Buckling of Columns

Columns are long structural members typically subjected to axial compressive forces. Unlike short compression members that fail by material yielding or crushing, long columns often fail by buckling.

Before applying column formulas, it is essential to understand the two primary modes of buckling instability in structural members:

Global Buckling is the primary focus of standard column analysis.

Local Buckling occurs when a section is susceptible to localized folding. Such a section is classified as a "slender element section" by structural codes (like NSCP/AISC) and its compressive strength must be significantly reduced.

For a long, slender column failing by global buckling, the critical load at which elastic buckling occurs is given by Euler's Formula. The buckling occurs about the weak axis, meaning the calculation must use the minimum moment of inertia ($I$).

The critical buckling stress can be determined by dividing the critical buckling load by the cross-sectional area.

Euler's formula assumes an ideally straight column with perfectly axial loading. In reality, columns have initial imperfections and loads are often applied eccentrically (off-center).

The Secant Formula is used for columns with a known load eccentricity $e$. As the load increases, the secant term grows non-linearly, leading to yielding at the extreme fibers before theoretical Euler buckling occurs.

Euler's formula is valid only for long, slender columns where failure is purely by elastic buckling (when $\sigma_{\text{cr}} < \sigma_{\text{Y}}$). For short columns, failure occurs by yielding ($\sigma_{\text{Y}}$). For intermediate columns, failure is a complex combination of yielding and buckling.

To bridge the gap between pure yielding and pure Euler buckling, empirical formulas are used. The American Institute of Steel Construction (AISC) defines a critical slenderness ratio, $C_{\text{c}}$, to distinguish between intermediate and long columns.

• Intermediate Columns ($KL/r < C_{\text{c}}$): The allowable stress is defined by a parabolic formula (often called the Johnson Parabola) to account for inelastic buckling and residual stresses.
• Long Columns ($KL/r \ge C_{\text{c}}$): The allowable stress is defined using Euler's elastic buckling formula with an appropriate factor of safety.

To achieve very high compressive strength or stiffness over long spans without resorting to massively thick single sections, engineers design built-up columns. The goal is to increase the total radius of gyration ($r$) about the weak axis by moving the constituent shapes away from the centroidal axis, thereby drastically reducing the overall slenderness ratio ($KL/r$).

Because the individual elements are connected intermittently (not continuously welded), the built-up column does not behave as perfectly as a solid section. The shear deformation in the lacing or battens causes an increase in the effective slenderness of the column.

Visualize how length, stiffness, and end conditions affect the critical buckling load and the buckled shape. Interact with the simulation below to observe these effects in real-time.