LEAP-ASIA-2019 Centrifuge Test Simulations of Liquefiable Sloping Ground

Abstract

Numerical simulations of a liquefiable sloping ground for LEAP-ASIA-2019 centrifuge tests are presented. The simulations are performed using a pressure-dependent constitutive model implemented with the characteristics of dilatancy, cyclic mobility, and associated shear deformation. The soil model parameters are determined based on a series of stress-controlled cyclic torsional shear tests of Ottawa F-65 sand with relative density D_r = 60% during calibration phase. Computational framework for the seismic response analysis is discussed and the computed results are presented for all selected centrifuge experiments during Type-C phase. Measured time histories of these experiments are reasonably captured. It is demonstrated that the pressure-dependent constitutive model as well as the overall employed computational framework has the potential to predict the response of the liquefiable sloping ground, and subsequently realistically evaluates the performance of an equivalent soil system subjected to seismically induced liquefaction.

1 Introduction

LEAP (Liquefaction Experiments and Analysis Projects) is an effort to facilitate validation and verification of numerical procedures for liquefaction-induced lateral spreading analysis of a liquefiable sloping ground (Kutter et al., 2015, 2018, 2020). As part of the ongoing LEAP, a new set of centrifuge tests have been performed in LEAP-ASIA-2019 (Ueda, 2018) to simulate the liquefaction induced lateral spreading phenomenon in a fully saturated sloping ground.

On this basis, the numerical simulation results of these new data during Type-C phase are presented. All the finite element (FE) simulations are performed using a pressure-dependent constitutive model (Parra, 1996; Yang, 2000; Yang & Elgamal, 2002; Elgamal et al., 2003; Yang et al., 2003; Khosravifar et al., 2018; Qiu & Elgamal, 2020a, b) implemented with the characteristics of dilatancy, cyclic mobility, and associated shear deformation. The soil parameters are determined based on a series of stress-controlled cyclic torsional shear tests provided in the calibration phase for matching the liquefaction strength curve of Ottawa F-65 sand with relative density D_r = 60%. To better capture the overall dynamic response of each selected centrifuge test during Type-C phase, two contraction parameters c₄ and c₅ controlling the rate of pore pressure build-up were adjusted based on observations from selected centrifuge test results.

The following sections of this chapter outline (1) computational framework, (2) specifics and calibration processes, (3) details of the employed FE modeling techniques, and (4) computed results of the selected centrifuge tests. Finally, a number of conclusions are presented and discussed.

2 Brief Summary of the Centrifuge Tests

A schematic representation of the centrifuge tests (El Ghoraiby et al., 2020) is shown in Fig. 17.1. The soil specimen is a sloping layer of Ottawa F-65 sand with 5° slope (target relative density D_r = 60%). The soil layer has a length of 20 m (in prototype scale) and a height of 4 m (in prototype scale) at the center. The specimen is built in a container with rigid walls. All centrifuge models were subjected to a target motion of ramped, 1 Hz sine wave base motion with amplitude 0.15g. Figure 17.2 shows the achieved base input motions for all selected centrifuge experiments with various relative densities in LEAP-ASIA-2019 (Ueda, 2018).

Fig. 17.1

Schematic representation of the LEAP-ASIA-2019 centrifuge test layout

Fig. 17.2

Selected base input motions

3 Constitutive Model of Soils

A two-dimensional FE mesh (Fig. 17.3) is created to represent the centrifuge test model, comprising 4961 nodes and 4800 quadrilateral elements (maximum size = 0.2 m). All numerical simulations for the selected centrifuge tests during Type-C phase are performed using the computational platform OpenSees. The Open System for Earthquake Engineering Simulation (OpenSees, McKenna et al., 2010, http://opensees.berkeley.edu) developed by the Pacific Earthquake Engineering Research (PEER) Center is an open source, object-oriented finite element platform. Currently, OpenSees is widely used for simulation of structural and geotechnical systems (Yang, 2000; Yang & Elgamal, 2002) under static and seismic loading.

