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Proc. IVth International Confe rence andField Workshop on T,~nds l ide s , 9 8 5 Tokyo

A New Method for Predicting the FailureTime of a SlopeFukuzonu, T.Sationrl It~nearuhCcntcr fo r Disaster PrrvcntionIhnraki , Japan

S m O P S I SA basic formula on the veloclty of surface displacement just before t h e f a l l u r e o f a slape ispropose d nd it is madc clear t h a t the curves wrltten by each p o i n t of t h e inv rs number of thev c l o c ~ t y f surface diaplacernent are divided into three k i n d s of forms, these a l e linear, convex a n dconcave and decrease u n ~ i o r r n l y ~n any case. In t h e c a s e o f a l i n e a r curve , t h e failure trme of aslope can be exactly predicted by the p o l n t at which the straignt line cr o sses the axls of absclssa.

n the case of a convex or concave curve, the .allure tlme c a n be roughly p c c d ~ c t e d y t he point atwhich the tangential line at each point of the curve crosscs the axls of a b s c ~ s s a . Furthermore, ~t1s clear that the curve writtcn by each t l m p r e d ~ c t e d oughly is linear. So t h e f a l l u r e t i m e ca nbe exactly predicted by t h e point t w hi c h t h e straight l ine crosses the axls of absclssa. Theavallabilrty 01 these methods IS s h o w n by a p p l i c a t i o n t o some types of slope failureINTRODUCTION

The n e w method f o r predicting t h e f a i l u r et i m e of a slope proposed i n thls pa pe r is amethod using t h e variations of surface d ~ s p l a c e -ment just b e f m r e t l t c failure.

Saito and Uezawa 1 9 6 1 performed m n y creeptests of so and proposed t h a t t h e increment O t h e loqarithm o f crecp r u p t u r e llfe dl 1 sproportional to the logarithm eE straln rate ( k )in the secondary creep. That is

is 1 / B . H u t t h e factor can be d e c i d e d not b e f o t eb u t after t h e f a t l u r e because t h e values ofa r e different a t each slope failure.Tn t h ~ w a p c r , a new method f o r predictingthe failure t ime u s i n g the inverse number ofvelocity of surface displacement 1s proposed.In the case o f cquatlontz), the failure time canbe exactly predicted by t h l s m et h o d . A l s o i nt h e case of equationtJ , i l e fallure tlme can bee x a c t l y p r e d i c t e d b y u s l n g more than two roughf a ~ l u r e ime known at different time.Il o g [ d e = c-m.log(E)-------------------- (1

BASIC F ORMJLA FOR VELOCITY OF SURFACEw h e r e c=2.33+0.59 a n d m=0.916 are constants. DIS PLn CEMENT JUST BEFORE F A I L U R EA ~ ~ u m i n ghat m e q u a l s a n d t; is t h e timeto failure a t each t i m e ( t ) in t h e stag e of Thc variations of surface displacement intertlary creep, the equatlon w s appllcd to slope failure caused by rainfall w s studied inprcdlct t h e f a l l u s e time of a slopa S a i t o , many k ~ n d s f e x p e r i m e n t a l models arp shown ~n1 9 6 5 ) . T h a t lo F i g . 1 b y Fukuzono and Terashlma(l982) a nd

log(L-k) = c-log( )--------------------I 2 40 l a t 40 m vex 30 flat 3o0con-where tl is t h e failure time and s strain rate.Based on these formulas Ill and 2 1 , Saitoproposed t w o w a y s w h i c h are r o u g h predlctasnmethods, by the strain rate I n the secondarycreep range, for long term and exact predlctlonby calculation and graphical a n a l y s l s using t h esurface dlsplacement or s t r a i n lust before t h ef a i l u r e i n t h e t e r t i a r y creep range. B u t I n t h et e r t i a r y c r e e p range, some slope failures whichare not represented by t h e equatio n(2) have beenreported by Yarnaguchl(l978). n these cases,the proport ional number o f t h r logarlthm of thetime to fallure and t h e logarlthm of the s t r a i nrate IS n o t un i t y as shown below:

where a nd B a r e cons tan ts .B a s e d on thc cguation(31, Yamaquchi(l978)has proposed that the f a i l u r e time can be exactlypredicted by multiplying the t i m e obtained bySalto s m e t h o d by the f a c t o r of correctlcn, t h a t

F i g . Profiles o f four typs ofexperimental models.

