This chapter provides a review and insight into the stress and strain analyses. Expressions for both stresses and deflections in mechanical elements are developed throughout the text as the subject unfolds, after examining their function and general geometric behavior. With the exception of Sections 3.12 through 3.18, we employ mechanics of materials approach, simplifying the assumptions related to the deformation pattern so that strain distributions for a cross section of a member can be determined. Afundamental assumption is that plane sections remain plane. This hypothesis can be shown to be exact for axially loaded elastic prismatic bars and circular torsion members and for slender beams, plates, and shells subjected to pure bending. The assumption is approximate for other stress analysis problems. Note, however, that there are many cases where applications of the basic formulas of mechanics of materials, so-called elementary formulas for stress and displacement, lead to useful results for slender members under any type of loading.

Our coverage presumes a knowledge of mechanics of materials procedures for determining stresses and strains in a homogeneous and an isotropic bar, shaft, and beam. In Sections 3.2 through 3.9, we introduce the basic formulas, the main emphasis being on the underlying assumptions used in their derivations. Next to be treated are the transformation of stress and strain at a point. Then attention focuses on stresses arising from various combinations of fundamental loads applied to members and the stress concentrations. The chapter concludes with discussions on contact stresses in typical members referring to the solutions obtained by the methods of the theory of elasticity and the general states of stress and strain.

In the treatment presented here, the study of complex stress patterns at the supports or locations of concentrated load is not included. According to Saint-Venant’s Principle (Section 1.4), the actual stress distribution closely approximates that given by the formulas of the mechanics of materials, except near the restraints and geometric discontinuities in the members. For further details, see texts on solid mechanics and theory of elasticity; for example, References 1 through 3.