Class Notes on Underground Excavations in Rock (2006)

ce.umn.eduUniversity of Minnesota

Department of Civil Engineering

[Last revision – June 06]

These notes areavailable for downloading atwww.cctrockengineering.com

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Class notes on Underground Excavations in Rock

Topic 0:

Table of contents

written by

Dr. C. Carranza-Torres andProf. J. Labuz

These series of notes have been written for the course Rock Mechanics II,CE/GeoE 4311, co-taught by Prof. J. Labuz and Dr. C. Carranza-Torresin the Spring 2006 at the Department of Civil Engineering, Universityof Minnesota, USA.




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List of topics covered in notes

1. Introduction to tunnelling. Methods and equipment

2. Review of some fundamental equations of solid mechanics

3. Elastic solution of a circular tunnel

4. Introduction to numerical modelling

5. Strength and inelastic deformation of rock

6. Elasto-plastic solution of a circular tunnel

7. Review of some fundamental equations of mechanics of beams

8. Elastic solution of a closed annular support

9. Convergence-Confinement Method of tunnel support design

10. Reinforcement in tunnels

11. Stability of shallow tunnels





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Topic 1:

Introduction to tunnelling. Methods and equipment

written by






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Tunnelling Methods

Sketch from ‘Underground rock excavation. Know-how and equipment’. Atlas CopcoTunnelling and Mining AB, S-105 23 Stockholm, Sweden.




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Drilling and Blasting Method

The drill and blast cycle:

1. Drilling and surveying

2. Charging with explosives

3. Blasting and ventilation

4. Loading and hauling

5. Scaling and cleaning

6. Rock bolting

(Sketch is adapted from Tamrock Corporation, www.tamrock.sandvik.com)




[UE-T1-4]Blasting patterns

(From Hoek E., 2000, ‘Rock Engineering’, Chapter 16, www.rocscience.com)




[UE-T1-5]Explosives

(www.austinpowder.com)




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Drilling rods and drilling bits

(www.mmc.co.jp/english/business/rocktool.html)




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Common equipment found in tunnelling sites

Drilling Jumbos

(www.atlascopco.com and www.tamrock.sandvik.com)




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Loaders and trucks

(www.toro.sandvik.com and www.casece.com)




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Excavators

(www.casece.com and www.hitachiconstruction.com)




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Bolting jumbos

(www.tamrock.sandvik.com)




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Scalers or breakers

(www.rockbreaker.com)




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Shotcrete Equipment

A- Manual sprayingB- Robotic spraying




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Robotic Sprayers

(Model shown is Sika PM500 PC – www.putzmeister.de)




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Lifters

Normet equipment (www.normetusa.com)




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Other equipment found in tunnelling sites

Improvement of tunnel front




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Mortar injection and backfilling equipment

(www.putzmeister.de)




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Pusher leg rock drills

(www.partshq.com)




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Images of tunnel excavation by traditional method

1. Video showing general description of the Gotthard Tunnel project,Switzerland (www.alptransit.com).

2. Video showing impressions of excavation at the Gotthard Tunnel,Switzerland (www.alptransit.com).

3. Video showing blasting at Gotthard Tunnel, Switzerland(www.alptransit.com).

4. Photographs at various tunnelling sites.




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Tunnel excavation by mechanized means

Classification based on the type of ground supportprovided by the machine

1. No ground support → Roadheader

2. Periphery → Open face Tunnel Boring Machine

or TBM (used in hard ground)

3. Front → EPBM (Earth Pressure BalanceMachine)

→ Slurry shield (used in soft ground below

the phreatic surface)

(A complete classification of mechanized methods of tunnel excavation can be foundinAFTES, 2000, ‘Recommendations for choosing mechanized tunnelling techniques’,available at www.aftes.asso.fr)




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Roadheaders

(www.vab.sandvik.com)




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Advance by TBM (Tunnel Boring Machine)

Sketch from ‘Underground rock excavation. Know-how and equipment’. Atlas CopcoTunnelling and Mining AB, S-105 23 Stockholm, Sweden.




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Advance by TBM (Tunnel Boring Machine)

(www.alptransit.ch)




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EPBM - Earth Pressure Balance (1)

Sketches from booklet ‘Hitachi Shield Machines’, Hitachi Construction MachineryCo., Ltd., www.hitachi-c-m.com




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EPBM - Earth Pressure Balance (2)





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Slurry shield (1)





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Slurry shield (2)





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Segmental lining used with EPBM and Slurry Shields

(adapted from AFTES, 2000, ‘The design, sizing and construction of precast concretesegments installed at the rear of aTunnel Boring Machine’, available at www.aftes.asso.fr)




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Microtunnelling

(www.lovat.com)




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Recommended references

• Hoek E., 2000, ‘Rock Engineering. Course Notes by Evert Hoek’.Available for downloading at ‘Hoek’s Corner’, www.rocscience.com

• U.S. Army Corps of Engineers, 1997, ‘Tunnels and shafts in rock’.Available for downloading at www.usace.army.mil

• AFTES Recommendations available in English. A series of PDF doc-uments on different topics related to tunnelling that can be downloadedat www.aftes.asso.fr

• For a series of short, clearly presented notes with recommendationsabout different aspects of tunnelling design with traditional methods(e.g., face drilling, blasting, rock reinforcement, etc.), seehttp://sg01.atlascopco.com/SGSite/default app.asp

•For information about GotthardTunnel (including videos, photographs,etc.) see www.alptransit.ch

• For information about on-going tunnel projects around the world seewww.tunnelintelligence.com

• Visit the web sites indicated in the previous slides on particular topicsof interest





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Topic 2:

Review of some fundamental equations ofsolid mechanics

written by






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Equilibrium of forces – Cartesian coordinate system (2D)

∂σx

∂x+ ∂τxy

∂y+ ρβx = 0 (1)

∂τxy

∂x+ ∂σy

∂y+ ρβy = 0 (2)

Note that τxy = τyx (from equilibrium of moments)




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Equilibrium of forces in cylindrical coordinate system (2D)

∂σr

∂r+ 1

r

∂τrθ

∂θ+ σr − σθ

r+ ρβr = 0 (3)

∂τrθ

∂r+ 1

r

∂σθ

∂θ+ 2

τrθ

r+ ρβθ = 0 (4)

Note that σrθ = σθr (from equilibrium of moments)




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Definition of Strains – Cartesian coordinate system (2D)

εx = −∂ux

∂x(5)

εy = −∂uy

∂y(6)

γxy = −(

∂ux

∂y+ ∂uy

∂x

)(7)




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Strains in cylindrical coordinate system (2D)

εr = −∂ur

∂r(8)

εθ = −(

ur

r+ 1

r

∂uθ

∂θ

)(9)

γrθ = −(

∂uθ

∂r− uθ

r+ 1

r

∂ur

∂θ

)(10)




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Elasticity equations – Isotropic material

General 3D case, for the normal component of the stresses,

σx = (λ + 2G)εx + λεy + λεz (11)

σy = λεx + (λ + 2G)εy + λεz (12)

σz = λεx + λεy + (λ + 2G)εz (13)

In the equations above λ is the Lamé’s constant and G is the Shearmodulus of the material.




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For the shear component of stresses

τxy = Gγxy (14)

τyz = Gγyz (15)

τxz = Gγxz (16)




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Relationship between elastic constants

λ = Eν

(1 + ν)(1 − 2ν)(17)

G = E

2(1 + ν)(18)

In the equations above E is the Young’s modulus and ν is the Poisson’sratio.

Note also the following relationships (to be used later when derivingelastic solutions)

2 (λ + G) = E

(1 + ν)(1 − 2ν)= 2G

1 − 2ν(19)

λ + 2G = (1 − ν)E

(1 + ν)(1 − 2ν)(20)




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Plane strain analysis




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Elasticity equations – plane strain

For plane strain conditions, we consider εz = γxz = γyz = 0 in equa-tions 11 through 16, and therefore these equations become

σx = (λ + 2G)εx + λεy (21)

σy = λεx + (λ + 2G)εy (22)

τxy = Gγxy (23)

Expressed in terms of E and ν, the equations are,

σx = E(1 − ν)

(1 + ν)(1 − 2ν)

[εx + ν

1 − νεy

](24)

σy = E(1 − ν)

(1 + ν)(1 − 2ν)

[ν

1 − νεx + εy

](25)

τxy = E

2(1 + ν)γxy (26)




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Elasticity equations – plane strain

The equations above can be inverted and expressed in terms of strainstoo,

εx = 1 + ν

E

[(1 − ν)σx − νσy

](27)

εy = 1 + ν

E

[(1 − ν)σy − νσx

](28)

γxy = 2(1 + ν)

Eτxy (29)

For plane strain problems, it can be shown that

σz = λ(εx + εy) = ν(σx + σy) (30)

The equations presented above are also valid for cylindrical coordinates,in such case σr ∼ σx, σθ ∼ σy, σrθ ∼ σxy, εr ∼ εx, εθ ∼ εy andγrθ ∼ γxy




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Example of simple elastic analysis:

Loading of unconfined body in plane strain

σy = py

εy = 1 − ν2

Epy

uy(y) = −1 − ν2

Epy y

uy(H) = −1 − ν2

Epy H

σx = 0

εx = −(1 + ν)ν

Epy

ux(x) = (1 + ν)ν

Epy x

ux(B/2) = (1 + ν)ν

Epy B/2




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Loading of confined body in plane strain

σy = py

εy = (1 + ν)(1 − 2ν)

(1 − ν)Epy

uy(y) = −(1 + ν)(1 − 2ν)

(1 − ν)Epy y

uy(H) = −(1 + ν)(1 − 2ν)

(1 − ν)Epy H

σx = ν

1 − νpy

εx = 0

ux(x) = 0

ux(B/2) = 0




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Gravity loading of confined body

σy = ρg(H − y)

εy = (1 + ν)(1 − 2ν)

(1 − ν)Eρg(H − y)

uy(y) = −(1 + ν)(1 − 2ν)

(1 − ν)Eρgy

(H − y

2

)

uy(H) = −(1 + ν)(1 − 2ν)

(1 − ν)Eρg

H 2

2

σx = ν

1 − νρg(H − y)

εx = 0

ux(x) = 0

ux(B/2) = 0




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Recommended References

• Brady B.H.G. and E.T. Brown, 2004, ‘Rock Mechanics for Under-ground Mining’, 3rd Edition, Kluwer Academic Publishers

• Jaeger J. C. and N. G.W. Cook, 1979, ‘Fundamentals of rock mechan-ics’, John Wiley & Sons

• Timoshenko S. P. and J. N. Goodier, 1970, ‘Theory of Elasticity’, 3rdEdition, Mc. Graw Hill, New York

• Chi P.C. and N. Pagano, 1967, ‘Elasticity, Tensor, Dyadic, and Engi-neering Approaches’ (Originally published by Nostrand Company, Inc.,Princeton), republished by Dover (1992)





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Topic 3:

Elastic solution of a circular tunnel

written by






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General form of Lamé’s solution

σr = σBr R2

B − σAr R2

A

R2B − R2

A

−(σB

r − σAr

)R2

AR2B

R2B − R2

A

(1)

σθ = σBr R2

B − σAr R2

A

R2B − R2

A

+(σB

r − σAr

)R2

AR2B

R2B − R2

A

(2)

ur = −1 − 2ν

2G

σBr R2

B − σAr R2

A(R2

B − R2A

) r −(σB

r − σAr

)R2

AR2B

2G(R2

B − R2A

) 1

r(3)




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Particular case of Lamé’s solution

Elastic solution of a thin annular ring (1)

We consider RA → R (1 − t/R), RB → R and r → R in equations (1)through (3). Also, we consider σB

r → ps and σAr → 0.

