Stability of orthotropic elastic–plastic open conical shells

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Abstract

The paper presents a derivation of the stability equation and the solution method of the problem for an orthotropic elastic–plastic open conical shell. The use is made of the constitutive relations of incremental plastic flow theory, elastic compressibility of the material, and Shanley concept of increasing load are taken into account in the consideration. A variation, strain energy method is used to derive the stability equation for bilayered open conical shell with nonuniform pre-critical stress distribution. The shell is free supported at the edges and the load acting the shell, in the form of longitudinal force and lateral pressure, is active one, i.e. unloading is not considered.

Introduction

Shell structures are very interesting from the design point of view and these are well recognized in the scientific literature [1], [2], [3]. A very significant problem in linear and nonlinear analysis of shell structures is stability and associated phenomena. One can find here multilayered thin-walled structures, and primarily shell structures [8], [9]. In order to obtain a proper design of new structures, or to check the carrying capacity of the existing structures comprehensive understanding of the phenomena on- and post-critical equilibrium paths is needed because of restrictive requirements during their design and manufacture. A typical bilayered shell structure consists of two load-carrying layers, where upper layer is made of metal or a composite, and lower (internal) layer is lining [5]. Such shell structures are featured by low weight and high flexibility.

A series of very interesting papers on orthotropic plate and shell structures were published last time. One can mention here a paper by Kołakowski and Kowal-Michalska [3] where interesting problems of the choice of the layers in orthotropic structures and stability problems in plastic state of stress were presented and discussed. One of the first monographs related with elastic stability of structures is a book written by Timoshenko and Gere [7] where analytical solutions are presented for bars, frames, plates, and shells. Now, we observe many directions in the development of the theory of stability, presented, among all, by Grigoluk and Kabanov [4] where we can find also a discussion on strain energy criteria from one can derive stability equations.

Section snippets

Basic assumptions and the structure geometry

Several basic assumptions are accepted at first. A general theory of thin-walled shells is obligatory; i.e. arbitrary strain state of the whole shell can be substituted by arbitrary (with no kinematic limitations) strain state of the shell basic surface and uniform strain state along the shell thickness. The shell layers are of different thickness and made of different materials. Strength properties of the both layers are of the same order, so, we can accept that Kirchhoff–Love hypothesis is

Constitutive relations of plastic flow theory for orthotropic elastic–plastic shells

We consider a bilayered orthotropic conical shell, see Fig. 1. The shell layers are laminated by 0° and 90° with the relation to s-axis (so-called transversal laminate, Fig. 2).

We accept that a small increment of strains dε for elastic–plastic materials consists of two parts, i.e. elastic and plasticdε=dεe+dεp.

The problems of elastic stability use the generalized Hooke's law. The elastic–plastic stability analysis requires the appropriate theory of plasticity to be used; we accept here the J2

Approximation of stress–strain curves in elastic–plastic strain states

A basic elastic–plastic model with linear stress hardening in plastic stress state is accepted (see Fig. 4). It approximates to some extent a real nonlinear σε diagram for two perpendicular directions of the orthotropy.

Following Fig. 4, we have the following for the x direction of orthotropy:{σ(x)=Exε(x),ω(ε)=0for0ε(x)σplxEx,σ(x)=σplx+Ext(ε(x)-εplx),ω(ε)=λ(1-εplxε(x))forσplxExε(x)ε¯(x).

Or, alternatively{ε(x)=σ(x)Exforσ(x)σplx,ε(x)=σ(x)Ext-εplx(Ex-ExtExt)forσ(x)σplx.

We change the indices x

Basic geometrical and physical relations for the considered shell

We assume the contact surface between the two layers to be the basis surface of the shell and we accept an orthogonal coordinate system s, ϕ coincided with principal curvatures of the shell. Axis z is perpendicular to basic surface of the shell with positive direction toward the centre of curvature (see Fig. 5).

We assume that the shell is shallow one and it is thin walled with constant thickness. According to the nonlinear theory of shells [10], [11] the strains and changes in curvatures for

Strain energy in the shell, boundary conditions and stability equation

A set of equations for the considered conical shell does not have an explicite solution. In order to get rid of difficulties appeared in a classical approach with Galerkin method implemented to get the solution (time-consuming calculations, complicated approximative functions for the displacements) we apply Ritz method to solve the equations.

When use is made of the virtual work principle we get the following equation for the shell in a deformed stress state:δUp=δ(W±+Lz)=0,where W± is strain

Algorithm of numerical calculations

Coefficients e˜i in stability equation (35) are variable and they depend on the external load, orthotropy parameters and on the stress–strain relation in elastic–plastic stress state. Numerical calculations will be performed iteratively using an algorithm of the elastic–plastic analysis. The numerical code is written in Fortran 95 computer language [14].

The result of program run is the determination of external load values (q and Na) to be the functions of shell deflection. Because the

Numerical examples and analysis of the results

The numerical calculations, aiming the determination of the equilibrium paths, and the determination of the upper and lower critical loads are performed for an opened orthotropic conical shell under lateral pressure q and longitudinal force Na (see Fig. 1).

The following input data are accepted for the analysis (Table 1, Table 2, Table 3):

A constant ratio of maximum deflection to the entire thickness of the shell w¯=3 was accepted; what fits within the scope of the theory.

The curves in Fig. 8,

FEM verification

Verification was already done previously by the authors using professional FEM system COSMOS/M, for an elastic–plastic stability problem of a conical shell discussed in the paper.

We present here, for example, some of the FEM results, related with the considered problem (Fig. 13).

The numerical results of the paper are as follows. The shell, being the subject of the comparison, was composed of two layers with the thicknesses t=10 mm and c=10 mm, respectively. The shell external load was lateral

Conclusions

The objective of this elaboration is the derivation of the stability equations, and the presentation of the problem of elastic–plastic stability loss for an open bilayered conical shell with nonhomogeneous pre-critical stress state under composed load. A special iterative solution procedure was elaborated implemented in a source code for the PC. The next step was the determination of the critical loads for the shell with different direction of the principal axes of orthotropy in the specific

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