Abstract |

Abstract | A simplified procedure is developed to consider the azimuthal orientation of buildings when estimating seismic risk. Two square-plan reinforced concrete building models are considered as a testbed, one with similar and one with dissimilar properties along the two principal horizontal axes. The fragility of both structures is analysed using a set of ground motion records rotated to multiple incidence angles to develop orientation-dependent fragility functions. It has been observed that, re-orienting all records so that these structures have the same azimuth vis-à-vis the corresponding epicentre leads to significant differences compared to assuming random orientations. Additional results stemming from single-degree-of-freedom oscillators further confirm such findings, showing a dependence to the proximity to the faults and the level of dissimilarity in the principal horizontal axes of the structure. The end results point to a non-negligible bias in assessment studies when a structure’s orientation with respect to governing rupture scenarios is not taken into account. It is shown that the median of fragility curves calculated for un-rotated incidence angles can be bias-corrected through shifted by an amount that depends on the azimuthal orientation and level of axes-dissimilarity of structures.

Abstract | A study is presented on the seismic fragility of a sample of 15 reinforced-concrete school buildings built between 1960 and 1980s in the province of Foggia, Southern Italy, for which near-perfect information is provided on geometrical and mechanical parameters. In particular, the focus is on the application of two probabilistic methods, employing different compromises in terms of complexity versus accuracy. First, a Eurocode 8 compatible nonlinear static approach is employed, which is augmented via the use of the regression expressions of the SPO2FRAG tool to incorporate record-to-record variability. Second, a nonlinear dynamic approach is used to directly account for record-to-record variability via a constrained multi-stripe analysis. In particular, in view of practitioners’ needs (analysis time, computational efforts, software used), a practical mode for application of the multi-stripe analyses is proposed, based on conducting few stripe analyses, in order to predict the behavior of structures in both the elastic and inelastic ranges of response. Both the static and the dynamic approaches are shown to be viable alternatives, offering fairly matching results at the individual building-level and even closer predictions at the level of ensemble regional fragility curves that incorporate both inter-building and intra-building uncertainties.

Abstract | Incremental dynamic analysis (IDA) is widespread to evaluate the seismic response of structures. It employs one records set scaled to multiple intensity levels (IMLs) to estimate the structural response distribution. However, the dominating earthquake scenarios differ with the intensity, and a single set introduces bias that increases with scaling. The response is, however, better estimated using multiple stripe analysis (MSA) by selecting multiple hazard consistent record sets at different IMLs via, for example, conditional spectrum (CS) method. Still, IDA remains a popular tool, and would benefit from a single “good” set that would be more amenable to scaling with minimal bias. We explore alternatives of a “multi-level” CS (CSML) scheme, whereby the seismic properties of multiple IMLs are combined to derive a single set. Combined with advanced IMs, CSML provides a viable trade-off between the more accurate and complex method of CS–MSA versus the conceptually and practically simpler IDA.

Abstract | We present correlation coefficient estimates between a number of ground motion intensity measures (IMs), as measured from the NGA-West2 database, with focus on the correlation of vertical–vertical and vertical–horizontal ground motion components. The IMs considered include spectral accelerations with periods from 0.01 to 10 s, peak ground acceleration, peak ground velocity, and significant duration (for 5%–75% and 5%–95% definitions). To facilitate their use, parametric equations are also fitted to the correlation models. Finally, the dependence of the obtained correlation coefficients to magnitude, distance, and Vs₃₀ is evaluated.

Abstract | The lack of empirical data regarding earthquake damage or losses has propelled the development of dozens of analytical methodologies for the derivation of fragility and vulnerability functions. Each method will naturally have its strengths and weaknesses, which will consequently affect the associated risk estimates. With the purpose of sharing knowledge on vulnerability modeling, identifying shortcomings in the existing methods, and recommending improvements to the current practice, a group of vulnerability experts met in Pavia (Italy) in April 2017. Critical topics related to the selection of ground motion records, modeling of complex real structures through simplified approaches, propagation of aleatory and epistemic uncertainties, and validation of vulnerability results were discussed, and suggestions were proposed to improve the reliability and accuracy in vulnerability modeling.

