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The slope modeling method with GIS support forrockfall analysis using 3D DDALu Zhenga, Guangqi Chenb, Yange Lib, Yingbin Zhangc & Kiyonobu Kasamab

a Sichuan University - The Hong Kong Polytechnic University Institute for DisasterManagement and Reconstruction, Sichuan University, Chinab Department of Civil and Structural Engineering, Kyushu University, Fukuoka, Japanc Department of Geotechnical Engineering, School of Civil Engineering, Southwest JiaotongUniversity, ChinaPublished online: 07 Feb 2014.

To cite this article: Lu Zheng, Guangqi Chen, Yange Li, Yingbin Zhang & Kiyonobu Kasama (2014) The slope modeling methodwith GIS support for rockfall analysis using 3D DDA, Geomechanics and Geoengineering: An International Journal, 9:2,142-152, DOI: 10.1080/17486025.2013.871070

To link to this article: http://dx.doi.org/10.1080/17486025.2013.871070

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The slope modeling method with GIS support for rockfall analysis using 3D DDA

Lu Zhenga*, Guangqi Chenb, Yange Lib, Yingbin Zhangc and Kiyonobu Kasamab

aSichuan University - The Hong Kong Polytechnic University Institute for Disaster Management and Reconstruction, Sichuan University, China;bDepartment of Civil and Structural Engineering, Kyushu University, Fukuoka, Japan; cDepartment of Geotechnical Engineering, School of Civil

Engineering, Southwest Jiaotong University, China

(Received 4 March 2013; accepted 25 November 2013)

Rockfall is the most frequent major hazard in mountainous areas. For hazard assessment and further countermeasure design, realistic and accurateprediction of rockfall trajectory is an important requirement. Thus, a modeling method to represent both geometrical parameters of slope and fallingrock mass is required. This study, suggests taking the advantages of discontinues deformation analysis (DDA) and geographical information system(GIS). In this study, after developing a three dimensional (3D) DDA program, firstly a special element named contact face element (CFE) wasintroduced into 3D DDA; secondly, effectively modeling tools with GIS support were developed. The implementation of CFE also improves theefficiency of both the contact searching and solution process. Then a simple impact model was devised to compare the 3D DDA implementeddirectly with a sliding model with theoretical analysis to verify the reliability of the modified 3D DDA program and investigate the parametersettings. Finally, simulations concerning rock shapes and multi-rocks were carried out to show the applicable functions and advantages of the newlydeveloped rockfall analysis code. It has been shown that the newly developed 3D DDA program with GIS support is applicable and effective.

Keywords: Rockfall; Slope modeling; 3D DDA; Contact Face Element; GIS

1. Introduction

Rockfall is a frequent and major hazard in mountainousareas of Japan and worldwide, as shown in Figure 1. Itrefers to quantities of rocks fragments from a cliff orboulders from a slope detached by sliding, falling, or top-pling that bounce roll and slide down over the slope surfacetill the finally come to rest (Evans and Hungr 1993).Compared with landslides, it is a natural downward motionwith a small volume. However, despite its limited volume,rockfall is characterized as high energy, mobility, and themost destructive mass movement (Guzzetti et al. 2002),making it a major cause of fatalities (Chau et al. 2004).Moreover, literature review (Dorren 2003) shows rockfallcould be activated by various trigger mechanisms and con-ditions. In most cases, rockfall occurs accidently (Masuyaet al. 2009). It indicates that rockfall is hardly predictableand usually occurs without any obvious warning. Therefore,rockfall is a potential high threat to both properties and liveswithin its runout. Thus, rockfall hazard and risk assessmentis particularly important.

