# Cross-Section Design of Two-Span Beam

### Technical Article

New 001552 2 January 2019
The cross-section class of a two-span beam will be designed in the following. In addition, the necessary cross-section designs will be performed. The global stability failure will be excluded due to sufficient stabilizing measures.

#### Design of Cross-Section Class

The area of the inner support of the two-span beam is governing for designing the cross-section class and performing the cross-section design.

Design for the web (ψ = -1)
[1] Table 5.2, cross-section parts supported on two sides
$\begin{array}{l}\mathrm c\;=\;230\;-\;2\;⋅\;(12\;+\;21)\;=\;164\;\mathrm{mm}\\\mathrm{exist}\;\mathrm c/{\mathrm t}_\mathrm w\;=\frac{164}{7.5}\;=\;21.87\\\mathrm{limit}\;\mathrm c/{\mathrm t}_\mathrm w\;=\;72\;⋅\;\mathrm\varepsilon\;=\;72\;⋅\;0.81\;=\;58.32\;>\;21.87\end{array}$

The web thus fulfills the requirements of cross-section class 1.

Design for the bottom chord (ψ = 1)
[1] Table 5.2, cross-section parts supported on one side
$\begin{array}{l}\mathrm c\;=\;\frac{240\;-\;(7.5\;+\;2\;\cdot\;21)\;}2=\;95.3\;\mathrm{mm}\\\mathrm{exist}\;\mathrm c/{\mathrm t}_\mathrm f\;=\;\frac{95.3}{12}\;=\;7.94\;>\;9\;⋅\;\mathrm\varepsilon\;=\;7.29\\\mathrm{limit}\;\mathrm c/{\mathrm t}_\mathrm f\;=\;10\;⋅\;\mathrm\varepsilon\;=\;10\;⋅\;0.81\;=\;8.10\;>\;7.94\end{array}$

The chords thus fulfill the requirements of cross-section class 2. The cross-section has to be assigned to cross-section class 2 because the class of the cross-section part which is evaluated the most unfavorable will be governing for the entire cross-section.

#### Shear Buckling of Web

It has to be checked if the shear buckling of the web has to be evaluated, regardless of the assignment of the cross-section class according to 6.2.6, Section (6) in [1]. The value η in Equation 6.22 in [1] is applied with 1.20.

$\frac{{\mathrm h}_\mathrm w}{{\mathrm t}_\mathrm w}\;=\;\frac{230\;-\;2\;⋅\;12}{7.5}\;=\;27.47\;<\;\mathrm{limit}\;\frac{\displaystyle{\mathrm h}_\mathrm w}{\displaystyle{\mathrm t}_\mathrm w}\;=\;72\;⋅\;\frac{\mathrm\varepsilon}{\mathrm\eta}\;=\;72\;⋅\;\frac{0.81}{1.20}\;=\;48.6$

According to EN 1993-1-5, Section 5, no shear buckling design is required.

#### Shear Designs

The cross-section design is carried out for the cross-section of class 2. Above the inner support, the beam is subjected to bending and shear force, at the location of the maximum sagging moment only to bending. The influence of the M-V interaction is checked before determining the ultimate limit state of the structure. If VEd is not more than 0.5 ⋅ Vpl,Rd, it is not required to reduce the moment resistance according to [1] Section 6.2.8 (2).

$\begin{array}{l}{\mathrm V}_{\mathrm{pl},\mathrm z,\mathrm{Rd}\;}=\;{\mathrm A}_\mathrm{Vz}\;⋅\;{\mathrm\tau}_\mathrm{Rd}\;=\;25.18\;⋅\;\frac{35.5}{\sqrt3\;⋅\;\;1.0}\;=\;516.09\;\mathrm{kN}\\\frac{{\mathrm V}_{\mathrm z,\mathrm{Ed}}}{\;{\mathrm V}_{\mathrm{pl},\mathrm z,\mathrm{Rd}}}\;=\;\frac{130.96\;}{516.09}\;=\;0.254\;<\;0.5\end{array}$

It is not necessary to reduce the moment resistance.

The following design has to be fulfilled for the design value of the acting bending moments MEd:
$\frac{{\mathrm M}_\mathrm{Ed}}{{\mathrm M}_{\mathrm c,\mathrm{Rd}}\;}\;\leq\;1.0$

The design value of the bending stress of a cross-section, loaded with uniaxial bending, is determined as follows for cross-sections of class 2:
$\begin{array}{l}{\mathrm M}_{\mathrm c,\mathrm{Rd}}\;=\;{\mathrm M}_{\mathrm{pl},\mathrm{Rd}}\;=\;\frac{{\mathrm W}_\mathrm{pl}\;⋅\;{\mathrm f}_\mathrm y}{{\mathrm\gamma}_{\mathrm M0}}\\{\mathrm M}_{\mathrm c,\mathrm{Rd}}\;=\;{\mathrm M}_{\mathrm{pl},\mathrm{Rd}}\;=\;\frac{\;2\;⋅\;{\mathrm S}_\mathrm y\;⋅\;{\mathrm f}_\mathrm y}{{\mathrm\gamma}_{\mathrm M0}}\;=\;\frac{2\;⋅\;372.3\;⋅\;35.5}{1.0}\;=\;264.33\;\mathrm{kNm}\\\frac{{\mathrm M}_\mathrm{Ed}}{\;{\mathrm M}_{\mathrm c,\mathrm{Rd}\;}}\;=\;\frac{155.76\;}{264.33\;}\;=\;0.59\;\leq\;1.0\end{array}$

#### Keywords

design, cross-section class, shear buckling

#### Reference

 [1] Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings; EN 1993-1-1:2010-12 [2] Kuhlmann, U.; Feldmann, M.; Lindner, J.; Müller, C.; Stroetmann, R.: Eurocode 3 - Bemessung und Konstruktion von Stahlbauten - Band 1: Allgemeine Regeln und Hochbau - DIN EN 1993-1-1 mit Nationalem Anhang - Kommentar und Beispiele. Berlin: Beuth, 2014 [3] Albert, A.: Schneider - Bautabellen für Ingenieure mit Berechnungshinweisen und Beispielen, 23. Auflage. Köln: Bundesanzeiger, 2018