Seismic Performance of Gravity Retaining Walls Considering the Passive Soil at the Front Side

Abstract

There are many reasons to make behavior and performance of retaining walls during earthquakes difficult to evaluate, for example complex response of soil under dynamic loads, insufficiency of pseudo-static analyses, and the necessity for dynamic analyses. In recent years, progressive attention to the concept of performance-based design alleviated the situation and seismic displacement of retaining walls has got a critical role in the designing process of these structures. Furthermore, to economize the process of seismic design and reduce uncertainties coupled with pseudo-static methods, establishing a relationship between these methods and performance-based approaches provides a valuable tool for designers of geotechnical structures. In the present paper, by focusing on gravity walls and establishing applicable curves, relationships between the pseudo-static factor of safety, seismic coefficient, and seismic displacement have been investigated. In order to achieve this goal, the upper bound of limit analysis and integral of Newmark have been used. The height of the soil in the front of wall has been introduced as a design parameter. Fourteen selected Iran’s earthquake records have been implemented in the process of analyses. The proposed curves can be used to determine the required embedment of a wall in the front soil and also for estimating the corresponding seismic displacement for specific values of the seismic coefficient and factor of safety. Using the charts presented in this paper, it can be inferred that for a particular factor of safety (FS), greater values of seismic coefficient (k_h) provide smaller seismic displacements, and also, in greater values of k_h, the difference between values of seismic displacement of various FS will get close. Furthermore, for a certain k_h, increasing FS generates decreasing seismic displacements.

Introduction

Throughout the world, it is witnessed that earthquakes cause serious damages to structures and people’s lives. Geotechnical structures play an essential role in diminishing a great part of these damages. Since gravity retaining wall is greatly used in many countries for lateral support, understanding their performances and behavior during an earthquake is highly recommended. The seismic performance of gravity retaining walls is considerably affected by permanent displacement caused by ground shaking, and conventional methods of seismic design of the walls cannot address the performance of walls. However, the emergence of the performance-based design minimized the number of uncertainties integrated with the conventional design methods. In these methods, not only the stability of structures is guaranteed but also their performances are considered in the designing process. During the twentieth century, several methods have been developed on stability analysis of retaining structures under seismic loads, and just over the last twenty years, the concept of performance-based design earned a proper amount of attention.

Until around 1960s, using the pseudo-static factor of safety (FS) was the most frequent method in seismic stability analysis of slopes and retaining walls. This approach compares the shear stresses required to maintain equilibrium with the available shear strength of soil [1]. The design seismic shear forces could be calculated by the pseudo-static method in which the forces derived by earthquake excitation were idealized by a specific seismic coefficient. This method is simple and, nevertheless, gives no noteworthy information about seismic displacements of retaining walls at the end of an earthquake, i.e., about their seismic performance.

In the 1960s, Newmark [2] developed the sliding block method, which, despite its simplicity, brought a useful criterion to assess the seismic performance of slopes. Newmark stated that when FS = 1, horizontal seismic acceleration of slopes is actually the yield acceleration of the system. Double-time integration of the difference between the yield acceleration and actual earthquake acceleration results in permanent seismic displacement, i.e., whenever the system acceleration exceeds the yield acceleration, the whole structure starts to move. Since then, many geotechnical authors tried to expand this method, e.g., extending to other geotechnical structures. By some simplifying assumptions, Richards and Elms [3] expanded Newmark’s method for retaining walls. Several numerical analyses took place after that to address retaining walls seismic displacement under various situations, to name a few, Nadim [4]; Nadim and Whitman [5], Rafnsson [6], and Rafnsson and Prakash [7]. Whitman and Liao [8] quantified the errors of Richards and Elms method discovered by other authors (Zarrabi-Kashani [9]) and modified the permanent seismic displacement formula generated by Richards and Elms. In 2006, Stamatopoulos et al. [10] proposed a model in which geometry changes of the backfill are accounted for in seismic displacement analyses. By using a kinematic approach of limit analysis and upper bound theorem, Michalowski [11] studied the displacement of slopes during seismic excitations considering multi-block analysis approach and acceleration hodographs. By using the limit equilibrium method, Trandefir et al. [12] analyzed the performance of reinforced retaining walls under earthquake loads and compared the displacement at the crest of the wall and backfill. Based on the limit analysis of upper bound theorem, Li et al. [13] studied the displacement of gravity retaining walls due to sliding and also the effect of vertical component of the seismic acceleration on the yield acceleration coefficient (k_y). Furthermore, they tried to consider wall roughness effects on the failure of the backfill soil. In 2013, Chowdhury and Singh [14] proposed an analytical solution to address the shortcomings of using Mononobe–Okabe [22, 23] in seismic evaluation of gravity retaining walls. Han et al. [15] investigated the effect of excess pore pressure in the rotational displacement of gravity quay retaining walls during earthquakes.

