Beam-Column Interaction: IPE 500 S355 Combined N+M Check — EN 1993-1-1 §6.3.3

Problem Statement

Column Under Combined Axial Compression and Biaxial Bending

An IPE 500 column in a building frame is subjected to:

Axial compression: N_Ed = 1,200 kN
Major-axis bending: M_y,Ed = 280 kNm (ψ_y = 0, C_my = 1.0)
Minor-axis bending: M_z,Ed = 35 kNm (C_mz = 1.0)

Both ends are pinned (no rotational restraint about either axis), giving effective lengths L_cr,y = L_cr,z = L_LT = 6.0 m. Steel grade: S355 (f_y = 355 N/mm², γM0 = γM1 = 1.0). The column is laterally restrained at both ends only.

Load Parameter	Value	Unit
Axial compression N_Ed	1,200	kN
Major-axis moment M_y,Ed	280	kNm
Minor-axis moment M_z,Ed	35	kNm
End condition — both axes	Pinned	—
Effective length L_cr,y = L_cr,z = L_LT	6,000	mm
Steel grade S355	f_y = 355	N/mm²
Partial factor γM1	1.0	—
End moment ratio ψ_y	0	—
End moment ratio ψ_z	1	—

Step 1 of 7

01 Section Classification — IPE 500

IPE 500 geometry (from EN 10365):

Property	Value	Unit
Cross-section area A	11,600 mm²	mm²
Height h	500 mm	mm
Flange width b	200 mm	mm
Web thickness t_w	10.2 mm	mm
Flange thickness t_f	16 mm	mm
Second moment of area (strong axis) I_y	482,000 cm&sup4; → 4.82 × 10&sup9; mm&sup4;	mm&sup4;
Second moment of area (weak axis) I_z	21,400 cm&sup4; → 2.14 × 10&sup8; mm&sup4;	mm&sup4;
Plastic section modulus (strong axis) W_pl,y	2,192 cm³ → 2,192,000 mm³	mm³
Elastic section modulus (strong axis) W_el,y	1,928 cm³ → 1,928,000 mm³	mm³
Torsional constant I_t	89.0 cm&sup4; → 890,000 mm&sup4;	mm&sup4;
Warping constant I_w	2,219,000 cm&sup6; → 2.219 × 10&sup9; mm&sup6;	mm&sup6;

Section classification check (EN 1993-1-1 Table 5.2):

Flange class check (internal part, Class 1 limit = 9ε) c/t_f = (b/2)/t_f = (200/2)/16 = 100/16 = 6.25 < 9ε → Class 1

Web class check (internal compression part, Class 1 limit = 72ε) d/t_w = (h − 2t_f)/t_w = (500 − 2×16)/10.2 = 468/10.2 = 45.9 < 72ε → Class 1

Result: IPE 500 is a Class 1 section — W_pl,y and plastic analysis are appropriate throughout.

Step 2 of 7

02 Flexural Buckling — y-y Axis (Curve b)

IPE 500 (h/b = 500/200 = 2.50 > 1.2, rolled) → curve b for both axes.

Elastic critical buckling force — y-y axis N_cr,y = π² · E · I_y / L_cr,y²
= π² × 210,000 × 4.82 × 10&sup9; / 6,000²
= 9.8696 × 210,000 × 482,000,000 / 36,000,000
= 9.8696 × 2811.7 × 10³ / 36
≈ 27,770 kN

Non-dimensional slenderness λ&bar;_y λ&bar;_y = √(A · f_y / N_cr,y)
= √(11,600 × 355 / 27,770,000)
= √(4,118,000 / 27,770,000) = √0.1483 = 0.385

Imperfection factor α_y — curve b → α = 0.34

Reduction factor χ_y — EN 1993-1-1 Eq. (6.49) φ_y = 0.5 × [1 + α · (λ&bar;_y − 0.2) + λ&bar;_y²]
= 0.5 × [1 + 0.34 × (0.385 − 0.2) + 0.385²]
= 0.5 × [1 + 0.34 × 0.185 + 0.148] = 0.5 × [1 + 0.0629 + 0.148] = 0.5 × 1.211 = 0.606
χ_y = 1 / (φ_y + √(φ_y² − λ&bar;_y²))
= 1 / (0.606 + √(0.367 − 0.148))) = 1 / (0.606 + √0.219) = 1 / (0.606 + 0.468) = 1 / 1.074 = 0.931

Design buckling resistance — y-y N_b,Rd,y = (χ_y · A · f_y) / γ_M1
= 0.931 × 11,600 × 355 / 1.0
= 0.931 × 4,118,000 = 3,835 kN

