anatomy of landslides

8/12/2019 Anatomy of Landslides

1/57

Theory of Slope Stability

Prepared for G 483/583 Anatomy of Landslides

(http://www.geol.pdx.edu/Courses/G483/)by

Kenneth M. Cruikshank Portland State Uni ersit!

"e ision #.$ (Spring %er& #'' )

0 m 5 0 0 m 1 0 0 0 m 1 5 0 0 m 2 0 0 0

S lu m g u l l i o n L a n d s l i d e , C o l o r a d o

ased on *ourse notes +!

Arvid M. JohnsonGeo,%esson -ngineering Geolog!

a+orator!Purdue Uni ersit!

#'' 0.1. Crui2shan2 and .1. ohnson

%hese notes *annot +e reprodu*ed withoutwritten per&ission


2/57

Table of Contents%a+le of Contents................................................................................................................................

ist of 5igures.....................................................................................................................................iiist of %a+les......................................................................................................................................i

ist of S!&+ols..................................................................................................................................iSo&e 6efinitions and Useful Con ersion 5a*tors..............................................................................%!pi*al 1aterial 7alues..................................................................................................................... iSu&&ar! of useful e uations........................................................................................................... iii

5a*tor of Safet!......................................................................................................................... iii6r! 9nfinite Slope............................................................................................................ iiiSu+&erged 9nfinite Slope................................................................................................ ii9nfinite Slope with Seepage parallel to slope.................................................................. iii9nfinite Slope with Seepage and %ree "oots................................................................... iii5ellenius &ethod............................................................................................................. iii1odified ishop &ethod................................................................................................ iii

Criti*al %hi*2ness....................................................................................................................... i6r! 9nfinite Slope............................................................................................................ iiiSu+&erged 9nfinite Slope................................................................................................ ii9nfinite Slope with Seepage parallel to slope.................................................................. iii9nfinite Slope with Seepage and %ree "oots................................................................... iii

Unit weight..................................................................................................................................ix6r! soil.............................................................................................................................ix1ixture of water and solids..............................................................................................ix

#. %heor! of Slope Sta+ilit!................................................................................................................#$. 9nfinite Slope..................................................................................................................................$

$.# - uili+riu& e uations.............................................................................................................$$.$ 6r! Soil..................................................................................................................................

$.3 9nfinite slope in standing +od! of water..................................................................................$.4 9nfinite slope with seepage parallel to the slope...................................................................#;$. 9n*orporation of strength *ontrolled +! tree roots...............................................................#$

3. 1ethod of sli*es............................................................................................................................#43.# 9ntrodu*tion..........................................................................................................................#43.$ 5ellenius 1ethod..................................................................................................................#eights andslide= las2a..............................................................................$'

andslides in sensiti e *la!s and *la!shales.....................................................................$'?ui*2 *la! slides..............................................................................................................$'

andslides in Utah...........................................................................................................3;Strain in andslides..........................................................................................................3;


3/57

1apping...........................................................................................................................3;Slides in @regon..............................................................................................................3;Slides in Portland.............................................................................................................3;

ppendix . 1ohrAs Cir*le..............................................................................................................3#Stresses on an in*lined plane ......................................................................................................3#1ohrBs Cir*le..............................................................................................................................33

ppendix . -xer*ises......................................................................................................................3-xer*ise # 6r! Soil..................................................................................................................3-xer*ise $ Standing water.......................................................................................................3eight of thrusting a+o e slip surfa*e in an+u nal!sish >eight of water ta+le in an+u anal!sis Phi Slope of root with respe*t to shear Eone %heta %angent to ar* in &ethods of sli*es anal!sis a&+da Coeffi*ient in 1orgenstern F Pri*e sta+ilit! anal!sis

D Dor&al for*e6ifferen*e +etween water pressure and nor&al for*e (D U)

%heta Slope angle %heta 7olu&e fra*tion of solids ( olu&e of solids / total olu&e) Sig&a Dor&al Stress on a slip surfa*ei Co&ponents of the stress tensor S Shear for*e on slip surfa*e %au Shear Stress on a slip surfa*ef 5ra*tion of failure strength of soil in an+u anal!sis% Shear for*et %hi*2ness of slidet* Criti*al thi*2ness of slideu Pore water pressureU Dor&al for*e exerted +! pore water pressure on +ase of sli*eH Height of an ele&entx >oriEontal distan*e to edge of sli*e in an+u anal!sis (fro& toe)! an+u: >eight of slip surfa*e (positi e down fro& datu&)! 9nfinite slope: 6istan*e (IdepthJ) &easured nor&al to slope! t >eight of line of thrusting in an+u anal!sisE >eight of ground surfa*e in an+u anal!sis


6/57

Some Definitions and Useful Conversion FactorsStress (1 ,#%,$): 5or*e / rea. Co&&on units are: t&osphere= +ar= Pas*al= pounds/ft$= pounds/in$=

in>g= d!nes/*&$ Force (1 % ,$): 1ass ti&es a**eleration. Co&&on units are: Dewton (S9) and d!ne (*gs).

Pascal . &easure of stress in S9 units= defined as one Dewton per &eter,s uared. Newton. &easure of for*e in S9 units. 6efined as &ass ti&es a**eleration. # Dewton is the for*ere uired to a**elerate # 2g of &ass at # &/s$.

Dyne. &easure of for*e in *gs units. %he for*e re uired to a**elerate # gra& at # *&/s$. Acceleration due to gravity: '.8;


7/57

Typical Material aluesDescription Unit Weight

Saturated ! Dry"#riction $ngle

Cohesion

Type Material Lb / ft 3 KN / m 3 Lb / ft 2 kPaC ohesi onl ess

Sand

Loose sand, uni%orm grain si&e 11'!(0 1(!1) 1'*+)

Dense sand, uni%orm grain si&e 1+0!10( 21!1 +2*)0

Loose sand, mi-ed grain si&e 12)!(( 20!1. +)*)0

Dense sand, mi-ed grain si&e 1+5!11. 21!1' +'*).

Gr a

el

/ra el, uni%orm grain si&e 1)0!1+0 22!20 +)*+

Sand and gra el, mi-ed grain si&e 120!110 1(!1 +0*)5

!l ast ed / br ok enr o" k

asalt 1)0!110 22!1 )0*50

Chal '0!.2 1+!10 +0*)0

/ranite 125!110 20!1 )5*50

Limestone 120!100 1(!1. +5*)0

Sandstone 110!'0 1 !1+ +5*)5Shale 125!100 20!1. +0*+5

C ohesi

e

C l ay

So%t bentonite '0!+0 1+!. *1+ 200 3 )00 10 3 20


8/57

4ery so%t organic clay (0!)0 1)!. 12*1. 200 3 .00 10 3 +0

So%t, slightly organic clay 100!.0 1.!10 22*2 )00 3 1000 20 3 50

So%t /lacial clay 110! . 1 !12 2 *+2 .00 3 1500 +0 3 5

Sti%% glacial clay 1+0!105 20!1 +0*+2 1500 3 +000 5 3 150

/lacial till, mi-ed grain si&e 1)5!1+0 2+!20 +2*+5 +000 3 5000 150 3 250

# o" k

ard igneous roc s/ranite, basalt, porphyry

1.0 to1(0

25 to +0 +)*)5 20000*1150000 +5000*55000

6etamorphic roc s7uart&ite, gneiss, slate

1.0 to1'0

25 to 2' +0*)0 )00000*'00000 20000*)0000

ard sedimentary roc sLimestone, dolomite, sandstone

150 to1'0

2+ to 2' +5*)5 200000*.00000 10000*+0000

So%t sedimentary roc sSandstone, coal, chal , shale

110 to150

1 to 2+ 25*+5 20000*)00000 1000*20000

>igher fri*tion angles in *ohesionless &aterials o**ur at low *onfining or nor&al stresses( fro& >all #''4= %a+le 46.#= p. 43 )


9/57

Summary of useful e!uations

Factor of Safety

$ry %nfinite Slope

)(

)()(+=

sintan*os

t

t C F (. a)

S&bmer'ed %nfinite Slope

)()(

)()()+=

sintan*os(

wt

wt t C

F (.##+)

%nfinite Slope (ith Seepa'e parallel to slope

)()()()+

=

sin

tan*os(

t

wt

t

t C F (.