Abstract | The buffer–gap–elastomeric bearings system used in precast girder bridges with relatively short piers is one of the efficient hybrid seismic isolation schemes. However, there are many uncertain environmental, material, and geometric parameters affecting gap size and, hence, the response of the bridge to earthquakes. To investigate, four different configurations of a three-span continuous bridge were designed for a high-seismicity site in Egypt, considering short or tall bridge piers of high or limited ductility. The performance of each bridge configuration was studied using nonlinear dynamic analysis under ground motions selected to be compatible with the site hazard. Two different treatments of parameter uncertainty were used: a mean parameter model, in which uncertainty was essentially disregarded, versus a random parameter model, represented by 60 realizations of each bridge generated via Latin Hypercube Sampling. Different intensity measures were also studied, showing that the geometric mean of multiple spectral ordinates over a period range that encompassed the vibration characteristics of both a closed-gap and open-gap model best satisfied both efficiency and sufficiency requirements. All in all, parameter uncertainty was shown to be an important issue for such bridges, requiring appropriate consideration for accurate assessment.

Summary | Pulse‐like records are well recognized for their potential to impose higher demands on structures when compared with ordinary records. The increased severity of the structural response usually caused by pulse‐like records is commonly attributed to the spectral increment around the pulse period. By comparing the building response to sets of spectrally equivalent pulse‐like and ordinary records, we show that there are characteristics of pulse‐like records beyond the shape of the acceleration response spectrum that affect the results of nonlinear dynamic analysis. Nevertheless, spectral shape together with the ratio of pulse period to the first‐mode structural period, T_p/T₁, are confirmed as “sufficient” predictors for deformation and acceleration response metrics in a building, conditioned on the seismic intensity. Furthermore, the average spectral acceleration over a period range, AvgSA, is shown to incorporate to a good proxy for spectral shape, and together with T_p/T₁, form an efficient and sufficient intensity measure for response prediction to pulse‐like ground motions. Following this latter route, we propose a record selection scheme that maintains the consistency of T_p with the hazard of the site but uses AvgSA to account for the response sensitivity to spectral shape.

Abstract | Non-linear dynamic response of SDOF systems enjoys widespread application in earthquake engineering, sometimes as a testing ground for cumbersome analytical procedures, but often as a direct proxy of first-mode-dominated structures, within the family of simplified, pushover-based methods for seismic structural assessment and/or design. This article presents DYANAS, a MATHWORKS-MATLAB^®-based graphical user interface that uses the OpenSees finite element platform to perform nonlinear dynamic analysis of single-degree-of-freedom (SDOF) oscillators. The scope of this open-source, freely distributed software is to serve as a tool for earthquake engineering research. The main advantages offered by the DYANAS interface are ease in the definition of the required analysis parameters and corresponding seismic input, efficient execution of the analyses themselves and availability of a suite of convenient, in-built post-processing tools for the management and organization of the structural responses. The types of dynamic analysis frameworks supported are incremental, multiple-stripe and cloud. Simultaneous consideration of pairs of uncoupled dynamic systems gives the possibility for intensity measures to refer to bidirectional ground motion. In the paper, an outline of the types of dynamic analysis frameworks typically used in performance-based earthquake engineering is provided, followed by a detailed description of the software and its capabilities, that include an array of post-processing tools. In order to properly place this software tool within its natural performance-based earthquake engineering habitat, some example applications are provided at the end of the paper.

Summary | The response of a non‐circular structure strongly depends on the orientation of the horizontal ground motion components vis‐à‐vis the structure’s principal axes. At the same time, the structural response is also a function of accelerogram characteristics that give rise to considerable record‐to‐record variability even when the incident angle is neglected. Therefore, when the structural orientation relative to the fault geometry is unknown and we have limited resources for estimating the distribution of structural response given the seismic intensity, the question arises as to whether it is preferable to use (1) few records rotated to multiple orientations, (2) many records, each at a random incident angle, or (3) some combination of the two. To this purpose, we subjected several single‐degree‐of‐freedom systems and one plan‐asymmetric multi‐degree‐of‐freedom structure to a pulsive and a non‐pulsive set of ground motions using different combinations of record set size and incident angle rotations. In all cases, the natural record‐to‐record variability of the (unrotated) waveforms clearly outweighed the influence of the record orientation. In addition, the choice of an intensity measure that utilizes the geometric mean of spectral accelerations in both horizontal axes at one or more periods of vibration was found to further enhance this difference, essentially nullifying the already small effect of the incident angle. In all cases, spending any significant proportion of the limited number of dynamic analyses to incorporate the effect of incident angle was detrimental to the fidelity of the estimated performance.

