Rock Strata’s Equal Thickness
Characteristic Research in Bending and
Toppling of Rock Slopes
Liangfu Xie, Echuan Yan, Liantong Gao, Juanjuan He
Faculty of Engineering, China University of Geosciences, Wuhan, China
ABSTRACT
In order to verify the applicability of the assumed condition that the thicknesses of the adjacent
rock strata are equivalent in anti-dip stratified rock slopes, the Zhongliang anti-dip rock slope
which is located in Chongqing, China is selected as an engineering example to verify the
assumed condition with numerical simulation. Firstly, a simplified two-dimensional anti-dip
slope model is built by the discrete element software (UDEC), and invoking the numerical
simulation of the anti-dip slope under different adjacent rock strata thickness ratios (1.0:0.1 ~
1.0:1.0). Then, bending and toppling deformation characteristics of the anti-dip rock slope
under different thickness ratios are comparatively analyzed to obtain the variation of
deformation with the thickness ratios. Finally, the conclusions of the research are further
validated by the engineering example of Zhongliang rock slope. The results show that: (1) The
closer the adjacent rock strata thicknesses are, the more prone is anti-dip slope to bending and
toppling; (2) Only when the adjacent rock strata thickness ratio is located in the range of 1.0:0.8
to 1.0:1.0, the anti-dip slope can be regarded as equal thickness distribution.
KEYWORDS:
Anti-dip slopes; Equal thickness; Bending and toppling; Numerical
simulation
INTRODUCTION
Bending and toppling deformation are the mainly failure mode of anti-dip rock slopes. The
research methods for anti-dip rock slopes include limit equilibrium method, physical model method
and numerical simulation method. Goodman and Bray (1976) first proposed an analysis method
(G-B) that based on the principle of limit equilibrium for anti-dip rock slopes, they pointed out that
the deformation region of the anti-dip rock slope can divided into the following three states:
stability, toppling, sliding. Duncan (1992) gave the basic analytical solution of G-B method. Sun
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Guangzhong (1988) systematically analyzed the deformation and failure mechanisms of the
anti-dip rock slope, and established the bending deformation and failure criterion. Zhang
Zhuoyuan(1993) analyzed the formation conditions and evolution of anti-dip slopes, and
established a rheology criterion of bending deformation. Chen Zuyu and Wang Xiaogang (1996)
improved the GB method by considering the connectivity rate of fracture surface. Huang Runqiu
(1994), Wang Xiaogang (1996), Luo Huayang (2000), Zuo Baocheng (2005) respectively used
physical model methods for anti-dip slope deformation mechanism research. In recent years, with
the rapid development of computer technology, many scholars studied the anti-dip slope mainly by
numerical simulation method. SJÖBERG (1999) used the discrete element software UDEC for
massive anti-dip mechanism researches of deformation and failure. Han Beichuan and Wang Sijing
(1999) studied the mechanical parameters on the impact of anti-dumping slope deformation based
on elastic-viscoplastic finite element model. Xu Peihua (2004) studied the three-dimensional
analysis by FLAC3D. Cai Yue (2008) used two-dimensional discrete element software studied the
effect of anti-dip slope stability factors.
Currently, scholars have achieved some researches for anti-dip slopes, but the results are nearly
all base on the assumption that the thicknesses of adjacent rock strata are equal, and in fact that the
thicknesses of adjacent rock strata have some differences. Therefore, it’s need to verify the
applicability of this assumption, this paper use two-dimensional discrete element soft UDEC for
anti-dip slopes numerical simulation analysis under different thickness ratios of adjacent rock
strata, and relying on an engineering example for further verification.
NUMERICAL SIMULATION ANALYSIS
This paper use two-dimensional discrete element simulation software (UDEC) to simulate the
deformation of anti-dip slopes. Firstly, a simplified anti-dip slope model is established by UDEC,
then comparing the differences of deformation and failure characteristics under various adjacent
rock strata thickness ratios (1.0:0.1 ~ 1.0:1.0) to verify the applicability of the assumptions that the
thicknesses of the adjacent rock strata are equivalent in the anti-dip stratified rock slope.
