Intact rock-mass strength (IRMS)

The failure criteria for rock is usually defined by a simple linear Mohr-Coulomb equation (Fig. l):

(1)

where τ and σ are shear and normal stresses, respectively, c is the value for internal cohesion, and ϕ is the angle of friction along the eventual failure plane (Reference HoekHoek 1970). A distinction is made betweenpeak shear strength, beyond which the rock deforms, and the residual shear strength of the deformed mass. These may be similar for soft rock, whereas residual strength may be as little as half the peak strength for hard igneous rock (Reference Hoek and BrayHoek and Bray 1974). Typical intact rock shear strengths and other properties are shown in Table I.

Fig. 1.

Linear Mohr-Coulomb relationships between shear stress and normal stress for intact rock mass (IRM); the progressive reduction in required shear stress is shown for dry discontinuous rock mass (DRM_d) where C = 0, and wetted discontinuous rock mass (DRM_w), where a given friction angle ϕ is reduced to an effective friction angle ϕ _r.

Table 1

Some typical rock-mass properties (after Reference Hoek and BrayHoek and Bray (1974), Reference KulhawyKulhawy (1975), among others)

Discontinuous rock-mass strength (DRMS)

Intact strength offers little scope for low-magnitude shear stress, of the order of 0.lMN m^-2 (Reference WeertmanWeertman 1979), at the glacier sole, and a substantial reduction in strength is required to permit large-scale erosion. Discontinuities Footnote ^* render rock mechanically and hydraulically defective to an extent where this is possible, by altering stress relationships in the rock mass and providing release planes requiring much lower disturbing forces (Reference Hoek and BrayHoek and Bray 1974). IRMS values become less meaningful in discontinuous rock and are replaced by DRMS.

The principal modification to the Mohr-Coulomb criterion lies initially in lower values for the parameters of cohesion c, and friction angle ϕ, and a comparison of typical values is made in Table I. The cohesion is now solely that of the discontinuity plane and is effectively zero, except in the presence of fill. How ever, the failure criteria are for dry rock and a further significant modification afforded by discontinuous rock mass (DRM) is high secondary permeability (Reference Witherspoon and GaleWitherspoon and Gale 1977). A swell as contributing to shear stress, water reduces normal stress σ to effective normal stress σ_η̃ by a value u (Reference Hoek and BrayHoek and Bray 1974), and for practical purposes the modified Mohr-Coulomb equation may be written as

(2)

or, in a simplified form, as

(3)

where ϕ_r is the effective friction angle. Roughness increases the friction angleϕ by an amountϕ _f, and the plane possesses, more correctly, bi-linear shear strength, which assumes the lower (residual) value upon shearing of asperities (Reference Witherspoon and GaleWither spoon and Gale 1977).

In practice, all rock mass possesses internal fractures which normally demonstrate a strongly preferred geometry, determined by geological(principally tectonic)history (Reference BartonAttewell and Farmer 1976). Marked planar anisotropy renders DRM liable to simple failure in a coaxial stress field and to complex failure, involving multiple planes, elsewhere. Compound failure along discontinuities and cracks propagated across rock bridges was reported in laboratory tests (Reference BrownBrown 1970). Brown also suggested that discontinuous rock-mass failure (DRMF) only shears through intact rock and ignores discontinuities under very high confining pressures (1400 MN m^-2), thus tending to confirm that glacier ice can only “ quarry” DRM and that failure of supposedly intact isotropic rock (Reference Broster, Oreimanis and WhiteBroster and others 1979) may be in error, overlooking pre-existing anistropy (Reference HoekHoek 1964).

Conditions of DRM stability and failure

Practical application of the modified failure criterion permits assessment of stability under gravitational and dynamic loading. Design in engineered slopes is particularly concerned discontinuity geometry (Reference HoekHoek 1973), (ii)individual block size and shape, and (iii)cohesion of any infilling material. Calculation of stability or predicted failure mode is then relatively straightforward. From the Mohr-Coulomb criterion, resistance to gravitational sliding is defined (after Reference Hoek and BrayHoek and Bray 1974) as

(4)

where A is the block/slope contact area, W is the block weight and ψ_d is the discontinuity angle. If shearing and resisting forces are exactly balanced, the block is said to be in limiting equilibrium for dry slopes when

(5)

and for wet slopes when

(6)

where V is the additional loading due to the weight of water behind the block, and u is the reduction of normal stress due to uplifting pressure of water under the block.

Failure modes

Probable failure modes on unstable slope scan be determined from the relationship between the excavated slope and discontinuity geometry. Three of four principal failure mechanisms recognized in slope engineering are considered here (Fig. 2). (Circular failure is disregarded for DRM with well-defined discontinuities.)

