Analysis of the Dynamic Active Earth Pressure from c-φ Backfill Considering the Amplification Effect of Seismic Acceleration

Abstract

This study extends the method of pseudo-dynamic analysis based on the Mononobe-Okabe (M-O) method by comprehensively incorporating the seismic acceleration response characteristics of backfill soil and the cohesive properties of the fill. The proposed method is adapted for backfill soils by incorporating the cohesion c and internal friction angle φ (including scenarios with non-horizontal backfill surfaces). Theoretical formulas for the active earth pressure coefficient and its distribution on rigid retaining walls under the most unfavorable conditions are derived. The rationality of the proposed formulas is preliminarily verified using model test data from the relevant literature. A detailed parametric sensitivity analysis reveals the following trends: The active earth pressure coefficient K_a increases with increases in the amplification factor f_a, wall backface inclination angle θ, backfill slope inclination i, lateral vibration period T, and horizontal seismic acceleration coefficient k_h; K_a decreases with an increasing internal friction angle φ and cohesion/unit weight ratio c/γH. The failure wedge angle α_a increases with increases in φ, θ, and c/γH, decreases with increases in f_a, the soil–wall friction angle δ, i, T, k_h, and the vertical seismic acceleration coefficient k_v. Calculations are carried out to further identify the critical tensile stress depth in cohesive backfill soils using c and φ. The proposed analysis highlights the necessity of considering the seismic acceleration amplification factor f_a, backfill cohesion c, and soil–wall adhesion c_w in active earth pressure calculations. This study recommends that the seismic design of retaining walls should involve appropriate evaluation of the the actual cohesion of backfill materials and fully account for the acceleration amplification effects under seismic loading.

1. Introduction

Retaining walls represent one of the most commonly used support structures in geotechnical engineering, and are widely employed for slope stabilization in construction projects, hydraulic engineering, and railway and highway engineering. The 2005 Fukuoka Prefecture West Offshore Earthquake caused severe damage to residential areas on Genkai Island. A survey of 218 retaining walls revealed that 83% of them were damaged, with 61% losing their functionality (collapsed or exhibiting deformation/cracks) [1]. Following the May 12th Wenchuan Earthquake in 2008, investigations along the Dujiangyan-Yingxiu section of National Highway 213 showed that three types of seismic damage—deformation cracking, sliding, and overturning—accounted for approximately 78% of all retaining wall failures. These failure cases demonstrate that the current seismic design theory for retaining walls still exhibits substantial room for improvement. Proposing appropriate seismic earth pressure theories to study the earth pressure behind retaining walls is a critical component of the seismic design of such structures, as the intensity and distribution of the earth pressure are key factors in retaining wall design [2].

The analysis of forces that act on retaining walls under seismic action is a classical problem in geotechnical engineering. Following the 1923 Great Kanto Earthquake in Tokyo and Yokohama, Japan, Okabe [3] and Mononobe et al. [4] conducted in-depth investigations into the failure modes of retaining walls during earthquakes. Building upon Coulomb’s static earth pressure theory, they proposed the Mononobe-Okabe method (referred to as the M-O method), also known as the pseudo-static method, for calculating the seismic earth pressure behind retaining walls. As a pseudo-static approach, the M-O method simplifies seismic loading as a static inertial force while ignoring its cyclic characteristics. The M-O method only provides the resultant seismic earth pressure acting on the wall, assuming a linear pressure distribution with the resultant force acting at one-third of the wall height (H) from the base. Additionally, the M-O method assumes that the backfill soil behaves as a rigid body, implying an infinite shear wave velocity and shear modulus of the soil—an assumption that is inconsistent with reality. The simplified hypothesis of there being a uniform acceleration field within the backfill also neglects the seismic wave amplification effect that occurs in soil masses. Consequently, the M-O method cannot accurately reflect the true dynamic response of the wall–soil system under seismic loading.

