Solution of viscously damped linear systems using a set of load-dependent vectors

This paper considers a solution method for viscously damped linear structural systems which are subjected to transient loading. The equations of motion of such systems are written in a first-order form. A solution subspace is generated using the damped dynamic matrix and the static deflection from the first-order form of the equations of motion. Two convenient bases, Lanczos vectors and Ritz vectors, are constructed from this subspace. An approximate solution is then obtained by superposition of the Lanczos vectors or the Ritz vectors. In contrast to the traditional mode superposition method using complex eigenvectors, the Lanczos vectors or the Ritz vectors are less expensive to generate than the complex eigenvectors, yet yield comparable accuracy. In addition, there is no need for a static correction since the static deflection is already contained in our solution subspace. Numerical examples are presented to show the potential of using the Ritz vectors to compute responses of damped dynamic systems.     

Multiple‐support response spectrum analysis using load‐dependent Ritz vectors

Load‐dependent Ritz (LDR) vectors are used in conjunction with the multiple‐support response spectrum (MSRS) combination rule for analysis of structures subjected to spatially varying earthquake ground motions. The LDR vector approach for MSRS analysis is motivated by the fact that LDR vectors in general are more accurate and computationally simpler than eigenvectors in mode superposition analysis, and because many researchers and engineers are using LDR vectors in linear structural dynamic analysis. Mode truncation rules for the original MSRS method are modified to apply to LDR vectors. Two methods for selecting LDR vectors for multicomponent MSRS analysis are introduced. Idealized models of two real bridges with differing structural characteristics are used to investigate the accuracy and efficiency of the two LDR‐MSRS methods in comparison with results obtained by the original MSRS method as well as an extended version that accounts for the static contribution of truncated modes. The results show that the LDR‐MSRS method is generally more accurate than the original MSRS method and at least as accurate as the extended MSRS method. Copyright © 2014 John Wiley & Sons, Ltd.     

Method for solving dynamic equilibrium equations based on displacement excitation

Mode superposition is a widely used method for solving the dynamic equilibrium equation in structural dynamic analysis. However, the accuracy of this method may be reduced when the dynamic equilibrium equations are set up using displacement excitation. A new method for developing solutions for dynamic equilibrium equations based on displacement excitation is introduced. The dynamic equilibrium equation is decomposed into two parts, namely displacement excitation and velocity excitation, and precise integration and mode superposition methods are combined to solve the equation. Ritz vectors are then used to calculate the static response of the truncated modes of the structure, and a method for determining the number of participating modes is obtained. Using multi-degree-of-freedom systems as two computational examples, the differences in the structural responses obtained from the displacement excitation and acceleration excitation are compared and analyzed. It is shown that the new solution method generates consistent accuracy between the displacement excitation and acceleration excitation.     

Dynamic analysis by direct superposition of Ritz vectors

The solution of the eigenvalue problem for large structures is often the most costly phase of a dynamic response analysis. In this paper, the need for the exact solution of this large eigenvalue problem is eliminated. A new algorithm, based on error minimization, is presented for the generation of a sequence of Ritz vectors. These orthogonal vectors are used to reduce the size of the system. Only Ritz vectors with a large participation factor are used in the subsequent mode superposition analysis. In all examples studied, the superposition of Ritz vectors yields more accurate results, with fewer vectors, than if the exact eigenvectors are used. The proposed method not only reduces computer time requirements significantly but provides an error estimation for the dynamic analysis. The approach automatically includes the advantages of the proven numerical techniques of static condensation, Guyan reduction and static correction due to higher mode truncation.     

Modified predictor–corrector numerical scheme for real‐time pseudo dynamic tests using state‐space formulation

This paper deals with an explicit numerical integration method for real‐time pseudo dynamic tests. The proposed method, termed the MPC‐SSP method, is suited to use in real‐time pseudo dynamic tests as no iteration steps are involved in each step of computation. A procedure for implementing the proposed method in real‐time pseudo dynamic tests is described in the paper. A state‐space approach is employed in this study to formulate the equations of motion of the system, which is advantageous in real‐time pseudo dynamic testing of structures with active control devices since most structural control problems are formulated in state space. A stability and accuracy analysis of the proposed method was performed based on linear elastic systems. Owing to an extrapolation scheme employed to predict the system's future response, the MPC‐SSP method is conditionally stable. To demonstrate the effectiveness of the MPC‐SSP method, a series of numerical simulations were performed and the performance of the MPC‐SSP method was compared with other pseudo dynamic testing methods including Explicit Newmark, Central Difference, Operator Splitting, and OS‐SSP methods based on both linear and non‐linear single‐degree‐of‐freedom systems. Copyright © 2004 John Wiley & Sons, Ltd.     

