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.     

微分求积法在结构动力分析中的应用

微分求积法（DQM）是1种求解微分方程初（边）值问题的数值方法,通常以较小的计算工作量即可获得较高的数值精度。这种方法应用于工程领域时多用来解决梁、板等结构的静力分析或结构特征值分析等问题,即对边值问题的微分方程的求解。结构动力分析属于初值问题,荷载和结构反应都具有特殊性,直接套用DQM求解边值问题并不能获得问题的解。本文尝试利用微分求积原理建立求解结构动力反应的具体方法。借鉴单元法的思想,将荷载持时划分为若干个时步,在每个时步内对动态荷载和结构反应进行离散,然后用DQM对时步逐个进行求解,得到体系在整个时域内的反应过程。通过对3种不同自振周期的线弹性单自由度体系在不同频率简谐激励下反应的计算,阐释了本文方法的可行性以及高精度、高效率的特点,通过数值试验确定了时步内相对较优的节点数,并为时步长度的选取提供了建议。     

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.     

Dynamic response of an elastic-viscoplastic system in modal co-ordinates

A dynamic analysis of elastic–viscoplastic systems, incorporating the modal co-ordinate transformation technique, is presented. The formulation results in uncoupled incremental equations of motion with respect to the modal co-ordinates. The elastic–viscoplastic model adopted allows the analysis not to involve yielding regions and loading/unloading processes. An implicit Runge–Kutta scheme together with the Newton–Raphson method are used to solve the non-linear constitutive equations. Stability and accuracy of the numerical solution are improved by utilizing a local time step sub-incrementing procedure. Applications of the analyses to multi-storey shear buildings show that good results can be obtained for the maximum displacement response by including only a few lower modes in the computation, but the prediction of the ductility factor response tends to underestimate the peak values when too few modes are used. In addition, stable and valid results can be obtained even with a sizable time step increment.     

三维土-结构动力相互作用体系分析的两步时域显式波动有限元过程

为减少直接分析三维大尺度复杂土-结构动力相互作用问题的计算量,提高计算效率,本文直接从波动方程出发,提出了较常规子结构法更简单的两步简化计算过程,即第一步简化上部复杂结构体系为集中质量杆系模型,并求基础处等效输入,第二步通过等效输入求上部结构各位置的动力反应.其中第一步计算主要采用集中质量显式有限单元法结合局部透射人工...     

Real‐time distributed hybrid testing: coupling geographically distributed scientific equipment across the Internet to extend seismic testing capabilities

Large‐scale testing and qualification of structural systems and their components is crucial for the development of earthquake engineering knowledge and practice. However, laboratory capacity is often limited when attempting larger experiments due to the sheer size of the structures involved. To overcome traditional laboratory capacity limitations, we present a new earthquake engineering testing method: real‐time distributed hybrid testing. Extending current approaches, the technique enables geographically distributed scientific equipment including controllers, dynamic actuators and sensors to be coupled across the Internet in real‐time. As a result, hybrid structural emulations consisting of physical and numerical substructures need no longer be limited to a single laboratory. Larger experiments may distribute substructures across laboratories located in different cities whilst maintaining correct dynamic coupling, required to accurately capture physical rate effects. The various aspects of the distributed testing environment have been considered. In particular, to ensure accurate control across an environment not designed for real‐time testing, new higher level control protocols are introduced acting over an optimised communication system. New large time‐step prediction algorithms are used, capable of overcoming both local actuation and distributed system delays. An overview of the architecture and algorithms developed is presented together with results demonstrating a number of current capabilities. Copyright © 2013 John Wiley & Sons, Ltd.     

结构动力反应分析中的一种显式输入反演方法

本文简要总结，评述了结构动力反应分析中两类输入反演方法（脉动函数法和直接时域法）的优缺点，在此基础上通过应用文献[1]中提出的3阶显式方法，获得了一个关于结构输入反演问题的新解法－显式直接时域法，新方法集合了脉冲函数法和直接时域法的优点，既可按显式求解动力微方程，又适用于非线性系统的动力反演问题，最后，通过算例对该方法进行了数值验证。     

A simple error estimator and adaptive time stepping procedure for dynamic analysis

A simple local error estimator is presented for time integration schemes in dynamic analysis. This error estimator involves only a small computational cost. The time step size is adaptively adjusted so that the local error at each time step is within a prescribed accuracy. It is found that the estimator performs well under various circumstances and provides an economical adaptive process. Attempts to estimate the global time integration error are also reported.     

