Plate Bending Problems Research Articles

Differential quadrature method for bending problem of plates with transverse shear effects

A differential quadrature (DQ) method for orthotropic plates was proposed based on Reddy’s theory of plates with the effects of the higher-order transverse shear deformations. Wang-Bert’s DQ approach was also further extended to handle the boundary conditions of plates. The computational convergence was studied, and the numerical results were obtained for different grid spacings and compared with the existing results. The results show that the DQ method is fairly reliable and effective.

Analysis of cracked plates and shells using “metis” finite element model

A special form finite element technique is presented for the calculation of displacement and stress field of cracked (or uncracked) plates and shells. The method based on a particular set of hybrid displacement finite element, namely “Metis elements”. Several numerical examples are considered to evaluate the performance of this model. For cracked plate in extension, the element HSM gives high accuracy in computation of stress intensity factors. For plate bending problem with or without crack, the element HSF presents also very good performance. The results are very close to the theoretical solutions of Timoshenko (without crack) and solutions of Chen, Sosa and Eischen (with crack). For the cylindrical shell problem, the present results are also in very good agreement with solutions proposed by Scordelis and Folias.

Complex potential formalisms for bending of inhomogeneous monoclinic plates including transverse shear deformation

This paper develops complex potential formalisms for the solution of the bending problem of inhomogenoeus anisotropic plates, on the basis of the most commonly used refined plate theories. Being an initial step in that direction, it works out such formalisms only in connection with the bending problem of shear deformable homogeneous plates as well as plates having a special type of inhomogeneity along their thickness direction. The adopted type of inhomogeneity is however still general enough to include certain classes of plates made of functionally graded material as well as the classes of cross- and angle-ply symmetric laminates as particular cases. The basic formalism, similar to that developed by Stroh in plane strain elasticity, is detailed in relation with the equilibrium equations of a generalized plate theory that accounts for the effects of transverse shear deformation and includes conventional, refined theories as particular cases. Some interesting specializations, related to the most important of those conventional plate theories, are then presented and discussed separately. Hence, the outlined formalisms provide, for the first time in analytical form, the general solution of the partial differential equations associated with the most commonly used refined, elastic plate theories.

Modal approaches for the stochastic finite element analysis of structures with material and geometric uncertainties

This paper presents two efficient modal approaches as an alternative to direct formulations for the time-harmonic dynamic analysis of structures with random material and shape parameters. In both approaches, the structural eigenproblem is solved and complemented by a sensitivity analysis to the random parameters. The modal perturbation stochastic finite element method (SFEM) then condenses the response sensitivities to assess the response variability. The mixed perturbation/Monte-Carlo SFEM assesses the response statistics by sampling the structural eigenmodes according to the perturbation estimation of their probability distribution functions (PDFs). Geometric uncertainties are handled through an appropriate shape parameterisation and a shape design sensitivity analysis. Two numerical applications examine both approaches in terms of accuracy, variability level and computational requirements. The applications involve a plate bending problem with random Young modulus or edge length and a plate with a random flatness default. Particular observations related to the influence of the parameter PDFs in simulation-based methods are also provided.

Stroh-like formalism for the coupled stretching–bending analysis of composite laminates

Due to the two-dimensional nature of thin plates, the lamination theory considering the composite laminates with in-plane and plate bending problems coupling each other is treated in this paper by using complex variable formulation. By following the steps of Stroh formalism for two-dimensional linear anisotropic elasticity, a displacement complex variable formalism developed by the other researchers was introduced and re-derived in a different but more Stroh-like way. In addition, a brand-new mixed formalism (mixed use of displacements and stresses as basic functions) is established to compensate the displacement formalism. In order to transfer all the related formulae and mathematical techniques of the Stroh formalism to these two formalisms, the general solutions for the basic equations of lamination theory and their associated eigenrelations have been purposely arranged in the form of Stroh formalism. Moreover, by using the presently developed mixed formalism, the explicit expressions for the fundamental matrix and eigenvectors are obtained first time for the most general composite laminates. Furthermore, letting the coupling stiffness vanish, the formalism has been reduced to the case of symmetric laminates and checked by a recently developed Stroh-like formalism for the plate bending problems. The comparison between Stroh formalism for two-dimensional problem, Stroh-like formalism for plate bending problem, displacement formalism and mixed formalism is then made at the end of this paper, and through their connection some useful relations are obtained.

ON MORTAR-TYPE TRUNC ELEMENT METHOD FOR A PLATE BENDING PROBLEM

In this paper, a mortar-type TRUNC element method for the discretization of a plate bending problem is established. All the subdomains are triangulated independently. At the interfaces, three mortar conditions are given to ensure the convergence of the method: the first is a pointwise condition for the solution, while the other two are projection conditions for the normal and tangential derivatives of the solution. An error estimate in the energy norm is obtained under the regularity assumptions that the exact solution [Formula: see text] and f ∈ L2(Ω). The error order is optimal as for the usual TRUNC element method. Finally, some numerical results are provided in order to show the validness of our theoretical results.

