ISRM-Is-1981-107_Probabilistic Consideration of Progressive Action in Weak Rock Masses

7/27/2019 ISRM-Is-1981-107_Probabilistic Consideration of Progressive Action in Weak Rock Masses

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P r oc e ed i n g s o f t h e In t er n at i o n al S y m p o s iu m o n W e ak R o c k / T o k y o /21-24 September 1981

Probabilistic consideration of progressive action

in weak rock masses

ROBIN N.CHOWDHURY

University of wollongong, Australia

DIMITRI A-GRIVAS

Rennsetaer Poly technlcal Lnstitute, USA

I NTRODUCTI ON

St abi l i t y i s a maj or consi der ati on i n many

pr obl ems of soi l and r ock mechani cs. Tr a-

di t i onal l y a f ac tor of s af et y agai ns t c om-

pl et e or si mul t aneous f ai l ur e i s est i mat ed

af t er cal cul at i ng r esi st i ng and di st ur bi ng

f or c es on t he bas i s of f i el d i nves t i gat i on,

t est i ng and anal ysi s. Fur t her devel opment

of t he s t at e- of - t hear t r equi r es t hat

at t ent i on be gi ven t o t he pos s i bi l i t y of

pr ogr essi ve act i on wi t hi n a par t i cul ar

geol ogi cal medi um.

Ther e ar e many ways i n whi c h ' pr ogr ess i veacti on' may occur and many f act or s t hat

i nf l uence changes l eadi ng t o such act i on.

The t er m must not be used synonymousl y wi t h

' pr ogr essi ve f ai l ur e' al t hough such f ai l ur e

may be a di r ect or i ndi r ect consequence of

' pr ogr essi ve act i on' i n some cases.Si nc e f ai l ur e of an ear t h mas s i s r ar el y

an i sol at ed or si mul t aneous event , consi d-

er a t i on of pr o gr e ss i ve ac t i on as an ai d t o

st abi l i t y anal ysi s i s not onl y concept ual l y

appeal i ng but al so makes pr act i cal sense.

Such an appr oach can enhance good engi neer -

i ng j udgement and f aci l i t at e speed andaCCur acy i n deci si on- maki ng.

Consi der i ng var i ous uncer t ai nt i es i n geo-mechani cs, a pr obabi l i st i c appr oach i s

advocat ed i n t hi s paper f or anal ysi ng chan-

ges i n st abi l i t y of a weat her ed r ock mass.

The over al l saf et y mar gi n of an ear t h mass

whi ch i s hi gh t o begi n wi t h, may decr ease

gr adual l y as secti ons of t he mass . f ai l and

l os e t hei r r es pec ti ve s af et y mar gi ns . Co-

nsi der at i on al so needs t o be gi ven a weat h-

er ed r ock mass as an assembl age of el ement s

whi ch ar e i nt er connect ed and hence i nt er -dependent .

I NFLUENCI NG FACTORS

Ther e ar e di f f er ences as wel l as si mi l ar i t -

i es bet ween t he engi neer i ng behavi our ofsoi l masses on t he one hand and r ock masses

on t he ot her . The r e spons e of al l geo-

l ogi cal medi a under l oadi ng or unl oadi ng

i s i nf l uenced by geomechani cal as wel l as

envi r onment al f ac tors . I n s t abi l i t y pr o-

bl ems , s hear s tr engt h and s hear s tr es s ar e

of di r ec t and pr i mar y i mpor t anc e. I t i s

of t en nec es s ar y t o gi ve par t i c ul ar at t ent -

i on t o st r ess- st r ai n behavi our , per meabi l i t y

and dr ai nage condi t i ons. Compr essi bi l i t y,

permeabi l i t y and t he devel opment and di ssi -

pat i on of exc es s por e wat er pr es s ur es ar e

much mor e i mpor t ant f or soi l s t han f orrocks.

