Displacement Ductility Demand Research Articles

Performance-Based Seismic Design for Bridge Bents Retrofitted with BRBs Using the Ductility Demand Spectra of SDOF Oscillators with Bilinear Fuses

ABSTRACT Reinforced concrete bridge bents with double columns in bridge substructures are commonly used, but it has suffered severe damage during strong earthquakes. The seismic retrofit with structural fuses is used to effectively mitigate seismic damage of bridge bents. To obtain the displacement ductility demand for dual systems including the primary structure (PS) and structural fuses (SF), the corresponding ductility demand factor spectra were developed. The interaction between the PS and SF is described by defining two parameters of dual systems: the yield displacement ratio (α) and the initial stiffness ratio (β), and the influence of α, β, and strength reduction factor on the ductility demand factor spectra are also analyzed. For seismic retrofitting in bridge bents, buckling-restrained braces (BRBs) are used as energy dissipation fuses, and the steel core lengath of BRBs with three configurations forms, including diagonal, inverted-V, and toggle systems, are also explored. Then, based on the structural fuses concept and ductility demand factor spectra of dual systems, a performance-based seismic design procedure is proposed for bridge bents retrofitted with BRBs to ensure that the bridge bents maintain the target ductility under design-level earthquakes. Finally, the procedure was utilized to design the bridge bents with BRBs, and the procedure’s feasibility was verified by nonlinear time-history analysis.

Ductility demands for stiffness-degrading SDOF systems under pulse-like ground motions of the 2023 Pazarcık (Kahramanmaraş) earthquake

This paper investigates the inelastic displacement ratios (IDRs) and displacement ductility demands of a wide range of single-degree-of-freedom (SDOF) systems subjected to pulse-like ground motions (GMs) of the 2023 Pazarcık (Kahramanmaraş) earthquake. A set of twenty-seven GMs characterized as pulse-like are utilized in the study. As-recorded velocity time histories of horizontal components are rotated over 90° at a step of 1° to attain the waveform with the largest peak ground velocity (PGV) over all horizontal orientations. Inelastic displacement ratio and displacement ductility spectra are computed through nonlinear response history analysis (RHA). Local amplifications of both spectra are observed at some periods. The results of this study show that large inelastic displacement and ductility demands are imposed on certain reinforced concrete (RC) structures. Finally, predictive models of the mean inelastic displacement ratio and mean ductility demand spectra (DDS) are developed based on the Gauss–Newton algorithm (GNA). The model provides a strong correlation between the computed and the estimated data, and sufficient convergence criteria. The results of this study collectively emphasize the necessity of integrating pulse-like GMs into future revisions of earthquake codes.

Full-scale static behaviour of prestressed geopolymer concrete sleepers reinforced with steel fibres

Profusive cracking leading to the deterioration of prestressed concrete sleepers has retained the sleepers from being exploited to their full extent. In addition, excess utilization of cement in producing the same has paved the way for developing its counterpart, geopolymer concrete (GPC), involving primary industrial residues as their precursors. This study presents a comprehensive investigation into the development and performance of high-strength geopolymer concrete sleepers for railway applications. The research focuses on the complete replacement of ordinary Portland cement with Fly ash and Ground Granulated Blast Furnace Slag in the geopolymer concrete mix, for mainline prestressed railway sleepers following Indian Railway Standards (IRS). Notably, the study includes experimental validations of long steel fibers as secondary reinforcement to the primary prestressing strands, aiming to enhance the ductility of the member. Experimental validations were performed on full-scale (1:1) sleeper members under static loading conditions. The main aim of this article was to assess the load carrying capacity at cracking and ultimate limits, rail seat bending moment, ductility indices, failure mechanism, electrical impedance, bond strength, and the eco-efficiency for the proposed prestressed geopolymer concrete sleeper members (with and without steel fibres) with reference to the conventional prestressed cement concrete sleepers. The considered GPC mix was found to achieve higher load and moment carrying capacity by about 8.3% and 8.54% respectively compared to the conventional concrete mix. The addition of steel fibres in the GPC increased the load and moment capacity of sleepers by 23% and 24.21% respectively. Additionally, the steel fibre reinforced GPC showed improved displacement ductility demand over the plain geopolymer concrete averaging 25%, 28.5%, and 3% corresponding to ultimate, failure, and residual ductility index respectively. Though inclusion of steel fibers improves the performance of sleepers in terms of load and ductility, they were found incompatible with regard to the electrical impedance property when used in conjunction with the pre-stressing strands.

Loading protocols for quasi-static cyclic testing of flexure dominated reinforced concrete circular bridge columns under crustal, subcrustal, and subduction earthquakes

Experimental procedures involving quasi-static cyclic tests have been the most prevalent for seismic performance assessments of structural and non-structural components and systems. However, there is a lack of consensus among experimentalists and researchers on a standard loading protocol. This study aims to propose a set of standard loading protocols representative of the seismic demands imposed on typical flexure dominated reinforced concrete bridge columns under uniaxial bending. Variable parameters: axial-load ratio, aspect ratio, longitudinal and spiral reinforcement ratios, concrete compressive strength, and steel yield strength were initially screened for significance in terms of their influence on the number of inelastic cycles and cumulative displacement ductility - two key reference parameters for the development of loading protocols. The number of inelastic cycles cumulative displacement ductility were then determined for reinforced concrete bridge columns having unique combinations of upper and lower levels of the significant parameters under different earthquake types (crustal, subcrustal, subduction) and displacement ductility demand levels (2, 4, 8). These target ductility demands intend to capture a wide range of structural responses including minimal and collapse damage states. Statistical analyses revealed that for the considered bridge columns, it is feasible to develop one representative loading protocol for each earthquake type and target displacement ductility demand level. Hence, nine loading protocols were proposed and then compared against equivalent loading protocols found in standards and literature. Standard protocols were found to impose unrealistic damage on bridge columns subjected to crustal and subcrustal earthquake hazards leading to inaccurate assessments. The proposed loading protocols for subduction earthquakes had a comparable number of inelastic cycles to those found in the literature for the same earthquake hazard and are representative of the seismic demands imposed on bridge columns.

Effective Stiffness and Seismic Response Modification Models Recommended for Cantilever Circular Columns of RC Bridges, First Part: Serviceability Limit State

This paper’s main aim was to explain the process of characterising the structural over-strength factor (R), seismic behaviour factor (Q), and the effective elastic stiffness, Keff, of cantilever-reinforced concrete (RC) urban bridge columns with solid circular cross-sections for use in seismic design under the Serviceability Limit State (SLS). Similarly, mathematical models have been proposed to determine the average values of effective stiffness and seismic response modification factors suggested for cantilever-reinforced concrete bridge columns at SLS. This is because multiple design codes stipulate that cantilever RC bridge columns must meet the SLS requirements. Therefore, to comply, the lateral displacement ductility demand must not exceed unity after a moderate or small earthquake. While the behaviour of the materials remains in the elastic range, this performance criterion can be conservative. If the materials undergo small deformations, the slight damage can be quickly repaired to meet the SLS.

Development of a new framework based on Gaussian regression process for rapid fragility analysis of 2-DoF base-isolated structures

This paper develops metamodels (or surrogate models) to predict the fragility parameters of structures isolated with friction pendulum (FP) isolators. An equivalent inelastic two-degree-of-freedom (2DoF) system is utilized to model the base-isolated structures.6000 system simulations are initially established based on the probability distribution of input parameters. Then, incremental dynamic analyses are carried out to achieve structural data for the fragility function generation. The developed dataset of the fragility parameters (median and dispersion) is divided into training and testing sets to train and testthe efficiency of the developed metamodels, respectively.Powerful machine learning tool, namely, Gaussian Process Regression (GPR) is utilized for mapping structural input parameters to output (or target) parameters, i.e., median and dispersion of the fragility functions. GPRs are kernel-based probabilistic models, with kernel functions constituting their core components. In this study, the predictive quality of the GPR is evaluated considering different kernel functions to obtain the most accurate and reliable predictions. The results indicate that automatic relevance determination (ARD) kernel functions are much more efficient than isotropic kernel functions. Statistical performance metrics indicate that, among the implemented kernel functions, ARD Matern 3/2 is preferable for both the superstructure and FP isolator. The proposed GPR-based prediction metamodels exhibit excellent performance in the prediction of the fragility parameters of the superstructure and isolator for all considered limit states. Using the developed metamodels, conducting a rapid seismic fragility assessment on 2DoF systems that are representative of base-isolated structures is possible. At last, metamodels are exploited in seismic reliability analysis to assess their precision for further investigations. The results of seismic reliability-based performance and relationships between the displacement ductility demand and strength reduction factor reaffirm the efficiency, sufficiency, proficiency, and practicality of the GPR-based generated metamodels.

A plastic design method based on multi-objective performance for high-strength steel composite K-shaped eccentrically braced frame

A plastic design method based on multi-objective performance (PDMP) for high-strength steel composite K-shaped eccentrically braced frame (HSS-KEBF) is developed in this study, which explicitly considers the different seismic fortification objectives of the structure and the shear distribution relationships of the brace-frame system. The performance objectives of the structure under three seismic hazard levels were obtained by combining different roof drift ratios and displacement ductility demands, and a trilinear capacity curve of HSS-KEBF considering the change of post-yield stiffness was proposed. The energy balance principle was used to calculate the elastic and elastic–plastic shear distributions of the brace-frame system under different performance objectives, to facilitate the elastic–plastic design of HSS-KEBF. A ten-story HSS-KEBF structure was designed using the proposed design method, and an elastic-plastic finite element model of the structure was established using OpenSees for analysis. The results show that the structural model achieves the expected performance objectives in terms of capacity curves, roof drift ratios, base shears, and failure modes. From the SLE level to MCE level, the base shear distribution ratio borne by the frame component increased from 36% to 46%, which satisfies the design idea of multi-aspect fortification.

Effect of nonlinear soil−structure interaction and lateral stiffness on seismic performance of mid−rise RC building

In this study, the nonlinear soil-structure interaction (SSI) is considered to analyse the mid-rise structure's seismic performance with proportional lateral stiffness of columns in the orthogonal direction. The model's geometry is created using a finite element-based software package (ABAQUS). Cap plasticity and hysteretic models are used to mimic the nonlinear behaviour of soil and structural components, respectively. Truncated soil mass is incorporated using the spring and dashpot at the boundaries node with a MATLAB code for the dynamic analysis. The methodology validation is carried out on the shallow foundation under the cyclic loading and found in good agreement. Static pushover analysis of 4-storey, 8-storey mid-rise structures and 12-storey dual RC frame system is carried out to estimate the yielding displacement of the building structure at the roof level, which is further used in the incremental dynamic analysis (IDA) of the SSI and fixed base system. IDA is carried out for earthquake intensities ranging from 0.1g to 0.6g. IDA results are represented in terms of displacement ductility demand, normalised peak settlement, base shear, normalised peak foundation sliding and drift ratio. From the IDA analysis, the fragility assessment of the building is carried out for both the cases (fixed and SSI base). The present study concluded that lateral stiffness and SSI significantly affect the seismic performance of mid-rise buildings.

Comparative study of the damage cost of reinforced concrete buildings with and without nonlinear viscous dampers subjected to seismic loading

In an optimal seismic design context, the seismic demand is characterized by hazard curves that can be obtained by simulation techniques, and the capacity of the structure is established by the designer following a predefined seismic code. The capacity of structures is generally characterized by the seismic design coefficient. Furthermore, the structure damage is evaluated based on certain well-defined damage indicators (e.g., displacement ductility). Thus based on the damage indicator, it is possible to estimate the cost of the associated losses. Furthermore, it is noted that the quantification of the damage costs associated with reinforced concrete (RC) structures with and without nonlinear viscous dampers under seismic loading is very scarce in the relevant literature. In this study, damage cost expressions, similar to those employed in the optimal seismic design criterion, were used to quantify and compare the damage cost on RC buildings with and without viscous dampers located in seismic-prone areas of Mexico. For the analysis, three RC buildings were designed according to Mexican seismic design regulations. The buildings under study were subjected to seismic actions characterized by actual seismic records, scaled according to simulated maximum ground motion accelerations. The damage to the structures caused by seismic action is calculated by means of a damage factor that is a function of displacement ductility demand. The cost of damage to the considered structures was estimated based on cost expressions that are a function of the damage factor. The analyses results indicate that the use of viscous dampers in concrete buildings subjected to seismic action can considerably reduce the associated damage costs with respect to buildings without such a damping system.

