Model Test for Falling-Rock Motion
Characteristics on Composite Slope
Section
1
Ye Si-qiao1, 2
Institute of Geotechnical Engineering, Chongqing Jiaotong University,
Chongqing, 400074 China
2
Postdoctoral workstation of Chongqing Bureau of Geology and Minerals
Exploration, Chongqing, 401121 China
Gong Shangqing
Institute of Geotechnical Engineering, Chongqing Jiaotong University,
Chongqing, 400074 China
Yang Zhuan-yun*
Sichuan College of Architectural Technology, Deyang, Sichuan, 618000,
China
*Corresponding Author, E-mail: [email protected]
Liu Hui
Sichuan College of Architectural Technology, Deyang, Sichuan, 618000,
China
ABSTRACT
Though falling-rock motion characteristic is the basis on research of falling-rocks and passive
treatment, the research accomplishments for the motion characteristics is lack in different
combinations of slope sections. It’s found from the model test for the composite slope sections in
different gradients that: (A) motion types on the slope surface mainly includes rolling and collision
bouncing, excluding sliding phenomenon. (B) The mass takes a non-significant effect on the
horizontal motion distance and transverse deviation ratio. The less the mass of the falling-rock, the
stronger the motion randomness is. The cylinder and sphere show little effect on the distance of
horizontal motion but significant effect on the transverse deviation ratio. The transverse deviation
ratio of cylinder falling-rocks is generally higher than that of the sphere ones, with stronger motion
scope and randomness compared to sphere ones. (C) The sphere falling-rocks show small deviation
ratio, all less than 0.35, while the cylinder ones indicate great deviation ratio, but still than 0.65.
Based on the result of the model test, it’s recommended that the range of the deviation ratio for the
steep-gentle-steep composite slope be within 0.05-0.35, while that for the gentle-steep-gentle slope
within 0.1-0.55. In practices, it’s suggested to take lower values of deviation ratio for falling-rocks
close to sphere or cube, and take higher values for falling-rocks close to bar-shape or cylinder.
KEYWORDS:
falling-rock, model test, motion characteristic, deviation ratio
INTRODUCTION
The motion characteristics of falling-rocks include: longitudinal and transverse motion
ranges, maximum speed during motion process of the falling-rocks (maximum kinetic energy),
maximum bouncing height, etc. Research on these characteristics is the major content of
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research on motion of falling-rocks, also the basis for dimension design and range setup of the
blocking structure of falling rocks.
Research on motion characteristics of falling-rocks mainly contain the back analysis of
practical falling rocks, model test, and on-site test, wherein, back analysis of practical falling
rocks is to make field survey on large amount of falling rock test-pieces, and propose
corresponding experiential theoretic methods to be consistent with the rockfall phenomenon, by
Chen Hongkai[1], Tang Hongmei[2], Liu Yongping[3], Hungr[4], Alejano[5].
Model test is to focus on the main affecting factors of falling rock motions according to
certain ideal hypothesis or emphasize the research of certain critical link of motion of falling
rocks, with repeated multiple tests. The model test is usually applied together with the
numerical simulation approach, like Yoichi Okura[6], Ya Nan[7], Zhao Xu[8].
The field test is the best method to determine motion locus of the falling rocks, which can
virtually reproduce the characteristics of falling rocks, but with large investment and serious
safety issue, by Hu Houtian[9], Huang Runqiu[10], Ye Siqiao[11, 12], Azzoni[13], and Joachim
Schweigl[14].
The above research methods can reveal some motion regularities of falling rocks, but few
studies touch upon the regularity of the motion characteristics on the composite slope
combining with different slope sections. The study analyzes the influence mechanism of the
mass and shape of falling rocks on the composite slope sections in the distance of horizontal
motion and transverse deviation ratio through the model test, providing theoretic basis and
parameter setting reference for calculation of the motion of falling rocks on the composite slope
section and passive control actions.
