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SLOPE STABILITY ANALYSIS
ONGCHEEZEN
Bachelor of Engineering with Honours TA (Civil Engineering) 710
2005058 2005
UNIVERSITl MALAYSIA SARA W A K
BORANG PENYERAHAN STATUS TESIS
Judu I _ _____ --SLO-P-F-S-TABI-L---IT--Y----AJ-NAL-Y-SIS--____ _
Ses i Pengajian 2004 - 20Q5
Saya ONGCHEE nN
(HU RUF BESAR)
mengaku rnrnbenarka l1 tc~ i s bull ini d i~ i mpan di Pusat Khidmat Mak lumll l Akademi k Uniycri ti Mol3)s lO Silrjvak dengiln syarat-syarat Jcgu na1Il sepert i bcrikul
I [ ~si s adalah ham ilik Universili M alaysia SJrltluk
2 Pusat Khiumat Makl ll11lat Akad~ll1i k Uni versili Malaysia Sara a~ di bcnarklln nHm huu t alillnn Unluf
Lujuan pengaj ian sahaja
3 Me-mhuH pengdigila n ulI wk membangunkan Pengkalan Dala Kandungan I c-mpaLan
4 PllaL Kh idmat maklumat Akad~mlk l nimiddotersiti Malaysia Saravak Jibcnarkan rnemhuat salinn Lc)is int
sebagai bahan perlu karan antara institu i pcngaj im Linggi
5 u Sil la n~1k )n ( i ) di kolak berkentian
D SlJLlT (Mengandungi Inakl um3t yang 1~rdJri h keselam3lan 8[lU kcpcntingan M() l aJ ~ja sepel1i )8 ng hmmlt ub Ji dulal11 A I fA RA HSI A RA SMI 197 )
D T ERHAO (Mengandu ngl makhlmat TERH A O )cmg tel ah di hnlukan ol th orga nisas ibadan di mana pellyelidi kan dijalankan)
~ T IO AK TERH AD
D i sa hkan~le h
~ (TAN DATfN GAN PENULl S) (TANDATANOAN lENYrt I)
AL AM A T middot11 r AP Dr Vishwas A Sawanc
17Ta llliJI1 Srimcwah
Jalan CIHkrawasih Nama Pen yelia 05460Alor SClar Kedh
Tarikh Tarikh
CATATAN fUli dim ak$udkJn Stb9gU t(Si~ bali Ijatah Doktor F lslrah Sarjana rl3n r)fl rjana Mud
lil3 hj ini SlILI ar ~ u T(RH O ~ ilu lamj)irkan SUrln datipadl pihak beru~11I1ani s si b(rkcnlla n tI(12 3 n
men ~ lIlakan seklrh ~C lJlI da1 Ifmpuh I~i ~ jrd pcrlu dikelaskan scbllga i I t LIT dan f IRJ IA O
The following Final Year Project Report
Titl e SLOPE STABILITY ANALYSIS
Name ONG CHEE ZEN
Malri x Number 7479
has been read and approved by
DR VISHWASSAWANT Date
Projec t Superv isor
Iusa KhidOlak1~~lIlJI [4 kademl) UNIVERSITl IvTAIAYSTA~ SRAWAJC
PKHIDMATMAKlUMAT AKADEMIK 9d Yll Ko Samarahan UWIMAS
11111111 1111111111111000137577
SLOPE STBILITY ANALYSIS
ONG CHEEZEN
This project is submitted in partial of fulfilment of the requirements for the degree of Bachelor of Engineering w ith Honours
(Civi l Engineerin g)
Faculty of Engineering UNlVERSlTl MALAY SIA SARA WA
2005
ACKNOWLEDGEMENT
First of all I would like to render my sincere thanks to my superv isor Dr
Vishwas Sawan for his va luable guidance His experience in the subject has drawn
up to thi s synopsis to an extent which cannot be expressed by more word s H is
sincere help at every stage o f thi s parametric synopsis has seen us through th is good
pi ece of work
I also want to thank my beloved father and mother fo r all their mora l and
financial supports within this year and also to my dearest broth ers and sisters for all
of their help and supportsLast but not least thank yo u to all my fr iends who have
shared their suggest ions and evaluations of thi s scri pt
ABSTRACT
Slope is an exposed ground surface that stands at an angle with the hori zonta l
If the component of gravity is large enough which mean driving force overcomes the
shea r strength of the soil along the rupture surface slope failure can occur There arc
few method developed for checking the safety of s lopes Among them th e
Morgenstern and Price method has been chosen for thi s parametric stud y
Morgenstren and Price Method is used for analyzing the slope stab ility in moment
eq uilibrium for general s lip s lope surface Normal and shea r forces acting on ve rti ca l
sides of the slice are al so taken into account Prop0l1ionality constant I~ between
shear and normal force is treated as unknown along with the factor of safety F The
va lues o f these two unkno wns (F and A) are evaluated using an iterati ve procedure A
so ftware program is written in Fortran90 which calculates the minimum factor of
sa fety for a certain slope angle and height with a so il parameters From the present
parametric stud y with a constant slope angle factor of safety increases with angle of
internal friction and cohesion Next for a given friction angle factor of safety
decreases wi th s lope angle Beside for a given cohes ion factor of safety dec reases
wi th s lope angle
)I
ABSTRAK
Cerun merupakan permukaan ysng berada dalam keadaan bersudut dengan ga ri s
mengufuk Iika da ya tarikan gravi ti yang cukup ini bermaksud daya tarikan sudah
mengatasi shear strength tanah pada permukaan gelangsar kagagalan cerun akan
berlakuTerdapat beberapa kaedah anal isis telah diperkena lkan bagi tujuan penentuan
kestabilan cerun Antaranya kaedah Morgenstern and Price telah dipilih untuk kajian
parametric Kaedah Morgenstern and Price digunakan untuk anal is is kestabilan
momen bagi permukaan cerun Oaya ri c ih dan normal bCI1indak ke atas arah menegak
bagi cerun Kadar pemalar Ie dengan da ya ri c ih dan daya normal dianggap seba ga i
anu-anu bagi fac tor keselamatan F N ilai F dan Ie daanali sis dengan kaeda h iterative
Satu peri s ian dituli skan dalam IROTRAN 90 untuk penggiraan minimum factor
