Previously, we discussed what buckling is and why slender members fail long before the steel itself yields, how to check flexural buckling of columns and how to check lateral torsional buckling of beams.

This guide covers the remaining member check: a beam-column carrying axial compression and bending at the same time, per Cl. 6.3.3 of Eurocode 3 (EN 1993-1-1), with the interaction factors from Annex B (Method 2).

❓ What is the combined compression and bending check?

It is a stability check for members that carries axial compression and bending at the same time (a beam-column) and has no restraint along the length being checked, so it is free to buckle sideways (flexural buckling) about either axis and to twist (lateral-torsional buckling).

Why it is needed: Compression and bending interact and increase the risk of buckling. A bending moment causes the member to deflect, while compression acts on the deflected member and increases the bending effect. This is why combined compression and bending must be checked together.

💡 The idea behind the calculation

The combined check measures how much of the member's capacity is used by compression and bending about each axis.

Each effect is expressed as a utilisation ratio, including the effect of buckling. These ratios are then combined using interaction factors (k), which account for how compression and bending influence each other. The final combined utilisation should remain within 100%.

The two equations are Eq. 6.61 & 6.62,EN 1993-1-1:

Buckling about the major axis (y-y)

\begin{aligned}\frac{N_{Ed}}{\chi_{y} \, N_{Rk} / \gamma_{M1}} + k_{yy} \, \frac{M_{y,Ed} + \Delta M_{y,Ed}}{\chi_{LT} \, M_{y,Rk} / \gamma_{M1}} + k_{yz} \, \frac{M_{z,Ed} + \Delta M_{z,Ed}}{M_{z,Rk} / \gamma_{M1}} &\leq 1.0\end{aligned}

Cl. 6.3.3, Eq. 6.61

Buckling about the minor axis (z-z)

\begin{aligned}\frac{N_{Ed}}{\chi_{z} \, N_{Rk} / \gamma_{M1}} + k_{zy} \, \frac{M_{y,Ed} + \Delta M_{y,Ed}}{\chi_{LT} \, M_{y,Rk} / \gamma_{M1}} + k_{zz} \, \frac{M_{z,Ed} + \Delta M_{z,Ed}}{M_{z,Rk} / \gamma_{M1}} &\leq 1.0\end{aligned}

Cl. 6.3.3, Eq. 6.62

The combine check depending on three componenets:

Axial utilisation ( $\frac{N_{Ed}}{\chi_{y} \, N_{Rk} / \gamma_{M1}}$ ): Flexural buckling factor χ Cl 6.3.1 and Compression Capacity N_Rk Cl 6.2.4
Major-axis bending utilisation ( $k_{yy} \, \frac{M_{y,Ed} + \Delta M_{y,Ed}}{\chi_{LT} \, M_{y,Rk} / \gamma_{M1}}$ ): Lateral Torsional Buckling factor χ_LT Cl 6.3.2 and Bending Capacity M_Rk Cl 6.2.5
Minor-axis utilisation ( $k_{yz} \, \frac{M_{z,Ed} + \Delta M_{z,Ed}}{M_{z,Rk} / \gamma_{M1}}$ ): Bending Capacity M_Rk Cl 6.2.5

k_yy, k_yz, k_zy and k_zz - Four interaction factors

The combined check uses four interaction factors k_yy, k_yz, k_zy and k_zz to account for how axial compression and bending interact. These factors captures several effects:

Shape of the moment diagram: It represented by C_m (C_my, C_mz, C_mLT).
Section class and shape: Based on class and shape of section, different formulas are used to obtain k factors
Torsional behaviour: an open I-section can twist and buckle laterally, a hollow section hardly does. This decides which table of factors applies.
Cross-coupling between axes: bending about one axis reduces the capacity about the other — represented by k_yz and k_zy.

Annex A vs. Annex B - obtaining interaction factors k_yy, k_yz, k_zy and k_zz

Eurocode 3 provides two approaches for obtaining the interaction factors. Annex A and Annex B use different formulae and calculation procedures, so they can give different levels of conservatism and require different amounts of calculation.

