In the previous chapter we have examined one geometrical requirement for plane failure in rock slopes: the basal discontinuity has to dip at a sufficiently steep angle to overcome friction. However, this is not the only condition for plane failure. The discontinuity has to be suitably oriented to allow detachment of the slipped rock mass and for its forward movement out of the slope. This condition is described as the requirement that the potential basal plane daylights on the slope.

Figure 45a illustrates the predicted direction of movement on a plane of weakness. The overlying rock will slide in the direction of dip of the plane of weakness. For plane failure to occur this down-dip direction has to be directed out of the surface slope. In Figure 45b, the discontinuity with down-dip direction labelled 2 will not give rise to movement because sliding in the direction opposite to the surface slope is prevented because of lack of space. Direction 1, on the other hand, corresponds to the down-dip of a plane that dips out of the surface slope, i.e. a plane that meets the daylighting condition for plane failure.

We use a stereogram to assess the daylight condition of any particular plane of weakness. On the stereogram we define the orientation field made up of planes that daylight. This daylight zone is bounded by a daylight envelope on the stereogram, a curve made up the poles of planes that are marginal in terms of the daylighting condition.

Both pitch and plunge are angles between a given line and the horizontal. The difference is that the plunge is measured in an imaginary vertical plane whereas the pitch is measured in the plane which contains the line (Fig. 12a). Consequently, on the stereogram (Fig. 12b) both angles are measured from the plotted line L to the primitive circle; the plunge is the angle in a great circle which is a diameter (vertical plane) whilst the pitch is measured in the great circle representing the dipping plane which contains the line.

Again the procedure is explained with the aid of an actual example. A lineation defined by aligned amphibole crystals pitches 35S on a foliation plane which dips 015/30SE (Fig. 12c).

1. 1 Plot the foliation plane 015/30SE as a great circle on the tracing paper (Fig. 12d, for method see pp. 20–1).

2. 2 Rotate the net under the overlay until one of its great circles coincides with that for the plotted plane on the tracing paper (Fig. 12e).

3. 3 Starting from the primitive circle, count out the angle of pitch (here, 35°) inwards along the great circle. This gives the plotted position of the line (Fig. 12e). Note that the pitch is 35S, the ‘S’ indicating that the pitch is measured downwards from the southern end of the strike line of the plane. This is why we start our counting from the southern end of the great circle (Fig. 12e).

1. 1 Rotate the line 30–060 about an E–W horizontal axis, through an angle of 100° with a sense of rotation which is anticlockwise facing east. What is this line's new orientation?

2. 2 Rotate the line 50–340 about axis 0–020. What is the minimum angle of rotation necessary to make the line horizontal?

3. 3 A fault striking 030° and dipping 60W has permitted a rotational movement of the block on its eastern side. The beds on the western side strike 067° and dip 35N. What are the strike and dip of the beds on the eastern side after a 40° rotation with a sense which is clockwise looking down the plunge of the rotation axis?

4. 4 A fold hinge plunges 58° towards 212° and another plunges 62° towards 110°. What is the angle between them?

5. 5 A fault plane with orientation 130/30N displaces a bed with orientation 118/70N. Find the plunge and plunge direction of the cut-off line.

6. 6 A fault plane (150/50W) has slickenside lineations on it which pitch at 60N. What are the plunge and plunge direction of these lineations?

7. 7 An upright fold has limbs 180/60E and 031/80W Calculate the inter-limb angle? What is the apparent angle between the limbs visible on a vertical cross-section which trends 170°.

8. 8 What is the apparent dip, on a vertical cross-section trending 040°, of beds with strike 020° and true dip 60E?

9. […]

A variety of sedimentary structures of linear type (e.g. the ripple marks on p. 6) provide evidence of the flow direction of currents at the time of sedimentation. Measurements of the plunge or pitch of structures are often made in the field with a view to estimating palaeocurrents.

In tilted and folded strata, however, the current directions cannot be ascertained directly from the trends of such linear structures. The raw field data need to be restored to their original orientation, i.e. their orientation prior to tilting/folding. Two different situations are distinguished.

