Nonlinear stability of unsymmetrically laminated angle-ply shear-deformable plates in biaxial compression

The generalized Karman–Reissner equations governing large deflection of unsymmetrically laminated angle-ply shear-deformable rectangular plates are presented in this paper, based upon Karman and Reissner plate theories. Much effort has been concentrated on derivation of the Karman–Reissner equations for the laminates taking account of transverse shear effects so that only two unknown functions for nonlinear analysis need to be treated to gain a precise insight into the complexities; otherwise high-order refined plate theories must be applied. An asymptotic series solution is constructed according to the Karman-type refined theory for postbuckling behavior of the plates with the boundary condition of four edges simply supported. Typical numerical examples are presented for comparison with other nonlinear analytical and experimental results. The effects of shear deformation, lamination angle and geometrical imperfection on buckling and postbuckling behaviors of the laminates subjected to a combination of biaxial compressive loadings are examined for industrial application.     

A differential quadrature nonlinear free vibration analysis of laminated composite skew thin plates

Using a differential quadrature (DQ) method, large amplitude free vibration analysis of laminated composite skew thin plates is presented. The governing equations are based on the thin plate theory (TPT) and the geometrical nonlinearity is modeled using Green's strain in conjunction with von Karman assumptions. To cause the impact due to nonlinear terms more significant, in-plane immovable simply supported, clamped and different combinations of them are considered. The effects of different parameters on the convergence and accuracy of the method are studied. The resulted solutions are compared to those from other numerical methods to show the accuracy of the method. Some new results for laminated composite skew plates with different mixed boundary conditions are presented and are compared with those obtained using the first order shear deformation theory based DQ (FSDT-DQ) method. Excellent agreements exist between the solutions of the two approaches but with much lower computational efforts of the present DQ methodology with respect to FSDT-DQ method.     

Postbuckling analysis of cross-ply laminated cylindrical shell panels under parabolic mechanical edge loading

Postbuckling equilibrium paths of simply supported cross-ply laminated cylindrical shell panels subjected to non-uniform (parabolic) inplane loads are traced in this paper. Love's shell theory with higher order shear deformation theory and von Kármán nonlinear strain–displacement relations are used in the mathematical formulation of the problem. In the first step, the plate membrane problem is solved to evaluate the stress distribution within the prebuckling range as the applied inplane edge load is non-uniform. The governing shell panel postbuckling equations are derived from the principle of minimum total potential energy using the above stress distributions. Adopting multi-term Galerkin's approximation, the governing equations are reduced into a set of non-linear algebraic equations. Newton–Raphson method in conjunction with Riks approach is employed to plot the postbuckling paths through limit points. Numerical results are presented for symmetric (0/90/0) crossply laminated cylindrical shell panels under parabolic inplane load, lateral distributed load and initial imperfections. Limit loads and snap-through behavior of shell panels are studied.     

Buckling and postbuckling of unsymmetrically laminated angle-ply plates in uniaxial and biaxial compression

A deflection-type perturbation technique is proposed to solve the nonlinear Karman equations of rectangular unsymmetrically laminated angle-ply composite plates with simply supported edges in uniaxial and biaxial compression. Higher-order asymptotic expansions for postbuckling behavior of the plates are constructed by this approach. Effects of initial imperfection, lamination angle and number of layers on buckling and postbuckling of the plates are examined. In some specific cases numerical results obtained from the present solution are in fairly good agreement with other theoretical predictions.     

Nonlinear vibration of laminated plates on an elastic foundation

This paper studies the effects of initial stresses on the nonlinear vibrations of laminated plates on elastic foundations. The nonlinear partial differential equations based on Mindlin plate theory are derived for the nonlinear vibration of laminated plates in a general state of nonuniform initial stress. Both rotary inertia and transverse stress are considered. Using the derived governing equations, the nonlinear vibration behavior of an initially stressed cross-ply plate on a Pasternak or Winkler elastic foundation is studied. The Galerkin's approximate method is applied to the governing partial differential equations to yield ordinary differential equations. The ordinary differential equations are solved by Runge–Kutta method to obtain the ratio of nonlinear frequency to linear frequency. The initial stress is taken to be a combination of a pure bending stress and an extensional stress in the plane of the plate. It is found that the foundation stiffness and initial stresses may result in a drastic change of the nonlinear vibration behavior. The frequency responses of nonlinear vibration are sensitive to the initial stress, vibration amplitude, modulus ratio and foundation stiffness. The effects of various parameters on the nonlinear vibration are discussed.     

