observational procedure of settlement prediction - asaoko (1978)

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6C-(/I(-~fSOILS AND FOUNDATIOJlS' VoL 18, No. 4, Dec, 197. Japanese Soety of. Soil Mechanics and Foundation Engn~~Dg

. OBSERVATIONAL'

PROCEDURE OFIi

SETTLEMENT. PREDICTIONi.

,

i

AxIRA AsAOKA *

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-

I ABSC1 .

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 observational 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' other is 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, .!

IGC: . E2 . .

ure~ent, settlement, statistical analysis

INTRODUCTION

The one-imenslnal consoldation theory is an important contribution of soil mechanics

to prctical foundaion engineering. Settlement' resulting from onHimensional compression .and one-dimensional drainage has been widely recogni~ed to be well explained by this theory. 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 the practical applicatiohs of this equ~tion have exhibited a tendency to utiliz only this characteristic of the so.ution. In fact. according to conven1onalsettlement 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. thei

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

I.. . .: .

JUL 19 Z~ i~:~i r~ Lioll lL.i::1\I

4:1.. :::: lO.. IV l..tJ~..Ulu..u"

"'~"'" '..r.:..~. ... "

::~. .. "

es;' -. . ..0. -~. ':~;-~.~'" ':': d~:"-I- .. ...

ASAO' ~. /

e~gihe~.. working :~n' ~itlehi~~t' pi-edicti~~ is: ~~uaiy . e~peted to have bad some previous

expeience. I

In this paper. a nw idetof settlement prediction is presented, the philosophy of which

is base on "Observational Proce\le." . The trend equation of time series data of settlement is derived nrst from the' ~n~dimenio'nal consolidtltion. equation, afte which the future 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 stochastic

pr, 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 problems including settlement 1ue to drainage from ::and piles.; i I

~

a

~~ EQATl~OF sn~-:iME~T10N ..

i,: (

iioship. .i . . .

i" this seon, ,he ori~ar.differ"ti equatin de.. giv.. a _1e,-tim re.(1)

. is a fundamental equatipn. 'Miksa's equation is adopted. that is.!

" "

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

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

f

. Cfi ; coeffcient of nslidation. . . . .~ in Eq. .(1) he upp& do. ~ 'l'epresents time-Iertiation' and the lower .scipt~ :l the , z. Even if the coeffients of b~tA. permiabiUty an~ depthdifferentiation with respett~ .

volume compressibilty va from time to time, the Eq. (1) is still efecive when Cfi remains coiitant (Miksa, ;1. 3). ''In,addition; Eq. (1). makes. it ~sy to express; settlement.For' these reasons Eq. (1) ~s adoptd'instead of Terzghi's eqution.. '. :

~

j"f

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

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

'" .

et,%)=T+.l(Lt)+..(Li')+..' 2! 'C" . 4! cv2.. .. ..' .. ':+.zF +W(l . %8 i)+ ..(L )+... . '.

. 31 ." c.t. I.;' C 5' -""T;:e(e, :=0).

.( 2) .. ~ .."'! ".i (

.' ..

and'

, : .' . '. . '. i. .F=eK(t, 2;:0)..'.' .'". ...... i . . .., '. ' '.'.~: T~e : ve:tica.l . sti~in ~ ~iI ~q.. (l)~' is'..'originii~'' . ~efin~ i~. a~ ~ul~i~"~ . .s.n~e,,'. ."&ti~( in. ....this

1

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

e

'by the ~uthor. - he results, . hwev~.. boine 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 nrainagcrt'on b~il top': b()t~D:bunaari~ .. . ~(Ti.~:::~C: ~:~~~)Fig. t::~o:w~. thisbou~8'ry ..~nditio~~ ~~~c~.i:~.,.~i:in~lat:ed...,:s :.:. ..: .'.' '.~ ...!"::'-~':' .'; .'.,. ....c .' I ". . - _. e((, ..iQ) =,.; !"..t .. ..... .' .. . ." ....~" ...,;, ,.! 3 ).

i

c r\ ......~, '

JUL 1 ~ ldl~ i ~ : c i r to L 101 IlL i:: I

o i ~O~ 1 ~~ i V 1 ~UwwU . UWV A11i i !

PREDICTION lJF. SE,LEMENT..r,

89

.nee drainaq~ :.,...,f.,-, -..:: ::.:'......:-:... .....'. top of clay

. " .". 11 --t-J bot,om of :~:.:.:.: :::: -: .:.... ~....';!:.:.... clayfre drainage

,1 :". top of clay

'H. . t .

J bottom of clay

impermable layerFig. 2. Upwar drainage .

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

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

. - ubsthutin of . (3) into Eq. (2) gives

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,

. .I'I '. . .', T=E: con~tant2) .. (OVle. ~ D,(4\~~'E)~(t, .,i=.0) =! : constant

(5)

F+..(HZ p)+.-(H: H 3! Cu 51 Cv F)+....= ~- ()

I (2) Upward d.ainage (Fig.

. Whe~ the bottom a the clay layer is. impermiale. boundary conditions are given as follows:

(7)

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

(8 ) (9 )

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

." ., F+- .' '1 (P'.'\." 1 -1 )' ,. '. --F '+'- (H' +...=0' .', (10)which is also a linear ordinary differential equation of F with constant coffcint......Now;.:foI' any'Wundary "condition, the 'settlement 'of clay stratum can be" expresse as. .._.. ~.,' ~ '.' I' ::. . 2!. ...u ' .1.. 4!. . cvs ..... "'.. ..' .' .

\.. ". I - pet) :;.)0' t(t. t)d% (11)in whi p(.) r+nts the settlement at tUne;.. Subsitutig the sOlution (2) into Eq.

Icl1 .I.... i I

(l~)l ',since T=f: onstant, we have .:::. .' .-: .;' ....IP('t):~fH+..(H!F)+irH'F)+~(H8 F)+.... ,-, (12) -; ~h~' se~u~~tial Jiff~rentiatjon of 'Eq. '.(12) .'wit~ respect .to t: we get. the' fon~~ing.set of

_ r. . . _ . . ...21 . 4fc ,. .' 61 . c Z ..' -'. .

eq~~~i~ns: I '. \. . : .., '. p=_(HiF)+ -F -F . - I 1 . . -MBA Cu 61 (R' ...) 1 1 21 41 ,,) 1+-c.,i +...".- -. ... ';....:.:..~.. _ a.J

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

( . ~- ~

. : .eft). "j . , . . '; . '1(:' 1. -.:. i'C'R:' (ll~l))' :"l(' -R: ""+iQ)' '. ".1.' ; , ". "'. ,JI3)

.. .". . .. . . . . ..

JUL 1S 25 15:22 F~ Cl~ll lClol:.. \ i I

b 1 ~ ~~~ 1 O~ tv ~u~~u. u~u &

-,.~. ';;~'~-':1 .. ~

90

ASAOJtI

Cmparing 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 P+3T~v P \ liH" P ..) ! H l) 4 ~14 aJ ~JHz )+m c.ll +...=~e+V '\.v-~e- aC

Similarly. from Eq. (0), we get

P+ IjH2 \+ 4T c. 1J.!)+...=Iie 2f c., )consolidation

D'n ~ (14b)

. : .' . 'Df'l.,v.Co.J

Eqs. (14) give a s

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