diederichs kaiser 1999a
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Stability of large excavations in laminated hard rock masses: the
voussoir analogue revisited
M.S. Diederichs *, P.K. Kaiser
Geomechanics Research Centre, F217 Laurentian University, Sudbury, Ont., Canada P3E 2C6
Accepted 8 November 1998
Abstract
The voussoir beam analogue has provided a useful stability assessment tool for more than 55 years and has seen numerous
improvements and revisions over the years. In this paper, a simpli®ed and robust iterative algorithm is presented for this model.
This approach includes an improved assumption for internal compression arch geometry, simpli®ed displacement determination,support pressure and surcharge analysis and a corrected stabilizing moment in the two dimensional case. A discrete element
simulation is used to verify these enhancements and to con®rm traditional assumptions inherent in the model. In the case of
beam snap-through failure, dominant in hard rock excavations of moderately large span, design criteria are traditionally based
on a stability limit which represents an upper bound for stable span estimates. A de¯ection threshold has been identi®ed and
veri®ed through ®eld evidence, which corresponds to the onset of non-linear deformation behaviour and therefore, of initial
instability. This threshold is proposed as a more reasonable stability limit for this failure mode in rockmasses and particularly
for data limited cases. Design charts, based on this linearity limit for unsupported stability of jointed rock beams, are presented
here summarizing critical span±thickness±modulus relationships. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction
Rockmass behaviour dominated by parallel lami-
nations is often encountered in underground exca-
vations in numerous geological environments. These
laminations can be the result of sedimentary layering,
extensile jointing, fabric created through metamorphic
or igneous ¯ow processes or through excavation-paral-
lel stress fracturing of massive ground. This structure
can be the dominant factor controlling the stability of
roofs in large civil excavations, in coal or other hori-
zontal mining stopes [1, 2] and can also dominate the
stability of inclined open stope hangingwalls such as
those encountered in hard rock mining in Canada [3, 4],in Australia [5, 6] and elsewhere.
In rare circumstances, the lamination partings rep-
resent the only joint set present in the rockmass. Roof
stability and de¯ection in this instance can be assessed
using conventional elastic beam de¯ection and lateral
stress calculations [7, 8]. It is more common, however,
to encounter other joint sets cutting through the lami-nations. These joints reduce and, in the extreme, elim-
inate the ability of the rockmass to sustain boundary-
parallel tensile stresses such as those assumed in con-
ventional beam analysis. However, where these joints
cut across the laminations at steep angles or where re-
inforcement has been installed, it is possible to assume
that a compression arch can be generated within the
beam which will transmit the beam loads to the abut-
ments (Fig. 1).
It was noted by Fayol [9] that underground strata
seemed to separate upon de¯ection such that each
laminated beam transferred its own weight to the abut-ments rather than loading the beam beneath. Stability
of an excavation in this situation, it was concluded,
could be determined by analyzing the stability of a
single beam de¯ecting under its own weight.
Conventional beam analysis, however, signi®cantly
underestimated the inherent stability of such beams.
Even intact laminations would crack at midspan as
predicted, but would, after additional deformation,
become stable again. The notion of the voussoir, while
traceable back to the architecture of ancient Rome [10],
International Journal of Rock Mechanics and Mining Sciences 36 (1999) 97±117
0148-9062/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.
P I I : S 0 1 4 8 -9 0 6 2 (9 8 )0 0 1 8 0 -6
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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was ®rst proposed by Evans [11] speci®cally to explain
the stability of a jointed or cracked beam. After gener-
ating a great deal of controversy when ®rst published,
the voussoir beam analogue has been generally
accepted and has since been reworked and presented
as a simpli®ed tool for stability analysis of excavations
in civil construction and in mining [10, 12±14]. Fig. 2
illustrates two example cases where the voussoir beam
analogue can be invoked to explain the inherent stab-
ility of a laminated hangingwall (Fig. 2a) and cross
jointed back (Fig. 2b) in hard rock environments.
The primary modes of failure assumed in the model
and veri®ed in laboratory tests by Sterling [15] are
buckling or snap-through failure, lateral compressive
failure (crushing) at the midspan and abutments, abut-
ment slip and diagonal fracturing (Fig. 3). Shear fail-
ure (Fig. 3c) is observed at low span-to-thickness
ratios (thick beams), while crushing (Fig. 3b) and
snap-through failure (Fig. 3a) is observed at higher
span-to-thickness ratios (thin beams). An examination
of the model data presented by Ran et al. [16] shows
that if the angle between the plane of the cross-cutting
Fig. 1. (a) Jointed rock beams; (b) voussoir beam analogue.
Fig. 2. Voussoir beams encountered at (a) Winston Lake Mine, Ontario and (b) mine access, Sudbury.
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joints and the normal to the lamination plane (and thenormal to the excavation surface) is less than one third
to one half of the eective friction angle of these
joints, then it is valid to apply the voussoir beam sol-
ution. For shallower cross-jointing, slip along these
joints and premature shear failure of the beam is
likely [16].
