Equivalent Linearization Procedure Research Articles

A critical assessment of the design displacements of friction pendulum systems

The equivalent linear static procedure suggested in seismic codes for determining the maximum displacements of seismic isolation systems is evaluated comprehensively for friction pendulum isolators with respect to the benchmark nonlinear response history analysis procedure for a set of radii of curvature and coefficients of friction. A large set of selected and scaled near field ground motions recorded on stiff and soft soils are employed in comparative evaluations. It has been observed that equivalent linear static procedure significantly overestimates the maximum isolator displacements for all practical combinations of radius of curvature and friction coefficient. Overestimation is larger for ground motions on soft soils. Alternatively, a practical description of inelastic displacement spectra is introduced where the hazard is defined in terms of the 1-second spectral acceleration accounting for the seismicity and soil class of the site, and friction pendulum isolation systems are characterized by their radius of curvature and nominal friction coefficient. Employing the proposed design spectra in design and production may lead to significant reductions of the isolator sizes. Moreover, realistic estimation of maximum isolator displacements also leads to realistic estimation of isolator stiffness, as well as the internal design forces of both the isolation system and the superstructure through response spectrum analysis under design spectrum.

The Use of SMAV Model for Computing Fragility Curves

Among the buildings with strategic role for civil protection purposes part of them were built before the introduction of modern seismic codes. Aiming to guarantee their operability in case of seismic events, they have to be assessed and, hopefully, seismically upgraded. This aspect arises the necessity of reliable low computational demanding numerical models and probabilistic strategies that can be easily adopted in case of the assessment of conspicuous building stock. Part of the authors proposed and validated, in previous works, a low-demanding seismic model calibrated on ambient vibrations (SMAV) that is engaged in this paper for calculating fragility curves. On the base of dynamic and geometric properties, the SMAV model simulates the dynamic response of buildings. The herein proposed procedure adopts the SMAV model and takes into account the direction of seismic input in a probabilistic manner. The use of the ASI scale as an Intensity Measure parameter is discussed and the results are compared to those obtained with others IM scales. Although SMAV implements an equivalent linear procedure the results are compared with those obtained by more sophisticated FEM models and nonlinear dynamic analyses with a satisfactory agreement. Finally, two real case studies are presented. The fragility curves for both the case studies are calculated and discussed. The results encourage the authors to the use of fragility curves based on SMAV model. Additionally, the fragility curves can be easily implemented in seismic risk assessment procedures at urban or territorial scale and, consequently, the proposed procedure may represent a key aspect in the reduction of seismic risk and rationalization of investments for the seismic upgrading of strategic buildings.

Machine learning-based prediction for maximum displacement of seismic isolation systems

This study developed machine learning (ML) models for predicting the peak lateral displacements of seismic isolation systems subjected to earthquakes. Six ML methods were employed for this purpose: linear, ridge, least absolute shrinkage and selection operator (lasso), artificial neural network (ANN), support vector machine, and random forest (RF). The isolated superstructure was assumed to be rigid, and the base isolation system was assumed to be firmly anchored to a bedrock foundation. A total of 234,360 data points were generated using OpenSees to train and test the predictive models. The input variables included three parameters representing the behavior of isolation systems (i.e., the ratio of the initial to post-yield stiffness, normalized characteristic strength, and post-yield period) and the average spectral accelerations at five selected periods (i.e., 1, 2, 3, 4, and 5 s) for the considered ground motions. The RF model exhibited the best performance among the six models. The coefficients of determination (R2) for the RF model were 0.9930 and 0.9498 for the training and testing datasets, respectively. In addition, the ANN model exhibited the second-best performance. The normalized characteristic strength was found to be the most influential variable in the investigation of the individual input variables for the prediction. Finally, a practical analysis of a hypothetical three-story base-isolated moment frame building was performed to demonstrate the effectiveness and limitations of the developed RF model in predicting the maximum displacements of the isolation systems. The prediction results were compared with those from a previous study and with the equivalent linear force procedure in American Society of Civil Engineers (ASCE) 7–16. A graphical user interface was created for the developed RF model for easy access by engineers. This study represents a pioneering step for the application of ML to the estimation of the responses of seismic isolation systems in practical design.

