fractures are one of the major features of rocks. most of the fractures are approximately parallel to each other and constitute what is called a 'set'. they may be closely or widely spaced. most rocks contain several fracture sets. the presence of fractures is the main cause of failure of rock slopes.

note that rocks may split:
     - along widely-spaced planes, following the grain of the rock;
     - along many closely-spaced planes, following the grain of the rock;
     - across the grain of the rock at a regular spacing and orientation. these
       fractures are called joints.

the grain of the rock is caused by layers of minerals, in different proportions or of different textures.

friction along the interfaces between the blocks governs the shear strength of the rock. shear strength is reduced when contact along the interfaces is lost.

rock strength is related to the number and weakness of fractures. strong rocks have fewer fractures or closed and cemented fractures.

open-jointed rocks are very weak because:
     - water movement and weathering take place preferentially along
        the joint planes;
     - there is a loss of frictional resistance along the interfaces.

the degree of weathering of the rock itself also controls the rock strength. a highly weathered rock may fail through the rock body rather than along the joints.

rocks are penetrated by fractures of various spacing and orientation.
rock strength is due to both:
     - spacing and weakness of fractures;
     - degree of weathering of the body of the rock.
fractures are the main cause of failure in rock slopes.

rock structures seen in a sample are related to those observed in the field. however, the rocks in the field vary from place to place because of differences in their composition, weathering conditions and fracturing.

notice the following features of the rock:
     - bedding;
     - orientation of structures;
     - fracturing and jointing.
the orientation of these planes controls the resistance of the rock to gravitational forces.

dip and strike
the geological compass permits to measure the dip direction of an inclined geologic plane and thus to define its position in the space. in the case of a vertical geological plane its strike define this position. horizontal geologic planes neither have dip nor direction of strike. figure 1 illustrates the above definitions.

figure 1: dip direction and strike of a geological plane

rock structures seen in a sample are related to those observed in the field. however, the rocks in the field vary from place to place because of differences in their composition, weathering conditions and fracturing.

notice the following features of the rock:
     - bedding;
     - orientation of structures;
     - fracturing and jointing.

the orientation of these planes controls the resistance of the rock to gravitational forces.

the planes control the movement of water through the rock, and hence weathering.
dip is the line of maximum slope of a rock plane. the angle of dip is measured with clinometer and the bearing of dip is measured with a compass. the bearing can be any figure from 000° to 360°. it is always written as a 3-figure number, e.g. 048. this distinguishes the bearing from the inclination, which cannot exceed 90°. a reading for the angle of dip, which appears to be greater than 90° means that the slope is in fact dipping in the opposite direction.

strike is the horizontal line contained in the plane of bedding, foliation, or jointing. the bearing of strike is measured with a compass. the figure is always given as a reading less than 180°.

in practice strike is measured first because a horizontal line needs to be established in order to find the maximum inclination of the dipping plane.

conventionally the bearing of dip is written first, followed by the angle of dip, e.g. 115/35.

note that the direction of dip is the direction towards which the plane is inclined. when you are measuring dipping surfaces in the field the procedure for orienting the compass differs slightly according to whether you are measuring the plane from on top or underneath. if you are measuring on top the back of the compass is placed against the rock and you measure away from the plane. if you are measuring underneath the front of the compass is placed against the rock and you measure into the plane.

introduction
stereographic projection is one of the convenient methods of projecting the linear and planar features. this method is used exclusively for the determination of the angular relationship among the lines as well as planes. in geotechnical engineering, it provides a quick and reliable picture of the discontinuities and their intersections. it is also used for the estimation of cut slope angle, for the preparation of hazard maps, and for the estimation of safety factors.

the stereographic projection is a projection of the sphere. the sphere is divided into two equal hemispheres by a horizontal (equatorial) plane and the upper and lower poles are fixed as shown in figure 1. the plane of projection is the equatorial plane itself. the circumference of the equatorial plane is called the primitive circle. for the stereographic projection one of the hemispheres is chosen. here the technique of projection on the upper hemisphere is discussed. the principle of projection is the same for the lower hemisphere.

