Modal equations of linear structures with viscoelastic dampers

Several types of energy dissipation devices using viscoelastic materials have been proposed to reduce vibration in structures subjected to wind and earthquake excitations. At constant temperature and small strain levels, the mechanical behaviour of Viscoelastic (VE) materials can be described using linear operators. In general, the stiffness and damping matrices of structures using VE devices are frequency dependent; this implies that the classical second-order differential equations for the modal co-ordinates are not a complete model for this type of structures. In this paper, the concept of modal coupling in the frequency domain is addressed, expressions for diagonalizable frequency-dependent stiffness and damping matrices are given, and an iterative technique for the computation of the response of viscoelastic structures is studied. Necessary and sufficient conditions for convergence of the technique are given and numerical examples are developed to illustrate the application of the method.     

Median floor acceleration spectra of linear structures with uncertain properties

Response parameters used to estimate nonstructural damage differ depending on whether deformation‐sensitive or acceleration‐sensitive components are considered. In the latter case, seismic demand is usually represented through floor spectra, that is response spectra in terms of pseudo‐acceleration, which are calculated at the floor levels of the structure where the nonstructural components are attached to. Objective of this paper is to present a new spectrum‐to‐spectrum method for calculating floor acceleration spectra, which is able to explicitly account for epistemic uncertainties in the modal properties of the supporting structure. By using this method, effects on the spectra of possible variations from nominal values of the periods of vibration of the structure can be estimated. The method derives from the extension of closed‐form equations recently proposed by the authors to predict uniform hazard floor acceleration spectra. These equations are built to rigorously account for the input ground motion uncertainty, that is the record‐to‐record variability of the nonstructural response. In order to evaluate the proposed method, comparisons with exact spectra obtained from a standard probabilistic seismic demand analysis, as well as spectra calculated using the Eurocode 8 equation, are finally shown. Copyright © 2017 John Wiley & Sons, Ltd.     

Flexibility-based linear dynamic analysis of complex structures with curved-3D members

A flexibility-based formulation of a new mass matrix for the dynamic analysis of spatial frames consisting of curved elements with variable cross-sections is presented. The main characteristic of such formulations is the exact equilibrium of forces at any interior point, with no additional hypotheses about the distribution of displacements, strains or stresses. Accordingly, the derived element mass matrix takes into account the exact stiffness and mass distribution throughout each element. In validation tests, results obtained with this method are compared with those obtained by other numerical or analytical formulations, showing the accuracy of the proposed method. The comparison of experimental results for a multispan arch bridge subjected to a dynamic load with those achieved by means of the proposed method are finally included to illustrate its efficiency in the treatment of complex structures. © 1998 John Wiley & Sons, Ltd.     

线性结构的非平稳随机地震反应与可靠度分析

基于Sigma振子过程完全非平稳地震荷载模型,建立线性结构随机地震反应分析的时域表达式,利用Simpson法则迭代计算结构随机地震反应的二阶统计特性,由穿越方法,给出具有较大设计安全界限结构地震可靠度的近似评估方法。一个抗弯钢框架的非平稳地震反应与可靠度分析,说明了本文方法的效率、精度和应用过程。     

System identification of linear MDOF structures under ambient excitation

This paper introduces the eigenspace structural identification technique for tall buildings subjected to ambient excitations that are stationary and where only the response time histories are measured. Based on the forward innovation model of the Kalman filter sequence, the actual response can be constructed as a function of the measured response time history with contamination of either displacement or velocity. The response time history is decomposed into subspace matrices using QR decomposition and Quotient Singular Value Decomposition (QSVD) techniques. These are then substituted into the least-square formulation to obtain the solution which is non-unique. Similarity transformation is applied to arrive at the desired solution employing the fact that eigenvalues of self-similar systems are identical. The advantages of this eigenspace technique are that it is non-iterative, initial estimates of the parameters to the identified are not required, well-established numerical algorithm of the decomposition techniques employed are available, and the method can handle MDOF systems efficiently. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimal design of supplemental viscous dampers for linear framed structures

