Keywords

1 Introduction

Liquefaction induced by earthquake is one of the important research topics in geotechnical earthquake engineering. During liquefaction of sand or sandy soil, the pore water pressure of saturated sand increases rapidly under dynamic loads, resulting in a decrease in the effective stress. When the effective stress approaches to zero, the soil bearing capacity reduces and the soil will liquefy and behave like liquid.

Liquefaction will cause severe damages to structures and foundations. For example, in 1995 Kobe earthquake [1], the soil liquefied due to the seismic motions, resulting in widespread lateral spreading and the collapse of the road surface. The liquefaction in 1999 Chi-chi earthquake [2], caused the settlement of buildings and severe lateral spreading.

A variety of site liquefaction evaluation methods have been proposed. In 1970, Seed simplified method [3, 4] was proposed to identify sand liquefaction phenomenon. In 1982, Tanimoto and Noda proposed the liquefaction potential index method to identify liquefaction [5]. Davis and Berrill proposed the energy method to identify sand liquefaction [6]. With the development of finite element method, numerical simulation has become an effective an effective method for liquefaction analysis, several constitutive models simulating sand liquefaction were proposed. Wobbes et al. [7] verified the accuracy of the finite element method (FEM) for simulating sand liquefaction based on shaking table test, and the results showed that the method was feasible. Subasi et al. [8] analysis of three soil profiles with different compactness with finite element method, and the simulation results yielded similar results to the empirical method. When adopting numerical simulation, the appropriate constitutive model should be selected for soil dynamic analysis, and the variation of pore water pressure and the accumulation of plastic strain during dynamic loading should be considered, which are the key factors to predict sand liquefaction.

In this paper, finite element method was used to analyze the liquefaction of the site under the excitation of earthquake, and the UBC3D-PLM constitutive model was used to simulate the liquefiable soil. To analyze the liquefaction characteristics, the surface and subsurface motions recorded in the dynamic analysis were analyzed by the Hilbert-Huang transform.

2 Case Study

In this paper, Treasure Island site was analyzed. The site is a man-made island located northwest of Yerba Island in the San Francisco area. In the 1989 Loma Prieta earthquake, liquefaction and lateral spreading were observed on the island. Bedrock motions recorded at Yerba Island were available in the PEER database [9].

Soil profile shown in Fig. 1 was used in the analysis, based on the site investigation, the Sandy fill shown in Fig. 1 was suspectable to liquefaction. Below the Sandy fill layer, there exists the Young Bay mud and Old Bay mud in sequence. The Rock fill was constructed adjacent to the seaside as used as supporting structure.

Fig. 1
A diagram of the soil profile has 5 layers. The bottom-most layer is old bay mud, followed by young bay mud, sandy fill, young bay mud, and rocky fill at the top.

The soil profile of Treasure Island

Full size image

3 UBC3D-PLM

UBCSAND was developed by Beaty and Byrne [10] for prediction of liquefaction behavior of sand, which has been implemented in Plaxis. P. V. UBC3D-PLM is a 3-D sand liquefaction model developed based on UBCSAND by Puebla et al. [11], which greatly improved the accuracy of calculation results under cyclic loading and earthquake motions. The model requires four primary parameters as inputs. The constant volume friction angle, φcv \({\varphi }_{cv}\), the peak friction angle, \({\varphi }_{p}\) φp, and cohesion, c, were evaluated from direct shear tests on material. In the absence of the test results, the SPT blow count of the liquefiable soil, (N1)60, can be used to estimate the input parameters.

3.1 Seismic Input

During the 1989 Loma Prieta earthquake, seismic waves were recorded on Yerba Buena Island, available in the PEER database. Conducting one-dimensional equivalent linear analysis, the input motion applied at the bottom of the model in the finite element analysis from the deconvolution analysis is shown in Fig. 2. The characteristics of input motion denoted as YBI 090 is shown in Table 1.

Fig. 2
A graph for acceleration versus time has a fluctuating signal for Y B I 0 90. The erratic pattern increases from about (7, 0.2) to (16, 0.2) and then again decreases and continues with less fluctuations.

Input motion (YBI 090 from deconvolution analysis)

Full size image
Table 1 The para maters of YBI 090
Full size table

3.2 Numerical Model

The mesh of finite element model based on the soil profile of Treasure Island is shown in Fig. 3. Three monitoring points are used to obtain the dynamic response of the site. Point A is located at the surface of the sandy layer (the elevation is 0.00 m), point B is located at the bottom surface of the sandy layer (the elevation −8.65 m), and point C is located at the bottom of the model. The Mohr-Coulomb constitutive model is used to model the behaviors of all layers of soil under static stress and the Hardening Small Strain constitutive model is used to model the behavior non-liquefiable soil in the dynamic analysis. Table 2 shows the material parameters as used in the Mohr-Coulomb model for modelling. The generation of the initial stresses is generated via the gravity loading process. Where, the left and right boundaries are set to normal fixed and defined as undrained and the bottom boundary is set to completely fixed and defined as undrained condition. In the dynamic analysis phase, Sandy fill was replaced by UBC3D-PLM sand liquefaction model, and the parameters were evaluated based on the literature [13]. The drainage condition is set to undrained-A, which represents the effective-stress analysis in Plaxis P. V. and the material parameters are shown in Table 3. The xmin and xmax boundary conditions are set to the free field boundary to absorb seismic waves and prevent seismic waves from being bounced at the boundary, and the bottom boundary is set as the compliance basis. At the same time, in order to avoid complete loss of shear strength of soil at the xmin and xmax and large deformation in the analysis results, 1 m-wide drainage boundaries using Mohr-Coulomb constitutive model are set at xmin and xmax, and the soil parameters in the drainage area was the same as the adjacent soil. To reduce the impact of boundaries on the analysis results, the model must be wide enough [14], so the model width was set to 100 m. In order to eliminate the influence of high-frequency components in seismic motion, the material damping ratio, ξ, was set to ξ = 5% and Rayleigh Damping was set based on the method by Hudson et al. [15].

