Shake Table Test and Numerical Simulation on Verification for Seismic Stability of Railway Embankments Taking into Account Damage Process

Abstract

This study aims to validate a seismic performance verification method for embankments that accounts for the damage process up to sliding failure. A centrifuge shake table test was conducted to observe the damage process during shaking. Results showed that sliding failure occurred as soon as shear strain at the toe of the slope exceeded the limit value of Damage level 3. Furthermore, the FEM using the GHE-S model combined with the multi-shear spring model successfully reproduced the damage process. These findings confirm the applicability of the proposed analysis method for both safety and reparability assessments of embankments.

1. Introduction

In the seismic design of embankments in the Japanese railway field, the settlement of an embankment calculated using Newmark's sliding block method (Newmark [1]) (hereinafter referred to as “Newmark method”), which assumes arc slip failure, is often used as a verification index for reparability (Railway Technical Research Institute [2, 3]) (hereinafter referred to as “the earth structure design standard” and “the seismic design standard”). This is because the Newmark method can easily calculate some reasonable sliding displacement. In general, it is supposed to verify reparability instead of safety because reparability verification is stricter than safety. Furthermore, many studies have been conducted to improve the accuracy of the Newmark method, for example, as described in Sarma [4].

On the other hand, because the Newmark method assumes rigid-plastic arc slip failure, it does not always accurately reproduce the damage process of embankments up to the point of sliding failure. This makes it difficult to properly evaluate seismic performance. For instance, Fujiwara et al. [5] compared the observed seismic displacement of railway embankments during the 2011 off the Pacific coast of Tohoku Earthquake with the settlement calculated using the Newmark method. As a result, although 13 embankments were actually undamaged, settlements exceeding 200 mm were calculated for five of them, which is generally regarded as the threshold for minor damage in the seismic design standard. They also cited the following factors as contributing to this discrepancy: errors in the setting of geotechnical material properties and input seismic acceleration, the influence of soft supporting ground, and differences in failure modes. In particular, the difference in failure mode is that the Newmark method assumes sliding failure, whereas the actual phenomenon is often limited to damage that does not lead to sliding failure, such as settlement, slope bulging, and cracking. In other words, the safety factor of the extreme balance method based on arc slip failure, which is assumed in the Newmark method, cannot evaluate this damage process leading up to sliding failure.

Based on the above background, we carried out centrifuge shake table tests to clarify the seismic damage process of embankments (e.g., Izawa et al. [6]). As a result, from digital images captured after each shaking, it was clarified that shear strain was concentrated at the toe of the slope and the sliding plane was formed from there as the damage process leading to sliding failure. It was also confirmed that the strain level at which sliding failure occurs can be explained by the damage level specified by the deformation characteristics of the embankment material, as shown in Fig. 1. In addition, a seismic safety performance verification method using shear strain at a toe of a slope was proposed that can take into account the damage process (hereinafter referred to as the “proposed verification method”). However, in the test, sinusoidal waves were input, and these tests results were observed based on image analysis using digital images captured before and after shakings. The damage process during earthquakes is not fully clarified.

Fig. 1　Seismic damage process leading to sliding failure in previous study [6]

In this study, we attempted to observe the damage process of embankments during earthquakes in the centrifugal acceleration field and to confirm the validity of the proposed verification method. In addition, a seismic response value calculation method using Finite Element Method (FEM) analysis was also discussed to estimate shear strain necessary for the safety verification in the proposed verification method. At this time, we combined the multiple shear spring model with the GHE-S model, which is often used in a ground response analysis in the Japanese railway field, in order to keep the calculations from becoming too complicated for practical seismic design. Finally, the results of the test and the FEM analysis were compared to validate the proposed verification method and the applicability of the calculation method using FEM analysis.

2. Centrifuge Shake table test

A centrifuge shake table test was carried out to clarify the damage process of an embankment with steep slope gradient during earthquakes. A beam type centrifuge with a diameter of 5.2 m and a shake table of the Nippon Koei Co. LTD. (Sreng et al. [7]) were used in this study. In particular, the damage process was observed in detail by capturing high-speed, high-resolution digital images of the embankment not only before and after shaking, but also during shaking. This test was carried out at a centrifugal acceleration field of 50 G. Hereafter, all values are expressed in prototype scale unless stated otherwise.