A methodology for the optimal design of supplemental viscous dampers for framed structures is presented. It addresses the problem of minimizing the added damping subject to a constraint on the maximal interstorey angular drift for an ensemble of realistic ground motion records while assuming linear behaviour of the damped structure. The solution is achieved by actually solving an equivalent optimization problem of minimizing the added damping subject to a constraint on a maximal weighted integral on the squared angular drift. The computational effort is appreciably reduced by first using one ‘active’ ground motion record. If the resulting optimal design fails to satisfy the constraints for other ground motions from the original ensemble, additional ground motions (loading conditions) are added one by one to the ‘active’ set until the optimum is reached. An efficient selecting process which is presented herein will usually require one or two records to attain an optimum design. Examples of optimal designs of supplemental dampers are presented for a 2‐storey shear frame and a 10‐storey industrial frame. The 2‐storey shear frame is required to withstand one given ground motion whereas the 10‐storey frame is required to withstand an ensemble of twenty ground motions. The resulting viscously damped structures have envelope values of interstorey drifts equal or less than the target drifts. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite volume upwind scheme for the solution of the linear advection–diffusion equation with sharp gradients in multiple dimensions

A finite volume upwind numerical scheme for the solution of the linear advection equation in multiple dimensions on Cartesian grids is presented. The small-stencil, Modified Discontinuous Profile Method (MDPM) uses a sub-cell piecewise constant reconstruction and additional information at the cell interfaces, rather than a spatial extension of the stencil as in usual methods. This paper presents the MDPM profile reconstruction method in one dimension and its generalization and algorithm to two- and three-dimensional problems. The method is extended to the advection–diffusion equation in multiple dimensions. The MDPM is tested against the MUSCL scheme on two- and three-dimensional test cases. It is shown to give high-quality results for sharp gradients problems, although some scattering appears. For smooth gradients, extreme values are best preserved with the MDPM than with the MUSCL scheme, while the MDPM does not maintain the smoothness of the original shape as well as the MUSCL scheme. However the MDPM is proved to be more efficient on coarse grids in terms of error and CPU time, while on fine grids the MUSCL scheme provides a better accuracy at a lower CPU.     

High precision locations of LP events on Mt. Etna: Reconstruction of the fluid-filled volume

During 1991–93 at Mount Etna, long-period (LP) events occurring in swarms characterized the evolution of the eruption. The presence of multiplets i.e. groups of events with similar waveform signatures, has been recognized within this activity. Traditional techniques for locating LP events do not allow obtaining reliable hypocenters, which have only succeeded in placing earthquakes in a roughly 1 km² area slightly east of the Mt. Etna Northeast Crater. Hypocenters have been relocated in two steps: the absolute location has been improved using Thurber’s code and a complex 3D velocity model; a highly precise relative location has been applied on multiplets to define the source geometry. 3D locations and high precision analysis suggest that during the 1991–93 eruption the resonator producing LP events was a part of the uppermost Northeast Crater conduit, measuring 210 meters in height and 45–50 meters in diameter.     

Supporting brace sizing in structures with added linear viscous fluid dampers: A filter design solution

Viscous fluid dampers have proved to be effective in suppressing unwanted vibrations in a range of engineering structures. When dampers are fitted in a structure, a brace is typically used to attach them to the main structure. The stiffness of this brace can significantly alter the effectiveness of the damper, and in structures with multiple dampers, this can be a complex scenario to model. In this paper, we demonstrate that the effects of the brace compliance on the damper performance can be modelled by way of a first‐order filter. We use this result to formulate a procedure that calculates the stiffness required by the supporting brace to provide a specified effectiveness of the damping action. The proposed procedure assumes that viscous dampers have been sized in a previous design step based on any optimal methodology in which, as is usually the case, the presence of supporting braces and their dynamic effects were neglected. Firstly considering a one degree‐of‐freedom system, we show that the proposed method ensures a desired level of damper efficiency for all frequencies within a selected bandwidth. Then the analysis is extended to the case of multi‐degree‐of‐freedom systems to show that the design criteria can be applied in a straightforward and successful manner to more complex structures. © 2014 The Authors. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd.     

