Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Stress Transformations
CHAPTER12
12.1 Introduction
In previous chapters, formulas were developed for normal and shear stresses that act on
specific planes in axially loaded bars, circular shafts, and beams. For axially loaded bars,
additional expressions were developed in Section 1.5 for the normal [Equation (1.8)] and
shear [Equation (1.9)] stresses that act on inclined planes through the bar. The analysis
presented in that section revealed that maximum normal stresses occur on transverse planes
and that maximum shear stresses occur on planes inclined at 45° to the axis of the bar. (See
Figure 1.4.) Similar expressions were developed for the case of pure torsion in a circular
shaft. It was shown that maximum shear stresses [Equation (6.9)] occur on transverse planes
of the torsion member, but that maximum tensile and compressive stresses [Equation (6.10)]
occur on planes inclined at 45° to the axis of the member. (See Figure 6.9.) For both axial and
torsion members, normal and shear stresses acting on specified planes were determined from
a free-body diagram approach. Such an approach, while instructive, is not efficient for the
determination of maximum normal and shear stresses, which are often required in a stress
analysis. In this chapter, methods that are more powerful will be developed to determine
(a) normal and shear stresses acting on any specific plane passing through a point of
interest, and
(b) maximum normal and shear stresses acting at any possible orientation at a point of
interest.
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