Wave Forces on Piles of Variable Diameter SUMMARY

Transcription

1 Wave Forces on Ples of Varable Dameter Bernard LeMehaute Ph.D., M. ASCE James Walker Ph.D., P.E., M. ASCE John Headland, A.M. ASCE John Wang Ph.D., M. ASCE SUMMARY A method of calculatng nonlnear wave nduced forces and moments on ples of varable dameter s presented. The method s based on the Morrson equaton and the lnear wave theory wth correcton parameters to account for convectve nertal effects n the wave feld. These correctons are based on the stream functon wave theory by Dean (974). The method permts one to take nto account the added wave force due to marne growth n the ntertdal zone or due to a protectve jacket, and can also be used to calculate forces on braces and an array of ples. INTRODUCTION Desgn of coastal structures such as pers requres calculaton of wave nduced forces on ples. The basc methodology s based on the applcaton of the Ary theory n the Morrson equaton. Ths has been a very useful tool, but the Ary theory does not account for many of the nonlnear dynamc and knematc effects whch fnte ampltude waves exhbt. Several nvestgators have used varous nonlnear wave theores and correctons to lnear theory. The stream functon wave theory by Dean (965) has been determned to be one of the most orgnally accurate theores (LeMehaute and Dean, 970). It has been used by Dean (974) to determne wave forces on ples. Dean has also used stream functon theory to develop a number of smple graphs whch gve the total wave force and moment about the mudlne on ple of unform dameter (SPM, 977). Many coastal engneerng applcatons requre calculaton of the wave loadng dstrbuton on the ple. On a per, the top of the ple Is n some cases fxed and therefore the moment about the mudlne s not a useful parameter, the desgner requres the load dstrbuton. Furthermore, the geometry of many marne structures are complcated by 2Charman, Dept. of Ocean Engneerng, RSMAS, Unversty of Mam, Fla. Chef Coastal Engneer, Moffatt & Nchol, Engneers, Long Beach, Ca..Coastal Engneer, Moffatt & Nchol, Engneers, Long Beach, Ca. Assocate Prof., Dept. of Ocean Engneerng, RSMAS, Unversty of Mam, Fla. 800

2 WAVE FORCES ON PILES 80 Photograph. Ple of Varable Dameter. the presence of marne foulng. Barnacles, mussels and other marne organsms can sgnfcantly ncrease the dameter of a ple n the ntertdal zone as shown n Photograph. Wooden ples are often protected by, or ther useful lfe extended by, a jacket wrapped around the ple. Both marne growth and protectve jackets ntroduce a varable dameter ple near the water surface. Because the dameter s ncreased near the free surface, wave forces are ncreased for a gven wave condton. Fgure schematcally shows a ple of three dameters whch could represent the stuaton of marne growth on a protectve jacket on a ple. The objectve of ths paper s to present a methodology for the coastal engneer to readly calculate the nonlnear, wave-nduced loadngs on ples of varable dameter. Consderng the large number of parameters, t s not possble to present an exact soluton to the problem n the form of a few graphs. Therefore an approxmate method has been developed n whch the velocty and acceleraton felds, and forces and moments are ntally obtaned at the ple locaton from the lnear wave theory. These parameters are ntegrated from the sea floor nto an arbtrary elevaton, z., whch can be the free surface. The free surface s gven by the nonlnear wave theory of Dean (974). Then correcton coeffcents are ntroduced to account for the nonlnear convectve nertal forces on the velocty and acceleraton felds. By so dong the method s, wth a few approxmatons, amenable to descrpton by a lmted number of graphs. The graphs reduce by an order of magntude the nterpolatons requred and permts the engneer to calculate the load dstrbuton to a hgh degree of accuracy wth a mnmal effort. Furthermore, the ple can comprse multple dameters.

3 802 COASTAL ENGINEERING 982 CREST PILE WITH NONUNIFORM DIAMETER P BOTTOM I m 7tf** I I I I Fgure. Defnton Sketch for a Ple wth Three Dameters.

