Block modelling of rock masses Concepts and application to dam foundations

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Block modelling of rock massesConcepts and
application to dam foundations
ALERT Geomaterials 19th Alert Doctoral School,
Aussois, 9-11 October 2008 Discrete Modelling of
Geomaterials

• José V. Lemos
• LNEC Laboratório Nacional de Engenharia Civil
• Lisboa, Portugal
• vlemos_at_lnec.pt

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Outline

• Lecture 1
• Rock mechanics modelling
• Discontinuum modelling Block models
• The UDEC / 3DEC approach
• Numerical issues
• Lecture 2
• Rock mechanics application issues
• Block model applications in rock engineering
• Dam foundations
• Earthquake analysis

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Rock masses

• Discontinuities at various scales
• micro-cracks, fractures, bedding planes,
  schistosity, joints, faults, ...
• Influence of discontinuities on rock mass
  behaviour
• deformability
• strength
• permeability
• Discontinuity properties
• geometrical orientation, planarity, roughness,
  connectivity, aperture, ...
• mechanical compliance, friction, dilation,
  state of weathering, filling materials, ...
• hydraulic, coupled effects, ...

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Planar and wedge failure modes
(Hoek 2000)
(Martin 2008)
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Modelling of rock masses

• Main approaches
• equivalent continuum constitutive models that
  represent the effect of the joints (anisotropic,
  ubiquitous joints, smeared cracks, ...)
• discontinuum represent discontinuities
  explicitly
• A model is always an idealisation of the physical
  system
• need to choose what to represent
• Criteria
• Scale of jointing with respect to
  structure/excavation
• Purpose of analysis (settlements, safety against
  failure, water flow, ...)
• In practice in a discontinuum model only
  selected joints are represented (major, extensive
  features joints that define failure modes of
  interest ...), the others taken into account
  through block properties

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Joint spacing vs. structure/excavation size
Wittke (1990)
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Behaviour of underground excavations Influence
of rock mass quality and in situ stresses
Examples of tunnel instability and brittle
failure (highlighted grey squares) as a function
of Rock Mass Rating (RMR) and the ratio of the
maximum far-field stress (s1) to the unconfined
compressive strength (sc), modified from Hoek et
al. (1995).
Martin, Kaiser, and McCreath (1999)
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Rigid block model of failure of rock slope (3DEC)

• - hard rock blocks, low stresses (blocks assumed
  rigid)
• modes of deformation and failure governed by
  discontinuities
• uncertainty about exact geometry and location of
  discontinuities
• large displacement modelling changes in contact
  conditions / connectivity

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History of development of UDEC / 3DEC codes

• 1971 Cundalls initial paper on block models
• A Computer Model for Simulating Progressive
  Large-Scale Movements in Blocky Rock Systems, in
  Proc. Symp. of the Int. Society for Rock
  Mechanics (Nancy, 1971), Vol. 1, Paper No. II-8.
• 1974 general rigid block model (machine
  language)
• 1978 Cundall et al. report (FORTRAN codes)
• RBM (rigid blocks)
• SDEM (uniform strain blocks)
• DBLOCK (deformable blocks with triangular FE
  mesh)
• 1980 UDEC 2D code (rigid and deformable
  blocks)
• 1988 3DEC 3D code (rigid and deformable
  blocks)
• latest versions
• 2004 Itasca UDEC version 4.0
• 2008 Itasca 3DEC version 4.1

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Rock mechanics
Onaping mine (Canada) OConnor et al. 2004
Lorig Calderon Rojo, 2002
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Micro-mechanical modelling with UDEC
UDEC model of lithophysal rock mass - assembly
of Voronoi blocks - failure in compression tests
of numerical samples
Damjanac et al. 2007
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Masonry structures ...
Modelling of irregular masonry structures with
UDEC (Roberti and Spina, 2001)
Pillar of Lisbon aqueduct
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Earthquake analysis of historical structures
Parthenon Pronaos model (Psycharis et al., 2003)
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Discrete element block modelling The 3DEC
approach

