where 
is the ratio of
excess pore pressure with respect to the initial effective vertical
stress, and L is the dimension of the analysis domain in horizontal
direction (Fig. 4a).
where H is the depth of the saturated soil layer (Fig. 4a).
Liquefaction risk assessment of a saturated soil deposit is presented next to illustrate the proposed Monte Carlo procedure. A loose to medium dense hydraulically placed sand deposit (Fig. 4a), with geomechanical properties as well as spatial variability characteristics estimated from the piezocone test results presented in Fig. 3a, is subjected to a horizontal earthquake motion. Response spectrum compatible acceleration time histories are used, considering two possible situations (Fig. 2c): type 2, for large epicentral distance, and type 4, corresponding to a site close to the epicenter.
Six sample functions of a stochastic vector field with probabilistic characteristics estimated from field data analysis are used to derive six sets of stochastic input parameters for the soil constitutive model used in the finite element program. Some of the numerical simulation results are presented in Fig. 4 in terms of:
where is the ratio of
excess pore pressure with respect to the initial effective vertical
stress, and L is the dimension of the analysis domain in horizontal
direction (Fig. 4a).
where H is the depth of the saturated soil layer (Fig. 4a).
Figure 4: Liquefaction strength assessment of a soil deposit.
The results of Monte Carlo simulations are shown in Fig. 4 by shaded areas representing ranges of predictions obtained from six sample functions. Results of deterministic finite element analyses, obtained using the average values of soil parameters, are also presented for comparison. It can be concluded that the amount of dynamically induced pore water pressure build-up and, consequently, the seismic response of saturated soil deposits are strongly affected by spatial variability of soil properties, as well as by the frequency content of seismic motion. For more details on these effects and their implications to geotechnical design, the reader is referred to Popescu et al (1997a).
