Damage identification in a concrete dam by fitting measured modal parameters

We propose a method, based on an inverse problem, to obtain numerically the material parameters that characterize the elasticity tensor of a body with linear elastic behavior, using accurate measurements of the first modal parameters, namely the natural frequencies and the modes of vibration (the eigenfrequencies and the eigenvectors). Appropriate functionals are defined, whose minimum points correspond to the unknown material parameters. To obtain these minimum points a highly nonlinear parametric optimization problem is solved. Its resolution involves specific mathematical tools like the derivative of the eigenvalues and eigenvectors with respect to the material parameters, the adjoint method, and gradient methods for the minimization of the functional. An application is presented, which considers a cracked dam in which is assumed the presence of transversely isotropic material in the cracked zone. The material parameters of the transversely isotropic material are obtained by minimizing the distance between the modal parameters (eigenfrequencies and eigenvectors) of a numerical model of the dam and the observed modal parameters physically measured in the dam. The algorithm is implemented in a C++ home made code with the aid of open-source libraries for scientific computation.     

Shape optimal design of arch dams including dam-water–foundation rock interaction using a grading strategy and approximation concepts

Optimal design of arch dams including dam-water–foundation rock interaction is achieved using the soft computing techniques. For this, linear dynamic behavior of arch dam-water–foundation rock system subjected to earthquake ground motion is simulated using the finite element method at first and then, to reduce the computational cost of optimization process, a wavelet back propagation neural network (WBPNN) is designed to predict the arch dam response instead of directly evaluating it by a time-consuming finite-element analysis (FEA). In order to enhance the performance generality of the neural network, a dam grading technique (DGT) is also introduced. To assess the computational efficiency of the proposed methodology for arch dam optimization, an actual arch dam is considered. The optimization is implemented via the simultaneous perturbation stochastic approximation (SPSA) algorithm for the various conditions of the interaction problem. Numerical results show the merits of the suggested techniques for arch dam optimization. It is also found that considering the dam-water–foundation rock interaction has an important role for safely designing an arch dam.     

A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations

In the present paper, a power series solution is developed for free vibration and damping analyses of viscoelastic functionally graded plates with variable thickness on elastic foundations. It is assumed that the material properties of the functionally graded material (FGM) vary in the transverse and radial directions, simultaneously. Therefore, the presented solution can be employed for the transversely-graded and radially-graded viscoelastic circular plates, as special cases. In addition to the edge conditions, the plate may be resting on a two-parameter elastic foundation. The complex modulus approach in combination with the elastic–viscoelastic correspondence principle is employed to obtain the solution for various edge conditions. A sensitivity analysis including effects of various edge conditions, geometric parameters, coefficients of the elastic foundation, parameters of the functionally graded material, and material loss factor is carried out. In the present paper, concept of the complex modal stresses of the viscoelastic plates is discussed in detail.     

Analytical solution considering the tangential effect for an infinite beam on a viscoelastic Pasternak foundation

In this study, the dynamic response of an infinite beam resting on a Pasternak foundation subjected to inclined travelling loads was developed in the form of the analytical solution wherein the tangential effect between the beam and foundation and the damping were taken into consideration. Three parameters were used to model the mechanical resistance of the viscoelastic Pasternak foundation, one of them accounts for the compressive stress in the soil, the other accounts for the shearing effect of soils, and the last one accounts for the damping of the foundation. By contrast, the Pasternak model is more realistic than the Winkler model that just considers the compressive resistance of soil. In the paper, the tangential effect between the beam and foundation was simulated by a series of separate horizontal springs, the damping was also considered to obtain the dynamic response under forced vibration. The theory of elasticity and Newton's laws were used to derive the governing equation. To simplify the partial-differential equation to an algebraic equation, the double Fourier transformation was used wherein the analytical solution in the frequency domain for the dynamic response of the beam is obtained. And its inversion was adopted to convert the integral representation of the solution into the time domain. The degraded solution was then utilized to verify the validity of the proposed solution. Finally, the Maple mathematical software was used for further discussion. The solution proposed in this study can be a useful tool for practitioners.     

末端质量作用下变长度轴向运动梁的振动特性

对带集中质量,变长度(或速度)轴向运动梁的振动特性采用两种精确方法求解.首先,对变长度轴向运动Euler(欧拉)梁横向自由振动方程进行化简,通过复模态分析得到本征方程,并在有集中质量的边界条件下得到频率方程,用数值方法求解固有频率和模态函数.然后,采用有限元方法建立运动梁自由振动的方程,求解矩阵方程得到复特征值和复特征向量,结合形函数得到复模态位移.最后,将两种方法的计算结果进行了分析和对比.数值算例的结果表明:不同的轴向运动速度和集中质量对变长度轴向运动梁的振动特性有显著影响,两种计算方法的结果接近且均有效.     