4 WAVE FORCES ON PILES 803 The force on an array of ples can be determned by calculatng the forces at varous phase angles. The methodology therefore permts the engneer to calculate wave forces on ples takng nto account many of the physcal propertes of waves observed n nature and the laboratory as well as specal crcumstances whch must be addressed n practcal desgn problems. The force and moment correcton factors are prmarlly greater than unty and therefore wll yeld forces and moment whch are greater than those calculated by lnear theory. Where force and moment correcton factors are less than unty, the correcton was retaned as unty for the sake of conservatsm. FORCES OK CYLINDRICAL PILES: BASIC FORMULATION A cylndrcal vertcal ple subjected to a tme dependent horzontal velocty, u(t), has a force, f, per unt length of cylnder whch s the sum of a drag force, f, and an nerta force, f : t = D+ f =l5 pc D u u + pc m^ () where p s the densty of sea water, D s the ple dameter, u s the partcle velocty, -j s the partcle acceleraton, C s the drag coeffcent, and C Is the nerta coeffcent, m When a cylndrcal ple s subjected to a water wave, one consders that these equatons hold true, provded u(t) s the horzontal component of the velocty feld at the ple locaton as f the ple dd not exst. The deformaton of the velocty feld by the ple, wave dffracton, the effect of the vertcal velocty component, the vertcal acceleraton component and the elastcty of the ple are neglected. The equaton s commonly called "Morrson's equaton". The total force, F, on the ple Is determned by ntegratng the unt forces from the sea floor to the water surface, S. F T " f 0 f dz (2) where subscrpt 6 refers to phase angle. Forces due to nonlnear waves over ples of varable dameter, D,, can be calculated usng equatons () and (2) usng expressons for u and du/dt from the lnear wave theory and adjustng them accordng to results obtaned from the nonlnear wave theory. Nonlnear correctons are taken from the stream functon theory as presented by Dean (974). Two basc correctons are made: the asymmetrc free surface correcton and the nonlnear correcton to the wave feld. Fgure shows a ple of three dameters, D., D and D_. The total force, F_» actng from the sea bottom (z=0j to the elevaton of the free surface (z = S fl ) s gven by:

5 804 COASTAL ENGINEERING 982 F T = V f(d l ) dz * f z 2 f(d 2 ) dz + f z 6 f(d ) dz (3) 3 T j. length actng on ple dameter D.. Graphs can be constructed whch ntegrate unt force and moment from the sea floor to an elevaton z. Then equaton (3) can be wrtten. F T = f Q l f(d x ) dz + / Q 2 f(d 2 ) dz -! Q l f(d 2 ) dz + / 0 6 f(d 3 ) dz - / Q 2 f(d 3 ) dz (4) Ths expresson can be wrtten: F T -F( Dl ) + F(D 2 ) - F(D 2 ) L + F(Vl S - F( VL Z D Z (5) where F(D.) sea bottom to a level z.. Use of equaton (5) requres evaluaton of the force F(D.) or a 2;L ple of arbtrary dameter, D, and elevaton above the bottom, z. Ths s done usng the lnear wave theory wth the approprate correcton coeffcents n order to account for the effects due to the nonlnear wave theory. Ths gves: H 2, ^-f H F( D ) ^PC D D? dk D z _ V pc raf d ~2 K TJZ. *I (6) T T l where H = wave heght, T = wave perod, and d = water depth. ELI., and K., are dmensonless drag and nertal coeffcents respectvely, obtaned by ntegraton from the mudlne to an elevaton z,, and < > n and <J> are correcton factors to the forces obtaned by the lnear theory. <f> s the correcton relatng to u and <J> s the correcton relatng to du/dt. X Accordng to lnear theory: K D lz = SM'Z cos6 cos6 (7) K l z = K IMIZ sln9 W The maxmum values K_. and K TW. ntegrated & from the mudlne to an,. DM'z IM'z elevaton z are:

