# Analysis of Lightweight Structure Mesh Topology of Geodesic Domes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Shaping Geodesic Dome Mesh

#### 2.1. The First Method of Dividing a Regular Octahedron

#### 2.1.1. Determining the Nodal Points of a Dome Shaped according to the First Method

#### 2.2. The Second Method of Dividing a Regular Octahedron

#### Determining the Nodal Points of a Dome Shaped according to the Second Method

- group 1—for layers ${n}_{w}=1,4,7,\dots ,\left(1.5n-3\right)$.

- group 2—for layers ${n}_{w}=2,5,8,\dots ,\left(1.5n-2\right)$. As in the case of group 1, first, the formula for the latitudinal ordinate was determined, followed by the formula for the angular ordinate of the k element in a given layer.

- group 3—for layers ${n}_{w}=3,6,9,\dots ,\left(1.5n-5\right).$ In this case, the formula was also determined first for the latitudinal ordinate and then for the angular ordinate of the k element in a given layer.

## 3. Combination of Loads

#### 3.1. Variable Loads

_{3}= 2.00 was assumed.

#### 3.2. Additional Dead Load of Geodesic Dome

_{k}= 0.58 [kN/m

^{2}].

## 4. Numerical Analysis Parameters

#### 4.1. General Remarks

#### 4.2. Material Properties

^{3}, (ii) Young’s modulus (E) 210 GPa, (iii) Poisson’s ratio (ν) 0.3, (iv) Kirchhoff module (G) 80.76 GPa, (v) partial safety factor (γ

_{M}) 1.0, and (vi) thermal expansion coefficient (α) 1.2 × 10

^{−5}. Supports were given as restraints, and nodes were assumed to be articulated (Figure 1).

#### 4.3. Numerical Models

## 5. Result from Numerical Analysis

#### 5.1. Internal Forces

_{w}∈ <2;8>, form tensioned rings; in the higher layers with n

_{w}∈ <9;14>, alternating compression and tension of the struts occurs, depending on the place on the ring. In the upper parts with n

_{w}> 14, only compressive forces occur. The struts resulting from the displacement of the side arms of the initial triangle along their entire length are compressed, obtaining maximum values of internal forces at the edges of the regular octahedron forming the dome.

#### 5.2. Strut Dimensioning

## 6. Comparative Analysis of the Obtained Results

#### 6.1. Geometric Parameters

#### 6.2. Internal Forces

#### 6.3. Support Reactions

#### 6.4. The Weight of the Domes

#### 6.5. Nodal Displacements

## 7. Discussion

## 8. Conclusions

- The mesh of struts obtained in the structure shaped according to the first subdivision method allows us to work on the computational model in computer programs efficiently. Also, it facilitates the possibility of grouping struts during assembly. The mesh obtained according to the second method may cause problems during the task implementation and may also result in difficulties with modeling the structure.
- To create the computational model of the first analyzed dome, it was necessary to divide it into n
_{w}groups based on the same division rules. In the case of the second dome, this division had to consider the shifting of the nodes, which resulted in the need to divide it into three additional groups of struts. Such difficulties may cause mistakes and complicate the generation of the computational model. Creating the dome according to the first subdivision method is, therefore, much more straightforward and minimizes the possibility of making mistakes. - The model obtained according to the first method gives the impression of continuity of the mesh of struts at the connections of the initial faces of the regular octahedron, while these connections in the dome shaped according to the second method cause the illusion of discontinuity of the mesh of struts, which may make assembly difficult and create a feeling of lack of aesthetics.
- In the first dome, a more even distribution of axial forces can be observed over the entire surface, and the values of extreme compressive forces are comparable to the maximum value of tensile forces.
- Another argument for using the first method of creating a mesh topology is the number of supports and, therefore, the value of the support reactions. Due to the more distributed weight of the dome on the ground, the foundations can be designed more economically. The second division method has completely different characteristics of transferring forces from external impacts on the foundations. The accumulation of extreme axial forces is located within the vertical struts extending from the foundation, and their values are due to the smaller number of supports. After minor changes in the static scheme by adding vertical struts in the support zone and thus increasing the number of supports almost twice, this would reduce the values of the maximum compressive forces and lead to a reduction in the cross-section in the fourth group of struts and, consequently, a reduction in the weight of the structure.
- In terms of usability, the first dome is less susceptible to vertical deflections, which indicates its greater stiffness in this direction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Meshes of geodesic domes generated from a regular octahedron using (

**a**) the first division method and (

**b**) the second division method.

**Figure 2.**Division of the triangular face of the initial regular octahedron according to the first method: (

**a**) n = 2, (

**b**) n = 3, and (

**c**) n = 4.

**Figure 4.**Division of the triangular face of the initial regular octahedron according to the second method: (

**a**) n = 2, (

**b**) n = 4, (

**c**) n = 6.

**Figure 5.**Roof shape coefficient for even snow load [48].

**Figure 6.**Roof shape coefficient for uneven snow load [48].

**Figure 7.**Maps of axial internal forces in the struts of a dome created according to the first method from a 4608-hedron.

