Elasto-Plastic Solution of A Circular Tunnelpdf
Elasto-Plastic Solution of A Circular Tunnelpdf
Elasto-Plastic Solution of A Circular Tunnelpdf
Topic 6:
written by
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-Torres
in the Spring 2006 at the Department of Civil Engineering, University
of Minnesota, USA.
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-3] [UE-T6-4]
If pi ≥ picr the problem is fully elastic (the solution is given by Lamé’s The critical internal pressure picr can be found as the intersection of
solution). the failure envelope and Lamé’s representation of the stress state in the
reference system σθ ∼ σ1 vs σr ∼ σ3 .
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-5] [UE-T6-6]
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-7] [UE-T6-8]
Note that in the equations above, the radius of the opening is Rp and the
internal pressure is picr . The extent of the failure zone is
picr s pi s
Rp = R exp 2 + − + (11)
σci mb m2b σci mb m2b
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-9] [UE-T6-10]
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-11] [UE-T6-12]
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-13] [UE-T6-14]
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-15] [UE-T6-16]
The critical internal pressure picr below which the failure zone develops The solution for displacements is
is
σc 1 r A1 r
picr = σo − ur = − A1 ur (1) (37)
2
(33) 1 − A1 Rp Rp
1 r A1 r
The extent Rp of the failure zone is − − u (1)
cr 1 − A1 Rp Rp r
pi − pi
Rp = R exp (34) Rp A2 − A3 r A1 r r r
σc − σc − + (1 − A1 ) ln
2G (1 − A1 ) 2 Rp Rp Rp Rp
The solution for the radial stresses field σr is given by the following
expression In the equation above, the coefficients ur (1), ur (1), A1 , A2 and A3 are
cr r the same coefficients defined by equations 27 through 30.
σr = pi + σc ln (35)
Rp
The solution for the hoop stresses field σθ is given by the following
expression
r
σθ = picr + σc ln + σc (36)
Rp
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
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The mesh used in the numerical models, the description of two particu-
lar problems of tunnel excavation in Hoek-Brown and Mohr-Coulomb
materials and the corresponding results (analytical and numerical) are
described in the following slides.
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
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Problem definition
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Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
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Problem definition
Solution for radial displacement
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ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
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Example of elasto-plastic analysis. Mohr-Coulomb material (2) Example of elasto-plastic analysis. Mohr-Coulomb material (3)
Solution for radial and hoop stresses Solution for radial displacement
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ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
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Effect of far-field loading on the shape of failure zone (1)
The chart is reproduced from Detournay and St. John (1988). As indi-
cated in the graph, Po is the mean far-field stress, Po = (σvo + σho )/2,
and So is the deviator far-field far-stress, So = (σvo − σho )/2. The chart is
valid for a Mohr-Coulomb failure criterion with friction angle φ = 30◦
and unconfined compression strength σc .
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-27] [UE-T6-28]
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-29] [UE-T6-30]
Books/manuscripts discussing elasto-plastic solutions for tunnel prob- Recommended references (2)
lems:
For elasto-plastic solution of cavities in Hoek-Brown materials:
• Brady B.H.G. and E.T. Brown, 2004, ‘Rock Mechanics for Under-
ground Mining’, 3rd Edition, Kluwer Academic Publishers. • Carranza-Torres, C. and C. Fairhurst (1999), ‘The elasto-plastic re-
sponse of underground excavations in rock masses that satisfy the Hoek-
• Hoek E., 2000, ‘Rock Engineering. Course Notes by Evert Hoek’. Brown failure criterion’. International Journal of Rock Mechanics and
Available for downloading at ‘Hoek’s Corner’, www.rocscience.com. Mining Sciences 36(6), 777–809.
• Hudson J.A. and Harrison J.P. (1997), ‘Engineering Rock Mechanics. • Carranza-Torres, C. (2004), ‘Elasto-plastic solution of tunnel prob-
An Introduction to the Principles’. Pergamon. lems using the generalized form of the Hoek-Brown failure criterion’.
• Jaeger J. C. and N. G.W. Cook, 1979, ‘Fundamentals of rock mechan- Proceedings of the ISRM SINOROCK 2004 Symposium China, May
ics’, John Wiley & Sons. 2004. Edited by J.A. Hudson and F. Xia-Ting. International Journal of
Rock Mechanics and Mining Sciences 41(3), 480–481.
• U.S. Army Corps of Engineers, 1997, ‘Tunnels and shafts in rock’.
Available for downloading at www.usace.army.mil
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com
[UE-T6-31] [UE-T6-32]
University of Minnesota These notes are University of Minnesota These notes are
ce.umn.edu available for downloading at ce.umn.edu available for downloading at
Department of Civil Engineering www.cctrockengineering.com Department of Civil Engineering www.cctrockengineering.com