Steel Section Design Theory - Eurocode 3

This is a concise theory reference for the design of steel sections to Eurocode 3 (EN 1993-1-1), covering the cross-section properties tabulated in this tool and the resistance checks the Capacity tab computes. It explains what each property means and how the key ultimate-limit-state (ULS) resistances are derived.

Section properties - what they mean

Every tabulated section carries the geometric properties below. They are the inputs to all strength and stability checks.

• Area ($A$, cm²) - the cross-sectional area, used for axial (tension/compression) resistance and self-weight.
• Second moment of area ($I_y$, $I_z$, cm⁴) - resistance to bending about the major (y-y) and minor (z-z) axes; governs deflection and buckling. Larger $I$ = stiffer.
• Radius of gyration ($i_y$, $i_z$, cm) - $i = \sqrt{I/A}$; sets the slenderness for buckling. The minor-axis $i_z$ usually governs column buckling.
• Elastic modulus ($W_{el}$, cm³) - $W_{el} = I / c$ (c = distance to extreme fibre); gives the elastic bending resistance (Class 3 sections).
• Plastic modulus ($W_{pl}$, cm³) - the first moment of area about the plastic neutral axis; gives the plastic bending resistance (Class 1 and 2 sections), which exceeds the elastic value.
• Torsion constant ($I_t$, cm⁴) and warping constant ($I_w$) - control torsional and lateral-torsional buckling behaviour.

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

Before any resistance is calculated, the section is classified 1–4 by the slenderness of its compression parts (flange outstand $c/t_f$, web $c/t_w$), scaled by the material factor:

• Class 1 - can form a plastic hinge with rotation capacity; full plastic resistance and plastic analysis allowed.
• Class 2 - can reach the plastic moment but with limited rotation; plastic resistance, elastic analysis.
• Class 3 - can reach the yield (elastic) moment but local buckling prevents the plastic moment; elastic resistance.
• Class 4 - local buckling occurs before yield; an effective (reduced) cross-section is used.

Bending resistance - $M_{c,Rd}$ (§6.2.5)

The cross-section bending resistance depends on class. For Class 1 and 2 it uses the plastic modulus; for Class 3 the elastic modulus:

where $f_y$ is the yield strength (e.g. 275 MPa for S275, 355 MPa for S355) and $\gamma_{M0} = 1.0$ (UK National Annex).

Lateral-torsional buckling - $M_{b,Rd}$ (§6.3.2)

An unrestrained beam in bending can buckle sideways and twist before reaching $M_{c,Rd}$. The buckling resistance applies a reduction factor $\chi_{LT}$ to the cross-section resistance:

The reduction factor $\chi_{LT}$ depends on the non-dimensional LTB slenderness:

where:

• $M_{cr}$ - the elastic critical moment for lateral-torsional buckling.
• $L_{cr}$ - the unrestrained (buckling) length of the compression flange.
• $C_1$ - the moment-distribution factor (1.0 for uniform moment, higher for a more favourable shape).

$M_{cr}$also depends on the section's torsional and warping stiffness. Shorter spans, more lateral restraint and a more uniform moment all raise $M_{b,Rd}$.

Compression / flexural buckling - $N_{b,Rd}$ (§6.3.1)

A column's resistance is the squash load $N_{pl,Rd} = A\, f_y / \gamma_{M0}$ reduced for buckling by the factor $\chi$:

The reduction $\chi$ follows the relevant buckling curve (a₀–d, Table 6.2) as a function of the non-dimensional slenderness:

Because $i_z < i_y$ for most I-sections, minor-axis buckling usually governs unless the column is braced about the weak axis.

Shear resistance - $V_{pl,Rd}$ (§6.2.6)

where $A_v$is the shear area (the web for I-sections). Shear and bending are checked together; high coexistent shear (>50% of $V_{pl,Rd}$) reduces the bending resistance.

Combined bending + axial - interaction checks

A member rarely carries pure bending or pure axial load - usually both, often biaxial. Two interaction checks apply, exactly as the Capacity tab computes them.

Cross-section interaction - Expr (6.41) §6.2.9

For Class 1/2 sections the plastic non-linear interaction governs:

where:

• $M_{N,y,Rd},\ M_{N,z,Rd}$ - axial-reduced moment resistances (the cross-section moment capacity after allowing for the coexistent axial force $N_{Ed}$).
• $n = N_{Ed}/N_{pl,Rd}$ - the axial utilisation ratio (clamped to 0–1).
• $\alpha,\ \beta$ - interaction exponents: for I/H sections $\alpha = 2$, $\beta = \max(5n,\,1)$; for CHS $\alpha = \beta = 2$; for solid/other shapes $\alpha = \beta = 1$ (linear). Class 3 sections use the linear elastic form.

Member stability - Expr (6.61) & (6.62) §6.3.3

When the member is also unrestrained (can buckle), both expressions must be satisfied - they combine column buckling with lateral-torsional buckling:

The interaction factors $k_{yy},\ k_{yz},\ k_{zy},\ k_{zz}$can be taken from either of two methods in EN 1993-1-1, presented in the standard's order below. This tool implements Annex B (Method 2), which is valid for all section types.

Annex A (Method 1)

Method 1 is only valid for doubly-symmetric sections. It splits each k-factor into auxiliary terms (it is not used by this tool):

Annex B (Method 2) - the factors this tool computes

For members not susceptible to torsional deformation (Table B.1), with the axial ratios $n_y = N_{Ed}/N_{b,y,Rd}$ and $n_z = N_{Ed}/N_{b,z,Rd}$:

$C_{my},\ C_{mz}$ are the equivalent-uniform-moment factors (Table B.3) - they depend on the moment shape over the member; a uniform moment gives the conservative value, a more favourable moment diagram a lower one.

Method 1 generally gives marginally higher capacity for doubly-symmetric members but is more involved; Method 2 is simpler and broadly applicable, which is why it is implemented here.

Section families compared

• UB / IPE (I-beams) - deep, efficient in major-axis bending; the workhorse beam. IPE is the European series (EN 10365); UB is the UK Blue Book.
• UC / HE (H-sections / wide-flange) - wider flanges, better minor-axis and axial performance; used as columns.
• Channels (PFC / UPN) - single-symmetric; common for purlins, bracing, and built-up members. The shear centre lies outside the web, so loading can induce torsion.
• Angles (EA / UA) - designed about principal axes (u-u, v-v); used in trusses and bracing.
• Hollow sections (CHS / SHS / RHS) - closed, high torsional stiffness, good in compression and architecturally clean.
• Cold-formed (Z-purlins, lipped channels) - thin-walled, light secondary members for roofs and cladding.

This is a summary for orientation; always design to the full provisions of EN 1993-1-1 and the relevant National Annex. Use the Capacity tab to compute section-specific resistances.