Plastic Design of Metal Thin-Walled Cross-Sections of Any Shape Under Any Combination of Internal Forces

Abstract

A short tribute to pioneers in the development of the plastic design of metal thin-walled cross-sections is presented. This large study investigates altogether fourteen steel and four extruded aluminum cross-sections in detail. Six groups of the cross-sections with various shapes consist of four I-shaped doubly symmetric sections with or without lips; three monosymmetric sections with an axis of symmetry z including T- and diamond sections; four monosymmetric channels with or without lips; two point-symmetric Z-sections; and four asymmetric sections. The four extruded aluminum cross-sections are an I 200a section, a diamond section, and closed oblique and irregular sections. For all 18 cross-sections, the plastic section moduli of three kinds were calculated, namely W_pl,y,nB and W_pl,z,nB for bimoment not considered as a constraint; W_pl,y, W_pl,z, and W_pl,w for bimoment considered as a restraint; and maximum values W_pl,y,max, W_pl,z,max, and W_pl,w,max. The values of cross-section plastic resistances N_pl, M_pl,y,Rd, M_pl,z,Rd, and B_pl are calculated in numerical examples too. The values of cross-section properties are calculated in different ways to verify the correctness of the results. The following methods of calculation are used: the rules given in Eurocode EN 1993-1-1:2022; MathCad programs; and freeware. Recommendations for educational institutes and designers in practice are given, including simple formulae for all cross-sectional properties for doubly and monosymmetric I-shaped sections, channels, and Z-sections. The formulae are presented in three tables containing formulae in dimensionless form convenient for parametrical studies and formulae for direct design. The background of the Eurocode rules given in EN 1993-1-1:2022 is explained together with recommendations for how to avoid the problems with using them.

1. Introduction

The very beginning of “Plastic Design” is connected to the following names: Gábor von Kazinczy (1889–1964) [1], Nicolaas Christiaan Kist (1867–1941), Hermann Maier-Leibnitz (1885–1962) [2,3,4], Hans Heinrich Bleich (1909–1985) [5], John Fleetwood Baker (1901–1985) [5], and many others.

A considerable amount of research has been conducted on the ultimate resistance of metal structures from 1930. Many old but very important and practically unknown publications of Russian and Ukrainian authors in the Russian language are referred to in [6]. The theoretical and experimental results for both steel I profiles and channels performed by the Ukrainian lady Streľbickaja are extremely important [7]. Her results replaced the incorrect formula in the several drafts of Eurocode of the second generation prEN 1993-1-1 during its preparation.

Several investigators (Figure 1) have applied plastic analysis to civil engineering structures. The ability of structural steel to deform plastically allows us to solve indeterminate structures in such a way that the reserve strength of its less heavily stressed portions can be fully utilized. The use of plastic analysis enables us to determine the true load-carrying capacity of structures. Plastic design is relatively simple. Most of the time-consuming solutions of equations necessary for an elastic analysis are eliminated. The sinking of supports, differences in the flexibility of connections, residual stresses, etc., that affect the elastic limit resistance of a structure have little or no effect upon the maximum plastic resistance. Plastic design leads to substantial savings and the more economic and efficient use of steel, as well as savings in the designer’s time. Plastic design will not replace all other verifications, such as fatigue, instability, and limiting deflection. In building designs, this is usually not the case. Plastic design has applications in continuous beams, industrial frames, and tier buildings.

Figure 1. Pioneers in plastic design development. From the left: Gábor von Kazinczy [1], Nicolaas Christiaan Kist, Hermann Maier-Leibnitz [2], Hans Heinrich Bleich, John Fleetwood Baker.

Important theoretical results relating to the plastic resistance of thin-walled cross-sections under a combination of internal forces were achieved by Kindmann [8,9], Osterrieder [10], and Vayas [11,12].

The novelty of this paper consists of the fact that the following are presented for the first time ever:

(a)

The plastic resistance of thin-walled cross-sections of arbitrary shapes, namely open, quasi-closed, and closed, under any combination of eight internal forces, N_Ed, M_y,Ed, V_z,Ed, M_z,Ed, V_y,Ed, B_Ed, T_w,Ed, T_t,Ed is investigated; the relevant freeware may be found here https://antonioagueroramonllin.blogs.upv.es/ (accessed on 15 November 2024).

(b)

The importance, calculation, and application of three types of plastic section moduli are studied, namely W_pl,nB validity when bimoment B_Ed is not considered as a constraint, W_pl validity when bimoment B_Ed is considered as a constraint, and W_pl,max. This was shown for all three plastic section moduli as follows: W_pl,y (bending about axis y-y), W_pl,z (bending about axis z-z), and W_pl,w (torsion about axis x), as well as for 14 different shapes of cross-sections. The mutual relationship of different plastic section moduli was explained on open, quasi-open, and closed thin-walled cross-sections with two axes of symmetry, one axis of symmetry, being point-symmetric, and cross-sections without an axis of symmetry.

(c)

All non-dimensionless properties of a monosymmetric I-section, channel, and Z-section, including all kinds of plastic section moduli which cannot be found in static tables and are not calculated by commercial programs, are determined. Non-dimensional properties are very convenient for parametrical studies and the optimalization of cross-sections.

(d)

Formulae for channel profiles in the plastic state loaded by a combination of M_y,Ed with positive bimoment B_Ed and M_y,Ed with negative bimoment B_Ed are used and are convenient for standards.

(e)

The influence of bilinear stress–strain diagrams without and with strengthening for both aluminum and steel I-shaped profiles is determined.

(f)

Problems in the second generation of Eurocode EN 1993-1-1:2022 and their solution by authors of the paper are provided.

(g)

Formulae of MathCad calculations which were removed from prEN 1993-1-3:2024 are included, this time also including an oblique hollow section. They are the result of Torsten Höglund, the main author of EN 1999-1-1:2023.

All numerical results were verified by different ways of calculation, also by using the computer programs of other authors.

The formulae for the plastic design of thin-walled members may be found in the modern standards EN 1993 and EN 1999—Eurocodes of the second generation and in the Slovak National Annex [13] and EN 1993-1-1:2022 [14].

2. Plastic Design in Metal Eurocodes EN 1993-1-1:2022 [14] and EN 1999-1-1:2023 [15]

2.1. Advantages of Plastic Analysis and Assumptions for Plastic Analysis Applications

The calculation of metal (steel and aluminum) structures according to the theory of plasticity has several advantages, of which at least the four most important ones should be mentioned. It performs the following:

(a)

It fully expresses the actual behavior of the structures;

(b)

It allows for the attainment of the same safety for a structure and its elements;

(c)

It often simplifies the analysis of current structures;

(d)

It is a source of substantial material and cost economies;

(e)

It provides a summary of important assumptions for the use of plastic design;

(f)

Steel and aluminum are regarded as an ideal elastoplastic material, the behavior of which is described by the idealized stress–strain diagram with sufficient accuracy (see Figure 2);

(g)

The material behaves identically in tension and simple compression;

(h)

The plastic resistance of the structural member at the cross-sections with a non-uniform distribution of longitudinal strains is exhausted due to the formation of plastic hinges, which are not capable of resisting any further increase in load;

(i)

The deformations of structures up to the limit of the plastic load-bearing capacity are so small that equilibrium of forces can be investigated for an undeformed structure. The expressions for virtual work can also be formulated with the assumption of small deformations;

(j)

The longitudinal strains of the cross-sections are distributed linearly in the cross-section elements. Vlasov’s hypothesis about the non-deforming shape of the cross-section is adopted;

(k)

For a complex state of stress, Huber–Mises–Hencky’s condition is usually applied as a condition of plasticity;

(l)

When the plastic load-bearing capacity or the deflections of statically indeterminate structures are being determined, it is assumed that the plastic strains are concentrated at the most highly stressed cross-sections—the plastic hinges. After the hinge mechanism has been formed, the curvature of elements between the hinges is supposed to remain unchanged;

(m)

Random imperfections are not taken into account;

(n)

Failure due to local and global instability of members is prevented by structural measures;

(o)

The structural connections are rigid to the extent that they are capable of transferring the redistributed effects.

The clauses of Eurocodes EN 1993-1-1:2022 [14] and EN 1999-1-1:2023 [15] regulating the exploitation of the plastic characteristics of steel and aluminum shall be respected, in addition to the above enumerated fundamental assumptions. In this paper, the properties of cross-sections which are not Class 1 or 2 are calculated to show the ability of the used software to investigate any complex shape of cross-sections.

Figure 2. Assumptions for plastic steel behavior:$f_{\mathrm{u}}/f_{\mathrm{y}}\ge 1.10$, ${\epsilon }_{\mathrm{u}}\ge 15\%$ [14].

2.2. EN 1993-1-1:2022 [14]

This Eurocode defines the following minimum ductility requirements for a plastic global analysis of steel structures:

(a)

The ratio of ultimate tensile strength to yield strength is $f_{\mathrm{u}}/f_y\ge 1.10$;

(b)

The elongation at failure is not less than ${\mathsf{\epsilon }}_{\mathrm{u}}\ge 15\%$.

NOTE: symbol ε_u was used in EN 1993-1-1:2005 [16] for something else; the ultimate strain ε_u corresponds to the ultimate strength f_u.

Steels conforming to one of the grades up to and including S700 listed in Table 5.1 and Table 5.2 in [14] may be assumed to satisfy the minimum ductility requirements for plastic global analysis. Plastic global analysis allows for the effects of material non-linearity in calculating the action effects of a structural system. The behavior should be modeled by either of the following two methods: (a) the plastic hinge method or (b) the plastic zone method. Plastic global analysis may be used if sufficient lateral restraint is provided in the vicinity of sections where a plastic hinge or a plastic zone can develop under the design loads.

For the determination of its plastic resistance, a cross-section should be classified according to one of the two following classes: (a) Class 1 cross-sections are those which can form a plastic hinge with the rotation capacity required from plastic global analysis without a reduction in the resistance; (b) Class 2 cross-sections are those which can develop their plastic bending moment resistance but have limited rotation capacity because of local buckling. The maximum width-to-thickness ratios for Class 1 and 2 compression parts should be obtained from Table 7.3 to Table 7.5 [14]. Cross-section requirements for plastic global analysis are given in clause 7.6 [14].

2.3. EN 1999-1-1:2023 [15]

The characteristic values for the heat-affected zone (0.2% proof strength, f_o,haz, and ultimate tensile strength, f_u,haz), reduction factors ρ_o,haz and ρ_u,haz (see 8.1.6), buckling classes A, B, and C (used in 8.1.4 and 8.3.1), and exponent, np, in the Ramberg–Osgood formula for plastic resistance should be taken from Tables 5.3 to 5.7 [15].

Plastic global analysis may be used only where the structure has sufficient rotation capacity at the actual location of the plastic hinge, whether this is in the members or in the joints. The member should satisfy the requirements specified in 7.4.3. When a plastic hinge occurs in a joint, the joint shall either have sufficient strength to ensure the hinge remains in the member or shall be able to sustain the plastic resistance for a sufficient rotation. When a plastic hinge occurs in a beam which is asymmetric perpendicular to the plane of bending (for example, a channel beam in y-axis bending), the beam should be prevented from rotation at the load application points and at the supports. Information on rotation capacity is given in Annex L [15]. Only certain alloys have the required ductility to allow for sufficient rotation capacity, see K.3 [15]. Plastic global analysis should not be used for beams with transverse welds on the tension side of the member at the plastic hinge locations, except for structures of alloys in 5xxx in temper O or H111 and welded with proper welding metal and it is documented that the properties in HAZ and the welding zone are not less than in the parent material. For the plastic global analysis of beams, guidance is given in Annex K [15]. Plastic global analysis should only be used where the stability of members can be assured, see 8.3 [15].

The plastic limit state is related to the strength of the section, evaluated by assuming a perfectly plastic behavior for a material with a limit value equal to the conventional limit of elasticity, f_o, without hardening. Global non-linear and rigid plastic analysis can be performed when all cross-sections where plastic hinges take place are of Class 1 or 2. Cross-sections should be classified depending on their local buckling resistance and rotation capacity. Class 1 cross-sections are those that can develop their plastic moment resistance forming a plastic hinge with the rotation capacity required for plastic analysis. Class 2 cross-sections are those that can develop their plastic moment resistance but have limited rotation capacity because of local buckling. An effective plastic modulus of the gross cross-section W_pl,haz obtained using a reduced thickness ρ_o,hazt for the HAZ material (see 8.2.5.2) is given in Table 4 [9]. Where plastic global analysis is used for the ultimate limit state, the effects of the plastic redistribution of forces and moments at the serviceability limit state should be considered. The following assumptions for the material behavior may be used: true stress–strain curve represented through the Ramberg–Osgood law (Formula (F.14) [15]) with exponent n = n_p from 3 and 4 of Table 5 [15]. This curve should be used if plastic resistance should be found.

