Bending Of Thin Elastic Plates Research Articles

Space-time generalized finite difference method for solving the thin elastic plate bending under dynamic loading

In this paper, the space-time generalized finite difference method (ST-GFDM) with supplementary nodes is applied to solve the thin elastic plate bending under dynamic loading. The proposed method treats the temporal dimension as another spatial dimension, thus the d-dimensional problem in space can be viewed as a new (d+1)-dimensional problem in the space-time domain. This method effectively reduces the complexity of numerical solution of the problem by simultaneously discretizing the temporal and spatial dimensions in the space-time domain. It avoids the temporal-difference measurement while maintaining all the advantages of the GFDM. The ST-GFDM leads a sparse linear system due to its local meshless feature, which is suitable for solving large-scale and long-time bending problems. Moreover, the supplementary nodes on the boundary are introduced to make the system well-determined. Several numerical examples are illustrated to demonstrate the accuracy and stability of the proposed method.

A GFDM with supplementary nodes for thin elastic plate bending analysis under dynamic loading

This work proposes a meshless generalized finite difference method (GFDM) with supplementary nodes for the solution of thin elastic plate bending under dynamic loading. The first- and second-order time derivatives in equilibrium equation are firstly discretized with the Houbolt method. The numerical solution of the resulting spatial problem at each time step is then calculated by the GFDM. The supplementary nodes on the boundary of the problem domain are introduced in the GFDM to yield the well-determined linear system of equations. Numerical example is provided to verify the accuracy and the stability of the developed approach, and these preliminary results reveal the potential of the method.

On the Boundary Value Problems of Bending of Thin Elastic Plates With Surface Effects

Abstract We modify classical thin plate theory by incorporating surface effects via the Gurtin–Murdoch surface model to accommodate the mechanical behavior of thin plates at the nanoscale. We formulate the corresponding Dirichlet and Neumann boundary value problems and establish uniqueness results in the appropriate function spaces. In addition, we obtain the fundamental solution of the governing system of equations, which is central to further studies concerning well-posedness analysis of the model by the boundary integral equation method. Finally, we validate our model by comparison with results in the existing literature.

Reduced-order strategy for meshless solution of plate bending problems with the generalized finite difference method

Abstract This paper presents some recent advances on the numerical solution of the classical Germain-Lagrange equation for plate bending of thin elastic plates. A meshless strategy using the Generalized Finite Difference Method (GFDM) is proposed upon substitution of the original fourth-order differential equation by a system composed of two second-order partial differential equations. Mixed boundary conditions, variable nodal density and curved contours are some of the explored aspects. Simulations using very dense clouds and parallel processing scheme for efficient neighbor selection are also presented. Numerical experiments are performed for arbitrary plates and compared with analytical and Finite Element Method solutions. Finally, an overview of the procedure is presented, including a discussion of some future development.

A Nonconforming Finite Element Approximation for the von Karman equations

In this paper, a nonconforming finite element method has been proposed and analyzed for the von Karman equations that describe bending of thin elastic plates. Optimal order error estimates in broken energy and H 1 norms are derived under minimal regularity assumptions. Numerical results that justify the theoretical results are presented.

Conforming finite element methods for the von Kármán equations

In this paper, we consider canonical von Karman equations that describe the bending of thin elastic plates defined on polygonal domains. A conforming finite element method is employed to approximate the displacement and Airy stress functions. Optimal order error estimates in energy, H1 and L2 norms are deduced. The results of numerical experiments confirm the theoretical results obtained.

Elastodynamics of FGM plates by mesh-free method

The paper deals with transient analysis of homogeneous as well as FGM (functionally graded material) thin and/or thick plates subjected to transversal dynamic loading. The Young modulus and mass density are allowed to be continuously variable through the thickness of the plate. The unified formulation is developed for plate bending problems including three various theories such as the classical Kirchhoff–Love theory for bending of thin elastic plates, the 1st and 3rd order shear deformations plate theories. Switching among these three theories is controlled by two key factors. From the derived equations of motion, the coupling between the bending and in-plane deformation modes is discovered in FGM plates with specification of the necessary conditions. For the numerical implementation, the strong formulation is proposed with meshless approximation of spatial variations of field variables. The semi-discretized equations of motion yield a system the ordinary differential equations which can be solved by standard time stepping techniques. The great attention is paid to the numerical study of coupling effects in FGM plates subjected to transversal Heaviside impact loading and/or Heaviside pulse loading. The achieved numerical results are thoroughly discussed and interpreted.

