Hollow Section Worked Example - SHS to Eurocode 3

A complete Eurocode 3 capacity check for a representative square hollow section (SHS) - classification, cross-section bending, lateral-torsional buckling and compression buckling, computed by the EC3 engine.

Hollow sections are closed profiles with very high torsional stiffness, and this worked example shows how that changes the Eurocode 3 checks. The section is classified - using the d/t ratio for CHS or the web and flange slenderness for SHS/RHS - and the cross-section resistances N_pl,Rd, M_c,y,Rd, M_c,z,Rd and V_pl,Rd are derived.

A square hollow section (SHS) has equal stiffness about both axes and a circular hollow section (CHS) has no weak axis at all, so neither develops lateral-torsional buckling: their buckling resistance equals the cross-section bending resistance, M_b,Rd = M_c,Rd. Only a rectangular hollow section (RHS) is checked for LTB. The compression buckling resistance N_b,Rd and the combined interaction (clauses 6.2.9 and 6.3.3) complete the check. Because closed sections are not susceptible to torsional deformation, the Annex B interaction factors take their simpler form. Every value matches the interactive hollow-section capacity tool.

Representative section SHS 140×140×8 in grade S355 is Class 1. Major-axis bending resistance M_c,y,Rd = 72.4 kNm, minor-axis M_c,z,Rd = 72.4 kNm, axial squash N_pl,Rd = 1477 kN, and compression buckling N_b,Rd = 572 kN at 6 m.

Section properties

Mass: 32.6 kg/m
Area, A: 41.6 cm²
Iᵧ (major): 1200.0 cm⁴
I_z (minor): 1200.0 cm⁴
iᵧ: 5.36 cm
i_z: 5.36 cm

Cross-section resistances

Class: 1
Mc,y,Rd: 72.4 kNm
Mc,z,Rd: 72.4 kNm
Npl,Rd: 1476.8 kN
Vpl,Rd: 426.3 kN

Cross-section & member checks - full derivation

Material

Yield strength

f_y = 355.0\ \mathrm{N/mm^{2}}\quad (S355)

Material factor

\varepsilon = \sqrt{\dfrac{235}{f_y}} = 0.8136

Cross-section classification (§5.5)

Web

c_w = h - 3t = 116.0\ \mathrm{mm},\ \dfrac{c_w}{t} = 14.50 \le 72\varepsilon = 58.58 \Rightarrow \text{Web Class 1}

Flange

c_f = b - 3t = 116.0\ \mathrm{mm},\ \dfrac{c_f}{t} = 14.50 \le 33\varepsilon = 26.85 \Rightarrow \text{Flange Class 1}

Governing

\text{Class} = \max(1,\ 1) = \mathbf{Class\ 1}

Cross-section resistances (§6.2)

Axial squash, Npl,Rd (§6.2.4)

N_{pl,Rd} = \dfrac{A f_y}{\gamma_{M0}} = \dfrac{4160 \times 355.0}{1.0 \times 10^{3}} = \mathbf{1,477}\ \mathrm{kN}

Bending, Mc,Rd (§6.2.5, W_{pl})

M_{c,Rd} = \dfrac{W_{pl} f_y}{\gamma_{M0}} = \dfrac{204000 \times 355.0}{1.0 \times 10^{6}} = \mathbf{72.4}\ \mathrm{kNm}

Shear, Vpl,Rd (§6.2.6)

V_{pl,Rd} = \dfrac{A_v f_y}{\sqrt3\,\gamma_{M0}} = \mathbf{426}\ \mathrm{kN}

Lateral-torsional buckling

Square hollow section

\text{No weak axis} \Rightarrow M_{b,Rd} = M_{c,Rd} = \mathbf{72.4}\ \mathrm{kNm}

Compression buckling, Nb,Rd (§6.3.1) - Lcr = 6 m

Assumption

\text{Member acts as a column, buckling length } L_{cr} = 6\ \mathrm{m}\ (\text{the minor } z\text{-axis governs})

Non-dimensional slenderness (z-axis)

\bar\lambda_{z} = 1.465

Reduction factor

\chi_{z} = 0.388

Buckling resistance

N_{b,z,Rd} = \chi_{z}\,\dfrac{A f_y}{\gamma_{M1}} = \mathbf{572.4}\ \mathrm{kN}

Axial + bending interaction (§6.2.9)

Axial ratio (worked at n = 0.3)

n = \dfrac{N_{Ed}}{N_{pl,Rd}} = 0.30

Reduced major-axis moment, M_N,y,Rd (§6.2.9.1)

M_{N,y,Rd} = M_{pl,y,Rd}\,\dfrac{1-n}{1-0.5a} = \mathbf{65.9}\ \mathrm{kNm}

Reduced minor-axis moment, M_N,z,Rd

M_{N,z,Rd} = \mathbf{65.9}\ \mathrm{kNm}

Biaxial interaction (Expr 6.41)

\left(\dfrac{M_{y,Ed}}{M_{N,y,Rd}}\right)^{\alpha} + \left(\dfrac{M_{z,Ed}}{M_{N,z,Rd}}\right)^{\beta} \le 1.0\quad(\alpha=2,\ \beta=\max(5n,1))

Member stability - combined buckling (§6.3.3, Annex B)

Worked at N_Ed = 172 kN, M_y,Ed = 29.0 kNm, M_z,Ed = 7.2 kNm

n_y = \dfrac{N_{Ed}}{N_{b,y,Rd}} = 0.300,\quad n_z = \dfrac{N_{Ed}}{N_{b,z,Rd}} = 0.300

Interaction factors (Annex B, Method 2)

k_{yy} = 1.116,\ k_{yz} = 0.670,\ k_{zy} = 0.670,\ k_{zz} = 1.116

Expression (6.61)

\dfrac{N_{Ed}}{N_{b,y,Rd}} + k_{yy}\dfrac{M_{y,Ed}}{M_{b,Rd}} + k_{yz}\dfrac{M_{z,Ed}}{M_{c,z,Rd}} = \mathbf{0.813}\ \le 1.0\ \checkmark

Expression (6.62)

\dfrac{N_{Ed}}{N_{b,z,Rd}} + k_{zy}\dfrac{M_{y,Ed}}{M_{b,Rd}} + k_{zz}\dfrac{M_{z,Ed}}{M_{c,z,Rd}} = \mathbf{0.679}\ \le 1.0\ \checkmark

Run SHS 140×140×8 with your own loads in the tool →

See the design theory behind these checks, or browse the SHS property table.

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