Keywords

1 Introduction

LEAP (Liquefaction Experiments and Analysis Projects, Manzari et al., 2014) is an international collaborative effort to verify and validate numerical liquefaction models. It serves to evaluate the state of the art for liquefaction model testing using centrifuge shaking tables and numerical prediction and simulation capabilities using advanced constitutive models that have been developed over the past two decades. The Tsinghua University numerical simulation team joined the project during LEAP-UCD-2017 (Wang et al., 2019), using a unified plasticity model for large post-liquefaction shear deformation of sand proposed by Wang et al. (2014). The model was shown to provide a good account of the cyclic mobility, and especially the post-liquefaction shear deformation of sand observed in element tests.

This chapter presents the calibration of the constitutive model and subsequent simulations of centrifuge shaking table tests under the LEAP-ASIA-2019 framework. During LEAP-ASIA-2019, high quality undrained cyclic torsional shear test data was provided to the simulation teams for model calibration. Type-C simulations of centrifuge tests were then conducted. The tests were categorized into Model A tests and Model B tests, which were initially designed to validate the “generalized scaling law” (Iai et al., 2005) for the “model of model” scenario. However, from the simulators’ point of view, rather than focusing on the “generalized scaling law,” which would be difficult to be proved either right or wrong by numerical simulations, the tests at different centrifugal acceleration g levels provide precious data to examine the performance of the constitutive models and numerical simulation methods under drastically different stress states.

2 Constitutive Model for Liquefiable Sand

2.1 Basic Features of the Constitutive Model

The unified plasticity model for large post-liquefaction shear deformation of sand proposed by Wang et al. (2014) is used in the simulations at Tsinghua University for LEAP-ASIA-2019, which is consistent with that for LEAP-UCD-2017. The model has been used to provide a unified description of sand of different conditions from pre- to post-liquefaction under both monotonic and cyclic loading on both element test and boundary value problem levels (e.g., Wang, 2016; Wang et al., 2016, 2017; Chen et al., 2018).

The model functions within the framework of elasto-plasticity, with the basic elastic and plastic constitutive equations following:

$$ d{\varepsilon}_{\mathrm{v}}^{\mathrm{e}}=\frac{dp}{K};\kern1em d{\mathbf{e}}^{\mathrm{e}}=\frac{d\mathbf{s}}{2G} $$
(19.1)
$$ d{\mathbf{e}}^{\mathrm{p}}=\left\langle L\right\rangle \mathbf{m};\kern1em d{\varepsilon}_{\mathrm{v}}^{\mathrm{p}}=\left\langle L\right\rangle D $$
(19.2)

Here, ε is the strain tensor, the volumetric strain is denoted by εv = tr(ε), the deviatoric strain tensor is e = ε − εv/3I. Superscripts e and p represent elastic and plastic, respectively. σ is the effective stress tensor, p = tr(σ)/3 is the mean effective stress, s = σ − pI is the deviatoric stress. K and G are the elastic bulk and shear moduli, respectively, L is the loading index, m is the deviatoric strain flow direction, and D is the dilatancy ratio. 〈 〉 are the Macaulay brackets with 〈x〉 = x for x > 0 and 〈x〉 = 0 for x ≤ 0.

The basic features of the model include the following: (a) dilatancy D is decomposed into a reversible part and an irreversible part following the findings of Shamoto and Zhang (1997) and Zhang (1997), providing explicit control for the dilatancy of sand under cyclic loading; (b) at the liquefaction state, no elastic strains are assumed to be generated while the dilatancy equations are still assumed functional, resulting in the generation of large yet bounded shear strain at the state of zero effective stress; (c) bounding surface plasticity is used to define the plastic modulus, where a maximum stress ratio surface serves as the bounding surface of the model; (d) critical state theory compliance and state dependency is achieved through incorporation of the state parameter (Been & Jefferies, 1985). This chapter only presents the very basic features of the model without explicitly going into its specific formulation, readers should refer to Wang et al. (2014) for the complete formulation of the model.

2.2 Calibration of Model Parameters

The calibration method for the model parameters has been documented by Wang et al. (2014). The four critical state parameters based on Vasko (2015) for Ottawa F65 sand were adopted. The model parameters are based on the results of the previous LEAP-UCD-2017 calibration results. However, adjustments were made to the parameters based on the undrained cyclic torsional shear test results provided in LEAP-ASIA-2019. As cyclic torsional shear tests can generally provide higher quality data related to liquefaction behavior, we placed more emphasize on simulating their results during this phase of LEAP. The model parameters are listed in Table 19.1. Note the parameter γd, r is kept at a default value of 0.05.

Table 19.1 Model parameters for the simulations
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Figure 19.1 compares the simulation results using the constitutive model directly with the corresponding cyclic torsional shear test results for Ottawa F65 sand at relative densities (Dr) of 50% and 60%, under cyclic stress ratio (CSR) of 0.19 and 0.18, respectively. Based on the results presented in Fig. 19.1, the model is able to capture the shear strain that is generated at liquefaction state very well, the decrease in effective stress under undrained cyclic loading and the eventual “butterfly orbit” of the shear stress-mean effective stress plot is also well represented. Generally, the model is able to simulate the liquefaction resistance. However, it is observed that the cyclic resistance ratio (CRR) curve for the constitutive model tends to be steeper than the measured results, especially at very low CSR.

