Breif guide of resistance of cross-section

To check if a steel member is structurally sufficient under certain axial force, moment and shear, a cross-section resistance check is required. In the article below, it provide a simple guide according to Eurocode 3 EN 1993-1-1:2005:

Section Classification
Axial Force Check [Tension and Compression]
Bending Resistance [Major and Minor Axis]
Shear Resistance
Combine Check [Bending and Shear, Bending and Axial Force and Bending, shear and axial force]

Please noticed that above check is for steel cross-section resistance under fully-restrained member. For un-restrained member, buckling effect need to be considered, which will discuss in another article.

Step 0: Gather Input

To design a steel section, there are three basic information you need to know:

Section Geometry (I section/RHS/CHS)
Steel Material Strength (f_y)
Design Action (Shear/Moment/Axial Load of the section)

Step 1: Section Classification

Here will based on the section geometry and the corresponding material strength, classify the corresponding Class (Class 1, 2, 3 or 4) subject to three checking conditions: Cl.5.5, EN 1993-1-1:2005

Compression
Bending
Compression and Bending

The details explanation and example of section classification can refer to other articles.

Step 2: Axial Resistance

For section under axial force, either tension or compression. Axial resistance check need to be carry out.

Step 2.1: Tension Resistance

The tension resistance of a steel section can be determined by:

Cross-section Tension Resistance

\begin{aligned}N_{t,Rd} = \dfrac{A\,f_y}{\gamma_{M0}}\end{aligned}

EN 1993-1-1 cl.6.2.3, eq. 6.6

where
f_y = Strength (N/mm²)
A = Gross Area (mm²)
γ_M0 = Material Factor = 1.0 from eurocode 3

Check if Pass

\begin{aligned}\dfrac{N_{Ed}}{N_{t,Rd}} ≤ 1\end{aligned}

EN 1993-1-1 cl.6.2.3, eq. 6.5

where
N_Ed = Design Axial Force

Step 2.2: Compression Resistance

The compression resistance of a steel section is similar to tension resistance for Class 1 - 3 Section, but an effective area will be used for Class 4, as shown below:

Cross-section Compression Resistance

\begin{aligned}N_{c,Rd} &= \dfrac{A\,f_y}{\gamma_{M0}} \hspace{0.1cm} (Class \hspace{0.1cm} 1,2,3) \\ &= \dfrac{A_{eff}\,f_y}{\gamma_{M0}} \hspace{0.1cm} (Class \hspace{0.1cm} 4)\end{aligned}

EN 1993-1-1 cl.6.2.3, eq. 6.10 and 6.11

where
A_eff = Effective Area (mm²)

Check if Pass

\begin{aligned}\dfrac{N_{Ed}}{N_{t,Rd}} ≤ 1\end{aligned}

EN 1993-1-1 cl.6.2.3, eq. 6.5

Step 3: Bending Resistance

For section with bending moment, either moment at major axis (y-axis) and/or minor axis (z-axis).

To carry out the capacity, it need to refer to the class subject to bending from section classification:

Cross-section Bending Resistance

\begin{aligned}M_{c,Rd} &= \dfrac{W_{pl}\,f_y}{\gamma_{M0}} \hspace{0.1cm} (Class \hspace{0.1cm} 1,2) \\ &= \dfrac{W_{el}\,f_y}{\gamma_{M0}} \hspace{0.1cm} (Class \hspace{0.1cm} 3) \\ &= \dfrac{W_{eff,min}\,f_y}{\gamma_{M0}} \hspace{0.1cm} (Class \hspace{0.1cm} 4)\end{aligned}

EN 1993-1-1 cl.6.2.5, eq. 6.13, 6.14 and 6.15

Check if Pass

\begin{aligned}\dfrac{M_{Ed}}{M_{c,Rd}} ≤ 1\end{aligned}

EN 1993-1-1 cl.6.2.3, eq. 6.12

If both moment at major and minor axis exist for the section, combination of resistance check need to used as below.