2.3.1. Plastic Section Modulus [15]

The cross-section is divided into n parts numbered as in Figure 3. The following procedure may be used for the plastic section modulus W_pl,y:

Figure 3. Cross-section subdivision parts. (a) Open cross-section with extra node (5) and plastic neutral axis, PNA; (b) cross-section with inclined parts and diamond shape; (c) compressed cross-section equivalent to (b), where thickness of inclined parts is measured horizontally; (d) stresses where parts 6 to 9 give the axial force and 1 to 5 (A_t, red area in tension) and 6 to 11 (A_c blue area in compression) give the moment. ${\mathit{f}}_{\mathbf{o}}$ is characteristic value of 0.2% proof strength. ${\mathit{\gamma }}_{\mathbf{M}1}$ = 1.1 in [15].

(1)

Add an extra node in the part that will be crossed by the plastic neutral axis (PNA);

(2)

Define the PNA so that the areas above and below the PNA are the same (=A/2);

(3)

Replace the thickness t_i below the PNA with the negative value −ti;

(4)

The plastic section modulus W_pl,y is then given by Formula (1). See also Appendix A below.

$$W_{\mathrm{pl},\mathrm{y}}={\displaystyle {\sum }_{i=1}^{n}0.5t_i(z_i+z_{i-1})\sqrt{{(z_i-z_{i-1})}^2+{(y_i-y_{i-1})}^2}}$$

(1)

For the plastic section modulus W_pl,z, change y and z, add a vertical PNA line, and add extra node(s) on this line.

2.3.2. Plastic Interaction Formula

The cross-section is divided into parts numbered as in Figure 3. A cross-section with more than one web (Figure 3b) can be concentrated as in Figure 3c. The following procedure may be used to find a point on the interaction diagram for the axial force versus bending moment:

(1)

Choose a node a (=9 in Figure 3c) above which there is compression stress;

(2)

Calculate the area Ac above this node a;

(3)

Add an extra node b (=5 in Figure 3c) on the tension side, placed so that the area A_t below this node is the same as A_c;

(4)

Set the thickness to −t_i for parts 1 to b and 0 for parts b + 1 to a;

(5)

Calculate the reduced plastic section modulus with Formula (1);

(6)

Calculate the area A_n = A − 2A_c;

(7)

A point on the interaction diagram is now given by M_Rd = W_pl,y f_o/γ_M1 and N_Rd = A_n f_o/γ_M1;

(8)

Repeat the procedure for the other nodes a;

(9)

If the bending moment resistance is searched for a specific axial force, then the node a should be chosen so that A_c = 0.5(A − N_Rd/(f_o/γ_M1)).

The Eurocode EN 1999-1-1:2007 [17] in the informative Annexes C and J, respectively, provides formulae for the calculation of (a) cross-section constants for thin-walled open cross-sections (C.1, J.1), (b) cross-section constants for open cross-sections with branches (C.2, J.2), and (c) a torsion constant and shear center of cross-sections with closed parts (C3, J.3). In EN 1999-1-1:2023 [15] contained in the informative Annex G, the same formulae as in the 2007 edition are included [17], as well as the formulae for the calculation of the shear area (G.9) [15], plastic section modulus, and the interaction formula.

The formulae in the above-mentioned Eurocodes were presented in the form of formulae especially convenient to use in MathCad software and similar programs, where the formulae are presented in the same format as the formulae in textbooks and codes. It is helpful to the users of MathCad software, but the formulae may be evaluated by any other program.

The formulations are especially convenient for cold-formed sections where the thickness is the same for all cross-section elements. These are automatically defined from the nodes.

As the elements are given with the midline dimension and the thicknesses, some errors may occur in the corners depending on how the corners are modeled. If they are modeled with one node, then there will be some overlapping of the area (red) and missing area (white), according to Figure 4a. The cross-section area will always be correct, but there will be a small error in I_y, I_z , and so on, because the area t/2 × t/2 (red in Figure 4a) will have a small error in the distance to the coordinate axis. This error is usually negligible in corners, but in sections with branches, the error may be notable, as the area is doubled in the overlapping area (red in Figure 4d). These errors can be overcome if the corners are modeled with two nodes, according to Figure 4b,c, with an element with a thickness of 0 between them.

Figure 4. Four possible kinds of nodes in corners and in connections between elements.

The examples of numbering nodes in MathCad formulae [15] are given in Figure 5 for cross-sections without branching and in Figure 6 for cross-sections with branches.

Figure 5. Numbering of nodes in cross-section without branching.

Figure 6. Nodes and parts in cross-section with branches.

(a)

Divide the cross-section into n parts. Number the parts 1 to n. Insert nodes between the parts. Number the nodes from 0 to n. Part i is then defined by nodes i − 1 and i. Give the nodes, coordinates, and (effective) thickness. j = 0 … n are nodes and i = 1 … n are cross-section parts. Note that y − z is an arbitrary coordinate system with the origin usually not in the center of gravity and that the principal axes are here denoted as $\xi$ and $\eta$.

(b)

Calculate the following geometrical magnitudes:

The area of cross-section parts by Formula (2) is expressed as

$$\mathsf{\Delta }{A}_i=t_i\sqrt{{\left(y_i-y_{i-1}\right)}^2+{\left(z_i-z_{i-1}\right)}^2}$$

(2)

The cross-section area by Formula (3) is expressed as

$$A={\displaystyle \sum }_{i=1}^n\mathsf{\Delta }{A}_i$$

(3)

The first moment of area with respect to the y-y-axis and the coordinate of the centroid by Formula (4) is expressed as

$$S_{\mathrm{y}0}={{\displaystyle \sum }}_{i=1}^n\left(z_i+z_{i-1}\right){\displaystyle \frac{\mathsf{\Delta }{A}_i}{2}}\hspace{1em}{z}_{\mathrm{gc}}={\displaystyle \frac{S_{\mathrm{y}0}}{A}}$$

(4)

The moment of inertia with respect to the new y-y-axis through the centroid by Formula (5) is expressed as

$$I_{\mathrm{y}}={{\displaystyle \sum }}_{i=1}^{\mathrm{n}}\left({(z_i)}^2+{(z_{i-1})}^2+z_i{z}_{i-1}\right){\displaystyle \frac{\mathsf{\Delta }{A}_i}{3}}-A{z}_{\mathrm{gc}}^2$$

(5)

$S_{z0}$, $y_{gc}$, and $I_z$ are determined in the same way by swapping the coordinates.

The product moment of area with respect to the original y- and z-axis and new axes through the centroid by Formula (6) is expressed as

$$I_{\mathrm{yz}}={{\displaystyle \sum }}_{i=1}^{\mathrm{n}}\left(2y_{i-1}{z}_{i-1}+2y_i{z}_i+y_{i-1}{z}_i+y_i{z}_{i-1}\right){\displaystyle \frac{\Delta A_i}{6}}-{\displaystyle \frac{S_{\mathrm{y}0}{S}_{\mathrm{z}0}}{A}}$$

(6)

Principal axes are given with standard formulae.

Section constants, which are usually not given in CAD programs, are the sectorial constants. For these, the sectorial coordinates given by Formula (7) are needed.

$${\omega }_0=0\hspace{1em}{\omega }_{0_i}=y_{i-1}{z}_i-y_i{z}_{i-1},\hspace{1em} {\omega }_i={\omega }_{i-1}+{\omega }_{0_i}$$

(7)

The mean of the sectorial coordinate by Formula (13) is expressed as

$$I_{\mathsf{\omega }}={{\displaystyle \sum }}_{i=1}^{\mathrm{n}}\left({\omega }_{i-1}+{\omega }_i\right){\displaystyle \frac{\mathsf{\Delta }{A}_i}{2}}\hspace{1em}{\omega }_{\mathrm{mean}}={\displaystyle \frac{I_{\mathsf{\omega }}}{A}}$$

(8)

The sectorial constants by Formulae (9)–(11) are expressed as

$$I_{\mathrm{y}\mathsf{\omega }}={{\displaystyle \sum }}_{i=1}^{\mathrm{n}}\left(2y_{i-1}{\omega }_{i-1}+2y_i{\omega }_i+y_{i-1}{\omega }_i+y_i{\omega }_{i-1}\right){\displaystyle \frac{\mathsf{\Delta }{A}_i}{6}}-{\displaystyle \frac{S_{\mathrm{z}0}{I}_{\mathsf{\omega }}}{A}}$$

(9)

$$I_{\mathrm{z}\mathsf{\omega }}={{\displaystyle \sum }}_{i=1}^{\mathrm{n}}\left({2\mathrm{z}}_{i-1}{\omega }_{i-1}+{2\mathrm{z}}_i{\omega }_i+z_{i-1}{\omega }_i+z_i{\omega }_{i-1}\right){\displaystyle \frac{\mathsf{\Delta }{A}_i}{6}}-{\displaystyle \frac{S_{y0}{I}_{\mathsf{\omega }}}{A}}$$

(10)

$$I_{\mathsf{\omega }\mathsf{\omega }}={{\displaystyle \sum }}_{i=1}^{\mathrm{n}}\left({({\omega }_i)}^2+{({\omega }_{i-1})}^2+{\omega }_i{\omega }_{i-1}\right){\displaystyle \frac{\mathsf{\Delta }{A}_i}{3}}-{\displaystyle \frac{I_{\omega }^2}{A}}$$

(11)

The shear center by Formula (12) is expressed as

$$y_{\mathrm{sc}}={\displaystyle \frac{I_{\mathrm{z}\mathsf{\omega }}{I}_{\mathrm{z}}-I_{\mathrm{y}\mathsf{\omega }}{I}_{\mathrm{yz}}}{I_{\mathrm{y}}{I}_{\mathrm{z}}-I_{\mathrm{yz}}^2}}, z_{\mathrm{sc}}={\displaystyle \frac{-I_{\mathrm{y}\mathsf{\omega }}{I}_{\mathrm{y}}+I_{\mathrm{z}\mathsf{\omega }}{I}_{\mathrm{yz}}}{I_{\mathrm{y}}{I}_{\mathrm{z}}-I_{\mathrm{yz}}^2}}, \left(I_y{I}_z-I_{yz}^2\ne 0\right)$$

(12)

The warping constant by Formula (13) is expressed as

$$I_w=I_{\omega \omega }+z_{\mathrm{sc}}\cdot I_{y\omega }-y_{\mathrm{sc}}\cdot I_{z\omega }$$

(13)

Torsion constants are given with standard formulae.

The sectorial coordinate with respect to the shear center, used for the polar cross-section constants, are given by Formula (14) as follows:

$${\omega }_{s_j}={\omega }_j-{\omega }_{\mathrm{mean}}+z_{\mathrm{sc}}\left(y_j-y_{\mathrm{gc}}\right)-y_{\mathrm{sc}}\left(z_j-z_{\mathrm{gc}}\right)$$

(14)

The maximum sectorial coordinate and warping modulus by Formula (15) are expressed as

$${\omega }_{\max }=\mathit{max}\left(\left|{\omega }_s\right|\right)\hspace{1em}{W}_{\mathrm{w}}={\displaystyle \frac{I_{\mathrm{w}}}{{\omega }_{\max }}}$$

(15)

The distance between the shear center and centroid by Formula (16) is expressed as

$$y_s=y_{\mathrm{sc}}-y_{\mathrm{gc}}, z_s=z_{\mathrm{sc}}-z_{\mathrm{gc}}$$

(16)

The polar moment of inertia with respect to the shear center by Formula (17) is expressed as

$$I_p=I_y+I_z+A\left(y_s^2+z_s^2\right)$$

(17)

The non-symmetry factor z_j according to Annex I [15] is hardly found in CAD programs. For the z axis, it is given by Formulae (18). For the other axis, $y_{\mathrm{j}}$ is given by swapping the coordinates.

$$z_{\mathrm{j}}=z_s-{\displaystyle \frac{0.5}{I_{\mathrm{y}}}}{{\displaystyle \sum }}_{i=1}^{\mathrm{n}}\left[{(z_{{\mathrm{c}}_i})}^3+z_{{\mathrm{c}}_i}\left({\displaystyle \frac{{(z_i-z_{i-1})}^2}{4}}+{y_{{\mathrm{c}}_i}}^2+{\displaystyle \frac{{(y_i-y_{i-1})}^2}{12}}\right)+y_{{\mathrm{c}}_i}{\displaystyle \frac{\left(y_i-y_{i-1}\right)\left(z_i-z_{i-1}\right)}{6}}\right]\cdot \mathsf{\Delta }{A}_i$$

(18)

where the coordinates for the center of the cross-section parts with respect to the centroid are given by Formula (19) as follows:

$$y_{{\mathrm{c}}_i}={\displaystyle \frac{y_i+y_{i-1}}{2}}-y_{\mathrm{gc}}\hspace{1em}{z}_{{\mathrm{c}}_i}={\displaystyle \frac{z_i+z_{i-1}}{2}}-z_{\mathrm{gc}}$$

(19)

NOTE: $z_j=0\hspace{0.33em}\left(y_j=0\right)$ for cross-sections with the y-y-axis (z-z-axis) being the axis of symmetry (Figure 5).

