Australian Combined Loading Design — AS 4100 Clause 8.4 Beam-Columns

Complete reference for the design of steel members subject to combined axial compression (or tension) and bending per AS 4100:2020 Clause 8.4. Beam-columns are the most common structural element in steel frames: virtually every column in a multi-storey building and every chord member in a truss subject to inter-node transverse loading is a beam-column.

Quick access: Column effective length factor | Lateral torsional buckling | Section compactness | Beam capacity calculator

Beam-Column Interaction Formula — Clause 8.4.2

The general interaction formula for a steel member subject to design axial compression N* and design bending moments M_x* and M_y* is:

N* / (phi N_c) + M_x* / (phi M_ix) + M_y* / (phi M_iy) <= 1.0

where:

The interaction formula is applied at every cross-section along the member, with the moments evaluated at the section under consideration. For members with moment gradient, the critical section is typically at mid-height (where the buckling mode amplitude is largest) or near the end (where applied moments are largest).

Reduced Moment Capacity Due to Axial Load — Clause 8.4.2.2

The presence of axial compression reduces the member's moment capacity. For compact doubly-symmetric I-sections, the reduced section moment capacity M_pr about each axis is:

For the major axis (bending about x-x):

M_prx = 1.18 x M_sx x (1 - N* / (phi N_s)) <= M_sx

For the minor axis (bending about y-y):

M_pry = 1.19 x M_sy x (1 - (N* / (phi N_s))^2) <= M_sy

where M_s is the nominal section moment capacity and N_s is the nominal section axial capacity (yield capacity, not buckling capacity).

For hollow sections (RHS/SHS/CHS), the reduced moment capacities are:

M_prx = 1.18 x M_sx x (1 - N* / (phi N_s)) <= M_sx M_pry = 1.19 x M_sy x (1 - N* / (phi N_s)) <= M_sy

The reduced moment capacity M_pr is then combined with the member slenderness reduction (from LTB and column buckling) to obtain the in-plane member capacity M_i:

M_i = M_pr x (1 - N* / (phi N_om))

where N_om is the elastic flexural buckling load for the relevant axis (pi^2 x E x I / L_e^2).

Moment Amplification (Second-Order Effects) — Clause 8.4.3

In sway frames, the first-order bending moments must be amplified to account for P-Delta effects (the additional moments produced by the axial load acting through the lateral displacement of the column). The moment amplification factor delta_s is:

delta_s = 1 / (1 - 1 / lambda_ms)

where lambda_ms is the elastic buckling load factor of the storey (ratio of the storey buckling load to the applied storey load).

For braced frames, the moment amplification factor for a column bent in single curvature is:

delta_b = C_m / (1 - N* / N_om) >= 1.0

where C_m is the equivalent uniform moment factor:

beta_m is the ratio of smaller to larger end moment (M_1/M_2), taken as positive when the moments produce single curvature and negative when they produce double curvature.

Biaxial Bending Interaction — Clause 8.4.5

When a member is subject to bending about both principal axes simultaneously (biaxial bending in addition to axial load), the general interaction equation becomes:

N* / (phi N_c) + M_x* / (phi M_ix) + M_y* / (phi M_iy) <= 1.0

An alternative and sometimes more accurate interaction expression for biaxial bending (where bending about the minor axis is significant) is:

(N*/(phi N_c))^gamma + (M_x*/(phi M_rx))^alpha + (M_y*/(phi M_ry))^beta <= 1.0

For compact I-sections: gamma = 1.4, alpha = 1.4, beta = 1.4 + 0.5 x alpha_y where alpha_y is the ratio of the factored minor-axis bending moment to the reduced section capacity.

Tension and Bending — Clause 8.3

For members subject to combined axial tension and bending, the interaction formula is simpler because buckling does not occur in tension:

N* / (phi N_t) + M_x* / (phi M_sx) + M_y* / (phi M_sy) <= 1.0

where phi N_t is the design tension capacity (lesser of gross section yield = 0.90 x A_g x f_y, and net section fracture = 0.90 x 0.85 x k_t x A_n x f_u per Clause 7.2).

Worked Example: Beam-Column in a Braced Frame

Problem: A 310UC158 Grade 300 column in a braced frame supports a factored axial compression N* = 1800 kN and a factored major-axis bending moment M_x* = 175 kNm from eccentric beam reactions. The column has L_ex = 3.8 m and L_ey = 3.8 m (k_e = 1.0 both axes, nominally pinned). Check the adequacy per AS 4100 Clause 8.4.

Given section properties (310UC158, Grade 300):

Solution:

Step 1: Section classification

Flange: b_f/(2t_f) = 311/(2 x 25.0) = 6.22. lambda_ep = 9 x sqrt(250/280) = 8.51. 6.22 < 8.51: Flange COMPACT.

Web in compression (column): d_1/t_w ~ (327 - 2 x 25.0 - 2 x 20)/15.7 = 237/15.7 = 15.1. lambda_ep compression = 45 x sqrt(250/300) = 41.1. 15.1 < 41.1: Web COMPACT.

Section is fully compact.

Step 2: Axial compression capacity (Clause 6.3)

Section capacity: N_s = k_f x A_g x f_y = 1.0 x 20,100 x 280 = 5,628 kN. phi N_s = 0.90 x 5,628 = 5,065 kN.

