Boundary element approach to the dynamic stiffness functions of circular foundations

BOUNDARY ELEMENT APPROACH TO THE DYNAMIC STIFFNESS FUNCTIONS OF CIRCULAR FOUNDATIONS E. ALARCON AND J. J. CANO AND J. DOMINGUEZ INTRODUCTION In he pas wen y yea s nume ous in es iga ions which deal wi h he compu a ion o dynamic s i ness (also called impedance) ma ices o massless igid ounda ions wi h di e en shapes ha e been epo ed. Taking ad an age o he simpli ica ions de i ed om axial symme y, ci cula and cylind ical ounda ions ha e been s udied in many cases. Analy ical solu ions o s i ness unc ions o ci cula ounda ions on a uni o m hal -space we e p esen ed o ho izon al and ocking mo ions by Vele sos and Wei1 and o e ical and o sional mo ions by Luco and Wes man.2 The de elopmen o abso bing bounda ies pe mi ed he use o he ini e elemen me hod o s udy he dynamic esponse o axisymme ic ounda ions on soils ha consis o one o se e al laye s based on a igid ock.3"5 An al e na i e app oach o he dynamic analysis o igid ounda ions is he use o bounda y in eg al equa ions which allow o he modelling o many di e en geome ies and soil p o iles. Bo h he di ec and he indi ec in eg al equa ion o mula ions ha e been used in dynamic soil-s uc u e in e ac ion p oblems.6"8 Dynamic s i ness unc ions o igid ci cula ounda ions on uni o m o laye ed soils ha e been compu ed by Apsel9 and Apsel and Luco7 using he indi ec o mula ion in combina ion wi h G een's unc ion o a ing load in a uni o m o laye ed hal - space, de i ed by he same au ho s.9"11 This G een's unc ion is w i en in e ms o in eg al o ms ha include p oduc s o wo Bessel unc ions. Owing o ha , i has o be e alua ed segmen ally and i s compu a ion becomes a he in ol ed. An auxilia bounda y whe e he unknown sou ces a e loca ed has o be de ined.7'9,12 The bounda y elemen me hod, based on he di ec in eg al equa ion o mula ion, has been used o compu e dynamic s i ness unc ions o ec angula 6 and s ip ounda ions in he equency domain13,14 as well as in he ime domain.15 In he p esen pape he bounda y elemen me hod is o mula ed o ime ha monic axisymme ic p oblems using he ull-space poin load undamen al solu ion. The app oach is applied o he compu a ion o he dynamic s i ness unc ions o igid ci cula ounda ions on laye ed iscoelas ic soils. The o mula ion o he BEM o axisymme ic elas os a ic p oblems was p esen ed by Ke manidis16 and C use e al11 They used he ing load undamen al solu ion. Howe e , a closed o m exp ession o he ing load does no exis in elas odynamics18 and when he same o mula ion is ollowed he compu a ions become in ol ed because o he di icul ies in he e alua ion o he undamen al solu ion a each in eg a ion poin . On he con a y, he poin load undamen al solu ion is simple and may be in eg a ed nume ically along he azimu hal co- o dina e wi hou any pa icula di icul y. As an example o he ange o si es o which he p oposed app oach can be applied, h ee di e en si es a e conside ed; a uni o m hal -space, a soil laye on a hal -space, and a soil consis ing o ou ho izon al laye s and a complian hal -space. The nume ical esul s compu ed by he p oposed app oach a e compa ed wi h esul s ob ained by di e en p ocedu es. BOUNDARY ELEMENT FORMULATION FOR ELASTODYNAMIC AXISYMMETRIC PROBLEMS Following he idea o Chapel,19 he ull-space poin load undamen al solu ion is used o sol