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The damage of a pipeline from a rockfall usually has serious economic andenvironmental impacts. Although its burial provides a better solution, question arisesfor the sufficient depth under known geotechnical site conditions (Wang and Cavers,2008). In any case the assumption that deeper burial offers greater protection is notsatisfactory mainly due to economic limitations and local geo-conditions.

The trajectory of a rockfall is quite complicate and can be comprised mainly fromsliding, rolling, bouncing and combinations of at least two of the abovementionedtypes of movements. Additionally rockfall is controlled by many parameters, such as:slope roughness, volume of blocks, and energy loss at impact; while the maximumrunout distance is directly associated to the previous factors as well as the initialvelocity of a rockfall.

Many researchers worked on 2D or 3D rockfall travel and many computer programssimulate the trajectories (Azzoni et al, 1995, Jones et al, 2000, Guzzetti et al, 2002),while for the scopes of this paper the simulation of a rockfall is accomplished through

the program RocFall v. 4.039 (Rocscience Inc, 2004). This program providesinformation regarding rockfall trajectory and specifically bounce height, kinetic energyand velocity during rockfall as well as during impact at the toe of the slope, etc. Thelast datum is very significant given that the value of kinetic energy just before impactat the toe of slope affects the depth of penetration of a rock boulder into the ground.However, this remark presupposes that the boulder will not rebound after its impact.

Having in mind a congener but quite simplified approach which was developed fromWang and Cavers (2008); this paper is aiming to present a methodology for burialdepth estimation and protection of pipeline against potential rockfall damage. Thedepth of rockfall penetration into the cover soil above pipeline provides an evaluationof the minimum depth of burial required to avoid puncture. This methodology is veryuseful for already placed and buried pipelines at the toe of rock slopes where noavailable data referring to rockfalls exist.

2. MAIN CHARACTERISTICS OF ROCKFALL

The goal for a rockfall protection measure is to determine the trajectory of theunstable rock blocks which detach from the rock slope. The variables which mainlyaffect the type and the positioning of the protection measures are: the maximum run-

out and also the bounce height, the transitional and rotational velocity of rock boulderduring fall. A detached block from a rock slope face may have one of the followingtypes of movement or combination of them: free falling, bouncing, rolling and sliding.In order to describe the aforementioned types of movements a mathematical modelis set up which should take into account the instant transition from one movement toanother or movement combinations.

Hungr and Evans (1988) divided the existing analytical formulations into twocategories: (a) rigorous methods and (b) lumped-mass methods. In rigorous methodsrock mass is considered as a blocky system; therefore block dynamic equilibriumequations should be solved, while the spatial movement of block striking a 3-D

topography is also studied. The size and shape of block is assumed beforehandknown and all block movements including rotation are considered. Additionally an



initial angular block velocity must be given. The block wings in the air with a ballistictrajectory and the transitional and rotational moments are transferred by block’simpact at the slope surface contact. According to Giani (1992) the impact changesthe moments through a very complex condition set which depends on the cornerblock shape at the surface contact, on the rotation angle at the impact point, on the

roughness of slope surface and on frictional deformations.

In lumped-mass methods, the single block is considered to be a simple point withmass and velocity. The point wings in the air and during impact with slope surface,the normal component of velocity is changing in sign and is reducing by the normalcoefficient of restitution R n and the tangential coefficient of restitution R t . Thesemethods take into account only the transitional moment and not the rotationalmoments, while the two restitution coefficients are assumed as general values whichinclude the characteristics of impact.

2.1 Rock block shape

Geological characteristics such as type of formation, number and orientation of jointsaffect the rock block shape. For example phyllite or even slate produces “flaky” boulders and sliding prevails in the main part of the trajectory since the impactusually occurs on the block face and the slope roughness has very low value. Theincreasing value of slope roughness induces toppling as well as rolling and bouncingmovements. On the contrary, limestone or sandstone with three set of joints producecubic boulders, which travels downwards mainly by bouncing, rolling or even theircombinations (Fig. 1, a,b after Gianni, 1992), while various tests worldwide haveshown that the rock boulder presents tendency to become more rounded as fallprogresses.

Fig. 1: a: impact on a block face, b: impact on a block corner (after Giani, 1992).