DESIGN OF MODEL TEST
Slope surface for test
Two types of composite slope shapes are designed: one is 0°-15°-60° steep-gentle-steep
slope, the other one is 15°-60°-15° gentle-steep-gentle slope, as shown in Figure 1 and Figure 2.
The related parameters for the composite slope section are shown in the Table 1.
a) steep-gentle-steep slope
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b) gentle-steep-gentle slope
Figure 1: Slope Shape
Table 1: Parameters for Composite Slope
Slope Section
AB
BC
CD
DE
Slope 60°~15°~60°
Gradient Length mm
60°
1800
15°
790
60°
1800
0°
4000
Paving
Smooth concrete
Soft layer
Smooth concrete
Weeds soft land
Slope 15°~60°~15°
Slope Section Gradient
AB
15°
BC
60°
CD
15°
DE
0°
Length mm
790
1800
790
4000
Paving
Soft layer
Smooth concrete
Soft layer
Weeds soft land
Figure 2: Spot Map for Composite Slope
Falling-rock test-pieces
Falling-rock test-pieces are made of C30 concrete pouring, with standard curing. The
detailed parameters are shown in Table 2.
A
E
3.56kg
D=150
d=80，H=40
Table 2: Parameters for test-piece
8.88kg
D=200
d=140，H=130
11.57kg
D=220
d=160，H=170
16.65kg
D=250
d=200，H=200
21.84kg
D=270
d=240，H=240
Note: A-sphere ball, E-cylinder, D-Diameter of sphere ball, d-diameter of base circle of cylinder;
the parameters in the table are in mm.
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a) Sphere A;
b) Cylinder E
Figure 3: Photo of falling-rock test-piece
Test procedures
Let the sphere and cylinder test-pieces roll from the top of the slope downward without
initial speed for 10 times, that is to say, each test-piece undergoes 10 repeated tests on the same
slope shape, in order to measure the horizontal motion distance and transverse deviation ratio of
the falling rocks. The horizontal motion distance is measured from the toe of the slope, and
from the bottom of 60° slope for the steep-gentle-steep slope, from the bottom of 15° slope for
the gentle-steep-gentle slope. The transverse deviation ratio is measured from the center line of
the composite slope.
TEST PHENOMENON
During falling process, the test-pieces indicate three motions including rolling, colliding,
and flying in the air, with rolling and impact as major motion type on the slope surface, without
sliding section. 1-2 times of collision bouncing occur. In the flying process, the falling rocks
usually roll along the centering, sphere one rolling along its own diameter, cylinder ones rolling
along the central axis.
Rolling
The falling rock usually rolls down along the test section surface in a regular and even way,
slightly deviating from the section axis, because the steep slope for the model test is made of the
smooth concrete plate showing even and consistent texture. The falling rock test-piece is
prepared in the regular shape and the initial motion is released in the rolling type most proper
for the falling rock test-piece. Therefore, the motion state on the surface is even and constant.
Collision bouncing
The collision bouncing will dramatically change the motion mode and direction of the
falling rock. The test-piece bounces for 1-2 times on both composite slopes, wherein there are 2
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bounces on the steep-gentle-steep slope, one occurring when the falling rock sharply rolls from
the high steep slope to the gentle slope, the other one occurring when the rock flying over the
lower steep slope to impact the ground and bounce. The gentle slope is one of the main
positions with collision bouncing. Many test-pieces impact the gentle slope to bounce and fly
over to the next steep slope. The lighter test-pieces, such as Sphere A2 and Cylinder E2 in
3.56kg, do not indicate obvious bouncing in the gentle section of the composite slope, but
continue rolling along the gentle slope after colliding, with one bouncing when impacting the
ground after flying over from the platform of the gentle slope. All the remaining test-pieces are
provided with two times of bouncing.
Each test-piece only has one bouncing on the gentle-steep-gentle slope, rolling from the
upper gentle slope swiftly downward along the lower steep slope, bouncing once after
impacting the lower gentle slope and then continuing the rolling state.