Then the solution for radial displacement results to be,

ur

R= −2 − 2ν − 2t/R + (t/R)2

2G (2 − t/R) t/Rps (4)




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Elastic solution of a thin annular ring (2)

Also, with the previous assumptions the solution for the radial and hoopstresses (at r = R) are, respectively

σr = ps (5)

σθ = 2 − 2t/R + (t/R)2

(2 − t/R) t/Rps (6)

Assuming the ratio t/R is small, the thrust Ts can be computed fromequation (6) as σθ × t , i.e.,

Ts = 2 − 2t/R + (t/R)2

2 − t/RR ps (7)

For thin annular rings, taking the limit limt/R→0 Ts, we get

Ts = Rps (8)

Equation (8) is the same equation obtained by applying the theory ofthin curved arches, and is a fundamental relationship used in the designof tunnel liners.




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Elastic medium loaded at infinity – no excavation

We consider RA → 0, RB → ∞, σAr → 0 and σB

r = σo in equations(1), (2) and (3).

Then Lamé’s solution results to be

σr = σθ = σo (9)

ur = −1 − 2ν

2Gσo r (10)




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Elastic excavated medium — loaded at infinity and inside the opening

We consider RA = R, RB → ∞ (or RA/RB → 0),σA

r = pi and σBr = σo in equations (1), (2) and (3).

The solution for stresses are,

σr = σo − (σo − pi)

(R

r

)2

(11)

σθ = σo + (σo − pi)

(R

r

)2

(12)




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Elastic excavated medium — loaded at infinity and inside the opening

The solution for the radial displacement is

uTOTr = uINI

r + uINDr (13)

where

uINIr = −1 − 2ν

2Gσo r (14)

uINDr = − 1

2G(σo − pi)

R2

r(15)

In the pre-stressed medium where excavation takes place the inducedcomponent of displacement has engineering significance only, thus ur =uIND

r , or

ur = − 1

2G(σo − pi)

R2

r(16)




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Lamé’s solution for a circular tunnel — graphical representation

The solution for stresses is,




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The stresses can be represented in a σθ vs σr diagram as follows (this isuseful for deriving the elasto-platic solution of a circular tunnel)




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The solution for displacements is




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Example of elastic analysis of a tunnel (1)




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Elastic solutions for tunnel problems — Historical perspective (1)

From Fairhurst, C. and C. Carranza-Torres, 2002 (see Recommended References)




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• Brady B.H.G. and E.T. Brown, 2004, ‘Rock Mechanics for Under-ground Mining’, 3rd Edition, Kluwer Academic Publishers.

• Jaeger J. C. and N. G.W. Cook, 1979, ‘Fundamentals of rock mechan-ics’, John Wiley & Sons.

• Savin G. N. ‘Stress Concentration Around Holes’, Pergamon Press,London, 1961.

• Fairhurst, C. and C. Carranza-Torres, 2002, ‘Closing the circle’. In J.Labuz and J. Bentler (Eds.), Proceedings of the 50 th Annual Geotech-nical Engineering Conference. St. Paul, Minnesota, February 22, 2002.University of Minnesota.





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Topic 4:

Introduction to numerical modelling

written by






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Classification of methods of analysis in geomechanics

Adapted from Potts D. et al., 2002, ‘Guidelines for the use of advanced numericalanalysis in geotechnical engineering’, Thomas Telford Publishing, London.




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Classification of numerical methods used in rock mechanics

1. Finite Element Method (FEM)

2. Boundary Element Method (BEM)

3. Finite Difference Method (FDM)

4. Discrete Element Method (DEM)

See, for example, Brady B.H.G. and E.T. Brown, 2004, ‘Rock Mechanics for Under-ground Mining’, 3rd Edition, Kluwer Academic Publishers




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Commercial and freeware software used in rock mechanics problems

1. Finite Element Method (FEM)→ Phase2 (www.rocscience.com)→ DEMON —available in reference (∗)

2. Boundary Element Method (BEM)→ Examine2D (www.rocscience.com)→ TWOFS/TWODD/TWOBI —available in reference (∗∗)

3. Finite Difference Method (FDM)→ FLAC/FLAC3D (www.itascacg.com)

4. Discrete Element Method (DEM)→ UDEC/3DEC/PFC/PFC3D (www.itascacg.com)→ DDA —available at www.ce.berkeley.edu/geo/research/DDA

(∗) Beer G. and J. O. ‘Watson, Introduction to Finite and Boundary Element Methodsfor Engineers’. John Wiley & Sons, 1992

(∗∗) Crouch S. L. and A. M. Starfield. ‘Boundary Element Methods in Solid Mechan-ics: With Application in Rock Mechanics and Geological Engineering’. George Allen& Unwin, London, 1983




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Example of analysis using FEM. Stress redistribution around tunnel

From Zienkiewicz O.C. and R.L. Taylor, 2000, ‘The Finite Element Method’, VolumeI: The Basis. 5th Edition. Butterworth-Heinemann




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Example of analysis using FEM. Rock-support interaction

From Wittke W., 1990, ‘Rock Mechanics. Theory and Applications with Case Histo-ries’. Springer-Verlag




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Example of analysis using BEM. Excavation near a fault

From Crouch S. L. and A. M. Starfield. ‘Boundary Element Methods in Solid Me-chanics: With Application in Rock Mechanics and Geological Engineering’. GeorgeAllen & Unwin, London, 1983




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Example of analysis using DEM. Tunnel in jointed rock mass

From Pande G.N., Beer G. and J.R. Williams, 1990, ‘Numerical Methods in RockMechanics’. John Wiley & Sons Ltd.




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Example of advanced numerical modelling

FLAC3D analysis of rockbolt loading behind a TBM

Modelling by C. Carranza-Torres in collaboration with Geodata Spa (www.geodata.it),Torino, Italy (2004). FLAC3D is developed and commercialized by Itasca(www.itascacg.com)




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FLAC3D analysis of subsidence due to EPBM excavation (1)

Modelling by C. Carranza-Torres in collaboration with Geodata Spa (www.geodata.it),Torino, Italy (2004) — FLAC3D is developed and commercialized by Itasca(www.itascacg.com)




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FLAC3D analysis of subsidence due to EPBM excavation (2)

Modelling by C. Carranza-Torres in collaboration with Geodata Spa (www.geodata.it),Torino, Italy (2004) — FLAC3D is developed and commercialized by Itasca(www.itascacg.com)




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Example of advanced modelling

FLAC3D thermo-mechanical analysis of underground repository (1)

Modelling by C. Carranza-Torres, B. Damjanac and T. Brandshug from Itasca Con-sulting Group, Minneapolis (2002) — FLAC3D is developed and commercialized byItasca (www.itascacg.com)




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FLAC3D thermo-mechanical analysis of underground repository (2)

Modelling by C. Carranza-Torres, B. Damjanac and T. Brandshug from Itasca Con-sulting Group, Minneapolis (2002) — FLAC3D is developed and commercialized byItasca (www.itascacg.com)




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UDEC analysis of stabilizing effect of rockbolts in granular material

Modelling by C. Carranza-Torres in collaboration with Dr. E. Hoek (2003). Descrip-tion of the physical model and animated version of the UDEC models available at‘Hoek’s corner’, ‘Discussion Papers’, www.rocscience.com — UDEC is developedand commercialized by Itasca (www.itascacg.com)




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Stability analysis of a large landslide — 3DEC analysis

Modelling by C. Carranza-Torres, in collaboration with Prof. M. Diederichs and Prof.J. Hutchinson, Geological Engineering Group (www.geol.ca), Queen’s University, On-tario (2006) — 3DEC is developed and commercialized by Itasca (www.itascacg.com)




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PFC2D/PFC3D modelling of forces generated by a block of rock

that breaks at impact with metal canister (1)

Modelling by C. Carranza-Torres in collaboration with Prof. C. Fairhurst (see Fairhurst,C. and C. Carranza-Torres, 2002, ‘Closing the circle’. In J. Labuz and J. Bentler (Eds.),Proceedings of the 50 th Annual Geotechnical Engineering Conference. St. Paul,Minnesota, February 22, 2002. University of Minnesota.) — PFC2D and PFC3D aredeveloped and commercialized by Itasca (www.itascacg.com)




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PFC2D/PFC3D modelling of forces generated by a block of rock

that breaks at impact with metal canister (2)

Modelling by C. Carranza-Torres in collaboration with Prof. C. Fairhurst (see Fairhurst,C. and C. Carranza-Torres, 2002, ‘Closing the circle’. In J. Labuz and J. Bentler (Eds.),Proceedings of the 50 th Annual Geotechnical Engineering Conference. St. Paul,Minnesota, February 22, 2002. University of Minnesota.) — PFC2D and PFC3D aredeveloped and commercialized by Itasca (www.itascacg.com)




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The Finite Element Method (FEM) – Basic steps

Note: Steps marked with ‘∗’ require user intervention

(Adapted from Desai and Christian, 1977, ‘Numerical Method in Geotechnical Engi-neering’, Chapter 1, John Wiley)




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FEM Analysis

Step 1: Problem definition

F1 = 15.81 kNFx1 = 5 kNFy1 = 15 kNα1 = 18.43◦

F2 = 22.36 kNFx2 = 10 kNFy2 = 20 kNα2 = 26.57◦

ρ = 2500 kg/m3

E = 10 GPaν = 0.25

σox = 200 kPa

σoy = 100 kPa




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FEM Analysis

Step 2: Selection of shape functions and discretization (1)




[UE-T4-21]

FEM Analysis





[UE-T4-22]

FEM Analysis





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FEM Analysis

Step 3: Derivation of element equations (1)

We will illustrate the analysis for the case of 3-Node triangular elements

The vector of nodal displacements {ue} and the vector of nodal forces{qe} are

{ue} =

uxi

uyi

uxj

uyj

uxk

uyk

{qe} =

qxi

qyi

qxj

qyj

qxk

qyk




[UE-T4-24]

FEM Analysis


The vector of (element) initial stresses {σ eo } and the vector of (element)

body forces {be} are

{σ eo } =

σox

σ oy

τ oxy

{be} =

{bx

by

}




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FEM Analysis


The objective of the Step 3 is to compute the relationship between thevector of nodal displacements {ue} and the vector of nodal/elementforces {qe}, {σ e

o } and {be}.




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FEM Analysis


For an elastic material, it can be shown that the following relationshipbetween the vectors {qe}, {ue} and {fe} holds,

{qe} = [

Ke] {ue} + {

fe} (1)

where [Ke] is the ‘stiffness’ matrix that depends on the shape functionand the elastic properties of the material in the element, while the {fe}is the ‘initial-loading/body-force’ vector, that depends on the vector ofinitial stresses

{σ e

o

}and the vector of body forces {be}.

(Any book on FEM in solid mechanics will include a demonstration the relation-ship above —e.g., see Zienkiewicz O.C. and R.L. Taylor, 2000, ‘The Finite ElementMethod’, Volume I: The Basis. 5th Edition. Butterworth-Heinemann; for a briefdemonstration, also see Brady and Brown, 2004, ‘Rock Mechanics for UndergroundMining’, 3rd Edition, Kluwer Academic Publishers)




[UE-T4-27]FEM Analysis


For example, for the element ‘a’ in the figure, equation (1), is written as{qa} = [

Ka] {ua} + {

fa}where the vectors and matrices in the equation above involve the nodesconnected to the element only.

