Theory. Single Soil Layer. ProShake User s Manual

Transcription

1 PoShake Ue Manual Theoy PoShake ue a fequency domain appoach to olve the gound epone poblem. In imple tem, the input motion i epeented a the um of a eie of ine wave of diffeent amplitude, fequencie, and phae angle. A elatively imple olution fo the epone of the oil pofile to ine wave of diffeent fequencie (in the fom of a tanfe function) i ued to obtain the epone of the oil depoit to each of the input ine wave. The oveall epone i obtained by umming the individual epone to each of the input ine wave. Thi ection decibe the baic mathematic of the poce fo a poblem involving a ingle oil laye, illutate how that poblem can be olved uing a widely available mathematical pogamming language, and ext the appoach to layeed ytem. Single Soil Laye The following paagaph decibe the bai of the analyi ued in PoShake and follow the geneal appoach of Kame (996). That efeence povide ubtantial backgound mateial and a moe detailed deciption of gound epone analye than i peented in thi Ue Manual. A B H To illutate the baic appoach ued in PoShake, conide a unifom oil laye lying on an elatic laye of ock that ext to infinite depth, a illutated to the ight. If the ubcipt and A B efe to oil and ock, epectively, the hoiontal diplacement due to vetically popagating hamonic -wave in each mateial can be witten a u ( t) Ae i ( t k ) B e i ( t k ), ω + + ω () u ( t) A e i ( t k ) B e i ( t k ), ω + + ω () whee ω i the cicula fequency of the hamonic wave and k i the complex wave numbe. No hea te can exit at the gound uface ( ), o τ(, ) (, ) γ (, ) t G t G u t (3) whee G G( + iξ ) i the complex hea modulu of the oil. Subtituting Equation () into Equation (3) and diffeentiating give ( i k ) i t ( ) G i k Ae i k ( ) B e ( ) e ω G i k A B e i ω t (4) which i atified when A B. Compatibility of diplacement and continuity of tee at the oil/ock bounday equie 35

2 PoShake Ue' Manual u( H) u ( ) (5) τ ( H) τ ( ) (6) Subtituting Equation () and () into Equation (5) ik ( H ik H) A e + e A + B (7) Fom Equation (6) and the definition of hea te (τ G u ) o The atio ik ( H ik H) ( ) A i G k e e i G k A B ik H ik H ( ) G k G k A e e A B (8) G k α G k whee α i known a the complex impedance atio. Solving Equation (7) and (8) imultaneouly give [( ) ik H ( ) ik H] A A + α e + α e (9a) [( ) ik H ( ) ik H] B A α e + + α e (9b) If a vetically popagating hea wave of amplitude, A, taveled upwad though the ock and the oil wa not peent, the fee uface effect at the ock outcop would poduce a bedock outcopping motion of amplitude A. If the oil wa peent, the fee uface motion amplitude would be A 4A ( α ) eik H ( α ) + ik H + e 36

3 PoShake Ue Manual The tanfe function, F(ω), defined a the atio of the oil uface amplitude to the ock outcop amplitude, i given by F( ω) ( α ) eik H ( α ) + + e ik H Obviouly, the tanfe function i a complex function. It can be ewitten uing Eule' Law a F( ω) cok H + iα in k () H Solution of Single Laye Poblem Becaue the tanfe function i defined a the atio of the oil uface amplitude to the ock outcop amplitude, the oil uface amplitude can be obtained a the poduct of the ock outcop amplitude and the tanfe function. Theefoe, the epone of the oil laye to a peiodic input motion can be obtained by the following tep:. Expe the input (ock outcop) motion in the fequency domain a a Fouie eie (a the um of a eie of ine wave of diffeent amplitude, fequencie, and phae angle). Fo an eathquake motion, thi Fouie eie will have both eal and imaginay pat.. Define the tanfe function (Equation ). The tanfe function will have both eal and imaginay pat. 3. Compute the Fouie eie of the output (gound uface) motion a the poduct of the Fouie eie of the input (bedock) motion and the tanfe function. Thi Fouie eie will alo have both eal and imaginay pat. 4. Expe the output motion in the time domain by mean of an invee Fouie tanfom. Thee tep ae coded into a pogam uing the mathematical poceing pogam MATLAB in the box located below. The yntax of a MATLAB pogam i imila to common language uch a FORTRAN and BASIC, but MATLAB contain high-level function that allow many complicated calculation and gaphic command to be poceed in a ingle line of text. The MATLAB pogam i well-commented, and hould be elatively eay to follow. 37

