Efficient 3-D reliability analysis of the 530m high

Engineering Geology 195 (2015) 269–281
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Engineering Geology
journal homepage: www.elsevier.com/locate/enggeo
Efficient 3-D reliability analysis of the 530 m high abutment slope at
Jinping I Hydropower Station during construction
Dian-Qing Li ⁎,a, Shui-Hua Jiang a,b, Zi-Jun Cao a, Chuang-Bing Zhou b, Xue-You Li c, Li-Min Zhang c
a
State Key Laboratory of Water Resources and Hydropower Engineering Science, Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan University, 8
Donghu South Road, Wuhan 430072, PR China
b
School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, PR China
c
Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history:
Received 4 January 2015
Received in revised form 10 June 2015
Accepted 14 June 2015
Available online 17 June 2015
Keywords:
Jinping I Hydropower Station
Rock slope
Reliability analysis
Slope stabilization
Shear strength reduction
Response surface
a b s t r a c t
Reliability-based slope stability analysis and design show promise, but few attempts have been made to study the
3-D reliability of high rock slopes. This paper aims to investigate the 3-D reliability of practical rock slopes and to
explore the effects of reinforcement measures on slope reliability. A non-intrusive stochastic finite difference
method is adopted to perform 3-D reliability analysis of high rock slopes, in which response surface functions
are adopted to establish the explicit functions between the factors of safety for multiple slope failure modes
and the significant random variables. The left abutment slope at Jinping I Hydropower Station in China is taken
as a case study. A parametric sensitivity analysis is carried out to identify the significant uncertain parameters.
The effects of two reinforcement measures, namely pre-stressed cables and shear-resistant concrete plugs, on
the reliability of the rock slope during construction are highlighted. The results indicate that the 3-D reliability
of complex rock slopes can be efficiently evaluated using a non-intrusive stochastic finite difference method. It
provides an effective tool for solving 3-D reliability problems of complex rock slopes in practice. The combined
support system with pre-stressed cables and three shear-resistant concrete plugs adopted in the Jinping I left
abutment slope can effectively restrain the slope deformation and ensure the slope stability during construction.
The disturbances induced by the excavation blasting during construction have a significant influence on the
engineered slope reliability.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
It is widely recognized that enormous uncertainties are present in
rock slope stability analysis owing to inadequate site characterization,
inherent spatial variability, and measurement errors in geological and
geotechnical parameters (Phoon and Kulhawy, 1999; Duncan, 2000;
Baecher and Christian, 2003; Zhu and Zhang, 2013; Li et al., 2014,
2015a; Jiang et al., 2014a, 2015). High rock slopes are traditionally
designed based on a global factor of safety which cannot account for
these uncertainties, and the reliability levels of such important slopes
are not known. Recently the design specifications for hydraulic structure
design are in the transition from traditional design to reliability-based
design (Duncan, 2000; Li et al., 2015b; Wang et al., 2015; Tang et al.,
2015). The reliability-based approach, allowing the systematic and
quantitative treatment of these uncertainties, has received an increasing
attention in rock slope engineering. The reliability levels of such high
rock slopes can be calibrated accordingly.
⁎ Corresponding author at: State Key Laboratory of Water Resources and Hydropower
Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, PR China.
E-mail address: [email protected] (D.-Q. Li).
http://dx.doi.org/10.1016/j.enggeo.2015.06.007
0013-7952/© 2015 Elsevier B.V. All rights reserved.
In the literature, many investigators studied the reliability of rock
slopes. For example, Low (2007) developed a practical EXCEL-based
First Order Reliability Method (FORM) procedure to analyze the reliability
of Sau Mau Ping slope. Park et al. (2012) adopted the point estimate
method to investigate the reliability of a rock slope in Gunwi-gun,
Korea. Rathod and Rao (2012) performed reliability analyses for rock
slope stability in the Subansiri Lower hydroelectric project. Lee et al.
(2011) and Tang et al. (2012) proposed a knowledge-based clustered
partitioning approach for the reliability analysis of rock slopes. Shen
and Abbas (2013) developed a random set distinct element method
for the reliability analysis of rock slopes. Vatanpour et al. (2014) quantified the influence of uncertainties of effective geotechnical parameters
on rock slope reliability in north-east Iran. Wang et al. (2015) applied a
reliability-based Robust Geotechnical Design (RGD) methodology to
the design of rock bolts for slope stabilization. Zheng et al. (2015)
developed a probabilistic block theory analysis procedure for the stability
analysis of a rock slope at a hydropower station in China.
The aforementioned studies on the reliability analyses of rock slopes
mainly focus on 2-D plane failure problems (Li et al., 2011; Rathod
and Rao, 2012; Tang et al., 2012; Wang et al., 2015) or wedge failure
problems (Low, 2007; Li et al., 2009; Lee et al., 2011; Park et al., 2012;
Vatanpour et al., 2014). Few attempts have been made to evaluate the
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D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
3-D reliability of practical rock slopes, especially for a rock slope with a
large height (e.g., higher than 500 m). It is well known that a 2-D plane
model cannot consider the anti-shear effect of both side sliding surfaces
or the joint reinforced effect of a stereoscopic reinforced system in a
practical rock slope. The height of slopes with wedge failure modes
studied in the literature is really small (around 20 m) in relation to
that of practical rock slopes. Thus, it is of great significance to perform
3-D reliability analysis of practical rock slopes which could be more
accurate and reasonable in slope safety evaluation. Additionally,
deterministic analyses of 3-D slope stability could be very complex and
time-consuming, because the factor of safety cannot be expressed as an
explicit function of the input parameters even though a limit equilibrium
method is utilized to calculate the factor of safety. The resulting
performance function for slope reliability analysis is an implicit function.
Theoretically, the direct Monte-Carlo Simulation (MCS) can be used for
such a purpose, but it could be computationally prohibitive for 3-D
slope reliability analysis. Therefore, an efficient reliability method should
be developed for the 3-D reliability analysis of complex rock slopes.
On the other hand, the reinforcement measures such as pre-stressed
cables, anchor bolts, and shear-resistant concrete plugs have been widely
used to improve the rock slope stability in hydroelectricity engineering
projects such as Jinping I and Dagangshan Hydropower Stations
(Song et al., 2010; Zheng et al., 2012; Qi et al., 2013). However,
some imperfections in installation, corrosion protection, workmanship,
human errors, and inadequate support may occur in engineering practice,
which may induce the failure of anchored rock slopes (Jiang et al., 2014b).