Simulation model
To save the simulation time, this paper use a simplified anti-dip slope model (Fgure 1), Where
α is the slope angle (70°), θ is the dip (70°), and d1 , d 2 are the adjacent rock strata
thicknesses.
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H=40m
L=60m
θ
α
d2
d1
Figure 1: Simplified model for simulation
Boundary conditions and parameters
In this simulation model (Fgure 1), where the slope surface is free, left and right sides are
horizontal boundary, the bottom is the vertical boundary. This paper does not consider the tectonic
stress, and only considers the gravity stress.
In this simulation model, rock is considered as elastic-plastic material. Mohr-Coulomb strength
criterion are used as failure criterion Physical and mechanical parameters of rock are from
laboratory tests and field-situ tests (Table 1)
Table 1: Physical and mechanics parameters
Rock Mass
Tensile
Rock Strata
Normal
Tangential
stiffness
stiffness
Cohesion
Friction
Modulus
Poisson
Density
σt/MPa
c/MPa
φ(°)
E/MPa
υ
/(Kg*m-3)
/(MPa*m-1)
0.126
0.225
37
4380
0.27
2640
18330
strength
Tensile
Friction
Cohesion
(MPa*m-1)
φ(°)
/MPa
/MPa
880
19
0.18
0
Simulation scenarios
For the study of influence that different thickness ratio ( d1/ d 2 ) of the adjacent rock strata on
bending and toppling deformation, the simulation program firstly fix one group of rock strata
thickness（ d1 ） to be 1.0m, then another group of rock strata thickness ( d 2 ) is successively
selected to ten different thickness values (0.1, 0.2, 0.3, ..., 1.0m). Then UDEC is used for anti-dip
rock slopes simulation under ten different strata thickness ratios（ d1/ d 2 ）.
Deformation and failure analysis
Simulate the various programs under the same number of iterative calculations by UDEC. Due
to space limitations, this paper only lists the horizontal displacement contours of the slope (Fgure
2-11).
strength
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Figures 2 to 11 show that the more similar the adjacent rock thicknesses (the smaller the ratio),
the more the anti-slope prone to bending and toppling; when the difference of the adjacent rock
strata thicknesses is large, the bending and toppling deformation occurs mainly in the front slope;
while the difference gradually decreases, the bending and toppling deformation area gradually
expanded and extended to the top of the slope. When the ratio of d1/ d 2 is in the range of
1.0:0.1~1.0:0.7, the toppling failure occurs mainly in the front and the middle; when the ratio of
d1/ d 2 is in the range of 1.0:0.8 ~ 1.0:1.0, the toppling failure extend to the top.
Figure 2: The horizontal displacement contour
Figure 3: The horizontal displacement contour
under adjacent rock strata thickness ratio of 1.0:0.1
under adjacent rock strata thickness ratio of 1.0:0.2
Figure 4: The horizontal displacement contour
Figure 5: The horizontal displacement contour
under adjacent rock strata thickness ratio of 1.0:0.3
under adjacent rock strata thickness ratio of 1.0:0.4
Figure 6: The horizontal displacement contour
Figure 7: The horizontal displacement contour
under adjacent rock strata thickness ratio of 1.0:0.5
under adjacent rock strata thickness ratio of 1.0:0.6
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Figure 8: The horizontal displacement contour
Figure 9: The horizontal displacement contour
under adjacent rock strata thickness ratio of 1.0:0.7
under adjacent rock strata thickness ratio of 1.0:0.8
Figure 10: The horizontal displacement contour
Figure 11: The horizontal displacement contour
under adjacent rock strata thickness ratio of 1.0:0.9
under adjacent rock strata thickness ratio of 1.0:1.0
On the basis of comparative analysis of the horizontal displacement contour, in order to further
reveal the slope bending and toppling deformation characteristics with variation ratios of the adjacent
rock strata thicknesses. Twenty displacement monitoring points are equidistant arranged on the
surface of the slope. By comparing the value of the monitoring points’ horizontal displacement (Figure
12), we can find that the bending and toppling deformation can be divided into three categories based
on the ratios of the adjacent rock strata thicknesses.
(1) When the ratio of the adjacent rock strata thicknesses located in the range of 1.0:0.8 to 1.0:1.0,
the displacement value of bending and toppling deformation is maximum, and the deformation is
distributed throughout the entire slope surface, which is scattered from the toe to the top of the slope.