Abstract | SPO2FRAG (Static PushOver to FRAGility) is introduced, a MATLAB^®-coded software tool for estimating structure-specific seismic fragility curves of buildings, using the results of static pushover analysis. The SPO2FRAG tool (available online at http://wpage.unina.it/iuniervo/doc_en/SPO2FRAG.htm) eschews the need for computationally demanding dynamic analyses by simulating the results of incremental dynamic analysis via the SPO2IDA algorithm and an equivalent single-degree-of-freedom approximation of the structure. Subsequently, fragility functions may be calculated for multiple limit states, using the intensity-measure-based analytical approach. The damage thresholds may also be random variables and uncertainty in estimation of the fragility parameters may be explicitly accounted for. The research background underlying the various modules comprising SPO2FRAG is presented together with an operational description of how the various functions are integrated within the software’s graphical user interface. Two illustrative SPO2FRAG applications are also offered, using a steel and a reinforced concrete moment resisting frame. Finally, the software’s output is compared with the results of incremental dynamic analysis as validation of SPO2FRAG’s effectiveness.

Abstract | Design of blast loaded structures is usually carried out following a deterministic rather than a probabilistic approach. The design load scenario would cover the plausible load conditions (typically some conservative estimate) that a structure would experience if an explosion occurs but the probability that the structure will satisfy the design performances for the considered scenario remains unknown. Applying a performance-based design framework typically requires arduous Monte Carlo simulations, but a probabilistic design could also be achieved by a single structural analysis when consistent safety factors are applied to the load and the structural resistance. Such a factor is proposed herein for the case of components subjected to impulsive blast loads. The dependence of the safety factor on the amount of explosive, stand-off distance and their variability is estimated numerically and provided by means of regression formulas. A design example using the proposed safety factor is carried out and Monte Carlo simulation is used for verification. The results confirm the validity of the proposed safety factor approach and its applicability for the performance-based design of blast loaded structures using the current design practice methods.

Summary | The seismic design of an eight‐story reinforced concrete space frame building is undertaken using a yield frequency spectra (YFS) performance‐based approach. YFS offer a visual representation of the entire range of a system’s performance in terms of the mean annual frequency (MAF) of exceeding arbitrary global ductility or displacement levels versus the base shear strength. As such, the YFS framework can establish the required base shear and corresponding first‐mode period to satisfy arbitrary performance objectives for any structure that may be approximated by a single‐degree‐of‐freedom system with given yield displacement and capacity curve shape. For the eight‐story case study building, deformation checking is the governing limit state. A conventional code‐based design was performed using seismic intensities tied to the desired MAF for safety checking. Then, the YFS‐based approach was employed to redesign the resulting structure working backwards from the desired MAF of response (rather than intensity) to estimate an appropriate value of seismic intensity for use within a typical engineering design process. For this high‐seismicity and high‐importance midrise building, a stiffer system with higher base shear strength was thus derived. Moreover, performance assessment via incremental dynamic analysis showed that while the code‐design did not meet the required performance objective, the YFS‐based redesign needed only pushover analysis results to offer a near‐optimal design outcome. The rapid convergence of the method in a single design/analysis iteration emphasized its efficiency and practicability as a design aid for practical application. Copyright © 2017 John Wiley & Sons, Ltd.

Summary | The use of a seismic intensity measure (IM) is paramount in decoupling seismic hazard and structural response estimation when assessing the performance of structures. For this to be valid, the IM needs to be sufficient;that is, the engineering demand parameter (EDP) response should be independent of other ground motion characteristics when conditioned on the IM. Whenever non‐trivial dependence is found, such as in the case of the IM being the first‐mode spectral acceleration, ground motion selection must be employed to generate sets of ground motion records that are consistent vis‐à‐vis the hazard conditioned on the IM. Conditional spectrum record selection is such a method for choosing records that are consistent with the site‐dependent spectral shape conditioned on the first‐mode spectral acceleration. Based on a single structural period, however the result may be suboptimal, or insufficient, for EDPs influenced by different period values, for example, peak interstory drifts or peak floor accelerations at different floors, potentially requiring different record suites for each. Recently, the log‐average spectral acceleration over a period range, AvgSA, has emerged as an improved scalar IM for building response estimation whose hazard can be evaluated using existing ground motion prediction equations. Herein, we present a recasting of conditional spectrum record selection that is based on AvgSA over a period range as the conditioning IM. This procedure ensures increased efficiency and sufficiency in simultaneously estimating multiple EDPs by means of a single IM. Copyright © 2017 John Wiley & Sons, Ltd.

