Rock Mechanics and Engineering Volume 3: Analysis ...

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Rock Mechanics and EngineeringVolume 3: Analysis, Modeling & DesignXia-Ting Feng

The numerical Discontinuous Deformation Analysis (DDA)method: Benchmark tests

Publication detailshttps://www.routledgehandbooks.com/doi/10.1201/9781138027619-4

G. Yagoda-Biran, Y.H. HatzorPublished online on: 23 Feb 2017

How to cite :- G. Yagoda-Biran, Y.H. Hatzor. 23 Feb 2017, The numerical DiscontinuousDeformation Analysis (DDA) method: Benchmark tests from: Rock Mechanics and Engineering, Volume3: Analysis, Modeling & Design CRC PressAccessed on: 13 May 2022https://www.routledgehandbooks.com/doi/10.1201/9781138027619-4

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-4Chapter 3

The numerical DiscontinuousDeformation Analysis (DDA) method:Benchmark tests

G. Yagoda-Biran1 & Y.H. Hatzor21Engineering Geology and Geological Hazards, Geological Survey of Israel, Jerusalem, Israel2Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Abstract:This chapter reviews benchmarking studies of 2D and 3D-DDAperformed bythe rock mechanics group at Ben-Gurion University of the Negev over the past decade.DDA fundamentals are briefly reviewed first, followed by a comprehensive review ofverification and validation studies. We conclude by presenting some limitations of 3D-DDA in its current formulation.

1 INTRODUCTION

Discontinuous Deformation Analysis (DDA) has been in use by the professional rockmechanics community for three decades, and has been successful in modeling discon-tinuous rock masses and masonry structures. DDA is an implicit, discrete element,numerical method proposed in the late 1980s (Shi, 1988, 1993; Shi&Goodman, 1985,1989), that provides a useful tool for investigating the dynamics of blocky rock massesand systems composed of multiple blocks, such as masonry structures.

In the 1980s Shi (Shi, 1988) proposed the two-dimensional DDA (2D-DDA) as aPhD thesis at UC Berkeley, and later on, the 3D-DDA was published (Shi, 2001). Areview of the essentials of DDA is provided by Jing (1998). Reviews of DDAwithin thescope of other numerical methods used today to solve problems in rock mechanics androck engineering are provided by Jing (2003), Jing and Hudson (2002) and Jing andStephansson (2007). MacLaughlin and Doolin (2006) published a comprehensivereview of 2D-DDA validations. The 3D-DDA, being a more recent development, hasnot been verified extensively; some new and useful validations of 3D-DDA are pre-sented in this chapter.

Many research groups have made modifications to the original code developed byDr. Shi (Shi, 1988, 1993; Shi & Goodman, 1985) in an attempt to better address someof the fundamental issues in DDA.

The contact algorithm has been the subject of many research groups. Lin et al. (1996)modified the original contact model of DDA, which is based on the penalty method, byadopting the Lagrange type approach. Ning et al. (2010) modified the contact algo-rithm of DDA by adopting the Augmented Lagrangian method. Later on Bao et al.(2014b) introduced another version of the augmented Lagrangian method, whichmakes use of advantages of both the Lagrangian multiplier method and the penaltymethod so as not to alter the structure of the system of equations. Bao and Zhao (2010,2012) have made some enhancements to the vertex to vertex contact. In the 3D-DDA

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-4 front, Beyabanaki et al. (2009) verified the 3D-DDAwith the analytical solution for thecase of a block on an incline, and modified the edge-to-edge contact constraints byusing the augmented Lagrangian method instead of the penalty method. By doing so,they claim to retain the simplicity of the penalty method, and reduce its disadvantages.Ahn and Song (2011) proposed a new contact-definition algorithm for 3D-DDA usinga virtually inscribed sphere installed in every contacting vertex, and reported anincrease in the efficiency and stability of the code, using their suggested algorithm.Mikola and Sitar (2013) developed a 3D-DDA formulation using an explicit timeintegration procedure, and a different contact detection algorithm. Wu et al. (2014)addressed the issue of contact-detection algorithm, claimed to be the most computa-tion-consuming stage in 3D-DDA simulations. They use a novel multi-shell coversystem, and claim that this method greatly reduces the contact detection volume anditerations. They report improved efficiency when their algorithm is implemented in the3D-DDA code.

In an attempt to overcome the DDA simply deformable blocks assumption andtherefore uniform distribution of stresses within blocks, Shi (1997) developed theNumerical Manifold Method, using superposition of a mathematical cover over thephysical mesh of the blocks. Bao and Zhao (2013) have integrated the advantages ofboth DDA and the finite element methods (FEM), and developed the hybrid nodalbased DDA (NDDA), thus improving the accuracy of stress distribution and allowingfor crack propagation within blocks. Jiao et al. (2012) developed a two-dimensionalcontact constitutive model to simulate the fragmentation of jointed rock. Several otherresearch groups have developed higher order DDA codes to address this issue (e.g.Grayeli & Mortazavi, 2006).

Considering friction and cohesion of block interfaces, Zhang et al. (2014a) added anadditional contact type determination for the edge-to-edge contact, in order to betteraddress the issue of discontinuities with both cohesion and friction. They verified andvalidated their improved code, and found it to be accurate. Zhang et al. (2015) replacedthe Mohr-Coulomb joint failure criterion in the original 3D-DDA code, which has aconstant friction coefficient, with rate-and-state friction laws. They tested their code bymodeling slide-hold-slide and velocity stepping tests, and reported that their numericalresults compared well with experimental results. Wang et al. (2013), in an attempt toovercome the cohesion treatment in the original DDA code, developed displacement-dependent interface shear strength. They report that landslides, simulated using themodified algorithm, exhibit a two-stage failure pattern: a relatively slow, downwardprogressive failure stage followed by a rapid, massive run-out failure stage.

Jiao et al. (2007) developed a viscous boundary for DDA based on the standardviscous boundary condition provided in the original DDA formulation, in order to dealwith dynamicwave propagation problems. Later on Bao et al. (2012) implemented newviscous boundary conditions to the 2D-DDA, in order to improve the absorbingefficiency. Zhang et al. (2014b) made two modifications to the original DDA code, inorder to investigate the seismic response of underground houses. They incorporatedviscous boundary conditions, and applied the seismic input as stress input from thebottom of the model. They report that the numerical solution of the modified DDA isclose to the theoretical solution. Ning and Zhao (2012b) coupled the superpositionboundary algorithm, to obtain a non-reflecting boundary, with the DDA, for wavepropagation modeling in infinite media.

