A long rectangular beam with cross section
A solution for the displacement and stress fields, using strong boundary conditions on the edges and
[Hint : Assume that the displacement can be expressed as a second degree polynomial (using the Pascal's triangle to determine the terms) ]
Step 1: Boundary conditions
Step 2: Assume a solution
Let us assume antiplane strain
Step 3: Calculate the stresses
The stresses are given by , and .
Step 4: Satisfy stress BCs
Thus we have,
Since and can be arbitrary, .
Hence, which gives us
Assume that the body force is zero. Then the equilibrium condition is . Therefore,
Therefore, the stresses are given by
Step 5: Satisfy displacement BCs
The displacement is given by
If we substitute , we cannot determine the constant uniquely.
Hence the displacement boundary conditions have to be applied in a weak sense,