# Methods to Reproduce In-Plane Deformability of Orthotropic Floors in the Finite Element Models of Buildings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simplified Analytical-Based Method

#### 2.1. Method

#### 2.2. Case Study

#### 2.2.1. Description

^{2}.

^{2}. The connection is made with steel U-shaped connectors welded to the top flange of steel profiles along its whole length. The inclined bottom chords and the vertical struts are, instead, made of circular hollow steel profiles with a diameter of 9 cm and a thickness of 6.5 mm.

^{2}. In this beam, there are 16 square holes of side 50 cm.

^{2}. The top flanges of HEA200 profiles are fully embedded in the topping RC slab of the floor, and the transverse reinforcement of the slab passes over the flanges, as shown in Figure 2. Hence, the slab is continuous all over the roof surface and bounded by the perimeter beams and the circular-shaped RC beam.

#### 2.2.2. Determination of Shell Element Properties through the Simplified Method

#### 2.2.3. Results

## 3. Theoretical Basis

_{iX}and D

_{iY}, which are parallel to the X and Y directions, respectively, and the rotation D

_{iθ}around the axis orthogonal to the floor middle plane—as shown in see Figure 5—and collected in the 3-vector

_{i}and F

_{iY}and by torque F

_{iθ}, collected in the 3-vector

_{1}, h∙E

_{2}, h∙G

_{12}, and ν

_{12}, where E

_{1}and E

_{2}are the Young’s moduli parallel and orthogonal to the RC joists’ direction, respectively (Figure 7), G

_{12}is the in-plane shear modulus, and ν

_{12}is the Poisson ratio.

_{1}, h∙E

_{2}, h∙G

_{12}, and ν

_{12}. With the influence of ν

_{12}on the in-plane behavior of the floor being initially neglected, the deformation patterns chosen to evaluate the equivalent membrane properties are those shown in Figure 9.

## 4. FE-Based Method

#### 4.1. Static Schemes

- MODE 1: shear deformation mode;
- MODE 2: extension along X axis, parallel to joists’ direction;
- MODE 3: extension along Y axis, orthogonal to joists’ direction.

#### 4.2. Application of Loads and Restraints to the 3D Model

#### 4.3. Operating Procedure

- Realization of the floor cell model, including the perimeter beams, using solid finite elements. Each element constituting the floor cell is modeled with its actual dimensions and with the elastic properties of the material of which it is made. This model is built to reproduce the actual behavior of the floor cell.
- Realization of the floor cell model, including the perimeter beams, using frame elements for the beams and 2D elements with the membrane behavior for the floor. The geometry is congruent with that of the 3D model. In particular, the side length of the 2D element is equal to the distance, in the 3D model, between the longitudinal axes of the perimeter beams orthogonal to that side. This condition guarantees that the distances between the longitudinal axes of the perimeter beams in the 2D model are equal to those in the 3D model.Perimeter beams are modeled with their actual cross-section and with the elastic properties of the concrete of which they are constituted, used also in the 3D model.The thickness of the 2D elements has to be chosen, and a homogenized material with orthotropic elastic behavior has to be defined, with initial values of the unknown equivalent elastic properties being set. Herein, for RC floors with joists, the thickness is set equal to that of the RC slab, while the initial values of the homogenized material elastic properties are set equal to those of the concrete constituting the real slab.
- Application to loads and restraints of the static scheme of MODE 1 to both the 2D and 3D model. The applied forces are equal to 707 kN so that the resultant load applied to each vertex node is equal to 1000 kN.Then, the values of the displacement components parallel to Y at vertex nodes B and C obtained from the two models are compared. If the differences between these displacement components are greater than a fixed tolerance, which is herein taken to be equal to 1%, it is necessary to modify the elastic shear modulus ${G}_{xy}$ of the 2D elements by raising or decreasing its value.Then, the analysis of the 2D model is carried out again and similarly to what was done previously, the comparison of displacement components is performed. Thus, the value of ${G}_{xy}$ is determined through an iterative procedure ended once the differences between the displacement components mentioned above are smaller than the fixed tolerance. The 2D model is then updated, with the value of ${G}_{xy}$ obtained at the end of the iterative procedure.
- Similarly to what done in step 3 for MODE 1, the 2D and 3D models of the floor cell are now analyzed under the static scheme of MODE 2. In this case, the load magnitude applied to nodes B and C is taken equal to 500 kN in order to apply, in X direction, a total force equal to 1000 kN.The values of the displacement components parallel to X at vertex nodes B and C obtained from the two models are compared. If the percentage difference between these displacement components is greater than a fixed tolerance, which is herein taken to be equal to 1%, it is necessary to modify the elastic modulus along X of the homogeneous material, ${E}_{x}$, by raising or decreasing its value. At the end of this second iterative procedure, the 2D element is characterized by the equivalent values of ${G}_{xy}$ and ${E}_{x}$.
- Finally, in consideration of the static scheme of MODE 3 and to search for the equality of the displacement components parallel to Y at vertex nodes C and D obtained from the two FE models, the equivalent value of ${E}_{y}$ is determined. Moreover, from the comparison of displacement components parallel to X at vertex nodes B and C obtained from the two FE models, the value of ${\nu}_{yx}$ is determined. Similarly to what was done for MODE 2, the load magnitude applied to nodes C and D can be taken to be equal to 500 kN.The considered deformation modes (MODE 1, MODE 2 and MODE 3) are not independent among them since the elastic parameter of the orthotropic material mainly involved in one mode influences the 2D element behavior also under the other deformation modes. Hence, further analyses are required to check if the differences between the displacement components, obtained from the two models in steps 3, 4, and 5, are still smaller than the fixed tolerance. If one of these steps is not satisfied, the parameter determined in that step has to be modified until convergence is achieved.

