Elasticity/Warping of rectangular cylinder

Example 3: Rectangular Cylinder

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In this case, the form of   is not obvious and has to be derived from the traction-free BCs

 

Suppose that   and   are the two sides of the rectangle, and  . Also   is the side parallel to   and   is the side parallel to  . Then, the traction-free BCs are

 

A suitable   must satisfy these BCs and  .


We can simplify the problem by a change of variable

 

Then the equilibrium condition becomes

 

The traction-free BCs become

 

Let us assume that

 

Then,

 

or,

 

Case 1: η > 0 or η = 0

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In both these cases, we get trivial values of  .

Case 2: η < 0

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Let

 

Then,

 

Therefore,

 

Apply the BCs at   ~~ ( ), to get

 

or,

 

The RHS of both equations are odd. Therefore,   is odd. Since,   is an even function, we must have  .

Also,

 

Hence,   is even. Since   is an odd function, we must have  .


Therefore,

 

Apply BCs at   ( ), to get

 

The only nontrivial solution is obtained when  , which means that

 

The BCs at   are satisfied by every terms of the series

 

Applying the BCs at   again, we get

 

Using the orthogonality of terms of the sine series,

 

we have

 

or,

 

Now,

 

Therefore,

 

The warping function is

 

The torsion constant and the stresses can be calculated from  .