Keywords

1 Introduction

Thanks to the new high-end computer processing units and the robust computational capabilities available, performing sophisticated geotechnical numerical simulations is possible for academic researchers and geotechnical professionals. Therefore, it is possible to model different soil types with complex constitutive stress-strain and stress-dilatancy relationships, which is considered to be the most important issue in studying the soil liquefaction phenomenon.

A verification and calibration process must be done towards the employed constitutive soil model to determine model parameters and check its capability of capturing the soil mechanical behavior in different loading conditions. The soil model parameters should first be estimated and calibrated against soil element tests through laboratory testing devices such as direct simple shear, triaxial, and hollow cylindrical torsional shear. The most common way to calibrate a soil model in any Finite Element Modeling (FEM) software is to model a single one element considering appropriate constraints and loading conditions to simulate a specific laboratory soil element test. Then, compare the computed and measured responses. This comparison checks the model’s ability to evaluate soil mechanical behavior and capture essential soil characteristics in simple loading conditions.

Eventually, this calibrated soil model should be validated against large-scale experiments that represent real geotechnical field problems. When simulating large-scale models, many other aspects are affecting the output simulation results rather than the main model parameters and their constitutive relations. The main target of this simulation type is to predict appropriate deformations, accelerations, and excess pore water pressure time histories in boundary value problems, which cannot be evaluated through the small-scale/element laboratory testing simulations. So that simulating real boundary value problems provide the final judgment on the soil model and the numerical technique adequacy in being used for earthquake geotechnical engineering practice or not. For physical modeling of real field geotechnical cases, centrifuge model testing is selected for fulfilling this purpose since they are confirmed to produce a reliable and accurate representation of geotechnical engineering systems (Madabhushi & Schofield, 1993; Schofield, 1998). Centrifuge model tests can provide high-quality readings and plenty of measurements that can be employed in validating the FEM tools.

The LEAP “Liquefaction and Analysis Projects” (Manzari et al., 2014; Kutter et al., 2015, 2018, 2020; Zhou et al., 2018) is a collaboration between different universities around the world that aims to study different liquefaction scenarios for various soil structures to evaluate and validate numerical simulation techniques. The project comprises experimental model tests that took place at different centrifuge facilities at the participating universities, as well as computational simulations that performed using various numerical methods and a wide range of constitutive models. The main target of this effort is to assess the available numerical simulation tools and constitutive soil models against practical geotechnical field applications in order to accurately predict soil liquefaction potential and associated hazardous effects.

Through the context, a calibration of the employed constitutive soil model through the OpenSees platform (McKenna, 1997, 2011; OpenSees, 2000) is deeply discussed. After that, Class-C simulations are performed after tuning the parameters’ selection of the employed soil model for the sake of getting better simulation results. The computed soil responses are assessed through detailed comparisons with the measured responses, besides highlighting the possibilities and drawbacks of the adopted numerical simulation technique.

2 Demonstration of the Selected Centrifuge Experiments for the FE Analysis and Simulation Types

2.1 Centrifuge Test Experiments

Seven centrifuge experiments were selected for this finite element simulation work from dozens of tests done at Rensselaer Polytechnic Institute (RPI), the University of California at Davis (UCD), Kyoto University (KyU), and Zhejiang University (ZJU). Two types of physical models are considered in these centrifuge experiments through a one-stage downscaling, namely Model A, and a two-stage downscaling, namely Model B, according to the Generalized Scaling Law (GSL) developed by Tobita et al. (2011). In Model A, 1G filed scaling law with a scaling factor μ is used in downscaling the prototype to a virtual model, whereas in Model B, a scaling factor η is applied to the virtual 1G model for downscaling to a new physical model by employing the conventional scaling law. Table 20.1 summarizes the centrifuge experiments with all related scaling factors, soils’ relative densities, and effective Peak Ground Acceleration (PGAeff) of the employed input motions.

Table 20.1 The selected LEAP-ASIA-2019 centrifuge experiments for numerical simulations
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A ramped sinusoidal input motion with a predominant period of 1 sec. is employed in all centrifuge experiments with different Peak Ground Accelerations (PGAs) ranging from 0.12 to 0.28 g. Figure 20.1 demonstrates the variations of input motions and the maximum spectral accelerations through the recorded response spectra for each input motion at each centrifuge facility.

