Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 485686, 11 pages

http://dx.doi.org/10.1155/2015/485686

## Semi-Analytical Finite Strip Transfer Matrix Method for Buckling Analysis of Rectangular Thin Plates

Mechanics Department, Nanjing Tech University, Nanjing, China

Received 16 June 2015; Revised 28 September 2015; Accepted 22 October 2015

Academic Editor: Alessandro Gasparetto

Copyright © 2015 Li-Ke Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Plates and shells are main components of modern engineering structures, whose buckling analysis has been focused by researchers. In this investigation, rectangular thin plates with loaded edges simply supported can be discretized by semi-analytical finite strip technology. Then the control equations of the strip elements of the buckling plate will be rewritten as the transfer equations by transfer matrix method. A new approach, namely semi-analytical Finite Strip Transfer Matrix Method, is developed for the buckling analysis of plates. This method requires no global stiffness matrix of the system, reduces the system matrix order, and improves the computational efficiency. Comparing with some theoretical results and FEM’s results of two illustrations (the plates and the ribbed plates) under six boundary conditions, the method is proved to be reliable and effective.

#### 1. Introduction

Buckling analysis is one of the important steps in the design of thin-walled structures which can be applied in different branches of engineering, including shipbuilding, civil architecture, and mechanical construction [1]. Thin plate, a main kind of thin-walled structure, is widely utilized to lighten engineering structures as well as save materials [2]. The reliability of one single plate lies in its stability chiefly, which has been studied by experimental or mathematical means [3].

In the early works, the vibration and buckling performances of rectangular plates loaded by in-plane hydrostatic forces for a wide variety of aspect ratios, boundary conditions, and loading magnitudes have been analyzed by numerical technology [4]. The solutions of the differential equations of the buckling Mindlin plate are obtained in discrete forms by applying numerical integrations [5]. And extensive numerical results have been presented for the critical buckling loads of simply supported, rectangular composite plates subjected to five types of loading conditions: () uniaxial, () hydrostatic biaxial, () compression-tension biaxial, () positive shear, and () negative shear [6]. By introducing an unified analytical solution technique for a multitude of combinations of boundary conditions, an analytical method is presented for the problem of elastic buckling of orthotropic rectangular plates [7]. If the biharmonic operator in the buckling control equations of rectangular plates is reduced by performing the Laplace’s operator and the finite difference method, the buckling load of the plate can be investigated [8]. Up to now, many methods have been used to analyze the buckling problems of rectangular plates, such as the extended Kantorovich method [9], differential quadrature procedures [10], asymptotic finite strip method [11], block GMRES method [12], first-order shear deformation meshless method [13], radial point interpolation method [14], untruncated infinite series technology [15], discrete singular convolution approach [16], and hierarchical Rayleigh-Ritz and finite element method [17].

Among these methods, finite element method is a powerful tool for engineering analysis, while the choices of the elements and the mesh sizes have significant influences on the results of buckling analysis [18]. When calculating the buckling problems of regular geometry shape structures, finite strip method can be regarded as an efficient way. And the arbitrary shaped plate may be discretized as many strip elements by the subparametric mapping concept [19]. To consider the transverse shear effect, the spline strip method has been proposed to analyze the buckling behaviors of rectangular Mindlin plates with linearly tapered thickness in one direction [20]. The buckling stresses and natural frequencies of rectangular laminated plates with arbitrary lay-ups and general boundary conditions can be predicted by the improved spline finite strip, which combines the super-strip concept [21]. Then a higher-order shear deformable plate finite strip element is developed and employed to investigate the critical buckling loads of composite laminated plates [22]. The spline finite strip method and multilevel substructuring procedures are combined for the buckling stress analysis of composite laminated, prismatic shell structures with general boundary conditions [23]. Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners is studied by finite strip method [24]. Notably, a comprehensive review of the finite strip method for structural analysis is given [25]. A layerwise B-spline finite strip method is developed for free vibration analysis of truly thick and thin composite laminated plates [26]. And a finite strip Fourier p-element is developed to analyze the natural vibration characteristics of the thin plate [27]. Based on the concept of the semienergy approach, the finite strip method can be proposed to analyze the buckling [28], postbuckling [29], dynamic buckling behaviors of the laminated plates [30], and stability phenomena of cylindrical shell structures [31]. The harmonic series satisfying the boundary conditions in the loaded ends of thin-walled members are generally employed in Semi-Analytical Finite Strip Transfer Matrix Method (SAFSM) [32]. And the buckling stress of the cold-formed sections can be analyzed by SAFSM [33], which has been widely used in computer softwares (such as THIN-WALL [34], CUFSM [35]) to develop the signature curves [36] of the buckling stress versus buckling half wavelength for thin-walled members. Furthermore, the constrained finite strip method innovated from SAFSM is developed and applied in cold-formed steel design [37]. Generalized constrained finite strip method for thin-walled members with arbitrary cross-sections can be used to analyze secondary modes [38] and primary modes [39]. The impacts of basis, orthogonalization, and normalization in constrained finite strip method are discussed for stability solution of open thin-walled members [40].

Transfer matrix method has been developed as an effective tool for vibration analysis of engineering structures, especially for chain connected system from topological perspective [41]. Then a combined finite element transfer matrix method is developed to study the statics [42] and dynamics [43] of structures. Later a new transfer matrix method based on the boundary element and transfer matrix technology is proposed for the vibration analysis of two-dimensional plate acted by uniform [44] and concentrated [45] loads. By introducing the numerical integration, nonlinear dynamics of structures [46], multi-rigid-body system [47], and multi-flexible-body system [48] can be dealt with by transfer matrix method. And a procedure which combines multiport transfer matrices and finite elements has been developed to resolve the acoustic phenomena of automotive hollow body networks [49]. It should be noted that three references enlighten us to start this investigation. The buckling analysis of the plate with built-in rectangular delamination has been implemented by strip distributed transfer function method [50]. And the transfer matrix method can be used to analyze the instability in unsymmetrical rotor-bearing systems [51] and tall unbraced frames [52].

Here by combining two powerful methods of the semi-analytical finite strip and the transfer matrix, an efficient technology named semi-analytical Finite Strip Transfer Matrix Method (FSTMM) for the buckling analysis of plates is developed. The text is organized as follows: In Section 2, the general theorem of the semi-analytical finite strip for buckling analysis of plates is shown. Section 3 presents semi-analytical Finite Strip Transfer Matrix Method for buckling analysis. In Section 4, some results calculated by FSTMM and other methods are given, which can validate the proposed method. The conclusions are included in Section 5.

#### 2. The Semi-Analytical Finite Strip Analysis

##### 2.1. Degree of Freedom and Shape Function

In classical Kirchhoff plate theory, the membrane or in-plane translations are neglected and only the -axis’s translation and the -axis’s rotation are considered as local displacements [1]. The global coordinate system () and local coordinate system () are actually the same in the plate model. In finite strip method (FSM), a thin plate, as shown in Figure 1, can be divided into many strip elements.