*Article* **Numerical Analysis of Steel Geodesic Dome under Seismic Excitations**

**Dominika Pilarska and Tomasz Maleska \***

Faculty of Civil Engineering and Architecture, Opole University of Technology, 45-758 Opole, Poland; d.pilarska@po.edu.pl

**\*** Correspondence: t.maleska@po.edu.pl; Tel.: +48-77-449-8568

**Abstract:** The paper presents the response of two geodesic domes under seismic excitations. The structures subjected to seismic analysis were created by two different methods of subdividing spherical triangles (the original octahedron face), as proposed by Fuli ´nski. These structures are characterised by the similar number of elements. The structures are made of steel, which is a material that undoubtedly gives lightness to structures and allows large spans. Designing steel domes is currently a challenge for constructors, as well as architects, who take into account their aesthetic considerations. The analysis was carried out using the finite element method of the numerical program. The two designed domes were analysed using four different seismic excitations. The analysis shows what influence particular earthquakes have on the geodesic dome structures by two different methods. The study analysed the maximum displacements, axial forces, velocities, and accelerations of the designed domes. In addition, the Time History method was used for the analysis, which enabled the analysis of the structure in the time domain. The study will be helpful in designing new structures in seismic areas and in assessing the strength of various geodesic dome structures under seismic excitation.

**Keywords:** geodesic dome; seismic response; dynamic analysis; seismic analysis

**Citation:** Pilarska, D.; Maleska, T. Numerical Analysis of Steel Geodesic Dome under Seismic Excitations. *Materials* **2021**, *14*, 4493. https:// doi.org/10.3390/ma14164493

Academic Editor: Davide Palumbo

Received: 13 July 2021 Accepted: 8 August 2021 Published: 10 August 2021

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**Copyright:** © 2021 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/).

#### **1. Introduction**

Strut domes are an effective two-curved cover, which is approximated by a mesh of struts. The use of less material and lower costs, combined with the possibility of covering very large areas, is a definite advantage of this type of structure. The stiffness of these three-dimensional systems justifies their ability to cover large spans with a small amount of construction material. Architects and engineers all over the world use a wide range of possibilities of connecting strut elements, constituting a mesh of dome covers. The aesthetic potential of this engineering system means that the structure has a valuable architectural aspect [1]. This aspect was important not only in steel geodesic domes but also in other spatial structures (e.g., made of concrete) [2].

From dome ceilings and full buildings to Arctic homes and artificial biomes, geodesic domes around the world continue to inspire and amaze both architecture enthusiasts and curious travellers. The elegant and aerodynamic form of geodesic domes creates expansive yet economical spaces that are ideal for greenhouses, arenas, sports facilities, entertainment halls, swimming pools, and other uses. For centuries, dome structures were used due to their thermal advantages, structural benefits, and availability of construction materials [1].

After Fuller [1] patented methods of dividing a sphere into spherical triangles in 1954 (based on an icosahedron as the initial solid), steel mesh domes of the geodetic type have almost completely replaced the use of other types of domes. In 1967, his design was shown to the world as a 'Biosphere', with a diameter of 76 m, constructed for Expo '67 in Montreal (Figure 1).

Fuller [1] believed that the geodesic dome was nature's perfect structure, enclosing the greatest space with the least amount of material. While remaining in tune with the

environment, the dome supports itself without the need for any internal columns or walls. The largest geodesic dome projects include the Fukuoka Dome (built in Fukuoka, Japan in 1993, 216 m), Nagoya Dome (Nagoya, Japan in 1997, 187 m), Louvre (Abu Dhabi, United Arab Emirates, 180 m), Tacoma Dome (Tacoma, WA, USA, 161 m), and the Superior Dome (Northern Michigan University, Marquette, MI, USA, 160 m).

**Figure 1.** Biosphere, Montreal from 1967 (Fulller [1]).

The advantage of Fuller's geodesic strut domes is the small number of struts required at different lengths. According to his patent, the domes were formed on the basis of the icosahedron, which requires the smallest number of groups of struts of equal lengths. There were many papers related to such geodesic domes. Significant and excellent achievements in lightweight, durable, and self-supporting structures have been attained by Makowski [3,4], Clinton [5,6], Tarnai [7–9], Huybers [10–15], Lalvani [16–19], Pavlov [20,21], Ramaswamy [22], Obr˛ebski [23], Szmit [24,25], and R ˛ebielak [26].

Although dome structures are economical in terms of consumption of construction materials compared to the conventional forms of structures [27], a more lightweight design can be obtained using optimisation methods. The optimum solution of the geometry design was obtained by Saka [28], Kaveh and Talatahari [29], Carvalho et al. [30], Saka and Carbas [31], Gholizadeh and Barati [32], Kaveh and Rezaei [33,34], Kaveh et al. [35], and Ye and Lu [36]. Other structures were analysed from the economical aspect, e.g., bridge [37,38], but in these cases, the methods and reasons of optimisation were totally different.

