I was working on a project in which I required to study the buckling of a miniature beam by both analytical equations and FEM. Since I did not find the equations anywhere in textbooks or on the internet, I developed the equations. Here, I am sharing the simplified version of that analysis.
Let’s assume there is a simply supported beam with length L. The modulus of elasticity and moment of area are E and I, respectively. A horizontal force P is applied at the middle of the beam. I need to find the critical load P which causes the buckling.
First, the support reactions are obtained as follows:
The deflection at the middle of the beam is assumed to be Δ. The magnitude of Δ will be calculated later.
Then, I need to look at the bending moment between the left support A and the midpoint.
by defining β as follows, the second order differential equation is solved.
Now we can apply some the boundary conditions of the beam to solve for the unknowns.
By replacing Δ, support forces can be simplified.
Now, we can look at the bending moment and deflection equation of the other half of the beam.
Then, we can apply the rest of boundary conditions as follows:
Deflection at Point B is zero.
Both equations must give us identical answers for the deflection and slope at the midpoint.
The equation above is the one that helps us get the critical force. We know A_0 cannot be zero so the expression in the bracket must be zero.
The equation above has infinite number of solutions. However, we need the first positive value. I used MATLAB to solve for β:
% x=beta*L/2 f=inline('x*cos(x)+sin(x)*(3-1/3*x^2)') fzero(f,3)