Ritz vectors and generation criteria for mode superposition analysis

A generation procedure of Ritz vectors to control the inclusion of static effect and the number of vectors in mode superposition dynamic analysis is presented. The original algorithm of the Ritz vectors¹⁵ is modified to improve stability in the generation procedure and to include the use of static residual. To reject unimportant Ritz vectors, cut-off criteria, which are based on the participation of mass distribution and spatial load distribution, are proposed. Numerical examples are presented to illustrate the effectiveness of the derived Ritz vectors over the eigenvectors and the performance of the cutoff criteria in the mode superposition dynamic analysis.     

Frequency dependent ritz vectors

In order to reduce the size of problems involving analysis of the dynamic response of structural systems, a transformatio based on appropriately selected Ritz shapes is commonly employed. The lower mode shapes may at times serve a effective Ritz shapes. However, the computation of mode shapes is a time consuming task; in addition, the mode shapes may not form the best basis for representing the spatial distribution of loads. The recently developed load dependent vectors, which are derived from a static solution for the applied loads, address some of the problems inherent in the use of mode shapes. However, both the natural mode shapes and the load dependent vectors fail to account for the frequency content of the loading, a parameter that may influence strongly the response, particularly for loading with a high frequency content. A procedure is presented here for the generation of frequency dependent vectors. A combination of load dependent and frequency dependent vectors will often form a very efficient basis for the representation of the response, as illustrated by several examples presented here.     

Use of ritz vectors in wave propagation and foundation response

The accuracy of a numerical method is demonstrated for the dynamic analysis of large complex finite element systems in which the spatial distribution of the loading is constant. The method is based on the use of a special class of Ritz vectors which were previously proposed and can be generated with minimum numerical effort. The purpose of this paper is to extend the use of these vectors to the solution of wave propagation and foundation response problems. The method is applied to one-, twoand three-dimensional problems in order to illustrate the efficiency and accuracy of the technique. Unless it is necessary to evaluate the very high-frequency behaviour of a structural system, it is shown that a small number of Ritz vectors will produce excellent results. Therefore, they can be very effective in the solution of three-dimensional soil-structure systems subjected to earthquake loading.     

Novel coupling Rosenbrock‐based algorithms for real‐time dynamic substructure testing

Real‐time testing with dynamic substructuring is a novel experimental technique capable of assessing the behaviour of structures subjected to dynamic loadings including earthquakes. The technique involves recreating the dynamics of the entire structure by combining an experimental test piece consisting of part of the structure with a numerical model simulating the remainder of the structure. These substructures interact in real time to emulate the behaviour of the entire structure. Time integration is the most versatile method for analysing the general case of linear and non‐linear semi‐discretized equations of motion. In this paper we propose for substructure testing, L‐stable real‐time (LSRT) compatible integrators with two and three stages derived from the Rosenbrock methods. These algorithms are unconditionally stable for uncoupled problems and entail a moderate computational cost for real‐time performance. They can also effectively deal with stiff problems, i.e. complex emulated structures for which solutions can change on a time scale that is very short compared with the interval of time integration, but where the solution of interest changes on a much longer time scale. Stability conditions of the coupled substructures are analysed by means of the zero‐stability approach, and the accuracy of the novel algorithms in the coupled case is assessed in both the unforced and forced conditions. LSRT algorithms are shown to be more competitive than popular Runge–Kutta methods in terms of stability, accuracy and ease of implementation. Numerical simulations and real‐time substructure tests are used to demonstrate the favourable properties of the proposed algorithms. Copyright © 2007 John Wiley & Sons, Ltd.     

Comparison between non-linear dynamic and static seismic analysis of structures according to European and US provisions

Several procedures for non-linear static and dynamic analysis of structures have been developed in recent years. This paper discusses those procedures that have been implemented into the latest European and US seismic provisions: non-linear dynamic time-history analysis; N2 non-linear static method (Eurocode 8); non-linear static procedure NSP (FEMA 356) and improved capacity spectrum method CSM (FEMA 440). The presented methods differ in respect to accuracy, simplicity, transparency and clarity of theoretical background. Non-linear static procedures were developed with the aim of overcoming the insufficiency and limitations of linear methods, whilst at the same time maintaining a relatively simple application. All procedures incorporate performance-based concepts paying more attention to damage control. Application of the presented procedures is illustrated by means of an example of an eight-storey reinforced concrete frame building. The results obtained by non-linear dynamic time-history analysis and non-linear static procedures are compared. It is concluded that these non-linear static procedures are sustainable for application. Additionally, this paper discusses a recommendation in the Eurocode 8/1 that the capacity curve should be determined by pushover analysis for values of the control displacement ranging between zero and 150% of the target displacement. Maximum top displacement of the analyzed structure obtained by using dynamic method with real time-history records corresponds to 145% of the target displacement obtained using the non-linear static N2 procedure.     