As it is mentioned, using seismic stability analysis was mainly focused on implementing FS before Newmark; even in the current state of the art, using FS is not obsolete at all. In recent years, a progressive gap has shaped between performance-based and conventional methods; however, some authors tried to establish a relationship between FS and displacement-based analysis. Such methodologies can help practitioners to have a better understanding of the performance of geotechnical structures, which was previously designed by traditional methods, and even decrease the amount of uncertainty involved in designing by these approaches, e.g., choosing k_h based on a rational methodology, not on judgment and expertise of designers. There are many pieces of researches centered on the relationship of seismic displacement of slopes and their pseudo-static FS; on the other hand, there are a relatively small number of papers in the technical literature. Bozbey and Gundogdu [16] undertook such studies to propose charts to link seismic displacements, k_h, and FS using strong motion records taken during earthquakes in Turkey. Nadi et al. [17] generated charts with the same functionality focusing on seismic records of Iran. Biondi et al. [18] studied the seismic displacement of gravity retaining walls due to sliding failure. They proposed a particular definition of FS, which is the ratio of limit acceleration coefficient to earthquake-induced peak ground acceleration (maximum horizontal acceleration) on a system of two-wedge gravity retaining wall. In this study, limit acceleration coefficient defined as the peak horizontal acceleration coefficient required to induce a permanent displacement equal to the limit values of displacement, which could be calculated based on recommendations and guidelines for various situations. Accordingly, the proper k_h based on allowable seismic displacement was provided.

Methodology

In order to calculate the seismic displacement of gravity walls, Newmark method is used in this paper. In this method, the yield acceleration (k_y) plays the main role in calculating the seismic displacement. Whenever applied excitation passes this threshold acceleration, the system begins to deform. In this paper, for calculating k_y, a formulation based on the upper bound theorem of limit analysis is proposed. This process is applied to various values of FS. The pseudo-static analysis of Mononobe–Okabe [19, 20] is considered as the method to calculate FSs against sliding failure. In the following, first, the characteristics of considered gravity retaining walls will be described, and then, formulations for calculating the seismic displacement of gravity retaining walls will be provided.

Characteristics of Considered Gravity Retaining Walls

Figure 1 shows a gravity retaining wall, which comprises three wedges. The soil mass with height H₁ that covers all the wall face, having the internal friction angle φ₁ and the unit weight γ₁, is taken as the first wedge, which is inclined at an angle θ with the horizontal line. Correspondingly, soil mass at the front of wall with height H₂, having an internal friction angle φ₂ and unit weight γ₂, is considered as the second wedge, which has got a vertical fill. The wall itself is introduced as the third wedge with unit weight γ₃, which is as high as the first wedge, and is inclined at angles λ₁ and λ₂ with the horizontal line next to the first and the second wedges, respectively. δ₁, δ₂, and δ₃ are interface friction angles of the wall with backfill, with the front soil, and with the soil at the base, respectively. It must be noted that all three wedges are considered dry.

Fig. 1

Failure mechanisms of active and passive blocks

Suggested Model for Permanent Seismic Displacements

In the current paper, a formulation based on the upper bound theorem of limit analysis is proposed to provide k_y, and based upon this value, Newmark’s sliding block is used as the method for calculation of seismic displacements of gravity retaining walls. In the following, assumptions are listed:

• All wedges are solid.

• The first and the second wedges follow the associated flow rule.

• The soil is dry and cohesionless.

Initiation of plastic deformation coincides with equality of stress state and yield condition of structure [11]. If the result of this condition forms a convex yield surface in the stress space, and the deformation is governed by the normality (or associative) flow rule, it can be concluded that in any kinematically admissible failure mechanism, the rate of internal work is not less than the rate of true external forces [21].

$$\mathop \smallint \limits_{V} \sigma_{ij}^{k} \dot{\varepsilon }_{ij}^{k} {\text{d}}V \ge \mathop \smallint \limits_{S} T_{i} \upsilon_{i} {\text{d}}S + \mathop \smallint \limits_{V} X_{i} \upsilon_{i}^{k} {\text{d}}V$$

(1)

The integral over entire volume V on the left side of Eq. (1) represents energy dissipated in entire mechanism, which is called the rate of internal work. Integral over entire boundary S on the right side introduces the external work rate of surface load T_i on S, which has got deformation velocity of υ_i (kinematic boundary condition). The external work rate of distributed forces X_i per unit volume (such as weight, and inertial) in the kinematically admissible velocity field $\upsilon_{i}^{k}$ is given by the latter integral in Eq. (1). Associated stress field $\sigma_{ij}$ marked with superscript k is compatible with the selected mechanism [11]. $\dot{\varepsilon }_{ij}$ is the strain rate compatible with real or virtual displacement rate $\upsilon_{i}$ or $\upsilon_{i}^{k}$ [22].

