New analytic buckling solutions of side-cracked rectangular thin plates by the symplectic superposition method

Highlights

• Symplectic superposition method for buckling of side-cracked plates is developed.

• New analytic buckling solutions of side-cracked rectangular plates are obtained.

• Comprehensive benchmark numerical and graphical results are presented.

• The present method provides a rational approach to exploring analytic solutions.

Abstract

Exploring new analytic buckling solutions of cracked plates is of much importance for providing benchmark results and implementing preliminary structural designs based on explicit parametric analysis and optimization. However, the governing higher-order partial differential equations as well as the geometric discontinuity across a crack essentially complicate the problems, make it more intractable to seeking analytic solutions. This paper presents a first attempt to extend an up-to-date symplectic superposition method to linear buckling of side-cracked rectangular thin plates. The problems are introduced into the Hamiltonian system, and a side-cracked plate is then divided into several sub-plates that are analytically solved by the symplectic superposition method, where the symplectic eigenvalue problems are formulated, followed by the symplectic eigen expansion. Some elaborated mechanical quantities are imposed on the edges of each sub-plate, with multiple sets of constants determined by the actual boundary conditions of the cracked plate, free edge conditions along the crack, and interfacial continuity conditions among the sub-plates. The final analytic solution of a side-cracked plate is obtained by integration of the solutions of the sub-plates. Comprehensive benchmark results are tabulated and plotted for buckling loads and mode shapes of typical plates with a side crack from a simply supported edge, a clamped edge, or a free edge. The crack length effect is also investigated by the analytic solutions obtained. Due to the rigorous mathematical derivations without predetermination of solution forms, the present method provides a rational approach to exploring more analytic solutions.

Introduction

Plates constitute a class of fundamental load-bearing components with applications in broad fields such as mechanical, ocean, aerospace, and civil engineering. Accordingly, mechanical behaviors have been one of the major concerns in studying such structures, with unremitting efforts and contributions made over the past decades [1], [2], [3]. Among various mechanics issues, stability analysis for rectangular plates has attracted continuous attention from both scientists and engineers because buckling is a prominent failure mode of the plate structures.

Analytic modeling is one of the main approaches to buckling analysis, which not only provides benchmark solutions, but is also very useful for giving a greater insight into the plate performances and facilitating preliminary structural designs via, e.g., rapid parametric analysis and optimization. However, it is well known that the analytic buckling solutions of rectangular plates by conventional methods are restricted to some specific cases such as the plates with at least two opposite edges simply supported. This is mainly attributed to the mathematical difficulties in meeting both the fourth-order partial differential equation and different complex boundary conditions. As a result, the analytic buckling solutions of many plates are unattainable by rigorous derivations. Furthermore, in engineering applications, cracks may exist in plates either due to inherent manufacturing defects or by sustained external loadings, and the actual buckling behaviors are quite different from those of intact plates because of stiffness degradation and geometric discontinuity across the cracks, which make it more intractable to pursue analytic solutions.

For intact plates, there have been some important analytic methods put forward, which are first reviewed here. Based on the Lévy method, different investigations [4], [5], [6] have been explored to study the buckling behaviors of rectangular plates with two opposite edges simply supported. Gorman [7] employed the semi-inverse superposition method to obtain buckling loads and free vibration frequencies for a family of elastically supported rectangular thin plates with uniform loading. Moslemi et al. [8] utilized the method of separation of variables to solve the buckling problems of transversely isotropic simply supported rectangular thick plates subjected to uniformly distributed loads. Using the Navier method, Khorshidi and Fallah [9] investigated the buckling of simply supported functionally graded nano-plates based on the exponential shear deformation theory, together with the nonlocal elasticity. Mantari and Monge [10] presented analytic solutions for buckling, free vibration and bending behaviors of simply supported functionally graded sandwich plates subjected to transverse and axial mechanical loads using a new optimized hyperbolic displacement under Carrera's unified formulation. Ullah et al. [11], [12], [13] derived analytic buckling solutions of clamped rectangular thin plates under compressive loads and clamped rectangular thick plates subjected to in-plane shear loads by the finite integral transform method. Li et al. [14], [15], [16] developed the symplectic superposition method to obtain analytic buckling solutions of rectangular thin plates with different combinations of simply supported, clamped and free edges.