Vibration of linear structures due to jump-discontinuous,non-interrupted,stochastic processes

The solution of the problem of vibration of linearly elastic structures, caused by stochastic excitation of non-interrupted, jump-discontinuous character, is considered in the paper. Two types of excitation are examined. They form an enhancement of the continuous model which is most often used. The solution in the form of expected values, correlation functions and spectral densities of the deflection of the structure is formulated. Examples illustrating the solution are presented.     

重力资料识别鄂尔多斯盆地线性构造方法研究

目的利用重力异常数据微弱信息挖掘技术识别线性构造的方法,并介绍其实现过程.方法采用水平梯度法和二次垂直导数等方法提取微弱信息,利用数字图象显示技术成图.结果识别出鄂尔多斯盆地线性构造多条,分析对比鄂尔多斯盆地区域地质构造资料及重力解释成果,具有较好的吻合性.为深入研究该地区线性构造补充了新的证据.结论该方法提取了区域重力异常资料中的微弱信息,弥补用等值线图示构造信息的不足,有助于线性构造的分析.水平梯度法(最大模)和二次导数处理方法与数字图象成图相结合取得良好的解释效果.     

Pounding of structures modelled as non‐linear impacts of two oscillators

A new formulation is proposed to model pounding between two adjacent structures, with natural periods T₁ and T₂ and damping ratios ζ₁ and ζ₂ under harmonic earthquake excitation, as non‐linear Hertzian impact between two single‐degree‐of‐freedom oscillators. For the case of rigid impacts, a special case of our analytical solution has been given by Davis (‘Pounding of buildings modelled by an impact oscillator’ Earthquake Engineering and Structural Dynamics, 1992; 21 :253–274) for an oscillator pounding on a stationary barrier. Our analytical predictions for rigid impacts agree qualitatively with our numerical simulations for non‐rigid impacts. When the difference in natural periods between the two oscillators increases, the impact velocity also increases drastically. The impact velocity spectrum is, however, relatively insensitive to the standoff distance. The maximum relative impact velocity of the coupled system can occur at an excitation period T_n^* which is either between those of the two oscillators or less than both of them, depending on the ratios T₁/T₂ and ζ₁/ζ₂. Although the pounding force between two oscillators has been primarily modelled by the Hertz contact law, parametric studies show that the maximum relative impact velocity is not very sensitive to changes in the contact parameters. Copyright © 2001 John Wiley & Sons, Ltd.     

On the use of the half-power bandwidth method to estimate damping in building structures

The objective of this note is to assess the accuracy of the half-power bandwidth method for the estimation of damping ratios in single- and multi-degree-of-freedom structures with linear viscous damping including those that do not possess classical normal modes. This is done by performing some numerical experiments involving single- and multi-degree-of-freedom structures. It is found that the use of the half-power bandwidth method in its classical form may lead to significant errors, while a third order correction to this classical form provides conservative and more reliable results. It is also found in conjunction with this correction that application of the half-power bandwidth method should be performed to the acceleration frequency response transfer function of the structure on a mode-to-mode basis and that its damping ratio estimates for higher modes should always be viewed with caution.     

Effective use of seismic response envelopes for reinforced concrete structures

By exploiting the theory of the response envelopes formulated by Menun and Der Kiureghian [Envelopes for seismic response vectors. I–Theory, J. Str. Engrg. 2000; 126(3); 467–473], an algorithmic approach for seismic analysis of reinforced concrete frames is presented. It aims to fill a gap between research on spectral analysis of structures and current design practice in which the use of seismic response envelopes, available since early 2000s, is hampered by the lack of efficient and robust implementations. The proposed strategy is based on customary features (such as modal shapes and response spectra) currently adopted in professional practice, and it takes advantage of recently published formulations for the evaluation of stress resultants in arbitrarily shaped reinforced concrete cross‐sections subjected to axial force and biaxial bending. Numerical applications are illustrated in order to show the procedure's efficiency and effectiveness. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control of linear and nonlinear asymmetric structures by means of passive energy dampers

