Page 244 - Mechanical Behavior of Materials

Section 6.3 Principal Stresses and the Maximum Shear Stress 245

y
σ
y σ
(a) (b) y
τ
τ xy xy
y
σ
x σ x
σ x z x σ z
z
z

(c)
θ
τ y ' y
σ σ
x x '
θ
σ z x
z
τ
xy
σ
y

Figure 6.7 State of generalized plane stress where two components of shear stress are zero.

with τ yz = τ zx = 0, we obtain σ 1 and σ 2 from Eq. 6.7 or from Mohr’s circle, and the third principal
normal stress turns out to be σ 3 = σ z . Also, since ordinary plane stress is simply a special case
of generalized plane stress where σ z = 0, the third principal normal stress in this case is simply
σ 3 = σ z = 0.

6.3 PRINCIPAL STRESSES AND THE MAXIMUM SHEAR STRESS

Consider any state of stress on an x-y-z coordinate system, as in Fig. 6.1. There is, in all cases,
an equivalent representation on a new coordinate system of principal axes, 1-2-3, where no shear
stresses are present, as illustrated by Fig. 6.8(a). The three normal stresses for the 1-2-3 coordinate
system are principal normal stresses, σ 1 , σ 2 , and σ 3 . Of these, one is the maximum normal stress
acting on any plane, another is the minimum normal stress acting on any plane, and the remaining
one has an intermediate value.
For x-y plane stress or generalized plane stress, the values of σ 1 and σ 2 and their directions
may be found, as described in the previous section of this chapter, by using Eqs. 6.6 and 6.7. In
Fig. 6.4, the directions of σ 1 and σ 2 are the 1-2 axes, with these directions being determined by the
θ n rotation from the original x-y axes. Further, the 3-axis is the z-axis, with σ 3 = σ z .
However, for the general three-dimensional case, the 1-2-3 axes are unique directions that may
all differ from the original x-y-z directions. The general procedure for finding σ 1 , σ 2 , and σ 3 and
the corresponding 1-2-3 axes will be considered in the next section of this chapter. However, before
proceeding with this somewhat advanced topic, it is useful to consider principal shear stresses and
the maximum shear stress, and also to revisit plane stress.