Free vibration analysis of isotropic and orthotropic triangular plates

A highly accurate and computationally efficient numerical method is developed for the flexural vibration of isotropic and orthotropic triangular plates. A set of two-dimensional orthogonal plate functions is used as an admissible displacement function in the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes for the plates. From a prescribed starting function satisfying the boundary conditions, the higher terms in the orthogonal plate functions are constructed using the Gram-Schmidt orthogonalization process. Natural frequencies and mode shapes are obtained for three triangular plates with different support conditions. The obtained numerical results are presented, and the isotropic case is verified with other numerical methods in the literature.     

Nonlinear analysis of rectangular orthotropic plates

The present study deals with the large deflections and nonlinear flexural vibrations of orthotropic rectangular plates. In this paper, generalized double Fourier series in terms of generalized beam eigenfunctions for deflection and force functions are presented. The influences of aspect ratio, material properties, transverse load, rotational edge restraint coefficients, and two types of in-plane conditions are studied. The results of deflection and frequency ratio for special cases are compared with those values given in the literature and good agreement is obtained.     

Postbuckling analysis of rectangular orthotropic plates

The postbuckling analysis of orthotropic simply supported rectangular plates of symmetric cross-section is studied and curves of load-shortening and effective widths are presented for various cases of orthotropic plates having different elastic properties. The results obtained are compared with results already published.     

Plastic analysis of perforated plates for orthotropic yield criteria

The concept of an equivalent orthotropic material is used in evaluating the collapse loads for perforated plates. Uniformly loaded circular plates are investigated. Limit pressure solutions are obtained for both the simply supported and rigidly built-in edge conditions. Statically admissible radial and circumferential bending moment fields are found and the associated velocity fields are shown to be kinematically admissible. Numerical results are given in dimensionless form covering arbitrary plate geometries over the entire range of material orthotropy.These solutions are quite useful in the plastic analysis and design of perforated plates used as pressure vessel heads, tube sheets, reactor core support plates and the like. Perforated plates have considerably higher effective yield strengths when subjected to equi-biaxial loading than when subjected to loading of arbitrary biaxiality and orientation with respect to the penetration pattern. Thus, the equivalent orthotropic plastic material concept is ideally suited to the analysis of such plates. The resulting limit load pressures are substantially lower than the values obtained using isotropic Tresca yield criterion based on the yield strength for equi-biaxial loading. The results are substantially higher than the values obtained using the maximum isotropic Tresca yield condition falling entirely within the orthotropic yield surface.     

Elastic large deflection analysis of isotropic rectangular Mindlin plates

Governing equations for the large deflection analysis of isotropic rectangular Mindlin plates are introduced and their solution using the DR algorithm is briefly outlined. Two computer programs, based respectively on interlacing and non-interlacing finite-differences, have been developed for the numerical solution of these equations. The programs have been verified by analysing a variety of thin and moderately thick plate problems for which alternative solutions are available. Sample results comparisons are presented in order to quantify the accuracy of the DR results. The non-interlacing finite-difference DR program is then used to compute new results for uniformly loaded square moderately thick plates with simply supported, clamped and combined simply supported and clamped edges.     

Stability of twisted orthotropic plates

Studied here is the stability of thin, rectangular, orthotropic plates which are in a state of tension and twist. A transfer matrix method is used to obtain numerical solutions to the linearized von Kármán plate equations, and to determine critical angles of twist per unit length which buckle the plate. Results are presented, in a compact nondimensional form, for a range of material, geometric and loading parameters. It is found that orthotropism significantly affects the stability of the plate.     