Summary | When performing loss assessment of a geographically dispersed building portfolio, the response or loss (fragility or vulnerability) function of any given archetype building is typically considered to be a consistent property of the building itself. On the other hand, recent advances in record selection have shown that the seismic response of a structure is, in general, dependent on the nature of the hazard at the site of interest. This apparent contradiction begs the question: Are building fragility and vulnerability functions independent of site, and if not, what can be done to avoid having to reassess them for each site of interest? In the following, we show that there is a non‐negligible influence of the site, the degree of which depends on the intensity measure adopted for assessment. Employing a single‐period (e.g., first‐mode), spectral acceleration would require careful record selection at each site and result to significant site‐to‐site variability of the fragility or vulnerability function. On the other hand, an intensity measure comprising the geometric mean of multiple spectral accelerations considerably reduces such variability. In tandem with a conditional spectrum record selection that accounts for multiple sites, it can offer a viable approach for incorporating the effect of site dependence into fragility and vulnerability estimates. Copyright © 2017 John Wiley & Sons, Ltd.

Abstract | Nonlinear static procedures, which relate the seismic demand of a structure to that of an equivalent single-degree-offreedom (SDOF) oscillator, are well-established tools in the performance-based earthquake engineering paradigm. Initially, such procedures made recourse to inelastic spectra derived for simple elastic-plastic bilinear oscillators, but the request for demand estimates that delve deeper into the inelastic range, motivated investigating the seismic demand of oscillators with more complex backbone curves. Meanwhile, near-source (NS) pulse-like ground motions have been receiving increased attention, since they can induce a distinctive type of inelastic demand. Pulse-like NS ground motions are usually the result of rupture directivity, where seismic waves generated at different points along the rupture front arrive at a site at the same time, leading to a double-sided velocity pulse, which delivers most of the seismic energy. Recent research has led to a methodology for incorporating this NS effect in the implementation of nonlinear static procedures. Both of the aforementioned lines of research motivate the present study on the ductility demands imposed by pulse-like NS ground motions on oscillators that feature pinching hysteretic behavior with trilinear backbone curves. Incremental dynamic analysis (IDA) is used considering one hundred and thirty pulse-like-identified ground motions. Median, 16% and 84% fractile IDA curves are calculated and fitted by an analytical model. Leastsquares estimates are obtained for the model parameters, which importantly include pulse period Tp. The resulting equations effectively constitute an R – μ – T – T p relation for pulse-like NS motions. Potential applications of this result towards estimation of NS seismic demand are also briefly discussed.

Summary | The potential of post‐tensioned self‐centering moment‐resisting frames (SC‐MRFs) and viscous dampers to reduce the economic seismic losses in steel buildings is evaluated. The evaluation is based on a prototype steel building designed using four different seismic‐resistant frames: (i) conventional moment resisting frames (MRFs); (ii) MRFs with viscous dampers; (iii) SC‐MRFs; or (iv) SC‐MRFs with viscous dampers. All frames are designed according to Eurocode 8 and have the same column/beam cross sections and similar periods of vibration. Viscous dampers are designed to reduce the peak story drift under the design basis earthquake (DBE) from 1.8% to 1.2%. Losses are estimated by developing vulnerability functions according to the FEMA P‐58 methodology, which considers uncertainties in earthquake ground motion, structural response, and repair costs. Both the probability of collapse and the probability of demolition because of excessive residual story drifts are taken into account. Incremental dynamic analyses are conducted using models capable to simulate all limit states up to collapse. A parametric study on the effect of the residual story drift threshold beyond which is less expensive to rebuild a structure than to repair is also conducted. It is shown that viscous dampers are more effective than post‐tensioning for seismic intensities equal or lower than the maximum considered earthquake (MCE). Post‐tensioning is effective in reducing repair costs only for seismic intensities higher than the DBE. The paper also highlights the effectiveness of combining post‐tensioning and supplemental viscous damping by showing that the SC‐MRF with viscous dampers achieves significant repair cost reductions compared to the conventional MRF.

