# Parametric Analysis of Tensegrity Plate-Like Structures: Part 2--Quantitative Analysis
## 4 Examples  
In this paper, the quantitative analysis of tensegrity plate-like structures is conducted. The plates built with modified Quartex modules for which qualitative analysis were performed in Part 1 [45] are considered. The coordinates of the structure nodes are the same ones as were used in [45] (the considered single module's dimensions allow it to fit into a unit cube). The qualitative analysis, which leads to identifying the characteristic features of tensegrity structures, does not depend on geometrical and mechanical characteristics, whereas the quantitative analysis does. Therefore, it was assumed that the structures are made of steel with the Young modulus \(E\ =\ 210\) GPa and density \(\rho\ =\ 7860\) kg/m\({}^{3}\). Halfen DETAN Rod System is used, and the following material and geometrical constants are assumed:  
* for cables (S460N): diameter \(\phi\ =\ 20\) mm, moment of inertia \(I\ =\ 7.85\cdot 10^{-9}\) m\({}^{4}\), cross-sectional area \(A\ =\ 3.14\cdot 10^{-4}\) m\({}^{2}\), load-bearing capacity (calculated with taking into account partial factor for structural resistance): \(N_{Rd}\ =\ 110.2\) kN; the maximum value of the tensile force \(N_{c,max}\) depends on the self-stress state,
* for struts (made of hot-finished circular hollow section, steel S355J2): diameter: \(\phi\ =\ 76.1\) mm, thickness: \(t\ =\ 2.9\) mm, moment of inertia: \(I\ =\ 4.47\cdot 10^{-7}\) m\({}^{4}\), cross-sectional area: \(A\ =\ 6.88\cdot 10^{-4}\) m\({}^{2}\), load-bearing capacity: \(N_{Rd}\ =\ 193.9\) kN; the maximum value of the compressed force is equal to \(N_{z,max}\ =\ -1\cdot 5\) (\(S\)--the initial prestress force).  
The influence of initial prestress level \(S\) on the displacements of the structure \(\mathbf{q}\) (the displacements are calculated using the second (II) and third (III) order theory) on the effort of the structure \(W_{\max}\) and on the global stiffness parameter \(GSP\) is considered. Starting from a single-module structure, more complex cases are sequentially analyzed. For all cases the self-stress state \(\mathbf{y}_{16}\) received in [45] for the single modified Quartex module is used.  
When applying the initial prestressing forces, the maximum effort of structures fluctuates between 0.86 and 0.96.  
The calculations were carried out using the original program. The procedure was written in the Mathematica environment, owing to which operations were simplified by using functions and commands implemented there. The calculation module is based on the finite element method. Defined in this work, the initially prestressed tensegrity element was used. The algebraic system of non-linear equations is solved by the Newton-Raphson method, implemented by the authors in the Mathematica environment. Firstly, the program finds the vector of nodal displacements using the second order theory. At this point, only the linear system of equations needs to be solved (in this case, the existing Mathematica toolbox was used). Next, the obtained vector of nodal displacements is used in the first iteration in order to find the results according to the third order theory. The program iterates as long as given precision is achieved (and solution is found) or the number of iterations is exceeded. If the solution is not found, the user must increase the number of iterations or change the initial vector of displacement, for example by changing the level of self-stress.  
The calculation procedure includes the analysis of geometrically non-linear rod systems and allows for the full analysis at any level of self-stress. The program makes it possible to freely define the geometry of the structure, material parameters, and loads, and then identify the self-stress state and track the behaviour of selected static and geometric parameters in the function of this state.