## Kinematic Analysis

Kinematic analysis is a method used to analyze the
potential for the various modes of rock slope failures (plane, wedge,
toppling failures), that occur due to the presence of unfavorably
oriented discontinuities (Figure 1). Discontinuities are geologic breaks
such as joints, faults, bedding planes, foliation, and shear zones that
can potentially serve as failures planes. Kinematic analysis is based on
Markland’s test which is described in Hoek and Bray (1981). According to
the Markland’s test, a plane failure is likely to occur when a
discontinuity dips in the same direction (within 20^{0}) as the
slope face, at an angle gentler than the slope angle but greater than
the friction angle along the failure plane (Hoek and Bray, 1981) (Figure
1). A wedge failure may occur when the line of intersection of two
discontinuities, forming the wedge-shaped block, plunges in the same
direction as the slope face and the plunge angle is less than the slope
angle but greater than the friction angle along the planes of failure
(Hoek and Bray, 1981) (Figure 1). A toppling failure may result when a
steeply dipping discontinuity is parallel to the slope face (within 30^{0})
and dips into it (Hoek and Bray, 1981). According to Goodman (1989), a
toppling failure involves inter-layer slip movement. The requirement for
the occurrence of a toppling failure according to Goodman (1989) is “If
layers have an angle of friction Φj, slip will occur only if the
direction of the applied compression makes an angle greater than the
friction angle with the normal to the layers. Thus, a pre-condition for
interlayer slip is that the normals be inclined less steeply than a line
inclined Φj above the plane of the slope. If the dip of the layers is σ,
then toppling failure with a slope inclined α degrees with the
horizontal can occur if (90 - σ) + Φj < α”.

Figure 1: Slope failures associated with
unfavorable orientation of discontinuities (modified after Hoek and
Bray, 1981).

Stereonets are used for graphical kinematic
analysis. Stereonets are circular graphs used for plotting planes based
on their orientations in terms of dip direction (direction of
inclination of a plane) and dip (inclination of a plane from the
horizontal). Orientations of discontinuities can be represented on a
stereonet in the form of great circles, poles or dip vectors. Clusters
of poles of discontinuity orientations on stereonets are identified by
visual investigation or using density contours on stereonets (Hoek and
Bray, 1981). Single representative orientation values for each cluster
set is then assigned. These single representative orientation values,
can be the highest density orientation value within a cluster set, or
the mean dip direction/dip of a pole cluster are calculated using
equations in Borradaille (2003).

Great circles for representative orientation values
along with great circle for slope face and the friction circle are
plotted on the same stereonet to evaluate the potential for
discontinuity-orientation dependent (Figures 2,3,4) (Hoek and Bray,
1981). This stereonet-based analysis is qualitative in nature and
requires the presence of tight data clusters for which a reasonable
representative orientation value can be assigned. The chosen
representative value may or may not be a good representation for a
cluster set depending on the tightness of data within a cluster. Tight
circular data is more uniform and has less variation making
representative values more meaningful than for cases where data shows a
wide scatter. Fisher’s K value is used to describe the tightness of a
scatter (Fisher, 1953). It is calculated as follows:

K = M – 1/M-|r_{n}|, where M is no. of data
within a cluster, and |r_{n}| is the magnitude of resultant
vector for the cluster set (Fisher, 1953).

High K values indicate tightly clustered data, i.e.
well-developed cluster set. If the cluster set is tight, the
representative values are more reliable and so is the stereonet-based
kinematic analysis. However, there are cases when a tight circular
clustering of discontinuity orientations does not exist. A different
quantitative approach that
does not require tightly clustered discontinuity sets can be performed
by ** DipAnalyst 2.0. DipAnalyst 2.0
**can also be used for the stereonet-based method.**
**

Figure 2: Stereographic plot showing requirements
for a plane failure (Hoek and Bray, 1981, Watts 2003). If the dip vector
(middle point of the great circle) of the great circle representing a
discontinuity set falls within the shaded area (area where the friction
angle is higher than slope angle), the potential for a plane failure
exists (figure created using RockPack).