92 Yagoda-Biran & Hatzor

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-4 Jiang et al. (2012) introduced the softening block approach for the simulation ofstage-wise sequential excavations in jointed rock masses. Their proposed modificationto the original code eliminates the need to remove the excavated blocks from thecalculation model. They present validations for their method, and report of accurateresults. Tal et al. (2014) improved the original numerical manifold method capabilityto model stability of underground openings embedded in discontinuous rockmasses byimplementing an algorithm which models the excavation sequence during simulations,starting with a domain with no opening at all and progressively adding openingsaccording to the planned construction phases. Later on this modification was imple-mented in the original DDA code and was applied in stability analysis of deep tunnelsexcavated in the columnar jointed basalts of the Baihetan hydroelectric project in SouthWest China (Hatzor et al., 2015).

Kim et al. (1999) and Jing et al. (2001) have made a modification where theycompute water pressure and seepage through rock mass, this way coupling fluid flowin fractures. More recent implementations of hydro-mechanical coupling in DDA wasdeveloped by Chen et al. (2013) and Ben et al. (2011). Koyama et al. (2011) combinedthe DDA and the finite element method for fluid flow simulation to model the interac-tion between solid particles movement and fluid flow. Jiao et al. (2015) have mademodifications to the original code in order to simulate the hydraulic fracturing process.In their proposed approach, they first perform the calculation of fluid mechanics toobtain seepage pressure near the tips of existing cracks, and then treat the fluid pressureas linearly distributed loads on corresponding block boundaries, by adding thesecomponents to the force matrix of the global equilibrium equation. They verifiedtheir new approach and found that it effectively simulates hydraulic rock fracturing.Morgan and Aral (2015) have also approached the subject of hydro-mechanicalcoupling for hydraulic fracturing. They combined the DDAwith finite volume fracturenetwork model for simulation of compressible fluid flow in fractures. The authorsclaim to improve the fluid, contact and coupling components to increase the accuracyof the solution, and report successful verification of their coupled H-M model againstanalytical and semi-analytical solutions.

Due to the first order displacement function used in DDA, the error, when computingblock rotations, may become excessive if the blocks are undergoing large rotations(Ohnishi et al., 1995).Wu et al. (2005) developed a post-contact adjustment method toovercome rotation errors when addressing rock fall problems in the original code. Liuet al. (2012) coupled the 3D-DDA with tetrahedron finite elements to overcome theproblem of block expansion under rigid body rotation. They validated their methodwith a physical model of a wedge, and got a very good agreement.

A model for cable bolt-rock mass interaction was integrated with DDA by Moosaviand Grayeli (2006). Other useful developments and applications of DDA are summar-ized in a series of ICADD proceedings (International Conference on Analysis ofDiscontinuous Deformation) published biannually since 1995.

In this chapter illustrative benchmark tests performed by members of the rockmechanics group at Ben-Gurion University of the Negev (BGU) during the past decadeare reviewed. These tests could be performed in every development of the DDA code,for verification purposes. We compare all benchmark tests reported here with analy-tical solutions, some developed at BGU and some adopted from existing publications.The authors do acknowledge other verification studies performed by other research

The numerical Discontinuous Deformation Analysis (DDA) method: Benchmark tests 93

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-4 groups, such as the extensive study of sliding blocks performed at NanyangTechnological University in Singapore (Ning & Zhao, 2012a), slope stability kine-matics (MacLaughlin et al., 2001) and analysis of three-hinged beams (Yeung, 1996)performed at U.C. Berkeley, and more, but here only verifications resulting from BGUresearch are discussed, for brevity.

2 DDA FUNDAMENTALS: A BRIEF REVIEW

DDA considers both statics and dynamics using a time-step marching scheme and animplicit algorithm formulation. The static analysis assumes the velocity of the differentblock elements is zero at the beginning of each time step, while the dynamic analysisassumes the velocity at the beginning of a time step is inherited from the previous one.The criterion for convergence in DDA is that there will be neither tension nor penetra-tion between the blocks at the end of a time step in the entire block system. These twoconstraints are applied using a penalty method, where stiff springs are attached to thecontacts. Extension or compression of the springs are energy consuming, therefore theminimum energy solution utilized in DDA assures no penetration or tension betweenthe blocks.

In the original DDA code a damping submatrix was not included in the equilibriumequations. There are two ways to introduce damping in the original code: the time stepmarching scheme introduces algorithmic damping (Doolin & Sitar, 2004), that isdetermined by the time step size used, and will be briefly discussed later, and kineticdamping. The latter can be applied by assigning a number lower than 1 for the dynamiccontrol parameter: a value of zero means the analysis is static and the velocity is zeroedat the beginning of each time step, a value of unity means the analysis is fully dynamicand the velocity at the beginning of a time step is inherited from the previous one, andany value between zero and one corresponds to the percentage of the velocity that isinherited from one time step to the next. For example, a value of 0.98 corresponds to2% kinetic damping.

For the sake of brevity, the DDA basic equations will not be reviewed here. Theinterested reader is encouraged to refer to basic DDA references (Shi, 1988, 1993; Shi& Goodman, 1985).

3 DDA VERIFICATIONS

3.1 Sliding block on an inclined plane

A block sliding on an inclined plane is a classic problem in rock mechanics, as it is asimple and intuitive model for some cases of rock slopes, and has a straightforwardanalytical solution. The DDA has been verified by several researchers with the analy-tical solution for a block on an inclined plane. Tsesarsky et al. (2005) and Kamai andHatzor (2008) have verified the 2D-DDA with the analytical solution for static anddynamic input, where in the latter a horizontal acceleration was added. Ning and Zhao(2012a) have verified the 2D-DDA for a block sliding on an inclined plane under bothhorizontal and vertical accelerations, and under different mechanisms of loading. Inthis subsection the analytical solution for a block sliding on an incline is reviewed, and


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the results of former verification studies of the 2D-DDA, as well as new verifications of3D-DDA are presented.

3.1.1 Single face sliding

Themodel of a block on an inclined plane is presented in Figure 1. The inclination angleof the slope is α, and the friction angle of the sliding interface is ϕ. The forces acting onthe block are its self-weightmg, the normal from the planeN and the frictional force atthe interface between the block and the plane, f. In the most generalized case, where anexternal force is applied on the block in the form of a harmonic function (Figure 1), thedownslope displacements of the block can be calculated by:

d tð Þ ¼ 12g sin α� cos α tan�ð Þt2 � A

ω2 sin ωtð Þ cosαþ sin α tan�ð Þ þ _d0t þ d0 ð1Þ

where g is the acceleration of gravity, tan ϕ is the friction coefficient of the slidinginterface, A and ω are the amplitude and angular frequency of the harmonic inputacceleration, respectively, _d0 is the initial velocity of the sliding block and d0 is its initialdisplacement.

3.1.1.1 2D-DDA

Kamai (2006) performed a 2D-DDA verification study of a block sliding on an inclinedplane subjected to gravity (A = 0), starting at rest ( _d0 ¼ 0, d0 ¼ 0), for an inclinationangle of α = 28°, and different values of friction angle. She obtained a good agreementbetween the two solutions (Figure 2), demonstrated by the relative numerical error,EN,defined as:

EN ¼ dA � dNdA

�� 100% ð2Þ

where dA and dN are the analytical and numerical displacements, respectively. Shefound that the relative numerical error increases with increasing friction angle, and waslower than 8% for ϕ = 25°, and at the order of 0.05% for ϕ = 5°.