Fig. 17.3

Finite element mesh (maximum size = 0.2 m)

Quadrilateral Four-node plane-strain elements with two-phase material following the u-p (Chan, 1988) formulation were employed for simulating saturated soil response, where u is the displacement of the soil skeleton and p is the pore water pressure. Implementation of the u-p element is based on the following assumptions: (1) small deformation and rotation; (2) solid and fluid density remain constant in time and space; (3) porosity is locally homogeneous and constant with time; (4) soil grains are incompressible; (5) solid and fluid phases are accelerated equally. Hence, the soil layers represented by effective stress fully coupled u-p elements (quadUP in OpenSees) are capable of accounting for soil deformations and the associated changes in pore water pressure.

3.1 Soil Constitutive Model

The soil is simulated by the implemented OpenSees material PressureDependMultiYield03 (Parra, 1996; Yang, 2000; Yang & Elgamal, 2002; Elgamal et al., 2003; Yang et al., 2003; Khosravifar et al., 2018); In this employed soil constitutive model (Fig. 17.4), the shear-strain backbone curve was represented by the hyperbolic relationship with the shear strength based on simple shear (reached at an octahedral shear strain of 10%). The low-strain shear modulus under a reference effective confining pressure ${p}_{\mathrm{r}}^{\prime }$ is computed using the equation $G={G}_0{\left({p}^{\prime }/{p}_{\mathrm{r}}^{\prime}\right)}^n$ where p^′ is effective confining pressure and G₀ is shear modulus at pressure ${p}_{\mathrm{r}}^{\prime }$. The dependency of shear modulus on confining pressure is taken as n = 0.5. The critical state frictional constant M_f (at failure surface) is related to the friction angle ϕ (Chen & Mizuno, 1990) and defined as M_f = 6sinϕ/(3-sinϕ). As such, brief descriptions of this soil constitutive model are included below.

Fig. 17.4

PressureDependMultiYield03 model response and configuration of yield domain in deviatoric strain space. (After Yang & Elgamal, 2002; Elgamal et al., 2003; Yang et al., 2003)

3.1.1 Yield Function

The yield function is defined as a conical surface in principal stress space:

$$ f=\frac{3}{2}\left(s-\left({p}^{\prime }+{p}_0^{\prime}\right)a\right):\left(s-\left({p}^{\prime }+{p}_0^{\prime}\right)a\right)-{M}^2{\left({p}^{\prime }+{p}_0^{\prime}\right)}^2=0 $$

(17.1)

where, s = σ ′  − p^′δ, is the deviatoric stress tensor, σ^′ is the effective Cauchy stress tensor, δ is the second-order identity tensor, p^′ is mean effective stress, ${p}_0^{\prime }$ is a small positive constant (0.3 kPa in this chapter) such that the yield surface size remains finite at p^′ = 0 for numerical convenience and to avoid ambiguity in defining the yield surface normal to the yield surface apex, a is a second-order deviatoric tensor defining the yield surface center in deviatoric stress subspace, M defines the yield surface size, and “:” denotes doubly contracted tensor product.

3.1.2 Contractive Phase

Shear-induced contraction occurs inside the phase transformation (PT) surface (η < η_PT), as well as outside (η > η_PT) when $\dot{\eta}<0$, where, η is the deviatoric stress ratio defined as $\sqrt{\frac{3}{2}s:s}/\left({p}^{\prime }+{p}_0^{\prime}\right)$ and η_PT is the deviatoric stress ratio at phase transformation surface (Fig. 17.4). The contraction flow rule is defined as:

$$ {\displaystyle \begin{array}{l}{P}^{{\prime\prime} }={\left(1-\frac{\dot{n}:\dot{s}}{\left|\left|\dot{s}\right|\right|}\frac{\eta }{\eta_{\mathrm{PT}}}\right)}^2\left({c}_1+{c}_2{\gamma}_{\mathrm{c}}\right){\left(\frac{p^{\prime }}{p_{\mathrm{a}}}\right)}^{c_3}{\left({c}_4{\eta}_{\mathrm{rv}}\right)}^{c_5}\\ {}{\eta}_{\mathrm{rv}}=\frac{\sqrt{\left[{\left({\sigma}_{11}-{\sigma}_{22}\right)}^{\mathbf{2}}+{\left({\sigma}_{22}-{\sigma}_{33}\right)}^{\mathbf{2}}+{\left({\sigma}_{11}-{\sigma}_{33}\right)}^{\mathbf{2}}\right]/2+{\tau}_{12}^2+{\tau}_{23}^2+{\tau}_{13}^2}}{\left({p}^{\prime }+{p}_0^{\prime}\right)}\end{array}} $$

