Chaoyang University of Technology -- Radial Consolidation -- APPLICATION OF TERZAGHI'S THEORY OF CONSOLIDATION TO PROBLEMS INVOLVING RADIAL FLOW

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1 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- UNIT 4 APPLICATION OF TEZAGHI'S THEOY OF CONSOLIDATION TO POBLEMS INVOLVING ADIAL FLOW Pepaed by D. oy E. Olso o Spig 989 Modified by Jiue Lai o Fall 4. Itoductio A vaiety of cases develop, i geotechical egieeig, whee it becomes ecessay to acceleate the ate of cosolidatio. Seveal examples ae:. A highway cosses ove a ive o a bidge. Fo some distace o each side of the bidge, the pavemet is suppoted o fill, which is udelai by a soft, silty clay that is quite compessible. Thus, the appoach embakmets ae likely to settle by sigificat amouts, wheeas the abutmets, which ae typically pile suppoted, will emai at essetially costat elevatio. If the fill settles afte the pavemet is placed, the a dageous bump ca develop at the poit whee the pavemet passes fom the fill aea to the bidge aea. The solutio to this poblem is typically to esue that fill settlemet is essetially complete whe the pavemet is placed. Ufotuately, the oe-dimesioal cosolidatio times may last fo decades.. Light stuctues ae goig to be placed at a site that has soft, compessible clays. A laye of fill will be placed fist ad the stuctue will be suppoted o shallow foudatios i the fill. Howeve, aticipated settlemets ae substatial, ad ae expected to be o-uifom acoss the site. The egiee decides to place a exta thickess of fill, with a pessue exceedig the pessue applied by the stuctue. Afte cosolidatio is essetially complete, the exta fill will be emoved ad the stuctue costucted. Miimal settlemets of the stuctue ae the aticipated. This pocess is usually called peloadig. Ufotuately, pedicted cosolidatio times fo the peload fill ae seveal yeas, makig this pocess uecoomical. 3. Oil stoage taks ae to be costucted at a site udelai by soft, compessible, clays. Calculatios idicate that apid fillig of the taks will esult i beaig capacity failues. Howeve, if the soil cosolidates duig fillig, the the facto of safety agaist a beaig capacity failue is quite lage. Calculatios idicate that it will ot be possible to fill the tak completely fo seveal yeas. The owe is uable to wait that log. The solutio to may poblems of these types, is to iset vetical, highly pemeable, elemets i the soil so the wate ca dai lateally to oe of these elemets ad the vetically to escape fom the soil system. Yeas ago, the pocess was to fom holes i the soil, by divig pipes o 5

2 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- by dillig, ad to fill the holes with sad, fomig sad dais. It is ow moe ecoomical to iset pe-fomed plastic daiage elemets, called wicks. Heeafte, we will use the tem wick to efe to ay vetical daiage elemet istalled i the soil. The zoe of ifluece of a sigle wick ca ofte be appoximated as a cylide of compessible soil with a impevious oute bouday ad with a cylidical wick, with a feely daiig suface, at the vetical axis. The cosolidatio pessue is applied to the top of the cylide ude coditios that ofte ivolve egligible lateal defomatio; thus, defomatio may be assumed oe dimesioal. Wate flow is thee dimesioal but fo may poblems of pactical iteest adial flow domiates. The othe assumptios ae the same as those made i deivig Tezaghi's oe-dimesioal theoy, i.e., the soil is satuated ad homogeeous, the poe wate is icompessible, Dacy's law is valid, thee ae o secoday effects, ad aveage soil popeties may be used without sigificat eo. I additio to the wick poblem, it is coveiet to coside the aalogous case of a laboatoy cosolidatio test i which the defomatio is oe dimesioal ad vetical, but the flow of poe wate is adial. 4. Diffeetial Equatio fo Oe Dimesioal Defomatio with adial Daiage The deivatio of the diffeetial equatio goveig oe dimesioal defomatio with adial daiage (edulic, 935) follows the same sequece of opeatios used to develop the equatio fo oe dimesioal cosolidatio. The diffeetial elemet of soil udegoig cosolidatio is show i Fig. 4.. Accodig to Dacy's law: q k u = dθ dz (4.) γ w dθ d dz Fig. 4. Diffeetial Elemet of Soil 5