Elastic buckling loads: N_omx = pi^2 x 200,000 x 411 x 10^6 / (3800)^2 x 10^(-3) = 56,140 kN N_omy = pi^2 x 200,000 x 125 x 10^6 / (3800)^2 x 10^(-3) = 17,080 kN

Note: I_x = A_g x r_x^2 = 20,100 x 137^2 = 377 x 10^6 mm^4 (approximate use 411 from tables). I_y = A_g x r_y^2 = 20,100 x 77.5^2 = 120.8 x 10^6 mm^4 (approximate use 125 from tables).

Non-dimensional slenderness: lambda_nx = sqrt(5,628 / 56,140) = 0.317 lambda_ny = sqrt(5,628 / 17,080) = 0.574

alpha_b = -0.5 (hot-rolled UB/UC section): alpha_cx = 0.658^(0.317^2) = 0.658^0.100 = 0.961 alpha_cy = 0.658^(0.574^2) = 0.658^0.329 = 0.872

phi N_cx = 0.90 x 0.961 x 5,628 = 4,870 kN phi N_cy = 0.90 x 0.872 x 5,628 = 4,419 kN

Minor axis buckling governs: phi N_c = 4,419 kN.

Step 3: Section moment capacity

M_sx = f_yf x S_x = 280 x 2,700 x 10^3 x 10^(-6) = 756 kNm phi M_sx = 0.90 x 756 = 680 kNm

Step 4: Reduced moment capacity due to axial load

M_prx = 1.18 x 756 x (1 - 1800 / 5065) = 1.18 x 756 x 0.645 = 575 kNm <= 756 kNm -- OK.

Step 5: In-plane member moment capacity

Check LTB for the column (braced frame, no lateral loading between floors): alpha_m = 1.0 + 0.35 x (1 - 0) = 1.0 (conservative, uniform moment assumption for column with end moments).

M_ix = 575 x (1 - 1800 / 56,140) = 575 x 0.968 = 557 kNm phi M_ix = 0.90 x 557 = 501 kNm

Step 6: Interaction check

N*/(phi N_c) + M_x*/(phi M_ix) + M_y*/(phi M_iy)

= 1800 / 4419 + 175 / 501 + 0

= 0.407 + 0.349 = 0.756 <= 1.0 -- OK.

The column is at 75.6% utilisation, governed approximately equally by axial compression (40.7%) and bending (34.9%).

Result: 310UC158 Grade 300 is adequate. Utilisation = 0.756 (75.6%). Minor axis buckling governs the axial capacity; in-plane member moment governs the bending interaction.

Frequently Asked Questions

What is the interaction formula for beam-columns per AS 4100?

The beam-column interaction formula per AS 4100 Clause 8.4.2 is N*/(phi N_c) + M_x*/(phi M_ix) + M_y*/(phi M_iy) <= 1.0. This linear interaction formula conservatively approximates the true non-linear beam-column behaviour. The formula checks that the combination of axial load and bending moments (about both axes) does not exceed the available capacities, with the moment capacities M_ix and M_iy reduced to account for the destabilising effect of the axial compression.

How does axial compression reduce the moment capacity of a steel section?

Axial compression reduces the available moment capacity in two ways. First, it increases the compressive stress on the compression flange, accelerating local and lateral-torsional buckling. Second, the axial pre-compression reduces the plastic moment capacity by consuming part of the section's force capacity. Per AS 4100 Clause 8.4.2.2, the reduced section moment capacity M_pr is calculated as M_pr = 1.18 M_s (1 - N*/phi N_s), which linearly reduces the plastic moment capacity as the axial load approaches the squash load of the section.

When is moment amplification required for beam-column design?

Moment amplification (second-order or P-Delta effects) is required whenever the axial load is a significant fraction of the elastic buckling load (N* > 0.10 N_om). In sway frames, the amplification factor delta_s = 1/(1 - 1/lambda_ms) can be substantial and must always be checked. In braced frames, the amplification factor delta_b = C_m/(1 - N*/N_om) is typically small when N* << N_om. For columns in braced frames with slenderness less than 40 and N*/N_om < 0.05, moment amplification can be neglected.

How does AS 4100 handle biaxial bending plus axial compression?

For biaxial bending (moments about both principal axes) plus axial compression, the three-term interaction formula N*/(phi N_c) + M_x*/(phi M_ix) + M_y*/(phi M_iy) <= 1.0 is used. Each term is computed separately and summed linearly. For members where the minor-axis bending is significant (>20% of the major-axis moment), an alternative exponential interaction formula may be used to achieve a more economical design. The linear interaction formula is conservative for biaxial bending.

Is the interaction formula different for tension and bending versus compression and bending?

Yes. For tension + bending (AS 4100 Clause 8.3), the interaction is N*/(phi N_t) + M*/(phi M_s) <= 1.0. Since tension does not cause buckling, the full section moment capacity phi M_s is used (no reduction for member buckling or LTB) and the section axial tension capacity is used. This is inherently less restrictive than the compression case, where buckling reduces both the axial and moment capacities.

Educational reference only. All design values must be verified against the current edition of AS 4100:2020 and the project specification. This information does not constitute professional engineering advice. Always consult a qualified structural engineer for design decisions.