Introduction to Elasticity/Airy example 1

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Example 1 - Beltrami solution[edit | edit source]

Given:

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)

Show:

Verify that the stresses when expressed in terms of Airy's stress function satisfy equilibrium.

Solution[edit | edit source]

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

or,

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.