Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method

The buckling of thin rectangular plates with nonlinearly distributed loadings along two opposite plate edges is analyzed by using the differential quadrature (DQ) method. The problem is considerably more complicated since it requires that first the plane elasticity problem be solved to obtain the distribution of in-plane stresses, and then the buckling problem be solved. Thus, very few analytical solutions (the only one available in the literature is for rectangular plates with all edges simply supported) have been available in the literature thus far. Detailed formulations and solution procedures are given herein. Nine combinations of boundary conditions and various aspect ratios are considered. Comparisons are made with a few existing analytical and/or finite element data. It has been found that a fast convergent rate can be achieved by the DQ method with non-uniform grids and very accurate results are obtained for the first time. It has also been found that the DQ results, verified by the finite element method with NASTRAN, are not quite close to the newly reported analytical solution. A possible reason is given to explain the difference.     

Wrinkling initiation and growth in modified Yoshida buckling test: Finite element analysis and experimental comparison

Wrinkling is one of the major defects in sheet metal products and may also play a significant role in the wear of the tool. The initiation and growth of wrinkles are influenced by many factors such as stress ratios, mechanical properties of the sheet material, geometry of the workpiece, contact condition, etc. It is difficult to analyze the wrinkling initiation and growth considering all the factors because the effects of the factors are very complex and the wrinkling behavior may show a wide scatter of data even for small deviations of factors. In this study, the bifurcation theory is introduced for the finite element analysis of wrinkling initiation and growth. All the above-mentioned factors are conveniently considered by the finite element method. The wrinkling initiation is found by checking the determinant of the stiffness matrix at each iteration and the wrinkling behavior is analyzed by successive iteration with the perturbed guess along the eigenvector. The effect of magnitude of perturbation on the wrinkling behavior can be avoided by the Newton-type iteration method. The finite element formulation is based on the incremental deformation theory and elastic-plastic material modeling. The finite element analysis is carried out using the continuum-based resultant shell elements considering the anisotropy of the sheet metal. For the verification of the analysis, the postbuckling of columns and circular plates are analyzed by finite element analysis using the bifurcation algorithm introduced in the study, and the results are compared with the exact solutions. In order to investigate the effects of geometry and stress ratio on the wrinkling initiation and growth, a modified Yoshida buckling test is proposed as an improved effective buckling test. In the modified Yoshida buckling test, the dimensions of the sheet specimen are varied to change the stress ratio and the degree of constraint. The finite element analysis is carried out for the modified Yoshida buckling test and compared with the experimental results.     

Buckling analysis of laminated composite rectangular plates reinforced by SWCNTs using analytical and finite element methods

In this paper, the buckling analysis of laminated composite plates reinforced by single-walled carbon nanotubes (SWCNTs) is carried out using an analytical approach as well as the finite element method. The developed model is based on the classical laminated plate theory (CLPT) and the third-order shear deformation theory for moderately thick laminated plates. The critical buckling loads for the symmetrical layup are determined for different support edges. The Mori-Tanaka method is employed to calculate the effective elastic modulus of composites having aligned oriented straight nanotubes. The effect of the agglomeration of the randomly oriented straight nanotubes on the critical buckling load is also analyzed. The results of analytical solution are compared and verified with the FEM calculations The critical buckling loads obtained by the finite element and the analytical methods for different layup and boundary conditions are in good agreement with each other. In this article, the effects of the carbon nanotubes (CNTs) orientation angle, the edge conditions, and the aspect ratio on the critical buckling load are also demonstrated using both the analytical and finite element methods.     

