# Optimal Design of Bubble Deck Concrete Slabs: Sensitivity Analysis and Numerical Homogenization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Table 1.**Differences between floors with HDPE inserts [25].

Bubble Deck | Cobiax |
---|---|

One spatial structure, upper and lower reinforcement combined | Single baskets with balls combined with only one (upper or lower) steel mesh |

Reinforcement to match the cartridges (balls) | Bottom and top reinforcement independent of inserts (balls) |

Plate thicknesses 25–60 cm | Thicknesses from 20–60 cm |

There is a danger of damaging the bullets | No danger of damaging the bullets |

## 2. Materials and Methods

#### 2.1. Representative Volume Element

#### 2.2. Numerical Homogenization Algorithm

#### 2.3. Sensitivity Analysis

## 3. Results

^{2}for the BD 340 variant and 3.90 kN/m

^{2}for the BD 320 variant.

#### 3.1. BD 340

^{2}. For the other variants of the bubble deck BD340, the weight of the plate changed and was 6.07 kN/m

^{2}(for H = 0.357 m), 5.22 kN/m

^{2}(D = 0.284 m), 5.75 kN/m

^{2}(for $\varphi $ = 0.014 m), and 5.65 kN/m

^{2}for bar spacings every 0.105 m.

^{2}. Reducing the section height by 5% reduced the slab weight to 5.25 kN/m

^{2}. Assuming a smaller diameter of the bars (ϕ = 0.010 m), the mass of the bubble plate was equal to 5.59 kN/m

^{2}, while the reduction in the bubble diameter by 5% compared to the reference model increased the weight of the plate and amounted to 6.05 kN/m

^{2}. Changing the spacing of the bars (0.095 m) did not significantly affect the weight of the bubble plate, which was 5.67 kN/m

^{2}.

#### 3.2. BD 230

_{1}= 0.189 m and D

_{2}= 0.242 m (increase in dimensions by 5.0% compared to the reference model). The concrete class was changed to C35/45 (increase in Young’s modulus by 6.25%). However, the diameter and spacing of the reinforcing bars increased by 5.0% and 16.67%, respectively, to 0.014 m and 0.131 m. The comparison of the stiffnesses obtained from the plates with a changed selected cross-sectional parameter is presented in Table 6.

^{2}. Increasing the dimensions of the holes caused a reduction in the mass of the bubble plate to 3.81 kN/m

^{2}(for D

_{1}= 0.189 m) and 3.71 kN/m

^{2}(for D2 = 0.242 m). For height H = 0.242 m, the weight of the plate was equal to 4.18 kN/m

^{2}; for reinforcement with diameter $\varphi $ = 0.014 m, the weight was equal to 4.01 kN/m

^{2}; and the weight was equal to 3.88 kN/m

^{2}when the bars were spaced 0.131 m.

_{1}= 0.171 m, D

_{2}= 0.219 m, respectively, and the distance between the reinforcements being equal to 0.119 m. The diameter of the rods was 0.010 m, which was a decrease in the value of the parameter by 16.67% compared to the initial model.

^{2}). Reducing the opening increased the amount of concrete in the element and thus increased the slab weight to 4.00 kN/m

^{2}for D

_{1}= 0.171 m and 4.09 kN/m

^{2}for D

_{2}= 0.219 m. The mass of the slab decreased with the change in the ceiling height (H = 0.219 m) and the diameter of the reinforcement $\varphi $ = 0.010 m to the values of 3.63 kN/m

^{2}and 3.81 kN/m

^{2}, respectively. The spacing of the bars equal to 0.119 m caused a slight change in the weight of the slab compared to the initial model and was 3.87 kN/m