10/57

Unit weight L

a is the unit weight of air (ta2en to +e ; in.# +) s is unit weight of solid parti*les w is the unit weight of water $ry soil

t L s N a (# ) (.# a)or

t L s M Dry Soil (. # +)

Mi*t&re of (ater and solids

t L s N (# ) w MSaturated Soil (.# *)


11/57

"# Theory of Slope Stability

9n the le*ture part of this *ourse we will dis*uss a ariet! of &ethods of anal!sis of slopesta+ilit! and insta+ilit!. 9t is essential that the engineering geologist +e inti&atel! fa&iliar with all ofthese &ethods +e*ause the! pro ide wa!s of deter&ining= relati el! una&+iguousl!= whether agi en slope is li2el! to slide or whether it will re&ain sta+le. Perhaps &ost i&portant= though= isthat fa*t that the &e*hani*al anal!sis of slope sta+ilit! pro ides us with 2nowledge of what para&eters *ontrol landslidingM the guesswor2 is entirel! re&o ed. t one ti&e it was *onsidereda**epta+le pra*ti*e for the engineering geologist to &a2e general state&ents a+out the effe*t of thegeolog!= egetation= the effe*t of intense rainfalls= or the effe*t of the fa*ing dire*tion of a slope(e g= north,fa*ing slopes are less sta+le than south,fa*ing slopes) on slope sta+ilit!. Do longer=though= is this a**epta+le. %he engineering geologist is expe*ted to 2now and understandtheoreti*al and pra*ti*al soil &e*hani*s= +etter than the usual Ci il -ngineer= and nearl! as well asthe spe*ialist= the Geote*hni*al -ngineer. %his *ourse is designed to supple&ent the theor! and pra*ti*e of soil &e*hani*s that !ou learn in *ourses taught in Ci il -ngineering. 9 assu&e that !ouha e had at least one *ourse in ele&entar! soil &e*hani*s. Oou *ertainl! *an *o&plete this *ourse

without ha ing had a *ourse in soil &e*hani*s= +ut 9 +elie e !ou will learn &u*h &ore if !ou 2nowso&e soil &e*hani*s +efore !ou ta2e this *ourse.Hith the a aila+ilit! of *o&puters and slope anal!sis software there is now e en less ex*use

for not doing the proper t!pe of slope sta+ilit! anal!sis. Oou *annot ha e *o&puter progra&s dothe thin2ing for !ou. %he +etter !ou understand the &e*hani*s of slope sta+ilit! the &ore effe*ti e!our use of these progra&s will +e. 9n this *lass we will use so&e si&ple progra&s. %he onl!differen*e +etween the progra&s we use and *o&&er*ial progra&s is the! are perhaps a little lessuser friendl!= and do not produ*e a wide range of plots. @ther than that= the! are full! fun*tional.Oou will get to see how little there is to slope sta+ilit!

9t is well 2nown in soil &e*hani*s that three general t!pes of para&eters deter&ine thesta+ilit! or insta+ilit! of a slope. @ne group *on*erns the strength of the soil. %he strength in*ludes*ohesion= fri*tion= interlo*2ing of grains= reinfor*e&ent= for exa&ple +! roots= and perhaps othefa*tors. nother group *on*erns the geo!etry of the soil. %his in*ludes the shape of the groundsurfa*e= the shapes of possi+le slide surfa*es= the pattern of la!ering within the soil= and the for&sof signifi*ant dis*ontinuities su*h as oints or shear Eones. %he other group of para&eters relates tthe pore"water pressure. %hese in*lude the pore,water pressure itself as well as the seepage for*esset up +! &o e&ent of water through the soil. @ur approa*h is going to +e to 2eep *ertain para&eters *onstant and to in estigate effe*ts of the re&aining para&eters. 9n this wa! we *anunderstand effe*ts of the arious para&eters. %hus= we will perfor& a series of in estigations inwhi*h we assu&e that the slope has si&ple geo&etr! and is er! long. %hese theoreti*alin estigations allow us to assess effe*ts of *ertain idealiEed pore,water pressure distri+utions andeffe*ts of tree roots on slope sta+ilit!. %hen we will assu&e that the failure surfa*e is a seg&ent of a

*ir*le= so that we *an readil! treat landslides where the thi*2ness of the sliding de+ris is a signifi*anfra*tion of the distan*e fro& the head to the toe of the landslide &ass. 5inall! we will introdu*erather *o&pli*ated and a**urate &ethods of sta+ilit! anal!sis that re uire the use of a *o&puter.%hese &ore *o&pli*ate &ethods allow us to treat &ost pro+le&s of slope sta+ilit!.


12/57

$# %nfinite Slope

2.1 Equilibrium equations

%he si&plest pro+le& of slope sta+ilit! that we *an anal!Ee= and that we should understand indetail +e*ause it is so +asi*= is that of the sta+ilit! of the so,*alled infinite slope. %he infinite slopesolution is also an exa*t solution= the &ethods of sli*es whi*h we exa&ine later are approxi&ations%he solutions used in the &ethod of sli*es are onl! approxi&ate. 9n an infinite slope solution wedeter&ine the *onditions under whi*h a la!er of soil of thi*2nesst will slip along a surfa*e=a,aB=that is parallel to the ground surfa*e= whi*h has a slope angle of. %he *ross se*tion of the infiniteslope is shown in5igure #. t this point !ou should stud! rele ant pages in a&+e F Hhit&an(#'


13/57

a*t. %he stresses ar! a*ross the width or depth of the ele&ent= and the different stresses onopposite sides are indi*ated +! pri&ed and un,pri&ed alues. 5or exa&ple=xx is the nor&al stressa*ting on the left,hand side of the ele&ent and is the nor&al stress a*ting on the right,hand side:

L xx N x (.3a)

Si&ilarl!= L x! N x (.3+) L !x N ! ( .3*) L !! N ! ( .3d)

5igure $. Unit ele&ent in a slide &ass. Dote that all the stresses shown are in their positi edire*tions= so that nor&al stresses are positi e if *o&pressi e.Dote that all the stresses shown in 5igure $ are shown in their positive dire*tions= so that nor&al stresses are positi e if *o&pressi e.

s alread! stated= the for*es= N and ' = are e ual to the *orresponding stresses ti&es theappropriate areas. %hus=

N xx L xx ! E M N L ! E (.4a)' x! L x! ! E M ' L ! E (.4+)' !x L !x x E M ' L x E (.4*)

N !! L !! x E M N L x E (.4d)So &u*h for definitions.

yy

9yy

9--

--

y-

-y

9-y

9y-

-

y

N yy

N9yy

N9--

N --

: y-

: -y

:9 -y

:9y-

W

$

& '

Stresses #orces


14/57

Dow let us appl! the e uations of &o&ent and for*e e uili+riu&= e s. (.$). Su&&ing&o&ents a+out the lower left,hand *orner (point A) in the ele&ent shown in5igure $ = with*ounter,*lo*2wise +eing a positi e &o&ent=

' ! ' x ( N xx N ) N ( N !! N ) (x ! E) L ;(. a)

in whi*h the weight of the ele&ent=( = is( L ( # y $) (. +)

%he le er ar&s for' !x and ' x! are Eero. Su+stituting e s. (.3) into (.4) and the resulting e s.(.4) into the result a+o e=

(!x N y) ( y # $) (x! N #) ( y # $)N ( y # $) ( y # $)

( y # $) ( *os() N sin() ) L ; (. *)

Dow= di iding ea*h ter& in e . (. *) +! the olu&e of the ele&ent ( # y $)= and ta2ing the li&it ofthe resulting e . (. *) asx ; and ! ;= we deri e the result that the &o&ents su& to Eero if

!x x! L ; (.


