Fundamental Period Of Vibration Research Articles

Dynamic Response of Symmetric and Asymmetric Setback Buildings

Numerical studies have been carried out to study the dynamic behaviour of a five-storey regular building, symmetric setback and asymmetric setback buildings having three bays along longitudinal direction and one bay along transverse direction. The objective of the study is to compare the irregularity indices with respect to different codal provisions such as IS:1893-2016, Euro Code 8-2004 and ASCE 7.05-2005 as well as with the expressions proposed by Karavasilis et al. (2008), and Pradip sarkar et al.(2010). These buildings are subjected to seismic excitations with time history analysis and the response parameters such as fundamental period of vibration of the structures, displacements and storey drifts were evaluated.

Effect of Earthquake Ground Motion Duration on the Seismic Response of a Low‐Rise RC Building

This paper investigates the effect of earthquake ground motion duration on the seismic response of a low‐rise reinforced concrete shear wall building. Two sets of spectrally equivalent ground motion sets were determined to isolate the effect of duration from other earthquake record characteristics. A numerical model that accounts for P‐delta effects and degradation of strength and stiffness of the structural elements was used. Detailed nonlinear dynamic analysis for both the design and collapse levels of shaking was performed, considering the spectral acceleration at the fundamental period of vibration with intensity measure and material strains as engineering demand parameters. The results showed that at the design level of shaking, slightly larger interstory drifts were obtained under the short‐duration events. However, the maximum values for interstory drifts were small, and minor damage is expected in the structure. When both seismic record sets were incrementally scaled until collapse, a slight increase in the material strains was found under the short‐duration seismic events. Overall, it is indicated that ground motion duration does not influence the seismic response of low‐rise buildings with low deformation capacity.

An Investigation of the Fundamental Period of Vibration for Moment Resisting Concrete Frames

The determination of fundamental period of vibration for structures is essential to earthquake design. The current codes provide empirical formulas to estimate the approximated fundamental period and these formulas are dependent on building material, height of structure or number of stories. Such a formulation is excessively conservative and unable to account for other parameters such as: length to width ratios, vertical element size and floors area. This study investigated the fundamental periods of mid-rise reinforced concrete moment resisting frames. A total of 13 moment resisting frames were analyzed by ETABS 15.2.2, for gross and cracked eigenvalue analysis and Extreme Loading for Structures Software® or ELS, for non-linear dynamic analysis. The estimated periods of vibration were compared with empirical equations, including current code equations. As expected, the results show that building periods estimated based on simple equations provided by earthquake design codes in Europe (EC8) and America (UBC97 and ASCE 7-10) are significantly smaller than the periods computed using nonlinear dynamic analysis. Based on the results obtained from the analyzed models, equations for calculating period of vibration are proposed. These proposed equations will allow design engineers to quickly and accurately estimate the fundamental period of moment resisting frames with taking different length to width ratios, vertical element size, floors area and building height into account. The interaction between reduction factor and the reduced period of vibration is studied, and it is found that values of maximum period of vibration can be used as an alternative method to calculate the inelastic base shear value without taking reduction factors in consideration.

Seismic performance of single layer steel cylindrical lattice shells

This paper examines the elastic and inelastic seismic behaviour of single layer steel cylindrical lattice shells. The main dynamic characteristics for this form of structure are firstly examined through a parametric assessment, which also leads to proposed expressions for estimating the fundamental period and mode of vibration. The seismic response of five typical shell configurations, representing a wide range of rise to span ratios, is then assessed within the linear elastic range under selected earthquake excitations. Particular focus is given to the relative influence of the horizontal and vertical seismic components on the internal actions. In order to provide a means for evaluating the underlying inelastic behaviour, a simple pushover approach, which is suitable for this structural form, is suggested using the forces obtained from the fundamental mode shape. The peak angle change is proposed as a damage parameter within the nonlinear analysis for characterising the inelastic global and local demands in shells of different geometries. Incremental dynamic analysis is subsequently carried out in order to evaluate the detailed nonlinear time history response. The results provide detailed insights into the influence of the horizontal and vertical excitations on the nonlinear seismic response, and illustrate the suitability of the peak angle change as an inelastic deformation measure for shells of different geometric configurations. The main findings from the linear and nonlinear assessments are highlighted within the discussions, with a view to providing guidance for performance based assessment procedures as well as simplified design approaches.