Conforming and nonconforming finite element methods for canonical von Kármán equations

The canonical von Karman equations which describe bending of thin elastic plates defined on polygonal domains are considered. Conforming and nonconforming finite element methods are employed to approximate the displacement and Airy stress functions. The results of the numerical experiments justify the theoretical results.

A fictitious domain method using equilibrated basis functions for harmonic and bi-harmonic problems in physics

In this paper we present a new meshless method to solve well-known problems in physics and engineering with either constant or variable material properties in 2D space. Harmonic and bi-harmonic problems are considered in this paper. The method constructs a set of basis functions, called Equilibrated Basis Functions (EqBFs), through a weighted residual integration over a fictitious domain embedding the main one. The bases can satisfy the governing partial differential equations approximately. Chebyshev polynomials of the first kind are employed to construct the EqBFs and exponential functions are used as the weights in the integrals. The parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1D ones over a normalized intervals. This reduces the computational efforts significantly. Either the EqBFs or the results of the integration process may be stored for further use. The validity of the results is examined through an extended patch test. A set of physical sample problems with variable/non-variable material properties; as the potential flow over a cylinder, steady-state heat conduction problems in an anisotropic inhomogeneous functionally graded material, potential and stokes flow through a channel of finite width obstructed by periodic cylinders, and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method. As a preliminary study, we show that the method may effectively be used in a domain decomposition approach.

Lateral variations of effective elastic thickness of the subducting Philippine Sea plate along the Nankai trough

We investigated lateral variations of effective elastic thickness of the subducting Philippine Sea plate along the Nankai trough. For this purpose, we applied two-dimensional thin elastic plate bending theory to several profiles almost perpendicular to the trough axis, correcting the effect of thick marine sediments in the Shikoku Basin. The 15 topographic profiles were selected from 61 profiles and were fit to respective theoretical deflection curves by minimization of the L1 norm. Classifying the Shikoku Basin into three regions, namely, the western, the middle, and the eastern regions, we obtained the results that the effective elastic plate thickness tends to decrease gradually from the east of the Ryukyu-Palau ridge to the Kinan seamount chain. This is consistent with heat flow data, seafloor age, formation process of the Shikoku Basin, length of the slab subducted from the Nankai trough, and spatial distribution of subcrustal earthquakes associated with subduction of the Philippine Sea plate.

Equivalent Boundary Integral Equations with indirect unknowns for thin elastic plate bending theory

Equivalent Boundary Integral Equations (EBIE) with indirect unknowns for thin elastic plate bending theory, which is equivalent to the original boundary value problem, is established rigorously by mathematical technique of non-analytic continuation and is fully proved by means of the variational principle. The previous three kinds of boundary integral equations with indirect unknowns are discussed thoroughly and it is shown that all previous results are not EBIE.

Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals

The typical Boundary Element Method (BEM) for fourth-order problems, like bending of thin elastic plates, is based on two coupled boundary integral equations, one strongly singular and the other hypersingular. In this paper all singular integrals are evaluated directly, extending a general method formerly proposed for second-order problems. Actually, the direct method for the evaluation of singular integrals is completely revised and presented in an alternative way. All aspects are dealt with in detail and full generality, including the evaluation of free-term coe cients. Numerical tests and comparisons with other regularization techniques show that the direct evaluation of singular integrals is easy to implement and leads to very accurate results. Copyright ? 1999 John Wiley & Sons, Ltd.

Elastodynamic Thin Plate Bending Analysis by Boundary Element Method with Laplace Transform

The boundary element method(BEM) is developed for the dynamic analysis of thin elastic plate bending problems with arbitrary boundary conditions. The formulation employs Laplace-transform technique, where the boundary integral equations are obtained on the Laplace transformed domain with the fundamental solutions derived from Kelvin's functions. The accuracy of the numerical results mainly depends on those of numerical estimation of the singular integral derived from the static term of the fundamental solutions. In the present paper, an BEM formulation based on a single boundary integral equation of the deflection, which employes a source point on both the boundary and outer region, is discussed in detail. A non-singular boundary integral equation is introduced on the transformed domain, which is obtained by superposition of the analyzed field and the referenced field with a uniform gradient of deflection. Numerical results obtained by the proposed method are compared with the analytical solutions and the other numerical solutions by means of several numerical examples. These examples also serve to illustrate the use of the proposed method.

