Keywords

1 Introduction

With major advances in computer science and technology, numerical modeling has made great strides in the field of earthquake geotechnical engineering. More than three decades ago, under the project called VELACS (Arulanandan & Scott, 1993, 1994), various institutes conducted centrifuge experiments to simulate ground response under liquefaction, and consistency with experimental results was verified for several numerical methods. At present, numerical simulation is a common practice to validate a design under static and dynamic loading conditions. To name a few, numerical models employed in practice are FLIP (Iai et al., 1992; Iai et al., 2013), LIQCA (Oka et al., 1999), PM4Sand (Boulanger & Ziotopoulou, 2015), and GEOASIA (Asaoka & Noda, 2007). Recently, particle methods, such as DEM (Cundall & Strack, 1979), SPH (Lucy, 1977; Monaghan, 1992), and MPS (Koshizuka et al., 1996), have been evolved and increasing users along with the advances in computing. However, they may not be advanced enough to be used in practice.

LEAP (Liquefaction Experiments and Analysis Projects) is an international collaboration project aiming at validating both experimental and analytical methods to study liquefaction-related phenomena (Manzari et al., 2015). So far, three exercises under the LEAP have been developed. All dealt with the dynamic response on saturated slopes. In LEAP-GWU-2015 (Manzari et al., 2018), six international institutions set up the model to be identical in density of the model ground and input motion in prototype scale and conducted centrifuge tests. The model ground consisted of a uniform sand layer of Ottawa F-65 sand, with a depth of 4 m in the middle, a length of 20 m, and a slope of 5 degrees (Fig. 1.1). As reported in Kutter et al. (2018), the strong correlation in the results prompted the development of a large-scale database for centrifuge modeling.

Fig. 1.1
Two schematic diagrams of the ground model construction at different positions after shaking. a. Parallel sensor position. b. Perpendicular sensor position. They have a uniform sand layer with a depth of 4 meters, a length of 20 meters, and a slope of 5 degrees.

Dimensions of the model and sensor position when shaken (a) parallel and (b) perpendicular to the centrifuge rotation axis. (After Kutter et al., 2018)

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Then the LEAP-UCD-2017 had been initiated and produced a sufficient number of tests and obtained the mean response of a specific sloping sand deposit (Kutter et al., 2020a). In their exercise, in 9 different international centrifuge facilities, 24 centrifuge experiments were conducted with various densities of the ground and peak ground acceleration (PGA) while keeping the same geometry as in the previous exercise. As the results, clear and consistent trends were found among the density of the ground, the input PGA, and the residual surface ground displacements.

Following the successful exercises, LEAP-ASIA-2019 set two objectives: (1) to validate the “generalized scaling law” (Iai et al., 2005) and (2) to increase the amount of experimental data to supplement the existing database developed by the “LEAP-UCD-2017.” Therefore, the model ground in prototype scale was intended to have the same geometries to that used in UCD-2017 (Fig. 1.1). With a new data set from 24 centrifuge tests in total (Table 1.1) from 10 international institutes, the previously observed results are expected to be confirmed and extended.

Table 1.1 Scaling factors employed in each institute
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In LEAP-ASIA-2019, as described later in detail, each institute conducted at least two tests: one applying conventional scaling laws (Model A) and the other applying GSL (Model B).

2 Generalized Scaling Law

With the development of computer technology, numerical analysis has come to be commonly used in the design of civil structures. Although numerical analysis can be used to analyze even hypothetically large structures, the results must be verified. Therefore, there is a growing need to verify the validity of numerical analysis through model experiments. Iai et al. (2005) applied a two-step similitude (Table 1.2) that combines a 1-g scaling law (Iai, 1989) and a centrifugal scaling law and found that it is possible to reproduce the behavior of a large prototype with a small centrifugal model, which was previously unfeasible. They name it the “generalized scaling law (GSL)” in dynamic centrifuge modeling.

