Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation

In this article, an exact analytical solution for buckling analysis of moderately thick functionally graded (FG) sector plates resting on Winkler elastic foundation is presented. The equilibrium equations are derived according to the first order shear deformation plate theory. Because of the coupling between the bending and stretching equilibrium equations of FG plates, these plates have deflection under in-plane loads lower than the critical buckling load acting on the mid-plane. The conditions under which FG plates remain flat in the pre-buckling configuration are investigated and the stability equations are obtained based on the flat plate assumption in the pre-buckling state. The stability equations are simplified into decoupled equations and solved analytically for plates having simply supported boundary condition on the straight edges. The critical buckling load is obtained and the effects of geometrical parameters and power law index on the stability of functionally graded sector plates are studded. The results for the critical buckling load of moderately thick functionally graded sector plates resting on elastic foundation are reported for the first time.     

Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.     

Thermo-Electro-Mechanical Buckling of Shear Deformable Hybrid Circular Functionally Graded Material Plates

Buckling analysis of perfect circular functionally graded plates with surface-bounded piezoelectric layers based on the first-order shear deformation theory is presented in this article. The material properties of the functionally graded (FG) layer are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituents. The plate is assumed to be under constant electrical field and two types of thermal loadings, namely, the uniform temperature rise and nonlinear temperature gradient through the thickness. Also, the stability of a plate under radial mechanical compressive force is examined. The equilibrium and stability equations are derived based on the first-order shear deformation plate theory using a variational approach. The boundary condition of the plate as an immovable type of the clamped edge is considered. Resulting equations are employed to obtain the closed-form solution for the critical buckling temperature for each loading case. The effects of electric field, piezo-to-host thickness ratio, and power law index of functionally graded plates subjected to thermo-mechanical-electrical loads are investigated. The results are compared with the classical plate theory and verified with the available data in the open literature.     

弹性地基上加热弹性圆板的热过屈曲及临界屈曲模态跃迁

基于von Kármán薄板理论建立了Winkler弹性基础上弹性圆板在均匀升温下的轴对称热过屈曲控制方程。这是一组以中面位移为基本未知量的非线性常微分方程,其中包含了温度载荷和弹性地基刚度两个参数。采用打靶法数值求解相应的非线性两点边值问题,获得了周边不可移简支圆板的热屈曲及热过屈曲响应。绘出了前三阶屈曲模态对应的临界温度载荷随地基参数连续变化的特性曲线,获得了反映临界热屈曲模态跃迁特性的地基参数值。给出了弹性圆板按一阶模态失稳后的热过屈曲平衡路径和平衡构形,分析了地基刚度参数对临界屈曲温度载荷以及过屈曲平衡构形的影响。     

Buckling analysis of a porous circular plate with piezoelectric sensor–actuator layers under uniform radial compression

This study presents the buckling analysis of a solid circular plate made of porous material bounded with piezoelectric sensor–actuator patches. The porous material properties vary through the thickness direction of the plate following a given function. The general mechanical nonlinear equilibrium and linear stability equations are derived using the variational formulations to obtain the governing equations of the piezoelectric porous plate. The buckling load is derived for solid circular plates under uniform radial compressive loading for the clamped edge condition. The effects of piezoelectric layers on the buckling load of the plate, piezoelectric layer-to-porous plate thickness ratio, feedback gain, and variation of porosity are investigated. The results are verified with the known results in the literature.     

前屈曲耦合变形对FGM圆板稳定性的影响

功能梯度材料结构沿厚度方向具有非均匀性,在其本构关系中会存在拉伸-弯曲耦合效应。在某些条件下,由于这个耦合效应的存在会引起前屈曲耦合变形,因此只要施加面内外载荷,就会伴随该载荷而产生耦合挠度。该文基于经典非线性板理论,导出了计及前屈曲耦合变形时功能梯度圆板稳定性问题的基本方程,并给出了判断功能梯度圆板是否发生屈曲现象的方法。用打靶方法对所得方程进行了数值求解,并利用数值结果研究了在不同边界条件和不同外因素下前屈曲耦合变形对功能梯度圆板稳定性的影响。     

Buckling analysis of orthotropic rectangular plate resting on Pasternak elastic foundation under biaxial in-plane loading

In this study, buckling of rectangular orthotropic plates resting on a Pasternak elastic foundation under biaxial in-plane loading by the power series method (the method of Frobenius) was analyzed. Similar to many studies, two opposite edges of loading are simply supported and two other edges are assumed clamped. In order to extract the characteristic equations of orthotropic rectangular plate under in-plane loading resting on a Pasternak elastic foundation, the classical plate theory, by considering the interaction between plate and foundation, is used. The results showed that in the aspect ratio of less than 2, the existing Pasternak foundation caused the buckling load to increase severely, but by increasing the aspect ratio, the effect of the foundation is negligible. Applying the in-plane load in the y-direction caused the buckling load to decrease, but by increasing the aspect ratios the effect of the load is negligible.     

Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates

In this article, an analytical method for buckling analysis of thin functionally graded (FG) rectangular plates is presented. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. Based on the classical plate theory (Kirchhoff theory), the governing equations are obtained for functionally graded rectangular plates using the principle of minimum total potential energy. The resulting equations are decoupled and solved for rectangular plate with different loading conditions. It is assumed that the plate is simply supported along two opposite edges and has arbitrary boundary conditions along the other edges. The critical buckling loads are presented for a rectangular plate with different boundary conditions, various powers of FGM and some aspect ratios.     

四边简支功能梯度矩形板的热屈曲分析

基于经典板理论,假设材料性质为板厚度方向坐标的幂函数,推导了功能梯度材料矩形板在热荷载作用下的平衡方程和稳定方程。给出了四边简支的功能梯度板在均匀受热时临界屈曲温度变化的封闭解,讨论了板的几何外形尺寸、相对厚度、梯度指数以及中面变形等因素对临界屈曲温度变化的影响。     

Effect of rotationally restrained and Pasternak foundation on buckling of an orthotropic rectangular Mindlin plate

This article, based on first-order shear deformation theory, presents the buckling analysis of a rotationally restrained orthotropic rectangular Mindlin plate resting on a Pasternak elastic foundation. Thus, the Mindlin–Reissner plate theory is employed for which the governing equations are solved by the Rayleigh–Ritz method. Uniformly distributed in-plane loads are applied to two simply supported opposite edges of the plate and the other two edges have rotationally restrained conditions without loading. Finally, the effects of plate parameters, such as foundation stiffness coefficients, aspect ratios, and ratio of elastic modulus in the x to y direction on the buckling loads are presented. The results show that the buckling load would increase when the ratio of the elastic modulus in the x to y direction increases and the plate is close to isotropic. The variation of buckling load versus changing ratio of elastic modulus in the x to y direction in the state of without elastic foundation and with clamp support is more than the rest of the state.     

Hygrothermo-mechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model

Hygrothermal and mechanical buckling responses of functionally graded (FG) plates resting on Winkler–Pasternak’s foundations are presented in this paper using a refined quasi-3D model. The effects due to transverse normal strain and shear deformation are both included. The present model exactly satisfies stress boundary conditions on the upper and lower surfaces of the FG plate without using shear correction factors. It is assumed that the material properties vary according to a power law of the thickness coordinate variable. The hygrothermal buckling equilibrium equations are derived from the principle of virtual work for FG plates resting on Winkler–Pasternak’s foundations with simply-supported boundary conditions. Two types of thermal and hygrothermal loading, uniform thermal and hygrothermal rise, linear thermal and hygrothermal distribution through the thickness are considered. Numerical results are presented to verify the accuracy of the present study. The effects played by Winkler–Pasternak’s parameters, plate aspect ratio, side-to-thickness ratio, gradient index, and loading type on the critical buckling of the FG plates are all investigated.     

Elastic-plastic buckling analysis of rectangular plates subjected to biaxial loads

In order to calculate the buckling load of a rectangular plate, the analytical approach is used in this study. The plate is assumed to be simply supported on four edges and loaded by uniform stresses along the edges. If the plate is slender, the buckling is elastic. However, if the plate is sturdy, it buckles in the plastic range. Then, the instantaneous moduli in the constitutive equations depend on the external loading. In this study, the elastic and plastic buckling equations are derived for rectangular plates under biaxial loading, and the corresponding interaction curves are presented. The influences of aspect ratios, load ratios and hardening factors on the buckling stresses are investigated for rectangular plates. From the plastic buckling analysis, the optimal combination of loads is given for the buckling strength.     

Elastic-plastic buckling analysis of rectangular plates subjected to biaxial loads

In order to calculate the buckling load of a rectangular plate, the analytical approach is used in this study. The plate is assumed to be simply supported on four edges and loaded by uniform stresses along the edges. If the plate is slender, the buckling is elastic. However, if the plate is sturdy, it buckles in the plastic range. Then, the instantaneous moduli in the constitutive equations depend on the external loading. In this study, the elastic and plastic buckling equations are derived for rectangular plates under biaxial loading, and the corresponding interaction curves are presented. The influences of aspect ratios, load ratios and hardening factors on the buckling stresses are investigated for rectangular plates. From the plastic buckling analysis, the optimal combination of loads is given for the buckling strength.     

Semi-analytical mechanical and thermal buckling analyses of 2D-FGM circular plates based on the FSDT

In the present paper, mechanical and thermal buckling analyses of two-directional functionally graded material (2D-FGM) circular plate are investigated. The motion equations have been derived based on the first-order shear deformation theory (FSDT) and power series method has been employed to solve the motion equations. Two different kinds of boundary condition including simply supported and fixed are considered. The material properties are assumed to vary in both transverse and radial directions according to power and exponential laws, respectively. Comparisons with available studies in the literature confirm the high accuracy of the current approach. The effects of geometrical parameters and 2D-FG power indices on the critical buckling load have been studied. It is shown that increase of modulus of elasticity of outer layers of plate due to higher presence of hard phase of FGM, in radius and thickness directions of the plate makes it possible to attain a more solid structure against mechanical buckling loads, while increase of coefficient of thermal expansion and coefficient of thermal conduction of outer layers of plate results in less stability against thermal buckling loads.     