Yet t he t r ansi t i on f r om soi l s t o r ocks i s

of t en a gr adual one. I n many s i t uat i onsweat her ed r ock masses behave l i ke soi l

mas ses . I t i s qui t e di f f i c ul t t o gener a l -

i se and t o pr ovi de uni f or m gui del i nes f or

al l ki nds of weat her ed r oc ks . Yet s ome

poi nt s ar e not ewor t hy i n t he c ont ext of

t hi s paper . Unl i ke har d unweat her ed r ocks,

weat her ed r ocks do not necessar i l y f ai l

al ong wel l - def i ned pl anar di scont i nui t i ese. g. j oi nt pl anes . Cur ved s l i p s ur f ac es

have been obser ved i n t he weat her ed zonesof r ock masses and especi al l y i n sof t and

f r ac tur ed r oc k mas s es . Br i t t l enes s and

s tr ai n- s of t eni ng ( and s tr engt h r educ ti on by

ot her means) may be si gni f i cant f or pr ogr e-

s s i ve ac ti on i n weat her ed r oc ks . Thes e

f ac tors c an enhanc e t he t endenc y f or pr ogr -

es si ve f ai l ur e as i n s oi l s . Cons i der i ng t he

f ac t t hat f ai l ur e may oc cur on s l i p s ur f ac es

of ar bi t r ar y s hape, t he i nf l uenc e of non-

uni f or m s t r es s and s t r ai n di s t r i but i on i sof t en si gni f i cant .

Qui t e apart f r om t he obvi ous geomechani calaspect s ment i oned above, i t i s desi r abl e

t o c ons i der envi r onment al f ac tors . The i n-

f l uenc e of s eepage, s ur f ac e r unof f ( and

er osi on) , under cut t i ng, f l oodi ng and ot herphenomena i s t o cause pr ogr essi ve changesi n a weat her ed r oc k mas s . Agai n i n t hi s

659


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r es pec t t her e i s a mar k ed s i mi l ar i t y bet -

ween soi l and weat hered r ock masses.

PROPOSED PROBABI LI STI C APPROACH - PROGRES-

SI ON OF SLOPE FAI LURE

I n t r adi t i onal det er mi ni st i c or pr obabi l i -s t i c appr oac hes onl y t he over a l l s ta bi l i t y

or s af et y of an ear t h mas s i s c ons i der ed.

I n r ecent wor k ( Chowdhur y, 1981) i t has

been suggest ed t hat an ear t h mass be con-

si der ed as an assembl age of el ement s, each

el ement i havi ng i t s own l ocal saf et y

mar gi n SMi ' As an exampl e, one may consi dera sl opi ng mass above a sl i p sur f ace di vi -

ded i nt o n v er t i c al s l i c es . The event of

f ai l ur e of t he i t h s l i c e may be r epr es ent -

ed by Fi and onl y t wo adj acent el ement s or

sl i ces wer e consi der ed at anyone t i me by

t he wr i t ers i n previ ous wor k ( Chowdhur y andA- Gr i vas , 1981) . I n gener a l , f ai l ur e of

t he i t h sl i ce wi l l depend on t he pr evi ous

f ai l ur e of al l ( i - I ) s l i ces al ong t he pat h

of pr ogr essi on of f ai l ur e, assumi ng f ai l -

~r e t o st ar t i n a gi ven mode. For exampl e,~t may s ta r t f r o m one ext r emi t y of t he

f ai l ur e pat h or s l i p s ur f ace ( i . e. t o e or

c r es t ) and pr o 9r e ss t o t he ot her ext r e mi t y

I n accor dance wi t h t he assumpt i on t hat

f ai l ur e can not j ump over unf ai l ed sl i ces

or segment s, one can wr i t e t he f ol l owi ng

equat i on of condi t i onal pr obabi l i t i es

P[ Fi/ Fl, F2 . . . . , F. l ] =P[ F. / F. 1] ( 1)1- 1. 1-

i n whi ch Fl, F2, . . . . , Fi ar e r espect i vel y

t he event s of f ai l ur e of sLi . ces 1, 2, . . . , i .

Ther ef or e, t he pr obabi l i t y wi t h whi ch t he

( i +l ) t h s l i c e f ai l s depends onl y on t he

f ai l ur e of t he i t h s l i ce. Fur t her mor e i t

ca~ be ar gued t hat t he pat h f ol l owed by

f a~l ur e up t o t he i t h s l i c e has no i nf l ue-

nc e on t he pr o babi l i t y of f ai l ur e of t he

( i +l ) t h sl i ce. The condi t i onal pr obabi l i t y

gi ven above i s anal ogous t o Markov memory

i n ~ Markov chai n model and i t may be con-ven~ent l y r ef err ed t o as a ' one- st ep memor y'

The pr obabi l i t y of l ocal f ai l ur e of sl i cei may be desi gnat ed as:

P[F.] = p[SM.


