# A Finite Element Method Integrated with Terzaghi’s Principle to Estimate Settlement of a Building Due to Tunnel Construction

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Methodology

#### 2.2. Fundamentals of the Finite Element Method Applied in Porous Media

- Mass conservation of flow. In steady flow conditions:

- 2.
- Constitutive law. The formulation of flow in a porous media with a free surface, in a general case, requires the integration of the Navier–Stokes equations [23]. By establishing the hypothesis of steady flow, Darcy’s law is obtained in the presented form [59]. In this type of non-transient problem, Darcy’s law is the constitutive equation:

- Essential or Dirichlet boundary conditions: In the problem in porous media with steady flow towards the interior of the tunnel, it consists of knowing the piezometric head h in a part of the boundary ${\mathsf{\Gamma}}_{h}$. In practice, it involves knowing the water table level:

- 2.
- Natural boundary conditions, also known as Neumann’s conditions, consist of knowing the derivative of flow through a portion of the boundary ${\mathsf{\Gamma}}_{{v}_{n}}$. The porous media flow problem is usually restricted to an impermeable boundary condition, such as in the case of this study: the interaction between soil and an impermeable rock layer:

#### 2.3. Implementation of Terzaghi’s Principle

_{s}, of a material, and it can be defined as [56,58]:

_{sk}, which is the Young’s modulus referring to their mineral skeleton obtained in the laboratory. This is acceptable, considering the geological uncertainty, however, assimilating E to D (constrained modulus) means a serious error.

#### 2.4. Study Case

## 3. Results and Discussion

#### 3.1. Effective Stress for the Three Methodological States

#### 3.2. Total Stress and Effective Stress in the Domain

_{sk}) of each type of compared soil [70]. For soil type 4 (i.e., mixed-grained sand, dense [54]), the stress in the left side of the tunnel reached 2.11 MPa, while in the right side it reached only 1.55 MPa.

#### 3.3. Field of Displacement and Settlement

#### 3.4. Sensitivity Analisys of the Mat Foundation Settlement

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Muhammed, T.A.; Karim, F.R. The Influence of Drop Panel’s Dimensions on the Punching Shear Resistance in Ultra-High-Performance Fiber-Reinforced Concrete Flat Slabs. Construction
**2022**, 2, 55–65. [Google Scholar] [CrossRef] - Coduto, D.P.; Kitch, W.A.; Yeung, M.R. Foundation Design, 3rd ed.; California State Polytechnic University: Pomona, CA, USA, 2016; pp. 3–4, 401–418. [Google Scholar]
- Yun, B.; Yang, Z.; Jiang, Z. Key protection techniques adopted and analysis of influence on adjacent buildings due to the Bund Tunnel construction. Tunn. Undergr. Space Technol.
**2014**, 41, 24–34. [Google Scholar] [CrossRef] - Boscardin, M.D.; Cording, E.J. Building response to excavation-induced settlement. J. Geotech. Eng.
**1989**, 115, 1–21. [Google Scholar] [CrossRef] - Zhou, S.; Di, H.; Xiao, J.; Wang, P. Differential settlement and induced structural damage in a cut-and-cover subway tunnel in a soft deposit. J. Perform. Constr. Facil.
**2016**, 30, 04016028. [Google Scholar] [CrossRef] - Di, H.; Zhou, S.; Xiao, J.; Gong, Q.; Luo, Z. Investigation of the long-term settlement of a cut-and-cover metro tunnel in a soft deposit. Eng. Geol.
**2016**, 204, 33–40. [Google Scholar] [CrossRef] - Krishna, S.S.; Lokhande, R.D. Study on the Effect of Surface Subsidence Due to Tunneling Under Various Loading Conditions. Geotech. Geol. Eng.
**2022**, 40, 923–943. [Google Scholar] [CrossRef] - Rallu, A.; Berthoz, N.; Charlemagne, S.; Branque, D. Vibrations induced by tunnel boring machine in urban areas: In situ measurements and methodology of analysis. J. Rock Mech. Geotech. Eng.
**2023**, 15, 130–145. [Google Scholar] [CrossRef] - Tian, X.; Song, Z.; Wang, J. Study on the propagation law of tunnel blasting vibration in stratum and blasting vibration reduction technology. Soil Dyn. Earthq. Eng.
**2019**, 126, 105813. [Google Scholar] [CrossRef] - Beben, D.; Maleska, T.; Bobra, P.; Duda, J.; Anigacz, W. Influence of Traffic-Induced Vibrations on Humans and Residential Building—A Case Study. Int. J. Environ. Res. Public Health
**2022**, 19, 5441. [Google Scholar] [CrossRef] - Rubio, H.; Garcia-Prada, J.C.; Castejón, C.; Laniado, E. Dynamic analysis of rolling bearing system using Lagrangian model vs. FEM code. In Proceedings of the 12th World Congress in Mechanism and Machine Science, IFToMM, Besançon, France, 17–21 June 2007; pp. 205–210. [Google Scholar]
- Kontoni, D.P.N.; Farghaly, A.A. Mitigation of train-induced vibrations on nearby high-rise buildings by open or geofoam-filled trenches. J. Vibroeng.
**2020**, 22, 416–426. [Google Scholar] [CrossRef] - Ruiz, J.F.; Soares, P.J.; Costa, P.A.; Connolly, D.P. The effect of tunnel construction on future underground railway vibrations. Soil Dyn. Earthq. Eng.
**2019**, 125, 105756. [Google Scholar] [CrossRef] - Umaru, I.; Alkali, B.; Alhaji, M.M.; Alhassan, M.; Adejumo, T.E.; Jagaba, A.H. Structural Design of Field Plate Load Test Equipment to Determine In situ Bearing Capacity and Settlement of Clayey Soil. Construction
**2023**, 3, 23–39. [Google Scholar] - Guerriero, V. 1923–2023: One Century since Formulation of the Effective Stress Principle, the Consolidation Theory and Fluid–Porous-Solid Interaction Models. Geotechnics
**2022**, 2, 961–988. [Google Scholar] [CrossRef] - Terzaghi, K. Erdbaumechanik; F. Deuticke & Leipzig U.: Wien, Austria, 1925. [Google Scholar]
- Lee, J.; Salgado, R. Estimation of footing settlement in sand. Int. J. Geomech.
**2002**, 2, 1–28. [Google Scholar] [CrossRef] - Moh, Z.C.; Ju, D.H.; Hwang, R.N. Ground movements around tunnels in soft ground. In Proceedings of the International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground, London, UK, 15–17 April 1996; Available online: http://www.maaconsultants.com/common/publications/1995/1995-021.pdf (accessed on 28 February 2022).
- Pinto, F.; Whittle, A.J. Ground movements due to shallow tunnels in soft ground. I: Analytical solutions. J. Geotech. Geoenviron. Eng.