For the wall shown in Fig. 1, two parallel lines at each boundary of first and second wedges show failure surfaces, and the third wedge (the wall) slides along its base. As it is shown in Fig. 1, $\dot{u}_{1}$, $\dot{u}_{2}$ and $\dot{u}_{3}$ are the velocities of the first, second and the third wedges; also, $\dot{u}_{31}$ and $\dot{u}_{23}$ represent the relative velocity between the wall and the first and the second wedges, respectively. Based on the associated flow rule, velocity vectors of wedges make specific angles with failure surfaces in kinematic boundary condition [23]. These angles between $\dot{u}_{1}$ and $\dot{u}_{2}$ and their corresponding failure surfaces are φ₁ and φ₂, and also their corresponding soil-wall failure surfaces are δ₁ and δ₂, respectively. The angle between velocity vector of the wall and the base is δ₃.

The velocity vectors of all three wedges form velocity hodographs. In Fig. 2, all possible shapes of velocity hodographs are shown.

Fig. 2

Shapes of velocity hodographs

Regarding Eq. 1, based on the upper bound theorem in the limit state, the rates of internal and external works in the soil wedges of Fig. 1 are as:

$$d \ge \left( {1 - k_{v} } \right)\left[ {\dot{E}_{{w_{1} }} + \dot{E}_{{w_{2} }} + \dot{E}_{{w_{3} }} } \right] + k_{h} \left( {\dot{E}_{{I_{1} }} + \dot{E}_{{I_{2} }} + \dot{E}_{{I_{3} }} } \right)$$

(2)

where d represents the dissipated energy in the whole soil-wall system and k_v is the vertical seismic coefficient. $\dot{E}_{{w_{1} }}$, $\dot{E}_{{w_{2} }}$ and $\dot{E}_{{w_{3} }}$, external work rates due to weights of the wedges, are:

$$\dot{E}_{{w_{1} }} = W_{1} \dot{u}_{1} \sin (\alpha - \phi_{1} )$$

(3a)

$$\dot{E}_{{w_{2} }} = - W_{2} \dot{u}_{2} \sin (\beta + \phi_{2} )$$

(3b)

$$\dot{E}_{{w_{3} }} = - W_{3} \dot{u}_{3} \sin \delta_{3}$$

(3c)

Correspondingly, $\dot{E}_{{I_{1} }} ,\dot{E}_{{I_{2} }}$ and $\dot{E}_{{I_{3} }} ,$ external work rates due to inertial forces derived from acting horizontal seismic acceleration, are given by:

$$\dot{E}_{{I_{1} }} = W_{1} \dot{u}_{1} \cos (\alpha - \phi_{1} )$$

(4a)

$$\dot{E}_{{I_{2} }} = W_{2} \dot{u}_{2} \cos (\beta + \phi_{2} )$$

(4b)

$$\dot{E}_{{I_{3} }} = W_{3} \dot{u}_{3} \cos \delta_{3}$$

(4c)

W₁, W₂, and W₃ are the weights of the back wedge, the front wedge, and the wall, respectively. α and β are the critical angles of back and front wedges, respectively. Assume the vertical seismic coefficient is a multiple of the horizontal seismic coefficient (k_v = xk_h) and also, assume as the failure occurs, the horizontal seismic coefficient equals yielding seismic coefficient (k_y = k_h). As long as the latter condition holds, Eq. (2) may be rewritten as

$$d = \left( {1 - xk_{y} } \right)\left[ {\dot{E}_{{w_{1} }} + \dot{E}_{{w_{2} }} + \dot{E}_{{w_{3} }} } \right] + k_{y} \left( {\dot{E}_{{I_{1} }} + \dot{E}_{{I_{2} }} + \dot{E}_{{I_{3} }} } \right)$$

(5)

Q₁ and Q₂ are the coefficients applied to velocity vectors $\dot{u}_{1}$ and $\dot{u}_{2}$ to be converted to $\dot{u}_{3}$. These coefficients, which are derived based on law of sines in each hodograph, are as follows:

$$\left\{ {\begin{array}{*{20}l} {Q_{1} = \frac{{\sin (\lambda_{1} - \delta_{3} - \delta_{1} )}}{{\sin (\alpha + \lambda_{1} - \phi_{1} - \delta_{1} )}}} \hfill & {(\delta_{3} \ge \phi_{1} - \alpha )} \hfill \\ {Q_{1} = \frac{{\sin (\pi - \lambda_{1} + \delta_{3} - \delta_{1} )}}{{\sin (\pi - \alpha - \lambda_{1} + \phi_{1} - \delta_{1} )}}} \hfill & {(\delta_{3} < \phi_{1} - \alpha )} \hfill \\ \end{array} } \right.$$

