Large Deflection Of Circular Plate Research Articles

A method for developing the equivalent continuum model of a single layer graphene sheet

A method is presented to develop the equivalent continuum model for a single-layered graphene sheet. This method integrates molecular dynamics method as an exact numerical solution with theory of shell as an analytical method. The force-depth results achieved from molecular dynamics simulation of nano-indentation of a single graphene sheet are compared with the formulation for large deflection of circular plates loaded at the centre. As a result, the effective Young's modulus and mechanical thickness of the sheet wall are independently obtained. The validity of this approach is verified by comparing finite element modeling of nano-indentation of a single graphene sheet with molecular dynamics results available in the literature. Presented results demonstrate that the proposed method could provide a valuable tool for studying the mechanical behaviour of single-layered graphene sheets, as well as efficiency of continuum theory in nano-structured material.

Hybrid changeable basis galerkin technique for nonlinear analysis of structures

Based on the asymptotical perturbation method and the Galerkin technique, the hybrid changeable basis Galerkin technique is presented for predicting the nonlinear response of structures. By the idea of changeable basis functions first proposed, it greatly reduces calculation and is easily used in other numerical discretization techniques, such as finite element method etc. It appears to have high potential for solution of nonlinear structural problems. Finally, the effectiveness of this technique is demostrated by means of two numerical examples: the large deflection of circular plates objected to uniform normal load and the large deflection of spherical caps under centrally distributed pressures.

On the method of orthogonality conditions for solving the problem of large deflection of circular plate

In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zang for solving the problem of large deflection of a circular plate, and find that the method could be regarded as the method of strained parameters in the singular perturbation theory. In terms of the parameter representing the ratio of the center deflection to the thickness of the plate, we make the asymptotic expansions of the deflection, membrane stress and the parameter of load as in Ref. [1], and then give the orthogonality conditions (i.e. the solvability conditions) for the resulting equations, by which the stiffness characteristics of the plate could be determined. It is pointed out that with the solutions for the small deflection problem of the circular plate and the orthogonality conditions, we can derive the third order approximate relations between the parameter of load and the center deflection and the first-term approximation of membrane stresses at the center and edge of the plate without solving the differential equations. For some special cases (i.e. under uniform load, under compound load, with different boundary conditions), we deduce the specific expressions and obtain the results in agreement with the previous ones given by Chien Wei-zang, Yeh kai-yuan and Hwang Chien in Refs. [1–4].

Large Deflection of Circular Plates with or without Hole at Center

The equivalent load concept is used in the analysis of large deflection (in the Von Karman sense) of circular and annular plates in which the nonlinear terms of lateral displacement are considered as an additional set of loads acting on the plate. the solution of a Von Karman-type plate is thus reduced to that of an equivalent plate undergoing small displacement. The results for uniformly loaded, simply supported and clamped circular and annular plates are obtained and found to be in good agreement with the available solutions. The method of solution used in this study is computationally efficient and offers an alternative technique for obtaining nonlinear solutions to annular plate problems. The present formulation can be extended to obtain elastic-plastic solutions for isotropic annular plates undergoing moderately large deflections.

Elastic large deflection of circular plates using graded finite-differences

Axisymmetric elastic large deflection circular plate analysis is briefly reviewed and the apparent popularity of uniform mesh finite-difference numerical solution techniques is noted. The inherent limitations of uniform meshes, especially the inability to accommodate local refinement, are discussed. The use of graded interlacing meshes is advanced as a more flexible alternative to uniform meshes and a description of a Dynamic Relaxation (DR) large deflection graded mesh circular plate analysis is presented in detail. The accuracy of several graded mesh solutions, compared with corresponding uniform mesh results, is explored via the large deflection analysis of uniformly loaded clamped and simply supported circular plates and similar plates with an interior concentric support. In general, it is established that deflections and stress couples are computed more accurately and stress resultants less accurately with graded meshes. The use of gradually graded meshes leads to the most consistently accurate results.

Perturbation method in the problem of large deflections of circular plates with nonuniform thickness

In this paper the perturbation method about two parameters is applied to the problem of large deflection of a cricular plate with exponentially varying thickness under uniform pressure. An asymptotic solution up to the third-order is derived. In comparison with the exact solutions in special cases, the asymptotic solution shows a precise accuracy.

Large deflection of circular plate under compound load

By means of the perturbation method, this paper presents an approximate solution for large deflection of clamped circular plate under uniform pressure together with a concentrated load at the center. The special case of vanishing central deflection is also discussed. In this paper, a load distribution function is introduced so as to make the compound loads depend on a single load parameter, and average angular deflection is used as the single displacement perturbation parameter.

Large deflections of circular plates of variable thickness

Large deflections of a clamped circular plate of variable thickness have been investigated following the equations of Banerjee and Dutta. Numerical results obtained have been compared with other known results.

Chien's solution and its asymptotic behavior in large deflection of circular plates

A general treatment and residual values of Chien's solution for large deflection of circular plates are given. By means of the above results, the asymptotic behavior of Chien's solution is studied. Two examples of uniform load and concentrated load are discussed, respectively.

