# Spoke Model for Calculating Reliability Index and Safety Factor of Slopes

Authors: Ping Wang; Dongyan Liu; Haibin Huang; Dongsheng Liu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Considering the disadvantages of the slice method commonly employed in reliability analysis of slopes, a novel method (Spoke model) was proposed for reliability analysis and safety factor calculation of slopes in this work based on geometrical relationship among slices. The safety factor and the coefficients of limit state function of slopes could be achieved with the Gaussian integral method. The minimum safety factor and the minimum reliability index, as well as their corresponding coordinates on the slip surface, can be calculated with the improved JC method and the searching method. A novel and practical method for reliability analysis of slopes has been achieved. With this method the slice process could be avoided, which helps to eliminate some calculation errors caused by oversimplified assumption. Moreover, the explicit expression of safety factor in this method requires no repeated iterative solution, which is employed in traditional slice methods, as well as can be developed into a limit state function required by calculation of the reliability index. It is demonstrated that this method works efficiently and succinctly in evaluation of reliability index and safety factor for soil slopes.


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## Body

## 1. Introduction

Each slope (including stable and unstable slopes) possesses an inherent tendency to degrade to a more stable form and eventually make the slope tend to horizontal. Mass movement (slope rupture, landslide, etc.) may take place as a result of the shear failure. The Spoke model proposed in this work is mainly focused on the stability analysis and reliability analysis of simple slopes, where the concept of simple slope discussed in this work is of the slope morphology as shown in Figure1, and the concept includes the following aspects: (1) the soil is homogeneous, and considering its geotechnical parameters it only includes weight density “w”, cohesion “c”, and internal friction angle “φ”; (2) the underground water are not taken into consideration.Figure 1
Illustration of a simple slope.


## 2. Literature Review

There are many methods to determine a slope slip surface. The center coordinate and the radius of a sliding circle were employed to determine the slip surface in one method [1, 2], while the center coordinate and the height of a sliding circle were introduced in another method [3]. A variety of methods have been applied to analyze the reliability of slopes. Based on traditional slice methods (such as Bishop method [3], Spencer model, Sarma model, and Janbu model [4, 5]), the reliability index can be achieved with the calculated safety factor F and the limit state function Z=F-1. In other cases the minimum reliability index was obtained with the limit state function on the slip surface where the safety factor reaches its minimum value [6]. These approaches are approximate methods for calculation of the reliability index, because all the expressions of safety factor in the slide methods are implicit, which require complicated iterative process when solving. Moreover, the slip surface corresponding to the minimum reliability index may not be the slip surface corresponding to the minimum safety factor [7].The parallel slide method was employed to analyze the stability of slopes in Bishop method. The safety factorFs is expressed as follows:
(1)Fs=∑(1/mai)[cibi+(Wi-uibi)tanφi]∑Wisinαi.Obviously, the equation is implicit where the parameterma contains Fs as well; therefore, it can be solved only with the iterative method. Since the Fs expression is implicit, the corresponding limit state function cannot be achieved when analyzing the stability of slopes with traditional slice methods.