6 WAVE FORCES ON PILES 805, ^2 2kz. snh 2kz. K I = L_ n + ] (9) Vl'z. 8 d L snh 2kz. J snh 2kd v J l 2 snh 2kz. K TM7 =h^~ rrr^ (0) IM z d cosh kd where k = wave number = 2r/L and L = wavelength. K. and K are gven by fgures 2 and 3, respectvely, as a functon of z/d and a/l. NONLINEAR FREE SURFACE F(D )j at z = S fl requres the knowledge of the free surface elevaton S as functon of phase angle, 0. Ths s done by applyng the free surface gven by the stream functon wave theory by Dean (974). The results are presented n the form of S fl /d as a functon of H/H, d/l and 9, n fgure 4. H, s the lmt wave heght as defnea by Dean (974). The use of fgure 4 s llustrated by an example. Gven d/l =.033, 6 = 20 and H/R =.75 enter fgure 4 vertcally wth a value of d/l =.033. As ndcated by the arrow n the fgure, one proceeds downward untl the lne of H/H =.75 s ntersected. A lne s then drawn horzontally towards the rght untl the 6 = 20 lne s ntersected. Now move vertcally downward to read S Q /d =.256. CORRECTIONS FOR NONLINEAR WAVE KINEMATICS The second correcton to lnear theory s due to the nonlnear wave partcle velocty and acceleraton felds. The correcton coeffcents $_ and (j> T ncorporate a number of nonlnear effects, most notably due to the convectve acceleraton terras. These nonlnear effects are a complex functon of relatve depth, d/l, wave phase angle, 0, the rato of the local heght to breakng heght, H/H,, and the rato of elevaton to water depth, z/d. These are gven by the rato of the forces obtaned by usng the values obtaned by a nonlnear wave theory to the correspondng value gven by lnear theory. To ncorporate all of these varables at each z would requre a large number of nomographs. Therefore a conservatve approach was adopted where the nonlnear correcton factors at the free surface were appled over the water column. Whle the nonlnear effects actually vary wth z/d and 0, the most mportant correctons are near the free surface. Therefore only a global nonlnear correcton s appled over the entre ple length, from the sea bottom to the nonlnear free surface. Also the phase varatons gven by cos6 and sn6 respectvely, are retaned n the general equaton. The error resultng from ths smplfcaton s that forces are generally over-predcted by a few percent at levels below the free surface. Fgures 5 and 6 present the nonlnear correcton factors <(> and $ as a functon of phase angle, 0, H/H_ and d/l. Note that the correctons approach unty for 0 = 30 and 50 for drag and nerta respectvely. For greater 0, correctons are less than unty, but a conservatve desgn procedure would be to use the correcton at unty for 0 > 30 for drag forces and 9 > 50 for nerta forces.

7 806 COASTAL ENGINEERING 982 -H D] * a. 6 rs H / X o S <^ <& L. < en UJ f- 4 v- ^ "Wy Vtt~ - x ^X^ v IS s% v ^S ^ 9, f v X * CM 2 < Q p/!z=pz

8 WAVE FORCES ON PILES 807 Z u. a: o or UJ z p/!z = p/z

9 808 COASTAL ENGINEERING 982 o g o ro O r o X V CO- M. eg o o s L^ 0 If) ^^^ 5 ID.x * y c3 V ft o / J n/ / h f?b o o o / / o/ / r<y / %l *0 2 oi VSTN 04/ ~-p "T 7 o N.7 N V s W y s l V N m w o PJ o * 2/ ~ _ ^ k L. w ^ w c ^ n *-- 3 A c C o "n I "SI - a. < - o N o 8

10 . WAVE FORCES ON PILES / 7r* Hb" 025 / -Rb-U.3U 9= 0 < ' A ^ / r*; -Hb" ' r;> A- -^ - v s v.. V,v >v :-^N V], _ >. " Hb" ' 00! N^ ^ v ^ ^ ^x :N. 0 ^ 0 r L '" n >Tb= 02 * -S /' f /.-',' "N v <s. /' Hb 0-75 / * - ***''* ^ H^' SU /' ' ^=^''00 - " / *v> / N **~ --- ^5 Lb.- * * """Hb' ^ a25 r 9=20 -'» N^ -- H L / ^ ""Hb075 Hb" <T< U,5U ' " S ">*- n A- ""~-^.4 f -H b I v - n / o.oo a.o o.02 d/lo Fgure Nonltnear Drag Force Cr.rv:-rt on Factor.

11 80 COASTAL ENGINEERING ^ ^ *> "«S, H. r' Hb ".00 e= o N ~' : *C H V>N 0.7! Hb" - D.53 sfv > V ^S N _ ' V ; N V N S k > sx ao n -/ -C >^7 fc = 0.25 Hb H _ -^. ^ S ^» *. j t / Hb V s ^"""N X ^.^s.^ -I e=2 0 =0.50 ^ ^>.^ ft].00 X. Nj :^ ; ; a B=0.7 5 *7 ^ :^ sr 0-25 /""Hh~ =0.50 /- -=0.75 3=J 0 H b. ^_ h F. V ^ / I < --H L b= 00 ***' _j - s aoos d/l 0 Fgure 6. Nonlnear Ine.rt.al Force Correcton Factor.