keselamatan dcngan sudut kecerunan dan ketinggian cerun seL-ta parameter tanah
tertentu Bagi kajian parametric ini diberi kecerunan sudut sebagai permalar factor
kese lamatan meningkat dengan sudut tanah dan cohesion meningkat Kem udian
diberi sudut tanah sebagai permalar factor keselamatan menurun dengan sudut cerun
men in gkat Selain itu diberi cohesion sebaga i permalar factor kese lamatan menurul1
dengan sudut cerun meningkat
III
bull bull bull bull I
n
TABLE OF CONTENTS
CONTENT age no
ACKNO WLEDGEMENT
ABSTR ACT II
ABSTRAK TAB LE OF CONTENTS IV
LIST OF TABLES V II
LIST OF FIG UR ES VIII
LIST OF NOMENLATURES x
CHAPTER I INTRODUCTION
11 GENERAL
1 2 HI STOR ICAL BA CKGROUN D 2
1 3 TY PE OF SLO PE 3
1 31 NAT URAL 3
132 EXCA VATED SLOPES 4
133 SL OPES OF EMB ANKMENTS AND EARTH DAMS 4
14 PROJ ECT OBJECTI VE 5
IV
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
UNIVERSITl MALAYSIA SARA W A K
BORANG PENYERAHAN STATUS TESIS
Judu I _ _____ --SLO-P-F-S-TABI-L---IT--Y----AJ-NAL-Y-SIS--____ _
Ses i Pengajian 2004 - 20Q5
Saya ONGCHEE nN
(HU RUF BESAR)
mengaku rnrnbenarka l1 tc~ i s bull ini d i~ i mpan di Pusat Khidmat Mak lumll l Akademi k Uniycri ti Mol3)s lO Silrjvak dengiln syarat-syarat Jcgu na1Il sepert i bcrikul
I [ ~si s adalah ham ilik Universili M alaysia SJrltluk
2 Pusat Khiumat Makl ll11lat Akad~ll1i k Uni versili Malaysia Sara a~ di bcnarklln nHm huu t alillnn Unluf
Lujuan pengaj ian sahaja
3 Me-mhuH pengdigila n ulI wk membangunkan Pengkalan Dala Kandungan I c-mpaLan
4 PllaL Kh idmat maklumat Akad~mlk l nimiddotersiti Malaysia Saravak Jibcnarkan rnemhuat salinn Lc)is int
sebagai bahan perlu karan antara institu i pcngaj im Linggi
5 u Sil la n~1k )n ( i ) di kolak berkentian
D SlJLlT (Mengandungi Inakl um3t yang 1~rdJri h keselam3lan 8[lU kcpcntingan M() l aJ ~ja sepel1i )8 ng hmmlt ub Ji dulal11 A I fA RA HSI A RA SMI 197 )
D T ERHAO (Mengandu ngl makhlmat TERH A O )cmg tel ah di hnlukan ol th orga nisas ibadan di mana pellyelidi kan dijalankan)
~ T IO AK TERH AD
D i sa hkan~le h
~ (TAN DATfN GAN PENULl S) (TANDATANOAN lENYrt I)
AL AM A T middot11 r AP Dr Vishwas A Sawanc
17Ta llliJI1 Srimcwah
Jalan CIHkrawasih Nama Pen yelia 05460Alor SClar Kedh
Tarikh Tarikh
CATATAN fUli dim ak$udkJn Stb9gU t(Si~ bali Ijatah Doktor F lslrah Sarjana rl3n r)fl rjana Mud
lil3 hj ini SlILI ar ~ u T(RH O ~ ilu lamj)irkan SUrln datipadl pihak beru~11I1ani s si b(rkcnlla n tI(12 3 n
men ~ lIlakan seklrh ~C lJlI da1 Ifmpuh I~i ~ jrd pcrlu dikelaskan scbllga i I t LIT dan f IRJ IA O
The following Final Year Project Report
Titl e SLOPE STABILITY ANALYSIS
Name ONG CHEE ZEN
Malri x Number 7479
has been read and approved by
DR VISHWASSAWANT Date
Projec t Superv isor
Iusa KhidOlak1~~lIlJI [4 kademl) UNIVERSITl IvTAIAYSTA~ SRAWAJC
PKHIDMATMAKlUMAT AKADEMIK 9d Yll Ko Samarahan UWIMAS
11111111 1111111111111000137577
SLOPE STBILITY ANALYSIS
ONG CHEEZEN
This project is submitted in partial of fulfilment of the requirements for the degree of Bachelor of Engineering w ith Honours
(Civi l Engineerin g)
Faculty of Engineering UNlVERSlTl MALAY SIA SARA WA
2005
ACKNOWLEDGEMENT
First of all I would like to render my sincere thanks to my superv isor Dr
Vishwas Sawan for his va luable guidance His experience in the subject has drawn
up to thi s synopsis to an extent which cannot be expressed by more word s H is
sincere help at every stage o f thi s parametric synopsis has seen us through th is good
pi ece of work
I also want to thank my beloved father and mother fo r all their mora l and
financial supports within this year and also to my dearest broth ers and sisters for all
of their help and supportsLast but not least thank yo u to all my fr iends who have
shared their suggest ions and evaluations of thi s scri pt
ABSTRACT
Slope is an exposed ground surface that stands at an angle with the hori zonta l
If the component of gravity is large enough which mean driving force overcomes the
shea r strength of the soil along the rupture surface slope failure can occur There arc
few method developed for checking the safety of s lopes Among them th e
Morgenstern and Price method has been chosen for thi s parametric stud y
Morgenstren and Price Method is used for analyzing the slope stab ility in moment
eq uilibrium for general s lip s lope surface Normal and shea r forces acting on ve rti ca l
sides of the slice are al so taken into account Prop0l1ionality constant I~ between
shear and normal force is treated as unknown along with the factor of safety F The
va lues o f these two unkno wns (F and A) are evaluated using an iterati ve procedure A
so ftware program is written in Fortran90 which calculates the minimum factor of
sa fety for a certain slope angle and height with a so il parameters From the present
parametric stud y with a constant slope angle factor of safety increases with angle of
internal friction and cohesion Next for a given friction angle factor of safety
decreases wi th s lope angle Beside for a given cohes ion factor of safety dec reases
wi th s lope angle
)I
ABSTRAK
Cerun merupakan permukaan ysng berada dalam keadaan bersudut dengan ga ri s