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i nvol ved i n f ai l ur e.

I nt ut i vel y one woul d expect t he over al l

pr obabi l i t y of pr ogr es s i ve f ai l ur e at

st age i ( whi ch may be desi gnat ed by Pf

( i

t o depend on t he f ol l owi ng quant i t i es: ~

1. The over al l pr obabi l i t y of f ai l ur e

bef or e any l oc al f ai l ur e i s c ons i der edi . e. P

2. T~e pr obabi l i t i es of l oc al f ai l ur e

of al l segment s or sl i ces consi der ed i n

t he SOl ut i on i . e. Pf l , Pf 2' . . . . Pf . . Not et hat t hese i ncl ude t he pr obabi l i t yl of

f ai l ur e i ni t i at i on l oc al l y i . e. POI ' i f

one can say t hat Pf l =POl . I f one knowst hat POI ( pr obabi l i t y of pr ogr es s i on f r om

s t at e z er o t o s t at e 1, s t at e z er o bei ng

t he s t at e of no f ai l ur e) i s di f f er ent f r om

Pfl, t hen POI must be cal cul at ed separ at el y. Not e t hat t he pr obabi l i t y POI may be

l nf l uenced markedl y by t ensi on cr acks,s t r es s c onc ent r at i on and other f ac tors .

On t he c ont r ar y Pf l i s es ti mat ed c onvent -10nal l y on t he bas i s of l oc al s af et ymar gi n.

3. The pr obabi l i t y of pr ogr es s i ve f ai l ur eConsi der i ng t wo sl i ces or segment s at a

t i me i . e. P12, P23' . . . . Pi - l ' i et c . Thi sl S by f ar t he mos t i mpor t ant s et of quant i t i es.

I n ot her wor ds t he over al l pr obabi l i t y

of pr o gr e ss i ve f ai l ur e at any s t a ge i s a

f unc ti on of s ever al quant i t i es or gr oups

of quant i t i es and one may wr i t e: -

( P 12, P23' . . . . p. 1 ) ]1.- ,1

(5)

The met hodol ogy f or quant i f yi ng Pf p( i )has yet t o be devel oped and i t i s desi r abl e

t hat t hi s s houl d be done. Yet , al r e ady t he

baSi c i nf or mat i on f or eval uat i on of pr o-

gr essi ve f ai l ur e pr obabi l i t y i s avai l abl e

and can be used f or deci si on- maki ng. The

Power f ul model of pr ogr essi ve f ai l ur e pr o-Posed by Chowdhur y and A- Gr i vas ( 1981) i s

not pl aced i n a ser i ousl y di sadvant aged

POSi t i on because Pf p( i ) can not , at pr esent ,be cal cul at ed st r i ctl y i n t he manner di s-

c us s ed above. I t i s appr opr i at e t o emp-

hasi se t he t r emendous i mpor t ance of t he

quant i t i es Pi i +l ( or Pi - l i ) par t i c ul ar l ywhen consi der ed i n combi nati on wi t h t he

Ot her quant i t i es ment i oned above. How-

ever , a somewhat si mpl i f i ed appr oach t o

t he comput at i on of over al l pr ogr essi ve

f ai l ur e pr obabi l i t y i s pr esent ed bel ow.

S! MPLI FI ED APPROACH TO QUANTI FY OVERALL

PROGRESSI VE FAI LURE PROBABI LI TI ES

Thenumer i cal val ues of t hese pr obabi l i t i es

can be f ound by assumi ng t he saf et y margi ns

of t he f ai l ed s l i c es al ong t he s l i p s ur f ac e

t o be equal t o zer o; and c ons i der i ng t hat

t he r emai ni ng saf et y mar gi n of t he sl ope

mas s i s t he s um of t he s af et y mar gi ns of

t he s l i c es not yet af f ec t ed by f ai l ur e or

r upt ur e. Thi s sum i s nor mal l y di st r i but edsi nce i ndi vi dual saf et y mar gi ns of sl i ces

are assumed t o be normal l y di st r i but ed.

