CHAPTER 9 Dynamic equilibrium equation method 9.1 METHOD OF
ANALYSIS This chapter deals with the dynamic stability analysis
method for rock slopes. Several ways of approaching this problem
have been proposed in rock mechanics literature. The methods can be
subdivided into empirical (Hoek, 1976), analytical and numerical
methods. Only analytical and numerical methods are dealt with in
this chapter. The examined numerical method is the Distinct Element
Method (DEM). The DEM incorporates the dynamic equilibrium
equations of the blocks which form the rock mass in the numerical
procedure. For this reason the DEM is well suited both for static
and dynamic analysis. The analytical methods can be subdivided into
the pseudostacic and dynamic method (overall displacement method).
Both methods are here dealt wth. 9.2 DISTINCT ELEMENT METHOD 9.2. I
Introduction The Distinct Element Method (Cundall, 1971) can be
applied to the mechanical behaviour analysis of a system of rock
blocks. The principal difference between the DEM features and other
discontinuous medium method features, such as the Goodman & Shi
block theory, discussed in the previous chapter are (Hart et al.,
1988): 1. The blocks can be subjected to large rotations and large
displacements and each movement is relative to the other block
movements; 2. The block interacting forces create a blocky sysiem
geometry rnodiflcal ion; 3. The solution procedure is explicit in
the time domain. The principal advantages of the DEM application
for slope stability problems are as follows:
296 Rock slope siability analysis 1. It is possible to model
both stable rock masses or unstable rock masses. When a block is
subjected to art unbalanced force1 it accelerates, by moving
towards a new position. When a force equilibrium is determined, the
system can be considered as being in a resting equilibrium or in a
constant velocity movem ent. 2. The forces are generated between
blocks in contact. Overlapping between blocks in contact is
possible. from a numerical simulation point of view. This
overlapping is small if compared with the block sizes. 3. The
computation sequence, beteen blocky system state and the subseq
uent state occurs for small time increments. The final solution is
reached when a stable equilibrium or a uniform continuous motion
situation is determined. The method was set up for the 2-D problem
analysis, but it has also been developed for the 3-D problem
analysis (Cundall. 1988) in order to describe, more reaJisticallv,
the mechanical behaviour of discontinuous rock masses. The
numerical formulation of the method was coded in a computcr program
(Cundall & Hart. 1985) which takes into account the blocky
system geometry. by considering polyhedral blocks. These blocks are
thvided into constant strain thetrahedra in order to simplify
computation. Every theirahedron is discretized by using a finite
difference grid scheme. The blocky system comctry of a DEM model
can be constructed by using, on the basis of rock mass
discontinuity features (Chapter 3), a joint network echnique, such
as one of those discussed in Chapter 6. or can be generated by usmg
specic computer programmes (Lemos, 1987; Bar oudi et al., 1990).
9.2.2 Theoretia1forinukaion of the method The rheortica[
formulation of the method is here iilustrared wiLh reference to the
2-D problem analysis. The problem domain is divided into a system
of blocks (Figure 9.1, Plate 9. 1). The block gcometry is defined
by spacing, persistence and orientation of rock mass
discontinuities. A block can interact or detach from the adjacent
blocks. The fundamental equations of the method ace: UtTU11 Fure
9.1. Disinci Element Method mode) eamp1cs (after Homand-Etienne ci
aL 1990).
