Inelastic Spectral Displacement Research Articles

Exploring the Use of Orientation-Independent Inelastic Spectral Displacements in the Seismic Assessment of Bridges

ABSTRACT Seismic intensity measures (IMs) provide a link between the seismic hazard and the dynamic response of structures subjected to earthquake shaking. The spectral acceleration at the first and usually dominant vibration mode, Sa(T 1), is a popular choice for building structures. Meanwhile, the IM selection for bridges is non-trivial since they do not typically possess a single dominant mode. Even for ordinary bridges with a dominant mode, the behavior can change significantly in each principal direction through the activation, or yielding, of its different components. This study examines the performance of a novel IM that incorporates ground motion directionality and structure non-linearity in this context: the nn th percentile of all rotation angles of the inelastic spectral displacement, Sd i,RotDnn. This evaluation is carried out within the context of an ordinary bridge structure and is compared with other conventional IMs used in regional risk assessment of bridges. The case study bridge utilized is a highway overcrossing located in California with two spans and a continuous prestressed reinforced concrete box girder deck section. A large ground motion set was selected from the NGA-West2 database, and incremental dynamic analysis was carried out on the structure to assess the IM performance to characterize collapse. The results indicate that Sd i,RotDnn performs very well compared to other IMs for the bridge structure and could be a prudent choice to characterize inelastic response of bridges with several possible mechanisms in different principal directions. Also, using the RotD50 definition, typically used in ground motion models, showed a 17.3% increase in efficiency compared to RotD100 definition typically used in engineering practice.

A ground motion model for orientation-independent inelastic spectral displacements from shallow crustal earthquakes

The peak inelastic displacement of a single-degree-of-freedom (SDOF) system is an efficient ground motion (GM) intensity measure (IM) for estimating seismic demands in structures and infrastructures. Most ground motion models (GMMs) that predict this IM are developed to estimate the arbitrary or geometric mean of the horizontal GM components obtained from the two as-recorded orientations. In this study, the median and maximum directional inelastic spectral displacements, denoted as Sdi,RotD50and Sdi,RotD100, respectively, were considered as IMs to develop GMMs. These orientation-independent measures of horizontal motion were calculated by rotating the GMs through all non-redundant rotation angles and have the advantage of removing the sensor orientation as a contributor to overall uncertainty. To do this, several SDOF systems were considered using a subset of the NGA-West2 database to fit GMMs for these IMs. A set of functional forms was developed using a mixed-effects regression approach. The predictor variables were the elastic period, T, the strength ratio, R, and a set of seismological GM causal parameters (i.e. magnitude, distance, soil condition, and so on). Comparison with existing GMMs that consider either inelastic spectral displacement or directionality is provided to highlight key differences and developments. In addition, the response directionality, defined as the ratio Sdi,RotD100/ Sdi,RotD50, was observed to be notably impacted by the strength ratio and elastic period, which has not been discussed in the literature to date. Overall, this GMM has the advantage of quantifying ground motion intensity via an efficient IM in Sdi, which can better highlight some fundamental issues surrounding the directionality of GMs and allow for more informative risk estimates.

A Region-Specific Ground-Motion Model for Inelastic Spectral Displacement in Northern Italy Considering Spatial Correlation Properties

Abstract The peak inelastic displacement of single-degree-of-freedom bilinear systems (Sdi) is an effective intensity measure linking ground-motion features to the inelastic response and subsequent structural and nonstructural damage of engineered systems. This study develops a region-specific ground-motion model for Sdi considering source, path, and site effects and explicitly accounting for the spatial correlation between intraevent residuals when the model parameters are estimated. The model is developed based on 2427 two-component horizontal ground-motion records from 85 events in northern Italy with magnitudes ranging from 4.0 to 6.4 and source-to-site distances less than 200 km. An exponential stationary and isotropic model is considered to represent the spatial correlation properties of Sdi (after scrutinizing the appropriateness of the underlying assumptions for such a model). Comparisons are performed with existing models in the literature in terms of Sdi estimates, as well as the (spatial correlation) effective range parameter. Two practical applications of the developed model are presented: one on estimating the spatial distribution of Sdi (as an essential ingredient for seismic loss assessments) and one on the engineering validation of region-specific ground-motion simulations. Challenges regarding such validations are also discussed.

