RBF-based meshless method for large deflection of thin plates

A simple, yet accurate, meshless method for the solution of thin plates undergoing large deflections is presented. The solution is based on the use of fifth order polynomial radial basis function to build an approximation for the solution of two coupled nonlinear differential equations governing the finite deflection of thin plates. The resulted nonlinear algebraic equations are solved using an incremental-iterative procedure. The accuracy and efficiency of the method is verified through several numerical examples.     

RBF meshless method for large deflection of thin plates with immovable edges

An efficient meshless formulation is presented for large deflection of thin plates with immovable edges. In this method, a fifth-order polynomial radial basis function (RBF) is used to approximate the solution variables. The governing equations are formulated in terms of the three displacement components u, v and w. The solution is obtained by satisfying three coupled partial differential equations and their boundary conditions inside the domain and over the boundary of the plate, respectively. The collocation procedure produces a system of coupled non-linear algebraic equations, which are solved using an incremental-iterative procedure. The numerical efficiency of the proposed method is illustrated through numerical examples.     

Analysis of large deflection of plates by the finite element method

The large deflection problem of a rectangular plate is analysed by using the finite element method and employing the iteration technique. In the present study, the stiffness matrix of a rectangular plate element for bending proposed by Greene is employed, and results of numerical examples duly justifies applicability of the present method.     

The analog equation method for large deflection analysis of membranes. A boundary-only solution

 In this paper the analog equation method (AEM) is applied to nonlinear analysis of elastic membranes with arbitrary shape. In this case the transverse deflections influence the inplane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear. The present formulation, being in terms of the three displacements components, permits the application of geometrical inplane boundary conditions. The membrane is prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of the membranes. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 21 November 2000     

The large deflection dynamic plastic response of rectangular plates

By assuming a kinematically admissible, time-dependent velocity field and considering the global equilibrium of all the forces acting on each rigid segment during large deflection of a plate, a complete theoretical investigation is undertaken herein to trace the large deflection dynamic plastic response (including the transient phase) of simply-supported or fully-clamped rectangular plates. This procedure deduces two modifying factors ƒ₁ and ƒ₂ which reflect the effect of the membrane forces and are employed to formulate the governing equations in the case of large deflections. The present prediction of the final plate deflection coincides excellently with the corresponding experimental results for deflections up to 5–10 times the plate thickness. This theory greatly improves the estimates based on the bending-only theory and provides a new way to trace the transient phase of dynamically loaded plates when the effect of membrane forces is significant.     

Elastic large deflection analysis of plates subjected to uniaxial thrust using meshfree Mindlin-Reissner formulation

A meshfree approach for plate buckling/post-buckling problems in the case of uniaxial thrust is presented. A geometrical nonlinear formulation is employed using reproducing kernel approximation and stabilized conforming nodal integration. The bending components are represented by Mindlin–Reissner plate theory. The formulation has a locking-free property in imposing the Kirchhoff mode reproducing condition. In addition, in-plane deformation components are approximated by reproducing kernels. The deformation components are coupled to solve the general plate bending problem with geometrical non-linearity. In buckling/post-buckling analysis of plates, the in-plane displacement of the edges in their perpendicular directions is assumed to be uniform by considering the continuity of plating, and periodic boundary conditions are considered in assuming the periodicity of structures. In such boundary condition enforcements, some node displacements/rotations should be synchronized with others. However, the enforcements introduce difficulties in the meshfree approach because the reproducing kernel function does not have the so-called Kronecker delta property. In this paper, the multiple point constraint technique is introduced to treat such boundary conditions as well as the essential boundary conditions. Numerical studies are performed to examine the accuracy of the multiple point constraint enforcements. As numerical examples, buckling/post-buckling analyses of a rectangular plate and stiffened plate structure are presented to validate the proposed approach.     

A dual reciprocity boundary element approach for the problems of large deflection of thin elastic plates

 A new numerical method, which is based on the dual reciprocity boundary element method, is developed for the large deflection of thin elastic plates whose behaviour is governed by von Kármán equations. In the proposed method, the nonlinear and coupled parts of von Kármán equations are transformed to a set of boundary integrals, and only are the boundary discretized into elements. Therefore, a `pure' boundary element approach for the problems of large deflection of thin elastic plates can be achieved. On the other hand, benefiting from the present method, the plate stresses can be calculated directly without integral and singularity. Several examples are given to demonstrate the efficiency and accuracy of the present method. Received 11 October 1999     

Large deflection initial failure analysis of angle-ply laminated plates

A large deflection initial failure analysis of square angle-ply laminated plates subjected to uniform pressure loading is described. Approximate numerical solutions of the Mindlin laminated plate equations are derived using finite differences in conjunction with the Dynamic Relaxation (DR) technique. The solutions are scaled to satisfy the Tsai-Hill lamina failure criterion. The effects of plate slenderness, in-plane edge restraint and lay-up on the initial failure pressure and associated plate centre deflection are quantified for ±45° laminated plates with simply supported and clamped edges. It is shown, particularly for thin plates, that in-plane edge restraint increases the initial failure pressure substantially, though changing the plate edge conditions from simply supported to clamped does not always increase the initial failure pressure.     

