ARMA 13-237

Mechanisms of Falling Rock Formation at Steep Slope due to Temperature
Perturbation.
S.S.S.S.M.Keppetipolaa*, Yoshiaki Fujiia, Nariaki Kodamaba Rock Mechanics Laboratory, Faculty of Engineering, Hokkaido University, Japanb Hakodate National College of Technology, Hakodate, Japan*Corresponding Author’s E-mail: sampath@eng.hokudai.ac.jp
Copyright 2013 ARMA, American Rock Mechanics Association
This paper was prepared for presentation at the 47th US Rock Mechanics / Geomechanics Symposium held in San Francisco, CA, USA, 23-26
June 2013.
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ABSTRACT: To clarify the mechanisms of formation of falling rocks, a 2D finite element modeling was performed tonumerically analyze natural stress development at a fracture tip for a rock mass (20 m x 20 m) with a rock beam of 3.0 m thickness
and 10 m long. The cyclic temperature variations between 10 to 40 oC (warm region) and -10 to 20oC (cold region) were assumed.
In warm region, the fracture closure due to difference in thermal expansion in the rock beam was prohibited inducing a verticalbending tensile stress (σy) in summer while in winter, the fracture opened due to difference in thermal shrinkage in the rock beam
inducing a horizontal tensile stress (σx) at the fracture tip. In the cold region σ y was also observed in winter due to fracture closure
by difference of freezing expansion in the rock beam was observed. The σ x developed during both regions were large enough forthe fracture propagation. The repeated occurrence of σ y would result in breakage of the rock beam from its bottom by either
subcritical or fatigue crack growth ensuing a rock fall at a steep rock slope.

1.

INTRODUCTION

2.

Fracture in steep rock cliffs can grow due to the thermal
deformation, weathering, intrusion of plant roots etc, androck beam can be formed [1-4]. Negishi and Nakajima(1994) explained toppling failure of columnar joints atSounkyo Japan by thermal deformation and rocks
trapped in the joints [5]. This is however a special caseand general reason for rock beam detachment from rockcliff which leads to rock fall has not been clarified yet.

NUMERICAL MODEL

A 2D finite element modeling was performed tonumerically analyze natural stress development at afracture tip for a rock mass (20 m x 20 m) with a rockbeam of 3 m thickness and 10 m long (Figures 1 and 2).
This 2D plane strain analysis is coupled with thefundamental heat transfer equation using the CrankNicolson method [6].
Cyclic air temperature variations in temperate region
between 10 to 40oC (warm region case) and -10 to 20oC
(cold region case) for a duration of 2 years wereassumed (Figure 3). The insulated condition was
basically assumed in which the heat flux was possible
only via the vertical cliff surface at 3 W/(m2K)
simulating thick soil or deep snow during winter. The
uninsulated condition in which heat flux was possible
via both cliff surface and the top of the steep rock slope
was also considered. Insolation and radiation were notconsidered.

The objectives of this study are to clarify thermal stressbehavior by which the vertical fracture is extended, amechanism of the breakage of rock beam from its
bottom and the influence of beam size in temperateregion for a steep rock slope.
The stress development occurred at the fracture tip of thesteep rock slope due to the cyclic air temperature
variation in warm and cold temperate regions with
influences of insulation conditions and beam size on
those will be shown.

1Table. 1. Physical, mechanical and thermodynamic parameters used in the analysis.
aassumed; m measured; *2 is used because of plane strain analysis.

Property

SymbolYoung’s modulus
Poisson’s ratio
Thermal conductivity
Specific heat
Density
Expansion coefficient
Effective porosity
Volumetric ice expansion due to freezing
Linear expansion strain due to freezing
Temperature at which freezing starts
Temperature at which freezing finishesEKcαv
fT1
T2ValueFormulam

110 (GPa)
0.2 a7.40 (W/ m K) m995 (J/ kg K) m
2636 (kg/m3) m
7.94  10-6 (K-1) a
0.8% m
9% a0.00036 a0 oC a–20 oC a(  v)/2*

Physical, mechanical and thermodynamic parameters
shown in Table 1 are used assuming a steep cliff
consisting of chert. It was assumed that rock expands byεf from T1 to T2 due to freezing of pore water, where T1
is the temperature at which freezing starts; T2 is thetemperature at which freezing completes (Figure 4).

