Equivalent Linear Approach Research Articles

An Anomaly in the Equivalent Linearization Approach for the Estimation of Inelastic Response

The estimation of inelastic displacement is an essential step in displacement-based design. Estimation of inelastic response using response spectra is a simpler and attractive alternative to nonlinear dynamic analysis. A number of methods based on different approaches are now available to estimate nonlinear response from design spectra. Equivalent linearization is one of the widely used approaches for this purpose. The approach is known to yield reasonably accurate results. However, an anomaly in the approach is observed, which results in significantly non-conservative estimates of inelastic displacement, particularly in displacement-controlled spectral range. This paper illustrates the anomaly using displacement modification factors and numerical examples of oscillators of different periods and ductility. The results from the equivalent linearization approach have been compared with the results of nonlinear dynamic analysis and yield spectrum approach.

Simulating stiffness degradation and damping in soils via a simple visco-elastic–plastic model

Stiffness degradation and damping represent some of the most well-known aspects of cyclic soil behavior. While standard equivalent linear approaches reproduce these features by (separately) prescribing stiffness reduction and damping curves, in this paper a multiaxial, 3D, viscoelastic – plastic model is developed for the simultaneous simulation of both cyclic curves over a wide cyclic shear strain range.The proposed constitutive relationship is based on two parallel resisting/dissipative mechanisms, purely frictional (elastic–plastic) and viscous. The frictional mechanism is formulated as a bounding surface plasticity model with vanishing elastic domain, including pressure-sensitive failure locus and non-associative plastic flow – which are essential for effective stress analysis. At the same time, the use of the parallel viscous mechanism is shown to be especially beneficial to improve the simulation of the overall dissipative performance.In order to enable model calibration from stiffness degradation (G/Gmax) and damping curves, the constitutive equations are purposely kept as simple as possible with a low number of material parameters. Although the model performance is here explored with reference to pure shear cyclic tests, the 3D, multiaxial formulation is appropriate for general loading conditions.

Approximate Method for Transverse Response Analysis of Partially Isolated Bridges

Current analysis procedures for seismically isolated bridges frequently use an equivalent (approximate) linearization approach to represent the response of nonlinear isolation/energy dissipation devices. Linearization allows standard linear elastic analysis methods, e.g., the response spectrum method, to be conveniently used for design purposes. The linearization approach is by nature an iterative method implying the need to repeatedly correct and analyze a numerical finite-element model. A further simplification could be achieved using closed form equations to represent (1) the structure displacement patterns and (2) the restoring forces from structural elements. The paper explores such a possibility with reference to partially isolated continuous bridges, i.e., bridges with isolation devices at piers and pinned supports at abutments. The role and effect of higher modes of vibration on the system response are discussed, and an approximate method is proposed to account for such effects. An improvement of the classical Jacobsen’s approximation for the effective viscous damping ratio is also proposed using the results of response history analyses. The latter are carried out on two-dimensional numerical models of five case studies, generated from a real existing bridge supposed to be isolated with friction pendulum devices. Comparison of approximate predictions with response history analysis results is presented and discussed. Nonlinear dynamic analyses of a three-dimensional numerical model of the existing bridge were also carried out for comparison purposes.

Displacement-Based Earthquake Loss Assessment Methodology Adopting Two-Dimensional Equivalent Linearization Approach

A simplified displacement-based earthquake loss assessment model (DBELA) on an urban scale is re-investigated by adopting two-dimensional equivalent linearization approach. Taken the reinforced concrete beam-sway frames as example, nonlinear dynamic time-history analyses are carried out to investigate the validity of DBELA methodology. Numerical comparison indicates that DBELA with two-dimensional equivalent linearization approach can give a precise vulnerability assessment of beam-sway concrete frames and the hysteresis rules of the equivalent linearization approach have a slight influence on the assessment results of DBELA.

Dual criterion equivalent linearization approach for yielding structures under earthquake excitation

AbstractIn order to estimate both maximum displacement and maximum inertia force of bilinear hysteretic system subjected to earthquake motions, an equivalent linearization approach with new effective parameters is presented. Effective mass and effective damping ratio as pair of effective parameters instead of the effective period and effective damping ratio in existing equivalent linear systems are introduced. Two error measures for displacement and inertia force are defined. Error distributions over a two‐dimensional parameter space of effective parameters are analysed, and the parameters are determined through a statistical approach with a dual optimization criterion for displacement and inertia force errors applicable to structural design. Single‐degree‐of‐freedom systems with bilinear hysteretic model; natural periods from 0.1 to 3.0 s; linear damping ratios from 3 to 50%; ductility ratios from 1.5 to 6; and post‐yield slope ratios 0, 0.05, 0.1 are considered. Analytical expressions for the effective parameters, and the ratio of the maximum inertia force to the maximum restoring force as functions of response ductility, elastic damping ratio and natural period of inelastic system are proposed for different site conditions and post‐yield slope ratios. Evaluation of proposed equations is performed, which reveals that the linear parameters lead to permissible error ranges. Copyright © 2010 John Wiley & Sons, Ltd.