Stimpson and Ahmed [17] also showed by physical
modelling that for thick beams, external loading (sur-
charge) can produce diagonal tensile ruptures (Fig. 3d)
extending from the upper midspan to the lower abut-
ments (parallel to the compression arch). While this
failure mode is partially the result of the loading con-
®guration, it may also be an important mechanismwhere weak or broken material exists above the beam
or where internal rockmass damage and bulking due
to stress overloads the surface beam. This paper deals
with thin laminations (span-to-thickness ratio greater
than 10) under the in¯uence of self-weight or moderate
surcharge loading. Therefore, only crushing and snap-
through failure are hereafter considered. Sliding stab-
ility is included in the analyses but does not control
limiting dimensions for thin beams.
The following is a summary of the voussoir analysis
procedure based on the iterative scheme proposed by
Brady and Brown [13], including a number of improve-ments and corrections by the authors. Most impor-
tantly, a more realistic yield threshold is introduced
for snap-through failure, to replace the ultimate rup-
ture limit originally proposed [11, 12]. This procedure
is summarized in several design charts and is veri®ed
using a discrete element simulation.
2. The voussoir model
Consider a laminated rock beam above an exca-
vation with a horizontal span given by S . The normalthickness of the single layer under analysis is T . For
an elastic beam, with no joints and with constant cross
section, a distribution of compression and tension,
symmetrical about the horizontal centreline of the
Fig. 3. Failure modes of the voussoir beam: (a) snap-through; (b)
crushing; (c) sliding and (d) diagonal cracking.
Fig. 4. Elastic beam with (a) ®xed ends and (b) simple (pin) supports.
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beam, is found across all plane sections within the
beam (Fig. 4a). The solution (Eqs. (1) and (2)) for the
maximum stress values, at the abutments, for com-
pression (bottom of beam) or tension (top of beam),
smax, as well as the maximum beam de¯ection, d, can
be easily calculated using closed form beam
equations [7]:
smax gS 2
2T 1
d gS 4
32ET 22
where E is the Young's modulus of the rock and g is
the speci®c weight. The maximum stress at the mid-
span is one half of the maximum stress at the abut-
ments. Therefore, for such a beam with ®xed ends and
distributed loading, yield is assumed when the maxi-
mum tensile stress, in the upper part of the beam at
the abutments, exceeds the tensile strength of the rock.
Vertical tensile fractures form at the abutments and
the beam becomes simply supported (assuming no slip
at the abutments), as shown in Fig. 4(b), with a maxi-
mum tensile stress at the midspan given by
smax 2gS 2
3T 3
This stress is now higher than the previous abutment
stress, and therefore higher than the rock tensile
strength. This leads to subsequent fracturing centred
about the midspan as shown by Stimpson and
Ahmed [17]. This process of progressive cracking at
the abutments, followed by cracking at the midspanand other parts of the beam can be responsible for a
¯urry of low-level seismic emissions (rock noise), often
encountered in newly developed underground spans at
low to moderate depth. This initial elastic phase fol-
lowed by progressive fracture and deformation of
laminated hangingwalls has been observed and is
described in detail by Milne [18].
It is highly unlikely that any thinly laminated roof
structure will remain in a completely continuous and
elastic state after excavation. The transition from con-
tinuous elastic beam to voussoir beam is normally
assured and assumed in most cases. Failure is not
inevitable in this situation, however. When the tensile
cracking, described above, or lamination-normal joints
crosscut the beam and render it incapable of sustaining
tensile stresses, a compression arch develops within the
beam, rising from the abutments to a high point at
midspan. For a half-span, this arch generates a
moment between the reaction force at midspan and at
the abutments (Fig. 5) which acts to resist the moment
imposed by self-weight. Horizontal stress symmetry
within the beam is lost, making a closed form solution
impossible without assuming an average thickness NT
for this arch. This is the case for the solution intro-
duced by Evans [11] and later by Beer and Meek [12].
In both cases N is assumed to be 0.5. Numerical exper-
imentation by Evans showed that this was an incorrect
assumption and that the equilibrium value was closer
to 0.7. Nevertheless, Evans chose the value, N =0.5,
to simplify the practical solution which he was present-
ing. Investigations by the authors of this paper have
shown that N is variable. While N is closer to 0.75 for
stable beams at equilibrium, N drops to below 0.5 as
the critical (unstable) beam geometry is approached.
In this formulation, the average thickness, NT , of the compression arch within the beam is initially
unknown, as is the ultimate moment arm, Z 0, between
Fig. 5. Voussoir beam (half-span shown) and nomenclature.
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the horizontal reaction forces, F , at the midpsan and
at the abutments. Considering the half beam (Fig. 5)
and building on the algorithm introduced by Brady
and Brown [13] the solution procedure begins with an
assumption of the initial moment arm prior to de¯ec-
tion, Z 0:
Z 0 T 1 À2
3
N 4
The length of the reaction arch (horizontal reaction
force locus) is given by
L S 8
3S Z 20 5
This equation is based on the assumption of a para-
bolic compression arch. Numerical [16] and
physical [17] experiments con®rm this assumption.
Note that this equation is correctly reported by Beer
and Meek [12] but is in error in the summary of Brady
and Brown [13].
