15.3 Design considerations for columns

Column Cross-Section Selection

Cross-Section Shapes and Factors Influencing Selection

Columns come in several common cross-section shapes: rectangular, circular, hollow structural sections (HSS), I-shaped (wide-flange), and built-up sections. Each shape has trade-offs, and the right choice depends on several practical factors:

• Load-carrying capacity: Select a cross-section that efficiently resists the applied loads. For example, hollow sections resist buckling well in all directions, while wide-flange sections are more efficient when buckling is restrained about one axis.
• Aesthetics: The column's visual appearance matters in exposed structures like building lobbies or bridges.
• Ease of fabrication: Some shapes are simpler to manufacture, cut, and assemble on site.
• Connection requirements: The cross-section must be compatible with the intended connection methods (bolting, welding). Wide-flange columns, for instance, offer flat surfaces that simplify bolted connections.

Strength and Stability Considerations

Strength considerations involve ensuring the column can withstand applied axial loads and bending moments without exceeding the material's yield stress or buckling stress. Stability requirements relate to the column's ability to resist buckling under compression, which depends heavily on two geometric properties:

• Moment of inertia ($I$): Measures the cross-section's resistance to bending. Higher values of $I$ directly improve buckling resistance because they increase the critical buckling load.
• Radius of gyration ($r$): Defined as $r = \sqrt{I/A}$, where $A$ is the cross-sectional area. A higher $r$ means a lower slenderness ratio, which reduces the risk of buckling.

Compact sections have a higher local buckling resistance compared to slender sections, making them better suited for columns under high compressive stress. The width-to-thickness ratios of cross-section elements (flanges and webs) control local buckling behavior and must stay within limits specified by design codes.

Slenderness Ratio Influence

Definition and Effects on Column Behavior

The slenderness ratio is defined as the effective length of the column divided by its least radius of gyration: $L_e/r$. This single ratio largely determines how a column will fail.

• Slender columns (high $L/r$) are governed by global (Euler) buckling, where the entire column deflects laterally as a single unit.
• Stocky columns (low $L/r$) fail by material yielding or local buckling of individual elements like flanges or webs.
• Intermediate columns fall between these extremes, and their behavior involves an interaction of yielding and instability. This is the range where most real columns land, and where design curves from codes become especially important.

As the slenderness ratio increases, the column's load-carrying capacity decreases because the risk of elastic buckling grows.

Buckling Load and Design Code Provisions

The critical buckling load is inversely proportional to the square of the slenderness ratio, as given by Euler's formula:

$P_{cr} = \frac{\pi^2 EI}{(KL)^2}$

where $E$ is the modulus of elasticity, $I$ is the moment of inertia about the buckling axis, $K$ is the effective length factor, and $L$ is the actual column length.

Design codes set upper limits on slenderness ratios:

• AISC (USA): Limits the slenderness ratio to 200 for main compression members.
• Eurocode 3 (Europe): Provides different limiting slenderness ratios depending on column type and material grade.

The effective length factor $K$ accounts for end restraint conditions and their influence on the buckled shape:

• Pinned-pinned: $K = 1.0$
• Fixed-fixed: $K = 0.5$
• Fixed-pinned: $K = 0.7$
• Fixed-free (cantilever): $K = 2.0$

These are theoretical values. In practice, true fixity is hard to achieve, so real $K$ values often fall between the idealized cases. Design codes provide alignment charts (nomographs) to estimate $K$ for columns in frames.

Design Codes and Standards

Codes and Their Provisions

Several design codes govern column design, including AISC 360 (USA), Eurocode 3 (Europe), and CSA S16 (Canada). While the details differ, they all specify:

• Material properties and characteristic strengths
• Load combinations and resistance/safety factors
• Design equations for axial compression, tension, and combined loading

For columns under combined axial load and bending, interaction equations check whether the combined demand exceeds the column's capacity. The AISC specification uses two equations, and you apply whichever controls based on the axial load ratio:

When $P_r / P_c \geq 0.2$, use AISC H1-1a:

$\frac{P_r}{P_c} + \frac{8}{9}\left(\frac{M_{rx}}{M_{cx}} + \frac{M_{ry}}{M_{cy}}\right) \leq 1.0$

When $P_r / P_c < 0.2$, use AISC H1-1b:

$\frac{P_r}{2P_c} + \left(\frac{M_{rx}}{M_{cx}} + \frac{M_{ry}}{M_{cy}}\right) \leq 1.0$

Here, $P_r$ is the required axial strength, $P_c$ is the available axial strength, and the $M$ terms represent required and available flexural strengths about each axis.

Local Buckling and Effective Length Considerations

Before a column can develop its full global buckling or yielding capacity, its individual plate elements (flanges, webs) must not buckle locally. Codes handle this through width-to-thickness ratio limits:

• AISC: Provides tables of limiting width-to-thickness ratios ($\lambda_r$ and $\lambda_p$) based on cross-section shape and applied stress. Sections are classified as compact, noncompact, or slender.
• Eurocode 3: Classifies cross-sections into Class 1, 2, 3, or 4 based on width-to-thickness ratios. Class 1 and 2 sections can develop full plastic capacity, while Class 4 sections require effective cross-section properties due to local buckling.

If a section's plate elements exceed these limits, the column's capacity must be reduced to account for local buckling, typically by using an effective cross-sectional area or reduced strength.

Imperfections and Eccentricities

Initial Imperfections and Their Effects

No real column is perfectly straight or stress-free. Two types of initial imperfections reduce load-carrying capacity:

Out-of-straightness is the deviation of the column's centerline from a perfectly straight line. Even a small bow introduces bending moments under axial load, which grow as the load increases (a second-order effect). Codes account for this with assumed imperfection magnitudes:

• AISC: Assumes an initial out-of-straightness of $L/1000$ for design purposes.
• Eurocode 3: Uses equivalent geometric imperfections tied to the column's assigned buckling curve (curves a, b, c, or d), which vary by cross-section type and fabrication method.

Residual stresses are internal stresses locked into the cross-section from manufacturing processes like hot rolling or welding. They cause portions of the cross-section to yield earlier than expected under load, reducing the effective stiffness and therefore the buckling capacity. Hot-rolled wide-flange sections, for example, typically have residual stresses on the order of 30-50% of the yield stress at the flange tips.

Eccentricities and Design Considerations

Eccentricities occur when the applied load doesn't line up with the column's centroidal axis, producing bending moments in addition to axial compression.

• Accidental eccentricities arise from construction tolerances, load misalignment, or connection geometry that shifts the load path slightly off-center.
• Intentional eccentricities are introduced by design, such as in moment-resisting frames where beams frame into one side of a column.

Both imperfections and eccentricities reduce the column's effective buckling strength. Design codes provide several methods to account for them: applying additional equivalent moments, using reduced material properties, or incorporating their effects directly into the interaction equations discussed above.

Axial Loads vs. Bending Moments

Combined Loading Effects

Columns in real structures rarely carry pure axial compression. Eccentricities, lateral loads (wind, seismic), and frame action all introduce bending moments alongside the axial force.

Bending moments cause additional compressive stress on one side of the column and tensile stress on the other. The combined stress on the compression side can exceed the yield stress or trigger premature buckling well before the pure axial capacity is reached. This is why the interaction equations (AISC H1-1a and H1-1b) are so central to column design: they check that the combined demand from axial load and bending stays within the column's capacity.

Second-Order Effects and Design Methods

When an axial load acts on a column that has already deflected (from bending moments or imperfections), the axial force creates additional bending moment equal to $P \times \delta$, where $\delta$ is the lateral deflection. This is the P-delta effect, and it amplifies the first-order moments.

The moment magnification factor $B_1$ approximates this amplification:

$B_1 = \frac{C_m}{1 - \frac{P_r}{P_{e1}}} \geq 1$

where $C_m$ is the equivalent uniform moment factor (accounts for the shape of the moment diagram), $P_r$ is the required axial strength, and $P_{e1}$ is the Euler buckling load for the member. Notice that as $P_r$ approaches $P_{e1}$, the magnification factor grows rapidly, which is why columns loaded near their buckling capacity are so sensitive to any bending.

Two primary design methods handle these effects:

1. Effective length method: Uses the $K$ factor to approximate the column's behavior within a frame. First-order moments are amplified by $B_1$ (and $B_2$ for sway effects). This method is simpler but less accurate for complex frames.
2. Direct analysis method: Models second-order effects and geometric nonlinearities directly in the structural analysis software. It uses $K = 1.0$ for all members and applies notional loads to represent imperfections. This approach is more rigorous and is the preferred method in current AISC practice.

Proper consideration of end restraints, bracing conditions, and lateral loads is essential regardless of which method you use. A column braced at mid-height, for example, has a very different effective length (and therefore capacity) than the same column unbraced over its full height.