Fig. 19.1
Four graphs. Two graphs of tau in kilonewtons per meter squared versus gamma in percentage for the computed and measured. Two graphs of tau in kilonewtons per meter squared versus p prime in kilonewtons per meter squared for the computed and measured. All graphs have fluctuation trends.

Undrained cyclic torsional shear test results compared with corresponding simulation results: (a) τ – γ for Ottawa F65 sand at Dr = 50% with CSR = 0.19; (b) τ – p′ for Ottawa F65 sand at Dr = 50% with CSR = 0.19; (c) τ – γ for Ottawa F65 sand at Dr = 60% with CSR = 0.18; (d) τ – p′ for Ottawa F65 sand at Dr = 60% with CSR = 0.18

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3 Detailed Specification of Numerical Simulation

OpenSees (McKenna & Fenves, 2001) was used to conduct the numerical simulations of this study, within which the constitutive model was implemented through a sub-stepping cutting plane algorithm (Wang et al., 2014). As the simulations were 2D plane strain and effective stress based, solid-fluid coupled quadUP elements were used (e.g., Yang et al., 2008). The element is a four-noded quadrilateral plane strain element that has two displacement degrees of freedom and one pore pressure degree of freedom at each node, and follows the u-p formulation proposed by Zienkiewicz and Shiomi (1984).

The mesh used in this study is illustrated in Fig. 19.2, with the same configurations as that of the LEAP-UCD-2017 study (Wang et al., 2019), which consists of 1280 elements and 1377 nodes. The few nodes that are not perfectly aligned with the rest in their respective rows were adjusted so that their coordinates matched those of the measurement sensors in the tests. Five centrifuge tests are simulated in this study, including two from KyU, two from RPI, and one from UCD, as listed in Table 19.2. For the current LEAP project, both Model A and Model B tests were conducted, with Model B being “model of models,” where the Model B tests should first be scaled to a “virtual 1G” model with the scaling laws for centrifuge tests, and then further scaled to its prototype through the so called generalized scaling law (Iai et al., 2005). In this study, the simulations are conducted in the 1G scale, which is the prototype for Model A and “virtual 1G” for Model B. The results are then all presented in the prototype scale, with the simulation results for Model B scaled with the “generalized scaling law.” Therefore, the test and simulation results for the prototype scale are compared. Note this scaling scheme does not provide validation for the generalized scaling law, as the transformation for both test and simulation from “virtual 1G” to prototype are exactly the same, and pure mathematical. However, the comparisons between Model A and Model B do provide validation for the ability of the numerical simulation method in providing unified representation of the liquefaction behavior under different stress states and different input motion frequency. For example, the RPI Model A and B tests were conducted at 23 g and 46 g, respectively, and the dominant input motion frequency were 1 and 0.6 Hz, respectively, at 1G (or “virtual 1G”).

Fig. 19.2
A graph of Y versus X plots the configuration of a mesh with 1280 elements and 1377 nodes.

Mesh configuration for the numerical simulations

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Table 19.2 Initial relative densities, centrifuge acceleration, and input motion PGA for tests simulated
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Krylov Newton solution algorithm along with a ProfileSPD approach is used to solve the system of equations (OpenSees manual). Newmark time integration scheme is adopted in the simulations. The boundary conditions are enforced using the penalty method with the penalty number of 1012. For the test of convergence, a norm of the displacement increment test is used with a 10−3 tolerance and maximum number of iterations of 50. Rayleigh damping with very mall values for α and β (both 0.002) are used. The same permeability corresponding to the density of the sand in each test, based on the test results provided in LEAP-UCD-2017. It is assumed that the permeability remains constant during the entire process, which is a simplification.

The input motions for the simulations follow the exact motions provided to us from the centrifuge tests. Prior to the seismic event simulations, each model is subjected to a gravity step to obtain the initial state of the model. The initial pore pressure and the vertical effective stress after spin-up and before shaking in the simulations are mostly similar to that reported by Wang et al. (2019) for LEAP-UCD-2017, and are not repeated here for brevity.

4 Simulation Results

4.1 Typical Simulation Results

Typical simulation results for acceleration, excess pore water pressure, and lateral displacement are presented in this section. Test RPI_A_B1_1 is chosen as a representative case for presenting the results.

Figure 19.3 plots the horizontal acceleration time histories at various depths in test RPI_A_B1_1. The input motion is presented in Fig. 19.3a. At AH1, with depth of 3.5 m, the simulated and test acceleration time histories match perfectly until about 8 s. After 8 s, strong spikes in the negative direction are observed in the test, while numerical simulation does not reflect this feature. The spikes in acceleration are often assumed to be related to the dilatancy and subsequent increase in effective stress of the soil during shaking. However, it is surprising that at 3.5 m depth for AH1, the dilatancy spikes would be so significant in the test, as the shear strain in sand is usually not expected to be so strong at large depths. At shallower depths for AH3 (1.5 m) and AH4 (0.5 m), strong dilatancy spikes in the negative direction are also observed in the tests, at earlier stages of shaking, after about 7 s. The acceleration time history becomes distinctly lopsided, with larger acceleration in the negative direction and smaller acceleration in the positive direction, whereas the input motion is mostly symmetric with respect to zero. The simulation results also captures this behavior. For the results in Fig. 19.3d for AH4, the simulated acceleration time history also shows significant spiking in the negative direction, while the peak acceleration in the positive direction becomes more fuzzy compared to that of the input motion. It should be noted that the measures spikes are generally stronger than those in the simulations.

