Pseudo-Static Design and Analysis of Seismic Earth Pressure for Cantilever Retaining Walls with Limitation Assessment

Abstract

By critically reviewing pseudo-static methods, it is demonstrated that approximating the earth pressure on a short heel’s vertical face (V-plane) using the Rankine solution for long-heel walls induces a negligible error. A finite element analysis is deployed to validate the pseudo-static results, with dynamic simulations incorporating 1–5 Hz sinusoidal seismic excitations to probe the resonance effects. The key results show that disregarding the impact of layered backfill placement on the initial stress states leads to non-conservative estimates of active earth pressure. Furthermore, the point of application of earth pressure rises significantly during strong shaking, and although the transient safety factors against sliding and overturning may fall below 1.0 during seismic events, the residual deformation analysis suggests that this does not necessarily lead to collapse. A significant amplification of bending moments and greater reductions in post-earthquake safety factors occur when the input frequency approaches the natural frequency of a wall. Finally, the paper proposes resonance prevention strategies for the seismic design of cantilever retaining walls, a methodology incorporating construction effects into the initial stress field modeling, and recommendations for selecting effective safety factors.

1. Introduction

Retaining walls are structures that support backfill soil or natural slopes, preventing deformation and instability. They are widely used in engineering practice [1,2,3,4,5]. The selection of wall type depends on a project’s requirements, soil conditions, material availability, and construction techniques. Cantilever retaining walls, typically constructed of reinforced concrete, feature an inverted T-shape comprising a vertical stem, toe slab, and heel slab. The backfill weight on the heel slab enhances stability, while the toe slab extends the anti-overturning lever arm. Their stability primarily relies on the soil mass above the heel, with the reinforcement designed to resist the bending and shear forces. This enables lightweight construction with smaller cross-sections, making such walls suitable for heights of 5–10 m, poor foundation soils, or scarce local rock resources. Their simple geometry facilitates their cost-effective fabrication and rapid construction. In urban railways, cantilever walls are often employed as shoulder-retaining structures to minimize land use.

As lightweight supporting structures, cantilever walls exhibit favorable seismic performance [6,7]. However, their dynamic response involves complex interactions between the seismic input, backfill behavior, and structural vibration, directly impacting the engineering safety. The accurate determination of dynamic earth pressure (seismic earth pressure)—the critical load for seismic design—remains a key challenge [8,9,10].

The pseudo-static method is the most widely adopted approach for seismic earth pressure calculation. The Mononobe–Okabe (M-O) theory [11,12], an extension of Coulomb’s earth pressure theory, incorporates horizontal and vertical seismic coefficients (k_h and k_v) as the body forces acting on a failure wedge. Recognized for its simplicity and clear mechanics, the M-O theory has been codified in numerous seismic design standards. Nevertheless, this method has inherent limitations: It disregards the temporal characteristics (time history, phase variations) and spectral properties of ground motions. The assumption of uniform seismic coefficient distribution along a wall height deviates from actual dynamic responses [13], particularly for tall walls or complex topographies. It fails to capture real-time evolution of earth pressure during earthquakes or predict the displacement responses [14,15]. These shortcomings have motivated the development of more refined analytical methods.

The seismic earth pressure mechanism involves complex multiphysics coupling, including soil–structure interaction, wave propagation, and soil nonlinearity. The pressure on cantilever walls is influenced by the seismic characteristics (amplitude, spectrum, and duration), soil properties, structural parameters, and boundary conditions [16,17,18,19,20]. Near-fault pulse-type motions with large amplitudes and long periods can cause severe damage [21]. Aghamolaei et al. [22] demonstrated via FEM that lateral displacements under near-field versus far-field earthquakes may differ by up to 85%. Seo et al. [23] developed probabilistic models linking wall displacement and backfill settlement to limit states, identifying backfill slope and motion characteristics as the major influencers.

The structural parameters—such as wall inclination, roughness, and heel/toe dimensions—significantly affect the seismic response. Wagner et al. [24] found that the Mononobe–Okabe (M-O) theory offers reasonable upper-bound estimates for rigid walls but overpredicts the loads for flexible ones. Chen et al. [25] derived pseudo-static solutions for narrow-backfill T-walls under translational mode using limit equilibrium and Rankine theory. Cakir et al. [26,27,28] employed 3D finite element models to analyze the backfill–structure–foundation interaction, highlighting the effects of wall flexibility, soil parameters, and ground motion. The seismic frequency content is critical, especially under soil-softening conditions. Jo et al. [29] compared the M-O and Seed–Whitman methods with centrifuge tests, noting that both overestimate the seismic pressure.

The failure surface plays a key role in the active earth pressure computation. Depending on the heel length and its interaction with the failure surface, two calculation systems exist [30]. For long-heel walls, Kloukinas et al. [31,32,33,34] applied a generalized Rankine theory. For short-heel walls, Greco [35,36,37] proposed a limit equilibrium model accounting for a tertiary slip surface that truncates the thrust wedge, making the Rankine theory inapplicable. Addressing the lack of a unified method, Kamiloğlu et al. [38] partitioned the backfill into three mechanical zones and derived a generalized seismic active earth pressure framework applicable to any heel length using optimization techniques.

