A S S E S S M E N T O F S L O P E S T A B I L I T Y 
A N D S T A B I L I Z A T I O N T E C H N I Q U E S 
T H R O U G H P R O B A B I L I S T I C A P P R O A C H 
D. C. A . M E T T A N A N D A 
This thesis was submitted to the Department of Civil Engineering of the University of 
Moratuwa in partial fulfilment of the requirements for the Degree of Master of 
Science 
LIBRARY 
UNIVERSITY C> MORATUWA. SRI 
M G H A T U W A 
Supervised by 
Dr. S. A. S. KULATHILAKA 
Department of Civil Engineering 
University of Moratuwa 
Sri Lanka 
T 
August 2002 
University of Moratuwa " 7 5 3 7 8 
75378 
7 5 3 7 8 
A B S T R A C T 
U n c e r t a i n t y a n d r a n d o m n e s s of d a t a is a ma jo r i s s u e a s s o c i a t e d in 
g e o t e c h n i c a l e n g i n e e r i n g . It is the re fo re d e s i r a b l e to u s e m e t h o d s a n d 
c o n c e p t s in e n g i n e e r i n g p l a n n i n g a n d d e s i g n t h a t c a n faci l i tate t h e 
e v a l u a t i o n a n d a n a l y s i s of u n c e r t a i n t y . F o r m a l p robab i l i s t i c a p p r o a c h 
p r o v i d e s a u se fu l f r amework to i n c o r p o r a t e t h e s e u n c e r t a i n t i e s in 
s l ope i n s t ab i l i t y p r o b l e m s . U n d e r t h e r e s e a r c h d e s c r i b e d in t h i s t h e s i s , 
n u m b e r of p robab i l i s t i c m o d e l s h a v e b e e n deve loped to e v a l u a t e t h e 
s t ab i l i t y of s l o p e s a n d the i r u s e in t h e e v a l u a t i o n of s tab i l i ty of s l opes 
a n d t h e e v a l u a t i o n of effect iveness of v a r i o u s s t ab i l i za t ion t e c h n i q u e s 
h a v e b e e n d i s c u s s e d . 
Two p r o b a b i l i s t i c ana ly t i ca l m o d e l s t h a t c a n be u s e d to e v a l u a t e t h e 
s t ab i l i t y of s l o p e s t h a t c a n fail e i t h e r a l o n g c i r c u l a r fa i lure s u r f a c e s or 
n o n - c i r c u l a r fai lure s u r f a c e s h a v e b e e n deve loped . T h e s e m o d e l s 
fo rmal ly recognize t h e u n c e r t a i n t i e s a s s o c i a t e d wi th v a r i o u s 
g e o t e c h n i c a l p a r a m e t e r s a n d p rov ide m e a n s to quan t i fy the i r effects 
o n t h e s tab i l i ty . The r e s u l t is given in t h e form of p robab i l i t y of 
u n s a t i s f a c t o r y p e r f o r m a n c e of t h e s lope . E a c h m o d e l deve loped w a s 
fac i l i t a ted to pe r fo rm c o m p u t a t i o n s in five different w a y s r a n g i n g from 
t h e o r e t i c a l l y s o u n d m e t h o d s to s o m e a p p r o x i m a t e m e t h o d s , a n d t h e 
r e s u l t s o b t a i n e d were c o m p a r e d w i th e a c h o t h e r . In a d d i t i o n , t h e 
r e s u l t s w e r e c o m p a r e d wi th t h e r e s u l t s o b t a i n e d by a c o m m e r c i a l 
so f tware , w h e r e v e r app l i cab le . In a d d i t i o n , two o t h e r p robab i l i s t i c 
a n a l y t i c a l m o d e l s were deve loped to a n a l y z e t h e s l o p e s s tabi l ized by 
soil n a i l i n g . 
It is s e e n t h a t t h e behav io r of t h e p robab i l i s t i c m o d e l s deve loped u n d e r 
t h i s r e s e a r c h pe r fo rm sa t is fac tor i ly , a n d it c a n be r e c o m m e n d e d to u s e 
t h e s e m o d e l s in r o u t i n e p r o b l e m s of s lope ins t ab i l i t y to provide m o r e 
r ea l i s t i c r e s u l t s i n c o r p o r a t i n g t h e u n c e r t a i n t i e s a s s o c i a t e d wi th 
various geotechnical parameters. Analyses of a number of examples 
and case histories discuss the uses of the probability of failure in 
decision making to evaluate the stability of slopes as well as the 
effectiveness of various stabilization techniques. It also emphasizes the 
importance of the appropriate failure mechanism and the appropriate 
deterministic model in the probabilistic analysis. Analysis of case 
histories provides an important discussion showing the inadequacy of 
the conventional factor of safety alone in evaluating the stability of 
slopes. 