Scaling of residual displacements in terms of elastic and inelastic spectral displacements for existing SDOF systems

Structures undergoing inelastic displacements during earthquake ground motions are known to sustain some amount of residual displacements, which may make them unusable or unsafe. In this study an attempt is made to estimate residual displacements for elastic-perfectly-plastic single-degree-of-freedom oscillators with a given lateral strength ratio. It is observed in the case of a class of ground motions that there are no trends in the dependence of residual displacement on the temporal features of the ground motion, and thus any estimation of residual displacements should be carried out only in the statistical sense. Statistical estimation of residual displacement spectrum via normalization with respect to inelastic or elastic spectral displacements is considered, and it is found that normalization with respect to inelastic spectral displacements is preferable. Expressions for residual displacement spectra are proposed for both types of normalizations and for the given-lateral-strength-ratio type oscillators.

Comparative earthquake loss estimations for high-code buildings in Istanbul

The paper presents the probabilistic and scenario based earthquake loss estimations for the case that the hazard and building inventory inputs are kept the same whereas the damage functions as well as the seismic demand estimation method are changed in an earthquake loss model. Spectral acceleration-displacement based damage assessments by alternating damage functions and inelastic demand evaluation methods are performed for high-code buildings in Istanbul. The buildings are mid- and high-rise, reinforced concrete, moment-resisting frames that are assumed to be designed in accordance with the provisions of Turkish Earthquake Resistant Design Code (1998). Three damage models, i.e. structural capacity and fragility curves, are employed for each building class: Expert judgment based capacity and fragility functions; HAZUS's high-code seismic design level capacity and fragility functions; and Capacity and fragility functions derived based on nonlinear analyses of code complying RC frames. Inelastic spectral displacement demands are computed with three methods: Capacity Spectrum Method, Modified Acceleration-Displacement Response Spectrum Method, and Displacement Coefficient Method. Analyses are realized under site-specific ground motions based on a state-of-the-art hazard model for eight return periods ranging from 100 to 2475 years as well as for an Mw = 7.5 scenario earthquake. Probabilistic loss curves for each case are developed. Estimated average annual losses (AAL) and loss ratios (AALR) are compared. Grid and district based maps illustrating the spatial distributions of estimated long term average losses per year and the loss ratios are presented. The estimated annualized loss ratios at district level in the city are compared to the earthquake insurance premium rates.

Ground motion scaling for seismic response analysis by considering inelastic response and contribution of the higher modes

This paper proposes a new scaling procedure to consider the inelastic response of structure along with the effect of higher modes in scaling the selected ground motions for seismic response analysis. This is done by obtaining the corresponding inelastic Single-Degree-of-Freedom (SDOF) system of the structure through performing a single-run modal pushover analysis with a load pattern consistent with the combined-modal-story-shear profile obtained from response spectrum analysis. Therefore, the effect of the higher modes and interaction between them in the nonlinear phase are reflected in the inelastic SDOF. The scaling process is performed such that the peak displacement of the inelastic SDOF system under the scaled record matches the inelastic spectral displacement (target displacement). The proposed procedure is evaluated through three regular and one irregular tall building. The results demonstrate the superiority of the proposed procedure in estimating the engineering demand parameters of structures with/without important higher modes effects.

Evaluation of a Simplified Method to Estimate the Peak Inter-Story Drift Ratio of Steel Frames with Hysteretic Dampers

In this paper, a simplified method is proposed to estimate the peak inter-story drift ratios of steel frames with hysteretic dampers. The simplified method involved the following: (1) the inelastic spectral displacement is estimated using a single-degree-of-freedom (SDOF) system with multi-springs, which is equivalent to a steel frame with dampers and in which multi-springs represent the hysteretic behavior of dampers; (2) the first inelastic mode vector is estimated using a pattern of story drifts obtained from nonlinear static pushover analysis; and (3) the effects of modes higher than the first mode are estimated by using the jth modal period, jth mode vector, and jth modal damping ratio obtained from eigenvalue analysis. The accuracy of the simplified method is estimated using the results of nonlinear time history analysis (NTHA) on a series of three-story, six-story, and twelve-story steel moment resisting frames with steel hysteretic dampers. Based on the results of a comparison of the peak inter-story drift ratios estimated by the simplified method and that computed via NTHA using an elaborate analytical model, the accuracy of the simplified method is sufficient for evaluating seismic demands.