**Figure 8.**Maps of axial internal forces in the struts of a dome created according to the second method from a 4704-hedron.

**Figure 9.**Division of struts into groups in a dome created according to the first method (4608-hedron).

**Figure 10.**Division of struts into groups in a dome created according to the second method (4704-hedron).

**Figure 12.**Maximal values of axial forces in the struts of modeled domes generated from 4608-hedron and 4704-hedron.

**Figure 13.**Maximal values of support reactions of modeled domes generated from 4608-hedron and 4704-hedron.

**Figure 14.**Weight of individual groups of struts and entire domes generated from 4608-hedron and 4704-hedron.

**Figure 15.**Vertical and horizontal displacements of nodes of domes generated from 4608-hedron and 4704-hedron.

${\mathit{n}}_{\mathit{w}}[-]$ | ${\mathit{\varphi}}_{\mathit{n}}[\xb0]$ |
---|---|

1 | 0.00 |

2 | 3.75 |

3 | 7.50 |

4 | 11.25 |

5 | 15.00 |

6 | 18.75 |

7 | 22.50 |

8 | 26.50 |

9 | 30.00 |

10 | 33.75 |

11 | 37.50 |

12 | 41.25 |

13 | 45.00 |

14 | 48.75 |

15 | 52.50 |

16 | 56.25 |

17 | 60.00 |

18 | 63.75 |

19 | 67.50 |

20 | 71.25 |

21 | 75.00 |

22 | 78.50 |

23 | 82.50 |

24 | 86.25 |

25 | 90.00 |

**Table 2.**Latitudinal coordinates ${\lambda}_{{n}_{w}}^{k}[\xb0]$ for the exemplary first and fifth layers (Figure 3).

$$\mathit{k}$$
| ${\mathit{\lambda}}_{1}^{\mathit{k}}[\xb0]$ Layer 1 ${\mathit{\lambda}}_{1}^{\prime}=3.75[\xb0]$ | ${\mathit{\lambda}}_{5}^{\mathit{k}}[\xb0]$ Layer 5 ${\mathit{\lambda}}_{5}^{\prime}=4.50[\xb0]$ |
---|---|---|

1 | 0.00 | 0.00 |

2 | 3.75 | 4.50 |

3 | 7.50 | 9.00 |

4 | 11.25 | 13.50 |

5 | 15.00 | 18.00 |

6 | 18.75 | 22.50 |

7 | 22.50 | 27.00 |

8 | 26.50 | 31.50 |

9 | 30.00 | 36.00 |

10 | 33.75 | 40.50 |

11 | 37.50 | 45.00 |

12 | 41.25 | 49.50 |

13 | 45.00 | 54.00 |

14 | 48.75 | 58.50 |

15 | 52.50 | 63.00 |

16 | 56.25 | 67.50 |

17 | 60.00 | 72.00 |

18 | 63.75 | 76.50 |

19 | 67.50 | 81.00 |

20 | 71.25 | 85.50 |

21 | 75.00 | 90.00 |

22 | 78.50 | – |

23 | 82.50 | – |

24 | 86.25 | – |

25 | 90.00 | – |

${\mathit{n}}_{\mathit{w}}[-]$ | ${\mathit{\varphi}}_{\mathit{n}}[\xb0]$ |
---|---|

1 | 0.00 |

2 | 2.14 |

3 | 4.29 |

4 | 6.43 |

5 | 8.57 |

6 | 10.71 |

7 | 12.86 |

8 | 15.00 |

9 | 17.14 |

10 | 19.29 |

11 | 21.43 |

12 | 23.57 |

13 | 25.71 |

14 | 27.86 |

15 | 30.00 |

16 | 32.14 |

17 | 34.29 |

18 | 36.43 |

19 | 38.57 |

20 | 40.71 |

21 | 42.86 |

22 | 45.00 |

23 | 47.14 |

24 | 49.29 |

25 | 51.43 |

26 | 53.57 |

27 | 55.71 |

28 | 57.86 |

29 | 60.00 |

30 | 62.14 |

31 | 64.29 |

32 | 66.43 |

33 | 68.57 |

34 | 70.71 |

35 | 72.86 |

36 | 75.00 |

37 | 77.14 |

38 | 79.29 |

39 | 81.43 |

40 | 83.57 |

41 | 85.71 |

42 | 87.86 |

43 | 90.00 |

**Table 4.**Latitudinal coordinates ${\lambda}_{w}^{k}[\xb0]$ for the first and seventh layers from group 1.

$$\mathit{k}$$
| ${\mathit{\lambda}}_{1}^{\mathit{k}}[\xb0]$ Layer 1 ${\mathit{\lambda}}_{1}^{\prime}=6.43[\xb0]$ | ${\mathit{\lambda}}_{7}^{\mathit{k}}[\xb0]$ Layer 7 ${\mathit{\lambda}}_{7}^{\prime}=7.50[\xb0]$ |
---|---|---|