DECLARATION 
The work included in this thesis in part or whole, has not been 
submitted for any other academic qualification at any 
institution. 
A C K N O W L E D G E M E N T S 
It i s w i t h a s e n s e of g r a t i t u d e t h a t I r eca l l a n d a p p r e c i a t e t h e 
a s s i s t a n c e r ece ived from different p e r s o n n e l to m a k e t h i s r e s e a r c h a 
rea l i ty . 
At t h e o u t s e t , I e x p r e s s m y hear t fe l t g r a t i t u d e a n d a d m i r a t i o n to m y 
s u p e r v i s o r Dr . A t h u l a K u l a t h i l a k a , S e n i o r L e c t u r e r , D e p a r t m e n t of 
Civil E n g i n e e r i n g , Unive r s i ty of M o r a t u w a . His i n v a l u a b l e c o m m i t m e n t 
a n d d e d i c a t i o n w a s a g r ea t e n c o u r a g e m e n t for m e . W i t h o u t h i s 
v a l u a b l e g u i d a n c e a n d u n t i r i n g effort, t h i s r e s e a r c h m i g h t n o t h a v e 
b e e n a s u c c e s s . 
N a t i o n a l B u i l d i n g R e s e a r c h O r g a n i z a t i o n of Sr i L a n k a e x t e n d e d i t s full 
c o o p e r a t i o n to t h i s p ro jec t by p rov id ing t h e i r d a t a a b o u t p r e v i o u s s lope 
f a i l u r e s in Sr i L a n k a w i t h o u t h e s i t a t i o n . I s h o u l d a p p r e c i a t e t h e 
c o o p e r a t i o n e x t e n d e d by t h e Di rec to r G e n e r a l of t h e N a t i o n a l B u i l d i n g 
R e s e a r c h O r g a n i z a t i o n , Sr i L a n k a by i s s u i n g t h e n e c e s s a r y a p p r o v a l to 
u s e t h e i r d a t a . In a d d i t i o n , Mr. R .M.S . B a n d a r a , H e a d L a n d s l i d e s 
Div is ion a n d Mr. D h a r m a s e n a , E n g i n e e r , L a n d s l i d e s Divis ion 
e n c o u r a g e d m e in d o i n g t h i s r e s e a r c h , a n d owe spec i a l t h a n k s for 
t h e i r s u p p o r t . 
Mr. R.M. A m a r a s e k e r a , Provincia l D i r ec to r (Uva Province) , R o a d 
D e v e l o p m e n t A u t h o r i t y w a s very he lpful in s h o w i n g t h e r e h a b i l i t a t i o n 
w o r k in p r o g r e s s a t l a n d s l i d e s i t e s , a n d a l s o p r o v i d i n g i n f o r m a t i o n for 
c a s e h i s t o r i e s a n a l y z e d in t h i s r e s e a r c h . His s u p p o r t is a l so 
a p p r e c i a t e d . 
1 
A s i a n D e v e l o p m e n t B a n k p rov ided f inanc ia l s u p p o r t for t h i s p ro jec t 
t h r o u g h t h e S c i e n c e a n d T e c h n o l o g y p e r s o n n e l d e v e l o p m e n t pro jec t , 
a n d o w e s spec i a l t h a n k s . 
G e o t e c h n i c a l E n g i n e e r i n g divis ion a n d t h e T r a n s p o r t a t i o n E n g i n e e r i n g 
Div is ion of t h e D e p a r t m e n t of Civil E n g i n e e r i n g s u p p o r t e d m e in 
p r o v i d i n g w o r k i n g s p a c e . C o m p u t e r Se rv i ces Divis ion p rov ided 
n e c e s s a r y a s s i s t a n c e r e g a r d i n g t h e p r o b l e m s a s s o c i a t e d w i th 
c o m p u t e r s . I w o u l d like to t h a n k all t h e staff m e m b e r s of t h e 
G e o t e c h n i c a l E n g i n e e r i n g Divis ion, T r a n s p o r t a t i o n E n g i n e e r i n g 
Div is ion a n d t h e C o m p u t e r Service Divis ion in t h e D e p a r t m e n t of Civil 
E n g i n e e r i n g for t h e i r a s s i s t a n c e e x t e n d e d for t h i s r e s e a r c h . 
F ina l ly , I w o u l d like to t h a n k Di rec to r P o s t g r a d u a t e S t u d i e s , D e a n 
F a c u l t y of E n g i n e e r i n g , H e a d D e p a r t m e n t of Civil E n g i n e e r i n g , 
R e s e a r c h C o o r d i n a t o r , D e p a r t m e n t of Civil E n g i n e e r i n g a n d t h e all t h e 
a c a d e m i c staff m e m b e r s of t h e D e p a r t m e n t of Civil E n g i n e e r i n g for 
g iving m e t h i s o p p o r t u n i t y to c o m p l e t e t h i s r e s e a r c h . 