Simplified estimation method of inelastic seismic demands of buildings with seesaw system using fluid viscous dampers

In this paper, the simplified estimation method, which is useful for structural reliability assessment and design, is proposed to estimate the inelastic seismic demands of buildings with a seesaw energy dissipation system (SEDS) using fluid viscous dampers (FVDs). The proposed method considers the inelastic spectral displacement and the inelastic first mode vector. In this method, the inelastic spectral displacement is estimated using a single-degree-of-freedom (SDOF) system equivalent to a multi-story frame with an SEDS using FVDs, and the inelastic first mode vector is estimated using a pattern of floor displacements obtained from nonlinear static pushover analysis (NSPA) of a multi-story frame, which considers the behavior of velocity-dependent dampers. Nonlinear time history analysis (NTHA) was performed on series of three- and six-story steel moment-resisting frames with an SEDS using FVDs and a corresponding equivalent SDOF system model. Also, NSPA was performed on an inelastic multi-story frame with an additional spring, which considers the viscous behavior of velocity-dependent dampers. Comparing the results of the peak inter-story drift ratio estimated by the proposed method with those computed via NTHA, the proposed method has sufficient accuracy for the evaluation of seismic demands.

Ground Motion Prediction Equation for Inelastic Spectral Displacement in Iran

This paper is devoted to investigate the inelastic displacement spectra compatible to Iran. Owing to inadequacy of code-compliant elastic design spectra to predict structural damage during sever earthquakes, different approaches are proposed to overcome this problem. Inelastic design spectrum is one of the most well-known methods introduced by researchers. In practice, attenuation relationships can be used in probabilistic seismic hazard analysis to obtain the inelastic design spectrum. In this paper a new ground motion prediction equation has been proposed for inelastic spectral displacement. In this regard, 806 horizontal ground motions are utilized with magnitudes ranging from 4 to 7 and epicentral distances less than 200 Km which are obtained from 330 earthquakes in Iran. According to the tectonic condition, Iran zonecan be divided into two parts: Zagrosand Alborz-central Iran. However, three equations have been presented for the whole country zone, Zagros and Alborz-central Iran zones, separately. The main parameters such as earthquake magnitude, site-source distance and site conditions have been related to the inelastic spectral displacement. Based on average shear wave velocity to a depth of 30m, sites have been categorized into three classes. For practical purpose, simplified equations have been proposed to predict inelastic spectral displacements for Iran.

Empirical Correlation between Inelastic and Elastic Spectral Displacement Demands

Inelastic spectral displacement demand is arguably one of the most effective, simplified means of relating earthquake intensity to building damage. However, seismic hazard assessment is typically conducted using empirical ground-motion prediction equations (GMPEs) that only provide indications of elastic spectral response quantities, which an engineer subsequently relates to inelastic demands using empirical relationships such as the equal-displacement rule. An alternative approach is to utilize relationships for the inelastic spectral displacement demand directly within the seismic hazard assessment process. Such empirical relationships are developed in this work, as a function of magnitude, distance, building period, and yield strength coefficient, for four different hysteretic models that are representative of a wide range of possible structural typologies found in practice. The new relationships are likely to be particularly useful for performance-based seismic design and assessment.

Impact of earthquake ground motion characteristics on collapse risk of post-mainshock buildings considering aftershocks

This paper investigates the influence of duration and frequency content of earthquakes on the collapse risk of post-mainshock buildings accounting for four damage states. The 5–95% significant duration Ds and the mean period Tm are selected as the index parameters to represent the duration and frequency content of ground motions, respectively. The modified Ibarra–Krawinkler hysteretic model is used in the structural models to capture strength and stiffness degradation associated with structural damages. The ground motion intensity is measured by inelastic spectral displacement (Sdi) to implicitly capture the spectral shape effect. Structural collapse capacities are determined using a suite of 62 records with a broad range of earthquake ground motion characteristics. The results demonstrate that both the duration and frequency content of ground motion play a significant role in structural collapse capacity. The degree of influence of aftershock characteristics on post-mainshock building collapse capacities becomes more significant as the structural damage level from the mainshock increases. As a result, post-mainshock structures with more serious damage states may be more fragile when subjected to the aftershocks with longer duration and lower frequency. Aftershocks are usually characterized by shorter duration and higher frequency than those of the corresponding mainshocks, thus the collapse risk of post-mainshock buildings may not be properly estimated using seeding scaled mainshock record as an aftershock compared to those using as-recorded mainshock–aftershock sequences.

