Complex Stress and Strain - Structural and Stress Analysis - page 398

σ II

σ I

σ III

σ III

σ I

F IGURE 14.18 Reduction of a complex three-dimensional
stress system

σ II

the most just three principal stresses, σ I , σ II and σ III , as shown, for a three-dimensional
element, in Fig. 14.18.

It now remains to relate, in some manner, these principal stresses to the yield stress
in simple tension, σ Y , of the material.

DUCTILE MATERIALS

A number of theories of elastic failure have been proposed in the past for ductile
materials but experience and experimental evidence have led to all but two being
discarded.

Maximum shear stress theory

This theory is usually linked with the names of Tresca and Guest, although it is more
widely associated with the former. The theory proposes that:

Failure (i.e. yielding) will occur when the maximum shear stress in the material is equal to
the maximum shear stress at failure in simple tension.

For a two-dimensional stress system the maximum shear stress is given in terms of the
principal stresses by Eq. (14.12). For a three-dimensional case the maximum shear
stress is given by

σ max − σ min

τ max = (14.37)

2

where σ max and σ min are the algebraic maximum and minimum principal stresses. At
failure in simple tension the yield stress σ Y is in fact a principal stress and since there
can be no direct stress perpendicular to the axis of loading, the maximum shear stress
is, therefore, from either of Eqs. (14.12) or (14.37)

σ Y
2

τ max = (14.38)