Example 1 - Beltrami solution[edit | edit source]
Beltrami's solution for the equations of equilibrium states that if
where is a stress function, then
Airy's stress function is a special form of , given by (in 33 matrix notation)
Verify that the stresses when expressed in terms of Airy's stress function satisfy equilibrium.
In index notation, Beltrami's solution can be written as
For the Airy's stress function, the only non-zero terms of are which can have nine values. Therefore,
Since for , the above set of equations reduces to
Now, is non-zero only if , and is non-zero
only if . Therefore, the above equations further reduce to
Therefore, (using the values of , and the fact that the order of differentiation does not change the final result), we get
The equations of equilibrium (in the absence of body forces) are given by
Plugging the stresses in terms of into the above equations gives,
Noting that the order of differentiation is irrelevant, we see that
equilibrium is satisfied by the Airy stress function.