# LEAP-2017 Centrifuge Test Simulation Using HiPER

Open Access
Conference paper

## Abstract

Simulations by effective stress analysis were carried out for nine centrifuge experiments of a gentle sloped ground. The constitutive equation is a hyperbolic model as a stress–strain relationship and a bowl model as a strain–dilatancy relationship. The experimental results largely vary depending on density and input acceleration. The analyses results can explain such a tendency of variability.

## Keywords

Effective stress analysis Slope Lateral spreading Numerical simulation

## 23.1 Outline of the Analysis Method

Analysis is done by FEM and the software used is “HiPER.” The integration scheme is implicit and the integration time interval is 0.002 s. The Newmark β a numerical integration method is adopted (β = 1/4). The method used for convergence is a modified Newton Raphson method. Convergence is determined as a relative force imbalance of 1.0E-3 or less. The maximum number of iterations is four; if convergence is not reached, the unbalanced force is carried over to the next step.

To ensure analytical stability, a low stiffness proportional damping (C = βK, where K is stiffness matrix and β = 1.0E−3) is used.

The u-p formulation is used and excess pore water pressure is evaluated at nodes (the Sandhu method (Sandhu and Wilson 1969)).

## 23.2 Outline of the Constitutive Equations

In this section, the three-dimensional stress–strain-dilatancy relationship is explained. A hyperbolic model extending in three dimensions is used for the stress–strain relationship, while the strain–dilatancy relationship is modeled with a bowl function. The hyperbolic stress–strain model parameters are determined from dynamic deformation tests (G/Gmaxγ, hγ relationships). The bowl model parameters are determined from liquefaction resistance tests (from the relationship between stress ratio and number of cycles).

### 23.2.1 Hyperbolic Model and Its Parameters

In multidimensionalizing the hyperbolic model, the shear stress versus shear strain relationships for the shear component and the axial difference component, respectively, are defined by the following equations.
$${\tau}_{xy}=\frac{G_{\mathrm{max}}\cdot {\gamma}_{xy}}{1+\frac{\gamma_{xy}}{\gamma_{\mathrm{r}}}},\kern0.62em {\tau}_{yz}=\frac{G_{\mathrm{max}}\cdot {\gamma}_{yz}}{1+\frac{\gamma_{yz}}{\gamma_{\mathrm{r}}}},\kern0.62em {\tau}_{zx}=\frac{G_{\mathrm{max}}\cdot {\gamma}_{zx}}{1+\frac{\gamma_{zx}}{\gamma_{\mathrm{r}}}}$$
(23.1a)
$$\begin{array}{c}\frac{\sigma_x-{\sigma}_y}{2}=\frac{G_{\mathrm{max}}\cdot \left({\varepsilon}_x-{\varepsilon}_y\right)}{1+\frac{\left({\varepsilon}_x-{\varepsilon}_y\right)}{\gamma_{\mathrm{r}}}},\kern0.75em \frac{\sigma_y-{\sigma}_z}{2}=\frac{G_{\mathrm{max}}\cdot \left({\varepsilon}_y-{\varepsilon}_z\right)}{1+\frac{\left({\varepsilon}_y-{\varepsilon}_z\right)}{\gamma_{\mathrm{r}}}},\\ {}\kern0.75em \frac{\sigma_z-{\sigma}_x}{2}=\frac{G_{\mathrm{max}}\cdot \left({\varepsilon}_z-{\varepsilon}_x\right)}{1+\frac{\left({\varepsilon}_x-{\varepsilon}_x\right)}{\gamma_{\mathrm{r}}}}\end{array}}$$
(23.1b)

When solving two-dimensional problems, the hyperbolic model is used for the shear components τxy − γxy and the axial difference components (σx − σy)/2 − εx − εy.

