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CHAPTER 2 5 EXPERIMENTAL STUDIES OF FORCES ON PILES by J . R . Morison, J . W. J ohnson and M. P. O'Brien Department of Engineering, University of California Berkeley, Calif. INTR ODUCTION In the design of a pile struoture exposed to surface waves of a given height and period, some of the faotors involved in the problem and studied herein are the sise, shape and spacing of the piles and the mo- ment distribution on uniform and non-uniform piles. Theoretical and ex- perimental
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  CHAPTER   2   5   EXPERIMENTAL   STUDIES   OF   ORCES   ON   ILES   by   J.   R.   Morison,   J.W.   Johnson   and   M.P.   O'Brien   Department   of   Engineering,   University of   California   Berkeley,   Calif.   INTRODUCTION   In   the   design   of   a   pile   struoture   exposed to   surface   waves   of a   given   height   and   period,   some   of   the   faotors   involved  in   the   problem   and   studied  herein   are   the   sise,   shape   and   spacing   of   the   piles   and   the  mo-   ment   distribution on   uniform   and   non-uniform   piles.   Theoretical   and ex-   perimental   investigations   have   shown   that   the   force   exerted   by   surface   waves   on   a  pile   consists   of   two   components   — a   drag   foroe   and   an   in-   ertia   foroe.   he   drag  foroe   is   proportional   to   the   fluid   density,   the   projected  area   and   the   square   of   the   fluid particle   velocity.   he   in-   ertia   force,   inoluding   the   virtual   mass,   is   proportional   to   the   fluid   density,   the volume   of   the   object and the   fluid   particle   acceleration.   The   virtual  mass   is   the   apparent   increase   of   the   displaced   mass   of   fluid   necessary to   account   for   the   increase   in   foroe   resulting   from   the   ac-   celeration   of   the   fluid   relative   to   the   object.   This   factor   is   included   in   the   coefficient   of   mass   term   in   the   foroe   calculations.   The   experimental   and analytical   approaches   to   the   pile   problem   presented   in   this   paper   have   been   based   on   the   total   moment about   the   bottom   of   the   pile   and   the  moment   distribution   over   the   length   of   the   pile.   n   order   to   calculate   a   theoretical   moment   it   is   necessary   to   obtain   from  the   experimental   results   two   empirioal   coefficients   —   a   drag   coefficient   and   a mass   coefficient   (Morison,   O'Brien,   Johnson and   Schaaf,   1950).   he   theoretical   equations   of   total  moment   corresponding   to   the   orest,   trough,   and   still-water   level   positions   along   the   surface   wave   are   used   to   compute   these   coefficients   from the  measured   total   moments   at   the   same   positions.   sing these   coefficients   and   the   theory,   a   comparison   to  experimental   results   is   made   by  comparing   the  maximum   moments,   the   phase   relationships   of   maximum   moments   to   the   surface   wave   orest,   and   oomparing   the   calculated   and   measured   total   moment time   histories.   comparison of   the   coefficients   obtained by   these   experi-   ments   to   other   published   coefficients   obtained   in   different   manners,   some   being   steady-flow  values,   shows   that   the   results   herein   are   of   the   right   order   of  magnitude   but   have   considerable   variability.'   Further   investigation   of   the   problems  would   clarify   the   reasons   for   the   scatter   of   the   ooeffioients.   Using   the   experimentally   determined ooeffioients,   the  moment   distributions   on   uniform   diameter   and   variable   diameter   round   piles   were   computed   and   compared   to   the   measured   distributions.   