J U L 1 9 Zøø ~ 1 ~ : ¿ i r ~ L 1 0 I 1r~: ,......1

1 L 1 0 I bId ::::".: 1 ..::;: I u 1 ::i.....C i D..O 1 r. t;.h..)

6C-(/I(-ò~f

SOILS AND FOUNDATIOJlS' VoL 18, No. 4, Dec, 197. Japanese Soety of. Soil Mechanics and Foundation Engn~~Dg

.

OBSERVATIONAL' PROCEDURE OF

SETTLEMENT. PREDICTIONIi

i.i,

AxIRA AsAOKA *

I ABSC1 .The use of anal~tical solution of consolidation equation for settlement prediction is not

always effective si,ce such conditions as an initial distribution of excess pore water pres-sure, drain lenKt,i final vertical strain of soils and the coeffcient of consolidation are130metimes quite ulceain in practical engieering problems. For this reason, an obser-

vational settlemen~ prediction is freshly presnted. First, the linea ordinary differential

equation derived i~ demonstrated to give a settlement-time relationship. After that, by

using the differeqce form of the equation (an autoregresive equation), observational

procedure of settlebient prediction is proposed. Two kinds of practical methods are pres-I ented. One is a ~raphical method. the advantage of which is its simplicity_ The' otheris the method bas~d on the Bayesian inference of the non stationary stochastic process,which can give a predictive probability'distribution of future settlement and then also pro~vide a preliminar~i theory for reliability-based design of settlement problems.

The propose m thodology is demonstrated to be also applicable for some other specialproblem 'ineludin settlement due to drainage from :sand piles. .

Key words: cohtlive soil, computer application. consolidation, graphical analysis, .!

ure~ent, settlement, statistical analysisIGC: . E2 . .INTRODUCTION

The one-imenslnal consolîdation theory is an important contribution of soil mechanicsto prctical founda¡ion engineering. Settlement' resulting from onHimensional compres-sion .and one-dimensional drainage has been widely recogni~ed to be well explained by thistheory. However,¡ for the theory to be als effective in the case of future settlement pre.diction, some other engineering judgement. has ben additionally indispensable.

It is known thaj a consolidation equation. a partial differntial equation of paabolic

type, produces a nique solution when a coeffcient of the equation and the initial and

boundary conditio s have been prior determined. This is common knowledge so that thepractical applicatiohs of this equ~tion have exhibited a tendency to utiliz only this char-acteristic of the so.ution. In fact. according to conven1Ïonalsettlement analysis, condi.tions such as an i~itial distribution of excess pore "rater presur, .' drain length, finaivericai strain of sOils. and the coeffcient of conslidation were naturlly ~nsider to begiven in advance d£ the analysis It is, however. als commonly accepted that the .eti~

mation of these cobditions usually goe with a high degIee of uncertainty_ Ther.efore. theiI.. . .: .

* Research Associat~ Depatment of -Civil Engineering, Kyoto' Univèrsity, Sakyo-ku. KyotWritten discussion, on this paper should be submitte bef-ore October I, 1979.

!

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JUL 19 Zøø~ i~:~i r~ Lioll lL.i::1

\

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e~gihe~.. working :~n' ~itlehi~~t' pi-edicti~~ is: ~~uaiïy . e~peted to have bad some previousexpeience. IIn this paper. a néw idetof settlement prediction is presented, the philosophy of which

is base on "Observational Proce\le." . The trend equation of time series data of settle-ment is derived nrst from the' ~n~dimenšio'nal consolidtltion. equation, afte which thefuture settlement is predic by using the put obserations. Two kinds of practical meth.ods ar propose. one~. i a graphical method, the advantage of which is its simplicity.The other is the method base on the Bayesian inference of a non stationary stochasticpr, whic can give a reol';"" proham,y distihutlon of f. uture settlement and thenalso provide a preliminary theory for reliabilty-base design of settlement problems.

The propose methodolo~ is demonstrated to be also applicable for some special prob-lems including settlement 1ue to drainage from ::and piles.

~~ EQATl~OF sn~-:iME~T10N ..i" this seon, ,he ori~ar.differ"ti equatin de.. giv.. a _1e,-tim re.iioÓship. .i . . .. is a fundamental equatipn. 'Mikåsa's equation is adopted. that is

. . :' .1 t=c~1Iin which I. e(t. z) : vertical strain (volumetric strin),

:t(~O) : time, . I.. . .z : depth from thî'toP of clay stratum and.. .:.

. Cfi ; coeffcient of nslidation. . . . .

