Mesh independent analysis of shell‐like structures

Mesh independent analysis is motivated by the desire to use accurate geometric models represented as equations rather than approximated by a mesh. The trial and test functions are approximated or interpolated on a background mesh that is independent of the geometry. This background mesh is easy to generate because it does not have to conform to the geometry. Essential boundary conditions can be applied using the implicit boundary method where the trial and test functions are constructed utilizing approximate step functions such that the boundary conditions are guaranteed to be satisfied. This approach has been demonstrated for two‐dimensional (2D) and three‐dimensional (3D) structural analysis and is extended in this paper to model shell‐like structures. The background mesh consists of 3D elements that use uniform B‐spline approximations, and the shell geometry is assumed to be defined as parametric surfaces to allow arbitrarily complex shell‐like structures to be modeled. Several benchmark problems are used to study the validity of these 3D B‐spline shell elements. Copyright © 2012 John Wiley & Sons, Ltd.     

Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first‐order shear deformation theory

In this study, free vibration of laminated skew plates was investigated. Discrete singular convolution (DSC) method is used for numerical solution of vibration problems. The straight‐sided quadrilateral domain is mapped into a square domain in the computational space using a four‐node element by using the geometric transformation. Typical results are presented for different geometric parameters and boundary conditions. It is concluded from the results that the skew angle have considerable influence on the variations of the frequencies with fibre orientation angle and number of layers in the laminate. The results obtained by DSC method are compared with those obtained by analytical and numerical approaches. It is shown that reasonable accurate results are obtained. Present work also indicates that the method of DSC is a promising and potential approach for computational mechanics. Copyright © 2009 John Wiley & Sons, Ltd.     

Free vibrations of shells of revolution filled with a fluid

This paper describes the method of determining the normal modes of vibrations and natural frequencies of elastic shells of revolution with an arbitrary meridian, partially filled with a fluid. The modes of vibration of the shells with fluids are determined as a linear combination of the natural modes of vibration in vacuum. The solution of the problem of hydroelastic vibrations has been obtained using the methods of the boundary element (BEM) and the finite element (FEM). Numerical investigations of vibrations of hemi-spherical shells conveying fluid have been conducted and analyzed. Illustrative examples are provided to demonstrate the accuracy and efficiency of the developed numerical procedure.     

Evaluation of stress intensity factors by the R-functions method

An attempt has been made to apply the novel R-functions method (RFM) to the linear elastic fracture mechanics (LEFM) problems. Essential feature of this method consists in a conversion of logical operations performed on sets (relevant to the sub-domains) into algebraic operations performed on elementary functions. The RFM is an analytical-numerical approach to the solution of the boundary value problems involving arbitrary domains that may be concave or/and multiconnected. The solution constructed by the R-functions method is realized in two phases. In the first one an analytical formula for the so-called general structure of solution (GSS) is designed in such a way that it satisfies the prescribed boundary conditions while a certain number of functions remains undetermined. In the second step a suitable numerical procedure is employed to evaluate these functions in order to satisfy the governing equation of the problem considered. The method was proved to be effective in elasticity problems, especially when fully computerized through the use of symbolic programming. The paper gives:

u

some basic information about the R-functions method,

formulation of the RFM designed for applications to the LEFM problems,

general structure of the solution for 2D cracked bodies with relevant types of singularity and boundary conditions taken into account,

an outline of the numerical approach and some illustrative examples.

The present work may be considered as an encouraging first step, but further significant effort is required before the R-functions method of treating the problems of fracture mechanics becomes a useful and efficient mathematical tool.     