(2) When the ratio of the adjacent rock strata thicknesses located in the range of 1.0:0.4 to 1.0:0.7,
the displacement value of bending and toppling deformation is secondary large, and the deformation is
distributed in the anterior and middle portions of the slope, and the toe to 3/4 slope height region is the
main deformation zone.
(3) When the ratio of the adjacent rock strata thicknesses located in the range of 1.0:0.1 to 1.0:0.3,
the displacement value of bending and toppling deformation is minimum, and the deformation is
distributed in the anterior portions of the slope, and toe to 1/2 slope height region is the main
deformation zone.
Figure 12: The horizontal displacement distribution under different adjacent rock strata
thicknesses ratios
ENGINEERING VERIFICATION
Project overview
In this paper, Zhongliang anti-dip rock slope as an engineering example to verify the above
research conclusions. Zhongliang anti-dip rock slope is located in Chongqing, China, and slope
height is about 600m, width is nearly 700m (Figure13).
Figure 13: Panorama of Zhongliang slope
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Actual deformation survey (fig 14) showed that the toppling and bending deformation mainly
occurs in the toe area to the middle zone, and does not extend to the top of the slope.
Figure 14: Bending and toppling phenomenon in the central of Zhongliang slope
Rock strata thickness statistics
Thirty-one randomly adjacent rock strata thicknesses were measured in the adit (fig 15, 16),
statistics result (fig 17) shows that the ratio of the adjacent rock strata thickness are mainly located in
the range of 1.0:0.3 to 1.0:0.8, and its average is 1.0:0.594, so the ratio of the adjacent rock strata
thickness can deemed as 1.0:0.6.
Figure 15: The adit located in the front of Zhongliang slope
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Percentage /%
Figure 16: Significant differences between the adjacent rock strata thicknesses in the adit
thickness ratios of adjacent rocks
Figure 17: Statistics result of the adjacent rock strata thickness ratios
Engineering numerical simulation analysis
The data in Table 1 are used as the physical and mechanical parameters of rock, using the UDEC
software for numerical simulation of slope under the adjacent rock strata thickness ratios of 1.0:0.6
and 1.0:1.0.
When the ratio is selected as 1.0:0.6, the result (fig 18) shows that the toppling and bending
deformation occurs mainly in the toe area to the rear zone, and does not extend to the top of the
slope.
When the ratio is selected as 1.0:1.0, the result (fig 19) shows that the toppling and bending
deformation occurs in the entire slope, and extends to the top of the slope.
Actual deformation survey results show that the numerical results use the ratio of 1.0:0.6 is more
accurate.
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Figure 18: The horizontal displacement contour of the slope under adjacent rock strata
thicknesses ratio of 1.0:0.6
Figure 19: The horizontal displacement contour of the slope under adjacent rock strata
thicknesses ratio of 1.0:1.0
CONCLUSION
Through the numerical simulation of anti-dip rock slopes under different adjacent rock strata
thicknesses ratios and engineering verification, this paper mainly get the following conclusions:
(1) The more similar the adjacent rock strata thicknesses, the slope more prone to bending and
toppling; when the difference of the adjacent strata thicknesses is large, the bending and toppling
deformation occurs mainly in the front slope; while the difference gradually decreases, the bending
and toppling deformation area gradually expanded and extended to the top of the slope.
(2) When the ratio of the adjacent rock strata thickness located in the range of 1.0:0.8 to 1.0:1.0,
the displacement value of bending deformation is maximum; when the ratio of the adjacent rock
strata thickness located in the range of 1.0:0.4 to 1.0:0.7, the displacement value of bending
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deformation is secondary large; when the ratio of the adjacent rock strata thickness located in the
range of 1.0:0.1 to 1.0:0.3, the displacement value of bending deformation is minimum.
(3) Only when the thickness ratio of adjacent rocks is located in the range of 1.0:0.8 to 1.0:1.0,
the anti-dip slope can be regarded as equal thickness distribution.
(4) Comparative analysis of numerical simulation of engineering and Actual deformation survey
results show that the calculated results under the assumption of equal adjacent thickness will cause
some errors.
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