~ in Eq. .(1) Ïhe upp& do. ~ 'l'epresents time-üIertiation' and the lower .scipt~ :l the

differentiation with respett~ . depth , z. Even if the coeffçients of b~tA. permiabiUty an~

volume compressibilty va from time to time, the Eq. (1) is still efecive when Cfi re-mains coiitant (Mikåsa, ;1. 3). ''In,addition; Eq. (1). makes. it ~àsy to express; settlement.For' these reasons Eq. (1) ~s adoptëd'instead of Terzghi's equåtion.. '. :

The solution of Eq. (1) I is expressed here by introducing two" unknown functioll. ottime, T and F, as follows:

et %)=T+.l(Lt)+..(Li')+..', 2! 'C" . 4! cv2

.. .. ..' .. ':+.zF +W(l . %8 i)+ ..(L ¡)+... . '.. 31 C 5' c.t.I.;' ." -""

4:1.. ::::ù lùO.. IV l..tJ~..Ulu..u" "'~"'" '..r.:..~. ... "

::~. .. "

ASAO' ~. /

~a

;iI

i ,: (

(1)

""

f

~

j

"f

.( 2) .'" .~.q. (2) shows tiiat

. '":; :: T;:e(e, :=0)..' .. . ~ .."'! ".i(

and', i. F=eK(t, 2;:0) '.' .'" . '..' . '. . '. : . i . . .., .. '. ' . .....

'.~: T~e : ve:tica.l . sti~in ~ ~iI ~q.. (l)~' is'..'originàii~'' . ~efin~ i~. a~ ~ul~i~"~ . .s.én~e,,'. ."&ti~( in. ....this

paper, e IS approximately regarded as a. Lagrangean stam" for simplièity. - .The' formulaof settlen.ent p~edicti~~~ b~.. ~~~~ading . ~'.á. an: E.?:~eriån stráin has. ~itea~.y '.1i~ . ~stabii~hed

'by the ~uthor. - îhe results, . hówev~.. bèoine a complex nonlinEiar probleJr and: therefore~~ey ~~e 'b:yond':~h~.s:b~~.i~f '.p~~~~c:r'~PP'!~c~~ii.itY:..'. '. :,' . :~..' .: " '. '. :....: .

The typical two bounde conditions ~re considere next~' ". .' '.' .. ..... . .. .' .

~.~.(ly nrainagcrtÍ'onï b~il top':åíö b()ïtó~D:bóunaari~ .. . ~(Ti.~:::~CÄ: ~:~~~)Fig. t::~o:w~. thisbou~8'ry ..~nditio~~ ~~~c~.i:~.,.£~i:in~lat:ed...,:s :.:. ..: .'.' '.~ ...!"::'-~':' .'; .'

.,. ....c .' I ". . - _. e((, ..iQ) =ö,.; !"..t .. ..... .' .. . ." ....~" ...,;, ,.! 3 ).

í

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JUL 1 ~ ¿ldló~ i ~ : c i r to L 101 IlL i:: Ic r\ ......~, '

o i ù ~O~ 1 ù~~ i V 1 ~UwwU . UWV A

11ii!

. . r ,

PREDICTION lJF. SE,LEMENT 89

'. top of clay

. " .". 11 --t--

J bot,om of clay:::: -: .:.... ~....';!:.:.... :~:.:.:.:fre¡ drainage

til'. .1. Drai~age from top åndbottom liundaries

.nee drainaq~:.,...,f.,-, -..:: ::.:'......:-:... .....

,1 :". top of clay

'H. . t .

J bottom of clay¡¡impermable layer

Fig. 2. Upwar drainage .

. e(t, z~'H) ~~ : constant (4)~here' H. represen . the thickness of clay: stratum and both i 'and~. the unknown bound-ar values respecti ely.

. - Šubsthutión of . (3) into Eq. (2) gives. . I '. ..', I' T=E: con~tant (5)

which shows that lPpe half of the solution (2) becomes constant. Then. from. Eq. (4),we have the lineflJ ordinary differential equation of ~nknown ~ with constant coeffcients,

F+..(HZ p)+.-(H: F)+....= ~-£ (ß)3! Cu 51 Cv HI (2) Upward d.¡ainage (Fig. 2) .. (OVle. ~ D,(4\~~'E). Whe~ the bottom a£ the clay layer is. impermiaõle. boundary conditions are given asfollows:

"

(7)(8 )

c

~(t, .,i=.0) =! : constant

£6(t. %=H) =0Since Eq. (7) is t e same as Eq. (3), it also follows that

T=i :-constatThen" calculating q. (8) from the solution (2),' we' get

.' '1 (¡P'.'\." 1 (H' )' ,. '.." ., F+- --F '+'- -1 +...=0' .', (10). .._.. ~.,' ~ '.' I' ::. . 2!. ...u ' .1.. 4!. . cvs ..... "'.. ..' .' .which is also a linear ordinary differential equation of F with constant coffciënt......Now;.:foI' any'Wundary "condition, the 'settlement 'of clay stratum can be" expresse asIcl1 .\.. ". I - pet) :;.)0' t(t. t)d% (11)in whi p(.) r+nts the settlement at tUne;.. Subsitutig the sOlution (2) into Eq.

(l~)l ',since T=f: ¡onstant, we have .

:::. .' .-: .;' ....IP('t):~fH+..(H!F)+irH'F)+~(H8 F)+.... ,-, (12)_ r. . . _ . . ...21 . 4fc ,. .' 61 . c Z ..' -'. .Ð-; ~h~' se~u~~tial Jiff~rentiatjon of 'Eq. '.(12) .'wit~ respect .to t: we get. the' fon~~ing.set ofeq~~~i~ns: I '. \. . : .., '.

. - I 1 . . -MBA ,,) 1 (R' ...) 1p=_(HiF)+ -F +- -F +...1 21 41 Cu 61 c.,iI . . . .'1(:' 1. .eft). i'C'R:' (ll~l))' :"l(' -R: ""+iQ)' '. ".1.' ; , ". "'. ,JI3)

p=~(HiF)+~ - F.' +" -: "F +.:.... . .'.:." .,21 41 Cu "_.': : ,..' 6;1' cui, .:;;. ..: ':~'.'; '.: ::. ~. .. ... . . . ... .". . ... : -.:. "j . , . . '; .¡

i

I

(9 )

~.. _ a.J ';....:.:..".- -. ...

( . ~- ~

b 1 ~ ~~~ 1 ÙO~ tv ¡ ~u~~u. u~u &

:..ï

JUL 1S 2ØØ5 15:22 F~ Cl~ll lClol -,.~. ';;~'~-':1.. ~

\i

I

I

Cómparing the set or ¡equations. Eq. ,(12) and Eqs. (13). with Eq.-(6) or Eq. .(10), the

unknown functionF fn be.eliminated. When Eq. ,6) is the case,' it follows that. i ~JHz \ liH" ..) ! H '\í.v-~e

P+3T~v P )+m c.ll P +...=~e+V í l)ó 4 ~14 aJ

Similarly. from Eq. (0), we get

P+ IjH2 ¡\+ 1J.!¡¡)+...=Iie2f c., ) 4T c.

Eqs. (14) give a settl ment-time relationship under the condition in which an external Ç7consolidation 'load is nstant. that is, Eqs. (14) are the reults derived from the station-

ary boundary conditio. s, Eq. -(3) and Eq. (7).It is easily ascertaiii that Eqs. (14) are absl1ltely stable for t_oo, that is. all the

eigenvalues of thes ebuations are negative rel number which ar dilIerent. from eachother. In succeeding Isections. however. we discuss the practical problems in wh~ch Cu. H

and boundary conditio~s of drainage and load are entirely unceain. Therefore, the expres-

sions of the solutions p£ Eqs. (14) by the use of prominent eigenvalues and boundry valuespthis is common in c0Itventional analysis, ca not be efficient for settlement prediction.

It should. be noted tlat the higher orer differential terms of Eqs. (14) are negligiblys.mall. Then, the foiOWing n-th order approximation equation is adopted as a masteequation of settlemen time relationship:. .. (ft). P+CiP+CtP+..'+CnP=C (15)wqere elf Ci, ..., Cn aiil Care regarded

as unknown constants. It wil be showD later thatEq. (15) is also appli~bie for some other settlement prediction problems. Introducingdiscrete time as I

\ t3=.4t.j, j=O;l, 2, ... 1I .dt: constant, . f

Eq. (15) can be reducbd to a difference form,I - n. . I PJ=

Pø+'J (3I1P3-11 . ~17)i

in which p, denotes p(tj). the sett1ement at the. time t=tj and the eoefIicients 90 and 8s

(s=1, 2. .... n) are uiikown plant parameters. Eq. (17) wil give us an idea of observa-tional settement prediPtion. . . .

For the convenience! sake of succing discussions, the fit order approximation equa-. Iuon, I;" "".",¡ned he - It initia condtion 'i. +. . C..;l=.....C.,_ -rl-~' "D ~ ~ ~.-ï p(t~Ql:;p. ~..e (19)in which the time. i~o, should be taken at the time after loading works sinc Eqs. (14)

have been derived frof the station houndar codition whü: do not.. from timeto time. When this the ca. Eq. (18) can be easily solved as .

j P~t)~Pf-(Pf-~O)exr(- :J' . -(20)

i\ PI:=C,thè final settlement. spinetimes refer to as the stable "State of p. -On the other hand.the fist order differenfe equation ~ expressed 'as '.' .

! : PJ=ßo+ßiPJ-iI

I

I

i

90 ASAOJt

- aC. : .' . 'Df'l.,v.Co.JD'n ~ (14b) er~l-

¿..