In cross-sections with branches, the Formulae in (2)–(19) can be used. However, follow the branching back (with thickness t = 0) to the next part with thickness t ≠ 0, see branch 3-4-5 and 6-7 in Figure 6.

3. Basis of Theory

The static or lower bound theorem is applied. The equilibrium and plastic condition are satisfied in order to achieve a plastification factor equal to or smaller than the plastic one.

The nomenclature of the plastic section properties is as follows:

(1)

Without considering bimoment as a constraint: W_pl,zy,nb and W_pl,z,nB.

(2)

Considering bimoment as a constraint: W_pl,y, W_pl,z and W_pl,w.

(3)

Maximum: W_pl,y,max, W_pl,z,max, and W_pl,w,max.

The interaction equations for the plastic resistance of cross-sections under N_pl,Rd, M_pl,y,Rd, and M_pl,z,Rd in Eurocode 3 are obtained without considering bimoment B_Ed as a constraint.

Method 1 has been implemented in two computer programs.

The software can be found in the following blog:

https://antonioagueroramonllin.blogs.upv.es/ (accessed on 15 November 2024)

(1)

thinwallsectiongeneral was used to compute values considering bimoment as a constraint and maximum values.

https://laboratoriosvirtuales.upv.es/webapps/thinwallsectiongeneral.html (accessed on 15 November 2024)

(2)

thinwallsectionopenclosed was used to compute values without considering bimoment as a constraint.

https://labmatlab-was.upv.es/webapps/home/thinwallsectionopenclosed.html (accessed on 15 November 2024)

3.1. Method 1 to Obtain Plastic Parameters

The following Table 1, Table 2, Table 3 and Table 4 explain how to obtain the section properties. The values obtained have to be divided by f_y. The cross-section is divided into n fibers of area A(i), coordinates y(i), z(i), and unit warping w(i). The normal stress σ(i) values at each fiber are optimized to maximize the different linear functions in the tables with the corresponding linear constraints; this problem can be solved using a simplex (linear programming).

Table 1. Case 1: computation of plastic parameters without considering bimoment B_Ed as a constraint.

Maximize	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)$	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)$	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)$
Constraints	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=0$	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=0$
	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=0$
	$-fy<\sigma \left(i\right)<fy$

Table 2. Case 2: computation of plastic parameters considering bimoment B_Ed as a constraint.

Maximize	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)$	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)$	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)$	$B={\displaystyle \sum _{i=1}^n}\omega \left(i\right)A\left(i\right)\sigma \left(i\right)$
Constraints	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=0$	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=0$
	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=0$	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=0$
	$B={\displaystyle \sum _{i=1}^n}\omega \left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$B={\displaystyle \sum _{i=1}^n}\omega \left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$B={\displaystyle \sum _{i=1}^n}\omega \left(i\right)A\left(i\right)\sigma \left(i\right)=0$	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=0$
	$-fy<\sigma \left(i\right)<fy$

Table 3. Case 3: computation of plastic parameter maximum.

Maximize	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)$	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)$	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)$	$B={\displaystyle \sum _{i=1}^n}\omega \left(i\right)A\left(i\right)\sigma \left(i\right)$
Constraints	$-fy<\sigma \left(i\right)<fy$

3.2. Method 1 to Obtain Interaction Diagram

To obtain the interaction diagram N, M_y, and M_z, the plastification factor ξ (the factor by which internal forces must be multiplied to reach the plastic resistance of a cross-section) can be obtained for each combination in case 1 of N₁, M_y1, and M_z1 and in case 2 of N₁, M_y1, M_z1, and B₁ by applying the statics or lower bound theorem.

Table 4. Computation of plastic factor ξ to plot interaction diagram.

	Case 1: Without Considering Bimoment B as a Constraint	Case 2: Considering Bimoment B as a Constraint
Maximize	ξ	ξ
Constraint	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{N}1$	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{N}1$
	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{M}\mathrm{y}1$	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{M}\mathrm{y}1$
	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{M}\mathrm{z}1$	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{M}\mathrm{z}1$
	$-$	$B={\displaystyle \sum _{i=1}^n}\omega \left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }B1$
	$-fy<\sigma \left(i\right)<fy$

Table 4. Computation of plastic factor ξ to plot interaction diagram.

	Case 1: Without Considering Bimoment B as a Constraint	Case 2: Considering Bimoment B as a Constraint
Maximize	ξ	ξ
Constraint	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{N}1$	$N={\displaystyle \sum _{i=1}^n}A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{N}1$
	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{M}\mathrm{y}1$	$My={\displaystyle \sum _{i=1}^n}z\left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{M}\mathrm{y}1$
	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{M}\mathrm{z}1$	$Mz=-{\displaystyle \sum _{i=1}^n}y\left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }\mathrm{M}\mathrm{z}1$
	$-$	$B={\displaystyle \sum _{i=1}^n}\omega \left(i\right)A\left(i\right)\sigma \left(i\right)=\mathsf{\xi }B1$
	$-fy<\sigma \left(i\right)<fy$

4. Numerical Results. Comparisons of Results of Various Ways of Calculation

The study comprises an investigation of 10 thin-walled profiles given in Table 5 with their dimensions, plus another four thin-walled cross-sections called diamond sections (Figure 3b,c), oblique (Figure 7a), irregular sections (Figure 7b), and sigma sections (Figure 5 and Figure 7c).

Table 5. Investigated cross-sections [15] and their midline dimensions and thicknesses under table.

	$I_w={\displaystyle \frac{h_{\mathrm{f}}^3{t}_1^3+b^3{t}_2^3}{36}}$		$I_w={\displaystyle \frac{h_{\mathrm{f}}^3{t}_1^3}{36}}+2{\displaystyle \frac{{\left(b/2\right)}^3{t}_2^3}{36}}$
	$e={\displaystyle \frac{3b^2{t}_2}{h_{\mathrm{f}}{t}_1+6b{t}_2}}$ $I_{\mathrm{w}}={\displaystyle \frac{h_{\mathrm{f}}^2{b}^3{t}_2}{12}}\cdot {\displaystyle \frac{2h_{\mathrm{f}}{t}_1+3b{t}_2}{h_{\mathrm{f}}{t}_1+6b{t}_2}}$		$e={\displaystyle \frac{h_{\mathrm{f}}^2{b}^{2}t}{I_y}}\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{c}{2b}}-{\displaystyle \frac{2c^3}{3h_{\mathrm{f}}^{2}b}}\right)$ $I_{\mathrm{w}}={\displaystyle \frac{b^{2}t}{6}}\left(4c^3+6h_{\mathrm{f}}{c}^2\right.$ $\left.+3h_{\mathrm{f}}^{2}c+h_{\mathrm{f}}^{2}b\right)-e^2{I}_y$
	$I_{\mathrm{w}}={\displaystyle \frac{h_{\mathrm{f}}^2{I}_{\mathrm{z}}}{4}}$		$e={\displaystyle \frac{h_{\mathrm{f}}^2{b}^{2}t}{I_y}}\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{c}{2b}}-{\displaystyle \frac{2c^3}{3h_{\mathrm{f}}^{2}b}}\right)$ $I_{\mathrm{w}}={\displaystyle \frac{b^{2}t}{6}}\left(4c^3-6h_{\mathrm{f}}{c}^2\right.$ $\left.+3h_{\mathrm{f}}^{2}c+h_{\mathrm{f}}^{2}b\right)-e^2{I}_y$
	$e={\displaystyle \frac{z_1{I}_1-z_2{I}_2}{I_1+I_2}}$ $I_{\mathrm{w}}={\displaystyle \frac{h_f^2{I}_1{I}_2}{I_1+I_2}}$ where $I_1={\displaystyle \frac{1}{12}}{b}_1^3{t}_1$    $I_2={\displaystyle \frac{1}{12}}{b}_2^3{t}_2$		$I_{\mathrm{w}}={\displaystyle \frac{h_{\mathrm{f}}^2{I}_{\mathrm{z}}}{4}}+{\displaystyle \frac{c^2{b}^{2}t}{6}}\left(3h_{\mathrm{f}}+2c\right)$
	$I_{\mathrm{w}}={\displaystyle \frac{h_{\mathrm{f}}^2{b}^3{t}_2}{12}}\cdot {\displaystyle \frac{2h_{\mathrm{f}}{t}_1+b{t}_2}{h_{\mathrm{f}}{t}_1+2b{t}_2}}$		$I_{\mathrm{w}}={\displaystyle \frac{b^{2}t}{12\left(2b+h_{\mathrm{f}}+2c\right)}}\times$ $\left(h_{\mathrm{f}}^2\left(b^2+2b{h}_{\mathrm{f}}+4bc+6h_{\mathrm{f}}c\right)\right.$ $+4c^2\left(3b{h}_{\mathrm{f}}+3h_{\mathrm{f}}^2+4bc\right)$ $\left.\left.+2h_{\mathrm{f}}c+c^2\right)\right)$

Dimensions in cm of profiles from the Table 5: No.1: short leg 7.5 × 1.3, long leg 9.35 × 1; No.2: flange 8 × 1.3, web 9.35 × 1; No.3: flanges 7.5 × 1.3, web 18.7 × 1; No.4: flanges 6.7 × 1.3, web 18.7 × 1.3, lips 2.35 × 1.3; No.5: flanges 8 × 1.3, web 18.7 × 1; No.6: flanges 6.7 × 1.3, web 18.7 × 1.3, lips 2.35 × 1.3; No.7: upper flange 10 × 1.3, bottom flange 6 × 1.3, web 18.7 × 1; No.8: flanges 6.7 × 1.3, web 18.7 × 1.3, lips 2.35 × 1.3; No.9: flanges 7.5 × 1.3, web 18.7 × 1; No.10: flanges 6.7 × 1.3, 18.7 × 1.3, lips 2.35 × 1.3.

Figure 7. Non-standard cross-sections: (a) oblique section, (b) irregular section, and (c) sigma section.

Dimensions of the diamond section (Figure 3c) are given in Table 6, those of the oblique section (Figure 7a) in Table 7, the irregular section (Figure 7b) in Table 8, and the sigma section (Figure 7c) in Table 9.

Table 6. Diamond cross-section dimensions (Figure 3c).

	Nodes, Coordinates y = 0, z [mm], Thicknesses t [mm]
	0		1		2		3		4		5		6		7		8		9		10		11
z	0		20		50		70		90		90.86		100		120		140		170		190		200
t		10		10		17		17		140		140		17		17		10		10		110

Table 7. Oblique cross-section dimensions (Figure 7a).

	Nodes, Coordinates y, z [mm], t = 0.4 [mm]
	0	1	2	3	4	5	6
y	3.775	20	70	66.225	50	0	3.775
z	6.538	34.64	34.64	28.102	0	0	6.538

Table 8. Irregular cross-section dimensions (Figure 7b).

	Nodes, Coordinates y, z [mm], Thicknesses t [mm]
	0		1		2					3		4				5				6			7					8			9		10
y	1.45		1.45		1.45					1.45		1.45				1.45				24.15			24.15					27.9			48.9		51.1
z	21.784		23.784		24.3					31.3		40.2				32.75				32.75			44.75					44.75			34.7		34.5
t		2.9		2.9			2.9				2.9			0				1.5			1.5				1.7				1.5			1.5
	Nodes, coordinates y, z [mm], thicknesses t [mm]
	10		11			12			13				14				15		16					17			18				19
y	51.1		56.7			59.3			63.55				63.55				63.55		63.55					63.55			1.45				1.45
z	34.5		37.5			37.5			34.6				18.043				16.3		7.3					0.85			0.85				21.784
t		1.5		1.5				1.6			2.9				2.9			3.3				2.9				1.7				2.9

Table 9. Sigma cross-section dimensions (Figure 7c).

	Nodes, Coordinates y, z [mm], t = 1.4 [mm]
	0	1	2	3	4	5	6	7	8	9	10
y	60	55	0	0	10	10	10	0	0	60	65
z	30	0	0	40	50	100	150	160	200	200	170

There are three groups of plastic section moduli in Table 10 as follows:

Table 10. Values of various kinds of plastic section moduli, including W_pl,y, W_pl,z [cm³] and W_pl,w [cm⁴].

