NUMERICAL MODELLING OF TWO-DIMENSIONAL HEAVE FOR SLABS- ON-GROUND AND SHALLOW FOUNDATIONS

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1 56 TH CANADIAN GEOTECHNICAL CONFERENCE 4 TH JOINT IAH-CNC/CGS CONFERENCE 2003 NAGS CONFERENCE 56 ème CONFÉRENCE CANADIENNE DE GÉOTECHNIQUE 4 ème CONFÉRENCE CONJOINTE AIH-CCN/SCG 2003 NAGS CONFÉRENCE NUMERICAL MODELLING OF TWO-DIMENSIONAL HEAVE FOR SLABS- ON-GROUND AND SHALLOW FOUNDATIONS Hung Q. Vu, Clton Assoctes Ltd., Ssktoon Delwn G. Fredlund, Proessor Emertus, Unverst o Ssktchewn, Ssktoon ABSTRACT: Ths pper presents numercl method or perormng two-dmensonl heve or slbs-on-ground nd shllow oundtons. The proposed method s n extenson o commonl used method or the predcton o onedmensonl heve to two-dmensons. The method s bsed on oedometer test results. Mesurements o sol suctons n the eld nd the swellng ndex wth respect to chnges n sol sucton re not requred. An exmple nvolvng concrete slb constructed on expnsve sols wth externl lods nd nltrton o wter nto the sol s nlzed to llustrte the proposed method. RÉSUMÉ: Cet rtcle présente une méthodenumérque pour smuler le gonlement dns deux dmensons pour les dlles sur le sol et les ondtons peu proondes. L méthode proposée est une extenson d'une méthode ordnrement utlsée pour l prédcton de gonlement dns un-dmenson ux deux-dmensons. L méthode est bsée sur les tests de oedometer. Les mesures de succons de sol dns le chmps et l'ndex de chngement de volume pr rpport ux chngements dns l succon de sol ne sont ps exgées. Un exemple clssque ssocé vec l déormton d'une dlle concrète construte sur les sols expnss grâce ux chrgements externes et l'nltrton d'eu dns les sols est nlsé pour llustrer l méthode proposée. 1. BACKGROUND The predcton o heve n two-dmensons requres denton o the ntl totl stresses nd mtrc sucton condtons, the elstct prmeters relted to chnges n net norml stress nd mtrc sucton condtons nd n ssumed Posson s rto (Hung nd Fredlund, 2002). Sol sucton condtons n the eld cn be mesured, estmted or ssumed. A sturted-unsturted seepge nlss cn be perormed to predct chnges n sol sucton. The sol propertes requred or trnsent seepge nlss re the sol-wter chrcterstc curve nd the coecent o permeblt uncton. The sol-wter chrcterstc curve nd the coecent o permeblt uncton or the volume chnge nlss nvolvng n unsturted sol re descrbed usng the Fredlund nd Xng (1994) equton nd the Leong nd Rhrdjo (1997) equton, respectvel. The net norml stress condtons n the eld cn be estmted rom totl stress theor nd the coecent o erth pressure t-rest. The elstct prmeter wth respect to net norml stress, E, nd elstct prmeter wth respect to sol sucton, H, cn be obtned b derenttng the equton or consttutve surce or the sol structure (Fredlund nd Rhrdjo, 1993) or clculted drectl rom the volume chnge ndex wth respect to net norml stress, C s, nd the volume chnge ndex wth respect to sol sucton, C m, (Hung nd Fredlund, 2002). Ths pper suggests tht commonl used method or the predcton o one-dmensonl heve (.e., the Fredlund, Hsn nd Flson method, 1980), cn be extended or the predcton o two-dmensonl heve. The Fredlund et l. method (1980) s brel revewed n the ollowng secton to serve s bckground nd the termnolog s set orth or the two-dmensonl procedure suggested n ths pper. A two-dmensonl exmple problem nvolvng slb-onground s used to llustrte the suggested procedure or heve nlss. The numercl solutons re obtned usng generl-purpose prtl derentl equton solver, FlexPDE 1. The results o the seepge nlss nclude the dstrbutons o sol sucton n the sol prole wth respect to tme or speced boundr condtons. The results o stress/deormton nlss nclude the dstrbuton