LEAP-2017 Centrifuge Test Simulation Using HiPER

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.

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)).

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/G_max–γ, h–γ relationships). The bowl model parameters are determined from liquefaction resistance tests (from the relationship between stress ratio and number of cycles).

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)

$$ {\displaystyle \begin{array}{c}\frac{\sigma_x-{\sigma}_y}{2}=\frac{G_{\mathrm{max}}\cdot \left({\varepsilon}_x-{\varepsilon}_y\right)}{1+\frac{\left({\varepsilon}_x-{\varepsilon}_y\right)}{\gamma_{\mathrm{r}}}},\kern0.75em \frac{\sigma_y-{\sigma}_z}{2}=\frac{G_{\mathrm{max}}\cdot \left({\varepsilon}_y-{\varepsilon}_z\right)}{1+\frac{\left({\varepsilon}_y-{\varepsilon}_z\right)}{\gamma_{\mathrm{r}}}},\\ {}\kern0.75em \frac{\sigma_z-{\sigma}_x}{2}=\frac{G_{\mathrm{max}}\cdot \left({\varepsilon}_z-{\varepsilon}_x\right)}{1+\frac{\left({\varepsilon}_x-{\varepsilon}_x\right)}{\gamma_{\mathrm{r}}}}\end{array}} $$

(23.1b)

When solving two-dimensional problems, the hyperbolic model is used for the shear components τ_xy − γ_xy and the axial difference components (σ_x − σ_y)/2 − ε_x − ε_y.

Here, G_max is the initial shear modulus, and γ_r is the reference strain. γ_r is obtained from the shear strength τ_f 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, h_max is the maximum damping ratio and G is shear modulus.

Three parameters are required to construct the hyperbolic model; G_max, h_max, and γ_r. Of these, G_max and γ_r are functions of effective stress. If G_max and γ_r at a certain reference effective stress σ′_mi are G_maxi and γ_ri, then G_max and γ_r 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 (G_maxi, h_max, γ_ri) are defined in Table 23.1 and Fig. 23.1. As shown in Eq. 23.4 above, G_max and γ_r depend on the effective stress. The formulation G_maxi, γ_ri is used when the mean effective stress σ′_m is 1.0 kN/m². These values can be simply determined from the stiffness reduction curve (G/G_max versus γ relationship) or the damping increase curve (h–γ relationship) obtained from the dynamic deformation tests.

Table 23.1 Parameters of the hyperbolic model

Fig. 23.1

Parameters of hyperbolic model: h_max, γ_r

2.2 Bowl Model (Dilatancy Model) and Its Parameters

To model deformation in three dimensions, in addition to the shear strains (γ_zx, γ_yx, γ_xy) representing simple shearing deformation and the axial shear strain differentials (ε_x − ε_y, ε_y − ε_z, ε_z − ε_x) 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 ε^s_v 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. ε_G is monotonic negative dilatancy (compressive strain). It is irreversible and represented as a hyperbolic function with respect to G^∗. ε_Γ is cyclic positive dilatancy (swelling strain) and is reversible and represented as an exponential function with respect to Γ. The ε_G component is the master curve and is the component that determines the basic dilatancy during cyclic shearing, while the ε_Γ 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 γ_xy, γ_yz, γ_zx, (ε_x − ε_y), (ε_y − ε_z), (ε_z − ε_x), and ε^s_v as axes.

Figure 23.2 illustrates the case of unidirectional cyclic shearing.

Fig. 23.2

Dilatancy 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 ε_v^s represented by Eq. 23.7 is the dilatancy component due to shearing, but an additional volumetric strain due to change in the effective stress σ′_m has to be considered. The total volumetric strain increment of the soil dε_v is then given by the following equation, where the shear component is dε_v^s and the consolidation component is dε_v^c.

$${\mathit{d\epsilon }}_{\mathit{v}}={{\mathit{d\epsilon }}_{\mathit{v}}}^s+{{\mathit{d\epsilon }}_{\mathit{v}}}^c$$

(23.8)

The consolidation term dε_v^c is given by the following equations, assuming one-dimensional consolidation conditions.

$${{\mathit{d\epsilon }}_{\mathit{v}}}^c=\frac{0.434\cdot C_s}{1\cdot e_0}\cdot \frac{{{\mathit{d\sigma }}^{\prime }}_m}{{{\mathit{\sigma }}^{\prime }}_m}\left(\text{for}{{\mathit{d\sigma }}^{\prime }}_m<0\right)$$

