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~BIREAU ;OF ~P~ECI~MATION

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!,,!ii ~i ̧ i~!i~i~i~i! iili~ ̧ ~i ~LL~ ~ ~ ~ii~i

~;i~:/-~!~. ~ . :%; : / / f -~) i : :;i k';~:;il ~ i :L- d , % ~ i' ~:i ~ d , d ~ . ~ - r ~ - ~ . ? : .~!/;I//~'i d ~ . k ; :, ~ ~ ,~',~ " ; - ~ . ~ ; ~ , - ~ , k ~ : ~ • ~ , 4 : ~ -~ ~ :~,~ - , ~ L : " ~'~;'~ • , , ~ ' , , " ~ :.::;,~ ~ ~

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-,,i:~ ~:i'?~: ~:> ,: ~i:i~'~:! ¸̧ "--~ '~: ! ~ / ~i " ~ ~'~':?~ "~ <~i~ ~ ~ •

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~'"~ :: .'::i ,.:5 ; _

~'OR D~'A'~II,:C.:Ti~ B~(ZC~AT~ K~E.,~x) BRIDGE PI~S

' B y ' : : T l ' i . : . t ~ e h b o c k : i'

~:,. , , : -

V •

" .. 'i~

,-.., , . , ,

";~;': • " " t i o n "for determImlng:.$h~:back~.tor •height ::~ive: Col)pl~tely ~ . e r ~ e 6 u s

, - . . . . : . ,~...- . -, . . ~ - '

. . . . . :~, ~ .... . '- ::In. order to th~,~ ~cme i i ~hg .-upon .tni~ .:.problem :.the. author-:.h~ s

.,water,.. and'. be6ause .an .exam% mathemati~l.:iin~erpr~tetlom ~:of .,.~uch ,::eom~li- : . : " " " " ' :": ")I~

fluld feasible arid ..... ":" ',::~ated ~..e~d o b s . emma.: of :. flow :18,. still rmot 5 i. ::;

• %~ii: probably~ rem~uhn .::so for..come ti~e :.to, come. ::I~r1:he~m~re,.l:,the fox,alas

. . . . . . L"

. ." .... O f r ~ ! ~ T " ~ - ~ n and d~Aubuiseon .ueed: by ~ h e : Rivsr: Ocnztructio~ Ad~.inl.~tr~-

~.... :: .,. '.~:..... ~sor~sa... %0 ". %n~, use of: rood.el tests..:.~y, ..:unn.e~.:-.the . . : ":::

of: difTerent- ms~,~bers and: f .~as~ @f::-l~iers..,In; ~.i~mode! ~:ch~el i:i, . : :: ':~

~.' ./i. ~'$%. ~s : thOusht.~-%ha%; m~nv: Of. the ." eltr~o1~ ..... " :.:'.i~

i::" .i!".:i:-:: clarified,' ..With:the ald, ~berefore,,:of-.:the:~ubiie~ Founa~tlon.:ofcGer~an: ~' ": " " ' "~' :'~ :" : " -

" ":"::'- ": ....." ' "ill~llstl~, .:e;. :~ seri¢ ~ ;~.... Ofi:ii~'~ st s-:~± c h ~ . ileluded .more :,~H e~ one hun,~a, ed ..diTfer- ....

. j.,en-t~:pler ;forms. ares umdert&ken 'i~ th~ old ,..r{ver ~l~.bor.~.ter~ .;~

- . . -.- . ,~.£ .... - cur, c, hc, tl..~n ~. Ccn_~t.r~'~;ion Of e ~, i~;-. ..... ,~hey • de,rended' re&~ .- c • " ' • "- "

~. ~.~.j ~ :4 _ ~..z=..,~., :../....: ..,~,~ .... . : . 4 : - _ •., . ~ , ..: . , ~ . • ... • • :- . • : . ,<:..

~ ~ - , ' , :

L

~,_iii~. ::., :,:.

• ' , : r'" i . " ,; ; . ~ . , '

, : , . . . . , . a c k ~ t . . : e r , ,, ,t ~ . ' , t h a t o a s e : J ~ O ~ t " . . . . . . . . :.~,"' "

'.', - , v~i'lo~ity .le~s .~h~n

. . . . j , ~ - : . , ' . , . , ' .. :' , ; . . . . ,

..:.:~ .,~: :- . . . . . .:.. . . . . . .,:- .. omp and.co~zClusive.

and: any

: p o s s i b l o :to :de~Y~.~£rom ::the~o : eXl~m~s a :g~eral ,.p~o~e~l~e :f~. ' ~:' ;:' .... ..:.i:::: " • :,. : / " "

. . i"

c e ~ :.,..,! .... . . . ':~i

' ' ? ; . : ". : ' . . . . i'::':" : : '.:: ,:.'0 :!; : : : "',:.. " "

. : ; . ,

V r~ . ' ,

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: " . ' 4 ,.. - . '

. ? _ _ _ _ - - ' : ~ 2 . 5-~: - ~ ~ 2 . •

71

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i ~t ~" ~ ~ j

"::" : "3. ...... :: I'": ' " /:' :"::" "

' . . , : ~. fl~()~:: thoae : e" £1owe, in, i,..~ _ ". ' t " .

• . . ~ .~.~....~:" .-: ,~....

.,~"

; 7

: " ,

° . .

~e : the ~Velo

" ~,. - -, icrs::for:.,mocez~;:bn

:: :::"~!~_ . ~ , :~hiCh :re :,generally :,~

• , :j

-:.:i;.. .,~ ,~. : :~

~ • . ;doee:not -> i -: .:~ roeo c deri~-e~ ;* : ~ ..i .~

:~:, ::i:/? : 'L -:̧

i~/?'i: ¸ :!/~ ii:~ii

~i!! . . . .

~. The width ,of ,the -~channel, B, .~, '~

~ . - •The , n u m b e r : o f ~ple~e, ::

r>" :5. The :thi'ek~ess :ofr.~the..

- ~ 6 . . T h e , h o r i z~n~a~ ~eross :,s~c~io~, ~ o f ~.,the :.plers.,,~cnara¢

". ~ it exerts only. s -very .insigh~C~ ~influence,!~n

• - - v;hich ,.cennot .e~en ~,be m.easured ~in~t~he :majority:for.