To address the limitations of the M-O method, Steedman and Zeng [5] proposed a pseudo-dynamic method, assuming that the seismic acceleration varies sinusoidally with the depth and time, and treating the backfill soil as a non-rigid body with a finite shear modulus and shear wave velocity. By considering the wave periodicity and phase variations, they derived a formula for calculating the seismic active earth pressure on retaining walls under seismic loads. Zeng et al. [6] validated the rationality of pseudo-dynamic analysis through comparisons between centrifuge model tests and theoretical results. The principal merit of the pseudo-dynamic approach resides in the explicit consideration of two seismological parameters—shear wave velocity and predominant frequency—which jointly govern the phase variation and cyclical behavior of seismic wavefields.; thereby, this method reflects the dynamic characteristics of backfill soil and its interaction with seismic waves [7]. In recent years, the pseudo-dynamic method has seen extensive application and development. Ghosh [8] incorporated vibration amplification effects to calculate the seismic active earth pressure behind non-vertical cantilever retaining walls with zero backfill slope angles. Further, Ghosh [9] applied the pseudo-dynamic method to cases with non-horizontal backfill but neglected the amplification of seismic acceleration along the wall height. Choudhury et al. [10] considered P-waves, S-waves, and Rayleigh waves, using limit equilibrium methods to estimate the seismic active earth pressure on rigid retaining walls. Compared to the existing pseudo-static and pseudo-dynamic methods, the seismic influence zone when accounting for Rayleigh waves was approximately 22% and 17% larger, respectively. Qin et al. [11,12] proposed a kinematic analysis method based on discretization, dividing backfill soil into finite elements to account for temporal and spatial effects of seismic acceleration through the consideration of P- and S-waves. Lei et al. [13] developed a novel framework by combining kinematic limit analysis with pseudo-dynamic methods to predict the 3D seismic active earth pressure in cohesive backfill with cracks. They found that 3D effects and the soil cohesion significantly reduce the active pressure, whereas cracks and seismic effects counteract this. Pain et al. [14] employed a modified pseudo-dynamic method for the rotational stability analysis of gravity retaining walls. Under various seismic conditions, wall–soil interactions may exhibit in-phase or out-of-phase behavior during seismic excitation, depending on the backfill properties, wall material characteristics, and input motion frequency content. Fathipour et al. [15] integrated lower-bound limit analysis, finite element discretization, and second-order cone programming to evaluate active/passive seismic earth pressure coefficients for retaining structures under pseudo-dynamic loads. They demonstrated that as the soil-wall frictional resistance increases, the interface friction angle exhibits progressively more pronounced influence on the limiting earth pressure coefficients. Beyond retaining wall analysis, the pseudo-dynamic method has been widely applied in seismic stability assessments of foundations [16,17,18,19,20,21], slopes [22,23,24,25], and other geotechnical systems.

Although the pseudo-dynamic method has been widely applied, the existing research outcomes mentioned above exhibit the following limitations when applied to the most common engineering conditions: (1) The insufficient consideration of local site effects: when the backfill behind the retaining wall is non-horizontal, the amplification characteristics of the seismic acceleration along the wall height cannot be adequately addressed. (2) The underestimation of cohesive effects: The influence of the backfill cohesion, wall–soil adhesion, and friction angles is not fully accounted for. While engineering practice often recommends sand or gravel as backfill materials, most natural backfills possess cohesion due to the presence of fine-grained components (i.e., c-φ backfill). (3) Limited experimental validation and practical applicability: There is a lack of sufficient experimental data to support theoretical findings. To address these issues, this study will begin by analyzing the seismic response characteristics of backfill soil behind retaining walls under earthquake action. Based on the pseudo-dynamic method, we will develop a generalized seismic earth pressure calculation framework that comprehensively incorporates the amplification (attenuation) effects of acceleration along the backfill height and the cohesive properties of the backfill. The rationality of the proposed method will be validated through experimental data.

2. Seismic Active Earth Pressure Analysis

2.1. Acceleration Response Characteristics of the Backfill Behind the Retaining Wall

Singh et al. [26] revealed through numerical simulations that, under seismic action, the peak acceleration at the top of a gravity retaining wall and in the backfill soil exhibited varying degrees of amplification compared to the input seismic peak acceleration when the peak ground acceleration (PGA) reached approximately 0.55 g. Beyond 0.55 g, however, the amplification weakened, with possible causes for this attenuation at higher accelerations (PGA > 0.55 g) including soil vibration densification and the formation of failure surfaces, as illustrated in Figure 1a. Kloukinas et al. [27], via large-scale shaking table model tests, observed that the peak seismic acceleration at the backfill surface significantly exceeded that which was output at the shaking table base, demonstrating pronounced acceleration amplification effects near free-face structures, as shown in Figure 2. Similarly, Tiwari et al. [28,29,30], in their model tests, identified acceleration amplification during seismic wave propagation from the shaking table to the backfill. Notably, the acceleration amplification became markedly significant when the seismic vibration frequency approached the natural frequency of the soil. The observed patterns of acceleration amplification in backfill soil can be summarized as follows: (1) Threshold-dependent amplification: When the input PGA is below a threshold (0.55 g–0.6 g), the acceleration at the backfill surface exhibits clear amplification relative to the base acceleration. Beyond this threshold, the amplification diminishes and the surface acceleration attenuates. (2) Nonlinear vs. simplified distributions: The acceleration amplification coefficient along the backfill height typically follows a concave-shaped nonlinear distribution. For engineering applications, this is often simplified to a linear distribution, which may lead to conservative estimates of the seismic earth pressure. Conversely, this simplification provides a safe upper limit for seismic earth pressure calculations in retaining wall design.

