# Some effects of non-linear stress/strain relations in rotationally symmetric thin shells and other engineering components

MacKenzie, A. C (1966) Some effects of non-linear stress/strain relations in rotationally symmetric thin shells and other engineering components. PhD thesis, University of Glasgow.

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## Abstract

This thesis is mainly concerned with rotationally symmetric thin shells made of materials obeying non-linear stress/strain laws, in particular an n-power law relating effective stress and effective strain. To formulate boundary value problems in such shells, it is convenient to have relations between edge forces and moments and mid-surface deformations. With the usual assumptions of thin shell theory, these relations can be obtained as integral expressions in the thickness coordinate, but the integrations cannot be performed analytically for all values of the index n. Simple approximate relations are thus suggested for thin shell which are rotationally symmetric both in geometry and loading; the approximate relations are compared with the exact relations computed numerically for the particular condition in which one curvature change of the mid-surface is zero. This condition applies in the analysis of circular cylindrical shells. The approximate relations are used to formulate boundary value problems in cylindrical shells, and a method of solution using an analogue computer is described. Solutions are given for a long, fixed end cylinder under uniform radial loading and under internal pressure. A feature of these solutions, in the form presented, is the small variation in the maximum values important variables with the index n. This suggests that the linear elastic (n = 1) solution may be used to make reasonable estimates of these maximum values for a range of values of n. As a preliminary to the work on shells, an examination was made of solutions obtained with an n-power law for deformations in a number of other engineering components. A simple method, based on the linear elastic solution, was devised for estimating deformations and is described in the present work. The method makes direct use of the stress/strain curve without need for determining material constants. Although it was derived for an n-power law, physical explanations for the method suggest that it might give reasonable estimates of deformations for non-linear laws other than the n-power law. Some available experimental results are given in support of this suggestion.

Item Type: Thesis (PhD) Doctoral Adviser: G DS MacLellan Mechanical engineering 1966 Enlighten Team glathesis:1966-72171 Copyright of this thesis is held by the author. 17 May 2019 12:40 17 May 2019 12:40 http://theses.gla.ac.uk/id/eprint/72171

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