Answer Happy

Determine the stress in a semi-infinite plate due to a normal
load, w force /unit length, acting on its edge given an Airy’s
stress function of the form ∅ = 𝐴𝜃 + 𝐵𝑟 2 + 𝐶𝑟𝜃𝑠𝑖𝑛𝜃 + 𝐷𝑟𝜃𝑐𝑜𝑠𝜃
exists, where A, B, C and D are constants. You may assume that the
bi-harmonic function of an Airy’s stress function in polar
coordinates is ∇ 4∅ = ( 𝜕 2 𝜕𝑟 2 + 1 𝑟 𝜕 𝜕𝑟 + 1 𝑟 2 𝜕 𝜕𝑟) 2 ∅ = 0
and that the stresses are given by 𝜎𝑟 = 1 𝑟 𝜕∅ 𝜕𝑟 + 1 𝑟 2 𝜕 2∅ 𝜕𝜃2
; 𝜎𝜃 = 𝜕 2∅ 𝜕𝜃2 ; and 𝜏𝑟𝜃 = 1 𝑟 2 𝜕∅ 𝜕𝜃 − 𝜕 2∅ 𝜕𝑟𝜕𝜃