## 2 Materials and Methods
### Methods  
#### 2.2.1 Finite Element Method (FEM)  
The Finite Element Method (FEM) is a numerical technique designed to solve complex engineering and physical problems by subdividing a large system into smaller, simpler parts known as finite elements. These elements are interconnected at nodes, forming a mesh that approximates the behaviour of the entire structure or system. The core concept of FEM \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Type** & **Param** & **Name** & **Description** & **Units** & **Levels** \\ \hline Input & P1 & Tip Height & Stiffener Tip Height & mm & [2, 15, 30] \\ Input & P2 & Separation & Diaphragm Separation & mm & [10, 25, 50] \\ Input & P3 & Web Plate Thickness & Leg Web Plate Thickness & mm & [8, 16, 32] \\ Input & P4 & Stiff Thickness & Stiffener Thickness & mm & [4, 8, 16] \\ Input & P5 & Line Pressure & Force in Z Component & N/mm & [\(-\)400, \(-\)300, \(-\)200] \\ Output & P6 & Equivalent Stress Max & Maximum Eq. Stress & MPa & Response \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and factorial design experiment. Figure 3: Experimental data design process. is to approximate a complex problem--such as stress, deformation, or heat transfer--by solving simpler, localised equations for each element. The solutions of these individual elements are then assembled to represent the entire system's behaviour. This approach is particularly effective in analysing structures [9; 10; 11; 12; 13; 14; 15] where obtaining analytical solutions is challenging. Here, FEM is used to investigate stress concentration at the tip of an angular profile stiffener welded at the centre of a thin plate, which is part of the leg web plate of a Ship-to-Shore (STS) container crane. Therefore, the first step is to discretize the continuous domain of the problem into triangles or quadrilaterals in 2D problems and tetrahedra or hexahedra in 3D problems. Within each finite element, the solution is approximated using simple functions (e.g., polynomials) defined over that element. These functions are typically called shape functions and are chosen to represent the behaviour of the solution within the element. FEM creates a system of equations by considering the contributions from all the finite elements. Each element contributes local equations based on physical laws. In this research, symmetry conditions have been utilised, as illustrated in Figure 1, and a stiffener has been isolated to exploit the symmetries of the model and the elastic problem. The stiffener has been delimited by half and at half the width between the stiffeners of the original panel (see Figures 1 and 2), resulting in dimensions \(a\times b\) equal to 962 mm \(\times\) 550 mm. To accurately capture the stress concentration at the stiffener's tip, a mesh refinement has been performed to model a portion of the plate within the region where the stiffener's tip is located, measuring 150 mm \(\times\) 200 mm, with a mesh refinement of 2 mm \(\times\) 2 mm grid. The tip is the final part or boundary of the stiffener where stress concentrations often occur due to the geometry and applied loads. The rest of the plate is modelled using a 10 mm \(\times\) 10 mm mesh grid. Therefore, that specific area of the stiffener needs a mesh refinement to accurately capture the stress concentration. The mesh type used was a mapped regular mesh using a four-node element with six degrees of freedom. Previous comparative evaluations of various FEM mesh models have demonstrated that a mesh size equivalent to the minimum element thickness is sufficient to accurately capture the anticipated behaviour. The use of a regular mesh, consisting of mapped quadrilateral elements, is justified by its superior ability to distribute stresses along gradients, thereby ensuring a precise determination of maximum stress values. The FEM model is formulated to determine the maximum Von Mises stress concentrated in the region of interest, specifically at the stiffener tip plate zone.