Example : Rotating Rectangular Beam

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A rotating rectangular beam

The body force potential is given by

 

Hence,

 

or,

 

The compatibility condition (in terms of stress) is

 

Plug   in to get

 

Since   is even in   and   and BCs are homogeneous, assume

 

Hence,

 

The traction BCs are

 

Apply BCs at  .

 

Therefore,

 

We then have,

 

Plug into compatibility equation

 

to get

 

or,

 

Apply BCs at  .

 

Strong BCs imply that

 

which cannot be true. So weak BCs on   need to be applied at  .

 

Hence,

 

or,

 

Hence,

 

The stress field is, therefore,

 

or,

 

The displacements can be found in the standard manner.

 
Stresses ( ) in a rotating rectangular beam
 
Stresses ( ) in a rotating rectangular beam