Abstract | Present building-specific loss assessment state-of-art involves the convolution of seismic hazard and building seismic demands. The latter is conditioned on spectral acceleration, Sa(T₁), at the building’s first mode as the ground motion intensity measure (IM) and is typically estimated by carrying out nonlinear dynamic analyses on a two-dimensional (2-D) model. By new proposals on the use of improved IMs that can introduce higher fidelity, the accuracy in loss estimation becomes an open question. In reply, we offer a uniform basis for comparing the loss estimates for a set of eight different scalar and vector IMs whose hazard can be predicted with existing GMPEs. Despite all eight being legitimate IMs, and despite the consistent use of conditional spectrum record selection, we find large differences in the estimated loss hazard. This points to the large uncertainty still lingering when connecting hazard to loss. Among the IMs considered here, the vector IMs and at least a scalar average of spectral accelerations showed a remarkable stability in their predictions for the three-dimensional (3-D) buildings, pointing to a potential for reliable applications.

Abstract | The advantages and disadvantages of using scalar and vector ground motion intensity measures (IMs) are discussed for the local, story-level seismic response assessment of three-dimensional (3-D) buildings. Candidate IMs are spectral accelerations, at a single period (Sa) or averaged over a period range (Sa_avg). Consistent scalar and vector probabilistic seismic hazard analysis results were derived for each IM, as described in the companion paper in this issue (Kohrangi et al. 2016). The response hazard curves were computed for three buildings with reinforced concrete infilled frames using the different IMs as predictors. Among the scalar IMs, Sa_avg tends to be the best predictor of both floor accelerations and inter story drift ratios at practically any floor. However, there is an improvement in response estimation efficiency when employing vector IMs, specifically for 3-D buildings subjected to both horizontal components of ground motion. This improvement is shown to be most significant for a tall plan-asymmetric building.

Abstract | A realistic assessment of building economic losses and collapse induced by earthquakes requires the monitoring of several response measures, both story-specific and global. The prediction of such response measures benefits from using multiple ground motion intensity measures (IMs) that are, in general, correlated. To allow the inclusion of multiple IMs in the risk assessment process, it is necessary to have a practical tool that computes the vector-valued hazard of all such IMs at the building site. In this paper, vector-valued probabilistic seismic hazard analysis (VPSHA) is implemented here as a post-processor to scalar PSHA results. A group of candidate scalar and vector IMs based on spectral acceleration values, ratios of spectral acceleration values, and spectral accelerations averaged over a period range are defined and their hazard evaluated. These IMs are used as structural response predictors of three-dimensional (3-D) models of reinforced concrete buildings described in a companion paper (Kohrangi et al. 2016).

Abstract | This paper addresses the prediction of the median peak floor acceleration (PFA) demand of elastic structures subjected to seismic excitation by means of an adapted response spectrum method. Modal combination is based on a complete quadratic combination (CQC) rule. In contrast to previous studies, in the present contribution closed form solutions for the correlation coefficients and peak factors entering the CQC rule are derived using concepts of normal stationary random vibration theory. A ground motion set, which matches the design response spectrum for a specific site and a target dispersion, is used to define the stochastic base excitation. The response spectrum method is tested for various planar and spatial generic high-rise structures subjected to this particular ground motion set. A comparison of the outcomes with the results of computationally more expensive response history analyses shows the applicability and accuracy of the proposed simplified method.

Abstract | Blast-loaded structures are presently assessed and designed following a deterministic approach, where only a set of structural analyses under worst-case design scenarios are carried out in order to verify each limit state. As a rational alternative, a conditional probabilistic approach is introduced to offer comprehensive risk assessment and to allow the design with user-defined confidence in meeting performance targets in view of uncertainties. To simplify the probabilistic consideration of the uncertain parameters, the determination of the blast hazard and the structural response are decoupled into the evaluation of blast hazard curves and structural fragilities curves, respectively, by introducing a single conditioning intensity measure. This is chosen to be the impulse density, shown to be sufficient for impulse-governed scenarios, achieving a reduction of the computational effort by several orders of magnitude without introducing bias. Furthermore a problem-specific safety factor formulation is introduced to incorporate the influence of uncertainties in a simple manner, akin to current engineering practice. As a proof-of-concept test, a steel built-up blast resistant door is subjected to an accidental detonation of mortar rounds in a military facility. The equivalent single degree of freedom model is adopted in order to conduct the structural analyses, while detailed finite element analyses are carried out for validation. Finally, the conditional approach risk analysis on the steel door is compared against the results obtained through the comprehensive (probabilistic) unconditional approach, showing the validity of both the proposed intensity measure and safety factor formulation.