Thi s i s because t he sum of nor mal var i at esi s al so a nor mal var i at e.

I f t he f ai l ur e al ong t he s l i p s ur f ac e

wi t h a t ot al of n s l i c es has r e ac hed s l i c e

i t her e ar e k = ( n- i ) s l i ces t hat cont r i but et o t he s af et y of t he s l ope.

The mean val ue and st andar d devi at i on of

t he net avai l abl e saf et y mar gi n ar e t henequal t o

n

SMk

= L SMk=i k

n

( Lk=i

and (6)

and t he pr o babi l i t y of f ai l ur e of t he s l opei s P [ SMk


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bei ng publ i shed separat el y.

The appr oach out l i ned i n t hi s paper ( and

especi al l y i n Eqs. 6 and 7) i s par t i cul ar l y

us ef ul i n s oi l s and r oc ks i n whi c h s tr ai n-

sof t eni ng i s not of pr i mar y i mpor t ance f or

pr o gr e ss i ve f ai l ur e . Ot her f ac t er s s uc has non- uni f or m s t r es s di s t r i but i on, s t r es s

c onc ent r a ti ons , t ens i on c r ac ks , j oi nt s andf i ssur es may be much mor e i mpor t ant . I n

any case, i nf or mat i on obt ai ned on t he

basi s of Eqs. 6 and 7 shoul d be suppl ement -

ed by cal cul at i on of ot her quant i t i es

di scussed i n t he pr evi ous sect i on andl i st ed i n Eq. 5.

EXAMPLE CALCULATI ONS

Tabl e 1 gi ves t he geomet r i cal and ot her

( cont i nued next col umn)

Tabl e 1. Dat a f or exampl e pr obl em

part i cul ar s of an exampl e pr obl em concer n-

i ng a s l ope. St at i s t i c al val ues of s hear

st r engt h par amet er s ar e al so shown. The

c al c ul at i ons of t he pr o babi l i t y of f ai l ur e

Pf p( i ) of t he s l ope as t he f ai l ur e r e ac hesand encompasses sl i ce i are shown i nTabl e 2.

c. eomet r i c al Dat a

Sl ope Type and: - Si mpl e Sl ope of 450

i ncl i nat i onI ncl i nat i on

Sl ope Hei ght : - 25f t . ( 7. 62m)

Cr i t i cal Sl i p Sur f ace: - Ci r cul ar Sl i p Sur f ace AB pass i ng t hr ough t oe A wi t hr adi us of 44. 12 f eet and di st ance OM ( f r o mori gi n t o chor d AB) = 38. 23 f t .

No. of Sl i ces: - 9 ver t i cal sl i ces.

Uni t Wei ght : - y = 120 pcf ( 18. 84 kN/ m3

)

Stati sti cal Val ues of Shear Str engt h Par amet er s-

( 4 ) 380), - kN/ m2)mean val ues ) J = 0. 78 = c = 500 psf ( 23. 94 TABLE 2 I Sc oef f . of var i at i on V = 10%, V

c = 50% ON NEXT PAGE) Jcor r el ati on coef f i ci ent r = - 0. 20

c , ) Jpor e wat er pressure as sumed t o be zero or negl i gi bl e.

Fact or of s af et y ( conven t i onal ) on t he cri t i cal sl i p sur f ace FS = 2. 43

DI SCUSSI ON OF RESULTS

These r esul t s ar e of consi der abl e i nt er est

al t hough t he met hod of cal cul at i on has no

di r ect l i nk wi t h t he compr ehensi ve pr obab-

i l i s ti c model of pr o gr es si ve f ai l ur e r ef er -

r ed t o ear l i er . Fi r s t l y, i t may be not e d

t hat t he pr o babi l i t y of f ai l ur e on t he

basi s of convent i onal l i mi t equi l i br i um

( i . e. consi der i ng t he event of si mul t aneous

f ai l ur e and i gnor i ng l oc al f ai l ur e) i s

negl i gi bl e or i nsi gni f i cant . The st andar d-i s ed nor mal var i at e i s c l os e t o 0. 5 and

