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

534
STRESS TRANSFORMATIONS
In Equation (12.27), the constants I 1 , I 2 , and I 3 are as follows:
I
I
1
2
x y z
2 2 2
x y y z z x xy yz zx
2 2 2
I = σσσ + 2 τ τ τ − ( στ + στ + στ )
3
= σ + σ + σ
= σσ + σσ + σσ − τ − τ − τ
x y z xy yz zx x yz y zx z xy
(12.28)
The roots of Equation (12.27)
can be readily estimated by
plotting a graph of the lefthand
side of the equation
as a function of σ.
I 1 , I 2 , and I 3 are stress invariants, because they are independent of how the coordinates x, y,
and z are oriented in a given state of stress. Recall that stress invariants for plane stress were
discussed in Section 12.7 and that the invariants I 1 and I 2 were given in Equation (12.9) for
plane stress, where σ z = τ yz = τ zx = 0. Equation (12.27) always has three real roots, which
are the principal stresses at a given point.
The roots of Equation (12.27) can be found by a number of numerical methods. One
simple method involves a procedure that begins by calculating the following constants:
Q = 1
2
I I − I − I
3 27
1 2 3 1 3
R = 1 I1 2 − 3I2
3
−1
Q
φ = cos
⎛ ⎞
⎜ −
⎝ 3
2R
⎠
⎟
Here, I 1 , I 2 , and I 3 are the three stress invariants and the angle φ is expressed in radians. The
roots of Equation (12.27) are calculated as follows:
σ
σ
a
b
⎡ φ I1
= 2R
cos
⎛ ⎞ ⎤
⎣
⎢ ⎝ 3 ⎠ ⎦
⎥ + 3
φ π
= R ⎡ ⎛ 2 ⎞ ⎤
+
⎝
⎜
⎠
⎟
⎣
⎢
⎦
⎥ + I1
2 cos 3 3 3
⎡ ⎛ φ 4π⎞
⎤ I1
σ c = 2R
cos +
⎝
⎜ 3 3 ⎠
⎟
⎣
⎢
⎦
⎥ + 3
The three roots (σ a , σ b , and σ c ) are then ranked algebraically. The algebraically largest root
(i.e., the root with the most positive value) is σ p1 , and the algebraically smallest root (i.e.,
the root with the most negative value) is σ p3 . The remaining root is σ p2 .
Often, only the algebraic values of the principal stresses are needed. However, there
are occasions in which the directions of the principal stresses are also desired. To determine
the direction cosines that define each principal plane, we begin by rewriting Equation (b)
in matrix form as
⎡ σx − σ pi τxy τxz
⎢
⎢ τxy σ y − σ pi τyz
⎢
τ τ σ − σ
⎣⎢
xz yz z pi
⎤ ⎧
⎥ ⎪
⎥ ⎨
⎥ ⎪
⎦⎥
⎩
li
mi
n
i
⎫
⎪
⎬ = 0
(c)
⎪
⎭
Here, i = 1, 2, or 3, σ pi = σ p1 , σ p2 , or σ p3 , and {l i , m i , n i } are the directions cosines corresponding
to the respective principal stresses σ pi . We next introduce a new term
k
i
=
1
a + b + c
2 2 2
i i i