Nonlinear Dynamic Time History Analyses Research Articles

A Comparative Study on Different Types of Passive Energy Dissipating Devices

Earthquake induced structural vibrations causes fatigue and reduction of performance of the tall steel framed structures. One of the effective ways for improving the seismic performance of a structure is the use of bracings, dampers and isolators. These devices change the stiffness and damping of the structures thereby reducing the seismic vibrations to a great extent. The main objective of this study is to compare the different types of passive energy dissipating devices in a 3 bay 10 storey steel frame building for better seismic response. The frame was modelled in SAP2000® and some of their responses to earthquake such as base shear, maximum joint displacement, inter storey drift were determined and compare during non-linear dynamic time history analysis in SAP2000®.The passive energy dissipating devices used in this study are diagonal bracing, v bracing, cross bracing, rubber base isolator and friction damper. At first a single storied shear frame model was tested in shake table and results obtained from SAP 2000® were compared. The close proximity validated the procedure. It is observed that addition of dampers to the system increases the stiffness to the frame. When compared, displacement for steel frame with dampers is reduced by 80% and that with different bracing systems are reduced up to 78% as compared to bare steel frame. For base isolators, base shear decreases up to 63.33% but an increase in the maximum joint displacement was observed. The comparison of performances of all types of dissipating mechanisms has been presented in the study.

Seismic Performance of Coupled Buildings Connected by Yield and SMA Dampers

Efficient attenuation and control of structural vibration during a seismic event is a critical factor for overall safety and serviceability of the adjacent buildings with little clearance between them. Without proper vibration control measures, adjacent buildings may undergo severe pounding and collapse leading to damage of life and property during an earthquake. Use of energy dissipation devices to connect adjacent buildings has been a popular method of vibration control and to stop mutual poundings. Among various available energy dissipation devices, Shape Memory Alloy (SMA) based dampers are getting popularity in recent time. Various studies are performed to show the efficiency of SMAs as passive vibration control device, but a holistic study of SMA dampers for non-linear connected buildings is still missing. Thus, here a comparative study on the efficiency of metallic yield and super-elastic SMA dampers as passive vibration control device is provided, for non-linear connect buildings. In the present study, non-linear dynamic time history analyses are performed to determine the response (peak floor acceleration and displacement, and floor residual displacement). Time history analyses confirm better acceleration and displacement (both peak and residual) control efficiency of SMA dampers over yield dampers. Finally, superiority of the SMA damper over the yield damper is established via parametric study under varying damper, building and ground motion parameters. Over yield damper, SMA damper reduces acceleration by 13 % to 56 %, displacement by 2 % to 47 %, and residual displacement by 15 % to 64 %.

Optimal selection of scalar and vector-valued intensity measures for improved fragility analysis in cross-fault hydraulic tunnels

Seismic intensity measures (IMs) are crucial because they are significantly connected to earthquake hazards and structural responses. It means that selecting appropriate IMs is necessary for underground structures under unfavourable earthquake hazard conditions. The seismic design code and dynamic analysis have conventionally employed the peak ground acceleration (PGA) as an optimal IM, mainly referring to the seismic design code of aboveground structures. However, seismic investigation of underground structures using PGA is still controversial. Simultaneously, directly employing the PGA as an ideal IM may increase the uncertainty associated with ground motions, which in turn increases the uncertainty for probabilistic seismic demand analysis of hydraulic tunnels. As a result, it is important to understand the roles played by different IMs while analyzing the fragility of hydraulic tunnels. This work aims to conduct a comprehensive analysis to identify the optimal scalar- and vector-valued IMs for fragility investigation of cross-fault hydraulic tunnels. Therefore, the 1080 complete nonlinear dynamic time history analysis by surrounding rock-concrete lining-fluid interaction system is conducted to express the earthquake-induced drift ratio of a cross-fault hydraulic tunnel by incremental dynamic analysis. Subsequently, a refinement process is performed to determine the optimal scalar-valued IMs by comparing the correlation, efficiency, practicality, and proficiency of the 7 examined IMs. Meanwhile, the probabilistic seismic demand model of optimum vector-valued IMs is also developed and compared, followed by the three scalar IMs. Eventually, the difference between the fragility curves of the tunnel produced using the optimal scalar- and vector-valued IM is compared. The results demonstrate that compared with scalar two-dimensional fragility curves, the vector three-dimensional fragility surface may increase the reliable probability and minimize the uncertainty in analyzing the structural response hazard curve. The generated vector fragility surface can also estimate the seismic fragility of identical cross-fault hydraulic tunnels in an approximative manner.