^{2}.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Krzyczkowski, D. Budownictwo; Wydawnictwo Politechniki Lubelskiej: Lwów, Poland, 1929. [Google Scholar]
- Mielnicki, S. Ustroje Budowlane; Spółdzielnia Wydawnicza “Meta”: Katowice, Poland, 1938. [Google Scholar]
- Markiewicz, P. Budownictwo Ogólne dla Architektów; Archi-Plus: Kraków, Poland, 2009. [Google Scholar]
- Derkowski, W.; Skalski, P. New concept of slimfloor with prestressed composite beams. Procedia Eng.
**2017**, 193, 176–183. [Google Scholar] [CrossRef] - Derkowski, W.; Walczak, R. Innowacje w stropach betonowych. In Proceedings of the 34th National Structural Designer Work Workshops, Szczyrk, Poland, 5–8 March 2019. [Google Scholar]
- Szydłowski, R. Concrete properties for long-span post tensioned slabs. In Proceedings of the 2nd International Symposium on Advanced Material Research (ISAMR 2018), Jeju Island, Republic of Korea, 16–18 March 2018. [Google Scholar] [CrossRef]
- Quraisyah, A.D.S.; Kartini, K.; Hamidah, M.S.; Daiana, K. Bubble Deck Slab as an Innovative Biaxial Hollow Slab—A Review. J. Phys. Conf. Ser.
**2020**, 1711, 012003. [Google Scholar] [CrossRef] - Shetkar, A.; Hanche, N. An Experimental Study on Bubble Deck Slab System with Elliptical Balls. Indian J. Sci. Res.
**2015**, 12, 21–27. [Google Scholar] - Tiwari, N.; Zafar, S. Structural Behaviour of Bubble Deck Slabs and Its Application: An Overview. Int. J. Sci. Res. Dev.
**2016**, 4, 433–437. [Google Scholar] - Bhowmik, R.; Mukherjee, S.; Das, A.; Banerjee, S. Review on Bubble Deck with Spherical Hollow Balls. Int. J. Civ. Eng. Technol.
**2017**, 8, 979–987. [Google Scholar] - Vakil, R.R.; Nilesh, M.M. Comparative Study of Bubble Deck Slab and Solid Deck Slab—A Review. Int. J. Adv. Res. Sci. Eng.
**2017**, 6, 383–392. [Google Scholar] - Mirajkar, S.; Balapure, M.; Kshirsagar, T. Study of Bubble Deck Slab. Int. J. Res. Sci. Eng.
**2017**, 7, 1–5. [Google Scholar] - Szymczak-Graczyk, A.; Ksit, B.; Laks, I. Operational Problems in Structural Nodes of Reinforced Concrete Constructions. IOP Conf. Ser. Mater. Sci. Eng.
**2019**, 603, 032096. [Google Scholar] [CrossRef] - Nowogońska, B. Diagnoses in the Aging Process of Residential Buildings Constructed Using Traditional Technology. Buildings
**2019**, 9, 126. [Google Scholar] [CrossRef] [Green Version] - Ksit, B.; Szymczak-Graczyk, A. Rare weather phenomena and the work of large-format roof coverings. Civ. Environ. Eng. Rep.
**2019**, 29, 123–133. [Google Scholar] [CrossRef] [Green Version] - Jamal, J.; Jolly, J. A study on structural behaviour of bubble deck slab using spherical and elliptical balls. Int. Res. J. Eng. Technol.
**2017**, 4, 2090–2095. [Google Scholar] - Konuri, S.; Varalakshmi, T.V.S. Review on Bubble Deck Slabs Technology and their Applications. Int. J. Sci. Technol. Res.
**2019**, 8, 427–432. [Google Scholar] - Surendar, M.; Ranjitham, M. Numerical and Experimental Study on Bubble Deck Slab. Int. J. Eng. Sci. Comput.
**2017**, 6, 5959–5962. [Google Scholar] [CrossRef] - Ali, S.; Kumar, M. Analytical Study of Conventional Slab and Bubble Deck Slab Under Various Support and Loading Conditions Using Ansys Workbench 14.0. Int. Res. J. Eng. Technol.
**2016**, 6, 5959–5962. [Google Scholar] - Mahalakshmi, S.; Nanthini, S. Bubble Deck. Int. J. Res. Appl. Sci. Eng. Technol.
**2017**, 5, 580–588. [Google Scholar] [CrossRef] - Mushfiq, M.S.; Student, P.G.; Saini, A.P.S.; Rajoria, A.P.N. Experimental study on bubble deck slab. Int. Res. J. Eng. Technol.
**2017**, 4, 1000–1004. [Google Scholar] - Hokrane, N.S.; Saha, S. Comparative Studies of Conventional Slab and Bubble Deck Slab Based on Stiffness and Economy. Int. J. Sci. Res. Dev.
