree of freedom system which duly takes into account the quasi - static component of bridge deck vibration. The fragility analysis includes uncertainties arising due to the variation in ground motion, material property, modeling, method of analysis, ductility factor and damage concentration effect. Probability of failure of the bridge deck is determined by the First Order Second Moment (FOSM) method of reliability. A three span double plane symmetrical fan type cable stayed bridge of total span 689 m, is used as an illustrative example. The fragility curves for the bridge deck failure are obtained under a number of parametric variations. Some of the important conclusions of the study indicate that (i) not only vertical component but also the horizontal component of ground motion has considerable effect on the probability of failure; (ii) ground motion with no time lag between support excitations provides a smaller probability of failure as compared to ground motion with very large time lag between support excitation; and (iii) probability of failure may considerably increase for soft soil condition.     

核电厂考虑结构参数不确定的 地震易损性分析

为考虑核电厂结构参数不确定对结构地震易损性的影响,基于一次二阶矩法(First-Order Second-Moment, FOSM)进行地震易损性分析。以核电结构中混凝土材料的密度、弹性模量,泊松比和抗拉强度为不确定参数,建立有限元模型,并与试验结果对比,以验证模型的准确性。基于有限元数值模拟方法,通过增量动力法计算核电厂模型在多条地震记录下不同峰值加速度的动力响应,同时基于FOSM得到参数不确定下的对数标准差,进而得到核电厂结构考虑参数不确定的地震易损性曲线。结果表明,结构参数的不确定对核电结构有一定的影响,未考虑参数不确定的地震易损性结果会低估结构的失效概率。该方法可为核电结构基于参数不确定下的易损性分析提供一定的理论依据与实用价值。     

大震下被动与智能隔震结构动力可靠度的对比

对被动及智能隔震结构在“大震”条件下的动力可靠度进行探讨。将被动及智能隔震体系均取作弹塑性模型,并用退化Bouc-W en滞变模型描述上部结构的恢复力,用非退化Bouc-W en模型描述隔震层的恢复力。采用虚拟激励法计算结构的随机响应,根据我国抗震规范中“大震不倒”的设防目标,采用各层最大层间位移峰值响应和累积滞变耗能构造双参数的随机疲劳累积损伤指数,作为功能状态指标。假定各层失效相关,用串联系统计算体系动力可靠度。通过数值算例,对比了被动隔震、智能隔震与非隔震体系的条件失效概率,从动力可靠度角度显示了智能隔震体系的减震优势。