D. C. A. M e t t a n a n d a 
u 
C O N T E N T S 
Page 
A c k n o w l e d g e m e n t s i 
C o n t e n t s iii 
List of F igures vii 
* List of Tables xi 
List of A n n e x e s xiv 
List of S y m b o l s xv 
Chapter 1 - INTRODUCTION 1 
1.1 Introduct ion 1 
1.2 Determinis t ic vs. Probabilistic Approaches 2 
1.2.1 Determinist ic Approach 2 
1.2.2 Probabilistic Approach 3 
i 1.3 Current State of the Probabilistic Evaluat ion of Slope 4 
Stabil ity 
1.3.1 Introduction 4 
1.3.2 Uncertaint ies of Geotechnical Parameters 5 
1.3.2.1 Types of Uncerta int ies 5 
1 .3 .2 .2 Methods of Quantif icat ion of Uncertainty 6 
1.3.3 Application of Probabilistic Concept s to Slope 7 
Stability Analysis 
1.3.3.1 Suitable Probability Dis tr ibut ions 7 
1 .3 .3 .2 Different Approaches of Probabilistic 10 
Analysis 
1.3.4 Role of Probabilistic Analys i s in Landsl ide Risk 12 
A s s e s s m e n t 
1.3.5 Role of Probabilistic Analys i s a s a tool in Decis ion 15 
Making 
1.4 S c o p e of the Project 17 
1.5 Out l ine of the Thes i s 18 
> 
111 
C h a p t e r 2 - DEVELOPMENT O F PROBABILISTIC M O D E L S 2 1 
2 . 1 Se lec t ion of a D e t e r m i n i s t i c Mode l 2 1 
2 . 2 A s s i g n m e n t of U n c e r t a i n t i e s 2 3 
C h a p t e r 3 - DEVELOPMENT O F T H E PROBABILISTIC M O D E L 2 5 
BASED ON BISHOP 'S SIMPLIFIED M E T H O D 
3 .1 B a s i s of t h e m e t h o d 2 5 
3 .2 A s s i g n m e n t of U n c e r t a i n t i e s 2 7 
3 . 3 De r iva t i on of Pa r t i a l Der iva t ives 2 8 
3 .4 D e v e l o p m e n t of t h e S p r e a d s h e e t s 3 0 
3 . 5 App l i ca t i on of t h e Model 4 2 
3 . 5 . 1 G e n e r a l 4 2 
3 . 5 . 2 T e s t E x a m p l e 1 4 2 
3 . 5 . 2 . 1 E x a m p l e 1(a) 4 3 
3 . 5 . 2 . 2 E x a m p l e 1(b) 4 7 
3 . 6 C o n c l u d i n g R e m a r k s 52 
C h a p t e r 4 - DEVELOPMENT OF T H E PROBABILISTIC MODEL 5 3 
B A S E D ON J A N B U ' S SIMPLIFIED M E T H O D 
4 . 1 B a s i s of t h e m e t h o d 5 3 
4 . 2 A s s i g n m e n t of U n c e r t a i n t i e s 5 5 
4 . 3 De r iva t i on of Pa r t i a l Der iva t ives 5 6 
4 . 4 D e v e l o p m e n t of t h e S p r e a d s h e e t s 5 8 
4 . 5 App l i ca t i on of t h e Model 6 9 
4 . 5 . 1 G e n e r a l 6 9 
4 . 5 . 2 E x a m p l e 2 6 9 
4 . 5 . 3 E x a m p l e 3 7 6 
4 . 6 C o n c l u d i n g R e m a r k s 8 3 
C h a p t e r 5 - PROBABILISTIC B A S E D A S S E S S M E N T O F 8 5 
DIFFERENT STABILIZATION M E T H O D S 
5 .1 I n t r o d u c t i o n 8 5 
iv 
5.2 Development of Probabilistic Models based on Bishop's 89 
simplified method to analyze Soil Nailed Structures 
5.2.1 Basis of the Method 89 
5.2.2 Assignment of Uncertainties 90 
5.2.3 Derivation of Partial Derivatives 90 
5.2.4 Development of the Spreadsheet 90 
5.3 Development of Probabilistic Models based on Janbu's 99 
simplified method to analyze Soil Nailed Structures 
5.3.1 Basis of the Method 99 
5.3.2 Assignment of Uncertainties 99 
5.3.3 Derivation of Partial Derivatives 100 
5.3.4 Development of the Spreadsheet 100 
5.4 Illustrative Example for the Stabilization using 107 
Drainage (Example 4) 
5.5 Illustrative Example for the Stabilization using Soil 108 
Nailing (Example 5) 
5.6 Illustrative Example for the Stabilization using both 109 
Drainage ans Soil Nailing (Example 6) 
5.7 Concluding Remarks 111 
Chapter 6 - APPLICATION TO SRI LANKAN CASE HISTORIES 113 
