A unified formulation of the piecewise exact method for inelastic seismic demand analysis including the P‐delta effect

The non‐linear analysis of single‐degree‐of‐freedom (SDOF) systems provides the essential background information for both strength‐based design and displacement‐based evaluation/design methodologies through the development of the inelastic response spectra. The recursive solution procedure called the piecewise exact method, which is efficiently used for the response analysis of linear SDOF systems, is re‐formulated in this paper in a unified format to analyse the non‐linear SDOF systems with multi‐linear hysteresis models. The unified formulation is also capable of handling the P‐delta effect, which generally involves the negative post‐yield stiffness of the hysteresis loops. The attractiveness of the method lies in the fact that it provides the exact solution when the loading time history is composed of piecewise linear segments, a condition that is perfectly satisfied for the earthquake excitation. Based on simple recursive relationships given for positive, negative and zero effective stiffnesses, the unified form of the piecewise exact method proves to be an extremely powerful and probably the best tool for the SDOF inelastic time‐history and response spectrum analysis including the P‐delta effect. A number of examples are presented to demonstrate the implementation of the method. Copyright © 2003 John Wiley & Sons, Ltd.     

Modified precise time step integration method of structural dynamic analysis

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can provide precise numerical results that approach an exact solution at the integration points. However, difficulty arises when the algorithm is used for non-homogeneous dynamic systems, due to the inverse matrix calculation and the simulation accuracy of the applied loading. By combining the Gaussian quadrature method and state space theory with the calculation technique of matrix exponential function in the precise time step integration method, a new modified precise time step integration method （e.g., an algorithm with an arbitrary order of accuracy） is proposed. In the new method, no inverse matrix calculation or simulation of the applied loading is needed, and the computing efficiency is improved. In particular, the proposed method is independent of the quality of the matrix H. If the matrix H is singular or nearly singular, the advantage of the method is remarkable. The numerical stability of the proposed algorithm is discussed and a numerical example is given to demonstrate the validity and efficiency of the algorithm.     

Numerical models for the dynamic response of submerged floating tunnels under seismic loading

The modeling of tethering elements of seabed anchored floating structures is addressed, with particular reference to the so‐called Archimedes Bridge (submerged floating tunnel, SFT) solution for deep water crossing; attention is devoted to the design solution encompassing slender bars as anchor elements. Two numerical tools are proposed: firstly, a geometrically nonlinear finite element (NWB model), developed in previous work, has been refined in order to capture the effect of higher flexural modes of anchor bars. Secondly, a 3D beam element, based on the classical corotational formulation (CR model), has been developed and coded. Both elements are implemented in a numerical procedure for the dynamic time domain step‐by‐step analysis of nonlinear discretized systems; seismic loading is introduced by generating artificial time histories of spatially variable seismic motion. An example of application of the NWB element is shown regarding the behavior of the dynamic model of a complete SFT. The model was subjected to extreme multiple‐support seismic loading. The seismic behavior is here illustrated and commented, especially in light of the effect of higher local vibration modes of the anchor bars. Finally, a comparison between the performances of the two modeling approaches is presented. Both harmonic and seismic excitations are considered in the test; the results justify the use of the simpler NWB approach, especially in the SFT design phase. Copyright © 2008 John Wiley & Sons, Ltd.     

Vibration of surface foundations of arbitrary shapes

Presented is a systematic procedure for generating impedance (or compliance) matrices for foundations with arbitrary shapes, resting on an elastic half-space medium. A technique to decompose prescribed harmonic tractions on the half-space medium is employed to solve analytically the differential wave equations in cylindrical coordinates. However, the interaction stresses due to the vibration of a foundation with arbitrary shape are described in rectangular coordinates, and assumed to be piecewise constant in the region of the arbitrary shape. A coordinate transformation matrix is introduced for the piecewise constant tractions in order to use the solution of the differential wave equations in cylindrical coordinates. Finite element modelling is assumed in rectangular coordinates for the foundation itself. The impedance matrix is then obtained for the finite element model, using a variational principle and the reciprocal theorem. A simple example of a rigid square plate resting on a half-space medium and subjected to vertical excitation is used to demonstrate the efficiency and effectiveness of the procedure. Some numerical aspects are investigated and some possible extensions of the procedure are also discussed.     

