Elastic stability of all edges simply supported,stepped and stiffened rectangular plate under uniaxial loading

In this paper using finite difference method the lower bound buckling load for simply supported stepped and stiffened rectangular plate with uniformly distributed compressive forces in one direction is discussed. The plate is divided into 400 rectangular meshes. The partial derivatives are approximated using central difference formula. Three hundred and sixty-one equations are formed and using MATLAB the least eigenvalue is obtained. The buckling coefficients are calculated for different types of stepped plates and the results are presented in tables and graphs for ready use by designers. Two separate tables are presented in comparison with the exact buckling factor for stepped rectangular plate by Xiang, Wei and Wang. The results give a close agreement.     

Local and post local buckling of stepped and perforated thin plates

The nonlinear mathematical theory for initial and post local buckling analysis of plates of abruptly varying stiffness based on the principle of virtual work is established. The method is programmed, and several numerical examples are presented to demonstrate the scope and efficacy of the procedure. Local buckling coefficients of perforated and stepped plates are obtained and the results are compared with known solutions. Post-buckling behaviour of perforated and stepped plates is studied for different geometries. The non-dimensional applied loads (P/P_cr), dimensionless lateral displacements and stress distribution of plates with varying stiffness in the post buckling region are given in several tables and graphs.     

The extended Melnikov method for non-autonomous nonlinear dynamical systems and application to multi-pulse chaotic dynamics of a buckled thin plate

The extended Melnikov method, which was used to solve autonomous perturbed Hamiltonian systems, is improved to deal with high-dimensional non-autonomous nonlinear dynamical systems. The multi-pulse Shilnikov type chaotic dynamics of a parametrically and externally excited, simply supported rectangular thin plate is studied by using the extended Melnikov method. A two-degree-of-freedom non-autonomous nonlinear system of the rectangular thin plate is derived by the von Karman type equation and the Galerkin approach. The case of buckling is considered for the rectangular thin plate. The extended Melnikov method is directly applied to the non-autonomous governing equations of motion to investigate multi-pulse Shilnikov type chaotic motions of the buckled rectangular thin plate for the first time. The results obtained here indicate that multi-pulse chaotic motions can occur in the parametrically and externally excited, simply supported buckled rectangular thin plate.     

Delamination buckling of a rectangular orthotropic composite plate containing a band crack

The delamination buckling problem for a rectangular plate made of an orthotropic composite material is studied. The plate contains a band crack whose faces have an initial infinitesimal imperfection. The subsequent development of this imperfection due to an external compressive load acting along the crack is studied through the use of the three-dimensional geometrically nonlinear field equations of elasticity theory for anisotropic bodies. A criterion of initial imperfection is used in determining the critical forces. The corresponding boundary-value problems are solved by employing the boundary-form perturbation technique and the FEM. Numerical results for the critical force are presented.     

有限位移理论的功的互等定理及其应用

提出了有限位移理论三维线弹性力学的功的互等定理.基于这一定理,导出了大挠度弯曲矩形板的功的互等定理.同时,应用简化矩形板的定理,直接得到了大挠度板条的功的互等定理.作为应用,计算了在均载作用下两端固定大挠度板条的弯曲和在均载作用下4边固定大挠度矩形板的弯曲.计算表明,根据弯曲薄板大挠度功的互等定理,大挠度弯曲矩形板可应用小挠度的相应基本解得以简单解决.     

矩形板的侧屈

本文研究矩形板的侧向屈曲问题.文中分别讨论了有集中力,均布荷载及集中力偶作用之下矩形板发生侧向屈曲时的最小临界荷载.文中使用了能量法.     

矩形板不对称侧向屈曲中若干问题

本文用能量法研究了矩形板不对称侧向屈曲的几个问题,文中讨论了具有不对称支承的矩形板分别在有集中力,均布荷载及集中力偶作用之下发生不对称侧向屈曲时的最小的临界荷载.     

Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

中厚板的弹性屈曲和后屈曲

本文采用Reissner假定考虑横向剪切变形的影响,导出弹性矩形板大挠度方程.本文讨论考虑横向剪切变形的矩形板的弹性屈曲和后屈曲.采用文[8]提供的摄动方法,给出了完善和非完善中厚板的后屈曲平衡路径,并与经典薄板理论结果进行了比较.     

Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation

In this paper the von Kármán model for thin, elastic, infinite plate strip resting on a linear elastic foundation of Winkler type is studied. The infinite plate strip is simply-supported and subjected to evenly distributed compressive loads. The critical values of bifurcation parameters and buckling modes for given frequency of longitudinal waves are found on the basis of investigation of linearized problem. The mathematical nonlinear model is reduced to operator equation with Fredholm type operator of index 0 depending on parameters defined in corresponding Hölder spaces. The Lyapunov-Schmidt reduction and the Crandall-Rabinowitz bifurcation theorem (gradient case) are used to examine the postcritical behaviour of the plate. It is proved that there exists maximal frequency of longitudinal waves depending on the compressive load and the stiffness modulus of foundation.     

悬臂板侧屈中的几个问题

本文用能量法研究悬臂矩形板侧向屈曲中的几个问题.文中分别讨论了有集中力,均布荷载,三角形分布荷载及集中力偶作用之下悬臂矩形板发生侧向屈曲时的最小临界荷载.     

中厚板热后屈曲分析

依据Reissner-Mindlin板理论考虑转动惯量和横向剪切变形影响,本文给出中厚板在(1)均布和非均布(线性)热荷载作用下;(2)单向压缩和均布热荷载共同作用下的后屈曲分析。采用摄动法导出完善和非完善中厚板的热屈曲载荷和热后屈曲平衡路径,并与经典薄板理论结果进行了比较。     

Nonlinear stability and vibration of pre/post-buckled multilayer FG-GPLRPC rectangular plates

The present study examines the nonlinear stability and free vibration features of multilayer functionally graded graphene platelet-reinforced polymer composite (FG-GPLRPC) rectangular plates under compressive in-plane mechanical loads in pre/post buckling regimes. The GPL weight fractions layer-wisely vary across the lateral direction. Furthermore, GPLs are uniformly dispersed in the polymer matrix of each layer. The effective Young's modulus of GPL-reinforced nanocomposite is assessed via the modified Halpin–Tsai technique, while the effective mass density and Poisson's ratio are attained by the rule of mixture. Taking the von Kármán-type nonlinearity into account for the large deflection of the FG-GPLRPC plate, as well as utilizing the variational differential quadrature (VDQ) method and Lagrange equation, the system of discretized coupled nonlinear equations of motions is directly achieved based upon a parabolic shear deformation plate theory; taking into account the impacts of geometric nonlinearity, in-plane loading, rotary inertia and transverse shear deformation. Afterwards, first, by neglecting the inertia terms, the pseudo-arc length approach is used in order to plot the equilibrium postbuckling path of FG-GPLRPC plates. Then, supposing a time-dependent disturbance about the postbuckling equilibrium status, the frequency responses of pre/post-buckled FG-GPLRC plate are obtained in terms of the compressive in-plane load. The influences of various vital design parameters are discussed through various parametric studies.     

3D Analyses of the symmetric local stability loss of the circular hollow cylinder made from viscoelastic composite material

The 3D approach was employed for investigations of the symmetric local stability loss of the circular hollow cylinder made from the viscoelastic composite materials. This approach is based on investigations of the development of the initial rotationally symmetric infinitesimal local imperfections of the circular hollow cylinder within the scope of 3D geometrically nonlinear field equations of the theory of viscoelasticity for anisotropic bodies. The numerical results of the critical force and critical time are presented and discussed. For comparison and estimation of the accuracy of the results given by the 3D approach, the same problem is also solved by using various approximate shell theories. The viscoelasticity properties of the plate material are described by the fractional–exponential operator. The numerical results and their discussion are presented for the case where the cylinder is made of a uni-directional fibrous viscoelastic composite material. In particular, it is established that the difference between the critical times obtained by employing 3D and third order refined shell theories becomes more non-negligible if the values of the external compressive force are close to the critical compressive force which is obtained at t = ∞ (t denotes a time).     

