Bending Of Thin Plates Research Articles

Trefftz Finite Element Method and Its Applications

This paper presents an overview of the Trefftz finite element and its application in various engineering problems. Basic concepts of the Trefftz method are discussed, such as T-complete functions, special purpose elements, modified variational functionals, rank conditions, intraelement fields, and frame fields. The hybrid-Trefftz finite element formulation and numerical solutions of potential flow problems, plane elasticity, linear thin and thick plate bending, transient heat conduction, and geometrically nonlinear plate bending are described. Formulations for all cases are derived by means of a modified variational functional and T-complete solutions. In the case of geometrically nonlinear plate bending, exact solutions of the Lamé-Navier equations are used for the in-plane intraelement displacement field, and an incremental form of the basic equations is adopted. Generation of elemental stiffness equations from the modified variational principle is also discussed. Some typical numerical results are presented to show the application of the finite element approach. Finally, a brief summary of the approach is provided and future trends in this field are identified. There are 151 references cited in this revised article.

Thermal stresses in microstretch elastic plates

We consider the linear theory of homogeneous and isotropic thermo-microstretch elastic solids. First, we present the basic equations which characterize the bending of thin plates. Then we establish a uniqueness result with no definiteness assumption on constitutive coefficients. Existence of solutions is proved under assumption that the internal energy density is positive definite. In the equilibrium theory we present a theorem of minimum potential energy. Finally, the effects of a concentrated heat source in an infinite plate are studied.

Structure and properties of the fundamental elastic plate matrix

AbstractThis work presents further development of the octet formalism established by the authors for the classical Kirchhoff anisotropic plate theory. The structure of the fundamental elastic plate matrix is fully explored and the explicit expression is provided. The matrices N2 and N3 are proved to be positive semi‐definite. Thus, H and L are positive definite. Further studies are concerned with a rotated coordinate system. The transform relation between the eigenvectors in the original and the rotated coordinate system is given. The fundamental elastic plate matrix associated with the eigenrelation referring to the dual coordinate systems, NΘ, is studied. The major properties that hold in the Stroh sextic formalism for generalized plane strain problems are also valid in the octet formalism for thin plate bending problems. In particular, we generalize a property in Stroh's formalism for any non‐semisimple matrix NΘ. We show a new property in the octet formalism. The non‐semisimple cases of NΘ are discussed. Finally, we make it transparent that the mixed/hybrid formalism of others is precisely one of sixteen permuted forms of the octet formalism.

A new orthogonality relationship for orthotropic thin plate theory and its variational principle

The thought how dual vectors are constructed in a new orthogonality relationship for theory of elasticity is generalized into orthotropic thin plate bending problems by using the analogy theory between plane elasticity problems and plate bending problems. Dual differential equations are directly obtained by using a mixed variables method. A dual differential matrix to be derived possesses a peculiarity of which principal diagonal sub-matrixes are zero matrixes. Two independently and symmetrically orthogonality sub-relatioships are discovered. By using the integral form for elastic bending theory of orthotropic thin plate the orthogonality relationship is demonstrated. By selecting felicitous dual vectors a new orthogonality relationship for theory of elasticity can be generalized into elastic bending theory of orthotropic thin plate. By using the integral form a variational principle which is relative to differential form and a whole function expression are proposed.

Solving high-order partial differential equations with indirect radial basis function networks

This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high-order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high-order derivatives. The proposed RBFN method is verified successfully through the solution of thin-plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number.

Mixed plate bending elements based on least‐squares formulation

AbstractA finite element formulation for the bending of thin and thick plates based on least‐squares variational principles is presented. Finite element models for both the classical plate theory and the first‐order shear deformation plate theory (also known as the Kirchhoff and Mindlin plate theories, respectively) are considered. High‐order nodal expansions are used to construct the discrete finite element model based on the least‐squares formulation. Exponentially fast decay of the least‐squares functional, which is constructed using the L2 norms of the equations residuals, is verified for increasing order of the nodal expansions. Numerical examples for the bending of circular, rectangular and skew plates with various boundary conditions and plate thickness are presented to demonstrate the predictive capability and robustness of the new plate bending elements. Plate bending elements based on this formulation are shown to be insensitive to both shear‐locking and geometric distortions. Copyright © 2004 John Wiley & Sons, Ltd.

