sinh 1 dp d sinh It will be shown in the next section that the use of Eq.2 results in a non-linear workhardening stress-strain curve for clay.

Transcription

1 57ème CONGRÈS CANADIEN DE GÉOTECHNIQUE 5ème CONGRÈS CONJOINT SCG/AIH-CNN 57TH CANADIAN GEOTECHNICAL CONFERENCE 5TH JOINT CGS/IAH-CNC CONFERENCE A NON-LINEAR ANALYSIS FOR PLANE-STRAIN UNDRAINED PRESSUREMETER EXPANSION TESTS Vncenzo Slvest, École polytechnqe de Montéal, Canada. Bad Benabdellah, École polytechnqe de Montéal, Canada. ABSTRACT Ths pape pesents the solton fo the ndaned expanson of a cylndcal cavty n a non-lnea wokhadenng sol. The solton s obtaned sng an nvese hypebolc sne law to epesent the elaton between appled adal pesse shea stan ndced at the cavty wall. As n the case of a lnealy elastc pefectly plastc (Tesca) mateal, thee paametes ae eqed to mplement the solton, the ntal hozontal pesse p o, the ndaned shea stength, the maxmm shea modls G max. The vales of these paametes ae estmated n a patcla case by lnea egesson analyss. Compasons ae also made wth soltons obtaned by means of powe law, smple hypebolc lnealy elastc pefectly plastc stess-stan elatonshps. RÉSUMÉ Cet atcle pésente la solton po l expanson non-danée d ne cavté cylndqe dans n sol non lnéae. La solton est obtene en tlsant ne lo sns hypebolqe nvese décvant la elaton ente la pesson adale et la défomaton de csallement. Comme dans le cas d n matéa de type Tesca, tos paamètes sont nécessaes, la pesson hozontale ntale p, la ésstance a csallement non-dané, le modle de csallement à l ogne G max. Ces paamètes sont estmés dans n cas patcle à l ade de égessons lnéaes. Des compaasons sont ass effectées avec des soltons obtenes en tlsant ne lo de pssance, ne elaton hypebolqe smple, et n modèle Tesca. 1. INTRODUCTION When the peconsoldaton pesse p of soft natal clays, whch s meased by means of constant stan ate (CRS) consoldaton tests, s plotted as a fncton of the stan ate sed, the esltng expemental elatonshp can be 2. STRESS-STRAIN RESPONSE descbed by an nvese hypebolc sne cve of the fom (Slvest et al. 1986): ' 1 snh p whee s the vale of p that wold be meased n a CRS consoldaton test pefomed at a stan ate s a vscosty coeffcent, s a tme coeffcent. Sch an eqaton s enconteed when sol defomaton s consdeed as a ate pocess (Mtchell 1993). Cosly enogh, when the appled total pesse p n a self-bong pessemete (SBP) test n clay s plotted as a fncton of the cent shea stan ndcted at the cavty wall, the esltng elatonshp s fond to be qte smla to that gven by Eq.1, except that the stan ate s eplaced wth the cent shea stan, that s: snh 1 p 1 [2] [1] It wll be shown n the next secton that the se of Eq.2 eslts n a non-lnea wokhadenng stess-stan cve fo clay. Fom the wok of Hll (195), Ladany (1972), Palme (1972) t s fond that the stess-stan cve of clay may be obtaned fom the expemental measement of adal pesse shea stan, by sng the followng elatonshp: dp d [3] whee s the shea stess ndced at the cavty wall, s the coespondng cent shea stan ( 1), dp/d s the slope of the adal pesse-shea stan cve. Intodcng Eq.3 nto Eq.2 yelds 1 [4] / 2 whee, now, s the vale of the pesse p (.e., epesents the hozontal geostatc stess p o, when the boehole s not dstbed po to the pefomance of the test), ae mateal paametes. Examnaton of ths eqaton shows that when s vey lage (.e., = 1), the shea stess s gven by [5] 2 1 1/ 2 Page 9

