A~rEi\ICA:\ SOCIETY OF CIVIL ENGli\EERS

T lL\ ;\' SAC T ION S

P:lp"r No. 32G9

\\>1:-'1) FORCES ON STRUCTURES

Fin.ll RC'port

T;lSk Committee all Wind Forces Committee all Loads aild Stresses

SI ruclural Division

FORE\\'OHD

TL(' pur" .. sI· ui this I'CI"rt is to assemble, eorrej;lte, and summarizl' ex!st­i;;~ ,:;;"rm."i'n; Ull the i.lct!)r:; til;lt ,klrrmillf' wind iorel's on slrllctul'l's. It Is inl.'I:," r: tv 11<' .. c<Jlllj.;,ct sonn'p of inI,)rmalion in a form lhat will be of prac­I1C;01 \:">\' : 0 (" i\' il ,. I;.,; 11('(' rs. :\0 new rl'Sl';l n' iJ IS ill'('S ('Ill ed, but all evaluat ion of ,:dstin~ d;;ta i:.li;., aecuIllj1anying compn:hC'nsivc fvrm has probably not been puhL::,:jt'U i)l~L'\'lU'J51y.

T~.;s rq)ort is tile resuit of a cumbined dfur! b~' all mcmbcrs of the Task Con:r .• itle,' t'x!elOding ()Vt~l" a !wl"iod uf sl,ver;d yearF>. A preliminary repnrt \\' •• 5 iJU:);isllCdas PrQce('di"L:;s Papers 1707 throuI-;h 1712 in the July 1958 issue of the "ourn:ll of thc Structural Dh'isioll, Subsequent issuE'S contained discus­sion tli;\t \O.':\S ta;,,'!] into account ill tlIP prf'paratioll uf this Final Report.

I. INTRODUCTION AND HISTORY

Tiwt('!'h;,i,':d 'J!(n·'.ll'!' of the past ISO VI' (since the ('arly 1800's) contains 1".(:.', r"';l (I'I.e,'s to stl"u('tural fJilul'Ps c;)uscd by wil~ct. ;.Iallv of these have b"l';; ;)I'i(:;~1' i;ti:ul'l'S a I;d perhaj)s thl' f'arlicst rCj>urteQ '.>:;\5 that of IhC',Dryhurr,h hi,I ... ·\·',d!OpcnsJOil b;-ilk" ilo Scotland in 1818. This beban a long iustory of dUfI­cull:. \vitil S\lSjWnSlOll hrict;;es primarily !'zmsl'd hy til!' dynamic e[fpcts of the Wind, It is Illt{"rt'stil<G t'J nolf' tll".t an E'yc'witncss to the failure, in 1836. of the Br;~~:;:l)n Ch;;in P;cr 13nci.:e (Bribhtun •. En~land) dCl'cri\)cd a catastroilhlc mo­; ion ";1"'["''''; to thZlt which destroyed the Wheeling Bridge (Wheeling, W. Va.) il; Hj~,t" al;d the Tacum<l i'<arrows I3rid~e (Tacoma, Wash.) in 1940. It Is ap­parf'l.t that the art of dcsiGning against wind had made little progress during this ill'riod.

1124

WI~~D FOHCI·:S 1125 :

F;Jilurcs·h.1VC· r,lIt 1)1'('11 ('Ilnfil"'cI to suspension brid;.:,cs. Pprhaps the mo!'.t :ra~ic failure d1le to wind W:lS lklt of \),1' Tay Bricl"~r in SrolL'wd in 1879 that rcsullrd in till' d,·"tils of sC\'C'llly-fin: I"-I's',ns. Thl' inquiry into the failurl' de­velopl'd t!.'~f;!d that the c{f"ctnf wind h:ldn;,t iW"11 cOlisinc),f'nal1rillo pro\'!si"11 had \)(,CI1 Illa(k to r.-sisllln'rt\lr!lil'!~ (If tbro structur('. Sim!laro\,f'rturnin~ fall­urI's occurred un til(' Smith A\'pll\lf' Brid;:" (St. Paul. Millll.) in 1904 and the Cllester i~ridl:(' (C!ll·stcr. Ill.) o\'l'r t!H' ~.lississippi Hiver in 19.;4. The latter two wpre, aplMI'clltly the r('su1l of t',Il'1oadocs. Dliiicuitil's havl' also bccil I'n­counterf'd with uscillatioll of I he hani:"rs nil arch hridg"s.

Thl're art' also nunwr1lUS exam!'l"s 'Jf Imildinl: faill1rrs due to wind. Many of tlwse have occurrl'd during ('rf'('1 I'HI, which may indicat!' a carelessnl'ss Oil

tl;c part of CI.,:i 1l(,E'rs with regard tn the pffecI s of wind. One of the Illllrp re­eel;! I'x:lIUph-s IS t hI' colbJlse of the Union Carbide Buildin;:: in Toronto, Canada ill 1958.

A:: i:nur;h civ il PI1!:incers han' had difficulties in df'si~:ning st r'lcturrs to rl'­sist w;l.d. t;il'Y iJ: .... e made many e£furts to understand tlll' problem beltl'r anll as a result. (',l,:s;(\crable cl;)1;1 have accumulated. Exp('rimcnts d:llE' \ .. ' .. G:lliii") (101)1 and ;o.:ewton (102) and :\I'P contilluillp: at thepresf'JlI' i'· The ilistory of the invcsti~aliuns preceding 1894 has be£'n fully , H. I'Jxh\' (105) ;u;o more bricfly oullilied by I1nbins Fleming (1J \01.

j)l'rtilH'l.t data c()verinf' ('n:::ineeri;l,: applic.ltiol.S were outlin£'d by W. Watters P:l::c,n. F. ASCE (112). ;'lore r,,{,('j',l :ldditiolls to the fund of knowled~e have res.:it("dir.dir,'ctlyfromthe c!t'\'dopmf'ntuf aircraft. The wind tunnel provides a facliity fol' 1;l<':1s\lrin~ t1;(" [orC(,,5 I'esull inl! from th!' wind and advanccs in the field uf fluid JT;1'l'!;a;;ics permit a rational interilrl'tation of wind-tunnel tests. Structural as well as aeronautical ('ilhineers .ire indebtl'd to thf' rl'search of the ;-';-atior.al Advisor\' C'Jli1mitlce for Aeronautics. 110W (1961) the National Aeronautics and Spac~ Adminis: ration (1\'ASA) •

• he early experln]('I.ts of N. V. Duci;e'min (10,1) and John SmC'aton (103) f .. ,rr.1I'd the has;s for gC'llerally accl'!'t cd formulas for wind forcC's on strllctures. Those oi B. na~cr (l05) in cOl1lwclion with the Firth of Forth Bridge in Scot­bl.c had an importaEt influl'nce on bridge sl1l'cificatiol1s. Wind-t'unr,el tests startl'd with tlnse of A. G. Eiffl'l (107) on flat plates. These were followC'd by tests on various shapes (lOS).

Tests on :11odf'1s of buihLngs bq:,;Jn with those of J. O. V. Irminger and C. :\okken:ved (405) in lSSS. These' were followed by others, notably by H. L. Drydr-n and G. C. Hill (·W2). Oll;cr noteworthy im'csti!-:ations are those made in COI1lH'CtiOl' with the Empire State nuHding (NC'w York. N. Y.) that Includl'd both modI' 1 and lido rxpC'rim"nts (411) (422). A former committl'p of ASCE ;))'('p;,reo ~ln ('xtc;;,sive rej10rt on wind braCing in steel buildings that also rl'c­om mended (;csign wind Iv;lcts (42:». :-'lore recent investil'ations Include those at the' Iowa L1~1 itl;te of Hydra'ulie Hcsl'arch, Al1H'S (434).

For vjwl1-framC' sl rue'lures. thc t ('sts 011 simple shapes were followed by tpsts on ::;o,:ds of truss!'s and compicte iJrld;,!C's by Ludwig Prandtl and A. Belz (lOai and hy O. Flachsb.lrt ;1I;c! H. Wl:itcr (111). The research follo\l,;ng the f:dhlre oi the Tacom;J :\ar1'o\':s nrid;:e has contributed to the knowledge of both aer()dY";lmic and aerostalic forces (113). More recent tests on complete bri<4;e models were c"Jl(;acted at Uw David Taylor Model Basin, Washin!;ton, D. C. for thc Highway Hest~arch I30ard (50G) and at the Massachusetts Institute of Tech­nulogy, Car.1bridt;e (50i) (509).

1 ~umel'als in parenlh .. ses ref .. r to corresponding Items in AppendiX I.

\'.'i~D FOHCES

Tilc .:I'c:l:.St cUlitfJ!',:L.q.s 11> th.' fi('!d of 1lI.'t n ornlo[:v h;I\'f' h""'l til<' obsf'r­V.lt; •. ,.:; ., ,,j ;",;dyC'{'~ of the W.':;U.{'r i1url'au. United Slatl's D"partlllcnt qf COIll­

:~ .• ';·d' (uSWHl. 01her ill\'.'sti~:.lI i"ns havf' providl'd data rm thO' naturt' of till w;:, .. -SU('.1 a:-; hUS, vdocities, v";ocilv variation with Iwi"ht, and similar plw­L ,Ill";;:! (20lJ) (213).

D.'s:'it(' thc hilurl's Ihat !l;1\'e oCclIl'rf'd and despite till' d;]ta that hav(' bl"'n ;Jcnll;;\,l.,tcd, Civil l·;;;;;".· •. ·'·5 still us(' hellcral wind loadings fl)r df'si~:ns that (~O .ilu1 :l~li<r ~i)l·l·iflCdl~v t(} tLcfurnl i){"jn~ considf'red and do not provid,.. accu­ran' ")I.Si.,U'l1t W.I!l":;:~:I;]gkll"\'.'l, .1!:,.' of wind forces, In view ()( th(';l\l";]rJ('l's :l1:ld.' in s~l'actl;;':tl t! •• on',. lilld tlll' us.' of mnre rdincd dl'sign methods hy st ruo.:­'Il;·.il ell;:;!; ,"",'S, thc ;.ccllracy (Jf th,' dt'u'rminati()11 of forcl's on structurcs h:.s b('Cu~.H' 11l0rl.1 i n1portLH,t.

T .. ,' wlild forces on any,,; ru(';urp, or on :l1,~' f'lplllcnt therf'of, rf'sult frolll 1::(' (i;if,,;,,·;lt i.1I pn'.,sl.res oliO such Pl"illl'iltS GHls('d by thp obstruction to thc :;-"c nu"'" (Jf the Wind, T;ll';,C'fnl"l', Illest' fo1'c('s are functions of the vP]ocity nf t!iC wi;.,;:(;;" ni :;:c nrie:.tatinn, :Irc:!, 'Inc! shapc of the elpl11ents, Data as to the Il.l:ur" (Ii 111<' Will,! cnn1<' f;'''I:! I Ill' science of 1ll.'t~'o1'nlo~y; that covering the .·('si.:;( :IlK,' of olJjl'cts tn; ;,l' i;, IW "f thp willa from thf' SCil'l;ce of fluid mechanics.

TIK ill': ,r ,~lll1at i"\i~ uf w;i.d forces OIl:1 stnlcture is baSically a dynamic proh­lem, ;illwl'\"f~r, it h'iS b('cf] USI •. :1 111';1<',icc to treat such forces as static loads, T;us ,.ppr.),lch is s"tisL,ctnry. 1'1'(1\";(:,,(/ that the re1atio['~<;hip fx>tween the time v;.1'Jallo)lI of tll(, Wil,d ,<lId th .. n .• ll1.',11 frl'qut'l'ocies of the structure is such as to C;I,15e an l'sse;,tially static r"Sp'lllSf', Hnwpver, even In such cases, the engi­/leer shOuld be cO';I,izant o( the dY:;;lmic character of the problem.

II. Nt. TURE OF THE WIND

I;';THODUCTIO;-';

.; .... ! w;:;d pnSSl'50".'S 1·~n('Lc p;H'r,:\' liy virtll(' of the vclocity and Ill:lSS of the nl l /vi:.;.: ail". jf ai, oL,;;1.l,·lt" i:;; Il~.lc('d ill the path of thl' wind 50 that the nlo\~il.~~ ;].1' is stoPPc-Gor is (>flt'ctcd fr')lll ;ts path, th(>11 alInr part of thekint'tic cl1cr­~.; of c:lei! fdame:;t "f mO\'illh ":1' is tr:lll.o;furnwd II:to the potential pnl.'rgyof iil'l'SStlrc, The O:ltCl1sit\' "f prcSSllI'f' at any point depPllds on the sh:lpe of the "iJ:;;;]c!I', the ali!:l,> of lJ;cid('J;('c of the wind, and the vl.'locity and density of the :J .1'.

If th .. ;iir \\",'1",' ;]I~":.ys '111i"t, it w'Jllld not he necessary for engineers to COIi­sid.:r Wind forces wilen (1o-si::Li l't; sl noci 1Jn's. Th,'y would nCl.'d tn ennsickr only ll." :·.~.ltlC pn'ssur(' I~jC tn lb,' wei~:ilt of till' air. HnwcV('r, thp air is sPld"m (~,,;;('i. i);.; tjq\\·s, sonp't1nl('s ',\"jth ~~l·(·.lt viol"I1C(', under t11(\ influPllcr of fnrc('s

,.·.,· ... 1";',1·.1 b:: !le.ll 11'<1111 till' s,m .. n,d 1!1" c(':.lriftl~al f"rees resll1tll1~ from thc ,· .• nh's r"ta,ioll, TIH' wind 011:1:; hI' dt'fjLt'das motiull of tl,e air caused by !:rav­it\", I;V rl<'flccti\'(' force'S dul' til l11t' (':ll'th's rotation, :lnd by centrifll[:al forcps CI:" : U the ('un'atllre (,f tb,' \"1 I,d p.l th, Th,'se iorces are oppos('d by ot h(' 1'5 ar is­i:'i: from friction a:;ri \"isC:ll.'.it·,·. T~l" air Ill'\·"r flows with a perfectly smooth ;,;,,1 str,'amline mution, Ilia ;;Jways with horizontal ana vertical fluctuatiol's \':1;]('11. wi;l'n su.k!ell and reJ..ti\""l:.hl'Jd, are called ~sts. The masses of air ir,volved in r,lIstiill'SS may Simultaneously, and in the same area, cover a wide range of sizes from vcry small to very brge.

- -- -~ ---

WL"D FOHCES 1127

This par~ of tIll' rl'l1':ll1. d(>als oniv with thl: wind vl'1<>cities that shnuld bt' cn;lshk red hy :-,1 ruclural en~;in('('rs, hisI'd on pro'SPllt (1961) inforllJation. It dn"s nut (ka I wit h the shape factors that ma~' 1,<, :'PI'J'npriat(' (or va rlo)us st nlc­lures bccause tlll'~'_' will IJl' ('valu<llcd su\)scquf'lltly: brief r(>cOmI1lCnl\aUons are r.;ivl'n for l:tiSt (actors.

ORIGIN OF STHO~G WI~DS

St:",~·:ical studi('s of til(' records ,A wind Ohsl'r\'alillns in differenl parts nf U", liliit.-d S::,;,'s ;;il!JW that tlie prob:lbilily of stron~ Willd is not th(>same inall t:l'''::Llph:c;!l ,11't':\S. TI:e nature uf thl' storms In which slroll!; winds occur Is ildtl'r un,io-rstn,)dnnw than when radar lll1daircrait wer.' Ilotavallabk (or me­tro ;-nl",.: ic a 1 illV('st i~al i(lll. There is sti II much I,) be lea rnl'd a h"\lt t he frt~uency ()i ;;il'III.;, wi;.ds :li-,d t;lC \';,riatiotlo( ""ind velocity wilh height, but engineers cailr",' w;,;t in:' 11,.'r[(-cl information; they must procced all existing knowledbc ai~d i:;:; 1,npl!c~1t 10::S.

Wind it);'C('5 act i:;:; 011 stn;<:\I;rI'S ar(' Siplificantly 1.lr[;(' only durinf: strong w;, .. :" .,IOli th,>,<,;,> "ccur only o\:ri,,:; sttlrms. In this rcpol't, th('re(ore, onlytropi­c., I c:;c j, ':i,'5, • ':-;1 ra IrOj1icai CyclcH.e,;, :ll1d tornado('s arc discussl'd bpcaus(' t hcse ;11',' ,;", St"l'IllS tl;;lt inc'ju(i" thc witHi furces thaI ,Ire uf interest to the structural ,"-'nr:.1"l\cr in his (h·si~l"'.s.

(;,':, ',',d C, ,'(',dalill,/ ,,( 1;/,' ,II'do_'I'/lcr,', - ThC' t:el1('ral circulation of the at­n~()Si':l('l'C is U1l' term apP];l'(l to tlit' av(,rage mo\'('nlf'l:ts of the atmosphere wLj ': h, 1 I; t 1:(' L. j ,I'!;e l;.titUl:CS cf tor J;,)rthf'rnlw misplH're a re fl" 1m west to east, ;lll<l i;lliw trOili"a! b,itlldes, alld SO nleti III rs ill tlw polar latitudes, are from east I,) w('st. T:lcrc is a p"imlal' imprcssioil that ti.is i1' a prcrlominanlly rq,'Ular ClIl.(ilt;on of flow tll;lt is occasiollally distnrbeo by storms or other local devia­li'H-,SI;;at ;1l'P ;iota nec'C'ssary P;ll'tllf a-.. ' !:C'neral cireuLlI ion. It is al1'o thought lii.lt the w.n'm air of tlll' tropical regions rises with a drlft toward the polcs a .. .; t;l;lt the <,,)Id :.ir of tile polar rt'~iOIlS sinks and drifts toward the tropics, 'lI.d th,ll :;;'S(' ll1l'ridiol.al IlhJVpml.'nts are likl'wise predominantly rl'hu1ar. But t:-.C'sC' ,)'.'o.!rsiul;'Lfieu\·iews arenotthose whichprc\'aii (;u; of 1961) among mc­tt-'(Ir" )~o::::if,; s a :,d physicists.

It is trlll~ th:lt ill th" t r')pies more hl'at is received from 1 he SUIl than is lost t::rry..;r;h l"ng-w<I\'e r;;(:iallOn into uuter spacl'. ilnd til;)t in the polar regions the revcrse is trul'. It IS also trul' that the dynamic ba!:tlicl' requires a trami(cr oi an;:.ular nlOlilcl1tum vertically ,\lid horizol,lally within the atmosplwre, and t:l;lt lilctl'.1I1SI"r "f kll,ptic rllel';,y is conlillu;tliyLlidn;: placl'. [\\It this dynam­ic b"l:tncr:. t!;,'rmlll and l11l'ch.u-,il'al, is llIuch too com].lt-x to he ;tchir\'C'd by stn',didd.r flrlW. It rpquil""s a turbulent svSlf'm in which the prn~:rcssioll o( Uw C',Tlulll'S ,Iilei :lli' IC\'c!OIll'S uf the middle bt ltudl's, for ('xample, arc cells o( t:.e Il['('(ssary sy:;tcm of turbulpllcc. If the velocities of t.hc winds that occur •• t a "ai·tlCulal' ;OCiltirm are avrr:lged over a long period. as. for example, a ;"1<'11.1 h. or a season, t1i1' aver;\;.;" thus obtai:.ed may be said to reprps{,llt the bL'.lcr:-.1 c;rn.lalioll of the atmosphpre at that 10cation. Bilt to the structural enr.;i:iceJ' tl.C' high vclocitiC's of the passlr.:; storms are of greater importance. A short a;;,.1:;S11' is theI'e(ure given herewith of those components of the general circubtion that give rise to strong Winds, namely extratropicai cycloncs. tropi­cal cyclones, and tornadoes.

: 12(\ W!i:n FORCLS

j'>ln;:rn!'U'(I! C". !"",'~,-A c~'d')I1C' is I",!"t df'scril",d by rdcl'('lIceto a syn­",':IC ",,-.;]1/,,,1' n,;,;'. wlwrc thl' isolJ:1rs ;11'(' lil,0s p( "'111al \):1r('l11elri(' pr('sslIr", ti; .. :, In tlw ;:"ill'd S:atl's, Is expl'4'ss('<1 in inclws of lll('rcl1ry r('duc('d Il. S(':l­

:"~"'l (;,,)1,::: i,'''s; til.' wint! <lir('ct ifln is sh~'wJ: hY;1rrf)WS, TI1l' circulation Is f'j)::'~,llv i;,',\'"rd in a c\luntprrlockwisc Sf'J1<;P ;uouml a c;'clon(', ;\l1d is spir;1l1y OU(W;i1',j in :\ ciockwisf' S('Il.';C :1rolll,d :111 antkvdnne,

Af. ;\ en!d ;'tir llL\SS ;ldv,1I1C(,S it is rf'tar,j( d \l'; fri('ljon c;1usf'd bytlw r:rol1r1el, T:',c '('I;Il'd,IL"" if' tr;1nslllittedupw;1rd hy till' ,'d,I'; \"lSC"l'itv of th(' :1ir, and the ',;;','),'r cold fro"t, to a hl'igLt nf s('\'('ral tho\1sand !Pct, runs ahrad of thr eolel ~ .. , • ,,: Ih,' ,',r.,)I;;,d. Till' il,t(,I'\'enill~ Vl'11ic;l\ di:;t,llH'P is fill('d with \l;arlll air ,l\"';' ",.rm ;,l'"ulidand I), <'.,U5(' the culn. or:; "ir ;l\oft is h(-;""if'r Ulan the warm a:. ;)('l,,'';'', ;;;"1'<' is crl';,[eo a cnqdition of ill~Lthility in which \'erlic.~l mixing o! ;;.(' :ii.L,i:"'; pLI(,(" HiJ.:h winovdocitiPs ,lrI' :Il't to oc('ur in this region be­e,lll,-C jl .• rl'ds of the f.lst-movin", IlppC'r air sink and r(';!ch I Ill' {:round before ;"fil":: a;:(,;~ tLei;' ('xref>S forw:,rd I< 10 ll1Pl1tu 111, urhecausethe hea\'~' LIst-moving u,'.<I" ;lir f.d:~ for',,'anl in a manner silllibr tu breakers on an ocean beach.

,\1 til" \\',.r~~, frollt tl:c warlll all' urdin:lril~' ac!\'ances over the cold air at a sll)w,'r L\i,' Un .. ;1, thl' cold front and UI)(kr conditions that prnll10te stauillty ;1I,(; a tC':,,~;(,l:C\" ;" Sl.i[::::lli,)n 01 tIl(' air, TIH'rC' is, therdore, littl", or no wind llf ;I,tcrl'st to tIl(' ~,~1-UCti...il~.11 ('n~j;.('t'r in this region.

T;,e l x; Lit r"pical cyclolwS :I:od ar.ticychmt's form a con! t.1Uf)US prnc('ssion ;:1tl1e mid . .l:"l.1titllllo-S, 1",J\'!i:g aiur,g pathst:latare :lpproximatel~' wcst to cast. Th"~' erOS'-1 :',., ~ 1.;[, d S;.,t,,;; at a rat<' of :11'1ut 2 C'yclones per w('ck. Each ey­e!· ,ne is us;];,il\' :1I~cC)m;1a"i('d 1):,' an ;11.lic~'clni'(, but this jlairir,}; is far from rl'h\;l.. )', ";,hp S\'lluj,tic w('.i~ht'r 1l1,1pS will ircqu('ntly show :tn Irrq,"1.Jlar clustc,r .,; r::d()I;t's ,1);1\ :~l.l:.' nh r,:c ir.tu a lal',.;cr cyelol,l' as thry move across the L;;'l~(.G S1;lt(".~; the sal':j"thlEr; iSlrdC of t)4I{'all~icyclnnes. The sha,lcnf thranti­Cy,~:rli~t")S ib ;,'Jt ;1;Y:J~\·s r(l~ll:lr arid their l"f'l;1tiri;l !cllhC' prpcpding and succC'cd-1;,1: c~'c llll:' 's b I I.";";; U((·. ;\I.'\'l'n Iwll'ss, tIl(' proccssi'lIl is a continuous phe­'l\lj~.('l/{lr .• I ."::j);'c:~lLy ill the \VI,itf'r ;lJ~d car!:; sjn-in(:. SJnle extratrop~c:lI cy­clfl:'('S r,,'rc3ist ~':'ou::,: li;e w(lrl<:' Som(' nrit:\n;,le as hurricanes (tropical cy­C:')Ll'S) til'lt L:l\"C I~)f":cd :~ol·tL\\"J.rJ fr')n1 the tropics, as \\'ill be discussed sub­f.l'q,~;t,:.t::,'. :',;, ,st of 111(' storm winds to which structure's inthe middle latitudes ;(re sul.j' ct.'do(:c"r I;; fill' southwest quadranl qf passi-l1f: cyclonic sturms in thc v.'inter ()r (';Ir~:: !-Ji)rin~.

Udnrllli;:.t('iy, Ii. 5"101(' parts of tllC' Uni:pd St;]t('s, therc hilS 1<J1'f: lJf'cn a tr'nd­("I,,-.,. to c()r:fusp the v..°(Jrd "'cycln!lf'" alIt! "tnrnario." Torn:lQP('s and hurricanes ~ll'.' CYc!·",jr 51 <)nllS, lJiI must rY"!fln!'s arc neither tornadops nOl' hurricancs, w:J;rh .1.:-. in; \ul,,\\(' i;,d('cd.

T.-,) "J( II! C.,.<I"I'.", - ';-:w ,',,;:1 i 'I"nt;11 arl'as of the Ullill'rl St :lIps ;( rc :l.f["ctC'd c;llPfly'lw Ihll!,0 hurr;c;ilw:, (t ,'op;(',,1 Cyc!cl/.f's) that ()ri~ii1:l.t(' ill the Atlantic 0"",1" ;'S LIZ' (,,1st .'5 1:1,' Cl;1P V"l'(le Islands (213) and i" the Western C:1rib­!"',lI'. Ti,"j'(' ar·' :11c.(j hurnc;l;i('S lhal ()ri;;i;"lIe in the Pacific Ocean of[ the c. ':1,'1 (,f :.1r'x :Cll ;tl~,d S',lUt hi r:, C .. ii fnr"L1, \J:lt t lH'sf' starn,s 1110\'C nul tu sea wilh­,1\,t :likclin:, Ihe UI>it,'d Stall's. Tile numilcr of hurrican('s that hav., reach£'d \:;" U;,.lrd S:atps irum 18H7 tu 195G is 143 with an averaf;c of 2 per yr and a ;';II;~:(' frniH 0 i.J G ;1('r yr (212).

',;1<.: llurrical~c "ropicai CI'dolle) is radica!l,. dU'ferent from the extratropi­ca 1 cyclune iii f'cv('ral ",'spects. and the assumption that any I>Jrt ion of the storm jI'}SSPSSI'S a s~l'ad? stat.' dynan:ic condition ill a closed system is not tenable Without a grl'at deal of additional information. It must be recognized also that,

WI:-;n FOHCES 1129

due to:!,(' limitrr\ numb!'r of stalilJlls llnthe coa~lal areas. and pal'lirularly the limited liumb(,l" of v('ssds _,t s('a that can m,lk(' ac\f"llI:tt(' r£'l"I't5. the spadn;; ;l.Ild curvatur(' of Ih,' Isollars :,lay be drawn onl.,' with cOllsicjl'rahle chance of ('rr,'r. TIll're III 11 1-;1 1)(' cOllsid,'rablc dollbt that th(' saml:' I"(-Iationship pr('valls bf'tw('('n the ~r;Hii('nt wind .wd the surface pn~ssurl? !;radil-nts, as that which ,'xists in till' I:'xtl'atrlJp'cal cycln:.f's.

A hllrrican(' has hCl'1l likl'll(,o (1) a h('at I:'ngillt' (209) that is constantly re­cd\'il.;.:: pncrr;y from tllf' walt'r v,ql'.r that is hdn;.: c:lI"l'ipd alcoft fr"m th(' trnpi­raloCl';lll. BII('r!;}' is r,'k,ls£'(\ in vast quant itie5 when the v:lp'.r is cond£'ma'd and falls in torrential rains. It is not impossiblP, therl'ior(', that }J('at ener~ iii ;.::r(,.1t qll:lnt iti4?s bC'cllnles .1vailai'lr' at r('btivl'iy low cleva! iOIlS, alld Is, at lea:;1 ill part, !r;lI,SfVrllll'O i:,to the ml'charlical ('ncrr;y nf wind vplocity.

TIl<' t1H'uri('s of the V. W. Eklllan (201) and C. G. Basshy (202j spiral~ are not ajlpropriatcfor 51\1(\\'111;: the variation of windvclocities with heil:ht in ma­tur .. ln;rric,lIl(,s, pal'Uv i'}rthei(\r('~()i:.; reasons andalsoh('cause of th£' rather n;lrrow spiral rain bar.(!s that charactf'rizl' the stl"UGture of typical hurricanes and tYi,houns. Tlw spiral bands arc' said to mer~e into a continuous ring as tiwy approach tL(' <:('I1.ral eye of th(' storlll and it is in tllis portion of tl]£' storm that tI1o-' s(ron~('st winds uccur. InDePi)ermann's typhoon model (214) the maxi­mum perij,her;tl wind vclocllies pr:cur :lh()\It 30 mll('s from the center of the eye ana deGrease r'lpiclly both tow:lrci and away from the eye:

"1;-. intense immatur4? storms. vdncities increase until a few minutes before the calm, hu: in the mature and ci('c:lyin,; stages till' hi;~hest winns may occur as much as two hours before and onc hOl,r aitf'r tlw central C;,:I:J ••• * III the br:;e Cape Verde storms, the diantHer of hurricane winds may exce('d 100 miles by t he time the storm has r('3ch('d the west­l'rn Atlantic.»

Fol' tlJ(> structlil'al en!;iIwer it is especi:llly iliterestin;: to llule that current (19Gl) informaUon re~ardint; matur(' hurncar;es seems to Incate tl1" most se­v('r,' and dangC'ruus turbule:lce b4?I,)w the m3in cloud formation. probably be­low 1,000 ft. It has been stated that tillS type of turbulcnce may lwcome ex­ccs;,j\'(, at 500 ft wah Winds over 80 knots (92 mph) and Ihat It t(,llc\S to dis­;'jl,w;,r rapidly as til(' altitudt, incrl'ases. Th('s,' observations are important to t1w slructural 4?ngint't'r Who must form some cstlmate vi th4? variation of wind \'r!. ,rity with heif;l.t licar the r;round. In the absence of detailed quantitative mC3Sllr(,IIH'111 s, made at Iwights that are important to structures lucatcd in the cO;,5ta1 ar"as th'lt han' ('xperiencL'd mature hurricanes, such qualitative ob­s('r\'3tions prm'icl(' usdul hackground matf'rial. It may 1)(' hup('d that quantita­tive rJ"St'J"\'atiol1s will he made uSln~ SOIJ1(, of the hi~:hpr towers in southern CII,lst;': al'eas. Oh~Pl'\'_IIioJls made ;]t the Brookh:lven l\ational l.aboratories, Dr00;~ha\',,,1, 1'. y" sUjljllr {!ood information re,:arcling the variation of wind w'lvc ity with hei!!ht duri nl;; decayinr: hurricanes (2 J 5) in coastal areas.

