Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation

The purpose of this paper is to present efficient and accurate analytical expressions for large amplitude free vibration and post-buckling analysis of unsymmetrically laminated composite beams on elastic foundation. Geometric nonlinearity is considered using Von Karman’s strain–displacement relations. Besides, the elastic foundation has cubic nonlinearity with shearing layer. The nonlinear governing equation is solved by employing the variational iteration method (VIM). This study shows that the third-order approximation of the VIM leads to highly accurate solutions which are valid for a wide range of vibration amplitudes. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams and the post-buckling load–deflection relation are studied.     

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.     

Analysis of large amplitude free vibrations of clamped tapered beams on a nonlinear elastic foundation

The purpose of this paper is to present efficient and accurate analytical expressions for large amplitude free vibration analysis of single and double tapered beams on elastic foundation. Geometric nonlinearity is considered using the condition of inextensibility of neutral axis. Moreover, the elastic foundation consists of a linear and cubic nonlinear parts together with a shearing layer. The nonlinear governing equation is solved by employing the variational iteration method (VIM). This study shows that the second-order approximation of the VIM leads to highly accurate solutions which are valid for a wide range of vibration amplitudes. The effects of different parameters on the nonlinear natural frequency of the beams are studied under different mode shapes. The results of the present work are also compared with those available in the literature and a good agreement is observed.     

弹性地基无限长梁动力问题的一般解

本文从Euler-Bernoulli梁出发,对弹性地基采用Winkler假定,建立了问题的数学模型.然后对空间变量和时间变量分别进行Fourier变换和Laplace变换,利用逆变换褶积积分,得到了弹性地基无限长梁一般动力问题的解析解.最后对自由振动,脉冲激励和运动载荷情况分别进行了讨论.     

Thermally induced vibrations of orthotropic rectangular plate resting on elastic foundation

Thermally induced vibration of orthotropic rectangular plate resting on elastic foundation has been investigated. The problem is solved in terms of double trignometric series. Complete solution is derived from the sum of two deflections—quasi-static and dynamic. The dynamic solution is obtained by the method of Laplace transform.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

Solution of free vibration equations of beam on elastic soil by using differential transform method

Free vibration differential equations of motion of one end fixed, the other simply supported and axial loaded beams on elastic soil is solved using differential transform method (DTM), analytical solution and frequency factors are obtained.     

Vibration and stability of non-uniform beams with abrupt changes of cross-section by using C1 modified beam vibration functions

A unified method is presented for the analysis of vibration and stability of axially loaded non-uniform beams with abrupt changes of cross-section. The beam may also be supported on Winkler elastic foundation, and both the axial force and the foundation stiffness can be varied arbitrarily. The method is based on the Euler–Lagrangian approach using a family of C¹ admissible functions as the assumed modes. The assumed modes comprise essentially the vibration modes of a single span hypothetical prismatic beam with the same end supports but without the intermediate supports, modified by piecewise C¹ cubic polynomials. The chosen admissible functions therefore possess both the advantages of fast convergence of the eigenfunctions and the appropriate order of continuity at the location of abrupt change of cross-section. The method allows extensive use of matrix notations and programming is rather straightforward. Numerical results also show that the method is versatile, efficient and accurate.     

DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges

This paper presents an efficient and accurate differential quadrature (DQ) large amplitude free vibration analysis of laminated composite thin beams on nonlinear elastic foundation. Beams under consideration have elastically restrained against rotation and in-plane immovable edges. Elastic foundation has cubic nonlinearity with shearing layer. We impose the boundary conditions directly into the governing equations in spite of the conventional DQ method and without any extra efforts. A direct iterative method is used to solve the nonlinear eigenvalue system of equations after transforming the governing equations into the frequency domain. The fast rate of convergence of the method is shown and their accuracy is demonstrated by comparing the results with those for limit cases, i.e. beams with classical boundary conditions, available in the literature. Besides, we develop a finite element program to verify the results of the presented DQ approach and to show its high computational efficiency. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams are studied.     

A new Orthonormal Polynomial Series Expansion Method in vibration analysis of thin beams with non-uniform thickness

In this article, OPSEM (Orthonormal Polynomial Series Expansion Method) is developed as a new computational approach for the evaluation of thin beams of variable thickness transverse vibration. Capability of the OPSEM in assessing the free vibration frequencies and mode shapes of an Euler–Bernoulli beam with varying thickness is discussed. Multispan continuous beams with various classical boundary conditions are included. Contribution of BOPs (Basic Orthonormal Polynomials) in capturing the beam vibrations is also illustrated in numerical examples to give a quantitative measure of convergence rate. Furthermore, OPSEM is adopted for the forced vibration of a thin beam caused by a moving mass. Dynamics of beams supported by flexible elastic base like free to free beam on elastic foundation are also regarded. Verifications are made via eigenfunction expansion method and GMLSM (Generalized Moving Least Square Method). The very close observed agreement between the results of the two recently mentioned methods and that of OPSEM can be regarded as a guarantee of validity for the newly introduced technique. In comparison with eigenfunction expansion method, the simplicity and handiness of OPSEM in coping with different boundary conditions of the beam can be considered as its benefit for engineering practitioners.     

Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

WFQEM‐based perturbation approach and its applications in analyzing nonlinear periodic structures

Based on the weak form quadrature element method, a perturbation approach is developed. Waves propagating in periodic beams on a nonlinear elastic foundation are studied by using the new proposed method. The feasibility and accuracy of the proposed method are verified by comparing the present results with those available in literatures in linear cases. Detailed modal analysis of the linear cases is conducted in order to obtain the dispersion relations of the nonlinear cases. The theoretical results show that the dispersion relations of the nonlinear cases are amplitude dependent. Furthermore, the effects of geometric parameters and degree of nonlinearity on the amplitude‐dependent dispersion relations are discussed in detail. This work provides a new method for analyzing the dispersion relations of nonlinear periodic structures and gives some useful guidelines for designing periodic beams or pipelines with nonlinear structure–foundation interaction. Copyright © 2016 John Wiley & Sons, Ltd.     

非线性弹性地基上的圆薄板的分岔与混沌问题

根据非线性弹性地基上圆薄板大幅度方程,弹性抗力有线性项,三次非线性项和抗弯曲弹性项。在周边固定的条件下,利用Galerkin法得到了一个非线性振动方程。在无外激励情况下,求出在平衡点处的Floquet指数。分析了其稳定性与可能发生的分岔条件。在外激励条件下,用Melnikov方法分析研究了可能发生的混沌振动。通过数字仿真给出了各种地基参数下混沌区域的临界曲线和相平面图。     

Size-dependent forced vibration of non-uniform bi-directional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass

In this study, general non-uniform material-varying micro-beam models under a moving harmonic load/mass are investigated. Material variation is modeled by combining axial and thickness material grading models using exponential, linear, parabolic and sigmoidal functions. Beam is assumed to be resting on an elastic foundation and in this linear foundation model, foundation modulus is assumed to vary axially with respect to space variable in a non-linear manner ignoring the effect of mass density of foundation on the behavior of micro-beam. Cross-section variation through the length is formulated for both thickness and width variation. Governing equations for such comprehensive beam model is achieved using Hamilton's principle in conjunction with modified couple stress theory to add the scale-effects and solved by discussing explicit and implicit finite element methods with using various-steps and Wilson-theta method. Current methodology is verified using previous studies on simplified problems. A comprehensive parametric study is presented in order to indicate the influence of each design, material and fundamental terms on the forced vibration behavior of such structures under a moving harmonic/constant load/mass. It is shown that by appropriately choosing the material variation in bidirectional functionally graded beams dynamic vibration behavior of such structures could change significantly. Moreover, it is shown that varying cross-section, elastic foundation and type of harmonic moving mass can change the dynamic reaction of the general micro-beam model. From the influence of modified couple stress term on mechanical behavior of such structures it is concluded that this term has crucial effect in varying the dynamic deflections and it is important to acknowledge it in analyzing such structures.     

Bending analysis of elastically connected Euler–Bernoulli double-beam system using the direct boundary element method

Double and multiple-Beam System (BS) models are structural models that idealize a system of beams interconnected by elastic layers, where beam theories are assumed to govern the beams and elastic foundation models are assumed to represent the elastic layers. Many engineering problems have been studied using BS models such as double and multiple pipeline systems, sandwich beams, adhesively bonded joints, continuous dynamic vibration absorbers, and floating-slab tracks. This paper presents for the first time a direct Boundary Element Method (BEM) formulation for bending of Euler–Bernoulli double-beam system connected by a Pasternak elastic layer. All of the mathematical steps required to establish the direct BEM solution are addressed. Discussions deriving explicit solutions for double-beam fundamental problem are presented. Integral and algebraic equations are derived where influence matrices and load vectors of double-beam systems are explicitly shown. Finally, numerical results are presented for differing cases involving static loads and boundary conditions.     

A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

弹性半空间地基上四边自由矩形板稳态振动的解析解

采用双重Fourier变换,分析得到弹性半空间地基受竖向稳态荷载作用下的积分变换解．与四边自由矩形板的振动解析解相结合,得出弹性半空间地基上四边自由矩形板稳态振动的解析解．还给出算例及参数影响分析．     

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.