AS 4100 Beam Design Guide — αm, αs, Mo, φMs
The complete reference for steel beam design to AS 4100-2020 (Australian Standard for Steel Structures). This guide walks through every parameter in the AS 4100 beam design workflow: section moment capacity φMs, member moment capacity φMb, the moment modification factor αm, the slenderness reduction factor αs, and the reference buckling moment Mo. Clause references are provided throughout for tracing back to the code.
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1. The AS 4100 Beam Design Framework
AS 4100 organises beam design around two capacity levels:
- Section capacity (φMs) — the bending strength of the cross-section itself, assuming the beam is fully restrained against lateral-torsional buckling (LTB). Governed by Clause 5.1 (bending) and Clause 5.2 (section classification).
- Member capacity (φMb) — the bending strength incorporating LTB effects for the actual unbraced segment length. Governed by Clause 5.6.
The member capacity can never exceed the section capacity:
φMb = αm × αs × φMs ≤ φMs
The design requirement is:
M* ≤ φMb
Where M* is the design bending moment from factored load combinations per AS 1170.
2. Section Moment Capacity — φMs
2.1 Section Classification (Clause 5.2)
Before computing φMs, the cross-section must be classified per Table 5.2 based on its flange and web slenderness. The yield slenderness limit λey determines the boundary between compact and non-compact behaviour:
λey = b/t × √(fy/250) must not exceed limits in Table 5.2
| Class | Behaviour | Capacity |
|---|---|---|
| Compact | Full plastic moment can develop with adequate rotation | Ms = fy × S (plastic) |
| Non-compact | Yielding at extreme fibres but local buckling prevents full plasticity | Ms = fy × Zeff |
| Slender | Elastic local buckling governs | Ms = fy × Ze × (λey/λe) |
For hot-rolled Australian sections (UB, UC, PFC, EA, UA), most standard sections are compact when used as beams.
2.2 Nominal Section Capacity Ms (Clause 5.2)
For a compact section:
Ms = fy × Ze
Where Ze is the effective section modulus. For doubly-symmetric compact I-sections, Clause 5.2.3 permits using the lesser of 1.5 × S (plastic modulus) and the yield moment:
Ms = min(fy × S, fy × Z × 1.5)
For Grade 300 steel (fy = 300 MPa, fu = 440 MPa) with a 610UB125 section (Z = 3,230 × 10³ mm³, S = 3,670 × 10³ mm³):
Ms = 300 × 3,230 × 10³ = 969 kN·m
Ms_plastic = 300 × 3,670 × 10³ = 1,101 kN·m (but 1.5 × Z limit applies: 1.5 × 3,230 = 4,845 cm³ > S = 3,670 cm³, so Ms = 1,101 kN·m)
2.3 Capacity Factor φ (Table 3.4)
For bending (yielding limit state): φ = 0.90
φMs = 0.90 × 1,101 = 991 kN·m
3. Member Moment Capacity — φMb (Clause 5.6)
When the compression flange is not continuously restrained, lateral-torsional buckling reduces the beam's capacity below φMs. The member capacity accounts for:
- The unbraced segment length between lateral restraints
- The moment distribution along that segment (via αm)
- The beam's lateral and torsional stiffness (via Mo)
3.1 When LTB Must Be Checked
LTB is required unless one of these conditions applies (Clause 5.6.4):
- The segment length L ≤ (80 × ry) / √(fy/250) with full lateral restraint at both ends and rotational restraint
- The beam is continuously restrained (e.g., slab cast on top flange)
- The beam bends about its minor axis only
- The beam is of hollow section type (CHS/SHS/RHS)
4. Reference Buckling Moment — Mo (Clause 5.6.1.2)
Mo is the elastic lateral-torsional buckling moment for a simply supported doubly-symmetric beam under uniform moment. It is the fundamental elastic stability parameter:
Mo = √[ (π² × E × Iy) / Le² × (GJ + π² × E × Iw / Le²) ]
Where:
- E = 200,000 MPa (elastic modulus)
- G = 80,000 MPa (shear modulus, ≈ E / 2.6)
- Iy = second moment of area about the minor axis (mm⁴)
- J = torsion constant (mm⁴)
- Iw = warping constant (mm⁶)
- Le = effective length for LTB = kt × kl × kr × L
The effective length factors account for:
- kt — load height factor (1.0 for shear centre loading, 1.4 for top flange loading)
- kl — lateral rotation restraint factor (1.0 for full restraint, 1.4 for no restraint at one end)
- kr — lateral rotation restraint factor at supports (1.0 for full restraint, 1.4 for free to warp)
Worked calculation for 610UB125, Grade 300, Le = 3.0 m:
Iy = 44.6 × 10⁶ mm⁴, J = 717 × 10³ mm⁴, Iw = 2.61 × 10¹² mm⁶
Mo = √[ (π² × 200,000 × 44.6 × 10⁶) / 3,000² × (80,000 × 717 × 10³ + π² × 200,000 × 2.61 × 10¹² / 3,000²) ]
Mo ≈ 3,450 kN·m
5. Slenderness Reduction Factor — αs (Clause 5.6.1.1)
αs translates the elastic buckling moment Mo into a design reduction that accounts for residual stresses and initial imperfections. It is based on the modified slenderness:
λs = √(Ms / Mo)
The three-region AS 4100 αs curve (Clause 5.6.1.1):
| Range | αs formula | Behaviour |
|---|---|---|
| λs ≤ 0.4 | αs = 1.0 | Short segments — no LTB reduction |
| 0.4 < λs ≤ 1.0 | αs = 1.0 − (λs − 0.4) / 0.6 × (1.0 − λs / 1.4) | Transition region |
| 1.0 < λs | αs = 1 / λs² | Slender — elastic buckling governs |
For singly-symmetric sections (e.g., channels, unequal flanges), use the more complex Appendix H formula which accounts for the monosymmetry parameter.
Continuing the 610UB125 example with Le = 3.0 m:
λs = √(1,101 / 3,450) = √0.319 = 0.565
Since 0.4 < 0.565 ≤ 1.0: αs = 1.0 − (0.565 − 0.4) / 0.6 × (1.0 − 0.565 / 1.4) = 1.0 − 0.275 × 0.596 = 0.836
6. Moment Modification Factor — αm (Clause 5.6.1.1)
αm accounts for the actual bending moment distribution along the unbraced segment. A uniform moment is the most severe case (αm = 1.0). A moment gradient — where the moment varies from a maximum at one end to zero or reversed at the other — delays LTB and gives αm > 1.0.
6.1 Standard αm Values (Table 5.6.1)
| Loading and Support | Moment Diagram | αm |
|---|---|---|
| End moments only, equal and opposite | M → ← M | 1.00 |
| Simply supported, uniformly distributed load | 1.13 | |
| Simply supported, central point load | 1.35 | |
| End moments βm = +0.5 (double curvature) | 1.75 | |
| End moments βm = −1.0 (pure moment) | 1.00 | |
| End moments βm = 0 (one end pinned) | 2.25 | |
| End moments βm = −0.5 | 2.50 | |
| Cantilever | 1.25 |
6.2 General αm Formula
For moment ratios between tabulated values, αm is computed from:
αm = 1.7 × Mmax / √(M2² + M3² + M4²)
Where Mmax, M2, M3, M4 are the absolute moments at the quarter-points of the segment, with M2, M3, M4 taken at the quarter-, mid-, and three-quarter points respectively.
6.3 Example: Beam with End Moments
For a beam with end moments M1 = +200 kN·m and M2 = +100 kN·m: βm = M2 / M1 = 100 / 200 = +0.5
αm = 1.75 + 1.05 × 0.5 + 0.3 × 0.5² = 1.75 + 0.525 + 0.075 = 2.35
The moment gradient more than doubles the LTB capacity compared to a uniform moment (αm = 1.0).
7. Complete φMb Calculation
Combining all parameters:
φMb = αm × αs × φMs ≤ φMs
Final worked example — 610UB125, Grade 300, Le = 3.0 m, simply supported UDL:
- φMs = 991 kN·m (from section 2.3)
- αs = 0.836 (from section 5)
- αm = 1.13 (UDL on simple span)
φMb = 1.13 × 0.836 × 991 = 936 kN·m
Check: φMb = 936 < φMs = 991 — LTB governs but with modest reduction.
For a fully restrained beam (continuous slab): φMb = φMs = 991 kN·m.
