#
New Interaction Formulae for Plastic Resistances of Z-Sections under Combinations of Bending Moments M_{y,Ed}, M_{z,Ed} and Bimoment B_{Ed}

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## Abstract

**:**

## Highlights

**What are the main findings?**

- -
- Exact interaction curves for the plastic resistances of Z-sections taking into account the combination of three internal forces, i.e., M
_{y,Ed}, M_{z,Ed}and B_{Ed}, are presented. - -
- Approximate interaction formulas for Z-section plastic resistances taking into account three internal forces, i.e., M
_{y,Ed}, M_{z,Ed}and B_{Ed}, is presented.

## Abstract

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

_{z,Ed}and bimoment B

_{Ed}, were created and analyzed. The exact interaction curves obtained by linear programming have enabled us to create and verify the proposed approximate interaction formulae. An interaction formula that takes into account these three internal forces is missing in the Eurocodes. A large parametric study was performed for rolled Z profiles. The differences between the values of the approximate interaction formulae and exact interaction curves were analyzed and summarized. The importance of correct analyses of Z-sections in Part 3 is described and examples of incorrect calculations in many publications are collected and corrected. Several researchers have analyzed plastic resistances of I-, H-, T- and U-sections under the combination of internal forces, but nobody has studied Z-sections which are frequently used as purlins. Another motivation is to show how to calculate the properties and normal stresses in the sections without axes of symmetry, which are frequently calculated in incorrect ways, when using non-principal axes of the section.

## 1. Introduction

_{y,Ed}and M

_{z,Ed}and bimoment B

_{Ed}.

_{y,Ed}in the bottom section fibers, M

_{z,Ed}, in the right section fibers, N

_{Ed}, and bimoment B

_{Ed}in the web of the Z-section.

#### 1.1. Overview of the Former Investigations

_{y}, M

_{z}, B, V

_{z}, V

_{y}, T

_{w}and T

_{t}. Designers may use a free computer program [13] based on the PIF (TSV) method.

#### 1.2. Calculation of the Factor ξ

_{Ed}, ξM

_{y,Ed}, ξM

_{z,Ed}, ξB

_{Ed}}, the lower bound theorem can be used.

- Method A, which divides the section into elements. The unknowns are the stresses in each element;
- Method B, which divides the section into three parts (two flanges and the web). In each part, the unknowns are the axial force and the bending moment about the strong axis.

#### 1.2.1. Method A (According to Osterrieder et al. [24])

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

**Step 2.**Considering linear constraints with Equations (1)–(5).

**Step 3.**Calculation of the maximum value of the factor ξ:

_{z,Ed}and V

_{y,Ed}, warping T

_{w,Ed}and St. Venant torsional moments T

_{t,Ed}, which cause the shear stresses τ

_{xyi}and τ

_{xzi}in each element. Then, the nonlinear constraint can be expressed by von Mises criterion (this criterion can be also linearized [24]).

#### 1.2.2. Method B (Kindmann and Frickel [12])

_{fl}, bottom flange—index B

_{fl}and web—index w. In Method B, the following quantities are calculated: factor ξ to accomplish plastic resistance, axial forces N {${N}_{Tfl,Ed}$; ${N}_{Bfl,Ed}$; ${N}_{w,Ed}$} and bending moments M in each Z-section part {${M}_{Tfl,z,Ed}$; ${M}_{Bfl,z,Ed}$; ${M}_{w,y,Ed}$}.

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

**Step 2.**Considering constraints.

_{fl}is the area of the flange,

_{w}is the area of the web.

**Step 3.**Calculation of the maximum value of the factor ξ:

#### 1.3. Research Significance

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

_{z,Ed}, and the bimoment B

_{Ed}are proposed. They are derived and presented in Section 3.

## 2. Interaction Formulae of the Authors’ Proposal (Approximate Curves)

_{Ed}= 0 for all combinations with M

_{y,Ed}and M

_{z,Ed}, and (b) to the vertical axis M

_{z,Ed}= 0 only for M

_{y,Ed}= 0.

## 3. Importance of Z-Section Analysis

## 4. Conclusions

_{z}, M

_{y}and B in an approximate way.