Table 1.2 List of the scaling factors in physical model testing (Iai et al., 2005)
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In a physical model test, the scaling factor is given in general form by selecting the basic physical properties as independent and deriving the scaling factors for the other properties by the governing equations of the analytical system. In the GSL, the “two steps” scaling law, a centrifuge model is considered as a scaled model of a 1-g model test. Figure 1.2 illustrates the concept of the generalized scaling law (Fig. 1.2b), where the prototype is scaled down via the scaling law of the virtual 1-g model test with a scale factor of η > 1. In the GSL, as shown in column (3) in Table 1.2, the geometrical scaling factor (μ) in the 1-g test and the scaling factor (η) in the centrifuge test are multiplied to obtain a scaling factor for GSL λ = μη.

Fig. 1.2
Three schematic concepts of the generalized scaling law. a. Conventional scaling law. b. Generalized scaling law for mu greater than 1. c. Generalized scaling law for mu less than 1.

Concept of the generalized scaling law: (a) conventional centrifuge scaling law (Model A) and generalized scaling law for (b) μ > 1 and (c) μ < 1 (Model B)

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Now, for both CU (Cambridge University) and RPI (Rensselaer Polytechnic Institute), the geometrical scaling factor for 1-g model was selected to be less than 1 (μ < 1), in which a size of the prototype is virtually increased as shown in Fig. 1.2c. Scaling up the 1-g model larger than the prototype and down to the centrifuge model is theoretically admissible. However, although it is theoretically feasible, its practicality is unknown. If verified, this could stretch the boundaries of centrifuge experiments.

To validate the GSL, physical parameters in prototype scale of Model A converted by the conventional centrifuge scaling law [e.g., μ = 1, η = 50] are compared with those of Model B obtained by multiplying the generalized scaling factor [e.g., μ = 2, η = 25].

3 Initial Conditions of the Model

As mentioned earlier, LEAP-ASIA-2019 required institutions to develop two minimum model tests. One is Model A, which is a model constructed using the conventional centrifuge scaling laws (Fig. 1.2a). The other, Model B, is the same prototype as Model A, but scaled by applying the GSL (Fig. 1.2b). Model A is built by the same procedures taken in LEAP-UCD-2017. Depending on the direction of excitation relative to the centrifuge rotation axis, a model was constructed, as shown in Fig. 1.1. Model B is built so that the viscosity of the pore fluid is the same as for Model A in the prototype scale, and the input PGA is adjusted when shaken. The two models are outlined below.

  • Model A: Same model as LEAP-UCD-2017 to fill gaps in the existing database.

  • Model B: A model for validation of the GSL. The same geometry as Model A with adjusted viscosity and input PGA in prototype scale.

Although the target input acceleration was a tapered sine wave of 1 Hz, it is important to note that the measured motion is contaminated with a variety of high-frequency components originating from characteristics of shaking table in each institute. Therefore, in order to standardize the PGA for each test and considering the relatively small effect of the high-frequency component on the model behavior, the concept of effective PGA “PGAeff” was used in this project (as a first approximation). Equation 1.1 defines PGAeff (Kutter et al., 2020b). Here, “PGA1Hz” denotes the filtered 1 Hz component of PGA, while “PGAhf” represents the contaminated higher-frequency components.

$$ {\mathrm{PGA}}_{\mathrm{eff}}={\mathrm{PGA}}_{1\mathrm{Hz}}+0.5\times {\mathrm{PGA}}_{\mathrm{hf}} $$
(1.1)

As an example, Fig. 1.3 depicts the filtered 1 Hz wave, filtered high-frequency wave, and a wave of the measured base acceleration of the model KyU_A_A1_1. For each test, different destructive and non-destructive motions were applied to the model; however, this paper reports the results of only the first destruction motion. Responses to other input motions can be described in the associated paper.

Fig. 1.3
Three graphs of acceleration versus time. a. Isolated 1-hertz signal. The highest wave is observed at negative 0.191. b. Isolated high frequency. The highest point is observed at 0.114. c. Achieved base motion. The highest point is observed at 0.304.

