SIMIU_E_SCANLAN_1996_3Ed!!!!!

Prévia do material em texto

WIND EFFECTS
STRUCTURES
ON
Vicw ol'C--hicago with Standard Oil Company (lncliana) huiltlilrpi lr('iu ('('ntr'r (Alt lritet'ts:
Pcrkins and Will, and Edward Durell Stone and Associirtt.s)
EMIL SIMIU
Nlsr Fellow, Building and Fire Research Laboratory, National lnstitute ofStandards and Technology, Gaithersburg, Maryland
ROBERT H. SCANLAN
Professor, Department of Civil Engineering, The Johns Hopkins
University, Baltimore, Maryland
WIND EFFECTS ON
STRUCTURES
Fundamentals and Applications
to Design
Third Edition
A Wiley-lnterscience publication
JOHN WILEY & SONS, INC.
New York / Chichester / Brisbano /
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oata .1lL *,{- I tg
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This text is printed on acid-free paper
Copyright O 1996 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work beyond
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Library of Congre ss Cataloging-in-Publication Data:
Simiu, Emil.
Wind effects on structures: fundamentals and applications to design
/ Emil Simiri, Robert H. Scanlan. - 3rd ed.p. ciTl.
Includes index.
ISBN 0471-12157-6 (cloth : alk. paper)
l. Wind-pressure. 2. Buildings-Aerodynamics. 3. Wind resistant
design. I. Scanlan, Robert H. II. Title.
TA654.5.S55 1996
624.1'76-dc2o
Printed in the United States of America
1098765432
96-5238
Soave sia il vento
tranquilla sia I'onda
ed ogni elemento
tranquillo risponda
ai nostri desir!
Cosi Fan Tutte
Act l, Scene V (Terzettino)
For Devra, Erica, and Michael paul.
PREFACE
'f 'lrt' tlrird edition of Wind Effects of Stuctures reflects the many developments
tlrrrt occurred during the last decade in the wind engineering field. The text has
lrt'en rcvised, updated, and augmented to include new information and/or ref-
crrnccs on, among other topics, windstorm damage and insurance, hurricane
rrricnrrnctcorology, aerodynamics of circular cylinders for large Reynolds num-
lrt'r's, c<lmputational fluid dynamics, tail-limited extreme value modeling of
rrorrtnrpical storm and hurricane winds by "peaks over threshold" methods,
t'rn;rirical aeroelastic models, progress and limitations in wind tunnel modeling,
tLrrrrping of flexible buildings, across-wind and torsional effects on tall struc-
Iule:;, low-rise buildings, behavior of roofing, power lines, and wind load
l:rt'lors. The material on suspended-span structures has been almost completely
rt'wlillcn. A new chapter on standards has been added, which includes a useful
rt'lt'rcncc to a diskette, appended to this book, containing an interactive com-
lrrrtt'r vcrsion of the ASCE 7-95 Standard provisions for wind loads.*f
Wc thank the many colleagues who used our book in their professional
prirt'ticc or as a primary text for teaching wind engineering and gave us the
lx'rrclit of their comments. Special thanks go to Dr. R. D. Marshall, whose
t'xpelicncc and judgment, particularly in the areas of wind tunnel-modeling
rrrrtl winrl-loading codification, are reflected in several portions of this book,
'll Il. St'lnlan was rcsponsiblc lirr rcwriting Sccrs. 5.3.3, 6.1.1, 6.5, 6.6, and 13.1. E. Simiu
rv;t: tt'slxrttsiblc lirr all othcr rcvisions antl ltltlitions. His contributions to this book are made in
lrr: privitlc c:tllrcity ancl cio not ncccssruily lrl)lcsont thc position of the U.S. Department of( onrnr'rcc ol ol tlrc Nirlional lnslilllle ol Slirntl:rnls irrrtl 'l'cchnokrgy.rllrr'tliskt:l1c's witttl loutlittg srtllw:trt', tt'lttrxlrrtcrl lirrrrr ltcl. l7-5, is in rhc public rkrrnain ancl
r: r()l iul AS('li llrrlllic:rliorr ol tkrr'ttttrr.ll
vll
vial l'llllA(,1 I() llll lllllll)ll)lll()l'l
ilu(l {o l)rrrli'ssor .l . Krrttrllr ol lltt' [ )trrvcrsily ol 'l'okyo, wll(r llirirrs(irkirrgly
.t,c.t c,l (lrt, tcx( ol' tlrc sr:t'oiltl crliti0rr rrrrtl krrrrlly lxrintccl tlut a nutnbcr ol'
typogrirlllriclrl ornrrs. wc irlso wislt lo (h:tttk rtttt'txlittlrs, charlcs Schmicg, Ira
tir,*irt y, arrtl Nancy Lin, krr lhcir-hcllllirl cooPt.r'rrtion._. .