12 W^VE FORCES ON PILES 8 TOTAL FORCE ON PILE OF VARIABLE DIAMETER Based on these assumptons the total force on a ple of three dameters can now be wrtten: F^pC^d [K DM Z ( Dl - D 2 ) + KD J Z2 (D 2 - D 3 ) + K DM g (D 3 )] ^ cose cose (2) +K I M ls e (D 3 )] *I Sln6 (3 > MOMENT CALCULATIONS Expressons analogous to the above force equatons are presented for calculaton of the wave moment on a ple of dameter, D, at an elevaton actng about the sea bottom. The moment expresson s: Mj, = f Q f z dz (4) Referrng to fgure and usng a smlar approach, the total moment, M_ f actng about the sea bottom can be wrtten as: WL-M(D) +M(D) Z -M(D) Z + M(D ) - M(D.) L (5) z 2 L J &, J e 2 where M = total moment actng about the sea bottom; M (D.) = wave moment actng on a ple of dameter D about the sea bottom o a level z. Equatons for M (D ) are gven: where: ^(D^ z = ^(D.) z co s e cos6 (7) W'.. - WL. sne < 8 >

13 82 COASTAL ENGINEERING 982 WVl,-^'; '2.. [ + 2(kz )2 g (Sr) cosh kd + 2k? snh 2kz. - cosh 2kz ] (9) W I ^fom [ + kz ± slnh kz - cosh kz 8r d (20 > T (D ) and T (D.) are gven as functons of z/d and d/l n fgures 7 and 8 respectvely and can be evaluated up to the free surface elevaton gven by fgure 4. 4* s the drag moment correcton for the velocty feld and s the nertal moment correcton for the acceleraton feld. Other varables are as prevously defned. The nonlnear moment correcton factors, and, are gven n fgures 9 and 0, respectvely, as functons or d/l, 0, and H/H,. MAXIMUM VALUES - EFFECT OF PHASE ANGLE The total maxmum force and total moment phase angle cannot be readly determned for a ple of varable dameter. The drag force s maxmum at 6 = 0, and for small dameter ples, the maxmum value s near 6=0; but as the dameter ncreases the nertal force becomes more prevalent and shfts the locaton of the maxmum total force toward The maxmum nertal force occurs at some unknown angle, but ts maxmum value can be determned as a functon of H/R and d/l only by applcaton of formula (3) n whch one takes z = 8 and one replaces <f> sn8 by A gven by fgure. A smlar method apples to calculate tne maxmum moment due to nerta. The correcton factor, gven by fgure 2. CONCLUSION The precedng outlnes the methodology for determnng nonlnear wave forces on ples of varable dameters. The methodology greatly smplfes the nterpolaton requred n usng stream functon theory and permts one to estmate wave forces and force dstrbuton over the ple column. The method presented s for a general case. The US Navy Manual DM26.2 (982) descrbes other cases n more detal to smplfy the wave force and moment calculaton for specal cases. By consderng 0 as a varable, the tme hstory of the total force and moments on ples of varyng dameter can be determned. Therefore the method s amenable to determne the total wave forces on structures supported by a number of ples. ACKNOWLEDGEMENT Ths work was conducted for the U.S. Navy under Contract N C-006 by Moffatt and Nchol, Engneers for preparaton of the Desgn Manual DM26.2, "Coastal Protecton."

14 WAVE FORCES ON PILES 83 -TT- ~T~ ', a: ql A 66d ' 6 LL- UJ a: CO LlJ 3 rf^«s^ ) $ ' V V X.. I v. c yx ^ s -' d p/!z = pz vnftv ow mv V W Wll -P * 'I 2 < cc

15 84 COASTAL ENGINEERING 982 -T > ' ' > A M3 a o u. UJ < ^ J ^V WP^ AVV$ svxk CO UJ D ^J XcoN ws AS Kl O U z (M UJ o 2. < O H 5ff 00 2 o n fe I P/!2 =P/2

16 WAVE FORCES ON PILES V "" s "--. ^ *%; W =025 Hb "^,.. =0.50 Hb CSN V /.. H =0 75 '" HB U ' 3 / /',.=l.00 / " V Hb ^ S s> N''N >;. v ^ V*k ^h ^S.0 IQ = <I»L )>- 2.0 # b =0.2 5 J ' - = -H_=o.50 -n«l > ' - / ^ Hb / / ' y /_ s. ' / / ** K- -, -- t=075 / s ^ *- Hb.0 t / s ^S / V * y ^s, ""Hb '* UU /,s -Hb" U '" fr =20 [I / - l C -> ^-RT 0-50 ] y ^ n b - 5 J- =0.25- t, T» Hb N ^ / " o.c d/l 0 Fgure 9. Nonlnear Drag Moment Correcton Factor.