Investigations of elastic vibration periods of tall reinforced concrete office buildings

The assessment of wind-induced vibration for tall reinforced concrete (RC) buildings requires the accurate estimation of their dynamic properties, e.g., the fundamental vibration periods and damping ratios. In this study, RC frame-shear wall systems designed under gravity and wind loadings have been evaluated by utilising 3D FE modelling incorporating eigen-analysis to obtain the elastic periods of vibration. The conducted parameters consist of the number of storeys, the plan aspect ratio (AR) of buildings, the core dimensions, the space efficiency (SE), and the leasing depth (LD) between the internal central core and outer frames. This analysis provides a reliable basis for further investigating the effects of these parameters and establishing new formulas for predicting the fundamental vibration periods by using regression analyses on the obtained results. The proposed constrained numerically based formula for vibration periods of tall RC frame-shear wall office buildings in terms of the height of buildings reasonably agrees with some cited formulas for vibration period from design codes and standards. However, the same proposed formula has a high discrepancy with other cited formulas from the rest of design codes and standards. Also, the proposed formula agrees well with some cited experimentally based formulas.

Seismic Response Modification Factorof Reinforced Concrete Frames Basedon Pushover Analysis

Response modification factor is an essential factor in seismic analysis to provide economic design of reinforced concrete structures. Base shear force is divided by the response modification factor to consider the ability of the structure to dissipate energy through plastic hinges. The current study investigates the effects of changing some parameters on response modification factor (R-factor). Four groups of reinforced concrete frames were studied with different number of bays, number of stories, load pattern, and fundamental period of vibration. All reinforced concrete frames were analyzed using SAP 2000 then the straining actions results were used at specific excel sheets which are developed to design reinforced concrete members according to the Egyptian code of practice ECP-203 and ECP-201. Frames were analyzed by nonlinear static analysis (pushover analysis) using SAP2000. A sum of thirty two systems of frames was analyzed. According to the results, every frame has its unique value of R-factor. Accordingly, many parameters should be mentioned and considered at code to simulate the actual value of R-factor for each frame. Response modification factor is affected by many factors like stiffness, fundamental period of vibration, number of bays, frame height, geometry of the structure, etc. The given values of R-factor at ECP-201 can be considered conservative; as the accurate values of R-factor is higher than the given values.

Stochastic response analysis of elastic and inelastic systems with uncertain parameters under random impulse loading

Response analysis of structural systems under impulse loading is extremely important for appropriate design against accidental loads such as blast. Herein, the response of elastic and inelastic single degree of freedom (SDOF) systems with uncertain parameters under random impulse loadings is investigated. Four different types of impulse loading profiles (rectangular-, half-sine wave-, and two triangular-shaped), having same duration and impulse, are applied to the SDOF systems with varying fundamental periods of vibration. Non-sampling stochastic simulation procedure based on the generalized polynomial chaos (gPC) expansion technique is used to model the dynamic response of the SDOF systems duly considering the uncertainties. Effects of uncertainty levels in the input parameters on the peak response and sensitivity of the peak response to the uncertain input parameters are investigated in detail. It is concluded that the propagation of uncertainties from the inputs to the response quantities is more prominent in stiffer systems. Amongst the impulse profiles considered, those pulses with sudden rise in force lead to higher deviations in response of the systems for a given uncertainty scenario. Further, it is observed that the shock response spectra of all the impulse loads are more sensitive to the uncertainties when time periods are less than or close to the loading duration. Furthermore, the gPC expansion-based simulation technique is observed to be an efficient alternative to computationally demanding conventional Monte Carlo (MC) simulation for quantifying the uncertainties.

