Relation Between the Iteration of Planar Retractable Plate Structure and Plane Symmetry Group

Relation Between the Iteration of Planar Retractable Plate Structure and Plane Symmetry Group

Aylin Gazi Gezgin (Ministry of Culture and Tourism, Turkey) and Koray Korkmaz (Izmir Institute of Technology, Turkey)
DOI: 10.4018/IJDIBE.2021070102
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Abstract

One of the most important issues in the design processes of retractable plate structure is to determine the most suitable shape of the plates that form an enclosure without any gaps or overlaps in both closed and open configurations of the structure. One of the approaches to find the most suitable shape of the plates is based on mathematical tessellation technique without using any kinematical or numerical analyses. Due to the usage of RPS on many different areas in architecture, it is just as important to be able to iterate them regularly. This study both focuses on the iteration of planar RPSs that are formed based on 1-uniform tessellation and develops a relation between iteration capacity of RPS and plane symmetry groups. By the help of developed relationship, it tries to realize whether this structure can be derived from 1-uniform tessellation and which tessellation should be selected before obtaining it.
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Introduction And Background

In 20th century, architecture become in need of kinetic structures to deal with some pressures such as changing weather conditions, different functional requirements and even for aesthetic reasons. With the increase of kinetic structure usage in architecture, necessity to cover them by variety of materials has emerged over time. Coming across many difficulties in covering these structures by fabric materials, utilization of rigid plates instead of fabrics has gained great importance. Throughout the time, rigid plates have been started to be used not only as covering materials but also as structures by themselves that are being called as retractable plate structure (RPS). RPS is a family of structures that are assembled by utilizing set of cover plates connected by revolute joints. The most important issue to design RPS is to determine the most suitable shape of the plates that form an enclosure without any gaps or overlaps in both closed and open configurations of the structure. Also there should not be any interference between the rigid plates during retraction or expansion motion.

In order to find the most suitable shape of the plates that match perfectly by forming assembly without any gaps or overlaps in their fully closed and open configurations, researchers use kinematical analysis and empirical design methods. The earlier attempt of the RPS started with covering pantographic retractable structure with rigid plates. Kassabian et al. proposed a method to cover pantographic bar structure by utilizing rigid panels and also display possibilities to employ them as a retractable roof (Kassabian et al., 1999). Thereby, they present a general solution for structures with many shapes to choose covering panels with triangular shapes. In addition, Jensen and Pellogrino have developed a method for covering any multi angulated bar structure with plates by finding extreme positions of pantographic structures (Jensen and Pellogrino, 2002). On the other hand, Jensen and Pellogrino showed that instead of covering angulated structures with rigid elements, angulated elements can be removed and connected directly with scissor hinges and develop a family of structures called radially retractable plate structures (RRPS) (Jensen and Pellegrino, 2002 and 2009). In their studies Jensen and Pellegrino have removed angulated elements and connected plates directly with revolute joints on exactly the same locations as in the original bar structure. During the study some empirical and analytical approach were developed to find the shapes of hinged plates (Jensen and Pellegrino, 2009). Moreover, another research focused on the question of whether all of the pivots beam must remain within the boundary of its corresponding plates or not by using an analytical approach and they have derived a set of conditions. By the help of this research, designers can choose and open a profile that suits their needs and then apply appropriate geometrical formulations to determine the edges of the cover plates (Lou, Mao, You, 2007). Chilton and friends developed swivel diagram with the combination of retractable plate structures and reciprocal structures. This diagram presents some advantages such as including fewer elements compared with RPS and also simpler joint configurations than deployable reciprocal plates, however, diagram generally utilizes triangular forms (Chilton et all, 2003). On the contrary to these researches, Gazi and Korkmaz presented a new design approach by utilizing mathematical tessellation technique without using any kinematical or numerical analysis (Gazi and Korkmaz, 2017), (Gazi Gezgin, 2016). By the help of that research, designers have gained the capability by having required conditions to easily determine, which uniform tessellation can be used for RPS design, also the same research proposed new methods to convert non-conditional RPS into the ones that fulfil required conditions. If the data obtained from that study is examined, it can be seen that some RPS that are produced by utilizing mentioned methods can be iterated infinitely and some of them cannot.

Finding the suitable form of the plate is very important issue for the design of retractable plate structure; however, if the designed RPSs are targeted to be used in architecture, such as on the wall, façade, roof or the area that will be covered with respect to the designer needs, solely determining the form is not sufficient in some cases. Thus, having the capability to be iterated limitlessly on architectural surface is another important issue for architecture.

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