The Seismic Response of Two Geotechnically Similar GRS-MB Walls During the Chi-Chi Earthquake: Insights from the Finite Displacement Method

Abstract

This study re-examines two geologically and geotechnically similar geosynthetic-reinforced soil walls with modular block facings (GRS-MBs) that exhibited markedly different seismic performances during the 1999 Chi-Chi earthquake (M_L = 7.3). Integrating a multi-wedge failure mechanism that captures soil–facing–reinforcement interactions with a nonlinear hyperbolic soil model representing shear stress–displacement behavior along the slip surface, the Force–equilibrium-based Finite Displacement Method (FFDM) provides consistent and robust displacement evaluations over a wide range of input seismic inertial forces. A systematic sensitivity investigation confirms that the FFDM framework responds to parameter variations in a physically meaningful manner, and that displacement predictions remain stable with respect to reasonable uncertainties in soil, reinforcement, and facing properties. The analysis clarifies why two similar GRS-MBs responded so differently during strong shaking and demonstrates the broader applicability of FFDM for displacement-based seismic assessment, including under shaking levels (e.g., k_h ≈ 0.3) that would drive conventional limit–equilibrium calculations to F_s < 1.0, a physically impossible state requiring shear resistance greater than the soil’s ultimate strength. A comparative evaluation of seismic displacement predictions using the Newmark method and FFDM shows that FFDM successfully generates displacement-based seismic resisting curves and reproduces field-observed displacements. In contrast, the Newmark method yields order-of-magnitude variability in predicted movements and may be unsuitable for displacement-sensitive engineered slopes where deformations on the order of several 10⁻³–10⁻² m are practically significant. For interaction-rich GRS-MBs with high values of k_hc, beyond the predictive capability of Newmark’s equation, FFDM offers a practical and physically grounded tool for seismic displacement assessment of reinforced soil structures.

1. Introduction

The seismic performance of soil retaining structures—including highway and railway embankments, bridge abutments, and slope protection systems—has received increasing attention following major earthquakes, such as the 1995 Hyogo Ken Nambu earthquake in Japan [1]. Similar post-earthquake investigations in New Zealand, Europe, Turkey, and Italy have also demonstrated the importance of integrating field evidence and modern remote-sensing tools for understanding seismic damage mechanisms [2]. These events have highlighted the need for reliable methods to evaluate the seismic-resisting capacity expressed by the displacement (or deformation) of the structure under earthquake excitation. Performance-based or displacement-based assessments of seismic capacity have a relatively short history compared with traditional force-based approaches. The force-based method evaluates seismic capacity through a safety factor (F_s) against a pseudo-static horizontal seismic force represented by the horizontal seismic coefficient (k_h), defined as λ·HPGA/g, where λ is an empirical scaling factor (typically 0.3–0.5), HPGA is the horizontal peak ground acceleration, and g is gravitational acceleration. However, several limitations of F_s-based seismic evaluation are well recognized: (1) increasing F_s beyond 1.5–2.0 often leads to overly conservative and cost-inefficient designs, with unclear implications for actual field performance; (2) the value of λ is empirically determined; and (3) the operational soil strength parameters, c and φ, are also selected empirically, requiring judgment that varies among practitioners.

A major shift toward displacement-based evaluation began with Newmark’s sliding-block method [3], which combines limit–equilibrium analysis with the dynamics of a rigid block sliding on a planar surface. Successful application of the original Newmark method requires calibration of a critical seismic coefficient (k_hc), obtained from the F_s–k_h curve at F_s = 1.0. This calibration depends on iterative adjustments to identify an operational internal friction angle that produces an acceptable F_s–k_h relationship. Numerous studies have attempted to improve the Newmark method in five main areas: (1) enhancing the accuracy of k_hc and horizontal slope displacement (Δ_h) through improved failure mechanisms [4,5,6,7,8,9,10,11]; (2) incorporating lumped-mass or soil–column dynamics to overcome rigid-block assumption [12,13,14,15,16]; (3) improving k_hc and Δ_h predictions by allowing evolving slip-surface geometry and/or strength degradation [17,18,19,20,21,22]; (4) accounting for interactions among multiple soil blocks within the sliding mass [23,24]; and (5) comparing or improving displacement-prediction equations [25,26,27]. Although these enhancements broaden the applicability of Newmark-type analyses, they introduce new challenges. A fundamental drawback remains: Newmark-type approaches predict zero displacement prior to yielding acceleration, in contrast to observed slope behavior, as emphasized in [28]. The lumped-mass approach requires additional parameters such as spring and dashpot constants, which often demand specialized laboratory or in situ testing not commonly available in routine design. Improvements related to slip-surface geometry are most relevant for large displacements on the order of 10⁻¹ m or more, whereas engineered transportation slopes typically require attention at much smaller displacement levels. The Newmark-based displacement-prediction equations simplify the double-integration step of Newmark’s method but still produce displacement estimates that vary by more than an order of magnitude, making them unsuitable for displacement-sensitive soil structures.

Other analytical approaches for evaluating seismic performance include: (1) finite-element and discrete-element methods that compute seismic displacements using nonlinear dynamic response analysis [29,30,31,32,33], and (2) energy-based theories for sliding-block displacement estimation [34]. Numerical analyses in item (1) require extensive effort for validation and calibration of input soil properties and remain research-oriented tools rather than practical design methods. The energy-based theory integrates wave-propagation concepts and earthquake energy release to estimate seismic displacement of a simple block with a friction angle φ on a planar slip surface. Because the method idealizes the slope as a simple block sliding on a planar surface, it is inherently unsuitable for evaluating complex, interaction-rich systems, such as GRS-MB walls, and remains oriented toward regional hazard mapping rather than engineered structures.