A semi-analytical solution for forced vibrations response of rectangular orthotropic plates with various boundary conditions

Orthotropic plates form an essential part of many marine, aerospace, and automobile structures. The recent increase in the use of composite materials for plate-type structures intensified the need for solutions utilizing orthotropic plates. These structural components, in many instances, are subjected to vibration. The main purpose of this paper is to present a semi-analytical analysis of the response of orthotropic plates to forced vibrations with different combinations of boundary conditions subjected to a general distributed excitation. The solution of the partial differential equation was reduced to an iterative sequential solution of ordinary differential equations using extended Kantorovich method. The efficiency and accuracy of the proposed method were examined through comparison with available literature and was in good agreement with them.     

Dynamic buckling of isotropic and orthotropic shallow spherical cap with circular hole

The axisymmetric dynamic behaviour of clamped isotropic and orthotropic spherical cap with central circular hole under a uniform step pressure of infinite duration is investigated. The critical loads are determined by using the rapidly converging Chebyshev series in a space domain and a Houbolt numerical integration scheme in time domain. Three different cases of openings are investigated and detailed numerical results have been obtained for the isotropic case and two different cases of orthotrophy, which may be useful for design engineers.     

Geometrically nonlinear axisymmetric vibrations of polar orthotropic circular plates

The geometrically nonlinear free vibrations of polar orthotropic circular plates are analyzed by the axisymmetric finite element. The formulation is based on the three-dimensional elasticity with all the nonlinear terms in the strains included and is considered to be more general and precise theoretically compared to the plate-theory-based nonlinear approaches. Numerical results with various geometries, orthotropies, amplitudes and boundary conditions by the present formulation are also shown and compared to those by other approaches.     

Buckling and bending behaviour of initially stressed specially orthotropic thick plates

The virtual work theorem is employed to derive the governing equations of specially orthotropic plates in a general state of non-uniform initial stress, where the effects of transverse shear are included. The equations were adjusted to a generic form by using appropriate transformations. Considering the transformations, introducing generalized parameters and employing a similarity parameter, comprehensive solutions are found to this problem; hence, the curves presented in the text are generic rather than specific. The thick plate equations are solved for a simply-supported rectangular specially orthotropic plate in a state of combined uniform uniaxial compression and bending. The results of the reduced set of equations agree with those obtained by previous investigators.     

Analytical solution of a two variable refined plate theory for bending analysis of orthotropic Levy-type plates

This paper presents analytical solutions of deflection and stress for orthotropic plates using a two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factor. Additional features of the theory are that it has strong similarity with classical plate theory in many aspects, and the number of involved variables is only two as against three in case of other shear deformation theories. The Levy-type solution procedure in conjunction with the state space concept is used to determine the closed-form solutions for orthotropic rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are performed to verify the validity of the present results. Finally, the effects of thickness ratio, modulus ratio and aspect ratio on the deflection and stress of orthotropic plates are investigated and discussed.     

A twelfth-order theory of antisymmetric bending of isotropic plates

In this paper the transverse shear and normal strain and stress effects on antisymmetric bending of isotropic plates are considered. A set of twelfth-order partial differential governing equations as well as a set of fourth-order ordinary differential equations for ƒ(z) and φ(z), which represent the transverse shear and normal effects, are derived from a mixed variational theorem. There exists coupling between the partial differential equations and the ordinary ones. In the homogeneous solutions for the former, besides an interior solution contribution, there exist two types of edge-zone solution contributions. One of them is similar to the edge-zone solution in the Reissner—Mindlin theory. The other one is an edge-zone solution consisting of a pair of conjugate functions. Two sample examples are calculated using the present theory. In the former the present two-dimensional theory obtains the three-dimensional exact solution. The latter gives the stress couple and maximum-stress concentration factors at the free edge of a circular hole in a large bent plate. The numerical results still approximate to exact solutions.     

Buckling of polar orthotropic annular plates under internal radial load and torsion

The buckling problem of clamped, polar orthotropic annular plates under internal radial load and torsion is studied theoretically with asymmetric modes taken into consideration. The problem is solved by means of the Galerkin method with the deflection function assumed in the form of cosine series in conjunction with a coordinate transformation. The critical combinations (λ_sc, λ_ic) of torsion and internal radial load are determined for a wide range of polar orthotropic material properties and various hole sizes of the orthotropic annulus. It is found that there are some ranges of λ_ic where λ_sc increases with λ_ic, principally when the internal radial load λ_ic > 0 (radial compression) acts on the annular plate.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.     