Performance Based Seismic Assessment of Masonry Infilled RCC Building with Diaphragm Discontinuity

In multistoreyed RCC framed buildings, critical damages are due to seismic ground excitations, which cause catastrophic failuresat the weaker locations. Buildings with two types of structural irregularities namely diaphragm discontinuity and open ground story are considered. Assessment of seismic vulnerability of these buildings is done by using Nonlinear Static Pushover Analysis (NSPA). Performance Based Seismic Design of masonry infilled RCC buildings with two different shape of openings in the diaphragm is considered here with Design Basis Earthquake(DBE) and Maximum Considered Earthquake(MCE) where by selecting appropriate performance criteria in terms of Inter-story drift ratio(IDR) and Inelastic displacement demand ratio(IDDR) are critically observed. The Equivalent Linearization Procedure of Pushover analysis presented in FEMA 440, which is a modification of Capacity Spectrum Method based on ATC-40 guidelines, is performed in ETABS-2016 to study the performance of R.C.C. buildings with diaphragm discontinuity, designed as per IS-1893-2016.

Prediction of earthquake-induced permanent deformations for concrete-faced rockfill dams

Prior studies highlight the importance of earthquake-induced permanent displacement for the safety assessment of the embankment dams. Concrete-faced rockfill dams (CFRDs) have become popular at especially seismically active regions with their high seismic energy absorption capacity. However, seriously damaged rockfill dams have been reported after some earthquake exposures. Even so researchers have focused on the dynamic response of embankments, few studies have been conducted to reveal the seismic behavior of rockfill dams. In this study, the issue under scrutiny is to establish a preliminary design procedure considering the earthquake-induced permanent displacement and acceleration response of CFRDs. For this purpose, numerical models are prepared with various geometric and material properties. Real earthquake records are used for seismic excitation. Dynamic analyses are conducted with a finite element method-based software. The equivalent linear analysis procedure is followed under two-dimensional plain-strain condition. Acceleration responses of dams are recorded at the centerline of the models. Then, the Newmark’s sliding block approach is utilized to calculate the permanent displacements from acceleration records. A procedure is developed for the engineers to foresee the likely permanent displacement at the preliminary design stage. To prove the reliability, the recommended procedure is applied to case histories, and then the results are compared.

A preliminary proposal towards the revision of Indian seismic code considering site classification scheme, amplification factors and response spectra

This paper aims to improve the site classification scheme and amplification factors for the Indian Seismic Code. The present code IS 1893 (1):2016 ignores the amplification of effective peak ground acceleration due to the existence of soft soil layers. This results in lower design spectral accelerations on soil sites compared to those adopted by codes in other nations. 124 well documented sites from India are considered for ground response analysis using the equivalent linear procedure of DEEPSOIL. Time histories recorded from India and abroad are made spectrum compatible with the Type I spectra of Indian code, corresponding to all the seismic intensities with Effective Peak Ground Acceleration of 0.1 g, 0.16 g, 0.24 g and 0.36 g. Statistical methods are used to construct mean, mean plus standard deviation and mean minus standard deviation response spectra and amplification factors for the sites considered. Before the statistical analysis, each amplification factor curve is normalized with the effective site period (Tg) which exhibits a “characteristic peak at values of T/Tg not far from 1”. Normalized acceleration response spectra as well as period dependent soil amplification factors and equations are proposed for the new classification system. It is observed that the stratigraphy, impedance contrast, sediment thickness and site period play an important role in determining the soil amplification factors and therefore, should be considered in the future site classification schemes of the Indian Seismic Code.

A new statistical equation for predicting nonlinear time history displacement of seismic isolation systems

This study develops a statistical relationship between the peak displacement of seismic isolation systems obtained from nonlinear time-history dynamic (NTH) analysis and the peak displacement predicted from an equivalent linear force procedure (ELF). Firstly, the effect of isolators’ mechanical property and excitation type and amplitude on the ratio between the peak displacement of an isolation system calculated from ELF procedure and the peak displacement determined from NTH are evaluated. The ELF procedure in ASCE/SEI 7–16 is employed to estimate the ELF peak displacement. The input ground motions for bidirectional NTH analysis are scaled following this code. The investigation of 540 numerical models subjected to four levels of excitation amplitude, represented by 50 pairs of ground motion, including near-field and far-field motions, reveals that both mechanical property of an isolation system and excitation type and amplitude affect the displacement ratio, where the post-yield stiffness of the isolation system and ground motion type have stronger influence than other parameters. These influences are then statistically processed. Based on the investigated data, a novel practical equation accounting for these parameters is proposed to predict the displacement ratio at different reliability levels.

