AECL-9566-2 ^ufek ATOMIC ENERGY &£& ENERGIE ATOMIQUE OF CANADA LIMITED \^Sj^ DU CANADA LIMITEE Research Company ^^^^^ Societe de Recherche UNDERGROUND RESEARCH LABORATORY ROOM 209 INSTRUMENT ARRAY: NUMERICAL MODELLERS' PREDICTIONS OF THE ROCK MASS RESPONSE TO EXCAVATION RESEAU ^INSTRUMENTS DE MESURE DE LA CHAMBRE 209 DU LABORATOIRE DE RECHERCHES SOUTERRAIN: PREDICTIONS NUMERIQUES DES GROUPES DE MODELISATION DE LA REACTION DE LA MASSE ROCHEUSE A L'EXCAVATION Volume II Compiled by / Compile et redige par P. A. Lang Whiteshell Nuclear Research Etablissement de recherches Establishment nucleaires de Whiteshell Pinawa, Manitoba ROE 1LO April 1989 avril
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AECL-9566-2 ^ufekATOMIC ENERGY &£& ENERGIE ATOMIQUEOF CANADA LIMITED \^Sj^ DU CANADA LIMITEEResearch Company ^^^^^ Societe de Recherche
UNDERGROUND RESEARCH LABORATORY ROOM 209 INSTRUMENT ARRAY:
NUMERICAL MODELLERS' PREDICTIONS
OF THE ROCK MASS RESPONSE TO EXCAVATION
RESEAU ^INSTRUMENTS DE MESURE DE LA CHAMBRE 209
DU LABORATOIRE DE RECHERCHES SOUTERRAIN: PREDICTIONS NUMERIQUES
DES GROUPES DE MODELISATION DE LA REACTION
DE LA MASSE ROCHEUSE A L'EXCAVATION
Volume II
Compiled by / Compile et redige parP. A. Lang
Whiteshell Nuclear Research Etablissement de recherchesEstablishment nucleaires de Whiteshell
Pinawa, Manitoba ROE 1LOApril 1989 avril
Copyright •'<=) Atomic Energy of Canada Limited. 1989
ATOMIC ENERGY OF CANADA LIMITED
UNDERGROUND RESEARCH LABORATORY ROOM 209 INSTRUMENT ARRAY:NUMERICAL MODELLERS' PREDICTIONS
OF THE ROCK MASS RESPONSE TO EXCAVATION
VOLUME II
Compiled by
P.A. Lang
T. Chan, P. Griffiths, B. Nakka,P.K. Kaiser1, D.H. Chan1, D. Tannant1, F. Pelli1, C. Neville1,
Gen hua Shi2, R.E. Goodman2 and P.J. Perie2
1Department of Civil EngineeringUniversity of AlbertaEdmonton, Alberta T6G 2E1
2Lawrence Berkley LaboratoriesBerkley, California 94620
Vhiteshell Nuclear Research EstablishmentPinava, Manitoba ROE 1L0
1989AECL-9566-2
- 331 -,
CONTENTS
VOLUME II
APPENDIX B
Page
NUMERICAL SIMULATION OF ROOM 209 INSTRUMENT RING,by P.K. Kaiser, D.M. Chan, D. Tannant, F. Pelli andC. Neville; University of Alberta, Department of CivilEngineering. 333
APPENDIX C PREDICTION OF ROCK DEFORMATIONS FOR THE EXCAVATIONRESPONSE EXPERIMENT - ROOM 209 IN THE URL, LAC DU BONNET,MANITOBA, by Gen hua Shi, Richard E. Goodman and PierreJean Perie; Lawrence Berkley Laboratories, Earth SciencesDivision. 581
- 333 -
APPENDIX B
NUMERICAL SIMULATION OF ROOM 209 INSTRUMENT RING
by
P. K. Kaiser, D. M. Chan, D. Tannant, F. Pelli and C. Neville
University of AlbertaDepartment of Civil Engineering
- 335 -
CONTENTS*
Page
ERRATA 337
Data Presentation Clarifications (Addendum to Report) 338
LIST OF FIGURES 352
Table of Contents 365
INTRODUCTION 367
1.1 General 3671.2 Research Team 3681.3 Scope of Interim Technical Report 3681.4 Finite Element Program Descriptions 370
DESCRIPTION OF ROOM 209 EXPERIMENT AND AVAILABLE DATA 372
DEVELOPMENT OF THE FINITE ELEMENT MODEL 373
3.1 Finite Element Mesh and Coordinate System 3733.2 Approximation of In Situ Stress Tensor for
Finite Element Analysis 3763.2.1 Selection of Representative State of Stress 3763.2.2 Simplified State of Stress 377
3.3 Selection of Rock Mass Parameters 3793.4 Selection of Fracture Zone Parameters 380
(Joint Element) 3813.4.1 Joint Properties and Characterization 3813.4.2 Linear-Elastic Joint Model 3843.4.3 Hyperbolic Normal Closure of Joint 388
3.4.3.1 Barton-Bandis Joint Model 3883.4.3.2 Selected Hyperbolic Normal Closure
Parameters 3893.5 Seepage Analyses in Fracture Zone 390
3.5.1 Fracture Permeability Characterization 3903.5.2 Finite Element Representation of the
Fracture Zone 391
continued...
Table of Contents provided by Technical Information Services
4.1 Introduction to Data Presentation 4034.2 Linear Elastic Rock Without Joint (I)
Uncoupled Analysis 4094.2.1 Assumed Parameters 4094.2.2 Comments to Figures 409
4.2.2.1 Displacements 4094.2.2.2 Stresses 410
4.3 Linear-Elastic Rock Mass with Fracture Zone (II)Uncoupled Analysis 4324.3.1 Assumed Parameters 4324.3.2 Comments to Figures 432
4.3.2.1 Displacements 4324.3.2.2 Stresses 433
4.3.3 Aperture of the Fracture Zone 4784.3.4 Uncoupled Pressure Head Distribution/Flow
Modelling 4784.4 Linear-Elastic Rock With Non-Linear Fracture
Zone (III) 493
5. LIMITATIONS OF APPLICABILITY 495
6. REFERENCES 497
7. Appendix A 501
8. Appendix B 505
9. Appendix C 531
10. Appendix D 569
- 337 -
ERRflTR
Numerical Simulation of Room 209 Instrument Ring
by P.K. Kaiser et al. (1987)
Page *
382
383
386
386
397
404
406
406-407
409
433
531
571-578
Text
These data was averaged
make simlifying assumptions
mechanical behavior of the
represent the behavior of
CSIR Cell
CSIR Stress Cell
Location of ExtensometerAnchors and AssociatedFinite Element Nodes
Nodal Point
to the anchor heads.
steps where simulated
...EXCAVATION STEPS
CSIR
Correction
were
simplifying
behaviour
behaviour
CSIRO
CSIRO
Location ofExtensometer An
Back-RotatedAnchor Location
collars
were
EXCAVATION STEAND RELATIVESHEAR IN FRACTUI
CSIRO(not back-rotated)
579 0.50(aperture change, last entry)
1.50
* Page numbers were adjusted by Technical Information Services
- 338 -
Data Presentation Clarifications (Addendum to Report)
1. Converence Plots
All converence plots show inward radial displacement along the crown (roof), springline,and invert (floor) for the symmetrical displacement fields only; i.e., none of these plotshave been back-rotated. These converences represent predictions in the directions ofmaximum and minimum stress in the x-y plane. (Sign Convention: inward displacementpositive)
2. Extensometer Plots
All extensometer plots presented in Figures 4.8 to 4.21 and 4.25 to 4.38 are calculatedusing the back-rotated displacement fields and thus are directly comparable to plots ofmeasured responses. All extensometer plots for the excavation stages J-l and J+l arepresented in Appendix C and are calculated using the symmetrical displacement fields andcannot be compared directly with measured responses. (Sign Convention: extensionalstrain positive)Note: 1 The displacement at anchor #4 in extensometer EXT02 is incorrect for
Figures 4.8b, 4.15b, 4.25b, and 4.32b. (Replacement figures areattached.)
2 Extensometer data for the full section excavation, no joint analysis (Figures4.15 to 4.21) were calculated before submission of the report but weremistakenly replaced by repeated plots of the pilot excavation, no joint analysis;i.e., they are repeated plots of the earlier figures. (The original plots areattached, for replacement.)
3. Shear Displacement Plots
All plots of relative shear displacement in the fracture at the roof and springline arepresented in Appendix C. These are calculated using the symmetrical displacement field,thus they are plots of relative shear displacement in the directions of maximum andminimun stress in the x-y plane. Back-rotation is required for comparison with fieldmeasurements. (Sign Convention: radial displacement positive inwards)
4. Stress Profiles
All stress profiles in Figures 4.43 to 4.60 are taken from the springline and from theroof/floor in the symmetrical stress fields. These plots show the variation in the stressmagnitudes in the directions of maximum and minimum initial stresses in the x-y plane.Back-rotation is required for comparison with field measurements. (Sign Convention:tensile stress positive, shear stress follows right hand rule)
- 339 -
5. Aperture Profiles
All aperture profiles presented in Figures 4.61 to 4.66 are taken from the springline andfrom the roof/floor in the symmetrical displacement fields. These plots show the variationin fracture apertures in the directions of maximum and minimum initial stresses in the x-yplane. Back-rotation is required for comparison with field measurements.
6. Fracture Fluid Pressure Profiles
All fracture fluid pressure profiles are presented in Figures 4.67 to 4.74. These profiles aretaken from the springline and from the roof/floor. Figures 4.67 to 4.70 incorporated thesymmetrical variations in aperture when the seepage analyses were performed.
7. Principal Stress Changes
All plots of principal stress changes are presented in Appendix D. These are calculatedfrom the back-rotated stress fields and are directly comparable to the measured stresschanges and orientations. (Sign Convention: positive stress change indicates a reduction incompressive stress or an increase in tensile stress)
8. Hydraulic Apertures
Table D. 1 presents the predicted changes in hydraulic apertures at the back-rotated packed-off intervals and should be directly comparable to the measured changes.