Figure 3: Stereographic plot showing requirements
for a wedge failure (Hoek and Bray, 1981, Watts 2003). If the
intersection of two great circles representing discontinuities falls
within the shaded area (area where the friction angle is higher than
slope angle), the potential for a wedge failure exists (figure created
using RockPack software).

Figure 4: Stereographic plot showing requirements
for a toppling failure (Goodman, 1989, Watts 2003). The potential for a
toppling failure exists if dip vector (middle point of the great circle)
falls in the triangular shaded zone (figure created using RockPack
software).

## Factor of Safety Calculations

Limit equilibrium analysis is used to calculate the
factor of safety (F.S.) of a slope against failure once the kinematic
analysis indicates the potential for failure. Factor of safety is the
ratio of the resisting forces (shear strength) that tend to oppose the
slope movement to the driving forces (shear stress) that tend to cause
the movement along a plane of discontinuity. The equation for F.S. is:

F.S. = (c +
s
tan
f)/t
(13)

Where: F.S. = factor of safety

c = cohesion

f
= angle of internal friction

s
= normal stress on slip surface

t
= shear stress

According to the limit equilibrium approach, a factor
of safety value equal to 1 represents limiting condition. A value
greater than 1 represents a stable slope, and a value less than 1
indicates an unstable slope. The desired value of factor of safety
depends upon the importance of the slope and the consequences of
failure. For heavily travelled roads, slopes are usually designed to
have a factor of safety equal to or greater than 1.3 under saturated
conditions, maximum loads, and worst expected geological conditions
(Canadian Geotechnical Society, 1992; Wyllie and Mah, 2004). The
equations derived for both plane and wedge failure, consider weight of
the sliding block, cohesion along plane of discontinuity, effect of
water present along planes of discontinuity and tension applied from a
rock bolt. Although the equations for factor of safety calculations of
plane and wedge failures are based on Equation (13), they vary due to
differences in the shape of the sliding block for the case of plane vs.
wedge failures. The methods for
calculating the factor of safety for plane and wedge failures with
corresponding equations to determine the resisting and driving forces,
including the effect of water pressure along discontinuities and
application of a rock bolt are given in Hoek and Bray (1981) and Wyllie
and Mah (2004).

**To calculate factor of safety for plane failure, DipAnalyst 2.0,**requires slope height, slope angle, discontinuity plane inclination, position and depth of tension crack, and ground water conditions along the discontinuity plane/tension crack (Figure 5).

For wedge failure,*
*
DipAnalyst 2.0*
*
follows
the “short solution” put forward by Hoek and Bray (1981). The solution
requires slope height, slope angle, orientation of the two intersecting
planes (Figure 6). The “short solution” does not consider the effect of
a rock bolt, and the presence of a tension track.

Figure 6: Components of wedge failure analysis.

REFERENCES

BORRADAILE, G., 2003,
*Statistics of Earth Science Data*:
Springer, New York, 351p.

CANADIAN GEOTECHNICAL SOCIETY, 1992, C*anadian
Foundation Engineering Manual*, BiTech Publishers Ltd.,

FISHER, R.,A., 1953,
*Dispersion on a sphere*:
Proceedings of the Royal society of London, A217, pp.295-305.

GOODMAN, R. E., 1989,
*Introduction to Rock Mechanics*:
John Wiley & Sons, New York, NY 562 p.

HOEK, E. and BRAY, J. W., 1981,
*Rock Slope Engineering*: The
Institute of Mining and Metallurgy, London, England, 358 p.

*
User’s Manual-ROCKPACK III for Windows*:
C.F.Watts and Associates, Radford, VA, 33 p.

*Rock slope Engineering: 4*^{th}*
Edition*, Spon Press,