26.2:
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.
26.2:
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.