For example, the vectors {qa}, {ua} and {fa} are, respectively

{qa} =

qax1

qay1

qax2

qay2

qax4

qay4

{ua} =

uax1

uay1

uax2

uay2

uax4

uay4

{fa} =

f ax1

f ay1

f ax2

f ay2

f ax4

f ay4




[UE-T4-28]

FEM Analysis


while the matrix [Ka] is

[Ka] =

Kax11 0 Ka

x12 0 Kax14 0

0 Kay11 0 Ka

y12 0 Kay14

Kax21 0 Ka

x22 0 Kax24 0

0 Kay21 0 Ka

y22 0 Kay24

Kax41 0 Ka

x42 0 Kax44 0

0 Kay41 0 Ka

y42 0 Kay44

Note: at this stage (Step 3) only the matrix [Ka] and the vector {fa}can be computed for each element, based on the geometry, materialproperties and loading (a finite element program will compute and storethe elements of these matrices and vectors for use in Step 4)




[UE-T4-29]

FEM Analysis

Step 4: Assembling the element properties to form global equations (1)

The (matrix) equation representing ‘global’ equilibrium of the systemcan be expressed as{

rG} = [KG] {

uG} + {fG}

(2)

The different vectors/matrices in equation (2) are described separately




[UE-T4-30]

FEM Analysis


The vector of nodal reaction forces{rG

}in equation (2) is

{rG} =

rGx1

rGy1

rGx2

rGy2

rGx3

rGy3

...

where {rGx1 = Fx1 (k)

rGy1 = −Fy1 (k)

{rGx2 = RxA (u)

rGy2 = RyA (u)

{rGx3 = 0 (k)

rGy3 = 0 (k)

{rGx4 = 0 (k)

rGy4 = 0 (k)

Note: ‘(k)’ means known quantity; ‘(u)’ means unknown quantity




[UE-T4-31]

FEM Analysis


The vector of nodal displacements forces{uG

}in equation (2) is

{uG} =

uGx1

uGy1

uGx2

uGy2

uGx3

uGy3

...

where {uG

x1 = ux1 (u)

uGy1 = uy1 (u)

{uG

x2 = 0 (k)

uGy2 = 0 (k)

{uG

x3 = ux3 (u)

uGy3 = uy3 (u)

{uG

x4 = ux4 (u)

uGy4 = uy4 (u)

Note: ‘(k)’ means known quantity; ‘(u)’ means unknown quantity




[UE-T4-32]

FEM Analysis


The vector of initial-stress/body-forces{fG

}in equation (2) is

{fG} =

f Gx1

f Gy1

f Gx2

f Gy2

f Gx3

f Gy3

...

where {f G

x1 = f ax1 + f e

x1

f Gy1 = f a

y1 + f ey1

{f G

x2 = f ax2 + f b

x2

f Gy2 = f a

y2 + f by2{

f Gx3 = f b

x3 + f cx3

f Gy3 = f b

y3 + f cy3

{f G

x4 = f ax4 + f b

x4 + f cx4 + f d

x4 + f ex4

f Gy4 = f a

y4 + f by4 + f c

y4 + f dy4 + f e

y4

Note: All quantities in the vector{fG

}are known




[UE-T4-33]

FEM Analysis


The ‘global’ stiffness matrix[KG

]in equation (2) is

[KG] =

KGx11 0 KG

x12 0 KGx13 0 . . .

0 KGy11 0 KG

y12 0 KGy13 . . .

KGx21 0 KG

x22 0 KGx23 0 . . .

0 KGy21 0 KG

y22 0 KGy23 . . .

KGx31 0 KG

x32 0 KGx33 0 . . .

0 KGy31 0 KG

y32 0 KGy33 . . .

......

......

......

where{KG

x11 = Kax11 + Ke

x11

KGy11 = Ka

y11 + Key11

{KG

x12 = Kax12

KGy12 = Ka

y12

{KG

x13 = KGx14 = 0

KGy13 = KG

y14 = 0

{KG

x21 = Kax21

KGy21 = Ka

y21

{KG

x22 = Kax22 + Kb

x22

KGy22 = Ka

y22 + Kby22

{KG

x23 = Kbx23

KGy23 = Kb

y23

{KG

x24 = Kax24 + Kb

x24

KGy24 = Ka

y24 + Kby24

{KG

x31 = 0

KGy31 = 0

{KG

x32 = Kbx32

KGy32 = Kb

y32




[UE-T4-34]

FEM Analysis


{KG

x33 = Kbx33 + Kc

x33

KGy33 = Kb

y33 + Kcy33

{KG

x34 = Kbx34 + Kc

x34

KGy34 = Kb

y34 + Kcy34

{KG

x41 = Kax41 + Ke

x41

KGy41 = Ka

y41 + Key41

{KG

x42 = Kax42 + Kb

x42

KGy42 = Ka

y42 + Kby42

{KG

x43 = Kbx43 + Kc

x43

KGy43 = Kb

y43 + Kcy43

{KG

x44 = Kax44 + Kb

x44 + Kcx44 + Kd

x44

KGy44 = Ka

y44 + Kby44 + Kc

y44 + Kdy44

Note: All quantities in the matrix[KG

]are known




[UE-T4-35]

FEM Analysis


As seen in previous slides, the ‘global’ equation (2) represents a systemof 2 × N equations with 2 × N unknowns (where N is the number ofnodes in the mesh).

This system of equations can be solved using direct or iterative methodsof matrix algebra (the most commonly used method is perhaps the GaussElimination Method).




[UE-T4-36]

FEM Analysis

Step 5: Computation of primary and secondary quantities (1)

Solution of the global equilibrium equation (2) defines the nodal dis-placement vector for all elements in the mesh.

Consider a point P of coordinates x and y inside an arbitrary elementfor which the vector of nodal displacements is {ue} (see Step 3).

The displacement vector {u} for the point can be computed as follows

{u} = [N] {ue} (3)

where

{u} ={

ux

uy

}

and [N] is a matrix that depends on the shape function chosen when themesh was created (Step 2).




[UE-T4-37]

FEM Analysis


For example, for the case of 3-Node triangular element with a linearshape function, the matrix [N] is

[N] =[

Ni 0 Nj 0 Nk 00 Ni 0 Nj 0 Nk

]where

Ni = [(xjyk − xkyj) + x(yj − yk) + y(xk − xj)

]/(2A)

Nj = [(xkyi − xiyk) + x(yk − yi) + y(xi − xk)] /(2A)

Nk = [(xiyj − xjyi) + x(yi − yj) + y(xj − xi)

]/(2A)

and A = 1

2det

1 xi yi

1 xj yj

1 xk yk

Note: the coefficients in the expressions Ni , Nj and Nk above are obtained from thecondition that the scalar function f (x, y) —see Step 2— inside the element is a linearfunction of the coordinates x and y of the point, i.e.,

f (x, y) = α1 + α2x + α3y

and that the known values of the function are recovered at the nodes. This impliesthat the coefficients α1, α2 and α3 in the expression above, must satisfy the followingsystem of equations

fi = α1 + α2xi + α3yi

fj = α1 + α2xj + α3yj

fk = α1 + α2xk + α3yk




[UE-T4-38]

FEM Analysis


The strain and stress vectors {ε} and {σ }, can be similarly computed atany point P based on the displacement vector of the element containingthe point, i.e.,

{ε} = [L] {u} = [L] [N] {ue} (4)

and

{σ } = [D] {ε} + {σo} (5)

where

{σ } =

σx

σy

τxy

{ε} =

εx

εy

γxy

According to basic equations of solid mechanics (see Topic 2 in theseseries of notes) the matrix [L] in equation (4) is formed by the followingdifferential operators that ‘affect’ the shape function [N],

[L] = ∂/∂x 0

0 ∂/∂y

∂/∂y ∂/∂x

Also, for an elastic isotropic material in plane strain conditions, thematrix [D] in equation (5) is

[D] = E(1 − ν)

(1 + ν)(1 − 2ν)

1 ν

1−ν0

ν1−ν

1 00 0 1−2ν

2(1−ν)




[UE-T4-39]

FEM Analysis

Step 6: Inspection of results

The figure above represents contours of total displacement ur and maximum princi-pal stress σ1, respectively for the problem outlined in Step 1. The views have beengenerated with the FEM code Phase2 (www.rocscience.com)




[UE-T4-40]

The Finite Element software Phase2 – Pre-Processing

Phase2 is developed and commercialized by Rocscience (www.rocscience.com)




[UE-T4-41]

The Finite Element software Phase2 – Processing





[UE-T4-42]

The Finite Element software Phase2 – Post-Processing





[UE-T4-43]

Basic steps for setting-up and solving a model in Phase2 (1)

1- Define the project settings (menu option Analysis/Project Settings. . . ).This controls basic aspects of the model to be created and solved —e.g.,plane strain or axi-symmetry problem, number of stages (of loading orexcavation) in the problem, system of units, etc.




[UE-T4-44]


2- Define the geometry of the problem (menu option Boundaries/. . . ).This normally involves creating excavations (option . . . /Add Excava-tion), external boundaries (option . . . /Add External) and materialboundaries (option . . . /Add Material Boundary )




[UE-T4-45]


3- Define the mesh (menu option Mesh/. . . ). This step can be sub-divided in three sub-steps:

3a - Choose the type of elements to use (e.g., 3 nodes, 6 nodestriangular elements, etc.). This is achieved with the menu optionMesh/Mesh Setup. . . .




[UE-T4-46]


3b - Discretize the boundaries of the model (the excavation,the external boundary, the material boundary, etc.). Note thatthe discretization of the boundaries defines the position of thenodes of the future mesh on these boundaries, and therefore, con-trols the density of elements when the mesh is actually gener-ated in the next sub-step. This is achieved with the menu optionMesh/Discretize, Mesh/Custom Discretize, etc.




[UE-T4-47]


3c - Mesh the model. This is achieved with the menu optionMesh/Mesh. Note that the mesh can be improved/modified (e.g.,density and shape of elements in the mesh) by using the optionsMesh/Increase Mesh Element Density and Mesh/Mapped Mesh-ing —this last option is useful to get regular (or mapped) meshes.




[UE-T4-48]


4- Define loading of the model (menu option Loading. . . ). Examples ofloading involve field loading (initial in-situ stresses before excavation)and distributed loading at the boundaries of the model (e.g., to representexternal loading such as surcharges).




[UE-T4-49]


5- Define the boundary restrains for the model (menu option Displace-ments. . . ). Besides options to apply restrains in the x, y or both, x andy directions, displacement boundary conditions (a fine value displace-ment) can be specified for the boundaries.




[UE-T4-50]


6- Specify material properties to be used in the model (menu optionProperties/Define Materials. . . ).




[UE-T4-51]


7- Assign material properties to different regions in the model (menuoption Properties/Assign Properties. . . ).




[UE-T4-52]


8- Solve the model (menu option Analysis/Compute).




[UE-T4-53]

Extracting results from a model with the program Interpret (1)




[UE-T4-54]





[UE-T4-55]





[UE-T4-56]


• All references and web sites mentioned in previous slides.

• If interested in the Finite Element Method, consider registering in thecourse CE 8401, ‘Fundamentals of Finite Element Method’, offered byProfessor H. Stolarski at the Department of Civil Engineering.

• If interested in the Boundary Element Method, consider registeringin the courses CE 8336, ‘Boundary Element Method’ (Parts I and II),offered by Professor S. Crouch and Professor S. Mogilevskaya at theDepartment of Civil Engineering.