'.lkrwing 
the Russian translation irl'oLrr trixrk, a Chinese translation was
publishcd b! Tongli University Prcss, Shanghai' We welcomed it while re-
[..tting the latter';disregard of intemational copyright norms'
Etuttl- Stvtu
RoeBnr H. ScnNl,qN
Rockville, MaryLand
Baltimore, Maryland
Mav 1996
PREFACE TO THE SECOND EDITION
lrr tlrc alnrost ten years that have elapsed since the writing of Wind Elfects on
.\tt'ttt'torrcs a number of significant advances have occurred in the wind engi-
Irr.e ring lield. These include the development of the following: improved micro-
rrrt'tcrlrological models, particularly for atmospheric flows over the ocean, which
:rle ol'interest in the design of offshore structures; procedures for the estimation
ol e:xtrcme winds from short records; new information on the modeling of
r'xlrcnlc winds in hurricane- and tornado-prone regions; improved procedures
lor r:stitnating the along-wind response of structures; new procedures for esti-
rrurting the across-wind and torsional response of tall buildings and the across-
rvirxl rcsponse of towers and stacks; simple and probabilistically rigorous meth-
,rrls lirr taking wind directionality effects into account in design; practical pro-
, t'rlrrrcs fbr the risk-consistent design of cladding for wind loads, which make
rt lxrssible to achieve more economical designs for any given safety level, or
s:rlr.r clcsigns for any given cost; methods for estimating the response of offshore
:itru('lurcs to wind in the presence of current and waves; and new information
t,n wirxl cffects on various types of structure, including trussed frameworks,
lrylrrbolic cooling towers, and semisubmersible platforms.
'l'hc tcxt has been expanded to reflect these and other advances. It now
rrrt'lrrtlcs fivc new chapters, as well as a new appendix that is intended to provide
tlrt.r'cirtlcr with a brief introduction to modern structural reliability concepts.
'Ilrc original chaptcr on wincl ttttttrc:ls wlts substantially revised, and much new
rrr;rtt'r'iirl was aclclcd to thc ollrrrl t'lt:tP(cr.s, plrrlicularly those on the atmospheric
lrrrlrtlirry laycr, cxtrcrttc wirttl t'liltrrrlokrgy, blull'body aerodynamics, aero-
t.l:rslic ltlrcrxlntir, tall builtlings, lrrrtl (orrrlrtLr ellecls. Most of the new material
r'orrsists ol'prlrclicirl tlcsigrr irtlotttr:tltotr ;tutl trtt'lltrxls. As in thc first edition, a
tx
I'lll ln(il l() llll :;l (:()Nl )ll)l ll{)l!
consistcnl cllirrt lurs lrr't'rt nt:rrlt'l() l)()lnl ottl rrttrl tlist'ttss (ltr: ttttccrtrtitttics,
limitatiorrs, rrrrtl crnrls irrlrt'r't'rrl irr vrrriorrs rllrllr, rrrt'llrtxls. luttl lcclrrriqttcs.
Thc aulhors woulcl likt'l() ('xl)r'('ss lltt'rr wlrrnr:rpptccirrtiott lo I)r. I{. D.
Marshall, who initiatcd antl tlcvclolletl tlrc wintl cngirrccring prograrn at thc
National Bureau of Standanls; I)r. N. lsyrrnrov ol'tlrc Univcrsity of Westcrn
Ontario, fbr contributions to Scct. 9.-l; rrrrtl l'rolcssor l). A. Recd of the Uni-
versity of Washington, forcontribu(ions to ('hrrp(cr ll. Spccial thanks are also
due, for valuable comments and criticisrrr, t<l l)rol'cssor E. A. Arens ol'the
University of California at Berkeley; Dr. R. l. Basu of H. G. Enginccring,
Inc.; Professor O. Ditlevsen of the Engineering Academy of Denmark; Dr.
B. R. Ellingwood of the National Bureauof Standards; Professor Y. Fu.jino ol
the University of Tokyo; Dr. M. P. Gaus of the George Washington Univcrsity;
Dr. P. S. Jackson of the University of Auckland; Dr. F. Mahmoodi of thc 3M
Company; and Professor B. J. Vickery of the University of Western Ontario.
However, the responsibility for any errors or omissions lies solcly with thc
authors. We also wish to thank our Editor, E. W. Smethurst, Editorial Super-
visor, Balwan R. Singh, Designer, Lee Davidson, and Production Supcrvisor,
Linda Shapiro, all of John Wiley & Sons, and Technical Editorof the Russian
translation (1984), Dr. B. E. Maslov.
The references to the authors' affiliations are for purposes of identification
only. The book is not a U.S. Government publication, and the views exprcssed
do not necessarily represent those of the U.S. Govemment or any ol' its agcn-
cies.