TOI"l/fu!u,'s ,-Torn~,dof's have the most violent wind \'elocities that occur in tile Unit('d States. ThC'lr occurreliCC is wid('sprcad in the eastern hall of tha­COlll",t~'r, Som£'times t1l1"Y are e,'cn fuund "elllbcdded" in southern hurricanes. Fortur,ately, the path of an individual tC,l'l1:ldu co\'('rs a rl:'latively small area aad althuugh about 700 were report4?d in the United States during 1954, the probability of occurrPllce in a hi~hly populated area, as in a city, is small. The peripheral wind velocities are frequently estimated to be in excess of 300 mph, based on the damage done. This is a dynamic pressure (velocity pres-

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Wlc,O FORCES

~I..l'~-.~ tlf :~::Jn l':"f. \t\~-r~ siJondi! ::to an ;r;er;)l:('dcsi~n prr'SSU(,C, 011 a srpJar(' flat ll~.'~(' oi :'J:\"l i/sL

;\1) pr"I"n,,!' i" ('\,,'!" 1)',I,i, Ihat stru('tur. s ar(' r\rsi>;ncd tf) withstand sll,h \- .. :"C;UI:S. :\t'v('rtlll·h':--,~. it is sunH:.'tinll'S a C;IUSL' of ~Ul1a7.C'Jll(,l~t tn f'l1t:illf'f'l-S

:;,;,1 fit!"1l.I.!!"('!'., app;ll"I I.I!'.-in Ilwdil'l'f'lP:llh of a tornado I:~\'(', if wpll d('si!',nl'd ;,nci w,'ll I"lilt, fiulf,T('U ,,101:. minor d;llll~~('. Scv('rE' (limal:" tn ~:ood \1\1ildini~s, wllt' .. it oceu,'s, S("'IllS II) resuit froJll til(' cxpj"sin? rl'l(,:lS(, of air prcssuH' within lr.lii(:lJ'~s wh(:J tile Vi'rv 1,)\\' pr,'ssure within th(' funnPl slldd('1l1y ('11\'('1-0pf'S t:.cnl Oil the dutsll;t'.

;\r) fllrtllCr cOllsi<ipr:d ion will hp ,:in'n h('rein to the SlliJjt'ct of t0rnatlpps. hC'(,JusC' 1 iH'rc IS n() d"/11;lIld f"r tilt' inclusion of their dfl'cts in speeificat ions.

THE GRADIE;\T W1;\0

1':,"1"(' is S"IlH' ,.r ;,:;;: ,tt which Ill!' i .. fllle;)CC of r~rnund frictiun. trallsll1Jt\{'d \];1\\':11'\i t:~r(fu'::l f'l:d\" \'j~ .. {·.,si~\\ h.ts a I:C~:ll~lblc cffl1ct ullth{'\Tlority of the wino

:is it ,','s,,"mb 1(1 ,;", jll ('s:,ur .. ~cr.!l;i(,lit. At thi::. h('u~ht tht' pr('ssure gradil'nt is ,~,(;'l to> "".iYIl.tllli(';,;l:, :>;)l:!i~('('J ,':~;li;;st two cnlllpOIWI:ls ansillg from ccntrifu­;';.; fl)r,;,c-oncrlut'tldiH' ,·"ta:iOil "f the earth and ti1eotherctucto thl' curvaf.ure (,; tilt"' win(! p,,: ll, '~'h(' wi; ,d velor it',' co m:lUlcd 011 th is basis is ter llH'd the "~rad­ii',.: wind," TiH'rt' "i1, . .,: he atram,itiunfr(lmthesuriace velocilytothe hradi<,nt VI'IO(,lty Olnd tilis is rd .. rreu to as Il".' variation uf wind vdncity with ht'i!;ht,

V,\lil\ ,';0:\ OF \VJ:':D VELOCITY WiTH HE1GHT

TL(' fJC'iors t'i~t! rJ1lt-: into ~ll(\ yarLtll\)n (If wllid velocity \\'ith height arr thr I Jrl'S~li rc ..:r.iij.t' II:' : 'Il' rn:l ss til1llSl::; of l he ;ti r, thp ;ll1~:ul;} r \'\--loc ity of th~ earth's r()t:l~iojl, llll,' bf'ogrdP:1,C lati~u,!t' at wili('h observations arC' nladf', the curvature "f Ih,' w;r~d p.ith, ,,,,<1 the cOf'fficlr'litof ,'dd:; viscosity of th(' air, The ('qllations ,'xjil'<'i'o"irii; t::L' rt Lilio;;s of these \'ari.lhles h.l',T i)('cl1 d('rived by Ekman (201) ;l;,d tllC r('sLI:t;;-,t: bl-;Iphic,d r,,'prl~[-,('lit~tiori (208) is kno\vJ1 astiH~ Ekn1an Spiral. ~: 1:1C :Ll'nry "f Lll' "mjxil~~ l,-,r,c;ill" is used instf'ad of the coefficient of ('dcf ... ViSC'J,,',\", \:,;, l·:;""Il,. (202) sjli!"~1 is ohtaincd. In tlus r<'port the theory of th~ L:kn;;lI, Spir;t1 wili 11<' "sed as a thr'o,·l'tic<ll l1lod('l ill presl'ntilig the results of nl,,<;el'v;dioliS in eXlr,.! ropir:ll cycIoIH's.

Wi:-;) VELOCITIES 1:-' THE C~ITEO STATES

',','""i ('00' c; mw:t lot' ],,,,,,,d Wi knf)\\'i('u(cp f)f th(' pr'oi13bilitv of occurl'('nC'c uf Ill.:o Wi ".j \'" bc JIll'S' I; Ita ill( d tr"llltlle sLll isti(,:ll an;lIy:sis (If wind \'ciocity rcc­(1,'.1:>. Tht'Ill')st v"l":: .. ;;,,us r. cords of :111S kinci arc take:l frol11 tllc oIJserva­I: ,;,S:tt lill' flrfit-Ilrckr sll.tilJilS of tlir' liSWI3. T;H'scd;,taar(' nothOl1logelleOIlS, :,' '\\"('\"'r, h'c'll'.ISl' 11,(' st:ll io .. s itan' bl'l'll located, ulltil rcc('nt years, in c it ips wl,\' l"(' t hl' t~('i~:Lt ~ If t hp inSi run}(lI1ts aLd the degree of exrxJsure were constantly \·:·",:.;iI'h willi the erection uf m'w, t<tll hllildil1t:s, and where the Jir flow was ,~'J1~H'tlnH's aiSlorl('<i bytlle buildings lIjlVn whiC'!, the instruments wt're mounted. Li I'('C";;t ::('ar5 th<' st.ltiol'S holve bet'll movccl to airports where a more uni­;'0;' i'I< u",:ree ,.f ,'xilUSUn! exists. Th<'rl'iore, the proper interpretation of these

lU",;-tillll' records I'l'ljuin'd expert ar.d painstakin~ study.

Wl;-;D FORCFS 1131

It is desirahh' I hat rodt' r('('oll1ll1elllbtions he b~·,t'd 011 Rome standard of re~('I'el'cc. Ruch as air flow ill Il'vC'1 open counlr: ... and t.hal the inIluf'llc(' of lo­cal sl.j('ldiq~ and uI,usual topugraphy be C'"al\l:1\l'd by Ihe designcr in E'ach in­divi.ll1:11 cas.',

l-'lgS. 1 (a) ami (b) a.c maps of the Unltl'ri Stales Ihat w,'rr prrp.lred b'1 H. C. S, Thom (216). Fig. 1 shows conlour lincs. ('ach of which gi\'cs the velocity of til,' i;;skht mile uf wind aftE'l' the obser\'E'd n'l"cit~, has b('('n J'('duCE'd In' the onc-sl'\'cnth-powC'r law to a hcight of 30 It aoov" I he ground. The contour lines wen' h.lsed on an avpral:e of ahlllt 15 y1' of USW13 rrcords at 14] airport sta­tions thrn,If',:lout :111' UnitC'd StatC's.

Tlip Wind 1'C'col'(:s w('rc C'x-trapo!:lled by Jllcans of statistical analysis to vr­l .. cities l:a\'inf: a 5J-yr probable perlodof rC'ClIl'Tl'IlCr in Figs. 1 (a) and a 100-yr pruiJahlc pcri0d iii Fi f:. 1 (b). On thp aVf'ra~:e for thc UnilC'd States the 100-yr willa is 1.08 times the 50-yr wind and the 200-yr \\ind is 1.16 timt's the 50-yr winri (21G). It should 1>1' nol£'lilhat thC'se al'l' ·probable" values anrl tl1.,t. over a JOl!~ iwriod oi lim.' tlipl'C may occur a 50-yr intC'rval which may include olle llr more has;c winds. h;"'j;;g- statlstlcally a "probahlf' period or recurrell('c" of 1 00 yr, :-;('vertht'ks"'. ll.is llll'.hod r,ives the desll~nlll!! engineer a ·conservative" ;I:od a "more CIIIlFl'rvativC''' basis for choosill'; the wind velocity that Is appro­p!'i:,1!' in tilc dcsi;:Jl oC each t::Jll' of structurf', takinG into account the wind sen­sit; \"ity of tlip Sll"uctllrf', and tlil' haz.lrd to UCI' and property that would accom­pany a failu!'C' of t1w st rilclul'C. Wind sensitivity depcl,ds on the importance of wiJ.d l()adin~ as cf,llOpared to the importance of all other kinds of loading In their effcct on til!' strength and stability of the structure.

Within :lliy ~;""n year th('rc is a 2~probahility that the 50-yr wind will be exce!:'ci.cd. This c:!IJes not mean that there is a 2'.tlchance of failure of the struc­tu;,\.' in allY ~i\,~il yC'ar, because It is also necessary to consider the pTohablli­tiC's im'(,lvC'd in the quality control of the structure. In determining the over­all "saCcty bctors" thCl'C arc many other variables, some of which cannot (as of 19(1) be <-'xalllinedstatistically, hut the subject of safety factors (217)(218) is not within thc sec'pe of this r('port.

STORM OBSERVA TIO:-;S

Wind [orecs a d in~ all st l'uctures arf' significantly large only durinlt strong winds, anri these uccur only during storms. It is essC'ntial to know the type of storms during whic!l the obscrvations were made because the variation or wind velocity with hpibht will be influC'nced hy the type of storm and the ter­rain. (or ".'atf'r area) ovcr which the wind has passed, Long-time averagcs that includC' hNl'rOhenC'ous llll'tcorological conditiuns can be very misleading. For c-xamplc, at 0111' l'nd of the possible rangC' is the tempC'raturC' bversion, during which the wi lid \'C'lolities are li~ht, the atmospherf' is so stable that tbprC' is little ,'ertlcal mixing, and thC' codficient of eddy viscosity is corre­sponrlin,:ly small. In that case thpre is litlle friction between the horizontal la),C'rs, ;,nd the velocity changes rapidly with height. It is not valid to use the rapid chanl~c llf velocity with height that occurs in an inverSion, when one is consideril.g .he \'aria~ion of storm winds with height. StOTlll observations should not be mL,ed with hctC'1'ogcneous obser\'ations, nor inland observations with coastal observations. Extratropical storms should be studied separately from hurricanes, and mature hurricanes separately from decaying ones (215).

11:.2 WI:-;n FORCES

;'. -. -

WIND FORe ES 1133

WJ~D FORCES

F:,!rafmf,j(',il C ... d,)IICS.-,C'lrY"S A and I3 in Fig. 2 are baspd on data tak .. n d,::-lI:i; a I~.od('r;lld:,'ir.t":',;l' sto,m at Ann Ario()r, Mich. on January 19, IH3~\, wlth the wi;;d ;)lowil1;, 3cr055 Liri,. 1~\'C'l, open farm land. at an inland location (21)8). Lin,~ C S;)OWS the COllOjlU!.''' g;radiPJlt velocity of 63 mph based on the pressure gradient and curvature uf tile wind path as obtained from a specially

FIe:. 2.-\\·j:-:n \'ELOCTTY FOH EXTR,\THOPICAL ST()jDl~

c<Jl.st;'ucted synoptic map. Curyes A and I3 arc Ekman Spirals fitted to the data uil'Otilll'dat five staliol.s on a2:;O-ft tower. The solJd port ion of the curve shows the ra"be of the d;,ta within which the fitting was done, while the dash part is extrapolated toward the gr3dlent level. Curve A Is for the fastest 5 min during

Wl:--<D FOHCES 1135

Lie storm, ;,,~r\ CIlr\"I' B is for the f;'Sfl'5t I mill th;!!, inrid"Ii!;llli', ()ccurrcn f!Ul'l:,;:ti;c s:lI~;e 5-Tllin pf'l'io<L Illcachc;lse't1H' ,·x!r.'1vll:tI<civOlluc of 'he grad­ic/;t wind is shnwll rl,r cIIll1p;lrisOIl with the cf).npl1t,'d \'alll!',

Th,' Ekil1:lll Sjliral postllLI!!'S stf';Hly st.",' dynamic condit iOlls wit hi nIh!' pnr­(if)l; uf tile' storm uJ;tif'r cl)llsidcratl<ln. If on!! accppts the llJl'ory uf the Ekman Spi rill as api,1kd til 1:1I"! S, it will be SPCIl ! hal the ti mp l'f'quin'd for the estab­l;shmt'nt of c(J;~tii: lOllS suitable for the applicltioll of this throry is somcwhere D('I wrcn 1 min alld S min.

Curves nand L arc iJ,lsrd on r\:Ita taken at thl' nrookhaven Lahnratori('s d\\ril'.~; the' very ;I01l'I1SC l'xtratropical !'torm ()f Nove'llIbcr 25, 1950. The wind \\':,5 hluwil1h ()\"E'r Uw f)ceallfrom I he C;Ist a nd occasionally from the Ilorthwpst. T!ie station is loc .. ttc'd st'veral miles inland from till' coast, but it is d('finitely in :\ cU.:1stal:1 rea. TIle' i:ltervpnin,; countryside Is sli~htly rolling with moderate 5mall-trC'l' gr'jwth. The ubsC'rvalions were' made with aerovanes mOllntedon a 410-ft !nwl'r. Tiw poin~s to which the curves are fitted arc shown with "plus· 1.1arks .

• LlI1e F sho',\,:, tll" ;.:r;ldi,':;t wind velocity cumputed irom the spacin[! and cur­vature o[ tile iSf,\)ars;'ls showll iii aSj)('ci:11pamphlct fortha! storm (207). The r.lI1:,(' of computed values {til' the ~r~,dient wind velocity was from 104 mph to 111) 11l;'h. ciC'prr;.:i",; 0;; wiwlhertbc curYalurf' (,f the wind path was based on the curvature of tlh~ is"b~,rs or on the wir,ci <:ireclioll as indicated by the arrows. Curve D is an E:':man Spiral [ia"d tu the m:Lximum 6-min average v('10c1l1es at the four n,'o.,i;;.ay(';) tower statinns. The ma..ximu;n vdocities shown in Fig. 2 ;-,re nut coincid",;I:1i for all stations, ;IS were the d3ta (or Curves A and B, hut tht·}' aretl1l'h:;:i.e,.;t valu{'sthatwer(' observpdat each station during the storm. The uiJsrl'v:nio,',s «.ffe.:<'d in time by as much as 2.5 hr.

CUT\'C E is a n Ekman Spiral fitted to the apr'aks" of this storm and (''(trapo­bled by <i:tsh 1;;:C5 to the gradient le\'e1. Althou~:h the fit of the spiral to the ol>st'rved ';C10cltWS is Yl~ry ~ood. the accompanying value of gradient velocity is not iii perfect agreement with th(' cO:iljluted value. The period of time nec­e5S:lry to establish cOl:ditions appropriate for tlle application of the Ekman Spiral thf'ory to these observations, lit'S somewhere between the period of the My.';! ks- and the pcri(,d of 6 min. This agrees with I he observations in the pre­\'ious storm wlH'n the appropriate period appe:U'ed to lie between 1 min and 5 min.

1. shuuld be not£'o in Ihf' application of thf' Ekman SpIral that not only are s.e~,dy Slate d:.'Ilamic conditions assumed, but also that this implies that the poI'lion of ti;e stnrm under ohsen'ation is a closed system, or at least that the add;t iOil of l'nf'r~y tn tilis systcm from an outside source is just exactly suffi­ckr.1 to hahnce the loss of energy thr()u~h frlction and Viscosity. During the early 1>la;;('s In the grov:th of an extratropicai cyclone it is probably true that the system 1S baining en('r~ (rom outside sourc/'s sufficient to promote its devdop;1lI';lt, al.d that during the mature stages of the cyclone the str'ady state conditions 111"Y be approximated. Durin~ the later star;es when the cyclone is d('Clyi:1~, t!l(' loss of ener,,'Y undoubtedly exceeds the gain from sources out­side th(' s\'stt'lll. Under th!'se conditions one would expect the theory of the Ekm;l:1 S;liral to fit the data best during the mature stages of the extratropical cyclone.

Both the storm at Ann Arbor and the one at Brookhaven are extratropical cyclolOes of continciltal origin. The center of the first storm traveled com­pletely across the United States and out over the St. Lawrence Gulf In approxl-

.. ---.'--~ .. --. --~'-------.-- ----.----

:11;I:pl:. :1 we;'I-to-I'a!"! ciJI'('rti"l1. Th(' sccond storm W:15 vrr\" romplicatf'd: i:s CClll.'r uri:'jD;lft'tI ill 1,l)rtill';lhtf'fn Tf~llJIPSSr~, tr;l\'p}£'d s(}u1h into G('()r~:ia. 1':1,,1 thr"u:;h till' C.II'nlh:.;::;. ""d IlI,rlll int" 11i(' norUlf'rn porli')11 of Virl~il1h. lkn' thr low pr,.ssurr ;,r",l dl\·iol,.d and the \~'('~,trrll ))r'l·tioll tr:l..-rlf'd into the OL:o :11'd Lake Eric r";':":5 \\1.,,1'" II I'Xf',:lIt,'rla rOl1lpl('l(~ le.op in the coursl' of ;.L .... l :10 hr. It tr;t';f'!"d 1111'11\1::11 1't';i:,s\'lv;lI,ia and int.o ~:('W Ynr;, Stalf' wlll'r(' it i"--\"';in,n verr intf'c:,f'. T:1f" \'f'~(:('itj('s intiH' ~,,'cond stnrrn. in which thp v;ind

i,.ls",'d oyer til!' oC'-,ln :Iild int"lhe ('O.lst:r1 ;]r(':ls, incn';lsed ml)I'I' rapidly with ;.ci ;i;t t; .. lliintll('flrstsfnrlll ".!,I'rf'lill' willdsp:lssedovprIe\!el f;,rmlanrl (215).

,\"(1:.'::, ,". -Thc ll't tel' s:;l1Ib. ,Is '](;""t('(l for Ilse Il('rein are rlr! illcd \\'h'-re II.(,}'

:.ro.[ ~ :'jW;} r. ill t 1.(' i::lIslrat iOIl" fir ,Ii the texl. and arc arr;1I1[:pd ;r Iphabl'lically. i,.r ('cl;,vl'l.if'IICC (If rderet1('p. in Appendix II.

j/;u"'/I'IJJ:,·s.-T;le basic wind \'elc.('ltics (fastrst milP of wind:lt 30 ft ah<l\'e ;!;c c .... ;·;;I'~(') 51.0"';1 in Fil;S.I(;,) and (hI includl' an an;d:;sis ofl'xistln" records (" b'.lh c;;tratroi,ic;]! ;11)(\ tropical (illlrricanl') winds. Thesp basic winds must :JP ;Is('d :\s a startin:: point b('fnre detprlllining :he variation of wind \'clocHy wj;:l I.l'igllt. rn;:lI'dless of wlwt her lllt' slructure under conslderal ion is located I;; ;l iI :\ 1',':\ that. is cb,.,sllled as inlal~d or ('o;lstai. and rrgardiess of whether the 5tO;';1I ~h ·llid lit' Cb5Siflcd as an extrat ropical or a trolJlcal (hUl'rirane) cy­cluJle'.

';':'l' \'ari:1tion of wind \'('lncitywitl.:1l.'i,.:ht r(,()uiresthat a distinction be made i)ctw'·.'n L.esc cla5~;('s of storm:; :Inri Idcations. In the case of coastal areas c;.;;'ilii; !J;lrl';caill'S, ;!1<' varialiol' of wir.d velocity with heibht is a subject on w: .irl: :l("'(illat ely contrnlieci ')i'sc)"\"I: iO:JS are fpw indeed and in which code rec­o:;.;~h':ld.t!inns I11I1S: th('refore be haSt'd Oil inferences drawn from what little <litla ;.r",tv;tiI;wlc(2i5J •

. ';t, ,~m wjl~ds :tj);ll''';iC~hill~ a cn;]st line han' a vdocity j)rofUe that is com­j1;,t.,,;L' ,.;·ith ,111' pn'ss'lre hr:tdi<·lit. ILe curv:lturc- of the wi;;d path, the rough­I;(,S<; ,.f Ili" w;,ter sUl'fa('e. and oUwr items th;lt affect air movements. As the w.;.c; .,I.o.;"cs t;lP co~.st linC' lhc~e arf' two main Influences that arc responsible fr,r slow;;.,; down its vl'l(J('it,:. First. it cncounters an incI'P;ts('d frictional dr:11; tl..lt ,J'I'cr;~:('s ;,n ;r;crt.'as(' (Of lurl"rlC'IlCf' and thiS, in turn, dUfllS(,S the drag up­w:,rds. S • .-con;!. (hI' supply (.f water \,;jp(,r is n<:llonger a\'aiLtlllf' from the ocean :1.:~d. cO:~S('<tl.jt';,11\·, (';1n laO 1.)nJ,:cr pruvidc a source of encrt:~' wherein \\'~lter \ .ii)"r i:; ('ol1ci('I~se(! i !',IO dr,ud droplets wi: h tLe accompanyin;.o: rel('ase of htcnt L(':.,. '::1'.' ;:rl'"l<'r the liist:l;:ce lhat til£' wind moves inland fr'll11 the coast line, ;:11' ;':l'e,,;,'r ...... :1 lx, the heif'.Lt to which tll(' i11crcascd turhuknce wiiI ue effective iii 1;',ulii;''';;;C:: t:I" wiLd prufile. It call he conclurkcl that in coastal areas, such as IlI'Of.J·;lia· ..... n. the' r:lpid :n(,I'(,;ls(' of wind n.'lociti('s with h('ir;ht need not be conlim;cd l.evcond GOO ft, nlnrl' thall which a constant velocity may bc used.

Ti ... Bl'",,;d;;n'cl1 data fnl' In;rricancs Carol and Eclna are plott£'d in Fi~. 3 t,.;:,-; 1,,1' with li;,' cllrv('s for tilp G-l1lin winds and the peaks of til<' storm of :->O\'('I1i;,pr 25. 1 ~'5(l, Thn'c 0.3-puwpl' ClI)"\'Cs are show:1 COl' cOlllp;lrison; they ;'('(':11 tl) J"di(';i1(' th;,t the ('XPOIll'Lt n.:> [;1\'(,S a conservative estimate of the v;,:';;lti():\ ,'f w;;"j \'( locity with !~rif:hl duril;g th('se thrce Brookhaven storms.

D .• t .. ior t he fast cst mile of wind n curdL'dat r-Iiami, Fla., by five hurricanes bPi wcen 19·.5 :\i.1i 19!i0 arc also 51·.ow;} in Fif~. 3. They were recorded, respcc­~i\'('ly. ;ll t:w City Officl' of the We .. thcr l3ureau where the anemometers were 229 ft a;.d 2.;9 ft abv\'c the street, and at the Airport Station 5 miles away where the anCIl','lI1wter was 50 it above the ground. The variation of wind velocity with hPi.;I.t, z. is very erratic, the ratios of Vz to VSO being 1.01, 1.14, 1.26,

WL'-;D F01\Cl:~S 1137

1.311 :\;;d 1.53 ;mcl tbl' COlTeSi"';nli;l[;valut.'sof x (til<' eXjl,1I1ent) bdngO.O, 0.082, 0.0;;3,0,0 05. ;Ult! (,'.28. Ol;ly!hel.ll"f:e!--(,';- thE' ('XlhHlClOtf: cOlJlpares atalldnsely Wit]; Ihos,' oiJtai;;.'d at Broo:d,;Iven. as Sil(,WIl bv the appropri:lt(' cUr\'cs for the currespo;](ti"bil;iSic wil.tis (V30) ill Fi::. 3. R. C. Gel;try puil.lS out (219) that

"Thf'sC d:<ta unfortunately cannot lJe uscd s;,tisfactorilv for det('rmlllllJ~ t!,C vari"tion of ~h(' wind with h,'i;;il( b,'c;JUSC (.f tho differcnt cxposun:> of t:letwu statiuI.s and tih'liisl:1l;ct'lwlw('clOthcm (ap;ll'oximately 5 mjJ('s).-

It mit:ll: also bc addl'd lhat in each pair tLe rcadir.o; are nol coincl(jpntal, that \l.c two st:l:io;,s are> in dlifercllt i1:lnS of the transition zone for winds blov.illg from tlj(' oc('an. and that t:.e anemumeter ;It lLl' Cit\' Oifice is mount cd on a tall office buil<~ing th.l.t aff('cts the wind dtiferently for different wind directions.

i

\1 -~

+ Nov. 2~. 1950. Idslt"st f, m.n ___ _

f l<.i,'st P('_HlS - - - -­

• Carol 1 1)[..-\ r.!r~n'i·""

1 [lIon ... \'-1 ... 1 5 0"1

2'!.el" .... 'v7:.~1:,<Tl

o [dpa j 154 t'J:r{'f11es

J mill .. \1 .. 1 5 CT 1

hiXIl--__ _ 7'){'t,....V2+'~J<r; ----

~ 6OL, ----.: --~­.;

~..) 400J ~ ______ .~

1

~ t

t 200L

"'13ml htjl,tc.'JfteS ---.-

~ (J !astt'!-t IT lie d ... 'nd. in (;Iphl

-~/~~~~;f=~· 9:~~:)-1 JO~;;;'~: ')/2:'/4H : 7~ i~O f1)! 76 (;'29 '11

10/5/4;; • r08 \50 ft\ I 86 (229"'

9/26/49

1~/17·1H/oO 80150 (1) 122 (ng") ==_~~~~ --,===~_~L~=

FIG. :1.-\\,1:\;) VELOCITY n)i{ CO,\ST.\L ,\ltEAS

I,ll of this ('n1I.h;tsizes Ill(' diainllty of oht:Ji;1in[~ rf'!i:,blequantit:Jtivf' informa­LC'I1 rcgal'C!;n~th(' variation uf v;ind velocity with hf'ightat any situation except where annDnmf'ters ar(' llloul:tC'don anexpos('d tnw('r. Inthf's('fivehurrlcanl's (!;c inc n'ase rJf mlLXil1lul1i wino sjwecl wi: h 179-ft incrl'ase of heildll varied from 1(;' to 53';. and only 011E' storm ill fl\'(' have a Wind profile comparahle to the s~orms at Brookha\·en. Althflu~;;, both :-1i:1mi and Brookhav('n are In transition ZOIi('S (If cuastal :1r('as. one is li~l'lyto experience mature hurrican('s, the other hurricanes in the decaying stage.

Fig. 1 (a) asSiplS a basic wind v('locity of 75 mph to 90 mph to trie Long Island area (faslest mile at 30-ft level. with a 50-yr period of recurrence). This is far greater than lhe extreme I-min winds shown in Fig. 3 for Carol

--- ~ - --.----~~ •.. ----- --

Wl:-<n FORCES

,I:~(: l-:'lJ:~1 :11 t i;(> :17 -;1 If":I'1 ;,'ld ('v('n ('xcl'('ds tlw l'xtrf'llll' 2-sC'c pC'aks for this ],":..1. TillS indicat.'s th;iI. III this coast:li arl'a, l'\'('11 strol1g-E'r winds !nIlSt he l':-.j~p('l ... d h'.th,' structural Plwi'll'prtllJnthosC' oi,sC'rved tillrillh" Carol and Edna. Tv,,'\) qUI>st iCJliS arise':

1. S:}()u!.l thp vprv npid incrl'asC' of wind YC'locity with hci[..:ht (x = 0.3), S:I"W:1 IJ:" t:1('s(' tWlI hUITir;];lf'~ in tl:<'ir decaying sl:Jgcs anrl lJv tilt: one intense' (x: ;·"l]'opil',ll stfJrll1, IJP llsed for f\lturc guidance in this coastal arpa when the SP('Cil;,,'d basic wind (75 lllph to 90 mph) is twice' as f;rf'at as the basic \'doci­tH';:; ;ICcOn1j""r;illh" the oilsl"rvati()ns (Carol'" 43.8 mph)?

2. S;;uL.ld it IX' us.'d in thnsp coastal areas that Illay be visited by hurri­C:Il;i';:; d~,ll'lllf: their mature st:lGes?

:1l'rJ)e:-t jtH'hl slatt's tint "110 Olll' knows h0w hlhh wind spC'C'ds can rise In trojlic~d s'torms" (220). Also, that a wind speed of 200 mph may \)(' attained

.;. \i\l E J,-~F,\:'TI:S·' ~l:;.i': uF \'.1:,1>, 1;\ ~llU:s PEH IIOUll, Fon VAHIOUS ii~;jl.lr<· Z,,'>ES Al.l(l\'E (:,lCW:>lJ

----- -~---,----~- -,----- -,-----------~

ii,';';ht Z,';)P.: fl;,SIC \\'1:,,\ Vcl,,,,;t}', in m,,;l (S,'c Figs. I(a) and (h) )

__ ~~ ____ ~...:~~~_ I~i--~---,(; ~0: ~;) i ~o !~]II)u:c-rI----:l-'I-::5-'-'1"'3-'O::-(oj Fnr Inl~nJ ,\n'3"

- -~~tll- ,-,.I --~ -:~;-~; -'-;5- -1-~;-:-~-~:-1--9-(l-rl-!-I--)~:-1 no I I :,', 1. I : -,~_ 70_1 __ ~~ ~ _ !llI~._!j.:)~_;_o~~I_.!Q.-J __ 1 ; I'~J~_-+-.

l:lll ttl -HI I) \.0. 11 ~JII 11111 110, 1,:, i 12., I 131) I 1·10 i

_.~,,(, tl/ __ 7_~II _____ :.,.;.l ____ 1.~.!n~_1_!~~_~~_i __ \:}\~___1J;l5_L.J)~L_.U=)(~~ __ _t--

7(10 t" :,"'10 ,jli" 11 0 , I~:, ! lJI) [' HI) : H.-, I IS5J' IGIi

~.(j~'~~~'_~'J_'_'.')_LI~'2_~J_I~l 1;;;; 14" i 150 l F,n ._1_0_'>_,-__ ""-__

w"verlile S"'l ;\Ild ;1[;] few Ir.ll1cir0d f('('t:ilJo\'(·thc h"rolll~,d" ill a very small area. i;,' :lisl) qC:0tl'S ~>, A. llu:',jH'~:>J the rfferllhat an <I\"'ra1':o (If 90 knots (103 mph) W,lS ,·j'''''l'I'"d l1l'"rthl' c,':',tor of 28 stormscillr;J,!'; 84 rI'cunnaissaflcl' flir:hts of tl.< ;],,,,ed ~;at('s ~:lV',' 1](';\1' ,lie l,000-ft lcn'i (22l). This a\,l'ra~C'\'alucshould be i:.Cl"',lS<'ci bv t\\'o f:iet(,rS of at !r'ast 1.3 each; one to obtain the ('xtrf'me stnrm ,I,',d .!w <Jthl'r to nl';;lil~ the fastest mile of wind \\'It III 11 that stnrm. This \\,(,;,:d ';]\'0 a "f:lslest mil,' 1)1 \\'il;ci" ui at least 175 mph at till' l,OOO-it level.