The field test [12] shows that in whatever shape, the falling rock embodies the main motion
mode of rolling bouncing. The sphere and cylinder falling-rocks hardly stop at the platform of
the gentle slope, which is consistent with the test phenomenon.
Fly in the air
The falling rocks fly in the air and roll along its axis center, the sphere rolling along its own
diameter, while the cylinder rolling along the central axis. Flying in the air takes place in the
steep-gentle-steep section, where the falling rocks roll downward from the starting point at the
initial speed of 0, bounce or roll out of the gentle slope after impacting the soft weeds slope, and
fly over to the next 60° slope, indicating an oblique projectile motion. In the gentle-steep-gentle
slope, the test-pieces fall from the top of the gentle slope, slightly break away from the slope
surface after rolling out of the gentle slope, with very short time of flying in the air, and
continue rolling to the lower 15° gentle slope along the surface.
ANALYSIS ON TEST RESULTS
The horizontal distance to the final stopping point can reflect the kinetic ability of the
falling rock, showing the range of falling-rock motion threats, which is not only usually the
reference to determine prevention actions of the falling rocks, but also the main parameter to be
established in the rock-falling prevention engineering design. The motion of falling rocks in the
range perpendicular to the motion cross-section is called as transverse deviation, which can
reflect the transverse range of the falling-rock motion threat, used for deciding the length and
position of the falling-rock blockage structure. Due to specific and unique features, the
transverse deviation distance can only represent the deviation characteristic of certain rock
falling motion, so it’s decided to adopt the deviation ratio defined by Azonni[13] (Figure 4) for
analysis of the deviation characteristics of the falling rocks.
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Figure 4: Deviation Ratio defined by Azonni[13]
Deviation ratio:
η=
D
2L
（1）
where: D-transverse affecting length, and L-Equivalent slope length.
Effect of the mass on the horizontal motion distance
a) Sphere A
b) Cylinder E
Figure 5: Horizontal Motion Distance of Falling-Rock on Steep-gentle-steep Slope
For each range of mass levels of sphere test-pieces, the horizontal motion distances have not
much difference between the minimum mass and the maximum one, and the average values
almost equal. For each range of mass levels of cylinder ones, there is no much difference as
well; the horizontal motion distances at each mass level are rather close in terms of minimum
value, maximum value, and average value.
On the steep-gentle-steep slope, with increase of the mass, the horizontal motion distance of
the sphere and cylinder shows no much change, which indicates that the mass has little effect on
horizontal distance.
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a) Sphere A
b) Cylinder E
Figure 6: Horizontal Motion Distance of Falling Rocks on Gentle-steep-gentle Slope
For each range of mass levels of sphere test-pieces, in terms of the minimum value, the
horizontal motion distances at each mass level change a lot without uniform regular change
trend; in terms of the maximum value, the horizontal motion distances at each mass level are
almost equal with little difference; in terms of average value, the horizontal motion distances at
each mass level change greatly. For each range of mass levels of cylinder test-pieces, the
horizontal motion distances do not show obvious regularity. Seen from the minimum value,
maximum value, and average value, the horizontal motion distances of each mass level
fluctuate, without consistent regularity. The mass takes an insignificant effect on the horizontal
motion distance.
On the gentle-steep-horizontal slope, with increase of the mass, the horizontal motion
distances of the sphere and cylinder show no obvious change, which indicates that the mass has
little effect on horizontal distance. In general, the mass takes no obvious effect on the horizontal
motion distance. With increase of the mass, the horizontal motion distance of the falling-rock
test-piece is provided irregular changes.