For instance, the sudden landslides of the left bank slope at Manwan Hydropower Station in 1989 and the failure of the right sea-route outlet
slope at Longtan Hydropower Station in 2002 were mainly caused by inadequate supports of the slope after the excavation blasting at the slope
toe (Tang and Gao, 2008). A major reason for the 2010 slope failure on
Freeway No. 3 in Taiwan was the failure of the anchor system induced
by corrosion (Lee et al., 2013; Wang et al., 2013). The effect of the reinforced system during the slope excavation on the reliability of practical
3-D slopes has not been explored in the literature.
The objective of this study is to investigate the 3-D reliability of
practical rock slopes and to explore the effects of two reinforcement
measures, namely the pre-stressed cables and shear-resistant concrete
plugs, on the deformation, stability and reliability of engineered rock
slopes. First, a parametric sensitivity study is performed to identify the
important uncertain parameters that have a significant effect on slope
stability. Then, a non-intrusive stochastic finite difference method
(NISFDM) is presented and implemented step by step. Thereafter,
deformation analysis and deterministic stability analysis of the 530 m
high left abutment slope at Jinping I Hydropower Station during
construction using the strength reduction method are carried out to
validate the numerical model. Finally, the reliability of the practical
rock slope is evaluated using the NISFDM. The disturbance effects
induced by the excavation blasting on the slope reliability are also
highlighted.
2. A non-intrusive stochastic finite difference method
It is widely accepted that the evaluation of 3-D reliability of rock
slopes could be time consuming. To efficiently evaluate the reliability
of complex 3-D rock slopes, a non-intrusive stochastic finite difference
method (NISFDM) is adopted for this purpose.
2.1. Parametric sensitivity analysis
Before applying the NISFDM, a parametric sensitivity analysis is
carried out to reduce the number of basic random variables involved
in slope reliability analysis because the computational cost for deterministic analyses of 3-D slope stability could be very large. For example, more
than 30 h are required to perform each deterministic finite difference
analysis of the reinforced slope stability studied in this paper on a desktop
computer with 4 GB RAM and four Intel Core i3 CPU clocked at 3.3 GHz.
Through the parametric sensitivity analysis, the important random
variables significantly affecting the slope stability can be identified
and the insignificant random variables are treated as deterministic
quantities. In this way, the computational effort of deterministic slope
stability analyses in reliability analysis can be reduced substantially. A
simple parametric sensitivity analysis method proposed by Shen and
Abbas (2013) is used. The sensitivity ratio of the ith random variable
Xi is given by
ηSR;i
h i.
f X U −f X L
f ðX Þ
i
i
.
¼
X
X Ui −X Li
ð1Þ
where i = 1, 2, ⋯, n, and n is the number of random variables; f (X) is the
output response of a rock slope such as stress or strain, displacement, and
factor of safety; X is the vector of the mean values; X iU and X iL are the
vectors of input samples at the mean value plus and minus two standard
deviations for the ith random variable, respectively; f (XiU) or f (XiL) is the
slope output response when the value of the ith random variable is taken
as XiU or XiL while the rest of the random variables remain at their mean
values. Then, the relative sensitivity α (Xi) of the ith random variable is
obtained as
α ðX i Þ ¼
ηSR;i
:
n
X
ηSR;i
ð2Þ
i¼1
The deterministic slope stability analyses will be conducted once for
the mean value and 2n times again. A total of (2n + 1) deterministic finite
difference analyses are required to perform the parametric sensitivity
analysis for the n random variables.
2.2. Construction of explicit performance function
Having identified the significant random variables, the next step
is to construct explicit functions between the factors of safety and
the significant random variables. In most cases, the factor of safety
of a 3-D rock slope cannot be expressed as an explicit function of the
basic random variables even though a limit equilibrium method is utilized
to calculate the factor of safety. The response surface function (RSF) based
on a quadratic polynomial chaos expansion is often used to explicitly
express the factor of safety in terms of the basic random variables
(Xu and Low, 2006; Zhang et al., 2011). As reported in the literature
(Bucher and Bourgund, 1990; Gomes and Awruch, 2004), the accuracy of
reliability results estimated from the response surface method is often
improved by increasing the number of terms (including cross terms) of
the quadratic polynomials, but a greater number of performance function
evaluations are needed with respect to the quadratic polynomials without
cross terms. Thus, to significantly improve the computational efficiency of
NISFDM, a single or multiple RSFs based on a quadratic polynomial chaos
expansion without cross terms are adopted to establish the explicit
functions between the factors of safety and the significant random
variables (e.g., Xu and Low, 2006; Zhang et al., 2011; Li et al., 2015c),
FS j ðX Þ ¼ a1; j þ
n
X
i¼1
bi; j X i þ
n
X
ci; j X 2i
ð3Þ
i¼1
where FSj(X) is the factor of safety for the jth failure mode, j = 1, 2, ⋯,
Ns, in which Ns is the number of the slope failure modes; X =
(X1, …, Xi, …, Xn)T is the vector of input random variables in the physical
space; and aj = (a1, j, b1, j, …, bn, j, c1, j, …, cn, j)T is the vector of (2n + 1)
unknown coefficients.
One key step of the NISFDM is the determination of the unknown
coefficients of the polynomial chaos expansion in Eq. (3) (Jiang et al.,
2014a, 2015). A sample design method with (2n + 1) combinations
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
where
min
j¼1;2;⋯;Ns
271
FS j ðÞ is the minimal factor of safety associated with
multiple (i.e., Ns) failure modes. Based on Eq. (4), a direct MCS with a
total of NMCS samples is performed to calculate the probability of slope
failure. In this case the evaluation of the factor of safety only involves
the evaluation of the algebraic expression in Eq. (3). Such a process
does not require deterministic slope stability analyses again because
the RSFs between the factors of safety and the basic random variables
have been constructed. In this way, the computational cost for each
random sample in the MCS is reduced substantially.
