Finite-Element Study for Seismic Structural and Global Stability of Cantilever-Type Retaining Walls

Bakr, Junied; Ahmad, Syed Mohd; Lombardi, Domenico

doi:10.1061/(ASCE)GM.1943-5622.0001505

Open access

Technical Papers

Aug 16, 2019

Finite-Element Study for Seismic Structural and Global Stability of Cantilever-Type Retaining Walls

Authors: Junied Bakr https://orcid.org/0000-0003-1441-0425, Syed Mohd Ahmad [email protected], and Domenico Lombardi https://orcid.org/0000-0003-4116-8347Author Affiliations

Publication: International Journal of Geomechanics

Volume 19, Issue 10

https://doi.org/10.1061/(ASCE)GM.1943-5622.0001505

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Abstract

This paper investigates the structural and global stability of a cantilever-type retaining wall under seismic loading using numerical modelling. A new and robust approach is proposed to compute the seismic earth pressure behind the stem and along a virtual plane passing the heel of the wall. The results show that under different earthquake characteristics and wall geometries, the seismic earth pressure forces may be out of phase, leading to different seismic responses of the wall. The critical scenario for the structural stability is observed when the maximum acceleration is directed toward the backfill soil, and the earthquake frequency content is close to the natural frequency of the wall. In contrast, the critical scenario for the global stability occurs when the maximum acceleration is directed with minimum frequency content. Further, the natural frequency of the wall does not affect the global stability of the wall. However, the duration of the applied earthquake acceleration does affect the global stability of the wall, whereas the structural stability remains unaffected by it. In contrast with the current understanding, the possibility of failure of a cantilever-type retaining wall by horizontal sliding is remarkably increased with time of the applied earthquake acceleration.

Introduction

A cantilever-type retaining wall, like the one presented in Fig. 1(a), is one of the most common types of retaining structures. For such walls, the most important design load that needs to be considered for a safe design under seismic conditions comes from the seismic earth pressure. For design purposes, a cantilever-type retaining wall is considered as a flexible structure and a design must address the structural and global stability arising because of the seismic earth pressure. As presented in Fig. 1(b) the seismic earth pressure will create a shear force, N_wall, and bending moment, M_wall, on the stem of the wall and also tend to cause the base slab to slide relative to the foundation layer and rotate about the toe, thereby overturning the wall [Fig. 1(c)]. For structural stability, the stem of the wall should be designed to resist the shear force, N_wall, and bending moment, M_w, whereas, for global stability, the wall should be designed such that it provides adequate sliding and overturning resistance. It is crucial to highlight that for both of these stability analyses assessments, an accurate estimation of the seismic earth pressure is important.

Fig. 1. Typical cantilever-type retaining wall–soil system: (a) schematic representation; (b) schematic of structural stability; (c) schematic of global stability; (d) body force diagram for structural stability analysis; and (e) body force diagram for global stability analysis.

In the current design practice, the cantilever-type retaining walls are designed by considering the concepts used for the design of rigid retaining walls; this is despite the important fact that the cantilever-type retaining walls behave as flexible structures, whereas the rigid retaining walls do behave as rigid structures. For a seismic case, the force-based Mononobe Okabe (M-O) method (Mononobe and Matsuo 1929)—a method primarily based on Coulomb’s earth pressure theory (Coulomb 1776)—is widely used to estimate seismic earth pressure for the retaining walls. Although the M-O method is a very straightforward method to use, it has a limitation of not replicating, and hence simulating, the real field behavior and, therefore, at times overestimating the seismic earth pressure (e.g., Nakamura 2006; Al Atik and Sitar 2009; Jo et al. 2017; Bakr and Ahmad 2018a). Extensive efforts have been made to study the seismic earth pressure behind a rigid retaining wall (e.g, Choudhury and Katdare 2013; Tang et al. 2014; Dey et al. 2017; Zhou et al. 2018; Rajesh and Choudhury 2017), whereas little attention has been paid to investigate the development of seismic earth pressure behind a cantilever-type retaining wall. Further, with regards to finding a critical case that needs to be considered for designing a cantilever-type retaining wall under seismic conditions, there are several contradictory evidences in the available literature. For example, the numerical study presented by Green et al. (2008) and the shaking table study presented by Kloukinas et al. (2015) show that a critical loading case for the seismic design of a cantilever-type retaining wall is when the earthquake acceleration is applied away from the backfill soil, thereby rendering the retaining wall–soil system in a passive earth pressure condition. This, however, is contradicted by the centrifuge-based study of Jo et al. (2017), who reported that a critical loading case for the retaining wall will be the one in which the earthquake acceleration is applied toward the backfill soil, thereby rendering the retaining wall–soil system in an active earth pressure condition. Cakir (2013) investigated the effect of frequency content of earthquake acceleration on the seismic response of a cantilever retaining wall in a three-dimensional space by using the finite-element method (FEM). This study found that the stresses behind the retaining wall and displacements at the top of the wall tend to increase with the decrease of frequency content of earthquake acceleration. Considering the importance of displacement-based methods, researchers have presented various approaches for the estimation of displacement of rigid retaining walls (e.g., Nadim and Whitman 1983; Madabhushi and Zeng 1998; Zeng 1998; Zeng and Steedman 2000; Paruvakat et al. 2001; Huang 2005; Choudhury and Nimbalkar 2008; Basha and Babu 2010; Huang et al. 2009; Ahmad and Choudhury 2010; Ni et al. 2017). Similarly, a few researchers like Green et al. (2008); Kloukinas et al. (2015); Bakr and Ahmad (2018b) have investigated the deformation mechanism of the cantilever-type retaining wall under the effect of seismic loading. However, very little information is available that details a relationship and effect of seismic earth pressure on the displacement of the cantilever-type retaining wall, thus leaving this important area of research and application poorly understood.

This paper presents a FE-based numerical modeling approach to study the seismic structural and global stability of a cantilever-type retaining wall–soil system, not only for a post-earthquake scenario, but also during an earthquake event. The present study identifies a critical scenario for the structural and global stability of a cantilever-type retaining wall–soil system by considering different earthquake characteristics and the effects of seismic earth pressure on the structural and global stability of a cantilever-type retaining wall–soil system. The study also captures the deformation mechanism of a cantilever-type retaining wall–soil system and investigates the effect of the natural frequency of a cantilever-type retaining wall–soil system on its structural and global stability. Recommendations that would help in providing a safer and economic design of a cantilever-type retaining wall for earthquake prone areas are also presented.

Seismic Structural and Global Stability of a Cantilever-Type Retaining Wall–Soil System

A cantilever-type retaining wall of height H and width of base slab b_slab, supporting a horizontal surface backfill to its full height, is presented in Fig. 1(a). As mentioned above, the retaining wall–soil system is to be analyzed for the seismic structural and global stability.

Seismic Structural Stability

For the seismic structural stability of the wall, the wall is considered to be subject to the seismic earth pressure force coming from the backfill soil, which is assumed to act behind the stem, and denoted as P_stem, and to the wall seismic inertia force, F_wa and F_wp; where F_wa and F_wp are the wall seismic inertia force acting away and toward the backfill soil, respectively [Fig. 1(d)]. For P_stem, F_wa, and F_wp, the following needs to be noted:

•

At a given point in time, F_wa and F_wp will not be acting simultaneously—it will either be F_wa or F_wp.

•

Depending upon the direction of the applied earthquake acceleration—that is, whether it is acting toward the backfill soil or away from it, the wall seismic inertia force will also either be acting toward the backfill soil, denoted as F_wp [Fig. 1(d)] or away from the backfill soil, denoted as F_wa [Fig. 1(d)].

•

Because the amplitude of the applied earthquake acceleration will vary with time, P_stem, F_wa, and F_wp will also be time-varying.

•

P_stem, F_wp, and F_wa will produce shear force, N_wall, and bending moment, M_wall, on the stem of the wall.

This paper presents a methodology to accurately predict N_wall and M_wall; to estimate the relative contributions of P_stem, F_wa, and F_wp on N_wall and M_wall; and to identify a critical case for the seismic structural stability of the wall that causes a maximum load case for the stem of the wall.

Seismic Global Stability

For the seismic global stability analysis of the wall, the wall is considered to be subject to [Fig. 1(e)]:

•

The seismic earth pressure force, coming from the backfill soil, which is assumed to act at a vertical plane passing through the heel of the wall, and denoted as P_vp.

•

The backfill seismic inertia force, F_sa and F_sp, where F_sa and F_sp are the backfill seismic inertia force acting toward and away from the wall, respectively.

•

The wall seismic inertia force, F_wa and F_wp.