Given the limitations of existing approaches, this study adopts the Force–equilibrium-based Finite Displacement Method (FFDM) proposed in [35,36] as an efficient, design-oriented tool for re-examining the seismic performance of two geosynthetic-reinforced soil walls with modular block facings (GRS-MBs). FFDM offers several key advantages: (1) it accommodates peak strength and post-peak degradation without requiring iterative derivation of operational strength parameters, consistent with the direction of recent improvements to Newmark-type methods; (2) it directly incorporates peak ground acceleration (HPGA/g) to represent the primary inertial force induced by ground shaking, reducing uncertainties associated with seismographic interpretation; and (3) the multi-wedge framework adopted in the FFDM explicitly incorporates an essential mechanism in interaction-rich GRS-MB systems, namely the interactions among adjacent components, such as soil–soil, soil–facing, and reinforcement–facing. This capability overcomes the common limitation of most improved Newmark-type methods, which still overlook or oversimplify interactions among soil blocks. In addition, GRS-MB walls often exhibit relatively high values of k_hc due to the presence of closely spaced reinforcement, which makes Newmark-type displacement-prediction equations unreliable, as demonstrated later in the case study. Building on these strengths, the FFDM-based multi-wedge analytical tool is used here to re-evaluate the two Chi-Chi GRS-MB walls to identify the mechanistic causes of their contrasting seismic responses, and to propose seismic-resisting curves (expressed by Δ_h vs. k_h) that resemble the monotonic pushover curves for displacement-based seismic performance evaluations in structural engineering [37].

Previous studies [35,36,38] focused on developing prototype versions of the FFDM framework and associated hyperbolic and post-peak soil models. Those efforts validated the FFDM by computing displacement along a known or observed slip surface, either in natural slopes subjected to groundwater-induced movement or in reinforced model slopes tested in a tilting box. No previous FFDM study has attempted to search for a critical failure surface, incorporate multi-wedge mechanisms, or evaluate reinforced slopes under high seismic loading. These earlier works, therefore, represent the first phase of FFDM development, in which the method was verified under controlled or predefined failure geometries. The present study represents a conceptual and methodological expansion into the second phase of FFDM development, implemented in a structured computer program and capable of identifying critical failure mechanisms and evaluating reinforced slopes under realistic seismic and hydraulic loading scenarios.

The remainder of this paper presents the analytical framework, summarizes the case histories, evaluates the FFDM results, and discusses implications for seismic design of reinforced soil structures.

2. Study Sites and Methodology

The 1999 Chi-Chi earthquake provided rare opportunities to uncover structural vulnerabilities that had been masked by facing elements or by inherent misdesign. Among the affected systems were two geosynthetic-reinforced soil walls with modular block facings (GRS-MBs), located on the upstream side of a 40 m wide highway, and constructed using similar backfill materials and reinforcement products. Despite their apparent similarity, the two walls exhibited markedly different seismic performances during the earthquake, prompting detailed post-event investigations.

2.1. GRS-MB Walls at Sites 1 and 3

The studied slopes [23] are steep-faced geosynthetic-reinforced soil walls with modular block facings (GRS-MBs) located in Taichung County, in central Taiwan. The wall at Site 1 experienced severe damage during the 1999 Chi-Chi earthquake (M_L = 7.3), as shown in Figure 1, Figure 2 and Figure 3. In contrast, a similar GRS-MB at Site 3, located approximately 2 km south of Site 1, exhibited only light damage, with facing displacement almost undetectable (Figure 4 and Figure 5). Although the overall deflection and displacement of the wall were minimal, the slight misalignment at the top of the facing column (Figure 5) suggests that a small amount of movement—on the order of a few centimeters (or tens of millimeters)—is an appropriate description of the post-earthquake condition of the Site-3 wall. At a nearby seismograph station (TCU052, N–S component), a peak horizontal ground acceleration of HPGA = 0.45 g (g: gravitational acceleration) was recorded.

For the two Chi-Chi sites examined here, the simultaneous occurrence of vertical and horizontal accelerations was analyzed comprehensively [23], using recordings from nearby seismographs during the 1999 earthquake. Their study showed that the ratio of vertical to horizontal peak ground acceleration around the major pulses was approximately k_v/k_h = 0.2 (k_v and k_h: vertical and horizontal seismic coefficients positive for upward and outward inertial forces, respectively). This ratio has been adopted directly in the present work, and the corresponding vertical inertial force is explicitly included in the FFDM calculations in the same manner as the horizontal inertial force. In reinforced soil retaining structures, the vertical component primarily modifies the normal stress acting on potential failure wedges. In the FFDM framework, using a positive k_v/k_h ratio means that upward and outward inertial forces are applied simultaneously to the wedges during strong shaking. As noted in [23], this upward inertial force represents the more critical condition because it reduces the effective normal stress and thus the mobilized shear resistance along the slip surface. The adopted k_v/k_h = 0.2, therefore, provides a realistic representation of the vertical shaking intensity for the studied sites and ensures that its influence on the stress state and displacement response is appropriately accounted for in the FFDM analysis.

2.2. Hyperbolic Soil Stress–Displacement Model

As illustrated in Figure 6, a hyperbolic model describing the normalized shear stress (τ/τ_f)–shear displacement (Δ) response along the potential failure surface was proposed in [35]). This model is an adaptation of the well-known hyperbolic relationship for shear stress–strain behavior developed in [39].