projection of a line
any required line 1 is moved parallel to itself in such a way that it passes through the centre of the sphere. In doing so, the line will pierce the sphere in two diametrically opposite points A and B (fig. 2). for the upper hemispherical projection, the upper point is projected down to the equatorial plane as the point of intersection A of the line joining the point A with the lower pole Z (fig. 2).

if the line is vertical, the projection will be at the centre 0 of the sphere. if the line is horizontal, the projection will be at the primitive circle in diametrically opposite points n and n (fig. 3). all the inclined lines will be projected between the primitive circle and its centre (fig. 2 and 4).

fig. 1: projection sphere


fig. 2: projection of an inclined line on the upper hemisphere


fig. 3: projection of vertical, horizontal, and inclined lines
on the upper hemisphere


fig. 4: stereographic projection of the horizontal, vertical,
and inclined lines of fig. 3


fig. 5: intersection of an inclined plane

fig. 6: projection of an inclined plane, intersecting
the upper hemisphere


fig. 7: the stereographic projection of an inclined plane of fig. 6


fig. 8: projection of a cone on the upper hemisphere


fig. 9: the stereographic projection of the cone


fig. 10: wuff stereographic net


fig. 11: method of projection for equal-area net

projection of a plane
a plane is projected in the same way as the line. we can imagine the plane containing several lines that pass through the centre of the sphere. each of the lines is projected on the equatorial plane and the trajectory of them will give the projection of the plane.

in general, a plane P is moved parallel to itself unless it passes through the centre of the sphere. the intersection of the plane with the sphere is a great circle with the radius equal to that of the sphere or primitive circle (fig. 5). then, each of the points, k, 1, m, n, etc from the upper hemispherical part of the great circle (fig. 7.5a), is joined by straight lines with the lower pole Z and the trace of the intersecting points, k, ii, m, n, etc on the equatorial plane, gives the projection (fig. 6 and 7).

any vertical plane passing through the centre will bisect both the hemispheres. its projection will be a straight line passing through the centre. any horizontal plane passing through the centre is projected as the primitive circle itself. inclined planes are projected as curves known as great circles with the ends at the diametrically opposite points in the primitive circle (fig. 7).

projection of a cone
let us consider a cone with a vertex at the centre of the sphere and horizontal axis of rotation (fig. 8).
it will intersect the sphere in a circle (fig. 8). by joining all the points of the circle from the upper hemisphere we get a projection of the cone as small circles (fig. 9).

by constructing a series of great circles and small circles on the upper hemisphere, we obtain the Meridional Stereographic, or Wulif Net, or Stereonet. in it the two types of curves are generally drawn every 2° interval (fig. 10).

figure 10 shows that the area on the net, for example, 10°x l0° in the centre of the net is smaller than the same area at the margin. it creates problems while doing the statistical analysis of the data, as the randomly distributed points on the Wulff Net will show a concentration of the points at the centre. that is, the random line would falsely show a weak preferred orientation in the vertical position. to overcome this problem a somewhat different type of projection is needed. this method of projection is called the Lambert Equal Area, or simply the Equal Area Projection. (fig 11). the small and great circles of the Wulif Net are modified to a series of curves and the result is the equal area or Schmidt Net (fig. 12). our further investigations will be carried out exclusively on the Schmidt Net to avoid confusion.

plotting techniques
when plotting on the stereonet it is important to visualize the net as the convex-upwards hemisphere and to imagine the curves inscribed on its outer surface. to make the plotting visual, several figures are given together with the plotting on the Equal Area Net. the techniques discussed here are modified after Ragan (1985).

