Residual Displacement Demands Research Articles

Post-earthquake repairability enhancement of BRBFs considering isotropic hardening with self-centering braces

This research's aim is to develop a novel peak and residual deformation-targeted design (PRBD) procedure for enhancing BRBF's post-earthquake repairability with self-centering braces (SCBs). To this end, the peak and residual displacement demands of improved BRBFs (denoted as RBRBFs) were studied by single-degree-of-freedom (SDOF) system analyses, where the isotropic hardening features of BRBs were properly considered. The hysteretic behavior's effects of SCBs on the RBRBF's peak and residual displacements were comprehensively investigated. Artificial neural network (ANN) models were developed for predicting the displacement demands of RBRBFs. Based on the developed ANN model, the PRBD was subsequently proposed. Two design cases were presented based on the developed PRBD method with different design objectives. The designed RBRBFs' performance was investigated via dynamic analyses. It can be found from the results that RBRBFs can achieve the desired peak and residual displacements under design earthquakes, validating the effectiveness of the developed PRBD. Moreover, the excellent post-earthquake repairability of RBRBFs with the partial self-centering feature was confirmed by showing residual displacements lower than 0.2% under maximum considered earthquakes.

Probability-based residual displacement estimation of unbonded laminated rubber bearing supported highway bridges retrofitted with Transverse Steel Damper

Predicting the post-earthquake residual displacement of isolated bridges is crucial in deciding their operational and recovery strategies after a seismic event. This study proposes a probability-based approach to estimate the residual displacement of a trilinear hysteretic system representing the unbonded laminated rubber bearing (ULRB) supported highway bridge retrofitted with transverse steel dampers (TSDs), considering the uncertainty of structural parameters and the variability of ground motions. To this end, this study systematically derives the quasi-static residual displacement of a trilinear hysteretic system, and presents the statistical distribution model for the related structural parameters, especially for the pier height and the TSD model parameters. After that, the dynamic residual displacement property of the ULRB supported highway bridge retrofitted with TSDs is investigated considering the uncertainty of various parameters, including model parameters of TSD and ULRB, the mass of superstructure, the height of piers, as well as the ground motions with and without pulse. The maximum possible residual displacement of the trilinear hysteretic system is then related to its quasi-static residual displacement and peak displacement demand. Finally, an innovative probability-based approach is developed in this study to estimate the conditional probability of the dynamic residual displacement of the trilinear system exceeding a threshold at the given ground motion intensity, and this innovative approach is further validated by comparing it with the incremental dynamic analysis (IDA) method. Results show that the dynamic residual displacement of the TSD retrofitted ULRB supported highway bridge system is random, and the ratio between the dynamic residual displacement and the maximum possible residual displacement approximately satisfies a uniform distribution bounded by zero and one. The probability-based approach can obtain a close result with the IDA method and significantly reduce computational efforts. This research represents a first step toward developing a rapid post-earthquake assessment approach for seismically isolated bridges and the seismic design of these bridges considering the residual displacement.

Residual displacement demand for non-degrading bilinear SDOF oscillators with self-centering viscous-hysteretic devices

The self-centering viscous-hysteretic device (SC-VHD) is a high-performance self-centering system that was recently developed to reduce structural residual displacement and damage during severe earthquakes. This study focuses on developing a residual displacement ratio C r spectrum and a residual displacement estimation method for non‐degrading bilinear SDOF oscillators with SC-VHDs. Statistical analysis is conducted to investigate the effect of design parameters on the constant-ductility residual displacement ratio C r spectrum under near-fault pulse-like earthquake ground motions (NFPE) and far-fault earthquake ground motions (FFE). Based on statistical data, a formula for the mean C r spectrum is proposed. Subsequently, an iterative calculation method for residual displacement is presented according to the equivalent linearization method and the proposed mean C r spectrum formula. Finally, using the iterative calculation method, the residual displacement of a two-story steel moment-resisting frame (MRF) with SC-VHDs is predicted, demonstrating its application. The results indicate that the mean C r spectrum under the NFPE is significantly larger than that under the FFE. Besides, the mean constant-ductility C r spectrum decreases with increasing post-yield strength ratio, ratio of preload to yield strength, ratio of loading stiffness to elastic stiffness increases, or ratio of unloading stiffness to loading stiffness, but increases with increasing viscous damping ratio. When estimating the C r spectrum, increasing the ductility ratio and decreasing the ratio of preload to yield strength can both reduce record-to-record variability. Both the formula for the mean C r spectrum and the iterative calculation method for residual displacement are validated by comparing with statistical data. • The influence of design parameters on the residual displacement C r spectrum is investigated. • A formula for the normalized mean C r spectrum is developed. • An iterative calculation method for residual displacement is proposed. • An example application is used to validate the calculation method.