o horzontl nd vertcl dsplcements tht occur due to chnges n ppled lod nd mtrc sucton. The numercl procedure cn be used or the nlss o wde vret o two-dmensonl heve problems ssocted wth expnsve sols. The results show tht t s possble to compute the moments nd shers n the concrete slb due to externl lodng to the slb nd swellng n the sol. 2. FREDLUND, HASAN AND FILSON METHOD (1980) FOR ONE-DIMENSIONAL HEAVE ANALYSIS The Fredlund et l. (1980) method s bsed on the constnt volume oedometer test results perormed on undsturbed smples. The bsc dt requred rom the lbortor test re the rebound or swellng ndex, C s, nd the corrected swellng pressure, P s. The dt must be corrected or the eects o compressblt o the pprtus pror to ts nterpretton. The mount o totl heve s computed rom chnges n vod rtos correspondng to the ntl nd nl stress sttes nd the swellng ndex, C s. The ntl nd nl stress sttes re projected onto the net norml stress plne, s shown n Fg. 1. The stress pth ollows constnt vod rto pth rom the n stu stress stte to the ntl stress stte (.e., the corrected swellng pressure, P s) on net norml stress plne, nd then ollows the

2 rebound curve rom the ntl stress stte to the nl stress stte. The equton or the rebound porton o the oedometer test dt s wrtten s ollows: FST e C log [1] s IST where: C s = swellng ndex wth respect to net norml stress mesured t sturton, FST = nl stress stte, nd IST = ntl stress stte. 3. PROPOSED METHOD FOR TWO-DIMENSIONAL HEAVE ANALYSIS The soluton o heve problem ssocted wth unsturted, expnsve sols nvolves the soluton o sturted-unsturted seepge model nd stressdeormton model. The models cn be ormulted bsed on the generl theor o unsturted sol behvour (Hung nd Fredlund, 2002). The use o dt rom oedometer tests nd concepts o mtrc sucton equvlent suggested n the Fredlund et l. (1980) method wll be presented or seepge nlss nd stress-deormton nlss. 3.1 Seepge nlss A seepge nlss s requred to predct chnges n the mtrc sucton condtons n the sol. Intl mtrc sucton condtons cn be estmted rom the corrected swellng pressure ssumng slope or the net norml stress versus sucton curve t constnt vod rto. Let us ssumng tht the slope cn be wrtten s the uncton,, nd or the ske o ths exmple t cn be tken s beng equl to the degree o sturton (Fg. 1), Eq. 2 becomes: P ( u ) ( u u ) [4] ' s w Fgure 1. Illustrton o totl stress pth or heve nlss nd ssumed reltonshp between mtrc sucton nd mtrc sucton equvlent The ntl stress stte, IST, or the corrected swellng pressure, P s, cn be ormulted s the sum o the overburden pressure nd the mtrc sucton equvlent (Fg. 1) s ollows: IST P ( u ) ( u u ) [2] ' s The nl stress stte must ccount or totl stress chnges nd the nl mtrc sucton condtons. The nl mtrc sucton condton cn ether be predcted or estmted. At low sucton condtons, the mtrc sucton equvlent cn be ssumed to be equl to the ctul mtrc sucton, nd the nl stress stte cn be estmted s ollows: FST ( u ) ( u u ) [3] where: = subscrpt ndctng nl condton, = subscrpt ndctng ntl condton, nd = chnge n totl stress due to ppled lods. w e w Thereore, the ntl mtrc sucton condtons requred or seepge nlss cn be estmted s ollows: ( u u ) w ' Ps ( u) where: = uncton set equl to the degree o sturton. 3.2 Stress-deormton nlss The stress-deormton nlss cn be perormed to predct dsplcements nd nduced stresses due to externl lods nd wettng. Deormton n the sol mss due to externl lods cn be ssumed to respond mmedtel, whle deormton due to wettng s tme dependent process. Thereore, the deormtons due to lodng nd wettng need to be nlzed ndependentl. The suggested stress pth or the stress-deormton nlss s llustrted n Fg. 2. The nlss s rst perormed to predct the dsplcements nd nduced stresses due to the lodng. The dsplcements due to chnges n mtrc suctons re then predcted or vrous elpsed tmes usng mtrc sucton proles obtned rom the seepge nlss. A set o ntl stress stte (IST), nl stress stte (FST) nd the swellng ndex (C s) rom the net norml stress plne cn be used or the ctul sucton stress pth s llustrted n Fg. 3. The n stu stress stte (IST) or two-dmensonl nlss cn be wrtten s ollows: [5]