Decomposition of the input acceleration to derive PGAeff: (top) filtered 1 Hz motion, (middle) filtered higher frequency, and (bottom) recorded base motion (KyU-A1). (Tobita et al., 2022)

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Figure 1.4 compares the initial conditions addressed in the LEAP-UCD-2017 and LEAP-ASIA-2019 series, namely, the PGAeff and relative density ranges. As mentioned earlier, one of the goals of LEAP-ASIA-2019 was to supply and add data to the database developed by the LEAP-UCD-2017 and to identify trends in Dr-PGAeff combinations. In Fig. 1.4, it can be seen that the PGAeff varies from 0.1 g to 0.4 g, and the relative density Dr_qc(2.0 m) varies from 45 to 85%. Their values assigned to each laboratory can be found in Tables 1.3a and 1.3b, where Dr_qc(2.0 m) is the relative density obtained through the correlation with tip resistance of CPT at a depth of 2 m. Note that in LEAP-UCD-2017, the CPT test results have been shown to be reliable (compared to estimates of mass and volume measurements) in estimating ground uniformity and the associated dry density, although they are indirect measurements (Vargas, 2020). Table 1.3c summarizes the surface lateral displacements for each of the tests for the first destructive motion. Detailed discussion will be made in Sect. 1.4.4.

Fig. 1.4
A scatterplot of P G A versus D R depicts the summary of tests developed for leap-U C D-17 and leap-Asia-19. The two plots are indicated by different shaded shapes. It is observed that the P G A subscript E F F varies from 0.1 grams to 0.4 grams and relative density D R varies from 45 to 85%.

Dr_qc(2.0 m) and PGAeff ranges covered by LEAP-UCD-2017 and LEAP-ASIA-2019. (Tobita et al., 2022)

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Table 1.3a Summary table of density measures for each of the tests (Tobita et al., 2022)
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Table 1.3b Summary table of input motions for each of the tests for the first destructive motion (Tobita et al., 2022)
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Table 1.3c Summary table of surface lateral displacements for each of the tests for the first destructive motion (Tobita et al., 2022)
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Kutter et al. (2018) found that tip resistance at medium depth (i.e., 2.0 m) showed good correlation with the relative density of the ground. Hence, based on LEAP-UCD-2017, the parameter qc (2.0 m) was used in the linear correlation with the dry density (Carey et al., 2020). As noted by Bolton et al. (1999) and Kutter et al. (2020b), values are influenced by container width “w” and CPT rod diameter (prototype scale “Dc”). Here, three correlations depending on the container size are derived: narrower containers (w/Dc = 20–25), deeper models (z/Dc = 11.0–14.5), and shallow models (z/Dc = 6.7–8.3). Vargas (2020) updated the correlations to include the results obtained in LEAP-ASIA-2019 and found that the power-type correlations fit well with the results achieved (see Eq. 1.2). Figure 1.5 shows the updated correlations for the three different test conditions.

Fig. 1.5
Three scatterplots of Q subscript C versus P subscript D. a. Correlation of the two plots, narrow models. b. Deep models, Z over D C = 11.0 to 14.5. c. Shallow models, Z over D C = 6.7 to 8.3. They have nine models each.

Relationship between the dry density ρd and the CPT tip resistance qc at 2.0 m with data from LEAP-UCD-2017 and LEAP-ASIA-2019: (a) narrow models (w/Dc = 20–25), (b) deep models (z/Dc = 11.0–14.5), and (c) shallow models (z/Dc = 6.7–8.3)

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In this paper, the relative density “Dr_qc(2.0 m)” was derived by the updated correlation equation, Eq. (1.2), by setting ρd _ max = 1757 (kg/cm3) and ρd _ max = 1492 (kg/cm3) (Carey et al., 2020).

$$ {\rho}_d=a{\left({q}_c(2.0m)\right)}^b\kern0.875em \left(\mathrm{kg}/{\mathrm{cm}}^3\right) $$
(1.2)

Figures 1.6, 1.7 and 1.8, respectively, show the initial conditions, Dr_qc(2.0 m), PGAeff, and viscosity, achieved in each institute. Figure 1.6 shows that the difference in relative density between Model A and Model B is less than 5%. From Figs. 1.6 and 1.7, it is noticed that, for CU and IFSTTAR, severe and difficult testing conditions were assigned with much low densities and high input PGA.

