7 - Exact stress analyses

Summary

Introduction

In this chapter we consider some more rigorous methods of predicting the stress distributions in granular materials. Many of these analyses involve the numerical solution of differential equations and the results are therefore subject to round-off errors. These can, however, be minimised by careful computation. Apart from this factor, most of the analyses are exact for a Coulomb material in incipient failure. One must, however, continuously bear in mind that the Coulomb model is a mathematical idealisation of the behaviour of real materials and that some materials may not conform, even approximately, to this concept. Furthermore, as emphasised in earlier chapters, analysis of incipient failure can only give the limits within which the stresses must lie. No attempt will be made to consider elastic–plastic analyses.

Many of the analyses presented in this chapter are relevant also to the field of soil mechanics. These analyses have been considered in detail by Sokolovskii, whose book (1965) summarises his work and provides the foundation for the topics of this chapter. Jenike and Johanson (1962) adapted these techniques to the problems of solids handling and their work has been extended by several workers, notably Hancock (1970), Home (1977), Wilms (1984) and Benink (1989). The present chapter is based heavily on these works to which the interested reader is referred for greater detail and tabulations of results.

In §7.2, we set up the basic equations for the stress distribution and in §7.3 and §7.4 we derive particular solutions relevant at great depth in cylindrical and conical bunkers.