EE
oo
V
oV)
c-•—X
0.30
0.25
0.20
0.05
0.00
Figure 4.8b
oi
5 10 15Distance Along Extensometer (m)
Roof Extensometer EXT02No Joint: Pilot Excavation
20
0.30
Ld
Figure 4.15a
5 10 15
Distance Along Extensometer (m)
Roof Extensometer EXT01No Joint: Full Section Excavation
I
20
E
oo
I
oCO
c
0.30
0.25
0.20-
0.15-
0.10-
0.05 :
0.00
Figure 4.15b
5 10 15Distance Along Extensometer (m)
Roof Extensometer EXT02No Joint: Full Section Excavation
u>
20
0.45
LULU
)
Col
lar
*
1o
iter
Ken
som
e0.40-
0.35-
0.30-
0.25-
0.20-
0.15-
0.10-
0.05-
0.005 10
Distance Along Extensometer (m)15
I
u>
I
20
Figure 4.16 Diagonal Extensometer EXT03No Joint: Full Section Excavation
u
E
6 0.3-
oinca>"xLLJ
0.2-
0.1-
0.0
Figure 4.20a
5 10 15Distance Along Extensometer (m)
SL Extensometer EXT07No Joint: Full Section Excavation
u>
20
EE
0.7-J
O 0.6-^oo
0.5J
c
0)
0.4-
O 0.3-
0)%Eocd)
0.2-
0.1-
0.0
Figure 4.17
5 10 15
Distance Along Extensometer (m)
5L Extensometer EXT04No Joint: Full Section Excavation
u>
i
20
"oO
cCD
£oV)c(D"xUJ
0.30
0.25 :
0.20 :
0.15-
0.10-
0.05-
0.00-
Figure 4.18
5 10 15Distance Along Extensometer (m)
Floor Extensometer EXT05No Joint: Full Section Excavation
I
u>
20
0.45
£ 0.40Co
lic
••
o
ite
r K
enso
me
0.35-
0.30-
0.25-
0.20-
0.15-
0.10-
0.05-
0.00-5 10 15
Distance Along Extensometer (m)
u>
20
Figure 4.19 Diagonal Extensometer EXT06No Joint: Full Section Excavation
£
o
£oV)
c
0.7-
0.6-
0.5-
0.4-
0.3-
0.2-
0.1-
0.0
Figure 4.20b
5 10 15Distance Along Extensometer (m)
SL Extensometer EXT08No Joint: Full Section Excavation
u>00I
20
0.45
EE,
Col
lar
•*
3:
cem
ent
o
som
ette
n:
XUJ
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
I
u>
5 10 15
Distance Along Extensometer (m)20
Figure 4.21 Diagonal Extensometer EXT09No Joint: Full Section Excavation
- 350 -
Q>
ID
(UJLU) '4'J'M
E
0.30
0.25
ou* ; 0.20
c
oc
Ul
0.15
0.10
0.05
0.00
Figure 4.32b
5 10 15Distance Along Extensometer (m)
Roof Extensometer EXT02Linear Joint 3x3: Full Section Excavation
Lnt—*
I
- 352 -
LIST OF FIGURES
General Figures Page'
Figure 3.1 Cross-Section of the Mesh 395
Figure 3.2 Enlarged Section of the Inner Mesh 396
Figure 3.3 Finite Element Mesh Near Instrumentation 397and Fracture Zone
Figure 3.4 Enlarged Section of the Altered Inner Mesh 398
Figure 3.5 Plan of Finite Element Model 399
Figure 3.6 Applied Initial Stresses and Coordinate System 400
Figure 3.7 Normal Stress - Normal Closure Relationship 401
Figure 3.8 Assumed Direction of Fluid Flow 402
Figure 4.1 Location of CSIRO Stress Cells 403
Figure 4.2 Location of Packed-Off Intervals 404
Figure 4.3 Location of Extensometer Anchors 405
Figure 4.4 Location of Convergence Measurement Points 408
Convergence Plots: No Joint (Non Back-Rotated)
Figure 4.5 No Joint : Development Room Excavation 411
Figure 4.6 No Joint : Pilot Excavation 412
Figure 4.7 No Joint : Full Section Excavation 413
Extensometer Plots: No Joint
Figure 4.8a Roof Extensometer EXT01 414No Joint: Pilot Excavation
* Page numbers were adjusted by Technical Information Services
- 353 -
Figure 4.8b
Figure 4.9
Figure 4.10
Figure 4.11
Figure 4.12
Figure 4.13a
Figure 4.13b
Figure 4.14
Figure 4.15a
Figure 4.15b
Figure 4.16
Figure 4.17
Figure 4.18
Figure 4.19
Figure 4.20a
Roof Extensometer EXT02No Joint: Pilot Excavation
Diagonal Extensometer EXT03No Joint: Pilot Excavation
SL Extensometer EXT04No Joint: Pilot Excavation
Floor Extensometer EXT05No Joint: Pilot Excavation
Diagonal Extensometer EXT06No Joint: Pilot Excavation
SL Extensometer EXT07No Joint: Pilot Excavation
SL Extensometer EXT08No Joint: Pilot Excavation
Diagonal Extensometer EXT09No Joint: Pilot Excavation
Roof Extensometer EXT01No Joint: Full Section Excavation
Roof Extensometer EXT02No Joint: Full Section Excavation
Diagonal Extensometer EXT03No Joint: Full Section Excavation
SL Extensometer EXT04No Joint: Full Section Excavation
Floor Extensometer EXT05No Joint: Full Section Excavation
Diagonal Extensometer EXT06No Joint: Full Section Excavation
SL Extensometer EXT07No Joint: Full Section Excavation
416
417
418
419
420
422
423
425
426
427
428
429
- 354 -
Figure 4.20b SL Extensometer EXT08No Joint: Full Section Excavation
Figure 4.21 Diagonal Extensometer EXT09 431No Joint: Full Section Excavation
Convergence Plots: Linear Joint (Non Back-Rotated)
Figure 4.22 Linear Joint 3x3 : Development Room Excavation 435
Figure 4.23 Linear Joint 3x3 : Pilot Excavation 436
Figure 4.24 Linear Joint 3x3 : Full Section Excavation 437
Figure4.73Constant Aperture : Full Section Excavation
100
15 20Springline Coordinate (m)
25
to
30 35
Figure 4.74 Constant Aperture : Full Section Excavation
- 493 -
4.4 Linear-Elastic Rock With Non-Linear Fracture Zone (III)
A non-linear finite element analysis was carried out in
order to study the effect of yielding in the fracture zone on
the tunnel behaviour. The rock mass was assumed to be linear
elastic with the same parameters as used in earlier analyses.
The following parameters were selected for the elasto-plastic
joint:
Ks1 = 4.6 MPa/mm
Ks2 = 4.6 MPa/mm
Kn = 10000 MPa/mm
f0 = 0.0
fr = 0.0
fp = 3.0 MPa
x = 1.5 mm
xr = 6.0 mm
<t> = 30°
i0 = 5.1°
oc = 127 MPa
where Ks1, Ks2 and Kn are shear and normal stiffnesses in the
linear elastic range; f0, fp and fr are parameters controlling
the position of the yield function at peak or in the post-peak
region; xp and xr represent the total accumulated plastic
relative shear displacement at peak and in the post peak
region; <j> is the friction angle; i0 is the dilation angle for
zero normal stress and oc is the unconfined compressive
strength of the joint asperities. The joint was assumed to be
linear elastic in the normal direction. This is because no
significant normal stiffness variation was expected in the
stress range of interest.
The pilot tunnel was excavated to one element slice before
the joint and one immediately after it. These two steps were
considered to be the most critical in terms of possible
- 494 -
yielding in the joint. However, no yielding was observed
because, for the given parameters, the linear elastic range was
not exceeded. The validity of these parameters and the
practical implications will have to be evaluated by comparison
with field observations.
- 495 -
5. LIMITATIONS OF APPLICABILITY
As outlined earlier, because of time and computing fund
limitations, only very selective cases were analyzed. While an
attempt was made to select the most appropriate rock mass
properties, it was found that insufficient data were available
to make an unconditional prediction. Hence, the results
presented must be viewed as a first approximation for
comparison with field observations. Many extreme
simplifications had to be made and further program development
and parametric studies will be required before an assessment of
our predictive capability can be made. For these reasons, we
cannot foresee that our predictions will agree accurately with
all field observations.
Our prediction is extremely limited and intended as a
basis for an evaluation of where further improvements are
needed.
The following specific examples are intended in support of
this statement:
a) The stress state could only be approximated;
b) Anisotropy was neglected;
c) The single fracture zone could not be orientated
accurately;
d) Effects of blast damage were neglected;
e) The influence of potential temperature changes was not
considered;
f) Only uncoupled analyses could be completed during Stage 1,
g) A single, constant pressure boundary was assumed for the
flow analyses ; and
- 496 -
h) A single, uniformly distributed, initial fracture aperture
was assumed.
Furthermore, in the late stages of our research effort, it
was realized that 3 x 3 instead of 2 x 2 integration schemes
were required for stable fracture zone aperture predictions.
This required a complete re-analysis during April 1987. No
critical data evaluation could be completed before the deadline
for this report. This will be undertaken in Stage 2 when
results are compared with field data. The final technical
report will present the verification of our prediction.
- 497 -
6. REFERENCES
Adina Eng., 1984. ADINAT - A finite Element Program for
Automatic Dynamic Incremental Nonlinear Analysis of
Tempertures. Adina Engineering, Inc. 71 Elton Avenue
Watertown, MA.
Bandis, S. 1980. Experimental Studies of Scale Effecects on
Shear Strength and Deformation of Rock Joints. Ph.D Thesis.
The University of Leeds.
Bandis, S.C., Lumsden, A.C., and Barton, N.R., 1983.
Fundamentals of rock deformation. International Journal of
Rock Mechanics and Mining Sciences & GeomechanicsAbstracts, 20, No. 6, pp. 249-268.
Barton, N. and Bakhtar, K. 1983. Rock joint description and
modeling for the hydrothermomechanical design of nuclear
waste repositories. CANMET, Mining Research Laboratories,
TRE83-10, 258p.
Barton, N., Bandis, S., and Bakhtar, K. 1985. Strength,
deformation and conductivity coupling of rock joints.
International Journal of Rock Mechanics and Mining Sciences
& Geomechanics Abstracts, 22, No. 3, pp. 121-140.
Barton, N. and Choubey, V. 1977. The shear strength of rock
joints in theory and practice. Rock Mechanics, 10, No. 1-2,
pp. 1-54.
Barton, N., Makurat, A., Vik, G., and Loset, F. 1985. The
modelling and measurement of super-conducting rock joints.
26th U.S. Symposium on Rock Mechanics, Rapid city, 1, pp.
487-495.
Carol, I. and Alonso, E.E. 1983. A new joint element for the
analysis of fractured rock. 5th Congress of the
International Society for Rock Mechanics, Melbourne, 2, pp.F147-F151.
- 498 -
Carol, I., Gens, A., and Alonso, E.E. 1985. A three dimensional
elastoplastic joint element. Proceedings of the
International Symposium on Fundamentals of Rock Joints,Bjorkliden, pp. 441-451.
Carol, I., Gens, A., and Alonso, E.E. 1986. Three dimensional
model for rock joints. 2nd International Symposium on
Numerical Models in Geomechanics, Ghent, pp. 179-189.
Gale, J.E., 1977. A numerical, field and laboratory study of
flow in rocks with deformable fractures. Inland Waters
Directorate, Water Resources Branch, Ottawa, 72, 145 p.
Jackson, R., Annor A., Wong, A.S. and Betournay, M., 1985.
Preliminary Results of Rock Joint Testing on URL Core.
Canmet - Energy Research Program, Mining Research
Laboratories, Division Report ERP/MRL 85-14(TR)
Kozak, E.T. and Davison, C.C., 1986. Hydrogeological Conditions
in a Vertical Fracture Intersecting Room 209 of the 240 m
level of the Underground Research Laboratory - 1.
Pre-excavation Conditions. AECL internal report.
Lang, P. A., Everitt, R. A., Kozak, E. T., Davison, C. C ,
1987. Room 209 Instrument Array - 1. Pre Excavation
Information for Modellers. AECL Report.
Tsang, Y.W. and Witherspoon, P.A. 1981. Hydromechanical
behavior of a deformable rock fracture subject to normal
stress. Journal of Geophysical Research, 86, No. B10, pp.
9287-9298.