Kamai and Hatzor (2008) then proceeded to loading the block with a sinusoidalhorizontal acceleration, as in Figure 1. In this case, downslope sliding willinitiate only when the yield acceleration, ay ¼ g tan �� αð Þ, is exceeded, at time

α

mgmA sin(ωt)

Nf

Figure 1 Schematics of the block on an inclined plane problem.


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θ ¼ sin�1ðtan �� αð Þg=AÞ � 1=ωð Þ, a model proposed by Newmark (1965) andGoodman and Seed (1966), and today largely referred to as ‘Newmark type’ analysis.In this case, Eq. 1 becomes (assuming that d θð Þ ¼ _d θð Þ ¼ 0):

d tð Þ ¼ g sin α� cos α tan�ð Þ 1=2t2 � θtþ 1=2θ

2� �� þ Aω2 ½ cos αþ sin α tan �ð Þðωcos ωθð Þ t � θð Þ � sin ωtð Þ þ sin ωθð ÞÞ� ð3Þ

The downslope displacements, d(t), are calculated while ay is exceeded for the firsttime at θ1, or the block’s velocity is positive. If neither condition is fulfilled, theblock is at rest, and will initiate sliding only once ay is exceeded again, at θ2, andso on.

Kamai and Hatzor (2008) have used this solution to verify the 2D-DDA. Theinclination angle they used for the slope was α = 20°, with different friction angles(for the entire set of results, please refer to their original paper). They obtained anexcellent agreement between the analytical and numerical solutions, with relativenumerical error lower than 1% for most of the simulation time (Figure 3), with afriction angle of 30°.

3.1.1.2 3D-DDA

(1) One direction of motionThe 3D-DDA mesh for the model of the block on an incline is similar to the one inFigure 1, and is constructed of a triangular prism base block, fixed in space, serving asthe incline, and a box as the sliding block.

This verification study of the block on an incline is performed in three steps: first theresponse of the block when subjected to gravity, starting at rest, is examined, then theblock is given initial horizontal velocity, and finally the block is subjected to one-dimensional horizontal sinusoidal acceleration.When subjecting the 3D block to initialvelocity and acceleration in one direction only, the problem is basically reduced to a 2Dproblem.

00

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.1 0.2

φ = 5°φ = 15°φ = 25°

0.3 0.4 0.5Time (s)

Dis

plac

emen

t (m

)

0.6 0.7 0.8 0.9 1

Figure 2 Block displacement vs. time for the case of a block on an incline – gravitational loading only.Comparison between analytical (curves) and DDA (symbols) solutions. After Kamai(2006).


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(a) Block starting at restIn this step the block is subjected to gravity alone (A = 0), starting at rest ( _d0 ¼ 0,d0 ¼ 0). In this case, Eq. 1 becomes:

d tð Þ ¼ 12g sin α� cos α tan �ð Þt2 ð4Þ

The numerical and physical parameters used can be found in (Yagoda-Biran &Hatzor, 2016). The downslope displacement history is compared for friction angleof 20° (remembering the inclination angle of the slope is 45°). Note the excellentagreement between the analytical and numerical solutions (Figure 4a), furtherdemonstrated by the low relative error in Figure 4b: after 0.2 s the numerical errordrops to below 1%.

(b) Block starting with initial velocityThe next step of the verification study is applying initial velocity _d0 to the sliding block.In this case, where no external forces are applied on the block, Eq. 1 becomes:

d tð Þ ¼ 12g sin α� cos α tan �ð Þt2 þ _d0t ð5Þ

An initial velocity of 1 m/s is applied horizontally in the dip direction. The agreementbetween the analytical and the numerical solution is good (Figure 5a), as demonstratedby the relative numerical error plotted in Figure 5b: less than 1% after 0.5 s of theanalysis.

6543Time (s)

Rel

ativ

e er

ror

(%)

Dis

plac

emen

t (m

)

2100.1

1

10

100

1

0

2

3

4

5

6

7

Figure 3 Verification of the dynamic case of a block on an incline for interface friction angle of 30°: top –comparison between analytical (curve) and DDA (symbols); bottom—relative error. AfterKamai and Hatzor (2008).


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(c) Block subjected to sinusoidal acceleration inputThe third step of the verification study is subjecting the block to one-dimensionalhorizontal sinusoidal acceleration, as in Figure 1. The amplitude and frequency usedfor the input acceleration are 2 m/s2 and 1 Hz, respectively.

Here the ‘Newmark’ type analysis is used as the analytical solution, as in the2D-DDA verification explained earlier. The friction angle of the interface betweenthe slope and the sliding block is set to ϕ = 50°, higher than the inclination angle α =45°, so block sliding will initiate only when the yield acceleration is exceeded. In Figure6a the downslope displacement histories calculated by the Newmark analysis and the

Time (s)

Rel

ativ

e er

ror (

%) 0 0.5 1 1.5 2

Time (s)

Dis

plac

emen

t (m

)

0 0.5a b

0 0.0001

0.001

0.01

0.1

1

10

123

54

DDA

Analytical

6789

10

1 1.5 2

Figure 5 a) Downslope displacement histories of a block on an inclined plane subjected to gravity andinitial velocity. b) The relative numerical error.

Time (s)

Rel

ativ

e er

ror (

%)

0 0.5 1 1.5 2

Time (s)

Dis

plac

emen

t (m

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0a b

0 0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

2

4

6

8

10DDA

Analytical12

14

1 2

Figure 4 a) Downslope displacement histories of a block on an inclined plane subjected to gravity alone,with interface friction of 20°. b) The relative numerical error.


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3D-DDA code are presented. The agreement between the two is good, and can again beexpressed in terms of relative error, presented in Figure 6b, which during most of theanalysis remains below 3%.

(2) Block sliding on an incline – loading in two directionsBakun-Mazor et al. (2009, 2012) have derived a semi-analytical 3D-formulationfor solving dynamic three dimensional displacements of single and double planesliding. In their paper, Bakun-Mazor et al. (2009) presented an analytical formu-lation which they called Vector Analysis (VA), based on the limiting equilibriumequations of vector forces acting on a block on an inclined plane. The dynamicequations of motion of their analytical solution have a discrete nature, thereforethe solution is considered semi-analytical. The formulation is explained in detailsin (Bakun-Mazor et al., 2009, 2012). They compared the 3D-DDA numericalresults to the results they obtained with their 3D solution, when subjecting theblock to sinusoidal accelerations in the horizontal dip and strike directions, withdifferent amplitudes and frequencies, and obtained a good agreement, where thenumerical error in the final position of the block after 6 seconds was approxi-mately 8% (Figure 7).