## 3. Spoke Model

To achieve the limit state equation, a novel slide model (the Spoke model) is proposed in this work. With this model we can get the limit state equation from which the reliability index of slopes can be obtained.Assuming that a slope is homogeneous, the height, slope angle, slope weight, angle of internal friction, and cohesion areH0, α0, w, φ, and c, respectively. The slip surface is supposed to be a circular arc surface (Figure 2). Point O is the center of the slope line. Points O, M, and N are on the same line. The angle between line NM and the slope line is ψ. Since OO′⊥MN and OO′=h, the arc NM is the slip line. Translate left the center O′, the arc NM, segments, and intersections with a distance of l. The coordinates (ψ,l,h) correspond to all possible slip surfaces, respectively.Figure 2
Spoke model.The slip soil can be divided into three partsS1, S2, and S3. Taking the second part S2, for example, the center angle Δθi has crossing points AFDG with the slip line and slope line. The area of the ith AFDG is ΔS2=SAFDG. The radius of slip circle is R0 which can be expressed in
(2)R0={h2+[H02sin(α0-ψ)]2}1/2.
Considering Δθi is a small value,
(3)SAFDG=(R02-O′D¯2)Δθi2,(4)O′D¯=O′B¯-BC¯-CD¯.
Assume that
(5)r=∠OO′C=θ-(α0-ψ)=θ+ψ-α0,(6)O′B¯=hcosr=hcos(θ+ψ-α0).
In triangle ▵OBC, according to Sine theorem, it can be achieved that
(7)BC¯sinψ=htanrsin(r+90°-ψ)=htanrcos(ψ-r).Combining (5) and (7),
(8)BC¯=htan(θ+ψ-α0)sinψcos(θ-α0).
Similarly, in triangle ▵OBC,
(9)CD¯=CP¯cos(θ-α0)lsinα0cos(θ-α0).
Combining (6), (8), (9), and (4),
(10)O′D¯=hcos(θ+ψ-α0)-htan(θ+ψ-α0)sinψcos(θ-α0)-lsinα0cos(θ-α0).
Combining (3) and (10),
(11)SAFDG=Δθi2{R02-[hcos(θ+ψ-α0)-htan(θ+ψ-α0)sinψcos(θ-α0)-lsinα0cos(θ-α0)]2}.Assuming that the area of theith slice is Si, bulk density and weight of the soil are  wi and Wi, respectively, as follows:
(12)Wi=wiSi.In the second part,Si is determined by (11). 						(13)Si=SAFDG.Similarly, theSi expression of the first and third parts can be achieved, respectively, as follows:
(14)Si=Δθi2{R02-R0sin[arcsin(h/R)+α0-ψ]cosθ},(15)Si=Δθi2{R02-R0cos[arccos(h/R)+α0-ψ]cosθ}.Shear forcesXi and Xi+1 pass through center O′, perpendicular to the slip surface. The sideward normal forces are Ei and Ei+1, respectively. According to the balance condition along the direction of each radial line,
(16)Wicosθi+Xi+1cos(Δθi2)-Xicos(Δθi2)+Eisin(Δθi2)+Ei+1sin(Δθi2)-Ni=0.SinceΔθi is a definitely small value,
(17)cos(Δθi2)=1,(18)Eisin(Δθi2)+Ei+1sin(Δθi2)=EiΔθi2+(Ei+ΔEi)Δθi2=EiΔθi.Combining (16), (17), and (18),
(19)Wicosθi+Xi+1-Xi+EiΔθi-Ni=0.In the limit equilibrium, the sum of all the force moments on the center should be zero; therefore,(20)∑WiRisinθi-∑TiR0+∑Qiei=0,
where Qi and ei are the inertia force of earthquake and its tension arm, respectively. Ri is the distance between center of the circle and the weight center of a slice. The shear stress at the bottom of the slice can be expressed as
(21)Ti=τli=τfFsli=ciliFs+(Ni-uili)tanφiFs.Combining (19), (20) and (21), the safety factor Fs  can be achieved. 						(22)Fs=R0∑[cili+(Wicosθi+Xi+1-Xi+EiΔθi-uili)tanφi]∑WiRisinθi+∑Qiei,
where the sum of all the force moments of the slide friction forces on the center of the circle resulting from shear stresses should be zero. Therefore,
(23)R0tanφ∑Xi=0,li=R0Δθi,Fs=R0∑[cR0Δθi+(Wicosθi+EiΔθi-uiR0Δθi)tanφ]∑WiRisinθi+∑Qiei.Compared with theFs expression in Bishop method, (23) possesses three obvious advantages as follows: (i) the shear stress between the soil strips is not involved in (23); (ii) it is an absolute expression without Fs in its right part, which requires no repeated iterative solution; (iii) (23) has a clear physical meaning. Fs is the ratio of the total resisting moment to the total sliding moment.When the slope is sliced infinitely, as well as the interstitial pressure and the seismic inertial force are not taken into consideration, the safety factor can be expressed in the integration form. Combining (13), (16), and (23),
(24)Fs=(cR02/w)∫θ0θ3dθ+R0tanφ×T∫θ0θ1S1R1sinθdθ+∫θ1θ2S2R2sinθdθ+∫θ2θ3S3R3sinθdθ,
where T=∫θ0θ1[S1cosθ+E1(θ)]dθ+∫θ1θ2[S2cosθ+E2(θ)]dθ+∫θ2θ3[S3cosθ+E3(θ)]dθ.Therefore,(25)f=cR02w∫θ0θ3dθ+R0tanφ×{∫θ0θ1[S1cosθ+E1(θ)]dθ+∫θ1θ2[S2cosθ+E2(θ)]dθ+∫θ2θ3[S3cosθ+E3(θ)]dθ}÷(∫θ0θ1S1R1sinθdθ+∫θ1θ2S2R2sinθdθ+∫θ2θ3S3R3sinθdθ),