**2017**, 5, 1396–1398. [Google Scholar] - John, R.; Varghese, J. A Study on Behavior of Bubble Deck Slab Using ANSYS. Int. J. Innov. Sci. Eng. Technol.
**2015**, 2, 2136–2139. [Google Scholar] [CrossRef] - Wilczyński, K.; Buziak, K.; Wilczyński, K.J.; Lewandowski, A.; Nastaj, A. Computer Modeling for Single-Screw Extrusion of Wood–Plastic. Compos. Polym.
**2018**, 10, 295. [Google Scholar] [CrossRef] [Green Version] - Available online: https://www.scribd.com/document/312500253/Prezentacja-stropow-Cobiax/prezentacjaJanFreczyński/ (accessed on 26 November 2022). (In Polish).
- Buannic, N.; Cartraud, P.; Quesnel, T. Homogenization of corrugated core sandwich panels. Compos. Struct.
**2003**, 59, 299–312. [Google Scholar] [CrossRef] [Green Version] - Garbowski, T.; Marek, A. Homogenization of corrugated boards through inverse analysis. In Proceedings of the 1st International Conference on Engineering and Applied Sciences Optimization, Kos Island, Greece, 4–6 June 2014; pp. 1751–1766. [Google Scholar]
- Hohe, J. A direct homogenization approach for determination of the stiffness matrix for microheterogeneous plates with application to sandwich panels. Compos. Part. B
**2003**, 34, 615–626. [Google Scholar] [CrossRef] - Tallarico, D.; Hannema, G.; Miniaci, M.; Bergamini, A.; Zemp, A.; Van Damme, B. Superelement modelling of elastic metamaterials: Complex dispersive properties of three-dimensional structured beams and plates. J. Sound Vib.
**2020**, 484, 115499. [Google Scholar] [CrossRef] - Marek, A.; Garbowski, T. Homogenization of sandwich panels. Comput. Assist. Methods Eng. Sci.
**2015**, 22, 39–50. [Google Scholar] - Xin, L.; Khizar, R.; Peng, B.; Wenbin, Y. Two-Step Homogenization of Textile Composites Using Mechanics of Structure Genome. Compos. Struct.
**2017**, 171, 252–262. [Google Scholar] [CrossRef] - Khizar, R.; Xin, L.; Wenbin, Y. Multiscale Structural Analysis of Textile Composites Using Mechanics of Structure Genome. Int. J. Solids Struct.
**2018**, 136–137, 89–102. [Google Scholar] [CrossRef] - Biancolini, M.E. Evaluation of equivalent stiffness properties of corrugated board. Comp. Struct.
**2005**, 69, 322–328. [Google Scholar] [CrossRef] - Garbowski, T.; Gajewski, T. Determination of transverse shear stiffness of sandwich panels with a corrugated core by numerical homogenization. Materials
**2021**, 14, 1976. [Google Scholar] [CrossRef] - Staszak, N.; Garbowski, T.; Szymczak-Graczyk, A. Solid Truss to Shell Numerical Homogenization of Prefabricated Composite Slabs. Materials
**2021**, 14, 4120. [Google Scholar] [CrossRef] - Staszak, N.; Gajewski, T.; Garbowski, T. Effective Stiffness of Thin-Walled Beams with Local Imperfections. Materials
**2022**, 15, 7665. [Google Scholar] [CrossRef] - Abaqus Unified FEA® Software. Available online: https://www.3ds.com/products-services/simulia/products/abaqus (accessed on 15 December 2022).
- Arslan, M.H.; Özkılıç, Y.O.; Arslan, H.D.; Şahin, Ö.S. Experimental and Numerical Investigation of the Structural, Thermal and Acoustic Performance of Reinforced Concrete Slabs with Balls for a Cleaner Environment. Int. J. Civ. Eng.
**2023**. [Google Scholar] [CrossRef] - Staszak, N.; Szymczak-Graczyk, A.; Garbowski, T. Elastic Analysis of Three-Layer Concrete Slab Based on Numerical Homogenization with an Analytical Shear Correction Factor. Appl. Sci.
**2022**, 12, 9918. [Google Scholar] [CrossRef] - Gajewski, T.; Staszak, N.; Garbowski, T. Parametric Optimization of Thin-Walled 3D Beams with Perforation Based on Homogenization and Soft Computing. Materials
**2022**, 15, 2520. [Google Scholar] [CrossRef] [PubMed] - Staszak, N.; Gajewski, T.; Garbowski, T. Shell-to-Beam Numerical Homogenization of 3D Thin-Walled Perforated Beams. Materials
**2022**, 15, 1827. [Google Scholar] [CrossRef] [PubMed]

