Error estimation of closed‐form solution for annual rate of structural collapse

With the increasing emphasis of performance‐based earthquake engineering in the engineering community, several investigations have been presented outlining simplified approaches suitable for performance‐based seismic design (PBSD). Central to most of these PBSD approaches is the use of closed‐form analytical solutions to the probabilistic integral equations representing the rate of exceedance of key performance measures. Situations where such closed‐form solutions are not appropriate primarily relate to the problem of extrapolation outside of the region in which parameters of the closed‐form solution are fit. This study presents a critical review of the closed‐form solution for the annual rate of structural collapse. The closed‐form solution requires the assumptions of lognormality of the collapse fragility and power model form of the ground motion hazard, of which the latter is more significant regarding the error of the closed‐form solution. Via a parametric study, the key variables contributing to the error between the closed‐form solution and solution via numerical integration are illustrated. As these key variables cannot be easily measured, it casts doubt on the use of such closed‐form solutions in future PBSD, especially considering the simple and efficient nature of using direct numerical integration to obtain the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Simplified estimation of the expected annual loss of reinforced concrete buildings

Performance‐based earthquake engineering procedures have now developed to the point that it is possible to evaluate a range of possible decision variables, including the expected annual monetary loss (EAL). Quantification of the EAL is considered to be particularly useful because it could assist with the identification of effective design and retrofit measures that consider seismic performance over a range of intensity levels. Recognizing, however, that existing procedures for the evaluation of EAL tends to be quite time consuming, this paper builds on a recent proposal to use simplified limit state loss versus intensity relationships to compute EAL via a closed‐form equation, without the need to compile an inventory of damageable components and with freedom in the choice of structural analysis method. Various developments to the simplified approach are made in this paper to allow consideration of loss thresholds, non‐uniform damage distributions and the impact of differences in seismic performance in orthogonal directions. In addition, means of accounting for uncertainties in the simplified EAL assessment are described. The work has focused on the assessment of EAL for reinforced concrete frame buildings with details representative of construction practice adopted in Italy in the 1950s through to the early 1970s. By comparing loss assessment results obtained using a refined methodology with those obtained using the new guidelines developed here for two case study buildings, it is concluded that the simplified approach works well. Future research should therefore aim to further validate the approach and extend it to other building typologies and construction eras. Copyright © 2017 John Wiley & Sons, Ltd.     

An efficient method for estimating the collapse risk of structures in seismic regions

Assessing the probability of collapse is a computationally demanding component of performance‐based earthquake engineering. This paper examines various aspects involved in the computation of the mean annual frequency of collapse (λ_c) and proposes an efficient method for estimating the sidesway collapse risk of structures in seismic regions. By deaggregating the mean annual frequency of collapse, it is shown that the mean annual frequency of collapse is typically dominated by earthquake ground motion intensities corresponding to the lower half of the collapse fragility curve. Uncertainty in the collapse fragility curve and mean annual frequency of collapse as a function of the number of ground motions used in calculations is also quantified, and it is shown that using a small number of ground motions can lead to unreliable estimates of a structure's collapse risk. The proposed method is shown to significantly reduce the computational effort and uncertainty in the estimate. Copyright © 2012 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.     

Improved seismic hazard model with application to probabilistic seismic demand analysis

An improved seismic hazard model for use in performance‐based earthquake engineering is presented. The model is an improved approximation from the so‐called ‘power law’ model, which is linear in log–log space. The mathematics of the model and uncertainty incorporation is briefly discussed. Various means of fitting the approximation to hazard data derived from probabilistic seismic hazard analysis are discussed, including the limitations of the model. Based on these ‘exact’ hazard data for major centres in New Zealand, the parameters for the proposed model are calibrated. To illustrate the significance of the proposed model, a performance‐based assessment is conducted on a typical bridge, via probabilistic seismic demand analysis. The new hazard model is compared to the current power law relationship to illustrate its effects on the risk assessment. The propagation of epistemic uncertainty in the seismic hazard is also considered. To allow further use of the model in conceptual calculations, a semi‐analytical method is proposed to calculate the demand hazard in closed form. For the case study shown, the resulting semi‐analytical closed form solution is shown to be significantly more accurate than the analytical closed‐form solution using the power law hazard model, capturing the ‘exact’ numerical integration solution to within 7% accuracy over the entire range of exceedance rate. Copyright © 2007 John Wiley & Sons, Ltd.     

Experimental multi‐level seismic performance assessment of 3D RC frame designed for damage avoidance

This paper experimentally investigates the application of damage avoidance design (DAD) philosophy to moment‐resisting frames with particular emphasis on detailing of rocking interfaces. An 80% scale three‐dimensional rocking beam–column joint sub‐assembly designed and detailed based on damage avoidance principles is constructed and tested. Incremental dynamic analysis is used for selecting ground motion records to be applied to the sub‐assembly for conducting a multi‐level seismic performance assessment (MSPA). Analyses are conducted to obtain displacement demands due to the selected near‐ and medium‐field ground motions that represent different levels of seismic hazard. Thus, predicted displacement time histories are applied to the sub‐assembly for conducting quasi‐earthquake displacement tests. The sub‐assembly performed well reaching drifts up to 4.7% with only minor spalling occurring at rocking beam interfaces and minor flexural cracks in beams. Yielding of post‐tensioning threaded bars occurred, but the sub‐assembly did not collapse. The externally attached energy dissipators provided large hysteretic dissipation during large drift cycles. The sub‐assembly satisfied all three seismic performance requirements, thereby verifying the superior performance of the DAD philosophy. Copyright © 2007 John Wiley & Sons, Ltd.     