• . cs icuia~ion ,~i s fl, erefore applicable

~,i For .~ery ~.r~ugh i.pierS:ia ,.~Rli ~Increese ~hould ~.be :edged. ~o the oelcuie~e~..i ~ '~"

; , . . ~ s m o o t h . 8 ~ c . ~ u ~ e ~ s u c h ~ s , c o n c r e t e ~ f i

": ;~" , r e d u c t i o n : . s h o u l d -be~ m e . d e ~ . , . . . . . . .

" " cui ~.t ed ~ back~vater ~ hei ght. -~iThi

.... • s t ~ c ~ c h a n n e l . , ~ ~ . . . . , , ~

"~ "~-~.

, !?~.

h

• ", .......... - ..... " '~ ~ ~ .i ~ , ..-, ',: • !':.:/",~

.': '" ~.b; -- ").~he,eont~c+:~on,.coe:~:~ic£en~, ~:.~., " : , .:~ .,ii'! :.'i :.. ,, - . . . . -' .;i~.:.' ~ ~ ,~.~

] m ~

~.: . .: ,,. ~: , .~.. . : ,, . . .. ~ ........ . . ..rL~., ~ • "~.~*~:'~::'

• :" " " ~-e.'com~ute-" .f~m :the ~Iveu~.-ouB_utitlea: (i} .to/.(5) ; c~.c . , 'd:" "~'~' ,:~d';~';r ~ ''d"

/;.::: "~Un ~%ion,, :,:o~.' the :,hot1. ~on~L ~.cro ss, ~ec t ion of". a -p~er ~nd .:mu~ ~:be/.det ermined ,'-'.:'

• " ~ _ __ ~ - _~o'21ows:, -~.:i'.:

: . . . . . i ,: : - " , ...... ." -: "" .{ • .a. iThe 'veZoclty ~hegd,i,~ko , :.is~foumd :;.in. ~.t.he usud~ .'~ay ~:from;,the ~.ave~ '. ~ : ' .....

i!: ' . " . . ~'ee .@eloci.t~. ,v -= ~ . , in ~e~:-.,he- tu~ob~t~lete~ :ehemm¢i,..ea~d, then :'~om. '-

!:'-. . ..b. ".The' cont'r~¢~ion coe.e~Icien~t::)C ~,,-,Is %he :ratlo ~oI~Lbhe,~re8 .,of, -: .... :

1 !

mF

i

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<-C",'.~ ~ '-'~ .4' ~,.' .... ~: ~,.;~~ :: : -,, ,.;I ~/~ ~-,.,,~;,,: ,~ ,~..:~. ~ '~?,~., ~ ': • .' .;,i:~ L~% w : '~:~/~:~%~ ~ :~' ~!%'+';,;" ",i;',-',~.~:::: !"~: "~'..~,:,~,, :¸ ,, r,~"~ :- "." ...... :,, <..::."::',~', "~ " "~-",.,..':. ~":~'~•~:,-~:~ t. ~ '~'~'~ ......... "" ' ~ ' ....... • ~ ...... • " ........ • ~ {" ;~~ " :,':. ! ~i '~ '~,~!:~,L~ i~. ~.,$~:/~?,:~_~,. • .... ,.i //; ~/>i ~ .... , ,? ,;..:,~ ,~ , ,.->~ • , ,, . . . . . . . . . . . . . . ~ , ....

? .,.. ..., . . • • L : ;;

/ ! , : ~ . ' : , ,

i n : , t h e ', u n o ~ s t ' r u ~ t ~ - < = h ~ ' e ! : , bY : , ~ h e ~ d e p t h , ' ~ o " ' " . . . . . . "

..... :,~ae. effect 'of ...the :flow~ratlo, .~o :: .- uu~

~.> " .! height ;.is..not ~:engirely,.dis~in~u~ish~le. ':.The: experi~entS:~.have~indlosted; i,.

• " ~hat for. the s~uue piers equ~l Velocity :.heads ~nd".i;~~ coon.~raOtlcn cco4~-- -"'./ ~/-.;" c

i: " " ficients,wlth l~cre~si~g val~e~ :of ~the ,flow :ratio, :,,t~e b~c~teri:hei~lit}: :": .: '-?.';i

.!:i~ ,.• :.••.:,...o., " between the .limits. o f O~ ..= 0.06-and OC ~,. O.36~ino~ea~s~-a~p~oxlmatelY_ . • ,.:

, . : , : . , " _ , . : : ; . , ~ " . . ' . ~ :, . . .' .

~~ :; a. :.The •.fo_l~ i~a_~i® ~-o ,, :.Is-the cooffici~nt ?.~y,.me~u's..df"~whieh . ." %.; }~.!;:~iJj

'~?~ .: the influ~mce,..of, the.~:geometrical ~:ahape~-of ~the pier;i,~ ~,e~esseo~ -",:~id,; ~ ' : :: " :

:;- : ~ : beckwater height :.is ,.:by .too: meane - t.he s~e ;for cc

~o~tractio~ -ceefficlem$ if ~he::~e.:of: the;.pi¢ . . . . . . . ,,~. .,

• ,trary, . it .v~les, between. ,~ide. llmlts, l-,~gelybeeause .:o1 .... ;i~:

' ' ' . . ,'Pier':nose or :i~..the pier.tail,-:ana"-to a;.'.le~ser el't~t ':;a~

:<';:'~: , "- primarily by .const.~ioting-..the-.effectlve-,,width ,: Of:the ~:.:cha~uel"::~:.lio~er~ " i... :::~:.,

, .:.the,:,.effective ~la~h'i...~oes :~o* ~;.alw~ys, Correspona~.to ~the 'Ci st~uC~-!ini~he:.'.i; .' i ' " :,

;": -cieer ~bet~ez tbu Piers.'or: piers :~,ud-abutments. :; "~ith :""::: -:"

;:i.":-;:' : . :':~ " , " : ?;"; " -"::~:';:: ;'":~": .... " ' :- ~ •.: ~,~0 i~ ~•e~ :~..~o.~e ~, "• ~owev~, ~ ~he" s~re~.,ii.~e:.~ :.~ii~-,~to-:.t~.•! - sl ~e:s .: or i~ ~ ! .': ..... .. :. ;--"•;~:

? • :,,;-plers, ~ud i.#he entlr, e.!,s~oe ::-between. piers ;is fi~ ,:

~: .... , .-tran.slticn :~etween ~.the '.. bgcy .of :the pier :amd .th, ~. .