Anisotropic thin plate bending problems by Trefftz boundary collocation method

Trefftz boundary collocation method is used to analyse the bending of anisotropic thin plates. Compared with the existing collocation methods employing complicated fundamental solutions in which the source points are located outside the plate, boundary collocation method presented in this article is relatively simple. It requires only the general solution of anisotropic thin plate bending problems and no singular integrals appear in the formulation. The collocation points are located along the plate edges. The implementation of the related computer codes is fairly straightforward. Numerical results are generally more accurate compared with existing solutions.

Analysis of general shaped thin plates by the moving least-squares differential quadrature method

The mesh-free moving least-squares differential quadrature (MLSDQ) method is proposed for solving the fourth-order, partial differential equation governing the bending of thin plates according to classical plate theory. The deflections of an arbitrary shaped plate are expressed in terms of the MLS approximation. The weighting coefficients used in the MLSDQ approximation are calculated through a fast computation of the shape functions and their derivatives. The discrete multiple boundary conditions and governing equations are solved by a least-squares approximation. Numerical examples are presented to illustrate the accuracy, stability and convergence of the present method. Effects of support size, order of completeness and node irregularity on the numerical accuracy are carefully investigated.

315 バックリングモードにおけるレーザ曲げ(OS13 レーザ応用加工)

Many studies on laser forming of thin plates have been done; however, few studies based on the buckling-mechanism have been made. Although characteristics of laser forming due to buckling mechanism have not been made clear, they are important for practical use because the forming of thin foils or plates is caused due to the buckling. In the present study, the bending of thin plates is made using 2 beams or line beam! bending direction is controlled.

Application of the Improved Immune Algorithm to Structural Design Support System

Genetic algorithms (GAs), based on a multipoint search method and a crossover operation, are useful search procedures for combinatorial optimization problems. They also are applied to many kinds of practical optimization problem. However, in general, the GAs have a tendency to decrease rapidly of population diversities in processes of searching. In order to address this drawback, some researchers have proposed new algorithms for maintaining the population diversity. On the other hand, immune algorithms (IAs) are optimization techniques that imitate immune systems in an organism. The IAs are able to obtain a multiple quasi-optimum solution while maintaining the population diversity compared with GAs. In this paper, in order to consider the application of the IAs to optimum structural design problems, we developed a useful design support system for reinforced concrete (RC) slabs by combining a three-dimensional (3D) nonlinear dynamic finite element method (FEM) and an improved immune algorithm (IA). For the analysis of impact failure behavior of RC slabs, layered nonlinear FEM models with thin plate bending elements, and 3D elastoplastic FEM models with eight-node isoparametric hexahedral elements were used. First, an existing IA was modified to make it suitable for RC slab design, and both genetic and immune algorithms were applied to optimization problems with multiple optimum solutions. This modified IA was also applied to quasi-optimization problems with multiple maximum values. By comparing the above two approaches, the usefulness of the IA for design problems with discrete parameters was verified. Next, indices for evaluating impact resistance were discussed, and design simulation was conducted, based on these indices, by combining the layered nonlinear FEM and the IA. From results obtained, the accuracy and usefulness of the IA-based design support system were discussed. For obtaining a more practical design support system, shear reinforcement will be taken into account by combining the IA with the 3D elastoplastic FEM. Furthermore, the effectiveness and future potential of the design support system were confirmed by investigating an effect of introducing a database of design plans.

