#
Improved Interaction Formula for the Plastic Resistance of I- and H-Sections under a Combination of Bending Moments M_{y,Ed}, M_{z,Ed}, and Bimoment B_{Ed}

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{y,Ed}, M

_{z,Ed}, and bimoment B

_{Ed}. It is shown in the graphical form that the proposed approximate formulae give almost identical values of relative plastic resistances of I-shaped sections to the exact solution, in which Pattern Search Algorithm, offered by MATLAB, is used. It is proven that the proposed approximate formulae are better then previous authors’ formulae, which are themselves better than the older Greek proposal and the newer German proposals. The approximate formulae may be used in the CEN Technical Specification (TS) or in Non Contradictory Complementary Information (NCCI) for supporting safe and economical design according to metal (steel and aluminum) Eurocodes.

## 1. Introduction

_{Ed,}M

_{y,Ed}, and M

_{z,Ed}. According to the Eurocode [1]: “The plastic resistance of cross sections should be verified by finding a stress distribution which is in equilibrium with the internal forces and moments without exceeding the yield strength. This stress distribution should be compatible with the associated plastic deformations” (cl. 6.2.1(6) in [1]). Similar clauses are missing in [2].

_{c,B,Rd}reduced by the bimoment B

_{Ed}may be calculated from Formula (8.29)”. Formula (8.29) is the second Formula (3) in this paper, and the history of its development is described below. The draft of the Eurocode [4] applies a similar formula (see formula (8.38) in cl. 8.2.7(6) [4]). Both drafts of the Eurocodes [3,4] contain the following the sentence: “For determining the plastic moment resistance of a cross-section due to bending and torsion, only torsion effects B

_{Ed}should be derived from elastic analysis” (see 8.2.7(6) in [3] or 8.2.7(5) in [4]).

_{y,Ed}and M

_{z,Ed}, and a bimoment, B

_{Ed}. In this paper, variant No.4 is developed, which is more exact than the previous variants, No.1, No.2, and No.3 published in [5].

#### 1.1. Linear-Interaction Formula

_{Ed}, the following form, if the bimoment B

_{Ed}is added:

#### 1.2. Proposals for Interaction Formula Containing My,Ed and BEd

_{z,Ed}, which needs to be taken into account when analyzing the behavior of unrestrained beams sensitive to lateral torsional buckling (Agüero et al. [22]).

#### 1.3. Calculation of Factor ξ

#### 1.3.1. Method A (According to Osterrieder et al.)

**Step 1.**Dividing section into elements.

**Step 2**. Considering linear constraints by Equations (5)–(9):

**Step 3.**Calculation of maximum ξ:

#### 1.3.2. Method B (According to Rubin and Kindmann et al.)

**Step 1.**Dividing section into three parts.

**Step 2.**Considering constraints:

**Step 3.**Calculation of maximum ξ:

#### 1.4. Research Significance

_{y,Ed}, the bending moment about the weak axis, M

_{z,Ed}, and the bimoment, B

_{Ed}, are compared for the beams with the I- and H-sections. The proposal is presented in Section 3.

## 2. Calculation of Exact Interaction Curves

**Method A is applied to obtain the interaction curves with parameter M**

_{z,Ed}**Step 1.**Calculation of M

_{pl,z,Rd}for the given N

_{Ed}= 0.0, M

_{y,Ed}= 0.0 and B

_{Ed}= 0.0.

_{z,Ed}:

**Step 2.**Calculation of B

_{Ed,γ}to achieve plastic resistance for the given N

_{Ed}= 0.0, M

_{y,Ed}= 0.0, M

_{z,Ed}= γ·M

_{pl,z,Rd}; γ [0,1]

_{pl,Rd,γ}:

**Step 3.**Calculation of M

_{y,Ed,γ,η}to achieve the plastic resistance for the given N

_{Ed}= 0.0, B

_{Ed}= ηB

_{pl,Rd,γ}, M

_{z,Ed}= γ·M

_{pl,z,Rd}; γ [0,1] and η [0,1].

_{yEdγ,η}:

**Application of Method B:**

_{Ed}= 0), the axial force at the web will be as follows:

_{y,Ed}is then:

## 3. Proposal for Improved Interaction Formulae (Variant No. 4)

$0\le \frac{{M}_{y,Ed}}{{M}_{pl,y,Rd}}<\delta $ | $\delta \le \frac{{M}_{y,Ed}}{{M}_{pl,y,Rd}}\le 1.0$ |

$\frac{\frac{{B}_{Ed}}{{B}_{pl,Rd}}}{1-\frac{{M}_{z,Ed}}{{M}_{pl,z,Rd}}}\le 1.0$ | ${\left(\frac{\frac{{M}_{y,Ed}}{{M}_{pl,y,Rd}}-\delta}{\left(\sqrt{1-\frac{{M}_{z,Ed}}{{M}_{pl,z,Rd}}}\left(1-\frac{{M}_{w,pl,y,Rd}}{{M}_{pl,y,Rd}}\right)+\frac{{M}_{w,pl,y,Rd}}{{M}_{pl,y,Rd}}\right)-\delta}\right)}^{\alpha}+{\left(\frac{\frac{{B}_{Ed}}{{B}_{pl,Rd}}}{1-\frac{{M}_{z,Ed}}{{M}_{pl,z,Rd}}}\right)}^{\beta}\le 1$ |