Integrating the current research and code practices, the deficiencies in seismic design calculations for cantilever walls manifest primarily in two aspects. (i) Limitations in failure mode analysis: Traditional methods (e.g., M-O) consider only a single failure mode, whereas actual earthquakes may induce multiple instabilities (e.g., secondary failure of surfaces in the backfill, wall–slope rotation/sliding). (ii) Inadequate soil–structure interaction modeling: Designs rely on rigid-foundation assumptions without fully accounting for elastic base–foundation interactions, causing internal force miscalculations. The seismic interaction between the retaining walls and the soil significantly influences the development of seismic damage. Under worst-case conditions, this interaction may induce resonance phenomena [39,40,41]. Currently, there is still insufficient in-depth research regarding how to incorporate resonance effects into the seismic design of retaining walls.

Therefore, to bridge this critical research gap, this study makes three fundamental advancements beyond conventional pseudo-static approaches: First, a unified seismic earth pressure formula applicable to both long- and short-heel cantilever retaining walls is developed, addressing the longstanding limitations of existing methods, which lack consistency across different structural configurations. Second, through numerical simulations that account for construction-induced initial stress fields, the proposed simplified method is rigorously validated, thereby enhancing its practical reliability—an aspect often overlooked in traditional analyses. Most importantly, by conducting finite element dynamic response analyses, an investigation is carried out into the effects of input seismic wave frequency, particularly resonance phenomena, on the seismic stability. This represents a significant departure from established design practices, as the current guidelines provide no provisions for resonance-induced amplification effects. The findings are expected to establish a framework integrating the frequency-dependent response into seismic design recommendations, offering transformative insights for cantilever wall design.

2. Seismic Earth Pressure Calculation for Long-/Short-Heel Walls

2.1. Formula Based on Rankine Theory for Long-Heel Walls

When the heel slab is sufficiently long, the failure surface does not interact with the vertical stem, allowing for full development of the Rankine limit state in the backfill. According to the lower-bound theorem of limit analysis, Kloukinas et al. [3,31,32,33] and Evangelista et al. [34] derived the dynamic active earth pressure on an idealized vertical plane (AD) passing through the heel (Figure 1):

$$P_{\mathrm{AE}}={\displaystyle \frac{1}{2}}\gamma K_{\mathrm{AE}}{h^{\prime }}^2$$

(1)

where γ is unit weight of backfill, h′ is height of vertical face at heel (h′ = h + b tan δ), h is wall height, b is heel length (heel ratio χ = b/h), and β is backfill slope angle (Figure 1). K_ae is the seismic active earth pressure coefficient, given by the following:

$$K_{\mathrm{ae}}={\displaystyle \frac{\cos \beta \cos (\beta +\theta )}{\cos {\delta }_{\mathrm{s}}\cos \theta }}\left[{\displaystyle \frac{1-\sin \phi \cos ({\Delta }_{1\mathrm{e}}-\beta +\theta )}{1+\sin \phi \cos ({\Delta }_{1\mathrm{e}}+\beta +\theta )}}\right]$$

(2)

with

$$\theta =\mathrm{atan}({\displaystyle \frac{k_{\mathrm{h}}}{1-k_{\mathrm{v}}}})$$

(3)

where θ is the seismic inclination angle, and k_h and k_v are the horizontal and vertical seismic acceleration coefficients, respectively.

$${\Delta }_{1\mathrm{e}}=\mathrm{asin}\left[{\displaystyle \frac{\sin (\beta +\theta )}{\sin \phi }}\right]$$

(4)

$$\delta (0)=\mathrm{atan}\left[{\displaystyle \frac{\sin \phi \sin ({\Delta }_{1\mathrm{e}}-\beta +\theta )}{1-\sin \phi \cos ({\Delta }_{1\mathrm{e}}-\beta +\theta )}}\right]$$

(5)

ω_α and ω_β represent the inclination angles of α-characteristic and β-characteristic, respectively:

$${\omega }_{\alpha }={\displaystyle \frac{\pi }{2}}-\phi -{\omega }_{\beta }, {\omega }_{\beta }={\displaystyle \frac{\pi }{4}}-{\displaystyle \frac{\phi }{2}}-{\displaystyle \frac{{\Delta }_{1\mathrm{e}}-\beta }{2}}-{\displaystyle \frac{\theta }{2}}$$

(6)

Applicability conditions:

$${\omega }_{\beta }\le \mathrm{atan}(b/h)=\mathrm{atan}\chi$$

(7)

Figure 1. Pseudo-static analysis of soil–wall system and characteristics of Rankine wedge. (a) Pseudo-static stress tensor in Mohr circle; (b) characteristics of Rankine wedge [34].

Figure 1. Pseudo-static analysis of soil–wall system and characteristics of Rankine wedge. (a) Pseudo-static stress tensor in Mohr circle; (b) characteristics of Rankine wedge [34].

