Diederichs Kaiser 1999b

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Tensile strength and abutment relaxation as failure control

mechanisms in underground excavations

M.S. Diederichs *, P.K. Kaiser

Geomechanics Research Centre, F217 Laurentian University, Sudbury, Ont., Canada P3E 2C6

Accepted 8 November 1998

Abstract

Classical assessment of instability potential in underground excavations are normally based on yield and rupture criteria for

stress driven failure and on limit equilibrium analysis of structurally controlled failure. While it is true that ultimate failure and

falls of ground can be an eventual consequence of stress fracturing and unfavourable structure within the rock mass, the timing

of such failure is often controlled by the presence of residual tensile capacity, in the form of rock bridges separating joint

segments and fractures and by the mechanisms of clamping and relaxation. Using crack and rock-bridge analogues in

conjunction with an updated voussoir beam model, this paper explores the inuence of residual tensile strength and boundary

parallel relaxation on the failure process. The impact on support design is also examined. In underground hard rock mines with

complex geometries and interacting openings, relaxation is identied as a key controlling factor in groundfall occurrence.

Empirical stability assessment techniques for underground tunnels and for mining stopes are updated to account for relaxation.

# 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction

Stability assessment of underground excavations isclassically divided into two procedural domains. These

two domains are based on a distinction between struc-

turally controlled and stress driven modes of instabil-

ity. It is the premise of this paper, however, that in

non-squeezing and non-bursting ground, structure and

stress serve merely as ground conditioning mechan-

isms. Gravity loading is ultimately responsible for

large scale groundfalls or for signicant loading of

support. When the rock mass jointing is non-persist-

ent, the rock bridges contribute to the stability of an

excavation through the rock mass residual tensile

strength or load bearing capacity. The failure to con-sider the rock mass self-supporting capacity and the

eects of abutment connement or abutment relax-

ation can lead to erroneous predictions of failure or of

support load.

Gravity induced groundfalls are common occur-

rences in underground excavations of all depths.

Numerous techniques can be applied to assess the po-tential for such groundfalls provided that an appropri-

ate failure mode is assumed. Typical failure modes

which can be analyzed include wedge fallout, slab or

plug failure, gravity driven caving and beam failure.

Models taking these failure modes into account are

generally usually utilized for associated with stability

assessments in low stress or near surface excavations.

The underlying assumption of most of these models is

that through-going joints or discontinuities are fully

persistent and that stability is controlled by structural

geometry and by friction (with or without dilation).

The inherent tensile or cohesive strength of moderately

jointed rock masses is often assumed to be negligible

in these models.

Alternatively, for excavations at depth or for shal-

lower excavations in weaker rock masses, linear elastic

stress analysis can be used to determine the location

and extent of problematic stress concentrations around

the openings. Failure criteria such as MohrCoulomb

and HoekBrown [1, 2] based on combined cohesive

and frictional strength components can be applied suc-

International Journal of Rock Mechanics and Mining Sciences 36 (1999) 6996

0148-9062/99/$ - see front matter# 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 4 8 - 9 0 6 2 ( 9 8 ) 0 0 1 7 9 - X

PERGAMON

* Corresponding author. 105 William St. W., Waterloo, Ont.,

Canada N2L IJ8. Tel.: +1-519-578-5327; e-mail: mdiederi@nickel.

laurentian.ca.


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cessfully if the rock mass yields with signicant plastic

deformation. These criteria, however, meet with only

limited success [3, 4] for failure prediction around exca-

vations in hard rock environments due to their insensi-

tivity to low connement behaviour. In brittle ground,

it has been shown [59] that under low connement

(such as that which exists adjacent to an opening), tan-

gential compressive stress above a dened threshold in-

itiates and propagates boundary parallel cracks orfractures.

Such rock mass damage can be responsible for

observed seismicity [9] and stress redistribution [10]

and was found to correlate well with a constant critical

deviator criteria for damage initiation, both in intact

and in moderately jointed rock [9, 11]. This initial

cracking or fracturing is generally parallel to the exca-

vation surface and therefore parallel to the major com-

pressive stress. This crack damage is strain dependant

and may be exacerbated by preferential deection and

dilation into an opening and by existing planes of

weakness such as foliation or meta-bedding resulting

eventually in moderately persistent planar laminationsin otherwise massive or moderately jointed rock

(Fig. 1).

Nevertheless, a prediction of rock mass stress in

excess of yield limits, based on elastic models, does not

inevitably correspond to actual catastrophic failure of

the underground opening. This is fortunate given the

depths encountered in modern mining, where much of

the ground above mining openings has reached a

damage or yield stress threshold at some point in the

mining sequence and yet, with the exception of highly

stressed ground experiencing rockbursting, shearing or

stress buckling, generally remains in place in the short

term. This is due to the rock mass' residual tensileload bearing capacity normal to the excavation bound-

ary and due to arching to the abutments.

Paradoxically, ultimate failure of this damaged ground

is often induced by changes in mine geometry which

reduce [1216] rather than increase the stresses across

the back or walls as might have been the case in Fig. 1.

It is the premise of this paper that the key to this

discrepancy between modeled or predicted failure and

the actual observed occurrence of failure in under-

ground excavations in hard rock is due to the domi-

nance of the rock mass's tensile load bearing capacity.

However, if combined with abutment relaxation, the

inuence of gravity can exceed the tensile load bearingcapacity of naturally jointed or stress fractured rock

masses leading to the type of failure illustrated in

Fig. 1. This paper will not attempt to explain the gen-

esis of natural fractures and associated rock bridges,

Fig. 1. Gravity collapse of moderately jointed to massive ground after stress induced fracturing.

M.S. Diederichs, P.K. Kaiser / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 699670


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nor will it examine the mechanics of formation for

boundary parallel stress cracks. The foregoing discus-

sion assumes that such damage is omni-present in

underground excavations and explains the possible

impact on ultimate stability.

In this paper we will discuss the nature of residual

tensile capacity in jointed or fractured rock masses

comparing its capacity to that of economical engin-

eered support systems. The eect of abutment relax-ation on rock wedge and blocky slab stability is

demonstrated. Dierent mining scenarios are described

which may lead to destabilizing relaxation. The vous-

soir arch analogue is used to further demonstrate the

importance of boundary normal tensile capacity and

abutment relaxation. The voussoir analogue is then

calibrated to correspond with empirical stability guide-

lines for rock mass stability. The eect of relaxation is

further demonstrated in this context.

2. Boundary-normal tensile strength of a damaged rock

mass

While tensile strength can often be justiably neg-

lected in cases of weak and highly fractured rock, the

residual strength of rock bridges in moderately jointed

hard rock masses must be considered. Extension joints

are rarely fully persistent at depth [1719]. Hence, it

would be realistic to expect the initial presence of

intact rock bridges. Foliation planes and bedding

planes are weaker then the surrounding rock but still

possess some limited tensile strength in many cases

until full parting occurs due to excessive deformations.

This can be seen in the case of progressive hangingwall

delamination reported by [20], for example, and par-tially explains why ground can remain unsupported in

spite of contrary limit equilibrium calculations based

on the stability of a fully bounded wedge, block or

slab. In spite of this initial and static stability, how-

ever, support may still be required to protect against

groundfalls induced by dynamic disturbance, time and

stress dependent tensile strength corrosion [21], and

mining induced deformations. Support in such con-

ditions can also prevent or inhibit the propagation of

fractures and the degradation of rock mass tensile ca-

pacity.

