A σ
n
ð Þ
X ¼
A
cos 45
∘
ð
Þ
À5MPa À 2MPa
½
σ
n
ð Þ
X ¼ σ
n
n cos 45
∘
ð
Þ
2.5 Principal Stresses and Principal Axes
The stress tensor given in the original Cartesian coordinate system changes values if
we change the orientation of the coordinate system, i.e., simply rotating it around the
origin O. Inclined plane represents the orientation of a new coordinate system.
Therefore, stress tensor becomes a function of the rotation angle with respect to
14
y
z
dy
dz
dx
σyx
mzx
x
σxy
σxy
σyx
myz
mzy
mzx
Fig. 2.8 Couple stresses
acting in conjunction with
linear shear stresses at a
point in x-y plane only
σ y
(n)
45
45
0
Y
X
2MPa
5MPa
10MPa
2MPa
σ x
(n)
A
Fig. 2.9 State of stress at a
point
2.5 Principal Stresses and Principal Axes
21
n
ð Þ
X ¼
A
cos 45
∘
ð
Þ
À5MPa À 2MPa
½
σ
n
ð Þ
X ¼ σ
n
n cos 45
∘
ð
Þ
2.5 Principal Stresses and Principal Axes
The stress tensor given in the original Cartesian coordinate system changes values if
we change the orientation of the coordinate system, i.e., simply rotating it around the
origin O. Inclined plane represents the orientation of a new coordinate system.
Therefore, stress tensor becomes a function of the rotation angle with respect to
14
y
z
dy
dz
dx
σyx
mzx
x
σxy
σxy
σyx
myz
mzy
mzx
Fig. 2.8 Couple stresses
acting in conjunction with
linear shear stresses at a
point in x-y plane only
σ y
(n)
45
45
0
Y
X
2MPa
5MPa
10MPa
2MPa
σ x
(n)
A
Fig. 2.9 State of stress at a
point
2.5 Principal Stresses and Principal Axes
21