Fig. 19.3
Four graphs of horizontal acceleration in gravity versus time in seconds. 1. It plots the input. 2. It plots the test and numerical simulation of A H 3. 3. It plots the test and numerical simulation of A H 1. 4. It plots the test and numerical simulation of A H 4. In all graphs, all the curves trend in a fluctuating pattern.

Comparison between test and simulation results for horizontal acceleration at various depths in RPI_A_B1_1: (a) input motion; (b) at AH1 with 3.5 m depth; (c) at AH3 with 1.5 m depth; (d) at AH4 with 0.5 m depth

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Figure 19.4 plots the excess pore water pressure at three different depths. At the base of the model (AH4), the sand does not reach initial liquefaction in both test and simulation, as zero effective stress is not achieved. However, the test results show greater fluctuations in pore pressure, which could be related to the dilatancy spikes observed at large depths in Fig. 19.3 At 2 m depth and above, the excess pore water pressure is able to reach the initial effective stress, suggesting that the sand within this depth reaches liquefaction during shaking. Strong fluctuations in the excess pore pressure are observed at P3 and P4 for both test and simulation. The dissipation process is also well simulated.

Fig. 19.4
Three graphs labeled a to c of excess pore pressure in kilopascals versus time in seconds for test and numerical simulations of P 1, P 3, and P 4. The test and numerical simulations of P 1, P 3, and P 4 trend in a fluctuating pattern at first and trend in a decreasing pattern after 30 seconds.

Excess pore water pressure histories at different depths in RPI_A_B1_1: (a) P1 with 4 m depth; (b) P3 with 2 m depth; (c) P4 with 1 m depth

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The lateral soil surface displacement time history at the center of the model in RPI_A_B1_1 is plotted in Fig. 19.5. It can be seen that general good agreement between simulation and test is achieved for the lateral deformation of the model. Again, stronger fluctuations are observed in the test compared with those in the simulations, especially after shaking terminates. The lateral displacement time histories at other locations also show good agreement between simulation and test, and are not presented for conciseness.

Fig. 19.5
A graph of the displacement of surface mark E 0 in meters versus time in seconds for the test and numerical simulation. The displacement of test trends in a fluctuating pattern. The displacement of the numerical simulation trends in an increasing pattern at first and then remains constant after 20 seconds.

Lateral soil surface displacement at the center of the model in RPI_A_B1_1

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4.2 Overall Comparison of Lateral Displacement Results

The residual lateral displacement in both simulation and test at each of the reported soil surface locations for all five tests are listed in Table 19.3. Based on the results from the RPI Model A and Model B, it can be seen that the simulation results for lateral displacements generally agreed with the test results, indicating that the constitutive model and numerical simulation method is able to provide a unified description of the seismic response of liquefiable sand under drastically different stress states and different input motion frequencies. The simulation results for the UCD test are also comparable with measurement.

Table 19.3 Initial relative densities and centrifuge acceleration for tests simulated
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In contrast, for the KyU tests, the simulation and test results are incomparable. Simulation results for KyU_A_A2 and KyU_A_B2_1 exhibit relatively large displacement, in comparison with the extremely small measured displacement. The small measured displacement is surprising, especially in comparison to UCD-A-A2–1, as although the input PGA is slightly smaller compared to that in UCD-A-A2-1, the soil is also much looser (Table 19.2).

5 Conclusions

This chapter presents the details and results for the simulations at Tsinghua University for the LEAP-ASIA-2019 project. Type-C simulations of five different centrifuge shaking table tests on mildly sloping liquefiable ground.

A unified plasticity model for the large post-liquefaction shear deformation of sand is used in this study. The model is calibrated based on the element test results provided to the simulators in the LEAP-UCD-2017 and LEAP-ASIA-2019 projects. Special emphasis was given to the calibrations against the undrained cyclic torsional shear test results provided in the most recent phase of the LEAP project. We find that cyclic torsional shear tests can provide a better basis for seismic liquefaction related model calibration compared with cyclic triaxial tests.

The details of the numerical simulations were presented in the chapter. Typical results for acceleration, excess pore pressure, and displacement for one of the tests were presented in detail, showing that the numerical simulation is able to capture the liquefaction related behavior in the tests well. Comparisons between the simulation and test residual displacement results for the five different tests showed complicated results, under the same model setup, especially constitutive model parameters. Although the simulations for the two RPI tests under different centrifuge acceleration levels with different stress states and input motion frequencies generally showed adequate agreement with test results, the simulation and test results for the KyU tests were incomparable.