The role of the stochastic equivalent linearization method in the analysis of the non-linear seismic response of building structures

The stochastic equivalent linearization method has been significantly improved during the last two decades, leading to rather efficient and accurate estimates of the first- and second-order statistical moments of the random response process, even when the non-stationarity of the excitation and the hysteretic degrading non-linearities of the structural system are taken into account. The purpose of this paper, apart from presenting a short survey of the most relevant methods developed in this area, indicating their main restrictions, is to call attention to the remarkable possibilities of the equivalent linearization technique as the most powerful approximate method to deal with the seismic response analysis of MDOF non-linear building structures, deserving to be considered by the engineering codes in the near future as an appropriate formulation for that purpose. To illustrate the real interest of this method, several applications concerning a simple shear-building structural model are presented, considering columns with non-linear restoring forces, either bilinear elastic or hysteretic, and the results obtained by some computer programs developed on the basis of the equivalent linearization technique are compared with the estimates achieved by digital simulation in order to check the level of accuracy. Moreover, these results are also used to evaluate limit violation (failure) probabilities, based on the vulnerability function concept.     

A hybrid time frequency domain formulation for non-linear soil–structure interaction

Methods that combine frequency and time domain techniques offer an attractive alternative for solving Soil–Structure-interaction problems where the structure exhibits non-linear behaviour. In the hybrid-frequency-time-domain procedure a reference linear system is solved in the frequency domain and the difference between the actual restoring forces and those in the linear model are treated as pseudo-forces. In the solution scheme explored in this paper, designated as the hybrid-time-frequency-domain (HTFD) procedure, the equations of motion are solved in the time domain with due consideration for non-linearities and with the unbounded medium represented by frequency-independent springs and dampers. The frequency dependency of the impedance coefficients is introduced by means of pseudo-forces evaluated in the frequency domain at the end of each iteration. A criterion of stability for the HTFD approach is derived analytically and its validity is sustained numerically. As is often the case, the criterion takes the form of a limit of unity on the spectral radius of an appropriately defined matrix. Inspection of the terms in this matrix shows that convergence can be guaranteed by suitable selection of the reference impedance. The CPU times required to obtain converged solutions with the HTFD are found, in a number of numerical simulations, to be up to one order of magnitude less than those required by the alternative hybrid-frequency-time-domain approach. © 1998 John Wiley & Sons, Ltd.     

Observations on non-linear dynamic characteristics of suspension bridges

For most forms of loading static and dynamic response of a suspension bridge is modelled adequately by linear analysis. By examining the two principal non-linearities of inclined and vertical hanger suspension bridges the limitations of linear analyses are shown. The inclined hanger configuration is shown to be the more strongly non-linear. Some examples are given for simplified planar analyses of this type of bridge to large amplitude vehicular, random, sinusoidal and propagating earthquake excitation, showing that the effect only becomes significant when deflections approach the extent of cable sag due to cable elasticity.     

A modal procedure for seismic analysis of non-linear base-isolated multistorey structures

A modal procedure for non-linear analysis of multistorey structures with high-damping base-isolation systems was proposed. Two different isolation devices were considered in the analysis: an high-damping laminated rubber bearing and a lead-rubber bearing. Starting from deformational properties verified by tests, the isolation systems were characterized using three different analytical models (an Elastic Viscous, a Bilinear Hysteretic and a Wen's Model) with parameters depending from maximum lateral strain. After non-linear modelling of isolation and lateral-force-resisting systems, the effects of material non-linearities were considered as pseudo-forces applied to the equivalent linear system (Pseudo-Force Method) and the formally linearized equations of motion were uncoupled by the transformation defined by the complex mode shapes. The modal responses were finally obtained with an extension of Nigam–Jennings technique to non-linear and non-classically damped systems, in conjunction with an iterative technique searching for non-linear contributions satisfying equations of motion and constitutive laws. Since the properties of the isolated structure usually change with maximun lateral strain of isolation bearings, the integration of a new set of governing equations was required for each design-displacement value. The procedure proposed was described in detail and then applied for the determination of modal and total seismic responses in some real cases. At first, a very good agreement between non-linear responses obtained with the proposed mode superposition and with a direct integration method was observed. Then a comparison of results obtained with the three different analytical models of the isolation bearings was carried out. At last, the exact modal response obtained with analytical models depending from the design displacement of the isolation bearings was compared with two different approximated solutions, evaluated using mode shapes and isolation properties, respectively, calculated under simplified hypothesis.© 1998 John Wiley & Sons, Ltd.     

