Tamara Al-Bittar

Probabilistic Analysis of Strip Footings Resting on Spatially Varying Soils and Subjected to Vertical or Inclined Loads

A probabilistic analysis of vertically and obliquely loaded strip footings resting on a spatially varying soil is presented. The system responses are the footing vertical and horizontal displacements. The deterministic computation of these system responses is based on numerical simulations using the software FLAC3D. Both cases of isotropic and anisotropic random fields are considered for the soil elastic properties. The uncertainty propagation methodology employed makes use of a nonintrusive approach to build up analytical equations for the two system responses. Thus, a Monte Carlo simulation approach is applied directly on these analytical equations (not on the original deterministic model), which significantly reduces the computation time. In the case of the footing vertical load, a global sensitivity analysis has shown that the soil Youngs modulus E mostly contributes to the variability of the footing vertical displacement, the Poisson ratio being of negligible weight. The decrease in the autocorrelation distances of E has led to a smaller variability of the footing displacement. On the other hand, the increase in the coefficient of variation of E was found to increase both the probabilistic mean and the variability of the footing displacement. Finally, in the inclined loading case, the results of the probability of failure against exceedance of a vertical and/or a horizontal footing displacement are presented and discussed.

Efficient sparse polynomial chaos expansion methodology for computationally expensive deterministic models

The sparse polynomial chaos expansion (SPCE) methodology is an efficient approach that deals with uncertainties propagation in case of high-dimensional problems (i.e. when a large number of random variables is involved). This methodology significantly reduces the computational cost with respect to the classical full polynomial chaos expansion (PCE) methodology. Notice however that when dealing with computationally-expensive deterministic models, the time cost remains important even with the use of the SPCE. In this paper, an efficient combined use of the SPCE methodology and the global sensitivity analysis (GSA) is proposed to solve such a problem. The proposed methodology is validated using a relatively non-expensive deterministic model.

BEARING CAPACITY OF STRIP FOOTINGS ON SPATIALLY RANDOM SOILS USING KRIGING AND MONTE CARLO SIMULATION

The probabilistic analysis of geotechnical structures presenting spatial variability in the soil properties is generally performed using Monte Carlo simulation (MCS) methodology. This methodology is not suitable for the computation of a small failure probability because it becomes very time-expensive in such a case due to the large number of simulations required to calculate the failure probability. For this reason, one needs to keep to a minimum the number of calls to the deterministic model when performing the probabilistic analyses. In order to overcome the shortcoming related to the excessive number of calls of the deterministic model when performing Monte Carlo simulations, Echard et al. (2011) proposed an Active learning reliability method (called AK-MCS) combining Kriging and Monte Carlo Simulation. The method was shown to be very efficient as the obtained probability of failure is very accurate needing only a small number of calls to the deterministic model. The probabilistic analyses presented in this paper were performed using AK-MCS methodology by Echard et al. (2011). The present study involves a probabilistic analysis at the ultimate limit state of a strip footing resting on a spatially varying soil using AK-MCS approach. The objective is the computation of the probability Pf of exceeding the ultimate bearing capacity of the footing under a prescribed footing load. The soil cohesion and angle of internal friction were considered as two non-isotropic non-Gaussian random fields. The deterministic model was based on numerical simulations using the finite difference code FLAC 3D . Some probabilistic results are presented and discussed.

