Elasto-plastic solution of a circular tunnelpdf

ce.umn.eduUniversity of Minnesota

Department of Civil Engineering

[Last revision – June 06]

These notes areavailable for downloading atwww.cctrockengineering.com

[UE-T6-1]

Class notes on Underground Excavations in Rock

Topic 6:

Elasto-plastic solution of a circular tunnel

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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Application examples of elasto-plastic solution of circular openings




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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.




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[UE-T6-19]

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

Problem definition




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Solution for radial and hoop stresses




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Solution for radial displacement




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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:

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

• 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.

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