The   com-   puted   results   are   shown   to   predict   the   moment   distribution   with rea-   sonable   accuracy   for   design   purposes.   x   Errors   occurred   in   Chapter   28,   Design   of   Piling   in   the   Proceedings,   First   Conference   on Coastal   Engineering   and   are   oorreoted   in  the  Ap-   pendix   of   this   Chapter.   340    EXPERIMENTAL   STUDIES   OF   ORCES   ON   ILES   The   effects   of   site,   shape   and   spacing   of   piles   were   obtained   ex-   perimentally*   heltering and   mutual   interference   effects   were   found   for   piles   arranged   in   rows   or   oolumns.   esults   are   presented   in   comparative   form   as  moment   ratios   with   respeot   to   a   single   cylindrical   pile.   enter   piles   in   rows   of   piles   aligned   parallel   to   the   wave   crests   showed   maximum   moments   that   were   higher   than   those   for   a   single   isolated   pile.   he   mo-   ment   depended  upon   the   relative   clearances.   oments   on   piles   arranged   in   oolumns   parallel   to   the   direction   of   the   wave   travel   showed   a   sheltering   effect   on the   oenter   piles   in   the   oolumns   with   moments   less   than   those   for   a   single   isolated   pile.   Moments   on   piles   such as  an   H   -   section   and a   flat   plate   section   were   larger   than   those   for   cylindrical   piles   of   the   same   projected  area.   THEORETICAL   CONSIDERATIONS   The   dynamic   force   on   an   object   in  fluid   moving   with   a   steady-   state   velocity   relative   to   the   object   is   given   by   the   expression   Fri   C D/3 A  u 2   1)   where   Cp   s   coefficient   of   drag*   p   ã   fluid   density.   A   projeoted area   of   object   perpendicular   to   the   velooity.   u   undisturbed   fluid   velooity   relative   to   the   object.   The   coefficient   of   drag   must   be   determined   experimentally.   t   inoludes   the   dynamic   effects   of   friotional   drag   and of   form   drag   resulting   from   the   disturbance   of   the   fluid   in   the   vioinity of   the   body.   In   steady   state   fluid flow the   drag  coefficient   is   related   to   the   flow   by the   Reynolds   number   given  by the   expression   where   Re   i-   2)   D   s   haracteristic   ength   f   he   bject*   y   a   inematio   viscosity   of   he   luid.   Ifhen   he   luid   s   n   on-steady   motion   past   n   bject,   the   c-   celeration   or   eceleration   f   he   luid   n   he   vicinity   f   he   bject   produces   oree   omponent.   Adding   his   oroe   ue   o   he   luid   nertia   to   he   riotional   orce,   the   otal   orce   s   given   by   he   xpression   (O'Brien   nd   Morison,   1950),   F»^C Dy oAu 2   +C M/ oV m ^   3)   where   C M   s   coefficient   of   mass.   V m   *   volume   of   the   displaced   fluid ã   ~   acceleration of   the   fluid   relative   to   the   object.   The   coefficient   of   mass   must   be   determined   experimentally.   This   total   force   does   not   include   any   hydrostatic   forces.   The   system   under   con-   sideration   is   essentially   in a   balanced   hydrostatic   field.   341    COASTAL   ENGINEERING   A   pile,   extending   vertically in a fluid   in   motion  due to   os-   oillatory   waves,   is   in   a   non-uniform   flow   field   with   respeot   to   time   and   to   the   submerged  pile   length.   onsider  a   pile   at   any   instant of   time.   qua-   tion   (3)   must   be   written  in   the   differential   form   and integrated   over   the   pile   length   in   order   to   obtain  the   total  resultant   foroe   on   the   pile.   n   Equation   (3)   the   area   A   is   D dS   and   the   displaced   volume   V m   is   (wD^/i)  dS.   Thus,   the   differential   foroe   on  the   pile   is   given by   the   expression   dF^ ^D^ +   C^iSf   ^ ds   4 )   where   D = pile   diameter.   