# Numerical Modelling of Structures Adjacent to Retaining Walls Subjected to Earthquake Loading

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Modelling

#### 2.1. Finite Element Discretisation

_{h}waves from the bedrock towards the ground surface following the recommendations of Semblat et al. [14] and Haigh et al. [15].

#### 2.2. Constitutive Models

- The variation of bulk and/or shear modulus is considered with mean confining effective stress.
- Cohesion can be included. The stress state will be cut off if the mean effective confining stress is more negative than the allowable cohesion. In the analyses presented in this, only a nominal cohesion of 100 Pa was used.
- The plastic potential can have a different slope with the yield surface.
- A smooth fit to the triaxial compression and triaxial extension state, so there is no corner or singularity in the π-plane [20].
- Strain hardening of the soil is incorporated in this model, so that the hardening of soil that occurs cycle by cycle during an earthquake load can be captured.

- As the Mohr-Coulomb model is a simple model, the shapes of the yield surface and plastic potential surface are similar.
- This model cannot be used for saturated soil conditions, as it does not capture the volumetric strains under cyclic loading, and hence, no excess pore pressure build-up occurs.

## 3. Static Equilibrium of the Retaining Wall

#### 3.1. Wall Deflections and Bending Moments

_{wall}used in these analyses. In Figure 4, the normalised wall deflections obtained for the case of the retaining wall only (i.e., “no structure”) are compared with the case when the sway frame structure was present on the backfill side (see Figure 3b). Clearly, the wall deflections increased substantially when the structure is in place behind the retaining wall. The wall tip deflection increased nearly by a factor of three.

^{3}) to obtain a nondimensioned variable. The same normalisation will be used for the results from dynamic analyses presented later. In Figure 5, it can be seen that the normalised bending moments increased substantially in the presence of the structure. Further, it can also be seen that the location of the peak bending moment was just below the excavation depth for the retaining wall-only case. This location of peak bending moment seems to shift downwards in the presence of the structure in the backfill. In general, the shapes of the bending moment curves are as expected in a static analysis.

#### 3.2. Horizontal Stresses and Earth Pressures

## 4. Dynamic Response of the Retaining Wall

#### 4.1. Acceleration Time Histories

#### 4.2. Dynamic Bending Moments

## 5. Soil Response

## 6. Dynamic Response of the Structure

## 7. Dynamic Earth Pressures

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Aarsleff. Available online: www.aarsleff.co.uk (accessed on 20 September 2020).
- Pelà, L.; Aprile, A.; Benedetti, A. Seismic assessment of masonry arch bridges. Eng. Struct.
**2009**, 31, 1777–1788. [Google Scholar] [CrossRef] - D’Amato, M.; Laterza, M.; Casamassima, V.M. Seismic performance evaluation of multi-span existing masonry arch bridge. Open Civ. Eng. J.
**2017**, 5, 1191–1207. [Google Scholar] [CrossRef] - Marcheggiani, L.; Clementi, F.; Formisano, A. Static and dynamic testing of highway bridges: A best practice example. J. Civ. Struct. Health Monit.
**2020**, 10, 43–56. [Google Scholar] - Madabhushi, S.P.G.; Patel, D.; Haigh, S.K. Geotechnical Aspects of the Bhuj Earthquake. In EEFIT Report; Madabhushi, S.P.G., Haigh, S.K., Eds.; Institution of Structural Engineers: London, UK, 2005; pp. 1–22. [Google Scholar]
- Zeng, X.; Steedman, R.S. On the behaviour of quay walls in earthquakes. Géotechnique
**1993**, 43, 417–431. [Google Scholar] [CrossRef] - Madabhushi, S.P.G.; Zeng, X. Seismic Response of Flexible Cantilever Retaining Walls with Dry Backfill. Geomech. Geoeng.
**2006**, 1, 275–290. [Google Scholar] [CrossRef] - Madabhushi, S.P.G.; Zeng, X. Simulating Seismic Response of Cantilever Retaining Walls with Saturated Backfill. J. Geotech. Geoenviron. Eng.
**2007**, 133, 539–549. [Google Scholar] [CrossRef] - Chan, A.H.C. A Generalised Fully Coupled Effective Stress Based Computer Procedure for Problems in Geomechanics; Swandyne User Manual: Swansea, UK, 1988. [Google Scholar]
- Cilingir, U.; Haigh, S.K.; Madabhushi, S.P.G.; Zeng, X. Seismic behaviour of anchored quay walls with dry backfill. J. Geotech. Geoenviron. Eng.
**2011**, 6, 227–235. [Google Scholar] [CrossRef] - Callisto, L.; Soccodato, F.M. Seismic design of flexible cantilevered retaining walls. J. Geotech. Geoenviron. Eng.
**2010**, 136, 344–354. [Google Scholar] [CrossRef] - Conti, R.; Viggiani, G.M. A new limit equilibrium method for the pseudo-static design of embedded cantilevered retaining walls. Soil Dyn. Earthq. Eng.
**2013**, 50, 143–150. [Google Scholar] [CrossRef][Green Version] - Yeganeh, N.; Bazaz, J.B.; Akhtarpour, A. Seismic analysis of the soil–structure interaction for a high rise building adjacent to deep excavation. Soil Dyn. Earthq. Eng.
**2015**, 79, 149–170. [Google Scholar] [CrossRef] - Semblat, J.F.; Brioist, J.J.; Luong, M.P. Mean Stress Dependent Damping of Seismic Waves in Sand; Proceedings of the Earthquake Geotechnical Engineering: Rotterdam, The Netherlands, 1999; pp. 95–100. [Google Scholar]
- Haigh, S.K.; Ghosh, B.; Madabhushi, S.P.G. A numerical investigation into effects of single and multiple frequency earthquake input motions. J. Soil Dyn. Earthq. Eng.
**2003**, 23, 691–704. [Google Scholar] - Desai, C.S.; Siriwardane, H.J. Constitutive Laws for Engineering Materials with Emphasis on Geologic Materials; Prentice Hall: Englewood Cliffs, NJ, USA, 1984. [Google Scholar]
- Wood, D.M. Geotechnical Modelling; Taylor and Francis: London, UK, 2001. [Google Scholar]
- Zhang, J.; Stewart, D.P.; Randolph, M.F. Modeling of Shallowly Embedded Offshore Pipelines in Calcareous Sand. J. Geotech. Geoenviron. Eng.
**2002**, 128, 363–371. [Google Scholar] [CrossRef] - Madabhushi, S.P.G.; Zeng, X. Behaviour of gravity quay walls subjected to earthquake loading. Part II: Numerical Modelling. JNL Geotech. Eng.
**1998**, 124, 418–428. [Google Scholar] [CrossRef] - Pande, G.N.; Zienkiewicz, O.C. Soil Mechanics, Transient & Cyclic Loads Constitutive Relations & Numerical Treatment; Wiley: Chichester, NH, USA, 1982. [Google Scholar]
- Ng, C.W.W.; Zheng, G.; Ni, J.; Zhou, C. Use of unsaturated small-strain soil stiffness to the design of wall deflection and ground movement adjacent to deep excavation. Comput. Geotech.
**2020**, 119, 103375. [Google Scholar] [CrossRef] - Madabhushi, S.P.G.; Haigh, S.K.; Houghton, N.E.; Gould, E. Development of a servo-hydraulic earthquake actuator for the Cambridge Turner Beam Centrifuge. Int. J. Phys. Model. Geotech.
**2012**, 12, 77–88. [Google Scholar] [CrossRef] - Mononobe, N. Earthquake-proof construction of masonry dams. In Proceedings of the World Engineering Conference, Tokyo, Japan, 19–20 October 1929; pp. 275–278. [Google Scholar]
- Okabe, S. General theory of earth pressures. J. Jpn. Soc. Civ. Eng.
**1926**, 12, 1083–1088. [Google Scholar]