τ/τ_f = ∆/(a + b × ∆)

(1)

where a = τ_f/k_initial, b = R_f, and R_f = τ_f/τ_ult

• k_initial: Initial shear stiffness of soils.
• τ_ult: Asymptote strength at infinite displacement.
• τ_f: Shear strength of soil according to the Mohr–Coulomb failure criterion.
• R_f: Failure ratio (=τ_f/τ_ult).

The shear strength of soils (τ_f) is defined by the Mohr–Coulomb failure criterion:

τ_f = c + σ‘_n × tan φ

(2)

• σ‘_n: Effective normal stress.
• c: Cohesion intercept.
• φ: Internal friction angle of soils.

Equation (1) can be viewed as a reciprocal formulation of the local safety factor FS_i at the base of the soil block (or slice) i:

FS_i = τ_{f i}/τ_i

(3)

The initial shear stiffness k_initial is given as a power function of effective normal pressure, proposed in [38]:

k_initial = K × G × (σ’_n/Pa)ⁿ

(4)

• K: Initial shear stiffness number (dimensionless).
• P_a: Atmospheric pressure (=101.3 kPa).
• G: Reference shear stiffness (=101.3 kPa/m).
• n: Pressure dependency exponent.

2.3. Multi-Wedge Failure Mechanism

Figure 7 illustrates the multi-wedge failure mechanism adopted in this study. This mechanism evolved from the classical “two-wedge” or “bi-linear” failure model, which has long been used to simulate the failure of engineered slopes with steep faces, such as mechanically stabilized earth (MSE) walls and geosynthetic reinforced slopes with rigid or near vertical facings. An enhanced version of the two-wedge mechanism was proposed in [23], comprising two active wedges (Wedges 1 and 2), a facing column (Wedge 3), and—when present—a passive wedge located in front of the toe of the facing (Wedge 4). This multi-wedge formulation provides deeper insight into the contribution of the facing system to overall slope stability. Its advantages are further amplified when combined with the FFDM, which enables displacement-based performance evaluation of the slope. Figure 8 presents the body forces and reaction forces acting on a generalized polygon wedge configuration. Triangular wedges, exemplified by Wedges 1 and 4, constitute special cases where the soil strength and reaction forces along one boundary are assumed to be negligible or zero.

By applying force equilibrium in both horizontal (x) and vertical (y) directions, the reaction force on the left side of each wedge can be expressed as:

R_e = f (R_r, R_b, T_e, T_r, T_b, C_e, C_r, C_b, φ_e, φ_r, φ_b, k_h, k_v, Q_v, Q_h)

(5)

C_e = c × l_e/FS_i, C_r = c × l_r/FS_i, C_b = c × l_b/FS_i,

φ_e = tan⁻¹ (tan φ/FS_i), φ_r = tan⁻¹ (tan φ/FS_i), φ_b = tan⁻¹ (tan φ/FS_i)

• C_e, C_r, C_b: Cohesive resistance along soil–block interfaces.
• l_e, l_r, l_b: Lengths of soil–block interfaces.
• φ_e, φ_r, φ_b: Mobilized internal friction angles at interfaces.
• T_e, T_r, T_b: Reinforcement force at interfaces.
• R_r, R_b: Reaction forces at interfaces.

To determine the reaction force acting on the left side of each wedge in the conventional F_s analysis, the computation advances sequentially from Wedge 1 to Wedge 3 (or Wedge 4, if present). A constant safety factor, F_s, is assumed and applied uniformly to all wedges. With this assumption, all terms on the right-hand side of the governing force–equilibrium relationship are known, except for the base reaction force R_b. Because both horizontal and vertical force equilibrium are imposed, the system is statically determinate. The F_s is updated through iterative equilibrium calculations, typically starting with an initial trial value of F_s = 1.0, and is adjusted until the virtual external force R_e on the left side of wedge 3 (or wedge 4) becomes negligibly small (R_e < 0.1 kN is used in this study).

In contrast, when the same governing relationship is solved within the FFDM framework, the safety factor for each soil block (FS_i) and the mobilized reinforcement force (T_r, T_b, T_e) depend on its displacement (Δ_i). The system remains statically determinate, but the FS_i is updated iteratively according to the displacement field at the block base and along the interfaces. The detailed computational procedure for the FFDM is summarized in Section 2.7, Algorithmic procedure for FFDM implementation.

2.4. Post-Peak Stress–Displacement Relationship

The post-peak segment of the shear stress–displacement (τ–Δ) curve is modeled using the Versoria (or Witch of Agnesi) curve [40,41], schematically illustrated in Figure 9. For efficient integration into the overall stress–displacement framework, a normalized local coordinate system (X–Y) is adopted. The X-axis origin (X = 0) aligns with the peak stress point (τ_f). The Y-axis origin (Y = 0) corresponds to the asymptotic value of the residual stress (τ_r). A distinctive feature of this curve is that the residual state (τ_r) is theoretically reached at an infinite Δ. In practice, however, a finite displacement (Δ_r), observed experimentally at the onset of the residual state (defined at X = 1), can be used to approximate τ_r with a negligibly small error (normally less than 1% error). The shear strength in the post-peak regime (τ_post-peak) can be expressed as:

τ_post-peak = τ_f − τ_f × (t − Y) = τ_f × (1 − t + Y)

(6)

Y = t³/(t² + X²)

(7)

X= (Δ − Δ_f)/(Δ_r − Δ_f)

(8)

t = (τ_f − τ_r)/τ_f

(9)

Δ_r = Δ_f × Δ_ratio

(10)

• t: Normalized strength reduction between peak and residual states.
• Y: Normalized post-peak shear stress.
• X: Normalized post-peak shear displacement.
• Δ_f: Shear displacement at peak stress state (obtained from Equation (1) by setting τ = τ_f).
• Δ_r: Shear displacement at the entrance of the residual state.
• Δ_ratio: Residual-to-peak displacement ratio.

