I-Section Worked Example - UB to Eurocode 3

A complete Eurocode 3 (EN 1993-1-1) capacity check worked through for a representative Universal Beam - classification, cross-section bending, lateral-torsional buckling and compression buckling, with every value computed by the same engine as the interactive tool.

This worked example follows the full Eurocode 3 design path for an I-section beam-column. It begins by establishing the material factor ε from the yield strength, then classifies the section by checking the web and flange slenderness against the Class 1–4 limits. The governing class decides whether the plastic modulus W_pl or the elastic modulus W_el is used for bending - a compact Class 1 or 2 Universal Beam develops its full plastic moment.

With the section classified, the cross-section resistances are derived: the axial squash load N_pl,Rd, the major- and minor-axis bending resistances M_c,y,Rd and M_c,z,Rd, and the shear resistance V_pl,Rd. Because an I-beam is weak about its minor axis, the unrestrained beam is then checked for lateral-torsional buckling: the elastic critical moment M_cr gives the LTB slenderness, the reduction factor χ_LT, and the buckling resistance M_b,Rd.

For columns and beam-columns, the compression buckling resistance N_b,Rd is derived about the governing (minor) axis. Finally the member is verified under combined axial force and bending using the cross-section interaction of clause 6.2.9 and the member-stability interaction of clause 6.3.3, whose factors k_yy, k_yz, k_zy and k_zz come from Annex B. Every number on this page is computed by the same engine as the interactive Universal Beam capacity tool, so you can reproduce and extend it with your own loads.

Representative section UB 457×191×89 in grade S275 is Class 1. Major-axis bending resistance M_c,y,Rd = 552.8 kNm, minor-axis M_c,z,Rd = 93.0 kNm, axial squash N_pl,Rd = 3135 kN, LTB resistance M_b,Rd = 360.0 kNm at 6 m, and compression buckling N_b,Rd = 954 kN at 6 m.

Section properties

Mass: 89.3 kg/m
Area, A: 114.0 cm²
Iᵧ (major): 41000.0 cm⁴
I_z (minor): 2090.0 cm⁴
iᵧ: 19.00 cm
i_z: 4.29 cm

Cross-section resistances

Class: 1
Mc,y,Rd: 552.8 kNm
Mc,z,Rd: 93.0 kNm
Npl,Rd: 3135.0 kN
Vpl,Rd: 818.3 kN

Cross-section & member checks - full derivation

Material

Yield strength

f_y = 275\ \mathrm{N/mm^{2}}\quad (S275,\ t_f = 17.7\ \mathrm{mm})

Material factor

\varepsilon = \sqrt{\dfrac{235}{f_y}} = \sqrt{\dfrac{235}{275}} = 0.9244

Cross-section classification (EN 1993-1-1 §5.5)

Web depth between fillets

d = h - 2t_f - 2r = 463.4 - 2(17.7) - 2(10.2) = 407.6\ \mathrm{mm}

Web slenderness

\dfrac{d}{t_w} = \dfrac{407.6}{10.5} = 38.82 \le 72\varepsilon = 66.56 \Rightarrow \text{Web Class 1}

Flange outstand

c_f = \dfrac{b - t_w - 2r}{2} = 80.5\ \mathrm{mm}

Flange slenderness

\dfrac{c_f}{t_f} = \dfrac{80.5}{17.7} = 4.55 \le 9\varepsilon = 8.32 \Rightarrow \text{Flange Class 1}

Governing class

\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{11400 \times 275}{1.0 \times 10^{3}} = \mathbf{3,135}\ \mathrm{kN}

Major-axis bending, Mc,y,Rd (§6.2.5, W_{pl,y})

M_{c,y,Rd} = \dfrac{W_{pl,y} f_y}{\gamma_{M0}} = \dfrac{2010000 \times 275}{1.0 \times 10^{6}} = \mathbf{552.8}\ \mathrm{kNm}

Minor-axis bending, Mc,z,Rd (§6.2.5)

M_{c,z,Rd} = \dfrac{W_{pl,z} f_y}{\gamma_{M0}} = \dfrac{338000 \times 275}{1.0 \times 10^{6}} = \mathbf{93.0}\ \mathrm{kNm}

Shear area, Av (§6.2.6)

A_v = A - 2bt_f + (t_w + 2r)t_f = 5154\ \mathrm{mm^{2}}

Shear resistance, Vpl,Rd

V_{pl,Rd} = \dfrac{A_v f_y}{\sqrt3\,\gamma_{M0}} = \dfrac{5154 \times 275}{\sqrt3 \times 1.0 \times 10^{3}} = \mathbf{818}\ \mathrm{kN}

Lateral-torsional buckling, Mb,Rd (§6.3.2) - Lcr = 6 m, C₁ = 1.13

Assumption

\text{Member acts as a beam, laterally unrestrained over } L_{cr} = 6\ \mathrm{m}\ (\text{if the compression flange is restrained, } M_{b,Rd} = M_{c,Rd})

Elastic critical moment

M_{cr} = C_1\,\dfrac{\pi^{2} E I_z}{L^{2}}\sqrt{\dfrac{I_w}{I_z} + \dfrac{L^{2} G I_t}{\pi^{2} E I_z}} = 452.6\ \mathrm{kNm}

LTB slenderness

\bar\lambda_{LT} = \sqrt{\dfrac{W_y f_y}{M_{cr}}} = \sqrt{\dfrac{2010000 \times 275}{452631022}} = 1.105

Reduction factor

\chi_{LT} = 0.651

Buckling resistance

M_{b,Rd} = \chi_{LT}\,\dfrac{W_y f_y}{\gamma_{M1}} = \mathbf{360.0}\ \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} = 1.611

Reduction factor

\chi_{z} = 0.304

Buckling resistance

N_{b,z,Rd} = \chi_{z}\,\dfrac{A f_y}{\gamma_{M1}} = \mathbf{954.1}\ \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{484.9}\ \mathrm{kNm}

Reduced minor-axis moment, M_N,z,Rd

M_{N,z,Rd} = \mathbf{93.0}\ \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 = 286 kN, M_y,Ed = 144.0 kNm, M_z,Ed = 9.3 kNm

n_y = \dfrac{N_{Ed}}{N_{b,y,Rd}} = 0.095,\quad n_z = \dfrac{N_{Ed}}{N_{b,z,Rd}} = 0.300

Interaction factors (Annex B, Method 2)

k_{yy} = 0.914,\ k_{yz} = 0.767,\ k_{zy} = 0.954,\ k_{zz} = 1.278

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{0.537}\ \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{0.809}\ \le 1.0\ \checkmark

Run UB 457×191×89 with your own loads in the tool →

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

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