Fig. 20.1
A multiple-line graph plots spectral acceleration in gravity versus period in seconds. 7 lines R P I A A 1 1, R P I A B 1 1, U C D A A 2 1, K Y U A A 2 1, K YU A B 2 1, Z J U A a 1 1 and Z J U a B 1 1 lines fluctuates between 0 to 0.5 along the period and peak at (1, 1.5) and drops to (10, 0). Values are approximated.

Response spectrum of the recorded input motions at different centrifuge facilities for the selected seven experiments with a magnified plot for the KyU recorded base motions

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The achieved base motions at RPI and UCD centrifuge facilities have the lowest wave oscillations at high frequencies, in comparison with other centrifuge facilities, which have no effects on the computed output results and do not cause any resonant with the soil’s natural frequency. In the ZJU centrifuge facility, slightly higher wave oscillations were noticed at frequencies ranging from 3.3 to 5 Hz; however, high wave oscillations arose in conducted centrifuge tests at KyU for the same frequency range, especially with Model B centrifuge experiments. These high wave oscillations can affect the simulation results negatively as well as may biasedly result in overestimating the simulated responses.

2.2 Brief Explanation of Numerical Predictions Classes Adopted in LEAP

The LEAP demands that the simulation teams provide the numerical simulations for the centrifuge experiments according to a specific criteria. The numerical simulation process should follow one of these classes according to the requirements of each LEAP event computational modeling task. Class A, B, and C are the standardized LEAP approaches for computational modeling and numerical predictions.

In Class A predictions, blind numerical simulation predictions are performed before commencing any centrifuge tests. In this type of simulations, the numerical predictors only know the soil’s main characteristics, the schematic of the centrifuge problem, and the target dynamic input motion for each centrifuge test. Calibration of the constitutive model parameters can be done towards previous/ongoing provided soil element tests. After that, these estimated model parameters are used in numerical simulations of centrifuge experiments.

Class B and C predictions are performed after finishing all centrifuge experiments. The recorded base motions form centrifuge experiments are the only information that is provided to the numerical modeler in Class B predictions. In contrast, all centrifuge measured responses are given to numerical modeler in Class C predictions. Class C simulations are meant to tune the utilized constitutive soil models through the centrifuge experiments to produce better and appropriate numerical predictions.

In LEAP-ASIA 2019 event, class-C simulations are required for being performed for a set of centrifuge model tests in order to address different possibilities and limitations of the available numerical tools and constitutive soil models. The following sections provide a comprehensive demonstration of the utilized soil model, numerical tool, FE model configurations and loading conditions. Thereafter, the computed output results are discussed and compared with measurements in order to highlight the main advantages and disadvantages of the employed numerical simulation technique.

3 The Employed Soil Constitutive Model

A multi-yield surface model with pressure dependency, namely PDMY02, was employed in the numerical simulations. The PDMY02 model (Elgamal et al., 2003; Yang et al., 2003) is based on the classical multi-surface plasticity work done by Prevost (1978, 1985) and is further extended to account for complex dilatancy characteristics. The model can capture the soil’s cyclic mobility and post liquefaction behaviors under different stress levels. The PDMY02 model employs a kinematic hardening rule to simulate the soil cyclic hysteretic response (Mróz, 1967; Yang et al., 2003). A concise demonstration of the main characteristics of the soil model is reviewed in the following sections.

3.1 Yield Surface Definition

The failure surface in the PDMY02 model is a Drucker–Prager conical surface in the principal stress space with a pressure dependency (Lacy, 1986; Elgamal et al., 2003; Yang et al., 2003). The inner nested yield surfaces defines the hardening zone and the outermost yield surface represents the failure zone (Prevost, 1978, 1985; Yang et al., 2003). The yield function is formulated based on the multi-yield-surface J2 plasticity model that is proposed by Prevost (1985) and further modified by Elgamal et al. (2003) and Yang et al. (2003) as follows:

$$ f=\frac{3}{2}\left(\tilde{s}-\left({p}^{\prime }+{p}_{\mathrm{res}}^{\prime}\right)\right):\left(\tilde{s}-\left({p}^{\prime }+{p}_{\mathrm{res}}^{\prime}\right)\right)-{M}^2{\left({p}^{\prime }+{p}_{\mathrm{res}}^{\prime}\right)}^2=0 $$
(20.1)