3 IST ( u ) ( u u ) [6] ve w where: ve = ( x + )/2 or K IST 1 0 ( u) ( u uw) 2 [7] where: K 0 = coecent o erth pressure t-rest, = slope set equl to the nl degree o sturton. The nl stress stte (FST) cn be wrtten s ollows: FST ( u ) ( u u ) [8] or ve K FST 1 0 ( u) ( ve u) ( u uw) 2 [9] w Fgure 3. Illustrton o the use o ntl stress stte (IST), nl stress stte (FST), nd swellng ndex obtned t net norml stress plne (C s) or sucton stress pth The coecent o lterl erth pressure t-rest, K 0, cn be selected rom tpcl vlues tht hve been bckclculted rom eld observtons o heve nd shrnkge (Ltton, 1994): K whenthe sol s dr nd crcked whenthe sol s dr nd crcks reopenng whencrcks reclosed nd suctons t [10] sted sttecondton whencrcks reclosed nd the sol s wettng whenthe sol s wettng nd s n hdrosttc stress condton whenthe sol s pprochng pssve erth pressure The elstct prmeter unctons cn be wrtten or twodmensonl plne strn condtons s ollows (Hung nd Fredlund, 2002): E (1 )(1 2)(1 e ) ( ve u ) [11] C s Fgure 2. Stress pth ollowed n the stress-deormton nlss H.605(1 )(1 e ) ( u uw ) [12] C 4 0 m It s suggested tht Posson s rto be ssumed to be constnt tht s estmted rom the coecent o erth pressure t-rest, K 0, s ollows: K 1 K 0 [13] 0

4 Fgure 4. Illustrton o the exmple problem nd the one-dmensonl soluton (Fredlund nd Rhrdjo, 1993) 3.3 Clculton o moments nd sher orces n the slb Assumng tht dsplcements t the edge o the slb re smll n comprson to ts thckness, the lods ppled on the slb cn be ssumed to be norml to the slb surce. The bendng moments cn be clculted rom the dsplcements or bendng stresses, nd the sher orce cn be clculted rom the moments. Tmoshenko nd Wonowsk-Kreger (1959) presented the ollowng equtons or computng bendng moments nd sher. M or E h v 3 2 c s (1 c ) x [14] Fgure 5. Illustrton o the exmple problem nd the boundr condtons or the seepge nd stressdeormton nlss 2 mxhs M [15] 6 Q M [16] dx where: v = vertcl dsplcement o the slb; M = bendng moment per unt length; Q = sher orce per unt length; h s = thckness o the slb; mx = mxmum bendng stress; E c = elstc modulus o concrete; nd c = Posson s rto o concrete. 4. EXAMPLE An exmple problem presented n Fredlund nd Rhrdjo (1993) ws nlzed to llustrte the suggested procedure or predctng two-dmensonl heve. The problem nd ts one-dmensonl soluton usng the Fredlund et l. method s presented n Fg. 4. The cl ler s 2 m n thckness. The ntl vod rto o the sol s 1.0, the totl unt weght s 18.0 kn/m 3, nd swellng ndex, C s, s 0.1. Onl one oedometer test ws perormed on smple tken rom depth o 0.75 m. Test dt showed corrected swellng pressure o 200 kp nd n ntl degree o sturton o 70%. It s ssumed tht the corrected swellng pressure s constnt throughout the entre sol ler. A totl heve o 114 mm ws predcted