_{y}, M

_{z}and B is justified in the introduction. The exact curves are obtained for different values of M

_{y}, showing the interaction between M

_{z}and B.

_{Ed}, M

_{y,Ed}, M

_{z,Ed}and B

_{Ed}, causing normal stresses, and later all eight internal forces causing normal and shear stresses.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature: (Alphabetically)

B_{Ed} | design value of the bimoment |

B_{pl,Rd} | design value of the plastic bimoment resistance |

b | width of a cross-section |

f_{y} | yield stress |

h | depth of a cross-section |

h_{w} | web depth |

M_{Bfl,z,Ed} | design value of the bending moment of the bottom flange about the z-axis |

M_{Bfl,z,Rd} | design value of the plastic bending moment resistance of the bottom flange about the z-axis |

M_{fl,z,Rd} | design value of the plastic bending moment resistance of the flange about the z-axis |

M_{pl,y,Rd}, M_{pl,Rd} | design value of the plastic moment resistance about the y-axis |

M_{pl,z,Rd} | design value of the plastic moment resistance about the z-axis |

M_{Tfl,z,Ed} | design value of the bending moment of the top flange about the z-axis |

M_{Tfl,z,Rd} | design value of the plastic bending moment resistance of the top flange about the z-axis |

M_{w,pl,y,Rd} | design value of the plastic moment resistance of the web about the y-axis |

M_{w,y,Ed} | design value of the bending moment of the web about the y-axis |

M_{w,y,Rd} | design value of the plastic bending moment resistance of the web about the y-axis |

M_{y,Ed} | design value of the bending moment about the y-axis |

M_{z,Ed} | design value of the bending moment about the z-axis |

N_{Bfl,Ed} | design value of the axial force of the bottom flange |

N_{Bfl,Rd} | design value of the plastic axial force resistance of the bottom flange |

N_{Tfl,Ed} | design value of the axial force of the top flange |

N_{Tfl,Rd} | design value of the plastic axial force resistance of the top flange |

N_{w,Ed} | design value of the axial force of the web |

N_{w,Rd} | design value of the plastic axial force resistance of the web |

T_{t,Ed} | design value of internal St. Venant torsional moment |

T_{w,Ed} | design value of internal warping torsional moment |

t_{f} | flange thickness |

t_{w} | web thickness |

V_{y,Ed} | design value of the shear force in direction of the y-axis |

V_{z,Ed} | design value of the shear force in direction of the z-axis |

y, z | section coordinates along the y- and z-axes |

γ_{M0} | partial safety factor for the resistance of cross-section whatever the class is |

σ | normal stress |

τ | shear stress |

ω | warping function |

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**Figure 1.**Lipped Z-section: (

**a**) local buckling in compression; (

**b**) distortion buckling in compression.

**Figure 3.**Resistances of Z200 section under the combination of ${N}_{Ed},\hspace{1em}{B}_{Ed}$ and ${M}_{z,Ed}$. Four small schemes indicate sign conventions of ${N}_{Ed}$ and ${B}_{Ed}$. Diagram will be the same if the sign of ${M}_{z,Ed}$ will be changed. In fact bimoments besides bending moments in flanges produce an axial force in the flanges that is in equilibrium with the axial force in web.

**Figure 4.**Exact (solid curves) and proposed approximate (dashed lines) interaction diagrams. Results for a Z—type section with the flange-to-web-area ratio ${\alpha}_{fw}=0.53$ and for the concomitant y—axis bending moment of ${m}_{y}=\{0.00,0.20,0.40,0.60,0.80\}$.

**Figure 5.**Exact (solid curves) and proposed approximate (dashed lines) interaction diagrams. Results for a Z—type section with the flange-to-web-area ratio ${\alpha}_{fw}=1.15$ and for the concomitant y—axis bending moment of ${m}_{y}=\{0.00,0.20,0.40,0.60,0.80\}$.

**Figure 6.**Exact (solid curves) and proposed approximate (dashed lines) interaction diagrams. Results for a Z—type section with the flange-to-web-area ratio ${\alpha}_{fw}=0.82$ and for the concomitant y—axis bending moment of ${m}_{y}=\{0.00,0.20,0.40,0.60,0.80\}$.