In rockfall hazard and risk assessment and further designand evaluation of rockfall countermeasures, the runout trajec-tory and velocity or energy along it are significant items. Atfirst glance, rockfall seems quite a simple process to model.After release, the rockfall trajectory is a combination of free

falling (flying), impacting/bouncing, rolling and sliding pro-cesses along the slope surface (Ritchie 1963, Lied 1977,Descoeudres 1997). These processes are controlled by well-known physical laws and can be described by simple equa-tions. However, rockfall dynamics is dominated by spatiallyand temporally distributed attributes, such as the location ofthe detachment point, geometry and mechanical properties ofboth rock block and slope. In addition, impact, as one rock–slope interaction, is the most complex, uncertain and poorlyunderstood stage of rockfall. Thus predicting rockfall move-ment behaviors, such as runout distance, distribution, trajec-tory, and velocity or energy, is a complicated operation with awide range of probabilistic calculation due to uncertainty ofparameters (Guzzetti et al. 2002).So far, empirical formulas (for example, Japan Road

Association 2000) are used to estimate the movement beha-viors of rockfall. Till recently, the rockfall inventory is insuffi-cient since most rockfall events were not actually reported(Chau et al. 2004). And the experimental investigations arealso insufficient for a thorough understanding of the phenom-enon or for statistical and parametric analysis (Volkwein et al.2011). Thus, virtually, as shown in Figure 2, that rockfalloccurred outside of preventive range, these empirical formulasare not applicable in many cases due to the extremely complexrock–slope conditions. Therefore, a computer code that has theability to efficiently manage and use both geometry and

*Correspondence author. Email: zhenglu@scu.edu.cn

© 2014 Taylor & Francis

Geomechanics and Geoengineering: An International Journal, 2014Vol. 9, No. 2, 142–152, http://dx.doi.org/10.1080/17486025.2013.871070

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mechanical parameters of rock and slope plays an importantrole in the prediction of rockfall movement behaviors.

Since discontinues deformation analysis (DDA) (Shi 1988)can analyze the dynamic displacement and deformation of anelastic block of any shape, for example, the rigid body dis-placement, rotation and deformation of a rock block, it isuseful for investigating the movement behaviors of rockfall.Previous research shows that rockfall problems can be simu-lated to some accuracy using two dimensional (2D) DDA(Ohnishi et al. 1996, Chen 2003, Shimauchi et al. 2006, Maet al. 2007). However, shown by field experiments (for exam-ple, Ushiro 2006), the rockfall movement behaviors are con-trolled by the detailed three dimensional (3D) shape of thefalling rock blocks and the complex 3D geometric features ofthe slope surface. It indicates that the 2D profile selection iscritical to obtain realistic analysis results using the 2D DDAmodel. More generally, the 2D models are limited in theirability to provide spatial distribution of rockfall trajectoriessince they lack lateral movements beyond the profile.Therefore, a technique to analyze the motion of rockfall on athree dimensional slope is needed, aiming to establish a morepractical simulation.

Although the basic formulas of 3D DDA had been derived(Shi 2001), there was no available program till now and themain problem of slope modeling arises in the 3D simulation. Inthe 2D DDA simulation, it is common that the slope can berepresented by a large block or many small artificial blocks.However, if the slope is represented by a single 3D block, therewill be a large number of faces in the slope block, whichmakes it difficult to deal with in contact detection. If theslope is represented by many small 3D artificial blocks, thegeneration of the slope blocks will become a difficult job.In this study, we describe a computer program designed to

help simulate rockfall trajectory based on 3D DDA with geo-graphical information system (GIS) support. The program wasdeveloped with the aim of resolving the problems mentionedabove. First the 3D DDA program was developed and then anew contact face element (CFE) was proposed in this study.Second, based on it, a new tool using GIS was developed tomodel the slope surface. Finally, an application of rockfallsimulation was carried out in order to verify its applicabilityand benefits on rockfall analysis.The details on the interpretation and implementation of 3D

DDA are beyond the scope of this paper. Without going intothe details, we refer the readers to the review articles by Shi(2001).

2. The 3D DDA program for rockfall with GIS Support

In the beginning of 2010, our new 3D rockfall simulationprogram based on 3D DDA was developed and it has beencontinuously upgraded since. In our last version, the programcan simulate spatially distributed 3D rockfall trajectories andcount statistics of rockfall movement behaviors incorporatingGIS support.

2.1 Assumptions

The model used in the proposed 3D rockfall simulation pro-gram is characterized by the following assumptions:

● The model deals with single block falls and mass falls,also called “fragmental rockfalls” (Evans and Hungr1993). The interactions among the falling blocks couldbe considered.