扭转作用下有缺陷柱形壳的屈曲和后屈曲性能

分析扭转作用下有缺陷柱形壳的屈曲和后屈曲性能。基于Karman-Donnell-Type非线性微分方程建立计算公式,采用壳屈曲的边界层理论进行分析,以获得能严格满足边界条件的解决方案。采用奇摄动技术,以确定屈曲载荷和后屈曲平衡路径。数值结果显示,目前的理论能对柱形壳的后屈曲性能进行较好评估。同时分析了几何参数对柱形壳的屈曲和后屈曲性能的影响。证实了扭转作用下柱形壳的后屈曲平衡路径并不稳定,并且相对更短的壳体具有更高的后屈曲平衡能力。最后,指出初始缺陷对柱形壳屈曲和后屈曲性能的影响。对具有初始横向挠曲的有缺陷壳体的分析结果显示：即便是非常小的缺陷,也确实会减少屈曲承载力,并使得后屈曲稳定性变差。扭转作用下柱形壳的屈曲和后屈曲性能显示出明显的缺陷敏感性。此外,如果缺陷更大,那么带来的影响也随之会变得更大。     

Buckling and postbuckling behavior of unstiffened slender curved plates under uniform shear

Buckling and postbuckling behavior of curved plates under in-plane shear are investigated. After revisiting classic elastic buckling results, the elastoplastic postbuckling behavior and the effects of curvature parameter and aspect ratio are simulated via geometrical and material nonlinear analyses. Imperfection sensitivity is studied for various imperfection shapes and magnitudes. An increase in curvature parameter raises the elastic buckling load, produces unstable buckling and reduces postbuckling reserves. The buckling load and shear capacity are higher in shorter plates. Small initial imperfections are found to have severe effects on the initial buckling load of plates with large curvature parameter, but little effect on ultimate postbuckling capacity.     

Nonlinear analysis of imperfect laminated thin plates under transverse and in-plane loads and resting on an elastic foundation by a semi-analytical approach

The large deflection and postbuckling behavior of imperfect composite laminated plates exposed to a combined action of transverse loads and in-plane edge compressions and resting on an elastic foundation are investigated in this paper by a semi-analytical approach. The formulations are based on the classical laminated plate theory (CLPT), and include the plate–foundation interaction effects via a two-parameter model (Pasternak-type) from which Winkler elastic foundation can be recovered as a limiting case. The present approach employs a perturbation technique, one-dimensional differential quadrature approximation and the Galerkin procedure to model the nonlinear performance of the plate with arbitrary combination of simply supported, clamped or elastic rotational edge constraints. Studies concerning its accuracy and convergence characteristics are carried out through some numerical illustrations. Effects of foundation stiffness, plate aspect ratio, total number of plies, fiber orientation, initial geometrical imperfection, the character of boundary conditions, and load patterns on the nonlinear behavior of the plate are studied. Typical numerical results are given in dimensionless graphical forms.     

均匀剪力作用下无加劲肋的细长曲面板的屈曲和后屈曲性能

研究平面内剪力作用下曲面板的屈曲和后屈曲性能。回顾经典的弹性屈曲分析,通过对几何和材料非线性的分析,模拟了弹塑性后屈曲性能、及曲率参数和长宽比的影响。通过各种不同大小和级别的缺陷的影响分析,研究缺陷敏感度。随着曲率的增大,弹性临界荷载增大,并且产生不稳定屈曲现象,以及使后屈曲承载力储备减少。微小的初始缺陷都会引起较大曲率的曲面板的初始屈曲,但是几乎不会影响板的极限承载力。     

Buckling and postbuckling behaviors of imperfect cylindrical shells subjected to torsion

Buckling and postbuckling behaviors of imperfect cylindrical shell subjected to torsion are investigated. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. A boundary layer theory of shell buckling is applied to obtain the analytic solutions that meet the boundary conditions strictly. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the current theory gives quite good estimates of the postbuckling paths of cylindrical shells. The effects of the geometric parameters on the buckling and postbuckling behaviors of the cylindrical shells are analyzed. It is confirmed that the postbuckling equilibrium paths of cylindrical shells subjected to torsion are unstable and the relatively shorter shells have higher postbuckling equilibrium paths. Finally, the effects of the initial imperfections on the buckling and postbuckling behaviors of the cylindrical shells are clarified. The illustrated results of the imperfect shells with different initial transverse deflections show that extremely small imperfections do indeed reduce the buckling loads and make the postbuckling equilibrium paths be lower. The buckling and postbuckling of cylindrical shells under torsion exhibit obvious imperfect sensitivity. Furthermore, the effects become greater following with the larger imperfections.     

