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.



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.

M.S. Diederichs, P.K. Kaiser / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 97±117 98



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.




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.




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.




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.




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.




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.




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).




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).




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].




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.




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.




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).




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).




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).




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.




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.




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.




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.




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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Diederichs Kaiser 1999a

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