Analysis of rock reinforcement using finite difference methods

Computers and Geotechnics 5 (1988) 1 23-149

ANALYSIS OF ROCK REINFORCEMENT

USING FINITE DIFFERENCE METHODS

B. Brady and L. Lorig Itasca Consulting Group, Inc. Minneapolis, Minnesota U.S.A.

ABSTRACT

Large-scale rock reinforcement offers the prospect of flexible and effective control of rock displacements around large open mine excavations. Although the technology of rock reinforcement is well developed, reinforcement analysis and design methods are not. Analysis is presented here for the mechanics of two models of rock reinforcement, one based on local action of a reinforcing element at a slipping joint, the other on spatially-extensive action in rock subject to diffuse deformation. The principles of some finite difference methods of analysis of stress and displacement are outlined. The performance of a code modeling spatially-extensive reinforcement is examined in a parameter study of stope hangingwall reinforcement using long, grouted cables.

INTRODUCTION

Ore extraction

based on mining methods

crater retreat stoping makes conflicting

demands on the mechanical

host rock mass.

On one hand,

with operational

safety and mine stability.

mize economic ore recovery,

such as longhole open stoping or vertical

it is necessary

frequently

tions of incipient

instability

volves achievement

of an economic

performance

to mine under conditions On the other,

of the

consistent

it is essential

to maxi-

implying mining large openings under condi-

of the near-field compromise

host rock.

Successful mining in-

between local ground control and ore re-

covery. In the past, the measures

available

to a mine design engineer to achieve effec-

tive ground control and to maximize ore recovery cious design of stope and pillar layouts,

in a stoping system involved judi-

development

of an appropriate

stope and

123

Computers and Geotechnics 0266-352X/88/S03.50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain

124 pillar extraction

sequence,

or high stope spans.

and use of various types of backfill

Although generally

cedures tend to be rather inflexible stope block design to subsequent Recently, forcement

considerable

cations of rock reinforcement

have subsequently

[i]), these pro-

due to the long lead times from

in mining has been concentrated

in open stoping operations. were in relatively

led to assessment

Previously,

small-scale

on rock reinthe main appli-

operations

such as rock

and some larger-scale

applications

and the crowns of cut-and-fill

of open stope crown pillars

long

extraction.

bolting of crowns and sidewalls of access drifts, in crusher excavations

(e.g., Pariseau

in application,

interest

for ground control

successful

in supporting

stopes

[2].

These applications

of long cable or tendon reinforcement

[3] and long, down-dip stope spans

steel tendons up to 20m long, with ultimate

[4].

for control

In these cases,

load capacity of 250-500 kN, may be em-

ployed. A notable aspect of large-scale nology has developed successful

translation uncertain

is that the operating

quite rapidly, with several field demonstrations

performance.

forcement mechanics

rock reinforcement

As indicated by a recent review

has not developed

of effective

exercises.

tech-

[3, 4, 5] of

[6], analysis of rock rein-

as rapidly, making both design optimization

reinforcement

and

designs from one mine site to another somewhat

The following material

is intended as a contribution

to recti-

fying this situation.

FORMAL ANALYSIS OF ROCK REINFORCEMENT

The function of rock reinforcement

is to mobilize

host rock mass which act to resist deformation

MECHANICS

loads in the interior of the

of the medium.

In performing

this

125 function, the reinforcement serves the purpose of controling displacements in the rock interior and at excavation surfaces.

An appropriate analysis of rock re-

inforcement must take due account of the way in which loads are mobilized in reinforcement elements, by relative displacement between rock units and components of the reinforcement. There are several different types of reinforcement, designed to operate effectively in a range of ground conditions.

One type is represented by a reinforcing bar

or bolt fully encapsulated in a strong, stiff resin or grout.

This system is charac-

terized by the relatively large axial resistance to extensions that can be developed over a relatively short length of the shank of the bolt and by the high resistance to shear that can be developed by an element penetrating a slipping joint.

A second

type of reinforcement system, represented by cement-grouted cables or tendons, offers little resistance to joint shear, and development of full-axial load may require deformation of the grout over a substantial length of the reinforcing element.

