Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

where a i , b i , and c i are the cofactors of the elements on the first row of the matrix in Equation
(c):
σ y − σ pi τ yz
a i =
τ σ − σ
yz z pi
535
gENERAL STATE OF STRESS
AT A POINT
τxy τyz
b i =−
τ σ − σ
xz z pi
τxy σ y − σ pi
c i =
τ τ
xz
Thus, the following are the direction cosines giving the orientation of principal stress i:
l
m
n
yz
= a k
i i i
= bk
i i i
= c k
i i i
Each set of direction cosines defines the normal to a plane upon which a principal stress
acts. Direction cosines must satisfy the relationship
l + m + n = 1
2 2 2
i i
(d)
It is clear that the direction cosines just derived will satisfy this requirement. Also, observe
that the trivial solution (l = m = n = 0) is not possible, given that the direction cosines satisfy
Equation (d). In addition, the following relations must be true if the planes are orthogonal:
ll + mm + nn = 0
12 1 2 1 2
ll + mm + nn = 0
23 2 3 2 3
ll + mm + nn = 0
31 3 1 3 1
In developing equations for maximum and minimum normal stresses, the special case will
be considered in which τxy = τyz = τ zx = 0. No loss in generality is introduced by considering
this special case, because it involves only a reorientation of the reference x, y, and z
axes to coincide with the principal directions. Since the x, y, and z planes are now principal
planes, the stresses σ x , σ y , and σ z respectively become σ p1 , σ p2 , and σ p3 . Solving Equation (a)
for the direction cosines thus yields
Sx
Sy
S
l = m = and n =
σ σ σ
p1 p2 p3
Substituting these values into Equation (d), we obtain
z
2
S S
2
x y Sz
+ + = 1
2 2
σ σ σ 2
(e)
p1
2
p2
The plot of Equation (e) is the ellipsoid shown in Figure 12.19. Observe that the magnitude
of σ n is everywhere less than that of S (since S 2 = σn 2 + τnt
2 ), except at the intercepts,
where S is σ p1 , σ p2 , or σ p3 . Therefore, two of the principal stresses (σ p1 and σ p3 of Figure
12.19) are the maximum and minimum normal stresses at the point we are examining. The
p3
S is the resultant stress acting
on the oblique plane of
Figure 12.19a. S x , S y , and S z are
the orthogonal components of
the resultant stress S.