A long, slender steel column is fixed at one end and free at the other. If its unbraced length is doubled, what is the precise impact on its critical buckling load?

The critical buckling load of a long, slender column is the maximum axial compressive force it can withstand before it suddenly deflects laterally and collapses, a phenomenon known as elastic buckling. This load is precisely determined by Euler's buckling formula: Pcr = (π² E I) / (K L)².

Here, Pcr is the critical buckling load, representing the specific axial load at which buckling occurs. E represents the Modulus of Elasticity, which is a material property indicating the column's stiffness or resistance to elastic deformation; for steel, this value is constant under normal operating conditions. I is the Moment of Inertia of the column's cross-section, a geometric property that quantifies its resistance to bending, with buckling typically occurring about the axis of minimum moment of inertia. L is the unbraced length, which is the actual physical length of the column between points of lateral support. K is the effective length factor, a dimensionless value that accounts for the column's end support conditions; it modifies the actual length L into an effective length (KL) that represents an equivalent pin-ended column (K=1) with the same buckling capacity.

For a column that is fixed at one end and free at the other, as described in the question, the theoretical effective length factor (K) is 2.0. This value reflects how the specific end constraints influence the column's overall stiffness against buckling.

If the unbraced length (L) is doubled, the new length becomes 2L. Substituting this new length into Euler's formula, the new critical buckling load (Pcr_new) can be calculated: Pcr_new = (π² E I) / (K (2L))². Simplifying this expression, we get Pcr_new = (π² E I) / (K² 4L²) = (1/4) [(π² E I) / (K L)²]. The term in the square brackets is the original critical buckling load (Pcr_original).

Therefore, when the unbraced length of a long, slender steel column fixed at one end and free at the other is doubled, its critical buckling load is reduced to one-fourth (1/4) of its original value.