where \( \tilde{s} \) is the deviatoric stress tensor, p′ is the mean effective pressure, and \( {p}_{\mathrm{res}}^{\prime } \) is a small residual pressure value assigned automatically by the model to move the yield surface towards the negative confining stress by \( {p}_{\mathrm{res}}^{\prime } \) when confining pressure equals zero. M is the shear stress ratio that defines the size of the yield surface. The Mf that represents the stress ratio of the outermost yield surface at failure is calculated through the soil friction angle that is obtained from the triaxial test (TX) and defined as \( {M}_{\mathrm{f}}=\frac{6\sin \left({\phi}_{\mathrm{TX}}\right)}{3-\sin \left({\phi}_{\mathrm{TX}}\right)} \) (Chen & Mizuno, 1990).

3.2 Stress-Strain Relationships

The stress-strain “τ − γ” backbone curve is defined by a hyperbolic relationship at a constant confinement pressure \( {p}_{\mathrm{r}}^{\prime } \) as follows:

$$ {\tau}_{\mathrm{oct}}=\frac{G_{\mathrm{oct},\mathrm{r}}\ {\gamma}_{\mathrm{oct}}}{1+\frac{\gamma_{\mathrm{oct}}}{\gamma_{\mathrm{r}}}} $$
(20.2)

where Goct, r is the small strain shear modulus at reference effective confining stress \( {p}_{\mathrm{r}}^{\prime } \) and γr is a calculated shear strain, which is computed internally and satisfies the following equation at failure for a given reference pressure \( {p}_{\mathrm{r}}^{\prime } \):

$$ {\tau}_{{\mathrm{oct}}_{\mathrm{f}}}=\frac{2\sqrt{2}\ \sin \left(\phi \right)}{3-\sin \left(\phi \right)}{p}_{\mathrm{r}}^{\prime }=\frac{G_{\mathrm{oct},\mathrm{r}}{\gamma}_{\mathrm{oct},\max }}{1+{\gamma}_{\mathrm{oct},\max }/{\gamma}_{\mathrm{r}}} $$
(20.3)

where γoct, max is the maximin reached shear strain (a user-defined value) that corresponds to the failure octahedral shear stress \( {\tau}_{{\mathrm{oct}}_{\mathrm{f}}} \). In the multi-yield surface plasticity principle, the hyperbolic presentation of the backbone curve is substituted by multi-linear segments that represent the domain of the evolved yield surfaces. Each segment represents a yield surface domain which is characterized by a tangent shear modulus (\( {G}_m=\frac{2\left({\tau}_{m+1}-{\tau}_m\right)}{\gamma_{m+1}-{\gamma}_m} \)) and size (\( {M}_m=\frac{3\ {\tau}_m}{\sqrt{2}\ \left({p}_{\mathrm{r}}^{\prime }+{p}_{\mathrm{r}\mathrm{es}}^{\prime}\right)} \)) at each m surface(s), as shown in Fig. 20.2.

Fig. 20.2
2 diagrams illustrate a 3 D plane diagram along sigma 1, sigma 2, and sigma 3. A cone shape expanded to form concentric circles along the diagonal p strike. A line graph illustrates tau oct versus gamma oct. A line increases from the origin to form a concave downward increasing trendline. Given G m bar 2.

Demonstration of the multi-yield conical surfaces in principal stress space and the octahedral shear stress-strain backbone curve with 10 yield surfaces

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The shear Goct and bulk B moduli are pressure dependent and being updated during the simulation with regards to the soil effective confinement stress p' according to Prevost (1985) and Elgamal et al. (2003) as follows:

$$ {G}_{oct}={G}_{oct,r}{\left(\frac{p^{\prime }+{p}_{res}^{\prime }}{p_r^{\prime }+{p}_{res}^{\prime }}\right)}^d\ \mathrm{and}\ B={B}_r{\left(\frac{p^{\prime }+{p}_{res}^{\prime }}{p_r^{\prime }+{p}_{res}^{\prime }}\right)}^d $$
(20.4)

where d is a material parameter and equals 0.5 for sandy soils (Kramer, 1996).