(17.2)

where c₁, c₂, c₃, c₄, and c₅ are non-negative calibration constants, γ_c is octahedral shear strain accumulated during previous dilation phases, p_a is atmospheric pressure for normalization purpose, and $\dot{s}$ is the deviatoric stress rate. In Eq. 17.2, η_rv is the shear stress ratio on load reversal point during cyclic loading, essentially representing the effect of previous shear stress on the subsequent contractive behavior. The $\dot{n}$ and $\dot{s}$ tensors are used to account for general 3D loading scenarios, where, $\dot{n}$is the outer normal to a surface. The parameter c₃ is used to represent the dependence of pore pressure buildup on initial confinement (i.e., kσ effect).

3.1.3 Dilative Phase

Dilation appears only due to shear loading outside the PT surface (η > η_PT with $\dot{\eta}>0$), and is defined as:

$$ {P}^{{\prime\prime} }={\left(1-\frac{\dot{n}:\dot{s}}{\left\Vert \dot{s}\right\Vert}\frac{\eta }{\eta_{\mathrm{PT}}}\right)}^2\left({d}_1+{\gamma}_d^{d_2}\right){\left(\frac{p^{\prime }}{p_{\mathrm{a}}}\right)}^{-{d}_3} $$

(17.3)

where d₁, d₂, and d₃ are non-negative calibration constants, and γ_d is the octahedral shear strain accumulated from the beginning of a particular dilation cycle (such as, stage 1–2 or 5–6 in Fig. 17.4) as long as there is no significant load reversal. Subsequently, dilation rate increases as the shear strain accumulates in a particular cycle. Furthermore, a significant unloading (such as stage 6–8 in Fig. 17.4) will reset γ_d to zero. Parameter d₃ in Eq. 17.3 reflects the dependence of pore pressure buildup on initial confinement (i.e., kσ effect).

3.1.4 Neutral Phase

When the stress state approaches the PT surface (η = η_PT) from below, a significant amount of permanent shear strain may accumulate prior to dilation, with minimal changes in shear stress and confinement (implying P^″ = 0). For simplicity, P^″ = 0 is maintained during this highly yielded phase until a boundary defined in deviatoric strain space is reached, and then dilation begins. This yield domain will enlarge or translate depending on load history. In deviatoric strain space, the yield domain (Fig. 17.4) is a circle with the radius γ defined as (Yang et al., 2003):

$$ {\displaystyle \begin{array}{l}\gamma =\frac{\gamma_{\mathrm{s}}+{\gamma}_{\mathrm{rv}}}{2}\\ {}{\gamma}_{\mathrm{s}}={y}_1{\frac{p_{\mathrm{max}}^{\prime }-{p}_n^{\prime }}{p_{\mathrm{max}}^{\prime}}}^{0.25}\int_t^0\mathrm{d}{\gamma}_{\mathrm{c}}\\ {}{\gamma}_{\mathrm{rv}}={y}_2{\frac{p_{\mathrm{max}}^{\prime }-{p}^{\prime }}{p_{\mathrm{max}}^{\prime}}}^{0.25}\mathrm{oct}\left(e-{e}_{\mathrm{p}}\right)\end{array}} $$

(17.4)

where, y₁ (non-negative) is used to define the accumulated permanent shear strain γ_s as a function of dilation history $\int_t^0\mathrm{d}{\gamma}_{\mathrm{c}}$ and allow for continuing enlargement of the domain, ${p}_{\mathrm{max}}^{\prime }$ is maximum mean effective confinement experienced during cyclic loading, ${p}_n^{\prime }$ is mean effective confinement at the beginning of current neutral phase, and 〈 〉 denotes MacCauley’s brackets (i.e., 〈a〉 =  max (a, 0)). The y₂ (non-negative) parameter is mainly used to define the biased accumulation of permanent shear strain γ_rv as a function of load reversal history and allows for translation of the yield domain during cyclic loading. In Eq. 17.4, oct(e − e_p) denotes the octahedral shear strain of tensor e − e_p, where e is current deviatoric shear strain, and e_p is pivot strain obtained from previous dilation on load reversal point.

