Buckling of the SSCF rectangular orthotropic plate subjected to linearly varying in-plane loading

The paper presents the solution of the buckling problem for an orthotropic rectangular plate having two parallel edges simply supported, one edge clamped and the remaining edge free (the SSCF plate). The plate considered is subjected to a linearly varying in-plane load that can take the form of uniform compression, combination of in-plane bending and uniform compression, or pure in-plane bending. The solution technique involves reduction of the relevant variational buckling equation to a one-dimensional form using the Kantorovich procedure and subsequent application of the generalised Galerkin method. The buckling problems are solved for isotropic and orthotropic plates with various aspect ratios. The analytical solution is verified using the finite-element analysis. The comparisons of computational results demonstrate the appropriateness and efficiency of the approach developed in this work for the calculation of critical loads of composite SSCF plates with various dimensional and stiffness parameters.     

Buckling analysis of orthotropic thin rectangular plates subjected to nonlinear in-plane distributed loads using generalized differential quadrature method

In the present work, buckling analysis of orthotropic thin rectangular plates with uniform thickness resting on Pasternak foundation are investigated for eight types of boundary conditions: SSSS, CCCC, SCSC, SSSC, SSCC, CCCF, SSFC, and CFCF. Based on classical plate theory, governing differential equation in buckling are solved numerically using generalized differential quadrature method (GDQM) to obtain critical buckling loads and corresponding modes. The kinds of nonlinear loading are presented in six cases including symmetrical and unsymmetrical distribution. In addition, the effects of aspect ratio, orthotropic moduli ratio and coefficients of foundation on the buckling load are illustrated. The present work is the first attempt to consider the influence of the nonlinearity of distributed in-plane bi-directional loading in determination of buckling load and representation of the corresponding shape modes. Some numerical examples are provided to demonstrate good accuracy of the GDQ method to evaluate the critical buckling load in case of nonlinear distributed bi-directional compressive loads. As shown, profile of distributed in-plane loading plays an important role on buckling behavior of the rectangular plate.     

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.     

Thermomechanical buckling of hybrid cross-ply laminated rectangular plates

A thermomechanical buckling analysis is presented for simply supported rectangular symmetric cross-ply laminated composite plates that are integrated with surface-mounted piezoelectric actuators and are subjected to the combined action of in-plane compressive edge loads, two types of thermal loads, and constant applied actuator voltage. The formulation of equations is based on the classical laminated plate theory and the von-Karman non-linear kinematic relations. The analysis uses an exact method to obtain closed-form solutions for the buckling load. The effects of applied actuator voltage, thermal and mechanical loads, plate geometry, and lay-up configuration of the laminated plates are investigated. The novelty of the present work is to obtain closed-form solutions for electro-thermomechanical buckling of hybrid composite plates, and to cover non-uniform temperature distribution loading. The results for various states are verified with known data in the literature.     

Buckling analysis of rectangular composite plates with rectangular cutout subjected to linearly varying in-plane loading using fem

A numerical study is carried out using finite element method, to examine the effects of square and rectangular cutout on the buckling behavior of a sixteen ply quasi-isotropic graphite/epoxy symmetrically laminated rectangular composite plate [0°/+45°/-45°/90°]_2s, subjected to various linearly varying in-plane compressive loads. Further, this paper addresses the effects of size of square/rectangular cutout, orientation of square/rectangular cutout, plate aspect ratio(a/b), plate length/thickness ratio(a/t), boundary conditions on the buckling bahaviour of symmetrically laminated rectangular composite plates subjected to various linearly varying in-plane compressive loading. It is observed that the various linearly varying in-plane loads and boundary conditions have a substantial influence on buckling strength of rectangular composite plate with square/rectangular cutout.     

Explicit local buckling analysis of rotationally-restrained orthotropic plates under uniform shear

The explicit closed-form local buckling solution of in-plane shear-loaded orthotropic plates with two opposite edges simply supported and other two opposite edges either both rotationally restrained or one rotationally restrained and the other free is presented. Based on the boundary condition of the other two opposite edges, two types of plates are considered: the RR (both the edges rotationally restrained) and RF (one edge restrained and the other free) plate elements. Different plate buckled shape functions are proposed, and the approximate explicit expressions for the buckling loads are derived using the Rayleigh–Ritz method for the plate with the generic rotationally-restrained (R) boundary conditions which can be reduced to two extreme cases, i.e., simply supported (S) and clamped (C). The accuracy of the derived explicit solutions is verified by comparing the predictions with the existing solutions and numerical finite element analysis, and excellent agreements are obtained. The effects of material and boundary restraining parameters on the local shear buckling behavior of the plate elements are discussed. The derived explicit formulas for the shear buckling loads are straightforward, efficient and reliable for preliminary engineering design and analysis of composite structures under primarily shear-dominant loading conditions.     