Section		I.a	I.b	I.c	II.	III.	IV.	V.	VI.	VII.	VIII.
		Bimoment Is Not Considered as a Constraint				Bimoment Is Considered as a Constraint			Wpl,max
No.	Shape	Wpl,y,nB	[Wpl,y]	(Wpl,y)	Wpl,z,nB	Wpl,y	Wpl,z	Wpl,w	Wpl,y,max	Wpl,z,max	Wpl,w,max
1	L	27.77	–	–	43.15	–	–	–	37.47	55.16	–
2	T	44.81	44.81	44.51	23.14	–	–	–	51.209	23.14	–
3	U	269.7 –	269.7 –	– –	74.14 –	241.4 240.6	74.14 73.06	592.6 591.1	269.7 269.7	81.31 81.12	631.7 610.3
4	U + lips inside	326.5	326.5	–	100.96	300.6	100.96	879.8	326.5	113.4	914.9
5	I doub. symm.	281.9	281.9	270.2	46.2	281.9	46.2	388.9	281.9	46.2	388.9
6	U + lips outside	340.8	340.8	–	100.9	298	100.9	758.7	340.8	113.4	769.3
7	I mono symm.	279.4	275.1	263.4	48.87	279.4	44.19	234.3	281.8	48.87	287.9
8	I + lips doub.s	376.4	376.4	361.2	78.	376.4	78.	703.	376.4	78.	703.
9	Z point symm.	166.4 – –	– – –	– – –	64.70 – –	166.4 162.9 –	64.70 63.22 –	683 687.5 –	269.7 269.7 269.7 *	77.8 73.13 –	760. 758.5 –
10	Z + lips point s.	205.8	–	–	90.9	205.8	90.9	976.5	326.4 326.5 *	107.4	1123.
11	Sect. + diamon.	219.9	214.7	–	160.	219.9	147.	307	230.	160.	415.
12	Oblique point s.	0.866	0.863	–	1.301	0.866	1.301	0.173	0.97	1.4	0.173
13	Irregu-lar sec.	6.05	6.014	–	9.954	5.955	9.933	7.022	6.09	10.05	7.227
14	Sigma asymm.	37.30	37.97	–	9.036	34.9	8.96	95.9	38.	10.5	104.

Key: index w—warping; the: The values with asterisk (*) in the boxes 9-VI and 10-VI were calculated using MathCad formulae.

(a)

W_pl,y,nB, [W_pl,y], (W_pl,y), and W_pl,z,nB (see columns I.a, I.b, I.c, II.) for the case when bimoment is not (index nB) considered as a constraint. The values in brackets [ ], ( ) in the columns I.b and I.c, respectively, are calculated using formula (G.39) given in Annex G, EN 1999-1-1:2023 [15]; here, it is Formula (1). Computer software MathCad was employed for bracket value calculations. Corner details in column I.b in the calculation of [..] values are (a) + (d) according to Figure 4. Corner details in column I.c in the calculation of (..) values are (b) + (c) according to Figure 4. See the consequences described before Figure 4.

(b)

W_pl,y, W_pl,z, and W_pl,w (see columns III., IV., V.) for the case when bimoment is considered as a constraint.

(c)

W_pl,y,max, W_pl,z,max, and W_pl,w,max (see columns VI., VII., VIII.) are also considered. These values may be achieved only if the given internal force exists with some other concomitant internal force. See Table 13a,b for the monosymmetric I-shaped section, Table 14a,b for the channel section, and Table 15a,b for the Z-section.

All W_pl values in Table 10 except “MathCad values” in columns I.b and I.c were calculated using computer programs described in Section 3. The model with overlaps in (a + d) corners according to Figure 4 is taken into account in the calculation of values given in the column I.b of Table 10. Other exceptions include the following: (1) the values in italics for the channel (No. 3) and Z-section (No. 9) were calculated using formulae designated for practice, which are given in Table 14a for the channel and in Table 15a for the Z-section; (2) the values with asterisks (*) in the boxes 9-VI (row No.–column No.) and 10-VI were also calculated using MathCad formulae; (3) only the value in box 11-I.b was calculated for the compressed equivalent shape of the cross-section given in Figure 3c. All other values for the diamond section (No.11) were calculated for the real cross-section shape in Figure 3b.

Values in bold in boxes 9-VI, 10-VI, and 12-VI are the resistances that can be reached in the case of sections being continuously laterally supported. The concomitant bimoment for the Z-section is zero (see Table 15a).

The way to obtain W_pl values and consequently the interaction diagrams is explained in Section 3, where the used software is also described.

The following are the facts (see Table 10):

(a)

For doubly symmetric sections (No. 5, 8):

W_pl,y = W_pl,y,nB = W_pl,y,max; W_pl,z = W_pl,z,nB = W_pl,z,max; W_pl,w = W_pl,w,max;

(b)

For monosymmetric sections with axis of symmetry z (No. 2, 7, 11):

W_pl,y = W_pl,y,nB < W_pl,y,max; W_pl,z < W_pl,z,nB = W_pl,z,max; W_pl,w < W_pl,w,max;

(c)

For monosymmetric sections with axis of symmetry y (No. 3, 4, 6):

W_pl,y < W_pl,y,nB = W_pl,y,max; W_pl,z = W_pl,z,nB < W_pl,z,max; W_pl,w < W_pl,w,max;

(d)

For point-symmetric sections (9, 10), Figure 8:

W_pl,y = W_pl,y,nB < W_pl,y,max; W_pl,z = W_pl,z,nB < W_pl,z,max; W_pl,w < W_pl,w,max;

(e)

Asymmetric sections are No. 1, 12, 13, and 14;

(f)

For non-warping cross-sections or for cross-sections with negligible warping through the element thicknesses, the calculation of W_pl,y, W_pl,z, and W_pl,w for bimoment B_Ed considered as a constraint is not possible. See No. 1 and 2, columns III.–V.

The results of programs https://laboratoriosvirtuales.upv.es/webapps/thinwallsectiongeneral.html (accessed on 15 November 2024) and https://labmatlab-was.upv.es/webapps/home/thinwallsectionopenclosed.html (accessed on 15 November 2024) are presented for sections from Table 5 and Table 10 in the following Figures as follows: in Figure 8 (section No.10 from Table 5), in Figure 9 (section No.9 from Table 5), in Figure 10 (section No.11 from Table 10), in Figure 11 (section No.12 from Table 10) and in Figure 12 (section 13 from Table 10). The results present interaction diagrams for given cross-sections under a combination of internal forces N_Ed, M_y,Ed, and M_z,Ed for the following cases: (a) without considering bimoment B_Ed as a constraint and (b) with bimoment B_Ed = 0 kNm² considered as a constraint. Interaction diagrams may be used for any values of N_Ed, M_y,Ed, and M_z,Ed because diagrams are accompanied by relevant A and W_pl values. In all interaction diagrams, the following sign convention of internal forces is accepted: the tension axial force N_Ed is positive, the bending moment M_y,Ed is positive when the bottom part of the cross-section is in tension and the upper part in compression, and the bending moment M_z,Ed is positive when the right part of the cross-section is in tension and the left part in compression. In the cross-section resistances, the calculation safety factor ${\gamma }_{\mathrm{M}0}=1.0$ is used for steel profiles as prescribed in [14] and ${\gamma }_{\mathrm{M}1}=1.1$ is used for aluminum profiles [15].

Figure 9. The Z-section without lips (No.9 in Table 5 and Table 10) with A = 38.2 cm². Interaction diagrams for a cross-section under a combination of N_Ed + M_y,Ed + M_z,Ed are calculated (a) without considering bimoment B_Ed as a constraint: W_pl,y,nB = 166.4 cm³, W_pl,z,nB = 64.7 cm³; (b) with bimoment B_Ed = 0 kNm² considered as a constraint: W_pl,y = 166.4 cm³, W_pl,z = 64.7 cm³.

Figure 10. The diamond section (No.11 in Table 5 and Table 10) with A = 49.07 cm². Interaction diagrams for a cross-section under a combination of N_Ed + M_y,Ed + M_z,Ed are calculated (a) without considering bimoment B_Ed as a constraint: W_pl,y,nB = 219.9 cm³, W_pl,z,nB = 160 cm³; (b) with bimoment B_Ed = 0 kNm² considered as a constraint: W_pl,y = 219.9 cm³, W_pl,z = 147 cm³.

Figure 11. The oblique section (No.12 in Table 5 and Table 10) with A = 0.72 cm². Interaction diagrams for a cross-section under a combination of N_Ed + M_y,Ed + M_z,Ed are calculated (a) without considering bimoment B_Ed as a constraint: W_pl,y,nB = 0.866 cm³, W_pl,z,nB = 1.301 cm³; (b) with bimoment B_Ed = 0 kNm² considered as a constraint: W_pl,y = 0.866 cm³, W_pl,z = 1.301 cm³.

Figure 12. The irregular section (No.13 in Table 5 and Table 10) with A = 4.395 cm². Interaction diagrams for a cross-section under a combination of N_Ed + M_y,Ed + M_z,Ed are calculated (a) without considering bimoment B_Ed as a constraint: W_pl,y,nB = 6.05 cm³, W_pl,z,nB = 9.954 cm³; (b) with bimoment B_Ed = 0 kNm² considered as a constraint: W_pl,y = 5.955 cm³, W_plz = 9.933 cm³.

Figure 8. The Z-section with lips (No.10 in Table 5 and Table 10) under bending moment M_y,Ed about the y-y axis with axes yz and principal axes uv. (a) The location of the plastic neutral axis (PNA) responds to stress distribution to obtain M_pl,y. In this case, the results are the same whether the bimoment constraint is considered or not; (b) the location of the PNA responds to the stress distribution to obtain M_pl,y,max. In the case when the beam is loaded in the z-direction and continuously laterally supported in the y-direction, the concomitant M_z,conc may appear.

5. Recommendations

The computer programs in Section 3 enable us to investigate the plastic reserve of cross-sections with any shape and under any combination of eight internal forces. Below are recommendations for frequent special partial cases.

5.1. Double Symmetric Steel I-Shaped Cross-Section of Class 1 or Class 2 Under Combination of Axial Force and Bending Moment N_Ed + M_y,Ed

The problems in 8.2.9.1 [14] solved by Baláž [13] are outlined. Using the approximate formulas (8.43) and (8.48) in [14] leads to the two following problems:

(a)

The approximate formula (8.48) after substitution into formula (8.43) has the form

$${\displaystyle \frac{{\mathit{M}}_{\mathbf{y},\mathbf{E}\mathbf{d}}}{{\mathit{M}}_{\mathbf{N},\mathbf{y},\mathbf{R}\mathbf{d}}}}={\displaystyle \frac{{\mathit{M}}_{\mathbf{y},\mathbf{E}\mathbf{d}}}{{\mathit{M}}_{\mathbf{p}\mathbf{l},\mathbf{y},\mathbf{R}\mathbf{d}}}}{\displaystyle \frac{\mathbf{1}-\mathbf{0.5}\mathit{a}}{\mathbf{1}-\mathit{n}}}={\mathit{m}}_{\mathit{y}}{\displaystyle \frac{\mathbf{1}-\mathbf{0.5}\mathit{a}}{\mathbf{1}-\mathit{n}}}\le \mathbf{1.0} \mathrm{but} {\mathit{m}}_{\mathit{y}}={\displaystyle \frac{{\mathit{M}}_{\mathit{y},\mathit{E}\mathit{d}}}{{\mathit{M}}_{\mathit{p}\mathit{l},\mathit{y},\mathit{R}\mathit{d}}}}\le \mathbf{1.0}, \mathit{n}={\displaystyle \frac{{\mathit{N}}_{\mathit{E}\mathit{d}}}{{\mathit{N}}_{\mathit{p}\mathit{l},\mathit{R}\mathit{d}}}}$$

(20)

Formula (20) leads, for large values of relative axial force n (if n approaches the value 1.0), to numerical instability and to looping of the automated calculation when designing the cross-section size. For the selected cross-sectional size, the verification may show a high value of utilization (e.g., more than 3.0). When increasing the size of the cross-section by one step, the verification of utilization can show a significantly uneconomical design. It leads to never-ending looping. The solution is to modify the EN Formula (20) before programming into the form (21), in which there is no division by the expression (1–n), which for n approaching 1.0 leads to division by a number approaching zero.

$$\mathbf{m}\mathbf{a}\mathbf{x}\left[\mathit{n}+\left(\mathbf{1}-\mathbf{0.5}\mathit{a}\right){\mathit{m}}_{\mathit{y}};{\mathit{m}}_{\mathit{y}}\right]\le \mathbf{1.0}, \mathit{n}={\displaystyle \frac{{\mathit{N}}_{\mathit{E}\mathit{d}}}{{\mathit{N}}_{\mathit{p}\mathit{l},\mathit{R}\mathit{d}}}}, {\mathit{m}}_{\mathit{y}}={\displaystyle \frac{{\mathit{M}}_{\mathit{y},\mathit{E}\mathit{d}}}{{\mathit{M}}_{\mathit{p}\mathit{l},\mathit{y},\mathit{R}\mathit{d}}}}, \mathit{a}=\mathbf{m}\mathbf{i}\mathbf{n}\left[\left(\mathit{A}-\mathbf{2}\mathit{b}{\mathit{t}}_{\mathit{f}}\right)/\mathit{A};\mathbf{0.5}\right]$$

(21)

The following is a numerical example:

HEA 300, S355, f_y = 355 MPa, γ_M0 = 1.0, N_Ed = 3795 kN, n = 0.95, M_y,Ed = 100 kNm, m_y = 0.204.