Dynamic equiiibiiui n equation method 297 The force-dtsplacement
law which relates the forces devetoped at block COn [3d to the
relai.i:e displacements; The motion equation which defines the
mOtiOn of each block, such as that due to unbalanced forces acting
on the block. The solution procedure is explicit in the time
domain. The term explicit refers to the fact that the unknown
quantities of the system equation are given as known ten-n
functions and hence the system solution does not require
transposition, elimination or back subsiut ion procedure use or
iterative technique use, as in an implicit formulation. In an
explicit formulation USC, a finite time is requircd for
informatiofl to propagate through a system of blocks. The Lime
ritervaI has to be chosen small enouQh so that information passes
between neighboring blocks at a speed less than is physically
possibic and the numerical procedure must be stable. The main
disadvantage of the explicit method is the small time step: On one
hand the time step value is connected to the block masses and the
discontinuit.y stiffncsscss. on the othcr hand, explicit procedure
allows one large diplacemcnts and non-linear or post-elastic
behaviour discontrnuous stnctures with no addit ional compu(ng
effort. 9.2.3 Block deforrnobiliry Blocks may be rigid or
deformable. In several applications, such as in the largest
298 Rock slope siohiliiy analysis pan of the rock slope
stability problems, the deformation of the single block can be
neglecicd. En the other cases, the block deformabdity can be
considered. Two numerical techniques can be used for these
purposes. In the first case the blocks are termed simply-deformable
and each block is allowed three degrees of freedom to deform
internally. In the second case the blocks are termed
fully-deformable and an arbitrary deformation of blocks is
permitted through internal block discretization in finite
differences zones. The fully-defomiable blocks are internally
discretized in finite difference triangles. The vertices of these
triangles are finite difference gndpoinis and the motion equation
can be written for each gridpoint as follows: f a, ii ds+F ,fl
where s is the surface enclosing the mass 1H lumped at the
gridpoint; il is the unit normal los; F. is the resultant of all
the external forces applied to the griclpoint; , is the
gravitational acceleration. Strains and rotations refer, for each
time step, to the model displacement. as: Ej= /(, + ,) /((t Since
incremental quantities are treated, it has to be pointed out that
the above reporied lav does not irnp)y a restriction as far as
small strains are concerned. The constitulive laws are applicd to
he deforrnable block in incremental form, in such a way as to be
easily implemented for non linear problem analysis. in the elastic
field: = + where ). and i are the Lame constants: are the elastic
increments of the stress tensor; are the incremental strains; is
the increment of volumetric strain and is the Kronecker delta
function (if i = j. 8, = l. otherwise it is nil). Non-linear and
post-peak strength behaviour law models can be readily incorp
orated into the rnathcniatical formulation of the procedure. In an
explicit formulation, after each time step, the Strain state in
each zone is computed and the correspondent stress slate must be
computed in order to proceed to the next time step. The stresses
are uniquely defined by the stress-strain mode; for each
linear-elastic, non-Linear and post-peak strength constitutive
model. 9.2.4 Disconiinuify behaviour model The discontinuity
deformability and strength features are represented by spring
slider systems with prescribed force-displacement relationships,
which allow the
Dynwuic equilibrium equorioii n7elhod 299 Fiure9.2.
Schematization of the mechanical I;:1 behaviour of diconiinuities
(after Cundall & 1-Tan. 1985). (a) our1 F n (a (b) (c) Cd) Ce)
Fitire 9.. Geometry of block coni;icis: a) Corncr tO cd; b) Edge to
edge: C) Length of the conlict. d) Rouncbn of the cointct corner (c
= center. d = knth of the rounded corner); e) DIfercm i pes of
contact between blccks (411cr Cundall & FTart. 985). evaluation
of shear and normal forces between blocks (Figure 9.2a). The overl
apping penetration amount beiweert Iwo adjacent blocks can be
determined by knowing the block geometry and the block centroid
translation and rotation. The force-displacement relationship which
can be used is: = = tsu where AF and iF5 are the incremental normal
and shear forces; and liu5 are the incremental normal and
tangential displacements and K and K5 are the not-ma) and shear
discontinuity stilfnessess (Figures 9.2b and 9.3a). (b) 1 1 It 1 14
I cfc d1 C
300 Rock slope stability analysis The contact between two block
edges (in 2-D fields, block edges substitute block faces) can be
schematized by the two corner edge contacts (Figure 9.3b). The
contact length I can be obtained as: = iii whilst the stress
increments can be obtained in terms of the discontinuity
stiffnessess expressed in the [Fl-3) units as: AG,, = kAu = k4u
(9.1) In the case of fully defot-mable blocks, some finte
difference gridpoints, which discretize block zones. may be placed
along the original edge positions (Figure 9.3c). These gridpoints
can be treated as new corners, since the block edge becomes able to
deform into a polygonal line. The expression for the stress
increment calculation does not change. The overlapping block
(Figure 9.3a-b) is only a mathematical way of computi ng relative
normal displacements. If discontinuity normal stiffness is
increased, the overlap displacement decreases. The stress
increments, determined at each time step with Equations 9. 1, are
added to the previous stresses and the constitutive criteria are
checked in the non-elastic field. If a no-tension material with a
shear strength behaviour (given by the Coulomb criterion) is
assumed. a k W1 cos k, W, sin 3 = N, (9.8) sin 3 +k,W cos + Nran )=
F1 cos 0 (9.9) K. W2 () (b) Figure 9.9. 3) Equilibrium of forces oi
a block at rest: h Forces on cliding block Caller Chang cial..
l9S4. C 1 )....