Here, Gmax is the initial shear modulus, and γr is the reference strain. γr is obtained from the shear strength τf by the following equation.
$${\gamma}_{\mathrm{r}}=\frac{\tau_f}{G_{\mathrm{max}}}$$
(23.2)
The h–γ relationship is given by the following equation.
$$h={h}_{\mathrm{max}}\cdot \left(1-\frac{G}{G_{\mathrm{max}}}\right)$$
(23.3)
where, h is hysteretic damping parameter, hmax is the maximum damping ratio and G is shear modulus.
Three parameters are required to construct the hyperbolic model; Gmax, hmax, and γr. Of these, Gmax and γr are functions of effective stress. If Gmax and γr at a certain reference effective stress σmi are Gmaxi and γri, then Gmax and γr satisfy the following equations.
$${G}_{\mathrm{max}}={G}_{\max i}{\left(\frac{\sigma_m^{\prime }}{\sigma_{mi}^{\prime }}\right)}^{0.5},\kern0.5em {\gamma}_{\mathrm{r}}={\gamma}_{ri}{\left(\frac{\sigma_m^{\prime }}{\sigma_{mi}^{\prime }}\right)}^{0.5}$$
(23.4)

As the effective stress varies, at each incremental calculation step, these parameters are calculated as they vary with time based on the above equations. At the same time, the shear stress versus shear strain relationship in Eq. 23.1 varies with time in accordance with the effective stress.

Applying the Masing rule to the hyperbolic model results in excessive hysteretic damping. Therefore, hysteretic damping h is adjusted using the method described by Ishihara et al. (1985).

The three parameters of the hyperbolic model (Gmaxi, hmax, γri) are defined in Table 23.1 and Fig. 23.1. As shown in Eq. 23.4 above, Gmax and γr depend on the effective stress. The formulation Gmaxi, γri is used when the mean effective stress σm is 1.0 kN/m2. These values can be simply determined from the stiffness reduction curve (G/Gmax versus γ relationship) or the damping increase curve (hγ relationship) obtained from the dynamic deformation tests.
Table 23.1

Parameters of the hyperbolic model

Parameters

Physical meaning of parameters

Gmax

Initial shear modulus. Gmax = ρVs2

hmax

Maximum damping ratio. As hmax increases the nonlinearity becomes stronger

γr

Reference strain. γr = τf/G0

The shear strain when G/Gmax = 0.5

### 23.2.2 Bowl Model (Dilatancy Model) and Its Parameters

To model deformation in three dimensions, in addition to the shear strains (γzx, γyx, γxy) representing simple shearing deformation and the axial shear strain differentials (εx − εy, εy − εz, εz − εx) representing axial deformation differential, definitions of resultant shear strain Γ and cumulative shear strain G are needed. They are defined by the following equations.
$$\Gamma =\sqrt{\gamma_{zx}^2+{\gamma}_{zy}^2+{\gamma}_{xy}^2+{\left({\varepsilon}_x-{\varepsilon}_y\right)}^2+{\left({\varepsilon}_y-{\varepsilon}_z\right)}^2+{\left({\varepsilon}_z-{\varepsilon}_x\right)}^2}$$
(23.5)
$${G}^{\ast }=\sum \Delta {G}^{\ast }=\sum \sqrt{{\Delta \gamma}_{zx}^2+{\Delta \gamma}_{zy}^2+{\Delta \gamma}_{xy}^2+\Delta {\left({\varepsilon}_x-{\varepsilon}_y\right)}^2+\Delta {\left({\varepsilon}_y-{\varepsilon}_z\right)}^2+\Delta {\left({\varepsilon}_z-{\varepsilon}_x\right)}^2}$$
(23.6)
The bowl model (as proposed by Fukutake and Matsuoka 1989, 1993) focuses on the resultant shear strain Γ and the cumulative shear strain G. Dilatancy εsv results from a certain soil particle repeatedly dropping into the valley between other soil particles (negative dilatancy), and rising up on other soil particles (positive dilatancy). This mechanism is represented by the superposition of positive and negative dilatancy using the following equation.
$${\varepsilon}_v^s={\varepsilon}_{\Gamma}+{\varepsilon}_G=\mathrm{A}\cdot {\Gamma}^{1.4}+\frac{{\mathrm{G}}^{\ast }}{\mathrm{C}+\mathrm{D}\cdot {\mathrm{G}}^{\ast }}$$
(23.7)

