The design of tunnel support in deep hard-rock mines under

The design of tunnel support in deephard-rock mines under quasistaticconditionsby C.R. Speers* and A.J.S. Spearingt

Synopsis

The induced elastic stresses in the sidewalls of gold-mining tunnelsat a depth of 4000 m in the Witwatersrand exceed the peakcompressive strength of even the best-quaUty quartzite rocks.Laboratory tests measuring post-yield stress versus strain responseshow that brittle fracture occurs, followed by a loss in strengthassociated with strain-softening behaViour. Numerical modellingbased on an assumed Mohr-COulomb criterion tlttedto the testresults shows that large ongoing deformations of thesidewalloccur, leading to total tunnel failure without adequate support.

A methodology is presented to enable a quick assessment to bemade of support requirements in terms of ground characte~ticcurves for numerically derived pressure versus displacement. Fromthese curves, estimates can be made of the force levels for cablebolting and of the thickness and strength reqUirements forshotcrete linings. As an example of its application, the methodologyis used in a comparison of the relative stabllityefflciencies of asquare-shaped tunnel and a horizontal ellipse-shaped tUnnel.

Introduction

After gradual extraction of the shallower gold-bearing reef, mining in South Africa is beingcarried out at increased depths (approaching4000 m). Spearing! has presented principlesrelating to the excavation and support of minetunnels with a desired final cross-sectionalarea of 14 m2 at a depth of 4000 m inWitwatersrand quartzite. Elastic stressanalysis of mining tunnels of typical shapes-square and rectangular-showed that there isa high potential for failure of the surroundingrock. This is most prevalent in the sidewalls oftunnels owing to the relatively low horizontaland high vertical field stresses at such depths(k-ratios of 0.5 are common). The extent ofthe potential failure into the sidewall can bereduced somewhat by reduction of the tunnelheight accompanied by an increase in the spanto maintain the same cross-sectional area. Ahorizontal ellipse showed promising results inthis regard, and preliminary proposals for thesupport of such a tunnel shape using cablebolting and shotcrete were presented. Thiscomprised rings of nine 4 m long cables at alongitudinal spacing of 1.25 to 1.5 m and

The Journal of The South African Institute of Mining and Metallurgy

shotcrete of 100 to 300 mm in thickness. Thiswas a significant development in that traditionalsupport comprised far longer bolts (longer than8.0 m was typical).

Although Spearing identified the potentialfor failure, he did not quantify the anticipatedrock-mass behaviour and the tunnel responseto such behaviour. He did, however, implicitlyexpect non-elastic/plastic behaviour in that heproposed a yielding type of cable bolting.

The objectives of this paper are to expandon the principles raised by Spearing, withgreater emphasis on the following specificrock-mechanics aspects:

~ the mechanical properties of typicalWitwatersrand quartzites, particularlythose relating to post-failure behaviour

~ the use of numerical modellingtechniques to predict the deformationresponse of tunnel excavations at depthsof 4000 m

~ the development of a methodology basedon numerical modelling in the design ofsupport to control such deformation andto define the degree of safety againsttunnel collapse.

The developed methodology is applied inan example presented in this paper.

Post-peak stress versus strain behaviourin Witwatersrand quartzites

Witwatersrand quartzites are typically 'hard',with a uniaxial compressivestrength (DCS) ofthe order of 100 to 250 MPa. Elastic stressanalysis of mine tunnels of common shapes atdepths of 4000 m has shown that the inducedsidewall stresses are between 175 and 200 MPa

*Sttffen, Robertson and Kirsten, P.D. Box 55291,

Northlands, 2116. Gauteng.

t Anglo American Corporation, P. D. Box 61587,

Marshalltown, 2107, Gauteng. Present address:

Rock Mechanics and Bacijill Consultanqy e.e.,

P.D. Box 2141, Honeydew, 2040.@ The South 4frican Institute if Mining and

Metallurgy, 1996. SA ISSN 0038-223X/3.00 +

0.00. Paper received Dec. 1994; revised paper

received Sep. 1995.