mengufuk Iika da ya tarikan gravi ti yang cukup ini bermaksud daya tarikan sudah
mengatasi shear strength tanah pada permukaan gelangsar kagagalan cerun akan
berlakuTerdapat beberapa kaedah anal isis telah diperkena lkan bagi tujuan penentuan
kestabilan cerun Antaranya kaedah Morgenstern and Price telah dipilih untuk kajian
parametric Kaedah Morgenstern and Price digunakan untuk anal is is kestabilan
momen bagi permukaan cerun Oaya ri c ih dan normal bCI1indak ke atas arah menegak
bagi cerun Kadar pemalar Ie dengan da ya ri c ih dan daya normal dianggap seba ga i
anu-anu bagi fac tor keselamatan F N ilai F dan Ie daanali sis dengan kaeda h iterative
Satu peri s ian dituli skan dalam IROTRAN 90 untuk penggiraan minimum factor
keselamatan dcngan sudut kecerunan dan ketinggian cerun seL-ta parameter tanah
tertentu Bagi kajian parametric ini diberi kecerunan sudut sebagai permalar factor
kese lamatan meningkat dengan sudut tanah dan cohesion meningkat Kem udian
diberi sudut tanah sebagai permalar factor keselamatan menurun dengan sudut cerun
men in gkat Selain itu diberi cohesion sebaga i permalar factor kese lamatan menurul1
dengan sudut cerun meningkat
III
bull bull bull bull I
n
TABLE OF CONTENTS
CONTENT age no
ACKNO WLEDGEMENT
ABSTR ACT II
ABSTRAK TAB LE OF CONTENTS IV
LIST OF TABLES V II
LIST OF FIG UR ES VIII
LIST OF NOMENLATURES x
CHAPTER I INTRODUCTION
11 GENERAL
1 2 HI STOR ICAL BA CKGROUN D 2
1 3 TY PE OF SLO PE 3
1 31 NAT URAL 3
132 EXCA VATED SLOPES 4
133 SL OPES OF EMB ANKMENTS AND EARTH DAMS 4
14 PROJ ECT OBJECTI VE 5
IV
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
The following Final Year Project Report
Titl e SLOPE STABILITY ANALYSIS
Name ONG CHEE ZEN
Malri x Number 7479
has been read and approved by
DR VISHWASSAWANT Date
Projec t Superv isor
Iusa KhidOlak1~~lIlJI [4 kademl) UNIVERSITl IvTAIAYSTA~ SRAWAJC
PKHIDMATMAKlUMAT AKADEMIK 9d Yll Ko Samarahan UWIMAS
11111111 1111111111111000137577
SLOPE STBILITY ANALYSIS
ONG CHEEZEN
This project is submitted in partial of fulfilment of the requirements for the degree of Bachelor of Engineering w ith Honours
(Civi l Engineerin g)
Faculty of Engineering UNlVERSlTl MALAY SIA SARA WA
2005
ACKNOWLEDGEMENT
First of all I would like to render my sincere thanks to my superv isor Dr
Vishwas Sawan for his va luable guidance His experience in the subject has drawn
up to thi s synopsis to an extent which cannot be expressed by more word s H is
sincere help at every stage o f thi s parametric synopsis has seen us through th is good
pi ece of work
I also want to thank my beloved father and mother fo r all their mora l and
financial supports within this year and also to my dearest broth ers and sisters for all
of their help and supportsLast but not least thank yo u to all my fr iends who have
shared their suggest ions and evaluations of thi s scri pt
ABSTRACT
Slope is an exposed ground surface that stands at an angle with the hori zonta l
If the component of gravity is large enough which mean driving force overcomes the
shea r strength of the soil along the rupture surface slope failure can occur There arc
few method developed for checking the safety of s lopes Among them th e
Morgenstern and Price method has been chosen for thi s parametric stud y
Morgenstren and Price Method is used for analyzing the slope stab ility in moment
eq uilibrium for general s lip s lope surface Normal and shea r forces acting on ve rti ca l
sides of the slice are al so taken into account Prop0l1ionality constant I~ between
shear and normal force is treated as unknown along with the factor of safety F The
va lues o f these two unkno wns (F and A) are evaluated using an iterati ve procedure A
so ftware program is written in Fortran90 which calculates the minimum factor of
sa fety for a certain slope angle and height with a so il parameters From the present
parametric stud y with a constant slope angle factor of safety increases with angle of
internal friction and cohesion Next for a given friction angle factor of safety
decreases wi th s lope angle Beside for a given cohes ion factor of safety dec reases
wi th s lope angle
)I
ABSTRAK
Cerun merupakan permukaan ysng berada dalam keadaan bersudut dengan ga ri s
mengufuk Iika da ya tarikan gravi ti yang cukup ini bermaksud daya tarikan sudah
mengatasi shear strength tanah pada permukaan gelangsar kagagalan cerun akan
berlakuTerdapat beberapa kaedah anal isis telah diperkena lkan bagi tujuan penentuan
kestabilan cerun Antaranya kaedah Morgenstern and Price telah dipilih untuk kajian
parametric Kaedah Morgenstern and Price digunakan untuk anal is is kestabilan
momen bagi permukaan cerun Oaya ri c ih dan normal bCI1indak ke atas arah menegak
bagi cerun Kadar pemalar Ie dengan da ya ri c ih dan daya normal dianggap seba ga i
anu-anu bagi fac tor keselamatan F N ilai F dan Ie daanali sis dengan kaeda h iterative
Satu peri s ian dituli skan dalam IROTRAN 90 untuk penggiraan minimum factor
keselamatan dcngan sudut kecerunan dan ketinggian cerun seL-ta parameter tanah
tertentu Bagi kajian parametric ini diberi kecerunan sudut sebagai permalar factor
