Nonlinear Time History Analysis Research Articles

Analytical plasticity‐based model for soil–structure interaction of lumped system on heterogeneous soil media

AbstractIn the present study, an analytical approach is developed for the soil–structure interaction for the linear, eq. linear and elastic‐perfectly plastic layered soil strata. The amplification factor is calculated at the top of a lumped structure incorporating layered soil, foundation, and super‐structure deformation. The equation of motion of the system under the normalized earthquake excitation is solved using the direct integration method. MATLAB script is developed to carry out the parametric study, and nonlinear time history analysis is performed. The present methodology is compared with the finite element model (FEM), which shows good agreement. The computational time is much less than the FEM method. A parametric study is carried out in terms of soil heterogeneity, mass ratio, slenderness ratio, excitation frequency, and material model. The results indicate that due to the heterogeneity of the soil, the response of the structure is increased significantly; with the increase in the mass ratio, the amplification factor decreases with excitation frequency up to resonance, and the effect of mass ratio is negligible at the higher frequency range; with the increase in slenderness ratio, the amplification factor increases up to resonance, however, at higher frequency ranges there is no effect of slenderness ratio.

Seismic design and performance evaluation of hybrid braced frames having buckling-restrained braces and self-centering viscous energy-dissipative braces

Hybrid braced frames (HBFs) emerge as an innovative seismic-resilient system, synergizing the strengths of buckling-restrained braces (BRBs) and self-centering viscous energy-dissipative braces (SCVDBs). This novel system has the potential to simultaneously manage three key seismic demand parameters: peak inter-story drift (PID), residual inter-story drift (RID), and peak absolute floor acceleration (PFA). Nevertheless, a dedicated seismic design method for HBFs is yet to be developed. This research proposes a performance-based seismic design (PBSD) method specifically tailored for HBFs. The method's core relies on energy decoupling in the HBFs using a proposed energy distribution factor (η). To validate the proposed design method, sets of 4-, 8-, and 12-story braced frames are designed using varied η. A comprehensive seismic performance evaluation is conducted, utilizing nonlinear static and nonlinear time-history analyses. The findings indicate that the HBFs, designed utilizing different values of η, consistently meet predefined performance objectives. Within the scope of this study, the η value of 0.5 is shown to be optimal, enhancing the structural resilience of HBFs against PIDs and PFAs while exhibiting excellent self-centering performance comparable to the self-centering viscous energy-dissipative braced frame (SCVDF). Additionally, the HBFs demonstrate superior control over PFA compared to pure buckling-restrained braced frame (BRBF) and SCVDF.

Advancing seismic resilience: Performance-based assessment of mid-rise and high-rise engineered cementitious composite (ECC) Buildings

Engineering Cementitious Composites (ECC) have received wide attention in recent years due to their enhanced mechanical properties and ductility. These properties offer an opportunity to design structures with significantly improved seismic performance having a low-damage and ductile response. However, existing research studies primarily focus on the performance of ECC at material and member scales, resulting in a knowledge gap regarding its response at the structural level. This study examines the system-level seismic response of buildings designed using ECC and compares their performance with those having conventional reinforced concrete (RC) members. For this purpose, two RC shear wall buildings (7-story and 24-story) were selected as a case study, and their elements were separately designed (for the combined gravity and lateral loads) as RC and ECC elements using the guidelines recommended by the Japan Society of Civil Engineering (JSCE) and ACI (American Concrete Institute)−318–19. The design results show that the requirement of longitudinal steel is reduced by a maximum of 24% in ECC flexural members and by 15% in compression members, in addition to a significant reduction in the required transverse reinforcement as compared to the corresponding RC members. Similarly, owing to improved tensile behavior, the ECC members also exhibited a higher shear capacity than RC members. The detailed nonlinear finite element models of the case study buildings (for both the design cases i.e., ECC and RC) were subjected to monotonic and reversed-cyclic pushover analysis, and nonlinear time history analyses (NLTHA) under a set of selected ground motions. It was observed that ECC structures exhibited significantly improved lateral capacity, revered-cyclic response, and overall seismic performance as compared to the corresponding RC structures. These results demonstrate that ECC can be effectively used to design various members for improved seismic performance of buildings.