5 rom one-dmensonl nlss when no externl lod ws consdered. The problem ws moded to show two-dmensonl behvor b plcng concrete slb o 100 mm thckness t the surce (Fg. 5). Dsplcements due to the externl lods, nd lekge o wter under the cover wll be predcted or vrous sucton condtons (.e., elpsed tmes) n the sol mss. A Young modulus o 10 MP nd Posson s rto o 0.15 ws used or the concrete slb. Fgure 5 shows the geometr nd boundr condtons or both the seepge nd stress-deormton nlses. Zero pore-wter pressure ws speced under the slb nd mosture lux equl to zero ws speced elsewhere long the boundres. A lod equl to 5 kp nd permeter lod o 15 kn/m were ppled on the surce nd t the permeter o the concrete slb. The sol s ree to move n vertcl drecton nd xed n the horzontl drecton t the let nd rght sdes o the domn. The lower boundr s xed n both drectons. The sol-wter chrcterstc curve nd the permeblt uncton presented n Hung nd Fredlund (2002) were ssumed or the seepge nlss (Fg. 6). A reltonshp between degree o sturton nd mtrc sucton ws ssumed to estmte the mtrc sucton equvlent rom mtrc sucton nd s presented n Fg. 7. Intl mtrc sucton condtons n the eld were estmted rom the corrected swellng pressure usng Eq. 5. Fgure 8 shows dstrbuton o overburden pressure, corrected swellng pressure, mtrc sucton equvlent nd the estmted ntl mtrc sucton condtons. E 25.8( ve u ) [17] The elstct prmeter uncton wth respect to chnges n mtrc sucton, H, cn be clculted or twodmensonl condton usng Eq. 12 or e 0 = 1.0; C m = 0.10; nd = 0.4 nd cn be wrtten s ollows: H 128.9( u uw ) [18] 5. COMPUTER RESULTS AND DISCUSSIONS Fgure 9 presents mtrc sucton proles t the centre o the slb or vrous elpsed tmes o wettng. Fgure 10 shows the mtrc sucton dstrbuton n the sol t d 45. It cn be seen tht under the speced boundr condtons, the mtrc suctons t d 45 pproches zero below centre o the slb nd bout 50 kp below the edge o the slb. Fgure 7. Assumed reltonshp between degree o sturton nd mtrc sucton Fgure 6. Assumed permeblt uncton nd sol-wter chrcterstc curve The coecent o erth pressure t-rest, K 0, equl to ws used to determne the ntl stress stte condtons (Eq. 10). A Posson s rto o 0.40 ws clculted rom K 0 usng Eq. 13. The elstct prmeter uncton wth respect to chnges n net norml stress, E, cn be clculted or twodmensonl condtons usng Eq. 11 or e 0 = 1.0; C s = 0.10; nd = 0.4 nd cn be wrtten s ollows: Fgure 8. Intl stress stte condtons

6 Fgure 11 shows contours o vertcl dsplcements due to lodng. About 4 mm o settlement due to lodng s predcted t the edge o the slb. The nduced net norml stress s used to clculte the nl net norml stress stte n the sol. The sol ws loded t the ntl net norml stress nd mtrc sucton condtons n the eld. Thereore, the sum o ntl net norml stress nd ntl mtrc sucton equvlent must be used long wth the swellng ndex obtned on the net norml stress plne or the predcton o dsplcements nd nduced stresses due to lodng. Fgure 12 presents the horzontl dsplcements versus depth t the edge o the slb ter lodng nd vrous nl pore-wter pressure condtons. A mxmum horzontl dsplcement o 17 mm ws computed or the 0.25 m depth t d 45. Insgncnt vlues o horzontl dsplcements were observed when the pore-wter pressure ws rsed unorml to zero throughout the entre sol prole. Fgure 11. Contours o vertcl dsplcements due to lodng Fgure 12. Horzontl dsplcements versus depth t the edge o the slb ter lodng nd vrous nl pore-wter pressure condtons Fgure 9. Mtrc sucton proles t the centre o the slb or vrous elpsed tmes Fgure 13. Vertcl dsplcements versus depth t centre o the slb ter lodng nd vrous nl pore-wter pressure condtons Fgure 10. Mtrc sucton condtons ter 45 ds o wettng Fgure 13 presents the predcted vertcl dsplcements versus depth t the centre o the slb ter lodng nd vrous nl pore-wter pressure condtons. A totl heve o 102 mm predcted or the cse when the nl pore-wter pressure s equl to zero nd ths compres well wth the totl heve o 114 mm predcted rom the one-dmensonl nlss. It must be noted tht the predcted one-dmensonl heve or ths exmple dd not consder the externl lod ppled on the slb.