An iterative linear procedure using frequency-dependent soil parameters for site response analyses

Local site amplification predictions that derive from the traditional equivalent linear (EL) procedure are deemed to be reliable when weak shaking is considered, while it is well recognized that significant discrepancies can occur for stronger earthquakes where soil nonlinearity dominates the site response. To better capture nonlinearity effects, frequency-dependent equivalent linear methods, commonly referred to as FDEL methods, were proposed by modifying the EL procedure with frequency-dependent soil parameters. However, these methods have never been widely used in earthquake engineering practice. Existing FDEL methods rely on generic spectral shape fitting parameters to approximate the variation of shear strain amplitudes in the frequency domain, which can lead to substantial inconsistencies in the model prediction when applied to a particular seismic event, with a specific signature. This study investigates a range of spectral seismic intensities to identify a proxy that could be used directly to update the shear strain spectra in accordance with the simulated ground motions. A new FDEL algorithm to compute effective shear strain as a function of frequency is proposed and tested on various scenarios. When simulating strong ground motions that exhibit higher levels of soil deformation, the frequency content of ground motions processed within the FDEL procedure is substantially improved as compared to the EL method, with results more consistent with those from NL analyses.

Evaluation of Seismic Demand for Substandard Reinforced Concrete Structures

Background: Reinforced Concrete (RC) buildings with no seismic design exhibit degrading behaviour under severe seismic loading due to non-ductile brittle failure modes. The seismic performance of such substandard structures can be predicted using existing capacity demand diagram methods through the idealization of the non-linear capacity curve of the degrading system, and its comparison with a reduced earthquake demand spectrum. Objective: Modern non-linear static methods for derivation of capacity curves incorporate idealization assumptions that are too simplistic and do not apply for sub-standard buildings. The conventional idealisation procedures cannot maintain the true strength degradation behaviour of such structures in the post-peak part, and thus may lead to significant errors in seismic performance prediction especially in the cases of brittle failure modes dominating the response. Method: In order to increase the accuracy of the prediction, an alternative idealisation procedure using equivalent elastic perfectly plastic systems is proposed herein that can be used in conjunction with any capacity demand diagram method. Results: Moreover, the performance of this improved equivalent linearization procedure in predicting the response of an RC frame is assessed herein. Conclusion: This improved idealization procedure has been proven to reduce the error in the seismic performance prediction as compared to seismic shaking table test results [1] and will be further investigated probabilistically herein.

An equivalent linear procedure for probabilistic displacement-based design of RC structures under earthquake

Starting from an already existing gradient-based method for the performance-based seismic design (PBSD) of reinforced concrete (RC) structures, in which performance objectives are formulated in terms of maximum accepted mean annual frequency (MAF) of exceedance for multiple limit states, some improvements are presented. The method allows for the computation of the values of selected independent design variables (beams and column cross-sections and reinforcement) leading to the attainment or non-exceedance of pre-defined MAFs for multiple limit states. General geometric, as well as capacity design constraints are duly accounted for. The procedure is applied to a 15-storey plane frame building and validation is conducted against results in terms of drift profiles and MAF of exceedance, obtained by multiple stripe analysis with records selected to match conditional spectra. Results show that the method is suitable for PBSD of RC structures with explicit targets in terms of desired risk levels.

The near-field method: a modified equivalent linear method for dynamic soil–structure interaction analysis. Part I: Theory and methodology

This paper presents a new direct modeling approach to analyze 3D dynamic SSI systems including building structures resting on shallow spread foundations. The direct method consists of modeling the superstructure and the underlying soil domain. Using a reduced shear modulus and an increased damping ratio resulted from an equivalent linear free-field analysis is a traditional approach for simulating behavior of the soil medium. However, this method is not accurate enough in the vicinity of foundation, or the near-field domain, where the soil experiences large strains and the behavior is highly nonlinear. This research proposes new modulus degradation and damping augmentation curves for using in the near-field zone in order to obtain more accurate results with the equivalent linear method. The mentioned values are presented as functions of dimensionless parameters controlling nonlinear behavior in the near-field zone. This paper summarizes the semi-analytical methodology of the proposed modified equivalent linear procedure. The numerical implementation and examples are given in a companion paper.