e ime ha monic axisymme ic p oblemsJ The axisymme ic ep esen a ion o he geome y and ield a iables is main ained. The bounda y o he gene a ing su ace o he body is disc e ized in o line elemen s and he poin load colloca ed a each node. The undamen al solu ion, in e ms o cylind ical co-o dina es, is in eg a ed along he bounda y elemen s o he gene a ing su ace and along he azimu hal co-o dina e. The basic BEM equa ion o ze o body o ces can be w i en in Ca esian co-o dina es as usual: CV+ Tud = U dT (1) whe e u and s and o he displacemen s and ac ion ec o s, espec i ely; U and T a e, espec i ely, he ma ices ep esen ing displacemen s and ac ions p oduced by he poin load applied a poin / and C is he independen coe icien ma ix such ha Ckl = (1/2)<5W when he su ace is smoo h a he poin i. The ela ion be ween Ca esian and cylind ical co-o dina es may be w i en o ec o s u and a a ce ain poin y as u = Qu (2) =Q cylind ical (Figu e 1) and he ans o ma ion Q COS& sind 0 sind 0 cos 0 0 0 1 (3) By subs i u ion o equa ion (2) in o equa ion (1) and p emul iplying by Q, he ollowing equa ion is ob ained: QiT C Q' u^ + J (Q*'-T TQ)ucd (Q*'-TUQ) cd (4) whe e QJ is he ans o ma ion ma ix o he colloca ion poin i. Equa ion (4) can now be w i en as C^+ j Tc ucd uc c d (5) which is he same as equa ion (1) w i en in cylind ical co-o dina es. [ The 9 co-o dina e o he colloca ion poin j is ixed du ing he in eg a ion p ocess and can be se equal o any desi ed alue; o ins ance $' = 0. In such a case T i.T Q UTQ ncos0+ 7 2sin$ !T21cos9 + 22sin9 31cosd+ 32sin0 nsind+ 12cos3 Tx 21sin»+ 22cos» T 31sin9+ 32cos9 T 3 23 33 (6) whe e he e ms Tkl a e he well-known Ca esian co-o dina e componen s o he ac ions in he / di ec ion a he in eg a ion poin , due o a uni poin load in he k di ec ion a he poin i. The same equa ion can be w i en o Uc. x i x 2 Figu e 1. Desc ip ion o co-o dina e sys ems When a ci cula ounda ion unde e ical o o sional mo ion is conside ed, he ield a iables a e axisymme ic. Assuming ha he bounda y is smoo h a i, equa ion (5) becomes 1 2 u i p 3 4 + 2 P ^cS+ ^sd o 31cd + 32s0 o T2ls$+T22c& 0 7-13 0 ^33 d& up "3 "z d 2 P l/nc&+ /12s3 0 U3! cd + U32 sd 0 U21sa+L/22c9 0 l>13 0 /33 dO ', '3 2 d (7) whe e T* is he bounda y o he gene a ing su ace (Figu e 2), s and c s and o sin and cos, espec i ely and a ze o has been placed o he skewsymme ic e ms, whe e he in eg als a ound he azimu hal axis a e null. I should be men ioned ha he o sional and e ical adial mo ions a e uncoupled in equa ion (7) and ha he ke nels o be in eg a ed along d a e known unc ions. When a ci cula ounda ion is unde ho izon al mo ion along Xx o ocking a ound X2, he ield a iables a e o he o m (Figu e 3) u u 3 ulpcos9 ; "id sin » ; lpcosd 3 ld sin $ (8) u2 = ulz cos 3 ; z = Xz cos $ whe e ulp, ul , ul , lp, x</> and j2 a e independen o 9. Ins ead o equa ion (7) he ollowing P (a) (b) Figu e 2. Gene a ing su ace o an axisymme ic domain: (a) gene a] p oblem; (b) su ace ounda ion on a hal -space X 2 X 1 X 3 X 2 Ho izon al X 1 X 1 Rocking Figu e 3. Ho izon al and ocking displacemen equa ion is ob ained: 1^ 2 u IP 0 u i + 2 P 2 nc2d + 12s&cd ns2o- 12sdcd o 0 2 31c29 + 32s9c9 31s2d- 32sdcd , 3 c» 0 33cd dd "IP «1S "l* d 2 P 2 l/n^S+l/jaSacd (/nS29-l/12sdc9 l/i 3 0 0 0 2 l/31c2d+l/32s9c0 l/31s29-U32sdcd U 33 c& c9 dd h. h* *i. d (9) Equa ion (9) can