FEA (hand calc) - beam buckling

[super sky]

Homework Statement

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

Homework Equations

KQ = PK_GQ
where K = stiffness matrix,
Q = displacement vector,
P = buckling load,
K_G = geometric stiffness matrix

det(K-PK_G)=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-K_G 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?

 

[KataKoniK]

Thank you for your post and for sharing your attempt at solving this problem. It seems like you have a good understanding of the problem and have made a good attempt at solving it. However, I would recommend double checking your calculations and inputs in MATLAB to make sure there are no errors.

Additionally, it is possible that the 7x7 matrix is too large to solve by hand in the given time frame. In this case, you may want to try using a numerical method, such as the finite element method, to solve the problem. This would involve breaking the structure into smaller elements and using numerical techniques to solve for the displacements and buckling load.

I hope this helps and good luck with your exam!