Proceedings of the ISRM International Symposium 3rd ARMS, Ohnishi & Aoki (eds)
ⓒ 2004 Millpress, Rotterdam, ISBN 90 5966 020 X
Earthquake response analysis of rock-fall models by discontinuous
deformation analysis
T. Sasaki, I. Hagiwara & K. Sasaki
Rock Engineering Lab., Suncoh Consultants Co. Ltd., Tokyo, Japan
R. Yoshinaka
Saitama University, Saitama, Japan
Y. Ohnishi & S. Nishiyama
School of Urban & Environment Engineering, Kyoto University, Kyoto, Japan
Keywords: rock-fall, DDA, seismic response analysis, dynamic problem
ABSTRACT: This paper describes rock-fall models of rock slopes caused by earthquakes using discontinuous
deformation analysis (DDA). It introduces viscous damping and the velocity-energy ratio for the rock-fall
models based on DDA theory. This paper proposes a method of inputting the earthquake accelerations in
DDA and examines the vibration characteristics of the rock slopes and the rock-falls of homogenous and
two-layered models. The results show qualitative agreement, and so the methods are applicable to physical
phenomena.
1 INTRODUCTION
Earthquakes are known to trigger rock-falls at rock
slopes and earthquakes are common in Japan.
However, it is very difficult to distinguish which
rocks will fall in investigations before earthquakes
and it depends on the experience and judgment of
the investigator. An earthquake response analysis by
DDA by Shi (2002) evaluated the stability of Yucca
Mountain tunnels combined with block theory. In
this case, the earthquake acceleration is input
directly for each block by inertia as the body forces.
Hatzor et al. (2002) compared an analytical solution
with the block DDA model, and showed that the
analytical error of DDA ranges from 5% to 10%.
They also recommended introducing a 5% damping
coefficient in the Mount Masada monument model.
Zaslavsky et al. (2002) pointed out that the spectral
response of an earthquake at the parts where the
topography changes, such as the edge of a cliff or
the entrance of a tunnel, is different from that
estimated using a point of the usual base rock.
Ishikawa et al. (2002) analyzed dynamic
characteristics of the ballast of railway foundations
and compared the results with experiments.
Tsesarsky et al. (2002) analyzed the frequency
characteristics of a single block and compared the
results with an experiment using a single block in
the slope model. Nishiyama et al. (2003) analyzed
the stability of a masonry-type retaining wall by
Proceedings of the ISRM International Symposium 3rd ARMS
dynamic response analysis.
These researches can be classified into two
categories by the input method of an earthquake
wave: the acceleration time history or the
displacement time history.
The present study investigates optimum values
of the parameters in some actual rock-fall problems
of natural large rock slopes by using DDA (Shi,
1984), focusing on the input of earthquake
acceleration time history, and estimates the energy
of the falling rocks depending on the efficiency of
countermeasure structures.
2
OUTLINE OF THEORY
The governing equation of the potential energy
sys
Π on large deformations of continuous and
discontinuous elastic bodies is given by:
n
Π
sys
= ∑Π
 i m i, j 
= ∑  Π + ∑ Π PL

i =1 
j =1
n
( block ) i
i =1
(1)
The first term on the right side of equation (1) is the
potential energy of the continuum part, and the
second term is the potential energy of the contact
between blocks. The first term is given by:
Π = F ( x, y)∫
i
V
1
ρc *
[
− σ ij δ (2 Dik Dkj − vk ,i vk , j) ]dV
0 τ ij δDij
2
ρ
(2)
− F ( x, y)∫ t ⋅ udΓ− F ( x, y)∫ [ρ(b& − u&&) − cu&] ⋅ dV
Γ
V
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The first term of equation (2) is the strain energy of
velocity field, the second term is the surface traction
energy, and the third term is the energy of the inertia
force and damping force, where, F ( x, y ) : shape
function, ρ : density before deformation, ρ :
0
c
density after deformation, τ ij : Kirchhoff stress
velocity, Dij : deformation velocity tensor, σ ij :
Cauchy stress, u&& : acceleration, u& : velocity, ρ :
unit mass, b: body force, c: viscosity coefficient,
t :surface traction force, V: volume of body, and Γ :
area of body.