3.3 The Model Flow Rule

Computation of the plastic shear strain increment is divided into a deviatoric component that follows an associative flow rule and a volumetric component that follows a non-associative flow rule. Based on Prevost and Elgamal considerations (Prevost, 1985; Elgamal et al., 2003), \( \tilde{Q} \) and \( \tilde{P} \) tensors are normal to the yield surface and the plastic potential surface, respectively. These tensors can be decomposed into two deviatoric and volumetric components, giving \( \tilde{Q}={\tilde{Q}}^{\prime }+{Q}^{{\prime\prime}}\tilde{I} \)and \( \tilde{P}={\tilde{P}}^{\prime }+{P}^{{\prime\prime}}\tilde{I} \). \( {\tilde{Q}}^{\prime } \)and \( {\tilde{P}}^{\prime } \) are the deviatoric parts that are following associative flow rule, whereas Q″ and P″ are the volumetric components that follow a non-associative flow rule.

The PDMY02 model considers a phase transformation surface (PT), which is first proposed by Ishihara (Ishihara et al., 1975). The contractive soil tendency is occurring if stress state is inside the PT surface when soil subjected to undrained shearing, whereas dilation happens when stress state is outside the PT surface. Volumetric plastic strains are computed during contraction and dilation phases according to Yang and Elgamal (2008, 2009) through the following formulae:

Contractive phase: (η < ηPT) or (\( \eta >{\eta}_{\mathrm{PT}}\ \mathrm{and}\ \dot{\eta}<0\Big) \)

$$ {P}^{{\prime\prime} }=-{\left(\ 1-\frac{\tilde{n}:\tilde{s}}{\tilde{\dot{s}}}\frac{\eta }{\eta_{\mathrm{PT}}}\right)}^2\left({c}_1+{\xi}_{\mathrm{c}}{c}_2\right){\left(\frac{p^{\prime }+{p}_{\mathrm{res}}^{\prime }}{p_{\mathrm{atm}}}\right)}^{c_3} $$
(20.5)

Dilative phase: (η > ηPT and \( \dot{\eta}>0\Big) \)

$$ {P}^{{\prime\prime} }={\left(\frac{\eta }{\eta_{\mathrm{PT}}}-1\right)}^2\left({d}_1+{\gamma}_d^{d_2}\right){\left(\frac{p^{\prime }+{p}_{\mathrm{res}}^{\prime }}{p_{\mathrm{atm}}}\right)}^{-{d}_3} $$
(20.6)

where n is a unit outer normal to an imaginary surface in the deviatoric stress space passing by the stress point s . η and ηPT are the current shear stress ratio and the shear stress ratio at the PT surface, respectively. c1 parameter mainly controls the plastic volumetric strain accumulation rate. d1 controls the accumulated volumetric shear strain per each dilation cycle. d2 and c2 account for fabric damage through ξc and γd which represent the total accumulative volumetric strain and the cumulative octahedral shear strain per each dilation cycle, respectively. c3 and d3 parameters account for overburden stress variation effects on contraction and dilation rates “Kσ effect”. When η equals ηPT, i.e., the neutral phase, the model starts to accumulate shear strains before starting the dilation phase at almost no changing either on shear stresses or soil confinement p′. For the sake of simplification, zero plastic potential component (P″ = 0) is maintained during this high yielding phase till reaching a boundary that defined in deviatoric space and dilation starts afterward. An initial isotropic domain is defined through this boundary as a circle in the deviatoric strain space, which will either enlarge or translate based on the loading history as demonstrated in Elgamal et al. (2003) and Yang et al. (2003).

4 Model Parameters Estimation and Calibration “Simulation Phase I”

4.1 The Initial Determination of the Model Parameters

The PDMY02 model parameters can be separated into four main groups. The first group is concerned with the stress-strain relationships in elastoplasticity, which comprises small strain shear and bulk moduli measured at a referenced confinement pressure “Goct, r, Br,” friction angle “ϕ,” and phase transformation angle “ϕPT.”

The second group is the flow rule user-input parameters (c1, c2, c3, d1, d2 and d3), described in section 20.3.3, that define the soil contractive-dilative behavior during undrained shearing. The third group is the liquefaction damage parameters (liq1 and liq2) that control the development of the liquefaction-induced perfectly plastic shear strain as a function of dilation history and load reversal history.

