THE SCHR(?)DINGER EQUATION IN THEORY OF PLATES AND SHELLS WITH ORTHORHOMBIC ANISOTROPY

This work is the continuation of the discussion of Refs.[1-5].In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection fororthorhombic anisotropic thin shells or orthorhombic anisotropic thin plates on Winkler’sbase are classified as several of the same solutions of Schr?dinger equation.and we canobtain the general solutions for the two above-mentioned problems by the method in Refs.[1]and[3-5].[B]The von Kármán-Vlasov equations of large deflection problem for shallow shellswith orthorhombic anisotropy(their special cases are the von Kármán equations of largedeflection problem for thin plates with orthorhombic anisotropy)are classified as thesolutions of AKNS equation or Dirac equation,and we can obtain the exact solutions forthe two abovementioned problems by the inverse scattering method in Refs.[4-5].The general solution of small deflection problem or the exact solution of largedeflection problem for the corrugated or rib-reinforced plates and shells as special c     

The Schrödinger equation in theory of plates and shells with orthorhombic anisotropy

This work is the continuation of the discussion of Refs. [1-5]. In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection for orthorhombic misotropic thin shells or orthorhombic anisotropic thin plates on Winkler’s base are classified as several of the same solutions of Schrodmger equation, and we can obtain the general solutions for the two above-mentioned problems by the method in Refs. [1] and [3-5].[B] The. von Karman-Vlasov equations of large deflection problem for shallow shells with orthorhombic anisotropy (their special cases are the von Harmon equations of large deflection problem for thin plates with orthorhombic anisotropy) are classified as the solutions of AKNS equation or Dirac equation, and we can obtain the exact solutions for the two abovementioned problems by the inverse scattering method in Refs. [4-5].The general solution of small deflection problem or the exact solution of large deflection problem for the corrugated or rib-reinforced plates and shells as special cases is included in this paper.     

THE RELATION OF VON KRMN EQUATION FOR ELASTIC LARGE DEFLECTION PROBLEM AND SCHRDINGER EQUATION FOR QUANTUM EIGENALUES PROBLEM

In this paper the solutions of von Karman for elastic large deflection problem areclassified as the several solutions of Schr(?)dinger equation for quantum eigenvaluesproblem,and we present the transfrom for elastic large deflection problem from non-linearequation into linear equation.Thus,we create favourable conditions of the adoption ofconverse scattering methnd and B(?)cklund transformation.We also discuss the largedeflection problem of long and narrow plate.We can study the non-linear transition of elastic thin plate by furnished method fromthis paper.     

FURTHER STUDY OF THE RELATION OF VON KF8B6RMN EQUATION FOR ELASTIC LARGE DEFLECTION PROBLEM AND SCHRDINGER EQUATION FOR QUANTUM EIGENVALUES PROBLEM

This work is the continuation and improvement of the discussion of Ref.[1]. We alsoimprove the discussion of Refs.[2-3] on the elastic large deflection problem by results ofthis paper.We again simplify the von Kármán equation for elastic large deflection problem,and finally turn it into the nonlinear Schr(?)dinger equation in this paper.Secondly,weexpand the AKNS equation to still more symmetrical degree under many dimensionalconditions in this paper.Owing to connection between the nonlinear Schr(?)dinger equationand the integrability condition for the AKNS equation or the Dirac equation,we can obtainthe exact solution for elastic large deflection problem by inverse scattering method.In otherwords,the elastic large deflection problem wholly becomes a quantum eigenvalues problem.The large deflection problem with orthorhombic anisotropy is also deduced in thispaper.     

Large deflection problem of thin orthotropic circular plate with variable thickness under uniform pressure

Basic equations for large deflection theory of thin orthotropic circular plate with variable thickness are derived in this paper. The modified iteration method is adopted to solve the large deflection problem of thin orthotropic circular plate with variable thickness under uniform pressure. If ε=0, then the solution derived from the result in this paper coincides completely with the result given by J. Nowinski (using perturbation method) for solving large deflection problem of thin orthotropic circular plate with constant thickness under uniform pressure.     

