Problem Of Elastic Plate Research Articles

Solving third- and fourth-order partial differential equations using GFDM: application to solve problems of plates

This paper describes the generalized finite difference method to solve second-order partial differential equation systems and fourth-order partial differential equations. This method is applied to solve the problem of thin and thick elastic plates.

A meshless collocation method for the plane problems of functionally graded material beams and plates using the DRK interpolation

A meshless collocation method is developed for the static analysis of plane problems of functionally graded (FG) elastic beams and plates under transverse mechanical loads using the differential reproducing kernel (DRK) interpolation, in which the DRK interpolant is constructed by the randomly distributed nodes. A point collocation method based on this DRK interpolation is developed for the plane stress and strain problems of homogeneous and FG elastic beams and plates. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach with excellent accuracy and has a fast convergence rate.

Analytical studies on stress concentration due to a rectangular small hole in thin plate under bending loads

In general means, the stress concentration problem of elastic plate with a rectangular hole can be investigated by numerical methods, and only approximative results are derived. This paper deduces an analytical study of the stress concentration due to a rectangular hole in an elastic plate under bending loads. Base on classical elasticity theory and FEM applying the U-transformation technique, the uncoupled governing equations with 3-DOF are established, and the analytical displacement solutions of the finite element equations are derived in series form or double integral form. Therefore, the stress concentration factor can then be discussed easily and conveniently. For the plate subjected to unidirectional bending loads, the non-conforming plate bending element with four nodes and 12-DOF is taken as examples to demonstrate the application of the proposed method. The inner force distribution is obtained. The solutions are adequate for the condition when the hole is far away from the edges and the thin plate subjected to any transverse loadings.

Solvability of Dynamic Contact Problems for Elastic von Kármán Plates

The existence of solutions is proved for unilateral dynamic contact problems of elastic von Karman plates. Boundary conditions for a free and clamped plate are considered.

An efficient co‐rotational formulation for curved triangular shell element

AbstractA 6‐node curved triangular shell element formulation based on a co‐rotational framework is proposed to solve large‐displacement and large‐rotation problems, in which part of the rigid‐body translations and all rigid‐body rotations in the global co‐ordinate system are excluded in calculating the element strain energy. Thus, an element‐independent formulation is achieved. Besides three translational displacement variables, two components of the mid‐surface normal vector at each node are defined as vectorial rotational variables; these two additional variables render all nodal variables additive in an incremental solution procedure. To alleviate the membrane and shear locking phenomena, the membrane strains and the out‐of‐plane shear strains are replaced with assumed strains in calculating the element strain energy. The strategy used in the mixed interpolation of tensorial components approach is employed in defining the assumed strains. The internal force vector and the element tangent stiffness matrix are obtained from calculating directly the first derivative and second derivative of the element strain energy with respect to the nodal variables, respectively. Different from most other existing co‐rotational element formulations, all nodal variables in the present curved triangular shell formulation are commutative in calculating the second derivative of the strain energy; as a result, the element tangent stiffness matrix is symmetric and is updated by using the total values of the nodal variables in an incremental solution procedure. Such update procedure is advantageous in solving dynamic problems. Finally, several elastic plate and shell problems are solved to demonstrate the reliability, efficiency, and convergence of the present formulation. Copyright © 2007 John Wiley & Sons, Ltd.

Compactly supported non-tensor product form two-dimension wavelet finite element

Some theorems of compactly supported non-tensor product form two-dimension Daubechies wavelet were analysed carefully. Compactly supported non-tensor product form two-dimension wavelet was constructed, then non-tensor product form two dimension wavelet finite element was used to solve the deflection problem of elastic thin plate. The error order was researched. A numerical example was given at last.

Optimal annular plates with respect to their stability under thermal loadings

In the paper the uni- and multimodal optimization problems of elastic annular plates with respect to their stability under thermal loading are investigated. We look for such a distribution of a thickness of a plate (circularly symmetric), which leads to the maximum increment |ΔT| of temperature, causing buckling of the optimal structure under the equality constraint of a constant volume of the material and under inequality constraints imposed on the minimal and maximal values of a plate thickness. The optimal solutions for different modes of supports and different ratio of the inner and outer radius are looked for using the method of moving asymptotes and the simulated annealing.

An asymptotic analysis of initial-value problems for thin elastic plates

An asymptotic technique is developed to analyse initial-value problems in application to the three-dimensional theory of thin elastic plates. Various sets of long-wave initial data are considered, with arbitrary distribution along the plate thickness. To account for the initial data, composite asymptotic expansions are employed, utilizing both two-dimensional low-frequency and high-frequency models. The former correspond to the classical theories of plate bending and extension, and their refinements, whereas the latter are associated with the long-wave motions occurring in the vicinities of thickness stretch and shear resonance frequencies. Six cases of iteration process are revealed depending on the symmetry of the initial data and their thickness variation. For each case, approximate two-dimensional initial conditions are derived, including higher-order corrections for the two-dimensional low-frequency refined plate theories. The validity of the proposed approach is justified by comparison with the exact solution of the model plane problem for initial data with uniform distribution along the thickness and sinusoidal distribution along the mid-plane. The methodology of the paper has potential for more general initial-value problems specified over a narrow domain, including many of those commonly met in physical applied mathematics.