Tilted strata – fold axis is non-plunging or of unknown plunge

In these situations the usual assumption made is that the tilted strata acquired their dip by being rotated about a horizontal axis parallel to the strike (Fig. 34a).

1. 1 Plot the present orientation of the measured linear structure and the strike line of the beds (= rotation axis) on the stereogram (Fig. 34b).

2. 2 Rotate the net under the stereogram until the centre of the net's small circles coincides with the position of the plotted strike line (= rotation axis).

3. 3 Rotate the linear structure to the horizontal. This is done stereographically by moving the plotted linear structure back along the small circle on which it lies as far as the primitive circle (L → L′, see p. 64). The linear structure is now horizontal and its original trend can be read directly from the stereogram (Fig. 34b).

The area 20 km south-west of Bristol, England, provides an example of the use of stereographic methods for the purpose of interpreting the geometrical characteristics of the folding.

The first stage involves the inspection of the data presented on a map (Fig. 40a). The contrasting dip directions in the northern and southern parts of the area indicate a fold in Lower Carboniferous strata. A boundary line can be drawn between the northern and southern dips; this is the axial trace, the line of outcrop of the axial surface. The convergence and closure of lithological boundaries in the east are further evidence for a fold and the dips of bedding in the vicinity of the closure suggest that the plunge of the fold is eastwards there. The combination of an easterly closure of beds on the map and an easterly plunge suggests that the fold is an antiform.

The axial trace on the map is seen to be curved (Fig. 40a); this could be due to the effects of topography on the exposure trace of the axial surface or it could be an expression of a real curvature of the axial surface. The slight swing in the strikes of the beds on moving from west to east favours the latter explanation. This curvature makes it possible that the total geometry is non-cylindrical, a situation which warrants the division of the area into sub-areas for the separate analysis of the orientation data.

Measurement of the attitude of folded bedding or other foliation, if analysed stereographically, can permit certain geometrical properties of the folds to be determined. These deductions can be made even in areas where rock exposures are scarce, and as a result folds are hardly ever seen. An example of the type of technique employed was described on p. 44 where, if the poles to bedding plot along a great circle, this is interpreted to mean that the folds involved are cylindrical.

In addition, the way in which the poles are distributed within the great circle girdle can suggest other features of the folding to be discussed below, e.g. tightness (inter-limb angle), curvature and asymmetry. Unfortunately there exist other factors which potentially influence the spread of poles within the great circle, in particular the distribution of the sites at which measurements of the bedding have been made. The pattern on the stereogram can be biased by this sampling effect. For this reason care must be taken with the deduction of fold shape from stereograms.

Fold tightness

The range of orientations of the folded surface is restricted in an open fold (e.g. Fig. 24c) but is greater in a tight fold (Fig. 24i). The stereograms resulting from open structures show a lower degree of spread of poles than for tight folds. In Figure 24b, 24c and 24f where fold profile shapes are illustrated together with representative stereograms, a pole-free part of the great circle can be identified.

The kinematics of wedge failure, like plane failure, can be analysed from data consisting of the angle of sliding friction, φ, and the orientation of the rock slope.

With plane failure, dangerous orientations of planes of weakness map onto the stereogram in terms of their plane normals (poles). In contrast, when considering wedge failure we consider the orientation of the direction of wedge sliding, parallel to the intersection line of two sets of discontinuities (Fig. 46a).

Friction cone

To overcome frictional resistance under dry conditions, the plunge of the intersection line of the two discontinuities must exceed the sliding friction angle. All intersection lines in dangerous (steep) attitudes lie inside a cone consisting of all lines with a plunge equal to φ (Fig. 46b). This is the friction cone which gives a small circle on the stereogram (Fig. 46c).

Daylighting

Intersection line 1 in Figure 46d allows the possibility of wedge failure because it plunges (or at least has a component of plunge) in the direction of the natural slope and has an angle of plunge less than the apparent dip of the slope in the plunge direction. Intersection line 2 (Fig. 46d) would not permit wedge failure because it has a component of plunge into the slope. Lines 3 and 4 are intermediate cases where the intersection line lies with the plane of the slope. Therefore, on the stereogram, the great circle representing the plane of the slope corresponds to the daylight envelope.