The last group is a miscellaneous collection of parameters that are estimated according to Yang and Elgamal (2009) recommendations. These parameters are pressure dependent coefficient, number of yield surfaces, and the maximin reached shear strain. A fixed value for soil permeability coefficient of “3 × 105 m/s” is selected for the calibration and centrifuge simulations. The permeability coefficient value has been selected as an average value from previous LEAP simulation work done (Ghofrani & Arduino, 2018; Ziotopoulou, 2018).

The parameters initially estimated through different correlations from a wide range of empirical formulae, previously done soil element testing for Ottawa-F65 sand by Bastidas (2016) and the PDMY02 online model documentation (Yang & Elgamal, 2009). Then, the model parameters values are tuned through a calibration process with the cyclic torsional shear tests conducted by Kyoto University (Ueda, 2018). Unfortunately, the number of drained monotonic triaxial or drained simple shear tests through previous LEAP events is minimal and not providing full data for calibration of the model basic elastoplastic parameters “Goct, r, Br, ϕ, ϕPT” (Beaty, 2018).

Regarding Ueda conducted tests (Ueda, 2018), the soil samples have slightly different relative densities for each relative density (Dr) group. (Dr 50%) and (Dr 60%) groups comprise samples’ relative densities ranges of 50.5–53.0% and 60.5–63.5%, respectively. Therefore, two data sets of model parameters are estimated herein for Ottawa-F65 sand, considering different relative density ranges of (50–55%) and (60–65%).

The soil shear and bulk moduli values are explicitly calculated for each soil relative density of each centrifuge test and are not fixed for each (Dr) group. This way contributes to adjusting the issue of using constant contraction and dilation model parameters for each relative density range and overcoming the small variances between relative densities for each (Dr) group, i.e., the variance in Dr from 54% to 57%. Table 20.2 represents the adopted model parameters of the Ottawa-F65 sand soils for different relative density ranges.

Table 20.2 The PDMY02 estimated model parameters
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4.2 Cyclic Torsional Shear Test Simulation (Simulation Phase I)

A single four-node element with mixed u-p formulation (QuadUP), which was implemented in OpenSees by Yang et al. (2008), is employed to simulate the undrained cyclic torsional shear test considering the appropriate boundary conditions and loading stages. The output computed results are compared with the measured responses from the soil element testing conducted by Ueda (2018) to calibrate the soil model and adjust selected parameters, as shown in Figs. 20.3 and 20.4.

Fig. 20.3
Eight graphs depict shear stress kilopascals against shear strain % and mean effective stress kilopascal. Computed and measured values are compared, depicting how shear stress and mean effective stress vary with shear strain % in different scenarios.

Computed vs. measured output results of undrained cyclic torsional shear test simulations for (Dr 50%) group under CSR values of 0.10, 0.13, 0.15 and 0.19; τ − γ relationship on the left and stress path on the right

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Fig. 20.4
Eight graphs depict shear stress kilopascals against shear strain % and mean effective stress kilopascals. Computed and measured values are compared, depicting how shear stress and mean effective stress vary with shear strain in different scenarios.

Computed vs. measured output results of undrained cyclic torsional shear test simulations for (Dr 60%) group under CSR values of 0.13, 0.15, 0.18, and 0.20; τ − γ relationship on the left and stress path on the right

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The output curves show the model capability in simulating the cyclic torsional shear test in terms of τ − γ relationship and stress path curves for all engaged tests with a fair determination of the number of loading cycles required to reach initial liquefaction for medium CSR levels, i.e., 0.15–0.18. However, the model cannot appropriately determine the number of loading cycles to reach initial liquefaction for lower and higher CSR levels.

Figure 20.5 summarizes the whole tests’ simulations to highlight the issue of estimating the proper number of loading cycles for triggering liquefaction. This issue is partially solved later in a new model version developed by Khosravifar et al. (2018) that has not been available in OpenSees till the time of commencing this simulation work.

Fig. 20.5
A line and scattered graph plots shear stress ratio tau per sigma strike 0 versus number of loading cycles to reach gamma D A = 7.5%. 2 lines measured D r 50% group and 60% group drops from (5, 0.20) to (300, 0.10).