3 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- The et time ate of volume chage of the elemet, q et, is: q k u u dq et = d = dθ dz + γ w (4.) whee dθ d dz is the volume dv. Fom the aalysis of Uit, the time ate of volume chage of the elemet is also: dq et dv e = (4.3) + e Equatios 4. ad 4.3 ae equated ad the coefficiet of compessibility fo oe dimesioal defomatio, a v, is used to covet fom a chage i void atio to a chage i excess poe wate pessue. The: u = c u u + (4.4) whee c is the adial coefficiet of cosolidatio ad is defied as: c k ( + e) a γ = (4.5) v w 4.3 Laboatoy Cosolidatio Tests The discussio of laboatoy tests is icluded hee because the mathematics is simplified compaed to cases ivolvig cetal dais, ad thus seves as a itoductio. Details of data eductio fo adial flow tests will be icluded i othe otes. A adial-flow laboatoy cosolidatio test may be pefomed usig a disc-shaped specime, as fo the usual oe-dimesioal cosolidatio test, but with daiage eithe though a oute poous ig o though a cetal cylidical dai. I this sectio, the case of a poous oute bouday will be aalyzed. The flat uppe ad lowe boudaies, which ae subjected to applied load, ae impevious. The hoizotal, flat boudaies, may be loaded though igid caps that foce the vetical stais to be uifom, the socalled equal stai case, o though flexible membaes, i which case the pessue is uifom but the stais ae ouifom, the socalled fee stai case Fee Stai Cosolidatio Tests Fo loadig though a flexible membae, the diffeetial equatio (Eq. 4.4) must be solved applyig the iitial coditio that: 5

4 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- u (,) = (4.6a) u i ad the bouday coditios: u (, t) = (4.6b) u(, t) = (4.6c) As fo the oe dimesioal poblem, it is assumed that vaiables ca be sepaated so that the excess poe wate pessue depeds o the multiple of two fuctios, F() ad G(t), which deped oly o adius ad time espectively: u = F() G(t) (4.7) Equatio 4.7 is iseted ito Eq. 4.4 which is the eaaged to yield: ( c ) G'(t) G(t) = F''() F() + F'() F() (4.8) Appaetly eithe side of Eq. 4.8 is a fuctio of eithe time o adius. Thus each side may be set equal to a costat, which is chose to be -A, ad solved. Fom Uit : G(t) = C exp (-A ct) (4.9) The tem cotaiig adius may be witte: d F() d + df() d + A F() = (4.) Equatio 4. is kow as Bessel's equatio. Its solutio may be witte i the fom: F() = C J (A) + C3Y (A) (4.) whee J ad Y epeset Bessel's fuctios of zeo ode of the fist ad secod kid espectively. The excess poe wate is the: u = [C 4 J (A)+C 5 Y (A)]exp(-A c t) (4.) The secod bouday coditio (Eq. 4.6c) is satisfied if C 5 is zeo. Thus: u = C 4 J (A)exp(-A c t) (4.3) To satisfy the fist bouday coditio (Eq. 4.6b), α must be a oot of: 53

5 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- whee: J (α ) = (4.4) A = α (4.5) Thee ae a ifiite umbe of oots coespodig to values of fom oe to ifiity. Tabulatios of these oots may be foud i hadbooks of mathematical fuctios (Abamowitz ad Stegu, 964). Thus: A = α / (4.6) Whe Eq. 4.6 is iseted ito Eq. 4.3, a liea equatio is obtaied that has a ifiite umbe of solutios, oe fo each value of. Thus, a geeal solutio is obtaied by summig the idividual solutios: α αct u = C J ( ) exp( ) (4.7) = It is coveiet to defie a adial time facto, T, as: ct T = (4.8) ad substitute it ito Eq The iitial coditio ow equies that: α u (4.9) i = CJ ( ) = Equatio 4.9 is a Fouie-Bessel seies, the coefficiets of which ae give by: Thus: C = α uij ( )d α J ( ) d = ui α J ( α ) (4.) u = = ui α J ( α α J ( ) exp( αt ) ) (4.) The degee of cosolidatio at adius is: 54

6 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- U u = = u = i α J ( α J ( α ), ) exp( α T ) (4.) The aveage degee of cosolidatio is: Itegatio yields: u πd u πd i uπd i U = (4.3) U = 4 = α exp( α T ) (4.4) The umeical elatioship betwee T ad U is peseted i Table 4. ad is plotted i Fig. 4.. Isochoes coespodig to vaious values of U ae plotted i Fig A fee stai laboatoy cosolidatio test ca be pefomed i the same geeal way that a stadad oe dimesioal test is pefomed except that volume chage will pobably be detemied by measuig the volume of fluid pumped ito o out of the cosolidatio cell athe tha with a dial idicato, ad a diffeet fittig method is eeded fo the estimatio of the coefficiet of cosolidatio. Table 4. elatioship Betwee the Aveage Degee of Cosolidatio ad Time Facto fo adial Flow to a Oute Feely Daiig Bouday Aveage Degee of Cosolidatio, % Fee Stai adial Time Facto Equal Stai adial Time Facto 55