• Main features
• blocks (polyhedra)
• rigid, with 6 degrees-of-freedom
• deformable, discretised into an internal element
  mesh
• mechanical interaction between blocks represented
  by sets of point contacts
• deformable contact approach general contact
  constitutive relations
• explicit solution algorithm employed for static
  and dynamic analyses
• large displacement analysis

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Modelling the behaviour of the block material

• rigid
• deformable
• internal FE mesh
• elastic or non-elastic behaviour
• provides internal stress state
• allows block failure

rigid polyhedral block
Rock mechanics modelling - mostly with
deformable elements
Terminology - UDEC/3DEC follow the
finite-difference (contour integral formulation)
tradition and refer to elements as zones - For
uniform strain triangles/tetrahedra it is
equivalent to FE formulation (the difference is
that the solution algorithm is explicit and
matrices are never formed)
deformable block with internal mesh of tetrahedra
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Contact representation points contacts vs.
joint elements

• Most DE codes represent the interaction between
  blocks by sets of contact points (instead of
  joint elements of FE codes)
• Point contact approach
• simple and versatile (face-to-face or true point
  interaction large displacements, ...)
• contact areas may be defined for stress
  calculations
• less rigorous stress distributions (more contact
  points needed)

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Contact representation in 3DEC
3D block models Types of elementary point
contact (sub-contacts) - vertex-to-face
(VF) - edge-to-edge (EE)
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Hard and soft contacts

• Most DE codes (including UDEC/3DEC) adopt a
  deformable contact (or soft contact) model
• contact stiffness parameters kn and ks are
  defined, in the normal and shear directions
• contact stiffness parameters are based on the
  concept of joint stiffness, which relates joint
  stresses with relative block displacements
• a small overlap results when the contact is in
  compression, as when zero-thickness joint FEs
  are used
• Some codes (e.g. DDA) adopt a rigid contact (or
  hard contact) model
• the condition of no overlap between blocks is
  enforced numerically

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DEM calculation cycle the explicit
time-stepping algorithm
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Explicit solution algorithm

• At each time step, two sets of calculations are
  performed
• (i) equations of motion of
• unbalanced nodal forces induce accelerations
• integration provides new velocities,
  displacements and positions
• update of
• (ii) application of constitutive laws to
• update
• assemble unbalanced forces for next step

- rigid blocks - deformable bock nodes
- contact displacements - element strains
- contacts - elements
- contact stresses and forces - element stresses
and nodal forces
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Solution of static and dynamic problems

• The central-difference explicit algorithm is used
  in UDEC / 3DEC for
• dynamic analysis
• real values of damping
• real block / nodal masses
• quasi-static analysis by means of dynamic
  relaxation
• high values of damping are used, so that kinetic
  energy is dissipated and the solution converges
  smoothly either to static equilibrium or to a
  failure mechanism
• as the dynamic response is not sought, inertial
  terms are not important, and block/nodal masses
  may be scaled to improve efficiency

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Rigid block motion (i)
Translational degrees of freedom (centre of mass
of the block) (i1,3)
sum of all forces applied to block
block mass
viscous damping

• contact forces
• function of relative block movements
• include stiffness terms

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Rigid block motion (ii)

• Rotational degrees-of-freedom (i1,3)
• (Eulers equations, expressed in the principal
  axes of inertia
• of the block)

sum of moments of all forces applied to block
principal moments of inertia
rotational velocities
contact force moments
applied force moments
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Explicit time integration Central-difference
algorithm

• Velocities evaluated at mid-point of time step
• New centroid velocity (at t?t/2) obtained
  explicitly from old velocity (at t-?t/2) and
  unbalanced force at t
• New centroid displacements
• New centroid positions

unbalanced force at t
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• Rotational equations
• For quasi-static analysis, the principal moments
  of inertia are scaled, allowing the same
  treatment as the translational equations
• For dynamic analysis, techniques to approximate
  the coupling tems in Eulers equations may be
  used, as well as more accurate integration
  algorithms (e.g. Simo Wong 1991, Munjiza 2004,
  ...)