15/57

L sin() (.'a) L *os() (.'+)

5ro& e uation. + we should also note that(.#;).

9ntegrating e s. (. )= we deri e=x! L y sin() N C; (.##a)

!! L y *os() N C# (.##+)in whi*hC ; and C # are ar+itrar! *onstants. >owe er= the shear stress and nor&al stress arene*essaril! Eero at the ground surfa*e= so that the *onstants are Eero=

x! L y sin() (.#$a)!! L y *os() (.#$+)

He should note that we ha e &ade no assu&ption *on*erning the rheologi*al properties ofthe &aterial in the slope. %hus e s. (.#$) are alid whether the &aterial is soil= water= la a or so&eother &aterial. >owe er= to sa! an!thing a+out slope sta+ilit!= we will need to introdu*e therheologi*al properties of the soil.

%he si&ple relationships in e s..#$ represent the e uili+riu& *onditions in an! &aterial. %he&aterial is out of e uili+riu& when these *onditions are not satisfied. %he uestion in slope sta+ilit!is how far out of e uili+riu& is the slope= and in what dire*tion is out of e uili+riu& is it sta+le ounsta+leR

2.2 Dry Soil

et us first *onsider the sta+ilit! of a slope underlain +! dr! soil= so that pore,water pressuresare Eero. 5or dr! soil= one generall! assu&es that the soil shears when the shear stress is e ual tothe shear strength of the soil. er! si&ple &odel of shear strength= whi*h wor2s re&ar2a+l! wellfor &ost soils= is Coulo&+Bs law of fri*tion$=

strength in shear L C N tan() (.#)

in whi*hC is *ohesion= is nor&al stress a*ting a*ross the surfa*e of shearing= and is the angle ofinternal fri*tion (generall! a+out 3;K for sand and # K to $ K for *la!). He generall! state thestrength in ter&s of a !ield *ondition= whi*h is

C N tan() (.$)

in whi*h is the shear stress a*ting on the surfa*e of failure in shear. %he a+solute alue sign isre uired +e*ause the shear strength e uall! resists positi e or negati e shear stress. %he Tless,thanor,e ualT== s!&+ol indi*ates that the shear stress applied to the soil &ust +e less than or e ual tothe shear strength of the soilM this is +! definition of shear strength.

$ lso *alled %ohr"Coulo!b aw in so&e texts= su*h as a&+ F Hhit&an (#'


16/57

5urther&ore= we generall! *o&pute a factor of safety against sliding= F = where the fa*tor ofsafet! is a &easure of the *loseness to *onditions of sliding that exist in a slope. %he fa*tor of safet!is defined as the ratio of the shear strength to the a*tual shear stress=

)tan(+

=C

F (.3)

%he fa*tor of safet! is greater,than one if the shear strength is greater than the shear stress= sothat the slope is sta+le= and it is e ual to one if failure is i&pending. 9f !ou *o&pute a fa*tor ofsafet! less than one= get out of the wa!

et us *o&pute the fa*tor of safet! for an infinite slope= %he shear stress is &axi&al at the +otto& of the soil (5igure #)= where y * t = so that fro& e . (.#$)=

x! L t sin() (.4a)!! L t *os() (.4+)

5urther= the shear stress is &axi&al on planes nor&al to the y,axes= so that

L x! L t sin() (.4*)%he nor&al stress a*ting on this plane is!! = so that

L !! L t *os() (.4d)**ordingl!= the expression for the fa*tor of safet! against sliding= e . (.3)= +e*o&es=

F L Mdry soil (. a)

9f the *ohesion=C = were Eero= F L Mdry soilMC L ; (. +)3

- uation (. +) represents a *o&&on notion for the angle of internal fri*tion. 9f we let thefa*tor of safet! +e one then e . (. +) +e*o&es

tan() L tan() or L (. *)%hus if a *ohesionless &aterial was piled into a *one= the angle of the *one will +e the angle

of internal fri*tion for the &aterial.5urther= if the thi*2ness of the potential slide is e ual to the *riti*al thi*2ness for sliding= the

fa*tor of safet! is one. 9n this *ase we *an sol e for the *riti*al thi*2ness +! setting F L l in e .(. a)=

# L (. d)

t sin() L C Nt *os() tan() (. e)

t * L M F L # Mdry soil (. f)

3 %his is the e uation at the top right,hand side of p. #'3 of a&+e and Hhit&an (#'


17/57

sin*e tan() L sin()/*os(). %he *riti*al thi*2ness=t *= is Eero if the *ohesion is Eero. %his &a!appear to +e a surprising result.

t this point !ou should *o&plete exer*ise # (p.3 ).

2. !nfinite slo"e in stan#ing bo#y of water

t

-

y

y

-

t

Ui

U

& ' C

#igure +8 De%inition diagram %or a submerged in%inite slope8

%he next pro+le& that we will *onsider is the infinite slope that is su+&erged +! a standing +od! of water ( a&+e F Hhit&an #'


18/57

y w *os() N f+#,V L L wsin() (.3+)(+e*ause y= of *ourse= is independent of #)= so that

f+#, L # w sin() NC o (.3*)in whi*hC o is a *onstant. %hus e . (.3a) +e*o&es=

u L w( # sin() y *os()) NC o (.3d) Dow= let the water pressure at the surfa*e of the soil &ass= at y L ;= +eui.

ui L w # sin() NC o (.4a)C o L ui w # sin() (.4+)

%hen we *an write e . (.3d) in the final for&=u L w( # sin() y *os()) Nui w # sin() (. a)

u L y w *os() Nui. (. +)%hus we ha e deri ed an expression for the pore,water pressure within the soil &ass= as well

as the water pressure within the standing +od! of water o erl!ing the soil &ass.t this point we return to the e uations for the stresses within the soil &ass= e s. (. d=.8+).

%hose e uations are for the total stresses= and are a result of the *o&+ined densit! of the soil andthe water *ontained in the soil. %hus= instead of using dr! unit weight= = we use total unit weight= t of the soil= and e s. (. d=.8+) +e*o&e

L t sin() (.


19/57

or L ( t w) sin() (.8+)

L t *os() (.8*)

%he *o&ponent ofxx that *o&es fro& the soil will +e a *onstant (as with the dr! soilderi ation).!! / y is left as a partial deri ati e sin*e!! aries with +oth # and y (again= this *an +eseen fro& exa&ining5igure 3). Oou should *o&pare these e uations with e . (.').

He should note two features of e s. (.8+=*). %he nor&al stress depends upon the unit weightof the &ixture of solids and water= whereas the shear stress depends upon the +uo!ant unit weight=( t w)= of the &ixture. 5urther= we ha e &ade no spe*ial assu&ptions in deri ing e s. (.8+=*). spe*ial assu&ption is introdu*ed onl! in a following step= when we assu&e that e . (.#) des*ri+esade uatel! the strength of the soil.