**Figure 1.**A structure in close proximity to a retaining wall [1] (courtesy Aarsleff Ground Engineering Ltd. (Newark, UK)).

**Figure 3.**Finite element (FE) discretization: (

**a**) a retaining wall only and (

**b**) a retaining wall with an adjacent structure on the backfill.

**Figure 6.**Horizontal stresses in the soil for: (

**a**) the “‘retaining wall-only” case and (

**b**) the “structure on the backfill” case.

**Figure 10.**Dynamic bending moment profiles at different time instants: (

**a**) during an earlier cycle and (

**b**) during a later cycle.

**Figure 16.**Distribution of maximum horizontal earth pressures acting on the retaining wall. M-O: Mononobe-Okabe.

**Figure 17.**Evolution of the horizontal earth pressures through: (

**a**) an earlier earthquake loading cycle and (

**b**) a later earthquake loading cycle.

Structural Element | Flexural Stiffness (EI) (MNm ^{2}/m) | Density (kg/m ^{3}) | Bearing Pressure (kPa) |
---|---|---|---|

structural columns | 66.68 | 2800 | - |

slab | Rigid | 5906.7 | - |

strip footing (left) | Rigid | 2800 | 83.0 |

strip footing (right) | Rigid | 2800 | 117.0 |

retaining wall | 34.02 | 2800 | - |

Parameter | Value | Definition | ||
---|---|---|---|---|

Dry Sand | Interface Elements | Structure/Retaining Wall | ||

Constitutive model | Mohr-Coulomb V | Slip | Elastic | Type of constitutive model used |

Young’s modulus | 50 MPa | 50 MPa | 70 GPa | Soil stiffness for static equilibrium |

Young’s modulus (dynamic) | 50 MPa | 50 MPa | 70 GPa | Soil stiffness for damping in dynamic analyses |

Poisson’s ratio | 0.3 | 0.3 | 0.15 | Links strains in horizontal and vertical directions |

Uniaxial yield stress | 100 Pa | - | - | Cohesion |

Friction angle (critical state) | 30º | 16.4º | - | To obtain critical state failure line |

Dilatancy angle | 2º | - | - | To obtain the peak friction angle |

Work-hardening modulus | 100 | - | - | The slope of the stress vs. yield strain |

Void ratio | 0.8 | - | - | For the calculation of material density |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guan, X.; Madabhushi, G.S.P. Numerical Modelling of Structures Adjacent to Retaining Walls Subjected to Earthquake Loading. *Geosciences* **2020**, *10*, 486.
https://doi.org/10.3390/geosciences10120486

**AMA Style**

Guan X, Madabhushi GSP. Numerical Modelling of Structures Adjacent to Retaining Walls Subjected to Earthquake Loading. *Geosciences*. 2020; 10(12):486.
https://doi.org/10.3390/geosciences10120486

**Chicago/Turabian Style**

Guan, Xiaoyu, and Gopal S. P. Madabhushi. 2020. "Numerical Modelling of Structures Adjacent to Retaining Walls Subjected to Earthquake Loading" *Geosciences* 10, no. 12: 486.
https://doi.org/10.3390/geosciences10120486