OF NATURAL SLOPES 
6.1 Stabilized Watawala Landslide 113 
6.1.1 Introduction 113 
6.1.2 Investigation of the Landslide 114 
6.1.3 Stabilization of the Watawala Landslide 117 
6.1.4 Extraction of Information 118 
6.1.5 Analysis 119 
6.1.6 Discussion of Results 121 
6.2 Slope at Marangahawela (Badulla District) 122 
6.2.1 Introduction 122 
6.2.2 Stabilization of the Marangahawela Slope 122 
6.2.3 Analysis 124 
6 . 3 C o n c l u d i n g R e m a r k s 1 2 6 
C h a p t e r 7 - PROBABILISTIC EVALUATION O F SAFETY O F 128 
CUT S L O P E S 
7 . 1 I n t r o d u c t i o n 1 2 8 
7 .2 Soil S t r e n g t h P a r a m e t e r s 129 
7 . 3 G r o u n d w a t e r C o n d i t i o n s 130 
7 . 4 Need for P robab i l i s t i c A p p r o a c h 130 
7 . 5 E x a m p l e C u t Slope 1 130 
7 . 6 E x a m p l e C u t S lope 2 134 
7 .7 C o n c l u d i n g R e m a r k s 138 
C h a p t e r 8 - CONCLUSIONS 139 
R E F E R E N C E S 145 
A N N E X E S 
> 
vi 
L I S T O F F I G U R E S 
Page 
F i g u r e 1.1 - N o r m a l D i s t r i b u t i o n 8 
F i g u r e 1.2 - G r a p h i c a l R e p r e s e n t a t i o n of Reliabi l i ty a n d 9 
Probabi l i ty of F a i l u r e 
F i g u r e 1.3 - L o g n o r m a l D i s t r i b u t i o n 9 
F i g u r e 3 .1 - F o r c e s a c t i n g o n a Slice 2 6 
F igu re 3 .2 - F l o w c h a r t s h o w i n g t h e overview of t h e 3 2 
S p r e a d s h e e t P r o g r a m (Bishop-Prob-Di rec t ) 
F i g u r e 3 . 3 - F l o w c h a r t for D e t e r m i n i s t i c W o r k s h e e t 3 3 
(Bishop-Prob-Di rec t ) 
F i g u r e 3 .4 - F l o w c h a r t for C a l c u l a t i o n s W o r k s h e e t (B ishop- 3 4 
Prob-Direct ) 
F i g u r e 3 . 5 - F l o w c h a r t for t h e P robab i l i s t i c W o r k s h e e t 3 6 
(Bishop-Prob-Di rec t ) 
F igu re 3 .6 - A s lope wi th va r i ed p h r e a t i c s u r f a c e s 3 9 
F i g u r e 3 .7 - F l o w c h a r t for S m a l l I n c r e m e n t P r o g r a m 4 0 
( B i s h o p - P r o b - S m a l l I n c r e m e n t ) 
F i g u r e 3 .8 - F l o w c h a r t for t h e c a l c u l a t i o n of F x + a n d Fx - 4 1 
( B i s h o p - P r o b - S m a l l I n c r e m e n t ) 
F i g u r e 3 .9 - E x a m p l e 1 G e o m e t r y 4 3 
F i g u r e 3 . 1 0 - R e s u l t s of E x a m p l e 1(a) 4 6 
F i g u r e 3 . 1 1 - G e o m e t r y of E x a m p l e 1 (b) 4 7 
F i g u r e 3 .12 - R e s u l t s of E x a m p l e 1 (b) 5 1 
F i g u r e 4 .1 - F o r c e s a c t i n g o n a Slice 5 4 
F i g u r e 4 . 2 - F l o w c h a r t s h o w i n g t h e overview of t h e 6 0 
S p r e a d s h e e t P r o g r a m ( J a n b u - P r o b - D i r e c t ) 
F i g u r e 4 . 3 - F l o w c h a r t for D e t e r m i n i s t i c W o r k s h e e t 6 1 
( J a n b u - P r o b - D i r e c t ) 
vii 
F i g u r e 4 . 4 - F l o w c h a r t for C a l c u l a t i o n s W o r k s h e e t ( J a n b u - 6 2 
Prob-Direc t ) 
F i g u r e 4 . 5 - F l o w c h a r t for t h e P robab i l i s t i c W o r k s h e e t 6 4 
( J a n b u - P r o b - D i r e c t ) 
F i g u r e 4 . 6 - F l o w c h a r t for S m a l l I n c r e m e n t P r o g r a m 6 7 
( J a n b u - P r o b - S m a l l I n c r e m e n t ) 
F i g u r e 4 . 7 - F l o w c h a r t for t h e c a l c u l a t i o n of F x + a n d Fx- 6 8 
( J a n b u - P r o b - S m a l l I n c r e m e n t ) 
F i g u r e 4 . 8 - E x a m p l e 2 G e o m e t r y 7 0 
