Keywords

1 Introduction

LEAP (Liquefaction Experiments and Analysis Projects) is an effort to formalize the process and provide data needed for validation of numerical models designed to predict liquefaction phenomena (Kutter et al., 2018).

After several LEAP projects (Zeghal et al., 2015; Kutter et al., 2018; Manzari et al., 2019), LEAP-ASIA-2018 simulation exercise aimed to validate the generalized scaling law proposed by Iai et al. (2005). To this end, this numerical simulation exercise was designed to assess the capability of constitutive models and the numerical modeling techniques in:

  1. 1.

    Capturing the response of saturated liquefiable soils (such as Ottawa F-65 sand) to cyclic shearing at different levels of confining stress, and

  2. 2.

    Simulating potential effects of confining stresses on the lateral spreading of liquefiable soils caused by earthquakes.

After a brief summary of the centrifuge tests (Sect. 16.2), this chapter describes the process followed in the calibration of the selected constitutive model (Sect. 16.3), covering the essential features of the constitutive model and showing the model parameters.

Section 16.4 describes the finite element analyses carried out and it is divided into three sub-sections. Section 16.4.1. defines the types of predictions, while Sect. 16.4.2 reports the calibration philosophy and the assumptions used in the calibration process. A comparison between the predicted and experimental cyclic laboratory tests and liquefaction resistance curves allowed verifying the accuracy of the calibration process. The adopted constitutive model has been calibrated on the results of the provided cyclic torsional shear tests for a relative density, Dr, equal to 50% and 60% under an initial effective confining stress of 100 kPa.

Section 16.4.3 details the numerical model of the centrifuge experiments, describing the main features of the numerical analysis platform used in the simulation, the model geometry and discretization, the applied boundary conditions, the solution algorithm employed, and some assumptions used in the analyses. Finally, Sect. 16.5 reports the results of the type-C simulations showing the comparisons between predicted and simulated time histories of acceleration and excess pore water pressure. Conclusions are provided in Sect. 16.6 about the lessons learned from the experienced simulation exercise.

2 Brief Summary of Centrifuge Experiments

The models were set up in a rigid box by dry pluviation. After saturation the models were positioned on the shaking table in the centrifuge to apply an input motion on the bottom of the box.

Centrifuge model tests were executed with reference to two different models:

  • Model A: models identical to LEAP-UCD-2017 simulation exercise (Manzari et al., 2019) whose response was used to confirm the trends obtained in the previous project (Table 16.1).

  • Model B: a model similar to Model A to validate the generalized scaling law proposed by Iai et al. (2005). Upon constructing the model to be tested, only the viscosity of pore fluid and the input acceleration were scaled (Table 16.2).

Table 16.1 Summary of centrifuge experiments, Model A in LEAP-ASIA-2018
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Table 16.2 Summary of centrifuge experiments, Model B in LEAP-ASIA-2018
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The orientation of the shaking table in the geotechnical centrifuge leads to two different models: the shaking direction of can be parallel to the rotation axis of the centrifuge beam or orthogonal to the rotation axis (see shaking direction in Tables 16.1 and 16.2).

The consequence of the two different model orientations is a different shape of the ground surface and water table (Fig. 16.1), due to the distribution of radial acceleration.

Fig. 16.1
Two sets of longitudinal and cross-section diagrams of centrifuge models with different shaking directions. a. Shaking parallel to the axis of the centrifuge. b. Shaking in the plane of spinning. The labels are rigid containers and water tables to cover the sand during spinning.

Schematic for LEAP-ASIA-2018 centrifuge model tests: (a) Sectional drawing for shaking parallel to the axis of the centrifuge; (b) Sectional drawing for shaking in the plane of spinning of the centrifuge

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The experimental tests reported in bold in Table 16.1 and 16.2 were simulated at the University of Napoli Federico II and, for sake of brevity, only the simulations of Model B types will be discussed in this chapter.