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Deformable block motion

• A deformable block is discretized into an
  internal FE mesh
• The motion of the block is defined by the motion
  of its nodes (or grid-points)
• (masses lumped at nodes)
• Equations of motion of node I

nodal mass
nodal force
nodal forces
contact
applied
element
Note Starting from the standard matrix FE
equation M a C v K u F, the above
equation may be obtained assuming diagonal mass
and damping matrices, and passing the
stiffness term to the r.h.s.
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Contact geometrical parameters

• Contact mechanics common to rigid and deformable
  polyhedra
• Assumptions
• convex polyhedra
• normal defined by the common-plane algorithm
  (points from block A block B)
• Relative velocity at contact point
• For a rigid block A

centre of mass velocity
rotational velocity
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Contact velocities Deformable blocks

• Vertex-to-face (VF) sub-contact
• (triangular faces)
• Edge-edge (EE) sub-contact

(J1,3)
linear shape functions (triangular face)
(J1,2)
(linear interpolation along edge)
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Contact displacements

• Increment in contact displacement
• (relative block displacement at the contact
  point)
• Normal component (a scalar)
• Shear component (a vector)

relative block vel.
Contact stresses

• Point contact assumption contact stress is
  function of relative block displacement
• 
  at the contact point
• trial elastic increments
• normal stress
• shear stress
• Contact constitutive model

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Contact forces

• Assignment of a contact area (Ac) to each point
  contact
• Face-to-face interaction calculate area of
  influence for each sub-contact
• True point interaction define a point
  interaction stiffness or a minimum contact
  area (on physical grounds)
• Contact forces
• Normal force Fn Ac sn
• Shear force (vector) Fsi Ac ss
• Force to be applied to blocks

contact area divided by point contacts
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Numerical stability of the central-difference
algorithm

• For an elastic system, the time step should be
  less than
• Calculation of natural frequencies is very time
  consuming
• The stiffness of nonlinear systems varies during
  analysis (e.g. contacts open and close)
• An upper-bound of the highest eigenfrequency is
  estimated, using expressions of the type
• Stiffness upper bounds may be obtained by
  Gerschgorins theorem (e.g., sum of absolute
  values of a row of stiffness matrix)

highest eigenfrequency
mass or inertial term
stiffness estimate
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Expedite time step estimates for rigid block
systems

• For a rigid block, translational motion
• Rotational motion
• The minimum time step for all blocks is chosen

stiffness sum add the contributions from each
contact
rotational stiffness sum add the
contributions from each contact
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Expedite time step estimates for deformable (FE)
blocks

• Time step estimate for a node
• Nodal stiffness estimate knode ke ktc
• contribution from elements that contain the node,
  ke
• obtained from Gerschgorins theorem given element
  stiffness matrix
• based on the condition that a P-wave should not
  traverse the element in less than one time step
• contribution from contacts on edges that include
  the node, ktc (as for rigid blocks)
• The minimum time step for all nodes in the system
  is employed

for a tetrahedron, it leads to
hmin
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Mass scaling for quasi-static analysis

• When the dynamic response is not sought, scaling
  of the inertial masses improve the convergence to
  the solution
• it attenuates the contrasts in block or element
  size and stiffness
• Common procedure
• assign the time step an arbitrary value
• (e.g. unit or the average time step)
• invert the time step expressions given before to
  calculate
• the scaled mass and moment of inertia for each
  rigid block
• the scaled nodal masses of deformable blocks

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Damping options for quasi-static analysis

• Motion for a single node
• Mass-proportional viscous damping
• Local damping (Cundall)
• To obtain faster convergence to the static
  solution
• the viscous damping constant a depends on elastic
  properties
• should be equivalent to critical damping at the
  dominant frequency
• an adaptive algorithm may be used to control a
• for local damping, ? 0.8 is suitable for most
  cases

unbalanced force
damping force
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Example of damping options Rigid block under
vertical load

• Rigid block
• 1x1 m, dens 1000 kg/m3
• Mass 1000 kg
• Joint
• kn 1000 MPa/m
• Total stiffness K 1000 MN/m
• Natural frequency
• ? sqrt(K/M) 1000 rad/s
• T 1/ f 2p/? 6.28e-3 s
• Static load 1 MN
• Static vertical displacement 1e-3 m