kese lamatan meningkat dengan sudut tanah dan cohesion meningkat Kem udian
diberi sudut tanah sebagai permalar factor keselamatan menurun dengan sudut cerun
men in gkat Selain itu diberi cohesion sebaga i permalar factor kese lamatan menurul1
dengan sudut cerun meningkat
III
bull bull bull bull I
n
TABLE OF CONTENTS
CONTENT age no
ACKNO WLEDGEMENT
ABSTR ACT II
ABSTRAK TAB LE OF CONTENTS IV
LIST OF TABLES V II
LIST OF FIG UR ES VIII
LIST OF NOMENLATURES x
CHAPTER I INTRODUCTION
11 GENERAL
1 2 HI STOR ICAL BA CKGROUN D 2
1 3 TY PE OF SLO PE 3
1 31 NAT URAL 3
132 EXCA VATED SLOPES 4
133 SL OPES OF EMB ANKMENTS AND EARTH DAMS 4
14 PROJ ECT OBJECTI VE 5
IV
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
Iusa KhidOlak1~~lIlJI [4 kademl) UNIVERSITl IvTAIAYSTA~ SRAWAJC
PKHIDMATMAKlUMAT AKADEMIK 9d Yll Ko Samarahan UWIMAS
11111111 1111111111111000137577
SLOPE STBILITY ANALYSIS
ONG CHEEZEN
This project is submitted in partial of fulfilment of the requirements for the degree of Bachelor of Engineering w ith Honours
(Civi l Engineerin g)
Faculty of Engineering UNlVERSlTl MALAY SIA SARA WA
2005
ACKNOWLEDGEMENT
First of all I would like to render my sincere thanks to my superv isor Dr
Vishwas Sawan for his va luable guidance His experience in the subject has drawn
up to thi s synopsis to an extent which cannot be expressed by more word s H is
sincere help at every stage o f thi s parametric synopsis has seen us through th is good
pi ece of work
I also want to thank my beloved father and mother fo r all their mora l and
financial supports within this year and also to my dearest broth ers and sisters for all
of their help and supportsLast but not least thank yo u to all my fr iends who have
shared their suggest ions and evaluations of thi s scri pt
ABSTRACT
Slope is an exposed ground surface that stands at an angle with the hori zonta l
If the component of gravity is large enough which mean driving force overcomes the
shea r strength of the soil along the rupture surface slope failure can occur There arc
few method developed for checking the safety of s lopes Among them th e
Morgenstern and Price method has been chosen for thi s parametric stud y
Morgenstren and Price Method is used for analyzing the slope stab ility in moment
eq uilibrium for general s lip s lope surface Normal and shea r forces acting on ve rti ca l
sides of the slice are al so taken into account Prop0l1ionality constant I~ between
shear and normal force is treated as unknown along with the factor of safety F The
va lues o f these two unkno wns (F and A) are evaluated using an iterati ve procedure A
so ftware program is written in Fortran90 which calculates the minimum factor of
sa fety for a certain slope angle and height with a so il parameters From the present
parametric stud y with a constant slope angle factor of safety increases with angle of
internal friction and cohesion Next for a given friction angle factor of safety
decreases wi th s lope angle Beside for a given cohes ion factor of safety dec reases
wi th s lope angle
)I
ABSTRAK
Cerun merupakan permukaan ysng berada dalam keadaan bersudut dengan ga ri s
mengufuk Iika da ya tarikan gravi ti yang cukup ini bermaksud daya tarikan sudah
mengatasi shear strength tanah pada permukaan gelangsar kagagalan cerun akan
berlakuTerdapat beberapa kaedah anal isis telah diperkena lkan bagi tujuan penentuan
kestabilan cerun Antaranya kaedah Morgenstern and Price telah dipilih untuk kajian
parametric Kaedah Morgenstern and Price digunakan untuk anal is is kestabilan
momen bagi permukaan cerun Oaya ri c ih dan normal bCI1indak ke atas arah menegak
bagi cerun Kadar pemalar Ie dengan da ya ri c ih dan daya normal dianggap seba ga i
anu-anu bagi fac tor keselamatan F N ilai F dan Ie daanali sis dengan kaeda h iterative
Satu peri s ian dituli skan dalam IROTRAN 90 untuk penggiraan minimum factor
keselamatan dcngan sudut kecerunan dan ketinggian cerun seL-ta parameter tanah
tertentu Bagi kajian parametric ini diberi kecerunan sudut sebagai permalar factor
kese lamatan meningkat dengan sudut tanah dan cohesion meningkat Kem udian
diberi sudut tanah sebagai permalar factor keselamatan menurun dengan sudut cerun
men in gkat Selain itu diberi cohesion sebaga i permalar factor kese lamatan menurul1
dengan sudut cerun meningkat
III
bull bull bull bull I
n
TABLE OF CONTENTS
CONTENT age no
ACKNO WLEDGEMENT
ABSTR ACT II
ABSTRAK TAB LE OF CONTENTS IV
LIST OF TABLES V II
LIST OF FIG UR ES VIII
LIST OF NOMENLATURES x
CHAPTER I INTRODUCTION
11 GENERAL
1 2 HI STOR ICAL BA CKGROUN D 2
1 3 TY PE OF SLO PE 3
1 31 NAT URAL 3
132 EXCA VATED SLOPES 4
133 SL OPES OF EMB ANKMENTS AND EARTH DAMS 4
14 PROJ ECT OBJECTI VE 5
IV
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
ACKNOWLEDGEMENT
First of all I would like to render my sincere thanks to my superv isor Dr
Vishwas Sawan for his va luable guidance His experience in the subject has drawn
up to thi s synopsis to an extent which cannot be expressed by more word s H is
sincere help at every stage o f thi s parametric synopsis has seen us through th is good
pi ece of work
I also want to thank my beloved father and mother fo r all their mora l and
