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.
Keywords
Effective stress analysis Slope Lateral spreading Numerical simulation23.1 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/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).
23.2.1 Hyperbolic Model and Its Parameters
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.}
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).
Parameters of the hyperbolic model
Parameters | Physical meaning of parameters |
---|---|
G_{max} | Initial shear modulus. G_{max} = ρV_{s}^{2} |
h_{max} | Maximum damping ratio. As h_{max} increases the nonlinearity becomes stronger |
γ_{r} | Reference strain. γ_{r} = τ_{f}/G_{0} The shear strain when G/G_{max} = 0.5 |
23.2.2 Bowl Model (Dilatancy Model) and Its Parameters
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.
Next, the consolidation term is taken into consideration in the stress–strain relationship and effective stress is modeled under undrained (constant volume) conditions.
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.
Parameters of bowl model
Parameter | Physical meaning of parameters |
---|---|
A | Parameter representing the swelling component ε_{Γ} of the dilatancy components. The larger the absolute value of A, the greater the cyclic mobility |
C, D | Parameters representing the compression component ε_{G} 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 e_{min} on the hyperbolic asymptotic line (maximum amount of compression) |
C_{s}/(1 + e_{0}) | C_{s} is the swelling index; e_{0} is the initial void ratio |
C_{c}/(1 + e_{0}) | C_{s} is the compression index; e_{0} is the initial void ratio |
X_{l} | The lower limit value of the liquefaction resistance. In the relationship between stress ratio τ/σ′_{m0} and the number of cycles N_{c}, it is represented by τ/σ′_{m0} when N_{c} is sufficiently large. When τ/σ′_{m0} > X_{l} excess pore water pressure arises |
23.3 Element Test Simulation
23.3.1 Determining Parameters by Test Simulation
Hyperbolic Model
Bowl Model
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.
Parameters for Ottawa sand
Parameter of constitutive equation | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
No | σ_{m0}(kN/m^{2}) | e | ρ_{d} (t/m^{3}) | V_{s} (m/s) | G_{0} (kN/m^{2}) | Reference strain (γ_{r}) | Hypabolic model | Bowl model | |||||||
G_{0i} (kN/m^{2}) | γ_{ri} | h_{max} (%) | A | C | D | C_{s} (1 + e_{0}) | C_{c} (1 + e_{0}) | X_{l} | |||||||
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
23.4 Centrifuge Simulation
23.4.1 Determination of Parameters
Soil density and void ratio of Ottawa sand
ρdmax (kg/m^{3}) | 1765 | ρdmin (kg/m^{3}) | 1480 |
emax | 0.766 | emin | 0.49 |
Parameters of constitutive equation
23.4.2 Summary of the Numerical Simulations
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.
In the dynamic analysis, horizontal acceleration and vertical acceleration were input simultaneously.
23.4.3 Results of Centrifuge Simulations
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
- 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
- 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
- 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
- Liu, K., Zhou, Y., & Chen, Y. (2017). Hardin equation of Ottawa sand-F65 for LEAP.Google Scholar
- Sandhu, R. S., & Wilson, E. L. (1969). Finite-element analysis of seepage in elastic media. Proceedings of ASCE, 95(EM3), 641–652.Google Scholar
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