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.     

Analytical solution of the dynamic behavior of non-homogenous orthotropic cylindrical shells on elastic foundations under moving loads

An analytical study on the dynamic behavior of an infinitely long, non-homogenous orthotropic cylindrical shell resting on elastic foundations subjected to combined action of the axial tension, internal compressive load and ring-shaped compressive pressure with constant velocity is presented. The problem is studied on the basis of the theory of vibrations of cylindrical shells. Formulas are derived for the maximum static and dynamic displacements, dynamic factors and critical velocity for homogenous and non-homogenous orthotropic cylindrical shells on Winkler or Pasternak elastic foundations and subjected to moving loads. A parametric study is conducted to demonstrate the effects of various parameters, such as Winkler or Pasternak foundations, the non-homogeneity and orthotropy of materials, the radius-to-thickness ratio and the velocity of the moving load on the dynamic displacements, dynamic factors and critical values of the velocity for cylindrical shells.     

Influences of elastic foundations and shear deformations on the buckling behavior of functionally graded material truncated conical shells under axial compression

This article presents an investigation on the buckling of functionally graded (FG) truncated conical shells under an axial load resting on elastic foundations within the shear deformation theory (SDT). The governing equations are solved using the Galerkin method, and the closed-form solution of the axial buckling load for FG conical shells on elastic foundations within the SDT is obtained. Various numerical examples are presented and discussed to verify the accuracy of the closed-form solution in predicting dimensionless buckling loads for FG conical shells on the Winkler–Pasternak elastic foundations within the SDT.     

On the effect of the backup plate stiffness on the brittle failure of a ceramic armor

The present investigation enquires the role of the backup plate mechanical properties in the brittle failure of a ceramic tile. It provides a full-field solution for the elastostatic problem of an infinite Kirchhoff plate containing a semi-infinite rectilinear crack (the tile) resting on a two-parameter elastic foundation (the backup plate) and subjected to general transverse loading condition. The backup plate is modeled as a weakly non-local (Pasternak type) foundation, which reduces to the familiar local (Winkler) model once the Pasternak modulus is set to zero. The same governing equations are obtained for a curved plate (shell) subjected to in-plane equi-biaxial loading. Fourier transforms and the Wiener–Hopf technique are employed. The solution is obtained for the case when the Pasternak modulus is greater than the Winkler modulus. Superposition and a two-step procedure are employed: First, an infinite uncracked plate subjected to general loading is considered; then, the bending moment and shearing force distribution acting along the crack line are adopted as the (continuous) loading condition to be fed in the solution for the cracked plate. Results are obtained as a function of the ratio of the Pasternak over the Winkler foundation stiffness times the tile flexural rigidity. It is established that the elastic foundation significantly affects the mechanical behavior of the elastic plate. In particular, the Winkler model substantially underestimates the stress state near the crack tip. Stress-intensity factors are determined, and they are employed as a guideline for increasing the composite toughness. The analytical solution presented in this paper may serve as a benchmark for a more refined numerical analysis.     

Mechanical buckling of two-dimensional functionally graded cylindrical shells surrounded by Winkler–Pasternak elastic foundation

In this research, buckling analysis of a two-dimensional, functionally graded, cylindrical shell that has been embedded in an outer elastic medium in the presence of combined axial and transverse loading based on third-order shear deformation shell theory is numerically investigated. Variations of the shell properties are considered to be continuous through length and thickness. Winkler–Pasternak foundation and simply supported boundary conditions have been applied. The problem has been solved using the generalized differential quadrature method. Geometrical, load, and foundation parameters beside functionally graded power indexes effects on the critical buckling load have been studied.     

Influences of elastic foundations and boundary conditions on the buckling of laminated shell structures subjected to combined loads

This article presents to study the stability of laminated orthotropic cylindrical and truncated conical shells resting on elastic foundations and subjected to combined loads with the clamped and simply supported boundary conditions. Here, axial tensile loads separately applied to the small and large bases of a laminated truncated conical shell, respectively. The basic relations, the modified Donnell type stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells on the Pasternak type elastic foundation. Applying Galerkin method, the critical combined loads of laminated orthotropic conical shells on the Pasternak type elastic foundation with different boundary conditions are obtained. The appropriate formulas for single-layer and laminated cylindrical shells on the Pasternak type elastic foundation made of orthotropic and isotropic materials are found as special cases. Finally, influences of the boundary conditions, the elastic foundation, the number and ordering of the layers and variations of the shell characteristics on the critical combined loads are investigated. The results are compared with their counterparts in the literature.     