City-scale buildings damage estimation based on broadband physics-based ground motion simulation of 2021 Ms 6.4 Yangbi, China, earthquake

This article proposed a computational workflow for city-scale building damage estimation based on broadband physics-based ground motion simulation. The dynamic stiffness matrix method in frequency-wavenumber domain is employed for broadband physics-based ground motion simulation of finite fault along with site response correction using equivalent-linear site response analysis approach. With simulated ground motion, seismic response of buildings, using multi-degree of freedom model, is simulated using nonlinear time history analysis, and building damage is evaluated based on a trilinear backbone curve of story’s nonlinear force-drift model. Main advantage of the framework lies in the employed dynamic stiffness matrix method for ground motion simulation, which can efficiently generate broadband ground motions in perfect parallel and naturally capture the physics of finite fault source, propagation path, and spatial variability characteristics of ground motion. To demonstrate the salient features of proposed workflow, the 2021 Ms 6.4 Yangbi earthquake is simulated with frequency resolution up to 20 Hz, and seismic damage of buildings in Yangbi county are estimated. Results show that most traditional wood-adobe structures suffer moderate to severe damage, and engineered seismic buildings (mostly RC frame) in Yangbi county suffer minor damage, which agrees with field investigation results.

Column Link Behavior in Eccentrically Braced Composite 3-Dimensional Frames

Eccentrically braced frames are renowned for their capacity to absorb seismic forces while offering greater adaptability. These frames incorporate bracings that are joined to the beams with an intentional offset, forming a connection within the beams. Nevertheless, there are drawbacks associated with implementing these beam connections when renovating frames. This paper seeks to enhance the design approach by introducing an eccentric link within the column of a composite structure. Eccentric braced frames (EBFs) are hybrid systems that offer both ductility in moment resisting frames (MRFs) and lateral stiffening in the concentrically braced system. The study examines composite frames with 5, 10, and 15 stories using eccentric X- and V-type bracings with an eccentricity of 0.5 m and 1 m. Three different earthquake zones are considered, based on Indian seismic code provisions: zone 3, zone 4, and zone 5. The structures are analyzed computationally by nonlinear time history analyses. The lateral load-resisting behavior of the structure with the same eccentricity in beam links and column links is compared. Then, the structure is subjected to a pushover analysis to study the performance characteristics such as capacity curve, lateral displacement, inter-storey drift, and plastification of the structure. As anticipated, compared to conventional moment resisting frames (MRFs) and concentrically braced frames (CBFs), eccentrically braced frames have better energy dissipation. Furthermore, the behavior of X-braced column links is found to be similar to the performance of beam links, but V-braced frames showed better performance in column link frames than in beam link frames. Also, the increase of the link length played a major role in the ductility of the frames.

Seismic behavior of a novel slotted steel plate shear wall under monotonic, cyclic, and time history analysis

Steel plate shear walls (SPSWs) are lateral load-resistant systems used as energy dissipation members in new or existing structures due to their stable hysteretic behavior. However, structures equipped with conventional SPSWs may have a higher initial stiffness, which causes them to be subjected to too much force during moderate earthquakes. In the present study, a novel slotted SPSW is developed to minimize the lateral forces of traditional SPSWs. The slotted SPSW consists of horizontal boundary elements (HBEs), vertical boundary elements (VBEs), and two layers of incline-slotted infill plates (ISIPs) connected by high-strength steel bolts. The proposed system's seismic behavior and damaged mode were compared with those of conventional SPSW using monotonic and cyclic analysis. The research results show that the slotted SPSW exhibits stable hysteretic behavior with a significant energy dissipation capacity. Furthermore, the slotted SPSW was applied to a three-story frame structure. Nonlinear time history analysis was conducted under far-field and near-field ground motion records to determine how a slotted SPSW system affects structure response. According to numerical results, the base force of the structure equipped with a slotted SPSW system was minimized to 37.0% and 49.6% under near-field and far-field earthquakes, respectively.