2.3. Implementation procedure
To facilitate the understanding of the NISFDM, details of each step of
the implementation procedure are summarized as below:
Fig. 1. The Jinping I left abutment slope after full excavation and support.
proposed by Bucher and Bourgund (1990) is adopted to determine
the unknown coefficients in Eq. (3). The factor of safety for each failure
mode is first evaluated at the following (2n + 1) sampling points:
n
o n
o n
o
μ X 1 ; μ X 2 ; ⋯; μ X n , μ X 1 kσ X 1 ; μ X 2 ; ⋯; μ X n , μ X 1 ; μ X 2 kσ X 2 ; ⋯; μ X n ,
n
o
⋯, and μ X 1 ; μ X 2 ; ⋯; μ X n kσ X n , where k is a sampling factor for
generating the sampling points; μ X i and σ X i are the mean and standard
deviation of the ith random variable Xi, respectively. In this way, a system
of linear algebraic equations can be established at the selected sampling
points in terms of the unknown coefficients, a ¼ ða1 ; a2 ; …; aNs ÞT . Then,
a regression based approach (e.g., Li et al., 2011) is used to compute the
unknown coefficients a. After that, the quadratic RSFs are obtained and
taken as the surrogate model between the factors of safety and the basic
random variables.
In engineering practice, blasting is often used for the excavation of
rock masses (e.g., Song et al., 2010; Zhou et al., 2011; Li et al., 2012).
The disturbance induced by rock blasting is significant during the slope
construction process especially for the high rock slopes (Zhou et al.,
2011). However, the disturbance effect on the overall slope stability is
often not directly incorporated in the numerical model, which will lead
to an unconservative slope safety assessment. As pointed out by Song
et al. (2010), the disturbance effect of the left abutment slope at Jinping
I Hydropower Station is mainly concentrated in the surrounding regions
of the spandrel groove and the excavated slope. The disturbance effect
resulting from the full excavation of the slope can be approximately
taken into consideration through dividing the factor of safety of the
slope under the natural condition by a factor of 1.05. To incorporate the
disturbance effect induced by the excavation blasting into the slope reliability analysis, a reduction factor, K, is introduced into the performance
function, which is larger than 1.0. Then, an explicit performance function
can be expressed as
GðX Þ ¼
min
j¼1;2;⋯;N s
FS j ðX Þ=K−1:0
ð4Þ
(1) Identify input random variables and determine their statistics such
as means, coefficients of variation (COVs), marginal distributions
and cross-correlation coefficients among the input random
variables.
(2) Construct the slope stability model including excavated bodies,
establish the finite difference grid, define the boundary conditions,
failure criteria, constitutive model of rock masses and weak structural features, and design slope support schemes.
(3) Perform a parametric sensitivity analysis to identify the significant
random variables. The factor of safety of the slope is calculated at
the (2n + 1) samples using the finite difference strength reduction
method that is widely employed to evaluate slope stability.
(4) Construct a single or multiple quadratic RSFs between the factors
of safety and the significant random variables. Establish an explicit
performance function for slope reliability analysis accounting for
the disturbance effects as discussed in Section 2.2.
(5) Perform MCS to generate a random sample matrix with
dimensions of n × NMCS in the independent standard normal
space. Then NMCS realizations of random samples in the physical
space are obtained using an isoprobabilistic transformation
(Li et al., 2011). The probability of slope failure is obtained
by substituting NMCS realizations of the physical samples into
the explicit performance function in Eq. (4).
Note that the NISFDM is an effective non-intrusive reliability analysis
method that does not require the user to modify existing deterministic
finite difference codes, which are treated as “black boxes”. The finite
difference strength reduction method is used for the deterministic
analyses that are decoupled from the probabilistic analysis. In contrast,
the existing deterministic finite difference (finite element) codes need
to be thoroughly modified for an intrusive reliability analysis method
(Stefanou, 2009). Additionally, it should be mentioned that the stochastic finite difference method is an intrusive reliability analysis method,
which is completely different from the random field method although
the former can incorporate the latter for modeling spatial variability of
rock and soil properties.
Table 1
Geomechanical parameters for rock masses.
No.
Rock mass
γ (kN/m3)
v
c (MPa)
φ (°)
1
Sandy slate in toppling
deformation zone
Strongly unloading sandy slate
Weakly unloading sandy slate
Thick-bedded sandy slate
Strongly unloading marble
Weakly unloading marble
Second block marble
Fresh sandy slate or marble
Reinforced concrete
27
1.0
0.30
0.4
30.96
27
27
27
27
27
27
27
27
2.0
3.0
9.0
2.0
4.0
11.0
21.0
21.0
0.28
0.35
0.30
0.30
0.27
0.25
0.3
0.167
0.6
0.9
1.5
0.6
0.9
1.5
2.0
1.0
34.99
45.57
46.94
34.99
45.57
46.94
53.47
45.0
2
3
4
5
6
7
8
9
E (GPa)
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D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
Table 2
Geomechanical parameters for structural planes.
No.
Structural planes
γ (kN/m3)
E (GPa)
v
c (MPa)
φ (°)
1
2
Fault f42-9
Lamprophyre dike X
(above elevation 1680 m)
Lamprophyre dike X
(below elevation 1680 m)
Deep fracture SL44-1
Fault f2, f5, f8
27
27
0.375
1.00
0.35
0.20
0.02
0.02
16.71
16.71
27
3.00
0.30
0.45
27.02
27
27
1.25
0.45
0.35
0.28
0.10
0.02
24.23
16.71
3
4
5
3. Engineering background
3.1. Slope overview
The Jinping I Hydropower Station is located in a deep valley of the
Yalong River in south-west China (e.g., Xu et al., 2009, 2011; Qi et al.,
2010, 2012; Song et al., 2010; Huang et al., 2010; Zhou et al., 2011). A
305 m high double curvature arch dam was constructed in a typical
V-shaped valley. The left abutment slope at Jinping I Hydropower
Station is one of the engineered rock slopes with the largest excavation
scale in rock engineering, which contains the following three main
features: (1) A large-scale model range with the relative height difference
of the natural slope reaching 1000–1700 m, the dip angle varying from
55° to 70° near elevation 1850 m, the total excavation height being
approximately 530 m (from elevations 2110 m to 1580 m), the maximum
excavation width being up to 350 m, and the total excavation volume of
materials reaching 5.50 million m3. (2) Very complicated geological structures in this area such as a V-shaped deep valley, high geo-stresses, strong
unloading induced stress releases, complex engineering and technical
conditions, inter-layer extrusion zones and deep fractures (Chengdu
Hydroelectric Investigation and Design Institute, 2003; Xu et al., 2011).