The R-function method for the free vibration analysis of thin orthotropic plates of arbitrary shape

Free flexural vibrations of homogeneous, thin, orthotropic plates of an arbitrary shape with mixed boundary conditions are studied using the R-function method. The proposed method is based on the use of the R-function theory and variational methods. In contrast to the widely used methods of the network type (finite differences, finite element, and boundary element methods), in the R-function method all the geometric information given in the boundary value problem statement is represented in an analytical form. This allows one to seek a solution in a form of some formulas called a solution structure. These solution structures contain some indefinite functional components that can be determined by using any variational method. A method of constructing the solution structures satisfying the required mixed boundary conditions for eigenvalue plate bending problems is described. Numerical examples for the vibration analysis of orthotropic plates of complex geometry with mixed boundary conditions for illustrating the aforementioned R-function method and comparison against the other methods are made to demonstrate its merits.

Modified three-dimensional formulations of bending problems of homogeneous plates and beams under complex fixing conditions

Modified three-dimensional formulations of bending problems of homogeneous elastic plates and beams are considered. Modification of the known three-dimensional formulations reduces to using additional constraints imposed on displacement functions. An advantage of the formulations proposed is that complex fixing conditions of plates and beams can be taken into account.

Numerical Weight Function Method for Structural Analysis

The weight function method was originally derived for two- and three-dimensional crack problems to calculate stress intensity factors for arbitrary loading conditions. In the present paper the weight function method has been generalized to the response analysis of structures basing upon the Maxwell-Betti's reciprocal theorem, where the weight function method is formulated and applied to two-dimensional elasticity and plate bending problems with illustrative examples. The method is very useful for the analysis of structures subjected to a variety of loading conditions, because once the weight function is calculated the responses such as displacements and stresses may simply be calculated by the integration of the inner product of the universal weight function and a load vector.

３．計算力学 Ｌａｇｒａｎｇｅ多項式に基づく三次元弾性問題のＥＦＧＭ解析

The element-free Galerkin method.(EFGM) is known as an alternative numerical approach that does not require the finite element sub-division. The authors have been already presented an EFGM, which is based on the Lagrange polynomial, and applied to finite displacement analyses of plates and beams. In this paper, we reformulate our EFGM for the elastic analysis of 3-D bodies. As a numerical example we adopt a bending problem of thick plates. Especially, we investigate the affection of numerical parameters to the 3-D analysis. In order to evaluate the efficiency of the present EFGM, FEM and Fourier analyses are also performed. It follows from numerical results that the present EFGM gives us excellent approximation for the linear elastic analysis of 3-D bodies.

Plate Bending Problem for a Finite Doubly Connected Domain with a Partially Unknown Boundary

The problem of bending of an isotropic elastic plate is solved for a finite doubly connected domain whose external boundary is a convex polygon and internal boundary is a smooth closed contour. The bending problem is reduced to the analytical solution of the Riemann—Hilbert problem for a ring. The deflection of the median surface and the shape of the plate's internal boundary are found

Generalized mixed variational methods for Reissner plate and its applications

Based on the mechanism of shear locking phenomenon and potential functional of Reissner plate bending problem, the generalized mixed variational principle for Reissner plate analysis is presented by parameterized Lagrange multiplier method. The proposed variational functional contains splitting factors which are able to adjust the shear potential energy and shear complementary energy components in it. The generalized mixed finite element formulation of bilinear quardrilateral element for Reissner plate bending analysis is established in terms of the new variational principle. The stiffness of the finite element model can be changed by the alteration of the splitting factors. Thus both the free of shear locking and higher accuracy are obtained by the choice of appropriate splitting factors. The most important is that this paper gives one self-adaptative way to choose the splitting factors for thin and moderately thick plates. This results in the comparative order of magnitude between the bending stiffness and shear stiffness for the arbitrary thickness. In the application of two-by-two exact Gaussian integration scheme to the proposed mixed element model, numerical examples show that free of locking is obtained even in the thin plate limit and high accuracy is given for moderately thick plate.

