Introduction to Elasticity/Fourier series solutions
Using the Airy Stress Function : Fourier Series Solutions
Useful for more general boundary conditions.
Substitute into the biharmonic equation. Then,
The hyperbolic form allows us to take advantage of symmetry about the plane.
Example of Fourier Series Technique
The traction boundary conditions are
The problem is broken up into four subproblems which are superposed. The subproblems are chosen so that the even/odd properties of hyperbolic functions can be exploited.
The loads for the four subproblems are chosen to be
The new boundary conditions are
Let us look at the subproblem with loads applied on the top and bottom of the beam. The problem is even in and odd in . So we use,
Hence if .
We can substitute and express the stresses in terms of Fourier series.
Applying the boundary conditions of we get
The first equation is satisfied if
Integrate the second equation from to after multiplying by .
All the odd functions are zero, except the case where .
Therefore, all that remains is
We can calculate and from equations (1) and (2), substitute them into the expressions for stress to get the solution.
We do the same thing for the other subproblems.
The Fourier series approach is particularly useful if we have discontinuous or point loads.