financial supports within this year and also to my dearest broth ers and sisters for all
of their help and supportsLast but not least thank yo u to all my fr iends who have
shared their suggest ions and evaluations of thi s scri pt
ABSTRACT
Slope is an exposed ground surface that stands at an angle with the hori zonta l
If the component of gravity is large enough which mean driving force overcomes the
shea r strength of the soil along the rupture surface slope failure can occur There arc
few method developed for checking the safety of s lopes Among them th e
Morgenstern and Price method has been chosen for thi s parametric stud y
Morgenstren and Price Method is used for analyzing the slope stab ility in moment
eq uilibrium for general s lip s lope surface Normal and shea r forces acting on ve rti ca l
sides of the slice are al so taken into account Prop0l1ionality constant I~ between
shear and normal force is treated as unknown along with the factor of safety F The
va lues o f these two unkno wns (F and A) are evaluated using an iterati ve procedure A
so ftware program is written in Fortran90 which calculates the minimum factor of
sa fety for a certain slope angle and height with a so il parameters From the present
parametric stud y with a constant slope angle factor of safety increases with angle of
internal friction and cohesion Next for a given friction angle factor of safety
decreases wi th s lope angle Beside for a given cohes ion factor of safety dec reases
wi th s lope angle
)I
ABSTRAK
Cerun merupakan permukaan ysng berada dalam keadaan bersudut dengan ga ri s
mengufuk Iika da ya tarikan gravi ti yang cukup ini bermaksud daya tarikan sudah
mengatasi shear strength tanah pada permukaan gelangsar kagagalan cerun akan
berlakuTerdapat beberapa kaedah anal isis telah diperkena lkan bagi tujuan penentuan
kestabilan cerun Antaranya kaedah Morgenstern and Price telah dipilih untuk kajian
parametric Kaedah Morgenstern and Price digunakan untuk anal is is kestabilan
momen bagi permukaan cerun Oaya ri c ih dan normal bCI1indak ke atas arah menegak
bagi cerun Kadar pemalar Ie dengan da ya ri c ih dan daya normal dianggap seba ga i
anu-anu bagi fac tor keselamatan F N ilai F dan Ie daanali sis dengan kaeda h iterative
Satu peri s ian dituli skan dalam IROTRAN 90 untuk penggiraan minimum factor
keselamatan dcngan sudut kecerunan dan ketinggian cerun seL-ta parameter tanah
tertentu Bagi kajian parametric ini diberi kecerunan sudut sebagai permalar factor
kese lamatan meningkat dengan sudut tanah dan cohesion meningkat Kem udian
diberi sudut tanah sebagai permalar factor keselamatan menurun dengan sudut cerun
men in gkat Selain itu diberi cohesion sebaga i permalar factor kese lamatan menurul1
dengan sudut cerun meningkat
III
bull bull bull bull I
n
TABLE OF CONTENTS
CONTENT age no
ACKNO WLEDGEMENT
ABSTR ACT II
ABSTRAK TAB LE OF CONTENTS IV
LIST OF TABLES V II
LIST OF FIG UR ES VIII
LIST OF NOMENLATURES x
CHAPTER I INTRODUCTION
11 GENERAL
1 2 HI STOR ICAL BA CKGROUN D 2
1 3 TY PE OF SLO PE 3
1 31 NAT URAL 3
132 EXCA VATED SLOPES 4
133 SL OPES OF EMB ANKMENTS AND EARTH DAMS 4
14 PROJ ECT OBJECTI VE 5
IV
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
ABSTRACT
Slope is an exposed ground surface that stands at an angle with the hori zonta l
If the component of gravity is large enough which mean driving force overcomes the
shea r strength of the soil along the rupture surface slope failure can occur There arc
few method developed for checking the safety of s lopes Among them th e
Morgenstern and Price method has been chosen for thi s parametric stud y
Morgenstren and Price Method is used for analyzing the slope stab ility in moment
eq uilibrium for general s lip s lope surface Normal and shea r forces acting on ve rti ca l
sides of the slice are al so taken into account Prop0l1ionality constant I~ between
shear and normal force is treated as unknown along with the factor of safety F The
va lues o f these two unkno wns (F and A) are evaluated using an iterati ve procedure A
so ftware program is written in Fortran90 which calculates the minimum factor of
sa fety for a certain slope angle and height with a so il parameters From the present
parametric stud y with a constant slope angle factor of safety increases with angle of
internal friction and cohesion Next for a given friction angle factor of safety
decreases wi th s lope angle Beside for a given cohes ion factor of safety dec reases
wi th s lope angle
)I
ABSTRAK
Cerun merupakan permukaan ysng berada dalam keadaan bersudut dengan ga ri s
mengufuk Iika da ya tarikan gravi ti yang cukup ini bermaksud daya tarikan sudah
mengatasi shear strength tanah pada permukaan gelangsar kagagalan cerun akan
berlakuTerdapat beberapa kaedah anal isis telah diperkena lkan bagi tujuan penentuan
kestabilan cerun Antaranya kaedah Morgenstern and Price telah dipilih untuk kajian
parametric Kaedah Morgenstern and Price digunakan untuk anal is is kestabilan
momen bagi permukaan cerun Oaya ri c ih dan normal bCI1indak ke atas arah menegak
bagi cerun Kadar pemalar Ie dengan da ya ri c ih dan daya normal dianggap seba ga i