Postbuckling behavior of rectangular orthotropic plates with two free side edges

The postbuckling behavior of rectangular orthotropic plates with two free side edges is studied based on a perturbation method. Both exact and approximate solutions of the second-order perturbation equations are given. Numerical results for plates having various elastic properties are also presented.     

A DXDR large deflection analysis of uniformly loaded square, circular and elliptical orthotropic plates using non-uniform rectangular finite-differences

A finite-difference analysis of the large deflection response of uniformly loaded square, circular and elliptical clamped and simply-supported orthotropic plates is presented. Several types of non-uniform (graded) mesh are investigated and a mesh suited to the curved boundary of the orthotropic circular and elliptical plate is identified. The DXDR method-a variant of the DR (dynamic relaxation) method-is used to solve the finite-difference forms of the governing orthotropic plate equations. The DXDR method and irregular rectilinear mesh are combined along with the Cartesian coordinates to treat all types of boundaries and to analyze the large deformation of non-isotropic circular/elliptical plates. The results obtained from plate analyses demonstrate the potential of the non-uniform meshes employed and it is shown that they are in good agreement with other results for square, circular and elliptical isotropic and orthotropic clamped and simply-supported plates in both fixed and movable cases subjected to transverse pressure loading.     

A study on the initial crack curving angle of isotropic/orthotropic bimaterial

In this paper, when the initial propagation angle of a branched crack is calculated from the maximum tangential stress criterion (MTSC) and the minimum strain energy density criterion (MSEDC), it is essential that you use stress components in which higher order terms are considered and stress components at the position in a distance 0.005 mm from the crack tip ( =γ . When an interfacial crack propagates along the interface at a constant velocity, the initial propagation angles of the branched crack are similar to the mode mixities (phase angle) and the theoretical values obtained from MTSC and MSEDC. The initial propagation angle of the branched crack depends considerably on the stress intensity factor K{IN2}.     

Analysis of clamped rectangular orthotropic plates subjected to a uniform lateral load

The increasing use of composite materials for plate-type structures intensified the need for solutions of orthotropic plates. One problem of interest is the clamped rectangular plate with reinforcing fibers parallel to the plate boundaries. There is no exact analytical solution for this problem. The purpose of this paper was to derive a closed-form approximate solution of high accuracy, even for third derivatives (shearing forces). This was achieved by utilizing the extended Kantorovich method. It was found that the convergence of this procedure is very rapid, in that the final form is reached after only four iterations, and that the final form of the generated solution is independent of the initial choice. To establish the accuracy of this solution, it was numerically evaluated for a plate made of Kevlar 49 Epoxy, and the results obtained were compared with the corresponding results by Whitney [Structural Analysis of laminated Anisotropic Plates. Technomic, Lancaster; Pennsylvania (1987)], as well as with the exact solution for a clamped infinite strip. It was found that with increasing plate aspect ratios, the deflections and bending moments generated approached closely those of the infinite strip, but those of Whitney did not.     

移动荷载作用下夹层板的动力学响应仿真分析

应用层合板理论将夹层板等效为一个正交各向异性板,分别计算出了实体金属夹芯、金属泡沫夹芯、正六边形蜂窝夹芯、金属波纹板夹芯四种夹层板的等效刚度,再应用正交各向异性板理论和模态叠加原理分析了这四种夹层板在移动荷载作用下的动力学响应,并对此做了比较分析,得出移动荷载作用下正六边形蜂窝夹层板相对于其它三种夹层板更具有优越性,而实体金属夹层板是这四种夹层板中承载能力最差的.所得结果对桥梁,公路的建设具有一定的指导意义.