9ntegrating e s. (.8+=*)=!x L x! L y ( t w) sin() (.'a)

in whi*h the ar+itrar! *onstant was set e ual to Eero= +e*ause the shear stress is Eero for y L ;.5urther=

!! L y t *os() N g+#, (.'+)

Hhere g+#, is an ar+itrar! fun*tion of #. >owe er= at y L ;=!! L ui= e . (. )= so that!! L y t *os() Nui (.'*)

in whi*hui L w # sin() NC i (.'d)

whereC i is a *onstant= whi*h we need not deter&ine= as we will show +elow. Dext we *onsider the possi+ilit! of sliding of a soil &ass of thi*2nesst (5ig. 3 ). %he shear

stress parallel to the +ase is L x! L t ( t w) sin() (.#;a)

and the effe*ti e nor&al stress at the +ase= y L t = is (e . (.'+) &inus (. +))= u L !! u L t ( t w) *os() (.#;+)

%he fa*tor of safet! against sliding is5 L (.##a)

so that=

5 L MSub!erged slope (.##+)

%he *riti*al thi*2ness=t *= deri ed for a fa*tor of safet! of one= is

t * L M F L # (.#$)


20/57

9f the *ohesion is negligi+le= the fa*tor of safet! redu*es to that of a dr! slope= F L M L ; (.#3)

as noted +! a&+e and Hhit&an (#'


21/57

Perhaps !ou find it interesting that this is the sa&e as e . (. +) withui= the pore,water pressure at the ground surfa*e= e ual to Eero. 9f so= &a!+e !ou should tr! to figure it out. 9t reall!is relati el! si&ple. %he pore,water pressure is independent of #= so that e s. (.


22/57

2.% !ncor"oration of strength controlle# by tree roots

S l i p s u r % a c e

S h e a r & o n e

#s

# n#

& '

#igure )8 ;ole o% tree roots in rein%orcing a soil mass8

1ar! "iesten+erg ("iesten+erg F So oni*2,6unford #'83) has done resear*h into the effe*tof roots of wood! egetation on the sta+ilit! of *ollu iu& on steep slopes. 9ndependentl!= the sa&eresults had +een deri ed +! Haldron (#' )= Huet al . (#' ') and Hu F Swanston (#'8').

5or a s&all slide that she studied in detail= 1ar! found that the a erage shear strength*ontri+uted +! the roots was a+out . 2D/&$ of the shear surfa*e= whereas the a erage strength*ontri+uted +! residual fri*tion= alone= was a+out ;. 2D/&. %he tree roots= therefore= in*reased fa*tor of safet! against sliding ',fold= in this *ase. ased on o+ser ations of &an! landslides of

arious thi*2nesses in the Cin*innati area= 1ar! has tentati el! *on*luded that tree roots *ansignifi*antl! in*rease the resistan*e to sliding for soil &asses up to a+out two &eters thi*2. >ere wewill +riefl! re iew her theoreti*al anal!sis= in order to deter&ine the effe*t of tree roots onresistan*e to sliding for er! long slopes. 5urther data on root strength is gi en +! %urner (in%urner F S*huster #''allet al (#''4= p. 43).

5igure 4shows nor&al= F n= and tangential= F s= for*es *ontri+uted at the failure surfa*e at theinstant a tree root is read! to +rea2 in tension. %he for*e re uired to +rea2 the tree root is F . %hus=

F s L F sin() (.#a)

F n L F *os() (.#+)

where is the angle of in*lination of the tree root (5igure 4).

Dow the a erage nor&al stress applied to the surfa*e of failure +! the tree roots is L (.$a)in whi*h A is the area of the part of the slip surfa*e penetrated +! a total ofn roots= F i is the for*ere uired to +rea2 ea*h root= andi is angle of in*lination of ea*h root. Si&ilarl!= the a eragetangential stress for the sa&e unit of area= A= is

L (.$+)


23/57

where= again= the su& is ta2en o er all the roots that penetrate that *ertain area of the slip surfa*e%he a erage strength *ontri+uted +! tree roots is to the shear resistan*e of the &aterial

r L tan() N (.3a)r L (.3+)

9n order to perfor& a sta+ilit! anal!sis= we add this strength= e . (.3+)= to that *ontri+uted +!the soil without roots= e . (.#). 1ar! noted= howe er= the roots that penetrated the slip surfa*e inthe s&all landslide *o&plex that she studied had +e*o&e distorted fro& a nearl! erti*al orientationto an orientation nearl! parallel to the slip surfa*e. 9n this *ase=i ';K and e . (.3+) si&plifies to

r L (.3*)sin*ecos(';K) L ;= and sin(';K) L #.

further ustifi*ation for using the for& (.3*) is that the uantit! in +ra*2ets in e . (.3+)&axi&iEes where

Cot() L tan (.3d)

that is=(i)&ax resistan*e L ';K . (.3e)

%herefore= if the angle of internal fri*tion is #;K= is 8;K= and in this *ase (.3+) is a *loseapproxi&ation to e .3*.

dding the strength *ontri+uted +! the tree roots= e . (.$*)= to that *ontri+uted +! the soilwithout roots= e . (.#)= we deri e

F L (.4a)

for the fa*tor of safet!= andt * L M F L # (.4+)

for the *riti*al thi*2ness.1ar! deter&ined tensile strengths of roots sa&pled fro& the +ase of the landslide that she

studied= and the results are presented in fig. #$ of her 1.S. thesis ("iesten+erg #'8#)= in ter&s offor*e (in Dewtons) re uired to +rea2 a root of a *ertain dia&eter (in &&). 5or exa&ple= for wood!roots a+out $ && in dia&eter= the for*e re uired to +rea2 the roots in tension ranges fro& a+out#;; to $;; Dewtons. %he ta+le +elow su&&ariEes so&e of her &easure&ents.


24/57

:able 18 Summary o% root strength data %or li e sugar maples8 #rom ;iestenberg 1('1,%ig8 12"8

W 9ndi*ates +est esti&ates of strengths.

9n order to use e s. (.4) for sta+ilit! anal!sis= of *ourse= !ou &ust +e a+le to &easure oresti&ate the nu&+er of roots of arious siEes that penetrate a gi en area of the slip surfa*e. %his=unfortunatel!= is diffi*ult to do in pra*ti*e.

Dow *o&plete exer*ise 4 (p.38).

(# Method of slices

.1 !ntro#uction

9n following paragraphs we will deri e the e uations essential for understanding two &ethodsof slope,sta+ilit! anal!sis= the 5ellenius and the &odified, ishop &ethods. He will not dis*uss howone *o&putes pore,water pressures along slip surfa*es= nor will +e dis*uss how one deter&inesrele ant soil properties to use in the anal!ses. 5or dis*ussions of these su+ e*ts= please see a&+e FHhit&an (#'


25/57

fter sele*ting the sli*es= we *o&pute resisting and dri ing &o&ents. @ne sli*e= the sixthshown in5igure = is shown in5igure as a free,+od! diagra& of a t!pi*al sli*e. %he free,+od!diagra& shows all rele ant for*es and &o&ents a*ting on the ele&ent. @ne for*e shown is( = theweight of the sli*e. 9t is deter&ined +! &ultipl!ing the area of the sli*e ti&es the unit of +readth inthe dire*tion nor&al to the page= ti&es the unit weight of the soil. >owe er= in this exa&ple we&ust use the unit weight for unsaturated soil a+o e the water ta+le= and the saturated unit weightfor the part of the sli*e +elow the water ta+le. nother for*e is the shear for*e=' i= whi*h= in +oththe 5ellenius and the &odified, ishop &ethods= is gi en +! the %erEaghi,Coulo&+ failure *riterion=

i L N ( u) tan() (.$a)or

' i L tan() (.$+)

in whi*hb is unit +readth (one &eter or one foot= depending upon the s!ste& of units for the*onstants)= #i is the width of the sli*e &easured horiEontall! (5igure )= is the slope angle ofthe tangent to the +otto& of the sli*e= N is the total for*e nor&al to the slip surfa*e= andu is the pore,water pressure at the +ase of the sli*e. %hus the uantit! b #i / *os(i)V is the area o erwhi*h a stress is a*ting.

