SlipSurface IQ Kinematic — theory

What SlipSurface IQ Kinematic computes, and where each method stops being valid. Angles are in degrees, lengths in m, forces in kN per metre of slope (planar and toppling) or kN (wedge), stresses in kPa and unit weights in kN/m³.

1. Orientations and the stereonet

A plane is given by its dip ψ (0–90°, measured down from the horizontal) and its dip direction α (0–360°, clockwise from north, the direction the plane dips towards). A line is given by its trend (azimuth) and plunge (down from the horizontal).

The unit normal of a plane (pointing down) and the unit vector of a line, in the frame x = east, y = north, z = up:

n = ( sin ψ · sin α,  sin ψ · cos α,  cos ψ )
l = ( cos(plunge) · sin(trend),  cos(plunge) · cos(trend),  −sin(plunge) )

The pole of a plane is its normal: plunge 90° − ψ, trend α + 180°. The intersection of two planes is the line along n₁ × n₂, taken to point down; its plunge and trend follow from the components above.

The stereonet is an equal-area (Schmidt) projection of the lower hemisphere. A direction with plunge p plots at the radius

r = √2 · sin( (90° − p) / 2 )         (the primitive circle has r = 1)

so that equal areas on the sphere give equal areas on the paper, which is what makes pole concentrations comparable. A plane plots as a great circle, a small circle (a cone about the vertical) as the friction limit.

2. Kinematic screening (Markland, Hocking, Goodman & Bray)

A mechanism is kinematically possible when the geometry allows the block to move out of the slope face; it says nothing about whether the strength lets it. Three conditions, with ψf the dip and αf the dip direction of the slope face, φ the friction angle and ±δ the lateral limit (default 20°):

Planar sliding. The plane must daylight in the face, dip more steeply than the friction angle and strike nearly parallel to the face:

φ  ≤  ψp  ≤  ψf          and          | αp − αf |  ≤  δ

Wedge sliding. The line of intersection of two planes must daylight and be steeper than φ, and trend nearly along the dip direction of the face. Its plunge must not exceed the apparent dip of the face in the direction of the line:

ψi_app = atan( tan ψf · cos(trend_i − αf) )
φ  ≤  plunge_i  ≤  ψi_app          and          | trend_i − αf |  ≤  δ

Flexural (block) toppling. Layers dipping steeply into the face topple when the sliding between the layers is possible. In pole space (Goodman & Bray 1976):

pole plunge (90° − ψ)  ≤  ψf − φ          and          | (α + 180°) − αf |  ≤  δ

the pole must lie inside the cone about the face direction bounded by the slope-minus-friction angle. The dashed circle drawn on the stereonet is this limit (φ for planar, 90° − φ for wedge, 90° − (ψf − φ) for toppling), and the red cloud is every orientation that satisfies the chosen criterion.

The check is conservative about cohesion: it ignores it, so a mechanism flagged as possible still has to be tested for stability.

Concentration of poles

The shading under the poles is the percentage of all poles that fall inside a counting cone around each point of the net. The cone follows Kamb (1959): it covers the fraction 9 / (n + 9) of the hemisphere, so its half-angle is acos(1 − 9 / (n + 9)) for n poles, narrowing as the sample grows, and it is held between 8° and 30°. Fewer than six discontinuities say too little to shade. The colour runs from blue (sparse) through green and yellow to red (the tightest clusters).

3. Probabilistic screening (Monte Carlo)

Each discontinuity set has a mean dip and dip direction and one standard deviation, applied to both and taken as normal. In every one of N trials the program draws a dip (kept between 0° and 90°) and a dip direction for every set, applies the criterion of the chosen mechanism, and reports

P_f = (number of trials in which at least one set, or pair of sets, is kinematically possible) / N

and, for each set (or pair, for a wedge, with its line of intersection recomputed from the drawn orientations every time), the share of trials in which that one is possible. The draws are seeded, so a run repeats exactly. The risk class is P_f < 5 % low, 5–15 % moderate, ≥ 15 % high. The result is only as good as the standard deviations: the scatter of a real site rarely justifies more than one significant figure.