Furthermore, the large number of deep fractures and faults, such as faults
f42-9, f5, f8 and lamprophyre dike X, could form huge latent unstable blocks
that impact the slope stability. (3) The use of a complex physical–mechanical model introduces complications into deriving the numerical solution of the factor of safety. The Jinping I left abutment slope after full
excavation and support is shown in Fig. 1.
These special features are rare in other hydropower projects in the
world, which make it very challenging for the stability analysis of the
particular rock slope. Recently, 2-D/3-D stability analysis of the slope
has been performed substantially with numerical methods such as the
limit equilibrium method (LEM) (Zhou et al., 2006; Song et al., 2010),
the finite element method (FEM) (Xu et al., 2011; Liu et al., 2013), the
strength reduction method (SRM) (Xu et al., 2009; Qi et al., 2010,
2012) and block theory (Huang et al., 2010; Sun et al., 2015). However,
to our best knowledge, 2-D reliability of the particular rock slope based
on limit equilibrium analyses was only studied by Tang et al. (2012),
and 3-D reliability of this slope taking into account the uncertainty in
the shear strength of rock masses and structural planes has never
been investigated. As reported in the literature (e.g., Xu et al., 2009;
Huang et al., 2010; Song et al., 2010; Zhou et al., 2011; Qi et al., 2012),
the overall stability of the left abutment slope is mainly controlled by a deformed and cracked rock mass, as concluded from the spatial distribution
and intersections of the weak structural features in the left abutment
slope. Thus, the wedge failure mode of the engineered slope below elevation 2110 m is selected as the case study for the 3-D reliability analysis in
this paper. The toppling failure mode of the natural slope above elevation
2110 m is beyond the scope of the present study.
3.2. Geomechanical parameters of rock masses and structural planes
It is very difficult to accurately determine the geomechanical parameters of a rock mass since it may exhibit extremely complex characteristics
due to long-term geological processes. In geotechnical practice, the
geomechanical parameters of a rock mass used in the numerical analysis
Strongly unloading sandy slate
Strongly unloading sandy slate
Sandy slate in toppling deformation zone
Sandy slate in toppling deformation zone
Weakly unloading sandy slate
Fresh sandy slate or marble
Thick-bedded sandy slate
Second block marble
Strongly unloading marble
Lamprophyre dike X (below elevation 1680 m)
Weakly unloading sandy slate
Weakly unloading marble
Faults f2, f5, f8
Lamprophyre dike X (above elevation 1680 m)
Lamprophyre dike X (above elevation 1680 m)
Fault f42-9
Fault f42-9
Deep fracture SL44-1
Strongly unloading sandy slate
Lamprophyre dike X (above elevation 1680 m)
Z
Y
O
X
Fig. 2. Schematic diagram of 3D finite difference model of the left abutment slope.
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
273
Table 3
Basic mechanical and geometric parameters of the pre-stressed cables.
Cable type
Pullout load Young's modulus Strand yield strength Number of strands Nominal cross-sectional area Hole diameter Cable diameter Anchored part length
(mm)
(mm)
(m)
(kN)
of strand (GPa)
(MPa)
(mm2)
DKDF-2000 2000
DKDF-3000 3000
180
180
1860
1860
12ϕ15.24 mm
19ϕ15.24 mm
of slope stability are usually derived from in-situ tests and/or laboratory
tests such as the direct shear test and triaxial test supplemented with
engineering judgment from similar projects (e.g., Miranda et al., 2009;
Xu et al., 2011). According to the geological investigation conducted by
Chengdu Hydroelectric Investigation and Design Institute (2003), the
geomechanical parameters of rock masses and structural planes are
summarized in Tables 1 and 2, respectively. These parameters are used
for the deterministic slope stability analysis.
3.3. Finite difference model
The variation of stress field within the left abutment slope during the
construction period is simulated using a finite difference model in this
study. To achieve a realistic evaluation of the stress field, the boundary
effect on studied area should be minimized. Therefore, the analysis
domain is selected as follows (Xu et al., 2009; Qi et al, 2012): 495 m
in width from 205 m at the upstream of section II-II to 10 m at the
downstream of section I-I, 1200 m in length in the direction perpendicular
to the river from the centerline of river to the deep internal part of
the slope and 1500 m in height from elevations 1300 to 2800 m. A
three-dimensional finite difference model is constructed in a Cartesian
coordinate system as shown in Fig. 2, in which the X-axis is along the
river, the Y-axis is perpendicular to the river, and the Z-axis trends
upwards. The finite difference grid consists of 75,647 grid points. The
rock masses and weak structural features are modeled by 46,875
eight-node hexahedral zones, 773 six-node wedge zones and 135,754
four-node tetrahedron zones. The materials of the rock masses and
structural planes have been represented by different colors. Following
Zienkiewicz et al. (1975) and Qi et al. (2012), a Mohr–Coulomb failure
criterion is adopted to represent the stress–strain behavior of the rock
masses and weak structural features. Movements are allowed vertically
on four lateral boundaries in the directions along the river and perpendicular to the river, while restraint is imposed in three directions on the base
boundary. Although faults f42-9, f5, f8, f2, deep fractures SL44-1–SL44-9 and
lamprophyre dike X are usually treated as interfaces between the rock
blocks, they are modeled as special zones with assigned shear strength
rather than as interfaces for simplicity. Additionally, the topography,
stratigraphic boundaries, unloading or weathering boundaries and the
excavation face are modeled properly.
The excavation process of the left abutment slope lasted for more
than 3 years, started in September 2005, advanced to the dam crest
2188.92
3465.79
ϕ 140
ϕ 165
ϕ 90
ϕ 110
12
12
(at elevation 1885 m) in June 2007 and the concrete cushion foundation
platform (at elevation 1730 m) in August 2008. The slope excavation
was completed when the dam foundation pit (at elevation 1580 m)
was excavated in August 2009. The slope excavation process is simulated
using FLAC3D (Itasca, 2007), which includes 25 steps. Step 0 represents
the stability of the natural slope. The slope is then cut to elevations
1960 m, 1885 m, 1855 m, 1810 m and 1780 m corresponding to steps
3, 5, 7, 10 and 12, respectively. Finally, the slope is excavated to elevation
1580 m in step 25. The materials that are to be excavated at various
stages of the excavations are represented using the null model available
in FLAC3D (Itasca, 2007).
