5.9 Numerical Examples
155
Hence, stress components are
σ x = −
3W
4h 3
x y
σ y = 0
τ xy = −
3W
8h
1 −
y
2
h 2
Example 5.2
Given the stress function φ =
H
π
z tan
−1
x
z
. Determine whether stress function φ
is admissible. If so determine the stresses.
Solution For the stress function φ to be admissible, it has to satisfy Biharmonic
equation. Biharmonic equation is given by
∂
4
φ
∂ x 4 + 2
∂
4
φ
∂ x 2 ∂z 2 +
∂
4
φ
∂z 4 = 0
( i )
Now,
∂φ
∂z
=
H
π
−
xz
x 2 + z 2
+ tan
−1
x
z
∂
2
φ
∂z 2 =
H
π
1
x 2 + z 2
2
2xz
2
− xz
2
− x
3
− xz
2
− x
3
∴
∂
2
φ
∂z 2 = −
2H
π
x
3
x 2 + z 2
2
Also,
∂
3
φ
∂z 3 =
H
π
8x
3 z
x 2 + z 2
3
∂
4
φ
∂z 4 =
H
π
8x
5
− 40x
3 z
2
x 2 + z 2
4
∂
3
φ
∂z 2 ∂ x
= −
2H
π
3x
3 z
2
− x
4
x 2 + z 2
3
∂
4
φ
∂z 2 ∂ x 2 =
H
π
64x
3 z
2
− 24xz
4
− 8x
5
x 2 + z 2
4
Similarly,