Structural health monitoring of the Ferrara University before and after the 2012 Emilia (Italy) earthquake, and after the damage repairs

In this article, a framework of building monitoring is developed and transient and permanent variations of the fundamental period of vibration caused by both damage and repair interventions are investigated. The buildings of the University of Ferrara (Emilia-Romagna Region, Northern Italy), struck by the 2012 Emilia seismic sequence, were monitored using both temporary and permanent equipment: the first one to perform ambient vibration tests and the second one to implement permanent real-time monitoring for earthquake recording. Three on-demand ambient vibration tests were performed at each floor of the buildings: the first dataset was acquired a few months before the mainshock occurred on 20 May 2012 (ML = 5.9, 6.8 km depth and 30 km epicentral distance); the second was acquired right after the end of the sequence, when the building showed slight damage (degree 1 according to the European Macroseismic Scale 98); finally, the third dataset was acquired in 2016 after the repair intervention. The data analysis clearly documented the permanent drop of the first vibration frequency as a symptom of the damage and its partial recovery that followed the repairs. The permanent real-time monitoring system, despite the fact that it was implemented using low-cost sensors, provided an insight into the intra-event frequency variation, allowing in turn a preliminary damage assessment.

Vibration of rectangular plates: fundamental mode and integer multiple of the fundamental period of vibration

This paper is concerned with a vibration analysis of the transverse vibrations of rectangular plates with freely-supported boundaries and corners fixed, in an imagistic way, to describe the deformation state for the specific natural frequency and mode shape for each degree-of-freedom. Two materials, i.e. aluminum and steel, showing the homogeneous properties in the direction of plate thickness were chosen. The exact solutions for free vibrations of the homogeneous rectangular plate were obtained and the fundamental mode and integer multiple of the fundamental period of vibration were provided. The comparisons show that the analytical model predicts natural frequencies reasonably well for a rectangular plate.

EVALUATION OF THE DYNAMIC CHARACTERISTICS FOR SEISMIC DESIGN OF MULTI-STORY BUILDINGS

The fundamental period of vibration is a critical structural dynamic characteristic in seismic design. Several expressions for the calculation of the fundamental period have been recommended by different building codes and previous studies. However, further studies are still needed to evaluate the design expressions used for the calculation of the fundamental periods and assess the need for further refinement. In this study, comprehensive fundamental period data from two sources is collected and compared with different formulas from building codes and previous studies. The first data set is obtained from 147 instrumented buildings with various lateral force resisting systems (LFRSs). The second set of period data are collected from the dynamic response simulations of selected structures. Different LFRSs are considered, including steel moment resisting frames (SMRFs), reinforced concrete moment resisting frames (RCMRFs), reinforced concrete shear walls (RCSWs), concentrically braced frames (CBFs), eccentrically braced frames (EBFs), masonry structures and pre-cast structures. The correlations between the derived period expressions with those recommended by the design provisions show that the code approach is conservative enough for SMRFs, CBFs, masonry buildings and pre-cast structures. For RCMRFs, EBFs and RCSWs, the design code is slightly unconservative for low-rise buildings. The outcomes of the study help to arrive at more efficient and cost-effective seismic design of buildings with different characteristics.

Evaluation of the Scaling Factor Bias Influence on the Probability of Collapse Using Sa(T1) as the Intensity Measure

Amplitude scaling is a common approach to modify recorded ground motions to achieve a desired intensity level. The possible bias introduced by scaling the amplitude of ground motions when using the first-mode spectral ordinate as the intensity measure is evaluated using intensity-based analyses. This study evaluates whether upward scaling introduces bias in lateral displacement demands, but more importantly, in the probability of collapse. The latter, which is of utmost importance, has received little attention in previous studies. Analyses were conducted using degrading single-degree-of-freedom and multiple-degree-offreedom systems with different fundamental periods of vibration and normalized strengths subjected to different sets of recorded accelerograms requiring different scale factors to reach a target intensity. The results demonstrate that this type of amplitude scaling introduces a bias in which lateral displacement demands and collapse estimates are increasingly overestimated with an increasing scale factor and that the bias is strongly dependent on the period and lateral strength of the system. Furthermore, the bias is considerably larger in collapse risk estimates.