On the bending of plates with transverse shear deformation and mixed periodic boundary conditions

AbstractThe method of matched asymptotic expansions is used to find a homogenized problem whose solution is an approximation to the solution of a mixed periodic boundary value problem in the theory of bending of thin elastic plates. A critical size for the fixed parts of the boundary is found such that the boundary condition of the homogenized problem is an intermediate case between that for the clamped edge plate and that for the free boundary plate.

Highly Accurate Analysis for Thin Elastic Plate Bending Problem by Boundary Element Method. 2nd Report, Application of Rigid Rotation Mode for Single Integral Equation Method.

Generally, the boundary element method (BEM) for thin elastic plate bending analysis uses two different types of integral equations for displacement and rotation. However, there is a different type of approach, based on a direct method, using only integral equations for displacement (single integral equation method). In this method, hyper-singularities do not arise, because there is no need to use integral equations for rotation. In the present paper, the regularizing operation using the rigid rotation mode, as suggested in the first report, is applied to the single integral equation method. Through the numerical results of a square plate under several boundary conditions, it is shown that results obtained using this method are more accurate in comparison with the without regularization.

Some nonsingular direct formulations ofboundary integral equations for thin elastic plate bending analysis

The boundary integral equations conventionally used for thin elastic plate bending analyses involve the Cauchy principal value integrals. Special care must be taken in evaluating them numerically when the boundary is discretized with higher order elements. In this paper boundary integral equations based on direct formulations are regularized by using the subtracting and adding-back technique. The boundary integral equations obtained both for deflection and rotation are nonsingular, and all the integrals in their discretized forms can be integrated accurately by using the standard Gaussian quadrature formula. Numerical implementation of the resulting regularized integral equations is presented, and the effectiveness of the proposed method is demonstrated with numerical examples.

Highly Accurate Analysis for Thin Elastic Plate Bending Problem by Boundary Element Method. 1st Report. Regularization for Hyper-Singular Integral by Rigid Rotation Mode.

The boundary element method (BEM) for thin elastic plate bending analysis uses two different integral equations for the displacement and the slope. It is difficult to integrate the integral equation for rotation accurately, because the equation has hyper-singularity. In the present paper, these boundary integral equations are nomalized by the superposition of the rigid rotation mode. Through some numerical results of square plates under several boundary conditions, it is shown that unknown nodal values along the boundary are more accurately obtained that those of the usual methods, and the accuracy of deflections, moments and shear forces at internal points has been clearly improved.

Application of complex path-independent integrals to problems of bending of thin elastic plates

Complex path-independent integrals have been already widely applied to problems of plane and antiplane elasticity for the determination of a variety of quantities of interest including stress intensity factors, loading intensities and the positions of geometrical characteristic lengths of singularities in the elastic field (like cracks, holes and inclusions). In this paper, we show that the same results apply also to the case of problems of thin isotropic elastic plates under bending where the complex-variable formulation is also valid. We make reference to the experimental methods which are appropriate for these integrals in an engineering environment and, finally, we apply this approach to the location of a circular hole in the problem of bending of a thin plate. Numerical results are also presented.

Regularized Boundary Integral Equation Formulations for Thin Elastic Plate Bending Analysis.

The boundary integral equations conventionally used for thin elastic plate bending analysis have the singularities of the Cauchy principal value order for which special care must be taken in numerical evaluation when the boundary is discretized by using the higher-order elements. In the present paper, the boundary integral equations are regularized up to an integrable order by using the subtracting and adding back technique. The obtained boundary integral equations both for deflection and rotation are weakly singular, and their discretized forms can be integrated accurately by the standard Gaussian quadrature formula. Numerical implementation of the resulting regularized integral equations is presented, and effectiveness of the proposed method is discussed through some numerical demonstrations.

Application of Boundary Integral Method to Continuous Plate Structures

Ship hulls are composed of complex plate structures. Such structures can be analyzed by analytical as well as numerical methods. Here it is demonstrated that boundary integrals can be used to analyze such structures by a seminumerical method. Boundary integrals have been used for analyzing single plate bending problems, and only recently have attempts been made to extend their use to complex plate structures. In this paper the Direct Boundary Integral Method for bending of thin elastic plates is extended to analysis of the continuous plate structures. The concepts of compatibility and equilibrium conditions along the common boundary have been imposed in conjunction with a boundary element formulation for each panel. Several examples of loading conditions are presented and compared with analytical solutions. The excellent agreement achieved illustrates the effectiveness of the extended formulation.