# Parametric Analysis of Tensegrity Plate-Like Structures: Part 1--Qualitative Analysis
## 1 Introduction  
Tensegrity is a term derived from the English language as a compound of two words: \"tension\"--stretching, and \"integrity\"--stability. The term describes structures composed only of compressed (struts, rods) and tensed elements (cables). Although tensegrities are rod-like structures, some specific mechanical and mathematical properties distinguish them from conventional systems. The components are in a self-equilibrium system of internal forces called the self-stress state. In the absence of self-stress, the tensegrity structures are unstable, in other words, geometrically variable. Stabilisation occurs only after introducing initial stresses. Their modification allows for the control of the static parameters of the structure, among others [1; 2; 3; 4; 5]. One of the multitudes of applications of the tensegrity principle in civil engineering are double-layer grids. Generally, the elements of a double-layer grid are organised into two parallel planes, which are connected by vertical and diagonal elements. In the horizontal projection, the elements are arranged in a regular pattern. Double-layer tensegrity grids (also called tensegrity plate-like structures or tensegrity plates) are usually built from basic tensegrity modules such as Simplex [6; 7; 8; 9; 10; 11; 12; 13] or Quartex [6; 14; 15; 16; 17; 18]. Adjacent modules in tensegrity plates can be connected in a contiguous configuration (struts are connected to each other) or a non-contiguous configuration (maintaining a discontinuous arrangement of compressed elements). Modules can be connected edge-to-edge, node-node or strut-cable. The first two-layer tensegrity grids were constructed by [PERSON]. In the patent [6], [PERSON] proposed a structure consisting of modified Simplex modules with node-node connections and two structures built with Quartex modules linked edge-to-edge. A similar design was found by [PERSON] [7] and [PERSON] ([PERSON]'s patent was rejected) [8] (Figure 1). Both of the abovementioned modules, i.e., Simplex and Quartex, are most often used to build tensegrity plate-like structures. An analysis of plates composed of Simplex modules was carried out, among others, by [PERSON] and [PERSON] [19], [PERSON] et al. [20] and [PERSON] [21]. [PERSON] and [PERSON] created the first experimental model of the tensegrity plate. It was built with thirty-three modified Simplex modules. The span of the structure was 9 m, and its area was 80 m\({}^{2}\). [PERSON] and [PERSON] compared the properties of tensegrity structures created from the same modules, but connected in different ways. The [PERSON] team proposed a method of obtaining tensegrity panels based on the geometry of traditional double-layer trusses, while [PERSON] studied constructions with single and double curvatures. Tensegrity plates composed of Quartex modules were considered, among others, by [PERSON] and [PERSON] [22], [PERSON] and [PERSON] [23] and [PERSON] et al. [24] (Figure 2). [PERSON] and [PERSON] used semidefinite programming (SDP) to determine the optimal topology of the tensegrity plate-like structure consisting of nine modules. [PERSON] and [PERSON] used a genetic algorithm to optimise the cross-sections of cables and struts of a structure composed of twenty modules, whereas [PERSON] et al. considered the application of double-layered tensegrity grids consisting of four or eight modules as roofing. Figure 1: Patents of Simplex modules: (**a**) [PERSON]’s [6], (**b**) [PERSON]’s [7], (**c**) [PERSON]’s [8]. A lot of papers about tensegrity plates are authored by [PERSON] [15; 25; 26]. [PERSON] proposed many tensegrity plate-like structures composed of Quartex modules connected together in various ways. [PERSON] used other tensegrity modules to build plates as well. He presented numerous interesting examples of tensegrity plate-like structures built from regular or modified octahedron modules. [PERSON] also proposed a temporary tensegrity plate-like structure, which could be folded and unfolded if necessary.