# Genetic Algorithm as a Tool for the Determination of the Self-Stress States of Tensegrity Domes
## 3 Results for Two-Dimensional Truss  
In order to compare the proposed methods, a simple two-dimensional truss (Figure 3a) is considered. The structure consists of six cables and two struts (\(\boldsymbol{n}=\boldsymbol{8}\)) and is characterized by eight degrees of freedom (\(\boldsymbol{m}=\boldsymbol{8}\)): \(\boldsymbol{q}=\left[ \begin{matrix}\boldsymbol{q_{3}}&\boldsymbol{q_{4}}& \boldsymbol{q_{5}}&\boldsymbol{q_{6}}&\boldsymbol{q_{9}}&\boldsymbol{q_{10}}& \boldsymbol{q_{11}}&\boldsymbol{q_{12}}\end{matrix} \right]^{T}\). The expansion matrix (1) takes the form \(\boldsymbol{B}\big{(}\in\mathbb{R}^{8\times 8}\big{)}\):  
\[\boldsymbol{B}=\left[ \begin{matrix}\boldsymbol{0.894}&\boldsymbol{0.447}& \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}& \boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0.894 }&-\boldsymbol{0.447}&\boldsymbol{0}&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{0}&-\boldsymbol{0.894}&\boldsymbol{0.447}& \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}& \boldsymbol{0}&-\boldsymbol{0.894}&-\boldsymbol{0.447}\\ -\boldsymbol{1}&\boldsymbol{0}&\boldsymbol{1}&\boldsymbol{0}&\boldsymbol{0}& \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{-1}&\boldsymbol{0}& \boldsymbol{1}&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{1}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{-1}& \boldsymbol{0}&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{1}&\boldsymbol{0}& \boldsymbol{0}&\boldsymbol{0}&-\boldsymbol{1}\end{matrix} \right]. \tag{12}\]  
In this case, the self-stress state \(\boldsymbol{S}=\left[ \begin{matrix}\boldsymbol{S_{1}}&\boldsymbol{S_{2}}& \boldsymbol{S_{3}}&\boldsymbol{S_{4}}&\boldsymbol{S_{5}}&\boldsymbol{S_{6}}& \boldsymbol{S_{7}}&\boldsymbol{S_{8}}\end{matrix} \right]^{T}\) can be simply determined by checking the static equilibrium of each node (Figure 3b):  
\[S_{2}=S_{3}=S_{4}=S_{1},\;S_{5}=S_{6}=S_{1}\text{sin}\,\boldsymbol{\alpha}, \;S_{7}=S_{8}=-S_{1}\text{cos}\,\boldsymbol{\alpha};\;\text{sin}\,\boldsymbol {\alpha}=\frac{2\sqrt{5}}{5}\,;\;\text{cos}\,\boldsymbol{\alpha}=\frac{\sqrt{ 5}}{5}. \tag{13}\]  
The forces (13) are normalized in such a way that the maximum compressive force in struts is equal to \(-\boldsymbol{1}\):  
\[\boldsymbol{S}=\left[\sqrt{5}\quad\sqrt{5}\quad\sqrt{5}\quad\sqrt{5}\quad \boldsymbol{2.0}\quad\boldsymbol{2.0}\quad\boldsymbol{-1.0}\quad\boldsymbol{-1. 0}\right]^{T} \tag{14}\]  
In turn, the spectral analysis of the compatibility matrix (5) leads to the following eigenvalues:  
\[\boldsymbol{\mu}=\left[ \begin{matrix}\boldsymbol{3.0}&\boldsymbol{2.86015}& \boldsymbol{2.30623}&\boldsymbol{2.0}&\boldsymbol{1.0}&\boldsymbol{0.69377}& \boldsymbol{0.13985}&\boldsymbol{0.0}\end{matrix} \right]^{T} \tag{15}\]  
The zero eigenvalue in (15)--\(\boldsymbol{\mu_{8}}=\boldsymbol{0}\) is responsible for the existence of one self-stress state considered as an eigenvector \(\boldsymbol{S}=\boldsymbol{y_{8}}(\boldsymbol{\mu_{8}}=\boldsymbol{0})\). The normalization of the eigenvector leads to:  
\[\boldsymbol{S}=\left[ \begin{matrix}\boldsymbol{2.23607}&\boldsymbol{2.23607}& \boldsymbol{2.23607}&\boldsymbol{2.23607}&\boldsymbol{2.0}&\boldsymbol{2.0}& \boldsymbol{-1.0}&\boldsymbol{-1.0}\end{matrix} \right]^{T}. \tag{16}\]  
Figure 3: (a) Geometry of two-dimensional truss; (b) equilibrium of nodes. The self-stress state (16) correctly identifies the type of elements (it is the same as (14)). Taking into account the geometric stiffness matrix (3), all eigenvalues of the stiffness matrix (2) are positive:  
\[\sigma=\left[ \begin{matrix}\mathbf{9.24}&\mathbf{7.26}&\mathbf{6.73}&\mathbf{6. 71}&\mathbf{3.16}&\mathbf{2.65}&\mathbf{2.63}&\mathbf{1.18}\end{matrix} \right]^{ \mathbf{T}}, \tag{17}\]  
which means that the stiffness matrix is positive definite and the structure is stable. Using the genetic algorithm, the selection of elements was completed manually.