Therefore the block’s face impact produces maximum stress above pipeline when therock boulder impacts the ground at the toe of the slope (Fig. 2, after Wang andCavers, 2008), while the second assumption may lead to maximum penetration asthe boulder may encounter the least resistance as shown in Fig. 3 (after Wang andCavers, 2008).



acceleration. Assuming that the seismic wave removes the rock fragment away fromthe slope equal to distance s and by equalizing dynamic mass inertia to kineticenergy, the initial velocity of a rock boulder is calculated as follows

sav samvm 2

2

1 2 (2)

2.4 Energy loss at impact

Imre et al., (2008) refer that the energy loss of a rock boulder at impact is controlledby the coefficient of restitution. The most commonly used definitions for thecoefficient of restitution are the normal (R n) and tangential (R t ) components where

in

rn

n

V

V R

(3)

it

rt t V

V R

(4)

where V rn and V rt are the magnitudes of the normal and tangential component of therebounding velocities respectively, whilst V in and V it refer to the magnitude of thesame components of the incoming velocities (Chau et al., 2002). Those coefficientsof restitution depend on the angle of impact and the impact energy level on slopesand also on the deformability of rock boulders and rock slopes. In case of a buriedpipeline the cover zone usually comprises from compacted sand and gravel wherethe deformability is very high.

2.5 Relation between penetration and impact velocity

As already presented the impact velocity depends on certain parameters, while thevelocity of the rock boulder at the moment of impact is crucial to the depth ofpenetration. As shown in Fig. 3, the falling block starts to encounter groundresistance when the rock boulder strokes the ground surface. As penetrationprogresses, the rock boulder decelerates, while its impact energy is absorbed, andthe stoppage corresponds to maximum penetration. It is assumed that the totalenergy loss is equal to work done by the resisting force over penetration distance,while energy produced as heat during impact and the dynamic energy between initial

impact and final position of rock boulder are ignored (Wang and Cavers, 2008). As aconsequence, the following equation can be written:

Rkin W E (5)

The kinetic energy at the moment of impact can be written as:

2

2

1mv E

kin

(6)



where m is the original mass of rock boulder and v is the impact velocity. Note thatduring rockfall the boulder remains to its original volume, while separation of boulderto several pieces during numerous impacts on slope is beyond the scope of thispaper.

The work done by soil resistance is a function of block’s resistance and penetration’s distance in the soil, as shown to the following equation:

H

R RdH W 0 (7)

where H is the maximum depth of penetration. It is reasonable to assume that aspenetration progresses the soil resistance increases. In order to calculate theresistance R the following two assumptions should be made (Wang and Cavers,2002):

(a) at the time rock boulder impacts ground, boulder forces soil to follow and aspenetration continues the boulder displaces soil out in both sides.

(b) the continuous penetration of boulder leads to soil failure.

It is reasonable to assume that this corresponds to bearing capacity of shallowfoundations under inclined loads, while maximum penetration corresponds tomaximum width B of the equivalent footing.

According to EN 1997-1 (2004) the design bearing resistance R for a rectangularfooting under inclined loading can be expressed as follows:

)''5.0''('

i sbi sb N qi sb N c A Rqqqqcccc

(8)

with the design values of dimensionless factors for:

)2/'45(tan2'tan e N q (9)

'cot)1( qc N N (10)

'tan)1(2 q N N (11)

φ usually takes values between 300 and 450, the inclination of the foundation base:

)'tan(

)1(

c

qq

c N

bbb

(12)

2)'tan1( abbq

(13)

α usually takes values between 300 and 450,

the shape of foundation:



'sin'

'1

L

B sq

for rectangular shape (14)

'sin1 q s for square or circular shape (15)

'

'3.01 L

B s

for rectangular shape (16)

7.0 s for square or circular shape (17)

1

)1(

q

qq

c N

N s s

for rectangular, square or circular shape (18)

the inclination of the load:

'tan

1

c

q

qc N

iii

(19)

m

qc AV

H i

'cot''1

(20)

1

'cot''1

m

c AV

H

i

(21)

where

'

'1

'

'2

L

B L

B

mm B

(22)

The cohesion of soil is equal to zero because pipeline’s cover soil comprises fromsand and gravel, as well as depth d of embedded foundation. The surface A’ isconsidered as:

''' L B A while '' L B (23)

Thus eq. 8 becomes:

i sb R 3''5.0

(24)

and from eq. 7:



H i sb N BdH i sb N BW

H

o

33

''5.0''5.0 (25)

Finally, from eqs. 5, 6 and 25 derives:

i sb N B

mv H

3

2

''

(26)

Thus eq. 26 can be easily solved by substituting the appropriate value of width B, aswell as by using the results of the followings tables, which present values for N γ , bγ ,and i γ according to φ, α and portion of horizontal to vertical loading. A typical value ofunit weight γ is 21kN/m3 while Sγ remains constant and being equal to 0.7.