The second term on the right side of equation (1)
is the potential energy of the contact between
discontinuous planes, and is evaluated by the least
squares method by using a penalty as follows:
*
Π PL =
i, j
1
1
2
j
i
j
i 2
k N [(u − u ) ⋅ n ] − k T [uT − uT ]
2
2
(3)
where, kN: penalty coefficient of the normal direction,
kT: penalty coefficient of the shear direction,
(u j − ui ) ⋅ n : amount of penetration in the normal
direction, uT: amount of slip in the shear direction, n:
direction cosine of the contact plane.
DDA (Shi, 1984) is formulated from equation (1)
using the kinematic equations based on Hamilton’s
principle and minimized potential energy expressed
by:
M u&& + C u& + K u = F
(4)
where, M: mass matrix, C: viscosity matrix, K:
stiffness matrix, F : external force vector, u&& :
acceleration, u& :velocity, and u: displacement of a
block center.
The viscosity matrix C in equation (4) can be
rewritten as follows in terms of viscosity η and
mass matrix M:
C = ηM
(5)
The physical meaning of viscosity η is the
damping of the rock itself, the viscosity of air around
the rock surfaces and the vegetation on the surface
of a rock slope.
The kinematic equation (4) is solved by
Newmark’s β and γ method (Hilbert,1993) by
using parameters β = 0.5 and γ = 1.0, and the
algebraic equation for the increase in displacement is
solved for each time increment by the following
three equations:
~
~
K ⋅ ∆u = F
(6)
where,
Proceedings of the ISRM International Symposium 3rd ARMS
2
2η
ρ
~
K =
M +
M + 0 [ K e + K s]
2
∆t
ρ
∆t
(7)
~ 2
F=
M ⋅ u& + (∆F − ∑ ∫ σdv) − Mα (t )
∆t
(8)
c
where, ∆u : incremental displacement, K e ：
stiffness matrix of linear term, K s ： initial stress
matrix caused by rigid rotations, α (t ) ： time
history of earthquake acceleration.
The relations between displacements, velocities
and accelerations at an arbitrary point of a block at
time t in step i are expressed by the following three
equations, respectively.
u i = [D i ] =
u& i =
2
2
∂ [ D ( t )]
∆ t ∂ [ D ( t )]
+ ∆t
2
∂t
2
∂t
∂[ D ( t )]
2
[D i ] − ∂[ D (t − ∆ t )] = 2 [D i ] − u& i −1
=
∂t
∆t
∂t
∆t
2
2
∂ [ D ( t )]
[ ] + 2 ∂ [ D ( t − ∆ t )]
=
2
2 D i
∆t
∂t
∂t
∆t
2
[ ] − 2 u& i − 1
=
2 D i
∆t
∆t
u&&i =
(9)
(10)
(11)
3 NUMERICAL EXAMPLES
The purpose of numerical studies of rock-fall models
is to distinguish the candidates of falling rocks on
the rock slopes and to evaluate the applicability of
the earthquake response analysis by using DDA, and
the method of inputting earthquake accelerations.
There are two methods in DDA of inputting the
earthquake
response
as
described
above:
displacement time response and acceleration time
response, as indicated by instruments. Analytically,
it is simpler to use the displacement time response
record for input in DDA, however, generally, it is
the acceleration time response of an earthquake that
is recorded.
The authors examined and analyzed both input
methods and compared the results. They propose
giving the acceleration response at the base points of
the rock slope block, which corresponds with the
large virtual mass at the base block of DDA models.
3.1 Homogeneous Rock Slope Model
Figure 1 shows the rock-fall models on a rock slope
in an earthquake. The height of the slope is assumed
to be 100 m, and ten rock blocks exist at the top of
the slope. Table 1 shows the analytical conditions
and material properties. The time interval used for
the numerical calculation is 0.001 second. The input
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earthquake wave is EL-Centro East-West and
Up-Down at the bottom point of the base block of
the rock slope at the same time. The boundary
conditions are defined such that the horizontal roller
is at the bottom and free at both sides of the base
block. Two values of the surface friction angle of the
rock slope are assumed, 35 and 20 degrees, and the
results for both are compared.