Discussion of Post-earthquake fast damage assessment using residual displacement and seismic energy: Application to Mexico City

A discussion on the article by Quinde et al. is presented. The methodology introduced by the authors is based on residual displacements computed from the inelastic response of elastoplastic single-degree-of-freedom systems subjected to synthetic ground motions representative of those recorded at soft soil sites in Mexico City. While the subject to the article is pertinent for post-earthquake assessment, this discussion points out and clarifies several issues about residual displacement demands and their estimation based on previously published investigations by the discussers.

On the aftershock polarity to assess residual displacement demands

A strong mainshock is able to trigger several aftershocks during a short time interval, causing additional damage to structures. The effect of the polarity of aftershock ground motions (positive and negative) is of great significance, particularly on the residual displacement (RD) demand. The positive and negative polarity occurs when the direction of aftershock is the same and the opposite of that of mainshock, respectively, while these are only two of the various situations of the incident angle of aftershock with respect to that of mainshock. This study evaluates the sufficiency of the positive and negative polarity in multiple earthquakes for the determination of critical residual displacements of structures by considering various relative differences between the incident angles of sequential earthquakes. The structures are schematized as bi-linear single-degree-of-freedom systems to develop constant-strength spectra in terms of the Normalized Residual Displacement (NRDθ). The results reveal that considering only the conventional aftershock polarity (positive and negative) may not necessarily lead to critical residual displacement demands, and also accounting for the variability of various incident angles for consecutive ground motions is necessary for the reliable evaluation of structures.

Study on normalization of residual displacements for single-degree-of-freedom systems

Residual displacement spectrum is one of the most important means to predict the permanent deformation of structures after the earthquake, and various normalizations of residual displacements have generally been used for construction of the spectrum. However, the issue regarding the merits and drawbacks of each normalization has not yet been investigated thoroughly. A comparison between two normalizations that relate the residual displacements to the elastic and inelastic displacements is made in terms of the effect of ground motion and structural characteristics by means of the results of nonlinear time history analysis. The statistical results reveal that the residual-to-peak-inelastic displacement ratios have the advantages of small dispersion, samples without any outliers, and relatively symmetric distribution, which benefits from the strong correlation between residual and peak inelastic displacements. Moreover, the residual-to-peak-inelastic displacement ratios are almost independent of site conditions, significant duration, and natural periods. Consequently, the peak inelastic displacements are superior to the elastic ones as an intermediate step for residual displacements estimation, provided that the peak inelastic displacements are estimated with a low uncertainty. For providing alternatives to estimate residual displacement demands, the constant-strength residual displacement spectra are developed for both normalizations.

Investigation on residual displacements for SDOF systems with various initial viscous damping models

Nonlinear time history analysis (NTHA) is widely used in estimation of post-earthquake residual displacement demands. However, the choice of initial viscous damping models in NTHA has attracted insufficient attention in study of residual displacements. The more versatile viscous damping model similar to the tangent-stiffness proportional Rayleigh damping commonly used in multi-degree-of-freedom (MDOF) systems is utilized in NTHA of single-degree-of-freedom (SDOF) systems. With different combinations of mass proportional and tangent-stiffness proportional damping terms, the significance of the choice of initial damping models is examined in terms of energy dissipation and residual displacement demands. The residual displacement ratio, defined as the ratio of residual displacements to elastic spectral displacements, is computed for constant-strength SDOF systems when subjected to an ensemble of 240 earthquake ground motions. The results are statistically organized to investigate the effect of initial viscous damping models on residual displacement demands. The difference in residual displacement ratios for various damping models is found to be significant, and dependent on natural periods, relative lateral strength, post-yield stiffness, unloading stiffness degradation and initial damping ratios. The effect of these factors on residual displacement demands of SDOF systems with tangent-stiffness proportional damping, which is confirmed to be most appropriate for nonlinear analysis, is also examined. Finally, the constant-strength residual displacement response spectra are constructed for seismic assessment of structures.