Wilson, C.R. and Witherspoon, P.A. 1970. An investigation of
laminar flow in fractured porous rocks. Dept. of Civil
Engineering, Institute of Transportation and Traffic
Engineering, University of California, Berkeley, 178 p.
Witherspoon, P.A., Wang, J.S.Y., Iwai, K., and Gale, J.E.,
1980. Validity of cubic law for fluid flow in a deformable
rock fracture. Water Resources Research, 16, No. 6, pp.
1016-1024.
- 501 -
7. Appendix A
SAFE - Soil Analysis by Finite Elementdeveloped byD.H. Chan
Department of Civil EngineeringUniversity of Alberta
Edmonton, AlbertaT6G 2G7
- 503 -
SAFE
SAFE (Soil Analysis by Finite Element) is a computer program developed at the University ofAlberta to analyze deformation of soil and rock structures. The program is written in FORTRAN IVlanguage and has been installed on different types of computer systems including the IBMsystem, the Amdahl MTS system and the CDC Cyber 205 vector computer system. The programhas been applied in analyzing a wide variety of geotechnical problems such as excavation, dam,shaft and tunnel constructions.
The initial development of the program was to analyze the post peak deformation of strainsoftening soil. But the program has now been extended to include 2 and 3 dimensional analysisusing total and effective stress formulation for fully undrained and drained conditions. A variety ofnon-linear elastic and plastic models with associated and non-associated flow rules are alsoavailable. Localized shear zone deformation can be modelled using the program and furtherdevelopments of the program are currently in progress.
The following is a list of the main features of the program SAFE.
Basic FormulationDisplacement finite element formulation assuming small strain and small deformation.
Element TypesTwo dimensional 3 to 6 nodes triangular, 4 to 8 nodes rectangular and three dimensional8 to 20 nodes solid elements.
Type of Analysis1. Plane stress, plane strain, axisymmetric and three dimensional analysis.2. Non-linear elastic hyperbolic model.3. Elastic perfectly or brittle plastic model using von-Mises, Tresca, Drucker-Prager,and Mohr-Coulomb yield criteria with associated or non-associated flow rule.4. Elastic plastic strain hardening and softening (weakening) model5. Elastic hyperbolic softening model.
Standard Features1. Prescribed concentrated point force or distributed pressure boundary condition.2. Prescribed displacement boundary condition.3. Changing material properties at any stage of the analysis.4. Program restart at any stage of the analysis.5. Newton Raphson and Modified Newton Raphson iterative scheme for non-linearanalysis.
- 50A -
6. Choice of 2 x 2 , 3 x 3 , 2 x 2 x 2 , and 3 x 3 x 3 integration scheme.7. Load increment subdivision for non-linear analysis.
Special Features1. Element birth and death option.2. Automatic application of stress relieve due to excavation.3. Skyline and extended skyline matrix equation solver.4. Choice of stress calculation for non-linear analysis:
Post Processing Programs for SAFE1. Finite element mesh and deformed mesh plotting.2. Stress and strain contour plotting.3. Displacement arrow plotting.
Current Development1. Improved undrained analysis.2. No tension analysis with crack model.3. Geogrid element and soil geogrkJ interface element.4. Anisotropic plasticity.5. Special shear band element with discontinuous shape function.6. Time depending strain softening.
- 5 0 5 -
8. Appendix B
DESCRIPTION OF JOINT ELEMENTimplemented and tested
byF. Pelli
Department of Civil EngineeringUniversity of Alberta
Edmonton, AlbertaT6G 2G7
- 507 -
APPENDIX BDESCRIPTION OF THE JOINT ELEMENT
B.I Introduction
For the purpose of modelling the behaviour of the discontinuous rock mass
a three dimensional isoparametric joint element was implemented into the
finite element computer program SAFE. The element is capable of modelling
several relevant features of rock joints such as dilatancy due to relative
shear displacement, opening-closure of the joint, non-linear normal and shear
behaviour (in pre- and post-peak region). The constitutive law implemented in
conjunction with the element is based on non-associated flow plasticity and
includes a strain-hardening/softening law.
The program SAFE (Soil Analysis by Finite Elements) was selected for the
implementation.
In the following paragraphs both element formulation and constitutive law
are described.
B.2 Element Formulation
The 3-D isoparametric joint element developed by Carol et ai. (1985) was
selected. The element (Fig. B.1) is defined by a variable number of nodal
points (8 to 16) and can be coupled with 3-D isoparametric solid elements.
The average position of the joint contact surface is described by a set
of 'mid-plane' points (4 to 8) located between each pair of nodes (Fig. B.2).
The relative displacement occurring at each of the mid-plane points is taken
to be equal to the difference of the displacements of the two adjacent nodes.
Geometry and relative displacements are interpolated between mid-plane
nodes by means of standard two dimensional isoparametric shape functions. Two
typical expressions for a corner and side node are:
- 508 -
N1 = 0.25 (1 - r) (1 - s) (-1 - r - s)
(B.1)
N 5 = 0.5 (1 - rz) (1 - s)
where: N^ ... shape function at node i
r,s,t ... local curvilinear coordinate system.
The orientation of the local axes (r,s,t) with respect of the global
coordinate system (x,y,z) can be established.
Two vectors, tangent to the element surface at a certain location are defined
as:
5r
5z5r
3s
as
ds
(B.2)
A third vector V^' perpendicular to the surface, can be found as the cross
product of V^ and V-,:
Dividing each of the vectors by its own magnitude results in three unit
vectors (v-, v~, _v,) (Fig. B.3).
The B matrix relating relative displacements to nodal displacements can
The element nodal force vector induced by initial stresses {a } can alsoo'
be written:
{F } = / [B] {a } dS (B.5)o o
s
Using the virtual work approach the following load-displacement relationship
is found:
- {FQ} - tK] {Au} (B.6)
where: {R} . . . externally applied load vector
. • . incremental displacement vector
For each load increment a series of equilibrium iterations is performed by the
- 510 -
program until the specified convergence criteria are met. The iterative
procedure used by the program SAFE is based on the Newton-Raphson method and
allows stiffness matrix updating after any iteration (i.e. as often as
specified by the user).
B.3 Behaviour in Compression
The behaviour of a real rock joint under normal compressive load was
approximated by a non-linear, elastic constitutive relationship. The
hyperbolic function proposed by Bandis et al (1983) was selected to relate
normal stress (o ) to normal relative displacement (V•)•
The hyperbolic function, graphically shown in Fig. B.4, takes the
following analytical form:
V .— 3 (B.7)
a - b A V.
where: a and b are function parameters.
Physical properties of the joint such as maximum closure (V) and initial
tangent modulus (K •) can be expressed in terms of a and b as follows:
Vm
(B.8)
Kni a
The slope of the curve at any of its points can be calculated by
differentiating Eqn. B.7 and the following expression is found:
-2
- 511 -
In the current version of the program the same curve is used to describe
loading and unloading. This is a reasonable assumption for the stress
conditions and stress history considered to be applicable in the area of
Room 209.
B.4 Behaviour in Shear
The constitutive law governing shear behaviour as well as dilatancy due
to shear relative displacement is based on non-associated flow plasticity
(Note: no dilation is assumed to occur as long as the shear stresses do not
exceed the elastic range).
Fornulation of the elasto-plastic model requires the definition of a
yield surface, a hardening law and a flow rule.
The hyperbolic yield function proposed by Carol et al. (1985) was
selected. The family of yield surfaces, schematically depicted in Fig. B.5,
can be expressed in the following form:
1 + T2
where: x , x ... shear stress components;
<t> . . . friction angle;
an . . . normal stress;
f . . . function parameter (Fig. B.5).
In order to follow the hardening/softening history of the joint the
parameter f (see yield function) is assumed to be a function of x, where x is
the total accumulated plastic rels. ~.ve shear displacement. Two polynomial
functions (2nd order up to the peak strength and 3rd order in the post-peak
- 512 -
range) were selected to describe the f - x relationship (Fig. B.6). The
hardening/softening law is defined by five parameters:
f 0 ' fp' fr' xp' xr
The plastic potential function (Q) was selected such as to have the following
partial derivatives:
2 n
where: i = dilation angle.
Note: the increment of plastic shear deformation is assumed to have the same
direction as the shear stress.
The dilation angle is expressed, to account for normal stress dependency
as follows:
tan i = tan iQ (1 - on/a )4 (B.12)
where: ±Q . . . dilation angle for an = 0
q . . . unconfined compressive strength of the rock irregularities
in the joint.
Eqn. B.12 given by Ladanyi and Archambault (1970), is graphically depicted in
Fig. B.7.
The parameter iQ is also a function of the total accumulated plastic
shear relative displacement. The iQ - x relationship is shown in Fig. B.8.
The dilation angle (iQ) decreases linearly with plastic deformation until the
- 513 -
ultimate strength is reached. After further straining, it remains constant.
B.5 Opening and Closure of Joint.
If the stress normal to the rock joint becomes tensile, joint opening is
expected to occur. An effective model must be able to simulate stress release
due to opening of the joint, open joint properties, and joint closure. The
isoparametric element simulates joint opening by completely releasing normal
and shear stresses if normal tensile stresses occur. The open joint is
assumed to be linear elastic having normal and shear stiffnesses close to
zero.
The normal relative displacement value at which opening occurs is stored
by the program. This value is then used as criterion to evaluate if the joint
is still open or closing. The closing joint resumes its elastic-plastic
properties as they were before opening occurred.
B.6 Testing of Joint Element
The element was tested to verify its behaviour under various loading
conditions:
(a) Tests on the linear elastic element were carried out by applying shear
and normal external loads. The results were found to be consistent with
the adopted linear elastic consitutive law.
(b) The non-linear elastic normal behaviour was tested by applying a normal
compressive load on the element. The results were compared with hand
calculations and were found to be consistent with the hyperbolic
stress-relative displacement relationship used. Both loading and
- 514 -
unloading conditions were considered. Convergence was achieved easily
except for those cases where high normal displacements (close to max
closure) were reached.
(c) The element was tested in the plastic range by applying initial normal
stress (in equilibrium with an external normal load) and shear relative
displacements. An elastic, perfectly plastic constitutive relationship
with constant initial dilation angle (ig) w a s assumed and the shear
stress and the normal dilation as calculated by the program were found to
correspond to those calculated by hand.
(d) Hie strain hardening-softening law with variable iQ was included in the
program and also tested. For this case, the shear stress was found to be
consistent with the final f value (in the post-peak region) and with the
accumulated plastic strain.
(e) Finally, the opening/closure feature was tested. The joint was opened
and then closed by means of applied nodal displacements in the normal
direction. The element was found to perform well under these conditions
but convergence during closure tended to be relatively slow.
B.7 References
Bandis, S.C., Lumsden, A.C., and Barton, N.R. (1983). Fundamentals of rock
joint deformation. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.
Vol. 20, No. 6, pp. 249-268.
- 515 -
Carol, I., Gens, A., and Alonso, E.E. (1985). A three dimensional
elastoplastic joint element. Proc. Int. Symp. Fundam. of Rock Joints,
Bjorkliden, pp. 441-451.
Ladanyi, B. and Archambault, G.A. (1970). Simulation of shear behaviour of a
jointed rock mass. Proc. 11th Symp. Rock Mech. AIME, pp. 105-125.
