# Assessing the Sustainability of Alternative Shaft Construction Methods
## 3 Numerical Model  
### Details of the Analysis  
To provide insights into the sustainability of the construction method employed in the excavation of the WA2 shaft, a numerical model was created using PLAXIS 2D v22.02 [38]. To incorporate all aspects of the problem in the simulation, a coupled consolidation analysis was performed while taking advantage of the axisymmetric nature of the problem. 15-noded triangular elements with 12 Gaussian integration points were employed for determining the displacements and stress states, respectively, in the solid elements, while for the diaphragm wall, 5-noded plate elements were employed. As can be seen in Figure 2, the mesh employed in the SBE analysis had a higher concentration of elements surrounding the diaphragm wall, where higher stress and strain variations are expected. Interface elements were used to simulate the soil-structure interaction and to prevent water flow through the diaphragm wall. Figure 1: Scheme of the stratigraphy and geometry of WA2 shaft. According to the information provided by the DPT administration [39], the WA2 shaft took approximately 5 months to excavate. To replicate the excavation process adequately, a linear advance rate was assumed in the numerical model, resulting in a value of 5 days per meter of excavation. An additional 15 days were assumed for the construction of the ring beam, which resulted in a total duration of 160 days. The simulation of the shaft excavation involved the following stages:  
1. Generation of the initial stress state (assumed as geostatic with K\({}_{0}\)-conditions);
2. Construction of the diaphragm wall (\"wished in place\");
3. Excavation to 14 m depth at a rate of 5 days/m;
4. Installation of the concrete ring beam (15 days);
5. Excavation to 29 m depth (base of the shaft) at a rate of 5 days/m. The initial ground conditions were generated using, as a reference, the parameters proposed by [PERSON] et al. [34]. A hydrostatic pore water pressure profile with the ground water table located at a depth of 2 m (bottom of UBrBC) was considered, together with the values of bulk unit weight (\(\gamma\)) and at-rest coefficient of earth pressure (K\({}_{0}\)), as presented in Table 1. As suggested by [PERSON] et al. [34] and [PERSON] et al. [40], the isotropic permeability (k) was considered for all layers, with values of 10\({}^{-8}\) m/s for the Dublin Brown Boulder Clay and 10\({}^{-9}\) m/s for all the other materials. In agreement with [PERSON] et al. [34], the behavior of the limestone was assumed to be linear elastic-perfectly plastic, with its stiffness being described by E = 3 GPa and \(\
u\) = 0.3, and its strength given by the Mohr-Coulomb failure criterion, with \(\varphi^{\prime}\) = 45\({}^{\circ}\) and \(c^{\prime}\) = 50 kPa, with non-associated plasticity characterized by \(\psi\) = 0\({}^{\circ}\). Similar to the model suggested by [PERSON] et al. [40] and [PERSON] et al. [41], and in contrast with the linear elastic-perfectly plastic model employed by [PERSON] et al. [34], the mechanical response of DBC units was described herein using an expanded generalized version of the Modified Cam-Clay (MCC) model [42]. In the present case, this model introduces the nonlinear Hvorslev surface of [PERSON] et al. [43] on the dry side and is combined with the nonlinear small strain stiffness formulation of [PERSON] et al. [44]. As in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & \(\gamma\) **(kN/m\({}^{3}\))** & **K\({}_{0}\)** & **k (m/s)** \\ \hline UBrBC & 21.0 & 1.50 & \(1\times 10^{-8}\) \\ UBkBC & 22.5 & 1.50 & \(1\times 10^{-9}\) \\ LBrBC & 22.0 & 1.35 & \(1\times 10^{-8}\) \\ LBkBC & 22.5 & 1.20 & \(1\times 10^{-9}\) \\ Limestone & 26.0 & 1.00 & \(1\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General soil parameters adopted in the analysis. Figure 2: Mesh employed in the SBE analyses.