%he nor&al for*e= N = a*ts through the *enter of the *ir*le= so that its le er ar& is Eero= and itexerts a Eero &o&ent with respe*t to the *enter. %he shear for*e=' has a le er ar& e ual to theradius of the *ir*le= so that the &o&ent resisting sliding for that sli*e si&pl! is=

& '

)

)

1

(',

.5

)+

2

Water :able

- i

Wii

C

$

D

; sin i"

i

i

i

:

N < u

: i

Niu i

Centralangle

#igure 58 De%inition diagram %or #ellenius and ishop methods8 $ simple slip sur%ace isassumed, and the total moments are determined by summing moments %or each slice8:he %actor o% sa%ety is de%ined as the ratio o% the sum o% the resisting moments di ided bythe sum o% the dri ing moments8


26/57

. ' 9 (.3a)

%herefore= the total resisting &o&ent is L . ' i (.3+)

where ' i is gi en +! e . (.$+).%he dri ing &o&ent arises entirel! fro& the weight of the sli*e. %he le er ar& for a sli*e is .

sin(i). Oou *an erif! this +! exa&ining5igure = where the le er ar& for sli*e < is illustrated.Oou should erif! that the angle +etween the erti*al and the radius for sli*ei is e ual toi and thatthe slope to the tangent to the +ase of sli*ei is e ual toi= the sa&e angle. %hus the dri ing&o&ent for sli*ei is

. ( i sin(i) (.4)

and the su& of the dri ing &o&ents is L . (. )

Su+stituting e . (.$+) into (.3+)= and the resulting e . (.3+) as well as (. ) into (.#)= we deri ethe expression for the fa*tor of safet!=

5 L (.


27/57

pressures= Hhit&an suggests using the +uo!ant unit weight of the sli*es if the pore,water is stati*(for details= see #'< = p. 4'#).

. &o#ifie# 'isho" &etho#

9n the &odified ishop &ethod of sta+ilit! anal!sis= the &ethod of *o&puting N is different.

9n this &ethod we su& for*es in the erti*al dire*tion= so that= for ea*h sli*e=( i L N i *os(i) N' i sin(i) (.#)

in whi*h' i is the shear for*e that is &o+iliEed= the shear strength of the soil= di ided +! the fa*torof safet!= F :

' i L (.$a)in whi*h

U i L (.$+)

is the nor&al for*e exerted +! the pore,water on the +ase of the sli*e= as in e . (.$+).

eti L N i U i (.$*)

then e . (.#) +e*o&es( i L (i NU i) *os(i) N' i sin(i) (.3)

Su+stituting e . (.$a) into (.3) and sol ing fori

(.4)

"ewriting e . (. ) using e . (.$*)=

5 L (. )9n order to *o&pute the fa*tor of safet!= we su+stitute e . (.4) into e . (. ). %he fa*tor of

safet! o**urs on +oth sides of the e ual sign= and *annot +e expli*itl! sol ed for= so we sol e e s.(.4) and (. ) +! iteration. %hat is= we guess F and then *o&pute N i. %hen we use those alues too+tain an i&pro ed esti&ate of 5 with e . (. ). He then use this new esti&ate of F to *o&pute N i and then again sol e for F using e . (. ). He repeat this pro*edure until the fa*tor of safet! *hanges +! insignifi*ant a&ounts.

Hhit&an F aile! (#'< ) indi*ate that a funda&ental assu&ption in the &odified ishop&ethod is that the resultant of for*es a*ting on the sides of ele&ents (su*h for*es are not e enshown in5igure ) are horiEontalM that is= on ea*h sli*e the shear for*es on ea*h side of the sli*eare e ual and opposite= so the! *an*el,when for*es are su&&ed in the erti*al dire*tion. 9n generathis is not true= so in general the fa*tor of safet! will +e in error.

Hhit&an has *o&pared fa*tors of safet! *o&puted with the &odified ishop &ethod withthose *o&puted with an a**urate &ethod. >e indi*ates that= in general= the error is less than Xand *o&&onl! is less than $X. >e indi*ates that &ore serious errors *an de elop if the fa*tor ofsafet! is less than one.


28/57

Oou should now *o&plete exer*ise (p.38). %his exer*ise onl! has a few sli*es= +ut it willgi e !ou a +etter understanding of how the pro*edure wor2s.

.$ (anbu an# &orgenstern)*rice metho#s of stability analysis

&

'

= a

= b

-

y

&

yt

y * yt

:

=

= < d=!d- -

: < d:!d- -

y < dy!d- d-" * y t < dy t!d- d-"

S

N

-

W

t

Width normal to -*y a-es assumed to be unity

> S ! l

> N ! l

l

y & h

#igure .8 De%inition diagram %or ?anbu and 6orgenstern*@rice methods o% stability8

an+u (#' 3) and 1orgenstern F Pri*e (#'< ) ha e de eloped si&ilar &ethods of sta+ilit!anal!sis +ased on a &ethod of sli*es that are &ore a**urate than the 5ellenius or the &odified,

ishop &ethods. %he! are &ore a**urate +e*ause the for*es +etween sli*es are spe*ifi*all!in*orporated in the differential e uations. %he onl! signifi*ant differen*e +etween the an+u and th1orgenstern,Pri*e &ethods are the spe*ial assu&ptions introdu*ed in order to &a2e thee uili+riu& e uations deter&inate. %he 1orgenstern,Pri*e &ethod assu&es a relation +etweennor&al and shear for*es a*ting on the sides of the sli*es. %hus= using the notation shown in5igure


29/57

f+#, L & # N! (.$)

in whi*h& and ! are *onstants. He will not follow the deri ation presented +! 1orgenstern,Pri*e(#'< ) rather we will use an+uBs &ethod of solution.

9n the an+u anal!sis= one assu&es the height of thrust= / = a+o e the +ase of the side of a

sli*e. @ne sele*ts a alue of (eta)= whi*h is the ratio +etween the height of thrust and the totalheight of the side of the sli*e. 5or passive *onditions= where the soil is +eing *o&pressed= should +e so&ewhat greater than ;.33= and foractive *onditions= where the soil is +eing extended= should +e so&ewhat less than ;.33. 9n general we first wor2 a pro+le& with of ;.33 e er!where=and then ad ust for the sides of ea*h sli*e a**ording to whether a*ti e or passi e *onditions o**urthere. Hhat is the logi* of *hoosing ;.33R

%he *oordinate s!ste& is shown in5igure


30/57

N L (.'a)S L = (.'+)

L tan() (.#;)

He will use e . (.#;) to eli&inate in other e uations.Su&&ing for*es in the #,dire*tion= Y F x L ;ZM

/ Y / N #Z NS *os() N sin() L ; (.##a)-li&inating S and N with e s. (.')=

L tan() (.##+)Using e . (.#;)=

(.#$)

- uation (.#$) will +e another +asi* e uation. Dow let us *o&pute the fa*tor of safet!. He i&agine integrating e . (.#$) o er the entirelength of the slide +lo*2. %he thrust at the left,hand end is / a and that at the right,hand end is / +

oth of these &a! +e Eero= or / a &a! +e the thrust against a retaining wall. %hus= integrating e .(.#$)=

/ + / a L tan() d # (.#3)t this point we introdu*e the shear strength of the soil= that is= the shear stress at failure=

f L N ( u) tan() (.#4)%he shear stress along the sliding surfa*e is assu&ed to +e so the fra*tion of the failure

strength of the soil= L (.# )where F is the fa*tor of safet!.

Su+stituting e . (.# ) into (.#3)= and sol ing for F =5 L (.#


31/57

and here we si&pl! repeat e uation (. = or e . 88 in an+u (#' 3= p. < ))

' L / tan(t) ( y $) (.#8+)

%he following uantities are 2nown: / += / a= =

5or ea*h sli*e we 2now: d( =d#=u=== y= $%he following uantities are to +e deter&ined: F =' = / =f

He ha e the *o&plete set of e uations that &ust +e sol ed in order to *o&pute the fa*tor ofsafet! against sliding. %he *o&puter ui*2l! sol es the e uations. %he &ethod of solution *an +eillustrated +! *onsidering two iterations:1st iteration : 5or the first iteration we assu&e thatd'0d# L ;. 9n this *ase it is *lear that e s. (.#


32/57

)eferences

+l"habetical listing of references

+ra&son= . H.= ee= %. S.= Shar&a= S. F o!*e= G. 1. #''


33/57

ro&head= -. D. #' 8. arge landslides in ondon Cla! at >erne a!= 0ent.uarterly 3ournal of /ngineering 1eology 11 = $'#,3;4.