(6a)

$$\left\{ {\begin{array}{*{20}l} {Q_{2} = \frac{{\sin (\lambda_{2} + \delta_{3} + \delta_{2} )}}{{\sin (\lambda_{2} + \delta_{2} + \phi_{2} + \beta )}}} \hfill & {(\phi_{2} + \beta \ge \delta_{3} )} \hfill \\ {Q_{2} = \frac{{\sin (\lambda_{2} + \delta_{3} - \delta_{2} )}}{{\sin (\lambda_{2} - \delta_{2} + \phi_{2} + \beta )}}} \hfill & {(\phi_{2} + \beta < \delta_{3} )} \hfill \\ \end{array} } \right.$$

(6b)

Combining Eqs. (6) and (5) yields

$$k_{y} = \frac{{d - (Q_{1} \dot{E}_{{w_{1} }} + Q_{2} \dot{E}_{{w_{2} }} + \dot{E}_{{w_{3} }} )}}{{(Q_{1} \dot{E}_{{I_{1} }} + Q_{2} \dot{E}_{{I_{2} }} + \dot{E}_{{I_{3} }} ) - x(Q_{1} \dot{E}_{{w_{1} }} + Q_{2} \dot{E}_{{w_{2} }} + \dot{E}_{{w_{3} }} )}}$$

(7)

Since the soil is cohesionless, it can be concluded that in the above formula d = 0. As the upper bound theorem has been used, the desirable value for k_y would be derived by minimizing Eq. (7), with respect to α and β.

Once the seismic acceleration outweighs the yield acceleration (k > k_y), the sliding of the soil-wall system initiates. In other words, plastic deformations induce inertial forces due to seismic acceleration. The sliding at the first, the second, and the third wedges induces accelerations $\ddot{u}_{1}$,$\ddot{u}_{2}$, and $\ddot{u}_{3}$, respectively. As it is shown in Fig. 3, acceleration vectors make angles Ψ₁, Ψ₂, and Ψ₃ with their corresponding failure surface, where Ψ_i is dilation angle. Relative acceleration and dilation angle between the first wedge and the wall equal to $\ddot{u}_{31}$ and Ψ₃₁, respectively, and between the second wedge and the wall are $\ddot{u}_{23}$ and Ψ₂₃, correspondingly. The current algorithm can consider 0 ≤ Ψ₁ ≤ φ₁, 0 ≤ Ψ₂ ≤ φ₂, 0 ≤ Ψ₃ ≤ δ₃, 0 ≤ Ψ₃₁ ≤ δ₃, and 0 ≤ Ψ₂₃ ≤ δ₂, but since it is assumed Ψ₁ = φ₁, Ψ₂ = φ₂, Ψ₃ = δ₃, Ψ₃₁ = δ₁, and Ψ₂₃ = δ₂, acceleration hodographs are as the same as velocity hodographs shown in Fig. 2. It must be noted that work rates due to inertial forces are negative; this stems from the fact that wedge acceleration and acceleration vector are in opposite directions. The new energy balance equation is as follows.

$$\begin{aligned} d & = - m_{1} V_{1} \ddot{u}_{1} \cos (\phi_{1} - \psi_{1} ) - m_{2} V_{2} \ddot{u}_{2} \cos (\phi_{2} - \psi_{2} ) \\ & \quad - m_{3} V_{3} \ddot{u}_{3} \cos (\delta_{3} - \psi_{3} ) + (1 - xk) \\ & \quad \times \left[ {\dot{E}_{{w_{1} }} + \dot{E}_{{w_{2} }} + \dot{E}_{{w_{3} }} } \right] + k(\dot{E}_{{I_{1} }} + \dot{E}_{{I_{2} }} + \dot{E}_{{I_{3} }} ) \\ \end{aligned}$$

(8)

Fig. 3

Displacement mechanism of a gravity retaining wall

The following equation derived by substituting Eqs. (5) and (7) into Eq. (8).

$$\begin{aligned} \ddot{u}_{3} & = \left( {k - k_{y} } \right)g\frac{{\left[ {m_{1} Q_{1} \cos (\alpha - \phi_{1} ) + m_{3} \cos (\delta_{3} ) + m_{2} Q_{2} \cos (\phi_{2} + \beta ) - x(\dot{E}_{{w_{1} }} + \dot{E}_{{w_{2} }} + \dot{E}_{{w_{3} }} )} \right]}}{{\left[ {Q_{1}^{2} m_{1} \cos (\phi_{1} - \psi_{1} ) + Q_{2}^{2} m_{2} \cos (\phi_{2} - \psi_{2} ) + m_{3} \cos (\phi_{3} - \psi_{3} )} \right]}} \\ & = C\left( {k - k_{y} } \right)g \\ \end{aligned}$$

(9)

where k is the seismic acceleration coefficient and C is a parameter representing geometrical and mechanical characteristics of the wedges. Since real acceleration acts on wedges, double integration with respect to the time on both negative and positive acceleration directions and then choosing the larger value result in seismic displacement of gravity retaining wall:

$$u_{3} = C{\iint }g(k - k_{y} ){\text{d}}t{\text{d}}t$$

(10)

A computer program was written in MATLAB [24] environment for the computation of permanent seismic displacement of a three-wedge gravity retaining walls system based on provided formula. The inputs are H₁, H₂, a, λ₁, λ₂, φ₁, φ₂, φ₃, δ₁, δ₂, δ₃, γ₁, γ₂, γ₃, θ₁, θ₂, k_h, k_v, FS, and an earthquake record. The outputs are α, β, k_y, C, and the seismic displacement. The optimization algorithm used in this program pursues the procedure developed by Michalowski [20] and also Farzaneh and Askari [25]. This optimization uses the substitution of various combinations of 0 < α, β < 90 in Eq. (7) and estimates the minimum value of k_y.

Pseudo-Static Factor of Safety

Okabe [20] and Mononobe and Matsuo [19] are architects of analyzing pseudo-static seismic forces acting on retaining structures known as Mononobe-Okabe method, which is a direct extension of Coulomb [26] approach [27].

As Mononobe–Okabe method is widely used in the practice, it is chosen as the method to calculate the pseudo-static factor of safety. As it is inferable by the name, the pseudo-static method, based on converting dynamic seismic forces to a static force, tries to evaluate the stability of geotechnical structures. For doing so, k_h and k_v are introduced as

$$k_{h} = \frac{{a_{h} }}{g}$$

(11)

$$k_{v} = \frac{{a_{v} }}{g}$$

(12)

where a_h, a_v, and g indicate horizontal seismic acceleration, vertical seismic acceleration, and gravity acceleration, respectively. It must be noted that coefficients of Eqs. (11) and (12) are dimensionless. Corresponding forces calculate as

$$f_{h} = k_{h} W$$

(13)

$$f_{v} = k_{v} W$$

(14)

where W is the weight of the (active or passive) soil blocks involved in the failure mechanism. The forces of Eqs. (4) and (5) add to static forces applying to gravity retaining walls.

P_ae and P_pe are forces acting on active and passive wedges, respectively, which are

$$P_{\text{ae}} = 0.5K_{\text{ae}} \gamma_{1} H_{1}^{2} (1 - k_{v} )$$

(15)

$$P_{\text{pe}} = 0.5K_{\text{pe}} \gamma_{2} H_{2}^{2} (1 - k_{v} )$$

(16)

where K_ae, dynamic active pressure coefficient, and K_pe, dynamic passive pressure coefficient, are Mononobe–Okabe’s coefficients which calculate as

$$K_{ae} = \frac{{\cos^{2} (\phi_{1} - 90 + \lambda_{1} - \eta )}}{{\cos \psi \cos^{2} (90 - \lambda_{1} )\cos (\delta_{1} + 90 - \lambda_{1} + \eta )\left[ {1 + \sqrt {\frac{{\sin (\delta_{1} + \phi_{1} )\sin (\phi_{1} - \theta_{1} - \eta )}}{{\cos (\delta_{1} + 90 - \lambda_{1} )\cos (\theta_{1} - 90 - \lambda_{1} )}}} } \right]^{2} }}$$

(17)

$$K_{pe} = \frac{{\cos^{2} (\phi_{2} - 90 + \lambda_{2} - \eta )}}{{\cos \psi \cos^{2} (90 - \lambda_{2} )\cos (\delta_{2} + 90 - \lambda_{2} + \eta )\left[ {1 + \sqrt {\frac{{\sin (\delta_{2} + \phi_{2} )\sin (\phi_{2} - \theta_{2} - \eta )}}{{\cos (\delta_{2} + 90 - \lambda_{2} )\cos (\theta_{2} - 90 - \lambda_{2} )}}} } \right]^{2} }}$$

(18)

β, λ₁, λ₂, θ₁, and θ₂ are shown in Fig. 4 and η = arctan(k_h/(1 − k_v)). It should be noted that the passive force values obtained by Mononobe-Okabe method are conservative [28]. In order to address the stability in the pseudo-static method, FS against sliding, the most frequent failure in gravity retaining walls, is given by the ratio of resisting to driving forces, which for the wall shown in Fig. 4 calculates as

$$FS = \frac{{R_{V} \tan (\delta_{3} ) + P_{pe} \cos (\delta_{2} + \lambda_{2} - 90)}}{{P_{ae} \cos (90 + \delta_{1} - \lambda_{1} ) + k_{h} W_{3} }}$$

(19)

Fig. 4

Active and passive forces on a retaining wall

In this formula, R_v, the resultant of vertical forces acting upon the wall, is:

$$\begin{aligned} R_{V} & = (1 - k_{v} )W_{3} + P_{ae} \sin (90 + \delta_{1} - \lambda_{1} ) \\ & \quad - P_{pe} \sin (\delta_{2} + \lambda_{2} - 90) \\ \end{aligned}$$

(20)

Model Verification by Comparison with Literature

For a gravity retaining wall, like the one shown in Fig. 1, with H₁ = 8 m, H₂ = 0, a = 0.3 m, γ₁ = 20 kN/m³, λ₁ = 90, λ₂ = 60.64, δ₁ = 0, θ₁ = 0, W_w = 556.8 kN/m, where W_w is the weight of the wall, α and k_y are calculated and compared to the ones predicted by Li et al. [13]. These results, which are listed in Table 1, show a great match between these models.

