Module 7: Steel Compression Members

Module 7: Steel Compression Members

Steel Compression Members

Unlike tension members, which simply stretch until they yield or tear, compression members (columns, struts) are subjected to instability. As a compressive load increases, a long, slender column will suddenly bow sideways and collapse long before the steel material itself actually yields or crushes. This phenomenon is known as buckling.

Euler Buckling and Residual Stresses

The capacity of a steel column is dictated by two primary factors: the theoretical elastic buckling stress (Euler's formula) and the physical reality of residual stresses locked into the steel during the hot-rolling and cooling process.

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Euler Buckling: The theoretical failure mode for perfectly straight, slender columns, depending purely on geometry and elastic modulus ( $E$ ). The Euler buckling stress is:

$$
F_e = \\frac{\\pi^2 E}{(KL/r)^2}
$$

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Residual Stresses: As a hot-rolled W-shape cools, the tips of the flanges cool faster than the intersection of the web and flange. This differential cooling creates internal "residual" compressive stresses at the flange tips.
Because these outer fibers are already under compression internally, they will yield prematurely when an external compressive load is applied. This premature yielding drastically reduces the stiffness ( $E$ ) of the column section, initiating inelastic buckling at loads lower than Euler's equation predicts.

Key Takeaways

Elastic buckling (Euler) dictates failure for long, slender columns.
Inelastic buckling (due to premature yielding from residual stresses) dictates failure for stockier columns.

Torsional and Flexural-Torsional Buckling

While standard W-shapes typically fail via flexural (Euler) buckling about their weak axis, asymmetric or open cross-sections are prone to twisting failures.

Twisting Limit States

Torsional Buckling: The member simply twists about its longitudinal shear center axis without bending laterally. This is most critical for highly symmetric but open cross-sections, like cruciform shapes.
Flexural-Torsional Buckling: A highly complex failure mode where the member simultaneously bends (flexure) and twists (torsion). This is the governing failure mode for asymmetric shapes (like single angles) or singly-symmetric shapes (like Tees or double angles) because their shear center does not align with their centroid, causing axial loads to induce torque.

Single-Angle Compression Members

Asymmetric Buckling

Single angles are unique because their principal axes (the axes of maximum and minimum moment of inertia, often denoted as Z and W) do not align with their geometric axes (X and Y).

When loaded in compression, a single angle naturally wants to buckle about its minor principal axis (Z-axis).
Because they are typically connected by only one leg (introducing eccentricity) and are asymmetric, they fail via a complex combination of flexural buckling and flexural-torsional buckling.
To simplify design, the code allows engineers to calculate an equivalent slenderness ratio ( $KL/r$ ) based on the geometric axes, provided specific end-connection and bracing criteria are met.

Effective Length and Slenderness

The critical parameter for any column is its slenderness ratio,

KL/r

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Effective Length Factor ( $K$ ): Accounts for the rotational and translational restraints at the ends of the column. A pinned-pinned column ( $K=1.0$ $K = 1.0$ ) will buckle more easily than a fixed-fixed column ( $K=0.5$ $K = 0.5$ ). Common theoretical and recommended (design) values include:
- Pinned-Pinned: Theoretical $1.0$ , Recommended $1.0$
- Fixed-Fixed: Theoretical $0.5$ , Recommended $0.65$
- Fixed-Pinned: Theoretical $0.7$ , Recommended $0.80$
- Fixed-Free (Cantilever): Theoretical $2.0$ , Recommended $2.1$
Radius of Gyration ( $r$ ): A geometric property ( $\sqrt{I/A}$ ) indicating how the cross-sectional area is distributed away from the centroidal axis. A higher $r$ means higher resistance to buckling. Columns will always buckle about their weak axis (the axis with the smallest $r$ , typically the y-axis for W-shapes) unless bracing prevents it.
Slenderness Ratio ( $KL/r$ ): The NSCP recommends a maximum slenderness ratio of 200 for compression members to ensure they are not too flimsy during construction and handling.

Key Takeaways

Columns buckle about their weakest unbraced axis (the axis yielding the largest $KL/r$ ).
The effective length factor ( $K$ ) drastically alters buckling resistance by modeling end-fixity.
The theoretical maximum slenderness ratio ( $KL/r$ ) for compression members is typically capped at 200.

Local vs. Global Buckling

Before a column can buckle as a whole (global buckling,

KL/r

), its individual plate elements (flanges or webs) must be sturdy enough not to buckle locally like a crushed soda can.

Local vs Global Buckling

Compact / Non-Slender Sections: The flanges and webs are thick enough relative to their width that they will not buckle locally before the entire column buckles globally.
Slender Element Sections: The width-to-thickness ratio ( $b_f/2t_f$ for flanges, $h/t_w$ for webs) exceeds the limits ( $\lambda_r$ ) set by the NSCP. If elements are slender, their effective area must be reduced (using a reduction factor $Q$ ), penalizing the column's total capacity.
Most standard W-shapes used for columns (like W14s and W10s) are specifically proportioned to be non-slender to avoid this complex capacity reduction.