The elastic modulus of the rock slope is assumed
to be 1 GPa and the penalty coefficient to be 10
GN/m3. The characteristic frequency of the rock
slope is about 2 Hz, and that of rock-falls ranges
from 10 to 20 Hz. These are approximately the same
as the single mass analytical solutions and the
influence of the penalty coefficient value shifts
towards the high-frequency side. The result depends
on the penalty value, and in this case was
equivalents 110% of the elastic modulus of the
block.
In order to get the same value between input and
output acceleration wave proportions, the mass of
the base block was defined as 10,000 times to avoid
the influences for the response by an additional mass
of base block, and the body force acting downward
was set to zero to eliminate up-down free vibrations
of the slope block as shown in Table 1.
Static analysis of this model shows that the blocks
on the 20-degree slope do not move because the
friction angle of the rock surface is defined 35
degrees.
block of EL-Centro earthquake. Figure 4 shows the
acceleration Fourier spectrum at the input point,
which perfectly coincides with the input acceleration
proportion.
Figure 2.
Acceleration time history of EL-Centro earthquake
Figure 3.
Acceleration response at input point of base block
Figure 4. Fourier spectrum at the input point
Figure 1. Rock slope model of homogeneous strata
Table 1.
Material properties of the models
Time interval
0.001sec
Name of input earthquake
EL-Centro-EW,UD
Elastic modulus, Poisson’s Ratio
1GPa,ν =0.25
Friction angle of the rock surface φ =35 and 20 degrees
Penalty coefficient
10GN/m3
Viscosity coefficient
0.02
Velocity / Energy Ratio
0.81
Figure
2 and Figure 3 show the
acceleration time
Unit mass of the base block
25000kN/m3(Virtual)
history
and
response
at
the
input
point
on the base
Unit mass of the rock blocks
25kN/m3(Actual)
Proceedings of the ISRM International Symposium 3rd ARMS
Figure 5.
Rock-fall after 5 seconds (φ = 35 degrees)
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Figure 5 shows the result when the rock surface
friction angle is 35 degrees, 5 seconds after the start
of earthquake motion. Blocks fall on the 20-degree
slope and receive acceleration from the base block
for 1.2 seconds. After that, about 50% of blocks
move on the 45-degree slope repeated with
collisions as shown in Figure5.
Figure 6 shows the position of falling rocks after
10 seconds. The front of the group of falling rocks
arrives at the bottom of the rock slope, and all the
following falling rocks also arrive there on a steep
slope side. The blue and red traces in the figure
show a falling rock (3) contacting the representative
slope in the central part of a group of falling rocks,
and a falling rock (9) from the top.
Figure 7 shows the position of falling rocks after
15 seconds. Most of the falling rocks arrive at a flat
area of a lower part of the slope. In addition, a large
mass of a falling rock (3) reaches the top.
Figure 8 shows the acceleration response of the
falling block (9) at the start of the earthquake. Figure
9 shows the velocity response of the falling rock (9).
The vertical axis shows the falling rock velocity, and
the horizontal axis shows the X coordinate along the
slope in the horizontal direction.
The falling blocks receive horizontal and vertical
accelerations from the base rock slope block before
sliding on the low-angle slope at first. When a
falling rock begins to slide on a slope, the falling
movement by gravity influences it, and the influence
of the earthquake wave diminishes.
In addition, acceleration by the earthquake from
the slope is propagated again when the falling rock
arrives at the flat area of the end of slope.
In the EL-Centro wave, the level of the vertical
acceleration component is large in the front and back
parts, but the horizontal acceleration component
reduces after 10 seconds to around 1/3, hence the
response of falling rocks reflects the acceleration
after 10 seconds.
The speed of block rotation is lower than 10% of
the plumb speed on the steep slope side and is
comparatively small, because the block is flat and
stable, but there is an influence of the earthquake
wave from the slope on the low-angle slope before
sliding and the flat area of the end of slope and the
speed of the rotation component increases to 30%
compared with natural rock-fall. Figure 10 shows the
positions of falling rocks after 15 seconds in the case
of 20-degree friction angle. In this case, the arrival
distance in the flat area of the falling rocks is large
compared with the case of 35 degrees, and so the
jump at the bottom of the slope is small.
