# A New Iterative Design Strategy for Steel Frames Modelled by Generalised Multi-Stepped Beam Elements

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## Abstract

**:**

## 1. Introduction

- (i)
- create a preset zone of the beam element in which plastic deformations develop, leaving the remaining part of the beam element in the elastic range;
- (ii)
- design the device such that its flexural stiffness and resistance can be suitably as-signed while remaining independent of each other, so avoiding any stiffness variation in the involved beam element.

## 2. Optimal Design Strategy

#### 2.1. Optimal Design Problem Formulation: First Step

- ${\mathcal{l}}_{i}$ $\left(i=l,r,b\right)$ length of the columns (left, right) and of the beam;
- ${b}_{i}$ $\left(i=l,r,b\right)$ cross-section width of the columns (left, right) and of the beam;
- ${h}_{i}$ $\left(i=l,r,b\right)$ cross-section depth of the columns (left, right) and of the beam;
- ${t}_{wi}$ $\left(i=l,r,b\right)$ web thickness of the columns (left, right) and of the beam;
- ${f}_{y}$ material yield stress;
- ${\xi}_{\mathcal{l}im}$ maximum admissible horizontal drift;
- ${p}_{z}$ uniformly distributed dead load;
- ${\mathit{F}}^{T}=\left|\begin{array}{cc}{F}_{A}& {F}_{B}\end{array}\right|$ cyclic load vector;
- ${c}_{c}$ cyclic load multiplier for serviceability limit state conditions;
- ${c}_{F}$ load multiplier for ultimate limit state conditions.

#### 2.2. Optimal Design Problem Formulation: Second Step

- ${\mathcal{l}}_{l1}$, ${\mathcal{l}}_{l2}$, ${\mathcal{l}}_{l3}$, ${\mathcal{l}}_{r1}$, ${\mathcal{l}}_{r2}$, ${\mathcal{l}}_{r3}$, ${\mathcal{l}}_{b1}={\mathcal{l}}_{b5}$, ${\mathcal{l}}_{b2}={\mathcal{l}}_{b4}$, ${\mathcal{l}}_{b3}$;
- ${b}_{l1}={b}_{l3}={b}_{lo}$, ${b}_{r1}={b}_{r3}={b}_{ro}$, ${b}_{b1}={b}_{b3}={b}_{b5}={b}_{bo}$;
- ${h}_{l1}={h}_{l3}={h}_{lo}$, ${h}_{r1}={h}_{r3}={h}_{ro}$, ${h}_{b1}={h}_{b3}={h}_{b5}={h}_{bo}$;
- ${t}_{wl1}={t}_{wl2}={t}_{wl3}={t}_{wl}$, ${t}_{wr1}={t}_{wr2}={t}_{wr3}={t}_{wr}$;
- ${t}_{wb1}={t}_{wb2}={t}_{wb3}={t}_{wb4}={t}_{wb5}={t}_{wb}$.

- ${b}_{l2}$, ${b}_{r2}$, ${b}_{b2}$, ${b}_{b4}$, ${t}_{fl1}={t}_{fl3}={t}_{fls},{t}_{fl2},{t}_{fr1}={t}_{fr3}={t}_{frs},{t}_{fr2}$,${t}_{fb1}={t}_{fb3}={t}_{fb5}={t}_{fbs},{t}_{fb2},{t}_{fb4},{\mathit{\xi}}_{c}^{k}$,

## 3. Multi-Step Beam Element Model Formulation

## 4. Application

^{®}14.0 environment, obtaining the optimal thicknesses reported in Table 1 and the related (minimum) volume ${V}_{opt}=0.1171$ m

^{3}, which determines a volume percentage reduction $\left({V}_{HEB240}-{V}_{opt}\right)/{V}_{HEB240}=15\%$. The time for conducting the proposed optimization procedures with a HP Intel Core i7-10700 CPU @2.90GHz equipped with 16 GB RAM is less than 2 s. It is easy to observe that even the volume of the structure realised with HEB220 turns out to be greater than the obtained optimal one (${V}_{HEB220}=0.1184$ m

^{3}).

