# Influence of Deep Foundation Pit Excavation on Adjacent Pipelines: A Case Study in Nanjing, China
## 4 Numerical Simulations
### Meshes and Boundary Conditions
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Soil Layer** & **C** & **\(\varphi\) (\(\circ\))** & **\(K_{0}\)** & **\(\psi\) (\(\circ\))** & **\(P^{ref}\)** & **\(m\)** & \(v_{ur}\) & **\(\text{E}_{00}^{ref}\)** & **\(\text{E}_{00}^{ref}\)** & **\(\text{E}_{00}^{ref}\)** & **\(R_{f}\)** & **\(G_{0}^{ref}\)** & **\(70.7\)** \\ \hline \(\backslash\) Miscellaneous fill & 25.0 & 15.0 & 0.74 & 0 & 100 & 0.65 & 0.42 & 5.5 & 29.2 & 4.9 & 0.84 & 28.1 & 1.5 \\ \(\backslash\) Plain fill & 27.3 & 20.5 & 0.65 & 0 & 100 & 0.75 & 0.39 & 6.1 & 32.0 & 5.6 & 0.79 & 30.2 & 2.3 \\ \(\backslash\) Silty powder clay & 18.3 & 30.2 & 0.5 & 0.2 & 100 & 0.64 & 0.33 & 3.8 & 37.5 & 3.5 & 0.90 & 47.4 & 2.9 \\ \(\backslash\) Powder fine sand & 1.6 & 35.4 & 0.42 & 5.4 & 100 & 0.56 & 0.30 & 8.9 & 42.8 & 6.9 & 0.64 & 90.5 & 3.5 \\ \(\backslash\) Fine medium sand & 0.8 & 33.3 & 0.46 & 3.3 & 100 & 0.73 & 0.32 & 12.6 & 56.2 & 10.7 & 0.72 & 108.2 & 3.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of parameters of HSS model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Support Elements** & **Cross-Section (mm)** & **Unit Weight (kN\(\cdot\)m\({}^{-3}\))** & **Elastic Modulus, \(E\) (GPa)** & **Poisson’s Ratio** \\ \hline Diaphragm walls & 1200 & 26 & 30 & 0.15 \\ Inner support & 1000 \(\times\) 1000 & 26 & 30 & 0.15 \\ Column pile & \(R\) = 500 & 24 & 30 & 0.22 \\ floor slab & 500 & 26 & 30 & 0.15 \\ Superstructure beams & (300 \(\times\) 400) & & & \\ and columns & and & 24 & 30 & 0.22 \\ Lower pile of building & 1000 \(\times\) 1000 & 24 & 30 & 0.22 \\ Gas pipeline & \(R\) = 600 & 75 & 206 & 0.2 \\ Water supply pipeline & \(R\) = 500 & 9.5 & 0.8 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Physical and mechanical parameters of enclosure structures, buildings, and pipelines. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Definition** & **Unit** \\ \hline \(\text{E}_{50}^{ref}\) & Reference secant modulus & MPa \\ \(\text{E}_{ood}^{ref}\) & Reference tangent modulus & MPa \\ \(\text{E}_{ur}^{ref}\) & Reference loading and unloading modulus & MPa \\ \(m\) & Stiffness stress level related power exponent & / \\ \(c^{\prime}\) & Effective cohesion & kPa \\ \(\varphi^{\prime}\) & Effective internal friction angle & \(\circ\) \\ \(\psi\) & Dilatancy angle & \(\circ\) \\ \(R_{f}\) & damage ratio & / \\ \(v_{ur}\) & Load–unload Poisson’s ratio & / \\ \(C_{0}^{ref}\) & Initial shear modulus under reference confining pressure & MPa \\ \(K_{0}\) & Static side pressure coefficient & \\ \(70.7\) & Shear strain threshold & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Constitutive parameters of HSS model. The boundary conditions of the model were set so that the upper part remained unconstrained, while constraints were applied to the other five surfaces. This approach minimized the influence of boundary conditions on the calculation results. The mesh size of the model, \(L_{e}\), was determined using the following formula:  
\[L_{e}=\frac{r_{e}}{20}\times\sqrt{(x_{\max}-x_{\min})^{2}+(y_{\max}-y_{\min})^{2} +(z_{\max}-z_{\min})^{2}}.\]  
where \(x_{\max}\), \(x_{\min}\), \(y_{\max}\), \(y_{\min}\), \(z_{\max}\), and \(z_{\min}\) are the geometric model's boundary dimensions. The relative element size coefficient, \(r_{e}\), was 0.5. The finite element mesh's basic soil element was a tetrahedron with ten nodes. There were 149,715 elements and 24,905 nodes in the finite element model. For numerical modeling analysis, the construction steps of foundation pit excavation were transformed into numerical analysis steps. The project's spatial location and the plan dimensions of the on-site pit were considered. Factors such as the impact of the surrounding buildings on pit excavation and the extent of ground settlement due to excavation were also taken into account. As a result, the influence area was determined to be five times the pit's excavation depth. Consequently, the final dimensions of the model were set at 350 m in length, 300 m in width, and 70 m in depth.