Comparsion of Annex A and Annex B

	Annex A (Method 1)	Annex B (Method 2)
Approach	Derives each k from first principles: elastic critical forces, plastic modulus ratio, torsional constants and more	Ready-made formulae in three tables (B.1, B.2, B.3)
Effort	Long — many auxiliary parameters (μ, w, n_pl, a_LT … plus C_yy etc.)	Short — hand-calculation friendly
Result	Usually less conservative, more economical	Slightly more conservative, easy to audit
Best for	Software, heavily loaded or optimised members	Hand checks, preliminary design, reports

📖 Glossary: parameters in the combined check

As many parameters are involved in the calculation, the glossary below lists each one with a short explanation. The step-by-step calculation guide follows right after it.

Common Glossary for the Combined Compression and Bending Check

Symbol	Name	Simple meaning
N_Ed	Design axial compression force	Factored axial load on the member
M_y,Ed, M_z,Ed	Design bending moments	Maximum factored moment about the major (y-y) and minor (z-z) axes between braced points
ΔM_y,Ed, ΔM_z,Ed	Moments from centroid shift	Only for Class 4: N_Ed × the shift of the effective centroid. Zero for Class 1–3
N_Rk	Characteristic compression resistance	f_y × A (Class 1–3) or f_y × A_eff (Class 4)
M_y,Rk, M_z,Rk	Characteristic moment resistance	f_y × W_pl (Class 1–2), f_y × W_el (Class 3), f_y × W_eff (Class 4)
χ_y, χ_z	Flexural buckling reduction factors	From Cl. 6.3.1: fraction of N_Rk left after buckling about y-y or z-z
χ_LT	Lateral-torsional buckling reduction factor	From Cl. 6.3.2; take 1.0 if the member is not susceptible to torsional deformation
λ̄_y, λ̄_z	Non-dimensional slenderness	How slender the member is about each axis (from the flexural buckling check)
N_cr,y, N_cr,z	Elastic critical buckling force	Load at which a perfect member would buckle about each axis
γ_M1	Partial safety factor	1.0 as recommended — check your National Annex
k_yy, k_yz, k_zy, k_zz	Interaction factors	Cover second-order effects and the coupling between axes; found from Annex A or B
C_my, C_mz, C_mLT	Equivalent uniform moment factors	Turn the actual moment diagram into an equivalent uniform moment (1.0 = uniform moment, the worst case)
ψ	End moment ratio	Smaller end moment ÷ larger end moment, from −1 to 1
α_s, α_h	Moment diagram ratios	Used in Table B.3 when lateral loads act between the supports
n_y, n_z	Axial utilisation	N_Ed / (χ_yN_Rk/γ_M1) and N_Ed / (χ_zN_Rk/γ_M1) — shorthand used in this guide
W_pl, W_el, W_eff	Plastic, elastic, effective section modulus	Choice depends on section class
A, A_eff	Gross and effective area	A_eff for Class 4
μ, w, n_pl, a_LT, b_LT …	Annex A auxiliary parameters	Only needed for Method 1 (Annex A); not used in Annex B

📍 Quick Calculation Guideline for the Combined Check (Annex B)

Given a beam-column with N_Ed, M_y,Ed and M_z,Ed and no intermediate restraint, determine whether it passes Cl. 6.3.3

The steps below follow Annex B (Method 2). Use the first-order moments between braced points — where the frame is sensitive to sway, the second-order (P-Δ) effects belong to the frame analysis.

Collect the design forces and section data

Take N_Ed, M_y,Ed and M_z,Ed (first-order moments between braced points), the steel grade f_y, and classify the section (Class 1–4).

Calculate the section resistances

N_Rk = f_y · A (for a Class 4 section: f_y · A_eff).

M_y,Rk and M_z,Rk = f_y · W_pl (Classes 1–2), f_y · W_el (Class 3) or f_y · W_eff (Class 4).

For Class 4, also find the centroid-shift moments ΔM_y,Ed and ΔM_z,Ed — they are zero for Classes 1–3.

