A web‐based methodology for the prediction of approximate IDA curves

A web‐based methodology for the prediction of approximate IDA curves, which consists of two independent processes, is proposed. The result of the first process is a response database of the SDOF model, whereas the second process involves the prediction of approximate IDA curves from the response database by using n‐dimensional linear interpolation. Such an approach enables user‐friendly prediction of the seismic response parameters with high accuracy. In order to demonstrate the capabilities of the proposed methodology, a web application for the prediction of the approximate 16th, 50th and 84th fractile responses of an RC structure was developed. For the presented case study, the response database was computed for a set of 30 ground motion records and the discrete values of six structural parameters. Very good agreement between the computed and the approximated IDA curves of the four‐storey RC building was observed. Copyright © 2012 John Wiley & Sons, Ltd.     

Incremental dynamic analysis for estimating seismic performance sensitivity and uncertainty

Incremental dynamic analysis (IDA) is presented as a powerful tool to evaluate the variability in the seismic demand and capacity of non‐deterministic structural models, building upon existing methodologies of Monte Carlo simulation and approximate moment‐estimation. A nine‐story steel moment‐resisting frame is used as a testbed, employing parameterized moment‐rotation relationships with non‐deterministic quadrilinear backbones for the beam plastic‐hinges. The uncertain properties of the backbones include the yield moment, the post‐yield hardening ratio, the end‐of‐hardening rotation, the slope of the descending branch, the residual moment capacity and the ultimate rotation reached. IDA is employed to accurately assess the seismic performance of the model for any combination of the parameters by performing multiple nonlinear timehistory analyses for a suite of ground motion records. Sensitivity analyses on both the IDA and the static pushover level reveal the yield moment and the two rotational‐ductility parameters to be the most influential for the frame behavior. To propagate the parametric uncertainty to the actual seismic performance we employ (a) Monte Carlo simulation with latin hypercube sampling, (b) point‐estimate and (c) first‐order second‐moment techniques, thus offering competing methods that represent different compromises between speed and accuracy. The final results provide firm ground for challenging current assumptions in seismic guidelines on using a median‐parameter model to estimate the median seismic performance and employing the well‐known square‐root‐sum‐of‐squares rule to combine aleatory randomness and epistemic uncertainty. Copyright © 2009 John Wiley & Sons, Ltd.     

Approximate incremental dynamic analysis using the modal pushover analysis procedure

Incremental dynamic analysis (IDA)—a procedure developed for accurate estimation of seismic demand and capacity of structures—requires non‐linear response history analysis of the structure for an ensemble of ground motions, each scaled to many intensity levels, selected to cover the entire range of structural response—all the way from elastic behaviour to global dynamic instability. Recognizing that IDA of practical structures is computationally extremely demanding, an approximate procedure based on the modal pushover analysis procedure is developed. Presented are the IDA curves and limit state capacities for the SAC‐Los Angeles 3‐, 9‐, and 20‐storey buildings computed by the exact and approximate procedures for an ensemble of 20 ground motions. These results demonstrate that the MPA‐based approximate procedure reduces the computational effort by a factor of 30 (for the 9‐storey building), at the same time providing results to a useful degree of accuracy over the entire range of responses—all the way from elastic behaviour to global dynamic instability—provided a proper hysteretic model is selected for modal SDF systems. The accuracy of the approximate procedure does not deteriorate for 9‐ and 20‐storey buildings, although their dynamics is more complex, involving several ‘modes’ of vibration. For all three buildings, the accuracy of the MPA‐based approximate procedure is also satisfactory for estimating the structural capacities for the limit states of immediate occupancy, collapse prevention, and global dynamic instability. Copyright © 2006 John Wiley & Sons, Ltd.     

Direct estimation of the seismic demand and capacity of oscillators with multi‐linear static pushovers through IDA

SPO2IDA is introduced, a software tool that is capable of recreating the seismic behaviour of oscillators with complex quadrilinear backbones. It provides a direct connection between the static pushover (SPO) curve and the results of incremental dynamic analysis (IDA), a computer‐intensive procedure that offers thorough demand and capacity prediction capability by using a series of nonlinear dynamic analyses under a suitably scaled suite of ground motion records. To achieve this, the seismic behaviour of numerous single‐degree‐of‐freedom (SDOF) systems is investigated through IDA. The oscillators have a wide range of periods and feature pinching hysteresis with backbones ranging from simple bilinear to complex quadrilinear with an elastic, a hardening and a negative‐stiffness segment plus a final residual plateau that terminates with a drop to zero strength. An efficient method is introduced to treat the backbone shape by summarizing the analysis results into the 16, 50 and 84% fractile IDA curves, reducing them to a few shape parameters and finding simpler backbones that reproduce the IDA curves of complex ones. Thus, vast economies are realized while important intuition is gained on the role of the backbone shape to the seismic performance. The final product is SPO2IDA, an accurate, spreadsheet‐level tool for performance‐based earthquake engineering that can rapidly estimate demands and limit‐state capacities, strength reduction R‐factors and inelastic displacement ratios for any SDOF system with such a quadrilinear SPO curve. Copyright © 2006 John Wiley & Sons, Ltd.     