Effect of the mass on transverse deviation ratio
a) Sphere A
b) Cylinder E
Figure 7: Transverse deviation distance of falling-rocks on steep-gentle-steep slope
For each range of mass levels of sphere test-pieces, in terms of the minimum value, with
increase of the mass, the transverse deviation ratio indicates no obvious change trend, with
strong randomness; in terms of the maximum value, the transverse deviation ratios at A4 level
are high, those in other cases have little difference; in terms of average value, the transverse
deviation ratios at each mass level have irregular trend and great volatility. For each range of
mass levels of cylinder test-pieces, seen from the minimum value, maximum value, and average
value, the transverse deviation ratios show inconsistent trend. Seen from the minimum value
and average value, the transverse deviation ratio increases with increase of the mass but
gradually decreases when the mass is up to E4. Seen from the maximum value, the transverse
deviation ratios of the first three mass levels are close to each other with increase of the mass,
but gradually decreases when the mass is up to and greater than E5.
On the steep-gentle-steep slope, the transverse deviation ratios of the sphere and cylinder
indicate no obvious regularity with increase of the mass.
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a) Sphere A
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b) Cylinder E
Figure 8: Transverse deviation ratio on gentle-steep-gentle slope
For each range of mass levels of sphere test-pieces, in terms of the minimum value,
maximum value, and average value, with increase of the mass, the transverse deviation ratio
indicates no obvious change trend, with large volatility and strong randomness; For each range
of mass levels of cylinder test-pieces, seen from the minimum value, maximum value, and
average value, the transverse deviation ratios show inconsistent trend. Seen from the minimum
value and average value, the transverse deviation ratio increases with increase of the mass but
gradually decreases when the mass is up to E5. Seen from the maximum value, the transverse
deviation ratios fluctuate slightly.
On the gentle-steep-gentle slope, the transverse deviation ratios of the sphere and cylinder
indicate no obvious regularity with increase of the mass.
The field test shows that with different masses from 3kg to 150kg, the deviation ratio of the
falling rock indicates the trend of decreasing with increase of the mass. In the meantime, the
falling rocks in smaller mass are provided with large range of fluctuation of the deviation ratio,
from 0 to 0.3. In the test, the masses of each falling rocks are no more than 22kg, so the
regularity is not obvious possibly due to small mass level. The model test by Yangwei[15] also
shows that when the mass of the falling rock is small, the motion deviation ratio of the falling
rock does not present obvious regularity.
It’s concluded from the statistical regularities of both slope shapes that the effect of the
mass on the transverse deviation ratio does not indicate explicit regularity.
Effect of the shape on horizontal motion distance
a) steep-gentle-steep slope
b) gentle-steep-gentle slope
Figure 9: Horizontal Motion Distance of Falling Rocks in Both Shapes
On the steep-gentle-steep slope, by comparison of the minimum value, maximum value and
average value, the horizontal motion distances of both cylinder and sphere are close; Comparing
the minimum value and maximum value, the ranges of fluctuation of the horizontal motion
distances in both shapes are consistent; on the gentle-steep-gentle slope, the difference in both
shapes is not big. The minimum values of both shapes are close. In terms of maximum
horizontal motion distances, the cylinder is longer than the sphere. In terms of the average
value, the cylinder is higher than the sphere. Comparing the maximum and minimum values, the
ranges of fluctuation of the horizontal motion distances in both shapes is different, where the
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range of fluctuation of the cylinder is obviously larger than the sphere, which indicates that the
cylinder has stronger randomness in the horizontal motion ability.
The field test [12] verifies that the range of fluctuation of the horizontal motion distances of
the cylinder falling-rocks is the largest among various shapes.
Effect of the shape on the transverse deviation ratio
a) steep-gentle-steep slope
b) gentle-steep-gentle slope
Figure 10: Transverse Deviation Ratio of Falling Rocks in Both Shapes
On the steep-gentle-steep slope, by comparison of the minimum, maximum and average
values, the transverse deviation ratio of the cylinder is larger than the sphere; Comparing the
minimum value and maximum value, the range of fluctuation of the transverse deviation ratio of
the sphere is within 0.03-0.29, while that of the cylinder is within 0.15-0.43; the average value
of the cylinder is 0.3 larger than that of the sphere of 0.17. For the gentle-steep-gentle slope,
comparing the maximum, minimum and average values, the cylinder is greater than the sphere,
while comparing the maximum and minimum values, the variation range of the cylinder is 0.170.63, that of the sphere is 0.08-0.34.