Get the buckling reduction factors

χ_y and χ_z from the flexural buckling check (Cl. 6.3.1), run for each axis.

χ_LT from the lateral-torsional buckling check (Cl. 6.3.2)

where χ_LT = 1.0 if the member is not susceptible to torsional deformation (e.g. hollow section).

Choose the table for k interactive factors

Section that not susceptible to torsion → Table B.1.

Section that susceptible (typically an open I- or H-section without torsional restraint) → Table B.2.

Find C_my, C_mz and C_mLT from Table B.3

The factors depend on the moment diagram between braced points — pick from the reference table below.

Use C_m = 1.0 for a sway buckling mode, and remember C_m = 1.0 is always a safe, conservative choice.

Moment diagram between braced points	C_m (C_my, C_mz, C_mLT)
Uniform moment (ψ = 1.0, the worst case)	1.0
Linear diagram from end moments, ratio ψ (−1 ≤ ψ ≤ 1)	0.6 + 0.4·ψ, but not less than 0.4
Simply supported, uniform lateral load, no end moments	0.95
Simply supported, point load at mid-span, no end moments	0.90
End moments combined with lateral load	Use Table B.3 of EN 1993-1-1 directly (depends on α_s or α_h)

Calculate the interaction factors k_yy, k_yz, k_zy and k_zz

Work with the shorthand axial utilisations: n_y = N_Ed / (χ_yN_Rk/γ_M1) and n_z = N_Ed / (χ_zN_Rk/γ_M1).

Pick each factor from the tables below — depending on the section class and on the table chosen in Step 4 (Table B.1/B.2).

Factor	Class 1 and 2	Class 3 and 4
k_yy	C_my [ 1 + (λ̄_y − 0.2)·n_y ] ≤ C_my (1 + 0.8·n_y)	C_my [ 1 + 0.6·λ̄_y·n_y ] ≤ C_my (1 + 0.6·n_y)
k_zz — I- and H-sections	C_mz [ 1 + (2·λ̄_z − 0.6)·n_z ] ≤ C_mz (1 + 1.4·n_z)	C_mz [ 1 + 0.6·λ̄_z·n_z ] ≤ C_mz (1 + 0.6·n_z)
k_zz — rectangular hollow sections	C_mz [ 1 + (λ̄_z − 0.2)·n_z ] ≤ C_mz (1 + 0.8·n_z)	C_mz [ 1 + 0.6·λ̄_z·n_z ] ≤ C_mz (1 + 0.6·n_z)
k_yz	0.6·k_zz	k_zz
k_zy — not susceptible to torsion (Table B.1)	0.6·k_yy	0.8·k_yy

For k_zy when the member is susceptible to torsional deformation (Table B.2):

Section class	k_zy
Class 1 and 2 & λ̄_z ≥ 0.4	1 − [ 0.1·λ̄_z / (C_mLT − 0.25) ]·n_z, but not less than 1 − [ 0.1 / (C_mLT − 0.25) ]·n_z
Class 1 and 2 & λ̄_z < 0.4	0.6 + λ̄_z, but not more than 1 − [ 0.1·λ̄_z / (C_mLT − 0.25) ]·n_z
Class 3 and 4	1 − [ 0.05·λ̄_z / (C_mLT − 0.25) ]·n_z, but not less than 1 − [ 0.05 / (C_mLT − 0.25) ]·n_z

For I- and H-sections and rectangular hollow sections with compression and major-axis bending only (M_z,Ed = 0), k_zy = 0 may be taken.

Calculate based on Eq. 6.61 and Eq. 6.62 and check

Eq. 6.61: N_Ed / (χ_y·N_Rk/γ_M1) + k_yy·(M_y,Ed + ΔM_y,Ed) / (χ_LT·M_y,Rk/γ_M1) + k_yz·(M_z,Ed + ΔM_z,Ed) / (M_z,Rk/γ_M1) ≤ 1.0

Eq. 6.62: N_Ed / (χ_z·N_Rk/γ_M1) + k_zy·(M_y,Ed + ΔM_y,Ed) / (χ_LT·M_y,Rk/γ_M1) + k_zz·(M_z,Ed + ΔM_z,Ed) / (M_z,Rk/γ_M1) ≤ 1.0

Evaluate Eq. 6.61 and Eq. 6.62. check if both are ≤ 1.0.