Analytical solution of a plate bending problem using symbolic algebra

AbstractThe estimation of critical voltage in a capacitive micro‐switch leads to a thin plate bending problem. This paper deals with the correct mathematical formulation taking into account deflection‐dependent electrostatic force. The relative simple geometry allows the reduction of the structure to a beam problem. With the use of Ritz‐method and the variational symbolic program Mathematica analytical closed form solutions have been obtained. Polynomial and trigonometric trial functions have been used to determine the deflection of the plate. It has been proved, that the critical collapse voltage is much overestimated, if constant electrostatic forces are assumed.

Multiple reciprocity method with two series of sequences of high-order fundamental solution for thin plate bending

The boundary value problem of plate bending problem on two-parameter foundation was discussed. Using two series of the high-order fundamental solution sequences, namely, the fundamental solution sequences for the multi-harmonic operator and Laplace operator, applying the multiple reciprocity method (MRM), the MRM boundary integral equation for plate bending problem was constructed. It proves that the boundary integral equation derived from MRM is essentially identical to the conventional boundary integral equation. Hence the convergence analysis of MRM for plate bending problem can be obtained by the error estimation for the conventional boundary integral equation. In addition, this method can extend to the case of more series of the high-order fundamental solution sequences.

Refined triangular discrete Mindlin flat shell elements

In this paper flat shell elements are formed by the assemblage of discrete Mindlin plate elements RDKTM and either the constant strain membrane element CST or the Allman’s membrane element with drilling degrees of freedom LST. The element RDKTM is a robust Mindlin plate element, which can perform uniformly thick and thin plate bending analysis. It also passes the patch test for thin plate bending, and its convergence for very thin plates can be ensured theoretically. The singularity of the stiffness matrix and membrane locking are studied for the present elements. Numerical examples are presented to show that the present models indeed possess properties of simple formulations, high accuracy for thin and thick shells, and it is free from shear locking for thin plate/shell analysis.

An exact solution for the bending of thin rectangular plates with uniform, linear, and quadratic thickness variations

In this paper, the analysis of the titled problem is based on classical thin-plate theory, and its numerical solution is carried out by using the small parameter method and Lévy-type approach. The thin rectangular plate considered herein is simply supported on two opposite edges. The boundary conditions at the other two edges may be quite general, and between these two edges the plate may have varying thickness. Closed-form solutions have been developed for the static response of isotropic rectangular plates with non-uniform thickness variation and subjected to arbitrary loading. The accuracy of the present model is demonstrated via problems for which the exact solutions and numerical results are available, and results are also compared with those obtained by using the finite-difference method.

403 厚板構造物の固体伝搬音に関する研究

Regarding the prediction of structure-borne sound in large structures consisting of thick plates, there are some difficulties in applying the conventional SEA based on thin plate bending and in-plane waves. This paper examines the applicability of wave intensity analysis to the prediction of the vibration energy level in an L-shaped structure assembled by thick plates. The bending wave in plates is thus expressed by Mindlin's theory. The new model is introduced to improve the accuracy of wave transmission coefficients for line junctions between thick plates. As a result of comparison of the present method with FEM, reasonable results are obtained for the vibration energy level in the L-shaped structure, within approximately 2dB up to 2kHz.

1452 オートローテーション状態における矩形薄板の飛行性能

Object of this study is to realize a flight of micro flyer that is bio mimetic flyer. We tried to test the performance of an autorotating thin plate. One experiment is a free flight of a rectangular paper. We measured the flight time and landing position from a launching. It is found that the thin bending plate with AR=7 is the most. The other experiment is carried out under micro gravity. We observed the takeoff of the thin aluminum plate. The lift force was estimated from the motion pictures.

Selected topics in the history of the two-dimensional biharmonic problem

This review article gives a historical overview of some topics related to the classical 2D biharmonic problem. This problem arises in many physical studies concerning bending of clamped thin elastic isotropic plates, equilibrium of an elastic body under conditions of plane strain or plane stress, or creeping flow of a viscous incompressible fluid. The object of this paper is both to elucidate some interesting points related to the history of the problem and to give an overview of some analytical approaches to its solution. This review article contains 758 references.