For buckling problems of cracked rectangular plates, however, one usually turns to numerical methods in view of the difficulties in obtaining analytic solutions. Markström and Storåkers [17] studied the buckling characteristics of cracked plates subjected to uniaxial tensile loads with the help of the finite element method (FEM). Sih and Lee [18] investigated tensile and compressive buckling of center-cracked thin plates by the FEM. Shaw and Huang [19] used the FEM based on Von Karman's linearize theory to calculate critical loads of center-cracked thin plates under tension. Riks et al. [20] presented a finite element procedure to analyze the buckling and post-buckling behaviors of center-cracked plates in tension where the loading direction is perpendicular to the crack faces. Brighenti [21], [22], [23], [24] analyzed the buckling phenomena of variously cracked rectangular thin plates under tension or compression as well as the effects of the relative crack length, crack orientation, material Poisson ratio and boundary conditions by the FEM. Baiz et al. [25] presented buckling analysis of uncracked and cracked isotropic shear deformable thin and thick plates using a quadrilateral element with smoothed curvatures and the extended FEM. Based on the Rayleigh–Ritz method, Kumar and Paik [26] used the hierarchical trigonometric functions to estimate the buckling loads of simply supported cracked thin plates under compressive and shear loads. Zeng et al. [27] used the moving least square (MLS) method with enriched basis functions to yield admissible functions for the Ritz method to deal with vibration and buckling analysis of a rectangular thin plate with a side crack. Based on the same procedure, Huang et al. [28] investigated the buckling and vibration of internally cracked thin square plates under uniaxial compressive and tensile loading. Milazzo et al. [29] presented a single-domain Ritz method for buckling and post-buckling analyses of moderately thick cracked plates. Doan Hong et al. [30] employed the phase-field method with the Reissner–Mindlin plate theory to simulate free vibration and buckling of plates with complex crack shapes. Besides, buckling analyses of functionally graded cracked plates were also explored by the FEM [31], [32], [33], extended isogeometric analysis [34,35], phase-field method [36,37], Ritz method [38], etc.

Compared to various numerical solutions, only a few analytic buckling solutions have been found for cracked plates. Stahl and Keer [39] derived the homogenous Fredholm integral equations for buckling and vibration problems of simply supported cracked rectangular thin plates. Vafai et al. [40] gave another integral equation for stability analysis of simply supported edge-cracked thin plates. Moreover, with the aid of the line-spring model that was proposed by Rice and Levy [41] and further developed by Israr et al. [42], several investigations focusing on vibration and buckling analysis of partially cracked rectangular thin plates under thermal environment have been explored [43], [44], [45], [46].

In this paper, for the first time, we further develop the symplectic superposition method to obtain new analytic linear buckling solutions of side-cracked rectangular thin plates. This method was proposed based on the superposition method and the symplectic elasticity approach [47], [48], [49], [50], [51], [52], [53], which is conducted in the symplectic space within the Hamiltonian-system framework rather than in the Euclidean space within the Lagrangian-system framework. Successful applications of the symplectic superposition method were previously achieved for bending [54], [55], [56], [57], buckling [14], [15], [16], and free vibration [58], [59], [60], [61], [62], [63] of intact plates and shells, but there have been no reports on its in-depth development for cracked plates due to the notably increased complexity.

The solution procedure of this study incorporates several key steps, i.e., obtaining the governing equation for buckling of a thin plate in the Hamiltonian system; decomposing a side-cracked rectangular thin plate into four sub-plates; solving the fundamental buckling problems of the sub-plates based on the symplectic approach, followed by superposition of the imposed constraints, where multiple sets of constants are determined by the edge conditions and interfacial continuity conditions of the sub-plates; integrating the solutions of sub-plates to obtain the final buckling solution of the side-cracked plate. Comprehensive buckling loads and mode shapes of side-cracked plates under different boundary conditions are presented with validations by available solutions in the literature, if any, and numerical results from the FEM software ABAQUS [64], revealing high accuracy and general applicability of the symplectic superposition method.