7 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- Fig 4. Aveage Degee of Cosolidatio (%) - equal stai theoy fee stai theoy adial Time Facto (dimesioless) T-U elatioships fo Laboatoy Cosolidatio Tests with adial Outwads Daiage Poe Wate Pessue (dimesioless) / Fig. 4.3 "Fee Stai" Isochoes fo adial Cosolidatio Test. The umbes o the lies ae aveage degees of cosolidatio Equal Stai Cosolidatio Tests I most laboatoy tests, a igid loadig cap is used. Thus the uppe bouday coditio is ot oe of uifom stess, as assumed fo the fee stai theoy, but athe oe of uifom vetical stai. Ude such coditios the soil ext to the poous ig begis to cosolidate at oce ad emoves suppot fom the loadig cap, causig a edistibutio of the cotact stess betwee the igid cap ad the soil. The cotact stess deceases ea the ig ad iceases ea the vetical axis though the cete of the specime

8 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- The total height ad adius of the soil specime ae L ad espectively ad is the vaiable adial coodiate. Fom Dacy's law: Q k = γ w u πl (4.5) whee Q is the total volume chage ad the othe tems ae as defied peviously. Fo equal vetical stais, ad assumig o lateal defomatios withi the sample: Q = ( π L) = π L (4.6) Equatios 4.5 ad 4.6 ae equated to obtai: u γ w = = C k L (4.7) whee C vaies with time but is idepedet of adius. Equatio 4.7 is itegated fo ay oe istat of time ad the bouday coditio u(,t) = is applied to obtai: u = C( ) (4.8) The aveage value of the excess poe wate pessue, u m, is defied as: πud u (4.9) m = πd Equatio 4.8 is substituted ito Eq. 4.9 ad itegated to obtai: Thus: C 4um = (4.3) Similaly: = um ( ) (4.3) u Q e π L e = π L = + e + e (4.3) 57

9 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- Equatios 4.5, 4.7, 4.3, ad 4.3 ae combied to obtai: e 8u = m k ( + e) γ w (4.33) The substitutio is made that: de de a v = = (4.34) dσ du to obtai: u 8u = m c (4.35) whee c is defied i Eq Itegatio ad isetio of the coditio u m (t = ) = q yields the solutio: u m = q exp( 8T ) (4.36) The excess poe wate pessue is: u = q( ) exp( 8T ) (4.37) ad the aveage degee of cosolidatio is: U = - exp(-8t ) (4.38) The time-settlemet cuve is calculated fom Eq covetig fom time facto to time (Eq. 4.8) ad fom the aveage degee of cosolidatio to settlemet (Eq..44, Uit ). The elatioship betwee T ad U is plotted i Fig. 4. ad umeical esults ae give i Table 4.. The iitial adial distibutio of excess poe wate pessue is foud fom Eq to be paabolic with u agig fom zeo at the cotact betwee the soil ad the ig to q at the cetal axis of the specime. If the ig wee smooth ad igid, the iitial defomatio would have to be oe dimesioal ad the iitial excess poe wate pessue would have to be uifom, ot paabolic. Howeve, as soo as cosolidatio begis, the cotact stess betwee the loadig cap ad the soil must be gadually edistibuted ad appoach the distibutio calculated with the equal stai theoy. As cosolidatio poceeds, the cotact stesses cotiue thei edistibutio ad appoach a uifom value as time appoaches ifiity. I compaig the fee stai ad equal stai theoies it may be oted that whe applied to a laboatoy cosolidatio test i which a igid, impevious cap ad a poous ig is used, the fee 58