Relative Dimensionless Quantities:$\mathit{f}\left({\mathit{\alpha}}_{\mathit{f}\mathit{w}}\right)$ | Cross-Section Properties |
---|---|

${\overline{W}}_{el,y}=\frac{{W}_{el,y}}{{h}_{f}{A}_{w}}=\frac{2+3{\alpha}_{fw}}{12}$ | ${W}_{el,y}=\frac{2+3{\alpha}_{fw}}{12}{h}_{f}{A}_{w}$ |

${\overline{W}}_{el,z}=\frac{{W}_{el,z}}{{b}_{w}{A}_{f}}=\frac{2+3{\alpha}_{fw}}{3\left(1+3{\alpha}_{fw}\right)}$ | ${W}_{el,z}=\frac{2+3{\alpha}_{fw}}{3\left(1+3{\alpha}_{fw}\right)}{b}_{w}{A}_{f}$ |

${\overline{W}}_{el,w}=\frac{{W}_{el,z}}{{b}_{w}{A}_{f}}=\frac{2+{\alpha}_{fw}}{6\left(1+{\alpha}_{fw}\right)}$ | ${W}_{el,w}=\frac{2+{\alpha}_{fw}}{6\left(1+{\alpha}_{fw}\right)}{b}_{w}{h}_{f}{A}_{f}$ |

${\overline{W}}_{pl,y}=\frac{{W}_{pl,y}}{{h}_{f}{A}_{w}}=\frac{1}{4}+\left(\sqrt{2}-1\right){\alpha}_{fw}$ | ${W}_{pl,y}=\left[\frac{1}{4}+\left(\sqrt{2}-1\right){\alpha}_{fw}\right]{h}_{f}{A}_{w}$ |

${\overline{W}}_{pl,y,\mathrm{max}}=\frac{{W}_{pl,y,\mathrm{max}}}{{h}_{f}{A}_{w}}=\frac{1}{4}+{\alpha}_{fw}$ | ${W}_{pl,y,\mathrm{max}}=\left(\frac{1}{4}+{\alpha}_{fw}\right){h}_{f}{A}_{w}$ |

${\overline{W}}_{pl,z}=\frac{{W}_{pl,z}}{{b}_{w}{A}_{f}}=\frac{1}{2}\left(1+\frac{1}{2{\alpha}_{fw}}-\frac{1}{16{\alpha}_{fw}^{2}}\right)$ | ${W}_{pl,z}=\frac{1}{2}\left(1+\frac{1}{2{\alpha}_{fw}}-\frac{1}{16{\alpha}_{fw}^{2}}\right){b}_{w}{A}_{f}$ |

${\overline{W}}_{pl,z,\mathrm{max}}=\frac{{W}_{pl,z,\mathrm{max}}}{{b}_{w}{A}_{f}}=1$ | ${W}_{pl,z,\mathrm{max}}={b}_{w}{A}_{f}$ |

${\overline{W}}_{pl,w}=\frac{{W}_{pl,w}}{{b}_{w}{h}_{f}{A}_{f}}=\frac{\left(4{\alpha}_{fw}-1\right){\left(1+{\alpha}_{fw}\right)}^{2}}{9{\alpha}_{fw}^{2}\left(1+2{\alpha}_{fw}\right)}$, if $2{\alpha}_{fw}\ge 1$ ${\overline{W}}_{pl,w}=\frac{{W}_{pl,w}}{{b}_{w}{h}_{f}{A}_{f}}=\frac{1}{2}$, if $2{\alpha}_{fw}\le 1$ | ${W}_{pl,w}=\frac{\left(4{\alpha}_{fw}-1\right){\left(1+{\alpha}_{fw}\right)}^{2}}{9{\alpha}_{fw}^{2}\left(1+2{\alpha}_{fw}\right)}{b}_{w}{h}_{f}{A}_{f}$, if $2{\alpha}_{fw}\ge 1$ ${W}_{pl,w}=\frac{1}{2}{b}_{w}{h}_{f}{A}_{f}$, if $2{\alpha}_{fw}\le 1$ |