Real‐time hybrid testing using the unconditionally stable explicit CR integration algorithm

Real‐time hybrid testing combines experimental testing and numerical simulation, and provides a viable alternative for the dynamic testing of structural systems. An integration algorithm is used in real‐time hybrid testing to compute the structural response based on feedback restoring forces from experimental and analytical substructures. Explicit integration algorithms are usually preferred over implicit algorithms as they do not require iteration and are therefore computationally efficient. The time step size for explicit integration algorithms, which are typically conditionally stable, can be extremely small in order to avoid numerical stability when the number of degree‐of‐freedom of the structure becomes large. This paper presents the implementation and application of a newly developed unconditionally stable explicit integration algorithm for real‐time hybrid testing. The development of the integration algorithm is briefly reviewed. An extrapolation procedure is introduced in the implementation of the algorithm for real‐time testing to ensure the continuous movement of the servo‐hydraulic actuator. The stability of the implemented integration algorithm is investigated using control theory. Real‐time hybrid test results of single‐degree‐of‐freedom and multi‐degree‐of‐freedom structures with a passive elastomeric damper subjected to earthquake ground motion are presented. The explicit integration algorithm is shown to enable the exceptional real‐time hybrid test results to be achieved. Copyright © 2008 John Wiley & Sons, Ltd.     

AN ITERATIVE SOLUTION METHOD FOR DYNAMIC RESPONSE OF BRIDGE–VEHICLES SYSTEMS

An iterative solution method is presented and illustrated to analyse the dynamic response of bridge–vehicle systems. The method consists in dividing the whole system into 2 subsystems at the interface of the bridge and vehicles; these 2 subsystems are solved separately; their compatibility at the interface is achieved by an iterative procedure with under-relaxation or with Aitken acceleration. The characteristics of this method are explained on a simplified system with 2 degrees of freedom (DOF). The numerical results for a simple example demonstrate the high performances of the proposed method: good convergence rate and high accuracy. Finally, the method is applied to a practical example: the linear dynamic response of the Yangtze-River Bridge at Wuhan under a moving train with 2 locomotives and 4 freight cars. The efficiency is attained because neither formation nor factorisation of the coefficient matrices for the equations of the system are needed at every time step in linear analysis. The Aitken acceleration technique is more efficient in systems with multi-degrees of freedom than the relaxation technique. The proposed method will be even more efficient in non-linear dynamic response because, in this case, the iterations are necessary whether the system is solved as a whole or not.     

A new family of generalized‐α time integration algorithms without overshoot for structural dynamics

A new family of generalized‐α (G‐α) algorithm without overshoot is presented by introducing seven free parameters into the single‐step three‐stage formulation for solution of structural dynamic problems. It is proved through finite difference analysis that these algorithms are unconditionally stable, second‐order accurate and numerical dissipation controllable. The comparison of the new G‐α algorithms with the commonly used G‐α algorithms shows that the newly developed algorithms have the advantage of eliminating the overshooting characteristics exhibited by the commonly used algorithms while their excellent property of dissipation is preserved. The numerical simulation results obtained using a single‐degree‐of‐freedom system and a two‐degree‐of‐freedom system to represent the character of typical large systems coincide well with the results of theoretical analyses. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

A methodology for dynamic analysis of linear structure-foundation systems with local non-linearities

An efficient computational technique is presented for the dynamic analysis of large linear structural systems with local non-linearities. The earthquake response evaluation for many practical structures belongs to this class of problems. The technique provides a rational approach to the earthquake-resistant design of structure-foundation systems with predetermined non-linearities occurring along the structure-foundation interface. Various possibilities for base isolation systems are naturally fitted within the proposed framework. In particular, we address uplifting of the structure as a natural base isolation concept. We use the dynamic substructuring technique and an efficient numerical algorithm which accommodates non-proportional damping as a consistent way to reduce significantly the computational effort, which is in sharp contrast to the vast majority of ad-hoc simplified models used for the same purpose. A numerical example which demonstrates the vibration isolation effect when the uplifting of the concrete gravity dam occurs is also presented.     

Stability analysis of MDOF real‐time dynamic hybrid testing systems using the discrete‐time root locus technique

The time delay resulting from the servo hydraulic systems can potentially destabilize the real‐time dynamic hybrid testing (RTDHT) systems. In this paper, the discrete‐time root locus technique is adopted to investigate the delay‐dependent stability performance of MDOF RTDHT systems. Stability analysis of an idealized two‐story shear frame with two DOFs is first performed to illustrate the proposed method. The delay‐dependent stability condition is presented for various structural properties, time delay, and integration time steps. Effects of delay compensation methods on stability are also investigated. Then, the proposed method is applied to analyze the delay‐dependent stability of a single shaking table RTDHT system with an 18‐DOF finite element numerical substructure, and corresponding RTDHTs are carried out to verify the theoretical results. Furthermore, the stability behavior of a finite element RTDHT system with two physical substructures, loaded by twin shaking tables, is theoretically and experimentally investigated. All experimental results convincingly demonstrate that the delay‐dependent stability analysis on the basis of the discrete‐time root locus technique is feasible. Copyright © 2014 John Wiley & Sons, Ltd.     