t he pr o babi l i t y of f ai l ur e Pf i s a ver y

smal l number ( out si de usual t abl es) . As

t he i ni t i al s l i c es ar e as s umed t o have

f ai l ed wi t h compl et e l oss of saf et y mar gi n,

t he val ue of t he over a l l pr o babi l i t y of

f ai l ur e Pf p( i ) r emai ns i ns i gni f i c ant . I t

i s onl y when f ai l ur e r eac hes t he f our t h

s l i ce t hat t he val ue of Pf p( i ) s t ar t s be-comi ng si gni f i cant . When onl y t he l ast

sl i ce r emai ns i n an unf ai l ed condi t i on, t he

pvobab iLi ty of over a l l f ai l ur e i s t he s ame

. s t he pr o babi l i t y of l Qc al f ai l ur e of

t hat s l i c e i . e. Pf p( 9) = Pf 9' The pr o babi l -

i t y of f ai l ur e wi l l , of c our s e, be 100%af t er even t he l as t s l i c e i s as sumed t o

have l os t al l t he s af et y mar g i n.

Thi s exampl e br i ngs i nt o shar p f ocus an

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Tabl e 2. Pr ogr ess i ve f ai l ur e pr obabi l i t i es Pf p( i )

Cal c ul at i on of over al l pr obabi l i t y of f ai l ur e of t he s l ope when " r upt ur e" has r eac heds l i c e i .

No. ofSl i ce

i

9L SM

k=i kSM.

1.

1

2

3

4

5

6

7

8

9

51. 10xl 0

51. 43xl 0

51. 69xl 0

51. 79xl 0

51. 55xl 0

51. 16xl 0

51. OOxl O 50. 785xl O

1. 52xl 05

51l . 938xl 0

510. 838xl 0

59. 408xl 0

57. 768xl 0

55. 978xl 0

54. 428xl 0

53. 268xl 0

52. 268xl O1. 51xl 05 .

44. 47xl 0

44. 63xl 0

44. 91xl 0

45. 70xl 0

45. 80xl 0

45. 83xl 0

46. 81xl 0

46. 90xl 0

49. 22xl O

* ver y c l os e t o 0. 5

**ver y smal l number ( out si de usual t abl es) .

i mpor t ant f eat ur e ( i n f act, a shor t comi ng)

of convent i onal l i mi t equi l i bri um model s

of s t a bi l i t y. Thi s i s t hat a hi gh over a l l

s af et y f ac tor al s o i mpl i es a hi gh l oc al

s af et y f act o r . Thus t he s l ope not onl y

has a hi gh over al l saf et y mar gi n t o begi n

wi t h but al s o l oc al s af et y mar gi ns ar e

Si gni f i cant . Consequent l y even whenei ght sl i ces have f ai l ed t he pr obabi l i t y

of f ai l ur e i s not hi gh. I n r eal i t y t hi s

may not be so. I t woul d be more meani ng-

f Ul t o bas e pr obabi l i s ti c c al c ul at i ons on

t he r esul t s ~f f i ni t e el ement anal ysi s

f r om whi ch st r ess- def or mat i on pat t er ns

can be i dent i f i ed whi ch t ake i nt o consi d-er a t i on t he hi s t o r y of s l ope f or mat i on i naddi t i on t o t he t ype of s l ope.

The r esul t s concer ni ng pr obabi l i t i es ofover al l f ai l ur e dur i ng t he p roc es s of

f ai l ur e pr ogr essi on woul d t hen be consi d-

er abl y di f f er ent and mor e r eal i s ti c t han

t hose based on convent i onal l i mi t equi l i -

br i um. whi l e i t i s not di f f i c ul t t o obt ai n

dat a f or c al c ul at i on of s af et y mar gi ns

f r om s tr es s and def or mat i on anal ys es , i t

woul d be advi sabl e t o i nt er pr et and modi f y

sUch dat a on t he basi s of obser vat i onal

i nf or mat i on concerni ng def or mat i on,

s tr ai ns , por e pr es s ur es and ot her par t i -

cUl ar s. I nst r ument at i on and per f ormance

moni t ori ng have al r eady assumed j ust i f i abl e1. mpor t ance i n geot echni cal engi neer i ng.

I n keepi ng wi t h t hi s t r end, cal cul at i onOf saf et y margi ns shoul d i ncl ude observa-t i onal dat a as par t of i t s bas i s .