anu-anu bagi fac tor keselamatan F N ilai F dan Ie daanali sis dengan kaeda h iterative
Satu peri s ian dituli skan dalam IROTRAN 90 untuk penggiraan minimum factor
keselamatan dcngan sudut kecerunan dan ketinggian cerun seL-ta parameter tanah
tertentu Bagi kajian parametric ini diberi kecerunan sudut sebagai permalar factor
kese lamatan meningkat dengan sudut tanah dan cohesion meningkat Kem udian
diberi sudut tanah sebagai permalar factor keselamatan menurun dengan sudut cerun
men in gkat Selain itu diberi cohesion sebaga i permalar factor kese lamatan menurul1
dengan sudut cerun meningkat
III
bull bull bull bull I
n
TABLE OF CONTENTS
CONTENT age no
ACKNO WLEDGEMENT
ABSTR ACT II
ABSTRAK TAB LE OF CONTENTS IV
LIST OF TABLES V II
LIST OF FIG UR ES VIII
LIST OF NOMENLATURES x
CHAPTER I INTRODUCTION
11 GENERAL
1 2 HI STOR ICAL BA CKGROUN D 2
1 3 TY PE OF SLO PE 3
1 31 NAT URAL 3
132 EXCA VATED SLOPES 4
133 SL OPES OF EMB ANKMENTS AND EARTH DAMS 4
14 PROJ ECT OBJECTI VE 5
IV
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
ABSTRAK
Cerun merupakan permukaan ysng berada dalam keadaan bersudut dengan ga ri s
mengufuk Iika da ya tarikan gravi ti yang cukup ini bermaksud daya tarikan sudah
mengatasi shear strength tanah pada permukaan gelangsar kagagalan cerun akan
berlakuTerdapat beberapa kaedah anal isis telah diperkena lkan bagi tujuan penentuan
kestabilan cerun Antaranya kaedah Morgenstern and Price telah dipilih untuk kajian
parametric Kaedah Morgenstern and Price digunakan untuk anal is is kestabilan
momen bagi permukaan cerun Oaya ri c ih dan normal bCI1indak ke atas arah menegak
bagi cerun Kadar pemalar Ie dengan da ya ri c ih dan daya normal dianggap seba ga i
anu-anu bagi fac tor keselamatan F N ilai F dan Ie daanali sis dengan kaeda h iterative
Satu peri s ian dituli skan dalam IROTRAN 90 untuk penggiraan minimum factor
keselamatan dcngan sudut kecerunan dan ketinggian cerun seL-ta parameter tanah
tertentu Bagi kajian parametric ini diberi kecerunan sudut sebagai permalar factor
kese lamatan meningkat dengan sudut tanah dan cohesion meningkat Kem udian
diberi sudut tanah sebagai permalar factor keselamatan menurun dengan sudut cerun
men in gkat Selain itu diberi cohesion sebaga i permalar factor kese lamatan menurul1
dengan sudut cerun meningkat
III
bull bull bull bull I
n
TABLE OF CONTENTS
CONTENT age no
ACKNO WLEDGEMENT
ABSTR ACT II
ABSTRAK TAB LE OF CONTENTS IV
LIST OF TABLES V II
LIST OF FIG UR ES VIII
LIST OF NOMENLATURES x
CHAPTER I INTRODUCTION
11 GENERAL
1 2 HI STOR ICAL BA CKGROUN D 2
1 3 TY PE OF SLO PE 3
1 31 NAT URAL 3
132 EXCA VATED SLOPES 4
133 SL OPES OF EMB ANKMENTS AND EARTH DAMS 4
14 PROJ ECT OBJECTI VE 5
IV
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
bull bull bull bull I
n
TABLE OF CONTENTS
CONTENT age no
ACKNO WLEDGEMENT
ABSTR ACT II
ABSTRAK TAB LE OF CONTENTS IV
LIST OF TABLES V II
LIST OF FIG UR ES VIII
LIST OF NOMENLATURES x
CHAPTER I INTRODUCTION
11 GENERAL
1 2 HI STOR ICAL BA CKGROUN D 2
1 3 TY PE OF SLO PE 3
1 31 NAT URAL 3
132 EXCA VATED SLOPES 4
133 SL OPES OF EMB ANKMENTS AND EARTH DAMS 4
14 PROJ ECT OBJECTI VE 5
IV
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
CHAPTER 2 LITERATURE REVI EW
21 INTROD UCTION 6
2 2 FELLEN IUS MET HOD ANALY SIS 7
2 3 SWEDISH CIRCLE SLI CES METHOD 9
24 BISH OP S SIMPLIFI ED METHOD OF ANA LYS IS OF 12
FINI TE SLOPE
25 JAN BU METHOD (NON-CIRCU LAR [Al LURE 14
SURFACE)
CHAPTER 3 METHOLOGY
31 GENERAL 17
32 MORGENSTERN AND PRICE METHOD 17
CHAPTER 4 RESULTS AND DISC USSION
41 PARAMETER AN D PR OPERTIES 23
CHAPT ER 5 CONCLUSION 29
REFERENC ES 30
v
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
APPEN DI X A DERIVAT IVE fOR MORG ENSTE RN
AN D PRI CE METH OD
APPEN DIX B fORT RAN PR OG RAM ON AN AL YSIS Of
SLO PE AN D STAB ILITY
APPEND IX C RES ULTS FO R FRO TRAN PROGRA M
v
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
LIST OF TABLES
TABLE PAG E
Table 2 1 Factor of safety which related to detail of s lope
Tab le 41 Result for FORTRAN Programmin g 2~
V II
7
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
FJGllRE
Figure 2 1
Figure 22
Figure 23
Figure 24
Fi gure 3 1
Figure 41
Figure 42
Figure 43
Figure 44
Fi gure 45
Figu re 46
LIST OF FIGURES
PAG E
Fe lleniu s method of slope stability analys is 8
Stab ility anal ys is by Swedish circle meth od 9
Bishops simplified method forces actin g on th e sli ce 12
Limited graph for the method Janbu 15
The forces acting on single slice 18
Factor Of Safety vs Angle of intern al fricti on cp at 25
Slope Angle 40deg
Lamda A vs An gle of internal fri cti on cp at Slope angle 40deg 25
Factor of Sa fety vs Sl ope An gle whi ch Angle of fri cti on 9 is 30deg 27
Lamda A vs Sl ope An gle which Angle of fr iction cp is 30deg 27
Factor of Sa fety F vs Sl ope An gle which Cohesion e u is 30 28
Lamda Avs Slope Angle which Co hesion Cu is 30 28
V III
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
LIST OF NOMENLATURES
F Factor of Safety
Lamda
Angle of internal friction
c Cohesion
Horizontal distance measured from centroid of circle
y Vertical distance measured from centroid of circle
H Height of slope