Fukuzono(l984). It has b e e n made clear that thel o g a r i t h m of accelelatlon o f s u r f a c e displacementincreases in proport ion to the loqarithm ofvel o ~y of s u r f a c e displacement as shownin Fig.2 a n d 3 . The relatlon l a

where x is downward surface dlsplacement: d /dt'is acceleration: d x / d t is v e l o c ~ t y ; and 6 areconst ant s .

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40 to 2 min.before fa i lu re3d FLAT.LCIAM( 4 0 to 2 min.before failure )a = 2 . 0

*..: /

Pig 2 The relatlmship bet- the velocityand the acceleration of surface displacementin th f m a l stage of slope f a l P mo the 40' and 30 lm slapes.

40 Iun,mmxldi 40 SAND pLAT25min. to 30sec 2lhnin. to 30sec.1 0 k o m fail? before f u u r e )- .

Fig.3 Ihe relationship etm the ve l a c i t yand the acceleration of surface d~spla tju s t before the fallure of the sand sl~pe ith40' mnvex nd 40 flat bases.

Equation ( 4 ) i s i n t e g r a t e d for t h e range ofa)O as follows:4a < l - - - e = 1- + (t+t &-----------d t 15

refore t h e relationship between t h e v e l o c io f surface displacement and the t ime t o falluj u s t before t h e f a i l u r e is generally represenby the equatlon ( 7 ) or ( 8 ) .A METHOD W R PREDICTTNG THE FAILURE TIMEEquatlon ( 7 ) is transformed a s follows:

where t t, and t are constants of i n t e g r a l .d was i n a range 1.5 to 2.2 il theexperiments. Also the values calculated from t h erelationship between the v e l o c l t y of surfacedisplacement and the time to failure i n a c t u a ls lape failure which were proposed by Yoshlda andYachl ( 1 9 8 4 ) a r e 1 . 7 0 to 2.13. Furthermore, theequation ( 7 1 is transformed as follows:

I dxlog tp-t = A-B .log -1 ---------------- 8)dtwhere 6=-log a (d-11 BLd-1

The form of the equation I ) becomes similarto the equation 3 1 , if is substituted f o r dx / d t .

where l/v is l/(dx/dt), that is inverse numbeof v e l o c i t y of surface displacement.The equatlon s hows t h t t h e c ur ve d r awneach p o i n t of t a n d l / v s l i n e a r i f 64=2,conv~f d ) 2 and concave if l d 2 and that t h valuun ~f or ml y ecreases. T y p i c a l figures of t h ecurve are illustrated n Pig.4. In any case,the curve ainF near to tp As t h e velocity isflnite, The relat ionship is n o t r e p r e s e n t e d bthe equation ( 9 ) at t h e near perlod of t,. Buthe period 1 s very short, so In t h ~ s aper ~tt r is assumed t o be t h e f a i l u r e time. me

. FailureLnL1w> ..

IH t -time: t

Fig. 4 Typical figures of the changesof the Inverse n of velocity of surfaced i s p l a m n t just before the falure.

n the case of d = 2 the failure time canexactly predicted by a point t at which t h estraight l ~ n ef inverse number of velocltycrosses t he a x i s of a b s c i s s a . Also using t hei n v e r s e numbers o f v e l o c i t y at two differenttimes, the failure tlme c a n be calculated asfollows;

I n t h e case of d = 2 , t h e failure t i mp r e d i c t e d by these methods equals t h e timep r e d i c t e d by Saito s method In the range of erof caLculation and drawlng, because t h e b a s i cformula h a s the same form.In the case of ds.2. the f a i l u r e time canroughly p r e d i c t e d by t h e po i n t at which thet angent line o f the curve of inverse number ovelocity at any time crosses the axis ofabscissa. Also the failure time can be e x a c tpredicted by the next method.