These

two types of reinforcement are identified, respectively, as local reinforcement and spatially-extensive reinforcement. The current analysis of reinforcement mechanics is an extension of the work by St. John and Van Dillen [7].

The notions pursued in that work reflect the earlier

studies of Fairhurst and Singh [8], Bjurstrom [9], and Haas [i0]. part of the analysis is provided by Lorig [ii].

A presentation of

The analysis is developed here in a

manner compatible with its implementation in any of the finite difference codes described subsequently.

126 Local Reinforcement In the analysis

of local reinforcement,

ized in the reinforcement involves deformation procedure that,

justified

attention

is focused on the loads mobil-

element by slip and separation at a joint.

of an "active length" of the element, by the experimental

in discontinuous

observations

rock, reinforcement

as shown in Fig.

of Bjurstrom

deformation

The analysis I., a

[9] and Pells

is concentrated

[12],

near an active

joint. DIRECTION OF SHEARING

/,// / /

/' /

DISCONTI

/

IAu~ FIGURE i.

Local reinforcement

action through an active length of bolt

The conceptual model of the local operation Fig 2(a), where local load and deformation springs,

of the active length is shown in

response

is considered

in terms of two

one parallel to the local axis of the element and one perpendicular

to it.

When shear occurs at the joint,

as shown in Fig. 2(b), the axial spring remains

parallel

of the active length, and the shear spring is taken

to the new orientation

to remain perpendicular

to the original axial orientation.

the joint are accompanied

by analogous

changes

Displacements

in the spring orientations.

normal to

127 Direction of Shearing

/

/

,,,/

~ioI Spring i

,/ ~

~xiol

Shear Spring

,,'

/ /

~

/

Conceptual model for local action of reinforcement

The loads mobilized

According

Spring

Shear Spring

(b)

(a)

placements

/"

,,/

i'

FIGURE 2.

t ,'A Us , ,'"~

in the element by local deformation

by the axial and shear stiffnesses to St. John and Van Dillen

at a slipping joint

are related to the dis-

of the bolt (K a and Ks, respectively).

[7], these can be estimated

from the expressions

!

developed

by Gerdeen at al.

[13]:

where k = [(112) Gg Ebl(d2/d I - 1)] 112

Eg = Young's modulus of grout, Gg = shear modulus of grout,

Ka = ~ k d I

(i)

K s = E b I B3

(2)

128 E b = Young's modulus of bolt,

d2,d I = hole and bolt diameter, respectively,

K

1/2

2 Eg K

, and (d2/d I - I)

I = second moment of area of the reinforcing element.

The qualification on the application of these expressions is that they appear to overestimate the stiffnesses slightly [ii].

If possible, K a and K s should be deter-

mined by laboratory testing on specimens of the relevant rock, grout and reinforcing material. The mobilized axial and shear forces are both assumed to approach limiting values asymptotically.

The so-called continuously-yielding model is described for axial

behavior (with an analogous expression for shear) by the expression

~F a = K a l~Ual f(F a)

(3)

where AF a is an incremental change in axial force due to an incremental axial displacement ~Ua, and f(F a) is a function defining the load path by which F a approaches its ultimate a value, Pult"

The expression for f(Fa) is conveniently defined by

129 a a a 2 f(F a) = IPult - Fal (Pult - Fa)/ (Pult)

(4)

The values for the ultimate axial and shear loads that can be sustained ideally should be determined ideally from appropriate laboratory tests on the rock-groutshank system, since it depends on factors such as grout properties, roughness of the grout-rock interface, adhesion between the grout and the shank, and the thickness of the grout annulus.

If measured values for the ultimate loads are not available, ap-

proximate values can be estimated (Bjurstrom [9], St. John and Van Dillen [7]) from the expressions: s Pult = 0.67 d12 (o b Oc)1/2

(5)

a Pult = ~peak ~ d2 L

(6)

where o b = yield strength of bolt, o c = uniaxial compressive strength of the rock, Tpeak = peak shear strength of grout or grout-rock interface, and L = bond length.

In Eq. (6), it is assumed that shear occurs at the hole boundary.