${\overline{W}}_{pl,w,\mathrm{max}}=\frac{{W}_{pl,w,\mathrm{max}}}{{b}_{w}{h}_{f}{A}_{f}}={\left(\frac{1+{\alpha}_{fw}}{1+2{\alpha}_{fw}}\right)}^{2}$ | ${W}_{pl,w,\mathrm{max}}={\left(\frac{1+{\alpha}_{fw}}{1+2{\alpha}_{fw}}\right)}^{2}{b}_{w}{h}_{f}{A}_{f}$ |

Combination of ${B}_{pl,Ed,\mathrm{max}}$ and concomitant ${N}_{Ed,conc}$: ${B}_{pl,Ed,\mathrm{max}}={W}_{pl,w,\mathrm{max}}{f}_{y}$, ${N}_{Ed,conc}=\frac{1}{{\left(1+2{\alpha}_{fw}\right)}^{2}}A{f}_{y}$, $A={A}_{w}+2{A}_{f}$ | |

${A}_{f}={b}_{w}{t}_{f}$, ${A}_{w}={h}_{f}{t}_{w}$, ${\alpha}_{fw}=\frac{{A}_{f}}{{A}_{w}}$, ${\omega}_{w}=\frac{{\alpha}_{fw}}{2\left(1+2{\alpha}_{fw}\right)}{b}_{w}{h}_{f}$, ${\omega}_{f}=-\frac{1+{\alpha}_{fw}}{2\left(1+2{\alpha}_{fw}\right)}{b}_{w}{h}_{f}$ |

**Table 2.**Numerical values for diagrams in Figure 4, bounded by straight lines $A{m}_{z}+B\ge \left|{m}_{\omega}\right|$ and $C{m}_{z}+D\ge \left|{m}_{\omega}\right|$.

$${\mathit{\mu}=\mathit{M}}_{\mathit{y},\mathit{E}\mathit{d}}/{\mathit{M}}_{\mathit{p}\mathit{l},\mathit{y},\mathit{m}\mathit{a}\mathit{x}}$$
| |||||
---|---|---|---|---|---|

0.000 | 0.200 | 0.400 | 0.600 | 0.800 | |

$\mathsf{\Psi}\left(\mu \right)$ | 1.161 | 1.020 | 0.925 | 0.863 | 0.824 |

$\mathsf{\Phi}\left(\mu \right)$ | 1.000 | 0.810 | 0.616 | 0.419 | 0.215 |

$\mathsf{\Psi}(-\mu )$ | 1.161 | 1.372 | 1.698 | 2.229 | 3.185 |

$\mathsf{\Phi}(-\mu )$ | 1.000 | 1.180 | 1.327 | 1.376 | 1.146 |

A | 1.000 | 0.879 | 0.797 | 0.744 | 0.710 |

B | 1.000 | 1.014 | 0.986 | 0.937 | 0.874 |

C | −1.000 | −1.182 | −1.463 | −1.920 | −2.744 |

D | 1.000 | 0.906 | 0.648 | 0.039 | −1.402 |

${\mathit{W}}_{\mathit{e}\mathit{l}}$ | Section | Z30 | Z40 | Z50 | Z60 | Z80 | Z100 | Z120 | Z140 | Z160 | Z180 | Z200 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${W}_{el.y}$ | DIN 1027 | 3.97 | 6.75 | 10.5 | 14.9 | 27.3 | 44.4 | 67.0 | 96.6 | 132 | 178 | 230 |

Table 1 | 1.47 | 2.57 | 4.13 | 5.87 | 11.3 | 18.7 | 28.5 | 42.4 | 58.4 | 80.1 | 103.9 | |

${W}_{el.z}$ | DIN 1027 | 3.80 | 4.66 | 5.88 | 7.09 | 10.1 | 14.0 | 18.8 | 24.3 | 31.0 | 38.4 | 47.6 |

Table 1 | 2.28 | 2.89 | 3.76 | 4.55 | 6.74 | 9.46 | 12.8 | 16.9 | 21.7 | 27.4 | 33.9 |