Asymmetric structures experience uneven deformation demand among different resisting planes and stories when subjected to earthquake excitation. Damage is focused in some elements jeopardizing structural integrity. These structures are common in professional practice because of architectural and functionality constraints. In this scenario the use of energy dissipation devices (EDD) has arisen as an advisable solution to balance and minimize structural damage. Procedures for the design of linear structures equipped with EDD have been widely proposed in the literature, few of them deal with the optimum spatial distribution of nonlinear systems. This paper evaluates and compares the optimized spatial damper distribution of linear and nonlinear systems. An optimization technique is presented based on control indexes called min–max algorithm. Then, this technique is compared with two simple methodologies: (i) the fully stressed design, which is an analysis‐redesign procedure, and (ii) the simplified sequential search algorithm (SSSA), which is a sequential method. It is pointed out that the SSSA is a fixed step coordinate descent type method. The examples considered show that the SSSA is a discrete approximation of the min–max algorithm, not only for linear but also for nonlinear structures equipped with linear and nonlinear EDD. Furthermore, it is found that the distribution of EDD obtained from a linear analysis is a good approximation of the nonlinear optimal solution. The SSSA is a reliable method that can be applied to achieve drift and torsional balance for design purposes; moreover, it can be implemented with conventional tools available in professional practice. Copyright © 2012 John Wiley & Sons, Ltd.     

On the use of biorthogonalization for solving systems of linear equations in numerical weather forecasting

Summary The paper deals with the solving of large systems of linear equations which are created when implicit and semi-implicit differential methods are used to solve the primitive equations of the atmosphere. Systems with similar properties are also created in solving boundary problems of multi-level filtered models. First, the specific properties of these systems, which are then used to construct an efficient method of solving them, are described.     

Effects of reinforcement slippage on the non‐linear response under cyclic loadings of RC frame structures

This paper discusses the importance of including the bond‐slip effects in assessing the response under cyclic loads of reinforced concrete frames. The discussion is based on analyses performed using numerical models which are simple, computationally efficient and capable of representing the salient features of reinforced concrete frames under both static and dynamic loads. The numerical models comprise a displacement‐based, reinforced concrete frame element with bond‐slip and a rigid beam column joint element with bond‐slip. Two applications illustrate the model accuracy and show the importance of including bond‐slip. The first application considers a reinforced concrete beam‐column subassemblage experimentally tested under cyclic loads. The second application considers the shaking table test of a two‐story one‐bay reinforced concrete frame In both cases the analytical results correlate well with the experimental results in terms of strength, displacement demands and hysteretic energy dissipation. Furthermore, the paper shows how the analyses that include bond‐slip yield a better correlation with the experimental results with respect to the analyses that assume a perfect bond. Copyright © 2003 John Wiley & Sons, Ltd.     

柴达木盆地航磁资料微弱信息增强技术研究及在线性构造识别中的应用

目的利用重磁异常数据微弱信息增强技术识别线性构造的方法,并介绍其实现过程.方法文中提出了一种梯级带滤波增强非线性滤渡技术与Tilt梯度及其水平导数的有效结合,增强放大微弱信息再识别提取线性构造,利用数字图象显示技术成图.结果识别出柴迭木盆地线性构造多条,分析对比柴达木盆地区域地质构造资料及重力解释成果,具有较好的吻合性.为深入研究该地区线性构造、成矿特征、寻找勘探靶区补充了新的证据.结论该方法提取了区域航磁异常资料中的微弱信息,弥补传统方法的不足,对断层边界及异常的边界划定更为准确.梯级带滤波增强技术与Tilt梯度及其水平导数相结合取得良好的解释效果.     

Theoretical models and recorded response in the estimation of cumulative seismic damage on non‐linear structures

A damage‐function model is proposed for the inelastic response of structures in terms of initial damage and of displacement amplitudes and secant stiffnesses of response cycles. The model is used to obtain an analytical closed‐form solution for the probability distribution of cumulative damage after an earthquake ground motion given the distribution prior to such excitation and information on the inelastic structural response. The formulation is applied to a reinforced concrete frame and the results show the capabilities of the method to yield updated distributions of damage. Copyright © 2000 John Wiley & Sons, Ltd.