4 PoShake Ue' Manual % EX73.M - A MATLAB cipt fo computing the eimic epone of a unifom % damped oil laye on elatic bedock. Input data coepond to % Example 7.3 in Kame, S.L. (996), Geotechnical Eathquake % Engineeing, Pentice Hall, 653 pp. % height54; % oil laye thickne v5; % oil hea wave velocity v5; % ock hea wave velocity ho_5; % oil unit weight ho_6; % ock unit weight x.5; % oil damping atio x.; % ock damping atio load ge.dat % load input motion nlength(ge); fo j:n+ a(j)ge(j-)/98; a().; dt.; df./(ndt); t.:dt:ndt; f.:df:ndf; afftfft(a)/n; abfftab(afft); fo j:n/+ b(j).abfft(j); ff(j)f(j); % hift and convet input motion to g % time tep % fequency incement % et up time vecto % et up fequency vecto % et up ingle-ided FAS alpha_(ho_v(+ix))/(ho_v(+ix)); % complex impedance atio h().; fo j:n/+ kh(j)pif(j)height/(v+xiv); h(j)./(co(kh(j))+ialpha_in(kh(j))); h(n+3-j)conj(h(j)); fo j:n/+ hab(j)ab(h(j)); ubplot(5,,) plot(t,a) ubplot(5,,) plot(ff,b) ubplot(5,,3) plot(ff,hab) fo j:n+ acc(j)afft(j)h(j); fo j:n/+ acc(j).ab(acc(j)); ubplot(5,,4) plot(ff,acc) atimeneal(ifft(acc)); ubplot(5,,5) plot(t,atime) % wave numbe x thickne % left half of tanfe function % ight half of tanfe function % modulu of tanfe function (fo plotting) % plot input motion (time domain) % plot FAS of input motion (fequency domain) % plot modulu of tanfe function (feq. domain) % compute output motion in fequency domain % compute FAS of output motion % plot FAS of output motion (fequency domain) % plot output motion (time domain) The vaiable defined in the fit pat of the MATLAB pogam coepond to Example 7.3 in Kame (996). Thi example conide the epone of a 54 ft thick oil laye of oil (v 5 ft/ec, γ5 pcf, and ξ5%) ovelying bedock (v 5 ft/ec, γ6 pcf, and ξ%). The MATLAB pogam geneate the plot hown below. 38

5 PoShake Ue Manual The fit of thee plot how a time hitoy of acceleation of the 4-ec input motion in the time domain. Immediately below thi i the Fouie amplitude pectum of the input motion - the Fouie amplitude pectum how the vaiation of amplitude with fequency fo each of the fequencie in the Fouie eie. The abcia of the econd, thid, and fouth plot ae fequency in H. The thid plot how the modulu (the quae oot of the um of the quae of the eal and imaginay pat) of the tanfe function. The tanfe function i clealy een to have a eie of local peak that illutate the natue of amplification that will take place at the natual fequencie of the oil laye; note that the geatet amplification will take place at the lowet natual fequency (whee the tanfe function eache it global maximum). The fouth plot how the Fouie amplitude pectum of the output (gound uface) motion which i numeically equal to the poduct of the input motion (econd plot) and the tanfe function (thid plot). The oigin of thi pectum i clealy een by compaing the econd, thid, and fouth plot - the peak in the fouth plot (the output motion) ae elated to the peak in the econd plot (the input motion) and the thid plot (the tanfe function). Finally, the lat plot how the output (gound uface) motion in the time domain a obtained by taking the invee Fouie tanfom of the output motion in the fequency domain. Multiple Soil Laye The baic appoach decibed in the peceding ection i alo ued to analye layeed oil depoit in PoShake - the only diffeence i that the tanfe function i diffeent fo a layeed oil depoit. The tanfe function fo a layeed oil depoit mut account fo the tanmiion and eflection of wave at boundaie between adjacent laye, much a thoe facto wee accounted fo at the oil/ock bounday in the peviou ection. 39

6 PoShake Ue' Manual Conide the oil depoit hown to the ight. Within a given laye, ay laye j, the hoiontal diplacement will be given by h h G G x x ik ik (, ) ( j j j j j ) u j i t j t A j e + B e e ω () At the bounday between laye j and laye j+, compatibility of diplacement equie that ϕ ϕ+ N N+ h ϕ hϕ+ h Ν Gϕ x j j Gϕ+ x j+ j+ GN x N N G x N+ N+ N+ j j j j A ik h ik h j+ + B j+ A j e + B j e () Continuity of hea tee equie that G j k j ik h A j B j A je j B ik h j e j + + G j+ k j+ (3) Note that Equation () and (3) ae analogou to Equation (7) and (8), epectively. Defining α j a the complex impedance atio at the bounday between laye j and j+, the wave amplitude fo laye j+ can be obtained fom the amplitude of laye j by olving Equation () and (3) ( ik h ) j( j) ( ik h ) j( j) A ik h j A j j e j j B e j j + + α + α (4a) B ik h j A j j e j j B e j j + α + + α (4b) At the gound uface ( ), the equiement that the hea te mut be eo mean that A B. Applying Equation (4) ecuively fo j,, 3,, N, the coefficient A j + and B j + can be elated to A j and B j by A j+ a j+ ( ω ) A (5a) B j+ b j+ ( ω ) B (5b) whee the function a j+ ( ω ) and b j+ ( ω ) epeent the effect of the wave inteaction that take place at all of the laye inteface above laye j+. Then, a tanfe function elating the motion at the top of any two laye, ay laye i and j, can be expeed a 4

