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.     

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.     

Adapting earthquake actions in Eurocode 8 for performance‐based seismic design

Performance‐based seismic design (PBSD) can be considered as the coupling of expected levels of ground motion with desired levels of structural performance, with the objective of achieving greater control over earthquake‐induced losses. Eurocode 8 (EC8) already envisages two design levels of motion, for no collapse and damage limitation performance targets, anchored to recommended return periods of 475 and 95 years, respectively. For PBSD the earthquake actions need to be presented in ways that are appropriate to the estimation of inelastic displacements, since these provide an effective control on damage at different limit states. The adequacy of current earthquake actions in EC8 are reviewed from this perspective and areas requiring additional development are identified. The implications of these representations of the seismic loads, in terms of mapping and zonation, are discussed. The current practice of defining the loading levels on the basis of the pre‐selected return periods is challenged, and ideas are discussed for calibrating the loading‐performance levels for design on the basis of quantitative earthquake loss estimation. Copyright © 2005 John Wiley & Sons, Ltd.     

Incorporating intensity bounds for assessing the seismic safety of structures: Does it matter?

The closed‐form solution for assessing the proportion of the mean annual frequency of limit‐state exceedance as a function of integration limits is introduced, in order to study whether or not the mean annual frequency of limit‐state exceedance is overestimated if the lower and(or) upper integration limit of the risk equation are(is) not selected in a physically consistent manner. Simple formulas for assessing the threshold value of the lower and upper integration limits are also derived. These formulas can be used to quickly assess the significant range of ground motion intensity that affects the mean annual frequency of limit‐state exceedance. It is shown that the threshold values of the integration limits depend on the median intensity causing a limit‐state, the corresponding dispersion and the slope of the hazard curve in the log domain. For several reinforced concrete buildings located in a region with moderate seismicity, it is demonstrated that the mean annual frequency of collapse can be significantly overestimated when assessed by integrating the risk equation over the entire range of ground motion intensity. Copyright © 2013 John Wiley & Sons, Ltd.     

Decision support for conceptual performance‐based design

Performance assessment implies that the structural, non‐structural, and content systems are given and that decision variables, DVs, (e.g. expected annual loss, mean annual frequency of collapse) are computed and compared to specified performance targets. Performance‐based design (PBD) is different by virtue of the fact that the building and its components and systems first have to be created. Good designs are based on concepts that incorporate performance targets up front in the conceptual design process, so that subsequent performance assessment becomes more of a verification process of an efficient design rather than a design improvement process that may require radical changes of the initial design concept. In short, the design approach could consist of (a) specifying performance targets (e.g. tolerable probability of collapse, acceptable dollar losses) and associated seismic hazards, and (b) deriving engineering parameters for system selection, or perhaps better, using the relatively simple design decision support tools discussed in this paper. 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.     

Post‐earthquake functionality of highway overpass bridges

Bridges are crucial to the transportation network in a region struck by an earthquake. Collapse of a bridge determines if a road is passable. Ability of a bridge to carry traffic load after an earthquake determines the weight and speed of vehicles that can cross it. Extent of system and component structural damage in bridges determines the cost and time required for repair. Today, post‐earthquake bridge evaluation is qualitative rather than quantitative. The research presented in this paper aims to provide a quantitative engineering basis for quick and reliable evaluation of the ability of a typical highway overpass bridge to function after an earthquake. The Pacific Earthquake Engineering Research (PEER) Center's probabilistic performance‐based evaluation approach provides the framework for post‐earthquake bridge evaluation. An analytical study was performed that linked engineering demand parameters to earthquake intensity measures. The PEER structural performance database and reliability analysis tools were then used to link demand parameters to damage measures. Finally, decision variables were developed to describe three limit states, repair cost, traffic function, and collapse, in terms of induced damage. This paper presents the analytical models used to evaluate post‐earthquake bridge function, decision variables and their correlation to the considered limit states, and fragility curves that represent the probability of exceeding a bridge function limit state given an earthquake intensity. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Seismic risk assessment considering cumulative damage due to aftershocks

