## 2 Self-Control: Geometrically Non-Linear Analysis  
Due to the infinitesimal mechanisms that occur in tensegrity systems, it is possible to control their properties by adjusting the values of pre-stressing forces in structural members. However, the adjustment of self-stress is not the only method of changing the properties of these structures. Thanks to their inherent smart features, they are capable of self-control: they stiffen under the applied external load, which causes displacements that are consistent with the infinitesimal mechanism modes. In order to examine this unique property of tensegrities, a geometrically non-linear analysis has to be used. Analysis according to the second order theory, which includes self-stress, but neglects geometrical non-linearities, allows identifying infinitesimal mechanisms and determining self-stress states of the structure, but does not take into account self-stiffening of the structure. Therefore, the analyses presented in this section are performed according to the geometrically non-linear theory, referred to by the authors as the third order theory. Three structures were analyzed:  
* Structure No. 1: a tensegrity-like structure consisting of two pin-jointed cables,
* Structure No. 2: a typical tensegrity module, three-strut simplex,
* Structure No. 3: a tensegrity plate-like structure built from fourteen three-strut simplex modules. Structural members were chosen from the rod system Detan by Halfen [28]. The following sections were used:  
* Structure No. 1: tension rods \(d\) = 30 mm made of steel S460N as cables,
* Structure No. 2 and 3: tension rods \(d\) = 20 mm made of steel S460N as cables and compression rods \(D\) = 76.1 mm (outer diameter) and \(t\) = 2.9 mm (wall thickness) made of steel S355J2 as struts. The calculations were performed in SOFiStiK software (educational license), using the finite element method [29; 30; 31; 32]. Structure No. 1 (Figure 1), built from two pin-jointed cables of a length of 1 m each, was loaded with a concentrated force \(P\) applied to the middle node. In order to eliminate the singularity of the problem, pre-stressing forces \(S_{0}\) were introduced. The scope of the analysis was to evaluate the vertical displacement of the loaded node of the structure and the value of internal forces in cables. The diagram (Figure 2) shows how self-stress forces (\(S_{0}\)) influence the analyzed displacement \(\delta\), with two levels of loading: \(P\) = 10 kN and \(P\) = 20 kN are considered. The calculations were performed  
Figure 1: Tensegrity-like structure with two pin-jointed cables (dimensions units (mm)). twice, according to the second and third order theory. When no self-stress is applied, the displacement determined with the use of the second order theory approaches infinity, which results from the infinitesimal geometrical instability of the system. Application of the geometrically non-linear analysis allows taking into account the self-stiffening of the structure under the applied external load. External load, similarly to self-stress, causes stiffening and pre-stressing of the structure. Its influence on the global stiffness of the system depends on the ratio between pre-stressing forces in cables and the forces caused by self-stiffening under the applied load. The diagram presented below (Figure 3) shows the relation between the final internal force (\(S_{0}\)) in the cable and the value of self-stress (\(S_{0}\)), with four levels of external load considered. The analysis was performed according to the geometrically non-linear theory. Comparison of the internal force in the cable with the pre-stressing force (curve obtained for \(P=0\) kN) indicates that self-stress forces have a bigger influence on the global stiffness of the structure when the values of external forces \(P\) are small. When \(P\) increases, external loading starts to play an important role in self-stiffening and pre-stressing of the system. Similar analysis was performed for Structure No. 2: a three-strut simplex module (Figure 4).