2.2. Distinguishing Long-Heel and Short-Heel Walls

Based on Equations (6) and (7), the critical heel length for distinguishing between long- and short-heel cantilever retaining walls can be derived. For example, consider a retaining wall with the following parameters: friction angle φ = 34°, wall–soil friction angle δ = 24°, unit weight γ = 18 kN/m³, and wall height h = 6.0 m. Figure 2 shows the critical heel length for distinguishing between long- and short-heel walls under different horizontal seismic acceleration coefficients. The corresponding backfill slope angles β are 0° and 15°; the vertical seismic acceleration coefficients are k_v = 0 and k_v = 0.5 k_h. From Figure 2, it is observed that as the horizontal seismic acceleration coefficient increases, the critical heel length decreases approximately linearly. A larger backfill slope angle β results in a smaller critical heel length. Comparing the cases with k_v = 0 and k_v = 0.5 k_h, k_v reduces the critical heel length, and this reduction increases with a higher k_h. It is evident that when calculating the seismic earth pressure for cantilever walls, determining whether the base slab is short- or long-heeled depends not only on the geometric dimensions but is also closely related to the seismic acceleration coefficients.

Using the finite element software PLAXIS 2D 2024 [42], a pseudo-static analysis was performed on cantilever retaining walls of different geometric dimensions. The calculation parameters are detailed in

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

Parameter	Backfill	Foundation Soil	Retaining Wall
Model	HS small	Hardening Soil	Viscoelastic
Unit weight ${\gamma }_s$ (kN/m3)	18.0	19.0	24.0
Secant modulus $E_{50}^{ref}$ (kPa)	3.60 × 104	4.68 × 104
Unloading/reloading stiffness $E_{ur}^{ref}$ (kPa)	1.08 × 105	1.404 × 105
Power for stress-level dependency of stiffness m	0.5	0.5
Cohesion $c_{ref}^{\prime }$ (kPa)	0	0
Friction angle $\varphi \prime$ (°)	34.0	40.0
Dilatancy angle $\psi$ (°)	0	0
Reference shear strain ${\gamma }_{0.7}$	0.0002
Shear modulus at very small strains $G_0^{ref}$ (kPa)	1.13 × 105
Poisson’s ratio ${\nu }_{ur}^{\prime }$	0.20	0.20	0.334
Failure ratio Rf	0.9	0.9
Young’s modulus E (kPa)			6.8 × 107
Interface reduction factor Rinter	0.65	0.65
Damping ratio	3.00%	3.00%	3.00%

. The numerical simulation procedure is detailed in Appendix A. Three types of cantilever walls were primarily selected (Type A, Type B, and Type C), all with a stem height of 5.5 m. The base slab lengths were 3.5 m (2.5 + 0.5 + 0.5 m), 5.25 m (4.0 + 0.5 + 0.75 m), and 7.0 m (5.5 + 0.5 + 1.0 m), corresponding to the heel-length-to-wall-height ratios (χ) of 0.45, 0.73, and 1.00, as shown in Figure 3.

Figure 4 shows shear strain contours of backfill failure surfaces for the three wall types under different accelerations, calculated via a pseudo-static analysis. From Figure 4, under static conditions (0 g horizontal acceleration), Type A exhibits three failure surfaces: Primary (α) and secondary (β) slip surfaces forming a V-shaped shear band through the heel, symmetric about the vertical line passing through the heel. A tertiary slip surface is formed by reflection where the β-surface intersects the cantilever tip, classified as a short-heel wall [30]. For Type B (increased base length), the β-slip surface passes exactly through the cantilever tip. Type C (further increased base length) represents a typical long-base cantilever wall [31,32]. Under a low horizontal seismic acceleration, Type A develops three slip surfaces and can be classified as a short-heel wall. When the horizontal acceleration is increased to 0.25 g, only two failure surfaces develop in the backfill at the limit state, and it can then be classified as a long-heel wall. This further proves that classifying a cantilever wall as short- or long-heeled depends not only on the geometric dimensions, but is also closely related to the seismic acceleration coefficients.

2.3. Calculation of Earth Pressure for Short-Heel Walls

Figure 5 shows a schematic diagram of the limiting equilibrium force analysis for short-heel cantilever walls based on the principle of limit equilibrium [31]. Figure 5a depicts the distribution of failure surfaces. The assumption of failure surfaces here slightly differs from that of Kamiloğlu et al. [30], which restricts the length of BE in sliding block BEF to be ≤BD, while this study imposes no such restriction.

As shown in Figure 5b, for sliding block BEF (analogous to Colomb’s earth pressure method), the interface BF has a wall–soil friction angle δ, generating thrust S₁. The failure surface EF in the backfill has a friction angle φ at the limit state, an inclination η to the horizontal, and an inter-surface force R₁. Based on the pseudo-static approach, block BEF is subjected to a body force W₁′, acting at angle ψ to the vertical. W₁′ is derived from the backfill unit weight, area of △BEF, and seismic coefficients. The force equilibrium perpendicular to R₁ yields S₁ as a function of W₁’.

As shown in Figure 5c, for sliding block ACBF, the thrust between the left wall AF and the soil is S₂, with a friction angle φ on the sliding surface; the force acting on the right sliding surface AC within the fill is R₂, with an inter-sliding surface friction angle φ under the limit state. The sliding surface AF is inclined at angle β to the horizontal, while the sliding surface AC is inclined at angle α to the horizontal. Blocks ADFB and ACD are subjected to body forces W₀′ and W₂′, respectively. With S₁ known, the force equilibrium perpendicular to R₂ gives the expression for S₂. S₁ and S₂ are functions of α, β, η, and ψ. Using block ADBF in Figure 5d, Greco [37] derived the thrust S_a on the vertical plane AD at the heel via force equilibrium.