As successi ve sl i ces ar e assumed t o havel ost t hei r l ocal saf et y mar gi n, one woul d

expec t t he over al l pr ogr es s i ve f ai l ur e

pr obabi l i t y t o i nc reas e muc h f as te r on t he

9

( Lk=i

-3

i

518. 564x10 4

18. 0183xl 04

17. 4132xl 04

16. 7067xl 04

15. 7044xl 04

14. 5940xl O413. 3789xl O11. 516x104

9. 22x104

+6. 431

+6. 0149

+5. 4027

+4. 4696

+3. 8066

+3. 0341

+2. 442+1. 9694

+1. 6377

**

0. 49993

0. 49866

0. 49277

0. 4755

0. 4555

- 57xl O_ 3

1. 34xl O_ 3

7. 23xl O_2

2. 45xl O

4. 5xl O- 2

bas i s of r eal i s ti c s tr es s es and obs er vat -

i onal dat a t han on t he basi s of si mpl e

l i mi t equi l i br i um. Notwi t hst andi ng t heser emar ks t he cal cul at i ons gi ven her e

est abl i sh a basi s f or assessi ng pr ogr essi ve

ac ti on. I n t hat s ens e s uc h c al c ul at i ons

may be of consi der abl e val ue t o t he geo-t echni cal engi neer who needs a l ogi cal and

quant i t at i ve bas i s f or exer c i s e of j udge-

ment and, i n gener al , f or deci si on- maki ng.

PROGRESSI ON CONSI DERED AS A MARKOV CHAI N( EXTENSI ON TO TI ME DOMAI N)

The Mar kov Chai n

Havi ng def i ned pr ogr essi on of f ai l ur e i n a

gener al way as a spat i al and cont i nuous

ext ens i on of t he f ai l ur e zone al ong a Sl i psur f ace, a pr obabi l i st i c model i s r equi r ed

t o eval uat e t he pr obabi l i t y of t he s l opebei ng i n any s tage of pr ogr es s i ve f ai l ur e

at a gi ven t i me. ( The st age may be i dent -

i f i ed by t he s l i c e i t o whi c h f ai l ur e has

pr o gr e ss ed f r o m one end of a s l i p s ur f ac e) .

The use of t he concept of Mar kov chai n,

named af t er Andr ei Andr evi ch Markov ( 1856-

1922) , appear s t o be c ons i s t ent wi t h t he

physi cal model di scussed i n t hi s paper and

wor k c i t ed ear l i er . A Mar kov pr oc es s wi t h

di scr et e st ages or l i mi t s may be consi der ed

and t he i ni t i al s t at e of a s l ope may ber e gar ded as t he f i r s t l i nk i n t he Mar kov

c hai n. Any s t at e i s des cr i bed by a s et or

vector of pr obabi l i t i es. Each el ement oft hi s vector denot es t he pr obabi l i t y wi t h

whi ch t he sl ope i s i n any di screte st age.

For exampl e, i n a t hr ee st age chai n t he

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6/6

vec to r ( 0. 9, 0. 06, 0. 04) i ndi c at es t hat t he

pr o babi l i t i es of bei ng i n t he f i r s t , s ec ond

or t hi r d s t a ge of f ai l ur e ar e r e spec t i vel y

0. 9, 0. 06 and 0. 04. Fr o m one s ta te t r a ns i -

t i on mayor may not oc cur t o t he n ext s ta te.

An abs or b i ng s ta te i s one f r o m whi c h t her e

i s no f ur t her t r a ns i t i on. F or exampl ewhen t he l as t s t age of f ai l ur e has been

r e ac hed t her e i s no f ur t her t r a ns i t i on and

t he pr o babi l i t y of bei ng i n s t age n i s one.

The Tr ansi t i on Mat r i x

The pr obabi l i t y vect or of st at e af t er one

or more t r ansi t i ons may be det ermi ned

us i ng t he model of Mar k ov c hai n. Let TIo

and TI denot e t he s t at e vect o r s af t e r 0

and mmt r ansi t i ons r espect i vel y. These may

be wr i t t en as f ol l ows:

(8)

i n whi c h n i s t he

ur e pr ogr essi on.

et c. ar e mut ual l y

number of st ages of f ai l -

Si nce TI lm' TI 2m' TI 3mexcl usi ve

T I .1. m

1

n

L T I .1.0and 1 (9)

Ac cor di ng t o t he model , t hes e vec to r s ar er el at ed by t he t r ans i t i on pr o babi l i t ymatr i x P as f ol l ows:

T I

m( 10)

i n whi c h P i s a s quar e mat r i x of s i z e n.