3 Constitutive Model of Soils

The constitutive model used in the simulation exercise is the PM4Sand model (Boulanger & Ziotopoulou, 2015). The PM4Sand (version 3.1) model follows the basic framework of the stress-ratio controlled, critical state compatible, bounding surface plasticity model for sands presented by Dafalias and Manzari (2004), who extended the previous work by Manzari and Dafalias (1997) by adding a fabric-dilatancy related tensor quantity to account for the effect of fabric changes during loading. The fabric-dilatancy related tensor was used to macroscopically model the effect that microscopically observed changes in sand fabric during plastic dilation have on the contractive response upon reversal of loading direction. The modifications were developed and implemented to improve the ability of the model to match existing engineering design relationships currently used to estimate liquefaction-induced ground deformations during earthquakes. These modifications are described in the manuals (version 1 in Boulanger, 2010, version 2 in Boulanger & Ziotopoulou, 2012, and version 3 in Boulanger & Ziotopoulou, 2015) and in the associated publications, as listed in the mentioned manuals.

The model is written in terms of effective stresses, with the conventional prime symbol dropped from the stress terms for convenience because all stresses are effective for the model. The stresses are represented by the tensor σ, the principal effective stresses are σ1, σ2, and σ3, the mean effective stress is p, the deviatoric stress tensor is s, and the deviatoric stress ratio tensor is r. The current implementation was further simplified by casting the various equations and relationships in terms of the in-plane stresses only. This limits the implementation to plane-strain (2D) applications, having the further advantage in its simplified implementation to improve the computational speed.

This constitutive model follows the critical state theory and uses the relative state parameter index (ξR) as defined by Boulanger (2010) and shown in Fig. 16.2. This relative parameter is defined by an empirical relationship for the critical state line:

Fig. 16.2
A graph of relative density versus mean principal effective stress. It includes three concave-down descending curves for Q = 10 and R = 1, Q = 10, R = 1.5, and Q = 9, R = 1. Q equals 10 and R equals 1 has the highest state.

Relative state parameter index and critical state line in the plane DR: p/pA

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$$ {\xi}_{\mathrm{R}}={D}_{\mathrm{R},\mathrm{cs}}-{D}_{\mathrm{R}} $$
(16.1)
$$ {D}_{\mathrm{R},\mathrm{cs}}=\frac{R}{Q-\ln \left(100\frac{p}{p_{\mathrm{A}}}\right)} $$
(16.2)

where DR,cs is the relative density at critical state for the current mean effective stress, instead, Q and R are two parameters that define the shape of critical curve.

Bounding, dilatancy, and critical surfaces are incorporated in PM4Sand following the form of Dafalias and Manzari (2004), respectively:

$$ {M}^{\mathrm{b}}=M\cdotp \exp \left(-{n}^{\mathrm{b}}{\xi}_{\mathrm{R}}\right) $$
(16.3)
$$ {M}^{\mathrm{d}}=M\cdotp \exp \left({n}^{\mathrm{d}}{\xi}_{\mathrm{R}}\right) $$
(16.4)
$$ M=2\cdotp \sin \left({\phi}_{\mathrm{cv}}\right) $$
(16.5)

where nb and nd are model parameters and ϕcv is critical state friction angle.

As the soil is sheared toward critical state (ξR = 0), the values of Mb and Md will both approach the value of M. Thus, the bounding and dilatancy surfaces move together during shearing until they coincide with the critical state surface when the soil has reached critical state.

A large portion of the post-liquefaction reconsolidation strains are due to the sedimentation effects which are not easily incorporated into either the elastic or plastic components of behavior. For this reason, in the PM4Sand, a post-shaking function was implemented. In a strongly pragmatic way, this function reduces volumetric and shear moduli, thus increasing reconsolidation strains to somehow simulate the sedimentation ones (not included in the model).

The post-shaking elastic moduli are determined by multiplying the conventional elastic moduli by a reduction factor Fsed as,

$$ {G}_{\mathrm{post}\hbox{-} \mathrm{shaking}}={F}_{\mathrm{sed}}\cdotp G $$
(16.6)
$$ {K}_{\mathrm{post}\hbox{-} \mathrm{shaking}}={F}_{\mathrm{sed}}\cdotp K $$
(16.7)

For more information on the Fsed, it is possible refer to Boulanger and Ziotopoulou (2015).

The model requires 27 input parameters, 3 of these are considered primary parameters while all the other parameters are suggested to be left with their default values. Table 16.3 reports the most important input parameters of the PM4Sand model, which were defined in the calibration process.