Experimental study on hybrid concrete-filled fiber reinforced polymer tube (HCFFTs) columns under simulated seismic loading

The concrete-filled fiber-reinforced polymer (FRP) tube (CFFT) system has been widely studied as a promising alternative to conventional reinforced concrete (RC) for the construction of bridge columns. While unreinforced CFFT columns are attractive for their ease of construction, their limited energy dissipation capabilities have restricted their application to seismic regions. The hybrid CFFT (HCFFT) system was introduced to address this shortcoming by incorporating steel reinforcement into the FRP shell in the form of 30-μm fibers. This system combines the ease of construction of conventional unreinforced CFFTs with the energy dissipation characteristics of lightly reinforced CFFT columns. This study reports the first experimental investigation on the seismic performance of HCFFT columns through large-scale testing of two columns under combined axial compression and cyclic lateral loading. Two shear loading conditions were studied. The flexural capacity, mode of failure, residual deformation, and energy dissipation capacity were evaluated. The low-shear specimen reached a drift ratio of 10.5% and the high-shear specimen reached a drift ratio of 6.5%. The specimens both reached displacement ductility demands above 5, demonstrating the system is appropriate for use as a bridge column in seismic regions. The curvature profiles for both specimens showed the hybrid shell prevented localized yielding of the steel fibers, which allowed for an efficient distribution of plasticity along the column height and enhanced displacement ductility. The results indicate that HCFFT columns can achieve high inelastic deformations under simulated seismic loading without substantial degradation of stiffness. It is anticipated that this large-scale experimental data will inform future research on the design and manufacturing process of HCFFT elements and encourage further experimental and analytical studies.

Perforated steel block of realizing large ductility under compression: Parametric study and stress–strain modeling

Abstract On one hand, the nature of linear elastic up to brittle rupture hinders the application of fiber-reinforced polymer (FRP) bars as reinforcement in the concrete member due to the displacement ductility demand of structures. On the other hand, FRP bars are equipped with many irreplaceable advantages as reinforcement in concrete structures. To resolve this contradiction, a possible solution is to use the so-called compression yield (CY) structural system and the ductility of a concrete beam incorporating a CY system comes from the compressive side rather than the tensile side. Thus, the development of material in the compressive side (CY material) with well-designed mechanical properties (strength, stiffness, and ductility) is a key challenge. In this study, the CY material is developed by perforating the mild steel block and then substantiated by the test results. Then, experimentally calibrated finite element models are used to conduct systematic parametric studies, based on which parametric equations are proposed to predict the stiffness and ultimate strength of the CY material. Finally, theoretical constitutive models are developed to predict the stress–strain response of perforated steel block under compression and a reasonably acceptable agreement is reached between the model predictions and the test results.

Low‐cycle fatigue performance of high‐strength steel rebars in concrete bridge columns

AbstractDesign codes often restrict the use of high‐strength steel (HSS) rebars in seismic applications due to their inferior low‐cycle fatigue performance when compared to conventional steel rebars. In this study, previously established low‐cycle fatigue life models of HSS rebars, namely ASTM A706 Grade 550 and ASTM A1035 Grade 690, were incorporated into a new cumulative damage model to identify conditions under which such rebars can achieve seismic performance comparable to that of benchmark ASTM A706 Grade 420 steel bars in concrete bridge columns. An ensemble of well confined flexure‐dominated circular concrete bridge columns located in high seismic regions was considered where the axial load ratio, and longitudinal and spiral reinforcement ratios were varied. The bridge columns were reinforced with ASTM A706 Grade 420, ASTM A706 Grade 550, and ASTM A1035 Grade 690 steel and numerically analyzed under different displacement ductility levels (2, 4, and 6), earthquake types (crustal and subduction), and ratio of hoop spacing to longitudinal bar diameter ratio (4 and 6). Their low‐cycle fatigue performances were compared based on the computed bar fracture and accumulated damage indices. Based on the reported observations, it is concluded that design codes are overly restrictive in not permitting the use of HSS in seismic application on the basis of inadequate low‐cycle fatigue life. As an alternative, this study proposes imposing certain limits on displacement ductility demand and ratio of hoop spacing to longitudinal bar diameter ratio to promote safe and efficient use of HSS rebars in concrete bridge columns.

