Analysis of rectangular laminated composite plates via FSDT meshless method

A meshless approach based on the reproducing kernel particle method is developed for the flexural, free vibration and buckling analysis of laminated composite plates. In this approach, the first-order shear deformation theory (FSDT) is employed and the displacement shape functions are constructed using the reproducing kernel approximation satisfying the consistency conditions. The essential boundary conditions are enforced by a singular kernel method. Numerical examples involving various boundary conditions are solved to demonstrate the validity of the proposed method. Comparison of results with the exact and other known solutions in the literature suggests that the meshless approach yields an effective solution method for laminated composite plates.     

Analysis of annular Reissner/Mindlin plates using differential quadrature method

The axisymmetric flexure responses of moderately thick annular plates under static loading are investigated. The shear deformation is considered using the first-order Reissner/Mindlin plate theory and the solutions are obtained using the differential quadrature (DQ) method. In the solution process, the governing differential equations and boundary conditions for the problem are initially discretized by the DQ algorithm into a set of linear algebraic equations. The solutions of the problem are then determined by solving the set of algebraic equations. This study considers the plate subjected to various combinations of clamped, simply-supported, free and guided boundary conditions and different loading manners. The accuracy of the method is demonstrated through direct comparison of the present results with the corresponding exact solutions available in the literature.     

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.     

Development of the meshless Hermite–Cloud method for structural mechanics applications

In this work, a novel true meshless numerical technique is proposed. It is termed the Hermite–Cloud method and is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is used instead. Another distinction is that the point collocation technique is used for the discretization of the governing partial differential equations. In this method, the Hermite theorem is employed for the construction of the interpolation functions. Through the constructed Hermite-type interpolation functions, we are able to generate the expressions of approximate solutions of both the unknown functions and the first-order derivatives, in a direct manner. A set of auxiliary conditions have also been developed so as to construct a complete set of PDEs with mixed Dirichlet and Neumann boundary conditions. Through several structural analysis examples, it is shown that the numerical results at the scattered discrete points generated by the Hermite–Cloud method are distinctly improved, for both the approximate solutions as well as the first-order derivatives.     

Buckling of antisymmetric cross- and angle-ply laminated plates

A system of three well-known equations of equilibrium governing the buckling response of arbitrarily laminated composite plates is reduced to a single eighth order partial differential equation in terms of a displacement function. This equation is then solved in closed form to predict the buckling response of antisymmetric cross- and angle-ply plates for different boundary conditions. The effect of various plate parameters including the effect of coupling between inplane extension and out of plane bending upon the buckling response of composite plates is discussed. The results are presented in nondimensional graphical form.     

Numerical differential quadrature method for Reissner/Mindlin plates on two-parameter foundations

Approximate solutions for the bending of moderately thick rectangular plates on two-parameter elastic foundations (Pasternak-type) as described by Mindlin's theory are presented. The plates are subjected to an arbitrary combination of clamped and simply-supported boundary conditions. An efficient computational technique, the differential quadrature (DQ) method, is employed to transform the governing differential equations and boundary conditions into a set of linear algebraic equations for approximate solutions. These resulting algebraic equations are solved numerically. In this study, the accuracy of the DQ method is established by direct comparison with results in the existing literature. The convergence properties of the method are illustrated for different combinations of boundary conditions. The deflections, moments and shear forces at selected locations are tabulated in detail for different elastic foundations. The efficiency and simplicity of the solution method are highlighted.     

Analytical solutions for buckling of multi-step non-uniform columns with arbitrary distribution of flexural stiffness or axial distributed loading

In this paper, the function for describing the distribution of flexural stiffness K(x) of a non-uniform column is arbitrary, and the distribution of axial distributed loading N(x) acting on the column is expressed as a function of K(x) and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of variable transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 14 cases. The analytical buckling solutions of one-step non-uniform columns are thus found. Then the obtained analytical solutions are used to derive the eigenvalue equation for buckling of a multi-step non-uniform column for several boundary supports by using the transfer matrix method. A numerical example shows that the proposed procedure is an efficient method for buckling analysis of multi-step non-uniform columns.     

Zigzag theory for buckling of hybrid piezoelectric beams under electromechanical loads

A new efficient coupled one-dimensional (1D) geometrically nonlinear zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electromechanical loads. The potential field is approximated layerwise as piecewise linear. The deflection is approximated to account for the normal strain due to electric field. The axial displacement is approximated as a combination of a global third-order variation and layerwise linear variation. It is expressed in terms of three primary displacement variables and a set of electric potential variables by enforcing exactly the conditions of zero transverse shear stress at the top and bottom and the conditions of its continuity at the layer interfaces. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of simply supported beams under electromechanical loads are presented. Comparisons with the exact 2D piezoelasticity solution establish that the present zigzag theory is very accurate for buckling analysis.     

