Vibration analysis of rectangular mindlin plates by the finite strip method

A finite strip analysis of the vibration of rectangular Mindlin plates with general boundary conditions is described. The normal modes of vibration of Timoshenko beams are used to represent the spatial variation along a strip of the deflection and the two cross-sectional rotations. For the crosswise representation equal-order polynomial interpolation is employed for each of these three basic quantities. The accuracy of the approach is demonstrated by the results of a number of applications to square plates with combinations of simply supported, clamped and free edges.     

Buckling of initially stressed mindlin plates using a finite strip method

The buckling of initially stressed Mindlin plates is considered using a thick finite strip method. The method is compared with a wide variety of published results and for both thin and moderately thick plates excellent accuracy is obtained. Some further results are obtained for initially stressed rectangular plates with two opposite edges simply supported and various support conditions on the remaining sides. In general, it is found that for moderately thick plates, Mindlin's plate theory gives lower buckling loads than those obtained using classical thin plate theory.     

Application of the collocation method to vibration analysis of rectangular mindlin plates

The flexural vibration analysis of rectangular Mindlin plates using the collocation method is described. The results obtained by the present method are compared with published results for plates with uniform thickness and two opposite edges simply supported. The comparison shows that the method yields very good results with a relatively small number of collocation points, and that estimates for the higher modes can be obtained without any difficulties. Furthermore, the method is applied to plates with linearly varying thickness, and new findings are presented for the frequencies of plates.     

Geometric nonlinear analysis of stiffened plates by the spline finite strip method

Geometric nonlinear analysis of stiffened plates is investigated by the spline finite strip method. von Karman’s nonlinear plate theory is adopted and the formulation is made in total Lagrangian coordinate system. The resulting nonlinear equations are solved by the Newton–Raphson iteration technique. To analyse plates having any arbitrary shapes, the whole plate is mapped into a square domain. The mapped domain is discretised into a number of strips. In this method, the displacement interpolation functions used are: the spline functions in the longitudinal direction of the strip and the finite element shape functions in the other direction. The stiffener is elegantly modelled so that it can be placed anywhere within the plate strip. The arbitrary orientation of the stiffener and its eccentricity are incorporated in the formulation. All these aspects have ultimately made the proposed approach a most versatile tool of analysis. Plates and stiffened plates are analysed and the results are presented along with those of other investigators for necessary comparison and discussion.     

Buckling of rectangular mindlin plates

The elastic buckling of rectangular Mindlin plates is considered using two related methods of analysis. These methods are the Rayleight-Ritz method and one of its piece-wise forms, the finite strip method. Arbitrary combinations of the standard boundary conditions of clamped, simply-supported and free edges are accommodated by the use in the assumed displacement fields of the normal modes of vibration of Timoshenko beams. The applied membrane stress field leading to buckling can comprise biaxial direct stress plus shear stress. A range of numerical applications is described for isotropic and transversely isotropic plates of thin and moderately thick geometry. The results obtained using the two methods compare closely to one another and to other published results where these are available. A direct relationship between unidirectional buckling stress and frequency of vibration is demonstrated for a category of plates having one pair of opposite edges simply supported.     

Bending analysis of rectangular moderately thick plates using spline finite element method

A bending analysis of rectangular, moderately thick plates with general boundary conditions is presented using the spline element method. The cubic B spline interpolate functions are used to construct the field function of generalized displacements w, φ_itxand φ_ity. The spline finite element equations are derived based on the potential energy principle. For simplicity, the boundary conditions, which consist of three local spline points, are amended to fit specified boundary conditions. The shear effect is considered in the formulations. A number of numerical examples are described for rectangular, moderately thick plates. Since the cubic B spline interpolate functions have sufficient continuity and are piecewise polynomial, so the present numerical solutions show not only that the method gives accurate results, but also that the unified solutions of thick and thin plates can be directly obtained; the trouble with the so-called shear locking phenomenon does not occur here.     

Stability analysis of skew orthotropic plates by the finite strip method

A stability analysis based on the Finite Strip Method is presented for skew orthotropic plates subjected to in-plane loadings. The straight sides of the plate are simply supported and the other two skewed sides are supported with any combination of fixed, free and simply supported boundaries. The plate is divided into strips, in contradistinction to elements in the Finite Element Method, and the displacement function is so chosen that it satisfies the boundary conditions and also the inter-strip compatibility conditions of an elemental strip. The energy expressions required to formulate the stiffness and stability coefficient matrices are formulated using smalldeflection theory. The buckling load intensity factor is evaluated for different aspect ratios of isotropic and orthotropic skew plates and the results of certain rectangular isotropic cases are compared with earlier investigations.     

Analysis of plates by finite strip method

New finite strips are developed for the analysis of plates. Based on Reissner's plate theory, the effect of shear deformation is included in the formulation. To eliminate artificial hardening, the shape functions for the strips are so chosen that there is no mismatched term along the interpolation functions for the interpolation parameters. Numerical examples are reported to demonstrate that the strips can work equally well in thick as well as thin plates.     

Geometrically nonlinear finite element analysis of tapered beams

A novel finite element methodology is developed capable of analyzing the geometrically nonlinear behavior of thin-walled framed structures composed of non-prismatic members. The pertinent element matrices are formulated on the basis of a modified version of the variational theorem of Hellinger and Reissner. Finite geometry changes are consistently described by using an updated Lagrangian (U.L.) formulation. Validity, accuracy and reliability of the proposed scheme are examined on the basis of several well-selected test examples.     

