%(as cxcn~plificd, my, by the velocity rcsponsc) arc brou~ht out rnthcr clcnr:y,and tiir.cct !)tiysical intc:pretations <strong>of</strong> thc output h;~ve bccrl possible.Through studies <strong>of</strong> the type pictured in Fig. 9, the rcl;~tivc rolcs <strong>of</strong> clnsticard rigid hndy actiLrlty nre distinguish:d. For both !on7 and short piles thevelocit:, pr<strong>of</strong>ile atpile top- rf?ve2l~first a period <strong>of</strong> strong elastic compression<strong>of</strong> the entire pilc, followed by a rigid M y deceleration as the compressedpile penetrates the ground; upon this deceleration pr<strong>of</strong>ile arc superimposedelastic oscillations <strong>of</strong> a 'minor" nature probably originating in the region <strong>of</strong>cap block-linnjmer interaction. (They are not due to w:lvcs ~unning up 2nddown thc pilc). In the case <strong>of</strong> long length, the pile 'compresses up* like a longspring under steady load, up to the point <strong>of</strong> rnasinium vclocity; thcn thc comprcsscd~)iiedescends into the soil more or less lilte a rigid tmdy. In the cxse<strong>of</strong> the short pilc, the rigid body action is the pr-incipal one, \vith only a relntivelysmall clarjtic effect exhibiting oscil1atio:ls about a niexn much Iiearerto zero. i"11esc considerntions esplnin \vhy the rizid botiy model <strong>of</strong> pile activit;(9) p~-ol,c(l adccjuate for a period <strong>of</strong> time in csplainin:: short pile ztctivitybut failcti to cs1)lain long pile activity atlcquntcly. Both cases nrc reasonnbly11nrdlc4, however, by the siniplified averngirlg nlctllods outlined above.For. the autliors, tlie present study has rcsu1tc.d in fnmiliariznlion with tlier~pitlly passing response pr<strong>of</strong>iles actually occurri~l; over a few tcns <strong>of</strong> n;illisecondsafter the han~n~cr inipnct. These rapidly cvolving pr<strong>of</strong>iles are no\rVboth bclievzblc and physically c:q)l:tin;tbls.Two particular facets <strong>of</strong> pile rcsponsc thnt h;~vc proved irltercsting, andpcrhnps unc:.:pcctd, are thc following:First, the onsct<strong>of</strong> the vclocitv response is a curve resembling the form 01thc curve <strong>of</strong> FT(!). That tllis is not merely coincidcntnl arid in fnct thnt the:wo cl;rves 21-c re1ati.d by :I constznt<strong>of</strong> pr.oportion;llits C,'-!:I is czsily denionstrable(8) when the responsc to I.-?(!) <strong>of</strong> an infinitely long elastic pile withno resistance is calcuhtcd. (A convenie:lt device for this d~rnonstration isthc us:: <strong>of</strong> the Laplace transform in solving tl~c elastic w;i\.c cquntio11 <strong>of</strong> thepile.) Thus, thc c:lrly part <strong>of</strong> the velocity currvc rc)~rcsents 'rvhat is going onc.iastic:iily ljcforc the soil rcsistnncc takes tiold.' R~~.ther. tvller~ ti;? cx~ierimcl:t:~llj.mcnsurci velocity 2nd ~ ~ . ( curvcs t ) part cornl~ari:; ti;< crfect <strong>of</strong> soilresistance c;idrntly has begun to bc fclt. Th~s occurs usually :it or near tiicxnln.?imum velocitv pe.llc, at :I tinie I, = L/C.Send, the pile wave sent clown Ilv the h71n!?,er blow.ci~ss.qntr.: . in I. themost pnrt, dircctly into tl~c soil on the tiownward pnssngc, anci little reflection<strong>of</strong>%c ;varVe occurs. cs unoer ttiese constderations thnt Lhc j~i~c mny oc vtsualizcdas -cornpr.esscd clown" and to 'actas n rigid LMtiy" for a tinie tlierealter(Fig. 9).The emphasis in the ,present paper has becn on thc prtdiction <strong>of</strong> dynnmicresponse for thc purpose <strong>of</strong> predicting static bearing capacity. Driving hammeriorces, as nvallablc, 11avc becn assumed adcc!unte to the job; hence noguidelines arc est:iblished in this work for deternlinin~: tllc necdcd niagnit~~fe<strong>of</strong> available driving force or energy. Ilowcvcr, it is clcar from recoids likethose shown in Fig. 2, thlt good knowldge <strong>of</strong> developed driving forces cnn beobtained by tllc methods us& and corrclatctl with othcr pile irJormation,such as displncemcnt, to detcrrninc d rivlng cn'eriy and required hamnicrc!iaractcristics.?'he hnmnlcr forccs shown in Fig. 2 nrc tliosc for :I p;irticulnr 1i:tnirnerimpact <strong>of</strong> cnch pile consiclcrcd in this jl:Ipc?r. A rccoid~d hhmrner force was-3ialtiercic n~casurcmer~tirnrler all conceival)le clrcu~nstances.It docs, however, prcsenl more than coincidental evidence that such a problcrnmaj. bc Well-posed and that