# Single Layer Tensegrity Structures This paper by Haoyu Yang, Ruiwei Liu, Ani Luo, Heping Liu and Chuanyang Li, describes a mathematical model of lightweight single-layer tensegrity structures on the basis of their geometric parameters. Connection matrix and configuration of the single-layer tensegrity structures are built using MATLAB software. The force balance equations of nodes of a three-bar tensegrity structure are established by introducing the force–density method, and the force–density relationship amongst the components is analysed. Thus, the configuration principle of single-layer tensegrity structure is verified. The force–density relationship between the components in the single-layer tensegrity structure is obtained. The change rule of the force–density relationship in different single-layer tensegrity structures is also analysed. Notably, p-1 stable configurations are present in the p-bar tensegrity structure. The force–density relationships of these p-1 configurations are in symmetrical distribution, that is, the j-th and (p-j)th configurations have the same force–density relationship. The lightweight nature of the structure is studied using the force–density relationship between the components, and the optimal structural parameter relationship is obtained when the structure has the lightest mass. Tensegrity structure is a new type of spatial structure system and is composed of discrete bars and continuous cables. Since the birth of tensegrity structure, scholars have carried out extensive research on it from different directions. Pellegrino and Calladine (1986) and Pellegrino (1990; 1993) proposed the classification of tensegrity structure. By using the self-stress mode number and displacement mode number of the structure, the geometric stability of the structure was determined by matrix analysis. The tangent stiffness matrix of the structure is obtained by the force balance equation and the physical meaning of each part of the tangent stiffness matrix is analysed (Guest 2006; 2011). On the basis of the tensegrity structure system classification theory proposed by Pellegrino and Calladine (1986) and Lazopulos et al. have made a more in-depth study on the determination of geometric stability of the IV class system (Lazopulos 2005a; b). Zhang et al.

(2010) proposed a double-sided “star” tensegrity structure which on the basis of the tensegrity prism structure. Oliveira and Skelton (2005; 2006) obtained the “tower” tensegrity structure by topological method, and completed the configuration and mechanical analysis of the structure (Masic and Skelton 2004). Luo et al. (2017) made a comprehensive and detailed study on the geometric stability and mobility of the tensegrity structure system (Luo 2000; Luo and Lu 2006). Luo et al. (2017) have systematically studied the theory of stable configuration of basic tensegrity units.

At present, the research on the tensegrity structure is limited to the analysis of its mechanical properties or kinematics. No relevant research is available on realising lightweight of the structure and finding the optimal configuration. In this study, the mathematical and mechanical models of the structure are constructed on the basis of the generalised coordinates of the nodes of the basic unit, and the relationship between the configuration and the force–density of the structure is obtained.
On the basis of the relationship of force–density between components, the parameter of material strength utilisation is introduced to measure the strength utilisation of components in the structure. A low-material strength utilisation indicates redundant dimensions and materials of the structure, a condition that requires light treatment.