Formula (20) [14] gives utilization factor U_EN = 3.574, requiring a much bigger size of HEA. The correct utilization factor according to the proposed formula (21) is only U = 1.128. If the bigger HEA 340 is chosen, formula (20) gives U_EN = 0.665, indicating a significantly uneconomical design, but the real utilization factor according to (21) is U = 0.933. Using EN formula (20) will lead to a never-ending loop.

In the same way, it is necessary to rearrange formulae (8.49), (8.50), and (8.43) in [14] to avoid numerical instability for the case of I-shaped cross-sections under axial force N_Ed and bending moment M_z,Ed. Baláž proposed [13,18], instead of EN 1993-1-1 [14] Formulae (8.49), (8.50), and (8.43), to use the following formulae:

$${\left(\mathit{n}-\mathit{a}\right)}^{\mathbf{2}}+\mathit{a}\left(\mathbf{2}-\mathit{a}\right)+{\left(\mathbf{1}-\mathit{a}\right)}^2{\mathit{m}}_{\mathit{z}}\le \mathbf{1.0}, \mathrm{if} \mathit{n}>\mathit{a}, \mathit{n}={\displaystyle \frac{{\mathit{N}}_{\mathit{E}\mathit{d}}}{{\mathit{N}}_{\mathit{p}\mathit{l},\mathit{R}\mathit{d}}}}, {\mathit{m}}_{\mathit{z}}={\displaystyle \frac{{\mathit{M}}_{\mathit{z},\mathit{E}\mathit{d}}}{{\mathit{M}}_{\mathit{p}\mathit{l},\mathit{z},\mathit{R}\mathit{d}}}}$$

(22)

$${\mathit{m}}_{\mathit{z}}\le \mathbf{1.0}, \mathrm{if} \mathit{n}\le \mathit{a}, \mathit{a}=\mathbf{m}\mathbf{i}\mathbf{n}\left[\left(\mathit{A}-\mathbf{2}\mathit{b}{\mathit{t}}_{\mathit{f}}\right)/\mathit{A};\mathbf{0.5}\right]$$

(23)

Formulae (8.51), (8.52), and (8.43) in [8], valid for rectangular hollow sections, must also be rearranged before programming in the same way as formulae (8.48) and (8.43), as in Equation (21), to avoid dividing by a value approaching zero in the case of the relative axial force n approaching 1.0.

(b)

Formula (20) gives results on the dangerous side for small values of the axial force (error is up to approximately 7%). This problem can be solved by replacing the approximate Eurocode formulae (8.43) and (8.48), which is (20) here, with more accurate formulae derived by Baláž [13]. For I-shaped sections, we present the exact m_y−n interaction formulae in two versions as follows:

(b1)

For a cross-section consisting of three rectangles, two independent dimensionless parameters appear in the formulae. The graphic interpretation is in Figure 13 and Figure 14. Formula (24) is valid for smaller values $\mathit{n}<\mathit{a}$, when the plastic neutral axis lies in the web of the I-section (see bottom left part of Figure 14). Formula (25) is valid for larger values $\mathit{n}>\mathit{a}$, when the plastic neutral axis lies in the flange of the I-section (see bottom right part of Figure 13). In the case of $\mathit{n}=\mathit{a}$, the plastic neutral axis is located in the point of contact of the flange with the web.

$$m_y=1-{\displaystyle \frac{n^2}{1-{\left(\frac{A_w}{A}-1\right)}^2\left(1-\frac{t_w}{b}\right)}}, \mathrm{if} n\le a={\displaystyle \frac{1-2b{t}_f}{A}}={\displaystyle \frac{A_w}{A}}\approx {\displaystyle \frac{h_w{t}_w}{h_w{t}_w+2b{t}_f}}, h_w=h-2t_f$$

(24)

$$m_y=1-{\displaystyle \frac{n^2-{\left(\frac{A_w}{A}-n\right)}^2\left(1-\frac{t_w}{b}\right)}{1-{\left(\frac{A_w}{A}-1\right)}^2\left(1-\frac{t_w}{b}\right)}}, \mathrm{if} n\ge a={\displaystyle \frac{1-2b{t}_w}{A}}={\displaystyle \frac{A_w}{A}}\approx {\displaystyle \frac{h_w{t}_w}{h_w{t}_w+2b{t}_f}}$$

(25)

(b2)

For the cross-section consisting of the midlines with thicknesses, only one independent dimensionless parameter appears in the formulae.

$$m_y=1-{\displaystyle \frac{n^2}{1-{\left(\frac{A_{w.st}}{A_{st}}-1\right)}^2}}, \mathrm{if} n\le {\displaystyle \frac{A_{w.st}}{A_{st}}}={\displaystyle \frac{h_f{t}_w}{h_f{t}_w+2b{t}_f}}, h_f=h-t_f$$

(26)

$$m_y={\displaystyle \frac{1-n}{1-0,5\frac{A_{w.st}}{A_{st}}}}, \mathrm{if} n\ge {\displaystyle \frac{A_{w.st}}{A_{st}}}={\displaystyle \frac{h_f{t}_w}{h_f{t}_w+2b{t}_f}}$$

(27)

Figure 13. Detail of Figure 14.

Figure 14. Geometrical interpretations of interaction m_y−n formulae for HEB 280 section under combination of axial force N_Ed with moment M_y,Ed. Comparison of (a) approximate red dotted bilinear function defined in EN 1993-1-1 [14] by Formulae (20) or (21) with (b) black solid line representing non-linear function defined by exact Formulae (24) and (25) [13].

For n = a/2, the approximate solution gives the value with an error ≤ 7% on the dangerous side.

The effect of axial force can be neglected if $n\le \mathrm{m}\mathrm{i}\mathrm{n} (0.25;{0.5h}_w{t}_w/A)$, which is approximately

$n\le \mathrm{m}\mathrm{i}\mathrm{n} (0.25;a/2)$ with $a=\mathrm{m}\mathrm{i}\mathrm{n} \left[\left(A-2b{t}_f\right)/A;0.5\right]$. S235 f_y = 235 MPa, safety factor γ_M0 = 1.0.

For the exact shape of the cross-section: N_pl,Rd = 3087 kN, M_pl,y,Rd = 360.6 kNm, M_pl,z,Rd = 168.6 kNm.

For the midline model: N_pl,Rd = 3015.3 kN, M_pl,y,Rd = 352.66 kNm, M_pl,z,Rd = 165.82 kNm.

The results for steel profiles: HEB 280 are presented in Figure 15 and UPE 360 are presented in Figure 16 and Figure 17.

Figure 15. HEB 280 section with A = 128.31 cm², W_pl,y = 1500.67 cm³, W_pl,z = 705.6 cm³, and W_pl,w = 9243.36 cm⁴ for midline model. Interaction diagrams for cross-section under combination of N_Ed + M_y,Ed + M_z,Ed calculated (a) without considering bimoment B_Ed as a constraint; (b) with bimoment B_Ed = 0 kNm² considered as a constraint.

5.2. Plastic Reserve of Steel Channel Cross-Section Under Bending Moment M_y,Ed

The maximum positive plastic moment in channel M_pl,y,max = W_pl,y,max f_y may be achieved only with negative concomitant bimoment B_Ed,conc. The beam that is continuously laterally supported cannot rotate; therefore, bimoment B = 0 kNm² and M_pl,y,max cannot be achieved. Only the value of the elastic–plastic moment M_pl,y = W_pl,y f_y may be achieved. The values of plastic reserves (shape factors) for I-sections, the channel, and similar sections are very small. For UPE 360, the shape factors are α_pl,y = 875.3/823.6 = 1.063 and α_pl,y,max = 982.6/823.6 = 1.193. For sections of this type under a single bending moment M_y,Ed, the utility of the plastic reserve is negligible.

Figure 16. Plastic reserves (shape factors α_pl,y and α_pl,y,max) of UPE 360 profile [9]. Above W_pl values are calculated for exact shape of cross-section. In paragraph 5.4, W_pl values are calculated for midline model with A = 76.52 cm², W_el,y = W_pl,y = 841.7 cm³, and W_pl,y,max = 959.37 cm³.

Figure 17. UPE 360 section. S235, f_y = 235 MPa, safety factor γ_M0 = 1.0. For midline model: N_pl,Rd = 1798 kN, M_pl,z,Rd = 44.24 kNm. Interaction diagrams of N_Ed + M_y,Ed + M_z,Ed calculated (a) without considering bimoment B_Ed as a constraint: M_pl,y,Rd = 225.45 kNm. 225.452 kNm in Table 12; (b) with bimoment B_Ed = 0 kNm² considered as a constraint: M_pl,y,Rd = 197.80 kNm. 199.52 kNm in Table 12.

5.3. Double-Symmetric Aluminum and Steel I-Shaped Profiles Under Combination of Bending Moment About y-y Axis and Bimoment M_y,Ed + B_Ed

The metal (steel and aluminum) Eurocodes of the first generation give interaction formulae valid for the I-shaped cross-sections, and the rectangular hollow sections included welded ones to allow the use of plastic resistance for Class 1 and 2 cross-sections, leading to a more economic design. These interaction formulae are valid only for combinations of axial load and bending internal forces. The influence of torsional bimoment is taken into account only in metal Eurocodes of the second generation. In both metal Eurocodes of the first and second generation, there are no formulae for the calculation of the plastic resistance of channel cross-sections, which are frequently used in practice. The steel Eurocode does not allow us to take into account the material strengthening. However, the aluminum Eurocode enables us to calculate the ultimate axial load N_u and ultimate bending moment M_u at the collapse limit state from the formulae given in Annex F.4 and F.5 in [15]. See Figure 18 and Figure 19.

Figure 18. Comparisons of the five stress–strain relationships given in [17] with continuous strength. The method (CSM) [19] applied in [20] for the non-heat-treatable wrought aluminum alloy EN AW-5083-O/H111. The incorrectly defined continuous model in E.2.2.1 [17] was later corrected thanks to findings in [20,21].

Figure 19. Bilinear stress–strain diagrams without and with strengthening. Definitions of η_ε and η_y [7].

Streľbickaja [7] investigated theoretically and experimentally I- and U-sections. She took into account the influences of five internal forces, namely two bending internal forces M_y,Ed and V_z,ed and three torsional internal forces B_Ed, T_w,Ed, and T_t,Ed. For the combinations of these internal forces, the following resistances were calculated: (i) elastic, (ii) plastic without strengthening, and (iii) plastic with strengthening. Streľbickaja’s analytical formulae were verified by computer programs [22,23]. The results comparisons show very good agreements and the geniality of Streľbickaja from Ukraine.

The resistance calculation according to Streľbickaja [7] is as follows:

$$W_{el,y}=t_f{b}_f{h}_f+t_w{h}_f^2/6=1.14\hspace{0.17em}\mathrm{x} 10\hspace{0.17em}\mathrm{x}\hspace{0.17em}18.86\hspace{0.17em}+\hspace{0.17em}0.7\hspace{0.17em}\mathrm{x}\hspace{0.17em}{18.86}^2/6=256.502\hspace{0.17em}{\mathrm{cm}}^3$$

(28)

$$W_{pl,y}=t_f{b}_f{h}_f+t_w{h}_f^2/4=1.14\hspace{0.17em}\mathrm{x} 10\hspace{0.17em}\mathrm{x}\hspace{0.17em}18.86\hspace{0.17em}+\hspace{0.17em}0.7\hspace{0.17em}\mathrm{x}\hspace{0.17em}{18.86}^2/4=277.251\hspace{0.17em}{\mathrm{cm}}^3$$

(29)

$$M_{el,y,Rd}=W_{y,el}{f}_y/{\gamma }_{M0}=256\hspace{0.17em}502\hspace{0.17em}\hspace{0.17em}\mathrm{x}\hspace{0.17em}110/1.0=28.215\hspace{0.17em}\mathrm{kNm}$$

(30)

$$M_{pl,y,Rd}=W_{y,pl}{f}_y/{\gamma }_{M0}=277\hspace{0.17em}251\hspace{0.17em}\mathrm{x}\hspace{0.17em}110/1.0=30.498\hspace{0.17em}\mathrm{kNm}$$

(31)

$$W_{el,w}=t_f{h}_{f}b{}_f{}^2/6=358.34\hspace{0.17em}{\mathrm{cm}}^4,B_{el,Rd}=W_{el,w}{f}_y/{\gamma }_{M0}=0.394\hspace{0.17em}{\mathrm{kNm}}^2$$

(32)

$$W_{pl,w}=t_f{h}_{f}b{}_f{}^2/6=537.51\hspace{0.17em}{\mathrm{cm}}^4,B_{pl,Rd}=W_{w,pl}{f}_y/{\gamma }_{M0}=0.591\hspace{0.17em}{\mathrm{kNm}}^2$$

(33)

For a given distance u, the size of the small elastic core 2e (Figure 20a) may be calculated from Equation (34).