• To learn all features of Phase2, attempt completing all 18 tutorialsavailable from the menu option Help Topics/Contents/Tutorials. To re-solve successfully the homework on this topic (Introduction to Numer-ical Modelling) distributed in class, complete at least the first tutorial(‘01 Quick Start Tutorial’).





[UE-T5-1]


Topic 5:

Strength and inelastic deformation of rock

written by






[UE-T5-2]

Strength of intact rock samples from triaxial tests

From Hoek E. and E.T. Brown (1980).




[UE-T5-3]Strength of intact rock. Hoek-Brown and Mohr-Coulomb models

The Hoek-Brown failure criterion is

σ1 = σ3 + σci

√mi

σ3

σci

+ 1 (1)

where σci is the unconfined compression strength of the rock and mi is afitting parameter determined from triaxial test results (see, for example,Hoek and Brown, 1980).

The Mohr-Coulomb failure criterion is

σ1 = Kφσ3 + σc (2)

where σc is the unconfined compression strength of the rock and Kφ isthe passive reaction coefficient (a function of the friction angle φ).




[UE-T5-4]

Strength of rock in terms of σ1 vs. σ3 and τs vs. σn components (1)




[UE-T5-5]

Strength of rock in terms of σ1 vs. σ3 and τs vs. σn components (2)

The following relationships, derived from geometrical considerations ina Mohr circle (see previous slide), allow to relate the shear and normalstresses with the principal stresses at the state of failure

σn = σ1 + σ3

2− σ1 − σ3

2

dσ1/dσ3 − 1

dσ1/dσ3 + 1(3)

τs = (σ1 − σ3)

√dσ1/dσ3

dσ1/dσ3 + 1(4)

Note: the equations above were presented in Balmer (1952), and arereferred to as Balmer’s equations in Rock Mechanics literature —seeHoek and Brown (1980).




[UE-T5-6]

Mohr-Coulomb failure criterion in σ1 vs. σ3 and τs vs. σn spaces

For a Mohr-Coulomb material, the failure criterion in terms of shear andnormal stresses is

τs = σn tan φ + c (5)

Balmer’s equations allow the following relationships between the pa-rameters Kφ and σc (in equation 2) and φ and c (in equation 5) to beobtained

Kφ = 1 + sin φ

1 − sin φ(6)

and

c = 1 − sin φ

2 cos φσc = σc

2√

Kφ

(7)

Deformability of intact rock. The elastic perfectly plastic model (1)

Deformability of intact rock. The elastic perfectly plastic model (2)




[UE-T5-9]

Plastic deformation. Flow rule (1)

According to plasticity theory —e.g., Hill (1950), Kachanov (1971)—the plastic strain (rate) vector is defined as the gradient of the potentialH(σ1, σ3), i.e.,

εp

1 = λ∂H

∂σ1(8)

εp

3 = λ∂H

∂σ3(9)

where λ is a positive scalar.




[UE-T5-10]

Plastic deformation. Flow rule (2)

We can consider, for example, a linear flow rule, for which the potentialH(σ1, σ3) is

H(σ1, σ3) = σ1 − σ3Kψ = 0 (10)

where Kψ is a function of the dilation angle ψ

Kψ = 1 + sin ψ

1 − sin ψ(11)

(Note the similarity of the coefficient Kψ in equation (11) with thecoefficient Kφ in equation 6)

From equations (8), (9) and (10)

εp

1 = λ (12)

εp

3 = −λKψ (13)

and therefore

εp

3 /εp

1 = −Kψ (14)

Thus, if the dilation angle is ψ = 0◦, then Kψ = 1 and εp

3 = −εp

1 andtherefore there is no material volume change in the plastic state.

If, for example, the dilation angle is ψ = 30◦ Kψ = 3 and ε3 = −3ε1,then the material shows significant volume expansion in the plastic state.

Note that a condition of mechanical stability requires that ψ ≤ φ.




[UE-T5-11]Plastic deformation. Flow rule (3)

For the triaxial test introduced in previous slides, the behavior of thematerial in the plastic state is as follows:

Note: the slopes indicated in the diagrams above are obtained from theanalytical solution of the elasto-plastic problem of material loading intriaxial conditions. The demonstration is simple but too lengthy to beincluded in these notes.




[UE-T5-12]

Application example

Triaxial compression test in Hoek-Brown material (1)




[UE-T5-13]

Application example





[UE-T5-14]

Application example





[UE-T5-15]

Strength of rock masses. Generalized Hoek-Brown failure criterion

For the implementation of the generalized form of the Hoek-Brownfailure criterion, see freeware software RocLab (www.rocscience.com)[The Help menu in RocLab provides a link to the reference Hoek,Carranza-Torres and Corkum (2002) where the equations above are de-scribed.]




[UE-T5-16]

Strength of rock masses. Charts for the determination of GSI (1)

General charts for determination of the Geological Strength Index (GSI)have been introduced in Hoek, Kaiser and Bawden (1995) and Hoek andBrown (1997). The chart above is from the freeware software RocLab,available at www.rocscience.com.




[UE-T5-17]

Strength of rock masses. Charts for the determination of GSI (2)

Marinos and Hoek (2001) and Hoek, Marinos and Marinos (2005) dis-cuss in detail the estimation of GSI for heterogeneous, undisturbed,sedimetary rock masses such as Flysch and Molasses. These commontype of sedimentary rocks are found in northern Greece, where more than600 km of tunnels are being completed as part of one of world largeston-going highway projects (www.egnatia.gr). The chart above is fromthe freeware software RocLab, available at www.rocscience.com.




[UE-T5-18]

Deformability of rock masses (1)

From analysis of in situ deformability measurements tests from under-ground excavation projects in China and Taiwan, Hoek and Diederichs(2005) propose the following equation for determining the rock massdeformability modulus Erm

Erm = Ei

[0.02 + 1 − D/2

1 + exp(60+15D−GSI

11

)]

(15)

In the equation above, Ei is the deformability modulus of the intact rock,GSI is the Geological Strength Index and D is the disturbance Factor.




[UE-T5-19]

Deformability of rock masses (2)

The diagram below (from Hoek and Diederichs, 2005) shows how theproposed expression plots together with the cases used to derive theexpression

The Hoek-Diederichs relationship is implemented in the freeware soft-ware RocLab (www.rocscience.com), as an alternative expression toanother relationship proposed by Serafim J.L. and Pereira (1983) —therelationship by Serafim and Pereira does not relate the deformability ofrock mass with the GSI, nor the factor D, but with another rating calledthe Bieniawski rock mass rating, RMR (this will be discussed in Topic12, ‘Classification systems for tunnel design’).




[UE-T5-20]

References mentioned in the slides (1)

• Hoek, E. & Brown, E. T. (1980), ‘Underground Excavations in Rock’.London: The Institute of Mining and Metallurgy.

•Balmer, G. (1952), ‘A general analytical solution for Mohr’s envelope’.Am. Soc. Test. Mat. (52), 1260– 1271.

• Hill R. (1950), ‘The Mathematical Theory of Plasticity’. OxfordScience Publications.

• Kachanov, L. M. (1971), ‘Foundations of the Theory of Plasticity’.North Holland Publishing Company.

• Hoek E., C. Carranza-Torres, and B. Corkum (2002), ‘Hoek-Brownfailure criterion – 2002 edition’. In Hammah R. et al. (Eds.), Proceed-ings of the 5th NorthAmerican Rock Mechanics Symposium: NARMS-TAC 2002. Toronto – 10 July 2002, pages 267–273.

• Hoek E., P. K. Kaiser, and W. F. Bawden (1995), ‘Support of Under-ground Excavations in Hard Rock’. Balkema, Rotterdam.




[UE-T5-21]

References mentioned in the slides (2)

• Hoek, E. and E. T. Brown (1997), ‘Practical estimates of rock massstrength’. Int. J. Rock Mech. Min. Sci., 34(8):1165–1186.

• Marinos, P.G. and Hoek, E. (2001), ‘Estimating the geotechnical prop-erties of heterogeneous rock masses such as Flysch’. Bull. Engg. Geol.Env. 60, 85-92.

• Hoek E., P.G. Marinos, V.P. Marinos (2005), ‘Characterization and en-gineering properties of tectonically undisturbed but lithologically variedsedimentary rock masses’. International Journal of Rock Mechanics &Mining Sciences, 42, 277–285.

• Hoek E., M.S. Diederichs (2006), ‘Empirical estimation of rock massmodulus’. International Journal of Rock Mechanics & Mining Sciences43, 203–215.

• Serafim J.L. and Pereira J.P. (1983), ‘Consideration of the geomechan-ical classification of Bieniawski’. Proc. Int. Symp. on EngineeringGeology and Underground Construction, Lisbon. 1(II): 33-44.




[UE-T5-22]

Other recommended references

• Brady B.H.G. and E.T. Brown (2004), ‘Rock Mechanics for Under-ground Mining’, 3rd Edition, Kluwer Academic Publishers.

• Hudson J.A. and Harrison J.P. (1997), ‘Engineering Rock Mechanics.An Introduction to the Principles’. Pergamon.

• Jaeger J. C. and N. G.W. Cook (1979), ‘Fundamentals of rock me-chanics’, John Wiley & Sons.

• Hoek E. (2000), ‘Rock Engineering. Course Notes by Evert Hoek’.Available for downloading at ‘Hoek’s Corner’, www.rocscience.com.

• Hoek, E., Marinos, P. and Benissi, M. (1998), ‘Applicability of the Ge-ological Strength Index (GSI) classification for very weak and shearedrock masses. The case of the Athens Schist Formation’. Bull. Engg.Geol. Env. 57(2), 151-160.

• Hoek, E. and Karzulovic, A. (2000), ‘Rock-Mass Properties for Sur-face Mines’. In W. Hustrulid et al. (Eds.), Slope Stability in SurfaceMining, pp. 59–67. Littleton, CO: Society for Mining, Metallurgicaland Exploration (SME).





[UE-T6-1]


Topic 6:

Elasto-plastic solution of a circular tunnel

written by






[UE-T6-2]

Application examples of elasto-plastic solution of circular openings




[UE-T6-3]

Elasto-plastic solution of a circular opening. Problem statement

If pi < pcri the problem is characterized by two regions:

1- Elastic region r ≥ Rp

2- Plastic region r ≤ Rp

If pi ≥ pcri the problem is fully elastic (the solution is given by Lamé’s

solution).




[UE-T6-4]

The critical internal pressure pcri (1)

The critical internal pressure pcri can be found as the intersection of

the failure envelope and Lamé’s representation of the stress state in thereference system σθ ∼ σ1 vs σr ∼ σ3.




[UE-T6-5]


Lamé’s solution for stresses, with σθ replaced by σ1 and σr replaced byσ3, is

σ1 = σo + (σo − pi)

(R

r

)2

(1)

σ3 = σo − (σo − pi)

(R

r

)2

(2)

Equating the last part of the right-hand side of the equations above wehave

σ1 = 2σo − σ3 (3)

The failure criterion of the material, defines the relationship betweenthe principal stresses σ1 and σ3 at failure, and can be written as follows

σ1 = f (σ3) (4)

where f is a linear function (of the coefficients Kφ and σc) in the caseof Mohr-Coulomb material, or a parabolic function (of the coefficientsmi and σci) in the case of Hoek-Brown material.




[UE-T6-6]


Equating the right-hand side of equations (3) and (4), making σ3 = pcri

—see diagram in previous slide— the critical internal pressure pcri is

found from the solution of the following equation

2σo − pcri = f (pcr

i ) (5)

The equation above, that can be solved in closed-form for commonlyused failure functions f , defines the critical internal pressure belowwhich the plastic zone develops around the tunnel —this critical internalpressure is also equal to the radial stress at the elasto-plastic boundary(see previous diagram).