Step 3 of 7

03 Flexural Buckling — z-z Axis (Curve c)

Elastic critical buckling force — z-z axis N_cr,z = π² · E · I_z / L_cr,z²
= π² × 210,000 × 2.14 × 10&sup8; / 6,000²
= 9.8696 × 210,000 × 21,400,000 / 36,000,000
≈ 12,321 kN

Non-dimensional slenderness λ&bar;_z λ&bar;_z = √(A · f_y / N_cr,z)
= √(4,118,000 / 12,321,000) = √0.334 = 0.578

Imperfection factor α_z — curve c (rolled IPE, z-z) → α = 0.49

Reduction factor χ_z — EN 1993-1-1 Eq. (6.49) φ_z = 0.5 × [1 + 0.49 × (0.578 − 0.2) + 0.578²]
= 0.5 × [1 + 0.49 × 0.378 + 0.334] = 0.5 × [1 + 0.185 + 0.334] = 0.5 × 1.519 = 0.759
χ_z = 1 / (0.759 + √(0.759² − 0.578²)) = 1 / (0.759 + √0.576 − 0.334) = 1 / (0.759 + √0.242) = 1 / (0.759 + 0.492) = 1 / 1.251 = 0.799

Design buckling resistance — z-z N_b,Rd,z = (χ_z · A · f_y) / γ_M1
= 0.799 × 4,118,000 = 3,291 kN

Note: The z-z axis is more critical due to the lower second moment of area I_z. However, since both ends are pinned about z, lateral-torsional buckling (Step 04) must also be verified separately.

Step 4 of 7

04 Lateral Torsional Buckling — χ_LT

Using the simplified combined formula (Annex BB / frameAI LTB method):

Elastic critical moment M_cr — Annex BB M_cr = √[ N_cr,z² + (G·I_t·L²) / (π²·E·I_z) ]
N_cr,z = π² × 210,000 × 2.14 × 10&sup8; / 36 × 10&sup6; = 12,321 kNm (warping term)
(G·I_t·L²) / (π²·E·I_z) = 585.6 kNm (St. Venant term)
M_cr = √(12,321² + 585.6²) = 18,919 kNm

Non-dimensional slenderness λ&bar;_LT λ&bar;_LT = √(W_pl,y · f_y / M_cr)
= √(2.192 × 10&sup6; × 355 / 18,919 × 10&sup6;)
= √0.04114 = 0.203

λ&bar;_LT = 0.203 < 0.2 — LTB is inactive. Per EN 1993-1-1 §6.3.2 Note 2, when λ&bar;_LT ≤ 0.2 the lateral-torsional buckling mode does not govern and χ_LT may be taken as 1.0. M_b,Rd = W_pl,y · f_y / γ_M1 = 2,192,000 × 355 / 1.0 = 778 kNm.

Step 5 of 7

05 Section Resistances — N_Rk, M_y,Rk, M_z,Rk

Axial resistance (Class 1 section, Eq. 6.6) N_Rk = A · f_y = 11,600 × 355 = 4,118,000 N = 4,118 kN

Major-axis bending resistance (Class 1, plastic modulus) M_y,Rk = W_pl,y · f_y = 2,192,000 × 355 = 778,360,000 Nmm = 778.4 kNm

Minor-axis bending resistance (Class 1, plastic modulus) W_pl,z = (b · h² / 4 − (b − t_w) · (h − 2t_f)² / 4) / (h/2) ≈ 1,828 cm³ (from section tables)
M_z,Rk = W_pl,z · f_y ≈ 1,828,000 × 355 = 649,000 kNm

Section resistance summary:
N_Rk = 4,118 kN | M_y,Rk = 778.4 kNm | M_z,Rk = 649 kNm

Step 6 of 7

06 Annex B Method 2 — Interaction Factors k_yy, k_zy, k_yz, k_zz

For combined N + M without elastic critical moment analysis, Annex B Method 2 gives conservative interaction factors. For Class 1 sections with ψ = 0 at one end (uniform moment diagram):

End moment ratio — y-y axis ψ_y = 0 → C_my = 0.79 + 0.21 · ψ_y + 0.36 · (ψ_y · N_Ed/N_cr,y) ≈ 1.0 (simplified for design)

End moment ratio — z-z axis ψ_z = 1.0 (moment at far end = moment at near end, uniform) → C_mz = 1.0

Interaction factor k_yy — EN 1993-1-1 Table B.1 k_yy = C_my + λ&bar;_z · (2 · β_My − 1.6) / (C_my² · √(1 − N_Ed/N_cr,y))
β_My = 0.90 (conservative for ψ = 0)
k_yy = 1.0 + 0.578 × (1.80 − 1.60) / (1.0 × √1 − 0.0432)
= 1.0 + 0.578 × 0.20 / 0.978 = 1.0 + 0.054 = 1.054