Seismic performance assessment of a multi-story bamboo frame structure

As a building material, engineered bamboo has caught attention around the world due to its advantages in energy conservation and environmental protection. The seismic performance of bamboo buildings needs to be evaluated to further promote the use of bamboo materials in building construction. We studied the seismic response of a 3-story bamboo frame structure numerically using nonlinear dynamic time history analysis. A simplified modeling method for bamboo column-beam joints was proposed in the numerical model. The hysteresis behaviour of the joint was simulated by Pinching 4 material in OpenSEES, with the parameters calibrated through test results. Comparative analysis shows that the proposed modeling method could reasonably reflect the pinching effect and the degradation of the joint hysteretic behavior. A total of 20 ground motions with three intensities were involved in the nonlinear dynamic analysis. The results demonstrate that the frame meets target performance levels, providing evidence for the further popularization of engineered bamboo structures.

Nonlinear Finite-Element Modeling of Precast Reinforced Concrete Moment-Resisting Frames

Modeling of precast reinforced concrete frames using three-dimensional solid elements and inelastic contact elements is a very difficult process. Therefore, several two-dimensional nonlinear models using inelastic frame elements and zero-length spring elements were introduced in recent decades. This paper aims to present a static and dynamic nonlinear time history analysis of cast-in-place and precast reinforced concrete frames and to predict the performance of these frames under earthquakes. One other objective is to simplify and improve the pinching effect and the energy dissipation characteristics of the precast beam-column joint model conducted by other analytical studies. All details of the nonlinear models and their specified parameters were introduced. The analytical model was verified with the available experimental tests of the beam-column joints and the moment-resisting frames. The results proved that the improved nonlinear beam-column joint model was better than the old model especially in estimating the load capacity, the pinching effect, and the energy dissipation characteristics. Generally, the nonlinear model can predict the seismic response of precast frames with acceptable accuracy.

Structural system design and earthquake response analysis of prefabricated pile-plate bridge

ABSTRACT With the advancement of building technology, the pile-plate structure, which originated in railway engineering, has been adopted into highway engineering. This study presents a new pile-plate structure connecting node. The addition of energy-dissipating components at the nodes and the use of non-shrinkage concrete at the joints demonstrate its novelty. The node’s ability to enter the plastic stage is enhanced, resulting in increased seismic performance. In this study, the seismic performance and energy consumption capacity of the pile plate joint were studied by numerical model, and the results showed that the stagnation curve of the pipe pile under low cycle reciprocating load was full and had good energy consumption capacity. Under the action of random seismic vibration, the displacement of the pile plate structure in one direction is up to 0.023 m, which meets the requirements of the specification. The innovative pile-plate joints are used in a nonlinear dynamic time-history analysis of the constructed bridge structure.

A negative stiffness dynamic base absorber for seismic retrofitting of residential buildings

In this study, a negative stiffness-based passive vibration absorber is developed and implemented as a seismic retrofitting measure for typical reinforced concrete (RC) residential buildings. The device, namely, the extended KDamper for retrofitting (EKD-R), is introduced at the base of the structure, between the foundation level and the first story of the building. The design of the EKD-R device and the selection of its properties are undertaken by incorporating a harmony search (HS) algorithm that provides optimized parameters for the mechanism, following constraints and limitations imposed by the examined structural system. Nonlinearities due to the plastic behavior of the structural members and soil–structure interaction (SSI) effects are modeled and taken into consideration during the process. Subsequently, a realistic case study of a benchmark three-story RC building is examined, and the performance of the EKD-R system is assessed. The building superstructure is designed according to Eurocodes. The structure–foundation system, along with the EKD-R, is explicitly modeled using finite elements (FE) that may realistically capture structural nonlinearities and SSI effects. The HS algorithm is employed, and optimized EKD-R components are obtained and implemented in the benchmark structure. Finally, a series of recorded real ground motions are selected, and nonlinear time-history dynamic analyses are conducted aiming to assess the behavior of the controlled system. Results indicate the beneficial role of the novel dynamic absorber, hence rendering the concept a compelling seismic retrofitting technology.