F i g u r e 4 . 9 - R e s u l t s of N o n - C i r c u l a r A n a l y s i s for E x a m p l e 7 3 
2 
F i g u r e 4 . 1 0 - R e s u l t s of C i r c u l a r Ana lys i s for E x a m p l e 2 7 4 
F i g u r e 4 . 1 1 - G e o m e t r y of E x a m p l e 3 7 6 
F i g u r e 4 . 1 2 - R e s u l t s of N o n - C i r c u l a r A n a l y s i s for E x a m p l e 82 
3 
F i g u r e 4 . 1 3 - R e s u l t s of C i r c u l a r A n a l y s i s for E x a m p l e 3 8 2 
F i g u r e 5.1 - F o r c e s a c t i n g o n a sl ice (B i shop ' s Model - 8 9 
Nailed Slope) 
F i g u r e 5 .2 - F l o w c h a r t s h o w i n g t h e overv iew of t h e 9 3 
S p r e a d s h e e t P r o g r a m (Soil Nai led S lope -
B a s e d on B i s h o p ' s Method) 
F i g u r e 5 .3 - F l o w c h a r t for D e t e r m i n i s t i c W o r k s h e e t (Soil 9 4 
Nailed Slope - B a s e d on B i s h o p ' s Method) 
F i g u r e 5.4 - F l o w c h a r t for C a l c u l a t i o n s W o r k s h e e t (Soil 9 5 
Nailed Slope - B a s e d o n B i s h o p ' s Method) 
F i g u r e 5 .5 - F l o w c h a r t for t h e P robab i l i s t i c W o r k s h e e t (Soil 9 7 
Nailed Slope - B a s e d o n B i s h o p ' s Method) 
F i g u r e 5 .6 - F l o w c h a r t for Nail R e s i s t a n c e W o r k s h e e t (Soil 9 8 
Nailed Slope - B a s e d o n B i s h o p ' s Me thod) 
F i g u r e 5 .7 - F l o w c h a r t s h o w i n g t h e overview of t h e 101 
S p r e a d s h e e t P r o g r a m (Soil Nai led S lope -
B a s e d on J a n b u ' s Method) 
F i g u r e 5 . 8 - F l o w c h a r t for D e t e r m i n i s t i c W o r k s h e e t (Soil 102 
Nai led S lope - B a s e d o n J a n b u ' s Method) 
F i g u r e 5 .9 - F l o w c h a r t for C a l c u l a t i o n s W o r k s h e e t (Soil 1 0 3 
Nai led S lope - B a s e d o n J a n b u ' s Method) 
F i g u r e 5 . 1 0 - F l o w c h a r t for t h e P robab i l i s t i c W o r k s h e e t (Soil 1 0 5 
Nai led S lope - B a s e d o n J a n b u ' s Method) 
F i g u r e 5 . 1 1 - F l o w c h a r t for Nail R e s i s t a n c e W o r k s h e e t (Soil 106 
Nai led S lope - B a s e d on J a n b u ' s Method) 
F i g u r e 5 .12 - G r a p h i c a l R e p r e s e n t a t i o n of R e s u l t s ( E x a m p l e 107 
4) 
F i g u r e 5 . 1 3 - P r o p o s e d Nail A r r a n g e m e n t for E x a m p l e 5 108 
F i g u r e 5 . 1 4 - G r a p h i c a l R e p r e s e n t a t i o n of R e s u l t s ( E x a m p l e 109 
5) 
F i g u r e 5 . 1 5 - P r o p o s e d Nail A r r a n g e m e n t for E x a m p l e 6 110 
F i g u r e 5 . 1 6 - G r a p h i c a l R e p r e s e n t a t i o n of R e s u l t s ( E x a m p l e 111 
6) 
F i g u r e 6 .1 - A G e o m o r p h o l o g i c a l M a p of t h e W a t a w a l a 1 1 5 
E a r t h s l i d e 
F i g u r e 6 .2 - P l an view of t h e W a t a w a l a E a r t h s l i d e 1 1 5 
F i g u r e 6 .3 - L o n g i t u d i n a l S e c t i o n Y-Y of t h e W a t a w a l a 116 
L a n d s l i d e 
F i g u r e 6 .4 - C o n c e p t u a l Model of t h e R e h a b i l i t a t i o n Work 1 1 7 
( W a t a w a l a Lands l ide ) 
F i g u r e 6 . 5 - G r a p h i c a l R e p r e s e n t a t i o n of R e s u l t s ( W a t a w a l a 120 
S lope - High W a t e r T a b l e S i tua t ion ) 
F i g u r e 6 . 6 - G r a p h i c a l R e p r e s e n t a t i o n of R e s u l t s ( W a t a w a l a 121 