Fuzzy‐Valued Intensity Measures for Near‐Fault Pulse‐Like Ground Motions

Abstract Two fuzzy‐valued (FV) structure‐specific intensity measures (IMs), one based on squared spectral velocity and the other on inelastic spectral displacement, are presented to characterize near‐fault pulse‐like ground motions for performance‐based seismic design and assessment of concrete frame structures. The first IM is designed through fuzzying structural fundamental period to account for the period shift effect due to stiffness degradation, whereas the second IM is developed to take into account higher mode contribution in high‐rise buildings by employing a fuzzy combination of the first two or three modes for the lateral loading pattern in pushover analysis. A benchmark study of three example reinforced concrete frame structures shows that for moderate‐ to medium‐period structures, both of the proposed IMs improve prediction accuracy in comparison with the existing IMs. For short‐period structures, the FV inelastic spectral displacement is the best.

Seismic fragility assessment of low-rise stone masonry buildings

Many historic buildings in old urban centers in Eastern Canada are made of stone masonry reputed to be highly vulnerable to seismic loads. Seismic risk assessment of stone masonry buildings is therefore the first step in the risk mitigation process to provide adequate planning for retrofit and preservation of historical urban centers. This paper focuses on development of analytical displacement-based fragility curves reflecting the characteristics of existing stone masonry buildings in Eastern Canada. The old historic center of Quebec City has been selected as a typical study area. The standard fragility analysis combines the inelastic spectral displacement, a structure-dependent earthquake intensity measure, and the building damage state correlated to the induced building displacement. The proposed procedure consists of a three-step development process: (1) mechanics-based capacity model, (2) displacement-based damage model and (3) seismic demand model. The damage estimation for a uniform hazard scenario of 2% in 50 years probability of exceedance indicates that slight to moderate damage is the most probable damage experienced by these stone masonry buildings. Comparison is also made with fragility curves implicit in the seismic risk assessment tools Hazus and ELER. Hazus shows the highest probability of the occurrence of no to slight damage, whereas the highest probability of extensive and complete damage is predicted with ELER. This comparison shows the importance of the development of fragility curves specific to the generic construction characteristics in the study area and emphasizes the need for critical use of regional risk assessment tools and generated results.

A Method for Estimating Spectral Displacements Using a Velocity Pulse

A method for computing spectral displacements using a single velocity pulse is presented. Bilinear systems with 5% damping were studied under 101 ground motions. Near-fault ground motions with known forward-directivity effects were not included. The systems covered a wide range of vibration periods and included six strength ratios. The results showed that for periods less than or equal to the characteristic period of the ground, spectral displacements were nearly always governed by the ground velocity pulse of largest amplitude. Based on these results, it is shown that spectral displacements can be estimated using a sinusoidal acceleration pulse whose elastic input energy is equivalent to that containing the peak ground velocity. Based on pulse duration and amplitude, a simplified method is proposed to estimate inelastic spectral displacements for periods less than or equal to the characteristic period of the ground. The proposed method provides estimates of spectral displacements that are in very good agreement with those computed from the actual ground motions, and it is shown to be as good a predictor as existing proposals.