Improved dynamic correction method in seismic analysis of both classically and non‐classically damped structures

A step‐by‐step approximate procedure taking into consideration high‐frequency modes, usually neglected in the modal analysis of both classically and non‐classically damped structures, is presented. This procedure can be considered as an extension of traditional modal correction methods, like the mode‐acceleration method and the dynamic correction method, which are very effective for structural systems subjected to forcing functions described by analytical laws. The proposed procedure, herein called improved dynamic correction method, requires two steps. In the first step, the number of differential equations of motion are reduced and consequently solved by using the first few undamped mode‐shapes. In the second step, the errors due to modal truncation are reduced by correcting the dynamic response and solving a new set of differential equations, formally similar to the original differential equations of motion. The difference between the two groups of differential equations lies in the forcing vector, which is evaluated in such a way as to correct the effects of modal truncation on applied loads. Copyright © 2001 John Wiley & Sons, Ltd.     

Ritz method for dynamic analysis of large discrete linear systems with non-proportional damping

Real and complex Ritz vector bases for dynamic analysis of large linear systems with non-proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.     

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.     

Experimental evaluation of piecewise exact solution for predicting seismic responses of spherical sliding type isolated structures

The use of base isolation for enhancing seismic resistibility has been proven as an efficient method in experimental and theoretical studies. It is usual to insert a flexible device in the horizontal direction to permit the most of relative deformation of a structure at this level. Because the rigidity of the superstructure is much higher than that of the base isolator underneath the structure, the behavior of the superstructure can be idealized as a rigid body during earthquakes. In general, hundreds of degrees of freedom and a step‐by‐step time history analysis are the basic requirements for calculating the seismic response of a base‐isolated structure under earthquakes. In order to develop a simple tool which can be easily adopted for calculating the seismic responses of the spherical sliding type isolated structures, a piecewise exact solution for predicting the seismic responses of base‐isolated structures has been derived in this study. The comparison between the experimental results conducted at the National Center for Research on Earthquake Engineering (NCREE) in Taiwan and the analytical results obtained from the piecewise exact solution show that the formulation derived in this study can predict the seismic responses of the base‐isolated structure with a very high accuracy. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic stiffness matrices for two circular foundations

A systematic procedure is presented for generating dynamic stiffness matrices for two independent circular foundations on an elastic half-space medium. With the technique reported in References 1–3, the analytic solution of three-dimensional (3D) wave equations satisfying the prescribed traction due to the vibration of one circular foundation can be found. Since there are two analytic solutions for two prescribed tractions due to the vibrations of two circular foundations, the principle of superposition must be used to obtain the total solution. The interaction stresses (prescribed tractions) are assumed to be piecewise linear in the r-directions of both cylindrical co-ordinates for the two circular foundations. Then, the variational principle and the reciprocal theorem are employed to generate the dynamic stiffness matrices for the two foundations. In the process of employing the variational principle, a co-ordinate transformation matrix between two cylindrical co-ordinate systems is introduced. Some numerical results of dynamic stiffness matrices for the interaction of two identical rigid circular foundations are presented in order to show the effectiveness and efficiency of the present method, and some elaborations for its future extensions are also discussed.     

Analysis of implicit HHT‐α integration algorithm for real‐time hybrid simulation

Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a successful real‐time hybrid simulation. The implicit HHT α‐algorithm is a popular integration algorithm for conducting structural dynamic time history analysis because of its desirable properties of unconditional stability for linear elastic structures and controllable numerical damping for high frequencies. The implicit form of the algorithm, however, requires iterations for nonlinear structures, which is undesirable for real‐time hybrid simulation. Consequently, the HHT α‐algorithm has been implemented for real‐time hybrid simulation using a fixed number of substep iterations. The resulting HHT α‐algorithm with a fixed number of substep iterations is believed to be unconditionally stable for linear elastic structures, but research on its stability and accuracy for nonlinear structures is quite limited. In this paper, a discrete transfer function approach is utilized to analyze the HHT α‐algorithm with a fixed number of substep iterations. The algorithm is shown to be unconditionally stable for linear elastic structures, but only conditionally stable for nonlinear softening or hardening structures. The equivalent damping of the algorithm is shown to be almost the same as that of the original HHT α‐algorithm, while the period elongation varies depending on the structural nonlinearity and the size of the integration time‐step. A modified form of the algorithm is proposed to improve its stability for use in nonlinear structures. The stability of the modified algorithm is demonstrated to be enhanced and have an accuracy that is comparable to that of the existing HHT α‐algorithm with a fixed number of substep iterations. Both numerical and real‐time hybrid simulations are conducted to verify the modified algorithm. The experimental results demonstrate the effectiveness of the modified algorithm for real‐time testing. Copyright © 2011 John Wiley & Sons, Ltd.     