Calculating the limit state (LS) exceedance probability for a structure considering the main seismic event and the triggered aftershocks (AS) is complicated both by the time‐dependent rate of aftershock occurrence and also by the cumulative damage caused by the sequence of events. Taking advantage of a methodology developed previously by the authors for post‐mainshock (MS) risk assessment, the LS probability due to a sequence of mainshock and the triggered aftershocks is calculated for a given aftershock forecasting time window. The proposed formulation takes into account both the time‐dependent rate of aftershock occurrence and also the damage accumulation due to the triggered aftershocks. It is demonstrated that an existing reinforced concrete moment‐resisting frame with infills subjected to the main event and the triggered sequence exceeds the near‐collapse LS. On the other hand, the structure does not reach the onset of near‐collapse LS when the effect of triggered aftershocks is not considered. It is shown, based on simplifying assumptions, that the derived formulation yields asymptotically to the same Poisson‐type functional form used when the cumulative damage is not being considered. This leads to a range of approximate solutions by substituting the fragilities calculated for intact, MS‐damaged, and MS‐plus‐one‐AS‐damaged structures in the asymptotic simplified formulation. The latter two approximate solutions provide good agreement with the derived formulation. Even when the fragility of intact structure is employed, the approximate solution (considering only the time‐dependent rate of aftershock occurrence) leads to higher risk estimates compared with those obtained based on only the mainshock. Copyright © 2016 John Wiley & Sons, Ltd.     

Analytical Solutions of Three‐Dimensional Contaminant Transport Models with Exponential Source Decay

Despite the availability of numerical models, interest in analytical solutions of multidimensional advection‐dispersion systems remains high. Such models are commonly used for performing Tier I risk analysis and are embedded in many regulatory frameworks dealing with groundwater contamination. In this work, we develop a closed‐form solution of the three‐dimensional advection‐dispersion equation with exponential source decay, first‐order reaction, and retardation, and present an approach based on some ease of use diagrams to compare it with the integral open form solution and with earlier versions of the closed‐form solution. The comparison approach focuses on the relative differences associated with source decay and the effect of simulation time. The analysis of concentration contours, longitudinal sections, and transverse sections confirms that the closed‐form solutions studied can be used with acceptable approximation in the central area of a plume bound transversely within the source width, both behind and beyond the advective front and for concentration values up to two orders of magnitude less than the initial source concentration. As the proposed closed‐form model can be evaluated without nested numerical computations and with simple mathematical functions, it can be very useful in risk assessment procedures.     

Uniform hazard versus uniform risk bases for performance‐based earthquake engineering of light‐frame wood construction

This paper investigates the implications of designing for uniform hazard versus uniform risk for light‐frame wood residential construction subjected to earthquakes in the United States. Using simple structural models of one‐story residences with typical lateral force‐resisting systems (shear walls) found in buildings in western, eastern and central regions of the United States as illustrations, the seismic demands are determined using nonlinear dynamic time‐history analyses, whereas the collapse capacities are determined using incremental dynamic analyses. The probabilities of collapse, conditioned on the occurrence of the maximum considered earthquakes and design earthquakes stipulated in ASCE Standard 7‐05, and the collapse margins of these typical residential structures are compared for typical construction practices in different regions in the United States. The calculated collapse inter‐story drifts are compared with the limits stipulated in FEMA 356/ASCE Standard 41‐06 and observed in the recent experimental testing. The results of this study provide insights into residential building risk assessment and the relation between building seismic performance implied by the current earthquake‐resistant design and construction practices and performance levels in performance‐based engineering of light‐frame wood construction being considered by the SEI/ASCE committee on reliability‐based design of wood structures. Further code developments are necessary to achieve the goal of uniform risk in earthquake‐resistant residential construction. Copyright © 2010 John Wiley & Sons, Ltd.     

Performance‐based seismic design via yield frequency spectra‡

Yield frequency spectra (YFS) are introduced to enable the direct design of a structure subject to a set of seismic performance objectives. YFS offer a unique view of the entire solution space for structural performance. This is portrayed in terms of the mean annual frequency (MAF) of exceeding arbitrary ductility (or displacement) thresholds, versus the base shear strength of a structural system having specified yield displacement and capacity curve shape. YFS can be computed nearly instantaneously using publicly available software or closed‐form solutions, for any system whose response can be satisfactorily approximated by an equivalent nonlinear single‐degree‐of‐freedom oscillator. Because the yield displacement typically is a more stable parameter for performance‐based seismic design compared with the period, the YFS format is especially useful for design. Performance objectives stated in terms of the MAF of exceeding specified ductility (or displacement) thresholds are used to determine the lateral strength that governs the design of the structure. Both aleatory and epistemic uncertainties are considered, the latter at user‐selected confidence levels that can inject the desired conservatism in protecting against different failure modes. Near‐optimal values of design parameters can be determined in many cases in a single step. Copyright © 2016 John Wiley & Sons, Ltd.     