To control the sustained deformation of the slope and ensure the
stability of the crushed rock masses within the shallow slope, a large
number of pre-stressed cables with design pullout loads of 2000 and
3000 kN and lengths of 40, 60 and 80 m are installed. For all the prestressed cables, the length of the anchored part is 12 m, the inclination
is 8°, and the horizontal and vertical spacings are 4 m × 4 m or
6 m × 6 m, respectively. A total of 2815 and 930 bunches of prestressed cables are simulated above and below elevation 1885 m, respectively. The basic mechanical and geometric parameters of the pre-stressed
cables are listed in Table 3. An arrangement of the pre-stressed cables
with the design pullout load of 3000 kN is shown in Fig. 3. The cable
elements in FLAC3D are utilized to simulate the stabilization effects
of the pre-stressed cables. For illustration, the code for a bunch of
pre-stressed cable of 2000 kN is programmed as “sel cable id=1
property emod 180e9 yt 4.071e6 xcarea 6.3585e-3 gr_coh 1.0e6
gr_fric 45.0 gr_k 15e9 gr_per 0.4396” in FISH language, in which
emod, yt, xcarea, gr_coh, gr_fric, gr_k, and gr_per denote the Young's
modulus of a strand (Pa), the tensile yield strength of a cable (N), the
cross-sectional area of a cable (m2), the cohesion of grout (N/m), the
friction angle of grout (°), the stiffness of grout (Pa) and the exposed
perimeter of grout (m), respectively.
To further improve the slope stability, three shear-resistant
concrete plugs (tunnels excavated and then backfilled with reinforced
concrete) with a size of 9 m (width) × 10 m (height) were constructed
at elevations 1883 m, 1860 m and 1834 m, respectively, to cut through
the crush zones of fault f42-9. The corresponding lengths are 110, 90 and
78 m, respectively. Fig. 4 shows the locations of the three shear-resistant
concrete plugs and the interrelation with the boundaries of the deformed
and cracked rock mass which include the downstream fault f42-9 as the
main slip surface, the deep fracture SL44-1 as the upstream cut surface,
Fig. 3. Sketch of the arrangement for pre-stressed cables of 3000 kN.
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D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
the Lamprophyre dike X forming the rear boundary and the fault f5
forming the excavation face. The stabilization and shear-resistant effects
of the three shear-resistant concrete plugs are modeled in FLAC3D by
using the entity elements assigned with the shear strength of reinforced
concrete as shown in Table 1. It should be pointed out here that there
inevitably exists model bias between the FLAC3D model and the field
condition, which may have a significant influence in the slope reliability
analysis. The model bias should be reduced by calibrating the numerical
model to reflect the field condition by the following means: Firstly, collect
as much data as possible relating to the specific field condition such as insitu test data, exploration information about rock mass structures, on-site
construction condition, reinforcement and seepage control measures, and
incorporate these data into the establishment of the numerical model.
Furthermore, determine proper boundary conditions and constitutive
models of the rock masses and weak structural features in accordance
with the field condition. Finally, collect some monitoring data including
stresses, strains, displacements and pore water pressures in different
areas of the studied slope to validate the numerical model.
creep, performance degradation of the reinforced system and so on as the
in-service time of the high rock slope goes on.
3.5. Deterministic stability analysis of rock slope using strength reduction
method
The finite difference strength reduction method (SRM) (e.g.,
Zienkiewicz et al., 1975; Qi et al., 2010) can not only account for
the effect of the stress–strain behavior of rock masses and weak
structural features, but also simulate the progressive failure path of
a slope without the assumption of the shape and location of the failure
surface in advance. Thus, the global factor of safety of the rock slope
during construction is herein evaluated by the SRM based on a bisection
method in FLAC3D (Xu et al., 2009; Qi et al., 2012). The convergence
criterion for the strength reduction analysis is the nodal unbalanced
3.4. Deformation analysis of rock slope during construction
As mentioned earlier, the reinforced system of the left abutment slope
is mainly composed of the pre-stressed cables and three shear-resistant
concrete plugs at elevations 1883 m, 1860 m and 1834 m, respectively,
as shown in Fig. 5. To account for the effect of the reinforced system on
the deformation and stability of the rock slope during construction, four
reinforcement conditions during slope excavation are considered as
follows: (1) Combined support with pre-stressed cables and three
shear-resistant concrete plugs; (2) The pre-stressed cables fail and only
the three shear-resistant concrete plugs perform well; (3) The three
shear-resistant concrete plugs fail and only the pre-stressed cables
perform well; (4) No reinforcements.
To evaluate the stabilization effects of pre-stressed cables and shearresistant concrete plugs on slope stability, the displacements at representative points, factors of safety, and probabilities of slope failure during
construction are taken as indices. Firstly, the horizontal displacements
perpendicular to the river of two representative points at the slope
surface, TPL14 (at elevation 1945 m) and TPL10 (at elevation 1990 m),
which are located in the region of the deformed and cracked rock mass
are analyzed. Fig. 6(a) and (b) show the variations of the horizontal
displacements at points TPL14 and TPL10 under various reinforcement
conditions, respectively. The time–history curves of the monitoring
horizontal displacements at these two points are also plotted in Fig. 6.
The starting monitoring dates for these two points were both January
24th, 2008. As far as point TPL14 is concerned, the horizontal
displacement is only 10.5 mm when the pre-stressed cables and
three shear-resistant concrete plugs are supported after full excavation
of the slope, which is approximately consistent with the monitoring
value of 12.5 mm. The horizontal displacement will increase to 18.3 mm
(or 27.7 mm) if all pre-stressed cables (or the three shear-resistant
concrete plugs) fail due to some causes. It will drastically increase to
80.35 mm if no stabilization of the slope is provided during excavation.
These results indicate that the combined stabilization measures with
the pre-stressed cables and three shear-resistant concrete plugs in the
left abutment slope are very effective in controlling the slope deformation. Compared with the pre-stressed cables, the three shear-resistant
concrete plugs can restrain the slope deformation more effectively.
