Uncertainty and reliability analysis applied to slope stability
Abdallah I.Huin Malkawi *,Waleed F.Hassan,Fayez A.Abdulla
Civil Engineering Department,Jordan University of Science and Technology,221-10Irbid,Jordan
Abstract
In this paper,reliability analysis of slope stability is prented using two methods of uncertainty ®rst-order cond-moment method (FOSM)and Monte Carlo simulation method (MCSM).The results of the methods are compared using four recognized methods of slope stability.The are Ordinary method of slices,simpli®ed Bishop's method,simpli®ed Janbu's method and Spencer's method.Two illustrative examples are prented in this paper:one is homogenous slope and the other is non-homogeneous layered slope.The study shows that the reliability index ( )is independent of the ed random number generator and a sample size of 700or greater is a good choice for MCSM.In the ca of homogeneous slope a good agreement is obrved between the calculated ( )using FOSM and MCSM for both the Ordinary and the Bishop's method.However,slight di erence in ( )is obrved between the two uncertainty methods whe-ther Janbu's method or Spencer's method is ud.In the ca of the layered slope good agreement is obtained between the two uncertainty methods for Ordina
ry,Bishop and Janbu methods.Similar to example 1,Spencer's method shows also slight di erence in ( )between FOSM and MCSM methods.Model uncertainty is addresd by evaluating the relative performance of the three slope stability Ordinary,Bishop and Janbu methods as compared to Spencer's method.#2000Elvier Science Ltd.All rights rerved.
Keywords:Slope stability;Factor of safety;Model uncertainty;Reliability index;First-order cond-moment;Monte Carlo simulation
1.Introduction
In geotechnical engineering analysis and design various sources of uncertainties are encoun-tered and well recognized.Several features usually contribute to such uncertainties,like:(1)tho associated with inherent randomness of natural process;(2)Model uncertainty re¯ecting the inability of the simulation model,design technique or empirical formula to reprent the system's true physical behavior,such as calculating the safety factor of slopes using limiting equilibrium methods of slices;(3)Model parameter uncertainties resulting from inability to quantify accurately梦见大洪水
0167-4730/00/$-e front matter #2000Elvier Science Ltd.All rights rerved.PII:
S0167-4730(00)00006-0
Structural Safety 22(2000)161±187
www.elvier.nl/locate/strusafe
*Corresponding author.Tel.:+962-2-295111,ext.2112;fax:+962-2-710-2299.E-mail address:mhuin@just.edu.jo (A.I.Huin Malkawi).
162 A.I.Huin Malkawi et al./Structural Safety22(2000)161±187
the model input parameters and(4)Data uncertainties including(a)measurement errors,(b)data inconsistency and non-homogeneity and(c)data handling.In slope stability computations,var-ious sources of uncertainties are encountered,such as geological details misd in the exploration program,estimation of soil properties that are di cult to he spatial variability in the®eld cannot be reproduced accurately,¯uctuation in pore water pressure,testing errors and many other relevant factors.
In a deterministic analysis,the factor of safety(F)is de®ned as the ratio of resisting to driving forces on a potential sliding surface.The slope is considered safe only if the calculated safety factor clearly
exceeds unity.Whereas,in a probabilistic framework the factor of safety is expres-d in terms of its mean value as well as its variance.Reliability analysis is therefore ud to asss uncertainties in engineering variables such as the factor of safety of slope stability.The reliability index( )is often ud to express the degree of uncertainty in the calculated factor of safety.Such uncertainty is usually assd by di erent approaches such as the®rst-order cond-moment method,point estimate method,and Monte Carlo simulation method.
Wu and Kraft[1],Cornell[2],Alonso[3],Tang et al.,[4],Venmarcke[5],Wol [6],Li and Lumb[7],Barabosa et al.,[8],among others,applied a probabilistic approach in analyzing slope stability using the®rst-order cond-moment method.Recently,a number of applications of probabilistic slope stability studies using other numerical approaches,such as Monte Carlo simulation or point estimate method,have been reported in the literature[9,10]
Christian et al.[11]ud the mean-®rst order reliability method,which is a simpli®cation of the more general®rst order reliability method.They found that the reliability analysis is especially uful in establishing design values of safety factor reprenting consistent risks for di erent types of failure.Tobutt[12]ud the Monte Carlo method as a nsitivity-testing tool for slope stability and also as a method for calculating the probability of failure of a given earth slope.
This paper prents a probabilistic bad approach by which the relevant sources of uncer-tainty involved in slope stability analysis can be modeled and analyzed.Two di erent methods are ud to quantify the uncertainty in the calculated safety factor.The are the®rst-order cond-moment method(FOSM)and Monte Carlo simulation method(MCSM).Two examples,repre-nting homogeneous and layered slopes,were lected to investigate how the probability of failure and the reliability index may vary among tho two methods.The results were obtained using four well-recognized methods of slope stability Ordinary method[13],Bishop's method[14], Janbu's method[15,16]and Spencer's method[17].Model uncertainty was addresd by evaluating the relative performance of the simpli®ed slope stability methods compared to Spencer's method.