In geotechnical applications of post-peak soil behavior, the generalized and widely accepted term “residual friction angle, φ_res” is preferred over the parameter t used in Equation (6). To maintain consistency with common geotechnical practice and to improve interpretability, t is reformulated based on Equation (9) in terms of φ_res as follows:

t = 1 − τ_r/τ_f = 1 − (σ‘_n × tan φ _res)/(c + σ‘_n × tan φ _peak)

(11)

This formulation is incorporated directly into the FFDM iterative procedure. When post-peak behavior is considered in the slope displacement analysis using SLOPE-ffdm 2.0 [41], Equation (6) is applied to update the available soil strength along the slip surface whenever the condition FS_i < 1 is detected during the iterative displacement computations. In this situation, the available shear strength τ_post-peak corresponding to intermediate states between “peak” and “residual” is represented through a “transient friction angle,” denoted as φ_transient:

φ_transient = tan⁻¹ (τ_post-peak/σ‘_n)

(12)

The value of τ_post-peak in Equation (12) is obtained from Equation (6), using the normalized displacement X and normalized stress Y defined by Equations (8) and (7), respectively. This ensures that the reduction from peak to residual strength is consistently captured within the displacement-driven FFDM framework.

2.5. Displacement Compatibility

A hodograph (displacement diagram) that satisfies displacement compatibility—as schematically illustrated in Figure 10—is derived following the approach in [42]. This formulation maintains displacement compatibility across soil interfaces and forms the basis for kinematic analysis of the sliding block system. The displacement at the soil wedge (or slice) i can be expressed as:

Δ_i = Δ₀ × f (α_i)

(13)

f (α_i) = cos (α₁ − 2Ψ)/[sin (α₁ − Ψ) × cos (2Ψ − α₁)]

(14)

2.6. Displacement Increment

To evaluate slope displacements resulting from changes in external or internal conditions—such as seismic loading, variations in the water table, or pore water pressure—two displacement values for each slice (Δᵢ) are calculated: one representing the state prior to the event and the other representing the state afterward. The displacement increments for slice i, induced by the change in stress conditions, are schematically illustrated in Figure 11 and defined as:

Δ_i = Δ^b_i − Δ^a_i

(15)

2.7. Algorithmic Procedure for FFDM Implementation

The FFDM analysis implemented in SLOPE-ffdm 2.0 follows the algorithmic sequence below:

(1)

Input slope geometry, soil strength parameters, water table position, and reinforcement or anchor strength data.

(2)

For each trial failure surface, assign an initial vertical displacement at the crest, denoted as Δ₀ (for example, Δ₀ = 0.001 m).

(3)

Compute the displacements of all wedges using the displacement-compatibility functions described in Equation (13).

(4)

Calculate the reinforcement or anchor forces T_r, T_b, and T_e at the pullout displacement corresponding to the local shear displacement Δ_i at the intersection between the reinforcement and the slip surface, following the pullout model in reference [36].

(5)

Calculate the peak shear strength τ_f and the initial stiffness k_initial using Equations (2) and (4). Then compute the local safety factors FS_i using Equations (1)–(3). If FS_i is greater than 1, or if post-peak strength is not considered, proceed directly to step (7).

(6)

For wedges where post-peak strength applies, compute the post-peak shear strength τ_post-peak and the transient friction angle φ_transient using Equations (6) and (12). Reset FS_i to 1.0. The values φ_transient and FS_i = 1.0 are used in Equation (5).

(7)

Starting from wedge 1 at the right-hand crest, use the known boundary value of R_r for wedge 1 as input and compute R_e, using Equation (5) together with the FS_i values obtained above.

(8)

Sequentially compute R_e for Wedges 2 through n, with the value of R_e for wedge i serving as the value of R_r for wedge i + 1.

(9)

The value of R_e for wedge n becomes P_ex, the virtual external force acting at the left boundary of the wedge at the toe of the slope.

(10)

Adjust Δ₀ and iterate steps (3) through (9) until P_ex satisfies the convergence criterion (e.g., within plus or minus 0.1 to 0.01 kN/m).

(11)

Iterate steps (2) through (10) for all trial failure surfaces and identify the critical failure surface associated with the maximum value of Δ₀.

2.8. Site Explorations

Ground explorations [23] included disturbed and undisturbed soil sampling, standard penetration tests (SPT), and direct shear tests using both small-size specimens (63 mm diameter × 42 mm) and large-size specimens (200 mm diameter × 113 mm). The resulting soil properties are summarized in

Table 1. Soil properties at Sites 1 and 3.

	Site 1	Site 3
Unified Soil Classification System (USCS)	ML, CL	SM, ML
Unit weight (kN/m3)	21.3	18.9
Water content (%)	16.5	12.1
Cohesion intercept (kPa)	0	0
Internal friction angle, φ	29.2–30.4° (1)	33.6–34.8° (1)
SPT-N value	8–10	6–7
Estimated φ from N-value	25–36° (2)	23–34° (2)

⁽¹⁾ Obtained from direct shear tests on large specimens (200 mm × 113 mm). ⁽²⁾ According to Dunham’s empirical equations [23].