be used o ep esen he adial and e ical displacemen s bu no he azimu hal displacemen o which an iden i y would be ob ained. The ampli ude o a unc ion such as ud = ux a sin 9 canno be compu ed wi h a colloca ion poin whe e he unc ion is ze o. Because o his, a di e en colloca ion poin whe e he shape unc ion —sin 3' has a uni ampli ude (»' 7 /2) is selec ed o he azimu hal equa ion: T o ,«.T Q'-'TQ T°2lc$-T°22sB ^ca+ ^sa ?,cs ?,s9+ ?2c» T°3is$+T°32c$ n 23 7^3 (10) whe e he supe index '0' indica es ha he colloca ion poin is a 8' ep esen s he h ee componen s is 7 /2. The equa ion ha 1 2 id i «u «i. + 2 P * J (7ilC2s+ 12 s9c9)dS 72 ( j^s+ jjsScSjdd -«/2 2 P * I (7-3, c29+T32s9c9)d9 c29+l/I2s9c9)d9 72 (C/?1c2»+{/o2s8c8)d8 -«/2 ["(U" c29+l/32s9c9)d9 J> s23- 12s&cS)d» «/2 ( ?jsa»- ?288o»)d» -«/2 Jo s29-7"32s9c9)d9 s29-l/,2s9c9)d9 72 0 „2 (l/?,s29-l/?2s9c9)d9 -«/2 T I *"" s2»-l/32$3c9)d3 c9)d3 :/2 ( ?3c»)ds -«/2 (733cS)d» u ip «i» u Xx d ^ c9)d9 n (l/?3c»)dS -*/2 (l/33c»)d» IP 'ia 'i. d (11) In gene al, non-axisymme ic bounda y condi ions can be analyzed using a plane model by ep esen ing he ield a iables by a Fou ie se ies along he azimu hal co-o dina e. The se ies is o he o m 00 U P (u*np cos n§ + ulp sin nS) « = o 00 uB = £ (-149 sin n9 + u^cos nO) (12) n = 0 ao U (u*zcos nO + wJJ sin nO) Fl = 0 e ms and a bounda y equa ion may be w i en o he ampli ude o he symme ic and he an isymme ic pa s o each mode. The equa ions o he ampli ude o he e ms o he o m sin n3 a e ob ained using a poin a 0' in as he colloca ion poin excep elemen ha con ains he colloca ion poin . In he nex sec ion, an in eg a ion p ocedu e o cons an elemen s con ained in planes x3 = cons an is p esen ed. This kind o elemen is he only one needed o he analysis o ci cula ounda ions on uni o m o laye ed soils. INTEGRATION OVER THE BOUNDARY ELEMENTS Cons an bounda y elemen s wi h one node pe elemen a e conside ed (Figu e 4). The bounda ies o he ci cula ounda ion p oblems analyzed a e pe pendicula o he X3-axis and he line bounda y co-o dina e T* coincides wi h he adial cylind ical co-o dina e p. Equa ions (7) and (11) can be w i en a e disc e iza ion as 4 N «£ + j-i p Pi [JT.d»]d,>. USAM*™ (13) Massless ounda ion 1 In eg a ion poin Colloca ion • • Colloca ion poin Figu e 4. Bounda y elemen disc e iza ion whe e N is he numbe o bounda y elemen s, pi indica es he segmen ha o ms he elemen j9 and Tc and Uc s and o he ma ices in equa ions (7) o (11). The double in eg als ex end o e one hal o he ci cula c owns ep esen ed by he line elemen s (Figu e 4). The colloca ion poin o equa ion (13) can be ei he a 9l =0 o a & n12 and o each e m he in eg a ion domain is ei he he elemen o which he colloca ion poin belongs (i=y) o a di e en one (i #7). The la e case is analysed i s . The in eg als a e done nume ically using a Gaussian quad a u e o mula a e a special co-o dina e ans o ma ion. The in eg a ion domain is shown in Figu es 5(a) and 5(b), he la e being o he case when he colloca ion poin is a S1 = — i /2. In o de o apply he Gaussian quad a u e he domain is ans o med in o a squa e in he dimensionless co-o dina es (— 1 ^ £ <; 1, — 1 ^ */ < 1). The adius p is ans o med in o by 9 1 2 l(Rx+R2) + i(R2 1 dp = ^(R2-Rx)d i *i)] (14) + A linea ans o ma ion o he same kind was es ed o he co-o dina e 0; howe e , he accu acy o he compu ed alues o he in eg als was poo e en when a la ge numbe o in eg a ion poin s was used. The accu acy was imp o ed, and sa is ac o y esul s we e ob ained, by using a quad a ic ans o ma ion o mula ha inc eases he numbe o in eg a ion poin s in he icini y o he X j-axis o he X2-axis in he cases o Figu es 5(a) o 5(b), espec i ely.20 The ollowing (a) X 1 (c) * 5 (b) Figu e 5. In eg a ion domains and coo dina es when he colloca ion poin is ou side he in eg a ion domain: (a) 9' = 0, (b) $' = - 7 /2 pa abolic ans o ma ion has been applied; n a =^<i + 02 n (15) dd=-(l + £)d£ when he colloca ion poin is a $' = 0 (Figu e 5(a)) and 4 n (16) d9=-(l+£)d£ when he colloca ion poin is a 01 7c/2 (Figu e 5(b)). In his way one hal o he o al numbe o in eg a ion poin s a e loca ed in he qua e o he domain close o he colloca ion poin . The numbe o quad a u e poin s is wo o he adial co-o dina e and wen y o he azimu hal co-o dina e. Thus, he in eg als in equa ion (13) become H*u P pi 2 20 dSJdp- X Z(7T/4)[(/?1+/?2) + nm(R 2 = 1 n=l Rl)1Tc( n.'lm)lR2-RllV+U">*Wm 2 20 G'j = | p Ucd» dp- £ ZinM i^ + R ) Pj L JS J m~lm*l + ^/?2-^i)]Uc(^,0[«2-^i][l+^]> ll> m (17) whe e Tc(^, jj and UJ&,, m) ep esen he alues o he ma ices in equa ions (7) and (11) compu ed a he (n, m) in eg a ion poin and wn, wm a e he weigh s o he Gaussian quad a u e o mula. When he in eg a ion is ca ied ou o e he elemen which con ains he colloca ion poin , a singula i y exis s in he undamen al solu ion and he in eg a ion scheme is di e en . Fi s o all, he pa con aining he singula i y is sepa a ed om he es o he undamen al solu ion o be in eg a ed. This is done by sub ac ing he s a ic undamen al solu ion om he dynamic one. The di e ence is non-singula a any poin o he in eg a ion domain and can be in eg a ed by means o he same kind o nume ical quad a u e o mula, combined wi h he quad a ic ans o ma ion o he co-o dina e $, as be o e. The coe icien s o equa ion (13) o which i==/ can be w i en as whe e H " and G|J a e he s a ic coun e pa s o H*" and G", espec i ely, and H&'= p{ Tc«Tc(s a ic)jddjdp= pj [Tc,dinda|dp (18) 2 20 X ( /4)l(Ri + R2) + im(R2 ~ *i)]TCld aK„ jm) = 1 n=l [«2-«l3[l+^]wnW G&= P< [Uc-Uc(s a ic)]d3Vdp = pi Uc,djnd9Vdp Pi IJB 2 20 l.(n/4 mRl+-R2) + im(R2-Rln = 1 11=1 Uc,din(^,^)[/?2-K1][l+^]wnw (19) whe e TCidin(<^„, m) and UK<6in(£H, jm) s and o he alues a £„, jm o ma ices o he same o m o hose in equa ions (7) and (11) bu wi h he di e ence be ween he dynamic and he s a ic undamen al solu ion ins ead o he dynamic undamen al solu ion (Tkldin = Tkl — ! , (s a ic) and *Aj.din = Uki — (A/(s a ic) ins ead o Tkl and Ukl, espec i ely). Figu e 6 shows he in eg a ion domains and he colloca ion poin s o his case. The p ocess is comple ed wi h he in eg a ion o he s a ic undamen al solu ion. The coe icien s Tu (s a ic) and Ukl (s a ic) a e known in e ms o he dis ance and i s de i a i es (Figu e 6): 2 . D2 = J{p2 + R2 - 2 Rpcos 3) R R + R2 2 /-, = (/? — pcos$)/ 2 = (psin9)/ (20) Figu e 13. Ci cula ounda ion on a mul ilaye ed soil on a hal -space. Bounda y elemen disc e iza ion CONCLUSIONS Dynamic s i ness unc ions o igid ci cula ounda ions on a uni o m o laye ed iscoelas ic soil ha e been compu ed by a bounda y elemen app oach ha makes use o he comple e space poin load undamen al solu ion. This solu ion is simple and easy o in eg a e o e he bounda y elemen s. The p