Etr,tlt- Stnaru
Ronp.nr H. Sr',rur.a.N
Rtx'kvilla, Mur.ylund
PREFACE TO THE FIRST EDITION
l'lrc wind loading of.civil engineering structures involves, in certain cases,t'rnsiderable complexities that must be taken into account in order to achieves;rl'c and serviceable.designs. Examples of wind engineering problems that.r:quire special attention include: the dynamic .".poni" of tati structures; thepcrformance of exterior glass and curtain walls, particularly in high_rise uuito_i'gs; the serviceability of pedestrian areas in clrtain types or t'uilt environ_rrcnts, the oscillations and flutter ofsuspension bridges; it. u.tion oftornadoesrrr nuclear power plants; the estimation of the piobability of occurence of('x(reme winds at a given site.
Motivated by the need to provide rational descriptions of the phenomenarrv<llved and to develop. appropriate analytical and design tools, a vast spe-t'i.lized literature-not always easily accessible-has emlrged in the last twotlcc.ades. wind Effects on structure^s is an attempt to preseni a synthesis of therrrrin trends of this literature in the form of a texi designed for use by advancedstrdcnts of engineering,and by practicing structural lngineers and architects.l'hc tcxt devleops its chosen _topics independentry anal as often as possible,llirrn fundamental principles. In addition, extensive references are provided to:r widc range of primary sources.
'l'he level of preparation assumctr .r' rhc rcader corresponds approximatelytr that of thc bachelor's degrec irr st'it'rt.c .r cngineering. a "onrirtent effortlr:rs bccn rnadc to avoid unncr'cssiuily t'l;rlxrrr(c rnathematical formulations.Silnplc n<ltions o1'probahirity rlrt.'ry. sr;rrrsr it.s :rrrtl thc theory of random pro_t't'sscs ctllpl<lyccl in tlloclcrlt wittrl t'ttlirrrt't'rirr11 :uurlysis havc been prescnted inlrpltt:ntliccs, irr which iltluitivc itl)p1ry.1q lrt.:, lr;rvt. lrct.rr slr-6ngly clnphlrsizcl.'l'lrc lilsl 1xrt1 ol'llrc lcxl tlist'rrs:,t.:, trrr.tr.nrolo;r11.:tl, rrrit.lrirrc,tcgr.gl69iclrl, lrnrlt'lirlr:r(ologit';rl :rspt't'ls ol'llrc wnlrl r.trlnorrrrrt.lrl lllrl luc ol.inlt,t.t.sl irr wilrrl
xt
xll plrEl ACt r() llll t llisll FDltloN
engineering. 'l'hc scconcl part proscnls l)itsic (:l(:tttL:ltts ol'acnrclyttittttic:s, struc-
tural dynamics, and aoroclasticity, lillkrwr:d by applications to thr: clcsign of
various types of Structures and structural tnctttbcrs. Scparate chaptcrs are de-
voted to a discussion of wind-induced discotttlir( in and around buildings, and
to assessments of the wind tunnel as a design tool.
Wind engineering is a new and rapidly developing field. Cunent procedures
for estimating wind effects, and the information on which they are based, should
therefore not be regarded as definitive. It is the authors' strong feeling that
areas of unceftainty must be carefully defined, and that the limitations inherent
in current procedures must be stated clearly. This has been done throughout
the text.
The division of responsibility for the work has been as follows: E. Simiu
has written Chapters 1-3, 5, 7,9-ll, and the Appendices, and R. H. Scanlan
has written Chapters 4, 6, and 8. * The authors have, however, shared editorship
and extensive critical exchange on all parts of the text.
The authors' sincere thanks are extended to the following persons who read
porlions of the manuscript and offered valuable criticisms: Professor H' A.
Fanofsky, Pennsylvania state University; Dr. N. J. Cook, Building Research
Establishment, U.K.; Dr. J. F. Costello, U.S. Nuclear Regulatory Commis-
sion; Dr. H. L. Crutcher, National Climatic Center, National Oceanic and
Atmospheric Administration; Dr. J. J. Filliben, statistical Engineering Labo-
ratory, National Bureau of standards; Dr. J. c. R. Hunt, Cambridge univer-
sity, U.K.; Dr. G. E. Mattingly, Institute forBasic Standards, National Bureau
of Standards; Dr. J. M. Mitchell, Environmental Data Service, National Oceanic
and Atmospheric Administration; Dr. R. N. Wright, center for Building Tech-
nology, National Bureau of Standards; and Professor J. T. P. Yao, Purdue
University. All of them should share the recognition for the many improve-
mcnts their comments brought about. The responsibility for all errors or im-
perfections rests, however, wholly with the authors. Many thanks are also due
to Devra Simiu and Robert N. Scanlan for careful reading and editing of the
text, and to Mrs. Sue Murray, Mrs. Rebecca Hocker, and Mrs. Nora Scanlan
for their capable typing effort. The authors also wish to express their indebt-
edness to the late R. S. Woolson, Editor, J. Frances Tindall, Editorial Super-
visor, Joel L. Bromberg, Editorial Assistant, and Debbie Oppenheimer and
Sandra Winkler, Production Supervisors, all of John Wiley & Sons.
