Kinematic Slope Stability

Scope. This page is the methodological companion to the Slope Stability user guide. The guide covers the dialog, the stereonet properties and the workflow; this page states what each test computes, with the formulas and the default values, so that a result can be understood, reported and reproduced.

Abstract

A kinematic analysis asks whether the geometry of a slope and of its discontinuities lets a block move: whether a plane is steep enough to slide on and daylights in the face, whether two planes form a wedge whose line of intersection does, whether steep planes dipping into the face let the rock between them bend over, and whether columns bounded by two sets lean far enough out of the face to fall. VRGS tests four failure modes — planar sliding, wedge sliding (Markland, 1972, with Hocking's 1976 refinement), flexural toppling (Goodman & Bray, 1976) and direct toppling (Hudson & Harrison, 1997) — Every test is a sign test on unit vectors, so it holds for any face orientation, overhangs included. A mode is scored as the percentage of a discontinuity set's planes, or of the lines where two sets' planes meet, that are critical. On a mesh or point cloud the slope face is fitted around every point at a chosen scale and each point reads its percentages from a table of face orientations; on the stereonet one face is analysed and its construction and critical zones are drawn, as Dips draws them.

1. Introduction

Structurally controlled failures of rock slopes are governed first by geometry. A block can slide on a plane only if the plane is steeper than its friction angle and is cut by the slope face, so that the block is free to move out of it; a wedge can slide only if the line along which its two planes meet does the same; a stack of steep layers dipping into the slope can topple only if the layers can slip past each other. These conditions are tested on a stereonet by the classical construction (Markland, 1972; Hoek & Bray, 1981; Goodman, 1980; Wyllie & Mah, 2004), and a result is usually reported as the percentage of the measured discontinuities, or of their intersections, that plot in a critical zone.

That result belongs to one slope face. On a photogrammetric or lidar model the face changes from place to place — a bench, a buttress, a gully, an overhang — and a single stereonet cannot say where along the outcrop a set becomes critical. VRGS therefore uses the same tests two ways:

1. As maps. The slope face is fitted around every point of a mesh or point cloud at a scale the user chooses, and the percentage critical for each mode is written as an attribute layer, with a classification into the dominant mode.
2. On the stereonet. For one face the user sets, the construction and the critical zones are drawn over the plotted planes, with the percentages set by set and pair by pair.

Both use the same tests, sets and percentages, so the key of the stereonet and the layers of the map agree for a face of the same orientation.

The analysis is kinematic. It says where the geometry permits a mode of failure; it knows nothing of cohesion, water pressure, block size, persistence or spacing, and a critical percentage is not a probability of failure or a factor of safety.

2. Conventions and notation

The frame is x east, y grid north, z up; angles are in degrees, and a direction is measured clockwise from grid north. The unit vector with dip $\delta$ towards direction $\alpha$ is

$$\mathbf n(\delta,\alpha) = (\sin\delta\sin\alpha,\ \sin\delta\cos\alpha,\ \cos\delta).$$

• A discontinuity plane has dip $\delta\in[0^\circ,90^\circ]$ and dip direction $\alpha$. Its upward normal is $\mathbf n = \mathbf n(\delta,\alpha)$, its horizontal dip direction $\hat{\mathbf h} = (\sin\alpha,\cos\alpha,0)$ and its down-dip vector $\mathbf d = \cos\delta\,\hat{\mathbf h} - \sin\delta\,\hat{\mathbf z}$. Its lower-hemisphere pole $\mathbf p = -\mathbf n$ plunges $90^\circ-\delta$ in the direction opposite the dip. A vertical plane ($\delta = 90^\circ$) dips both ways, and both senses of $\hat{\mathbf h}$ are tried wherever a test uses the dip direction.
• The slope face has dip $\psi_f\in[0^\circ,180^\circ]$ and dip direction $\alpha_f$, the direction it looks. Its outward normal is $\mathbf f = \mathbf n(\psi_f,\alpha_f)$ and its horizontal direction $\hat{\mathbf h}_f = (\sin\alpha_f,\cos\alpha_f,0)$. A face dipping more than $90^\circ$ overhangs, and its outward normal points down.
• A line — here always a line of intersection of two planes — is the downward unit vector $\mathbf l$, with trend $\tau$, plunge $\beta$ (so $-l_z = \sin\beta$) and horizontal direction $\hat{\mathbf h}_l$.
• Parameters: the friction angle $\phi$ of the discontinuities and three lateral limits, $\lambda_p$ for planar sliding (default $20^\circ$), $\lambda_t$ for flexural toppling ($30^\circ$) and $\lambda_d$ for direct toppling ($20^\circ$).

