Strength of Materials — Columns and Struts At what load does a slender column just begin to buckle (lose lateral stability) under axial compression?

Introduction:
Columns fail by buckling long before the material reaches its crushing strength. The precise load at which a perfectly straight, slender column just becomes laterally unstable is a fundamental idea in strength of materials and design of compression members.

Given Data / Assumptions:

• Member: long, slender column.
• Loading: concentric axial compression.
• Ideal behavior: linearly elastic, small initial imperfections neglected for definition.

Concept / Approach:
The onset of buckling is predicted by Euler's theory. The load that makes a column neutrally stable (neither straight nor bent without additional load) is called the critical load. Many texts also use the term crippling load for the same quantity in the elastic range. Buckling load is a generic phrase, but the standard technical term is critical load.

Step-by-Step Solution:
For an ideal pin–pin column: P_cr = (pi^2 * E * I) / (L^2)For other end conditions: replace L with L_e (effective length).At P = P_cr, any small lateral deflection can persist without additional load, indicating the threshold of instability.

Verification / Alternative check:
Design codes (e.g., using KL/r) identify the same threshold as the transition from stable shortening to lateral instability.

Why Other Options Are Wrong:
buckling load: colloquial; not the standard precise term in formulas.crippling load: used in some texts, but critical load is the accepted technical name.any one of these: terminology matters in design; only critical load is unambiguous.

Common Pitfalls:
Confusing crushing strength with buckling strength; slender columns fail at much lower loads.Ignoring end conditions (effective length) when computing P_cr.

Final Answer:
critical load