An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions

An analytical spectral stiffness method is proposed for the efficient and accurate buckling analysis of rectangular plates on Winkler foundation subject to general boundary conditions (BCs). The method combines the advantages of superposition method, stiffness-based method and the Wittrick–Williams algorithm. First, exact general solutions of the governing differential equation (GDE) of plate buckling considering both elastic foundation and biaxial loading is derived by using a modified Fourier series. The superposition of such general solutions satisfy the GDE exactly and BCs approximately, which guarantees the rapid convergence and high accuracy. Then, based on the exact general solution, the spectral stiffness matrix which relates the coefficients of plate generalized displacement BCs and force BCs is symbolically developed. As a result, arbitrary BCs can be prescribed straightforwardly in the stiffness-based model. As an efficient and reliable solution technique, the Wittrick–Williams algorithm with the J₀ problem resolved is applied to obtain the critical buckling solutions. The accuracy and efficiency of the method are verified by comparing with other methods. Benchmark buckling solutions are provided for plates with all possible boundary conditions. Also, dependence of various factors such as foundation stiffness, load combinations and aspect ratio on the buckling behaviors are investigated.     

FEM Analysis of the Three-Dimensional Buckling Problem for a Clamped Thick Rectangular Plate Made of a Viscoelastic Composite

The buckling instability of a thick rectangular plate made of a viscoelastic composite material is studied. The investigation is carried out within the framework of the three-dimensional linearized theory of stability. The plate edges are clamped and the plate is compressed through the clamps. Moreover, it is assumed that the plate has an initial infinitesimal imperfection, and, as a buckling criterion, the state is taken where this imperfection starts to increase indefinitely at fixed finite values of external compressive forces. From this criterion, the critical time is determined. The corresponding boundary-value problems are solved by employing the three-dimensional FEM and the Laplace transform. The material of the plate is assumed orthotropic, viscoelastic, and homogeneous. Numerical results related to the critical time are presented.     

An evolutionary method for optimization of plate buckling resistance

Optimization of plate buckling resistance is very complicated, because the in-plane stress resultants in the prebuckled state of a plate are functions of thickness distribution. This paper discusses the problem of finding the optimum thickness distribution of isotropic plate structures, with a given volume and layout, that would maximise the buckling load. A simple numerical method using the finite-element analysis is presented to obtain the optimum thickness distribution. Optimum designs of compression-loaded rectangular plates with different boundary conditions and plate aspect ratios are obtained by using the proposed method. Optimum designs from earlier studies and the methods of buckling analysis used to attain these results are discussed and compared with the designs from the proposed method. This paper also examines the reliability of the optimality criterion generally used for plate buckling optimization, which is based on the uniform strain energy density.     

四边固定变厚度正交各向异性矩形板的非线性非对称弯曲问题的一致有效渐近解

利用“修正的两变量法”和“混合摄动法”,引进4个小参数,对厚度线性变化的正交各向异性矩形板的非线性非对称弯曲问题进行了研究,得到了挠度函数W(x,y)和应力函数Φ(x,y)对ε1为N阶及对ε2为M阶的一致有效渐近解．     

Thermal buckling analysis of point-supported laminated composite plates in unilateral contact

This paper deals with the thermal buckling analysis of point-supported thin laminated composite plates. The analysis is performed for rhombic and rectangular plates and two cases of bilateral and unilateral buckling. In the unilateral buckling, it is assumed that the plate is in contact with a rigid surface and lateral deflection is forced to be only in one direction. The element-free Galerkin (EFG) method is employed to discretize equilibrium equations. Point supports are modeled in the form of distinct restrained circular surfaces through developing a numerical procedure based on the Lagrange multiplier technique. The unilateral behavior of the plate is incorporated in the analysis by using the penalty method and the Heaviside contact function. The final system of nonlinear algebraic equations is solved iteratively. Two types of point support arrangements are considered and the effect of different parameters such as number of point supports, plate aspect ratio and lamination scheme on the buckling coefficient of composite plates is investigated.     

Analytical determination of residual stresses in a butt-weld of two thin rectangular plates

Based upon the direct integration method, the technique for analysis of residual stresses in a thin rectangular plate consisting of two butt-welded plates is presented. The material of plate is assumed to be elastic-plastic. The welded joint is modeled by a locally distributed field of longitudinal and transversal residual strains. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)