Improved methods to transform frequency‐dependent complex stiffness to time domain

A method to transform the frequency‐dependent complex stiffness to the impulse response in the time domain was proposed in the previous paper. However, there is a problem in that the accuracy and the convergence of the transformed impulse response are not good in some cases. Moreover, the hysteretic damping was not considered in the previous study although it is essential for practical purposes. In this paper, transform method improvements are proposed. First, the accuracy and the convergence are improved by taking the concept of virtual mass into account. Then, a more improved method for transforming the complex stiffness with large hysteretic damping to the time domain is proposed using the least square method. It is well known that the rigorous transform of the hysteretic damping is impossible because it is non‐causal. So this method is thought to be an approximate causalization process. Copyright © 2006 John Wiley & Sons, Ltd.     

Time‐harmonic response of non‐homogeneous elastic unbounded domains using the scaled boundary finite‐element method

The scaled boundary finite‐element method is extended to simulate time‐harmonic responses of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. The unbounded domains and the elasticity matrices are transformed to the scaled boundary coordinates. The scaled boundary finite‐element equation in displacement amplitudes are derived directly from the governing equations of elastodynamics. To enforce the radiation condition at infinity, an asymptotic expansion of the dynamic‐stiffness matrix for high frequency is developed. The dynamic‐stiffness matrix at lower frequency is obtained by numerical integration of ordinary differential equations. Only the boundary is discretized yielding a reduction of the spatial dimension by one. No fundamental solution is required. Material anisotropy is modelled without additional efforts. Examples of two‐ and three‐dimensional non‐homogeneous isotropic and transversely isotropic unbounded domains are presented. The results demonstrate the accuracy and simplicity of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

Operator‐splitting method for real‐time substructure testing

It has been shown that the operator‐splitting method (OSM) provides explicit and unconditionally stable solutions for quasi‐static pseudo‐dynamic substructure testing. However, the OSM provides only an explicit target displacement but not an explicit target velocity, so that it is essentially an implicit method for real‐time substructure testing (RST) when the velocity‐dependent restoring force is considered. This paper proposes a target velocity formulation based on the forward difference of the predicted displacements so as to render the OSM explicit for RST. The stability and accuracy of the resulting OSM‐RST algorithm are investigated. It is shown that the OSM‐RST is unconditionally stable so long as the non‐linear stiffness and damping are of the softening type (i.e. the tangent stiffness and damping never exceed the initial values). The stability of the OSM‐RST for structures with infinite tangent damping coefficient or stiffness is also proved, and the stability of the method for MDOF structures with a non‐classical damping matrix is demonstrated by an energy criterion. The effects of actuator delay and compensation are analysed based on the bilinear approximation of the actuator step response. Experiments on damped SDOF and MDOF structures verify that the stability of the OSM‐RST is preserved when the experimental substructure generates velocity‐dependent reaction forces, whereas the stability of real‐time substructure tests based on the central difference method is worsened by the damping of the specimen. Copyright © 2005 John Wiley & Sons, Ltd.     

Simple formulas for the dynamic stiffness of pile groups

Simple formulas are derived for the dynamic stiffness of pile group foundations subjected to horizontal and rocking dynamic loads. The formulations are based on the construction of a general model of impedance matrices as the condensation of matrices of mass, damping, and stiffness, and on the identification of the values of these matrices on an extensive database of numerical experiments computed using coupled finite element–boundary element models. The formulations obtained can be readily used for the design of both floating piles on homogeneous half‐space and end‐bearing piles and are applicable for a wide range of mechanical and geometrical parameters of the soil and piles, in particular for large pile groups. For the seismic design of a building, the use of the simple formulas rather than a full computational model is shown to induce little error on the evaluation of the response spectra and time histories. Copyright © 2009 John Wiley & Sons, Ltd.     

Consistent lumped-parameter models for unbounded soil: Frequency-independent stiffness,damping and mass matrices

A systematic procedure to develop consistent (symmetric) stiffness, damping and mass matrices with real coefficients to represent any unbounded soil is developed. These property matrices are based on the lumped-parameter models of Reference 1. Either stiffness and damping matrices corresponding to first-order differential equations involving the internal degrees of freedom and those on the structure-soil interface result or, alternatively, in addition mass matrices are introduced, corresponding to second-order differential equations, which reduce the number of internal degrees of freedom by a factor 2. The stiffness, damping and mass matrices can easily be incorporated in a general-purpose structural dynamics program working in the time domain, whereby the structure can even be non-linear.     

