# An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Beam-Like Model

#### 2.1. Kinematics of Frames and Beams

#### 2.2. The Inter-Storey Shear and Torsional Stiffness

#### 2.3. The Equations of Motion of the Proposed Non-Uniform Beam-Like Model

#### 2.4. Eigen-Problem of the Non-Uniform Beam-Like Model

#### 2.5. Shear and Torsional Beam Stiffness Optimization

#### 2.6. Dynamic Response

## 3. Applications

^{2}, while all the remaining beams have a 110 × 23 cm

^{2}size. The columns were assumed fully fixed at the base. An equivalent floor thickness of 9.32 cm at each level was considered. The material properties were characterized by a Young’s modulus of 29,962 MPa, a Poisson ratio equal to 0.2 and a mass density of 25 kN/m

^{3}.

#### 3.1. Type 1: Building with Planar Asymmetry and Vertical Uniform Distribution of Mass and Stiffness

#### 3.2. Type 2: Building with Planar Symmetry and Vertical Non-Uniform Distribution of Mass and Stiffness

#### 3.3. Type 3: Building with Planar Asymmetry and Vertical Non-Uniform Distribution of Mass and Stiffness

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Conceptual representation, starting from (

**a**) the 3D structure; (

**b**) the generic kth floor (or inter-storey) up to the (

**c**) sub-beam element; and the (

**d**) the proposed beam-like model.

**Figure 6.**Four storey building benchmark (Type 1): (

**a**) column cross section plan, and (

**b**) the FEM model.

**Figure 7.**Four-storey benchmark. Modal shapes comparison between the (

**a**) FEM and (

**b**) beam-like models.

**Figure 8.**Four-storey benchmark. Santa Venerina earthquake. Displacements time history along the (

**a**) x and (

**b**) y direction.

**Figure 9.**Four-storey benchmark. L’Aquila earthquake. Displacements time history along the (

**a**) x and (

**b**) y direction.

**Figure 10.**Four-storey benchmark. Maxima floor displacements in the (

**a**) Santa Venerina and (

**b**) L’Aquila earthquakes.

**Figure 11.**Eight storey building benchmark (Type 2): (

**a**) the column cross section plan, and (

**b**) the FEM model.

**Figure 12.**Eight -storey benchmark. Modal shapes comparison between the (

**a**) FEM and (

**b**) beam-like models.

**Figure 13.**Eight-storey benchmark. Santa Venerina earthquake. Displacements time history along the (

**a**) x and (

**b**) y direction.

**Figure 14.**Eight-storey benchmark. L’Aquila earthquake. Displacements time history along the (

**a**) x and (

**b**) y direction.

**Figure 15.**Eight-storey benchmark. Maxima floor displacements in the (

**a**) Santa Venerina and (

**b**) L’Aquila earthquakes.

**Figure 16.**Six storey building benchmark (Model 2): (

**a**) the column cross section plan and (

**b**) the FEM model.

**Figure 17.**Six-storey benchmark. Modal shapes comparison between the (

**a**) FEM and (

**b**) beam-like models.

**Figure 18.**Six-storey benchmark. Santa Venerina earthquake. Displacements time history along the (

**a**) x and (

**b**) y direction.

**Figure 19.**Six-storey benchmark. L’Aquila earthquake. Displacements time history along the (

**a**) x and (

**b**) y direction.

**Figure 20.**Six-storey benchmark. Maxima floor displacements in (

**a**) Santa Venerina and (

**b**) L’Aquila earthquakes.

**Figure 21.**Six-storey benchmark. Maxima floor displacements in the L’Aquila earthquake for a different number of shape functions adopted in the Rayleigh–Ritz discretization.

Record | State | Date | Hour | Mw | Dist. (km) | PGA (cm/s ^{2}) | PGV (cm/s) | PGD (cm) | Lat. (°) | Long. (°) |
---|---|---|---|---|---|---|---|---|---|---|

Santa Venerina | Italy | 2018-12-26 | 02:19:17 | 4.9 | 4.5 | −547.932 (N) | 37.128 (N) | 5.696 (N) | 37.644 | 15.116 |

L’Aquila | Italy | 2009-04-06 | 01:32:40 | 6.1 | 4.9 | 644.246 (E) | −42.720 (N) | 6.789 (E) | 42.342 | 13.380 |

Type 1 | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|

FEM | 0.571 | 0.564 | 0.514 |

Proposed beam-like model | 0.571 | 0.564 | 0.514 |

Type 2 | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|

FEM | 0.941 | 0.917 | 0.812 |

Proposed beam-like model | 0.941 | 0.917 | 0.812 |

Type 3 | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|

FEM | 0.854 | 0.837 | 0.766 |

Proposed beam-like model | 0.854 | 0.837 | 0.766 |

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**MDPI and ACS Style**

Greco, A.; Fiore, I.; Occhipinti, G.; Caddemi, S.; Spina, D.; Caliò, I. An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings. *Appl. Sci.* **2020**, *10*, 3212.
https://doi.org/10.3390/app10093212

**AMA Style**

Greco A, Fiore I, Occhipinti G, Caddemi S, Spina D, Caliò I. An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings. *Applied Sciences*. 2020; 10(9):3212.
https://doi.org/10.3390/app10093212

**Chicago/Turabian Style**

Greco, Annalisa, Ilaria Fiore, Giuseppe Occhipinti, Salvatore Caddemi, Daniele Spina, and Ivo Caliò. 2020. "An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings" *Applied Sciences* 10, no. 9: 3212.
https://doi.org/10.3390/app10093212