# Stability Assessment of Earth Retaining Structures under Static and Seismic Conditions

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## Abstract

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## 1. Introduction

## 2. Seismic Active Thrust Using the Pseudo-Dynamic Method

_{p}/V

_{s}= 1.87 is assumed to be valid for the most geological materials. The period of lateral shaking is defined as T = 2π/ω, where ω is the angular frequency. A planer rupture surface inclined at an angle, α with the horizontal is considered.

_{h}g, and harmonic vertical seismic acceleration of amplitude k

_{v}g, where g is the acceleration due to gravity. The acceleration at any depth z and time t below the top of the wall can be expressed as,

_{b}is the unit weight of the backfill. The total horizontal and vertical inertia forces acting within the failure zone ABD is expressed as:

_{ae}(t) is obtained by resolving the forces on the wedge. Considering the equilibrium of the forces and hence P

_{ae}(t) is expressed as follows,

## 3. Seismic Active Thrust Using the Modified Pseudo-Dynamic Method

_{l}and η

_{s}= Viscosities; $\overline{u}$ = Displacement vector and κ = div ($\overline{u}$). The solution of a plane wave propagating vertically in a Kelvin-Voigt homogeneous medium for the computation of earth pressure under seismic condition is obtained. The proposed simplified form of Equation (1) is given below

_{s}= shear strain; η

_{s}= viscosity, G = shear modulus. For a horizontal harmonic base shaking of angular frequency ω; ${\eta}_{s}$ = $\frac{2G\xi}{\omega}$ where $\xi $ = damping ratio. Equation (9) is solved for a harmonic horizontal shaking. At the ground surface (z = 0), shear stress is zero. Assuming a base displacement ${u}_{hb}={u}_{h0}\mathrm{cos}\left(\omega t\right)$, the horizontal displacement is obtained as

_{s}, S

_{s}, C

_{sz}, and S

_{sz}depend on normalized frequency (ωH/V

_{s}) and damping ratio (ξ).

_{h}= Horizontal seismic acceleration coefficient at the base.

_{h}, G, and η

_{s}are replaced by u

_{v}, E

_{c}(= λ + 2G) and η

_{p}= (η

_{l}+ 2η

_{s}), respectively. Similarly, Equation (10) is solved for the case of harmonic vertical shaking. At the ground surface (z = 0), normal stress is zero. Assuming a base displacement ${u}_{vb}={u}_{v0}\mathrm{cos}\left(\omega t\right)$, the vertical displacement is obtained as:

_{p}, S

_{p}, C

_{pz}, and S

_{pz}depend on normalized frequency (ωH/V

_{p}) and damping ratio (ξ).

_{v}= Vertical seismic acceleration coefficient at the base. The total horizontal and vertical inertial forces acting within the active wedge ABD are obtained following the same procedure as described in the pseudo-dynamic method of analysis, however the only difference is that the expression of acceleration is used as given in Equations (17) and (21). The value of shear and primary wave velocity and damping ratio (ξ

_{s}) of the backfill material should be used.

#### 3.1. Stability of Rigid Retaining Wall Under Seismic Conditions Using the Modified Pseudo-Dynamic Method

_{hw}(z,t) and a

_{vw}(z,t) are obtained by putting the value of shear and primary wave velocity and damping ratio (ξ

_{w}) of the wall material in Equations (17) and (21). Using D’Alembert’s principle for inertial forces acting on the wall as shown in Figure 1, the weight of the wall required for equilibrium against sliding under seismic conditions is obtained as:

_{IE}(t) is the dynamic wall inertia factor given by

_{T}is defined as

_{I}is defined as

_{W}proposed for the design of the wall is defined as

_{w,static}is the weight of the wall required for equilibrium against sliding under static condition.

#### 3.2. Results and Discussion

_{W}. Figure 2a shows that the stability factors are in phase and they attain their maximum values almost at the same time of the instance at t/T = 0.5. Figure 2b shows the variation of the stability factors F

_{I}, F

_{T}, and F

_{W}for the given input parameters. The maximum value of F

_{W}is 1.41 and occurs at t/T = 0.081, whereas the maximum active thrust occurs at t/T = 0.402 and the value of F

_{W}is 1.02. It is clear that when F

_{W}has attained the maximum value, other two partial factors may not have attained their peak values.