Optimal implicit staggered‐grid finite‐difference schemes based on the sampling approximation method for seismic modelling

We propose new implicit staggered‐grid finite‐difference schemes with optimal coefficients based on the sampling approximation method to improve the numerical solution accuracy for seismic modelling. We first derive the optimized implicit staggered‐grid finite‐difference coefficients of arbitrary even‐order accuracy for the first‐order spatial derivatives using the plane‐wave theory and the direct sampling approximation method. Then, the implicit staggered‐grid finite‐difference coefficients based on sampling approximation, which can widen the range of wavenumber with great accuracy, are used to solve the first‐order spatial derivatives. By comparing the numerical dispersion of the implicit staggered‐grid finite‐difference schemes based on sampling approximation, Taylor series expansion, and least squares, we find that the optimal implicit staggered‐grid finite‐difference scheme based on sampling approximation achieves greater precision than that based on Taylor series expansion over a wider range of wavenumbers, although it has similar accuracy to that based on least squares. Finally, we apply the implicit staggered‐grid finite difference based on sampling approximation to numerical modelling. The modelling results demonstrate that the new optimal method can efficiently suppress numerical dispersion and lead to greater accuracy compared with the implicit staggered‐grid finite difference based on Taylor series expansion. In addition, the results also indicate the computational cost of the implicit staggered‐grid finite difference based on sampling approximation is almost the same as the implicit staggered‐grid finite difference based on Taylor series expansion.     

Simplified analysis of mid‐story seismically isolated buildings

The mid‐story isolation design method is recently gaining popularity for the seismic protective design of buildings located in the areas of high population. In a mid‐story isolated building, the isolation system is incorporated into the mid‐story rather than the base of the building. In this paper, the dynamic characteristics and seismic responses of mid‐story isolated buildings are investigated using a simplified three‐lumped‐mass structural model for which equivalent linear properties are formulated. From the parametric study, it is found that the nominal frequencies of the superstructure and the substructure, respectively, above and below the isolation system have significant influences on the isolation frequency and equivalent damping ratio of a mid‐story isolated building. Moreover, the mass and stiffness of the substructure are of greater significance than the superstructure in affecting the dynamic characteristics of the isolated building. Besides, based on the response spectrum analysis, it is noted that the higher mode responses may contribute significantly to the story shear force of the substructure. Consequently, the equivalent lateral force procedure of design codes should carefully include the effects of higher modes. Copyright © 2010 John Wiley & Sons, Ltd.     

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 1: normal modes

Based on the Hilbert–Huang spectral analysis, a method is proposed to identify multi‐degree‐of‐freedom (MDOF) linear systems using measured free vibration time histories. For MDOF systems, the normal modes have been assumed to exist. In this method, the measured response data, which are polluted by noises, are first decomposed into modal responses using the empirical mode decomposition (EMD) approach with intermittency criteria. Then, the Hilbert transform is applied to each modal response to obtain the instantaneous amplitude and phase angle time histories. A linear least‐square fit procedure is proposed to identify the natural frequency and damping ratio from the instantaneous amplitude and phase angle for each modal response. Based on a single measurement of the free vibration time history at one appropriate location, natural frequencies and damping ratios can be identified. When the responses at all degrees of freedom are measured, the mode shapes and the physical mass, damping and stiffness matrices of the structure can be determined. The applications of the proposed method are illustrated using three linear systems with different dynamic characteristics. Numerical simulation results demonstrate that the proposed system identification method yields quite accurate results, and it offers a new and effective tool for the system identification of linear structures in which normal modes exist. Copyright © 2003 John Wiley & Sons, Ltd.     