2.1 Outline of the test

Figure 2 shows the schematic view of the model embankment, together with an arrangement of sensors. The slope of the left side embankment was set to be 1:1.0, which is steep enough to induce a sliding failure during shakings, since the purpose of this study is to capture the damage process of embankments leading to sliding failure. The slope of the other side was 1:2.5 so as not to cause a sliding failure. The embankment height was set to 5 m.

Fig. 2　Schematic view of model embankment

The embankment material used was Edosaki sand with a degree of compaction (Dc) of 95%. The physical and mechanical properties of the embankment material are shown in Tables 1 and 2. Figure 3 shows the shear stress-shear strain relationship obtained from a monotonic loading test performed on the embankment material. This test was carried out using a hollow torsional shear testing machine under a confining pressure of 50 kPa. Additionally, a proposed definition of damage levels and limit value are shown. The shear strain at the point of the maximum shear stress (τ_max = 48 kPa), which can be defined as the failure of the embankment material, is 7.5% (hereinafter referred to as “γ_fail”). This value is equivalent to the limit value γ_cr3 of Damage level 3. Damage level 2, associated with γ_cr2, can be also defined as the point where the shear stress begins to increase linearly toward the failure point (γ_fail, τ_max). For the sake of convenience, this is defined as the point at which shear strain reaches approximately half of γ_fail. The limit value γ_cr1, associated with Damage level 1, shows the undamaged region, and is set at the point where shear stress reaches half of τ_max (24 kPa), which corresponds to the elastic region E₅₀. In this scheme, Damage level 1 signifies no damage, Damage level 2 denotes minor damage, Damage level 3 suggests relatively severe damage, and Damage level 4 is equivalent to failure. Since these limit values are defined in terms of strain, they can be considered as valid for both at a centrifugal acceleration field and 1 G field. Furthermore, these limit values were also examined under higher confining pressures of 100 kPa and 150 kPa. In the embankment material used, it was confirmed that the limit values could be set at approximately the same strain level without being affected by confining pressure.

Table 1　Physical properties of Edosaki sand

Soil particle density : Gs	2.68
Mean grain size : D50	0.34 mm
Effective grain size : D10	0.15 mm
Uniformity coefficient : Uc	2.6
Coefficient of curvature : U'c	1.00
Fine content : Fc	4.4%
Optimum water content : wopt	14.6%
Maximum dry density : ρdmax	1.707 g/cm3

Table 2　Physical and Mechanical properties of Edosaki sand (Dc = 95%)

Dry density : ρd	1.622 g/cm3
Degree of compaction : Dc	95%
Compression index : Cc	0.055
Consolidation yield stress : Pc	318.1 kPa
Cohesion : c	9.5 kPa
Internal friction angle : φ	32.5 deg.

Fig. 3　Shear stress-strain relationship of Edosaki sand and proposed damage level

In capturing digital images during shaking at a centrifugal acceleration field of 50 G, it is necessary that a camera possesses the ability to withstand forces equivalent to 50 times its own weight. A high frame rate is also necessary since an elapsed time is 1/50 during the tests. There are several constraints, such as lens selection to ensure an adequate field of view within confined spaces, as well as appropriate lighting arrangements. In addition, when capturing images at high-speed, it is desirable to fix a camera on a shake table to maintain a constant positional relationship between the model embankment and that camera. Hence, as in a previous study (Ibuki et al. [8]), a high-speed camera was fixed to the shake table as shown in Fig. 4 to capture digital images of an embankment during shaking. The camera speed was set to 100 frames per second, allowing for the acquisition of images at intervals of 0.5 seconds in a 50 G centrifugal acceleration field (equivalent to 0.01 second intervals in the model scale). An image analysis technique was applied to these images to calculate displacements and strain distribution of the embankment during shaking, thereby a seismic damage process of the embankment was observed in detail. In the image analysis, the mesh was approximately 250 mm by 250 mm (equivalent to 5 mm by 5 mm in the prototype scale). In addition, it was confirmed that the calculated displacement and the displacement measured with a sensor were generally in agreement.