2.2. Analysis of Active Earth Pressure Forces on a Retaining Wall Under Seismic Loading

Real seismic ground motions exhibit significant three-dimensional spatial characteristics, where P-waves, S-waves, and surface waves form complex wave field interferences during their propagation. Due to this complexity, the sinusoidal wave model, as the most fundamental mathematical representation of ground motion, maintains an irreplaceable position in earthquake engineering. This simple periodic function is widely applied in various fields, including seismic building design, earthquake early warning system development, and ground motion parameter studies. Within the theoretical framework of the Fourier transform method, any complex ground motion time history can be decomposed into the superposition of sinusoidal waves with different frequencies, amplitudes, and phases. This mathematical property establishes sinusoidal waves as the fundamental unit for the frequency-domain analysis of ground motions. As shown in Figure 2, the height of a rigid retaining wall is H, while the retaining wall slope angle is θ. The backfill cohesion is c and the internal friction angle is φ. The unit weight of the backfill is γ. The friction angle between the backfill and the retaining wall is δ. The angle between the failure surface of the backfill and the horizontal bottom plane is α. The backfill slope angle is i. Under sinusoidal shaking conditions, the amplitude of the horizontal and vertical seismic acceleration at the bottom is k_hg and k_vg, respectively. The following basic assumptions are proposed: (1) the sliding surface is planar, and the failure wedge passes through the heel of the retaining wall; (2) the seismic acceleration varies with the time and depth; (3) the backfill soil is homogeneous and isotropic, with a constant shear modulus along the wall height; (4) the retaining wall is assumed rigid, with its deformation negligible.; (5) the seismic acceleration amplification (attenuation) coefficient of the backfill soil follows a linear distribution along the elevation; (6) the seismic vibration of the retaining wall–soil system is characterized as sinusoidal steady-state vibration. The detailed derivation process of Equations (1) and (2) is provided in Appendix A, and the definitions of all symbols are listed in Appendix B.

The lateral active force acting on the retaining wall P_ae(t) can be defined as follows:

$$P_{\mathrm{ae}}(t)={\displaystyle \frac{1}{2}}\gamma H^2{K}_{\mathrm{ae}}(t)-K_{\mathrm{ac}}cH+{\displaystyle \frac{2K{c}^2}{\gamma }}$$

(1)

The active earth pressure coefficient K_a(t) can be defined as follows:

$$K_{\mathrm{a}}(t)={\displaystyle \frac{2P_{\mathrm{ae}}(t)}{\gamma H^2}}$$

(2)

From Equations (A1)–(A18) in Appendix A, it can be observed that the active earth pressure coefficient K_a(t) is a function of dimensionless expressions like α, t/T, H/λ, H/η, and c/γH.

At a certain time t in the shaking period, there is a corresponding sliding surface with an inclination angle α and an active earth pressure coefficient K_a(t). The peak active earth pressure coefficient coincides with the critical stress state of backfill soil under seismic loading, with the corresponding angle α defining the inclination of the critical failure plane.

In special cases, when c ≠ 0, φ ≠ 0, θ = 0, δ = 0, k_h = 0, k_v = 0, α = π/4 + φ/2, K_ae(t) reduces to tan² (π/4 – φ/2), K_ac reduces to 2tan (π/4 – φ/2), and K becomes 1, then Equation (1) can be simplified as follows:

$$P_{\mathrm{ae}}(t)={\displaystyle \frac{1}{2}}{\tan }^2({\displaystyle \frac{\pi }{4}}-{\displaystyle \frac{\phi }{2}})\gamma H^2-2\tan ({\displaystyle \frac{\pi }{4}}-{\displaystyle \frac{\phi }{2}})cH+{\displaystyle \frac{2c^2}{\gamma }}$$

(3)

It is interesting to find that Equation (3) is the same as the Rankine equation for cohesive backfills in the active earth pressure state. Furthermore, Equation (3) can be reduced to the common Rankine equation for cohesionless backfills when c = 0.

2.3. Computation of Active Earth Pressure Using the Proposed Calculation Formula

The dynamic earth pressure coefficients were calculated by the software MATLAB-R2015a, as shown in Figure 3a. The calculation results from MATLAB-R2015a were verified by the theoretical solutions under special conditions.