;.:,:•,,,:, . .:if,lt is:,,~e,up of s,.sharp.edge or a • ..

,~ i I

t h e n , ..ae' e;~ ~ e e u l . % iioT

. f om :the,

. . . . . . . i i , ' . ; , ~ . ~ :3 : : ' ~ " " "

• k /i : •

.... . ":7 -:'~..,i ~"~ • /i~ -~, .~:

I." - . . ~ . . .

,. ever, .~doei~no% : assist

: :~regton ; in v~Iilch z~lle~s r=~' ~ fOxx~sd:i and)in.,i,~ni 6h;:~h e ;d treet i'o~:;..o£ :f.low

..", I s a ~ ' .~ Imes . .~a~ '~ . . . . ' '

e

> ¢ , . :,tion:.

~': :'3~ . i :~ems

o

:~"~! ...... ' " ,.'dtest_o=~,e J.~pns

channel:.,~i es which. :do :not :,.~.~In !1~ , e .::

.:-:~o1~nst:ream; ..-~en i~ In ~.%hi:a,:oa'se ~ ~:'ho%wevep,

, e ai:tto:

!~in ,the~eoi~

. )~ . : ~ , ' °

+

. . . 7 ¸ . . .

C

,' . .

'i2"

. ..,

ii~:'~=~i ¸ :: : ~;i

. . e v '~. ~s T : r ' z

.., .

: .,, :. ~ :,.,.5 ̧ .]

• <

.... .

,si~ a'~ ed ...by .:them ~ :Wh£ Ch~ i ~: considerably - " " '?:> ::::: ~:i'

~io:~ : a* ~.a ::'~ixed ,'wall. :if;likeWise; :t~e :piling :.:up ~!~f : " " ": :"

~iO0: '.e~t ~Rdln~ .:~rolll-,tbe :,~ler" nose :~tO : i~s :-tet~ ~.@~::- b e,:: ~ ::-

, . .- ,;';::,,.,i!.~ ; , : ' , . ,

:i

~A

" less :the ei~eet the nose of::~e pie:

:,:-! . -ion:It2 =

"- : about heal :.the ":Sl)aee cas :f@r<a.

~1-o~e~s on ::both ~ilts-:sides': Rmoun.ts.:,,.t :: ...... ~ ....... ....... '

of the_pier. The ~iling up:,:of :,me~: L " ' , "

• pier ,is .: consl~erable in ~,~hls :~c~se.

:!lets c~; be ;:avoided " by:.round£ng~,or :.,

~the i-wet er.. at ":

: .l~rcent-, of:,~

..... : . : ". ., : . : % L %

: . . re.:

::role.~ia - _ 7 -̧. ::

~.:C ̧ :... :....!

., . . . . . . :~,~. :. ~ ....

:fer the :~ost part, :p~oduced ~b~

- % . *.-," 2"* J

. . . . . ¢ , . .

*:!Lbe10~ ~be p i e r . * " * '~:-

L"

, . . , ~ •

!i~ii! • ¸ i , ..... L r :

i: ~ i o n . d o p 'into-vo~tSee wh~ :i :mh~ ,:!'=::: . ....

" ~ ~ L ~' "* ~ ~ F " " ' " . ~ l e r *they mm~o.ut , t o ~Immst .the Tu.l.l.~m2d~h o f , . t ~ e . : p i e r , : ~ t s ~ T ~ . , " "::*: r "~*: .... :'':'~:::.:~:" ~

I

L< , .

lers:~hu.~ c~usi~e, sharp/iuo~ease In..the~bac~er i'

. . . . ' ~ - . : ' . . . . i~ e

- ,..

]

:,'; . . , . • .-

" :L:'

.!

T .

. ,,..

:. are :of. tO a:

olear~r ipfcture :c~e,

() ~ - ,

• ' ::,: . . . . . . ' - . 4"..

":sipat.~on, . .Sinoe-.m..theoretical .:solut !om-.:-..af ::~:~rOb:l

..... -,.:. at..~ope.ess, .-an:,:: s~nCe:::~o~serva~mo:zs...at :::actual ?bz~'~

: .:,~h@:wholes:~aord%na~.< ,:dlff~e:ilti ea,,~,c~v.uot:,~:.::e~e

.cons!dorlng -:~he Impos~Ibi.li~ of:~r~vi.4 s . the..'st~t~

-- ~v~ . . -.-..::..

. . .... .o: .th.~. pler, ,on::the.-..ba,,~at~.,..~:"' .... ~: o~,y- :' "~ '.'5": " ' " ~" i . . . . . ::~'~' ".:. ."::

-:: './' °m~thod :- :,,#h~t of :mo~_ml-: t~s~.s. :In,i order :,to ,.o~tain :;prao~i-,

__ " " :," "3 " ..... .': : :'

:: t.~ese te st.s ~ im spi~e :iof :!.~h • !-,aume~ou~ : variablee :,~,-.~e ~taki

~:i:: :~...: : .. : : .:,eratf~n,; ::%h,:,whol~ b,ck4.~a~er.~.groble= .was :i'~es~fgate~:i-~oi

• - \ .F: . . - .

~:: ::derived for:.tlmls-"st~u-iard..~ier, ~whi:eh: ,... . -. .-Sives ,.the i-baok~mt~

" :~u "terms of.>t~e -.va!, oi }y gea¢, :.k o of -the :.unO bst~uc.ge~:, st~

,:... ' : ~..ne! oontrac~ion .-.r.a~.~oo,- :o~ -; :.and ..ths::~at~m,.- t~.i..}. " " '< . < ~ - ~ - . . , . . . . . " . . - ~ ...... .~ .,~'..:.~.. . "-,

• ...:.. 0 '• ,head ,belo-~. :~.h~ ..cs~+.m.~.tion .:.~o i.the :d?~@h IOTI: , ~ 4~ ~ . ~:~

~ ! ~!~il '~ ~i ~ i:'~:!i~".: "':~

~ F~I , /

'" :ehape~.~a~ studled :bY which th~ f o ~ u t ~ deT~v~d ,or ~the S ~ ~

, : ~, oo~ be made 'applicable .%o ,any other pier :forum by m~ItipIimatiOni"' ~"2 ~.:,: , ..:

"~, ~ . The~e remtl%~ we~.e .further :e~p~Icated, as the .~s~s :show.,....,.