The eval uat i on of t hi s mat r i x woul d be t he

mai n t ask i n such an appr oach. Once t hi s

mat r i x i s known, si mpl e mat r i x mul t i pl i c-

at i on may be used t o det er mi ne t he st at e

af t er any speci f i ed number of t r ansi t i ons.

Not e t hat t r a ns i t i ons c or r es pond t o t i me

and i t woul d be nec es s ar y, t her ef or e, t oknow t he physi cal pr obl em i n t he t i me

domai n as wel l as i n i t s ot her aspect s

( i . e. geomet r y, st r engt h par amet er s et c. ) .

For exampl e pore wat er pr essur e may change

wi t h t i me, st r engt h may change wi t h t i me

or ot her c ondi t i ons af f ec ti ng s ta bi l i t y

may c hange. The nat ur e of t he P mat r i x i n

t hi s c as e i s qui t e di f f er ent f r om t hat of

t he t r ansi t i on mat r i x ( wi t hout t i me domai n)consi dered ear l i er ( Chowdhur y and A- Gr i vas,1981) .

Ti me of f ai l ur e may i t sel f be r egar ded

as a r andom var i abl e and i s i ncor por at edi n t he Mar kov model as t he number of t r ans-

i t i ons. I f t he t ot al l i f e of a sl ope can

be speci f i ed on t he basi s of exper i ence

or on desi gn consi der at i ons t hen a cor r es-

pondence between t he t otal mean number of

t r a ns i t i ons t o c ompl et e f ai l ur e and bet -

ween sl ope l i f e can be est abl i shed as a

s c al e. Thi s s c al e may t hen be us ed t o

s t udy t he t i me r at e of s pat i al pr o pagat i on

of f ai l ur e . I n s ome c as es i nc r eas e or

dec r eas e wi t h t i me of t he l oc al s af et ymargi ns wi t h changi ng condi t i ons ( e. g.

f al l or r i s e i n p or e pr es s ur e, r h eol ogi c al

dec r eas e of s tr engt h wi t h t i me) c an be

expl i ci t l y consi dered.

Many soi l s exhi bi t st r ai n- sof t eni ng be-

havi our and i t may be necessary t o i ncl ude

t hi s ef f ec t i n r el evant expr e ss i ons . As

a c ons equenc e of s t r a i n- s of t eni ng, t he

post - f ai l ur e saf et y mar gi n of any segment

wi l l be r educed and t he t r ansi t i onal

pr obabi l i t i es wi l l change accordi ngl y.

I t i s wor t hwhi l e t o ment i on her e t hat

t he cor r el at i on bet ween t wo adj acent sl i ceswi l l not be s o s t r ong i n a s oi l whi c h has

mar ked st r ai n- sof t eni ng behavi our t han i n

one whi ch does not have such behavi our .

Thi s i s because t he saf et y mar gi n of a

f ai l ed sl i ce wi l l depend on par amet ers

( cr ' r ) whi l e t hat of t he unf ai l ed s l i c ewi l l depend on par amet er s ( c and ) .

REFERENCES

A- Gr i vas, D. & Chowdhur y, R. N. 1981, Pro-

gr es si ve f ai l ur e of s t r a i n- s of t eni ngsoi l sl opes under undr ai ned condi t i ons.

( under r evi ew f or publ i cat i on) .

Chowdhur y, R. N. 1981, Probabi l i st i c Appr o-

ac hes t o pr o gr es si ve f ai l ur e, t wo

r epor t s, Uni ver si t y of Wol l ongong.

Chowdhur y, R. N. &A- Gr i vas , D. 1981, A

pr obabi l i st i c model of pr ogr essi ve f ai l -

ur e of s l opes ( under r evi ew f or

publ i cati on) .

Skempt on, A. W. 1964, Long- t erm st abi l i t y

of cl ay sl opes, Ranki ne l ecture, Geo-

t echni que, Vol . 14, No. 1, 77- 101.

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