Elastic-Plastic Cross-Section Design

Technical Article

The following article describes designing a two-span beam subjected to bending by means of the RF-/STEEL EC3 add-on module according to EN 1993-1-1. The global stability failure will be excluded due to sufficient stabilizing measures.

Structural System and Loading

Figure 01 - Structural System, Loading, Internal Forces

Figure 02 - Girder Section HEA 600, S235

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
$\begin{array}{l}\mathrm c\;=\;590\;-\;2\;⋅\;(25\;+\;27)\;=\;486\;\mathrm{mm}\\\mathrm{existing}\;\frac{\mathrm c}{{\mathrm t}_{\mathrm w}}\;=\;\frac{486}{13}\;=\;37.38\\\mathrm{limit}\;\frac{\mathrm c}{{\mathrm t}_{\mathrm w}}\;=\;72\;⋅\;\mathrm\varepsilon\;=\;72\;⋅\;1\;=\;72\;>\;37.38\end{array}$
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
$\begin{array}{l}\mathrm c\;=\;\frac{300\;-\;(13\;+\;2\;\cdot\;27)}2\;=\;116.5\;\mathrm{mm}\\\mathrm{existing}\;\frac{\mathrm c}{{\mathrm t}_{\mathrm f}}\;=\;\frac{116.5}{25}\;=\;4.66\;<\;9\;⋅\;\mathrm\varepsilon\;=\;9\;⋅\;1\;=\;9\end{array}$
The chords thus fulfill the requirements of cross-section class 1. The cross-section has to be assigned to cross-section class 1.

Elastic-Plastic Cross-Section Design

$\begin{array}{l}{\mathrm W}_{\mathrm{pl},\mathrm y}\;=\;2\;\cdot\;\left(30\;\cdot\;2.5\;\cdot\;28.25\;+\;27\;\cdot\;1.3\;\cdot\;16\right)\;=\;5,360\;\mathrm{cm}^3\\{\mathrm M}_{\mathrm{pl},\mathrm y,\mathrm{Rd}}\;=\;{\mathrm W}_{\mathrm{pl},\mathrm y\;}\;\cdot\;\frac{{\mathrm f}_{\mathrm y}}{{\mathrm\gamma}_{\mathrm M0}}\;=\;5,360\;\cdot\;\frac{23.5}1\;=\;125,960\;\mathrm{kNcm}\;=\;1,259.6\;\mathrm{kNm}\\{\mathrm A}_{\mathrm v,\mathrm z}\;=\;\mathrm A\;–\;2\;\cdot\;\mathrm b\;\cdot\;{\mathrm t}_{\mathrm f\;}+\;({\mathrm t}_{\mathrm w}\;+\;2\;\cdot\;\mathrm r)\;\;\cdot\;{\mathrm t}_{\mathrm f}\;=\;226\;–\;2\;\cdot\;30\;\cdot\;2.5\;+\;(1.3\;+\;2\;\cdot\;2.7)\;\cdot\;2.5\;=\;92.75\;\mathrm{cm}^2\\{\mathrm V}_{\mathrm{pl},\mathrm{Rd}}\;=\;{\mathrm A}_{\mathrm v,\mathrm z}\;\cdot\;\frac{{\mathrm f}_{\mathrm y}}{\sqrt3\;\cdot\;{\mathrm\gamma}_{\mathrm M0}}\;=\;92.75\;\cdot\;\frac{23.5}{\sqrt3\;\cdot\;1}\;=\;1,258.41\;\mathrm{kN}\end{array}$

The cross-section design is performed for the cross-section of class 1. Above the inner support, the beam is subjected to bending and shear force, at the location of the maximum moment in the span 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).

$\frac{{\mathrm V}_{\mathrm z,\mathrm{Ed}}}{{\mathrm V}_{\mathrm{pl},\mathrm z,\mathrm{Rd}}}\;=\;\frac{853.55}{1,258.41}\;=\;0.678\;>\;0.5$

It is necessary to reduce the moment resistance.

For I-cross-sections with the same flanges and uniaxial bending about the principal axis, it is allowed to determine the reduction of the design value of the plastic moment resistance due to the shear loading as follows:

$\begin{array}{l}\mathrm\rho\;=\;\left(\frac{2\;\cdot\;{\mathrm V}_{\mathrm{Ed}}}{{\mathrm V}_{\mathrm{pl},\mathrm{Rd}}}\;-\;1\right)^2\;=\;\left(\frac{2\;\cdot\;853.55}{1,258.41}\;-\;1\right)^2\;=\;0.127\\{\mathrm A}_{\mathrm w}\;=\;{\mathrm h}_{\mathrm w}\;\cdot\;{\mathrm t}_{\mathrm w}\;=\;\left(59\;-\;2\;\cdot\;2.5\right)\;\cdot\;1.3\;=\;70.2\;\mathrm{cm}\\{\mathrm M}_{\mathrm{pl},\mathrm V,\mathrm{Rd}}\;=\;\left({\mathrm W}_{\mathrm{pl},\mathrm y}\;-\;\frac{\mathrm\rho\;\cdot\;\mathrm A_{\mathrm w}^2}{4\;\cdot\;{\mathrm t}_{\mathrm w}}\right)\;\cdot\;\frac{{\mathrm f}_{\mathrm y}}{{\mathrm\gamma}_{\mathrm M0}}\;=\;\left(5,360\;-\;\frac{0.127\;\cdot\;70.2^2}{4\;\cdot\;1.3}\right)\;\cdot\;\frac{23.5}{1\;\cdot\;100}\;=\;1,231.32\;\mathrm{kNm}\\{\mathrm M}_{\mathrm{Ed}}\;<\;{\mathrm M}_{\mathrm{pl},\mathrm V,\mathrm{Rd}}\;\rightarrow\;1,068.36\;<\;1,231.32\;\mathrm{kNm}\end{array}$

Keywords

Design Cross-section class Elastic Plastic

Reference

[1]   Eurocode 3: Design of steel structures - Part 1‑1: General rules and rules for buildings; EN 1993‑1‑1:2010‑12
[2]   Manual RF-/STEEL EC3. (2018). Tiefenbach: Dlubal Software.
[3]   Albert, A.: Schneider - Bautabellen für Ingenieure mit Berechnungshinweisen und Beispielen, 23. Auflage. Köln: Bundesanzeiger, 2018

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