Computed vs. measured number of loading cycles to reach a double amplitude shear strain (γDA) of 7.5% in cyclic torsional shear tests

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5 Specifications of the Finite Element Model for Centrifuge Experiments Simulations

A 2D plane strain analysis was conducted through the OpenSees platform for modeling the soil slope in a rigid container. With Model A simulations, the soil model parameters and the slope geometrical dimensions are in prototype scale. On the other hand, the Model B simulations are performed after downscaling the FE model components to the model scale according to the Generalized Scaling Law (GSL). The figure below represents a schematic view of the modeled centrifuge test with markers and sensor locations for measuring different soil response components (Fig. 20.6).

Fig. 20.6
A chart illustrates a vertical cross-section of a test setup, detailing markers, sensors, and structures. Key features include markers 1 to 6, different points A H 1 to A H 10, water table, P W P transducer, accelerometer, and lateral displacement surface markers. The measurements provide dimensional information for the components.

Schematic view of the centrifuge test in prototype scale showing the different sensor locations. (Elbadawy et al., 2022)

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5.1 FE Model Description

A 2D finite element mesh comprised 1377 nodes and 1280 QuadUP elements for Model A and Model B simulations with elements spatial size ranging from (0.2*L to 0.3*L) m, where L equals “1” when modeling in prototype scale and equals μη when modeling in virtual model scale.

The element size range is selected according to Kuhlemeyer and Lysmer (1973) suggestion to consider an element size less than or equal to one-eighth of the wavelength associated with the maximum frequency component of the input wave. The chosen element size range ensures that the mesh can resolve waves up to a frequency of 50 Hz. This way is also adopted by previous researchers in their computational simulations of dynamic analysis of previous LEAP events (Ghofrani & Arduino, 2018).

The quadUP element is used in this simulation to model the cyclic response of soil in partially drained and undrained conditions. The right and left vertical model sides are fixed in the horizontal direction, and the model base is fully fixed in vertical and horizontal directions. Water pressure is applied on the slope surface nodes with prescribed PWP values and appropriate nodal loads during the simulation. Top mesh nodes are fixed for pore water pressure to allow flow across the model surface, whereas model bottom and side boundaries are not allowed for water drainage. Figure 20.7 shows the FE mesh discretization and the model boundary conditions.

Fig. 20.7
A mesh diagram illustrates a structural simulation with fixed pore water pressure at top nodes, selective water drainage, and pressure from a water table on surface nodes. Nodes have varied constraints, impacting the simulation's behavior.

The FE mesh discretization layout is showing the boundary conditions and the different sensor locations. (Elbadawy et al., 2022)

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The FE analysis is initiated by considering a linear elastic material behavior when applying gravity loads and self-weights. After that, the model is turned to the nonlinear elastoplastic stage that considers all material nonlinearity and plastic behaviors. In order to achieve a successful transition, few analysis steps should be run to ensure that the soil’s internal variables are adjusted to the new stress-state and to avoid numerical errors for the subsequent simulations. The initial state of stress output results from gravitational loading is presented in Fig. 20.8. The postprocessing contour plots were generated using the Scientific Tool Kit for OpenSees “STKO” (ASDEA-Software, 2019).

Fig. 20.8
3 chromatic graphs illustrate variations. a. sigma x kilopascals from 0.00 to negative 25.00, b. sigma y kilopascals from 0.00 to negative 50.00, and c. Pore Water Pressure in kilopascals from 0.00 to 50.00.

Output contour plots of the initial stress analysis. (a) horizontal stress distribution, (b) vertical stress distribution, and (c) pore water pressure distribution

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After that, dynamic loading is subjected to the FE model base by application of the recorded acceleration time history for each centrifuge test. The numerical simulation is performed according to the Newmark-Beta step by step integration method in the time domain with γ = 0.5 and β = 0.25 (Newmark, 1959, 1972). A modified Newton-Raphson solution algorithm is selected for the numerical computations with a Krylov subspace accelerator to accelerate the convergence process (Scott & Fenves, 2010).

The material damping emerging from the soil nonlinear hysteresis behavior is accompanied by a small initial stiffness proportional damping coefficient equals 0.003 as prescribed by Parra (1996) and Yang and Elgamal (2009) in order to ensure stable numerical simulations during high shearing phases. The following section will discuss the output results from the simulation of the centrifuge experiments, besides spotlighting the main findings.