Like F_wa and F_wp, depending upon the direction of the applied earthquake acceleration, the backfill seismic inertia force will also either be acting away from the wall, denoted as F_sp [Fig. 1(e)] or toward the wall, denoted as F_sa [Fig. 1(e)]. In addition, at any given point in time, the wall–soil system will be subject to either F_sp or F_sa; that is, they will not both be acting simultaneously. It is important to highlight that the seismic earth pressure force, P_vp, is computed along the virtual plane because the global stability of the cantilever-type retaining wall is maintained, in addition to the weight of the wall, W_w, by the weight of the backfill soil above the base slab, W_s. This paper presents a methodology to accurately predict the deformation mechanism of the wall to compute the relative displacement between its base slab and foundation soil; to estimate the contribution of P_vp to the aforementioned relative displacement; and to identify a critical case for the global stability of the wall.

For both the seismic structural and global stability analyses, the effects of applied earthquake acceleration characteristics (i.e., its amplitude and frequency content) and the natural frequency of the wall–soil system have been studied in detail.

Finite-Element Model of the Cantilever-Type Retaining Wall–Soil System

An FE model of the cantilever-type retaining wall–soil system has been developed by using the PLAXIS2D 2016 software (Brinkgreve et al. 2016). To validate the FE model and compare the results, the geometry of the model and the material properties were chosen to be similar to the one used by Jo et al. (2014) for their centrifuge tests. The FE model is presented in Fig. 2(a), in which the wall has a height of 5.4 m and sits on 9-m-thick foundation soil. The wall retains a horizontal surface backfill soil to its full height, and the confines of the model in the horizontal direction are large enough so as to exclude any boundary effects. For the purpose of analysis and for comparison of the results of the proposed model with the centrifuge test results of Jo et al. (2014), the stem of the retaining wall has been considered to be of a uniform stiffness. No embedment has been considered in front of the cantilever-type retaining wall. This is a slight deviation from actual practice, but this simplification facilitates comparison of the present model with a centrifuge-based study of Jo et al. (2014), who, in their experimental model, did not consider any embedment, and simplifies numerical modeling to concentrate on the soil behavior. Further, because of impartial mobilization of passive resistance, a no-embedment case does not lead to unrealistic results. For the FEM, the backfill and foundation soil is modeled by using six-noded triangular elements with 2 degrees of freedom at each node and three Gauss integration points, whereas the wall is modeled by using the plate elements of the PLAXIS2D 2016 library (Brinkgreve et al. 2016).

Fig. 2. (a) FEM of the cantilever-type retaining wall–soil system chosen for the present study; (b) 1989 Loma Prieta earthquake acceleration–time history (data from PEER 2018); and (c) frequency content for (b).

Wall–Soil Interaction

The interaction between the stem of the wall and backfill soil, and between the base slab of the wall and foundation soil, has been modeled by using the six-noded interface elements of the PLAXIS2D 2016 library (Brinkgreve et al. 2016). For the chosen interface element, the interface roughness was controlled by using an interface strength-reduction-factor, R_inter, where R_inter = 0 for a perfectly smooth interface, R_inter = 1 for a perfectly rough interface, and 0 < R_inter > 1 for a partially rough interface. To simulate the interface properties similar to the Jo et al. (2014) centrifuge tests, for the present study, a partially rough interface was considered, such that R_inter = 0.334 between the stem of the wall and backfill soil and 0.5 between the base slab of the wall and foundation soil.

Boundary Conditions

For the FE model, the lateral boundaries were restrained against any horizontal movement, whereas at the base boundary, both the horizontal and vertical movements were restricted [Fig. 2(a)]. Further, absorbing lateral boundaries were considered so that the effects of reflection of the seismic waves were negligible for the boundaries.

Constitutive Models for the Backfill and Foundation Soil, and the Material of the Wall

During a seismic analysis, a soil depicts a strong nonlinear stress–strain behavior, which is accompanied with a gradual reduction in the shear modulus, and a simultaneous increase in hysteretic damping with an increase in the shear strain amplitude (Brinkgreve et al. 2016). Thus, it is imperative that the chosen constitutive model for the present study should replicate the aforementioned soil behavior. In PLAXIS2D 2016 (Brinkgreve et al. 2016), the best model that can replicate all of the above aspects of the constitutive behavior of a soil is the hardening-soil-with-small-strain model (denoted as HSsmall model from this point hereafter in the present paper, and the same model was chosen for modeling the backfill and foundation soils. However, because the HSsmall model is almost linear for a very small strain with no hysteretic damping that would cause unrealistic resonance during a numerical analysis, Rayleigh damping was used in the present study by adopting the procedure outlined by Rajasekaran (2009), which suggests

[C] = α [M] + β [K]

(1)

where [C], [M], and [K] = damping, mass and stiffness FE matrices of the wall–soil system; and α and β = Rayleigh parameters, computed as (Rajasekaran 2009)

{\begin{matrix} α \\ β \end{matrix}} = \frac{2 ζ_{s, wall}}{ω_{z 1} + ω_{z 2}} {\begin{matrix} ω_{z 1} \times ω_{z 2} \\ 1 \end{matrix}}

(2)

where ζ_s,wall = damping ratio of the soil and wall, respectively; and ω_z1 and ω_z2 = first two natural circular frequencies of the FE model. ABAQUS software (ABAQUS 2013) was used to compute the first two natural circular frequencies of the FE model, and for the model presented in Fig. 2(a), they were found to be ω_z1 = 28.27 rad/s and ω_z2 = 39.93 rad/s. The constitutive behavior of the material of the wall was simulated by using a linear viscoelastic constitutive model.

Seismic Loading

The above FE model was subjected to a seismic loading, which, for this study, consisted of a real acceleration–time history of the 1989 Loma Prieta earthquake (PEER 2018), having a peak ground acceleration (PGA) of 0.264g [Fig. 2(b)] and dominant frequencies (f) of about 0.7, 2.2, and 2.7 Hz [Fig. 2(c)]. To investigate the effects of applied earthquake acceleration amplitude and frequency content (f) on the seismic response of the wall–soil system, scaled uniform sinusoidal acceleration–time histories were also considered with three amplitudes of 0.2, 0.4, and 0.6g, and scaled frequencies of 0.5, 2, and 4 Hz. These acceleration–time histories were applied at the base of the FE model, as presented in Fig. 2(a). The effect of vertical seismic acceleration was neglected in the present analysis. Notionally, the vertical seismic acceleration may contribute to the vertical seismic inertia and, as a consequence, may slightly affect the performance response of the cantilever-type retaining wall. However, arguments of several past studies (like Green et al. 2008; Cakir 2013; Kloukinas et al. 2015; and Jo et al. 2017) as well as that of Bakr and Ahmad (2018a) suggest that it is primarily the horizontal seismic acceleration that contributes to the seismic earth pressure force and affects the stability of a retaining wall. The same has been observed through the results of the present study as well as shown later in the results section of this paper.

Construction Sequence Simulation and Results from the Static Analysis

As a first step to run the analysis, the geostatic stresses need to be distributed in the wall–soil system. This has been performed by considering the construction sequence of a typical cantilever-type retaining wall. To run the FE simulation, the input properties used in the study were chosen as per Table 1. The wall was assumed to be constructed in six stages, as presented in Fig. 3, in which the initial stage relates to the placement of the foundation soil and installation of the wall. The subsequent four stages simulate the placement of the backfill soil in lifts of thickness 0.22 H for each stage, whereas the last stage simulates the placement of the backfill soil in a lift of thickness 0.11 H. The backfill soil layers was placed in the FE model and simulated by using the physical parameters measured after the compaction process in the centrifuge test conducted by Jo et al. (2014). Therefore, the compaction process was already taken into account during the static analysis. The soil behavior was also simulated by using the Hssmall model. This model can replicate the volumetric mechanism (cap). The main role of the volumetric mechanism (cap) is to close the elastic domain and simulate the densification/compaction of the material. With the progressive placement of the backfill soil, the wall will be subject to displacement and rotation, as presented in Fig. 3. It can be noted from Fig. 3 that during the placement of the backfill soil in lifts of 0.22-H thickness, the maximum deformations are mobilized in the backfill soil above the base slab because the stem as well as the base slab of the wall rotate toward the backfill soil; this trend continues until Stage 4. In Stage 5, the movement of the stem away from the backfill soil appears to be more than the backfill soil in the upper part (the negative sign in Fig. 3 means that the displacement is away from the backfill soil), perhaps because of the development of the earth pressure thereby causing an elastic deformation of the stem. However, in the lower parts of the stem, it is observed that the backfill soil moves more than the stem of the wall. From Stages 4 and 5 of Fig. 3, a clear formation of a failure plane originating from the heel of the wall and extending up to the ground level at an inclination of an angle less than 45° to the horizontal is also observed. The deformation behavior of the wall matches with what has been observed in the real field observations reported by Bentler and Labuz (2006). The deformation of the wall is also predicted at different durations of earthquake and discussed later in the seismic analysis. After the simulation of the construction stages and consequent distribution of the geostatic stresses in the FE model, the static earth pressures are computed by the FE model, both behind the stem, p_{stem(static),} and along the virtual plane, p_vp(static), which are then compared with the Rankine solution as well as with the Jo et al. (2014) centrifuge test results. The earth pressure comparison is presented in Figs. 4(a and b) for walls of heights 5.4 m and 10.8 m, respectively. From Figs. 4(a) and 6(b), it is observed that the FE model predictions are in an excellent agreement with the Jo et al. (2014) centrifuge test results. In addition, it is observed that p_stem(static) and p_vp(static) values obtained by the FE model in the top ¾ H are very close to the static active earth pressure values obtained by Rankine’s theory, whereas for the bottom ¼ H, the earth pressure values are between the static active value, also obtained by Rankine’s theory, and the at-rest value. Jo et al. (2014) explained this phenomenon was due to the interlocking grain fabric of the dry pluviated sand. The numerical modeling simulation of the present study, also showed similar results, but these are attributed to the complicated deformation mechanism during the construction process of a cantilever-type retaining wall. This is depicted very clearly in Fig. 3. As can be observed from Fig. 3 for the last stage of construction process, the wall rotates as a rigid body about the heel toward the backfill soil while the stem rotates about the base away from the backfill soil because of the elastic deflection. Therefore, the upper part of the stem has a sufficient lateral movement away from the backfill soil, causing the development of active earth pressure. However, for the lower section of the wall, the rotation of wall as rigid body about the heel toward the backfill soil is dominant, and the stem has insufficient lateral yielding away from the backfill compared with the upper part of the stem causing that the lateral earth pressure to remain close to the at-rest state.