Fig. 2.

Failure modes and their related discontinuity stereonet (equatorial equal-area lower hemisphere projection).

Plane failure (of single or multiple slabs)occurs when

(7)

and wedge failure occurs when

(8)

where ψ_s and ψ_d are the slope and discontinuity angles and ψ_i is the angle of the slope of intersection of two planes defining a wedge. (The excavated slope must permit the potential failure planes to “daylight”.) Failure, particularly in the case of planar slides, requires the presence of other release surfaces provided by other discontinuities.

Toppling failure occurs when the primary plane D₁ dips into the slope and is apparently stable, but also when a secondary stable plane D₂ is so spaced as to define a block whose centre of gravity overhangs a pivot point (Reference Freitas and Wattersde Freitas and Watters 1973).

It is emphasized that, although actual slope stability may be complex, practical application of theoretical analysis is generally successful (Reference HoekHoek 1973). Also, whilst its primary application is for gravitational loading, an extension of principles to dynamic glacier loading may be appropriate and theoretical modifications to stress relationships induced by glacier ice are now proposed.

(9)

Rock mass confined by ice

This is the more difficult to analyse because of the relative inaccessibility of the ice-rock contact, and also because DRM is inherently stable in its primary valley-floor position. Equations (7) and (8) do not apply, and discontinuity-stress relationships show

Although gravitational load does not meet limiting equilibrium requirements, two factors increase shear stress. First, ice flow generates a low-magnitude shear stress of about 0.1 MN m^-2 (Reference WeertmanWeertman 1979) enhanced for an ice-rock mix. Second, melt water at the ice-rock mass contact penetrates discontinuities, contributing a force V and reducing normal stress by a value u (Equation 6). However, it is recognized that a complex feedback relationship exists here, where by secondary penne ability afforded by DRM may alter critically the basal pressure-melting balance. Two further qualifications are made to this domain. Once begun, block removal permitting low-angled planes to “ daylight”, and also removing the restraining presence of the block, may reduce sliding resistance sufficiently under dynamic load, where ψ_d, j< ϕ. More significantly, the primary sub glacial loading mode assumes alow-angled bedrock floor; in practice this is probably unnecessarily conservative, since steep slopes occur frequently beneath ice for short distances, and DRMF may approach more closely that of the unconfined domain discussed below.

Rock mass unconfined by ice

Glacier erosion in this domain is in two parts: (i) an initial amount of vertical or horizontal excavation in the confined domain, cutting the slope beyond the limiting equilibrium, followed by (ii) unconfined DRMF of a mode determined by slope-discontinuity geometries. Equations (7) or (8) must apply. The amount of confined excavation need not be large, as will be shown later. Progressive failure in one mode may destabilize other blocks unmoved by initial disturbance, and very large volumes ofrock mass may be involved in such compound translation failures; this is also shown infield examples presented later.

Water derived from ice-melt and also precipitation beyond the glacier margin assumes a considerable significance in destabilizing unconfined rock. Reference BartonBarton (1973) states that water may reduce the friction angleϕ by as much as 30°, and therefore the presence of water throughout the back-wall zone of cirque glaciers must be considered to make a major contribution to DRMF without recourse to freeze-thaw mechanisms.

Again, further qualifications may be made with respect to the destabilizing mechanism. (i) Unconfined failure can only apply to glaciers confined within bedrock channels, limiting the mechanism almost entirely to valley and cirque glaciers, and thus accounting in part for their significantly greater erosive power. (ii) Once initiated, progressive failure is controlled primarily by the DRM geometry; unlike engineering applications, where the required slope and discontinuity geometries mayor may not coincide favourably, it is suggested that glacier erosion will always show close conformation. Principal applied stress will seek out the most closely related potential failure planes, and hence structure controls glacier erosion.

Geo mechanical principles, modified for the glacier environment, are now investigated for application to glacier-eroded rock mass in Snowdonia, North Wales, after reviewing the general structural geology of the region.

Confined failure

Slope-failure criteria do not apply so readily here as stated earlier. At the three chosen sites, DRMF was compound, being induced dynamically in otherwise stable rock mass by basal shear, and then, once block separation began, local small-scale slab, wedge, and toppling failures occurred along destabilized discontinuities.