2.Limiting equilibrium methods
Most problems in slope stability are statically indeterminate,and as a result,some simplifying assumptions are made in order to determine a unique factor of safety.Due to the di erences in assumptions,various methods have been developed.Among the most popular methods are pro-cedures propod by Fellenius,Bishop,Janbu and Spencer referred to before[13±17].Some of the methods satisfy only overall moment,like the Ordinary and simpli®ed Bishop methods and are applicable to a circular slip surface,while Janbu's method satis®es only force equilibrium and is appli
cable to any shape.Spencer's method,however,satis®es both moment and force equilibrium
and it is applicable to failure surfaces of any shape.It is considered as one of the rigorous and accurate methods for solving stability problems.Table 1prents a summary of static equilibrium conditions in limit equilibrium methods of slices considered in this study.In order to formulate the algorithm to solve for the factor of safety bad on the above-mentioned methods,one should consider the forces acting on a typical slice as shown in Fig.1.
For the Ordinary method of slices,which is considered the simplest method of slices,the factor of safety is directly obtained.The method assumes that the inter-slice forces are parallel to the ba of each slice,thus they can be neglected and the factor of safety is given as follows:
Table 1
Summary of static equilibrium conditions in di erent limit equilibrium methods of slices Method
Force equilibrium Moment equilibrium
1st Direction
品茶的精美句子
2nd Direction Ordinary or Fellenius Yes No Yes Bishop's simpli®ed Yes No Yes Janbu's simpli®ed Yes Yes No Spencer
Yes
Yes
Yes
Fig.1.Depicts forces acting on a typical slice.
A.I.Huin Malkawi et al./Structural Safety 22(2000)161±187
163
F
n
i 1
c H b W os Q os "À W os À k h W sin n ÀU X b t n0H n
i 1
W W os Q os" sin À
n
i 1
W sin Q sin" os Àh R
n
i 1
k h W os Àh R
I
All terms and symbols appearing in Eqs.(1)±(5)are de®ned in Fig.1.
In Bishop's method the factor of safety is determined by trial and errors,using an iterative process,since the factor of safety(F)appears in both sides of Eq.(2).The inter-slice shear forces are neglected,and only the normal forces are ud to de®ne the inter-slice forces.The factor of safety is given as follows:
F n
i 1
c H b
1
os sin t n0H
F
WÀc H b t n
F假如我是班主任
ÀU X b W os Q os" t n0H n
i 1
W W os Q os" sin À
n
i 1
W sin Q sin" os Àh R
n
i 1后天八卦方位图
k h W os Àh R
P
Similarly,for Janbu's method the factor of safety is determined also by an iterative procedure through varying the e ective normal stress on the failure surface.The inter-slice shear forces are ignored and the normal forces are derived from the summation of vertical forces.The resulting factor of safety is given below:
F
n
i 1
c H b
1
少年中国梦歌曲os sin t n0H
F
WÀc H b t n
F
Àub os W os Q os" t n0H os
n
i 1
U X b sin Wk hÀW sin ÀQ sin"
n
致我们失去的青春
i 1
1
os sin t n0H
F
WÀc H b sin
F
Àub os W os Q os" sin
Q
In Spencer's method,the e ect of inter-slice forces is included and both moment and force equi-librium are explicitly satis®ed.This eventually will lead to an accurate calculation of the factor of safety.The factor of safety is determined through an iterative procedure,slice by slice,by varying F an
d until force and moment equilibrium are satis®ed.
The force equilibrium equation is as follows:
164 A.I.Huin Malkawi et al./Structural Safety22(2000)161±187
Z Z v FW sin Àc H b ÀW os t n0H sin À t n0HÀF os À
U X b t n0H Wk h FÀt n0H t n os sin À t n0HÀF os À
Q F sin À" À os À" t n0H
sin À t n0HÀF os À
W F sin À" À os À" t n0H
sin À t n0HÀF os À
R
While the moment equilibrium equation is given by:
h Z v
h vÀ
Z v
b
t n
Z v
t n
b
b
t n À
b
t n
hW sin
Z os
hQ sin"
Z os
À
h k h W
Z os
S
The iteration is terminated when the calculated values of Z and h di er within an acceptable tolerance from the known values of Z and h at the boundary.中国古代科举制度
The computer software called stability analysis of slopes using Monte Carlo technique(SAS-MCT)developed by Huin Malkawi and Hassan[18]is ud in this study to perform slope sta-bility analysis using the above mentioned method of slices.The program performs probabilistic approach in analyzing slopes stability problem;reliability analysis is performed using Monte Carlo simulation technique.It generates a large number of di erent soil parameter ts and cal-culates the safety factor for each random t.Then,the generated factors of safety are ud to construct the associated probability distribution.The corresponding reliability index( )and the probability of failure(P f)of the slope can be obtained.
3.Methods of reliability evaluation
3.1.First-order cond-moment(FOSM)method
我的寒假生活作文The FOSM method estimates the uncertainty of the factor of safety of a slope against instability as a function of the variances of the stochastic input variables,such as soil friction angle,soil cohesion,and soil unit weight.It us Taylor's ries expansion to estimate the local uncertainty of the factor of safety at a lected expansion point.
Consider that the factor of safety(F)can be expresd as a function of stochastic input para-meters X
s as
F g X g x1Y x2Y F F F Y x n T
in which X is an n-dimensional vector of stochastic input parameters;the superscript(T)is the matrix or vector transpo;and g()reprents a functional relationship for the safety factor.In the context of the prent study,g(X T)is the factor of safety formulation using the four
A.I.Huin Malkawi et al./Structural Safety22(2000)161±187165