..~,,,. ,.;.~ by the facis %hat.'the Tor~ 'index was not solely,dapenden% on %he :/sha~e ', ":,.2:. :

'"- of-th~,:pler so as to ~m~aln a,oonnt~mt-for e .:given pier ~oz~ ~butr,, '-'O~'-. '.'.' .-., "

.,,, . the contra~'~, varled,.w!th the chanmel oout.~ao~io~,, T~w~ver, the ~:r~e~-..~, :,::,/.'~,

':,2, " .m~ts also showed :that, ~t~tr~ the .~n~e .of a~cuP.acy:.at:%aLued in ",~he . :~°~

:" .... Me, ~nd ,the flow ~ t i o , . :~.~ " .A ,~onvex l~n~-shape ~.~hose n ,!f~ ~he "~ : i {

:-..-, this fozm~'ofl pler •was boarded by %we oirg~m ,~ey~:lad~i~l..:s:~.fa~e/;'::, :...' .,:,,./:'. :'i .~..:'!~!J

" ~' it e~uld be"~inlshed..to ~y aesired de~ree ~ ' "" ,. o~.:ac~-~raOy.on..a lathe. " ' :"" ...... " '

.., " ~e mater%al for the .at;mda=6 let ;,was pOlished .ste~i ~ r .:[' .'... , ~'+ ' ~'

~/..~:-..... ~. the o~her-ple~s and .'a saoond ~e~ of :standard pie~s ~ e ~ '~ ( i e ~ o ~ . ~ , . . ,, -,.~... , , ..... ~, , ,.:,~ ...... ,~ ..,.~

• c:,~ '.poI~'~h~6, seasoned 'plt~m p i n e , so ~ h a t they a t t a ! n e ~ a l ~ e t . ~ e ~ k ~ , ~.~ ::~•'" ,~ .::' ~"'" ' " u " _ ' - ' , . ~

;:~ de~m~e of s~oo~hnese as '.the steel p L e r . -A~eumiuga :pier.,wi~h :of ~th~ " .

: : . " me%ers ( 9 . 8 4 / f e e t ] and a ; i ~ h o f :~0 meters i ( 6 ~ . 6 fee%;) ~ aad u s i n ~ , a . . . . . ~

1

)!

.: ?.,.:., :.

,~ -:

1.,, > J " ' t .

- , , . .

• h ~,£olI~ ~In8 .'fo~mu!. for g.ho: ,baok~%er, holght '!,was J,derlv~

• - ; from: i~I ' d i f f e ~ a n t ': t e o t s < , o n standard pler~.for purely .:.streamin8 t'f~owi

- ' - "" " 0

: : : .

a " , ,,. /

ii ""

• i ~ o r o t h e r : p l e r i t o ~ s ~ e ~ d ' f o r < < ~ u r e l j : s t l ~ e m ( - m g ' f l o w , t b e . : b a o k , _: . . . . . . ' >

i, : ...... " ; i , . I t e r . . . h e i g h t ~. I s d e t e ~ a i n e d ' s s ' . f o l l o w e : ' : < ! ..." " f , : % .

.., .The .fo~a*.~In f o r t h e . s h ~ p o ! . Index , ~ ~ , :was iTo~nd. :::P~om':a . . . . . .~ , ,

. ,i~aphlcal ,analysis-of ghe t e s t s ~n-.all.f~e,:6f~::pl~rs ~h.,.'Oh~am@l, ' ,'~;2, i';,.:,ii 2

, ,e~tzaotion,v,.~jins:fro~ ~,'=.0~,00 ~ ~ . = : O ; , ; ~ 6 , " r S J O , d , : ; , " i

T h e . T i ~ a l i . f o r m u ] . , ~ ,.which :Is ::val£d ,.within the' given am~ I ~ , .'< - :~ • ~ -. : " " ' ' " ' . . . . "

::~ . . . . . . ~:oontraotlon i S : " :;ii

! 'F : . . . . , .~- 0.~ ~& +-:(~ ';_ .. . ? "

"?: -"~n - : this -~-formul~) ' ~ o " ~'s:..a .COnsgmUt'!:£or;~my'ple~,jeL~gi2~:~% :'? ':: , . ,

" ' " ' - , " , , . " ' ~ ' ~ - : . i

" " ':" ,:fom~m, o o e f f l o l e m t ,for,the..pior.umeer.-oo~sideratlon w h ~ n : . g h e . : o h ~ e l } : ~ g , . " a

!~ii *',:i;c''' t m ~ o t i o n OC ~ i s . O . T~i~.~ I d e a l . i v a l u o . i ~ ' c e l ! e ~ - t h e l l m t ~ I r ~ ; form . . in- , , . " ', : , "

:d~. ~ n • ~..:is,~!otted.sgai~t thel;,:oha~nel :eontraetloa~ o~ ~., " " . .,._.~,:~:

:~!."i/":-: " ~e,-cur~s- ob%alnod: ere :.,app,,.~oxlmat~y !~ straJ:~ht.!.l~es ~'2~'e r.~ e :?Of" " :

it;?::.'.!:i: .tNo:o~ina~z.:ch~el.ooutr~etioms,, or . : . .between oC ~ .Oa08 :amd ::.(~ ;~..;.:0~:~, . "i

: ; : , . . w h l o h , ~ m m , , e x t e n ~ e d , :-p~t~s~th1~-t~e point "O( - : !i N := I. ,:i..:.< ": '"" 2::". " ' ' ii~

- , : g ~ o , c ' ~ , S ,_, em '~ . . : be :calcula~ed::for.>m~y~vsiue ot g, : ' b y . m e ~ s " :~' ,. )~ . '< !

.... . . . . , -:of<fo:r u1.e , . . . . . . : : . : , - : o c , ,

~,~

Av "

Ji'• -

F' . -

l

In order,to determine ~o, for .:~ny ~ivan pier for~n, a pre- " " ~

vious determination of ~, was required for various values of the =~ '

channel' contraction, O~ , ~he form coefficient was determlned for at ..

least.t~to ~differ~t::channel:oontracti~s and usually f~ ;OC ....= .Ol15

piers)" and for ,~ = '0,30 .(4' piers): sothat the liu;iting form~ef- \

!fici~t could be 'calculated:f rosa the a v e r a g e of txvo. observati~us. ".