Biaxial Bending Check

\begin{aligned}\dfrac{M_{y,Ed}}{M_{c,y,Rd}} + \dfrac{M_{z,Ed}}{M_{c,y,Rd}} ≤ 1\end{aligned}

EN 1993-1-1 cl.6.2.1, eq. 6.2

Step 4: Shear Resistance

The shear resistance of a steel section is determined by below, without any need of section classification:

Plastic shear resistance

\begin{aligned}V_{pl,Rd} = \dfrac{A_v\,(f_y/\sqrt{3})}{\gamma_{M0}}\end{aligned}

EN 1993-1-1 Cl.6.2.6(2), eq. 6.18

where
A_v = Shear Area

Section	Shear area A_v
Rolled I/H section — load parallel to web	$A_v = A - 2b_f t_f + (t_w + 2r)t_f \ \ge \eta\,h_w t_w$
RHS — load parallel to depth h	$A_v = A\,\dfrac{h}{h+b}$
RHS — load parallel to width b	$A_v = A\,\dfrac{b}{h+b}$
CHS — any direction	$A_v = \dfrac{2A}{\pi}$

Check if Pass

\begin{aligned}\dfrac{V_{Ed}}{V_{c,Rd}} ≤ 1\end{aligned}

EN 1993-1-1 cl.6.2.3, eq. 6.12

Step 5: Combine Check

As axial force, moment and shear force may be applied to same cross-section at the same time, so other than simple separate resistance check, we also need to carry out the combine check.

Step 5.1: Bending and Shear Check

Combine Bending and Shear Check

1. Check the shear force to shear resistance ratio,

\frac{V_{Ed}}{V_{pl,Rd}}

Check if reduction factor required

\begin{aligned}\frac{V_{Ed}}{V_{pl,Rd}} &< 0.5, no \hspace{0.1cm} combined \hspace{0.1cm} check \hspace{0.1cm} required \\\frac{V_{Ed}}{V_{pl,Rd}} &> 0.5, reudction \hspace{0.1cm} factor \hspace{0.1cm} \rho \hspace{0.1cm} required\end{aligned}

EN 1993-1-1 Cl.6.2.8(3), eq. 6.29

2. Calculate shear reduction factor, ρ

Shear Reduction Factor

\begin{aligned}\rho = (\frac{2V_{Ed}}{V_{pl,Rd}} - 1)^{2}\end{aligned}

EN 1993-1-1 Cl.6.2.8(3)

3. Apply reduced factor to material yield stregth

Reduced yield strength

\begin{aligned}(1 - \rho) f_{y}\end{aligned}

EN 1993-1-1 Cl.6.2.8(3), eq. 6.29

4.1 For I section at Class 1,2 at Major Axis, the reduced bending moment can be calculated as:

Reduced plastic resistance for I section

\begin{aligned}M_{y,V,Rd} = \frac{[W_{pl,y} - \frac{\rho {A_{w}}^{2}}{4t_{w}}]f_{y}}{\gamma_{M0}}\end{aligned}

EN 1993-1-1 Cl.6.2.8(5), eq 6.30

4.2 For other condition:

Cross-section Bending Resistance

\begin{aligned}M_{V,Rd} &= \dfrac{W_{pl}\,f_y(1-\rho)}{\gamma_{M0}} \hspace{0.1cm} (Class \hspace{0.1cm} 1,2) \\ &= \dfrac{W_{el}\,f_y(1-\rho)}{\gamma_{M0}} \hspace{0.1cm} (Class \hspace{0.1cm} 3) \\ &= \dfrac{W_{eff,min}\,f_y(1-\rho)}{\gamma_{M0}} \hspace{0.1cm} (Class \hspace{0.1cm} 4)\end{aligned}

EN 1993-1-1 cl.6.2.5, eq. 6.13, 6.14 and 6.15 & 6.12

Step 5.2: Bending and Axial Check

When the section is under bending and axial, below check need to perform based on the Class subject to compression and bending:

Class 1 and 2 - I section - Specific Axis (y/z) Bending and Axial Force Check

1. Check Condition to determine if the bending resistance of specifc axis need to apply reduction

Y axis of I SectionCheck if reduction factor required

\begin{aligned}N_{Ed} &< 0.25 N_{pl,Rd} \hspace{0.5cm} and \\N_{Ed} &< \frac{0.5h_{w}t_{w}f_{y}}{\gamma_{M0}}\end{aligned}

EN 1993-1-1 Cl.6.2.9.1(4), eq. 6.33 & 6.34

Z axis of I Section:Check if reduction factor required

\begin{aligned}N_{Ed} &< \frac{h_{w}t_{w}f_{y}}{\gamma_{M0}}\end{aligned}

EN 1993-1-1 Cl.6.2.9.1(4), eq. 6.35

2. If reduction required, the corresponding Bending Resistance with Axial Reduction Factor will be:

Y axis of I Section:Bending Resistance with Axial Reduction Factor

\begin{aligned}M_{N,y,Rd} = M_{pl,y,Rd}(1-n)/(1 - 0.5a)\end{aligned}

EN 1993-1-1 Cl.6.2.9.1(5), eq. 6.36

Z axis of I Section:Bending Resistance with Axial Reduction Factor

\begin{aligned}for \hspace{0.2cm} n ≤ a: M_{N,z,Rd} &= M_{pl,z,Rd} \\for \hspace{0.2cm} n > a: M_{N,z,Rd} &= M_{pl,z,Rd}[1 - (\frac{n-a}{1-a})^{2}]\end{aligned}

EN 1993-1-1 Cl.6.2.9.1(5), eq. 6.37, 6.38

where
n = N_Ed/N_pl,Rd
a = (A - 2bt_f)/A

By comparing

M_{N,Rd} > M_{Ed}

, it can determined if the section is sufficient with the combine design action.