Evn SrvItu
RoeEnr H. ScaNI-aN
Washington, D.C.
Princeton, New Jersey
June,1977
*Chaptcrs 4, (1, ilnrl ll ol llrc lirst ctlilioil trrrrr:spond in lltt'sctotttl irrtrl llrinl oditions to Clhilptcrs
4, 6, antl 13. l,9r'llrc st't.orrtl t.tlition, l{. ll. Scunltn lrirs tevist'rl lltc r'ltirplcr ott witttl lttttncls. antl
Ii. Sirrriu lrls bcrlr n.s;xrrrrihh'lor tlrt'rrllrr'r'r'cvisirttts ittxl ttrltliliorts l() tltc t('xl.
CONTENTS
INTRODUCTION
PART A THE ATMOSPHERE
1 ATMOSPHERIC CIRCULATIONS
2 THE ATMOSPHERIC BOUNDARY LAYER
3 EXTREME WIND CLIMATOLOGY
PART B WIND LOADS AND THEIR EFFECTS ONSTRUCTURES
I Fundamentals
4 BLUFF-BODY AERODYNAMICS
5 STRUCTURAL DYNAMICS
6 AEROELASTIC PHENOMENA
7 WIND TUNNELS
B WIND DIRECTIONALITY EFFFC IS
5
33
9'1
135
195
216
273
308
xlll
xiv (,( 
'N 
il N tl
ll Applications to Design
9 BUILDINGS: WIND LOADS, STRUCILJHAL RESPONSE'
AND DESIGN OF CLADDING ANt] TIOOFING
1O SLENDER TOWERS AND STACKS WII'H CIRCULAR
CROSS SECTION
11 HYPERBOLIC COOLING TOWERS
12 TRUSSED FRAMEWORKS AND PLATE GIRDERS
13 SUSPENDED-SPAN BRIDGES, TENSION STRUCTURES,
AND POWER LINES
14 OFFSHORE STRUCTURES
15 WIND.INDUCED DISCOMFORT IN AND AROUND
BUILDINGS
16 TORNADO EFFECTS
17 STANDARD PROVISIONS FOR WIND LOADING
APPENDIX A1 ELEMENTS OF PROBABILITY THEORY AND
APPLICATIONS
APPENDIX 42 RANDOM PROCESSES
APPENDIX 43 ELEMENTS OF STRUCTURAL RELIABILITY
APPENDIX A4 PRESSURE COEFFICIENTS FOR BUILDINGS
AND STRUCTURES
INDEX
ABOUT THE DISK
327
383
404
420
446
487
511
551
576
591
629
643
665
676
684
INTRODUCTION
l'lrc rlcvclopment of modem materials and construction techniques has resultedrrr rhc cmergence of a new generation of structures that are often, to a degreerrrrkrr'wn in the past, remarkably flexible, low in damping, and light in weight.srrt'' structures, as welr as uu.iou, nou.r typ", of rigid structures, exhibit anrrrt're'scd susceptibility ro rhe action of wina. acirdt;;,r,1;has becomerr('(('ssilry to develop tools enabling the designer to ".ti*ai"'*ind effects with;r lrig'cr degree of refinement than-was previ,ously ,"quiJ. wini 
"ngrn"".rngrs rlr. discipline that has evolved, p.imarityau.ing ilr"iurt-e."d; fiom efforts;rrrrctl a[ developing such tools. rsu! uvvsuwD'
It is the task of the engineer to ensure that the performance of structures:'rr'jt:crcd to the action of wind will be adequate during their anticipated rifelr.rrr rhc standpointoftroth^structural safety and serviceibilitv. io'u"tieve thist'r(,, rhc designer needs information regarding (r) tt" winJ'environment, 12;tlrt' re l.tirn between that environment arid the iorces it induces on,h" r,*",u.",;rrrrl ('1) thc behavior of the structure under the action of these forces.
rHE WIND ENVIRONMENT
lrrlr.rrr(i<ln <ln the wincr cnvinrnrrcnt needed in design includes elements de_r r,u'cd lhrrn .mctconrlogy, nticr()nlcte()tl)l()!,y, and climatology.Merc.nrl'gy pnlvirrc.s lr trcsr.r'iPr i.r,,,,,,i'"^prunation of the basic f.eatures of'rlrttosltltcrr-ic lklws. Such ll.rrlrrt.s rrrirv lrt.ol.c.gnsirlcrabl" ;j;;ifi.;nce fiom a.'lttttlttt':tl tlcsigrr vic:wP.irrl. li,r't'rrrrrrIlt'. irr llrc t.usc.l,rr",nrnu,ir, the prcs_t'ttt t' ol'ir rcgiorr ol' low irlrnosplrt,r,, 1,,,.r,n,,,.,. ;tl lllc cctllcl- <ll. lhcl slrlrrrr is lrl:rt'tr''l'rrrrj.r irrrp.rrirrrr'e irr tir,, ,r,.,,,j.,, ,r rrrrr.rt.lrr.rx)w(rr prirrrrs.