[UE-T6-7]

Solution for the elastic region (r ≥ Rp)

The solution for stresses and displacements in the elastic region is knownfrom Lame’s solution

σr = σo − (σo − pcr

i

) (Rp

r

)2

(6)

σθ = σo + (σo − pcr

i

) (Rp

r

)2

(7)

ur = − 1

2G

(σo − pcr

i

) R2p

r(8)

Note that in the equations above, the radius of the opening is Rp and theinternal pressure is pcr

i .




[UE-T6-8]

Solution for the plastic region (r ≤ Rp). Hoek-Brown material (1)

A closed-form (exact) solution is possible when the coefficient a is equalto 0.5 in the generalized Hoek-Brown criterion.

The failure criterion to be considered is

F = σ1 − σ3 − σci

√mb

σ3

σci

+ s = 0 (9)

With the failure criterion (9), the critical internal pressure pcri is obtained

from the solution of equation (5) and results

pcri = σci mb

16

1 −

√1 + 16

(σo

σci mb

+ s

m2b

)

2

− s σci

mb

(10)

The extent of the failure zone is

Rp = R exp

[2

(√pcr

i

σci mb

+ s

m2b

−√

pi

σci mb

+ s

m2b

) ](11)




[UE-T6-9]


The solution for the radial stress is

σr = mbσci

(√

pcri

σci mb

+ s

m2b

+ 1

2ln

(r

Rp

))2

− s

m2b

(12)

The solution for the hoop stress is

σθ = σr + σci

√mb

σr

σci

+ s (13)

The solution for the radial displacement is

ur = 1

1 − A1

[(r

Rp

)A1

− A1r

Rp

]ur(1) (14)

+ 1

1 − A1

[r

Rp

−(

r

Rp

)A1]

u′r(1)

−Rp

2G

(σci mb

4

) A2 − A3

1 − A1

r

Rp

[ln

(r

Rp

)]2

−Rp

2G(σci mb)

[A2 − A3

(1 − A1)2

√pcr

i

σci mb

+ s

m2b

− 1

2

A2 − A1A3

(1 − A1)3

]

×[(

r

Rp

)A1

− r

Rp

+ (1 − A1)r

Rp

ln

(r

Rp

)]




[UE-T6-10]


where the coefficients ur(1) and u′r(1) are

ur(1) = −Rp

2G

(σo − pcr

i

)(15)

u′r(1) = Rp

2G

(σo − pcr

i

)(16)

and for a linear flow rule, the coefficients A1, A2 and A3 are

A1 = −Kψ (17)

A2 = 1 − ν − νKψ

A3 = ν − (1 − ν)Kψ

with

Kψ = 1 + sin ψ

1 − sin ψ(18)

where ψ is the dilation angle.




[UE-T6-11]

Solution for the plastic region (r ≤ Rp). Mohr-Coulomb material (1)

The Mohr-Coulomb yield condition is

F = σ1 − Kφσ3 − σc = 0 (19)

In the equation above the coefficient Kφ is related to the friction angleφ according to

Kφ = 1 + sin φ

1 − sin φ(20)

The unconfined compression strength σc is related to the cohesion c andthe coefficient Kφ as follows

σc = 2c√

Kφ (21)

The critical internal pressure pcri below which the failure zone develops

is

pcri = 2

Kφ + 1

(σo + σc

Kφ − 1

)− σc

Kφ − 1(22)

The extent Rp of the failure zone is

Rp = R

[pcr

i + σc/(Kφ − 1

)pi + σc/

(Kφ − 1

)]1/(Kφ−1)

(23)




[UE-T6-12]


The solution for the radial stresses field σr is given by the followingexpression

σr =(

pcri + σc

Kφ − 1

) (r

Rp

)Kφ−1

− σc

Kφ − 1(24)

The solution for the hoop stresses field σθ is given by the followingexpression

σθ = Kφ

(pcr

i + σc

Kφ − 1

) (r

Rp

)Kφ−1

− σc

Kφ − 1(25)

The solution for the radial displacement field ur is given by the followingexpression

ur = 1

1 − A1

[(r

Rp

)A1

− A1r

Rp

]ur(1) (26)

− 1

1 − A1

[(r

Rp

)A1

− r

Rp

]u′

r(1)

−Rp

2G

A2 − A3Kφ

(1 − A1)(Kφ − A1)

(pcr

i + σc

Kφ − 1

)

×[(A1 − Kφ)

r

Rp

− (1 − Kφ)

(r

Rp

)A1

+ (1 − A1)

(r

Rp

)Kφ

]




[UE-T6-13]


where the coefficients ur(1) and u′r(1) are

ur(1) = −Rp

2G

(σo − pcr

i

)(27)

u′r(1) = Rp

2G

(σo − pcr

i

)(28)

and for a linear flow rule,

A1 = −Kψ (29)

A2 = 1 − ν − νKψ

A3 = ν − (1 − ν)Kψ

with

Kψ = 1 + sin ψ

1 − sin ψ(30)

where ψ is the dilation angle.




[UE-T6-14]

Solution for the plastic region (r ≤ Rp). Tresca material (1)

A Tresca material is a particular case of Mohr-Coulomb material inwhich the friction angleφ is equal to zero. In such case the coefficient Kφ

becomes one (see equation 20), and singularities appear in the solutionfor stresses and displacements listed earlier (equations 22 through 26).

The solution for Tresca material can be obtained by taking the limit ofthe expressions for the Mohr-Coulomb failure criterion (equations 22through 26) when Kψ → 1, applying L’Hospital rule, as needed.

The resulting expressions for Tresca material are given in the followingslides.




[UE-T6-15]


The Tresca yield condition is

F = σ1 − σ3 − σc = 0 (31)

where the unconfined compression strength σc is related to the cohesionc as follows

σc = 2c (32)

The critical internal pressure pcri below which the failure zone develops

is

pcri = σo − σc

2(33)

The extent Rp of the failure zone is

Rp = R exp

[pcr

i − pi

σc

](34)

The solution for the radial stresses field σr is given by the followingexpression

σr = pcri + σc ln

(r

Rp

)(35)

The solution for the hoop stresses field σθ is given by the followingexpression

σθ = pcri + σc ln

(r

Rp

)+ σc (36)




[UE-T6-16]


The solution for displacements is

ur = 1

1 − A1

[(r

Rp

)A1

− A1r

Rp

]ur(1) (37)

− 1

1 − A1

[(r

Rp

)A1

− r

Rp

]u′

r(1)

−Rp

2G

A2 − A3

(1 − A1)2σc

[(r

Rp

)A1

− r

Rp

+ (1 − A1)r

Rp

ln

(r

Rp

)]

In the equation above, the coefficients ur(1), u′r(1), A1, A2 and A3 are

the same coefficients defined by equations 27 through 30.




[UE-T6-17]

Application examples of the exact elasto-plastic solutions and

comparison with numerical models

The closed-form solutions presented earlier for Hoek-Brown and Mohr-Coulomb materials will be compared with results given by the finitedifference numerical software FLAC (www.itascacg.com).

The mesh used in the numerical models, the description of two particu-lar problems of tunnel excavation in Hoek-Brown and Mohr-Coulombmaterials and the corresponding results (analytical and numerical) aredescribed in the following slides.




[UE-T6-18]




[UE-T6-19]

Example of elasto-plastic analysis. Hoek-Brown material (1)

Problem definition




[UE-T6-20]


Solution for radial and hoop stresses




[UE-T6-21]


Solution for radial displacement




[UE-T6-22]

Example of elasto-plastic analysis. Mohr-Coulomb material (1)

Problem definition




[UE-T6-23]


Solution for radial and hoop stresses




[UE-T6-24]


Solution for radial displacement




[UE-T6-25]Effect of far-field loading on the shape of failure zone (1)




[UE-T6-26]

Effect of far-field loading on the shape of failure zone (2)

The chart is reproduced from Detournay and St. John (1988). As indi-cated in the graph, Po is the mean far-field stress, Po = (σ o

v + σoh )/2,

and So is the deviator far-field far-stress, So = (σ ov −σo

h )/2. The chart isvalid for a Mohr-Coulomb failure criterion with friction angle φ = 30◦and unconfined compression strength σc.




[UE-T6-27]


(The solution above is presented in Detournay and Fairhurst, 1987).




[UE-T6-28]


Displacements at the springline and crown of the tunnel

(The solution above is presented in Detournay and Fairhurst, 1987).




[UE-T6-29]

Recommended references (1)

Books/manuscripts discussing elasto-plastic solutions for tunnel prob-lems:


• Hoek E., 2000, ‘Rock Engineering. Course Notes by Evert Hoek’.Available for downloading at ‘Hoek’s Corner’, www.rocscience.com.

• Hudson J.A. and Harrison J.P. (1997), ‘Engineering Rock Mechanics.An Introduction to the Principles’. Pergamon.

• Jaeger J. C. and N. G.W. Cook, 1979, ‘Fundamentals of rock mechan-ics’, John Wiley & Sons.

• U.S. Army Corps of Engineers, 1997, ‘Tunnels and shafts in rock’.Available for downloading at www.usace.army.mil




[UE-T6-30]


For elasto-plastic solution of cavities in Hoek-Brown materials:

• Carranza-Torres, C. and C. Fairhurst (1999), ‘The elasto-plastic re-sponse of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion’. International Journal of Rock Mechanics andMining Sciences 36(6), 777–809.

• Carranza-Torres, C. (2004), ‘Elasto-plastic solution of tunnel prob-lems using the generalized form of the Hoek-Brown failure criterion’.Proceedings of the ISRM SINOROCK 2004 Symposium China, May2004. Edited by J.A. Hudson and F. Xia-Ting. International Journal ofRock Mechanics and Mining Sciences 41(3), 480–481.




[UE-T6-31]


For elasto-plastic solutions of cavities in Mohr-Coulomb materials, in-cluding cases of non-uniform far-field stresses:

• Detournay E. and C. St. John (1988), ‘Design charts for a deepcircular tunnel under non-uniform loading’. Rock Mechanics and RockEngineering, 21:119–137.

• Detournay E. and C. Fairhurst (1987), ‘Two-dimensional elasto-plasticanalysis of a long, cylindrical cavity under non-hydrostatic loading’. Int.J. Rock Mech. Min. Sci. & Geomech. Abstr., 24(4):197–211.

• Detournay E. (1986), ‘Elastoplastic model of a deep tunnel for arock with variable dilatancy’. Rock Mechanics and Rock Engineering,19:99–108.

• Carranza-Torres, C. (2003), ‘Dimensionless graphical representationof the elasto-plastic solution of a circular tunnel in a Mohr-Coulombmaterial’. Rock Mechanics and Rock Engineering 36(3), 237–253.




[UE-T6-32]


Some classic papers/books on the topic of elasto-plastic solutions oftunnel problems:

• Brown E.T., J. W. Bray, B. Ladanyi, and E. Hoek (1983), ‘Groundresponse curves for rock tunnels’. ASCE J. Geotech. Eng. Div.,109(1):15–39.

• Duncan-Fama (1993). ‘Numerical modelling of of yield zones inweak rocks’. In J. A. Hudson, E. T. Brown, C. Fairhurst, and E. Hoek,editors, Comprehensive Rock Engineering. Volume 2. Analysis andDesign Methods., pages 49–75. Pergamon Press.

• Salençon J. (1969) ‘Contraction quasi-statique d’une cavité a symétriesphérique ou cylindrique dans un milieu élastoplastique’. Annls PontsChauss. 4:231–236.