In the process of finding the maximum value of K_a(t), α ∈ (0, π/2 + θ), t/T ∈ (0, 1), each variable is discretized into 100 increments, as shown in Figure 3b. The corresponding active earth pressure coefficient K_a(t) is calculated for each point (α_i, t_j/T), and the maximum value K_a-max, along with its associated parameters (α_m, t_n/T), is identified. This K_a-max is defined as the active earth pressure coefficient, and the corresponding α_m is regarded as the rupture angle of the failure surface. The calculation of the earth pressure distribution along the wall height employs a discretization interval of 0.05 H (where H is the total wall height). In this paper, the values of H/λ and H/η were kept equal to 0.30 and 0.16, respectively, unless otherwise stated; these values are the same as those presented in [5,8,31,32].

3. Comparison and Validation of the Proposed Calculation Method

3.1. Validation 1

A comparison between the results obtained using the present study and those obtained with the commonly used M-O method with f_a = 1.0, φ = 30°, θ = 20°, i = 15°, k_v = 0, and c/γH = 0 is shown in Figure 4. The M-O method gives the linear distribution of the dynamic earth pressure as the failure wedge is assumed to be rigid. In the active condition shown in Figure 4a, the dynamic earth pressure predicted by the M-O method is more conservative than the result predicted by the present study. In the active condition, the difference between the two methods increases as k_h increases.

The normalized dynamic earth pressure distributions are shown in Figure 4b. The distribution curves obtained by the present study and the M-O method are tangent at approximately z/H = 2/3. The difference between the two distribution curves increases as the backfill–retaining wall friction angle δ increases.

3.2. Validation 2

A comparison between the present method and both the previously published Iskander method [33] and the Shukla method [34] of applying active earth pressure coefficients K_a for c-φ backfills is shown in Figure 5. It should be pointed out that the above three methods are different in their theoretical derivation. Specifically, in the Iskander method, the parameter δ increases nonlinearly with seismic acceleration. In this study, δ is set to φ for comparison purposes, while in the Iskander method, the average δ value approximates φ. In addition, the Iskander method converts the earth pressure into a horizontal component and then calculates the horizontal earth pressure coefficient. Therefore, it is difficult to directly compare the results obtained by the present study with those obtained by the Iskander method. To enable direct comparison, the horizontal earth pressure coefficient K_aH in the proposed method is calculated as K_a multiplied by β, where β = cos(δ + θ).

Looking at the overall trend, it can be seen that the results obtained by the present method are not much different from those obtained from the Iskander and Shukla methods, as shown in Figure 5, with φ = 30°, δ = φ, i = 0°, θ = 0°, f_a = 1.0, and c/γH = 0.05. When k_h is larger than 0.5, the K_aH obtained by the Iskander method is the largest, that obtained by the present method is the second largest, and that of the Shukla method is the smallest.

A comparison of the K_aH from the present study with that obtained using the other methods for different φ values with f_a = 1.0, δ = φ, θ = 20°, i = 15°, k_v = 0.5 k_h, and c/γH = 0.05 is shown in Figure 6. When k_h is small, the K_a obtained by the present method is larger than that of the Iskander method. With an increase in k_h, the K_a obtained by the present method gradually becomes smaller than that of Iskander method.

3.3. Validation 3

Nakajima et al. [35] conducted large-scale shaking table tests to investigate the seismic earth pressure on retaining walls. The input vibration was a 5 Hz sinusoidal wave, with the maximum acceleration amplitude being incrementally increased by approximately 50–100 Gal until significant displacement of the retaining wall occurred. The test configuration comprised a 0.60 m high retaining wall with a vertical interface between the wall and backfill soil. The backfill material was Inagi sand, which was placed horizontally behind the wall. Key computational parameters are listed in

Table 1. Model test conditions and calculation parameters.

	Backfill	Unit Weight of Backfill	Residual Strength
			φres	cres	c/γh
Case 3	Inagi sand, w = 12%	15.6 kN/m3	32.5°	2.2 kPa	0.24
Case 4	Inagi sand, w = 0%	13.6 kN/m3	39°	0 kPa	0

, where it is shown that Case 3 considered cohesive backfill soil and Case 4 adopted cohesionless backfill (cohesion = 0). An amplification factor fa = 1.1 was incorporated in the calculations. The pseudo-dynamic method assumes that the backfill soil reaches a limit state; thus, residual strength was adopted in the computations to better align with practical conditions.

A comparison between the computational results of the proposed method and experimental data is presented in Figure 7. The results are consistent with the experimental observations. However, when k_h < 0.7, the shaking table test results exceed the computational values. This discrepancy may arise because the experimental conditions did not fully attain the assumed active stress state, leading to an overestimated earth pressure. When the input acceleration exceeds 0.7 g, the backfill soil reaches the critical state of active earth pressure in the model tests, resulting in excellent agreement with the computational results.

