Modern Steel Construction • June 2021 • Compressions Member Design: A Primer

Richard M. Drake (rick.drake@fluor.com) is a senior fellow in structural engineering, and Erik Espinoza (erik.espinoza@fluor.com) is a director in structural engineering, both with Fluor Enterprises, Inc.

Last month's SteelWise article, also written by Drake and Espinoza, focused on tension member design. You can read it in the Archives section at www.modernsteel.com.

Here, we’ll highlight key steps for designing compression members according to the provisions of the 2016 AISC Specification for Structural Steel Buildings (ANSI/ AISC 360, aisc.org/specifications). Note that we’ll focus solely on W-shape columns, though much of the material is readily applicable to other rolled shapes as well.

A compression member is any member that is loaded with an axial compression force. Compression members are commonly located in building columns, structural bracing, and top chords of trusses. Loads transfer to a building column from the columns above and from beams framing into it. If the center of gravity of the loads coincides with the center of gravity of the column, then the column is considered concentrically loaded, though in practice, columns are seldom concentrically loaded. Ideal concentrically loaded columns do not exist since all columns have accidental eccentricities resulting from material imperfections, end connections, initial crookedness of rolled shape, eccentric loads, and residual stresses (see Figure 1).

Application of external load (P) at eccentricity (e) introduces flexural stress. If the member is short, the lateral deflection is small, and the eccentricity will introduce negligible flexural stresses—and the column can take a lot of axial load before buckling. If the member is long, the lateral deflection is large, and the eccentricity can introduce significant flexural stresses—and the column will take much less axial load before buckling.

Critical buckling. Consider an ideal concentrically loaded column, making the following assumptions:

• The column is braced against lateral translation (sidesway) but allowed to rotate at each end.
• The column is perfectly straight
• The load is applied along the column’s centroidal axis.
• The column material behaves elastically.

If the axial load (P) is gradually applied, then the column will eventually buckle into the deflected shape of a simply supported beam. AISC calls this limit state flexural buckling. The axial load that forces the buckled shape is the critical buckling load (P_cr). The column buckles before the axial stress level reaches the material yield stress. The column is entirely elastic and is a function of its flexural stiffness (EI), as shown by Euler’s critical load formula:

Dividing both sides of the equation by the column crosssectional area will convert this relationship to the critical stress, as used in the Specification. For mathematical convenience, define the radius of gyration (r) as:

Yield strength (F_y) and tensile strength (F_u) have no effect on the critical stress. A36 and A992 steels have the same modulus of elasticity and will buckle at the same load for a given column size and support condition. If this elastic critical buckling stress (F_cr) exceeds the material yield strength, then the critical stress equation is not applicable.

Every column has an X-axis and Y-axis, each with its own I (area moment of inertia), r and L (length). Every column will flexurally buckle about the axis with the highest slenderness ratio (L /r) and therefore the lowest critical stress (see Figure 2).

The concept of effective length is simply a mathematical method of replacing a given column with an equivalent pinnedend column braced against side sway. In other words, the elastic flexural buckling length is equal to KL, where K is the effective length factor and L is the column length between supports.

Effective length factors can be derived by repeating the critical buckling load differential equation derivation with different boundary conditions, and they can also be determined graphically (see Figure 3).

Specification Commentary Table C-A-7.1 provides a summary of theoretical effective length factors based on upper bound derivations. It also provides recommended values that should be used in design in recognition that member end boundary conditions are rarely fully pinned or fully fixed. Note that in the Specification, KL is replaced with L_c.

Structural steel shapes are comprised of rectangular plate elements, each with its own aspect ratio (λ). See Figure 4 for an example W-shape.

If an individual plate element has a high aspect ratio, it may become unstable and experience local buckling before flexural buckling of the overall section can occur. The section plate elements are classified into two types based on their boundary conditions and unstiffened and stiffened elements. Unstiffened elements are supported along only one edge parallel to the normal compressive force and will buckle like a cantilever beam (see Figure 5). Stiffened elements, on the other hand, are supported along both edges parallel to the normal compressive force and will buckle like a fixed end beam (see Figure 6).

Specification Section B4 includes a classification system to identify the compressive members that may experience local buckling before flexural buckling. Limiting width-to-thickness ratios (λr) for local buckling have been developed based on elastic plate buckling theory and are listed in Specification Table B4.1a for members subject to axial compression.