Fig. 1.6
A graph depicts the D R for models A, conventional and B, G S L. Model A is denoted by circles and B by triangles. The range from low to high for D R is labeled above the graph. It is observed that the difference in relative density between models A and B is less than 5%.

Relative density, Dr_qc(2.0 m), computed by the updated correlation equation based on the CPT measurements. Sorted by the value of Dr_qc(2.0 m): lower (left) to higher (right). (Tobita et al., 2022)

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Fig. 1.7
A graph depicts the effective P G A subscript E F F for the first destructive motion. The models A and B are indicated by circles and triangles. It is observed that for C U and I F S T T A R, the conditions provided with low density and high input P G A.

Effective PGAeff of the first destructive motion. Sorted by the order of Dr_qc(2.0 m) (Fig. 1.6). (Tobita et al., 2022)

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Fig. 1.8
A graph depicts the achieved viscosity for models A and B denoted by circles and triangles. In model A, I F S T T A R has a high rate of 62 and in model B, K Y U A 2 has a high rate of 39. It is observed that model A has the maximum range.

Achieved viscosity of pore fluid. Sorted by the same order of Dr_qc(2.0 m) (Fig. 1.6)

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For the validation of the GSL, the diffusion process of excess pore pressure should also match in the prototype scale. Therefore, the pore fluid viscosity must be properly scaled. Methylcellulose solutions are common in centrifuge modeling and were also used in this study (Adamidis & Madabhushi, 2015; Stewart et al., 1998). Each institute was asked to carefully measure the viscosity of the pore fluid because it is known to be sensitive to fluid temperature. Some laboratories used cup-and-bob viscometers for the measurement, while others used capillary or oscillating viscometers. Figure 1.9 compares the achieved viscosities of the pore fluid of Model A and Model B. Both are judged to be in good agreement.

Fig. 1.9
A graph of model B versus model A compares the range of achieved viscosity of the pore fluid. It compares both values. It is observed that both have an enhanced agreement.

Comparison of the achieved viscosity for Models A and B

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4 Test Results

4.1 Penetration Resistance

To measure the stiffness and strength of the ground, the small CPT developed for LEAP-UCD-2017 (Kutter et al., 2020a) was used in most tests. In the penetration resistance profiles shown in Figs. 1.10a and 1.10b, CPT1 is the CPT tip resistance measured before the first shaking, CPT2 was measured after the first excitation (and before the second), and CPT3, if plotted, corresponds to before the third excitation. Tip resistances of IFSTTAR, UCD, and ZJU slightly increase after shaking. In other institute, the increments are insignificant. The profiles of CU, IFSTTAR, NCU, and ZJU whose Dr_qc(2.0 m) is less than approx. 60% show, as expected, low resistance.

Fig. 1.10a
Eleven graphs depict the C P T profiles for model A. The value of P subscript D 0 changes for all the profiles. It includes C U 2, Ehime 2, I F S T T A R-A-A 1-1, Kaist-A-A 1-1, K Y U-A-A 1-1, K Y U-A-A 2-1, N C U-A, U C D-A-A 1-1, U C D-A-A 2-1, Z J U-A-A 1-1. It includes a plot for effective cone depth versus Q C for C P T 1, 2, and 3.

All CPT profile for Model A. (Tobita et al., 2022)

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Fig. 1.10b
Ten graphs depict the C P T profiles for model B. The value of P subscript D 0 changes for all the profiles. It includes Ehime-A-B 1-1, I F S T T A R-A-B 1-1, Kaist-A-B 1-1, K Y U-A-B 1-1, K Y U-A-B 1-2, K Y U-A-B 2-1, N C U-A-B 1-1, U C D-A-B 1-1, Z J U-A-B 1-1. It includes a plot for effective cone depth versus Q C for C P T 1, 2, and 3.

All CPT profile for Model B. (Tobita et al., 2022)

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CPT tip resistance increases with depth in the range of 5–15 MPa in this study. Figure 1.11 compares qc value. The qc values of Models A and B show good agreements at shallow depths, and the divergence becomes more pronounced with depth. Figure 1.12 shows that Model B tends to slightly overestimate the tip resistance at all depths.