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adaptive viscous damping (3DEC AUTO option)
no damping
viscous damping (10 of critical)
local damping
viscous damping (critical)
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Solution of quasi-static problems by explicit
algorithms (dynamic relaxation)

• Advantages
• low computational time per time step, as no
  solution of a system of equations is required
• robust for strongly non-linear behaviour
• appropriate for systems with variable geometry
  and connectivity, e.g., with block contacts
  created and deleted during the analysis
• Disadvantages
• run times for convergence depend on mesh and
  material properties (system dynamic properties)
• large stiffness contrasts in the system (due to
  mesh size or stiffness) reduces efficiency of
  dynamic relaxation technique
• need to avoid very stiff elements
• high contact stiffness
• high block moduli
• small, distorted elements

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Rayleigh damping for dynamic analysis

• Damping matrix

C a M ß K
dashpot damping relative block motion
dashpot damping global motion
Fraction of critical damping as a function of
frequency
stiffness
mass
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Rayleigh damping Stiffness-proportional
component (i)Numerical implementation in an
explicit code

• The mass-proportional component (aM) is included
  in the equations of motion
• The stiffness-proportional component (ßK) is
  included in the stress calculations
• contact viscous force
• element viscous stress
• for sliding contacts or element at yield, viscous
  damping forces are usually switched off

elastic increment of contact force
elastic increment of element stress
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Rayleigh damping Stiffness-proportional
component (ii)Effect on numerical stability of
the explicit algorithm

• Stiffness-proportional damping time step
  required for stability needs to be reduced
• This reduction in time step may be severe and
  make analysis impracticable
• using only mass-proportional damping is an option
  in some cases, but the response will show high
  frequency noise
• may need to improve mesh
• eliminate small blocks and narrow or distorted
  elements
• reduce excessive stiffnesses

fraction of critical damping at highest frequency
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Some numerical implementation issues ...
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Contact detection Cell method

• Model space is divided into cells
• Each block is classified into each of the cells
  it overlaps
• Only blocks that fall inside the same cell need
  to be checked against each other
• In order to minimise the number of times the
  contact verification routines are invoked,
  movement indicators are stored for each block, so
  that a new check is only triggered after a
  pre-defined threshold is reached.
• References Cundall (1987), Williams (1999),
  Munjiza (2004), ...

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Definition of contact normal corner rounding
in UDEC

• UDEC for contact purposes, corners are
  approximated
• by a circular arc tangent to the edges
• Corner-corner interaction has a well-defined
  normal, easily calculated
• Transition from corner-corner to corner-edge is
  smooth

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Definition of contact normal in 3DEC the
common-plane concept

• Cundall (1987) introduced the concept of
  common-plane to define the normal of the
  interaction between convex polyhedra
• The common-plane (c-p) is determined iteratively
  in order to
• Maximize the gap between the c-p and the closest
  vertex
• Minimize the overlap between the c-p and the
  vertex with the greatest overlap
• The c-p logic is also helpful in contact detection

non-convex blocks may be formed by rigidly
joining sub-blocks
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Large displacements analysis issues (i)
Contact type changes

• Numerical implementation should provide smooth
  transitions (in normal orientation, contact
  forces, ... ) for
• Changes in contact types
• (e.g. face-face to vertex-face)
• Changes in the vertex involved in a vertex-face
  contact

3D change from VF to 2 EE (view of the contact
plane)
2D from VF to FV
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Large displacements analysis issues (ii)
Corner interlocking problem

• In small displacement analysis, it is usually
  assumed that blocks A and B do not interact
• In large displacement analysis, when geometry is
  updated, a very small sliding displacement
  prevents movement in the cross-joint, which may
  be unrealistic (rock block vertices are not sharp
  and infinitely strong ...)
• Options corner smoothing (e.g. UDEC rounding),
  tolerances to define a minimum offset for
  mechanical interaction, ...

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Outline

• Part 2 Block model applications in rock
  engineering
• Rock mechanics application issues
• Dam foundations
• Hydromechanical analysis
• Failure analysis
• Earthquake analysis
• Modelling requirements

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Rock mechanics problems

• Physical problems
• Mechanical
• Hydraulic
• Thermal
• Models need to include
• Representation geological features
• Joint / block generation routines
• Interaction with structures
• Structural elements
• Rock bolts
• Tunnel support elements (beams, shells, ...)