**2014**, 140, 04013040. [Google Scholar] [CrossRef] - Pinto, F.; Zymnis, D.M.; Whittle, A.J. Ground movements due to shallow tunnels in soft ground. II: Analytical Interpretation and Prediction. J. Geotech. Geoenviron. Eng.
**2014**, 140, 04013041. [Google Scholar] [CrossRef] - Wang, F.; Gou, B.; Zhang, Q.; Qin, Y.; Li, B. Evaluation of ground settlement in response to shield penetration using numerical and statistical methods: A metro tunnel construction case. Struct. Infrastruct. Eng.
**2016**, 12, 1024–1037. [Google Scholar] [CrossRef] - Korsawe, J.; Starke, G.; Wang, W.; Kolditz, O. Finite element analysis of poro-elastic consolidation in porous media: Standard and mixed approaches. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 1096–1115. [Google Scholar] [CrossRef] - Larese, A.; Rossi, R.; Oñate, E. Finite Element Modeling of Free Surface Flow in Variable Porosity Media. Arch. Computat. Methods Eng.
**2015**, 22, 637–653. [Google Scholar] [CrossRef] - Sandström, C.; Larsson, F.; Runesson, K.; Johansson, H. A two-scale finite element formulation of Stokes flow in porous media. Comput. Methods Appl. Mech. Eng.
**2013**, 261, 96–104. [Google Scholar] [CrossRef] - Liu, W.K.; Li, S.; Park, H.S. Eighty years of the finite element method: Birth, evolution, and future. Arch. Comput. Methods Eng.
**2022**, 29, 4431–4453. [Google Scholar] [CrossRef] - Callari, C.; Armero, F.; Abati, A. Strong discontinuities in partially saturated poroplastic solids. Comput. Methods. Appl. Mech. Eng.
**2010**, 199, 1513–1535. [Google Scholar] [CrossRef] - Callari, C.; Casini, S. Tunnels in saturated elasto-plastic soils: Three-dimensional validation of a plane simulation procedure. In Mechanical Modelling and Computational Issues in Civil Engineering; Maceri, F., Frémond, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2005; pp. 143–164. Available online: https://link.springer.com/content/pdf/10.1007/3-540-32399-6.pdf#page=153 (accessed on 9 March 2022).
- Zhang, L.; Wu, X.; Ji, W.; AbouRizk, S.M. Intelligent approach to estimation of tunnel-induced ground settlement using wavelet packet and support vector machines. J. Comput. Civ. Eng.
**2017**, 31, 04016053. [Google Scholar] [CrossRef] - Burland, J.B.; Burbidge, M.C.; Wilson, E.J.; Terzaghi, K. Settlement of foundations on sand and gravel. Proc. Inst. Civ. Eng. Civ.
**1985**, 78, 1325–1381. [Google Scholar] [CrossRef] - Maugeri, M.; Castelli, F.; Massimino, M.R.; Verona, G. Observed and computed settlements of two shallow foundations on sand. J. Geotech. Eng.
**1998**, 124, 595–605. [Google Scholar] [CrossRef] - Jiménez, J.A.; de Justo, J.L. Geotecnia y Cimientos: Propiedades de Los Suelos y de Las Rocas, 2nd ed.; Editorial Rueda: Madrid, Spain, 1975; pp. 181–217. [Google Scholar]
- Shi, X.S.; Yin, J.; Zhao, J. Elastic visco-plastic model for binary sand-clay mixtures with applications to one-dimensional finite strain consolidation analysis. J. Eng. Mech.
**2019**, 145, 04019059. [Google Scholar] [CrossRef] - van Rijn, L.C.; Barth, R. Settling and consolidation of soft mud–sand layers. J. Waterw.
**2019**, 145, 04018028. [Google Scholar] [CrossRef] - Feng, X.; Gourvenec, S. Consolidated undrained load-carrying capacity of subsea mudmats under combined loading in six degrees of freedom. Géotechnique
**2015**, 65, 563–575. [Google Scholar] [CrossRef] - Gu, X.; Chen, F.; Zhang, W.; Wang, Q.; Liu, H. Numerical investigation of pile responses induced by adjacent tunnel excavation in spatially variable clays. Undergr. Space
**2022**, 7, 911–927. [Google Scholar] [CrossRef] - Basile, F. Effects of tunnelling on pile foundations. Soils Found.
**2014**, 54, 280–295. [Google Scholar] [CrossRef] - Gokuldas, S.; Banerjee, S.; Nimbalkar, S.S. Effects of tunneling-induced ground movements on stability of piled raft foundation: Three-dimensional finite-element approach. Int. J. Geomech.