Markovian modeling of seismic damage accumulation

Analysis of civil structures at the scale of life‐cycle requires stochastic modeling of degradation. Phenomena causing structures to degrade are typically categorized as aging and point‐in‐time overloads. Earthquake effects are the members of the latter category this study deals with in the framework of performance‐based earthquake engineering (PBEE). The focus is structural seismic reliability, which requires modeling of the stochastic process describing damage progression, because of subsequent events, over time. The presented study explicitly addresses this issue via a Markov‐chain‐based approach, which is able to account for the change in seismic response of damaged structures (i.e. state‐dependent seismic fragility) as well as uncertainty in occurrence and intensity of earthquakes (i.e. seismic hazard). The state‐dependent vulnerability issue arises when the seismic hysteretic response is evolutionary and/or when the damage measure employed is such that the degradation increment probabilistically depends on the conditions of the structure at the time of the shock. The framework set up takes advantage also of the hypotheses of classical probabilistic seismic hazard analysis, allowing to separate the modeling of the process of occurrence of seismic shocks and the effect they produce on the structure. It is also discussed how the reliability assessment, which is in closed‐form, may be virtually extended to describe a generic age‐ and state‐dependent degradation process (e.g. including aging and/or when aftershock risk is of interest). Illustrative applications show the options to calibrate the model and its potential in the context of PBEE. Copyright © 2015 John Wiley & Sons, Ltd.     

Probabilistic performance‐based optimum design of seismic isolation for a California high‐speed rail prototype bridge

Previous comparison studies on seismic isolation have demonstrated its beneficial and detrimental effects on the structural performance of high‐speed rail bridges during earthquakes. Striking a balance between these 2 competing effects requires proper tuning of the controlling design parameters in the design of the seismic isolation system. This results in a challenging problem for practical design in performance‐based engineering, particularly when the uncertainty in seismic loading needs to be explicitly accounted for. This problem can be tackled using a novel probabilistic performance‐based optimum seismic design (PPBOSD) framework, which has been previously proposed as an extension of the performance‐based earthquake engineering methodology. For this purpose, a parametric probabilistic demand hazard analysis is performed over a grid in the seismic isolator parameter space, using high‐throughput cloud‐computing resources, for a California high‐speed rail (CHSR) prototype bridge. The derived probabilistic structural demand hazard results conditional on a seismic hazard level and unconditional, i.e., accounting for all seismic hazard levels, are used to define 2 families of risk features, respectively. Various risk features are explored as functions of the key isolator parameters and are used to construct probabilistic objective and constraint functions in defining well‐posed optimization problems. These optimization problems are solved using a grid‐based, brute‐force approach as an application of the PPBOSD framework, seeking optimum seismic isolator parameters for the CHSR prototype bridge. This research shows the promising use of seismic isolation for CHSR bridges, as well as the potential of the versatile PPBOSD framework in solving probabilistic performance‐based real‐world design problems.     

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 reliability of traditional and innovative concentrically braced frames

Structural engineering problems are always affected by many sources of uncertainty, such as aleatory of material properties, applied loads and earthquake intensity, therefore, seismic assessment of structures should be based on probabilistic methods. Since PBSD (Performance‐based Seismic Design) philosophy was formulated, many researches have been conducted in this field in order to develop simple and accurate procedures for evaluating structural reliability. An important contribution has been provided by Jalayer and Cornell, who have developed a closed‐form expression to evaluate the mean annual frequency of exceeding a defined limit state. In this paper, by assuming the record‐to‐record variability as the only source of uncertainty, the seismic reliability of concentrically braced frames designed according to traditional and innovative methodologies is investigated, and a comparison between their performances is presented. In particular, two design methodologies have been applied: Eurocode 8 provisions and a new design methodology based on a rigorous application of ‘capacity design’ criteria. The innovative reduced section solution strategy, based on the reduction of cross sections at bracing member ends, has also been analysed. Copyright © 2011 John Wiley & Sons, Ltd.     

Closed‐Form Approximations for Two‐Dimensional Groundwater Age Patterns in a Fresh Water Lens

Simple closed‐form approximations are presented for calculating the steady‐state groundwater age distribution in two‐dimensional vertical cross sections of idealized fresh water lenses overlying salt water, for aquifers that are vertically semi‐infinite and of finite thickness. The approximations are developed on the basis of existing one‐dimensional analytical solutions for travel‐time calculation in fresh water lenses and approximate streamline formulations. The two‐dimensional age distributions based on the closed‐form solutions match convincingly with numerical simulations. As expected, notable deviations from the numerical solution are encountered at the groundwater flow divide and when submarine groundwater discharge occurs. Ratios of recharge over hydraulic conductivities are varied to explore how the magnitude of the deviations changes, and it is found that the approximate closed‐form solutions perform well over a range of conditions found in natural systems.     