Additionally, the slope deformation induced by the slope excavation can
be captured in the numerical model, except that a fall back of the horizontal displacement near the end of 2008 cannot be effectively reflected. This
can be attributed to the fact that the numerical model cannot accurately
model the complex structural features within the slope and on-site
construction conditions. The uncertainties in input parameters also
cannot be fully taken into account in the numerical analysis. Additionally,
a continual steady increase of monitoring displacements after full excavation of the slope can be observed in Fig. 6, which can be attributed to rock
Location of the
deformed and cracked
rock mass
(a) Location of the deformed and cracked rock mass
Lamprophyre dike X
Deep crack SL44-1
Fault f42-9
Shear-resistant concrete plug
at elevation of 1883 m
Shear-resistant concrete plug
at elevation of 1860 m
Shear-resistant concrete plug
at elevation of 1834 m
(b) Layout of the deformed and cracked rock mass
Fig. 4. Location and layout of three shear-resistant concrete plugs and boundaries of the
deformed and cracked rock mass.
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
275
the natural slope and the slope excavated to elevations 1960 m, 1885 m,
1855 m, 1810 m, 1780 m and 1580 m, respectively. For the case of the excavated slope without reinforcements in Fig. 7, the factor of safety almost
remains the same at about 1.5 before the slope is excavated to elevation
1885 m in June 2007. The reason is that the marble outside the faults f5
force: the sum of forces acting on a node from its neighboring elements. A
simulation is considered to have converged when the ratio of the nodal
unbalanced forces is less than 10−5 under a specified calculation step
(e.g., 5500). Fig. 7 compares the factors of safety of the rock slope under
various reinforcement conditions. The seven points in each curve denote
Excavation and
support area
(a) Excavation and support area
2815 bunches of pre-stressed cables
above elevation 1885 m
Shear-resistant concrete plug
at elevation 1883 m
Shear-resistant concrete plug
at elevation 1860 m
Shear-resistant concrete plug
at elevation 1834 m
930 bunches of pre-stressed cables
below elevation 1885 m
(b) Arrangement of pre-stressed cables and three shear-resistant concrete plugs
Fig. 5. Layout of the reinforced system of the left abutment slope.
276
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
Fig. 6. Variation of horizontal displacement perpendicular to the river during construction.
1.55
2200
1.50
2100
2000
1.40
1900
1.35
Excavation process line
1800
1.30
1.25
1.20
1.15
2005/9/30
1700
Support with cables and shear-resistant plugs
Only three shear-resistant plugs perform well
Only pre-stressed cables perform well
No reinforcements
2006/3/30
2006/9/30
2007/3/30
2007/9/30
1600
2008/3/30
2008/9/30
Excavation date
Fig. 7. Variation of factor of safety of slope during construction.
2009/3/30
1500
2009/9/30
Excavation elevation /m
Factor of safety
1.45
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
Table 4
Results of 3D slope stability analysis.
Methods
Nature Support after
No reinforcements
Source
slope
full excavation after full excavation
SRM
SRM
SRM
LEM
FEM
Block theory
Block theory
1.483
1.277
1.45
1.202
1.35
1.413
1.56
1.405
1.385
1.35
–
1.36
–
1.43
1.186
1.152
1.05
1.130
–
–
1.104
This study
Xu et al. (2009)
Qi et al. (2012)
Zhou et al. (2006)
Liu et al. (2013)
Huang et al. (2010)
Sun et al. (2015)
The symbols “–” denote that the results are not available.
Contour of Shear Strain Increment
Plane: on
Magfac = 0.000e+000
Gradient Calculation
3.5458e-005 to 2.0000e-002
2.0000e-002 to 4.0000e-002
4.0000e-002 to 6.0000e-002
6.0000e-002 to 8.0000e-002
8.0000e-002 to 1.0000e-001
1.0000e-001 to 1.2000e-001
1.2000e-001 to 1.4000e-001
1.4000e-001 to 1.6000e-001
1.6000e-001 to 1.8000e-001
1.8000e-001 to 1.9033e-001
Interval = 2.0e-002
277
and f8 could support the deformed and cracked rock mass like a “rock
wall”, and the excavation process plays a role in reducing the surcharge
loads and improving the slope stability. When the slope is excavated to
elevation 1810 m in March 2008, the factor of safety decreases to 1.264
due to gradual exposure of the boundaries of the deformed and cracked
rock mass as the excavation is proceeding. When the slope is excavated
to elevation 1780 m in May 2008, the fault f42-9 is fully exposed at the
excavation face. Consequently, the factor of safety dramatically decreases
to 1.186, which is below the minimal factor of safety of 1.25–1.3 for
the slope in a very important project under the permanent condition
specified by the Design Specification for the Slope in Water Conservancy
Contour of Shear Strain Increment
Plane: on
Magfac = 0.000e+000
Gradient Calculation
2.2621e-005 to 2.5000e-002
2.5000e-002 to 5.0000e-002
5.0000e-002 to 7.5000e-002
7.5000e-002 to 1.0000e-001
1.0000e-001 to 1.2500e-001
1.2500e-001 to 1.5000e-001
1.5000e-001 to 1.7500e-001
1.7500e-001 to 2.0000e-001
2.0000e-001 to 2.2160e-001
Interval = 2.5e-002
Fig. 8. Contours of shear strain increment for the slope excavated to elevation 1780 m.
278
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
Fig. 8 (continued).
and Hydropower Project (DLT5353-2006) (Song et al., 2010). These
results indicate that the slope stability cannot be guaranteed in this
case. The status of slope safety will be adversely impacted if other factors
such as excavation disturbances and unloading, rainfall, and earthquakes
are also incorporated. Therefore, some reinforcement measures are
urgently taken to ensure the slope safety.
As discussed earlier, the combined support using pre-stressed cables
and three shear-resistant concrete plugs can be used to reinforce the
rock slope. In this case, the factor of safety of the excavated slope
increases to 1.498, which indicates that the combined stabilization
measures can effectively ensure the Jinping I left abutment slope stability.