Material | $\mathit{E}$$\text{}\left(\mathit{G}\mathit{P}\mathit{a}\right)$ | $\mathit{\nu}$$\text{}(-)$ | $\mathit{\rho}$$\text{}\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ |
---|---|---|---|

steel | 210 | 0.3 | 7900 |

concrete | 32 | 0.2 | 2400 |

Stiffness | RVE BD340 | RVE BD230 |
---|---|---|

A11 (${10}^{6}$ MPa mm) | 6.85 | 4.35 |

A22 (${10}^{6}$ MPa mm) | 6.85 | 4.35 |

A33 (${10}^{6}$ MPa mm) | 2.48 | 1.57 |

D11 (${10}^{10}$ MPa mm^{3}) | 9.92 | 3.05 |

D22 (${10}^{10}$ MPa mm^{3}) | 9.92 | 3.05 |

D33 (${10}^{10}$ MPa mm^{3}) | 3.58 | 1.06 |

R44 (${10}^{6}$ MPa mm) | 1.68 | 0.957 |

R55 (${10}^{6}$ MPa mm) | 1.68 | 0.957 |

**Table 4.**Stiffness for RVE BD340 in case of increasing the value of individual cross-section parameters.

Stiffness | C35/45 | H = 0.357 m | D = 0.284 m | $\mathit{\varphi}$ = 0.014 m | Bars Every 0.105 m |
---|---|---|---|---|---|

A11 (${10}^{6}$ MPa mm) | 7.25 | 7.45 | 6.33 | 7.04 | 6.82 |

A22 (${10}^{6}$ MPa mm) | 7.25 | 7.45 | 6.33 | 7.04 | 6.82 |

A33 (${10}^{6}$ MPa mm) | 2.64 | 2.71 | 2.27 | 2.48 | 2.48 |

D11 (${10}^{10}$ MPa mm^{3}) | 10.48 | 11.69 | 9.42 | 10.28 | 9.86 |

D22 (${10}^{10}$ MPa mm^{3}) | 10.48 | 11.69 | 9.42 | 10.28 | 9.86 |

D33 (${10}^{10}$ MPa mm^{3}) | 3.81 | 4.27 | 3.39 | 3.58 | 3.58 |

R44 (${10}^{6}$ MPa mm) | 1.79 | 1.85 | 1.51 | 1.69 | 1.68 |

R55 (${10}^{6}$ MPa mm) | 1.79 | 1.85 | 1.51 | 1.69 | 1.68 |

**Table 5.**Stiffness for RVE BD340 in case of reducing the value of individual cross-section parameters.

Stiffness | C25/30 | H = 0.323 m | D = 0.256 m | $\mathit{\varphi}$ = 0.010 m | Bars Every 0.095 m |
---|---|---|---|---|---|