10 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- stai theoy cotais the pope iitial excess poe wate pessue but makes o povisio fo the edistibutio of stess, wheeas the equal stai theoy edistibutes the stess but cotais a ulikely iitial distibutio of excess poe wate pessue. Methods to be used i fittig the theoy to laboatoy data wee peseted by Tautwei, Olso ad Thomas (98) ad will ot be cosideed hee. 4.4 Dais (Wells) with Fee Stai This aalysis will apply to the case of a cylidical colum of homogeeous soil cotaiig a coaxial colum of feely daiig soil of the same compessibility. Wate flow is assumed to be adial towad the dai well. No flow of wate occus acoss the oute suface of the soil colum. Equatios descibig the cosolidatio of the soil ae obtaied by solvig Eq. 4.4 usig the followig bouday coditios: u( w,t) = u( e, t) = (4.39a) (4.39b) ad the iitial coditio: u(,) = u, (4.39c) i w e whee w is the adius of the daiage well ad e is the exteal adius of the cosolidatig colum of soil. The aalysis is too legthy to be peseted. The solutios (Bao, 944, 948; Moa et al., 958) ae: u q A B = exp( αn T ) = α (N C D ) (4.4a) whee: A = J o (α )Y (α )-Y o (α )J (α ) B = J o ( α w (4.4b) )Y o (α )-Y o ( α w )J o (α ) (4.4c) C = [J o (α N)Y o (α )-Y o (α N)J o (α )] (4.4d) D = [J o (α )Y (α )-Y o (α )J (α )] (4.4e) ct T = (4.4f) ad: 59

11 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- U = = α (N 4A )(N C exp( αn T ) D ) (4.4) ad: J, J ae Bessel fuctios of the fist kid of ode zeo ad oe espectively. Y, Y ae Bessel fuctios of the secod kid of ode zeo ad oe espectively, α ae oots of the equatio: ad: J (α N)Y (α ) - Y (α N)J (α ) = (4.4) N = e / w (4.43) The T -U cuves fo a age of values of N ae peseted i Fig. 4.4 (Moa et al., 958). cuves togethe with Eqs..44 ad 4.8 ae used to detemie the time-settlemet cuves. These - Aveage Degee of Cosolidatio (%) N=4 N= N=5-9 Fig adial Time Facto T-U elatioship fo adial Iwads Daiage Calculated usig the Fee Stai Theoy 6

12 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio Dais (Wells) with Equal Stai The geomety is the same as that just utilized fo the fee stai theoy. I this case, howeve, the assumptio is made that the suface is loaded with a igid plate so that the suface settlemet must be uifom. The aalysis is simila to that used to develop the equatios fo cosolidatio of soil i a laboatoy cosolidatio test with igid, impevious caps ad a pevious ig except that the daiage ow occus towad the cetal dai well ad thee is o flow acoss the oute bouday of the soil. The solutios ae as follows (Bao, 944, 948; Moa et al., 958): u = q N F N l( ) - ( w ) w - exp(- F T ) (4.44) ad: U = - exp(- F T ) (4.45) whee: F = N N - l(n) - 3N - 4N (4.46) ad the othe tems ae as defied fo the fee stai theoy. The time settlemet cuve is agai defied i usig Eqs..44 ad 4.8. The elatioships betwee T ad U (Eq. 4.45), fo a age of values of N, ae peseted i Fig Aveage Degee of Cosolidatio (%) adial Time Facto Fig. 4.5 T-U elatioships fo adial Cosolidatio to a Cetal Cylidical Dai Based o the Equal Stai Theoy. Numbes show ae values of N, the dai spacig facto

13 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- Accodig to the equal stai theoy the iitial excess poe wate pessue vaies fom zeo at the edge of the dai well to a maximum at the oute edge of the zoe of ifluece of the well. Fo N = 5, the maximum poe wate pessue is about.q. This distibutio is substatially diffeet fom that ecouteed with the fee stai theoy ad aises the questio as to which distibutio is the moe likely i the field. The actual distibutio of excess poe wate pessue i the field must deped o such factos as the elative compessibilities of the wick ad the suoudig soil, o the factio of the total suface aea occupied by the dais (wicks), ad o the actual igidity of the stuctue applyig the load. Howeve, fom a pactical poit of view, the eos associated with defiig umeical values fo the coefficiet of adial cosolidatio, ae likely to be cosideably moe tha diffeeces betwee the two theoies. Fo had solutios, the equal stai theoy is vey much easie to apply ad thus has bee used fo essetially all of the field desigs that have bee epoted i the liteatue. Numeical aalyses typically use the fee-stai appoach (see late discussio). 4.6 Sad Dais with Smea Zoes Istallatio of wicks i the field, by divig o by static foce, must distub the soil ea the dais. The distubace is likely to vay with distace ad pobably with depth; it cetaily vaies with the soil type ad details of the placemet method. The distubed zoe is likely to exet cosideable ifluece o the time ate of settlemet, ad pehaps eve o total settlemet. Ay attempt to take ito accout the distubace i a igoous mathematical mae would lead to geat aalytical poblems ad esult i equatios that would pobably be too complex fo pactical applicatio. Futhe, the "exact" equatios would cotai soil paametes ad geometic factos that could ot be evaluated fo pactical wok. Thus a compomise must be made to obtai tactable solutios yet oes that do ot yield aswes too much at vaiace with field obsevatios. I the ed, the available solutios have bee used oly to get a cude idea of distubace effects ad have ot, to my kowledge, bee used fo pactical desig. Bao (944, 948) assumed that the zoe of distubace could be appoximated as a homogeeous cylidical shell of soil suoudig the dai well. He assumed that this soil would cosolidate so apidly that its cosolidatio time could be igoed. Thus, the model he aalyzed cosists of a cylidical feely daiig well, of adius w, suouded by a cylidical shell of icompessible distubed soil, of oute adius s ad pemeability k s, ad the a shell of udistubed homogeeous soil extedig to adius e. Fee stai solutios fo this case wee developed by Bao (944, 948) ad discussed by Moa et al. (958). They ae too complex fo use i omal egieeig desig ad will ot be eviewed. Applicatio of the assumptios of the equal stai theoy to the case of sad dais with smea zoes leads to the followig solutios (Moa et al., 958): u = q ζ l - - s s e + k k ( N -S s N )l(s) exp(-t ζ ) (4.47) 6

