Nonlinear analysis of bimodular composite plates under compression

The present work deals with the buckling and post-buckling behaviour of bimodular laminated composite plates. The fiber-governed bimodular constitutive model is adopted for the analysis and the geometrical nonlinearities are accounted for by using the consistent small strain and moderate rotation theory for a shear deformable Mindlin plate. Governing equations of the problem are developed and the finite element formulation is given using lagrangian C°-elements. Some numerical examples are developed to analyze the buckling and post-buckling behaviour of laminated bimodular plates.     

高阶剪切变形板理论下FG-GRC板的屈曲和弯曲分析

基于三阶剪切变形板理论(TSDPT)和正弦剪切变形板理论(SSDPT),研究了功能梯度石墨烯增强复合材料(FG-GRC)板的屈曲和弯曲行为,并通过与一阶剪切变形板理论(FSDPT)计算结果的比较,分析了TSDPT、SSDPT与FSDPT在FG-GRC板屈曲和弯曲力学行为研究过程中的差异。材料的有效杨氏模量通过修正的Halpin-Tsai微观力学模型估算,有效泊松比通过混合律确定。利用最小势能原理推导出了包含五个未知量的控制方程,并获得了简支FG-GRC矩形板弯曲挠度和临界屈曲载荷Navier形式的解析解。数值结果表明：与TSDPT和SSDPT相比,FSDPT明显高估了FG-X型FG-GRC板的临界屈曲载荷而明显低估了其弯曲挠度,且略微低估了FG-O型FG-GRC板的临界屈曲载荷而略微高估了其弯曲挠度,而UD型和FG-A型FG-GRC板在三种理论下的计算结果几乎完全一致;TSDPT和SSDPT在计算FG-GRC板的弯曲挠度和临界屈曲载荷时结果十分相近;当板的总层数NL小于10层～15层时,弯曲载荷比率和临界屈曲载荷比率的变化非常显著,当总层数NL超过10层～15层时,弯曲载荷比率和临界屈...     

Linear buckling analysis of orthotropic inhomogeneous rectangular plates under uniform in-plane compression

Summary A linear buckling analysis is carried out for orthotropic inhomogeneous rectangular plates under uniform in-plane compression. It is assumed that material inhomogeneities of Young's modulus and shear modulus of elasticity are continuously changed in the thickness direction with the power law of the coordinate variable, while Poisson's ratio is assumed to be constant. The buckling equation can be successfully constructed as the linearized von Kármán plate model by introducing the newly defined position of the reference plane which enables us to easily deal with the problem. The critical buckling loads of the simply supported rectangular plate are presented using the derived fundamental relations. Effects of material inhomogeneity, material orthotropy, aspect ratio, width-to-thickness ratio and load ratio are discussed.     

The influence of mechanical couplings on the compressive stability of anti-symmetric angle-ply laminates

The compressive stability of anti-symmetric angle-ply laminated plates with particular reference to the degrading influence of membrane–flexural coupling is reported in this paper. The degree of membrane–flexural coupling in the laminated composite plates is varied, essentially, by altering the ply-angle and the number of plies in the laminated stack for a given composite material system. The coupled compressive buckling solutions are determined in the paper using the finite strip method of analysis and the buckling displacement fields of the strip formulation are those which are able to provide zero in-plane normal movement at the edge boundaries of the laminated plates.

Results are given for anti-symmetric angle-ply laminated plates subjected to uniaxial compression and these have been obtained from fully converged finite strip structural models. Validation of the finite strip formulation is indicated in the paper through comparisons with exact solutions where appropriate. Increasing the number of plies in the laminated system is seen to reduce the degree of coupling and the critical stress levels are noted to tend towards the plate orthotropic solutions. The ply-angle corresponding to the optimised buckling stress for any particular laminate is noted in the paper to be influenced by the support boundary conditions at the plates unloaded edges. For any particular laminate the minimum critical buckling stress and corresponding natural half-wavelength of the buckling mode are shown to be highly sensitive to ply-angle variation.

Some post-buckling results are presented in the paper and these have been determined using the finite element method of analysis. The influence of membrane–flexural coupling is shown to be significant throughout the compressive post-buckling history of the laminated plates. The optimised ply-angle with regard to the critical compressive buckling stress of square simply supported anti-symmetric angle-ply laminates is shown to be less effective in the post-buckling range with regard to post-buckled compressional stiffness.     

Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory

A two-dimensional finite element model is presented to perform the linear static analysis of laminated orthotropic composite plates based on a refined higher order shear deformation theory. The theory accounts for parabolic distributions of transverse shear stresses and requires no shear correction factors. A finite element program is developed using serendipity element with seven degrees of freedom per node. The present solutions are compared with those obtained using three-dimensional elasticity theory and those obtained by other researchers. The theory accurately predicts displacements and transverse shear stresses compared to previously developed theories for thick plates and are very close to three-dimensional elasticity solutions. The effects of transverse shear deformation, material anisotropy, aspect ratio, fiber orientation and lamination sequence on transverse shear stresses are investigated. The error in values of transverse shear stresses decreases as the number of lamina increases, for a plate of same thickness. An increase in degree of anisotropy results in lower values of deflection in the plate. For cross-ply plate an increase in anisotropy results in an increase in effective stress whereas for angle-ply plate the effect is almost negligible. Through thickness variation of transverse shear stresses are independent of anisotropy. The maximum effective stress increases exponentially at lower values of anisotropy and reaches to an asymptotic value at higher values. The stacking sequence has a significant effect on the transverse deflections and shear stress. Rectangular plates experience less effective, in-plane and transverse shear stresses compared to square plates.     

Large deflections of laminated composite plates

A semi-energy approach is formulated for a stability study on laminated composite plates with simply supported edges subjected to uniaxial compression. The analytical approach incorporates the solution of von Karman's fundamental equation for large deflections of plates and consideration of the energy in the buckled-plate system. The effect of coupling responses on the initial buckling load and the immediate post-buckling region is discussed. The results for the initial buckling load are compared with results obtained by the well-known exact solution of Jones. The results for post-buckling behaviour are compared with the experimental data of Rahman.     

Levy-type solution for buckling analysis of orthotropic plates based on two variable refined plate theory

This paper presents closed-form solution for buckling analysis of orthotropic plates using two variable refined plate theory. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Governing equations are derived from the principle of minimum total potential energy. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, loading condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the critical buckling load of orthotropic plates are investigated and discussed in detail.     

Elastic large deflection analysis of plates subjected to uniaxial thrust using meshfree Mindlin-Reissner formulation

A meshfree approach for plate buckling/post-buckling problems in the case of uniaxial thrust is presented. A geometrical nonlinear formulation is employed using reproducing kernel approximation and stabilized conforming nodal integration. The bending components are represented by Mindlin–Reissner plate theory. The formulation has a locking-free property in imposing the Kirchhoff mode reproducing condition. In addition, in-plane deformation components are approximated by reproducing kernels. The deformation components are coupled to solve the general plate bending problem with geometrical non-linearity. In buckling/post-buckling analysis of plates, the in-plane displacement of the edges in their perpendicular directions is assumed to be uniform by considering the continuity of plating, and periodic boundary conditions are considered in assuming the periodicity of structures. In such boundary condition enforcements, some node displacements/rotations should be synchronized with others. However, the enforcements introduce difficulties in the meshfree approach because the reproducing kernel function does not have the so-called Kronecker delta property. In this paper, the multiple point constraint technique is introduced to treat such boundary conditions as well as the essential boundary conditions. Numerical studies are performed to examine the accuracy of the multiple point constraint enforcements. As numerical examples, buckling/post-buckling analyses of a rectangular plate and stiffened plate structure are presented to validate the proposed approach.     

Buckling behavior of a central cracked thin plate under tension

The buckling characteristics of cracked plates subject to uniaxial tensile loads are analysed by the aid of the finite element method. Owing to the fact that crack buckling behavior is affected by the in-plane stress distribution around a crack, to get more accurate results, pre-buckling in-plane stress fields are analysed by the finite element method. For the critical loads calculation, the finite element approach adopted is based on Von Karman's linearize theory for buckling of plates subjected to pre-buckling state of plane stress. Several singular elements based on the Willian series are used in this plate bending approach. In this study, the effect of crack length, the effect of boundary condition, the effect of Poison's ratio and the effect of biaxial force on critical loads are analysed and discussed. Furthermore, the effect of initial imperfection is also discussed. There is a good agreement between other researcher's work and present results.     