3.2 Boundary and Loading Conditions

The boundary and loading conditions for dynamic analysis of the liquefiable sloping ground (Fig. 17.3) under a base input motion are implemented in a staged fashion as follows:

• 1. Gravity was applied to activate the initial static state with the following: (i) linear elastic properties (Poisson’s ratio of 0.47 to lower the initial locked shear stress), (ii) nodes on both side boundaries (vertical faces) of the FE model were fixed against longitudinal translation for complicity, (iii) nodes were fixed along the base against vertical translation only to avoid superfluous unrealistic initial locked shear stress at the model base, (iv) water table was specified with related water pressure and nodal forces specified along ground surface nodes, and flow of water was restricted to across the container boundaries.

• 2. Nodes were fixed along the base against longitudinal translation.

• 3. Soil properties were switched from elastic to plastic and the internal variables of the constitutive model were adjusted to this stress state (Fig. 17.5 before shaking).

• 4. Dynamic analysis is conducted by applying an acceleration time history to the base of the FE model.

Fig. 17.5

Initial state of soil due to gravity (before shaking): (a) Pore water pressure; (b) Vertical effective stress ${\sigma}_{\mathrm{yy}}^{\prime }$; (c) Horizontal effective stress ${\sigma}_{\mathrm{xx}}^{\prime }$

The FE matrix equation is integrated in time using a single-step predictor multi-corrector scheme of the Newmark type with integration parameters γ = 0.6 and β = 0.3025 presented in early studies (Chan, 1988; Parra, 1996). The equation is solved using the modified Newton-Raphson method, i.e., Krylov subspace acceleration (Carlson & Miller, 1998) for each time step. For the convergence criterion, a test of energy increment is used with 10⁻⁶ and maximum number of iterations of 50. Furthermore, the constraints are imposed using Transformation method in OpenSees. Finally, a relatively low-level of initial stiffness proportional damping (coefficient = 0.003 leading to 1% damping ratio at frequency = 1 Hz) with the main damping emanating from the soil nonlinear shear stress-strain hysteresis response (Parra, 1996) was used to enhance numerical stability of the liquefiable sloping system.

4 Determination of Soil Model Parameters

To predict dynamic response of centrifuge tests in LEAP-ASIA-2019, the employed soil constitutive model parameters are calibrated for matching the liquefaction strength curve of the Ottawa F-65 sand with relative density D_r = 60%, which will be further used in Type-C simulations. For that purpose, a total of four undrained stress-controlled cyclic torsional shear tests for Ottawa F-65 sand are performed and the laboratory results are provided in calibration phase (Ueda, 2018). The permeability of the Ottawa F-65 sand is about 1.1 × 10⁻⁴ m/s determined from El Ghoraiby et al., 2020. On this basis, the calibrated soil model parameters are listed in Table 17.1.

Table 17.1 Sand model parameters in calibration phase and Type-C simulations

Figure 17.6 shows the comparison results of computed and experimental liquefaction strength curves. The data plotted in Fig. 17.6 is composed of the number of cycles until a 7.5% double amplitude (i.e., 3.75% single amplitude) of strain is achieved versus the applied cyclic stress ratio. It can be seen that the computed results are in good agreement with the laboratory data. An example of undrained stress-controlled torsional shear test with CSR = 0.2 is illustrated in Fig. 17.6b–e. The computed results reasonably match the stress path and shear stress-strain response of laboratory test results (Ueda, 2018).