System identification of the Vincent Thomas suspension bridge using earthquake records

The Vincent Thomas Bridge in the Los Angeles metropolitan area, is a critical artery for commercial traffic flow in and out of the Los Angeles Harbor, and is at risk in the seismically active Southern California region, particularly because it straddles the Palos Verdes fault zone. A combination of linear and non‐linear system identification techniques is employed to obtain a complete reduced‐order, multi‐input–multi‐output (MIMO) dynamic model of the Vincent Thomas Bridge based on the dynamic response of the structure to the 1987 Whittier and 1994 Northridge earthquakes. Starting with the available acceleration measurements (which consists of 15 accelerometers on the bridge structure and 10 accelerometers at various locations on its base), an efficient least‐squares‐based time‐domain identification procedure is applied to the data set to develop a reduced‐order, equivalent linear, multi‐degree‐of‐freedom model. Although not the main focus of this study, the linear system identification method is also combined with a non‐parametric identification technique, to generate a reduced‐order non‐linear mathematical model suitable for use in subsequent studies to predict, with good fidelity, the total response of the bridge under arbitrary dynamic environments. Results of this study yield measurements of the equivalent linear modal properties (frequencies, mode shapes and non‐proportional damping) as well as quantitative measures of the extent and nature of non‐linear interaction forces arising from strong ground shaking. It is shown that, for the particular subset of observations used in the identification procedure, the apparent non‐linearities in the system restoring forces are quite significant, and they contribute substantially to the improved fidelity of the model. Also shown is the potential of the identification technique under discussion to detect slight changes in the structure's influence coefficients, which may be indicators of damage and degradation in the structure being monitored. Difficulties associated with accurately estimating damping for lightly damped long‐span structures from their earthquake response are discussed. The technical issues raised in this paper indicate the need for added spatial resolution in sensor instrumentation to obtain identified mathematical models of structural systems with the broadest range of validity. Copyright © 2003 John Wiley & Sons, Ltd.     

The dynamic response of seabed anchored floating tunnels under seismic excitation

In this paper a procedure for analysing the seismic response of seabed anchored floating tunnels is presented. The first step of the research was the development of an ‘ad hoc’ finite element for modelling the behaviour of anchor elements, with particular reference to the problem of transverse oscillations under time varying axial loads. The element was subsequently inserted in a step‐by‐step procedure for the numerical analysis of non‐linear response to multiple‐support seismic input; the procedure encompasses simplified modelling of fluid–structure and soil–structure interaction effects. An example of an application is given concerning two 4680 m long floating tunnels with different seabed profiles. Copyright © 2000 John Wiley & Sons, Ltd.     

一种基于多项式外推的局部透射边界位移解(外行波为非平面波情形)

一般情况下,通过人工边界向外透射的波动常为非平面波。对于近场非平面波在人工边界上的透射问题,不能采用简单的平面波透射方法。本文针对近场非平面波在人工边界上的透射特点,分析了非平面波沿人工边界法向视波速的变化规律及其近场失稳机制,推导了适合近场非平面波的、人工边界点位移解的不等步距线性外推公式。     

Near‐optimal piecewise linear fits of static pushover capacity curves for equivalent SDOF analysis

The piecewise linear (‘multilinear’) approximation of realistic force‐deformation capacity curves is investigated for structural systems incorporating generalized plastic, hardening, and negative stiffness behaviors. This fitting process factually links capacity and demand and lies at the core of nonlinear static assessment procedures. Despite codification, the various fitting rules used can produce highly heterogeneous results for the same capacity curve, especially for the highly‐curved backbones resulting from the gradual plasticization or the progressive failures of structural elements. To achieve an improved fit, the error introduced by the approximation is quantified by studying it at the single‐degree‐of‐freedom level, thus avoiding any issues related to multi‐degree‐of‐freedom versus single‐degree‐of‐freedom realizations. Incremental dynamic analysis is employed to enable a direct comparison of the actual backbones versus their candidate piecewise linear approximations in terms of the spectral acceleration capacity for a continuum of limit‐states. In all cases, current code‐based procedures are found to be highly biased wherever widespread significant stiffness changes occur, generally leading to very conservative estimates of performance. The practical rules determined allow, instead, the definition of standardized low‐bias bilinear, trilinear, or quadrilinear approximations, regardless of the details of the capacity curve shape. Copyright © 2012 John Wiley & Sons, Ltd.     

