Microcomputer Solution of Large Structural Dynamic Problems

Microcomputers are currently used to solve linear problems in structural dynamics. However, realistic engineering applications require a large amount of processing operations such that computer execution time and storage requirements can become critical. It is shown in this paper that the solution of large three-dimensional linear dynamic problems using microcomputers is best accomplished by working with a reduced system of equilibrium equations expressed in generalized coordinates derived from a new load-dependent subspace Ritz-transformation method instead of the classical eigensolution procedure. Numerical applications on the microcomputer earthquake response analysis of a three-dimensional model of nearly 2000 dynamic degrees of freedom is used to illustrate the relative efficiency of the proposed solution algorithm as compared to the subspace iteration method.     

结构地震反应分析中的反应谱理论

介绍了结构地震反应分析中的地震反应谱、设计反应谱等三种反应谱理论,并阐述了他们之间的相互联系和功能区别,论述了几种多自由度体系线性动力反应分析方法,为工程抗震设计提供了便捷、有效地分析手段。     

Reliability of linear structures with parameter uncertainty under non-stationary earthquake

The present work aims towards the development of a general framework of time varying unconditional reliability evaluation of linear elastic multi degree of freedom structures with uncertain parameter subjected to the generalized earthquake ground motion, a non-stationary process both in amplitude and frequency content. The formulation is developed in double frequency spectrum to derive the generalized power spectral density function of the structural responses. The time varying reliability is evaluated using conditional crossing rate following the Vanmarcke’s modification. The perturbation based stochastic finite element method is utilized in deriving unconditional reliability. An idealized three dimensional dam structure subjected to El Centro (1940) earthquake is taken up to elucidate the proposed unconditional time varying reliability computation procedure based on the maximum top displacement and base shear criteria. The results are presented to compare the change in reliabilities of the uncertain system with that of deterministic system and associated variance of the reliability due to parameter uncertainty.     

基础转动结构随机地震响应分析的复模态法

对多自由度基础转动结构随机地震响应问题进行了系统研究。针对用第一振型将上部结构展开所得方程为非经典阻尼和非对称结构情况 ,用复模态法解耦 ,获得了以第一振型表示的结构地震响应的解析解 ,对单自由度体系 ,此解即为结构响应的精确解。该方法可用于隔震结构、加层减震结构、带TMD减震结构的随机响应分析与优化设计     

Evaluation of modal incremental dynamic analysis,using input energy intensity and modified bilinear curve

In this paper, a technique for the study of nonlinear performance of structures in different levels of earthquakes is developed. In this method, the Incremental Dynamic Analysis (IDA) curves are not achieved from nonlinear dynamic analysis of multi‐degree‐of‐freedom (MDF) structure. However, the procedure of constructing these curves is based on modelling of the entire structure with several single‐degrees‐of‐freedom (SDF) structures and evaluating them through the modal pushover analysis method. An innovative idea for approximating pushover curves that is based on error distribution is introduced in this investigation. Furthermore, the total input energy applied towards the SDF oscillator, which gets converted into MDF structure, is used as intensity measure to impose the different levels of scaled earthquakes. This technique possesses all the advantages of the IDA method in studying the performance of structures in different levels of earthquake. In addition, it benefits from easy usage, high solving speed and less computational CPU time and in attendance with the modified bilinear curves. In this paper, six earthquake records have been applied over four different 4, 8, 12 and 16‐storey structures. The structural responses derived by this method, in the format of multi‐modal incremental curves, containing maximum displacement, drift, hinge rotation and hysteretic energy at the end of the earthquake, have been compared with the IDA method. Comparison of the results is found to have acceptable precision and reveals good agreement between two methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimization and performance of metafoundations for seismic isolation of small modular reactors