Variable Factors Affecting Progressive Destruction of Composite Steel Tall Building

In recent years, the presence of progressive collapse in tall buildings induced a catastrophic event which attracted the majority of the community’s attention. The purpose of this paper is to develop a 3D numerical analysis of tall building under column loss. A composite steel frame building with 25 stories with five spans in both directions is proposed. The building has 3 m story height and 8 m span in both directions. The building is designed through the commercial software SAP2000 software against wind loads based on Eurocode 1-2005. The focus here is to investigate various parametric studies under abrupt column loss of multi-story composite building. The effect of composite slab is considered with full composite action between beam and slab. The findings of a parametric formulation incorporating important parameters for the progressive collapse design technique are given and confirmed using nonlinear dynamic time history analyses. The assessment of results has been introduced based on deformation, axial force in columns, equivalent plastic strain, major moment and axial force in the considered beams above the column loss. Next, a probabilistic analysis has been performed to assess the behavior of composite steel buildings against column loss. The study investigates the critical column loss and pinpoints the location of the next critical column. The results show that the concrete grade, position of the removed column, beams cross-section, and place of bracings have a significant effect in the response of the building rather than the steel grade and bottom reinforcement density. The removal of exterior column has the significant increase of the axial force percentage by 111.4% for the corner column. The corner column removal gives the maximum equivalent plastic strain with a value of 0.00449. Furthermore, the results reveal the potential impact of uncertainty on the structural elements of the considered buildings through the progressive collapse analysis. The vertical displacement above the column is fitted with mean value of 0.0251387 m and with a coefficient of variation 0.01664.

Wind-Induced Response Characteristics and Equivalent Static Wind-Resistant Design Method of Spherical Inflatable Membrane Structures

The wind-induced responses and wind-resistant design method of spherical inflatable membrane structures are presented in this paper. Based on the wind pressure data obtained from wind tunnel experiments, the characteristics of wind-induced responses are studied via nonlinear dynamic time–history analysis, considering the influences of spans, rise–span ratios, internal pressures, wind velocities, and cable configurations. The results show that with the increment of wind velocity, the position of the maximum displacement changes from the top to the windward region, which usually leads to the exceedance of the displacement limitation. Under high wind velocity, enhancing the internal pressure can effectively reduce deflection. However, the membrane stress will also increase. Particular attention should be paid to checking the strength. The restraint effect of cross cables on wind-induced response is better than radial cables. Furthermore, an equivalent static analysis method for the wind-resistant design of spherical inflatable membrane structures is developed. The empirical formulas and recommendation values of gust response factors and nonlinear adjustment factors are provided for engineering reference.

Seismic vulnerability analysis for shallow-buried underground frame structure considering 18 existing subway stations

The variability of structural engineering parameters is not accounted for in available literature on seismic vulnerability of underground frame structures. This study investigates the seismic vulnerability of 18 existing subway stations with different site condition, structural stiffness, reinforcement ratio and buried depth employing two-dimensional (2D) finite element analysis considering soil-structure interaction. The peaks of structure inter-story drift ratio under different earthquake intensity of different ground motion records are obtained from non-linear dynamic time-history analysis. Taking the inter-story drift ratio as the damage measure (DM) and the peak ground acceleration (PGA) as the ground motion intensity measure (IM), the seismic vulnerability curves of 18 subway stations are constructed. The comparison of seismic fragility curves of different subway stations demonstrates that site conditions are critical to the post-earthquake structure probability of failure. Furthermore, representative seismic fragility curves are established based on the data collected form the computed results of 18 subway stations, which can aid in evaluating the seismic performance of existing urban underground frame structures in China. The results of this study can provide reference for the performance-based seismic design of urban underground frame structures.

Evaluation of Intermediate Reinforced Concrete Moment Frame Subjected to Truck Collision

In this study, the progressive collapse of reinforced concrete structures due to vehicle collision to the columns of the ground floor was modeled and examined. For this purpose, a four-story reinforced concrete building with the intermediate moment frame system was designed using ETABS software followed by the simulation of impact loading by SAP2000 software. Performing non-linear time history dynamic analysis, the critical forces required to the column failure were determined via trial and error by considering different live load contribution. Then, the corresponding critical velocities for 4, 8, and 12 ton vehicles were determined. Finally, the progressive collapse of the building was examined by the sudden removal of the column. The results showed that by increasing the percentage of live load contribution, the force and critical velocity for the instability and damage of the column will decrease. Furthermore, comparing the perimeter and corner columns showed that the corner columns are the most critical columns for occurrence of the progressive collapse. In addition, during the assessment of the progressive collapse, it was found that the number of damaged springs in the corner column removal scenario is less than that of the perimeter column removal scenario.