Probabilistic seismic demand analysis of a slender RC shear wall considering soil–structure interaction effects

Reinforced concrete shear walls are often used to resist the lateral loads imposed by earthquakes. Accurate evaluation of the seismic demands on shear walls requires adequate considerations of the nonlinear behavior of structural and foundation elements, the interaction between them, and the uncertainty and variability associated with earthquake ground motions. This paper presents a comprehensive probabilistic seismic demand analysis of a typical mid-rise slender shear wall in western US with a flexible foundation and evaluates the significance of soil–structure interaction (SSI) effects on their damage probability. Utilizing realistic numerical models for the shear wall and its foundation, the nonlinear time history analyses were conducted with a large number of recorded ground motions. Response quantities such as maximum inter-story drift ratio, base shear, foundation displacement and rotation are monitored and related to the intensity measure of ground motions (i.e. the inelastic spectral displacement S d i ) for the cases with and without considering the SSI effects. Subsequently, the fragility functions of the shear wall are derived and the impact of SSI effects is investigated. It is found that the SSI generally reduces the damage probability of the shear wall, especially when soil nonlinearity is taken into account. The sensitivity of various seismic demands to soil parameters is also discussed. Under strong ground shakings, SSI effects on the maximum inter-story drift are most sensitive to the friction angle of the soil. It is suggested that the damages in foundation and surrounding soil should also be considered in order to systematically evaluate the SSI effects on damage probability of shear wall buildings.

Ground-Motion Prediction Equations (GMPEs) for inelastic displacement and ductility demands of constant-strength SDOF systems

The objective of this paper is to present ground-motion prediction equations for ductility demand and inelastic spectral displacement of constant-strength perfectly elasto-plastic single-degree-of-freedom (SDOF) oscillators. Empirical equations have been developed to compute the ductility demand as a function of two earthquake parameters; moment magnitude, and source-to-site distance; one site parameter, the ground type; and three oscillator parameters, an undamped natural period, critical damping ratio, and the mass-normalized yield strength. In addition, a comparative study of the proposed model with selected previous studies and recommendations of Eurocode 8 is presented. Proposed equations can easily be incorporated in existing probabilistic seismic hazard analysis (PSHA) software packages with the introduction of an additional parameter. This leads to hazard curves for inelastic spectral displacement, which can provide better estimates of target displacement for nonlinear static procedures and an efficient intensity measure for probabilistic seismic demand analysis (PSDA). Proposed equations will be useful in performance evaluation of existing structures.

Structural performance assessment under near‐source pulse‐like ground motions using advanced ground motion intensity measures

AbstractThis paper demonstrates the effectiveness of utilizing advanced ground motion intensity measures (IMs) to evaluate the seismic performance of a structure subject to near‐source ground motions. Ordinary records are, in addition, utilized to demonstrate the robustness of the advanced IM with respect to record selection and scaling. To perform nonlinear dynamic analyses (NDAs), ground motions need to be selected; as a result, choosing records that are not representative of the site hazard can alter the seismic performance of structures. The median collapse capacity (in terms of IM), for example, can be systematically dictated by including a few aggressive or benign pulse‐like records into the record set used for analyses. In this paper, the elastic‐based IM such as the pseudo‐spectral acceleration (Sa) or a vector of Sa and epsilon has been demonstrated to be deficient to assess the structural responses subject to pulse‐like motions. Using advanced IMs can be, however, more accurate in terms of probabilistic response prediction. Scaling earthquake records using advanced IMs (e.g. inelastic spectral displacement, Sdi, and IM 1I&2E; the latter is for the significant higher‐mode contribution structures) subject to ordinary and/or pulse‐like records is efficient, sufficient, and robust relative to record selection and scaling. As a result, detailed record selection is not necessary, and records with virtually any magnitude, distance, epsilon and pulse period can be selected for NDAs. Copyright © 2008 John Wiley & Sons, Ltd.

Probabilistic seismic demand analysis using advanced ground motion intensity measures

AbstractOne of the objectives in performance‐based earthquake engineering is to quantify the seismic reliability of a structure at a site. For that purpose, probabilistic seismic demand analysis (PSDA) is used as a tool to estimate the mean annual frequency of exceeding a specified value of a structural demand parameter (e.g. interstorey drift). This paper compares and contrasts the use, in PSDA, of certain advanced scalar versus vector and conventional scalar ground motion intensity measures (IMs). One of the benefits of using a well‐chosen IM is that more accurate evaluations of seismic performance are achieved without the need to perform detailed ground motion record selection for the nonlinear dynamic structural analyses involved in PSDA (e.g. record selection with respect to seismic parameters such as earthquake magnitude, source‐to‐site distance, and ground motion epsilon). For structural demands that are dominated by a first mode of vibration, using inelastic spectral displacement (Sdi) can be advantageous relative to the conventionally used elastic spectral acceleration (Sa) and the vector IM consisting of Sa and epsilon (ε). This paper demonstrates that this is true for ordinary and for near‐source pulse‐like earthquake records. The latter ground motions cannot be adequately characterized by either Sa alone or the vector of Sa and ε. For structural demands with significant higher‐mode contributions (under either of the two types of ground motions), even Sdi (alone) is not sufficient, so an advanced scalar IM that additionally incorporates higher modes is used. Published in 2007 by John Wiley & Sons, Ltd.

