AS 4100 Column Design Guide — αa, αb, αc, Ns

The complete reference for steel column design to AS 4100-2020 (Australian Standard for Steel Structures). This guide covers the column buckling formulation: nominal section capacity Ns, member capacity Nc, the compression member constants αa, αb, αc, and the form factor kf. All clause references are provided and a fully worked example is included.

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1. The AS 4100 Column Design Framework

AS 4100 Clause 6 governs the design of members in compression. The design requirement is:

N* ≤ φNc

Where:

• N* is the design axial compressive force from factored load combinations (AS 1170)
• φ is the capacity factor = 0.90 (Table 3.4)
• Nc is the nominal member capacity (Clause 6.3.3)

The member capacity Nc ≤ Ns (the nominal section capacity) because buckling reduces the compressive resistance of slender members.

2. Nominal Section Capacity — Ns (Clause 6.2)

2.1 Form Factor kf (Clause 6.2.2)

The form factor accounts for local buckling of individual plate elements that make up the cross-section before overall member buckling occurs. For each plate element:

λe = (b/t) × √(fy/250)

The element is fully effective if λe ≤ λey (the yield slenderness limit from Table 6.2.4). If λe > λey, the effective width is:

be = b × (λey / λe)

The form factor is the ratio of effective to gross area:

kf = Σ(be × t) / Ag  ≤  1.0

2.2 Yield Slenderness Limits (Table 6.2.4)

2.3 Ns for Standard Sections

For hot-rolled UB and UC sections in Grade 300 steel, the flange and web are typically compact under uniform compression, giving kf = 1.0:

Ns = kf × An × fy = 1.0 × Ag × fy

For the 200UC46.2 section (Ag = 5,880 mm², fy = 300 MPa):

Ns = 5,880 × 300 / 1,000 = 1,764 kN

3. Member Capacity — Nc (Clause 6.3.3)

The member capacity incorporates overall flexural buckling:

Nc = αc × Ns  ≤  Ns

Where αc is the member slenderness reduction factor, computed from the modified slenderness λn:

λn = (Le/r) × √(kf) × √(fy/250)

And the compression member constant parameters:

η = 0.00326 × (λn − 13.5)  ≥  0
ξ = [(λn/90)² + 1 + η] / [2 × (λn/90)²]

αc = ξ × [1 − √(1 − (90 / ξ × λn)²)]

Key parameters:

• Le = ke × L — the effective length
• r = radius of gyration (ry for major axis, rz for minor axis)
• kf = form factor
• fy = yield stress (MPa)

4. Compression Member Constants — αa, αb, αc (Table 6.3.3)

The AS 4100 column curve is parameterised by three constants that calibrate the curve to experimental data for different section types. These are NOT the same as the slenderness reduction factor αc; they are the curve-defining parameters.

4.1 Table 6.3.3(1) — Hot-Rolled Sections

4.2 Table 6.3.3(2) — Cold-Formed and Other Sections

4.3 Physical Meaning of αa, αb, αc

• αa defines the plateau extent — how far the curve stays at αc ≈ 1.0 before buckling reduction begins. Higher αa means more capacity for intermediate-slenderness columns. Hot-rolled sections have αa = 13.5, meaning columns with λn ≤ 13.5 have negligible buckling reduction.
• αb controls the slope of the transition from the plateau into the post-buckling region. Most sections use αb = −0.5, giving a smooth transition.
• αc (the table constant, distinct from the αc reduction factor) controls the asymptotic behaviour for very slender columns. The product αa × αb enters the η parameter and shifts the entire curve. A more negative αb × αa produces a steeper post-buckling drop-off.

5. Effective Length

5.1 Basic Effective Length Factor ke (Clause 4.6.3)

The effective length Le = ke × L, where L is the actual member length.

5.2 Chart Method (Figure 4.6.3.3)

For frames with known end stiffness, ke is determined from the stiffness ratios:

γ1 = (Σ Ic/Lc) / (Σ Ibg/Lbg)  at end 1
γ2 = (Σ Ic/Lc) / (Σ Ibg/Lbg)  at end 2

Where Ic and Lc are the column properties and Ibg and Lbg are the beam/girder properties at each joint. ke is read from Figure 4.6.3.3(a) for braced frames or Figure 4.6.3.3(b) for sway frames.