● Air drag and block fracturing are not taken into account.● The kinematic approach, fully adopted from 3D DDA,

treating the falling blocks as polyhedron blocks, allowsthe modeling of free fall (flying), impacting/bouncing,rolling, and sliding processes in a 3D framework. Theinfluences of the shape, the size and the angular momen-tum of the rock on the rockfall movement behaviors arefully taken into consideration.

● The small deformation of the slope surface is not takeninto account.

Figure 1. Rockfall occurred on a coastal road.

Figure 2. Rockfall occurred outside of the preventive range.

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2.2 Input data

The described model requires the following input data to runrockfall simulations. The data can be managed and obtainedbased on GIS support, as shown in Figure 3:

● A digital terrain model (DTM) in raster format withoutresolution restrictions in Layer 3.

● A grid for restitution coefficients number and the list ofcorresponding materials of the model which is chosen tocompute the loss of energy by the means of Rv, which is apost adjustment coefficient that represents the energy lossin terms of velocity, when the falling rock impacts on theslope surface in Layer 2.

● A grid for dynamic frictional characteristics and the list ofcorresponding joint materials, which are defined in DDAsimulation, used to compute the loss of energy by themeans of friction angle f which is used to calculateenergy dispersion by friction force, when the fallingrock is rolling or sliding along the slope surface inLayer 2.

● A list of rockfall sources in 3D points and the amount ofrock material and the list of corresponding geometriesand materials, which are defined in DDA simulationincluding the initial velocities in Layer 1.

● A list of tree locations in 3D points and the list ofcorresponding geometries in Layer 1.

2.3 Contact face element

In 2D DDA simulation, it is common to model a whole slopeand runout area as a large single block. Sometimes, a problemcan be caused in calculation due to a very large base block andmany small falling rocks, especially, when the detailed shapeof a slope is modeled.

To solve this problem, an alternative way is to divide the basepart into many artificial blocks, especially when different physi-cal properties are considered for different areas of the slope.

The base blocks are commonly fixed since the effect of theslope deformation can be negligible. And good results can beobtained by 2D DDA simulation comparisons with the fieldobservations.However, it is difficult to use the same ways in 3D slope

modeling as those in 2D mentioned above since it couldbecome much more complicated.

● If the whole base part is modeled by a big block in 3Dthere would be a large number of faces in the slope block,as shown in Figure 4a, and it would be difficult to dealwith in contact detection.

● If the base part is divided into many artificial blocks, it isnecessary to generate the artificial 3D blocks, which is adifficult job (Figure 4b).

With the described assumptions the problems mainly concernthe movement behaviors of rock blocks, the small deformationof the slope block can be ignored. Thus, in 3D DDA simula-tion, the function of the slope is limited to the surface bound-ary only. Based on this assumption, a special element namedcontact face element (CFE) has been introduced into 3D DDA,which plays the role of a fixed block. A CFE is a line segmentin 2D and a plane in 3D without any physical property. It canhave contacts with other blocks but no displacement and nodeformation will be produced.Each CFE is a fixed and rigid triangle in 3D DDA. Then a

slope can be easily modeled using a network of CFEs asshown in Figure 5.Based on CFE, we developed a slope modeling method

combining the GIS support to DDA and modifying the DDAimplemented from the theory of Shi (2001) as followings.

2.3.1 Incorporation of GIS

GIS is a computer system for managing spatial data. GIScontains facilities for constructing and importing digital eleva-tion models (DEMs) and triangulated irregular networks(TINs). GIS can be easily utilized to overcome the difficultiesin slope topography information acquisition and modeling.The slope topography is represented by a raster format in

GIS as shown on the left in Figure 6. The black dot points onFigure 3. Data management and access from GIS support.

(a) Single block (b) Artificial blocks

Figure 4. Slope modeling approaches.

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grids present the elevation. In this form, a slope surface hasbeen arranged in square cells in X and Y directions, x� andy� coordinate can be computed out with its index. Theelevation data in Z can be acquired in the elevation matrixby index.