6 Result and Discussion

Class C simulation output results are presented in this section comprising time histories of excess PWP, acceleration, and surface horizontal displacement at different sensor locations. For achieving computed model responses as similar to centrifuge experiments, a slight change was made to the initially calibrated model parameters. These modifications were done towards c1 and liq2 parameters, as presented in Table 20.3.

Table 20.3 The modified values of PDMY02 model parameters for class-C simulations
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Increasing the c1 model parameter results in generating significantly higher excess pore water pressure and magnifying the horizontal surface deformations. Increasing liq2 model parameter enhances the model predictions of shear strain accumulations, especially in case of the model subjected to initial static shear stresses, which provides proper estimations of the permanent horizontal surface displacements.

The selection of these values for c1 and liq2 parameters aims to employ a unified set of model parameters for each relative density group in centrifuge simulations. The PDMY02 estimated model parameters for (Dr 50%) group are used in simulating centrifuge tests done by ZJU and KyU, whereas PDMY02 estimated model parameters for (Dr 60%) group are selected for RPI and UCD centrifuge simulations.

The output responses of middle array sensors “P1, P2, P3, P4, AH1, AH2, AH3, and AH4” and two top surface displacement recorders are selected for the comparisons between computed and measured output results for the selected Model A and Model B centrifuge tests. The figures below show the time histories of excess pore water pressure (PWP), horizontal acceleration, and horizontal surface displacements.

6.1 Model A Output Results

From Figs. 20.9 and 20.10, excess PWP and horizontal surface displacements are in good agreement with the experiment output responses despite little overestimation in the displacement output time histories with the UCD_A_A2_1 output result. The acceleration time histories reveal some dilation spikes that cannot be captured by the employed soil model. These spikes are very high in case of modeling the ZJU_A_A1_1 centrifuge test, as shown in Fig. 20.11.

Fig. 20.9
12 line graphs compare simulation and experimental results for parameters such as excess pore water pressure kilopascals, acceleration in gravity, and horizontal displacement in millimeters over time for specific points P 4, P 2, P 6, A H 1, A H 2, A H 3, A H 4, Point 3, and Point 5.

Simulation vs. experiment output results of RPI-A-A1-1; time histories of excess pore water pressure, horizontal acceleration, and horizontal surface displacement (measured PWP data of sensor P3 is not available and exchanged with P6 output results)

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Fig. 20.10
12 line graphs depict the comparison between simulated and experimental data for parameters such as acceleration in gravity, excess pore water pressure in kilopascals, and horizontal displacement in millimeters at different points, providing insights into the system's behavior over time.

Simulation vs. experiment output results of UCD_A_A2_1; time histories of excess pore water pressure, horizontal acceleration, and horizontal surface displacement

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Fig. 20.11
12 line graphs depict the comparison between simulated and experimental data for parameters such as acceleration in gravity, excess pore water pressure in kilopascals, and horizontal displacement in millimeters at different points, providing insights into the system's behavior over time.

Simulation vs. experiment output results of ZJU_A_A1_1; time histories of excess pore water pressure, horizontal acceleration, and horizontal surface displacement

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Much higher negative PWP spikes were monitored in all sensor locations in the ZJU_A_A1_1 centrifuge test. The simulated excess PWP generation and dissipation rates in the ZJU_A_A1_1 test matched perfectly with the measured output responses, regardless of the absence of the negative excess PWP spikes in the simulated responses. These intensive negative PWP spikes have resulted from the damage that occurred in the soil fabric when reaching the liquefaction stage, which is why large surface horizontal deformation values are generated in measured and computed time histories.

Despite the accurately computed output results in the previous simulations, KyU_A_A2_1 computed response has deviated from the centrifuge experiment measurements, as presented in Fig. 20.12. This issue is very noticeable in the excess PWP and surface horizontal displacement time histories. On the contrary, the computed and measured acceleration time histories entirely coincide throughout the test duration.

Fig. 20.12
12 line graphs depict the comparison between simulated and experimental data for parameters such as acceleration in gravity, excess pore water pressure in kilopascals, and horizontal displacement in millimeters at different points, providing insights into the system's behavior over time.

Simulation vs. experiment output results of KyU_A_A2_1; time histories of excess pore water pressure, horizontal acceleration, and horizontal surface displacement

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6.2 Model B Output Results

The Model B centrifuge experiments were simulated in the model scale. With RPI_A_B1_1 test, Fig. 20.13 shows computed output responses similar to experimental recorded results for the excess PWP and acceleration time histories despite slight disparities regarding the dissipation rate of the excess PWP. The simulated horizontal surface displacements are a bit lower than the experiment outputs, although they are still following similar centrifuge measured trends.