S lope - Low W a t e r T a b l e S i tua t ion ) 
F i g u r e 6 .7 - P l a n View of M a r a n g a h a w e l a Si te 123 
F i g u r e 6 .8 - P l an view of t h e s u b s u r f a c e d r a i n s a n d 123 
d r a i n a g e well 
F i g u r e 6 . 9 - C r o s s - s e c t i o n of t h e M a r a n g a h a w e l a S lope 124 
F i g u r e 7 .1 - G e o m e t r y of t h e C u t S lope 1 131 
tx 
4-
F i g u r e 7 .2 - P r o p o s e d Nail A r r a n g e m e n t a n d t h e F a i l u r e 133 
S u r f a c e s c o n s i d e r e d for C u t S lope 1 
F i g u r e 7 . 3 - G e o m e t r y of C u t S lope 2 135 
F i g u r e 7 .4 - P r o p o s e d Nail A r r a n g e m e n t a n d t h e F a i l u r e 137 
S u r f a c e s c o n s i d e r e d for C u t S lope 2 
F i g u r e 8 .1 - Var ia t ion of P robab i l i ty of F a i l u r e w i t h A n a l y s i s 140 
M e t h o d s ( S u m m a r y of E x a m p l e s - B i s h o p ' s 
Method) 
F i g u r e 8.2 - Var ia t ion of P robab i l i ty of F a i l u r e w i t h Ana lys i s 141 
M e t h o d s ( S u m m a r y of E x a m p l e s - J a n b u ' s 
Method) 
F i g u r e A l . 1 
F i g u r e A 1.2 
F i g u r e A2 .1 
F i g u r e A2 .2 
F i g u r e A 2 . 3 
F i g u r e A2 .4 
F i g u r e A 2 . 5 
F i g u r e A3 .1 
F i g u r e A8 .1 
F i g u r e A8 .2 
F i g u r e A 8 . 3 
F i g u r e A8 .4 
F i g u r e A 8 . 5 
- Typical P robab i l i ty D i s t r i b u t i o n of F a c t o r of II 
Safety 
- E s t i m a t i o n of P iezomet r i c Line p o s i t i o n s in a III 
Mon te Car lo t r ia l 
- Cri t ical F a i l u r e S u r f a c e c o n s i d e r e d in E x a m p l e V 
1(a) 
- Flow Net for E x a m p l e 1 (b) VII 
- Cri t ical F a i l u r e S u r f a c e for E x a m p l e 2 IX 
- Cri t ical C i r c u l a r F a i l u r e S u r f a c e for E x a m p l e 3 XII 
- Flow Net for E x a m p l e 3 XIII 
- C h a r t for f0 in J a n b u ' s s implif ied m e t h o d XVI 
- S u b s u r f a c e Profile of t h e W a t a w a l a S lope CVII 
Locat ion of I n s t r u m e n t s a t W a t a w a l a CVIII 
Lands l ide 
- Identif ied F a i l u r e S u r f a c e s a t W a t a w a l a CVIII 
Lands l i de 
- P l an View of t h e D i r ec t i ona l D r a i n s CIX 
- P l an View of t h e I n s t a l l a t i o n of E d u c t o r D r a i n s CIX 
x 
L I S T O F T A B L E S 
Page 
T a b l e 1.1 - Rel iabi l i ty I n d e x a n d Probab i l i ty of F a i l u r e V a l u e s 15 
for S l o p e s ( c o n s t a n t COV) 
T a b l e 1.2 - Coefficient of Var i a t i on of F (to h a v e c o n s t a n t 16 
* p r o p o r t i o n b e t w e e n (3 a n d F) 
T a b l e 1.3 - A c c e p t a b l e v a l u e s of p robab i l i t y of fa i lure for 17 
N a t u r a l S l o p e s 
T a b l e 3 .1 - D e t e r m i n i s t i c Soil P a r a m e t e r s for E x a m p l e 1(a) 4 3 
T a b l e 3 .2 - Coeff icients of Var i a t ion v a l u e s for c a n d <|> 4 4 
T a b l e 3 . 3 - R e s u l t s for E x a m p l e 1(a) 4 5 
T a b l e 3 .4 - D e t e r m i n i s t i c Soil P a r a m e t e r s for E x a m p l e 1(b) 4 7 
T a b l e 3 . 5 - Coeff icients of Var i a t i on v a l u e s for c' a n d <J>' 4 8 
T a b l e 3 .6 - M e a n a n d Coefficients of Va r i a t i on of u 4 9 
* T a b l e 3 .7 - R e s u l t s of E x a m p l e 1(b) 5 0 
T a b l e 4 . 1 - D e t e r m i n i s t i c Soil P a r a m e t e r s for E x a m p l e 2 7 1 