FI,:. 1 (a) ';C)i.'S not specify u:lsic \'elocitil's in C'xoess of 120 mph, but 1'\'011

,Lis va:;;(' would require the faslest mi:e uf wind at 600 It to be taken at 2llS I1ljl!1 jf the 0.3 cXjlUI1C,;t were' used. Also, for the be;-.dit of engir.cl'rs who may :~cC'lIIH' lH'cd of 5till hit.';lwr i',lSic 'winds in unusual situations, as ill Fig. 1 (b), i,;IS ic WJIlc:s as hi gil as 130 IlJi,h are il,cluded in Table 1(b). The accompanying wind sjH'ed at 600 ft wuulci Iic('ci to be 320 mph if 0.3 were used as an exponent.

W1;-';D FOrtC ;·:s 1139

T;w vcluciti"s (If 295 mph ;lnd 320 mph "t thl"' GOl1-ft 11"'\'1'1 arc hdicvcd to be cxcc:isivc, ))"l lil('r~' 'Ipl'car to be no well-conlrolit'd ,:,J,s''\"\';lti'll1s in mature hurrjc~nc:i, Stich as ClOse which werc madc 011 th(" tf)w,'r at nrookh.lvf'n for the two decayin:~ hurricancs. Consequent ly, a prlHit'nt compromise bet ween reck­lessness and ('xtr:! \';o;::lI1t conl'ervatis III requires a careful consideration o[ any rcl,'vallt materjal frum which rcasonable Inler('lices Illay bl' drawn.

Cna,,/al ~li'.,tls. -Till.! anl'wers to the two questions p'Jscd previously will be aJortc-d as follows:

1. til coastal an';,s whf'rc the basic wind V30 ~ GOmph. the variation of wind velocity with ill?if'ht

[ Z JO.3 V z = V 30 30 ••••••••••••••••• (1)

will b(' adopted for lll'it::hts up to the 600-it level, higher than which a constant vele.city wili be usni.

2. In coastal areas wh('re the basic wind V30 ~ 60 mph. variahle ('xponents­will be adopted 1'u("h that V z is ncvcr greater than 200 mph at 600 It in Table 1 (11):

••••••••••••••••• (2)

in which x v;].ri"s fr'JIl1 0.3 whcll V30 = GO mph to 0.143 whf'n V30 = 130 mph. Cur';es arc ~;,;OU'il in Fij!. 3 for some nf th,' intermediate values of x.

II. tl;(' presc.lt (E"J i) statp of idormat illn it seems undesirable to try to Jist ;"1,:u;sh l)('tW"'<'11 the wi:,d \"t~locities which, in coastal arc'as, are appropriate ~or stl'UClur.'s local ed :It the coast line and those located brther inland. It is r,'comlllPndeci tha! co;].sLlI areas be considpred to extend at least 30 miles in­land from a w(' Ii-clefi ned coast 1i;1C (222).

Fnr :lllsltuatiol15 which, inthejuor;ment of thE' engineer, deservetheclassi­fiC;ll;n;l of "coastal arC';l,' :he wind velocities shown in Table 1 (b) should be \'Sl'd to;.':(,j;,'1' "'ith thC' "basic velociti('s· shown in FIgs. 1 (a) or (b l. Special anal:.sis siJoulrl \)(' mad,! for promontories and in areas where the coast line is pr("cij)ituus~

/J:!(JI/.l ~1 rC(ls. -r'O)' all inland a rC':ls Table 1 (a) should be used together with Fi;;s.l (a) or (hi. Tlor-se vailles are minimum and should be increased where highl'l' \);\SIC v,-locitif's have \)("(')1 rC'furdC'd locally, or an' likely to occur, clue to mou;,tains, t.:()q;('~;. I)r Cor oth.'r unusual rca SOilS.

Th!'!"l' is a consi(h-rable d,'crease in the d~nslty of the air at locations high ab(J\'e sC!a l,'yd and consequ('ntly thep' is a r('duction in the dynamiC pressure (\'eloci(:>' pn'ssurl') as cnmjl:tl"l'd to the 1';11)1(' wind specd and temperature at Sf'a lcvci. If. lornvi'ver, the [as:.'st mill' of wind, for which the contoursfJC Figs, ~ (al and (b) a;'e o;',.wn. should occur at a ("mperatlll'l~ t;rcatly below that of the s~ar,d;li"d al mn1'phere (59 0 F), the lnss in cif'l1sity due to elevation may be c'/nlj)lc,dy C(/IT.p,:n::;;,~('d by the, incrc;Jsc in density due to lower temper;.ture. For ('x.lmpl", G. 1'. Brekke ~ives (222) a tahle which shows that, at 4,000 ft­e}..,V;.tj011 a:Hi at Sl.dld:lrd temperature, there is a decrease of 14%i:1 the den­sity compared to sea-Ievd clevati0n, but if the temperature were reduced to -lO"F there would bl! no over-all reduction in density of the atmosphere. Consequently, it is recummended that the engineer make no reduction in wind