Ca+rera= . G. #' . F 1i22elsen= P. -. #''nternational Sy!posiu! on 7andslides (edited +! Senneset= 0.)3. . . al2e&a= %rondhei&= Dorwa!= # 3',# 44.

Crandell= 6. ". F 7arnes= 6. . #'ough= . -.= Go*2e= . @. F ion= %. #'8#. -ngineeringgeolog! of the Cin*innati area. 9n:1eological Society of A!erica Annual %eeting Field'rip 1uideboo& Fie#d $ri% &o. 18 . Geologi*al So*iet! of &eri*a= oulder= C@= 43, ;.

5le&ing= ". H.= ohnson= ". . F S*huster= ". . #'88a. %he rea*ti ation of the 1anti landslide=Utah. 9n:'he %anti5 Utah5 landslide 1311 . United States Geologi*al Sur e! ProfessionalPaper= Hashington= 6.C.= #,$$.


34/57

5le&ing= ". H.= ohnson= ". .= S*huster= ". . F Hillia&s= G. P. #'88+.'he %anti5 Utah5landslide United States Geologi*al Sur e! Professional Paper1311 = ane+erg= H. C. #''#+. Pore pressure diffusion and the h!drologi* response of nearl! saturated=thin landslide deposits to rainfall. 3ournal of 1eology 99= 88ansen= H. ". #'< . -ffe*ts of the earth ua2e of 1ar*h $ #'olEhausen= G. F Soto= . -. #'8 . &ethod of e aluating the relati e sta+ilit! ofground for hillside de elop&ent. /ngineering 1eology 12= 3#',33ut*hinson= . D. #'8


35/57

9 erson= ". 1. #'';. Groundwater flow fields in infinite slopes.19otechni)ue 40(#)= #3',#43.9 erson= ". 1. #''3. 6ifferential e uations go erning slip,indu*ed pore,pressure flu*tuations in a

water,saturated granular &ediu&. %ath!atical 1eology 2! (8)= #;$ ,#;48.9 erson= ". 1. F a>usen= ". G. #'8'. 6!na&i* pore,pressure flu*tuations in rapidl! shearing

granular &aterials.Science 24( = 'irs*hfeld= ". C. F Poulos= S. .)Casa*rande . ohn Hile! F Sons= Dew Oor2= 4 ,8C progra!s for co!puting displace!ent5 strains5 and

tilts fro! )uadilateral !easure!ents United States Geologi*al Sur e! @pen,file report87-343 = #' p.

ohnson= . 1. F >a&pton= 1. . #'


36/57

0eefer= 6. 0. #'84. andslides *aused +! earth ua2es.1eological Society of A!erica -ulletin 9! (4)= 4;ntroduction to the !echanics of continuous !ediu!. Prenti*e,>all= 9n*.=-nglewood Cliffs= Dew erse!.

1i*halows2i= ". . #'' . Sta+ilit! of slopes: i&it anal!sis approa*h. 9n:Clay and shale slopeinstability (edited +! >ane+erg= H. C. F nderson= S. .). .eviews in /ngineering1eology X. Geologi*al So*iet! of &eri*a= oulder= Colorado= #,eights andslide= n*horage= las2a. 3ournalof the Soil %echanics and Foundations Division5 Proceedings of the A!erican Society ofCivil /ngineers 93(S1 4)= 3$ ,3 3.

S2e&pton= . H. #'


37/57

S2e&pton= . H. #'8 . "esidual strength of *la!s in landslides= folded strata and the la+orator!.19otechni)ue 3! (#)= 3,#8.

S2e&pton= . H. F Pelt!= 6. . #'elens ro*2slide,a alan*he of #8 1a! #'8;. 19otechni)ue 33= $43,$ 3.

7onder inden= 0. #' $. n anal!sis of the Portuguese +end landslide= Palos 7erdes >ills=California. 9n: Depart!ent of 1eology5 Stanford University.

7onder inden= 2. #'8'. %he Portugese +end landslide. /ngineering 1eology 27= 3;#,3 3.

Haldron= . . #' . %he shear resistan*e of root,per&eated ho&ogeneous and stratified soil.SoilScience Society of A!erica 3ournal 41= 843,84'.Hatr!= S. 1. F -hlig= P. . #'' . -ffe*t of test &ethod and pro*edure on &easure&ents of

residual shear strength fro& entonite fro& the Portuguese end landslide. 9n:Clay and shale slope instability (edited +! >ane+erg= H. C. F nderson= S. .). .eviews in /ngineering 1eology X. Geologi*al So*iet! of &eri*a= oulder= Colorado= #3,38.

Hhit&an= ". 7. F aile!= H. . #'< . Use of *o&puters for slope sta+ilit! anal!sis. 3ournal of theSoil %echanics and Foundations Division5 Proceedings of the A!erican Society of Civil /ngineers 93(S1,4)= 4 ,4' .

Hie*Eore2= G. 5.= Gori= P. .= ]er= S.= 0appel= H. 1. F Degusse!= 6. #''nternational Sy!posiu! on 7andslides (edited +! Senneset= 0.)1. . .al2e&a= %rondhei&= Dorwa!= 4##,4#.= 1*0innell= H. P.= 999 F Swanston= 6. D. #' '. Strength of tree roots in a landslide onPrin*e of Hales 9sland= las2a.Canadian 1eotechnical 3ournal 1( = '#,33.

Hu= %. >. F Swanston= 6. D. #'8'. "is2 of landslides in shallow soils and its relation to*lear*utting in southeastern las2a. Forest Science 2( = 4' , #;.


38/57


39/57

,eferences by to"ic

9n this se*tion so&e papers are grouped +! su+ e*t. -a*h su+ e*t *an &a2e a s&all reading pro e*t in itself. @nl! the author and date for referen*es are listed in this se*tion. %he full referen*eare gi en in the a+o e se*tion.

Con"ept of resid&al and peak stren'ths

(S2e&pton #'ane+erg #''#a)(>ane+erg #''#+)(5le&ing F ohnson #''4)(5le&ing et al. #'8#)(>ane+erg F G[2*e #''4)

T&rna'ain -ei'hts Landslide+ ,laska

(>ansen #'< )(7oight #' 3)(Seed F Hilson #'< )

Landslides in sensiti e "lays and "layshales

(Do+le #' 3)(Crawford F -den #'< )Cloughet al (#' )

ro&head (#' 8).( awren*e et al. #''


40/57

(%orran*e #'83)(Hie*Eore2 et al. #''oexter et al. #'8 )

Slides in 0re'on

Slides in Portland

(Cornforth F 1i22elsen #''


41/57

&ppendi+ Mohr,s Circle

>ere we will *onsider stress +oundar! *onditions and how these *an +e represented +! the1ohr *ir*le. 1ohrAs *ir*le has found extensi e used in &e*hani*s= and is *o&&onl! en*ountered instru*tural geolog!= ro*2 &e*hani*s and engineering geolog!. 1ohrAs *ir*le pro ides a graphi*al

wa! of relating stresses on ar+itraril!,oriented surfa*es to the prin*iple stresses. 9t is also used indefining para&eters su*h as *ohesion and angle of internal fri*tion.9n order to see where 1ohrAs *ir*le *o&es fro&= +elow we pose two uestions and pro*eed

to answer the&. %he first uestion (5igure ) we will pose is given a surface inclined at an angle to two &nown perpendicular stresses what is the !agnitude of the stresses parallel to5 and nor!alto this surfaceR

%he se*ond uestion (5igure 8) is given a surface inclined at an angle to the principle stress directions5 what is the !agnitude of the stresses parallel to5 and nor!al to this surfaceR s!ou *an see the uestions are er! si&ilar. %he 2e! in the se*ond uestion is that we are using prin*iple stress dire*tions= so that the planes to whi*h our surfa*e is in*lined ha e no shear stressesa*ting on the&. 9n the first uestion there are shear stresses a*ting on all of the planes. %he answerto the se*ond uestion will lead us to the e uations that when graphed gi e 1ohrAs *ir*le. He willthen see how to use the 1ohrAs *ir*le to graphi*all! sol e the e uations we deri e. Pro+le& # is a&ore general pro+le&.