lN il i( )t )l,o l l( )N
MiCrornCtcgnrftlgy lrltr.lrrlrls (p rlt'sr'rilrc llrr' rlt'lltrlt'rl sltlt('ltllc ol ltltttospllt't ic
flows near the gnruncl. 'l'oltics ol rlirt't'l ('()n('('ltt {o lltc sttrtcltrrltl tlcsigrrt't'ilt
clude the variation of nrcan spcorls willr heip,lrt irltovt: gtttttlttl, tlrc tle:sct-illtitlll
of atmospheric turbulence, and thc tlcl'rctttltttcc ol' (ltc tttcitrt spectls ancl tll'
turbulence upon roughness of terrain.
Climatology, as applied to the wind cnvirotrtttcttt, is ctlnccrncd with thc
prediction of wind conditions at given geographical locations. Probability state-
ments on future wind speeds may be conveniently summarizcd in wind maps,
such as are currently included in various building codes.
WIND-INDUCED FORCES ON STRUCTURES
A structure immersed in a given flow field is subjected to aerodynamic forces
that, in general, may be estimated using available results of aerodynamic theory
and experiments. However, if the environmental conditions or the properties
of the structure are unusual, it may be necessary to conduct special wind tunnel
tests.
Aerodynamic forces include drag (along-wind) forces, which act in the di-
rection of the mean flow, and lift (across-wind) forces, which act perpendic-
ularly to that direction. If the distance between the elastic center of the structure
and the aerodynamic center (i.e., the point of application of the aerodynamic
force) is large, the structure is also subjected to torsional moments that may
significantly affect the structural design.
STRUCTURAL RESPONSE TO WIND LOADS
Because the aerodynamic forces are dependent on time, the methods of struc-
tural dynamics may have to be employed to determine the response. Further-
more, the random character of this dependence requires that elements of the
theory of random vibrations be applied to the analysis. In certain cases. it may
be necessary to perform an aeroelastic analysis, that is, a study ofthe interaction
between the aerodynamic and the inertial, damping, and elastic forces, with
the purpose of investigating the aerodynamic stability of the structure.
From the foregoing it is seen that the design of modern structures subiected
to wind loads requires the use of information and methods derived from a broad
spectrum of disciplines. It will not be suggested here that complete answers to
the questions involved exist at the present time. However, considerable prog-
reSS has been made toward an understanding of some of these questions. As a
result, procedures and techniqucs have been devclopccl lhat have significantly
improved the designcr's irhility to ostinratc thc cll'ccts ol wirttl l-rorn thc stand-
point of both strcrrgtlr ltrrtl scrvicr:lrbility. lt is lht: itittt ol tlris lcxt to prescnt
thesc proccdurcs llul tt't'lrrrit;rrt's, to llrovirlc lltr-: lrirt'kgnrttrrrl trtlttcrial rccluircd
firr rrntlcrsllrrrtlirrg, llrcir r':rliorr:rlc, rrrul lo cxrttttittt' t'rilit':rlly llrt'ir clrplrhilitics irs
wcll rrs tlrt'ir lirtritlrli()ttri it,'i th'si1'.rr (txrls.
PART A
THE ATMOSPHERE
CHAPTER 1
ATMOSPHERIC CIRCULATIONS
Wirrrl, or the motion of air with respect to the surface of the earth, is funda-
rrrt'ntally caused by variable solar heating of the earth's atmosphere. It is ini-
trirlt:tl, in a more immediate sense, by differences of pressure between points
ol t't;ual elevation. Such differences may be brought about by thermodynamic
;rrrrl rncchanical phenomena that occur in the atmosphere nonuniformly both in
tilil(' irnd space.
'l'hc energy required for the occurrence of these phenomena is provided by
tlrt' sun in the form of radiated heat. While the sun is the original source, the
.,()rr'('c of energy most directly influential upon the atmosphere is the surface
,l thc carth. Indeed, the atmosphere is to a large extent transparent to the solar
rrtli:rtion incident over the earth, much in the same way as the glass roof of a
I'rccnltouse. That portion of the solar radiation that is not reflected or scattered
lr:rt'k into space may therefore be assumed to be absorbed almost entirely by
tlrc crrfth. The earth, upon being heated, will emit energy in the form of ter-
rcslrirrl radiation, the characteristic wave lengths of which are long (of the order
ol l0p) compared to those of heat radiated by the sun. The atmosphere, which
r' lrrlgcly transparent to solar but not to terrestrial radiation, absorbs the heat
r:rtlilrtccl by the earth and re-emits some of it toward the ground.