Table 16.3 Input parameters of the PM4Sand model
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4 Finite Element Analyses

Numerical analyses carried out with the abovementioned constitutive model are presented in this section, starting from the description of the calibration process, geometry definition, and boundary conditions of the model.

4.1 Definition of Type A, B, and C Predictions

Numerical predictions could be of different classes: “Class A” are true predictions of an event made prior to the event, “Class B” are predictions made during the event, and “Class C” are predictions made after the event (Lambe, 1973).

In this chapter, only Class C predictions are reported with reference to both simulations of cyclic laboratory tests and centrifuge experiments.

4.2 Determination of Model Parameters

The model parameters obtained from the calibration process are listed in Table 16.4, which also include some parameters kept at their default value.

Table 16.4 Parameters of the PM4Sand model based on the cyclic torsional test data
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The model parameters are obtained by using the results of the provided cyclic torsional shear tests, as described in the following about the calibration procedure. The simulated liquefaction resistance curves for γDA = 7.5% (i.e., the number of cycles required to reach a 7.5% double amplitude shear strain) are compared with the laboratory test results in Fig. 16.10a, b for Dr = 50% and 60%, respectively. The following trends are observed from the curves:

The approach used in the calibration of the constitutive model parameters is hereafter explained.

The PM4Sand constitutive model is calibrated on the results of laboratory element tests. PM4Sand has 27 input parameters (6 primary and 21 secondary) but only 3 of them are required as independent inputs: the initial relative density (Dr), the shear modulus coefficient used to define the small strain shear modulus (G0), and the contraction rate parameter used for the calibration of the undrained shear strength (hp0). Basically, these three parameters were calibrated against the experimental data. The initial relative density has been set equal the value of relative density used in the cyclic torsional tests, Dr = 0.5 and 0.6.

The value of the shear modulus coefficient G0 was determined as a function of the relative density using the follow relationship:

$$ {G}_0=167\cdotp \sqrt{46\cdotp {D}_{\mathrm{r}}^2+2.5} $$
(16.8)

The parameter hp0 scales the plastic contraction rate and is the primary parameter for the calibration of undrained cyclic strength. It is calibrated using an iterative process, in which undrained single-element simulations are conducted to match with the experimental liquefaction triggering curve by keeping the other parameters fixed.

With reference to the secondary parameters of the model, some with a clear physical meaning have been defined on the available experimental data, while the others have been left with their default values.

Shear strength parameters are computed from the monotonic triaxial test data, available on the NEES Hub (https://www.re3data.org/repository/r3d100010105).

Drained triaxial compression tests, carried out by Vasko (2015) on loose and dense specimens, were used to define the critical state line in the plane q: p′ and the constant volume friction angle, \( {\phi}_{\mathrm{c}}^{\prime } \). As well known, the evaluation of critical state conditions in triaxial tests is a very complex issue, being such a test intrinsically affected by a number of experimental limitations (localization, bulging, shear stresses on the rough porous stones, difference between local and external displacements, etc.). One of the best ways to evaluate the final state is therefore the one that analyses dilatancy trend at the end of the tests. Based on all the elaborations of the available experimental data, the best fit of this parameter is the following:

$$ {\phi}_c={32}^{{}^{\circ}} $$
(16.9)

Minimum and maximum void ratios, emax and emin, have been defined as mean values of the experimental measurements carried out in the LEAP-UCD-2017 Simulation Exercise (Manzari et al., 2019).

To sum up, the model parameters for static loading conditions were defined on the physical properties and tests results provided for the considered sand.

Conversely, the model parameters for cyclic loading conditions were defined using experimental data of cyclic torsional tests (Dr = 50% and 60%).

The material parameters used to perform the simulations of the laboratory tests are those reported in Table 16.4 for each relative density. Every cyclic test is simulated imposing the prescribed CSR and computing the number of cycles, NL, to induce liquefaction. Liquefaction condition has been defined according to the stress-based approach, i.e., ru = 95%, where ru is the excess pore pressure ratio (\( {r}_{\mathrm{u}}=\varDelta u/{\sigma}_{\mathrm{m}0}^{\prime } \) ratio between the excess pore water pressure increment induced by cyclic loading and the initial effective confining pressure applied during the test, \( {\sigma}_{\mathrm{m}0}^{\prime } \)).

