Bending Of Thin Plates Research Articles

Finite element method for calculation of bending of micropolar elastic thin plates

Finite element method for calculation of bending of micropolar elastic thin plates

A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates.

A unified analytic solution approach to both static bending and free vibration problems of rectangular thin plates is demonstrated in this paper, with focus on the application to corner-supported plates. The solution procedure is based on a novel symplectic superposition method, which transforms the problems into the Hamiltonian system and yields accurate enough results via step-by-step rigorous derivation. The main advantage of the developed approach is its wide applicability since no trial solutions are needed in the analysis, which is completely different from the other methods. Numerical examples for both static bending and free vibration plates are presented to validate the developed analytic solutions and to offer new numerical results. The approach is expected to serve as a benchmark analytic approach due to its effectiveness and accuracy.

Bending of thin periodic plates

We show that nonlinearly elastic plates of thickness h → 0 with an e-periodic structure such that e^2 h → 0 exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity: in general, their effective stored-energy density is “discontinuously anisotropic” in all directions. The proof relies on a new result concerning an additional isometric constraint that deformation fields must satisfy on the microscale.

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.

Evaluation of the mechanical properties of SS-316L thin foils by small punch testing and finite element analysis

Abstract Thin foils having thickness values of 200 μm and less are commonly applied in the food industries, medical applications and more. Small punch technique (SPT) is a promising mechanical testing method for specimens thicker than 250 μm, in which a formulation correlating the measured parameters to standard tensile properties was previously reported. The current research is focused, for the first time, on the correlation between SPT and tensile mechanical properties of SS-316L thinner specimens in the range of 100–200 μm. It is demonstrated by finite-element-analysis, that the mechanical response of thin foils having thicknesses in the range of 25–500 μm can be divided into three categories. For specimens thicker than 300 μm, thin plate bending equations that were applied previously for thick specimens, are still valid, while for thinner specimens this theory fails to provide adequate correlation between SPT and tensile yield stress. For specimens thinner than 50 μm it was identified that equations derived from membrane solution should be employed rather than classical plate theory. For intermediate thickness values in the 50–300 μm range, a “transition-zone” was identified between plate and membrane-like mechanical responses. For the lower region, 50–100 μm, an analytical expression correlating the measured SPT parameters and the tensile yield stress is currently proposed.

Axisymmetric bending of strain gradient elastic circular thin plates

Axisymmetric bending of strain gradient elastic circular thin plates is studied, adopting Kirchhoff’s theory of plates. Based on the principle of minimum potential energy, the governing equation with its boundary conditions is derived through a variational method. It turns out that new terms depending upon the thickness are introduced. Those terms are missing from the existing strain gradient elastic circular thin plate theories; however, they strongly increase the stiffness of the thin plate. In addition, the static deformations and the natural frequencies of two circular gradient elastic microplates, clamped and simply supported at edges, are analyzed. This article may contribute to the design of the microstructures.

Development of DKT Finite Element Formulation for Thermal Bending of Thin Plate

The finite element formulation for thermal bending analysis of a thin plate due to the temperature gradient through its thickness is presented. The formulation is developed for the Discrete Kirchhoff Triangle (DKT) element to analyze the plate bending behavior under the thermal loading. The finite element load vector is derived in closed-form expressions so that the numerical integration is not required. Solution accuracy of the developed DKT finite element is evaluated by several examples. Performance of the developed DKT finite element is also compared with the effective nonconforming triangular element (BCIZ element). Results show that the developed DKT finite element formulation provides high solution accuracy for analysis of plate bending problem under thermal loading.

Stochastic spline fictitious boundary element method for analysis of thin plate bending problems with random fields

Mathematical formulation and computational implementation of the stochastic spline fictitious boundary element method (SFBEM) are presented for stochastic analysis of thin plate bending problems with loadings and structural parameters modeled with random fields. Two sets of governing differential equations with respect to the mean and deviation of deflection are derived by including the first order terms of deviations. These equations are in similar forms to those of deterministic thin plate bending problems, and can be solved using deterministic fundamental solutions. The calculation is conducted with SFBEM, a modified indirect boundary element method (IBEM), resulting in the means and covariances of responses. The proposed method is validated by comparing the solutions obtained with Monte Carlo simulation for a number of example problems and a good agreement of results is observed.