MARCH/APRIL 1996 47 ~

Design of tunnel support

for a square tunnel, and between 150 and 200 MPa for a

rectangular tunnel. Maximum sttesses in the corners of the

excavations are as high as 250 MPa. The same analysis for a

horizontal-ellipse tunnel shows sidewall sttesses of the sameorder, but over a smaller area of extent owing to the smaller

overall height of the tunnel.

The analysis shows that even the best-quality quartzite

will be over-stressed, and the post-peak behaviour is therefore

of interest. Traditionally, such post-peak behaviour is notconsidered in rock-mechanics tests measuring the UCS2.

......A.::I

70::1:to-

i 80w~ 50(I)

~ 40ii5

~ 30a:~ 200()

10-'~)( 0~ 0z~

35

: 30::I....::1: 26to-~if! 20a:to-(I)

W~ 10(I)

w:I: 60()

l\

\'. ~,\ .,'. \',\ .,'. \','" \',

'.. '-",-'-'-'-......

0.6 1.0

PLASTIC STRAIN (,,)

00

-------0.6 1.0

PLASTIC STRAIN (,,)

However, with more specialized tests using a servo-controlledmachine that is programmed to a constant strain rate, thepost-peak stress versus strain in rock specimens can bemeasured.

Several typical Witwatersrand quartzites have been testedin such a machine under triaxial conditions at varying cellpressures, giving the strength-strain curves shown inFigure 1. From these curves, equivalent Mohr-COulombshear-strength parameters of cohesion, friction angle, anddilation angle were derived by a curve-fitting procedure.

35

1.5

,-------------------------a. 30 /11

; 25 j-'-'-'-'-'-'-'-'-'-i 20 !cz 15

u_n_u_n_u_.g .."".t3 10 ,/

~ 5 !0

0

- 'Dirty'

0.5 1.0PLASTICSTRAIN(11)

1.5

36

-a. 303

26w-'i 20cz 150~ 10-'i3

6

r ,~_. -',, ." '.\ ," "

"-"- "'- .., ----------."-. n_u_n1.6

00 1.50.5 1.0

PLASTIC STRAINnu

Pebble conglomerate

-.- Silaceous

tv1ARCH/APRIL 1996

-n- ArgillaceouB

~ 48

figure 1-straln-softenlng behaviour of typical Wltwatenrand quartzites (after Kersten3)



The behaviour shown in Figure 1 depicts brittle fracturefollowed by the rapid decrease in strength typical of strain-softening behaviour in which the friction angle remainsconstant after full mobilization whilst the cohesion drops topractically zero at a strain of 1.0 per cent.

The Mohr-Coulomb criterion pre-supposes threeconditions4.

> that a major shear fracture exists at peak strength> that the direction of fracturing is at a specific angle to

the maximum principal stress (Figure 2)

> that there is a linear relationship between the shearstrength and the normal stress on the plane of thefracture.

Observations in the fields have shown that theseconditions are not always satisfied, particularly theorientation of fracturing. Stacey6 found that a criterion basedon tensile (extensional) strain fitted the extent andorientation of the observed fractures in areas of lowconfinement, Le. close to the surface of the excavation.

The appropriateness of the extensional strain criterion toareas of higher confinement away from the surface of theexcavation is open to question, as is also the fundamentalbasis of the criterion7.

It is concluded therefore that no single criterion can orshould be used exclusively in the study of a given problem ofrock stability.

!0, """, """",,, .....

~01 Minor principal

stress

~ = 45° + 0/2

0 = friction angle

rFigure 2-Qrientation of failure: Mohr-Coulomb criterion

The Joumal of The South African Institute of Mining and Metallurgy

In this paper, the Mohr-Coulomb criterion is used for thefollowing reason. The problem of support design is not theprediction of initial fracturing (at the surface of the excavation)but, rather, the prediction of the total surface deformation,which itself is a manifestation of the deep-seated plasticstrains in the body of the rock under high stress confinements.It is believed that the shear criterion will predict these strainswith sufficient engineering accuracy. It is therefore assumedin this paper that the Mohr-Coulomb parameters can be usedin numerical modelling to predict tunnel-excavation responsein the quartzites shown in Figure 1 at a depth of 4000 m.This approach will be tested by in situ monitoring at a laterstage.