3.1.2 Double face sliding with 3D-DDA

Double face sliding refers to the classic problem in rock mechanics – the wedge failure.Bakun-Mazor et al. (2009) verified the 3D-DDA with their VA solution for a wedgefailure (Figure 8a). The input acceleration to the wedge was a sine function actingparallel to the line of intersection between the boundary planes along which the wedge

Time (s)

Rel

ativ

e er

ror (

%)

0 0.5 1 1.5 2

Time (s)

Dis

plac

emen

t (m

)

0 0.5a b

0 0.00001

0.0001

0.001

0.01

0.1

100

10

1

1000

10000

100000

0.05

0.1

0.15

0.2

0.25DDANewmark analysis

0.3

1 1.5 2

Figure 6 a) Downslope displacement histories of a block on an inclined plane subjected to gravity and1-D sinusoidal input function. b) The relative numerical error.


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slides. The plunge of the line of intersection was 30° below the horizon, and the frictionof the sliding interfaces was set to 20°. They obtained a good agreement between the3D-DDA and their VA formulation where the relative numerical error remained below7% for the entire simulation (Figure 8b).

0 0.5 1

1

Time (s)1.5

0.1

10

1000.00

0.40

0.80

1.20Vector Analysis3D-DDA

ba

sliding

block

Rel

ativ

e er

ror

(%)

Dis

plac

emen

t (m

)

Figure 8 Dynamic sliding of a wedge: comparison between 3D-DDA and VA solutions. (a) The wedgemodel in the 3D-DDA. (b) Wedge response to one component of horizontal sinusoidal inputmotion and self-weight. Lower panel presents the relative error. After Bakun-Mazor et al.(2009).

01

10

1000

0.4

0.2

0.6

0.8

DDA

Vector analysis1.2

1

1.4

1 2Time (s)

Rel

ativ

eer

ror

(%)

Dis

plac

emen

t (m

)

3 4

Figure 7 Comparison between vector analysis and 3D-DDA for 2D horizontal input motion simulta-neously. The relative error is plotted in the lower panel where the VA is used as a reference.After Bakun-Mazor et al. (2009).


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-4 3.2 Failure mode mapping for a block on an inclined plane

A block on an incline, of which sliding failure was reviewed in the previoussection, has actually four possible modes: it can stay at rest, it can slide, it cantopple, or it can slide and topple simultaneously. The actual failure mode iscontrolled by three factors, when the block is subjected to gravity: the inclinationof the slope α, the slenderness of the block δ, and the friction angle of theinterface, ϕ (see Figure 9a). The boundaries between the modes were modifiedover the years (Ashby, 1971; Bray & Goodman, 1981; Hoek & Bray, 1981;Yeung, 1991) (see Figure 9b), and recently mapping the failure mode when theblock is subjected to an external pseudo-static earthquake inertia force F (seeFigure 9a), has been demonstrated (Yagoda-Biran & Hatzor, 2013). Yagoda-Biran and Hatzor (2013) have found that when adding a horizontal pseudo-staticforce, with its resultant with the weight vector forming an angle β with thevertical direction (see Figure 9a), the mode of the block is now controlled bythe slenderness of the block δ, the friction angle of the interface ϕ, and a newangle ψ = α + β, rather than α.

3.2.1 2D-DDA

The first comparison between 2D-DDA and analytically derived failure mode chart fortoppling and sliding was performed by Yeung (1991) at U.C. Berkeley, and the resultsare discussed extensively in his PhD dissertation. Inconsistencies found in his compar-ison actually led him to develop a new equation for the boundary between toppling andsliding and toppling, turning it from static to dynamic in nature.

In this section, the agreement between the DDA and the mode chart can’t bediscussed in terms of relative error, since in this case only the first motion of theblock is of interest, so the degree of agreement can be thought of as binary – eitherthe solutions agree, or they don’t.

α

β

α

δ

δ

δ=α

φ = δ

φ=α

φ

00

10

20

30

40

50

60

70

80

90

10Toppling

Stable Sliding

Toppling+sliding

a) b)

mg

F

20 30 40 50 60 70 80 90

Figure 9 a) Schematics of the block on an incline and the angles controlling its mode. b) An example forthe failure mode chart for gravitational loading, at ϕ = 30°. After Yagoda-Biran & Hatzor(2013).


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-4 3.2.1.1 Comparison between 2D-DDA and modified failure mode chart under pseudo-static loading

Yagoda-Biran and Hatzor (2013) used the 2D-DDA to verify their newly developedfailuremode chartwhen a pseudo-static inertia force is considered. They put forth a set ofrules to determine the firstmotion of the block as obtainedwithDDA, and compared it tothe prediction of their modified mode chart. The slope angle α used was 10° for most ofthe simulations, and they used awide range of slenderness and friction. The change in theangle ψwas controlled by changing the magnitude of the applied pseudo-static force. Anexcellent agreement was obtained between the two solutions, with the DDA returningthe modes as predicted by the analytical mode chart for 106 out of the 110 simulations.

3.2.2 3D-DDA

In the case of 3D-DDA, the numerical solution was compared for both static andpseudo-static cases.

3.2.2.1 Gravitational loading

Since the 3D-DDA has not been verified many times in the past, Yagoda-Biran andHatzor (2013) first verified the 3D-DDA with the previously published failure modechart as derived by (Ashby, 1971; Bray & Goodman, 1981; Hoek & Bray, 1981;Yeung, 1991), using the same criteria for determining the first motion of the block asused for the 2D comparison. The range for αwas between 14 and 50°, ϕwas 20° inmostof the simulations, and δ was between 11 and 50°. An excellent agreement wasobtained between the two solutions; 49 out of the 51 DDA simulations produced thefailure mode predicted by the analytical mode chart.

3.2.2.2 Pseudo-static loading

After verifying the 3D-DDA with the original failure mode chart for gravitationalloading, Yagoda-Biran and Hatzor (2013) proceeded with comparing the 3D-DDAand their newly developed mode chart incorporating a pseudo-static force, using thesame criteria for determining the first motion of the block as used for the 2D compar-ison. The slope angle α used was 10° for most of the simulations. The block slendernessrange δ was between 6 and 70°, and the range of input friction angle ϕ between 6 and80°. The change in the angle ψwas controlled by changing themagnitude of the appliedpseudo-static force. All 89 simulations performed with the 3D-DDA returned failuremodes as predicted by the analytical mode chart.

3.3 Block rotation under dynamic excitation

Makris & Roussos (2000) studied the problem of the dynamic rocking of a free-standing column subjected to a sinusoidal input acceleration and their analyticalsolution can be reviewed in (Makris & Roussos, 2000; Yagoda-Biran & Hatzor,2010). The free body diagram for the problem is shown in Figure 10a. The analyticalsolution assumes that no sliding occurs at the base of the rocking block.