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he e q u a t i o n ( 7 ) 1 s differentiated by t andr ea r ranged as follow;

T h i s equatlon shows t h a t the c u r v e drawn fareach v a l u e of (l/vl/(dllJv)/dt) and t islinear and uniformly decreases. o the failure

t rm e can be exactly predicted by the time atwhlch the c u r v e is expected to cros s t h e axis o fabscissa. Also using t h e i nv e r s e number ofvelocity and t he increment of oncs at twod i f f e r e n t tlmes t h e failure time can bec a l c u l a t e d by t h e next e q u a t io n ;

F u r t h e r m o r e , a grayhlcal method bascd on thecurve of i nvers e numbex of veloclty is shown I nFig.5. F i r s t , a tangent llne is drawn at a nypoint Q which is In a vertlcal d ~ r e c t i o n o thea x i s nf a b s c i s s a from point TI,on t h e curve. Tciis t h e polnt t which the tangent l l n e crossesthe a x i s of abscissa. is plotted at the pointw h i ch is apart from the abscissa In a v e r t i c a ld i r e c t i o n for t h e same distance o f m< n thes a m e way , P is plotted. The abscissa t, of thepoint Tr a t which a s t r a i g h t l l n e connecting P,and P crasses the a x l s of abscissa is t h e f a ~ l u r et i m e .

C u r v e of inverse number

Time: tF i g . 5 Graphical method f o rpredicting t h e time of slope failurei n the case of d f 2

The theory is v e r i f i e d as f o l l o w s ;(ma)mJ qua ls (m -J in Fig .-m m nd T, 0 TI are substituted for TI Tb ndm T h a t is- - - -P I T , T , O P J T x T ~ O -----T O = - (13PI TI - Pa Tr---Also , ~ i ; ~ , = ~ , / t a n ~ , ,AT p T t ~ T ~ Q , q / t a n 8 z-(I41As t h e lines of Q, tIand Q,Tczare t h e tangent Ilneson the curve of inverse number o f veloclty t t h epoint Q,and Q, (times are t I a nd t~):

('/v) I t a n ~ ~ ~ z - - - - l ~ )a n B = - r dA l so , m~thm=4m=tl,V P ~ / V ) , ,z = l / ~ ) I---- I 6 ) The equations ( 1 4 1 , ( 1 5 ) and 1 6 ) aresubstituted f o r the equation 1 3 and after thetransformation, equa t i on / 12 i s obtained.

Another graphica l method 1s shown in Fig.6In this method, polnts PI and P, are p l o t t e d n avertical direction n o t at p o l n t s T , and T2but a tpoints TC, nd T

Ti m e : tF i g . 6 Another graph i ca l methodfor predlctlnq the time o f slopefailure In t h e case of 642 .

APPLfCATIOH FY R PAST SLOPE FAILUREThese proposed methods were applied to somk i n d s of slope failure executed exper imenta l l yI n t h e p a s t to estzmate t h e availab~lity. An

example w h l c h was a p p l i e d to the failure of the4 0 loam slope w l t h a flat bottom boundaryparallel to t h e sur fa c e is shown in Flg.7. T h ecurve of inverse number o f velocity of surfacedisplacement 1s convex in t h e period f rom 60r n l n u t e s to 10 minutes before failure and l x n e a rfrom 1 0 mlnutes t o 2 0 seconds before Eallure.u s i ng the tangent llne of t h e curve the failuretime can be predicted to w i t h i n an error of 1m i n u t e in t h e period from 10 minutes to 20seconds be fo r e failure. A l s o a curve obtained

O . Ry a methodT f o r t h e case4 of d f

6 0 5 0 40 3 0 2 0 oT l m e to failure minu tes ) FailureF i g 7 a r ~ a t i o n f the Inverse number ofv e l x l t y and surface displacement In th failureof the 40 loam slope with a flat bttm h d a r yparallel to the surface.

