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 (