Page 245 - Mechanical Behavior of Materials

246 Chapter 6 Review of Complex and Principal States of Stress and Strain

3
(a) 3
σ 3 45° (c)

σ τ2
2
σ
3 2 2
45° τ 2 σ
(b) σ 1 2
σ σ
τ1
τ τ1 1
1
3 1 σ τ2
σ (d)
2 3
σ 1

1 σ τ3 τ 3 45° 2
σ τ3

1

Figure 6.8 Principal normal stresses and principal axes (a), and principal shear stresses (b), (c),
(d). In (b), rotation of the unit cube 45 about the axis of σ 1 gives the planes where τ 1 acts.
◦
Similar rotation about σ 2 gives the τ 2 planes (c), and about σ 3 the τ 3 planes (d).

6.3.1 Principal Shear Stresses and Maximum Shear Stress
In Fig. 6.8(a), if the equivalent state of stress is found for a 45 rotation about any of the 1, 2, or 3
◦
axes, a shear stress is encountered that is the largest for any rotation about that axis. The three shear
stresses that result are called the principal shear stresses, τ 1 , τ 2 , and τ 3 . These are each accompanied
by normal stresses that are the same on the two shear planes, σ τ1 , σ τ2 , and σ τ3 , respectively. For
the planes containing each pair of principal axes, 1-2, 2-3, and 3-1, we have a state of generalized
◦
plane stress, so that relationships similar to Eq. 6.12 apply for each 45 rotation. Hence, the three
principal shear stresses and the accompanying normal stresses are given by

|σ 2 − σ 3 | |σ 1 − σ 3 | |σ 1 − σ 2 |
τ 1 = , τ 2 = , τ 3 = (6.18)
2 2 2
σ 2 + σ 3 σ 1 + σ 3 σ 1 + σ 2
σ τ1 = , σ τ2 = , σ τ3 = (6.19)
2 2 2