B.8 Oopy of Paper by Carol et al. (1985)
- 516 -
Figure B.1 Joint Element
mid-plane
mid-nodes
Figure B.2 Mid-plane and Mid-nodes
mid-plane
Figure B.3 Global and Local Coordinate Systems
- 517 -
Figure B.4 Hyperbolic o -v Relationship
Figure B.5 Yield Function
- 518 -
Figure B.6 Hardening-softening Law
Figure B.7 Plastic Potential Function
- 519 -
Figure B.8 i 0 - * Relationship
- 520 -
Proceedings ol the international Symposium on Fundamentals of Rock Joints / B/orkltden / 15-20 September 1985
A three dimensional elastoplastic joint elementI. CAROLA GENSE.E. Al.ONSOUniversiim Politecnica tie Barcelona, Spain
ABSTRACT: A nt-w elastoplastic joint model for three-dimensional analysis has. beendeveloped. Firstly an elastoplastic constitutive law that incorporates wellaccepted empirical properties of joints is presented. The model prediction:; art*compared with published results and some field shear test data. A 16 node isoparametrie is then described and applied to the modelling of one of the field shear tests.
1.INTRODUCTION
In order to obtain a good estimate of tin-.k-fornaLii li ty an-: s.ifety conditions ofm<iny large structures, such as archdams, on fractured rock it is imperativeto iiiodc-l threedimensional effects. Inthis type of problems finite elementmethods, which combine continuum elementswith joint elements representing surfacesof discontinuity provide a powerfulanalysis tool.
The authors are currently involved inthe safety evaluation of a large archdam in the Pyrinean area (Canelles dam)and have already used a jointed finiteelement model under plane strainconditions to explore its possibilities(Alonso and Carol, 1985). The resultsof this previous work and the markedthreedimensional effects of the dam-rockinteraction prompted the development ofa surface type joint element to be usedin conjunction with available three-dimensional finite element computercodes.
This development, which is reported inthis paper, required the considerationof two different aspects: 1) The selec-tion of a realistic constitutive lawincorporating relevant and well acceptedexperimental facts and ?) The develop-ment of a threodimensional joint elementcompatible with existing computers codesinndl inij isoparametric "brick" elements.
A literature rcvi •••-• revealed that very
few threcdimensional elements have beendeveloped. Its application to r-.W prol-lems seems also to be very lir.itc:. WinDillon and Ewing (1981) concisely J^crina plastic threedimensional joint elemenrbut no details are available concerningits formulation and its use in boundaryvalue problems. Heuze and Barbour (1982)develop an axisymmetric joint element asan extension of a twodimensional formula-tion. It was concluded that the develop-ment of the proposed surface jointelement should start from basic consid-erations of joint behaviour in order togeneralize them and be able, finally, tointegrate them in a properly definedsurface element.
2.CONSTITUTIVE LAWA primary objective was to incorporate,within a rigorous elastoplastic frame-work, the accepted empirical behaviourof rock joints. In order Co build anelastoplastic model it is necessary toestablish:
a) The elastic behaviour
b) The yield surfacec) The hardening lawd) The flow rule
2.1.Elastic behaviourConcerning the elastic behaviour boththe normal (to the joint) behaviour andthe shear behaviour should be defined.The normal displacement u was splitted
- 521 -
in two parts: an stress-relate.) compo-nents ( u* ) .mi a .|i.un,.-t r lei 1 term,associated to di latency effects (uP| :
u - u e l • UP (1}
The dilatancy term uP is describedby means of the plastic formulationdeveloped later on.
The u e l term may be analyzed fromnormal compression tests. Availableresults indicate a nonlinear ;-u°lrelationship with a marked "locking"character (the joint tends to be infinitely rigid for increasing nor~.il stress, ~;) . A relationship propose! by Goodmanand St. John (1977), somewhat simpli-fied,
" " C ''u *uc-l' (2)
rocw a s u s e d , a s a s t a r t i n g p o i n t . , t - j
d o r c r i b e t h e n o r m . i l c o m p r e s s i o n o f '.;•:•
T o m t . I r . t i n : , o q u . i t . i o n u,,,,. i s t h<
r i . i x i m u r . s h r t e n i n ; o f t h e l o i r . : i n . i u - - . ' - i
b y t h-. i: i- •.: . t i e . • . : . . ; .- l t . • • l r , -< .>.,!. , t . l ! l ! • . Til- •. ) . ' f f : . . ' l . - , : - ' 1- . . J ! ' . u . i l ly
f - u n i i '.<.• :,•• ::, i l l e r b u t ~lz'< • • o n e ,
a n ! : r . : ) , : • v • t h - r - - l . i ' i :. ' . ; : '-:!*•'•: I ;;. t..
wii: c h
• • iifrn i
T h i " i s i l l u s t r a t e d in Fi'J-
•>liow;. ,I pi M -j{ r h " 'iiri"ii' : .:.
for 5-..-V. u l v.'ili.o:- •..: t ('.. : ! .. .
tin.: u,.c • ! r-:. Th:\. t a c t or I iit.i--
n'jjnerical 3i f f I C J 11 les siiic*1 th<- u ' *
i n t e r c e p t o f t h e t-inger.t tc; t h e • - i" •
c u r v e ( w h i c h is u s e d l n t h e i t e r a t i v e
folution process di".;cr ibc.J I J V I I -aybfco:::e ^c^itivo (joint open.';1. Forthis rea — ri n two parameter law (t=l)
was selected in this work. It is be-lieved that it provides a sufficientlygood description of normal (elastic)behaviour for practical purposes.
With regard to the elastic shearstiffness terms, constant, or stressdependent moduli may be specified.Accordingly, the elastic matrix Erelating the stress vector acting on
the joint (see f i g .
and the r e l a t i v e el.ist
vector ("s t ra in")
is given by
v '
(u<-J) 0
K.,
'::.- : l.iy. -merit1 ' 1 . V j ' 1 , v ^ 1 !
(•1)
Fig. 1- ExaiTiple plor of t h erelationship showing a "tadjle" ;.»Jir,tfor value; of t less thian 1
where K,r i:> the shear stiffnei.i, and Kr.
l s obt .J l n<- 1 i ro;:: c^udt lor. ' 3 :
Fit,- - r ~ !_ . -*»•"Y.
It is- possible, by changing t.".e matrixn, to uso more complex elastic behaviourincorporating, for instance, shear andnormal behaviour coupling, anisotropyor nonlinearity.
2.2.Yield surfaceIt has been found that a simple two-parameter hyperbolic yield function fitswith acceptable accuracy the shearstrength of natural and artificial joints.For a single shear component the law isgiven by.
t = B • " ' •2-a-C (6)
where a and B are constants. The constantB happens to be the slope of theassymptote when o•*>••. On the other handa gives the distance between theorigin and the C-axis intercept of theassymptote (see Fig.4).
A few comparisons between jointstrength data and the proposed envelope
- 522 -
are shown in Fig. 3. Fig 3a. reproducesthe results used by Hoek (1983) tosubstantiate his proposal of a jointstrength criterium and shows the compa£ison between the Barton and Hoek crite-ria and the equation (6) to a number ofstrength data points of moderatelyweathered greywacke (15x15 cms samples)
tested by Martin and Millar (1974).Any of the three mentioned criteria aregood enough for practical purposes. InFig. 3b. the results of several testson 13x10 cms plaster replicas tested byPeek (1981) are presented together withthe best approximation of the law givenby equation (6).
o)Stresses
blrelotive displacements
Fig.2-Definition of stresses and "strains"
in the joint.
The range of normal stresses is verysmall but unlike the Martin and Millar(1974) results, multistage testing couldbe avoided in this case. Again the compa£lr.on is judged quite acceptable. Finally,Fig. Jo shows unpubll sir; ! shear strengthdata obtained in the Hcx-k portable iristru'•nt (Houk <IIK1 Uray, 1974) on 0 11,6 cms
sampler of clay-banded tertiary marlsin the vicinity of Caspc Jam, Spain.Curve (1) shows the test results of apreviously induced tension fracturethrough the intact material. Curve (2)reflects the strength of the clayeybands, again in a pre-splitted joint.The test followed a multistage techniquesimilar to the one described by Martinand Millar (1974).
Fig. 3-Comparit,on of equation (6) withjoint strength data, a) Moderatelyweathered greyvacko. Grade 3, Test sample7 (Martin and Millar, 1974). b) Plasterreplicas of a natural joint (Peek, 1981).c) Tertiary clay banded marls of Caspedam, Spain.
In all the caws presented the fit wasobtained by moans of a minimun squareliiii'iir rogr'bsion between T'/" and 0(set1 equation 0 ) . It is concluded thatthe proposed '.lio/ir strength equation is•is <JOCK1 .f. otliT;-. already existing, nutit offer:. ';oirio ri'lviintageo. Fif-t of all
- 523 -
it bears a direct- relatlonship with theMoht-Coulomb linear criteria and its twuparameters tg0 and a have equivalentmeanings. Unlike Mohr-Coulomb, law,however, equation (6) shows a continuous,first derivative in the origin and thisis an advantage if it is to be used asa yield function in a numerical model.On the other hand the Barton (1974) wellknown relationship presents rapidoscillations for small values of ; winchis an undesirable situation for numericalmoit.-1 ling- Finally, the criteria propose,jby Hoek (lrJ03) is perhaps too compl ieatf-,3anj, for large value;;, tends to a non~rejii.tic jcrj (tangent) friction angle(horizontal ainjTiptotf) .Frun t rto numerical point of view,
equation (C) offers an additional advan-)••: There exists analytical solution for
point of intersection between the1 : f ' i n : l ; •-:: ''.-j'j-jt i u n fi- ai,-\ t 11- -
;• i'. :,. I r. t h i s w j y \ !.>• i .i:.il
v seir :. t .r I h' ;..:n' :r.
d .", (d (8)
th--
. ' . i •
b- -I;.iv b>.
a v j i i < ? : an:! s ' ± : . ; i ' u t ' - J by a f a : t .m<jl<-o p e r a t i o n . B ' i s e j on e q u a t i o n J.) th<-f o l l o w i n g y i e l ' i f u n c t i o n h a s ( n
•->'
Tht zones F '0, F 0 and the bouniaryF=0 are represented in fig. 4. Theconstants a ani tg*l were selected asthe hardening paranc-ters of the model.
K m . •J-Vn-1
-'. i .Hardening lawIn order to fellow the h.iriienin; Inof the joint the following mfertiilplastic vari.ille was chosen
where dv\ and dv . arc the plasticcomponents of the two shear relativedisplacements. Any function relatinga and tg 0 with C may be used, bearingalways in mind the adaptability toexperimental results.
In the present work the- polynomialvariations, shown in Fig. 0 for a, havebeen used. They allow the modelling of,i peak shear strength for total accumu-lated "strains" given by r,p and a strainsoftening behaviour up to residual conthtion.i for a plastic "strain" given byC . The proposed formulation may beextended to include amsotropic effectsand to model reversal effects.
Fig. rj-Definifion of hardening l.jv forparameter a.
i.4.Flow ruleThe plastic potential muit be able tomodel twu distinct effects: the influ-ence of norrnal stress on the intensityof dilatancy and its variation withplastic deformation.
