**
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 (n^{th}) slice. Later, it can be a
summation of all the slices. For one strip, the disturbing moment about centre O
(figure 4 &5) is Wx_{n }and the driving moment is W_{n}rsinα_{n}.

Resisting movement = shear strength x length of slice x radius of slip circle

= s_{n}L_{n}
r = (c + σ_{n} tanф)L_{n}r

Where ,

Therefore,

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 n |

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

Where,

=average shear strength of soil,

=average shear stress developed along the potential surface.

For n^{th} slice considered in the figure 7, W_{n} is its the
weight. The forces N_{r} and T_{r} are the normal and the
tangential components of reaction, R. P_{n} and P_{n+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 T_{n} and T_{n+1}. It is
assumed that the resultants of P_{n} and T_{n} are equal in
magnitude to the resultants of P_{n+1} and T_{n+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