Rapid peak seismic response prediction of two-story and three-span subway stations using deep learning method

A deep learning-based rapid peak seismic response prediction model for the most common two-story and three-span subway stations is proposed in this study. The established model predicts the peak seismic responses of subway stations with a data-driven fashion and using limited information. The prediction model extracts the features of ground motions using one-dimensional convolutional neural network (1D-CNN) and then integrates the information of subway stations (i.e., the seismic fortification intensity, buried depth, and shear wave velocity) through a fully connected neural network for regression, resulting in peak seismic responses, namely the peak floor acceleration (PFA) and maximum inter-story drift ratio (MIDR). The model is trained using 19,200 samples obtained from the nonlinear time-history analyses (NLTHAs) of the designed 48 typical subway station structures. Furthermore, the external model verification was performed on 960 additional samples. For the predictions of PFA and MIDR, the coefficient of determination (R2) values are 0.967 and 0.986, respectively, and the damage states of subway stations are further evaluated, achieving an accuracy of 95.0%. These indicates that the model has good predictive performance and generalization ability. Moreover, the prediction model demonstrates a significantly higher computational efficiency compared to numerical simulation methods.

Investigating earthquake-induced irregularity based on post-earthquake running performance analysis of track-bridge system

In evaluating post-earthquake damage to track-bridge systems, this study conducts nonlinear time history analysis on the CRTS II ballastless track simply-supported beam system subjected to transverse earthquake loading and explores the characteristics of residual displacement and stiffness degradation of the track-bridge system under transverse earthquakes. The effect of the earthquake-induced stiffness degradation of track-bridge systems on the dynamic performance of high-speed trains is investigated, and earthquake-induced dynamic irregularities are constructed to quantify the overall stiffness degradation degree of the track-bridge system. Furthermore, a reconstruction method for the earthquake-induced dynamic irregularity characteristic curve is proposed, considering probability guarantee rates. The research results indicate that the earthquake-induced dynamic irregularity can effectively quantify the running performance degree of high-speed trains under earthquake-induced stiffness degradation conditions. The transverse acceleration and derailment coefficient of high-speed trains increase as the ground motion intensity rises. The amplitude of the characteristic curve grows significantly with the increase in pier height.

Effects of Deck-Abutment Pounding on the Seismic Fragility Curves of Box-Girder Highway Bridges

ABSTRACT Earthquake-induced pounding in bridges is a complex contact phenomenon in which the dynamic responses of structures, including collisions between deck and abutments, are strongly related to structural properties and earthquake excitation. The goal of this study is to develop and compare the seismic fragility curves of overall system and individual components of regular and irregular box-girder highway bridges in two cases: with pounding and without pounding. For this purpose, four levels of altitudinal irregularity, ranging from regular to highly irregular, are considered. To extend the results for all bridges in the same class, different sources of uncertainties related to earthquakes, structural geometries, and material properties are taken into account. The analytical fragility curves have been developed based on nonlinear time history analyses in OpenSees finite element software for the cases with and without pounding effects. The process has been repeated for each two-, three-, and four-span classes at four irregularity levels. The fragility function parameters for the two cases with and without pounding have been compared for all classes considered in this study. Using fragility functions, this paper clarifies the interactive roles of irregularity and pounding between deck and abutments for seismic vulnerability of multi-span box-birder highway bridges. The results indicate that collisions often show an adverse effect on all structural components. It has also been observed that the detrimental effect of pounding on seismic fragility is more apparent in irregular bridges when compared with regular ones. In addition, the study introduces a conversion coefficient to clarify the effects of pounding on the fragility of bridge components and the overall system. This coefficient can be applied in both conventional analytical methods like static or simplified analysis and technical earthquake models like HAZUS, adjusting fragility values for pounding and irregularity effects.

Effects of modeling assumptions on prediction of seismic demands of a single-story RC structure

This study investigates the seismic response of a single-story reinforced concrete (RC) structure subjected to strong ground motions. Effects of modeling assumptions used to idealize the reinforcement slip in beam-column joints and column footing are studied. For this purpose, the numerical model of a single-story, single-bay, three-dimensional asymmetric RC structure, tested on a shake table and its symmetric version, is analyzed. Two modeling approaches are considered to show the significance of rotation associated with the reinforcement slip. Bidirectional nonlinear time-history analyses are conducted using symmetric and asymmetric 3D models. In these analyses, 30 pairs of near-field ground motions, selected and scaled to match the target spectrum, are used. The findings indicate that the neglection of reinforcement slip in numerical model may result in an underestimation of almost 50% in story drift ratio demands.