The solution of large deflection problem of thin circular plate by the method of composite expansion

In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method. the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky’s membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coondinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.     

THE SOLUTION OF LARGE DEFLECTION PROBLEM OF THIN CIRCULAR PLATE BY THE METHOD OF COMPOSITE EXPANSION

In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method. the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky's membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coondinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.     

LARGE DEFLECTION PROBLEM OF THIN ORTHOTROPIC CIRCULAR PLATE WITH VARIABLE THICKNESS UNDER UNIFORM PRESSURE

Basic equations for large deflection theory of thin orthotropic circular plate withvariable thickness are derived in this paper.The modified iteration method is adopted tosolve the large deflection problem of thin orthotropic circular plate with variable thicknessunder uniform pressure.Ifε=0 ,then the solution derived from the result in this papercoincides completely with the result given by J.Nowinski(using perturbation method)forsolving large deflection problem of thin orthotropic circular plate with constant thicknessunder uniform pressure.     

THE SOLUTION OF DEFLECTION OF ELASTIC THIN PLATE BY THE JOINT ACTION OF DYNAMICAL LATERAL PRESSURE, FORCE IN CENTRAL SURFACE AND EXTERNAL FIELD ON THE ELASTIC BASE

In this paper the Euler equation of the deflection of elastic thin plate is reduced to the equation with Schrǒdinger form by the principle of quantum electro-dynamics.Then we can obtain the general solution of deflection of elastic thin bending plate by the joint action of dynamical lateral pressure, force in central surface and external field on the elastic base.     

ON THE GENERAL EQUATION AND THE GENERAL SOLUTION IN PROBLEMS FOR PLASTODYNAMICS WITH RIGID-PLASTIC MATERIAL

This work is the continuation of the discussion of refs.[1-2].We discuss thedynamics problems of ideal rigid—plastic material in the flow theory of plasticity in thispaper.From introduction of the theory of functions of complex variable under Dirac-Paulirepresentation we can obtain a group of the so-called“general equations”(i.e.have twoscalar equations)expressed by the stream function and the theoretical ratio.In this paperwe also testify that the equation of evolution for time in plastodynamics problema is neitherdissipative nor disperive,and the eigen-equation in plastodynamics problems is a stationarySchr(?)dinger equation,in which we take partial tensor of stress-increment as eigenfunctionsand take theoretical ratio as eigenvalues.Thus,we turn nonlinear plastodynamics problemsinto the solution of linear stationary Schr(?)dinger equation,and from this we can obtain thegeneral solution of plastodynamics problems with rigid-plastic material.     

The singular perturbation method applied to the nonlinear stability problem of a shallow spherical shell (II)

In this paper we consider the nonlinear stability of a thin elastic circular shallow spherical shell under the action of uniform normal pressure with a clamped edge. When the geometrical parameter k is large, the uniformly valid asymptotic solutions are obtained by means of the singular perturbation method. In addition, we give the analytic formula for determining the centre deflection and the critical load, and the stability curve is also derived. This paper is a continuation of the author’s previous paper[11].     

STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH LARGE OPENINGS

Based on Donnell's shallow shell equation, a new method is given in this paper to ana-lyze theoretical solutions of stress concentrations about cylindrical shells with large openings. With themethod of complex variable function, a series of conformal mapping functions are obtained from dif-ferent cutouts' boundary curves in the developed plane of a cylindrical shell to the unit circle. And,the general expressions for the equations of a cylindrical shell, including the solutions of stress concen-trations meeting the boundary conditions of the large openings' edges, are given in the mapping plane.Furthermore, by applying the orthogonal function expansion technique, the problem can be summa-rized into the solution of infinite algebraic equation series. Finally, numerical results are obtained forstress concentration factors at the cutout's edge with various opening's ratios and different loadingconditions. This new method, at the same time, gives a possibility to the research of cylindrical shellswith large non-circular openings or with nozzles.     

AXISYMMETRIC SPHERICAL SHELL WITH VARIABLE WALL THICKNESS

This paper is engaged in reaeareh of the problem of axisymmetric spherical shell with variable wall thickness. The solutions for the problem are given for the spherical shell segment which does not contain the pole of sphere and the point of zero wall thickness.     

