Introduction to Elasticity/Sample final 5

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Sample Final Exam Problem 5[edit | edit source]

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:[edit | edit source]

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