Fig. 3
A schematic represents a soil profile with multiple layers. It has directional arrows A, B, and C at different levels.

Finite element analysis model

Full size image
Table 2 Material parameters as used in the Mohr-Coulomb model for modelling
Full size table
Table 3 Main parameters as sued in UBC3D-PLM for dynamic analysis
Full size table

The motion was applied at the bottom of the model, converted to a specified line displacement in Plaxis P. V. The x-direction displacement components have to be divided by a factor of two, considering that the signal that downward going waves needs to be absorbed.

4 The Results of Element Analysis

The step-by-step construction stages of finite element analysis are as follows, Phase 1. In the initial phase, the initial static stress is generated by gravity load process and the steady groundwater seepage is used to generate the static pore pressure. Phase 2. In this stage, plasticity analysis was used to simulate the static conditions of the model to remove the unbalanced static stress. Phase 3. The dynamic analysis was performed in this phase by using the UBC3D-PLM constitutive model for the liquefiable soil, and the line displacement was activated.

4.1 Dynamic Response

The acceleration time histories at different elevations (Point A, B and C) are shown in Fig. 4. It can be seen from Fig. 4a that the recorded value of the signal at the bottom of the model (point C) is the average value of the input signal, as the coefficient of 0.5 was specified when the signal was as input of within motion and a complaint boundary condition was used. It can be seen from Fig. 4b the PGA at the ground surface is 1.10 m/s2. From the point C to point A the peak value amplified from 0.84 to 1.10 m/s2.

Fig. 4
Two graphs for acceleration versus time have a fluctuating wave signal. A has 2 points, 0.42 at (11.5, negative 0.4), and 0.84 at (11, negative 0.8). B has 3 points, 0.42 at (12, negative 0.5), 0.89 at (12, negative 0.8), and 1.10 at (12, negative 1.1). Values are approximated.

Time history of acceleration

Full size image

4.2 Excess Pore Water Pressure Ratio

The time history curve of excess pore water pressure at point B is plotted in Fig. 5, and according to Fig. 5, the soil liquefied at the time 11.95 s.

Fig. 5
A graph of excess pore water pressure ratio versus time has an increasing trend for point B. The curve starts at (0,0) and hikes to (40,1). Values are approximated.

Excess pore water pressure versus time

Full size image

4.3 Hilbert-Huang Transform

The Hilbert-Huang transform is a technique for non-stationary signal processing. The signal is decomposed in the form of intrinsic mode function (IMF) by empirical mode decomposition (EMD) [17, 18]. The two constraints of natural mode function are as follows: (1) the number of extreme points and zeros should be equal or not more than one difference; (2) The upper and lower envelope should be locally symmetric with respect to the time axis, and then the corresponding Hilbert spectrum can be obtained by applying Hilbert transform to each IMF.

In this paper, the Hilbert-Huang transform was conducted for analyzing the nonlinear response of the site. The Hilbert spectrum of the recorded motion of Point A, B and C. Figure 6 shows the distribution of the two acceleration time history signals at monitoring points A, B and C in the time-frequency-amplitude spectrum. For the bottom of the model (point C), the maximum amplitude was 0.23 corresponding with the time of 12.1 s, and the corresponding frequency was 0.8 Hz. For the bottom of the sand layer (point B), the maximum amplitude occurred at 12.4 s with a value of 0.24, corresponding with a frequency of 0.6 Hz. For the surface corresponding to the top of the Sandy layer (point A), the maximum amplitude occurred at the time of 13.0 s with a peak value of 0.55 and a corresponding frequency of 3.2 Hz. The differences between the three Hilbert-Huang spectra shows that the amplification of the amplification occurs from a depth of 8.65 underground to the surface, and the low-frequency content is shifted due to liquefaction in the process of transmission to the surface, at the same time, the high-frequency content is identified at the surface.

Fig. 6
3 three-dimensional graphs for instantaneous energy, frequency, and time. A for point A has a point with coordinates (13.0,3.2,0.55). B for point B has a point (12.4,0.6,0.24). C for point C has a point (12.1,0.8,0.23). Values are approximated.

Hilbert spectrum

Full size image

5 Conclusions

In this paper, the finite element software was used to simulate the liquefaction within the Treasure Island site, in which UBC3D-PLM was selected for the sand liquefaction constitutive model, and the calculation results were analyzed by Hilbert transform.

The UBC3D-PLM constitutive model can simulate the dynamic behavior of cyclic and liquefiable soil and generate pore water pressure. For the motion YBI 090, the sand layer liquefies in 11.95 s, and the PGA is amplified from 0.84 to 1.10 m/s2 on the surface. The amplification of the peak value is found from the depth of 8.65 m underground to the surface, and the low-frequency content is shifted due to liquefaction in the process of transmission to the surface, and the high-frequency content is identified at the surface.