Amrican Journal of Applid Scincs 4 (1): 95-956, 7 ISSN 1546-939 7 Scinc Publications An Invstigation on th Effct of th Coupld and Uncoupld Formulation on Transint Spag by th Finit Elmnt Mthod 1 Ahad Ouria, 1 Mohammad M. Toufigh and 1 Ali Nakhai 1 Civil Enginring Dpartmnt, Univrsity of Krman, Krman, Iran Abstract: -D transint spag bnath a dam as invstigatd by th finit lmnt mthod. Govrning quations r considrd in coupld and uncoupld mthods. At first stag, th fluid continuity quation for comprssibl porous mdia considrd as uncoupld. Bcaus of th occurrnc of spag forcs, and thir intraction ith th fluid por prssur, particularly in high comprssibl soils, uncoupld mthods sm to b far from ral conditions. Thrfor forc quilibrium quations r takn into account by coupling ith th fluid continuity quation. Finit lmnt formulation basd on Galrkin mthod. Rsults of to mntiond mthods r compard ith stady-stat spag rsults. Comparing th rsults of coupld and uncoupld modls shod that th tim rquir to rach stady-stat condition in coupld modl considrably shortr than th uncoupld modl. Coupld analyss shos that th ffctiv strss du to spag forcs r smallr than ons calculatd by th uncoupld mthod. Ky Words: Transint spag, coupl, uncoupl, F.E.M INTRODUCTION Spag problm is on of th most important issus in th dsign and construction of dams and hydraulic structurs. Spag forcs chang strss stat, dformation and chang prmability of soil lmnts hich ould affct th amount of th spag rat and stability of structur. In routin analyss, stady-stat conditions commonly ar considrd ignoring th tim rquird to rach th stady-stat condition. But actually, moving atr trough soil voids is a tim consuming procdur. Incras of th atr tabl hight in a dam rsrvoir is gradual and tim dpndnt, so transint analysis is ssntial to yild rliabl rsults. In addition, in uncoupl systms thr is only on dgr of frdom in th govrning quation hich is th hydraulic potntial. So in ordr to dtrmin dformations of th structur du to spag forcs it is ssntial to considr lmnt quilibrium quations bsids th fluid continuity quation. In this cas a st of partial diffrntial quations including th continuity and quilibrium quations must b solvd at th sam tim. Considring spag problm in th coupld form is similar to th coupld consolidation problm cpt in loading hich contains hydraulic potntial only. In this rsarch th spag problm is invstigatd by to coupld and uncoupld mthods. Biot s coupld consolidation quations ar adoptd to invstigat th spag problm along ith th traditional uncoupld transint spag quation. Galrkin mthod is usd to finit lmnt formulation. F.E.M formulation for uncoupld systm: Fluid flo through porous mdia is govrnd by hydrodynamic quations considring th intraction of th fluid in motion ith th porous mdia nsuring th continuity of th fluid. Continuity is nsurd by rquiring that th nt volum of atr floing pr unit of tim into or out of an lmnt of soil b qual to chang pr unit of tim of th volum of atr in that lmnt. Diffrnc of th quantity of th atr that lavs or intrs to an lmnt is qual to chang of th lmnt volum. Thrfor th uncoupld fluid continuity quation in th transint stat ould b as follos [1] : v v y (ndv) t (1) Whr; v and v y ar flo vlocitis in horizontal and vrtical dirctions rspctivly and, n, is th porosity. Th abov quation can b rrittn in th folloing form: k u k y u u mv t () Corrsponding Author: Ahad Ouria, Civil Enginring Dpartmnt, Univrsity of Krman, Krman, Iran, Phon: 98914311564 95