Fig. 3. Construction sequence of a typical cantilever-type retaining wall–soil system and the associated contours of horizontal displacement.

Fig. 4. Comparison of static earth pressure profiles: (a) for wall height, H = 5.4 m; and (b) for wall height, H = 10.8 m.

Table 1. Soil and wall parameters used for the present study

Parameter	Symbol	Unit	Value
Soil
Dry unit weight	γ_s	kN/m³	14.23
Angle of shearing resistance	ϕ′	degrees	40.00
Dilatancy	ψ	degrees	10.00
Young’s modulus at 50% failure strength, corresponding to $p^{ref}$	$E_{50}^{ref}$	MPa	46.80
Odometeric modulus, corresponding to $p^{ref}$	$E_{od}^{r e f}$	MPa	46.80
Modulus for unloading–reloading conditions, corresponding to $p^{ref}$	$E_{ur}^{ref}$	MPa	140.40
Small strain shear modulus, corresponding to $p^{ref}$	$G_{o}^{ref}$	MPa	113.00
Reference shear strain, corresponding to 70% of $G_{o}^{ref}$	γ_0.7	—	0.0002
Poisson’s ratio for unloading–reloading conditions	ν_ur–s	—	0.20
Damping ratio	ζ_s	%	3.00
Reference confining pressure	$p^{ref}$	kN/m²	100.00
Stiffness stress-level dependency constant	K	—	0.50
Failure ratio	R_f	—	0.90
Wall
Unit weight	γ_wall	kN/m³	26.60
Modulus of elasticity	E_wall	GPa	68.00
Second moment of area of the stem	I_stem	m⁴	8.873 × 10^–4
Poisson’s ratio	ν_wall	—	0.334
Damping ratio	ζ_wall	%	3.00

Seismic Analysis

After performing the aforementioned static analysis, a seismic analysis was carried out for the FE model, which, as mentioned above, was carried out by applying a seismic loading in the form of an acceleration–time history at the base of the FE model. Through the seismic analysis, apart from obtaining P_stem, P_vp, F_wa, F_wp, F_sa, F_sp, N_wall, and M_wall, the acceleration and sliding response of the wall–soil system has also been obtained and its importance is discussed in the sections below. It is to be highlighted that for both the static and seismic analyses, a mesh-size sensitivity analysis was carried out until a convergent solution was achieved. The total earth pressure force was chosen to be the parameter for the sensitivity analysis, and based on this, the final FE mesh comprised of 4,418 elements with 9,164 nodes.

Acceleration Response of the Cantilever-Type Retaining Wall–Soil System

The acceleration response of the wall–soil system is presented in Fig. 5 for the time duration of 3–7 s. This is the duration in which the intensity of the applied earthquake acceleration was concentrated [Fig. 2(b)] and, hence, it was chosen for presenting the results from the seismic analysis. From Fig. 5, it is observed that the acceleration response for the top of the stem and top of the backfill soil match each other. This implies that at the top of the FE model, the stem of the wall and backfill soil move together; that is, they are in phase with each other. It is also observed that the acceleration of the top of the stem and backfill soil is higher than the acceleration at the base of the wall, thereby implying a possible amplification of acceleration toward the top of the FE model. The acceleration response of the wall–soil system, when it is subject to a uniform sinusoidal acceleration–time history of different amplitudes and frequency contents, is presented in Fig. 6. From Fig. 6(a) it is observed that when the amplitude of the applied earthquake acceleration is 0.2g with the frequency content of 0.5 Hz, the amplitude of the acceleration response for both the top of the stem and backfill soil matches the amplitude of the applied earthquake acceleration itself. However, when the frequency content of the applied earthquake acceleration is increased by four times to 2 Hz while the amplitude of the applied acceleration is kept same at 0.2g, the amplitude of the acceleration response for the top of the stem and backfill soil amplifies to a value close to 0.4g, as presented in Fig. 6(b). On a further increase of the frequency content to 4 Hz, with the amplitude of the applied acceleration remaining the same at 0.2g, as presented in Fig. 6(c), the amplitude of the acceleration response for the top of the stem is much higher than that for the backfill soil. Similarly, from Figs. 6(d and g), it is observed that for an applied acceleration with 0.5 Hz frequency, the amplitudes of acceleration for the top of the stem and backfill soil have the same amplitude as the applied acceleration amplitudes of 0.4g and 0.6g, respectively. However, for an applied acceleration with a frequency content of 2 Hz, as presented in Figs. 6(e) and 8(h), the amplitude of acceleration for the top of the stem is higher than the acceleration for the top of the backfill soil. When the frequency content of the applied acceleration is further increased to 4 Hz for applied acceleration amplitudes of 0.4g and 0.6g, as presented in Figs. 6(f) and 8(j), respectively, it is observed that the amplitude of acceleration for the stem amplifies to a maximum value of 1.8g (i.e., it becomes more than the amplitude of the applied acceleration). On the other hand, the amplitude of acceleration for the top of the backfill soil seems to deamplify, and its maximum value becomes less than the amplitude of the applied earthquake acceleration. This behavior—of acceleration amplification for the top and a deamplification for the bottom of the FEM—reflects a nonlinear soil behavior that deamplifies a strong earthquake, resulting in a higher dissipation of the seismic energy. It is to be noted that similar deamplification behavior of soil for strong earthquakes has also been reported by Griffiths et al. (2016) and Stamati et al. (2016).

Fig. 6. Acceleration response for top of the wall and backfill soil for the uniform sinusoidal acceleration–time history of different amplitudes and frequency contents: (a–c) a = 0.2g; (d–f) a = 0.4g; and (g–i) a = 0.6g.

Wall and Backfill Seismic Inertia Forces: F_wa, F_wp, F_sa, and F_sp

The wall and backfill seismic inertia forces—F_wa, F_wp, F_sa, and F_sp—are estimated by using the following procedure: first, elemental acceleration, a_we and a_se, is obtained by the FE model: for the wall, a_we is obtained for all the elements of the base slab and stem, whereas for the backfill soil, a_se is obtained only for those elements that lie in the middle of backfill soil above the base slab. The corresponding masses of the elements (for the case of the wall) and masses of the horizontal strips (for the case of the backfill soil) are multiplied with the elemental accelerations to get the elemental seismic inertia force. These elemental seismic inertia forces are summed together, both for the wall and backfill soil, to get the seismic inertia forces F_wa, F_wp (for the wall) and F_sa, F_sp (for the backfill soil). As presented in Fig. 7, the wall and backfill seismic inertia forces are dependent upon the applied earthquake acceleration. It is observed from Fig. 7 that the maximum wall and backfill seismic inertia forces are acting in an active direction (F_wa = 18.3 kN/m and F_sa = 54.2 kN/m) when the applied earthquake acceleration has a maximum value and is applied toward the backfill soil at t = 3.9 s. However, the maximum wall and backfill seismic inertia forces are acting in a passive direction (F_wp = 36.7 kN/m and F_sp = 88.5 kN/m) when the applied earthquake acceleration has a maximum value and is applied away from the backfill soil at t = 4.5 s. It is observed from Fig. 7 that the wall seismic inertia force (i.e., F_wa, F_wp) and backfill seismic inertia force (i.e., F_sa, F_sp) are in phase, which implies that the wall and backfill soil move as one entity. This is an extremely important finding because this will significantly affect the development of the active state in the wall–soil system when the earthquake acceleration is applied toward the backfill soil, as discussed below.