In jointed or in massive rock masses at depth, under

the low normal connement conditions encountered

adjacent to an excavation, surface parallel cracks can

develop as a result of compressive stress. This micro-

cracking parallel to the maximum principal stress has

limited eect on the ultimate compressive strength

since it is an inherently stable fracture process [22].

The same cracks, however, subjected to tensile loading

normal to the plane of the crack (as occurs above a

horizontal excavation roof), propagate easily when a

critical tensile strain is exceeded and have a profound

impact on the strength, particularly if the compressive

stress parallel to the boundary is maintained. Under

such compressive stress, boundary interaction also

plays a role in the enhanced propagation of near-exca-

vation cracks [23]. It can be assumed however, that in

many cases the initial jointing or parallel fracturing is

not complete and that intermittent rock bridges remain

to be exploited by gravity.The ratio of tensile (sT) to uniaxial strength (UCS)

of intact rock is signicantly lower than for other ma-

terials such as steel. Typical tensile values range from

one tenth to one twentieth of the UCS for hard

rocks [24]. These relatively low values, combined with

the conservative assumption of fully persistent jointing

lead to the common neglect of the rock mass tensile

strength as a practically signicant factor in excavation

stability. Grith theory [25] has become a classic

means of explaining the relatively low tensile strength

of heterogeneous and imperfect solids such as rock,

describing how small imperfections such as cracks

serve to concentrate tensile stresses locally resulting ina reduction in macroscopic sample strength. The as-

sociated stress intensity relationships for internal and

external cracks in plates [2628] can be extended for

isolated cracks and rock bridges in three dimensions

(Fig. 2).

For a given mode I stress intensity factor at crack

extension KIC, the tensile strength for a partially

cracked solid can be computed for circular non-inter-

acting cracks (Fig. 2a and d) of radius c, with crack

normals oriented at angle g to the direction of tensile

loading [28]:

sT KIC

p2pc

p 1cos2g

1

Using Kemeny and Cook's solution [28] for the so-

called external crack, the tensile strength of a cylinder

of rock with a total cross sectional area, A, containing

and a circular rock bridge of radius a (surrounded by

a planar, annular crack) is given by

sT KIC

2apa

pA

1

cos2g2

If N/V is the number of regularly distributed cracks or

rock bridges in a unit cubic volume (V= 1 m3), then

the total coplanar cross sectional area (cracked anduncracked) associated with the crack or rock bridge, A

is

A 1N2a3

3

If (Ac*)A is the area of the crack and (Aa

*)A is the area

of the rock bridge (where Ac* and Aa

* are the ratios of

cracked and intact area, respectively, to the total cross

M.S. Diederichs, P.K. Kaiser / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 6996 71


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sectional area):

cI

A*c

pN2a3

s4

aIA*

apN2a3

s 5where Aa

*= 1 Ac*.It is possible now to express the tensile strength with

respect to the percentage of cracked cross sectional

area for cracks perpendicular to loading (cos2g=1):

sT p2

KIC

N1a3pA*c

qvuut for A*c ( 1 6

or inversely, with respect to percentage of intact rock

bridges:

sT 2KICA*a 3a2N1a3

pp

sfor A*a ( 1 7

These equations for isolated cracks and isolated rock

bridges, analyzed as `external cracks' by Kemeny and

Cook [28], serve as limiting cases for progressive crack

damage as illustrated in Fig. 3 using a unit stress

intensity factor. The actual tensile strength varies line-

arly with KIC as per Eqs. (6) and (7).

Fig. 3 clearly illustrates the signicant reduction in

theoretical tensile strength with small amounts of dis-

tributed micro-cracking (Ac= 0 to 0.1). It is acceptable

to presume that even intact rock samples normally

contain a signicant amount of internal damage and

therefore exhibit tensile strengths which are lower than

those for a perfect solid. Fig. 3 also illustrates that thetheoretical tensile strength is apparently insensitive to

the areal extent of cracking for the intermediate range,

Ac= 0.1 to 0.9. For values of Ac>0.9 or Aa


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superposition of a circular crack or bridge within a

square associated area.In order to fully examine the role of tensile strength

as a support mechanism it is also necessary to examine

the impact of crack damage on rock stiness. This is

achieved using the Grith locus for tension [29]. By

assigning spherical averages to the orientation terms in

an expression for the strain energy due to an individ-

ual crack and incorporating the result into a variation

of Betti's reciprocal theorem, Kemeny and Cook [28]

calculate the ratio of Young's modulus for a solid with

randomly distributed non-interacting cracks:

Ec

E0 1

1 16a45w1 n210 3ua2 u 9

and for rock bridges in a randomly cracked solid:

Ec

E0 1

1 p2a15w1 n5 4namm 1 10

Eq. (9) can be simplied for the case of mono-direc-

tional cracks perpendicular to the direction of tensile

loading:

Ec

E0 1

1 16a3w1 u2 11

while Eq. (10) can be simplied for regularly spaced

parallel rock bridges within an axisymmetric (cylindri-cal) volume:

Ec

E0 1

1 p2w H1 n2amm 1 12

where w= N(c)3/V is the crack density for isolated

cracks, w H = N(c H)3/V is the crack density associatedwith isolated rock bridges, N is the number of cracks

or rock bridges, V= 1 is the unit volume in the study

and a is the average radius of rock bridges surrounded

by a cracked annulus of width c H such that:

c H 1

A*a

qpN2a3

p 13

m aac H

A*a

q1

A*a

q 14The value of c H represents the average edge separationfor an array of equivalent cylindrical (and not cubic)

rock bridge volumes. The denition is dierent than

that obtained for c = f(a) from Eqs. (4) and (5) anddiers from the three dimensional formulation of

Kemeny and Cook [28]. These modied terms are

necessary to obtain a three-dimensional solution for

rock bridges which is behaviourally compatible with

the original two-dimensional formulation [28].

Eqs. (11) and (12) combined with Eqs. (6) and (7),

dene the Grith locus and the stress/strain relation-

ships for isolated cracks (Fig. 4) and isolated rock

bridges (Fig. 5). In Fig. 4, the cracks nucleate at points

arranged in a regular cubic array with the initial separ-

ation dened by Eq. (3). These cracks then propagate

in a plane perpendicular to loading. In Fig. 5 the

cracks have coalesced leaving a regular array of intact

and progressively shrinking rock bridges with the same

separation as the original cracks. The total number of

cracks, N, remains constant. The genesis of natural

and induced joints or fractures is undoubtedly sensitive

to initial boundary condition and response and may

not necessarily progress to full rupture in a single

stage, terminating at some point in this sequence. At

issue here is the renewed response of non-persistent

Fig. 3. Tensile strength versus normalized cracked cross-sectional area.



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cracks and rock bridges upon exposure to excavation

induced gravity loading.

In Fig. 4, it is interesting to note that while a drastic

loss in crack propagation stress does occur in the pre-

sence of very small cracks, practically detectable re-

duction of the reloading modulus does not occur until

the cracks occupy more than 10% of the cracked area

and does not become signicant until the cracked area

exceeds 20%. This latter stage is the point at whichthe crack diameter exceeds the minimum separation

between the edges of neighbouring cracks (Inset in

Fig. 4). In other words, this is the point at which the

ratio of inter-crack edge separation to crack length

falls below 1:1. Similarly, a cracked area of 10% corre-

sponds to an edge separation of approximately 2 crack

lengths. This is consistent with the theoretical work of

Pollard and Segall [30] who show that the strain per-

turbation due to a crack decays considerably at an

edge separation of one crack length and becomes insig-

nicant for an edge separation of 2 crack lengths.

At a level of crack density beyond the limit of 10

to 20% cracked area, it is also dicult to reconcile

the Grith locus for isolated cracks with the locus

for isolated rock bridges as end members of a transi-

tional cracking process. It is even more dicult then,

to attach any physical meaning to the residual tail of

the isolated crack locus (Fig. 4) which can be calcu-

lated for high crack densities corresponding as shown

to cracked areas in excess of 100%, a nonsensical

result.

Fig. 5 illustrates the theoretical stressstrain re-

lationships and crack propagation strengths for a

highly cracked solid. In these two plots, the cracks

have coalesced and are converging on remnant (circu-

lar) rock bridges, a more realistic end process for in

situ fracturing. There is a stage between approximately

15% intact area and 1% intact area (corresponding to

85 and 99% cracked area, respectively) through which

the rupture occurs at a near-constant level of overallextension strain. For a single crack (N=1), a unit

fracture toughness and with E=10 GPa and u=0.2

for the intact rock, this corresponds to a range of re-

sidual tensile strength between 0.3 and 0.03 MPa at a

strain limit of approximately 0.004%. For N=1000

(indicating a highly distributed cracking process), this

threshold strain is approximately 0.012%, spanning re-

sidual strengths of 1 to 0.1 MPa for the same range of

cracked areas. Okubi and Fukui [31] performed direct

tensile tests on numerous rock types using an extre-

mely high resolution servo-control and reported re-

sidual tensile strengths within these ranges of

magnitude. Of particular interest for support design,

given an assumption for crack or rock bridge distri-

bution (N), is the constant nominal strain limit which

can be expected of residual rock bridges. The threshold

is small indicating that if failure is to occur due to

gravity loading, only relatively sti support systems

such as resin grouted rebar on a tight pattern can pre-

vent the loss of rock bridges and residual strength.

The minimum residual strengths 30 to 100 kPa, pre-

Fig. 4. Example of Grith locus and vertical stiness relationships for a rock sample containing isolated horizontal cracks.



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dicted in Fig. 5 are also of interest, however, since

they are similar in magnitude to the holding capacities

of conventional support systems used in mining.

Pattern support in the form of regularly spaced

mechanical rockbolts, grouted rebar, split sets or

cablebolts serve primarily to resist gravity and to

`hold' the rock mass adjacent to an excavation. These

support congurations also contribute to the inherent

integrity of the rock mass, resisting dilation and

increasing frictional resistance of rough fractures and

discontinuities. In stress-damaged rock, sti or actively

pre-loaded reinforcement can also serve the role of

preventing the failure of rock bridges and thus the

complete loss of rock mass tensile strength. In their

primary role, however, these support systems can pro-

vide distributed tensile load capacities ranging from 20

to 300 kPa, respectively, for sparse and for economi-

cally limiting support patterns as shown in Table 1.

The demand equivalence in terms of vertical height of

supported rock (assuming no other stabilizing mechan-

ism) is given for comparison. Table 1 also illustrates

the very small relative cross sectional areas of intact

rock (rock bridges) required to achieve the same tensile

load capacity using the example values from Figs. 3

and 5.

The main dierence between the supporting mechan-

isms of articial tendons and internal rock bridges is

the magnitude of the rupture strains in the direction of

tensile loading. While steel tendons such as rockbolts

and cables typically yield at strains of approximately 1

to 2% for average grades of steel [33], reasonably

intact rock specimens (Fig. 4) begin to rupture at ten-

sile strains of less than 0.1% and small rock bridges

(Ac*< 10%) such as those modeled in Fig. 5 rupture

at strain levels below 0.01%. The convex shape of the

locus in Fig. 5 also indicates that any tensile rupture

Fig. 5. Example of Grith locus and vertical stiness relationships for rock sample containing isolated horizontal rock bridges.

Table 1

Support patterns [32] and equivalent rock bridge area (N=125)

Support type Support

pattern(mm)

Equivalent

pressure(kPa) [30]

Maximum

supportedthickness (m)

Capacity

equivalent rockbridge area

(% cross section)

Rockbolts 22 20 0.7 0.1Rebar 1.31.3 60 2.0 0.4Single strand cablebolts 22 65 2.2 0.4Double strand cablebolts 22 130 4.3 1.2Double strand cableholts 1.31.3 300 10 4.0

1% rockbridge area is equivalent to a 1010 cm rockbridge per 1 m2 total area.



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involving rock bridges will be sudden, unstable or cat-

astrophic and can be considered brittle constitutive

behaviour for the purpose of comparison to most arti-

cial support systems. Only very sti reinforcement

can be considered eective in the preservation of the

rock mass tensile strength.

If tensile rupture of rock bridges under gravity load-

ing is considered a small-strain brittle phenomenon,

then it is appropriate to consider residual tensilestrength as a supporting mechanism for gravity loaded

blocks as illustrated, for example, in Fig. 6. Consider,

for example, a design factor of safety of 2 against

gravity loading for the 45 degree wedge. This dictates

a maximum span of 4 m if full persistence (intact

area=0%) is assumed and a 22 m pattern of singlestrand cablebolts is installed. If it could be determined

that rock bridges exist across the bounding planes of

the wedge and that the relative intact area across these

planes is merely 1%, then a maximum span of 11 m

would be acceptable without the need for support.

This result demonstrates that the stability of an exca-

vation or the factor of safety can be profoundly inu-enced by the presence of small rock bridges within the

jointed rock mass and thus draws into question the

conventional assumption of full joint persistence in

limit equilibrium analyses. It also points to the import-

ance of blast control for the preservation of such rock

bridges in moderately jointed rock masses.

3. Tensile strength corrosion and time dependency

A caution is warranted with respect to long-term

rock bridge strength. Time dependent failure of exca-

vations is an issue of particular concern in mining and

in particular, with respect to support costs for tempor-

ary openings. Spans which can remain safely unsup-

ported upon excavation can fail suddenly and

catastrophically after a long period of stability with lit-

tle displacement warning. Subcritical crack growth [34]

and humidity and chemically assisted stress

corrosion [21, 35, 36] are recognized as controlling fac-

tors for the time-dependent rupture of cracked solids.

According to Atkinson and Meredith [34] waterinduced stress corrosion and subcritical crack growth

can occur in soda lime glass at stress intensities (K0) of

approximately 0.3 times the critical intensity factor

(KIC). This ratio could be as low as 0.2 for minerals

such as quartz with a corresponding reduction in ten-

sile crack propagation stress or strain for a given crack

geometry. Such crack growth under constant strain

can lead, over time, to a critical cracked area (Fig. 5)

resulting in unstable propagation and ultimate rupture.

In particular, the humidity uctuations or chemically

laden air in underground mines exacerbate crack

instability. Rock bridges, in loose and permeable

ground, which are acting as a signicant supportingcomponent and are strained are particularly suscep-

tible. Stress corrosion of rock bridges leading to un-

stable crack propagation may also be a source of

delayed microseismicity around underground exca-

vations.

4. Abutment relaxation as a destabilizing mechanism

In order for a fractured or jointed rock mass to sup-

port itself against gravity driven failure, it must have

the ability to transfer load to the abutments through a

frictional or arching mechanism. Both mechanisms rely

Fig. 6. Example of residual tensile strength of rock bridges ( N=125/m3) acting as eective wedge support (gravity loaded 45 8 prismatic wedge).



9/28

on the existence of stable abutments and on boundaryparallel connement which can result from the in situ

stresses or from deection-induced compression.

