Compression Members (Columns)

Compression members, primarily columns but also compression truss chords and bracing struts, are structural elements subjected to axial compressive forces. Unlike tension members, which generally fail by yielding or rupture, the strength of compression members is almost always governed by stability, specifically buckling failure.

Buckling is a sudden loss of stability where a member deflects laterally under compressive load before the material itself reaches its yield stress. The critical load depends primarily on the member's flexural stiffness ($EI$), length ($L$), and end support conditions ($K$).

The Euler formula assumes a column is pinned at both ends. Real columns have various support conditions (fixed, free, etc.) which alter the effective buckling length.

Theoretical and recommended design values for $K$ based on idealized boundary conditions:

• Pinned-Pinned: $K = 1.0$
• Fixed-Fixed: $K = 0.5$ (Theoretical), 0.65 (Design value)
• Fixed-Pinned: $K = 0.707$ (Theoretical), 0.80 (Design value)
• Fixed-Free (Cantilever): $K = 2.0$ (Theoretical), 2.10 (Design value)

The critical buckling axis is the one with the highest slenderness ratio $KL/r$. For standard W-shapes, this is almost always the weak y-axis ($I_y \ll I_x$), unless the y-axis is braced at intermediate points.

For columns in continuous frames (where the ends are neither perfectly fixed nor perfectly pinned), $K$ must be determined using alignment charts found in AISC Appendix 7.

• Sidesway Prevented (Braced Frames): The frame is braced against lateral translation. $K$ ranges from 0.5 to 1.0.
• Sidesway Uninhibited (Moment Frames): The frame relies on the bending stiffness of columns and beams for lateral stability. $K$ ranges from 1.0 to infinity.
• G-Factor: The relative stiffness ratio ($G$) at each end of the column is calculated: $G = \frac{\sum (EI/L)_{\text{columns}}}{\sum (EI/L)_{\text{beams}}}$. The $G$ values at the top ($G_A$) and bottom ($G_B$) are used to read the $K$ factor from the nomograph.

The theoretical Euler buckling formula assumes a mathematically perfect, perfectly straight column loaded exactly at its centroid. Real structural steel columns deviate from this ideal in two critical ways:

1. Residual Stresses: As discussed in the introduction, uneven cooling induces internal stresses that cause premature, localized yielding before the entire cross-section reaches the yield point.
2. Initial Out-of-Straightness: The AISC Code of Standard Practice allows columns to have a slight initial curvature or "sweep" (typically limited to $L/1000$ between points of lateral support).

When a compressive load is applied to a column that is already slightly bent, the initial curvature acts as an eccentricity. This immediately induces bending moments (a $P-\delta$ effect) from the very start of loading, accelerating the onset of buckling. The empirical AISC column curves specifically account for both residual stresses and a maximum allowable initial out-of-straightness of $L/1500$ in their derivation.

The Euler formula assumes a perfectly straight, perfectly elastic column. Real columns have initial crookedness and residual stresses from uneven cooling during manufacturing. These imperfections cause the column to yield prematurely, reducing its strength below the theoretical Euler load. The AISC Specification uses two equations based on the slenderness ratio to account for this inelastic behavior.

First, calculate the theoretical Elastic Buckling Stress ($F_e$):

Case 1: Inelastic Buckling (Short/Intermediate Columns)

If $\frac{KL}{r} \le 4.71\sqrt{\frac{E}{F_y}}$ (or $F_y/F_e \le 2.25$):

This represents the non-linear portion of the column curve where residual stresses cause partial yielding before buckling. The vast majority of building columns fall into this category.

Case 2: Elastic Buckling (Slender Columns)

If $\frac{KL}{r} > 4.71\sqrt{\frac{E}{F_y}}$ (or $F_y/F_e > 2.25$):

This is the Euler formula, multiplied by a 0.877 reduction factor to account for initial out-of-straightness.