1" ,;0 W1;\D FOHCTS

P;'c' ,s"" r",' 1 ',1,,<'<1 'Ill nriuccri d .. , i ~', t '.' ;It hi :~il <'If'\,'I (ioll;; unless he has i nvest i~~atf'd ti; .. prr",.dl1!i 1 v :il.lt th,'h-,trs( :ni1" of wi;irill:;"riinthedf'Slf:n mayu': coincident With.1 si~,nlllt·;tfll l"('riqrll'lll 'Jf rt'j~ljH·r.ltpr('.

~~~L(l ~,""iq\ lOr :hl'S(' j"1 CUI:-,lrLPIH1.ltln))sh;lS fW-('lllildltrd tn thos(l gel1('ral enD­

s:';, r;]t;,,::s tl..lt are COllllllon t,-, most jlrohlt-lllS ill the S('Jf'ction (If :1ppropri;]te ~ il:d .~.~ :f,'(,lt ii'S [,}r th(' (ksiGJl qf l!rid;:cs. hni ;tiin:,~s. to\\Trs. ;)Ild siillilar stnlc­h;res. It h,\,; i,l'rn ;lSS11/1wd I Int hjl~hly spl'cj;llizl'd pruhlf'nls will rf'ceive the ;]~tei.:ion qf ";;:::n('I'n, with "olllpl'iellce til 111;11; .. or direct the nf'cessary modi­fil..~~lt inI..·~ "1' ,HiditiOl:S to t Itt' I"eet llllllH'l1tiat inns.

1. All rC('(1:11nlt~;-jdati0ns h('rf'ill :lrelJasrdon ronliition5 ill open, level coun­L'y as ,I .~t:;;:<',lrcl of n'f,'r"llcc, and the illfl1lC'IlCC of shicldin~, or of opflections :i:]'" c: •. ,,1;;,,:liJ:,; cr.;e tll uliUSU.tl t"JlO~rapllV ur to laq;e oiJstnlctions shall be ~'\";i::;,i~t'd :J\" tJ;l' t;l,~:il,ei.'r i:l 1'.tCd iLdivic;ual C~lS(,.

2, ,':" "":l, ,'t;n' .::-"Ui](;" (til": ;S, thc l]('i;:ht in Opl'lJ, Jf'vt:'! country or its ('(iL:\,.il, ;it, ;It w;'h'h rliij"Jr ol,structiol1s are uninlpOrLl.nt, ana which approxi­Il •• ,t.'s : :,1' l]I'i,::.( of ar"m',llld"rs now (19G 1) in liSP by the USWB in the Cirst­nl"t; "II' ~: ,\! l(l:l.s :l t :1 irpnrls, ~ll.t 1: i-l(1 Ld-:f'jj at 30 ft.

J. ;l.,~,.c w.,-,d "I';"""Y sh;lll Ill' lit'fl;lf'O as the f.lslpst-milc-of-wind, re­duc,'ci to ;, L,,!!::',t nf 30 ft: 'IS "h,,\\')1 ill Fi,~s, 1 Cd ar;d (h) for the various (;eo-1:;'.lilhic.1: .lr('a~; 'l[ t:,p L'"il"d SLII.'s. Fi,:. 1 (~.) shows the hasic winds that h.;\'(· ;] Sl:,: :<iclllv (it-1'j .... ·,i ;'Jl"j]'ahlf' pt'riud of rpcurrencp of 50 yr and Fig. J ;ltj a ;))'"h.li);" i""ri(>d "f !'f''-Url','llce of 100 ~T.

'" :. :",sic wlI:d \''';uc;(\' ",i,ll tI 50-rl' I"'riod of recurrence (Fig. Ha) ;;,.l:; ;h' "''''(; fur ,.Ii pl'rm.l:w;,t structures ('XCl'jl! thosl' that. ill the judgment \,; ; :1\' f'Lr:i;jl'cr. ha\'c a hL.:h \~('~:rf'e ni wind Sf'n5itjvity and an ullusually high (k,', ... : 0; h;],,,rd "f Lf,' ;Ind 1'1'r)jHTtr ill cas" flf f:1ilu,e. In the latter case a 1(,'1-\'1' p,:rilu uf rl'n .. ,r('l~ce siI()\lld l)e us('d (Fif';. ](b). In th(' else of tcm­~)'J~·d:-\· ~ti'UCl:;r" s. Si,t'h as thus(' used durjn~~ construction only, b3Sic \\inds !: I' .. ,,:\-: '.'i'l Icit:,·s (",,,.,; to thre"-qlJ:ll'[( rs of t!J(' vclocities shown ill Fig. l(a) nl.~l:: LIl' \jSl'd a~ thp cis\r~li'lIi tlf the Cllt:in(,Clr.

L: I'fJ"".ec'lio;; with th" ftln ,;()ll~,; rl'coJ:111]('mlations, it should be noted that th,' s"/c"l lOll (If (;. si;~11 wllid \"('iucilles IS directly related to the s('Jection or a1:',",\',I;')C ,,1.<,c.S"S ~.:ld <i,'si~:n prrlc('ciures ll"c;]\Ist;thcse combine to determine t:lI' w;er-ail factt1r ()f s;tld~', Thesl' recommended vciocities ar!' !Jeliev('d to h(, rC;!:'();Iabk fnr l;S(' ;n e'l;;;unctitln withtlH' ;]llowaille str('sses givl'nby most c, ":":" h ,· ...... vcr. th, lipl erlJlll:;llion "r loads for a p;t!'t!cuL!r structure should 1),1 a t),lrt uf tll~' t·:, .. ~I;-'l'l'r'S r('sponsii)ilily.

J, II; sp!'c ;,.1 ;',C:ltiuIlS such as OIl promontories, on mountains, in gort;es, Ill' "!" "tiler IIl:US!,;t1 1f';]SUiiS, w/.('1'c recor~,<.; ur ('xperienci.' indicate that the ha:,ic v";"(,,ti('s ".HI] in Fi,:s, 1 (a) or (b) aH' Inadl'quate. hif;her basic veloci­tic!'-) rn;1~,· b(' 1;',.,'(:;It the discrl-~iu;1 ()f the elll:illt'er.

!J. Ta: ,: I.' ;''') Sil:l II Ill' u,<.;cd in it.l:lnd a n';]s to ,,!.Jtll i n the V;] riat ion of wind vdc'cit:, w;th III i::1:t z.fil'r Fq:. 1(~tl or (h) 1;;]15 been used to obtain the minimum 1,.l",C w;nd \'p,oC'H:,' for til.lt ;]re:t. Fill' e;tch b •• sic Wii,d velocity. a stepped ap­proxiln:Jtiot] t.l th(' olw-sevcnth-;Xlwer law has be('n used in Table Ha) to a lll'i"l:l of 1,000 ft, above which it constal,! vl'locity has b('en used.

WIXD FORCES 1141

'. T:tlllc l(h) shall be used in coastal areas to ubtain til(' "arlation I)f wind . v('locit]' with hei~ht after Fj~. 1 (a) ur (IJ) has bcen used to Obt.lill the mini­

mum h:tsic winci vdocily for that area. For t'ach basic wH,d velocity, a stepped approximation to a power law has been used for the variation of wind velocity with heil!ht. The power law up to a height of 600 fl Is as follows: (a) Where :h(' b;:.sic wind \,elocity is eq'.I.ll to or less than 60 mph, an exponent of 0.3; (il) where the basic wind veluci:y is gr('ater than 60 mph. an exponent vary­in;; from 0.3 to 0.143 at 130 mph. Above 600 it a constant velocity is used In each case. Coastal areas shall be considered to extend at least 30 milE's in­land from a well-defined coast line along the oceans, great lakes, or other large bodies of water.

/TI' / , / ' "-

/ t "'-

/1,1 k / 1"-

/ "-/ II "-/ II '

~ kl /1 , / I "'-

/ I! "-/' l: , I:

/11', !, "-// I, "-

/ I' , : I

l'IG. 4.-W!:\1)-LOADl:>:G VARl·\TIOl'~ TO PRODUCE MAXIMU;\I ;\IO;\IE;>;TS A:-iD SIIEARS ll' A G\.;YEIJ TOWEH

8. Appropriate ~st factors Sh.111 be used fnr structures that are small ('1i,Jubh to hp r('sjlOr.siv(' to ~u!-ts im'olving l('s5 than 1 mile of passini~ wind. The i,'il!-ot factor (gust vt'l')city .;. the velocity of the fastest mile of wind) should he-ar snme 1"('1.11 iOIl to the minimum size of \,'1.15t necessary to envelope the Stl"u,'lUre and its accompanying pattern oC flow (223). A gust factor of 1.3wlll allow for gusts of apllroxim:ltC'l~ one-second duration (224) that. in a !lO-mph oosic wind. would have a length downwind of about 130 n. This will be adequate for Signboards and small r('sidencl's. A gust factor of 1.1 will allow Cor gusts

. of approximately 10-seconOs duration that, in a 90-mph basic wind, would have a length downwind of about 1,300 it; this factor is adequate Cor structures having a horizontal dimenSiOn, transverse to the wind, of about 125 ft.

------- -." ----

9. Gun'd !o''\''crs shail hI' d.:si,"\f'd for 1:\1' mn\"it\~ load ('ffpcts of af'rial jf'ts (208) and rksCf'l\ning Plst Ironts. as inrill'atrd ill Fir.. 4. In FiG. 4

(3)

in wl..Lh ql': Is thl' d)" nam ic I'rf'~sllrf' of t h" f 1''''' 5t r!'am at hei~ht z and {l denotes deJ;sity qf the air.

HI. In th'HI(' I!:"o;~raphical art'as \\'h"r,' icl' storms may h(' followed by stron~ winds. :1t least a ~ illCh thickllrss of icl' shaJl iJf' asslImf'd on :Ill structural mcmlwrs.

III. FUNDAMENTAL CONSIDERATIONS

I~TnODUCTIO:-<

Willd florcps on <l bnoy il11llwl's<'d in <l wino stre-am rt'sult frr .. m changes in \'('~v('il~' :Il'Olil,cj the body. OJ; biunt ObJlTts thl' dra~ is primarily due to sepa­ratJ()n at ti,., l'd,::f's. W:lkl' formatioll. and th(' cOllsequent difff'rf'lIce bf!twcen sllrf.II'(' ;'n':;sul"<'S on th!' upslr!'am :lnd oowl1strralll faces. Additional forces, !-;/.'ller:1.jj~' r .. i';'Ir, r.'sult fl',)111 friction lwtwcc'n the :Ill' and the body. Usually, for pllrl~)sl's ( .. f slructur:J,i design, it is sufficient ly a('curat(' to consider these fOI'(,('5 as an ~"luivalt'nt static pr('55Ul"<', In some cases, however, the time n,ri;l~ion of forc.: nn:st he consider('d. This may ue riu,e to ullstrady velocities or to altel'l,ating ~orc('s th;\1 1'l'5UI1 from periodic vortf'X formation. The latter applies mal;]ly to fkxihle structur'"s such as stacks, suspension bridges, trans­missini' lilles. individual Illl'mhel's of framed structures, and tower guys.

EQtiIYALE~T S"ATIC FO:1CES

11,1\';",: s.'Il'C'lt'd a lk:;i;:n wind vdocity (Sl'dion II) the el,;ineer must then rlCkl'lilii;,' :i,l- rl'sultilig furct's (111 the structur!'. The variables that determine th(' n~:1~~;)':'\;'~l' of thcsp forc('s are dis\~uss('d her~in.

Whl'll any object is imll1f'rsC'd ill a wind stream that may be cOllsidered as a l.u:ni>C'T of par~.llel filamcnts. it is n('cessary that some of thf!se filaments c).a;;,;': th.'lr dirf'ction and their \,elocity in order to pass the object. For an iJeal fhulL by Df'rnoulli's theorem,

~ p V~ + Po = ~ p y2 + P

i:1 whicb Y n is thr> "docHyof thf'> appro;\chillg fre(' stn-am. Po denotes the stat­ic prt.'Sblirp (Jf thf' ap;}roaeiling (r("e stream, and p is the static pressure at a

jlui;;l on the Objl'Ct. The terlll ~ p Y~ is the -dynamic pressure- of the frf'e

s;rcam •• lId is r,,'sip1aled hy q. The term} p y2 is the dynamic pressure at

some iiuint OJ, the obj<'ct. Thus the slim of the dynan1ic and static pressures i!> a cur.stant at all points. If thl' dynamic pressure chall~es by a certa in amount, t;le st;,tic pressure mllst chanbc by the samc alllount but in the opposite direc­tion. Whc.'c tl;c velocity oecomes zero, as at the center of the iront of a plate, the incre;,5C ill static pn'ssure must f'qual the dynamiC pressure in the ap-

\VI:-;n FOHCES 1143

i)r0:1chini~ S~ l'l':lnl. At the t'O(:cs :lno in thi' Icc of the plate, the velocity be­C()I~~('S b1"catl.'i· t~i:ll, ~h.,t ()l the str('~lrn awl a Ilf'!:ati'~'c pr('S5UrC is dpvfiloped (tll.ll ':;, ;.,:;s th .. n ;.: IW1Si)!lCr;c prt'SSUl"('). These ch:lll(:;f's in prf'ssurf' pro­(;"ce <ii'a"; .. r :wt f,':-ce ('Il til(' niljlTt. Iii ad(li:ion, there :Ire friction forces, but these ai'" l~"J\l"',lily sm:ili for structural ('\('nil'nts.

H is c(,r,vedcnt to rpsulve the wil:d ('[if'ct mto two force comp')nf'nts and a rno.nent ;i;:ci tu use ciimens;unless coefficients to ddinl' the magnitude of the fnccc's. TLus,

0;';".:; (forcr parallel to wind)

'.;it (1,)rcf' ',I)rmal to wind)

ia which en i,.; tl](' dn h cc,,.fiicient. CL repn'sellts thp lilt codflcipnt, CCP is the l'Cl·('i.t;:jci~y cocfticl,-'nt. A dc'riolt..'s the exr"j~pd area, and h is some con­Y('nil';,: d.:I~(l;ls;on. The n\dl-:1C'nt ar"r~l is th.' dlstanC(' fronl sonlC reference poi;;t to t:;" C(,llt( l' pi ;1",'SSIll',' a:ld the j1"mll:llt is thp distance tim('s the re­StdLl;,l ft",Ct,. Except ;,,1' S:;:.ll1lctnc,li sltu;]tioi\s, the center of pressurl? is not u;;,;,0I1:: ,It tl.c ('l':,lruici uf ttw etf"lscd area. T:lt' ';altles of the coefficients r.lust 1)('d,'tCl'lllln('dp'ip,'ri"leIlLllh-u('CaUSl' the SC;Clwe of fluid m('chanics has not dC'vc!upf'd to the' i,nllii. ;l.t wLich thf'c)fctical dt'terniinations can be made. ex­cC'~)~ pcr!l;l , )S i:1 t he ~lIn~)icst of C;\scs.

Thl' 11l.q::,j:~d.iC (If t:,C' f\Jr('}:.ltL·.~: cc;c~ici(Tlts dt'p(,i~~:S on (a) the r:conlt:trical form of t::(' Oiljl'ct, (0) i:s ol'iI.'J,talion in the wi:,d sln'am, (c) friction effpet1', arid (<iI, to ;l li:llil..cd extL';lt. th<: s~7e of thl' n\)jC'cL

(;," )/",' In"" I Fo rill, - An o!J,l'ct that is • sl ,1';1I1l11IlCd" symm et ricall y with the ();n:Clw,; Ul th'_ ',\,';Ld. produces littlc chaJ~c'e ill the ';elocity of al,y of tiw wind f11 .. n1f'"t1'; hrl1C(" tlH' prin<'i,),l.l rffccts ansc from fr;cticol1; the drar; is small a l.d t he ;li~ ( il',;;le<:t .w; U.(' pffccts of alte rnat i JOC; \'o~tices) is zero. An air foil .s S) sin;wd th;lti\~its tilpthe veioci:v is hi~h, rf'sultinl~ ill a redurtiua of pres­s .. r,'-thcl't'hy I'rpr!',iCilli; " lark" lilt. As CDI:1i';\rui tQ strc:lmlincd ohjects, ~~rc ... ;'I'r vl'l()(' ~ty cLa:l~~('S a:~c ill\'ol\,pd V..-lH'J1 the object is "b:UIll- v;ith sh~rr cur;;ers aI":, CO:1Scql.l'nt:: .. , hi",her dr«:; coefficients r"5ult. The wind forces on ,1llY ')['J"c; ;ll't' ~tff"ct(>d as conditions change tilf' pattern alid velocity oi the w;l.d.

~; lh,'; hin rCl't;lJ\i~ljbr piate ill Fig. 5(a) is ill:i:\itcly long, that is, the aspect r,l~lU ,;I = rati" ',f 1"10,,1.11, L, to width, b) i.<; i:1Lllite, the wind can escape only on~r t::L' :Wi) (1:,;,'5 a:ld the prCSSlll,.,distributiollS lire as shown. The summa­L'lj', <If pr"t-~'jr<'s 0'; tilf' two faces ('qUills the total dr:l(:;, th:lt in this casf" Is 1.:1" q A Ol' Cn = 1.~13.

1·'C;l" a .c.::(I',~lrf' pl:llP. t!~~H is,,\ = 1. t:l(:' v:i:,d ('.1;1 l_·c,cape aruui~d all four sides, WIth t!, .. rl'slolt th,lI t1.e suction Oil thc rt::\r, al,d h('l,ce, tilt' dral', is reduced. Ti.., vllri .. t,u!; of dr;i~ cucfiicient with as,lt'ct rati,) is shnwn in Table 2(a). It should he 110tco that iI relalively hi~h aspect ratio is required for a suustantial illcrl',lSL' j:; dr.l,; l!rea:f'r than that for a squ:lrc pl;\;p.

For a c;r,'u[,1r di"" :hl' air flow is unifurl11 arnuno the periphery. Conse­qIiCII:,y tllt' dra,: coeffiCIent is reduceci, as COnl;Klred WIth a square plate, to 1.1l.

Whl'll a I'P<.:t.lll,~ular platl' is placl'o ontlie (,:roulCd (Fit'. 5(b)) the air can no loli~('r [lo>w arou;.d the b .. Uura eo::;'}. TlLis has Ilttle efiect on the front facp pressures out reduces the rf'ar huet ion conSiderably, especially in the case of

"---' ,.r 04'_1/, /1

CJ;q-lIoj

O g 'l ~'I - ~.; - i :::; :t ~ -,,",. I,

~ PrE ;:;~ rr?":_~. ' $_ ,,,:n,.,,, 'd'" ~J (a) fLAT PLATE, Ir;Tlilt:LY LONG

WInd .......

---~--

Wind --~ ~

+ • ," '7 >/7:'" .r77777!:"";~" ... ·77T......-r),,"7,·~'-T."""'7 /

f..-- b --~! (el PARALLE.LOF'IPED

Y.

1.4

(h) RE (T ANGL'~ ~q f'~·\ r E ON GflOLND

/ Te':t 'es~!!" (," :- J~ -'YI

III t' " 'Irl::' 1

'-~--' , I

},. • 3 },. • ~ },."1 1---'-la,,-,,~ 1+- I

--- L II ... 10 t> ., ~ 08 Q

..... '0 06 ~

-;0 >

_ . ., 05

\ CCP

: -- ~-,- 04 ~ '-

-r- C ~ - ~- G 3 : , I J ~

'-"~--'-. ----[j' "02 ,. , I ' I : i ,

.r 'V ! I ~ 'Ol 70 5<J 30 10 0,90 70 50 30 100,90 70 50 30 10 0

Cd) INCLINED PLATE Angle ct, In de"rt'~c;

FIG. 5.-Wl:-:lJ PRESSUHES 0:-; ELE~IE:-;TAHY BODIES

... ... ol­ol·

~ >-1 7. tI "'l o ~.

n :iJ

W1"J1 F()RCES 1145

l(jll~: plat('!;. As a result. Ill<' ;lSp<:ct r;]iin h;lS little df"t'[ ill this rase ;1nd the dl"~q.: c,wfrlci<:nt is Iwart'." lh" ~.lnj(' a<.; lqr a ~fJ\l;lrC'. lrC'Piy suspcnrif'o pbtc. It siluuid hl' IIC,I"'l how('\'f'I'.IIl.1t t:IP(iJ';Jl~ in t:lis C;J~'t' (!;'p<:nds on the dp(!rpc of (k;'e!01'1il I' ,it of lhe bOllnd.1 rv lavel'lI rst I'l';lllllhat cI"ppnris, ill turn, on thf' [.:rnuno­sudacc' C'Jndili"n. Cunst.''lllc!lti';, til<.' pr""sure dis! rilJuti"n may, in some cases, he quite diff,'rcllt lrum tlMI sil,'wll ill Fi,:. 5(lJ).

In all cases the suet iOI, Ull tile rca r is the b q.:('r factor In tljf' total drag. \\'1\('11. for any r('ason. tillS ~ucti'Jn is rrduced. th(' ..cfecl on the dral! is apl're­ciaiJ'£'. As all ('x;ll1lpie,collsiderthf' parallelopipedin Fa::;. S(c). As till' width. h. increases r('lative to the Ilf'ibht. thl' object become!! more "strf'amlined- and the drag Ct'l'lfieienl dr'erf'ases.

If the plate Is perioratf'd t h(' dra~ b('eomes a fUllctioll of the solidity ratio, <;>, or the r;;.tio of solid to enclosed area, as well as the aspect ratio (see Scc­tiun V).

D;'ag; and liIt coefficients for s('veral typical struct ural shapes are given in Ta'Jle 3. Extensive data on the drag of elemental shapes are available (1l2) (311) (315).

TABLE 2.-DRAG COEFFrCIE);TS

, I I i '(b) Rp,·tanv.ular I (e) Parallelo-I (,,) Flat Plate in ;o,;orm"l Wind" I Plate on pip('diJ

i ~1 __ ~G~I~'o~u~n~d~a __ ~ __________ __ I , I ) r b ·h L Valut' of /I. ,---t Va ue 0 I

' •. 0 2.0 !,i.O :10 '21l l"O : '" 1.0 110 i .., ,0.111.016.0 --J- ,. . -'---+'---f--

-l'~~ltrl', fl'u-~t ra~T0.7r,:- , -~. -j--,-. --i~~GO·:G.:J-8~1---il 0.[,.,0'. I' !I ! I ' I : : t ,I

S",=-,~'~~('''r race_+IJ.:l~l __ . ___ , ____ L_L_~I I.:lkL~~~ _ __L' __ -''___

Va:lI,' \If Cn : 1.1~ 1.1~, 1.20i I ~::ll 4:ll,6(;11.9~ 1.10 l.20 1.2f1 l,!!'j 2.03 0.90 _____ ~ ___ -------..l _-----i.- ! I __________ _

Wll"lI two ;'latf's or Llttic(' f;tfUi:tI:res form the obstrnction, the lceward p:;,t,' is "slllddcci" by ti:e windward U:ll'. When the silac in;; is irJlnite, it Is c\'l\kl~t that l!re Case is t wo s<'parate plates. The dq::;ree of shieldin~ Is a func­tion c,f the sp:,cin; hctweell the structures, the soliday ratio and the angle of att.tc~ eilhe.- hori7.onLll, vertical, or both (sel' Section V).

O;'iclltafiull of /III' (Jil;,"'/ ill /he ll'i,;r/ St,.clIlI1,-IIl gelleralthe wind does not s: rii;c an Immersed ub)f'ct perpf'l1dicuiar til any of Its axes. Even for a strue­t':r~ 1war Ulf' ~round till' wino may h;]. ... c a vertical ar.r,le of attack, possibly as Im:ch as 20 0 from tile horizontal. The horizontal ali!;lp (yaw) may be of any arnount.

Fit::. 5 (d) shows a platt' inclined to the wind. For aspect ratios of 1, 3, and 6 the variati.)n of pressur£' coefficients with the angle of attack is given. The prt's.';ure distribut Ions on the two f.lces are. of c'-JUrse, not uniform and the cen­ter uf pressure' is not at the plate centroid. Similar information for various struetl'fes is biven In Sectior,s IV and V.

Fric/ion Forces. - In the examples given previously, the forces were a result of the chanbe of surface pressure. Additional forces result from friction 00-

wpm FOi\CES

T ,\ "i.e: :1. -DIU" COE F FICi ~;"TS n'R STHVCTTfIL·\ L SIT A PES \\1TII ),. =00 (I 11)

, --" ~ -------------- -------

2 OJ o

_________ -1 _______ _

1% 201

,- -----204 i

---1~-- j

o

o

o

----.. ---------------

20 03

_______ L __ _

i.R3 207

; I _________________________ ~ ____________ I

\

1.99 -0()<1

i --- ------------ --- .. ------- r----------

162 .. 048

----_.---------

201 o

--_ ...... - -.. _-- -------<--------1-________ _ t

199 - l.l9

219 o

Wll\D FOllCES 1147

t'.;CPI> til<' air ;1,](-\ tlif' illllllcrsedulJjPct. Ol1Yi r)USiy. tilisfriction varics withthc 5ur;,lCl' rt1u~~hlless of tlH'imlllPrsed \YJd:.', hilt. f"r.most cil'jJPIlI:illccring slruc­t l.r, s, t:.is is of ir'ss C(lIlSCqUPr.cp thall th,., Sh.l!," ,,[ llH' body, so that it bccomes possihle to cOl1s](h>r thc c[Jeet of lieynoicLs !lumher, R, only.

The lkyllolds IlUmiH'I' IS the ratio of thl' inprtiai fore .. to th" viscous fnrce which a fj.,id btn'.tlll exprts on an objPct, TIlf' inf'rtial force is thp prfJduct of dyn;lll,ic prt'sf'urp and area, and the viscous force is the viscous drag or fric­tion bctwcpn the r;uid and thp ubject.

Vh /.J.'P

Vh

F ••••••••••.••• (5)

ill which ,0 is t:,C' kinf'lll:1tic viscusity, h is a characteristic dimension of the olJ)('ct, "!ld /1 is tloc oynamic viscosity. For standard air (density = 0.002378 slu;,:s. tcnlj'Ji·ra:ure " 59 "F) R = 6380 V h where V is in fcct pcr second and h is in fed. Th,'re arc (lilly minor \'ariatiulls with changes from thcs(' conditions.

Tile Y:Jri:diulls of draR coefficient with R for some elementary objects are shown in Fi,;. G. Some comnJ('nts on r'ig. 6 aI''':

a. Expprimer,tal ciilta arC' larkin,.; for R much p'e;tter than 106. For com­parise/no a circular stack of 20-ft diameter "ith a vplocity of 60 mph has an R -\';l!up uf11.3 x 106. TtJ('re is little rea SOil to bclieve that there is Illaterial Ch;tll[:P il; U',e dr:tg codficicnt above' R = 106•

b. '1l1e t;rag coC'fficien:s for streamjinC'd objects (such as cylinders. tanks, , and so furtll) bh()v, a large decrease at the critical R of about 0.5 x 106.

c. ;\0 such chalege occurs for bodi('s with sharp corners such as bridges and !Jui:<iill~;s. In these cases. the boundary layer is forced to separate from Uw body at the sharp c()rner instead of s('parating at variab~e distances behind tl,t· !c.lo;nc; ('('gP. Hl'nce. R has little dfpct 011 "hlunt- objects. An exception to this is till' dr0p in cOl'lficients for fbt platcs ill the range R = 1,000 to 3,000 that may have sume effcct ill tests on slllall models at low velocities.

E,I,'ct of Si.e£' oj Object. -Most tpsts havc been made on comparatively small lllocl<'ls. Pa;un (112) cites examples in which tilC're was a moderate increase in dra;,: cO"ificients with increas(' in dimenSions. even for blunt objccts. For l'xamrk, fr!l' large s'1uares CD beCOllH'S 1.28 in place of 1.12 as shown In Table 2(:11. The diffprence is small comjl;tred with the other uncertainties of wind fnrC"fl ,uKI may be Illerf'l:; the result of cxpf'rilllental error.

Ur-:STEADY WI!'D FORCES

St l'ucl II res l1l;ty be sl1bject to winel-induced oscillations of large amplitude anrj im'olving 5f'\'cre stressps when dampinl!; is small and one or more of the fo.l"win" conditions cxist:

a. The struct\irp, or stnlctul'al componcnt, is rplativciy flexible or flexibly sUj)portnl:

b. all ('XCitjll~ force is proctuced bv the wind; and c. the cross sf'Cliun of thf structurp. or structural component. is such as

tCI promot .. till' discilarge of vOl'tices ur to exhibit unstable 11ft or moment char­actt)risl ;cs; or

d. the structure, or structural component, is free to oscillate in a bending j,]f),\C al,d a tursiollal mooe Simultaneously. Under this condition the two modes may COUi1le in a motion compounded from buth modes and appear as an eccentric

WIND FOHC r.s

ttll·;'llji: 1;1 ",vi1if'li fhl,l ('t'Jitpr of l"(It.If;r)11 h,l~ 11l(1\'('rl upwind by an alnqUnt dr'tcr­j~~l:.'-·~. ;,y tLf~ r,.'~at j\.(' !': l"t'il!",th oi the' l)f'J;c1ir~~: llilldf': alJl~ lhp coupled frequency

Lc'!"> L.·~WI"';, Ih;.~ Iii ll)l'sillll and ill "{,I:dllli~' Till:; tri"' ,,{ motinn that Is known a~. i:uli.('r n~;\:,·, ujldrr SOJlle condltiqns, beCOrtH '1111t!' violent, reaching a cata­struphic '''IOll)~ilHd(' iii a fc\\· cyc:t'S ,d osclllatlOIl.

'':;:,'';1 th0 sourc,' uf ~xcitatlnn takes the form of " !:'ust the chanhe In wind vdoc;ty :odd/Jill C~h;:;{'S a S1!:I,ificant oscilIatil)ll b,'causc the tillle is lIsuallv l"n,; w;~ll rl'sp,'ct tu tiie natl;ral frl'qucllcy of thf' structure. A possible excep­tIOn ;U'ISt'S wllh fret' standinl' multiplp tall stacks in line. It has been noted

~4r-,,------~-----------------------r----------------------~----------' , I

22~'-- - --~--. I I

70 r--'

;0 jr/ 10' lU~ la'

FIG, ';.-111(.\(; ('(Jj';FFlClI':;-';';'S FOI{ SPIIFHi:S ,\:"\0 I.O;-';G CYU:-;OEIlS (112)

:i •. lt t!.('fin .. t stack in !jlw may be at rl'st wh"reasthp secund in line may show St;'Oll" nwU"'l. CUiiSldlT;ltic,n1;; of f;usts ill selecting the basic design wind ve­lucil:; };;1\'" iJcpn cii,;cusseci (Section il).

V'H'lt'X f"rm.it.um; I~lily caU1;;e perimiic alterJ.atin;:; forces perpendicular to the w.nti din'eti')1l resulting in vill.-.I:i'-)I. IIf tile structure. This phenomenon hilS been rf'p(';1k,il:; oUservt·d 1)11 cyli:lciric:d obj.~cts. Of particular interest to the civil ell;;illl'er is tlle rase of vilJraUtog; stacks.

Certain cruss sectiuns develup negative slopes over certain portions of the lift or l1i(lment curves plotted 3bal"st angle of attack of the wind; and even

Wi~;n FORCES 1149

~h\"l[,i1 th,' winri n1;)Y sh' ,w a I] :\11'",:1" (If ;tt t;'", kin ("1" tli r('ctir)T1 the lift or moment d['"'ct lllay art ill the opp<,sile dirf'c\:"Il, TillS ",,"riiti,]11 may ca\lse disastrous motio .. on ccrt;lin typf'S of pbte b1J'(\t<r stif["lwd S\ls"'~l1sioll bridges (see Sec­tion V1).

IV, ENCLOSED STRUCTURES

Rt'f'''i!ni?"ill!:tlut tllf' civil (,!ll:int'er needs 10 kll"'" t1lnn~ ,d'''lll the ;)croclvlla­m;c cI.,ll'.H'tPl'ist iI'S "f sl nlcturcs, lids SIC, i'1I] ;j ('"d), ;"s t lIP work of (l;)st ;lilt! l'rcs('lli (1 Of; 1) illV"st i;:,\lors ami C"lrrroiatf'S (1;[1:1 rl'btin:; tu sh'qH' or drag ('0-,<ii<:;'~'nts for ,-"anr.us I",ilrlin,; f.,,-ms. Illdudrod is :\ t;dll,' [rOil] the SWiss lIulld­irl~ Cud"til;)t ('.1I,tail's IiH' btt~c;1 ;11l1i most C:-'le::SIVC d:.u put'lisher!. This S('C­t ;,,;1 siloulrl ;Issi!"t thl' civil ('1:~inCf'r ill eElaillishi!ll:; nl(ore lo~ical design criteria for p.Ojler eCO;1011,\' and ~aI('ty Iii iJuilrliJl!:s.

A review of In:tIP,' tHlildi;1~ ("()(!,'s and sppcific:JtiOllS (or tilt' structural d('sl~n 0; iJ\;;Ic!i"~s rc've;lls that surprisi"r:ly little interest is Llkcll In the propel- ap­pi;"':Jljlln lof Willt! fu.c"s <'Vl'il thuu,:h there ar(' eXperimcnt.!1 data avaiLlble on wl.; " II tu \n.i lei;, C(llIS is t C:it t ih'ury "ml wh icll wi 11 COITe btl' much of the, ,bse rve<i i,l,f'ilv;llt'l1a. "Or PX;I,nple, b\lilr1il'~ cudes usually sl1ecif ... a certain number of i~!llnds i)l'r s'lllare [.,ot w;.h il() r('[u-,'Ilc<'tothe wind vf'locity implied or huild­j;,,, s;;apc ;]"sLO:lwd, j:'urLhe;-mcJre, the pr,'ssurl' r:in'll is not a real pres~ure but ra.;,cr a \'~.Ilit' that, whea 1Il\,;lipLcli by the projected arpa ill pl('\'atioll. is i:l:._ nd"d to h iYc till' tot;)l forc(' un tiol' ltl;ildi nh, Thi s forr ,! is the rcsult of pres­sur.' (,ll (',t' wil;,;w;)rd ;ll;d suCt;011 Oll tl;e leew.lrd side. It fiC('glS re;)~oI<able to eXjwct lLc Ill",i..:r;: en~il1ecr to usc Illore refired methods. FortuEately. some (>f :~~t~ ~lf)re rt'cc'nt blliLdil~~ cc,dc rt~~;i~ion.':i arc takinr~ steps if: this direction.

ill orc'c. ttl ,',;,aolish a It)c:;ic~ll i',ISis for desk", all eX;)mi;l;)tiOIl of all a';ail­a:11,· n:sl'arcil d:,LI. to 1~:'3G, w:los made by Suhcoll\ll.:(t.,c 31 of ASCE (419). T:ll' .-,'sldts uf t:.is st",;y WI'rf' sabsc'lt;c ",ly i;;corpor.ltl'd ir,to a final reilflrt of tile S .. JJCOHlll-..iti..L't? (423) lhat reC(JJ1l111t.:'l1dt.'d ;lett.al pr(,SStirCs and sUCtiO~1S to be' 1.. 5(' c; in dC5i~:\ raUlcr tll'1Il sha,l(' cocfficicl.t s that could be lIsed after the (;",", ;~11 ";":"city h:lcl beel; scl('cte,:' It is bclie\-c'd that it wUldt! be more helpful to tilL' ch'U cil;~in('cr to start with ;l:'jJropriate sl;;)pe coefiiciellts for structures So t;;;l~ CO;)SiS~l'nt rlf'si[;il loa(,s could bc oLtain.'d for areas subjected to winds that nt:1y be :-;reatt'r thall the avera,~es establis:lf'd fur t!lC country as a whole. For the eastern coast and the soutlwast section of the United Slates, where wir.ds of hurricane velocity are COm:1lOIl. it is believed that a more detailed k,-,u',vled;;;e of tl;e af'rodynamic chariicteristics of structures would be most useful in "rrivil1;:: at lo;:ical critf'na for dcsic:;n. At the same time engineers ;;!lOuld be cognizanl of economy in desihn of Slructures and proper factors of safety.

l:SE OF COEFFICIE~TS

Tilc local pressure at any point on the surface of a building is equal to the d]'l;amic pressure (q) times the pressure coefficient, Cpo li one desires the

._, -.,- - -----------_.---

wr"n FORCES

: .. l;;.! ~on'" lin :\ \l<;f:dlJ1,: use may he marl .. of a shape r)r r\ra~: cf)dfklpnt, CD' t;j;il, W;,l'io nlul:ip!kd II}' (I ~i\'e~ the a\'era[;e fon-c Illr Ul1lt of devatlon ar('a.