- ${\mathcal{l}}_{l1}={\mathcal{l}}_{l2}={\mathcal{l}}_{r1}={\mathcal{l}}_{r2}=220\mathrm{m}\mathrm{m}$;
- ${\mathcal{l}}_{l3}={\mathcal{l}}_{r3}=3560\mathrm{m}\mathrm{m}$;
- ${\mathcal{l}}_{b1}={\mathcal{l}}_{b5}={\mathcal{l}}_{b2}={\mathcal{l}}_{b4}=220\mathrm{m}\mathrm{m}$;
- ${\mathcal{l}}_{b3}=4120\mathrm{m}\mathrm{m}$;
- ${b}_{lo}={b}_{ro}={b}_{bo}=220\mathrm{m}\mathrm{m}$;
- ${h}_{lo}={h}_{ro}={h}_{bo}=220\mathrm{m}\mathrm{m}$;
- ${t}_{wl}={t}_{wr}={t}_{wb}=9.5\mathrm{m}\mathrm{m}$.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${a}_{i}\left(i=l,b,r\right)$ | coefficients depending on the shape of the yield boundary domain |

${b}_{i}$ $\left(i=l,r,b\right)$ | cross-section width of the columns (left, right) and of the beam |

${c}_{c}$ | cyclic load multiplier for serviceability limit state conditions |

${c}_{F}$ | load multiplier for ultimate limit state conditions |

${f}_{y}$ | material yield stress |

${h}_{i}$ $\left(i=l,r,b\right)$ | cross-section depth of the columns (left, right) and of the beam |

$\mathbf{k}\left({x}_{iG,k}\right)$ | stiffness matrix of the generic Gauss cross-section |

${\mathcal{l}}_{i}$ $\left(i=l,r,b\right)$ | length of the columns (left, right) and of the beam |

${n}_{f}$ | number of strips |

${n}_{iG}$ | number of Gauss i-th cross-sections |

${p}_{z}$ | uniformly distributed dead load |

${\mathbf{q}}_{e}$ | vector of nodal displacements |

${t}_{fi}$ $\left(i=l,r,b\right)$ | flange thicknesses of the columns (left, right) and of the beam |

${t}_{wi}$ $\left(i=l,r,b\right)$ | web thickness of the columns (left, right) and of the beam |

${u}_{x}\left(x\right)$ | axial displacement function |

${u}_{z}\left(x\right)$ | transversal deflection function |

${w}_{k}$ | weight of each Gauss point |

${x}_{iG,k}$ | abscissa of Gauss k-th cross-section |

${A}_{i}$ $\left(i=l,r,b\right)$ | cross-section area of the columns (left, right) and of the beam |

${A}_{f}$ | area of strip |

$\mathbf{B}\left(x;{\beta}_{ix,k},{\beta}_{iz,k}\right)$ | matrix dependent on the stiffness decay parameters |

$\mathit{C}$ | compatibility matrix |

$\mathit{D}$ | internal stiffness matrix of beam element |

${\mathit{D}}_{m}$ | internal stiffness matrix of the multi-step beam element |

$E$ | Young’s modulus of the material |

${\mathit{F}}^{T}=\left|\begin{array}{cc}{F}_{A}& {F}_{B}\end{array}\right|$ | cyclic load vector |

${F}_{A},{F}_{B}$ | perfect cyclic loads |

${I}_{i}\left(i=l,b,r\right)$ | moment of inertia of the columns (left, right) and the beam |

${\mathbf{K}}_{e}\left({\beta}_{ix,k},{\beta}_{iz,k}\right)$ | multi-step element stiffness matrix |

${N}_{i,1}\left(x\right),\dots ,{N}_{i,6}\left(x\right)$ | axial/transversal displacement shape functions |

${\mathit{Q}}^{*}$ | perfectly clamped generalised elastic stress response vector |

${\mathbf{Q}}_{e}$ | vector of nodal forces |

$U\left(x-{x}_{j}\right)$ | Heaviside (unit step) generalised function |

$V$ | volume of the structure |

${W}_{i}^{p}$ $\left(i=l,r,b\right)$ | cross-section plastic modulus of columns and beam |

${\alpha}_{ix,j}$ | normalized variation of the current axial stiffness of the columns and beam segments |

${\alpha}_{iz,j}$ | normalized variation of the current flexural stiffness of the columns and beam segments |

${\beta}_{ix,k}$ | parameters representing the axial plastic state of the k-th Gauss cross-section |

${\beta}_{iz,k}$ | parameters representing the flexural plastic state of the k-th Gauss cross-section |

$\mathsf{\alpha}\left({z}_{f}\right)$ | vector, dependent on the distance ${z}_{f}$ |

${\gamma}_{br}$ | safety coefficient |

${\gamma}_{y/e}$ | coefficient representing the ratio between yield limit bending moment and elastic limit bending moment |