Seismic Earth Pressure Forces: P_stem and P_vp

The seismic earth pressure force behind the stem, P_stem, and seismic earth pressure force at the virtual plane, P_vp, have been estimated by adopting the following procedure. First, the elemental lateral stresses are obtained from the FE model for all those elements of the backfill soil that are in contact with the stem of the wall as well those that are along the virtual vertical plane. These elemental lateral stresses are multiplied with the corresponding element heights, to get the elemental seismic earth pressure forces. These elemental seismic earth pressure forces are summed together to get P_stem and P_vp, and their variation with time is presented in Figs. 8(a and b), respectively. From Fig. 8(a), it is observed that at the beginning of the seismic analysis (i.e., at time t = 0 s), P_stem is about 53 kN/m, which is between the static active and at-rest earth pressure force; at time t = 3.9 s, when the applied earthquake acceleration has a maximum value and is applied toward the backfill soil, P_stem has a maximum value of 112 kN/m, whereas it attains a minimum value of about 45 kN/m when the applied earthquake acceleration has a maximum value but is applied away from the backfill soil at time t = 4.5 s. As discussed above, the cantilever-type retaining walls are designed by considering the same concepts as used for the design of rigid retaining walls. However, the aforementioned present study results—which are in contrast with the observation of Nakamura (2006), who observed that for a rigid retaining wall, the maximum seismic earth pressure force is developed when the applied earthquake acceleration is maximum but applied away from the backfill soil—show that P_stem is maximum when the applied acceleration is applied toward the backfill soil. Thus, an active state is not developed behind the stem despite the fact that the acceleration is applied toward the backfill soil, and, consequently, the retaining wall moves away from the backfill soil. The present study observations are also in contrast with what was observed for the behavior of a cantilever-type retaining wall modeled via a numerical model by Green et al. (2008) and via an experimental work by Kloukinas et al. (2015)—both of these studies reported that a maximum value of P_stem is same to what is observed for a rigid retaining wall. Fig. 8(b) shows the variation of P_vp with time. It is observed that at the beginning of the seismic analysis (i.e., at time t = 0 s), P_vp is about 60 kN/m, which, like P_stem, is between the static active and at-rest state earth pressure force. At time t = 3.9 s, when the applied earthquake acceleration has a maximum value and is applied toward the backfill soil, P_vp has a minimum value of 61 kN/m, whereas it attains a maximum value of about 165 kN/m when the applied earthquake acceleration has a maximum value but is applied away from the backfill soil at time t = 4.5 s. These observations are similar to the observations of a rigid retaining wall as observed via a centrifuge test carried out by Nakamura (2006). Thus, it can be said that at time t = 3.9 s, when the applied earthquake acceleration has a maximum value and is applied toward the backfill soil, a maximum load case is developed behind the stem of the wall, whereas a minimum load case is developed at the vertical virtual plane. On the other hand, at time t = 4.5 s, when the applied earthquake acceleration has a maximum value and is applied away from the backfill soil, a minimum load case is developed behind the stem, whereas a maximum load case is developed at the vertical virtual plane. This clearly points to the fact that P_stem and P_vp attain maximum and minimum values at different times, thus suggesting that there is a phase difference between these two quantities. Therefore, for the purpose of structural and global stability, they have to be assessed individually.

Fig. 8. Seismic earth pressure forces: (a) at the stem, P_stem; and (b) at the vertical plane, P_vp.

It is also observed from Figs. 8(a and b), that at time t = 30 s, that is, at the end of the seismic analysis, there is a residual P_stem and P_vp of about 88 kN/m and 100 kN/m, respectively. These residual seismic earth pressure forces are attributed to the densification of backfill soil during the earthquake, thereby implying that the chosen HSsmall constitutive model caters to the simultaneous densification and settlement during a seismic analysis. Figs. 9(a, c, and e) show the effect of varying earthquake amplitude and frequency content on P_stem, and the same for P_vp is presented in Figs. 9(b, d, and f). From Figs. 9(a–f), it is observed that for all amplitudes and frequencies of the applied earthquake, P_stem is maximum when the earthquake acceleration is applied toward the backfill soil, whereas P_stem is minimum when the earthquake acceleration is applied away from the backfill soil. On the other hand, P_vp is maximum when the earthquake acceleration is applied away from the backfill soil, and P_vp is minimum when the earthquake acceleration is applied toward the backfill soil. It is further observed that P_stem and P_vp increase significantly when the amplitude of the applied earthquake is increased from 0.2g to 0.4g, whereas on further increasing the amplitude to 0.6g, P_stem and P_vp do not change with the same proportion as before—again indicating a possible deamplification of the acceleration response of the backfill soil for a strong earthquake. Both P_stem and P_vp are moderately sensitive to the number of acceleration cycles; on the other hand, P_stem is highly sensitive to the natural frequency of the wall. It is observed that P_stem increases with an increment in the frequency content of earthquake acceleration and its maximum value is predicted when the earthquake acceleration is applied with a maximum frequency content and its value is close to the natural frequency of the wall. These observations are in contrast to the observations reported by Cakir (2013), who, based on an FE study, found that the stresses behind the wall tend to increase when the frequency content of earthquake acceleration decreases. However, P_vp appears to be not significantly affected by the natural frequency of the wall.

Fig. 9. Seismic earth pressure forces, P_stem and P_vp, for the uniform sinusoidal acceleration–time history of different amplitudes and frequency contents: (a and b) f = 0.5 Hz; (c and d) f = 2 Hz; and (e and f) f = 4 Hz.

Phase Difference between the Seismic Earth Pressure Force Increments, ΔP_stem and ΔP_vp, and Wall and Backfill Seismic Inertia Forces, F_wa, F_wp, F_sa, and F_sp

This section details the phase difference between various forces acting on the wall under seismic conditions. In order to clearly understand the contribution of the seismic earth pressure force, it is discussed in terms of the seismic earth pressure force increments, ΔP_stem and ΔP_vp, defined as ΔP_stem = P_stem – P_stem(static) and ΔP_vp = P_vp – P_vp(static), where P_stem(static) and P_vp(static) = static earth pressure force acting at the stem and vertical virtual plane, respectively. Figs. 10(a and b) and 10(c and d) show the variation of seismic earth pressure force increments, ΔP_stem and ΔP_vp, and wall and backfill seismic inertia forces, F_wa, F_wp F_sa, and F_sp, for the top ⅓H_stem and bottom ⅓ H_stem, respectively. From Fig. 10(a and b) it is observed that the seismic earth pressure force increment ΔP_stem and wall seismic inertia force, F_wa and F_wp, for the top ⅓ H_stem are out of phase from each other, whereas for the bottom ⅓ H_stem, ΔP_stem is in phase with the wall seismic inertia forces. The reason for this disparity could be that the stem of the wall is monolithically fixed with the base slab and thereby not allowing any relative displacement between the stem and backfill soil. Similarly, from Figs. 10(c and d) it is observed that the seismic earth pressure force increment ΔP_vp and soil seismic inertia forces, F_sa and F_sp, do not act in phase.

Fig. 10. Phase difference between (a and b) ΔP_stem, F_wa, and F_wp for the top and bottom ⅓ H_stem; and (c and d) ΔP_vp, F_sa, and F_sp for the top and bottom ⅓ H.

Shear Force, N_wall, and Bending Moment, M_wall

Figs. 11(a and b) respectively show the shear force N_wall and bending moment M_wall time history predicted at the stem of the wall. Studying Figs. 11(a and b) in conjunction with Fig. 8, it is observed that the shear force N_wall and bending moment M_wall have the exact same trend as the seismic earth pressure force P_stem. In addition, at time t = 3.9 s, when the applied earthquake acceleration has a maximum value and is applied toward the backfill soil, both N_wall and M_wall attain their maximum values of about 120 kN/m and 220 kN·m/m, respectively, whereas they attain their minimum values of about 25 kN/m and 64 kN·m/m, respectively, at time t = 4.5 s, that is, when the applied earthquake acceleration has a maximum value and is applied away from the backfill soil. Thus, it can be argued that a critical case for the structural stability of a cantilever-type retaining wall is when the maximum acceleration is applied toward the backfill soil. Figs. 12(a, c, and e) and 12(b, d, and f) show the effect of varying earthquake amplitude and frequency content on N_wall and M_wall. It is observed that N_wall and M_wall show the same trends as were observed for the P_stem and also that both N_wall and M_wall are highly sensitive to the amplitude of the applied earthquake when its value is between 0.2g and 0.4g. For an applied earthquake acceleration of amplitude > 0.4g, N_wall and M_wall do not remain as sensitive as before—again concreting the fact that deamplification effects creep in for strong earthquakes. It can also be noted that N_wall and M_wall, like P_stem and P_vp, are not sensitive to the number of acceleration cycles where the maximum values of shear force and bending moment are still having the same rate with increasing of acceleration cycles.