Fig. 4　Location of high-speed camera

The input acceleration wave was the Spectrum I ground motion as indicated in the seismic design standard. Although the acceleration amplitude was not accurately reproduced, the characteristics of Spectrum I were generally reproduced.

2.2 Test Results

Figure 5 shows time histories of the maximum shear strain at the toe of the slope calculated from the image analysis using digital images captured between 5 and 30 seconds, together with horizontal displacement HD2, vertical displacement VD1 and the input acceleration. The timing of image capture is plotted in time history of the input acceleration. Furthermore, the strain distributions obtained from image analysis at ➀～➃ are shown in Fig. 6. It was confirmed that the horizontal displacement obtained from the image analysis increased rapidly at around 29 seconds, and the model embankment experienced sliding failure as shown in Fig. 7. Additionally, the maximum shear strain at the toe of the slope increased rapidly at around 21 seconds. As illustrated in Fig. 6, the shear strain is concentrated at the toe of the slope during shaking. In particular, after 19.1 seconds, the strain is concentrated near the sliding plane as shown in Fig. 7, which is a so-called potential sliding plane. The red dotted line in Fig. 5 is the limit value of Damage level 3 (7.5%), defined from the deformation characteristics of the embankment material shown in Fig. 3. Similar to the damage process observed in the previous study (Izawa et al. [6]), when the maximum shear strain at the toe of the slope exceeds the limit value of Damage level 3 and reaches Damage level 4, the toe of the slope locally leads to shear failure and progressive failure begins, resulting in overall sliding failure. In addition, Fig. 8 shows that the volumetric strain tends to expand in the surface layer of the embankment and slightly compress in the central part. However, the magnitude of the compressive volumetric strain is only about 1%. Therefore, if compaction is sufficient and uniform, it is unlikely that the volume change will occur in the center of the embankment.

Fig. 5　Time histories of maximum shear strain at the toe of the slope and displacements

Fig. 6　Maximum shear strain distribution obtained from image analysis

Fig. 7　Moment sliding failure occurs

Fig. 8　Volumetric strain distribution

3. Seismic response value evaluation

The above test results indicate that an embankment with a steep slope may lead to sliding failure during earthquakes. Thus, safety verification against sliding failure is required in the seismic performance verification. Since the occurrence of sliding failure could be judged by whether or not shear strain at the toe of a slope reaches Damage level 3 of an embankment material, a seismic response value calculation method is necessary for safety verification that can evaluate shear strain accumulated at the toe of a slope. Therefore, a method based on parameters obtained from seismic design was discussed with a practical method in mind.

3.1 Outline of the numerical simulation

Here, as a practical response value calculation method, the effectiveness of the Finite Element Method (FEM) was verified. In the FEM analysis, a constitutive model was used, which was combined with the GHE-S model (Nogami et. al. [9]), a nonlinear model of a ground that is standard in the seismic design of the Japanese railways, and the multiple shear spring model (Towhata and Ishihara [10]). In this model, a number of shear springs applied to the GHE-S model take into account a region of shear stress - axial differential stress relationship and shear strain - axial differential strain relationship. For example, using two shear springs allows evaluation of shear behavior at θ = 45° and 135°. Furthermore, by using four shear springs, θ = 22.5°, 67.5°, 112.5°, and 157.5° can be evaluated. In this study, the shear behavior was evaluated at 5-degree intervals by setting 18 shear springs. When carrying out dynamic analysis using the GHE-S model, the GHE parameters are generally set by fitting shear stiffness - strain relationship obtained from dynamic shear tests. On the other hand, in this calculation of an embankment, here, it is necessary to properly evaluate the shear behavior of an embankment material up to around Damage level 3 to 4 (about 10% strain) in order to perform safety verification. Therefore, the GHE parameters were set to match the region corresponding to Damage level 3 of the embankment material used, as shown in Fig. 9 and Table 3(b). The historical damping was set by fitting the historical damping-shear strain relationship up to the 1% shear strain level. However, it was confirmed that the historical damping did not significantly affect the analytical results of the embankment. The analysis parameters used in the study are listed in Table 3(a). Here, the reference strain γ_r is calculated from the shear strength τ_f and the initial shear stiffness G₀ as γ_r = τ_f ⁄ G₀. The shear strength parameters were set as well, as shown in Table 3, for the central part of the embankment, while for the slope surface, the cohesion was reduced considering the drying of the embankment material. In addition, joint elements were placed between the surface layer of the embankment and the supporting ground, because the toe of the slope was observed to slide toward the front of the slope during shaking.