Effects of Sediment Thickness upon Seismic Amplification in the Urban Area of Chiapa de Corzo, Chiapas, Mexico

Local variations in soil type and (to a lesser extent) topography determine seismic amplification, thereby establishing spatial patterns in the damages caused by large earthquakes, such as the two that occurred in Chiapas, Mexico on 6 October 1975 and 7 September 2017. The latter, of Mw 8.2, was the region’s strongest during the last 100 years. Based upon damage patterns, an analysis was made to identify the effects of sediment thickness and type upon seismic amplification in the urban zone of Chiapa de Corzo, Chiapas. Salgado et al., (2004) had shown that the fundamental periods of soil vibration in the city vary from0.14 s to0.39 s. From those periods, and using the model presented in Newmark and Rosenbleuth (1976), with an average shear-wave velocity of 150 m/s, the variation of the sediment thicknesses that cause site effects in the area was determined. The thicknesses ranged between 7.12 and 14.6 m. Finally, through the use of a map prepared with GIS software, a spatial correlation was made with the historical earthquake damage recorded within the city.

Influence of slab spatial composite effect on dynamic behaviour of composite frame structures under earthquake excitation

The slab spatial composite effect plays a significant role in the seismic behaviour of composite frame structures. In this work, the influence of the slab spatial composite effect on the dynamic behaviour of composite frame structures was numerically studied. The self-developed fibre beam–column element program COMPONA-MARC was used to establish models considering and neglecting the slab spatial composite effect for nonlinear static analysis and elastic–plastic time history analysis under various seismic inputs. The maximum roof displacement, inter-storey drift ratio, normalised displacement indices, deformation mechanism, and failure mode were considered to investigate the structural performance. Comparisons indicated that, despite the contributions of the slab spatial composite effect to the structural lateral strength and stiffness, considering this effect may produce more unsafe results. On one hand, it reduces the fundamental natural vibration period and amplifies the effect of seismic excitation; on the other hand, it changes the structural deformation mechanism and failure mode. The influence of slab spatial composite effect on strong column–weak beam mechanism in different storey drift direction angles is also investigated in the paper.

Fundamental period of vibrations influencing characteristics of torsional irregularity in reinforced concrete buildings

This paper presents an analysis of irregular reinforced concrete buildings with respect to fundamental period of vibrations. During earthquake, reinforced concrete building with asymmetric building system experience extensive damage due to this torsional irregularity. Torsional irregularity of reinforced concrete buildings has been analysed with taking the strength and stiffness eccentricities as the main parameters of the rectangular in-plan building system. Displacement demand in reinforced concrete building has been determined in terms of fundamental period of vibrations and the behaviour of the building system was presented using the normalised displacement of buildings. The hypothetical model has been analysed has been set up using varies values of fundamental period of vibrations using Ruaumoko program and the lateral displacement of the building were extracted using Fortran program. The results indicated through the damage index showed that the fundamental period of vibration plays an important role in earthquake studies on reinforced concrete building since earthquake induced excessive displacement to the building system under the increment of the fundamental period of vibrations.

Empirical Period-Height Relationship for reinforced Concrete Moment Resisting Buildings in India

The fundamental period of vibration of buildings is the unique parameter that has a major influence on the earthquake induced lateral forces. Many design codes provide simple empirical period -height relationships for the determination of the fundamental period of vibration but using these equations time period and the resulted base shear are conservatively estimated. In most design cases the results obtained using conservative estimates formulae are largely deviating from the required engineering accuracy. Therefore a relationship is sought that will give accurate prediction of period of vibration for structural assessment of Indian reinforced concrete moment resisting frame buildings. On the basis of fundamental period of vibration of an Indian reinforced concrete moment resisting frame (RC MRF) building, obtained by finite element analysis, new equation has been proposed. In this study 21 existing reinforced concrete moment resisting frame buildings in India, with varying length, width and height are considered. The fundamental time periods of all the selected buildings were calculated using finite element method and IS 1893(Part 1) : 2002 based empirical formula. The comparison of proposed equation with Indian code shows that finite element analysis yielded 100 % higher periods as compared to the code approach.