Effectiveness of Lateral Load Resisting Systems for Open Ground+20 Storied RC Framed Structure

Abstract: In a highly populated country like India, problem of parking of the vehicles arises, this problem leaves no option for design of open ground storey buildings. Since there are no infill walls in ground storey, stiffness in the upper storey is much more than the ground storey. The columns in the ground storey are heavily stressed, therefore it is required that the ground storey columns must have sufficient strength and adequate ductility, the increased base shear is resisted entirely by the columns of ground storey only. These buildings are vulnerable due to the sudden lowering of stiffness and strength in the ground storey. This results in the attraction of more earthquake forces for the lower time periods, which also results in snapping of lateral ties in column, crushing of core concrete, buckling of longitudinal bars and finally shear failure in open ground storey columns due to lateral earthquake forces. Solution for this problem is to prevent the failure of open ground storey columns due to lateral earthquake forces by providing the lateral load resisting system. Many times, stiffness of walls is not considered while designing, this results in inaccurate designing of elements. An Open Ground +20 storied RC frame subjected to strong motion earthquakes viz. Duzce in Turkey (12/11/1999), Erzincan in Turkey (13/03/1992), Imperial valley at El-centro (19/05/1940), Landers (28/06/1992) and Nahanni in Canada (23/12/1985) creating soft storey effect at ground storey so it should be provided with lateral load resisting systems viz, Shear wall, steel bracing, lead rubber bearing base isolator with different configurations. Performance of equivalent diagonal strut provided to structure is compared with brick work modelled as actual brick work, equivalent diagonal struts, and considering only the mass of brick work. Time History Analysis is used in a RC framed building using ETABS Version18 and SAP2000 Version20 software in comparison with Response quantities Roof displacement, soft storey check, Base shear, overturning moment and storey drift. In this research, equivalent diagonal struts are provided as brick masonary, which shows accurate behavior of structure under strong ground motions as mass and stiffness both are considered during analysis. When Open ground storied structure is subjected to strong ground motions, stiffness at ground storey is drastically reduced. When structure is assigned with Lateral Load Resisting Systems, shear wall with configuration shear wall at ground storey and 1st storey, stiffness at ground storey is increased to 81% and 82% in X and Y direction respectively within permissible limits. Hence it can be concluded that when there is soft storey effect at open ground, structure becomes hazardous in presence of strong ground motions. When structure is subjected to strong ground motions, the vulnerability condition is high. Since we have performed non-linear dynamic time history analysis, more accurate results can be obtained as non-linear analysis considers vertical irregularities

Seismic behavior of linked column system as a steel lateral force resisting system

In previous research, the seismic behavior of linked column frame (LCF) system as a dual structural steel frame system that Consists of the primary linked column (LC) and the secondary moment frame (MF) systems has been evaluated. In this research, based on the high capacity of the linked column system in lateral load resisting, this system has been used as the only lateral load resisting system in the building frame. Studies have shown that this approach not only eliminates the main disadvantages of the LCF system, but also preserves the ability to repair structures after an earthquake. Therefore, the designed models of the linked column system have been evaluated using nonlinear static pushover analysis, nonlinear dynamic time history analysis, incremental dynamic analysis and endurance time method analysis. In the evaluation of the seismic behavior of the linked column system, design approach, structural stability and collapse mechanism, ductility, the seismic performance factors of response modification coefficient (R), overstrength factor (Ω0) and deflection amplification factor (Cd), the values of interstory drift ratio at the DBE and MCE seismic hazard levels, maximum seismic capacity and fragility curves, permanent displacement after the earthquake and the weight of steel used in the structure have been investigated. Each of the analysis results shows that the linked column system has a suitable seismic behavior against earthquakes. Therefore, in a preliminary evaluation, the linked column system is introduced as a steel lateral load resisting system that provides the ability to repair structures after an earthquake.