**2020**, 20, 04020104. [Google Scholar] [CrossRef] - Gepp, J.E.; de Santayana, F.P.; Martínez, Á.P. Bases del Anejo Nacional Español del Eurocódigo EC-7 (proyecto geotécnico). Hormigón y Acero
**2014**, 65, 47–62. [Google Scholar] [CrossRef] - Dirección General de Arquitectura, Vivienda y Suelo. Código Técnico de la Edificación. Documento Básico. Seguridad Estructural. Cimientos. CTE-SE-DBSE-C; Ministerio de Fomento. Gobierno de España: Madrid, Spain, 2019; Available online: https://www.codigotecnico.org/pdf/Documentos/SE/DBSE-C.pdf (accessed on 15 March 2021).
- Wu, Y.; Dong, L.; Shu, X.; Yang, Y.; She, W.; Ran, Q. A review on recent advances in the fabrication and evaluation of superhydrophobic concrete. Compos. B Eng.
**2022**, 237, 109867. [Google Scholar] [CrossRef] - Luciani, A.; Peila, D. Tunnel waterproofing: Available technologies and evaluation through risk analysis. Int. J. Civ. Eng.
**2019**, 17, 45–59. [Google Scholar] [CrossRef] - Su, K.; Zhou, Y.; Wu, H.; Shi, C.; Zhou, L. An analytical method for groundwater inflow into a drained circular tunnel. Groundwater
**2017**, 55, 712–721. [Google Scholar] [CrossRef] [PubMed] - Dan, M.M.; Tonnizam, M.E.; Komoo, I.; Madun, A.; Talib, M.A.; Ramadhansyah, P.J.; Taib, A.M.; Hasbollah, D.Z.; Yusof, Z.M.; Noorasyikin, M.N. Physico-mechanical characteristics of tropical granite boulders in weathered heterogeneous zones for geotechnical design purposes. Phys. Chem. Earth Parts A/B/C
**2023**, 129, 103311. [Google Scholar] [CrossRef] - Nikvar, A.; Katibeh, H.; Farhadian, H. Numerical analysis of steady-state groundwater inflow into Tabriz line 2 metro tunnel, northwestern Iran, with special consideration of model dimensions. Bull. Eng. Geol. Environ.
**2016**, 75, 1617–1627. [Google Scholar] [CrossRef] - Park, K.H.; Owatsiriwong, A.; Lee, J.G. Analytical solution for steady-state groundwater inflow into a drained circular tunnel in a semi-infinite aquifer: A revisit. Tunn. Undergr. Space Technol.
**2008**, 23, 206–209. [Google Scholar] [CrossRef] - Yoo, C. Interaction between Tunneling and Groundwater—Numerical Investigation Using Three Dimensional Stress–Pore-water pressure Coupled Analysis. J. Geotech. Geoenviron. Eng.
**2005**, 131, 240–250. [Google Scholar] [CrossRef] - Zienkiewicz, O.C.; Taylor, R.L. The Finite Element Method for Solid and Structural Mechanics, 6th ed.; Elsevier: Amsterdam, The Netherlands, 2005. [Google Scholar]
- Zienkiewicz, O.C.; Taylor, R.L.; Nithiarasu, P. The Finite Element Method for Fluid Dynamics, 7th ed.; Elsevier: Amsterdam, The Netherlands, 2013. [Google Scholar]
- González, J.A.; Lee, Y.S.; Park, K.C. Stabilized mixed displacement–pressure finite element formulation for linear hydrodynamic problems with free surfaces. Comput. Methods Appl. Mech. Eng.
**2017**, 319, 314–337. [Google Scholar] [CrossRef] - González, J.A.; Park, K.C.; Felippa, C.A. FEM and BEM coupling in elastostatics using localized Lagrange multipliers. Int. J. Numer. Methods. Eng.
**2007**, 69, 2058–2074. [Google Scholar] [CrossRef] - Bathe, K.J. Finite Element Method. In Wiley Encyclopedia of Computer Science and Engineering; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2008; pp. 2–4. [Google Scholar] [CrossRef]
- Kattan, P.I. MATLAB Guide to Finite Elements: An Interactive Approach; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2010. [Google Scholar] [CrossRef]
- Geuzaine, C.; Remacle, J.F. Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods. Eng.
**2009**, 79, 1309–1331. [Google Scholar] [CrossRef] - Terzaghi, K.; Peck, R.B.; Mesri, G. Soil Mechanics in Engineering Practice, 3rd ed.; John Wiley & Sons: Toronto, ON, Canada, 1996; pp. 22, 83–89. [Google Scholar]
- Wang, J.; Ye, X. A weak Galerkin finite element method for the Stokes equations. Adv. Comput. Math.