I.1 ATMOSPHERIC THERMODYNAMICS
1.1.1 Temperature of the Atmosphere
'lir illtrstlutc thc nrlc ol'thc lctttgx'r'lrltul'tlistribution in the atmosphere in the
pttxlttt'liott ol'wincls, a sirtrplilit'rl tttrxlt'l tll'atrnosphcric circulation will be
n lMolit't il illo ollrot,l All()Nti
prescntcd. In this rnoclcl thc cll'ccls tll'tltc vctlicitl vrtriittion tll'itir tcttlpcraturc,
of the humiclity ol'thc air, ol'lho nrlirliorr ol lltc eirrllt, artd ol'l'riction will be
ignored, ancl the surfhcc of thc carlh will bc rrssrttttctl to ho unilorm and smooth.
It will be recalled that thc axis ol'nrllliou ol tlrc carlh is inclined at approx-
imately 66'30' to the plane of its orbit an)rrrKl tltc sun (planc of the ecliptic).
Therefore, the average annual intcnsity ol'stllitr ladiation and, consequently,
the intensity of terrestrial radiation and thc lcnlpcralurc of the atmosphere will
be higher in the equatorial than in the polar rcgions. To explain the circulation
pattem that arises as a result of this tempcraturc difl'crence, Humphreys Il-l]
proposed the following ideal experiment (Fig. l.l.l).
Assume that the tanks A and B are filled with fluid of uniform temperature
up to level a and that tubes 1 and 2 are closed. If the temperature of the fluid
in A is raised while the temperature in B is maintained constant, the fluid in
A will expand and reach the level b. The expansion entails no change in the
total weight of the fluid contained in A. The pressure at c remains therefore
unchanged, and if tube 2were opened, there would be no flow between A and
B. If tube I is opened, however, fluid will flow from A to B, on account of
the difference of head (b - a).Consequently, at level c the pressure in A will
decrease while the pressure in B will increase. Upon opening tube 2, fluid will
now flow through it from B to A. The circulation thus developed will continue
as long as the temperature difference between A and B is maintained.
If tanks A and B are replaced conceptually by the column of air abovc the
equator and above the pole, it can be seen that, in the absence of other efl'ects,
FIGURIT l.l.l, ('in'ulittiotr l)irllcnr tluc to tcmpcrattrtc tlillercrtcc bctween two col-
r.rrrrns trl'llrritl. lirrrrrr /'/ry,rlr',r tl tlrt Ait hy W. .1. lltrtrrplttcys. ('opyright 1929, l94O
hy W..l . llrrntlrlrreys. llst,rl wrllr lrcrrrrissiort ol Mc(irrrw llill lixrk ('otrtpany.
W0r rn
I I AIM( )r;t,ilt ilt(; il il ttM()llyNAMt(il;
FIGURE 1.1.2. Simplified model of atmospheric circulation.
rrn atmospheric circulation would be developed that could be represented as in
f iig. 1.1.2. In reality, the circulationof the atmosphere is vastly complicated
by the factors neglected in the above model. The effect of these factors will
bc discussed later in this chapter.
The temperature of the atmosphere is determined by the following processes
lt-2, I-3, l-4, l-5, l-61:
o Solar and terrestrial radiation, as discussed previously in this chapter.
o Radiation in the atmosphere.
o Compression or expansion of the air.
r Molecular and eddy conduction.
o Evaporation and condensation of water vapor.
1.1.2 Radiation in the Atmosphere
As a conceptual aid, consider the action of the following model. The heat
rrrtliated by the surface of the earth is absorbed by the layerof airimmediately
:rlrovc the ground (or the surface of the ocean) and reradiated by this layer in
two parts, one going downward and one going upward. The latter is absorbed
Iry the next higher layer of air and again reradiated downward and upward.'l'lrc transport of heat through radiation in the atmosphere, according to this
t'onccptual model, is represented in Fig. 1.1.3.
1.1.3 Compression and Expansion
Alrrrosphcric pressurc is pnrtlucctl by tlrr-: wcight of the overlying air. A small
rrr:rss (or particlc) of clry irir rrurvirrg vr.rtit'lrlly thus experiences a change of
ptl'srittro to which thcro ctlrtcsltottrls lt cltirnp,e ol'lcrnpcrature. To determine the
l;rltcr. lho oc;uation ol'stirle lirr'1x'r'lt'r'l llits('s iul(l thc lirst law <lf thcrrnodynamics
lrlc ttsctl:
n lM():;l ,lll lrl{i (:llr(:t,t Ail{}t..ti:
Ilcirl rir.lrillrll Irt(,
lriltlr'r !lr.r r.--f --
l,'l(;tlRl,l 1.1.3. Transport of heat through radiation in the atmosphere.
pu-RT
dq:c,flT*pdu
(l. r. r)
(1.1.2)
In these expressions p is the pressure, er the specific volume, R the spccific
constant for dry air, Z the absolute temperature, dq the amount of heat trans-
ferred to the particle, and c, the specific heat at constant volume.