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Rock mass modelling Rigid vs. deformable blocks

• Rigid blocks are suitable for many hard rock /
  low stress problems (near surface problems,
  slopes, ...)
• Deformable blocks provide a better representation
  for most rock mechanics problems (heterogeneous
  deformability, non-linear behaviour,) and more
  information (internal stresses, )
• In computational terms
• The advantage of rigid blocks may be significant
  in explicit dynamic analysis
• For most static problems a coarse-mesh deformable
  block model is almost as fast as a rigid block
  model
• Simple elements (tetrahedra) simplify contact
  geometric calculations higher order elements are
  useful in special cases (structural bending,
  plasticity, ...)

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Contact models

• In rock mechanics, joint stiffness (normal and
  shear) has a physical meaning
• defined in terms of stress (normal or shear) over
  relative block displacement
• represents the local deformation associated with
  contact between rough, non-matching surfaces
• may be measured in lab experiments
• Deformable contacts are usually adopted
• global deformability important
• for dynamics, need to model
• wave propagation
• natural frequencies (e.g. masonry analysis)
• Constitutive models
• Mohr-Coulomb (simplest requires less data)
• Barton-Bandis
• Many others Plesha, Saeb-Amadei, ...
• UDEC, 3DEC user-defined contact models,
  programmed in C

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Rock joint constitutive models normal direction
Barton-Bandis model
constant stiffness model
cyclic behaviour
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Rock joint constitutive models shear behaviour

• elasto-plastic type of shear behaviour

peak-residual behaviour using Mohr-Coulomb
criterion
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Barton-Bandis model
peak shear strengh
shearing of rough joints induces dilatant
behaviour - under confined conditions increases
normal stress - increases apertures and
hydraulic conductivity
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Representation of joint persistence

• Detailed analysis of rock bridge fracture may be
  addressed with PFC (particle models)
• How to apply block models for larger scale
  analysis ?

(D.C. Wyllie 1999)
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• - UDEC
• - Deformable blocks (elastic)
• Barton-Bandis joint model
• Modelling excavation sequence
• Rock bolt support

Gjovik cavern, Norway (Barton et al. 1994)
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Combination of equivalent continuum and
discontinuum representations in slope model
fine mesh block with elasto-plastic model in
shear zone
Lorig Calderon Rojo, 2002
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Concrete dam foundations

• Representation of concrete structure
• Gravity monolith
• Arch dam
• Modelling scenarios
• Operating conditions (effect of water loads,
  temperature variations, creep, ...)
• Failure scenarios (major earthquakes, flood, ...)
• Hydromechanical analysis
• fracture flow
• grouting
• drainage system
• Failure modelling
• dam-rock interface (foundation joint, arch
  abutments)
• failure mechanisms involving rock mass joints
• constitutive models for rock joints

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3DEC modelling of concrete damfoundation problems

• Rock mass representation
• Deformable blocks (with tetrahedral FE meshes)
• Dam representation
• Gravity dam standard deformable block with
  tetrahedra
• Arch dam special blocks with 20-node brick FEs
• good bending behaviour with only 1 element
  through thickness
• include vertical joints in the concrete arch
  (i.e. model cantilevers as separate blocks)

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Elastic model to calculate displacementsduring
first filling of reservoir
(all blocks are joined elastic continuum model)
comparison of monitoring data and numerical
displacements
Funcho dam
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Comparison with experiment on blocky physical
model at LNEC

• Experimental-numerical comparison
• crest displacements
• failure mechanism due to sliding on horizontal
  joint with lower friction

low friction joint plane
dam crack after sliding
Cambambe dam
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Classical analysis of concrete arch foundation
failure

• Analytical / graphical methods (1960s)
• stability of rock wedge
• - dam loads
• - water loads
• - water pressure in rock joints

Londe (1973)
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Hydromechanical behaviour fracture flow model