Whr; k and k y ar prmability cofficints in horizontal and vrtical dirctions, u is th atr prssur and m v is th comprssibility cofficint of th soil. Equation, is a -dimnsional consolidation P.D.E hich can b solvd by th finit lmnt mthod. Using linar shap functions ith rctangular lmnts, th final form of th Equation discrtizd by th F.E.M ould b as follos [] : KP du KPu PM dt (3) Whr; N N ( k ky ) ddy (4). And: PM Ni N jd. dy (5) In th abov quations, N i is th linar shap function. Using th Crank-Nicolson s mthod, tim intgration of quation 3 involving linar intrpolation and fid tim stps t ould yild unconditionally stabl and convrg rsults according to Equation 6 [3] : PM t KP u ( t t ) t PM KP u Am. J. Applid Sci., 4 (1): 95-956, 7 ( t) (6) Aftr finding govrning matri quations for a singl lmnt, th assmbld matrics for all th lmnts can b obtaind and boundary conditions can b introducd. Thrfor valus of unknon variabls can b calculatd at tim t t t basd on knon paramtrs at tim t t. Th initial conditions at tim t all ar knon. F.E.M formulation for coupld systm: Biot formulatd th thory of coupld solid-fluid intraction hr th soil sklton is tratd as porous lastic solid and th laminar por fluid is coupld to th solid by th conditions of comprssibility and of continuity. Thus Biot s govrning quations ar combination of Equation 1 and lmnt quilibrium quations. For to-dimnsional quilibrium in th absnc of body forcs considring spag forcs, th gradint of ffctiv strss must b augmntd by th gradints of th fluid prssur as follos [] : 951 σ τ σ y z τ z u u (7) Th strss-strain rlations basd on gnralizd Hook s la for plan strain condition can b rittn as follos [4] : G σ G ε K δ ( ε11 ε ε 33) (8) 3 Whr, K and G ar bulk and shar modulus rspctivly. Continuity quation can b obtaind using Equations 1 and hr th volum chang of th lmnt is rittn in trms of displacmnt componnts: k u k y u d dt u v ( ) (9) Whr u is dfind bfor, u and v ar horizontal and vrtical componnts of displacmnts. As is usual in th displacmnt mthod in solid mchanics, strss and strain ar rplacd ith displacmnt componnts so, final coupld variabls ar th por atr prssur, and horizontal and vrtical displacmnts. Sam as th uncoupld mthod, using th linar shap functions ith rctangular lmnts for th solid body and th por fluid, th final form of th st of P.D.E hich is combining Equations 7 and 9 discrtizd by th F.E.M can b as follos [] : KMr Cu C T dr dt KPu (1) Whr: KP is alrady dfind and, r is th displacmnt vctor [u,v]; and also KM and C ar as follos: KM KMy KM T KMy KMyy Whr: N N KM ( R11 R33 ) ddy KMy KMyy ( R 1 ( R R33 R33 ) ddy(1) ) ddy (11)
Am. J. Applid Sci., 4 (1): 95-956, 7 In th abov quations, R is th strss strain rlationship matri according to Equation 8. And also: C C (13) Cy Whr: j C Ni ddy (14) j Cy Ni ddy For intgration of th Equation 1 ith rspct to tim, Crank-Nicolson mthod is implmntd, thrfor: θkmr θcu 1 θc r θ T 1 1 tkpu ( θ 1) KMr ( θ 1) Cu T 1 θc r θ( θ 1) tkpu (15) In abov quations, ifθ 1, th systm ill b absolutly stabl ithout any oscillatory rsults. Thrfor th final form of th Equation 15 in fully implicit typ of tim-intgration ill b as follo: KM T C C r1 t KP T u 1 C r u (16) Aftr finding govrning matri quations for a singl lmnt, th assmbld matrics for total lmnts can b obtaind and boundary conditions can b introducd. Solving such quations at any tim, horizontal and vrtical dformations (u,v) and th fluid prssur at various nodal points can b found and strain valus for ach lmnt can b calculatd. Aftr calculation of primary unknons, scondary unknons such as flo flus, vlocitis and ffctiv strsss ould b dtrmind. Calculation of th ffctiv strss in coupld modl ould b don using th Equation (8) aftr calculating nodal displacmnts. In th cas of th uncoupld mthod, bcaus of th absnc of displacmnts, classical formulas ar usd to dtrmin spag forcs. Vrtical and horizontal componnts of ffctiv strss du to th spag forc ould b calculatd as [1] : σ and z ' h v ( ) z σ ' h dz h d h, c c (17) (18) In Equation 18, c is th position in hich th horizontal gradint of flo has its maimum valu. Thrfor horizontal spag forc in lft and right sids of c has qual of amount and dirction. So th horizontal spag forc in th location of c is zro. Location of c in ach lvation