1 | 0.00 | 0.00 |

2 | 6.43 | 7.50 |

3 | 12.86 | 15.00 |

4 | 19.29 | 22.50 |

5 | 25.71 | 30.00 |

6 | 32.14 | 37.50 |

7 | 38.57 | 45.00 |

8 | 45.00 | 52.50 |

9 | 51.43 | 60.00 |

10 | 57.86 | 67.50 |

11 | 64.29 | 75.00 |

12 | 70.71 | 82.50 |

13 | 77.14 | 90.00 |

14 | 83.57 | – |

15 | 90.00 | – |

**Table 5.**Latitudinal coordinates ${\lambda}_{w}^{k}[\xb0]$ for the second and eighth layers from group 2.

$$\mathit{k}$$
| ${\mathit{\lambda}}_{2}^{\mathit{k}}[\xb0]$ Layer 2 ${\mathit{\lambda}}_{2}^{\prime}=6.59[\xb0]$ | ${\mathit{\lambda}}_{8}^{\mathit{k}}[\xb0]$ Layer 8 ${\mathit{\lambda}}_{8}^{\prime}=7.71[\xb0]$ |
---|---|---|

1 | 2.20 | 2.57 |

2 | 8.78 | 10.29 |

3 | 15.37 | 18.00 |

4 | 21.95 | 25.71 |

5 | 28.54 | 33.43 |

6 | 35.12 | 41.14 |

7 | 41.71 | 48.86 |

8 | 48.29 | 56.57 |

9 | 54.88 | 64.29 |

10 | 61.46 | 72.00 |

11 | 68.05 | 79.71 |

12 | 74.63 | 87.43 |

13 | 81.22 | – |

14 | 87.80 | – |

**Table 6.**Latitudinal coordinates ${\lambda}_{w}^{k}[\xb0]$ for the third and ninth layers from group 3.

$$\mathit{k}$$
| ${\mathit{\lambda}}_{3}^{\mathit{k}}[\xb0]$ Layer 3 ${\mathit{\lambda}}_{3}^{\prime}=6.75[\xb0]$ | ${\mathit{\lambda}}_{9}^{\mathit{k}}[\xb0]$ Layer 9 ${\mathit{\lambda}}_{9}^{\prime}=7.94[\xb0]$ |
---|---|---|

1 | 4.50 | 5.29 |

2 | 11.25 | 13.24 |

3 | 18.00 | 21.24 |

4 | 24.75 | 29.12 |

5 | 31.50 | 37.06 |

6 | 38.25 | 45.00 |

7 | 45.00 | 52.94 |

8 | 51.75 | 60.88 |

9 | 58.50 | 68.82 |

10 | 65.25 | 76.76 |

11 | 72.00 | 84.71 |

12 | 78.75 | – |

14 | 85.50 | – |

**Table 7.**Values of external pressure coefficients [39].

Curve | h/d | f/d | ${\mathbf{c}}_{\mathbf{p}\mathbf{e},10}$ |
---|---|---|---|

A | 0 | 0.5 | +0.80 |

B | −1.20 | ||

C | 0.00 |

Group | Dome Created according to the First Method 4608-Hedron | Dome Created according to the Second Method 4704-Hedron |
---|---|---|

1 | from −100 to −50 | from −100 to −50 |

2 | from −50 to 0 | from −50 to 0 |

3 | from 0 to 50 | from 0 to 50 |

4 | from 50 to 100 | from 50 to 120 |

Group | Dome Created according to the First Method 4608-Hedron | Dome Created according to the Second Method 4704-Hedron |
---|---|---|

1 | RO 30 × 4 | RO 25 × 3.6 |

2 | RO 38 × 3.2 | RO 42.4 × 3.2 |

3 | RO 44.5 × 6.3 | RO 48.3 × 4.5 |

4 | RO 44.5 × 5.6 | RO 54 × 6.3 |

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**MDPI and ACS Style**

Bysiec, D.; Jaszczyński, S.; Maleska, T.
Analysis of Lightweight Structure Mesh Topology of Geodesic Domes. *Appl. Sci.* **2024**, *14*, 132.
https://doi.org/10.3390/app14010132

**AMA Style**

Bysiec D, Jaszczyński S, Maleska T.
Analysis of Lightweight Structure Mesh Topology of Geodesic Domes. *Applied Sciences*. 2024; 14(1):132.
https://doi.org/10.3390/app14010132

**Chicago/Turabian Style**

Bysiec, Dominika, Szymon Jaszczyński, and Tomasz Maleska.
2024. "Analysis of Lightweight Structure Mesh Topology of Geodesic Domes" *Applied Sciences* 14, no. 1: 132.
https://doi.org/10.3390/app14010132