7. Excavation design in massive elastic rock

7. Excavation design in massive elastic rock

7.1 General principles of excavation design

• Mining excavation 1) Service openings

- Mine access, ore haulage drive, airway, crusher chambers… - Duty life: mining life of the orebody

2) Production openings- Ore sources, stopes, drill headings, stope access… - Duty life: life of stope (as short as a few months)

• Excavation design in massive elastic rock- The simplest design problem in mining rock mechanics

7.1 General principles of excavation design

• Two points in mine design - Existence of an extensive damage zone or failure rock near the

boundary of the opening is common even in successful mining practice

- Basic mining design objective is to avoid large, uncontrolled displacement of rock in the excavation boundary.

• Logical framework for mine excavation design in massive rock- Refer to Fig 7.2

7.2 Zone of influence of an excavation

• Zone of influence - A domain of significant disturbance of the pre-mining stress field by an excavation- Depends on excavation shape and pre-mining stresses

• Stress distribution around a circular hole (Kirsch equations)- General case:

( ) ( )

( ) ( )

( )

2 2 4

2 2 4

2 4

2 4

2 4

2 4

1 1 1 1 4 3 cos 22

1 1 1 1 3 cos 22

1 1 2 3 sin 22

rr

r

p a a aK Kr r r

p a aK Kr r

p a aKr r

qq

q

s q

s q

s q

é ùæ ö æ ö= + - - - - +ê úç ÷ ç ÷

è ø è øë ûé ùæ ö æ ö

= + + + - +ê úç ÷ ç ÷è ø è øë ûæ ö

= - + -ç ÷è ø


- Hydrostatic stress case:2

2

2

2

1

1

0

rr

r

apr

aprqq

q

s

s

s

æ ö= -ç ÷

è øæ ö

= +ç ÷è ø

=

1.04

0.96

, 6I IID a³

1) Openings of the same radius

-(within ±5% from the field stresses)


2) Openings of the different radius- Genera rule: openings lying outside one another’s zones of influence

can be designed by ignoring the presence of all others- Boundary stresses around II can be obtained by calculating the state of stress at the center of II which is adopted as the far-field stresses in the Kirsch equations, prior to its excavation.


3) Elliptic opening - General shapes of openings can be represented by ellipses inscribed in the opening cross sections.

- Zone of influence of an elliptic excavation:( ) ( )

( )( )

( ) ( )

( )( )

1/ 2

1/ 22 2

1/ 2

1/ 22

2 3 2

or

1 2 3 2

or

1

, 100 2 ,= 1, if < 1, and =1 , if > 1

I

I

I

I

W H A q q K q

W H A K q Kq

H H A K q q q

H H A K q

q W H A cK K K

a

a

a

a

a a

é ù= + - +ë û

é ù= + +ë û

é ù= + - +ë û

é ù= + +ë û= =

7.3 Effect of planes of weakness on elastic stress distribution

• Elastic analysis for the excavations with discontinuities - In some cases, provides a perfectly valid basis for design - or a basis for judgment of engineering significance of a discontinuity.

• Basic assumption of discontinuities- Zero tensile strength- Non-dilatant in shear- Shear strength follows

tannt s f=


• Case 1: Horizontal discontinuity passing through the opening center - : no slip, are principal stresses.- The plane of weakness (discontinuity) has no effect on the elastic stress distribution

( ) ( )

( ) ( )

( )

2 2 4

2 2 4

2 4

2 4

2 4

2 4

1 1 1 1 4 3 cos 22

1 1 1 1 3 cos 22

1 1 2 3 sin 22

rr

r

p a a aK Kr r r

p a aK Kr r

p a aKr r

qq

q

s q

s q

s q

é ùæ ö æ ö= + - - - - +ê úç ÷ ç ÷

è ø è øë ûé ùæ ö æ ö

= + + + - +ê úç ÷ ç ÷è ø è øë ûæ ö

= - + -ç ÷è ø

0 for all 0r r atqs q= = andrr qqs s


• Case 2: Vertical discontinuity passing through the opening center 1) At- : no slip, are principal stresses.

2) At - De-stressed zone from Eq.6.24

21 0

2 2or11 3

2

BKp Kq

K W aq qK H H

Kh aK

sæ ö

= - + =ç ÷è ø

æ ö= = =ç ÷- è ø-æ öD = ç ÷

è ø

0 for all 90r r atqs q= = o andrr qqs s

13K ³

13K <


• Case 3: Horizontal discontinuity passing through the opening - Normal & shear stresses at the intersections on boundary

- Slip occurs when - Intersection regions are either de-stressed or at low confiningstress

( )

( )

2

tan slip occurrs

sin cos cos tansin

tan tan or 0cos

n

qq qq

qq

t s f

s q q s q fq f

q f sf

=

® =

-= =

2cossin cos

n qq

qq

s s qt s q q

=

=

or 0qqq f s= =


• Case 4: Arbitrary discontinuity passing through the opening center- Normal & shear stresses on the weak plane

- Slip does not occur if

2

2

2 4

2 4

1.5 12

2 30.5 12

n

r

p ar

p a ar r

qq

q

s s

t s

æ ö= = +ç ÷

è øæ ö

= = + -ç ÷è ø

19.6f > o

0.357 tan19.6= o


• Case 5: Horizontal discontinuity not intersecting the opening- Normal & shear stresses on the weak plane

