Channel Worked Example - PFC to Eurocode 3

A complete Eurocode 3 capacity check for a representative parallel flange channel (PFC) - classification, bending, lateral-torsional buckling and compression buckling, every value from the EC3 engine.

Parallel flange channels (PFC) are single-symmetric, so this worked example covers the same Eurocode 3 checks as an I-section but with the channel’s distinctive behaviour in mind. After establishing ε and classifying the section, the cross-section resistances N_pl,Rd, M_c,y,Rd, M_c,z,Rd and V_pl,Rd are derived from the plastic or elastic modulus according to the governing class.

Because a channel’s shear centre lies outside the web, a load applied through the web induces torsion as well as bending - so the lateral-torsional buckling check (M_cr, λ̄_LT, χ_LT, M_b,Rd) is important whenever the compression flange is unrestrained. The compression buckling resistance N_b,Rd and the combined axial-plus-bending interaction (clauses 6.2.9 and 6.3.3, with the Annex B k-factors) complete the verification. All values are produced by the same engine as the interactive PFC capacity tool.

Representative section PFC 260×75×28 in grade S275 is Class 1. Major-axis bending resistance M_c,y,Rd = 90.2 kNm, minor-axis M_c,z,Rd = 17.1 kNm, axial squash N_pl,Rd = 965 kN, LTB resistance M_b,Rd = 37.0 kNm at 6 m, and compression buckling N_b,Rd = 91 kN at 6 m.

Section properties

Mass: 27.6 kg/m
Area, A: 35.1 cm²
Iᵧ (major): 3620.0 cm⁴
I_z (minor): 185.0 cm⁴
iᵧ: 10.10 cm
i_z: 2.30 cm

Cross-section resistances

Class: 1
Mc,y,Rd: 90.2 kNm
Mc,z,Rd: 17.1 kNm
Npl,Rd: 965.3 kN
Vpl,Rd: 307.7 kN

Cross-section & member checks - full derivation

Material

Yield strength

f_y = 275\ \mathrm{N/mm^{2}}\quad (S275,\ t_f = 12\ \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 = 260 - 2(12) - 2(12) = 212.0\ \mathrm{mm}

Web slenderness

\dfrac{d}{t_w} = \dfrac{212.0}{7} = 30.29 \le 72\varepsilon = 66.56 \Rightarrow \text{Web Class 1}

Flange outstand

c_f = b - t_w - r = 56.0\ \mathrm{mm}

Flange slenderness

\dfrac{c_f}{t_f} = \dfrac{56.0}{12} = 4.67 \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{3510 \times 275}{1.0 \times 10^{3}} = \mathbf{965}\ \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{328000 \times 275}{1.0 \times 10^{6}} = \mathbf{90.2}\ \mathrm{kNm}

Minor-axis bending, Mc,z,Rd (§6.2.5)

M_{c,z,Rd} = \dfrac{W_{pl,z} f_y}{\gamma_{M0}} = \dfrac{62000 \times 275}{1.0 \times 10^{6}} = \mathbf{17.1}\ \mathrm{kNm}

Shear area, Av (§6.2.6)

A_v = A - 2bt_f + (t_w + 2r)t_f = 2082\ \mathrm{mm^{2}}

Shear resistance, Vpl,Rd

V_{pl,Rd} = \dfrac{A_v f_y}{\sqrt3\,\gamma_{M0}} = \dfrac{2082 \times 275}{\sqrt3 \times 1.0 \times 10^{3}} = \mathbf{308}\ \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}} = 38.1\ \mathrm{kNm}

LTB slenderness

\bar\lambda_{LT} = \sqrt{\dfrac{W_y f_y}{M_{cr}}} = \sqrt{\dfrac{328000 \times 275}{38050486}} = 1.540

Reduction factor

\chi_{LT} = 0.411

Buckling resistance

M_{b,Rd} = \chi_{LT}\,\dfrac{W_y f_y}{\gamma_{M1}} = \mathbf{37.0}\ \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} = 3.005

Reduction factor

\chi_{z} = 0.095

Buckling resistance

N_{b,z,Rd} = \chi_{z}\,\dfrac{A f_y}{\gamma_{M1}} = \mathbf{91.5}\ \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{83.5}\ \mathrm{kNm}

Reduced minor-axis moment, M_N,z,Rd

M_{N,z,Rd} = \mathbf{17.1}\ \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 = 27 kN, M_y,Ed = 14.8 kNm, M_z,Ed = 1.7 kNm

n_y = \dfrac{N_{Ed}}{N_{b,y,Rd}} = 0.039,\quad n_z = \dfrac{N_{Ed}}{N_{b,z,Rd}} = 0.300

Interaction factors (Annex B, Method 2)

k_{yy} = 0.917,\ 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{0.482}\ \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 PFC 260×75×28 with your own loads in the tool →

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

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