Page 233 - Mechanical Behavior of Materials

6

Review of Complex and

Principal States of Stress

and Strain

6.1 INTRODUCTION
6.2 PLANE STRESS
6.3 PRINCIPAL STRESSES AND THE MAXIMUM SHEAR STRESS
6.4 THREE-DIMENSIONAL STATES OF STRESS
6.5 STRESSES ON THE OCTAHEDRAL PLANES
6.6 COMPLEX STATES OF STRAIN
6.7 SUMMARY

OBJECTIVES

• For plane stress, develop the equations for transformation of axes, and apply these to
determine principal normal and shear stresses. Include graphical representation by Mohr’s
circle, as well as extension to generalized plane stress.
• Explore three-dimensional states of stress, with emphasis on principal normal stresses,
principal axes, principal shear stresses, and maximum shear stress.
• Review complex states of strain, applying the fact that the mathematics and analysis
procedures are analogous to those for stress.

6.1 INTRODUCTION

Components of machines, vehicles, and structures are subjected to applied loadings that may include
tension, compression, bending, torsion, pressure, or combinations of these. As a result, complex
states of normal and shear stress occur that vary in magnitude and direction with location in the
component. The designer must ensure that the material of the component does not fail as a result
of these stresses. To accomplish this, locations where the stresses are the most severe must be
identified, and then further analysis of the stresses at these locations is needed.
At any point in a component where the stresses are of interest, it is first necessary to note
that the magnitudes of the stresses vary with direction. By considering all possible directions, the

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