A Nonstationary Mathematical Model for Acceleration Time Series

The choice of nonstationary stochastic models for the study is fully justified by the limitation of acceleration time series number. The three acceleration time series under consideration are used to generate a new, artificial series of ten per historical one using autoregressive moving average model. Subsequently, the average of nonlinear is utilized for the ten acceleration time series in order to obtain the spectral response of a system with single degree of freedom. Modeling of acceleration time series involves critical estimation of metrics that characterize nonstationary acceleration time series. Thus, for the stiffness degrading systems and bilinear systems, the metrics of hysteretic energy demand and displacement ductility demand during displacement are used. The applicability of artificially generated acceleration time series for the qualitative description of information was shown. More specifically, ARMA (2,2) showed the best results in the study for three accelerated time series through nonlinear response analysis. In addition, as a result, normalized hysteretic energy demand, empirically valid displacement ductility relationships, and model parameters were proposed.

Shaking table investigation of inelastic deformation demand for a structure isolated using friction-pendulum sliding bearings

This study aims at investigating experimentally if seismically isolated structures can manifest inelastic behavior when they are subjected to strong earthquake ground motion excitation. A steel structure is seismically isolated with four friction pendulum bearings and subjected to an ensemble of recorded earthquake ground motion excitations using the shaking table of ETH Zurich Laboratory. The structure is designed to concentrate its inelastic behavior on a mechanical clevis connection comprising two hinges and a pair of replaceable steel coupons. The use of this configuration enables the adjustment of the strength of the structure and thus the parametric investigation of its displacement ductility demand µ for a wide range of earthquake ground motion excitations. The experimentally derived values of the ductility demand µ for the seismically isolated structure are compared with the analytically determined values obtained through the use of Ry-μ-Tn relations for fixed-based structures and the corresponding relations developed for seismically isolated structures.

The September 7, 2017 Tehuantepec, Mexico, earthquake: Damage assessment in masonry structures for housing

Mexico is located among five tectonic plates. Therefore, it is frequently disturbed by the action of several moderate and strong earthquakes, affecting both the population and its infrastructure. The origin, causes, and effects of the typical observed damage patterns during the September 7, 2017 Tehuantepec earthquake (Mw = 8.2), in masonry houses and apartment buildings, located in cities and towns of the Mexican states of Chiapas and Oaxaca in a radius of about 250 km from the epicenter are reported and discussed in this paper. The high structural vulnerability and the severity of the earthquake resulted in the death of 96 people and more than 110,000 houses damaged, where more than 41,000 were considered total losses (collapse, beyond repair or tagged for demolition). In order to analytically evaluate the seismic vulnerability of typical masonry structures ranging from one to five stories, displacement ductility demand spectra (DDDS) were computed for both earthquake ground motions records and artificial acceleration records for a wide range of base shear capacities. Special attention is given to relate the observed damage with conditions of structural irregularity. Effects of flexible diaphragms, structural pounding between adjacent structures and site effects are also addressed. From the information presented, it can be concluded that most of the observed typical damage it is mostly due to poor design, inadequate construction processes, poor quality of construction materials and low wall density ratios. Finally, it was clearly observed that DDDS is a useful tool for the simplified seismic vulnerability assessment of a large portfolio of existing structures under a given earthquake scenario.

Residual drift spectra for RC bridge columns subjected to near-fault earthquakes

Permanent displacement of a bridge column can be directly measured during the inspection after near-fault earthquakes. However, the engineer needs to estimate the expected residual drift at the design stage to determine if the bridge seismic performance is satisfactory. The most direct method to estimate the residual displacement is nonlinear response history analysis, which is time consuming and cumbersome. Alternatively, an attractive but indirect method is generating estimated residual displacement spectra that depend on displacement ductility demand, column period, site conditions, and earthquake characteristics. Given the period and the expected displacement ductility demand for the column, the residual drift response spectra curves can be utilized to estimate the residual drift demand. Residual drift spectra that are applicable to RC bridge columns in different parts of the United States were developed based on nonlinear response history analyses using a comprehensive collection of recorded and synthetic near-fault ground motions and were linked to one-second spectral acceleration (S1) of the AASHTO maps. It was also found that the residual drift ratio is below one percent when S1 is less than 0.6 g.