Non-classical dual methods in equilibrium problems for thin elastic plates

A dual functional is constructed for the Dirichlet problem for plates with transverse shear deformation, which makes it possible to obtain bilateral error estimates for approximate solutions. In the absence of an elastic foundation, the classical dual functional is maximized in spaces of functions that satisfy certain differential restrictions. The non-classical dual functional proposed here is maximized in spaces of functions with no such restrictions, regardless of whether or not an elastic foundation is present. A numerical example is given to illustrate the efficiency of this dual method.

Mixed Formulations of Bending Problems for Homogeneous Elastic Plates and Beams

Mixed formulations of bending problems for homogeneous plates (beams) are proposed, whose essence is that the deformation of a plate (beam) near its fixed boundary is described by the three‐dimensional elasticity equations, and the remaining part by the conventional equations of plate (beam) bending. At the interface between these regions, the solutions of these equations are joined. The mixed formulation allows one to describe the three‐dimensional stress state in the neighborhood of the fixed boundaries of plates (beams) and take into account the complex nature of the fixing conditions. Finite‐element implementation is more efficient for the mixed formulations of plate (beam) bending problems than for the well‐known three‐dimensional formulations.

On modelling of dynamic problems for elastic periodic plates

AbstractThe subject of analysis are thin elastic plates with a periodic structure along the midplane. The aim of this contribution is to formulate a new effective 2D‐model of the plate, i.e. a 2D‐model represented by equations with constant coefficients. It is assumed that the period lengths are of an order of the mean plate thickness.

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.

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.

Boundary Integral Equations in Dynamic Problems for Elastic Plates

Initial-boundary value problems with Dirichlet and Neumann conditions arising in the theory of bending of plates with transverse shear deformation are reduced to time-dependent boundary integral equations by means of layer potentials. The solvability of these equations is then investigated in Sobolev-type spaces.

Γ-Convergence of external approximations in boundary value problems involving the bi-Laplacian

In addition to the various uses it was introduced for, the theory of Γ-convergence offers a rather natural setting for discussing and developing nonorthodox approximation methods for variational problems. For certain boundary value problems involving the bi-Laplacian, sequences of discrete functionals are here defined and are shown to Γ-converge to the corresponding functionals of the continuous problems. The minimizers of the discrete functionals provide converging approximations to the solution of the limit problem in question. Thus, we obtain approximation schemes that are nonconforming, but direct, and that can be treated by current algorithms for symmetric and positive definite functionals. The class of problems considered in this paper includes the Stokes problem in fluid dynamics, the loading problem of 2-D-isotropic elastostatics, and some boundary value problems of the Kirchhoff–Love theory of plates. Also discussed is an extension of the discretization method that seems suitable for treating more general boundary value problems of elastic plates, but whose convergence is conditional to a conjecture that remains to be proved. A relevant application to the so-called Babuška paradox is presented.

Unilateral Problems for Nonlinearly Elastic Plates and Shallow Shells

Using the topological degree for pseudo-monotone operators of type (S+), we establish a general existence result for variational inequalities of von Karman type, which model unilateral problems for nonlinearly elastic plates. Then, we give a reduced operatorial form of Marguerre von Karman equations for nonlinearly elastic shallow shells and get a new existence result for this model.

Combined Displacement-Iraction Boundary Wlue Problems for Elastic Plates

The solvability is investigated of the equations of bending of plates when either the first two displacement components and the transverse shear force or the bending and twisting moments and the deflection are prescribed on the boundary of the middle section domain. These problems are then reduced to boundary integral equations, whose solutions are sought in spaces of distributions.

A boundary element method applied to the elastostatic bending problem of beam-stiffened plates

The present paper proposes a basic formulation for the static bending problem of beam-stiffened elastic plates. This problem has been so far analyzed using the Timoshenko theory in which the equivalent shear force and bending moments are assumed to act on the beam stiffener. Since fourth-order derivatives of unknown displacements are included in the formulation, in its numerical implementation fourth-order polynomials must be used as the interpolation functions. In this paper, the interactive forces and moments between the plate and the stiffener are treated as line distributed unknown loads. In the numerical implementation of the formulation, these forces can be approximated using a suitable family of interpolation functions. The formulation is presented in detail and a computer code is developed. The numerical results obtained by the computer code are discussed, whereby the usefulness of the proposed solution procedure is demonstrated.

Numerical instability of magnetic damping problem of elastic plate

Numerical instability occurs in an analysis of a vibration with magnetic damping, or an electromagnetic and structural coupled problem. In this paper, the numerical instability of the coupled analysis is examined by the finite element in time. It is confirmed that the simultaneous method is unconditionally stable even if the magnetic field and the time increment are large. For the staggered method, we obtain the conditions where the numerical instability occurs.

Boundary Element Method Applied to the Elastostatic Bending Problem of Stiffened Plates.

The present paper proposes a formlation simpler than previous investigations for the static bending problem of beam-stiffened elastic plates. This problem has been analyzed so far using the Timoshenko thin plate theory in which the equivalent shear force and bending moments are assumed to act to the beam stiffener. Since they include at most fourth-order derivatives of unknown displacements, at least fourth-order polynomials must be used as the interpolation functions in numerical implementation of the formulation. In this paper, the forces and moments acting between the plate and the stiffener are treated as line distributed unknown loads. The numerical implementation of the formulation, therefore, these forces can be interpolated using any type of interpolation functions. In this study, however, the resulting set of integral equations are discretized using constant elements for interpolation of these forces. The numerical results obtained by the computer code developed in this study are discussed, whereby the versatility of the proposed analysis method is demonstrated.