Residual displacement estimation for soft soils: Application to Mexico city lake-bed

Current seismic regulations do not assess the seismic performance of structures during a given seismic event, and therefore do not require an estimation of their residual displacement or level of structural damage. Nevertheless, this information is relevant after intense ground motions to assess the possibility of re-occupying a building. This paper develops simple expressions to estimate the maximum displacement demand from the residual displacement of single-degree-of-freedom systems subjected to long-duration narrow-banded ground motions generated in soft soil sites (particularly, those located in Mexico City). To overcome the limited scope of previous studies, more than 35,000 seismic simulations were performed. The influence of the lateral strength of the single-degree-of-freedom systems is considered. A simplified relation between the residual and maximum displacement demands is established.

Prediction of residual displacement ratios for simple structures built on soft-soil sites of Mexico City

This short communication discusses the prediction of residual displacement demands through residual ratios (i.e., the ratio of residual displacement demand to the maximum displacement demand) for single-degree-of-freedom (SDOF) systems subjected to earthquake ground motions recorded at soft soil sites of Mexico City. A functional form to obtain estimates of mean residual ratios for elastoplastic and stiffness-degrading SDOF systems on this type of sites is presented and its ability to capture the empirical spectral trend of residual ratios is illustrated in this paper. It is shown that the introduced equation provides good estimates of the empirical data.

Residual displacement ratio demand of oscillators representing HSSF-EDBs subjected to near-fault earthquake ground motions

This research investigates the post-earthquake residual displacement ratio demand of high strength steel frames with energy dissipation bays (HSSF-EDBs) under near-fault earthquake ground motions. The study focuses on the ultimate stage of the structure where the HSS frame develops sufficient inelastic deformations. The nonlinear force-displacement behaviour of a HSSF-EDB is idealised by a classical trilinear kinematic model, and the idealised hysteretic behaviour is assigned to oscillators for representing the novel system. To facilitate performance-based seismic design of HSSF-EDBs, the post-earthquake residual displacement demand is related to the yield displacement characterising yielding of energy dissipation bays using the trilinear oscillator analogy, and a non-dimensional residual displacement ratio is proposed. A large number of inelastic spectral analyses of trilinear oscillators covering a reasonable spectrum of hysteretic parameters are performed to quantify the influence of the essential variables on the residual displacement ratio demand of the systems. The results show that the residual displacement ratio of trilinear oscillators representing HSSF-EDBs subjected to near-fault earthquake ground motions is affected by hysteretic parameters in various yielding stages. To offer a practical tool for estimating the residual displacement demand of HSSF-EDBs in the preliminary design phase, empirical formulas quantifying the post-earthquake residual displacement ratio demand of trilinear oscillators are proposed. Finally, the effectiveness of using trilinear oscillators for quantifying post-earthquake residual displacement demand of HSSF-EDBs is emphasised by a comparative study on seismic responses of a prototype HSSF-EDB system and equivalent oscillators under near-fault ground motion samples.

Residual displacement ratios of structures under mainshock-aftershock sequences

Residual displacement demand is a key indicator of seismic damage and plays a pivotal role in post-earthquake decision making on damaged structures. This parameter can be effective as an estimator of the residual capacity of structural systems damaged under seismic sequences. Residual displacement ratio (Cr) is a conventional approach to predict the residual displacement demand from maximum linear displacement. This study investigates the residual displacement ratios of structures as bi-linear SDOF systems subjected to mainshock-aftershock sequences. Constant-strength spectra based on Cr are developed taking into account the effects of ground motion parameters (such as site condition, magnitude, epicentral distance, and duration), and of structural characteristics (such as strength modification factor and post-yield stiffness ratio). Several analytical equations are proposed to predict the residual displacement ratio as function of the elastic vibration period and the strength modification factor for different levels of the post-yield stiffness ratio and also various aftershock intensities. The proposed equations can be employed for seismic assessment of structures against mainshock-aftershock sequences.