Differentiating the first relation and substituting the quantity p der thus ob-
tained in the second relation, there results
dq:(c,,+R)dT*udp
Comparing this relation with
(r.r.3)
dq : crdT (l .t.41
which cxpresses the first law of thermodynamics in the particular case of an
isobaric (constant pressure) process (co is the specific heat at constant pressure),
it is easy to see that c,, * R : cr. It is therefore possible to write, if the
equation of state is used once more,
dq : cpdT - (l.l.s)
Processes for which dq : O are referred to as adiabatic. For such processes,
the previous relation becomes
dnRT -:_p
qI_4oo _o
T cpP
which, alicr irr(cgltrtion, yicltls tlrc cquation
,1,
't',, (';,)""
(1. 1.6)
I I n l M( 
'r,t 
,ilt ilt{ . I ilt nM( )t )yNn Mt(.1
kttowtt lrs Poissorr'r, or llrt'rlry rrtli:rbalic c(pr:rrr()n. lior tlly ;lt, ll/t.,, o.llitJ.
A lirrrriliar e,x:ttttplr'ol tlrt'e'llecl ol'prcssrrrc r'llrnp,c orr llrt' lcrrrPcrirltrrt: is llrt'
lrt':rtirrg ol'cornprt'ssr'tl :ur irr lr tirc pultrp.
ll', in thc itltttospltr:t't'. tlrc verlical rnotiorr ol'irrr rrir'pirrccl is sullicicntly
rrrpid, thc hcat cxcltiurgc ol'that parcel with its cnvir'onnrcnt tnay bc consiclcrcdt. bc negligiblc and thc assumption dq : 0 is appnrximatcly correct. lt then
Iolkrws from Poisson's equation that since ascending air experiences a pressure
rltrcrcase, its temperature will also decrease. The temperature drop of adiabat-
it'rrlly ascending dry air is known as the dry adiabatic lapse rate and is ap-
proximately 1"C/100 meters in the earth's atmosphere.
consider a small mass of dry air at position 1 (Fig. 1.1.4). Its elevation and
l('nrperature are h1 and z', respectively. If the particle moves vertically upward
irt some reasonable speed, its temperature change will effectively be adiabatic,
regardless of the lapse rate (temperature variation with height above ground)
Plcvailing in the atmosphere. At position 2, while the temperature of the am-
lricnt air is 22, the temperature of the element of air mass is T) : T, - (hz -/r,)'y,,, where 7o is the adiabatic lapse rate. Since the pressure of the element
rrrrtl of the ambient air will be the same, it follows from the equation of state
tlrirt to the temperature difference T5 - T, there corresponds a difference of
tlt:rrsity between the element of air and the ambient air. This generates a buoy-
rrrrcy force that, if rz I Ti, acts upwards and thus moves the element farther
;rway from its initial position (superadiabatic lapse rate, as in Fig. 1.1.4), or,
tl 'l'2 ) Tl, acts downwards, thus tending to retum the particle to its initial
lxrsition. The stratification of the atmosphere is said to be unstable in the first
r'rrsc and stable in the second. If T2: Ti,that is, if the lapse rate prevailing
rrr llre atmosphere is adiabatic, the stratification is said to be neutral.
A simple example of the stable stratification of fluids is provided by a layer.l water underlying a layer of oil, while the opposite (unstable) case would
lurvc the water above the oil.
I: Lapse rate prevailing
in the atmosphere
Adiabatic lapse rateII
2(h2, r;)
(l l7) lrl(ltlRl'l I l..l= l,rl,i.( r,rt(.s
1 0 A I MO:lt,l tf ntc otttct,l A t t( )Nr;
1.1.4 Molecular and Eddy Conductlon
Molecular conduction is a dill'usion pft)ccss thlt cll'ccls a translbr ol'hcat. It is
achieved through the motion of individual rnoloculcs and is negligiblc insofar
as atmospheric processes are concerncd. Hddy hoat conduction involves the
transfer of heat by actual movement of air in which hcat is stored.
1.1.5 Condensation and Evaporation of Water Vapor
The pressure of moist air is, according to Dalton's law, equal to the sum p of
the partial pressure e of the water vapor and that of the dry air, p - e. It has
bccn established experimentally that if the pressure e exceeds some value E,
known as the saturation vapor pressure, condensation of the excess moisture
will occur, and that the saturation pressure E increases exponentially as the
tcmpcrature of the moist air increases.
An elementary mass of ascending unsaturated moist air (i.e., for which
e/E < l) will experience a temperature drop that can be shown to be essentially
equal to the dry adiabatic lapse rate. As the element ascends and its temperature
decreases, its saturation pressure will also decrease. If the element reaches a
level at which the ratio e/E becomes unity, condensation will normally occur.