%he e uations we will deri e will also allow us to *al*ulate the orientation and &agnitude of prin*iple stresses gi en the stress state on two ar+itraril! oriented surfa*es. %hus we *an get a lot o&ileage fro& the following deri ation. fter *o&pleting the deri ation we will go on to explore the properties of the 1ohr *ir*le.

Stresses on an incline# "lane

Gi en a surfa*e in*lined at an angle to two 2nown perpendi*ular stresses what is the&agnitude of the stresses parallel to= and nor&al to this surfa*eR

nn

ns

--

&-

&&

-&

$ C

#igure 8 De%inition diagram %or a" %orcesand b" stresses8 nn is normal to someinclined plane, ns are parallel to theinclined plane8

9t is assu&ed that the ele&ent shown in5igure is in e uili+riu& ^ that is= it is nota**elerating ^ so all for*es will su& to Eero. Using for*e e uili+riu& we will +e a+le to deri erelationships +etween all the for*es in the pro+le&. %his is a er! routine and standard pro*edure in


42/57

&e*hani*s. l&ost all e uations in &e*hani*s start with su&&ing for*es (see for exa&ple= ohnson#' ;= 1al ern #'


43/57

nn L xx sin$() NEE *os$() $xE sin() *os() ( .#.##a)ns L (EE xx) sin() *os() NxE(*os$() sin$()) ( .#.##+)

&ohr-s Circle

He now *onsider a slightl! &ore spe*ialiEed for& of the e uations that will !ield 1ohrAs*ir*le. %he uestion posed here is: gi en a surfa*e in*lined at an angle to the prin*iple stressdire*tions= what is the &agnitude of the stresses parallel to= and nor&al to this surfa*eR

#igure '8 De%inition diagram %orthe deri ation o% 6ohrAs circle8

--

-y

2

1

$

C

%his is a er! si&ilar *ase to that *onsidered a+o e onl! in this *ase we 2now the prin*ipal

stresses=# and $= so there are no shear stresses a*ting on those planes. He ha e defined a positi e #, and y,*oordinate s!ste& as shown in5igure 8. He assu&e that the +od! is in e uili+riu&= so westart +! su&&ing for*es in the #,dire*tionY5x L ;Z

xx # C *os() $ C sin() L ; ( .$.#a)xx # C/ *os() $ C/ sin() L ; ( .$.#+)

re*ogniEing thatC/ L *os() and C/ L sin() ( .$.$)

He get

xx L # *os$() N$ sin$() ( .$.3a) Dow we su& for*es in the !,dire*tion. 5ollowing the sa&e pro*edure as a+o e= !ou *an show

that !ou will get (the reader should perfor& the operations)x! L # $ (*os() sin()) ( .$.3+)

Using the following trigono&etri* identities*os$() L _(#N*os($)) ( .$.4a)


44/57

sin$() L _ (#, *os($)) ( .$.4+)sin()*os() L _ (sin($)) ( .$.4*)

He end up with the following relationshipsxx L (# N$)/$ N (# , $)/$ *os($) ( .$. a)

x! L (# $) sin($) ( .$. +)

%hese e uations ha e the for& of a *ir*le. %hese are the e uations for 1ohrs *ir*le= the!relate the stress on an! plane to the orientation of the plane with respe*t to the prin*iple stresses.

2

-y

--2 11


45/57

&ppendi+ '# -+ercises

E ercise 1 / Dry Soil

ssu&e the following properties for the dr! soil:

Cohesion= C L 48;; Pa 9nternal fri*tion angle, L $;K Unit weight= L $;=;;; D & ,3

9 suggest !ou do all these *al*ulations in a spreadsheet. %he spreadsheet will +e &ore useful if !ouha e single *ells where !ou enter alues forC == = and later the oid ratio=. ll *al*ulationsshould lin2 +a*2 to these *ells. Changing the alues of these *ells will *ause the spreadsheet tore*al*ulate and re,plot !our results allowing !ou to easil! generate plots for different *ases. He will +e adding to this wor2sheet as ti&e goes on so tr! to &a2e the wor2sheet as general as possi+le= itwill +e*o&e a useful tool for future *al*ulations= e en outside this *lass.

#. Plot the *riti*al thi*2ness=t *= as a fun*tion of the slope angle= where the slope angle ranges fro&;K to ';K. Oou will also want to *onstru*t a se*ond plot showing the range of slope angleswhere the *riti*al thi*2ness *hanges rapidl!.

$. Hhat general *on*lusions *an !ou rea*h fro& the plots !ou generated in part #. 5or exa&ple=!ou will want to use the *riti*al thi*2ness e uation to explain wh! the *ur e has this shape.9s this shape li&ited to this set of nu&+ers= or do all possi+le *ur es ha e this shapeR Hhat*ontrols the *riti*al parts of this *ur e that is what is the &a or *ontrol on the *riti*althi*2ness. Part of the *ur e &a2es no sense ph!si*all!= dash that part and explain wh! it isin alid.

3. Suppose that we expe*t failure at the soil,ro*2 interfa*e= and that the soil is 3 & thi*2 and the

slope angle is $ K. Hhat is the fa*tor of safet! against slidingR >ow does the fa*tor ofsafet! *hange if !ou onl! ar! the soil thi*2ness= for exa&ple= what is the fa*tor of safet!against sliding at half the thi*2ness of the soilR

4. 1an! landslides in *ollu iu& in Cin*innati are a+out one &eter thi*2 and o**ur on slopes withslope angles of a+out $ K. Plot the relation +etween *ohesion (C ) and angle of internal fri*tion() that would pro ide a fa*tor of safet! of one under su*h *onditions. Using a range of internalfri*tion angles (sa! K to 4 K)= what is the differen*e in the plotsR %he &axi&u& alue of angof internal fri*tion that !ou should use is $ K wh!R


46/57

E ercise 2 / Stan#ing water

5or this exer*ise we ha e a su+&erged slope. ssu&e that the porosit! of the soil is $;X andthat the soil is saturated with water (no air in pore spa*es). %he dr! unit weight= the *ohesion andthe angle of internal fri*tion are assu&ed to +e the sa&e as those gi en a+o e. 9n this pro+le&= inexer*ise # is t in e uation (#.3.# a).#. Plot the *riti*al thi*2ness=t *= as a fun*tion of the slope angle= where the slope angle ranges fro&

;K to ';K. Oou will also want to *onstru*t a se*ond plot where the slope angle ranges fro&;K to a+out 3 K. Plot the relation +etween *riti*al thi*2ness and slope angle on the sa&egraph as in exer*ise #.

