Nonconventional approaches to static problems for noncircular cylindrical shells in different formulations

A numerical-analytical method for solving the plane problem of elasticity is proposed. Systems of nonorthogonal functions are used. The method involves the minimization of a quadratic form that is equal to the integral of the sum of squared residuals of the solution and given forces. An explicit expression for stresses is derived. Bessel's inequality and the convergence of the solution are proved. The accuracy of the boundary conditions is estimated. The stress and strain distribution in the plate depending on the maximum magnitude of distributed forces and the size of their localization area is analyzed numerically. New quantitative and qualitative features of the stress distribution in the plate are established Keywords: boundary-value problem, numerical-analytical method, plane problem of elasticity, stress and strain distribution, new quantitative and qualitative features Introduction. Various analytic and numerical approaches are used [3,4,12,[19][20][21] to determine the stress state of structural members (plates and shells of different shapes). Love [22], Fadle [15], and Papkovich [8] proposed the method of homogeneous solutions (eigenfunctions) to satisfy boundary conditions, but did not follow it up with numerical calculations. Gaydon and Shepherd [17] expanded the first ten eigenfunctions in a special orthogonal system and calculated the stresses. The superposition method is widely used to solve a plane boundary-value problem [3,5]. The method of homogeneous solutions was used in [1] to analyze the stress distribution in a plate under a normal parabolic load. The analytic approach employing eigenfunctions was further developed in the papers [8,9], which give a theory of a numerical-analytic method for the determination of the coefficients of two boundary forces expanded into series in a complete system of nonorthogonal functions. This method was used in [10] to determine the stress-strain state (SSS) of a half-strip. Note that Ostrogradsky was the first to use the eigenfunction expansion to solve boundary-value problems in mechanics [7]. The relevant literature is reviewed in [2, 5, 23]. We will use the theoretical method from [8,9] for the numerical determination of the SSS of a rectangular plate. We will also prove that the solution converges and analyze the accuracy of satisfying the boundary conditions.

Numerical-analytical method to determine the stress state of an elastic rectangular plate

Revenko

2008

Stress-strain analysis of closed nonthin orthotropic conical shells of varying thickness

Grigorenko

Avramenko

2008

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Thermoelastoplastic deformation of noncircular cylindrical shells

Merzlyakov

2008

A method to determine the nonstationary temperature fields and the thermoelastoplastic stress-strain state of noncircular cylindrical shells is developed. It is assumed that the physical and mechanical properties are dependent on temperature. The heat-conduction problem is solved using an explicit difference scheme. The temperature variation throughout the thickness is described by a power polynomial. For the other two coordinates, finite differences are used. The thermoplastic problem is solved using the geometrically nonlinear theory of shells based on the Kirchhoff-Love hypotheses. The theory of simple processes with deformation history taken into account is used. Its equations are linearized by a modified method of elastic solutions. The governing system of partial differential equations is derived. Variables are separated in the case where the curvilinear edges are hinged. The partial case where the stress-strain state does not change along the generatrix is examined. The systems of ordinary differential equations obtained in all these cases are solved using Godunov's discrete orthogonalization. The temperature field in a shell with elliptical cross-section is studied. The stress-strain state found by numerical integration along the generatrix is compared with that obtained using trigonometric Fourier series. The effect of a Winkler foundation on the stress-strain state is analyzed Keywords: thermoelastoplasticity, noncircular cylindrical shell, Kirchhoff-Love hypotheses, linearization method, explicit difference scheme, Godunov's discrete orthogonalization, cylindrical shell of elliptical cross-section Introduction.Methods and elastic problems of designing noncircular cylindrical shells with arbitrary cross-section and arbitrary thickness are addressed in [3-6, 8]. These methods were further developed and some problems were solved in [14][15][16][17][18][19][20][21]29]. The thermoelastoplastic stress-strain state (SSS) of this class of inelastic shells is analyzed below. To calculate thermal stresses, we will preliminarily solve the nonstationary heat-conduction problem for shells that transfer heat to the environment by convection.1. Problem Formulation. Basic Equations. Let us determine the thermoelastoplastic SSS of a cylindrical shell with arbitrary cross-section and thickness varying in two directions. The shell can be coupled with an elastic foundation so that there can be no separation between them. At time zero, the shell, which is unstressed at temperature Ò 0 , is subjected to mechanical and thermal loads that do not cause buckling. We will formulate a noncoupled quasistatic problem and use the geometrically nonlinear theory of shells to solve it. The meridian and thickness of the shell and the applied loads permit accepting the Kirchhoff-Love hypotheses. The physical and mechanical characteristics of the shell material are assumed temperature-dependent.The position of points on the mid-surface of the shell is defined by the longitudinal coordinate s (s 0 £ s £ s N ) and the circumferential coordi...