Two horizontal directions are within $\lambda$ of each other when $\hat{\mathbf a}\cdot\hat{\mathbf b}\ge\cos\lambda$. Comparing vectors rather than differences of angles means dip directions of $359^\circ$ and $1^\circ$ are $2^\circ$ apart, with no special case at north. A level face ($\psi_f = 0^\circ$ or $180^\circ$) looks no way; the tests that need its direction are then never critical.

3. The failure modes

3.1 Planar sliding

A plane is critical for planar sliding (Hoek & Bray, 1981) when all three hold:

1. it is steeper than the friction angle, $\delta > \phi$;
2. its dip direction is within the lateral limit of the face's, $\hat{\mathbf h}\cdot\hat{\mathbf h}_f \ge \cos\lambda_p$;
3. it daylights: its down-dip vector points out of the face, $\mathbf d\cdot\mathbf f > 0$.

For a face dipping less than $90^\circ$ the third condition is the familiar apparent-dip test: with $\Delta$ the angle between the two dip directions,

$$\mathbf d\cdot\mathbf f = \cos\delta\,\sin\psi_f\cos\Delta - \sin\delta\cos\psi_f > 0 \iff \tan\delta < \tan\psi_f\cos\Delta ,$$

that is, the plane is flatter than the face in its own dip direction. Under an overhang $\cos\psi_f < 0$, and every plane dipping out of the face within the lateral limit daylights; a vertical plane daylights only there.

3.2 Wedge sliding

Two planes with normals $\mathbf n_a$ and $\mathbf n_b$ meet along

$$\mathbf l = \pm\frac{\mathbf n_a\times\mathbf n_b}{\lVert\mathbf n_a\times\mathbf n_b\rVert},$$

taken downward. $\lVert\mathbf n_a\times\mathbf n_b\rVert$ is the sine of the angle between the planes, and planes within $10^\circ$ of parallel form no wedge. Lines are formed only between planes of two different sets.

Markland's test (Markland, 1972): the line is critical when it plunges more steeply than the friction angle and daylights,

$$\beta > \phi \quad\text{and}\quad \mathbf l\cdot\mathbf f > 0 .$$

There is no lateral limit: the two planes confine the block to the line.

Hocking's refinement (Hocking, 1976). A block that passes Markland's test does not always slide along its line. When the dip direction of one of the planes lies between the trend of the line and the direction the face looks, the block slides down that plane alone and lifts off the other. With $x$ the signed angle from $\tau$ to $\alpha_f$ and $d_a$ the signed angle from $\tau$ to $\alpha_a$, both in $(-180^\circ,180^\circ]$, plane $a$ lies between when

$$d_a \ne 0,\qquad \operatorname{sign} d_a = \operatorname{sign} x,\qquad \lvert d_a\rvert < \lvert x\rvert ,$$

and likewise for plane $b$. When the face or the line is vertical, or a plane is level, nothing lies between and the block slides along the line. Either way the block is free to move, so a wedge counts when its line passes Markland's test, on the maps and in the stereonet's Wedge column. Hocking's test says only how it slides; the stereonet reports the blocks that slide on one plane separately (below).

One-plane sliding (Dips' secondary zone). A wedge whose line is too flat to slide along can still slide on one of its planes. VRGS counts a line in this zone when

$$\beta \le \phi,\qquad \mathbf l\cdot\mathbf f > 0,\qquad \mathbf l\cdot\mathbf n_\phi < 0,$$

where $\mathbf n_\phi = \mathbf n(\phi,\alpha_f)$ is the upward normal of the friction plane, the plane dipping at the friction angle towards the face: the line daylights and lies below that plane. It is counted as sliding on one plane when one of its two planes is steeper than $\phi$ and daylights (§3.1 without the lateral limit). Together with the Markland-critical lines that Hocking's test sends down one plane, which the Wedge column counts as well, these make the 1 plane column of the stereonet key.