Probabilistic nonlinear analysis of CFR dams by MCS using Response Surface Method

This paper presents the probabilistic analysis of concrete-faced rockfill (CFR) dams according to the Monte Carlo Simulation (MCS) results which are obtained through the Response Surface Method (RSM). ANSYS finite element program is used to get displacement and principal stress components. First of all, some parametric studies are performed according to the simple and representative finite element model of dam body to obtain the optimum approximate model. Secondly, a sensitivity analysis is performed to get the most effective parameters on dam response. Then, RSM is used to obtain the approximate function through the selected parameters. After the performed analyses, star experimental design with quadratic function without mixed terms according to the k = 1 is determined as the most appropriate model. Finally, dam-foundation-reservoir interaction finite element model is constituted and probabilistic analyses are performed with MCS using the selected parameters, sampling method, function and arbitrary factor under gravity load for empty and full reservoir conditions. Geometrically and materially nonlinearity are considered in the analysis of dam-foundation-reservoir interaction system. Reservoir water is modeled by fluid finite elements based on the Lagrangian approach. Structural connections are modeled as welded contact and friction contact based on Coulomb’s friction law. Probabilistic displacements and stresses are presented and compared with deterministic results.     

Frequency-dependent vibration analysis of symmetric cross-ply laminated plate of Levy-type by spectral element and finite strip procedures

This research describes spectral finite element formulation for vibration analysis of rectangular symmetric cross-ply laminated composite plates of Levy-type based on classical lamination plate theory (CLPT). Formulation based on SFEM includes partial differential equations of motion, spectral displacement field, dynamic shape functions, and spectral element stiffness matrix (SESM). In this paper, vibration analysis of composite plate is investigated in two sections: free vibrations and forced vibrations. In free vibrations, natural frequencies are calculated for different Young’s moduli ratios and boundary conditions. In forced vibrations, plate vibrations are investigated under high-frequency concentrated impulsive loads. The resulting responses due to spectral element formulation are compared with those of (time-domain) finite element and analytical formulations, whenever available. The results demonstrate the superiority of SFEM with respect to FEM, in reducing computational burden, simultaneously increasing numerical accuracy, specifically for excitations of high-frequency content. By reducing the time duration of impulsive loads, and consequently increasing the modal contribution of higher modes in the transient response of plate, the accuracy of FEM responses decreases substantially accompanied with a high volume of computations, while the accuracy of the SFEM response results is very high and simultaneously, with a low computational burden. Practically, SFEM follows very closely exact analytical solutions.     

Comparison of coupled and decoupled modal approaches in seismic analysis of concrete gravity dams in time domain

Different methods are available for dynamic analysis of concrete dams. Among these, modal approach is highly popular due to the efficiency of the method. This becomes more significant if the response is to be calculated for several earthquake ground motion records as required in most practical cases. In this study, two different modal approaches have been considered for dynamic analysis of concrete gravity dam–reservoir systems. These are coupled modal method and decoupled modal technique. The former approach utilizes the coupled modes of the system. It is well known that calculation of these modes involves some complications due to its corresponding unsymmetrical eigenproblem. However, the response at each step can be obtained very efficiently in this method. The latter technique, relies on the decoupled modes of the system, which are easily obtained by standard eigenvalue solvers. The equation of motion is also solved with reasonable efficiency in this approach. In the latter part of this paper, analysis of a typical dam–reservoir system is performed by both methods mentioned above. These analyses are compared from accuracy and efficiency point of view.     

Damage analysis of arch concrete dams subjected to underwater explosion

The nonlinear behavior of arch concrete dams subjected to underwater explosions was investigated. Pressure waves were spherically modeled using a 3-D finite element method. The nonlinear behavior of concrete was modeled using the concrete damage plasticity model. In addition, absolute wave formulation was used to develop a fully-coupled model incorporating the propagation of the shock wave and water–dam interaction. Analysis of an arch concrete dam subject to underwater explosion was performed upon model verification. The dynamic response of the dam subject to the underwater explosion was studied for different sizes of explosions at different depths. The results showed that the closer the point to the explosive source, the sooner the impression and damage was caused by the blast. It was clear that the middle part of the dam facing the explosion was the first location to incur damage and then cracks extended to the downstream face and abutments. Finally, cracks extended to the bottom of the dam. The results of the dynamic analysis and dissipated energy by the loss of elements’ resistance indicated that the time slice of 150 ms was appropriate to analyze arch concrete dam structure subject to an underwater explosion. In addition, for a given amount of explosives, an increase in the depth of explosion corresponded to an increase in the maximum displacement of the upstream face of the dam. The results also confirmed that the damage distribution to the arch concrete dam subject to an underwater explosion depended upon the amount and depth of explosive materials.     