A Posteriori Error Control at Numerical Solution of Plate Bending Problem

The classical approach to a posteriori error control considered in this paper bears on the counter variational principles of Lagrange and Castigliano. Its efficient implementation for problems of mechanics of solids assumes obtaining equilibrated stresses/resultants which, at the same time, are sufficiently close to the exact values. Besides, it is important that computation of the error bound with the use of such stresses/resultants would be cheap in respect to the arithmetic work. Following these guide lines, we expand the preceding results for elliptic partial differential equations and theory elasticity equations upon the problem of thin plate bending. We obtain guaranteed a posteriori error bounds of simple forms for solutions by the finite element method and discuss the algorithms of linear complexity for their computation. The approach of the paper also allows to improve some known general a posteriori estimates by means of arbitrary not equilibrated stress fields.

Nonsingular boundary element flexural analysis of stiffened plates

This paper concerns a nonsingular formulation to deal with thin plates reinforced by beams using a BEM-FEM combination. The nonsingular formulation developed for the Kirchhoff׳s plate bending problem is extended to include the effect of the stiffener beams. In the proposed model, the plate is modelled by BEM using quadratic elements with transverse deflection and rotations in two orthogonal directions as the degrees of freedom (w, w,x and w,y). The beam is modelled by FEM using Timoshenko beam theory with quadratic elements and taking into account the transverse shear and bending and twisting moments (three degrees of freedom per node). As the plate and beam are modelled using C0 quadratic interpolation with the same set of three nodal values, these can be easily coupled. The nonsingular BEM formulation for thin plate bending problems is presented and the coupling of the BE equations of the plate and the FE equations of the beam is done by using the compatibility and equilibrium equations. A computer program has been developed using C++ with object oriented programming concepts. Several numerical examples are presented to illustrate the accuracy and efficiency of the present formulation.

A unified approach for embedded boundary conditions for fourth‐order elliptic problems

SummaryAn efficient procedure for embedding kinematic boundary conditions in the biharmonic equation, for problems such as the pure streamfunction formulation of the Navier–Stokes equations and thin plate bending, is based on a stabilized variational formulation, obtained by Nitsche's approach for enforcing boundary constraints. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the higher continuity requirements typical of these problems. Variationally conjugate pairs weakly enforce kinematic boundary conditions. The use of a scaling factor leads to a formulation with a single stabilization parameter. For plates, the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners. The single stabilization parameter is determined from a local generalized eigenvalue problem, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic B‐splines, providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter. Copyright © 2014 John Wiley & Sons, Ltd.

Upper bound limit analysis of plates using a rotation-free isogeometric approach

This paper presents a simple and effective formulation based on a rotation-free isogeometric approach for the assessment of collapse limit loads of plastic thin plates in bending. The formulation relies on the kinematic (or upper bound) theorem and namely B-splines or non-uniform rational B-splines (NURBS), resulting in both exactly geometric representation and high-order approximations. Only one deflection variable (without rotational degrees of freedom) is used for each control point. This allows us to design the resulting optimization problem with a minimum size that is very useful to solve large-scale plate problems. The optimization formulation of limit analysis is transformed into the form of a second-order cone programming problem so that it can be solved using highly efficient interior-point solvers. Several numerical examples are given to demonstrate reliability and effectiveness of the present method in comparison with other published methods.

Hybrid stress and analytical functions for analysis of thin plates bending

In this paper, two efficient elements for analyzing thin plate bending are proposed. They are a triangular element (THS) and a quadrilateral element (QHS), which have 9 and 12 degrees of freedom, respectively. Formulations of these elements are based on hybrid variational principle and analytical homogeneous solution of thin plate equation. Independent fields in hybrid functional are internal stress and boundary displacement field. The internal stress field has been calculated using analytical homogeneous solution and boundary field is related to the nodal degree of freedoms by the boundary interpolation functions. To calculate these functions, the edges of element are assumed to behave like an Euler-Bernoulli beam. The high accuracy and efficiency of the proposed elements are demonstrated in the severe tests.