Alternative closed‐form solutions for the mean rate of exceedance of structural limit states

Two new closed‐form expressions representing the mean rate of exceedance of a given limit state are presented herein. These proposals overcome limitations that were identified with the original formulation of the well‐known SAC/FEMA approach. The new expressions involve new parametric functions for the modeling of the seismic hazard data and for the demand evolution for increasing values of the earthquake intensity measure. Given the carefully selected parametric form of these functions, mathematical tractability is able to be maintained to establish two new closed‐form solutions representing the mean rate of exceedance of a given limit state. The function proposed for the hazard exhibits nonlinear behavior in log‐log space and is able to represent the actual hazard data over a wider range of earthquake intensity levels. The function proposed for the demand evolution addresses issues related to the inadequate performance of the SAC/FEMA approach when force‐based demand parameters such as the shear force are considered. To illustrate the applicability of the new closed‐form solutions, the probability of occurrence of several limit states is determined for a reinforced concrete structure. Copyright © 2013 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.     

基于耐震时程法的进水塔结构抗震性能分析

进水塔结构作为水电站的进水建筑物,其抗震性能对水电站的安全运行有重要意义.以沙牌水电站的高耸独立进水塔结构为研究对象,应用3条耐震时程曲线对其进行抗震性能分析与评估,并将计算结果与天然地震动下的增量动力分析结果进行比较,对比分析进水塔塔顶位移、基底剪力及地震损伤历程.实例研究表明:耐震时程法可以较好地分析结构的抗震响应...     

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.     

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.     

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.     

Alternative non‐linear demand estimation methods for probability‐based seismic assessments

Alternative non‐linear dynamic analysis procedures, using real ground motion records, can be used to make probability‐based seismic assessments. These procedures can be used both to obtain parameter estimates for specific probabilistic assessment criteria such as demand and capacity factored design and also to make direct probabilistic performance assessments using numerical methods. Multiple‐stripe analysis is a non‐linear dynamic analysis method that can be used for performance‐based assessments for a wide range of ground motion intensities and multiple performance objectives from onset of damage through global collapse. Alternatively, the amount of analysis effort needed in the performance assessments can be reduced by performing the structural analyses and estimating the main parameters in the region of ground motion intensity levels of interest. In particular, single‐stripe and double‐stripe analysis can provide local probabilistic demand assessments using minimal number of structural analyses (around 20 to 40). As a case study, the displacement‐based seismic performance of an older reinforced concrete frame structure, which is known to have suffered shear failure in its columns during the 1994 Northridge Earthquake, is evaluated. Copyright © 2008 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.     

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.     

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.     

Identification of critical ground motions for seismic performance assessment of structures

A method is established to identify critical earthquake ground motions that are to be used in physical testing or subsequent advanced computational studies to enable seismic performance to be assessed. The ground motion identification procedure consists of: choosing a suitable suite of ground motions and an appropriate intensity measure; selecting a computational tool and modelling the structure accordingly; performing Incremental Dynamic Analysis on a non‐linear model of the structure; interpreting these results into 50th (median) and 90th percentile performance bounds; and identifying the critical ground motions that are close to these defining probabilistic curves at ground motion intensities corresponding to the design basis earthquake and the maximum considered earthquake. An illustrative example of the procedure is given for a reinforced concrete highway bridge pier designed to New Zealand specifications. Pseudodynamic tests and finite element based time history analyses are performed on the pier using three earthquake ground motions identified as: (i) a Design Basis Earthquake (10% probability in 50 years) with 90 percent confidence of non‐exceedance; (ii) a Maximum Considered Event (2% probability in 50 years) representing a median response; and (iii) a Maximum Considered Event representing 90 percent confidence of non‐exceedance. Copyright © 2006 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.     

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.     

Self‐centering structural systems with combination of hysteretic and viscous energy dissipations

This paper presents an innovative set of high‐seismic‐resistant structural systems termed Advanced Flag‐Shaped (AFS) systems, where self‐centering elements are used with combinations of various alternative energy dissipation elements (hysteretic, viscous or visco‐elasto‐plastic) in series and/or in parallel. AFS systems is developed using the rationale of combining velocity‐dependent with displacement‐dependent energy dissipation for self‐centering systems, particularly to counteract near‐fault earthquakes. Non‐linear time‐history analyses (NLTHA) on a set of four single‐degree‐of‐freedom (SDOF) systems under a suite of 20 far‐field and 20 near‐fault ground motions are used to compare the seismic performance of AFS systems with the conventional systems. It is shown that AFS systems with a combination in parallel of hysteretic and viscous energy dissipations achieved greater performance in terms of the three performance indices. Furthermore, the use of friction slip in series of viscous energy dissipation is shown to limit the peak response acceleration and induced base‐shear. An extensive parametric analysis is carried out to investigate the influence of two design parameters, λ₁ and λ₂ on the response of SDOF AFS systems with initial periods ranging from 0.2 to 3.0 s and with various strength levels when subjected to far‐field and near‐fault earthquakes. For the design of self‐centering systems with combined hysteretic and viscous energy dissipation (AFS) systems, λ₁ is recommended to be in the range of 0.8–1.6 while λ₂ to be between 0.25 and 0.75 to ensure sufficient self‐centering and energy dissipation capacities, respectively. Copyright © 2010 John Wiley & Sons, Ltd.