8. Shear Capacity — φVv (Clause 5.11)
Shear must be checked alongside bending. For an unstiffened web:
Vv = 0.6 × fy × Aw
Where Aw = d × tw for rolled I-sections (d = depth between flanges, tw = web thickness).
For the 610UB125: d = 572 mm, tw = 11.9 mm
Aw = 572 × 11.9 = 6,807 mm² Vv = 0.6 × 300 × 6,807 = 1,225 kN φVv = 0.90 × 1,225 = 1,103 kN
The shear-bending interaction per Clause 5.12:
If V* ≤ 0.6 × φVv: No interaction — φMs is unreduced
If V* > 0.6 × φVv: Reduced moment capacity applies
9. Deflection Limits (AS 1170.0 Table C1)
Serviceability deflection limits must be verified independently of strength:
| Element | Total Load | Live Load |
|---|---|---|
| Floor beams | L/250 | L/360 |
| Roof beams (no ceiling) | L/200 | L/250 |
| Roof beams (with ceiling) | L/250 | L/360 |
| Purlins and girts | L/200 | — |
| Crane runway beams | L/600 | — |
For a 9.0 m floor beam:
- Total deflection limit: 9,000 / 250 = 36 mm
- Live load deflection limit: 9,000 / 360 = 25 mm
10. Step-by-Step AS 4100 Beam Design Procedure
- Determine design loads — compute M* and V* from AS 1170 factored combinations
- Propose trial section — select a section from the OneSteel/InfraBuild tables
- Classify section — check flange and web slenderness (Table 5.2)
- Compute φMs — section moment capacity including classification limits
- Determine unbraced length Le — including kt, kl, kr factors for each segment
- Compute Mo — reference buckling moment for each segment
- Compute αs — slenderness reduction factor
- Establish αm — from moment diagram at quarter-points
- Compute φMb — member moment capacity = αm × αs × φMs
- Verify — M* ≤ φMb for all segments
- Check shear — V* ≤ φVv; check interaction if V* > 0.6 φVv
- Check deflection — verify against AS 1170.0 Table C1 limits
- Check web bearing and buckling at supports (Clause 5.13)
- Document — record all assumptions: effective lengths, restraint conditions, k-factors
11. Common Pitfalls in AS 4100 Beam Design
- Wrong effective length: Using the full span length when intermediate lateral restraints exist. Each segment between restraints is checked separately.
- Ignoring load height: kt = 1.4 for top-flange loading drops Mo significantly. Always check whether the load is applied at the top flange, shear centre, or bottom flange.
- Overlooking segment end restraint: The kl and kr factors depend on whether the segment ends provide both lateral and torsional restraint. A brace that restrains the compression flange laterally but not torsionally requires kr > 1.0.
- Applying αm without checking the condition: αm > 1.0 from Table 5.6.1 assumes that the maximum moment occurs within the segment and that intermediate lateral restraints exist at the segment ends. A moment diagram that peaks outside the segment cannot use the full αm.
- Forgetting shear-moment interaction: At supports where both M* and V* are high, check the interaction limit per Clause 5.12.3.
- Section not compact under bending: While most UB/UC sections are compact for bending, some universal beams with slender flanges (610UB101, 610UB113) may be non-compact — check Table 5.2 limits.
- Yield stress variation: For thicker flanges (t > 17 mm in Grade 300), fy drops to 280 MPa. Always verify the applicable fy for the actual flange and web thicknesses per AS/NZS 3679.1.
12. Verification Checklist
Use this checklist to verify any AS 4100 beam design:
- Design loads M* and V* confirmed from AS 1170 factored combinations
- Section properties verified from current manufacturer tables (not outdated catalogues)
- Steel grade and fy confirmed for actual plate thicknesses
- Section classification checked — flange and web slenderness limits satisfied
- Unbraced segment lengths correctly identified with restraint conditions documented
- Effective length factors kt, kl, kr explicitly stated and justified
- Mo calculated with correct Iy, J, Iw values for the relevant axis
- αm selected from correct moment diagram case (not assumed)
- φMb checked against φMs — member capacity does not exceed section capacity
- Shear capacity checked; interaction verified if applicable
- Deflection limits checked for both total and live load
- Web bearing and buckling checked at supports and point loads
- All assumptions documented for peer review