Spline finite strip analysis of the buckling and vibration of composite prismatic plate structures

A spline finite strip capability is described for predicting the buckling stresses and natural frequencies of vibration of prismatic plate structures which may be of composite laminated construction with arbitrary lay-ups. The plate structures may have general boundary conditions. The capability embraces analyses based on the use of first-order shear deformation plate theory and of classical plate theory, and utilizes substructuring procedures which include the use of superstrips. The theoretical development is not detailed since the present paper reports a very direct extension of a theoretical study developed for the analysis of single plates in an earlier paper in this Journal. A considerable range of buckling and vibration applications is documented and comparison of spline finite strip numerical values of buckling stresses and frequencies is made with results generated using the semi-analytical finite strip method and, in some cases, the finite element method. Buckled and vibrational mode shapes are presented for some applications.     

An elastic-plastic buckling solution using the incremental theory

A finite difference numerical procedure is used in considering the elastic-plastic buckling of rectangular plates which can be classified as being either statically determinate or indeterminate depending upon the loading condition along the boundary. The incremental theory of Prandtl-Reuss is used with assumed material properties expressed by a Ramberg-Osgood stress-strain relationship.Previous solutions have not attempted to investigate the indeterminate stress effects upon the plastic buckling of rectangular plates. In this paper, the author has dealt with this problem. In addition, the effect of Poisson's ratio upon plastic buckling is considered. Furthermore, for certain loading cases a comparison is made between the buckling values found using the deformation and incremental theory.     

Buckling and vibration of stiffened plates subjected to partial edge loading

The buckling and vibration characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element method. The initial stresses are obtained considering the pre-buckling conditions. Buckling loads and vibration frequencies are determined for different plate aspect ratios, edge conditions and different partial non-uniform edge loading cases. The non-uniform loading may also be caused due to the supports on the edges. The analysis presented determines the stresses all over the region for different kinds of loading and edge conditions. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The vibration characteristics are discussed and the results are compared with those available in the literature. Buckling results show that the stiffened plate is less susceptible to buckling for position of loading near the supported edges and near the position of stiffeners as well.     

Thermally induced buckling of functionally graded hybrid composite plates

In this paper, thermal buckling analysis is performed on hybrid functionally graded plates (FGPs) with an arbitrary initial stress. The governing equations are derived using the average stress method, including the effect of transverse shear deformation. Then, an eigenvalue problem is formed to evaluate thermal buckling temperatures for simple supported initially stressed ceramic-FGM-metal plates. The effects of functionally graded material (FGM) layer thickness, volume fraction index, layer thickness ratio, thickness ratio, aspect ratio and initial stress on the thermal buckling temperature of hybrid FGPs are investigated. The results reveal that the volume fraction index, initial stresses and FGM layer thickness have significant influence on the thermal buckling of hybrid FGPs.     

Thermal buckling analysis of annular FGM plate having variable thickness under thermal load of arbitrary distribution by finite element method

In this paper, a finite element formulation is developed for analyzing the axisymmetric thermal buckling of FGM annular plates of variable thickness subjected to thermal loads generally distributed nonuniformly along the plate radial coordinate. The FGM assumed to be isotropic with material properties graded in the thickness direction according to a simple power-law in terms of the plate thickness coordinate, and has symmetry with respect to the plate midplane. At first, the pre-buckling plane elasticity problem is developed and solved using the finite element method, to determine the distribution of the pre-buckling in-plane forces in terms of the temperature rise distribution. Subsequently, based on Kierchhoff plate theory and using the principle of minimum total potential energy, the weak form of the differential equation governing the plate thermal stability is derived, then by employing the finite element method, the stability equations are solved numerically to evaluate the thermal buckling load factor. Convergence and validation of the presented finite element model are investigated by comparing the numerical results with those available in the literature. Parametric studies are carried out to cover the effects of parameters including thickness-to-radius ratio, taper parameter and boundary conditions on the thermal buckling load factor of the plates.     