Table 1 Comparison of yield acceleration (k_y) and critical back-soil wedge failure angle (α)

Another gravity retaining wall with the following characteristics is used to compare the values of k_y and seismic displacement derived from proposed method over other methods: H₁ = 4 m, H₂ = 0, a = 0.3 m, γ₁ = 21.6 kN/m³, λ₁ = 90, λ₂ = 62.3, φ₁ = 33, δ₁ = 22, δ₃ = 23.3, θ₁ = 0, W_w = 130.08 kN/m. Values of seismic displacement have been calculated under Northridge 1994 earthquake, which its peak ground acceleration (PGA) and magnitude are 0.334 g and 6.69, respectively. Table 2 provides the results. It shows that the proposed model outcomes are close to the results provided by Mojallal and Ghanbari [29] and also Whitman and Liao [8].

Table 2 Comparison of yield acceleration (k_y) and seismic displacement

For a gravity wall with H₁ = 8 m, H₂ = 0, a = 1 m, γ₁ = 18 kN/m³, λ₁ = 90, λ₂ = 60.255, φ₁ = 26, δ₁ = 11, δ₃ = 18, θ₁ = 0, W_w = 634 kN/m, Stamatopoulos et al’s [10] model predicts α = 54.2 and k_y = 0.016; on the other hand, the proposed model estimates α and k_y equal to 54.10 and 0.0172, respectively. This comparison shows that the proposed model and Stamatopoulos et al’s [10] model are in very close agreement.

A Methodology Linking Seismic Displacement, Pseudo-Static Factor of Safety, and Horizontal Seismic Coefficient

In the current paper, by using records chosen by Miraboutalebi et al. [31] from seven earthquakes in transversal and longitudinal directions (fourteen records in total) recorded in Iran with various PGAs, values of seismic displacement of gravity retaining walls are calculated. Of course, a comprehensive conclusion for every location cannot be provided by only this number of records, this said, these records have got acceptable extensiveness and acceptability. Table 3 provides information on the records.

Table 3 Characteristics of earthquake records of Iran [32]

In order to establish a relationship between seismic displacement, k_h and FS, geometric and mechanical characteristics of the wedges are introduced as follows. Internal friction angles of the backfill, front and the base soils are identical, and their values are 25, 30, 35, and 40 degrees. Also, δ₁ = δ₂ = δ₃ = 1/3φ, 2/3φ. Moreover, for the wall shown in Fig. 1, θ = 0, 10, λ₁ = 70, 90 λ₂ = 80, k_v = 0, and a = 0.0375H₁. Before introducing other parameters, it is worth-mentioning that after the development of the pioneer solution denoted as Mononobe–Okabe formulae, the effect of the vertical inertia force due to the vertical component of the seismic acceleration has been always accounted in the evaluation of the seismic active and passive thrust. Furthermore, most of the pseudo-static approaches proposed in the literature for the seismic analyses of gravity retaining walls account for both k_h and k_v. Conversely, since the effect of the vertical component of the seismic acceleration is generally neglected in a displacement-based analysis carried out with a Newmark-type approach, in this framework, it is well known that its influence is almost negligible from a practical point of view. Accordingly, it is assumed that k_v = 0. It must be mentioned that the recommended seismic FS for gravity retaining walls in many references is 1.1. For more profound analyses, FS = 0.9, 1, 1.1, and 1.2 are considered too, and H₂/H₁, C/γH₂ and γ₃/γ₁ are non-dimensional parameters derived from formulations. In order to analyze H₂/H₁ variation with k_h for considered values of FS, C/γH₁, ²₂  = 0, γ₃/γ_1,2 = 1.22, where γ₂ = γ₁ are obtained. The height of the soil in the front of the wall (H₂) can be calculated for a given characteristic of soil-wall system and a certain FS and k_h. Consequently, all geometric and mechanical parameters necessary for calculating k_y, α and β are given. With these parameters in hand, by using Newmark integration, seismic displacement of the gravity retaining wall for different records will be available too. The above-mentioned explanations can be depicted as a flowchart shown in Fig. 5.

Fig. 5

The procedure of calculating the seismic displacement

Numerical Results

Based on the described algorithm in the previous sections, for the wall shown in Fig. 1 with H₁ = 8 m, λ₁ = 90, λ₂ = 80, a = 0.0375H₁, φ₁ = φ₂ = φ₃ = 40, δ₁ = δ₂ = δ₃ = 2/3φ, θ₁ = 10 for various factors of safety, Fig. 6a shows H₂/H₁ variations with k_h; Fig. 6b, c illustrates variations of logarithm of average seismic displacement and also the summation of logarithm of average and standard deviations of seismic displacement with k_h. Figure 7 shows similar curves for a wall with λ₁ = 70, λ₂ = 80, a = 0.0375H₁, φ₁ = φ₂ = φ₃ = 35, δ₁ = δ₂ = δ₃ = 2/3φ, θ₁ = 0. These curves help to link the seismic displacement of the gravity retaining wall to horizontal seismic coefficient and factor of safety. The dashed line in Figs. 6b, c and 7b, c shows the allowable seismic displacement equal to 2% of wall height [33].

