## 3 Mathematical Modelling, Theoretical Aspects
### Indirect Variational Method for the Study of the Conditions and Nature of Equilibrium  
The potential energy is defined as a function of a function, that is, \(\pi=\pi(v(x))\), a functional presented in Expression (45). The problem consists of determining the condition of the minimum functional \(\pi\) of one function \(v\) of one variable \(x\)[20]. \[\pi[v(x)]=\int_{0}^{l}\Phi(x,v,v^{\prime},v^{\prime\prime})dx \tag{45}\]  
In Expression (45), \(v(x)\) is presumed to be continuous, with continuous first four derivates and also appropriate boundary conditions. The mathematical form of the total potential energy is presented in Expression (46) [23]. \[\pi=U-V \tag{46}\]  
In Expression (46), \(U\) denotes the strain energy, \(V\) represents the potential of the external forces or the mechanical work of external force applied in joints. In this study, \(U\) is defined in Expression (47) through two energetic quantities. \[U=L_{ef}-L_{ext}(Q) \tag{47}\]  
It is assumed that the equilibrium position corresponds to an extreme value of the total potential energy, a condition presented in Equation (48) [23]. \[\delta\pi=\{0\} \tag{48}\]  
Equation (48) represents the condition for the Euler equation, with the proper boundary conditions. In mechanics, Equation (48) is generally equivalent to the equilibrium condition [23]. For the study of the nature of equilibrium, there are considered two stages of deformation, for the intensity \(\lambda\) of the axial force: stage 1, which is defined by displacements \(v(x)\) and total potential energy \(\pi(v(x)\lambda)\); stage 2, which corresponds to displacements \(v(x)+\delta v(x)\) and total potential energy \(\pi(v(x)+\delta v(x),\lambda)\). The variation of the total potential energy by passing from stage 1 to stage 2 can be expressed in Expression (49) [23]. \[\Delta\pi=\pi(v(x)+\delta v(x),\lambda)-\pi(v(x),\lambda) \tag{49}\]  
In Expression (49), \(\Delta\pi\) is a continuous n-time differentiable function defined on the interval \([0\ L]\), which is expanded into a Taylor series, presented in Expression (50) [20]. \[\Delta\pi=\delta\pi+\delta^{2}\pi+\delta^{3}\pi+\ldots=\delta\pi+\delta^{2} \pi+T_{s}(\pi) \tag{50}\]  
In Expression (50), \(\delta\pi\) set the first-order derivative of the total potential energy, \(\delta^{2}\pi\) denotes the second-order derivative of the total potential energy and \(T_{s}(\pi)\) representsthe higher-order derivative of the total potential energy, terms which are neglected in the modelling process [23]. The beam-type element in this study has a single degree of freedom \(v(x)\), which defines the system's state. The terms obtained after the expansion into a Taylor series of \(\Delta\tau\), are presented in Expressions (51a)-(51c) [20]. \[\delta\pi=\frac{1}{1!}\sum_{i}\sum_{j}\frac{\partial\pi(v,\lambda)}{\partial v_ {i}}\partial v_{i} \tag{51a}\] \[\delta^{2}\pi=\frac{1}{2!}\sum_{i}\sum_{j}\frac{\partial^{2}\pi(v,\lambda)}{ \partial v_{i}\partial v_{j}}\partial v_{j}\] (51b) \[\delta^{3}\pi=\frac{1}{3!}\sum_{i}\sum_{j}\sum_{k}\frac{\partial^{3}\pi(v, \lambda)}{\partial v_{i}\partial v_{j}\partial v_{k}}\partial v_{i}\partial v _{j}\partial v_{k} \tag{51c}\]  
With respect to the condition of equilibrium, presented in Equation (48), and to the fact that the potential of the external forces, for the case of constant loads, is expressed as \(V=\sum_{1\leq i\leq n}F_{i}v_{i}\), where \(F_{i}\) represents a force associated with displacement \(v_{i}\), which is a linear displacement function, whose high-order derivative is canceled, the variation of the total potential energy between stage 1 and 2 is equivalent to the second-order derivative of the strain energy, presented in Expression (52) [20].