Elsevier

Computers & Structures

Volumes 108–109, October 2012, Pages 93-99
Computers & Structures

Buckling analysis of functionally graded plates under thermal loadings using the finite strip method

https://doi.org/10.1016/j.compstruc.2012.02.011Get rights and content

Abstract

In the present article, a finite strip method is applied for analyzing the buckling behavior of rectangular functionally graded plates (FGPs) under thermal loadings. The material properties of FGPs are assumed to vary continuously through the thickness of the plate, according to the simple power law distribution. Derivations of equations are based on the classical plate theory (CPT). The fundamental equations for rectangular plates of functionally graded material (FGM) are obtained by discretizing the plate into some finite strips. The solution is obtained by the minimization of the total potential energy and solving the corresponding eigenvalue problem. In addition, numerical results for a variety of functionally graded plates with different boundary conditions are presented and compared with those available in the literature. Moreover, the effects of geometrical parameters and material properties on the FGPs’ buckling temperature difference are determined and discussed.

Introduction

Functionally graded materials (FGMs) are advanced high-performance, heat-resistant materials being able to withstand ultra-high and extremely large temperature gradients. FGMs are microscopically inhomogeneous in which the mechanical properties vary smoothly and continuously from one surface to the other one. This is achieved by gradually varying the volume fraction of the constituent materials. The special characteristics of FGMs, which are usually made of metal and ceramic, make them preferable to the conventional composite materials, which are subject to delamination, for engineering applications. These novel materials were first introduced by Koizumi in 1984 [1] and then developed by other scientists [2], [3]. Most of the researches on FGMs have been restricted to thermal stress analysis, fracture mechanics, and optimization while little work has been done to consider the stability analysis and vibrational behavior of FGM structures. Some research works related to the subject of the present paper are introduced below.

Different viewpoints are reported with respect to the condition of a FG plate in the pre-buckling stage. Some researchers believe that the FG plates cannot be assumed to be flat under in-plane loads. Nonetheless, some others have studied the buckling problem of FG plates with the flat plate assumption in the pre-buckling configuration. Naderi and Saidi [4] studied the pre-buckling configuration of the functionally graded plates and discussed the aforementioned viewpoints and the conditions of accuracy of each viewpoint were reported. Saidi and Baferani [5] performed buckling analysis of moderately thick functionally graded annular sector plate based on the first order shear deformation theory. They analytically found the critical buckling temperature for two types of thermal loadings, i.e. uniform temperature rise and gradient through the thickness. The radial edges of the sector were assumed to be simply supported whilst there were arbitrary boundary conditions along the circular edges. Birman [6] studied the buckling problem of functionally graded composite rectangular plates subjected to uniaxial compression. Two classes of fibers are used in hybrid composite material. Linear equilibrium equations for a symmetrically laminated plate, which are uncoupled, have been derived and then solved to obtain the critical buckling load for simply supported edges condition. Feldman and Aboudi [7] carried out elastic buckling analysis of functionally graded plates subjected to axial load and also investigated the optimal spatial distribution of the volume fraction to improve buckling resistance. Javaheri and Eslami [8] studied buckling of functionally graded plates subjected to uniform temperature rise. They used an energy method and reached a closed-form solution. Javaheri and Eslami [9] studied thermal buckling of functionally graded plates based on the classical plate theory for different types of thermal loadings (i.e. uniform, linear and non-linear temperature change across the thickness). The results show that critical temperature differences for the functionally graded plates are generally lower than the corresponding values for homogeneous plates. They also [10] used classical plate theory for the buckling analysis of functionally graded plates under in-plane compressive loading. Liew et al. [11], [12] performed post-buckling analysis of functionally graded plates subjected to thermo-electro-mechanical loading and also considered the thermal post-buckling of these plates. Yang and Shen [13] examined the nonlinear bending and post-buckling behaviour of functionally graded plates subjected to combined transverse and in-plane loads using a semi-analytical approach. Woo et al. [14] presented an analytical solution for the post-buckling behaviour of moderately thick FGM plates and shells under thermal and mechanical loading. Recently, mesh-free methods have been widely applied to a variety of engineering analyses due to their flexibility. Studies include thin shell analysis [15], [16], the large deformation analysis of nonlinear structures [17] and the static and vibration analysis of FGMs [18], [19], [20]. Sofiyev [21] studied the buckling analysis of circular truncated conical and cylindrical shells constructed from FGM subjected to combined axial extension loads and hydrostatic pressure. He analytically found the critical combined buckling loads for above structures with or without elastic foundations. Zenkour and Sobhy [22] used the sinusoidal shear deformation plate theory to find the critical buckling temperature difference of various symmetric FGM sandwich plates subjected to three types of thermal loadings, i.e. uniform, linear and non-linear distribution through-the-thickness.