(16)

-~

where

(21)

\'" ,,. Mjvr-u. -~. Sel~ yt-l~-iS \,,"eo'

J U L 1 9 2ØJò 5 15: 22 r ~ L 1 ~ I IlL 1 :: IF. \,

¡

The coeffcients. .~a and Pl. have the following values.i

I Pj=Pj-I=PI,into Eq. (21). wJ obtain

01.. ::;:.. 1 "'0.. i V ¡, ..U..-'U I ....'- ""

PREOJCTION OF SETTLEME1'91 I

r. Substituting the stable. iitate,(22)

L

r

80p¡= l-ßi .

Furthermore, by \the recursive operation with respect to j, Eq. (2~) yields

. I.. PJ~ l~Ïi.-ll~Ïi.. - ~.l (P.)J ...which can be coipate with Eq. (20). From this. it follows that .

t = - H6cr..4t : the case of both top and bottom drainage'.

.de - . .~Pi= - '"c, 2c f d.=:- H: 4t : the case 0 upwsr drainage (26)

which suggests t at Pi doe not depend on the boundary values, f and ~; 81 is independ-

ent of load.

-7 ~'Äcá ~\~. . "23)

(24).

(25)

rr GRAHICAL ~ETTEMENT PREDICTION

We first exami e the effectiveness of the first order approximation equation. Assumethat we have had 71+1 settlement òbservations. (Po. Pu .'., p".) generated by a constantexternal load. U~ing these observations we can plot 71 points, (Pii. P&-l) for k=l, 2, .... ".on the (PJ. PJ-I) co-ordinate. From this, we can visually see whether all these points lieon a straight linel as Eq. (21) suggests or not. Figs. 3 and Fig. 4 show the actual resultsobtained from th~ settlement observations of extensive reclaimed lands in Japan. The da.tafor Figs. 3 are from the reclaimed land at Kobe Port. Fig. 4 is the rearranged resultfrom the data published by Aboshi (1969). In every.Ctse of these figure, settlement wasconsidered to be \ resulting from' one-dimensional consölidation. Through these figures.1 I .'.

-+----- ¿r---------------------1968\169 '70 '71 '72 '73 l74 '7$ '76 . '77 '.7S6 +2 6 12 6 12 6 12 6 12 \5 12 6 12 .6 12 G 12 6 12 6 12

I

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a.~

11)

cie

Eo.. io

+It:Gl

~.-+I

t 20'. C/ .

30l"

Fig. 3.(a) Settlement observations of the recaimed laud at Kobe Port

.-;t ~~

,J U L. .I.. ~ ~ LJ -' .l... Co '- . ., '-.... I.. .. '- .. .. I u.. W ....W .. W.... I V .. ~U..~U I U..V'"

.,:. 30~i

í

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gu

..

Kobe Port ~0.1 /¿J-?

.."I J7I .,,-;"'..' if.v

.';'. , i

. . I. / i,

, . / .\..

, .l' . ,. /. "i,," I

100I

200 30ÇI

300

..

CC

~

et

./lCobe Port: ''No. 2 .~/

..;-.~-; .

.."/. ../.'/

..~..y.' /: /

..- /. /. /1//

200'" 200ij

100 100

oo o 100 200Pk-i, emPkrl'..

300

i

. .!

Kobe port: No.3 //. Ii

~.", :l

.:~. ...."../

...../.. /'."/

V. . '.11~

..../. / ., :...../ ... /.. /'

1,,/

. '

200

.1. . eu

......

~

. Q.~

I . ioo

I.. '.

o ioa 200 300. .. Pk-1, em ,.'....... - ..... .. Fig. 3.(b) pj~pj-i relationship , .:

a time iiitèrval~ '.dt wäl taken to be 3 .nl~iiths (9i-92 days).' '.. . .i.' ." .As shown .in the figures, -the fit order difference. equation, Eq. (21),., gives a good nt-

ting to the observations, which enables us to predict future settlement by graphical meth-

od. This procur iJ ilustrated in Fig. 5, . Since the rough estimates of. ß. and Piare Biven by the inter~Pt and the slope of a ntted straight line,. it is possibl~ to predictsettlement of anY f,i.tur~ time, j, by using Eq. (24), Moreover, the intercepting:.point of

this line with the 45°-1 ne represents the final settlement, because Pr is Biven by Pj'~PJ-i.

If necessary, the unkn wn coefcient of time factor.. co'/H2, ciin be easily calc"Úlated. back.ward by Eq. ~25) or (~). .. . :.