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Yagoda-Biran and Hatzor (2010) compared between results from 2D-DDA simula-tions and theMakris and Roussos (2000) solution. They selected a geometry of a blockwhere b = 0.2 and h = 0.6m, and used an acceleration input function of the form€ug tð Þ ¼ ap sin ωpt þ ψ

� �, with changing amplitude ap and ω = 2π from t = 0 to t = 0.5 s.

For this specific geometry and frequency of motion, the analytical solution shows thatthe block will not topple with ap ¼ 5:43m=s2, but will topple with ap ¼ 5:44m=s2.Yagoda-Biran and Hatzor (2010) found the value of contact spring stiffness (i.e. theoptimal penalty parameter) that will give the same results, in terms of stability-failure,in the 2D-DDA, and then compared the rotation time histories of the column calculatedby the analytical and DDA solutions. They obtained an excellent agreement betweenthe two solutions, where the relative error drops below 10% after about 0.3 s ofsimulation, and below 1% after about 0.5 s (Figure 10b). Yagoda-Biran and Hatzor(2010) found that the error grows larger and the DDA deviates from the analyticalsolution as soon as the first impact between the rocking column and the fixed baseoccurs. They explained it by the way damping is implemented in the two solutions.While in the analytical solution the motion during impact is energetically damped dueto conservation of angular momentum following the constant value of the coefficient ofrestitution (Makris & Roussos, 2000), in DDA oscillations at contact points arerestrained due to inherent algorithmic damping (Doolin & Sitar, 2004; Ohnishiet al., 2005b).

3.4 Block response to shaking foundation

In the preceding sections the dynamic response of the blocks was a result of directdynamic input to the blocks centroid. In this sub-section dynamic loadings will beinduced by displacement input to a foundation block, a state that resembles the trueloading mechanism during an earthquake.

010−5

b

00’

R

ü(t)a

2h

2b

hα

b

c.m.

θ

−1−0.5

00.5

1

0

0.2

0.4

100

105

0.5Rel

ativ

e er

ror (

%)

Acc

eler

atio

n (g

)θ/

α1 1.5 2 2.5

Time (s)3 3.5 4 4.5 5

Figure 10 a) Free body diagram and sign convention used in this paper for the rocking block problem. b)upper panel: input accleration. Middle panel: comparison of analytical (curve) and DDA(symbols) solution for dynamic column rotation (b=0.2 m, h=0.6 m). lower panel: relativenumerical error. After Yagoda-Biran & Hatzor (2010).


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-4 3.4.1 2D-DDA

Kamai and Hatzor (2008) verified the 2D-DDA with a semi-analytical solution for ablock responding to induced displacements at its foundation. The model comprises ofthree blocks as follows (Figure 11): a stationary base block (0), an intermediate block(1) to which the input displacements are applied, and an overlying very flat block (2)which responds to the induced displacements in Block 1.

Block 1 is subjected to a horizontal displacement input function in the form of acosine, starting from zero:

d tð Þ ¼ D 1� cos 2πftð Þ� �

ð6Þ

The only force acting on Block 2, other than its weight and the normal from Block 1, isthe frictional force, which determines the acceleration of block 2. For full derivationof equations please refer to Kamai and Hatzor (2008). Kamai and Hatzor (2008)defined a set of inequalities and boundary conditions that determine the magnitudeand direction of the acceleration of Block 2, as a function of the acceleration of block1 and the relative velocity between the blocks. Kamai and Hatzor (2008) comparedthe displacements computed with 2D-DDA and the analytical solution for changingvalues of friction angle of the interface between blocks 1 and 2, and for changingamplitudes of input displacement. They found that generally the relative numericerror stayed below 5%, and was more sensitive to friction coefficient than amplitudechanges (Kamai & Hatzor, 2008). Example of the response of block 2 is presented inFigure 12.

3.4.2 3D-DDA

The verification of the case of a responding block to moving foundation in threedimensions is based on the one-dimensional verification described by Kamai andHatzor (2008), with the exception that here the displacements, velocities and accelera-tions are vectors.

3.4.2.1 The semi-analytical solution

The model used for the verification study is presented in Figure 11. Each of the twomoving blocks, blocks 1 and 2, has time dependent displacements d(t), velocities _d tð Þand accelerations €d tð Þ. The displacement induced to block 1, d1, is in the form of acosine function:

y

xzBlock 0

Block 1Block 2

Figure 11 The model used in the DDA verification of block response to induced displacements.


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-4

d1 tð Þ ¼ A 1� cos 2π�f t� ��

ð7Þ

where A and f are the amplitude and frequency of motion, respectively. The forcesacting on block 2 are its weight, m2g, the normal from block 1, N = m2g, and thefrictional force between the two blocks, μ*m2g, where μ is the friction coefficient.Newton’s second law of motion yields that the acceleration of block 2 is j €d2j ¼ μ � g.Following Kamai andHatzor (2008), the direction of the frictional force, and thereforeof €d2, is determined by the direction of the relative velocity between the two blocks,_d� ≡ _d1 � _d2, defined by the unit vector of the relative velocity, _d�, When j _d�j ¼ 0, theacceleration of block 2 ( €d2) is determined by the acceleration of block 1 ( €d1). When theacceleration of block 1 exceeds the yield acceleration μ*g, over which block 2 no longermoves in harmony with block 1, the frictional force direction is determined by the

direction of _d�, but the magnitude of €d2 is equal to μ*g. This rationale can be for-

mulated as follows:

If _d�� ¼ 0 . . . . . . . . . . . . and €d2

�� μ � g . . . . . . then €d2 ¼ €d1

and €d2�� 4μ � g . . . . . . then €d2 ¼ μ � gð Þ � €d1

(8)

If j _d�j≠0 …… then €d2 ¼ μ � gð Þ � _d�This set of conditions and inequalities was applied with a time step of 0.0001 s. Sincethe analytical solution is calculated numerically, it is actually a semi-analyticalsolution.

21.51Time (s)

Rel

ativ

e er

ror (

%)

Upp

er b

lock

disp

lace

men

t (m

)

0.500.01

0.10

1.00

0

0.2

0.4

0.6

0.8

1

1.2

Figure 12 Response of Block 2 to displacement input of f=1Hz, and amplitude of 0.5 m. Curve –analytical, symbols – DDA. After Kamai & Hatzor (2008).


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3.4.2.2 The numerical model

The actual model used for the 3D-DDA is shown in Figure 11. Block 2 was designed tobe very flat, so as to avoid rotations during motion. The physical and numerical controlparameters used in the verification analyses can be found in (Yagoda-Biran & Hatzor,2016).