The given e~perimautal approach hadthe advantage thatl-an:ao.

curate Investigati~.f~ .,each s ingle pier shape! by, varying all +three + . ~.~

variables, namely ko, ~ , and ~ , was.not, required.since ~it was carried ~

out with a ve~'.signifioant..loss of-time [.or _three fonas.of piers:rill- ...~. • ~:

fering essentially from.one, another,, namely, I'~ms K, F, an~ :'A..~(see

table I i i ) . : .

M.any"checks.have sh~km .thaithe above ~empirlcal method~of at- :,

tack gives.:satisf'actory results through~t .the giv~ range of channel ':-":

"c ont ra~t~oa, . -

~. ~he ~.resuit.~ of~ the determimat i~n of the: ideal form :linden,, ~e,., i

. -for .th~-.,n~merous,,different pier ~ shapes ~ shown in .tables -I .to .IIl.~ra~ ~in- " - " •

areal, by ,the. number ~ '" ;~ - die , accompan~,,~Ug each ~ier form. Furthermore, ;%gese

~. values of :'~ .are .in terms.:of :"the index/'or the. standard pier!ta~.as , ~: :~ ....

".~umi.ty. :~For.a.'clear. comparisontheF~are : also. r~presented.:by '~%he ~l~gth . . . . ~ ~ ~'

~ .: ~', ~.:: a ' , r" '0-- .:~e.:.1~ee~l~ne..,band in..',the c e n t e r o f , each ~ p i e r , • :,.; ;

~ : ~ ~ : t i m e t e r s ( , ~ ' ; i 8 . } ~ ' ~ i ? ) ; C n l y . o n e . . f o r m " " " " .... :. of.uier.was tested...~th'. " ....... ,,~

i

, ~i/~ ~ .

i~5, .6, .~and ~12 @emtime1~ers i(0,147,. " " •

,,respectively/,iin o:'der to x%est : .- . ...~.:,

/

• • ,,.

• . :~he 'val~-~i: : , . ; , , . . . . . .

":7 '•- ~

~,~f :: ~hei:model test: results':to ,::the~:proto t~rpe. , • T~I s ~ Invest l@~tion le4 : .~,

to.,•entiT ;: satiafa -con01uslons' th~ou out :the r~x~@ of %e~t ":

! In most, oases the! l~ngth of ,pier was ~chosen e%ual to ~B0 ~: oentl '~ :' , ,¢ •

m~.tera i(7.97 Inohes):: In:oNer ~o b r i N Into ahatp :r~aif ;the'::In•flmemoe . :~ :

" Of:%he form:of-the:pier,:nose'om the.slzeofr~he l~mlti,~,formilm~ex:'s~

thus :on, ~h~ size of the backz~ater, iln numerous :'eases t!~e 10mgth ~f ~ ~,

: - ,pier-was alsovaried, keeping the width .con~tanti..in,:order:.to';establIsh

the imflueno~ Of the length of.the .nose and also the,tail:on:the)height ,

of' baek~ater . . . . , • ~ ~ i

AUr absolutely~:~encral .Tormula:for -~ cannot be .set~ :'~ . . . . . :., ~'°.

sines the:pler . sect to~s • ~M~S~ were :too..numerous and there were ~,too ~mauy ".";~!~i

:~varl~bles dependin~ on ~ the pier shape. However, :. for~8 single grouF - " • ;ii

' of:plots i,.it .was, posslble: to : @erlve ~a sufficlently, accurate, aud :simple " ,.~. ~ ":":i

, , f o o t , I n : f o r ~ e t e r m i n l n ~ i t h e l t ~ L i t ~ a ~ form Jm~x ~o • ' . , . ~

Formulas for-beveled, rounded, dia~ond-~haped, lens-ahape~.,. ~ !

and ~elllptlcal piers as well. ~s :.'for 9lets ~ith a trl~g;ilar, :hell-lens ~. !

:~shaped,:~ud,~eil~ptleal moses.are shown~in~table:l, iThe ~ in~eme~::a~e • ' 'i•

::.e%pl~@ssed.~i~ :evel~ a- case •as a fumc~Ion of th:, nose len~h~number ~ ~i

• ( the! ratio of the •length of the nose, ~•la ~, to:~the :thlekaess~ of :~the r ~ Je~

~. :~ ~u e ~ slmilar iTashien, :formula~~for ~ t~u~ ~il'~ereAit . '~JL~)~IpS~iOf : " &

:~++~, +.piers are g~ven in~ ~a~e: in wh~ch,+kuepins +the, sha~e, of +the,nose

o

.... end. tail- the.same, :the length of ~, the p~er.was ~arled ~ In:order ~to .:find /~ :ii

-~.~he efZect of the/.plerilength on th~:baek-eater .helf~ht. Yor these g~ups ..

the:~Imlt[n~ form_Incex ~.o ~:i~-expressed as ~a.:functlon of the:~ier .

len~thnttm~er' J,~. --~the ~atlo:-of,.%he.~tot~l leugth:Of::pier, : i ~ . . ..... . :, to , .... . ."i:;.

. % '

•!i

h-

i!ir '̧ i :iiii

• . /.j ,"

!,the -pier ',thickness. , " b On r aC ~ O~t I'F'OZ :the ~,~ndeter~inate :nature~.of':~tlm ~'" "

"~ .-.curve : for one~ of'the groups ~ of~ piers .an ~analy~is ,:of ttwo.s~ibgrou~e

was undertaken fin order ~to~establish a ~simple !formUla .of,,suf~i:~ient ~,ae- ;:-'-': ~,

euraey.

~Por ',those .forms :of piers for'~which.no formula~was denive~/.~it •

is.easy to ascertain ~the'J~ index with.::su2ficient accuracy h~ ~ ÷ ~ ^ =

• ,~.lation mud a- critical 'analysis ~of ~the pier ~shapelin ~ question, ~f@r i%he

ntlnerous experiments ~ .cover a i

Table I also 'includes~,the. limitin~ if otto ,index" ~for ila,~ai~tgle, :

It-so.happens that .the fish-form pier, 1,i", possesses ~a:smaller .~ ,value

: ,-. ~.than~any~ other form aud .therefore .in'dicates a smaller.~bnc_~ater effect, ~'

~" "" providing the road head is~:.directed ~,upst~;enm. This is.uxPlained by ::~e

, ,:..... reiat£~ely . ,.. under.:rollerz . ' i, .,. '.: . ' "'

Values,of ,~o for a !ar~e n~nber °£'-.unsy~e~ica! -piers ,of

!~:~,,:~. equal len~h~-andl width s,~'~ siren in table I/i° ,The "~o values .determined

~i~[[ for ~th~se tmSymmetrica i 9i~. r form~., hays made:.i~ ,possible to~ear.ry out,-an ~ " / i

.... exter.d~d inv~zti~ation of the bac~#ater produced~by ~he.front and,.back ~- i- .

halves of~a ~i~gle plot..

i . T hese obscrvati'ons made-wit}~ the ,greatest premis'ior, ~have .~h~a -

u ,

•. ~ ?"

