# Simulation-Based Investigations of the Load-Bearing Behavior of Concrete Hollow Sphere Slabs Exposed to Fire

^{*}

## Abstract

**:**

_{2}emissions; therefore, the potential for mass reduction in structural components must be fully exploited. However, the design regulations for weight-minimized components, particularly slabs with internal voids, are often not explicitly covered by standards, such as the fire design standard relevant to this paper. (2) Methods: Based on the design guidelines for statically determinate structures in Eurocode 2-2 and DIN 4102-4, a solid slab and a concrete slab with concrete hollow spheres are designed and evaluated with regard to their weight and flexural capacity when subjected to fire. The temperature profiles within the slab cross-section exposed to fire are simulated using ABAQUS finite element software, considering the physically nonlinear, temperature-dependent material behavior of concrete and steel. Using these results, the strain distribution corresponding to the maximum flexural moment is iteratively determined at the weakest cross-section, which exhibits the largest void. (3) Results: All components show sufficient flexural capacity for the target fire duration of 90 min. (4) Conclusion: In the context of this study, the design guidelines according to Eurocode 2-2 and DIN 4102-4 are proven to be fully applicable also for concrete hollow sphere slabs.

## 1. Introduction

## 2. Methods and Tools

#### 2.1. Thermal Analysis

#### 2.1.1. Temperature on Fire-Facing Side

^{2}/s calculated as $\lambda /\left({c}_{p}\ast \rho \right)$ where $\lambda $ is the thermal conductivity in W/(mK), c

_{p}is the specific heat in J/(kgK), $\rho $ is the bulk density in kg/m

^{3}, and the variables x, y, and z are the space coordinates in meters.

_{,}and bulk density must be considered [8].

^{2}K and the emissivity ${\epsilon}_{m}$ as 0.7.

#### 2.1.2. Temperature-Dependent Material Properties of Concrete and Steel

#### 2.1.3. Simulation Parameters

#### 2.2. Mechanical Analysis

#### 2.3. Fire Resistance Classes

## 3. Test Program

#### 3.1. Structural System, Load Assumptions, and Time of Fire Exposure

^{2}, a partition wall surcharge of 1.2 kN/m

^{2}, and a live load for, e.g., museum use of 5.0 kN/m

^{2}. The slab height is determined from the provided boundary conditions and on the basis of the verification of the bending slenderness [14].

#### 3.2. Test Specimens

## 4. Results

#### 4.1. Solid Slab

#### 4.2. Hollow Sphere Slab

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Unit—temperature—time curve (reproduced from [8]).

**Figure 2.**Calculated values of the temperature-dependent thermal material properties of concrete (reproduced from [8]).

**Figure 3.**Calculated values of the temperature-dependent thermal material properties of steel (reproduced from [10]).

**Figure 4.**Temperature profiles for a slab provided in [5], Annex A, compared to simulation results with standard parameters (

**left**) and adjusted parameters (

**right**).

**Figure 5.**Stresses and strains in case of fire and pair of forces used to determine the flexural moment for a cross-section of the solid slab with the following abbreviations: ${\epsilon}_{th}$—thermal expansion, $\epsilon $—total strain, ${\epsilon}_{\sigma}$—stress-generating strain, $\sigma $—stress, S—center of gravity line, F

_{1}—compression force, F

_{2}—tension force, e—distance of forces, and M—flexural moment.

**Figure 7.**Cross-sections of the solid and hollow sphere slabs dimensioned according to [5] (all dimensions in mm).

**Figure 8.**Cross-sections of the solid and hollow sphere slabs dimensioned according to [6] (all dimensions in mm).

**Figure 9.**Left: Mesh of the solid slab with the following features: Mesh size: 38 mm; Element shape: Hex. Right: Mesh of the hollow sphere slab with the following features: Mesh size: 38 mm; Element shape: Hex.

**Figure 11.**Temperature profile of the solid slab after 30, 60, and 90 min fire exposure with node temperature in °C with dimensions based on [5].