Figure 20. Normal stress distribution in (a) I-shaped section (e = 0 mm is possible) and (b) channel under combination of positive bending moment M_y,Ed and negative bimoment B_Ed (sign convention: see Figure 21). (i) Elastic limit state defined by σ_max = f_y—dotted line (c—compression; t—tension); (ii) plastic limit state without strengthening defined by σ_max = f_y—dashed line; (iii) plastic limit state with strengthening defined by ε_max = (1 + ηε)ε_y—solid line with related ε distribution [7,20].

The sign convention of bimoment B_Ed is clear from Figure 20 and Figure 21.

$$e(u)=\left(1+{\displaystyle \frac{b_f}{2u}}\right){\displaystyle \frac{h_f}{2{\mathrm{\eta }}_{\epsilon }}}$$

(34)

Figure 21. Sign convention. Combination of moment M_y,Ed with (a) positive and (b) negative bimoment B_Ed [7,20].

(i)

The elastic limit state is defined by the following linear interaction formula:

$${\displaystyle \frac{M_{y,Ed}}{M_{el,y,Rd}}}+{\displaystyle \frac{B_{Ed}}{B_{el,Rd}}}\le 1.0$$

(35)

From Figure 20a, the plastic resistances may be derived as a function of the distance u, defining the position of the neutral axis in the I-section under the interaction of the bending moment M_y,Ed and bimoment B_Ed. For the plastic limit state, the following are calculated:

(ii)

Without strengthening η_y = 0 (Figure 18 and Figure 19):

$$M_{y,Ed}(u)=\left[2t_{f}u{h}_f+\left({\displaystyle \frac{h_f^2}{4}}-{\displaystyle \frac{e{(u)}^2}{3}}\right)t_w\right]f_y,B_{Ed}(u)=\left({\displaystyle \frac{b_f^2}{4}}-u^2\right)t_f{h}_f{f}_y$$

(36)

(iii)

With strengthening η_y = 0.25 (Figure 18):

$$M_{y,Ed,s}(u)=M_{y,Ed}(u)+\left[t_f{b}_f{h}_f+\left({\displaystyle \frac{h_f^2}{6}}-{\displaystyle \frac{e{(u)}^2}{3}}-{\displaystyle \frac{h_f}{6}}e{\left(u\right)}^2\right)t_w\right]{\displaystyle \frac{u}{u+0.5b_f}}{\mathrm{\eta }}_y{f}_y$$

(37)

$$B_{Ed,s}(u)=B_{Ed}(u)+{\displaystyle \frac{t_f{b}_f^3{h}_f}{12\hspace{0.17em}\left(\hspace{0.17em}u+0.5b_f\right)}}\hspace{0.17em}{\mathrm{\eta }}_y{f}_y$$

(38)

The obtained numerical results [20] are given in Table 11. For the case without strengthening, the results are compared with the numerical results of the computer program DUENQ [22], which uses the simplex method, and the computer program QST-TSV-3Blech [23] based on Kindmann’s and Frickel’s method, called the partial internal forces (PIFs) method (original German name, the Teilschnittgrößenverfahren (TSV) method), valid for the cross-sections consisting of three rectangles [9].

Table 11. Resistances of I 200a-section under combination of bending moment M_y.Ed and bimoment B_Ed [20].

	Source	Quantity	Results for Various u [cm] and e [cm] Values
stress–strain diagram with strengthening [7]	Streľbickaja [7]	u [cm]	5.0	4.667	3.667	2.667	1.667	0.667	0.0
		e [cm]	1.347	1.395	1.592	1.936	2.694	5.725	–
	Equation (38)	BEd,s [kNm2]	0.0	0.174	0.382	0.546	0.667	0.747	0.776
	Equation (37)	My,Ed.s [kNm]	39.10	35.29	29.74	23.96	17.82	10.62	0.0
stress–strain diagram without strengthening	Streľbickaja Equation (36) [7]	BEd [kNm2]	0.0	0.076	0.273	0.423	0.526	0.581	0.591
		My.Ed [kNm]	30.45	28.87	24.13	19.36	14.54	9.159	0.0
	Equation (40) [20]	My.Ed [kNm]	30.50	28.93	24.20	19.46	14.69	9.92	Mw,Rk = 6.82
	Program QST-TSV-3Blech [23]	My.Ed [kNm]	29.69	28.12	23.40	18.66	13.9	9.157	0.0
	Program DUENQ [22]	My.Ed [kNm]	29.0	27.5	22.6	18.0	12.9	7.6	0.0

Table 11 and Figure 22a show excellent agreements between the 66-year-old analytical solution [7] and the numerical results of modern computer programs [22,23]. The solution by Streľbickaja [7] can also take into account material strengthening. The computer programs [22,23] cannot perform this. Equations (36)–(38) were the basis for their modification by Baláž [20], who created an interaction formula which is convenient for design standards. It may be written in the following different forms:

$${\left({\displaystyle \frac{M_{y,Ed}-M_{pl,y,w,Rd}}{M_{pl,y,f,Rd}}}\right)}^2+{\displaystyle \frac{B_{Ed}}{B_{pl,Rd}}}=1.0, M_{pl,y,Rd}=M_{pl,y,f,Rd}+M_{pl,y,w,Rd}$$

(39)

$$M_{pl,y,w,Rd}=W_{pl,y,w}{\displaystyle \frac{f_y}{{\gamma }_{M0}}}, W_{pl,y,w}={\displaystyle \frac{1}{4}}{t}_w{h}_w^2, B_{pl,Rd}=W_{pl,\omega }{\displaystyle \frac{f_y}{{\gamma }_{M0}}}, W_{pl,\omega }={\displaystyle \frac{1}{4}}{t}_f{h}_f{b}^2$$

(40)

$${\rho }_{My,B}=1-\left(1-\sqrt{1-{\displaystyle \frac{B_{Ed}}{B_{pl,Rd}}}}\right){\displaystyle \frac{M_{pl,y,f,Rd}}{M_{pl,y,Rd}}} \mathrm{if} M_{y,Ed}\le M_{pl,y,w,Rd}, {\mathit{B}}_{\mathit{E}\mathit{d}}={\mathit{B}}_{\mathit{p}\mathit{l},\mathit{R}\mathit{d}}$$

(41)

$$M_{pl,y,B,Rd}={\rho }_{My,B}{M}_{pl,y,Rd}=M_{pl,y,Rd}-\left(1-\sqrt{1-{\displaystyle \frac{B_{Ed}}{B_{pl,Rd}}}}\right)M_{pl,y,f,Rd}$$

(42)

$${\rho }_{B,My}=1-{\left(1-{\displaystyle \frac{M_{pl,y,Rd}-M_{y,Ed}}{M_{pl,y,f,Rd}}}\right)}^2\le 1.0$$

(43)

$$B_{pl,My,Rd}={\rho }_{B,My}{B}_{pl,Rd}=\left[1-{\left(1-{\displaystyle \frac{M_{pl,y,Rd}-M_{y,Ed}}{M_{pl,y,f,Rd}}}\right)}^2\right]B_{pl,Rd}$$

(44)

The meaning of symbol ρ_My,B is a reduction factor by which the plastic bending moment resistance M_y,pl,Rd should be reduced for the given values B_Ed/B_pl,Rd and M_pl,y,f,Rd. Similarly, ρ_B,My gives the value by which the plastic bimoment resistance B_pl,Rd should be reduced for the given values M_y,Ed, M_y,pl,Rd, and M_pl,y,f,Rd.

The plastic resistance of an I-section without strengthening was solved in [24]. The formula valid for an I-section under a combination of a bending moment and bimoment was solved by Mirabelle in [24]. It was replaced by Baláž’s formula [25], which is now in [14], because Mirambell’s formula has several drawbacks.

The form of original formula proposed by Mirambell [24] is as follows:

$$M_{c,B,Rd}=\sqrt{1-{\displaystyle \frac{{\sigma }_{w,Ed.\max }}{\frac{\alpha f_y}{4{\gamma }_{M0}}}}}{M}_{c,Rd}=\sqrt{1-{\displaystyle \frac{{\sigma }_{w,Ed,\max }}{\frac{5f_y}{4{\gamma }_{M0}}}}}{M}_{c,Rd}=\sqrt{1-{\displaystyle \frac{{\sigma }_{w,Ed,\max }}{\frac{1.25f_y}{{\gamma }_{M0}}}}}{M}_{c,Rd}$$

(45)

Mirambell’s formula (45) is not convenient for standard purposes, because Eurocodes prefer to use internal forces in formulae, not stresses. Moreover, it is incorrect at a first glance, because M_y,Ed = 0 kNm does not give for α = 5 the value B_Ed,max = B_pl and differs a lot from exact results (Figure 22b).

Mirambell’s incorrect formula [24], which was in the draft prEN 1993-1-1, was replaced by Baláž’s correct formula [25], which is now in EN 1993-1-1:2022 [14].

$$M_{pl,y,B,Rd}=\sqrt{1-{\displaystyle \frac{6}{\alpha }}{\displaystyle \frac{B_{Ed}}{B_{pl,Rd}}}}{M}_{pl,y,Rd}={\rho }_{My,B}{M}_{pl,y,Rd}$$

(46)

Figure 22. I-shaped profiles under combination of bending moment M_y,Ed and bimoment B_Ed: (a) aluminum I 200a old Russian section, material: EN AW-5083-O/H111, f_o = 110 MPa, f_u = 270 MPa, γ_M1 = 1.0 is used instead of γ_M1 = 1.1 [13]. Elastic and plastic resistance without and with strengthening. Details of calculation based on continuous strength method (CSM), see [20]. W_pl values are in Figure 23a; (b) steel HEA 240 section, material: S235, f_y = 235 MPa, γ_M0 = 1.0 [25]; elastic and plastic resistance without strengthening. W_pl values are in Figure 23b.

Figure 23. Interaction diagrams of M_y,Ed + M_z,Ed + B_Ed for midline model: (a) aluminum I 200a section: A = 36.0 cm², W_pl,y = 277.25 cm³, W_pl,z = 57.0 cm³, W_pl,yw = 537.51 cm⁴; (b) steel HEA 240 section: A = 73.95 cm², W_pl,y = 716.45 cm³, W_pl,z = 345.6 cm³, W_pl,yw = 3 767.504 cm⁴.

The Formula (46) gives for values α = 4, 5, 6 the maximum bimoment values B_Ed,max = (4/4) B_el = (2/3) B_pl = 0.667 B_pl; B_Ed,max = (5/4)B_el= (5/6) B_pl= 0.833 B_pl; and B_Ed,max = (3/2) B_el = 1.0 B_pl, respectively, because B_pl= (3/2) B_el = 1.5 B_el. See the intersections of the ends of curves with the horizontal axis in Figure 22b.

The case with α = 6 is very close to the more exact solution by Streľbickaja [7], defined by Equations (46) and (48) being on the safe side. Equation (46) with α = 6 was recommended by Baláž [20,25] for standard purposes and finally was accepted in EN 1993-1-1:2022 [14] in the form:

$$M_{pl,y,B,Rd}=\sqrt{1-{\displaystyle \frac{B_{Ed}}{B_{pl,Rd}}}}{M}_{pl,y,Rd}={\rho }_{My,B}{M}_{pl,y,Rd}$$

(47)

The 66-year-old exact analytical solution (Equations (46) and (48)) by the Ukrainian lady Streľbickaja [7], giving the same values as the computer program DUENQ [22] and slightly simplified by Baláž [25] for standard purposes, finally replaced the incorrect solution [24] accepted for prEN 1993-1-1.

The plastic resistance of an IPE-section without strengthening under the interaction of two bending internal forces M_y,Ed and V_z,Ed and three torsion internal forces B_Ed, T_w,Ed, and T_t,Ed was solved in [26].