On the global large deflection and postbuckling of symmetric angle-ply laminated plates

This paper presents studies of the global nonlinear deflection and postbuckling behavior of symmetric angle-ply laminated plates. A new transformation between on-axis and off-axis modulus is established to make a link between the stiffness of angle-ply plates and the global constants in orthotropic transformed space. All the linear and nonlinear terms in the plate equations are properly nondimensionalized, such that the results in this paper are generic rather than specific. The results indicated that with increasing generalized rigidity ratio D^*, the deflection is reduced and the buckling coefficient is increased, and naturally the postbuckling deflection is decreased. The generalized Poisson's ratio only has a small effect on the problems. The information presented in the text should be helpful to whoever is seeking a better understanding of the correlation between composite material properties and nonlinear behavior of laminated plates.     

Dual boundary method for assembled plate structures undergoing large deflection

In this paper, the dual boundary element method is combined with a multiregion formulation to simulate plate assembly undergoing large deflection. The incremental load approach is used to treat the geometrical nonlinearity, and radial basis functions are used to approximate the derivatives of the large deflection terms. The dual reciprocity method is used to transfer all the domain integrals to the boundary. Once the solution at the boundary is obtained for the assembly, a J ?integral for large deflection is implemented to extract the fracture parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

The analog equation method for large deflection analysis of heterogeneous orthotropic membranes: a boundary-only solution

In this paper, the analog equation method (AEM) is applied to nonlinear analysis of heterogeneous orthotropic membranes with arbitrary shape. In this case, the transverse deflections influence the in-plane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear with variable coefficients. The present formulation, being in terms of the three displacement components, permits the application of geometrical in-plane boundary conditions. The membrane may be prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation, the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method, and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of heterogeneous anisotropic membranes. The method has all the advantages of the pure BEM, since the discretization and integration are limited only to the boundary.     

Failure analysis of composite laminates under large deflection

This paper presents the results of the application of a procedure, developed by Reddy & Reddy [ Composites Science and Technology 1992, 44, 227–255], for the evaluation of the first ply-failure load in multilayered composite plates. The procedure, which is based on the use of the finite element method (FEM) and which is suitable for the analysis of generally loaded plates, uses the non-linear von Karman formulation and, therefore, allows comparison of the failure loads in both the linear and the geometrically non-linear behaviour. Nevertheless, the use of the Newton-Raphson technique in searching the non-linear equilibrium points restricts its application to the case of plates without limit-point behaviour. The displacement model adopted in the FEM formulation is the traditional firstorder Reissner-Mindlin plate model that takes the shear deformation effect into consideration. Concerning the failure criteria, the analysis is based on a tensor polynomial criterion to which all other polynomial and independent criteria are brought back as particular cases. The study refers to the failure analysis of thin and thick plates under a uniformly distributed transverse load. Furthermore, a comparison of the failure criteria when the shear stresses are evaluated by means of the constitutive equations and by means of the local equilibrium equations is carried out. Finally, adopting a very simple degradation model of the mechanical properties to account for the stiffness decrease consequent to the failure, the qualitative behaviour of plates after the first non-catastrophic failure is also presented.     

Finite element large deflection analysis of shallow shells

A finite element formulation is presented to study the non-linear buckling of arbitrary shallow elastic thin shells with general boundary conditions and subjected to conservative pressure loading. Pre and post buckling behaviour of a large number of shallow and semi deep doubly curved shells is studied in detail. Unsymmetrical bifurcation paths of a shallow spherical shell subjected to uniform inward pressure are also investigated.     

A generalized collocation method for stress analysis of cracked plates

A generalized collocation method for stress analysis of cracked plates is developed in this paper. The Papkovich-Neuber general solution has been used instead of the Williams stress function. The expressions obtained are very suitable for computer programming. The trial functions can be used for force-boundary conditions as well as displacement-boundary conditions. In this method the conditions on all boundaries are satisfied approximately. As more trial functions are used, more complicated boundary value problems may be analysed. The accuracy and versatility of the method are demonstrated by several interesting examples.     

Techniques based on genetic algorithms for large deflection analysis of beams

A couple of non-convex search strategies, based on the genetic algorithm, are suggested and numerically explored in the context of large-deflection analysis of planar, elastic beams. The first of these strategies is based on the stationarity of the energy functional in the equilibrium state and may therefore be considered weak. The second approach, on the other hand, attempts to directly solve the governing differential equation within an optimisation framework and such a solution may be thought of as strong. Several numerical illustrations and verifications with ‘exact’ solutions, if available, are provided For communication     

A numerical green's function approach for boundary elements applied to fracture mechanics

The most accurate boundary element formulation to deal with fracture mechanics problems is obtained with the implementation of the associated Green's function acting as the fundamental solution. Consequently, the range of applications of this formulation is dependent on the availability of the appropriate Green's function for actual crack geometry. Analytical Green's functions have been presented for a few single crack configurations in 2-D applications and require complex variable theory. This work extends the applicability of the formulation through the introduction of efficient numerical means of computing the Green's function components for single or multiple crack problems, of general geometry, including the implementation to 3-D problems as a future development. Also, the approach uses real variables only and well-established boundary integral equations.