It’s concluded from the comprehensive analysis that the deviation ratio of the cylinder is
greater than the sphere, and the variation range is also larger. But for whatever shape, the
average deviation ratio is always below 0.65.
The field test by Ye Si-qiao [11] also indicates that the average value of transverse deviation
ratio of the cylinder is higher than that of the sphere, and the variation range is also larger. The
deviation ratios on the field test are all below 0.3, and it’s found from Yang Wei[15] model test
that the deviation ratios of the falling rock are within 0.5, but test result in the study is greater
than the above two results, the reason of which is that the proportion relation of the model test is
different from the field test condition, with relative large difference in the obstacles on the slope
surface.
SUGGESTIONS ON TAKING VALUE OF DEVIATION
RATIO
At present, the researchers haven’t got generally valid conclusion. The research on
transverse deviation of the motion trail of the falling rocks done by Azzoni[13] shows that the
transverse motion range of falling rocks is within 10%-20% of the slope length. The deviation
ratio defined in the study is within 0.05-0.1. The steeper the slope, the smaller the transverse
motion range is.
There are little cases for value setting in the specifications. During design of the falling-rock
passive control system by the previous national transport bureau of Ministry of Railways,
experiential regularity for the length of the protection area is defined: Expansion of 5m or
10m on both ends of the protection area starting from rockfall 2D calculating section is
implemented as the safety reserve. Ye Siqiao suggests that the deviation ratio is determined
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according to the falling-rock hazard level at the tunnel portal section, with 0.3, 0.25, and 0.2
taken respectively for Level 1, Level 2, and Level 3.
Based on the above study results, it’s found with the aid of the model test that the shape of
the composite slope and the shape of falling rocks have obvious effect on value setting of
deviation ratios. The shape of the composite slope is taken as the basis for classifying the value
ranges. It’s recommended to set values according to the following principles:
(A) Make sure 95% of the test data are within the value range;
(B) The value shall present the effect of different slope shapes on the deviation ratio.
It’s suggested that the deviation ratio on the steep-gentle-steep slope is within 0.05-0.35,
while on the gentle-steep-gentle slope is within 0.1-0.55. For actual value setting, when the
falling rock is close to sphere or cube, lower values shall be taken, and when close to bar or
cylinder, higher values shall be taken.
CONCLUSIONS
(A)During motion process on the composite slope, the falling rocks present rolling and
collision as major motion type, without sliding.
(B)The horizontal motion distance and transverse deviation ratio of the lighter falling rocks
indicate strong randomness. With increase of the mass, the horizontal motion distance and
transverse deviation ratio show non-significant regularity.
(C)The cylinder and sphere shapes have little effect on the horizontal motion distance but
significant effect on the transverse deviation ratio. The transverse deviation ratio of the cylinder
falling-rocks is higher than that of the spherical ones, and the variation range is larger as well,
showing stronger randomness. The deviation ratio of the sphere is small, less than 0.35, while
that of the cylinder is large, less than 0.65.
(D)In the actual engineering, the deviation ratio on the steep-gentle-steep slope shall be
within 0.05-0.35, and on the gentle-steep-gentle slope within 0.1-0.55. When the falling rock is
close to sphere or cube, lower values shall be taken, and when close to bar or cylinder, higher
values shall be taken.
ACKNOWLEDGEMENTS
This paper was supported by National Natural Science Foundation of China (Grant No.
51108488), Foundation for University Key Teacher by Chongqing, and Chongqing Municipal
Natural Science Foundation(Grant No. CSTC2010BB4265), the authors express gratitude to
them.
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