To help visualization of the rotation of a line about a horizontal axis, take a pair of dividers (Fig. 32a), open them slightly, hold one leg horizontally and spin it between your fingers. This leg represents the rotation axis and retains this orientation throughout. The other leg represents the line which rotates; note how it changes orientation during rotation. It sweeps out a cone in space. This cone has a horizontal axis which corresponds to the rotation axis.

Cones project to give small circles (Fig. 32b). Therefore, a given line rotating about a fixed axis moves to take up different positions on the same small circle (Fig. 32c). The rotation axis R occupies the centre of the small circle defined by the line which rotates.

To rotate a line L about a horizontal axis, R

1. 1 Plot lines L and R on a stereogram (Fig. 32c). (The method for plotting lines is explained on pp. 22–5.) R is horizontal and therefore plots as a point on the primitive circle.

2. 2 Rotate the net beneath the stereogram until the point at the centre of all the net's small circles is positioned beneath R on the stereogram (Fig. 32d).

3. 3 The small circle on which L lies describes its path during rotation. For example, in Figure 32d, L lies on the α = 30° small circle.

4. 4 The new orientation of L (labelled new in Fig. 32d) is found by moving it around the small circle by an amount governed by the required angle of rotation (50° in Fig. 32d) in a direction governed by the required sense of rotation.

Figure 20a shows an example of the type of data frequently displayed on stereograms. The points represent the long axes of clasts measured in a till. The reason for measuring and plotting these data is not to show the orientations of individual axes but to analyse the pattern of orientations shown by the whole sample. The pattern of preferred orientation of till clasts is indicative of the flow direction of the ice. Before we can recognize such patterns confidently we need to know whether the stereographic projection faithfully represents the true clustering of directions in space.

One way of checking for possible distortion is to plot data that are known to be devoid of any preferred orientation. Figure 20b shows 2000 directions randomly chosen by the computer, plotted using the stereographic (Wulff) net (Fig. 20d). These directions are not evenly distributed across the stereogram (Fig. 20b) as we would expect but are more crowded in the central part of the net. This crowding is the direct result of the method of projection used. Our conclusion must be that the stereographic projection introduces an artificial preferred orientation of line directions, crowding the projected directions in the centre of the stereogram.

It is easy to understand this effect when we see how cones of identical size but different orientations are projected stereographically The small circles in Figure 20f represent two such cones.

In outcrop-scale folds it is often possible to measure the orientations of geometrical features such as the fold hinge line and fold limbs directly with a compass/clinometer in the field. In these instances the stereographic projection is used to manipulate these data, including rotation (see pp. 64–9) or calculation of the interlimb angle (pp. 46–7). For folds which are larger than outcrop-scale, the stereographic projection can be employed as well for the estimation of the orientation of the fold axis and axial plane.

Are the folds cylindrical?

The fold axis is defined with respect to folds which are cylindrical. Cylindrically folded surfaces (Fig. 22a, 22b) are surfaces with the form that could be swept out by a straight-line generator moving in space but remaining parallel to itself. Cylindrical folds have the property that their shape remains constant in serial sections. The fold axis is the orientation of the generator line of such surfaces. It is parallel to the hinge lines (the lines of sharpest curvature) of individual folds in the surface.

A cylindrical fold can be easily recognized from the measurements of orientations of the folded surfaces (e.g. bedding) taken at a variety of locations across the fold (Fig. 22d). The normals to the bedding planes in a cylindrical fold are all parallel to a single plane, the profile plane. Therefore, to check whether a fold is cylindrical or not:

1. 1 Plot on a single stereogram all the readings of the bedding (or other surfaces which have been folded) as poles (Fig. 22e, 22f).

2. […]

Consider a plane P and a line L which is not parallel to it (Fig. 19a). Imagine now a distant light source which is shining directly onto the plane, i.e. the plane is facing the light. The line L will cast a shadow onto the plane A. This shadow defines a line L′ in the plane P. We say that L′ is the orthogonal projection of L on plane P. Constructions involving orthogonal projection find a none of important applications in structural geology.