Artificial bee colony-based neural network for the prediction of the fundamental period of infilled frame structures

The artificial bee colony (ABC) algorithm is a recently introduced swarm intelligence algorithm for optimization, which has already been successfully applied for the training of artificial neural network (ANN) models. This paper thoroughly explores the performance of the ABC algorithm for optimizing the connection weights of feed-forward (FF) neural network models, aiming to accurately determine one of the most critical parameters in reinforced concrete structures, namely the fundamental period of vibration. Specifically, this study focuses on the determination of the vibration period of reinforced concrete infilled framed structures, which is essential to earthquake design, using feed-forward ANNs. To this end, the number of storeys, the number of spans, the span length, the infill wall panel stiffness, and the percentage of openings within the infill panel are selected as input parameters, while the value of vibration period is the output parameter. The accuracy of the FF–ABC model is verified through comparison with available formulas in the literature. The results indicate that the artificial neural network, the weights of which had been optimized via the ABC algorithm, exhibits greater ability, flexibility and accuracy in comparison with statistical models.

Seismic Evaluation of Steel Moment Resisting Frames (MRFs)—Supported by Loose Granular Soil

Soil underneath a structure might affect the behavior and the overall response of the structure in seismic events. The role of loose soil conditions and the inclusion of soil-structure interaction (SSI) in the analysis are important issues that need to be addressed. Since steel structures are light, two configurations designed as spatial and perimeter are considered to study the effect of soil on the steel structural frames for the same building. The paper provides a parametric analysis on the influence of SSI on the overall performance of MRFs (Moment Resisting Frames) according to the provisions of Saudi Building Code (SBC) [1]. A case study has been developed in which spatial and perimeter moment resisting frames of 12, 6 and 3 stories residential buildings are designed using Saudi Building Code (SBC) prescriptions. A modal response spectrum analysis has been carried out to see the influence of SSI on the fundamental period of vibration, top story displacement and inter-story drift limitations. Moreover, a static non-linear analysis has been performed to investigate the performance of frames, thus allowing to identify the influence of SSI on the structural design of steel MRFs.

Fundamental vibration frequency prediction of historical masonry bridges

It is very common to find an empirical formulation in an earthquake design code to calculate fundamental vibration period of a structural system. Fundamental vibration period or frequency is a key parameter to provide adequate information pertinent to dynamic characteristics and performance assessment of a structure. This parameter enables to assess seismic demand of a structure. It is possible to find an empirical formulation related to reinforced concrete structures, masonry towers and slender masonry structures. Calculated natural vibration frequencies suggested by empirical formulation in the literatures has not suits in a high accuracy to the case of rest of the historical masonry bridges due to different construction techniques and wide variety of material properties. For the listed reasons, estimation of fundamental frequency gets harder. This paper aims to present an empirical formulation through Mean Square Error study to find ambient vibration frequency of historical masonry bridges by using a non-linear regression model. For this purpose, a series of data collected from literature especially focused on the finite element models of historical masonry bridges modelled in a full scale to get first global natural frequency, unit weight and elasticity modulus of used dominant material based on homogenization approach, length, height and width of the masonry bridge and main span length were considered to predict natural vibration frequency. An empirical formulation is proposed with 81% accuracy. Also, this study draw attention that this accuracy decreases to 35%, if the modulus of elasticity and unit weight are ignored.

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.

Concurrent effects of inertial and kinematic soil-structure interactions on strength-ductility-period relationship

As a key stage of performance-based seismic design of structures, strength-ductility-period relationship is used to determine either inelastic strength or ductility ratio of the structure with a specific fundamental vibration period. On the other hand, period, strength, and ductility demands of structures are significantly affected by inertial and kinematic soil-structure interactions. Hence, this study addresses strength-ductility-period relationship of structures with embedded foundations subjected to non-pulse earthquakes. To achieve this goal, a soil-foundation-structure interaction system well suited for parametric study was used: the soil beneath the rigid foundation was considered as a half-space, the foundation was modelled using a stack of discretised embedded disks, and the structure was modelled as an equivalent single-degree-of-freedom oscillator. A number of soil-foundation-structure interaction systems defined according to key interaction dimensionless parameters were analysed subjected to a suit of 40 non-pulse ground motions. The results show that ductility ratio spectra are significantly influenced by inertial and kinematic interaction effects. It is also found that equal displacement rule can be either valid or invalid under different soil-foundation-structure interaction effects. Further, a practice-oriented mathematical expression was recommended for ductility ratio of soil-foundation-structure interaction systems, which can find use in performance evaluation of existing structures with embedded foundation.