Anisotropic elastic plates with holes/cracks/inclusions subjected to out-of-plane bending moments

The problems of anisotropic plates containing holes, cracks and/or inclusions have been studied extensively for two-dimensional deformations. Although the correspondence between the two-dimensional problems and the plate bending problems has been observed long time ago, without clarifying the involved mathematical details one still cannot get the solutions of the plate bending problems directly from the solutions of the corresponding two-dimensional problems. Based upon the correspondence relation, recently we developed a Stroh-like formalism for the bending theory of anisotropic plates. By this newly developed formalism, most of the relations for bending problems can be organized into the forms for two-dimensional problems. Thus, by using the Stroh-like formalism, the analytical solutions for problems of anisotropic plates with holes/cracks/inclusions subjected to out-of-plane bending moments can be obtained directly from the solutions of the corresponding two-dimensional problems. The stresses (or resultant bending moments) around the hole/crack/inclusion boundaries are also given explicitly in this paper. Since the explicit closed-form solutions for the present cases have not been found in the literature, comparison is made with some special cases of which the analytical solutions exist, which shows that our solutions are exact, simple and general.

Explicit Expressions of the Fundamental Elasticity Matrices of Stroh-Like Formalism for Symmetric/Unsymmetric Laminates

AbstractBased upon our recent development of Stroh-like forma lism for symmetric/unsymmetric laminates, most of the relations for bending problems can be organized into the forms of Stroh formalism for two-dimensional problems. Through the use of Stroh-like formalism, the fundamental elasticity matrices Ni, S, H and L appear frequently in the real form solutions of plate bending problems. Therefore, the determination of these matrices becomes important in the analysis of plate bending problems. In this paper, by following the approach for two-dimensional problems, we obtain the explicit expressions of the fundamental elasticity matrices for symmetric and unsymmetric laminates, which are all expressed in terms of the extensional, bending and coupling stiffnesses of the composite laminates.

Direct evaluation of singular integrals in boundary element analysis of thick plates

This work presents the derivation of the asymptotic expansions for two dimensional elasticity and plate bending problems fundamental solutions, applied to the direct evaluation of BEM singular integrals. Interesting conclusions arise from the resulting analytical expressions, regarding the actual order of singularity of the kernel functions. The expansions were tested for a number of plate bending benchmarks, showing good agreement to analytical solutions for thin and thick plates. The convergence behavior for constant, linear and quadratic elements is analyzed and compared with other integration techniques.

On the finite displacement analysis of quadrangular plates

In this paper, a numerical method for the linear and geometrically non-linear static analysis of thin plates is presented. The method begins with the elasticity equations pertaining to strain components, stresses, displacement components, strain energy and work due to externally applied loads. The plate geometry is defined by a quadrangular boundary with four straight edges and the natural coordinates in conjunction with the Cartesian coordinates are used to map the geometry. The matrix equation of equilibrium is derived using the work–energy principle with the displacement fields expressed by algebraic polynomials, the coefficients of which are then manipulated to satisfy the kinematic boundary conditions. To validate the results from the present method, square plates having all sides fully fixed and all sides simply supported without in-plane movement are analysed. Comparison is made for the uniformly loaded square plate with the results obtained by Levy who solved the non-linear plate bending problem using the Th.von Ka’ rma’ n’ s equations. Rhombic plates are examined and numerical results corresponding to these cases are presented in this paper. Very good comparison of the results regarding deflection and bending stresses with other sources available in the literature is found.

Error estimates for isoparametric mixed finite element solution of 4th order elliptic problems with variable coefficients

Considering the combined effect of boundary approximation and numerical integration, error estimates for the isoparametric mixed finite element solution of fourth order elliptic problems with variable coefficients in convex domains, which, in the particular case of aniso-/ortho-/ isotropic plate bending problems, gives a direct and simultaneous approximation to bending moment tensor field Ψ=(ψ ij ) 1≤i,j≤2 and displacement field `u', have been developed. Results of numerical experiments justify the theoretical results.

Element functions of discrete operator difference method

The discrete scheme called discrete operator difference for differential equations was given. Several difference elements for plate bending problems and plane problems were given. By investigating these elements, the ability of the discrete forms expressing to the element functions was talked about. In discrete operator difference method, the displacements of the elements can be reproduced exactly in the discrete forms whether the displacements are conforming or not. According to this point, discrete operator difference method is a method with good performance.

The least‐squares finite element method in elasticity. Part II: Bending of thin plates

AbstractA least‐squares finite element method (LSFEM) for bending problems of thin plates is developed. This LSFEM is based on the first‐order deflection‐slope‐moment‐shear force formulation. Four compatibility conditions are added into the first‐order system; thus, the method can accommodate all kinds of equal‐order interpolations. Numerical experiments on various examples show that the method achieves an optimal rate of convergence for all eight variables. Copyright © 2002 John Wiley & Sons, Ltd.

Minimum principles for the bending problem of elastic plates with voids

In the context of a linear theory for the bending of porous elastic plates, we explore minimum properties of suitable functionals for homogeneous and non-homogeneous mixed problems.