Nonlinear time-history analysis of the Evolution Tower resting on cellular raft due to earthquake loads

This study includes an investigation of using a cellular raft over piles on the seismic response of both a twisting and regular tower using the direct method, demonstrating the variance between a solid raft and a cellular raft. Both towers are 52 stories high and made of reinforced concrete. They were also built on a reinforced concrete piled-raft foundation. The soil model is thought to be multi-layered and has the same profile as the zone under investigation (New Mansoura, Egypt). All structural properties (dimensions, section properties, materials) are equal to allow a fair comparison of output response for both towers. The sole difference between the two towers is their elevation. The study is carried out under seismic loads using nonlinear time-history analysis. All analyses are carried out employing finite-element software (Midas GTS NX). Thus, seven distinct earthquake records with full 3D models were used for time-history research. According to the findings and discussions, it is concluded that adopting a cellular raft can reduce the dynamic response of the towers. Considering the soil–structure interaction, the maximum inter-story drift ratio decreases by 8.17% for the twisting tower and 5.58% for the regular tower while using the cellular raft. Furthermore, the regular tower is more effective than the twisting tower at resisting lateral loads.

Study on rocking response of base-isolated structure considering the tensile effect of isolators

This paper carries out the parametric study of the rocking response of base-isolated structures by considering the tensile effect of isolation bearings. Experimental investigations are conducted to analyze the compression-shear and tension-shear behavior of lead rubber bearings (LRBs).The results reveal significant differences between the tension-shear and compression-shear behavior of LRBs. Building upon the experimental findings, the equations of motion for the base-isolated structure are established using the two-spring extension model proposed by Ryan and Chopra, with particular emphasis on incorporating the tensile effect of LRBs. Important parameters are selected based on practical engineering considerations within the governing equations. Nonlinear time-history analysis is performed to obtain numerical results, leading to the conclusion that the tensile effect of the isolation bearings should not be overlooked, especially for base-isolated structures with a slenderness ratio greater than 2. Furthermore, the number of isolation bearings also has a greater impact on the rocking response of base-isolated structures compared to the mechanical behavior of bearings. Finally, the paper analyzes the impact of variations in selected parameters and proposes the limiting values of slenderness ratios for base-isolated structures based on the requirements of the Chinese Standard for seismic isolation design of buildings (GB/T51408-2021).

Numerical Studies on Seismic Response of Structural Systems Using a Response-Spectra Based Intensity Measure

This paper proposes a novel intensity measure (IM) based on the geometric mean of acceleration response spectral ordinates to assess the probabilistic performance of structures subjected to seismic loading. Instead of relying solely on the fundamental period, the proposed IM is evaluated across a fixed period range for all structural systems, which allows consideration of higher mode effects and period changes due to nonlinearity. The proposed seismic IM is evaluated using two established indices sufficiency and efficiency. Sufficiency quantifies the independence of an engineering demand parameter at a specific intensity level with ground motion characteristics such as seismic magnitude [Formula: see text] and distance from site to fault plane [Formula: see text]. It is calculated by linear regression analysis, or using gradient-based relative sufficiency measures. On the other hand, efficiency is measured as dispersion across ground motions at a given intensity level for any physical response quantity. It helps to reduce computational demand for failure probability assessment by considering a smaller number of records compared to an inefficient IM for similar confidence levels. The effectiveness of the proposed IM along with 10 other IMs is demonstrated on single degree of freedom systems with various fundamental periods by performing nonlinear time history analysis using a far-field ground motion record set. The study is also extended to five degree of freedom lumped mass stick models, 2D models (4-, 8-, and 12-story archetype steel frames), and 3D reinforced concrete shear wall building model. The results indicate that the proposed IM limits dispersion to within 10% for long-time period structures, and demonstrates improved sufficiency across different structural systems. For example, gradient of the proposed IM with respect to magnitude [Formula: see text] and site-to-source distance [Formula: see text] for a 12-story steel frame is reduced by 42.9% and 94%, respectively, compared to spectral acceleration at fundamental time period. Potential application of this research lies in efficiently conducting seismic reliability assessment and design for structural systems.

Influence of Fluid Viscous Damper Stiffness on the Floor Acceleration Response of Steel Moment-Resisting Frames Under Far-Field Ground Motions

ABSTRACT The implementation of fluid viscous dampers has been proven as a feasible method to improve the seismic performance of buildings. However, the different variables involved in the damper’s design affect the structural response, especially in terms of seismic demand on non-structural elements. This paper presents a parametric study on the floor acceleration response of three case study buildings equipped with fluid viscous dampers through nonlinear time-history analysis. The results show that the different design parameters of fluid viscous dampers significantly modify the floor acceleration response, with some damper configurations yielding floor accelerations larger than those of the buildings without dampers.