A comparative computational investigation on the effects of randomly distributed general corrosion on the post-buckling behaviour of uniaxially loaded plates

Post-buckling behaviour and ultimate strength of imperfect corroded steel plates used in ships and other related marine structures are investigated. Nonlinear elastic-plastic large deflection finite element analyses are performed on corroded steel plates. General corrosion wastage is considered to be distributed randomly on either one or both surfaces of the analyzed plates. The effects of general corrosion are introduced into the finite element models using a random thickness surface model. The effects of corroded plate parameters on the plate post-buckling and ultimate strengths are evaluated in detail. It was realized that the aspect ratio and thickness (slenderness) of the corroded plates affects their strength characteristics. Age of the plate models affects mainly their post-buckling-strength regimes and degrades their buckling/ultimate strength. Also, nonlinear post-buckling characteristics of the plates suffering either one-side or both-side random corrosion exhibit some differences. Finally, simple empirical formulations are proposed in order to give rough estimations of the ultimate strength of randomly corroded plates.     

Buckling of symmetrically laminated rectangular plates under parabolic edge compressions

Buckling analysis of symmetrically laminated rectangular plates with parabolic distributed in-plane compressive loadings along two opposite edges is performed using the Rayleigh-Ritz method. Classical laminated plate theory is adopted. Stress functions satisfying all stress boundary conditions are constructed based on the Chebyshev polynomials. Displacement functions for buckling analysis are constructed by Chebyshev polynomials multiplying with functions that satisfy either simply supported or clamped boundary condition along four edges. Methodology and procedures are worked out in detail. Buckling loads for symmetrically laminated plates with four combinations of boundary conditions are obtained. The proposed method is verified by comparing results to data obtained by the differential quadrature method (DQM) and the finite element method (FEM). Numerical example also shows that the double sine series displacement for simply supported symmetrically laminated plates having bending-twisting coupling may overestimate the stiffness, thus providing higher buckling loads.     

铁路起重机伸缩臂模糊可靠性分析

在载荷随机的情况下，把铁路救援起重机基本臂模型离散为390个板壳单元，然后利用摄动随机有限元计算最大应力单元主应力的均值和方差。在强度为确定性变量的情况下，把结构从安全状态到失效状态的过滤性考虑成一模糊事件，采用半梯形偏小隶属函数，计算QTJS160铁路起重机伸缩式吊臂的模糊可靠度。     

随机载荷作用下轴对称结构模糊可靠度计算方法

利用摄动随机有限元法,推导在载荷随机情况下轴对称结构的位移的均值和方差的计算公式,进而计算出单元应力的均值和方差。在强度为随机变量的情况下,把结构从安全状态到失效状态的过渡性考虑成一模糊事件,采用线性隶属函数,计算了轴对称结构的模糊可靠度。计算结果表明,把载荷处理为随机载荷,把失效事件处理为模糊事件,更符合实际情况,为结构可靠性优化设计提供更可靠的信息。     

Linear and nonlinear buckling analysis of a locally stretched plate

Uniformly stretched thin plates do not buckle unless they are in special boundary conditions. However, buckling commonly occurs around discontinuities, such as cracks, cuts, narrow slits, holes, and different openings, of such plates. This study aims to show that buckling can also occur in thin plates that contain no defect or singularity when the stretching is local. This specific stability problem is analyzed with the finite element method. A brief literature review on stretched plates is presented. Linear and nonlinear buckling stress analyses are conducted for a partially stretched rectangular plate, and various load cases are considered to investigate the influence of the partial loading expanse on the critical tensile buckling load. Results are summarized in iso-stress areas, tables and graphs. Local stretching on one end of the plate induces buckling in the thin plate even without geometrical imperfection.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

Stability of laminated composite plates subjected to various types of in-plane loadings

The aim of the present investigation is to examine the stability characteristics of laminated plates subjected to various types of in-plane loadings. Towards this, a rectangular four node finite element, having fourteen degrees of freedom per node, based on a simple higher-order shear deformation theory is developed. The theory employed herein involves four dependent variables. The element is found to be free of shear lock problems. A series of numerical problems are solved to study the effect of various parameters such as lay-up, side to thickness ratio, aspect ratio, type of loadings (uniaxial/biaxial/positive and negative shear/tension—compression/compression—tension) and boundary conditions. Some interesting observations regarding the considerable difference in the buckling resistance of angle-ply plates when subjected to positive and negative shear loading are made.     

