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.
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- (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,
Expanding out,
Rearranging,
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
and