5.4. Steel Channel Cross-Section Profile Under Combination of Moment M_y,Ed and Bimoment B_Ed

Channel Cross-Section Resistance Calculation According to Streľbickaja [7]

Streľbickaja investigated channels in the same way as I-sections (see paragraph 5.3). The authors verified her analytical formulae by computer programs [22,23]. The comparison of results confirm again the geniality of Streľbickaja. In the following paragraph, the authors show the results related to all three kinds of channel resistances. The combination of the bending moment M_y,Ed and negative bimoment B_Ed was taken into account. In [7], both cases b1) and b2) (Figure 20b and Figure 21) were solved. The a) case with positive bimoment B_Ed (Figure 21) was not solved in [7].

$$W_{el,y}=t_f{b}_f{h}_f+\hspace{0.17em}\hspace{0.17em}{t}_w{h}_f^2/6=1.7\hspace{0.17em}x\hspace{0.17em}10.4\hspace{0.17em}x\hspace{0.17em}34.3\hspace{0.17em}+\hspace{0.17em}1.2\hspace{0.17em}x\hspace{0.17em}{34.3}^2/6=841.722{\mathrm{cm}}^3,$$

(48)

$$W_{pl,y,\max }=t_f{b}_f{h}_f+\hspace{0.17em}\hspace{0.17em}{t}_w{h}_f^2/4=1.7\hspace{0.17em}x\hspace{0.17em}10.4\hspace{0.17em}x\hspace{0.17em}34.3\hspace{0.17em}+\hspace{0.17em}1.2\hspace{0.17em}x\hspace{0.17em}{34.3}^2/4=959.731{\mathrm{cm}}^3,$$

(49)

$$M_{el,y,Rd}=W_{el,y}{f}_y/{\gamma }_{M0}=841.722\hspace{0.17em}{\mathrm{cm}}^{3}235\hspace{0.17em}\mathrm{MPa}/1.=197.805\hspace{0.17em}\mathrm{kNm}$$

(50)

$$M_{pl,y,Rd,\max }=W_{pl,y,\max }{f}_y/{\gamma }_{M0}=959.371\hspace{0.17em}{\mathrm{cm}}^{3}235\hspace{0.17em}\mathrm{MPa}/1.0=225.452\hspace{0.17em}\mathrm{kNm}$$

(51)

Position of the shear center in the elastic state:

$$a_{y,el}=t_f{h}_f^2{b}_f/\left(2W_{el,y}\right)=3.746\hspace{0.17em}\mathrm{cm}$$

(52)

Positions of the shear center in the plastic state without and with strengthening, respectively:

$$a_{y,pl}=t_f{h}_f^2{b}_f/\left(2W_{pl,y}\right)=3.287\hspace{0.17em}\mathrm{cm},\hspace{1em}{a}_{y,pl,s}=a_{y,el}\left(1+{\mathrm{\eta }}_y\right)W_{el.y}/\left(W_{pl,y}+{\mathrm{\eta }}_y{W}_{el,y}\right)=3.37\hspace{0.17em}\mathrm{cm}$$

(53)

The warping coordinates at the flange end and at the point of the flange–web intersection, respectively:

$${\omega }_1=\left(b_f-a_{y,el}\right)h_f/2=114.11{\mathrm{c}\mathrm{m}}^2, {\omega }_2=a_{y,el}{h}_f/2=64.25{\mathrm{c}\mathrm{m}}^2$$

(54)

$$I_w=\left[\left(b_f-a_{y,el}\right)t_f{\omega }_1^2+\left(t_f{h}_w/2+t_w{a}_{y,el}\right){\omega }_2^2\right]2/3 =172.4\mathrm{x}{10}^3{\mathrm{c}\mathrm{m}}^6$$

(55)

$$W_{el,w}=I_w/\max \left({\omega }_1,{\omega }_2\right)=1510\hspace{0.17em}{\mathrm{cm}}^4, B_{el,Rd}=W_{el,w}{f}_y/{\gamma }_{M0}=3.549\hspace{0.17em}{\mathrm{kNm}}^2$$

(56)

$$W_{pl,w}=\left[t_f{b}_f^2+t_f{b}_f{h}_f/2-t_w^2{h}_f^2/\left(16t_f\right)\right]h_f/4=2878\hspace{0.17em}{\mathrm{cm}}^4,B_{pl,Rd}=W_{pl,w}{f}_y/{\gamma }_{M0}=6.763\hspace{0.17em}{\mathrm{kNm}}^2$$

(57)

(i)

The elastic limit state is defined by the linear interaction formula (see Equation (35)).

From Figure 20b, the plastic resistances may be derived as a function of the distance u, defining the position of the neutral axis in the channel under a combination of the bending moment M_y,Ed and bimoment B_Ed. For the plastic limit state, the following are determined:

(ii)

Without strengthening η_y = 0 (Figure 19), formula (58) is formula (38.26) in [7]:

$$M_{y,Ed}\left(u\right)=\left[\left(2u-b\right)t_f{h}_f+t_w{h}_f^4/4\right]f_y,\hspace{1em}{B}_{Ed}\left(u\right)=M_{y,Ed}\left(u\right)a_{y,pl}+\left(b_f^2/2-u^2\right)t_f{h}_f{f}_y$$

(58)

(iii)

With strengthening η_y = 0.25 (Figure 19):

For b₁ case (Figure 20b):

$$M_{y,Ed,s}\left(u\right)=M_{y,Ed}\left(u\right)+\left[W_{el,y}-t_f{h}_f{b}_f^2/\left(2u\right)\right]{\eta }_y{f}_y$$

(59)

$$B_{Ed,s}=M_{y,Ed}\left(u\right)a_{y,pl,s}+\left(b_f^2/2-u^2\right)t_f{h}_f{f}_y+\left[2b_f^3/\left(3u\right)-b_f^2\right]{\eta }_y{f}_y{t}_f{h}_f/2$$

(60)

For b₂ case (Figure 20b):

$$M_{y,Ed,s}\left(u\right)=M_{y,Ed}\left(u\right)+\left[u{W}_{el,y}-t_f{h}_f{b}_f^2/2\right]{\eta }_y{f}_y/\left(b_f-u\right)$$

(61)

$$B_{Ed,s}=M_{y,Ed}\left(u\right)a_{y,pl,s}+\left(b_f^2/2-u^2\right)t_f{h}_f{f}_y+\left(2b_f/3-u\right){\eta }_y{f}_y{t}_f{h}_f{b}_f^2/\left[\left(b_f-u\right)2\right]$$

(62)

For u = b_f/2, Equations (61) and (62) are equal to Equations (59) and (60).

In this paper, the results of the analytical solution [7] for the mparison of results confirm agb) case (negative B_Ed, Figure 20b and Figure 21) are presented and compared with the results of the computer program [23].

The obtained results and comparisons of analytical [7] and computer results [23] are presented in graphical form in Figure 24. The comparisons of these results show acceptable agreements. For negative bimoment (b) case in Figure 20 and Figure 21), the final result is the simple interaction formula (63) derived by Baláž [20]. Formula (63) is limited to M_y,Ed ≤ M_pl,y,Rd, being on the safe side. The solid line on the right end is compared with “full black circles” obtained by the computer program [23] (see Figure 24). For the a) case (positive B_Ed, Figure 21), only the results of the computer program [23] are shown.

$${\displaystyle \frac{B_{Ed}}{B_{pl,Rk}}}=K_1{\left({\displaystyle \frac{M_{y,Ed}}{M_{pl,y,Rk}}}\right)}^2-K_2{\displaystyle \frac{M_{y,Ed}}{M_{pl,y,Rk}}}-1$$

(63)

where

$$K_1={\left(4a_{wf}+1\right)}^2/\left(16a_{wf}^2+8a_{wf}-1\right), K_2=2/\left(16a_{wf}^2+8a_{wf}-1\right),a_{wf}=A_f/A_w=b_f{t}_f/\left(h_f{t}_w\right)$$

(64)

Figure 24. A comparison of channel resistances—results of different ways of calculation. In the right part of the figure, there is a quadrant for the combination +M_yd,Ed with +B_Ed. In the left part of the figure, there is a quadrant for the combination +M_yd,Ed with −B_Ed. We can compare these two quadrants with the upper part of Figure 25, where there are all 4 quadrants including combinations −M_yd,Ed with −B_Ed and −M_yd,Ed with +B_Ed seen in the bottom part of the diagram in Figure 25.

Figure 25. UPE 360 section. Interaction diagrams of M_y,Ed + M_z,Ed + B_Ed. For midline model: N_{pl, Rd} = 1798 kN, M_pl,y,Rd = 197.80 kNm, M_pl,z,Rd = 44.24 kNm, B_pl,Rd = 6.795 kNm², M_pl,y,Rd,max = 225.45 kNm with B_Ed,conc = −1.0357 kNm², B_pl,Rd,max = −7.3304 kNm² with M_y,Ed,conc = 47.31 kNm. We can compare these values with the results in Table 12.

It is interesting to show (i) the value B_Ed,conc associated (concomitant) with the maximum bending moment M_pl,y,Rd,max and (ii) the value M_Ed,conc associated (concomitant) with the maximum bimoment B_pl,Rd,max > B_pl,Rd. See the diagram for the plastic resistance without strengthening in Figure 24. Both computer programs DUENQ [22] and QST-TSV-3Blech [23] gave very similar values, which are presented in Table 12.

Table 12. UPE 360 section plastic resistances without strengthening for negative bimoment B_Ed (case b) in Figure 20b and Figure 21). We can compare these results with values given in Figure 17a,b and Figure 25 [20].

Computer Programs	Maximum Internal Force	Corresponding Value
QST-TSV-3Blech [23]	Mpl,y,Rd,max = 225.452 kNm	BEd,Mmax = BEd,conc = −1.0144 kNm2
DUENQ [22]		BEd,Mmax = BEd,conc ≈ −1.0500 kNm2
QST-TSV-3Blech [23]	Bpl,Rd,max = │−7.12│ kNm2 > Bpl,Rd = 6.763 kNm2	MEd,Bmax = MEd,conc = 43.31 kNm
DUENQ [22]	Bpl,Ed,max ≈ │−6.68│ kNm2 > Bpl,Rd = 6.763 kNm2	MEd,Bmax = MEd,conc ≈ 43.31 kNm
QST-TSV-3Blech [23]	Mpl,y,Rd = 199.52 kNm	BEd,Mpl = BEd,conc = 0 kNm2
DUENQ [22]	Mpl,y,Rd = 192.00 kNm

For positive bimoment (a) case in Figure 21), the linear interaction formula defined by Equation (65) may be used (see left part of Figure 24), being on the safe side.

$${\displaystyle \frac{M_{\mathrm{y},\mathrm{E}\mathrm{d}}}{M_{\mathrm{p}\mathrm{l},\mathrm{y},\mathrm{R}\mathrm{d}}}}+{\displaystyle \frac{B_{\mathrm{E}\mathrm{d}}}{B_{\mathrm{p}\mathrm{l},\mathrm{R}\mathrm{d}}}}\le 1.0$$

(65)

For the investigated UPE 360, formula (65) may be used in the interval $-0.1\le B_{\mathrm{E}\mathrm{d}}/B_{\mathrm{p}\mathrm{l},\mathrm{R}\mathrm{d}}\le 1.0$, and formula (63) may be used after multiplication by factor −1.0 in the interval $-1.0\le B_{\mathrm{E}\mathrm{d}}/B_{\mathrm{p}\mathrm{l},\mathrm{R}\mathrm{d}}\le -0.1.$

In the frame of a large parametrical study, the other sizes of UPE-sections were investigated too. The obtained results confirmed the possibility to use the simple interaction formulae for channels defined by Equations (63) and (65), derived for standard purposes.

The elastic limit state is defined by the linear interaction Formula (35). The graphical interpretation of Equation (35) in Figure 24 may be described as follows:

$${\displaystyle \frac{M_{\mathrm{y},\mathrm{E}\mathrm{d}}}{M_{\mathrm{e}\mathrm{l},\mathrm{y},\mathrm{R}\mathrm{d}}}}\pm {\displaystyle \frac{B_{\mathrm{E}\mathrm{d}}}{B_{\mathrm{e}\mathrm{l},\mathrm{R}\mathrm{d}}}}\le 1.0, M_{\mathrm{e}\mathrm{l},\mathrm{y},\mathrm{R}\mathrm{d}}=M_{\mathrm{p}\mathrm{l},\mathrm{y}\mathrm{R}\mathrm{d}}, B_{\mathrm{e}\mathrm{l},\mathrm{R}\mathrm{d}}=0.525B_{pl,Rd}, {\displaystyle \frac{M_{\mathrm{y},\mathrm{E}\mathrm{d}}}{M_{\mathrm{p}\mathrm{l},\mathrm{y},\mathrm{R}\mathrm{d}}}}\pm {\displaystyle \frac{B_{\mathrm{E}\mathrm{d}}}{{0.525B}_{\mathrm{p}\mathrm{l},\mathrm{R}\mathrm{d}}}}\le 1.0$$

(66)

Table 13a,b, Table 14a,b and Table 15a,b contain formulae of all cross-sectional properties for (a) monosymmetric I-shaped sections, (b) channels, and (c) Z-sections, including relevant concomitant internal forces. The calculation of the approximate values is based on the midline model with thicknesses, taking into account (a) + (d) details of element connections (Figure 4). The formulae are presented in both (a) dimensionless form, enabling parametrical studies, and (b) formulae for direct design. They are proposed for educational purposes and the quick design of structures in practice as well. W_pl values relate to the fibers located in the midline, not to the surface of the section.