Verification of system identification utilizing shaking table tests of a full‐scale 4‐story steel building

This paper verifies the feasibility of the proposed system identification methods by utilizing shaking table tests of a full‐scale four‐story steel building at E‐Defense in Japan. The natural frequencies, damping ratios and modal shapes are evaluated by single‐input‐four‐output ARX models. These modal parameters are prepared to identify the mass, damping and stiffness matrices when the objective structure is modelled as a four degrees of freedom (4DOF) linear shear building in each horizontal direction. The nonlinearity in stiffness is expressed as a Bouc–Wen hysteretic system when it is modelled as a 4DOF nonlinear shear building. The identified hysteretic curves of all stories are compared to the corresponding experimental results. The simple damage detection is implemented using single‐input‐single‐output ARX models, which require only two measurements in each horizontal direction. The modal parameters are equivalent‐linearly evaluated by the recursive Least Squares Method with a forgetting factor. When the structure is damaged, its natural frequencies decrease, and the corresponding damping ratios increase. The fluctuation of the identified modal properties is the indirect information for damage detection of the structure. Copyright © 2015 John Wiley & Sons, Ltd.     

An improved capacity spectrum method for ATC‐40

In order to account for the non‐linear behavior of structures via non‐linear static procedure, the capacity spectrum method has been adopted by ATC‐40 for evaluation and retrofit of reinforced concrete buildings. For elastic‐perfectly‐plastic SDOF systems, the accuracy of the capacity spectrum method depends only on the acceleration response spectrum chosen to form the demand spectrum and the adopted model for calculating the equivalent viscous damping ratios. According to this method, the pseudo‐acceleration response spectrum (PS_a) is used to create the demand diagram. It is found that the ATC‐40 procedure, using its Type A hysteretic model, may be inaccurate especially for systems with damping ratios greater than 10% and periods longer than 0.15sec. In order to improve the accuracy of the capacity spectrum method, this study proposes to use the real absolute acceleration response spectrum (S0._a) instead of the PS_a to establish the demand diagram. The step‐by‐step procedure of the improved method and examples are implemented in this paper to illustrate the calculations of earthquake‐induced deformations. In addition, three selected models of equivalent viscous damping are also compared in this paper to assess the accuracy of the model used in the ATC‐40 procedure. Results show that the WJE damping model may be used by the capacity spectrum method to reasonably predict the inelastic displacements when the ductility demand (μ) of the structures is less than 4, whereas the damping model proposed by Kowalsky can be implemented when μ>4.0. Alternatively, the damping model proposed by Kowalsky may be used to calculate the equivalent viscous damping for the entire range of ductility. Copyright © 2003 John Wiley & Sons, Ltd.     

On the accuracy of simplified methods for the analysis of isolated bridges

To foster the use of seismic isolation in structures, existing guidelines strive to formulate design methods which are simple and accessible to non‐specialized engineers. On the other hand, not all of the simplifying provisions adopted by the norms can be said to have been adequately tested to provide a consistent level of accuracy. The study attempts, in particular, to elucidate three aspects related to the methods of analysis for linear or linearized isolated bridges on which little or no advice can be found in the norms. The first one is about the way one has to account for the fact that damping matrices of isolated bridges are never of proportional type. The present study demonstrates, through a number of typical applications, that classical modal analysis, using real modes and the diagonal terms of the modal damping matrices, still provide a fully acceptable approximation. The second and third aspects are related to the use of linearization expressions extended to the analysis of hyperstatic bridges. Parametric analyses conducted in the study show that none of the formulas in current use gives satisfactory results for both the displacement and the force responses, a requirement for a reliable design of an isolated bridge. How to use the equivalent linear parameters, and in particular the isolators equivalent damping ratios, in the context of a modal analysis, is treated next. This problem is seldom if ever mentioned in the norms where at most a formula is given for constructing modal damping ratios based on the damping ratios of the isolators. A rational, approximate procedure is discussed in this paper, applicable to all types of structures with non‐proportional damping, which in the case of bridges can be shown to reduce to the expression provided in the Japanese bridge design guidelines. Copyright © 2001 John Wiley & Sons, Ltd.     

Implementation and application of the unconditionally stable explicit parametrically dissipative KR‐α method for real‐time hybrid simulation

In real‐time hybrid simulations (RTHS) that utilize explicit integration algorithms, the inherent damping in the analytical substructure is generally defined using mass and initial stiffness proportional damping. This type of damping model is known to produce inaccurate results when the structure undergoes significant inelastic deformations. To alleviate the problem, a form of a nonproportional damping model often used in numerical simulations involving implicit integration algorithms can be considered. This type of damping model, however, when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion for an RTHS with too large of a time step can result in spurious high‐frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the parametrically controllable numerical energy dissipation available in the recently developed unconditionally stable explicit KR‐α method. This paper reviews the formulation of the KR‐α method and presents an efficient implementation for RTHS. Using the method, RTHS of a three‐story 0.6‐scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Copyright © 2014 John Wiley & Sons, Ltd.     