${\epsilon}_{x}\left(x\right)$ | axial strain |

$\kappa $ | function for axial/transversal displacement shape functions |

${\xi}_{\mathcal{l}im}$ | maximum admissible horizontal drift |

${\xi}_{max}$ | horizontal drift in serviceability limit state conditions |

${\mathit{\xi}}_{c}^{k}$ $\left(k=+,-\right)$ | the vectors of nodal displacements |

${a}_{1},{a}_{2,}{c}_{1},{c}_{2},{c}_{3},{c}_{4}$ | integration constants |

${g}_{2}\left(x;{\beta}_{ix,k}\right),{g}_{3}\left(x;{\beta}_{ix,k}\right),$ ${f}_{3}\left(x;{\beta}_{iz,k}\right),{f}_{4}\left(x;{\beta}_{iz,k}\right),{f}_{5}\left(x,{\beta}_{iz,k}\right)$ | functions for the multi-step beam model formulation |

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**Figure 1.**Steel frame (green: plate; red: bolts; blue: reduced section portion): (

**a**) overall view; (

**b**) detail 1, sketch of the beam-column connection; (

**c**) detail 2, sketch of the column base.

**Figure 3.**Geometric characteristic of the multi-stepped beam element frame: (

**a**) lengths of the different portions of the beam elements; (

**b**) specific dimensions of the portions at the left column base; (

**c**) specific dimensions of the portions at the left beam end.

**Figure 7.**Yield domains (black point element AC, red point element AB, blue point element BD): (

**a**) frame made of HEB 220 profiles; (

**b**) frame made of HEB 240 profiles.

**Figure 9.**Yield domains of columns and beam for the frame designed in the first step: (

**a**) common yield domain of the columns, where the black point indicates the more stressed cross-section of the left column, the blue point indicates the more stressed cross-section of the right column; (

**b**) yield domain of the beam, where the red point indicates the more stressed cross-section of the beam.

**Figure 11.**Dimensionless results for the frame designed in the second step in ULS conditions (black line, yield domain; red line, elastic domain of the end sections): (

**a**) black point, element AC; blue point, element BD; (

**b**) red point, element AB.

**Figure 13.**Response to cyclic loads of frame equipped with a standard profile available on the market.

${\mathit{t}}_{\mathit{f}\mathit{l}}$ | ${\mathit{t}}_{\mathit{f}\mathit{r}}$ | ${\mathit{t}}_{\mathit{f}\mathit{b}}$ |
---|---|---|

18.361 | 18.361 | 13.320 |

Left Column | ||||||

${b}_{lo}$ | ${h}_{lo}$ | ${t}_{wl}$ | ${t}_{fls}$ | ${b}_{l2}$ | ${h}_{l2}$ | ${t}_{fl2}$ |

220 | 220 | 9.50 | 21.791 | 139.819 | 216.570 | 18.361 |

Right column | ||||||

${b}_{ro}$ | ${h}_{ro}$ | ${t}_{wr}$ | ${t}_{frs}$ | ${b}_{r2}$ | ${h}_{r2}$ | ${t}_{fr2}$ |

220 | 220 | 9.50 | 21.791 | 139.819 | 216.570 | 18.361 |

Beam | ||||||

${b}_{bo}$ | ${h}_{bo}$ | ${t}_{wb}$ | ${t}_{fbs}$ | ${b}_{b2}$ | ${h}_{b2}$ | ${t}_{fb2}$ |

220 | 220 | 9.50 | 16.339 | 162.188 | 214.159 | 10.499 |

${b}_{b4}$ | ${h}_{b4}$ | ${t}_{fb4}$ | ||||

162.188 | 214.159 | 10.499 |

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## Share and Cite

**MDPI and ACS Style**

Benfratello, S.; Caddemi, S.; Palizzolo, L.; Pantò, B.; Rapicavoli, D.
A New Iterative Design Strategy for Steel Frames Modelled by Generalised Multi-Stepped Beam Elements. *Buildings* **2024**, *14*, 2155.
https://doi.org/10.3390/buildings14072155

**AMA Style**

Benfratello S, Caddemi S, Palizzolo L, Pantò B, Rapicavoli D.
A New Iterative Design Strategy for Steel Frames Modelled by Generalised Multi-Stepped Beam Elements. *Buildings*. 2024; 14(7):2155.
https://doi.org/10.3390/buildings14072155

**Chicago/Turabian Style**

Benfratello, Salvatore, Salvatore Caddemi, Luigi Palizzolo, Bartolomeo Pantò, and Davide Rapicavoli.
2024. "A New Iterative Design Strategy for Steel Frames Modelled by Generalised Multi-Stepped Beam Elements" *Buildings* 14, no. 7: 2155.
https://doi.org/10.3390/buildings14072155