Fig. 1.11
A set of four graphs depict the comparison of Q subscript C value for models A and B. a. Q C at G L-1.5 meters. b. Q C at G L = 2.0 meters. c. Q C at G L = 2.5 meters. d. Q C at G L = 3.0 meters. The C P T tip resistance increases with depth in the range of 5 to 6 megapascals.

CPT tip resistance measured at the specified depth. Sorted by the same order of Dr_qc(2.0 m) (Fig. 1.6). (Tobita et al., 2022)

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Fig. 1.12
A set of four graphs of model B versus model A at different Q C values. a. Q C = 1.5 megapascals. b. Q C = 2. c. Q C = 2.5. d. Q C = 3. It is observed that model B tends to slightly overestimate the tip resistance at all depths.

Comparison of CPT tip resistance measured at the specified depth. (Tobita et al., 2022)

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4.2 Response of Acceleration

Figures 1.13a (Model A) and 1.13b (Model B) plot all the acceleration time histories for the center column (AH1–AH4) and the average of the two lower time histories (AH11 and AH12). Before conducting the Model B test, calibration of the shaker was made in each institute to have the identical input motion with Model A in prototype scale. Figures 1.14a and 1.14b show better agreements on the PGAeff (Fig. 1.14b) than the measured PGA (Fig. 1.14a) between Models A and B. This indicates that the identical input motions in terms of PGAeff were input in both models for each institute.

Fig. 1.13a
A set of graphs depicts the acceleration rate for model A. It includes time histories for the center column A H 1 to A H 4 and the average of two lower time histories A H 11 to A H 12. All plots have similar trends, approximately.

All records of measured acceleration for Model A. (Tobita et al., 2022)

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Fig. 1.13b
A set of graphs depicts the acceleration rate for model B. It includes time histories for the center column A H 1 to A H 4 and the average of two lower time histories A H 11 to A H 12. All plots have similar trends approximately.

All records of measured acceleration for Model B. (Tobita et al., 2022)

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Fig. 1.14
2 graphs of the comparison of models B and A for the first destructive motion. a. P G A measured. The value of R square = 0.8206. b. Estimated P G A underscore E F F. The value of R square = 0.9189. It is observed that a better agreement on P G A-E F F than measured P G A between the two models.

Consistency of the input PGA for Models A and B: (a) measured PGA and (b) processed PGAeff. (Tobita et al., 2022)

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Response acceleration varies with the input seismic motion and soil density. In the records at shallow depth, dilatancy spikes can be seen, which might be caused by the gradual ground movement in the downstream direction. KAIST_A_A1_1 and KAIST_A_B1_1, in which higher Dr_qc(2.0 m) and larger PGAeff were given, show good agreements between two models. Also, NCU and ZJU show good agreements.

Time histories of acceleration response of Models A and B depicted in Fig. 1.15a for higher target density and higher PGAeff for KyU experiment (Tables 1.3a and 1.3b) show dilatancy spikes in common in the record shallower than 2.5 m (AH2), and their wave shapes at the same depth are quite similar. On the other hand, in time histories of acceleration response depicted in Fig. 1.15b for lower target density and lower PGAeff for KyU experiment, dilatancy spikes appear only on the record of AH4 in Model B. As shown in the figure, the larger input acceleration amplitude of Model B may have caused this difference. It should be noted that due to the characteristics of the shaking table, a given acceleration level may not be achieved in the designated centrifugal field.

Fig. 1.15
A set of graphs depicts the acceleration rate versus time. a. Higher P G A-E F F and D R-Q C 2.0 meters series. It compares K Y U-A-A 1-1 in model A and K Y U-A B 1-1 in model B. b. Lower P G A-E F F and D R Q C 2.0 meters series. It compares K Y U-A-A 2-1 in A and K Y U-A-B 2-1 in B.