. Table 1 indicates that Site 1 exhibits slightly higher N-values (8–10) than Site 3 (6–7), which would suggest higher friction angles at Site 1 (25–36°) compared with Site 3 (23–34°). However, the results from large-specimen direct shear tests show the opposite trend: soils at Site 3 exhibit higher friction angles (33.6–34.8°) than those at Site 1 (29.2–30.4°). Overall, the site explorations indicate that soils at Sites 1 and 3 have comparable strength characteristics. Based on these findings, a cohesive intercept of c_peak = 5 kPa is adopted for soils with an in situ water content of 16.5% in the subsequent analysis. Accordingly, two representative strength conditions are defined: a “low soil strength” case, with c_peak = 0 and φ_peak = 30.4°, and a “high soil strength” case, with c_peak = 5 kPa and φ_peak = 35°, consistent with the data summarized in Table 1. The low soil strength case with c_peak = 0 and φ_peak = 30.4° is consistent with the c = 0 and φ = 30.4° used in the post-earthquake analysis [23]. In addition, a strength reduction factor f_inter-wedge = 1.0 is adopted, indicating that full soil strength is mobilized along the interfaces between soil wedges in the multi-wedge analysis.

2.9. Input Soil and Reinforcement Parameters

The input parameters summarized below were selected based on published experimental databases and site-specific material information. Their uncertainty ranges and influence on computed seismic displacements are further examined through the sensitivity analyses in Appendix A. The sensitivity analysis in Appendix A quantifies the influence of both high-sensitivity and low-sensitivity parameters and shows that the soil-uncertainty-induced variations in the computed seismic displacements are well bounded. The results confirm that, despite the variability of individual input parameters, the overall conclusions drawn in this study remain robust and reliable.

(1)

Peak soil strengths (c_peak, φ_peak): Three sets of soil strengths are used to represent possible variations in in situ conditions. The low-strength case uses c_peak = 0 and φ_peak = 30.4°. The high-strength case uses c_peak = 5 kPa and φ_peak = 35°. The integrated pre- and post-peak soil strength model uses c_peak = 5 kPa and φ_peak = 35°, and c_res = 0 kPa and φ_res = 31°. The adopted residual-to-peak friction ratio φ_res/φ_peak = 0.88 is close to the median of the range 0.82–0.93 reported from a series of direct shear tests on various soil types [43,44].

(2)

Post-peak soil stress–displacement model: A residual displacement ratio Δ_ratio = 2.0 is adopted, consistent with the post-peak soil stress–displacement relationships reported in [38,43,44]. For medium to dense sands, reported values of Δ_ratio typically range from 2.0 to 6.0.

(3)

Shear stiffness number K: Stiffness values of K = 200 and K = 350 are adopted based on a database of direct shear tests [45]. These values represent approximately the lower-bound and median stiffnesses for soils with φ = 35°.

(4)

Pressure dependency exponent n: Approximate median values of n = −0.1 and n = 0.2, obtained from the same database [45], for sands with φ = 30°and 35° are used here.

(5)

Failure ratio R_f: A majority of data on soils with φ ≥ 30° indicates that R_f values typically fall within the range 0.79 and 0.87 [45]. A median value of R_f = 0.83 is used throughout the main analysis.

(6)

Displacement-dependent reinforcement pullout model: The mobilized reinforcement force in the backfill is computed using the hyperbolic reinforcement pullout model described in [36,46]. The peak adhesion at the soil–reinforcement interface is taken as c_s-r = 0, and the peak interface friction angles are φ_s-r = 24° and 28°, approximately 0.8 φ_peak. The hyperbolic pullout model parameters are the median values for geogrid pullout from silty sands, including the initial pullout stiffness number K_t = 12, stress-dependency exponent n_t = −0.1, and strength ratio R_t = 0.7, as summarized in [46].

(7)

Tie-break strength of reinforcement: In the FFDM hyperbolic pullout model, the tie-break strength T_tie-break = 75 kN/m is assigned based on the available design and analysis documents for the studied walls.

(8)

Inter-wedge strength ratio: The inter-wedge strength ratio f_inter-wedge, defined as the ratio of failure shear strength to the shear strength available at the soil block–block interface in the force–equilibrium calculations, is set to 1.0. This choice is consistent with field observations indicating that no permanently open tension cracks developed at the interface between reinforced and unreinforced zones at the investigated sites.

Both the hyperbolic curves and the integrated curves combine a pre-peak hyperbolic model with a post-peak strength degradation model, as presented in Figure 12. The conditions of “high soil strength” with c_peak = 5 kPa, φ_peak = 35°, and c_res = 0, φ_res = 31°, associated with hyperbolic parameters of K = 350, n = 0.2, and R_f = 0.83, are used. These soil strength parameters are determined from the results of ground exploration and from

Table 2. Material properties used in the FFDM analysis for the Chi-Chi GRS-MB walls.