In:the case of multi-staged loading, the straight line, Pj=ßo+(JIPJ-i, wil be moved upas is¡ ;iw-slIo~~'.u; Fig\ S., wh~iid~hë~ ~t~iement is r~iàiivelY .smail ~~~pare.d to the:' thick-nes :of clay layer, the ~hifted hne becomes almost parallel to the initial line. because .piis d~texmined not from !the exteral load but from the thickness. H and the cóeffcient of

cons~lidatio:a, Cir ! .,.; :Tlii§ graphical mèthir-~qûìtë'useful in' šittiatÎöñs . in .. w liich "siiñpliëíty-îs-mó~¡':vah1abie

than strictness i . :but .fQI~wings ai. noteworthy 0 .. ~

Ii

\

i

i

I . Ul,''''~.. \'-y

300

JUL 19 2Ø:05 15: 22 r~ C 1 ~ I IlL 1 0 I~~ .f'. Iold ~~d 1 ..::~ i U 1 ::~~~C i C..O 1 . . IU..

I

I

I 140 160..Rk-1' emFr~ . 4. Settlement observations at Shin~Ube thermoelectrie

power statioii(after Aboshi (1969))

I

( 1 ) To obtain Po, we are generally 'compelled to extrapolate the fitted line to the (J1i-i=0 a.xis. So, if t~e banking period is so lopg anØ then Po becomes so large, the extra-

polation should be ,performed very caefully_ . . ¡.(2) The accura~y of the graphical method depends .mainly on a time inteal,. ¿e..

The longer At beccyes, the higher the acc~racy is.. For ~his purpose, the reanangementof obsevations is s9metiines recommended. . If.. as' an example, '\e observe settlement

ever week and ,de \ is taken to be 10 weeks, then, letting p(kJ denote the setlement' 'ofthe'.k-th week, we have the following iO..time-series of observations, that is;

I

Ii1

I,

(

5

.. llOO~ iQl ,m ....i+I

~r~oo

Ii

\

i

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Ii

I

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.JlI

S

1.0i

.1. i

150

PiiDICTION"OF .SBTTL£J4ENT

~~

-¡

93':'.

,;

/ -!- 1.6 i:~ Durat.~on,/ 8.5 t: ~year

2i S

.. .'

. ,;..

. !

. .. . .. i:. . . .. . .. . -

:wo

18Q

i:o160~~ /////////120/ ieo 200

.140

!l~II¡,.1

!:

. !

.1

¡1

i

II

..(

:1: I

~

::11

";."

:,:'1'i-

111

94

Lloll lLlol

\

\

\p~ . .\ /\ ~/\i "~7~";l finalplacement of' / settlementaddi tional fi1i . //r""g ...! .........'7' .\ ...... // final settlement

IJ" / resulting from" t1 / ~ initial fillY ,.' /rI /I /\ ///.i /1/

b 1.. ::::.. 1 ..::~ i u 1 O~....O i o~o 1I ... u

JUL 19 2ØØ~ l~:¿~ r~ ."" .,.¿,..al'. . '~~:-'r

ASAOU

/

IlIIill

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li

\

1

i

l1

45° (Pk-1:

1 S. Grphic.. of ..ia-ent pri

I

i

. I (10) p(9), p(19J. p(29)

,

Since Po and Pi are indepedent of an absolu:të'time, all the time-series data wil be plot.ted on the same straight line.

( 1 ) p(O), p(lO) , p(20) ,

(2) pll), p(ll) , p(21) ,

:.

REIABILITY ANALYSIS

The solution of Eq. (18), the fist order approximation equation, is predicted here byusing Bayesian techniquet under the conditions in which Ci and C are unknown at thetime t=O. One definite ~dvantage that the probabilistic approach has over the graphicalmetho.d ~s th.at the predi9t~ valu~ is g~ven with its reliabilty. .

As is imphed by the pievious discussions, the :frst order antoregressive model,Pi=Bo+PipJ-i+07J ('Z)

is chosen here as being a\ suitable one for settlement prediction, in which 1Ij is the randomvariable with zero mean fnd unity deviation. Iii .order to make succive calculations

simple. the followings m 8t be assumed: one is that 1Ij is a Gaussian random variable andthe other is I r 1: j=k .: E(llj71~):: 1 0 ; j~k (28)where E( J denotes an ~xpectation opetion. It may be unecessary to state that Eq.(28) does not suggest tpe statistical independency between Pj and Pt. Thr . unknownparameters, Po, 8i and (llare sometimes wrtte as fJ for simplicity, that is,

fJ=l (110' Pi' 0)1 : unkown parameter vector.

Now suppose that the sbt of observations,

\ . pti= (Pa, PJ, .... pJ -(29)Ii

I

¡

rUL 19 2005 15:23 FR Cloll lCl~1bId ~::d 1 ..~~ I U 1 ::t:~..D i O~C 1

(

\i

I. ihas ben already obta~ned. We, then. ha"\e the posterior Probabilty Distribution Function(pdf) of unknown fJ .$ follows:

\i

e (6) :\ a-ptiori pdf of 0. p~n.110, P.1_I):\Gaussian pdf with mean. (Bo+ßiPJ-I)' and variance at.