A thin-plate bending equation of second-order accuracy

A thin-plate bending equation of second-order accuracy

Development of the large increment method in analysis for thin and moderately thick plates

Many displacement-based quadrilateral plate elements based on Mindlin-Reissner plate theory have been proposed to analyze the thin and moderately thick plate problems. However, numerical inaccuracies of some elements appear since the presence of shear locking and spurious zero energy modes for thin plate problems. To overcome these shortcomings, we employ the large increment method (LIM) for the analyses of the plate bending problems, and propose a force-based 8-node quadrilateral plate (8NQP) element which is based on Mindlin-Reissner plate theory and has no extra spurious zero energy mode. Several benchmark plate bending problems are presented to illustrate the accuracy and convergence of the plate element by comparing with the analytical solutions and displacement-based plate elements. The results show that the 8-node plate element produces fast convergence and accurate stress distributions in both the moderately thick and thin plate bending problems. The plate element is insensitive to mesh distortion and it can avoid the shear locking for thin plate analysis.

Dynamic model of micropolar elastic thin plates with independent fields of displacements and rotations

A general model of dynamic bending of isotropic micropolar elastic thin plates with independent fields of displacements and rotations is presented. The model has been justified asymptotically based on the solutions for special cases subject to simplifying assumptions. The model incorporates transverse shear deformations. Neglecting transverse shear, a model of the dynamics of micropolar elastic thin plates is also constructed. Then, we study free and forced oscillations and derive the natural frequencies, the amplitudes of the forced oscillations and the resonance conditions for micropolar elastic hinge-supported rectangular and circular plates. Finally, the basic characteristic features of micropolar plates are numerically analysed for different values of various elastic and inertial constants of the micropolar material.

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.

An applied electro-magneto-elastic thin plate theory

The physical variational principle (PVP) in a static magneto-elastic field without source current is discussed first, and then, the PVP in a general electromagnetic field is derived. It is especially useful in the plate vibration problem. Using the PVP and the pseudo total stress principle, the electro-magneto-elastic thin plate bending theory in first order and a Mindlin-type plate bending theory for a moderate thickness plate are easily obtained. This method significantly simplifies the derivation of the governing equations of the thin and moderate plates with nonlinear electromagnetic behavior. The Maxwell stress is naturally included in the governing equation. Using the governing equation of a thin plate, the analyses of the stability and vibration of a plate in an external homogeneous magnetic field are discussed.

Level set-based topology optimization of thin plate structure for maximizing stiffness under out-of-plane deformation

The topology optimization method using the level set method and incorporating a fictitious interface energy derived from the phase field method was proposed by part of the authors. The method has been applied to several structural and multidisciplinary design problems. However, the method has not been applied to the plate bending structural design problem using the two-dimensional plate bending element model yet, regardless that the thin plate structures are widely used in engineering applications that require lightweightness. This paper extends the topology optimization method to the thin plate structure for maximizing stiffness under out-of-plane deformation by using the thin plate bending elements based on Reissner-Mindlin theory. The structural design problem using the thin plate bending elements is formulated as the mean compliance minimization under volume constraint problem. Through simple numerical examples, effects of the proposed method are illustrated. At first, the method significantly reduces computational cost for the thin plate maximizing stiffness design problem in comparison with the three-dimensional solid model. The obtained optimum configuration is shown to be equivalent to that of the three-dimensional solid model. Then, it is shown that the advantages of the method such as high convergence property, low initial design dependency, and the effect of the regularization parameters on the complexity of the configuration are also held in the plate bending element model.

A new strain based brick element for plate bending

This paper presents the development of a new three-dimensional brick finite element by the use of the strain based approach for the linear analysis of plate bending. The developed element has the three essential external degrees of freedom (U, V and W) at each of the eight corner nodes as well as at the centroidal node. The displacement field of the developed element is based on assumed functions for the various strains satisfying the compatibility equations and the static condensation technique is used for the internal node. The performance of this element is evaluated on several problems related to thick and thin plate bending in linear analysis. The obtained results show the good performances and accuracy of the present element.