However, since usually the cell is not planar, a cell will needto be divided into two triangles to describe the topography.Therefore, the representation of the slope surface turns to aterrain regular network (TRN) (Guzzetti et al. 2002) as shownon the right side of Figure 6.

Usually, the topography is prepared initially by a contourmap which is composed by extracted polyline files as shown inFigure 7a. The polylines are the direct description of terrain butare hardly able to be used in slope modeling directly since theydon’t with intersect each other and cannot form faces. The firststep we need to carry out is to transform the polygon files intoa TIN map. Although there are lots of methods to create a TINfrom the vertices of polygons (Lee 1991), in this paper thescope is only limited by the incorporation of GIS. A TINmodel shown in Figure 7b is already applicable to be used.However, for efficient data storage, access and slope modeling,it is necessary to transform the TIN into a raster type as shown

in Figure 7c. The TIN's Xmin; Ymin is defined as the defaultorigin, and the Xmax; Ymax as the upper right corner of theextent to generate a raster covering the full extent of theTIN. Then resolution or cell spacing is defined to interpolatecell z-values from the input TIN to create the raster data. WithGIS support an arbitrary complex slope surface can be mod-eled using a network of CFEs as shown in Figure 7d.

2.3.2 Improvement in contact searching

For a slope with 2� ðn� 1Þ � ðm� 1Þ faces, where n is thecolumn number of the raster data and m is the row number;when only one rock falls down, the searching for possiblecontact blocks needs to compare the boundary of rock withthe boundary of each face, that is 2� ðn� 1Þ � ðm� 1Þ times.If using the TRN map of CFEs, during the search for possiblecontacts, the number of computation times could decreasefrom 2� ðn� 1Þ � ðm� 1Þ to 1. The effectiveness of search-ing for possible contact blocks would be largely improved. Thecontact searching can be illustrated as following.Supposing that the lower left index of cell is ðiþ 1; jþ 1Þ,

the triangles of two parts are:The lower triangle is constructed by taking the upper left,

lower right, lower left elevation points and has an index:i� jþ 1; and the upper triangle is constructed with the upperleft, upper right and lower right elevations of raster data; itsnode indices are: i� jþ 2;

Figure 6. Generation of Contact Face Elements from Raster data of GIS.

(a) Original DEM (b) Converting DEM to TIN

(c) Converting TIN to Raster (d) Slope modeling from Raster data

Figure 7. Slope modeling with GIS support.

(a) Single block (b) Network of Contact Face Element

Figure 5. Contact Face Element and its application in slope modeling.

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It indicates that the lower triangles are always generated andstored with odd numbers in the global list; and the upper trianglesare at even numbers. Each triangle in the list is a CFE.

Since the projection of CFEs in xy� plane is continuousand indexed, the contact searching can be improved usingindex searching as shown in Figure 8.

The possible contacts between a rock block and CFEs can beestimated within the region presented by x� and y� indices.Denote xmin, xmax, ymin, and ymax are the minimum and maximumcoordinates of a block projected in xy� plane domain; xll, yll arex� and y� coordinates of lower-left corner respectively,cellsize is the interval in both x� and y� direction.

imin ¼ xmin�xllcellsize ; imax ¼ xmaxþxll

cellsize

jmin ¼ ymin�yllcellsize ; jmax ¼ ymaxþyll

cellsize

)(1)

The possible contact between two rock-blocks can also beestimated using this map.

2.3.3 Improvement of the solution process

The basic principles can be found in Shi (2001) in a morecomprehensive derivation. Here, only the sub-matrices of con-tact are discussed.

In 3D DDA, the fundamental contact candidates can bepresented by point-to-face contact and edge-to-edge contact(not in a plane). Contact forces are generated by contactsprings to push the penetrating blocks to the target surfacethrough the shortest distance.