Fig. 6

k_h variation with a H₂/H₁, b average seismic displacement and c summation of average and standard deviation of seismic displacement for different values of safety factors (H₁ = 8 m, λ₁ = 90, λ₂ = 80, a = 0.0375H₁, φ = 40, δ = 2/3φ, θ₁ = 10)

Fig. 7

k_h variation with a H₂/H₁, b average seismic displacement and c summation of average and standard deviation of seismic displacement for different values of safety factors (H₁ = 8 m, λ₁ = 70, λ₂ = 80, a = 0.0375H₁, φ = 35, δ = 2/3φ, θ₁ = 0)

Yet, for a wall with λ₁ = 90, λ₂ = 80, a = 0.0375H₁, φ₁ = φ₂ = φ₃, δ₁ = δ₂ = δ₃ = 1/3φ, 2/3φ, θ₁ = 0, Fig. 8 shows k_y variation with H₂/H₁ for φ = 25,30,35,40.

Fig. 8

k_y variation with H₂/H₁ for various φ values. a δ = φ/3, b δ = 2φ/3

Discussion

To shed more light on the procedure of using these types of curves, examples are provided as follows: assume all of the wedges’ parameters are given as same as the ones contributed to represent Fig. 6. By using Fig. 6a and a certain k_h, one can generate H₂/H₁ values for various values of FS. Consider a situation in which the geotechnical designer wants to determine the height of the soil in the front of a wall to have the optimum performance. This type of calculation may help the designer to have a deeper understanding of the height of fill in front of gravity retaining walls. The same procedure applies for determining FS based on a certain H₂/H₁ and k_h: For instance, consider FS = 1 and k_h = 0.2, H₂/H₁ equals 0.15 and vice versa, Fig. 6b, c brings seismic displacement into the circle, in such a way that one can decide on a proper k_h to use in the pseudo-static analyses or on the other hand, estimating displacement during an earthquake (with k_h in hand) for a gravity wall. In other words, a designer can estimate the seismic displacement of a gravity wall just by having the factor of safety without undertaking time-consuming and expensive dynamic analysis. As an example, for the wall shown in Fig. 1, considering FS = 1.1, H₂/H₁ = 0.185 and k_h = 0.2, by using Fig. 6a and c, the seismic displacement of the wall is equal to 72 mm.

The designer can also find the proper k_h based on allowable seismic displacement. There are many recommendations for allowable seismic displacement of gravity walls which can be found in various guidelines; for example, AASHTO LRFD Bridge Design Specifications suggest that horizontal wall displacement can be allowed as much as 250k_h0 in millimeters. Similarly, Eurocode 8 [34] recommends 300k_h0 in millimeters for allowable wall displacement. In both of these codes, the k_h0 is equal to F_PGA PGA. F_PGA is site factor at zero period on acceleration response spectrum which the values provided in tables for different site classes, and PGA is peak ground acceleration on rock.

Assuming H₁ = 8 m and using allowable displacement as 2% of wall height [33] in Fig. 6c, the desired k_h for FS = 1.1 approximately equals 0.14. Even the height of the wall at the front side of the wall can be determined based on allowable seismic displacement. In the process of choosing the permissible displacement, designers must keep in mind that different amount of displacement is required to fully mobilize the active and the passive soil thrust, which, according to the soil type for passive force, is 1.25–10 times larger than the corresponding displacement for active force [35]. This may help them to even choose more reasonable values of allowable seismic displacement.

Based on Fig. 6, for FS = 1.1, an increase in k_h values, from 0 to 0.3, results in H₂/H₁ = 0.04–0.22. That is to say, in order to maintain the passive force against mobilized active force, H₂ must increase. In addition, for a special k_h, as it is inferable from Fig. 6a, larger H₂/H₁ is necessary to satisfy greater FS.

Figure 7 shows similar curves for a wall with λ₁ = 70, λ₂ = 80, a = 0.0375H₁, φ₁ = φ₂ = φ₃ = 35, δ₁ = δ₂ = δ₃ = 2/3φ, θ₁ = 0, which, at FS = 1.2, a rise in k_h from 0.057 to 0.3, declines the average values of seismic displacement from 416.3 to 13.04 mm. Generally stated, in a particular FS, using greater k_h values will provide smaller values for seismic displacement. In greater values of k_h, the difference between values of seismic displacement of various FS will get close. Furthermore, for a certain k_h, increasing FS generates decreasing seismic displacements. It must be noted that the three charts shown in Figs. 6 and 7 must be considered simultaneously, and without taking into account the first chart (Figs. 6a, Fig. 7a), others are irrelevant. Figure 8 states that for a constant H₂/H₁, higher values of φ leads to better seismic stability.