Fig. 11. (a) Shear force, N_wall; and (b) bending moment, M_wall.

Fig. 12. Shear force, N_wall, and bending moment, M_wall, for the uniform sinusoidal acceleration–time history of different amplitudes and frequency contents: (a and b) f = 0.5 Hz; (c and d) f = 2 Hz; and (e and f) f = 4 Hz.

Effect of Wall Seismic Inertia Forces, F_wa and F_wp, and Seismic Earth Pressure Force, P_stem, on the Shear Force, N_wall, and Bending Moment, M_wall

A free-body diagram of the stem of a cantilever-type retaining wall–soil system showing various forces acting on it during an earthquake is presented in Fig. 13. From the discussion in the preceding sections, it is clear that along the height of the stem, H_stem, there is an amplification of acceleration toward the top of the stem—the same is presented in Fig. 13. In addition, the variation of the seismic earth pressure along the height of the stem may be considered to be linear, as presented in Fig. 13. It is clear that N_wall and M_wall will vary along the height of the stem and with the time of the earthquake. Further, both of these quantities are dependent upon the wall seismic inertia forces, F_wa and F_wp. For an accurate seismic structural stability analysis of the wall, it is crucial that the contribution of wall seismic inertia forces should be evaluated at different locations along the height of stem and at different times of the earthquake. Assuming that the stem behaves as a cantilever beam, having a fixed connection with the base slab, the shear force, N_w, and bending moment, M_w, time history can be computed by using the dynamic Euler–Bernoulli beam theory as shown by Eqs. (3) and (4), respectively:

N_{wall} (z, t) = E_{wall} I_{stem} \frac{\partial^{3} u_{x} (z, t)}{\partial z^{3}} = \int_{0}^{t} \int_{0}^{H_{stem}} (- m \frac{\partial^{2} u_{x} (z, t)}{\partial t^{2}} + p_{stem} (z, t)) d z d t

(3)

M_{wall} (z, t) = E_{wall} I_{stem} \frac{\partial^{2} u_{x} (z, t)}{\partial z^{2}} = \int_{0}^{t} \int_{0}^{H_{stem}} \int_{0}^{H_{stem}} (- m \frac{\partial^{2} u_{x} (z, t)}{\partial t^{2}} + p_{stem} (z, t)) d z^{2} d t

(4)

where N_wall(z, t) and M_wall(z, t) = shear force and bending moment on the stem for time t and at height z along the stem; E_wall = Young’s modulus of the wall; I_stem = second moment of area of the stem; u_x(z, t) = horizontal elastic deflection of the stem for time t and at height z along the stem;

\partial^{2} u_{x} (z, t) / \partial t^{2}

= predicted horizontal acceleration for time t and at height z along the stem =

a_{stem} (z, t)

;

a_{stem} (z, t)

= acceleration of the stem for time t and at height z along the stem; and

p_{stem} (z, t)

= seismic earth pressure for time t and at height z along the stem. It is to be noted that z is measured above the base slab along the height of the stem, H_stem. Eqs. (3) and (4) have been integrated numerically; and thus, if N is the number of elements of the stem, then Eqs. (3) and (4) could be rewritten

N_{wall} (z, t) = \sum_{1}^{N} - m_{n} \times a_{stem_n} (t) + \sum_{1}^{N} P_{stem_n} (t)

(5)

M_{wall} (z, t) = \sum_{1}^{N} - m_{n} \times a_{stem_n} (t) \times z_{n} + \sum_{1}^{N} P_{stem_n} (t) \times z_{n}

(6)

where

m_{n}

= mass of the nth element of the stem;

a_{stem_n} (t)

= acceleration of the nth element of the stem for time t;

z_{n}

= height of the nth element along the stem; and P_{stem_n}(t) = seismic earth pressure force for the nth element for time t. Eqs. (5) and (6) can be further simplified as

N_{wall} (z, t) = - γ_{wall} \times b_{stem} \times \sum_{1}^{n} (z_{n} - z_{n - 1}) \times a_{stem_n} (t) + \sum_{1}^{n} P_{stem_n} (t)

(7)

M_{wall} (z, t) = - γ_{wall} \times b_{stem} \times \sum_{1}^{n} (z_{n} - z_{n - 1}) \times a_{stem_n} (t) \times z_{n} + \sum_{1}^{n} P_{stem_n} (t) \times z_{n}

(8)

where

γ_{wall}

= unit weight of the wall; and b_stem = width of the stem. From Eqs. (7) and (8), it is clear that the shear force and bending moment depend upon the wall seismic inertia force and the seismic earth pressure force. Their effects are discussed next for the top and bottom ⅓H_stem as well as for the mid-height of the stem of the wall.

Fig. 13. Free-body diagram of the stem of a cantilever-type retaining wall–soil system showing various forces acting on it during an earthquake.

Effect of Stem Seismic Inertia Force, F_{stem_a} and F_{stem_p}, on Shear Force, N_wall, and Bending Moment, M_wall, for the Top ⅓ H_stem

Table 2 shows the relative contributions of static earth pressure force for the stem P_{stem(static),} seismic stem inertia force F_{stem_a}, F_{stem_p}, where F_{stem_a}, F_{stem_p} are the stem seismic inertia force acting away and toward the backfill soil, and seismic earth pressure force increment at the stem, ΔP_stem [=P_stem – P_stem(static)] on shear force N_wall and bending moment M_wall of the stem. As observed, at the beginning of the seismic analysis (i.e., time t = 0 s), only P_stem(static) causes N_wall and M_wall because other values are 0; however, at time t = 3.9 s, when the applied earthquake acceleration has a maximum value and is applied toward the backfill soil, P_stem(static), F_{stem_a}, and ΔP_stem all contribute to N_wall and M_wall. In addition, these N_wall and M_wall for time t = 3.9 s act away from the backfill soil (i.e., in a direction opposite to the direction of the applied earthquake acceleration and they have negative values, as shown in Table 2). When the applied earthquake acceleration has a maximum value but is applied away from the backfill soil at time t = 4.5 s, then, like before, P_stem(static), F_{stem_p}, and ΔP_stem all contribute to N_wall and M_wall. However, unlike the previous case, P_stem(static) and ΔP_stem produce N_wall and M_wall in the same direction as the direction of the applied earthquake acceleration and they have negative values, as shown in Table 2, whereas F_{stem_p} produces N_wall and M_wall in a direction opposite to the direction of the applied earthquake acceleration and it has a positive value, as shown in Table 2.

Table 2. Effect of stem seismic inertia force, f_{stem_a} and f_{stem_p}, on shear force, n_wall, and bending moment, m_wall, for the top ⅓H_stem, the mid-height of the stem, and the bottom ⅓H_stem

Time [t (s)]	N_wall (kN/m)	Contribution to N_w (kN/m) due to			M_wall (kN·m/m)	Contribution to M_w (kN·m/m) due to
Time [t (s)]	N_wall (kN/m)	P_stem(static)	F_{stem_a} or F_{stem_p}	ΔP_stem	M_wall (kN·m/m)	P_stem(static)	F_{stem_a} or F_{stem_p}	ΔP_stem
Top ⅓ H_stem
0.0	−0.72	−0.72	0.01	0.00	−3.68	−3.68	0.02	0.00
3.9	−2.65	−0.72	−1.08	−0.85	−14.50	−3.68	−8.89	−1.93
4.5	−0.88	−0.72	3.72	−3.88	−7.68	−3.68	17.22	−21.22
30.0	−2.20	−0.72	0.00	−1.48	−13.11	−3.68	0.00	−9.43
Mid-height of the stem
0.0	−9.45	−9.45	0.02	0.00	−36.32	−36.40	0.08	0.00
3.9	−26.79	−9.45	−4.68	−12.68	−110.68	−36.40	−24.79	−49.49
4.5	−10.51	−9.45	11.33	−12.39	−42.61	−36.40	47.94	−54.15
30.0	−22.70	−9.45	0.00	−13.52	−88.95	−36.40	0.00	−52.55
Bottom ⅓ H_stem
0.0	−52.97	−53.10	0.04	0.00	−92.34	−92.45	0.11	0.00
3.9	−122.90	−53.10	−11.86	−58.20	−217.60	−92.45	−29.34	−95.78
4.5	−28.17	−53.10	21.75	3.18	−63.90	−92.45	64.53	−35.98
30.0	−88.56	−53.10	0.00	−53.46	−173.90	−92.45	0.00	−81.45

Note: For all forces, a negative value indicates that the force is acting away from the backfill soil, whereas a positive value indicates that the force is acting toward the backfill soil. Similarly, for the moments, a negative value indicates that the moment has a counterclockwise-sense, whereas a positive value indicates that the moment has a clockwise-sense.