Take the point-to-face contact between a polygonal block iand a CFE j shown in Figure 9 for example. Assume that P1 isa point before deformation that moves to point P0

1 after defor-mation. ðxrock ; yrock ; zrockÞ and ðurock ; vrock ;wrockÞ are the coor-dinates and displacements of P1, P2P3P4 is the CFE face, andðxi; yi; ziÞ are the coordinates of Pi; i ¼ 2; 3; 4. According to theassumption of CFE, there is no displacement of P2;P3;P4.P0 is

the projected point of P1 on P2P3P4, which is not changedafter deformation.Here, d0 is the penetration distance, n

*is the outer normal

direction, pn is the stiffness of normal spring, ps is the stiffnessof shear spring, and l

*

p is the projected direction on P2P3P4

of P0P01.

● Sub-matrix of normal spring.

Since CFE acts as a displacement constraint only. The normaldisplacement increment comes only from the rock block:

d*

n ¼ δþ dδ ¼ n*

xrock þ urockyrock þ vrockzrock þ wrock

0@

1A ¼ n

*ðd0 þ TrockDrockÞ

(2)

where δ is the normal penetration; Trock is the displacementtransformation matrix of the rock block formed by the firstorder approximation displacement functions; Drock is 12� 1submatrice, the displacement variables of rock blockThe potential energy contribution from the normal spring is:

�n ¼ 1

2pnd

2n (3)

By minimizing the total potential energy:

pnðn*TrockÞT ðn*TrockÞ ! Kii½ � (4)

� pnd0n*n*T Trock½ �T! Fi½ � (5)

● Sub-matrix of normal spring.

Figure 8. Scheme of contact searching with Contact Face Element network.

Figure 9. Point-to-Face contact model (modified from Shi 2001, Jiang andYeung 2004).

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Correspondingly, the shear displacement increment is:

d*

s ¼ d* � d

*

n ¼ ð1� n*n*T Þd* (6)

The potential energy contribution from the shear spring is:

�s ¼ 1

2psd

2s (7)

By minimizing the total potential energy:

ps ð1� n*n*T ÞTrock

h iTð1� n

*n*T ÞTrock

h i! Kii½ � (8)

� psd0 ð1� n*n*T ÞTrock

h iT! Fi½ � (9)

● Sub-matrix of friction force.

When the state of contact is sliding, a pair of friction forcesparallel to the sliding direction is applied with the same mag-nitude and opposite directions. The magnitude and directionsof the friction force are obtained from the previous iteration.

F ¼ pn δ0j j tanfþ c (10)

where δ0 is the normal penetration after the previous iteration,f is the friction angle, and c is the coherence.

Therefore, the potential energy of the pair of frictional forcesis given by

�f ¼ F

l*

p

��� ��� urock vrock wrock½ � l*T

p ¼ F ½Drock �T ½M �� �

(11)

where

M ¼ 1

l*

p

��� ��� Trock½ �T l*T

p (12)

By minimizing the total potential energy:

� F½M � ! Fi½ � (13)

As shown above, since

● the CFEs are not taken into matrix;● and the sub-matrices between a rock block and a CFE are

only located in main diagonal.

The solution process can be largely enhanced.

2.4 Coping with natural variability and uncertainty in theinput data

As pointed out before, parameters of rockfall vary largely innature and are difficult to define precisely. On the other hand,the DDA theory from Shi (2001) is a deterministic approachwithout uncertainties. Our program provides a way to copewith the natural variability and local uncertainty by adding tothese values a random component.

● Slope topography.

The normal distribution with mean value and standard devia-tion σ is given to grid elevations to represent the roughness andkeeps the whole inclination unchanged, as shown in Figure 10.

● Mechanical parameters of slope surface.

The normal distribution randomness is also given to Rv and thefriction angle, respectively.The iteration number can be defined by the user.

2.5 Output data

Currently, the program outputs results are the trajectory pointof centroid coordinate Cx;Cy;Cz

� �and the rigid movement

velocity Vx;Vy;Vz;Ryz;Rzx;Rxy

� �of each step respectively.

Figure 11 shows a simple example of our program results forrockfall events occurring in a channel without coping varia-bility and uncertainty.

3. Program performance

To evaluate the reliability and the applicability of our programwe performed tests aimed at:

● Comparing the results of the simple but most importantrockfall process: impacting/rebounding with 3D DDAimplemented from the theory of Shi (2001) and investi-gating the parameter settings.

Figure 10. Slope surface with roughness.