5.3 Conservative Defaults

When the actual frame stiffness cannot be evaluated, use:

• ke = 1.0 for braced frames (pinned-pinned assumption)
• ke = 2.0 for sway-permitted frames

6. Worked Example — 200UC46.2 Column

Design problem: Internal column in a braced multi-storey frame. Simply connected top and bottom (pinned-pinned). Storey height = 4.0 m. Steel grade: 300PLUS (fy = 300 MPa for tf ≤ 11.5 mm).

Loading:

• Dead load: NG = 420 kN (axial from floors above)
• Live load: NQ = 280 kN (imposed from floors)

AS 1170 factored load: N* = 1.2 × 420 + 1.5 × 280 = 504 + 420 = 924 kN

Step 1 — Section Properties (200UC46.2)

Step 2 — Form Factor kf

Flange outstand: be = (bf − tw − 2r) / 2 = (203 − 7.3 − 2 × 10) / 2 = 87.9 mm λe_flange = (be / tf) × √(fy/250) = (87.9 / 11.0) × √(300/250) = 7.99 × 1.095 = 8.75 λey_flange = 16 (hot-rolled flange outstand) — OK, compact

Web: b = d − 2tf − 2r = 203 − 22 − 20 = 161 mm λe_web = (161 / 7.3) × √(300/250) = 22.05 × 1.095 = 24.2 λey_web = 45 — OK, compact

kf = 1.0 (all elements compact under uniform compression)

Step 3 — Nominal Section Capacity Ns

Ns = kf × Ag × fy = 1.0 × 5,880 × 300 / 1,000 = 1,764 kN

φNs = 0.90 × 1,764 = 1,588 kN

Step 4 — Effective Length

Braced frame, pinned-pinned: ke = 1.0 Le_y = Le_z = 1.0 × 4,000 = 4,000 mm

Step 5 — Modified Slenderness

Major axis: λn_y = (Le_y / ry) × √(kf) × √(fy/250) = (4,000 / 88.9) × 1.0 × 1.095 = 49.3 Minor axis: λn_z = (4,000 / 51.3) × 1.0 × 1.095 = 85.4 ← governs

Step 6 — Member Slenderness Reduction Factor αc_z

Using αa = 13.5, αb = −0.5 (hot-rolled UC, Table 6.3.3):

η = 0.00326 × (85.4 − 13.5) = 0.00326 × 71.9 = 0.234 ξ = [(85.4/90)² + 1 + 0.234] / [2 × (85.4/90)²] = [0.901 + 1 + 0.234] / [2 × 0.901] = 2.135 / 1.802 = 1.185

αc_z = 1.185 × [1 − √(1 − (90 / (1.185 × 85.4))²)] = 1.185 × [1 − √(1 − (90/101.2)²)] = 1.185 × [1 − √(1 − 0.791)] = 1.185 × [1 − √0.209] = 1.185 × [1 − 0.457] = 0.644

Step 7 — Member Capacity Nc

Nc_z = αc_z × Ns = 0.644 × 1,764 = 1,136 kN φNc_z = 0.90 × 1,136 = 1,022 kN

Step 8 — Design Check

N* = 924 kN ≤ φNc_z = 1,022 kN → OK (utilisation = 924/1,022 = 0.904)

The column satisfies the AS 4100 compression requirement with 90% utilisation.

7. Behaviour Across Slenderness Ranges

Stocky Columns (λn ≤ 13.5)

For λn ≤ αa (13.5 for hot-rolled), η = 0 and the buckling reduction is negligible. Nc ≈ Ns. Design is governed by the cross-section squash load. These are short columns in low-rise construction or basement levels where storey heights are small relative to section depth.

Intermediate Columns (13.5 < λn ≤ 90)

This is the typical range for multi-storey building columns. As λn increases from 15 to 90, αc drops from approximately 0.95 to 0.55. The reduction is due to inelastic buckling — the column buckles at a stress between the proportional limit and the yield stress, with partial yielding at the extreme fibres.