## 4. Conclusions

_{y,Ed}, M

_{z,Ed}, and bimoment B

_{Ed}, was investigated. The paper suggested an improvement to a previous proposal, published by the authors in [5], in which the three variants, No.1, No.2, and No.3, of approximate-interaction formulae were investigated. The proposed formulae may be called variant No.4. It was shown in graphical form that the proposed variant No.4 of the approximate Formula (37) gives almost identical values of the relative plastic resistances of I-shaped sections to the exact results obtained by linear programming (Figure 1 and Figure 2).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

## References

- EN 1993-1-1:2005 and Corrigendum AC (2006) and Corrigendum AC (2009) and Amendment A1 (2014); Eurocode 3: Design of Steel Structures. Part 1.1: General Rules and Rules for Buildings. European Union: Brussels, Belgium, 2005.
- EN 1999-1-1:2007 and Amendment A1 (2009) and Amendment A2 (2013); Eurocode 9: Design of Aluminium Structures. Part 1.1: General Structural Rules. European Union: Brussels, Belgium, 2007.
- prEN 1993-1-1:2020; Eurocode 3: Design of Steel Structures. Part 1.1: General Rules and Rules for Buildings. European Union: Brussels, Belgium, 2020.
- prEN 1999-1-1:2022-01-14; Eurocode 9: Design of Aluminium Structures. Part 1.1: General Structural Rules. European Union: Brussels, Belgium, 2022.
- Agüero, A.; Baláž, I.; Koleková, Y.; Moroczová, L. New interaction formula for the plastic resistance of I- and H-sections under combinations of bending moments M
_{y,Ed}, M_{z,Ed}and bimoment B_{Ed}. Structures**2021**, 29, 577–585. [Google Scholar] [CrossRef] - Vayas, I. Interaktion der plastischen Grenzschnittgrößen doppelsymmetrischer I-Querschnitte. Stahlbau
**2000**, 69, 693–706. [Google Scholar] [CrossRef] - Ludwig, C. Assessment of the interaction conditions of i-shaped cross-sections. Eurosteel
**2021**, 4, 1085–1095. [Google Scholar] [CrossRef] - Baláž, I.; Koleková, Y. Resistances of I- and U-sections. Combined bending and torsion internal forces. In Proceedings of the EUROSTEEL, Copenhagen, Denmark, 13–15 September 2017; pp. 1–10, Paper No. 13_12_ 772 on USB. [Google Scholar]
- Mirambell, J.E.; Bordallo, E. Real torsion and its interaction with other internal forces in EN 1993-1-1—A new approach. Steel Constr.
**2016**, 9, 240–248. [Google Scholar] - Osterrieder, P.; Kretzschmar, J. First-hinge analysis for lateral buckling design of open thin-walled steel members. J. Constr. Steel Res.
**2006**, 62, 35–43. [Google Scholar] [CrossRef] - Yang, Y.B.; Chern, S.M.; Fan, H.T. Yield surfaces for I-sections with bimoments. J. Struct. Eng.
**1989**, 115, 3044–3058. [Google Scholar] [CrossRef] - Kindmann, R.; Frickel, J. Grenztragfähigkeit von häufig verwendeten Stabquerschnitten für beliebige Schnittgröβen. Stahlbau
**1999**, 68, 817–828. [Google Scholar] [CrossRef] - Rubin, H. Zur plastischen tragfähigkeit von 3-blech-querschnitten unter normalkraft, doppelter biegung und wölbkrafttorsion. Stahlbau
**2005**, 74, 47–61. [Google Scholar] [CrossRef] - Wolf, C.; Frickel, J. QST-TSV-3Blech. Program. Lehrstuhl für Stahl- und Verbundbau. Prof. Dr.-Ing. R. Kindmann. Ruhr-Universität Bochum. 2002. Available online: https://www.kindmann.de/downloads/file/1-rubstahl-programme (accessed on 13 July 2022).
- Dlubal Software GmbH. Programm SHAPE-THIN 8. German name DUENQ, 8.13.01.140108 x64. 2018. Available online: https://dlubl.com/en/products/cross-section-properties-software/shape-thin (accessed on 13 July 2022).
- Agüero, A.; Gimenez, F. Thinwallres. 2022. Available online: https://labmatlab-was.upv.es/webapps/home/thinwallres.html (accessed on 13 July 2022).
- Baláž, I.; Kováč, M.; Živner, T.; Koleková, Y. Plastic resistance of H-section to interaction of bending moment M
_{y,Ed}and bimoment B_{Ed}. In Proceedings of the 2nd International Conference on Engineering Sciences and Technologies, High Tatras Mountains, Tatranské Matliare, Slovakia, 29 June–1 July 2016; CRC Press, Taylor & Francis Group, A Balkema book. 2017; pp. 33–38. [Google Scholar] - Baláž, I.; Kováč, M.; Živner, T.; Koleková, Y. Plastic resistance of IPE-section to interaction of bending internal forces M
_{y,Ed}, V_{z,Ed}and torsion internal forces B_{Ed}, T_{ω,Ed}and T_{t,Ed}. In Proceedings of the 2nd International Conference on Engineering Sciences and Technologies, High Tatras Mountains, Tatranské Matliare, Slovakia, 29 June–1 July 2016; CRC Press, Taylor & Francis Group, A Balkema book. 2017; pp. 27–32. [Google Scholar] - Baláž, I.; Koleková, Y. Plastic resistance of Aluminium I-profile under bending and torsion according to continuous strength method. In Proceedings of the 23rd International Conference Engineering Mechanics, Svratka, Czech Republic, 15–18 May 2017. [Google Scholar]
- Baláž, I. Resistance of I- and channel sections according to continuous strength method. In Proceedings of the 22nd Conference of Structural Engineers, Piešťany, Slovakia, 16–17 March 2017. (In Slovak). [Google Scholar]
- Baláž, I.; Koleková, Y. Plastic Resistance of I- and U-Section under Bending and Torsion. In Stahbau, Holzbau und Verbundbau; Jubilee Publication in Honour of Mrs. Prof. Kuhlmann on the Occasion of Her 60th Birthday; Ernst & Sohn, A Wiley Brand: Hoboken, NJ, USA, 2017; pp. 203–209. [Google Scholar]
- Agüero, A.; Pallarés, F.J.; Pallares, L. Equivalent geometric imperfection definition in steel structures sensitive to lateral torsional buckling due to bending moment. Eng. Struct.
**2015**, 96, 41–55. [Google Scholar] [CrossRef] - Kindmann, R.; Frickel, J. Elastische und Plastische Querschnittstragfähigkeit Grundlagen, Methoden, Berechnungsverfahren, Beispiele. Mit CD-ROM: RUBSTAHL Lehr- und Lernprogramme; Ernst & Sohn, Wiley Company: Berlin, Germany, 2002. [Google Scholar]
- Kindmann, R.; Ludwig, C. Plastische tragfähigkeit von gewalzten und geschweißten I-Querschnitten. Stahlbau
**2014**, 86, 890–904. [Google Scholar] [CrossRef] - Wolf, C.; Frickel, J. QST-TSV-I. Program. Lehrstuhl für Stahl- und Verbundbau. Prof. Dr.-Ing. R. Kindmann. Ruhr-Universität Bochum. 2002. Available online: https://www.kindmann.de/downloads/file/1-rubstahl-programme (accessed on 13 July 2022).