This paper aims to study the seismic mitigation of a typical nuclear small modular reactor (SMR) where extreme loading conditions are considered by the safe shutdown earthquake. For this purpose, to reproduce the main dynamic properties of the reactor's reinforced concrete system, a detailed structural model was synthetized, also taking into account the presence of the reactor pools. Thus, to protect the reactor from strong earthquakes, finite locally resonant multiple degrees of freedom metafoundations were developed; and resonator parameters were optimized by means of an improved frequency domain multivariate and multiobjective optimization procedure. Also, the stochastic nature of the seismic input was taken into account. It is proposed: (i) a linear metafoundation endowed with multiple cells, linear springs, and linear viscous dampers; and (ii) a foundation equipped with additional nonlinear vertical quasi-zero stiffness (QZS) cells. QZS cells were obtained by horizontally precompressed springs in an unstable state with vertical springs in parallel. With this arrangement, additional flexibility and dissipation against nonsymmetrical modes of the SMR and vertical seismic loadings are proposed. It was shown in both cases, how each metafoundation was successfully optimized via a sensitivity-based parameter grouping strategy and a hybrid grid searching algorithm. Thus, the performance of the optimized metafoundations was assessed by means of frequency and time history analyses; and finally, results were compared with an SMR endowed with both rigid foundation and conventional base-isolation solutions.     

弹性多自由度系统地震输入能量的研究

本文介绍了基于能量抗震设计方法的基本思路和基本概念。根据振型分解法推导了弹性多自由度系统与单自由度系统弹性地震输入能量间的关系。研究表明,N个自由度弹性系统的地震输入能量可表示为N个弹性单自由度系统弹性输入能量的线性组合,组合系数与多自由度系统的振型和质量分布有关。通过时程分析算例,以Eleentro波输入验证了这种方法的正确性。算例结果表明,对于自由度较多的系统,可以只取前几阶振型计算即可得到足够精确的结果。     

在用RC框架结构基于位移的抗震性能评估

根据能力谱方法的基本原理 ,建立了在用RC框架结构基于位移抗震性能的评估方法。该方法采用了振型分解法建立的多自由度体系和等效单自由度体系之间的转换关系 ,以及由现行《建筑抗震设计规范》的反应谱曲线建立的结构抗震要求曲线。用本文方法对清华大学主楼 1 0层钢筋混凝土框架结构进行了抗震性能评估 ,同时进行了小震和大震作用下的弹塑性时程分析。本文方法得到的顶点位移角、层间位移角和塑性铰分布与时程分析的结果比较符合     

单层厂房考虑扭转的弹塑性地震反应分析

本文研究了单层厂房在地震动作用下屋盖保持完整的非弹性地震反应。具有不同高度的每一屋面作为一个刚性盘体,并将其质量集中于重心。这样,对应于每一质量有三个自由度(两个平移和一个转动)。 按照上述模型编制了一个电算程序研究了等高和不等高单层厂房在单向或双向地面运动作用下的非弹性地震反应。     

Performance‐based plastic design and collapse assessment of diagrid structure fused with shear link

A diagrid structure fused with shear link (DSSL) is an innovative earthquake resilient structural system. The DSSL combines the steel diagrid structural system with shear links to dissipate the earthquake energy with the goal to minimize structural repair and downtime after strong earthquake shaking. The SLs are placed between diamond‐shaped grid units and decoupled from the gravity system. To facilitate the design of the proposed DSSL system, the performance‐based plastic design (PBPD) procedure is extended to design a prototype building utilizing DSSL. Detailed finite element model is developed to simulate the non‐linear dynamic response of the structure under a range of earthquake shaking intensities. The results of non‐linear dynamic analyses show that the DSSL has excellent seismic performance and can be efficiently designed using PBPD. Lastly, detailed collapse risk assessment of the prototype building is performed using the FEMA‐P695 methodology. The result shows that the PBPD‐designed DSSL has adequate margin against collapse. Hence, it can be used as an effective seismic force resisting system.     

