# Different Configurations of the Non-Minimal Prismatic Tensegrities
## 2 Theory
### Simple Model of Behavior During Prestressing  
Here, a simple model of the behavior during prestressing is presented in order to explain how prestressing coefficients \(c_{\rm i}\) are limited. Prestressing every member can be understood as inserting it between two supports, in which one is rigid and the other one is flexible with a defined stiffness \(k\) (see Figure 7). The stiffness \(k\) originates from the stiffness of the assembled tensegrity structure and is a function of the material and cross-section parameters, as well as the geometry of other members. It cannot be determined directly, yet for the sake of this example it will be assumed as constant value for understanding.  
Figure 7: Simple model of prestressed member, where three possibilities exists—the member is unstressed (**a**), the member is compressed (**b**) and the member is tensioned (**c**) after assemblage.  
Prestressing coefficients can be generalized as follows (in regard to Equation (12)):  
\[c=L_{n}/L_{g}=(L_{g}+\Delta_{\text{mod}})/L_{g} \tag{14}\]  
where \(L\) is the length of the considered member, and \(\Delta_{\text{mod}}\) is the length modifier. Indexes \(n\) and \(g\) were explained previously. For positive values of \(\Delta_{\text{mod}}\), the member is lengthened. Please note that this modification has no effect on the prestressing of the member before it is installed between the supports or when is installed for \(\Delta_{\text{mod}}=0\) (natural lengths are equal to geometrical ones--see Figure 7a). If the inserted member length is different than the distance between supports (Figure 7b,c), the member experiences external force due to the reaction of the flexible support. Assuming small strains, i.e., \(\varepsilon=(L-L_{\text{n}})/L_{\text{n}}=\Delta_{\text{L}}/L_{\text{n}}\), as well as determining the \(\Delta_{\text{mod}}\) from Equation (14), the force \(R\), which is externally interacting with the member, is a function equal to (see Figure 7b,c for spring reaction)  
\[R(c,\varepsilon)=k\cdot(\Delta_{\text{mod}}-\Delta_{L})=k\cdot L_{g}\cdot(c- 1-\varepsilon c) \tag{15}\]  
For positive values of \(\Delta_{\text{mod}}\), the \(\Delta_{\text{L}}\) values are always negative--the member, which is initially longer than the distance between supports, is starting to be compressed (Figure 7b). Therefore, it is identified as a bar. If the inserted member is shorter than the distance between supports, it starts to be tensioned--the member is identified as a cable (Figure 7c). In order to define the limitations of prestressing coefficients, the limit state is induced:  
\[R(c,\varepsilon)=EA\varepsilon_{f} \tag{16}\]  
where \(E\) is the modulus of elasticity, \(A\) is the cross-section area and \(\varepsilon_{\text{f}}\) is the strain at failure of the material. Substituting (15) into (16) one obtains  
\[c_{\text{max}}=(EA\varepsilon_{f}/kL_{g}+1)/(1-\varepsilon) \tag{17}\]  
For typical steel with \(E=210\) GPa and \(\varepsilon_{\text{f}}=0.2\%\); a member with a cross-section made from CHS 26.9x2.3, i.e., \(A=178\) mm\({}^{2}\); member length equal to \(L_{\text{g}}=2000\) mm; as well as assuming that strains in the member are equal to \(\varepsilon=0.04\%\); Figure 8 presents the maximal prestressing coefficient in a function of stiffness.  
Figure 8: Maximal values of prestressing coefficients due to material limitations, depending on the stiffness of the tensegrity system.  
The solution shown in Figure 8 enables to understand how the prestressing coefficients' maximal values are dependent on the general stiffness of the tensegrity structure. As can be seen, the greater the stiffness, the lower the values of the maximal prestressing coefficients. This is a simple example. However, in general, in order to verify the load-bearing capacity of members, a full analysis based on Equation (13) needs to be performed.