The solution proceeds on the assumption that for
stability to be obtained, the moment generated at the
abutment due to self-weight of the beam, given by
M W gTS 2
86
must be exactly compensated, as the beam de¯ects a
distance d at the midspan, by an opposite or resisting
moment, M R. This moment is generated by the oppos-
ing horizontal reaction forces in the beam centre and
at the abutments separated by a distance Z = Z 0Àd:
M R FZ f maxNTZ 2
7
where f max is the maximum stress acting in the beam
(at the bottom edge of the abutment and at the top
edge of the midpsan section).
In Eq. (6), the speci®c weight, g, of the rockmass
can be replaced with an eective speci®c weight, ge,
given by
ge g cos a 8
where a is the dip (angle from the horizontal) of the
lamination plane. This analogue considers only the
component, of the driving weight, oriented normal to
the beam. For inclined beams, therefore, the eect of
beam-parallel loads due to settling is not considered.
In this model, as in reality, an inclined roof is more
stable under gravity loading than a horizontal roof.
In addition, active support pressure can be con-
sidered in the equation by further adjustment of the
eective speci®c weight:
gep ge À p
T 9
for the case of a uniformly applied support pressure,
p, illustrated in Fig. 6(a).
If the support pressure is applied in a triangular dis-
tribution varying from zero at the abutments to p at
the midspan (Fig. 6b) as in the case of passive rock-
bolts, then use
g*e ge À2 p
3T 10
A parabolic distribution of support pressure (Fig. 6c)
can be applied as
g*e ge À7 p
9T 11
To introduce a distributed surcharge, q, replace (+ p)
with (Àq) in Eqs. (9)±(11) as appropriate. For example,
in the case of a distributed surcharge loading ranging
parabolically, as in Fig. 6(d), from 0 to q (due to bro-ken rock above the beam for example) the modi®ed
eective speci®c weight becomes
g*e ge 7q
9T 12
Assuming a triangular horizontal stress distribution
within the compression arch (of thickness NT ) at the
abutments and at the midspan as illustrated in Fig. 5,
the following equilibrium equation is obtained for
f max:
Fig. 6. External beam loading due to (a) uniformly distributed sup-
port pressure; (b) linearly varying support pressure; (c) parabolically
varying support pressure and (d) parabolically varying surcharge
loading.
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f max g*e S 2
4NZ 13
In order to calculate the elastic shortening of the arch
(and thereby calculate the central de¯ection) an
assumption must be made about the internal distri-
bution of compressive stress within the beam. Previous
authors [11±13] have based their calculation on the
assumption of a quasi-linear variation of stress (Fig. 7)along a constant arch section NT , yielding an average
stress along the reaction line, f av, given by
f av f max
2
2
3
N
2
14
Careful examination of this assumption shows it to be
in error. It is reasonable to expect that at some point
the entire beam section must be under compression
and that at the point where the reaction line crosses
the centreline of the beam, this stress is constant across
the entire beam section T . It is also reasonable toassume that the variation of stress along the reaction
line is not linear. Numerical experimentation by the
authors of this work and examination of numerical
results obtained by others [16, 19] con®rms that the
distribution of stress along the centreline of the arch is
in fact parabolic (Fig. 7). This yields the following cor-
rected equation for average stress in the beam:
f av f max
3
2
3 N
15
Elastic shortening of the arch can then be obtained
from
DL L f av
E
L
E f max
2
9
N
3
16
where E is the rockmass modulus in the direction par-
allel to the beam.
This shortening leads to a downward displacement
of the arch midspan and a new reaction moment arm
given by
Z
3S
8
8
3S Z 2
0 À DL
s 17
The de¯ection, d, at midspan is given by (Z À Z 0) and
a negative value for the term under the square root
sign in Eq. (17) indicates that the critical beam de¯ec-
tion has been exceeded. In other words, as de¯ection
increases, the resisting moment, M R, a product of increasing reaction force, F , and decreasing moment
arm, Z , passes through its maximum without achieving
equilibrium with the weight moment M W. In this case,
snap-through failure would occur for the speci®ed
value of compression arch thickness, NT . If there are
no values for N (between 0 and 1) for which a stable
solution can be obtained, ultimate collapse of the
beam is assumed to occur.
In order to ®nd a solution for the equilibrium pos-
ition of the beam, Evans [11] chose to maximize the
product of the resisting moment and the moment arm,
Z , although by his own admission, this choice was
somewhat arbitrary. In the procedure of Brady andBrown [13] a two-variable relaxation technique is
employed to solve for N , the arch thickness and Z , the
®nal moment arm. The equilibrium solution corre-
sponds to the unique pair (N , Z ) which results in ana-
lytical equality through the sequence of equations
summarized in the forgoing discussion. This approach
was found by the authors to be highly unstable and
convergence was often dicult. Fortunately it can be
seen through examining the results of this process that
equilibrium solution also corresponds with the mini-
mization of f max, the maximum stress at the abutments
and midspan.In the approach presented here, N is varied in incre-
ments (e.g. 0.01) over its ®nite range (0 to 1). Z is
modi®ed in an iterative fashion and a convergent sol-
ution is thereby obtained with only a few steps for
each value of N . For a stable beam with a span well
below the critical limit for the given geometry and
rockmass properties, a solution for Z is possible for all
values of N . Equilibrium corresponds to the minimum
value of f max. As the critical span is approached, the
percentage of N values which yield rational results for
Fig. 7. Conventional assumptions for compressive stress variation
inside the beam compared with the parabolic variation proposed in
this paper.