正交各向异性带钢瓢曲变形行为样条有限元研究

带钢在轧制过程由于晶粒的择优取向而产生各向异性,为了研究这种各向异性对带钢板形瓢曲行为的影响规律,针对生产中常见的板形瓢曲缺陷,根据正交各向异性薄板小挠度理论和屈曲问题的驻值势能原理,建立了带钢在纵向延伸塑性应变(初应变)作用下以样条函数为基础的前屈曲样条有限元计算模型,获得了屈曲临界塑性应变、临界波长与带钢各向异性程度的关系;根据大挠度理论和虚功原理,应用样条有限元获得了正交各向异性带钢后屈曲路径。计算分析了带钢正交各向异性程度对中浪、边浪、四分之一浪、边中复合浪屈曲变形前屈曲临界条件及后屈曲变形过程的影响规律。     

Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation

The main purpose of this study is to investigate buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. In-plane compressive loads may be applied to either radial, circumferential, or all edges of circular/annular sector plates. Based on the classical plate theory (CPT), critical buckling loads and fundamental frequencies of the circular/annular sector plates under simply-supported and clamped boundary conditions are obtained by using differential quadrature method (DQM). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is considered to be constant. Convergence study is carried out to demonstrate the stability of the present method. To confirm the excellent accuracy of the present approach, a few comparisons are made for limited cases between the present results and those available in literature. Critical buckling load and fundamental frequency parameters of the circular/annular sector thin plates are computed for different boundary conditions, various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Post-buckling of elastic annular plates at multiple eigenvalues

The buckling and post-buckling problem of elastic annular plates is treated at multiple eigenvalues making strong use of symmetry properties of the problem in deriving bifurcation equations (amplitude equations of the critical modes) from the von Karman plate equations by means of the Ljapunov-Schmidt method. Further, a restricted generic unfolding of the bifurcation equations is made and some practically interesting results concerning the post-buckling behaviour are presented.     

Post-buckling of elastic annular plates at multiple eigenvalues

The buckling and post-buckling problem of elastic annular plates is treated at multiple eigenvalues making strong use of symmetry properties of the problem in deriving bifurcation equations (amplitude equations of the critical modes) from the von Karman plate equations by means of the Ljapunov-Schmidt method. Further, a restricted generic unfolding of the bifurcation equations is made and some practically interesting results concerning the post-buckling behaviour are presented.     

Multiple-parameter reduced basis technique for bifurcation and post-buckling analyses of composite plates

A multiple-parameter reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analyses of composite plates subjected to combined loadings. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the stability boundary. The plate is discretized by using displacement finite element models and the analysis region is reduced by exploiting the special symmetries exhibited by the response of the plate. The vector of unknown nodal displacements is expressed as a linear combination of a small number of path derivatives (derivatives of the nodal displacements with respect to path parameters), and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the stability boundary of the plate. In the second stage, a nonllnear solution in the vicinity of the stability boundary is obtained by using a bifurcation buckling mode as a predictor, and a set of reduced equations is generated. In the third stage, the reduced equations are used to trace post-buckling paths corresponding to various combinations of the load parameters. The potential of the proposed approach is discussed and its effectiveness is demonstrated by means of a numerical example of laminated composite plate subjected to combined compressive and shear loadings.     

Energy-release-rate evaluation for delamination growth prediction in a multi-plate model of a laminate composite

In this paper, numerical methods for the evaluation of the energy-release-rate along a delamination periphery under conditions of local buckling of the delaminate, as well as global buckling of the entire laminate, are presented. A multi-plate model, using independent Reissner-Mindlin plate models for each of the delaminated and undelaminated plies, with Reissner-Mindlin constraints for relating the degrees of freedom of the delaminated plates to those of the undelaminated plate at the crack front, is used to model the laminate with embedded delaminations. Explicit expressions, in terms of finite element nodal or Gauss-point variables, are derived for the pointwise energy release rate in terms of the J-integral and the Equivalent Domain Integral in the context of a typical multi-plate model for characterising the delamination growth. A finite element method with a 3-noded quasi-conforming shell element, and an automated post-buckling solution capability, is used for conducting the numerical analyses in this paper. Using these numerical results, mechanisms of multiple buckling modes and their effect on the propagation of embedded delaminations in plates, are studied.This work has been supported by a grant from FAA to the Center of Excellence for Computational Modeling of Aircraft Structures, Georgia Institute of Technology. The first author also acknowledges the support by the INDO-US Science and Technology Fellowship under the auspices of US Agency for International Development.     

