###### Abstract  
The resistances of HVH sections (H--horizontal upper flange, V--vertical web, H--horizontal bottom flange) were investigated by the authors in the frame of the large parametrical studies focused on the complete HVH-section categories: channels, Z-sections, I-sections, and H-sections. The three different variants of the approximate formulae for the calculation of the elastic and plastic resistances of I-shaped sections loaded by several internal forces were previously published by the authors. In this paper, the improved approximate formulae are presented for the plastic resistance of I- and H-sections loaded by bending moments \(M_{\mathbf{y,Ed}}\), \(M_{\mathbf{x,Ed}}\), and bimoment \(B_{\mathbf{Ed}}\). 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. I-shaped sections; plastic section resistance; bending moments; bimoment; interaction formula +
Footnote †: journal: Applied Sciences  
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 \(B_{\text{Ed}}\) should be derived from elastic analysis\" (see 8.2.7(6) in [3] or 8.2.7(5) in [4]). The need for an interaction formula for the calculation of the plastic resistance of the I-shaped section was solved by the authors in [5], in which three variants of the formula were presented, taking into account the influence of three internal forces: two bending moments, \(M_{\text{y,Ed}}\) and \(M_{\text{z,Ed}}\), and a bimoment, \(B_{\text{Ed}}\). 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]. In Table 1 in [5], the authors present three approximate formulae variants. Table 1 in [5] shows that variant No. 3 is the best. The purpose of this paper is to present variant No. 4, which is better than variant No. 3. Variant No. 4 is defined by the formulae in Table 1, below, and by the equations in (37). The approximate results agree well with the exact results obtained by linear programming (Figures 1 and 2). Figures 1 and 2 show also comparisons with the older results obtained by [PERSON] (Equations (9) and (21c) in [6]), and to the brand new results obtained by [PERSON] (Equation (29) in [7]). We can see that the results of the proposed variant No. 4 fit the exact curves well (Figures 1 and 2). The proposals of [PERSON] and [PERSON] substantially differ within some intervals of the exact results. 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 [PERSON] [6]; and (iii) the results of [PERSON] [7]. The parameters of the curves from top to bottom are \(M_{\text{z,Ed}}/M_{\text{pl}\cdot\text{z-Rd}}=0\); \(0.2\); \(0.4\); \(0.6\); \(0.8\). It is worth mentioning that [PERSON] informed us that he used paper [8]. He forgot to include [8] in the list of references in [7]. Moreover, [PERSON] mistakenly referred to the formula in Table 6[7] as [PERSON]'s formula and referred to [26] ([9] here). The formula in this form is not found in [26]. In fact, it is [PERSON]'s formula, taken by [PERSON] from [8], and not from [26]. [PERSON]'s formula was accepted for the last draft of prEN 1993-1-1 [2], where it replaced [PERSON]'s original, incorrect formula (see Figure 3b in [8]).