Fig. 9　GHE skeleton curve calibrated to monotonic loading test

Table 3　Analytical parameters

(a) physical properties				(b) GHE parameters
	Embankment	Supporting ground		C1(0)	1.0
Unit volume weight : γ	16.9 kN/m3	18.0 kN/m3		C2(0)	0.2
Poisson ratio : ν	0.499	0.499		C1(∞)	0.11
Initial stress stiffness : G0	10,179 kN/m2 *	165,306 kN/m2		C2(∞)	1.0
Cohesion : c	3.0 kPa (Surface) 9.5 kPa (Central part)	-		α	0.986
Internal friction angle : φ	32.5 deg.	-		β	1.38

*When the confining pressure is 1 kPa.

In this calculation, first, an embankment was analyzed under self-weight as an elastic body with uniform shear stiffness. The shear stiffness was set based on confining pressure near the center of the embankment (50 kPa). Secondly, the shear stiffness G₀ was modified to be expressed as in equation (1), which takes into account the dependence of each element on confining pressure σ_m obtained from the self-weight analysis.

  

$$G_0=\mathrm{10,179}{{\sigma }_m}^{0.5}$$(1)

Finally, dynamic analysis was carried out taking over the stress state obtained by nonlinear self-weight analysis using the shear stiffness considering the dependence on confining pressure.

3.2 Numerical simulation results and discussions

Figure 10 shows time histories of the settlement, horizontal displacement, and maximum shear strain at the toe of the slope calculated using FEM. It can be seen that the results reproduce an increasing trend in the test. Since the FE model does not take into account the deformation after shear failure, the behavior after sliding failure (around 29 seconds) is not consistent with the test. However, the maximum shear strain at the toe of the slope reaches Damage level 3 in around 16 seconds. Thus, it can be judged from the analysis result that the embankment might lead to sliding failure. Comparing the timing when the toe of the slope reaches Damage level 3 between the test and the analysis, although the analysis reaches Damage level 3 a little earlier, the timings are generally in agreement. In addition, Fig. 11 shows the distributions of the maximum shear strain after 10, 15, 16, and 20 seconds. As in the test, the damage process was adequately reproduced. Altogether, the shear strain was concentrated at the toe of the slope and the potential sliding plane.

Fig. 10　Time histories obtained from the FEM analysis and the test results

Fig. 11　Calculated distribution of maximum shear strain

In the proposed verification method, safety against sliding failure is evaluated based on whether the shear strain at the toe of the embankment, where deformation is concentrated, exceeds the limit value of Damage level 3 (γ_fail) of the embankment material. The FEM employed in this study not only reproduces the damage process leading up to sliding failure, but also enables precise calculation of time history of shear strain at the toe of the slope. Therefore, the proposed FEM is considered applicable as a seismic response value calculation method for verifying safety against sliding failure. Furthermore, based on the settlement at the embankment crest obtained from the FEM analysis, it is also considered possible to assess the reparability of the embankment. Although the present model does not account for volumetric strain, good agreement between test and analysis is partly attributed to the fact that volumetric strain was negligible in the test. Given that railway embankments are typically subjected to sufficient compaction, the proposed model is also considered applicable for evaluating reparability.

4. Conclusions

This study aimed to validate the proposed performance verification method for embankments that can take into account the seismic damage process up to sliding failure. A centrifuge model test was carried out to observe detailed damage process during shaking, and a seismic response value calculation method was proposed using a finite element method. The following findings were obtained:

1)　A measurement system capable of high-speed, high-resolution imaging during shaking in a centrifuge acceleration field was developed. By conducting deformation analysis using digital images, it became possible to observe the seismic damage process of the embankment during shaking.

2)　When the shear strain at the toe of the slope exceeded the limit value of Damage level 3 (γ_fail), as defined by the shear stress-shear strain relationship of the embankment material, a sliding plane developed, leading to sliding failure. This observation supports the validity of the proposed performance verification method.