Fig. 20.13
12 line graphs depict the comparison between simulated and experimental data for parameters such as acceleration in gravity, excess pore water pressure in kilopascals, and horizontal displacement in millimeters at different points, providing insights into the system's behavior over time.

Simulation vs. experiment output results of RPI_A_B1_1; time histories of excess pore water pressure, horizontal acceleration, and horizontal surface displacement (measured PWP data of sensor P2 is not available and exchanged with P6 output results)

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Through Fig. 20.14, there is a good agreement between the computed output results and the centrifuge measured responses, which can be depicted through the time histories of excess PWP and horizontal displacements. Soil dilative behavior appears again in the ZJU_A_B1_1 experiment through negative spikes in excess PWP and acceleration time histories (Fig. 20.15).

Fig. 20.14
12 line graphs depict the comparison between simulated and experimental data for parameters such as acceleration in gravity, excess pore water pressure in kilopascals, and horizontal displacement in millimeters at different points, providing insights into the system's behavior over time.

Simulation vs. experiment output results of ZJU_A_B1_1; time histories of excess pore water pressure, horizontal acceleration, and horizontal surface displacement

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Fig. 20.15
12 line graphs depict the comparison between simulated and experimental data for parameters such as acceleration in gravity, excess pore water pressure in kilopascals, and horizontal displacement in millimeters at different points, providing insights into the system's behavior over time.

Simulation vs. experiment output results of KyU_A_B2_1; time histories of excess pore water pressure, horizontal acceleration, and horizontal surface displacement

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With KyU_A_B2_1 centrifuge simulation, the output computed PWP results are slightly better than the results obtained in Model A simulations; however, the displacement time histories are still highly deviated from the measured centrifuge responses. One of the expected reasons for getting overestimated horizontal surface displacements is the high wave oscillations that generated at frequencies near the natural soil frequency, which may cause resonance effects on the simulated model (see Sect. 20.2.1). Moreover, the model’s flow rule may generate over estimated responses at low to medium CSR levels, as described in Sect. 20.4.2, which might increase the PWP and horizontal displacement output results.

7 Conclusion

Extensive class-C numerical simulations were performed by means of the Finite Element Analysis method to model seven centrifuge tests of mildly sloping ground composed of Ottawa-F65 sand in a rigid container under ramped sinusoidal input motions with different PGAs. Numerical simulations were performed in prototype scale and model scale for simulating Model A and Model B centrifuge experiments, respectively. An elastoplastic multi-yield surface model with pressure dependency, namely PDMY02, was engaged in this simulation work through the OpenSees platform.

The PDMY02 model calibration process, “Simulation Phase I”, was carried out by simulating the cyclic torsional shear tests for soil samples with different relative densities composed of Ottawa-F65 sand under different CSR values. The model parameters were first estimated through geotechnical correlations and previous soil laboratory tests and then calibrated with the provided cyclic torsional shear laboratory tests.

The PDMY02 model provides fair computed responses for all cyclic torsional shear test simulations despite estimating inaccurate counts of loading cycles to reach initial liquefaction for very low and very high CSR levels. Prior to commencing Class C simulations, minor changes were made to model parameter values to deliver proper numerical predictions of the simulated centrifuge experiments. The computed results revealed that the PDMY02 model is capable of predicting the liquefaction potential and lateral spreading of the mildly sloping ground problem. The computed time histories of excess PWP, accelerations, and displacements are in good agreement with the measured responses in all centrifuge tests despite getting disparate simulated responses with experiments that took place at the Kyoto University centrifuge facility.

A prominent drawback when using the PDMY02 model appeared when calibrating the model parameters to predict the appropriate number of loading cycles to trigger liquefaction under a wide range of CSR values. The model’s flow rule should be adjusted to accurately predict the number of loading cycles to reach initial liquefaction under different stress levels in order to achieve successful model calibrations and obtain better numerical simulation predictions. Generally, the PDMY02 model was found to be adequate for investigating the liquefaction potential of mildly sloping grounds and providing proper seismic predictions under different dynamic motion intensities.