The liquefaction strength curves, obtained from the simulated cyclic torsional tests, are hereafter plotted and compared with the experimental results (Fig. 16.3). Table 16.3 reports the numerical values of the simulation results, i.e., the cyclic stress ratio, CSR, versus the number of cycles until excess pore pressure ratio, \( {r}_{\mathrm{u}}=\varDelta u/{\sigma}_{\mathrm{m}0}^{\prime } \), achieved 95% for each simulated test.

Fig. 16.3
A graph of C R R versus N subscript L. It has four plots for experiment and simulation values with D R equals 50 and 60 percentages. Both have negtively correlating simulation. 50% plots for experiment and lines of simulation nearly merge.

Liquefaction strength curves obtained from experimental and simulated cyclic torsional tests on Ottawa F65 Sand

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It can be observed how the adopted calibration provides a good prediction of the experimental cyclic resistance curve for high/medium values of the cyclic resistance ratio (CRR), while underestimation of the experimental cyclic strength is observed for low values of CRR (Table 16.5).

Table 16.5 Predicted liquefaction strength curves from cyclic torsional test
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Figures 16.4 and 16.5 compare the model performance on the prediction of the soil volume element response as obtained by cyclic torsional shear tests, for two different values of CSR and relative density. In both cases, the calibrated constitutive model is able to correctly simulate the shear stress-strain cycles and also the time histories of shear stress, strain, and excess pore water pressure.

Fig. 16.4
Six graphs with measured and computed curves. a. Shear stress versus shear strain. b. Shear stress versus mean effective stress. c, d, e, and f. Shear stress, shear strain, E P W P, and E P W P ratio versus number of cyclic loading. The curves oscillate or increase, and are mostly merged.

Torsional tests Dr = 50%. CSR = 0.127; Number of cycles until ru = 95% is achieved =25.5. (a) Shear stress-strain cycles, (b) stress path, (c) shear stress, (d) shear strain, (e) excess pore water pressure, and (f) excess pore pressure ratio as function of the number of cycles

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Fig. 16.5
Six graphs. a. Shear stress versus shear strain. b. Shear stress versus mean effective stress. c, d, e, and f. Shear stress, shear strain, E P W P, and E P W P ratio versus number of cyclic loading. The measured and computed curves mostly merge.

Torsional tests Dr = 60%. CSR = 0.174; Number of cycles until ru = 95% is achieved =13. (a) Shear stress-strain cycles, (b) stress path, (c) shear stress, (d) shear strain, (e) excess pore water pressure, and (f) excess pore pressure ratio as function of the number of cycles

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4.3 Initial and Boundary Conditions and Input Motions

The simulations are carried out by using PLAXIS (Brinkgreve et al., 2016) as the analysis platform. PLAXIS is a 2D commercial Finite Element Method (FEM) code that includes several constitutive models. Among them, the PM4Sand model (Boulanger & Ziotopoulou, 2015) has been adopted as constitutive model in the simulation exercise.

The main reason to use PLAXIS rather than other platforms where such a constitutive model is implemented is that this numerical code, although not specifically oriented to solve boundary value problems in earthquake geotechnical engineering, is quite well widespread in the community of geotechnical practitioners (Fasano et al., 2019b). Hence, it was for this team interesting to check the possible benefit of a rigorous validation of numerical simulation procedures implemented in PLAXIS through experimental data, in order to apply those procedures to a boundary value problem involving soil liquefaction.

Figure 16.6 shows the mesh density and the boundary conditions. The mesh consists of 443 15-noded triangular elements. The nodes located at the base are constrained in y direction while an acceleration time history in direction x is applied on the base and lateral boundaries. The nodes on the ground surface allow full drainage from the base to the top of the scheme.

Fig. 16.6
A rectangular mesh diagram with a decreasing line on top having downward arrows that increase in height from left to right. The other 3 sides have rightward arrows.

Finite element model with applied boundary conditions

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The Newmark time integration scheme is used in the simulations where the time step is constant and equal to the critical time step during the whole analysis. The proper critical time step for dynamic analyses is estimated in order to accurately model wave propagation and reduce error due to integration of time history functions.