If shear oc-

curs at the bolt-grout interface, the appropriate diameter in Eq. (6) is d I. Considering Eqs. (i-6), it is observed that any increment of relative displacement at a joint can be used to determine incremental and then total forces parallel and transverse to the axis of the reinforcement element.

From the known orientation

of the element relative to the joint, these forces can be transformed into components acting normal and transverse to the joint.

In this form, they can be introduced into

130 an appropriate

finite difference

code which simulates the behavior of a jointed rock

mass.

Spatially-Extensive

Reinforcement

In a rock mass subject to a diffuse mode of deformation, model of reinforcement model.

is mechanically

Because local resistance

nificant,

a one-dimensional

performance

of a reinforcing

reinforcement

representation

constitutive component.

equivalent

is considered

model is adequate for describing Following

of the reinforcement

involves division of the complete cable or tendon of equivalent masses to the nodes of the rein-

of a segment

to the shear stiffnesses,

is represented

by a spring of stiffness

limited by a plastic yield condition.

of the spring representing

the elastic deformability

the limiting shear resistance of the slider representing pacity of either the grout annulus

itself,

Thus, characterization

formal description

com-

after the fashion shown in Fig. 3. The finite difference

Axial extension

rock contact.

the axial

the general notions of action of

tion between the tendon and the rock is modeled by a spring-slider stiffness

insig-

of separate segments of a long reinforcement

into separate units, and assignment forcement.

than the local deformation

to shear in the reinforcement

[7], the deformation

ponent may be considered,

more appropriate

the spatially-extensive

of the performance

unit, with the

of the grout,

the ultimate

the tendon-grout

Interac-

contact,

and with

shear load caor the grout-

of the system requires

of the elastic and limiting performance

of each component

of the

system. The performance

of the tendon segment

For the elastic-plastic ties and dimensions

response,

of the tendon:

in axial extension

the bar stiffness

is shown in Fig. 4(a).

is related simply to the proper-

131

EbA Ka ='~-'-

(7)

where A and L are the cross-sectional area and length of the tendon segment.

reinforcing ~ elmnt (mteel)~

/

_~~grout

annulos

///~// / /

/ ~-

" /

~

~slider

(cohcsive

a x i a l s t i f f n o s s of steel

/

/~shear

stiffness of qrout

f

FIGURE 3.

~e

Conceptual model of spatially-extensive reinforcement in rock subject to diffuse deformation

yield condition is related directly to the yield strength of the tendon and

the cross-sectional area.

If, after yield, the segment is subjected to a phase of

unloading, the unloading stiffness is taken to be equal to the loading stiffness, as sho~

in Fig. 4(a). The elastic-perfectly plastic performance of the tendon segment represents the

simplest possible constitutive model.

With minimal increase in complexity, a kine-

matic hardening model or a continuous weakening model analogous to that described by Eqs. (3) and (4) could be introduced.

132

/ ~

ond

/ -,-t U

/

~ O~

gl 2{

,.

u D i s p l a c e m e n t of C a b l e R e l a t i v e to S u r r o u n d i n g Material

Relative

Segment

(a)

FIGURE 4(a). 4(b).

Displacement

Ends

(b)

Axial performance of a reinforcement segment; Shear performance of grout annulus between tendon and borehole surfaces

The elastic-perfectly plastic performance of the grout annulus is in Fig. 4(b).

of

represented

As a result of relative shear displacement u a between the tendon sur-

face and the borehole surface, the shear force mobilized per unit length of cable is related to the stiffness Kbond; i.e., fa = Kbond Ua

Usually, Kbond would be measured directly in laboratory pull-out tests. tively,

(8)

Alterna-

it may be calculated [7] from the expression:

2~ Gg

Kb°nd

in (l+2t/d I )

where t is the thickness of the grout annulus.

(9)

133 The ultimate load capacity of the grout is defined by Eq. (6), with length L of unity, and Tpeak given by ~peak = ~I Qb

(i0)

where ~I is approximately one-half of the smaller of the uniaxial strengths of the grout and the rock, and Qb is a factor defining the quality of the bond between the grout and the rock. For perfect bond quality, Qb = I.

Calculation of the loads generated in reinforcement requires determination of the relative displacement between the rock and a node of a reinforcement segment.