_{W}for different sets of normalized frequencies (ωH/V

_{s}and ωH/V

_{sw}) keeping the other input parameters the same as in Figure 2a. The acceleration distribution along the depth is nonlinear in nature. It is worth mentioning that for the value of ωH/V

_{s}= 1.885 and ωH/V

_{sw}= 0.0754, the inertia force in the active wedge acts in the opposite direction to that mentioned in Figure 1, which in turn reduces the active thrust on the wall. On the contrary, the inertia force in the wall acts simultaneously in the direction shown in the Figure 1. The amplitude and phase of acceleration vary with depth in the modified pseudo-dynamic method (Pain et al. [52]). In the modified pseudo-dynamic method, the amplification of acceleration depends on the soil properties and no simplifying assumption of any amplification value is necessary. Seismic inertia forces are functions of normalized frequency (ωH/V) and damping ratio (ξ). At frequencies above the normalized fundamental frequency (ωH/V

_{s}= π/2), part of the active wedge moves in one direction while another part moves in the opposite direction (See Figure 3 for ωH/V

_{s}= 1.885). However, the active wedge moves in the same direction when ωH/V

_{s}< π/2. This is one of the unique aspects of the modified pseudo-dynamic method.

_{h}) exceeds 0.2. The values reported by Pain et al. [52] are higher than the values of Baziar et al. [41] for the given set of input parameters. This is attributed to the fact that the effect of amplification of acceleration is not considered by Baziar et al. [41]. This difference may change depending on the variation in input parameters.

## 4. Effect of Soil Arching on the Stability of Retaining Structures

#### 4.1. Analytical Model

_{aw}) is defined as:

_{c}) within the backfill, the lateral earth pressure within this depth is assumed to be zero. By integrating Equation (33) with respect to y from H

_{c}to H, the lateral active earth pressure force is obtained as given below:

#### 4.2. Results and Discussion

#### 4.3. Comparison with Other Studies

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Caltabiano, S.; Cascone, E.; Maugeri, M. Seismic stability of retaining walls with surcharge. Soil Dyn. Earthq. Eng.
**2000**, 20, 469–476. [Google Scholar] [CrossRef] - Caltabiano, S.; Cascone, E.; Maugeri, M. Static and seismic limit equilibrium analysis of sliding retaining walls under different surcharge conditions. Soil Dyn. Earthq. Eng.
**2012**, 37, 38–55. [Google Scholar] [CrossRef] - Huang, C.C. Seismic displacements of soil retaining walls situated on slopes. J. Geotech. Geoenviron. Eng.
**2005**, 131, 1108–1117. [Google Scholar] [CrossRef] - Huang, C.C.; Wu, S.H.; Wu, H.J. Seismic displacement criterion for soil retaining walls based on soil shear strength mobilization. J. Geotech. Geoenviron. Eng.
**2009**, 135, 74–83. [Google Scholar] [CrossRef] - Shukla, S.K. Dynamic active thrust from c–ϕ soil backfills. Soil Dyn. Earthq. Eng.
**2011**, 31, 526–529. [Google Scholar] [CrossRef] - Conti, R.; Viggiani, G.M.B.; Cavallo, S. A two-rigid block model for sliding gravity retaining walls. Soil Dyn. Earthq. Eng.
**2013**, 55, 33–43. [Google Scholar] [CrossRef] - Choudhury, D.; Ahmad, S.M. Stability of waterfront retaining wall subjected to pseudo-static earthquake forces. Ocean Eng.
**2007**, 26, 291–301. [Google Scholar] [CrossRef] - Choudhury, D.; Ahmad, S.M. Design of waterfront retaining wall for the passive case under earthquake and tsunami. Appl. Ocean Res.
**2007**, 29, 37–44. [Google Scholar] [CrossRef] - Choudhury, D.; Ahmad, S.M. Stability of waterfront retaining wall subjected to pseudodynamic earthquake forces. J. Waterw. Port Coast. Ocean Eng.
**2008**, 134, 252–260. [Google Scholar] [CrossRef] - Ahmad, S.M.; Choudhury, D. Pseudo-dynamic approach of seismic design for waterfront reinforced soil-wall. Geotext. Geomembr.
**2008**, 26, 291–301. [Google Scholar] [CrossRef] - Shamsabadi, A.; Xu, S.Y.; Taciroglu, E. A generalized log-spiral-Rankine limit equilibrium model for seismic earth pressure analysis. Soil Dyn. Earthq. Eng.
**2013**, 49, 197–209. [Google Scholar] [CrossRef] - Zhang, F.; Leshchinsky, D.; Gao, Y.F. Required unfactored strength of geosynthetics in reinforced 3D slopes. Geotext. Geomembr.
**2014**, 42, 576–585. [Google Scholar] [CrossRef] - Yang, X.-L.; Li, Z.-W. Upper bound analysis of 3D static and seismic active earth pressure. Soil Dyn. Earthq. Eng.
**2018**, 108, 18–28. [Google Scholar] [CrossRef] - Mylonakis, G.; Kloukinas, P.; Papantonopoulos, C. An alternative to the Mononobe–Okabe equations for seismic earth pressures. Soil Dyn. Earthq. Eng.
**2007**, 27, 957–969. [Google Scholar] [CrossRef] - Li, X.; Wu, Y.; He, S. Seismic stability analysis of gravity retaining walls. Soil Dyn. Earthq. Eng.
**2010**, 30, 875–878. [Google Scholar] [CrossRef] - Veletsos, A.S.; Younan, A.H. Dynamic response of cantilever retaining walls. J. Geotech. Geoenviron. Eng.
**1997**, 123, 161–172. [Google Scholar] [CrossRef] - Gazetas, G.; Psarropoulos, P.N.; Anastasopoulos, I.; Gerolymos, N. Seismic behaviour of flexible retaining systems subjected to short-duration moderately strong excitation. Soil Dyn. Earthq. Eng.
**2004**, 24, 537–550. [Google Scholar] [CrossRef] - Psarropoulos, P.N.; Klonaris, G.; Gazetas, G. Seismic earth pressures on rigid and flexible retaining wall. Soil Dyn. Earthq. Eng.
**2005**, 25, 795–809. [Google Scholar] [CrossRef] - Jung, C.; Bobet, A.; Fernández, G. Analytical solution for the response of a flexible retaining structure with an elastic backfill. Int. J. Numer. Anal. Methods Geomech.
**2010**, 34, 1387–1408. [Google Scholar] [CrossRef] - Kloukinas, P.; Langousis, M.; Mylonakis, G. Simple Wave Solution for Seismic Earth Pressures on Nonyielding Walls. J. Geotechn. Geoenviron. Eng.
**2012**, 138, 1514–1519. [Google Scholar] [CrossRef] - Richards, R.; Elms, D.G. Seismic behavior of gravity retaining walls. J. Geotech. Eng. Divis. ASCE
**1979**, 105, 449–464. [Google Scholar] - Whitman, R.V.; Liao, S. Seismic design of gravity retaining walls. In Proceedings of the 8th World Conference on Earthquake Engineering, San Francisco, CA, USA, 21–28 July 1984; Volume 3, pp. 533–540. [Google Scholar]
- Saran, S.; Viladkar, M.N.; Reddy, R.K. Displacement dependent earth pressures. Indian Geotech. J.
**1985**, 17, 121–141. [Google Scholar] - Saran, S.; Viladkar, M.N.; Tripathi, O.P. Displacement dependent earth pressures in retaining walls. Indian Geotech. J.
**1990**, 20, 260–287. [Google Scholar] - Bakr, J.; Ahmad, S.M. A finite element performance-based approach to correlate movement of a rigid retaining wall with seismic earth pressure. Soil Dyn. Earthq. Eng.
**2018**, 114, 460–479. [Google Scholar] [CrossRef] - Steedman, R.S.; Zeng, X. The influence of phase on the calculation of pseudo-static earth pressure on a retaining wall. Géotechnique
**1990**, 40, 103–112. [Google Scholar] [CrossRef] - Choudhury, D.; Nimbalkar, S. Pseudo-dynamic approach of seismic active earth pressure behind retaining wall. Geotech. Geol. Eng.