Effect of Stem Seismic Inertia Force, F_{stem_a} and F_{stem_p}, on Shear Force, N_wall, and Bending Moment, M_wall, for the Mid-Height of the Stem

Table 2 shows the relative contributions of static earth pressure force for the stem P_stem(static), stem seismic inertia force F_{stem_a}, F_{stem_p}, and seismic earth pressure force increment ΔP_stem on N_wall and M_wall for the mid-height of the stem. At the beginning of the seismic analysis (i.e., time t = 0 s), only P_stem(static) causes N_wall and M_wall. However, at time t = 3.9 s, when the applied earthquake acceleration has a maximum value and is applied toward the backfill soil, P_stem(static), F_{stem_a}, F_{stem_p}, and ΔP_stem all contribute to N_wall and M_wall, but the effect of F_{stem_a} on N_wall and M_wall is reduced at the top ⅓ H_stem. In addition, all of these quantities act away from the backfill soil (i.e., in a direction opposite to the direction of the applied earthquake acceleration and they have negative values, as shown in Table 2). When the applied earthquake acceleration has a maximum value but is applied away from the backfill soil at time t = 4.5 s, like at the top ⅓ H_stem, P_stem(static), F_{stem_p}, and ΔP_stem all contribute to N_wall and M_wall. However, P_stem(static) and ΔP_stem produce N_wall and M_wall in the same direction as the direction of the applied earthquake acceleration and they have negative values, as shown in Table 2, whereas F_{stem_p} produces N_wall and M_wall in a direction opposite to the direction of the applied earthquake acceleration and it has a positive value, as shown in Table 2.

Effect of Stem Seismic Inertia Force, F_{stem_a} and F_{stem_p}, on Shear Force, N_wall, and Bending Moment, M_wall, for the Bottom ⅓ H_stem

Table 2 shows the effect of static earth pressure force P_stem(static), stem seismic inertia force F_{stem_a}, F_{stem_p}, and seismic earth pressure force increment ΔP_stem on the N_wall and M_wall for the bottom ⅓ H_stem. At the beginning of the seismic analysis (i.e., time t = 0 s) for the top ⅓ H_stem and mid-height of stem, only P_stem(static) causes N_wall and M_wall; however, at time t = 3.9 s, when the applied earthquake acceleration has a maximum value and is applied toward the backfill soil, the contribution of F_{stem_a} to N_wall and M_wall is very small compared with the effect of P_stem(static) and ΔP_stem. In addition, all of these quantities act away from the backfill soil and they have negative values, as shown in Table 2. When the applied earthquake acceleration has a maximum value but is applied away from the backfill soil at time t = 4.5 s, observations for the bottom ⅓ H_stem are similar to the top ⅓ H_stem and mid-height of the stem.

From the above, it can be concluded that when the earthquake acceleration is applied toward the backfill soil, then for the top half of the wall, the wall seismic inertia force (F_{stem_a} or F_{stem_p}) has a major contribution to the shear force N_wall and bending moment M_wall, whereas for the bottom half of the wall it is the combination of static earth pressure force P_stem(static) and seismic earth pressure force increment ΔP_stem that contribute to the shear force, N_wall, and bending moment, M_wall. When the earthquake acceleration is applied away from the backfill soil, the stem seismic inertia force produces shear force, N_wall, and bending moment, M_wall, in the direction of the static earth pressure force and the increment of seismic earth pressure force, causing the shear force and bending moment to attain a minimum value of less than the static value.

Relative Displacement of the Wall and Backfill Soil with respect to the Foundation Soil, Δ_w–f and Δ_s–f

For the wall and backfill soil, relative displacement profiles were constructed for the following pairs:

•

The base of the wall and a reference point 0.5 m below the base of the wall in the foundation soil [i.e., between base_wall and P₁ in Fig. 1(a)].

•

The center of gravity of the backfill soil and a reference point 0.5 m below the base of the wall in the foundation soil [i.e., between s_CG and P₂ in Fig. 1(a)].

These relative displacement profiles for the above pairs are presented in Figs. 14(a and b). It is observed from Fig. 14(a) that a maximum relative displacement of about 0.035 m between the wall and foundation (Δ_w–f) occurs at time t = 3.9 s, which is the same time at which P_vp is minimum [Fig. 8(b)]. In addition, the relative displacement between the backfill and foundation soil (Δ_s–f) achieves its maximum value of about 0.025 m for time t = 3.9 s, and remains constant until the end of the seismic analysis. Thus, from the above two observations, it can be said that the wall and backfill soil move as a single entity and P_vp is causing the sliding of the wall–soil system. Fig. 15 shows the effect of varying earthquake amplitude and frequency content on the relative displacement between the wall and foundation (Δ_w–f). As the amplitude of the applied earthquake acceleration increases from 0.2g to 0.6g, Δ_w–f increases, whereas with an increase in the frequency content of the applied earthquake acceleration from 0.5 Hz to 4 Hz, the Δ_w–f decreases. This is in contrast to what has been observed for the shear force N_wall and bending moment M_wall, which, as described in the preceding sections, attain maximum values when both the frequency content and amplitude of the applied earthquake are maximum (Fig. 12). It is also interesting to note that Δ_w–f is about 0.2 m for an applied earthquake amplitude of 0.6g and a frequency content of 4 Hz [Fig. 15(c)], whereas the same for an applied earthquake amplitude of 0.4g and a frequency content of 2 Hz [Fig. 15(b)] is about 0.25 m, thereby suggesting that the frequency content of the applied earthquake is a more dominating factor than its amplitude that contributes to Δ_w–f. From the above observations, it can be argued that for the global stability of a cantilever-type retaining wall a low-frequency content of the applied earthquake creates a critical case scenario, whereas, for the structural stability of the cantilever-type retaining wall a high-frequency content of applied earthquake creates a critical case scenario. The results also show that the sliding of the wall–soil system is highly sensitive to the number of acceleration cycles (and the duration of the applied earthquake acceleration), which is in contrast to what has been observed for the structural stability where the shear force N_wall and bending moment M_wall are not sensitive to the acceleration cycles.

Fig. 14. (a) Wall–foundation relative displacement, Δ_w–f; and (b) soil–foundation relative displacement Δ_s–f.

Fig. 15. Wall–foundation relative displacement, Δ_w–f, for the uniform sinusoidal acceleration–time history of different amplitudes and frequency contents: (a) f = 0.5 Hz; (b) f = 2 Hz; and (c) f = 4 Hz.

Deformation Shapes of the Cantilever-Type Retaining Wall–Soil System at Various Times during the Duration of the Earthquake

Fig. 16 presents the deformation and contours of horizontal displacement of the cantilever-type retaining wall–soil system at times t = 3.9, 4.5, and 30 s (i.e., at the end of the seismic analysis). Because the dynamic analysis was carried out after the completion of all the five stages of construction of the retaining wall, the deformation and contours of horizontal displacement of the cantilever-type retaining wall–soil system presented in Stage 5 of Fig. 3 can also be said to be the results for t = 0 s of the dynamic analysis. For Figs. 16(a–c), it is important to highlight that the deformation shape of the stem and base slab shown is measured relative to the original position of the retaining wall, that is, at the start of the dynamic analysis at t = 0 s. Stage 5 of Fig. 3 shows that at t = 0 s, the stem rotates by 0.02° away from the backfill soil while the base slab heel has a clockwise rotation of 0.065°—thereby suggesting that the stem and base slab rotate in opposite directions to each other). However, at time t = 3.9 s—Fig. 16(a)—when the earthquake acceleration has its maximum value and is applied toward the backfill soil, the stem rotation is predicted as 0.217° away from the backfill soil while the base slab toe has a counterclockwise rotation of 0.014°—thereby suggesting that between t = 0 and 3.9 s, both the stem and base slab rotate in the same direction. In addition, as described in the previous section, at time t = 3.9 s, the wall slides as a rigid body away from the backfill soil by about 0.025 m. At time t = 4.5 s, as presented in Fig. 16(b), when the earthquake acceleration has its maximum value and is applied away from the backfill soil, the stem rotates back toward the backfill soil (but is still away from its original position) by about 0.017° while the base slab toe rotates clockwise (but still has a rotation of about 0.01° compared with its original position at t = 0 s). The wall slides toward the backfill soil; however, it is still away from the backfill soil by about 0.017 m as compared with its original position. At the end of the seismic analysis at time t = 30 s, the stem has a residual rotation of 0.204° relative to its original position and the base slab toe has a residual counterclockwise rotation of 0.038° in to the foundation soil while the wall has a residual sliding away from the backfill soil of about 0.035 m, as presented in Fig. 16(c).

