Large Deflections Of Thin Plates Research Articles

Existence of polynomial solutions of degree 4 of the Monge-Ampère equation. Large deflections of thin plates

We provide necessary and sufficient conditions for the solvability of a simplest Monge-Ampère equation, assuming that both the right-hand side and the solution are polynomials of degree 4. We give a constructive method of solution of the basic system of algebraic equations corresponding to the Monge-Ampère operator under the above conditions on the prescribed polynomial. Applications to large deflections of thin plates are presented. Bibliography: 9 titles.

Large Deflections of Elastic Rectangular Plates

It is known that elastic large deflections of thin plates are governed by von Karman nonlinear equations. The analytical solution of these equations in the general case is unfeasible. Samuel Levy, in 1942, showed that large deflections of the rectangular plate can be expressed as a double series of sine-shaped harmonics (deflection harmonics). However, this method gave no way of creating the computer algorithm of solving the problem. The stress function expression taken in the Levy's method must be revised to find the approach that takes into account of all possible products of deflection coefficients. The algorithm of solving the problem for the rectangular plate with an arbitrary aspect ratio under the action of the lateral distributed load is reported in this paper. The approximation of the plate deflection is taken in the form of double series proposed by Samuel Levy. However, the expression for the stress function is presented in the form that incorporates products of deflection coefficients in the explicit form in distinction to the Levy's expression. The number of harmonics in the deflection expression may be arbitrary. The algorithm provides composing the system of governing cubic equations, which includes the deflection coefficients in the explicit form. Solving the equation system is based on using the principle of minimum potential energy. A method of the gradient descent is applied to find the equilibrium state of the plate as the minimum point of the potential energy. A computer program is developed on the basis of the present algorithm. Numerical examples carried out for the plate model with 16 deflection harmonics illustrate the potentialities of the program. The results of solving the examples are presented in the graphical form for the plates with a different aspect ratio and may be used under designing thin-walled elements of airplane and ship structures.

Nonlinear Vibration Analysis of Stochastic Thin Plates Based on Lindstedt-Poincar\'{e} Perturbation Method

Nonlinear vibration computational problem of stochastic thin plates in large amplitude was investigated here. Based on the nonlinear theory of large deflection of thin plates, we derived the governing equations of nonlinear free vibration of isotropic thin plates. And solved the governing equations by Lindstedt- Poincare Perturbation Method Presented herein are asymptotic analytical solutions for the frequency function of the thin plates with fixed parameters. Based on the theory of random field, after mathematical treatments of the probability density function of random vector expressions, such as determining the interval of integrating, substituting variable and transforming integral order, then the probability density function of nonlinear vibration frequency of stochastic thin plates can be obtained directly. And combined with Monte Carlo Simulation analysis and our previous work (1), the result of the method in this paper is validated, which testifies that the method in this paper is effective and of preferable precision.

Large Deflection of Thin Plates in Convex or Concave Cylindrical Bending

Considered in this paper is a special case relating to the large deflection of a thin beam. One end of the beam is fixed (i.e., clamped) to a rigid wall, whereas the other end is placed on a flat surface of arbitrary orientation. In previous studies, unique and non-unique solutions to the deflected shape were derived for cases in which the curvature of the beam experiences at least one change in sign. In this paper, a special case is examined in which the curvature of the beam does not change sign. Experimental results from photographs of deflected beams are presented to support the numerical predictions. An excellent agreement was found between the photographed and the predicted shapes.

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.

Large deflection of thin plates in cylindrical bending – Non-unique solutions

Considered in this paper is the large deflection of a thin beam. One end of the beam is fixed (clamped) to a rigid wall, while the other end is placed on a flat surface of arbitrary orientation. It is shown that under certain conditions, the solution to the deflected shape of the plate is not unique. Conditions for the existence of multiple solutions are identified. Numerical methodologies are developed to obtain the multiple solutions. Experiments were conducted to verify the numerical predictions. Excellent agreements are found between the predicted deflection and the experimental measurements.

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.

Large Deflection of Thin Plates Under Certain Mixed Boundary Conditions—Cylindrical Bending

Considered in this paper is the large deflection of a thin beam. One end of the beam is fixed (clamped) to a rigid wall, while the other end is placed on a flat surface of arbitrary orientation. Under the assumption that the axial deformation of the neutral axis is negligible, a closed form analytical solution to the deflection curve is obtained in terms of elliptical functions. The analytical solution is shown to have certain scalability properties with respect to the beam length and cross-section. By using appropriate bending rigidity, the same solution can be used for a thin plate under large cylindrical bending. In addition, a finite element analysis using nonlinear shell elements is also conducted showing the axial strain of the neutral axis to be less than one percent of the overall deformation. Therefore, it is valid to assume that the axial strain is negligible.