3 10 Rock slope siabiliry analysis then the substitution of
Equations 9.6-9.9 in equation: F1 =W1/g. gives: = (k - L) cos(
(9.l0) COS 3. By using the results obtained in step 2 and starling
from the beginning of the seismic event, the first positive motion
acceleration ?. which corresponds to the sLartin of the slidin
motion at the time ,. is determined. if .i, is the first positive
motion acceleration, then .i,1 at lime i, must be neatie, except
the particular casc in which.1_ = 0. Time !,at wliich. = 0 must
then be computed. The motion velocity . will start to increase from
zero from this lime. By linear Interpolation one obtains: (.t,.?1_
) At the time in which the acceleration induced by the seismic
event exceeds the yield acceleration, the sliding block velocity
increases from zero and the motion displacement soccurs. 4. The
motion velocity .tr at the time i. can he computed by assuming a
linear varia(ofl of the acceleration as: = .v/2 ( By knowing.t. the
motion veIocityx1, at the time!,,1 can similarly be calculated as:
.t,+ ((..j ()/2) The value of ., , in this equation, is obtained
from Equation 9.10. All the vclociies, in the selected time, can be
calculated by using the same procedure. The resistance to the
uphill movement can be assumed as being indefinitely large without
causing serious errors, as Newniark pointed out. The rigid block
can only slide downhill with positive velocity regardless of the
direction of the accelerat ion. if the velocity passes from
positive to negative, the times at which the velocity is equal 10
zero arid the sliding block starts to move, must be found. TI, in
the interval between two selected times, the velocity passes from
positive to negative, the Lime at hich the velocity is zero must be
found. If, for example. .t1 is positive and 2 is negmive, the
displacement increases until the velocity becomes zero. The time at
which .v becomes zero can be calculated by using he following
relation:
Dynamic equilibrium equa non method 3 I I I .. (x 2 .v 1X1, + ;)
. .,I + I i. I + I ii. I 2(i,, i,, ) which, by solving fori,1 i+I
gtves: + 4! . 2(A14.? 1)k, iJ-j I ______________________ (.v1,2 )
(I,,, Ii,. ) However, if, during a Lime interval, ihe velocity
changes from positive to negative and the acceleration changes from
negative roposicwe. the Lime .. and the Lime , should also be
computed (Figurc 9. 10). This occurs because the velocity cannot be
negative (uphill sliding is not possible). Thus the velocity
remainc nil between the timc I,,, and the lime i,,, 2 The block
will slide again, between the time and ,, , whenever ihe motion
acceleration becomes positive , will be the time at which the
movement svill acain occur. The time i, 2 can be expressed as:
x,,1(t,2i, i) +Ii+l (x,,,x1, ) Two non-coneculive displacements can
be computed during the time which passes from t, to I,, 2
1-lowever, the calculailons for these two separaced displacements
are required only in the case in which the motion accelation is (a)
x t. (b) Figure 9.10. Shding mass motion wilh pOsi(ive e)Ocits in
,, + I and posiive accelerat,on in + 2: a) cgatie acceleration in I
+ I and posi,iv ins + 2; b) If a negative velocily is possible: C)
if a negative velocity ic not possible (aftcr Chan ci al 19S4). x
t. 1 ? t t (C)
312 Rock slope stability analysis negative at the time t.) and
positive at the time t,,. Othenxise. only the lime r,,, will be
required and (he movement will cease a time t, A similar situation
can also occur when x, , is positive. This is the case when at. ti
me the acceleration is negative and the velocity is positive and
atlirne t,.., when both the acccleration and the velocity are
positive. The velocity at time 2 (Figure 9.fla-c) needs to be
checked in order to evaluate the above quoted situations. 5. The
displacementx, , between (ime 1. and Lime t,. can be calculated as:
+ .t, (t, i) + [(2., + ) ((4. t,)2116 (9.11) Thus the block overall
displacement can be dctemiined by applying Equation 9.11 to all the
times of the seismic event. The principal factors that should be
considered in order to apply the above given method to the rock
slope stability analysis in seismic conditions are: The Coulomb
criterion, expressed in terms of cohesion and friction angle is not
always available for joint shear strength in dynamic conditions:
The joint water pressures and the interactive block moveinen-vater
pressur e behaviour should be incorporated in a dynamic rock slope
inalycis model; Complex geometry shaped rock blocks, such as
teirahedra should be consId e red. Since the jOints are in some
ways rough, the JRC-JCS model ol shear joint behaviour (see Chapter
4) can be applied to follow the shear displacement of ngid x 141 n2
t (b) x x Fisure 9. Ii. Slidma mass motion with positive vetnciic t
(c) bothini,+ I andv,+2:a) Negativecceleraiionini+ 1 and positi.e
in i. + 2: b) Ahays positive velociiy: c) Negative vetoIy between n
and i , if a ncative elocIty is possibic: d NI vclocity between I,,
and I,,, if a negative velocity is not possible (alter t (