Thermal buckling load optimization of laminated composite plates

In this study, the applicability of the Modified Feasible Direction (MFD) method on the thermal buckling optimization of laminated plates subjected to uniformly distributed temperature load is investigated. The objective function is to maximize the critical temperature capacity of laminated plates and the fiber orientation is considered as design variable. The first-order shear deformation theory is used in the mathematical formulation. For this purpose, a program based on FORTRAN is used for the optimization of laminated plates. Finally, the effect of aspect ratio, antisymmetric lay-up, boundary condition, material anisotropy, ratio of coefficients of thermal expansion, and hybrid laminates on the results is investigated and the results are compared.     

Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers

The free vibration behavior of quadrilateral laminated thin-to-moderately thick plates with carbon nanotube reinforced composite (CNTRC) layers is studied. The governing equations are based on the first-order shear deformation theory (FSDT). The solution procedure is based on transforming the governing differential equations from an arbitrary straight-sided physical domain to a regular computational one, and discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. Four different profiles of single walled carbon nanotubes (SWCNTs) distribution through the thickness of layers are considered, which are uniformly distributed (UD) and three others are functionally graded (FG) distributions. The fast rate of convergence of the presented approach is numerically demonstrated and to show its high accuracy, wherever possible comparison studies with the available results in the open literature are performed. Then, the effects of volume fraction of carbon nanotubes (CNTs), geometrical shape parameters, thickness-to-length and aspect ratios, different kinds of CNTs distribution along the layers thickness and different boundary conditions on the natural frequencies of laminated plates are studied.     

Dynamic stiffness analysis of laminated composite plates

An analysis is presented for the vibration and stability problem of composite laminated plates by using the dynamic stiffness matrix method. A dynamic stiffness matrix is formed by frequency dependent shape functions which are exact solutions of the governing differential equations. It eliminates spatial discretization error and is capable of predicting several natural modes by means of a small number of degrees of freedom. The natural frequencies and buckling loads of composite laminated plates are calculated numerically. The effects of the boundary conditions, the number of layers, the orthotropicity ratio, the side to thickness ratio, and the aspect ratio are studied. It is also illustrated that connected composite plate structures can be handled without difficulty by the present method.     

Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection

Nonlinear thermomechanical post-buckling of an imperfect functionally graded material (FGM) circular plate, subjected to both mechanical load and transversely non-uniform temperature rise, is presented. The material properties of FGM plates are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on von Kármán's plate theory, equilibrium equations governing a large axi-symmetric deformation of the FGM circular plate under thermomechanical loads are derived. In the analysis, the geometric imperfections of the plate are taken into account. By using a shooting method the nonlinear ordinary differential equations with immovably clamped boundary conditions are solved numerically. Responses for the nonlinear thermomechanical post-buckling responses of the FGM plate are obtained. Numerical examples are presented that relate to the performances of perfect and imperfect, homogenous and graded plates. Characteristic curves of the post-buckling deformation of the imperfect FGM circular plate varying with thermal loads, imperfection parameters and volume fraction index are plotted. And then effects of the load parameters, materials constitution, and the geometric imperfection of the plate on the deformation are discussed in detail.     