14 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- i which q is the aveage suface pessue ad: Futhe: ζ = N N -S l(n S ) S 4N + k N -S k s N l(s) (4.48) S = s w (4.49) U = - exp(- T ζ ) (4.5) The time-settlemet cuve is easily calculated usig Eqs..44, 4.8, ad 4.5. Aalyses show that ay sigificat decease i the pemeability of the smea zoe, compaed to the adial pemeability of the udistubed soil, will esult i a dastic icease i the time equied to achieve ay specified amout of cosolidatio uless, of couse, the thickess of the smea zoe appoaches zeo. 4.7 Sad Dais with Smea Zoes ad Well esistace I pevious aalyses it was always assumed that the coefficiet of pemeability of the sad i the daiage well exceeded that of the suoudig soil by such a lage amout that o cosideatio eed be give to the possibility of excess poe wate pessues existig i the well itself. It is impotat to have a theoy that takes ito accout the fiite coefficiet of pemeability of the daiage well, ot because the theoy will be used egulaly i egieeig desig, but athe to establish what coefficiets of pemeability the daiage wells must have to esue adequate daiage. Bao (944, 948) applied the assumptios of the equal stai theoy to this poblem with the added assumptio that the coefficiet of pemeability of the soil suoudig the dai is zeo i the vetical diectio. He obtaied the followig solutios: whee: u (,z,t) = q ξ l( ζ ) - - s s e + (k k ) N -S s N l(s) +-ξ exp(-ξt ζ ζ = exp[β(z-l)]+exp(-βz) +exp(-βz) ) (4.5) (4.5) β = (N -S ) (k w /k ) e ζ (4.53) 63

15 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- ζ = N N -S l(n S ) S 4N + k k s N -S N l(s) (4.54) ad L is the thickess of the homogeeous compessible statum cotaiig the dais. The symbol u(,z,t) is used to deote specifically that the excess poe wate pessue vaies with adius, depth, ad time. The aveage degee of cosolidatio is foud by itegatig the excess poe wate pessues as follows: U = - L ql u(z,t)dz (4.55) whee: u (z,t) = q exp(- ξt ζ )(56) Equatio 4.55 is itegated gaphically. To demostate the effects of well esistace, Eq was solved fo the case of o smea zoe ad N = fo a age of values of L/ w,k w, ad T. epesetative data ae show i Table 4.. If the compessible laye is vey thi, e.g., L/ w =, Table 4.. Ifluece of Well esistace o the Time ate of Cosolidatio whe S= ad N= L/ w = 5 T kw/k= if satisfactoy daiage is achieved if the dai well has a hydaulic coductivity of the ode of to 3 times that of the soil. As the compessible laye becomes thicke thee is a lage ifluece of well esistace but a atio k w /k of 5 would appea to peclude ay sigificat well esistace fo the age of values of L/ w omally ecouteed i the field. If smalldiamete dais ae used i deep deposits it appeas that cae should be take to avoid sigificat amouts of well esistace. Sice smea zoes ted to educe the iflow ito the wells, the existece of smea zoes makes well esistace less of a poblem. 64