3.4.2.3 One direction of motion

The first step was inducing displacements to block 1 in the x-direction (see Figure 11),similar to the work reported by Kamai and Hatzor (2008). This was done withchanging amplitudes, frequencies and friction angles, which can be viewed in(Yagoda-Biran & Hatzor, 2016). One example can be seen in Figure 13, where theinput motion is of amplitude of 0.5 m, frequency of 2 Hz and friction coefficient of 0.6.The two solutions compare very well (Figure 13a), and the relative numerical errorstays below 1% after about 0.3 s into the simulation.

3.4.2.4 Two directions of motion

In the second step of the verification study, displacements were induced to block 1in the x and y directions (Figure 11), each with different amplitude and frequency.Some results are presented in Figure 14. where the resultant horizontal displace-ment vs. time is presented in a). Figure 14c is a 3D plot of the x and y displace-ments vs. time, presented as the vertical axis. Again, the agreement between the3D-DDA and the analytical solution is good, as expressed by the numerical errorin Figure 14b, which remains below 10% for the entire simulation. A deviationbetween the numerical and analytical solutions is observed with increasing simu-lation time.

Times (s)Times (s)

Dis

plac

emen

t (m

)

0.10

0.2

0.4

0.6

0.8

1

1.2

0a b

1 2 3 4 5

10 1 2 3 4 5

1.0

Rel

ativ

e er

ror

(%)

Figure 13 a) Comparison between the analytical solution (curve) and 3D-DDA solution (symbols), foramplitude of A = 0.2 m, friction coefficient of 0.6, and frequency of 2 Hz for the input motion.b) relative numerical error.


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3.2.4.5 Three directions of motion

The third verification step was subjecting block 1 to sinusoidal displacements inall three directions: x, y and z. Adding input in the vertical direction affects theresponse of block 2 as it changes the normal force between the two blocks, andtherefore the frictional force between them. This in turn changes the acceleration

0

0 1 2Time (s)

Time (s)3 4 5

AnalyticalDDA

0c0.05

y displacement (m)

Tim

e (s

)

x displacement (m)0.10.15

0.2

0

1

2

3

4

0.10.2

0.30.4

Dis

plac

emen

t (m

)

0 0.0001

ba

0.001

0.01

0.1

11 2 3 4 50

10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Rel

ativ

e er

ror

(%)

Figure 14 a) Comparison between analytical (curve) and 3D-DDA (symbols) solutions. The amplitudeand frequency for the input displacements are 0.3 m and 2 Hz, 0.2 m and 4Hz for the x and ydirections, respectively. b) relative numerical error. c) x and y displacements as a function oftime (vertical axis).


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-4 of block 2, €d2. Applying time-dependent displacements in the z direction isactually equivalent to time-dependent changes in g: when block 1 has positive z

acceleration ( €d1k > 0), it is added to g. When €d1k is negative, it is subtracted fromg. The analytical solution in this case assumes no other effect of the verticaldisplacement of block 1 on the horizontal displacement of block 2. The induceddisplacement function is now:

d1 tð Þ ¼ 0:1 1� cos 2π2tð Þ� �

� iþ 0:1 1� cos 2π4tð Þ� �

� jþ 0:1 1� cos 2πtð Þ� �

� kð9Þ

In Figure 15 results of the verification study with three components of induceddisplacements are presented. Figure 15a presents the resultant horizontal (x-yplane) displacement vs. time, for different values of the k-normal contact springstiffness. The black curve is the analytical solution, while the other curves are the3D-DDA numerical solutions for different values of k. The range of contact springstiffness that best fits the analytical solution is between 1*107 and 1*109 N/m,with stiffness of k = 1*107 N/m, or 0.0003 E*L, being the optimal selection,where E is the Young’s modulus of the block and L is the length of the line acrosswhich the contact springs are attached. When considering 3D-DDA, it might bemore relevant to compare k to E*A, where A is the area across which the contactsprings are attached. In this case, k = 1*107 is ~0.0001 E*A, not much differentfrom E*L. Figure 15b demonstrates this with the relative numerical error. Notethat for the results obtained with k = 1*107 N/m, the relative error stays below3% for the entire analysis, and the error is well below 10% for k = 1*108 and1*109 N/m as well.

Time (s)

Rel

ativ

e er

ror

(%)

Dis

plac

emen

t (m

)

Time (s) ba0

0 0.01

0.1

1

10

100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5

Analyticalk=10^7 N/mk=10^8 N/mk=10^9 N/m

k=10^7 N/mk=10^8 N/mk=10^9 N/m

0 1 2 3 4 5

Figure 15 a) Comparison between analytical and 3D-DDA solutions. The best fit is obtained withcontact spring stiffness of k = 1*107 N/m, but the overall trend of the analytical solution ismaintained for all values of stiffness. b) Relative numerical error for the different solutionspresented in a).


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-4 3.5 Shear wave propagation

Bao et al. (2014a) tested the ability of the 2D-DDA to correctly simulate wave propa-gation. They modeled a set of stacked horizontal layers (Figure 16a), generated a shearwave by inducing horizontal displacements at the base, and compared the waveformsat the measurement point in the model (Figure 16a). Because the DDA blocks are linearelastic and un-damped, the analytical amplitude in the tests is the same amplitude asthat of the incident wave at the rigid base. After performing sensitivity analysis to theblock size and time step size, theymanaged to successfully preserve thewave form in theDDA model (Figure 16b).

Bao et al. (2014a) also validated the DDA with SHAKE program for site responseapplications. SHAKE (Lysmer et al., 1972; Schnabel et al., 1972) is a program thatanalyzes the 1-dimensional response of a stack of linear-elastic layers in the frequencydomain. The stack can be composed of several layers with varying properties, and it issubjected to seismic motion through its base. Although in this chapter we reviewverification studies, we choose to present this validation part, because the SHAKEprogram has been verified many times, its accuracy is well established for the under-lying assumptions and boundary conditions, and validating the DDA with SHAKEfor wave propagation applications seems to be an important step. Bao et al. (2014a)generated a stack of 15 layers in 2D-DDA, each with different mechanical properties(Figure 17a), and applied a real earthquake time history at its base. They modeled thesame sequence of layers in SHAKE, and compared the spectral amplificationsobtained with the two approaches at the top of the layer stack, with respect to itsbase. A very good agreement between the two methods was obtained (Figure 17b),both for homogeneous and inhomogeneous media, both for frequency and ampli-tude. Their results suggest the DDA can be used to model wave propagation throughdiscontinuous media, provided that the numerical control parameters are wellconditioned.

Time (s)

Hor

izon

tal d

ispl

acem

ent (

m)

50 m

50 m

–0.03ba

–0.02

–0.01

00 0.02

Measurement point

Input

Input

18.5m

Measurementpoint

0.04

0.01

0.02

0.03

Figure 16 Wave propagation through a stack of layers with DDA. a) DDA model. b) One-cyclesinusoidal incident wave time history used to induce vertical shear wave propagation inthe DDA, and wave forms as obtained at the measurement point. After Bao et al. (2014a).