Thermal Postbuckling of Imperfect Shear-Deformable Laminated Plates on Two-Parameter Elastic Foundations

Thermal postbuckling analysis is presented for a simply supported, shear-deformable composite laminated plate subjected to uniform or nonuniform parabolic temperature loading and resting on a two-parameter (Pasternak-type) elastic foundation. The initial geometric imperfection of the plate is taken into account. Reddy's third-order shear-deformation plate theory with von Karman nonlinearity is used. The governing equations also include the plate-foundation interaction and thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, symmetric cross-ply laminated plates resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influence played by a number of effects, among them foundation stiffness, transverse shear deformation, plate aspect ratio, fiber orientation, thermal load ratio, and initial geometric imperfections, is studied. Typical results are presented in dimensionless graphical form.     

Thermomechanical buckling of hybrid cross-ply laminated rectangular plates

A thermomechanical buckling analysis is presented for simply supported rectangular symmetric cross-ply laminated composite plates that are integrated with surface-mounted piezoelectric actuators and are subjected to the combined action of in-plane compressive edge loads, two types of thermal loads, and constant applied actuator voltage. The formulation of equations is based on the classical laminated plate theory and the von-Karman non-linear kinematic relations. The analysis uses an exact method to obtain closed-form solutions for the buckling load. The effects of applied actuator voltage, thermal and mechanical loads, plate geometry, and lay-up configuration of the laminated plates are investigated. The novelty of the present work is to obtain closed-form solutions for electro-thermomechanical buckling of hybrid composite plates, and to cover non-uniform temperature distribution loading. The results for various states are verified with known data in the literature.     

Numerical investigation of the response of sandwich-type panels using thin-walled tubes subject to blast loads

The response of a novel lightweight panel design under blast loading is numerically investigated. The sandwich-type panel uses thin-walled square tubes as the core material with mild steel outer plates. A parametric study is carried out with ABAQUS/Explicit to examine the effects and interaction between design variables in three different tube layouts. Tube position, thickness and aspect ratio as well as top plate thickness are varied. Buckling stability and absorption performance are shown to be highly sensitive to tube placement due to interaction effects between the top plate and tubes. For each panel an optimal tube positioning is obtained corresponding to nearly perfect axial progressive symmetric tube buckling. Tube thickness is shown to influence the onset of buckling and hence affects the stability of the core, while energy absorption performance is also highly configurable. Tube aspect ratio shows only a small effect on core buckling stability and energy absorption. Top plate thickness influences absorber performance significantly while having a small effect on buckling stability. A simple theoretical analysis is presented and shows reasonable agreement with the numerical simulations.     

Buckling analysis of shear deformable plates by boundary element method

In this paper, the derivation and numerical implementation of boundary integral equations for the buckling analysis of shear deformable plates are presented. Plate buckling equations are derived as a standard eigenvalue problem. The formulation is formed by coupling boundary element formulations of shear deformable plate and two dimensional plane stress elasticity. The eigenvalue problem of plate buckling yields the critical load factor and buckling modes. The domain integrals which appear in this formulation are treated in two different ways: initially the integrals are evaluated using constant cells, and next, they are transformed into equivalent boundary integrals using the dual reciprocity method (DRM). Several examples with different geometry, loading and boundary conditions are presented to demonstrate the accuracy of the formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Buckling of a single-layered graphene sheet on an initially strained InGaAs thin plate

The elastic buckling behavior of a defect-free single-layered graphene sheet deposited on a strained InGaAs substrate is investigated. Such a buckled sandwich structure can be formed by local etching of an initially strained InGaAs substrate. We numerically investigated the necessary buckling conditions for a single-layered graphene sheet of circular geometry on an initially strained InGaAs thin plate. A criterion for buckling for various axisymmetric buckling shapes was obtained. It is shown that for a thin circular InGaAs plate with a monolayer graphene sheet of radius 80 nm and thickness 4 nm three axisymmetric buckling shapes can be obtained. For an initial value of the elastic deformation of the plate of 3%, the in-plane strain in graphene can reach a value of 1%. This deformation is shown to be distributed inhomogeneously along the radius of the graphene monolayer.     

Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method

Here, asymmetric free vibration characteristics and thermoelastic stability of functionally graded circular plates are investigated using finite element procedure. A three-noded shear flexible plate element based on the field-consistency principle is used. Temperature field is assumed to be a uniform distribution over the plate surface and varied in thickness direction only. Material properties are graded in the thickness direction according to simple power law distribution. For the numerical illustrations, aluminum/alumina is considered as functionally graded material. The variation in critical buckling load is highlighted considering gradient index, temperature, radius-to-thickness ratios, circumferential wave number and boundary condition of the plate.