Comparison of response acceleration for Models A and B for KyU experiments: (a) higher and (b) lower PGAeff and Dr_qc(2.0 m) series

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As mentioned in Sect. 1.2, 1-g scaling factor employed in RPI experiment is μ = 0.5 < 1, in which a virtual 1-g model is twice as large as a prototype. As shown in Fig. 1.16, response accelerations of Models A and B have some similarities except for the record of AH1. Response acceleration of AH1 of Model B shows dilatancy spikes which is not clearly appearing in the record of Model A.

Fig. 1.16
Fifteen graphs depict the acceleration rate versus time. They compare R P I-A-A 1-1 in model A and R P I-A-B 1-1 in model B. It is observed that the plots for both models have similarities except for the record of A H I.

Measured acceleration records of RPI experiment which employed the 1-g scaling factor μ = 0.5 < 1 for Model B

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4.3 Response of Excess Pore Water Pressure

Figure 1.17a, b plots all records of the excess pore water pressures. In each plot, the first half is representing the excess pore pressure buildup during excitation period, and the second half is those during dissipation period. Figure 1.17a shows that the excess pore water pressure in the downstream lower corner may not have reached the initial effective stress. Both Figs. 1.17a and 1.17b show that the records IFSTTAR_A_A1_1, KAIST_A1_1, KYU_A1_1, and ZJU_A1_1 have dilatancy spikes in the negative direction. For medium-density soils with moderate PGAeff, dilatancy spikes tend to be smaller as in KyU and UCD. For loose ground, records of CU_A_B1_1 show small spikes.

Fig. 1.17a
A set of graphs depicts the measurements of excess pore water pressure for model A. It is observed that the excess pore water pressure in the downstream lower corner has not reached the initial effective stress and displays a negative direction.

All records of excess pore water pressure measurements for Model A. (Tobita et al., 2022)

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Fig. 1.17b
A set of graphs depicts the measurements of excess pore water pressure for model B. The records for I F S T T A R-A-A 1-1, Kaist-A 1-1, K Y U-A 1-1, and Z J U-A 1-1 exhibit dilatancy spikes in the negative direction.

All records of excess pore water pressure measurements for Model B. (Tobita et al., 2022)

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Figure 1.18 shows results of KyU experiments and compares the response of excess pore water pressure. Figure 1.18a is the case with relatively higher PGAeff and Dr_qc(2.0 m) series, while Fig. 1.18b is for lower case. As in Fig. 1.18a, both in Models A and B, the curves of buildup and dissipation phase show a relatively good match. However, as shown in Fig. 1.18b, no negative spikes are seen in the buildup phase. This is, as mentioned earlier, due to the difference in the input acceleration amplitude (Fig. 1.15b).

Fig. 1.18
A set of graphs depicts the comparison between models A and B in the measurement of excess pore water pressure. They compare all K Y U records. a. Higher P G A-E F F and D R Q C 2.0 meters series. A good match is observed. b. Lower P G A-E F F and D R Q C 2.0 meters series. No negative spikes.

Comparison of excess pore water pressure responses of Models A and B in the KyU experiments: (a) higher and (b) lower PGAeff and Dr_qc(2.0 m) series

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Figure 1.19 compares results obtained by RPI experiments which employed the 1-g scaling factor μ = 0.5 < 1. The responses are similar as shown in the figure. This demonstrates the applicability of the GSL even with the 1-g scaling factor less than unity.

Fig. 1.19
A set of graphs depict the comparison between models A and B in the measurement of excess pore water pressure. They compare all R P I records with 1-G scaling factor mu = 0.5 less than 1. The responses are similar in nature.

