A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations

Free vibration analysis of functionally graded sandwich beams with general boundary conditions and resting on a Pasternak elastic foundation is presented by using strong form formulation based on modified Fourier series. Two types of common sandwich beams, namely beams with functionally graded face sheets and isotropic core and beams with isotropic face sheets and functionally graded core, are considered. The bilayered and single-layered functionally graded beams are obtained as special cases of sandwich beams. The effective material properties of functionally graded materials are assumed to vary continuously in the thickness direction according to power-law distributions in terms of volume fraction of constituents and are estimated by Voigt model and Mori–Tanaka scheme. Based on the first-order shear deformation theory, the governing equations and boundary conditions can be obtained by Hamilton’s principle and can be solved using the modified Fourier series method which consists of the standard Fourier cosine series and several supplemented functions. A variety of numerical examples are presented to demonstrate the convergence, reliability and accuracy of the present method. Numerous new vibration results for functionally graded sandwich beams with general boundary conditions and resting on elastic foundations are given. The influence of the power-law indices and foundation parameters on the frequencies of the sandwich beams is also investigated.     

Dynamic response of shear-deformable axisymmetric orthotropic circular plate structures solved by the DQEM and EDQ based time integration schemes

The dynamic response of shear-deformable axisymmetric orthotropic circular plate structures is solved by using the DQEM to the spacial discretization and EDQ to the temporal discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.     

DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media

Based on the Porous Media Theory presented by de Boer, the governing differential equations for a layered space-axisymmetrical fluid-saturated porous elastic body are firstly established, in which the suitable interface conditions between layers are presented. Then, a differential quadrature element method (DQEM) is developed, and the DQEM and the second-order backward difference scheme are applied to discretize the governing differential equations of the problem in the spatial and temporal domain, respectively. In order to show the validity of the present analysis, the dynamic response of a fluid-saturated porous medium is analyzed, and the obtained numerical results are directly compared with the existing analytical results. The effects of the numbers of the elements and grid points on the convergence of the numerical results are considered. Finally, the dynamic characteristics of a layered fluid-saturated elastic soil cylinder subjected to a water pressure or a dynamic loading are studied, and the effects of material parameters are considered in detail. From the above numerical results, it can be found that the DQEM has advantages, such as little amount in computation, good stability and convergence as well as high accuracy, so it is a very efficient method for solving the problems in soil mechanics, especially such problems with discontinuities.     

Differential quadrature element method for static analysis of shear deformable cross‐ply laminates

In this paper, the two‐dimensional differential quadrature element method (DQEM) is developed for the static analysis of symmetric cross‐ply laminates using the first‐order shear deformation plate theory. In this study, the laminated plate, which may contain different discontinuities in loading, geometry, material, and boundary conditions, is first divided into several simple plate elements and then the differential quadrature method (DQM) is applied to each simple element. Compatibility conditions are derived to connect the plate elements so that the overall matrix equation system for the whole plate is obtained and solved. The reliability of the DQEM for solving the titled problems is examined carefully through convergence and accuracy studies and finally some numerical test examples are given to demonstrate the applicability and flexibility of this method for practical use. The methodology presented here has overcome some critical drawbacks of the global DQM but is different from the Quadrature Element Method (QEM) since only one grid point is employed to represent the interface point. Copyright © 1999 John Wiley & Sons, Ltd.     

Free and forced vibrations of plates by boundary and interior elements

A direct boundary element method is developed for the dynamic analysis of thin elastic flexural plates of arbitrary planform and boundary conditions. The formulation employs the static fundamental solution of the problem and this creates not only boundary integrals but surface integrals as well owing to the presence of the inertia force. Thus the discretization consists of boundary as well as interior elements. Quadratic isoparametric elements and quadratic isoparametric or constant elements are employed for the boundary and interior discretization, respectively. Both free and forced vibrations are considered. The free vibration problem is reduced to a matrix eigenvalue problem with matrix coefficients independent of frequency. The forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the plate dynamic response to arbitrary transient loading. The effect of external viscous or internal viscoelastic damping on the response is also studied. The proposed method is compared against the direct boundary element method in conjunction with the dynamic fundamental solution as well as the finite element method primarily by means of a number of numerical examples. These examples also serve to illustrate the use of the proposed method.     

Time-dependent creep fracture using singular boundary elements

 This paper presents a procedure for modelling singular crack tip regions of creeping, cracked structural components using singular boundary elements. These special boundary elements correctly simulate the time-dependent singular behaviour of stress and strain fields at the crack tip of creeping materials. The investigated structural components are considered to undergo time-dependent, two-dimensional creep deformation and to be subjected to remote loading conditions. The deformation of the components is assumed to be described by the elastic power law creep model. Examples of various crack problems are investigated to illustrate the efficiency of the proposed singular boundary elements for analysing creep stress and strain distribution problems and for determining some important creep fracture parameters. The effectiveness of the proposed approach is demonstrated and its accuracy is compared with the results obtained by finite element solutions for different creep conditions. Received: 27 February 2002 / Accepted: 28 May 2002 The authors are grateful to Professor D.E. Beskos for encouragement and helpful discussions during the course of this work.     