Seismic response of RC buildings subjected to fling‐step in the near‐fault region

AbstractFling‐step and forward directivity are the major consequences of near‐fault ground motions as they can impose unexpected seismic demands on structures located in the vicinity of the fault. The pernicious effect of forward directivity on the seismic behavior of structures has been studied widely. However, not much research has been conducted to investigate the influence of fling‐step that is related to a large co‐seismic displacement. Moreover, the inconsistent results reported in the literature create a scientific challenge about the effect of fling‐step on the seismic behavior of long‐period structures. In this paper, the effect of fling‐step is studied by comparing the displacement ductility demand in various single degree of freedom systems with different natural frequencies and strength reduction factors, subjected to long‐period ground motions (generated and as‐recorded) with and without fling‐step. Subsequently, two 11‐ and 20‐story reinforced concrete buildings are considered and the effect of removing the fling‐step on their maximum inter‐story drifts is studied. The results indicate that the ratio of the fundamental period of the structure to the fling‐pulse period plays an important role and the demands imposed on those systems without fling‐step may increase or decrease based on the ground motions type and structural characteristics. Also, a similar trend in the displacement ductility demand was observed in this condition.

Seismic reliability-based design of hardening and softening structures isolated by double concave sliding devices

This study proposes seismic reliability-based design relationships in terms of behavior factors and displacement demands for softening and hardening structures equipped with double concave sliding bearings (i.e., DFPS). An equivalent 3dof system is adopted, that is characterized by a hardening or softening post-yield slope, representative of the superstructure response, and by velocity-dependent laws to model the frictional responses of the two surfaces of the DFPS. The yielding characteristics of the superstructures are defined for increasing behavior factors, provided by the codes, in compliance with the seismic hazard of L'Aquila (Italian site) and with the life safety limit state as provided by NTC18. Considering several natural ground motions and different elastic and inelastic structural properties under the hypothesis of modelling the friction coefficients of the two surfaces of the DFPS as random variables, incremental dynamic analyses are developed to assess the seismic fragility. From the convolution integral of the fragility curves with the seismic hazard curves related to L'Aquila (Italian site), assuming a lifetime of 50 years, the corresponding reliability curves are computed. Precisely, seismic reliability-based linear and multi-linear regressions relating the displacement ductility demand to the behavior factors for the softening and hardening superstructures together with seismic reliability-based design (SRBD) curves for the two surfaces of the double concave sliding bearings are provided as the design relationships.

Constant yield displacement procedure for seismic evaluation of existing structures

The starting point of the proposed procedure for seismic evaluation of existing structures is that the yield displacement of a structure in flexure is constant and that it depends only on the yield strain of the yielding material and the geometrical characteristics of the structure, not on the yield strength of that structure. The fundamental vibration period of the structure is, thus the dependent variable derived from the estimated yield strength and yield displacement of the structure. To facilitate an evaluation of the maximum inelastic deformation of an existing structure using a corresponding single-degree-of-freedom system approach, a new relation between the yield strength (defined using a new yield strength reduction factor) and the displacement ductility demand of a corresponding single-degree-of-freedom system is proposed. This relation is consistent with the constant yield displacement assumption and characterizes the relevant properties of the structure using the yield strain of its yield material, its aspect ratio and its size. The proposed Constant-Yield-Displacement-Evaluation (CYDE) procedure for seismic evaluation of existing structures has four steps. Given an existing structure, its seismic hazard environment, and an estimate of its strength, the CYDE procedure estimates the displacement ductility demand, i.e. the maximum inelastic displacement, the structure may experience at the examined seismic hazard levels. The proposed CYDE evaluation procedure is similar to the current constant-period procedures, but provides a more realistic estimate of the displacement ductility demand for stiff structures, enabling a more accurate seismic assessment of numerous existing structures.

Seismic reliability-based design of inelastic base-isolated structures with lead-rubber bearing systems

In this paper, a seismic reliability-based approach is proposed to design inelastic steel moment frame structures isolated by lead-rubber bearing (LRB) systems. An equivalent two-degree-of-freedom system is assumed in which a bilinear behaviour is assigned to both the superstructure and the base. Furthermore, uncertainties associated with the equivalent superstructure mass, stiffness, and yield properties are taken into account by employing proper probability density functions. The proposed design approach is twofold: 1) Reliability curves that return the key design parameters of the inelastic base-isolated structure including: the period of the superstructure, the target base displacement, and the ductility-dependent strength reduction factor for a given target reliability. 2) Regression equations, which estimate the displacement ductility demand of the inelastic superstructure and the optimal design properties of the lead-rubber bearing system including: the total initial stiffness and the total yield force. These regression equations are calibrated against a large set of optimally-designed base-isolated buildings using Genetic Algorithm.