How do you design columns for biaxial bending with slenderness effects?

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Columns are essential structural elements that support axial loads and bending moments. However, designing columns for biaxial bending with slenderness effects can be challenging, as both the strength and stability of the column need to be considered. In this article, you will learn the basic concepts and steps involved in biaxial bending of columns, and how to use the interaction equations and charts from the AISC Specification to check the design adequacy of steel columns.

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1 What is biaxial bending?

Biaxial bending occurs when a column is subjected to bending moments about both the major and minor axes of its cross-section, in addition to the axial load. This can happen due to eccentric loading, lateral forces, or frame geometry. Biaxial bending reduces the axial load capacity of the column, as the bending stresses add to the axial stress in some regions of the cross-section, and subtract from it in others. Biaxial bending also affects the buckling behavior of the column, as the bending moments can induce additional deflections and rotations.

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2 How do you account for slenderness effects?

Slenderness effects refer to the tendency of a column to buckle under compressive loads, depending on its length, cross-section, and end conditions. Slender columns are more prone to buckling than short columns, and require a lower design strength to prevent instability. To account for slenderness effects, you need to calculate the effective length factor (K) and the slenderness ratio (KL/r) for each axis of the column, based on the bracing and support conditions. The effective length factor (K) reflects the degree of restraint at the column ends, and ranges from 0.5 for fixed-fixed columns to 2.0 for pinned-pinned columns. The slenderness ratio (KL/r) is the ratio of the effective length (KL) to the radius of gyration (r) of the cross-section, and indicates how slender the column is relative to its stiffness.

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3 How do you determine the nominal strength?

The nominal strength of a column is the maximum axial load that the column can resist without exceeding the yield stress or the critical buckling stress, whichever is lower. The nominal strength depends on the cross-sectional shape, material properties, and slenderness ratio of the column. To determine the nominal strength, you need to use the appropriate column curve from the AISC Specification, which relates the nominal strength to the slenderness ratio for different cross-sectional shapes and steel grades. The column curve is based on the Euler formula for elastic buckling, modified by a reduction factor (Q) to account for inelastic behavior and residual stresses.

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4 How do you check the interaction equations?

The interaction equations are used to check whether the combined effects of axial load and biaxial bending are within the design limits of the column. The interaction equations compare the actual axial load (P) and bending moments (Mx and My) to the nominal strength (Pn) and flexural strength (Mnx and Mny) of the column, and ensure that the ratio of the actual to the nominal forces does not exceed a certain value. The value depends on the load combination and the resistance factor (phi) specified by the AISC Specification. For example, for LRFD load combinations, the interaction equation is:

phi*P/Pn + phi*Mx/Mnx + phi*My/Mny <= 1.0

The flexural strength (Mnx and Mny) can be calculated by multiplying the nominal strength (Pn) by the eccentricity (e) of the axial load relative to the centroid of the cross-section.

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5 How do you use the interaction charts?

The interaction charts are graphical tools that simplify the process of checking the interaction equations. The interaction charts plot the axial load ratio (phi*P/Pn) versus the bending moment ratios (phi*Mx/Mnx and phi*My/Mny) for different values of slenderness ratio (KL/r) and cross-sectional shape. The interaction charts show the regions where the column is adequate or inadequate for a given load combination and resistance factor. To use the interaction charts, you need to locate the point that corresponds to your column's axial load ratio and bending moment ratios on the chart, and check whether it falls inside or outside the shaded area. If it falls inside, the column is adequate; if it falls outside, the column is inadequate.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you design columns for biaxial bending with slenderness effects?

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1 What is biaxial bending?

2 How do you account for slenderness effects?

3 How do you determine the nominal strength?

4 How do you check the interaction equations?

5 How do you use the interaction charts?

6 Here’s what else to consider

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