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4 DISCUSSION

4.1 Numerical control parameters

The user-defined numerical control parameters have a significant effect on the resultsof DDA simulations. In this section a discussion of these parameters and suggestionsas to optimal selection of them will be made. It is however important to stress thatalthough some general guidance for optimal selection of the parameters is presentedhere, given the effect they have on the results of the simulation, their calibrationshould be performed whenever possible, and routine sensitivity analyzes are highlyrecommended.

4.1.1 Normal contact spring stiffness

The normal contact spring stiffness, k, is the stiffness of the virtual springs assigned atthe dynamically formed contacts. In many sensitivity analyzes it was found that thevalue of k significantly affects the results of the simulation.

Shi (1996), in his usermanual, recommended that as a rule of thumb, k = E * L,whereE is the Young’s modulus of intact block material and L is the average block diameter.However, it is sometimes reported that the optimal value does not follow this rule. Intheir study of wave propagation with DDA, Bao et al. (2014a) found that a k valuelower by 1.5 orders of magnitude than Shi’s rule of thumb is optimal. They suggestedthat the condition of the interface between the blocks might have an effect on the valueselected: a weathered interface might effectively lower the Elastic modulus, thereforelower the optimal stiffness value. This observation is supported by Yagoda-Biran andHatzor (2010) who modeled a physical problem somewhat similar to the one modeledby Bao et al. (2014a) and concluded that a k value of about 2 orders ofmagnitude lowerthan Shi’s rule of thumb (Shi, 1996) would be optimal. In other cases however, it seems

252015Frequency (Hz)

Top/

base

am

plitu

de

10

DDA

SHAKE

50Base

Inputa b

15 mTop

15 m

5

10

15

20

25

30

Figure 17 a) The model of the stacked layers in DDA. b) Spectral ratios as obtained with DDA andSHAKE. After Bao et al. (2014a).


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-4 that the optimal stiffness value even further deviates from Shi’s rule of thumb, such asthe case presented in section 3.4.2. In this case, the optimal stiffness that results in thesmallest numerical error is 2 to 4 orders of magnitude lower than Shi’s recommenda-tion. In this case however the simulation is in 3D-DDA, while Shi’s recommendationswere given as a guide for 2D-DDA, therefore generalizationmight not be appropriate inthis case.

4.1.2 Time step interval

Selecting an appropriate time step interval is an important issue with DDA simulations,and should be a balance between accuracy and computational efficiency. As reviewedin Bao et al. (2014a), various authors proposed different rules of thumb for optimaltime step size in relation to the wave period: less than 1%of the primary wave period infinite element simulations of nonlinear sound wave propagation (Kagawa et al., 1992),5% of the shortest period of incident waves in Newmark time integration scheme(Moser et al., 1999), or smaller than 2/π of the un-damped period of vibration of thesystem, in order to avoid bifurcation in the DDA solution (Doolin & Sitar, 2004).

In order to ensure the stability of the numerical solution, the time step interval shouldbe smaller than the fraction of the period that is equal to the ratio between the sidelength of the element along the direction of wave propagation path, and the wave-length, according to the Courant–Friedrichs–Levy condition (Courant et al., 1967).This condition however does not ensure accuracy. In the next section we discuss howthe choice of time step interval is reflected in the systems’ damping.

4.1.3 Damping

In the original DDA code a damping submatrixwas not incorporated in the equilibriumequations. Therefore, if the original code is to be used without modifications, dampingcan be introduced artificially by means of either kinetic damping or algorithmic damp-ing. Kinetic damping is applied when the transferred velocity to the consecutive timestep is reduced by some measure. Any value between 0 and 1 of the user defineddynamic control parameter would correspond to the percentage of velocity transferredfrom one time step to the following, that is, a dynamic parameter of 0.97 correspondsto 3% damping. When studying dynamic deformation of jointed rock slopes, Hatzoret al. (2004) reported that a 2% kinetic damping is required to obtain stable solutionwith the 2D-DDA version they used at the time. But if true and accurate displacementsare required, then no kinetic damping should be introduced at all. This can be doneprovided that all other numerical control parameters are properly conditioned.

Algorithmic damping (Doolin & Sitar, 2004) is associated with the time integrationscheme used for integrating second order systems of equations over time. Numericaldamping stabilizes the numerical integration scheme by damping out the unwanted highfrequency modes. For the Newmark time integration scheme used in DDA (with β = 0.5and γ = 1.0, see Ohnishi et al., 2005a), it also affects the lower modes and reduces theaccuracy of integration scheme to first order. In DDA, the numerical damping that isassociatedwith the time integration scheme increases with increasing time step size. If thetime step is small enough, the numerical damping phenomenon is insignificant. Bao et al.(2014a) suggested away to utilize this time step size dependence of algorithmic damping,


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-4 and obtained an equivalent damping ratio by seeking the time step size that will result inexactly the same damping ratio thatwould have been assumed otherwise in the structuralanalysis. They inspected the damped oscillations of the free end of a cantilever beammodeled with DDA with different time step intervals, and obtained an equivalentdamping ratio, using the algorithmic damping inDDA as a function of time step interval.Then they modeled a stack of horizontal layers in DDA, subjected to earthquakedisplacements at the foundation, with a time step size of 0.001s, which correspondedin that case to 2.3%damping. They compared the amplification and resonance frequencyobtained with the DDA model, to those of an equivalent SHAKE (Lysmer et al., 1972;Schnabel et al., 1972) model with an input of 2.3%damping, and obtained an extremelygood agreement between the two methods (see section 3.5).

4.2 Limitations of 3D-DDA

4.2.1 Constructing a 3D-DDA mesh

Modeling three dimensional multi-block structures in 3D-DDA is an elaborate andchallenging task. The block cutting code in 3D-DDA does not have a user friendlygraphic interface, and does not accept three-dimensional blocks as input, but rathertwo-dimensional triangles, of which the blocks are constructed. For example, in orderto build a rectangular face of a box, two triangles will be required, with three verticeseach. Therefore, to form a simple box, one needs to input the vertices for 12 triangles.The accuracy of the input coordinates is of great importance as well: if the coordinatesof two adjacent triangles do not exactly coincide, the code will not be able to cut ablock. This requires sophisticated pre-processing with suitable software. When model-ing problems involving only of a few blocks, the process of building the mesh might betolerable, but when constructing meshes which consist of many blocks (see section4.2.2) the task becomes difficult and exhausting. It is therefore recommended to usecomputer aided design (CAD) software to construct a 3D-DDA mesh, and use afunction of the CAD software to export the model. In the work presented here insection 4.2.2we built themodel in a CAD software, exported the nodes, and thenwrotethe model’s nodes to a file readable by the DDA. The scope of this chapter does notallow for a full presentation of the process, but the interested reader is welcome tocontact the corresponding author for more information.