continu& cffort In th!s direction can resultin even cvccllcnt and rapid dynnnlic predictions <strong>of</strong> pile static bearing capacity.The full rangc <strong>of</strong> applicnbility <strong>of</strong> the method as well as its limitations arcobjccts oi continuing study by the \vriters.A portion <strong>of</strong> thc thcorctical work and the entire ex~~crimcntal work rel?ortedhere wns sponsored at the former Casc Institute <strong>of</strong> Tcch~ology by theOhio Highway 1)cpnrtmcnt and the U.S. Bureau <strong>of</strong> Public Roads. The opinions,find~ngs, and conclusions cvpresscd in this paper arc those <strong>of</strong> the writers andnot necessarily <strong>of</strong> the Shtc <strong>of</strong> Ohio or the i3urcnu <strong>of</strong> F'ubiic Roads.he writers arc lizrtfculnrly rndebtcd to the rocicwcrs <strong>of</strong> tlie origin31manuscript <strong>of</strong> thc pnper for thcir mxny constr,ictive sagzestions and comments.APPENDIX I.-THEORY OF THE EL.AfT1C PILE---The basic Eovernink: equ:ttioti is :he clnssic;c!, onr-dimcnsionni elnsticwnvc ecluatioii, \v~-ittrr, for t!lc pile \rith n icsist:i:~cr! tcsrn RI /(il E) includedin :r.hic!l c = :.f:r$ i.? thcs o.:ivc vclocit:; in tlli. pile. n;::lcri:ll, 1 is .the ])ilc materi:iicicnsity, I: is tlic riioduli~s <strong>of</strong> 01:lsticily <strong>of</strong> tkis 1:1;1tcri;~l, :tncl ..\ is thc pilecr-oss-scctiori:ll :Ire:t. (I'ilc tic:td wcic!~f Is ri~.!:!(.c.tc.~! In Eq. 5. It is :I ric(!ligib!cfraction <strong>of</strong> thc u1tiril:itc 11ilc st;ltic c:~]jac:ity).Tl~c bo~~ritinry contiit inns :trc,\ ,; 5 - F- (.I = 0) (! .r 0) .................. (Gn)n ,. y41)'I'!le initial co~~ditions arc quiescent.Y(O) + SLY, 0) = 0 (0 := .r 5 L) . ................. (70)
tile cove rning cquation becotnes .k- 3In particular, at x= 0, the pile topNow let the soil resist'ance take the formn,(x,O = R,(.Y,!) + D i(7,i) ........................ (10)in which D = d A E ................................. (11)for an appropriate 'damping coefficient" (I to be defined latcr.Taking the Fourier cosine trznsiorm <strong>of</strong> Eq. 9 yieldsThe folm and application <strong>of</strong> the soil resfstance force law gixrcn by Eq. 10needs further e1ucid:ltion. 111 the presctlt worlr the constant contribution %(x, 1):rill be defined and applied as an uactive" force as followsin which the barred quantities together 'vith the subscript c rcprescnt the finiteFo~lrier cosine transform- r L71- Y%, (11, I ) = J z ( s, 1) cos - A\- ...................... (13)0 LIt is retnal.ked in pnssing that t!l~ rosine transform is e~:iplo:rcci bc~nusc <strong>of</strong>its approprintcness to thc presrnt bounti:lry contiitions.Solutions <strong>of</strong> Eq. 12 is given by- - : II(,l, T) c-?'(t-+.......Z, (?I, 1) = & (91, t = I ---------- sin ;3(1 - 7) (!T (14)for )I = 1, 2, 3, ...cZ/lin which : = - 20 ,'jin which C, is constant for all ,Y 2nd I 2 1,. The tlme I, will be arbitmrilytnken as f, 2 L/C thc time for an elnstic wave to tr:lvel fro111 top to bttonl <strong>of</strong>the pile; however thc exact value c!loscn for !, cnn be considered a parameter<strong>of</strong> the solution as used here.nnc! ci is to be defined aswith [ = ratio <strong>of</strong> damping to critical, so th.it e:xc'h mode <strong>of</strong> the response h:isthe same da~nplny rat10 5:For the case ?I = 0 (Eq. 12) yields the rigid bdy contributlonfrom which Tc and yC are obtnined by successive time integrations.The net solution for Z(,Y, f), the total tl1sl)laccmcnt 01 a pint on the pile,is given by*0- -117vZ(U.I) = "'(') + 2 1 {,(11,!) cos - .................. (16)L LWtlile many definitions <strong>of</strong> i;,(y, 1 ) tll~gtit 1.1~ nl:~de, 2nd iurt!icrcs;~erin:entationin this direction is vcry wortll;vl~ilc, thr: ;il,ove definition has provenadequate in c.unmples :;tt~tlired to date. In othcr :-:ords, a tot:\l stirtic soil rcsistance<strong>of</strong> arnount C, is nssunicd tobc,!:in actink: inst:intly oncc time 1, = 1.l~hasbeen rcnchcd. We llotts that sotnr: 1:iw <strong>of</strong> resistnllcc G'!lich tlcfincs the time<strong>of</strong> coming into actiorlis necess:iry in th~..~~r~set~tcont(~xt; Tills si~i~lehas worked well in : variety <strong>of</strong> cs;lnll)lcs. Under t!lc stated :~ssurilption theresistance R,(y, :) 11;is tlic forni indic:~trd In Fig. 11.One slight n~odiiicatiotl <strong>of</strong> thc :ibovc schrnrc for rcsistance I
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