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● Comparing the results of the sliding process with theore-tical analysis.

Based on the fundamental tests above, we also carried out tworockfall tests to show the advantages of our program comparedwith other rockfall codes:

● Rockfall analysis considering different rock shapes.● Rockfall analysis considering multi-rocks falling.

3.1 Verification of bouncing motion

The simulation of bouncing motion is the greatest challenge inmodeling the interactions between a falling block and theslope’s surface. Of the four types of motion, the bouncingphenomenon is the least understood and the most difficult topredict. The bouncing motion can be directly simulated byDDA in which elastic deformation and rigid body movementare taken into account.

A simple example of impacting and reboundingwas introducedto investigate the reliability of modified 3D DDA. As shown inFigure 12, the right-upper part shows the geometrical patterns ofthis example. The parameters of the upper block are listed inTable 1. Simulation cases were simulated to investigate penaltysettings when�t ¼ 0:01s, 0:005s, taking E ¼ 5� 109Pa, andp ¼ 109N=m as an example. As shown in Figure 12 the blue solidline shows the simulation results from 3D DDA implementedfrom Shi (2001), while white circles are the simulation resultsfrom our modified 3D DDA using CFE. The result shows that thetwo programs can obtain nearly the same rebounding velocitywhen considering a completely elastic impact.

Then we investigated how the penalty value used in DDAaffected the accuracy in terms of rebound velocity ratio. If theimpact/rebound interaction is elastic collision, the theoreticalanalysis result of rebound velocity ratio is 1.0. Figure 13shows rebound velocity ratio varies with Young’s Moduluswhen �t ¼ 0:01s. Figure 13 shows that the materials withE < 109Pa are largely affected by their plastic deformation

during impact. Figure 13 also shows that the accuracy isaffected by the penalty method used in DDA. However,p> 109N=m could obtain enough accuracy for application.According to our results, we suggest adopting the penaltyvalue p ¼ 109N=m when �t ¼ 0:01s. By investigation, thesuggested parameter value p ¼ 109N=m is appropriate when�t ¼ 0:01s, 0:005s. The following verification of the slidingprocess used the same setting parameters as shown in Table 2.

Figure 12. Verification of Contact Face Element idea by rebound model(original 3D DDA means the one implemented directly from the theory ofShi (2001)).

Table 1. Cases using to investigate the parameters settings

Parameters Value

Density (kg/m3) 2.50E + 03Young’ Modulus (Pa) Varied from 1.00E + 06 to 1.00E + 11Poisson’s Ratio 0.1Penalty (N/m) Varied from 1.00E + 07 to 1.00E + 12Time Step (s) Varied from 0.01 to 0.005

Figure 13. Rebounding velocity ratio vs. Young’s modulus.

Figure 11. Rockfall displacements output in terms of centroid.

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The DDA directly from Shi (1988, 2001) is derived basedon the assumption of a completely elastic collision withoutenergy losses when rocks impact on the slope. That is, therebound velocity will equal to the incident velocity. However,because part of kinematical energy may be absorbed as plasticdeformation of the soft layer on the slope surface, or convertedto the other types of energy such as heat, the rebound velocitybecomes much lower. The phenomenon of energy loss incollision has been shown by many experiments. If the collisionenergy loss is not considered, the falling rock will travel muchlonger and faster, and jump much higher than the observationsby in situ experiments.

The collision energy loss is considered similar to Chen(2003) using a so called post adjustment method. The velocityof the falling rock is modified after a contact between a rockand a CFE is experienced:

~Vx~Vy~Vz

0@

1A ¼ Rv

Vx

Vy

Vz

0@

1A (14)

where, ðVx;Vy;VzÞ is the velocity of a rock right after collision;Rv represents the energy loss in terms of velocity, from theobserved data cited in Chen (2003), the range of Rv is approxi-mately ranged from: 0.6~0.8; ð~Vx; ~Vy; ~VzÞ is the modified initialvelocity used in our simulation.

