Page 246 - Mechanical Behavior of Materials

Section 6.3 Principal Stresses and the Maximum Shear Stress 247

τ τ τ
τ (b)
(a) 2 + (σ, τ)
τ 3
τ 1 1
σ σ
0 0 σ σ σ
3 2 1
σ 1-2 plane
3 3 2-3
plane
1-3 plane
σ
2
σ
1

Figure 6.9 Mohr’s circles for a three-dimensional state of stress.

The maximum shear stress for any plane in the material is the largest of the three principal shear
stresses:

τ max = MAX(τ 1 ,τ 2 ,τ 3 ) (6.20)

Mohr’s circles may be applied for the rotations of Fig. 6.8 about each of the principal axes.
The three circles that result are shown in Fig. 6.9. Two of the circles lie inside the largest one, and
each is tangent along the σ-axis to the other two. The radii of these circles are the principal shear
stresses, τ 1 , τ 2 , and τ 3 , and the centers are located along the σ-axis at the points given by the three
σ τi values. Also, each plane where one of these principal shear stresses occurs is seen to be a 45 ◦
rotation away from the corresponding planes of principal normal stress, which is consistent with the
previous discussion and with Fig. 6.8.

Example 6.3
For the following state of stress, determine the principal normal stresses, the principal axes, and
the principal shear stresses:

σ x = 100, σ y =−60, σ z = 40 MPa
τ xy = 80, τ yz = τ zx = 0MPa

Also determine the maximum normal stress and the maximum shear stress.
Solution Since there is only one nonzero component of shear stress, we have a state of
generalized plane stress, and the stress normal to the plane of the nonzero shear stress is one