# Different Configurations of the Non-Minimal Prismatic Tensegrities
## 2 Theory  
### Basic Assumptions and Limitations  
Members of minimal prismatic tensegrities can be divided into three groups; these are as follows (see Figure 4a,b for an example of the quarter prism): _2t_ horizontal cables with the length denoted as \(l\) (black members 1-2, 2-3, 3-4, 1-4 and 5-6, 6-7, 7-8, 5-8), \(t\) bars with their length denoted as \(b\) (dark blue thick members 1-5, 2-6, 3-7 and 4-8) and \(t\) cross cables with their length denoted as \(s\) (light blue continuous line members 1-8, 2-5, 3-6 and 4-7). The non-minimal prismatic tensegrities possess one more group--additional members denoted as \(a\) (light blue dashed line members 1-7, 2-8, 3-5 and 4-6). Another thing is the \(\alpha\) angle. For all prismatic tensegrities, this angle is a function of the number of bars \(t\). Therefore, the initial angle \(\alpha\) is (see Figure 4b)  
\[\alpha=\pi/2+\pi/t, \tag{1}\]  
Since it will be more convenient to work on a different rotational angle, the angle \(\theta\) is introduced. This angle denotes a similar rotation as the \(\alpha\) angle. However, for \(\alpha\) equal to the value presented in Equation (1), the angle \(\theta\) is equal to zero. Figure 3: Quartex: (**a**) minimal (\(\alpha=135^{\circ}\)), (**b**) non-minimal (\(\alpha=135^{\circ}\)), (**c**) non-minimal with doubled number of bars (\(\alpha=130^{\circ}\)) and (**d**) non-minimal with doubled number of bars (\(\alpha=110^{\circ}\)). Also, an assumption on symmetry is made, where all members of the same group possess the same initial lengths. Moreover, during prestressing the members of the same group are prestressed simultaneously; i.e., the change in the length is applied to all group members. This implies the simplification of energy equations, making them able to be derived and written in an acceptable form. Due to symmetry, the description of the prism is dependent on only three variables: \(\theta\), \(r\) and \(b\), where \(r\) is the radius of the circle circumscribed on the bases of the prism (Figure 4c). If the prism is not regular, multiple variables would need to be introduced to define the current configuration. For example, eight radii \(r\), eight lengths \(b\) and eight \(\theta\) angles would need to be introduced, in general, to define the nodes of the structure. This, however, would also require us to perform the form-finding procedure, since not whichever of these parameters would create a stable structure. Therefore, the symmetry assumption is crucial, especially for the possibility of writing the energy-based equations in an analytical form. The angle \(\theta\) has its limitations (Figure 4c). It possesses a minimal and maxima value. When increasing the rotation of the upper bases (\(\theta>0^{\circ}\), \(\alpha>135^{\circ}\)), locking of bars with each other occurs at  
\[\theta_{\text{max}}=\pi-(\pi/2+\pi/t)=\pi/2-\pi/t, \tag{2}\]  
while the locking of the bars and additional members occur when the rotation is decreased (\(\theta<0^{\circ}\), \(\alpha<135^{\circ}\)):  
\[\theta_{\text{min}}=2\pi/t-\alpha=2\pi/t-(\pi/2+\pi/t)=\pi/t-\pi/2 \tag{3}\]  
In other words, the change in angle \(\theta\) during prestressing is limited by these two bounds. Also, in this study the material is linearly elastic, and strains are assumed to be small, although the displacements, as in every tensegrity, may be large. It is also worth mentioning that non-minimal prismatic tensegrities possess three self-stress states, whereas the minimal ones possess only one--see [61]. This means that there are three possible ways of prestressing these structures, as well as infinite linear combinations of these three states, and each of them may or may not include all members of the structure. However, prestressing all members of a single group ensures that all of the three self-stress states are affected. In other words, prestressing whichever of the presented group enables us to prestress the whole structure in a symmetric way.