Mechanism and optimum design of shared tuned mass damper for twin‐tower structures connected at the top by an isolated corridor

The concept of shared tuned mass damper (STMD) for twin towers linked by a sky corridor using flexible joints is proposed in this paper. The analytical expressions of the transform functions and random earthquake responses of the flexibly connected structures are derived using a three‐degrees‐of‐freedom model system. The seismic reduction mechanism of the STMD is revealed using comparative analysis between the structures with STMD and those connected using a viscoelastic damper. The effect of the nondimensional parameters such as the frequency ratio of the two primary structures, mass ratio, tuning frequency ratio of the corridor, and damping ratio of the passive control devices on the structural seismic response is investigated. Optimum parametric analysis is performed using the principle of minimizing the displacements of both towers, and the optimal parameter formulas are established. Numerical analysis is conducted to verify the control effectiveness of the connected multi‐degree‐of‐freedom system subjected to the El Centro earthquake ground motion. The results show that the earthquake responses of the towers can be effectively reduced if the parameters of the flexible connecting elements are appropriately selected for a certain corridor mass. Moreover, a remarkable seismic reduction effect can be achieved if the towers have similar dynamic properties.     

Accelerated convergence of receptance matrices: comments on a paper by Y. T. Leung

Some results are given which amplify a recent paper by Leung,¹ concerned with combining the advantages of matrix condensation and modal acceleration for linear vibrating systems. The receptance matrix for the active degrees of freedom is expressed in terms of the reduced mass and stiffness matrices associated with those degrees of freedom, together with a series based on the interior modes corresponding to the passive degrees of freedom.     

七层混凝土小型空心砌块房屋振动台试验研究

按相似理论设计制作了 1/ 6比例七层混凝土小型空心砌块房屋整体模型。通过在三向六自由度振动台上进行的模拟地震振动台试验 ,研究了模型结构的动力特性 ;研究模型结构在 7度多遇烈度、 7度基本烈度、 7度罕遇烈度、 8度罕遇烈度地震动作用下的自振特性加速度、位移、应变反应 ;研究模型结构的开裂烈度、开裂部位、震害发展情况。并根据振动台试验的结果 ,按相似关系推算七层混凝土小型空心砌块房屋原型结构的地震反应。     

附加油阻尼器的非线性滞回结构系统的等价线性化方法

以安装油阻尼器的单自由度非线性滞回结构系统为研究对象,建立地震作用下该非线性结构系统的等价线性化方法。鉴于结构系统具有双非线性特征,其中油阻尼器的阻尼力与速度呈双线性关系,主体结构具有双线性滞回特征,将主体结构等价为线性Kelvin模型,附加体系分两步等价:将双线性黏滞单元等价为线性黏滞单元,得到含线性黏滞单元的附加体系与主体结构的Kelvin模型并联构成的等价模型A;基于模型A进一步求解系统的等价刚度和等价阻尼比,建立系统的等价线性化Kelvin模型,即模型B。通过共振曲线分别对模型A和模型B的精度进行了验证。考虑地震动的随机性,通过动力非线性时程分析对等价模型的阻尼比公式进行了修正。将等价线性化方法应用到性能曲线中,建立非线性结构系统的减震性能曲线绘制方法,结合性能曲线分析了油阻尼器对非线性结构的减震效果。结果表明:模型A与模型B的共振曲线与原结构均吻合,说明在非线性结构系统中将双线性黏滞单元等价为线性黏滞单元的方法可行;修正后的等价阻尼比可较好地预测地震作用下非线性结构系统的最大位移。     

Optimum structural design based on seismic reliability and response distribution

An analytical approach to the optimum structural design parameters is developed on the basis of seismic reliability and inter-story response distribution. An earthquake-like stationary random excitation being assumed, probabilistic earthquake response is estimated as the solution of a simple simultaneous algebraic equation. Based on this probabilistic earthquake response, the optimum parameters are determined by selecting seismic structural reliability and an inter-story response distribution index as objective functions. The validity of this approach is demonstrated by examining the perspective and contour lines of the objective functions regarded as a function of design parameters for elastic structural systems with six degrees of freedom. The possible application of this approach is also presented to the determination of elasto-plastic structural parameters with bi-linear hysteretic characteristics. Other aspects of the approach's validity are also examined from the response distribution viewpoint for the elastic structural system with recorded strong earthquake motions. The influence of the higher modal response and the earthquake's predominant angular frequency on the optimum parameters are also discussed.     

