Example 3 : Torsion of a cylinder[edit | edit source]
Suppose that the cross-section of the cylinder is bounded by the
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
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%.