车路耦合非线性振动高阶Galerkin截断研究

将移动车辆模型化为运动的两自由度质量-弹簧-阻尼系统,道路模型化为立方非线性黏弹性地基上的弹性梁,并将路面不平度设定为简谐函数．通过受力分析,建立车路非线性耦合振动高阶偏微分方程．采用高阶Galerkin截断结合数值方法求解耦合系统的动态响应．首次研究不同截断阶数对车路耦合非线性振动动态响应的影响,确定Galerkin截断研究车路耦合振动的收敛性．研究结果表明,对于软土地基的沥青路面,耦合振动的动态响应,需要150阶以上的截断才能达到收敛效果．并通过高阶收敛的Galerkin截断研究了系统参数对车路耦合非线性振动动态响应的影响.     

Analysis of flow characteristics of granular material unloaded on nonlinear vibration inclined platform

In this study, the repeated discontinuous friction between granular material and contact platform and structural nonlinearity of inclined vibration platform giving rise to the vibration flow-aiding unloading is a complicated process, which has significant effects on the dynamic behaviors and flow characteristics of granular material. A simplified mathematical model of the inclined vibration platform and granular material is deduced by mechanical properties. Based on the equations of motion and a good degree of accuracy and applicability of the process with calculated data reported in the literature, the approximate analytical solution and flow properties are investigated by using the modified incremental harmonic balance method and numerical integration method. Moreover, the influences of friction coefficient, excitation amplitude, nonlinear stiffness and inclined angle on the complicated dynamic behaviors are explored and discussed. It is shown that the different motion paths of granular material on inclined vibration platform are observed depending on the different parameters. The increasing friction coefficient has complicated effects on the nonlinear dynamic behaviors of the granular material. The excitation amplitude and nonlinear stiffness can effectively control the flow characteristics of granular material at low excitation frequency but the inclined angle presents opposite property. The research may contribute to improve unloading efficiency, predict the motion state of granule and provide a theoretic foundation for further design the unloading system.     

Snap-through buckling of a shallow arch resting on a two-parameter elastic foundation

In this paper, we study the static and dynamic snap-through of a shallow arch resting on a two-parameter elastic foundation under a point load moving at a constant speed. The deformation of the arch is expressed in Fourier series. We extend the previous works when the arch resting on a certain model of two-parameter elastic foundation. Each model of foundation has a different definition for the second foundation parameter, where the model is discussed in this paper is Pasternak model. For quasi-static analysis, it is noted that the first four modes in the expansion are sufficient to predict the response of the arch. Similarly, when the point load moves with a significant speed, we integrate the equation of motion numerically using the first modes in the expansion.     

The variability of dynamic responses of beams resting on elastic foundation subjected to vehicle with random system parameters

This paper investigates the variability of dynamic responses of a beam resting on an elastic foundation, which is subjected to a vehicle with uncertain parameters, such as random mass, stiffness, damping of the vehicle and random fields of mass density, and the elastic modulus of the beam and stiffness of elastic foundation. The vehicle is modeled as a two-degree-of-freedom spring-damper-mass system. The equations of motion of the beam was constructed using a finite element method. The mass and elastic properties of the beam, and the stiffness of foundation are assumed to be Gaussian random fields and were simulated by the spectral represent method. Masses, stiffness of the spring, and the damping coefficient of the vehicle are assumed as Gaussian random variables. The numerical analyses were performed using the finite element method (FEM) in conjunction with the Monte Carlo simulation (MCS). The variability of dynamic responses of the beam were investigated with various cases of random parameters. For each sample, the equations of motions were solved with the Wilson-q integral method to find dynamic responses. The influence of random system parameters and their correlation on the response variability is discussed in detail.     

Spatial Random Material Property Model for Vibration of Composites

Investigation of vibration and buckling of thin walled composite structures is very sensitive to parameters like uncertain material properties and thickness imperfections. Because of the manufacturing process and others, thin walled composite and other structures show uncertainties in material properties, and other parameters which cannot be reduced by refined discretization. These parameters are mostly spatial distributed in nature. Here I introduce a semivariogram type material property model to predict the spatial distributed material property (like young's modulus) over the structure. The computation of semivariogram parameters needs the local material properties over a prespecified gird. The material properties at each grid have been obtained by considering a statistically homogeneous representative volume element (RVE) at each gird. According to random nature of the spatial arrangement of fibers, the statistically homogeneous RVE is obtained using image processing. The effective material properties of the RVE have been obtained numerically with the help of periodic boundary condition. The methodology is applied to a composite panel model and modal analysis has been carried. The results of the modal analysis (eigen values and mode shapes) are compared with experimental modal analysis results which are in good agreement. Using the presented material property model we can better predict the vibration characteristics of the thin walled composite structures with the inherent uncertainties. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