Behaviour of a hydrodynamic finite element model

A finite element model which solves the vertically integrated momentum and continuity equations is described. Linear triangular elements are used to describe the geometry and parameter variations. The Galerkin method of weighted residuals is employed to cast the equations in a form amenable to numerical solution. The model is based on a fully implicit formulation using finite differences for the temporal derivatives.Means of evaluating the non-linear terms of the governing equations are described, and model results are presented for a frictionless tidal channel. The example is chosen such that the non-linearities have a large influence on the solution, and as a result the linearization scheme significantly affects the model's behaviour.Suppression of the non-linear instabilities generated by the convective terms in the momentum equations is examined for the case of flow around a 180° bend. Both the imposition of artificially high roughness coefficients and the use of an effective eddy viscosity are examined in terms of their ability to damp the oscillations which arise for this example.Finally, model results are presented for a case study involving determination of remedial measures to improve flow conditions at a river outfall in Southern Ontario.     

Predictive optimal control for seismic analysis of non-linear and hysteretic structures

In this paper, an effective active predictive control algorithm is developed for the vibration control of non-linear hysteretic structural systems subjected to earthquake excitation. The non-linear characteristics of the structural behaviour and the effects of time delay in both the measurements and control action are included throughout the entire analysis (design and validation). This is very important since, in current design practice, structures are assumed to behave non-linearly, and time delays induced by sensors and actuator devices are not avoidable. The proposed algorithm focuses on the instantaneous optimal control approach for the development of a control methodology where the non-linearities are brought into the analysis through a non-linear state vector and a non-linear open-loop term. An autoregressive (AR) model is used to predict the earthquake excitation to be considered in the prediction of the structural response. A performance index that is quadratic in the control force and in the predicted non-linear states, with two additional energy related terms, and that is subjected to a non-linear constraint equation, is minimized at every time step. The effectiveness of the proposed closed-open loop non-linear instantaneous optimal prediction control (CONIOPC) strategy is presented by the results of numerical simulations. Since non-linearity and time-delay effects are incorporated in the mathematical model throughout the derivation of the control methodology, good performance and stability of the controlled structural system are guaranteed. Copyright © 1999 John Wiley & Sons, Ltd.     

The effect of magnetite particle size on paleointensity determinations of the geomagnetic field

The Thellier method for paleointensity determinations has been applied to prepared samples containing magnetites whose mean particle sizes range from single domain, SD, to multidomain, MD. Linear (ideal) PNRM-PTRM curves are obtained for samples containing SD and submicron magnetite particles. However, for MD particles non-linear (concave-up) PNRM-PTRM curves are observed such that a linear approximation to the lower blocking-temperature data leads to apparent paleointensities that are higher than the actual paleofield; however, the ratio of the end-points, NRM/TRM, yields the correct (laboratory) intensity. The non-linear (concave-up) PNRM-PTRM curves for the MD particles are explained in terms of the lack of symmetry of the domain-wall movements during the two heatings of the Thellier experiment. Low stabilities with respect to alternating fields and with respect to temperature cycles below magnetite's isotropic temperature are diagnostic in detecting samples most likely to exhibit non-linearities due to the MD effect.     

Non-parametric identification of a class of nonlinear multidegree dynamic systems

A non-parametric identification technique is presented for chain-like multidegree-of-freedom non-linear dynamic systems. The method uses information about the state variables of non-linear systems to express the system characteristics in terms of two-dimensional orthogonal functions. The technique is applied to a model of a steel frame that has been extensively investigated both analytically and experimentally. The method can be used with deterministic or random excitation to identify dynamic systems with arbitrary non-linearities, including those with hysteretic characteristics. It is also shown that the method is easy to implement and needs much less computer time and storage requirements compared to the Wiener-kernel approach. The method is shown to have low sensitivity to the effects of additive noise in the experimental data.     

Effect of non-linear soil–structure interaction on seismic response of tall slender structures

Some structures may be very massive and may have to be located on relatively soft soil. In such cases, the soil adjacent to the structure behaves in a non-linear fashion and affects the response of the structure to the dynamic loading. An approximate hybrid approach to analyse soil–structure systems accounting for soil non-linearities has been developed in this paper. The approach combines the consistent infinitesimal finite-element cell method (CIFECM) and the finite-element method (FEM). The CIFECM is employed to model the non-linear (near-field) zone of the soil supporting the structure as a series of bounded media. The material properties of the bounded media are selected so that they are compatible with the average effective strains over the whole bounded medium during the excitation. The linear zone of soil away from the foundation, the far-field, is modelled as an unbounded medium using the CIFECM for unbounded media. The structure itself is represented by the FEM. The proposed method is used to model the dynamic response of a one-mass structure and a TV-tower supported on a homogenous stratum and excited by an earthquake. It was found that the secondary soil non-linearity might increase or decrease the base forces of tall slender structures depending on the type of structure, frequency content of the input motion and the dynamic properties of the near-field soil.     

A refined method of recovering potential coefficients from surface gravity data

Summary A new method for computing the potential coefficients of the Earth's external gravity field is presented. The gravimetric boundary-value problem with a free boundary is reduced to the problem with a fixed known telluroid. The main idea of the derivation consists in a continuation of the quantities from the physical surface to the telluroid by means of Taylor's series expansion in such a way that the terms whose magnitudes are comparable with the accuracy of today's gravity measurements are retained. Thus not only linear, but also non-linear terms are taken into account. Explicitly, the terms up to the order of the third power of the Earth's flattening are retained. The non-linear boundary-value problem on the telluroid is solved by an iteration procedure with successive approximations. In each iteration step the solution of the non-linear problem is estimated by the solutions of two linear problems utilizing the fact that the non-linear boundary condition may be split into two parts; the linear spherical approximation of the gravity anomaly whose magnitude is significantly greater than the others and the non-linear ellipsoidal corrections. Finally, in order to solve the problem in terms of spherical harmonics, the transform method composed of the fast Fourier transform and Gauss Legendre quadrature is theoretically outlined. Immediate data processing of gravity data measured on the physical Earth's surface without any continuation of gravity measurements to a reference level surface belongs to the main advantage of the presented method. This implies that no preliminary data handling is needed and that the error data propagation is, consequently, maximally suppressed.     

Performance of composite steel/concrete members under earthquake loading. Part I: Analytical model

This paper presents an advanced non-linear model developed for the analysis of composite steel/concrete frame structures subjected to cyclic and dynamic loads. The formulation consists of beam-column cubic finite elements accounting for geometric non-linearities and material inelasticity. The non-linear cyclic concrete model considers confinement effects and the constitutive relationship for steel includes the effect of local buckling and variable amplitude cyclic degradation. The model is calibrated and compared with experimental data from cyclic and pseudo-dynamic tests conducted by the writers on a new ductile partially-encased composite beam-column. The accuracy and efficiency of the developed model are demonstrated through the correlation between the experimental results and analytical simulations. In a companion paper, the model is used to conduct parametric studies leading to important conclusions for ductility-based earthquake-resistant design.     

Verification of two-dimensional non-linear analysis of sand-structure system by examining the results of the shaking-table test

The verification of a two-dimensional non-linear analysis based on a hybrid constitutive model, which consists of the Ramberg-Osgood model extended to two-dimensional problems and a dilatancy model, is discussed through a comparison of the shaking-table test results for a one-storey structure standing on a dry sandy deposit with those for the same on a saturated sandy deposit. Since the relationships G vs. γ and h vs. γ for shear strains of 10^?5?10^?3 can be obtained accurately by an inversion analysis of the resonance curve of the sandy deposit the proposed non-linear method can represent well the observed non-linear response of the dry or the saturated sandy deposit including the structure. This method, however, should be applied carefully to assess the response of a soil-structure system when the three-dimensional interaction affects significantly the response of the model. An equivalent linear analysis using a sway-rocking model is applied to simulate the non-linear ground motion including the three-dimensional interaction, and it is found that the sway-rocking model can also represent well the non-linear response of the system.     

Solution of the transport equations using a moving coordinate system

A convection-diffusion equation arises from the conservation equations in miscible and immiscible flooding, thermal recovery, and water movement through desiccated soil. When the convection term dominates the diffusion term, the equations are very difficult to solve numerically. Owing to the hyperbolic character assumed for dominating convection, inaccurate, oscillating solutions result. A new solution technique minimizes the oscillations. The differential equation is transformed into a moving coordinate system which eliminates the convection term but makes the boundary location change in time. We illustrate the new method on two one-dimensional problems: the linear convection-diffusion equation and a non-linear diffusion type equation governing water movement through desiccated soil. Transforming the linear convection diffusion equation into a moving coordinate system gives a diffusion equation with time dependent boundary conditions. We apply orthogonal collocation on finite elements with a Crank-Nicholson time discretization. Comparisons are made to schemes using fixed coordinate systems. The equation describing movement of water in dry soil is a highly non-linear diffusion-type equation with coefficients varying over six orders of magnitude. We solve the equation in a coordinate system moving with a time-dependent velocity, which is determined by the location of the largest gradient of the solution. The finite difference technique with a variable grid size is applied, and a modified Crank-Nicholson technique is used for the temporal discretization. Comparisons are made to an exact solution obtained by similarity transformation, and with an ordinary finite difference scheme on a fixed coordinate system.     

