13.1 Sensitivity Analysis
Principles of static equilibrium to evaluate the balance of driving and resisting forces are used in limit equilibrium methods. The factor of safety is defined as the resisting forces divided by the driving forces, or alternatively as the shear strength divided by the calculated shear stresses. In traditional slope stability analysis methods analysis, single fixed values (typically, mean values) of representative samples or strength parameters or slope parameters are used. The factor of safety is generally calculated for a slope to assess its stability by using single value of soil properties & slope parameters. The deterministic analysis is unable to account for variation in slope properties and parameters and other variable conditions. In reality, each parameter has a range of values and that value with affect the stability of slope. Therefore, geotechnical properties of Slope parameters have always pose some uncertainties in the simulation. This uncertainty imposes a limit on our confidence in the response or output of the model.
Sensitivity analysis of involves a series of calculations in which each significant parameter is varied systematically over its maximum credible range in order to determine its influence upon the factor of safety. It investigates the robustness of a study when the study includes some form of mathematical modelling. In Sensitivity analysis the factor of safety is calculated using upper & lower bound values for these parameters during critical design. It increased understanding or quantification of the system (e.g. understanding relationships between input and output variables); and when Input is subject to many sources of uncertainty including errors of measurement, absence of information and poor or partial understanding of the driving forces and mechanisms. Good modeling practice requires that the modeler provides an evaluation of the confidence in the model, possibly assessing the uncertainties associated with the modeling process and with the outcome of the model itself. Figure 1show the identification of sliding and toppling blocks on various field conditions
Figure 1: Identification of sliding and toppling blocks (a) geometry of block on inclined plane (b) conditions for sliding and toppling of block on an inclined plane.
The sensitivity analysis is able to account for variation in slope properties and different geo-technical conditions. The stability of a slope depends on many factors such as water pressure, slope height, slope angle, shear strength, strength shear joint material etc. These factors not only help in designing the slope but also help in understand the failure mechanism the sensitivity analysis is It determines the effect of various input parameters on stability of slope under different geo-mining conditions.. Sensitivity analysis is an interactive process adopted to simulate slope instability more realistically and determine the influence of the different parameters on the factor of safety. It indicates which input parameters may be critical to the assessment of slope stability, and which input parameters are less important.
In sensitivity analysis, a common approach is that of changing one-factor-at-a-time (OAT), to see what effect this produces on the output. This appears a logical approach as any change observed in the output will unambiguously be due to the single factor changed. Furthermore by changing one factor at a time one can keep all other factors fixed to their central or baseline value. This increases the comparability of the results (all ‘effects’ are computed with reference to the same central point in space) and minimizes the chances of computer program crashes, more likely when several input factors are changed simultaneously. The conventional variation by a fixed percentage of the initial parameter value is problematic for two reasons. However, to carry out sensitivity analyses for more than three parameters is cumbersome, and it is difficult to examine the relationship between each of the parameters. Consequently, the usual design procedure involves a combination of analysis and judgment in assessing the influence on stability of variability in the design parameters, and then selecting an appropriate factor of safety.
The results will depend on how the initial value was chosen because a small initial value leads to a small variation and a greater initial value to a larger variation. On the one hand, if the model response to parameter variations is nonlinear, then, if the initial parameter value is located nearby the upper or lower bound of the valid parameter range, the variation can lead to inadmissible values beyond the bounds of the range.
Therefore, the parameter variation is considered in which the parameters are not varied by a fixed percentage of the initial value but by a fixed percentage of the valid parameter range. Sensitivity is expressed by a dimensionless index I, which is calculated as the ratio between the relative change of model output and the relative change of a parameter (Figure 2). The sensitivity index (I) as defined by Lenhart15 is expressed in equation 1. Sensitivity index is calculated and analysed the effect of input parameter on the factor of safety or stability of internal dump slope.
Mathematically, the dependence of a variable y from a parameter x is expressed by the partial derivative δy/δx. This expression is numerically approximated by a finite difference: let y0 is the model output calculated with an initial value y of the parameter x. This initial parameter value is varied by ±Δx yielding x1= x0 -Δx and x2=x0+ Δx with corresponding values y1 and y2.
The finite approximation of the partial derivative δy/δx then is
To get a dimensionless index, I’ has to be normalized. The expression for the sensitivity index I then assumes the form
Figure 2: Schematic of the relation between an output variable y and a parameter x.
The sign of the index shows if the model reacts co-directionally to the input parameter change, i.e. if an increase of the parameter leads to an increase of the output variable and a decrease of the parameter to a decrease of the variable, or inversely.