Deformable Elasto-Plastic Object Shaping using an Elastic ...
Elasto-plastic solution of a circular tunnelpdf
Transcript of 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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-2]
Application examples of elasto-plastic solution of circular openings
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
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[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).
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
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[UE-T6-5]
The critical internal pressure pcri (2)
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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-6]
The critical internal pressure pcri (3)
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).
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
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[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 .
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[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)
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-9]
Solution for the plastic region (r ≤ Rp). Hoek-Brown material (2)
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
)]
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-10]
Solution for the plastic region (r ≤ Rp). Hoek-Brown material (3)
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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[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)
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-12]
Solution for the plastic region (r ≤ Rp). Mohr-Coulomb material (2)
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
]
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-13]
Solution for the plastic region (r ≤ Rp). Mohr-Coulomb material (3)
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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-15]
Solution for the plastic region (r ≤ Rp). Tresca material (2)
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)
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-16]
Solution for the plastic region (r ≤ Rp). Tresca material (3)
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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-18]
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
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[UE-T6-19]
Example of elasto-plastic analysis. Hoek-Brown material (1)
Problem definition
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-20]
Example of elasto-plastic analysis. Hoek-Brown material (2)
Solution for radial and hoop stresses
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
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[UE-T6-21]
Example of elasto-plastic analysis. Hoek-Brown material (3)
Solution for radial displacement
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-22]
Example of elasto-plastic analysis. Mohr-Coulomb material (1)
Problem definition
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-23]
Example of elasto-plastic analysis. Mohr-Coulomb material (2)
Solution for radial and hoop stresses
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-24]
Example of elasto-plastic analysis. Mohr-Coulomb material (3)
Solution for radial displacement
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-25]Effect of far-field loading on the shape of failure zone (1)
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
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[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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-27]
Effect of far-field loading on the shape of failure zone (3)
(The solution above is presented in Detournay and Fairhurst, 1987).
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-28]
Effect of far-field loading on the shape of failure zone (4)
Displacements at the springline and crown of the tunnel
(The solution above is presented in Detournay and Fairhurst, 1987).
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[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
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-30]
Recommended references (2)
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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-31]
Recommended references (3)
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.
ce.umn.eduUniversity of Minnesota
Department of Civil Engineering
These notes areavailable for downloading atwww.cctrockengineering.com
[UE-T6-32]
Recommended references (4)
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.