• TypicalIy, water flow is mainly through the rock
  mass discontinuities
• Modelling assumptions
• fracture flow ( cubic law) flow rate
• hydraulic aperture
• mechanical effects of water in joints
• pressure reduces effective stress
• water stiffness kwj Kw / a
• Representation of discrete fracture network
• 3DEC fracture flow model Damjanac (1996)

mech. disp.
kwj
kn
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Hydromechanical analysis of gravity dam
fracture flow model
monitoring (Kovari 1989)
Albigna dam model Gimenes Fernández (2006)
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Joint water pressures for dam failure analysis
Fracture flow model vs. equivalent continuum
(porous medium model)

• Flow in rock is mostly through joints, but ...
• there is often insufficient data to characterize
• joint geometry
• joint conducting apertures
• effects of grouting on joint conductivity
  (representation of grout curtains)
• 3D fracture flow models are
• complex with multiple sets, small spacing
• sensitive to joint connectivity
• computational demanding
• Water pressures on the rock joint planes for
  failure analysis with a discontinuum model may be
  evaluated approximately by
• Standard design pressure diagrams
• Hydromechanical analysis by equivalent continuum
  (using permeabilities measured in situ at
  various points)
• Simplified pressure distribution based on
  monitored pressures

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Flow in foundation heterogeneous continuum model
equivalent continuum model - hydraulic head
contours
piezometers
grout curtains
D drains
Alqueva dam model L.B. Farinha (2006)
69
Model of arch dam Analysis of failure involving
foundation rock mass joints

• Model generation steps
• Dam FE mesh (20 node brick elements) with
  vertical joints
• Create valley geometry
• Insert major faults at their location
• Insert selected discontinuities from each joint
  set, in order to create possible failure
  mechanisms

• Analysis steps
• In situ stress state
• Dam construction
• Reservoir filling
• Joint water pressures
• Strength reduction to find safety factor

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3DEC model generation - Geometry of valley and
ground surface (all blocks joined no real
discontinuities)
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Major rock mass discontinuites
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Model with rock discontinuities- major faults
(at their location)- a few joints representative
of the 3 joint sets (with their orientation)
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Modelling failure involving foundation rock mass
joints
Loads - self weight - water pressure on
dam - uplift pressures on dam-rock interface
- water pressure on rock joints (simplified
distribution)
In situ stress - case 1 analysis with
gravity loading - case 2 horizontal/vertical
stress ratio k1 (along valley axis) k0.5
(normal to valley axis)
(following in situ measurements)
Safety evaluation procedure - progressive
reduction of strength (rock joint friction)
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Displacements on horizontal plane (y-50) (max.
16 mm) Development of failure mode for a strength
reduction factor F2
75
Simplified model to assess failure mechanism
involving the dam-rock interface
Discontinuities - dam contraction joints -
dam-rock interface
Rock mass elastic
Loads - Water loads - Design diagrams of water
pressures on foundation joints
Safety assessment - reduce foundation joint
strength up to failure
Mesh of rock mass deformable blocks
onset of failure mode
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Modelling concrete dams under intense seismic
action

• Modelling issues
• Arch dams vertical contraction joints
• Dam-water hydrodynamic interaction
• Uplift pressures along concrete-rock interface
  and water pressure in rock joints
• Dynamic boundary conditions for dam-rock system
• absorbing boundaries free-field analysis
• Failure mechanisms
• concrete compressive failure (arch dams)
• sliding on concrete lift joints (gravity dams)
• lift joint cracking and toppling of blocks into
  reservoir
• failure along concrete-rock interface
• sliding on rock mass joints

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Pacoima dam (California)

• Concrete arch dam (built 1929) withstood 2 large
  earthquakes (accelerations above 1g measured at
  the abutments)
• 1971 San Fernando
• 1984 Northridge

Abutment block sliding - 0.2 m in 1971 - 0.4 m
in 1984
78
Sefirud concrete buttress dam (Iran)(Ahmadi et
al. 1992)

• 1990 earthquake (M7.6) at 5 km from dam
• Damage in concrete lift joints at the upper
  elevations
• Shear displacement of 20 mm (monolith 15)