3.2 Verification of sliding motion

The sliding friction is defined by means of the normal compo-nent with respect to the boundary surface of the block's weightaccording to Coulomb's law. The displacement of a slidingblock under gravity on an inclined plane can be easily derivedas follows.

d ¼ 1

2gðsin θ � cos θ tanfÞt2 (14)

where θ is the slope angle and f is the angle of kinetic friction.Equation (14) is valid only when f is less than θ.

It also can be simulated by DDA directly. An inclined planeslope with slope angle of 30� was modeled as shown in theright-upper part of Figure 14. A rectangular block sliding onthe slope was simulated by DDA.

Two cases of the friction angle 10� and 20� were analyzedrespectively. The results are shown in Figure 14 together with the

analytical solutions from Equation (14). The blue solid and reddash-dot lines present the theoretical results for friction angle 10�

and 20� respectively and the blue circle and red square dots aresimulation results from modified DDA with CFE.The DDA results are in quite good agreement with theore-

tical solutions. Therefore, DDA can be used to simulate thesliding motion.

3.3 The Influence of rock shape

In most previous rockfall analysis codes, the falling rock wastreated as a lumped mass of a point that did not take the rockshape into consideration directly (Dorren 2003, Volkwein et al.2011). However, the lateral displacement, which is very impor-tant (Crosta and Agliardi 2004) in 3D rockfall analysis, wouldbe influenced by mechanical and geometrical parameters ofboth slope and falling rock as shown by rockfall field andlaboratory experiments (for example, Ushiro et al. 2006).Thus, the lumped mass assumption is not realistic. Based on3D DDA, our program has the advantage that the influence ofrock shape on rockfall trajectories can be considered directly.The simulation model of slope surface and rock block is

shown in Figure 15. The slope surface consists of two parts.The inclination part is h ¼ 57:7m high, 100m long with anaverage slope angle θ ¼ 30�. The plane part is located in xy�plane and is 150 m long. The width of the slope surface is100 m. The roughness obeys normal distribution and averagesat �x ¼ 0:1m and standard deviation σ ¼ 0:1. The grid length iscellsize ¼ 5m. Two types of rock blocks were used in the simu-lation, one is a regular 20-faced polyhedron (red) and the other isa cube (blue). Both rock blocks have the same volume of 1 m3.Table 3 shows the mechanical parameters used for the rockfragments, restitution and friction coefficients between therocks and slope and shows the penalty and time step values.

Table 2. Parameters settings with suggested Penalty and Time step

Parameters Value

Density (kg/m3) 2.50E + 03Young’ Modulus (Pa) 1.00E + 10Poisson’s Ratio 0.1Penalty (N/m) 1.00E + 09Time Step (s) 0.01

Figure 14. Verification of modified 3D DDA using Contact Face Elements insliding process analysis.

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The simulation results are shown in Figure 16. The red dash-dot lines are the trajectories of regular 20-faced polyhedronrock blocks, while the blue solid lines present the trajectoriesof the cubic ones falling from the same position. After 10iterations it shows the trajectories are dispersed due to theroughness of the slope surface. The lateral displacementcould be represented by the dispersion angle. If denoted, β asthe denoted dispersion angle, W as the dispersion width and Las the falling length along the cross section plane:

β ¼ arctanW

2L

� �(15)

Using Equation (15), we can obtain the dispersion angleβr20 ¼ 5:6� and βcube ¼ 11:6�respectively for regular

20-faced polyhedral and cubic falling rocks. The results showthat the rounder the rock block is, the smaller the dispersion is.It shows there is an obvious effect from the detailed shape ofthe rock block. Our program can obtain more realistic resultswhen there is detailed rock shape information.

3.4 Multi-rocks falling analysis

According to literature review (Dorren 2003, Volkwein et al.2011), previous rockfall analysis codes considered only singleblock falls. However, our program, based on 3D DDA, cansimulate mass rockfalls with consideration of collisionsbetween falling rocks. An example was given using the fol-lowing data.Figure 17 shows the topographical map of the mountain

used for rockfall simulation. At the foot of the slope, there isa national road passing along. To prevent the road and trafficfrom rockfall disasters, there is an embankment with a heightof 3:5m planned to be constructed at the foot of the mountain.It is considered that rock fragments containing 41 blocks witha total mass of 63; 957:365kg and volume of 25:600m3 fallfrom the upper part of the slope, at an average height ofZ ¼ 48:259m. The volume of the rocks range from 0:170m3

to 1:330m3 with an average of 0:625m3.

