132
5 Two-Dimensional Problems in Cartesian Co-ordinate System
Thus,
σ x =
∂
2
φ
∂ y 2 −
and σ y =
∂
2
φ
∂ x 2 −
(5.25)
For two-dimensional problems with body forces, the equilibrium equations are
given by
∂σ x
∂ x
+
∂τ xy
∂ y
+ F x = 0
(5.26)
∂σ y
∂ y
+
∂τ xy
∂ x
+ F y = 0
(5.27)
Substituting Eq. (5.24) in Eqs. (5.26) and (5.27), we get
∂
∂ x
(σ x − ) +
∂τ xy
∂ y
= 0
(5.28)
∂
∂ y
σ y −
+
∂τ xy
∂ x
= 0
(5.29)
But compatibility equation for strains is given by
∂
2
ε x
∂ y 2 +
∂
2
ε y
∂ x 2 =
∂
2
γ xy
∂ x ∂ y
(5.30)
For plane stress problems, the stress–strain relations are
ε x =
1
E
σ x − νσ y
ε y =
1
E
σ y − νσ x
ε z = −
ν
E
σ x + σ y
γ xy =
τ xy
G
= 2
(1 + ν)
E
τ xy
Substituting the above stress–strain relations in Eq. (5.30) and then simplifying,
we get
∂
2
∂ y 2
σ x − νσ y
+
∂
2
∂ x 2
σ y − νσ x
= 2(1 + ν)
∂
2
τ xy
∂ x ∂ y
(5.31)