⚠️ Common mistakes to avoid

Checking compression and bending separately and adding the ratios — the second-order amplification between them is exactly what the interaction equations capture
Taking χ_LT = 1.0 for an open I-section that can twist — it is only valid when the section is not susceptible to torsional deformation
Picking the wrong table in Step 4 — the two tables differ only in k_zy, but that factor matters
Forgetting the ΔM terms for a Class 4 section, or using W_pl where W_eff is required
Feeding second-order moments into the check — Annex B expects first-order moments between braced points; sway effects belong to the frame analysis
Checking the whole member instead of each segment between restraints, with its own moment diagram and buckling length
Forgetting the γ_M1 value and the method choice of your National Annex

🙋 Frequently asked questions

1. What is the combined compression and bending check in Eurocode 3?

It is the stability check for a beam-column — a member under axial compression and bending at the same time. Eurocode 3 checks it with the two interaction equations (Cl. 6.3.3, Eq. 6.61 and 6.62), and Annex B provides simple formulae for the interaction factors in them.

2. Why can't I check N and M separately and add the utilisation ratios?

Because compression and bending make each other worse: the bending bows the member, and the compression acts on that bow, producing additional second-order moment. The interaction equations capture this amplification through the interaction factors k — adding two separate ratios would miss it.

3. What is a beam-column?

A member that carries axial compression and bending simultaneously. Nearly every real frame column is one: wind, eccentric load paths and frame action always introduce some moments on top of the axial force.

4. When can χ_LT be taken as 1.0 in the combined check?

When the member is not susceptible to torsional deformation — for example hollow sections, or a section restrained against twisting. Otherwise χ_LT comes from the lateral-torsional buckling check of Cl. 6.3.2.

5. What is the difference between Annex A and Annex B?

Both provide the interaction factors k. Annex A (Method 1) derives each factor from first principles — usually less conservative and suited to software. Annex B (Method 2) provides ready-made formulae in Tables B.1–B.3 — slightly more conservative but hand-calculation friendly. Each country's National Annex decides which method(s) are allowed.

6. What are the ΔM terms in the interaction equations?

They are additional moments caused by the shift of the centroid of the effective cross-section, and apply only to Class 4 sections (ΔM = N_Ed × the centroid shift). For Class 1–3 sections the ΔM terms are zero.

📋 Key takeaways

The combined check (Cl. 6.3.3) is the stability check for beam-columns — members under axial compression and bending at the same time.
It uses two interaction equations: Eq. 6.61 (major axis y-y) and Eq. 6.62 (minor axis z-z) — both must be ≤ 1.0.
Each equation adds three utilisations: axial (with χ), major-axis bending (with χ_LT) and minor-axis bending — tied together by the interaction factors k.
Annex B (Method 2) gives the hand-friendly k factors: choose Table B.1 or B.2 by torsional susceptibility (they differ only in k_zy), and C_m from Table B.3.
Check each segment between restraints on its own, with first-order moments and the buckling length of that segment.

📄 Need a beam-column check in your design report?

The combined check usually decides which capacity of a real frame column you are allowed to use. If you want a ready-to-submit design report with the combined compression and bending check, the buckling checks, section classification and the full cross-section check in Eurocode 3, feel free to try our application — what you need is just the design data values, and a ready-to-submit PDF report will be generated in a minute.

The ready-to-submit PDF Report is not generated by AI but programmed by chartered engineer, which is accurate, no hallucination and same input same output.

#steel design #glossary

References

Eurocode 3: Design of steel structures — Part 1-1: General rules and rules for buildings (EN 1993-1-1) —CEN (European Committee for Standardization) ·2005

CivilSimple Team

The CivilSimple Team writes practical engineering guides for the profession and the curious. All articles are reviewed for technical accuracy before publication.