• Panet M. (1995), ‘Calcul des Tunnels par la Méthode de Convergence-Confinement’. Press de l’École Nationale des Ponts et Chaussées.





[UE-T7-1]


Topic 7:

Review of some fundamental equations ofmechanics of beams

written by






[UE-T7-2]

Equilibrium of forces and bending moments in a beam

dQ

dx+ py = 0 (1)

dN

dx+ px = 0 (2)

dM

dx+ Q = 0 (3)

The equations above express equilibrium conditions for forces in the i)vertical and ii) horizontal directions and iii) bending moments, respec-tively.




[UE-T7-3]

Relationships between bending moment and deflection

M = −Kd2uy

dx2(4)

where K = EI/(1 − ν2) for plane strain and K = EI for plane stressconditions (I is the moment of inertia of the beam section —per unitlength of beam in the out-of-plane direction, in the case of plane strain).




[UE-T7-4]

Relationships between thrust and axial displacement

N = −Dduy

dx(5)

where D = Eh/(1 − ν2) or D = Eh for plane-strain or plane-stressconditions, respectively (h is the height of the beam section).




[UE-T7-5]

Solution of beam problems

For given values of px and py, we have five unknowns —the quantitiesN , Q, M , ux and uy.

We have five equations —equations (1) through (5)— to solve for thefive unknown functions.

Particular solutions are obtained by application of the appropriate bound-ary conditions.




[UE-T7-6]

Solution of beam problems. Application Example (1)

Since px = 0 and N = 0 at x = 0 and x = L, from equation (2) wehave N = 0; also, from equation (5), ux = 0.

Therefore the unknown functions in the problem are uy, M and Q.




[UE-T7-7]


Combining equations (1), (3) and (4), the following differential equationfor vertical the displacement is obtained,

Kd4uy

dx4+ py = 0 (6)

Four boundary conditions are needed to solve the 4th order differentialequation. These are

at x = 0 → uy = 0 (7)

x = L → uy = 0

x = 0 → M = 0 or d2uy/dx2 = 0

x = L → M = 0 or d2uy/dx2 = 0




[UE-T7-8]


Solution of the differential equation (6) with boundary conditions (7)gives

uy = − pyx

24K

(L3 + x3 − 2Lx2) (8)

or expressed in dimensionless form,

uy

L

K

pyL3= − 1

24

x

L

[1 +

(x

L

)3 − 2(x

L

)2]

(9)

For plane stress conditions, K = EI , where E is the Young’s modulusand I is the the moment of inertia of the beam section. Note, for arectangular section of width b and height h the moment of inertia isI = bh3/12.

With the solution for the vertical displacement (equation 8), using equa-tions (4) and (3), the solution for bending moment and shear force are,respectively

M

pyL2= −1

2

x

L

[1 − x

L

](10)

Q

pyL= 1

2

x

L

[1 − 2

x

L

](11)




[UE-T7-9]





[UE-T7-10]

Equilibrium of forces and bending moments for a circular ring

dQ

dθ− N + prR = 0 (12)

dN

dθ+ Q + pθR = 0 (13)

dM

dθ+ QR = 0 (14)

The equations above express equilibrium conditions for forces in the i)radial and ii) tangential directions and iii) bending moments, respec-tively.




[UE-T7-11]

Relationships between bending moment and deflection (circular ring)

M = − K

R2

d2ur

dθ2(15)

where K = EI/(1 − ν2) for plane-strain and K = EI for plane-stressconditions (I is the moment of inertia of the beam section —per unitlength of beam in the out-of-plane direction, in the case of plane strain).




[UE-T7-12]

Relationships between thrust and axial displacement (circular ring)

N = −D

(ur + duθ

dθ

)(16)

where D = Et/(1 − ν2) or D = Et for plane-strain or plane-stressconditions, respectively (t is the thickness of the section).




[UE-T7-13]


• Flügge, W. (1967), ‘Stresses in Shells’. Springer-Verlag NewYorkInc.

• Den Hartog J.P. (1961), ‘Strength of Materials’. Dover Publications,Inc. New York.

• Pflüger, A. (1961), ‘Elementary Statics of Shells’. Second Edition,F.W. Dodge Corporation, New York.

• Timoshenko, S. (1955), ‘Strength of Materials’. Third Edition, VanNostrand. New York.





[UE-T8-1]


Topic 8:

Elastic solution of a closed annular support

written by






[UE-T8-2]

Problem Statement

The mean loading is

q = qx + qy

2(1)

while the ratio of horizontal to vertical load is

k = qx

qy

(2)

Note: Plane strain conditions assumed. Analysis considers a ‘slice’ ofannular ring of unit length in the z direction.




[UE-T8-3]

Sign convention and nomenclature

Note: Sign convention for bending moment, shear force, thrust, radialand tangential displacements is in agreement with the sign conventiondiscussed in notes ‘Review of some fundamental mechanics of beams’(Topic 7).




[UE-T8-4]

Solution of thrust, bending moment and shear force

The scaled thrust is

N

qR= 1 − k − 1

k + 1cos 2θ (3)

The scaled bending moment is

M

qR2= −1

2

k − 1

k + 1cos 2θ (4)

The scaled shear force is

Q

qR= −k − 1

k + 1sin 2θ (5)




[UE-T8-5]

Solution of radial and tangential displacement

The scaled displacement in the radial direction is

ur

qR

E

1 − ν2= − 12

12(t/R) + (t/R)3− 2

(t/R)3

k − 1

k + 1cos 2θ (6)

The scaled displacement in the tangential direction is

uθ

qR

E

1 − ν2= 4 + (t/R)2

4(t/R)3

k − 1

k + 1sin 2θ (7)




[UE-T8-6]

Particular case of uniform loading

If qx = qy = q, then k = 1 and the solution for the thrust results

N = qR (8)

This is the same expression found as a particular case of Lamé’s solu-tion (see equation 8, page T3-4, in notes ‘Elastic solution of a circulartunnel’).

In the case of uniform loading, the bending moments and shear forcesare both zero, i.e.,

M = Q = 0 (9)

The tangential displacement is also zero (i.e., uθ = 0) and the radialdisplacement takes the simple form

ur = −1 − ν2

E

12 qR

12(t/R) + (t/R)3(10)




[UE-T8-7]

Thrust. Graphical representation




[UE-T8-8]

Bending moment. Graphical representation




[UE-T8-9]

Shear force. Graphical representation




[UE-T8-10]

Radial displacement. Graphical representation




[UE-T8-11]

Tangential displacement. Graphical representation




[UE-T8-12]

Plane strain analysis of composite sections. Problem statement




[UE-T8-13]

Plane strain analysis of composite sections. Homogenized section

The thickness and Young’s modulus of the equivalent (homogenized) section are,respectively

heq = 2

√3CACI

CA

(11)

Eeq =√

3

6

CA2

√CACI

(12)

where

CA = n (A1E1 + A2E2) (13)CI = n (I1E1 + I2E2) (14)




[UE-T8-14]

Plane strain analysis of composite sections.Distribution of thrust to original components

N1 = N

n

A1E1

A1E1 + A2E2(15)

N2 = N

n

A2E2

A1E1 + A2E2(16)




[UE-T8-15]

Plane strain analysis of composite sections.Distribution of bending moment to original components

M1 = M

n

I1E1

I1E1 + I2E2(17)

M2 = M

n

I2E2

I1E1 + I2E2(18)




[UE-T8-16]

References

• Flügge, W. (1967), ‘Stresses in Shells’. Springer-Verlag New YorkInc., 1967.

• Den Hartog J.P. (1961), ‘Strength of Materials’. Dover Publications,Inc. New York.

• Pflüger, A. (1961), ‘Elementary Statics of Shells’. Second Edition,F.W. Dodge Corporation, New York.

• Timoshenko, S. (1955), ‘Strength of Materials’. Third Edition, VanNostrand. New York.

Note: the solutions presented in previous slides were derived by oneof the authors (Dr. Carranza-Torres). Equations presented here donot appear explicitly in the references above —or other references theauthors may be aware of.





[UE-T9-1]


Topic 9:

Tunnel support systems. Technologies and design.The Convergence-Confinement Method

written by






[UE-T9-2]

Classification of tunnel supports in terms of time of installation (1)




[UE-T9-3]

Classification of tunnel support in terms of time of installation (2)




[UE-T9-4]

Common support systems used in tunnel construction

• Steel ribs (or steel sets) and lattice girders.

• Shotcrete or sprayed concrete.

• Cast-in-place concrete.

• Prefabricated segmental lining (used with mechanized excavation).

Note: Rockbolts do not fall into the category of support systems but intothe category of reinforcement systems —they will be treated separatelyin these series of notes.




[UE-T9-5]

Steel ribs and lattice girders. Technological aspects (1)

Bracing bars, wood or steel plates are normally installed between steelsets and lattice girders.

For squeezing ground, sliding joints and sliding arches are emplacedbetween segments conforming the steel section.




[UE-T9-6]


See explanation in the next slide.




[UE-T9-7]


Description photographs in previous slide

The previous slide shows photographs of tunnel sections supported with steel sets.Photograph (a) shows wood blocks used between the steel ribs and the rock (shotcreteis seen ahead of the steel sets). Photograph (b) shows steel sets failing under extremeground loading. Photograph (c) shows heavy steel sets used while traversing a faultzone (note the bracing bars between steel sets).

The photographs have been taken Dr. Evert Hoek, Rock Mechanics Consultant(www.rocscience.com/hoek/Hoek.asp) at various underground sites. (a) Drainagetunnel at Chuquicamata mine, Antofagasta, Chile. (b) Drifts at Sullivan mine, BritishColumbia, Canada. (c) Headrace tunnel for at Victoria Hydroelectric Scheme, SriLanka.




[UE-T9-8]Steel ribs and lattice girders. Technological aspects (4)





[UE-T9-9]



The photographs in the previous slide show the use of circular steel sets (with slidingjoints and shotcrete) as a mean of supporting a tunnel in highly squeezing ground atthe Yacambu-Quibor project, Lara State, Venezuela. The Yacambu-Quibor tunnel isa ∼24 km hydraulic tunnel of mean diameter ∼4 m with maximum overburden of1,200 m excavated in low strength phyllites and schists. The tunnel has been called‘the most difficult modern tunnel ever to excavate’ —excavation has been taking placesince the late 70s (by late 2004, ∼3.5 km of tunnel were still to be excavated).

The photographs in the previous slide have been taken by Drs. Mark Diederichs, BrentCorkum and Carlos Carranza-Torres, during a visit to the project in 2004, togetherwith Dr. Evert Hoek and Dr. Rafael Guevara (members of the panel of experts in theproject).









[UE-T9-11]



The photographs in the previous slide show the sequence of construction of steel setsand sliding joints used as primary support in the Yacambu-Quibor tunnel, Lara State,Venezuela. Photograph (a) shows the steel section before being bent into a curvedsegment (note the steel plates welded to the central flange of the section, to avoidbucking during the process of bending). Photograph (b) shows the steel section duringan early stage of bending in the press. Photograph (c) shows the curved segment afterfurther pressing (note that the oscillations of the upper and lower flanges in photograph(b) have been removed). Photograph (d) shows the different segments comprising thesteel section alienated for assembly. Photograph (e) shows the final assembly of thecircular steel set. Note the sliding joints installed between different segments.

These photographs have been taken by Drs. Mark Diederichs, Brent Corkum andCarlos Carranza-Torres, during a visit to the project in 2004, together with Dr. EvertHoek and Dr. Rafael Guevara (members of the panel of experts in the project).