Soil Hyperbolic Model		Pullout of Reinforcement		Facing and Connection Strengths
c	0, 5 kPa	cs-r	0	cb-r	2.5 kPa
φ	30.4°, 35°	φs-r	24°, 28°	φb-r	40°
K	200, 350	Kt	12	cb-b	45 kPa
n	−0.1, 0.2	nt	−0.1	φb-b	35°
Rf	0.83	Rt	0.7	cback	0 kPa
Ψ	0°, 15°	Ttie-break	75 kN/m	φback	30°, 35°
				cbase	0 kPa
Post-peak model		Post-peak model		φbase	30°, 35°
cpeak	5 kPa			Post-peak model
φpeak	35°
cres	0	Not available		Not available
φres	31.0
Δr/Δf	2.0

c_s-r and φ_s-r: the adhesion and friction angles, respectively, at the soil–reinforcement interface. c_b-b and φ_b-b: the adhesion and friction angles, respectively, at the facing block–block interface. c_b-r and φ_b-r: the adhesion and friction angles, respectively, at the facing block–reinforcement interface. c_back and φ_back: the adhesion and friction angles, respectively, at the back-face of the facing. c_base and φ_base: the adhesion and friction angles, respectively, at the base of the facing. T_tie-break: the tie-break strength of reinforcement.

, as discussed previously. The use of these soil parameters is further supported by the consistent and physically meaningful trends observed in the sensitivity investigation presented in Appendix A.

2.10. FFDM Displacement Analysis

In the Type 4 (multi-wedge) FFDM analysis implemented in SLOPE ffdm 2.0, more than 18,000 trial facing surfaces are evaluated to identify the critical failure mechanism, defined by the minimum safety factor (F_s) and the maximum vertical displacement at the slope crest (Δ₀), for a specified pseudo-static seismic coefficient k_h (=HPGA/g). The seismic displacement at the base of the facing (Δ₃) is then computed using Equation (11). The horizontal component of Δ₃, denoted Δ_3h, is used to evaluate the seismic resistance behavior of the wall. To ensure the robustness of the analytical procedure and the hyperbolic stress–displacement model used in the subsequent analyses, a systematic sensitivity investigation is conducted and presented in Appendix A.

The facing and reinforcement properties, except the friction angle-related terms, are kept identical for all cases.

3. Results

3.1. Results of Comparative Study on Sites 1 and 3

Figure 13 presents the F_s–k_h curves obtained from conventional slope stability analyses using SLOPE ffdm 2.0. For both Sites 1 and 3, two soil strength conditions were examined: a low-strength case, with c = 0 and φ = 30.4°, and a high-strength case, with c = 5 kPa and φ = 35°. The two strength conditions produce distinctly different F_s–k_h relationships, with the high-strength case yielding k_hc values nearly twice those of the low-strength case. In addition, Site 3 exhibits a markedly different set of F_s–k_h curves compared with Site 1, despite their similar reinforcement, soil, and facing properties. This contrast will be examined further later.

Figure 14 summarizes the Δ_3h–k_h relationships computed using the FFDM for Site 1, where Δ_3h denotes the horizontal displacement at the base of the facing column. Three input conditions were analyzed: low soil strength with a hyperbolic model, high soil strength with a hyperbolic model, and high soil strength with a post-peak model. For each condition, two dilation angles (Ψ = 0° and Ψ = 15°), representing large-displacement and peak states, respectively, for the soil with φ = 30°, were used to evaluate the influence of Ψ on the displacement distribution along the failure surface. The resulting Δ_3h–k_h curves show limited sensitivity to Ψ. The low-strength hyperbolic model predicts extremely large Δ_3h values at k_h ≈ 0.25, whereas the high-strength hyperbolic model yields negligible displacements (approximately 0.007–0.008 m) at k_h = 0.45. The curve with integrated pre- and post-peak model exhibits yielding behavior beginning around k_h ≈ 0.15–0.20, and approaches Δ_3h ≈ 0.4 m at k_h = 0.45. This behavior contrasts sharply with the low-strength hyperbolic model, which predicts unrealistically large displacements for k_h > 0.25. In this regard, the curve with integrated post-peak provides a more realistic representation of the observed performance than the low-strength hyperbolic model.

Figure 15 presents the corresponding results for Site 3. Compared with Site 1, the Δ_3h–k_h curves for Site 3 exhibit more numerically stable behavior. All three input conditions show consistent trends. The low-strength hyperbolic model predicts Δ_3h = 0.07–0.08 m at k_h = 0.45, comparable to the displacement predicted by the integrated post-peak model (Δ_3h = 0.06–0.07 m). In contrast, the high-strength hyperbolic model produces much smaller displacements (0.003–0.005 m) at the same seismic coefficient, indicating a highly stable response. Overall, regardless of the assumed soil strength model, all curves for Site 3 indicate a stable condition, with wall movements that would go unnoticed in the field unless a displacement monitoring system were used.

3.2. Factors Influencing the Failure Mechanism of Site 3

To investigate the primary cause of the higher stability of the GRS-MB at Site 3 compared with Site 1, the gravity concrete wall located behind the GRS-MB at Site 3 was removed from the slope profile, and is referred to as “Site 3, no gravity wall.” In contrast, the actual configuration is referred to as “Site 3, with gravity wall”. Figure 16 shows that the F_s values obtained from conventional stability analyses for the no-gravity-wall case are significantly smaller than those for the with-gravity-wall case. This difference arises because the no-gravity-wall condition allows the critical failure surface to extend freely into the backfill, whereas the presence of the gravity wall constrains the failure mechanism by forcing the critical surface to pass outside the wall (as shown in Figure 17). The resulting difference in k_hc between the two cases exceeds 50%. For reference, two dotted curves representing Site 1 are also included in Figure 16. The difference in F_s between these dotted curves and the curves for the no-gravity-wall case reflects factors other than the presence of the gravity wall, including reinforcement configuration and the slight difference in wall height. These factors account for differences in k_hc on the order of 13–38%, which are comparatively smaller than the influence of the gravity concrete wall.