EFFECT OF SOIL SPATIAL VARIABILITY ON THE DYNAMIC BEHAVIOR OF A SLOPE

The analysis of earth slopes situated in seismic areas has been extensively investigated in literature using deterministic approaches where average values of the soil properties were used. In this paper, a probabilistic dynamic approach is presented for the slope stability analysis. In this approach, the effect of the soil spatial variability on the dynamic responses was investigated. The soil shear modulus G was modeled as an anisotropic non-Gaussian random field. The deterministic model was based on numerical simulations using the dynamic option of the finite difference code FLAC2D. Notice that when dealing with dynamic problems, the deterministic model becomes very time-consuming. For this reason, one needs to keep to a minimum the number of calls to the deterministic model when performing the probabilistic analyses, the probabilistic method generally employed when considering spatially varying soil properties being the Monte Carlo Simulation (MCS) methodology. This method is known to be not suitable for the computation of the small failure probabilities encountered in practice because it becomes very time-expensive in such cases due to the large number of simulations required. In order to overcome the shortcoming related to the excessive number of calls of the deterministic model when performing Monte Carlo simulations, Echard et al. (2011) proposed an Active learning reliability method (called AK-MCS) combining Kriging and Monte Carlo Simulation. The method was shown to be very efficient as the obtained probability of failure is very accurate needing only a small number of calls to the deterministic model. The probabilistic dynamic analyses presented in this paper were performed using AK-MCS methodology by Echard et al. (2011). The failure probability was computed for the point located on the toe of the slope. For a given realization of the random field, failure is considered to be achieved at the toe if the maximal acceleration Amax at this point, as computed bu FLAC2D software, exceeds a prescribed threshold value. Some probabilistic results corresponding to different values of the maximum threshold value are presented and discussed.

Probabilistic Analysis at Ultimate Limit State of Strip Footings Resting on a Spatially Varying Soil Using Subset Simulation Approach

The failure probability of geotechnical structures with spatially varying soil properties is generally computed using Monte Carlo simulation (MCS) methodology. This approach is well known to be very time-consuming when dealing with small failure probabilities. One alternative to MCS is the subset simulation approach. This approach was mainly used in the literature in cases where the uncertain parameters are modelled by random variables. In this paper, it is employed in the case where the uncertain parameters are modelled by random fields, because the spatial variability of the soil properties has proven to greatly affect the behavior of geotechnical structures and to induce a significant change in the variability of their responses. This is illustrated through the probabilistic analysis at the ultimate limit state (ULS) of a strip footing resting on a one-and two-layer purely cohesive soil with a spatially varying cohesion. The soil cohesion parameter was modeled as an anisotropic non-Gaussian (log-normal) random field using a square exponential autocorrelation function. The Expansion Optimal Linear Estimation (EOLE) method was used to discretize this random field. The deterministic model was based on numerical simulations using the finite difference software FLAC 3D .

Influence of soil spatial variability and stochastic Ground-Motion on the dynamic behavior of a slope

A probabilistic dynamic approach is used for the slope stability analysis. In this approach, the effect of both the soil spatial variability and the variability of the Ground-Motion (GM) time history on the dynamic responses (amplification, permanent displacement) are studied and discussed. The soil shear modulus G is considered as an isotropic non-Gaussian random field. The simulation of random acceleration time histories based on a real target accelerogram is done using a fully nonstationary stochastic model in both the time and the frequency domains. The deterministic model is based on numerical simulations. An efficient uncertainty propagation methodology which builds up a sparse polynomial chaos expansion for the dynamic responses is used. The probabilistic numerical results have shown that: (i) the decrease in the autocorrelation distance of the shear modulus leads to a small variability of the dynamic responses; (ii) the randomness of the earthquake GM has a significant influence on the variability of the dynamic responses; (iii) the probabilistic mean values of the dynamic responses are more critical than the deterministic ones.

Probabilistic Analysis of Shallow Foundations on Rocks Obeying Hoek-Brown Failure Criterion

A probabilistic analysis using Polynomial Chaos Expansion method is presented to compute the probability density function (PDF) of the ultimate bearing capacity of a strip footing resting on a rock mass and subjected to a vertical/inclined load. The rock is assumed to follow Hoek-Brown failure criterion. The kinematic approach of the limit analysis theory is used. The results of the vertical load case have shown that: (i) the Geological Strength Index GSI and the uniaxial compressive strength of the intact rock σc have the most significant weight in the variability of the ultimate bearing capacity, (ii) the non-normality of the input variables has a significant effect on the shape of the PDF of the ultimate bearing capacity and, (iii) a negative correlation between GSI and σc leads to less spread out PDF. Finally, it was shown in the inclined load case that the variability of the ultimate bearing capacity decreases with the increase of the footing load inclination.