**Figure 13.**Temperature profile of the hollow sphere slab after 30, 60, and 90 min fire exposure with node temperature in °C with dimensions based on [5].

**Figure 14.**(

**a**) Strain due to thermal expansion, (

**b**) Total strain, (

**c**) Stress-generating strain, (

**d**) Compressive stress at the mesh nodes of the hollow sphere slab after 90 min fire exposure with dimensions based on [5,6]; The points of the scatter plot represent the data for a mesh element output at its centroid.

**Figure 15.**Dead weight—Flexural Capacity diagram of a solid slab (S) and hollow sphere slab (H) after 30, 60, 90 min fire exposure.

**Table 1.**Analysis parameters according to [5] and adjusted values chosen for the simulation.

Analysis Parameter | Value from Eurocode 2-2 | Adjusted Values |
---|---|---|

Thermal conductivity | Lower limit value | Lower limit value |

Specific heat | Moisture content 1.5% | 0% |

Emissivity | 0.7 | 0.8 |

Convective heat Transfer coefficient | 25 W/m^{2}K | 55 W/m^{2}K |

Type | Weight | Design Moment |
---|---|---|

kg/m^{2} | kNm/m | |

Solid slab DIN 4102-4 | 920.7 | 256.1 |

Solid slab Eurocode 2-2 | 909.2 | 254.1 |

Hollow sphere slab DIN 4102-4 | 574.6 | 195.0 |

Hollow sphere slab Eurocode 2-2 | 563.0 | 193.0 |

**Table 3.**Flexural resistances of the different slab types after 30, 60, and 90 min of fire exposure.

Type | Flexural Resistance | kNm/m | |
---|---|---|---|

Fire Duration | 30 min | 60 min | 90 min |

Solid slab DIN 4102-4 | 357.9 | 333.7 | 258.0 |

Solid slab Eurocode 2-2 | 352.8 | 316.8 | 257.1 |

Hollow sphere slab DIN 4102-4 | 344.5 | 326.1 | 253.5 |

Hollow sphere slab Eurocode 2-2 | 340.0 | 303.0 | 248.5 |

**Table 4.**Utilization of flexural resistance of the different slab types after 30, 60 and 90 min of fire exposure (with design moment cf. Table 2).

Type | $\frac{\mathbf{Design}\mathbf{Moment}}{\mathbf{Flexural}\mathbf{Resistance}}$ | % | |
---|---|---|---|

Fire Duration | 30 min | 60 min | 90 min |

Solid slab DIN 4102-4 | 71.6% | 76.7% | 99.3% |

Solid slab Eurocode 2-2 | 72.0% | 80.2% | 98.8% |

Hollow sphere slab DIN 4102-4 | 56.6% | 59.8% | 76.9% |

Hollow sphere slab Eurocode 2-2 | 56.8% | 63.7% | 77.7% |

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**MDPI and ACS Style**

Miller, O.; Gericke, O.; Nigl, D.; Kovaleva, D.; Blandini, L.
Simulation-Based Investigations of the Load-Bearing Behavior of Concrete Hollow Sphere Slabs Exposed to Fire. *Fire* **2022**, *5*, 197.
https://doi.org/10.3390/fire5060197

**AMA Style**

Miller O, Gericke O, Nigl D, Kovaleva D, Blandini L.
Simulation-Based Investigations of the Load-Bearing Behavior of Concrete Hollow Sphere Slabs Exposed to Fire. *Fire*. 2022; 5(6):197.
https://doi.org/10.3390/fire5060197

**Chicago/Turabian Style**

Miller, Olga, Oliver Gericke, David Nigl, Daria Kovaleva, and Lucio Blandini.
2022. "Simulation-Based Investigations of the Load-Bearing Behavior of Concrete Hollow Sphere Slabs Exposed to Fire" *Fire* 5, no. 6: 197.
https://doi.org/10.3390/fire5060197