Nonlinear transient response of orthotropic thin rectangular plates on elastic foundations

This paper presents geometrically nonlinear transient analysis of rectilinearly orthotropic thin rectangular plates resting on Winkler and Pasternak foundations for uniformly distributed step function and sinusoidal pulse loadings. The orthogonal point collocation method in the space domain and Newmark-β scheme in the time domain are employed. Immovable clamped and simply-supported plates are analysed. An approximate method is used to predict the maximum dynamic response to step loads from the results for static loads and is found to yield sufficiently accurate results.     

Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation

In this study, the mechanical buckling of functionally graded material cylindrical shell that is embedded in an outer elastic medium and subjected to combined axial and radial compressive loads is investigated. The material properties are assumed to vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Theoretical formulations are presented based on a higher-order shear deformation shell theory (HSDT) considering the transverse shear strains. Using the nonlinear strain–displacement relations of FGMs cylindrical shells, the governing equations are derived. The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. The boundary condition is considered to be simply-supported. The novelty of the present work is to achieve the closed-form solutions for the critical mechanical buckling loads of the FGM cylindrical shells surrounded by elastic medium. The effects of shell geometry, the volume fraction exponent, and the foundation parameters on the critical buckling load are investigated. The numerical results reveal that the elastic foundation has significant effect on the critical buckling load.     

弹性地基板广义边值问题的边界元法

本文利用Ｈａｎｋｅｌ变换导出了弹性地基板弯曲问题的基本解，该基本解对于Ｗｉｎｋｌｅｒ地基、Ｐａｓｔｅｒｎａｋ地基和弹性半空间地基模型具有统一的表达形式。在此基础上，建立了适用于弹性地基板广义边值问题的边界积分方程组，最后文中给出了若干数值算例。     

Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment

The main aim of this paper is to investigate the nonlinear buckling and post-buckling of functionally graded stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load by analytical approach. Shells are reinforced by closely spaced rings and stringers in which material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. Based on the classical shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived. Using Galerkin method with three-term solution of deflection, the closed form to find critical torsional load and post-buckling load–deflection curves are obtained. The effects of temperature, stiffener, foundation, material and dimensional parameters are analyzed.     

Boundary element formulations for Mindlin plate on an elastic foundation with dynamic load

Boundary element method (BEM) for a shear deformable plate (Reissner/Mindline's theories) resting on an elastic foundation subjected to dynamic load is presented. Formulations for both Winkler and Pasternak foundations are presented. The boundary element formulation in Laplace domain is presented together with complete expressions for the internal point kernels (i.e. fundamental solutions). Quadratic isoparameteric boundary elements are used to discretise the boundary of plate domain. Time domain variables are obtained by the Durbin's inversion method from transform domain. Numerical examples are presented to demonstrate the accuracy of the boundary element method and the comparisons are made with other numerical technique.     

Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin and Toorani–Lakis theory. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the general formulation and the Classical Reissner–Mindlin and Classical Toorani–Lakis theory are presented. New results are presented in order to investigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the free vibrations of anisotropic shells of revolution with a free-form meridian.     

Size-dependent vibration analysis of a three-layered porous rectangular nano plate with piezo-electromagnetic face sheets subjected to pre loads based on SSDT

ABSTRACT

In the present research vibration of a porous rectangular plate which is located between two piezo-electromagnetic layers based on two variables sinusoidal shear deformation plate theory and according to nonlocal theory is investigated. The plate is resting on Winkler–Pasternak foundation and was subjected to pre loads. The motion equations have been obtained using Hamilton principle and are solved using analytical Navier's solution method. The effects of porosity coefficient, pores distribution, nonlocal parameter, pre load values, foundation constants and geometric size of the plate have been discussed in details. The results can be used to design more efficient sensors and actuators.     

