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Steel beam-column design to SANS 10162-1

A 203 × 203 × 46 H-section in S355, 4 m long, carrying 600 kN and 40 kNm: section class, column buckling, lateral-torsional buckling with M_cr, shear and the cl. 13.8 interaction. Every number below comes from the free STATIX steel member calculator and was checked by hand.

1. The problem

Input	Value
Section	203 × 203 × 46 H-section (universal column), hot-rolled
Steel	S355: f_y = 355 MPa, E = 200,000 MPa, G = 77,000 MPa
Length	4.0 m, pinned both ends, K = 1.0 about both axes, braced frame
Design loads (ultimate, already factored)	C_u = 600 kN, M_ux = 40 kNm (major axis), M_uy = 0, V_u = 120 kN
Lateral restraint	none between the ends: unbraced length for LTB = 4.0 m
Moment gradient	ω₂ = 1.0 (uniform moment, the safe value); ω₁ = 1.0

All resistance factors are φ = 0.9. Axes follow SANS 10162-1: x-x is the major axis, y-y the minor axis.

N-M interaction envelope for major-axis bending with the design point at 40 kNm and 600 kN, and the H-section shape, from the STATIX steel member calculator — The free steel member calculator with these inputs: the N–M envelope for major-axis bending, from C_r = 1,069 kN to M_r = 136 kNm, with the design point (40 kNm, 600 kN) at 89%.

2. Section properties

Property	Calculator library	SAISC Red Book
A	5,870 mm²	5,880 mm²
h × b × t_w × t_f	203.2 × 203.6 × 7.2 × 11.0 mm	203.2 × 203.2 × 7.3 × 11.0 mm
I_x / I_y	45.68 / 15.48 × 10⁶ mm⁴	45.6 / 15.4 × 10⁶ mm⁴
Z_e,x / Z_pl,x	450 / 497 × 10³ mm³	449 / 497 × 10³ mm³
Z_e,y / Z_pl,y	152 / 231 × 10³ mm³	151 / 230 × 10³ mm³
r_x / r_y	88.2 / 51.4 mm	88.1 / 51.2 mm
J	203 × 10³ mm⁴	225 × 10³ mm⁴
C_w	I_y(h − t_f)²/4 = 143 × 10⁹ mm⁶	142 × 10⁹ mm⁶

The two sets agree to within 1%, except the torsion constant J, which the library has about 10% below the Red Book. A lower J gives a lower M_cr, so the library value is on the safe side. With the Red Book values every result below changes by less than 1%.

3. Section class (SANS 10162-1 cl. 11, Table 4)

For the flange of an I-section the width b is half the full flange width (cl. 11.3.1c). The web width is the clear distance between the flanges (cl. 11.3.2c).

Flange: b/t = (203.6/2) / 11.0 = 9.25 Class 1 limit 145/√355 = 7.70, class 2 limit 170/√355 = 9.02, class 3 limit 200/√355 = 10.61 9.02 < 9.25 ≤ 10.61 → flange is class 3 Web: h_w/t_w = (203.2 − 2 × 11.0) / 7.2 = 25.2 Class 1 limit (1100/√355)(1 − 0.39 C_u/φC_y) = 51.1 → web is class 1 Section: class 3 (the worst element governs)

Class 3 means the section reaches first yield but not the full plastic moment. Bending resistance is based on the elastic modulus Z_e, not Z_pl. Many H-sections in S355 are class 3 for this reason, while the same section in 300WA would be class 2.

4. Axial compression (cl. 13.3.1)

C_r = φ A f_y (1 + λ²ⁿ)^−1/n, n = 1.34 for hot-rolled sections λ = (KL/r) √(f_y/(π²E)) Minor axis: KL/r_y = 4,000 / 51.4 = 77.9 F_e = π² × 200,000 / 77.9² = 325.3 MPa, λ_y = √(355/325.3) = 1.045 C_ry = 0.9 × 5,870 × 355 × (1 + 1.045^2.68)^−0.746 = 1,875.5 × 0.570 = 1,069 kN Major axis: KL/r_x = 4,000 / 88.2 = 45.3, λ_x = 0.608 C_rx = 1,875.5 × 0.840 = 1,575 kN Torsional buckling (cl. 13.3.2): f_ez = (π²EC_w/L² + GJ) / (A r₀²) = 544 MPa → C_rz = 1,343 kN, does not govern C_r = 1,069 kN (minor axis) ≥ 600 kN OK (56%)

n = 2.24 applies only to welded three-plate members with flame-cut flange edges and to hot-formed or stress-relieved hollow sections. A rolled H-section always takes n = 1.34. The calculator uses 1.34 for every section, which is safe for the others.

5. Lateral-torsional buckling (cl. 13.6)

The compression flange is free over the full 4 m, so the moment resistance is limited by lateral-torsional buckling. For a class 3 section cl. 13.6(b) uses the yield moment M_y.