Compression members are classified as non-slender element sections if all elements have aspect ratios less than or equal to the limiting width-to-thickness ratios.

Compression members are classified as slender-element sections if any elements have aspect ratios that are greater than or equal to the limiting width-to-thickness ratios. Compression tests on short W-shapes (called column stubs) show that all fibers on the cross section are not stressed at the same level. Residual stresses cause early yielding, followed by inelastic behavior.

Residual stresses. Residual stresses are the stresses that remain in a member after it has been rolled into a finished shape (see Figure 7). Sources of residual stresses in structural steel include uneven cooling, which occurs after hot rolling of structural shapes. Note the following:

• Tension residual stresses occur in the regions that cool the slowest Other causes of residual stress include cold bending or cambering during fabrication, punching of holes, cutting, or welding during fabrication.

When a compression load is applied to a column, the parts of the column with residual compressive stresses will reach the material yield stress before the rest of the section and go into the plastic range of behavior. The stiffness of the column will be reduced and become a function of the part of the column cross section that is still elastic. As the applied load increases, the column will buckle inelastically because part of the cross section has reached the yield stress before flexural buckling occurs.

• Columns with smaller slenderness ratios tend to buckle inelastically before elastic buckling can be achieved (see Figure 8).

• Short columns fail by yielding. Nominal compression strengths are predicted using yield stress theory.

• Long columns fail by elastic buckling. Nominal compression strengths are predicted by considering elastic buckling theory.

• Intermediate columns fail by inelastic buckling. Nominal compression strengths are predicted using empirical formulae. Most practical (economical) columns end up in this range.

When determining the nominal strength of a compression member (Pn), it is necessary to first classify the shape for axial compression. Members in axial compression that are classified as non-slender element sections can reach flexural buckling of the entire cross section before local buckling of any elements. Members in axial compression that are classified as slender element sections will experience local buckling of one or more elements before flexural buckling of the entire cross section.

Non-Slender and Slender Element Sections. Specification Section E3 applies to non-slender element sections as defined in Specification Section B4 for members in axial compression. There are multiple useful formulas that apply to these elements:

Determine the critical stress (F_cr) using the appropriate equation, depending on the column slenderness ratio L(c)/r.

Specification Section E7 applies to slender element sections as defined in Section B4 for members in axial compression. The procedure is not included here for brevity.

Column load tables. AISC has created over 200 pages of column load tables to tabulate the calculated available strength for common column shapes and sizes, reasonable effective lengths, and common material strengths. These tables are located in Part 4 of the AISC Steel Construction Manual (aisc.org/manual).

For each shape, the tables consider the most common material yield stress only, including:

Note that while the double and single angles are listed as 36 ksi in the 15th Edition of the Manual, they will be updated to 50 ksi in the 16th Edition.

You must use Specification formulas for other shapes and yield stresses. Tables are limited to Lc/r <= 200 because AISC prefers that you not exceed this.

Tables assume that weak-axis buckling will govern the column design and are calculated based on L_cy in feet. If L_c is not the same for both axes, then the table may still be used to determine the available strength by converting the X-axis effective length, L_cx, to an equivalent Y-axis effective length, L_cy(equiv).

Every column will buckle about the axis with the greater slenderness ratio, represented by the larger of L_cy orL_cy(equiv). Enter the column load tables with the larger of L_cy or L_cy(equiv) and select the precalculated value for available strength. Table values consider the classification of sections for local buckling and are correct for all sections of Specification Chapter E, including Section E3 (non-slender) and Section E7 (slender).

Super tables. AISC has created nearly 100 pages of load tables to tabulate the calculated available strength for W-shapes ranging in size from W44 to W4 with F_y = 50 ksi. These tables are located in Part 6 of the Manual. AISC calls Manual Table 6-2 its “super table” because it combines some of the best design strength features of the Manual beam and column design aids. Although these tables were created to facilitate the design of members in combined flexure and axial compression, they are very useful for evaluating W-shape columns that are not included in the column load tables.

Although this primer is intended to summarize the nominal compression strength requirements in the Specification, designers are cautioned that the choice of member cross sections and connection details may introduce eccentricity and moment when designing compression members. In those cases, the designer should consult Specification Chapter H for combined flexure and axial forces.