Fig. 2. Finite elements mesh configuration andElement where stress values are examined.

Fig. 1. Model showing rock mass (20 m x 20 m)with 10 m rock beam.

2

Warm region case

40oTemperature ( C)60

200-20Cold region case-40
Sp Su Fa Wi Sp Su Fa Wi Sp(b) Fracture closure/deformation and σy distribution
in summer.

Fig. 3. Cyclic air temperature variations.

Strain, εTref (Reference/Initial temperature)εf1αα1T10T2()Temperature, T(c) Fracture opening/deformation and σx distribution
in winter.

(+)

103

82.5

6Opening

y1.5

4x20Sp

21Opening (mm)Stress (MPa)Fig. 4. Model for expansion due to freezing of porewater.

(d) Temperature distribution when y maximum insummer.

0.5
SuFa

WiSp0Fig. 5. Numerical result for 20 m x 20 m warm
region.

(a) Stresses at the fracture tip and fracture
opening and closure with time.

3

During winter, a fracture opening was observed inducing
a horizontal tensile stress (σx) at the fracture tip. σxdevelops with the fracture opening due to the shrinkage
difference between outer strip and the inner strip (Figure6b).

3. RESULTS
a. Warm region case (Figure 5)
During summer, a vertical bending tensile stress (σy)
appeared due to the fracture closure prohibition that
occurs as a consequence of the thermal expansion
difference of the rock beam (Figure 6a).

σx gradually increased with the fracture opening,
reached its maximum (1.9 MPa) and thereafter showeda gradual decrease. The observed σy was minute. σx arelarge enough for the fracture propagation forming therock beam. The stresses are larger in the uninsulated
condition (Figure 7a).

Rock beamyb. Cold region case (Figure 8)
x

Fracture closing

During both summer and winter, the vertical bending
tensile stress (σy) was developed at the fracture tip due tothe thermal or freezing expansion of the rock beam and afracture closure prohibition (Fig. 6a). The maximum σyis 5.4 MPa and the repeated occurrence of this stressdevelopment at the fracture tip will result either
subcritical or fatigue crack growth ensuing a breakage ofthe rock beam from its bottom.

Fracture closureprohibition(a) Fracture closure prohibition
development at the fracture tip.

andthestressIn winter σx behaved a little bit complicated but the peakvalue was almost the same as in the warm region caseand was large enough for fracture propagation. σx and σyin summer were larger but σy in winter was slightly
smaller in the uninsulated condition (Figure 7b).

Rock beam`
4. EFFECT OF BEAM SIZE ON THE STRESSDEVELOPMENT

xa. Warm region
The vertical bending tensile stress (σy) was observed
during summer while the horizontal tensile stress (σx)
was observed during winter.
The maximum vertical bending tensile stress (σy) showeda gradual increase with increasing beam size and reached
its maximum at a beam length and thickness of 20 m and
6 m respectively. Thereafter, σy decreased.

Fracture opening(b) Fracture opening due to the shrinkage difference.

Fig. 6. Mechanism of thermal deformation.

σx was comparatively lower than σy . σx showed a gradualincrease with increasing beam length, reached its
maximum value at a beam length of 32 m and decreased(Figure 9a).

σy showed a gradual increase up to its maximum (4.9MPa) and thereafter decreased gradually. The highest
temperature gradient in the rock beam with respect to the
air temperature variation was observed at maximum σy(Figure 5d). The observed horizontal tensile stress (σx)
was minute. The repeated occurrence of σy development
at the fracture tip will result either subcritical or fatigue
crack growth ensuing a breakage of the rock beam fromits bottom.

These behaviors were observed because the maximum
temperature gradient reached around beam size of 20-32
m for the used parameters and temperature variation.
The significant decrease of σy was due to the overburdenpressure.

