Page 243 - Mechanical Behavior of Materials

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

location and radius of the circle:

σ x + σ y
σ 1 = + τ 3 = 60 + 40.3 = 100.3MPa Ans.
2
σ x + σ y
σ 2 = − τ 3 = 60 − 40.3 = 19.7MPa Ans.
2
The resulting state of stress is the same as previously illustrated in Fig. E6.1(b). Note that the
◦
counterclockwise rotation 2θ n on the circle corresponds to a rotation of θ n = 14.9 in the same
direction in the material.
The diameter corresponding to the original state of stress must be rotated clockwise to obtain
◦
the equivalent representation that contains the principal shear stress. Since this is 90 from the
σ-axis, the angle of rotation is
◦
2θ s = 90 − 2θ n = 60.26 ◦ (CW) Ans.
◦
so that θ s = 30.1 clockwise. The coordinates of the ends of this vertical diameter give the same
state of stress with τ 3 as previously shown in Fig. E6.1(c).
As already noted for Ex. 6.1, we have σ max = 100.3MPa (Ans.), but we cannot determine
τ max at this point.

6.2.4 Generalized Plane Stress

Consider a state of stress where two components of shear stress are zero, such as τ yz = τ zx = 0.
Such a situation is illustrated in a three-dimensional view in Fig. 6.7(a). It can also be illustrated
by a diagram in the x-y plane, where the z-direction is normal to the paper, as shown in (b). The
freebody of a portion of this unit cube is shown in (c). This freebody is similar to that employed
previously for plane stress—specifically, Fig. 6.3. The only difference is the presence of the stress
σ z . The equations of equilibrium in the x-y plane are the same as before.
Hence, all of the equations previously developed for the x-y plane apply to this case as well.
This includes the equations for the principal normal stresses in the x-y plane, σ 1 and σ 2 , and also
those for the principal shear stress and the accompanying normal stress, τ 3 and σ τ3 .Itissimply
necessary to note that σ z remains unchanged for all rotations of the coordinate axes in the x-y
plane. Moreover, since Mohr’s circle was also derived from the same equilibrium equations, it can
also be employed for the x-y plane.
Since this category of stress state is closely related to plane stress, we will call it generalized
plane stress. For example, such a state of stress occurs in a thick-walled tube with closed ends loaded
by internal pressure and torsion, as illustrated in Fig. A.6(a) in Appendix A. Note that, for the r-t-x
coordinate system shown, two shear components are zero, with the only nonzero shear being τ tx .
The three normal stresses, σ r , σ t , and σ x , generally have nonzero values. (The only exception is that
σ r = 0at R = r 2 .)
As will be apparent from the discussion of three-dimensional states of stress given later in this
chapter, there are always three principal normal stresses, σ 1 , σ 2 , and σ 3 . For generalized plane stress