fig. 12: equal area or schmidt net

plotting A line (fig. 13)
given a line with trend = 40° and plunge = 46°:
   1. with the tracing paper on the Equal Area Stereonet trace the primitive
     circle and make a small tick mark and label it N; this is the first step
     in all works on the stereonet;
   2. to locate the trend of the line, count off 40° clockwise from N and make
     the second small mark on the primitive circle at this point;
   3. revolve this trend mark about the centre of the net to the north point
     (or any other straight line such as East-West and North-South) of the net;
   4. count off 46° from the primitive, inwards along the diametrically
     opposite (Southern) end of theNorth-South straight line and mark
     the point P; and
   5. restore the overlay to the starting position and recheck by visualization.

visualization: hold a pencil over the centre of the stereonet in the direction of the given trend direction at the angle of plunge. visualize its intersection with the upper hemisphere in the southwest quadrant.

plotting a plane (fig. 14)
given a plane with dip direction = 324° and angle of dip = 40°
   1. with the overlay in place (fig. 14a) and the north index marked N,
     locate a point on the primitive representing the dip direction of the plane
     by counting 324° clockwise or 36° counter clockwise. (fig.14b).
   2. turn the dip direction mark on the overlay about the centre of the net
     (fig. 14c) until it is exactly over the west (or east) direction (fig. 14d).
     count off 40° from the primitive, inwards along the diametrically opposite
     end (i.e., the eastern end) of the East-West straight line of the net, and
     trace the required great circle (fig. 14d).
   3. restore the overlay to the starting position and recheck by visualization
     (fig. 13e).

remarks: do not forget to begin counting off inwards from the diametrically opposite end of the marked position. if you count off directly inwards from the marked position, it will be the projection on the lower hemisphere. visualization: hold the right hand palm upwards, over the centre of the stereonet with the fingers pointing towards 324° + 90° = 414° = 54° NE and the plane of the hand inclined 40° to the northwest (324°). the plane of the hand in this position can be imagined to extend into the upper hemisphere and intersect its surface. its trace cuts the southeast quadrant and this is where the final plot must be.

fig. 13: plotting a line


fig. 14: plotting a plane

plotting a line contained in a plane
given a plane (sandstone bed) with a dip direction of 130° and a dip amount 55° and a line (intersection of the vertical cut slope with the sandstone bed) with a trend of 78° and lying in that plane. find the plunge of the line (apparent dip of the bed along that cut slope).
note: the apparent dip is the angle between the line of intersection of the sandstone bed with the vertical cut slope and any horizontal line parallel to the slope.
   1 with the overlay in place (fig. 15), and the north index marked N, locate
     a point on the primitive representing the dip direction of the plane
     by counting 130° clockwise from N.
   2. turn the dip direction mark about the centre of the net until it is exactly
     over the east (or west) direction (fig. 15). Count off 55° from the primitive,
     inwards along the diametrically opposite end (i.e., the western end)
     of the East-West straight line, and trace the required great circle.
   3. restore the overlay to its starting position and count off 78°
     from N and make a tick mark.
   4. rotate the overlay about the centre and coincide the tick mark with
     the nearby diameter and locate the required point (line) in the previously
     traced great circle. In the same position count off the plunge of the line
     which is equal to the angle between the tick-marked point in the primitive
     and point (line) located in the great circle.
   5. restore the overlay to the starting position and recheck
    by visualization (fig. 15).
result: the plunge is 41°

plotting a pole
like any other projection, the stereographic projection reduces the dimensions of the object by one. here, a line becomes a point, and the plane is reduced to a curve. it is important to notice the fact that every plane has a unique normal which can define its angular relations unequivocally. therefore, it becomes possible to reduce further the dimension of the plane and to represent it on the stereonet by a point called the pole. to visualize it, hold the left hand as in figure. 14 and hold a pencil between the fingers so that it is perpendicular to the plane of the hand. the line of the pencil will intersect the upper hemisphere at a point in the northwest quadrant.
given a plane with a dip direction of 60°, and a dip angle of 72°, plot the plane and locate the pole to it (fig. 16).
   1. with the overlay in place and the north index marked N, locate a point
     on the primitive circle equal to the dip direction of the plane by counting
     off 60° clockwise from N and tick mark it.
   2. rotate the overlay about the centre until the tick mark is over
     the East-West diameter of the net and, by counting off 72° inwards
     from the opposite end of the diameter, trace the required great circle.