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Z decrease, converging to a single rational solution
pair (N , Z ) at the absolute critical limit.
Finally. taking Z and Z 0 (as determined using the
equilibrium value of N ), the de¯ection, d, at the mid-
span is simply Z À Z 0. The complete procedure for the
determination of stability and equilibrium de¯ection is
summarized in Fig. 8.
The factor of safety with respect to crushing (com-
pressive failure) at the abutments and at midspan is
given by the ratio of uncon®ned compressive strength
of the rock with respect to the maximum compressive
stress calculated in the model:
FXSXcrush UCS
f max
18
Fig. 8. Flow chart for the determination of stability and de¯ection of a voussoir beam.
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The factor of safety with respect to vertical sliding, of
an unsupported beam under self-weight, along joints
at the abutments is given by
FXSXslide f maxN
geS tan f 19
A numerical buckling limit, B.L., is introduced herewhich is the percentage of values of N within the
range of 0 to 1 for which a solution (i.e. a real value
of Z ) cannot be obtained. Fig. 9 illustrates the
decrease in normalized equilibrium arch thickness, N ,
and the increase in buckling limit, B.L., with increasing
span/thickness ratio. N consistently drops below 0.5 as
the critical span/thickness ratio is approached and
reaches a limit of 0.35 immediately prior to failure
(snap-through). Ultimate collapse and the critical span
versus thickness relationships proposed by Evans [11]
and by Beer and Meek [12] correspond to a buckling
limit of 100%. In other words, stability is impossible if there is no arch thickness which yields an equilibrium
solution. For a more conservative approach, however,
a threshold can be speci®ed for the buckling limit in
order to obtain design dimensions for the beam.
In order to obtain a reasonable yield threshold for
the buckling limit, the relationship between midspan
displacement, d, and thickness, T , was considered.
Fig. 10 illustrates summary results for several beam
stinesses. The de¯ection/thickness relationships are
log-linear for beam geometries with ample thickness.
As the thickness is reduced (or the span is increased)
the relationship becomes non-linear and eventually
becomes unde®ned at ultimate failure. Ultimate failureoccurs at a buckling limit of 100%, corresponding to a
displacement equivalent to approximately 0.25Â T .
The onset of non-linearity consistently occurs at a dis-
placement equivalent to approximately 0.1Â T . This
`yield' point also consistently corresponds to a buck-
ling limit, B.L., of 35%.
The yield threshold determined in Fig. 10 more clo-
sely corresponds to the critical displacement limit of
approximately 0.15 T obtained by Mottahed and
Ran [19] through numerical modelling. Beyond this
displacement the jointed beam model used in their ex-
perimentation began to exhibit unstable behaviour andfailure became imminent. This limit was independent
of span and the modulus.
The choice of buckling limit aects the calculated
critical span or the critical thickness (limiting case for
Fig. 9. Variation of N and buckling limit with span to thickness ratio.
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beam stability) as illustrated by the example in Fig. 11.
These limits are obtained by including the analysis out-
lined in Fig. 8 inside a simple iterative bisection algor-
ithm to ®nd the critical value of a speci®ed input
parameter (in this case, thickness) which gives a value
(span) straddling the interface between stability and
failure.
The displacement limit of 0.1Â T (B.L. = 35%) is
used forthwith, as a yield limit, due to its convenience
and to account conservatively for uncertainties in the
approach. This limit is important for monitoring appli-cations since it is independent of the span and rep-
resents a universal rule of thumb for determining the
signi®cance of measured displacements. Displacements
of less than 10% of the eective lamination thickness
(determined using a borehole camera or from mapping
data) can be con®dently assessed as being within the
elastic limit of the voussoir beam, independent of rock
modulus or eective span.
Fig. 11 indicates critical limits for snap-through fail-
ure of the beam. Crushing failure is also considered in
the discussion which follows. Typically thin hard rock
beams (signi®cant compressive strength) under their
own self-weight will tend to snap-through before they
crush (Fig. 12). If an otherwise stable beam is sub-
jected to surcharge loading, however, as is the case in
many of the laboratory experiments described in the
literature, then crushing failure will result (as may
abutment sliding or diagonal cracking). The in¯uence
of surcharge loading on the stability of the beam as
well as on the failure mode is demonstrated in Fig. 13.
The transition line (snap-thru/crushing) indicateswhich failure mode controls the critical limits as
shown.
3. Field evidence of snap-through limit
3.1. Mount Isa Mine
A limiting displacement (for linear voussoir beha-
viour) of 0.1 to 0.15 times the mapped bedding thick-
Fig. 10. Example determination of beam `yield' limit (example span = 20 m).
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ness is observed in the data, presented by Milne [18],
from numerous extensometers installed in the stope
hangingwalls at Mount Isa Mine in Australia. The dis-
placement data from one of the extensometers is sum-
marized in Fig. 14. In this case, the extensometer was
installed in the hangingwall, slightly above the stope
midspan and prior to mining of a 50 m high stope at adepth of 870 m. The hangingwall consists of laminated
shale with bedding plane partings observed every 20 to
25 cm. The stope was mined via full face longitudinal
retreat. Measurements were recorded as the stope was
progressively mined past and beyond the location of
the extensometer.
The radius factor shown on the horizontal axis of
Fig. 14 is a scale and shape index developed by
Milne [18]. The details of this index are summarized
elsewhere [20]. For illustrative purposes, the radius fac-
tor describes the harmonic average of distance, from a
point in a excavation surface or face, to all abutments.The radius factor, R.F., of the hangingwall at the
extensometer location increases as the stope is widened
along strike (constant height and thickness). The maxi-
mum R.F. prior to back®lling was approximately 14.
Milne [18] identi®es four displacement zones with
respect to increasing span including:
1. Elastic rock behaviour.
2. Stable voussoir beam behaviour.
3. Unstable voussoir beam behaviour
Fig. 11. Eect of buckling limit on critical beam geometry for `snap-thru' failure of a horizontal beam ( E rm= 10 GPa, g= 0.03 MN/m3).
Fig. 12. Typical critical limits for the stability of slender beams (axis
scales are identical for both plots).
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4. Ultimate failure.
Stage (1) is not presented in Fig. 14 due to the
scale. Stages (2) through (4) are clearly apparent in
Fig. 14. The lamination thickness, T , is determined to
be relatively consistent and equal to 0.2 to 0.25
m [18]. The onset of non-linear or unstable voussoir
displacement (interface between stages (2) and (3)
occurs at a surface displacement of approximately
20 to 30 mm (0.08Â T to 0.15Â T ), re¯ecting the
proposed yield limit for de¯ection of 10% of the
beam thickness. A further transition to rapidly
increasing displacement occurs at around 55 to 60
mm. This is consistent with the ultimate collapse
limit of 0.25 T . No extensometer anchor displace-
ments are recorded beyond 80 mm although it is
assumed that had mining and recording continued
beyond a R.F. of 14, without back®lling, this stope
wall would have experienced signi®cant surface fail-
ure. This corresponds (given uncertainties regarding
the local parting thickness) with the predictions from
the voussoir analogue.
Fig. 13. In¯uence of surcharge loading on beam stability and failure mode.
Fig. 14. Hangingwall response to mining compared with voussoir limits. Data from Milne [18].
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3.2. Winston Lake Mine
A second case study from Winston Lake Mine in
Ontario, Canada is used here to demonstrate the utility
and validity of the voussoir analogue. The typical
stope geometry is similar to that in the previous
example except that the stopes are smaller (20 m high)
and less steeply dipping (458 to 608) and in this case,
shallower (550 m). Winston Lake Mine uses a modi®ed
version of the longitudinal retreat technique called
modi®ed AVOCA, in which the stope is blasted in full
height slices perpendicular to the strike of the ore
body. Rock ®ll is introduced by end dumping some
distance back from the active face, creating an open
stope of variable open strike length. The ®ll is com-
pressible rock®ll and cannot be considered tight ®ll
until signi®cant closure has occurred. As a stability
analogue, therefore, the two dimensional beam is valid
in this case.
As previously illustrated in Fig. 2(a), the hanging-
wall is composed of blocky chert with surface parallel jointing along foliation planes and two orthogonal sets
of cross-joints. Cablebolt support is installed from a
remote drift to supply full coverage at the hangingwall.
Fig. 15 shows the layout of drifts, stopes, cablebolts
and instrumentation for an experiment investigating
cablebolt performance in hangingwalls and the impact
of stress change [21, 22]. Extensometer data [23] from
this study is used here to illustrate the voussoir beam
behaviour of the chert hangingwall. Borehole camera
data [24] is also used here to estimate the eective
thickness of continuous partings parallel to the hang-
ingwall as the wall de¯ects.
The rockmass quality as indicated by the index
Q [25] is approximately 7 to 28 which, using the
relationship [26]
E rm 25 log10Q 20
indicates a rockmass modulus in the range of 20 to 35
GPa. The data set, collected from numerous sources
contains a large scatter and according to variability
limits proposed by Barton [27] this rockmass quality
could have a modulus as low as 10 GPa and as high
as 50 GPa. The lower modulus could apply if the rock-
mass is relaxed due to undercutting as was the case in
some of the stopes [28].
The displacement performance of the hangingwall
can be predicted by the voussoir analogue. The results
for this case are shown in Fig. 16. The equilibrium dis-
placement is plotted as a function of lamination thick-
ness and rockmass modulus. The critical displacement
limits for yield (10% of the thickness) and for collapse
(25% of the thickness) are overlain on the plot. The
joints at Winston Lake were not fully continuous and
the eective lamination thickness, therefore, evolved as
the hangingwall de¯ected and as joints propagated
Fig. 15. Plan layout and typical cross sections through study area, Winston Lake Mine.
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along the foliation planes or connected with other
joints. The borehole camera surveys [24] can be used
to crudely estimate the eective beam thickness at any
point in space and time. The measured displacements
are plotted against the estimated lamination thickness
in Fig. 16 for speci®c points along dierent extens-
ometer-borehole camera clusters (Fig. 15).
The results show that the rockmass behaviour corre-
lates well with the predicted displacements. The data
with higher beam thicknesses are from earlier in themining. As more partings form, the eective thickness
decreases and the points move to the left on Fig. 16. It
is interesting that while the earlier points correlate
with a rockmass with low modulus, the later data
seems consistent with a stier rockmass. This can be
linked to two possible mechanisms.
