Introduction to Elasticity/Warping of rectangular cylinder

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Example 3: Rectangular Cylinder[edit | edit source]

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: or [edit | edit source]

In both these cases, we get trivial values of .

Case 2: [edit | edit source]

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 .