Numerical modelling of tunnel-deformation response

Numerical methods of analysis are based on calculations ofstress versus strain, usually in the form of a computerprogram and, as such, are relevant in cases where theinduced stresses around a tunnel excavation are importantsuch as those featuring in this paper.

A number of computer packages are available for use,and the package selected must be related to the requirementsof the problem requiring analysis. This study required aprogram that can model tunnel support in various ways andthat can accept, without the occurrence of numerical instability,large deformations due to strain-softening rock behaviour.The program FLAC8,which is based on a procedure of finite-difference calculation, is well suited to such an application.

Common mining supports such as cables and shotcreteand their interaction with a rock mass can be modelled byuse of FLAC.The cable element allows the user to specifypre-tensioning, grouting, and the length and spacing of eachcable, either with or without a faceplate. Shotcrete is modelledas a beam element that is, in effect, totally bonded to therock. Support pressure can also be modelled as an appliedpressure to the periphery of the tunnel excavation or to partof it.

All the plasticity models in FLACuse the same formulation.This consists of a yield function and a plastic-flow functionto the deforming model, a shear-yield function, and a non-associated shear-flow rule (Le. dilation can be modelled),which are based on Mohr-Coulomb elasto-plasticity. Thedifference, however, in the strain-softening model lies in theability of the cohesion, friction, and dilation to soften afterthe onset of plastic yield whereas, in the Mohr-Coulombplasticity model. the cohesion, friction, and dilation areassumed to remain constant. The user defines the cohesion,friction, and dilation as piecewise linear functions of theplastic strain. The code determines the total plastic strain ateach time increment, and causes the cohesion, friction, anddilation to conform to the user-defined functions.

Some initial modelling of tunnels of various shapeswithout support at a depth of 4000 m, with the (FLAC)Mohr-Coulomb strain-softening constitutive material havingpebble quartzite parameters, showed that sidewall deforma-tions of up to 1 m are possible. Such deformations wouldpractically close a tunnel or render it unserviceable. Suchorders of closures have already been measured in existingshallower tunnels. For example, a 3 m by 3 m haulage at adepth of 2300 m in argillaceous quartzite with a DCSof90 MPa excavated in the 1980s is still in places experiencingclosure rates of up to 50 mm per week9.

MARCH/APRIL 1996 49 <Ill!


The large sidewall displacements predicted during thisinitial modelling show that a yielding type of support wouldbe required, as proposed initially by Spearing!.

Support design based on numerical modelling

Several papers published overseas and locally have describedthe use of numerical methods in the design of tunnelsupport!()-!2. The basic approach followed by the authors of

these papers was to choose a trial support system, input itsconstitutive behaviour into the numerical model, and check ifthe trial support can bring the excavation into equilibriumwithout over-stressing the support elements. If not, strongeror denser support is required and remodelled.

This approach, although successful, has proved somewhattime-consuming, particularly since a number of differentcombinations of support can be used to stabilize the excava-tion without necessarily indicating the optimum solution. Inaddition, computing times using FLACand a PC486 computercan easily exceed a day.

An alternative approach, one that is presented in thispaper, is to derive from the numerical model a relationshipbetween support pressure on the sidewall and displacementof the corresponding sidewall. This relationship is essentiallythe well-known ground reaction, ground characteristic curve,or required support line4.

With the large deformation capability of FLAC,varioussidewall pressures can be inputted, even those that are toolow to prevent the finite-difference grid from distortingexcessively. The calculated sidewall displacement at stabili-zation can then be plotted against each applied pressure toprovide a curve of pressure versus displacement. Where thepressures are too low, the grid will continue to deform untilthe iterative solution cannot proceed any further owing topoor geometric considerations. The critical pressure that juststabilizes the grid can be determined quickly, and can bereferred to as the pressure at a factor of safety of unity, orthe critical pressure.