Summary | 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.

Abstract | Traditional or historic masonry structures occur in large populations throughout the world, particularly in preserved historical city clusters. Being non-engineered and aging these structures are in urgent need of assessment and seismic repair/rehabilitation. However, traditional masonry presents important challenges to computational modeling, owing to complexity of structural system, material inhomogeneity, and contact interactions that collectively can only be addressed through detailed 3D nonlinear representation. In this article, a simple performance assessment model is developed in order to address the need for preliminary assessment tools for this class of structures. The objective is to be able to rapidly identify buildings that are at higher risk in the event of a significant earthquake, potentially justifying a second round of more detailed evaluation. The proposed model defines the characteristics of a Single Degree of Freedom representation of the building, formulating consistent 3D shape functions to approximate its fundamental mode of vibration considering both in-plane and out-plane wall bending as a result of insufficient diaphragm action. Parametric expressions for the dynamic properties are derived in terms of the important geometric, material, and system characteristics, and are used to express local demand from global estimates. Acceptance criteria are established both in terms of deformation and strength indices to guide retrofit. An application example of the proposed assessment methodology is included to demonstrate the ability of the model to reproduce the essential features of traditional masonry buildings under seismic action.

Abstract | We discuss the current state-of-the-art on the assessment of systems (structures and lifelines) subjected to seismic loading. Severe earthquakes are random events that may have a devastating outcome. Therefore, the treatment of seismic loading and its effects has many layers due to the random nature of the problem and also its consequences. We attempt to provide a wide approach to the problem, referencing the major contributions in the field and discussing the most suitable methods for tackling the challenges involved. Our study extends from modeling seismic hazard to the modeling and analysis of structures and lifelines. Emphasis is given to probabilistic performance estimation frameworks and the rational treatment of uncertainty, a major element of contemporary earthquake engineering.

Abstract | The applicability of nonlinear static procedures for estimating the seismic demands of typical regular RC moment-resisting frames is evaluated. This work, conducted within the framework of the ATC-76-6 project, shows the degree to which nonlinear static methods can characterize global and local response demands vis-à–vis those determined by nonlinear dynamic analysis for three RC moment-frame buildings. The response quantities (engineering demand parameters) considered are peak story displacements, story drifts, story shears, and floor overturning moments. The single-mode pushover methods evaluated include the N2 and the ASCE-41 coefficient methods. Multi-modal pushover methods, such as modal pushover analysis and the consecutive modal pushover method, were also evaluated. The results indicate that the relatively good performance of the single-mode methods observed for low-rise buildings rapidly deteriorates as the number of stories increases. The multi-modal techniques generally extend the range of applicability of pushover methods, but at the cost of additional computation and without ensuring the reliability of the results.

Summary | The piecewise linear (‘multilinear’) approximation of realistic force‐deformation capacity curves is investigated for structural systems incorporating generalized plastic, hardening, and negative stiffness behaviors. This fitting process factually links capacity and demand and lies at the core of nonlinear static assessment procedures. Despite codification, the various fitting rules used can produce highly heterogeneous results for the same capacity curve, especially for the highly‐curved backbones resulting from the gradual plasticization or the progressive failures of structural elements. To achieve an improved fit, the error introduced by the approximation is quantified by studying it at the single‐degree‐of‐freedom level, thus avoiding any issues related to multi‐degree‐of‐freedom versus single‐degree‐of‐freedom realizations. Incremental dynamic analysis is employed to enable a direct comparison of the actual backbones versus their candidate piecewise linear approximations in terms of the spectral acceleration capacity for a continuum of limit‐states. In all cases, current code‐based procedures are found to be highly biased wherever widespread significant stiffness changes occur, generally leading to very conservative estimates of performance. The practical rules determined allow, instead, the definition of standardized low‐bias bilinear, trilinear, or quadrilinear approximations, regardless of the details of the capacity curve shape. Copyright © 2012 John Wiley & Sons, Ltd.