**2016**, 42, 155–174. [Google Scholar] [CrossRef] - Roy, E. Geotechnical Investigation Methods: A Field Guide for Geotechnical Engineers; CRC Press: Boca Raton, FL, USA, 2007; pp. 212–213. [Google Scholar]
- Remacle, J.F.; Henrotte, F.; Carrier-Baudouin, T.; Béchet, E.; Marchandise, E.; Geuzaine, C.; Mouton, T. A frontal Delaunay quad mesh generator using the L∞ norm. Int. J. Numer. Methods Eng.
**2013**, 94, 494–512. [Google Scholar] [CrossRef] - París, F. Teoría de la Elasticidad; Escuela Superior de Ingenieros Industriales, Grupo de Elasticidad y Resistencia de Materiales: Seville, Spain, 1998; pp. 85–86, 134–135. [Google Scholar]
- Bear, J. Dynamics of Fluids in Porous Media; Dover Publications, Inc.: New York, NY, USA, 1972; pp. 32–33. [Google Scholar]
- Sillerico, E.; Ezquerro, P.; Marchamalo, M.; Herrera, G.; Duro, J.; Martínez, R. Monitoring ground subsidence in urban environments: M-30 tunnels under Madrid City (Spain). Ingeniería e Investigación
**2015**, 35, 30–35. [Google Scholar] [CrossRef] - Alielahi, H.; Feizi, D. Numerical Study on Dynamic Effects of Soil-Tunnel-Structure Interaction. Int. J. Civ. Eng.
**2021**, 19, 1339–1355. [Google Scholar] [CrossRef] - Oreste, P.P. A numerical approach to the hyperstatic reaction method for the dimensioning of tunnel supports. Tunn. Undergr. Space Technol.
**2007**, 22, 185–205. [Google Scholar] [CrossRef] - Vermeer, P.A.; Ruse, N.; Marcher, T. Tunnel heading stability in drained ground. Felsbau
**2002**, 20, 8–18. Available online: https://structurae.net/en/literature/journal-article/tunnel-heading-stability-in-drained-ground (accessed on 28 February 2022). - Srivastav, A.; Pandey, V.H.R.; Kainthola, A.; Singh, P.K.; Dangwal, V.; Singh, T.N. Numerical analysis of a collapsed tunnel: A case study from NW Himalaya, India. Indian Geotech. J.
**2022**, 52, 132–144. [Google Scholar] [CrossRef] - Gong, C.; Ding, W.; Mosalam, K.M.; Günay, S.; Soga, K. Comparison of the structural behavior of reinforced concrete and steel fiber reinforced concrete tunnel segmental joints. Tunn. Undergr. Space Technol.
**2017**, 68, 38–57. [Google Scholar] [CrossRef] - Bheel, N.; Tafsirojjaman, T.; Liu, Y.; Awoyera, P.; Kumar, A.; Keerio, M.A. Experimental study on engineering properties of cement concrete reinforced with nylon and jute fibers. Buildings
**2021**, 11, 454. [Google Scholar] [CrossRef] - Gravina, R.J.; Li, J.; Smith, S.T.; Visintin, P. Environmental durability of FRP bar-to-concrete bond: Critical review. J. Compos. Constr.
**2020**, 24, 03120001. [Google Scholar] [CrossRef] - Yu, L.; Xia, J.; Gu, J.; Zhang, S.; Zhou, Y. Degradation Mechanism of Coal Gangue Concrete Suffering from Sulfate Attack in the Mine Environment. Materials
**2023**, 16, 1234. [Google Scholar] [CrossRef] [PubMed] - Zheng, G.; Cui, T.; Cheng, X.; Diao, Y.; Zhang, T.; Sun, J.; Ge, L. Study of the collapse mechanism of shield tunnels due to the failure of segments in sandy ground. Eng. Fail. Anal.
**2017**, 79, 464–490. [Google Scholar] [CrossRef] - DeJong, J.T.; Westgate, Z.J. Role of Initial State, Material Properties, and Confinement Condition on Local and Global Soil-Structure Interface Behavior. J. Geotech. Geoenviron. Eng.
**2009**, 135, 1646–1660. [Google Scholar] [CrossRef] - Sadeghi, J.; Esmaeili, M.H. Safe distance of cultural and historical buildings from subway lines. Soil Dyn. Earthq. Eng.
**2017**, 96, 89–103. [Google Scholar] [CrossRef] - Wu, Y.; Wang, K.; Zhang, L.; Peng, S. Sand-layer collapse treatment: An engineering example from Qingdao Metro subway tunnel. J. Clean. Prod.
**2018**, 197, 19–24. [Google Scholar] [CrossRef] - Zhao, Y.; Chen, X.; Hu, B.; Wang, P.; Li, W. Evolution of tunnel uplift induced by adjacent long and collinear excavation and an effective protective measure. Tunn. Undergr. Space Technol.
**2023**, 131, 104846. [Google Scholar] [CrossRef] - Dirección General de Ferrocarriles. Orden Circular nº4/2007. Criterios Para el diseño de Revestimientos, Soleras y Contrabóvedas en Túneles Ferroviarios; Ministerio de Fomento, Gobierno de España: Madrid, Spain, 2007; Available online: https://www.mitma.gob.es/recursos_mfom/ordenc42007mf.pdf (accessed on 15 March 2021).