16 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- If the field coditios coespod to the fee stai assumptios, well esistace will be moe of a poblem because of the highe ate of cosolidatio at low values of the time facto. Howeve, such well esistace would appea to be compesated fo by the commo use of the equal stai aalysis. 4.8 Time Depedat Loadig Wicks ae ofte used i cases whee it is impotat to have a majo pat of cosolidatio completed duig o shotly afte costuctio. Thus, the costuctio time epesets a majo pat of the time duig which settlemets ae to be calculated ad the assumptio of istataeous loadig at time zeo leads to sigificat eos. The method of aalysis used i Uit 3 ca be applied to the case of time depedet loadig of sites povided with sad dais. The loadig diagam is assumed to be that show i Fig. 3.4 (Uit 3). The equatios fo the excess poe wate pessues obtaied usig the equal stai theoy, Eqs. 4.44, 4.47, ad 4.5, ca all be put ito the fom: u = Bq exp(-at ) (4.57) Applicatio of the pocedues of Uit 3 the leads to the solutios: ν u (,t) = Bq c AT c [-exp(-at )] (4.58a) ν > u (,t) = Bq c AT c [exp(at c ) - ]exp(-at ) (4.58b) whee: q c = applied aveage suface pessue at the ed of costuctio, T c = c t c / ad t c is the costuctio time (Fig. 3.4, Uit 3). Similaly, the equatios fo the aveage degee of cosolidatio (Eqs ad 4.5) ca be put ito the fom: U = - exp(-at ) (4.59) ad applied to the time depedet loadig to obtai: ν U = T c T - A[ ] - exp(-at ) (4.6a) ν > U = - AT c [ exp(at c ) - ] exp(-at ) (4.6b) 65

17 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- Solutios ae easily obtaied by evaluatig the factos A ad B fom the equatios ad chats peseted peviously. These solutios, ad the esultig T-U cuves, wee published by Olso (977). 4.9 Combied Vetical ad adial Flow Cases i the field that ivolve adial flow also ivolve vetical flow. The diffeetial equatio fo combied vetical ad adial flow, fo istataeous loadig, is: u = c v u + c z u u ( + ) (4.6) Caillo (94) demostated that vaiables i Eq. 4.6 could be sepaated i such a way that the degee of cosolidatio fo puely vetical flow (U v ) ad puely adial flow (U ) could be calculated as if thee was oly vetical o adial flow, espectively, ad the the solutios combied i a simple equatio: U = - (-U v )(-U ) (4.6) whee U is ow the degee of cosolidatio fo combied vetical ad adial flow. By calculatig values fo U v ad U fo effects of time depedet loadig, we ca ow solve cases with time depedet loadig ad combied vetical ad adial flow. 4. efeeces Abamowitz, M. ad I. A. Stegu (964), Hadbook of Mathematical Fuctios, Natioal Bueau of Stadads, Applied Mathematics Seies pp. Bao,. A. (944), "The Ifluece of Dai Wells o the Cosolidatio of Fie-Gaied Soils," U.S.Amy Cops of Egs. Dist., Povidece. Bao,. A. (948), "Cosolidatio of Fie-Gaied Soils by Dai Wells," Tas., ASCE, Vol. 3, pp Caillo, N. (94), "Simple Two- ad Thee-Dimesioal Cases i the Theoy of Cosolidatio of Soils", Jou. of Math. ad Physics, Vol., No., pp. -5 McKilay, D. G. (96), "A Laboatoy Study of ates of Cosolidatio i Clays with Paticula efeece to adial Poewate Daiage," Poc., Fifth Ite. Cof. o Soil Mech. ad Foud. Eg., Vol. 3, pp Moa, Pocto, Meuse, ad utledge (958), "Study of Deep Soil Stabilizatio by Vetical Sad Dais," Publ. PB569, U.S.Dept. of Commece. 66

18 Depatmet of Costuctio Egieeig Advaced Soil Mechaics Chaoyag Uivesity of Techology -- adial Cosolidatio -- Olso,. E. (977), "Cosolidatio ude Time-Depedet Loadig," Jou., Geotechical Egieeig Div., ASCE, Vol. 3, pp edulic, L. (935), "De hydodyamische spaugsausgleich i zetal etwassete tazylide," Wasse-witschaft,., pp Tautwei, S. J.,. E. Olso, ad. L. Thomas (98), "adial Flow Cosolidatio Testig," Poc., Teth Ite. Cof. o Soil Mech. ad Foud. Eg., Stockholm, Vol., pp