Euler's Formula for Pin-Ended Columns

In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.

To calculate the critical load, envision column PQ as a vertical beam. Consider point O, situated on the elastic curve of the beam, at a distance x from the free end P. With the application of the load, point O gets deflected by a distance y from its original vertical position. At this point, the bending moment at point O can be described by the second derivative of its deflection, y, with respect to x, symbolizing a pivot towards understanding the beam's behavior under stress.

Where f is defined as,

This equation has a general solution having sine and cosine terms. The boundary values of the system give the coefficients of the solution.

The solution requires that the sine term be zero, giving the expression for critical load. This expression is known as Euler's formula.

Substituting Euler's formula back into the differential equation gives the expression for the elastic curve of the column after buckling.

Here, it is important to note that Euler's formula is derived from the assumptions that before loading, the column be perfectly straight, homogeneous, and isotropic and that the axial load is applied perfectly along the vertical axis.