From the perspective of the FFDM, differences in seismic displacement behavior are also evident. Figure 18 shows that, for a given value of k_h, the resulting Δ_3h values follow the sequence Site 1 > Site 3 (no gravity wall) > Site 3 (with gravity wall). This ordering is the reverse of the F_s–k_h relationships shown in Figure 16. Figure 19 presents results similar to those in Figure 18 but based on the soil model integrating pre- and post-peak soil strengths. The sequence of Δ_3h among the three cases remains the same as in Figure 18, with a minor exception at k_h = 0.25–0.30, where Δ_3h for Site 1 becomes slightly smaller than that for Site 3 (no gravity wall).

3.3. Comparisons Between Newmark’s Chart and the FFDM

To compare the effectiveness of the FFDM with Newmark’s method, a Newmark chart compiled from [3,25,47,48,49] is presented in Figure 20. In this chart, the seismic displacement of slopes is normalized by v_max (maximum velocity) and HPGA of selected seismographs, and the critical seismic coefficient k_hc is normalized by k_m (design value of seismic coefficient, or seismic coefficient corresponding to an anticipated ground shaking). Among these curves, those associated with the El Centro and Chi-Chi earthquakes reported in [49] are selected for comparison. These two curves approximately represent the lower and upper bounds of the reported range and are used in the following comparative study.

Table 3. Comparisons of Δ_3h for Site 1 (k_hc = 0.162, k_m = 0.45 for Newmark’s chart; with post-peak model, K = 350, n = 0.2, R_f = 0.83, Ψ = 0–15°, and k_m = 0.45 in the FFDM analysis).

Curve and Earthquake	X	Y	vmax (m/s)	HPGA (g)	Newmark’s Δ3h (m)	FFDM’s Δ3h (m)
Chi-Chi [49]	0.356	3.5	1.28	1.018	0.575	0.456–0.460
El Centro [49]	0.356	1.0	0.369	0.318	0.044

compares the predicted Δ_3h for Site 1 under a ground shaking intensity of k_m = 0.45 using Newmark’s chart and the FFDM. Newmark’s chart estimates Δ_3h values ranging from 0.044 to 0.575 m for the Chi-Chi and El Centro earthquakes, respectively. This range spans more than a factor of 13, covering conditions from lightly damaged to severely damaged. In contrast, the FFDM predicts Δ_3h = 0.456–0.460 m for k_m = 0.45 (Figure 14). The FFDM-predicted displacement of Δ_3h = 0.456–0.460 m is consistent with the observed displacement of 0.47 m at Site 1, where part of the facing column collapsed.

For Site 3, the FFDM predicts seismic displacements of Δ_3h = 0.042–0.048 m under k_m = 0.45 (Figure 15). This displacement range is consistent with the misalignment observed at the top of the facing column during the post-earthquake investigation, as shown in Figure 5. When applying Newmark’s chart to Site 3, the parameter X = 0.361/0.45 = 0.802 falls beyond the upper limit of the charted range, even when the lowest possible soil strength is used. This illustrates the difficulty of applying Newmark’s chart to reinforced soil structures such as GRS-MBs, which typically exhibit higher k_hc values than conventional soil slopes.

4. Discussion

(1)

The distinctive responses of the two GRS-MB walls (Sites 1 and 3), despite their similar geological and geotechnical conditions, are attributed primarily to the presence of the gravity concrete wall behind the reinforced zone (Figure 4). The concrete gravity wall impeded the development of the slip surface and strongly influenced the critical failure mechanism in both force-based F_s analyses and displacement-based FFDM analyses. Secondary factors include the smaller vertical reinforcement spacing (S_v = 0.6 m) used at Site 3 and its smaller wall height (2.6 m) compared with Site 1 (3.2 m).

(2)

For a unified interpretation of the seismic resistance of the two sites, the FFDM response curves, based on the soil model integrating pre- and post-peak strengths, provide a consistent explanation. These curves indicate Δ_3h = 0.456–0.460 m at k_h = 0.45 for Site 1 (Figure 14), and a much smaller Δ_3h ≈ 0.042–0.048 m for Site 3 (Figure 15), consistent with the observed performance.

(3)

The use of high soil strength with a hyperbolic model may underestimate seismic displacement. This is evident for Sites 1 and 3, as shown in Figure 14 and Figure 15, where Δ_3h values on the order of 10⁻³ m are predicted under k_h = 0.45. These results arise from the inherent approximation of the hyperbolic model, which does not capture soil behavior as accurately as the model incorporating both pre- and post-peak strengths. The discrepancy, therefore, reflects limitations of the hyperbolic curve itself rather than uncertainty in the selected soil parameters.

(4)

For the two GRS-MB walls examined, the FFDM does not require a pre-determined critical acceleration (k_hc). Instead, seismic displacements are computed directly through a displacement-driven formulation that enforces compatibility and equilibrium at each iteration. In these case histories, this feature allowed the FFDM to capture deformation patterns that were not represented in conventional approaches, such as Newmark’s method, which relies on a pre-determined critical acceleration and does not explicitly model the evolution of soil deformation. The constitutive model adopted in the FFDM enabled a more detailed representation of pre- and post-peak soil behavior than was possible with the operational strength parameters used in limit-equilibrium-based methods. These observations suggest potential advantages of the FFDM for reinforced soil structures similar to those studied here.