Relationship between the Seismic Earth Pressure and Wall Displacement

In order to understand the relationship between the seismic earth pressure and wall displacement, Figs. 10 and 16 should be studied together. It can be observed from Fig. 16 that the deformation mechanism of the cantilever-type retaining wall is quite complicated and includes the sliding of the cantilever-type retaining wall relative to the foundation layer, rotation of cantilever-type retaining wall as a rigid body because of the foundation soil deformability, and the rotation of the stem due to its elastic deflection. At time t = 3.9 s, when the earthquake acceleration is applied toward the backfill soil and has a maximum value [Fig. 2(b)], the cantilever-type retaining wall slides away from the backfill soil to a maximum distance, the stem deflects by a maximum value away from the backfill soil, and the retaining wall rotates about the toe by a maximum value away from the backfill soil [Fig. 16(a)]. At the same time, the seismic earth pressure force increments behind the upper half of the stem ΔP_stem and along the virtual vertical plane ΔP_vp reach a minimum value and they are close to the static earth pressure [Figs. 10(a, c, and d)]. Thus, it can be said that the seismic earth pressure forces in these locations reach the active states and their values do not change with increasing acceleration level and wall movement. However, at the same time, t = 3.9 s, it can be observed that the seismic earth pressure force increment, ΔP_stem, in the lower half of the stem reaches the maximum value and does not reach the active state [Fig. 10(b)] and its value increases with increasing earthquake acceleration and wall movement. Similarly, at time t = 4.5 s, when the earthquake acceleration is applied away from the backfill soil and it has a maximum value, the cantilever-type retaining wall slides by a maximum value toward the backfill soil, the stem deflects by a maximum value toward the backfill soil, and the retaining wall rotates about the toe by a maximum value toward the backfill soil. At the same time, the seismic earth pressures force increment, ΔP_stem, in the upper half of the stem and along the virtual vertical plane reach a maximum value, and they partially mobilize the passive state [Figs. 10(a, c, and d)]. Thus, it can be said that the seismic earth pressure in these locations partially mobilize the passive state and their value increases with increasing earthquake acceleration and wall movement. However, at the same time, t = 4.5 s, it can be observed that the seismic earth pressure force increment, ΔP_stem, in the lower half of the stem reaches the minimum value [Fig. 10(b)] and its value remains close to the static earth pressure and does not change with increasing earthquake acceleration and wall movement.

Effect of Height of the Cantilever-Type Retaining Wall–Soil System, H, and the Natural Frequency, f_n, on Bending Moment, M_wall, and Wall–Foundation Relative Displacement, Δ_w–f

All of the aforementioned analysis has been carried out for a wall of height H = 5.4 m. In order to assess the influence of the height and thereby the natural frequency of the wall on the structural and global stability under seismic loading, a wall of a different height of H = 10.8 m is also analyzed using the aforementioned FEM, and the comparison of results is presented herewith. From Figs. 17(c and e) it is observed that for walls of height H = 5.4 m and 10.8 m, the bending moment, M_wall, is maximum (330 kN·m/m for H = 5.4 m and 1,750 kN·m/m for H = 10.8 m, respectively) when the applied earthquake acceleration has a frequency content of 4 Hz (for H = 5.4 m) and 2 Hz (for H = 10.8 m), which are the frequencies close to the natural frequency of the respective wall–soil systems. However, from Figs. 18(a and d), which show the relative displacement between the wall and foundation soil, Δ_w–f, it is observed that for the wall of heights H = 5.4 m and 10.8 m, the maximum Δ_w–f is predicted when the applied earthquake acceleration has a frequency content of 0.5 Hz. In addition, for both H = 5.4 m and H = 10.8 m, it is observed that with increasing frequency content of the applied earthquake acceleration, the relative displacement between the wall and foundation soil, Δ_w–f, is decreased. From the above results, it is found that the structural stability is affected by the natural frequency of the wall, and a critical scenario will be when the natural frequency of the wall is equal to the frequency of the applied earthquake acceleration. On the other hand, it can be safely argued that the height of the wall does not affect the nature of the results for the global stability and that the critical case will always occur when the frequency content of the applied earthquake acceleration has a minimum value.

Fig. 17. Bending moment, M_wall, for a uniform sinusoidal acceleration amplitude of 0.4g and different frequency content (a–c) for wall height, H = 5.4 m; and (d–f) for wall height, H = 10.8 m.

Fig. 18. Wall–foundation relative displacement, Δ_w–f, for a uniform sinusoidal acceleration amplitude of 0.4g and different frequency content (a–c) for wall height, H = 5.4 m; and (d–f) for wall height, H = 10.8 m.

Conclusions and Recommendations

•

This paper presents a unique FE-based numerical modeling approach to study both the seismic structural and global stability of a cantilever-type retaining wall–soil system.

•

The study successfully identifies critical scenarios for the structural and global stability of a cantilever-type retaining wall–soil system by considering earthquake characteristics, analyzing the effects of seismic earth pressure and natural frequency, and capturing the deformation mechanism.

•

For the seismic structural and global stability analyses of a cantilever-type retaining wall, the FEM has been innovatively used, in which the structural and global stability of a cantilever-type retaining wall is analyzed by considering the seismic earth pressure, computed at the stem (P_stem) and along a virtual plane (P_vp), and wall and backfill seismic inertia forces. It is noted that P_stem contributes to the structural stability, whereas P_vp contributes to the global stability. It is also observed that P_stem and P_vp are out of phase during the entire duration of the earthquake.

•

A critical case for the structural stability occurs when the earthquake acceleration is directed toward the backfill soil and its frequency content is close to the natural frequency of the retaining wall. In contrast, for the global stability, the critical case occurs when the earthquake acceleration is maximum applied toward the backfill soil having the smallest frequency content.

•

The structural stability of the cantilever-type retaining wall is found to be highly dependent on the natural frequency of the cantilever-type retaining wall relative to the applied earthquake frequency content, whereas the global stability does not appear to be affected by it.

•

For the critical structural stability case, it has been observed that the wall seismic inertia force has a significant contribution to the shear force and bending moment in the top half of the height of the stem. For the lower half of the height of the stem, the seismic earth pressure contributed significantly to the shear force and bending moment, whereas the contribution coming from the wall seismic inertia force was very nominal.

•

When the earthquake acceleration is applied away from the backfill soil and has its maximum value, the wall seismic inertia force acts in a direction opposite of the seismic earth pressure, thereby causing a reduction of the shear force and bending moment.

•

The number of acceleration cycles of the applied earthquake acceleration moderately affects the seismic earth pressure behind the stem as well as the shear force and bending moment, whereas the relative displacement between the wall and soil is observed to be highly sensitive to this. It is also observed that the shear force and bending moment profiles match with the profiles of the seismic earth pressure behind the stem.

Based on the aforementioned conclusions, the following recommendations can be made:

•

The seismic structural and global stability of a cantilever-type retaining wall should be checked separately, in which, the structural stability should be checked by considering the maximum earthquake acceleration anticipated at the construction site and the frequency content of earthquake acceleration being equal to the natural frequency of the structure, whereas for the global stability, a check should be made by considering maximum anticipated earthquake acceleration with a minimum frequency content.

•

The stem seismic inertia force should be considered for the structural design of the upper half of the stem, whereas for lower half part, it could be neglected—thus proposing an economic yet safe design.

•

The seismic earth pressure is crucial for the structural stability especially for the lower half of the stem and should be considered in the structural design. However, for global stability, seismic earth pressure is not significant, and thus only static earth pressure, wall seismic inertia force, and backfill seismic inertia force should be considered as the total driving force causing sliding instability to the wall–soil system.

•

The effect of site characteristics should be considered during the analysis of the seismic structural and global stability of a cantilever-type retaining wall (i.e., the amplification of acceleration response of backfill soil at low acceleration level and de-amplification of acceleration response of backfill soil at high acceleration level).