Optimization methods were then used to find values of α, β, η, and ψ that minimized the sliding safety factor. Minimizing the overturning safety factor (operationally targeting minimal eccentricity [43]) yielded identical α, β, η, and ψ values as Greco’s sliding-minimization approach. Note: If the solved y ≥ h (Figure 5a), the short-heel condition is invalid (S₁ = 0), the failure surfaces reduce to two, and the wall should be treated as long-heeled.

As shown in Figure 6a, when the heel of the cantilever retaining wall is almost nonexistent, its two slip lines nearly coincide. When b = 0 (χ = b/h = 0), the V-plane coincides with the wall surface, and the earth pressure on the V-plane can be entirely calculated using the Coulomb earth pressure coefficient K_a^C. As illustrated in Figure 6d, when the cantilever wall is determined to be a long-heel cantilever wall, the earth pressure on the V-plane can be fully calculated using the Rankine earth pressure coefficient K_a^R. When the heel of the cantilever wall gradually shortens, as shown in Figure 6c, although it is classified as a short-heel cantilever wall, since the V-plane lies entirely within the Rankine slip zone, the earth pressure on the V-plane can still be entirely calculated using K_a^R. When the heel shortens further, as in Figure 6b, the earth pressure coefficient on the V-plane becomes a combination of K_a^C and K_a^R. In summary, as the heel length of the cantilever wall increases (b increases and χ = b/h consequently increases), the earth pressure coefficient on the V-plane gradually transitions from the Coulomb coefficient K_a^C to the Rankine coefficient K_a^R.

Considering different backfill friction angles φ and slope angles β, the calculation zones for the earth pressure coefficient and the corresponding parameter values are shown in Figure 7. In Figure 7a, line AB represents the boundary between long-heel and short-heel conditions, determined using Equations (3), (4), (6) and (7). The intercept of AB with the χ-axis, OB, corresponds to χ_LS. Line AC in Figure 8a represents the 100% Rankine earth pressure line, corresponding to Figure 7c. It can be derived that

$$OC={\displaystyle \frac{{\chi }_{LS}}{2-{\chi }_{LS}\tan \beta }}$$

(8)

Thus, in the region above line AC (yellow zone) in Figure 7a, calculating the earth pressure on the V-plane using Rankine theory is valid. Further, if line AD represents the partial Rankine earth pressure line, and if the portion of the V-plane applicable to Rankine theory constitutes the vast majority, calculating the entire V-plane using Rankine theory would introduce negligible error. This implies that the region suitable for Rankine-based calculation can be reasonably extended to the area above line AD. Let η be the proportion of the V-plane within the Rankine zone. It can be derived that

$$OD={\displaystyle \frac{{\chi }_{LS}}{2+2(\frac{1-\eta }{\eta })-{\chi }_{LS}\tan \beta }}$$

(9)

For example, when η = 80%, substituting it into the above equation yields the following:

$$OD={\displaystyle \frac{{\chi }_{LS}}{2.5-{\chi }_{LS}\tan \beta }}$$

(10)

The critical k_h values (corresponding to the length of OA) for different φ and β are shown in Figure 7b. The values of χ at k_h = 0 (where χ = χ_LS corresponds to OB, η = 100% corresponds to OC, and η = 80% corresponds to OD) for different φ and β are shown in Figure 7d. The variation in χ with k_h (corresponding to line AB) for different φ and β is also provided in Figure 7d. Therefore, whether for long-heel or short-heel cantilever walls, if the wall falls within the region above line AD in Figure 7a, the earth pressure on the V-plane can be calculated using Rankine theory with sufficient accuracy. Figure 7b–d consider numerous working conditions, and in practice, the values of OA and OD can be determined via interpolation from the data in these figures.

To further validate the above conclusions, the earth pressure coefficients on the V-plane under various working conditions were calculated and compared. In Figure 8, as b increases from 0.1 m to 2.0 m, K_a gradually shifts from K_a^C toward K_a^R. In Figure 8a,b, where δ < φ, K_a^R consistently serves as the upper limit for K_a. Using K_a^R for the calculations is conservative from a safety perspective. In Figure 8c, since δ = φ, K_a^C and K_a^R are essentially equal when the horizontal seismic acceleration coefficient k_h < 0.5, making K_a = K_a^R appropriate. When k_h ≥ 0.5, the configuration corresponds to a long-heel cantilever wall, and K_a = K_a^R applies. Figure 8d shows the variation in the earth pressure coefficient on the V-plane with the horizontal acceleration coefficient for different β values (at b = 1.0 m), which also conforms to the above pattern. Additionally, in practical engineering, the heel length of a cantilever retaining wall should not be too short, as this would undermine the wall’s reliance on the overlying soil to enhance its sliding stability. General structural requirements specify that χ (b/h) = 1/4–1/2. Therefore, using Rankine theory for the regions above line AD in Figure 7a ensures sufficient accuracy.

It should be noted that Rankine’s theory is generally not applicable to multi-layered backfill soils, as it assumes soil homogeneity, whereas layered soils exhibit the following differences: variations in c, φ, and γ values across the strata, and weak interlayers that induce composite failure surfaces. For cohesive soils, tension-induced vertical cracks develop, resulting in zero earth pressure within the crack zone; thus, effective earth pressure distribution requires a separate calculation. Application of Rankine’s theory to such conditions necessitates modifications.