4. Parametric Study

A parametric sensitivity analysis was conducted on the active earth pressure coefficient and earth pressure distribution. The calculation parameters are listed in detail in

Table 2. Calculation parameters for the parametric study.

ID	fa	δ/°	φ/°	θ/°	i/°	H/λ	kv/kh	c/γH
1	1.0, 1.4, 1.8	15	30	20	8	0.30	0	0, 0.05
2	1.0	0, 15, 30	30	20	15	0.30	0	0, 0.05
3	1.4	15	25, 30, 35	20	8	0.30	0	0, 0.05
4	1.4	15	30	−20, 0, 20	8	0.30	0	0, 0.05
5	1.4	15	30	20	0, 8, 15	0.30	0	0, 0.05
6	1.4	15	30	20	8	0.45, 0.30, 0.15	0	0, 0.05
7	1.4	15	30	20	8	0.30	0, 0.5, 1.0	0, 0.05

, with each working condition considering a series of horizontal acceleration coefficients kh ranging from small to large values.

4.1. Effect of Acceleration Amplification Factor f_a

Figure 8 shows the effect of the acceleration amplification factor f_a on the dynamic earth pressure coefficients K_a, the failure wedge angles α_a, and the normalized dynamic earth pressure distribution of the rigid retaining wall filled with cohesionless backfills (c/γH = 0) and cohesion backfills (c/γH = 0.05), with φ = 30°, δ = φ/2, θ = 20°, i = 8°, and k_v = 0.

It can be observed in Figure 8a that the dynamic active earth pressure coefficient K_a increased with the increase in the acceleration amplification factor f_a. The larger the horizontal acceleration coefficients k_h, the stronger the effect of the amplification factor f_a on K_a was. When the c/γH of a backfill increases from 0 to 0.05, K_a decreases significantly overall. For example, K_a decreases from 0.541 to 0.424 when k_h = 0, which is a decrease of 21.6%. The differences in the active earth pressure coefficient with different f_a values tend to increase as k_h increases, as shown in Figure 8b.

In Figure 8c, the failure wedge angle α_a in the active condition is decreased as f_a changes from 1.0 to 1.8. The failure wedge angles increase in active conditions when the c/γH of a backfill increases from 0 to 0.05. The differences in the failure wedge angle with different f_a values tend to increase as k_h increases.

Figure 8d shows the typical normalized dynamic active earth pressure distribution for different values of f_a with k_h = 0.2 and k_v = 0. The critical tension depth z_c is defined as the distance from the acting point of zero active earth pressure to the top of the wall. In the case where c/γH = 0, the critical tension depth is 0. The dynamic active earth pressure at the bottom of the wall increases from 0.739 to 0.899 as f_a increases from 1.0 to 1.8, corresponding to an approximate 21.7% increase. When c/γH increases from 0 to 0.05, the critical tension depth increases significantly. For c/γH = 0.05, the normalized critical depth z_c/H shows a slight increase with rising f_a, with an average value of 0.117.

4.2. Effect of Backfill-Wall Friction Angle δ

Figure 9a indicates that as the backfill–wall friction angle δ increases from 0° to 30°, K_a increases significantly when k_h > 0.1. With the increase in k_h, the difference in K_a induced by δ is increased, as shown in Figure 9b.

In Figure 9c, the failure wedge angle decreases as δ changes from 0 to φ = 30° in both active and passive conditions. The failure wedge angle α_a becomes larger when c/γH increases from 0 to 0.05, as shown in Figure 9c. The failure wedge angle with different δ values tends to be the same when k_h increases to a certain value.

Figure 9d shows the effect of δ on the normalized seismic earth pressure distribution in the cases where the c/γH values are 0 and 0.05, with f_a = 1.0, φ = 30°, θ = 20°, i = 15°, k_h = 0.2, and k_v = 0. When δ is 0 and c/γH is 0, φ/2 and φ, the values of the earth pressure distribution at the bottom of the wall, are 0.765, 0.813, and 0.967, respectively. As c/γH increases, the critical tension depth increases significantly in the active condition shown in Figure 9d. When δ = 0 and c/γH = 0.05, the critical depth increases with δ, yielding z_c values of 0.111, 0.127, and 0.145, respectively.

4.3. Effect of Internal Friction Angle φ

Figure 10 provides the influence of the internal friction angle φ on K_a, α_a, and normalized dynamic active earth pressure distributions with f_a = 1.4, δ = φ/2, θ = 20°, i = 8°, and k_v = 0.