**Figure 2.**Illustration of the two-dimensional finite element mesh for the three methodological states. Cross-section of the macroscale domain (200 m-wide by 55 m-deep). (

**a**) State 1: before the tunnelling, and (

**b**) states 2 and 3: after the tunnelling.

**Figure 3.**Cross-sectional distribution (selection of 100 m-wide by 55 m-deep) of the pore-water pressure (MPa). (

**a**) State 1: before the tunnelling, (

**b**) state 2: tunnel without groundwater inflow, and (

**c**) state 3: tunnel with groundwater inflow.

**Figure 5.**Cross-sectional distribution (domain of 200 m-wide by 55 m-deep) of the total stress (${\sigma}_{yy}$, in MPa). Comparison according to the combination of the methodological state and the load step (q, in MPa): (

**a**) state 1 for q = 0.40 MPa, (

**b**) state 1 for q = 0.80 MPa, (

**c**) state 3 for q = 0.40 MPa, and (

**d**) state 3 for q = 0.80 MPa.

**Figure 6.**Total stress (${\sigma}_{yy}$, in MPa) distribution around the crown and invert of the tunnel cross-section. Illustration of decompression: (

**a**) q = 0.10 MPa and (

**b**) q = 0.80 MPa.

**Figure 7.**Cross-sectional distribution (selection of 100 m-wide by 55 m-deep) of the total stress (${\sigma}_{yy}$, in MPa). Comparison according to soil type for q = 0.80 MPa. (

**a**) Soil 1: uniform sand, loose, (

**b**) soil 2: uniform sand, dense, (

**c**) soil 3: mixed-grained sand, loose, and (

**d**) soil 4: mixed-grained sand, dense.

**Figure 8.**Cross-sectional distribution (selection of 100 m-wide by 55 m-deep) of the effective stress (${{\sigma}^{\prime}}_{yy}$, in MPa). Comparison according to load step (q, in MPa): (

**a**) q = 0.10 MPa, (

**b**) q = 0.30 MPa, (

**c**) q = 0.60 MPa, and (

**d**) q = 0.80 MPa.

**Figure 9.**Cross-sectional distribution (selection of 100 m-wide by 55 m-deep) of the effective stress (MPa). Comparison according to state: (

**a**) ${{\sigma}^{\prime}}_{xx}$ to state 1, (

**b**) ${{\sigma}^{\prime}}_{xx}$ to state 2, (

**c**) ${{\sigma}^{\prime}}_{xx}$ to state 3, (

**d**) ${{\sigma}^{\prime}}_{yy}$ to state 1, (

**e**) ${{\sigma}^{\prime}}_{yy}$ to state 2, and (

**f**) ${{\sigma}^{\prime}}_{yy}$ to state 3.

**Figure 10.**Cross-sectional distribution (selection of 100 m-wide by 55 m-deep) of the shear stress (${\tau}_{xy}$, in MPa). Comparison according to state: (

**a**) state 1 and (

**b**) states 2 and 3.

**Figure 11.**Cross-sectional distribution (domain of 200 m-wide by 55 m-deep) of the influence area of groundwater inflow to the tunnel due to ${\Delta {\sigma}^{\prime}}_{yy}$.

