Introduction to Elasticity/Energy methods example 3

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Example 3 : Torsion of a cylinder[edit | edit source]

Suppose that the cross-section of the cylinder is bounded by the curves:

A statically admissible Prandtl stress function for this cross-section is

with the restrictions that is twice continuously differentiable and .

We seek to derive the best approximate Prandtl stress function of this form by minimizing the complementary energy.

You can show that the complementary energy per unit length of the cylinder can be expressed as

Plugging in the given form of and after some algebra, we get

Taking the variation of , and after considerable manipulation, we get

Now, if we consider the cross-section to be rectangular, then we have , , and . Therefore, the above equation reduces to

Therefore, the function that minimizes satisfies the equation

or,

with the static admissibility conditions .


The general solution of the above equation is

Therefore, from the BCs,

We can substitute back into to get the approximate Prandtl stress function for this problem. The error between the exact and this approximate solution is generally less than 1%.