Imperfection sensitivity and stability of an elastic-plastic conical shell under axisymmetrical load

Abstract

Theoretical fundamentals are presented for large deformation stability of a conical shell under assumptions of elastic-plastic properties following the Prandtl-Reuss incremental plastic flow theory and initial shape imperfections of the order of the shell thickness. The problem is reduced to a set of two nonlinear partial differential equations of the fourth order with two unknowns, i.e. the deflection function w and the stress function w. Solution is obtained as a nonlinear relation between the load and deflection parameter. Numerical results are presented in diagrams. IntroductionThe subject of the present paper is large deformation stability analysis of a free-supported elastic-plastic conical shell under uniform hydrostatic pressure. The shell is thin-walled with a constant thickness and it is assumed that its initial shape imperfections are of the order of the thickness. The material of the shell follows a linear stress-strain relation with linear stress hardening. It also obeys the plastic flow incremental rule corresponding to the Huber-MisesHencky (H-M-H) yield criterion, thus incompressibility follows in the plastic region. Problems related to geometrically linear analysis of such conical shells have been discussed in [21]. Most problems analysed there are related to unilayered shells under rotary symmetrical loads. Specific features of the stability of elastic-plastic shells were described in the literature for the elastic-plastic small deformations theory (Nadai-Hencky model), for the incremental plastic flow theory (Prandtl-Reuss model) associated with Shanley concept of growing load, and for the Engesser-Karman approach, s. [1-3, 7, 8, 11-13, 15, 26-28]. Opened and sandwich conical shells under axial and lateral loads were investigated to determine the bifurcation loads and the equilibrium paths, [13,[21][22][23]25].In the study [4], an elastic stability analysis of liquid-filled conical tanks having various imperfection shapes was considered, and the axisymmetric imperfection shape was found to be the critical one. By extending the analysis to include nonlinear material behaviour, it was found that for conical tanks of practical dimensions, yielding precedes the elastic buckling and that thus inelastic buckling governs the failure of these structures. The aim of another work, [5], was to study the effect of different imperfection shapes on the inelastic stability of liquid-filled conical tanks and to determine the critical imperfection shape that would lead to the minimum inelastic limit load. The study was carried out numerically, using a self-developed shell element and applied to simulate a number of conical tanks with imperfection shapes in the form of Fourier series of equal coefficients. The Fourier analysis of the buckling modes has indicated that the existence of an axisymmetric imperfection will lead to the critical inelastic limit load for conical tanks.The finite element method (FEM) is widely used in the stability analysis. Two-dimensional discretization procedures employing finite eleme...

A survey of buckling of conical shells subjected to axial compression and external pressure

“…Several investigations on the influence of initial imperfection can be found in Refs. [118][119][120][121][122][123][124]. In [119], the importance of conducting initial geometric measurements of shells was highlighted.…”