Hello,

I am working on the derivation of ideal buckling of columns under fixed supports at both ends.

The general equation that describes the deflection of the column is:

Where A,B,C,D are constants, k is also a constant ( E = modulus of elasticity and I is second moment of inertia of the member)

I have four boundary conditions for a fixed-fixed support that are:

where L is the full length of the column. These boundary condition simply mean that the deflection at each support is 0 and the slope of the deflection curve is also 0.

The purpose of this exercise is to determine the critical axial load P that will cause buckling.

Ideally I want to solve for k, then isolate for P which is gives critical axial load. The solution to this is:

where n is any natural number, which determines the shape of the deflected shape. For simplicity we will take n=1.

I have generated this matrix to solve:

I want the non trivial solution, that is the 4x4 matrix = 0.

I have forgotten how to solve such matrices...

However, in class the prof briefly mentioned that we use Cramer's method to solve

This included finding the determinant of the matrix until you get a numerical equation.

I computed the determinant of the above and got:

If the above is true, how would you explicitly solve for k?

Any guidance is appreciated.