Width-to-Thickness Ratios

For a compression element to be non-slender, its width-to-thickness ratio must be less than the limit

\lambda_r

. For unstiffened elements (e.g., flanges of W-shapes),

\lambda_r = 0.56 \sqrt{E/F_y}

. For stiffened elements (e.g., webs of W-shapes),

\lambda_r = 1.49 \sqrt{E/F_y}

Built-Up Compression Members

Laced and Battened Columns

When standard shapes cannot provide enough buckling resistance, two or more shapes (like channels or angles) can be connected side-by-side with open space between them, joined by lacing (diagonal flat bars) or battens (horizontal plates).

This drastically increases the radius of gyration ( $r$ ) without adding significant weight.
Modified Slenderness ( $KL/r_m$ ): Because the member is not solid, the lacing/battens can deform slightly under load (shear deformation). This reduces the column's overall stiffness. To account for this, the standard slenderness ratio must be mathematically increased to a modified slenderness ratio ( $KL/r_m$ ) before calculating the critical stress.
The individual elements (e.g., the channels) must also be checked to ensure they do not buckle independently between the lacing points.

Design of Axially Loaded Columns (Non-Slender Sections)

The NSCP defines the critical buckling stress (

F_{cr}

) based on a slenderness threshold that separates inelastic (intermediate columns) from elastic buckling (long slender columns). This threshold is

4.71\sqrt{E/F_y}

Critical Stress Equations

Inelastic Buckling: When

\frac{KL}{r} \le 4.71 \sqrt{\frac{E}{F_y}}

, the column fails by inelastic buckling, meaning parts of the cross-section have yielded due to residual stresses before the whole member buckles. The critical stress is:

F_{cr} = \left[ 0.658^{\frac{F_y}{F_e}} \right] F_y

Elastic Buckling: When

\frac{KL}{r} > 4.71 \sqrt{\frac{E}{F_y}}

, the column fails entirely within the elastic range (Euler buckling) before yielding occurs anywhere in the cross-section. The critical stress is purely dependent on geometry and stiffness:

F_{cr} = 0.877 F_e

Where

F_e

is the elastic (Euler) buckling stress:

F_e = \frac{\pi^2 E}{(KL/r)^2}

Checklist

Determine the nominal compressive strength:

$$
P_n = F_{cr} A_g
$$

Checklist

Where $F_{cr}$ depends on the slenderness parameter $KL/r$ :

Fcr Equations

Inelastic Buckling ( $KL/r \le 4.71\sqrt{E/F_y}$ ):
$F_{cr} = \left[ 0.658^{(F_y/F_e)} \right] F_y$
Elastic Buckling ( $KL/r \gt 4.71\sqrt{E/F_y}$ ):
$F_{cr} = 0.877 F_e$

Local Buckling and Compactness

Before analyzing global column buckling, the individual plate elements (flanges and webs) of the cross-section must be evaluated for local buckling.

Width-to-Thickness Ratios ( $\lambda$ )

To prevent premature local buckling, the NSCP limits the width-to-thickness ratio (

b/t

for flanges,

h/t_w

for webs) of elements under uniform compression. If an element's ratio exceeds a specific limit (

\lambda_r

), the section is classified as slender, and its nominal compressive strength is severely reduced. Most standard W-shapes used as columns are dimensioned specifically to be non-slender, ensuring global buckling governs rather than local crippling.

Key Takeaways

Compression capacity is primarily dictated by the slenderness ratio ( $KL/r$ ). The column will buckle about the axis with the largest $KL/r$ .
Local Buckling must be checked using width-to-thickness ratios ( $\lambda_r$ ). Sections must be classified as compact, non-compact, or slender.
The NSCP column equations use a transition point ( $4.71\sqrt{E/F_y}$ ) to separate inelastic buckling (governed by residual stresses and yielding) from elastic Euler buckling.
The critical buckling stress ( $F_{cr}$ ) is the singular value determining nominal compressive strength ( $P_n = F_{cr} A_g$ ).

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Steel Compression Members

Euler Buckling and Residual Stresses

Checklist

Checklist

Torsional and Flexural-Torsional Buckling

Twisting Limit States

Single-Angle Compression Members

Asymmetric Buckling

Effective Length and Slenderness

Checklist

Local vs. Global Buckling

Local vs Global Buckling

Width-to-Thickness Ratios

Built-Up Compression Members

Laced and Battened Columns

Design of Axially Loaded Columns (Non-Slender Sections)

Critical Stress Equations

Checklist

Checklist

Fcr Equations

Local Buckling and Compactness

Width-to-Thickness Ratios (λ\lambdaλ)

Width-to-Thickness Ratios ( $\lambda$ )