Con-

sider the constant strain triangular zone shown in Fig. 5(a), with x-components of displacements Uxi, for example, at the corners i(i=l,3).

The natural coordinates n i

of a reinforcement node p lying within the triangle are given by the relative areas of the triangular areas defined in Fig. 5(b).

Thus,

2 r~grldpoint constant strain / finite difT.~ /

i

\ \

~reinforcement nodal point

(a) FIGURE 5 ( a ) .

B a s i s of n a t u r a l

1

(b) coordinates

for interpolating

rock displacements

1 34

nl

Ai = -A

~ = 1,3

where A is the area of the triangle with corners

The displacement at the corners,

ux

where summation

An expression

1,2,3.

uxP at node p is interpolated

using the natural coordinates P

(11)

linearly from the displacements

as weight factors;

= n i Uxi

i.e.,

i = 1,3

(12)

is implied on repeated subscripts.

analogous

to Eq. (12) relates the y-component

of rock displacement

at a

segment node to the zone corner displacements. In the finite difference each calculation new orientation displacement

analysis,

cycle to calculate

incremental

incremental

nodal displacements,

of an element can be determined.

at a node can then be determined

node and the absolute displacement Having determined

forms of Eq. (12) are used in

The axial component of relative

from the absolute displacement

of the adjacent

ing condition defined by Eq. (i0). terface at a node is distributed

of a

rock.

the relative nodal displacement,

from Eq. (8) and the active length adjacent

from which the

the axial force is determined

to the node, taking account of the limit-

P The force F~, Fy mobilized at the grout-rock

in-

to the zone corners using the natural coordinates

of

the node as the weight factor--i.e., P Fxi = n i F x

where Fxi are forces assigned to the zone corner.

etc.

(13)

135 EXPLICIT FINITE DIFFERENCE METHODS

The formulations of reinforcement mechanics described previously have been developed specifically for inclusion in dynamic relaxation, finite difference methods of analysis of deformable body problems. on the original work of Cundall [14].

The methods used by these authors are based

Recent developments of the methods for discon-

tinuous rock masses are described by Cundall [15] and for continuous media by Cundall and Board [16].

The computer codes implementing the developments are identified as

MUDEC [17], a two-dimensional distinct element code, 3DEC [15] a three-dimensional distinct element scheme (both of which are applicable to discontinua), and FLAC [16], a two-dimensional code for analysis of continua subject to large-strain, elasto-plastic or strain-softening deformation. Each of the codes defined above is an explicit finite difference technique for solution of the governing equations, taking account of the initial and boundary conditions and the constitutive equations of the medium.

An explicit procedure is one

in which the problem unknowns can be determined directly at each stage from the difference equations,

in a stepwise fashion, from known quantities.

Advantages of such

an approach are that large matrices are not formed, making limited demands on computer memory requirements, and that locally complex constitutive behavior such as strain softening does not result in numerical instability.

An offsetting disadvan-

tage is that the iterative solution procedure may sometimes consume an excessive amount of computer time in reaching equilibrium. A finite difference scheme is developed by dividing a body into a set of convenient, arbitrarily-shaped quadril~teral zones, as shown in Fig. 6(a).

For each repre-

sentative domain, difference equations are developed based on the equation of motion

136 . . . .

~ (a) FIGURE 6.

scuom no tm uroitn (b)

Finite difference grid (a), and representative zones involved in contour integrals for a gridpoint (b)

and the constitutive equations.

Lumping of part of the mass from adjacent zones at a

gridpoint or node, as implied in Fig. 6(b), and a procedure for calculating the outof-balance force at a gridpoint, provide the technique for establishing and integrating the equations of motion. The Gauss Divergence Theorem is the basis of the method for determining the outof-balance gridpoint force.

In relating stresses and tractions, the theorem takes

the form: 8°ij = lim -1 ~xi A~0 A

f

cij nj as S

where x i = components of position vector, oij = components of stress tensor, A = area, dS = increment of arc length, nj = unit outward normal to dS.

(i = 1,2)

(13)

137 An approximate expression may be then established for the P~S of Eq. (13) involving summation of products of tractions and areas over the sides of a polygon, to yield a resultant force on the gridpoint.