When we have no information at the time, t:=O, e(fJ) can be given by a diHuse prior pdf,. I .. . I ~((J)o: ;

If we have some in£~~mation at that time, t(fJ) can be expresse by the more informa-tive one than Eq. (3ip. When we take eel)) as a Gauss-gamma joint pdf, the poterior

pdf, E(O\p'). also beqomes a Gauss-gamma joint pd.Next derived is the\ pdf of p~ conditioiied by (J and PO. This pdf is calculated from. I

p(p,IO, flo) =f...f ii P(p:.IO, P.-I). dp¡dPi...dpJ-I (32)\ 1.1(j .By using the niatheitical inductive method, the p(p.110, Po) is demonstrated to be a Gaus.

sian pdf, the mean~-MJ' and the variance, Zl, of which are given by

\ Mj==.J-(~- Po"cßi)J 1\ 1-81 1-81 'j

and . f i =d 1-Blu~ J l-"Pi-respectively. Altho h both M;and XJ are monotone increaing functions of j, sinceinBi":O, they are als bounded from above:

i- M Po 1I un j~ 1-8J-- iI lim XJ=dJJ 1-1312I J....

Now, what we should. examine here is the meanig of d. If we take a time series obser.vation pl, at only onelspOt on a clay stratum. XJ represets the randoni £1ucuation of 'PJfrom the trend MJ. IHowever, we can generally observe settlement at various spotS locat-ed within a certain ~ea. Let lpl be a time series observation of settlement at spot "l",and let i

I pi=(ip', Ipl, .... .pl) ~35)in which m is the number of obseation spots.. In this case, if a settlement proce can- . - ibe assumed to be spatially stationar, the d estimated by the posterior pdf.

. I ~(lJlpi)o:E\O) ñ Ìip(ipJiJ, &PJ-1)I i-u-i . .ca be considered to ìfclude the property of unequal settlement charcterized by the spa.tial interval between observation spots, 1:;1, 2, .", m. When this is the ease, introducingthe concept of an ini~al distribution of Po, p(Po), we obtain

p(rJ10) = IJ pl..Po) 11/(P.10; PIl-i) .dpodpi...d(J,-1

I 0)which is different fro~Eq. .(32). An initil distribution of p, p(po) wil represent the

distribution of uneq~ isettlement just after loading. Provided that p(Po) can be expresed

\,

PREICTION OF SETLEME9S

iE (Olp') ciH(1) n p(pJIl1, P,-I)'-1

(30)

where

(31)

(33)

(34)

(\. .

-(36)

.(37)

i

l

.\

I.1

iI:I'~ i

: i:

. .

::

I.,.-

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as a Gaussiaii pdf, p(pJl6) becoinés 'also a Gaussian pdf just as op(p,,\8,. Põ)~o :." '...,: . . '.:Now back to our prediction problem. We have already ben able to derive. the predi~

tive ~f of PJ. condition¥ only by the set of observations, pi, as follOWS;. \ . 1.0i

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Dember,"19'2 at Kobe Port, :No.3 0 .

~6.~..-In 5.i:g~ 4........ 3..rl.0.2 2.o.,.ii l.

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Fi. '1. Pösteri~ marg¡al of t1 bý. . _. the same :data. .of ~ig.. 6

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Fig.: 8. . .Predictiye pd of thë settlement ~f Decem-':. ...... :ber;.1916 from .the' data;

of Dembet. i97:f'. 0 '.¡

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I dfJ.=:dßodßiM.Reliabilty of settlemtnt prediction wil be defned from Eq. (38). For example, the probeabilty of taking PJ Between Po and p" at time j can be given by. I f'.

. I Prob. (Pci~PJ~p"i(Jl) = J,. p(pJlpl)dPJ -(39)The calculation of Et (38) or Eq. (39) can not be performed without an electronic com-

puter because of the ¡multiple inegrations in these equations. .

Figs. 6, 1 and 8 arl the numerical results of the applications of Eq. '(30) and Eq. (38)to the observations of Kobe Port No.3, which are ilustrated in Figs. 3. To obtain theresults, both e. difIusb prior pdf and the data observed until Decmber, 1972 were em-ployed. The predictipn was conducted towards the settlement of December,

1976, the realîzed

value of which was 281 cm. Fig. 8 is the predictive pdf of the settlement. Figs. 6 and

7 are the marginal p~f's of Po. Pi and (J of Eq. (30), respectively. For these calculations

At was taken to be SO-31 days.

I1

\ SOME SPECIAL PROBLEMSIn this section we aerive a general prediction formula of future settlement through in.

vestigations of three Fpecial problems.