Nonlinear dynamic instability and dynamic response of stiffened laminated composite plates subjected to in-plane pulsating patch loading

In this article, the nonlinear dynamic instability of stiffened laminated composite plates is studied in the finite element (FE) framework subjected to uniform in-plane harmonic patch loading. The harmonic load is applied to the two opposite sides of the stiffened plate. The linear and nonlinear time-history response analysis is also studied. The skin and the stiffener are modeled using an eight-node isoparametric degenerated shell element and a three-node curved beam element, respectively. A system of matrices is developed by considering the Green–Langrange strain–displacement relationship. In the linear case, the Bolotin method is used to analyze the dynamic instability region (DIR). The nonlinear instability behavior of the laminated composite stiffened plate is studied by applying the Incremental Harmonic Balance Method (IHB). The Newmark-β method is used to solve the linear and nonlinear time-history response equations to understand the instability behavior of the stiffened plates. The effect of the parameters such as the length of the in-plane loading patch, varying number of stiffeners in x -direction and the position of the patch on the nonlinear vibrations and nonlinear dynamic response is examined.

Seismic response of two-storey steel frame equipped with metallic yielding and viscous dampers under far- and near-field earthquakes: Shaking table tests and numerical investigations

Seismic energy dissipation technology is widely used in building structures to mitigate earthquake damage; two typical energy dissipation devices (EDDs), viscous fluid dampers (VFDs) and metallic yield dampers (MYDs), are often used. To investigate the effects of different types of EDDs on seismic performance, shaking table tests were conducted on a two-storey moment-resistance steel frame (MRSF). This study examined three types of damping schemes: VFDs only, MYDs only, and a combination of both. The test results showed that different damping schemes had different effective stiffnesses and damping ratios. The basic frequency and damping ratio of the MRSF with hybrid dampers increased by 36.18% and 38.41%, respectively, compared with those with VFDs only. Compared with the other two controlled MRSFs using a single type of EDDs, the MRSF equipped with hybrid dampers can effectively coordinate the control of interstorey drifts. To further analyse the impact of pulse-type ground motions (GMs) on the dynamic responses of the controlled MRSFs, numerical models were established for all test models, and the nonlinear time-history analysis (NTHA) was performed. The numerical results fitted well with the test results and showed that the hybrid damping scheme can take full advantage of the two types of EDDs. The mean dissipation energy ratio of the dampers was 86% with GMs with an intensity of 0.80 g. There was a significant difference in the seismic damping effect of the controlled MRSFs with different types of seismic excitations. The acceleration control effect of the MRSF with only MYDs was best with near-field pulse-type GMs, with up to a 30% reduction over the MRSF with only VFDs. The acceleration control effect of the hybrid-controlled MRSF was best with near-field non-pulse GMs; the 1st-storey acceleration control of the hybrid damping MRSF was reduced by up to 42.4% compared to the VFD. The displacement reduction rates of the three controlled MRSFs subjected to pulse-type seismic excitation were lower than those of the three MRSFs subjected to non-pulse-type seismic excitations.

Seismic response and performance prediction of steel buckling-restrained braced frames using machine-learning methods

Nowadays, Buckling-Restrained Brace Frames (BRBFs) have been used as lateral force-resisting systems for low-, to mid-rise buildings. Residual Interstory Drift (RID) of BRBFs plays a key role in deciding to retrofit buildings after seismic excitation; however, existing formulas have limitations and cannot effectively help civil engineers, e.g., FEMA P-58, which is a conservative estimation method. Therefore, there is a need to provide a comprehensive tool for estimating seismic responses of Interstory Drift (ID) and RID with novel approaches to fulfill the shortcomings of existing formulas. The Machine Learning (ML) method is an interdisciplinary approach that makes it possible to solve these types of engineering problems. Therefore, the current study proposes ML algorithms to provide a prediction model for determining the seismic response, seismic performance curve, and seismic failure probability curve of BRBFs. To train ML-based prediction models, Nonlinear Time-History Analysis (NTHA) and Incremental Dynamic Analysis (IDA) were performed on the 2-, to 12-Story BRBFs subjected to 78 far-field ground motions, and 606944 data points were prepared for different prediction purposes. The results indicate that the considered approaches are justified. For instance, the proposed ML methods have the ability to predict the maximum ID, maximum RID and maximum roof ID with the accuracy of even 98.7%, 95.2%, and 93.8%, respectively, for the 4-Story BRBF. Moreover, a general preliminary estimation tool is introduced to provide predictions based on the input parameters considered in the study.