u:' ;;.li.:iC pl'C'5SUre is tlw rrllc;uct of une-half the air r1f'nsily and the square of till' rI'Su:t:ll,t (!,~sl!:n vdocity, th .. units bellll; seU-conslsl"nl, anrl rrprpsents :he kJnrtic f'nc.~y 1>1'1' unit vol'll1l~ 'J! mrwlng air. For standard air (0.07651 Ib p,'r cu ft, corn'spondin!; to 15~C at 760 mill of nlprrury) ami \'('locity V ex­jJr('ssed i II mill'S per huur, thl! dynamic prt'SSl1re in pounds per square foot Is ~;\'en by

2 fj = 0.002558 V ••••••••••••••••• (6)

,\5 an ";,:1I"0Ip1.' of thl' lIse of codfici(>nts, for the external forces on bulld­in[,s wit:l jl!;II,P sul'faces 1·.l1l'1llal tn the y:lnd the total combined an'rage pres-5 .. ,,' Iln the outside of the wilidwa;'d and Ipeward faces of an average building m;IY l.'l' 1.3 q of whicll O.S q is :he pr,'ssure on the windward wall and 0.5 q the Sllctinn rJll tl1l'leeward wall (423). The \'alue 1.3 represents the shape coeU1-cil'nt for a typical buildint-: with vertical walls normal to the wInd direction. Th~'r('fol'l',

-2 p = q x Cn = q x 1.3 = 0.0033 V •••••••••••• (7)

in which p is :;n an'rahP p.'cssurf' that. when multiplied by the projectl'd eleva­tJIlOl ar('a. !:iv(">s th,.. t'ltal force on the building. On the basis of F.q. 7 the wind v(,jncily l'OrresixlIJriiJ.,: to a tOl:!1 Wil,d pressu,c of 20 psf is 77.8 miles per hour •

. T;l\~ Ii:; ;1:1 '1lic j n-ess,;rf' fur this n·:ocity is 15.5 pst. It Sliot:;ci bo.) lIot('d lilat t:." v;llues gin'lI abo\'c for avcra~e wall pressur('s

:\1'1' the I,et [,)rcI'S 011 th(' w.llls only iil the c:\se in w!lich the building is airtit-:ht. S\,('il a eillic;ition will ran:'lY arise. ioil)rmally, air lcakar.e due to small Ol>('n­il'~:s ai'ouml Windows, doors, 5kylihhts. and ('aV(,5 w111 give rise to a nct iatf'r­n;.] p;'('ssur(' or suction d('p"nding on whether thl' opE'ninr,s are chiefly in the wil.dward or l('C'ward s"I'fac(,5. WIlPre oppnings are of subst:l.nt ial size, such as ha:l~~,rs, the i •• ternal wind prl'ssure may he of cOllsiclerablf' ma~nitude. In the dc'sl!!n ui r(JoC ;\I,d walls the prc:'ssur(> codCicipnts should include the sum­mation (,f inl"I'nal all'\ ('xt('r11;11 pressurE's for the particular geometric con­fi~;\Jr~'t ~(jn (Ii U,c bLructllrC'.

EX,)Pl'illlC'nial vaju('s of avera~e and local prcssure and shape coefficients arC' avail,I;1:.' f,)r a \'aril:ty of building confl!,'lJralions. Some of these are pre­sented "I;d liiscllsSE'd hereill. Althou!;h thc numbpr of possible building shapes is olJvioul'I:; io.flllite, It is ft'asibll', on the basis of the available coeffiCients, to make reasonable prl'dictir)ns of wind force on most structures.

AV ERAGE PRESSUHE COEFFICIE"TS

I: is r.,ssihlP tndraw SO)Il1~ [::l'neral conclusions regardil1(! pressure cot'ffl­ckllts fur comn,on building typl'S of typical proportions; these are summar-17..,·(j lll'J'('wil :1. For more unusualshapcs the test data described in a latcr part of this st'ction wiil be helpful.

Flnl-Toi>/>"d /luildi'l.(s.-Considerfirst a iJol(-shapC'Q building with the wind direction pt'l'j1endicuIar tf) one of the sides. The average coefficicnt for the front fal'(, is abuut 0.9 regardless of building proportions. The rear suction, huw(,\,pr, depl'l1lls on the ratio of the various dimensions. For example, con-

WT" D FO I\C 1-:S 1151

si(it'ril~!~:l. i,uildingtilat is squan' in pl.l". tlo.' sll('til1n codficilmt is about 0.3 if til!" 1;C';':ht-widlh I'atio is 1/4. alY)ut n.5 fur a ratio of unity and ab)utO.6 if the r;;thl is 2.5 or !;rc:l.ter. An incrcase· in tilp If·q:th or til£' hUilding (pC'l'l)C'ndicu­lar 10 til<' willd) h.15 ahout the S;llllf dfpcl ;H' all illcr(,:1sc in height. The side w:111s ('xpl'riellce a suction or abuut n.R q fflr BlOSt proportions. The roof suc­tion is g"l1f'rally laq;er nn thp windward sid!? bllt til(' a\'pra!;c codficient {or the pntir(' ro,·fis n\)ol1t -n.5 [01' all("it;ht-width r:1tio of 1/4 and -O.S for aralio of 2.5 or 1;11':.:<'1'.

If tlic wind is not perpendicular to :1 fal''' 01' if thc plan shape of the build­ing IS r!·"t:1nglilar thc tcst data dpscribed subscqupntly Will be lIseful.

(;tllr/ I'd lloof.\, -M;1I1Y inn'stil!ations havc \>"C'n made of the wind forces on typical gahlt'd-roof buildings. SOliit' (If these nrc summarized herewith and sho· .... nin Fi;:. 7.

ASC E S ... hC'lll1l111ittpC' 31 rccomm('nded (419) .1 \'era~£' pr('s5ure coe(£lclents as {"llows: On th._' windward wall, O.B. and on the lccward wall, -0.5; on the lcew.lrd sir'lill', -0,6, and on the windward slopl' a value varying with the pitch as sho"m in Fig. 7. Thpse \'al\1es aI'£' obviously averages because no consld­I'ratioa was ~ivcn to U;e proportil'ns fJf the builliin-g.

Thl' D •• l.ish n'lildin~ Cnde r£'l'ommends (427) the following: On the wind­wa.rd w:111, 1.n, and on thc 11.'ew:1rd wall, -0.2; on the leeward slope, -0.2, and on the windward s10pe a linear variation with pitch as snown in Fig. 7. It will bc' notf'd that the ncba.ti\'e pressun-s arc considl'rably less than those recom­n.C':lcir-d l'iscw:lere. Hf)wc\'cr, the 'llicillional rl'quiremC'nt is gil'en that all walls and ro"fs shall be anchored to n'~;st a suction of -0.8 q. Again. no reference is madt' to tiw proportions o[ the l.uilding.

Till' Swiss Building Code r('colllmcnds (438) the following: For the wind­ward wall, 0.9. and for the lC'eward wall, -0.5; Cor till' leeward slope, -0.7, and fur the windward slope, the variation with tJitch shown in Fig. 7. Recognizing I1w cHert of the dinH'nsions o[ tilc huilclinr" thC' Swiss Code recommends the f<Jrq.!"ing valj;('s oilly within the f"Howing Un,its: Ratio of wall height to build­in~: rit'pth (paralll'l to wil.e! direcLun), 0.67 tu 1.5, and ratio of building width (pcrpC';;c,ic\,lar to wind) to dppth, 0.4 to 2.5.

Als-) shown in Fig. 7. as a matter of historical intercst, is the Duchemin fl' rn-.IlL! til .. t (or many yC'ars was used as thl' prcssure all the windward slope. Tio;s fOl'llml.a assull.l'd positive pressurcs for all roof ant;les and neglected suc­tion nn llH' IN'wart"! sloiJe. The more recl'nt data as indicated by the three rec­omn1l'nd .• tions n(Jtcd prE'viously shnw appreciahle sUl:tion 011 the leeward slope am: a11>(1 on lnf' windward 51'Jpe for small rllur pitches.

Of t/i{' data J!:i"'!H pn'\,iously the Swiss Code is probahly thl' most accurate I.ecausl: it is based on thl' most recent and ext£'nslve series of tests; however, it is I'I';lsnnably ci'-,sc to the l'arlier ASCE recommendation. All of the coeffi­Ci(,l.:S .\1'", a\'l:rabes for the various wall or roof areas. More detailed local prl'SSllre cocfficients arc ~iven III the test results described subsequently in this sc .... t iun.

UtilI' 'tied Roo/.<; , -The situatioll with regard to arched roofs is more complex. Pcrha,Js the most bl'ller;ll iniormation is that contained in the Subcommittee 31 Rl'port (oi19) that is silown.gr.lphically in Fi~s. 8 and 9. Pressure coeffi­cients are given for [oLir sebments or circlllar arches: The cel.ter half and the lep' .. /ard and windw.u'd quarters. (Subcommittee 31 used variaule segments depending on the rise-span ratio but in the Final Rcport (423) these were changed to the Simpler procedure given herein.) These are given as a func-

1152 Wl:';u FORCES

" -- - -1 0

OUCh~"T'rrI

OP

o l

0(,

'.'

" or;~--!

I ' 04

01[-

' .. , '"

riG, 7,-I';XTiCH:'-:AI. ('\lri--,'-icrr:;I'TS YOH WiND PHr;sSL;HES A:\O SI'(:T;O~~s (',\ sy,J:\IEnUCAL GABLE HOOFS

WL-.:n FOHCi·:S 1153

tiU:l ,,; the rise-span ratio, r. Fi!:. 8 is rOll' :111 arch Si'ril1f:i;~,: from the rrc,ul1d a:-.l: 1-'i.~. (l for DIll' supported allOv£, 1;1''''1110. fr)T eX·\llli)k, 011 vertical w;llls. In thl' ;;,t:( I' case the vertical-wall cfwiiicil'llts cnuld reasonably be lak{'n as n.8 0;1 ti,l' fr01.~ :1"d -0.5 Oil thl' rear. };otl'that in Fig. 9 alternate valtH's of Pl"l'S­sure and suction are~ivcnfurthe willdward/'ieplll'llt and for r = 0.25 and 0.30. AL"0 notc that ill all cases sucti'Jn occurs over a major portion of the roof.

These recomlll~ndations for ruund,>(\ roofs arc obviuusly rather approxi­mate. Huwev\iir, they appear reasonable in view of moP' recent tests (438) (434). An important factor not cOllsid{'red in Fit:. 9 is the h{'!i;ht above ground.

,---10r-----------~----.-----------_.----r_--,_--~

06 --·_04

~·IG. 8.-\\'C>iD PitF.,s,sun: or:-;THiRUTIO:': Olli ItOU)liDED ROOr'S ST:\HTI;.G FHO:'l GI\O{;I'n U:VEL

In ~enera 1, rrJof suction increases or pressure decreases, especially on the windw;trd segment, with incrf'asing hei::;ht of the st;pports above ground. These mta ;tre for a transvers(' wind. If the wind is parallel to the arch axis the roof pressures are similar to those on it flat-roofed building.

DETAILED TEST RESULTS

The Tt'C " ,r, from various sources giV(,ll pl'cviouslyare obviously of limit.:G \i,~". • 01' j' "oJ not consider oblique wind directions, the variation of pressure over the surfaces or buildings of unusual shape. In many cases it is

..

==

House

TA!lU: ·1 (:1).- \'.'1':0 [,lli>~'L'!!r, l'll'- !'r:"!"\'!" c', ,~ iiut."",", ,\\ll CI 0:0;1:0 i:.\l.I,S

'--r- -'T' Int~rrul i'ress',lr,·

Cuc:ilicicnt, Cl!l l E~t,'rrl.11 I'rC'~,;u:'e C,,,:([i· {('fO~, Cl"~

--r' ---'I' .-.... 1-1 ..... ··-1-·- -I f ~ I OP"'"=-:'~":· I I ~ ~ L on Slde

Structure: Ct A B c Il

cj-='= 1--'-0:' 0~9-'1'~~: .

E: r G 11: 30S I'

'I :Il" I "" -;. .\ B .§~ :.; 6

c

R(y)fs O· to 10', !, ,L ~ 1 I 1

nr.L,~~~,,*-,. _t_ lis, 0.8 -0.51-0 •7 1-0 .5 j -0.71-0,61-0.51-0.61 ;0,21' 0.71-0 ,41-0 ,.;

\. n>"~':1n ___ . _ -15' 0 .. ; -O.JL 0.5[-0.,1 -O.~I-O.:> -0.", -I).-\! '0.2 0.4:-0.4 10"\

o·j~:;a;}D~J ··'.5·I--'~'-~· c,:;:." ;::'::~;. .. : - r ~- .-~ . -~~-. o'.l;.i~=~=---=-~::'-· --0:- .-~-;;-~]:; -=0', r---"---T

~1>.! '" '1" 1'l"'I"_I;" '·'·,.T., _L -Ij' 0.1; -0,5 0.-\

"m\ C 90' -OJl -0.5 0.9 ... "" ~'<:'1 - -'-'f'- - _---'I __ L-_..L-_-'-_-L.

h.b 1.=2.5 2 ~ oi "[G: O· __ .L_ A ll:.j..J B '-l

~~ __ L o

House

45'

-0,7 -0.6 -0,6

-0.3 -0,9 -0,7

-0.4 -0,8, -0,2

For m, C;c -1,5

-0.5 -0.5

-0,6

-0.8 -0,4

/-O.fi

0.:1

I ().~

.... .... '" ~

:s ;;: :J "'1 o f5 t"1 ::tl

T.-\fll.r:l (i).- \'.I\PI'!ll.'Tl:l-: (·('".FF!'·'I·;::r:-. F' ':,01',: .. '.:::, CI.OS!lJ ,W;I.i'r.:V;

-- " C 'i[lci'c e :, C~ .. : OJ 'L" :Ilr~' (". __

f>.:t.":nl.ll t I',.; - - - -" -r- -- -r- -l ~ I l' 'K __ -.- [ I I r.' FiG I -=1 : _ J_ __ .. ____ _

.... [) " J. _L_. !AiB CI J '-_ <lI .. _L.l _ .. -l '. \\'11[ "pen . (I) OilL 101,'; .• _ _ '~'." .

.. -'1' I I, ' -- -·-·-·r I I !.! O'.-(".! U.". ' I ~1-J7 -0.01 . I ! ' - -(). 7 O. 'I o. , " I I _ 0 7 ' O• , O.R -0"1' I " 0.', _')." . ,

' , -I).j -0. _ , I I I • I U 4. 0,11 I 0 .... , . I. ")

-Ii' 0.7 -0 .. , I "I I -0 3~ '.2 -O,(j~ 0'_1 II I 1 ():l 0,-1 -1).4 I ' 1_ 60' 0.3 I -0,7 O,7l1 • I _ I "' -0,1 -0,5 -0.2 -0 J,

'I i 0.{ _I) ') ... 0.) l -fJ. ~ ! - 0 c 0.9 I -0. . L _ --' ___ _ Il/j:_I~~~_L __ L ____ . __

r- i (II) On" e:l' _ _ __ __ i ----- -1 '- -- ~ I l' ~ _. ~=-:-;l1--};--rr:-- -----l 1 1-------1----;- 1_ 7 1-0,3 1-0,2'1-

0,. -0.4

1.-0"1:

o II I II _ 0--0.4. O. I .' O· 0,9 -0" - "I I I 1-031 0,7 -0.'" O.d I

- 0.71-0.4. ! I - 0 f· -0. J __ ~

I I ". 0,5 0.., _0< -0,' 0,9 -0 .• : '. ' I u 00' .no ',9 , , , o· _L__, II ~ it 60' 0,' 0,9 'I', 03 ! -0,,1-0,' 0,9 -0,' I 0"

==+~J l-iJi" , . f' J~~ 'rn_~,IJ __ ._

Structure

Build·nr: '>P':'>'l 1,,)11 (."~ ,~~

Roo13C',flhL 124

If'''':-l; (G /:::: •. :;

I" " "",~ ~.H'''Hjlr Je:., ,J. . . .

I wall op,'n

... ..... <" C.

-::: "

Z U "I o :::J ()

"'1 ill

------------ 3oo-1::=C.'=~=---:j- 0I':3 ! -0.5 i -0,jl'-O·7fO·~~-:5Fo·~r~0,2f-O~~1-0 .. ip~G-House ~~::.~._i-:<1'T --, - 45' 0.6 -0.:; I 0.4 -0.41-0"11-0 .. ; 1-0,ti -0.7 .:0.2 I 0")1-,).4 I 0.:1

~ (I:T'-------"- 90' 0- O' 09 1 ~, I I I I h_b.t..=2 5 2 5 ~ ~Jf:G; 1-.=> 1-"'-' I . -0.4 -fJ.7L- ll .- i -fJ.7 -O.~ .:0.2 i -0.4,-0 .-! O.'i

OO-l-A~~~~ ___ 1. 45"1--~':'-c~ : --;'2 --- __ ..L_. - -- .-- --,- -1--- - -.- -

-O.G

-o.r, House h.b I. = 2_5:2.5

50<Jy ?;"~~=.-.-.:,-=- -·~:T-;;T~~.;,rO-;Tl~~~i]--;; i-o~r~~ ~"!ti'~:=~'~ 45' I 0.6 -0.5t 0.4 -0..1 0.3 1 -0.1 1-0.5

,J:.I~J f 90· ,-0.5 -0 .. 5 0.9 -0.4 -0.81-0.2 1-0.8 -0.2

~ :~T:::·:::T.~:~-,:0.1 -0.4 -0.;lO.8 o A 4 B 04 r- - - -_- __ - __ ---' __ _

F H i I • o -- - 75" For m, Cpu = -1.2

-0.3 -0.4 -0.4 -O.S, -0.8 -0.3,-0.3 ,:0.2 0.81

-0.2 -0.3 o· 0.9 o::.-fC:l- -___ ~Th '"~ Z::-;.i· .,-

Clos~1 hall '\.., m C /" __

Roofs O· to 3·, h b.L ~ 1.4.4 '" " E ~~ I _.~~:: .. ::r.~ ""1

0·L..-/ f J~ B __ 1

. -r-- --r::-:-ffi' [--I I r-1:: ,,0.S .0.3 I .o.~ ·0.' .,. to., ·0.2 I·'·~ I :0.: 0''1"0'' .,., 4" 0.5 -O"ll~~ -~.~~.9 -~ -0.6 -0.31 :,0._ 0.4,-0,l 0.4

Closed hall

h:b L -1.8.16

15" • For 0, side C, Cpe = -0.8

o • • .n-~~ ___ .-F~rJ~~ ~Pt) = -2.0 n, Cpt) = -1.0

t'O. b -.j 0' 0.8 -0.5 -0.5 ,-0.5 0.2 0.2' -0.6 -0.6 300 - -- ------,

m, 10.~~_ ~;9l)o----rh 450 0.5 -0.5 0.4 -0.3 0.1 -0.1 1-0.81-0.5 t' ~ E G f 90' -0.3 -0.3 0.9 -0.3 -0.3 -0.1 -0.5 -0.1

0·_·- A B ~ ~ ____ . __ .

~D --~ and For ro, C~e = -1.0 90·

• F H l~

j - i j

·0.2 0.7 -1l.4/-0.4

~ 0.2 I 0.4. -O.~ I ~.~ :.o.~~.}-o.~

~ o 'rj o ::::J (")

8

... ... c.n c.n

-~~-- ----~.-u~~-.':.;'.;~:;:'~;;:;'~'--o. r.~-~-. -~. I.---o-'(~?r:~~~.:r.·:~;,~o:'~~:-~:'I :,~,~.-.,) .. ~r _. . .... _- .... -rG~~~~ . G_~~ W ' [' I 0.5 1 -0.·; 0.5: -0.-1 I 0 . -0.,1 -0,.;' \) !

"'H ,,- ,- ,- Ie H ~. ']1 I ' I I Iii 'J7'7c'C>7'"1-".-rt~'-~-:"-;'''\ri).J';' tj~~~_ ~ __ J 0.7l:0~6 ~:;J~ __ :lj~-,Jj -0,\ -<)':l-~J ~ r---=-G~= -t~ ~ -=-:..-_c_~, -1 I (Il) Two end w:tll:; "l',,'~ 1

.! fO ! I l[:l o· r;;1-=;~;-r~-~;1 ~o~4T----T-~--ro~ II-O-'-7t-o:~r~ 0"-- L ____ ' ---I !..:j il -- II - .. : 450! 0.5' -0"1 -0.11-0 ,S l : 1-0,3 I -0.-1 -0,8 i -0.3 i f-:-'l I, I, • el,o 6O+.od~:jO'J~05.J ... i ,010:' 'O:SL~

=-r-=-: ...:J __ l.:-_.:,:-=!j 60°1 End fkt, c; 0.7, d = -v.G, End [Jet e = 0.6, f = -O,~ ________ __ ___ ~ __________ . _______ L ___________ . ____ _._ ____________ ----------4-----

h·~O.5h I Interrlal Pr,'-,_;C1re

Clo-sed budcH" t> With rooh'ent:3

Root> 20' hbL=143

---~-------- .~

G~ jH J Coe[f,cient, Cpi ' with \','nt,;

F ,-, ~/ - t h' - r - --I - ----\t~r:i~T!--hf I IF &. ,J F &: J (Onl, .J ,,,.t,

:Y.1'V r. ,,' - -"ct :" .-: \ : "t,_ I I open clo:Oc'd opt'n I ')"L'I~ rn. 5" ;'-::::r-;;:;-, - - ,- -- I - -- - - -- r- -T-- r -.- r -- -r- -Gr- -1' - 1--- -'1---r --

~ r:/. 0 f 0° o.~ l_0.5 -0.7

Go.7 1 -o.~ I 0.6 -1.0 -0.6 1' -0.51 -0.'; I -0.2 "O,~ 0.3 I -1).,1 'A ,_"0 - • I 'I I I - I r ! r- Inw I _ _ Ii· I _ _ I i

0 • .1- nwtit ~ ~ 45: 0.4 -O.~ 0.4 -0." 1-0.3 1 o.:! -1.3 -U -1.0,-0., ! -0." I .: I).~ I 0.: -0.9 tt.' 11':'1::'" :".~: "'J:" ~- 1.0 .• ·0.3 :":'L'.::'l" 1"- l°.::.l~o'_

o -- --- ~~ }o-or m, Cpe = -1.2; n, Cpe = -2.4

--.- ----- -> ,-------------- - -~--.-----~---

~

~ t1 "1 o ;:J () .,.. :n

~ ... U1 --.J

TAHU:'; (c),- " :';!J PI:f>:,Ull' i'l11:i'F!I'll';"TS n.:: T.\I.I. /'.t:ll.l'!\(~S. I:u(ll':" 1.',:1· 1'.\~,;\f;l-: Y.":'

1" r' . I -. Extt.'rl ,-:1 Plt'!>':lJr\..' C'l v ·:f:l'1;..::':, CPt'

·_--L I _ .. _,---I I, c;

i -,

I

, 0 !I , I '"::: ;;

Intel'll',l I'rc',;3ure Co~ff1C1C :It, Cpi

Op~nings ~!.il ":: 10

---r--Structt.re

I I I k"=,

a A II C 0 E F: G II I § -:; R "

I I ' I '" .:" B co"'

Tall bUlldlllbl C!Oi!"~ j fl')!)f\ O· to l"­hio/"25l'l

j I ' ''''''nl''' 'F I I :;.; t : C

I Ie' I! ! I i~ ; T -1·'~ ,,: .o~ i·o~' ro.; t .~.;!:.~r-:T:r;.:~; :-, --:-;.:::i­

~~ Ibt "'jl 0,5: -0,5 ! -0,9]-0,6 j -<I •• ; .0..1 -0·-1 -0.' i ,0.2 0.'1-0 .. ; I -O.l i

0.-1- ~- m~~(.~ L."· .. _°·'1 -ULO.' . :'~5_-~L'.d~0. 71-0.'1 ! ':'L-'~ :":'L~' : m

f '::'6'i.H 45° For m, Cpe = -1,0; n, Cpc = -0,8 .L.n-~".l l .. _ .... '--'-- .-_ .. --- --_. -~:. -o-:-;r~o"~I-~:~- [~~~ . '::;r:;:~:-:'5

30· - r-(f --. -l , "..I ' ~y r.~j .cI 45' 0,6 I -0,5 0,4 -0,4 -0.3 -0,4 I -0,5

Tall bUlldln;:s a 71fW:- '~?C. '11"""). 90' -0,6 I -0,6 0,9 -0,4 -0,71 -0,51 -0,7 closed i E C G ~ :

hb'L=2'12 m '-'ri..'Br1 - r .---- ---

-~:~ t-j' ,;p.! -~·i--=·;i

-0,6 ,':0,21 0,5 -0,4 I 0,31

-0,5 I :O'lo~~.~:.~1. .0,5 L_ ... _.

0·- .. --At!' B "-l

f!~H--j· o

0'

---- ----------------"'---

• For m, Cp\! -1.2

--------------------~

... ... ';''1 OJ

~ ';~

c ~ o () C<! cr.

-------- --- -~"~-·! .• jO: r -r ···r- -"1" <' r- "r '·r AN- -1 0' I 0.9, -0.; -0.', -n.,' -0.5 -o~" -u.; -0.-) I +0.2 o.sl -0.4 -fl.i -IJ.·I

...... \; :-,.~ ...... ""';;-''T~ : I I - I Iii C 45" O.'i -O,(i 0.·\ -0.4 '_\.2 -1).7' -1.1

1-0.7 .:0.2 0.4 -O.-\~ 0.:1

1. _n.,

Sr.ect rvel /' m __ r;:r".1 - -'1 , ,I!, hbL.12412 / ~ ill I! 90' -0.-1 -0.:1 0.9 -0.2. -0.:1 0 : -U.l

1

0 .':ll.2.: -0.2, -0.\1. 0.0, 0

Ped<cd rcol

h.b.L-115

" "j'EG . I ' I. I I I I I O"-i.~--- ~A~.-- B ~ 1180" -0.4 O.S -0.7 -0.7' 0.1 11 O.I! 0.2 0.2 t' .':.0.21 -0.3 0. 711 -0.';1 0

" 1 L---- -1___ - J" ____ L --- __ J - - ---[fIt!] ____ - ~ • o 45" For o1. Cpc = -1.4

;~--'--- - T-I-- i -r-[-- r--r -LT .. , I \-'-' 60.-~r)".-:r~-3l i o· 0':1 -0.:;, -O.G I -O.G. 0.6\ O.G -0." -0.51':0.21 o.~ -0.41 -O,,';! o.~

Al _ B .. .,t- 45" 0 .... 1 -O.S [ 0..11 -0.5 0.21 -0.11 -1,0 -0,,81 ~o,21 0.1 -0,7 O,:ll 0

~Fl"'.'T":".'.\n. ,,, I I I 1

I~" mtr;'- ~-I: ')0" -u.41 -0.,1, 0.9 -0,3 -0.·1 0 I -0.4 0 i ,:0.2 -0,1 -0.1 0., I -0.1

o.I-~Jd i l'ao~~:'j O'l ~:6)_-~6Jo'I_'_":{01"toL~' _..'::L:~'I':~ ----- - ~-- - --- -- - -- - -- ---- - . ----~- - ... ---------~- .-----.--- "--.,,

IlltU ___ i. j ~5 FOI 01, Cpc -1.3

Clo~ed connecting PJ5s.1ge bet ... "en large "'JI'5

h:bLal:l:lO

[~~J;-i r"" Ac~m-d=_o~~u;e~~~ 0.] O.S' -1.2l-1.4 -1.5 i-o.5) 0.7j -1,1 -1.31'

C;;, - f --- ---1\11- ----~ I ,I I a N ~ I

___ . =-~.-~-:=-.::--=--=:.---~--_-~~~ -__ =--=" .c-==---=:c==~-_= _,, ___ =_~ ________ -_~_ -=-_c::...--===-=---==:-..::_ ~-= _-=-=::.=.-::..-..::c .=1=

:e s:: tJ "l o :0 () t'1 Ul

... ..... <.n <D

T.\III.E .\ (ti)." \\"1.,IJ 1'!~r.SSI::U'; CU~. F FIe [,.: ... .; ,"0 .. :"111-: 1.1'1-:[\ 1'~llOF:;

Structure

Roof.; 30' ,,= 0·, 45', A 0 fGil len.;lh

"'-90', A-O PHt len£:h l.'

Effect of train<i or s~ore.1 ma!erl,l' Roof ~ 30' Q~0'-45··180·, A·D full kn£ttl

a = 90', A,D part length r.'

~xtt:rn3.1 I'rt..";:~sur,: Cot:(f).c:t.:r:t l Cpt.;'

-------,~ +--1 -00

--A\. fe

l ' = 0 5 b _ ..... : , .•

Il 'v fj;~'_. c;'"''.''' '>.. lTJT,\, . m __ 45~

90·

Cf

Tj -. r-I I ' I ~-IT-_~_~ __ ~ .. ~~ J ! K : _ .. ~~! __

1 j I! I o 6 1 -1 0 . -0; -0 91 - , I • I . I . , . I : i

O 1 0 3 0 ,' .. 'J ' ! • ,- • '" • J -v,, I I !

End Surfac"s

0.:1: -1).:1 /'~TKt~~i !-r":b La ~ t t:-::1 L

---- i -f ',_ R • o· -- --- o..l . b .... 1 )0 ~-- - ---

-0.31!"O.~ 1-0.31-o.~, O.~ i -1).,1: ___..:. .. _l. _1 ____ L..--l

For m, C;e,top 1':.", -l.r.d MT

'.---,----_. -------

1l~0_5b _ A'~C ~":rB-~~:-'r .. J rh'~O 81: /' \.. ,,~." c]"' ;:.!. .... ,.

a ~. rK~ ~ "'~f '. -

J ;,/'. I • 00

- ___ '" A kC)l_';~l :--- L'·b vr-'P;'I- J·

1J!J_i TR90'

I 45'

90'

-1.0. Cpc-,bottom -0.2

Tangential actin!: friction, R = 0.05 q: 1.' b

0T'~-U.j i oj -0.1 I 0.;:, /-0. 8 I 0.) i -0.4 -0.5 -O,-l i-O.5

I -,).31- U,61 0.-1 I,o.6l -- --.. -

___ J-__ . __

, O·

45·

90·

1 ~O·

45· For m, C;c,top = -1.5: C"c,oott,)m 0.5

90· End surface frictii)n load, see 3bo,e

-'------------ -----

..... ..... ~ :::>

:;:

3 "'1 o :<l () t'1 en

----- -----.-. h.O~5b ---A-;"([~-~.-~~----I- ;-1---;:~:1:~;~ ·o~;f--f--T-··-

~\t:~J.~;"~-~.~," 45· 1-0~3 0.1;-0.3 i ".1/' I I I Roo' +10· a lKflTrl r [.'.b 90· i -0.3 0 1-0.3 I 0 O.S I-O.';! '1.3 I -D.4 a~0··45·, A·O f~lIlp.n,th • J ~ A ,;l\l._C I r f---J.- - _______ -L-___ - - -,----- L __ .. ~90·, A·O pJrt lerith r.' o.l ., • Rgo' cor m, p~,top = - .v; Cpe,bouom = u ...

, I" b"}-L o - -- " -t· 1 - • O· J ~. C· 1 _. • •

J-rli#i£M O·and~IO~ Tangent:.u actin);; (nctl,)n, H = 0.1 q' L b

h=OSb. A-\_,C,O. "R J • O· 1-1.3 0.5,-0,6' 0.71 ----.- .---.----. ------------- .------ - . --- ---J--r~----'-'-

~::'B::-~l/'rLJ-~'~Og;, 4:;· -0.5 O-l:-03 031 ... (Heet of traIns )f stQr,!j ··lJterral / ~ _ -:" J:, r; '" ~. j. .... I 0 !' I . i . I

Rool T 10' a ~ [fA'K .... I~ (-.b I 90 -0,3 0 -0.3 I 0 :

a-0··45°·160', A-:.J t',III~nbl" 0.-1-- ~ 1j~L':~1 --t' b~OO -0.4 -0.3t~~.31 ________ _ a:90·,A·OPJrtle1;;:hI.' o.l ~, R9Q' •••

J. tt;;;t:f, L O· For m, Cpc,top = -1,6: epe.bottom = O.!; m7 M:I 1

and 1 <;0· End surface frictin" \o'tcl. set' ~!~)I:"

------- ---------h~~-~:~-~-~i~ -.. no' - ,. I '.3 10°'1 02 i,=-;;rl-------

Roof-10· a .,. ~ L' b 90· -0,1 0.1 -0.1 1 0.1' O• 4~· A 0 1 I' I "'h I'" A: Cr---· _.- --.. LJ. ___ . -_.---- ._-.-

A"':-E~'" ". I 0 1-0.2 0.1 ;-0.3

aD - J. • L.I I ent.~ • .0 t. j~ •• a .. 90", A.O pJrt I~ng!h L' 0 -.+- ~ ~! R9Q' O· For m, Cpe,top = 004, Cp<.:,bottom = -1,5

-Lm 0·and90· T.Ullo:entia\ acting friction, R = 0.1 q: L:b

Effect 01 train:; or stcred mdtonal Roo' -10· a=0·-4~·-180', A·D 'ulilengtl) a·90', A·O part le"g:~ L'

h~0.5b"'T~ -A ...J B DO

I "'T~;~~ O • .J.. ~ "~C

- M -,:: o.l L_b ;

--1-.. Ro' O·

;'!!..'=O 811 45°

90°

--L'·b 180' !.

10' 0-

O· and 180· --------- ---- ------------ ----- - -- - ---- - ------ --- .

---r-------0.1;1

~:2ll --0.3 -----.

- -,

-0.7 0.8 -0.6 I -0.4 0.3 -0.21 -0.1 0.1 -0.1 I -0,4 -0.2 -0.6 J

• For m, Cpe,top • -1.1; Cpl!,bottum = 0.9

Tangential actin gfrict!on, R = 0,1 q:L:b

~ Z t:1 '":l o f5 ~

-.... e> ....

T,\Ul~," ("),- \\'r~f) l'i~~> '! ::1 Cuf:Ff't1 !r,',~r~ ()'; 1:1 {t~'~, ,'''II L;i~:\~;n-';i,\:~r..;

r f:'{t,..'rll.d f'rL';i,f!rt.: C":..f~, "','!!t. "f.'" lI,t· rl!.t! i J f' '_\'4J'L'

COcIf1C;'; ::, Cpi -I

1 T' I-II [I'-I! I II h f_O~":~_':~"-CJ I AlB len f: I f" G III J K; L I ~I :-; 1 niP , Q 1 3 i I· r I

1 1 I I 11:;; AlB! c

i 1 ! I I I i I!~ ! I -------;F~\~-L'··----I -: ;.- r 1--1--1-- - l-i--T'T !--r-'--~--'-'-------' -t--

~llj I" .I~· O~I-O··i-{J.4i 0"1 " . .;1-1.'0/-0°1'''':; -I) ,,-.),4 -,),3, 0.3 1"0.2 0.' -0,2 -0.3 _\J ',-..:,.. ' __ , /30, I I 1 ' I , I . I

6011...... .... ,,-,' 1'- -, I "5· 1),-),-0.41 0.5 -J.3! !) .. ~!) '1' ~I).j -0 .• ).~,-o.41 ),~ -".)1 I ·O.l ,0.4 -0.3 0.4 IA 8 .-~ , I 1 'I 'I I' , 1 - •

S ..... xfh 'co' ~-~ --c~. -,~-' ! 90-'-0.4 -0.4! 0.~,-O.3, .).:11-0·1 -u II- il .! -0,'11'-0.4 -0) -0.) : \1 :'),2 1 -0.3 -0.3 I 0." J, .•. ' '" --- I I I I 1 I ' F'IC~"~::':;~1~'''d r!1;r~I({f~1 ., 1100-1-0.31 O'9I'-0.31,o:JI-().~:-')31-,).· .. ,).4!-0.1,-O.6:-0';'-1).11 L :0.2 1-0.2 0.9' -0.2

RQ.Olqbl.m!~;}~I,~ml, - - 1 I 1 _. _________ L __ L ______ ---->-----

hbL-145 A·· ,. ~ -8 ..;

I rOil' r t~ 111'\R. :,1,1 Ii,' an,~' UJL lU_l 1""

f'or nl. C;~ -C -1.3; n, C;3 . -2.0

1 0 I

t.. ---b --."

---~--- - --- -T--l---r - r - i - r - -1- -r' ,--.--- r- -fl--T-T--r--1---r- --E---r ---1-035h,1 0' 0.91-0':; -0.6 -Q.6 -0.3 -0"1 0, 0 ~I-l 0,-0,4 -0.5 -0';1' ·0.2 0.3 -O.~ ".0.;

30' ~-F-'::~- i , I I 1 I I 1 I I I I 1 ' ---tc-~!· ~_i 45' O.:;f-O.5 0"1-0.4, -0.6 -0.5 -0.5 '-0.5 1-0'-'1 0.':; (0.5 -0.5 I: :0.2 0.4 I -OA 1 0.4

~I;'~~: J4 'f. .. m-~'~)iL I I 90-[-0.51- 0 .:;1 0,9 -0A!-0.Jo.~!-,)-u.~I-o.41-o.4'-1,ul-O'~1 11 , : : :0.2 -0 ~ 1 -0.1 L 0.'1 O,-L-AY '~'B -.l ~--r-~-L--J ----- L_~ i_I - ___ J -- -~- _..L ____ ~__ ----- -----I F I H _.. ,i O· , • •

-::' =_\J K _ ~lU"L! f"vr m. epc • -1.1. n. Cpe • -1.5 M- 0 I 90' "-b-~ --------------- - ----- ---~----- ----

.... ~.

0> N

:5 ~ t:l 'Tj o :n () t:l en

-[ --~-: --;~-:"-In .. -.-:: -.,,~:~):---~ ! In E5';; ~ ; ~ --- - --- --. --- ... --.

I :: '"='. 5

I ~! ~: ::th 'W.~~'l: ,we" :, . ~ .3 'n I ""oir.·ll,.~ -cat- ~ ':'Lte \.

I ' J:.'" ~,: \' "I'<'" op,'r. ,)~.!n . I I ... '" I i I I S I I I

a-;;--b/6-1tl'- -l-J--T-~ --r-l-- r r-1 -/T- i----I-'r"-~-r-+-~r-: I. yo_J If -{I 00 0 •• -O.~ -0,.1 -0.1,-0,1 -0.5,-0.8 I-O.~ -u.t -0.1 I I I !O.2 I 0.4 -0.1' -I..

H."~,."h "Cft:~ . .;ri.t<:~~~:_! ,30" 0.01-0,31 0.2 -0.4,-0.1 _O'41·ry'7['_O'9to.~I_O'4': i :' !O.2 I O.~ 0.,;: 0.7 S'·OI;tr'wf.c. Y i .l_~!'=:'~J ~ I 90' -o .. l'-O'~ll O.~ -o.~l II J i-o." L'tJ,1 I-O .. ij-O.3,-O.11-0.11 !O:~ I -1.0 I O.~ i o.~ ~:~:6:1n~'~it~-~-f~-lB~ , __ ',_i,_ .... -._. -, . .1.'_ -- - .- -- --~- ---------.--.---.----.

.\,.0 I b 1 P __ '_ -J l 300 For m, Cpu· -I.i with CpC,m1!1 • -2.5 DL..1' __ J __

--;::0-;;- --;. ti°~",:, +t --[ ~;-.-;;;~-;.;~>ttu;,:; r-;; -]~~ (runt and hark ur " . .11 +-! --".08b-j BDF~ • ..,l" - -r-~-r--l'-T-T ~'-T":"-1'--'r-~' -~ --- ---..,--

~Jl'~'~f.)" 001'1.1) 0.9 -1..)10.9 ' -1).1 0.91-0•71 U.9: o).~I-O.:; 1 0 .9 -u.5 I r I G "'II' "o~"",,·-;.,··nr'- 4S' -1.11/ O.~I-o.711 0.41-0.5, vA -0.5: 0.3,' O·~W'-O.GI O'~'llM I 1 ~~;~;~,~~' t· ~:r:.:1r~·. 135' -G.4 '1.1\,-0.7 -I.O,-U.:,-I.I; ·v.~ 1-1.0 .-1.1 ~.6 ,-1,0, O'~I I, I i

Sid.. '\.! I !AljJJ:,:'K ~ '1.0' :O.6~:~-~6 :~.3.t~:til~O~3_!:.~.6_.:~~ O.9.l~ 0.9 L: _1 __ _ Root ;1 _~ or. f" I t4

0':'-2 .~I!~: it olj' For mO· c~.tf)P • -2.0: ~~,but:"m • 1.0

1 C! G I "'~ 600; For row' Cpe,k • -1.0: Cpe,l • 1.0

- I·M 1 -=::~ __ , 900 ....:.._2~.g~nt~ actl':'l: frlct'22:..~J> • 0.1 q, b: L, RW • 0.1 q: b: L

-r--O 14 () !!....---=---------TD'!_ FO· PI-P. A

Tffi15~ ". '1 '; i d'i In which PI • working prcs.ur,'. A • '4 d-, Pa • C.,., q, :lnd ... I, J . - d-· ~ C. -I 0

'=====- ,..~. ,~':':"'~"!~ __ =_L---=~~:...='7:-----

SmOOlh clowd rewn.Olf

--r"--I

_ 1- : _ ...... --

~ Z t;)

d ~ ~ fIl

... ... Cf> W

TABLE ,I (1),-'.·,l~:lll"(::~:il'l:l' L:UEfT!C!I';:;'[,,; Fuft HI',Sr:lt,,'O![t,';, ST'v'r:", ~,!.\STS, PUL!:s, nC'!'i.~~, ,\:\u '.\':r'.r:~

Stru\.·tt~ r't..'

-T-----~-----'- ---

Value of h, .j ---1- ------1--------I 7 25

a IS.,ho:l'c t Va~ It) d l./d I --- [ --- --J -~, ____ ~ __ 14 _. _ 5('

----- l' -!

1 __________ ----- ____ L Value of Cpe

I,O'l ----r--- -

1.0 I : ,I)

Cyhnd.:or, SV' J ~ lo),! ---'-.' '"'l,;

.1 p= P S~;th >UrfpJ~'~SC r&-' -;--.....,i ... j~~hl,~; : 7 ,e e p.: q \ '\.

P ~C q \--=-N h/d-I

~i! {~ ~ d¥2;''''O~\,j/\ \ "~I! rd ; 1'--. ~-~ < -," , 'J"- ~7"-1" / , :~J + a /+.:::: .. ·.1 1'.

",\p .,..0.-.) _1-_ 51",-- -- -+- J

Sphere, S,Jr~l>~ I ",J!j

Smoo!r, sur" J(t j

.1 p: Pi - 1'e P, z C?" 'I

FLJ~ Cl

J,rq > II 3 Ib 1/2 (I ;.

J):;JQ. ,\= ~ ,,2

C~.O? '" \ •

81 r ~fi''''' y'rl ~ YJ ' 1Jt"}{N/'I'! 1/, I I' -L _->- '/L /, ' , I' i' I"-~ 'f7/7';-/,;/!:,-~J .• -f 1 I' ~ .. j~ ", ,~1 I- HIt~:~~ I qJl St.lck work,rg Cpl~"O I

Closed reservoils Pi = ... or~ in;! press Stack closeJ ('PI = - 0 8

--------_._---------------- -------_._------------------

0' 1.0

15' 0,9

30' 0.5

45' -0.1

60' -0.7

i 5° -1.1

90' -1,2

lOS· -1.0

120' -0,6

135' -0.2

150' 0,1