Design Strength

While flexural (Euler) bending buckling is the primary mode for closed shapes (HSS) and doubly symmetric open shapes (W-shapes), other buckling modes can govern, especially for asymmetric shapes.

• Torsional Buckling: The column simply twists about its longitudinal shear center axis without bending. This mode can theoretically govern for doubly symmetric shapes with very weak torsional resistance, like cruciform (+) shapes or some built-up sections, but rarely governs for W-shapes unless their unbraced length for torsion is much larger than for flexure.
• Flexural-Torsional Buckling: A simultaneous combination of twisting and lateral bending. This is the primary buckling mode for singly symmetric shapes (like Tees, WT, and double angles) and unsymmetric shapes (like single angles). The shear center and centroid do not coincide, causing bending forces to induce torsion.

When these modes govern, the elastic buckling stress $F_e$ in the AISC equations must be calculated using complex formulas (AISC Section E4) that account for the torsional constant ($J$) and warping constant ($C_w$) of the section, rather than just the weak-axis $KL/r$.

The cross-section itself is composed of plate elements (flanges and webs). These individual elements can buckle locally (wrinkle) under compression before the entire member buckles globally.

• Non-Slender: Elements can reach $F_y$ before local buckling. No strength reduction required. Most standard hot-rolled W-shapes are non-slender for compression.
• Slender: Elements buckle elastically before reaching $F_y$. A reduction in global capacity is required.

AISC Table B4.1a provides limiting width-to-thickness ratios ($\lambda_r$) to classify sections. For example, a W-shape flange is checked using $b_f/2t_f \le 0.56\sqrt{E/F_y}$.

If a cross-section has slender elements, the full gross area cannot be effectively utilized because the slender parts will buckle locally and shed load to the stiffer corners. This reduces the overall buckling capacity.

AISC Section E7 requires the calculation of an effective area $A_e$, which is smaller than the gross area $A_g$. Alternatively, in newer codes, an effective yield stress is often used via reduction factors.

• $Q = Q_s \times Q_a \le 1.0$
• $Q_s$ (Unstiffened Elements): Reduction factor for elements supported along only one edge (like the outstanding flanges of a W-shape, angle legs, or tee stems). Calculated based on the width-to-thickness ratio.
• $Q_a$ (Stiffened Elements): Reduction factor for elements supported along two edges (like the web of a W-shape or the walls of an HSS). Calculated using an effective width $b_e$.

Steel columns must transfer their massive axial loads safely into the concrete foundation. A steel base plate is welded to the bottom of the column to spread this load over a larger area of the weaker concrete, preventing the concrete from crushing.

Once the required base plate area ($A_1$) is determined, the required thickness ($t_p$) is calculated by assuming the base plate acts as a cantilever beam extending outwards from the column flanges/web, resisting the upward bearing pressure from the concrete.

Steel columns must transfer their massive axial loads safely into the concrete foundation. A steel base plate is welded to the bottom of the column to spread this load over a larger area of the weaker concrete, preventing the concrete from crushing.

Once the required base plate area ($A_1$) is determined, the required thickness ($t_p$) is calculated by assuming the base plate acts as a cantilever beam extending outwards from the column flanges/web, resisting the upward bearing pressure from the concrete.

When standard rolled shapes are insufficient, two or more shapes can be connected together (e.g., two channels back-to-back, or four angles in a box). The individual shapes must act as a single, unified member to achieve the desired flexural stiffness ($EI$). They must be connected at intervals to prevent them from buckling independently between connection points.

• Lacing: Diagonal bars zigzagging between the main members. They act like the web of a truss, resisting shear forces induced during column buckling. They are highly effective.
• Batten Plates (Tie Plates): Horizontal plates placed at intervals. They act like rigid frame connections (Vierendeel truss action). They are less efficient than lacing and require heavier main members because they induce local bending moments.

Furthermore, the global slenderness ratio of the built-up member must be modified to account for shear deformation in the connectors.