Fig. 17.6

Torsional shear tests: (a) Liquefaction strength curve; (b)-(e) Undrained stress-controlled cyclic soil response with CSR = 0.2. (Qiu & Elgamal, 2020a, b)

5 Computed Results of Type-C Simulations

This section presents the simulation results of selected eight centrifuge tests during Type-C phase. The achieved base input motions of selected centrifuge tests with different relative densities at various facilities are displayed in Fig. 17.2. The material parameters (Table 17.1) are calibrated primarily for matching the liquefaction strength curve of Ottawa F-65 sand with D_r = 60% in calibration phase. To better capture the overall dynamic response of centrifuge tests of RPI-A-A1-1, RPI-A-B1-1, UCD-A-A1-1 and UCD-A-A2-1, the contraction parameters c₄ = 7.0 and c₅ = 4.0 in Table 17.1 are adjusted to 0.6 and 0.1, respectively. The experimental results of RPI-A-A1–1 (D_r = 64%) are selected as the calibration basis for adjusting the material parameters c₄ and c₅. As such, the numerical simulations of RPI-A-B1-1, UCD-A-A1-1 and UCD-A-A2-1 are performed based on the material properties from calibration phase (Table 17.1) with the adjusted parameters c₄ and c₅ obtained from the calibration of RPI-A-A1-1.

As observed from the four centrifuge test results of KyU, the pore pressure transducers are not providing consistent results with those from RPI and UCD. As such, contraction parameters c₄ and c₅ are further adjusted to 0.2 and 0.1, respectively, to better capture the much slower pore pressure build-up rate in these centrifuge tests of KyU. In the following results, numerical simulations of four KyU centrifuge tests are performed based on the material properties from calibration phase (Table 17.1) with the newly adjusted parameters c₄ and c₅.

5.1 Acceleration

Figure 17.7 depicts the computed and experimental acceleration time histories at locations AH1-AH4 (Fig. 17.1). It can be seen that the computed accelerations reasonably match those from the measurements in RPI-A-A1-1, UCD-A-A2-1, KyU-A-A21, and KyU-A-B2-1 (Fig. 17.7). Both the computed results and measurements showed a consistent trend of acceleration spikes due to dilation. However, there are significant differences in the rest of simulations compared to the experimental data. As seen in this Fig. 17.7, the accelerations spikes are overpredicted by RPI-A-B1–1, UCD-A-A1-1, and are underpredicted by KyU-A-A1-1, KyU-A-B1-1, mainly due to the different relative densities at various facilities.

Fig. 17.7

Measured and computed acceleration time histories

5.2 Excess Pore Pressure Ratio

Figure 17.8 illustrates the time histories of excess pore pressure ratio r_u. It can be seen that the computed excess pore pressure r_u reasonably matches with those from the centrifuge tests of RPI-A-A1-1, RPI-A-B1-1, and UCD-A-A2-1. However, in calibration of UCD-A-A1-1, the computed results showed a relatively faster pore pressure build-up compared to the measurements.

Fig. 17.8

Measured and computed time histories of excess pore pressure ratio

In four tests of KyU, the much slower pore pressure build-up rate exhibited in measurements is well captured by further adjusting contraction parameters c₄ and c₅ (as discussed above) in numerical simulations. As such, both the computed results and measurements (Fig. 17.8) showed a consistent trend of negative spikes due to dilation. In addition, the dissipation of computed pore pressure generally following the trend of measurements after shaking.

5.3 Displacement

Figure 17.9 displays the computed horizontal displacement time histories and measurements at the midpoint of the ground surface. It can be seen that the computed results of RPI-A-A1-1, UCD-A-A2-1, and the four tests of KyU are in good agreement with those from measurements. However, some discrepancies are seen between the simulation results and experimental measurements in tests of RPI-A-B1-1 and UCD-A-A1-1. This higher amount of experimental permanent displacement of RPI-A-B1-1 can be better captured by further adjusting the parameter y₂ in numerical simulation (as discussed below).