抗震结构总输入能量及其影响因素分析

采用时程分析法综合考虑各种影响因素分析了单自由度弹性和弹塑性体系、多自由度弹性和弹塑性体系剪切型、弯曲型以及剪弯型结构在罕遇地震作用下的总输入能量反应及其受各类因素影响的规律，得出了上述结构的总输入能量谱曲线、提出了多自由度体系及弹塑性能量分析简化分析方法的建议，为寻求基于能量准则的结构抗震设计方法和完善结构抗震双得破坏准则提供了一定的依据。     

Triangular and rectangular plane elasticity finite elements

A triangular and a rectangular strain based elements are developed for the general plane elasticity problems. The triangular element has the two essential external degrees of freedom at each of the three corner nodes and at a mid-side node. It is based on assumed independent linear direct strains and constant shearing strain. The rectangular element has the same degrees of freedom at each of the four corner nodes as well as at the centroidal node. This element is based on linear variations of all the three components of strains.These elements are used in various forms, including statical condensation and satisfying equilibrium to obtain solutions to two dimensional elasticity problems where the contribution of the shearing stress on the deformation can be significant.     

Wavelet-Based Generation of Energy- and Spectrum-Compatible Earthquake Time Histories

Abstract:  A method of modifying earthquake ground motion based on the wavelet transform is proposed, to take into account the effects of linear/nonlinear response spectra, frequency content, and ground motion energy. A wavelet-based procedure has been used to decompose recorded ground motion into finite wavelet coefficients, and then, with matrix processing, the coefficients have been suitably substituted and scaled to match the response spectra and total energy of earthquake ground motions. The proposed method has been verified by modifying five recorded accelerograms such that they are compatible with the same linear/nonlinear and energy spectra.     

Localized identification of shear building with embedded foundation in frequency domain

For studying the behavior of structure in an earthquake, it is advisable to model the structure as a multi‐degrees of freedom system, consisting of numerous single‐degree of freedom substructures and pay attention to soil–structure interaction. System identification is divided into two categories: namely time domain method and frequency domain approach. In this paper, a localized substructure identification of shear building considering the soil–structure interaction is presented using a frequency domain approach. In order to deal with noise‐corrupted data, a spectral smoothing technique with Parzen's window reduction method is adapted. It is shown that better convergence and accuracy can be achieved by the present technique. As a result, it is shown that by taking into account the soil–structure interaction more realistic results can be obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

On the modal incremental dynamic analysis

In this article a new technique for the dynamic response of structures is investigated. This applied procedure can predict the approximate seismic performance of the structures and it is fast, inexpensive and results are reasonably acceptable. In fact, this novel method logically combines two different techniques, ‘incremental dynamic analysis (IDA)’ and ‘modal pushover analysis (MPA)’, presented by other researchers. This method will take advantage of both methodical ideas such as equivalent single degree of freedom of multi‐degree structures and the implementation of different scaled level of an earthquake record to the provided equivalent SDF structure. Using this procedure, simple approximate curves that present a realistic linear and non‐linear seismic behaviour of the structure due to the applied scaled level of earthquakes can easily be extracted. In this investigation, several four‐, eight‐ and 12‐storey structures are specified as the example models and are dynamically analysed. Next, three different scaled earthquakes, El Centro, Northridge and San Fernando, are applied to each example problem. The results of the presented technique, modal incremental dynamic analysis (MIDA), are then compared with the IDA method. Comparison of the results reveals good accuracy in building seismic demands evaluation. Finally, it is also shown that the MIDA method is simple enough to be carried out on most personal computers and the authors believe this technique will serve design engineers working in real design conditions. Copyright © 2005 John Wiley & Sons, Ltd.