Since in the current paper the finite strip method is utilized to analyze the buckling of functionally graded plates, it is worth providing a brief review on some of the related works accomplished by implementing finite strip method (FSM). Cheung [23], among others, may be considered as the pioneer who first presented the concept of finite strip method. The finite strip method can be considered as a kind of finite element method in which a special element called strip is used. The basic philosophy is to discretize the structures into longitudinal strips and interpolate the behavior in the longitudinal direction by different functions, depending on different versions of FSM and in the transverse direction by polynomial functions. Subsequent to Cheung, other researchers developed different variations of the method and applied them to different problems. To name a few, Graves Smith and Sridharan [24] proposed the first FSM buckling formulations for isotropic plate structures under edge loading. Plank and Wittrick [25] and Plank and Williams [26] used a finite strip method to study the buckling behavior of isotropic stiffened plates subjected to combined loading. Their results were related to the infinitely long plates pertaining to local buckling. Loughlan [27] used a FSM approach to study the buckling of composite blade stiffened square box sections under pure compression and bending. Dawe and Peshkam [28] re-examined the buckling characteristics of composite blade stiffened panels, which had been initially analyzed by Stroud et al. [29] using a FEM approach, by using strips of higher orders so called superstrips.

In the field of geometrically non-linear analyses, the first two authors of the current paper and their co-workers have made several contributions by developing different versions of finite strip methods, namely full-energy semi-analytical FSM, and full-energy spline FSM. The developed methods are extensively applied to the analysis of the geometrically non-linear response of flat [30], [31] or imperfect [32] rectangular composite laminated plates of arbitrary lay-up configurations, while the plates are subjected to progressive end-shortening in their plane. In a fairly recent study [33] the full-energy semi-analytical FSM is employed by the first two authors of the current paper to investigate the large deflection behaviour of functionally graded plates subjected to lateral pressure loading.

More recently, Ovesy and Ghannadpour have developed a new exact finite strip for analyzing the buckling and initial post-buckling analysis of isotropic flat plates [34] and isotropic I-sections [35]. In their analysis, the new strip was developed based on the concept that it is effectively a plate, and thus the Von-Karman’s equilibrium equation was solved exactly to obtain the general form of out-of-plane buckling deflection mode for the corresponding plate/strip and the Von-Karman’s compatibility equation was subsequently solved exactly to obtain the general form of in-plane displacement fields in the post-buckling region. They have also developed a new exact finite strip for the buckling analysis of symmetrically laminated composite plates and plate structures [36].

In the current paper, the application of the FSM is extended to the analysis of buckling behavior of functionally graded plates subjected to the three types of thermal loadings, namely; uniform temperature rise, linear temperature change across the thickness, and nonlinear temperature change across the thickness. This task is fulfilled by defining a strip which is called a functionally graded strip. Here, the functionally graded strips are developed on the basis of the classical plate theory. The plate is discretized into some functionally graded strips and the solution is obtained by invoking the principle of minimum total potential energy. In representing the out-of-plane deflection function of each strip, the harmonic functions are used in longitudinal direction whereas in the transverse direction the quartic Lagrange–Hermite interpolation functions are used. Whilst the theoretical development of the current FGS is restricted to the strip with the simply supported boundary conditions on its ends, it offers no limitation in terms of application of different boundary conditions (i.e. clamp, free etc.) on the longitudinal edges of the strip. The material properties of the functionally graded plates are assumed to vary continuously through the thickness of the plate, according to the simple power law distribution. After convergence studies, numerical results for a variety of functionally graded plates with different boundary conditions are given and compared with the available results, wherever possible. Additionally, the effects of geometrical parameters and material properties on the buckling temperature difference of FGM plates are determined and discussed. In this study, it is the first time that the concept of FGS is introduced for thermal buckling analysis whilst the corresponding numerical results are presented for different boundary conditions.

Section snippets

FGS material properties

Functionally graded materials are one of the advanced high-temperature materials capable of withstanding extreme temperature environments. FGMs are microscopically inhomogeneous new materials in which the mechanical properties vary smoothly and continuously from one surface to the other one. This is achieved by gradually varying the volume fraction of the constituent materials. Typically, these materials are made from a mixture of ceramics and metal or a combination of different metals. Suppose

Results and discussions

To illustrate the FSM approach, a ceramic-metal functionally graded plate which consists of aluminum and alumina is considered. The surface c of the plate is assumed to be pure alumina (i.e. pure ceramic), while the surface m of the plate is assumed to be pure aluminum (i.e. pure metal). Young’s modulus and thermal expansion coefficient are selected as being 380 GPa and 7.4×10−6/°C for alumina and 70 GPa and 23×10−6/°C for aluminum, respectively. Poisson’s ratio is chosen to be constant, 0.3,

Conclusion

In the present study, the application of the FSM is extended to the analysis of buckling behavior of functionally graded plates subjected to the three types of thermal loadings, namely; uniform temperature rise, linear temperature change across the thickness, and nonlinear temperature change across the thickness. Theoretical development is based on the classical plate theory and with the assumption of power law composition for the material. The figures show that the results of the present study

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