A boundary element method for solving 3D static gradient elastic problems with surface energy

 A boundary element methodology is developed for the static analysis of three-dimensional bodies exhibiting a linear elastic material behavior coupled with microstructural effects. These microstructural effects are taken into account with the aid of a simple strain gradient elastic theory with surface energy. A variational statement is established to determine all possible classical and non-classical (due to gradient with surface energy terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution with surface energy is explicitly derived and used to construct the boundary integral equations of the problem with the aid of the reciprocal theorem valid for the case of gradient elasticity with surface energy. It turns out that for a well posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. All the kernels in the integral equations are explicitly provided. The numerical implementation and solution procedure are provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Two numerical examples are presented to illustrate the method and demonstrate its merits. Received: 9 November 2001 / Accepted: 20 June 2002 The first and third authors gratefully acknowledge the support of the Karatheodory program for basic research offered by the University of Patras.     

A boundary/domain element method for analysis of building raft foundations

In this paper, a new boundary/domain element method is developed to analyse plates resting on elastic foundations. The developed formulation is then used in analysing building raft foundations. For more practical representation, the considered raft plate is treated as thick plate with free edge boundary conditions. The soil or the elastic foundation is represented as continuous media (follows the Winkler assumption). The boundary element method is employed to model the raft plate; whereas the soil is modelled using constant domain cells or elements. Therefore, in the present formulation both the domain and the boundary of the raft plate are discretized. The associate soil domain integral is replaced by equivalent boundary integrals along each cell contour. The necessary matrix implementation of such formulation is carried out and explained in details. The main advantage of the present formulation is the ability of analysing rafts on non-homogenous soils. Two examples are presented including raft on non-homogenous soil and raft for practical building applications. The results are compared with those obtained from other finite element and alternative boundary element methods to verify the validity and accuracy of the present formulation.     

Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation

In this paper, thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded (FG) beams on nonlinear elastic foundation are investigated. Nonlinear governing partial differential equation (PDE) of motion is derived based on Euler–Bernoulli assumptions together with Von Karman strain–displacement relation. Based on the Galerkin’s decomposition method, the nonlinear PDE governing equation is reduced to a nonlinear ordinary differential equation (ODE). He’s variational method is employed to obtain a simple and efficient approximate closed form solution for the resulted nonlinear ODE. Comparison between results of the present work and those available in literature shows accuracy of the presented expressions. Some new results for the thermo-mechanical buckling and nonlinear free vibration analysis of the FG beams such as the effects of vibration amplitude, material inhomogeneity, nonlinear elastic foundation, boundary conditions, geometric parameter and thermal loading are presented to be used in future references.     

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Based on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem. The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices. By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.     

A 3D Computational Model of RC Beam Using Lower Order Elements with Enhanced Strain Approach in the Elastic Range

A procedure has been described to carry out three-dimensional elastic analysis of reinforced concrete beam employing finite element technique, which uses lower order elements. The proposed procedure utilizes 8-noded isometric solid /hexahedral elements HCiS18 with enhanced assumed strain (EAS) formulation, recently developed in the literature, to predict load-deformation and internal stresses produced in case of a simply supported RC beams in the elastic regime. It models the composite behaviour of concrete and reinforcements in rigid /perfect bond situation and their mutual interaction in bond-slip condition considering continuous interface elements at the material level. Although, bond-slip relation are very much non-linear in behaviour even at the beginning of the loading condition, predictions from the proposed model /procedure are found to be very close to the experimental observations as far as accuracy is concerned in the elastic range. The sole purpose of this paper is to demonstrate the general applicability and to explore the potentiality of using lower order solid elements in the 3D finite element analysis with an aim of developing a general analytical method for the study of reinforced concrete beam in the elastic range.     

A compact analytical method for vibration of micro-sized beams with different boundary conditions

In this article, a compact analytical method for vibration analysis of gradient elastic beams is presented to solve any combination of boundary conditions. The general frequency determinant for microbeams with general restraints are derived by using Stokes’ transformation. The main advantage of this determinant is capability of considering any possible combination of boundary conditions. By assigning proper values to spring parameters in the general frequency determinant, the solutions can also be determined for the rigid boundary conditions. Numerical results are presented to investigate the influences the material length scale parameter, rotational, and translational springs on the free vibration behavior of microbeams. Some of the present results are compared with the previously published results to establish the validity of the present formulation. The microbeams with restrained boundary conditions exhibit significant size dependence when the length of the microbeam approaches to the material length scale parameter.     

Boundary element analysis of reinforced shear deformable shells

In this paper, stiffened shear‐deformable shells are analysed using the boundary element method. Coupled boundary integral equations are presented for describing curved shells under general loading conditions. The equations are based on boundary integral equations for plane stress and plate bending, with coupling terms arising from the curvature of the shell. Domain integrals are transformed into boundary integrals using the dual reciprocity technique. Stiffeners are modelled as curved beams, continuously attached to the shell. Numerical solutions calculated using the present method are compared with finite element results in two examples. Copyright © 2002 John Wiley & Sons, Ltd.     

