## 2 Materials and Methods
### Mechanical Engineering Analysis
#### 2.2.8 Linear Static Analysis  
By way of introduction, and based on the numerous existing publications [27; 28; 29], which detail the linear static analysis within the finite element method framework, its fundamentals are explained below for a better understanding by the reader. Linear static analysis can be performed once it is verified that none of the natural frequencies of the different vibration modes is zero. For a better understanding of this stage and the functions integrated into the software, which displays other magnitudes in the post-processing stage, they are explained below, starting from the equation used to obtain the displacement vector (\(u\)):  
\[u=N\delta^{e} \tag{4}\]  
where for each element, \(N\) is the shape function, and \(\delta^{e}\) is the nodal displacement vector, providing the 3D relative position and rotation to the reference coordinate axes. With this, the value of the unit deformations (\(\varepsilon\)) is obtained:  
\[\varepsilon=\partial u=B\delta^{e} \tag{5}\]  
where the relationship between displacements of the element's nodes and the unit deformations at any point within the element are given by the matrix \(B\), which defines the unit deformation field existing inside the element, and \(\delta^{e}\) is the nodal displacement vector. Furthermore, linear static analysis is based on fundamental assumptions such as static equilibrium conditions, linear elasticity, and small deformations. Besides, [PERSON]'s law establishes the following:  
\[\sigma=E\cdot\varepsilon \tag{6}\]  
where the stress, given by \(\sigma\), is related to the strain, given by \(\varepsilon\), by the Young's modulus or elastic modulus of the material, \(E\). The assumption of small deformations implies that the original geometry does not change because the changes in the ensemble's geometry are negligible. Furthermore, the static equilibrium conditions require that the sum of forces and moments acting on the system be zero. Additionally, linear static analysis is governed by equilibrium equations where internal forces must be in equilibrium, stress-strain relationships derived from the linear static behavior of materials, and strain compatibility, ensuring they conform to the boundaries and interactions between components. All of this is defined under differential equations for an axial load \(P\) and an internal pressure \(p\) acting on the various elements where the load is applied. Besides, moment equilibrium, axial equilibrium, and transverse equilibrium are represented by Equations (7), (8), and (9), respectively:  
\[\frac{dM}{dx}-V=0 \tag{7}\]  
\[\frac{dN}{dx}+q=0 \tag{8}\]  
\[\frac{dV}{dx}+pA=0 \tag{9}\]where bending moment is given by \(M\), normal force by \(N\), the distributed force by \(q\), and the shear force by \(V\). Regarding the stress-strain relationship for axial strain, it is given by the following:  
\[\varepsilon_{x}=\frac{du}{dx} \tag{10}\]  
where \(u\) represents axial displacement. The strain for bending deformation is given by the following:  
\[\varepsilon_{xx}=\frac{d^{2}v}{dx^{2}} \tag{11}\]  
where \(v\) represents transverse displacement. On the other hand, to achieve the normal and transverse stresses, these stress-strain relationships are considered using [PERSON]'s law:  
\[\sigma_{x}=E\cdot\varepsilon_{x} \tag{12}\]  
\[\tau_{xy}=G\cdot\tau_{xy} \tag{13}\]  
where the shear modulus is given by \(G\) and the shear strain by \(\tau_{xy}\). Thus, considering the applied boundary conditions and loads, the stresses, displacements, and deformations at each point can be determined. As previously mentioned, for greater accuracy of results, a mesh convergence analysis must be performed, which consists of refining the mesh based on the von Mises stress results until the maximum value does not vary substantially (relative error between iterations less than 10%), thereby optimizing the model to avoid high computational cost. Furthermore, to conduct a proper analysis, the stress envelope must be established.