**2006**, 24, 1103–1113. [Google Scholar] [CrossRef] - Ghosh, P. Seismic active earth pressure behind non-vertical retaining wall using pseudo-dynamic analysis. Can. Geotech. J.
**2008**, 45, 117–123. [Google Scholar] [CrossRef] - Kolathayar, S.; Ghosh, P. Seismic active earth pressure on walls with bilinear backface using pseudo-dynamic approach. Comput. Geotech.
**2009**, 36, 1229–1236. [Google Scholar] [CrossRef] - Ghanbari, A.; Ahmadabadi, M. Pseudo-dynamic active earth pressure analysis of inclined retaining walls using horizontal slices method. Sci. Iran. Trans. A Civ. Eng.
**2010**, 17, 118–130. [Google Scholar] - Ghosh, S.; Sharma, R.P. Seismic Active Earth Pressure on the Back of Battered Retaining Wall Supporting Inclined Backfill. Int. J. Geomech.
**2012**, 12, 54–63. [Google Scholar] [CrossRef] - Handy, R.L. The arch in soil arching. J. Geotech. Eng.
**1985**, 111, 302–318. [Google Scholar] [CrossRef] - Paik, K.H.; Salgado, R. Estimation of active earth pressure against rigid retaining walls considering arching effect. Géotechnique
**2003**, 53, 643–653. [Google Scholar] [CrossRef] - Goel, S.; Patra, N.R. Effect of arching on active earth pressure for rigid retaining walls considering translation mode. Int. J. Geomech.
**2008**, 8, 123–133. [Google Scholar] [CrossRef] - Khosravi, M.H.; Pipatpongsa, T.; Takemura, J. Experimental analysis of earth pressure against rigid retaining walls under translation mode. Géotechnique
**2013**, 63, 1020–1028. [Google Scholar] [CrossRef] - Rao, P.; Chen, Q.; Zhou, Y.; Nimbalkar, S.; Chiaro, G. Determination of active earth pressure on rigid retaining wall considering arching effect in cohesive backfill soil. Int. J. Geomech.
**2015**. [Google Scholar] [CrossRef] - Nakamura, S. Reexamination of Mononobe–Okabe theory of gravity retaining walls using centrifuge model tests. Soils Found.
**2006**, 46, 135–146. [Google Scholar] [CrossRef] - Athanasopoulos-Zekkos, A.; Vlachakis, V.S.; Athanasopoulos, G.A. Phasing issues in the seismic response of yielding, gravity-type earth retaining walls—Overview and results from a FEM study. Soil Dyn. Earthq. Eng.
**2013**, 55, 59–70. [Google Scholar] [CrossRef] - Al Atik, L.; Sitar, N. Seismic Earth Pressures on Cantilever Retaining Structures. J. Geotech. Geoenviron. Eng.
**2010**, 136, 1324–1333. [Google Scholar] [CrossRef] - Tiznado, J.C.; Rodríguez-Roa, F. Seismic lateral movement prediction for gravity retaining walls on granular soils. Soil Dyn. Earthq. Eng.
**2011**, 31, 391–400. [Google Scholar] [CrossRef] - Baziar, M.H.; Habib, S.; Moghadam, M.R. Sliding stability analysis of gravity retaining walls using the pseudo-dynamic method. Proc. Inst. Civ. Eng. Geotech. Eng.
**2012**, 166, 389–398. [Google Scholar] [CrossRef] - Choudhury, D.; Nimbalkar, S. Seismic rotational displacement of gravity walls by pseudodynamic method. Int. J. Geomech.
**2008**, 8, 169–175. [Google Scholar] [CrossRef] - Zeng, X.; Steedman, R.S. Rotating block method for seismic displacement of gravity walls. J. Geotech. Geoenviron. Eng.
**2000**, 126, 709–717. [Google Scholar] [CrossRef] - Pain, A.; Choudhury, D.; Bhattacharyya, S.K. Computation of rotational displacements of gravity retaining walls by pseudo-dynamic method. In Proceedings of the 4th GeoChina International Conference: Sustainable Civil Infrastructures: Innovative Technologies for Severe Weathers and Climate Changes, Jinan, China, 25–27 July 2016. [Google Scholar]
- Basha, B.M.; Babu, G.L.S. Optimum design of bridge abutments under high seismic loading using modified pseudo-static method. J. Earthq. Eng.
**2010**, 14, 874–897. [Google Scholar] [CrossRef] - Bellezza, I. A new pseudo-dynamic approach for seismic active soil thrust. Geotech. Geol. Eng.
**2014**, 32, 561–576. [Google Scholar] [CrossRef] - Bellezza, I. Seismic active soil thrust on walls using a new pseudo-dynamic approach. Geotech. Geol. Eng.
**2015**, 33, 795–812. [Google Scholar] [CrossRef] - Harop-Williams, K. Arch in soil arching. J. Geotech. Eng.
**1989**, 15, 415–419. [Google Scholar] [CrossRef] - Wang, Y.Z. The active earth pressure distribution and the lateral pressure coefficient of Retaining wall. Rock Soil Mech.
**2005**, 26, 1019–1022. [Google Scholar] - Li, J.; Wang, M. Simplified method for calculating active earth pressure on rigid retaining walls considering the arching effect under translational mode. Int. J. Geomech.
**2014**, 14, 282–290. [Google Scholar] [CrossRef] - Yuan, C.; Peng, S.; Zhang, Z.; Liu, Z. Seismic wave propagation in Kelvin-Voigt homogeneous visco-elastic media. Sci. China Ser. D Earth Sci.
**2006**, 49, 147–153. [Google Scholar] [CrossRef] - Pain, A.; Choudhury, D.; Bhattacharyya, S.K. Seismic stability of retaining wall-soil sliding interaction using modified pseudo-dynamic method. Geotech. Lett.
**2015**, 5, 56–61. [Google Scholar] [CrossRef] - Tsagareli, Z.V. Experimental investigation of the pressure of a loose medium on retaining wall with vertical backface and horizontal backfill surface. Soil Mech. Found. Eng.
**1965**, 91, 197–200. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**) Variation of stability factors F