The differential equation of motion is:

P

a~i

aaij

~t

~axj + Pgi

(14)

aij nj dS + gi

(15)

and introducing Eq. (13) yields

aui

1 [

= at m

S

where m = pA.

If a force F i is applied at a gridpoint due, for example, to reactions mobilized by reinforcement

or contact forces between blocks, Eq. (15) becomes

8ui = ~ (Fi + I sij nj dS) + gi 8t m S i = - Ri

m

+ gi

(16)

Equation (16) indicates that the acceleration at a gridpoint can be calculated explicitly from an integration of the surface tractions over the contour of the region surrounding the gridpoint.

138 When the acceleration equations

of a gridpoint

can be used to calculate

time interval

has been determined,

gridpoint velocities

central difference

and displacements

after a

At:

.(t + &t/Z) .(t - At/2) ui = ui +

1

F

l Ri _ [ -- + gi J At m

(17)

(t + At) (t) .(t + At/2) xi = xi + ui At

hen

a pseudo-static

problem

is being analyzed,

(18)

viscous d ~ p i n g

terms are introduced

in Eqs. (17) and (18). Calculation strain i n c r ~ e n t s cr~ents

of changes

and their introduction

are determined

Divergence

in the state of stress proceed through calculation in the constitutive

from the velocity gradients.

equations.

of

Strain in-

Noting that, from the Gauss

Theorem,

~xj

A

u i nj dS

(19)

s

and that the M S

of Eq. (19) can be evaluated as a su~mlation over the contour of a

polygon surrounding

a gridpoint,

strain increments

are determined

from the expres-

sion

Aeij = ~

8xj + 8x i

i At

(20)

139 Finally, the stress increment in the time interval At is calculated directly from the existing state of stress, the strain increments and the material constants k s for the medium, through an appropriate constitutive equation:

Aoij ffi f(Aeij, oij, k s)

(21)

In the codes identified above, the form of the constitutive function f may represent isotropic and transversely isotropic elasticity, Mohr-Coulomb plasticity, strain softening, and anisotropic plasticity defined by ubiquitous joints. Equations (13-21) are solved sequentially through a series of timesteps of duration At.

Thus, the procedure is a time-based integration of the governing equations,

to yield the displacement and state of stress at a set of collocation points in the medium.

It is noted that, for the spatially-extensive reinforcement discussed previ-

ously, simple equations analogous to each of Eqs. (13-21) are developed for reinforcement nodes, and forces mobilized by relative displacement between rock and tendon are introduced in Eq. (16).

For local reinforcement, the equations defining the

action of reinforcement are applied to determine surface forces, also introduced in Eq. (16).

A DEMONSTRATION PROBLEM

The solution of a simple problem is provided to demonstrate the application of a model of rock reinforcement,

implemented in the code called FLAC.

The particular

model of reinforcement is the spatially-extensive one, corresponding to long, grouted, untensioned cable anchors.

140 The problem considered static stress field,

is a circular hole of Im diameter,

of magnitude

i0 MPa.

The problem parameters

erate a zone of failed rock around the excavation could be identified

readily.

excavated

in a hydro-

were chosen to gen-

so that the effect of reinforcement

The rock mass was modeled as an elasto-plastic

with shear and bulk elastic moduli of 4 GPa and 6.7 GPa, respectively,

medium,

and Mohr-

Coulomb plasticity defined by cohesion of 0.5 MPa, angle of friction of 30 ° , and dilation angle of 15 ° . oriented steel cables,

Reinforcement

of the medium consisted of a series of radially

of 15=~ diameter,

grouted into 50m~-diameter

had a Young's modulus of 200 GPa, and a yield load of 1 G N . Kbond and Sbond were 45 GN/m/m and 94 kN/m.

holes.

The steel

Values assigned to

These properties

with a Young's modulus of 21.5 GPa and a peak bond strength

correspond

to a grout

(~peak) of 2 MPa.

The problem was analyzed as a quarter plane, taking account of the symmetry about both the x- and y-axes.

The near-field

problem geometry

is illustrated

in Fig.