• " "that. with~-piers ~having~: a jlength.ratlo of 20:5~there .~is ,no: mutual iinflu,

en~e of'the t~;o pi0r halves ' " :. : -:v~rth mentionin 5 as far :as baoLn~at ....

,~ ~erned~ s o~.tl~t :it is possible ~ "- ~ .~." . • . ~o o.t.r~:.~e the bs.ci~ater..ac~ti

L.. ~-~_, ,umsymmetric.a! pier from. the kn~..m pe~form;..nc e of~the.two.halves:!forming

i~i . . a pier[by simpl= addition. The c~:2arlson of the bac~teri::ac~ion

~j~ . "" p~'oau~ed bY single h~ives, of" piers is .partieularl}, :instructive S[inee , : -.7 ~ i . • .

iii ! ik i ̧ '. , L ~L: ~'

i~::i:~ ,..),,-,. .,

~,i ~ - , . I

= / ~ / , ~ , . , , & / _

'"it ~c~fords,;)~)rthwhil~ ,oon~l=sloma. as:, to the oauses of. backwater. The', :. ' ' I ' 1 : ~

~valleble spaCe-.im.%hi's p~per preciudes a eetaiiea .stu~, 'sO the ~Za- -

t i on b e t w e e n beek~mterlhelg~t and ple~ shape. : ., • . .;.::-. - ...... : .... '. "A:

.... ~; " After .~6 has been ,dete~Ine~, I% is possible in all oases ..... ._ )i

• " . ~o ca!oula~ethe backwater heightZ 1 from ~his . ~ a l u e of ,~ 'o and the :.. ~-.

v ~ u e ~ ks, ' OC , and :t.U . This oale~letlom is sufflo~ently as . . . . .." :~'

., , .:. au..~ate .if attention is .paid .tO-insure that the discharge at the piers "" "-

Isi~t~zlY :s~:¼r~aming. .If a hhange fmom stresm~g to . ~oo t i ng flow.-ooO~s,

~atlx~y di.~fer~t Causes of .baakwater format!ore e~ter'.and, in th£~- ' ~"::;~

.. caoe, it is necessary t o :mse .the ferules for backwater ,caleulatlo~

-~-~: ~.-... o ther repor%~ :(reference ~,, formulas 6 to. i 0 , ~nd reference':.;;3,, fo~_~] i . !~i i L~

las 8, 1.%. i2, .an~ i~). .. " : .... ,~.

>, lj ~ ~'" ' v - 0 "" "

"" .... A graphic~l proce~u.-e 16 developed la table~nleh 9ermlts ~ .=

'~'~'"" ,': c o a v e n i ~ :~e~tnatioa of t h e b a c k - w a t e r h e l g h % and" , a l s o J J ~ I c a ~ "

rD~e of purely

) f~ow,. " . "

iii " i:" "i: . . . . . i ",.," , ;i -i '.~I~ ~

.~ " ,In order :~o fired ~he bae~ater heD~ht, ZI, for s~ ~:" , ' 'mr' L' " ~L'" ~'~::::"

flo~ iby foz~,~la (4), I¢ ts e . . . . " ' "; ' : ::: n oe ~seznj ,to.mu! ~ l,ply- g~aphl tally. X-he .. ~w~., i ; ~ '

. . . . . . . . . p e e n t i ~ . Z tb_ t w o e r e s s i o n = . . . . - ; . . . . . . , r" ~'- ~:" : b . . . . . . . . . 1~ ' r . . . . . x~ con~aineg : ,: , ..... ;,. i;:

i.~... /., , . / . \ .I-:,-:--. ' .>~ , . '~ . cc ( ~o --., ~ ' :o.q 0~ :)-~-- ,~ I ~C I : 0 , , -~ -~) .~o ( s ) :

!i}¢ . % = , . ,. _ : :

' ~,~ q. : the d!achar~e in c~blo fe~ p ~ r f o o ' U wldth o£ ~ucontraeted '

• oh e~ e I. '

x ,

:":~+i':: i::!:,++:+i,::!.::: ~ L/ :L:' :i'': : +

: ~ +++

1' 'il .

" +

I L I I

: I l

iI \ "

! . / )~ x + H ' x

, /

: |.,+.+.+,+~

+,I ,+I +

+ i+ . i

, . ' t

iil ": t •, 2. .

2

, I

h? * •

___--Z.! +

' : ~ !": :U.: I . • ,

• . - _ _ : > , + , , , . . . . , . , . , . ,,., ,. . . . . .

/:," . r

'I t " .I '.' I' ! - , | '.

I I ~r • ,

i .. i i , - , : .: 71

u + .+. ++++,..., i . •

,i "+ ~I i

, i ~ • ~I':~ ' ' : :-.~i'~ . -, .:.l.,5

, I , " :+i' s +. :::

21 : ~ +I . ~ : " :

:1 i ~ _ / o . :~.! .. ,

,' "~ ',~I '

~ I - ' - " - - ; " + I ' . ~"" +r~" :: : :'" : [+'" ++

+

t + :: + . . - . - . . . . . . ~ , ~ - _ : - = , . ...+--?..--. ~ _ _ , -~-_:

. , +

. ' ] ~ , . ~ . + + ,,+++ , - . . . . . . , , , . .+ ..+ ~., +: ..+,, + . . + ~ - + . ? + ~ : - +-- .+++ ~ , ~ , , .~y+.+ :++ :? ++.+~ ~ : : . . ,++, '_

i! i,i/ .... k -i

. . ~.~ - .. :

?~. "~ ,.. " - -: . .

'i . . . . The/longer of i:the %w~ Lfa-~or~l :~, . is fO1~d' ~aS iS

':" in ~8~ap 'I In/the ,:upper .-fell'hand co~-~aer ..o~ " "~

'-. Ordinate cori-~spondLq~, to,:tho :abscissa :~.