_{I}, F

_{T}, and F

_{W}with time/Period of lateral shaking (t/T) for (ωH/V

_{s}< π/2) and (

**b**) Variation of stability factors F

_{I}, F

_{T}, and F

_{W}with time/Period of lateral shaking (t/T) for (ωH/V

_{s}> π/2).

**Figure 3.**Distribution of non-dimensional horizontal acceleration versus non-dimensional depth of the wall at the instance of maximum value of F

_{W}.

**Figure 5.**Variation of active earth pressure distribution with the cohesion of backfill soil ([Rao et al. 2015], with permission from ASCE).

**Figure 6.**Variation of active earth pressure distribution with soil friction angle ([Rao et al. 2015], with permission from ASCE).

**Figure 7.**Comparison between the predicted and experimental data ([Rao et al. 2015], with permission from ASCE).

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**MDPI and ACS Style**

Nimbalkar, S.; Pain, A.; Ahmad, S.M.; Chen, Q.
Stability Assessment of Earth Retaining Structures under Static and Seismic Conditions. *Infrastructures* **2019**, *4*, 15.
https://doi.org/10.3390/infrastructures4020015

**AMA Style**

Nimbalkar S, Pain A, Ahmad SM, Chen Q.
Stability Assessment of Earth Retaining Structures under Static and Seismic Conditions. *Infrastructures*. 2019; 4(2):15.
https://doi.org/10.3390/infrastructures4020015

**Chicago/Turabian Style**

Nimbalkar, Sanjay, Anindya Pain, Syed Mohd Ahmad, and Qingsheng Chen.
2019. "Stability Assessment of Earth Retaining Structures under Static and Seismic Conditions" *Infrastructures* 4, no. 2: 15.
https://doi.org/10.3390/infrastructures4020015