where Sjdθ (j=1,2,3)  in (24) and (25) is the integrated area element, Rj (j=1,2,3) is the distance between the slice center and the center of the circle, and  SjRjdθ (j=1,2,3) represents the quadrature. Combining (11)–(15), they could be expressed as the functions of the parameters H0, α0, ψ, l, and h as follows:
(26)Sjdθ=(R02-rj2)dθ2,Rj=2(R02+R0rj+rj2)3(R0+rj),SjRjdθ=(R03-rj3)dθ3,(j=1,2,3),
where
(27)r1=R0sin[arcsin(h/R0)+α0-ψ]cosθ,r2=hcos(θ+ψ-α0)-htan(θ+ψ-α0)sinψcos(θ-α0)-lsinα0cos(θ-α0),r3=R0cos[arccos(h/R0)+α0-ψ]cosθ.In (24) and (25), the upper/lower limit of integral θi(i=0,1,2,3) represents the central angle between the line O′H and another segment whose two endpoints are the center of circle and the endpoint of the slide slope segment in the circle arc, which can be expressed as the functions of the parameters H0, α0, ψ, l, and h as follows:
(28)θ0=-arccos(hR0)+α0-ψ,θ1=-arcsinR0cos[arcsin(h/R0)+α0-ψ]-(s-l){R02+(s-l)2-2R0(s-l)cos[arcsin(h/R0)+α0-ψ]}1/2,θ2=arcsinR0sin[arccos(h/R0)+α0-ψ]-(s+l){R02+(s+l)2-2R0(s+l)cos[arcsin(h/R0)-α0+ψ]}1/2,θ3=arccos(hR0)+α0-ψ.In (26), (27), and (28),
(29)R0={h2+[H02sin(α0-ψ)]2}1/2,(30)s=H0sinψ2sin(α0-ψ)sinα0.In (24) and (25), E(θ) can be achieved by solving differential equations. The differential equation in the tangential direction of the ith slice slope can be expressed as follows:
(31)ΔEi=-Wisinθ+Nitanφ.Combining (20) and (31),
(32)ΔEi=-Wisinθi+(Wicosθi+EiΔθi)tanφ+ΔXitanφ.To simplify the calculation, letΔXi=ΔEitanφ, μ=tanφ, and (32) can be developed into a differential equation:
(33)dE(θ)-μ1-μ2E(θ)dθ=w(μcosθ-sinθ)f(θ)dθ,
where f(θ)dθ is the area element, which can be expressed as follows with (26):
(34)f(θ)dθ=Sjdθ=(R02-rj2)dθ2.
Let α=μ/(1-μ2). The general solution of the linear differential (33) is
(35)E(θ)=eαθ[∫w(μcosθ-sinθ)f(θ)e-αθdθ+C],
where C is an arbitrary constant. Considering the complexity of f(θ), it is definitely complicated to solve the indefinite integral of the transcendental function. To simplify the calculation, f(θ) is replaced with a proper constant M.Within the interval θ>0 in (26), rj is replaced with OO′¯. Equation (32) can be developed into
(36)E(θ)=eαθM[∫w(μcosθ-sinθ)e-αθdθ+C].With(37)∫e-αθcosθdθ=11+α2e-αθ(-αcosθ+sinθ)+C1,∫e-αθsinθdθ=11+α2e-αθ(-αsinθ-cosθ)+C2,
(36) can be developed into
(38)E(θ)=wM1+α2×{[(1-αμ)cosθ+(α+μ)sinθ]+C3e-αθ+C4}.
Assuming E(θ)=0 when θ=θ3,
(39)E(θ)=wM1+α2{-eα(θ-θ3)[sinθ3(1-αμ)cosθ+(α+μ)sinθ]-eα(θ-θ3)[(1-αμ)cosθ3+(α+μ)sinθ3]}.When∂E(θ)/∂θ=0, we can obtain the maximum value of broadside normal stress Em(θ) and its corresponding angle θ=θm. The normal stress would reduce to zero gradually from the cut-off point toward both directions. It can be analyzed that θm is in the interval θ>0. Combining E=E(θm)/2 and (24), (25), the safety factor can be calculated, and the limit state function can be determined.Subsequently, combine (24)–(30). As for a certain slope (i.e., α0 and H0 are fixed.), each group of (ψ,l,h) would result in its corresponding safety factor and limit state function. Namely, safety factor and limit state function can be expressed as a function containing variables ψ, l, and h, respectively as follows:
(40)Fs=g1(ψ,l,h),f=g2(ψ,l,h,c,φ,w).Considering the random variability of soil parameters, we could obtain the reliability index of each group of(ψ,l,h). By varying the numerical values of (ψ,l,h), the minimum values of reliability index and safety factor can both be obtained. The calculation of structure reliability index has been discussed in many reported works [9–13]. In a typical failure mode of a slope (Figure 3), the slip surface passes through both the top plane and bottom plane. Based on geometrical condition and the constraint relationship among ψ, l,  and h, inequalities (41) can be obtained theoretically as follows:
(41)0≤ψ<α0,|l|≤H0sinψ2sinα0sin(α0-ψ),H02cos(α0-ψ)≤h≤(H0/sinα0)-((s-l)/cos(α0-ψ))2tan(α0-ψ).Figure 3
Area of most possible slip surface.Pan Jia-zheng has stated that the center of critical slip surface is in the area formed by two circulars with radius ofH0/2tanα0 and 3H0/4tanα0, respectively, central normal line and perpendicular bisector of the slope surface (Figure 3) [14]. By varying the values of ψ, l, and h within this area, we can achieve the minimum values of reliability index βmin, and safety factor Ksmin, and their corresponding (ψ,l,h).