M_y = Z_e,x f_y = 450 × 10³ × 355 = 159.8 kNm, φM_y = 143.8 kNm M_cr = (ω₂ π / KL) √[ E I_y G J + (πE/KL)² I_y C_w ] E I_y G J = 200,000 × 15.48×10⁶ × 77,000 × 203×10³ = 4.84 × 10²² (πE/KL)² I_y C_w = (157.1)² × 15.48×10⁶ × 143×10⁹ = 5.46 × 10²² M_cr = 1.0 × π/4,000 × √(10.30 × 10²²) = 252.1 kNm M_cr = 252.1 > 0.67 M_y = 107.0 kNm, so M_rx = 1.15 φ M_y (1 − 0.28 M_y/M_cr) = 1.15 × 143.8 × (1 − 0.28 × 159.8/252.1) = 165.3 × 0.823 = 136.0 kNm (not more than φM_y = 143.8) M_ux = 40 kNm ≤ 136.0 kNm OK (29%)

ω₂ = 1.0 is the value for uniform moment, the worst case. If the moment varies linearly between the ends, cl. 13.6(a) gives ω₂ = 1.75 + 1.05κ + 0.3κ² ≤ 2.5. Here κ is the ratio of the smaller to the larger end moment, positive for double curvature and negative for single curvature. Uniform moment is κ = −1 and gives ω₂ = 1.0. A beam with one end moment zero is κ = 0 and gives 1.75. Getting the sign wrong turns 1.0 into 2.5.

6. Minor-axis bending (cl. 13.5)

There is no minor-axis moment in this example, but the resistance is still reported. For a class 3 section cl. 13.5(b) uses the elastic modulus about both axes.

M_ry = φ Z_e,y f_y = 0.9 × 152 × 10³ × 355 = 48.6 kNm

7. Shear (cl. 13.4.1.1)

A_v = h t_w = 203.2 × 7.2 = 1,463 mm² (rolled section) h_w/t_w = 25.2 ≤ 440 √(k_v/f_y) = 440 √(5.34/355) = 54.0 → no shear buckling f_s = 0.66 f_y = 234.3 MPa V_r = φ A_v f_s = 0.9 × 1,463 × 234.3 = 308.5 kN ≥ 120 kN OK (39%)

k_v = 5.34 is the value for a web without transverse stiffeners.

8. Axial load and bending (cl. 13.8.3)

The section is class 3, so cl. 13.8.3 applies (all sections except class 1 and class 2 I-shapes). Its coefficients are 1.0 on both moment terms. Class 1 and 2 I-shapes use cl. 13.8.2 with 0.85 and β.

C_u/C_r + U_1x M_ux/M_rx + U_1y M_uy/M_ry ≤ 1.0 Amplification (cl. 13.8.4): C_ex = π²E I_x/L² = π² × 200,000 × 45.68×10⁶ / 4,000² = 5,636 kN U_1x = ω₁ / (1 − C_u/C_ex) = 1.0 / (1 − 600/5,636) = 1.119 (a) Cross-section: C_r at λ = 0 = φAf_y = 1,875.5 kN, M_rx = φM_y = 143.8 kNm, U_1x ≥ 1.0 600/1,875.5 + 1.119 × 40/143.8 = 0.320 + 0.311 = 0.63 (b) Overall member: C_r = C_ry = 1,069 kN, M_rx = 143.8 kNm 600/1,069 + 1.119 × 40/143.8 = 0.561 + 0.311 = 0.87 (c) Lateral-torsional buckling: C_r = C_ry = 1,069 kN (weak axis), M_rx = 136.0 kNm (cl. 13.6), U_1x ≥ 1.0 600/1,069 + 1.119 × 40/136.0 = 0.561 + 0.329 = 0.89 ← governs 0.89 ≤ 1.0 OK

Case (c) governs, as it usually does for an unrestrained column in bending: the weak-axis buckling load and the LTB moment are both the lowest values of their kind. U_1x = 1.119 means the 40 kNm grows by about 12% from the member's own deflection under 600 kN.

In case (b) the calculator uses the weak-axis C_r. For bending about the strong axis only, cl. 13.8.2(b) lets you use C_r = C_rx = 1,575 kN, which gives 0.381 + 0.311 = 0.69. The calculator's 0.87 is on the safe side, and case (c) governs either way.

9. Result

Check	Result
Section class	Class 3 (flange b/t = 9.25 > 9.02)
Axial compression	600 ≤ 1,069 kN (56%)
Major-axis bending with LTB	40 ≤ 136.0 kNm (29%), M_cr = 252 kNm
Minor-axis bending resistance	48.6 kNm (no demand)
Shear	120 ≤ 308.5 kN (39%)
Interaction, cl. 13.8.3(c)	0.89 ≤ 1.0
Mass	46.1 kg/m × 4.0 m = 184 kg

The member works at 89%. The two cheap ways to more capacity are a lateral restraint at mid-height (halves the LTB and weak-axis lengths) and a real ω₂ if the moment is not uniform.

Run this example yourselfThe free STATIX steel member calculator opens with exactly these inputs. Change the section, grade, length or loads and every step updates, with a printable calc sheet.
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