# Abstract is considered unstable if the safety factor

Abstract
This report is based on the study on the STB software, it contains an explanation of the theory behind it (Bishop’s method), and two examples with a result comparison and analyzation from using this software.
Contents
Abstract 1
Introduction 2
Bishop’s method 2
Koppejan’s modification 3
Study of STB’s program codes 4
Examples 6
Analysis 8
Conclusion 8
Reference 9
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Introduction
Slope stability is the balance of shear stress and shear strength of soil covered slopes to withstand and undertake movements. A formerly stable slope can be initially affected by elementary factors, cause the slope to become conditionally unstable. These factors can decrease the stability of the slope and cause slope failure, which leads to mass soil movements. Such movements can be caused by increase in shear stress, such as loading, momentary forces and horizontal pressures. In addition, a decrease in shear strength caused by weathering, deviations in pore water pressures, and organic materials.
STB or Slope Stability is an analytical software used to determine the stability factor of a slope made of non-homogenous soil, using Bishop’s method. The procedure in slope stability analysis is to determine the safety factor of numerous potential slip surfaces, and to have the smallest calculated value of safety factor as the critical safety factor for the slip surface. A slope is considered unstable if the safety factor is smaller than 1. However, safety factors greater than 1 like 1.5 are usually required in the designs of ditches and dams.
Bishop’s method
Bishop’s method of slope stability analysis is based on the consideration of moment equilibrium of the soil mass above an estimated circular slip surface. The part of the slope within the circular curve is divided into slices.
Coulomb’s relation: ?_f=c+?’tan?
?_f is the maximum shear stress acting at the lower boundary of a slice, c is the local cohesion, ?’ is the normal effective stress, and ? is the angle of internal friction.
Now assuming the actual shear stress is acting upon the lower boundary of a slice is ?_f/F, where F is the stability safety factor. Assumed that F is the same for all slices.
?=1/F(c+?’tan?)

Slip circle method (Verruijt, 1995)
The equilibrium of moments with respect to the center of the slip circle is expresses by the sum of moments of each slice’s weight with respect to the circle’s center equals to the sum of the shearing forces moments at the bottom of the slices. The horizontal distance between the center and a slice is Rsin?, the area of the slice’s bottom part is bcos?, and this equilibrium condition is expresses as:
???hbRsin?=???bR/cos?
? is the unit weight of the slice, with h height and b base, the weight of each slice is ?hb .
F=(???((c+?^’ tan?)/cos?)?)/(???hsin?) If all slices have the same width.
Bishop’s method assumes that the forces transferred between end-to-end slices are only horizontal. Then it follows from the vertical equilibrium of a slice,
?h=?^’+p+tan?
By using the equation for shear stress:
?^’ (1+(tan? tan?)/F)=?h-p-c/F tan?
Substituting this equation into the equation for stability safety factor F:
F=(??(c+(?h-p)tan?)/(cos?(1+(tan? tan?)/F)))/(???hsin?)
?h is the total weight of a slice. In an inhomogeneous soil, the total unit weight may be the sum of the weight of numerous sections with different soil types, vertically. The upper part of the slice may contain dry soil, as the lower part (under the water table) may contain wet/saturated soil. The strength parameters c and ? apply to the bottom of the slice, which is the slip surface. Values of c and ? should be taken at the slice surface in an inhomogeneous soil.
Koppejan’s modification
With Koppejan’s modification, the maximum shear stress acting at the slice’s bottom is:
?_f=(c+(?h-p)/tan?)/(1+tan? tan??/F)
If F=1, shear stress is infinitely large for ?=?-1/2 ?, because this leads to tan? tan??=-1. Such values for ? usually occur near the lower end of the slip circle. The friction angle is large if the circle is deep. The shear stress is negative for larger values of ?, this means that the direction of the shear stress is not acting against the direction of the slip. This creates an unrealistic value of stability factor. Therefore, A.W. Koppejan suggested that the value of ? in the formula for shear stress should be cut off at 1/2?-1/4 ?, known as the modified Bishop’s method.
Study of STB’s program codes
STB is a computer program that runs calculations described in a set of codes. The coding language consists of lists of notations and parameters with values including positions and coordinates, series of “if” functions, and formulas.
We know that the safety factor
F=(??(c+(?h-p)tan?)/(cos?(1+(tan? tan?)/F)))/(???hsin?)
fa:=a/b f=a/b
So we can tell a=??(c+(?h-p)tan?)/(cos?(1+(tan? tan?)/F)) , and b=???hsin?
a:=a+(cc+(sk-pk)*tf)/(co*(1+ta*tf/f))
b:=b+sk*si
So if (cc+(sk-pk)tf)/(co(1+(ta*tf)/r)) = (c+(?h-p)tan?)/(cos?(1+(tan? tan?)/F)) , and sk*si = ?hsin?
then we can identify that
cc=c
sk= ?h
pk= p
tf= tan?
co=cos?
ta= tan?
r=F
si=sin? (Left: from codes, Right: from formula)
From codes:
ta:=(x-xc)/(yc-yb)
co:=sqrt(1.0/(1.0+ta*ta))
si:=co*ta
From the codes, we can tell that the whole code is to determine the value of the slope stability safety factor f, and f=a/b, a is the resisting moment, which is expressed as a+(cc+(sk-pk)*tf)/(co*(1+ta*tf/f)), and b is the distributing moment, which is expressed as b+sk*si . From these expressions, we can tell that cc is the local cohesion, sk is the product of the unit weight and the height of each slice, pk is p, tf is tan?, co is cos?, ta is tan?, is a value of stability factor that must be determined with an initial estimation, and si is sin?.
The codes also contain commands that converts degrees to radians.
First of all, when the shape and all the other data of the slope is being put in, the program cuts the slope into slices, measures the different angles and dimensions such as the widths and heights of the slices and their distances from the circle center, then calculates the weights of each slices and all the others forces acting on them, and their shear stresses at their lower boundaries.
Values of soil properties input into the software such as density, local cohesion and angles of internal friction are used to determine the weights, forces and stresses.
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Examples
Two modified models are run and analyzed in this section.
Model 01 F=1.774

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Model 02 F=2.148

Analysis
In model 1, the slope is steeper, there are more variations of soil types and they are closer to the slope, all of these factors give the slope a stability factor of 1.774.
In model 2, the slope is relatively less steep to the slope in model 1, there are less variations to soil type near the slope, soil type 1, 3, 5, 6, 7 and 8 are further away from the slope, leaving only soil type 2 and 4 near the slope. Secondly, all the soil types in model 2 have greater values of Wd and Ws, and the water table in the slope (soil type 2) is on the same level as the sea/water level, so there are more unsaturated soil above the water table in the slope in model 2. All of these differences give model 2 a stability factor of 2.148.
There is a stability factor difference of 0.374 between the two models. The slope in model 2 overall is a more stable slope compared to model 1.
Conclusion
Soil types and soil properties influence the slope stability. Having a slope with more consistent soil type (less variation) increases the stability of the slope. A less steep slope or a slope with a lower gradient is relatively more stable than a steeper one. A higher water table increases pore water pressure in the soil and decreases effective stress, which decrease the slope stability.
STB can do thousands of calculations in a second, with greater accuracy compared to human calculation, so it is recommended for academic uses, especially for comparing hand calculation results to STB’s results. It helps academics to get quick results and to understand slope stability.
However, STB only does analysis on a theoretical 2D slope, assuming the slope is the same all the way into the Z-direction. Secondly, soil types and properties are organizely laid out with clear separations between different soil types. Thirdly, the water table in STB is in a specific position. All these are far from reality. In a realistic scenario, the slope face and soil types can vary across the z-direction, the soil properties can vary and the water table changes positions depending on the weather. To sum up, this version of STB developed by Arnold Verruijt is not recommended for actual industrial uses.
On the other hand, there are existing infrared laser and ultrasound slope monitoring systems and technologies used by the industry for determining and monitoring slope stability at a higher accuracy, which are more reliable, accurate and practical then this STB software.