An Empirical Ground-Motion Attenuation Relation for Inelastic Spectral Displacement

This article presents an empirical ground-motion prediction model (attenuation relation) for inelastic (as opposed to elastic) spectral displacement ( Sdi ) for ground motions without forward directivity effects. It is a function of two earthquake parameters, moment magnitude ( M w) and the closest distance to rupture ( R rup), and two bilinear oscillator parameters, an undamped elastic period ( T ) and a yield displacement ( dy ). The dy is introduced via the predicted median strength-reduction factor (![Graphic][1] ), a proxy for the ratio of elastic spectral displacement ( Sde ) to dy , which is identical with the familiar strength-reduction factor ( R ). The proxy ![Graphic][2] recognizes that R can only be estimated indirectly because it implicitly contains the random variable, Sde , which cannot be known a priori ; therefore, the median estimate or predicted median ( Ŝde ) from a conventional (elastic) ground-motion prediction model is used instead to calculate ![Graphic][3] = Ŝde / dy . For enhanced generality, the inelastic spectral displacement prediction model here is based on a ratio concept, that is, the total model is a (any) conventional elastic prediction model coupled with a new inelastic displacement ratio prediction model, with proper statistical correlation between the two. We empirically consider the dependence of this ratio on source and path effects (i.e., M w and R rup), and find that M w is significant, but R rup is not. The resulting prediction model can easily be added to existing probabilistic seismic-hazard analysis (psha) software packages with only one extra structure-specific parameter, dy of the oscillator. In practical engineering applications, this will likely have been estimated from the conventional static pushover analysis of the multi-degree-of-freedom (mdof) structure under consideration. The resulting psha product is a hazard curve for Sdi , the inelastic spectral displacement of a nonlinear oscillator. Such a curve can provide a more direct hazard- based target displacement for nonlinear static procedures (Federal Emergency Management Agency [fema] 356, 2000) and/or a basic input function for new probabilistic seismic-demand analyses that is based on Sdi (as opposed to Sde ) as an efficient and sufficient intensity measure. This new attenuation relationship will be particularly useful in evaluating the performance of existing structures and specified designs with known lateral strength. In particular, unlike most past studies, it does not pre-fix the ductility level. [1]: /embed/inline-graphic-1.gif [2]: /embed/inline-graphic-2.gif [3]: /embed/inline-graphic-3.gif

A static predictor of seismic demand on frames based on a post-elastic deflected shape

Predictors of seismic structural demands (such as inter-storey drift angles) that are less time-consuming than nonlinear dynamic analysis have proven useful for structural performance assessment and for design. Luco and Cornell previously proposed a simple predictor that extends the idea of modal superposition (of the first two modes) with the square-root-of-sum-of-squares (SRSS) rule by taking a first-mode inelastic spectral displacement into account. This predictor achieved a significant improvement over simply using the response of an elastic oscillator; however, it cannot capture well large displacements caused by local yielding. A possible improvement of Luco's predictor is discussed in this paper, where it is proposed to consider three enhancements: (i) a post-elastic first-mode shape approximated by the deflected shape from a nonlinear static pushover analysis (NSPA) at the step corresponding to the maximum drift of an equivalent inelastic single-degree-of-freedom (SDOF) system, (ii) a trilinear backbone curve for the SDOF system, and (iii) the elastic third-mode response for long-period buildings. Numerical examples demonstrate that the proposed predictor is less biased and results in less dispersion than Luco's original predictor. Copyright © 2006 John Wiley & Sons, Ltd.