The critical pressure at which a tunnel 'collapses' isprobably dependent to some extent on the fineness of thegrid and the boundary conditions. For the study, a quarter-symmetry grid of 2500 zones (50 by 50) was used. Thesymmetry axes were fIXedagainst normal displacements. Theright-hand boundary was placed at a distance of 50 m fromthe centre of the tunnel, which is 11.8 times the equivalentdiameter (4.225 m) of the excavation with an area of 14 m2.This boundary was fIXedagainst horizontal displacements.The top boundary was also placed at a distance of 50 m, buta pressure was applied at this level equal to the overburdenstress at this level.

The horizontal displacement of the tunnel springline wasused to monitor whether the FLACrun was stabilizing, sincethis part of the tunnel periphery was most highly stressedand showed the highest displacements. The runs, however,were always continued for up to 90 000 timesteps or until theFLACgrid became 'unstable' the message returned being badgeometry. Further analysis is required to assess the effects ofthe mesh fineness on the results.

Once the curve of pressure versus displacement has beendefined, together with the critical pressure, decisions can bemade on the allowable displacement, and the corresponding

~ 50 MARCH/APRIL 1996

required pressure can be read off the curve. This pressure canbe referred to as the design pressure. The ratio of the designpressure to the critical pressure is defined as the factor ofsafety of the tunnel excavation.

Once the design pressure has been defined, the pressurecan be manually converted into equivalent forces for cablesor other rock reinforcement. Also, the design pressure can beused in manual computation of the required surface supportbetween the rock reinforcement. For example, shotcreteacting as a beam in bending and shear, or mesh acting as acatenary, can be designed.

The development of pressure-versus-displacement curvesis illustrated in the following case study.

Case study: Horizontal ellipse versus square tunnelprofiles

As part of a study!3, the relative performances, as regardssupport requirements, of a square tunnel and a horizontalellipse had to be compared. Each had a cross-sectional areaof 14 m2 and was excavated at a depth of 4000 m (thereforeat assumed field stresses of 104 MPa vertical and 52 MPahorizontal) in a strain-softening pebble quartzite having theproperties given in Figure 1 and shear and bulk moduli of29 and 33 GPa respectively.

The following relationships were used.

EShear modulus, G = 2(1 + v)

E-

3(1-2v)'Bulk modulus

where E is Young's modulus and u is Poissons's ratio.The rock properties used in the FLACruns were as follows:

Unconfined compressive strength 57.7 MPaBulk modulus 6.7 GPaShear modulus 4.0 GPa.

The moduli values are lower than the laboratory values inorder to reflect the highly stressed rock mass at depth, andcorrespond loosely to typical underground displacements.More reliable estimates of these parameters will be obtainedby in situ monitoring at a later stage.

By use of the FLACprogram (an input file is given in theAddendum), initially without any applied support pressureto the inside of the tunnel excavations, it was possible toidentify the portion of the tunnel periphery that requiredsignificant support to prevent collapse of the model mesh.Various pressures were then applied, and the sidewalldisplacement at the springline level was computed (thespringline is defined as the mid-height of the tunnel sidewall).

The required support lines for each proftle in terms ofsidewall pressure versus horizontal displacement (dilation) ofthe springline are shown in Figure 3. The following can beseen.

~ If the sidewall pressure drops below 250 kPa, thesquare tunnel will not stabilize. If the sidewall pressuredrops below 460 kPa, the elliptical tunnel will notstabilize. These pressures are the critical pressures atwhich the factor of safety of the excavation is assumedtobeunity.



Table I

Comparison of support pressures, showing the effect of tunnel shape

Elliptical tunnel

Total forcekN/(m of tunneQ

4399244417111124

*Pressure-versus-displacement relationships are shown in Figure 3Total horizontal force = Pressure X 3.74 m (square)

= Pressure X 2.44 m (ellipse)Six individual horizontal forces are required for the square to prevent collapse of the tunnel between the forces in the absence of surface support.For the ellipse. four at least are required.