3.2 Two-layered Model
Generally, a bedrock slope is not uniform and tends
to present a multilayered structure. Figure 11 shows
the positions of falling rocks after five seconds in a
model when the base is on a slope. The earthquake
wave is input at the center of the bottom end of the
base block.
Figure 6. Rock-falls after 10 seconds (φ = 35 degrees)
Figure 7.
Rock-falls after 15 seconds (φ = 35 degrees)
Figure 8. Acceleration response of block (9)
Figure 9. Velocity along horizontal coordinate of block (9)
Proceedings of the ISRM International Symposium 3rd ARMS
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As for the dimensions of the model, the right and
left size are doubled to give a symmetrical part of
case 1. The input earthquake wave and bedrock
properties of matter are more similar to the single
strata case.
Figures 12 and 13 show the state after 10 seconds
and the arrival position of falling rocks after 15
seconds. The arrival distance of the falling rocks is
larger than in the single strata case.
Figure 14 shows the vibration property for the
slope of block No.2 itself. This result assumes a
damping coefficient of 2%, and the characteristic
frequency of the slope is around 3 Hz, which almost
matches a one-spring mass model of the theoretical
solution.
Figure 15 shows the vibration property of a falling
rock of block No.10 under gravity. The
characteristic frequency of the falling rocks is
around 10–20 Hz. Figure 16 shows the acceleration
response of the slope block (2).
Figures 17 and 18 show the velocity response of
the falling rock. In comparison with the single strata
case, the velocities are about 10% faster. In addition,
the velocity is affected by the vibration of the slope
block as shown in Figure 16 while a falling rock
passes by on the slope block.
As a result, the vertical vibration is large, and so the
arrival distance of the falling rocks increases after
arrival at the bottom of the slope.
Figure 13. Rock-falls after 15 seconds
Figure 14. Vibration characteristics of the slope under gravity
Figure 10.
Rock-falls after 15 seconds (φ = 20 degrees)
Figure 15. Vibration characteristics of a rock under gravity
Figure 11. Rock-falls after 5 seconds
Figure 12. Rock-falls after 10 seconds
Proceedings of the ISRM International Symposium 3rd ARMS
Figure 16.
Acceleration response of the slope of block (2)
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property of the slope block in two levels.
In future, we will clarify the frequency
characteristic of an input earthquake wave and the
vibration characteristics, and the relation with the
material property of the slope and falling rocks.
The authors examined the basic vibration
characteristics of a slope model of multi-layer
ground, and its applicability, by using the presented
methods. The results must be confirmed with actual
vibration characteristics by earthquake wave
records.
Figure 17. Velocity along horizontal coordinates of block (4)
ACKNOWLEDGMENT
The authors thank Dr. Gen Hua Shi for many
informative discussions.
REFERENCES
Figure 18. Velocity along horizontal coordinates of block (10)
4 CONCLUSIONS
In this study, the authors presented an earthquake
response analysis method and used it to analyze the
slope stability of two kinds of models by DDA. An
external earthquake force was shown to trigger
rock-falls.
In order to get the same response between input
and output acceleration of earthquake record, the
authors employed a large number of virtual mass for
the base block to avoid the influences by the
additional mass of the base block. And to adjust the
base block mass, we can be control the
characteristics of vibrations of rock slope with zero
gravity force. The vibration characteristics of the
falling blocks are governed by its mass and the
gravity forces principally and the secondary, the
frequency characteristics of earthquake accelerations
and the friction angle of the block surfaces are
influenced for the block motions during earthquakes.
The authors also compared the characteristic
behavior of falling rocks with varying friction angle
in the model, and showed that the model can
qualitatively express physical phenomena reasonably
well.
In the case of the two strata model, the velocities
of the falling rocks and arrival distance were larger
than with the single strata model due to the vibration
Proceedings of the ISRM International Symposium 3rd ARMS
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