Problems encountered from the use (or misuse) of Rayleigh damping

Rayleigh damping is commonly used to provide a source of energy dissipation in analyses of structures responding to dynamic loads such as earthquake ground motions. In a finite element model, the Rayleigh damping matrix consists of a mass‐proportional part and a stiffness‐proportional part; the latter typically uses the initial linear stiffness matrix of the structure. Under certain conditions, for example, a non‐linear analysis with softening non‐linearity, the damping forces generated by such a matrix can become unrealistically large compared to the restoring forces, resulting in an analysis being unconservative. Potential problems are demonstrated in this paper through a series of examples. A remedy to these problems is proposed in which bounds are imposed on the damping forces. Copyright © 2005 John Wiley & Sons, Ltd.     

A rate‐dependent constitutive model of high damping rubber bearings: modeling and experimental verification

In this study, a constitutive model of high damping rubber bearings (HDRBs) is developed that allows the accurate representation of the force–displacement relationship including rate‐dependence for shear deformation. The proposed constitutive model consists of two hyperelastic springs and a nonlinear dashpot element and expresses the finite deformation viscoelasticity laws based on the classical Zener model. The Fletcher–Gent effect, manifested as high horizontal stiffness at small strains and caused by the carbon fillers in HDRBs, is accurately expressed through an additional stiffness correction factor α in the novel strain energy function. Several material parameters are used to simulate the responses of high damping rubber at various strain levels, and a nonlinear viscosity coefficient η is introduced to characterize the rate‐dependent property. A parameter identification scheme is applied to the results of the multi‐step relaxation tests and the cyclic shear tests, and a three‐dimensional function of the nonlinear viscosity coefficient η with respect to the strain, and strain rate is thus obtained. Finally, to investigate the accuracy and feasibility of the proposed model for application to the seismic response assessment of bridges equipped with HDRBs, an improved real‐time hybrid simulation (RTHS) test system based on the velocity loading method is developed. A single‐column bridge was used as a test bed and HDRBs was physically tested. Comparing the numerical and RTHS results, advantage of the proposed model in the accuracy of the predicted seismic response over comparable hysteretic models is demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.     

Time‐domain analysis of unbounded media using mixed‐variable formulations

Formulation of a matrix‐valued force–displacement relationship which can take radiation damping into account is of major importance when modelling unbounded domains. This can be done by means of fundamental solutions in space and time in connection with convolution integrals or by means of a frequency dependent boundary element representation, but for discrete frequencies Ω only. In this paper a method for interpolating discrete values of dynamic stiffness matrices by a continuous matrix valued rational function is proposed. The coupling between interface degrees of freedom is fully preserved. Another crucial point in soil–structure interaction analysis is how to implement an approximation in the spectral domain into a time‐domain analysis. Well‐known approaches for the scalar case are based on the partial‐fraction expansion of a scalar rational function. Here, a more general procedure, applicable to MDOF‐systems, for the transformation of spectral rational approximations into the time‐domain is introduced. Evaluation of the partial‐fraction expansion is avoided by using the so‐called mixed variables. Thus, unknowns in the time‐domain are displacements as well as forces. Copyright © 2001 John Wiley & Sons, Ltd.     

Extremely large displacement dynamic analysis of elastic–plastic plane frames

This paper presents a numerical analysis of large displacement responses of elastic–plastic plane frames under static and dynamic loads, by applying the vector‐form intrinsic finite element (VFIFE or V‐5) method. The VFIFE method defines the structure into a number of mass points, and applies Newton's second law and the internal force equilibrium to describe the motions of each mass point. By tracing the motions of all the mass points, it can analyze the large geometrical and material nonlinear changes during the motion of the structure without using the geometrical stiffness matrix and iterative procedures. Three different numerical examples are presented to demonstrate both the capability and the accuracy of the VFIFE method in a nonlinear dynamic analysis of frame structures with extremely large displacement. Copyright © 2011 John Wiley & Sons, Ltd.     