Effects of soil non-linearity on the seismic response of restrained retaining walls

Retaining structures characterised by high rigidity and various kinematic constraints, such as bridge abutments and basement walls, do not permit limit-equilibrium conditions to be developed. Therefore, according to contemporary norms and geotechnical design practice worldwide, they are most frequently designed utilising the elasticity-based methods, which usually lead to substantially increased dynamic earth pressures. The present paper aims to examine how and to what extent the potentially developed non-linearity of the retained soil may affect: (a) the dynamic distress of a rigid fixed-base retaining wall, and (b) the seismic response of the retained soil layer. For this purpose, a parametric study is conducted which is based on two-dimensional dynamic finite-element analyses using various idealised or real seismic excitations scaled to several intensity levels. Soil non-linearity is realistically taken into account via the commonly used equivalent-linear approach. The results of the present study demonstrate that potential non-linearity of a soil layer retained by a rigid fixed-base wall alters the soil amplification pattern behind the wall and leads to dynamic earth pressures usually lower than those proposed by seismic norms.

Shooting-Based Complete Bifurcation Prediction for Aeroelastic Systems with Freeplay

In recent years, there have been several applications of numerical continuation approaches to aeroelastic systems with freeplay. While some of these have been successful, the general application of the method to such systems remains problematic. Numerical continuation can fail in the presence of complex bifurcations, numerous nearby periodic solution branches, and other factors. In this paper, a three-part procedure for applying numerical continuation to aeroelastic systemswith freeplay is proposed, designed to ensure that the complete periodic behavior is identified, even for systemswith very complex bifurcation diagrams. First, the equivalent linearization approach is used todetermine approximations to theperiodic solutions of thenonlinear system.Then, a shooting-based technique is applied separately to each linearized approximation in order to pinpoint the nearest exact periodic solution. This process results in a cloud of periodic solutions, representing points on all the solution branches and sub-branches. Finally, a branch-following shooting procedure is applied to this cloud of points in order to obtain a complete description of every branch of periodic solutions. The methodology is applied to a simple aeroelastic system with three degrees of freedom and freeplay in the control surface. This system has been often studied but never fully characterized. It is shown that the proposed method succeeds in describing the complete bifurcation behavior of the system and explaining its limit cycle response.

A Site Specific Study on Evaluation of Design Ground Motion Parameters

Design ground motions are usually developed by one of the two approaches: site-specific analyses or from provisions of building codes. Although contemporary codes do consider approximately the site effects, they provide more conservative estimates. Hence it is preferred to carry out site specific analysis which involves both the seismic hazard analysis and ground response analysis. This article presents a site specific analysis for a seismically vulnerable site near Ahmedabad, Gujarat. The seismic hazard analysis was carried out by DSHA approach considering seismicity and seismotectonics within 250km radius. The site is predominantly characterized by deep stiff sandy clay deposits. Extensive shear wave velocity measurement by cross hole test is used for site classification and ground response analysis. The ground response analysis was carried out by equivalent linear approach using SHAKE2000. It is found that the deep stiff soil site considered is found to amplify the ground motion. The site specific response spectra obtained from RRS analysis is compared with the codal provision which reveals high spectral acceleration in site specific spectra for mid period range.

Evaluation of Approaches to Characterize Seismic Isolation Systems for Design

Current design codes generally use an equivalent linear approach for preliminary design of a seismic isolation system. The equivalent linear approach is based on effective parameters, rather than physical parameters of the system, and may not accurately account for the nonlinearity of the isolation system. This article evaluates an alternative normalized strength characterization against the equivalent linear characterization. Considerations for evaluation include: (1) ability to effectively account for variations in ground motion intensity; (2) ability to effectively describe the energy dissipation capacity of the isolation system; and (3) conducive to developing design equations that can be implemented within a code framework.