Measured excess pore water pressure records of RPI experiment which employed the 1-g scaling factor μ = 0.5 < 1 for Model B

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The maximum excess pore pressure ratios of P1 to P4 are compared in Fig. 1.20. The maximum excess pore pressure is almost 1 in most of the tests, except for CU, IFSTTAR, ZJU, and KAIST, which have low density or large PGA. As for the second round of tests developed at Kyoto University, a significant difference was found in the excess pore water pressure ratio (i.e., KyU_A_A2_1 and KyU_A_B2_1), despite having similar Dr. (56% and 58%, respectively) and PGAeff (0.118 and 0.126, respectively) values. At this facility, it has been found that for “small” PGA levels at a “low” gravity level, a significant increase in the high-frequency contents of the input acceleration (i.e., additional high-frequency components are induced in Model B than in Model A) was induced by the shaking table, causing significant differences in the PGA values (0.134 and 0.163, respectively), when similar PGAeff values are achieved. From this, if the acceleration record is contaminated with high-frequency components of relatively large amplitudes, the definition of PGAeff might need to be modified to better represent the demand; however, further research is required to clarify this point. One of the possible causes of this excessive high-frequency amplitude might be due in part to the vertical component induced by the rocking motion at large cyclic amplitude under lower centrifugal accelerations. This is a known restriction of a shaker, and in some institute, its use at low centrifugal acceleration is prohibited.

Fig. 1.20
Four graphs of R U max for models A and B. They compare the maximum excess pressure ratios of P 1 to P 4. The maximum excess pore pressure is almost 1 in most of the tests except for C U, I F S T T A R, Z J U, and Kaist which have low density.

Comparison of the excess pore water pressure ratio for Models A and B. Sorted by the same order with Dr_qc(2.0 m) (Fig. 1.6). (Tobita et al., 2022)

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Figure 1.21 compares the max. Excess pore water pressures measured at different depths. While a consistent relationship is observed, some tests are far from a one-to-one relationship.

Fig. 1.21
Four graphs depict the comparison of model A and B for various depth values. a. R U max P 1. b. P 2. c. P 3. d. P 4. It is observed that there is no relationship between the models.

Comparison of the maximum excess pore pressure ratios for Models A and B. (Tobita et al., 2022)

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4.4 Response of Ground Surface Deformation

The surface displacements of the markers after the first destructive motion are summarized in Table 1.3c, and are plotted in Figs. 1.22a and 1.22b for Models A and B, respectively, with arrows. As expected, soft ground (Dr_qc(2.0 m) < 55%) (CU, ZJU) and large PGAeff (PGAeff = 0.35) (IFSTTAR) resulted in large displacements (about 300–600 mm). Figure 1.23 compares the lateral displacements averaged over all markers in Models A and B. Figure 1.24 shows the displacement in the x-, y-, and z-directions. When the displacement in the x-direction is large (more than 250 mm for Model B), a large discrepancy between the displacement of Model A and Model B is observed, indicating the limit of GSL application. Care should be taken when applying the GSL in such severe cases. Interested reader should refer to Tobita et al. (2022) for discussion on the limitation of the GSL. Displacements in y-direction show small fluctuation. This appears to be a random error that occurred during the model construction and measurement process. On settlement (negative values in the z-direction), agreements for Models A and B can be seen.

Fig. 1.22a
A set of numerous scatterplots depict the surface displacements of the markers for model A. All records display different values. Here C U, Z J U, large P G A-E F F, and I F S T T A R have large displacements for about 300 to 600 millimeters.

Displacement vectors at the location of markers – after the first destructive motion for Model A. (Tobita et al., 2022)

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Fig. 1.22b
A set of numerous scatterplots depict the surface displacements of the markers for model B. All records display different values. Here C U, Z J U large P G A-E F F, and I F S T T A R have large displacements for about 300 to 600 millimeters.

Displacement vectors at the location of markers – after the first destructive motion for Model B. (Tobita et al., 2022)

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Fig. 1.23
A graph depicts the X-displacement average of all markers for all records. They compare models A and B denoted by circles and triangles. It is observed that both models have a better agreement.

Lateral displacements of Models A and B (average of all markers) after the first destructive motion. Sorted by the order of Dr_qc(2.0 m) (Fig. 1.6). (Tobita et al., 2022)

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Fig. 1.24
Three graphs depict the comparison between models A and B under various displacements. a. X-displacement average all markers. b. Y-displacement. c. Z-displacement. When the displacement in the X-direction is large for more than 250 millimeters, a large discrepancy between the displacements of models A and B is observed.