Abstract | Throughout the world, buildings are reaching the end of their design life and develop new pathologies that decrease their structural capacity. Usually the ageing process is neglected in seismic design or seismic risk assessment but may become important for older structures, especially, if they are intended to be in service even after they exceed their design life. Thus, a simplified methodology for seismic performance evaluation with consideration of performance degradation over time is presented, based on an extension of the SAC/FEMA probabilistic framework for estimating mean annual frequencies of limit state exceedance. This is applied to an example of an older three-storey asymmetric reinforced concrete building, in which corrosion has just started to propagate. The seismic performance of the structure is assessed at several successive times and the instantaneous and overall seismic risk is estimated for the near collapse limit state. The structural capacity in terms of the maximum base shear and the maximum roof displacement is shown to decrease over time. Consequently, the time-averaged mean annual frequency of violating the near-collapse limit state increases for the corroded building by about 10% in comparison to the typical case where corrosion is neglected. However, it can be magnified by almost 40% if the near-collapse limit state is related to a brittle shear failure, since corrosion significantly affects transverse reinforcement, raising important questions on the seismic safety of the existing building stock.

Abstract | Analytical, closed-form solutions are derived for the computation of equivalent constant rates of limit-state exceedance for structures under seismic loads whose capacity is degrading with time. Seismic guidelines currently designate constant, time-independent probabilities or mean annual frequencies of exceedance that are assumed to remain invariable for the entire design life. These are at odds with the time-dependent, ever-increasing exceedance rates of ageing structures. Based on the concept of social equity and discounting of societal investments, the equivalent constant rate provides a basis for judging the safety of structures with time-dependent capacity by allowing comparisons with the code-mandated rates of limit-state exceedance. Starting from the simple SAC/FEMA solution and assuming a power-law degradation of capacity with time and a linear change in the combined epistemic and aleatory variability of capacity, we provide general solutions for the equivalent constant rate and for the limiting case of the average rate over the design life of the structure. The solutions are formulated both in the demand-based and in the intensity-based format, the latter being suitable for all limit-states, even close to global collapse. By using a 7-story reinforced concrete building as an example, we demonstrate the accuracy and the practicality of these approximations for the assessment of an existing structure.

Abstract | Incremental dynamic analysis has recently emerged to offer comprehensive evaluation of the seismic performance of structures using multiple nonlinear dynamic analyses under scaled ground-motion records. Being computer-intensive, it can benefit from parallel processing to accelerate its application on realistic structural models. While the task-farming master–slave paradigm seems ideal, severe load imbalances arise due to analysis non-convergence at structural instability, prompting the examination of task partitioning at the level of single records or single dynamic runs. Combined with a multi-tier master–slave processor hierarchy employing dynamic task generation and self-scheduling we achieve a flexible and efficient parallel algorithm with excellent scalability.

Abstract | Approximate methods based on the static pushover are introduced to estimate the seismic performance uncertainty of structures having non‐deterministic modeling parameters. At their basis lies the use of static pushover analysis to approximate Incremental Dynamic Analysis (IDA) and estimate the demand and capacity epistemic uncertainty. As a testbed we use a nine‐storey steel frame having beam hinges with uncertain moment–rotation relationships. Their properties are fully described by six, randomly distributed, parameters. Using Monte Carlo simulation with Latin hypercube sampling, a characteristic ensemble of structures is created. The Static Pushover to IDA (SPO2IDA) software is used to approximate the IDA capacity curve from the appropriately post‐processed results of the static pushover. The approximate IDAs allow the evaluation of the seismic demand and capacity for the full range of limit‐states, even close to global dynamic instability. Moment‐estimating techniques such as Rosenblueth’s point estimating method and the first‐order, second‐moment (FOSM) method are adopted as simple alternatives to obtain performance statistics with only a few simulations. The pushover is shown to be a tool that combined with SPO2IDA and moment‐estimating techniques can supply the uncertainty in the seismic performance of first‐mode‐dominated buildings for the full range of limit‐states, thus replacing semi‐empirical or code‐tabulated values (e.g. FEMA‐350), often adopted in performance‐based earthquake engineering.

Abstract | 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.