Figure 15. Bi-planar slope model with toughness to investigate the influenceof rock shapes.

Table 3. Parameters settings to investigate the influence of rockshapes

Parameters Value

Density (kg/m3) 2.50E + 03Young’ Modulus (Pa) 1.00E + 10Poisson’s Ratio 0.1Restitution coefficient 0.8Friction coefficient (�) 15Time Step (s) 0.005

Figure 16. Trajectories of rocks with different shapes.

Figure 17. Multi-rocks falling analysis.

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The mean restitution coefficient between the rocks and slopeis 0.8, which is the upper band from Chen (2003) with stan-dard deviation 0.05. There is medium roughness and no vege-tation since a previous landslide had occurred. The slopematerial is soft rock. Thus, according to field experiments(Japan Road Association 2000), the mean coefficient of fric-tion is 0.11 ~ 0.20, which approximately equals to f ¼ 10�

with standard deviation 0.05. Table 4 presents the mechanicalparameters and gives the penalty and time step values.

The calculated trajectories are shown in Figure 18. It can beshown that most of the falling rocks finally stop on a gentlestep at the height Z ¼ 33:605m. However, there are 10 leadingrocks which reach the road due to collisions among the rocks.

It shows that the collisions among rocks should be takeninto consideration to obtain accurate results in simulating massrockfall event.

4. Conclusions

We have developed a new 3D rockfall trajectory programbased on a 3D DDA program with the support of GIS in thisstudy. According the limitations of slope modeling mentionedin the text, after developing a 3D DDA code, we added a new

slope modeling method into the 3D DDA codes with GISsupport for the following:

● A new CFE has been proposed in this study.● Based on it, a new tool using GIS has been developed to

model the slope surface.

Our program can take both the advantages of DDA and GIS.And then a special element named CFE has been introducedinto 3D DDA.

● Spatial distributed data can be efficiently managed andobtained from GIS.

● 3D slope surface modeling techniques using GIS havebeen incorporated into the program.

● The efficiency of the contact searching and solution pro-cess has been improved.

Based on it, we have carried out fundamental simulation testsusing an impacting/rebounding process model comparing the3D DDA implemented from the theory of Shi (2001) and asliding process model comparing with theoretical analysis. Theresults verify that our modified 3D DDA is reliable to analyzerockfall problems. The parameter settings (penalty and timestep) used in DDA has also been investigated. By these inves-tigations, we suggest p ¼ 109N=m is appropriate value usedwhen �t ¼ 0:01s, 0:005s.Furthermore, two applications of our new program have

been carried out to show its advantages compared with pre-vious rockfall codes.

● Rockfall analysis considering different rock shapes.We tested the program using different rock block shapes.The results show a large variation if the shapes of fallingrock are taken into consideration.

● Rockfall analysis considering multi-rocks falling.We have simulated a case of mass rockfalls. It shows thatthe leading rocks travel farther due to collisions amongthe falling rocks.

However, due to the limitation of time, the evaluation worksare based on only a few cases. Further evaluation should becarried out with incorporation of engineering uncertainties inprobabilistic analysis.

Acknowledgements

The authors would like to thank Dr. Genhua Shi, for his guidance, discussionsand suggestions on developing the code.

Funding

This study has received financial support from the Global EnvironmentResearch Found of Japan (S-8) and from Grants-in-Aid for ScientificResearch (Scientific Research (B), 22310113, G. Chen) from Japan Society

Table 4. Parameters setting for multi-rocks falling simulation

Parameters Value

Density (kg/m3) 2.50E + 03Young’ Modulus (Pa) 1.00E + 09Poisson’s Ratio 0.2Restitution coefficient 0.8Friction coefficient (�) 10Time Step (s) 0.001

Figure 18. Trajectories of multi-rocks falling and stopped by countermeasure.

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for the Promotion of Science. This financial support is gratefullyacknowledged.

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