!= " . .. :.j

ii-. .. . . . . :the ~o ~curv~. " ~n a simi~l,--r .way , . f e c t o r B ,i.~!,

.:.~ ,..:: @w~!)h "~II :lhl ~ %he ~,1.ower irig~ht,h.and ,corner .i

" !i

• through P-~I ~-intersect in ,graph ,i"I~

cozmer.). ,'This ,point; .pIIIi,ipsr~i~ .cI~-" :" ~

~ ~ : . : : t B

L: '~':!i

- /

, • scale :.ra%io. .

L

::': ' f.t, i:a~a ,~h~ acceleration of gravity ~g, the,

• : The~,elo~, ~th~ pro due% :of -~ .~ud ,B .:2ha~ a dimenelon:~.of

:-i' :- .' " .~ i :the .~aek~a~er 2,height, Zl." --

11!2 -

, ~,i ~, ~. < ' .

.. mile ~back~ater ~helght is :pro~or.tionai ......... '

sonsi%i~e by iinoreasind ,the s&ale :.n, ~ime~. T~e"/.~ue ~ do ~mms~.i:,~h~, b e ......

~. ~lltiplled-:.:By :n/end., ,the ~j~: •

• ':: ~./:i . To .~ genuine s~,-re~.m- : " ,~i!./!

~er~me~t

:"I

::" . ? The:':,plers i--,'studig~::.we:~e: . . .. , " : ., ,, ::/:

:~., ,,~ae~:poi~te~a~cb:L~pe~.::,pler , :F .,: ":: - ::""/: ..... . ,. " :

• -., ,:

: .~1~o~ler :':had:a ~:temg@~:~velo ei~y: i~,h~n r, the ::ways ::veio city ~ i~d- ~hus} bed. . . . . ,:

.... >~£0~1.:0~ : the~.:chc:mgl,.~contr~._~ti~n, " 0(... : ~ ~ ....:''.,'_[~a~ ~,fo~owin~' _ --..: i. - .'i--o:;::e'""~:-,¢.:~ ::~£)rt~Q~t-'i

~ were ~_derive~':

iii::i. .... :- :~i. /Fo:

.: . , • ; .u ̧ .., ":

"I

another :con s~4erably :'.:end -In ,.:con sequenee, of :.~hich ~POSe r / -

!

:,,g :eal;~,y :fj.'o~: one ~ ano,th

' . . . . . . . . ' . . . . . . . . . . . . . . . . . . . . . . " i l ¸~ i: ' " ' ~ ' "

... ./ :i: i/:: :i~i~:: I:~ ..... :"~:,: i.%he~,l~nit~a~g .~ l~,v .ra~ios ..: c~mi~u%ed fro~.equ~t'fons :.7.:%o 9 ~,O'/~fere ':' • :~: "' ..... ' -

'- li%~le':f~om,.one e~oth'er. The i~i~ingf, low:ra~ios-.for ~ohannel~ :. . ~..."::

:.~'~..: .: ,. ~tlon ~be~ween ~-=, 0.06 ,.and ~= 40.36 be~en wh~h .th~, ':~ I~ '":"::: ~ : i i " . , , , : • , , : . ~ . h ~ , . • • . : :

~;i" :~- :equat ions: :for ",the .. determinnt ion :oC".%he ;bac lo~ater ,he£'ght ::shoul~; "

i!i' i i i ? . ,eral, be.~:a~piie'd, ~ .... i " ... :dif.~er Im%~ e ~l~om .one another., ::Therefore, ~the;l,imit- ."':

,.be ,oh rained ,~'th ~ C f i c i e n % ":: " " ,.,: . .... :ing~f:iow. ~ v . f o r - -other , :shapes ::of ~ p i e r s .o~n.. " " :' ~

, . :!:

.::.-~: Al!::~,orms .~ of piers.:~ike ly:to i be :met :in .practice.

::. - : " : m a j . o r ~ . ~ y . o f ::oases . 1 " l u o t u ~ e : between.: : , 6 a n ~ Z o ~ , . ~ e ~ W

S|iO~

i-: .:; :: ~: : :.;;:-~. :,olles",as:!::~l'l :,;as-,whether

• . ~ - . . . . : ' • , :

a s , . a b s c i s s a e ~ ' a s ~ I n , ~

In.:.this ' ~caph are :~f@~-,~N :l~a!ti~g ifOnn t n ~ e x e s ; " , ; ; ~ , . •

~,, . : i~ : e q u ~ t i o ~ s :V -: t o :8.- ' :- ' :' •

" Furt~.~rmore, in graph !II ~(~low i~ht-han~:o~ner) .are .:ourves

i . f o r !~the• ~ depth: ~of : f i o v i ; d O,. :wt~;H :~a lue

!p!otte~ .as :oz d~na~es .aria values~of B: as ~ n b s o l s ~ a e . :~B m~

- -.exceed~d:bv~the.:given A,,~alue ! i£ , ; s t i , semi~

':' ~:~.:. : :' .,i:~ ..,i~"~; " "': .',"

:~ :i!~ • ' :•

!i .

,:do~ ,•end Oe-sh..:linos :in ~,the !,!

--,..: ." .ul~n~ ,:lime " is ~rs'~ zTro~,,. ~.,.

- T •

• ~ • •/ 7•• • .i •"

~I':!./ ,: :/i~i'~ ̧ :

~io: .~'~ ,, ~andi,, eCl'~o- "

~ler:~thc~n '~the : .....

~ I' " ' " " ' ~*'hether ~,thi~ '_"f6 .,aCtuSlly. ::the ~i.ease i:io!~

: In. graph iII ":cOrtes oondin~ :to ,i.the

Investlge.ted. From ~th-i.s -poi'nt! :o

S

• ..correspOndsr:-:.t,~ ~%h~ dead%h-of :.the_unobs%~e.ted,:.:tlow. : ' :

. . . . . . ~ .~e, ~ ~ I ~th.rougm .'_the ~ point, ~P :!~bus ~fo~

. ,.no% ,occur.• :if .%1~ ili~,~i.~, ~i.~ ~ exceeded,..pe.r% .~i' :,tn~,.wa:

,':ahootizg flow. ~a~6 the~.c!.,Ic~l~tio~ :.Of ~the backi~ater ~he~

,S .'.fle~ ::.:do.es ~ . .:. :~: ' ::,:.•:~i.:/:i '~

~' ~us ~-PI I"! i es :.to :,t he~:!eft

lies~to ,~ue left ..... ~ ~ .... - . :-. ~':

Th. re ~ o r e, ~.the, ~e.~ermination of ;th s i!backwaterihe~.ht ~:~i for

~e

....