A11 (${10}^{6}$ MPa mm) | 6.45 | 6.30 | 7.40 | 6.70 | 6.92 |

A22 (${10}^{6}$ MPa mm) | 6.45 | 6.30 | 7.40 | 6.70 | 6.92 |

A33 (${10}^{6}$ MPa mm) | 2.33 | 2.27 | 2.70 | 2.48 | 2.48 |

D11 (${10}^{10}$ MPa mm^{3}) | 9.35 | 8.36 | 10.31 | 9.65 | 9.97 |

D22 (${10}^{10}$ MPa mm^{3}) | 9.35 | 8.36 | 10.31 | 9.65 | 9.97 |

D33 (${10}^{10}$ MPa mm^{3}) | 3.36 | 2.97 | 3.74 | 3.58 | 3.58 |

R44 (${10}^{6}$ MPa mm) | 1.58 | 1.55 | 1.84 | 1.68 | 1.69 |

R55 (${10}^{6}$ MPa mm) | 1.58 | 1.55 | 1.84 | 1.68 | 1.69 |

**Table 6.**Stiffness for RVE BD230 in case of increasing the value of individual cross-section parameters.

Stiffness | C35/45 | H = 0.242 m | D_{1} = 0.189 m | D_{2} = 0.242 m | $\mathit{\varphi}$ = 0.014 m | Bars Every 0.131 m |
---|---|---|---|---|---|---|

A11 (${10}^{6}$ MPa mm) | 4.60 | 4.76 | 4.20 | 4.07 | 4.51 | 4.31 |

A22 (${10}^{6}$ MPa mm) | 4.60 | 4.76 | 4.20 | 4.07 | 4.51 | 4.31 |

A33 (${10}^{6}$ MPa mm) | 1.66 | 1.72 | 1.50 | 1.46 | 1.57 | 1.57 |

D11 (${10}^{10}$ MPa mm^{3}) | 3.22 | 3.60 | 2.95 | 2.97 | 3.21 | 3.00 |

D22 (${10}^{10}$ MPa mm^{3}) | 3.22 | 3.60 | 2.95 | 2.97 | 3.21 | 3.00 |

D33 (${10}^{10}$ MPa mm^{3}) | 1.13 | 1.28 | 1.03 | 1.03 | 1.06 | 1.06 |

R44 (${10}^{5}$ MPa mm) | 10.16 | 10.47 | 9.41 | 8.49 | 9.60 | 9.57 |

R55 (${10}^{5}$ MPa mm) | 10.16 | 10.47 | 9.41 | 8.49 | 9.60 | 9.57 |

**Table 7.**Stiffness for RVE BD230 in case of decreasing the value of individual cross-section parameters.

Stiffness | C25/30 | H = 0.219 m | D_{1} = 0.171 m | D_{2} = 0.219 m | $\mathit{\varphi}$ = 0.010 m | Bars Every 0.119 m |
---|---|---|---|---|---|---|

A11 (${10}^{6}$ MPa mm) | 4.10 | 3.95 | 4.51 | 4.64 | 4.24 | 4.40 |

A22 (${10}^{6}$ MPa mm) | 4.10 | 3.95 | 4.51 | 4.64 | 4.24 | 4.40 |

A33 (${10}^{6}$ MPa mm) | 1.47 | 1.41 | 1.63 | 1.67 | 1.57 | 1.57 |

D11 (${10}^{10}$ MPa mm^{3}) | 2.88 | 2.56 | 3.14 | 3.13 | 2.93 | 3.09 |

D22 (${10}^{10}$ MPa mm^{3}) | 2.88 | 2.56 | 3.14 | 3.13 | 2.93 | 3.09 |

D33 (${10}^{10}$ MPa mm^{3}) | 0.998 | 0.869 | 1.10 | 1.09 | 1.06 | 1.06 |

R44 (${10}^{5}$ MPa mm) | 8.98 | 8.77 | 9.74 | 10.66 | 9.55 | 9.57 |

R55 (${10}^{5}$ MPa mm) | 8.98 | 8.77 | 9.74 | 10.66 | 9.55 | 9.57 |

Stiffness | C25/30 | H = 0.323 m | D = 0.256 m | $\mathit{\varphi}$ = 0.010 m | Bars Every 0.095 m |
---|---|---|---|---|---|