2 Snce 1 fo most soft clays, as ndcated below, then when = 1. Ths, the mateal paamete epesents the ndaned shea stength of the clay,. Dffeentaton of Eq.4 wth espect to allows to detemne the tangent shea modls G: d G d / 2 [6] G max 2 1 I 1/ 2 [13] The cve descbed by ths eqaton epesents a nonlnea wokhadenng constttve elatonshp, as shown n Fg.1. Sbstttng = nto ths eqaton pemts to fnd the maxmm shea modls G max : d d G max [7] G max The second mateal paamete s ths eqal to: G G S max max I [8] whee I s the gdty ndex of the clay. Becase = = I fom above, Eq.2 may also be wtten as: p p snh 1 I I 2I The lmt pesse p lmt s fond by pttng = 1 n ths eqaton leadng to: p p snh 1 I [1] In addton, as the gdty ndex I, whch s eqal to, anges between a mnmm vale of abot 5 to a maxmm vale of 15 fo soft clays, t follows that I 1. As a conseqence, when a SBP test s psed to stans of even as low as 1%, the nvese hypebolc sne tem n Eq.9 may be appoxmated by an exponental fncton of the fom (Mtchell 1993): snh 1 ln yeldng p p S ln 2I o [9] [11] [12] whch shows that the ndaned shea stength can be obtaned fom the gadent (.e., slope) of a plot of total appled pesse vess the natal logathm of the cent shea stan ndced at the cavty wall. Snce = = I as befoe, the stess-stan cve of Eq.4 becomes: Fge1. Non-lnea wokhadenng stess-stan cve. 3. APPLICATION 3.1 Invese Hypebolc Sne Low The elatonshps gven by Eqs.9 13 wee appled to selbong (SBP) test data epoted by Bolton Whttle (1999). The eslts ae compaed wth those obtaned by these athos sng a powe law fncton. Fthe compasons ae caed ot by assmng a smple hypebolc stess-stan cve as well as a lnealy elastc pefectly plastc esponse. The expemental adal pesse vess shea stan cve s shown n Fg.2. The data was obtaned by scannng dgtzng the ognal eslts of Bolton Whttle (1999). In ode to get the best ft to the data, the vale of the ndaned shea stength was fst detemned fom the slope, at lage stan, of the lnea poton of the p-log elatonshp of Fg.2. Ths yelded = 178 kpa, as also fond by Bolton Whttle (1999). Secondly, Eq.9 was tansfomed nto: p p I S snh [14] Lnea least-sqaes analyses wee then caed ot by assmng dffeent vales fo p. Eq.14 was ths appoxmated by the staght lne elatonshp: Page 1

3 yˆ ˆ ˆ a bx [15] kpa [16] / 2 Expanson pesse, p Examnaton of the cve descbed by ths eqaton, whch s shown n Fg.3, ndcates that the shea stess eaches a hozontal asymptote fo lage stan. As fo the vales of the lmt pesse p lmt, calclated on the bass of Eq.1, Table 1 shows that these ae qte nsenstve to the choce of the paametes â bˆ. Indeed, the lmt pesse vaes fom kpa to kpa Expemental data Invese hypebolc law Powe law Smple hypebolc law 4 1E-4 1E Shea stess, 2 16 Fge2. Radal pesse-logathm of cent shea stan elatonshp. ŷ whee s the estmated mean of snh [(p - p ) / ] fo each x =, â s the ntecept, bˆ s the slope (.e., the estmated mean of I ). It was fond that the vales of the paamete p epoted n the fst thee ows of Table1 eslted n a vey satsfactoy ft to the data. Examnaton of the entes n ths table ndcates that the choce of p = 46 kpa yelds a slghtly bette ft, becase t gves the hghest vale fo the coeffcent of coelaton (.e., =.99916). In ths case, the estmated mean vale of the gdty ndex I eqals Howeve, as t wold be expected that snh [(p p ) / ] n Eq.14 shold be zeo when =, mplyng a staght lne appoxmaton passng thogh the ogn (.e., â = n Eq.15), addtonal analyses wee pefomed by focng the egesson lne to pass thogh the ogn. The eslts whch ae epoted n the last fo ows of Table 1 ndcate that the choce of p = 45 kpa yelds, ths tme, a slghtly bette ft than the othe vales of p. The soltons gven n Table 1 ae, howeve, pactcally eqvalent n that they all gve a vey good ft to the data, fo the whole ange of shea stans (.e.,.1353). Ths notwthstng, t was fond that the ntal pat of the adal pesse vess shea stan elatonshp of Fg.2 (.e., fo.2) was bette appoxmated by sng p = 47 kpa wth ethe â = bˆ =I =34.6 (=.9976) = â.17 =I bˆ = 32.7 (=.9998). The cve shown n Fg.2 coesponds to Eq.9 wth = 178 kpa, p = 47 Kpa, â =, bˆ = I = As these paametes mply G max = I = kpa (54.2 MPa), the coespondng stess-stan cve s: Fge3. Stess-stan cves. Shea stan, 3.2 Powe Law Repesentaton Powe law Invese hypebolc sne law Smple hypebolc epesentaton Palme's appoach Elastc-plastc model The adal pesse vess shea stan elashonshp of Fg.2 was appoxmated sng the powe law of Bolton Whttle (1999). These athos poposed that the stessstan cve be epesented by: y [17a] y [17b] whee ae mateal paametes, wth 1 y = ( /) 1/ s the shea stan at yeld. It shold be also noted that complete soltons fo the expanson of both sphecal cylndcal cavtes, based exactly on ths same assmpton, wee pblshed by Ladany Johnson (1974), Ladany (1975). The powe law fncton n Eq.17 descbes a non-lnea elastc pefectly plastc (Tesca) model. The coespondng adal pesse vess shea stan cve s: p p y [18a] Page 11

4 p p 1 ln y y [18b] In addton, the lmt pesse s fond by pttng = 1 nto Eq.18b. The vales of the paametes wee detemned by Bolton Whttle (1999) sng nload / eload loops. On the bass of the last two loops epoted by these nvestgatos, whch eslted, espectvely, n ln 1 = =.5758, wth = ln 2 = =.573, wth =.9996, the coespondng aveage vales of ln ae: ln = o = 4181 kpa = As y = ( /) 1/ fom Eq.17, the shea stan at yeld s eqal to.178 fo = 178 kpa, = 4181 kpa, = Howeve, Bolton Whttle (1999) etaned =.57 y =.86, coespondng to = 4698 kpa. In addton, these athos fond that p = 449 kpa, epesented a satsfactoy geostatc stess vale. Eq.18 was, theefoe, sed wth p = 449 kpa, = 178 kpa, y =.86, = 4698, =.57 to plot the adal pesse vess shea stan elatonshp. Ths cve s also shown n Fg.2. Compason between the nmecal elatonshp based pon Eq.18 wth that obtaned sng the nvese hypebolc sne ndcates that the two ae qte smla. The coespondng stess-stan cve, based pon Eq.17, s epoted n Fg.3. Compason between the cves shown n ths fge ndcates that the stan-stess elatonshp obtaned by means of the nvese hypebolc sne law s stffe than the one based pon the powe law, n the ange.1.7. Fthemoe, n ode to detemne whethe the mateal paametes cold be also obtaned fom the ntal poton of the pesse-expanson cve (.e., fo y.187), wthot havng to se the nload / eload loops, Eq.18 was fst tansfomed nto: p ln ln ln p [19] then appoxmated by the egesson lne of Eq.15, whee ŷ s now the estmated mean of ln (p-p ) fo each vale of x = ln, â = ln s the ntecept, = bˆ s the slope. Regesson analyses wee agan pefomed by assmng dffeent vales fo p. The eslts ae epoted n Table 2. In ths table ae also gven the vales of y = ( /) 1/ Examnaton of the vaos entes ndcates that the choce of p =42 kpa constttes the best ft to the data wth â = ln = o = 438 kpa, bˆ = =.535, = Recallng that the eslts etaned by Bolton Whttle (1999) wee = 4698 kpa, =.57, y =.86 wth =.9996, t appeas that these coespond to a choce of p close to 43 kpa n Table 2. As fo the vales of the lmt pesse p lmt, detemned on the bass of Eq.18b fo = 1, Table 2 ndcated that these vay between kpa kpa. 3.3 Smple Hypebolc Repesentaton In ode to detemne whethe a smple hypebolc stessstan cve of the fom: Gmax 1 I [2] cold also be sccessflly sed, ths eqaton was sbsttted n Eq.2 to obtan the adal pesse vess shea stan cve. Ths eslted n: p p ln 1 I [21] Fo vey lage, I 1, Eq.21 may be appoxmated by: p p ln I [22] ndcatng, once agan, that the ndaned shea stength can be obtaned fom the slope, at lage stan, of a plot of total pesse vess the natal logathm of the shea stan. In ode to cay ot lnea egesson analyses, Eq.21 was fst tansfomed nto: p p S e 1 I [23] then appoxmated by the egesson lne of Eq.15, whee ths tme ŷ s the estmated mean vale of [e (p-p)/s -1] fo each vale of x =, â s the ntecept, bˆ = I s the slope. Reslts of these egesson analyses ae smmazed n Table 3 fo dffeent vales of p. Althogh the calclated vales of the coeffcent of coelaton epoted n ths table ae qte hgh, t was fond that none of the egesson eqatons cold satsfactoy appoxmate the adal pesse vess shea stan elatonshp fo shea stan Ths s de to the pesence of the negatve â tem. As a conseqence, addtonal egesson analyses wee pefomed by focng the egesson lne to pass thogh the ogn (.e., â = ). The eslts ae also epoted n Table 3. Examnaton of the entes shown ndcates that the choce of p = 45 kpa constttes the best ft to the data ( =.99793), yeldng bˆ = I = These paametes (.e., p = 45 kpa, I = 669.5, = 178 kpa G max = I = kpa) wee then ntodced nto Eqs.2 21 to plot both the stessstan cve the adal pesse vess shea elatonshp, as shown n Fgs.2 3. Examnaton of the Page 12