Table 1: Values of N γ according to φ.

φ ( ) Ν γ

30 20,09

35 45,23

40 106,06

45 267,74

Table 2: Values of bγ according to φ and α .

φ ( ) α ( ) 2)'tan1( abbq

30

30 0.487

35 0.41940 0.356

45 0.299

35

30 0.401

35 0.327

40 0.261

45 0.203

40

30 0.314

35 0.238

40 0.172

45 0.116

45

30 0.227

35 0.151

40 0.091

45 0.046

Table 3: Values of i γ .

Portion of H/V i γ

0.01 0,975

0.05 0,880

0.1 0,768

0.15 0,666



3. APPLICATION OF ABOVE DESCRIBED METHODOLOGY

As it was mentioned before, rockfalls occur in many places around the world andmay affect various man-made activities. The examination of case studies of rock

boulders which have penetrated to the ground (see Fig, 4a 4b, 5, 6, 7a and 7b) at theend of their trajectories as well as the study of Wang and Cavers (2008), whopresented a very simplified approach for rockfall ground penetration, have been themotive for developing the above presented methodology.

The following analyses were executed through RocFall program (Rocscience Inc,2004) simulating the trajectories of many rock boulders (Fig. 8) which can fall fromthe crest of a slope, while at its toe a buried pipeline is hypothetically placed. Theprogram uses a statistical approach for rock paths meaning that every path isdifferent from the previous. In the analyses the following assumptions were taken intoaccount:

(a) two thousands (2000) rock boulders were thrown to eliminate any statisticalerrors.

(b) the entire slope comprises of clean hard bedrock, where the mean value ofcoefficient of normal restitution R n is 0.53, with a standard deviation of 0.04,while the mean value of coefficient of tangential restitution is equal to 0.99, witha standard deviation of 0.04. The soil above pipeline is simulating as soil whereR n is 0.3 with standard deviation 0.05, R t is 0.8 with standard deviation 0.05and friction angle is 400 with standard deviation 50.

(c) the engineering geological study concluded that the more probable volumes of

detached rock boulders vary from 0.5 to 1m3

. In the analyses the maximumvolume of 1m3 is presenting, corresponding to 2500kgr of total mass. It is alsoassumed that the rock boulder is not changing its volumes during impacts.

(d) in order to begin a rockfall an initial velocity is adopted. By assuming seismicground acceleration equal to 0.24g and a horizontal movement of rock fragmentaway from the slope s equal to 0.05m during earthquake, then from eq. 2derives initial horizontal velocity equal to 0.155m/s.

The results of the analyses are presented in Figures 8 to 14, while some remarksshould be mentioned:

- The portion of the rotational energy is less than 15% of the total kinetic energy,

while Fig. 14 presents rotational velocity with very low values. Therefore therotational kinetic energy can be ignored to the aforementioned calculations.

- Figs. 8 and 9 clearly demonstrate that there is no bouncing after rock boulder’simpact above pipeline, allowing the use of eq. 26 which supposes penetrationinstead of bouncing.

Finally having in mind the previous assumptions and using eq. 26 with B = 1m, thedepth of penetration for a rock boulder of 1m3 is equal to 1.53m.



Fig. 4a, 4b: The 2008 San Severino rockfall event: Deep impact mark caused on theroad by a 7 m3 block (in Budetta, 2009).

Fig. 5: Impact of rockfall on ground (in

Perret et al, 2004)

Fig 7a: A spheroidal boulder with abounce mark (in Kobayashi et al,

1990)

Fig. 6: A large boulder slid down a 45°

slope and landed on a pipeline. Afterimpact, the boulder tipped forward (in

Wang and Cavers, 2008)

Fig 7b: A disk shaped boulder (inKobayashi et al, 1990)



Figure 8: Rock paths of 2000 rock boulders

Figure 9: Distribution of bounce height along a rockfall trajectory (figure out of scale).

Figure 10: Distribution of total kinetic energy along a rockfall trajectory (figure out ofscale).