T a b l e 4 . 2 - Coeff icients of Var i a t i on v a l u e s for c' a n d <])' 7 1 
( E x a m p l e 2) 
T a b l e 4 . 3 - R e s u l t s of N o n - C i r c u l a r Ana lys i s for E x a m p l e 2 7 3 
T a b l e 4 . 4 - R e s u l t s of C i r c u l a r Ana lys i s for E x a m p l e 2 7 4 
T a b l e 4 . 5 - D e t e r m i n i s t i c Soil P a r a m e t e r s for E x a m p l e 3 7 7 
T a b l e 4 . 6 - Coefficient of Var i a t i on v a l u e s for c' a n d 7 8 
/ ( E x a m p l e 3) 
T a b l e 4 . 7 - M e a n a n d Coefficients of V a r i a t i o n of u for 7 9 
E x a m p l e 3 
T a b l e 4 . 8 - R e s u l t s of N o n - c i r c u l a r A n a l y s i s for E x a m p l e 3 8 0 
T a b l e 4 . 9 - R e s u l t s of C i r c u l a r Ana lys i s for E x a m p l e 3 8 1 
T a b l e 5.1 - R e s u l t s s h o w i n g t h e I m p r o v e m e n t of S tab i l i ty by 107 
D r a i n a g e (Example 4) 
f 
xi 
A 
T a b l e 5 . 2 
- R e s u l t s s h o w i n g t h e I m p r o v e m e n t of S t a b i l i t y b y 1 0 9 
S o i l N a i l i n g ( E x a m p l e 5) 
T a b l e 5 . 3 - R e s u l t s of E x a m p l e 6 1 1 1 
T a b l e 6 . 1 - R e s u l t s for H i g h W a t e r T a b l e S i t u a t i o n ( W a t a w a l a 1 1 9 
S l o p e ) 
T a b l e 6 . 2 -- R e s u l t s for L o w W a t e r T a b l e S i t u a t i o n ( W a t a w a l a 1 2 0 
S l o p e ) 
T a b l e 6 . 3 - P r o b a b i l i t y of F a i l u r e for M a r a n g a h a w e l a S l o p e 1 2 5 
( w i t h p r o b a b i l i s t i c p a r a m e t e r s e t 1) 
T a b l e 6 . 4 - P r o b a b i l i t y of F a i l u r e for M a r a n g a h a w e l a S l o p e 1 2 5 
( w i t h p r o b a b i l i s t i c p a r a m e t e r s e t 2) 
T a b l e 7 . 1 - S t r e n g t h P a r a m e t e r s of R e s i d u a l S o i l s - R a t n a p u r a 1 2 9 
M C A r e a 
T a b l e 7 . 2 - P e a k S t r e n g t h P a r a m e t e r s for C u t S l o p e 1 1 3 1 
T a b l e 7 . 3 - So i l N a i l A r r a n g e m e n t for C u t S l o p e 1 1 3 2 
T a b l e 7 . 4 - R e s u l t s for C u t S l o p e 1 - w i t h t h e g i v e n w a t e r t a b l e 1 3 3 
T a b l e 7 . 5 -• R e s u l t s for C u t S l o p e 1 - w i t h w a t e r t a b l e r e d u c e d 1 3 4 
b e l o w f a i l u r e s u r f a c e s 
T a b l e 7 . 6 -- R e s u l t s of C u t S l o p e 1 - w i t h t h e p r o p o s e d n a i l 1 3 4 
a r r a n g e m e n t 
T a b l e 7 . 7 - P e a k S t r e n g t h P a r a m e t e r s for C u t S l o p e 2 1 3 5 
T a b l e 7 . 8 - So i l N a i l A r r a n g e m e n t fo r C u t S l o p e 2 1 3 6 
T a b l e 7 . 9 - R e s u l t s for C u t S l o p e 2 - w i t h t h e g i v e n w a t e r t a b l e 1 3 7 
T a b l e 7 . 1 0 - R e s u l t s of C u t S l o p e 2 - w i t h t h e p r o p o s e d n a i l 1 3 7 
a r r a n g e m e n t 
T a b l e A 2 . 1 - S l i c e D e t a i l s for E x a m p l e 1 (a) VI 
T a b l e A 2 . 2 - C o o r d i n a t e s of P h r e a t i c S u r f a c e fo r E x a m p l e 1 (b) VII 
T a b l e A 2 . 3 - S l i c e D e t a i l s of E x a m p l e 1 (b) VIII 
T a b l e A 2 . 4 - F a i l u r e S u r f a c e C o o r d i n a t e s for E x a m p l e 2 IX 
T a b l e A 2 . 5 - S l i c e D e t a i l s for E x a m p l e 2 ( N o n C i r c u l a r F a i l u r e X 
S u r f a c e ) 
f 
xii 
T a b l e A 2 . 6 - Sl ice De t a i l s for E x a m p l e 2 (C i rcu la r F a i l u r e XI 
