Engineering Transactions, 16, 3, pp. 363-383, 1968

### Ścisłe równania i obliczanie skończonych sprężysto-plastycznych ugięć ortotropowych powłok obrotowo-symetrycznych

Z. Waszczyszyn
Politechnika Krakowska
Poland

Exact geometric equations and those of internal equilibrium are derived on the grounds of the assumptions of the theory of thin shells by introducing Lagrangian and Eulerian coordinates. As the fundamental functions are assumed the displacement u and the angle o between the tangent
to the meridian of the deformed shell and the geometric axis of the shell. The independent variable is a Lagrangian coordinate g or curvilinear coordinate M1.
The physical equations of the strain theory are generalized to the case of an initially orthotropic elastic plastic material. The coefficients of anisotropy a are resultants of the coefficients a and a for the elastic and perfectly plastic material, respectively. The strain-hardening is taken into consideration, in the σi, εi-plane by assuming an arbitrary function Φ(εi) taken from experiment. The longitudinal forces ni and moments mg are related to the strains εi by means of the integrals Bij (k).
A semi-inverse solution method is proposed in which the boundary-value problem is replaced by a Cauchy problem. The algorithm proposed in [26] is generalized to the computation of finite
elastic-plastic deflections of orthotropic shells of variable thickness. The physical relations are inverted by the iteration method, by evaluating the integrals Bij (k) by means of quadrature formulae. The set of differential equations thus obtained is solved numerically by the Runge-Kutta method.
The theory is illustrated by a numerical example of computation of a conical shell, of small rise and variable wall thickness.

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