# FEA (hand calc) - beam buckling

## Homework Statement

http://www.imageupload.org/thumb/thumb_62195.jpg

## Homework Equations

KQ = PKGQ
where K = stiffness matrix,
Q = displacement vector,
P = buckling load,
KG = geometric stiffness matrix

det(K-PKG)=0

## The Attempt at a Solution

Elements 1 (A-B), 2 (B-C) and 3 (vertical) each have element stiffness matrices where K1=K2 and K3 = (L)'(K1)(L) where L is a 6x6 transformation matrix. When assembled, the global stiffness matrix is a 12x12

The geometric stiffness matrix can also be assembled to a 12x12

Boundary conditions (where each node (in order of A,B,C,D) has 3 DOF: displacement in x, displacement in y, rotation about z) are:
Q2 = 0
Q4 = Q5 = 0
Q8 = 0
Q10 = 0

So eliminating columns and rows 2,4,5,8,10 reduces the K-KG matrix to a 7x7, which is still too large to solve by hand in half an hour (which is the time per question in the exam this question's from). Also, I tried solving det(__)=0 for the 7x7 matrix using matlab and unless I've made mistake in inputting it, it was unable to find a solution whereas it found a few solutions for the 12x12, some of which were zero.

Am I missing something?

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