Secondary Compressibn due to Creepi

If in the simplest ~se, the secondary compression of clay stratum can be approximately

described by a single iVoigt model subject to a constant external load. Then in this case

the secondary settiemfnt, fJn, is considere to satisfyfJii+CPn;:p. (40).the 6rst order linear ~ifferential equation with constant coeffcients. Since the solution ofEq. (40) is given as ï .

\ PI=(Jjn-(PfJ-pon)ex~- ~)

in "..hich Pan is the iJitial condition of Eq. (40) and (J/n==P, then, the total settlement canbe expressed as £011015:

i P= (Pri+(Jfn) - (Pfr-POi)ex~ - ;J

\ -(Pfl-fJoix)exp( - ~)I .where the settlement ¡resulting froIl Terzaghi's consolidation theory is denoted by thelower subscript, J, ahd is exp¡;essed also by the solution of the "frst order appr-oximation.

i

'equation, Eq. (20). It may be easily noticed that p of Eq. (42) is the formal solution ofa stable 2 nd order lidear ordinary differential equation with constant coeffcients. Then,by introducing a dHfetence form, the next 2 nd order autor-egressive equation.I t

L fJJ=ßo+ ~ ßi()J-_is suggested as being uitable for settlement ;~diction including creep factor.

\i

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"/

PREDICTION OF SETTLEMENT97

p(PJlpt)= f p(PJlfJ. Po)E(O\pt)dfJ.. -(38)

where

(41)

(42)

~43)

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b i ~ ~~ù 1 ~O~ i V ~ ~u~wu l u~v & f;

98. ASAO"K

,

'Søttúmen of Clay Seraturi with Sand Seams -In an ordinary soil explobition. a thin sand seam "Sometimes goe unnoticed. However,

if the sand se allows frJe drainage, this overlooking causes a great error in settlement-

time prediction. . Fig. 9 sh~ws the simple cae of such a situation, in which the total settle-ment of both stratum I an~ n is gÏ\,en by

I pet) =Pi(t)+Pn(e) (44)where pi(t) and Pu(t) reprJsent settlements of stratum I and stratum n respectively. Sup-pose that the sand seam of! Fig. 9 allows free drainage. Furthermore, for simplicity, letPiCe) and PnCt) be expresSC by the form of Eq. (20). When this is the case, Eq. .(44)

suggests that the total :;ettlement pet) can also be predicted in the same way as is givenin the discussion of the seJondary compression due to creep, that is, by the 2 nd order

autoregressive equation. I

Sand Drains t,1 ; .. In order to reduce the ti e for consolidatioh, vertical sad drains are sometimes used.

, As is well known, if the ual verical strain hypothesis is approximately satisfied, the

¿e~Tee of consolidation cai be expresse by a single mode, that is,,., U=l-exp(.b) (45)wh~re an eigenvalue À is determined from both a horizontal consolidation coeffcient andgeometrical dimensions of knd piles. According to the tradition in soil engineering field,settlement can be treated Js being proportonal to the degree of consolidation. Eq. (45)shows, therefore, that the settlement by sand drains can be predicte by the first order

autoregressive equation.

General Prediction FormulaThe above discussions i1p1Y that a general"settlement prediction model is given by

pJ:=/3o+:E P&Pj_&. (46)¡-l

An actual settlement may ~ govered by many different factors. However, if those £ac~tors can be mathematically¡ formulate as an eigenvalue problem, prediction formula of Eq.

(46) is quite general. IIt should be noted that Eq. (46) is the same form as that of Eq. (17), although thei

physical meanings of the ioeffcieiits are quite differ.ent.

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." .'.-'.: :.".:: .., - -~ "'d.-..' -,'- '- ,-- \

Tol?Clay i I

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cla,'

:tol ~at'1-

clayey peateç;it

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F~. 10. Soil profile and embankment crossetion of Iwaiizawa tet fiU

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I PRECTION OF SELEMENTi

The'statistical patameter identification of Eq. ~46) for n~2 from past obsations is.howevei, hopeless * presnt, since the settlement proces is not statioD1y, that is E(PjJ

~ const. An analytical expression for a predictive pdf of PJ .by Bayesia statistic is for.mally possible in t~1e same way as is pr.esented in the case of n=l in the previous setion.

The results: howevb-, include super multiple integrations. Therefore, the author recom-

mends the least sqJue estimates of (30 and (i, (s=1, 2, ...) if the long te obsrvation is

available and the c~n5iderabiy large time interal, Llt=tj-tJ-i, can be adopted.

Practically speaking, the nrst order prediction model, Eq. ~21) or Eq. (Z7) should betaken first. When \there exists only one prominent eigenvalue, the :frst order appoxim-

h ¡~-g.! U76I'N

99

50

(, . 100

È 150Q

~..c: 200..~...o 250..ll"'

300

350

~ 1977e , 10 11 2 1 2 3 . $ 6 1 e

-.

'...........

...

'.. ....... .. . ..00 . .

Fig. 11. settilment observations at sand-drain test section of Xwamizw~ test fiB

\ 400I

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..1300~~I

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Placemeni: ofaddi tiona), fillfroi Apr. to June.1977 ~i

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200 250 350 400300Pk-1, em

i

Fig. 121 Application of proposed graphical method to obserationsOf\ lwamiuwa tes fi

I

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:1;

JUL 1:: ¿kl.J:: 1:::C:41 .i -

r-l' L1::1 1iII

1 L 1:: I o 1 ~ ~~~ i U~~ i V A ~U~~~ i ~~~ & . ~

¡ .

;(. .

.,,.1

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iiI

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100 A'AOKAtion method wil give sJtisfactory results. If the model doe not give a good approxima-tion to obser'l8tio~s. thd second order predicuon model, Eq. (46) with n::2, is to be triedne~t. I .A Practical E~åmpk 0) A CampI.", BouY Vale P,oI

The applicabilty of t~e proposed settlement prediction method to a complex multilayered.

clay deposit was examin¡led by using actual data provided from the test fill which was con-

strcted during 1975-19'8 at lwamizawa in Hokkido as a portion of the Do-o exprsswayby Japan Highway Publ~c Corporation. The soil profile and the embankment croSS sectionat the test secti~n is shbwn in Fig. 10. As shown in the figure, the clay deposit consistsof 7 subsoil layers inclu~ing clay, peat and silty clay stratum. Sand piles were driven tilthe elevation of -13 m ~ or the purpose of vertical drain through the piles. Measuredsettlements under cente~ of embankment are giiren in Fig. 11. The embankment was left.alone two times with nò additional fill, the first time is from the later part of November,1976 to the midst of Aø.rU, 1977 å,nd the o~her time began in the last of May, 1977. Thepropose graphical settl~ment prediction method was applied to the observations obtained.during these two period~. The results is ilustrated in Fîg. 12 which shows the remarkable'

"~pplicabilty of the first I order autoregressive prediction medeL. For the ilustration of Fig.

12, the time interval,4t, is taken to be 15 days.I

\ CONCLUSIONSThe observ~tional me~hod for settlement prediction is freshly' proposed in this paper.

For one-dimensional consolidation problems and sand-drain problems, the first order autO.regressive equation, Eq.I(2l) or Eq. (27) is demonstrated to be a suitable one for settlementprediction. For this pre~iction formula, both the graphical and reliabilstic approaches werpresented. \

The proposed method is considere to have a great advantage to predict settlement of

an arbitrary future tim, under situations in which a coeffcient of consolidation and bothinitial and boundary Conditions are quite uncertain. Such situations ar considered to beso common in engineeri+g practice that the propOsed method wil ha-ve a certain availabilty.As for the unknown coiditions. they can be guessed backward, if necssar, from settlementobservations by using the estimated coeffcients of the first order autoregressive equation. i

The higher order aut~regressive equation, Eq. (46), suggests to provide more precise!settlement prediction a~d to be applicable to some special problems, that is, one is con.

cerned with seondaryepmpression and the other is the settlement of multilayered stratumwith sand seams. !

I.i

I ACKNOWLEDGMENTTh: auth~r is extremrly greatful to Professor Yoshimi Nagao of the Kyoto University

for his continuous encouragement. The author also wishes to express his thanks to Pro.fessor Minoru Matsuo Óf the Nagoya University for his valuable comments on the "Obser-vational Procedure." . \ .

1ij R.EFERENCES

1) Aboshi, H. (1969): oil Mechanics, Chap. 4 (edited by Mogami, T.). Tokyo, -Gibo-do, pp.

46-465 (in Japanese).2) Japan Highway Public Corporation (1971: Technical Report of lwamizawa Test Fill -(in Japa-

nese).

IUL 1:: c.~~;: 1;:;.c:i ,.'" '-l::li 1'-101

I

01 Ù ~~ù 1 W~~ i V ~ WUW~U l UwU ¿. . ..:.¡~~

.

i

I

:3) Matsuo. M. and \Asaoka, A. (1978): "Dynamic desirn philosophy of soils based on the Bayesianreliabilty predicttion." Soils and Foundations, Vol.S, No.4. pp.1-11.

4) Mikasa, M. (i9é): Consolidation of Soft Clay, Tokyo, Kajima-shuppan-kai (in Japanese).

.5) Ten:aghi, K. add Fröhiich. O.K. (1936): 1'eorie der Setiung von Tonschichten, Leipzig.

6) Ten:aghi, K. (~48): Theoretical Soil Mechanics, New York, John Wiley and Sons.

'1) Terzaghi, K. a~d Peck, R. B. (1967): Soil Mechanics in Engineering Practice. New York, John

Wiley and Sons.'I (Received May 22 1978)I

I

I

PREDICION OF SETTLEMEN 101

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