余弦分布压力下矩形薄板的屈曲

针对不同支承条件,两对边受半余弦非线性分布压力下弹性矩形薄板的屈曲问题,进行了分析研究。对于只产生对称变形的矩形薄板,基于辛弹性力学的平面矩形域理论,给出了精确的面内应力分布。运用Galerkin法分析计算了半余弦分布压力下矩形薄板的屈曲载荷。根据各种不同支承矩形薄板弯曲的位移边界条件,借助于符号运算软件Maple,编写了相应的用户计算程序。对九种不同支承组合下的弹性矩形薄板进行了计算,得到了不同长宽比矩形薄板的屈曲载荷系数。通过与已有文献结果的比较表明,该文求解方法是有效和精确的。基于所给出的结果,可望为解决矩形薄板在非线性分布载荷下的屈曲分析提供一种新的研究方法。     

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.     

Buckling of composite plates by global–local plate theory

The bifurcation buckling problem of laminated composite plates is formulated within the framework of a multilength scale plate theory. This theory is a combination of single-layer and layer-wise theories. It is generated by representing the displacement as the sum of global and local effects that introduce a coupling between the two length scales.Comparisons between the presently predicted buckling loads of homogeneous and orthotropic laminated plates and the exact solutions show a very good correlation. Furthermore, the theory accurately predicts the buckling load of symmetric cross-ply plates as compared with the results of a layer-wise approach. This accuracy is achieved with reduced computation expense.The global–local plate theory is general enough to incorporate delamination effects. As a result of the inclusion of these effects, the buckling loads of plates with imperfect interlaminar bonding are predicted.     

A generalized higher-order theory for buckling of thick multi-layered composite plates with normal and transverse shear strains

The onset of buckling in square laminated multi-layered composite plates, subject to unidirectional in-plane loads, is investigated within the framework of a generalized higher-order shear deformation theory suitable to capture significant transverse shear and thickness-wise deformation effects. The displacement field is expanded in a Taylor series of the thickness coordinate with arbitrary polynomial degree; in turn, the series coefficients, expressed as a superposition of admissible functions, are determined according to the Rayleigh–Ritz method. Truly higher-order polynomial terms, along with a sufficient number of in-plane admissible functions, are shown to be necessary for convergence towards the fundamental buckling load multiplier. As a by-product, reduced-order models are identified for various plate geometries and lamination schemes. The sensitivity of the lowest buckling load with respect to the nondimensional parameters (the thickness ratio, the ratio between the elastic moduli, the ply angle) is investigated. In particular, the attention is focused on the cross-over phenomenon between the lowest two buckling eigenvalues in multi-layered composite square plates with different lamination schemes. The presented results shed light onto the buckling behavior of thick shear-deformable multi-layered plates.     

Explicit local buckling analysis and design of fiber–reinforced plastic composite structural shapes

Pultruded fiber–reinforced plastic (FRP) composite structural shapes (beams and columns) are thin-walled open or closed sections consisting of assemblies of flat plates and commonly made of E-glass fiber and either polyester or vinylester resins. Due to high strength-to-stiffness ratio of composites and thin-walled sectional geometry of FRP shapes, buckling is the most likely mode of failure before material failure. In this paper, explicit analyses of local buckling of rectangular orthotropic composite plates with various unloaded edge boundary conditions (i.e., (1) rotationally restrained along both unloaded edges (RR), and (2) one rotationally restrained and the other free along the unloaded edges (RF)) and subjected to uniform in-plane axial action at simply-supported loaded edges are first presented. A variational formulation of the Ritz method is used to establish an eigenvalue problem, and explicit solutions of plate local buckling coefficients in term of the rotational restraint stiffness (k) are obtained. The two cases of rotationally restrained plates (i.e., the RR and RF plates) are further treated as discrete plates of closed and open sections, and by considering the effect of elastic restraints at the joint connections of flanges and webs, the local buckling of different FRP shapes under uniform axial compression is studied. The approximate expressions of the rotational restraint stiffness (k) for various common FRP sections are provided, and their application to sectional local buckling predictions is illustrated. The explicit local buckling formulas of rotationally restrained plates are validated with the exact transcendental solutions. The analytical predictions for local buckling of various FRP profiles based on the present discrete plate analysis and considering the elastic restraints of the flange–web connections are in excellent agreements with available experimental results and finite element eigenvalue analyses. A design guideline for local buckling prediction and related performance improvement is proposed. The present explicit formulation can be applied effectively to determine the local buckling capacities of composite plates with elastic restraints along the unloaded edges and can be further used to predict the local buckling strength of FRP shapes.     