Residual Displacement Demand Evaluation from Spectral Displacement

In this study, residual displacement demands are investigated for SDOF systems with period range of 0.1-3.0 s for near-field and far-field ground motions. The effects of stiffness degradation and post yield stiffness ratio on residual displacements are investigated. The modified-Clough model is used to represent structures that exhibit significant stiffness degradation when subjected to reverse cyclic loading. An elastoplastic model is used to represent non-degrading structures. For inelastic time history analyses, Newmark’s step by step time integration method was adapted in an in-house computer program. Based on time history analyses, a new simple equation is proposed for residual displacement demand of a system as a function of structural period (T), ductility (µ), strain hardening ratio (α) and spectral displacement (Sd).

Residual displacement estimation of simple structures considering soil structure interaction

As the residual displacement and/or drift demands are commonly used for seismic assessment of buildings, the estimation of these values play a very critical role through earthquake design philosophy. The residual displacement estimation of fixed base structures has been the topic of numerous researches up to now, but the effect of soil flexibility is almost always omitted. In this study, residual displacement demands are investigated for SDOF systems with period range of 0.1-3.0 s for nearfield and far-field ground motions for both fixed and interacting cases. The elastoplastic model is used to represent nondegrading structures. Based on time history analyses, a new simple yet effective equation is proposed for residual displacement demand of any system whether fixed base or interacting as a function of structural period, lateral strength ratio and spectral displacement.

Residual displacement ratios of SDOF systems subjected to ground motions recorded on soft soils

120 earthquake ground motions recorded on soft soil sites were employed to assess, through response history analysis of simplified systems, the residual displacement demand, Cr, defined as the ratio of the residual displacement to the maximum elastic displacement. Single degree of freedom systems were considered and the lateral strength ratio was parametrized to account for varying structural inelasticity. Four hysteretic laws were chosen to represent degrading and non-degrading performance while variation in the post-yield stiffness was considered. The analysis scheme enabled assessing the relationship of the residual displacement with the aforementioned structural characteristics. The residual displacement demand was found to be sensitive in the post-yield stiffness ratio. An equation was finally introduced to accommodate the reliable estimation of residual displacement ratio, the latter being beneficial for evaluating the seismic performance of existing structures built on soft soil sites.

Estimation of residual displacement ratios for simple structures built on soft-soil sites

Seismic demands computed from earthquake ground motions recorded at soft soil sites have shown different spectral trends with respect to those computed from earthquake ground motions gathered from accelerographic stations placed on firm sites. Therefore, an examination of the spectral trend of residual displacement ratios, Cr, which allow the direct estimation of lateral residual displacement demands from their maximum linear elastic demands, for elastoplastic single-degree-of-freedom systems subjected to earthquake ground motions recorded at soft soil sites of the lake-bed of Mexico City and the San Francisco Bay Area is presented in this technical note. A functional form to obtain estimates of mean residual displacement ratios for this type of sites is introduced and its ability to capture the empirical spectral trend of Cr is illustrated in this paper.

Residual displacement demands of conventional and dual oscillators subjected to earthquake ground motions characteristic of the soft soils of Mexico City

This paper presents a statistical evaluation of residual displacement (RDs) demands in conventional and dual single-degree-of-freedom (SDOF) oscillators subjected to a set of 220 earthquake ground motions recorded at soft soil sites of Mexico City. A dual SDOF oscillator represents a highly-dissipating energy system (e.g. buckling-restrained braces, BRBs) acting in parallel with a conventional system (e.g. flexible moment-resisting frames). To provide a context, RDs were normalised with respect to the corresponding: a) maximum transient displacements, and b) elastic spectral displacement. The effects of post-yield stiffness ratio (as affected by P-Δ effects), normalised period of vibration with respect to the predominant period of the ground motion, the type of hysteretic response, maximum displacement ductility, lateral strength ratio, type of transition (from elastic-to-plastic response), and damping ratio on normalised RDs were examined for the conventional oscillators. In addition, the effects of the stiffness ratio, lateral strength ratio, displacement ductility, post-yielding stiffness ratio, and type of hysteretic response of the primary and secondary parts of dual SDOF oscillators on normalised RDs were also evaluated. From the results of this investigation, it was observed that for dual SDOF systems the amplitude of normalised RDs is small when the primary part remains elastic. On the contrary, if the primary part exhibits inelastic response, normalised RDs might increase significantly and the post-yielding stiffness ratio of the secondary part plays a key role for constraining them (i.e. while a positive value reduces RDs significantly; a negative value is highly detrimental). Also, it was found that the type of hysteretic response of the primary part of a dual system has a significant effect on normalised RDs (e.g. it was found that a primary part with self-centring capacity, acting in parallel with a highly dissipation system (e.g. BRBs) as secondary part, is very effective for diminishing RDs). A discussion section is also offered to highlight the new findings of this study and differences on normalised RD demands between soft and stiff soils.

