AE/ME 6222 Theory of Elasticity

Stresses, strains and displacements

1. The rectangular plate -a

edges y ==b being traction-free. The plate is loaded only by its own weight in the

negative y-direction. The body force potential V=pgycan describe the gravitational

force. For this problem, the Airy's stress function is given by (x,y) = Ax²y + By³.

a. Find a solution for the stress field assuming uniform shear tractions on x==a

Use strong boundary conditions on x==a and weak conditions on y = zb.

(Hints: find and show A=P3; B = P8; ₂ = ₁ = 0; x = px)

2

b. Find and show that the displacements corresponding to this stress field are

(1+ v) pg(a²-x²)

given by: ₂ = 0; u =

=-

E

c. Do the resulting expressions permit the built-in boundary condition at x = ta to

be satisfied in the strong sense?