Theoretical Background for Slope Stability
The term, slope stability, can be defined to assess the degree of resistance to the collapse of the slope based on its factor of safety. The factor of safety is a value that results from dividing the shear strength by the shear stress of the soil of a given slip surface. In other words, it can be defined as the ratio of the forces that block the activity of the slope to the forces that trigger it. Therefore, theoretically, if the calculated factor of safety is above 1.0, the target slope can be judged as safe. Nevertheless, when analyzing the actual behavior of a slope, its soil properties are examined through an on-site survey. However, even after a significant on-site survey, uncertainties about properties that cannot be considered (such as strength parameters), external loads that trigger the activity of the slope, and failure models are still present. Owing to such uncertainties, the concept of the allowable factor of safety was introduced [
24], and the factor of safety specified in design standards is the allowable factor of safety. Therefore, in practice and for a slope to be deemed safe, its calculated factor of safety must be not above 1.0, but above the allowable safety factor specified in the design standards.
The analysis of the safety of a slope against earthquakes can be divided into two stages. First, the slope response analysis stage, in which the forces that act on the slope when seismic waves are applied, along with the slope acceleration, speed, and displacement, are analyzed. Second, the safety analysis stage, in which the slope safety is analyzed by calculating the factor of safety of the slope where deformation has occurred.
The slope response analysis regarding earthquakes can be divided into the pseudo-static analysis (or equivalent static analysis), which considers seismic loads as static, displacement analysis, which uses the Newmark’s sliding block, and dynamic analysis, which uses the finite difference method and the FEM.
The limit-equilibrium-method based pseudo-static analysis is widely used in practice. The pseudo-static analysis has the advantage of being simple to use, as it evaluates the factor of safety of slopes by converting transient seismic loads into static loads that act in only one direction. However, loads that act on slopes within the vibration duration change constantly and act within short periods of time; thus, calculating the factor of safety by applying them as constant static inertial forces has been reported as being very uncertain [
25].
To overcome these limitations of the pseudo-static analysis method, Newmark [
26] proposed the activity displacement method. This method presented a displacement analysis method based on the similarity of the movement of a sliding block placed on an inclined plane, with the permanent displacement of a slope occurring owing to seismic loads. When the ground’s vibration acceleration exceeds the yield acceleration, the active soil moves, and permanent deformation occurs until the vibration speed of the active soil and the vibration speed of the ground become the same. Therefore, permanent displacements of a slope can be found by multiple integrations of the difference between the ground’s vibration acceleration and yield acceleration.
Finally, the time history analysis, which is a dynamic analysis method, consists of finding the solution of the equation of motion when dynamic loads act on the target slope, calculating its behavior (displacement, member force, etc.) at any period of time by using the dynamic characteristics and applied loads [
27,
28]. One of the disadvantages of this method is that the time required for the analysis is much longer than in the other methods because it uses the dynamic characteristics of the target slope and actual seismic wave data. However, it provides high accuracy because the analysis is performed using actual models and input values. The equation of motion used in this method is as follows [
28]:
where
is the mass matrix,
is the damping matrix,
is the stiffness matrix, and
is the dynamic load generated by seismic waves, whereas
are the acceleration, speed, and displacement, respectively.
The calculation of the slope deformation in the dynamic analysis must be consecutively followed by the slope’s factor of safety analysis stage, so that the analysis of the slope stability regarding seismic waves can be performed.
Generally, the analysis methods used in the slope stability analysis are broadly classified into the elastic or elasto-plastic analysis, which considers the ground deformations by means of numerical analysis methods, such as the FEM, and the limit equilibrium method, which analyzes only the mechanical equilibrium relationship of the critical plane in which failure occurs [
25]. Fundamentally, the limit equilibrium method not only performs a slope stability analysis, but also explains geotechnical problems, such as soil pressure and bearing power. Moreover, as a method, it deals with the target ground as a single body and considers the forces and moment equilibrium conditions of any failure surface.
Numerical analysis methods, such as the FEM, are actually difficult to use and require substantial analysis time; thus, in practice, the limit equilibrium method is widely used because of its relatively easier analysis. Several methods can be used for analyzing the factor of safety of a slope, including the Fellenius method, Bishop simplified method, Spencer method, Janbu simplified and precision solution methods, and the Morgenstern-Price method [
10,
25]. In this research, the Bishop simplified method was used to perform the slope stability analysis [
29].
According to research conducted on the numerical difference between factors of safety calculated using the various analysis methods, such a difference is very small [
30]. However, as an exception, it was reported that the Fellenius method shows a maximum difference of 60% when compared to other methods [
31]. In the slope stability analysis, it can be said that the geometric conditions and accurate calculation of slope strength constants have a bigger impact than the numerical analysis method. Moreover, during the slope stability analysis, it is more desirable to compare the factors of safety during active failure depending on the state of the expected active surface considering geotechnical aspects.
In this research, to analyze the stability of a slope regarding earthquakes, the time history analysis, which is a dynamic analysis, was used to perform a response analysis on the member force, acceleration, speed, and displacement occurring in the slope owing to seismic waves. In addition, the Bishop simplified method was used to calculate the factor of safety of the slope in which deformations occurred [
29]. For the time history analysis, QUAKE/W software [
32], which is widely utilized for slope analysis, was employed, and for the slope factor of safety calculation, SLOPE/W software [
33] was used.