Page 270 - Mechanical Behavior of Materials

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

(c) Show that the maximum normal stress and the maximum shear stress are located at the
inner radius for any rotating annular disc with constant thickness.

Section 6.4
6.33 Rework Prob. 6.2 by solving the cubic equation and finding the direction cosines for the
principal axes. Also, show that your direction cosines are consistent with the axes rotations
from Eq. 6.6.
6.34 Rework Prob. 6.11 by solving the cubic equation and finding the direction cosines for the
principal axes. Also, show that your direction cosines are consistent with the axes rotations
from Eq. 6.6.
6.35 Consider the special case where normal stresses σ x , σ y , and σ z are present, but where the
only nonzero shear stress is τ xy , so that τ yz = τ zx = 0. For determining principal normal
stresses, show that the solution of the cubic equation (Eq. 6.26 or 6.27) corresponds to the
two equations represented by Eq. 6.7 and the third equation σ z = σ 3 . Also show that for this
special case the direction cosine for σ 3 is perpendicular to the x-y plane, that is (l 3 , m 3 ,
n 3 ) = (0, 0, 1).
6.36 An element of material is subjected to the following state of stress: σ x =−40, σ y = 100,
σ z = 30, τ xy =−50, τ yz = 12, and τ zx = 0 MPa. Determine the following:
(a) Principal normal stresses and principal shear stresses.
(b) Maximum normal stress and maximum shear stress.
(c) Direction cosines for each principal normal stress axis.
6.37 to 6.43
Proceed as in Prob. 6.36, but use the indicated stresses from Table P6.37.

Table P6.37
Problem Stress Components, MPa
No. σ x σ y σ z τ xy τ yz τ zx
6.37 0 0 0 0 100 100
6.38 0 0 50 0 300 300
6.39 100 0 0 50 50 50
6.40 100 −100 0 0 50 50
6.41 65 −120 −45 30 0 50
6.42 25 50 40 20 −30 0
6.43 10 20 −10 −20 10 −30

6.44 Consider the state of stress σ x = 90, σ y = 130, σ z =−60, and τ xy = τ yz = τ zx = 0MPa.
Employ the cubic equation (Eq. 6.26 or 6.27) and answer the following:
(a) Determine the principal normal stresses and the maximum shear stress.
(b) Show that, for such a special case, where τ xy = τ yz = τ zx = 0, the principal normal
stresses are always simply σ 1 , σ 2 , σ 3 = σ x , σ y , σ z . Also, show that the x-y-z axes are
coincident with the 1-2-3 principal axes.