Performance-Based Seismic Design of Corrugated Steel Plate Shear Walls

Corrugated steel plate shear walls (CoSPSWs) consist of corrugated steel wall plates and a steel boundary frame, which could be adopted as seismic-resistant systems for high-rise buildings. In this study, a performance-based seismic design (PBSD) method was proposed for CoSPSWs, in which an ideal yield mechanism and target drifts were selected as performance objectives to evaluate and control the inelastic behavior of the system. Two 10-story CoSPSW structures were then designed with the PBSD method and the traditional method respectively, and static pushover analyses as well as nonlinear dynamic time-history analyses were conducted on both structures. It turned out that the CoSPSW structure designed with the PBSD method presented the ideal yield mechanism, all inter-story drifts were well below the target drift of 2.5%, and the drifts distributed more smoothly under rare earthquakes. Structural and nonstructural fragility curves of both CoSPSW structures were obtained through probabilistic seismic demand analysis using Incremental Dynamic Analyses. The results showed that for the structural repair states of RS1 to RS5, the 25th percentile PGA values of the fragility curves of the CoSPSW structure designed with the PBSD method were 220%, 98%, 84%, 51 % and 13% higher than the CoSPSW structure designed with the traditional method, respectively. For the nonstructural damage states of Slight to Complete, the 25th percentile PGA values of the fragility curves of the CoSPSW structure designed with the PBSD method were 145%, 98%, 36%, and 7% higher than the CoSPSW structure designed with the traditional method, respectively. Therefore, CoSPSW structure designed with the PBSD method had lower seismic vulnerability as well as probability of nonstructural damage.

Seismic assessment of irregular RC frames with tall ground story incorporating nonlinear soil–structure interaction

This paper is dedicated to evaluate the nonlinear Soil–Structure Interaction (SSI) impact on the earthquake-based response of vertically irregular reinforced concrete (RC) moment resisting frames. The main contribution of this paper is to evaluate the seismic performance of RC structures with 5 and 10 stories considering a number of key parameters including irregularity ratio, soil type and the vertical safety factor (FSv) of the foundation. In this regard, the beam on nonlinear Winkler foundation (BNWF) scheme is used for the modeling of shallow foundations resting on a semi-infinite sandy medium. Vertical irregularity is considered by using a tall ground story in the RC frames. The models are subjected to 15 earthquake ground motions for conducting nonlinear dynamic time-history analyses to capture the seismic demands of soil–structure systems. The results demonstrate that considering SSI influence reduces the seismic demands in the lower stories, which is more evident in the case of soft soils (site class E). Moreover, a rational relationship between the maximum inter-story drifts and the response of the foundations is observed. It is also concluded that for both regular and irregular structures, lower values of the FSv lead to a considerable decrease in the structural demands (story drifts) and an impressive growth in transient and residual foundation settlements.

Modal Pushover Analysis for Seismic Performance Assessment of Underground Structures

ABSTRACT Pushover analysis as an efficient nonlinear static analysis method has been widely applied to the seismic inelastic performance evaluation of above-ground structures. Although it has the main advantage of high computational efficiency compared to nonlinear dynamic time-history analysis (THA), the assumption that the fundamental mode dominates structural response limits its application to long-period structures. Improved pushover analysis has been developed to consider the higher mode effect, named modal pushover analysis (MPA), but all research mainly focused on buildings and bridges. Since the mechanical behaviors of the underground structures and above-ground structures under seismic exaction are not exactly the same, so far, MPA suitable for seismic performance assessment of underground structures still has not been developed. Therefore, the aim of this investigation is to develop MPA for the seismic performance assessment of underground structures. A prototype subway station structure and a soft site are used to study the applicability of the developed MPA and compare it with the traditional pushover analysis procedure and nonlinear THA. Numerical analysis results of the subway station illustrate the applicability and potential of the developed MPA for underground structures.