Abstract | A methodology based on incremental dynamic analysis (IDA) is presented for the evaluation of structures with vertical irregularities. Four types of storey‐irregularities are considered: stiffness, strength, combined stiffness and strength, and mass irregularities. Using the well‐known nine‐storey LA9 steel frame as a base, the objective is to quantify the effect of irregularities, both for individual and for combinations of stories, on its response. In this context a rational methodology for comparing the seismic performance of different structural configurations is proposed by means of IDA. This entails performing non‐linear time history analyses for a suite of ground motion records scaled to several intensity levels and suitably interpolating the results to calculate capacities for a number of limit‐states, from elasticity to final global instability. By expressing all limit‐state capacities with a common intensity measure, the reference and each modified structure can be naturally compared without needing to have the same period or yield base shear. Using the bootstrap method to construct appropriate confidence intervals, it becomes possible to isolate the effect of irregularities from the record‐to‐record variability. Thus, the proposed methodology enables a full‐range performance evaluation using a highly accurate analysis method that pinpoints the effect of any source of irregularity for each limit‐state. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract | 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.

Abstract | 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.

Abstract | We are presenting a practical and detailed example of how to perform incremental dynamic analysis (IDA), interpret the results and apply them to performance-based earthquake engineering. IDA is an emerging analysis method that offers thorough seismic demand and capacity prediction capability by using a series of nonlinear dynamic analyses under a multiply scaled suite of ground motion records. Realization of its opportunities requires several steps and the use of innovative techniques at each one of them. Using a nine-story steel moment-resisting frame with fracturing connections as a test bed, the reader is guided through each step of IDA: (1) choosing suitable ground motion intensity measures and representative damage measures, (2) using appropriate algorithms to select the record scaling, (3) employing proper interpolation and (4) summarization techniques for multiple records to estimate the probability distribution of the structural demand given the seismic intensity, and (5) defining limit-states, such as the dynamic global system instability, to calculate the corresponding capacities. Finally, (6) the results can be used to gain intuition for the structural behavior, highlighting the connection between the static pushover (SPO) and the dynamic response, or (7) they can be integrated with conventional probabilistic seismic hazard analysis (PSHA) to estimate mean annual frequencies of limit-state exceedance. Building upon this detailed example based on the nine-story structure, a complete commentary is provided, discussing the choices that are available to the user, and showing their implications for each step of the IDA.

Abstract | Incremental dynamic analysis (IDA) is a parametric analysis method that has recently emerged in several different forms to estimate more thoroughly structural performance under seismic loads. It involves subjecting a structural model to one (or more) ground motion record(s), each scaled to multiple levels of intensity, thus producing one (or more) curve(s) of response parameterized versus intensity level. To establish a common frame of reference, the fundamental concepts are analysed, a unified terminology is proposed, suitable algorithms are presented, and properties of the IDA curve are looked into for both single‐degree‐of‐freedom and multi‐degree‐of‐freedom structures. In addition, summarization techniques for multi‐record IDA studies and the association of the IDA study with the conventional static pushover analysis and the yield reduction R‐factor are discussed. Finally, in the framework of performance‐based earthquake engineering, the assessment of demand and capacity is viewed through the lens of an IDA study. Copyright © 2001 John Wiley & Sons, Ltd.

Abstract | The present work describes a procedure for the numerical evaluation of the classical integral-transform solution of the transient elastodynamic point-load (axisymmetric) Lamb’s problem. This solution involves integrals of rapidly oscillatory functions over semi-infinite intervals and inversion of one-sided (time) Laplace transforms. These features introduce difficulties for a numerical treatment and constitute a challenging problem in trying to obtain results for quantities (e.g. displacements) in the interior of the half-space. To deal with the oscillatory integrands, which in addition may take very large values (pseudo-pole behavior) at certain points, we follow the concept of Longman’s method but using as accelerator in the summation procedure a modified Epsilon algorithm instead of the standard Euler’s transformation. Also, an adaptive procedure using the Gauss 32-point rule is introduced to integrate in the vicinity of the pseudo-pole. The numerical Laplace-transform inversion is based on the robust Fourier-series technique of Dubner/Abate-Crump-Durbin. Extensive results are given for sub-surface displacements, whereas the limit-case results for the surface displacements compare very favorably with previous exact results.