(i) Cwm Stwlan

Excavation for the upper dam foundations of the Ffestiniog pumped-storage hydro-electric scheme, constructed on the bedrock threshold of a glacial cirque, revealed considerably disturbed, hard, un weathered rhyolite dislocated along pre existing discontinuities to a depth of 13 m across a front 150 m wide, which necessitated design modifications. Reference AndersonAnderson (1969:193) considered the dislocation to have been caused by glacier drag across the threshold: “ ... facilitated by the presence of five faults in the part most affected and by joints almost at right angles to the rock-lip ... The affected zone does not tail off but ends abruptly on both sides. The limits may be partly related to the faults. but they may also mark the width within which the glacier was thick enough to exert drag ”.

(ii) Ogwen

Bedrock floor in a major glacier-breached watershed is shown in Figure 5. Ice flowing from left to right first abraded the rock, followed by “ quarrying” (displaced blocks show striation son one surface only) which removed some blocks altogether and displaced others; there after, gravitationally loaded secondary failure further dislocated the rock mass, at least in part sub-glacially since many blocks are missing. Failure surfaces were entirely controlled by the discontinuity geometry, and primary and modified confined failure is indicated by the displacement of debris down-glacier and down-slope.

Fig. 5

Ogwen valley floor. Sub-glacier confined DRMF, with secondary failure evident in excavated sections. Slab failure occurred along two planar sets (a, b) and toppling failure away from (c).

(iii)Nant Peris

Figure 6 shows an example of toppling failure towards the valley floor generated by the removal of adjacent rock mass under glacier confinement close to the valley floor. Cliff elements such as these are common, with at least the greater part of the failed rock mass absent from the toe; by comparison, the few remaining instable blocks which have recently toppled from the now unconfined face are all present at the toe, and exhibit less-weathered contact planes.

Fig. 6

Toppling failure in Nant Peris; destabilization caused the parting of blocks along arrowed discontinuities.

Unconfined failure

CWm Graianog

This cirque basin affords one of the finestsite concentrations of all modes of rock-massfailure in Snowdonia. It is excavated inFfestiniog grit, and failure modes are discussedwith reference to a standard structural presentation (Reference Hoek and BrayHoek and Bray 1974) shown in Figure 7.

Fig. 7.

Bedrock map and discontinuity stereonet for Cwm Graianog.

The north sidewall consists almost entirely of a spectacular series of D₁ surfaces (Fig.8) and the slope angle is effectively the same as the discontinuity dip of 38-40° to the south-east. Slab failure down D₁ was released along D₂, which frequently possesses an injected quartz fill, and D₃ and the vertical extent of individual slabs is limited by D₄ with the same strike, but opposite dip, as D₁. slab towards the western end of the basin may have failed to the full height of the rock wall(200 m) and across a width averaging 60 m. The remaining rock wall is clean, being devoid of residual blocks, overhanging elements below which rock mass have been released, and stable laterally confining units. With a principal D₁ spacing of 3 01, this would have yielded a single failure of 36 000 m³ of the side wall, all of which was removed by the glacier. D₁ planes in the wall at this point are marked uniquely by large-scale bedding-plane ripples (Fig. 9), and the only debris blocks so marked are found several hundred metres away in, and resting upon, cirque moraines in positions where they could have been deposited only by glacier ice.

Fig. 8.

Cwm Graianog (Ffestiniog grit). North side wall (right) with D1 planar surfaces; head wall (1eft) with D¹- D₂ wedging and D2 planar surfaces.

Fig. 9.

Cwm Graianog. Single D1 major planar slide release surface, (showing ripple marks).

The irregular strength and spacing of D₅ renders it less apparent as a control, but D₂ and D₄ become more important as the back wall is approached. The “curved” transition is effected by D₁- D₂ wedge failure (ψ_i=39°, orientated at 130°) producing a series of buttresses, and the extent to which the geometry predetermined failure on excavated slopes is completed by the superimposition of topp1ing failure released from D₁ and low-angled D₄ (itself below tile assumed friction angle) on more complex slab and wedge failure in the south wall. A rapid assessment of current stability (after Reference HoekHoek 1970) suggests that D₁ has a factor of safety F≥ 1 for typical assumed mechanical values, which has two important implications. (i) Excavation at the toe would rapidly cause D₁ to “ daylight” and generate further slab failure; this is considered to be typical of the effect of glacier erosion. (ii) Other modifications to Mohr-Coulomb parameters would result in failure; locally small contemporary slides are evident, considered to be the smoothing effect of weathering on rough nesses along the D₁ planes.

There is a dramatic decline in side-wall height at the Ll an virn slates-granophyre contact; it is suggested that the lower ORMS of slates permitted greater excavation of the side wall, and it is further noted here and elsewhere that cleavage planes do not appear to have provided significant failure surfaces during glacier erosion.