3. Influence of Backfilling Process on Earth Pressure on Cantilever Retaining Walls

Conventional pseudo-static methods for calculating the earth pressure on retaining walls typically rely on simplified stress field distributions, such as the initial K₀ stress state. However, in practical engineering, (i) the friction at the wall–soil interface induces soil-arching effects, causing deflection of initial stress trajectories [44,45]; and (ii) during backfill compaction, the wall undergoes minor deformations under progressively increasing fill loads and compaction-induced loads [46,47,48], resulting in deviations between the actual initial stress fields and the theoretical assumptions. Limited research exists on how such deviations affect the earth pressure in cantilever retaining walls. Therefore, we conducted pseudo-static calculations and dynamic numerical simulations, comparing the results with Equations (2) and (5) to validate the rationality and reliability of the cantilever wall earth pressure calculation method proposed by Kloukinas et al. [33].

3.1. Comparative Analysis of Considering Initial Stress Effects from Layered Backfilling Versus Ignoring Initial Stress Effects in Pseudo-Static Methods

The calculation model and parameters are detailed in Table 1. The construction simulation process and its corresponding contour plot of horizontal displacement are detailed in Figure 9. As the backfill is placed in layers, the retaining wall undergoes displacement and rotation. Figure 9 indicates that during the initial backfilling stages (Stage 1 and Stage 2), the wall deflects toward the left. When the backfill height exceeds half the wall height (Stage 3 and Stage 4), the upper section begins to deflect away from the backfill direction, rotating toward the right. This deformation alters the initial stress field used for the subsequent earth pressure calculations. As shown in Figure 10, the influence of layered backfilling generates two preliminary shear stress bands near the wall heel, deviating significantly from the ideal K0 stress state. This phenomenon may cause discrepancies in the seismic earth pressure calculations for cantilever retaining walls.

As shown in Figure 11a, the earth pressure coefficients on the computational model’s surface obtained by the pseudo-static method exhibit good overall agreement with those derived from Equation (2), though the pseudo-static values are slightly higher. While the earth pressure coefficients on the V-plane differ minimally from those on the stem face of the cantilever retaining wall, critical discrepancies emerge in the seismic performance: Equation (2) suggests that the wall can withstand seismic loads exceeding 0.65 g; however, when considering the influence of initial stress fields and wall displacements, its seismic capacity is reduced to approximately 0.45 g.

Figure 11b compares the friction angles obtained from the pseudo-static method with those calculated using Equation (5). Under static conditions, the friction angle on the V-plane calculated by Equation (5) is 0°, while the friction angles obtained from the pseudo-static method are 4.3° (Type C), 7.1° (Type B), and 8.8° (Type A), respectively. In general, the friction angles on the surfaces first increase and then decrease, with minor differences in the peak values. The friction angles on the vertical faces of the three types of cantilever retaining walls obtained by the pseudo-static method are essentially the same, gradually decreasing as the acceleration coefficient k_h increases.

Figure 11c shows the relationship between the height of the earth pressure resultant point obtained from the pseudo-static method and the horizontal seismic acceleration coefficient. It can be observed that the resultant point on the V-plane is generally higher than that on the vertical face of the cantilever retaining wall. The resultant point on the V-plane is located approximately at 0.33 h′, while the resultant point on the vertical face of the cantilever retaining wall first decreases and then slightly increases as the horizontal acceleration coefficient k_h increases, remaining within the range of 0.28–0.30 h′. In terms of the influence of cantilever retaining wall types, the order of the heights of the earth pressure resultant points is as follows: Type C > Type B > Type A.

Figure 11d presents the relationship curves between the failure surface angles ω_α, ω_β, and ω_α + ω_β obtained from the pseudo-static method and the horizontal seismic acceleration coefficient. The figure shows that the initial stress state also significantly affects the angles of the failure surface, with the values of ω_α + ω_β being greater than those calculated by Equation (6).

Since the height h of the cantilever retaining walls in Figure 3 is identical and the mechanical parameters of the backfill material are the same, the earth pressure coefficients and friction angles on the V-plane for Type A, Type B, and Type C calculated using the seismic earth pressure formula (2) proposed by Kloukinas et al. [33] should be identical. Comparing these with the pseudo-static method results that consider the initial stress state during backfill construction, it is evident that the active seismic earth pressure coefficients calculated using Equation (2) represent the lower bound of the values obtained using the pseudo-static method. This indicates that if the initial stress state is not considered, the active earth pressure coefficients calculated using Equation (2) may be unsafe.

3.2. Comparative Analysis of Dynamic Finite Element Simulation Considering Initial Stress Effects from Layered Backfilling Versus Pseudo-Static Method

The numerical simulation procedure is detailed in Appendix A, with the model’s validation and verification provided in Appendix B.