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a graphical m e t h o d in t h e case of d Z is s h o w n i nthc same f i q u r e As O: 1 s not constant a l l t hetime, the c u r v e i s n o t l ~ n c a r . B u t the failuretime predlcted by t h c curv e is more e x a c t t h a nthe time p r e d l c t e d by t h e crossing p o i n t of t h etangent line of t h e curve of inverse number o fveloc3 ty and the axis o f abscissa and t h e alarmfor evacuation c a n be g i v e n more safely.Another example which was a p p l ~ e d o thef a l l u r e of the 30 l o m s l o p e with a flat bottomboundary parallel to t h e surface is s h o w n inFiq.8. In this c a s e t h e curve 1s linear a t 30mlnutes b e f o r e failure. So the faxlure time canbe exactly p r e d l c t e d f rom approximately 3m i n u t e s befo re allure by u s l n g o n l y the curveof inverse number o f v e l o c i t y

2 0 T 4 0 30 FLAT. LOAHInverse number of

2 1 1 150 40 30 2 lo Fa,,. ,T ime te f a i lu r e [ m i n u t e s 1

fig. 8 Variatlm o f the inverse n m k r ofvelocity and surface dsplacement In the failureof the 30 Iwm slope with, f l a t h t t w n k m d a r yparallel t.o the surface

O t h e r examples which were applied to t h ef a i i u r e of the sandy sol1 slope a r e shown i nFig.9-11. Usuallyn the duration from the beginningof sur face m o v e m e n t to the final failure in sandysoil slope i s s h a r t c r than t h e one i n loam slopeand t h e p a t t e r n is vcry complex N e v e r t h l e s s t h ef a l l u r e t i m e c a n be suff icient ly p r e d l c t e d byt h e curve of inverse number o f v e l o c ~ t y ,

h Inverse n m k r ofY + 2 0 2

4 0 30 20 o pailbreTime to failure (m i n u t e sR g . Variations of the i n v e r s e nu*r ofvelcxrlty and surface aspla-nt In t hef a l l u r e of rhe 40 sandy sol1 slqw with aflat h t t m boundary ~ r a l l e l o t th surface

1/':Inverse number ofv e l o c i t y

Tlme to failure m i n u t e s a i L u f cFig. lO Variatlms of t h ~nver* cudxrof velrxlty and surface d_lsplacemnt in thefailurc of the 40 sandy sol1 slope w i t ha convex htt n boundary.

I n v e r w n u m b e r ,w \1 l D pYE: ;o%s

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Mater1al:KAATO LOAM

Concrete foundatlo

30 LOAM FLAT AMateria1:KANTO LOAMWidth:400, Uni+.:cm---- F i n a l s l l d i n g ,qsurf ace*waswing p i n &

o f surf ce

30'G C

Pig.12 Fmfiles of s p mxlels by loam of 40and 30 in surface gradient w i t h a flat bottmboundary parallel kc the surface

Ptrrto.1 - ::e:- 3f 5P.e ? a lope v:c lk f y r e t h e kyinnlna of e x p e r k n t

Artificial r a l n f a l l was s u p p l i e dcontinuously by a large scale rainfall s ~ m u l a t o rln National Research C e n t e r f o r DisasterPrevention at an intensity of 2 D mm/h for theb o t h slope m o d e l s .Surface displacement were m e a s u r e d by m e a n sof extensometers having t h e accuracy of 0.1 mmat rneasurlng points I n each model as are s h o w nIn F i q . 1 2

T a b l e I n l t l a l condrtions and m a j o rresults of experiments4 o 0 s l W 30 slope

O r y density Ig/cm3) 0.49 0.49f n l t i a l w a t e r con ten t ) 100 1 99.8Rainfall ~ntensity [mn/h) 2 20Failure t m mm.) 280.6 500.7

4 0 SLOPE MODELThe 40 slope model undeswent small scalcollapse 4 h o u r s 40 mlnutes 7 se onds a f t e rb e g i n n i n g of r a i n f a l l . A final slldlng surfa1 s drawn by a broken llne in Pig. 12.Three curves of the i n v e r s e n u m b e r ofveloclty of surface displacement obtained undexperiment at t h e measuring point 2 3 and 4s h o w n I n Flg. 13. T h e inverse numbers. ofveloclty a t e a c h t i m e w e r e c a l c u l a t e d by adlgltal c o m p u t e r from the d i g ~ t a l alue lntowhlch a A D c o n v e r t e r chang ed t h e o u t p u t s ofextensometers and were p l o t t e d continuously bya X-Y plotter.