4.2.2 The L’Aquila case study

The DDA in its two dimensional formulation has been used several times as a tool toestimate historical seismic hazard (Kamai & Hatzor, 2008; Yagoda-Biran & Hatzor,2010). While attempting to use the 3D-DDA in a similar manner, we have stumbledupon limitations that suggest the 3D-DDA, at its current formulation, is still not readyfor solving reliably dynamic problems involving a large number of blocks. We use thecase study of L’Aquila to illustrate this problem.

The city of L’Aquila, the capital of the Abruzzo region, Italy, suffered strong groundmotions during the Mw 6.3 earthquake of April 6, 2009. Over 300 people were killed,and many of the buildings in the old city were severely damaged and evacuated. Sincethere are several strong ground motion accelerographs in the city, this seemed as an


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excellent case study to check the validity of the DDA for solving complicated dynamicproblems in three dimensions.

We searched the old city of L’Aquila for small, simple buildings that can be easilymodeled with the 3D-DDA.We found a small masonry structure that was damaged bythe earthquake, but did not collapse (Figure 18). Naturally, the observed damagecannot be modeled correctly with 2D-DDA, and a 3D approach is required. Themodel of the structure in 3D-DDA was comprised of 197 blocks.

When trying to run forward analyzes with the model, subjecting it to the accelerationtime series as recorded in a station located about 500meters away from the structure, thesolution converged only when kinetic damping of at least 3% was used, and a time stepsize no larger than 10-5 s. With less kinetic damping and longer time steps a stablesolution could not be obtained, probably due to limitations of the contact algorithm in itscurrent form, which does not allow the system to converge when a large number ofcontacts must be solved in every iteration. The choice of such a small time step inevitablyleads to extremely long CPU time, especially when running simulations with long “real”run time of tens of seconds. The use of small time step intervals in the 3D-DDA, smallerthan the ones used in similar simulations in 2D-DDA, was observed many times. Forexample, the 3D simulations in Yagoda-Biran and Hatzor (2013) used a time step sizetwo orders of magnitude smaller than the time step size used in similar 2D-DDAsimulations. For the case of the responding block to induced displacements, the timestep size used in the 3D-DDA case in section 3.4 is one order of magnitude smaller thanthe one used in 2D-DDA (Kamai, 2006). For the case of a block on an incline, the timestep size used in the 2D-DDA case is 0.002 s (Kamai & Hatzor, 2008), while the 3D-DDA simulation in section 3.1.1.2 used a time step size of 0.0001 s.

Furthermore, we noticed that the displacements of the blocks were several orders ofmagnitude smaller than expected. These results led us to investigate what effect doesthe coupling of kinetic damping and small time step has on the numerically obtained

Figure 18 a) The damaged masonry structure in the city of L’Aquila, Italy following the Mw 6.3earthquake that struck the region, b) snapshot of the corresponding 3D-DDA mesh.


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cumulative displacements. We conducted a numerical experiment, where the timedependent displacements of a mass driven by a constant force were computed, withkinetic damping of 3% and different time step sizes. As observed in Figure 19, whenusing kinetic damping, the cumulative displacement decreases with decreasing timestep, an effect also observed in the same experiment performed with the 3D-DDA (seeYagoda-Biran&Hatzor, 2016). This effect was not observedwhen no kinetic dampingwas applied: the time step size had no effect on the cumulative displacements.Furthermore, decreasing the kinetic damping by even 2%, down to 1% damping, didnot change the results significantly: the displacements were still highly restrained.

It is thus evident that a combination of a very small time step interval and kineticdamping of a small percentage significantly decreases the cumulative displacement duringthe simulation rendering the numerical results inaccurate and unrealistic. It is also evidentthat a small increase in kinetic damping coefficient will not make a great difference whenvery small time steps are used. Ideally, it would be preferable to use zero kinetic damping indynamic DDA simulations, as the displacements per time step are reduced with increasingtime step size anyhow due to the inherent algorithmic damping in DDA.

In cases such as these, where displacements are the desirable output, these limitationsof the current version of the 3D-DDA code are not tolerable, and this effect makes the3D-DDA, in its current form, ill-suited for dynamic simulations ofmulti-block systems.A completely new contact algorithm has recently been proposed by Shi (2013), but hasnot yet been implemented in executable codes which are available to us for testing.After the new contact algorithm is implemented 3D-DDA should be tested again for itsapplicability for multi-block systems and multiple contacts using benchmark testssimilar to those presented in this chapter.

time (s)

disp

(m)

00

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2dt = 0.1 sdt = 0.01 sdt = 0.001 sdt = 0.0001 s

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 19 Cumulative displacement of a mass subjected to constant force, under 3% kinetic damping anddifferent time steps, as calculated semi-analytically. After Yagoda-Biran & Hatzor (in review).


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-4 5 SUMMARY AND CONCLUSIONS

In this chapter the validity of dynamic analysis with 2D and 3D-DDA is verified byreviewing published verification studies. As the numerical discrete element DDAmethod is becoming more popular in rock mechanics and engineering geologyresearch worldwide, it becomes supremely important to verify the accuracy andapplicability of the method before it is accepted and established as a standardanalytical approach in the practice. The DDA has been proven to accurately solveproblems involving block translations (block on an incline, double face sliding) androtations (rocking of a free standing column) when the loading is applied at thecenter of the block, as well as translations when the loading is applied at thefoundation (responding block). It has been proven to accurately solve large displa-cements, as well as wave propagation problems, that involve small displacements,despite the simply deformable blocks assumption and the first order approximation.This accuracy however is highly dependent on an educated selection of the numer-ical control parameters, first and foremost the penalty parameter otherwise knownas the contact spring stiffness, as well as the time step size. A wise selection of thetime step size would balance between small computation times and high accuracy. Awise contact spring stiffness selection would ensure accuracy of the solution; theblock size becomes an issue primarily when dealing with wave propagationproblems.

Naturally, 2D-DDA cannot be used in cases where out-of-plane deformations areexpected in the physical problem. In such cases using the 3D-DDA would seem moreappropriate although as much as we have experimented with 3D-DDA in its currentformulation we have concluded that obtaining a stable solution to dynamic problemsinvolving a large number of blocks is a very challenging task.

ACKNOWLEDGMENTS

We wish to thank very sincerely Dr. Gen-hua Shi for his support of our research overthe years by coming to visit us in our laboratory and by providing access always tomostupdated versions of the 2D and 3D-DDA codes, his ongoing code modifications thatmany times resulted from our fruitful interaction, and for stimulating discussions. Wealso wish to thank sincerely the graduates of our rock mechanics group Drs. MichaelTsesarsky, Ronnie Kamai, Dagan Bakun-Mazor and Huirong Bao for their friendship,collaboration, stimulating discussions and for sharing their ideas and data with us.Financial support of Israel Science Foundation through grant No. ISF-2201, ContractNo. 556/08 is thankfully acknowledged.

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