Table 13. Plastic properties of monosymmetric I-shaped section. $b_{ub}={\displaystyle \frac{b_u}{b_b}}$, $t_{ub}={\displaystyle \frac{t_u}{t_b}}$, $A_{ub}={\displaystyle \frac{A_u}{A_b}}={\displaystyle \frac{b_u{t}_u}{b_b{t}_b}}$, $A_{wb}={\displaystyle \frac{A_w}{A_b}}={\displaystyle \frac{h_f{t}_w}{b_b{t}_b}}$, $h_f=h-{\displaystyle \frac{t_u+t_b}{2}}$.

(a)
Relative Plastic Section Moduli	Plastic Section Moduli
(b)

Table 14. Plastic properties of channel section. $A_f=b_w{t}_f$, $A_w=h_f{t}_w$, ${\alpha }_{fw}={\displaystyle \frac{A_f}{A_w}}$ [27].

(a)
Relative Plastic Section Moduli	Plastic Section Moduli
(b)
(a)
Relative Plastic Section Moduli	Plastic Section Moduli
(b)

Table 15. Plastic properties of Z-section. $A_f=b_w{t}_f$, $A_w=h_f{t}_w$, ${\alpha }_{fw}={\displaystyle \frac{A_f}{A_w}}$ [28].

(a)
Relative Plastic Section Moduli ${\overline{W}}_{pl}\left({\alpha }_{fw}\right)$	Plastic Section Moduli $W_{pl}$
${\overline{W}}_{pl,y}={\displaystyle \frac{W_{pl,y}}{h_f{A}_w}}={\displaystyle \frac{1}{4}}+\left(\sqrt{2}-1\right){\alpha }_{fw}$	$W_{pl,y}=\left[{\displaystyle \frac{1}{4}}+\left(\sqrt{2}-1\right){\alpha }_{fw}\right]h_f{A}_w$
${\overline{W}}_{pl,y,\max }={\displaystyle \frac{W_{pl,y,\max }}{h_f{A}_w}}={\displaystyle \frac{1}{4}}+{\alpha }_{fw}$	$W_{pl,y,\max }=\left({\displaystyle \frac{1}{4}}+{\alpha }_{fw}\right)h_f{A}_w$
${\overline{W}}_{pl,z}={\displaystyle \frac{W_{pl,z}}{b_w{A}_f}}={\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{1}{2{\alpha }_{fw}}}-{\displaystyle \frac{1}{16{\alpha }_{fw}^2}}\right)$	$W_{pl,z}={\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{1}{2{\alpha }_{fw}}}-{\displaystyle \frac{1}{16{\alpha }_{fw}^2}}\right)b_w{A}_f$
${\overline{W}}_{pl,z,\max }={\displaystyle \frac{W_{pl,z,\max }}{b_w{A}_f}}=1$	$W_{pl,z,\max }=b_w{A}_f$
${\overline{W}}_{pl,w}={\displaystyle \frac{W_{pl,w}}{b_w{h}_f{A}_f}}={\displaystyle \frac{\left(4{\alpha }_{fw}-1\right){\left(1+{\alpha }_{fw}\right)}^2}{9{\alpha }_{fw}^2\left(1+2{\alpha }_{fw}\right)}}$,$2{\alpha }_{fw}\ge 1$ ${\overline{W}}_{pl,w}={\displaystyle \frac{W_{pl,w}}{b_w{h}_f{A}_f}}={\displaystyle \frac{1}{2}}$, if $2{\alpha }_{fw}\le 1$	$W_{pl,w}={\displaystyle \frac{\left(4{\alpha }_{fw}-1\right){\left(1+{\alpha }_{fw}\right)}^2}{9{\alpha }_{fw}^2\left(1+2{\alpha }_{fw}\right)}}{b}_w{h}_f{A}_f$, if $2{\alpha }_{fw}\ge 1$ $W_{pl,w}={\displaystyle \frac{1}{2}}{b}_w{h}_f{A}_f$, if $2{\alpha }_{fw}\le 1$
${\overline{W}}_{pl,w,\max }={\displaystyle \frac{W_{pl,w,\max }}{b_w{h}_f{A}_f}}={\left({\displaystyle \frac{1+{\alpha }_{fw}}{1+2{\alpha }_{fw}}}\right)}^2$	$W_{pl,w,\max }={\left({\displaystyle \frac{1+{\alpha }_{fw}}{1+2{\alpha }_{fw}}}\right)}^2{b}_w{h}_f{A}_f$
Combination of maximum plastic bimoment $B_{pl,Ed,\max }$with concomitant axial force $N_{Ed,conc}$: $B_{pl,Ed,\max }=W_{pl,w,\max }{f}_y$, $N_{Ed,conc}={\displaystyle \frac{1}{{\left(1+2{\alpha }_{fw}\right)}^2}}A{f}_y$
Maximum plastic bending moment about the y-y axis $M_{pl,y,Ed,\max }=W_{pl,y,\max }{f}_y$ is concomitant with the maximum plastic bending moment about the z-z axis $M_{pl,z,Ed,\max }=W_{pl,z,\max }{f}_y$.
(b)
${\overline{W}}_{el,w,f}={\displaystyle \frac{W_{el,w,f}}{b_w{A}_f}}={\displaystyle \frac{2+{\alpha }_{fw}}{6\left(1+{\alpha }_{fw}\right)}}$, $W_{el,w,f}={\displaystyle \frac{2+{\alpha }_{fw}}{6\left(1+{\alpha }_{fw}\right)}}{b}_w{h}_f{A}_f$, ${\overline{W}}_{el,w,we}={\displaystyle \frac{W_{el,w,we}}{b_w{A}_f}}={\displaystyle \frac{2+{\alpha }_{fw}}{6}}$, $W_{el,w,we}={\displaystyle \frac{2+{\alpha }_{fw}}{6}}{b}_w{h}_f{A}_f$, Note: indexes w—warping (except bw, ywu, ywb); we—web. $\alpha ={\displaystyle \frac{1}{2}}$atan$\left({\displaystyle \frac{2I_{yz}}{I_z-I_y}}\right)$, $u\left(s\right)=y\left(s\right)\cos \left(\alpha \right)+z\left(s\right)\sin \left(\alpha \right)$, $v\left(s\right)=z\left(s\right)\cos \left(\alpha \right)-y\left(s\right)\sin \left(\alpha \right)$, $I_u={\displaystyle \frac{I_y+I_z}{2}}+{\displaystyle \frac{1}{2}}\sqrt{{\left(I_y-I_z\right)}^2+4I_{yz}^2}$, $I_v={\displaystyle \frac{I_y+I_z}{2}}-{\displaystyle \frac{1}{2}}\sqrt{{\left(I_y-I_z\right)}^2+4I_{yz}^2}$, $C_1=2{\alpha }_{fw}{\left({\displaystyle \frac{b_w}{h_f}}\right)}^2$, $C_2={\displaystyle \frac{1}{4}}\left(1+6{\alpha }_{fw}\right)$, $I_u={\displaystyle \frac{1}{6}}\left[C_1+C_2+\sqrt{C_1^2+C_2^2+\left(3{\alpha }_{fw}^2-{\alpha }_{fw}\right){\left({\displaystyle \frac{b_w}{h_f}}\right)}^2}\right]h_f^2{A}_w$, $W_{el,u}={\displaystyle \frac{I_u}{v_{\max }}}$, $I_v={\displaystyle \frac{1}{6}}\left[C_1+C_2-\sqrt{C_1^2+C_2^2+\left(3{\alpha }_{fw}^2-{\alpha }_{fw}\right){\left({\displaystyle \frac{b_w}{h_f}}\right)}^2}\right]h_f^2{A}_w$, $W_{el,v}={\displaystyle \frac{I_v}{u_{\max }}}$

6. Conclusions

The advantages of plastic analysis and assumptions for plastic analysis applications are explained together with the newest metal Eurocodes requirements for the global plastic analysis of steel structures according to EN 1993-1-1:2022 [14] and of aluminum structures according to EN 1999-1-1:2023 [15].

There is necessity to distinguish among three kinds of plastic section moduli. They are calculated (a) without considering bimoment B_Ed as a constraint W_pl,nB; (b) considering bimoment B_Ed as a constraint W_pl; and (c) considering maximum values W_pl,max. These are explained for the first time ever in Section 3 and in the several illustrative numerical examples provided.

The results of this large parametrical study focused on 18 metal thin-walled cross-sections with various shapes, namely doubly symmetric, monosymmetric with an axis of symmetry z, monosymmetric with an axis of symmetry y, point-symmetric, and asymmetric open and closed cross-sections, are given in Table 10 and Table 11. The values of different plastic section moduli are obtained by different ways of calculation using our own computer programs (a) created in MathCad 15, (b) freeware https://laboratoriosvirtuales.upv.es/webapps/thinwallsectiongeneral.html, (c) freeware https://labmatlab-was.upv.es/webapps/home/thinwallsectionopenclosed.html (accessed on 15 November 2024), and with (d) dimensionless formulae for I-shaped, channels, and Z-sections (Table 13, Table 14 and Table 15) also contain for the first time ever all three kinds of plastic section moduli, which are not possible to calculate by any commercial programs. More details about the methodology and development of the above freeware may be found in our previous papers [29,30,31]. Comparisons of values show excellent agreements and correctness of results. Two of the above freeware allow us to solve for the first time ever the plastic design properties of thin-walled cross-sections with any open, quasi-closed, or closed shape with two, one, or without axis symmetry under any combination of eight internal forces, which are N_Ed, M_y,Ed, M_z,Ed, B_Ed, V_z,Ed, V_y,Ed, T_t,Ed, and T_w,Ed.

Altogether, 15 interaction diagrams present results valid for various shapes of cross-sections under combinations of three internal forces, namely N_Ed + M_y,Ed + M_z,Ed or M_y,Ed + B_Ed + M_z,Ed. They are a product of own freeware, which can be found in the blog https://antonioagueroramonllin.blogs.upv.es/ (accessed on 15 November 2024).

Eurocode EN 1993-1-1:2022 [14] offers only the possibility to perform the plastic design of limited simple shapes of cross-sections under combinations of N_Ed + M_y,Ed, N_Ed + M_z,Ed or N_Ed + M_y,Ed + M_z,Ed. The authors identified important problems in the Eurocode rules related to I-shaped sections under combinations of N_Ed + M_y,Ed and N_Ed + M_z,Ed and provided suggestions on how to solve these problems. Instead of approximate Eurocode formulae valid for I-shaped sections under combinations of N_Ed + M_y,Ed, the exact formulae (24)–(27) are proposed. The draft of prEN 1993-1-1 contained Mirambell’s formula (45) [24] for I-shaped sections under combinations of M_y,Ed + B_Ed, which was incorrect. In the Eurocode EN 1993-1-1:2022 [14], it was replaced by the correct Baláž’s formula (47) based on Streľbickaja’s theoretical and experimental results [7].

Eurocodes do not offer the possibility to perform the plastic design of channels under the very frequent combination of bending and torsion internal forces M_y,Ed + B_Ed. Baláž proposes for this combination simple formulae convenient for standard purposes for both (a) positive bimoment B_Ed (65) and (b) negative bimoment B_Ed (63). See Figure 24 or [20]. This proposal was verified by three different computer programs, namely by (a) DUENQ [22], (b) QST-TSV-3Blech [23], (c) the interaction diagram in Figure 25 (see also Table 12). It is the first time that designers may design channels under the combination of a moment and bimoment with very simple formulae. The influence of material strengthening is also shown for the first time ever for both I-profiles and channel sections (Figure 19 and Figure 20).

Appendix A presents a detailed example of very useful MathCad program by Torsten Höglund for the calculation of the plastic section modulus of point-symmetric closed oblique cross-sections. It can serve as a guide for any other shape of cross-section.

The future direction of our research will focus on the influence of shear on cross-section plastic resistances with any shape under any combination of eight internal forces.