Estimating Residual Seismic Displacements for Bilinear Oscillators

AbstractAn important consideration in the design of structures for seismic effects is residual displacement. However, few design specifications have residual displacement criteria. As performance-based design gains popularity, the need arises to estimate residual displacements in structures from strong ground shaking. A total of 16,896 nonlinear analyses of single-degree-of-freedom bilinear oscillators were performed to estimate residual displacement ratios. Comparisons were made to previously developed models for estimating residual displacement demand.

Seismic Upgrading of Steel Moment-Resisting Frames by Means of Friction Devices

In recent decades, several passive energy dissipation systems have been conceived in order to minimize the damage in structural and non-structural components of either new or existing buildings. In this study, the use of friction damped tension-compression diagonal braces for seismic upgrading of a steel moment resisting frames is investigated. To this aim, nonlinear time history analyses have been carried out on a set of representative frames with and without friction damped braces. In the nonlinear time history analyses, two sets of natural accelerograms compatible with seismic hazard levels of 10% and 2% probability of exceedance in 50 years have been considered. Under these records, the structural response has been comparatively investigated in terms of the maximum inter-storey drift ratio, maximum storey acceleration, residual drift ratio and displacement demand for the friction device. The results clearly highlighted that the application of friction damped braces allows reducing the damages to the main structural elements, thus significantly improving the seismic behaviour of the frame.

On the influence of strong-ground motion duration on residual displacement demands

This paper summarizes results of a comprehensive analytical study aimed at evaluating the influence of strong ground motion duration on residual displacement demands of single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems. For that purpose, two sets of 20 earthquake ground motions representative of short-duration and long-duration records were considered in this investigation. While the influence of strong ground motion duration was evaluated through constantstrength residual displacement ratios, Cr, computed from the nonlinear response of elastoplastic SDOF systems, its effect on the amplitude and height-wise distribution of residual drift demands in MDOF systems was studied from the response of three one-bay two-dimensional generic frame models. In this investigation, an inelastic ground motion intensity measure was employed to scale each record, which allowed reducing the record-to-record variability in the estimation of residual drift demands. From the results obtained in this study, it was found that long strong-motion duration records might trigger larger median Cr ratios for SDOF systems having short-to-medium period of vibration than short strong-motion duration records. However, taking into account the large record-to-record variability of Cr, it was found that strong motion duration might not be statistically significant for most of the combinations of period of vibration and levels of lateral strength considered in this study. In addition, strong motion duration does not have a significant influence on the amplitude of peak residual drift demands in MDOF systems, but records having long strong-motion duration tend to increase residual drift demands in the upper stories of long-period generic frames.

Statistical modeling of joint probability distribution using copula: Application to peak and permanent displacement seismic demands

Reliability analysis of structures requires statistical models of multi-variate random data that are nonlinearly interrelated. The current reliability methods that use the Nataf transformation and the linear correlation matrix may encounter difficulties in dealing with such situations. A copula approach can offer a general and flexible way of describing nonlinear dependence among multi-variate data in isolation from their marginal probability distributions, and serves as a powerful tool for modeling and simulating nonlinearly-interrelated data. In this study, an introduction as well as illustrative application of the copula theory is given in the context of structural reliability. The numerical example deals with the performance evaluation of existing structures subjected to earthquake loading in terms of both peak and residual displacement demands. The joint probability distribution modeling of peak and residual displacement seismic demands based on the copula theory is demonstrated, and the developed statistical models are used to examine the effects of nonlinear dependence on seismic reliability assessment.