混凝土重力坝多参数弹性位移反演分析不唯一性理论探讨

反演分析是现场监测⁃反演分析⁃工程实践检验⁃正演分析及预测的闭环系统的重要环节,而参数反分析是工程实践中研究最多的反分析问题.针对混凝土重力坝多参数反演分析是否具有唯一性,基于均质地基上重力坝在水压力作用下的位移解析解建立目标函数,进而以目标函数和非空凸集构建一个凸规划问题,然后通过分析目标函数的Hesse矩阵是否是正定矩阵,验证目标函数是否是严格凸函数,从而辨识构建的凸规划问题是否具有唯一全局极小点.对坝体和岩基弹性参数的不同组合方案分析表明,当采用理论值与实测值的差值的l1范数作为目标函数时,目标函数的Hesse矩阵均不能保证为正定矩阵,即混凝土重力坝多参数弹性位移反演分析凸规划问题不具有唯一全局极小点,反演分析不唯一.     

Transfer matrix method for the free and forced vibration analyses of multi-step Timoshenko beams coupled with rigid bodies on springs

Multi-step Timoshenko beams coupled with rigid bodies on springs can be regarded as a generalized model to investigate the dynamic characteristics of many structures and mechanical systems in engineering. This paper presents a novel transfer matrix method for the free and forced vibration analyses of the hybrid system. It is modeled as a chain system, where each beam and each rigid body with its supporting spring are dealt with one element, respectively. The transfer equation of each element is deduced based on separation of variables method. The system overall transfer equation is obtained by substituting an element transfer equation into another. Then, the free vibration characteristics are acquired by solving exact homogeneous linear equations. To compute the forced vibration response with modal superposition method, the body dynamic equations and augmented eigenvectors are established, and the orthogonality of augmented eigenvectors is mathematically proved. Without high-order global dynamic equation or approximate spatial discretization, the free and forced vibration analyses of the hybrid system are achieved efficiently and accurately in this study. As an analytical approach, the present method is easy, highly stylized, robust, powerful and general for the complex hybrid systems containing any number of Timoshenko beams and rigid bodies. Four numerical examples are implemented, and the results show that this method is computationally efficient with high precision.     

Static and dynamic analysis of two-layer Timoshenko composite beams by weak-form quadrature element method

To improve the efficiency in predicting the dynamic mode and static response of the two-layer partial interaction composite beams, this paper utilizes the differential quadrature technique to approximate derivatives of the primary unknowns with adaptive order of precision, rather than the low and constant order of interpolation used in the conventional finite element method (FEM). A degree-of-freedom-adaptive weak-form quadrature element (WQE) for dynamic analysis is formulated and implemented based on the principle of virtual work. For the purpose of comparison, a parabolic displacement-based finite element is also provided, thus (1) the predicted deflections and natural frequencies of the composite beams are verified; (2) the smoothness of the internal forces and stresses generated by WQE method and FEM are compared, and (3) the convergent rates of higher order free vibration modes are also examined. Numerical results show that the efficiency of the proposed WQE method has, on the one hand, significantly triumphed over that of FEM on analyses including static response, natural frequencies and higher order free vibration modes, on the other hand, the smoothness of results, including internal forces and stresses, is greatly refined.     

Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation

In this paper, post-buckling and nonlinear vibration analysis of geometrically imperfect beams made of functionally graded materials (FGMs) resting on nonlinear elastic foundation subjected to axial force are studied. The material properties of FGMs are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The assumptions of a small strain and moderate deformation are used. Based on Euler–Bernoulli beam theory and von-Karman geometric nonlinearity, the integral partial differential equation of motion is derived. Then this partial differential equation (PDE) problem, which has quadratic and cubic nonlinearities, is simplified into an ordinary differential equation (ODE) problem by using the Galerkin method. Finally, the governing equation is solved analytically using the variational iteration method (VIM). Some new results for the nonlinear natural frequencies and buckling load of the imperfect functionally graded (FG) beams such as the effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogeneity are presented for future references. Results show that the imperfection has a significant effect on the post-buckling and vibration response of FG beams.     

Interval modal superposition method for impulsive response of structures with uncertain-but-bounded external loads

This paper is concerned with the problem of the dynamic response of structures with uncertain-but-bounded external loads. Based on the theory of complex modal analysis, and interval mathematics, a new non-probabilistic method-interval modal superposition method is proposed to find the least favorable impulsive response and the most favorable impulsive response of structures. Through mathematical analysis and numerical calculation, comparisons between interval modal superposition method and probabilistic approach are made. Instead of probabilistic density distribution or statistical quantities, in the presented method, only the bounds on uncertain parameters are needed, Numerical examples indicate that the width of the region of the dynamic response yielded by the interval modal superposition method is larger than those produced by probabilistic approach while the interval modal superposition method will required less computation effort.