Calculating the projection of a line L onto a plane P

To calculate the direction of L′ use is made of the fact that lines L′, L and N (the normal to plane P) all lie in the same plane (Fig. 19b). This plane intersects plane P along the sought direction L′. Details of this construction are as follows.

1. 1 Plot the great circle for the plane P and its pole, N (Fig. 19c).

2. 2 Plot line L and then fit a plane (great circle) through N and L (Fig. 19c).

3. 3 L′ is given by the point of intersection of the great circle drawn at Stage 2 and that of plane P (Fig. 19c).

Application: slip direction of faults

The idea of projecting a line onto a plane is relevant to the problem of predicting the movement direction on a potential fault plane.

This construction is used frequently. It allows, for example, the calculation of inter-limb angles of folds and the angle of unconformity between two sequences of beds.

The solution using the stereographic projection is easy to understand as soon as it is appreciated what is actually meant by the angle between two planes. Figure 17a-17d helps explain this. Planes A and B cut each other to produce a line of intersection, L. The apparent angle between the pair of planes A and B depends on the cross-section chosen to view this angle. For example, the angle α (in Fig. 17a) observed on section plane C (which is perpendicular to the line of intersection) is different to the angle β seen on the oblique section plane (Fig. 17c). In fact, α is the true or dihedral angle between the planes A and B since the dihedral angle between a pair of planes is always measured in the plane which is perpendicular to the line of their intersection.

Determining the dihedral angle between a pair of planes (A, B) stereographically

Method using great circles

1. 1 Plot both planes as great circles (labelled A and B in Fig. 17b).

2. 2 The line of intersection L of these planes is given directly by the point of intersection of the great circles (see p. 26).

3. […]

Stage 1- - the structural line or plane is projected onto a sphere Let us start by projecting a structural line. Figure 6a shows such a line as it is observed, e.g. in the field. Alongside it we imagine a hollow sphere. The line is now shifted from the outcrop and, without rotating it, placed at the centre of the sphere (Fig. 6c). Now the length of the line is increased downwards until the line meets the sphere's surface. The point where the line and sphere meet is called the spherical projection of the line. This point is always located in the lower half of the sphere and its exact position depends on the orientation of the line.

A plane (Fig. 6b) is projected in the same way. It is translated to the sphere's centre and extended down until it touches the surface of the lower hemisphere (Fig. 6d). The line of contact is now a circle; a circle on the sphere with the same radius as the sphere itself. Such circles on a sphere which are produced by the intersection of a plane which passes through its centre are called great circles.

Thinking of the Earth as an approximate sphere, lines of longitude are great circles and so is the equator (Fig. 6e). However, there are other circles on the globe such as the polar circles and the tropics which do not qualify as great circles because they have a smaller radius than the Earth.

There are two ways of describing the orientation of a geological line:

Plunge and plunge direction In this system the orientation of the line is described with reference to an imaginary vertical plane which passes through the line (Fig. 4a). The angle of tilt of the line measured in this vertical plane is called the angle of plunge. The plunge is measured with a clinometer which is held upright and with the edge of the instrument aligned with the linear structure (Fig. 4c). The plunge direction is parallel to the strike of the imaginary vertical plane passing through the plunging line (Fig. 4a). It is measured by placing the edge of the lid of the compass along the lineation and, with the plate of the compass held horizontal, measuring the compass direction of the down-plunge direction (Fig. 4d).

These measurements are written as

angle of plunge–plunge direction.

For example, a linear structure with orientation 30–068 is tilted down or ‘plunges’ at an angle of 30° towards bearing (compass direction) 068° (Fig. 4a). Line 0–124 could equally be written 0–304 because a horizontal line can be said to plunge in either of two directions 180° apart. A linear structure with a plunge of 90° is vertical and its direction of plunge is not defined.

Pitch Many linear structures are developed on planar structures; for example, slickenside lineations are to be found on fault planes.