is approimatly at th cntr of th dam and varis slightly. In th lft sid of c (upstram) th horizontal componnt of th ffctiv strss is tnsil and in th othr sid is comprssiv. RESULTS AND DISCUSSION For both coupld and uncoupld formulations, sparat finit lmnt cods ar dvlopd. In ordr to compar th ffct of th coupld and uncoupld formulations on th spag calculations, th transint spag bnath a concrt dam in invstigatd by both mthods. Th gomtry of th finit lmnt modl and matrial proprtis ar illustratd in Fig.1. Watr tabl in rsrvoir riss to 3 mtr gradually during 9 days. At first stag of this study, dvlopmnt of th flo nt for coupld and uncoupld analysis is invstigatd. Th rsultd quipotntial lins for coupld and uncoupld analysis ar shon in Figurs and 3 rspctivly. h EmPa ν.5 6m 4m 6m.5 k.1 m day ky.1 m day Fig.1: Finit lmnt msh and matrial proprtis 95
Am. J. Applid Sci., 4 (1): 95-956, 7 Fig.: Watr prssur distribution rsultd from coupld analysis Fig.3: Watr prssur distribution rsultd from uncoupld analysis It can b sn from Fig. that th atr prssur distribution achivs to stady-stat condition aftr 6 days approimatly. Also a symmtrical flo nt is obsrvabl. Equipotntial lins ar moving to th ground surfac as th tim passs. Incras of th atr prssur in don stram is du to spag forcs inducd in th upstram and this is bcaus of th coupling ffct. Bcaus th spag forc in th upstram acts as a comprssiv load and comprsss th soil. In th othr hand lo prmability of th soil sklton dlays th atr prssur dissipation procss in th don stram and incrass th atr prssur and also dcras ffctiv strss. Fig.3 shos th flo nt dvlopmnt bnath th dam rsultd from th uncoupld modl. Equipotntial lins ar moving toards don stram as th tim passs. It can b sn in Fig.3 that aftr 15 days th 953 stady-stat condition is not rachd. Also in th cas of uncoupld modl, unlik th coupld modl th flo nt is not symmtric until th stady-stat condition. Outlt flo rat from don stram vs. tim is illustratd for coupld and uncoupld mthods in Fig.4. q(m3/day/m).6.4. Coupld Uncoupl 3 6 9 1 15 Tim (Day) Fig.4: Outlt flo rat for uncoupld and coupld analysis
Am. J. Applid Sci., 4 (1): 95-956, 7 Z (m) -1 - -3-4 Fig.5: Vrtical Effctiv Strss (kpa) 4 6 8 1 1 3 6 9 18 36 75 15 Variation of th vrtical ffctiv strss in donstram to from coupld mod As mntiond prviously for th atr prssur distribution in Figurs and 3, it can b infrrd from Fig.4 that th coupld analysis rachs to stady-stat conditions fastr than th uncoupld on. Final valus of th flo rat for both cass ar th sam. In Fig.5, th variation of th ffctiv strss in donstram to du to spag forcs calculatd by th coupld modl is plottd in various tims. Effctiv strss calculatd by th uncoupld modl basd on Equation 17 in th donstram to is plottd in Fig.6. Z (m) -1 - -3-4 Fig.6: Vrtical Effctiv Strss (kpa) 4 6 8 1 1 3 6 9 18 36 75 15 Variation of th vrtical ffctiv strss in donstram to from uncoupld modl Comparing Fig. 5 and Fig. 6 sho that th amount of th ffctiv strss calculatd basd on th spag gradint is ovrstimatd comparing to th coupld rsults. It s du to ffct of th horizontal ffctiv strss hich is omittd in Equation 17. Uplift prssur bnath th dam in various tims, ar plottd in Fig. 7 and Fig. 8 for coupld and uncoupld modls rspctivly. Uplift Prssur (kpa) 3 5 15 1 5 Fig.7: Uplift Prssur (kpa) - -16-1 -8-4 4 8 1 16 3 5 15 1 5 Fig.8: X(m) 3 6 9 18 36 75 15 Day Uplift prssur distribution bnath th dam from coupld modl 3 6 9 18 36 75 15 Day - -16-1 -8-4 4 8 1 16 X(m) Uplift prssur distribution undr dam from uncoupld modl As shon in Figurs 7 and 8, uplift prssur rsults from coupld and uncoupld modls ar diffrnt in amount and distribution in various tims but thy ar qual at stady-stat condition. In th coupld modl flo nt is symmtrical about cntr of dam but in th cas of th uncoupld modl it is not symmtrical cpt at stady-stat condition. In Fig.9, uplift forcs for both coupld and uncoupld analyss ar shon. Uplift Forc (kn) 8 6 4 Fig.9: 3 6 9 1 Tim (Day) Coupld Uncoupld Total uplift forc in coupld and uncoupld modls 954