3.3 Flexural toppling

Steep layers dipping into a slope topple by bending when they can slip past each other. Goodman (1980), after Goodman & Bray (1976), states the condition for a plane dipping straight into the face as

$$(90^\circ-\delta) + \phi < \psi_f ,$$

within a lateral limit of straight in. VRGS applies it as Dips does, with the slip-limit plane: the plane dipping at $\sigma = \psi_f - \phi$ towards the face, with upward normal $\mathbf s = \mathbf n(\sigma,\alpha_f)$. A plane is critical when $\sigma > 0$, its pole trends within the lateral limit of the face direction, $-\hat{\mathbf h}\cdot\hat{\mathbf h}_f \ge \cos\lambda_t$ (the plane dips into the face within $\lambda_t$ of straight in), and its pole lies below the slip-limit plane:

$$\mathbf p\cdot\mathbf s = \sin\delta\sin\sigma\cos\Delta - \cos\delta\cos\sigma > 0 ,$$

with $\Delta$ the angle between the pole's trend and $\alpha_f$. Straight into the face ($\Delta = 0$) this is $-\cos(\delta+\sigma) > 0$, which is Goodman's condition. To the side it reads $\tan(90^\circ-\delta) < \tan\sigma\cos\Delta$: the pole must plunge less than the slip-limit plane's apparent dip in its direction, which is stricter than comparing the pole's plunge with $\sigma$. On the stereonet the critical poles lie between the slip-limit plane's great circle and the primitive, within the lateral limits.

3.4 Direct toppling

Direct (block) toppling needs columns and a base (Hudson & Harrison, 1997). The columns are bounded by two sets whose line of intersection plunges steeply into the slope, so the columns lean out of the face; the base is a plane too gentle for the columns to slide on. VRGS uses the zones of Dips:

• Zones 1 and 2 — direct toppling. A line of two sets' planes is critical when it trends into the slope within the lateral limit of straight in, and lies no further from vertical than the face dip:

  $$\hat{\mathbf h}_l\cdot(-\hat{\mathbf h}_f) \ge \cos\lambda_d \quad\text{and}\quad 90^\circ-\beta \le \psi_f \;\;(\text{i.e. } -l_z \ge \cos\psi_f).$$

  A vertical line leans no way and passes the lateral test. Under an overhang every line within the lateral limit passes the second test.

• Zone 3 — oblique toppling. A line within the friction angle of vertical that leans into the slope but outside the lateral limit: $90^\circ-\beta < \phi$ and $0 < \hat{\mathbf h}_l\cdot(-\hat{\mathbf h}_f) < \cos\lambda_d$.

• Base planes. A plane that can be the base of the columns without sliding: it dips more gently than the friction angle, $\delta < \phi$, and either dips out of the slope, $\hat{\mathbf h}\cdot\hat{\mathbf h}_f > 0$, or is level. On the net the pole of such a plane lies within $\phi$ of the centre on the side away from the face, which is where the zones above lie: inside the lateral limits it is in the direct-toppling zone (Dips' zone 2) whenever $\phi \le \psi_f$, and outside them in the oblique zone.

The Direct Toppling map layer holds the percentage of lines in zones 1 and 2. Oblique toppling and base planes are reported on the stereonet.

4. Discontinuity sets and percentages

4.1 Sets

A set is the planes in one folder. The maps take the orientation folders ticked in the dialog, each with the planes directly in it; the stereonet takes the folders of the plane interpretations it plots, one set per folder as the maps do, whatever the folders are called. Lineations are left out, and on the stereonet so are faults, scanline fractures, polylines, DFN fractures and mesh poles.

Auto-cluster copies the planes it clusters into folders below the one it read them from and leaves the originals there. A folder holding every plane of some folder below it — a copy being the same dip and dip direction in single precision and the same fitting circle — is therefore not a set by default: the dialog leaves it unticked the first time, and the stereonet leaves its poles out. Taken as a set, its planes would meet copies of their own set's in lines that no two sets form. One plane in common is not enough, since compass readings in whole degrees repeat by chance. When the ticked folders share planes anyway, the run warns how many.

Before testing, each set's plane normals are merged into bins $1^\circ$ wide in polar angle and azimuth. A bin holds the weighted mean direction of its planes and their number, $w_b$, so the cost of the tests does not grow with the number of measurements. Lines of intersection are formed from every pair of plane bins from two different sets, weighted by the number of plane pairs, $w_a w_b$, and binned in the same way. When a pair of sets has more than $10^6$ pairs of plane bins, its planes are re-binned at $2^\circ$, $4^\circ$ and $8^\circ$ until it has fewer.