Abutment deformation or displacement leading to tan-

gential stress relaxation in the adjoining roof or hang-

ingwall can be an important catalyst for failure in

jointed or fractured ground [37]. In the context of this

discussion `relative relaxation' is synonymous with

connement reduction while `absolute relaxation' or

simply `relaxation' is used to describe a state of com-

plete connement loss, parallel to the excavation

boundary. Relaxation displacement results in the open-

ing of joints and fractures and is equivalent in elastic

models to the generation of boundary parallel tensilestresses. Of particular interest are situations in which

an entire wall is engulfed in such a `tensile' or relaxed

zone. This scenario is common in mining where high

in situ stress ratios and complex and extreme exca-

vation geometries are present.

It is important to note rst, however, that while the

analyses here are based on simple analogues for relax-

ation in situ, the processes leading to abutment and

boundary parallel relaxation are complex and depend

on rock properties and geological setting, mining

method and sequence, backll procedures, etc. The

objective of this discussion is to convince the reader of

the importance and ubiquity of relaxation in complex

mining environments and to qualitatively and semi-

empirically accommodate the concept into design.

For purposes of illustration, reconsider the simple

example of a gravity driven (non-sliding) wedge above

a horizontal excavation. It can be shown [38] for mod-

erate spans that a two dimensional symmetrical wedge

with an semi-apex angle signicantly less than the in-

herent friction angle (including dilation) is unlikely to

be liberated if the lateral compressive stresses acrossthe back are signicant greater than zero as illustrated

in the example of Fig. 7. In three dimensions, a kine-

matically feasible pyramid shaped wedge is inherently

stable, for a practical excavation span and under mod-

est connement, if its sides and edges fall entirely

within an inverted vertical cone with an angular radius

equal to the friction angle of the joints. This explains

why, in conned hard rock environments with rough

and unweathered joint surfaces, failures of wedges stee-

per than 458, a typical hard rock joint friction angle,

are rare [39]. On the other hand, Fig. 7 also shows

that if subsequent and surrounding mining activities

contribute to a signicant reduction in connement,the factor of safety drops rapidly.

In low stress situations near surface or where signi-

cant relaxation has occurred due to back deection or

changes in mine geometry, steep wedges (and of course

shallow wedges as well) which would normally have

been clamped in place can be liberated and destabilized

causing serious and often catastrophic failures (Fig. 8).

At depth in the Canadian shield, for example, where

the far eld horizontal stress is 1.5 to 2 times the verti-

cal stress, isolated mining drifts will have high conn-

ing stresses across the back as shown by the example

boundary element analysis in Fig. 9(a). Such a drift

will tend to exhibit fewer groundfalls than excavations

adjacent to mining blocks (even after discounting

those failures directly induced by stope blasting). In

the latter case the stress ow is disrupted by large sub-

vertical stopes creating back relaxation (indicated by

horizontal elastic tension parallel to the back) and

groundfall potential in the access drifts (Fig. 9b).

Stopes which are aligned normal to the major principal

compressive stress can, in extreme cases, generate

Fig. 7. Example of clamped wedge stability.



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extensive tensile stresses in elastic models which reect,

in reality, tangential hangingwall relaxation (Fig. 9c).

In tabular stopes in hard rock mines, inclined hang-

ingwalls, composed of blocky or laminated ground,

which are inherently stable under design conditions,

can be brought to the point of failure both in reality

(Figs. 10 and 11) and in simulation (Fig. 12) by abut-

ment softening or inelastic displacement resulting in

stress relaxation within the hangingwall. Fig. 13 shows

the results of stress change monitoring in the hanging-wall, in the vicinity of the cablebolt array shown in

Fig. 10, as the stope was mined past the array. All

directions show a signicant measured reduction in

stress in the hangingwall due to stress shadowing and

also due to relative abutment relaxation. The model in

Fig. 12 is a discrete element model with elastic deform-

able blocks representing the hangingwall and sur-

rounding rock mass. The stope in Fig. 12 was highly

stable until the lower hangingwall abutment was sof-

tened (using viscous constitutive behaviour [12]) to

account for the undercutting due to the crosscut

shown in Fig. 10. This undercut and the induced relax-

ation in the hangingwall resulted in failure both in the

model and in reality [12].

Gravity driven failure of stress fractured ground can

also be triggered by relaxations caused by mining of

adjacent panels. Stress driven or structurally controlled

failures can induce unraveling and large scale caving

of adjacent fractured ground [13]. The creation of

intersections in jointed or fractured rock masses

reduces the arching ability or the rock mass in the

back by extending outwards the zone of deformation.

This is equivalent to relaxing the abutments of a tun-

nel span. Barton et al. [40] suggest that the destabiliz-

ing impact of intersection creation is equivalent to a

minimum 50% reduction in eective rock mass qual-

ity, a factor conrmed by Diederichs and

Hutchinson [39]. Other examples of relaxation mechan-

isms are illustrated in Fig. 13.

The inuence of low tangential connement across

excavation spans is considered explicitly in the Qclassication system [40]. For an excavation near sur-

face or with near-zero stress conditions, the impact on

stability, relative to moderate depth and connement,

is equivalent to one order-of-magnitude reduction in

rock mass quality, Q. In other words, there is a corre-

sponding reduction of up to 60% in allowable unsup-

ported span, due to relaxation, for a given rock mass.

As discussed in the following sections, relaxation can

be included in the modied stability graph analysis [41]

for stope stability. Relaxation causing near zero stress

conditions tangential to excavation spans can be

shown to reduce the self-supporting capacity of an ex-

cavation in structured or fractured ground.

Furthermore, there is conclusive evidence [42, 43]

that relative stress relaxation tangential to excavation

surfaces can drastically impact on the performance of

frictional support systems such as plain strand cable-

bolts. The relaxation measured in Fig. 11, which led to

the failure in Fig. 10, was also linked to measured re-

duction in cablebolt interface bond strength. The

increase in gravity demand coupled by the reduction in

Fig. 8. Liberation and failure of steep wedges due to (a) low stress near surface and (b) localized deection-induced tangential relaxation.



11/28

support capacity explains the often catastrophic nature

of relaxation induced groundfalls in mining.

Erosion of residual tensile strength with time and

with dynamic disturbance as well as relaxation due to

mine geometry changes or rock movements can be im-

portant mechanisms causing the delayed failure and

collapse of excavations. In order to further illustrate

the impact of residual tensile strength on excavation

stability, to demonstrate the destabilizing impact of

abutment relaxation and to provide a simplied tool

to assess the impact of these factors for at back or

hangingwall design, the voussoir beam analogue for

Fig. 9. (a) High induced compressive (horizontal) stresses aligned parallel to back of isolated mining drift (sH/sV=2); (b) induced tension or

relaxation parallel to drift back adjacent to large subvertical stope and (c) induced hangingwall relaxation (expressed as equivalent elastic tension

parallel to face) due to long stope axis perpendicular to major principle stress. Stresses are in MPa.

Fig. 10. Failure of jointed hangingwall due to relaxation of lower abutment above crosscut (after Ref. [39]).



12/28

stability of laminated ground will be utilized in Section

5. Again it should be noted that the simplicity of the

following analogue is not intended to explicitly rep-

resent the normally complex mechanisms of relaxation.

It is a tool to obtain an understanding of the impact

of tensile strength and relaxation and to modify

empirical recommendations accordingly.

5. The voussoir mechanism of self support

The voussoir beam forms in laminated or blocky

ground when the tensile strength is reduced to zero in

the radial or normal direction (Fig. 14) due to

through-going fractures perpendicular to the beam.