~ With decreasing pressures below 500 kPa, the sidewalldisplacement of the square profile increases so rapidlythat the tunnel becomes dysfunctional.

~ For a given pressure above the highest critical pressure,the sidewall displacement of the elliptical profile is lessthan that for the square.

~ The sidewall pressure for the elliptical tunnel has to beapplied over a height of 2.4 m on the sidewall, comparedwith a height of 3.6 m for the square tunnel.

~ If a design criterionof 500 mm maximumallowable

2000

1800

1800

..D.~ 1400Q...

..J

..J

<

~ 1200etnz0 1000wa:~tntn

~ 800D.

~a:0D. 800D.~tn

400

200

,

\

\

\

\

\

\

\

\

\

\

\

~

\EIliPse---'

\\\

~-~

00

sidewall displacement is adopted, then an appliedpressure of 700 kPa is required for the elliptical tunnel,compared with 1150 kPa for the square tunnel. Thesepressures are the design pressures. In terms of totalhorizontal sidewall force, the elliptical tunnel requires aforce of 1700 kN/m longitudinally, compared with4300 kN/m longitudinally for the square tunnel.

Table I gives the required sidewall support pressures andtotal horizontal forces for various allowable sidewall displace-ments. Also given are the number and value of individual

[[]~

~

Square

EIUp,.

-

400 1200 1400


200 800 1000

SIDEW ALL DISPLACEMENT,4Io(mm)

Figure 3--Support lines required in terms of sidewall pressure versus sidewall displacement

MARCH/APRIL 1996 51 ....


forces required to provide the total horizontal force and toprevent collapse of the tunnel between these forces. in theabsence or ineffectiveness of surface support such as meshand lacing or shotcrete. It can be seen from Table I that thetotal horizontal support force is higher for the square tunnel.

Since the cost of tunnel support is related directly to therequired force capacity. it can be inferred that the cost ofsupporting the square tunnel could be approximately 2.5times the cost of supporting the ellipse in order to realize thesame level of sidewall deformation.

Design methodology

Based on the case study. the following site-specificmethodology is recommended.(1) By direct measurement. extrapolation from similar

cases. or three-dimensional elastic modelling. obtainan estimate of the field stresses acting prior to thetunnel excavation.

(2) Obtain the strain-softening behaviour of the specific

rock by carrying out triaxial compression tests in aservo-controlled machine that tests at a constant strainrate. and then fit equivalent Mohr-Coulomb parametersto the laboratory-obtained curves of stress versusstrain.

(3) Model the tunnel excavation using the two-dimensionalFLACprogram. and run it initially without any supportin the large strain mode. This procedure will allowidentification of that part of the excavation peripherythat will require support to prevent the FLACgrid fromcollapsing through excessive distortion.

(4) Apply various pressures to the part of the excavation

periphery identified in (3). which gives thecorresponding wall displacement at stabilization. Inthis way. a pressure-versus-wall displacement can bedeveloped that is the numerically derived groundcharacteristic curve.

(5) Choosing an allowable maximum deformation level andusing this curve. calculate equivalent force levels invarious chosen directions and at various longitudinalspacings in the tunnel. These forces represent therequired working loads of the rock-reinforcementelements being considered. It should be noted that theinstalled support must be able to accommodate thedeformation level selected without failing (Le. thefactors of safety of the individual support elementsmust all be greater than unity).

(6) In the same way. compute the shotcrete thicknesses

and mesh strengths. using the design pressure derivedfrom the characteristic curve and the spacings of therock-reinforcement elements.

(7) Undertake final FLACruns in the usual manner. using

cables and shotcrete dimensioned on the basis of thecharacteristic curve. Such runs will be far fewer thanotherwise required since the required support pressure/force levels have already been determined. and moreoptimum cost-effective solutions can be obtained.

~ 52 MARCH/APRIL 1996

Conclusions

Based on this paper. the following conclusions can be drawn.

(1) The use of numerical stress analysis in the design oftunnel support has traditionally involved a trial-and-errorbasis. which may not necessarily give an optimum design.