Cross-section geometrical properties | ||||||||

$h$ | $b$ | ${t}_{w}$ | ${t}_{f}$ | ${h}_{f}$ | ${b}_{w}$ | ${A}_{f}$ | ${A}_{w}$ | $A$ |

mm | mm | mm | mm | mm | mm | cm^{2} | cm^{2} | cm^{2} |

200 | 80 | 10 | 13 | 187 | 75 | 9.75 | 18.70 | 38.20 |

Relative dimensionless quantities as a function of ${\alpha}_{fw}=\frac{{A}_{f}}{{A}_{w}}=0.5215$ | ||||||||

${\overline{W}}_{el,y}$ | ${\overline{W}}_{el,z}$ | ${\overline{W}}_{el,w}$ | ${\overline{W}}_{pl,y}$ | ${\overline{W}}_{pl,y,\mathrm{m}\mathrm{a}\mathrm{x}}$ | ${\overline{W}}_{pl,z}$ | ${\overline{W}}_{pl,z,\mathrm{m}\mathrm{a}\mathrm{x}}$ | ${\overline{W}}_{pl,w}$ | ${\overline{W}}_{pl,w,\mathrm{m}\mathrm{a}\mathrm{x}}$ |

0.297 | 0.463 | 0.276 | 0.466 | 0.771 | 0.865 | 1.000 | 0.5027 | 0.555 |

Physical properties of the cross-section | ||||||||

${W}_{el,y}$ | ${W}_{el,z}$ | ${W}_{el,w}$ | ${W}_{pl,y}$ | ${W}_{pl,y,\mathrm{m}\mathrm{a}\mathrm{x}}$ | ${W}_{pl,z}$ | ${W}_{pl,z,\mathrm{m}\mathrm{a}\mathrm{x}}$ | ${W}_{pl,w}$ | ${W}_{pl,w,\mathrm{m}\mathrm{a}\mathrm{x}}$ |

cm^{3} | cm^{3} | cm^{4} | cm^{3} | cm^{3} | cm^{3} | cm^{3} | cm^{4} | cm^{4} |

103.86 | 33.88 | 377.71 | 162.94 | 269.75 | 63.22 | 73.125 | 687.47 | 758.48 |

Bimoment–Axial Force Combination Example in Figure 3 | |||||
---|---|---|---|---|---|

${B}_{pl,\mathrm{E}\mathrm{d},\mathrm{m}\mathrm{a}\mathrm{x}}$ | ${B}_{pl,\mathrm{R}\mathrm{d}}$ | ${N}_{\mathrm{E}\mathrm{d},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}$ | ${N}_{pl,\mathrm{R}\mathrm{d}}$ | $\frac{{N}_{\mathrm{E}\mathrm{d},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}}{{N}_{pl,\mathrm{R}\mathrm{d}}}$ | $\frac{{B}_{pl,Ed,\mathrm{m}\mathrm{a}\mathrm{x}}}{{B}_{pl,\mathrm{R}\mathrm{d}}}$ |

kNm^{2} | kNm^{2} | kN | kN | - | - |

1.782 | 1.605 | 215.12 | 897.70 | 0.24 | 1.11 |

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## Share and Cite

**MDPI and ACS Style**

Agüero, A.; Baláž, I.; Koleková, Y.; Lázaro, M.
New Interaction Formulae for Plastic Resistances of Z-Sections under Combinations of Bending Moments M_{y,Ed}, M_{z,Ed} and Bimoment B_{Ed}. *Buildings* **2023**, *13*, 1778.
https://doi.org/10.3390/buildings13071778

**AMA Style**

Agüero A, Baláž I, Koleková Y, Lázaro M.
New Interaction Formulae for Plastic Resistances of Z-Sections under Combinations of Bending Moments M_{y,Ed}, M_{z,Ed} and Bimoment B_{Ed}. *Buildings*. 2023; 13(7):1778.
https://doi.org/10.3390/buildings13071778

**Chicago/Turabian Style**

Agüero, Antonio, Ivan Baláž, Yvona Koleková, and Mario Lázaro.
2023. "New Interaction Formulae for Plastic Resistances of Z-Sections under Combinations of Bending Moments M_{y,Ed}, M_{z,Ed} and Bimoment B_{Ed}" *Buildings* 13, no. 7: 1778.
https://doi.org/10.3390/buildings13071778