### 3.1. Example Analysis

Example (I) (see  [15]). There is a slope whose height and slope angle are 4.5 m and 20°, respectively, with a sand bed beneath which helps to drain water away adequately. The natural unit weight of soil is r=16 kNm−3. The average efficacious shear strength and variation coefficient are μc′=2.159 kPa, δc′=30%, μtanφ′=0.312, and δtanφ′=2.5%, respectively. Find out the failure probability of the given slid surface. The safety factor was calculated approximately to be Ks=1.25 with a vertical slice method in [15]. The failure probability was calculated to be Pf=0.002 (β=2.901) with statistics formulas.With the novel method (Spoke model) proposed in this work, the safety factor is calculated to beKs=1.252. If c and tanφ′ are regarded as normal random variables, and they are not interrelated, the corresponding reliability index is calculated to be 2.883 with the Spoke model.Example (II) (see [8]). There is a homogeneous simple slope whose height (h) is 50 m, slope proportion (m) is 3.25, bulk density of soil (r) is 19.62 kNm−3, coefficient of friction is tanφ=0.2, and adhesion stress (c) is 58.66 kPa.Work out the location of the most dangerous slide arc and minimum value of safety factor.The calculated results with the Spoke model proposed in this work and the corresponding results in [8] are listed in Table 1.Table 1
Calculation Results for Example (II).
Corresponding coordinate
x (m)
y (m)
r (m)
Result achieved in [8]
K
s
min
=
1
.
28
59.21
-
84
.
72
152.13
Corresponding coordinate
ψ (°)
l (m)
h (m)
Result in this work
K
s
min
=
1.31
2.18
6.35
115.22Example (III) (see [7]). There is a clay slope whose height is 10.0 m, and the proportion of slope is 1 : 2. The average value and standard deviation of c and tanφ are 20 kPa, 0.1817 and 4.0 kPa, 0.05, respectively, and all of them are considered to be of normal distribution and not interrelated. The unit weight of soil is 16.0 kNm−3, considered as a fixed value. Find out the minimum value of safety factor and reliability index of the slope.With the Spoke model in this work, we takeΔψ=0.2°, Δl=0.05×H0sinψ/2sinα0sin(α0-ψ), and Δh=0.05×[(H0/sinα0-(s-l)/cos(α0-ψ))/tan(α0-ψ)-H0/cos(α0-ψ)] as the step length to search solution. The results are listed in Table 2.Table 2
Calculation Results for Example (III).
Corresponding coordinates of the minimum value
ψ (°)
l (m)
h (m)
Safety index (variation coefficients ofc and tanφ are 0.2 and 0.275, resp.)
K
s
min
=
1.38
7.4
1.32
11.09
β
min
=
1.60
7.6
1.34
11.13
Safety index (variation coefficients ofc and tanφ are both 0.15)
K
s
min
=
1.38
7.4
1.32
11.09
β
min
=
2.37
6.4
1.45
10.98According to the analysis of Examples (I) and (II), the calculated results with Spoke model and traditional slice methods are subequal. Example (III) indicates that minimum safety factor and minimum reliability index are not achieved on the same slip surface. In addition, the variability of soil parameters affects the minimum reliability index and the position of its corresponding slip surface but possesses little influence on the minimum safety factor and the position of its corresponding slip surface.It is worth noting that the expression of safety factor has a clear physical meaning in this work. With the calculation program built, it is convenient for data input, and the calculation process is quick. With this method for reliability analysis of slopes, the limit state function can be achieved which directly results in the reliability index.