It can be observed that the dynamic active earth pressure coefficient K_a decreased with the increase in the internal friction angle φ, as shown in Figure 10a. As shown in Figure 10b, it can also be observed that the difference in K_a under various internal friction angles increases with the increase in k_h. The existence of cohesion (c/γH = 0.05) reduces K_a.

In Figure 10c, the failure wedge angle α_a in the active condition is increased as φ increases from 25° to 40°. When c/γH changes from 0 to 0.05, α_a tends to increase. This trend becomes more and more obvious with further increases in k_h.

The typical normalized dynamic active earth pressure distribution for different values of φ with k_h = 0.2 and k_v = 0 is shown in Figure 10d. When φ increases from 25° to 40°, in the case c/γH = 0, the dynamic active earth pressure at the bottom of the wall decreases from 0.949 to 0.625, which represents a decrease by about 34.1%. When the internal friction angles φ are 25°, 30°, and 35°, respectively, in the case of c/γH = 0.05, the normalized critical depth z_c/H is almost constant, with an average of 0.117.

4.4. Effect of Retaining Wall Slope Angle θ

Figure 11 shows the K_a, α_a, and normalized dynamic active earth pressure distribution for variations in θ, c/γH, and k_h with f_a = 1.4, φ = 30°, δ = φ/2, i = 8°, and k_v = 0. In Figure 11a, K_a increases significantly as θ increases from −20° to 20°. The magnitude of the increase remains nearly constant with increasing k_h. When k_h = 0.2 and c/γH = 0, K_a increases from 0.401 to 0.857 as θ changes from −20° to 20°. Comparing the cases of c/γH = 0 and c/γH = 0.05 in Figure 11b, it is found that the existence of cohesion (c/γH = 0.05) reduces K_a significantly.

In Figure 11c, the failure wedge angle α_a in the active condition is increased as θ changes from −20° to 20° when k_h is less than 0.25. When k_h is larger than 0.25, the α_a values obtained with different θ values are almost the same. The failure wedge angle increases when c/γH changes from 0 to 0.05, and the difference amplitude rises as k_h increases.

Figure 11d shows the typical normalized dynamic active earth pressure distribution for different values of θ with k_h = 0.2 and k_v = 0. When θ increases from −20° to 20° in the case of c/γH = 0, the dynamic active earth pressure at the bottom of the wall changes from 0.373 to 0.810. When θ is −20°, 0° and 20°, in the case of c/γH = 0.05, the corresponding normalized critical depth z_c/H is 0.274, 0.174, and 0.116, respectively. It can be found that θ has a significant impact on the normalized critical depth.

4.5. Effect of Backfill Slope Angle i

In Figure 12a, K_a increases significantly as i increases from 0° to 15°. The magnitude of the increase with different i values is increased as k_h increases. When k_h = 0.2 and i = 0°, 8°, and 15°, the corresponding K_a increases sequentially, reaching values of 0.699, 0.857, and 1.142, respectively. As shown in Figure 12b, the existence of cohesion (c/γH = 0.05) reduces K_a.

In Figure 12c, α_a is decreased as i changes from 0° to 15°. When the normalized cohesion c/γH changes from 0 to 0.05, α_a tends to increase with same i. This trend becomes more and more obvious with further increases in k_h.

Figure 12d shows the typical normalized dynamic active earth pressure distribution for different values of i with k_h = 0.2 and k_v = 0. When i increases from 0° to 15°, in the case of c/γH = 0, the dynamic active earth pressure at the bottom of the wall increases from 0.670 to 1.039. When i is 0°, 8°, and 15°, in the case of c/γH = 0.05, the corresponding normalized critical depth z_c/H is 0.105, 0.116 and 0.140, respectively.

4.6. Effect of Period of Lateral Shaking T

Figure 13 shows the effect of a lateral shaking period T on the K_a, α_a, and normalized dynamic active earth pressure distributions with f_a = 1.4, φ = 30°, δ = φ/2, θ = 20°, i = 8°, and k_v = 0 in the cases of c/γH = 0 and c/γH = 0.05. The different H/λ (H/λ = 0.45, 0.30, and 0.15) values are shown in Figure 13 to represent the different T values, since T directly affects the value of H/λ. If the period corresponding to H/λ = 0.30 is t₀, then the periods corresponding to H/λ = 0.45 and 0.15 are 0.67 t₀ and 2.00 t₀, respectively. As V_p/V_s can be taken as 1.87 in most geological materials [31,32], the value of η/λ is also taken as 1.87 in this paper.