弯剪复合载荷作用下复合材料层合板屈曲的强度校核方法

基于复合材料各向异性板的大挠度方程，采用Ritz法和最小势能原理推导出了正交各向异性复合材料层合板在面内弯曲载荷作用下临界屈曲弯矩的解析解；给出了面内弯剪复合载荷作用下复合材料层合板屈曲的强度校核方法，并将本文方法所得结果与用有限元法（FEM）所得结果进行了比较。结果表明，本文方法与有限元方法相吻合，并且形式简洁，便于工程应用。     

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.     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation

In this article, an exact analytical solution for buckling analysis of moderately thick functionally graded (FG) sector plates resting on Winkler elastic foundation is presented. The equilibrium equations are derived according to the first order shear deformation plate theory. Because of the coupling between the bending and stretching equilibrium equations of FG plates, these plates have deflection under in-plane loads lower than the critical buckling load acting on the mid-plane. The conditions under which FG plates remain flat in the pre-buckling configuration are investigated and the stability equations are obtained based on the flat plate assumption in the pre-buckling state. The stability equations are simplified into decoupled equations and solved analytically for plates having simply supported boundary condition on the straight edges. The critical buckling load is obtained and the effects of geometrical parameters and power law index on the stability of functionally graded sector plates are studded. The results for the critical buckling load of moderately thick functionally graded sector plates resting on elastic foundation are reported for the first time.     

A state of stress and displacement of elastic plates using simple and mixed shear deformation theories

This paper presents accurate two-dimensional solutions for bending response of four types of single-layer orthotropic rectangular plates. The plates considered are of the type having two opposite sides simply-supported, and two other sides having combinations of simply-supported, clamped, and free-boundary conditions. Analytical solutions for deflections and stresses of rectangular plates are developed by means of the simple (SFPT) and mixed (MFPT) first-order shear deformable plate theories. The present MFPT not only shows improvement on predicting frequencies, critical buckling loads, deflections and in-plane stresses, but also accounts for variable transverse shear stress distributions through the thickness. This puts into evidence the important role played by MFPT in the modeling of homogeneous plate theories, which in contrast to SFPT does not require the incorporation of a shear-correction factor. For illustrative purposes, sample free vibration, stability, and bending problems for simply supported orthotropic plates are considered and comparisons of the obtained results are made with the exact and higher-order shear deformation theory results given in the literature.     

Vibration and buckling of rectangular composite plates with variable fiber spacing

This is a summary of the first known work which analyzes the structural behavior of composite plates having nonuniformly spaced fibers. The present investigation is limited to single layer composites having parallel fibers. This results in a plate which is macroscopically orthotropic, but nonhomogeneous. The free vibrations and buckling of such plates subjected to inplane boundary loadings are studied. A plane elasticity problem must first be solved to determine the inplane stresses caused by the applied boundary loading, and these stresses become input to the vibration and buckling problem. Both problems are dealt with by the Ritz method. Numerical results are obtained for six nonuniform distributions of E-glass, graphite and boron fibers in epoxy matrices in simply supported, square plates. The redistributions are seen to increase the buckling load by as much as 38%, and the fundamental frequency by as much as 21%.     

On a new symplectic geometry method for exact bending solutions of orthotropic rectangular plates with two opposite sides clamped

A novel symplectic geometry method is presented for exact bending solutions of orthotropic rectangular thin plates with two opposite sides clamped. In the proposed mathematical method, it starts with the basic governing equations for the bending of orthotropic plates, and there are no predetermined functions, which overcome the deficiency of conventional semi-inverse methods; therefore, the straightforward implementation procedure serves as a completely rational model in plate bending analysis. The proposed method can be extended to more problems of plates such as buckling and vibration. Tabulated accurate results are included in this paper, which are expected to be valuable for future comparison.     

Buckling analysis of orthotropic rectangular plate resting on Pasternak elastic foundation under biaxial in-plane loading

In this study, buckling of rectangular orthotropic plates resting on a Pasternak elastic foundation under biaxial in-plane loading by the power series method (the method of Frobenius) was analyzed. Similar to many studies, two opposite edges of loading are simply supported and two other edges are assumed clamped. In order to extract the characteristic equations of orthotropic rectangular plate under in-plane loading resting on a Pasternak elastic foundation, the classical plate theory, by considering the interaction between plate and foundation, is used. The results showed that in the aspect ratio of less than 2, the existing Pasternak foundation caused the buckling load to increase severely, but by increasing the aspect ratio, the effect of the foundation is negligible. Applying the in-plane load in the y-direction caused the buckling load to decrease, but by increasing the aspect ratios the effect of the load is negligible.