Figure 12 presents time–history curves of earth pressure coefficient, friction angle, and resultant force elevation on the computational surface from the dynamic simulation. The input seismic acceleration is defined as follows:

$$g=2(1-\cos {\displaystyle \frac{\pi t}{3}})\sin (6\pi t) (\mathrm{m}/{\mathrm{s}}^2)$$

(11)

A 6-second-duration sinusoidal wave at 3 HZ frequency was the input. The selection of sinusoidal waves in this study represents a simplification of actual seismic motions. The primary rationale was that sine waves constitute the fundamental building blocks of complex ground motions via Fourier transform, serving as the most direct and effective tool for revealing the core dynamic characteristics (e.g., natural frequencies and resonance phenomena). The 3 Hz frequency was specifically chosen because it approximates the fundamental natural frequency of the cantilever retaining wall–backfill system analyzed in this study (see Section 4 for details); this frequency falls within the predominant frequency range of actual ground motions recorded during seismic events. Equation (11) indicates a peak acceleration of 4 m/s² for the seismic wave. The horizontal seismic coefficient kₕ = 0.4 was derived from the peak acceleration. According to the calculation method proposed by Kloukinas et al. [33], the earth pressure coefficient Kₐₑ for the V-plane is 0.72, with a friction angle δ₀ = 33.7°.

The analysis of Figure 12a reveals that as the seismic acceleration reaches its peak, the earth pressure coefficient on the V-plane increases from its static values (0.30–0.35) to approximately 1.0—significantly exceeding the 0.72 value calculated from Equation (2). Post-earthquake, the coefficient decreases below 0.72, but remains higher than the initial static values. Figure 12b demonstrates that during seismic excitation, the V-plane friction angle for all three cantilever wall types fluctuates between 33.7° (maximum) and approximately −30° (minimum), indicating directional reversal. After shaking cessation, it stabilizes at 6.8–7.8°. The vertical face friction angle exhibits smaller variations (>0° throughout) with a decreasing trend, confirming consistently downward friction forces from the backfill to the wall. Figure 12c shows the time–history curve of the resultant force elevation on the computational surface. From the initial stage to the end of the earthquake, the resultant force elevation (z/h′) on the computational plane exhibits an overall increasing trend. For example, at the V-plane of Type A cantilever wall in Figure 12c, the initial static resultant force location z/h′ is 0.33 and the post-earthquake resultant force location is 0.37; furthermore, at the peak input seismic motion phase, z/h′ even exceeds 0.44. At the wall surface plane of Type A cantilever wall, the static resultant force location z/h′ is 0.31 and the post-earthquake resultant force location is 0.36. Furthermore, at the peak input seismic motion phase, z/h′ even exceeds 0.55 at the wall surface plane.

This indicates that under seismic conditions, the earth pressure application point is significantly higher than that under static conditions. Consistent findings have been reported in references [4,24]. This elevation increase adversely affects walls’ structural stability and global anti-overturning stability, which aligns with the observed overturning failure modes in seismic damage investigations of retaining walls. In the pseudo-static method calculations, the resultant force acts at z/h′ = 0.33, failing to reflect this phenomenon. The calculation results may be non-conservative.

4. Seismic Response of Retaining Walls Under Different Frequency Input Waves

To investigate the influence of input seismic wave frequencies on the seismic response characteristics of cantilever retaining walls, numerical simulations were performed for the cases specified in

Table 2. Summary of dynamic simulation cases with variable-frequency synthetic seismic waves.

Height of the Retaining Wall h	Input Frequency of the Seismic Wave f
6 m	f = 1, 3, 5 Hz
8 m	f = 1, 2, 3, 4, 5 Hz
10 m	f = 1, 3, 5 Hz

. The analysis considered three wall heights h (χ = 0.55) with a backfill slope angle β = 0°. The computational parameters are listed in Table 1. The input wave had a 6-second duration, calculated as follows:

$$g=2(1-\cos {\displaystyle \frac{\pi t}{3}})\sin (2\pi ft) (\mathrm{m}/{\mathrm{s}}^2)$$

(12)

where f denotes the frequency (Hz) of the input sinusoidal wave.

Figure 13 shows the time–history curves of the anti-overturning stability safety factor F_t, the anti-sliding stability safety factor F_s, and the dynamic bending moment for cantilever retaining walls subjected to seismic waves with multiple distinct input frequencies. From Figure 13a, it can be observed that as the seismic load cycles repetitively, the anti-overturning stability safety factor F_t of the cantilever retaining wall also exhibits cyclic fluctuations. Ultimately, at the end of the seismic motion, compared to the pre-earthquake conditions, F_t shows significant reductions across all working conditions. When the cantilever wall height h = 8 m and the input seismic wave frequency f = 3 Hz, F_t decreases from 3.086 before the earthquake to 1.714. The seismic action substantially reduces the wall’s anti-overturning stability. On the other hand, during the earthquake, the F_t time–history curve repeatedly drops below 1.0. After shaking cessation, F_t stabilizes at a safety factor greater than 1.0 that satisfies engineering safety requirements. This indicates that during seismic action, structural instability cannot be simply confirmed whenever the safety factor falls below 1.0 (as in static analysis methods).

Figure 13b shows that the anti-sliding stability safety factor F_s of the cantilever retaining wall follows similar patterns. Compare the time–history curves of dynamic bending moment at the cantilever base of retaining walls under different input frequency seismic waves in Figure 13c. The bending moment at the wall base exhibits cyclic fluctuations following seismic waves. Post-earthquake, compared to the pre-seismic conditions, the base bending moment shows significant increases, indicating seismic moment increments ΔM_d. These increments vary substantially with the input seismic frequencies. For example, the seismic moment increments ΔM_d of 10 m high retaining wall at f = 1, 3, and 5 Hz are 522.6, 678.9, and 315.2 kN·m/m, respectively.

