One of the classical problems in the Calculus of Variations is the
buckling of a beam. You will analyze a beam under stress (possibly
with several different boundary conditions). The critical load for a
beam is determined by an eigenvalue analysis of the energy equation
for the beam. Discuss the techniques from the Calculus of variations
used in this problem. You might want to show how the Rayleigh
Quotient can be used to determine the buckling load. If time permits,
then show how this point of stress just prior to the buckling of the
beam leads to instabilities in the finite element methods for
approximating solutions.
A reference below provides an introduction to this problem. You
may want to check other texts on the Calculus of variations to give
you alternative perspectives. There are a collection of locations on
the internet that should help you make progress on this problem. For
more mathematical analysis of the buckling beam, examine the
websites:
http://webserver.aero.usyd.edu.au/structures/mos/Mosch08Con.html
For more information on the finite element method applied to thin
beams consider the following websites:
http://www.ce.luth.se/abb/master/Hellingeng.html
http://www3.sympatico.ca/peter_budgell/Modeling_issues.html
References: