5.5 Ordinary slip circle method
Slip circle method of circular failure analysis uses the theory of limiting equilibrium. It solves a two-dimensional rigid body stability problem using potential slip surface of circular shape. This method is used to investigate the equilibrium of a soil mass tending to move down the slope under influence of gravity.
The trial slip circle is drawn and the material above the assumed slip surface is divided into a number of vertical strips or slices. In the ordinary slip circle. the forces between slices are neglected and each slice is assumed to act independently as a column of soil of unit thickness and width. The weight of each slice is assumed to act at its centre. The factor of safety is assumed to be the same at all points along the slip surface. The surface with the minimum factor of safety is termed the critical slip surface. Such a critical surface and the corresponding minimum factor of safety represent the most likely sliding surface.
Initially, the moment can be calculated for only one (nth) slice. Later, it can be a summation of all the slices. For one strip, the disturbing moment about centre O (figure 4 &5) is Wxn and the driving moment is Wnrsinαn.
Resisting movement = shear strength x length of slice x radius of slip circle
= snLn r = (c + σn tanф)Lnr
Following similar approach, the driving and the resisting forces are calculated separately for all the slices and finally the factor of safety of the slip circle is determined by the expression given below:
Figure 4: Calculation of factor of safety for nth slice
Figure 5: Dividing the slip circle into vertical slices
Stability analysis using the method of slice can also be explained with the use of figure 6 in which AC is an arc of circle representing the trial failure surface. The soil above the trial failure surface is divided into several vertical slices. Various forces actin on typical slice considering its unit length perpendicular to the cross section are shown in the figure23. In this case the factor of safety can be defined as
=average shear strength of soil,
=average shear stress developed along the potential surface.
For nth slice considered in the figure 7, Wn is its the weight. The forces Nr and Tr are the normal and the tangential components of reaction, R. Pn and Pn+1 are the normal forces that act on the sides of the slice. Similarly, the shearing forces that act on the sides of the slice are Tn and Tn+1. It is assumed that the resultants of Pn and Tn are equal in magnitude to the resultants of Pn+1 and Tn+1, and that their lines of action coincide.
Figure 6: Geometry of circular failure in slope
Figure 7: Geometry of circular failure in slope
For equilibrium consideration
The resisting shear force
The normal stress
Now, for equilibrium of the trial wedge ABC, the moment of the driving force equals to the moment of the resisting force about point O. Thus