- Slip does not occur if 24f > o

( ) ( )2

2

1 1 cos 22 2

1 cos 2

n rr rr

apr

qq qqs s s s s a

a

= + + -

æ ö= -ç ÷

è ø

( )2

2

1 sin 22

sin 2

rr

apr

qqt s s a

a

= - -

=

7.4 Excavation shape and boundary stresses

• Elliptic opening

- q = W/H- ρ : radius of curvature

- : curvature of an ellipse

- Larger curvature makes higher stress concentration

( ) 2322

1

yx

yxyx

¢+¢

¢¢¢-¢¢¢=

r

( )

÷÷ø

öççè

æ+-=÷÷

ø

öççè

æ+-=

÷÷ø

öççè

æ+-=+-=

BB

AA

WKKpqKKp

WKpqKp

rs

rs

2121

2121


• Ovaloidal opening- Applying the boundary stress

of an ellipse inscribed in theovaloid

-

- By B.E.M: sA= 3.60p, sB = -0.15p

( )p

HHHp

pH

Hp

B

A

17.023

25.015.0

96.32

325.01

2 -=÷÷ø

öççè

æ+-=

=÷÷ø

öççè

æ ´+-=

s

s


• Square opening with rounded corners- Applying the boundary stress of an ellipse whose curvature isthe same as those of the rounded corners

-

- By B.E.M: sA= 3.14p- Boundary stress is dominatedby the local geometry:St Venant’s Principle

( )( )

p

BBpA

53.3

2.0124.02211

21

=

÷÷

ø

ö

çç

è

æúû

ùêë

é --+-=s


• Effect of changing the relative dimensions- Sidewall stress 2.5p → 1.7p- The maximum boundary stress can be reduced if the opening

dimension is increased in the direction of the major principal stress.


• Effect of local geometry of an opening- Width/height = 2/3- A, B, C are highly stressed due to their high level of curvature.- D is at low state of stress (tensile failure)- Rock mass in compression may behave as a stable continuum whilein a de-stressed state, small loadscan cause large displacement ofrock units.

7.5 Delineation of zones of rock failure

- Estimation of the extent of fracture zones provides a basis forprediction of rock mass performance, modification of excavation design, or assessing support and reinforcement requirement.

- The solution procedure suggested here examines only the initial, linear component of the problem. For mining engineering purposes, the suggested procedure is usually adequate.

• Extent of boundary failure- Applicable compressive strength at boundary is sci.- Tensile strength of rock mass is taken to be zero.


- Case of a circular excavation havingsci of 16 MPa:

- Change in shape, installation ofsupport/reinforcement, or increase the height of the opening can be used.

[ ][ ] ( )

[ ] ( )

1 2(1 )cos 2

7.5 1.3 1.4cos 2 16 compressive

26 26 or 154 2067.5 1.3 1.4cos 2 0 tensile

79 101 or 259 281

p K Kqqs q

q

q qq

q q

= + + -

+ ³

® - £ £ £ £

+ £

® £ £ £ £

o o o o

o o o o


• Extent of failure zones in rock mass- Close to the boundary (within a radius):

the constant deviator stress criterion is useful.

- Example of an circular opening inLac du Bonnet granite: the maximumdeviator stress contour of 75 MPapredicted well the failure domain.


- General cases including interior zones of rock mass:Hoek-Brown criterion with scd

1) Principal stress contour methoda. Calculate various values of b. Contour plots of s1 and s3 are

superimposed.c. Find the intersections of s1 and s3

isobars satisfying the failure criterion.

( )f13 ,ss


2) Direct comparison with the failure criterion a. Calculate the state of stress (principal stress)b. Compare with the failure criterion.c. Display failure locations throughout the rock mass.

7.6 Support and reinforcement of massive rock

• Explanation of the effect of support(1) Elastic rock medium

- Stress before support:

- Stress after support

- Support pressure does not significantly modify the elastic distribution around an underground opening

MPa0.8,MPa2.170 -== BA ss

MPa0.741

19821

198190

MPa0.16181981191

321

321

-=÷øö

çèæ ´

´+-+=

+=

=÷øö

çèæ +-+=

+=

BBB

AAA

sss

sss


(2) Elastic rock mass with a failed rock annulus- Rock mass strength is assumed to follow Coulomb’s criterion:

- Equilibrium equations in fractured rock (c.f. Sec.2.11)

Massrock Fractured : sin1sin1

MassRock : sin1cos2

sin1sin1

3131

03131

ssffss

ssff

ffss

d

Cbc

f

f

=®-+

=

+=®-

+-+

=

( )

( )11

11

1

11

or

,

1

--

--

÷÷ø

öççè

æ=÷

øö

çèæ=

÷øö

çèæ=÷

øö

çèæ=®

-=-

=

d

ie

de

i

d

i

d

irr

rrrrrr

ppar

arpp

ardp

arp

rd

rdrd

qq

qq

ss

ssss


- Stress in elastic zone (c.f. Eqn.6.8)

- At the inner boundary of the elastic zone (r = re)

- Applying to Coulomb’s criterion ( )

-

2

2

12

2

2

2

12

2

1,1rrp

rrp

rrp

rrp ee

rree +÷÷

ø

öççè

æ-=-÷÷

ø

öççè

æ+= ssqq

11,2 ppp rr =-= ssqq

031 Cb += ss

bCppCbppp

+-

=®+=-1

22 01011

( )

( )11

0

12

-

úû

ùêë

é+-

=®d

ie pb

Cpar

arpCpp

e

if

99.15.0,05.0,35At 0

=®==== off

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7. Excavation design in massive elastic rock