**Figure 12.**Cross-sectional distribution (selection of 100 m-wide by 55 m-deep) of the settlement (in m) around the mat foundation and the tunnel (to state 2, i.e., without groundwater inflow). Comparison according to soil type for q = 0.80 MPa. (

**a**) Soil 1: uniform sand, loose, (

**b**) soil 2: uniform sand, dense, (

**c**) soil 3: mixed-grained sand, loose, and (

**d**) soil 4: mixed-grained sand, dense.

**Figure 13.**Cross-sectional distribution (selection of 100 m-wide by 55 m-deep) of the settlement (in m) around the mat foundation and the tunnel (Terzaghi’s principle). Comparison according to soil type for q = 0.80 MPa. (

**a**) Soil 1: uniform sand, loose, (

**b**) soil 2: uniform sand, dense, (

**c**) soil 3: mixed-grained sand, loose, and (

**d**) soil 4: mixed-grained sand, dense.

**Figure 14.**Settlement in the mat foundation (d

_{y}) versus horizontal distance (x) from the left end of the mat foundation. Comparison according to load step: (

**a**) settlement without groundwater inflow, (

**b**) settlement due to Terzaghi’s principle, and (

**c**) settlement with groundwater inflow. The axis scales have been kept linear to facilitate the visualisation of nonlinearities.

Soil Type | ||||
---|---|---|---|---|

Parameter used in the model | 1. Uniform sand, loose | 2. Uniform sand, dense | 3. Mixed-grained sand, loose | 4. Mixed-grained sand, dense |

Porosity, η (%) | 45.9 | 33.8 | 40.1 | 30.1 |

Void ratio, e (unitless) | 0.85 | 0.51 | 0.67 | 0.43 |

Water content, w (%) | 32.0 | 19.0 | 25.0 | 16.0 |

Dry density, ρ_{d} (mg/m^{3}) | 1.43 | 1.75 | 1.59 | 1.86 |

Specific gravity, Gs (unitless) | 2.65 | 2.65 | 2.66 | 2.66 |

Wet density, ρ_{wet} (mg/m^{3}) | 1.58 | 1.81 | 1.69 | 1.91 |

Saturated density, ρ_{sat} (mg/m^{3}) | 1.89 | 2.09 | 1.99 | 2.16 |

Young’s modulus, E (MPa) ^{a} | 25 | 80 | 40 | 100 |

Poisson’s ratio, ν (unitless) | 0.3 | 0.3 | 0.3 | 0.3 |

Bulk modulus, K_{s} (MPa) ^{b} | 21 | 67 | 33 | 83 |

Permeability, k (m/s) ^{c} | 10^{−2}–10^{−5} | 10^{−2}–10^{−5} | 10^{−2}–10^{−5} | 10^{−2}–10^{−5} |

**Table 2.**Effective stress (${{\sigma}^{\prime}}_{yy}$ ) and effective stress variation (${\Delta \sigma \prime}_{yy}$ ) at preeminent tunnel and mat foundation locations as a function of the load step (q) for soil type 4.

Location of Nodes | Load Step, q (MPa) | $\mathbf{Effective}\mathbf{Stress},{\mathit{\sigma}}_{\mathit{y}\mathit{y}}^{\prime}$ (MPa) | $\mathbf{Effective}\mathbf{Stress}\mathbf{Variation},{\Delta \mathit{\sigma}}_{\mathit{y}\mathit{y}}^{\prime}$ (MPa) | ||||
---|---|---|---|---|---|---|---|

State 1 | State 2 | State 3 | $\mathbf{States}1\to 2$ | $\mathbf{States}2\to 3$ | |||

Tunnel | Crown | 0.10 | 0.25 | −0.17 | 0.01 | −0.41 | 0.17 |

0.80 | 0.42 | −0.16 | 0.01 | −0.57 | 0.17 | ||

Invert | 0.10 | 0.38 | −0.29 | 0.03 | −0.67 | 0.32 | |

0.80 | 0.61 | −0.29 | 0.03 | −0.91 | 0.32 | ||

Mat foundation | Extreme left | 0.10 | 0.17 | 0.16 | 0.18 | −0.02 | 0.02 |

0.80 | 0.06 | 0.02 | 0.04 | −0.04 | 0.02 | ||

Extreme right | 0.10 | 0.16 | 0.31 | 0.34 | 0.14 | 0.03 | |

0.80 | 0.14 | 0.35 | 0.38 | 0.21 | 0.03 |

**Table 3.**Maximum settlement caused by tunnelling (with and without groundwater inflow) for soil type 4 (mixed-grained sand, dense) at prominent tunnel and mat foundation locations as a function of the load step (q).