Investigation on the accuracy of the N2 method and the equivalent linearization procedure for different hysteretic models

In this paper, an extensive parametric study was carried out to evaluate the dynamic response of single degree of freedom (SDOF) systems with elasto-plastic and flag-shape hysteretic behaviour for three different dissipation capacities. Three sets of natural accelerograms were used, each one composed by at least 7 records which are on average spectrum-consistent in pseudo-acceleration, spectral displacement or both of them. All sets were also employed to draw the inelastic spectra for different ductility values. Such rigorous spectra were then compared with the approximated curves calculated using the N2 method and the Equivalent Linearization Procedure (ELP) based on the use of overdamped elastic spectra.The analyses demonstrate a general accuracy of the N2 method, which is mostly based on the ductility of the system, even for the hysteretic behaviour characterised by reduced energy dissipation. Larger discrepancies were found for systems with lower damping ratios and reduced fundamental periods. The ELP, which mainly depends upon the dissipated energy, led instead to overall slightly larger discrepancies than the N2 method, particularly for not dissipative and ductile systems, whereas the approximation is generally acceptable for elasto-plastic systems.

Seismic response analysis of Ganga sand deposits using shake table tests

An accurate estimation of seismic behavior of a soil deposit is important from various aspects, namely, predicting the influence of the deposit in altering the earthquake waves propagating through that medium, quantifying the amplification or attenuation of the seismic shaking intensities, development of site-specific design spectra for seismic design of infrastructure to be built on that deposit, evaluating the possibility of ground failure and so on. The present study focuses on dynamic response analysis of local Ganga sand through performing a number of shake table experiments using a flexible laminar container. Uniform, dry Ganga sand of 30 % relative density exhibited acceleration amplification of 20, 40 and 68 % for input motions with 1, 2 and 5 Hz frequencies with amplitudes 0.2, 0.36 and 0.56 g, respectively. Further, analytical investigation on seismic response of the same soil deposit has been carried by assuming the behavior of the soil deposit as: (1) linear and (2) equivalent linear. The acceleration amplifications are estimated as 1, 4 and 28 % using the linear analysis and 3, 30 and 72 % using the equivalent linear analysis for the 1, 2 and 5 Hz motion, respectively. Compared to the linear analysis, the equivalent linear analysis predicts the surface amplification of the experiment with better accuracy (a maximum deviation being less than 6.7 %). This may due to the strain dependent behavior of the soil bed during strong shakings, which is compatible with the assumptions of the equivalent linear analysis procedure.

Seismic response and permanent displacement of landfills with liner interfaces and various foundation types

The slip displacement along liner systems and the seismic responses of geo-structures significantly impact the seismic design of municipal solid waste (MSW) landfills. The influences of various factors, such as local underlying foundation conditions, the properties of the MSW, the properties of the foundation soils, and the shear strength of the liner interface, should be considered. In this paper, a preliminary single-degree-of-freedom model was developed to analyze the slip potential. This model demonstrated that the type of foundation significantly affects the seismic response of a landfill. A 2-D time-domain finite difference model was established, and the seismic responses and displacements were calculated for three different foundation types. In addition, the nonlinearity of the MSW and the underlying soil were considered by applying the equivalent linear procedure. The calculated results show that the slip along the liner interface exhibits two different modes: slip–stick and slip–slip. Furthermore, the shear strength of the interface and the foundation type significantly influence the overall seismic responses and the slip displacements of the landfills. Additionally, greater interface shear strengths may effectively reduce the slip displacement.

Strain effects on kinematic pile bending in layered soil

The kinematic bending of single piles in two-layer soil is explored to account for soil stiffness degradation and associated damping increase with increasing levels of shear strain, a fundamental aspect of soil behaviour which is not incorporated in current simplified seismic design methodologies for pile foundations.A parametric study of a vertical cylindrical pile embedded in a two-layer soil profile to vertically-propagating S waves, carried out in the time domain by a pertinent beam-on-dynamic-Winkler-foundation (BDWF) model, is reported. Strain effects are treated by means of the equivalent-linear procedure which provides soil stiffness and damping ratio as function of shear strain level. Whereas the approach still represents a crude representation of the actual soil behaviour to dynamic loading, it is more realistic than elementary solutions based on linear visco-elasticity adopted in earlier studies.The paper highlights that soil nonlinearity may have either a detrimental or a beneficial effect on kinematic pile bending depending on the circumstances. The predictive equations for kinematic pile bending in visco-elastic soil recently developed by the Authors are extended to encompass strain effects. Numerical examples and comparisons against experimental data from case histories and shaking table tests are presented.