fig. 15: plotting a line contained in a plane


fig. 16: plotting the pole to a plane

   3. as the pole is everywhere located 900 from the plane, count off
     90° towards the tick-marked point from the great circle and locate
     the pole (fig. 15).
   4. restore the overlay to its original position and visualize the plotting
     by holding the left hand over the centre of the net and tilting it towards
     NE (60°) by 72°, while the fingers will point towards
     60°-90° = -30° = 330° NW (the strike). In this position, the plane will
     be plotted in the SW quadrant and the pole in the NE quadrant.

note that it is easier to plot the pole by rotating the overlay so that the dip direction mark coincides with any nearby diameter and by counting off the dip amount (here, 70°) from the centre of the net towards the tick mark (dip direction). check the previous results with this method.
result: the position of the pole is 240°/18°.

plotting the line of intersection of two planes (fig 17)
given two intersecting planes (joints), having dip amounts of 400 and 60° and a dip direction of 120° and 260° respectively, it is required to find the plunge and the trend of the line of intersection (wedge axis).
   1. with tracing paper over the stereonet trace the primitive circle and mark
     the north point. measure off the dip direction of 120° clockwise from N
     and mark this position on the primitive.
   2. rotate the overlay about the centre of the net until the dip direction mark
     lies on the East-West diameter of the net. measure 40° from the primitive
     and trace the required great circle.
   3. rotate back the tracing paper to its original position and repeat the same
     for the second plane.
   4. the point of intersection of two great circles defines the required line.
     rotate the overlay until the intersection of the two great circles lies along
     the East-West diameter of the stereonet and measure the plunge of the
     line of intersection. also mark with a tick on the overlay at the diametrically
     opposite direction (trend) along the East-West line on the primitive circle.
   5. rotate the tracing back to its original position and count off the trend
     of the line.

result: The trend and plunge of the line are 183° and 22° respectively.

fig. 17: plotting the line of intersection of two planes

location:                           altitude:

sketch of the slope face:
make a sketch of the slope face and colour it. also, show the north direction, landmarks, length of rock outcrop, foliation, and joints.

describe the fractures:
provide rough dip direction (i.e. N, NNE, NE, NEE, E, etc) and angle of dip (less than 10 degrees, 10-30 degrees, 30-60 degrees, 60-80 degrees, and more than 80 degrees).
foliation or bedding F: ... . .. .. . .. .. . . .. .. . .. .. .. . .. .. . .. . .. . .. .. .. . .. .. . .. .. . ..
joint set one: J1:
joint set two: J2:
joint set three: J3:

measurement of joint spacing:
measure the perpendicular distance between the foliation/joint sets and write down.
foliation or bedding F: .. .
joint set one: J1:
joint set two: J2:
joint set three: J3:

measurement of joint roughness:
draw the joint profile as seen along the joint plane of about 1m length.
foliation or bedding F:   
joint set one: J1:    
joint set two: J2:     
joint set three: J3:   

determination of rock weathering grade:
use the table of rock weathering grade and the classify the rock into:
completely weathered:   
highly weathered:           
moderately weathered:   
slightly weathered:          
fresh rock:                       

determination of rock type:
classify the rock into the main rock type according to its strength.
     hard rock (Gneiss, Quartzite, Granite)
     moderately soft rock ((Dolomite, Limestone, Marble, Sandstone)
     soft rock (Schist, Phyllite, Slate)
     very soft rock (Shale, Mudstone)

comparison of joint orientation for stability of slopes:
compare the slope with the block diagrams of and find out the position of the joints with respect to canal alignment.

bearing and amount of natural slope:

bearing and amount of cut slope:

signs of instability:
     a. Cracks
     b. Gully erosion
     c. Soil erosion
     d. Rill erosion
     e. Uphill tilted slopes
     f. Drunken trees

comments on the rock slope: