6.11 The Effect of Circular Holes on Stress Distributions in Plates
187
∴ f (r ) = A e
2t
+ B e
4t
+ C e
−2t
+ D
∴ f (r ) = A r
2
+ B r
4
+
C
r 2 + D
The stress function may now be written in the form:
φ =
A r
2
+ B r
4
+
C
r 2 + D
cos 2θ
The stress components σ
r and σ
θ may now expressed as
σ
r =
1
r 2
∂
2
φ
∂θ 2 +
1
r
∂φ
∂r
∴ σ
r = −
2 A +
6C
r 4 +
4D
r 2
cos 2θ
and
σ
θ =
∂
2
φ
∂r 2
∴ σ
θ =
2 A + 12Br
2
+
6C
r 4
cos 2θ
and
τ
r θ =
1
r 2
∂φ
∂θ
−
1
r 2
∂
2
φ
∂r ∂θ
∴ τ
r θ =
2 A + 6Br
2
−
6C
r 2 −
2D
r 2
sin 2θ
The boundary conditions are,
(a) At r = a, σ
r = 0
(b) At r = b, σ
r =
p
2
cos 2θ
(c) At r = b, τ
r θ = −
p
2
sin 2θ
(d) At r = a, τ
r θ = 0.
Therefore, we have on substitution in stress components,
2 A +
6C
a 4 +
4D
a 2 = 0
2 A +
6C
b 4 +
4D
b 2 = −
p
2
2 A + 6Ba
2
−
6C
a 4 −
2D
a 2 = 0