The ®rst is that as the voussoir beam de¯ects the
compression in the arch increases. The rockmass mod-
ulus of a jointed rockmass is normally pressure
dependent [29] and therefore, in this case displays a
strain-stiening behaviour. In addition, plain strand,
cement grouted cablebolts tend to require up tobetween 30 and 40 mm of slip to develop their peak
frictional capacity [14]. If the eective embedment
length of the strand, equal to the lamination thickness,
is less than the critical length required to break the
steel, the cablebolt can provide this capacity over rela-
tively large rockmass displacements. It can be seen
that late in the experiment, as mining progresses
farther past the instrumentation sites, the de¯ection
beyond 40 mm follows a more stable trend than that
predicted by the model. Cablebolts must be contribut-
ing to the stability of the hangingwall by applying an
eective support pressure.
4. Numerical veri®cation of snap thru mechanism
A discrete element model [30] was employed to
further verify the analogue. Other authors have per-
formed similar analyses with mixed results. A recent
paper by So®anos [31] describes a simulation using®nite elements which appears to show that the voussoir
model presented by Brady and Brown [13], which is
similar to the model described here, poorly predicts
the simulated displacements and beam stresses, primar-
ily due to an over-prediction of the eective thickness
of the compression arch at the midpsan and at the
abutments. The numerical simulations of So®anos [31]
predict a normalized arch thickness of less than 0.3 at
equilibrium and less than 0.1 near failure. These are
low compared to the values 0.75 for equilibrium and
0.3 to 0.4 at failure, predicted by the iterative
approach in this paper. Numerical simulations by Ranet al. [16] indicate an arch thickness more consistent
with the predictions in this paper.
Clearly, something is amiss. The answer appears to
be the numerical boundary conditions and discretiza-
tion used in the model. It was found during the course
of this research that a UDEC model [30] composed of
rigid blocks and elastic joints (no tension) and a model
composed of deformable blocks and elastic joints (no
tension) both seemed to exhibit similar behaviour to
the model of So®anos [31] if the beam had ®xed sup-
Fig. 16. Measured displacements and observed lamination thicknesses (data points), compared with voussoir predictions, Winston Lake Mine.
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ports or was bounded by rigid blocks as abutments.
This creates problems with discretization which do not
seem to permit the formation of a stress triangle
(0 < 1) at the abutments. Instead, reaction forces are
concentrated at the bottom corner of the abutments
leading to over-prediction of the moment arm and in-
accurate de¯ection predictions.
The model ultimately used for this paper utilized de-
formable elastic blocks (discrete blocks with internal
®nite dierence zones), elastic joints (no tension) and
¯exible (elastic and internally discretized) but very sti
abutment blocks. This provided better displacement
compatibility at the abutments. The results of this
model correlate very well with the voussoir analogue
as will be discussed presently. Simulations included
single beams only since the purpose of the model was
a comparison with the analogue model. Hatzor and
Benary [32] have recently examined the in¯uence of
multiple layers with interbed friction. The single beam
prediction represents the worst case failure initiation
potential in cases where lamination thickness and mod-
ulus is uniform.
The beam was ®rst allowed to de¯ect elastically
while maintaining a non-zero tensile strength within
the joints. If this initial elastic de¯ection is not per-
mitted, sliding will occur before arch stresses are gener-
ated. It is assumed that excavation is a gradual process
such that beam deformations occur before the abut-
ments are fully liberated. After elastic equilibrium was
achieved, the joint tensile strength was set to zero and
the beam was allowed to continue deforming until
either equilibrium or failure occurred. The joints had
no cohesion but had a frictional strength. The initial
elastic deformation stage allowed lateral stress and
frictional strength to accumulate in the joints, particu-
larly at the abutments. Without this stage the beam
would simply slip past the abutments under its own
weight.
Several con®gurations were tested, using dierent
rock stinesses, E i, dierent joint normal stiness,
Fig. 17. Exaggerated displacement pro®le (top) for a typical UDEC voussoir beam simulation and contours of internal horizontal compressive
stress (bottom).
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JKN, and joint spacings, s j. The rockmass modulus,
E rm for comparison with voussoir predictions, was cal-
culating using the relationship:
1
E rm
1
E i
1
JKNs j
21
The de¯ection pro®le and the internal stress distri-
bution are shown in Fig. 17 for one example simu-
lation. The parabolic nature of the compression arch isimmediately apparent. Fig. 18 compares the lateral
stress distribution at the abutment and at the midspan
with the assumed triangular distribution and the pre-
dicted voussoir arch thickness (0.65 T). The distri-
bution at midspan correlates very well, while the
numerically simulated abutment stress distribution is
more non-linear, resulting in a higher maximum stress
near the bottom edge of the beam. In addition, the lat-
eral stress variation along a parabolic path, ABC, in
the UDEC model is compared with the assumed para-
bolic variation introduced in this paper. Again the
match is acceptable for engineering application of the
method.