对带几何缺陷的FGM圆板的非线性热力学后屈曲现象的分析

带有几何缺陷FGM圆板的非线性热力学后屈曲现象是由力学荷载及横向不均匀温度升高导致的。基于组成材料的体积分数导致的能量分布法则,FGM板的材料特性假定在板厚度方向呈梯度变化。根据VonK偄rm偄n平板理论,推导出FGM板在温度荷载作用下的轴对称大变形的计算公式。在此分析中同时考虑了板的几何缺陷。通过运用目标法,解决了那些带有固定边界条件的非线性方程。数值算例中考虑了是否带有几何缺陷、均质或非均质的钢板。带缺陷FGM圆板的后屈曲变形特征曲线根据温度荷载、缺陷参数和体积分数的变化绘制而成。文中同时讨论了荷载参数、材料构成及板的几何缺陷对变形的影响。     

Inelastic buckling behavior of stocky plates under interactive shear and in-plane bending

Inelastic buckling and postbuckling behavior of stocky plates under combined shear and in-plane bending stresses are investigated and compared to slender plates. Aluminum and steel plates having various slenderness ratios are modeled and analyzed by means of (i) numerical nonlinear finite element method and (ii) theoretical p-Ritz energy method; and both results are compared to the classic interaction equation. It is observed that whereas in slender plates, elastic buckling occurs prior to the material's proportional limit load, stocky plates buckle in an inelastic way within the post-yield stage. In contrast to slender plates with considerable postbuckling reserves, the buckling of stocky plates is immediately followed by softening. In addition, it is shown that the classic interaction equation overestimates buckling loads; and therefore, a modified equation that can safely be applied to both stocky and slender plates is proposed.     

Nonlinear dynamic stability of composite plates with piezoelectric layers subjected to periodic in-plane load

Nonlinear dynamic stability characteristics of composite plates subjected to periodic in-plane load are investigated via the finite element method with dynamic response analysis. Piezoelectric actuator layers are embedded at the top and bottom of the laminated composite plate. The theoretical formulation is based on the modified first order shear deformation theory (MFSDT) incorporating the von Kármán type nonlinear strains. The structural system is considered to be undamped. The nonlinear governing equations obtained are solved using the Newmark's direct integration method coupled with the direct iteration method. The boundaries of dynamic instability regions are obtained using Bolotin's approach. Effects of in-plane forcing frequency and applied voltage on the characteristic features of dynamic stability behaviour are investigated using both linear and nonlinear dynamic response analyses.     

Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM

Large amplitude free vibration analyses of tapered Mindlin rectangular plates with elastically restrained against rotation edges are investigated using different differential quadrature method (DQM). The governing equations are based on the first-order shear deformation plate theory in conjunction with Green's strain and von Karman assumption. The spatial derivatives are discretized using DQM and the harmonic balance method is used to transform the resulting differential equations into frequency domain. A direct iterative method is used to solve the nonlinear eigenvalue system of equations. The convergence of the method is shown and their accuracy is demonstrated by comparing the results with those of the limiting cases, i.e. nonlinear free vibration analysis of plates with classical boundary conditions and also linear free vibration analysis of tapered plates. The effects of the elastic restraint coefficient at the edges and the geometrical parameters on the ratio of the nonlinear natural frequency to linear natural frequency of plates with linearly and bi-linearly varying thickness are studied.     

Buckling of laminated skew plates

A general high precision triangular plate bending finite element has been extended to the buckling analysis of laminated skew plates. This procedure involves development of the transformation matrix between global and local degrees of freedom for nodes lying on the skew edges and suitable transformation of the element matrices. The accuracy of the present formulation has been verified against literature values. New results are obtained for antisymmetric angle-ply and cross-ply laminated skew plates. In this analysis, the critical buckling loads for different skew angles with various lamination parameters, such as number of layers, fibre orientation angle, different boundary (simply supported, clamped) and loading (uniaxial, biaxial) conditions, have been presented.     

First-order generalised beam theory for arbitrary orthotropic materials

This paper presents the formulation of a Generalised Beam Theory (GBT) developed to analyse the structural behaviour of composite thin-walled members made of laminated plates and displaying arbitrary orthotropy. The main concepts and procedures involved in the available isotropic first-order GBT are revisited and adapted/modified to account for the specific aspects related to the member orthotropy. In particular, the orthotropic GBT fundamental equilibrium equations and corresponding boundary conditions are derived and their terms are physically interpreted, i.e., associated with the member mechanical properties. Moreover, different laminated plate material behaviours are dealt with and their influence on the GBT equations is investigated. Finally, in order to clarify the concepts involved in the formulated GBT and illustrate its application and capabilities, a thin-walled orthotropic beam is analysed and the results obtained are thoroughly discussed.