On the other hand it has been acceptedthat the increment of plastic deformationin tanqenti.il direction follows thedirection of the shear stress. Howevera dilatancy angle i is defined to modelpl.istic dilat.incy effects. Therefore,i! y V ,". ] , " j) is the plastic potentialit gradient along *.,Tn and : mustit l sf y:
8o . 3p J . t g i (9)1
Tli>- v.in.ition of cillatancy ,inqle withrnMl Lre.-, n..iy lie specified hy some(•') f unct ion . The Laiianyi nr>.I
- 524 -
Archambault (1970) expression gives
tgi=tgic. (l-j/qu) * (jo)
where l; is the dilatancy angle for0 = o and q u the unconfined compressivestrength of the rock irregularitiesdescribing the joint.
Fig. 6-Variation of initial diljtancyingle with the lntprna) plastic vari.i-
In or:i"r to introduce tin.- influence ofaccumulated plastic deformation, theangle iQ was assumed to decreaselinearly with Cup to a constant "resid
ual" value (see Fig. 6i. The .ihapetnc {• ti-ntial O in th'.C" i '• shown it1. Fig. 7.
Fig. 7-Plaitic potential
?.NUMERICAL IMPLEMENTATION OF THECONSTITUTIVE LAW
In tlii implementation of a constitutivelaw in a general numerical procedure,situations may arise that have not beenspecifically contemplated in the formu-
lation of the model or that requiresome specifical numerical treatment.In the case of the rock joint model itis necessary to distinguish, firstlywhether the joint is open or closed.In fact, the constitutive law describedabove is only valid for a closed joint.If the joint is open it will be neces-sary to ensure that no stresses aretransmitted across it.
In the case of a closed joint it isnecessary again to distinguish betweentwo cases; an opening joint or a closingjoint, since they require differentnumerical treatment. If the joint isin the process of opening an "initialstress" approach can he used directly.However, if the- joint is closing thereis the possibility that the- normal dis-placement imposed by the "initialstress" ni'-thod is larger (in absolutet-.T'v.; th.jn t h' • trKiXiriU.". ii i splaceniontuivii :.••/ th' ii:r;-..il stress - norm-ilil :,pld' e::i.-nt r< lationshlp (eq. i). Il,trial case tliv corivorg'-nco can notbe achieved. The u»e of a conventional"initial deforir.dtion" procedure is notacceptable either because it couldlead to difficulties if a tangentialstress higher than the peak tangentialstress was imposed at some stage. Toovercome' this difficulty a "xixed"procedure is adopted in which thevariables imposed in each incrementcirv t Sie normal stress an-i the two' an:jent i .11 "strains", '. , ' v i ait i' vFinjlJy, in c-v:r\on with all ••Irtst"-
pl.is!;/.- -..:'•:•, it is also n e c s : J: yto Jist i r.-j.ji«ifi elastic stress incrementsfr-ori e lastoplast I C ones.
In Fig. H, a flow chart is presentedshowing the various paths that can befoll'.jwe.-j by the analysis depending onth< lifferent states of the joint.
4.APPLICATION OF THE CONSTITUTIVh LAW
TO FIKLD TKST DATATli'.- ,iv,ii labi 1 l ty of 3o:n<- g->..J qiiiilityfield test results has al lewe.' to checkthe usefulness of the constitutive lawdescribed above for modelling adequatelythe actual behaviour of joints. Thetests were performed on '.he beddingplfiries of 'i fairly massive limestoneusing the ,iri ing'.-ment depicted in I'm.9. There was .i especial intercut inmodelling the behaviour of this discon-
- 525 -
t inuities because th<> arch Jari to beanalyzed is founded in this limestone
( " n . H j .
us» Homes atucs 1
TCP 9 THE Ci.*S !I t MUCHCNI Of 6" v' t !
x it..141" i
r..r..«•Situ, sac siacss «.ms j
S. .e
Kig. fl b.
Th'_' parajaoters defining the normalstress/normal displacement elasticrelationship (eq. 3) can be independently determined from the appropiate expo£imentai curves. The values of umc =0.9x 10-3m and c = -98 KN/m2 yieldeda sufficient approximation to the fieldtest d.ita.
Of more interest is the definitionof the shear stress/shear displacementbehaviour because the behaviour at dif-
Kig. c<-Conputation flow chart for thepropo'-'.1!: constitutive law. a) Open joint.,b) Closed joint, "mixed" procedure, c!Closei joint "initial stress" procedure.
ffroiu jtrcsa levels has to be adequate-ly uri/dictei. Firstly, the elastic partot thi-- iy.o<iel must be completed by th"dcten::inantion of IC, the shear stressshear displacement elastic modulus. Avalue of ICj. = 473OOOKN/m3 was dJopte.ibased on the mean slope of the earlyparts of the test curves of Fig. 11.
The definition of the family ofyield surfaces and the hardening lawrequires the determination of
- initial yield surface (ao, 0O)
- peak yield surface (ap# 0_)
- residual yield surface (ar, t*r)- residual plastic variable at peak f.p- internal plastic variable at residual
state ; r.
The values for ap, 0p, ar and (!r wereobtained by imposing the best possiblefit between tin1 constitutive lawhyperbolic functions (eq. 7) and themeasured peak and residual strengthvalues. The parameters thus determinedwere:
l<(0.i» KN/mJ C , 3r>.2i%"M i \ ( , KN/m:! fir - \'-.?(,o
i tit iTi'st lnrj t<-> not" th.it I Iv
;,-Milfll da',1 all«wi»! ,'.., -n (••• JI.I ic
ap
a,-It is
- 526 -
equal to 0r without loss of accuracy.This appears to imply that for thejoints studied, the strain-softeningprocess did not affect the frictionangle defined according to equation 7.
Fig. 3-Arr<inge.i>ent for field sheartest.
It was attractive to make theassumption that 0 did not change duringstrain hardening either and that onlythe degree of mobilization of the parameter a varied. Therefore, the a valuesof ao=o and 0O=35,26° were selected todefine the initial yield surface. InFig. 10 the initial, peak and residualyield surfaces are shown.The values of the internal plastic
variable at peak and residual states(Op and ',r) were fixed at 3 mm and9 mm respectively. They were based onthe displacement required to achievepeak and residual conditions in thefield tests.In Fig. II the very good agreement
between the model prediction and thetest results can be observed.Unfortunately the data concerning
the normal displacements during shear,necessary to obtain dilatancy parame-
ters, was not quoted in the originalreport presenting the field test data.A conservative value of io=9° wasadopted, which was the same one usedin the previous analysis (Alonso andCarol, 190S). Finally compression teston rock specimens gave an averagecompression strength of 50 MPa.
Fig. 10-Initial, peak and residualyield surfaces of limestone beddingplane (Canelles dam).
5.THE JOINT ELEMENT
5.1.GeometryA 16-node isoparametric joint elementhas been developed (Fig. 12). Eight"midplane" points, PMt, located betweeneach pair of opposite joint nodes definethe average position of the jointcontact surface (Fig. 13). A curvilinearcoordinate system (s,t) is defined inthis average surface by means ofquadratic twodimensional shape functionsNi. Two typical expressions for corner(Nj) and side (N5) nodes arc
1/4 -d-s) • (l-t)-(-l-s-t)
$= 1/2 (II)
The global coordinates of any pointin the joint surface arc then definedby the vector r(s,t)*(x(s.t), y(s,t),z(s,t)) where
sx(s.t)xyN.(s.t).xPMi n ; )
and y and z have^similar expressions.The vectors it and -j£ define a
tangent plane at any joint in the joint
- 527 -
Model predictions
Actual measurements
< 4 6 7 8 9Refcitn* horizontal displacement (mm)
Fig. 11-Experimental results and r^odcl predictions for the shear behaviour of
limestone- b.riding plan* 'Candles 3an) .
surface (Fig. !••,. Thes-= two vectorsallow the determination of th' n:.'rr::j:vector n and the parameter J, necessaryto perform the numerical integration.The following expressions hold
G =
nx "y nz
[ t2x C2y C2
(16!
J =•dsdt
- 1n = T
(13)
(14)
where
JL 3?
3s
and dA is the differential vector ofjoint surface area. Knowing it andselecting a tangential to the surfaceunit vector T\ it is possible tocompute the third reference vectort2 = rf -~t, and to establish a local refe£ence system at any point in the surface.The following matrix characterizes thissystem
Fig. 12-Three dimensional joint element
- 528 -
Fig. 13-Definition of mid-surface points(PMi). Local and global coordinates.
5.2.Linear formulationKnowing the relative displacements
'*xPMi# 'vPMi' '*zPMi' at t n e mldsur£'iCC
points of the joint, PMi, it is possibleto obtain the relative displacements atany point in the midsurface by means ofisoparametric interpolation. For instance
( s , t > =8
(s,t)-'xPMi (I7)
and similarly for the remaining twocomponents. The vector of relativedisplacements is then given by
Txyz(s,t) = (fx(s,t), iy(s,t>, Vs,t)>-
= N(s,t>- *el (18)
where the matrix N contains the shapefunctions and oe| is the element nodaldisplacement vector. The vector ofrelative displacement in the localcoordinate system, C^, is needed todescribe constitutive behaviour. It isgiven by
*j(s, t) = B (s, t) • ?el (19)where the geometric matrix B is givenby
B «{ - B , | - B 2 | - B 3 | - B 4 | B , | B 2 | B 3 | B 4 |
I - 5 s I - » e I - S T I - 5 e I » s l s 6
where
(20)
(21)
Fig. 14-Local reference coordinatesystem.
Using the standard virtual work approachthe element nodal force vector ? ej, cannow be established:
«„el -cel+ Fel (22)
where Kej is the element stiffnes matrixwhich can be obtained if an "elastic"constitutive matrix D is defined in thejoint and Fej is the element nodal forcevector induced by initial stresses ^y.
?el = 'el 5* • * ] • " * (23!
5. 3.Extension to nonlinear behaviourAn iterative procedure is needed tctake into account the nonlinear jointbehaviour. The method adopted is indi-cated in Fig. IS in terms of twogeneralized stress and "strain" axis?j and U . Beginning with an initialelastic p matrix a stress-strain statelike 1 {ifj1,!^1) is obtained in anypoint within the joints. The integrationof the constitutive model using the"initial stress" or "mixed" proceduresdescribed previously, leads to a dif-ferent stress strain state (I'). The"initi.il" stress to be relaxed in thenext iteration is givi>n by
- 529 -
,01 o-a, oi - '- Di
(24)
where the matrix D l can be computed indifferent ways (initial, tangent, secantor any suitable combination). A newlinear solution takes the joint stressesand strains to a state like 2 and fromthen on the iterative procedure isrepeated until convergence is finallyreached.
Fig. 15-IteraLivt- procos? ;n th" t. -• j spac-
6.MODELLING LARGE SCALE FIELD SHEAR TESTSWhen checking the usefulness of theconstitutive laws against some fieldtest results, the assumption was madethat those results represented the actualbehaviour of an clement of joint subjectto uniform stresses and displacements.In fact, the test was performed usingthe arrangement of Fig. 9 in whichuniforms conditions do not necessarilyprevail. Taking advantage of theincorporation of the joint element in a3-Dimensional Finite Element Programmeit is possible to analize the actualtests and compare the predicted resultswith the experimental data.
The test was modelled as an clasticbrick element resting on a jointelement (Fig. 16). The selected dimen-sions wore the saim; as in tho actualfield test.