3D source simulation method for static fields in inhomogeneous media

In this paper, an integral approach based on the source simulation method (SSM) for the solution of three dimensional (3D) static electric fields is presented. This is an extension of the well‐known charge simulation method to the electric and current static fields. The formulation allows the solution of 3D electrostatic or static current field problems, where several perfectly conducting bodies (electrodes) are placed in inhomogeneous media. Only electrode surfaces and dielectric interfaces need to be discretized by triangular elements. Matrix coefficients are accurately evaluated by using closed integral forms of 1/r and of grad(1/r). Dirichlet and Neumann conditions can be imposed on the different conducting bodies; in particular, floating potential problem can also be solved. The method of images allows ground plane to be taken into account. The procedure has been applied to several practical cases and it represents an efficient tool for the evaluation of lumped circuit parameters such as capacitances of 3D conducting bodies and ground resistances of grounding systems. Copyright © 2006 John Wiley & Sons, Ltd.     

基于弹性力学的裂缝梁自由振动分析

基于弹性力学平面应力理论,采用Chebyshev-Ritz法分析裂缝梁的自由振动特性。将梁分成三个子梁,取边界函数与Chebyshev多项式的乘积作为每个子梁的位移试函数,保证解的快速收敛性,并使该方法适用于不同的几何边界条件。用里兹法列出每个子梁的振动特征方程,并根据各子梁在界面上的位移连续性条件得到整个裂缝梁的振动特征方程。计算结果与文献数据和有限元分析吻合很好。最后分析了裂缝深度和梁的高跨比对动力特性的影响。     

Dynamic analysis in solid mechanics by an alternative boundary element procedure

A new procedure for eigenvalue and transient dynamic analyses in solid mechanics is presented here using a boundary integral approach. Contrary to previous work, it is shown in this paper that by applying a special class of co-ordinate functions one can obtain the mass matrix using boundary values. As a consequence, the free vibration case is reduced to an algebraic eigenvalue problem, while transient vibrations can be solved by using a direct time integration procedure.     

圆柱壳-圆锥壳组合结构振动分析的新方法

提出了一种区域分解法来分析不同边界条件下圆锥壳-圆柱壳-圆锥壳组合结构的自由振动和强迫振动特性。首先将组合壳体的位移边界与固定边界分开,将其分解为圆柱壳、圆锥壳子结构;为能获取组合壳体的高阶振动特性,进一步将圆柱壳、圆锥壳子结构分解为自由的圆柱壳段和圆锥壳段。采用分区广义变分和最小二乘加权残值法将各壳段分区界面上的位移和转角协调方程引入到组合壳体的势能泛函中,使组合壳体的振动分析问题,归结为在满足分区界面位移和转角协调条件下的无约束泛函变分问题。圆柱壳段和圆锥壳段位移变量的周向和轴向分量分别采用Fourier级数和Chebyshev多项式展开。算例表明:区域分解法计算出的不同边界条件下组合壳体自由振动和强迫振动结果与有限元软件ANSYS结果非常吻合;该方法具有高效率、高精度和收敛性好等优点。     

Three-dimensional structural vibration analysis by the Dual Reciprocity BEM

The dual reciprocity boundary element method (DR/BEM) is employed for the analysis of free and forced vibrations of three-dimensional elastic solids. Use of the elastostatic fundamental solution in the integral formulation of elastodynamics creates an inertial volume integral in addition to the boundary ones. This volume integral is transformed into a surface integral by invoking the reciprocal theorem. A general analytical method is described for the closed form determination of the particular solutions of the displacement and traction tensors corresponding to any radial basis function employed in the transformation process. The simple but effective 1+r radial basis function is used in the applications of this paper. Quadratic continuous and discontinuous 9-noded boundary elements are used in the analysis. Free vibrations are studied by solving the corresponding eigenvalue problem iteratively. Harmonic forced vibration problems are solved directly in the frequency domain. Transient forced vibration problems are solved by integrating the equations of motion stepwise with the aid of various algorithms. Interior collection points are also used for assessing the accuracy of the method. Two numerical examples involving free and forced vibrations of a sphere and a cube are presented in detail.     

Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions

The objective of this work is to present a Haar Wavelet Discretization (HWD) method-based solution approach for the free vibration analysis of functionally graded (FG) spherical and parabolic shells of revolution with arbitrary boundary conditions. The first-order shear deformation theory is adopted to account for the transverse shear effect and rotary inertia of the shell structures. Haar wavelet and their integral and Fourier series are selected as the basis functions for the variables and their derivatives in the meridional and circumferential directions, respectively. The constants appearing in the integrating process are determined by boundary conditions, and thus the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations. The proposed approach directly deals with nodal values and does not require special formula for evaluating system matrices. Also, the convenience of the approach is shown in handling general boundary conditions. Numerical examples are given for the free vibrations of FG shells with different combinations of classical and elastic boundary conditions. Effects of spring stiffness values and the material power-law distributions on the natural frequencies of shells are also discussed. Some new results for the considered shell structures are presented, which may serve as benchmark solutions.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

Efficient boundary element analysis of sharp notched plates

The present paper further develops the boundary element singularity subtraction technique, to provide an efficient and accurate method of analysing the general mixed-mode deformation of two-dimensional linear elastic structures containing sharp notches. The elastic field around sharp notches is singular. Because of the convergence difficulties that arise in numerical modelling of elastostatic problems with singular fields, these singularities are subtracted out of the original elastic field, using the first term of the Williams series expansion. This regularization procedure introduces the stress intensity factors as additional unknowns in the problem; hence extra conditions are required to obtain a solution. Extra conditions are defined such that the local solution in the neighbourhood of the notch tip is identical to the Williams solution; the procedure can take into account any number of terms of the series expansion. The standard boundary element method is modified to handle additional unknowns and extra boundary conditions. Analysis of plates with symmetry boundary conditions is shown to be straightforward, with the modified boundary element method. In the case of non-symmetrical plates, the singular tip-tractions are not primary boundary element unknowns. The boundary element method must be further modified to introduce the boundary integral stress equations of an internal point, approaching the notch-tip, as primary unknowns in the formulation. The accuracy and efficiency of the method is demonstrated with some benchmark tests of mixed-mode problems. New results are presented for the mixed-mode analysis of a non-symmetrical configuration of a single edge notched plate.     

Vibration of non-ideal simply supported laminated plate on an elastic foundation subjected to in-plane stresses

In this paper, the effect of non-ideal boundary conditions and initial stresses on the vibration of laminated plates on Pasternak foundation is studied. The plate has simply supported boundary conditions and is assumed that one of the edges of the plate allows a small non-zero deflection and moment. The initial stresses are due to in-plane loads. The vibration problem is solved analytically using the Lindstedt–Poincare perturbation technique. So the frequencies and mode shapes of the plate with non-ideal boundary condition is extracted by considering the Pasternak foundation and in-plane stresses. The results of finite element simulation, using ANSYS software, are presented and compared with the analytical solution. The effect of various parameters like stiffness of foundation, boundary conditions and in-plane stresses on the vibration of the plate is discussed. Dependency of non-ideal boundary conditions on the aspect ratio of the plate for changing the frequencies of vibrations is presented. The relation between the shear modulus of elastic foundation and the frequencies of the plate is investigated.     

Frictional contact formulation using quadratic programming

A new solution procedure for contact problems in elasticity with prescribed normal tractions on contact surface has been proposed in this paper. The procedure is based on the boundary element method and quadratic programming. It is next used in a two step solution algorithm for the analysis of contact problems with friction. Several numerical examples are presented and compared with results obtained using alternative solution methods.This research has been supported in part of the National Science Foundation Presidental Young Investigator award MSS-9057055 with D. Oscar Dillon as the program manager, and by the U.S. Department of Energy, under contact DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc., by an appointment to the Oak Ridge National Laboratory Postdoctoral Research Associates Program administered jointly by the Oak Ridge Institute for Science and Education and Oak Ridge National Laboratory.     

