Dynamic Prediction of Pile Static Bearing Capacity - vulcanhammer ...

tI5114 f b hTarch, 1969 Shl 2TITE THEORETICAL PILE MODEI,Consider the pile mode! of length L shown in Fig. 1, where .Y representsthe rigid body motion of the pile, x the distance coordinate nlong the pile of asystem of ases moving with the pile arid : the elastic deflecklon of the pint vawny from its rest position. Let concentrated forces, functions of time, FT(l)and FB (11, act upon the top and bottom respectively of the pile, and let RL(x, 1)be the distributed resistive force of the soil along the length of the pile.The top force F~(I) = the hammer force transmitted to the pile. FB (f) = aconcentrated reactive force due to the soil resistance; R=(I-, t) is a distributedfrictional force. h the model to be cons!dered here, ~ ~ ( will 1 ) be takenas the actual input force of the hnmmer as measured at the pile top for example,by a suitable force transducer. (See Rcfs. 8 and 9 and Appendix 11.)--'ainwhich E(1) is thc finite Fourier Cosine transform of the rigid body contri- (11..,--% @.- ., .,,1 !tmtion xlti the strrnnlation of [, ternls covers the transfor~ncd elastic contri- ,:;!j:.:.!I : I#,-,-,:: 'bution. When the displacement Z isohtninc4 froni Eq. 2, successfve numerical.... ;::'''.-'.;,,;!:d:.iidifferentinttons yield the theoreticnl velocity and acceleration, respectively. . . ..,411 terms of Eq. 2 arc c.qltcit]y derived in Appendix I. Itwill merely be re- ,,I-,iitcrated here that these terms are all functions of the hamnler force F~(I) ,--~ Ill:SlS rlVE FOIICI:SFB (t) will be !.?ken as zel-o in the present shldy, its cffcct (if any) bclng includedIn the dis!riht+ resistive force liL(.r,t), which will be dlscusscdsubsequently.The basic governing equation for the pile is t!~e elastic wave ec!u:ltiori. Dchilsof this are considered in Appendix I. At the pile to)) (r = 0) the total dynar??icdisplacctnent under the hnnirner blow is e>:presscd by % (0, I)in which Z =A'+ 5 ................................ (1)is the sum 01 rigid body (X) and elastic (!,) con!ril~utior~s. The nct result(solution of the wave equation) may be stated in the formTIME IN MILLISECONDSConin~enls on lllc fonn nssurncd for these laws are also made in Appendix1. Esscntlally, ln lhc ex:~niples Ire:itcd in the prcscnt work, the laws are takenas (a) ;r siniplc dl~tril~l~il Coulonlb rcsistnncc nlonl: thc sicics and 01) =0, i.c. no point bc:~ring.In cnlculntion, thc Coulomb rcsistnncc was taken to be very small until thetlnle 1, , corrcslmild\n~ closely to Ihc 11c:tk bl ttlt: ~ne;tsurc.ci vc.locity responsecurve; thc fill1 reslat;lncc value W:ls cn~ployed thcr.c:tflcr. This full rcsistnnccv;\lue is callrtl C, ;~rvl Llic dctcrnlir~ition of ;i proper \.nluc for it is the prin-