(2) The approach presented in this paper requires thederivation of the support line or the ground reaction/characteristic curve in the form of sidewall pressureversus horizontal displacement on the tunnel sidewall.The pressure required to control the displacement to apredetermined level can be read off the derived curve.from which the equivalent required cable forces can becalculated. as well as the shotcrete thickness and meshstrength.

(3) The numerical model is based on uniform. elastic.strain-softening rock behaviour in which largedeformations due to elastic over-stressing are expected.A comparison should be made with field results beforeconclusions are drawn as to the general applicability ofsuch an approach.

Acknowledgements

The idea for this paper came after the application ofnumerical analysis to a particular mining-tunnel problem13.The permission of the Gold & Uranium Division of the AngloAmerican Corporation and of Steffen. Robertson & Kirsten topublish this paper is gratefully acknowledged.

The assistance of Mr M.I. Forsyth of Mining StressSystems (Pty) Ltd. Skeerpoort. South Africa. in conductingthe computer runs is gratefully acknowledged.

References

1. SPEARING,A.).s. A possible fast track tunnelling method at depth.

Tuncon '92-Design and construction o/tunnels. Maseru (Lesotho),

1992. pp. 3-8.2. INTERNATIONAL SOCIETY POR RoCK MECHANICS COMMISSION ON STANDARDIZATION OF

LABORATORYANDFIELDTESTS.Suggested methods for determining the

uniaxial compressive strength and deformability of rock materials.

Int.! Rock Mech. Min. Sci.. vol. 16. 1979. pp. 135-140.

3. KERSTEN,R.W.O. Anglo-Vaal. private communication, 1992.

4. BRADY,B.H.G., and BROWN,E.T. Rock mechanicsJor underground mining.

London, George Alien & Unwin, 1985.

5. STACEY,T.R., and DE)ONGH,C.L. Stress fracturing around a deep level bored

tunnel.! S. Aft. Inst. Min. Metall., vol. 78. 1977. pp. 124-133.

6. STACEY,T.R. A simple extension strain criterion for fracture of brittle rock.

Int.!. Rock Mech. Min. Sci & GeomechAbstr., vol. 18. 1981. pp. 469-474.

7. STACEY,T.R. Boring in massive rocks-rock fracture problems. Pretoria,

South African National Council on Tunnelling Seminar, 1989.

8. ITASCA.FLAC-fast Langrangian analysis of continua. Version 3.0/3.2,

1992.9. LEWIS,A. Anglo-Vaal, private communication, 1993.

10. SPEERS,C.R. Support for tunnels subjected to changing rock loads: a

comparison of design methods. Tunnelling and Underground Space

Technology, vol. 7, no. 1. 1992. pp. 25-32, England.

11. KIRSTEN,H.A.D., and BARTLETT,P.). Research-determined support charac-

teristics and support-design method for tunnels subject to squeezing

conditions.j S.1fr.lnst. Min. Metall., vol. 92.1992. pp. 195-214.

12. SPEERS,C.R. Application of numerical methods in the design of primal)'support for rock tunnels. Tuncon '92-Design and Construction 0/Tunnels. Maseru (Lesotho), 1992. pp. 163-171.

13. SPEARING,A.).s., SPEERS,C.R., and FORSYTH,M.I. Design concepts and

support considerations for highly stressed tunnels in deep hard-rock

mines.j S. Aft. Inst. Min. Metall., vol. 94.1994. pp. 93-102. .

The Jaumal af The South African Institute af Mining and Metallurgy


Addendum: FLAC simulation data file (for reference)

titOptimal tunnel shape: Run 4-Ellipse,a=2.55m b=1.75mgr 50,50model ss*; Generate geometry*gen 0,-4000 0,-3950 50,-3950 50,-4000 rat 1.05 1.05 ;=1,51 j=l,51;ellipsetable 1 0.0, - 3998.2500 0.25,-3998.2584 0.5,-3998.2840 0.75,-3998.3274table 1 1.0, - 3998.3902 1.25,-3998.4747 1.5,-3998.5848 1.75,-3998.7272table 1 2.0~ - 3998.9144 2.25,-3999.1765 2.4,-3999.4086 2.5,-3999.6552table 1 2.5~, - 4000.0gen table 1t ~orrect coordinates,of ~tart and end points1nl x=O.O y=-3998.25 1=1 J=7ini x=2.55 y=-4000.0 i=10 j=lmark i =1 j=7mark i =10 j=1*windo -1,6 -4001,-3994*iset boundary conditions