Notation

The following symbols are used in this paper:

a: acceleration (g);
a_n: acceleration of the nth element (g);
a_we: elemental acceleration for the elements of the base slab and stem (g);
a_se: elemental acceleration for the soil elements lying above and in the middle of the base slab (g);
a_{stem_n}(t): acceleration of the nth element of the stem for time t (g);
a_stem(z, t): acceleration of the stem for time t and at depth z along the stem (g);
b_slab: width of the base slab (m);
b_stem: width of the stem (m);
[C]: damping FE matrix of the system;
D_r: relative density of the soil (%);
E_wall: Young’s modulus of the wall (kN/m²);
$E_{50}^{ref}$: Young’s modulus at 50% failure strength of soil, corresponding to $p^{ref}$ (kN/m²);
$E_{od}^{ref}$: odometric modulus, corresponding to $p^{ref}$ (kN/m²);
$E_{ur}^{ref}$: modulus for unloading–reloading conditions, corresponding to $p^{ref}$ (kN/m²);
F_sa: backfill seismic inertia force acting toward the wall (kN/m);
F_sp: backfill seismic inertia force acting away from the wall (kN/m);
F_{stem_a}: stem seismic inertia force acting away from the backfill soil (kN/m);
F_{stem_p}: stem seismic inertia force acting toward the backfill soil (kN/m);
F_wa: wall seismic inertia force acting away from the backfill soil (kN/m);
F_wp: wall seismic inertia force acting toward the backfill soil (kN/m);
f: frequency of applied earthquake acceleration (Hz);
$G_{o}^{ref}$: small strain shear modulus, corresponding to $p^{ref}$ (kN/m²);
g: acceleration due to gravity (m/s²);
H_stem: height of the stem (m);
I_stem: second moment of area of the stem (m⁴);
[K]: stiffness FE matrix of the system;
k: stiffness stress-level dependency constant;
[M]: mass FE matrix of the system;
M_wall: bending moment on the stem (kN·m/m);
M_wall(z, t): shear force on the stem for time t and at height z along the stem (kN·m/m);
m_n: mass of the nth element of the stem (ton);
N_wall: shear force on the stem (kN/m);
N_wall(z, t): shear force on the stem for time t and at height z along the stem (kN/m);
p^ref: reference confining pressure (kN/m²);
P_stem: seismic earth pressure force at the stem (kN/m);
P_{stem_n}(t): seismic earth pressure force for the nth element for time t (kN/m);
P_stem(static): static earth pressure force at the stem (kN/m);
P_vp: seismic earth pressure force at the virtual plane (kN/m);
P_vp(static): static earth pressure force at the virtual plane (kN/m);
p_stem(static): static earth pressure at the stem (kN/m²);
p_stem(z, t): seismic earth pressure for time t and at height z along the stem (kN/m²);
p_vp(static): seismic earth pressure acting at the virtual plane (kN/m²);
R_f: failure ratio;
R_inter: interface strength-reduction factor;
W_s: weight of the backfill soil above the base slab (kN);
W_w: weight of the wall (kN);
u_x(z, t): horizontal elastic deflection of the stem for time t and at height z along the stem (m);
z: height of the stem above the base slab (m);
z_n: height of the nth element along the stem (m);
α, β: Rayleigh damping parameters;
γ_s: unit weight of the soil (kN/m³);
γ_wall: unit weight of the wall (kN/m³);
γ_0.7: reference shear strain, corresponding to 70% of $G_{o}^{ref}$ ;
ΔP_stem: seismic earth pressure force increment at the stem (kN/m);
ΔP_vp: seismic earth pressure force increment at the virtual plane (kN/m);
Δ_s–f: relative displacement between backfill and foundation soil (m);
Δ_w–f: relative displacement between wall and foundation soil (m);
ζ_s, ζ_wall: damping ratio for the soil and wall, respectively (%);
ν_wall: Poisson’s ratio of the wall;
ν_ur–s: Poisson’s ratio for unloading–reloading conditions of the soil;
φ′: angle of shearing resistance of the soil (°);
ψ: dilatancy of the soil (°); and
ω_z1, ω_z2: first two natural circular frequencies of the FE model (rad/s).

Acknowledgments

The first author would like to thank the Ministry of Higher Education and Scientific Research in Iraq for supporting their studies and funding this research. All the authors also thank the School of Mechanical, Aerospace and Civil Engineering, the University of Manchester, for providing facilities to conduct the research. The critique of the two anonymous reviewers is gratefully acknowledged by the authors—this helped greatly in revising and improving the manuscript.

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Information & Authors

Information

Published In

International Journal of Geomechanics

Volume 19 • Issue 10 • October 2019

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

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Received: Jun 19, 2018

Accepted: Apr 12, 2019

Published online: Aug 16, 2019

Published in print: Oct 1, 2019

Discussion open until: Jan 16, 2020

Authors

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Junied Bakr https://orcid.org/0000-0003-1441-0425

Formerly, Ph.D. Student, School of Mechanical, Aerospace and Civil Engineering, Univ. of Manchester, Manchester M13 9PL UK. ORCID: https://orcid.org/0000-0003-1441-0425.

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Syed Mohd Ahmad [email protected]

Lecturer in Geotechnical Engineering, School of Mechanical, Aerospace and Civil Engineering, Univ. of Manchester, Manchester M13 9PL UK (corresponding author). Email: [email protected]

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Domenico Lombardi https://orcid.org/0000-0003-4116-8347

Lecturer in Geotechnical Engineering, School of Mechanical, Aerospace and Civil Engineering, Univ. of Manchester, Manchester M13 9PL UK. ORCID: https://orcid.org/0000-0003-4116-8347.

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Abstract

Introduction

Seismic Structural and Global Stability of a Cantilever-Type Retaining Wall–Soil System

Seismic Structural Stability

Seismic Global Stability

Finite-Element Model of the Cantilever-Type Retaining Wall–Soil System

Wall–Soil Interaction

Boundary Conditions

Constitutive Models for the Backfill and Foundation Soil, and the Material of the Wall

Seismic Loading

Construction Sequence Simulation and Results from the Static Analysis

Seismic Analysis

Acceleration Response of the Cantilever-Type Retaining Wall–Soil System

Wall and Backfill Seismic Inertia Forces: Fwa, Fwp, Fsa, and Fsp

Seismic Earth Pressure Forces: Pstem and Pvp

Phase Difference between the Seismic Earth Pressure Force Increments, ΔPstem and ΔPvp, and Wall and Backfill Seismic Inertia Forces, Fwa, Fwp, Fsa, and Fsp

Shear Force, Nwall, and Bending Moment, Mwall

Effect of Wall Seismic Inertia Forces, Fwa and Fwp, and Seismic Earth Pressure Force, Pstem, on the Shear Force, Nwall, and Bending Moment, Mwall

Effect of Stem Seismic Inertia Force, Fstem_a and Fstem_p, on Shear Force, Nwall, and Bending Moment, Mwall, for the Top ⅓ Hstem

Effect of Stem Seismic Inertia Force, Fstem_a and Fstem_p, on Shear Force, Nwall, and Bending Moment, Mwall, for the Mid-Height of the Stem

Effect of Stem Seismic Inertia Force, Fstem_a and Fstem_p, on Shear Force, Nwall, and Bending Moment, Mwall, for the Bottom ⅓ Hstem

Relative Displacement of the Wall and Backfill Soil with respect to the Foundation Soil, Δw–f and Δs–f

Deformation Shapes of the Cantilever-Type Retaining Wall–Soil System at Various Times during the Duration of the Earthquake

Relationship between the Seismic Earth Pressure and Wall Displacement

Effect of Height of the Cantilever-Type Retaining Wall–Soil System, H, and the Natural Frequency, fn, on Bending Moment, Mwall, and Wall–Foundation Relative Displacement, Δw–f

Conclusions and Recommendations

Notation

Acknowledgments

References

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Wall and Backfill Seismic Inertia Forces: F_wa, F_wp, F_sa, and F_sp

Seismic Earth Pressure Forces: P_stem and P_vp

Phase Difference between the Seismic Earth Pressure Force Increments, ΔP_stem and ΔP_vp, and Wall and Backfill Seismic Inertia Forces, F_wa, F_wp, F_sa, and F_sp

Shear Force, N_wall, and Bending Moment, M_wall

Effect of Wall Seismic Inertia Forces, F_wa and F_wp, and Seismic Earth Pressure Force, P_stem, on the Shear Force, N_wall, and Bending Moment, M_wall

Effect of Stem Seismic Inertia Force, F_{stem_a} and F_{stem_p}, on Shear Force, N_wall, and Bending Moment, M_wall, for the Top ⅓ H_stem

Effect of Stem Seismic Inertia Force, F_{stem_a} and F_{stem_p}, on Shear Force, N_wall, and Bending Moment, M_wall, for the Mid-Height of the Stem

Effect of Stem Seismic Inertia Force, F_{stem_a} and F_{stem_p}, on Shear Force, N_wall, and Bending Moment, M_wall, for the Bottom ⅓ H_stem

Relative Displacement of the Wall and Backfill Soil with respect to the Foundation Soil, Δ_w–f and Δ_s–f

Effect of Height of the Cantilever-Type Retaining Wall–Soil System, H, and the Natural Frequency, f_n, on Bending Moment, M_wall, and Wall–Foundation Relative Displacement, Δ_w–f