Non-linear seismic analysis of base-isolated RC frame structures

In this paper, seismic analysis of plane RC frame structures with High Damping Rubber Bearings (HDRBs) base-isolation systems is performed in the non linear range. For RC members, a modified version of hysteretic Park model is used. For HDRB isolators, a new hysteretic model is presented, which is able to accurately predict the mechanical response in the large strain range. The dynamic equilibrium equations are solved making use, at each time step, of a block iterative Newton–Raphson scheme: the frame is divided into superelements (beams and columns) with master nodes at the extremities and internal local nodes for the computation of relations between end moments and relative rotations at superelement extremities. The effectiveness of HDRB base-isolation systems to reduce inelastic deformations in the RC superstructures is investigated through some numerical examples. Copyright © 1999 John Wiley & Sons, Ltd.     

Dynamic analysis of sliding structures with unsynchronized support motions

The dynamic analysis of sliding structures is complicated due to the presence of friction. Synchronization of the kinematics of all the isolation bearings is often granted to simplify the task. This, however, may lead to inaccurate prediction of the structural responses under certain circumstances. Stepped structures or continuous bridges with seismic isolation are notable examples where unsynchronized bearing motions are expected. In this paper, a logically simple and numerically efficient procedure is proposed to solve the dynamic problem of sliding systems with unsynchronized support motions. The motion equations for the sliding and non‐sliding modes of the isolated structure are unified into a single equation that is represented as a difference equation in a discrete‐time state‐space form and the base shear forces between the sliding interfaces can be determined through simple matrix algebraic analysis. The responses of the sliding structure can be obtained recursively from the discrete‐time version of the motion equation with constant integration time step even during the transitions between the non‐sliding and sliding phases. Therefore, both accuracy and efficiency in the dynamic analysis of the highly non‐linear system can be enhanced to a large extent. Rigorous assessment of seismic structures with unsynchronized support motions has been carried out for both a stepped structure and a continuous bridge. Effectiveness of friction pendulum bearings for earthquake protection of such structures has been verified. Moreover, evident unsynchronized sliding motions of the friction bearings have been observed, confirming the necessity to deal with each of the bearings independently in the analytical model. Copyright © 2000 John Wiley & Sons, Ltd.     

Partial eigensolution of damped structural systems by Arnoldi's method

An efficient numerical algorithm is developed to solve the quadratic eigenvalue problems arising in the dynamic analysis of damped structural systems. The algorithm can even be applied to structural systems with non-symmetric matrices. The algorithm is based on the use of Arnoldi's method to generate a Krylov subspace of trial vectors, which is then used to reduce a large eigenvalue problem to a much smaller one. The reduced eigenvalue problem is solved and the solutions are used to construct approximate solutions to the original large system. In the process, the algorithm takes full advantage of the sparseness and symmetry of the system matrices and requires no complex arithmetic, therefore, making it very economical for use in solving large problems. The numerical results from test examples are presented to demonstrate that a large fraction of the approximate solutions calculated are very accurate, indicating that the algorithm is highly effective for extracting a number of vibration modes for a large dynamic system, whether it is lightly or heavily damped.     

Assessing the seismic behavior of steel moment frames equipped by elliptical brace through incremental dynamic analysis (IDA)

Incremental dynamic analysis and nonlinear static pushover analysis are carried out on a performance-based design to determine the seismic demands and capacities of an elliptic braced moment resisting frame (ELBRF). The objective is to assess ductility, overstrength and response modification factors in a modern steel-braced structural system based on incremental dynamic analysis. This integrated system is connected to a beam and column with an appropriate length while providing enough architectural space to allow for an opening without having the common problems associated with architectural spaces in braced systems. Several different classes of buildings are considered on soil type II. Linear dynamic analysis, nonlinear static pushover analysis and incremental nonlinear dynamic analysis related to 12 records from past earthquakes are carried out using OpenSees software. The factors of ductility, overstrength and response modification are calculated for this system. The values of 9.5 and 6.5 are found and suggested only for the response modification factor for ELBRF systems in allowable stress and ultimate limit state methods, respectively. The fragility curves are plotted for the first time for this type of bracing, which contributes to the assessment of building seismic damage.     

不同阻尼特性材料组合结构的弹塑性动力时程响应计算

本文提出一种不必形成与存储总阻尼矩阵的非线性结构动力时程分析的算法,用于计算非比例、非经典阻尼结构弹塑性动力响应。该算法的软件实现简单,对现有比例阻尼结构分析程序稍加修改即可。基于此算法的计算机程序满足了由两种阻尼特性材料建造的建筑结构非线性动力时程分析的需要。