$. Suppose that we expe*t failure at the soil,ro*2 interfa*e= and that the soil is 3 & thi*2 and theslope angle is $ K. Hhat is the fa*tor of safet! against slidingR >ow does the fa*tor ofsafet! *hange if !ou onl! ar! the soil thi*2ness= for exa&ple= what is the fa*tor of safet!against sliding at half the thi*2ness of the soilR

3. Co&pare the results fro& part # and $ in this exer*ise with the results fro& parts # and 3 in-xer*ise #.

4. >ow do !ou expe*t the angle of internal fri*tion and *ohesion of a dr! soil to *hange when it issaturatedR >ow large is the *hange in ea*hR >ow do !ou thin2 this will affe*t the results of!our *al*ulationsR


47/57

E ercise / See"age

#. Plot a third *ur e for *riti*al thi*2ness as a fun*tion of slope angle on the graph *onstru*tedfor exer*ises # F $. Use the sa&e para&eters as the pre ious exer*ises.

t

y

-

B

W a t e r : a b l e

#igure 108 De%inition diagram %or e-ercise +8

$. Sol e the pro+le& analogous to that treated a+o e= +ut this ti&e for a water ta+le that is adistan*ew (&easured nor&al to the slope) +elow the ground surfa*e= where ; w t . Plot*ur es on the diagra& *onstru*ted for assign&ent #= for alues ofw0t of ;= ;.$ = ;. = ;. and#.;. Hrite out the for&ula that !ou de elop. Show that forw0t L # the e uation +e*o&es thee uation for a dr! slope= and that forw0t L ; the e uation +e*o&es that for seepage parallel tothe slope (the e uation !ou used in part #).

3. ppl! the for&ula !ou de eloped for part $ to a landslide at 1*0el e! "oad in Cin*innati.\nder G[2*e has &ade a detailed stud! of the landslide and following are so&e of the rele antdata. %he landslide was a*ti e at the ti&e the &easure&ents were ta2en.

%he landslide is a+out &eters long (fro& head to toe)= and a+out #; &eters deep(&easured erti*all!).

%he a erage slope of the ground surfa*e is a+out 8K. 6uring drilling of three +oreholes through the slide &ass it was noted that the soil= whi*h

*onsists of a+out & of till o erl!ing a+out 3 & of gla*ial la2e,*la!= was rather dr! untilwe had drilled through the la2e *la! and through one or two li&estone +eds in the +edro*2 underl!ing the la2e *la!. %hen the *uttings +e*a&e er! wet (&udd!) and aftera few hours water rose in the +orehole to a le el of ;. to #. & fro& the ground

surfa*e (depending on the ti&e of !ear).Using these data= !ou are to *o&pute the residual angle of internal fri*tion of the la2e *la!in ol ed in the sliding.


48/57

E ercise $ / Tree ,oots

1ar! "iesten+erg deter&ined and esti&ated the following para&eters for a landslide that shestudied in Cin*innati:

L ; L 3 K L #$K w L '.8 2D/&3

t L ;. & t L #'.< 2D/&3 (#/ ) L . 2D/&$

Oou are to deter&ine the fa*tor of safet! against sliding.Hhat is the effe*t of roots on the slide &ass. Hhat is the fa*tor of safet! without rootsR >ow

do roots enter the fa*tor of safet! *al*ulation what is what soil propert! do the! *hange andhow would we &odel their effe*t. %hat is= if we were to do a +a*2 *al*ulation for &aterial para&eters= what propert! would root strength +e indistinguisha+le fro&.


49/57

E ercise % / Fellenius &etho#

9n order to Tget the feelT for *o&putation of sta+ilit! fa*tors= deter&ine the fa*tor of safet!against sliding for the slope shown +elow:

1.

+)

35 285 m

h > 5825 m

+825 m

- >1185 m 1185 m 1185 m28,),

1

#igure 118 De%inition diagram %or e-ercise 58

%he water ta+le is assu&ed to +e at the upper surfa*e of the slide= and= using the solution for pore,water pressures in an infinite slope= we *o&pute

r u L (u/ h) ;.3 ;.4in whi*hh is the height of the sli*e (see figure +elow) and is the saturated unit weight of the

soil. %he para&eters are=

L #< 2D/&$ L $4 2D/&3 L #4K

:able 28 Some o% the important parameters %or each slice in e-ercise 58

5or pro+le&s other than this one= of *ourse= !ou would use &an! &ore than three sli*es. %he purpose is for !ou to +e a+le to ui*2l! sol e a pro+le& approxi&atel!M the &ethod is what isi&portant right now= not the result.

Sli*e #i(&)

hi(&)

( i(2D)

i ui(2D/&$)

# ##. $.


50/57

5irst= *o&pute the fa*tor of safet! using the 5ellenius &ethod ( nswer is approxi&atel! F L#.#< at R). %hen= start with an esti&ated fa*tor of safet! of #.; and iterate the &odified, ishopsolution three ti&es in order to esti&ate the fa*tor of safet! +! that &ethod.

s a se*ond exer*ise= use the 5ellenius &ethod to T+a*2 *al*ulateT the residual strength of thsoil in ol ed in the slide shown in the figure on pg. RRRRRR. %his is a &ethod that we *o&&onl! uin anal!Eing existing landslides. %he residual *ohesion=r is generall! nearl! Eero= so we set it e ualto Eero. %hen we set the fa*tor of safet! e ual to #.; and *o&pute the angle of residual fri*tion=r .

9t is presu&a+l! *lear= upon exa&ination of e . ($.3. )= that we *an sol e dire*tl! for theresidual fri*tion=

(RRRRR)

in whi*hi L Di Ui and Di L H i *os(i)

for the 5ellenius &ethod= andU i is gi en +! e . (RRRRRR).


51/57

E ercise 0 / (anbu &etho#

"A+$ ,Gi en the following landslide geo&etr!:

10 m

Slip Sur%ace

Water table

Emper ious layer

nd the following a erage &aterial properties:

erage total unit weight L $; 2D/&3

"esidual Cohesion L 2Pa "esidual 5ri*tion ngle L # K#. Cal*ulate the a erage fa*tor of safet!= and the distri+ution of for*es within the landslide. >ow

sensiti e is !ou solution to ar!ing these para&etersR$. ssu&e that !ou ha e h!drostati* pore water *onditions at the slip surfa*e. >ow would !ou

i&pro e on this esti&ate of pore water pressure *onditions3. >ow would one ha e to either load or unload the upper ` of the slide in order to get a fa*tor

of safet! of approxi&atel! #.


52/57

"A+$ ,,#. "ead the following paper:

au&= ". .= and 5le&ing= ".H.= #''#. Use of longitudinal strain in identif!ing dri ing andresisting ele&ents of landslides.1eological Society of A!erica -ulletin 103 (8):##$#,##3$.

. Hhat is the topi* of the paper= and wh! is it i&portantR

. >ow would !ou identif! the arious parts of the slide in surfa*e &appingR Use infor&ationfro& so&e of the other papers we are reading in *lass. Cite referen*es for !ourinfor&ation.

C. Using either the spreadsheet ( an+u.xls) or progra&s ( an+u3#.*pp/ an+u3#.exe= an+u3;.+as) pro ided perfor& a an+u anal!sis of the lani,Pat! slide. Hat*h !our*oordinate s!ste& and sli*e nu&+ering

6. Hhat assu&ptions were &ade in the anal!sis of this slideR re the! reasona+leR

-. Hould !ou perfor& a different anal!sisR 9f so= what would !ou do differentl!R5. Use infinite slope theor! to anal!Ee the lani,Pat! slide. >ow different is the fa*tor of

safet! fro& that *al*ulated using the an+u &ethodR Hh! do !ou thin2 the results aresi&ilar/dissi&ilarR

G. Hhat would !ou do to sta+iliEe the slide. a*2 this up with results fro& !our anal!sis.


53/57


54/57

(anbu 1.c"" 1 "ages3

1.1


55/57

(anbu 4.bas 1$ *ages3

1.1


56/57

(anbu. ls "ages3


57/57

&ppendi+ D# /anbu 0"12(3 .aper on Slo"e Stability Com"utations

anatomy of landslides

Documents