7, where the extent of the failure zone that develops both in the absence and presence of the reinforcement placement

is also indicated.

around the excavation

tion near-field

rock is both unreinforced

forcement.

for the unreinforced Examination

Independent

analysis can

case corresponds

closely to

of the distribution

of the

stresses o r and o t indicates the function of the radial rein-

It generates

a higher gradient

a higher magnitude

in the o t distribution.

transition closer to the excavation duced.

and reinforced.

solution given by Bray [18].

radial and tangential

of stress and dis-

are shown in Fig. 8 for the cases where the excava-

confirm that the stress distribution the closed-form

The distributions

of o r in the fractured zone, resulting

in

The effect is to shift the elastic-plastic

boundary--i.e.,

the zone of yielded rock is re-

141 It is recognized that the density of reinforcement used in this demonstration problem is greater than would be applied in mining practice.

Nevertheless,

it con-

firms, qualitatively, the mode of action of reinforcement and suggests that even small changes in boundary displacements induced by reinforcement may imply significant improvement in ground control.

I

I

~

l

grouted

I?

i

cabli~/1

anchors/F

•

!%/i

(a) FIGURE 7.

(b)

Yield zones about a circular excavation without (a) and with (b) reinforcement (. = yield)

|

142 1.6

9

1.,~ 1.2

Unreinforced

E 1.C "u --~a.0 " 8

/ ~ . . . . O't R e i n f o r c e d " " " L~ ~ : ~ = - ~ ~e ~ Infor'ced

45 0 . 6

Kzj~F

e

(b)

~E6

:

Z

• _~ 5 ~ Unreinforced ~ 4 x~u,,,~,,,,~, ~

O'r U n r e i n f o r c e d

"~

RO.4

"~ 2

0.2

n-

O

i

I 0"3

0'1

i

I 0"5

i

I 0.7

i

I 0"9

I

0

i

I 0"3

0"1

Radial distance

FIGURE 8.

i

I 0"7

i

3, 0"9

Distributions of stress (a) and displacement (b) around a circular excavation for unreinforced and reinforced near-field rock

Control of stope hangingwall tions of large-scale

REINFORCEMENT

spans is one of the primary prospective

rock reinforcement.

problems are properly analyzed

difficult

such analyses may be fairly ex-

to interpret.

Thus, a scoping study based

on two-dimensional

plane analysis usually should be considered

to a comprehensive

three-dimensional

[4].

the stope hangingwall

problem.

reinforcement

study

The problem geometry and field stresses are shown in Fig.

The inclined open stope had a down-dip

20m strike span.

as a necessary prelude

analysis of a stope wall reinforcement

The problem considered here resembles reported by Greenelsh

applica-

Although many stope wall reinforcement

in three dimensions,

pensive to execute and sometimes

In this study,

span of about 90m, a width of 15m, and a

the field principal

of section are 27 HPa and 18 MPa, respectively stope.

i

Radial distance

STOPE HANGINGWALL

9.

I 0"5

The rock mass properties were taken as:

stresses

in the vertical plane

normal and parallel to the dip of the

T

143 18 WPo

/ 27 MPo

/ FIGURE 9.

r /

Problem geometry for stope hangingwall reinforcement study

cohesion

9.6 MPa 25 °

dilation angle tensile strength shear modulus bulk modulus

i0 ° 0 12 GPa 20 GPa

The purpose of this study was to determine what layout of hangingwall reinforcement was likely to be most effective in hangingwall control.

With this in mind, foul

analyses have been conducted for the problems indicated in Fig. i0.

(The inclined

stope geometry has been rotated to vertical for convenience in presentation.) ure 10(a) is the reference condition, with no stope wall reinforcement. shows the zone of failure around the stope.

Fig-

The plot

The reinforcement represented in Fig.

10(b) resembles the layout used in the field study [4], except that six tendons in each fan of reinforcement are employed here, compared with eight (in four boreholes) used in the field test.

144 The reinforcement performance

pattern shown in Fig. 10(c) is designed to assess the

of reinforcement

oriented predominately

compared with the radial orientation length of tendon reinforcement ces in performance

from the stope wall,

in the plane analysis was 150m, so that any differen-

from headings developed

in the boundary

forcement was required

FIGURE i0.

in the hangingwall

the advantage

in the immediate hangingwall,

(a)

In Fig. 10(d), the tenside of the stope,

and outside the zone of failure surrounding

pattern was intended to determine

reinforcement

to the stope wall,

For both cases (b) and (c), the

should be related to tendon orientation.

dons were installed

inforcement

in Fig. 10(b).

subparallel

the stope.