Models of unit thickness (T = 1) with several dier-
ent rockmass moduli were analyzed with increasing
span, in increments of 5 m, up to snap-through failure
(blocks fall and beam disintegrates). Crushing failure
was not investigated in this analysis. Fig. 19 illustrates
the calculated equilibrium displacements in the UDEC
models and the associated voussoir predictions. Themodeled de¯ections match the predicted relationships
well, with the exception of a slight over-prediction by
the voussoir model at higher stinesses. In addition
the UDEC model failed in each case immediately
beyond the critical span predicted by the voussoir ana-
logue.
The maximum stress occurred inevitably at the bot-
tom edge of the beam at the abutment. The midspan
maximum stress correlated fairly well with predictions
(Fig. 20) while the abutment stress was consistently
Fig. 18. Predicted variation of horizontal stress along arch axis (ABC) compared with UDEC results (top) and predicted and simulated stress dis-tributions at abutment and at midspan (bottom).
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Fig. 19. Midspan displacements (data points) from UDEC beam simulations compared with voussoir predictions ( T =1 m).
Fig. 20. Maximum compressive stress in UDEC beams compared with voussoir predictions (T =1 m).
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higher. If this is real and not an artifact of the model-
ling, it is unlikely to devalue the predicted crushing
limit for the voussoir analogue, since the area over
which the increased stresses occur in the UDEC model
is quite limited and any resulting initial crushing
would be highly localized. These numerical results,
therefore, give us con®dence that the voussoir model
can be used to predict the de¯ection and stability of
rock beams which form in laminated ground.
5. Stability guidelines
The results of this work can be summarized into
normalized stability charts for use in design of under-
ground openings. Parametric modelling has shown
that the snap-through stability limits for critical span
and thickness can be related to the normalized rock-
mass modulus, E *, which is equal to the modulus
divided by the eective speci®c gravity, S.G.*(E *= E /
S.G.*). Similarly, the limits for crushing failure can be
related to the normalized compressive strength
(UCS* = UCS/S.G.*). The eective speci®c gravity,
S.G.*, for a beam with a dip of a can be obtained
from Eq. (22).
SXGX
* SXGXcos a 22
In order to simplify the estimation of rockmass mod-
ulus based on rockmass classi®cation, the test data for
®eld modulus presented by Barton et al. [27] and by
Bieniawski [33] was reexamined by Hutchinson and
Diederichs [14] to account for the eect of con®nement
and sample disturbance. The results have been further
modi®ed here to provide convenient functions for
modulus based on rock quality and con®nement
(Fig. 21). These functions can be used to estimate
rockmass modulus for the voussoir beam. For most
excavations where the voussoir analogue is valid, the
lower bound moduli should be used.
Finally the compressive strength must be speci®ed.
Crushing failure is local in nature and therefore relates
to the strength of the intact rock. However, in accord-
ance with recommendations by Martin [34] regardingthe strength of intact rock in situ, one third to one
half of the laboratory UCS is used for this analysis.
The summary design limits based on yield (buckling
limit = 35%) are presented in Fig. 22. If a beam thick-
ness is speci®ed, then a critical span is obtained for
snap-through failure and for crushing failure. The de-
sign span is the lesser of the two values. Likewise, if a
span is speci®ed, the critical thickness is the greater of
the two.
Actual spans are rarely fully two dimensional in
nature. In fact the two-dimensional beam represents a
lower bound model for determining critical span. Anupper bound is obtained from the model for a square
span. Again, Brady and Brown [13] derive a relation-
ship for a square span by assuming that the span fails
as four triangular panels. The weight of each panel
creates a moment about the abutment edge:
M W gTS 3
2423
which is resisted by a reaction moment
Fig. 21. Simpli®ed relationships for rockmass modulus as a function of rock quality and con®nement.
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M R f maxNTSZ
224
This equation, however, is in error due to the assump-
tion of a constant moment arm, Z , through all sections
of the triangular panel. The de¯ection at the hinge
point is zero and therefore, the moment arm must
vary from Z = Z 0Àd, at the midspan, to Z 0 at the
corners. Eq. (24) must therefore be revised to
M R f maxNTS Z Z 0
425
Combining Eqs. (23) and (25) yields a modi®ed re-
lationship for f max:
f max geS 2
6N Z Z 026
Eq. (26) is then substituted for Eq. (13) in the original
analysis. One ®nal adjustment is necessary, however.
For the square span, Eq. (16), for the shortening of
the beam must be replaced by
Fig. 22. Stability guidelines for jointed rock beams (tunnel spans). Eective speci®c gravity, S.G.* =S.G. cos a.
Fig. 23. Stability guidelines for jointed rock plates (square spans). Eective speci®c gravity, S.G. * =S.G. cos a.
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DL 1 À n L
E f max
2
9
N
3
27
where n is the Poisson's ratio for the rockmass.
The results for the square span are summarized in
Fig. 23.
A few limitations with respect to these charts should
be noted:
. These charts can be used with con®dence in lami-
nated ground where the lamination thickness is
known and where the modulus can be estimated
with some degree of reliability.
. It is imprudent to rely on such an assumptive
method for very large spans (>100 m for steeply
inclined spans and >60 m for horizontal spans)
where other in¯uences, not considered here, may
govern stability. Consistent excavation quality is
also assumed. Over-blasting, or uneven excavation
surface geometry will negatively impact roof stab-
ility.. The assumption in these charts is that the joints are
rough enough to provide frictional resistance under
low to moderate con®nement (i.e. no slickensides or
low friction coating) and that the span to thickness
ratios are greater than 10. Sliding failure along joints
at the abutments or within the beam is not con-
sidered.
