Euler's critical buckling load formula, derived from elastic column theory, has a limited range of validity: it applies only when a column's slenderness ratio (effective length divided by radius of gyration) exceeds a material-dependent limiting value, beyond which the predicted critical stress would exceed the material's proportional limit or yield stress. Columns are classified as long (Euler's formula applies), short (failure governed by yielding, critical stress equals yield stress), or intermediate, for which Rankine's empirical formula combines the direct-compressive (yield) load and Euler's buckling load to estimate the critical buckling load. This belongs to structural/mechanical engineering (mechanics of materials), extending elastic stability theory of columns to account for material strength limits that Euler's original formula omits.
Euler's critical buckling load formula, derived from elastic column theory, has a limited range of validity: it applies only when a column's slenderness ratio (effective length divided by radius of g…