Here A, C, and D are parameters. εG is monotonic negative dilatancy (compressive strain). It is irreversible and represented as a hyperbolic function with respect to G. εΓ is cyclic positive dilatancy (swelling strain) and is reversible and represented as an exponential function with respect to Γ. The εG component is the master curve and is the component that determines the basic dilatancy during cyclic shearing, while the εΓ component is a oscillating component associated with it. 1/D is the asymptotic line to the hyperbolic curve, corresponding to a relative density of 100%.

The mechanism of such a bowl model is the movement of soil particles in seven-dimensional strain space with γxy, γyz, γzx, (εx − εy), (εy − εz), (εz − εx), and εsv as axes.

Figure 23.2 illustrates the case of unidirectional cyclic shearing. Fig. 23.2Dilatancy in unidirectional cyclic shearing

Next, the consolidation term is taken into consideration in the stress–strain relationship and effective stress is modeled under undrained (constant volume) conditions.

The volumetric strain εvs represented by Eq. 23.7 is the dilatancy component due to shearing, but an additional volumetric strain due to change in the effective stress σm has to be considered. The total volumetric strain increment of the soil v is then given by the following equation, where the shear component is vs and the consolidation component is vc.
${\mathit{d\epsilon }}_{\mathit{v}}={{\mathit{d\epsilon }}_{\mathit{v}}}^{s}+{{\mathit{d\epsilon }}_{\mathit{v}}}^{c}$
(23.8)
The consolidation term vc is given by the following equations, assuming one-dimensional consolidation conditions.
${{\mathit{d\epsilon }}_{\mathit{v}}}^{c}=\frac{0.434\cdot {C}_{s}}{1\cdot {e}_{0}}\cdot \frac{{{\mathit{d\sigma }}^{\prime }}_{m}}{{{\mathit{\sigma }}^{\prime }}_{m}}\phantom{\rule{0.62em}{0ex}}\left(\text{for}\phantom{\rule{0.24em}{0ex}}{{\mathit{d\sigma }}^{\prime }}_{m}<0\right)$
(23.9)
${{\mathit{d\epsilon }}_{\mathit{v}}}^{c}=\frac{0.434\cdot {C}_{c}}{1\cdot {e}_{0}}\cdot \frac{{{\mathit{d\sigma }}^{\prime }}_{m}}{{{\mathit{\sigma }}^{\prime }}_{m}}\phantom{\rule{0.86em}{0ex}}\left(\text{for}\phantom{\rule{0.24em}{0ex}}{{\mathit{d\sigma }}^{\prime }}_{m}>0\right)$
(23.10)
Here, Cs is the swelling index, Cc is the compression index, and ${e}_{0}$ is the initial void ratio. If undrained conditions (v = 0) are assumed in Eqs. 23.9 and 23.10, the following equation is obtained.
${{\mathit{d\epsilon }}_{\mathit{v}}}^{s}+\frac{0.434\cdot {C}_{s}}{1+{e}_{0}}\cdot \frac{{{\mathit{d\sigma }}^{\prime }}_{m}}{{{\mathit{\sigma }}^{\prime }}_{m}}=0$
(23.11)
If the mean effective stress in the initial shearing stage is σm0, and if the above equation is integrated under the initial conditions (σm = σm0), the following equation is obtained.
${{\mathit{\epsilon }}_{\mathit{v}}}^{s}+\frac{{C}_{s}}{1+{e}_{0}}\cdot log\phantom{\rule{0.24em}{0ex}}\frac{{{\mathit{\sigma }}^{\prime }}_{m}}{{{\mathit{\sigma }}^{\prime }}_{m0}}=0$
(23.12)
From Eq. 23.12, the effective stress under undrained shear σm is given by the following equations.
${{\sigma }^{\prime }}_{m}={{\sigma }^{\prime }}_{m0}\cdot 10{}^{\alpha }\phantom{\rule{0.12em}{0ex}},\phantom{\rule{0.74em}{0ex}}\alpha \equiv \frac{-{{\mathit{\epsilon }}_{\mathit{v}}}^{s}}{{C}_{s}/\left(1+{e}_{0}\right)}$
(23.13)
Therefore, substituting α from the above equation, the effective stress reduction ratio is given by the following equation:
$$\left(\frac{\sigma_{m0}^{\prime }-{\sigma}_m^{\prime }}{\sigma_{m0}^{\prime }}\right)=1-10{}^{\alpha }$$
(23.14)
In order to suppress the occurrence of dilatancy under small shear amplitude, a spherical region with shear strain radius Γ = Re is considered within the strain space, and within this region there is no G. Re is given as follows, with reference to Fig. 23.3. Fig. 23.3Lower limit of liquefaction resistance Xl and shear strain Re. (a) Liquefaction resistance curve and lower limit of liquefaction resistance Xl. (b) Effective stress path and Xl. (c) Skeleton curve of hyperbolic model and Re
Figure 23.3a shows the liquefaction resistance curve and the relationship between liquefaction resistance and the lower limit value Xl. Xl is the liquefaction resistance after a very large number of cycles, in other words, it represents the stress ratio at which liquefaction does not occur even however many cycles occur. Also, for simplicity, it is assumed that the excess pore water pressure does not arise (Pw = 0) for repeated stress ratios less than Xl. From Fig. 23.3b, c, setting the shear strain when the stress ratio is Xl equal to Re, then Re is given by the following equation.
$${R}_e=\frac{X_l\sigma {\hbox{'}}_{m0}}{G_{\mathrm{max}}-\frac{X_l\sigma {\hbox{'}}_{m0}}{\gamma_{\mathrm{r}}}}$$
(23.15)