Large Deflections of Circular Plates of Variable Thickness

The large deflection of a clamped circular plate of variable thickness under uniform load has been investigated using von Karman’s equations. Numerical results obtained for the deflections and stresses at the center of the plate have been given in tabular forms.

Nonlinear stability of thin elastic circular shallow spherical shell under the action of uniform edge moment

In this paper, nonlinear stability of thin elastic circular shallow spherical shell under the action of uniform edge moment is considered by the modified iteration method to obtain second and third approximations to decide the upper and lower critical loads. Results are plotted in curves for the engineering use and are compared with results of Hu Hai-chang's. We also investigate the neighbour situation of the critical point, i.e. the double points of the upper and lower critical loads and denote the range of validity of the second approximation. In the end, we obtain the special case, the design formulas of rigidity and stress as well as the corresponding curves as ν=0.3 of large deflection of circular plate under the same load. These results are also compared with Huang Tse-yen's.

Nonlinear Analysis of Plates and Shells by the Incremental Procedure Using a Mixed Model of the Finite Element Method

A formulation of the mixed finite element method is proposed in this paper which may be suitable for analysis of the instability problems of plates and shells. A plane triangular element is applied to the problems of a large deflection of square plates with initial deflection, a snap-through of shallow square caps, a large deflection of circular plates and a compression buckling of square plates with the results obtained by previous workers. And it can be concluded that, although the expression of the obtained stiffness matrix is quite simple sufficiently accurate results can be obtained by using the present method.

Large deflection of circular plates with affine imperfections

The axisymmetric large deflection behaviour of imperfect, tapered, circular plates is analysed by the method of Dynamic Relaxation. The analysis is restricted to plates with linear thickness tapers and affine imperfections, the latter being sympathetic to the applied uniform lateral pressure. In general, the results of the analysis show that the deflections and stresses do not vary greatly with variations in the taper ratio, but that they reduce significantly as the magnitude of the imperfection increases.

Application of Chebyshev polynomials to the nonlinear analysis of circular plates

Chebyshev polynomials have been used to solve the problem of the large deflection of circular plates and the results have been compared with existing results. The nonlinear equations are treated as linear differential equations with variable coefficients and the analysis with Chebyshev polynomials results in a solution of linear algebraic equations. It is shown that the present analysis in the form of Chebyshev series has the property of rapid convergence. This technique has also been applied to postbuckling behaviour of circular plates with initial curvature.

Large Deflection of a Circular Plate on Elastic Foundation under Symmetrical Load

ABSTRACT The large deflection of a clamped circular plate on elastic foundation under nonuniform but symmetrical loads has been investigated following Berger's approximate method. The deflections are obtained in the form of an infinite series involving Bessel functions. Graphs are plotted for deflections, bending moments, and bending stresses for various values of foundation modulus and load functions.

Axisymmetric Large Deflections of Circular Plates Subjected to Thermal and Mechanical Loads

Summary This paper is concerned with the nonlinear axisymmetric analysis of circular plates with in-plane edge restraint. Both temperature and mechanical loads are accommodated as an ex­ tension of investigations performed for the isothermal mechanical loading problem.1-4 An exact mathematical formulation within the framework of the von Karman large-strain displacement relations5 is developed. The equilibrium equations and boundary conditions are then derived by utilizing the calculus of variations for arbitrary axisymmetric temperatures and normal distributed loading. The satisfaction of equilibrium and compatibilhVy equations requires the solution of two simultaneous nonlinear ordinary differential equations subject to the prescribed boundar}^ conditions. Analytical solutions of such equations are apparently not possible and therefore numerical procedures must be em­ ployed. A finite-difference procedure utilizing relaxed iterations, developed by Keller and Reiss,4 and employed by them for the solution of isothermal problems with apparently unlimited loadparameter ranges, is used here for combined thermomechanical problems. Numerical results are presented for the special case of a simply supported circular plate with radially immovable boundaries, subject to a uniform pressure and an arbitrary tem­ perature variation through the thickness (no planform variation). These results have been obtained for a large range of temperature and load parameters. However, because of space limitations, only a limited number of data are presented in this paper.

Large deflections of circular plates under uniform and concentrated central loads

Die gewohnlichen von Karman-Foppl-Gleichungen fur grosse Durchbiegungen einer kreisformigen Platte unter kombinierter, gleichmassiger und konzentrierter Belastung werden in Potenzreihen gelost.

Large Deflections of Circular Plates

Abstract This paper contains a solution of von Kármán’s equations for a uniformly loaded, simply supported circular plate. The method used to obtain the solution is the perturbation procedure. Series expansions for the deflection and stresses in the plate are obtained. The legitimacy of these expansions is demonstrated in the Appendix. Critical values of stress and deflection are presented in graphical form. Furthermore, tables of coefficients for the afore-mentioned series are presented if anyone desires to extend the results which are presented here.