Stability analysis of anisotropic laminated composite plates by finite strip method

The finite strip method has been applied to the stability analysis of rectangular shear-deformable composite laminates. However, for the plates with two opposite simply supported sides, the existing analysis was restricted to the symmetrical cross-ply laminates under compression loading.In the present study, by selecting proper displacement functions and including the coupling between different series terms, the finite strip method is extended to the stability analysis of any anisotropic laminated plates under arbitrary in-plane loading. Furthermore, a number of numerical results are presented to show the effects of thickness, fibre orientation and stacking sequence on the buckling loads.     

基于边光滑有限元法的剪切变形板几何非线性分析

为改善在计算板的几何非线性问题时有限元法系统过硬的数值缺陷,提高计算精度,在考虑剪切变形的yon Karman假设下,基于全拉格朗日描述方法,将边光滑有限元法应用于板的几何非线性分析.计算公式基于1阶剪切变形理论,并采用离散剪切间隙有效地消除剪切自锁.在三角形单元的基础上进一步形成边界光滑域,在每个光滑域内对应变进行光...     

A unified approach for the analysis of bridges,plates and axisymmetric shells using the linear mindlin strip element

In this paper a finite strip formulation which allows to treat bridges, axisymmetric shells or plate structures of constant transverse cross section in an easily and unified manner is presented. The formulation is based on Mindlin's shell plate theory. One dimensional finite elements are used to discretize the transverse section and Fourier expansions are used to define the longitudinal/circumferential behavior of the structure. The element used is the simple two noded strip element with just one single integrating point. This allows to obtain all the element matrices in an explicit and economical form. Numerical examples for a variety of straight and curve bridges, axisymmetric shells and plate structures which show the efficiency of the formulation and accuracy of the linear strip element are given.     

Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element

Summary This paper presents a state of the art review on geometrically nonlinear analysis of shell structures that is limited to the co-rotational approach and to flat triangular shell finite elements. These shell elements are built up from flat triangular membranes and plates. We propose an element comprised of the constant strain triangle (CST) membrane element and the discrete Kirchhoff (DKT) plate element and describe its formulation while stressing two main issues: the derivation of the geometric stiffness matrix and the isolation of the rigid body motion from the total deformations. We further use it to solve a broad class of problems from the literature to validate its use.     

Geometrically nonlinear finite element behaviour using buckling mode superposition

Within the context of finite element analysis a method is presented for analysing the geometrically nonlinear static behaviour of thin-type structures under one-parametric, conservative loading. The method is applicable in cases of mild geometrical nonlinearities, which are characterized by the existence of a linear bifurcation situation which is “close” to the considered situation. In this case the true nonlinear behaviour in the initial loading range can be quite well approximated by a linear combination of the eigenvectors of the linear buckling problem where the coefficients of combination are nonlinear functions of the load parameter and are easily computable from the eigenvalues, eigenvectors and the applied loading. A comprehensive treatment of this method of linear buckling mode superposition is given, including theoretical derivations, physical interpretation, software implementation, computational considerations and selected application examples which illustrate the basic nature of the method.     

Analysis of piezolaminated plates by the spline finite strip method

This paper deals with the development of a family of higher order B-spline finite strip models applied to the static and free vibration analysis of laminated plates, with arbitrary shape and lay-ups, loading and boundary conditions. The lamination scheme can be such that the embedded and/or surface bonded piezoelectric actuating and sensing layers are included. The structure is discretised in a specified number of strips, and the geometry and displacement components of each strip are represented by interpolating functions that are products of linear or cubic B-spline, and linear or quadratic Lagrange functions along the y and x orthonormal directions. The accuracy and relative performance of the proposed discrete models are compared and discussed among the developed and alternative models.     

Nonlinear analysis of skew plates using the finite element method

In this paper finite element analysis of the large deflection behaviour of skew plates has been done. A high precision conforming triangular plate bending element has been used. The central deflection, bending and membrane stresses have been reported for simply supported and clamped rhombic plates. The variations of these quantities have been studied for different skew angles.     

A three-dimensional nonlinear analysis of cross-ply rectangular composite plates

The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropie composite plates are presented. Individual laminae are assumed to be homogeneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusiodal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those predicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement     

Free vibration analysis of plates using least-square-based finite difference method

In this paper, we apply the two-dimensional least-square-based finite difference (LSFD) method for solving free vibration problems of isotropic, thin, arbitrarily shaped plates with simply supported and clamped edges. Using the chain rule, we show how the fourth-order derivatives of the plate governing equation can be discretized in two or three steps as well as how the boundary conditions can be implemented directly into the governing equation. By analyzing vibrating plates of various shapes and comparing the solutions obtained against existing results, we clearly demonstrate the effectiveness of LSFD as a mesh-free method for computing vibration frequencies of generally shaped plates accurately.     

Application of the spline finite strip to the analysis of shear-deformable plates

A spline finite strip is proposed to analyse thick isotropic or laminated composite plates. The formulation is based upon the principle of virtual work and the third-order plate theory developed by Reddy. The variational functional requires the satisfaction of C₁,-continuity of the assumed vertical deflection variable which can be easily fulfilled by the present method. The proposed spline finite strip is a conforming element with a smaller number of unknowns at each node compared to other existing elements based on the third-order theory. For the analysis of thin isotropic or laminated plates, the present element shows no sign of shear locking. A number of computational examples are given to demonstrate the efficiency and the accuracy of the present method.     

A comparison of the linear,quadratic and cubic mindlin strip elements for the analysis of thick and thin plates

The behaviour of the linear, quadratic and cubic elements of the Mindlin plate strip family for thick and very thin plate analysis is investigated in this paper. Selective integration techniques are used to ensure the good behaviour of the elements when dealing with thin plates. Numerical results showing the convergence and accuracy of the elements for the analysis of plates of a wide range of thicknesses are given. The general performance of the three elements is discussed in detail. In particular, the linear element with a single integration point seems to be the best value strip element for practical purposes.