Airy stress function

∂σ x

∂x+∂τ yx∂y

= 0

∂τ xy∂x

+∂σ y

∂y= 0

τ xy = τ yx

σ x =∂2φ∂y2

σ y =∂2φ∂x2

τ xy = −∂2φ∂x∂y

A

BCD

E G I

F H

a

b

c

d e

f

g

h

i

Timoshenko and Goodier Theory of elasticity