#### 4.4. Application of the Method to a Case Study

- Floor with a 408 × 408 cm
^{2}plane size; - Perimeter beams with a 30 × 52 cm
^{2}cross-section; - A 4 cm thick RC slab;
- RC joists with a 12 × 20 cm
^{2}cross-section; - Ceiling bricks 48 × 15 × 20 cm
^{3}in size.

#### 4.4.1. Material Properties

_{12}. The transmission of shear stresses between the bricks and the RC elements of the floor, i.e., the slabs and the joists, relies on the grip due to the scratches on the bricks’ surfaces. Since, under horizontal actions, ceiling bricks are not fully adhered to the joists and the slab, it is cautiously assumed that the transmission of shear stress between the bricks and the surrounding concrete is negligible; hence, G

_{12}= 0.1 MPa.

_{1}is the Young’s modulus parallel to bricks’ holes, and E

_{2}is the Young’s modulus orthogonal to the holes.

#### 4.4.2. FE Models

^{3}were used, while for the perimeter beams, solid FE with dimensions 6 × 5 × 4 cm

^{3}or 5 × 6 × 4 cm

^{3}were used, respectively, for beams parallel to the X or Y direction. In the intersection regions, solid elements 5 × 5 × 4 cm

^{3}in size were used. The beams were modeled with the extrados coplanar to that of the slab. Out-of-plane displacements of the floor cell were restrained at the slab top surface.

#### 4.4.3. Discussion of Results

_{x}, is equal to 42,000 MPa, roughly 33% higher than that of concrete (31,476 Mpa). Since in the 2D model, the thickness of membrane elements is equal to that of the floor’s slab, it follows that the 33% increase is due to the axial in-plane stiffness of joists and ceiling bricks.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Connection between the floor and the top beams of the radial trusses (dimensions are in cm).

**Figure 11.**Parallelepiped constituting the intersection between two adjacent beams in the floor cell model made by solid finite elements.

**Figure 12.**Mesh nodes in a vertical cross-section of the parallelepiped, to which loads and restraints are applied at intersection regions.

**Figure 13.**Most used floor typology in the set of existing Italian RC schools considered in the survey.

**Table 1.**Elastic properties of the homogenized orthotropic material constituting the equivalent solid bricks used in the 3D model to reproduce the real hollow ceiling bricks.

E_{1}(MPa) | E_{2}(MPa) | E_{3}(MPa) | G_{12}(MPa) | G_{23}(MPa) | ν_{12} | ν_{21} |
---|---|---|---|---|---|---|

6970 | 4000 | 4000 | 0.1 | 0.1 | 0.2 | 0.115 |

**Table 2.**Elastic properties of the homogenized orthotropic material of the equivalent 2D elements obtained for the considered case study.

E_{x}(MPa) | E_{y}(MPa) | G_{xy}(MPa) | ν_{xy} | ν_{yx} |
---|---|---|---|---|

45,200 | 35,300 | 14,500 | 0.2 | 0.26 |

**Table 3.**Elastic properties of the homogenized orthotropic material of the equivalent 2D elements obtained for the case study described in Section 2.2.

E_{x}(MPa) | E_{y}(MPa) | G_{xy}(MPa) | ν_{xy} | ν_{yx} |
---|---|---|---|---|

51,935 | 31,950 | 33,500 | 0.2 | 0.26 |

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**MDPI and ACS Style**

Frappa, G.; Pitacco, I.; Baldassi, S.; Pauletta, M.
Methods to Reproduce In-Plane Deformability of Orthotropic Floors in the Finite Element Models of Buildings. *Appl. Sci.* **2023**, *13*, 6733.
https://doi.org/10.3390/app13116733

**AMA Style**

Frappa G, Pitacco I, Baldassi S, Pauletta M.
Methods to Reproduce In-Plane Deformability of Orthotropic Floors in the Finite Element Models of Buildings. *Applied Sciences*. 2023; 13(11):6733.
https://doi.org/10.3390/app13116733

**Chicago/Turabian Style**

Frappa, Giada, Igino Pitacco, Simone Baldassi, and Margherita Pauletta.
2023. "Methods to Reproduce In-Plane Deformability of Orthotropic Floors in the Finite Element Models of Buildings" *Applied Sciences* 13, no. 11: 6733.
https://doi.org/10.3390/app13116733