4

35

82.5

4
321Su

Fa

Wi

21xSp

Sp

2
1.5

4

0.5

0Su

Fa

Wi0Sp

(a) Stresses at the fracture tip and fracture opening
and closure with time.

3

y

2.5

Opening

62

4

1.5

2

x

1

0

0.5

-20Sp

SuFa

WiOpening (mm)10Stress (MPa)Openingy0(a) Warm region.

86Opening (mm)x10

Stress (MPa)Opening

6Opening (mm)Stress (MPa)108 y64
20-2-4SpSp(b) Fracture closure/deformation
distribution in summer.

(b) Cold region.

andσxFig. 7. Stresses at the fracture tip and fracture
opening and closure with time.

b. Cold region
The vertical bending tensile stress (σy) was observed
during both summer and winter while the horizontal
tensile stress (σx) was observed only during winter.
The σy developed up to a beam length of 6 m wasdominant during winter. σy in summer and σx behave in asimilar way as for warm region. The maximum σx washowever at beam length of 9 m (Figure 9b).

(c) Fracture opening/deformation and σy distribution
in winter.
Fig. 8. Numerical result for 20 m x 20 m coldregion.
58

max (MPa)6(ii) The vertical bending tensile stresses in summer andhorizontal tensile stress are larger when heat transferfrom the top is not insulated.

y420(iii) The stresses show peaks at specific beam sizes
which depend on thermal and mechanical properties ofrock mass and weather conditions.

xNumerical investigation on the influences of theproperties and weather conditions on the beam sizes atwhich the stress peaks are shown and field investigations
on the specific beam sizes which may control the fallingrock size with joint spacing will be carried out further.

1 2 4 10 20 40 100
Beam length (m)(a) Warm region.
8

yWintery Summer

6

max (MPa)The results shown in this paper as well as results infuture will help deeply understand the mechanisms ofrock falls and reduce human and economical damages bythem.

406. REFERENCES

x2[1] Ishikawa, M., Kurashige, Y., Hirakawa, K., 2004.
Analysis of crack movements observed in an alpine
bedrock cliff. Earth Surface Processes andLandforms, 29, pp. 883–891.
[2] Matsuoka, N., 2001. Direct observation of frost
wedging in alpine bedrock. Earth Surface Processesand Landforms, 26, pp. 601–614.
[3] Matsuoka, N., 2008. Frost weathering and rock wall
erosion in the southeastern Swiss Alps: long-term(1994–2006) observations. Geomorphology, 99, pp.
353–368.
[4] Mufundirwa, A., Fujii, Y., Kodama, N, and Kodama,
J. (2011), Analysis of natural rock slope
deformations under temperature variation: A casefrom a cool temperate region in Japan, ColdRegions Science and Technology, Vol. 65, No. 3, pp.
488–500.
[5] Negishi, M., and I. Nakajima, I., 1994. Mechanismof Toppling of Long Columnar in Sounkyo Welded
Tuff Study on Slope Failure in Cold Region (2ndReport). J. Japan Society of Engineering Geology
35(1).
[6] Reddy, J.N., 1993. An introduction to the finiteelement method. McGraw-Hill, 2nd edition, pp. 295–403.

1 2 4 10 20 40 100
Beam length (m)(b) Cold region.
Fig. 9. Maximum stress with beam length for fixed
length to thickness ratio at 10/3.

5. CONCLUDING REMARKS
2-D FEM stress analyses considering thermal
deformation including freezing expansion for steep cliffs
subjected to cyclic air temperature change in temperateregion were carried out to clarify the mechanisms offalling rock formation. The key point of the analyses isthat it considers freezing expansion.
It was already known that the cyclic air temperaturechange in temperate regions could induce tensile stressunder which rock fractures can grow. The analyses inthis paper further quantitatively showed the magnitudeof the tensile stress and its variation with the rock beamsize.
This paper also newly clarified the followings mainly on
the mechanisms by which falling rock blocks are formed.
(i)
Vertical bending tensile stress can develop andrepeated occurrence of the stress can break the beamfrom its bottom.

6