jJ Slope angle
w Weight of small slices
r Density of soil
IX
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
CHAPTER 1
INTRODUCTION
11 GENARAL
A n exposed grou nd surface that stands at an angle with the hor izo ntal is called
an unres tra in ed s lope The s lope can be natural or man made If the gro und surface is
not hori zo ntal a component of gravity will tend to move the so il do wnward If the
component of gravity is large enough s lope failu re can occ ur The driving force
overcomes the resistance from tile shear strength o f the soil along the rupture surface
Slope fa ilures can cause eno nnou s damage to propeny and loss of human lives
In man y cases c ivil engineers are expected to make calcu lations to check the
safety of natural s lopes s lopes o f excavat ion and com pacted emban kments Thi s
check in vo lves determi ning and comparing the shear stress developed along the most
likely rupture surface with the shear strength of the so il This process is called s lope
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
stability ana lys is The most like ly rupture surface is the critical plane that has the
min imum factor of safety
The stability analysis of a slope is not an easy task Eva luation of var iables
such as the so i I stra tifi cati on and its in-place shear strength parameters by prove to be
formid ab le task Seepage through the slope and the choice of a potential slip surface
add to complexit y of the prob lem
12 HISTORICAL BACKGROUND
The devel opment of I imit eq uilibrium methods based on the plast ic
eq uilibrium of tr ial failure surface began in Sweden in 19 16 iJllowin g the fa ilure of a
number of quay walls at Gothenburg Harbor Petterson (195 5) and Hull (1916) in
separate publications reported that th e fail ure surfaces in the so ft clays o f Gothengurg
Harbor c losely resembled arcs of circles Over the next fe w years the fri cti on circle
method of ana lysis was devised results from simpl e undrained shear tests were used
with reasonable success in predicting sta bility ( So Or ltp = 0 ana lysis) and the method
of slices was introduced (Fellenius 1927) The concept of pore water pressure and the
effecti ve stress method of ana lys is were introduced by Terzaghi ( 1967) Improved
so il strength measurements resulted from better samp ling techniques the
development of the tri axi al shear test and the measurement of pore pressures
Improved methods of analys is that include the side forces between slices were
developed beginn i ng wi th Fellen i us (1936) and Bi shop ( 1955) More rigorous
2
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
analytical methods usually in vo lving the use of digital computer are avai labl e
Howeve r despite the use of more rigorous methods of analys is and improved so ilshy
testing techniques may uncertainti es remai n in predicting the stability of slope These
uncertainties are primarily associated with the measurement of so il strength (Johnson
1975) and the prediction of pore pressures
13 TYPE OF SLOPE
131 NAT URAL
The routes by which a natural (not man-made) slope has reac hed its presen t state
may be split in to two main categor ies shy
Those which are made up of a series of long-term processes many of which are
still active
2 Those which are made up of processes which act for a shoti duration so much
fewer active processes can y on at present
This second ca tegory of slope takes much more investigation to d iscern the
original cause of slope formati on and an understanding of these processes is essential
for a successful engineering in vestigation and to know how to deal with prob lems of
instability which may arise
3
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
132 EXCA VA TED SLOPES
It is critical to pay attenti on to the pore wate r pressures as th ey tend to
increase over time This means that c heap undrained shear strength tests are only
usefu I if looking at very sho rt term stabilit y
The geo logica l sequence and hi story must be known so we are sure if there
are ex ist ing tectonic shea rs Excavations are more susceptible to th e effects o f
tectonic shears than e mbankments because embankments ra ise the normal e ffect ive
stresses on potenti a l sliding surfaces and these offset th e increased leve ls of shear
stress they impose
133 SLOPES OF EMBANKMENTS AND EARTH DAMS
Embankments are cons tru cted by placing and compacting success ive laye rs of
a fill material onto a foundation soil Construction causes the total stress in the
embankment layers themselves and a lso in the foundat ion so i I to increase The
initial pore water pressure (u o) depends primarily on the placement water co ntent of
the fill
At the e nd of construction of a n embankme nt the Facto r of Safety is lower
than in th e long term Thi s is because water dissipates after the end of construction
with the pore water pressure decreasing to the final lo ng term va lu e Thi s assumes
the permeability of the compacted fill layers is low so not much dissipation takes
4
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
place during construction The construction pe ri od is usually quite short Stability