Summary and Conclusions

In this paper, a logical methodology was presented to establish a relationship between the pseudo-static factor of safety (FS), horizontal seismic coefficient (k_h), and seismic displacement of gravity retaining walls. In order to determine seismic displacement of gravity retaining walls, a three-block failure mechanism was considered and the height of the soil in the front of the wall is taken into account. In order to determine the seismic displacement of the soil-wall system firstly, the yielding acceleration coefficient (k_y) was presented. To do so, a formulation based on the upper bound theorem of limit analysis was proposed. Then, using k_y along with the Newmark method the seismic displacement of gravity wall was calculated. In this study, some of strong motions data recorded in Iran are used to carry out the Newmark displacement analysis, characterized by PGAs varying between 410 and 1006 cm²/s/s. Based on the provided algorithm, a series of charts presented which show the relationship between k_h, the ratio of the soil in front of the wall (H₂) to wall height (H₁) and average seismic displacement resulted from the aforementioned records. The various applications of these charts were described: Using these types of curves, one may generate H₂/H₁ values for different values of k_h and FS. Moreover, one can decide on a proper k_h to use in the pseudo-static analyses or estimate displacement during an earthquake (with a particular k_h) for a gravity wall. The effect of various values of internal frictional angle of soil on yielding acceleration coefficient was also investigated.

Appendices

Appendix A

According to Fig. 9, wedge dimensions calculate as

$$m_{5} = \frac{{H_{1} }}{{\tan \lambda_{1} }}$$

(21)

$$m_{6} = \frac{{H_{1} }}{\tan \alpha }$$

(22)

$$m_{1} = (m_{5} + m_{6} )\cos \theta = H_{1} \left( {\frac{1}{{\tan \lambda_{1} }} + \frac{1}{\tan \alpha }} \right)\cos \theta$$

(23)

$$m_{7} = m_{1} \tan \theta = H_{1} \left( {\frac{1}{{\tan \lambda_{1} }} + \frac{1}{\tan \alpha }} \right)\sin \theta$$

(24)

$$m_{2} = \frac{{m_{7} }}{\tan (\alpha - \theta )} = \frac{{H_{1} \sin \theta }}{\tan (\alpha - \theta )}\left( {\frac{1}{{\tan \lambda_{1} }} + \frac{1}{\tan \alpha }} \right)$$

(25)

$$m_{3} = \frac{{H_{1} }}{\sin \alpha }$$

(26)

$$m_{8} = (m_{1} + m_{2} )\sin \theta = H_{1} \sin \theta \left( {\frac{1}{{\tan \lambda_{1} }} + \frac{1}{\tan \alpha }} \right)$$

(27)

$$m_{4} = \frac{{m_{8} }}{\sin \alpha } = H_{1} \left( {\frac{\sin \theta }{\sin \alpha }} \right)\left( {\frac{1}{{\tan \lambda_{1} }} + \frac{1}{\tan \alpha }} \right)$$

(28)

$$m_{9} = m_{5}$$

(29)

$$m_{10} = \frac{{H_{1} }}{{\tan \lambda_{2} }}$$

(30)

$$m_{11} = \frac{{H_{2} }}{\tan \beta }$$

(31)

$$m_{12} = \frac{{H_{2} }}{{\tan \lambda_{2} }}$$

(32)

Then, the weights of wedges are given as below.

$$W_{1} = \gamma_{1} (m_{5} + m_{6} )\left( {\frac{{H_{1} }}{2} + \frac{{m_{8} }}{2}} \right)$$

(33)

$$W_{2} = (m_{11} + m_{12} )\left( {\frac{{H_{2} \gamma_{2} }}{2}} \right)$$

(34)

$$W_{3} = (2a + m_{9} + m_{10} )\left( {\frac{{\gamma_{3} H_{1} }}{2}} \right)$$

(35)

Fig. 9

Parameters used in wedges’ weights calculation

Appendix B

The accelerograms of the earthquake’s records used in the paper are provided here. These records are all obtained from International Institute of Earthquake Engineering and Seismology (IIEES) [32] of Iran (Figs. 10, 11, 12, 13, 14, 15, 16).

Fig. 10

Abbar a longitudinal and b transversal accelerograms

Fig. 11

Avaj a longitudinal and b transversal accelerograms

Fig. 12

Bam a longitudinal and b transversal accelerograms

Fig. 13

Deihook a longitudinal and b transversal accelerograms

Fig. 14

Meymand a longitudinal and b transversal accelerograms

Fig. 15

Tabas a longitudinal and b transversal accelerograms

Fig. 16

Zanjiran a longitudinal and b transversal accelerograms