Euler's Formula for Pin-Ended Columns

Consider a column PQ, pin-connected at both ends, with a centric axial load applied at end P. Buckling occurs if this load surpasses the critical load.

To calculate the critical load, envision the column as a vertical beam. Now, consider a point O on an elastic curve of the beam at a distance x from the free end P and having the deflection y from the vertical.

The bending moment at point O can be written as the second derivative of its deflection with respect to the distance x. Rearranging the terms, a second-order differential equation is obtained, expressing a solution in terms of sine and cosine functions.

The first boundary condition requires that the coefficient B be zero. The second condition requires either coefficient A or the sine term to be zero.

Making the sine term zero yields the axial load expression, with the lowest value being the critical load; this is Euler's formula.

By substituting Euler's formula into the differential equation, an equation for the elastic curve after buckling is obtained.