Sur face) 
T a b l e A 2 . 7 - C o o r d i n a t e s of t h e Cr i t ica l F a i l u r e S u r f a c e for XII 
E x a m p l e 3 
T a b l e A 2 . 8 - C o o r d i n a t e s of P iezomet r ic Su r f ace for E x a m p l e 3 XIII 
T a b l e A 2 . 9 - Sl ice De ta i l s for E x a m p l e 3 (Non C i r c u l a r F a i l u r e XIV 
Surface) 
> T a b l e A 2 . 1 0 - Slice De ta i l s for E x a m p l e 3 (Ci rcu la r F a i l u r e XV 
Surface) 
xiii 
L I S T O F A N N E X E S 
Page 
Annex 1 - Monte Carlo Analysis Performed by SLOPE/W I 
Annex 2 - Details of the Examples V 
Annex 3 - Chart for/o in Janbu's simplified method XVI 
Annex 4 - Sample set of Spreadsheets - Model Based on XVII 
Bishop's Method 
Annex 5 - Sample set of Spreadsheets - Model Based on LII 
Janbu's simplified Method 
Annex 6 - Sample set of Spreadsheets - Model Based on LXXXIX 
Bishop's Method for Slopes Stabilized by Soil 
Nailing 
Annex 7 - Sample set of Spreadsheets - Model Based on XCVII 
Janbu's simplified Method for Slopes Stabilized 
by Soil Nailing 
Annex 8 - Additional Details of the Stabilizzed Watawala CVII 
Landslide 
L I S T O F S Y M B O L S 
Mean Factor of Safety 
Mean Value of "i"th Random Variable 
Angle of the Nail to the Horizontal 
Angle of Internal Friction 
Bulk Density 
Standard Deviation 
Reliability Index 
Effective Angle of Internal Friction 
Standard Deviation of Effective Angle of Internal Friction 
Standard Deviation of Effective Cohesion 
Standard Deviation of Factor of Safety 
Standard Deviation of Uncorrected Factor of Safety 
Slice Angle of "i",h Slice 
Lognormal Reliability Index 
Undrained Angle of Internal Friction 
Standard Deviation of Pore Pressure 
Small Increment applicable ti "i"th Random Variable 
Standard Deviation of "i"lh Random Variable 
Width of "i"'h Slice 
Cohesion 
Effective Cohesion 
Coefficient of Variation 
Coefficient of Variation of Factor of Safety 
Coefficient of Variation of Angle of Internal Friction 
Coefficient of Variation of Cohesion 
Coefficient of Variation of Pore Pressure 
Undrained Cohesion 
Diameter of a Nail 
Side Normal Force acting on "i"th Slice (Left Boundary) 
Side Normal Force acting on "i"lh Slice (Right Boundary) 
Factor of Safety 
Uncorrected Factor of Safety for Janbu's Simplified Method 
Janbu's Correction Factor for Factor of Safety 
Bond Coefficient 
Yield Strength of Nail Material 
HCV - Highest Conceivaoie Value 
hi - Average Height of the "i"* Slice 
1 - Effective Length of a Nail 
LCV - Lowest Conceivable Value 
n - Total number of Nails passing through a particular Slice 
N - Total number of samples 
Ni' - Effective Normal Force on "i"lh Slice 
Pr - Probability of Failure 
Pr(LN) - Probability of Failure assuming Lognormal Distribution for FOS 
Pf(N) - Probability of Failure assuming Normal Distribution for FOS 
Qi - Surcharge acting on "i"th Slice 
SM - Safety Margin 
T m , i - Shear Force acting along the Failure Surface on the "i"th Slicce 
T n - Force Mobilized in Nails 
T N , i - Nail Force acting on the "i"th Slice 
Ui - Pore Pressure Force on "i"th Slice 
U i - Pore Pressure acting on "i"th Slice 
V p - Variance of Factor of Safety 
Wi - Weight of V t h Slice 
X i - "i" t h random variable 
N o t e : 
S y m b o l s u s e d in t h e f lowchar t s a r e d e s c r i b e d in t h e s a m e f lowchar t s 
j