It was subject first to a normalload of fij';KN, •"-t.ing uniformly on the
top face of the brick element. Afterwardsa uniform horizontal displacement wasapplied to the vertical face (5-7-9-11)of the same element. The parameters ofthe joint model were those determined insection 4 whereas the intact rock wasmodelled elastically with E « 10000 MP a
and v = 0.2.
Fig. 16-Discretization of field sheartest.
The predicted load-horizontal displace-ment curve is presented in Fig. 17together with the field result. Againthe agreement is very good demonstratingthat the sample dimensions chosen forthe test were adequate.
7.CONCLUSIONSAn elastoplastic three dimensionalconstitutive law to model the behaviourof joints has been developed. It incor-porates some well accepted empiricalfacts of joint behaviour.
A new equation for the definition ofthe yield surfaces is proposed. In ad-dition to some numerical advantages, itfits very well available experimentaldata.
The proposed constitutive law yieldspredictions in good agreement withexperimental data from field sheartests.
A 16-node isoparametric joint element,for use in three-dimensional finiteelement programs, is developed. Whenapplied to the modelling of one largescale shear test it shows a goodagreement between computed and experi-
- 530 -
mental results. Joint strength characteristics of aweathered rock. Proceedings of the3rd Congress of the ISRM, Denver, vol.2A, pp. 263-270.
Peek, R. (1981). Roughness - shearstrength relationship. Jnl. of theGeot. Engng. Div. Tech. Note, vol.107, n? GT5, pp. 672-677.
Van Dillen, D.E. and Ewing, R.D. (1901).EMI-IES: A Finite Element Code for RockMechanics Applications. Proc. of the22nd Syrap. on Rock Much. MIT, pp.373-37?.
Fig. 17-Coroparison of computed andexperimental results of field shear test.
REFERENCES
Alonso, E.E. and Carol, I. (19B5).Foundation analisis of an arch da.T..Comparison of two modelling techniques:No tension and jointed rock material.
Rock Mechanics and Rock Engineering.
In press.
Barton, N. (1974). Estimating the shearstrength of rock joints. Proc. ofthe 3rd Congress of the ISRM, Denver,vol. 2A, pp. 219-220.
Goodman, R.E. and C. St. John (1977).Finite element analysis for discont^nuous rocks. In "Numerical Methods inGeotechnical Engineering". Ed. by C.S. Desai and J.T. Christian. McGraw-Hill.
Heuzc, F.E. and Barbour, T.G. (1982).
New models for rock joints and inter-
faces. Jnl. of Geot. Engng. Div.,
vol. 108, n? GTS, pp. 757-776.
Hoek, E. (1983). Strength of jointed
rock masses. Twenty third Rankine
Lecture, Geotechnique, vol. 23, n? 3,
pp. 185-224.
Hoek, E. and Bray, J.W. (1974). RockSlope Engineering. The Institutionof Mining and Metallurgy. London.
Ladanyi, B. and Archambault, G.A. (1970)Simulation of shear behaviour of ajointed rock mass. Proc. llth Symp.Rock Mech. AIME, pp. 105-125.-
Martin, G.R. and Millar, P.J. (1974).
-M1 (six
9. Appendix C
SUPPLEMENTAL EXTENSOMETER DATAFOR
INTERMEDIATE EXCAVATION STEPS
0.30
10 15 20 25Distance Along Extensometer from Collar (m)
30
u>I
Figure C.1 Roof Extensometers EXT01-02Linear Joint 3x3 : Pilot Excavated to 23.5 m
E
oo
c
Q>>O
5oCO
c
"x
to
10 15 20 25Distance Along Extensometer from Collar (m)
Figure C.2 Diagonal Extensometers EXT03-09Linear Joint 3x3 : Pilot Excavated to 23.5 m
EE
oo
Q)
Q>
%
EoW
c<D
~xLU
0.7-
0.6-
- 0 .1 -10 15 20 25
Distance Along Extensometer from Collar (m)
Figure C.3 SL Extensometers EXT07-08-04Linear Joint 3x3 : Pilot Excavated to 23.5 m
30
U>
35
0.45 J
"oo
cEQ>
%EoW
c<D
-0 .10-
i
10 15 20 25
Distance Along Extensometer from Collar (m)30 35
Figure C.4 Diagonal Extensometer EXT06Linear Joint 3x3 : Pilot Excavated to 23.5 m
0.30-
10 15 20 25Distance Along Extensometer from Collar (m)
30 35
Figure C.5 Floor Extensometer EXT05Linear Joint 3x3 : Pilot Excavated to 23.5 m
0.5-
0.4
0.3
E 0.2^
1 o.HE0)O 0.0QLOT
Q.2 -0.1-
| -O2-Da:
-0.3-
-0.4
-0.5
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
0 5 10 15 20 25Distance from the Tunnel Wall (m)
Figure C.6 Full Section Roof at the FractureLinear Joint 3x3 : Pilot Excavated to 23.5
30 35
1.75
1.50
1.25E
jment
on
spl<
Q
oa:
i
0.75
0.50
0.25
n
-0.25-
-0.50
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
Figure C.7
10 15 20 25
Distance from the Tunnel Wall (m)
Full Section SL at the FractureLinear Joint 3x3 : Pilot Excavated to 23.5 m
30
Cn
i
35
0.30
10 15 20 25Distance Along Extensometer from Collar (m)
30
s
35
Figure C.8 Roof Extensometers EXT01-02Linear Joint 3x3 : Pilot Excavated to 25.2 m
E
oo
0)
Q)
O
c
10 15 20 25Distance Along Extensometer from Collar (m)
30 35
Figure C.9 Diagonal Extensometers EXT03-09Linear Joint 3x3 : Pilot Excavated to 25.2 m
1 J0\
.t. C
olw
.r,em
ent
o
"5C
0.7-
0.6-
0.5-
0.4-
0.3-
0.2-
0.1-
oV)
S o.o
-0.110 15 20 25
Distance Along Extensometer from Collar (m)30
Figure C.10 SL Extensometers EXT07-08-04Linear Joint 3x3 : Pilot Excavated to 25.2 m
E
oo
otoc
"x
10 15 20 25Distance Along Extensometer from Collar (m)
30
i
35
Figure C.11 Diagonal Extensometer EXT06Linear Joint 3x3 : Pilot Excavated to 25.2 m
0.30
E
jg"oo
c
a>
"5o(0
cUJ
o.oo
-0.05-
-0.10-10 15 20 25
Distance Along Extensometer from Collar (m)30
I
35
Figure C.12 Floor Extensometer EXT05Linear Joint 3x3 : Pilot Excavated to 25.2 m
0.5
0.4-
0.3
-0.5
o = Displacement @ Z = 24.0 m^ = Displacement @ Z = 24.1 mo = Relative Displacement
10 15 20 25Distance from the Tunnel Wall (m)
30 35
Figure C.13 Full Section Roof at the FractureLinear Joint 3x3 : Pilot Excavated to 25.2 m
1.75 -J
E
0)O
_gQ.OT
o
oQ:
o = Displacement @ Z = 24.0 m= Displacement @ Z = 24.1 m
o = Relative Displacement
-0.25
- 0 . 5 0 -10 15 20 25
Distance from the Tunnel Wall (m)
Figure C.14 Full Section SL at the FractureLinear Joint 3x3 : Pilot Excavated to 25.2 m
10 15 20 25Distance Along Extensometer from Collar (m)
Figure C.15 Roof Extensometers EXT01-02Linear Joint 3x3 : Slash Excavated to 23.5 m
30 35
J,
"oo
c<D
oCOc0)
00
10 15 20 25Distance Along Extensometer from Collar (m)
30 35
Figure C.16 Diagonal Extensometers EXT03-09Linear Joint 3x3 : Slash Excavated to 23.5 m
I
in
i
10 15 20 25Distance Along Extensometer from Collar (m)
30 35
Figure C.17 SL Extensometers EXT07-08-04Linear Joint 3x3 : Slash Excavated to 23.5 m
E
oo
c
Q>
0)
oincQ)
UJ
i
10 15 20 25Distance Along Extensometer from Collar (m)
30 35
Figure C.18 Diagonal Extensometer EXT06Linear Joint 3x3 : Slash Excavated to 23.5 m
0.30
10 15 20 25Distance Along Extensometer from Collar (m)
30 35
Figure C.19 Floor Extensometer EXT05Linear Joint 3x3 : Slash Excavated to 23.5 m
o = Displacement @ Z = 24.0 m^ = Displacement @ Z = 24.1 mo = Relative Displacement
10 15 20 25Distance from the Tunnel Wall (m)
t
30 35
Figure C.20 Full Section Roof at the FractureLinear Joint 3x3 : Slash Excavated to 23.5 m
E
0>O
"5.CO
b"6T3O
-0.25
-0.50
o = Displacement @ Z = 24.0 m^ = Displacement @ Z = 24.1 mo = Relative Displacement
10 15 20 25Distance from the Tunnel Wall (m)
30 35
Figure C.21 Full Section SL at the FractureLinear Joint 3x3 : Slash Excavated to 23.5 m
u.ou-
J , 0.25 -i
^ 0.20-O
•
^ 0.15-
| 0.10-0) :
2 0.05-V .0)
g 0.00^ow£ -0.05-"x :Ul :
-0.10
r0 5 10 15 20 25
Distance Along Extensometer from Collar (m)
Figure C.22 Roof Extensometers EXT01-02Linear Joint 3x3 : Slash Excavated to 25.2 m
30
I
Ul
I
35
0.45
"5O
c
O(OCQ>
"xUl
LnUitn
10 15 20 25Distance Along Extensometer from Collar (m)
30 35
Figure C.23 Diagonal Extensometers EXT03-09Linear Joint 3x3 : Slash Excavated to 25.2 m
-0.110 15 20 25
Distance Along Extensometer from Collar (m)35
Figure C.24 SL Extensometers EXT07-08-04Linear Joint 3x3 : Slash Excavated to 25.2 m
E
oo
Q)
o
Eowc0)
en
10
Distance Along Extensometer from Collar (m)
Figure C.25 Diagonal Extensometer EXT06Linear Joint 3x3 : Slash Excavated to 25.2 m
E
oo
c<D
EQ)>O
"5oCO
c
~xUl
0.05-
0.00
-0.05-
-0.1010 15 20 25
Distance Along Extensometer from Collar (m)30
I *Ln09
35
Figure C.26 Floor Extensometer EXT05Linear Joint 3x3 : Slash Excavated to 25.2 m
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
10 15 20 25Distance from the Tunnel Wall (m)
30 35
Figure C.27 Full Section Roof at the FractureLinear Joint 3x3 : Slash Excavated to 25.2 m
1.75
1.50
1.25
1-
0.75-
0.50
E
cq>
o>o_go.CO
QTi 0.25T3Oa:
-0 .25-
-0.50
h
Ab.
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
0 5 10 15 20 25
Distance trom the Tunnel Wall (m)
Figure C.28 Full Section SL at the FractureLinear Joint 3x3 : Slash Excavated to 25.2 m
30 35
00rsj
0.5
(LU
LU
)
jmen
t
oo
Dis
plR
adia
l
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
10 15 20 25Distance from the Tunnel Wall (m)
Figure C.29 Full Section Roof at the FractureNo Joint : Pilot Excavation
1.75
= Displacement @ Z = 24.0 m= Displacement @ Z = 24.1 m
o = Relative Displacement
-0.5010 15 20 25
Distance from the Tunnel Wall (m)
Figure C.30 Full Section SL at the FractureNo Joint : Pilot Excavation
0.5
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
10 15 20 25Distance from the Tunnel Wall (m)
Figure C.31 Full Section Roof at the FractureNo Joint : Full Section Excavation
?
c
EQ>UDQ.10
b
Radi
al
u .