Here σm0 is the mean effective stress in initial shear. When the amplitude of the stress ratio is Xl or less, positive excess pore water pressure does not arise.

There are six parameters in the bowl model (A, C, D, Cs/(1 + e0), Cc/(1 + e0) and Xl) as shown in Table 23.2. The values of these parameters are determined from a liquefaction resistance curve.
Table 23.2

Parameters of bowl model

Parameter

Physical meaning of parameters

A

Parameter representing the swelling component εΓ of the dilatancy components. The larger the absolute value of A, the greater the cyclic mobility

C, D

Parameters representing the compression component εG of the dilatancy components. 1/C is the slope of the dilatancy in the initial stage of shear. 1/D is calculated from the minimum void ratio emin on the hyperbolic asymptotic line (maximum amount of compression)

Cs/(1 + e0)

Cs is the swelling index; e0 is the initial void ratio

Cc/(1 + e0)

Cs is the compression index; e0 is the initial void ratio

Xl

The lower limit value of the liquefaction resistance. In the relationship between stress ratio τ/σm0 and the number of cycles Nc, it is represented by τ/σm0 when Nc is sufficiently large. When τ/σm0 > Xl excess pore water pressure arises

Table 23.2 and Fig. 23.4 give the meanings of the bowl model parameters. These parameters are determined by fitting to the liquefaction resistance curve. Fig. 23.4Liquefaction resistance curves and bowl model parameters C and Xl