In Figure 13a, it can be seen that K_a increases significantly as H/λ decreases from 0.45 to 0.15, increases significantly as T increases from 0.67 t₀ to 2.00 t₀. For low-period lateral shaking, portions of backfill above the failure surface behind the rigid retaining wall may be moving in opposite directions. The long wavelengths associated with higher-period motion cause the backfill above the failure surface behind the rigid retaining wall to move essentially in phase. This may be the reason why the active earth pressure increases as T increases. The existence of cohesion (c/γH = 0.05) reduces K_a, as shown in Figure 13b.

In Figure 13c, the failure wedge angle is decreased as H/λ decreases from 0.45 to 0.15. As k_h increases, the difference in the failure wedge angle with different H/λ values tends to increase. When c/γH changes from 0 to 0.05 and T remains the same, α_a tends to increase.

Figure 13d shows the typical normalized dynamic active earth pressure distribution for different values of H/λ with k_h = 0.2 and k_v = 0. When H/λ is 0.45, 0.30, and 0.15 in the case of c/γH = 0.05, the corresponding normalized critical depth z_c/H is 0.147, 0.116, and 0.097, respectively. As can be seen from Figure 13d, the normalized earth pressure distribution obtained by the present study is nonlinear. When H/λ decreases from 0.45 to 0.15, the degree of nonlinearity of the distribution curves increases. It can be inferred that, as T → ∞ or H/λ → 0, the distribution curves will become straight lines. This is if the backfill is assumed to be rigid, which is the same assumption of the Mononobe-Okabe method.

4.7. Effect of Horizontal and Vertical Seismic Acceleration Coefficients k_h and k_v

It can be observed in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 that K_a increases as k_h increases. The failure wedge angle shows significant decreases with increases in k_h.

In Figure 14a,b, K_a decreases slightly for k_h < 0.2 and c/γH = 0 when k_v = 0, but increases significantly for k_h > 0.2.

In Figure 14c, the failure wedge angle is decreased as k_v decreases from 0 to k_h. The difference in the failure wedge angle with different k_v tends to increase as k_h increases. When c/γH increases from 0 to 0.05 with k_v held constant, α_a tends to increase, and this trend becomes more pronounced with higher k_h.

Figure 14d shows the typical normalized dynamic active earth pressure distribution for different values of k_v with k_h = 0.2. When k_v increases from 0 to 0.2 in the case c/γH = 0, the dynamic active earth pressure at the bottom of the wall decreases from 0.844 to 0.767, which represents a decrease of about 9.1%. For kv = 0, 0.25, and 0.5 with c/γH = 0.05, the normalized critical depth z_c/H is 0.116, 0.130, and 0.146, respectively.

4.8. Effect of Normalized Backfill Cohesion c/γH

As mentioned above, in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, K_a is decreased and α_a is increased due the presence of cohesion (c/γH = 0.05).

In c-φ backfill, the critical tension depth z_c should be clarified first in the active condition. In the generally static condition, the depth of the tension crack z_c can be computed as $z_{\mathrm{c}}={\displaystyle \frac{2c}{\gamma }}\tan ({\displaystyle \frac{\pi }{4}}+{\displaystyle \frac{\phi }{2}})$, based on Rankine’s analysis [36]. The normalized tension crack z_c/H is 0.173 in the static condition and larger than that in dynamic condition with φ = 30° and c/γH = 0.05. In the dynamic condition, the critical tension depth is affected by many factors based on the present study. Figure 15 shows that the normalized critical tension depth z_c/H varies with k_h, i, and θ when f_a = 1.0, φ = 30°, δ = 0°, k_v = 0, and c/γH = 0.05. In all cases, z_c/H first decreases, and then increases again as k_h increases. When θ remains constant, the normalized critical tension depth z_c/H increases as i increases. When i remains constant, the normalized critical depth z_c/H decreases as θ increases.

When i = θ = 0, the normalized critical depth z_c/H calculated based on the present study is 0.173, which is the same as that obtained using Rankine’s analysis. Generally, the z_c/H values calculated by the present study are less than those calculated by the Rankine’s formula, as show in Figure 15. Note that z_c/H is derived from Rankine’s formula as an approximate estimate in this dynamic analysis, ensuring the proposed method’s applicability to practical engineering scenarios. The actual value of z_c/H can be determined by setting Equation (3) as equal to zero and solving it. This equation is a little complex and can only obtain a numerical solution. The active earth pressure coefficient obtained by employing such an approximate value of z_c will be slightly conservative. According to Equation (3), the value of z_c has no effect on the distribution of the active earth pressure since the active earth pressure is determined by taking the derivative of the active force with respect to z, and z_c is independent of z. Therefore, the critical depth z_c can be determined by the obtained active earth pressure distribution curve first, then the actual dynamic active earth pressure coefficients can be determined by the numerical integration of the earth pressure distribution.