A DISCUSSION ON“THE LARGE DEFLECTION PROBLEM OF CIRCULAR THIN PLATE WITH VARIABLE THICKNESS UNDER UNIFORMLY DISTRIBUTED LOADS”

“The large deflection problem of circular thin plate with variable thickness under uniformlydistributed loads”has been solved by using the small parameter method and modified iterationmethod jointly in the ref.[1].The solution of ref.[1]and its special results are correct.But,theprocedure in the ref.[1]is almost same as those of perturbation method in essence[2].The loadwas assumed to be     

CALCULATION OF ORTHOTROPIC SHALLOW CONOIDAL SHELL ROOFS

The present paper is to introduce the method for the calculation of the roof withorthotropic shallow conoidal shells which are always used in the industrial buildings. Sincethe middle surface of the shallow conoidal shell having the variable curvatures and twists,therefore the system of fundamental equations, which are obtained by us, are beingpossessed of the variable coefficients.If we want to find the exact analytical sohutions, it will be involved in greatmathematical difficulties. This paper gives the approximate solutions of the orthotropicshallow conoidal shell roof which is simply supported at all edges and under the uniformlydistributed load.In this paper the method of small parameter is used.     

Analytical solution of rectangular plate with in-plane variable stiffness

The bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form, and Poisson’s ratio is constant. A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation, and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads. The validity of the present solution is shown by comparing the present results with those of the classical solution. The influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examples. The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness.     

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 byProf.Chien Wei-zang for solving the problem of large deflection of a circular plate,and findthat the method could be regarded as the method of strained parameters in the singularperturbation theory.In terms of the parameter representing the ratio of the centerdeflection to the thickness of the plate,we make the asymptotic expansions of thedeflection,membrane stress and the parameter of load as in Ref.[1],and then give theorthogonality conditions(i.e.the solvability conditions)for the resulting equations,bywhich the stiffness characteristics of the plate could be determined.It is pointed out thatwith the solutions for the small deflection problem of the circular plate and theorthogonality conditions,we can derive the third order approximate relations between theparameter of load and the center deflection and the first-term approximation of membranestresses at the center and edge of the plate without solving the differential equ     

Progressing step by step and integrating calculation of overcritical deformation of spherical shallow shells

In this paper, the problem of second buckling of the spherical shallow shell is calculated by use of the method of progressing step by step and integrating. The result is more exact than that of first approximate analysis for over-critical deformation of spherical shallow shell. It has been solved that the solution of second approximate analysis in this problem can’t be found. The calculating example in this paper shows that the solution of progressing step by step and integrating converges to second approximate solution.     

Large deflection problem of a clamped elliptical plate subjected to uniform pressure

In this paper, the perturbation solution of large deflection problem of clamped elliptical plate subjected to uniform pressure is given on the basis of the perturbation solution of large deflection problem of similar clamped circular plate (1948), (1954). The analytical solution of this problem was obtained in 1957. However, due to social difficulties, these results have never been published. Nash and Cooley (1959) published a brief note of similar nature, in which only the case λ=a/b=2 is given. In this paper, the analytical solution is given in detail up to the 2nd approximation. The numerical solutions are given for various Poisson ratios v=0.25, 0.30, 0.35 and for various eccentricities λ=1, 2, 3, 4, 5, which can be used in the calculation of engineering designs.     

DEFORMATIONS AND STABILITY OF SPHERICAL CAPS UNDER CENTRALLY DISTRIBUTED PRESSURES

In this paper the deformations and stability in large axisymmetric deflection of spherical caps under centrally distributed pressures are investigated. The Newton-apline method for solving the nonlinear equations governing large axisymmetric deflection of spherical caps is presented. The buckling behavior is studied for a cap with fixed geometry when the size of the loaded radius is allowed to vary, and for a fixed loaded radius when the shell geometry is allowed to vary. The influence of the buckling modes on the criticalloads is analysed. Numerical results are given for v=0.3.