Figure 11: Distribution of transitional kinetic energy along a rockfall trajectory (figureout of scale).

Figure 12: Distribution of rotational kinetic energy along a rockfall trajectory (figureout of scale).

Figure 13: Distribution of transitional velocity along a rockfall trajectory (figure out ofscale).



Figure 14: Distribution of rotational velocity along a rockfall trajectory (figure out ofscale).

4. CONCLUSIONS

A procedure for evaluating rockfall impact and penetration above buried pipeline wasderived having in mind the simplified methodology of Wang and Cavers (2008). Bothprocedures are based on bearing capacity of soils, however in this article was takeninto account the transitional velocity of rock boulder before impact and not the verticalvelocity and also the inclination of the rock boulder during impact. The abovedescribed procedure is very useful to estimate the height of overburden granular soilabove pipeline. Nevertheless rockfall is a statistical process taken into account manyparameters and it is always possible a “failure” to occur (exciding of mass, differenttrajectory etc). Thus a combination of other protection measures should also bedesigned according to “engineering judgment”.

REFERENCES

Azzoni, A., La Barbera, G., Zaninetti, A., 1995. Analysis and prediction of rockfallsusing a mathematical model. Int J Rock Mech Min Sci & Geomech Abstr 32, 709-724.

Budetta P., 2009. Application of the Swiss Federal Guidelines on rockfall hazard: acase study in the Cilento region (Southern Italy). Landslides, DOI 10.1007/s10346-010-0247-3

Chau, K.T., Wong, R.H.C., Wy, J.J., 2002. Coefficient of restitution and rotationalmotions of rockfall impacts. Int J of Rock Mech. Min. Sci. 39, 69-77.

Chen C.H., Ke C.C, Wang C.L., 2009. A back propagation network for theassessment of susceptibility to rock slope failure in the eastern portion of theSouthern Cross-Island Highway in Taiwan, Environ Geol. 57, 723-733.

Cook, R., Doornkamp, J., 1990. Geomorphology in environment management: a new

introduction. Oxford University Press, Oxford.



Giani, G.P., 1992. Rock Slope Stability Analysis. A.A. Balkema, Rotterdam, TheNetherlands.

Guzzetti, F., Crosta, G., Detti, R., Agliardi, F., 2002. STONE: a computer program forthe three-dimensional simulation of rockfalls. Comp and Geosci, 28, 1079-1093.

Hungr, O., Evans S.G., 1988. Notes on dynamic analysis of flowslides. In BonnardCh. (ed.), Landslides, Proc. 5th Int. Symp., Lausanne, Balkema, Rotterdam, pp. 685-690.

Imre, B., Räbsamen, S., Springman S.M., 2008. A coefficient of restitution of rockmaterials. Comp and Geosci, 34, 339-350.

Jones Cl, Higgins JD., Andrew RD., 2000. Colorado Rockfall Simulation Program,Version 4.0 (For Windows). Technical report sponsored by Colorado Department ofTransportation. Report No: OMBNo. 0704-0188, p 127.

Kobayashi, Y., Harp, E.L., Kagawa, T., 1990. Simulation of rockfalls triggered byearthquakes. Rock Mech and Rock Eng., 23, 1-20.

Lin, C.W., Liu, S.H., Lee, S.Y., Liu, C.C., 2006a Impacts on the Chi-Chi earthquakeon subsequent rain-induced landslides in central Taiwan. Eng Geol 86(2 –3), 87 –101.

Lin, J.C., Petley, D., Jen, C.H., Koh, A., Hsu, M.L., 2006b. Slope movements in adynamic environment—a case study of Tachia River, Central Taiwan. Quaternary Int147:103 –112.

Perret, S., Dolf, F., Kienholz, H., 2004. Rockfalls into forests: Analysis and simulationof rockfall trajectories — considerations with respect to mountainous forests inSwitzerland. Landslides, 1:123 –130

Rocscience Inc, 2004. RocFall. Risk analysis of Falling Rocks on Steep Slopes.

User’s Guide. Toronto, Ontario, Canada.

Roth, R.A., 1983. Factors affecting landslide susceptibility in San Mateo County,California. Bull Assoc Eng Geol 20(4), 353 –372.

Wang B., Cavers DS., 2008. A simplified approach for rockfall ground penetrationand impact stress calculations. Landslides 5:305-310.

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