**Figure 1.**IPE 300. Comparison of the exact curves obtained by linear programming with: (i) the results of the authors (Table 1 below, Equation (37)); (ii) the results of Vayas [6]; and (iii) the results of Ludwig [7]. The parameters of the curves from top to bottom are M

_{z,Ed}/M

_{pl}.

_{z}.

_{Rd}= 0; 0.2; 0.4; 0.6; 0.8.

**Figure 2.**HEB 200. Comparison of the exact curves obtained by linear programming with: (i) the results of authors (Table 1 below, Equation (37)); (ii) the results of Vayas [6]; (iii) and the results of Ludwig [7]. The parameters of the curves from top to bottom are M

_{z,Ed}/M

_{pl}.

_{z}.

_{Rd}= 0; 0.2; 0.4; 0.6; 0.8.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Agüero, A.; Baláž, I.; Koleková, Y.
Improved Interaction Formula for the Plastic Resistance of I- and H-Sections under a Combination of Bending Moments *M*_{y,Ed}, *M*_{z,Ed}, and Bimoment *B*_{Ed}. *Appl. Sci.* **2022**, *12*, 7888.
https://doi.org/10.3390/app12157888

**AMA Style**

Agüero A, Baláž I, Koleková Y.
Improved Interaction Formula for the Plastic Resistance of I- and H-Sections under a Combination of Bending Moments *M*_{y,Ed}, *M*_{z,Ed}, and Bimoment *B*_{Ed}. *Applied Sciences*. 2022; 12(15):7888.
https://doi.org/10.3390/app12157888

**Chicago/Turabian Style**

Agüero, Antonio, Ivan Baláž, and Yvona Koleková.
2022. "Improved Interaction Formula for the Plastic Resistance of I- and H-Sections under a Combination of Bending Moments *M*_{y,Ed}, *M*_{z,Ed}, and Bimoment *B*_{Ed}" *Applied Sciences* 12, no. 15: 7888.
https://doi.org/10.3390/app12157888