Fig. 17.9

Measured and computed displacement time histories

5.4 Computed Response of RPI-A-B1-1

An additional numerical simulation is performed to better capture the horizontal permanent displacement of RPI-A-B1-1. In this scenario, parameter y₂ = 0 (Table 17.1) is adjusted to 1.0, in order to reproduce a higher accumulation of shear deformation during earthquake loading. For comparison purposes, the computed results with material parameters y₂ = 0 and 1.0 are displayed in one figure (Figs. 17.10 and 17.11). As seen in Fig. 17.10, the computed permanent displacement (y₂ = 1) is accumulating faster than that of scenario y₂ = 0 after 11 s, and eventually matches the experimental result (about 0.3 m) at end of shaking.

Fig. 17.10

Adjusted parameter y₂ = 1.0 to capture horizontal permanent deformation of RPI-A-B1–1: (a) Displacement time history; (b) Horizontal displacement contour; (c) Vertical displacement contour; (d) Shear strain contour. (Qiu & Elgamal, 2020a, b)

Fig. 17.11

Adjusted parameter y₂ = 1.0 to capture horizontal permanent deformation of RPI-A-B1–1: (a) Shear stress-strain; (b) Mean effective stress-shear stress

Figure 17.10b shows the horizontal displacement contour with arrows displaying the direction of ground movement. It can be seen that the horizontal displacements of soil ground at deeper depths are also high due to the liquefaction of underlying soil layers (Fig. 17.8). Vertical displacement contour at end of shaking is illustrated in Fig. 17.10c. As seen in this figure, the upslope soil settled about 0.2 m and ground heave in downslope reached about 0.08 m. In accordance with the deformation contour, Fig. 17.10c shows the shear strain γ_xy contour with a peak value of about 12% at deeper depth of the liquefiable sloping ground.

Figure 17.11 depicts the computed shear stress versus mean effective stress, and shear stress versus shear strain for integration points near the locations of pore pressure transducers (P1–P4). It can be seen that the computed results with adjusted parameter y₂ = 1.0 reproduce a larger cycle-by-cycle accumulation of shear deformation in downslope direction than that of scenario y₂ = 0.

6 Conclusions

The numerical simulation results of centrifuge model tests (Type-C phase) conducted by various facilities in LEAP-ASIA-2019 for a liquefiable sloping ground are presented. All the numerical simulations are performed using a calibrated pressure-dependent constitutive model (PressureDependMultiYield03) implemented with the characteristics of dilatancy, cyclic mobility, and associated shear deformation. The soil parameters are determined based on a series of stress-controlled cyclic torsional shear tests (provided in the calibration phase) for matching the liquefaction strength curves of Ottawa F-65 sand with relative density D_r = 60% in LEAP-ASIA-2019. The computational framework and staged analysis procedure are presented as well. The primary conclusions can be drawn as follows:

• 1. The unintended inconsistencies of centrifuge test results in Type-C phase may indeed hinder the comparisons between numerical simulations and measurements. To better capture the overall dynamic response of selected centrifuge tests, contraction parameters c₄ and c₅ (controlling the pore pressure build-up rate) are suggested to be adjusted to simulate the inconsistent contractive behavior exhibited by Ottawa F-65 sand of various relative densities.

• 2. Although the centrifuge tests conducted by various institutions are not providing completely consistent results, measured time histories are reasonably captured by the numerical simulations in LEAP-ASIA-2019, using the same soil constitutive model parameters. The good agreement between computed and measured results for each centrifuge test demonstrated that the PressureDependMultiYield03 soil model as well as the overall employed computational methodology have the potential to predict the response of liquefiable sloping ground, and subsequently realistically evaluate the performance of analogous soil systems subjected to seismically induced liquefaction.

• 3. Permanent deformations of the liquefiable sloping ground are captured to a reasonable level. However, further adjustment of damage parameters y₁ and y₂ is much helpful to quantify the permanent displacement. As such, additional experimental data sets are needed related to large post-liquefaction shear deformation accumulation. As such new data sets become available, parameters y₁ and y₂ of the PressureDependMultiYield03 material can be better calibrated and applied in the prediction of liquefaction-induced lateral spreading.