165' 0.3

180' 0.4

0."

0,1

-0' I "

-1.2

0.8 I O,~ 0,1 0,1

-I) 8 ; -0 9 , i ' -1,7 1-1,9

• ') ? ., --1.6 -_,_ --",)

-1.7 -~.2 I "r -... )

-I,~ -1.7 -1,9

-0.7 -0.8 -0.9

-0,5 -0.6 -0,7

-0.4 -0,5 -0,6

-0,4 -0,5 1-0 ,6

-0.4 -0,5 -{) ,6

....

..... C> ,t,

~ 9 c;l .., o ::J () 1': Ul

Upright cyl,nder

FDcCuQA

d.f'i~23Ibl/2 Acdh

hid:

Jl-l d . r-1

~mJJ, 2~

Structure

---~-------------------p._------

~ ::l "1 o ;:0 <; ~ C/l

.... .... en '-"

WI:-:D FOiiCES

ad\' j ',;, b:l' [, 'I' tIl(' (,I1,:ln('cr (1) LI KP ,jlh'ant;q:" of thp mallY tp~\ rpsulls a \·ailahlp.. P";':',ljlS (hp (',Irliest tc~t" UII buildillh ["rIns wpre thu"e I,y Trllllll,:t:'r and ;-\,);';'u'i,(vni (405). Till' most "lI.iPII!;ivp h'sts rOl1clllctpd in the United Slales are ,m.ll,;!hly those conducted at thl' Iowa Institute d Hydraulic Rcsearch (434). Tlh':;C :\re v('ry useful in that th"y not only orlllsi(\(>r a variety of building shap"s hut alsc) glv<' contours of pressure for the cxternal surfaces.

Tilt' no')st extcnsive sprieR of tests condl.lcteo to date (as of 1961) and the most t.sdul from the engineer's viewpoint are those conducted b}' J. Ackeret of tile Jlihtitute for A('rodynamics of Zurich. Th('se have since been included in

, , 'I 1.0 ~ ---,- t -- ,.- '-, -------'--+--~_+

I I I .

171.---~- ,- ~-'--~-.;,..:....c..::-........ -_+__t I ' :

"- () 1 '~-1 ".\J 2 I ,

"'"0 n ~ --1-L-! ,.".04 , , ,.=d ~ " I

--0\---

Oh

I I

T ___ l~~--~-~-L~----~--~--~--~~--L---~--J o 01 0.2 01 04 05 06 0.7 0.8 09

r''1ck}nal po:.r1 nf cP'"trala"g'" I} me~su,(''' cit."Ckw,'>e

Fie;. ~J.-WI:-<(l l'i\F.SSl:IIE IlISTllii\lfTIOl' 0:-; r,OUNPEU HOOFS iU-;STI!"G ON EU;V,\TElJ ::iTlll:CfliiU-:S

10

li;e Swiss Bllil(iin;~ Code (438). necau~pthe Cummittee believes this to be the i ,('st infurI,latlfHl availahle, the t.'st rt'sults for building forms are reproduced, I :lI'oliC:h lloe rourtt'f'y uf Ac;wr('t, in Tab1c 4. Earh set of data is fur a partlcu-1;.1' b'.lici..;ng Illndd. The values give;1 are local pressure coefficients for vari­ous parts of tile building surface and for various wind dircctiol;s. Also !;iYcn an' coef[icicnts for internal pressure dcpej',diIOg 011 the location of openillbs. 'l'la' ~ot.d furce in any direction can. of course, be obtained by summing the prt's:-.url'S Ull the appropriate surfaces. The prf'ssure coefficients given for m and Ii take into account the large local pressures that occur near toof edges.

WI;-';D FORCES 1167

Th,'y shoulct Ue used in the design of incal ('lpmenls hul not for n\'cr-all rOQf loadlli[:.

SUMMARY

With the tcst data available (inc111ding both that rC'produced here and that ;:!vell iathe rdcrl'l1ces) and the inforrnatiull Oil windvclocitiesgiven in Section 11 the engllleer should h(> ahle to make re:lsonahle precUctions of wind forces on most building f .. rms. It is not necessary, therefore, to rely on the rather crude approximations containE'd i 11 many building codes.

V. PLATE GIRDERS AND TRUSSES

1;-';,' HODUCTION

111 C();~: rast to the prect'ding sretion (Section IV) 011 e'nclosed structurcs (pri!11:ll'i;y IJUilciill,:s),lilis section deals with open struct\lres such as brldg-es, tow,'rs, or nlh'!r exp'Isl'J framewol':(s. For purposps of discussion open struc­ture's may bo' divided into (wu classes: (a) I •• rge shect areas such as plate girrkrs or sip. boan!s t hat have ala n:p surface exposed to the wind but small d"pti~ and (b) trusses such as those' in bridges and tower:". When placed in p.lirs ;II1c\ combined with a road drck classes (a) and (b) may be considered :0 Indudc platl' I:il'cier and truss l)rid~:o's.

O~ t:t'Il''l':l! illterest in the consideration of such structures arc the eifrcts n~ aSi)('C't rat ,0, H"j'nolds number. solidity ra.lin, al;d shieldln~. Aspect ratio, vi' ratio of Ipn!;tlJ to Width, is of particular impnrtance i1, connection with class (;t l. As discussed i" SI'ctinn 1:1 a nd shown in Table 2 (a), the dra,.: coeH icient \'.lriesfrllllll.12 fill' a squarl'pL.:cto1.98 for an infinitely long plate; the same b true fur a platl' gircit'r with flanges. It should be noted tilat a plate may, in effect.. be L"u!lsidel"cd inii .. ite1y long if flow is prevented around the ends. For lxa:npk, ti,c pl.lte ,;irrk-I's of a deck bridge should llol'l1lally be' considered 1n­fLute in It'll!:! h l)t'cause flow around the pm!:> is pr("'vel~ted by the abutments. The aspect rillio of a truss is generally less important. This is true because there is sum\.' (l'.IW thrnu~h the st l'ucture and the {I,lW pattern around the edges uf thc lo! al pnclosf'd ar!'a is of lc:;s illlpoJ'lan(~e. In this case the aspect ratio of the i .. dividua 1 truss I"llemb!' 1'5 is a morf' s igni flcant paral1lf'ter.

1;1 gelll'r;d, t iJe (,ffect of the n"ynolds number may be il~nored with re;::ard to tile opell-tn1f' 51 ~'ucturcs cOllsidered in this sf'ction, thal is, the drag coeffi­cient mav bf' aSSlIlllf'dcollstant and il-,dependent of wind velocity. This is true b<'Cl'i'-H' such strur!;ireS are gcnerally comprised of members havin~ sharp cdg,'s (~l'e' S,'cti011 IU). A possihle exceptiun w0uld be atruss made upof pipe or solid rOllnn me nll}prs.

TIll' slJ::dity rati .. , 0, df'fin!'d as the solideievation area ctivlded by the total enclnsf"'(; (<<'vatlOn area, is obviously a paramete'r of imporlance. Atruss with " = 1 h"{'Ulll{'S a illate girder and thc th'ag codfieit'llts rllr the two clements are Hi('l'lic;d. 011 the other hand, the drag on a truss ~ith very sm:l.ll solidity rat io if; n1f'reiy the sum of the drags on the individual members, that are es­sentially independent of each other. Between these extremes the flow through the openings in the element is accelerated and contracted into "jets.· This obviously affects the drag on the total element and also that on other elements downstream.

W1:-':D FORCES 1167

Thl'Y shoulo be used in the design o[ Inca I demellts hut not for n\'er-all roof 10aciu1t:,

SUMMARY

With the tpst d.lla availahle (including' 1l0th that rf'prociuced here and that ;:!vell inUIt' r'>[f'r"lic(5) and the inforrnalillll on wind velocities given In Section 11 the enpnf'cr should hI' able to makf' reasonahlE' predictions of wind forces on most building r"rllls. It is not necessary, therefore, to rely on the rather crude al'proxi mations contained in l11any uuildlng COUl's.

V. PLATE GIRDERS AND TRUSSES

1:-':',' RODUCTION

In c');;:r;l5t to tllf! preceding sretion (Sf!ction IV) on C'nclosed structurf'S (prirn:Hiiy lJuildill,:s), this sl'ctiondt'aIs with open structurcs such as bridges, towl'rs, or nth'!r eXp"f:l'U [1':II11('wor;(s. For purp0sPs of discussionopenstruc­lurC's may Ill' divioed into twu classes: (a) l."lrl!c shcC't areas such as plate hirrkrs or sip. boards: hat have ala r!:C' surf.1Ct' C'xjluseu to the wind but small d"pth. and (h) trusses such a!' thc'sf' in bridges and towers. When placed In p.Lirs .Inc! "OlllbillCd \\'Ith a road df'ck classes (a) and (b) may be considered :rl Inrlulie plat<· girder ahd truss bric!~t·s.

Of t;l'llI'l'a: illt<'l'f'st in the consideration of such stru('tllres arc the e[ff'CIS (If aSjlcct rat iU, R"Ylloltis number. so;idity ratio, ana Shielding. Aspect ratio • .,1' rat II) of II'l1l;th to width. is of partkular importance iii ('ollnecti()]] with class L l. As disclissed i;; Sf'ction LI and shown in Table 2 (a). the drab coefficient v,lricsirom 1.12 for a squal'f'p\;':etoI.98 for all infinitely long platl'; the same i;, true fur a plat" gircif'r with flangcs. It should be noted that a plate may, In effect, ue t'vilsictcl'l'd infinitely Ion!! if flow is prevented around the ends. For uampl.', tile jll.lte ~iT(krs of a deck bridge should nOl'mally bf' considered In­fiilill~ in kl1!:II\ 1)4,'('ause ilf)w around the {'lids is pr{,vcl~ted by the abutments. The aspcct r:ltlO o[ a t russ is generally less important. This is true because th!'re is SUl11~' fl'lW thrOUGh the st l'ucture alld the {I.1W pattern around the edges uf the total f'nclnsf'd ar£>a is of less implll·tan('e. In this case the aspect ratio uf the i"ciividual Ir\lss 11lembC'rs is a morf' Significant paranlC'ter.

1<1 g{'nf'r:d. the (,[fect nf the n'.'ynolds number may be il~llored with rer,ard to the IJpell-!Yi'f' 5I,'uctUl'(;S consi<tel'ed in this sf'ction, that is, the draf!; coeffi­cient Jl1a~: b,' asslIlllr'<icollstant and independent of wind velocity. This is true bf'C:I\is(' such sl rut'lures are gf'ncrally comprised of members havinb sharp edgl's (M'(' Sl'ctiUII Ill). A possihle exception wpuld be atruss made upof pipe or solid rOlJnri me mbl'rs.

TIll' s'l!idily raUl', {'o, ddinpd as the solid elevation area divided by the total cnc10sf'(; ( .. 'vat Ion area, is obviously a parameter of importance. A trt:ss with .;. = 1 1Jt'r.VIlH'S a j'lla!c girder and the dl'ag coeffiCients fur the two elements ar~' Hier,lic:il. On the other hand, the drag on a truss with very small solidity ratio is I1wrely the sum o[ the drags on the individual members, that are es­sentially independent of each other. Between these extremes the flow through the openings in the element is accelerated and contracted Into -jets.· This obviously affects the drag on the total element and also that on other elements downstream.

11i13 WI~J) FOl{CLS

F{)I- ,. s\ru, IlJrc cOIl~istll1!: of:l "(lll1hln:llion (If tru"ses ;jnrl phtl's, or Ix.th, U", .,;'(",,! pf llo.' ~hiI'Jdillr: (,f (1l1e IIH'IJl\1,'r h)' allother I~ ari'r"~l:Ibl", For ex­an,j';'" til(' .1r,1<: P:' Illf' lecw;.rd tr'.lSR (,f a brl(L~., !11ust \)(' kss tha!1 that on the windwal'd I rlls~; to a dq:rec that d"p','nds nil tilt' sl':lcin!:. F:;1~h upwind member C;.'IS(,S a ·S:I,Hj",,'" or wi(lLning wake with I('S5CII("<I velocity and ;\ llIemher ly­iiI':; wI:hin thiS w,li",' w.mld l\;IvP rp(ruced dr,,~. In this connection the wind di­.-(,("lion is uUvitJl.sly illll'ortant. With zcr'J allr:lr's of attack and yaw each up­Wli"! m{,lllbf'r siliplds its d"wllwind coulllt'I'part but with small ,'allies of these ail~lr5 the downwind m( miJers are more t'xpns"d. Thp amount o[ shipldln(;also (i"lKlids nn th,! sulidify ratio (511).

SI!\'G"IE GinDEH..<; Ai-<D TRllSSES

As Il(lled }ll-"\'iollsly a pla!c glni!'r "With fbllges may be cnnsidered to be a fi;l~ phiI', T: .. 'rdorf' the dra!: r()('Uiclcnl dep('llds onl\" all the a~p.'ct ratio as 1I10,h fif-d by t h" pr!,f,enC't' (lf abut nH'lIt S nr nthf'r ollstructlons at t hp b()lItldari£'s.

l'he dr:I:: Oil a sin[::ll' I 1'1.1"" dl'pcI:ds on tlw ;1 "I'ect ratio of the indivIdual mem­\)o'rs. F(.r SIO,;;;! ~:nlidily ratios (Ipss th:1n 0.15) lhe coeificipnt (hased on the ~;fl;;d ,II"';I! :ijlPI''':t.'h('s 2.0 b('(,;(lIsl', ill thi" case, the truss m('mlJers must be lnl,: ;11;.1 Ii,Hr .. \\" with [low prc'V'>I;tCc\ ar,'lIl1d Ill!' ('mjs-that illiplies an illfinite a"i)(~Cl r;l~ itl. lnl!;i:; connect lOll it should he noted that truss IlwmiJ('rs for whIch the d"i,:h ,,:;d wiri!h al'(' i1(J.-maIiy of thf' salllc order han' essentially the same ';1"',: cOt':fwicllt ;,5 flat jll.lll'S. Fur l;'!'he solidity ratios (grl':Jter thall 0.9) the ",»,:'f'(";,.",,: approaches tll,lt fqr a s"lId pl.ltf' havir;" the s;tnll' ovcr-all aspect Llti, '. IlL'lw.'cll :;,e5(, tWil ('Xll't :n.'s IlH' codfiei(,lIt is Ipss casily determln£'d 1o'.',',\\1<;c; Iii' di.'rt iv(' ;ISjll'Cl ra t jo of tlll' Il1l':"!'f'rs is \Incerta ill and the drag on t;;c l;USSd ;)l.lt"s i.e; st~1l1ficallt, l\lf'st t.'sts (111) (507) (50G) (512) for common lnl:i" f",lI<l,::: ra:ius indicatl' ("\Ie;ficien!s bl'twef'n 1.6 and 1.7, The actual ar­rai,:',l'in " I.{ (Jf I~l('illi..)t'rs d'-lf'S JJ)~ api)ear sif!Iufic.1nt.

1',1;',')1\ Sllr:t:pf,ts, (:ill) as a ('ol;Sl'rv;tt i .... c ;Ipproximatio:l. that the drar; co­diICl";,! b,.s,'(; ')ll tlil' toLd l'ncll)Sed ar{'.t be taken as 1.8 ~~ up to 6 = 0,6 and thcrt ,\;l' 1- a 11:1"ar inlcqnl;.tinn to the vahle at 6 = 1.0 correspundlng to the aSj)('I', LIlia f •• r the (,Iltire endosed area. Pal!on also slI.,gests (511) a more I"c:i,,('.\ pr'JCl'dure based on an dfccth'{' solidity that is grl'atl'r than the true soiid,ty dile to LIC contractIOn of flow through the openings.

T:O!.' ;urt,:0in.: appli,,;" to a wi:;d din'ctionpcrpcntiicLilartothe plate ortruss. As till.' d;rp<:lifHl departs from thl' Ilormal, the dra:; in that directi(ln decreases <I1tr.nllgh t he dl;fl'rf'llce is npt impnrlant for small angles. For design purposes 011(' wlluld nOl-llJ.lily bl' cOllcenled v.'ith a wind acting perpendicular to an area.

PA.RS OF GIRDEHS OR THUSSES

W;wn two clement san' ,llarpd in lin<.' with the wil.d the tot.d drag depends on the !'i',H'ing (,f tllf' ell'Illf'IlL,. If till' sjMcing is zero or very small tlie drat: is ,ilt" SOW,,' as Oil a singh! l'lUllt'llt, wherf',ls if tile 5pacin~ were infinite the t",tal fur"'" W01ild be twice as much as 011 a sin;,:h~ Illen.iter.

Fig. 10 c"l.la.ns some experimental results relalint; the total drag coefflciellt to Lie Si)acill'; of clements. Curves A, n. and C arc for solid plates, the first bdl;;.::for circularci1sits and th"lattertwo[or rectangularpiates, At zero spac­in;:: lhe coefficient plott\.'u is that for a single plate of the same shape. At small

W1;<;0 FOHCES 1169

101---I A C.fClltiH 01\,"-:' 107

~--, ----I : '

051--:-...._--_ .. _-r ---.--------.

A ?: rE'. t~n~1! .1' ~L.ltf"'i 107 C tl 5 I ('01111 ti:,rdp,,, III

D Trl'~"f'~ Q.l!' ,'7 III T rll";~,"'<; Q' Hi'> iiI Trur,c;pc;, ~ ""(\ 1('0 II;

,; l"J<;<.e'S Q "u/'Jn 507 OL-____ ~ ____ ~ ____ ~ ______ ~ ____ ~ ____ ~ ____ _J

o :; 1\ S 6 7

FIG. lO.-I·;r'VECT PI.' :;1',\l'I:~G 0:--; iii;:: ililAG OF P:\!H~ 01.' ELE;\l;':;';TS

JD r, ---------------------------~---------~ I l 'C'~:(I"d 1 ! _-.':.~.--~ ___ ,._. ___ --- ./".0 . ~._-.-:::-: .. :;;: .. ;..::..-:::-.:;;~~... - - - ~/~ -0 5 1

2.'~- .-.:;-.-,-:-, ---.-.-.----~--- ---- </"-0751 '....--- " '''. ------ ...... 'i/h-. i (JO '. : --- --., -... ---~ ~~ ... - ,>h~L5°1

701----. ____ , ___ .. ___ . ___ '~. ' __ 0_ , " , o· , I"'\. ' ~ 1- ___ . __ ~_ _ _ __ ;. ____ ~,,~.-_.

~ I~.l--.---.:-.-, --.. -\~--f-~ .-.--- --_. I---~~

If):----~ _ </J.r,?} ____ , .. ______ ~

·-------t-~---------------~--4

05~---~----------~----~---~--------------~ o ;0 20 40 50 70

nG. ll.-EFI'ECT OF YAW 0:-; P.\IHS OF THUSSES (512)

1170 \VIi'll) FORCES

spaclnf: (~ lrss th:111 2) the total dr,l;: d('c reas('s. Actually the drar: on th(' front

lI\('mbf'r incl'I';II:I('S sli!:htly b('call~(' thl' lr('w;trd memb('r callses an Incre:1se in sllclir.n hrilind till' windward ml'lIlhrr. At tht' san1(, timf', howrvI'r, the drag 0'11 til!' il'f'ward 1lH'\11ber is nf'~~'lt i\'p h<'c;llIse the sllction hetw('rn IlJ('mbers is 1:1r[:rl' than thaI behind lh!' le('wart\ member. As lhe spadll;': Incr('asl's further, tht' tot a I ura;.: increases and would rv('ntua Ily ('qual twicr that at zero spacing.

Whf'I] cOllsir\('rinr: pairs of trusses, the solidity rat io is of Importance be- • c;;use till' rti:-;tanl'e downstream III whIch sh\plding 15 effectlv(' d('jll'ncis on the ) sizE' of the individual memhcrs. This is illustrated in Fit:. lOin which cllrv!'s D throu;.:h G ar{' f<.lr various solidity ratios. As would be expected, the effecl ~ of shield;;;;.: dies out at smaller spacings as the solidity d('creases. It should be :lIlt ('J that most bridge trusses have solidities of less than 0.4 and spacings less :han twice the truss depth. On the basis of this and other data Pagon (513) ;. has s\.lg.:esled that a fair approxima\ion for thf' drag coefficient on a pair of trllSses is given by

C IT 1 s = 'fl~- + ()I' -D ,~ .. h (8)

ilO£' rf'st.lts shnwn in Fih. 10 are for a wind directtnn normal to the plane oC tI.e ekl'lcnts. If till' wind is at an anr:lc the effect of shielding is, of course, reciucC'li (Ill) (506) (507) and t!Jet .. tal dra~ all th,! two clements may be greatly illcr,,;uo,,'o. The {~('neral effect is thus to rl'duce the importance oC spacing abo,'e a c('rlai:, critical mi .. imum. It has UI'f'1l shown (507) that Cor a particular \V,IIT.'n truss (0 = 0.25) the cW[prencc ill total dr:tb coefficient betweens/h = 0.27 and s'h = 1.22 is 40% for a normal wind but only 15% for an angle of attack of 20°. The same tC'lIdency is shown in Fig. 11 in which experimen­tal rira;.: (,Ot'fiirients (for force normal to trusses) for a pair of Pratt lrussE'S (0 = 0,27) .11'1' plottE'd; in tltis case th(' wind direction is varied. Of course shiddil1'; would be further reduced if angles of attack and yaw occurrcdsimul­lall('()us\y.

EFFECT OF BlUnGE DECKS

Till' ;1r1<llti011 of a railway or highway floor system to a pair of girdprs or trll:;~l'S de" rl~· complicate6 t he flow and \tPllce the ncier minatioil of wind Corces. EssrlOtiaily ;1 (h'ck is Siluil.lr to a Hat pLitt' but with a greatly rou~hened lower surbcr due to I hI' presence of strill!-:l'rs. ThE' p/'{'ssures on a smooth flat plate ar(' showlI in Fi~. 5(d). The ('ff('ct of thr rou:;hness on the low('r surface Is to rNard the flow bdow thE' plate and accelerate the Clow above. In general this pro due ('s lift.

f;l1Iokr pictures an' aYailabl(' (113) that dl'pirt the flow around typical decks. Frpi1l these il is possihle to conclude tll:-,t for reasonable an"les oC attack the dra,! 011 all hut the windward striager is very small. At thp same time there is, of course, little drag on the smooth upper surface and, helice, the total hori­zU:1tal Con·e is almost entirely a function of the frontal elevation area. These conclutiions must be modified if there are openings in the deck betwecn string­ers. Each otJening becomes an -escape vent" that tcnds to deflect the horizon­tal wind velocity, thllS producing additional drag.

'. • I t:

. r "

I

" I

I ~

WIl\D FOHCI::S 1171

Tl;o' most 1;1;;'''1':-' ! <'f,-cel r,f til" hril~:1' dec); Is the /llodHica'irlll of th" flow p;lI,lcrii ;Jr,".:"d t:;" \,;,.i,.:rll:'Sl'S or ;::irdi'rs. The drar; 011 till' l"ew,Ird "'Pl11ent ii> '):";;<>lI:'::; .Iff,'ci,·.1 Loll. ia arldition, thp pn'ssun's behInd tllP front £'ienlf'nl, ;11;(; :.o'".T UiP(;!';!,: ":. t!I.lt ('1o';I1('nt. are also J:\o)<\ifl('d. Decallse of the obvious \;ncl'rtaini it.'R it 1;; d;.fl.'\j4L tu prcdicl tij(' \\"'lId (ItrCf'R nn a J~ivcfl hrid~:c section. Wiall-lllllll('; test d;,:,1 "r£' a\'aila'Jk fol' a "ariet}' uf bridge sectiolls. H(lwever, tllL'Se inait'att' tllat th.' 1i!'.1;; cot'ificiell: is very spnsiliv(' to th£' spat b 1 arran!;f'­n;Pl1t of roadw'l), C:"c;';. siclt-walks, railways. and so forth, and whe!'" wind forces are jllljl')I·1.llIt, special willl.l-tunlwi tests .11'1' ar\visahl('.

PLATE GinDER nHlDGES

Hcport5 h.lvl' l.H .. 'en Ill:lde (50~) «(310) (50C) (113) of wind-tunnel t£'sts ondeck and through llritl,.:l' sectlOIlS c()llr;islin~ of girders In various numbprs together

30" ----------------~--------------_r------~------~

V .. lut' '~f -a I R,tllr('!')d fH"!~!P~ -----I ~i-- 'Z'J', --~-~I'::rve1) ._---1

l ~~~ .... _1 I

~-, --;:0' -- - -'-+- \4-.-... ----I ? '.-

,- -;--- --- --.- --- - ,- , -_.. 1 ...... I ,t I

t- .. -- ~"l-"':':"'~-~~C=-=--=-_~-:::-=-:-- -- .. -! :~ i·,I- ... - --~~-" ------! --j

L' I ___ \ __ ~~'::~~~~~_~-_~~_. __ I r-~~.h"'.~'J-f~O~C"'- :\ 1',

! (<I ",~w(l (.u""~",l \ I

i rJi- bT-, c;!~ ~_.~ _V~-:c-2QO ---- -r---·--I ------1

1

~ -. -~ -~ .-~ --\ " . -----:----~- .---, L-+..----'----i-

n ,.1 !,' I) 3 4 5 6

P,..,hl1 of r~r<l(iOt:: '0 (jPvfl1. silt

With 1".111 "!"'I. ;"id do!'wti d"t:kH. Typical of Ihese r('sults Jre those' plotted in F.::. 12 (:)(J~I). Two lJrid.:i' typcs al'(' shown: A two-hircil'r thr'Jugh railway bl'id;:e w;ti. Ojll'1l fluor. a\lll a twn-~ircier cicci< hie:hway bridge with solid floor. Ti,'" liH::1W.IV l)rjd~;L' ;.lso has a sidpwali< projection with railing outsidf' the gird­er;;. ';'"" 1'o'I>\i;t6 ar!' plotted as horizontal drag cocfLcient (based on frontal area i "'t'rSlls S 'h. The angle of attack. (J, is positivc when the wind has an up­w~lrLi CC/illilor.('nt.

It Illay be oilsf'!'vcd in Fi~. 12 that tllf'rc is no systematic variation in drag with the s /h ratio. The i1reSC'l1ce of the floor system disrupts the orderly in­crease with slh tl!;\l occurs with pairs of girders alone.

11';"2 wn~/) FOHCES

1"h£' diff.'rf"lI'£' in dr"~, \H'tw('t'n~h(' tW<l l:;pps of hrldlJ's shown In Fig. 12 Is ilot ,:re"l for 7,('ro ;11;,:1(' of attack (hori;:"lllal wind). As (''\pf'cted, til£' opC'n­j]"')f nliw,ty brirlr;e h,lS slil:hlly grra!('r drag \wcaw;e tlip [£,('ward r,lrdpr is nHlrl' cxposed. ilnw('\ I'r, (()r an::lls o( etltack n!lv'r thetll ?ero the (liff£'rence is Illore prtlI1UUllcl'J. The railwav brirl::c sho\'~; a cOlIsld('r<l1>le Increase In dr:J:; with :lI1::le of attack alld it lllaK"s littk diffcrclI('e wl1f'tllf'r th£' angle is pIu" or ndnus. On the other halld, the dnr; un the hi~hway hring£' increases ulllv siil',htly for positivI' ~11!:1£' of atLt('k :Jllti ~('tu:Jily decrc;ts£'s for a ner;ative ;;,,;;1(', This occurs bf'c;.use tlic dcck shidds the gin]ers wh£'n th" wind has a d'lWllWard componcnt alld in 110 casC' is flow perrnltt('d betwI'en !:irders. This res~ilt is, of course, affected by the sidewalk projection on these models.

j{!'sullS fur three-girder and four-1;irriPr bridges arc also available (509). Thl'se indicate t!1;}~ then' is no appreci:lble increase in drag as the number of prdcrs is increased. Hesults are also given for deck railw;}y and through hlgh­w:ty bridGPs. Comp,trisons with currellt dcslgn specifications Indicate that !iF'se are probably cOllservative.

T('stshave al.sobeen madp (610) that indicate greater drag on !,;irderbrlciges with opcn or partly open cit'cks, Hvwever, as discussed In Section Vl, the open­deck spalls h;lYe better cilaract('ristics with rl'gard to oscillation, that Is Impor­tant in the cas!' of suspended spans,

THUSS BRIDGES

neports havf' been made (507) (50G) (113) of a variety 0f tests on complete :russ bl'idi:l' Illorif'is. Tilp addition of a floor deck (op('n or closrd) to a pair of t .. usscs 1.;}5, in gPllcral, a minor d[ITt because the trussC's have a depth romco ~n'rttpr than that of tr.(' fluor s:,slc'll1; in fact, it has uc£'n shown (507) that when the floor lies between chords (as in a thrOUGh structure) the total dral: m;lY UP r('duced from that of t',Wl trusses alone. This is truC' because the s:.ll·lding of thC' l('ew;}rd truss by the floor is mure important than the drag on the floor itself. TI.erefore, if the floor is tillls positioned the data discussed prC'viously fIJI' pairs of trusses alone may be used,

If for some r('asoll the floor is displaced vertically from tllr chords. the drag is. of COli 1''; (', increased. IYlth i>eC:ltls(' tile flonr is less shielded by the front truss and bec:\use the real' truss is less shi~lded by the floor (507),

FFFECT OF AKGLES OF ATTACK Ai"D YAW Oi" B1UDGES

The' £'t["r( pi angle of ~ttack (VPliic;ti ~Ilde) un the horizontal drab for ~iruer i1rid,:cs is <1pprecialJlc, as showil ill F:g. 12 :Il:d ;lS discussed preViously. Th" c[fC'('t ·.lli tl'USS IJri~;cs i;.; due to a reti'lced shiC'ldi;lg of the leeward truss th"t is ollly slightly a(fccleel by til'! [l.)or systl.'tlI. Therefore, the for('going dI5,:ussion for pairs of trusses would al'p1/. l'(~rh;lps th(' most difficult prob­lplll is tbe sC'Iection of a proppr d,'si~n v:Jlue for the anhle of attack. This is \'<'1':,' uncrrtain be('ause it dCjlC'nds, tu a hrbe extent, on local tOf)l)graphy, al­though certain types of storms produce vertical wind components. It is prob­a01(' th;.t "altles in excess of !lSn should not be conslderc'd for desibll.

The effect of angle of yaw (horizontal angle) is not appreciable on girder bridges but is on truss bridges, agaIn because of reduced shieldin~ of Lle lee­ward truss. In the latter case yaw is not as important as angle of attack be-

-----,- --.---~.-~

\Vf:'iD FOHCES 1173

(',llIS" y.IW d'""s !lot fC(illl'f' shil'ldinl: on Ih(' horizontal chC)rcls. Rpsults are ;l\'.til.lilte (jOG) for :1ll,:!cS 1)[ all.lck and yaw ill combill:1tirlll t/1111. illtuit iyely. Is the situation PI"'c!UC;I1,: Ill;tximum drag.

I ,1FT A~D OVF.HT1;H;\I:--iG l\!OMEl'<T OJ'-; BIUDGES

D .• ta 1o;1\'f' bern ",';ai,wd Cj(lG) (:;07) on lift f0rC'('s and momcnts for both

i',irekr ;t;,(! tn:ss Sp.IIIS. Tit" two dfccts sh.)uld he considered tobHhcr since i;it, altliull~h s"Lc.nm of illlpo,rLonc'21w itsdf. is of primary importance witlll'e­~:.1l'(1 to ')H'rturni;,,:. FIJI' rai!v".lY o'r other open-cleck bl'idf,:es, IUt forces are sm,II; (, '\'o'a wit h ;]lll:!e of .la;jC· U 1,:;d the overturninf,: moment Illay be computed Sln:p]v Ii;; !lit' dfl','t uf ti.(' horizon(;;,! force acting at the centroid of the eleva­tiOli :Ir"a. Fur su;id-cit'ck hricl,:l's. however. the lift is appreciable, increasing with widlh of deck. With upward angle of allack tile effE'ct on the overturning lllonH'llt 1Jecomt's sir;nificant.

THUSSEO TOWERS

p, ,'ioaiJS till' r.l')sl CO[]~i)lde (!iSCtlssiOll of wind forccs on towers was pre­s.-.n('d hy £.Cohen, F.ASeI-:, and H. Perrin. ;\1. ASCE (510). Aftcr a study of a \'ailablL' test dat •• it was Stl'~bl'stcd that the following expressions for drag coeUicient be used: for square towers.

CD = 4 - 5 ~ ••••••••••••••••••• (9a)

a.nd fur tri~ulgljlar towers,

CD = 3.65 - 4.65 (I •••••••••••••••• (9b)

P;\;1:011 (:ill) has Slli:gpsted tilC formula

(3 CD = \ 2¢ ................... (10)

for "'1I1:1l'e towers which docs !lot ciifi('r grl',ltlv from the Cohell-Pcrrin value iii tile si,;r;ificll:<t r;tnge of Q. All of these ('xp;'essions are for a wind normal to 01,(' [ace. For square towers it has lH'ell sllggestL'd tllat CD be multiplied b-j

iLl + O.t. 01 to a1l0\\' for yaw (510), For tuwers Ill,ll;" up of round members then' are littlc data available but

tIl(' (,piniull is r,eliPLdly held th:1! tll" drag coeffici('nts for such structures may b., takell as twu-thinis tl1;\( for stn'ctun,s having anh'"Ular members.

SU:-'lMARY

Drl;,: ,'n'-';';'i,i"nts [f,r npf'n structur"s cOllsisti;I,: (If combinations of struc­tUI'al d('lI.,';,ls ii, \'.Iriu\;s spali:d C""flb-UraticlI,s can reliably be determined oilly l,y WiLd-tunliel tests. Huwcn'r. considerable data arc avaibble and a dc­si;':lh:r is ,,;[,'1. "il!e to find test results for a structure sufficicntly similar to the olie ulic!,'r considcrati,)J1. It shuuld be notl'd. hllWE'Ver, that apparently minor val' iat ions iii confl{;uration may h;l\'c major effects on the wind forces. There-