Class 1 and 2 - RHS/CHS - Specific Axis (y/z) Bending and Axial Force Check

For RHS or CHS, a reduction factor from axial need to apply to the Bending resistance M_pl,Rd, the reduced resistance can be found in below:

Both axis of CHS/RHS: Bending Resistance with Axial Reduction Factor

\begin{aligned}M_{N,y,Rd} &= M_{pl,y,Rd}(1 - n)/(1 - 0.5a_{w}) ≤ M_{pl,y,Rd} \\M_{N,z,Rd} &= M_{pl,z,Rd}(1 - n)/(1 - 0.5a_{f}) ≤ M_{pl,z,Rd}\end{aligned}

EN 1993-1-1 Cl.6.2.9.1(5), eq. 6.39, 6.40

where
n = N_Ed/N_pl,Rd
a_w = (A - 2bt)/A ≤ 0.5
a_f = (A - 2bt)/A ≤ 0.5

By comparing

M_{N,Rd} > M_{Ed}

, it can determined if the section is sufficient with the combine design action.

Class 1 and 2 - RHS/CHS/I section - Biaxial Bending and Axial Force Check

For RHS or CHS or I section, where biaxial bending (M_y & M_z) and axial force N_Ed exist, first determined the section's corresponding M_N,y,Rd and M_N,z,Rd, and carry out check as below:

Bi-axial bending and axial force check

\begin{aligned}[\frac{M_{y,Ed}}{M_{N,y,Rd}}]^{\alpha}+[\frac{M_{z,Ed}}{M_{N,z,Rd}}]^{\beta} ≤ 1\end{aligned}

EN 1993-1-1 Cl.6.2.9.1(5), eq. 6.39, 6.40

where
α = 2 ,β = 5n ≥ 1 for I section
α = 2 , β = 2 for CHS
α = β = 1.66/(1 - 1.13n²) ≤ 6 for RHS

Class 3 - RHS/CHS/I section - Specific Axis (y/z) or Biaxial Bending and Axial Force Check

For section subject to bending and compression classified as Class 3,

Compression and Bi-axial bending check

\begin{aligned}\frac{N_{Ed}}{N_{c,Rd}} + \frac{M_{y,Ed}}{W_{el,y}f_{y}} + \frac{M_{z,Ed}}{W_{el,z}f_{y}} ≤ 1\end{aligned}

EN 1993-1-1 Cl.6.2.9.2, eq. 6.42

where
N_c,Rd = 2 ,β = 5n ≥ 1 for I section

Step 5.3: Bending, Shear and Axial Force Check

For section under bending, shear and axial force at the same time, it will:

Determine the shear force to resistance ratio $\frac{V_{Ed}}{V_{pl,Rd}}$ and shear reduction factor $\rho$ required, refer to Step 5.1 - Bending and Shear Check.
If $\frac{V_{Ed}}{V_{pl,Rd}} > 0.5$ , replace f_y to $(1-\rho)f_{y}$ and carry out checking according to Step 5.2 - Bending and Axial Check.

📝Wrap Up and Summary

To check if a steel section is adequate, above is a quick guideline for different combination of design action and sections:

Section Classification
Axial Force Check [Tension and Compression]
Bending Resistance [Major and Minor Axis]
Shear Resistance
Combine Check [Bending and Shear, Bending and Axial Force and Bending, shear and axial force]

Further work examples will be providing in other articles.

Again, please noticed that above steel cross-section check are for fully-restrained member where no buckling effect need to be considered. For un-restrained member, buckling effect need to be considered, which will discuss in another article.

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References

Eurocode 3: Design of steel structures — Part 1-1: General rules and rules for buildings (EN 1993-1-1) —CEN (European Committee for Standardization) ·2005
DESIGNERS’ GUIDE TO EN 1993-1-1 EUROCODE 3: DESIGN OF STEEL STRUCTURES —L. GARDNER and D. A. NETHERCOT ·2005

CivilSimple Team

The CivilSimple Team writes practical engineering guides for the profession and the curious. All articles are reviewed for technical accuracy before publication.