## 23.3 Element Test Simulation

### 23.3.1 Determining Parameters by Test Simulation

#### Hyperbolic Model

The initial shear stiffness Gmax is calculated from the following equation based on research by Liu et al. (2017).
${G}_{\mathrm{max}}=6.35\frac{1}{0.3+0.7{e}^{2}}{\sigma }_{m}^{0.5}$
(23.16)
Here γr and hmax are standard values for sand based on the shear strain dependence of stiffness ratio and damping for Toyoura sand (G/Gmaxγ, h–γ relationship). Since the experiment of Ottawa sand was not carried out, it replaced it with Toyoura sand experiment. Figure 23.5 shows the G/Gmaxγ, hγ relationship used in calculations. Fig. 23.5G/Gmax–γ, h–γ relationship used in calculations (hyperbolic model)

#### Bowl Model

The parameter D is determined from the following equation since 1/D is the asymptote of dilatancy. The average of emin is 0.49.
$\frac{1}{D}=\frac{{e}_{0}-{e}_{\mathrm{min}}}{1+{e}_{0}}$
(23.17)

The lower limit value of liquefaction resistance Xl is determined based on the liquefaction resistance curve. The liquefaction resistance curve is calculated using values for standard sand for the other parameters. Parameter C is adjusted while comparing the test results and the calculation results so as to match the whole liquefaction resistance curve with the measured values.

The parameters for Ottawa sand set by this procedure are given in Table 23.3.
Table 23.3

Parameters for Ottawa sand

Parameter of constitutive equation

No

σm0(kN/m2)

e

ρd (t/m3)

Vs (m/s)

G0 (kN/m2)

Reference strain (γr)

Hypabolic model

Bowl model

G0i (kN/m2)

γri

hmax (%)

A

C

D

Cs (1 + e0)

Cc (1 + e0)

Xl

1

100

0.515

1.7442

274

130751

0.0005

13075

5.0E−5

27

−1.5

35.0

61

0.0060

0.0061

0.20

2

100

0.542

1.7126

271

125585

0.0005

12558

5.0E−5

27

−0.8

9.0

30

0.0060

0.0061

0.16

3

100

0.585

1.6656

266

117689

0.0005

11769

5.0E−5

27

−0.6

6.0

17

0.0060

0.0061

0.09

### 23.3.2 Results of Element Test Simulations

Figure 23.6 shows the simulations of liquefaction resistance curves. They explain the experiment results for the three selected densities. Fig. 23.6Liquefaction resistant curves
Figure 23.7 shows the effective stress path and stress–strain relationship at these three different densities. In the dense case, strong cyclic mobility is present. Fig. 23.7Effective stress path and stress strain relationship. (a) e0 = 0.515. (b) e0 = 0.542. (c) e0 = 0.585

## 23.4 Centrifuge Simulation

### 23.4.1 Determination of Parameters

The model parameters (that is to say the G/G0γ, hγ relationship and the liquefaction resistance curve) are determined by element test simulations of Phase I except for G0 and D. G0 and D are reset by the following procedure for each centrifuge test. From the density of sand (Table 23.4), the relative density Dr is obtained, and the void ratio e is calculated from Dr. The initial shear stiffness G0 is calculated from Eq. 23.16 based on the research of Liu et al. 2017. G0i is the value at unit mean stress (σm = 1.0 kN/m2). Parameter D is determined from Eq. 23.17 since 1/D is the asymptote of dilatancy. The average of emin is 0.49.
Table 23.4

Soil density and void ratio of Ottawa sand

 ρdmax (kg/m3) 1765 ρdmin (kg/m3) 1480 emax 0.766 emin 0.49
Table 23.5 shows the parameters set by this method. The Poisson ratio of the ground is set to 0.33. The bulk modulus of water is Kw = 2.2E + 6 kN/m2. For soil permeability, k = 0.015 cm/s is used from experimental results.

### 23.4.2 Summary of the Numerical Simulations

The FEM mesh is shown in Fig. 23.8. The FEM model has 2013 nodes and 1920 elements. A plane strain four-noded quadrilateral element with displacements and pore pressure degrees of freedom at each node is employed. Four integration points are used in each element. The u-p formulation is used. The excess pore water pressure is evaluated at the nodes (the Sandhu method). Fig. 23.8Boundary conditions and time history output points (black square box acceleration, red square box pore water pressure)

The nodes located at the base of the model are fully constrained in the x and y directions while the nodes on the side walls are constrained laterally. The nodes on the free surface allow full drainage and have a fixed pore water pressure.