17 86 COASTAL ENGINEERING ^ ^ ^ 9«I0 N 0" t- 00 :: '^ V > s ^ > v X v.' X N,. v N N s, S- b = 0-50 k %- 2 * X ^V, t s ^ -.k- -25 e=2o V a.? ""?' ^"..^-0.50 s.' r * ^ > :: **. '. ' s --?-^ / A v *. ^ "Hb" L-v 0 - Nl 75 N (/ t -^=' 00 V N., S V V. >. V s. 4 Hl= 025 /' < "S > = * r S, / / '..' < "#,a75 JG-30 J V», ** s 'v V ft $> HF=025 -#=0.50 ~ r r )=50 o t- I,'fr 075 L -- - ;=SL / X d/l 0 Fgure 0. Nonlnear Inertal Moment Correcton Factor.

18 WAVE FORCES ON PILES 87 ft"- 00 ^^ / ^ - fk = ^, ^: > V. ^ s Hb U 50 H ^ r ^5*r* '* CrrfZ d/l 0 Fgure. Maxmum Nonlnear Inertal Force Correcton k 0J5 ^ <N.- - 5C X S, ^ >^ H _ 5.25 V Nj "V S < '. - ^x.,^ < "«x^ '*o>'- ^ ^ ^T; *J I d/lo Fgure 2. Maxmum Nonlnear Inerta], Moment Correcton.

19 88 COASTAL ENGINEERING 982 REFERENCES Dean, R.G., "Stream Functon Wave Theory; Valdty and Applcatons", Santa Barbara Specalty Conference, Santa Barbara, Calforna, October, 965. Dean, R.G., "Evaluaton and Development of Water Wave Theores for Engneerng Applcaton", Specal Report No. Volumes I and II, U.S. Army Corps of Engneers Coastal Engneerng Research Center, Fort Belvor, Va., November, 974. Dean R.G. and LeMehaute B., "Expermental Valdty of Water Wave Theores," Paper presented at the 970 ASCE Structural Engneerng Conference, Portland, Oregon, Aprl 8, 970. U.S. Army Corps of Engneers, Coastal Engneerng Research Center, Shore Protecton Manual, 3d ed., Vols. I, II, and III, U.S. Government Prntng Offce, Washngton, DC, 977. U.S. Navy, Desgn Manual DM26.2, "Coastal Protecton," Naval Facltes Engneerng Command, May 982. C = drag coeffcent C = nerta coeffcent m d = water depth D. = ple dameter l f(d.) = wave force per unt length of ple of dameter D. F(D.) = wave force on a ple of dameter D. F = total wave force g = 32.2 H = wave heght k = 2TT/L K n j - lnear drag force coeffcent evaluated at an elevaton z above the sea bottom K = maxmum lnear drag force coeffcent evaluated at an elevaton z above the sea bottom K T = lnear nerta force coeffcent evaluated at an elevaton z above sea bottom

20 WAVE FORCES ON PILES 89 K I = maxmum lnear nerta force coeffcent evaluated at an elevaton z. above the sea bottom L = wavelength M(D.) - wave moment on a ple of dameter D. Sg = free surface elevaton at arbtrary wave phase angle 9 T = wave perod z. = elevaton above the bottom l r_ = lnear drag moment coeffcent evaluated at an elevaton z above the sea bottom T = maxmum lnear drag moment coeffcent evaluated at an elevaton z above the sea bottom T = lnear nerta moment coeffcent evaluated at an elevaton z above the sea bottom r T w = maxmum lnear nerta moment coeffcent evaluated at an IM z elevaton z above the sea bottom 8 = wave phase angle p = densty of water <f> = nonlnear drag force correcton factor (f> - nonlnear nerta force correcton factor Y_ = nonlnear drag moment correcton factor = nonlnear nerta moment correcton factor