Nonlinear vibration and instability analysis of functionally graded CNT-reinforced cylindrical shells conveying viscous fluid resting on orthotropic Pasternak medium

In this study, nonlinear vibration and instability of embedded temperature-dependent cylindrical shell conveying viscous fluid resting on temperature-dependent orthotropic Pasternak medium are investigated. The equivalent material properties of nanocomposites are estimated using rule of mixture. Both cases of uniform distribution and functionally graded distribution patterns of reinforcements are considered. Based on orthotropic Mindlin shell theory, the governing equations are derived. Generalized differential quadrature method is applied for obtaining the frequency and critical fluid velocity of a system. The effects of different parameters, such as distribution type of single-walled carbon nanotubes (SWCNTs), volume fractions of SWCNTs, and Pasternak medium are discussed.     

Analysis of plates on Pasternak foundations by radial basis functions

This paper addresses the static and free vibration analysis of rectangular plates resting on Pasternak foundations. The Pasternak foundation is described by a two-parameter model. The numerical approach is based on collocation with radial basis functions. The model allows the analysis of arbitrary boundary conditions and irregular geometries. It is shown that the present method, based on a first-order shear deformation theory produces highly accurate displacements and stresses, as well as natural frequencies and modes.     

Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches

In this paper, nonlinear dynamic response of rectangular laminated composite plate resting on nonlinear Pasternak type elastic foundations is investigated. First-order shear deformation theory (FSDT) is used for modeling of moderately thick plates. The plate formulation is based on the von Karman nonlinear equation. The resulting nonlinear governing equations for transient analysis of laminated plates on elastic foundation are integrated using the discrete singular convolution-differential quadrature coupled approaches. The nonlinear governing equations of motion of plate are discretized in space and time domains using the discrete singular convolution and the differential quadrature methods, respectively. The validity of the present method is demonstrated by comparing the present results with those available in the open literature. The effects of the foundation parameters, boundary conditions and geometric parameters of plates on nonlinear dynamic response of laminated thick plates are investigated.     

Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations

Exact solutions for functionally graded thick plates are presented based on the three-dimensional theory of elasticity. The plate is assumed isotropic at any point, while material properties to vary exponentially through the thickness. The system of governing partial differential equations is reduced to an ordinary one about the thickness coordinate by expanding the state variables into infinite dual series of trigonometric functions. Interactions between the Winkler–Pasternak elastic foundation and the plate are treated as boundary conditions. The problem is finally solved using the state space method. Effects of stiffness of the foundation, loading cases, and gradient index on mechanical responses of the plates are discussed. It is established that elastic foundations affects significantly the mechanical behavior of functionally graded thick plates. Numerical results presented in the paper can serve as benchmarks for future analyses of functionally graded thick plates on elastic foundations.     

Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium

As a first endeavor, the axisymmetric free and forced vibrations of circular single- and double-layered nanoplates under initial in-plane radial stresses and embedded in an elastic medium are investigated. The governing equations are derived by decoupling the nonlocal constitutive equations of the Eringen theory in polar coordinates in conjunction with the classical plate theory. The elastic medium is modeled as a two-parameter elastic foundation (Pasternak type). Galerkin’s method is employed to solve the resulting equation for vibration frequencies and dynamic response. The effects of small scale together with the other parameters such as initial in-plane load, Winkler and shear elastic foundation coefficients and the radius of the nanoplate are investigated. It is shown that the corresponding natural frequencies obtained by nonlocal elasticity theory are very different from those predicted by classical elasticity theory when the radius of the nanoplate is less than an approximate limit value.     

Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin and Toorani–Lakis theory. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the general formulation and the Classical Reissner–Mindlin and Classical Toorani–Lakis theory are presented. New results are presented in order to investigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the free vibrations of anisotropic shells of revolution with a free-form meridian.     

Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations

A nonlinear bending analysis is presented for a simply supported, functionally graded plate resting on an elastic foundation of Pasternak-type. The plate is exposed to elevated temperature and is subjected to a transverse uniform or sinusoidal load combined with initial compressive edge loads. 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. The formulations are based on a higher-order shear deformation plate theory and general von Kármán-type equation that includes the plate-foundation interaction and thermal effects. A two step perturbation technique is employed to determine the load–deflection and load–bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded plates with two constituent materials resting on Pasternak elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The results reveal that the characteristics of nonlinear bending are significantly influenced by foundation stiffness, temperature rise, transverse shear deformation, the character of in-plane boundary conditions and the amount of initial compressive load. In contrast, the effect of volume fraction index N becomes weaker when the plate is supported by an elastic foundation.