Site response to multi-directional earthquake loading: A practical procedure

Site response to earthquake loading is one of the fundamental problems in geotechnical earthquake engineering. Most site response analyses assume vertically propagating shear waves in a horizontally layered soil–rock system and simply ignore the effect of site response to vertical earthquake motion, although actual ground motions are comprised of both horizontal and vertical components. In several recent earthquakes very strong vertical ground motions have been recorded, raising great concern over the potential effect of vertical motion on engineering structures. Being a step toward addressing this concern, this paper presents a simple and practical procedure for analysis of site response to both horizontal and vertical earthquake motions. The procedure involves the use of the dynamic stiffness matrix method and equivalent-linear approach, and is built in the modern MATLAB environment to take full advantages of the matrix operations in MATLAB. The input motions can be specified at the soil–bedrock interface or at a rock outcropping. A detailed assessment of the procedure is given, which shows that the procedure is able to produce acceptable predictions of both vertical and horizontal site responses.

Development and application of an advanced capacity spectrum method

In this study an advanced capacity spectrum method (CSM), incorporating inelastic response history analysis is proposed. By employing inelastic response history analysis, the errors introduced due to the use of equivalent linear approaches are eliminated. The available variants of the CSM and the proposed method are used for the evaluation of the seismic response of a woodframe structure. The results are compared to experimental test data. It is observed that more accurate results are obtained with the method proposed in this paper, while the existing CSM approaches exhibited several shortcomings. These shortcomings include failure to predict the displacement demand, incompatibility between the demand and capacity diagrams and non-convergence. It is also observed that updating the bilinear idealization of the structural system according to the selected trial performance point on the capacity diagram significantly improves the accuracy of the CSM.

Rate-dependent soil behavior in seismic site response analysis

One-dimensional site response analysis is widely used in estimating local seismic site effects. The soil behavior in the analysis is often assumed to be independent of the rate of seismic loading. Laboratory test results, on the other hand, indicate that cyclic cohesive soil behavior is influenced by the rate of loading. Three models of rate-dependent dynamic soil behavior were derived based on available laboratory data. The models were implemented and evaluated in a modified one-dimensional equivalent linear site response analysis approach. Results show that rate-dependent shear modulus and damping can have a pronounced influence on propagated weak ground motion but a secondary influence on propagated strong motion. Rate dependence of the damping ratio has a greater impact on the computed response than rate dependence of the shear modulus. This paper highlights the relevance of the compatibility between frequencies at which dynamic soil properties are measured and their use in site response analysis.

A hypoplastic model for site response analysis

Site response analyses must take into account the nonlinear behavior of soils. This is typically achieved using an equivalent linear approach or using numerical analyses with an appropriate constitutive model. In this work a family of hypoplastic models is proposed for use in site response analyses. These nonlinear models use a rate-type tensorial equation and are capable of reproducing plastic deformation of soils for cyclic loading under both drained and undrained conditions. A methodology for the calibration of hypoplastic parameter for dynamic loading is proposed. The hypoplastic constitutive models are implemented in a finite element code and the site response of the Lotung downhole array site is used to validate the use of hypoplastic models for site response analysis. The hypoplastic models reproduce accurately site response at the Lotung site. The advantages and disadvantages of the hypoplastic models compared to other models are discussed.

Nonlinear Analysis of Local Site Effects on Seismic Ground Response in the Bam Earthquake

The influence of local geologic and soil conditions on the intensity of ground shaking is addressed in this study. The amplification of the ground motion due to local site effects resulted in severe damage to dwellings in the Bam area during the 2003 Bam Earthquake. A unique set of strong motion acceleration recordings was obtained at the Bam accelerograph station. Although the highest peak ground acceleration recorded was the vertical component (nearly 1 g), the longitudinal component (fault-parallel motion) clearly had the largest maximum velocity as well as maximum ground displacement. Subsurface geotechnical and geophysical (down-hole) data in two different sites have been obtained and used to estimate the local site condition on earthquake ground motion in the area. The ground response analyses have been conducted considering the nonlinear behavior of the soil deposits using both equivalent linear and nonlinear approaches. The fully nonlinear method embodied in FLAC was used to evaluate the nonlinear soil properties on earthquake wave propagation through the soil layer, and compare with the response from the equivalent linear approach. It is shown that thick alluvium deposits amplified the ground motion and resulted in significant damage in residential buildings in the earthquake stricken region. The comparison of results indicated similar response spectra of the motions for both equivalent and nonlinear analyses, showing peaks in the period range of 0.3–1.5 s. However, the amplification levels of nonlinear analysis were less than the equivalent linear method especially in long periods. The observed response spectra are shown to be above the NEHRP building code design requirements, especially at high frequencies.