The symmetrical distribution of compression and ten-

Fig. 11. Stress changes monitored in the hangingwall during the excavation in Fig. 10 (after Ref. [12]).

Fig. 12. UDEC simulation of failure in Fig. 10. Softening of lower abutment in model was required to induce failure (after Refs. [12, 20]).



13/28

sion through a cross-section of the elastic beam is

replaced by a compressive arch (Fig. 15a) which varies

in thickness but is typically between 0.5 and 0.75 times

the beam thickness, T, at midspan for a highly stable

beam [44]. The compression arch thickness reduces to

below 0.5T as the critical span is approached as orig-

inally suggested by Evans [45]. Failure of the beam is

by snap-through or by crushing at the upperside of the

beam at midspan where the compressive stress is the

highest.

The problem is statically indeterminate but can be

solved in an approximate fashion using simplifying

assumptions [45, 46] or accurately through a iterative

procedure introduced by Brady and Brown [38] and

modied by Diederichs and Kaiser [44]. Only a brief

summary is presented here. The basis of the solution is

to balance the moment generated at the abutment by

the self-weight of the half-span with the opposing

moment generated by the oset reaction force, F, at

the midspan. (Fig. 15b). Two key independent

unknowns are the thickness of the compressive arch,

NT, and the moment arm between the reaction resul-

tants at the abutment and at midspan, Z. An iterative

solution which minimizes the peak compressive stress,

fmax, at the abutments and midspan results in the equi-

librium solution for N and Z. The procedure presented

in Diederichs and Kaiser [44] is followed here with the

inclusion of support pressure or rock mass tensile

strength and the incorporation of abutment relaxation.

The reaction stresses at the abutment and at the

midspan section are assumed to be triangular such

that the reaction force, F, acts at the one-third point

Fig. 13. Relaxation due to (a) and (b) unfavourable stress ratio; (c) and (d) changes in mining geometry (excavation step 2 creates stress shadow

around excavation 1); (e) abutment yield; (f) intersection (roof relaxes in the direction of tunnel branches); (g) undercut; (h) concave geometries.



14/28

of the distribution (Fig. 15b). The reaction distri-

butions at the midspan and at the abutments are

assumed to be identical such that the initial moment

arm is given by

Z0 T1 23

N 15

The reaction arch along which F acts is parabolic such

that the length of the arch can be approximated by:

L S 83S

Z20 16

Elastic shortening of the arch DL, due to compression

leads to a downward deection at midspan and a new,

reduced moment arm Z:

Z

3S

8

8

3SZ20 DL

s17

The deection, D, at midspan is given by (Z Z0) anda negative value for the term under the square root

sign in Eq. (17) indicates that the critical beam deec-

tion for the assumed thickness, NT, has been exceeded.

Prior to calculating the shortening of the arch due

to deection and compression, it is possible to intro-duce an symmetrical displacement dA, acting in oppo-

site directions at each abutment. This displacement

yields a reduced initial moment arm, Z0*:

Z*0

3S

8

8

3SZ20 2dA

s18

for substitution into Eq. (17).

For equilibrium, consider only the right hand half

of the beam and equate moments about the abutment

reaction point (right side of the arch) generated by the

self-weight of the half beam and the reaction at the

midspan. The peak compressive strength a half spanfor a horizontal beam is given by:

fmax gS2

4NZ19

The shortening of the arch, dL, is calculated by assum-

ing a parabolic distribution [43] of stress along the

length of the reaction arch:

DL LE

fmax

2

9 N

3

20

Eqs. (19) and (20) are for the span of a tunnel which is

Fig. 14. Voussoir beam formed by subvertical joints and surface par-

allel laminations.

Fig. 15. (a) Nomenclature for voussoir beam calculation; (b) Solution parameters for the voussoir arch.



15/28

much longer than it is wide. For a square span, these

equations can be replaced with the following:

fmax gS2

6N

Z0 Z 21

DL LE

fmax

2

9 N

3

1 n 22

For evenly distributed support pressure p distributed

evenly over the length of the beam, a solution can be

obtained by substituting an equivalent unit weight, g *:

g* g pT

23

into Eq. (19). If g*

= 0, then the beam is fully sup-ported and no voussoir deection can occur. For tri-

angular distributions of support pressure varying from

0 at the abutment to p at the midspan, use p *=2/3p

in Eq. (23).

An iterative solution is needed to nd the equili-

brium value for Z corresponding to NT, the thickness

of the compression arch. Discrete interval values of N

are taken over the range of 0 < N< 1. The parametric

pair (N, Z(N)) which minimizes fmax give the equili-

brium solution for the stable beam. The limit of stab-

ility is determined when no solution is possible for

Eq. (18) for any possible value of N and the beam fails

by `snap through'. For the two-dimensional case thispoint corresponds to a deection at the midspan equiv-

alent to approximately 25% of the thickness.

Diederichs and Kaiser [44] propose a more conserva-

tive stability threshold based on the onset of snap-

through instability (deviation from a linear deection

thickness relationship). This limit corresponds to a

deection of 10% of the thickness and is dened by

the parametric set which yields a minimum of 35%

invalid values of N in the range 0 to 1 (i.e. no real sol-

ution for Eq. (17)). The range of unstable displace-

ments (10 to 25%) of the thickness corresponds to the

results of numerical experimentation (H15%) obtained

by Mottahed and Ran [47].

The peak stress in the beam, fmax, is compared with

the unconned compressive strength, UCS. In ad-

dition, failure can occur through snap thru, through

crushing or through vertical shear at the abutments if

the reaction pressure is insucient to generate ade-

quate friction. Summary design charts, based on the

rst two failure modes, for horizontal beams and

square slabs are given in Fig. 16.

6. Boundary normal tensile strength and the voussoir

beam

From Fig. 16 it is immediately apparent that thicker

laminations are more stable than thinner laminations.

This is because the compression arch is able to rise

higher within a thicker beam increasing the resisting

moment and reducing the necessary de ections

required to reach equilibrium. A condition for thetransmission of compressive stresses within the arch is

that there is continuity through the beam thickness.

This may not be satised in partially separated lami-

nations with small and sparse rock bridges. A pro-

portion of the beam's weight, however, can be

transferred to the next beam above through tensile

stress in the rock bridges. This is analogous to a sup-

port pressure acting from above the beam. If the dis-

tributed capacity of the rock bridges is equal to the

self weight of the rock beam it can be considered fully

supported and fully connected to the beam above

forming a beam of twice the thickness. This analogue

can be used to generate a ground reaction curve asshown in Figs. 17 and 18. The array of linear curves

represent the elastic response for varying lamination

thicknesses as indicated on the left-hand axis. The

required support pressure or boundary normal resist-

ing stress is indicated on the right-hand axis. The com-

posite curve for a laminated rock mass is shown as the

dark curve, obtained as the upper limit for required

tensile support capacity.

Fig. 17 illustrates the minimum support require-

ments or alternatively the minimum internal boundary

normal tensile strength required to limit the beam

deections and to achieve stability. The response

curves for typical support patterns are shown for com-parison. The minimum lamination thickness is

assumed to be 0.1 m and the composite curve rep-

resents the results of stacking or of partial delamina-

tion (rock bridges). In this case both the support

pressure and the tensile resistance are assumed to be

constantly distributed across the span. For tensile

strength this is likely to be a valid assumption due to

the small strain nature of rock bridge rupture. The ca-

pacity requirements (support or tensile demand)

increased with reduced rock mass modulus as shown

in Fig. 18. These demand values are very low for the

span in question, falling in the range of residual tensile

strength presented in Figs. 3 and 5 for small rock

bridges.