ice paid o using such a simple undamen al solu ion is ha he soil ee su ace and he laye in e aces ha e o be disc e ized. Howe e , he disc e iza ion o he soil su ace and he laye in e aces can be educed o a limi ed zone so ounding he ounda ion and is au oma ically done by he code. App oaches based on G een's unc ions o a laye ed hal -space7,9-11 ha e he ad an age o equi ing only he disc e iza ion o he soil ounda ion in e ace bu ha e he disad an age o dealing wi h a undamen al solu ion ha is no known in closed o m bu in e ms o in eg al o ms ha include p oduc s o wo Bessel unc ions and ha e o be e alua ed segmen ally by elabo a ed nume ical in eg a ion p ocedu es. The p oposed app oach is easie o implemen han hose o he 1.0 k hh 0.5 0.0 1.0 hh 0.5 0.0 V 1 ^—J p^4 i.d k 0.5 0.0 | • j >s>^ | "«».• ^ .2 .4 .6 .8 1 1.2 1.4 .2 .4 .6 .81.1.21.4 1 .0 k 0.5 • 0.0 1 .5 c 1.0 0.5 .2 .4 .6 .8 1. 1.2 1.4 Luco B.E.M Figu e 14. S i ness coe icien s o ci cula ounda ion on a mul ilaye ed soil on a hal space app oaches. The amoun o compu e ime equi ed o sol e a pa icula p oblem is smalle o one case o he o he depending on he numbe o laye s in he soil p o ile. To illus a e he capabili ies o he p oposed app oach, dynamic s i ness unc ions o a uni o m hal -space and wo laye ed soils ha e been ob ained. The esul s ob ained o he hal - space ha e been alida ed by compa ison wi h analy ical solu ions; hose o he laye ed soils by compa ison wi h he esul s o an app oach based on nume ical solu ion o se s o in eg al equa ions and, in he case o a single soil laye on a hal -space, also by compa ison wi h o he bounda y elemen esul s. The compa isons indica e a good deg ee o accu acy. ACKNO WLEDGEM ENTS The wo k desc ibed he ein is pa ially suppo ed by a g an p o ided by he local go e nmen o Andalusia, Spain. The au ho s exp ess hei since e app ecia ion o his suppo . APPENDIX In eg als o he s a ic pa o he undamen al solu ion o e he elemen whe e he poin load is loca ed, a e gi en in his Appendix. Te ms co esponding o e ical o o sional mo ion o he ounda ion (equa ion (7)): J Pi Jo 2/4 +^3-cos0+—cosa-^3-(l+cos29) pdOdp 2/4 7,cos9d8+ /.cos&d& + /?2 J,cos&d9 VI- R j /3(l+cos2d)d3 2 0 (24) whe e A = (l/16); p(l - ) and B = 3 - 4 . G,2,2c,s,= 2| I l/22c>$lpd3dp = 2,4j J !-_+£_ JpdSdp 2/4 B /jCOsSdd + K /3sin2Odd (25) G^C.«=2J , /33c.,,pdddp=2/l [ [*p<&dp = 2AB 1,6$ (26) P i J O J Pi G il _ i%i _ iu _ in __ iu _ in _ ( 12c,s — °13c,s — u23c,s — u21c,« — u31c,s — u32c,* — u (27) «?!*..«-2 J j 13c>4 pd3dp = 2^J ^(pcosd-/?)Jpdadp 2AB | /3cos9d9-.R |/2d9 (28) #5iU=2J 31c,$ pdddp = 2,4 J -^(p-Rcosd)"|pd»dp 2^B| I Add-/? YcosOdS ,• ] £7*ii _ H*" — *7*" — 77*" — 77*" —£7*" _ 77*" 13 llcs — /722c,s — "33c,s — A 12c,s ~ "21c,s — "23c,$ — "32c,s Te ms co esponding o ho izon al o ocking mo ion (equa ion (11)): H- G?M = 2A B /1cos2»d9+ /4cos2»dd 0 JO + R2 J2cos23-R J3(l+cos29)cos3d9 G 2Ci l = 2i4 B Ixsm2&d$ + R2 /2sin2dd»-/? | /3sin29cos9dd L Jo Jo Jo G 22c, s — 2-4 j^ l j" cos2 a da + 7* 73sin2acosada G*2!ic,s = 2i4 73 71sin2ada+ 74sin2ada —7? 73 sin2 a cosa da G3''3c,s = 2/IB 7lCos»dd u13c,sl — u23c,s — u31c,s — °32c.s — u ?3c.. = 2i4£ /3cos2&d9-K /2cos»d9 0 JO /!&,„ = 2,4 J? /3 sin2 Odd J o H i^=-2AB ^cosOdO-K J2cos20d3 /*« #*"c.s = 2/4B/? 72sin29d9 77*ii 77*" 77*" 77*" 77*" A 11 lics — /322c,si — /733c,s "** " 12c.s — n 21c,s — u