This

of a more uniform distribution

of re-

compared with the local concentrations

in cases (b) and (c).

10m

of

About 200 meters of tendon rein-

for this layout•

(b)

(c)

(d)

Four cases to assess hangingwall reinforcement: (a) no reinforcement, showing yield zones; (b) radial layout; (c) longitudinal layout; and (d) uniform distribution

145 The properties Kbond = 4.5 GN/m/m,

of the reinforcement

used in the analysis were E b ffi 200 GPa,

and Sbond = 500 kN/m.

tendon in a 50mm-diameter

borehole,

These correspond

to a 15mm-diameter

steel

with a grout of 20 MPa uniaxial compressive

strength. It is suggested

that the effectiveness

of the reinforcement

may be evaluated

from the relative control exercised by the reinforcement

on displacements

hangingwall

deflection over the stope

surface.

Figure Ii is a plot of hangingwall

span for the cases (a) through (d) illustrated hangingwall,

the analysis

distribution

uneven shape of the wall deflection

However,

subject to large deflection mid-helght

can be taken as an index of failure.

of wall deflection

reinforcement

tern being marginally more effective. of the hangingwall.

are remarkably

the existence

similar, with the radial pat-

this pattern

is completely

ineffective

control rock

failure may occur at about the

consistent with the observed

field behavior of the host rock mass for the field study [4]. flection for the uniform distribution

sequence

in each case of zones of destressed

This is qualitatively

The

for the radial reinforce-

Both have achieved quite substantial

implies local hangingwall

of the wall span.

and it is in-

curve reflects the five-stage extraction

The distributions

ment and the longitudinal

For the unreinforced

indicated that the wall rock was destressed,

ferred that this displacement

used for the stope.

in Fig. I0.

at the

of wall reinforcement in controlling

The plot of wall de-

[case (d)] indicates

hangingwall

displacements.

that

146 o

+

70

~/~,~

Bo

design

<>design(c) A design(d)

90

t 0

i

I

20

a &O

E

t

f

60

80

distance from stope base (m) FIGURE ii.

Hangingwall deflections for wall reinforcement patterns defined in Fig. 10

Three notable observations may be recorded concerning these analyses.

First,

the density of reinforcement (measured, for example, in kg of steel per m 3 of rock) is quite low in these mining applications, compared with what would be applied around civil excavations subject to extensive failure of wall rock.

Second, and somewhat

surprisingly, the relatively uniform distribution of sparse reinforcement is quite ineffective in achieving stope wall control.

Finally, it appears that, where rein-

forcement is necessarily sparse on average, concentration into properly selected zones may maximize the value of the installed reinforcement in achieving control of wall rock displacements. Figure 12 is a plot of the calculated distribution of axial force along the tendon marked A in Fig. 10(b).

The uneven distribution reflects the fairly coarse dis-

147 cretization

of the problem domain.

this distribution

However,

it is interesting

to compare the form of

of mobilized tendon load with that given [4] for measured

forces in

the field study. SO

20

-

0

0

FIGURE 12.

2

4

Distribution

6

B

~0

12

14

of axial force along a reinforcing

tendon

DISCUSSION AND CONCLUSIONS

This paper has shown how two different mechanisms be described behavior.

analytically

and incorporated

The performance

of rock mass reinforcement

into finite difference models of rock mass

of the model of one particular

type of rock reinforcement,

based on long steel cables or tendons grouted into boreholes, by reference to a simple excavation wall reinforcement

problem.

a field reinforcement

has been demonstrated

shape, and a more realistic open stope hanging-

The behavior of the rock predicted

study appears to be qualitatively

field behavior of the host rock mass, providing modeling technique.

can

from the analysis of

consistent with the observed

some assurance of the validity of the

148 The studies reported here suggest substantial benefit may be derived from application of the reinforcement models in analysis of reinforcement performance and design.

