Learning topic
Stability of Compressed Members
Learn column buckling and stability: Euler critical load, effective length and end conditions, slenderness ratio, weak-axis buckling, and applicability limits.
Stability of compressed members is their ability to preserve the initial equilibrium configuration under compression. A slender column may buckle laterally before its average compressive stress reaches the material strength limit.
Section roadmap
The section begins with the mechanism and modes of buckling. It then introduces radius of gyration, slenderness ratio and the weak buckling axis, followed by Euler's formula for long elastic columns.
Separate topics cover critical-stress curves and inelastic buckling, column end conditions and effective length, practical design using a buckling reduction factor, and special cases such as imperfections, local buckling and non-prismatic columns.
Core equations
For an ideal elastic column,
\[P_{cr}=\frac{\pi^2EI_{min}}{(KL)^2}.\]
The slenderness ratio and Euler critical stress are
\[\lambda=\frac{KL}{r_{min}},\qquad r_{min}=\sqrt{\frac{I_{min}}{A}},\qquad \sigma_{cr,E}=\frac{\pi^2E}{\lambda^2}.\]
Real columns contain initial imperfections, eccentricities, residual stresses and non-ideal restraints. Euler's formula is therefore a fundamental theoretical model, while design resistance must be determined using the applicable structural design standard.
About this topic
Buckling is a sudden side-way deflection mode occurring in slender compression members when axial loads reach a critical threshold Pcr. This section covers stable, unstable, and neutral equilibrium states, factors influencing critical load capacity, and column stability verification principles.