1.75
1.50 J
4 OC
1.Z5 -
1
0.75
0.50
0.25
0
-0.25
-0.50
1i«
%\\
«l),\
" * ^ m+ __^ ^ ^
t o o o O O O O O 0 0
1 ' " . » • • • ! I ' 1
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
-o----9 s r — - • ? • • - • • • i g
10 15 20 25Distance from the Tunnel Wall (m)
in
30 35
Figure C.32 Full Section SL at the FractureNo Joint : Full Section Excavation
0.5
0.4
0.3
E
"cQ>E<UOOQ.W
Q
dial
o
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
o 5 10 15 20 25Distance from the Tunnel Wall (m)
Figure C.33 Full Section Roof at the FractureLinear Joint 3x3 : Pilot Excavation
ini
30 35
E
c
<D
<DQ.
b"oO
OH
1.75
1.50
1.25
1
0.75
0.50
0.25
0
-0.25
-0.50
'&.
oA
Displacement @ Z = 24.0 mDisplacement @ Z = 24.1 mRelative Displacement
0 5 10 15 20 25Distance from the Tunnel Wall (m)
Figure C.34 Fuii Section SL at the FractureLinear Joint 3x3 : Pilot Excavation
35
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
10 15 20 25Distance from the Tunnel Wall (m)
30 35
Figure C.35 Full Section Roof at the FractureLinear Joint 3x3 : Full Section Excavation
E
1.50
1.25
1ca>E 0.75uoH 0.50
-R 0.25ooon
-0.25
-0.50
*
V
o = Displacement @ Z = 24.0 mA = Displacement @ Z = 24.1 mo = Relative Displacement
o 5 10 15 20 25
Distance from the Tunnel Wall (m)
Figure C.36 Full Section SL at the FractureLinear Joint 3x3 : Full Section Excavation
30 35
10. Appendix D
PRINCIPAL STRESS CHANGES AND HYDRAULIC APERTURESAT STRESS CELL AND PACKER LOCATIONS
N
i -
J-1
\\-
Pilot
• MeshCoerdinott Sysftm
Slosh
Jnt
lCSIR
t
Pilota = a1A =cr2
o = cr3
Slash• = a1o = a2• = a3
8 18 20 22 2410 12 14 16
Distance (m)
Figure D.1 Principal Stress Changes @ N1 vs Excavation Stage
26
N
0 0 -
Q)CDC
oCOV)
CM-
O- l
J-1 J+1
toI
COI
Jnt
lCSIR
t
PilotD = a1
A = a2
O = CT3
Slash• = cr1
o = c r 2
• = cr3
8 10 12 14 16 18 20 22 24
Distance (m)
Figure D.2 Principal Stress Changes @ N2 vs Excavation Stage
26
J-1 J+1
.. „Pilot
Coordinol* System
Slosh
Full
Jnt CSIRt
Pilot
A = ( 7 2
o = CT3
Slash
o = (72
• = (73
6 8 20 22 2410 12 14 16 18
Distance (m)
Figure D.3 Principal Stress Changes @ S1 vs Excavation Stage
26
J-1
-II-
J+1
••
Pilot
Coordinate Sy$l«m
Slosh
Jnt
tCSIR
t
Pilota = a1A =cr2
o = cr3
Slash• = a1o = cr2
• = CT3
8 18 20 22 2410 12 14 16
Distance (m)
Figure D.4 Principal Stress Changes @ S2 vs Excavation Stage
26
0 0 -
t o -
Q)CM-
J-1 J+1Pilot
• MtshCoordinote Sytlcm
Slosh
2!
I
00I
O
Jnt CSIR
l t
Pilota = a1A = a2o = a3
Slash- = cr1o = a 2• = cr3
8 10 12 14 16 18 20 22 24 26
Distance (m)
Figure D.5 Principal Stress Changes @ R1 vs Excavation Stage
CO-
f
Q>
C
0)
CM -
O-l h
toI
I
J-1 J+1Pilot
• MeihCoordinol*
Slosh
Jnt
lCSIR
t
Pilota = <J1
A = ( 7 2
o = cr3
Slash• = a1o = a2• = (73
8 18'
I
20 22 24 26
Figure D.6
10 12 14 16
Distance (m)
Principal Stress Changes @ R2 vs Excavation Stage
N
8 10 12 14 16 18 20 22 24 w 26
Distance (m)
Figure D.7 Principal Stress Changes @ F1 vs Excavation Stage
0 0 -
t o -
Q_
Q>
OinV)
CM-
J-1 J+1
-II-
Pilot
* MtshCoordinol* Sytltm
Slosh
00I
Jnt CSIR
t
Pilota = <J1
A = c r 2
o = a 3
Slash• = a1o = CT2
• = cr3
8 10 12 14 16 18 20 22 24 26
Distance (m)
Figure D.8 Principal Stress Changes @ F2 vs Excavation Stage
- 579• / .520
Table 0.1 Hydraulic Apertures
Packed-offInterval
F1§ F2ou *2 RIS • R2| S si
•S N1N2
F1F2
O "g R1
•Q.« R 2>
J « S1^ S S2
N1N2
F1F2
« 5 R2
3 2 si•M • S2
N1N2
Hydraulic Aperture'(um)
50.1050.20
49.9249.90
50.1750.04
50.0050.34
49.0948.64
49.5949.76
50.3750.23
50.4150.26
48.7748.97
49.4249.63
50.6850.34
50.3551.50
Aperture Change
(wn>
0.100.20
-0.09-0.10
0.170.04
0.00.34
-0.91-1.36
-0.41-0.24
0.370.23
0.410.26
-1.23-1.03
-0.58-0.37
0.680.34
0.350.50
'The initial aperture prior to excavation of thedevelopment room was assumed to be 50 urn.
- 581 - 58a,
APPENDIX C
PREDICTION OF ROCK DEFORHATIONS FOR THE
EXCAVATION RESPONSE EXPERIMENT - ROOM 209 IN THE
URL, LAC DU BONNET, MANITOBA
by
Gen hua Shi, Richard E. Goodman, Pierre Jean Perie
Lawrence Berkley LaboratoriesEarth Sciences Division
- 5 8 3
CONTENTS*
Page
Introduction 585
Predictions of Response - Extensometers: 585
Prediction of response - strain changes 586
Prediction of Response - Fluid Pressure Test in Hole Rl 587
The Discontinuous Deformation Analysis 587
Analysis of Face Advance Coefficients 588
Analysis of fluid flow in the water-filled fracture 589
Discussion 590
References 591
Figures 592
Appendices 606
* Table of Contents provided by Technical Inforaation Services
- 585 -
PREDICTION OF ROCK DEFORMATIONS FOR THE EXCAVATION RESPONSEEXPERIMENT - ROOM 209 IN THE UNDERGROUND RESEARCH LABORATORY, LACDU BONNET, MANITOBA.
by: Gen hua Shi, Richard E. Goodman, and Pierre Jean Perie,Earth Sciences Division, Lawrence Berkeley Laboratory
June 8, 1987
Introduction
As part of the excavation response experiment at theUnderground Research Laboratory, a series of extensometers,stress meters, and other instruments were set in place andmonitored as the 209 room was advanced. This group was involvedin the planning of the experiment and was asked to provide aprediction of instrumental readings. The expertise beingdeveloped in the group relates to the mechaui.es of discontinuousrocks. Although the experiment in question involves mining inrock relatively free of joints, some slabbing has beenexperienced following advance of the face in the URL so thebehaviour does not qualify as purely linear elastic andcontinuous. While the discontinuous deformation analysis methodthat was used for this work is intended for blocky rock, it didprove possible to apply it to this experiment. This predictionreport 'is intended to convey an experiment with a new andpromising type of model, rather than as a test of the capacity topredict rock response with the best available techniques.
The development of discontinuous deformation analysis isvery recent and, in fact, this document reports its first use inforward modelling on any real project. The recency of itsdevelopment to this stage limited the kinds of instrumentalresponses it could address. A prediction was made for theextensometers radiating in the plane perpendicular to the 209room at chainage 17.85m, and to the final readings of the stressmonitors in a plane perpendicular to the room axis at chainage27.2. Further a prediction was made of water flow in the fracturethat would be measured by water injection in borehole Rl aftercompletion of mining through the fracture.
The report first presents the predictions for eachextensometer, for the stress monitors, and for the pump test.Then it describes the analytical methods and assumptions thatwere introduced, followed by general comments and observations.Output listings of displacements and stresses and other workingmaterials are appended to the report.
Predictions of Response Extensometers:
The extensometer readings are predicted in Figures 1 through7. Each figure presents the relative displacement, in mm, betweenthe farthest anchor position and each of the other five anchors,corresponding to advance of the face. It was assumed that theextensometers were installed when the face was an average of 0.61
- 586 -
meters ahead of the extensometer ring. We treated the crosssection as circular with an average radius of 1.8 meters; thusthe initial position, describing zero response, was at z/a =0.17, where z is the longitudinal distance from the extensometerring to the face at any time. The curve of relative displacementversus face advance is based on calculations for successiverounds corresponding to the advance of the central bore, andtherefore the curves are continuous with discontinuous slopes.The response to excavation of the annular enlargement around thiscentral pilot drift - - the "slash" was not modelled. If ourpredictions were accurate in total displacement, their paths tothat final value should be initially too high and then too flat.
Each Figure of the extensometer predictions has five curves,corresponding to the first five anchor positions. The highest ofthe curves gives the relative displacement between the collar andthe end anchor (#1); the second highest curve gives relativedisplacement between anchors #5 and #1; the next highest curvedescribes relative displacements between anchors #4 and #1,followed by #3 and #1. Finally the relative displacement betweenanchors #2 and #1 are given by the lowest of the five curves.The depths of these anchors for each extensometer are listed inTable 1. The analysis used uniform, average anchor positionsrather than the actual values of each anchor.
TABLE 1Actual anchor depths corresponding to each of Figures 1 - 7 andanchor depths used to develop the points plotted on the curves
Figure Extensometer Anchor
123456
2O9-O18-ExtO4II II II QO
II II II Q 1
• I tl II QQ
• I 05 0.30
16.03 *15.8114.95 *15.0115.0115.04
9.01 15.01
All Values used: 0.20 1.50 3.00 5.00 9.00 15.00
* note renumbering of certain anchors
Prediction of response strain changes
Triaxial strain cells were embedded in the rock in boreholesahead of the face. These instruments lie in the planeperpendicular to the axis of the 209 room at chainage 27.2,beyond the fracture. The strain cells respond to changes instrain from their initial condition. These strain changes canthen be converted to stress changes. The predicted strainchanges in each triaxial strain cell are referred to coordinatesas follows: x is horizontal and y is upward in the planeperpendicular to the tunnel axis (section C-C in Figure 3-5 of
37-194-161-150-679-147-681-131
-73-68
-115-115-262-21
-470-55
- 587 -
the furnished documents). The predicted strain changes relatedto mining of the room completely past the instrument station. Thepredictions are listed in Table 2.