The limiting tensile strength (required for stability)

as a function of span is summarized in Fig. 19. In

Fig. 19, the upper and lower bounds for tensile

strength are dened by initial yield (buckling

limit= 35%, deection = 10% of thickness) and absol-

ute collapse (buckling limit = 100%). Fig. 20 illustrates

the impact of modulus.



16/28

Again, it is important to note the extremely small

values of tensile strength required for stability of large

spans. This is not surprising in light of the large num-

ber of large spans found in natural caverns, overhangs

and arches. The strength required to resist gravity

loading is commonly found even in jointed rock due to

the lack of complete persistence and the presence of

rock bridges. If the self supporting and arching ability

of moderate quality rock masses is considered, as in

the voussoir analogue, the tensile strength require-

ments for stability are further reduced.

Support is still required, however, in many appli-

cations in moderately jointed rock, particularly those

with longer or undetermined stand-up time require-

ments. In mining it is commonly and correctly held as

imprudent to leave fractured ground unsupported (if

human trac is a concern) in spite of the fact that re-

sidual tensile strength may create a self-supporting

opening over the short term. The dynamic nature of

mining combined with the eects of moisture on long-

term tensile strength can lead to delayed failure ofunsupported ground. The presence of rock bridges,

however, provides signicant capacity reserve at small

deformations, before tensile strength degradation is

allowed, thus reducing rst pass or short term support

requirements in some cases. Very sti support such as

grouted rebar can serve to reduce this degradation,

leading to economic benets through the use of shorter

tendon support elements designed to preserve tensile

Fig. 16. Voussoir design charts for horizontal tunnel and square span ( g=30 kN/m3); Maximum stable span for a given thickness is the mini-

mum value determined from Erm or UCS limit.



17/28

strength and allowing voussoir-like arching, carrying

load to the abutments.

7. Abutment relaxation and the voussoir beam

Another dominant factor in the reduction of stand-

up time for unsupported (and inadequately supported)

excavations is relaxation, parallel to the excavation

face, which is inuentially, if not mechanistically,

equivalent to an outward displacement of the abut-

ments. Fig. 21 illustrates the physical implementation

of relaxation for the voussoir beam analogue. The

quoted displacements in the following gures apply to

a single abutment. An equal and opposite displacement

is assumed at the other abutment. Before beam deec-

tion in this schematic and in the classic Voussoir ana-

logue, the beam is assumed to be stress free,

corresponding to a zero local in situ stress. The relax-

ation displacement shown is based on this datum and

Fig. 17. Composite ground reaction curve for laminated rock mass based on voussoir calculations: (span= 20 m; E=10 GPa, g=30 kN/m3;

UCS= 25 MPa); response curves for typical support patterns are shown for comparison; internal tensile strength refers to boundary normal

strength.

Fig. 18. Ground reaction curve: relationship between rock mass modulus and tensile support demand (span= 20 m; buckling limit,

B.L. = 100%).



18/28

does not include additional displacement incurred as

initial compressive in situ stress and strain reduce to

zero.

In other words, the displacement pictured here is

equivalent to the presence of tangential tension in the

back of an excavation in an elastic continuum such as

that simulated in Fig. 9. In Fig. 9b, the average ten-

sion adjacent and parallel to the back of the 10 m

wide access drift was approximately 5 MPa. Using an

assumed rock mass modulus of 10 GPa, for example,

the equivalent symmetrical abutment displacement, dA,

as per Fig. 21 is equated as 2.5 mm. In Fig. 9c, the

average tangential tension (10 MPa) across the 50 m

hangingwall yields an equivalent dA, at both the topsill

and the bottomsill, of approximately 25 mm. These

examples represent only a rst approximation and will

likely underestimate the true analogous relaxation dis-

placement since stress shedding is not taken into

account in the elastic model. Admittedly it is very di-

cult to accurately estimate dA for many situations.

This should not, however, be a deterrent to the con-

sideration of relaxation in design.

The impact of this abutment deformation or relax-

ation is to increase the beam deection required to

generate sucient compressive stress and resisting

moment for equilibrium. This additional deection

brings the beam closer to failure. This eect is illus-

trated in the ground reaction diagram in Fig. 22. The

individual curves are generated in a similar fashion to

the composite curve in Fig. 17, assuming that thin

laminations would stack and create a composite beam

Fig. 19. Limiting demand for boundary normal tensile strength in

laminated ground (modulus = 10 GPa; collapse B.L.= 100%; yield

B.L. = 35%).

Fig. 20. Eect of modulus on minimum tensile strength demand (B.L. = 100%).

Fig. 21. Schematic abutment relaxation and destabilization of a

voussoir beam.



19/28

with a thickness determined by the available tensile

strength (rock bridges) or articial support. For com-

parison, an abutment relaxation of 10 mm over a half-

span of 10 m is equivalent to a tensile strain (parallel

to the excavation face) of 0.1% or 10 MPa of analo-

gous tensile stress using a modulus of 10 GPa. This is

realistic for deep mining excavations.

This abutment relaxation can occur as a result of

changing geometry around the excavation in question,

by unfavourable stress ratios or by increased deec-

tions due to the creation of intersections or undercuts

(Fig. 13). The latter is a source of large and unfavour-

able abutment deformations and is responsible for nu-

merous failures observed by the authors and by

others [12]. Figs. 23 and 24 show the corresponding

increase in support or tensile strength demand due to

abutment relaxation. The term yield in these gures

corresponds to midspan deections equivalent to 10%

and a buckling limit, B.L., of 35% as discussed earlier.

The term collapse (B.L. = 100%) describes the least

conservative solution. Figs. 23 and 24 show that very

little tensile strength (normal to laminations) can com-

pensate for abutment deformation, but that the maxi-

mum span to ensure the marginal stability of

Fig. 22. Ground reaction curves for laminated ground illustrating the impact of abutment relaxation (modulus= 10 GPa, B.L. = 35%;

span=20 m).

Fig. 23. Impact of abutment relaxation on the minimum demand for boundary normal tensile strength (B.L.= 100%).



20/28

unsupported openings (stopes) decreases signicantlyin the presence of relaxation if additional strength or

support is not available.

8. Critical span reduction by relaxation

For unsupported beams, there exists a critical relax-

ation at which failure (snap-thru) occurs. For a rock

mass with a modulus of 10 GPa, Fig. 25 shows the

critical relaxation (as a function of span and thickness)

for yield and collapse of a horizontally laminated roof

(without rock bridges). Critical relaxation varies, of

course, with rock mass modulus as illustrated inFig. 26a). In order to compare the absolute relaxation

displacements in Figs. 25 and 26 to modeled strains

and equivalent tensile stresses in elastic stress analysis

models, Fig. 26b) provides a conversion chart.

Remember that the displacements indicated are

`measured' from the state of zero stress and do not

include recovery of the initial compressive strain pre-sent in situ.

9. Relaxation and the modied stability graph

To explore the eect of relaxation on empirical

stope design methods, it is desirable to apply the vous-

soir analogue for relaxation to more general exca-

vation design. This can be achieved by calibrating

voussoir analysis to an existing empirical design limit.

The stability graph method was rst proposed by

Mathews et al. [41] and later modied and calibrated

for Canadian mining conditions by Potvin [48].Additional data has been added [16,49] and the

method is now a popular tool in Canadian open stope

design. The characterization of the rock mass by

means of a stability number NH is based initially on theQ-system [40]. The stress reduction factor is dropped

(set to 1) as is the joint water factor (since deep mines

Fig. 24. Same simulation as in Fig. 23 showing limits for yield (B.L.= 35%) and collapse (B.L. = 100%).