The value of both the analytical models of rock reinforcement and the computa-

tional models of reinforcement and rock mass performance is that a basis is provided for analysis of results of field studies of reinforcement and for logical transfer of results from one field site to another.

A further particularly attractive applica-

tion of the techniques is in parameter studies of reinforcement patterns, to determine the most effective pattern, in relative terms, for the particular site conditions.

The analysis of various hangingwall reinforcement patterns reported here is

an example of such a parameter study and its prospective benefits in reinforcement design. REFERENCES

I.

2.

3.

4.

5.

6.

Pariseau, W. G., Fowler, M. E., Johnson, J. C., Poad, M. and Corp, E . L . Geomechanics of the Carr Fork Mine East Stope In: Ge0mechanics Applications in Underground Hardrock Mining, Ed. by W. G. Pariseau, S.M.E. of A.I.M.E, New York (1984) 3-38. Fuller, P. G. Pre-reinforcement of cut-and-fill stopes In: Applications of Rock Mechanics to Cut and Fill Mining, Eds. O. Stephansson and M. J. Jones, Inst. Min. Metall., London (1981) 55-63. Matthews, S. M., Tillmann, V. H. and Worotnicki, G. A modified cable bolt system for the support of underground openings In: Proc. Aus. Inst. Min. Metall. Conf.~ Broken Hill (1983) 243-255. Greenelsh, R. W. The N663 stope experiment at the Mount Isa Mine Int. J. Min. Eng. 3 (1985) 183-194. Gale, W. J. and Fabjanczyk, M. W. Application of field measurement techniques to the design of roof reinforcement systems in underground coal mines In: Proc. 13th Cong. Common. Min. Metall. Inst.~ Singapore (1986) 135-141. Brady, B.H.G. and Brown, E. T. Rock Mechanics for Underground Mining, Geo. Allen and Unwin, London (1985).

149 7.

8.

9.

I0.

Ii.

12.

13.

14.

15.

16.

17.

18.

St. John, C. M. and Van Dillen, D. E. Rockbolts: a new numerical representation and its application in tunnel design In: Rock Mechanics - - T h e o r y - Experiment - Practice (Proc. 24th U.S. Sym. on Rock Mech. r 19837 , AEG, New York (1983) 13-26. Falrhurst, C. and Singh, B. Rockbolting in horizontally laminated rock Eng. and Min. J. (February 1974), 80-90. Bjurstrom, S. Shear strength on hard rock joints reinforced by grouted untensioned bolts In: Proc. 3rd Int. Cong. Rock Mech. (1979) Vol. 2, 1194-1199. Haas, C. J. Analysis of rock bolting to prevent shear movement in fractured ground Min. Eng. (June 1981) 698-704. Lorig, L. J. A simple numerical representation of fully bonded passive reinforcement for hard rocks Computers and Geotechnics 1 (1985) 79-97. Pells, J.P.N. The behaviour of fully bonded rockbolts In: Proc. 3rd. Int. Cong. Rock Mech., Vol. 2 (1974) 1212-1217. Gerdeen, J. C., Snyder, V. W., Viegelahn, G. L. and Parker, J. Design criteria for rockbolting plans using fully resin-grouted non-tensioned bolts to reinforce bedded mine roof USBM (1977) OFR 46(4)-80. Cundall, P. A. Explicit finite difference methods in geomechanics In: Numerical Methods in Engineering (Proc. EF Conf. on Num. Methods in Geomech), Ed. by C. S. Desai (1976) Vol. i, 132-150. Cundall, P. A. Formulation of a three-dimensional distinct element model ~ part I: a system tc detect and represent contact in a system consisting of many polyhedral blocks Submitted to: Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. (1988). Cundall, P. A. and Board, M. P. A microcomputer program for modeling large-strain plasticity problems Submitted to: 6th Int. Conf. on Num. Methods in Geomech., Innsbruck (1988). Cundall, P. A. and Hart, R. D. Development of generalized 2-d and 3-d distinct element programs for modeling jointed rock Misc. Paper SL-85-I, U.S. Army Corps of Engineers (1985). Bray, J. W. A study of jointed and fractured rocks -- part II Felsmechanik und Ingenieungeologie 4 (1967), 197-216.