Stochastic hybrid numerical method for transient analysis of stress field in functionally graded thick hollow cylinders subjected to shock loading

This investigation aims to study the random stresses in a functionally graded (FG) thick hollow cylinder with uncertain material properties subjected to mechanical shock loading using a hybrid numerical method. The mechanical properties are considered to vary across thickness of FG cylinder as a nonlinear power function of radius. The stresses are obtained by solving Navier equation and using Galerkin finite element and Newmark finite difference methods. The Monte Carlo simulation is used to generate the random mechanical properties for the problem. The failure probabilities and time history analysis of stresses are determined for various coefficient of variation considering various grading patterns of mechanical properties. The presented hybrid numerical method is effective, with high capability for stochastic analysis of dynamic and transient analysis of FG structures with various boundary conditions.     

Buckling analysis of circular and annular composite plates reinforced with carbon nanotubes using FEM

The critical compressive load in the buckling of circular and annular composite plates reinforced with carbon nanotubes (CNTs) is calculated using finite element method. The developed model is based on the third-order shear deformation theory for moderately thick laminated plates. Effects of CNTs orientation angles and thickness-to-inner radius ratio on the buckling of composite plates are discussed. The results are compared with those obtained by analytical method based on classical plate theory. The finite element method shows lower values for critical buckling load because of the elimination of shear strain in the classical plate theory.     

基于材料-结构协同设计的层合板多级优化方法

提出一种以材料参数与结构参数为变量的复合材料层合板多级优化设计新方法。以单层纤维含量、层厚和铺层角为设计参数,建立了刚度和强度约束下的结构减重设计模型,基于有限元分析,运用所提出的三级优化策略完成结构轻量化设计。本文给出了不同载荷类型和位移边界条件下层合板优化算例,设计结果验证了方法的有效性。     

DSC-element method for free vibration analysis of rectangular Mindlin plates

A novel DSC-element method is proposed to investigate the free vibration of moderately thick plates based on the well-known Mindlin first-order shear deformation plate theory. The development of the present approach not only employs the concept of finite element method, but also implements the discrete singular convolution (DSC) delta type wavelet kernel for the transverse vibration analysis. This numerical algorithm is allowed dividing the domain of Mindlin plates into a number of small discrete rectangular elements. As compared with the global numerical techniques i.e. the DSC-Ritz method, the flexibility is increased to treat complex boundary constraints. For validation, a series of numerical experiments for different meshes of Mindlin plates with assorted combinations of edge supports, plate thickness and aspect ratios is carried out. The established natural frequencies are directly compared and discussed with those reported by using the finite element and other numerical and analytical methods from the open literature.     

Stochastic perturbation-based finite element for deflection statistics of soft core sandwich plate with random material properties

The effect of random variations in material properties of laminated sandwich plates on the transverse deflections is investigated. An improved higher-order plate model is proposed earlier by the authors, which satisfies the transverse shear stress continuity conditions at the layer interfaces including the zero transverse shear stress conditions at the plate top and bottom surfaces. The theory assumes the variation of in-plane displacements to be cubic with discontinuities in the transverse shear strains at the layer interfaces, while the transverse displacements varies quadratically across the core thickness, thereby including transverse normal deformation of the soft core. The core is considered to behave as a 3-D elastic medium. To obtain the second-order statistics of deflections of sandwich plate, a stochastic C⁰ finite element (FE) based on the first-order perturbation technique is developed, where the lamina properties are considered as basic random variables while the other system properties are assumed to be deterministic. The performance of the improved stochastic laminated sandwich model is demonstrated through comparison of mean and standard deviations (SDs) of deflections obtained through independent Monte Carlo simulations and by comparison with results available in literature.