Alternatively, if only the three shear-resistant concrete plugs or only the
pre-stressed cables perform well, the factor of safety of the excavated
slope will decrease to 1.342 or 1.306, respectively. It can be observed
that the three shear-resistant concrete plugs also have a more significant
effect on the slope stability in comparison with the pre-stressed cables.
Table 4 compares the results of 3-D slope stability analyses using the
SRM, LEM, FEM and block theory. The results obtained from this study
are consistent with those reported in the literature (e.g., Zhou et al.,
2006; Xu et al., 2009; Qi et al., 2012; Huang et al., 2010; Liu et al., 2013;
Sun et al., 2015). These results further demonstrate the validity of the
3-D numerical model in this study.
Fig. 8 presents the contours of shear strain increments for a representative cross section II1-II1 of the left abutment slope under four different
reinforcement conditions when the slope is excavated to elevation
1780 m in May 2008. The failure path of the rock slope can be approximately determined from the shear strain increments (Zienkiewicz et al.,
1975; Xu et al., 2009). As expected, the sliding surface mainly coincides
with the bottom of the deformed and cracked rock mass. Only a single
shallow failure mode is observed in Fig. 8. The combined stabilization
measures taken in the Jinping I left abutment slope can effectively control
the development of the plastic zones of the slope. In addition, the plastic
zones do not cut through the rock masses when the pre-stressed cables
perform well.
4. Three-dimensional reliability analysis of rock slope
4.1. Statistics of input geomechanical parameters
To evaluate slope reliability, the statistical parameters of the uncertain
geomechanical parameters should be determined first. They are often
derived from in-situ tests, laboratory tests and expert experience based
on similar projects in the study area. Ideally, the parameters should be determined based on sufficient laboratory or field tests. In this study, the
ranges of the COVs of shear strength parameters are adopted based on
the values reported in the literature (e.g., Phoon and Kulhawy, 1999;
Baecher and Christian, 2003; Tang et al., 2012). The statistics of the basic
random variables are summarized in Table 5. Following Hoek (1998)
and Tang et al. (2012), lognormal distributions are used to model the
distributions of geomechanical properties to avoid negative values.
4.2. Reliability analysis results
A parametric sensitivity study is first carried out to identify uncertain
parameters which have a significant effect on the slope reliability. With
this purpose in mind, the mean value plus and minus two standard
deviations for each random variable are adopted when studying the
sensitivity of the parameters, while keeping the rest of the random
variables at their mean values. Considering the 12 random variables
in Table 5, a total of 25 deterministic finite difference analyses are
conducted for the parametric sensitivity analysis. For simplicity, Fig. 9
presents the relative sensitivities of the 12 input random variables at
seven key excavation steps (i.e., steps 0, 3, 5, 7, 10, 12 and 25) for the
case of the excavated slope without reinforcements. It can be observed
that cohesion c6 and friction angle φ6 of the class III2 rock mass
(i.e., weakly unloading sandy slate and marble) are the most sensitive
random variables, followed by cohesion c5 and friction angle φ5 of the
class IV1 rock mass before the slope is excavated to elevation 1810 m in
March 2008. As explained earlier, the marble outside faults f5 and f8 can
prevent the slope from sliding before fault f42-9 is fully exposed at the
excavation face. Thus, the shear strength parameters of the class III2
rock mass have a significant effect on the slope stability. When the
slope is excavated to elevation 1780 m, the relative sensitivities of the
friction angle φ1 of fault f42-9, cohesion c3 and friction angle φ3 of the
deep fracture SL44-1 become larger. For example, when the slope is
excavated from the crown to elevation 1780 m, the values of α (φ1), α
(c3) and α (φ3) increase from 0.053 to 0.2, 0.024 to 0.11 and 0.038 to
0.2, respectively, while the value of α (φ6) decreases from 0.46 to 0.15.
These results indicate that the shear strength of the sliding surfaces of
the deformed and cracked rock mass begins to affect the slope stability
significantly once the fault f42-9 is fully exposed at the excavation face.
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
279
Table 5
Statistics of the basic variables of the rock slope.
Materials
Random
variables
Mean
COV
Distribution
type
Fault f42-9
c1/MPa
φ1/°
c2/MPa
φ2/°
c3/MPa
φ3/°
c4/MPa
φ4/°
0.02
16.7
0.02
16.7
0.1
24.228
0.4
30.96
0.3
0.2
0.25
0.15
0.22
0.14
0.2
0.12
Lognormal
Lognormal
Lognormal
Lognormal
Lognormal
Lognormal
Lognormal
Lognormal
c5/MPa
φ5/°
0.6
35
0.18
0.10
Lognormal
Lognormal
c6/MPa
φ6/°
0.9
45.57
0.15
0.08
Lognormal
Lognormal
Lamprophyre dike X
(above elevation 1680 m)
Deep fracture SL44-1
Class IV2 rock mass
(Sandy slate in toppling
deformation zone)
Class IV1 rock mass
(Strongly unloading sandy
slate and marble)
Class III2 rock mass
(Weakly unloading sandy
slate and marble)
Fig. 10. Variation of probability of slope failure with sampling factor, k.
To identify the significant uncertain parameters for the slope
reliability analysis, a threshold value for the relative sensitivity should
be specified. The threshold value is usually within a range of 5%–10%
(Shen and Abbas, 2013). A threshold value of 1% is selected for conservative purposes. The random variables with the relative sensitivities
below the threshold value are treated as deterministic quantities in
the slope reliability analysis. In this way, the numbers of random variables in the slope reliability analysis are reduced from 12 to 9, 8, 9, 9,
11, 10 and 9 for the natural slope and the slope excavated to elevations
1960 m, 1885 m, 1855 m, 1810 m, 1780 m and 1580 m, respectively.
In agreement with investigations by Huang et al. (2010) and Qi et al.
(2012), the slope stability below elevation 2110 m is mainly controlled
by a single wedge failure mode as observed in Fig. 8. Thus, only one
response surface function needs to be constructed. The probability of
slope failure is then calculated using a direct MCS with NMCS = 106
samples in this study. To explore the effect of the sampling factor, k,
for the construction of the response surface function on the slope
reliability results, Fig. 10 presents the variation in the probability of
slope failure with the sampling factor, k, for the natural slope and the
slope excavated to elevation 1780 m without reinforcements,
respectively. It can be observed that k has some effect on the probability
of slope failure, but it is less significant than one order of magnitude.