5 pedcted vales of the total pesse p n Fg.2 ndcates that they oveestmate the actal tend at low modeate shea stan (.e., fo.3). Concenng the hypebolc stess-stan cve plotted n Fg.3, t appeas that t s ntally mch stffe than both the powe law nvese hypebolc sne elatonshps. As an addtonal pont concenng the stess-stan cves epoted n Fg.3, dscete data ponts, obtaned fom a dect applcaton of Palme s appoach (.e., Eq.2), wee scaled fom the eslts epoted by Bolton Whttle (1999) ae also nclded n Fg.3. It appeas that the latte ponts follow qte closely the powe law appoxmaton. As fo the lmt pesse p lmt, t was detemned by pttng = 1 nto Eq.21 the eslts ae also epoted n Table 3. Examnaton of the vaos entes shows that t vaes n a vey small ange, fom kpa to kpa. Fthe, compason between Eqs shows that fo the lmt pesse to be nqe, the gdty ndex obtaned fom hypebolc model shold be appoxmately eqal to twce the vale detemned on the bass of the nvese hypebolc appoach. Ths s pefectly bone ot by the eslts pesented n Table Lnea Elastc Pefectly Plastc Response Fnally, n ode to examne the possblty that a smple lnealy elastc pefectly plastc (Tesca) stess-stan elatonshp of the fom: G, y = / G [24a], y = / G [24b] mght be appopate fo the clay, these eqatons wee sbsttted n Eq.2 to obtan the well-known pesseexpanson elatonshps (Gbson Andeson, 1961): p p G, y = / G [25a] p p 1 ln y, y [25b] Agan, the lmt pesse may be fond by pttng = 1 nto Eq.25b. Instead of pefomng two sepaate egesson analyses, that s, one fo elastc phase of defomaton the othe fo the plastc esponse, t was deemed pefeable to obtan by teaton a good ft to the expemental pesse-expanson cve. Ths was done by adjstng the vales of the paametes p G max. Two sch soltons ae epoted n Table 4 ae compaed n Fg.4 wth the expemental pesseexpanson cve. Examnaton of the data shown n ths fge ndcates that the solton whch coesponds to p = 496 kpa, = 178 kpa, G max = kpa, y =.515, compaes well wth the expemental eslts. In addton, the lmt pesse p lmt whch was calclated on the bass of Eq.25b fo = 1 vaes between kpa kpa. A stess-stan cve based pon the solton jst mentoned s also shown n Fg.3. Examnaton of the vaos elatonshps epoted n ths fges ndcates that althogh the smple lnealy elastc pefectly plastc solton s mch softe than the pevosly obtaned elatonshps, t nevetheless gves a good ft to the expemental pesseexpanson cve of Fg.4. Expanson pesse, p Expemental data Elastc plastc model (p =496 kpa, = 178 kpa, G max = kpa) Elastc plastc model (p =56 kpa, = 178 kpa, G max = kpa) 4 1E-4 1E Shea stan, Fge.4: Compason of expemental data wth lnealy elastc pefectly plastc esponses. 4. DISCUSSION It appeas at fst sght that the nvese hypebolc sne solton obtaned n ths stdy that of Bolton Whttle (1999) ae qte smla. Indeed, lnea egesson analyses yelded almost dentcal vales fo the coeffcent of coelaton. Thee s, howeve, a sght dvegence that ases between the two appoaches. Whle the vale of G max s fnte n the pesent appoach as fond fom Eq.7, that detemned by sng the powe law epesentaton s nfnte. Indeed, dffeentaton of Eq.17a gves: Page 13