As shown in Figure 14, the residual displacement of the cantilever retaining wall (h = 6.0 m) after an earthquake, compared to its pre-earthquake position, was limited to approximately 5 cm under seismic excitations. This displacement is well below the h/100 threshold (0.06 m). The wall also experienced a slight rotation (2.5 × 10⁻³ rad), but did not fail according to the design codes [49,50]. As shown in Figure 13a,b, during dynamic action, transient instances where the structural safety factor falls below 1.0 do not necessarily indicate imminent structural instability. However, they exert a pronounced influence on the structure’s post-earthquake residual deformations.

Figure 15 summarizes the pre-earthquake/post-earthquake safety factors and dynamic bending moment increments at the cantilever bases under the different input frequencies. The key observations are that the safety factors (F_t, F_s) decreased with an increasing frequency (1–3 Hz) and then exhibited a reduced decline (3–5 Hz), and the bending moment increments initially increased (1–3 Hz) then decreased (3–5 Hz). A modal analysis of the retaining wall–backfill system was performed using ABAQUS 6.14 finite element analysis software. The first two natural frequencies of the system were obtained, as illustrated in Figure 16. It was found that the first and second natural frequencies of the wall system were near 3 Hz (for a 6 m high cantilever wall, the corresponding first and second natural frequencies of the system were 3.285 Hz and 3.768 Hz, respectively; for an 8 m high cantilever wall, the corresponding first and second natural frequencies of the system were 2.876 Hz and 3.434 Hz, respectively; for a 10 m high cantilever wall, the corresponding first and second natural frequencies of the system were 2.545 Hz and 3.148 Hz, respectively). When the input seismic wave approached the natural frequency of the wall system itself, the system’s seismic response intensified, the post-earthquake safety factor decreased more substantially, and the bending moment increment on the cantilever increased.

5. Design Recommendations

The resonance amplification factor of seismic earth pressure on retaining walls results from a dynamic interaction, which occurs when the natural frequency of the wall–soil system approaches the predominant frequency of the ground motion. This leads to significant amplification of wall displacement and earth pressure, which is particularly severe for flexible retaining walls and soft-soil sites. Pseudo-static methods generally fail to explicitly account for this amplification effect. It is difficult to provide a universally applicable numerical range for the amplification factor because it depends critically on the specific conditions mentioned above. For rigid retaining walls, under non-resonant conditions, the results from pseudo-static methods may closely approximate those from dynamic analyses, with an amplification factor close to 1.0. When approaching resonance, the displacements and earth pressure may be amplified by a factor of 1.5 to 3.0 or higher, especially under long-duration earthquakes with low-frequency content. For flexible retaining walls, even under general seismic motions without strict resonance, the displacement response is typically much larger than that predicted by pseudo-static methods due to their inherent flexibility. Consequently, the corresponding earth pressure amplification factor frequently reaches 2.0 or more, and may exceed 3.0 or even higher under resonant conditions.

The actual amplification factor varies widely (potentially ranging from near 1.0 to over 3.0) depending on complex factors, such as the wall stiffness, soil properties, seismic motion frequency spectrum, and duration. In seismic designs involving flexible walls or long-period sites, it is essential to fully consider the risks posed by resonance amplification, either through dynamic time–history analyses or by employing validated dynamic amplification factors, to avoid underdesign. For a soil layer with thickness H overlying rigid bedrock, the site amplification factor can be readily obtained by solving the viscous–elastic one-dimensional wave equation [51], as explicitly given in Equation (13):):

$$A_a={\displaystyle \frac{1}{\sqrt{{\cos }^2(\omega H/V_s)+{\left[\xi (\omega H/V_s)\right]}^2}}}$$

(13)

where A_a is the amplification factor, ξ represents the material damping ratio, ω is the input motion circular frequency (rad/s), and V_s is the soil layer shear wave velocity. For a semi-infinite horizontal soil layer, f₀ = V_s/4H is used [51]. Using ω = 2πf and substituting it into Equation (13), we obtain

$$A_a={\displaystyle \frac{1}{\sqrt{{\cos }^2(\pi f/2f_0)+{\left[\xi (\pi f/2f_0)\right]}^2}}}$$

(14)

where f is the predominant frequency of the input seismic wave, and f₀ is the fundamental natural frequency of the wall–soil structural system. Using the parameters of wall height h = 6.0 m, backfill slope angle β = 0°, backfill friction angle φ = 34°, and wall–soil interface friction angle δ = 24°, a comparison of the dynamic active earth pressure coefficients obtained by different calculation methods is shown in

Table 3. Comparison of dynamic active earth pressure coefficients derived from different computational methods.

kh	Coulomb Method	Rankine Method	Present Method (Considering Resonance Effects, f0 = 3 HZ)
			Aa = 1.14	Aa = 1.52	Aa = 1.00
0.1	0.363	0.329	0.375	0.500	0.329
0.2	0.451	0.389	0.444	0.592	0.389
0.3	0.563	0.495	0.565	0.753	0.495
0.4	0.712	0.656	0.748	0.998	0.656
0.5	0.919	0.895	1.020	1.361	0.895

.