[UE-T9-13]



The photographs in the previous slide show views steel sets used in the Driskos tunnelof Egnatia project, Greece (www.egnatia.gr), a tunnel excavated in weak rock. Photo-graph (a) shows shotcrete being applied in the vicinity of the (top heading) front. Notethe forepoling and fiberglass reinforcement used in the front, as a means of stabilizingthe front during excavation. Photograph (b) shows the complete section after the lowerbench has been excavated and supported.

The photographs described above have been taken by Prof. Paul Marinos (from theDepartment of Geotechnical Engineering, School of Civil Engineering, National Tech-nical University ofAthens, http://users.ntua.gr/marinos/) who is a member of the Panelof Experts in the Egnatia project.









[UE-T9-15]



The photographs in the previous slide show views of lattice girders used in tunnels of theEgnatia project, Greece (www.egnatia.gr). During excavation, lattice girder sectionsare delivered in ‘segments’ to the front of the tunnel, where they are assembled andinstalled.

The photographs have been taken by Prof. Evert Hoek (and independent rock mechan-ics consultant, www.rocscience.com/hoek/Hoek.asp) who is a member of the Panel ofExperts in the Egnatia project.









[UE-T9-17]



The photographs in the previous slide show the use of steel sets in tunnels. In pho-tograph (a) steel plates are emplaced between steel ribs. In photograph (b) bracingbars are emplaced between steel ribs (in this case, a wire mesh has also been installedbefore shotcreting the space between rock and steel sets).

The photographs were taken by Ing. Luca Perrone, Tunnel Design Engineer, GeodataSpa., Torino, Italy (www.geodata.it). Photograph (a) is at the portal for the St. Martinde la Porte tunnel (∼1,400 m), in France —this is an access tunnel for the futureTorino-Lyon railway system (to formally start construction this year). Photograph(b) is at the front of Traffic Release Tunnelling System (∼1,400 m), Western KualaLumpur, Malaysia.




[UE-T9-18]


To learn more about the system see:

Chapter 5, ‘Design of Steel Ribs and Lattice Girders’ in document‘Tunnels and shafts in rock’, U.S.Army Corps of Engineers, 1997 (avail-able for downloading at www.usace.army.mil).

‘Use of arches in the construction of underground works’, DocumentNo 27, 1978, Recommendations from AFTES (available for download-ing at www.aftes.asso.fr).

For use of sliding joints and sling arches, see Chapter 12, ‘Tunnelsin weak rock’, in document ‘Rock Engineering. Course Notes by EvertHoek’ (available for downloading at ‘Hoek’s Corner’,www.rocscience.com).

To find supliers of the system in the market see:

American Commercial Inc. (www.americancommercial.com) — seepages ‘Steel ribs’, ‘Liner Plates’ and ‘Lattice Girders’.

Tunnel Builder (www.tunnelbuilder.com). Go to ‘Suppliers’ andchoose ‘Support’.

InfoMine, Mining Intelligence andTechnology. (www.infomine.com).Go to ‘Suppliers’ and search for ‘Steel Ribs’, ‘Lattice Girders’, etc. (askeyword).




[UE-T9-19]

Shotcrete or sprayed concrete. Technological aspects (1)

Shotcrete is frequently applied on a wire mesh bolted to the rock face(wire mesh acts as reinforcement).

Steel fibers are sometimes added to the shotcrete mixture to increasethe strength of the shotcrete.




[UE-T9-20]Shotcrete or sprayed concrete. Technological aspects (2)





[UE-T9-21]



The photographs in the previous slide shows shotcrete used as support for an under-ground excavation. Photograph (a) shows a drift supported by steel sets near the front.A robotic sprayer is applying shotcrete on top of a wire mesh between steel sets. Pho-tograph (b) and (c) show shotcrete with fiber reinforcement (the fibers are the steelwires embeded in the mortar).

The photographs have been taken by Prof. Mark Diederichs, from the Geological En-gineering Group at Queen’s University (www.geol.ca), also an independent consultant,at Kidd Creek Mine, near Timmins, Ontario.




[UE-T9-22]



Document ‘Standard practice for shotcrete’, U.S. Army Corps ofEngineers, 1993 (available for downloading at www.usace.army.mil).

‘Sprayed Concrete — Technology and Practice’, Document No 1,1974, Recommendations from AFTES (available for downloading atwww.aftes.asso.fr).

‘Design of sprayed concrete for underground support’, Document No164, 2001, Recommendations from AFTES (available for downloadingat www.aftes.asso.fr).

Chapter 15, ‘Shotcrete support’, in document ‘Rock Engineering.Course Notes by Evert Hoek’ (available for downloading at ‘Hoek’sCorner’, www.rocscience.com).


American Commercial Inc. (www.americancommercial.com) — seepages ‘Hany’ and ‘Aliva’.


InfoMine, Mining Intelligence andTechnology. (www.infomine.com).Go to ‘Suppliers’ and search for ‘Shotcrete’.




[UE-T9-23]

Cast-in-place concrete. Technological aspects (1)

Traditionally, the use of cast in place concrete as a tunnel supportmethod has followed standard technological practices in general civilengineering works (e.g., standards regarding material component mix-tures, additives, curing, etc.).

For the case of final support, considering that the concrete structureworks mostly in compression, the use of plain concrete (i.e., massiveunreinforced concrete) is also a standard practice in tunnel construction—see ‘The use of plain concrete in tunnels’, recommendation byAFTES(full reference in the last slide on this topic).




[UE-T9-24]Cast-in-place concrete. Technological aspects (2)

The photographs above show views of cast-in-place concrete support used in TunnelTazon (6700 m), Central Railway System, Caracas, Venezuela. The photographs havetaken by Ing. Luca Perrone, Tunnel Design Design Engineer, Geodata Spa., Torino,Italy (www.geodata.it).




[UE-T9-25]

Cast-in-place concrete. Technological aspects (3)


Document ‘Standard practice for concrete for civil works structures’,U.S. Army Corps of Engineers, 1994 (available for downloading atwww.usace.army.mil).

‘The use of plain concrete in tunnels’, Document No 149, 1998,Recommendations from AFTES (available for downloading atwww.aftes.asso.fr).



InfoMine, Mining Intelligence andTechnology. (www.infomine.com).Go to ‘Suppliers’ and search for ‘concrete’.




[UE-T9-26]Pre-fabricated concrete blocks. Technological aspects (1)

The photographs above show views of pre-cast concrete blocks used as support intunnels of the Light Rail System at the Minneapolis-St.Paul International airport.The photographs have been reproduced from the article ‘Design and Constructionof Minneapolis-St.Paul International Airport Precast Concrete Tunnel System’, byJohnson R.M. et al., published in Precast-Prestressed Concrete Institute Journal, Vol.48, No 5, September/October 2003.




[UE-T9-27]

Pre-fabricated concrete blocks. Technological aspects (2)


Chapter 5, ‘Construction of Tunnels and Shafts’ in document ‘Tunnelsand shafts in rock’, U.S. Army Corps of Engineers, 1997 (available fordownloading at www.usace.army.mil).

‘The design, sizing and construction of precast concrete segmentsinstalled at the rear of a tunnel boring machine (TBM)’, Document No147, 1998, Recommendations from AFTES (available for downloadingat www.aftes.asso.fr).



American Commercial Inc. (www.americancommercial.com) — seepage ‘Charcon Segment’.




[UE-T9-28]

Types of analyses used in the design of tunnel support (1)

• Analyses that focus on structural behavior —e.g., structural frameswith ‘dead’ load, representing the action of the ground on the structure.

- From the models above, thrust, bending moments and shear forcesare computed, and based on their magnitudes, the structural sectionsdesigned (e.g., given appropriate dimensions).

- Main drawback of the approach: how to quantify realistically the val-ues of qx and qy? —and in the second case, how to quantify realisticallythe stiffness of springs representing the ground?




[UE-T9-29]


• Analyses that focus on rock-support interaction —e.g., pre-stressedelastic or elasto-plastic ground that ‘unloads’ onto the support.

- The main difference between approaches in this category lies on thetype of models considered for the ground and for the interface betweenground and support (e.g., elastic material, elastic-perfectly plastic ma-terial, frictional or frictionless interface between ground and support,etc.).




[UE-T9-30]


• Rock-support interaction analyses (continuation):

- Few (mechanically sound) closed-form solutions are possible in thiscategory. When the geometry of the tunnel and support are circular,and the materials are elastic, Einstein and Schawrtz (1979) present anelegant solution of the rock support interaction problem (see, list ofreferences).

- A semi-rigorous graphical-analytical approach is the Convergence-Confinement Method of support design. The method is based on strongrestrictive assumptions (see next slides), but it provides a basis for reduc-ing a complex 3D problem (increasing support loading with tunnel faceadvance) into simpler 2D (plane-strain) problem —see list of references.

- The most powerful approach in this category is the use of numeri-cal models (e.g., finite elements, finite difference methods). In thesenumerical models, the support can be represented by linear ‘structuralelements’ (a type of element supported by commonly used codes, thatdoes not require discretization of the structure along its thickness) or bynormal material elements (e.g., elastic-isotropic solid elements).




[UE-T9-31]

The Convergence-Confinement Method. Generalities

• The Convergence-Confinement Method is a 2D simplistic approachfor resolving the 3D rock-support interaction problem associated withinstallation of support near a tunnel front.

• The methodology allows estimation of the load that the rock masstransmits to the liner once the ‘supporting’ effect of the tunnel front onthe section analyzed has disappeared (the face has moved away fromthe section).




[UE-T9-32]

Basic assumptions of the Convergence-Confinement Method

Tunnel is circular.

Far-field stresses are uniform (or hydrostatic).

Material is isotropic and homogeneous —e.g., elastic or elasto-plastic.

Support is axi-symmetric —e.g., shotcrete layer forms a closed ring.

Effect of the tunnel front in the vicinity of the tunnel section regardedas a ‘fictitious’ support pressure.




[UE-T9-33]

Basic ‘ingredients’ of the Convergence-Confinement Method

• Ground Reaction Curve (GRC):

The Ground Reaction curve is the graphical representation of the rela-tionship between radial convergence and internal pressure for a circulartunnel excavated in a medium subject to uniform (hydrostatic) far-fieldstresses.

• Support Characteristic Curve (SCC):

The Support Characteristic curve is the graphical representation of therelationship between support radial displacement and uniform pressureapplied to the extrados of a circular (closed) support.

• Longitudinal Deformation Profile (LDP):

The Longitudinal Deformation Profile is the relationship between radialdisplacement and distance to the front for a circular tunnel excavated ina medium subject to uniform (hydrostatic) far-field stresses.




[UE-T9-34]

Ground reaction curve (GRC)

[Note: Positive radial displacement means inward radial displacement in theConvergence-Confinement Method.]




[UE-T9-35]

Construction of Ground reaction curves

- The elasto-plastic solutions described in the notes for Topic 6, ‘Elasto-plastic solution of a circular tunnel’, can be used to construct GroundReaction Curves.

- Construction of GRC requires computing the values of radial displace-ment for various values of internal pressure to outline the curve in theprevious slide.

- For an elasto-plastic material, the radial displacement for the criticalinternal pressure pcr

i (point C in the previous slide), and the radial dis-placement for various values of internal pressure in the interval [pcr

i , 0](between points C and M in the previous slide) must be computed —note that the upper most point of the GRC (point C in the previous slide)has the coordinates pi = σo and uw

r = 0.

- In the case of complex material behavior, numerical models can also beused. To construct the GRC with numerical models, radial convergenceof the tunnel wall is recorded for decreasing values of internal pressure,in the interval [σo, 0].