Am. J. Applid Sci., 4 (1): 95-956, 7 Fig.1: Horizontal and vrtical ffctiv strsss du to th spag forc in th coupld modl Fig.11: Vrtical and horizontal ffctiv strss du to spag forcs form uncoupld modl Vrtical and horizontal componnts of th ffctiv strss du to spag forcs form coupld analyss at stady-stat condition ar illustratd in Fig.1. In Fig. 1, th ngativ sign rfrs to comprssiv strss. Stady-stat ffctiv strss componnts from uncoupld analysis from Equations 18 and 19 ar shon in Fig. 11. Comparing Figurs 1 and 11 shos that th vrtical componnt of ffctiv strss calculatd using th coupld and uncoupld modls hav good corrspondncs. Valus of uncoupld modl basd on Equation 18 ar mor than ons rsultd from th uncoupld modl hich is calculatd basd on calculatd displacmnts. Th horizontal ffctiv Fig.1: Dformd msh from coupld modl 955 strss calculatd using th Equation 19 idly diffrnt from on rsultd from coupld modl. But its distribution and rang hav a maning full rlation ith th coupld rsults. On advantag of th coupld modl is its capability of th calculating of th soil dformations du to spag forcs. It ould b vry important in nonlinar spag analysis ith variabl prmability. Also th coupld modl ould b usd to dtrmin additional strsss inducd in th dam structur du to its foundation dformations. In th coupld mthod, nodal displacmnts ar primary unknons and thir valus obtain from solutions dirctly. Fig.1 shos th dformd msh du to spag forcs in coupld analyss assuming lastic bhavior of soil. In this
Am. J. Applid Sci., 4 (1): 95-956, 7 figur, dashd msh shos initial shap and th displacmnt of th msh is magnifid tims. Th maimum displacmnt of th ground surfac is about.m. CONCLUSION In this papr th ffct of th coupld and uncoupld formulations on th rsults of th confind spag trough a dam foundation is invstigatd. Th transint stat of spag is du to th chang of th atr lvl in rsrvoir and also porous mdia comprssibility. Comprssibility of soil causs th problm to bcom as a consolidation problm. Thrfor uncoupld modl is tratd as consolidation phnomna. Sinc th fluid flo producs spag forcs in th soil body and bcaus of th intraction btn th spag forcs and th por fluid prssur, anothr modl basd on th Biot s coupld consolidation thory is dvlopd to invstigat this intraction. For both cass th finit lmnt tchniqu implmntd using th Galrkin mthod. In th coupld modl, th ffctiv strsss ar calculatd basd on th lmnt displacmnt componnts hich ar primary unknons of th modl unlik th uncoupld modl hr th ffctiv strsss ar calculatd by th classical formulas of th soil mchanic basd on flo gradints in an indirct mannr. Thrfor th ffctiv strss calculatd by th coupld modl is mor rliabl comparing to th uncoupld modl. Th rsults of this study shod that th final valus of th outlt flo rat calculatd using th coupld and uncoupld mthods is qual. Calculatd ffctiv strss by th uncoupld modl is gratr than th coupld modl. Thrfor th safty factor of piping in don stram calculatd by th uncoupld modl is undrstimatd. REFERENCES 1. Karl Trzaghi, Rolph B. Pck, and Golamrza Msri, 1996. Soil mchanics in nginring practic. 3rd d, John Wily, N York, pp 13-41. Smith I. M., and Griffiths D.V., 3. Programming th Finit Elmnt. 3rd Ed, John Wily, pp: 319-44 3. Erik G. Thompson, 5. Introduction to th Finit Elmnt Mthod: Thory, Programming and Applications. John Wily, pp: 15-4. S. P. Timoshnko, and J. N. Goodir, 197. Thory of lasticity. McGra-Hill, pp: 15-33 5. Toufigh, M. M,.. Spag ith nonlinar prmability by last squar F.E.M, Intl J. Enginring., 15(): 15-134. 6. Toufigh, M. M., and B. Shafii, 1996. Finit lmnt consolidation modl for subsidnc problm basd on Biot's thr dimnsional thory. Indian Gotchnical Journal., 6(3): 1-8. 956