$\mathrm{A}11\text{}(\%)$ | 5.84 | 8.39 | −7.81 | 2.48 | −0.73 |

$\mathrm{A}22\text{}(\%)$ | 5.84 | 8.39 | −7.81 | 2.48 | −0.73 |

$\mathrm{A}33\text{}(\%)$ | 6.25 | 8.87 | −8.67 | 0.00 | 0.00 |

$\mathrm{D}11\text{}(\%)$ | 5.70 | 16.78 | −4.49 | 3.18 | −0.55 |

$\mathrm{D}22\text{}(\%)$ | 5.70 | 16.78 | −4.49 | 3.18 | −0.55 |

$\mathrm{D}33\text{}(\%)$ | 6.28 | 18.16 | −4.89 | 0.00 | 0.00 |

$\mathrm{R}44\text{}(\%)$ | 6.25 | 8.93 | −9.82 | 0.30 | −0.30 |

$\mathrm{R}55\text{}(\%)$ | 6.25 | 8.93 | −9.82 | 0.30 | −0.30 |

Stiffness | C25/30 | H = 0.219 m | ${\mathrm{D}}_{1}\text{}=\text{}0.171\text{}\mathbf{m}$ | ${\mathrm{D}}_{2}\text{}=\text{}0.219\text{}\mathbf{m}$ | $\mathit{\varphi}$ = 0.010 m | Bars Every 0.119 m |
---|---|---|---|---|---|---|

$\mathrm{A}11\text{}(\%)$ | 5.75 | 9.31 | −3.56 | −6.55 | 3.10 | −1.03 |

$\mathrm{A}22\text{}(\%)$ | 5.75 | 9.31 | −3.56 | −6.55 | 3.10 | −1.03 |

$\mathrm{A}33\text{}(\%)$ | 6.05 | 9.87 | −4.14 | −6.69 | 0.00 | 0.00 |

$\mathrm{D}11\text{}(\%$) | 5.57 | 17.05 | −3.11 | −2.62 | 4.59 | −1.48 |

$\mathrm{D}22\text{}(\%)$ | 5.57 | 17.05 | −3.11 | −2.62 | 4.59 | −1.48 |

$\mathrm{D}33\text{}(\%)$ | 6.23 | 19.39 | −3.30 | −2.83 | 0.00 | 0.00 |

$\mathrm{R}44\text{}(\%)$ | 6.17 | 8.88 | −1.72 | −11.34 | 0.26 | 0.00 |

$\mathrm{R}55\text{}(\%)$ | 6.17 | 8.88 | −1.72 | −11.34 | 0.26 | 0.00 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Staszak, N.; Garbowski, T.; Ksit, B.
Optimal Design of Bubble Deck Concrete Slabs: Sensitivity Analysis and Numerical Homogenization. *Materials* **2023**, *16*, 2320.
https://doi.org/10.3390/ma16062320

**AMA Style**

Staszak N, Garbowski T, Ksit B.
Optimal Design of Bubble Deck Concrete Slabs: Sensitivity Analysis and Numerical Homogenization. *Materials*. 2023; 16(6):2320.
https://doi.org/10.3390/ma16062320

**Chicago/Turabian Style**

Staszak, Natalia, Tomasz Garbowski, and Barbara Ksit.
2023. "Optimal Design of Bubble Deck Concrete Slabs: Sensitivity Analysis and Numerical Homogenization" *Materials* 16, no. 6: 2320.
https://doi.org/10.3390/ma16062320