Equivalent damping of SDOF structure with Maxwell damper

To predict the maximum earthquake response of an SDOF structure with a Maxwell fluid damper or supplemental brace-viscous damper system using the seismic design response spectrum technique, a new approach is presented to determine the first- and second-order equivalent viscous damping and stiffness, the peak responses, and the damper force of the above structure. Based on the fact that the dynamic characteristics of a general linear viscoelastically damped structure are fully determined by its free vibration properties and the relaxation time constants of a Maxwell fluid damper and supplemental brace-viscous damper system in engineering practice are all small, the method of improved multiple time scales and the equivalent criterion in which all free vibration properties are the same are used to obtain the first- and second-order equivalent viscous damping and stiffness of the above structure in closed form. The accuracy of the proposed method is higher and significantly better than that of the modal strain energy method. Furthermore, in the parametric range of the requirements of the Chinese "Code for Seismic Design of Buildings", the error of the proposed second-order equivalent system for the above-mentioned engineering structure is not more than 0.5%.     

Stability and accuracy analysis of outer loop dynamics in real‐time pseudodynamic testing of SDOF systems

Real‐time pseudodynamic (PSD) testing is an experimental technique for evaluating the dynamic behaviour of a complex structure. During the test, when the targeted command displacements are not achieved by the test structure, or a delay in the measured restoring forces from the test structure exists, the reliability of the testing method is impaired. The stability and accuracy of real‐time PSD testing in the presence of amplitude error and a time delay in the restoring force is presented. Systems consisting of an elastic single degree of freedom (SDOF) structure with load‐rate independent and dependent restoring forces are considered. Bode plots are used to assess the effects of amplitude error and a time delay on the steady‐state accuracy of the system. A method called the pseudodelay technique is used to derive the exact solution to the delay differential equation for the critical time delay that causes instability of the system. The solution is expressed in terms of the test structure parameters (mass, damping, stiffness). An error in the restoring force amplitude is shown to degrade the accuracy of a real‐time PSD test but not destabilize the system, while a time delay can lead to instability. Example calculations are performed for determining the critical time delay, and numerical simulations with both a constant delay and variable delay in the restoring force are shown to agree well with the stability limit for the system based on the critical time delay solution. The simulation models are also used to investigate the effects of a time delay in the PSD test of an inelastic SDOF system. The effect of energy dissipation in an inelastic structure increases the limit for the critical time delay, due to the energy removed from the system by the energy dissipation. Copyright © 2007 John Wiley & Sons, Ltd.     

LMS‐based structural health monitoring of a non‐linear rocking structure

A structure's health or level of damage can be monitored by identifying changes in structural or modal parameters. This research directly identifies changes in structural stiffness due to modelling error or damage for a post‐tensioned pre‐cast reinforced concrete frame building with rocking beam column connections and added damping and stiffness (ADAS) elements. A structural health monitoring (SHM) method based on adaptive least mean squares (LMS) filtering theory is presented that identifies changes from a simple baseline model of the structure. This method is able to track changes in the stiffness matrix, identifying when the building is (1) rocking, (2) moving in a hybrid rocking–elastic regime, or (3) responding linearly. Results are compared for two different LMS‐based SHM methods using an L ₂ error norm metric. In addition, two baseline models of the structure, one using tangential stiffness and the second a more accurate bi‐linear stiffness model, are employed. The impact of baseline model complexity is then delineated. The LMS‐based methods are able to track the non‐linearity of the system to within 15% using this metric, with the error due primarily to filter convergence rates as the structural response changes regimes while undergoing the El Centro ground motion record. The use of a bi‐linear baseline model for the SHM problem is shown to result in error metrics that are at least 50% lower than those for the tangential baseline model. Errors of 5–15% with this L ₂ error norm are fairly stringent compared to the greater than 2 × changes in stiffness undergone by the structure, however, in practice the usefulness of the results is dependent on the resolution required by the user. The impact of sampling rate is shown to be negligible over the range of 200–1000Hz, along with the choice of LMS‐based SHM method. The choice of baseline model and its level of knowledge about the actual structure is seen to be the dominant factor in achieving good results. The methods presented require 2.8–14.0 Mcycles of computation and therefore could easily be implemented in real time. Copyright © 2005 John Wiley & Sons, Ltd.