(23.9)

$${{\mathit{d\epsilon }}_{\mathit{v}}}^c=\frac{0.434\cdot C_c}{1\cdot e_0}\cdot \frac{{{\mathit{d\sigma }}^{\prime }}_m}{{{\mathit{\sigma }}^{\prime }}_m}\left(\text{for}{{\mathit{d\sigma }}^{\prime }}_m>0\right)$$

(23.10)

Here, C_s is the swelling index, C_c is the compression index, and $e_0$ is the initial void ratio. If undrained conditions (dε_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 σ′_m0, and if the above equation is integrated under the initial conditions (σ′_m = σ′_m0), the following equation is obtained.

$${{\mathit{\epsilon }}_{\mathit{v}}}^s+\frac{C_s}{1+e_0}\cdot log\frac{{{\mathit{\sigma }}^{\prime }}_m}{{{\mathit{\sigma }}^{\prime }}_{m0}}=0$$

(23.12)

From Eq. 23.12, the effective stress under undrained shear σ′_m is given by the following equations.

$${{\sigma }^{\prime }}_m={{\sigma }^{\prime }}_{m0}\cdot 10{}^{\alpha },\alpha \equiv \frac{-{{\mathit{\epsilon }}_{\mathit{v}}}^s}{C_s/\left(1+e_0\right)}$$

(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 = R_e is considered within the strain space , and within this region there is no dε_G. R_e is given as follows, with reference to Fig. 23.3.

Fig. 23.3

Lower limit of liquefaction resistance X_l and shear strain R_e. (a) Liquefaction resistance curve and lower limit of liquefaction resistance X_l. (b) Effective stress path and X_l. (c) Skeleton curve of hyperbolic model and R_e

Figure 23.3a shows the liquefaction resistance curve and the relationship between liquefaction resistance and the lower limit value X_l. X_l 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 (P_w = 0) for repeated stress ratios less than X_l. From Fig. 23.3b, c, setting the shear strain when the stress ratio is X_l equal to R_e, then R_e 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 σ′_m0 is the mean effective stress in initial shear. When the amplitude of the stress ratio is X_l or less, positive excess pore water pressure does not arise.

There are six parameters in the bowl model (A, C, D, C_s/(1 + e₀), C_c/(1 + e₀) and X_l) 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

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. 23.4

Liquefaction resistance curves and bowl model parameters C and X_l

3 Element Test Simulation

3.1 Determining Parameters by Test Simulation

3.1.1 Hyperbolic Model

The initial shear stiffness G_max is calculated from the following equation based on research by Liu et al. (2017).

$$G_{\max }=6.35\frac{1}{0.3+0.7e^2}{\sigma }_m^{0.5}$$

(23.16)

Here γ_r and h_max are standard values for sand based on the shear strain dependence of stiffness ratio and damping for Toyoura sand (G/G_max–γ, 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/G_max–γ, h–γ relationship used in calculations.

Fig. 23.5

G/G_max–γ, h–γ relationship used in calculations (hyperbolic model)

3.1.2 Bowl Model

The parameter D is determined from the following equation since 1/D is the asymptote of dilatancy . The average of e_min is 0.49.

$$\frac{1}{D}=\frac{e_0-e_{\min }}{1+e_0}$$

(23.17)

The lower limit value of liquefaction resistance X_l 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

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. 23.6

Liquefaction 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. 23.7

Effective stress path and stress strain relationship. (a) e₀ = 0.515. (b) e₀ = 0.542. (c) e₀ = 0.585

4 Centrifuge Simulation

4.1 Determination of Parameters

The model parameters (that is to say the G/G₀–γ, h–γ 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 D_r is obtained, and the void ratio e is calculated from D_r. The initial shear stiffness G₀ is calculated from Eq. 23.16 based on the research of Liu et al. 2017. G_0i is the value at unit mean stress (σ_m = 1.0 kN/m²). Parameter D is determined from Eq. 23.17 since 1/D is the asymptote of dilatancy. The average of e_min is 0.49.

Table 23.4 Soil density and void ratio of Ottawa sand

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 K_w = 2.2E + 6 kN/m². For soil permeability, k = 0.015 cm/s is used from experimental results.

Table 23.5 Parameters of constitutive equation

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. 23.8

Boundary 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. 23.9

Initial stress calculated by linear analysis

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

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. 23.10

Horizontal 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. 23.11

Fivefold 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. 23.12

Stress–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.

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.