Figure 1. Schematic of tunnelling-induced soil–pile interaction A pile deformation diagram shows a shear layer, initial pile, deformed pile, springs, ground surface, and dimensions H, L, D, R, x, z, and x 0. The diagram shows a ground surface line, a shear layer, an initial pile, and a deform... More about this image found in Schematic of tunnelling-induced soil–pile interaction A pile deformatio...

Figure 2. Physics-informed extreme learning machine framework for soil–pile interaction (ELM, extreme learning machine) A workflow diagram shows an E L M model using input z tilde, fixed weights, hard constraints, loss functions, optimisation, and predictions w, theta, M tilde, and Q tilde. Th... More about this image found in Physics-informed extreme learning machine framework for soil–pile interacti...

Figure 3. Comparison of pile lateral deflections and bending moments between the physics-informed extreme learning machine ( PIELM ) and the boundary element method ( BEM ): (a) lateral displacements; (b) bending moments A two-panel plot compares lateral displacement and bending moment along z,... More about this image found in Comparison of pile lateral deflections and bending moments between the phys...

Figure 4. Comparison of pile lateral deflections, bending moments, and shear forces between the physics-informed extreme learning machine (PIELM) and the finite-difference method (FDM) under different boundary conditions: (a) lateral deflection; (b) bending moment; (c) shear force for the pile fre... More about this image found in Comparison of pile lateral deflections, bending moments, and shear forces b...

Figure 5. Schematic representation of data-monitoring locations A pile monitoring layout shows sensor groups S 1 to S 4 beside a pile, plus longer vertical layouts labelled S 5, S 6, and S 7. The layout shows a vertical pile beside 4 staggered sensor groups labelled S 1, S 2, S 3, and S 4. Eac... More about this image found in Schematic representation of data-monitoring locations A pile monitoring ...

Figure 6. Distribution of absolute errors with data-monitoring locations: (a) lateral displacements for series 1–4 and 7; (b) bending moments for series 1–4 and 7; (c) lateral displacements for series 5–7; (d) bending moments for series 5–7 A 4-panel absolute error plot compares sensor layouts ... More about this image found in Distribution of absolute errors with data-monitoring locations: (a) lateral...

Figure 7. Pile deflection under different noise levels: (a) 10%; (b) 20% (FDM, finite-difference method; PIELM, physics-informed extreme learning machine) A two-panel displacement plot compares F D M and P I E L M curves for w across z, with 10 per cent noise in Panel A and 20 per cent noise in... More about this image found in Pile deflection under different noise levels: (a) 10%; (b) 20% (FDM, finite...

Figure 8. Distribution of absolute errors with data-monitoring numbers: (a) lateral displacements; (b) bending moments A two-panel absolute error plot compares sensor layouts S 8 to S 13 for w and M across z from 0 to 25 metres. Both panels use vertical axis z: metres, ranging from 0 to 25 and... More about this image found in Distribution of absolute errors with data-monitoring numbers: (a) lateral d...

Figure 9. Evolution of relative L ₂ error with data-monitoring point numbers: (a) lateral displacements; (b) bending moments A two-panel line chart compares relative L 2 errors for w and M across data-monitoring points, with 0, 5, 10, and 20 per cent noise. Both panels use the horizontal axis Number of data-monitoring points, ranging from 0 to 10. Panel A has a logarithmic vertical axis labelled Relative L 2 for w, ranging from 1 times 10 to the power of negative 4 to 1. The 0 per cent noise curve starts near 0.04, drops steeply to about 0.001 at 5 points, and ends near 0.0008 at 10 points. The 5 per cent noise curve starts near 0.04, dips to about 0.02, and stays near 0.02 to 0.03 after 2 points. The 10 per cent noise curve starts near 0.04, stays around 0.03 to 0.04, and changes little after 2 points. The 20 per cent noise curve starts near 0.04, rises after 2 points, and ends near 0.08 at 10 points. Panel B has a logarithmic vertical axis labelled Relative L 2 for M, ranging from 0.001 to 1. The 0 per cent noise curve starts near 0.15, drops to about 0.02 at 5 points, and stays near 0.02 at 10 points. The 5 per cent noise curve starts near 0.15, stays near 0.08 to 0.12 after 2 points, and ends near 0.08 at 10 points. The 10 per cent noise curve stays near 0.12 to 0.16 across the data-monitoring points. The 20 per cent noise curve stays highest, ranging from about 0.16 to 0.30. More about this image found in Evolution of relative L ₂ error with data-moni...

