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Research Article
An Optimization of the Analytical Method for Determining the
Flexural Toppling Failure Plane
Xin Qu
1
and Fangfang Diao
2
1
School of Civil and Architecture Engineering, Anyang Institute of Technology, Anyang 455000, China
2
School of Foreign Languages, Anyang Institute of Technology, Anyang 455000, China
Correspondence should be addressed to Xin Qu; xqu1987@163.com
Received 8 October 2019; Revised 23 December 2019; Accepted 3 January 2020; Published 28 March 2020
Academic Editor: Harry Far
Copyright ©2020 Xin Qu and Fangfang Diao. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
According to the results of the physical model tests, the failure plane of an anaclinal layered rock slope was a linear-type plane at an
angle above the plane normal to the discontinuities, and the failure mode of rock strata was bending tension. However, the shear
failure occurred near the slope toe, the effects of the cohesion of the discontinuities on the stability of the slope, and the
contribution of tangential force to cross-section axial force were neglected in such studies. Moreover, none of the experts had
developed a rigorously theoretical method for determining the angle between the failure plane and the plane normal to the
discontinuities. is paper was initiated for the purpose of solving the problems described above. With the cantilever beam model
and a step-by-step analytical method, an optimization of the analytical method for determining the flexural toppling failure plane
based on the limit equilibrium theory was developed and the corresponding formulations were derived. Based on the present
computational framework, comparisons with other studies were carried out by taking a slate slope in South Anhui in China and a
rock slope facing the Tehran-Chalus Road near the Amir-Kabir Dam Lake in Iran. Furthermore, the sensitivity analyses of the
parameters used in the calculation process of the failure angle of the slate slope in South Anhui in China were performed. e
results demonstrated that the failure plane and the safety factor of the stability obtained with the presented method were credible,
which verified the proposed method. e dip angle of the slope, the dip angle of the rock stratum, and the friction angle of the
discontinuities were the controlling factors for the overall failure of the slate slope in South Anhui in China.
1. Introduction
Toppling failure, being a kind of typical instability mode for
rock slopes, widely exists in domestic and foreign water
conservancy, hydropower, highways, and open-pit mine
slope engineering. e toppling instability of slopes has
caused great harm to people’s lives, property safety, and
engineering construction [1–4]. However, due to the cog-
nition limitations of the public with regard to anaclinal
layered slopes, these slopes had not been able to arouse
people’s attention until anaclinal layered toppling landslides
frequently occurred. Currently, more and more experts are
devoting themselves to the study of anaclinal layered top-
pling slopes. In [5], the authors divided the toppling failures
into three groups, i.e., block toppling, flexural toppling and
block-flexural toppling (block-flexural toppling can be
treated as a recombined form of the above two types).
Numerous significant results had been found for block
toppling, and relatively complete theoretical analysis
methods were developed [6–12], while researchers contin-
ued to be preoccupied with flexural toppling [13–22].
e limit equilibrium method is one of the most common
and effective approaches in studies of flexural toppling failure.
However, before using the limit equilibrium method for
mechanical analyses, the position and shape of the failure
plane of a slope must be determined. us, many experts put
forward various methods to study this problem, and they
carried out physical model tests to verify their methods. In
[16], the authors first proposed a stability analysis method for
slopes and underground openings under various loading
conditions against the flexural toppling failure, derived by
employing the equations of the limit equilibrium and the
Hindawi
Advances in Civil Engineering
Volume 2020, Article ID 5732596, 12 pages
https://doi.org/10.1155/2020/5732596
boundary conditions. ey verified the method by carrying
out base friction tests and found that the failure plane was a
linear-type plane, emanating from the toe of the slope and
perpendicular to the discontinuities, i.e., the angle between
the failure plane and the plane normal to the discontinuities
(called the failure angle in this study) was equal to 0°. In
[13, 23] and [24], the authors improved Aydan and Kawa-
moto’s theory of flexural toppling failure in open excavations
through centrifuge tests and found that the failure angle was
10°. By using the principles of compatibility and the equations
of equilibrium along with the governing equations of elastic
deformation for the beams, the authors of [25, 26] derived
equations for determining the intercolumn forces in rock
masses with the potential for flexural toppling failure. Since
this model did not allow for slippage between layers, this
method might yield very low safety factors for short layers
near the slope toe, which might only be true for small
deflections. In [15], the authors carried out a series of
model tests on a single column and slopes under dynamic
loading and found that the failure angle was in the range of
0°to 15°. In [22], the authors developed a new UDEC
Trigon approach for simulating the flexural toppling fail-
ure. e simulated results showed that the inclination of the
total failure surface was around 13°above the plane normal
to the discontinuities.