7 Fgure 14 shows vertcl dsplcements o the slb ter lodng nd vrous nl pore-wter pressure condtons. Fgures 15 nd 16 presents contours o horzontl nd vertcl dsplcements, respectvel. Totl heves o 88 mm nd 51 mm were predcted or d 45 t the centre nd the edge o the slb, respectvel. A derentl heve o 37 mm ws observed t d 45. When the porewter pressure ws rsed unorml to zero throughout the entre sol prole, the derentl heve o the slb s mnml. The results o the stress-deormton nlss lso nclude the dstrbuton o the resultng stresses n the slb. Fgure 17 shows the predcted lexurl stresses t the top nd bottom o the slb t d 10 ter wettng. Fgure 18 presents the lexurl stresses t the top o the slb ter lodng nd vrous elpsed tmes o wettng. Fgure 17. Flexurl stresses t top nd bottom o the slb t d 10 Fgure 14. Vertcl dsplcements o the slb ter lodng nd wettng wth tmes Fgure 18. Flexurl stresses t top o the slb ter lodng nd vrous elpsed tmes o wettng Cumultve bendng moments n the slb ter lodng nd vrous elpsed tmes o wettng re presented n Fg. 19. It ws ssumed tht the permeter lod ws ppled ter the concrete ws hrdened. The bendng moments predcted or ths exmple t erl perods o wettng (.e., less thn 25 ds) re due mnl to the lodng o the slb. The bendng moments resulted rom wettng took plce n ltter perod. Fgure 15. Contours o horzontl dsplcements t d 45 Fgure 16. Contours o vertcl dsplcements t d 45 Fgure 19. Cumultve bendng moments o the slb ter lodng nd vrous elpsed tmes o wettng

8 6. SUMMARY The method or the predcton o two-dmensonl heve s proposed bsed on generl theor o unsturted sols usng conventonl oedometer test results. Intl sol sucton condtons n sols cn be estmted rom the corrected swellng pressure. Chnges n sol suctons cn be estmted through sturted-unsturted seepge nlss. Sol propertes obtned n net norml stress plne long wth the concepts o mtrc sucton equvlent cn be used or the stress-deormton nlss or both externl lods nd chnges n sol sucton. Clculted dsplcements, lexurl stresses nd bendng moments o the concrete slb ppers to be resonble nd consstent wth those generll observed n the eld. Totl heve predcted rom two-dmensonl nlss ppers to gree well wth the nltcl one-dmensonl heve predcted usng the Fredlund et l. (1980) method or heve predcton. REFERENCES Fredlund, D.G., nd Rhrdjo, H Sol mechncs or unsturted sols. John Wle & Sons, New York, 560 p. Fredlund D.G., Hsn J.U., nd Flson H The Predcton o Totl Heve. Proc., Fourth Int. Con. on Expnsve Sols, Denver, CO., June 16-18, Vol. 1, pp Fredlund, D.G. nd Xng, A Equton or the Sol- Wter Chrcterstc Curve. Cndn Geotechncl Journl, 31(3): Leong, E.C. nd Rhrdjo, H Permeblt Functons or Unsturted Sols. Journl o Geotechncl nd Geoenvronmentl Engneerng, ASCE, pp: Ltton, R.L Predcton o movement n expnsve cl. Vertcl nd horzontl deormtons o oundton nd embnkments, Geotechncl Specl Publcton No. 40, ASCE, New York, Vol. 2, pp Hung, Q.V., nd Fredlund, D.G Usng Volume Chnge Indces or Two-dmensonl Swellng Anlss. Proceedngs o the 55 th Cndn Geotechncl nd 3 rd Jont IAH-CNC nd Groundwter Speclt Conerences. Ngrlls, Ontro, pp Tmoshenko, S., nd Wonowsk-Kreger, S Theor o Pltes nd Shells, McGrw-Hll, 580 p. 1 FlexPDE s propretr product o PDE solutons Inc., 2120 Spruce W, Antoch, CA 94509, USA