1174 wrr-;D FORCES

fore, in casps III which wind In:ld!\ "rpo( prim:lry Importance, wInd-tunnel tests rue advisable.

VI. STRUCTURES SUOJECT TO OSCILLATION IN WIND

tNTHODUCTION

The dynamic effect of windon mnd!'rn struc(ul"I'S has r('ce\ved ever-Increas­ill:.": :lttf'I1tiOIl since I he dramatic coILlI'S(' of th~ Tacoma Narrows Brldg(' In 1940, Prior to thfR I)('rloo the prevailln~; o('sign practice depended wh"l1y OIl

static wind Inads !Jaapd on an arhitrary force constant, that gpnerally Ignored the tillar,' of thp at ructure and the aclual value of the wind velocity and dlrec­tlOl1 at the hitt'.

I:lcn';l1;;;;~ly, the civil f'ngill('ering prof('ssloll has been forced to consIder the l'os5ilJdilv of wlnd-inducl'd dynamic response on certain types of fl('xlble structures tli;.t h.ld formerly been desihned solely on the basIs of static Con­sider;ltions.

The notahle contribution (112) of Pagon, provldpd a firm basis (or the so­lution of the problt·11J of wind effects on structures, but did not receIve the at­tention it dl.'served.

The prnh!nn of the dynamiC effect of wind on modern structures may be divided ilito fnur principal cat('gories, depending on the shape and flexibility of the structure, All of these categories fall under the general classification of sf'lf-£'xci(ed oscillations:

1. lIi;h-frequE'ncy osci1i.ttlon~ may be excited on slender structures such as transmission and telephone lines. tower guys, and some types of girder­stiffened suspension bridges as a result of resonance (generally subharmonlc) between the rate of \'ortl]." shedding and some natural frequency of the struc­ture.

2. Cylindrical bcltiies (Til(' tcrm "cylinder- is used throughout to denote a body wllOse cross-sectional sl1,\pe Is the same al e\'el}' section along (he span) tii;owing a ne[::ltiv(' s :')pe o\'er certain portions of the curve of the lift force plc1ttcda{:a.instan"le of att:lck of the windmayos!'illate in abendin~ mode (FiG. 13 (a)). Sleel-coatf'ci t ransl1Jlssion lines and some types of girder-stiffened suspcnslon br1dges faU in tllis classi {ication. II the structure is free to oscll­lat e in tursion, a nf'gatlve slope, when moments are plotted against angle of attack, wi.ll inJ.icate the possibility of torsional instability (Fig. 13(b)).

3, Thin flat phtes placed horizontally in a hori7.0ntal wind, or the deck of a ;irdcr-st;f:enl'd suspension bridge, may be subject tu flutter if the natural frequencies in bending and torsion arc not tou dissimilar. The wind force causes a lift that acts eccentrically, thus i"rlucin~ a twisting moment. The couilling o~ these two modes may result in a motion that is catastrophic.

4. There have been some occasions in which a buffetinlj action has developed 011 a structure due to the effect of some other structure relatively close up­Wind. Such a condition has been reported on multiple tall stacks in line.

SELF-EXCITED OSCILLATIONS

Experience h<iS indicated that the most sIgnificant manifestations o( wind­induced oscillation faUin the classification of ·self-excited- motion that may

1175

be c!f>scrih,...d hy til(' ("llowi,,!.: statelllent (r,O~):

111 :1 Rcif-ex('itl'doscill.llIlllI th,... allel'nalin!: force that su~talns the mo­tion is creatl'nor controlled hythe lllulirlll ilsl'lI; U the motion stops, the <Ilternatill t: furce disappears (GO!).

In the cas" of vnrtex excitat inn of st ructurl's 01' structural components that an' fl'l'e to oscillat(' ill Ollf' or more bendilU':: IIr torsion;ll modI'S, the following criteri;, m;IY be staled (614).

a. Whi1t' the structure is at rest the \"1)1'1(')( frequency is controlled hy the wind.

b. At cl'rtain discrete wind velocities. dcsif;l1ateo as ·crltical vclocities,· the vortex frequency will dthcr coincldc with. or he a multiple of one of thc frec;.uenc;.,s of motion of the structure; such a coincidencc results In self­excited oscilla~ion.

c. Beyond the critical wind vt'locities thc oscillating structure and not the wind velocity controls the vortex frequency. In such cases the range of oscil-

e"

-08

(al LIFT (bl MOMENT

FIG. 13,-STATIC LIFT I\:-;n l\IOl\m..-T GHAPHS

latioll I'xtelltLs over c.erlain finite ranges of wind vclocity. ThE> lower limit of e;tch ran[.:e is :\ critical ve-loclty. The upper limills notas well defined as the critical vdoclly. How('\'C'r, bet we-en the upper limit and the next critical ve­locity tile- structurE> is prac.tically at rest.

When the "ortex fr!'qllPllcy coincide'S with a natural frequency of the struc­ture, hal"mDnic I'xcilatiull results. IC the vortex frerlucncy is some multiple of a lIatul"al ir.:qu('llcy u( the structure, the reSUlting excita! ion Is subharmonlc.

In \'('I"Y [1!'Xible st l'ucturps, such as girder-stiffened suspension brtd;;es, SOI1H.' of the lower i.wmiinl; modes llIay ri!'velop twice, but at tht' same frequency. The first appear-anc!' will be harmonically f'xdted, whereas tl:c second appcar­ance will be of subharmonic origin. The torsional mode also shows two appear­ances at the same frequency. The first is harmonic and the sccond 15 subhar­monic. However, thc second appearance may be catastrophic and supprcsses thc development of any but the lower two or three bending modes.

I •

117" \\1;-';D FORCES

r;,.,kr cOI1(iitions of \'ery low d:lmpinc:, which can only llf' obtained In the Lth"rat(ll'Y. l1Inl'e than tWf) ai'IH';[rill.ces of a torsional nlOoe may develop for Cl'I·\.lill Cl'I'SS-sl'ctional sh.lpcs (i)14).

t:XPEHI~lEl\TAL VEHlFICA110:; OF VOHTEX EXCITATIO~

Ti!f' pi"liceri, : i!lVI'st i~:;lti<J11 d II:" as.nll::"I1,llldc st~iJi1ity of a suspf'nsion bri,,:'" '.\',15 Illadl iJy Thco(tOl"C \'Oil i{.lrma'l, lIull. ~:. ASCE. and Louis Dunll at the Cd;:",r"ja II",titlltC of TI'cill"ol"q'. Pasildella. Calif .. in 1942 (1314). The ri!:i(;, ~pril,g-sl1l'p()rt('d modf'! lIsI'd in thcse tests was a hlr;hly idcalized H. sectiul, (j·'i,:. J ;, si:llllar to tile sllspendPfi structure of the orig;inal Tacoma ~~:lrr(I·.\".s Bridi::e.

VlIrt.'x frl'qul':H'il's at I:le wi"d',"ard ~ircter ",'('1'1' measurcd by means of a h0t-w'l·c anell10ll1iCtt'r with thc 1110<i('j iH'ld slat\nnary in the wind. and also when thr model W.l::; all(lw,'d to oscillate.

If t:lt':'Jscillatiolis are a n'sultLif \'nrtex shcddinl.:. it <11!';l)followstlntalincar 1''';;11 if>llsilip · ... ·ii! exist lJetwl'Cl1 lh,' frr'fJ.ucncjcs and the vcl<)C'itics at which mo­tlln stans. A nUllllJl'r of tests were conduct,'r! in which the sprill~ con!';tants ',\"1'" \";,rir"d in onlo'r tn detprmin" Ihis rf'lation b('twel'n frc'luency and critical wj"d \·docity. Th., critical wind vel(lcity. V c. is th;lt \'cloclty beyond whieh a ii ni tp Illation ex isl s ..

",'h .. rpsults of ol1e spr[rs (If I<'sts are Sh(lWl1 in Fib' 15 that indica!" that as ~,V"ii ;IS v,'rticalmotir)ll starts the v"nex frequency has a well-r!efinl'd constant \"';\:(': Ilut IS, the mudd cnntJ'''ls till' vortcx frequency. As the wind vl'Ioclty IS ilil'l'l':tsed a POi:lt is l"';tciu'(; :11 \\');ich the mnuc:is no lonr:er capable of con-1[(';: itw till' vorU'x fr"'i\;l'LCY "lId ti,t' wind again iJPcol11cs the conI rollll1,~ fac­t"r. S.lllil;ll·ly, in torsional (\scil;,,~inns the vortex' fr('qul'ncy also bccol1ll's C";lSI:1:1: ;IS I11,)tjoll starts. blot ill llus instance it shifts suddenly to a hir:her \'al"," as th,· wir,d \'(~locit\' eXC<:','ds cPl'tain limits.

I'l Fi!;. 16. fO desil~nates tlll'fl'l'qul'ney at which the modc'l movcs vertically as a tj"it, .n,d fl is the f1'\,(;l1f'I,C',' of vertic<1! n;otion about the z-a.xis. These two I11oti'>!'!, arc known ;,s the "funciamclltal" mo(;c and the "single-noded· mode, l'rsw'c li \'l·lr.

Tile r; rst arp,'arance nf ClI' three moJdcs was relatively weak. At higher W;I'ri vl'lrJeities tl1e eye:., of mode d,·\'t:lopl1lt'nt was rcpcatedwith a g-reatly In­C'r(';!seci a~lI)lit\lGc ia tilf' vcrtical Il1f)de and a catastrophic second appearance vi t~.e torsi'lnal modf'.

Th," criticll ',\"ind Vl';ucity as ilill:lcti()n [)f frl'que,',ey is ShO',"1l in Fi,;. 17. It is 5l'rIO li',lt a lillcar reiatlollslti;) exists for each of the two valU('s Lit' critical willd v(';'Jcity. Also, the test points indicate that the iirst critical velocity Is jllst i.,ll[ of thf' second.

La".:r il;vf'stir:alions at the Univf>rsity of Washinf,rton, involving a 50-scale flexible model uf t Iw origil;;ll Tacoma i"arrows Brid;;e, confirmed the relation­s:.ips discussrd herewit;l. Tile.se tests also duplicated faithfully all of the modes of m,)tion obscrved on the ori{:;inal brld~e. inciudinr: the fiaal catastrophic mo­tiun that destroyed this structure on ~ovelllber 7, 1940 (614).

MECHA"lSM OF VORTEX SHEDDU;G

Ciyculay Cylilldcrs.-Cylinders of circular craBS section are of special in­terest and have bccn the subject of considerable attention in recent years. Above

32 r I

". I ~ I

~ 24 1-"0 I

~ I ~ 16 l-Co. f

~ 8 1 ;;. I-

~ oL

WIND FOHCES

y Y 4 • I I "205 in. I L=; x I V E--'--I _ r

I 'T . . I..- --- 2825 In --_ ~ 10 In .j

FIG. 14.-IDF.ALlZED SIJSPr::-;OED STRUCTURE ,

FIG. 15.-TEST RESULTS

1177

O~----~~~~------~~----~------~--~~--~L-~ ____ ~o b 8 10 12 14 16 18 20 22

VeklClty, 10 feet ppf secOl'd

FIG. It;.-THREE MODES OF AMPLITUDE RESPONSE

WIND FORCF,S

IH~--------------r------'----------------------,

! . I Ii ~ .. /1 .6:---- !...---+--+---t----- ----r-~

.J---~ I --9~~-----'------

i

B - !-¥~: ----j--

'ltv/i· o~~----~------~----~------~----~----~ o 8 12 16 20 24

Cntlcal v~toc,t)'. '1"1 '~t!t per second

FIG. li'.-CHiTfCAL \\l;>;D VELOCITY AS A FC;\'CTIO:-l OF FaEQUENCY

,~ ~~'~'0- - --(A - --c---~'\ ~"t .. 02':;\ .~ -----_- ·---It-07I1C,{,.ri-I'2l1

- --c- - - -c- - - 1- c- - - - -(<.I) -1-13{1,--

FiG. I '.-HEYNOLDS Nl;;\1l3ER Vt:HSUS DHAG COEFFICIEKT AND STROUHAL NUMBER

\'.1:--;1) FORCES 1179

R "'25,000 the wakrbecrmrsturblllcnt:1t the IX1!nt of f'epal"ation. In thl~ ranbe IIH' \"or~('x filirrlctJng Is pcrlodlc but the velnc\tlf's fluctuate In an Irrq:u!ar ;)lam,PI" duf' til t rubuknct' In the flnw. AA a result thf> ma~nlt udes of the oscll­bUll': ilit lorn's are somewhat random, a:;<\ become marl' so ·,,;!th Incr('asln{; Ikvnolds numhers (6:l3).

'T:;(' localized vortex strf'ct In Flf':. 18(a) Is that theorized hy von K;\rman and lllay be consld('rNI as reasonably ;lccuratl' for sL'( or seven diameters do ..... nst~eam (618). The data In Fi~:I[1(a) and curVf> Bare applicahle up to R "" 3 x 105 where the flow chan~('s from bm Inar to turiJulf'nt and t he sf'para-110:1 pair.ls moY(' downwind on the cyUnd!'r. At this critical point there Is a sharp drop in thr drar; Cl :ll'lIt, an abrupt rls(' In thE' Strouhal number 5 (curve D), and the wake narrows with rC[';\llar vortex formation no lonf,er vlsl­hh'.

Tile value R ",,3 x 105 1s Imown asthe critical Reynolds number (curve B) ami \'alu,'s bC'l,)w this nUllll>f'r al'e called suhcrltical, whereas higher numbers arc tcrmf'd supercrillcal. In the 5upercrlt!cal range the Strouhal number In­creases sharply. but the pl;cnonwllon Is aperiodic and Its ('fflclency Is hnpatred. ThE' S:.rouhal I.umbcr, wl.lcills 0.20 at R "" 2 x 105, has cl1mbed (615) to 0.43 at R '" 1.5 x lOG. The ('ff('cto{ surface roughness on a clrcular cyUnder (635) is shown ii] curve C, Fi~:. 18.

In .eccl.t years a !;i;:nlflcant numher of tall stacks have oscillated in wind a:;d some have suff('r('d cataslrop:l1c damaf,f' (605) (624) (6~5) (626). All of these structures werf' of circular cross section with diamet e.s up to 18 ft. Th,':; a:l oscillated in the supercritical rallt:(' but generally In an erratic or s;-lasn.')cuc m:-\l:ner, and always at their lowest natural frequency. Their per­[or:1.a:1('e il;dicates that in splle of the aperiodic sheddlni~ of vortices at values of R "" 1! x lOG thf' ·proces:;rs in UlC wake are essentially unchanged at Rey­nolds numbf'rs a\)<)ve critical- (619).

Cyiil,'ders with Sharp I~rlgcs. - When a c:;linner Involves sharp ed(!es that fix tl:e point of detachment (Jf the vortex, the re!'Ular process of vortex shedding is jlOdcj1pncient of Rl'\'nolds number above R "" 103, as shown In Fig. 18, curve E, for a flat plate ~,orl11al to the wind. The Strouhal number for a variety of sh~ "c; of stnlctur:-.l interest is shown In TaLle 5.

7he effect of w;r.d Impinging on a flat plate at an angle (Fi~. lS(b» was in­vpstir::Jted by A. Fn;:e and F. C .• Tohansen (603). It w •• s found that diD sin a rClllairwd subst:lIJtially constant with an average value uf 5.32, as the anble of iIlcid(·llC., (T was varied jwtwcen 30 n and !lO".

EIFel of R/)/II/riillg Shm"p Edges. -Ill I,cncral the effect of roun(Ung sharp ed;-;cs on a cyllndpr tpnds to promote a sl1bcrltical region of R, but until the ral;O of corner radius to transverse I11r<lsurcment between edges excecds a value of 0.20, the (·ffect of corner rounding Is not sl~niflcant (615).

MorlZ/\'illg Willd -Exci tol Oscill n/ion, -Many of the situations In which wind­ir.d"c('d oscillations arp of a serious character involve ctrcular cross sections. It will l-p instructivP to examine thrcc devices that have been developed, two of wl;ic;1 have been founn to be effcctive or. existing structures.

Prior to these developments only two effective devices have been known:

1. Alteration of the desi{;n to increase the mtural frequency thus raising the critical wind velocity to a level beyond that antiCipated at the site.

2. Increasing the effective damping through the installation of mechanical damping devices that act to reduce the amplitude.

1180 Wi:-';O FORCES

T,\RLE ;'.-STHOlillAL :-;t:~mr:R FOR A VAllIET'i OF SHAPES

----.-- ----- -------- ---,-----

I 10 nlln in mm

Value 01 S i S Wind:

----.----~!-----+-__t.-----~-.·20 I •• 10

0120 I I I -----~ ,

I .,.

-

i !125-=r-L.I __ ...... L_.

I I ~ f 1125 r! ,

--H ' -"'1" c' u' •

__ L_ -0137

r---

~-50 , --SO....., I ' ,-1.0 '-H I 125J-' , '

Ii, : -_L. I 1__ 0.120 I I !125 '-1 )1-12.5 1

f<2S., . "-50~ I ------. --- ~-- --- __ l ____ I----_ ~ I , I-I . I

""I )J ,.\.O

i .,.

"I-, ~: ii-, ---;

-,-_I

-r-- I

LL: ~-50-~

____ • ____ . ______________ 1 ____ -

,-1.5 I+--:

:"'LI : "" I · I t i ' ~-50-..: / i

--~- -.-- ~--;-;;-----~-=----~:-----'---I-O

--:-i_fJ I LI _--.1 0140 'I I tiL ~ j 1

I ,-.-50 -- ..i 0153 t ~25":'2~ "1"25 ~

1-1.0

0147

O.ISO

0145

0142

., Orol

"

Oi37

0.121

0143

----7""-------- 1--t---1,-------------

'I .-iO :~145t I 1-10 , , .

---~125-t-1 -' --- _ ! J _ I

0135

! f-o--50- ... I 0168

--~-----,-~-----l ''',) ----,-; i-- 50- ., I ~

, I Oi45 _' ~______ I

0.160

J---___ I

---, 1 ____ '-1_'-_1.0_

1---100 ---~ I

r=l 0

Cv',nde, II!\lJO < R < 19100 , -I -'---1 0_114

jO I

'" I 0200 N

.I. .L..

r-2~--1 I I t--SO-i 0145

..-l

l181

III prac.ticc bntll of t h('~c 1I1<'t 110 (Is an' ,J'nerali:; unsaf isfactorv bpc .11IS" !,:aills

arc likf'ly to be quite limiter!. This i<i ('sp.'t'ially true wilprc all incrcasc In dampinr: is immlvcd. bC(';llISC si:;nificant ::ailll'; l1Ia~' only bc acilic\'cd thr011[;11 elal~)rate au:x ilia ry syst ems tlllt al'e prac\ ieal onl:; on i ree-standing slructures such as tall slac.ks.

In 1956,1'. Price re\'caled the results of an ('xtC'nsive Investigation at Stan­!ol'd Univcrsity, Stan!on~ Calif., usln!: a i)(~rf"rated circular shroud that sur­rO\indcd a circular cylindrical member (629). Wind-tunnel tpsls proved the shroud to he an e((('ctive vibration supressor at subcrltical and transitional values of Reynold,> number (up to R = 4 x 105 ).

10 ~----~---______ ----_------. T I'

0.9 __ ~' 0 ___ 1 ____ J __ 1 __ _

08----.L"---~----1 -L-o 7 1 - --.-~ .... g _ _'AL_.,. '\' ,., • J · =-l--= (f T ,",~ .-['. J

~ 05\1---';- .; \ I ----0.4 --;--.~--:

03i L---- i ! . __ _ I '" 1 Hole Shroud

dldmeter gap S 0.2 r-- () Plain cyhnd~r

I • Shroudf!d C),iind .. r

o I 1--- '" Shrouded cyh"d.' 025 in.

025 in

o 7~.n

075 in.

025 in.

o 7~ In.

312!'>

2708

oL-______ -L ________ ~ ______ -L ________ L-______ ~

o 100 200 300 400 500 Value of R X 10-)

FIG, 19.-CYU:.;nEH OHAG COEFFICIENT VERSUS HEY:-;OLOS l\U;\lfl£1t WITH D1A!\lETEIl OF 6 I~;.

Fif;. 19 shows the marked effect of thrre shroud designs applif'd to a 6-ln. circuiar cylin(!cr, with various combinations of shroud gap and perforation hole di:lmeter. A striking (eature regarding the (>flect o( the shroud lies in the fact that the drag was drastically reduced in I he tranSitional range of Reynolds num­b('r :lnd tilat [or each of the three shrouds mentioned the drag coefficient re­mair,ed constant into the supercritical range upto the limit of the wind velocity available.

Fig. 20 shows a plot of double amplitude Y2 versus dynamic pressure q for the plain cylinder. At q = 4 a double amplitude of about 0.039 In. was ob&erved.

116:! WIND FORCES

Thls :llllplltudr. [(':1 ofi to O.Oi 7 1;1. ~lt q ~ 8 and therl'artt'r Increased at a unl­for ... r~,le to 'l GOIlLl(' al~j;)l!tu,lc n! 0.069 11;. at q = 20.

L:;ll'l" of the thrc(l shrota:s mentioncd in Fig. 19 proved to be effective In EHliJpreSSWb vortex exc itation ;.t subcritical. transitional, and slIpercrltlcal Rcy nolds numb(,Ts. The shroud \'Irtua 11 y eliminated per lodlc osc illation of a stiff model at lransJtlonal and supercritlcal Reynolds numbers.

Q

v '0 .. ~

-;, ;>

1.0,..., ------..,-----:-,---.,..----,-. ---,

I I 01--- l_l_~l , !

06L--: - \ --+-1 -4----1

[' Ii: I ! I I

04 ---'-- -...... ----4 -----<--.-----',----1

I ! ! I 'I I'

02 ---J----l---!---H--! I i I Ii, I I

'(a)! : i , I

i '\ 1

°1 , I

' i i o OlD f----.. -----, ____ + __ ---1

I : \ i I

I !! 00\l81--~-----~-+----+

l I J ' 00(''''-- t--:-\--

>, I I I \ ~ I I l \ ~ I- I ; \ ,

O(J(l4t---I--t--\-~--I I \ ' I I !

\. I, I o 002 i/-----t----tl----·+!----1/: ! ~ (b) I I I'

I I GL---~---~----~---~-----'~--~ o 4 8 12 16 20 24

OVflamlC p,e"SUfP.., q. in pounds per square toot

FIG. 20.-0HAG COi-:n'ICIJ::;o.;T AND TTP DOUBLE A~IPLITUDE VERS\jS Dyz..;iI~IIC PHJ::SSURE

\VG,:n FORe ,.::-. 1183

I;, 195i. C. Scruton ami D. E .. J. WalslH' published (631) the results of an eXlwl"il11cntal inn?sti:;atlon at tilt' "'alioll;u Physical Lab()l"atory. (Teddin~on, Engl:tnd). that accomplishrd th£' S;lllle rr<;ult as th.'lt acllievC'd by Price, but in a manner that promises to be mOr£~ universally acceptahle on all sizes and len!;ths of m<>mUers of circular cross sf'ctlnn, and is applicable to tapered members.

~~l~ '0 ~ .,. >

Il~--------------------~--------------------~

"'-10 "" --,------

" I \ Instabilltv boufldary (or stack. 'r'--- f,tt~d .'lIth strakes

---- \---+l-----------j 9-

J ,I

....... - I -"'"

I

6~------------------~·--------~~--------~

o 5r--------~~~====--~~----~

o

4'~----------------~--------------~ o 10 20

Non-dimensIonal d.1mping c~fhcu~nt, 2 MI, .D

FIG. 21.-~TAiliLlTY DL\GHA~lS FOR A FLEXUlLE :liODEL STACK OF CmCl'LAH SECTION WITH A:>iD WITH­OUT STHAKES

Th;s device involves the use of helical strakes that are spiraled around the circular cylinder. The strakcs are rectanGular in cross section and their ef­ficiency Is dependent on their number. iwight, and pitch.

A 4 ~ -in. diameter aluminum tube, 61 in. long, was spring-mounted hori­

zontally in a wind tunnel with Creedom to oscillate in a vertical direction only. ,

11 R't WI;.;n FORCES

I 7i;(' ,;( 1.1::('<' W. re mark pf _I -in. wid., ruhul'r of \';lrifll1S Iwi::llis. With three

,'qu.tll:; "1',1'(" "idr;!1.,d str:lkes of pitch 15 nand iwil:lit 0.12 D thC' oscillations u;;.!"!' ',\'111, \\""n' <"";V bardy <li<;cel'nil1l.,. TIl<' nl:lXil1l11111 alllplitude found with U;c i'iain ~: ,H'i,; was 0.75 D; with tit<' "LI('j; fitted with stl'akes It was only 0.05 D. Fig. 21 shu',I's l;lC boundary of In:;tabilily for a barC' tube as COnlp:lrcd with a Luue i.tt.,d with ~.trak .... s, In Fh:. 21. 1\1 rC'presf'nts mass [If'r unit IplI~th.

l\Iured,'t:1ilcd experiments (f,32) III 1953 by L. Woot4:ate and .J. F. M. M.IY­hrl'}, f'xt"lllil'd th(' ori;:lnal invC':lti[:.ltion and su:~g('sted an optimum arrange­l1Ient "r thi- .... " stra;(('R with a pitch of 5 D ami a hc\r:ht of 0.09 D.

nllrll1t~ 19;'9, i"\'C'sti~al;lIl;s of til(' t'ffrctiveness of spiraled spoilers of cir­cuI.lrcro;.ss"f'tiol1 appll .... d to clrcuiarcylinders. was COll1pkff'Cl by W. Weaver (1;331;\t ti;r> LIII,'I,ill Lahor;llory of M:lss:lchusetts Instiluteof Technology. The pur;",:,>., nf l:,I'Sl' studies WaS the evaluation of this tY!lf' of spoil('r as a means of clJnJlllat;I'f~ wiiill-inducC'd motion on circular :llumilllllll tubular memhers of nd" r ,I'il ('I;:,;, s;1ace frames ill which excessive vll.rations and fatigue failures h:1v(' ch'currcci.

T:.i;; L,':"stipl j,)n cr:mfil'n1ed. in !!f'iJeral, tllf' flndin~s at the National Physi­c:lI L.\I""',ltol-:;. L.ut throu"h the l1Sf' of spollers of circular cross section that makf' up tile ar,tl'lllla structure. It was also revealed that tht' spoiler need not ue ajlj'!H'd o\"('!' the (rit.d length of the nlC'mher if the oscUlation was confined to the fuudanH'lllal mock.

It W.IS Cl.ll1dllded that thc spnil('r5 of circular cross section were most ef­f.'ctivc with fOI.r wiI,dilI!;s. Tile I;lOst cffC'ctin' spoiler diameter was found to 0(' 3 0'32. Th ... optimum pitch was dl'termln"'d to be 12 D.

Two ex:unpil's will 5llOW the most int I'rC'stin,: (-{fect of adclir;!; ciz:cular spiral wincli(:~_:s If) 5tl'\.C:\,I,:11 nJrmuers of circular cross section (633).

Fi". 22 S:H1\\,5 r('sponse Cllr\'CS oh"alnf'd on as-in. diamcter spring-supported ri;,;d tube wlwrt, the lion-dimcli~iolial deflection Ym /D Is plotted a~alnst 'l!.i"d VE'!,)CI:V. The addition of eight windillhs, 3/1G-in. diamf'ter. at a pitch of 12 D actl'U :lS :l .. tkpu;:ers and lncr~ased thc ma.-"imulll Ylll iD from 4.6 to 5.76 (23 i:1. t') n.s Ln.!.

Tllf' U.';f' (if i,.,,; l' windinf;s. 5/1 G-in. diameter. at the same pitch showed strong f,iJoi:er "etin;l. with the mcmbf'r at rest for V = 10 mph. and reaching Yn/D = 0.3 at 60 Illl':', TIll' d;llllping in these tests as t;iven by the logarithmic dec­n'mel;t was 'i = 0.022.

II; Fig. 23 th,' (,(fect of sjloilf'rs over various mid-portions of the span of a flt'xural llll'miJ,'r supported at huth cnd5, is illustrated. With 20~ oHhe lcngth sl.'rved with optilllum s;x)ilcrs (1/4 in. diamc!er at 10 D) the vibration ampli­tudes were supprcs,.,ed If) apt'roximately 10%. A spoiler length of 40% reduced tnfO :1ll'pliLuctf' to aUtJut 2',;;.

These i:;\'('f;tkatirms oUer the first truty practical solution for the problem of vnl",'x-i,·,ciuc. d \'il11'atioll o[ structures of circubr cross section. However. it should be I;utpd t h:,t spoilers of circular cross st'ction are not effective in the r:1I.,:e oi i<upercritical Reynolds numher.

SC1'utC)iI has reported that the :-;PL spiral strakes, of rectangular cross sec­tiOll, Lave l>een successfully applied to a 250-ft tapered welded steelstac"k that haci oscillatcci ill thp wind. At the end of 18 months tlus stack had shown no further oscillation. It would appear that the spiral spoilers of rectangular

Wr:-.;D FORCES 11B5

cro!'s section are effective at slIpf'rcritic;)i Reynolds numbers as a result 1)f the sharp edges presented by the spiraled st rakes.

DUFFETING OSCl I ,LATlON

Oscillation due to buffeting may develop wh(,11 a structure lies in the wake of an obstruction that !!heds a vortex trail in a more or less regular fashion. The obstruction may arise from the topography of the region, but most fre-

o

v

" o

: , - --0-- - Eight windioi~, Ii In. diameter @ 12 D

-.---.- Four wlnd'"g!.. i~ in. diameter., 12 D -

VI!'I()(' ltv. \', '" mijps per hOlif

FIG. :!2.-ill-:SPO~S;:; 0F A :;-l~. DI.UIETl:lt SPiU~G-SUPP()RTED CYLn."DER .... 1TH IIOTH SPOl L F. itS A:-> [) '\:-;TI-~ ,'Ull. t; HS ATT A C iI E D

qucnl1y takes the form of another structure, or on complex structures one ele­ment of the s~rllcturt' may become the source of a buffeting action on another dow::wir.d portion.

The buffeting action is generally irregular, resulting in short bursts of am­plitude and has rarely been catastrophic except in the case of some aircraft components.

wr:-;D FORCES

01;(' i"tc:rcstil.:: m:lI;if"sLltioll of this pher;omE'non occurred in connrction w:t!-. 1-,"ot,111 ,'t,leks 11 it ill cil:l.ll1C'tcr, spaced 48 it apart (605). 'W1wn thl.:' wind hlew at an allglr of IG" to ll.c alincllwnt of the starks and at a vclocity of 53 mph, tlot~ downwind stack swayed in a direction transverse to the wind where­as the uiJwind stack remain!'!! at rt'sl. The rate of vortex shedcUng was almost exartly equal to the natural fl'equency of the stack, but the amplitude of vi bra-

FIG, ?3,-llF.Si',):';SE OF ol-;:\, [)IA~IETEll FLEXUR1\], CYL­I);IJt:H WITH SPu;LdlS OVER VARIOeS PROl'Oll­TiO);S Of TliE ;\;I·;;\1BER

linll was sm:J Ii due to reIa! i\'c 1y hil:h damping because thE' stacks were supported on ;, ronf structure,

EVi(it'ntly. due 10 the stack spacing, a vortex discr,a.ged from stack No.1 could r(';>.ch the opposite side of slack No.2 just as a vortex was detachedfrom this slack. The e[(ective vortex from stack No, I, being of opposite rotation

wn;n FORCES 1187

from that dlsc!lart;C'r! from stack No.2, caul'lE'd a n('utrall7.lng ('{fect. ThlR would rC'sult in an effective vortp.x cii<;char>:(' from only one side of the stack and consequently !;cnel'ated a pprlodic force.

F,\CTOns WHICH CONTROL SUSPENSION BRIDGE DEHAVIOR IN THE WIND

The a('rotinlarnic forces that act on a bridGe In the wino df.'pend only on the velucity and ;Ul"('ctJ.m of the wind and the size, shape, and motion of the' bridge (623). Prnp('rtll's of the brldge, including Its clastic forces and its mass and motiulI, can be computed and rE'duced to modC'lscale, and wind conditions brack­etin!! all probable conditions at the site can bl' Imposl'd upon an elastically­support.'d sectivll mudel.