Based on the active earth pressure coefficients listed in Table 3 and selecting the horizontal seismic acceleration coefficient kₕ = 0.4 (the input ground motion is given in Equation (11)), the earth pressure values calculated by the different methods were compared with the numerical simulation results. It can be observed from Figure 17 that, after considering the resonance effect, the active earth pressure coefficient proposed in this study closely matches the numerical simulation results. A comparative summary of the root mean square errors (RMSEs) of earth pressure values calculated by the different methods and the numerically simulated values is presented in

Table 4. Comparative summary of root mean square error (RMSEs) of calculated earth pressure values using different methods and numerically simulated values.

Earth Pressure	Method	RMSE (kPa)
Dynamic earth pressure	Rankine method—FEM	23.96584
	Coulomb method—FEM	21.72183
	Present method—FEM	15.39561
Residual earth pressure	Rankine method—FEM	18.92327
	Coulomb method—FEM	16.37654
	Present method—FEM	10.08094

. In contrast, both the Coulomb and Rankine earth pressure coefficients underestimate the pressure.

The seismic design of cantilever retaining walls must transcend static conceptual paradigms and adopt a dynamic design philosophy. Implementing a systematic approach encompassing critical frequency avoidance, initial stress field regulation, and post-earthquake performance control can significantly enhance their structural seismic resilience.

(1)

Critical frequency ranges and dynamic response avoidance

The stability of cantilever retaining walls under seismic loading is highly dependent on the match between the ground motion frequency characteristics and the structural natural frequencies. These walls exhibit significant sensitivity to seismic waves within specific frequency bands. When subjected to ground motion inputs within a 1.5–5 Hz range (equivalent to medium-to-high frequency vibrations), the backfill soil experiences intense dynamic responses, leading to a substantial increase in the active earth pressure. This critical frequency range notably overlaps with the dominant frequency components of most moderate-to-strong earthquakes, necessitating special attention in seismic design.

During design, hazardous frequency responses can be mitigated by adjusting the wall’s geometric stiffness (e.g., increasing the toe slab length, optimizing the reinforcement ratio) and improving the backfill soil properties (e.g., adding fiber materials, using lightweight fill). These measures can shift the fundamental frequency below 0.7 f₁ or above 1.3 f₂, where f₁ and f₂ are the first two natural frequencies of the retaining wall–backfill system.

(2)

Quantification of initial backfill stress field and design considerations

The initial shear stress level governs the dynamic softening threshold. An initial shear stress ratio exceeding 0.4 promotes significant stiffness degradation and, in saturated sands, rapid pore water pressure buildup, greatly reducing wall stability. The small-strain shear modulus Gₘₐₓ, which depends on the mean effective stress, directly affects the fundamental frequency of the soil-structure system. Layered compaction induces non-uniform stresses, with locked-in zones at the top of each layer and relaxed stresses at the bottom, leading to initial earth pressures that deviate from K₀ conditions and cause early wall–backfill interaction.

To address these effects, the following measures are recommended: (i) finite element simulations of construction processes to predict post-construction stress fields; (ii) in situ verification via pressuremeter tests to calibrate the stress assumptions; (iii) base reinforcement techniques, such as using reinforced concrete shear keys or soil mixing, to increase the interface stiffness.

(3)

Control standards for post-earthquake effective safety factors

It is recommended that seismic stability verification for cantilever retaining walls satisfy dual safety criteria: instantaneous stability during seismic shaking and control of residual deformations after an earthquake. The instantaneous stability verification must consider the coupled effects of seismic inertial forces and dynamic earth pressures. The seismic stability verification for cantilever walls should include the following: checking the sliding and overturning stability under seismic conditions (check the sliding stability with a minimum recommended factor of safety (F_S ≥ 1.2 under seismic conditions)); evaluating the overturning stability with a required factor of safety (F_t ≥ 1.5) to account for amplified moments under dynamic conditions, consistent with the major codes, such as Eurocode 8 [52] and AASHTO [49] guidelines; assessing post-earthquake residual deformations; and verifying compliance with the minimum safety factors per the applicable codes.

6. Conclusions

This study systematically reviews the calculation methods for cantilever retaining walls and introduces a unified formula for the seismic earth pressure applicable to both long-heel and short-heel configurations. The formula is validated via pseudo-static and dynamic finite element analyses, revealing the following key findings:

(1)

The distinction between short-heel and long-heel walls depends on the geometry and seismic coefficients, but is less critical in seismic earth pressure estimation. Using Rankine’s theory for short-heel walls—treating them as long-heel types—yields small deviations that decrease with higher seismic acceleration. This method overestimates the horizontal pressure and underestimates the vertical pressure, reducing the safety factors but offering a conservative design.

(2)

Ignoring the backfill construction effects leads to inaccurate initial stress fields. The simulations incorporating layered construction show that construction-aware models yield higher failure surface angles and lower active earth pressure coefficients, indicating that conventional pseudo-static methods may produce unsafe estimates.

(3)

The pseudo-static method oversimplifies the earth pressure distribution as triangular, with the resultant at a 1/3 height. Under real seismic loads, the resultant rises significantly, impairing stability and overturning resistance—a critical factor in seismic design.

(4)

The dynamic simulations reveal that the safety coefficients for sliding and overturning frequently fall below 1.0 during seismic events but recover afterward. For walls with natural frequencies of 2.5–3.8 Hz, input frequencies between 1 and 3 Hz induce resonance effects, amplifying seismic responses, reducing post-seismic safety margins, and increasing stem bending moments.