Solution of random structural system subject to nonstationary excitation: transforming the equation with random coefficients to one with deterministic coefficients and random initial conditions

This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator.     

Dynamic response of non-linear building-foundation systems

An analysis is made of the steady-state response of a bilinear hysteretic structure supported on the surface of a viscoelastic half-space. The method of equivalent linearization is used to solve the equations of motion, and simplified approximate formulas are obtained for the fundamental resonant frequency of the system and for an effective critical damping ratio. Numerical results indicate that for non-linear hysteretic structures compliance of the soil foundation may lead to larger displacements than would occur if the base were rigid. This behaviour differs from that generally observed for linear systems, for which the effect of soil-structure interaction is to reduce the rigid-base response.     

Towards a seismic design method for plane steel frames using equivalent modal damping ratios

A method for seismic design of plane steel moment resisting frames based on the use of equivalent modal damping ratios is developed. The method determines the design base shear of the structure through spectrum analysis using rationally obtained equivalent modal damping ratios instead of the crude strength reduction (behavior) factor. An equivalent linear structure, which retains the mass and initial stiffness of the original non-linear structure and takes into account geometrical non-linearity and inelasticity in the form of equivalent, time-invariant, modal damping ratios is established. The equivalent damping ratios for the first few significant modes are numerically computed by first iteratively forming a frequency response transfer function modulus until it satisfies certain smoothness criteria and then by solving a set of non-linear algebraic equations. Thus, design equations providing equivalent damping ratios as functions of period and allowable deformation and damage are constructed using extensive numerical data coming from plane steel moment resisting frames excited by various seismic motions. These equations can be used in conjunction with a design spectrum, appropriately constructed for high damping values, and modal synthesis tools to calculate the seismic design forces of the structure. The proposed method is illustrated by numerical examples. It is concluded that unlike the usual approach of seismic codes employing a single common value of the strength reduction factor value for all modes, the proposed approach working with deformation and damage dependent equivalent modal damping ratios leads to more accurate and deformation and damage controlled results.     

Non-linear soil-structure-interaction analysis using green's function of soil in the time domain

The contribution of the (linear) unbounded soil to the basic equation of motion of a non-linear analysis of soil-structure interaction consists of convolution integrals of the displacement-force relationship in the time domain and the history of the interaction forces. The former is calculated using the indirect boundary-element method, which is based on a weighted-residual technique and involves Green's functions. As an example of a non-linear soil-structure-interaction analysis, the partial uplift of the basemat of a structure is examined. As the convolution integrals have to be recalculated for each time step, the computational effort in this rigorous procedure is substantial. A reduction can be achieved by simplifying the Green's function by ‘concentrating’ the region of influence. Alternatively, assuming a specified wave pattern, a coupled system of springs and dashpots with frequency-independent coefficients can be used as an approximation.     

A computational procedure for the implementation of equivalent linearization in finite element analysis

This paper deals with the practical implementation of the statistical equivalent linearization method (EQL) in conjunction with general FE‐analysis to evaluate non‐linear structural response under random excitation. A computational procedure is presented which requires the non‐linear part of the system to be subdivided into suitable sub‐domains (elements). Each element is independently linearized using only a minimum number of co‐ordinates. A local co‐ordinate system is introduced using linear transformations of the global (master) degrees of freedom. Restoring forces and non‐linear constitutive laws are defined by the local co‐ordinates of each element. The linearization coefficients are further transformed back to establish the global linearized system. The procedure has, on one hand, the ability to use any desired linearization criterion and, on the other hand, it can be combined with highly developed procedures to determine the response of arbitrary large FE‐models. To illustrate the applicability of the procedure, two different non‐linear systems are analysed under bi‐directional earthquake excitation. Copyright © 2000 John Wiley & Sons, Ltd.