An analysis of the behaviour of shallow geodesic lattice domes was presented by Guan et al. [39]. Barbieri et al. [40] analysed the dynamic behaviour of a geodesic dome in aluminium alloy through numerical models obtained using the finite element method. Szaniec and Zielinska [41] presented the results of a dynamic analysis of an existing reticulated dome under wind loads. The calculation model of the structure was constructed using the finite element method. The dome was subjected to the standard wind pressure, assuming that it operates harmonically. Satria et al. [42] considered the dynamic behaviour

of a new type of two-way single layer lattice dome with nodal eccentricity. Fu [43] presented a static analysis and design of tensegrity domes. New forms of the tensegrity domes were proposed.

Studies of geodesic domes under seismic loads have rarely been investigated. A few papers on this subject can be found. In the paper of Cai et al. [44], the dynamic characteristics of a space beam string structure was determined. The aim of the paper was to determine the structure's response to seismic excitation using the Time History method and the modal method. These methods were used to determine the structure's response to the given seismic excitation. In the paper by Takeuchi et al. [45], the response evaluation method of domes and cylindrical shell roofs with substructures was shown. In addition, the response amplification factors approach was proposed. Furthermore, Nakazawa et al. [46] focused on methods for evaluating responses under seismic loads to metal roof spatial structures. In addition, papers by Kato and Nakazawa [47], Li et al. [48], and Qin et al. [49] focused on the assessment of lightweight structure responses exposed to earthquakes. In addition, the dynamic stability and failure probability analysis of dome structures under stochastic seismic excitation was presented by Li and Xu [50].

It should be added that the most of the analysis on geodesic domes relate to space frames, the basis of which is the icosahedron, which is the development of Fuller's patent. Davis [51] showed that the octahedron might seem to be a better option for geodesic domes than the icosahedron. However, the problems arise because more subdivisions were required, and thus, more different lengths were required.

The paper presents the numerical analysis of the geodesic domes under seismic excitations. The developed structures were created on the basis of the regular octahedron, which was a polyhedron that has not been considered in great detail so far [52–54]. Two different methods for the subdivision of the spherical triangle proposed by Fuli ´nski [1,55] were used to design the two geodesic domes under seismic analysis. The numerical analyses were carried out for four different seismic excitations. Different times and intensities of the seismic records were presented. The results obtained from numerical analysis were compared with two structures differing in subdivision methods and under four different excitations. The presented analysis was the first step for further consideration of the optimisation of the geodesic dome. It is very important to take into account the wider possibilities of using this type of structure to cover large areas. It should be added that the geodesic domes around the world continue to inspire and amaze both architectural enthusiasts and curious travellers.

#### **2. Description of Numerical Modelling**

#### *2.1. Subdivision Methods for Spherical Dome (Strut) Structures Based on the Regular Octahedron*

The two developed geodesic domes under the seismic excitations were shaped in accordance with two methods proposed by Fuli ´nski [55]. The structures were designed on the basis of being regular octahedra, subdividing their equilateral faces into smaller sub-faces and taking the resulting face vertices to define the nodes of the structural grid, while the edges of the sub-faces define the axes of the structural members. Both of the methods used lead to the division of the initial triangle of the octahedron into smaller triangles by frequency (V), i.e., the number of subdivisions. The subdivision process naturally leads to the generation of a three-way grid on every face of the basis octahedron. The central projection of this grid's vertices on the octahedron's circumscribed sphere leads to a polyhedron approximating the sphere in which only the grid's nodes lie on the sphere's surface. More parts give smoother spheres. The mentioned methods were developed in detail in the paper by Pilarska [54]. Figure 2 shows the difference in the shaping of geodesic domes using the first and second subdivision methods.

#### *2.2. Tested Models*

Two geodesic strut domes were subjected to numerical analysis, taking into account the seismic excitations. These structures were designed according to the two different methods of creating geodesic domes proposed by Fuli ´nski [55], which are presented in Section 2.1 and described in detail in the papers by Pilarska [52–54]. The basis for generating the meshes of both structures was a regular octahedron. It was an initial triangle and was divided into as many parts to finally obtain domes characterised by a similar number of struts. Including the first method, after dividing the initial triangle of the regular octahedron into 19 parts, 2888 hedra were obtained, i.e., a structure consisting of 761 nodes and of 2204 struts (Figure 3a). The analysed dome was 49.97 m wide and 25.0 m high. Using the second method, after dividing the initial triangle of the regular octahedron into 22 parts, 2904 hedra were modelled. This dome consists of 749 nodes and 2156 struts (Figure 3b), with a width of 50.0 m and a height of 25.0 m.

**Figure 2.** Methods of subdividing the initial triangle edge, according to Fuli ´nski.

**Figure 3.** View of numerical model: (**a**) method 1 and (**b**) method 2.