4.2 Percentages

For a set $S$ and a mode tested on planes, the percentage critical is

$$P(S) = 100\,\frac{\sum_{b\in S,\ b\ \text{critical}} w_b}{\sum_{b\in S} w_b},$$

and for a pair of sets $(S,T)$ and a mode tested on lines,

$$P(S,T) = 100\,\frac{\sum_{\text{critical lines}} w_a w_b}{\sum_{\text{all lines}} w_a w_b},$$

where the denominator counts every pair of planes that meets in a line, critical or not.

A map holds the highest percentage over the sets (pairs of sets), $P = \max_S P(S)$ or $\max_{S,T} P(S,T)$, so a set that is fully critical is not diluted by a set that never is. The stereonet lists every set (pair) and All: the critical share of every plane of every set, or of every line of every pair of sets, counted together. Lines between two planes of the same set are never formed, so they are not in All either.

4.3 Classification

The Kinematic Mode layer takes, at every point, the mode with the highest percentage among those tested — planar sliding (1), wedge sliding (2), flexural toppling (3) or direct toppling (4), ties going to the earlier — or none (0) when that percentage is zero or below the classification threshold (default 10%).

5. The slope face at a scale

On a mesh or point cloud the face is the local surface at the scale of the failures being assessed, the radius $r$ — not a triangle, and not the whole outcrop.

1. Samples. The live points are merged into cubes of side $r/8$. Each occupied cube is one sample at the mean position of its points, carrying the sum of their outward hints (below). The work then depends on the area and the radius rather than the point density.
2. Plane fit. For every sample $c$, the samples within $r$ are weighted by the biweight $w = (1 - d^2/r^2)^2$, which falls smoothly to zero at the radius so the face does not jump as samples cross it. The face normal is the eigenvector of the smallest eigenvalue of their weighted covariance. A sample with fewer than 6 neighbours, or whose neighbourhood is a line or a point (second eigenvalue under $10^{-9}$ of the first), gets no face.
3. Outward sense. A fitted plane has two sides. The side is taken from the hints of the neighbourhood: on a mesh the area-weighted vertex normals, which follow the triangle winding; on a point cloud the camera-oriented normals of an SfM dense cloud (the Nx, Ny, Nz layers), which point back towards the cameras. With $\mathbf H$ the weighted sum of the hints and $L$ the weighted sum of their lengths, the normal is turned to agree with $\mathbf H$ when $\lvert\mathbf n\cdot\mathbf H\rvert > 0.1\,L$. Otherwise — no hints, or hints that disagree within the radius — the face is unsigned and is folded to look upward, so an overhang among them reads as a slope facing the other way. The dialog's flip option reverses the hints, for a mesh whose normals point into the rock.
4. Every point takes the face of its cube, as a dip $\psi_f\in[0^\circ,180^\circ]$ and a dip direction.

The suggested radius is 2% of the model's diagonal but no less than twenty point spacings; a radius under four point spacings is raised to four, so a cube holds about a point or more.

6. The table of face orientations

Once the sets, the friction angle and the limits are fixed, every percentage depends on the face orientation alone. VRGS therefore evaluates them once per orientation on a regular grid — face dip from $0^\circ$ to $180^\circ$ (both ends included) by dip direction from $0^\circ$ to $360^\circ$ (wrapping), $1^\circ$ apart — and every point reads its face from the table by bilinear interpolation between the four nodes around it. Hocking's test and the lateral limit of direct toppling depend on the face direction only, so each column of the table sorts the lines once for all its dips. After the faces are fitted, the cost is independent of the number of points.

7. The stereonet construction

The stereonet draws, for the face set in its properties, the construction of Dips on a lower-hemisphere net, equal-angle or equal-area. Planar sliding and flexural toppling are analysed on poles, wedge sliding on lines of intersection, and direct toppling on both, in zones they share.