## 3.1. Example Analysis

Example (I) (see  [15]). There is a slope whose height and slope angle are 4.5 m and 20°, respectively, with a sand bed beneath which helps to drain water away adequately. The natural unit weight of soil is r=16 kNm−3. The average efficacious shear strength and variation coefficient are μc′=2.159 kPa, δc′=30%, μtanφ′=0.312, and δtanφ′=2.5%, respectively. Find out the failure probability of the given slid surface. The safety factor was calculated approximately to be Ks=1.25 with a vertical slice method in [15]. The failure probability was calculated to be Pf=0.002 (β=2.901) with statistics formulas.With the novel method (Spoke model) proposed in this work, the safety factor is calculated to beKs=1.252. If c and tanφ′ are regarded as normal random variables, and they are not interrelated, the corresponding reliability index is calculated to be 2.883 with the Spoke model.Example (II) (see [8]). There is a homogeneous simple slope whose height (h) is 50 m, slope proportion (m) is 3.25, bulk density of soil (r) is 19.62 kNm−3, coefficient of friction is tanφ=0.2, and adhesion stress (c) is 58.66 kPa.Work out the location of the most dangerous slide arc and minimum value of safety factor.The calculated results with the Spoke model proposed in this work and the corresponding results in [8] are listed in Table 1.Table 1
Calculation Results for Example (II).
Corresponding coordinate
x (m)
y (m)
r (m)
Result achieved in [8]
K
s
min
=
1
.
28
59.21
-
84
.
72
152.13
Corresponding coordinate
ψ (°)
l (m)
h (m)
Result in this work
K
s
min
=
1.31
2.18
6.35
115.22Example (III) (see [7]). There is a clay slope whose height is 10.0 m, and the proportion of slope is 1 : 2. The average value and standard deviation of c and tanφ are 20 kPa, 0.1817 and 4.0 kPa, 0.05, respectively, and all of them are considered to be of normal distribution and not interrelated. The unit weight of soil is 16.0 kNm−3, considered as a fixed value. Find out the minimum value of safety factor and reliability index of the slope.With the Spoke model in this work, we takeΔψ=0.2°, Δl=0.05×H0sinψ/2sinα0sin(α0-ψ), and Δh=0.05×[(H0/sinα0-(s-l)/cos(α0-ψ))/tan(α0-ψ)-H0/cos(α0-ψ)] as the step length to search solution. The results are listed in Table 2.Table 2
Calculation Results for Example (III).
Corresponding coordinates of the minimum value
ψ (°)
l (m)
h (m)
Safety index (variation coefficients ofc and tanφ are 0.2 and 0.275, resp.)
K
s
min
=
1.38
7.4
1.32
11.09
β
min
=
1.60
7.6
1.34
11.13
Safety index (variation coefficients ofc and tanφ are both 0.15)
K
s
min
=
1.38
7.4
1.32
11.09
β
min
=
2.37
6.4
1.45
10.98According to the analysis of Examples (I) and (II), the calculated results with Spoke model and traditional slice methods are subequal. Example (III) indicates that minimum safety factor and minimum reliability index are not achieved on the same slip surface. In addition, the variability of soil parameters affects the minimum reliability index and the position of its corresponding slip surface but possesses little influence on the minimum safety factor and the position of its corresponding slip surface.It is worth noting that the expression of safety factor has a clear physical meaning in this work. With the calculation program built, it is convenient for data input, and the calculation process is quick. With this method for reliability analysis of slopes, the limit state function can be achieved which directly results in the reliability index.


## 4. Conclusion

The explicit expression of safety factor in Spoke model proposed in this work requires no repeated iterative solution, as well as can be developed into a limit state function required by calculation of the reliability index. The Spoke model avoids complicated repeated iterative solution as required in traditional slice methods as well as eliminates the errors arising from oversimplified assumption.According to the example analysis, the slip surface with minimum reliability index is not in coincidence with, but similar to, the one with minimum safety factor. We can find out the minimum safety factor and its corresponding coordinates first, then search the minimum reliability index nearby.The minimum reliability index and location of its corresponding slip surface are related with soil parameters such as weight density “G,” cohesion “c,” and internal friction angle “φ.” The minimum safety factor and location of its corresponding slip surface have no relation with variability of soil parameters.This work only discusses the calculation analysis of reliability of simple slopes. The failure surface is supposed as a circular arc, and some simplifications are utilized on the resistance calculation of slip surface caused by the stress between slices. The influence of simplified treatment in this work on calculation results could be a direction of further research.



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