A meshless method for large deflection of plates

The nonlinear integro-differential Berger equation is used for description of large deflections of thin plates. An iterative solution of Berger equation by the local boundary integral equation method with meshless approximation of physical quantities is proposed. In each iterative step the Berger equation can be considered as a partial differential equation of the fourth order. The governing equation is decomposed into two coupled partial differential equations of the second order. One of them is Poisson's equation whereas the other one is Helmholtz's equation. The local boundary integral equation method is applied to both these equations. Numerical results for a square plate with simply supported and/or clamped edges as well as a circular clamped plate are presented to prove the efficiency of the proposed formulation.

Large Deflection of Thin Plates in Pressure Sensor Applications

The present paper examines the large deflections of a clamped circulate thin plate for pressure sensor applications and establishes a simple solution using the singular perturbation technique. The perturbation solution for the slope of the lateral deflection is in a closed form in terms of the load-induced radial stress resultant. The nonlinearity has a strong stiffened effect. The nondimensional load-induced stress resultants are functions of the nondimensional initial stress resultant, lateral load, and Poisson’s ratio.

Timoshenko's plate equation as a singular limit of the dynamical von Kármán system

We consider the full nonlinear dynamic von Kármán system of equations which models large deflections of thin plates and show how the so-called Timoshenko and Berger models for thin plates may be obtained as singular limits of the von Kármán system when a suitable parameter tends to zero. We also show that in the case where the plate is of infinite measure this limit process gives the usual linear plate model. Therefore the nonlinear term of the system vanishes asymptotically when the domain has infinite measure. Strong convergence is also discussed: It holds under additional compatibility conditions on the initial data. Our results extend a previous work by the authors on the corresponding 1− D models.

Analysis of Floating Roofs by ODE-Solver Method

An ODE-solver-based method for the solution of the deflection and stresses of the deck plate of an oil storage tank floating roof, either with or without a central pontoon, is proposed in this paper. Based on the bending theory for the large deflection of thin plates, a system of standard ordinary differential equations (ODEs) and associated boundary conditions for the problem is derived with various factors taken into consideration. The ODE system is then solved directly and efficiently by using a standard general-purpose ODE solver with the solution accuracy satisfying the user-specified error tolerance. Two numerical examples are given to show the remarkable performance of the proposed method.

Large deflection analysis of plates under thermal loading

A global method of generalized differential quadrature is presented to solve the problems of large deflections of thin plates under thermal loading. By applying the generalized differential quadrature, the drawbacks of existing analytical and numerical solution techniques for the problem of non-linear bending of heated plates, such as complexity, lack of generality and computational inefficiency, have been overcome. The weighting coefficients for the approximation of derivatives required in differential quadrature formulation are calculated in a very simple way yet without any restriction on the choice of grid points. Numerical implementation of the method is straightforward and different boundary conditions can be easily incorporated. Examples are given to demonstrate the accuracy and efficiency of the proposed method. To validate the method, a case of large deflection of plates without thermal loading is also considered and the results compare well with existing analytical and numerical solutions.

General mathematical theory of large deflections of thin plates with some holes

General mathematical theory of large deflections of thin plates with some holes

The large deflection of thin plates with overhang

This investigation concerns the application of a special method of successive approximations to the solution of a practically motivated, nonlinear problem of the bending of a uniformly loaded circular plate supported on a concentric circle. Numerical results are obtained for a range of plate-radius/support-radius ratios, and the former are found to be in good agreement with the results of experiments.

Postbuckling Behavior of Rectangular Plates With Small Initial Curvature Loaded in Edge Compression

Abstract The solution of Marguerre’s fundamental equations for large deflections of thin plates with slight initial curvature is presented for the case of a rectangular plate subjected to edge compression. The problem is solved under eight different boundary conditions, combining two kinds of loading conditions and four kinds of supporting conditions. Numerical solutions are obtained for square plates with and without initial deflection, and the connections of deflection, edge shortening, and effective width of the plate with applied loads are clarified. The solutions here obtained include as special cases those investigated by Levy and Coan.

LARGE DEFLECTION OF A CIRCULAR PLATE WITH A CIRCULAR HOLE AT THE CENTER

The equation for the large deflection of thin plates established by Th. von Karman has been well known for many years. But so far there are only a few problems been studied with numerical certainty. S. Levy was the first to apply this equation to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy got the solution of the simply supported rectangular plate also under uniform load by the method of double trigonometric series. These two methods used nearly the same procedure of determining the numerical value of the coefficents. But their numerical works are too cumbersome. Lately, W. Z. Chien treated Way's problem again by means of the perturbation method and obtained excellent results.In this paper, the problem of large deflection of a circular plate with a circular hole at the center is treated with the perturbation method.Recently, C. A. AлeKceeB worked out the same problem with the membrance theory, but his results differ greatly from the practical case of a thin plate. The reason of this is chiefly due to his neglecting the effect of bending. The results obtained in this paper are compared with those of AлeKceeв and discussed. We conclude that under concentrated load, the bending effect is momentous and therefore it cannot be neglected. The problem can be extended to other boundary conditions with the maximum deflection YM as parameter.