Fig. 25. Critical relaxation limits for unsupported voussoir beam (Erm = 10 GPa).



21/28

are typically dry). Three new factors A, B and C are

dened to account for induced stress, relative joint

orientation and gravity loading respectively such that:

NH RQDJn

JrJa

A B C Q H A B C 24

where A, B and C are dened in Fig. 27.Excavations of dierent geometries are equated

through the use of a shape factor or hydraulic radius,

HR, which describes the geometry and size of the

stope face in question. For a rectangular stope face of

dimensions a b:

HR AreaPerimeter

ab2a 2b 25

Using this relationship the hydraulic radius, HR, of a

tunnel roof (with length much longer than the span) is

given as one half of the span. For a square excavation,

the hydraulic radius is equivalent to one quarter of the

span. The hydraulic radius expression, therefore,

denes that a horizontal tunnel is equally as stable as

a square excavation surface of twice the linear span

(measured orthogonal to the face edge). This assump-

tion creates some diculties which will be discussed.A calibration database of 189 case histories invol-

ving unsupported stopes has been assembled by

Potvin [48] and Nickson [49]. The database and the

resulting no-support limits are shown in Fig. 28.

In order to link the voussoir analogue to the modi-

ed stability graph, it is necessary to relate the vous-

soir parameters to rock mass quality and the modied

stability number NH (Fig. 29). In the absence of com-prehensive case data, an iterative procedure was used,

bounded by logical upper and lower parametric limits.

Fig. 26. (a) Eect of modulus on critical relaxation and (b) equivalent relaxation stress.

Fig. 27. Stability parameters for the modied stability graph (after Ref. [39]).



22/28

The values of A, B and C are taken to correspond to

the voussoir assumptions such that A = 1 (low stress),

B = 0.3 (surface parallel jointing and C= 2 (horizontal

roof). this yields a relationship between N and Q H:

Q H Na0X6 26

Next a simplied relationship between rock mass mod-ulus and Q H in a moderately relaxed setting proposedby Diederichs and Kaiser [44] and a similar relation-

ship for unconned compressive strength (UCS) are

assumed:

Erockmass GPa 5

Q HpI6X5

N

p27

UCS MPa 20

Q H

pI25X8

N

p28

These relationships are used here, not as engineeringrecommendations, but rather as reasonable baseline

assumptions in order to create a generalized model to

Fig. 28. Unsupported stope data from Refs. [48, 49] and no-support limits for modied stability graph (after Ref. [39]).

Fig. 29. Variation with respect to NH of voussoir parameters used in simulation (for model calibration purposes only).



23/28

assess the eects of relaxation. In addition, while intact

UCS is independent of Q, it is related to A. A relation-

ship (Eq. (28)) is included in this calibration since in

hard rock masses, increased RQD and lower Q is in

general associated with weaker rock such as shists,

rhyolites, talcs and meta-shales, for example. The eec-

tive UCS values resulting from this relationship rep-

resent approximately one half of the nominal lab

strength [9]. Using these relationships, for E and UCS,

in analyses of voussoir stability for dierent hydraulic

radii and lamination thicknesses, it is possible to derive

the following relationship for NH as a function of criti-cal lamination thickness for unsupported openings (in

metres):

NH 150 Thickness3 29While this calibration procedure is somewhat subjec-

tive and is conned in its application to the assump-

tions described, it yields reasonable parametric ranges

for thicknesses between 0.1 to 1 m. `Yield' calculations

were used for the voussoir simulation (buckling

limit= 35%), corresponding to a long-term no-support

limit for span. The calibration resulting in the para-

metric set in Fig. 29 is based on the upper no-support

limit in Potvin's [48] stability graph. This limit corre-

sponds to the onset of instability or yield. Fig. 29 is

not for general application and is intended for model

calibration purposes only. The resultant calibration is

shown in Fig. 30 for both the square and the long

(tunnel) span.

The hydraulic radii for critical square openings are

uniformly 0.77 time that of critical tunnel spans for

equivalent N values. This results from a relative over-

prediction of square span stability inherent in the de-

nition of hydraulic radius. The denition of HR infers

that a square span is as stable as a long tunnel of half

(0.5 times) the linear span. Voussoir predicts that a

square opening of span, S, is as stable as a long tunnel

of span 0.65S. The hydraulic radii, therefore, of critical

square spans (as predicted by voussoir simulation) are

0.77 times the HR of critical tunnel spans (for a uni-

form set of voussoir parameters). This discrepancy cor-responds closely to a correction (approximately 0.72)

proposed by Milne [14] using a radius factor based a

harmonic average distance from the centre of the span

to the perimeter. It can be shown that the hydraulic

radius for a circular face is equal to that for the corre-

sponding circumscribed square. Failure to account for

the additional mass and deformation due to the cor-

ners of the square is partly responsible for this error.

For the purposes of this calibration, the square span is

used to determine HR. HR's derived from tunnel

spans (2D voussoir calculations) should be multiplied

by 0.77.

Now that the voussoir simulation is calibrated tothe stability graph it is possible to simulate the impact

of abutment relaxation on the no-support limit. The

stress parameter A does not consider relaxation, unlike

the stress reduction factor, SRF, which it replaced.

This factor monotonically increases from a low of 0.1

at high stress to a maximum of 1 at low or zero

induced stress. A is set to 1 in this simulation (no

stress) so that the impact of relaxation can be exam-

ined explicitly. In the calibrated voussoir calculations,

outward abutment deection corresponds to relax-

ation. Fig. 31 shows the new no-support limits for

varying degrees of abutment relaxation. It is apparent

Fig. 30. Voussoir calibration for square span. Linear tunnel spans

require geometric correction.

Fig. 31. Revised no-support limits for NH resulting from abutmentrelaxation (negative relaxation values in mm correspond to stope

wall compression).



24/28

that very little relaxation displacement is required to

signicantly increase the potential for instability.

Fig. 32 shows the same results plotted against the

log of the hydraulic radius. Using the modulus re-

lationship in Fig. 29 and the relaxation-stress equival-

ence calculated in Fig. 27(b), it is possible to express

the no-support limit alternatively as a function of

induced average tensile stress tangential to the span (as

derived from elastic models). For the voussoir ana-

logue, relaxation is dened with respect to the position

of the abutments under zero lateral stress (or strain).

Therefore the relaxation displacement used in the

voussoir analysis corresponds directly to equivalent

tensile stress (parallel to the span) in an elastic model.

The eect of relaxation in terms of the equivalent ten-

sile stress is shown in Fig. 33. For competent rock

masses, modeled elastic tensile stress magnitudes, of

appreciable spatial extent, in excess of 30 MPa are

rare. If inelastic compressive strain has occurred adja-

cent to the excavation, as in the case of joint slip or

abutment crushing, full relaxation (tension) may not

be necessary to cause failure. In previously stressed

Fig. 32. No-support limits resulting from abutment relaxation plotted with respect to log HR: function shown is for Potvin's [48] upper no-sup-

port limit.

Fig. 33. Equivalent elastic tensile stresses corresponding to the relaxation displacements in Fig. 32.



25/28

ground around excavations, any signicant reduction

in lateral compression in the surrounding regional rock

mass may result in relaxation induced groundfall.

