Uncertainty in Determination of Seismic Yield Coefficient of Earth Slopes

The landslides occurred due to an earthquake have ever caused great losses of life and property. Earth slope stability is influenced by many factors that should be taken into account for a comprehensive assessment of it. However, slope seismic analyses are complicated due to the necessity of concerning the dynamic stresses caused by an earthquake and the soil resistance changes in seismic conditions. Seismic slope instability can arise from two significant factors including the increase of inertia forces and the decrease of soil shear strength. Moreover, the earthquake motions can produce considerable dynamic shear and vertical stresses in earth slopes. If these stresses are added to the static stresses existing in the soil mass, they may lead to slope instability [1]. The methods of analyzing the earth slopes stability can be pseudo-static and stress-strain analyses or simplified integration method (Newmark rigid-block method). In the pseudo-static analysis, slope seismic safety factor is determined in a very similar way as what is done in static equilibrium analysis. In the stress-strain analysis, first of all, soil mass is divided into the finite elements. Afterwards, by using numerical methods, the stresses and strains caused by external and internal forces in soil mass is calculated. Although this method is accurate and gives a realistic model of the earth slopes, it requires the special expertise and precise input information. The rigid-block model assumes that permanent deformation initiates when the earthquake-induced accelerations acting on a slide mass exceed the yield resistance on the slip surface. The resistance is quantified by the seismic yield coefficient (ky). At this point, the slide mass breaks away from the rest of the underlying slope and sliding occurs at a constant rate of acceleration equal to ky. During this time, the velocity of the ground is greater than the velocity of the slide mass. Sliding continues until the following conditions are met: (1) accelerations fall below ky, and (2) velocity of the slide mass and the underlying ground coincide [2]. In deterministic computation methods, input data such as viscosity, inner friction angle, inclination of slope, etc. is considered as fixed values; therefore, by using these methods, the probabilistic variation of a parameter cannot be considered. In other words, deterministic methods do not include the uncertainty related to the effective design parameters. In the probabilistic method, instead of using a fixed value for a parameter, the probability distribution function (Probability Density Function) of it is used. Due to the possibility of performing fast extensive numerical calculations by computer, the numerical simulation methods such as Monte Carlo simulation have been employed in slope stability analysis. Assuming homogeneity of materials, along with, absence of underground water on one hand, and considering that the slope seismic yield coefficient is a function of the angle of slope β, height of slope h, soil bulk density γ, the coefficient of soil cohesion c and internal friction angle φ, on the other hand, a closed-form relationship between the above-mentioned parameters can be achieved using the extensive results of limit equilibrium analysis and artificial intelligence methods such as genetic algorithm [4]. Any change in each of the stated parameters can lead to a change in slope seismic yield coefficient. The uncertainty in calculation of slope seismic yield coefficient can be taken into account by utilizing genetic algorithm and probability distribution functions of input parameters.
Due to the fact that coseismic slope deformation is a function of seismic yield coefficient (ky) and other effective parameters, using the statistical analysis, the statistical distribution and the standard deviation of ky can be assessed. In the present study, the following steps were taken for studying the variation of ky: Implementing a pseudo-static analysis of earth slopes having various physical and mechanical parameters, determining the relationship between slope seismic yield coefficient and other given parameters using curve fitting and genetic algorithm method (evolutionary strategy method), simulating the effective parameters on ky by means of Monte Carlo method based on a specified statistical distribution [9]. The Monte Carlo simulation procedure is as follows: - Providing a deterministic model for solving the problem. - Selecting the random variables including soil characteristics, slope geometry and proper probability density functions. - Generating the input parameters by taking into account the density functions and random variables dependency (parameters’ dependency) to each other. - Using simulated numbers related to all independent parameters, along with defined deterministic model, the value of slope seismic yield coefficient and probability density function of it are obtained. Consequently, the following results were concluded: 1. If normal distribution is selected for the parameters affecting ky, slope seismic yield coefficient will be normally distributed. 2. The coefficient of variation of ky has a direct relationship with coefficient of variation of parameters affecting it. If slope seismic yield coefficient is increased, the acceleration coefficient will be decreased. 3. With selecting the minimum amount of coefficient of variation for effective factors, the range of coefficient of variation of slope seismic yield coefficient is between 5 and 11 percent. 4. With considering the maximum amount of coefficient of variation for effective factors, the range of coefficient of variation of slope seismic yield coefficient is between 25 and 59 percent. 5. With taking into consideration the minimum amounts of the coefficient of variation for effective parameters, and increasing mean value of slope yield acceleration from 0.1 to 0.35, its variation coefficient (COV%) decreases from 11 percent to 5 percent. 6. With taking into account the maximum values of the coefficient of variation for the effective parameters, and increasing mean value of slope yield acceleration from 0.1 to 0.35, its variation coefficient (COV%) decreases from 59 percent to 25 percent. In this study, coefficient of variation of ky is calculated assuming that the properties of a soil layer changes similarly in all points of it. If these properties vary from point to point, there is a need to employ random field method (stochastic finite element) in order to simulate the spatial variation of soil properties.