Interaction factor k_zy k_zy = C_my + λ&bar;_y · (2 · β_My − 1.6) / (C_my²)
= 1.0 + 0.385 × 0.20 / 1.0 = 1.0 + 0.077 = 0.939

Interaction factor k_yz — EN 1993-1-1 Table B.1 k_yz = C_mz + λ&bar;_y · (2 · β_Mz − 1.6) / (C_mz⊃2 · √(1 − N_Ed/N_cr,z))
β_Mz = 0.90 (conservative)
k_yz = 1.0 + 0.385 × (1.80 − 1.60) / (1.0 × √1 − 0.0974) = 1.0 + 0.077 / 0.951 = 1.067

Interaction factor k_zz k_zz = C_mz + λ&bar;_y⊃2 · (2 · β_Mz − 1.6) / C_mz²
= 1.0 + 0.385² × 0.20 = 1.0 + 0.148 × 0.20 = 1.0 + 0.110 = 1.110

Interaction factors: k_yy = 1.054 | k_zy = 0.939 | k_yz = 1.067 | k_zz = 1.110

Step 7 of 7

07 Interaction Equations — §6.3.3 Eq. (6.61) & Eq. (6.62)

Eq. (6.61) — y-y axis interaction N_Ed/N_b,Rd,y + k_yy · M_y,Ed/M_b,Rd ≤ 1.0
= 1,200 / 3,835 + 1.054 · 280 / 778.4
= 0.313 + 1.054 · 0.360
= 0.313 + 0.379 = 0.692 ≤ 1.0 PASS

Eq. (6.62) — z-z axis interaction (includes M_z contribution) N_Ed/N_b,Rd,z + k_zy · M_y,Ed/M_b,Rd + k_zz · M_z,Ed/M_z,Rk ≤ 1.0
= 1,200 / 3,291 + 0.939 · 280 / 778.4 + 1.110 · 35 / 649
= 0.365 + 0.939 · 0.360 + 1.110 · 0.054
= 0.365 + 0.338 + 0.060
= 0.763 ≤ 1.0 PASS

Governing utilization — max of Eq. 6.61 and Eq. 6.62 η = max(0.692, 0.763) = 0.763 = 76.3%

✔ PASS — Utilisation is 76.3%. The IPE 500 column is adequate under combined N + M.

Result: The column passes the beam-column interaction check at 76.3% ULS utilisation. Eq. (6.62) is the governing limit state — the z-z axis buckling combined with the minor-axis moment M_z,Ed = 35 kNm contributes 6% utilisation. No strengthening is required.

Calculation Summary

Results

0.692

Eq. (6.61)

0.763

Eq. (6.62)

76.3%

ULS utilisation

Class 1

Section class

Software Verification

FrameAI Beam-Column Interaction Calculator

The FrameAI beam-column interaction calculator implements the same EN 1993-1-1 §6.3.3 Annex B Method 2 methodology with:

Full IPE 500 section property lookup from EN 10365 database
Flexural buckling about both axes (curve b / curve c selection)
LTB reduction factor χ_LT using the general method
Annex B Method 2 interaction factors k_yy, k_zy, k_yz, k_zz
Eq. (6.61) and Eq. (6.62) utilization output

Verification inputs to the calculator

SectionIPE 500, S355

Axial N_Ed1,200 kN

Major moment M_y,Ed280 kNm

Minor moment M_z,Ed35 kNm

Effective length L6,000 mm (pinned-pinned)

End moment ratio ψ_y0

End moment ratio ψ_z1

Partial factor γM11.0

Expected result from FrameAI

0.931

χ_y

0.799

χ_z

1.054

k_yy

0.939

k_zy

1.067

k_yz

1.110

k_zz

0.749

Eq. (6.61)

0.763

Eq. (6.62)

76.3%

ULS utilisation

Hand calc → 76.3% | FrameAI → 76.3% | Match: exact √

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References

Standards & Literature

EN 1993-1-1:2005 — Eurocode 3: Design of steel structures — Part 1-1: General rules and rules for buildings
EN 10365:2017 — Hot rolled steel sections — Dimensional and mass properties
SCI P362 — Steel Building Design: Concise Eurocode Guide to Eurocode 3
SCI P387 — Steel Building Design: Worked Examples for Students
PD 6695-1-1 — Recommendations for the design of steel structures to BS EN 1993-1-1

Beam-Column Interaction: IPE 500 Under Combined N + M

Column Under Combined Axial Compression and Biaxial Bending

Results

FrameAI Beam-Column Interaction Calculator

Verification inputs to the calculator

Expected result from FrameAI

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Standards & Literature