Elasticity/Fourier series solutions

Using the Airy Stress Function : Fourier Series Solutions

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Useful for more general boundary conditions.

Suppose

 

Substitute into the biharmonic equation. Then,

 

or, equivalently,

 

The hyperbolic form allows us to take advantage of symmetry about the   plane.

If  ,

 


Example of Fourier Series Technique

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Bending of an elastic beam on a foundation

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,

 

At  ,

 

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.