# Finite Difference Method for Beam Buckling ```CE579 – Structural Stability and Design
Homework No. 6, 7, 8
Due Date: April 15, 2013
Problem No. 1
A] Consider the differential equation for lateral torsional buckling of beams subjected to uniform
bending moment.
f iv - l1f &cent;&cent; - l2f = 0
Solve this differential equation analytically for the boundary condition: (i) fix-fix for torsion.
Use the solution to compute the critical moment for a W27x94 beam section with length equal to
200 ry.
Compare the solution with that obtained numerically using the finite difference method shown in
the handout in class. Please note that the handout for the UNIFORM moment case already used
fix-fix end conditions not simply supported.
Compare the solution with that using the lateral torsional buckling equation from the AISC
specification. In that equation, account for end condition effects using the effective length
concept.
Problem No. 2
Use the finite difference method and inverse iteration eigenvalue calculation methods to
numerically solve and determine the elastic critical buckling moment for the W27x94 beam
problem described above, but for the case of NON-UNIFORM moment. Assume the moment
diagram to be varying linearly (zero at one end, and maximum value at the other end) like
explained in class.
Assume eight stations along the length of the beam.
Calculate the effective Cb value using the results from your implementation with those for the
case of UNIFORM moment calculated analytically.
Comment on your results, and suggest ways to improve / enhance the methodology.
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