Cloud to IDA: Efficient fragility assessment with limited scaling

Incremental dynamic analysis (IDA) leads to curves expressed in terms of structural response versus intensity, commonly known as the IDA curves. It is known that implementation of IDA usually involves significant computational effort and most often significant scaling of the original records to various intensity levels. Employing as the performance variable the critical demand to capacity ratio (DCR) throughout the structure, which is equal to unity at the onset of the limit state, facilitates the identification of the intensity values at the onset of a desired limit state and hence the implementation of the IDA procedure. Employing the structural response to un‐scaled records and the corresponding regression‐based response predictions (a.k.a., the “Cloud Analysis”) helps in identifying the range of intensity values corresponding to demand to capacity ratio values in the vicinity of unity. The Cloud to IDA procedure for structural fragility assessment is proposed on the premise of exploiting the Cloud Analysis results to obtain the IDA curves both with minimum number of analyses and minimum amount of scaling. The transverse frame of a shear‐critical 7‐story older RC building in Van Nuys, CA, which is modeled in Opensees with fiber‐section considering the flexural‐shear‐axial interactions and the bar slip, is employed as a case study. It is demonstrated, by comparing the results to IDA and other state of the art non‐linear dynamic procedures based on no scaling or spectral‐shape‐compatible scaling, that the Cloud to IDA procedure leads to reliable results in terms of structural fragility and risk for the prescribed limit state.     

A critical examination of seismic response uncertainty analysis in earthquake engineering

The last decade of performance‐based earthquake engineering (PBEE) research has seen a rapidly increasing emphasis placed on the explicit quantification of uncertainties. This paper examines uncertainty consideration in input ground‐motion and numerical seismic response analyses as part of PBEE, with particular attention given to the physical consistency and completeness of uncertainty consideration. It is argued that the use of the commonly adopted incremental dynamic analysis leads to a biased representation of the seismic intensity and that when considering the number of ground motions to be used in seismic response analyses, attention should be given to both reducing parameter estimation uncertainty and also limiting ground‐motion selection bias. Research into uncertainties in system‐specific numerical seismic response analysis models to date has been largely restricted to the consideration of ‘low‐level’ constitutive model parameter uncertainties. However, ‘high‐level’ constitutive model and model methodology uncertainties are likely significant and therefore represent a key research area in the coming years. It is also argued that the common omission of high‐level seismic response analysis modelling uncertainties leads to a fallacy that ground‐motion uncertainty is more significant than numerical modelling uncertainty. The author's opinion of the role of uncertainty analysis in PBEE is also presented. Copyright © 2013 John Wiley & Sons, Ltd.     

Developing efficient scalar and vector intensity measures for IDA capacity estimation by incorporating elastic spectral shape information

Scalar and vector intensity measures are developed for the efficient estimation of limit‐state capacities through incremental dynamic analysis (IDA) by exploiting the elastic spectral shape of individual records. IDA is a powerful analysis method that involves subjecting a structural model to several ground motion records, each scaled to multiple levels of intensity (measured by the intensity measure or IM), thus producing curves of structural response parameterized by the IM on top of which limit‐states can be defined and corresponding capacities can be calculated. When traditional IMs are used, such as the peak ground acceleration or the first‐mode spectral acceleration, the IM‐values of the capacities can display large record‐to‐record variability, forcing the use of many records to achieve reliable results. By using single optimal spectral values as well as vectors and scalar combinations of them on three multistorey buildings significant dispersion reductions are realized. Furthermore, IDA is extended to vector IMs, resulting in intricate fractile IDA surfaces. The results reveal the most influential spectral regions/periods for each limit‐state and building, illustrating the evolution of such periods as the seismic intensity and the structural response increase towards global collapse. The ordinates of the elastic spectrum and the spectral shape of each individual record are found to significantly influence the seismic performance and they are shown to provide promising candidates for highly efficient IMs. Copyright © 2005 John Wiley & Sons, Ltd.     

Assessing uncertainty in estimation of seismic response for PBEE

State‐of‐the‐art approaches to probabilistic assessment of seismic structural reliability are based on simulation of structural behavior via nonlinear dynamic analysis of computer models. Simulations are carried out considering samples of ground motions supposedly drawn from specific populations of signals virtually recorded at the site of interest. This serves to produce samples of structural response to evaluate the failure rate, which in turn allows to compute the failure risk (probability) in a time interval of interest. This procedure alone implies that uncertainty of estimation affects the probabilistic results. The latter is seldom quantified in risk analyses, although it may be relevant. This short paper discusses some basic issues and some simple statistical tools, which can aid the analyst towards the assessment of the impact of sample variability on fragility functions and the resulting seismic structural risk. On the statistical inference side, the addressed strategies are based on consolidated results such as the well‐known delta method and on some resampling plans belonging to the bootstrap family. On the structural side, they rely on assumptions and methods typical in performance‐based earthquake engineering applications. Copyright © 2017 John Wiley & Sons, Ltd.     

Seismic performance sensitivity and uncertainty analysis of gravity dams

Uncertainties in structural engineering are often arising from the modeling assumptions and errors, or from variability in input loadings. A practical approach for dealing with them is to perform sensitivity and uncertainty analysis in the framework of stochastic and probabilistic methods. These analyses can be statically and dynamically performed through nonlinear static pushover and IDA techniques, respectively. Of the existing structures, concrete gravity dams are infrastructures which may encounter many uncertainties. In this research, probabilistic analysis of the seismic performance of gravity dams is presented. The main characteristics of the nonlinear tensile behavior of mass concrete, along with the intensity of earthquake excitations are considered as random variables in the probabilistic analysis. Using the tallest non‐overflow monolith of the Pine Flat gravity dam as a case study, its response under static and dynamic situations is reliably examined utilizing different combinations of parameters in the material and the seismic loading. The sensitivity analysis reveals the relative importance of each parameter independently. It will be shown that the undamaged modulus of elasticity and tensile strength of mass concrete have more significant roles on the seismic resistance of the dam than the ultimate inelastic tensile strain. In order to propagate the parametric uncertainty to the actual seismic performance of the dam, probabilistic simulation methods such as Monte Carlo simulation with Latin hypercube sampling, and approximate moment estimation techniques will be used. The final results illustrate the possibility of using a mean‐parameter dam model to estimate the mean seismic performance of the dam. Copyright © 2014 John Wiley & Sons, Ltd.     

Incremental dynamic analysis with consideration of modeling uncertainties

Incremental dynamic analysis (IDA) has been extended by introducing a set of structural models in addition to the set of ground motion records which is employed in IDA analysis in order to capture record‐to‐record variability. The set of structural models reflects epistemic (modeling) uncertainties, and is determined by utilizing the latin hypercube sampling (LHS) method. The effects of both aleatory and epistemic uncertainty on seismic response parameters are therefore considered in extended IDA analysis. The proposed method has been applied to an example of the four‐storey‐reinforced concrete frame, for which pseudo‐dynamic tests were performed at the ELSA Laboratory, Ispra. The influence of epistemic uncertainty on the seismic response parameters is presented in terms of summarized IDA curves and dispersion measures. The results of extended IDA analysis are compared with the results of IDA analysis, and the sensitivity of the seismic response parameters to the input random variable using the LHS method is discussed. It is shown that epistemic uncertainty does not have significant influence on the seismic response parameters in the range far from collapse, but could have a significant influence on collapse capacity. Copyright © 2008 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 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.     

Prediction of the median IDA curve by employing a limited number of ground motion records

A methodology has been proposed which can be used to reduce the number of ground motion records needed for the reliable prediction of the median seismic response of structures by means of incremental dynamic analysis (IDA). This methodology is presently limited to predictions of the median IDA curve only. The reduction in the number of ground motion records needed to predict the median IDA curve is achieved by introducing a precedence list of ground motion records. The determination of such a list is an optimization problem, which is solved in the paper by means of (1) a genetic algorithm and (2) a proposed simple procedure. The seismic response of a simple, computationally non‐demanding structural model has been used as input data for the optimization problem. The presented example is a three‐storey‐reinforced concrete building, subjected to two sets of ground motion records, one a free‐field set and the other a near‐field set. It is shown that the median IDA curves can be predicted with acceptable accuracy by employing only four ground motion records instead of the 24 or 30 records, which are the total number of ground motion records for the free‐field and near‐field sets, respectively. Copyright © 2007 John Wiley & Sons, Ltd.     