Figure 23.9 shows the initial stress calculated by linear analysis. The initial stress is determined by linear self-weight analysis. For stiffness, the value at 4 m of depth is used. Fig. 23.9Initial stress calculated by linear analysis

In the dynamic analysis, horizontal acceleration and vertical acceleration were input simultaneously.

### 23.4.3 Results of Centrifuge Simulations

Horizontal displacement time histories of the ground surface at the center are shown in Fig. 23.10. The amount of lateral spreading of the ground surface was of the same order in experiment and analysis. For UCD3, Ehime2, and CUD1, the simulated values and experimental values show good correspondence. The analysis values are rather smaller for CU 2, ZJU 2, and NCU 3. The experimental show large variation depending on density and input acceleration. The simulations successfully explain this tendency. Fig. 23.10Horizontal displacement time histories of ground surface at the center. (a) Simulations, (b) experiments (displacements from ACC and surface marker)
Figure 23.11 shows fivefold enlarged deformation plots and contours of excess pore water pressure ratio. For KAIST1, KyU3, and UCD1, liquefaction does not occur and deformations are small. In the other six cases, the ground liquefies and deforms in the downstream direction. Fig. 23.11Fivefold enlarged deformation and excess pore pressure ratio distribution (contour) at 25 s. (a) CU2, (b) Ehime2, (c) KAIST1, (d) KAIST2, (e) KyU3, (f) NCU3, (g) UCD1, (h) UCD3, and (i) ZJU2
According to Fig. 23.12, shear strain accumulates in the direction in which the initial shear stress acts. The reverse warping tendency of the stress–strain relationship due to cyclic mobility is not noticeable. Fig. 23.12Stress–strain and effective stress path at P3. (a) CU2, (b) Ehime2, (c) KAIST1, (d) KAIST2, (e) KyU3, (f) NCU3, (g) UCD1, (h) UCD3, and (i) ZJU2

Generally, horizontal deformation is large in the upper layer at the center of the slope. Vertical deformation is dominated by sinking on the upper side of the slope. Deformation peaks at the end of the excitation, and then recovers slightly during the process of dissipating excess pore water pressure. Excess pore water pressure is fully dissipated after 600 s.

## 23.5 Conclusion

We simulated the lateral spreading of sloping ground using a hyperbolic model and a bowl model for comparison against experimental results. The parameters of the constitutive equation were determined based on simulation of element tests. Experimental results show large variation with density and input acceleration. The simulations successfully explain this tendency to variability. The lateral spreading of the ground surface was of the same order in the experiments and simulations.

## References

1. Fukutake, K., & Matsuoka, H. (1989). A unified law for dilatancy under multi-directional simple shearing. In Proceedings of Japan Society of Civil Engineers, No. 412/III-12 (pp. 143–151).Google Scholar
2. Fukutake, K., & Matsuoka, H. (1993). Stress-strain relationship under multi-directional cyclic simple shearing. In Proceedings of Japan Society of Civil Engineers, No. 463/III-22 (pp. 75–84).Google Scholar
3. Ishihara, K., Yoshida, N., & Tsujino, S. (1985). Modelling of stress-strain relations of soils in cyclic loading. In Proceedings of 5th International Conference for Numerical Method in Geomechanics, Nagoya (Vol. 1, pp. 373–380).Google Scholar
4. Liu, K., Zhou, Y., & Chen, Y. (2017). Hardin equation of Ottawa sand-F65 for LEAP.Google Scholar
5. Sandhu, R. S., & Wilson, E. L. (1969). Finite-element analysis of seepage in elastic media. Proceedings of ASCE, 95(EM3), 641–652.Google Scholar