## 3 Mathematical Modelling, Theoretical Aspects
### Stiffness Matrix Definition for Beam-Type Element with Elastic Restrains  
The characterization of a structure or of an element from a structure, presented in Figure 2, through the geometric aspect, refers to the establishment of the deformed shape and defining the parameters which can define that specific deformed shape. The geometric characterization of structural elements highlights the fact that the deformed shape can be defined if the displacement field is known, displacements which can be translations or rotations. As a consequence of the geometric aspect, it is possible to choose the displacements as independent parameters to describe the elastic equilibrium state. The matrix-based formulation of the stability problem assumes the definition of the displacement field, which represents a function approximation problem. The task is to find a function that closely matches the displacement field. In the case of a bi-dimensional stability problem, for the beam-type element presented in Figure 2, the approximation problem assumes the approximation of the displacement field through a polynomial function of degree \(n-1\), presented in Expression (16). \[v(x)=\alpha_{1}+\alpha_{2}\cdot x+\alpha_{3}\cdot x^{2}+\ldots+\alpha_{k}\cdot x ^{k+1}+\ldots\alpha_{n}\cdot x^{n-1}=\sum_{k=1}^{n-1}\alpha_{k}x^{k+1}=[F]\{\alpha\} \tag{16}\]  
\[\left\{ \begin{array}{c}\
u(0)=\alpha_{1}+\alpha_{2}\cdot 0+\alpha_{3}\cdot 0^{2}+ \ \ldots\ +\alpha_{k}\cdot 0^{k+1}+\ \ldots\ +\alpha_{n}\cdot 0^{n-1}=\
u_{i}\\ \varphi(0)=\ \alpha_{2}+2\cdot\alpha_{3}\cdot 0^{1}+\cdots+(k+1)\cdot\alpha_{k} \cdot 0^{k}+\cdots(n-1)\cdot\alpha_{n}\cdot 0^{n-2}=\varphi_{i}\\ \
u(l)=\alpha_{1}+\alpha_{2}\cdot i+\alpha_{3}\cdot l^{2}+\ \ldots\ +\alpha_{k}\cdot l^{k+1}+\ \ldots\ +\alpha_{n}\cdot l^{n-1}=\
u_{j}\\ \varphi(l)=\ \alpha_{2}+2\cdot\alpha_{3}\cdot l^{1}+\cdots+(k+1)\cdot\alpha_{k} \cdot l^{k}+\cdots(n-1)\cdot\alpha_{n}\cdot l^{n-2}=\varphi_{j}\\ \alpha_{5}=a_{5}\\ \alpha_{6}=a_{6}\\ \ldots\\ \alpha_{n}=a_{n}\end{array} \right. \tag{20}\]  
The linear system of Equations presented in (20), is figured as a matrix equation in Equation (21). \[[H]\{\alpha\}=\{a_{e}\} \tag{21}\]  
In Equation (21), the matrix \([H]\) represents the coefficient matrix of the limit displacement boundary condition system. By expressing the vector of the polynomial function coefficients and by introducing it in Expression (16), the interpolation function matrix \([N]\) can be defined, as in Expression (22) and the polynomial function which describes the displacement field can be defined as in Expression (23). \[v(x)=[F][H]^{-1}\{a_{e}\}=[N]\{a_{e}\} \tag{22}\]  
\[v(x)=N_{1}(x)a_{1}+N_{2}(x)a_{2}+\ldots+N_{k}(x)a_{k}+\ldots+N_{n}(x)a_{n}=[N] \{a_{e}\} \tag{23}\]  
In Expression (23), \([N]=[N_{1}(x)\ N_{2}(x)\ldots N_{k}(x)\ldots N_{n}(x)\ ]_{1\leq k\leq n}\) represents the nodal interpolation function matrix, which links the displacements of the beam-type element with the nodal displacements, \([a_{e}]=[a_{1}\ a_{2}\ldots a_{k}\ldots a_{n}]^{T}_{1\leq k\leq n}\) is the vector of nodal and non-nodal displacements. Given Expression (23), Figure 3 is highlights the significance of non-nodal parameters/displacements through 4 interpolation functions of a degree \(n-1=9\) polynomial function. Given Expression (23) and the geometric meaning of the differential relations between displacements, presented in Figure 4, the slope of the deflected beam axis \(\varphi(x)\), expressed in Expression (24), the strain \(\varepsilon(x)\), expressed in Expression (25), and the reaction forces induced in the beam-type element \(\sigma(x)\), expressed in Expression (26), can be defined as a function of nodal and non-nodal displacements \(\{a_{e}\}\). Figure 3: Interpolation functions \(N_{k}(x)\), for \(k=[1\ 3\ 7\ 10]\). \[\varphi(x)=\frac{dv(x)}{dx}=\frac{d[N]}{dx}\{a_{e}\}=[S]\{a_{e}\} \tag{24}\]  
In Expression (24), \([S]\) represents a matrix that links the slope of the deflected beam axis \(\varphi(x)\) with the nodal and non-nodal displacement vector \(\{a_{e}\}\).