Comparing the adequacy of alternative ground motion intensity measures for the estimation of structural responses

Practical methods for the probability‐based seismic assessment of structures make use of estimates of demands produced by earthquakes of different intensities. The uncertainties associated with these estimates are highly dependent on the variable adopted as the intensity measure (IM, e.g., PGA, spectral acceleration, etc.). This generates the need to compare the efficiency of an originally adopted IMwith that of a new candidate. This implies comparing the dispersion of the demand measure (DM, e.g., maximum interstorey drift ratio, ductility demand, etc.) conditional to each of the two IMs. In order to obtain the demand estimates in a conventional way, a full set of dynamic response analyses should be performed for each IM under scrutiny, i.e., multiple records scaled at several fixed values of each IM. The procedure developed here serves to accelerate this comparison avoiding the effort required to evaluate the dynamic responses of the structure for all the ground motion time histories considered every time that a new IM is adopted. For this purpose, use is made of available results of analyses performed for a different (i.e., the original) IM. Two methods are proposed: the direct method involves performing a regression of the results obtained from the original analyses, taking the candidate IM as the independent variable. The indirect method involves rebuilding the probability density function of the DM given a defined value of the candidate IM by means of the total probability theorem, using the results of the original analyses and certain data relating the two IMs. The proposed methods have been tested by application to several SDOF systems with different periods and different cyclic‐response backbone curves. The conditions affecting their approximation are explored, and some criteria to improve them are identified. The procedure can also be used to determine the optimum value of a parameter to be used in a parameter‐based IM. Copyright © 2004 John Wiley & Sons, Ltd.     

Post‐earthquake bridge repair cost and repair time estimation methodology

While structural engineers have traditionally focused on individual components (bridges, for example) of transportation networks for design, retrofit, and analysis, it has become increasingly apparent that the economic costs to society after extreme earthquake events are caused at least as much from indirect costs as direct costs due to individual structures. This paper describes an improved methodology for developing probabilistic estimates of repair costs and repair times that can be used for evaluating the performance of new bridge design options and existing bridges in preparation for the next major earthquake. The proposed approach in this paper is an improvement on previous bridge loss modeling studies—it is based on the local linearization of the dependence between repair quantities and damage states so that the resulting model follows a linear relationship between damage states and repair points. The methodology uses the concept of performance groups (PGs) that account for damage and repair of individual bridge components and subassemblies. The method is validated using two simple examples that compare the proposed method to simulation and previous methods based on loss models using a power–law relationship between repair quantities and damage. In addition, an illustration of the method is provided for a complete study on the performance of a common five‐span overpass bridge structure in California. Intensity‐dependent repair cost ratios (RCRs) and repair times are calculated using the proposed approach, as well as plots that show the disaggregation of repair cost by repair quantity and by PG. This provides the decision maker with a higher fidelity of data when evaluating the contribution of different bridge components to the performance of the bridge system, where performance is evaluated in terms of repair costs and repair times rather than traditional engineering quantities such as displacements and stresses. Copyright © 2009 John Wiley & Sons, Ltd.     

Seismic loss estimation of non‐ductile reinforced concrete buildings

Non‐ductile reinforced concrete buildings represent a prevalent construction type found in many parts of the world. Due to the seismic vulnerability of such buildings, in areas of high seismic activity non‐ductile reinforced concrete buildings pose a significant threat to the safety of the occupants and damage to such structures can result in large financial losses. This paper introduces advanced analytical models that can be used to simulate the nonlinear dynamic response of these structural systems, including collapse. The state‐of‐the‐art loss simulation procedure developed for new buildings is extended to estimate the expected losses of existing non‐ductile concrete buildings considering their vulnerability to collapse. Three criteria for collapse, namely first component failure, side‐sway collapse, and gravity‐load collapse, are considered in determining the probability of collapse and the assessment of financial losses. A detailed example is presented using a seven‐story non‐ductile reinforced concrete frame building located in the Los Angeles, California. Copyright © 2012 John Wiley & Sons, Ltd.     