fix x i=1fix x ;=51fix y j=l*apply syy=-106.65e6 j=51*set grav=10*r.Set properties

prop bu=6.666e9 sh=4e9 dens=2700 ctab=2 ftab=3 dtab=4 tens=16e6tab 2 0,17.5e6 0.001

616.0e6 0.002~13.5e6 0.004,11.0e6 0.006,7.5e6

tab 2 0.008~4.0e6 O. 1~0.5e6 0.01~,0.5e6 0.1~0.00001tab 3 0,27.~ 0.001,32.~ 0.002

632.5 0.004

i32.~ 0.006,32.5

tab 3 0.009,32.5 0.016

32.5 O. 15~32.5 O. ~32.5tab 4 0,0 0.001,20 O. 02,20 0.00~~20 0.000,20tab 4 0.008,6.5 0.01,5.5 0.015,5.~ 0.1,5.5*iInitialise stress regime

ini syy=-108.0e6 var=06

1.35e6ini sxx=-54.0e6 var=O, .675e6ini szz=-54.0e6 var=0,0.675e6*iSet histories

his nsteps=10his unbalidisplacements and velocities in h/w,f/w and s/wnist yd i=l j=7hist yv i=1 J=7hist xd i=10 j=lhist xv ;=10 J=livert;cal and horizontal stresses in h/w,f/w and s/wnist sxx ;=1 j=7hist syy i=l ~=7hist sxy i=l J=7hist sxx ;=10 j=lhist syy ;=10 ~=1hist sxy i=10 J=l*

The Journal of The South African Institute of Mining and Metallurgy MARCH/APRIL 1996 53 <Ill


iSolve to equilibrium

solve step=30 force=100 clock=1000*;Initialise displacements*set 1argeini xdis=O ydis=Omo null reg=1,1;Run 200 steps plastically to simulate failure before support is effectivesolve 5=200pl hi 4 holdpl hi 2 holdpl gr red plas hold*solve s=10000 f=100 c=1000*tit*Ellipse:unsupported*ret*apply Rressure=2.0e6 from 8,6 to 10,1apply xforce=137ge3 i=8 j=5apply yforce=1157e3 i=8 J=5apply xforce=173ge3 i=10 j=3apply yforce=466e3 i=10 j=3titEllipse:4 s/w normal forces each 1800kNpl gr red apply yel holdsolve s=10000 f=100 c=1000save ellip48.savret;Put in cablesstr c b g 8

65.

e 4.29A-3996.98 se=12 p=2str c b 9 1 3 e 5.3~ -3998.834 se=12 p=2*str p=1 a=864e-6 e=260e9 kb=le9 sb=1382e3 yi=362e3 *Y32 yield,483kN ult*str p=2 a=491e-6 e=200e9 kb-le9 sb=1194e3 y;=221e3 *Y25 yield 221kN~295kN ult*str p=2 a=442.2e-6 e=198e9 kb=le9 sb-1392e3 yi=795e3 *3 strands 15 Immstr p=2 a=1031.8e-6 e=198e9 kb=le9 sb=3142e3 yi=1855e3 *7 strands 15 7mm*str p=l a=201e-6 e=200e9 kb=1e9 sb=1194e3 yi=91e3 *Y16 yield 91kN,121kN ulttitEllipse:4 s/w reinforcing (7/15,7mm strands), each 1855kN capacitypl hold grid red cab yellsolve 5=9000 f=100 c=1000save ellip47.savret

~ 54 MARCH/APRIL 1996 The Joumal of The South African Institute of Mining and Metallurgy

Documents

The design of tunnel support in deep hard-rock mines under