Analytical models of steady-state plumes undergoing sequential first-order degradation

An exact, closed‐form analytical solution is derived for one‐dimensional (1D), coupled, steady‐state advection‐dispersion equations with sequential first‐order degradation of three dissolved species in groundwater. Dimensionless and mathematical analyses are used to examine the sensitivity of longitudinal dispersivity in the parent and daughter analytical solutions. The results indicate that the relative error decreases to less than 15% for the 1D advection‐dominated and advection‐dispersion analytical solutions of the parent and daughter when the Damköhler number of the parent decreases to less than 1 (slow degradation rate) and the Peclet number increases to greater than 6 (advection‐dominated). To estimate first‐order daughter product rate constants in advection‐dominated zones, 1D, two‐dimensional (2D), and three‐dimensional (3D) steady‐state analytical solutions with zero longitudinal dispersivity are also derived for three first‐order sequentially degrading compounds. The closed form of these exact analytical solutions has the advantage of having (1) no numerical integration or evaluation of complex‐valued error function arguments, (2) computational efficiency compared to problems with long times to reach steady state, and (3) minimal effort for incorporation into spreadsheets. These multispecies analytical solutions indicate that BIOCHLOR produces accurate results for 1D steady‐state, applications with longitudinal dispersion. Although BIOCHLOR is inaccurate in multidimensional applications with longitudinal dispersion, these multidimensional multispecies analytical solutions indicate that BIOCHLOR produces accurate steady‐state results when the longitudinal dispersion is zero. As an application, the 1D advection‐dominated analytical solution is applied to estimate field‐scale rate constants of 0.81, 0.74, and 0.69/year for trichloroethene, cis‐1,2‐dichloroethene, and vinyl chloride, respectively, at the Harris Palm Bay, FL, CERCLA site.     

Seismic reliability assessment of classical columns subjected to near‐fault ground motions

A methodology for the performance‐based seismic risk assessment of classical columns is presented. Despite their apparent instability, classical columns are, in general, earthquake resistant, as proven from the fact that many classical monuments have survived many strong earthquakes over the centuries. Nevertheless, the quantitative assessment of their reliability and the understanding of their dynamic behavior are not easy, because of the fundamental nonlinear character and the sensitivity of their response. In this paper, a seismic risk assessment is performed for a multidrum column using Monte Carlo simulation with synthetic ground motions. The ground motions adopted contain a high‐ and low‐frequency component, combining the stochastic method, and a simple analytical pulse model to simulate the directivity pulse contained in near source ground motions. The deterministic model for the numerical analysis of the system is three‐dimensional and is based on the Discrete Element Method. Fragility curves are produced conditional on magnitude and distance from the fault and also on scalar intensity measures for two engineering demand parameters, one concerning the intensity of the response during the ground shaking and the other the residual deformation of the column. Three performance levels are assigned to each engineering demand parameter. Fragility analysis demonstrated some of the salient features of these spinal systems under near‐fault seismic excitations, as for example, their decreased vulnerability for very strong earthquakes of magnitude 7 or larger. The analysis provides useful results regarding the seismic reliability of classical monuments and decision making during restoration process. Copyright © 2013 John Wiley & Sons, Ltd.     

Performance‐based earthquake engineering assessment of a self‐centering,post‐tensioned concrete bridge system

Highway bridges in highly seismic regions can sustain considerable residual displacements in their columns following large earthquakes. These residual displacements are an important measure of post‐earthquake functionality, and often determine whether or not a bridge remains usable following an earthquake. In this study, a self‐centering system is considered that makes use of unbonded, post‐tensioned steel tendons to provide a restoring force to bridge columns to mitigate the problem of residual displacements. To evaluate the proposed system, a code‐conforming, case‐study bridge structure is analyzed both with conventional reinforced concrete columns and with self‐centering, post‐tensioned columns using a formalized performance‐based earthquake engineering (PBEE) framework. The PBEE analysis allows for a quantitative comparison of the relative performance of the two systems in terms of engineering parameters such as peak drift ratio as well as more readily understood metrics such as expected repair costs and downtime. The self‐centering column system is found to undergo similar peak displacements to the conventional system, but sustains lower residual displacements under large earthquakes, resulting in similar expected repair costs but significantly lower expected downtimes. Copyright © 2010 John Wiley & Sons, Ltd.