Comparison of fracture properties of two wood species through cohesive crack simulations

In the present work fracture (Mode I) was induced through three point-bending tests in two wood species used in timber construction: Maritime pine ( Pinus pinaster Ait.) and Norway spruce ( Picea abies L.). Load–displacement curves were experimentally determined and corresponding Resistance-curves ( R-curves) obtained using an equivalent linear elastic approach. An inverse method is presented to identify cohesive crack properties of a cohesive crack bilinear model used to simulate fracture in both wood species, combining experimental data and a developed genetic algorithm. Good agreement between numerical and experimental load–displacement and R-curves was obtained. Conclusions are drawn from finite element simulations regarding the extent of the numerical cohesive zone computed for each studied wood.

Impact damper for controlling friction-driven oscillations

Control of friction-driven oscillations by using an impact damper is investigated. An experimental setup of a single degree-of-freedom friction-driven oscillator with an attached impact damper is designed and fabricated. An approximate nonlinear mathematical model of the friction-driven oscillator is derived. The values of unknown parameters in this mathematical model are obtained by using a least square error method so as to match the experimentally obtained response. An approximate solution of the general steady-state response of the friction-driven oscillator with an attached impact damper is derived analytically by using a piecewise equivalent linearization approach assuming two non-symmetric impacts per cycle. The effects of mass ratio, coefficient of restitution and clearance on the performance of the impact damper have been investigated analytically and the results are verified by numerical integration. Experimental results obtained for steel-on-steel, aluminum-on-aluminum and aluminum-on-rubber impacts are reported.

Nonlinear analysis of rock-fill dams to non-stationary excitation by the stochastic Wilson- θ method

The objective of this investigation is to predict the nonlinear response of a rock-fill dam subjected to non-stationary excitation by using the finite element method in conjunction with the equivalent linearization approach. The nonlinear behavior is based on the equivalent linear method which considers the nonlinear variation of soil shear moduli and damping as a function of shear strain. Linear and non-linear analyses are performed by the stochastic Wilson- θ method. Nonlinear response of a rock-fill dam disturbed by non-stationary random excitations is compared with the linear analysis. Meanwhile, some statistical responses obtained by the linear and nonlinear analyses are checked by the Monte Carlo simulation technique. Finally, the effect of deconvolution ground motion on the nonlinear response of dam-foundation system is investigated in this paper. Two different earthquake input mechanisms are used in the dynamic analysis: the standard rigid-base input and the deconvolved-base-rock input models.

Estimating the Effects of Tunneling on Existing Pipelines

A method is presented for estimating the maximum bending moment for continuous (or rigidly jointed) pipelines affected by tunnel-induced ground movement. The estimation can be made based on the knowledge of tunnel and pipeline geometries, the stiffness of soil and pipeline, and tunnel-induced ground deformation at the pipeline level. The method takes account of soil nonlinearity by an equivalent linear approach, in which the stiffness of the soil is evaluated based on an average deviatoric strain developed along the pipeline. The approach is conservative and promises that the bending moment is not underestimated. The validity of the method as an upper bound approximation is evaluated against centrifuge test results.

Equivalent linearization applied to earthquake excitations and the [formula omitted] relationships

This paper uses the equivalent linearization approach to derive the approximate harmonic equivalent stiffness and damping for bilinear systems in the context of earthquake resistant design. Whereas the resulting stiffness is identical to that in published works, the damping herein depends on the harmonic output frequency as well. This additional dependence produces an efficient method, which, although deterministically derived, predicts quite well the maximum output displacements for earthquake excitations. Two methods for choosing output frequencies are described. The results from these two methods are compared to those from several available approximation methods for 16 separate cases for an ensemble of 20 earthquake time history records. For the bilinear systems studied the methods presented in this paper give in most cases more accurate results. These results can also be used for evaluating the strength reduction factor R for given ductility or vice versa–the R – μ – T 0 relationship–and hence improve the accuracy of the simplified procedures now used for displacement-based design.

A finite element analysis on random vibration of nonlinear shell structures

The objective of this investigation is to study the random vibration of a nonlinear geometrically shell structure by using the finite element method in conjunction with the equivalent linearization approach. When the shell structure is subjected to excessive loadings, the large deformations of the shell structure must be considered. In that sense, the stiffness of the governing equation of the shell structure is related to deflection; therefore, it is nonlinear and difficult to solve. In this study, the applied loadings to the shell structure are assumed to be a nonstationary random excitation to characterize many physical loadings such as earthquake, wind, aerodynamic and acoustic loadings. The equivalent linearization and the finite element method are adopted to perform the nonlinear random vibration analysis of the shell structures, which can be quite nonuniform and complex in geometry or nonhomogeneous in material. These obtained statistical dynamics responses are very useful for estimating the structure safety and reliability. Meanwhile, some statistical responses obtained by the present approach are checked by the Monte Carlo simulation technique.