TABLE 2Predicted strain changes for complete mining past
Sign Convention: positive normal strain is extensile;negative normal strain is contractive; shear strain isconsistent with this convention (as for example, inTheory of Elasticity, by Timoshenko and Goodier).
Prediction of Response - Fluid Pressure Test in Hole Rl
Hole Rl pierces the water-filled fracture crossing room 209between approximate chainages 21 to 23. This hole was tested todetermine the permeability before mining through the fracture.The fluid flow was found to be 2.05 L/min. Figure 8 showsequipotential lines and flow directions for flow from borehole Rlinto the fracture, with discharge into the tunnel. Thepermeability was calculated from reported results of flow testsprior to mining. This Figure allowed calculation of the totalflow from the hole at a pressure of 1000 kPa in the hole andatmospheric pressure in the tunnel. The discharge amounts to 2.4L/min.
The Discontinuous Deformation Analysis
Discontinuous deformation analysis was originally developedby Shi and Goodman for back calculation of a blocky rock masswhose displacements were measured at discrete points.Displacements, rotations, and strains were found for each blocksuch that the indicated measurements were matched with minimumerror. Interpenetrations of one block by another were disallowedby a numerical process based upon kinematic rules. In a thesis inpreparation, Shi has advanced this method for forward calculationusing constitutive relations within the blocks. At present,friction and cohesion are zero between blocks. Each block canstrain, rotate and displace and the balance of forces and momentsin each block is established. Interpenetrations of blocks areprohibited. The solution is achieved iteratively. The entire load
- 588 -
is achieved in a number of load steps. In the present problem,the loads, derived from the initial stress state, were applied inten steps. To achieve force and moment equilibrium without anyinterpenetration of blocks, and without any tension betweenblocks, each load step required an average of four iterations.
The mesh used for the computation of the 209 room is shownin Figure 9. Four rings of blocks, delimited by radial andtangential joints, surround the excavation. The annular jointsbetween each ring are intended to model slabbing joints formed asa result of excavation. This idealized discontinuous medium isplaced in a state of initial stress with Sigma 1 equal to 31.4MPa directed clockwise 17.4 degrees from horizontal, and Sigma 3equal to 15.2 MPa (both compressive) as shown on Figure 3-3 ofthe furnished documents. The region was assumed to be in a stateof plane strain. To initiate the computation, the outer boundarywas fixed and the inner boundary was freed.
Figure 10 shows the deformed shape of the mesh after allload steps. Figure 11 compares the final and the initial blockcorner positions. On these figures, the deformations are drawnwith a fifty-fold exaggeration of scale. The numerical values ofthe x and y displacements of every corner are listed in theoutput, which is appended. On the basis of this information, thex and y displacements of each extensometer anchor position weredetermined and then rotated into the direction along the lengthof the extensometer. The resulting values represent the totaldisplacement associated with changing the region from one withouta tunnel to one with a tunnel. In order to consider the effectof face advance, the displacements were multiplied by faceadvance coefficients determined independently, as described below.
Analysis of Face Advance Coefficients
In the Proceedings of the 11th Symposium on Rock Mechanics,De la Cruz and Goodman presented mathematical results developingthe relationships between face advance and radial displacement,as a function of theta, for a linearly elastic and continuousmedium. If an instrument capable of measuring radialdisplacements is placed near the face of a tunnel and the tunnelface is then advanced more than four radii ahead, the instrumentwill record displacments amounting to something less than half ofthe total displacements associated with the plain strainsolution. De la Cruz and Goodman's formula is expressed in termsof a parameter, Z (=z/a), expressing the ratio of the initialdistance from the instrument to the tunnel face. The formulaextends only from Z equal zero to Z » 0.5. To express the fullrange of the displacement/advance relationship, a logarithmicextrapolation was performed. The results were then combined withthose from the published paper to yield Figure 12, computed forthe conditions of room 209. Figure 13 shows the same dataplotted on a logarithmic scale for Z. This establishes thejustification for the logarithmic extrapolation from Z of 0.5 togreater values. Figure 12 expresses the radial deformation in mm.for values of Z from .17 to 4, from which value onward, the
- 589 -
deformation is zero. Since this computation was made for alinearly elastic material, the proportion of total displacementcannot be determined with reference to Figure 11, as thatcomputation is for a discontinuous material. Consequently, thetotal displacement for creating the excavation was computed fromthe Kirsch Solution (using the formula presented by Goodman in"Introduction to Rock Mechanics". This amount of totaldisplacement, which is less than that given by the discontinuousdeformation analysis, is presented for each extensometerorientation in Figure 11. For a given value of Z, each curve willhave undergone a difference in radial displacement indicated bysubtracting the abscissa corresponding to Z from the initialabscissa at Z = 0.17. The ratio of this displacement differenceto the total radial displacement, given by the appropriatestraight line, establishes the proportion of total displacementwhich would be measured by an extensometer when the face hadadvanced to position Z.
Figure 14 shows the ratios as computed in this way as afunction of the face position. These curves establish thecoefficient multipliers to convert the computed discontinuousdeformation results to a three dimensional solution. Theincrements of face advance established the coefficientmultipliers given in Table 3.
TABLE 3EXCAVATION ADVANCE MULTIPLIERS
Proportion of total displacementtheta = 0,180 45,225 90,180 135,315
Stage
01234 etc
distance(m)
0.311.22.75.6
>7.2
Z
13
>4
(=z
.17
.67
.50
.11
.00
0.185.295.393.413
0.221.352.407.407
0.246.391.403.403
0.188.299.399.413
Analysis of fluid flow in the water-filled fracture
The finite difference method was used to create a mesh inthe plane of the fracture. All flow is confined to this plane.The flow is generalized LaPlacian in the plane, with theconductivity varying from point to point. The conductivity wasback-calculated from the flow experiments by running the programwith a mesh representative of the conditions of the flowexperiments. The boundary conditions were then changed to thoseshown in Figure 8 and the flow balance established. In principle,it is possible to input varying conductivity according to thestresses derived from the stress/deformation analysis of thesurrounding rock but no such runs were completed for this effort.
- 590 -Discussion
The methods used are intended for discontinuous rock. Thismedium is not really optimum for testing such techniques but itwill be interesting to see how these predictions compare withconventional finite element analyses using linear elasticity. Asnoted, the real case is neither linearly elastic and continuous,nor truly discontinuous so every method represents a compromise.
The discontinuous deformation analysis used here is adeveloping technique which is very new. It is in a primitivestate here and is not realizing its full potential. It is hopedthat a more complete analysis by this technique will be possibleat the time of further excavation response experiment stages.
The problem faced by the excavation advance is threedimensional. A two dimensional approach has been adapted byadjustments with coefficient derived from a generic threedimensional study by de la Cruz and Goodman made 17 years ago.The use of this method is premised on the assumption that thesame coefficients apply to all values of the radial distanceequally; it is not known if this is a valid assumption. It isalso assumed that the coefficients can be applied withoutmodification to a non-linear, non-elastic, discontinuousanalysis; this may also be questionable.
The introduction of water in the crack was done at theeleventh hour, as most of the energies of the modellers weredevoted to completion of the discontinuous deformation analysis.A more comprehensive effort at hydraulic modelling, coupled withthe discontinuous deformation analysis, will be directed tofuture modelling problems.
- 591 -
References
de la Cruz, Rodolfo, and Goodman, R.E., (1970), Theoretical basisof the borehole deepening method of absolute stress measurement,Proc. 11th Symp. on Rock Mechanics (AIME), pp. 353 - 375/
Goodman, R.E. (1980), Introduction to Rock Mechanics,(John Wiley)(Equation 7.2a and its derivation).
Shi, Gen-hua, and Goodman, R.E. (1986) Two-dimensionaldiscontinuous deformation analysis, International Journal ofNumerical and Analytical Methods in Geomechanics, (John Wiley)
DISPLACEMENT: MM
to
I
10 10
TUNNEL ADVANCE: M
Fig.l DISPLACEMENT OF 209-018-EXT04
DISPLACEMENT: MM
1 -
.8 .
10 IB
TUNNEL ADVANCE: M
Fig.2 DISPLACEMENT OF 209-018-EXT03
DISPLACEMENT: MM
1 .
.B .
I
10r IIB 20
TUNNEL ADVANCE: M
F i g . 3 DISPLACEMENT OF 209>-018-EXT01
DISPLACEMENT: HM
.B ,
I
10 IS 20
TUNNEL ADVANCE: M
Fig.4 DISPLACEMENT OF 209-018-EXT09
DISPLACEMENT: MH
.8 .
T I10 15 20
TUNNEL ADVANCE: H
Fifl.5 DISPLACEMENT OF 209-018-EXT07
DISPLACEMENT: MM
1 .
.0 .
IB IB 20
TUNNEL ADVANCE: M
Fig.6 DISPLACEMENT OF 209-01B-EXT06
DISPLACEMENT: HH
t J
.8 .
I
to ts
TUNNEL ADVANCE: M
Fig.7 DISPLACEMENT OF 209-018-EXT05
- 599 -
- 600 -
O)
- 601 -
- 602 -
2.0 -i
total displacement: theta - 0 deg.
135 deg
oOu>
Fig. 12
Z = * / a (dimensionless)
2.0 -i
total displacement, theta = 0 deg.
135 deg
45 deg.
10 z = / a (log scale)
oo
Fig. 13
0.50 -i
c0)
.2Q.
'"5 0.30 -iO
o0.20 :
0 0.10 -
OCL2 0.00CL 0.00
90
o
Face advance functionURL excavation response experimentroom 209
Fig. 141.00 2.00
m/a3.00 4.00
i5,00
- 606 -
Appendices
Output of discontinuous deformation analysis: The blocknumbers are defined on Figure 9 in the main report. The cornernumbers in each block are defined in the first figure of theappendix. Following that is a deformed mesh for each of the firstnine iterations. The results at the end of the tenth iterationare given in Figure 10 in the main report. Tables of values fordisplacements, strains, and stresses, are given for corner ineach block.
In these listings of output numbers, the symbols are asfollows: Uo, Vo, AZ are respectively the x and y displacement androtation pf, the block centroid. EX, EY, and GXY are respectivelythe x, y, and xy strains in the block. Strains are constantthroughout each block. CX, CY, and TXY are respectively thenormal stress parallel to x, the normal stress parallel to y, andthe shear stress x,y. The displacements of the corners of eachblock are given with the following symbols: X and Y arerespectively the x and y coordinates of the corner; U and V arerespectively the x and y displacements of the corner. The cornersare listed numerically in each block. The dimensions of theoutput are meters and MPa.
- 607 -
- 608 -
- 609 -
&
- 610 -
- 611 -
- 612 -
in
- 613 -
(0
I
_ 614 -
\
\\
XT
l
- 615 -
\
bSi
. 616 -
o>
(0
NUMBER OF BLOCKS:48NUMBER OF MEASURED POINTS:24NUMBER OF CONTACT POSITIONS:480NUMBER OF 6*6 SUBMATRICES:369NUMBER OF MEASURED STRAINS0TOTAL ITERATION NUMBER:42YONG'S MODULUS & POISSON RATIO:50000INITIAL STRESS:-30.8-163TOTAL DEFORMATION AND STRESS OF