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Dynamic boundary conditions energy radiation
into far-field

• Time domain analysis viscous boundaries (Lysmer
  Kulhmeyer 1969)
• Normal and shear stress reactions
• provided by the viscous boundary
• Applying wave input at viscous boundary
• stress record to be applied at boundary
• obtained from desired velocity input

Requirement for accurate wave propagation mesh
with 7-10 elements per wavelength of highest
frequency of interest
80
Boundary conditions for seismic analysis
lateral free-field 1D model equivalent
continuum
base absorbing (viscous) boundaries seismic
input applied as stress wave
81
Free-field model in 3D

• Free-field calculations
• 1D meshes at the corners
• 2D meshes on model sides

82
Dam-reservoir interaction

• Modelling
• dam-water interaction
• (Westergaard added masses)
• no-tension vertical joints in the concrete arch

effect of water level and joints on 1st frequency
Cabril dam
83
Rock joints hydrodynamic pressure effects
Stiffness of water in joints kjw Kw / a - if
aperture a is small, then kjw may be very large
(larger than joint normal stiffness kn ) -
joint water pressures large variation during
earthquake
84
Seismic analysis of gravity dam
sliding due to seismic action
joint water flow rates (steady state)
85
Shaking table tests of a gravity monolith on a
jointed foundation (LNEC)

• Model dam block and 4 foundation blocks
• Model height 1.5 m
• Joint friction f 32
• Loading
• constant horizontal force on dam (hydrostatic
  pressure)
• 1-D horizontal dynamic input motion at base block

J.P. Gomes (2006)
86
Dynamic test with dam-foundation joint locked

• Dam-foundation top joint may be locked to allow
  sliding on lower joints

• Final position after sliding-rotational
  mechanism
• (20 mm permanent displ.)

87
Numerical model (3DEC)

• Deformable blocks
• E 4.8 GPa
• Internal tetrahedral mesh
• Mohr-Coulomb joints
• kn 10 GPa/m
• kt 4 GPa/m
• Applied shaking table motion record at the base
  of model
• Stiffness-proportional damping of 1 at 10 Hz.

88
Displacement history of dam block- Experimental
results vs. numerical model
J.P. Gomes (2006)
89
Earthquake analysis of arch dam
El Frayle dam, Peru (INADE/Autodema)

• Dam built in 1958
• Height 74 m
• Central cantilever thickness 1.5 to 6.2 m
• Narrow valley with steep slopes
• Andesitic rock mass with dominant vertical
  jointing

Downstream view of arch and concrete buttresses
supporting left bank abutment
Lemos et al. (2004)
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Analysis of left bank failure modesunder seismic
action
Simplified representation of left bank
discontinuities to test failure mechanism -
Selected vertical joints with orientation of 2
main joint sets - Horizontal joints

• Joint water pressures set at monitored values
  (constant during seismic analysis)

• Hydrodynamic forces on dam upstream face
  represented by Westergaard added masses

(deformable blocks internal mesh not shown)
92
3DEC model - Dam and buttresses
Dam deformable blocks separated by vertical
contraction joints
93

• Material and joint models
• Dam and rock blocks linear elastic behaviour
• Concrete E 30 GPa
• Rock E 10 GPa
• - Joints Mohr-Coulomb model

94
Earthquake input
- Pacoima dam record (1971 San Fernando
earthquake) scaled to PGA of 0.6g

• - Applied in the upstream-downstream direction
  (at the model base, as an upward propagating
  shear wave)
• Absorbing boundary conditiions at model base
  (viscous boundary formulation)

95
Permanent displacements after seismic analysis
max. displ. 0.9 mm
Stress history at crest arch
Opening of dam vertical joint at crest
96
Closing remarks A few issues in practical rock
mechanics modelling ...

• Modelling of geological features/joint structure
• representation of joints/blocks for failure
  analysis
• detailed particle models vs. global block models
• Hydromechanical modelling
• discrete fracture networks / fracture flow
  application issues
• grouting analysis and effects
• Coupled processes
• Dynamic analysis
• joint behaviour
• field data on response to intense ground motion
• Modelling methodologies for safety control and
  risk assessment