4. Limit equilibrium

4.1 Planar sliding (Hoek & Bray 1981)

A block of unit thickness slides on a plane of dip ψp under a face of dip ψf and an upper slope ψs, with ψs < ψp < ψf. A tension crack of depth z, at the distance b behind the crest (or on the face), closes the block; without one the plane runs to the upper slope. With the weight W = γ·A (A the area of the block per metre), the seismic coefficients kh (horizontal, towards the outside) and kv (vertical, downwards) and the length L of the plane,

N = W (1 + kv) cos ψp − kh W sin ψp − U − V sin ψp
S = W (1 + kv) sin ψp + kh W cos ψp + V cos ψp
FS = ( c L + N tan φ ) / S

where U is the uplift on the plane and V the thrust of the water in the crack. The water is the Hoek & Bray model: the crack holds a depth zw = p·z (p the filled fraction), the pressure is triangular in the crack and falls linearly along the plane to zero at the toe,

V = ½ γw zw²          U = ½ γw zw L

and without a crack the pressure on the plane rises from zero at both ends to a maximum at the middle. A support force T at the angle θ below the horizontal has the components T cos(θ + ψp) along and T sin(θ + ψp) normal to the plane; an active (tensioned) support reduces S and adds to N, a passive one adds to the resistance. The force needed for a target factor of safety FS_t is

T = ( FS_t · S − R₀ ) / ( FS_t · cos(θ + ψp) + sin(θ + ψp) tan φ )          (active)

where R₀ is the resistance and S the driving force without the support, with the most economical angle tan(ψp + θ) = tan φ / FS_t. If the effective normal force N is not positive the block lifts off the plane, and the program says so. With no support, no water and no earthquake the factor of safety reduces to the closed form of the book, (c L + W cos ψp tan φ) / (W sin ψp).

4.2 Wedge sliding (Hoek & Bray, vector method)

Two discontinuities J1 and J2, the slope face and the upper slope make a tetrahedral wedge. The program builds it from the four planes (volume, areas and the line of intersection by vector algebra) and resolves the forces as vectors in 3-D:

• the resultant of the active forces is R = W (down) + the water thrusts U₁, U₂ (normal to each plane, pushing the wedge off it) + the seismic force + an active support force. The water is dry, filled (the Hoek & Bray model, the average pressure on a plane γw·Hw/6), a percentage of it, or a pressure given for each plane;

• with the unit normals n₁, n₂ (pointing into the wedge) and the unit vector s down the line of intersection, the contact forces and the driving force follow from R + N₁ n₁ + N₂ n₂ = S s, a 3×3 linear system;

• if N₁ > 0 and N₂ > 0 the wedge slides on both planes, along the line of intersection:

  FS = ( N₁ tan φ₁ + c₁ A₁ + N₂ tan φ₂ + c₂ A₂ ) / S
  

• if contact is lost on one plane the wedge slides on the other alone, along the line of steepest descent of that plane in the resultant (and only if the movement is away from the plane that has lifted); if it lifts off both, it falls (FS = 0) when R points downwards and is stable when it points into the rock.

For a wedge with no water, support or earthquake this is the Hoek & Bray closed-form solution; the program matches it exactly (dry 1.696, flooded 1.065 for the worked example of the book).

4.3 Block toppling (Goodman & Bray 1976; Wyllie & Mah 2004)

The rock is idealised as a row of rectangular blocks of width Δx and heights y_n formed by discontinuities dipping into the slope at ψd, standing on a stepped base of general angle ψb; the base of each block is inclined at ψp = 90° − ψd. From the top block down to the toe the program carries the thrust P_n of each block on to the next. For block n there are two candidate thrusts that the block above must give for it to stay in place:

P_topple = [ P_(n+1) (M − Δx tan φ) + ½ W (y sin ψp − Δx cos ψp) + ½ kh W (y cos ψp + Δx sin ψp)
             + V_u h_u / 3 − V_d h_d / 3 + U x_U ] / L
P_slide  = P_(n+1) + (V_u − V_d) − [ (W cos ψp − kh W sin ψp − U) tan φ − (W sin ψp + kh W cos ψp) ] / (1 − tan² φ)

with M and L the lever arms of the thrust from the pivot (the lower front corner of the block) and the block's own geometry, V_u and V_d the water thrusts on the rear and front joints and U the uplift on the base. The block topples if P_topple is the larger and the block is tall enough to topple (y / Δx > cot ψp), otherwise it slides; if both are zero or negative it is stable, and the thrust passes on. The water follows RocTopple: a triangular pressure on each joint, V = ½ γw h², with h the filled fraction of the smaller of the two adjacent block heights, and a trapezoid under each block. The thrust P₀ left at the toe block is zero when the slope is in equilibrium.

The factor of safety is the one by which the friction has to be reduced for equilibrium to be just reached: the program finds by bisection the friction angle φ_req for which P₀ = 0, and gives

FS = tan φ / tan φ_req

(0 when equilibrium is not reached even at φ = 45°, ∞ when it holds without friction). The sliding equation has a factor 1 − tan² φ, so results for φ ≥ 45° are not reliable and are flagged. The anchor at the toe needed for the factor of safety is found by the same equations with the friction reduced to tan φ / FS_t.

5. Support: bolts and anchors

A support force T of a given trend and plunge, tensioned or passive, enters the equilibrium of section 4 as a force with components normal and along the surface (for the wedge the direction giving the largest gain in FS per unit of force can be searched for, and the minimum force for a target FS found by a scan followed by a bisection, since FS(T) need not be monotonic when the failure mode changes). The force the slope needs is turned into a bolt pattern:

• the capacity of a bolt is its working tensile capacity (for example about 0.6 of the yield of the bar); the number of bolts is T divided by it, per metre of slope for planar failure and in total for a wedge, and the square spacing s is chosen within the limits you give and rounded to a step;

• the bond length in the stable rock, for the working capacity T_b, the hole diameter d and the grout–rock bond strength τ:

  L_bond = T_b · FS_bond / ( π d τ )          (not less than the minimum bond length)
  

• the total length is the free length (from the face to the sliding surface along the bolt direction) plus the bond length plus the allowance at the head.

A chosen design is then checked the other way: the capacity of the pattern against the force the slope needs, and the factor of safety it gives.

6. What is not covered

• Rotational and circular failure: use SlipSurface IQ LE for those.
• The shape of the wedge is the ideal tetrahedron of the vector method; a wedge cut by more than the two planes, or by a plane that does not cross the face, is outside the method.
• Water is the Hoek & Bray model of a filled tension crack with a linear fall; real seepage needs a groundwater analysis.
• Dynamic loading is the pseudo-static coefficient alone.
• A non-planar base (stepped, rough) of a toppling block, and the interaction of the blocks that have already sheared.

References

• Markland, J.T. (1972). A useful technique for estimating the stability of rock slopes when the rigid wedge sliding type of failure is expected. Imperial College Rock Mechanics Research Report 19.
• Hocking, G. (1976). A method for distinguishing between single and double plane sliding of tetrahedral wedges. Int. J. Rock Mech. Min. Sci. 13, 225–226.
• Goodman, R.E. & Bray, J.W. (1976). Toppling of rock slopes. Proc. Specialty Conf. on Rock Engineering for Foundations and Slopes, ASCE, Boulder, 2, 201–234.
• Hoek, E. & Bray, J.W. (1981). Rock Slope Engineering, 3rd ed. Institution of Mining and Metallurgy.
• Kamb, W.B. (1959). Ice petrofabric observations from Blue Glacier, Washington, in relation to theory and experiment. J. Geophys. Res. 64, 1891–1909.
• Wyllie, D.C. & Mah, C.W. (2004). Rock Slope Engineering: Civil and Mining, 4th ed. Spon Press.