6 d d 2 1 G [26] whch, when evalated at =, leads to G max =, snce 1. Sch a patcla behavo of the powe law epesentaton at the ogn s thoght to be nappopate fo clay. As fo the smple hypebolc stess-stan cve, t s shown that t s mch stffe than the nvese hypebolc sne law. In addton, t s ndcated that althogh the lnealy elastc pefectly plastc cteon fts easonably well the expemental pesse-expanson elatonshp, t nevetheless fals to epesent the non-lnea stess-stan esponse of the mateal at small stans. As a fnal pont woth of dscsson, t appeas fom the eslts shown n Fg.2 3, that whle any of the cveftted elatonshps s moe o less adeqate to epesent the expemental pesse-expanson date, the deved stessstan cves ae howeve all qte dffeent. Ths epesents a fomdable task fo the geotechncal engnee becase the deved stess-stan cve s dependent pon an assmed elatonshp fo the pesse-expanson cve. It s ths mpossble to make an objectve assmpton, even f one makes se of statstcal methods. The dffclty s lnked to the fact that the stess-stan cve s obtaned fom the dffeentaton of the pesse-expanson elatonshp. Howeve, the evese poblem, that s, the task of obtanng the pesse-expanson cve fom a known stess-stan elatonshp, s mch smple, becase of the ntegaton pocede. 5. CONCLUSIONS Ths techncal pape pesents a method to obtan the stessstan cve of clay fom ndaned plane-stan pessemete tests. The expemental adal pesse vess shea stan cve s appoxmated by an nvese hypebolc sne fncton. The esltng stess-stan cve can be descbed as a non-lnea wokhadenng sol model, havng a fnte modls at the ogn. Becase the stess-stan cve was obtaned sng Palme s appoach, t was not necessay to sepaate the sol esponse nto elastc plastc components. Compaed to the stess-stan cve based pon a powe law epesentaton, that obtaned n ths stdy was stffe. 6. ACKNOWLEDGMENTS The athos expess the gattde to the Natonal Reseach Concl of Canada fo the fnancal sppot eceved n ths stdy. 7. REFERENCES Bolton, M.D., Whttle, R.W A non-lnea elastc/pefectly plastc analyss fo plane stan ndaned expanson tests. Géotechnqe 49(1): Gbson, R.E., Andeson, W.F In st measement of sol popetes wth the pessemete. Cvl Engneeng Pblc Woks Revew, 56: Hll, R The mathematcal theoy of plastcty. Oxfod Unvesty Pess, London. Ladany, B In st detemnaton of ndaned stessstan behavo of senstve clays wth the pessemete. Canadan Geotechncal Jonal, 9(3): Ladany, B Beang capacty of stp footngs n fozen sols. Canadan Geotechncal Jonal, 12(3): Ladany, B., Johnson, G.H Behavo of ccla footngs plate anchos embedded n pemafost. Canadan Geotechncal Jonal, 11(3): Mtchell, J.K Fndamentals of Sols Behavo. 2 nd Edton, John Wley Sons, Inc., New Yok. Palme, A.C Undaned plane-stan expanson of a cylndcal cavty n clay: a smple ntepetaton of the pessemete test. Géotechnqe 22(3): Slvest, V., Yong, R.N., Solé, M., Gabel, F Contolled-stan, contolled-gadent, stad consoldaton testng of senstve clays. Poceedngs, Symposm on Consoldaton of Sols: Testng Evalaton; Fot Ladedale, Fla; ASTM STP 892, R.N. Yong F.C. Townsend, Eds.; Amecan Socety fo Testng Mateals, Phladelpha, pp It was also fond that a smple hypebolc stess-stan cve eslted n a mch stffe esponse compaed to both the nvese hypebolc sne law the powe law epesentatons. As fo the smple lnealy elastc pefectly plastc (Tesca) esponse, t faled to capte the pononced non-lnea stess-stan behavo at small stans. Page 14

7 8. APPENDIX A Table 1: Reslts of egesson analyses on nvese hypebolc sne law. Regesson analyses p â b ˆ Gmax / Coeffcent. of coelaton, p lmt * * * * *Regesson lne foced to pass thogh the ogn. Table 2: Reslts of egesson analyses on powe law appoxmaton. P â=ln Regesson analyses bˆ Coeffcent of coelaton, y P lmt (kpa Table 3: Reslts of egesson analyses on smple hypebolc law appoxmaton p â b ˆ G / S max Coeffcent. of coelaton, p lmt * * * *Regesson lne foced to pass togh the ogn Table 4: Reslts of lnealy elastc pefectly plastc esponse appoxmaton p G max / y p lmt Page 15