3)　To reproduce the test results, a two-dimensional dynamic FEM was proposed by combining the GHE-S model with the multiple shear spring model. By fitting the model parameters to align with the shear stress-strain relationship around Damage level 3 obtained from the laboratory test, it was confirmed that the proposed model can accurately capture the embankment response up to the onset of sliding failure. This confirms the applicability of the proposed calculation method for verifying the seismic safety of embankments against sliding failure.

References

• [1]  Newmark, N. M., “Effects of earthquakes on dams and embankments,” Geotechnique, Vol. 15, No. 2, pp. 139-160, 1965.
• [2]  Railway Technical Research Institute. The Design Standards for Railway Structures and Commentary (Earth Structure), Supervised by Ministry of Land, Infrastructure and Transportation, Maruzen, 2009 (in Japanese).
• [3]  Railway Technical Research Institute, The Design Standards for Railway Structures and Commentary (Seismic Design), Supervised by Ministry of Land, Infrastructure and Transportation, Maruzen, 2012 (in Japanese).
• [4]  Sarma, S. K., “Seismic Stability of Earth Dams and Embankments,” Geotechnique, Vol. 25, No. 4, pp. 743-761, 1975.
• [5]  Fujiwara, T., Nakamura. H., Taniguchi, Y., Takasaki, H. and Kaneda, J., “A study about factors which caused railway embankment damages by circular slip analysis,” Journal of Japanese Society of Civil Engineering A1, Vol. 71, No. 4, pp. I_87-94, 2015 (in Japanese).
• [6]  Izawa, J., Doi, T., Suzuki, A. and Kojima, K., “Seismic Design of Embankments in Consideration of Damage Process during Earthquakes,” Quarterly Report of RTRI, Vol. 63, No. 1, pp. 56-63, 2022.
• [7]  Sreng, S., Okochi, Y., Kobayashi, K., Tanaka, H., Sugiyama, H., Kusaka, T., Miki, H. and Makino, M., “Centrifuge model tests of embankment with a new liquefaction countermeasure by ground improvement considering constraint effect,” 6th International Conference on Earthquake Geotechnical Engineering, 2015.
• [8]  Ibuki, R., Uemura, K., Doi, T., Izawa, J. and Sreng, S., “High speed observation on damage process of embankments during earthquakes in a centrifugal acceleration field,” 10th Physical Modelling in Geotechnics, pp. 912-915, 2022.
• [9]  Nogami, Y., Murono., Y. and Morikawa, H., “Nonlinear hysteresis model taking into account S-shaped hysteresis loop and its standard parameters,” Proceedings of 15th World Conference on Earthquake Engineering, 2012.
• [10]  Towhata, I., Ishihara, K., “Modelling soil behavior under principal stress axes rotation,” 5th International Conference on Numerical Methods in Geomechanics, pp. 523-530, 1985.

Authors

	Ryuichi IBUKI Senior Researcher, Soil Dynamics and Earthquake Engineering Laboratory, Center for Railway Earthquake Engineering Research Research Areas: Seismic Behavior of Earth Structures
	Tatsuya DOI, Ph.D. Senior Researcher, Soil Dynamics and Earthquake Engineering Laboratory, Center for Railway Earthquake Engineering Research Research Areas: Seismic Behavior of Structures and Earth Structures
	Jun IZAWA, Ph.D. Senior Chief Researcher, Head of Soil Dynamics and Earthquake Engineering Laboratory, Center for Railway Earthquake Engineering Research Research Areas: Soil Liquefaction, Seismic Behavior of Surface Ground, Seismic Behavior of Earth Structures
	Kentaro UEMURA, Dr.Eng. Researcher, NIPPON KOEI CO., LTD., Geotechnical Experiment Group, Testing & Experiment Center, Research and Development Center Research Areas: Geotechnical Experiment, Soft Soils and Liquefaction, Seismic Resistance of Retaining Wall, Soil Improvement
	Sokkheang SRENG, Dr.Eng. Chief Specialist, NIPPON KOEI CO., LTD., Geotechnical Experiment Group, Testing & Experiment Center, Research and Development Center Research Areas: Geotechnical Experiment, Shear Strength of Coarse-grain Material, Soft Soils and Liquefaction, Seismic Resistance of Embankments