A full Rayleigh damping formulation has been considered in the simulation and the coefficient αRAY and βRAY are equal to 0.02513 and 6.366* 10−3, respectively.

The soil properties are not changed during the simulations.

Table 16.6 shows the list of model parameters used in the five simulations (A and B models). The model parameters are the same obtained from the calibration, some of them are just updated to take into account for the different relative density used in the experiments.

Table 16.6 Parameters of the constitutive model
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5 Results of Type-C Simulations

Numerical analyses carried out with the abovementioned constitutive model are presented in this section, starting from the description of the calibration process, geometry definition, and boundary conditions of the model.

The results of the Type-C simulations for model B are here reported (RPI_A_B1_1 and KyU_A_B2_1), while all simulation results can be found in Fasano et al. (2019a).

Figures 16.7 and 16.8 report the comparison between recorded and simulated time histories of acceleration and excess pore water pressure for the centrifuge test RPI_A_B1_1.

Fig. 16.7
Fourteen graphs of a of g versus t in seconds with horizontally moving oscillating merged curves for centrifuge and numerical results. A V 1 and A V 2 have small amplitude oscillations in the near center. A H, 1 to 12 curves have high amplitudes in the near center with varying degrees.

Acceleration time histories for RPI_A_B1_1 (centrifuge results in black lines and numerical results in grey lines)

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Fig. 16.8
10 graphs of P W P versus t in seconds. The centrifuge and numerical results lines nearly merge in P 6, P 4, P 8, P 9, and P 10 with mostly ascending trends. P 5, P 3, P 7, P 2, and P 1 curves have horizontal or ascending centrifuge lines. All curves mostly oscillate in the near center.

Excess pore pressure time histories for RPI_A_B1_1 (centrifuge results in black lines and numerical results in grey lines)

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The comparison shows that the amplitude of the simulated time histories is quite similar to the experimental one in centrifuge test, except for some spikes recorded by the sensors located in the center of the box (Fig. 16.7).

A reasonable prediction of pore water pressure is also provided by the numerical simulation, even though some underestimation is related to the deepest sensors from the surface of the box (Fig. 16.8).

Figures 16.9 and 16.10 report the comparison between recorded and simulated time histories of acceleration and excess pore water pressure for the centrifuge test KYU_A_B2_1.

Fig. 16.9
Fourteen graphs of a of g versus t in seconds with horizontally moving oscillating merged curves for centrifuge and numerical results. A V 2 is horizontal. A V 1 has small amplitudes. A H 1 to 12 curves have high amplitudes in the near center with varying degrees.

Acceleration time histories for KYU_A_B2_1 (centrifuge results in black lines and numerical results in grey lines)

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Fig. 16.10
10 graphs of P W P versus t in seconds. The centrifuge and numerical results lines have ascending and descending trends. Curves in P 6, P 4, P 8, P 9, and P 10 are nearly merged. P 5 and P 7 have numerical results curves. Centrifuge in P 3 and P 2 are higher.

Excess pore pressure time histories for KYU_A_B2_1 (centrifuge results in black lines and numerical results in grey lines)

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The amplitude of the simulated time histories catches the experimental one better than the previous test, even though a probable inversion of the sign of the measured accelerations (Fig. 16.9).

With reference to the pore water pressure, an underestimation of the maximum values is provided by the numerical simulation, while the general trend is adequately reproduced (Fig. 16.10).

6 Conclusions

The numerical simulations carried out at the University of Napoli Federico II in the framework of the LEAP-ASIA-2018 Simulation Exercise were described in the paper. A bounding surface plasticity model for sand was calibrated on the results of cyclic torsional shear tests. The calibration correctly reproduced the cyclic strength of the tested sand and the pre-failure behavior observed in terms of shear stress-strain cycles, even though the limitations related to the adoption of plane-strain conditions in the numerical model. Centrifuge test simulations have been performed by means of a finite element code, PLAXIS, largely adopted in the community of geotechnical practitioners. Type-C simulations highlighted the capability of the numerical model to reasonably predict the time histories of acceleration and excess pore water pressure measured during the experimental tests.