S   *   distance   above   the   bottom   into   fluid.   Equation   (4)   may  be   integrated   if   CD,   Cy,   and   u,   nd   du/dt   are   known   as   funotions   of   time   (t),   or   the   phase   angle,   and of   the   position S.   Taking   S   ã   (d   +   y   +77)   where   d   s   depth of   still water,   a   depth  below   the   mean   water   surface   to   the   mean   particle   position   (measured   negatively   downward),   and   77*   vertical   particle   displacement  about   the   mean   position,   and  assuming that  the   horizontal   particle   velooity  is   sero   when   7   ã   0,   then   the   horizontal   velooity   and  acceleration   of   the   fluid   in   wave   action   are   given   by   the   expressions   (Stokes,   1901),   Cos   6   5)   and   Sin   8   6)   u t   T -   Slnn   -ã,ã,-   L   where   H   s  wave height.   T   s  wave   period.   L   *   wave   length   8   Zirt/T,   angular   position   of   partiole   in   its   orbit   measured   counter-clockwise   from  the  crest   position   at   time   t   *   0.   The   coefficients   C D   and   C M   depend   upon the   state   of   the   fluid   motion   with  respect   to   the   object   motion*   ittle   is   known about   either   of   the   coefficients   in   aooelerated   systems.   s   a first approximation   they   are   considered   as   constant   with   respect to   time   and   position   to   enable   integration of   Equation   (4).   hus,   C D   and   CJJ   become   overall   co-   efficients.   This   study   is   based   on   the   total   moment  about   the   bottom   of   the   pile,   or   the   total  moment   contributed   by   the   wave   motion   above   any   level,   Si,   above   the   bottom.   his   moment   is   given   by   the   expression   u   „-_,.   27TS   7T   H   Co8h   -TT   T   Sinh^Jl-   du   . m   j>   Cosh^£   2   7TT[   L   dt   $ 2   Sinn   2,rd L   Mi   S s   (S-Si)   dP   7)   342    (8)   EXPERIMENTAL   STUDIES   OF   FORCES   ON   PILES   In   order   to   simplify   the   calculations   of   the   first   few   experiments  made,   it   was   assumed   that   the   wave   elevation above   or   below   mean   water   level   contributed   little   to   the   total moment   about   the   bottom;   hat   is,   77   at   the   surface  was   small   compared   to   d.   ence   in   Equation   (7)   the  wave   surface   S g   is   reduced   to   d   and   S   «   d   +   y.   y   making   the  neoessary   substi-   tutions   into   Equations   (4)   and   (7)   and   integrating,   we   have   Pi   =   -rrp   £JUi   |±C D   kl   Cos 2   9   +   Cjfc,   S£   in   9\   Mi   =   P   ^5^   {   C D   k 3   Cos 2 0   +   C^   2JL   Sin   9   -£^c i|cos 8   Sin*]}   9)   Hie   line   of   aotion   of   the   resultant   total   thrust,   Fj,   above   the   level,   Si   is   given   by   the   expression   Mi   7   * ~   10)   where   ^d.   477^   +   glnh   jrd   .   inh   4^ ki   7Td   \i 16   \   Sinh   rj—   Sinh   TTd_  .   iQh   ^Si   2   ,   u   27r d   Sinh  -r—   (12)   £(lZ±f-   i *mi+   47rd   Sinh   ±2_*   H£i   Sinh   fjrSi  .   Coshi^+Coshi^   (13)   q&L   Sinh   Z£L   -   osh   3p*   2   Sinhifi   Equation   (9)   for   the   total   moment   contains   sine   and   cosine   terms   which  are   functions   of   the   angular   position,   9»   Thus,   a phase   angle   is   indicated   which   depends   upon the   relative   magnitude   of   the   sine   and   oosine   terms,   he   wave   equations   (5)   and   (6)   are referenced   at  a   wave   crest   at   time   t s   0.   he   phase   angle,   /3 of   the   maximum   moment   in   relationship   to   the   wave   crest   is   determined   by  differentiating   Equation   (9)   with   respect   to   9   and   setting   the   results   equal   to   seroj   thus,   27TSi   k2   v   $ .   sin 1   {   Zl3d±LZ±L2L)   16)   ã   TTS1   k v   J   16 >   8HC D U --TT^ -r 343  
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