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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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