Slender Columns (λn > 90)

At high slenderness, elastic (Euler) buckling governs. αc approaches the Euler curve asymptotically:

αc_Euler = π² × E / (λn² × 250)

For λn = 150: αc_Euler = π² × 200,000 / (150² × 250) = 0.351

8. Biaxial Buckling

When Le_y ≠ Le_z, both axes must be checked independently. The governing case is the one producing the lowest αc.

For a column with different effective lengths in each direction (e.g., girts on one face provide intermediate lateral restraint along the minor axis but not the major axis):

Check: N* ≤ min(φNc_y, φNc_z)

9. Combined Compression and Bending (Clause 8.4)

When a column also carries bending moments (beam-column), the interaction formula from Clause 8.4.2.2 applies:

N* / (φNc) + M*_x / (φMbx) × (1 − N*/Nombx) + M*_y / (φMby) × (1 − N*/Nomby)  ≤  1.0

Where Nom_b is the elastic buckling load for the relevant axis. The moment amplification term (1 − N*/Nom) accounts for P-δ effects. For braced frames where N*/Nom < 0.15, the amplification may be neglected.

10. Step-by-Step AS 4100 Column Design Procedure

1. Determine N* — axial compression from AS 1170 factored load combinations
2. Propose trial section — select UC, UB, SHS, or CHS from manufacturer tables
3. Determine steel grade and fy — verify for actual flange and web thicknesses
4. Classify section for compression — check each plate element for compact/non-compact/slender
5. Compute kf — form factor (Clause 6.2.2); kf = 1.0 if all elements compact
6. Compute Ns — Ns = kf × An × fy (Clause 6.2.1)
7. Determine effective lengths Le_y and Le_z — including ke factors, brace points, and frame type
8. Compute λn_y and λn_z — modified slenderness for each axis
9. Obtain αb_y and αb_z — slenderness reduction factors using αa, αb, αc from Table 6.3.3
10. Compute φNc_y and φNc_z — member capacity; governing = min(φNc_y, φNc_z)
11. Verify — N* ≤ min(φNc_y, φNc_z)
12. Check combined actions — if M* is significant, apply Clause 8.4 interaction
13. Check base plate — bearing pressure on concrete per AS 3600
14. Document — ke factors, restraint assumptions, load path

11. Common Pitfalls in AS 4100 Column Design

• Using the wrong ke: Assuming ke = 1.0 in a sway-permitted frame can underestimate effective length by 100% or more. Always confirm whether the lateral load resisting system provides sway restraint.
• Checking only major axis: For standard UC sections, the minor axis frequently governs because iz is roughly half of iy. Both axes must be checked.
• Ignoring thickness-dependent fy: Grade 300 has fy = 300 MPa for t ≤ 17 mm but drops to 280 MPa for t > 17 mm. Heavy columns (310UC158, etc.) may operate at reduced fy.
• Applying kf incorrectly: kf applies to the gross area, not the net area at bolt holes. Bolt hole deductions are accounted for separately in An.
• Overlooking effective length differences by axis: If lateral braces (e.g., girts at mid-height) restrain only the minor axis, Le_z ≠ Le_y. Compute separately.
• Forgetting second-order effects: For slender columns in sway frames, the amplified moment method (Clause 4.7) must be used before the interaction check.

12. Column Design Verification Checklist

• Design loads N* confirmed from AS 1170 combinations (1.2G + 1.5Q, etc.)
• Section properties from current InfraBuild/OneSteel tables
• Steel grade and fy verified for actual flange and web thicknesses
• Form factor kf computed for each plate element; kf = 1.0 confirmed or justified
• Effective length factors ke_y, ke_z explicitly stated with frame type (braced/sway)
• Both major and minor axis λn computed; governing axis identified
• αb reduction factor computed using correct αa, αb, αc from Table 6.3.3
• φNc verified against N* for governing axis
• Combined compression + bending interaction checked if applicable
• Base plate and connection design compatible with column end conditions
• All assumptions documented for peer review