Generally, there is no rigid rule for selecting k (Xu and Low, 2006).
Additionally, note that the factors of safety obtained from the parametric
sensitivity analyses can be further used to construct RSFs if k = 2.
Consequently, the computational cost is further reduced. Thus, k = 2 is
used in this study for illustration according to Zhang et al. (2011).
Fig. 11 compares the probabilities of slope failure between the case
where only the three shear-resistant concrete plugs perform well and
the case where no reinforcements are used. It can be observed that the
three shear-resistant concrete plugs can improve the slope reliability
substantially. For the case of the reduction factor K = 1.05 and the
slope reinforced with the three shear-resistant concrete plugs, the
probability of slope failure decreases from 5.3 × 10−5 for the natural
slope to 3.1 × 10−5 and 1.2 × 10−6 for the slope excavated to elevations
1885 m in June 2007 and 1780 m in May 2008, respectively. In contrast,
the probability of slope failure dramatically increases from 3.1 × 10−5 to
1.27 × 10−2 when the slope is excavated from elevations 1885 to 1780 m
for the excavated slope without reinforcements. The latter is about 400
times greater than the former. Such a high probability of failure cannot
meet the slope safety requirement of above average performance level
(i.e., pf = 1.0 × 10−3) as suggested by the U.S. Army Corps of Engineers
(1997). Additionally, the probability of slope failure significantly increases
when the reduction factor K increases from 1.05 to 1.1, which indicates
that the disturbance induced by rock bursts is an important factor
affecting slope reliability. Therefore, effective measures should be taken
to minimize disturbance effects during excavation blasting.
0.6
α (c1 )
0.4
0.3
α (c2 )
α (ϕ2 )
α (c3 )
α (ϕ3 )
α (c4 )
α (ϕ4 )
α (c5 )
α (ϕ5 )
α (c6 )
α (ϕ6 )
Excavation process line
2200
2100
2000
1900
1800
0.2
1700
0.1
1600
0.0
2005/9/30
2006/3/30
2006/9/30
2007/3/30
2007/9/30
2008/3/30
2008/9/30
Excavation date
Fig. 9. Variation of relative sensitivities of input random variables.
2009/3/30
1500
2009/9/30
Excavation elevation /m
Relative sensitivity α
0.5
α (ϕ1 )
280
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
2200
0.1
2000
1E-3
Above average level for pf =1.0×10
-3
1900
1E-4
1800
1E-5
1E-6
Reduction factor K = 1.05
Only three shear-resistant plugs perform well
No reinforcements
Reduction factor K = 1.1
Only three shear-resistant plugs perform well
No reinforcements
1E-7
2005/9/30
2006/3/30
2006/9/30
2007/3/30
2007/9/30
1700
Excavation elevation /m
Probability of failure
2100
Excavation process line
0.01
1600
2008/3/30
2008/9/30
2009/3/30
1500
2009/9/30
Excavation date
Fig. 11. Variation of probability of slope failure during construction.
It can be seen from Fig. 11 that the probability of slope failure will be
significantly increased if no reinforcements are used during slope
excavation. To further explain such a finding from a perspective of
probabilistic analysis, the probability density functions (PDFs) of the
factor of safety for the case where only the three shear-resistant concrete
plugs perform well and the case where no reinforcements are used are
plotted in Fig. 12 for the slope excavated to elevation 1780 m. Note that
the mean values of the factor of safety are 1.346 and 1.172 for the
aforementioned two cases, respectively, while the COVs of the factor of
safety almost remain the same. These results imply that a significant
increase in the probability of failure for the slope without reinforcements
is mainly attributed to the decrease in the mean value of the factor of
safety rather than to the increase in the variability of the factor of safety.
5. Conclusions
This paper has conducted 3-D reliability analysis of the left abutment
slope at Jinping I Hydropower Station in south-west China using a nonintrusive stochastic finite difference method. The effects of the reinforced
system, namely pre-stressed cables and three shear-resistant concrete
plugs, on the reliability of a practical rock slope during construction are
investigated systematically. Several conclusions can be drawn from this
study:
(1) The 3-D reliability of complex rock slopes can be efficiently
evaluated using a non-intrusive stochastic finite difference
method. A major advantage of the non-intrusive stochastic finite
difference method is that it does not require the user to modify
existing deterministic finite difference codes, such as FLAC3D. It
provides an effective tool for solving 3-D reliability problems of
complex rock slopes in practice.
(2) The parametric sensitivity analysis approach can effectively
identify the significant uncertain parameters in the slope stability
analysis. The shear strength parameters of the sliding surfaces of
the deformed and cracked rock mass significantly affect the slope
stability once the fault f42-9 is fully exposed at the excavation
face. Additionally, the deterministic analyses of slope stability
used for the parametric sensitivity analysis can be used again for
the slope reliability analysis. Thus, the computational efforts can
be reduced substantially.
(3) The combined support system with the pre-stressed cables and
three shear-resistant concrete plugs adopted in the Jinping I left
abutment slope can effectively restrain the slope deformation
and ensure the slope stability during construction. The horizontal
displacement and probability of slope failure will increase greatly
if the reinforced structures, especially the three shear-resistant
concrete plugs, do not take effect after the slope has been
excavated to elevation 1780 m. The shear-resistant concrete
plugs play a more important role in improving the slope reliability
than the pre-stressed cables. Additionally, the excavation disturbance also affects the slope reliability significantly.
(4) The statistics of shear strength parameters of rock masses and
structural planes are not accurately determined due to the limited
in-situ data available, which may impede the wide applications of
reliability methods to evaluate the stability of high rock slopes.
Additionally, to improve the computational accuracy of the
slope reliability analysis, the numerical models need to be calibrated to sufficiently reflect the field condition.
Acknowledgments
Fig. 12. Probability density functions of factor of safety.
This work was supported by the National Science Fund for Distinguished Young Scholars (Project No. 51225903), the National Basic
Research Program of China (973 Program) (Project No. 2011CB013506),
D.-Q. Li et al. / Engineering Geology 195 (2015) 269–281
the National Natural Science Foundation of China (Project No. 51329901)
and the Natural Science Foundation of Hubei Province of China (Project
No. 2014CFA001).
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