. "!.' .,'.,/:i ¸

¸ i ¸I/i i ii i !ii~/~:/ F' !7,¸¸:"" :i, ¸ L.~

• ~;t M.. ." :,

. is

i i I ~' , .... ~,The ee~a~,, ' " ' ' t

:i..~. ~ ~:i:}:' , i~ i~ ~.o,,. i.~ ~i~,

i'i.: :. : . ,

,~. q~

. . . . ~.;!., ,. ';

deci~al pl~ees Deca~

t~

#h@".lim:

the

~g 0": accur-

~he l~elUa~" of ' '- ~" .......... ' ~ " . . . . " ~ i v e ~ t o t. o. declmei ~pleces a s found"...CI ;.: , . <-~

r . . , .L , . , . . . .

/:~i::":::":;: :: *, i:~;,hei~ht i = ~he model e x p e r i m e n t S , i n ~e~era.1; , ~ O ~ t ' e a : {@ O ~ e - ~ e ~ t U " ~ . "

: . as:..deter~nlne::l by£a~rouS du~lieelte t e ~ s . The data Obte£1ed by ithe: .v i , . . : ?' ?¢

" J ~ " i; C~ ser ie~ of che~}." % e s ~ " f requent ly 6 a v e 'the: s~ne r e e . a ~ & s ss the " e 6 r l i e r k.~, < • " r e e f s . ~ a r e ! y dla the ,dtf I '~remae e-~ot~% to . ~ %hen:'O;02 ram, "~,hile ..... i. ~,.s

' ; ;~ f o r :{ ~he .S%~n~erd p i e r , s~n~&e observe- : , . - . ,:.

!i "~ ~%,!c~nS en ,~ceurccy of ebou:b o~z.e-~w~;ntl;e~h ~.~;. ~as probebly g.b%eined" " : ~

i%. :,, ,~ .~:..

~VJ : .

~a

" ~r~cl~ .,%:

• , ~R~a~.t to 'knowitbe:ibeigh'~,Of'~ba"6kwa-b'Bf

.i ~! i

Bf~o.,t ,.r~-obt~,Inu~g,,.ucn ~.reF,'c, aec~Lwac~,-"gn t.,..~ ~a.~~;;: ' ' ~ .... " " "" '~

~w'~of 'the .~fae~J ~tliaS ' ....... ' " ~=,=~i~o~, ' .'it ~i:~ no~ "~ :'~ '~ :"; :.. i i

Of,a pier sccu~-.

[ ."

; ~.~'.~er heigh:~ cau~c by ~ ut~tlonsry "bo<~ in flo~,iiDg .w~.ter stan(~ I.~ .

!iii~ ,~" . . . . .

-~ .~ :.>water a.t br.tdges. -. .. :".. :;

: . .... minlng...~ne .~imensions co~ ~br~.&~e ~'~,-,?,,~. )but ol

~gener~,i.~a¥ .to ghe furthel:~nee.of e, ithcore~ic~l ~olutiom ~.of /:low

:p~enomena- -r ~, . .. - ~ . , ._t ,io ,on15 .~eoe~ce ,ry .to ~ec~l,l :.,h~t ttle-impoztent oz'obl e~

,of the resi~tence..of moving :ei~ips is -clo.~ely .relatedZo .$Le~t .of ,,bridge ' ~ '- ",I

" ...... :,thO :s~e :relatlen as :2:,:. ' ..: :.. . ,

ii:il .̧

....... -, c eura; an~ !.In"

i :":-""" :' . .conse~.uenee oe ;~.~hic~:.!larv.e :.baeks'~ter;~elwh+.~ .:~~ :.in ~ev ,~:

ii i :invest: streaming ":fio'~:::.at 'b~d:d~ ipier'~.e.ah :b,

:.- ~fn i a :z:ore ,exte'-:aed .fo:~;:, ~:i ..... tud • :is " : ~ ..... ~ . .nor,. an ~entlr~ly :,~Or.~hle~e 'll~,k :. :

?; ; t . . . . . " :: /' ; ......... , • :

:-":in he:,iong.:ch~in.:cf research ~;hidh :has,.beeome necessa~ I :to <cle~I/~ ,,the ' " :~ '.'

- ¢o~plie~$ed :relhtlon~ involved in the 'flow uf '~ater. ,-It ~has as

~he• ~roble~. - '. .-' ~ ..... ' - :

"" :As :far es,~a pr~ ,.'t}:ca~. ~,l.~u:ica~ioa ,i' s." ..'concerned, :....

. ,con,..:,,t i,. ,he ..concAu.lon :,tb

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;:il , .: ?,,,. ::oomacn¢ o# :Pro:~osed :brioge,~ ,in :mos~;.-cases;::has,_ :now ;become .... • ., @

,,-..:~ ...This ;~ceede con~incir.gly :from this d n : e ~ t : i g e t i o n . . ':.

:.?:",., . . - . . . _,_~c.:,.. a,,,~,ow~ed~|-ent.;~.:~"~: --- -- .:due :t~, ,~Dr,.~Bos,~:-,who:..o~v~ied :out :. : .: .... '

. e ~o~e .-obaer, vaticns .'on :thg -:.r,Oa el: :na ~he c • : alculatlons..:.

'" ..unaer the instz~,e.tlcn~; :of :the .author, .an}f :-.Who ~ai~o. ,played •an.i.~es~glal /:. :4" .

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t :o~ the/~raphleal procedure. : ' "

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~:xie+~'i:becauee of thb det:r[~!ent~il:~:action-.o~ , t h e bRc~Zer ..:.:,. : .~ ,

at ~bridge.,~, whlch,:ha s e:aaad ~%o i¢he ~dli'~ic~t't~es or :so ~m~ny ~ri~e,...~:ena ....

m.,a~y ¢~e~ to co~tiy.,¢n~,c~e s zn ~e~,~,'Cn.ana .e,~:~.:h~.,, ,:led. to :the ~ab~tu- ,-

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