Free vibration analysis of thin arbitrarily laminated anisotropic plates using boundary-continuous displacement Fourier approach

A boundary continuous displacement based Fourier series solution to the boundary-value problem of free vibration of an arbitrarily laminated thin rectangular plate is presented. This powerful approach is employed to solve a system of three highly coupled partial differential equations arising from the Kirchhoff hypothesis as applied to an anisotropic laminate, with the SS2-type simply supported boundary conditions prescribed at all four edges. The accuracy of the computed eigenvalues (natural frequencies) is ascertained by studying the convergence characteristics of the lowest seven natural frequencies, and also by comparison with the computed degenerate FEM (finite element methods) results. Other important numerical results presented include variation of the response quantities of interest with geometric and material parameters, such as fiber orientation angle and longitudinal-to-transverse modulus ratio.     

Investigation into nonlinear vibrations of composite plates using the <Emphasis Type="Italic">R</Emphasis>-function theory

We have developed an effective approach to the solution of problems on geometrically nonlinear vibrations of orthotropic multilayer plates of irregular shapes in a classical statement based on the use of the R-function theory, Ritz variational method and Bubnov-Galerkin method. Using the proposed method, problems of vibrations of both multilayer rectangular plates and plates of complex geometries have been solved. The effect of the domain geometry and boundary conditions on the amplitude-frequency characteristics has been investigated. __________ Translated from Problemy Prochnosti, No. 5, pp. 101–113, September–October, 2007.     

斜拉桥拉索的参数振动有限元分析

为探讨斜拉桥拉索发生大幅参数振动的条件,以及在振动过程中拉索的振幅和张力波动的特点,本文利用作者建立的几何非线性结构非一致激励响应分析方法,进行了斜拉索的参数振动有限元分析。分析表明：桥面的面内激励易使斜拉索的振动包含面外对称振型,从而出现组合振动现象;而桥面的面外激励只激起面外对称振动,不易引起斜拉索的面内振动;增加斜拉索的阻尼可抑制其振幅,缩小其发生大幅振动的频率范围,但其张力波幅却相应增加。所得结果与以往研究者的解析结果及实验观测现象相一致。     

Effects of local damage on vibration characteristics of composite pyramidal truss core sandwich structure

The effects of local damage on the natural frequencies and the corresponding vibration modes of composite pyramidal truss core sandwich structures are studied in the present paper. Hot press molding method is used to fabricate intact and damaged pyramidal truss core sandwich structures, and modal testing is carried out to obtain their natural frequencies. A FEM model is also constructed to investigate their vibration characteristics numerically. It is found that the calculated natural frequencies are in relatively good agreement with the measured results. By using the experimentally validated FEM model, a series of numerical analyses are conducted to further explore the effects of damage extent, damage location, damage form on the vibration characteristics of composite pyramidal truss core sandwich structures as well as the influence of boundary conditions. The conclusion derived from this study is expected to be useful for analyzing practical problems related to structural health monitoring of composite lattice sandwich structures.     

Statistical modelling of predicted non‐stationary vehicle vibrations

This paper describes a method to predict the vertical vibrations of road vehicles from measured pavement profiles. It discusses the limitations of current methods used for analysing and simulating vehicle vibrations and shows that more accurate characterization and simulation of the transport environment must take into account the non‐stationary nature of road vehicle vibrations. Vertical vibrations for typical transport vehicles under various operating conditions and pavement profiles are predicted using a computer model of the vehicle characteristics and analysed to produce the spectral and statistical characteristics. The paper also presents an improved method to compute the vibration intensity by using a dynamic segmentation data reduction technique. The effectiveness of the procedure to characterize the non‐stationarity of random vehicle vibrations is demonstrated. Finally, the paper deals with the statistical distribution of the vibration intensity and demonstrates how it can be adapted to a technique for the simulation the non‐stationary nature of random vehicle vibrations. Copyright © 2002 John Wiley & Sons, Ltd.