(5)

For the GRS-MB walls analyzed in this study, the critical seismic coefficient k_hc, derived from limit–equilibrium analyses, was relatively high, making direct application of Newmark’s chart difficult. For example, at Site 3, k_hc remained as high as 0.361 even under the lowest plausible soil strength. Given the recorded HPGA of 0.45 g at station TCU052, the resulting k_hc/k_m ratio exceeded the upper limit of Newmark’s chart. This illustrates a practical challenge when applying Newmark’s method to reinforced soil structures with relatively high stability margins, at least for the two cases examined here. In these specific situations, the FFDM provided an alternative means of estimating seismic displacement by explicitly incorporating interaction effects and pre- and post-peak soil behavior.

(6)

In these case histories, Newmark’s chart provided only order-of-magnitude displacement estimates and did not resolve the small but engineering-significant movements (several 10⁻³ to 10⁻² m) observed in the field. In contrast, the FFDM is capable of generating displacements in the 10⁻³ to 10⁻² m range in response to small to intense seismic inertial forces. For the wall systems examined here, this capability was essential for interpreting performance where millimeter- to centimeter-scale movements governed serviceability.

(7)

For the two GRS-MB walls studied, the FFDM served as a useful complement to Newmark’s sliding-block method. The two approaches differ fundamentally in how they treat soil behavior and seismic loading: Newmark’s method relies on operational strength parameters and double integration of ground accelerations, whereas the FFDM incorporates additional deformation-related soil parameters to represent progressive strength and stiffness changes. In these case histories, the simplified inertial loading used in the FFDM, combined with its mechanistic soil modeling, provided displacement estimates that aligned with observed performance. Nevertheless, uncertainties in deformation-related soil parameters and variability in ground-motion characteristics remain important considerations, underscoring the need for continued calibration of the FFDM against additional well-documented case histories before broader generalization.

5. Conclusions

The contrasting seismic responses of the two geologically and geotechnically similar GRS-MB walls (Sites 1 and 3) during the 1999 Chi-Chi earthquake can be mechanistically explained through force-based F_s analysis and displacement-based FFDM. Although the walls shared comparable backfill soils and reinforcement materials, the FFDM revealed the primary factor governing their divergent behaviors: the presence of a concrete gravity retaining wall behind the GRS-MB at Site 3, which restricted the development of the critical slip surface and fundamentally altered the failure mechanism. Secondary influences included the smaller reinforcement spacing at Site 3, which enhanced internal stability, and its slightly smaller wall height. These findings illustrate how site-specific structural constraints can dominate seismic performance even when soil and reinforcement conditions are nominally similar.

Using these site-specific conditions, the FFDM successfully reproduced the observed seismic displacements when the high in situ soil strength was combined with a post-peak strength model. In contrast, lower-bound and upper-bound hyperbolic soil models tended to overestimate or underestimate the observed displacements. This highlights the importance of incorporating post-peak soil behavior and mechanistic soil parameters (c, φ, K, n, and R_f) when modeling reinforced soil structures under strong shaking, and underscores the need for a more comprehensive database of deformation-related soil parameters—particularly those governing peak and post-peak response.

For these two case histories, Newmark’s displacement method produced a wide range of predicted displacements due to its reliance on operational soil strength parameters and its sensitivity to ground-motion variability. These characteristics limited its ability to resolve the small but engineering-significant movements observed in the field. For the GRS-MB walls examined, the FFDM provided a more consistent interpretation of seismic displacement by combining simplified inertial loading with deformation-related soil parameters. While these findings demonstrate the usefulness of the FFDM for the specific wall systems studied, broader validation across additional case histories is needed to fully establish its general applicability.

As demonstrated in both the main analysis and the sensitivity study, the FFDM remains applicable at strong shaking levels (e.g., k_h ≥ 0.3) that would cause conventional limit-equilibrium calculations to yield F_s < 1.0—a non-existent equilibrium state implying mobilized shear resistance greater than the soil’s ultimate strength. This capability is fundamental to the FFDM’s usefulness for displacement-based seismic assessment. The sensitivity investigation further shows that most database-derived soil, reinforcement, and facing parameters influenced predicted displacements only on the order of a few 10⁻³ m, while the parameters exerting the greatest influence, such as the internal friction angles of soils and the friction angle at the base of facing, were well constrained by site exploration. These results confirm that the main conclusions are robust with respect to reasonable uncertainties in input properties.

For the studied walls—and, potentially, for similar reinforced soil systems—the FFDM offers a practical balance between simplified inertial loading and deformation-related soil behavior, enabling displacement estimates that align with observed field performance across a wide range of shaking intensities. Because the FFDM produces displacement-based seismic resistance curves during the design phase, it may serve as a useful component of seismic resilience planning for reinforced soil structures, particularly those resembling the systems analyzed here. Despite these advantages, several limitations remain. In the short history of reinforced-soil seismic case studies, the lack of a comprehensive database for deformation-related soil parameters, the need for engineering judgment when selecting input soil properties, and the reliance on prescribed failure–surface patterns all constrain current modeling accuracy. These limitations also indicate that the present findings should be viewed as case-specific until additional well-documented seismic case histories become available.

Looking forward, automated back-analysis systems, AI-assisted estimation of soil parameters, AI-based seismic displacement prediction frameworks, and self-updating slope displacement monitoring systems functioning as digital twin models represent promising directions for improving the reliability, adaptability, and resilience of displacement-based seismic design.