Design procedure and seismic performance assessment of structures equipped with bar-fuse dampers

The application of metallic yielding dampers (MYDs) to enhance the seismic performance of a structure has been a focus of attention in recent years. One recently introduced MYD is the bar-fuse damper (BFD). Its advantages include ease of installation, simple construction method, cost effectiveness (including replacement and repair costs) and shorter required repair time after strong seismic events. However, the most important benefits of the BFD are its high energy dissipation capacity, which can improve collapse safety and enhance the seismic performance of a structure. Despites its benefits, BFDs have been less studied than other types of MYDs. The present research provides a simple and efficient method for designing structures equipped with BFDs. Moreover, the mechanical characteristics of differently shaped BFDs made with different types of steel has been investigated using finite element analysis. Finally, the seismic performance of three model structures was evaluated using nonlinear time-history analysis. The results indicate that structures equipped with BFDs experience less inter-story drift and are less likely to undergo severe damage compared to similar bare moment-frame buildings.

Seismic behaviour of post-earthquake composite frame structures with different damage levels

Since earthquakes pose a huge threat to buildings, the post-earthquake safety assessment is quite important, especially in earthquake-prone areas. This paper investigates the seismic performance of post-earthquake composite frame structures with different damage levels. Two-stage nonlinear time history analysis is carried out for a ten-story steel–concrete composite frame and the peak ground acceleration (PGA) of the seismic waves is used as the main parameter characterizing the seismic intensity and damage level. In the first stage, the seismic waves with the PGAs of 0.20 g, 0.30 g and 0.40 g are applied to the frame to result in three different damage levels. The peak inter-story drift ratios, residual inter-story drift ratios and plastic hinge ratios of the structure in the first-stage earthquakes are analysed, which presents a uniform distribution pattern for the three damage levels. The composite frame is evaluated as structurally safe at all three damage levels based on the limits of residual inter-story drift ratios given in current standards. In the second stage, the seismic waves with the PGA of 0.40 g are used to explore the residual capacity of the structure subjected to strong earthquakes. The dynamic responses of the intact structure and the structure with different damage levels are compared. The results indicate that despite the recognised structural safety, pre-existing seismic damage can enlarge the lateral deformation, reduce the internal forces and resistance capacity and increase the plastic hinges when the composite frame is exposed to the second-stage earthquakes. Generally, the damage caused by earthquakes with the PGA of 0.40 g will have an obvious influence on the seismic behaviour of the post-earthquake structure, in which situation further safety assessment and structural repairs are necessary.

Evaluation of equivalent linearization methods of MDOF isolated structures in response spectrum analysis

In the seismic design of isolated structures, equivalent linearization of the hysteretic isolators is normally considered. There exist many equivalent linearization methods and evaluation of these methods has been widely studied. However, most of the previous studies concern a single degree of freedom system only, and the used analysis method is time history analysis. In such a case, the evaluation heavily depends on the ground motions selected. Note that the response spectrum analysis is a commonly used method for seismic design of structures. Therefore, it is more valuable to evaluate these equivalent linearization methods based on the response spectrum analysis. In this study, a multiple degree of freedom model is adopted due to the contribution of higher modes of the superstructure to the seismic responses, and the procedure of the response spectrum analysis in conjunction with the equivalent linearization method is presented in detail. Then, considering a series of parameters of isolators, a statistical study is performed to evaluate the accuracy of these equivalent linearization methods by comparing with the nonlinear time history analysis. Moreover, the effect of the damping reduction factor for the response spectrum on the accuracy of the equivalent linearization methods is discussed in terms of probability in a desirable error range. The analytical results show that the accuracy of the equivalent linearization methods is associated with the earthquake intensity. Meanwhile, for different response quantities the accuracy of each equivalent linearization method varies significantly. In addition, to estimate the seismic responses with a better accuracy in the response spectrum analysis, a suitable damping reduction factor should be considered.