Mode	Curves	Primary zone (red)	Secondary zone (yellow)
Planar sliding	Slope face; friction cone — the poles of planes dipping at $\phi$, the circle of plunge $90^\circ-\phi$; daylight envelope — the poles of the planes whose dip vector lies in the face; lateral limits at $\alpha_f\pm\lambda_p$	Poles critical by §3.1	—
Wedge sliding	Slope face; friction cone — the lines plunging at $\phi$; friction plane — the great circle dipping $\phi$ towards the face	Lines passing Markland's test	Between the slope face and the friction plane: one-plane sliding (§3.2)
Flexural toppling	Slope face; slip-limit plane — the great circle dipping $\psi_f-\phi$ towards the face; lateral limits at $\alpha_f\pm\lambda_t$	Poles critical by §3.3	—
Direct toppling	Slope face; friction cone (plunge $90^\circ-\phi$); slope angle cone — the lines as far from vertical as the face dip, plunge $90^\circ-\psi_f$, drawn for faces under $90^\circ$; lateral limits at $\alpha_f\pm\lambda_d$	Zones 1 and 2	Zone 3: oblique toppling, and oblique base planes

A point of the net lies in a zone when the zone's test holds for the vector it plots. The zones are traced by marching squares over a grid of 400 cells across the net: the test is evaluated at every node, each crossing is refined by bisection along its cell edge, and the outlines are closed into rings and filled with the even-odd rule, so a hole in a zone stays a hole. The outline lies on the true boundary to well within a cell; a corner is cut by up to half a cell, and a part of a zone narrower than a cell can be missed.

The zones are drawn for the orientations as measured. With Remove Regional Dip on, the plotted poles are rotated and would sit on zones drawn for other orientations, so the overlay is not drawn.

8. Assumptions and limitations

• Kinematic only. Cohesion, water pressure, block size, persistence, spacing, roughness and external loads are not considered. A percentage says how much of a set's measured orientation distribution is kinematically free to fail, not how likely failure is.
• One friction angle applies to every set, and the lateral limits are rules of thumb: $20^\circ$ for planar sliding (Hoek & Bray, 1981) and $30^\circ$ for flexural toppling (Goodman & Bray, 1976) are the usual values. Wider limits are more conservative.
• Sets are the user's. The analysis takes the folders as given; a mixed folder mixes the percentages. Every plane of one set is paired with every plane of another, so a pair of sets forms $n_S n_T$ intersections, not only the ones exposed together.
• The face depends on the radius. A small radius follows benches and noise; a large one smooths them into the overall slope. Points with too few neighbours get no value.
• Unsigned faces. A point cloud without camera-oriented normals cannot tell a face from its back, and its faces are folded to look upward, so overhangs are misread.
• Overhangs. The criteria are defined for slopes; the vector tests simply carry them past vertical. Under a face dipping more than $90^\circ+\phi$ the slip-limit plane is itself past vertical, so every plane dipping into the face within the lateral limit counts as critical for flexural toppling, however gently it dips. Treat overhanging faces as a hazard in their own right rather than reading the mode from the map.
• The stereonet plots poles. In wedge sliding and direct toppling the zones apply to lines of intersection, which the net does not plot; the key's percentages count the lines.

References

1. Goodman, R. E. (1980). Introduction to Rock Mechanics. Wiley, New York.
2. Goodman, R. E., & Bray, J. W. (1976). Toppling of rock slopes. Proceedings of the Specialty Conference on Rock Engineering for Foundations and Slopes, ASCE, Boulder, Colorado, 2, 201–234.
3. Hocking, G. (1976). A method for distinguishing between single and double plane sliding of tetrahedral wedges. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 13(7), 225–226.
4. Hoek, E., & Bray, J. W. (1981). Rock Slope Engineering (3rd ed.). Institution of Mining and Metallurgy, London.
5. Hudson, J. A., & Harrison, J. P. (1997). Engineering Rock Mechanics: An Introduction to the Principles. Pergamon, Oxford.
6. Markland, J. T. (1972). A useful technique for estimating the stability of rock slopes when the rigid wedge slide type of failure is expected. Imperial College Rock Mechanics Research Report 19.
7. Wyllie, D. C., & Mah, C. W. (2004). Rock Slope Engineering: Civil and Mining (4th ed.). Spon Press, London.

Related

• Slope Stability — the user guide: the dialog, the layers, the stereonet properties and troubleshooting.
• Stereonet User Guide — plotting, contouring and grouping the orientations the analysis reads.
• Orientation — the Orientation branch of the Interpretation tree, where the set folders live, and Auto-cluster Selected Orientations.
• Project Properties — the project's friction coefficient, from which the default friction angle is taken.