Elastic and inelastic drift performance optimization for reinforced concrete buildings under earthquake loads

This paper presents an effective optimization technique for the elastic and inelastic drift performance design of reinforced concrete buildings under response spectrum loading and pushover loading. Attempts have been made to develop an automatic optimal elastic and inelastic drift design of concrete framework structures. The entire optimization procedure can be divided into elastic design optimization and inelastic design optimization. Using the principle of virtual work, the elastic drift response generated by the response spectrum loading and the inelastic drift response produced by the non‐linear pushover loading can be explicitly expressed in terms of element sizing design variables. The optimization methodology for the solution of the explicit design problem of buildings is fundamentally based on the Optimality Criteria approach. One ten‐story, two‐bay building frame example is presented to illustrate the effectiveness and practicality of the proposed optimal design method. While rapid convergence in a few design cycles is found in the elastic optimization process, relatively slow but steady and smooth convergence of the optimal performance‐based design is found in the inelastic optimization process. Copyright © 2004 John Wiley & Sons, Ltd.     

Kinematic transformations for planar multi‐directional pseudodynamic testing

The pseudodynamic (PSD) test method imposes command displacements to a test structure for a given time step. The measured restoring forces and displaced position achieved in the test structure are then used to integrate the equations of motion to determine the command displacements for the next time step. Multi‐directional displacements of the test structure can introduce error in the measured restoring forces and displaced position. The subsequently determined command displacements will not be correct unless the effects of the multi‐directional displacements are considered. This paper presents two approaches for correcting kinematic errors in planar multi‐directional PSD testing, where the test structure is loaded through a rigid loading block. The first approach, referred to as the incremental kinematic transformation method, employs linear displacement transformations within each time step. The second method, referred to as the total kinematic transformation method, is based on accurate nonlinear displacement transformations. Using three displacement sensors and the trigonometric law of cosines, this second method enables the simultaneous nonlinear equations that express the motion of the loading block to be solved without using iteration. The formulation and example applications for each method are given. Results from numerical simulations and laboratory experiments show that the total transformation method maintains accuracy, while the incremental transformation method may accumulate error if the incremental rotation of the loading block is not small over the time step. A procedure for estimating the incremental error in the incremental kinematic transformation method is presented as a means to predict and possibly control the error. Copyright © 2009 John Wiley & Sons, Ltd.     

EM MODELLING USING SURFACE INTEGRAL EQUATIONS1

The theory by which the Surface Integral Equation method may be applied to the solution of electromagnetic transmission boundary value problems is presented. For a 3D target of arbitrary electrical property contrast with its host medium excited by an arbitrary time-harmonic source, two integral equations are derived which need to be simultaneously solved for tangential electric and magnetic source density on the target's surface. If the target is 2D, though still excited by an arbitrary source (the 2½ D case), the problem is best solved in the transform domain for a number of different wavenumbers in the target's strike direction. Then a set of four simultaneous scalar integral equations needs to be solved for the components of the surface source density transforms in the target's strike direction and in the direction of the tangent vector to the target's cross-sectional contour. Examples are presented in which the 2½D problem is solved numerically using the method of moments with piecewise linear basis functions. Although the results generally compare well with analytical solutions, or solutions obtained numerically by other means, errors appear in the calculation of the real response of these targets to excitation by a magnetic dipole source at low frequencies. This is attributed to ill-conditioning of the system resulting from a non-unique solution at zero frequency.     

Modal analysis of vibrating viscoplastic composite beams on multiple supports

A vibration analysis for viscoplastic, shear deformable composite beams is presented, where non-linear strains consistently are treated as an additional loading of the linear elastic structure due to fictitious sources of eigenstresses. Therefore, linear solution methods such as mode superposition and Green's functions become applicable, and a fast, computer-oriented solution strategy is developed. The non-linear structural response is found as the sum of the corresponding linear elastic response and the solution due to the additional loading by the inelastic strains, where the sources of eigenstresses are calculated from the material's law in a time-stepping procedure. The method is demonstrated using a composite sandwich beam with thin surface layers on multiple supports, which are excited independently. Elastoplastic material behaviour of endochronic type as well as a viscoplastic material's law are considered simultaneously.     

Time domain dynamic analysis of piles under impact loading

In this paper, time domain dynamic analysis of piles under impact loading is presented. For this purpose a hybrid boundary element technique is implemented. Linear beam column finite elements are used to model the piles and resulting governing equations are solved using an implicit integration scheme. The continuum is assumed to be elastic and an efficient step-by-step time integration scheme is implemented by using an approximate half space integral formulation. By enforcing displacement equilibrium conditions at each time step, a system of equations is generated which yields the solution. Results of this time domain formulation under linear material behavior are compared with Laplace domain results to validate the methods.