States of Stress

Abstract

The six components σx, σy, σz, τxy, τxz and τyz of the state of stress at a point in terms of a given cartesian coordinate system were defined in Chap. 2. In a state of plane stress, the only nonzero components are σx, σy and τxy. Let the x ‐ y plane lie in the plane of the page, with the z axis directed out of the page. Let a second cartesian coordinate system x′y′z′ be superimposed on the xyz system, then its orientation by rotating it about the z′ axis through a counterclockwise angle θ. By passing an oblique plane through a cubic element with its faces perpendicular to the xyz system and applying the equilibrium equations, expressions are obtained for the stress components σx′, σy′ and τx′y′ in terms of θ. Because of their importance in design, the maximum and minimum values of the normal stresses and the maximum value of the magnitude of the shear stress on any plane through the point are discussed. A graphical procedure known as Mohr’s circle, which helps in visualizing and interpreting the equations for determining the components σx′, σy′ and τx′y′ in terms of θ, is described. Attention is then given to the problem of determining the maximum and minimum normal stresses and the magnitude of the maximum shear stress for a general state of stress. The normal stresses are given by a cubic equation whose coefficients are functions of the components of stress. Its three solutions are called the principal stresses. The maximum magnitude of the shear stress is given by an expression in terms of the principal stresses. A special case called triaxial stress, in which the only nonzero stress components are σx, σy and σz, is discussed. Two applications giving rise to interesting states of stress, bars subjected to combined loads and pressure vessels, are then covered. The chapter closes with a discussion of the tetrahedron argument, which allows determination of the normal and shear stresses acting on any plane through a point in terms of the state of stress at the point.