Residual displacements for Models A and B: (a) x-, (b) y-, and (c) z-directions. (Tobita et al., 2022)

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5 Updated Correlation among Dr_qc(2.0 M), PGAeff, and Ux

Kutter et al. (2020b) found that in lateral spreading, residual surface displacement is primarily a function of the intensity of shaking and the relative density of the sand; indeed, based on the LEAP-UCD-2017 results, a good correlation was obtained between these three variables. The three variables for a better agreement are “Ux2” for residual displacement averaged over the two central markers, PGAeff for shaking intensity, and Dr_qc(2.0 m) for density of the ground.

This correlation is based on the curve proposed by Yoshimine et al. (2006) to estimate the maximum shear strain generated and the factors of safety against liquefaction proposed by Idriss and Boulanger (2008). The equation of the regression is shown in Eq. (1.3) (Kutter et al., 2018).

$$ {U}_{x2}={b}_2{\left[{b}_1-\frac{{\left({D}_r\_{q}_c(2.0m)-0.125\right)}^{n_3}+0.05}{1.3 PG{A}_{\mathrm{eff}}}\right]}^{n_1} $$
(1.3)

where b1, b2, n1, and n3 are the regression parameters.

Based on the results of LEAP-UCD-2017 and LEAP-ASIA-2019, the correlation equation is updated to incorporate and build a reliable and large-scale database for centrifuge model. Figure 1.25 shows a surface plot of Eq. (1.3) with the values for a median response b1 = 1.756, b2 = 100, n1 = 4, and n3 = 3.245 derived from the entire LEAP-UCD-2017 and LEAP-ASIA-2019 experimental data (Vargas, 2020). In addition to some tests being out of trend (considered outliers and excluded), a significant improvement in the correlation can be observed with the R2 value increasing from 0.75 (LEAP-UCD-2017 data only) to 0.90 (this study).

Fig. 1.25
A three-dimensional graph depicts the U X 2 versus P G A-E F F and D R%. The correlations are based on 17 tests of leap-U C O-2017,17 tests of leap-Asia-2018, and updated D R-C P T power correlation. The values of R square = 0.90148, S X = 1505.6365, and S Y = 1786.1223.

Updated Dr_qc(2.0 m)-PGAeff-Ux correlations based on 17 tests in LEAP-UCD-2017 and 17 tests in LEAP-ASIA-2019 (results of Model B are included). (Tobita et al., 2022)

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6 Conclusions

Following the LEAP-UCD-2017, centrifuge model tests with a wider range of input conditions were conducted at ten institutions in LEAP-ASIA-2019. In addition to the conventional centrifuge model test (Model A), a model test (Model B) was conducted to validate the generalized scaling law. This was the first multi-institutional attempt to validate a generalized scaling law for saturated sandy slope deposits under a wide range of initial conditions. A detailed discussion is expected to be provided by the associated papers from each institute. A brief overview of test results from each institute is presented here so that the reader can locate the results of interest.

The correlation between tip resistance and initial relative density of the ground at 2.0 m deep was updated by using a power-type correlation, which depends on the size of the container and the diameter of the rod. This is because the achieved value of qc (2.0 m) was found to be strongly influenced by the distance from the boundary and the diameter of the rod.

Based on the results obtained in LEAP-UCD-2017 and LEAP-ASIA-2019, the correlation surface between Dr_qc(2.0 m), PGAeff, and Ux2 was updated to incorporate new findings and build a reliable large-scale centrifuge model database. As a result, the correlation was greatly improved with a fit coefficient R2 of 0.90.

One facility reported that for “small” PGA levels at a “low” gravity level, the high-frequency component of the input acceleration is significantly increased by the shaking table, resulting in large differences in PGA values even when similar PGAeff values are obtained. One of the causes of this excessive high-frequency amplitude may be partially due to the vertical component, which induces oscillatory motion with a large periodic amplitude at low centrifugal acceleration of the shaking.

When the displacements in the x-direction (down slope direction) are larger, say more than 250 mm in Model B, because of low density and high PGA, significant discrepancies between Models A and B were found. This is considered to indicate the limitation of GSL, and caution should be exercised when applying GSL under such severe conditions.