Introduction to Elasticity/Sample final 5
Sample Final Exam Problem 5 
Assuming that plane sections remain plane, it can be shown that the potential energy functional for a beam in bending is expressible as
where is the position along the length of the beam and is the beam's deflection curve.
- (a) Find the Euler equation for the beam using the principle of minimum potential energy.
- (b) Find the associated boundary conditions at and .
Taking the first variation of the functional , we have
Integrating the first terms of the above expression by parts, we have,
Integrating by parts again,
Using the principle of minimum potential energy, for the functional to have a minimum, we must have . Therefore, we have
Since and are arbitrary, the Euler equation for this problem is
and the associated boundary conditions are