Q:

Restoring force in circular rod

Can somebody explain how to calculate the restoring force in a uniform circular rod with known Young's modulus and diameter. I would need the restoring force in a specified distance from the origin with also a specified deviance.

A:

The force of a stress uniformly distributed over an area A is
$$
F = \sigma \times A
$$
Stress = Young's modulus * strain
$$
\sigma = E \times \epsilon
$$
Strain = displacement / Length
$$
\epsilon = {u \over L}
$$
so putting it together, along with the area of a circle:
$$
F = E \times {u \over L} \times \pi \times d^2/4
$$
That gives positive force for a positive displacement. If you want the restoring force applied to the thing that causes the displacement, negate it.
The force is the same at any distance along the bar.