Three-mode pneumatic management of marine U-tank systems

This paper deals with a new pneumatic control strategy for the roll damping enhancement of marine U-tank stabilizers. The proposed technique consists in a three-mode operation, where the control is active only within a limited resonant range around the ship natural frequency, whereas the control valves are kept closed in the remaining frequency range. Moreover the connection valve between the two air chambers is either closed or partially opened for the low or high frequencies, respectively. The pressurized air for the active control is fed by a turbo-blower set aboard and operates accelerating the motion of the water mass in the U-duct. The theoretical analysis is conducted in the hypothesis of harmonic excitation and includes an equivalent linearization procedure for the turbulent air flow through the upper valve regulating the efflux from the one to the other tank. The application of the laws of thermodynamics to the open air system overhanging the water in each tank chamber reveals the negligibility of the heat transfer through the tank wall in comparison with the mass transfer effects. As a practical result, the mathematical model permits developing a suitable optimization process with the aim at the best damping operation, which however must be subjected to two fundamental constraints: the system stability and the pre-imposed limit power absorbable from the control equipment. Comparing with natural damped passive systems, a reduction of the order of 20–40% can be roughly attained for the resonant roll amplitude, depending on the supplied control power.

Predicting the Seismic Response of Capacity-Designed Structures by Equivalent Linearization

An equivalent linearization procedure is developed for predicting the inelastic deformations and internal forces of capacity-designed structures under earthquake excitations. The procedure employs response spectrum analysis, and mainly consists of the construction of an equivalent linear system by reducing the stiffness of structural members that are expected to respond in the inelastic range. These members are well defined in structures designed with capacity principles. Maximum modal displacement demands of the equivalent linear system are determined either from the equal displacement rule, or from independent nonlinear response history analysis of SDOF systems representing inelastic modes. Predictions obtained from the proposed equivalent linearization procedure are evaluated comparatively by using the results of nonlinear response history analysis as benchmark, linear elastic response spectrum analysis and conventional pushover analysis. The deformations and capacity controlled actions of a 12-story symmetrical plan concrete frame and a 6-story unsymmetrical plan concrete frame are obtained by each method under 96 strong ground motions. It is observed that the proposed procedure results in better accuracy in estimating the inelastic seismic displacement response parameters and capacity controlled forces than the other two approximate methods.

Simulating the seismic response of embankments via artificial neural networks

Geotechnical earthquake engineering may generally be considered as an “imprecise” scientific area due to the unavoidable uncertainties and the simplifications adopted during the design process of geostructures. Therefore, relatively accurate predictions using advanced soft computing (SC) techniques can be tolerated rather than solving a problem conventionally. Artificial neural networks (ANNs), being one of the most popular SC techniques, have been used in many fields of science and technology, as well as, into an increasing number of earthquake engineering applications on structures and infrastructures. In this work the implementation of ANNs is focused on the simulation of the seismic response of a typical embankment. The dynamic response of the embankment is evaluated utilizing the finite-element method, where the nonlinear behavior of the geo-materials can be taken into account by an equivalent-linear procedure. In the present study, this extremely time-consuming process is replaced by properly trained ANNs.

EPC procedure for PDF solution of non-linear oscillators excited by Poisson white noise

The stationary probability density function (PDF) solution of the responses of non-linear stochastic oscillators subjected to Poisson pulses is analyzed. The PDF solutions are obtained by the exponential-polynomial closure (EPC) method. To assess the effectiveness of the solution procedure numerically, non-linear oscillators are analyzed with different impulse arrival rates, degree of oscillator non-linearity and excitation intensity. Numerical results show that the PDFs obtained with the EPC method yield good agreement with those obtained from Monte Carlo simulation when the polynomial order is 4 or 6. It is also observed that the EPC procedure is the same as the equivalent linearization procedure under Gaussian white noise in the case of the polynomial order being 2.

Stability of tailings dams under static and seismic loading

The stability of tailings dams has drawn much attention over the past few decades as a significant number of tailings dam failures have been recorded worldwide. The present study focuses on the investigation of the behavior and the stability of this kind of earth structure under static and dynamic loading. To accomplish this task, elaborate two-dimensional numerical simulations are conducted, utilizing two widely used geotechnical finite element codes. Initially, numerical analyses are performed for three typical types of tailings dams to obtain the potential modes of slope instability by determining the corresponding factors of safety under static conditions. Subsequently the dynamic distress of tailings dams is thoroughly investigated in terms of inertial accelerations developed. Based on the models examined under static loading, dynamic analyses are performed to examine the effects of local site conditions on the seismic response of tailings dams, and therefore on their inertial distress. Emphasis is given to the special characteristics of the ground motion, whereas the material nonlinearity of both soil and tailings is taken into account by an efficient equivalent-linear procedure.