Page 252 - Mechanical Behavior of Materials

Section 6.4 Three-Dimensional States of Stress 253

The shear stress due to the torsion is

6
T 1200 × 10 N·mm
τ tx = = = 205.3MPa
2
2
2πr avg t 2π(305 mm) (10 mm)
Since we have a state of generalized plane stress, the principal normal stresses are

2

σ t + σ x σ t − σ x 2
σ 1 ,σ 2 = ± + τ tx = 450 ± 254.3 = 704.3, 195.7MPa
2 2
σ 3 = σ r , σ 3 = 0 (outside), σ 3 =−20 (inside) Ans.
From Eqs. 6.18 and 6.20, the maximum shear stress is

|σ 2 − σ 3 | |σ 1 − σ 3 | |σ 1 − σ 2 |
τ max = MAX , ,
2 2 2
τ max = 352.1MPa (outside), τ max = 362.1MPa (inside)

Note that σ 1 and σ 3 give the controlling choice in each case. The larger value for the two locations
is, of course, the final answer—specifically, τ max = 362.1MPa (Ans.).

6.4 THREE-DIMENSIONAL STATES OF STRESS

In the general three-dimensional case, all six components of stress may be present: σ x , σ y , σ z , τ xy ,
τ yz , and τ zx . This general case can be analyzed to obtain transformation equations that permit values
of the stress components to be evaluated for any choice of coordinate axes in three dimensions. This
is accomplished by considering the freebody of a portion of the stress cube of Fig. 6.1 as cut off by
an oblique plane in Fig. 6.13. Equilibrium of forces is then applied to this cube portion, as shown in
Fig. 6.14.
The stresses on the cube portion are shown in Fig. 6.14(a) and some needed geometry in (b)
and (c). The stresses on the original x-y, y-z, and z-x planes are the same as in Fig. 6.1. On
the new oblique plane, there is a normal stress σ and a shear stress τ. In (b), the normal to the
oblique plane, which is the direction of σ, is described by angles θ x , θ y , θ z to the x, y, z axes,
respectively. The cosines of these angles are useful, l = cos θ x , m = cos θ y , n = cos θ z , and are
called the direction cosines. By applying equilibrium of forces to this cube portion, σ and τ can be
evaluated in terms of the stresses on the original x-y-z coordinate system for any direction (l, m, n)
of the normal. Further analysis can be performed to find the maximum and minimum values of σ,
and also an intermediate value, which are found to be accompanied by zero τ and to have orthogonal