Jld;(!/'i(lr of a Gil'd.'r-Sf~(rl'ned SpC/ion.-In all tests of oBclllatln~ models oC ;::lrocl"-stU[en('ci secUons, except extremely shallow girders, the motion has b('c'll ('inwr a pure vcrtical modl' or a pure torsional mod(' at substantially the natural in'quency oi vibration (614 )(623). For such a section the vi (fb)­ratios and the ch.-.ract('r of oscillation (restrict!'d or catastrophic) are Influ­(,I1C('l\ primarily by t:H' ratio, h:b, of the dCilt h of the (!Irder to the 1111dth oC the britt,:!'. (The dominilnt charactE'rlstic of the behavior of a bridGe In the wind is the consistent relationship bctw!'en the frl'quency N [Ill cycles per second] of the oscillation of th{' structure and the velocity V [In feet per second] of the wind that causes it. It is convenl('nt to divide this .. ratlo hy h, the width of the bridi:e. ill fcpt, rC':1dl'ring the ratio non-dinwnslonal and thus applicable to the bridge or It.s 1I10<l1'l.) Det;; 11s such as gitder flan~es, 5t ringers, and curbs, and the positin:\ oC the deck with rl'(erence to the girders also have Important modiiyillb" influences. This Influence of details Is not always predictable.

It is char.lcteristic of girder sections that thC'y may osc.111ate at restricted amplitud('s at modf'rat(' wind velocities and with a variety of successive modl's and corrpsj.oi:ding frequencll's as the velocity changes. In general, depending on the s!"1pe, bnth vertical and torsional modes may reappear at hl;;her veloci­ties (but at their ch.lIacteristic frequenclt's), and some of these may be cata­strophiC.

As the h III rat io d('c reases below about 1/7 the glrder-stUfened section 5hQWS Incrl~asillb evidence of the coul'h'd type oC oscillation, namely flutter.

/l.'iWl"if)J' of TI"'I.~s -S'fl/f,',/cd Sccli,ms. -All oscUlations that have been ob­scrved on models of truss-stiffened 'bridges and the observed small motion of the Gohil'n G;Jte Bricl;:e as well have D<>C'n of the couple type (611). Often the mode illlpears to be almost a pure torsion but the frequency 15 reduced toward that oC t~e corr":';i'QIldinJ:: vertical mode. In many cases (Includin~ all of the catastroilhic motions ob::;C'rved) the vertical mode component has sufficient amplitude LO shift thc' axis of rotatirll1 considerably to the windward from the centt'r li:le oC the structure. In some cascs th(' axis of I'otation has be!'n prac­tically at the windward truss or even wl;'ll beyond It.

It is characteristic that coupled typt's of oscillation occur at ('onslderably higher velQcities than do the pure vertical and torslor.al types of the girder sC'ct ions. It is a pparcnt also that the ae rodynamic forces on the truss-stiffened section are detprmincd primarily by the deck and truss members adjacent thereto. In model t~sts the removal of th(' bottom chord and web members of a deck tl"UBS section has affected the motion only slightly. These members are bcne!i('ial aerodynamically in that they add a little to the aerodynamic damp­ing.

1168 v"l:-:D FOHCfS

DrcallS(, ('onpir'd motion requires a blf'nding of vertical and torsional mo­tions in a manner to be mutually excltlnf' llnner wind action, it depp.nds upon the clastic factors af[('ctinr, the relative frequencies of the vprtical and tor­sional modes. The use of latt:'ral systems In the planes or the top and bottom chords increases the torsional stiffness and thus may 50 widely separate the vertical and torsional' frequencies that the critical vdocity of coupling and flut­ter falls heyond any probable wind velocity.

AC IO;OWLEDGMENTS

In additioIl to the task committee membf'rs, many former members made valuable contributions' to the effort representpd herewith. These incluued C. £i. Gronquist, F. ASCE, G. E. Howe (deceased), F. A. Reichert, S. Mitchell, F. ASCE, and J. Karol, F. ASCE.

Respectfully submitted

E. Cohen F. B. Farquharson .J. J. Kozak H. G. Lorsch W. W. Pagon E. P. Ser,ner, Jr. R. H. Sherlock T. W. Sin,:;ell H. C. S. Thom G. D. Woodruff J. M. Big~s, Chairman Task Committee on Wind Forces Committee on Loads and Stresses Structural DiVision, ASCE

APPENDIX I.-BIBLIOGRAPHY

1. mSTOHY

101. "Discorsi a Ou(' t-;u()ve Scipnze,· by Galileo, Bo1ol'na, 1655.

102. "Philosophiae Naturalis Prillcipia Mathematica" by Sir Issac Newton, 1687.

103. "An i::xperimentallnv!'stigation ConcernIng the :-:atural Powers of Water aloe! Wind,· by John Smeaton, Philosophical Tl'ansactions, Royal Soc., 1759.

104. "Recherches Rxperilllental('s sur les Lois de la Resistance des Fluides,· by N. V. Duchemin, Memorial de I' Artlllerie, Vol. 5, 1842, p. 65.

WL\;D FOnCES 1189

lOS. "The Forth Rrldi;e,· by B. Bakf'r, Em:inf''prl/1[:, Vol. 38, August, 1884, p. 213. _ .. ---

106. "Wind Prel'Sllre in F.n!~IIl",p.rl/1r: COllstrurtloll," by W. H. Bixby, En~lneer­~:; l\('~"'- RC'curli. Vol. 33. 189:'. p. 1 ;5.

107. "l\ouvdles Rechf'rches sur la Hcsi;;tance dC' I' Air f't Ia Pressl('ln du Vent sur 1<.'5 Han~ar!'l de Diri[:eablf's." by A. G. Elif"I, Paris, 1914, Trans. by J. C. Hunsaker, lIoughton-MifClln Co., New York.

108. "Erp'bnlsse der Aerodynamischen Vers\lchsanstalt (A VA) zu Geottln­gel;," by LudwJg Prandtl and A. B('tz, Vol. I. 1920; Vol. n. 1923; by O. Flachsbart. Vol. ill, 1927; Vol. IV. 1932, Berlin.

109. "Wind Prf'!'Isure on Structures," by H. L. Drydf'n and G. C. Hill, Sci. Pa­pcr 523, Natl. Bur. Stds .• 1926.

110. "Wind 8:.resses in BuUc'Jn~s," by Robins Fleming, John Wiley and Sons, Inc., New York, N. Y., 1930.

Ill. ":.Yf)ddlversuche ueber die Belastun{! von Gltterfachwerken durch Wind­kraf'fte," by O. Flachsb:ut and H. WintE-r, Der Stahlbau. No.9, p. 65; No. 10, 1934, p. 73; No. S, 1935, p. 65; Trans. by SancUa Corp. AFSWP-464.

112. "Aero(::;namlcs and the Civil Engineer." by W. W. Pagon, Engineering News-Record, Vol. 112, 1934, p. 348; Vol. 113, 1934, pp. 41, 456, 814; Vol. 114, H;35, pp. 562, 665, 742: Vol. 115. 1935, p. 601.

113. "Aerod::namic Stability of Suspension Bridges," by F. B. Farquharson, univ. or W .. shlngton Bulletin No. 116, Part I, 1949; Part II. 1950, by F. C. Smith and G. S. Vincent; Part III, 1952, by F. B. Farquharson; Part IV. 1954, by F. B. Farquharson; Part V, 1954, by G. S. Vincent, Seattle, Wash.

2. NATURE OF THE WIND

201. "On the Influence of the Earth's Rotation on Ocean Currents," by V. W. Io.kman, Arkiv. Mat. Astr. Fysik, 1905. -

202. "A Gf'neralization of the Theory of the Mlx1n!; Length with Applications to Atmospheric and Oceanic Turbulence," by C. G. Rossby. Meteorolo~l­cal Papers, Vol.!, No.4, Mass. Inst. of TeCh., Cambridge, Mass., 1932, p~:-j.--

203. "Hurricanes," by I. R. Tannehill, Princeton Univ. Press, Princeton. N. J., 1930.

20·.. .. At mo~phaerische Turbulenz,· by H. Lettau, Akad. Verlags!;es, Leipzig, 1939.

205. "Effective Gust Structure at Low Altitudes,· by Philip Donely, NACA, TR 692, 19';0.

206. "Physics of the Air,· by W. J. Humphreys, 3rd Ed., McGraw-Hill Book Co. Inc., New York, 1940.

207. "Report on the Storm of November 25, 1950,· by Earnest J. Christie, W\.'ather Bur. Offlce, New York, N. Y., December, 1950.

Ij.90 WIND FORCES'

206. ""ariation of Wi •• d V('loclty apd Gusts with Height,· by R. H. Sherlock, T.",il1!";\ctlons, ASCE, Vol. 118, 1953, p. 463.

209. "Ibrrlca;ies," by R. H. Simpson, 5cientUlc American, Vol. 190, No.6, June, 1954, p. 34.

210. "Wind Velocities During Hurricanes," by R. C. Gentry, Transact1on~. ASCE, Vol. 120. 1955, p. 169.

211. "A Comparison of SLx Great Florida Hurricanes, .. by D. C. Buntlng, Wcath-erwise, Vol. 8, ~o. 3, June, 1955, pp. 64-67, 78-79, 82. ---

212. "illirr!canes a\~d Tropical Storms 1887-1956,· Weatherwise, Vol. 10, No. 4, August, 1957, pp. 132, 133, 139.

213. "WlH're Hurricanes Bq:in,· by W. H. Hagl'ard, Weatherwlse, Vol. 12, No. -i, Aur,ust, 1959. pp. 145-150.

214. "Coll1,lendiulll of ;o,ietcorolo~:y; Tropical Cyclones," by G. E. Dunn, Amer. ;o,Xeteorologlcal Soc., Boston, Mass., 1951, pp. 887-889.

215. Closure to "~a~ure of the Wlnd," by R. H. Sherlock, Proceedings, ASCE, Vol. 66, ~o. STi, July, 1960, Fig. 11.

216. .. Di~tribut!on8 of Extreme Winds in the United States," by H. C. S. Thorn, ~rCJceeC;inr.s, ASCE, Vol. 80, No. ST4, April, 1960, Figs. 2, 3, 4.

217. "Tile SaIety of Structures," by A. M. Freudenthal, Transactions, ASCE, Vol. 112, 1947, p. 125.

218. "Synopsis of First ProGress Report of Comn~ittee on Factors of Safety,· by O. G. Julian. Proc. Papl'r ~~o. 1316, ASCE, Vol. 83, No. ST4, July, 1957: Discussion by T. Au, Proe. Paper No. 1442, ASCE, Vol. 83, No. STG, ~ovemh('r, 1957; Milos VlIrllcek, Jan Suchy, Ernest Basler, Rene r~. Walther, Proc. Paper No. 1522, ASCE, VoL 84, No. ST1, January, H~58: E. Ruble, R:,,"lTIond ArChibald, Proceedin~s, ASCE, Vol. 86, No. 51'2, February, 1959; Closure, by O. G. Julian, Proc('edings, ASCE, VoL 66, ~o. ST2, February, 1959.

219. Discussion by R. C. Gentry of -Nature of the Wlnd,· by R. H. Sherlock, Pro(('cdin!?s, A~CE, Vol. 85, No. ST3, March, 1959.

220. "Tropical l.!eteorolob),,· by Herbert Riehl, McGraW-Hill Book Co. Inc., ~ew York, 1954, p. 288.

221. "On the Lo\\'- Level Wind Structure of Tropical Storms," by L. A. Hughes, ~!,m;al of Mctl'orololEl, Vol. 9, 1952, p. -i22.

222. "Wind Pressures in Various Areas of the United States," by G. N. Brekke, nuilJint; r.:atcrialB and Structures Report 152, Natl. Bur. of Stds., April. ]959, pp. 4, 7.

223. "Gust Factors for the Design of Buildings,· by R. H. Sherlock, Publica­tions VoL 8, Intcrnatl. Assoc. for Bridge and Structural Engrg., Zurich, S,,1tzerland, 1947, Fig. 11.

224. Ibid., Fig. 19.

we;D FonCES 1191

225. "D.llbs Torn;ldo of April ~. 10!i7.· b:; S. G. D1g1E'r and E. P. SrL~ner, Jr., Raw Data H(>p'>rt No.2, Ikpt. of Oce;lnor:raphy and Meteorology, Agrlc. and Mech. Coliege of Texa~, College Station, Tex., Angust, 1957.

3. FlJ1ImAMENTAL CONSIDERATIONS

301. "Aerod:,namlc6," by L •• T. flri~gs and H. L. Dryden, International Criti-cal Tablr5, Vol. L 192G, pr. 402-411. --

302. uTurlmlent Flnw,· by L. Frandt!, NACA TM 435, 1927.

303. "On the Flow of Air Behlnd:1I1 Indlned Flat Plate of Infinite Span.· by A. F;h~C and F. C. Johansen. R. and M. 1104, British A. R. C., 1927; and Pr.?_c:ecl:~:;.s, Hoyal Soc., Vol. 116, No. 773. Series A, September, 1927, pp.l,U-19 •.

304. "The Structure of Vortex Sheets," by A. Fage and F. C. Johansen, R. and 111. 1143, British A. R. C., 1927; and Philosophical Magazine, Vol. ~ No. 28, Series 7, February, 1(128, pp. 417-4'11.

305. "Applied Hydro- and AE'romechanics,· by L. Prandtl and O. G. Tletjens, Trans. by J. P. Den li,U'-l()I~, McGl'aw-Hill Book Co. Inc., New York, 1934.

306. "Aerodynamic Theory,· Ed. by W. F. Durand, Springer, Berlin, 1934-5.

307. "Variation of the Wind-Pressure Distdbutlon on Sharp-Edged Bodies,· by Chr. N.;kh:ent vee!. l~'lboratorium for Bygningasstatik, Denmark Tekniske Hr.-jskole, 1936, and SC-4204 (TR).

308. ":':ndrrn Df'vclopments in Fluid Dynamics,· Ed. by S, Goldstein, Oxford, F. •• !;land, 1938.

309. ·Vll).ations Proc!uced by Wind," by Chr. N1>kkentved, Dansk Selskab for BYg'llingssl;;.tik, CopenhagPIL, DelUllark. 1941.

310. "H::drodynamics,· by H. Lamb, 6th Ed., Dover, New York, 1946.

311. "L' Action. du Vent sur les Constructions,· by L. Blanjean, Ossature Me­tallique, No.2, F('bruary, 1949.

312. "On the Drag and Sheddinr: Frpquf'l1CY of Two-Dimensional Bluff Bodies,· by A. Roshko, NACA TN 3169, Waghington, July, 1954.

313. "Vortex Formation and Resistance in Ppriodic Motion,· by J. S. McNown and G. H. Keulegan, Proceedings, ASCE, Vol. 65, No. EMl, January, 1959.

314. "Vortex Systems in Wakes,· by L. Rosenhead, Advances in Applied Me­chaniCS, Vol. III, Academic Press, New York, pp. 185-195.

315. "Fluid-Dynamic Drag,· by S. F. Hoerner, published by the author, Mid­land Park, N. J., 1958.

4. FORCES ON ENCLOSED STRUCTURES

401. "Windloads on Buildings,· by A. Smith, Journal of the Western C",clety of Engineers, 1912, 1914.

1192 WIND FORCES

402. .. Wi ncl Pressures 01' Structur('s." by H. L. Dryden and G. C. Hill, Sci. P;lj)"I' 5::3, r;atl. Bu1'. Sld<;., 1£l26.

4(,3. "Wind 1'rl'ssure 011 Circular Cylinders ami Chimneys.· by H. L. Dryden ami G. C. Hill, Rese;lrch Pap(>r jl;n. 221, Nat!. Bur. Stds., 1930.

404. "Wind Pn'SSdrt' on En.;incerinE: Structures," by R. Katzmayr, O(>sterr. In.~. Arch. V('rein, Vienna, Vol. 62, No. 21-22, 1930, p. 175; jl;o. 23-24, iJ. !rH. .

405. "Wind Pn'ssures on BuildillhS," by J. O. V. Irminger and C. N6kkentved, COllenll:q;cl:, Denmark, 1930, 1936.

405. "Wind 1'rl's";lIr(>s on a l\1ndcl of a Mill Bufldinh •• hy H. L. Dryden and G. C. Hill, U. S. flllr. Stcis. lksean:h 1'apl"'r No. 301, 1931.

407. "Wind LO:lds on Airs;,;;> Hangars, "uy H. Sylvester, 6th jl;atl. Aeronautical ~;('ctinb of AS"lE, 1932.

408. "WJl](! 1'n'ssure on Buildings," by Chr. ~~t.>kkeIltved, Pub!. Int. Assoc. r;ri(4~(' and Str\lctural El1br~ .. V .. 1. I. 1932, p. 355; Vol. II, 1934, p. 257, ZUI'il'h, S'~itzerland.

409. "Wind Prcssul'e on a Model of the Empire State Bulldll1g," by H. L. Dry­d"ll and G. C. Hill, Hesearcli Paper jl;o. 545, jl;atl. Bur. Stds., 1933.

410. "Wind Pr('s~:ure on Bui1diIl~s," uy A. Bailey, Journal, Inst. Clv. Engrs., Sl'l!'ct. E:l"I(:. 1' •• j1CI' :-;0. 139, 1933. ---

411. .. LJ;u('l.c.! of jI;('ir:hixJl'int-r Structur('s on the Wind Prcssure on Tall Build­i;0,:5," by C. L. liarris, Pa. Sl;]te Call. Enbrg. E.xperimf'nt Sta. Ser. Bull. :-;0. 43. lC13-i, also :-;;,tl. Bur. Stds., Journal of Reticarch, Vol. 12, No.1, 1 ~3-i, p. 103.

412. "L' Act ion du V('llt Sur Ll.'s Constructions,· by F. BI ron, Science et Indust. C,mstr., V'Jl. 18, jl;f). Iii, 1931, p. 131.

413. "L' Actio;, du Vent Sur L(>s n.1~hnents ct Constructiuns,· by R. Pris, Trav. ArchitPct. Construct., Vul. 20, 1936, p. 71.

414. "Wind 1'r('ss\lr(' Distribution on Sharp-Edged 8odJcs,· by H. Sylvester, B:~gnin~sstl\LSke :'i~i.dl'lelse12 Vol. 8, 1936, p. 41.

415. "L' Azione d"l Vcnte suU" Costruzlonl.· by Renato Glovannozzl, Aero-[('enka, V"I. 16. 1936, p. 413. --

416. "The Effect of Wind Loads on Frames With Semi-Rigid Conncctlol1s," by J. F. Daker and E. F. \Villiams, Final Hep()rt. Steel Struct. Res. Comm., LOlld(111, Er .. :la.ld. 1936.

417. "1kr Winddruck auf Sch()rnsteine mit Kr('isqllcrschnltt: by J. Ackeret, Schwdz. Bauzeitullg, Vol. 108, 1936, p. 25.

418. "Wind Pressurcs on the Akr<Jll AirShip Dock,· by K. Arnstein and W. Kl~'m;Jercr, J'Jurnal of the Aeronautical Sciences, Vol. 3, 1936, p. 88.

419. "Wind- Bracing in Stl'el Buildings," FIfth Progress Report, Sub-Committee :-;0. 31, Committee on Steel. Structural Division, Proceedings. ASCE Vol. 62, :'larch, 19JG, p. 397.

WIi-iD FonCES 1193

420. "Acrod:;n;Wlics oi Ulf' PerispiJcrf' alld Tr}'lon at the World's Fair," by A. Klemln. E. B. Schaefer ~lId L. H, Burer. Jr., Transactions, ASCE, Vol. 104, 1~3J, p. 1449. ------

421. "Th£' Bi,", Storm," Eel. by L. H. Tyl£'r, Th£' City Printing Co., Ncw Haven, Con n., 1938.

422. "Wind Forces on a Tall Building," by J. C. Rathbun, Transactions, ASCE, Vol, 105, 1940, p. 1.

~23. "Winci Bracinr- in Steel Builclip!:s," Final Report, ASCE Sub-Committee Nu. 31, Committee on St!'e!, Transactions, ASCE, Vol. 10!i. 1940, p. 1713.

424. "Wind PrC's5ure on Buildings Including Effects of Adjacent Buildings,­U}' A. B,lii!'Y and N. D. G. Vincent, Journal, Inst. Ch'. Engrs., Vol. 20, 1943, p. 243.

{25. "'Wind Load." on B\lildil\~s," by H. Ferrin~on, Structural Engineer, Vol. 21, 1943, p. 497; Vol. 22, 1941, p. 15.

426. "Aerodynamic Characteristics of Circular Arch Roof Structures," by J. H. Ciss.?] al.d L. r.r. Lpt;atski, Eng. Res. Prof. M-518, Volv. of Mich., Ann Arbor Mich., 1944.

42i. N0rnwr for Bygningsi{Qnstruktloner 1. Belastnlngsforskrlfter, June, 1948 (Danish Standard).

428. ... Protection of Small Buildings i\f;alnst Hi~h Velocity Winds," by R. A. Ti;'Hll,lsvn, B.Ill. 28, Fla. Engrg. and Industrial Experiment Sta., Galnes­"ilie', Fia., 1949.

429. "L' Action du Vent Sur Les Constructions, - by L. BIanjean, Ossature Me­talli~ue, Brussels, Belgium, February, 1949.

430. "Wind war'.s on Structures," by Michael R. Horne, Journal, Inst. Clv. Engrs., Vol. 33, January, 1950, p. 155.

431. "Wind Stresses in SC'mi-Rigid Connectlolls of Stcel Framework," by B. Soun,cilni koff, Trans.lctions, ASCE, Vol. 115, 1950, p. 382.

432. "Wind-Load Standards In Europe," by John W. T. Van Erp, Proc. Sepa-at£' ~o. 42, ASeE, Ye,I. 76, November, 1950. .

433 ... Lateral Forces of Earthquake and Wincl," hy Arthur W. Anderson, et a!, .!'~·oc~_Scpar,~tf' :,;()~, ASCE, Vol. 77. April, 1951.

43,1. "Wind Tunllcl Stucii('s of Vrcssllrf' Distribution on Elementary BuUding F"rms," by N. Chien, Y. Feng, H. Wang, and T. Llao, Iowa Inst. of Hydr. Hescarch, State Vniv. lOf Iowa, Amcs. Iowa, 1951.

435. "Wind Pressure on Elementary Building Forms Evaluated by Model Tests," hy J. W. Howe, Civil Ent;ineering, May, 1952, p. 42.

436. "Determining Basic Wind Loads." by George F. Collins, Proc. Paper No. 825, ASCE, Vol. 81, November, 1955. ;

437. "Minimum Design Loads in BuIlding and Structures," Amer. Stds. Assoc., A5S.l, 1955.

1l!14 WL~D FORCES

438. S,'hweizl'rischer Int;enil'lIr tInd Archlt(>lcten Ver!'!n, T('chnit=;rhc Normen No. 1(;0, 1956.

5. FORCES ON PLATE GIRDERS AND TRUSSES

501. ".\;>"ll(':.tlo:J of the Th£'ory of Free .Tet:"," by A. Belz and R. Peterson, N~tl. Ali .... COI.,m. for Aeronautics. TNG67 (translation).

502. "Winddrllck iW!' Fachw£'rktuerme yon quadratischcm Querschnltt," by R. Katzlll'I)'r ;,.ad H. Seitz, Bauinr.f'nit'ur. Vul. 15, 1934, pp. 215-221.

503. .. B.'(pl'r i ments on a Static 1>10del of a S£'ction of the Seyern Bridgf', • hy D. H. Williams, H. L. Mxon, and W. C. Skelton, Natl. Physical 1...10., Eng­la;](I. A('ro /209.

504. "T<.'sts of PlatP Gircler BridGes in thl' Duplex Wind Tunnel," by D. H. ',v.llh;.ls. H. L. NLxon, and W. C. Sk('lton, Nall. Physical 1...11>., Ent;land, A",·u/216.

505. "?:n~ pi r iea. For n.ul.ls for the TUli;;el Blockage Correction for Wind Force ;'~f'asur('ments 0 .. Latticl' Structur£'s. and some Wind Load CoeCIicll'nts ior Lattices l>bsts." by C. Scruton, Natl. Physical Lab., NPL/ Aero/355 • . Pcrmission has bCl'll t;iv!':1 to publish this unpublished data with thIs note: "7:.e date in NFL! Aero/3S5 was obtainl'd In behalf of th(' British Insu­bh,d Callendel"s Construction Co., Ltd., and is now pubUshed by per­mission of til(' Dil'e"tor, Nat ional Physical Laboratory, England.-

50G. "'lnv('st igation of Wind Forces (Jr. Highway Drldcies," by G. S. Vincent. Hit,.:hway "{'search Bo,lrcl Special R('port 10, No. 272, 1&5~.

507. "Wind Loads on Truss [\rid;;cs,· by J. M. Bibgs, Transactions, ASCE, Vol. 119, 1954, p. 8';'9.

SOd. "'Wind-LJuc('d Vihratinns of a Pipe- Line SURpension Bridge and Its Cure," b:; Il. C. Baird, Tl'a:1s3ctions, ASl\IE, Vol. 77, No.6, August, 1955.

509. "Wi:·.d I.I·,lds on Girder Bridges,· by J. M. BIggs, S. Namyet, and J. Ada­chi, Tr;I.J~sactio;:s, ASCE, Vol. 121, 1956, p. 101.

510. "'DiCSit=r. of Multi-Level Guyed Towers; Wind Loading,· by Edward Cohen and Henl'! Perrin, Proc. Paper No. 1355, ASCE, Vol. 83, No, ST5, Sep­temb.::!r, 1957.

511. "Wind Forces on Structures: Plate Girders and Trusses," by W. Watters Pa;;on, Proc. Pa;l('r No. 1711, ASCE, Vol. 84, No. ST4, July. 1958.

512. "'Wind Resistance of Lattice Girder Bridges," Inst. of Civ. Engrs., Lon­don, England, 1948.

513. CLosure to ·Wind Forces on Structures: Plate GIrders and Trusses," by W. Watters Fagon, Proceedinbs, ASCE, Vol. 85, No. ST7, September, 1959.

6. STRUCTURES SUBJECT TO OSCILLATION

601. "truer eine besondere Art del' Tonerregung,· by V, Strouhal, Ann. der Physik, Vol. 5, 1678.

WDiD FOlKES 1195

602. "Viler den M(>chanismus des Fhl."slr:kf'its und Luftwloderslamles,· by Theodore von K.arman <lnd H. nubach, Phys. ZeUs., Vol. 13, 1912.

603. uThe Flow of Air Al)<)ut an lnelin/'u Fl:1t Plat(' of Infinit(' Span,· by A. Fage and F. C. Johansen, Proce('di n<':5, nnyal Suc., Vol. Al16, London, England, 1927. -----.--

604. "The Eff('cts of Turbul<:'llce and Surfar.c nour.hness onl;\e DraJ of Circu­lar CyliniJprs,· by A. Fage and J. W. Warsap, British ARC, Rand M No. 1283, 1929.

605. "Vibration Probl('ms in Tall Stacks S<lIven by Aerodynamics,· by W. Wat­ten; PaGon, En~illeerlnG News-Ik('fJrd •. luly 12, 1934.

606. "L1.bora:ory Studies of Conductor Vlhratlons,· by Joseph S. Carroll, Trar.sactions, AlEE, Vol. 55, 1936, p. 543.

6G7. .. Rigidity a;od Aerodynamic Stabi lity of Suspension Bridges,· by D. B. Steinman, Transactions, ASeE, Vol. 110, 1945, p. 439.

608. "Problems of Aerodynamic and Hydrodynamic StabUity,· by D. B. Stein­man, Builetin 31, Unlv. of Iowa Studies In Engrg., Iowa City, Iowa, 1947.

609. ":.t('ch:inical Vibrations,· by J. P. Den Hartog, McGraw-Hlll Book Co. Inc., New York, 1947.

610. "Wind Tunnel Tests of Suspension Bri(lf;e Section Models,· by F. J. Maher, D. Frederick, E. H. Estes, and D. B. Stl'inman, Bulletin, Va. Poly tech. Inst. Bq;rb". E.'Cperiment Sta .. Vol. ~1, No.6, Series No. 69, 1948.

611. "Dynamic Instability of Tl'11Ss-StiIfenerl Suspension Bridges under WInd Action," uy F. Bleich, Transactions, ASCE, VoL 114, 1949.

612. "Aerodynamic Theory of Bridge Oscillations,· by D. B. Steinman, Trans-actions, ASCB, Vol. 115, 1950, p. 1180. ---

613. "The M,,~hemati('al Theory of Vihrationln Susrenslon Brldg~s, ,. by Fred­rich Bleich, C. B. McCulloubh, R. nosecrans, and George S. Vincent, Bur. of Pub. Rels., 1950.

614. "Aerodynamic Stability DC Suspension Bridges," by F. B. Farquharson, Univ. of Washinbton En~:rb' E.xperiment Sta. Bulletin, 116, Part 01, 1952.

615. "Inw Sreed Drag of Cylinders of Various Shapes," by N. K. Delany and ~. E. Sorenson, NACA TN 3038, November, 1953.

616. "Thl' Il1\'esti~ation of ;""Jdeis of the Original Tacoma Narrows Bridge under the Action of WiI1d," by F. B. Farquharson, Univ. of Washln:,,'ion Engrr;. Experiment Sta. lIulletin 116, Part IV, 1954.

617. "Extended Studies,· by G. S. Vince;]t, Univ. of WaShington Engrg. E.'Cperl­ment Sta. Bulletin 116, Part V, 1954.

G 18. ., Aa IntroJuct\nn to the Theory of Acroelastlclty," by V. C. Fung, John WUey .aad Sons, Inc., Nl'w York, 1955.

619. "On the Wake and Drag of Bluff Bodies,· by A. Roshko, Journal of the Aerol::lutical SCiences, Vol. 22, No. 2, February, 1955.

620. "Wind Effects on Bridges and Other Flexible Structures,· Notes on Ap­plied SCience, No. 11, Natl. PhySical Lab., England, 1955.

1l~6 wnm FOnCES

621. "A('r~)j::namic BuffetiI1f,""f Pridt-:('~." by C. Scruton, The Engineer, Vol. 1%, :-Iay 13, 1955, pp. f3:J4-6f37.

622. ''',vind-Indllccd Vibrations of a Pipeline Suspension Bridr,e, and Its Cure,· by R. C. iJaird, ASME Paper :-10. 54-PET-12. July, 1955.

623. "Ar'rodynamlc Stab1lity of Bridr:('s.· Report of th(' Advisory Board on the ;nve:;ti~ationf; of Suspension Bricib"s, Transactions, ASCE, Vol. 120, 1955, p. 721.

624. "Factors l:uillencinr, the Dynamic Dehavior of Tall Stacks under the Ac­t il'n of WineL· ily l>i. S. Ozkcr and J. C. Smith, Transactlons, ASl\iE, Vol • • a. No.6. 1956.

625. "Vihrations of Steel Stack::!." by W. L. Dickey and G. B. W(lodrulf, Trans-adions, ASeE, Vol. 121, 1~156. ---

626 ... itnsonant Vibrations of Stl'el StackS," by E. A. Dockstader, \V. F. Swiger, al.d E. Ireland. Transactions, ASCE, Vol. 121. 1956.

627. "Progress Rf'port on thl' In\'esti~ation of Galloping Transmission Line C(lOUuctors." by A. T. Edwards and A. l\.fudeysld, Transactions, AlEE P:lper No. 55-213, 1956.

628. "Wind Tunnel L,vesti:;ation of Conductor Vibration wIth Use of Rigid Mo­d.~:s," u)' ". D. Farqu:1arsoil and Rohert E. ?IclIugh. Jr., Power Appara­!.::~s :1:1d S~'stems. AIEE, October, 1956, pp. 871-8.8.

629. "S..IiliJre5SitJn of the Fluid-!nciuced Vibration of Circular Cylinders,· by Pd.:r Price, Proc. 1';;1)(''' ~;ll. 1030, ASCE, Vol. 82, No. E:\,3, July 1956.

630. "Wind Ind\,('ed Vi:Jration of Cylindrical Structures,· by Joseph Penzlen. p;'nc. jl.1ppr !"). 1141, ASCF.. Vul. 83, No. E:\ll, January, 1957.

631. "A :.:cal\s fur /woldinb Winci F.xcited Oscillations of Structures with Cir­culal" or !\f'arly Circlil<Jr Cross Sections," by C. Scruton and D. E. J. \'1aI5:1e, N?L/Aero/335, Octuber, 1957.

632. "Fur.her Experiments on the Use of Helical Strakes for Avoiding Wind Excited Oscillations of Structure with Circular or Near Circular Cross SectIOn," by L. Woodgate and J. F. M. ~Iabrey, NPL/A£'ro/381, June, 195!).

633. "I::;,jleri:ncntal In .... esti'"ation of Wind-Induced Vibration in Antenna Mcm­u.:r5.· by W. WeaYer, Group RCjlOrt ~o. 75-4 Lincoln Lab., Mass. 1nst. of Tccll .. CMllbridt;c, ~L'ss., AUl~st 26, 1959.

634. "Effect of Flow Illddence and Reynolds Number as Low-Speed Aerody­namic Chal"aClcristies of Several ~oncircular Cylinders with Appllcation to Di.rectiona.l Stability and Spinning,· by Edward C. Paulhamus, NASA, TH R-29, 1959.

635. "ARC Reports and Memoranda. • by A. Fage and J. W. Warsap, No. 1283, 1930.

WI~D FORCES 1197

APPEN;)IX II.-NOTATION

The following letter symbols conform essentially with • American Standard Letter Symhols for Structural Analysis" (ASA Z10.8-1949), prepared by a com­mittee of the American Standards Association, with Society representation, and approved by the Association In 1949:

A exposed area;

b width;

C coefficient;

C C P eccentricity cneliicient;

CD dr:lb coefficient;

Cr. lilt cot'fficient;

C P pressure coefficient;

C Pc extE'rn .. l pressure coefficient;

C Pi internal pressure coefficient;

D di;,meter;

d distance between vortexes;

frec;uer.cy;

fO = frequency at which model moves vertically as a Wlit;

f 1 frequency of vertical motion about the z-axis;

h hei.;ht or depth;

L length;

span;

M = mass per unit length;

p static pressure at a point;

Po static pel'ssure of the approaching free stream;

<i dynamiC pressure of the frE:e stream;

qz dynamiC pressure of the free stream at height z;

R Reynolds number;

r rise-span ratio;

S Strouhal number;

s spacing;

V wind velocity;

1198 Wlj'<D FOHCES

Vc crillcal wind v£'IocHy;

Vo wind vdocity of the appr'Jachlnb free stream;

V 7. wind vcloeity at heif:ht z;

V 30 wind velocity at h£'ibht of 30 it;

V 50 wind vdocity at hd(;ht of 50 ft;

x exponent in Eq. 2;

Y deflection;

Ym d,'flcction (If tube; .

Y2 dQubie ampUt\lde;

z devation;

a an,:le of .. ttack of wind on object;

6 lOf;ari: hmic decrt'ment;

8 celltral angle:

>. lcr • .;th-width ratio, or aspect ratio;

1.1. dynamic viscosity;

II '" kil~ematic viscosity;

p density of air; and

9 solidity ratio.