船体开孔薄板面内剪切屈曲特性研究

船体板材不可避免地存在不同形式的开孔,开孔破坏了结构的连续性,对结构的强度、稳定性具有重要影响,因此研究船体板开孔结构的屈曲特性对保证船舶安全十分重要。在面内载荷作用下,通过画框型剪切夹具、3D全场变形测量-分析系统等对两种不同形式的船体开孔薄板进行剪切屈曲试验,获得了圆形开孔板和方形开孔板的临界屈曲载荷、全场位移/应变信息和拉力-伸长率曲线等;考虑试验夹具的影响,基于Abaqus对不同形式的开孔板进行数值仿真,通过对开孔板进行特征值屈曲分析和非线性屈曲分析,获得了两种不同形式船用开孔薄板的屈曲、后屈曲力学响应信息。通过数值仿真与试验结果的对比研究,验证了数值仿真方法的有效性、准确性。在此基础上,重点剖析载荷-伸长率关系、典型时刻板面全场位移、临界屈曲载荷以及开孔边缘的应力分布响应特征,明确了面积等效情况下圆孔和倒圆角方孔对方形薄板剪切稳定性的影响。为船用薄板面内剪切稳定性的试验和仿真研究以及大型船体结构的设计优化、力学性能评估等提供有益参考。     

A Fully Discretized Nonlinear Finite Strip Formulation for Pre-Buckling and Buckling Analyses of Viscoelastic Plates Subjected to Time-Dependent Loading

A semi-analytical fully discretized finite strip method is developed to investigate the pre-buckling and local buckling of viscoelastic plates with different boundary conditions subjected to time-dependent loading. The mechanical properties of the material are considered to be linear viscoelastic by expressing the relaxation modulus in terms of Prony series. The fully discretized finite strip equations are developed using a two-point recurrence formulation, which leads to a computationally superior formulation. Time history of maximum deflection of plates with different end conditions is calculated. The effects of thickness, length of plate, and transverse loading on critical buckling load are also studied.     

Buckling Analysis of Laminated Composite Plates Using Higher Order Semi—Analytical Finite Strip Method

Rectangular plates made of laminated composite material because of the advantageously high strength and stiffness to weight ratio are used frequently as structural component in various branches of engineering, chief of which are aerospace and marine engineering. Design concepts of these plates that lead to the increase in the buckling load can directly lower the structural cost and/or weight. The finite strip method is one of a number of procedures which can be used to solve the buckling problem of plate structures. In the present work the main concern is with the buckling behavior of plates with simply supported ends subjected to uni-axial pure compression loads. The solution is sought by implementing the higher order semi-analytical finite strip method which incorporates additional degrees of freedom for each nodal line by using Reddy’s higher order plate theory. Therefore the current method is more universal in dealing with different plate thicknesses. In addition, in this semi-analytical finite strip method, all the displacements are postulated by the appropriate harmonic shape functions in the longitudinal direction and polynomial interpolation functions in the transverse direction. The solution is based on the concept of principle of minimum potential energy and an eigen-value analysis is subsequently carried out. From the presented results it can be concluded that the higher order semi-analytical finite strip method is very reliable for the preliminary design of composite plates especially in the case of buckling analysis of relatively thick plates.     

Buckling of shear deformable laminated composite plates

A shear deformable finite element is developed for the buckling analysis of laminated composite plates. The finite element formulation is based on Mindlin's theory in which shear correction factors are derived from the exact expressions for orthotropic materials. A variety of problems on uniaxial and shear bucklings of laminated composite plates are solved. The effects of material properties, plate aspect ratio, length-to-thickness ratio, number of layers and lamination angle on the buckling loads of symmetrically and antisymmetrically laminated composite plates are investigated. Optimal lamination arrangements of layers for maximizing the buckling loads of the plates are determined.