# Different Configurations of the Non-Minimal Prismatic Tensegrities
## 1 Introduction
These are as follows: \(T\)--the structure is a truss; \(S\)--there is a self-stress state; \(C\)--tensile elements have no rigidity in compression (cables); \(M\)-there are infinitesimal mechanisms stiffened by the self-stress state; \(D\)--the extremities of compressed components do not touch each other, and bars constitute a discontinuous set; and \(I\)--the set of compressed components is included inside the set of tensile components. A total of four categories are considered. An \"abstract tensegrity\" possesses all of the features, a \"pure tensegrity\" possesses all of the features excluding \(D\) and a \"structure with tensegrity features\" possesses all of the features \(T\), \(S\) and \(C\), as well as at least one of the features \(M\), \(D\) or \(I\). A structure which does not meet the above criteria is considered as \"not a tensegrity structure\". For example, structures presented in Figure 1b-d are \"abstract tensegrities\" (\(T\), \(S\), \(C\), \(M\), \(D\) and \(I\)), while the structure presented in Figure 1e is a \"structure with tensegrity features\" (\(T\), \(S\), \(C\), \(D\) and \(I\)), though only additional cross cables were attached. However, this addition did remove the infinitesimal mechanism. The literature shows that many structures are often not pure tensegrities and lie between trusses and tensegrities. This study also lies in an interesting and increasingly transient area between typical trusses and tensegrity trusses. For example, many deployable trusses are currently designed and analyzed [12; 50; 51], especially for space exploration as  
Figure 1: Prismatic systems: (**a**) the unstable, theoretical initial configuration, (**b**) the triplex, (**c**) the quarter, (**d**) the pentex and (**e**) the non-minimal triplex (the first from a group of non-minimal prismatic systems). Drawn numbers indicate nodes. was mentioned before. On the other hand, trusses for a static, high load-bearing capacity and efficiency are created by a combination of bar and cable members, such as in [52]. Practical applications require elasticity in design, regardless of precise definitions. Also, please note that many structures labeled as tensegrities in the cited articles are called so, even though from the mechanical point of view of the presented classification they are not or are somewhere in between. Therefore, modified structures analyzed in this article, such as those presented in Figure 1e, will be called tensegrities. However, when the classification matter will be important, this will be detailed more accurately. Up to date, tensegrities were not widely used in civil engineering. In case of prismatic tensegrities, the only real-scale civil engineering realization is the Rostock Tower in Germany--see Figure 2a. This structure is built from six triplex modules attached to each other by bar endings. In order for the structure to obtain the required stiffness to withstand the crucial wind load, the cross cables needed to be prestressed with a force of 1100 kN [53]. The Rostock Tower can be categorized as a \"pure tensegrity\" (\(T\), \(S\), \(C\), \(M\) and \(I\)). The other closest realizations connected to prismatic tensegrities are the White Rhinos--the first was raised in 2001 and the second in 2012 [54; 55]. The Rhinos are approximately 10 m tall supports for a membrane roof. The static scheme is closest to the non-minimal triplex modules, i.e., triplex modules with three additional cables (see Figure 1e). Creating such a scheme was beneficial from the point of view of structural stiffness; i.e., it eliminates the infinitesimal mechanisms. However, it categorizes these structures into \"structures with tensegrity features\" (\(T\), \(S\), \(C\), \(D\) and \(I\)). Other examples are two sculptures created by Mr. [PERSON]--The Needle Tower I (Figure 2c) and the Needle Tower II--see [29]. Both are self-supporting towers with heights equal to 18 and 30 m, respectively.