Causal and functional relations in fluvial geomorphology

The concept of causality has been a fundamental principle of scientific investigation for many centuries. Recently, however, there has been a significant reappraisal of the philosophical basis of the concept of causality and this has important implications for future geomorphological research. It is now recognized that there are two approaches to the study of relations between variables. Conventionally knowledge was divided into classes and, on the basis of time precedence, causal relations were defined. While this approach enables static systems to be analysed it does not apply to dynamic systems which exhibit mutual causality. This traditional view of knowledge became obsolete with the introduction of the concept of the continuum. Algebraic techniques now enable the solution of systems of simultaneous equations, provided that there are as many equations as unknowns. Significantly this approach applies to dynamic as well as static systems. Further advances in fluvial geomorphological research are dependent on the adoption of mathematical reasoning. This will concentrate research on the mode and rate of operation of channel processes and feedback mechanisms. Given this information it will be possible to develop a static process-response model to explain and predict channel response to a set of input conditions, and a dynamic model to simulate channel development through time and space.     

Generalized force vectors for multi‐mode pushover analysis

A generalized pushover analysis (GPA) procedure is developed for estimating the inelastic seismic response of structures under earthquake ground excitations. The procedure comprises applying different generalized force vectors separately to the structure in an incremental form with increasing amplitude until a prescribed seismic demand is attained for each generalized force vector. A generalized force vector is expressed as a combination of modal forces, and simulates the instantaneous force distribution acting on the system when a given response parameter reaches its maximum value during dynamic response to a seismic excitation. While any response parameter can be selected arbitrarily, generalized force vectors in the presented study are derived for maximum interstory drift parameters. The maximum value of any other response parameter is then obtained from the envelope of GPAs results. Each nonlinear static analysis under a generalized force vector activates the entire multi‐degree of freedom effects simultaneously. Accordingly, inelastic actions develop in members with the contribution of all ‘instantaneous modes’ in the nonlinear response range. Target seismic demands for interstory drifts at the selected stories are calculated from the associated drift expressions. The implementation of the proposed GPA is simpler compared with nonlinear response history analysis, whereas it is less demanding in computational effort when compared with several multi‐mode adaptive nonlinear static procedures. Moreover, it does not suffer from the statistical combination of inelastic modal responses obtained separately. The results obtained from building frames have demonstrated that GPA is successful in estimating maximum member deformations and member forces with reference to the response history analysis. When the response is linear elastic, GPA and response spectrum analysis produce identical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Modal time history analysis of non-classically damped structures for seismic motions

The step-by-step modal time history integration methods are developed for dynamic analysis of non-classically damped linear structures subjected to earthquake-induced ground motions. Both the mode displacement and mode acceleration-based algorithms are presented for the calculation of member and acceleration responses. The complex-valued eigenvectors are used to effect the modal decoupling of the equations of motion. However, the recursive step-by-step algorithms are still in terms of real quantities. The numerical results for the acceleration response and floor response spectra, obtained with these approaches, are presented. The mode acceleration approach is observed to be decidedly better than the mode displacement approach in as much as it alleviates the so-called missing mass effect, caused by the truncation of modes, very effectively. The utilization of the mode acceleration-based algorithms is, thus, recommended in all dynamic analyses for earthquake-induced ground motions.     

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 2: Complex modes

A method, based on the Hilbert–Huang spectral analysis, has been proposed by the authors to identify linear structures in which normal modes exist (i.e., real eigenvalues and eigenvectors). Frequently, all the eigenvalues and eigenvectors of linear structures are complex. In this paper, the method is extended further to identify general linear structures with complex modes using the free vibration response data polluted by noise. Measured response signals are first decomposed into modal responses using the method of Empirical Mode Decomposition with intermittency criteria. Each modal response contains the contribution of a complex conjugate pair of modes with a unique frequency and a damping ratio. Then, each modal response is decomposed in the frequency–time domain to yield instantaneous phase angle and amplitude using the Hilbert transform. Based on a single measurement of the impulse response time history at one appropriate location, the complex eigenvalues of the linear structure can be identified using a simple analysis procedure. When the response time histories are measured at all locations, the proposed methodology is capable of identifying the complex mode shapes as well as the mass, damping and stiffness matrices of the structure. The effectiveness and accuracy of the method presented are illustrated through numerical simulations. It is demonstrated that dynamic characteristics of linear structures with complex modes can be identified effectively using the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

Corrective eccentricities for assessment by the nonlinear static method of 3D structures subjected to bidirectional ground motions

Nonlinear static methods are reliable in the evaluation of the seismic response of planar structural schemes, but they are not very effective in the assessment of three-dimensional building structures. The authors of this paper have recently proposed a nonlinear static approach for asymmetric structures, which is an improvement on that stipulated by seismic codes. This method is based on the observation that the distribution of the maximum dynamic displacements of the deck can be enveloped by two pushover analyses performed by applying the lateral force with two eccentricities with respect to the center of mass of the deck. These eccentricities, named “corrective eccentricities”, are defined so that the two corresponding pushover analyses provide displacements that are equal to those evaluated by nonlinear dynamic analysis at the two sides of the deck. In this paper, the corrective eccentricities are determined for a wide set of single-story systems. The equations for their analytical evaluation are determined and their reliability is demonstrated. Finally, the analysis of a multi-story structure is conducted to show how the method can be applied to real buildings. Copyright © 2012 John Wiley & Sons, Ltd.     

某大型复杂结构地震响应与控制研究

针对某大型复杂结构的地震响应及控制策略进行了研究。首先,建立该结构的整体有限元模型,采用Ritz向量法对结构进行模态分析,研究该复杂结构的振动特性,基于位能加权平均法求解了该复杂结构等效阻尼比;随后,研究了不同布置型式下粘滞消能支撑的受力特点和计算模型;最后.对该结构进行减振设计,分析了不同消能支撑布置形式下的减振效果。结果表明,采用消能减震装置能有效降低复杂结构的地震响应,且并不改变结构体型;同等减震效果前提下,采用人字形支撑比采用斜撑更为经济。本文工作可供类似的大型复杂结构减震设计参考和借鉴。     

非比例阻尼线性体系地震反应计算的振型分解反应谱法

以非比例阻尼线性体系地震反应计算实数形式的一般解答为基础,推导得到了非比例阻尼线性体系水平地震作用计算的多种形式,建立了非比例阻尼线性体系地震反应计算振型分解反应谱法的基本过程与步骤。最后,以一个五层剪切型结构为例,通过与各种常用直接积分方法计算结果的比较,证实了本文非比例阻尼线性体系地震反应计算实数形式的一般解答的高精度与可靠性。通过对多种形式地震作用所得地震效应的比较,证实了非比例阻尼线性体系地震反应振型分解反应谱方法的可靠性及可行性。     

A precise integration approach for dynamic impedance of rigid strip footing on arbitrary anisotropic layered half-space

The precise integration method (PIM) is proposed for the dynamic response analysis of rigid strip footing resting on arbitrary anisotropic multi-layered half-space. In the frequency domain, the governing equation of wave motion is converted into dual vector form of first-order ordinary differential equations which is solved by PIM. Each layer is divided into a large number (say, 2^N) of mini-layers of equal thickness, within which characteristic matrices are assumed to vary following the Taylor series expansion to the fourth order. As a result, any desired accuracy of the displacements and stresses can be achieved by PIM. In addition, dual vector form equation makes it quite easily to combine two adjacent mini-layers into a new one. Each pass of combination reduces the total number of mini-layers by a half. The computational effort for the evaluation of the dynamic impedance of rigid strip footing can be reduced to a great extent. Numerical examples are provided to validate the efficiency and accuracy of the proposed approach.     

A Rosenbrock‐W method for real‐time dynamic substructuring and pseudo‐dynamic testing

A variant of the Rosenbrock‐W integration method is proposed for real‐time dynamic substructuring and pseudo‐dynamic testing. In this variant, an approximation of the Jacobian matrix that accounts for the properties of both the physical and numerical substructures is used throughout the analysis process. Only an initial estimate of the stiffness and damping properties of the physical components is required. It is demonstrated that the method is unconditionally stable provided that specific conditions are fulfilled and that the order accuracy can be maintained in the nonlinear regime without involving any matrix inversion while testing. The method also features controllable numerical energy dissipation characteristics and explicit expression of the target displacement and velocity vectors. The stability and accuracy of the proposed integration scheme are examined in the paper. The method has also been verified through hybrid testing performed of SDOF and MDOF structures with linear and highly nonlinear physical substructures. The results are compared with those obtained from the operator splitting method. An approach based on the modal decomposition principle is presented to predict the potential effect of experimental errors on the overall response during testing. Copyright © 2009 John Wiley & Sons, Ltd.