Location of Nodes | Load Step, q (MPa) | Settlement without Infiltrations, ${\mathit{d}}_{0\mathit{y}}$ (cm) | Settlement Due to Terzaghi’s principle, ${{\mathit{d}}^{\prime}}_{\mathit{y}}$ (cm) | Settlement Total with Infiltration, ${\mathit{d}}_{\mathit{T}\mathit{y}}$ (cm) | |
---|---|---|---|---|---|

State 2 | State 3 | State 3 | |||

Tunnel | Crown | 0.10 | 7.01 | 0.94 | 7.95 |

0.80 | 10.90 | 1.46 | 12.36 | ||

Invert | 0.10 | −6.19 | −0.83 | −7.02 | |

0.80 | −8.20 | −1.10 | −9.30 | ||

Mat foundation | Extreme left | 0.10 | 0.32 | $\approx 0$ | 0.32 |

0.80 | 0.43 | 0.01 | 0.44 | ||

Extreme right | 0.10 | 4.28 | 0.14 | 4.42 | |

0.80 | 7.82 | 0.26 | 8.08 |

Location of Node | Load Step, q (MPa) | Settlement Due to Terzaghi’s Principle, ${{\mathit{d}}^{\prime}}_{\mathit{y}}$ (cm) | ||||
---|---|---|---|---|---|---|

Soil Type 1 | Soil Type 2 | Soil Type 3 | Soil Type 4 | |||

Tunnel | Crown | 0.10 | 12.76 | 1.40 | 5.29 | 0.94 |

0.40 | 16.33 | 1.75 | 6.69 | 1.16 | ||

0.80 | 21.10 | 2.22 | 8.55 | 1.46 | ||

Invert | 0.10 | −11.13 | −1.24 | −4.64 | −0.83 | |

0.40 | −12.98 | −1.42 | −5.36 | −0.94 | ||

0.80 | −15.44 | −1.66 | −6.32 | −1.10 | ||

Mat foundation | Left end | 0.10 | 0.10 | $\approx 0$ | 0.04 | $\approx 0$ |

0.40 | 0.11 | $\approx 0$ | 0.05 | $\approx 0$ | ||

0.80 | 0.14 | 0.01 | 0.06 | 0.01 | ||

Right end | 0.10 | 1.94 | 0.21 | 0.80 | 0.14 | |

0.40 | 2.75 | 0.29 | 1.12 | 0.19 | ||

0.80 | 3.82 | 0.40 | 1.54 | 0.26 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rodríguez, C.A.; Rodríguez-Pérez, Á.M.; López, R.; Hernández-Torres, J.A.; Caparrós-Mancera, J.J.
A Finite Element Method Integrated with Terzaghi’s Principle to Estimate Settlement of a Building Due to Tunnel Construction. *Buildings* **2023**, *13*, 1343.
https://doi.org/10.3390/buildings13051343

**AMA Style**

Rodríguez CA, Rodríguez-Pérez ÁM, López R, Hernández-Torres JA, Caparrós-Mancera JJ.
A Finite Element Method Integrated with Terzaghi’s Principle to Estimate Settlement of a Building Due to Tunnel Construction. *Buildings*. 2023; 13(5):1343.
https://doi.org/10.3390/buildings13051343

**Chicago/Turabian Style**

Rodríguez, César A., Ángel M. Rodríguez-Pérez, Raúl López, José Antonio Hernández-Torres, and Julio J. Caparrós-Mancera.
2023. "A Finite Element Method Integrated with Terzaghi’s Principle to Estimate Settlement of a Building Due to Tunnel Construction" *Buildings* 13, no. 5: 1343.
https://doi.org/10.3390/buildings13051343