4.9. Summary

As summarized in

Table 3. Trend in seismic active earth pressure coefficients K_a and failure angle α_a with various parameters.

	fa	δ	φ	θ	i	T	kh	kv	c/γH
Ka	↑	↑	↓	↑	↑	↑	↑	*	↓
αa	↓	↓	↑	↑	↓	↓	↓	↓	↑

* The trend changes as the horizontal seismic acceleration coefficient k_h increases. ↑: Increasing trend, ↓: Decreasing trend.

and mentioned above, the impacts of many factors are considered in detail in this study. The active earth pressure coefficient K_a increases as f_a, θ, i, T, and k_h increase and decreases as φ and c/γH increase. The failure wedge angle α_a increases as φ, θ, and c/γH increase and decreases as f_a, δ, i, T, k_h, and k_v increase.

In current seismic design practices for retaining walls, the seismic amplification coefficient of backfill soils and the influence of soil cohesion are generally regarded as secondary factors and have been routinely overlooked. However, as demonstrated in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 and Table 3, the present parametric sensitivity analysis reveals that both the seismic amplification effect of backfill materials and their cohesive properties constitute non-negligible influencing parameters. Although these two factors exhibit contrasting effects on the increase in the active earth pressure coefficient—one acting as a positive contributor and the other as a negative suppressor, potentially creating mutual offsetting effects on the seismic-induced increase in the active earth pressure coefficient—it is imperative that reliable assessments be grounded in comprehensive computational methodologies that properly account for the combined influence of these critical parameters.

5. Conclusions

This paper derives a calculation method for determining seismic active earth pressure based on the pseudo-dynamic method. Considering the most general engineering conditions (including the cohesive backfill, acceleration amplification (attenuation) effects behind the wall, non-horizontal backfill surface, and non-vertical retaining wall backface), the proposed method enhances the practical relevance and applicability of seismic active earth pressure calculations, demonstrating significant engineering value. Subsequently, the rationality and reliability of the derived seismic active earth pressure calculation formula were validated through model test data from the relevant literature.

The most unfavorable working conditions are calculated by the compiled program, and the influence of various factors as f_a, δ, φ, θ, i, T, k_h, k_v, and c/γH on the earth pressure coefficient, failure wedge angle, earth pressure distribution, and critical tension depth are considered in detail. Under special conditions, the earth pressure calculation formula of this paper can be degraded to the same formula as the Rankine theory. The critical depth z_c behind the rigid retaining wall needs to be determined before calculating the seismic active earth pressure due to the existence of cohesion in c-φ backfill. The critical depth decreases with an increase of θ, and increases with an increase in i and c.

The acceleration amplification effect of the backfill soil behind the retaining wall significantly influences the magnitude of the seismic active earth pressure. When the amplification factor f_a increases from 1.0 to 1.8 (under conditions of c/γH = 0, φ = 30°, δ = φ/2, θ = 20°, i = 8°, and k_v = 0), the seismic active earth pressure coefficient rises from 0.9745 to 1.5852, marking a 63% increase. Conversely, the presence of cohesion notably reduces the seismic active earth pressure. For instance, when c/γH = 0.05 (with φ = 30°, δ = φ/2, θ = 20°, i = 8°, and k_v = 0), the seismic active earth pressure coefficient drops to 1.1615, representing a 27% reduction compared to that of cohesionless backfill (1.5852). Current seismic design practices for retaining walls inadequately account for these two factors. This study recommends that the seismic design of retaining walls should appropriately evaluate both the acceleration amplification effect under seismic loading and the actual cohesion of backfill materials.

The earth pressure calculation method proposed in this study can be extended to scenarios that involve layered backfill behind retaining walls. However, the fundamental assumptions of the model are incompatible with flexible retaining wall configurations. The following aspects require further improvement: (1) The formula derivation assumes a straight-line sliding surface for the soil wedge behind the wall, simplifying the inclination angle α. Actual engineering scenarios may involve curved sliding surfaces, which could affect the accuracy of the seismic active earth pressure results. (2) The amplification factor of the seismic acceleration of the backfill behind the retaining wall may have a coupling relationship with the magnitude of the seismic wave acceleration input at the base. This effect is not considered in this paper. (3) The simplification of seismic loading into equivalent sinusoidal waves based on the predominant frequency requires further in-depth research on the equivalence of dynamic responses in retaining walls under these two excitation patterns.