Modeling the seismic response of thin concrete walls using the non-linear Beam-Truss Model

In recent years, the construction of buildings with thin reinforced concrete (RC) walls has increased significantly in Colombia and other Latin American countries, mainly due to their speed of construction and economy. The reinforcing steel of most of these walls is commonly arranged in a single layer, without any effective confinement. In cases where boundary elements are included, adequate confinement is not guaranteed due to their reduced thickness. Software tools allow verifying the behavior of these buildings under representative seismic events. This article studies the behavior of thin RC walls using the Non-linear Beam-Truss Model (NL-BTM). The commercial software Perform-3D, widely used in professional practice, has been selected for this research. The calibration of the modeling parameters is carried out using different types of walls tested experimentally. The results are then extrapolated for modeling representative buildings in Bogotá. The geometric and mechanical characteristics of the walls correspond to those reported in actual tests, and those of the building are the characteristics indicated in representative building plans. Numerical modeling is performed through non-linear static pushover analysis and non-linear dynamic time-history analysis. The study proposes recommendations and limitations in using the NL-BTM model for modeling thin RC walls in actual buildings.

Seismic performance analysis of steel frames with FCP composite external wall

Fiber cement panel (FCP) composite external wall is a new type of exterior wall, to study its effect on the seismic performance of the steel frame, the hysteretic loading test was carried out and the numerical simulation method of FCP composite external wall was proposed. The feasibility and accuracy of the simulation method were verified by comparing it with experimental data. A three-bay, six-story steel frame model was established, and the nonlinear dynamic time-history analysis of the frame with and without FCP composite external wall was conducted respectively. The analysis results showed that the frame structure with FCP composite external wall has good seismic performance under the rare earthquake, and the FCP composite external wall can effectively improve the stiffness of the structure and reduce the top-story displacement, base shear force, and inter-story displacement angle of the steel frame under earthquakes.

Seismic Performance of RC Split-Foundation Frame Structures with Steel Braces in the Mountainous Area

Reinforced concrete (RC) split-foundation frame structures are widely used in mountainous areas. There are two grounding terminals with different heights in the split-foundation frame structures. Under the earthquake, the local damage phenomenon of the structures is severe due to the uneven stiffness distribution. It is one way to improve the seismic performance of the split-foundation frame structures to install steel braces in the upper embedding floor. According to the deformation characteristics of the upper embedding floor, three types of steel braces arrangement for the split-foundation frame structures were proposed: horizontal braces, V-type braces, and V-[Formula: see text] braces. Then based on the ratio of lateral stiffness of steel braces and concrete frame columns, the section of steel brace was designed. A quasi-static test was carried out to analyze the influence of steel braces on the failure mode of the RC split-foundation frame. The reasonable seismic waves were selected to carry out the nonlinear dynamic time history analysis. Then the seismic response and failure mode of the split-foundation frame structures with different braces arrangements were compared. On this basis, fragility analysis was carried out, and the earthquake-induced collapse resistance capacity of split-foundation frame structures with steel braces was preliminarily evaluated. The results show that the horizontal steel brace can improve the phenomenon of damage concentration of the split-foundation frame structures. Furthermore, setting V-shaped braces can significantly reduce the damage degree of the upper embedding end columns and change the phenomenon of locally severe damage of the split-foundation frame structures. Moreover, according to the design method proposed in this paper, setting steel braces can improve the seismic collapse resistance capacity of the split-foundation frame structures.

Simple approach to evaluate the influence of seismic residual displacements on post-liquefaction settlements of RC-frames

Residual permanent displacements generally characterize the status of structures at the end of severe earthquake events because of large inelastic deformations demand. When the structure is founded on soils susceptible to liquefaction phenomenon, this status could particularly influence the subsequent damages due to the occurrence of liquefaction induced vertical settlements. As a consequence, the correct prediction of the effects of earthquake induced liquefaction on the built environment must consider the residual displacements induced by the ground shaking. To this aim, the paper proposes a simplified approach to evaluate the seismic response of RC frames based on a four-steps procedure composed of a set of nonlinear static analyses reproducing a sequential process of a ground shaking followed by a liquefaction induced ground settlements. The novelty of the proposal is to carry out sequential nonlinear static analyses by preserving the status of the RC frame at the end of the seismic event in terms of residual displacements and then to directly consider their influence on the subsequent analysis simulating the occurrence of post-liquefaction vertical settlements.A parametric study has been developed by means of numerical analyses, referring to two RC frames derived from literature, in order to highlight the relevance of the seismic residual displacements on the damage induced by the subsequent vertical settlements due to soil liquefaction. The proposed simplified approach has then been validated through more sophisticated nonlinear dynamic time-history analyses. The results presented in the paper underline the efficacy of the proposed approach, the simplicity in implementing the four-steps procedure and the lower computational effort with respect to nonlinear dynamic time-history analyses.