7 PoShake Ue Manual Fij ( ω) ai ( ω) + bi( ω) a j ( ω) + b j ( ω) (6) Thi tanfe function can become quite complicated, but it i ued in exactly the ame way a the much imple tanfe function developed fo the ingle laye cae. In fact, the MATLAB pogam that illutated the ingle laye cae could be ued to compute the epone fo a multilayeed poblem by changing only one line - the line whee the tanfe function i defined (with the comment % left half of tanfe function). Equivalent Linea Analyi The nonlinea and inelatic behavio of oil i well etablihed in geotechnical engineeing. The nonlineaity of oil te-tain behavio mean that the hea modulu of the oil i contantly changing. The inelaticity mean that the oil unload along a diffeent path than it loading path, theeby diipating enegy at the point of contact between paticle. Rigoou analyi of the mechanical epone of oil to any type of loading, dynamic o othewie, would equie that the te-tain behavio of each element of oil be tacked diectly in the time domain. The method of analyi ued in SHAKE (and PoShake) cannot allow fo nonlinea te-tain behavio becaue it epeentation of the input motion by a Fouie eie and ue of tanfe function fo olution of the wave equation ely on the pinciple of upepoition - which i only valid fo linea ytem. To appoximate the actual nonlinea, inelatic epone of oil, τ an equivalent linea appoach can be utilied. In the equivalent G ec linea appoach, linea analye ae pefomed with oil popetie that ae iteatively adjuted to be conitent with an effective level of hea tain induced in the oil. In the equivalent linea appoach, the hea modulu i taken a the γ ecant hea modulu which, a hown to the ight, appoximate an aveage hea modulu ove an entie cycle of loading. A the level of hea tain inceae, the ecant hea modulu deceae. The elationhip between ecant hea modulu and hea tain amplitude can be chaacteied by mean of a modulu eduction cuve. The natue of thi cuve, which ha an odinate of modulu atio ( G/G max ) and an abcia of log (hea tain), ha been well etablihed fo many oil. PoShake ha a libay of modulu eduction elationhip that can be elected in the Input Manage. The olution algoithm ued in SHAKE (and PoShake) aume vicou oil damping which it epeent uing a complex hea modulu. Vicou damping implie behavio that would be chaacteied by elliptical te-tain loop. Becaue actual te-tain loop ae eldom elliptical, an equivalent damping atio i ued - the equivalent damping atio i equal to the damping atio that would be computed baed on the aea within the hyteei loop, the ecant hea modulu, and the maximum hea tain. The elationhip between thi equivalent damping atio and hea tain i chaacteied by mean of a damping cuve. The natue of thi cuve, which ha an odinate of damping atio and an abcia of log (hea tain), ha been well etablihed fo many oil. PoShake ha a libay of damping 4

8 PoShake Ue' Manual cuve that can be elected in the Input Manage. In an equivalent linea analyi, the fit iteation i pefomed uing hea modulu and damping atio that coepond to ome initially etimated level of hea tain. In PoShake, the fit iteation i baed on an aumed hea tain of.%. Following the fit iteation, the effective hea tain, defined a γ eff Rγ γ max whee R γ i a tain eduction facto often taken a R γ M i computed. The hea modulu and damping atio coeponding to γ eff i then ued fo the next iteation. Thi poce i epeated until the computed effective tain doe not change much fom one iteation to the next. At thi point, the equivalent linea poce i aid to have conveged. While the equivalent linea appoach allow the mot impotant effect of nonlinea, inelatic oil behavio to be appoximated, it mut be emphaied that it emain a linea method of analyi. The tain-compatible hea modulu and damping atio emain contant thoughout the duation of an eathquake - when the tain induced in the oil ae mall and when they ae lage. Pemanent tain cannot be computed and poewate peue cannot be computed. Howeve, the equivalent linea appoach ha been hown to povide eaonable etimate of oil epone unde many condition of pactical impotance. 4