一种分析薄壁箱梁剪滞效应的高精度法

考虑了三个不同的剪滞纵向位移差函数以反映薄壁箱梁不同宽度翼板的剪滞变化幅度 ,提出一种分析薄壁箱梁剪滞效应的高精度法。应用能量变分原理 ,导出了箱梁受横向荷载作用下的剪滞控制微分方程和边界条件 ,获得闭合解 ,通过高阶有限条法计算验证了本文方法的正确性。并探讨了不同纵向位移差函数对剪力滞的影响。     

大挠度后屈曲倾斜梁结构的非线性力学特性

基于弹性梁的几何非线性大挠度屈曲理论,建立两端固定对称倾斜支撑梁结构的大挠度后屈曲控制微分方程,采用几何非线性隐式变形协调关系来表达强非线性超静定边值问题,得到描述倾斜梁大挠度后屈曲行为的精确解析解.采用数值方法求解含有第一、二类椭圆积分的强非线性微分方程,给出不同倾角梁结构从初始屈曲到后屈曲并发生两态跳转过程中的位形曲线及非线性刚度.根据最小能量原理和挠曲线拐点个数,分析对称屈曲模态与非对称屈曲模态之间相互跳转的内在联系及其对结构非线性刚度突变的影响,得到了屈曲模态之间的转换条件.跳转过程的数值仿真表明,倾斜支撑梁结构发生大挠度后屈曲时具有明显的双稳态特性且只出现低阶(1、2阶)屈曲模态,仿真计算结果与试验结果相一致.     

Buckling of columns: Allowance for axial shortening

This paper investigates the effect of axial shortening on (i) the elastic buckling of columns with a continuous elastic restraint, (ii) the elastic buckling of rotating columns and (iii) the free vibration of columns under a static axial load. These column problems can be solved in a unified approach because the resulting energy functional is similar. The field differential equation is derived by minimizing the energy functional with respect to the lateral displacement function via calculus of variations. The buckling load or fundamental frequency may be obtained by analytically solving the two-point boundary-value problem. It was found that the boundary conditions and the restraint parameter or angular velocity parameter affect the influence of axial shortening on the buckling load. In vibrating columns, tensile forces enhance the effect of axial stretching on the fundamental frequency.     

An analysis of the static and dynamic instability of thick cylinders

A general solution of Bolotin's differential equations for the dynamic stability of a homogenous isotropic medium is given. This takes the form of displacement functions which express the solution as a sum of dilatational and distortional effects. Using these functions, solutions are found for the vibration of cylinders of finite thickness when initial axial stresses are present. The behaviour of solid rods, simple vibration and simple buckling are all seen to be special cases of the general solution. The results are compared with approximate formulae for the buckling of thin cylinders and it is shown that the known solution for the natural frequency of unloaded cylinders is a particular case.     

Exact solution for the free vibration of axially-loaded pretwisted rods

For a pretwisted rod, in which torsional and flexural effects are decoupled, both vibration and buckling behaviour may be described by a pair of fourth-order linear ordinary differential equations. By considering the free vibration of axially-loaded pretwisted rods, a superset of the buckling and vibration equations may be obtained, and these equations may be solved analytically. Such solutions indicate that the relationship between the natural frequency and the applied load is effectively independent of the pretwist angle, for compressive loads and moderate tensile loads.     

Natural frequencies and buckling of pressurized nanotubes using shear deformable nonlocal shell model

The small-scale effect on the natural frequencies and buckling of pressurized nanotubes is investigated in this study. Based on the firstorder shear deformable shell theory, the nonlocal theory of elasticity is used to account for the small-scale effect and the governing equations of motion are obtained. Applying modal analysis technique and based on Galerkin’s method a procedure is proposed to obtain natural frequencies of vibrations. For the case of nanotubes with simply supported boundary conditions, explicit expressions are obtained which establish the dependency of the natural frequencies and buckling loads of the nanotube on the small-scale parameter and natural frequencies obtained by local continuum mechanics. The obtained solutions generalize the results of nano-bar and -beam models and are verified by the literature. Based on several numerical studies some conclusions are drawn about the small-scale effect on the natural frequencies and buckling pressure of the nanotubes.     