may also depend on the shea r strength of the foundation soil
14 PROJ ECT OBJECTIVE
The main objective of the project is to study the merits and demerits o f
available classical meth ods of slope stability analys is and select a more rational
method which considers the parameter like pore pressure and hori zo ntal pressure A
parametric stud y is aimed to find out the effect of soil parameters and s lope
parameters
5
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
CHAPTER 2
LITERATURE REVIEW
21 INTRODUCTION
Over the past decade the slope stab ility analysis had been exam ined
ex tensively us ing numerical methods particularly integral equation or boundary
element method Such methods developed out with formulas to show how the slope is
affected by the parameter of the characteri stic ofso i All solution is simplified by the
factor of safety Next the pore water pressure of soil had been cons idered By
considering pore water pressure of soil at the potential sliding slope the simplified
formulas for fac tor of safety had been generated
6
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
Table 2 1 Factor of safety whi ch related to detail of slope
FACTOR OF SAFETY DETAILS OF SLOPE
lt O Unsa fe
10-125 Questionable sa fety
125-14 Satisfactory for rout ine cuts and fills Questionable for
dams or where failure would be catastrophic
gt 14
-Satisfactory for dams
[n this chapter foll owing methods are rev iewed
l Fellenius method of analysis
2 Swedish circle slices method
3 Bishop s simplified method of analysis of finite slope
4 Janbu method
22 FELLENIUS METHOD OF ANALYSIS
Referrin g to Figure 2 1 the total weight W o f the so il above the failure sur face
that cause the instab ilit y in slope The weight W is given by the area between below
the slope and until the failure surface of radius R T he moment o f the driv ing force
M d about the center of circle to cause slope in stability is given as follows
(2 1)
7
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
x=Rsin a 1- ~a
L- - I RI I
I wl I I
Figure 21 Fellenius method of slope stability analysis
In which x is horizontal di stance between centre of the circl e and the centre of
gravity o f weight W
The res istance to sliding is derived from the cohesion mobilized along the potential
surface of sliding If Cm is the mobilized cohesion then the res isting moment M is
gi ven by
M = c LR = c R a (22)r III In
a is the angl e subtended by th e assumed failure arc of length L
For equilibrium M = Md Hence
W x= c Ra
W x (23) em = R1a
8
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
The factor of safety against sliding FS is given by the rati o of shear strength of so il
Tf ( Undrained shear strength c in thi s case) to the mobili zed shear strength mshy
FS= 2=~ = cRa _(24) 1 till W X
23 SWEDISH CIRCLE SLICES METHOD
Swidish Circl e Sli ces Method is a genera l method that can be used for both
cohes ive and c-ltp soi ls The soil above the tri al failure surface is div ided into several
vel1ica l slices _Usuall y 6 to 12 slices are recomll1ended_ The width of each sl ices need
not be the same But a new slice should be considered at the interface of the changed
so iI layer
The Ylelhod of Slice
RS lna
-------r-1
Forepound Aetins On Eltllh Slilt middot
N T wsina wcasa
r~ N R T
Figure 22 Stability analySiS by Swedish circle method
9
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
Figure 22 shows the forces that act on a typical sli ce for uni t perpendicular
length
Let W be the weight of the s li ce The forces N and T are the normal and
tangential components of the reaction R respective ly Po and P0+ 1 are the normal
forces that act on the sides of the slice For simplicity the pore water pressure is
assumes to be zero It is also assu med that both the norm al and tangential forces on
the sides of the s li ces are eq ual in magni tude and their line of action coincides As
such these forces will be cancelled out and not shown in the figure
For equi librium consideration
N = Wcosa (25)
The res isting shear force can be expressed as
T ( L (c + cr tan ltIraquoL T = T L = --= (26)
m FS FS
N Wco sa Now norma l stress (J = - = --- (27 )
LII Ln
For equilibrium of the trial wedge the moment of the dri ving fo rce about 0 equals
the moment of the resisting force about that po int Therefore
w cosa tan tp )c + -shy( Ln (28) I W sin a r= I
FS
I _ (CLn + wcosatan tp )L f smar - L r (29) FS
10
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
LeL +Wcosa tanipFS = ~ (210)
LW sina
Note that L is approximately equa l to ~ B is the width of the nIh s lice cosa
Steadv state seepage
For steady state seepage through slopes as is th e situation in many practical
cases the pore water pressure must be considered when effecti ve stress paramete rs
are used
For the nth s lices th e average pore water pressure at the bottom ofthe s li ce is
equal to u = h I Total force caused by the pore water pressure at the bottom of nth
s li ce is eq ual to u x L Thus the equation for factor ofmiddotsafety wo uld be modifi ed as
LeL + (W cos a - ilL )tan ipFS = ~~~=-----~--- (2 II )
LW sina
II