. It is assumed that there is no frictional or cohesive
resistance along the interfaces between the lami-
nations. This represents a worst case assumption
since such resistance increases the stability of the
beams. Snyder [1] showed that limited friction (gen-
erated by the active pressure created by mechanical
rockbolts) resulted in a small increase in stability,
while cohesive resistance (no slip) generated by fully
grouted bolts (or rock bridges) had a signi®cant sta-
bilizing eect.
. These charts are only valid if no low to mid angle
jointing is present. In this case or in the case of
thinly laminated ground, support is necessary in
order to create a reinforced beam. Rockbolts or
(preferably [1]) grouted rebar should be approxi-
mately equivalent in length to the desired beam
thickness [10]. Assuming a lower-bound rockmass
modulus, a reinforced thickness can be estimated
from Figs. 22 and 23 which will ensure adequate
stability. A factor of safety of 1.5 to 2 with respect
(i.e. use half the span or twice the thickness) to is
adequate for most situations [14].
. This method is not suitable for poor rockmasses
with a low RQD rating (<50) and more than three
joints sets.
. This technique is designed to predict the onset of
roof instability. Thus only the ®rst (lowermost) lami-
nation is considered and not a composite beam
structure. This is based on the assumption that, pro-
vided the beam thickness is stability of the whole
roof is controlled by the stability of this ®rst beam.
. The in¯uence of the gravity load component parallel
to inclined laminations is ignored in order to achieve
a tractable solution. This leads to the apparently
reasonable conclusion that inclined layers (dip = a)
are signi®cantly more stable than horizontal layers.
The de¯ecting beam must still achieve the same
Fig. 24. Examples of parallel weight correction to critical span for inclined beams.
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moment equilibrium (between the perpendicular
weight component and the compression arch reac-
tion force) regardless of initial (gravitational) com-
pression within the beam. The beam does, however,
settle under this imposed loading. The eect of this
additional compression can be considered implicitly
by imposing an initial shortening of the beam, DdC,
equivalent to the compression due to gravity (paral-
lel to the beam) given by Eq. (28). The beam must
®rst close this gap before additional compressioncan be created during beam de¯ection. This gap is
used to calculate a new initial moment arm in
Eq. (29). This value is substituted for Z 0 in Eq. (17).
DdC S 2g sin a
E 28
Z *0
3S
8
8
3S Z 20 À DdC
s 29
Examples of the in¯uence of this correction on criticalspan are shown in Fig. 24. Given the bevy of simplify-
ing assumptions already inherent in the model, the
impact of this correction is of limited practical signi®-
cance for rockmass moduli greater than one GPa and
can be neglected in the interest of producing the sim-
pli®ed and uni®ed charts in Figs. 22 and 23. On the
other hand, according to this simpli®ed correction,
non-critical equilibrium de¯ections are signi®cantly
aected as in the examples shown in Fig. 25, with
increases, at one half of the critical span, exceeding
100% for near-vertical beam (dip = 8 08).
Undoubtedly, the in¯uence of the parallel weight com-
ponent is more complex than described here, leading
to assymetrical beam de ection and other compli-
cations. A program of numerical experimentation is
warranted to investigate this in¯uence and provide
more accurate stability predictions for inclined beams.
Such a program is beyond the scope of this paper.
6. Conclusions
An improved iterative approach to a classic ana-
logue for stability assessment of laminated ground has
been presented with several improvements and correc-
tions including an improved assumption for lateral
stress distribution and arch compression, the appli-
cation of support pressure and surcharge loading, sim-
pli®ed displacement determination and a robust
iteration scheme.
A linearity limit or yield limit has been identi®ed,
corresponding to a midpsan displacement of approxi-mately 10% of the lamination thickness. This displace-
ment limit appears to be independent of rockmass
modulus and is a useful guideline for performance
monitoring and observational design. Beyond this dis-
placement, stability cannot be assured. The yield limit
and the methodology in general has been veri®ed using
®eld evidence and numerical simulations.
This limit is used to present stability charts for de-
sign. These design charts have been normalized with
respect to eective speci®c gravity which is a function
Fig. 25. Corrected equilibrium midspan de¯ections for stable steeply dipping beams (dip = 808), accounting for parallel weight component.
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of rock density, excavation dip angle and surcharge
loading or support pressure. Two failure modes are
presented for thin laminations. These are snap-through
and crushing. Other failure modes apply to thick
beams and are not considered. Critical span of the
beam is determined by the critical failure mode (mode
which gives the lowest critical span).
More work is needed to account for the boundary
parallel component of weight in steeply dipping vous-soir beams. While geometrical limits for stability are
marginally aected by the neglect of this component,
equilibrium displacement predictions may be signi®-
cantly in error for steeply dipping beams.
Acknowledgements
This research has been funded by the Natural
Science and Engineering Research Council (Canada).
A special thanks is due to Doug Milne (currently at
the University of Saskatchewan) for supplying rawextensometer data for the Mount Isa case example.
Thanks are also due to Sean Maloney of the
Geomechanics Research Centre and to Winston Lake
Mine and Noranda Technology Centre.
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