Effects of two alternative representations of ground‐motion uncertainty on probabilistic seismic demand assessment of structures

A probabilistic representation of the entire ground‐motion time history can be constructed based on a stochastic model that depends on seismic source parameters. An advanced stochastic simulation scheme known as Subset Simulation can then be used to efficiently compute the small failure probabilities corresponding to structural limit states. Alternatively, the uncertainty in the ground motion can be represented by adopting a parameter (or a vector of parameters) known as the intensity measure (IM) that captures the dominant features of the ground shaking. Structural performance assessment based on this representation can be broken down into two parts, namely, the structure‐specific part requiring performance assessment for a given value of the IM, and the site‐specific part requiring estimation of the likelihood that ground shaking with a given value of the IM takes place. The effect of these two alternative representations of ground‐motion uncertainty on probabilistic structural response is investigated for two hazard cases. In the first case, these two approaches are compared for a scenario earthquake event with a given magnitude and distance. In the second case, they are compared using a probabilistic seismic hazard analysis to take into account the potential of the surrounding faults to produce events with a range of possible magnitudes and distances. The two approaches are compared on the basis of the probabilistic response of an existing reinforced‐concrete frame structure, which is known to have suffered shear failure in its columns during the 1994 Northridge Earthquake in Los Angeles, California. Copyright © 2007 John Wiley & Sons, Ltd.     

Seismic intensity measures for above‐ground liquid storage tanks

A series of scalar and vector intensity measures is examined to determine their suitability within the seismic risk assessment of liquid storage tanks. Using a surrogate modelling approach on a squat tank that is examined under both anchored and unanchored support conditions, incremental dynamic analysis is adopted to generate the distributions of response parameters conditioned on each of the candidate intensity measures. Efficiency and sufficiency metrics are used in order to perform the intensity measure evaluation for individual failure modes, while a comparison in terms of mean annual frequency of exceedance is performed with respect to a damage state that is mutually governed by the impulsive and convective modes of the tank. The results reveal combinations of spectral acceleration ordinates as adequate predictors, among which the average spectral acceleration is singled out as the optimal solution. The sole exception is found for the sloshing‐controlled modes of failure, where mainly the convective period spectral acceleration is deemed adequate to represent the associated response due to their underlying linear relationship. A computationally efficient method in terms of site hazard analysis is finally proposed to serve in place of the vector‐valued intensity measures, providing a good match for the unanchored tank considered and a more conservative one for the corresponding anchored system.     

Development of empirical and analytical fragility functions using kernel smoothing methods

Fragility functions that define the probabilistic relationship between structural damage and ground motion intensity are an integral part of performance‐based earthquake engineering or seismic risk analysis. This paper introduces three approaches based on kernel smoothing methods for developing analytical and empirical fragility functions. A kernel assigns a weight to each data that is inversely related to the distance between the data value and the input of the fragility function of interest. The kernel smoothing methods are, therefore, non‐parametric forms of data interpolation. These methods enable the implicit treatment of uncertainty in either or both of ground motion intensity and structural damage without making any assumption about the shape of the resulting fragility functions. They are particularly beneficial for sparse, noisy, or non‐homogeneous data sets. For illustration purposes, two types of data are considered. The first is a set of numerically simulated responses for a four‐story steel moment‐resisting frame, and the second is a set of field observations collected after the 2010 Haiti earthquake. The results demonstrate that these methods can develop continuous representations of fragility functions without specifying their functional forms and treat sparse data sets more efficiently than conventional data binning and parametric curve fitting methods. Moreover, various uncertainty analyses are conducted to address the issues of over‐fitting, bias, and confidence intervals. Copyright © 2014 John Wiley & Sons, Ltd.     

Non‐ergodicity and PEER's framework formula

A framework formula for performance‐based earthquake engineering, advocated and used by researchers at the Pacific Earthquake Engineering Research (PEER) Center, is closely examined. The formula was originally intended for computing the mean annual rate of a performance measure exceeding a specified threshold. However, it has also been used for computing the probability that a performance measure will exceed a specified threshold during a given period of time. It is shown that the use of the formula to compute such probabilities could lead to errors when non‐ergodic variables (aleatory or epistemic) are present. Assuming a Poisson model for the occurrence of earthquakes in time, an exact expression is derived for the probability distribution of the maximum of a performance measure over a given period of time, properly accounting for non‐ergodic uncertainties. This result is used to assess the approximation involved in the PEER formula for computing probabilities. It is found that the PEER approximation of the probability has a negligible error for probabilities less than about 0.01. For larger probabilities, the error depends on the magnitude of non‐ergodic uncertainties and the duration of time considered and can be as much as 20% for probabilities around 0.05 and 30% for probabilities around 0.10. The error is always on the conservative side. Copyright © 2005 John Wiley & Sons, Ltd.