# Pre-Rationalized Parametric Designing of Roof Shells Formed by Repetitive Modules of Catalan Surfaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Early Stage of Rationalized Design

## 3. Geometric Properties of Catalan Surfaces

- oblique ruled surfaces of second order—hyperbolic paraboloid

## 4. Parametric Pre-Designing of Roof Shells—Results

#### 4.1. Geometric Modeling by Means of Grasshopper Scripts

#### 4.1.1. Hyperbolic Paraboloid

#### 4.1.2. Cylindroid and Conoid

#### 4.2. Form Optimisation by Means of Galapagos

#### 4.3. Optimization Based on Structural Analysis

^{2}and wind pressure of 1 kN/m

^{2}, and applied different load combinations, one of them is shown in Figure 12.

#### 4.4. Discussion

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Typical examples of Catalan surfaces; respectively from the left: a cylindroid, a conoid, a hyperbolic paraboloid.

**Figure 2.**Single hyperboloid paraboloid form creation: (

**a**) Grasshopper script for a single form; (

**b**) result.

**Figure 4.**Building covers composed of hyperbolic paraboloid units: (

**a**) with the same direction; (

**b**) with different directions.

**Figure 9.**Building covers composed of conoid units: (

**a**) with the same direction; (

**b**) with different directions.

**Figure 11.**The assembly of load and supports in a simulation model in order to check roof shell behavior under self-load.

**Figure 13.**The best simulation result—behavior under dead and live loads for the shell roof number 2.

Kind of Thse Urface | Type of the Compoud Roof Shell | |
---|---|---|

Hyperbolic paraboloid | ||

1 | 2 | |

Conoid | ||

3 | 4 | |

Cylindroid | ||

5 | 6 | |

7 |

Kind of the Surface | The Scheme of the Mesh and Supports’ Location | Dimmentions (m) | Mass (kg) | Displacement (m) |
---|---|---|---|---|

Hyperbolic paraboloid 1 | a = b = 4.00 | 3021.93 | 0.0004 | |

h = 1 | ||||

Hyperbolic paraboloid 2 | a = b = 4.00 | 3021.93 | 0.0000 | |

h = 1 | ||||

Conoid 3 | a = 4.75 | 2993.24 | 0.0042 | |

b = 3.35 | ||||

h = 1 | ||||

Conoid 4 | a = 4.75 | 3993.52 | 0.0035 | |

b = 3.35 | ||||

h = 1 | ||||

Cylindroid 5 | a = 3.47 | 2901.03 | 0.0033 | |

b = 4.61 | ||||

h = 1 | ||||

Cylindroid 6 | a = 4.69 | 3113.00 | 0.0033 | |

b = 3.41 | ||||

h = 1 | ||||

Cylindroid 7 | a = 4.67 | 3107.00 | 0.0008 | |

b = 3.47 | ||||

h = 1 |

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**MDPI and ACS Style**

Dzwierzynska, J.; Prokopska, A.
Pre-Rationalized Parametric Designing of Roof Shells Formed by Repetitive Modules of Catalan Surfaces. *Symmetry* **2018**, *10*, 105.
https://doi.org/10.3390/sym10040105

**AMA Style**

Dzwierzynska J, Prokopska A.
Pre-Rationalized Parametric Designing of Roof Shells Formed by Repetitive Modules of Catalan Surfaces. *Symmetry*. 2018; 10(4):105.
https://doi.org/10.3390/sym10040105

**Chicago/Turabian Style**

Dzwierzynska, Jolanta, and Aleksandra Prokopska.
2018. "Pre-Rationalized Parametric Designing of Roof Shells Formed by Repetitive Modules of Catalan Surfaces" *Symmetry* 10, no. 4: 105.
https://doi.org/10.3390/sym10040105