[UE-T9-36]Example of Ground Reaction Curve

The example above are discussed in Carranza-Torres and Fairhurst (2000).




[UE-T9-37]

Support characteristic curve (SCC)

[Note: Positive radial displacement means inward radial displacement in theConvergence-Confinement Method.]




[UE-T9-38]

Construction of SCC (1)

The elastic solution described in the notes for Topic 8, ‘Elastic solutionof a closed annular support’, for the particular case of uniform loading,can be used to construct a Support Characteristic Curve. From thosenotes we saw that the radial convergence of the closed annular ringexpressed as a function of the pressure applied on the extrados of thering was

usr = 1 − ν2

s

Es

12R ps

12(ts/R) + (ts/R)2(1)

where E is the Young’s modulus and ν is the Poisson’s ratio (for anexplanation of the other variables see previous slide).

Therefore, the stiffness Ks of the support, that represents the slope ofthe elastic part of the Support Characteristic curve (see previous slide)is

Ks = Es

1 − ν2s

12(ts/R) + (ts/R)2

12R(2)




[UE-T9-39]

Construction of SCC (2)

The relationship between the thrust Ts and the pressure ps applied onthe extrados of the support is (see notes for Topic 8, ‘Elastic solution ofa closed annular support’)

Ts = R ps (3)

If the ultimate compressive strength of the material is σ maxs , considering

that the normal stress on a radial section of the support is σs = Ts/t ,then the maximum value of support pressure pmax

s that makes the supportyield is (see figure in previous slide)

pmaxs = ts

Rσ max

s (4)




[UE-T9-40]

Support Characteristic Curves for various support systems (1)

The maximum support pressure is,

pmaxs = σcc

2

[1 − (R − tc)

2

R2

]

The elastic stiffness is,

Ks = Ec

(1 + νc)R

R2 − (R − tc)2

(1 − 2νc)R2 + (R − tc)2

where

σcc is the unconfined compressive strength of the shotcrete or concrete[MPa]

Ec is Young’s Modulus for the shotcrete or concrete [MPa]νc is Poisson’s ratio for the shotcrete or concrete [dimensionless]tc is the thickness of the ring [m]R is the external radius of the support [m] (taken to be the same as the

radius of the tunnel)

Note: The equations above are from Hoek and Brown (1980), ‘Underground Excava-tions in Rock’. The notation has been changed to make it consistent with the notationused in previous slides. For typical ranges of parameters to use in these equationssee the above mentioned reference. These equations and typical parameters are alsosummarized in Carranza-Torres and Fairhurst (2000) —see list of references.




[UE-T9-41]



pmaxs = 3

2

σys

SR θ

AsIs

3Is + DAs [R − (tB + 0.5D)] (1 − cos θ)(5)


1

Ks

= SR2

EsAs

+ SR4

EsIs

[θ(θ + sin θ cos θ)

2 sin2 θ− 1

]+ 2SθtBR

EBB2(6)

where

B is the flange width of the steel set and the side length of the squareblock [m]

D is the depth of the steel section [m]As is the cross-sectional area of the section [m2]Is is the moment of inertia of the section [m4]Es is Young’s modulus for the steel [MPa]




[UE-T9-42]


σys is the yield strength of the steel [MPa]S is the steel set spacing along the tunnel axis [m]θ is half the angle between blocking points [radians]tB is the thickness of the block [m]EB is Young’s modulus for the block material [MPa]R is the tunnel radius [m]





[UE-T9-43]



pmaxs = Tbf

sc sl


1

Ks

= sc sl

[4 l

πd2bEs

+ Q

]




[UE-T9-44]


The parameters in the equations in the previous slide are

db is the bolt or cable diameter [m]l is the free length of the bolt or cable [m]Tbf is the ultimate load obtained from a pull-out test [MN]Q is a deformation-load constant for the anchor and head [m/MN]Es is Young’s Modulus for the bolt or cable [MPa]sc is the circumferential bolt spacing [m]sl is the longitudinal bolt spacing [m]





[UE-T9-45]Example of Support Characteristic Curves





[UE-T9-46]

The advancing front

The objective of the Convergence-Confinement method is to determinefinal load in the support section A-A′, installed at time t0, once the effectof the tunnel face has disappeared, at time tD.

The figure above is from Carranza-Torres and Fairhurst (2000) —see list of references.




[UE-T9-47]

Longitudinal Deformation Profile (LDP)

The figure above is from Carranza-Torres and Fairhurst (2000) —see list of references.




[UE-T9-48]

Equations for the definition of LDP

With reference to the diagram in the previous slide, the equation pro-posed by Dr. M. Panet (see list of references) based on the analysis ofresults from finite element axi-symmetric elastic models is

ur

umaxr

= 0.25 + 0.75

[1 −

(0.75

0.75 + x/R

)2]

(7)

With reference to the diagram in the previous slide, the equation pro-posed by Dr. E. Hoek based on the analysis of actual data and resultsfrom numerical models is

ur

umaxr

=[

1 + exp

(−x/R

1.10

)]−1.7

(8)

The equations above are discussed in Carranza-Torres and Fairhurst (2000) —see listof references.




[UE-T9-49]

Use of numerical models to construct the LDP (1)

Numerical models of a longitudinal section of circular tunnel (includingthe front region) can be used to compute LDPs. The material constitutivemodels used in these numerical models should be the same used toconstruct the GRCs. The most efficient way of setting up and runningthese models is as 2D axi-symmetric numerical models (commercialcodes like Phase2 and FLAC do have an axi-symmetry option).

The figure in the next slide shows: (a) an axi-symmetric mesh in the finitedifference code FLAC (www.itascacg.com); (b) a 3D representation ofthe actual problem that the axi-symmetric mesh represents; (c)the LDPobtained from the model.




[UE-T9-50]

Use of numerical models to construct the LDP (2)




[UE-T9-51]

Ground-support interaction analysis. Final support pressure

The final support pressure pfinali is obtained from the superposition of

the GRC and the SCC (see point P in the diagram below). The LDPdefines the ‘starting point’ of the SCC (point S, of horizontal coordinateuA-A′

r ). This point is the horizontal projection of point A on the GRC.The vertical coordinate of point A is pA-A′

i and represents the fictitioussupport pressure provided by the tunnel front at the time of installationof the support at section A-A′.

A proper support design according to the Convergence-Confinementmethod is one for which the ratio of the maximum support pressurepmax

i and the final support pressure pfinali is larger than a factor of safety,

F.S., chosen for the design (normally F.S.∼ 1.5 ).




[UE-T9-52]Example of Ground-support interaction analysis





[UE-T9-53]

Illustration of Convergence-Confinement analysis (1)

The purpose of this exercise is to verify that the characteristics of the shotcrete liner(thickness, strength, distance to the front) for this tunnel are appropriate.




[UE-T9-54]


The Ground Reaction Curve (GRC) will be computed with Lamé’s solution (see equa-tion 16, in notes on Topic 3, ‘Elastic solution of a circular tunnel’), i.e.,

ur(pi) = 1

2G(σo − pi) R (9)

The Support Characteristic Curve (SCC) will be computed with equations for elasticloading of a closed annular ring (see equations 8 through 10 in notes on Topic 8,‘Elastic solution of a closed annular support’, and equations 1 through 4 in notes onTopic 9, ‘The Convergence Confinement Method’). Thus the relationship betweenradial displacement and support pressure is,

ur(pi) = uIr + 1 − ν2

c

Ec

12R pi

12(tc/R) + (tc/R)2(10)

and the maximum pressure that makes the ring of shotcrete (of compressive strengthσcc) yield plastically is

pmaxs = tc

Rσcc (11)

In equation (2), uIr is the horizontal coordinate of the intersection of the SCC with

the horizontal axis, that will be computed in this example using the expression forLongitudinal Deformation Profile (LDP) for elastic materials proposed by Dr. Hoek—see slides ‘Equations for definition of LDP’ in this note, i.e.,

uIr = umax

r

[1 + exp

(−x/R

1.10

)]−1.7

(12)

In the equation above, umaxr is the coordinate of the intersection of the GRC with the

horizontal axis, that for the case of elastic ground considered here is computed withequation (1) above, considering pi = 0, i.e.,

umaxr = σo

2GR (13)




[UE-T9-55]


The following slides shows the LDP, GRC and SCC for the properties considered inthis example, constructed with the equations described earlier. The following valuesare obtained from application of the mentioned equations and graphical constructionof LDP, GRC and SCC:

umaxr = 3.9 mm (from GRC)

uIr = 0.679 × umax

r = 2.65 mm (from LDP)

uFr = 0.308 × umax

r = 1.2 mm (from LDP)

pFs = 0.32 MPa (from GRC, see Note below the diagram)

pmaxs = 0.583 MPa (from SCC)

pfinals = 0.131 MPa (from intersection of GRC and SCC)

ufinalr = 3.39 mm (from intersection of GRC and SCC)

From the values above, the factor of safety FS for the shotcrete liner is found to be

FS = pmaxs

pfinals

= 0.583 MPa

0.131 MPa= 4.45

Since FS � 1.5, the proposed shotcrete liner is acceptable.

Note: The final thrust in the liner can be computed as T finals = Rpfinal

s and results tobe T final

s = 0.39 MN/m.




[UE-T9-56]





[UE-T9-57]





[UE-T9-58]The program Rocsupport (1)

Rocsupport implements the Convergence-Confinement Method (creation of GRC,SCC and LPD) through a user-friendly graphical interface. The code allows to per-form deterministic and probabilistic analyses of tunnel support design. Rocsupport isdeveloped and commercialized by Rocscience (www.rocscience.com).




[UE-T9-59]

The program Rocsupport (2)

Rocsupport implements the Convergence-Confinement Method (creation of GRC,SCC and LPD) through a user-friendly graphical interface. The code allows to per-form deterministic and probabilistic analyses of tunnel support design. Rocsupport isdeveloped and commercialized by Rocscience (www.rocscience.com).




[UE-T9-60]


For technological aspects of tunnel support systems, see all references(including web sites) mentioned in the slides.

For a rigorous solution of the problem of rock-support intereaction inthe case of a circular tunnel lined by an elastic closed ring in an elasticground subject to non-hydrostatic far-field stresses, see:

• Einstein, H. H. and C. W. Schwartz (1979), ‘Simplified analysis fortunnel supports’. ASCE J. Geotech. Eng. Div., 105(4):449–518.

For tunnel support design and Convergence-Confinement method:

• Hoek, E. & Brown, E. T. (1980), ‘Underground Excavations in Rock’.London: The Institute of Mining and Metallurgy.


• Hoek E., 2000, ‘Rock Engineering. Course Notes by Evert Hoek’.Available for downloading at ‘Hoek’s Corner’, www.rocscience.com.

• U.S. Army Corps of Engineers, 1997, ‘Tunnels and shafts in rock’.Available for downloading at www.usace.army.mil.




[UE-T9-61]


• ‘The Convergence-Confinement Method’, Document No 170, 2002,Recommendations from AFTES (available for downloading atwww.aftes.asso.fr).

• Panet M. (1995), ‘Calcul des Tunnels par la Méthode de Convergence-Confinement’. Press de l’École Nationale des Ponts et Chaussées.

• Carranza-Torres, C. and C. Fairhurst (2000), ‘Application of the con-vergence confinement method of tunnel design to rock-masses that sat-isfy the Hoek-Brown failure criterion’. Underground Space, 15(2),2000.

Documents

Class Notes on Underground Excavations in Rock (2006)