3:OO 13:30 , I L4.00 , I+. Time ~ a l - u rpig.13 Variations of the inverse n m h r of~ l c e i t y nder falPure of t k 40' loam slupe.

collapse 8 h o u r s 2 0 r n i n u t ~ s 1 seconds a f t ~ rbeqinnlng o f rainfall. A f i n a l sliding surfacis drawn by a broken Line I n Fig. 12.T h r e e curvnR oE inverse number of velocitof s u r f a c e d i ~p l a c e r n en tnbtained under exper imat t h e m e a s u r ~ n g oints 2 3 a n d 4 are shownIn Fig. 14. Ths va lues of e a c h t i m e w e r eobtained t h e same way as in t h e case of 4 0 slmodel. I n this case , the curves were linear E30 minutes before failure. I n splte of t h e fathat t h e gradients of t h e three curves weredifferent each othcr, a l l curves came toward asame p o i n t on the a x j s oP ab s c j s s a that i s t hfailure t i m e . The variation of t h e lnversenumber of velocity at measuring polnt 3 was

l l c u r v e s were convex, so t h e fai lure ticould not be predicted exactly, b u t roughly.I f one of the methods proposed in t h e case ofwas a p p l i c d it mlght be posslble to predict tfailure time e x a c t l y .30- SLOPE WDRL

The 3 0 slope model u n d e r w e n t midd l e scal

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CONCLUSION

fig.14 Varlations of the inverse n-r ofvelocity under farlure of the 30 loam slope.

monitored w i t h t h e v i e w o f f a i l u r e on atelevision. These are shown in Photograph and3 . Using t h e s e me t h o d s t h e exac t failure t imec ou l d be predicted continuously fromapproximately 30 minutes before fallure and t h eexperiment was closed successfully,

M . 2 V s w of X Y p lo t t e r plotL r r>vr~r.-r-.n W r of velaclty under e xpe r mnt .

f n t h i s paper new method for predlctithe failure t l m e o f a slope has been proposed.The major d l s t i n c t l v e f eature of t h e rnpthod i sthat the i n v e r s e number of velocity and no t thdisplacement or strain rate in sur face 1s usedpredict the failure trme. The f a i l ur e t i m e c abe easlly predicted by using a curve drawncontinuously by t h e lnverse numbers of velocitof s u r f c e displacement at each tame. Also i ncases that c a n n o t be exactly pre d~c ted y Saitmethod t h e rough fallure trme can be e a s i l ypredicted by us ing the tangent llne of the curand exactly predicted by s t r a ~ g h t i n e drawnby the rough predicte d time.It is hoped that the proposed method willapply in practical s l o p e f a i l u r e .

The experiments were performe8 by uslng alarge s c a l e rainfall s l r n u l a t o r i n NationalResearch Center f o r i s a s t e r PreventionH Aok i , a t e c h n l c l a n assisted In t h e operatioo f t h e simulator. T h l s assistance i s gratefulacknawledgod.

REFERENCESFukuzono, T and Terashima, H . (1982)

xperimental study of t h e process of f a i l u r eln cohesive soil slope caused by rainfall (iJapanese), the Report o f the National ResearCenter f o r Disaster Pceventlan, 2 9 pp. 1 0 3 -

Fukuzono T 1984 ) A method f o r predicting tfailure time of a sandy soil slope u s ~ n g h ei n v e r s e number o f velocity [in Japanese), Prof 23rd Meeting of Japan Landslide Society,pp. 80- 1.Saito. M. and U e z a w a H . ( 1 9 6 1 ) Failure of sodue to creep, Proc. of 5 t h I C. S. M F. E .

pp. 315-318S a i t o , M. I 9 6 5 Forcasting the time of

occurrence of a slope failure, Prec. of t hI . C . S. M. F. E . pp 5 3 7 5 4 1 .Yamaguchi , S 1 9 7 8 ) Some notices ofcountermeasure for landslide and s lope failu( ~ napanese), Landslide Prevention and SlopStability 2 Soqo Doboku Laboratory, pp. 14-Yoshida, T and Yachi, M ( 1 9 8 4 ) On the velociof landslide (in J a p a n e s e , Proc. of 23rdMeeting of Japan Landslide Society, pp 136139.

m . 3 Y l c w of t r l c v ~ ~ l o nhowing inversen m k r of w l m l t y untkr exprirrent.