Fundamental Solutions in Problems of Bending of Anisotropic Plates

Fundamental solutions based on the engineering theory of bending of thin anisotropic plates (Kirchhoff–Love hypotheses) are constructed for anisotropic, in particular, orthotropic plates of canonical plan form (half-plane, quadrant, band, half-band, rectangle, and unbounded plate with an elliptic hole)

A BENDING THEORY OF POROUS THERMOELASTIC PLATES

This article is concerned with a plate theory for thermoelastic materials with voids. We establish the field equations for the bending of thermoelastic thin plates made from an isotropic and homogeneous material in the context of the dynamic linear theory. Then, we present a uniqueness theorem with no definiteness assumptions on the constitutive coefficients. Finally, by means of the logarithmic convexity method, we obtain continuous dependence results and study the instability of the solutions.

ANALYSIS OF ELASTODYNAMIC DEFORMATIONS NEAR A CRACK-NOTCH TIP BY THE MESHLESS LOCAL PETROV–GALERKIN (MLPG) METHOD

The Meshless Local Petrov-Galerkin (MLPG) method is used to analyze transient deforma- tions near either a crack or a notch tip in a linear elastic plate. The local weak formulation of equations govern- ing elastodynamic deformations is derived. It results in a system of coupled ordinary differential equations which are integrated with respect to time by a Newmark family of methods. Essential boundary conditions are imposed by the penalty method. The accuracy of the MLPG so- lution is established by comparing computed results for one-dimensional wave propagation in a rod with the an- alytical solution of the problem. Results are then com- puted for the following two problems: a rectangular plate with a central crack with plate edges parallel to the crack axis loaded in tension, and a double edge-notched plate with the edge between the notches loaded by compres- sive tractions. Stresses at points near the crack/notch tip computed from the MLPG solution are found to agree well with those obtained from either the analytical or the finite element solution of the same problem. The index of stress singularity is ascertained from a plot of log (stress) vs. log ( r) where r is the distance from the crack tip. It is found that, for the double-edge notched plate, the mode-mixity of deformations near a notch-tip in an or- thotropic plate can be adjusted by suitably varying the in-plane moduli of the material of the plate. The varia- tion of shear stress with r exhibits a boundary layer effect near r= O. tegrals appearing in the local weak formulation of the problem. Atluri et al. (1999) have pointed out that the Galerkin approximation can also be adopted that leads to a symmetric stiffness matrix. Atluri and Zhu (2000) solved elastostatic problems by the MLPG method, and Lin and Atluri (2000) introduced the upwinding scheme to analyze steady convection-diffusion problems. Ching and Batra (200 1) enriched the polynomial basis functions with those appropriate to describe singular deformation fields near a crack tip and used the diffraction criterion to find stress intensity factors, the J-integrals and sin- gular stress fields near a crack tip. Gu and Liu (2001) used the Newmark family of methods to study forced vi- brations of a beam. The problem of bending of a thin plate has been studied by Long and Atluri (2002). War- lock et al. (2002) have analyzed elastostatic deforma- tions of a material compressed in a rough rectangular cavity analytically by the Laplace transformation tech- nique and numerically by the MLPG method. Atluri and Shen (2002a,b) have demonstrated the use of different weight functions and have compared their performance with that of the Galerkin finite element method. By choosing a Heaviside step function as the test function, they eliminated the domain integration in the local weak form. Thus only boundary integrals over local subdo- mains remained in the local weak form. For elastostatic problems, this was shown to be more efficient than both the finite element and the boundary element methods. The paper is organized as follows. Section 2 gives the MLPG formulation including the local weak form, the moving least squares approximation, the discrete gov- erning equations and the time integration scheme. Cal- culations of the dynamic stress intensity factors from the near-tip stress fields are also described. Numerical ex- amples are presented in Section 3. The MLPG results are compared with either analytical or finite element so- lutions. Section 4 summarizes the conclusions.