# In Search of Lightweight Deployable Tensegrity Columns
## 3 Description of the Columns  
The tensegrity columns considered in this paper are based on three regular tensegrity modules: three-strut simplex (Figure 1a), X-module (Figure 1b), and expanded octahedron (Figure 1c).  
The properties of the modules are given in Table 1. It can be noticed that the three-strut simplex and X-module are similar from the point of view of geometrical characteristics and mechanical features characterized by the number of infinitesimal mechanisms and the number of self-stress states that stabilize them. The expanded octahedron has more structural elements and more infinitesimal mechanisms; however, it was chosen because of its potentially high load carrying capacity.  
There are three ways in which the columns can be constructed:  
* modular system--the structure is composed of regular tensegrity modules joined in nodes without any additional elements,
* non-modular system--the structure is based on regular modules, but they are connected in such a way that they create a new tensegrity system,
* modular system with extra cables--the structure is created similarly to the modular system, but some additional cables connecting adjacent modules are needed. Contrary to the module members, these extra cables are not prestressed.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **3-Strut Simplex** & **X-Module** & **Expanded Octahedron** \\ \hline Number of struts & 3 & 3 & 4 \\ Number of cables & 9 & 9 & 12 \\ Number of joints & 6 & 6 & 8 \\ \hline Number of infinitesimal mechanisms & 1 & 1 & 3 \\ Number of self-stress states & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Properties of regular tensegrity modules.  
Figure 1: Regular tensegrity modules: (**a**) three-strut simplex; (**b**) X-module; (**c**) expanded octahedron.  
Based on the described tensegrip modules and methods of constructing complex structural systems, four columns are proposed:  
* simplex (Figure 2a)--a modular structure based on the three-strut simplex module,
* overlapped simplex (Figure 2b)--a non-modular structure based on the three-strut simplex module,
* X (Figure 2c)--a modular structure based on the X-module,
* octahedron (Figure 2d)--a modular structure with extra cables, based on the expanded octahedron module.  
All columns consist of four modules and have a total height of 4 m. They can all be inscribed into a cuboid wit a square base of an edge of 1 m.  
In the analyses, a uniform state of self-stress was adopted in all columns, which means that the prestressing forces are the same in each module of the given column. The level of self-stress is indicated by the factor \(S_{0}\), which is a multiplier of forces introduced to the structural members. There are three nodal supports in each column that eliminate the six degrees of freedom corresponding to the rigid body motion. The loading is applied to the nodes of the upper surface (the ones that support for example a ceiling plate) in the form of concentrated forces equal to: \(P_{1}=8.33\) kN in the simplex and overlapped simplex columns and \(P_{2}=12.50\) kN in the X and octahedron columns. The described forces correspond to a uniformly distributed surface load \(q=1\) kN\(/\)m\({}^{2}\) acting on the area of 25 m\({}^{2}\). In addition, the self-weight of the structures is included. The values of external loading and prestressing forces were adopted so that the forces in the cables and struts were lower than 90% of the maximum load specified for the cables and struts. For structural members, catalog elements used in typical systems were used:  
* cables--\(\phi=10.0\) mm made of steel S460 with a maximum load \(N_{\text{Rd}}=71.1\) kN,  
Figure 2: Tensegrip columns: (**a**) Simplex. (**b**) Overlapped simplex. (**c**) X. (**d**) Octahedron.  
* struts--hollow cross-section \(\phi=54.0\) mm, wall thickness 2.6 mm, made of steel S355J2 with a maximum load \(N_{\text{Rd}}=84\) kN.