Flexural-Torsional Buckling and Vibration Analysis of Composite Beams

In this paper the general flexural-torsional buckling and vibration problems of composite Euler-Bernoulli beams of arbitrarily shaped cross section are solved using a boundary element method. The general character of the proposed method is verified from the formulation of all basic equations with respect to an arbitrary coordinate system, which is not restricted to the principal one. The composite beam consists of materials in contact each of which can surround a finite number of inclusions. It is subjected to a compressive centrally applied load together with arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting problems are (i) the flexural-torsional buckling problem, which is described by three coupled ordinary differential equations and (ii) the flexural-torsional vibration problem, which is described by three coupled partial differential equations. Both problems are solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the method can treat composite beams of both thin and thick walled cross sections taking into account the warping along the thickness of the walls. The proposed method overcomes the shortcoming of possible thin tube theory (TTT) solution, which its utilization has been proven to be prohibitive even in thin walled homogeneous sections. Example problems of composite beams are analysed, subjected to compressive or vibratory loading, to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. Moreover, useful conclusions are drawn from the buckling and dynamic response of the beam.     

The Timoshenko beam model of the differential quadrature element method

A new numerical approach for solving Timoshenko beam problems is proposed. The approach uses the differential quadrature method (DQM) to discretize the Timoshenko beam equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of Timoshenko beam structures. The resulting overall discrete equation can be solved by using a solver of the linear algebra. Numerical results of the DQEM Timoshenko beam model are presented. They demonstrate the DQEM numerical method.     

On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

The static behaviour of spherical shallow shells bonded with piezoelectric actuators and subjected to electrical loading are studied in this paper by using the differential quadrature element method (DQEM). Geometrical non‐linear effects are considered. Detailed formulations for the DQ circular spherical shallow shell element and the DQ annular spherical shallow shell element are given for the first time. Numerical studies are performed to evaluate the effects of actuator size, thickness and boundary conditions. Very accurate results are obtained by the DQEM. Based on the results reported in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions for smart materials and structures exhibiting geometric non‐linear behaviours. Thickness effects cannot be neglected when the actuator thickness is comparable to that of the base material. Snap‐through may occur when the applied voltage reaches a critical value even without mechanical loading for certain geometric configurations. Copyright © 2001 John Wiley & Sons, Ltd.     

Nonlinear dynamic response and vibration of imperfect shear deformable functionally graded plates subjected to blast and thermal loads

Based on Reddy's higher-order shear deformation plate theory, this article presents an analysis of the nonlinear dynamic response and vibration of imperfect functionally graded material (FGM) thick plates subjected to blast and thermal loads resting on elastic foundations. The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. Numerical results for the dynamic response and vibration of the FGM plates with two cases of boundary conditions are obtained by the Galerkin method and fourth-order Runge–Kutta method. The results show the effects of geometrical parameters, material properties, imperfections, temperature increment, elastic foundations, and boundary conditions on the nonlinear dynamic response and vibration of FGM plates.     

解析型Winkler弹性地基梁单元构造

该文采用Winkler弹性地基梁理论确定了弹性地基梁的挠度方程解析通解; 根据最小势能原理建立了解析型Winkler弹性地基欧拉梁及铁摩辛柯梁的单元刚度及等效节点荷载; 得到了解析型弹性地基欧拉梁单元AWFB-E及铁摩辛柯梁单元AWFB-T。同时,论文还采用传统里兹法求得了相应的Winkler弹性地基欧拉梁及铁摩辛柯梁单元刚度矩阵,得到了里兹法弹性地基欧拉梁单元RWFB-E及铁摩辛柯梁单元RWFB-T。对该文构建的两类单元与一般梁-基体系有限元分析结果及理论解析解进行了对比。对比结果表明,传统里兹法由于其多项式形函数无法精确模拟弹性地基梁变形,因此其结果与理论解析解有误差,但随着单元数量增多其误差减小; 采用解析型单元进行计算时,无论单元数量多少,得到的均为“真实”解,说明解析试函数法求得的位移形函数比一般的多项式形函数精确,得到的弹性地基梁单元具备解析型、精确性的特点,可应用于解决实际工程问题。     

Shear deformation effect in flexural–torsional vibrations of beams by BEM

In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.     

A simple finite element for beams on elastic foundations

A simple "routine" beam on elastic foundation finite element using a polynomial displacement function has been developed which yields acceptably accurate deflection, shear and bending moment values for prismatic or non-prismatic beams of elastic material resting on foundations with varying or nonlinear subgrade reactions. Limited extension of the formulation to an "exact" finite element using the exact displacement function of a beam on elastic foundation has also been carried out. The subgrade is represented by a non-homogeneous solid medium to include nonlinear parameters if required. The iterative solution is extended to cases where the beam may uplift because the foundation is a no tension material. The model is also suitable for calculating the elastic deflections, membrane. and bending stress resultants for axisymmetrically loaded variable thickness shells of revolution. A computer program called FEBEF [finite element: beam on elastic foundation] incorporating the routine finite element has been prepared for the solution of beams on elastic foundations and axi symmetrically loaded shells of revolution.