Scrutiny of non-linear differential equations Euler-Bernoulli beam with large rotational deviation by AGM

The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams and geometrically nonlinear beam deflection. In this study, solving the nonlinear differential equation governing the calculation of the large rotation deviation of the beam (or column) has been discussed. Previously to calculate the rotational deviation of the beam, the assumption is made that the angular deviation of the beam is small. By considering the small slope in the linearization of the governing differential equation, the solving is easy. The result of this simplification in some cases will lead to an excessive error. In this paper nonlinear differential equations governing on this system are solved analytically by Akbari-Ganji’s method (AGM). Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration solving. The solution of the problem can be obtained very simply and easily. Furthermore, to enhance the accuracy of the results, the Taylor expansion is not needed in most cases via AGM manner. Also, comparisons are made between AGM and numerical method (Runge-Kutta 4th). The results reveal that this method is very effective and simple, and can be applied for other nonlinear problems.     

A unified generalized thermoelasticity; solution for cylinders and spheres

A new unified formulation for the generalized theories of the coupled thermoelasticity based on the Lord–Shulman, Green–Lindsay, and Green–Naghdi models is proposed in this paper. The unified form of the governing equations is presented by introducing the unifier parameters. The formulations are derived and given for the anisotropic heterogeneous materials. The unified equations are reduced for the isotropic and homogeneous materials. Transforming the governing equations into the Laplace domain, they are analytically solved in the space domain for a hollow sphere and cylinder, where a parameter is introduced to consolidate the solution for the sphere and cylinder in a unified form. A thermal shock load is applied to the inner surface of the sphere and cylinder and the results are presented using a numerical inversion technique of the Laplace transform. The results are validated with the known data in the literature.     

Postbuckling of symmetrically laminated, moderately thick, axisymmetric shallow spherical shells

The paper deals with the buckling and postbuckling behaviour of cylindrically orthotropic, axisymmetric laminated, moderately thick shallow spherical shells under uniformly distributed normal loading. Considering the effects of transverse shear, the governing equations of equilibrium for the shells are derived and expressed in terms of normal deflection , slope qf and stress function gy. An iterative Chebyshev series solution technique is employed for the buckling and postbuckling analyses. Critical loads are estimated and the effects of boundary conditions, material properties, shell parameter, base radius to thickness ratio and number of layers on the postbuckling behaviour are shown.     

The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads

The unified equations to obtain the exact solutions for piezoelectric plane beam subjected to arbitrary mechanical and electrical loads with various ends supported conditions is founded by solving functional equations. Comparing this general method with traditional trial-and-error method, the most advantage is it can obtain the exact solutions directly and does not need to guess and modify the form of stress function or electric displacement function repeatedly. Firstly, the governing equation for piezoelectric plane beam is derived. The general solution for the governing equation is expressed by six unknown functions. Secondly, in terms of boundary conditions of the two longitudinal sides of the beam, six functional equations are yielded. These equations are simplified to derive the unified equations to solve the boundary value problems of piezoelectric plane beam. Finally, several examples show the correctness and generalization of this method.     

Stress and electric potential fields in piezoelectric smart spheres

Piezoelectric materials produce an electric field by deformation, and deform when subjected to an electric field. The coupling nature of piezoelectric materials has acquired wide applications in electric-mechanical and electric devices, including electric-mechanical actuators, sensors and structures. In this paper, a hollow sphere composed of a radially polarized spherically anisotropic piezoelectric material, e.g., PZT_5 or (Pb) (CoW) TiO₃ under internal or external uniform pressure and a constant potential difference between its inner and outer surfaces or combination of these loadings has been studied. Electrodes attached to the inner and outer surfaces of the sphere induce the potential difference. The governing equilibrium equations in radially polarized form are shown to reduce to a coupled system of second-order ordinary differential equations for the radial displacement and electric potential field. These differential equations are solved analytically for seven different sets of boundary conditions. The stress and the electric potential distributions in the sphere are discussed in detail for two piezoceramics, namely PZT_5 and (Pb) (CoW)TiO₃. It is shown that the hoop stresses in hollow sphere composed of these materials can be made virtually uniform across the thickness of the sphere by applying an appropriate set of boundary conditions.     

A post-buckling analysis for isotropic prismatic plate assemblies under axial compression

This paper presents a post-buckling analysis for prismatic plate assemblies made of isotropic materials. The structures are assumed to consist of a series of long flat strips rigidly connected together at their edges, subjected to longitudinal in-plane compressive load. The buckling load and corresponding buckling mode of the structure are first obtained as the results of transcendental eigenvalue problems, which arise when exact solutions to the member differential equations are used to form the stiffness matrix of the plate assemblies. The other post-buckling field functions are also obtained analytically as exact solutions to the member differential equations. Results for the load end-shortening and load–deflection relationships for long prismatic plate assembly examples are obtained and compared with results obtained by other authors.