Field evidence conrms the impact of relaxation as

illustrated in Fig. 34. A subset of the data from

Greer [50] and reanalyzed by Bawden [51] is plotted in

the lower half of Fig. 34, corresponding to horizontal

stope backs. Bawden [51] reports that these backs areunder high compression (factor A = 0.1 to 0.4). The

data seems to correspond well to the original stability

limits proposed by Potvin [48]. Data for the inclined

hangingwalls are plotted in the upper portion of

Fig. 34. Bawden [51] reports that many of these faces

have tangential tensile stresses according to elastic

models. In spite of this relaxation eect, the stress fac-

tor A must be set to its maximum (=1) in accordance

with the stability graph procedure. Note that Mathews

et al. [41] in their original discussion of this method

state that it is not valid for cases of tangential relax-

ation. In the data of Fig. 34, the method overpredicts

the stability of these hangingwalls. Adjusted stability

limits corresponding to average elastic tangential ten-

sion values of 4, 8 and 16 MPa are superimposed on

Fig. 34. While the magnitude of tension in these cases

is not documented, the value shown is not unreason-

able for typical hangingwall geometries in elastic ana-

lyses as indicated in Fig. 9(c). If the stability limit is

shifted up as indicated by these stress-dened curves,

the unstable stopes can be appropriately captured.

This is equivalent to a corresponding reduction in NH

for the aected data points.

Using the apparent shift in NH, illustrated in Figs. 33and 34, as a result of tangential tensile stress or relax-

ation, it is possible to derive an approximate function

for a revised stress factor A which reects the inuence

of relaxation. The modeled tensile stresses are rst nor-

malized with respect to a range for nominal UCS ofintact lab samples which is typical of hard rock (e.g.

200 MPa at NH = 10). The relationship illustrated inFig. 35 can be obtained from an examination of

Figs. 33 and 34. An average shift of N (applied

through the A factor), for a practical range of HR, is

determined for dierent values of tension in order to

provide for a simple and general correction to A. The

general equation for factor A adjustment:

A 0X9e11sTaUCS for sT`0 30for tangential boundary stress values (sT) in tension, is

obtained as an exponential best t. For practical pur-

poses a linear t with extrapolation limits may also be

adequate given the uncertainties in relating equivalent

relaxation displacement with elastic tensile stress.

It is important to realize that other failure mode

such as wedge fallout are likely to be even more sensi-

tive to relaxation than the voussoir beam. The destabi-

lizing eect of relaxation, therefore, may be

signicantly greater than that illustrated in Fig. 35 and

Eq. (30). It is important to note that this relationship,

Fig. 34. Correlation of relaxation adjustment for upper no-support limit (data from Refs. [50, 51]); Stope backs were under moderate to high

compression while elastic models predicted tension in hangingwalls.



26/28

derived through the voussoir analogue is valid only forlow to moderate relaxation or tension. For higher

values of tension, it has been suggested that A drops

to zero and that the method, as a whole, becomes

invalid [41]. The best t coecient 0.9 in Eq. (30) cre-

ates an intercept which indicates an initially vertical

slope to the relationship. In fact it is likely that the

left-hand decline (Fig. 35) in the A factor begins at

compressive stresses below several MPa [16] particu-

larly if wedges are present. It should be emphasized,

again, that Eq. (30) represents an upper bound for the

A factor and that the impact of relaxation can be

much greater than that indicated.

The results presented here can also be applied to theno-support limit of the Q-system [40] as initially dis-

cussed in Ref. [37]. Using the calibration presented

here and plotting against span and Q, instead of HR

and NH, and assuming dry conditions and an initialSRF of 2, Fig. 33 can be replotted with respect to the

Q-system as shown in Fig. 36. The equivalent tunnel

spans are calculated by dividing the stable HR values

from the square (voussoir) spans by 0.77 (long span

correction according to voussoir analysis). Note that

the base-line no-support limit for tunnels (long spans)

corresponds to the limit for ESR = 3, while that for

the square span corresponds to ESR= 5. Barton [52]

Fig. 35. Revised denition of stress factor, A, used to determine the modied stability number, NH. For consistency with compressivefunction [48], relaxation is reected as the normalized maximum tension parallel to the face near midspan.

Fig. 36. Tangential elastic tensile stresses corresponding to abutment relaxation and the shift in the Q no-support limit; Q limits are calculated

from NH limits in Fig. 33; Barton's [52] ESR limits are plotted for comparison.



27/28

dened this range of stability suitable for temporary

mine openings. Hutchinson and Diederichs [39] rede-

ne the limits ESR = 3 to correspond to open stope

backs and limited access drifts while ESR = 5 is suit-

able for non-entry mining stopes and stope walls.

These denitions are consistent with the results shown

in Fig. 36. Note again, however, the eect of relax-

ation and the additional design conservatism which

must be implemented to compensate for this relax-ation.

10. Conclusions

The presence of geometrically unfavourable jointing

or rock mass damage due to elastic predictions of

overstress does not, in itself, dictate that failure will

take place. Neglecting the possibility of dynamic pro-

cesses such as compression-induced buckling, degra-

dation of residual tensile strength or abutment

relaxation is often required before failure can take

place.Rock masses are inherently discontinuous due to

natural jointing or induced fracturing. It is often erro-

neous to assume that this fracturing is fully persistent.

In massive to moderately jointed rock residual tensile

load bearing strength arising from incomplete fractur-

ing or from rock bridges separating non-persistent

joining is a key factor in the control of ultimate grav-

ity driven failure of jointed or stress damaged ground.

Very little rock bridge cross-sectional area (less than

1% in most cases) is required in hard rocks to replace

most articial support systems. The time dependency

of this residual tensile strength due to stress corrosion

and atmospherically induced crack growth controlsstand-up time and mandates the use of support sys-

tems in most underground excavations. Sti support

such as grouted rebar can suppress dilation strains and

preserve some of this internal tensile strength, contri-

buting to a reduction in both short term and long

term support requirements. There is great economic

advantage to selecting, as part of a multi-component

support system, sti elements which can preserve the

rock mass internal tensile capacity. Careful blast

damage control is also an obvious advantage. Early

installation of shotcrete or spray-on linings in a dry

rock mass at depth can reduce the impact of atmos-

pheric eects and time dependent stress corrosion on

stressed rock bridges.

Boundary parallel relaxation is another dominating

factor in delayed mining induced failure of spans in

underground excavations, signicantly shifting conven-

tional no-support limits so that smaller spans or ad-

ditional support are required. Dangerous abutment

relaxation can occur even at depth, driven by un-

favourable stress ratios, complex mine geometries,

abutment damage, intersection development and

undercutting. Abutment relaxation increases support

demands by reducing the natural ability of the rock

mass to transfer loads to the abutments through arch-

ing. In addition, boundary normal stress relaxation

has been shown to reduce the capacity of frictional

support systems, exacerbating ground control pro-

blems.

A voussoir beam analogue was used to illustrate theimportance of internal boundary normal tensile

strength and of abutment relaxation in controlling the

stability of spans in laminated rock masses. The results

of the voussoir simulation were used to modify empiri-

cal stope design limits, accounting for abutment relax-

ation. A few millimetres of hangingwall or back

abutment relaxation can signicantly shift the no-sup-

port limit, inducing failure in previously stable spans.

It is therefore important to sequence development and

stope extraction properly to minimize this relaxation

and to minimize the size of secondary stopes. The cre-

ation of high relaxation geometries, such as hanging-

wall undercutting, must be avoided.

Acknowledgements

This work was funded by the Natural Science and

Engineering Research Council of Canada.

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