Page 256 - Mechanical Behavior of Materials

Section 6.4 Three-Dimensional States of Stress 257

where
I 1 = σ x + σ y + σ z
2
2
I 2 = σ x σ y + σ y σ z + σ z σ x − τ xy − τ 2 yz − τ zx (6.29)
2
2
2
I 3 = σ x σ y σ z + 2τ xy τ yz τ zx − σ x τ yz − σ y τ zx − σ z τ xy
These quantities are called stress invariants, as they have the same values for all choices of

coordinate system. For example, σ x + σ y + σ z = σ + σ + σ = constant. Hence, the sum of the
z
x
y
normal stresses I 1 for the stress state as represented on the original x-y-z coordinate system is the

same as the sum for the equivalent representation on any other coordinate system, x -y -z , including
the coordinate system given by the principal directions, 1-2-3.
Determining values for the principal normal stresses thus consists of finding the three roots
of the cubic equation in one of the forms just given. In doing so, it is common practice to assign
the subscripts 1, 2, and 3, in order, to the maximum, intermediate, and minimum values. However,
this convention is not a necessity, and it is useful in working numerical problems to relax this
requirement and allow the numbers to be assigned as convenient. We will write all equations
involving principal stresses in general form, so that it is not necessary to assume that the subscripts
are assigned in any particular order.
6.4.3 Directions for the Principal Normal Stresses
Now consider finding the directions for the principal normal stresses—that is, the principal axes
1-2-3 of Fig. 6.8(a). To proceed, the values of the principal normal stresses, σ 1 , σ 2 , σ 3 , first need to
be determined, as described previously. Then one of these is substituted as σ i into Eqs. 6.25, which
are then solved simultaneously with

2
2
2
l + m + n = 1 (6.30)
i
i
i
to give the values of l i , m i , and n i . Note that Eq. 6.30 is required by geometry, and that only two of
the three elements of Eq. 6.25 will be found to be independent, so that the third will not aid in the
solution. To find all three principal axes, the process is repeated for each of σ 1 , σ 2 , and σ 3 .
In presenting the direction cosines, it is conventional to minimize negative signs. This can
be accomplished by replacing one or more sets of direction cosines by its negative, which is
merely a vector pointing in the opposite direction along the same line. For example, (l 1 , m 1 , n 1 ) =
(0.300, −0.945, −0.130) can be replaced by (−0.300, 0.945, 0.130). Also, the three sets of direction
cosines should represent a right-hand coordinate system. This can be accomplished by checking the
vector cross product.

(l 1 , m 1 , n 1 )×(l 2 , m 2 , n 2 ) = (l 3 , m 3 , n 3 ) (6.31)

If this is not obeyed, then replace one direction cosine vector with its negative so as to both satisfy
Eq. 6.31 and minimize negative signs. Alternatively, the first two direction cosine vectors can be
obtained by solving Eqs. 6.25 and 6.30, and the third from Eq. 6.31, so that the right-hand system
convention is automatically satisfied.