Figure 10. Comparison with field test data of Lee et al. (1994) (FDM, finite-difference method; PIELM, physics-informed extreme learning machine) A lateral displacement plot compares F D M, P I E L M, monitoring data, and validation data against depth, with a tunnel axis at 15 metres. The horizontal axis reads Lateral displacement: millimetres and ranges from 0 to 18. The vertical axis reads z: metres and ranges from 0 to 30, increasing downward. The F D M dashed curve starts near 0 millimetres at 0 metres, rises to about 11 millimetres near 12 metres, reaches about 12 millimetres near the tunnel axis at 15 metres, then decreases to about 0 millimetres near 27 metres. The P I E L M curve starts near 0 millimetres at 0 metres, rises to about 10.5 millimetres near 15 metres, then decreases to about 0 millimetres near 27 metres. Monitoring data points appear at about 2 metres and 2 millimetres, 10 metres and 6 millimetres, 20 metres and 7 millimetres, and 28 metres and 1 millimetre. Validation data points range from about 5 to 25 metres and about 2 to 10 millimetres. A circular tunnel section is labelled R, with text reading R equals 4.125 metres. More about this image found in Comparison with field test data of Lee et al. (1994) (FD...

Figure 11. Comparison with centrifuge test data of Loganathan et al. (2000) : (a) test 1; (b) test 2; (c) test 3 (FDM, finite-difference method; PIELM, physics-informed extreme learning machine) A 3-panel lateral displacement plot compares F D M, P I E L M, monitoring data, and validation data for tunnel cases with R equals 3 metres. The 3 panels have a horizontal axis labelled Lateral displacement: millimetres, ranging from 0 to 12, and a vertical axis labelled z: metres, ranging from 0 to 21 and increasing downward. Each legend lists F D M, P I E L M, monitoring data, and validation data. Panel A shows a tunnel axis at 15 metres and R equals 3 metres. Its F D M curve starts near 3 millimetres at 0 metres, rises to about 7 millimetres near 14 metres, then drops to about 5 millimetres at 18 metres. Its P I E L M curve starts near 3 millimetres at 0 metres, rises to about 6 millimetres near 14 metres, then drops to about 5 millimetres at 18 metres. Panel B shows a tunnel axis at 18 metres and R equals 3 metres. Its F D M curve starts near 2 millimetres at 0 metres, then rises steadily to about 8 millimetres at 18 metres. Its P I E L M curve starts near 3 millimetres at 0 metres, dips slightly near 4 metres, then rises to about 7 millimetres at 18 metres. Panel C reads Tunnel axis equals 21 metres and R equals 3 metres. Its F D M curve starts near 2 millimetres at 0 metres, then rises to about 7 millimetres at 18 metres. Its P I E L M curve starts near 3 millimetres at 0 metres, then rises to about 5 millimetres at 18 metres. More about this image found in Comparison with centrifuge test data of Loganathan et al....

Figure 12. Discretisation of pile for finite-difference method A 3-panel lateral displacement plot compares F D M, P I E L M, monitoring data, and validation data for tunnel cases with R equals 3 metres. The 3 panels have a horizontal axis labelled Lateral displacement: millimetres, ranging f... More about this image found in Discretisation of pile for finite-difference method A 3-panel lateral d...

Figure 1. Longitudinal ground displacements with distance from excavation face, modified from Cao et al. (2020) A staged tunnelling sequence showing ground displacement development above a tunnel boring machine and tunnel lining installation. The staged tunnelling sequence illustrates ground displacement development during tunnel boring machine excavation and lining installation across 5 construction stages. A curved settlement profile extends downward from the ground surface above the advancing tunnel alignment. Vertical dashed lines separate Stage 1 through Stage 5 along the tunnel path. The tunnel boring machine, labelled T B M, advances leftward beneath the surface while completed tunnel lining segments extend behind the excavation front. An arrow labelled Tunnelling indicates excavation direction. Ground displacement progressively increases as excavation advances and stabilises after tunnel lining installation. More about this image found in Longitudinal ground displacements with distance from excavation face, modif...

Figure 2. Geometry and location of the ancient pagoda and tunnels: (a) 3D view of the project; (b) section view of the pagoda and tunnels (unit of measurement: m) A schematic showing the Xiangjisi Temple pagoda above twin metro tunnels with tunnel depth and spacing dimensions. The schematic il... More about this image found in Geometry and location of the ancient pagoda and tunnels: (a) 3D view of the...

Figure 3. Features of the Xiangjisi temple pagoda: (a) south facade; (b) gourd-shaped finial; (c) carved body A multi-panel view showing the Xiangjisi Temple pagoda and detailed architectural carvings on the tower structure. The multi-panel view presents the ancient pagoda at Xiangjisi Temple ... More about this image found in Features of the Xiangjisi temple pagoda: (a) south facade; (b) gourd-shaped...

Figure 4. Geological profile according to cone penetration test A set of soil investigation plots showing cone tip resistance, sleeve friction, and geological profile with tunnel location. The set of soil investigation plots presents cone penetration test data and a geological profile surround... More about this image found in Geological profile according to cone penetration test A set of soil inve...

Figure 5. Instrumentation and monitoring layout (all units in m): (a) layout of monitoring points; (b) points around the pagoda; (c) deep horizontal displacement inclinometer tubes; (d) pore water pressure piezometers A four-panel monitoring layout showing settlement, displacement, inclinometer... More about this image found in Instrumentation and monitoring layout (all units in m): (a) layout of monit...