e above theoretical investigations greatly enriched the
understanding of the deformation and failure mechanism of
such slopes. e significant achievements can be summarized
as two viewpoints, i.e., the failure plane of an anaclinal layered
rock slope was a linear-type plane at an angle above the plane
normal to the discontinuities, and the failure mode of rock
strata was bending tension. However, these studies still have
some issues needed to be resolved. For instance, the failure of
rock strata was not only the result of bending tension as shear
failure often occurred near the slope toe, which was supported
by field investigations and theoretical analysis. Furthermore,
the effects of the cohesion of the discontinuities on the sta-
bility of anaclinal layered rock slopes against flexural toppling
failure were neglected, which resulted in underestimation of
the stability. In addition, the contribution of tangential force
to cross-section axial force was neglected, which caused the
calculated tensile stress to be overestimated. Last but not least,
the failure angle was obtained only from the laboratory data or
numerical simulation results, and none of the experts had
developed a rigorously theoretical method for determining
the failure angle.
In order to solve the above problems, an optimization
of the analytical method for determining the flexural
toppling failure plane based on the limit equilibrium
theory [27–29] was developed in this study. e basal
failure plane was considered as a plane at which the stress
of the slope arrived at the state of limit equilibrium, i.e.,
the plane, at which the residual sliding force at the toe of
the slope was equal to zero, was the basal failure plane.
With a cantilever beam model and a step-by-step ana-
lytical method, the corresponding formulations were
derived in this study. Furthermore, the controlling factors
for the overall failure of an anaclinal layered rock slope
were obtained after the sensitivity analyses of the
parameters used in the calculation process of the slope
failure angle were performed.
2. Geological Geometrical Model of an Anaclinal
Layered Slope
e geological geometrical model of an anaclinal layered
slope is shown in Figure 1, where His the slope height, βis
the dip angle of the slope, ηis the dip angle of the rock strata,
η
0
is the natural slope angle, αis the dip angle of the plane
normal to the discontinuities, θis the dip angle of the failure
plane, θ
r
(called failure angle) is the angle between the basal
failure plane and the plane normal to the discontinuities, β
0
is the difference between βand α,bis the rock thickness, and
h
i
is the contact height between stratum iand stratum i+ 1.
According to the geometrical conditions, equation (1) can be
obtained as follows:
α�π
2−η,
β0�β−α,
θ�α+θr.
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(1)
3. Searching Failure Plane
3.1. Searching Principle. According to the results of the
physical model tests [13, 15, 16, 22–26], the failure plane
of the failed strata was a linear-type plane at an angle
above the plane normal to the discontinuities. However,
none of the experts had developed a rigorously theoretical
method for determining the failure angle. In this study,
the basal failure plane was considered as a plane at which
the stress of the slope arrived at the state of limit equi-
librium, i.e., the plane, at which the residual sliding force
at the toe of the slope was equal to zero, was the basal
failure plane.
Due to the limitation of the computer’s accuracy, it was
unable to guarantee the existence of the failure angle that
made the residual sliding force at the toe of the slope equal to
zero. An attempt was made to control the error of the failure
angle to be in an acceptable range by taking a small size of
searching step, and thus, the residual sliding force at the toe
of the slope was close enough to zero. e searching di-
rection is shown in Figure 2.
Obviously, the above problem was actually an optimi-
zation problem with the failure angle θ
r
as a variable pa-
rameter, which can be written as follows:
F�min fjθr
fjθr
�max P0, T0
,1≤j≤nn, 0≤θr≤β0.(2)
Here, P0, calculated by the shear failure mode, is the
residual sliding force at the toe of the slope, T
0
, calculated by
the tension failure mode, is the residual sliding force at the
toe of the slope, β
0
is the difference between the dip angle of
the slope and the plane normal to the discontinuities, and nn
2Advances in Civil Engineering
is the searching times. Taking 10
6
as the value of the
searching times, the error of failure angle is less than 10
−4
,
which is in an acceptable range.
Δθr�β0
nn
θr� (j−1)Δθr.
(3)
Here, Δθ
r
is the size of the searching step. According to
the geometrical conditions, the contact height h
i
between
stratum iand stratum i+ 1 can be written as follows:
hi�
ib tan β0−tan θr
,1≤i<ntp,
ib tan β0−tan θr
−ib −H
sin β
cos β0
tan β0+cot β1
, i ≥ntp.
⎧⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(4)
β1�η+θ0.(5)
Here, ntp is the number of the first stratum on the top of
the slope (numbered from the toe to the top of a slope).
After h
i
was obtained through equation (5), the me-
chanical analyses of rock strata can be performed based on
the slope geometry and the limit equilibrium theory [27–29],
which are presented in Sections 3.2 and 3.3.
3.2. Failure Modes. Flexural toppling failure is one of the
main types of toppling failures in anaclinal layered rock
slopes, and its failure mechanism is due to the bending of
layers with or without cross joints. For flexural toppling-type
anaclinal layered slopes, although bending tension failure
occurred in most of the rock strata, bending tension failure
was unlikely to occur in several strata with a small slen-
derness ratio at the toe of the slopes [22, 30]. Loaded by self-
weight and thrust provided by the upper and underlying
strata, the strata with a small slenderness ratio were likely
subjected to shear failure according to actual field investi-
gations [31, 32] and theoretical analysis [33, 34]. In this
study, the failure modes of the rock strata in flexural top-
pling-type anaclinal layered slopes were shear failure and
bending tension failure. e specific failure modes of the
rock strata (tension or shear) depended on the types of stress
(tension or shear) that reached their critical state earlier.
3.2.1. Tension Failure. If tension failure occurs in stratum i,
it should meet the criterion of the maximum tensile stress
theory [35], which can be expressed as follows:
Plane normal to the
discontinuities
e upper part of stratum
i above failure plane
H
Basal failure plane
η0
h
i
b
β
0
α
θr
θ
β
η
Figure 1: Geological geometrical model of an anaclinal layered slope.
Plane normal to
the discontinuities
Basal failure plane
Shear failure zone
Bending fracture failure zone
Searching direction
θ
r
β
0
θ
β
α
η
Figure 2: Searching process.
Advances in Civil Engineering 3
σmax �σt,(6)
σmax �Mib
2I−Ni
b,(7)
where σ
t
is the tensile strength of column i,M
i
is the bending
moment exerted on the center of column i,Iis the second
moment of inertia, and N
i
is the axial force exerted on
column i.
As shown in Figure 3, the bending moment exerted on
column ican be written as follows:
Mi�hi
2wisin α+Tiχhi−1
2Qib−Ti−1χhi−1−1
2Qi−1b, (8)
where
wi�hibc,(9)
hi�hi+hi−1
2,(10)
χ∈[0,1],(11)
I�b3
12.(12)
Here, wiis the weight of the part of stratum iabove the
failure plane, h
一
iis the equivalent height of stratum iabove
the failure plane, cis the unit weight, T
i
is the normal force
provided by the part of stratum i+ 1 above the failure plane,
Q
i
is the tangential force provided by the part of stratum i+ 1
above the failure plane, Ti−1is the normal force provided by
the part of stratum i−1 above the failure plane, Q
i−1
is
the tangential force provided by the part of stratum i−1
above the failure plane, h
i−1
is the contact height of
stratum iand stratum i−1, ϕ
i
is the friction angle of the
discontinuities, and χis the height of the thrust line. As
the derivation and verification of χ= 0.6 obtained in [13]
were extremely rigorous, the value of χwas equal to 0.6 in
this manuscript.
For the calculation of the axial force exerted on column i,
the tangential forces (Q
i
and Q
i−1
), neglected in previous
studies, were reconsidered here, which can be written as
follows:
Ni�wicos α+Qi−Qi−1.(13)
For the calculation of the tangential force (Q
i
and Q
i−1
),
the cohesion of the discontinuities (c
i
), neglected in previous
studies, was reconsidered here, which can be written as
follows:
Qi�Titan ϕi+cihi,(14)
Qi−1�Ti−1tan ϕi+cihi−1.(15)
From equations (6)–(15), the value of Ti−1can be de-
termined as follows:
Ti−1�Tiχhi−2btan ϕi/3
+hiwisin α/2 −b2σt+bwicos α
/6 −2cihib/3 +cihi−1b/3
χhi−1+btan ϕi/3
.(16)
3.2.2. Shear Failure. If shear failure occurs in stratum i, it
should meet the Mohr–Coulomb criterion, which can be
expressed as follows:
τ�σtan ϕ+c. (17)
Here, τis the shear stress, σis the normal stress, ϕis the
friction angle of the rock strata, and cis the cohesion.
Multiplying the bottom length b/cos θrof the stratum on
both sides of equation (18), equation (19) can be obtained as
follows:
τb
cos θr
�σb
cos θrtan ϕ+cb
cos θr
.(18)
Namely,
S�Ntan ϕ+cb
cos θr
.(19)
Here, Sand Nrepresent the tangential force and the
normal force of stratum i, respectively.
As shown in Figure 4, the tangential force and the
normal force on the bottom of stratum iabove the failure
plane can be written as follows:
S�wisin θ+Qisin θr+Picos θr−Qi−1sin θr
−Pi−1cos θr,(20)
N�wicos θ+Qicos θr−Pisin θr−Qi−1cos θr
+Pi−1sin θr.(21)
Qi
Qi–1
Ti–1
Ti
hi
hi–1
χhi
χhi–1
Wi
θ
γ
θ
α
Figure 3: Force analysis chart of stratum ifor tension failure.
4Advances in Civil Engineering
For the calculation of the tangential force (Q
i
and Q
i−1
),
the cohesion of the discontinuities (c
i
), neglected in previous
studies, was reconsidered in this study, which can be written
as follows:
Qi�Pitan ϕi+cihi,(22)
Qi−1�Pi−1tan ϕi+cihi−1.(23)
Here, Piis the normal force provided by the part of
stratum i+ 1 above the failure plane and Pi−1is the normal
force provided by the part of stratum i−1 above the failure
plane.
From equations (17)–(23), the value of Pi−1can be de-
termined as follows:
Pi−1�Pi−cos θ(tan ϕ−tan θ)wi+cos θrtan ϕ−tan θr
cihi−hi−1
+cb/cos θr
cos θr1+tan ϕitan θr
+tan ϕcos θrtan θr−tan ϕi
.(24)
3.2.3. Criteria for Determining Failure Modes. If Pi−1>Ti−1,
it indicated that the thrust causing the shear failure in
stratum iwas less than that causing bending fracture failure.
In this case, we believed that stratum iwould be subjected to
shear failure rather than bending fracture failure, and vice
versa.
3.3. Amount of Failed Strata. During the calculation process,
if max (Pi−1, Ti−1)<0, failure would not occur for stratum i,
and the interaction force between stratum iand stratum i−1
would be zero. If max (Pi−1, Ti−1)>0, it indicated that failure
occurred for stratum i. Furthermore, Pn�Tn�0, which
meant no force provided by the upper stratum was exerted on
the last stratum. After the failure modes of the rock strata were
determined, the residual sliding force at the toe of the slope,
max (P
0
,T
0
), can be derived with the step-by-step method.
en, the failure angle θ
r
was changed constantly and the
above process was repeated until the residual sliding force at
the toe of the slope was less than zero for the first time.
Comparing the absolute value of the residual sliding force of
the last step and that of the penultimate step, the smaller
value was the desired result, recorded as F. e failure angle
corresponding to Fwas the final failure angle. In this case,
the amount of total failed strata, the amount of shear failed
strata and the amount of bending fracture strata, recorded as
nm,ns, and nt, respectively, can be obtained.
4. Numerical Example
4.1. Slate Slope in South Anhui. A slate slope in South Anhui
in China [31] was taken as an engineering example. Table 1
and Figure 5 show the parameters and the geometry profile
of the slope [31], respectively.
Using the proposed method developed in this paper, the
final results were presented as follows.
Toppling failure occurred in the slate slope in South
Anhui in China, which was consistent with field observa-
tions and the results reported in [30, 31]. e angle between
the failure plane and plane normal to discontinuities, θ
r
, was
7.9766°, and the amount of total failed strata was 38. Shear
failure occurred in strata 1–3, while bending tension failure
occurred in the remaining failed strata. e safety factor of
stability was 0.7573, and the residual sliding force was
26.2239 MN. e failure angle obtained in [30] was 13°, and
the amount of failed strata was 37. Shear failure occurred in
strata 1–4, and bending tension failure occurred in the
remaining failed strata. e safety factor of stability calcu-
lated with Aydan and Kawamoto’s method [16] was 0.9343,
and the residual sliding force was 4.6781 MN (see Table 2).
Qi–1
Pi–1
Pi
Wi
Qi
θ
γ
θ
Figure 4: Force analysis chart of stratum ifor shear failure.
Advances in Civil Engineering 5
e safety factor of stability calculated with Majdi and
Amini’s method [26] was 0.5430. e result comparisons of
the failure planes are shown in Figure 6, and the convergence
curves of the safety factor of the stability are shown in
Figure 7.
According to the data presented in Section 4.1, it was
found that the safety factor of the stability of the slate slope in
South Anhui in China obtained with the proposed method
was smaller than that calculated with Aydan and Kawa-
moto’s method [16]. For Aydan and Kawamoto’s method
[16], the plane normal to the discontinuities was considered
to be the failure plane. However, the overall failure of the
slate slope in South Anhui in China occurred before the
damage developed to the plane normal to the
Table 1: Calculation parameters of rock mass of the slate slope in South Anhui in China [31].
Parameters Values Units Parameters Values Units Parameters Values Units
χ0.6 η63 °c0.4 MPa
b4 m η
0
0°c
i
0.01 MPa
H100 m c27 kN m
−3
σ
t
1.5 MPa
n40 ϕ
i
18 °
β55 °ϕ45 °
Failure plane
Tensioned
crack
Figure 5: Schematic diagram of a slate slope in South Anhui in China [31].
Table 2: Result comparisons of the slate slope in South Anhui in China obtained using different methods.
Slate slope θ
r
(°) ns nt nm F(MN) F
S
Aydan and Kawamoto [16] 0 0 40 40 4.6781 0.9343
Zheng et al. [30] 13 2 35 37 — —
Majdi and Amini [26] — — — — — 0.5430
Presented method 7.9766 3 35 38 26.2239 0.7573
Note. “—” indicates that no specific result is given.
0 50 100 150
0
20
40
60
80
100
e presented method
Zheng et al. [30]
Aydan and Kawamoto [16]
Slope height (m)
Slope width (m)
Figure 6: Slope failure planes of the slate slope in South Anhui in China obtained using different methods.
6Advances in Civil Engineering
discontinuities. erefore, the safety factor of stability of the
slate slope in South Anhui in China calculated with Aydan
and Kawamoto’s method [16] was overestimated. As the
effects of the cohesion of the discontinuities on the stability
of the slate slope in South Anhui in China were neglected for
the method developed in [30], the failure plane obtained in
[30] was above the actual failure plane. ereupon, the
failure plane and the safety factor of the stability of the slate
slope in South Anhui in China obtained with the presented
method were credible.
4.2. Rock Slope Facing the Tehran-Chalus Road. A rock slope
facing the Tehran-Chalus Road near the Amir-Kabir Dam
Lake in Iran [36] was taken as another example. Table 3 and
Figure 8 show the parameters and the geometry profile of the
slope [36], respectively.
Using the proposed method developed in this paper, the
final results were presented as follows.
e rock slope facing the Tehran-Chalus Road near the
Amir-Kabir Dam Lake in Iran was stable, which was con-
sistent with field observations and the results reported in
[36]. e safety factor of stability obtained with the proposed
method was 3.1666, and the residual sliding force was
−6.1078 MN. e safety factor of the stability calculated with
Majdi and Amini’s method [26] was 3.5990. e safety factor
of the stability calculated with Aydan and Kawamoto’s
method [16] was 3.1082, and the residual sliding force was
−7.3278 MN. e safety factor of the stability obtained in
[36] was 2.6037, and the residual sliding force was
−3.1683 MN. e result comparisons of the safety factor of
the stability and the residual sliding force are shown in
Table 4, and the convergence curves of the safety factor of the
stability are shown in Figure 9.
According to the data presented in Section 4.2, it was found
that the safety factor of the stability of the rock slope facing the
Tehran-Chalus Road, obtained with the proposed method, was
larger than that, obtained in [36]. In [36], the authors assumed
that the joints completely cut some of the rock strata and no
tensile stress existed in these rock strata, which was relatively
rare in the actual slope. erefore, the safety factor of stability of
the rock slope facing the Tehran-Chalus Road obtained in [36]
was underestimated. On the one hand, the effects of the co-
hesion of the discontinuities and the cohesion of rock strata on
the stability of a slope were neglected for Aydan and Kawa-
moto’s method [16], resulting in the underestimation of the
safety factor of stability of the slope. On the other hand, the
potentially shear failure of stratum 1 at the toe of the slope was
neglected for Aydan and Kawamoto’s method [16], resulting in
the overestimation of the safety factor of stability of the slope.
e safety factor of stability of the rock slope facing the Tehran-
Chalus Road obtained with Aydan and Kawamoto’s method
was almost equal to that obtained with the presented method
due to the combined effects of the two aspects. For the Majdi
and Amini’s method [26], the effects of the cohesion of rock
strata, the cohesion of the discontinuities, the friction angle of
the rock strata, and the friction angle of the discontinuities on
the stability of anaclinal layered rock slopes against flexural
toppling failure were neglected, leading to a decrease of the
accuracy of the safety factor of the stability. ereupon, the
safety factor of the stability of the rock slope facing the Tehran-
Chalus Road obtained with the presented method was credible.
5. Parametric Sensibility Analyses for the
Failure Angle
e failure angle θ
r
is an important indicator for measuring
the degree of instability of anaclinal layered slopes. e
position of the failure plane becomes higher as the failure
angle θ
r
increases, which indicates that it takes less time to
arrive at the state of limit equilibrium, and thus, the slope is
more unstable. During the computational process, many
parameters may influence the failure angle θ
r
. In this section,
the parametric sensibility analyses for the failure angle θ
r
are
discussed in detail. All these parameters are classified into
two groups, i.e., the geometrical parameters of a slope and
the mechanical parameters of rock mass. e geometrical
parameters of a slope contain the height of the thrust line χ,
the rock thickness b, the slope height H, the dip angle of the
slope β, the dip angle of the rock stratum η, and the natural
slope angle η
0
. e mechanical parameters of the rock mass
contain the tensile strength σ
t
, the cohesion of rock strata c,
the cohesion of the discontinuities c
i
, the friction angle of the
rock strata ϕ, the friction angle of the discontinuities ϕ
i
, and
the unit weight c. When the value of one parameter changed,
the others remained the same as in Table 1.
5.1. Sensitivity Analyses of Geometrical Parameters of the Slate
Slope in South Anhui. e value of one geometrical parameter
of the slate slope in South Anhui was changed constantly, and
the corresponding failure angles calculated through the above
process are listed in Tables 5 and 6. e failure angles changed
with the changes of the geometrical parameters of the slope,
and the changing laws are shown in Figure 10.
From the data in Tables 5 and 6 and Figure 10, some
conclusions can be drawn as follows:
(1) e relatively sensitive parameters for the failure
angle θ
r
in the geometrical parameters of the slate
slope in South Anhui were the dip angle of the slope
βand the dip angle of the rock stratum η, which was
0.76 0.78 0.81 0.83 0.85 0.88 0.90 0.93 0.95 0.98 1.00
0.00
2.63
5.25
7.87
10.49
13.11
15.74
18.36
20.98
23.60
26.22
Safety factor
e presented method
Aydan and Kawamoto [16]
Residual sliding force (MN)
Figure 7: Convergence curves of the safety factor of stability of the
slate slope in South Anhui in China obtained using different methods.
Advances in Civil Engineering 7
Table 3: Calculation parameters of rock mass of rock slope facing the Tehran-Chalus Road [36].
Parameters Values Units Parameters Values Units Parameters Values Units
χ0.6 η47 °c1.11 MPa
b2.31 m η
0
29 °c
i
0.1 MPa
H20.45 m c26.5 kN m
−3
σ
t
5.5 MPa
n16 ϕ
i
30 °
β85.3 °ϕ45 °
Figure 8: Rock slope facing Tehran-Chalus Road near the Amir-Kabir Dam Lake in Iran.
Table 4: Result comparisons of rock slope facing the Tehran-Chalus Road obtained using different methods.
Tehran-Chalus Aydan and Kawamoto [16] Majdi and Amini [26] Amini et al. [36] Presented method
F
S
3.1082 3.5990 2.6037 3.1666
F(MN) −7.3278 — −3.1683 −6.1078
Note. “—” indicates that no specific result is given.
1.00 1.22 1.43 1.65 1.87 2.08 2.30 2.52 2.73 2.95 3.17
–7.33
–6.60
–5.86
–5.13
–4.40
–3.66
–2.93
–2.20
–1.47
–0.73
–0.00
Safety factor
e presented method
Aydan and Kawamoto [16]
Residual sliding force (MN)
Figure 9: Convergence curves of the safety factor of stability of the rock slope facing the Tehran-Chalus Road obtained using different
methods.
Table 5: Sensibility analyses of the geometrical parameters of the slate slope in South Anhui.
Parameters Values θ
r
(°) Parameters Values θ
r
(°) Parameters Values θ
r
(°)
χ
0.5 6.4058
b(m)
2 10.3104
H(m)
80 5.2464
0.6 7.9766 3 8.7094 90 6.7824
0.7 9.2014 4 7.9766 100 7.9766
0.8 10.1287 5 6.9476 110 9.0023
0.9 10.9281 6 5.8870 120 9.7852
8Advances in Civil Engineering
consistent with the result obtained in [30]. e effect
degree of these two sensitive parameters for the
failure angle θ
r
was β>η. For the anaclinal layered
rock slopes, their failure was mainly associated with
the upper parts of the rock strata above the plane
normal to the discontinuities. When the dip angle of
the slope or the dip angle of the rock stratum became
larger, the possible failure zones of the slope in-
creased, and thus, the slope was more unstable.
(2) When β= 45°or η= 55°, the slope was stable. is
result further verified Goodman and Bray’s view-
point [5] that only if inequality (25) was satisfied, the
overall failure of an anaclinal layered rock slope
would occur.
β−90°−η
>ϕi.(25)
(3) e thickness bis an important indicator for mea-
suring the ability to resist the bending deformation
[37]. However, the thickness bwas not the most
sensitive parameter for the failure angle θ
r
in the
geometrical parameters of the slate slope in South
Anhui according to the results obtained in this study,
i.e., it was not the controlling factor for the overall
failure of the slate slope, which was consistent with
the opinion of [30].
(4) When the geometrical parameters of the slope, ex-
cept for the rock thickness b, became larger, the
failure angle θ
r
increased, which indicated that the
slope was more unstable.
5.2. Sensitivity Analyses of Mechanical Parameters of Rock
Mass of the Slate Slope in South Anhui. e value of one
mechanical parameter of the rock mass of the slate slope in
South Anhui was changed constantly, and the corresponding
failure angles calculated through the above process are listed
in Tables 7 and 8. e failure angles changed with the
changes of the mechanical parameter of the rock mass, and
the changing laws are shown in Figure 11.
From the data in Tables 7 and 8 and Figure 11, some
conclusions can be drawn as follows:
Table 6: Sensibility analyses of the geometrical parameters of the slate slope in South Anhui.
Parameters Values θ
r
(°) Parameters Values θ
r
(°) Parameters Values θ
r
(°)
η
0
(°)
0 7.9766
β(°)
45 —
η(°)
55 —
5 8.1398 50 2.5753 60 4.9877
10 8.3440 55 7.9766 63 7.9766
15 8.5585 60 13.6030 65 9.9294
20 8.7808 65 19.4932 70 14.5796
Note. “—” indicates that the slope is stable.
0.5 0.6 0.7 0.8 0.9
6.4058
7.5364
8.6669
9.7975
10.9281
χ
θr (°)
(a)
2 3 4 5 6
5.887
6.9928
8.0987
9.2045
10.3104
θr (°)
b (m)
(b)
80 90 100 110 120
5.2464
6.3811
7.5158
8.6505
9.7852
θr (°)
H (m)
(c)
0 5 10 15 20
7.9766
8.1776
8.3787
8.5797
8.7808
θr (°)
η0 (°)
(d)
60 63 65 68 70
4.9877
7.3857
9.7836
12.1816
14.5796
θr (°)
η (°)
(e)
50 53.75 57.5 61.25 65
2.5753
6.8048
11.0343
15.2637
19.4932
θr (°)
β (°)
(f)
Figure 10: Sensitivity analyses of geometrical parameters of the slate slope in South Anhui.
Advances in Civil Engineering 9
(1) e most sensitive parameter for the failure angle θ
r
in
the mechanical parameters of the rock mass of the slate
slope in South Anhui was the friction angle of the
discontinuities, ϕ
i
, which verified Goodman and Bray’s
viewpoint [5] that the intense interlayer movement was
the precondition for the toppling failure.
(2) e tensile strength σ
t
is an important indicator for
measuring the ability to resist the bending tension
failure [38]. e cohesion cand the friction angle of
the rock strata ϕare the major parameters of shear
failure [39]. However, these three parameters were
proven not to be the most sensitive parameters for
the failure angle θ
r
according to the results obtained
in this study, i.e., they were not the controlling
factors for the overall failure of the slate slope, which
was consistent with the opinion of [40].
(3) When the mechanical parameters of the rock mass,
except for the unit weight c, became larger, the
failure angle θ
r
became smaller, which indicated that
the slope was more stable.
6. Discussion
Based on the above analyses, it was clear that the dip angle of
the slope, β, the dip angle of the rock stratum, η, and the
Table 7: Sensibility analyses of the mechanical parameters of the rock mass of the slate slope in South Anhui.
Parameters Values θ
r
(°) Parameters Values θ
r
(°) Parameters Values θ
r
(°)
c
i
(MPa)
0 8.1830
σ
t
(MPa)
1.0 9.1689
c(MPa)
0.2 10.3858
0.01 7.9766 1.2 8.6778 0.3 8.9295
0.02 7.7669 1.5 7.9766 0.4 7.9766
0.03 7.5544 1.8 7.3296 0.5 7.2976
0.04 7.3382 2.0 6.9286 0.6 6.7945
Table 8: Sensibility analyses of the mechanical parameters of the rock mass of the slate slope in South Anhui.
Parameters Values θ
r
(°) Parameters Values θ
r
(°) Parameters Values θ
r
(°)
ϕ(°)
30 8.5268
ϕ
i
(°)
10 13.1107
c(kN·m
−3
)
23 6.9042
35 8.3731 15 10.0839 25 7.4735
40 8.1866 18 7.9766 27 7.9766
45 7.9766 20 6.4187 29 8.4535
50 7.7720 25 1.5086 31 8.9040
10 14 18 21 25
1.5086
4.4091
7.3096
10.2102
13.1107
θr (°)
ϕi (°)
(a)
30 35 40 45 50
7.772
7.9607
8.1494
8.3381
8.5268
θr (°)
ϕ (°)
(b)
23 25 27 29 31
6.9042
7.4042
7.9041
8.404
8.904
θr (°)
γ (kN·m–3)
(c)
0.2 0.3 0.4 0.5 0.6
6.7945
7.6923
8.5901
9.488
10.3858
θr (°)
c (MPa)
(d)
0 0.01 0.02 0.03 0.04
7.3382
7.5494
7.7606
7.9718
8.183
θr (°)
ci (MPa)
(e)
1 1.3 1.5 1.8 2
6.9286
7.4887
8.0488
8.6088
9.1689
θr (°)
σt (MPa)
(f)
Figure 11: Sensitivity analyses of the mechanical parameters of the rock mass of the slate slope in South Anhui.
10 Advances in Civil Engineering
friction angles of the discontinuities, ϕ
i
, were the most
sensitive parameters for the failure angle θ
r
, which indicated
that these three parameters were the controlling factors for
the overall failure of the slate slope in South Anhui in China.
e effect degree of these three controlling factors for the
overall failure of the slate slope was β>η>ϕ
i
.
As we all know, the failure of a slope depends on the
stress distribution and the strength of the rock mass. e
stress distribution of a slope is mainly affected by the
geometrical morphology of the slope, and the strength of the
rock mass is mainly controlled by the characteristics of the
rock mass discontinuities. For an anaclinal layered slope, the
dip angle of the slope, β, and the dip angle of the stratum, η,
are the two main factors of the geometrical morphology. e
friction angle of the discontinuities, ϕ
i
, is a significant pa-
rameter for measuring the ability to resist movement be-
tween layers. Hence, these three parameters are the
controlling factors for the overall failure of the slate slope in
South Anhui in China.
7. Conclusions
As the shear failure occurred near the slope toe, the effects of
the cohesion of the discontinuities on the stability of ana-
clinal layered rock slopes against flexural toppling failure
and the contribution of tangential force to cross-section axial
force were neglected in previous studies, and none of the
experts had developed a rigorously theoretical method for
determining the angle between the failure plane and the
plane normal to the discontinuities for a slope, an optimi-
zation of the analytical method for determining the flexural
toppling failure plane based on the limit equilibrium theory
was developed and the corresponding formulations were
derived in this study. After that, the effectiveness of the
proposed method was verified by taking a slate slope in
South Anhui in China and a rock slope facing the Tehran-
Chalus Road near the Amir-Kabir Dam Lake in Iran, and the
sensitive analyses of the parameters used in the calculation
process of the slope failure angle were performed as well.
Eventually, some conclusions can be drawn as follows.
e proposed method, developed based on the limit
equilibrium theory, is feasible to determine the flexural top-
pling failure plane, which was verified by the case history study.
If the geometrical parameters of a slope, except for the
rock thickness, increase, the slope will be more unstable. If the
mechanical parameters of the rock mass, except for the unit
weight, increase, the slope will be more stable. e dip angle of
the slope, the dip angle of the rock stratum, and the friction
angle of the discontinuities are the controlling factors for the
overall failure of the slate slope in South Anhui in China.
It should be noted that the results obtained in this study
only applied to the ultimate failure of anaclinal layered
slopes and can be used for preliminary evaluation of the
stability.
Data Availability
All data generated or analyzed during this study are included
in this article.
Conflicts of Interest
e authors declare that there are no conflicts of interest
regarding the publication of this paper.
Acknowledgments
is study has been financially supported by the Doctoral
Scientific Research Fund of the Anyang Institute of Tech-
nology (Grant no. BSJ2018009). e authors would like to
thank LetPub (http://www.letpub.com) for providing lin-
guistic assistance during the preparation of this manuscript.
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12 Advances in Civil Engineering
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