Sample Final Exam Problem 5

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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.

 
Beam bending problem
  • (a) Find the Euler equation for the beam using the principle of minimum potential energy.
  • (b) Find the associated boundary conditions at   and  .

Solution:

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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