6.15 Numerical Examples
199
φ =
P
π
r θ cos θ
Determine the stress components σ r , σ θ and τ r θ .
Solution: The stress components, by definition of ϕ, are given as follows
σ r =
1
r
∂φ
∂r
+
1
r 2
∂
2
φ
∂θ 2
(i)
σ θ =
∂
2
φ
∂r 2
(ii)
τ r θ =
1
r 2
∂φ
∂θ
−
1
r
∂
2
φ
∂r ∂θ
(iii)
The various derivatives are as follows:
∂φ
∂r
=
P
π
θ cos θ
∂
2
φ
∂r 2 = 0
∂φ
∂θ
=
P
π
r (−θ sin θ + cos θ )
∂
2
φ
∂θ 2 = −
P
π
r (θ cos θ + 2 sin θ)
∂
2
φ
∂r ∂θ
=
P
π
(−θ sin θ + cos θ)
Substituting the above values in Eqs. (i), (ii) and (iii), we get
σ r =
1
r
P
π
θ cos θ −
1
r 2
P
π
r (θ cos θ + 2 sin θ )
=
1
r
P
π
θ cos θ −
1
r
P
π
θ cos θ −
1
r
P
π
2 sin θ
∴ σ r = −
2
r
P
π
sin θ
σ θ =
∂
2
ϕ
∂r 2 = 0
τ r θ =
1
r 2
P
π
r (−θ sin θ + cos θ ) −
1
r
P
π
(−θ sin θ + cos θ )
∴ τ r θ = 0
Therefore, the stress components are