# Parametric Analysis of Tensegrity Plate-Like Structures: Part 2--Quantitative Analysis
## 2 Mathematical Description
### Model of Tensegrity Structure  
The subject of consideration is a \(n\)-element space truss \((e\;=\;1,2,\ldots,n)\), with \(m\) degrees of freedom \(\mathbf{q}\big{(}\in\mathbb{R}^{m\times 1}\big{)}\):  
\[\mathbf{q}\;=\;[\;\;q_{1}\;\;\;q_{2}\;\;\;\ldots\;\;\;q_{m}\;\;]^{T}\text{.} \tag{8}\]  
The incremental static equilibrium equation for the structure takes the form:  
\[\mathbf{K}_{T}(\mathbf{q})\Delta\mathbf{q}\;=\;\Delta\mathbf{P}+\mathbf{R} \text{,} \tag{9}\]  
where \(\mathbf{K}_{T}(\mathbf{q})(\in\mathbb{R}^{m\times m})\) is the tangent stiffness matrix of a structure:  
\[\mathbf{K}_{T}(\mathbf{q})\;=\;\mathbf{K}_{L}+\mathbf{K}_{\text{G}}(S+N)+ \mathbf{K}_{NL}(\mathbf{q})\text{;}\;\;\;\;\;\;\;\mathbf{K}_{NL}(\mathbf{q}) \;=\;\mathbf{K}_{u1}+\mathbf{K}_{u2}\text{,} \tag{10}\]  
where \(\mathbf{K}_{\text{G}}(S+N)\) is the geometric stiffness matrix and \(\mathbf{K}_{NL}(\mathbf{q})\) is the non-linear displacement stiffness matrix. The residual force vector \(\mathbf{R}\big{(}\in\mathbb{R}^{m\times 1}\big{)}\) in (9) results from the aggregation. In equilibrium, it is equal to zero \((\mathbf{R}\;=\;0)\), whereas in a process of iteration, a norm \(\|\mathbf{R}\|\) is the \"distance\" from the equilibrium state. The iterative process converges if \(\|\mathbf{R}\|\to 0\). To solve the system of non-linear Equation (9), numerical iterative or incremental-iterative techniques [51; 52; 53; 54; 55] should be used. In this paper, the Newton-Raphson method [55] was applied. Influence of the Initial Prestress Level on Static Properties of Tensegrity Structures (Quantitative Analysis)  
The quantitative analysis, in the case of tensegrity structures, is a parametric analysis leading to the determination of the impact of the initial prestress level \(S\) on the behaviour of the structure. The first and crucial aspect of the parametric analyses is the determination of the prestress range, which is an individual feature of the structure. The value of the minimum prestress level \(S_{min}\) depends on the proper distribution of normal forces in the structure's elements, i.e., tensile forces must be present in cables and compressive forces must be present in struts. In some cases, the external load causes a different distribution of normal forces, which can be corrected only by introducing an appropriate level of the initial prestress. In turn, the value of the maximum prestress level \(S_{max}\) depends on the load-bearing capacity of the most stressed elements. The static parametric analysis contains assessment of the impact of the initial prestress level on the displacements of the structure \(\mathbf{q}\) and the effort of the structure:  
\[W_{\max}\ =\ N_{\max}/N_{Rd}, \tag{11}\]  
where \(N_{\max}\) is the maximal normal force, and \(N_{Rd}\) is the load-bearing capacity. Normal forces \(\mathbf{N}\) are determined as a function of the initial prestress forces \(S\):  
\[\mathbf{N}\ =\ \mathbf{y}_{i}S, \tag{12}\]  
where \(\mathbf{y}_{i}\) is the normalised vector of self-stress state determined in the qualitative analysis, which was performed in Part 1 [45]. Due to the fact that the assessment of the behaviour of the displacements is not sufficient for determination of structural rigidity, it seems reasonable to introduce a parameter that will determine the impact of self-stress state on the total rigidity of the structure at a given load. In the literature on tensegrity structures, however, any parameter characterizing the change in rigidity has not been found, so the so-called global stiffness parameter (\(GSP\)) is proposed in this paper.