Above this level, water vapor contained in the air element will continue to
condense. In the process, heat of condensation is released. This is equal to the
heat that was originally required to change the phase of water from liquid to
vapor, that is, the latent heat of vaporization stored in the vapor.
The heat of condensation contributes to the mechanical work involved in
the expansion of an ascending particle, which before saturation was performed
only at the expense of the internal energy. The temperature drop of the saturated
adiabatically ascending element of air is therefore slower than for dry or moist
unsaturated air (Fig. 1.1.5). By furnishing energy that increases the temperature
\ to,r, {or saturation)
\diabatic 
lapse rate
l,'l( ;t llll,) l. l.-5. l illtt'(s ol' corttlcrtsirlion ulxrr Lrlrsc nrlc l.'l(;tlltll 1.2.1. Vcrtic:rl Irr':,:.rrr.s ()r irr e lcrrL)nrrrry nrirss .l uir.
\ lM( )t;t,nt nt( , I tytrn( )t)yNn Mt(;ti I I
ol'a parliclc: witlr n.spcr'l lo wlrat it woultl hirvc lrccrr urrtkrr tlrv arliabirtit.
cottclititlns, tltc hcitl ttl cttttdcttsation hclps sul)lxrt1 corrvct.liorr ol thc 1ir t9
lrighcr levcls ol'thc ltlntospltcrc. 'l'his lactor is irrrpollirrr( irr tlrc gcncsis ol'cqlain
typcs ol winds.
1.2 ATMOSPHERIC HYDRODYNAMICS
'l'hc motion of an elementary mass of air is determined by Newton's second
l:rw
DF : ma (1.2.1)
where m is the mass, a is the acceleration, and D F is the sum of forces acting
'rr the elementary mass of air. It is the purpose of this section to briefly describetho forces F and some of their effects upon the motion of air.
1-2.1 The Horizontal Pressure Gradient Force
('onsider an infinitesimal volume of air dx dy dz (Fie. r.2.r), and let the mean
l)rcssuresacting on the lower and upper face be p and p + (0pl0z) d1, respec_tivcly. In the absence of forces other than pressures, the net vertical force
rrcting on the volume dx dy dz will be -(0pl0z) dx dy da, or -)pllzperunitvolume. similarly, the net forces per unit volume acting in the "r and y directionwill be denoted -\pl0x and -0pl0y, respectively. The resultant of these forces
is called the horizontal pressure gradient and is denoted -0pl0n, where n isthc normal to some contour of constant horizontal pressure. The horizontal
l)rcssure gradient is the driving force that initiates the horizontal motion of air.
12 n lMolil'l ll lil(; cllt(;l/ln ll()N:i
Htglr llrcsltrtrr
An
I o,r".,,.,, ol pressurc gradrent
Low presure
FIGURE 1.2.2. Direction of prcssurc gradient fbrce.
'l'hc nct lirrce per unit mass exerted by the horizontal pressure gradient, (l/p)
t)1tl\n, is oltcn referred to as the pressure gradient force (p is the air density.)
Air subjccted solely to the action of pressure gradient forces will move from
rcgions of high pressure to regions of low pressure. The direction of the pres-
sure gradient force is indicated in Fig. 1.2.2, in which the isobars (lines con-
tained in the same horizontal plane and connecting points of equal pressure)
are also shown.
1.2.2 The Deviating Force Due to the Earth's Rotation
If defined with respect to an absolute frame of reference, the motion of a particle
not subjected to the action of an external force will follow a straight line. To
an observer on the rotating earth, however, the path described by the particle
will appear curved. The deviation of the particle motion from a straight line
fixed with respect to the rotating earth may be attributed to an apparent force,
the Coriolis force, the vector expression of which is [1-7]
F,.:2m(v x a) (t.2.2)
where m is the mass of the particle, or is the angular velocity vector of the
earth, and v is the velocity of the particle relative to a coordinate system rotating
with the earth. Fc is perpendicular to crr and to v, is directed according to the
vector multiplication (right-hand) rule, and has the magnitude 2nlc,rl lvl sin a,
where a is the angle between o and v.
Let N (Fig. I .2 .3) be the norlh pole, and consider an element of air moving
in a straight line in space along the direction NP. If the motion starts from N
at time t : O, at time / the particle arrives aI P, and the position of the meridian
along which the motion started is NP'. To an observer on the earth, it appears
that the element is deflected westward by an amount P'P.
It can thus be seen that, in thc Northern Hemispherc, owing to the rotation
of the earth, a wind initially dircctcd along a mcridian vcors to the right of its
initial dircction; (lurl is, il tlirr:ctctl northwarcl il vc:crs lowurtl thc oast (bccomcs
a wcstcrly wintl). ll tlilct'tctl sottlltwatd il vccrs lowrrnl llrc wcst (bccorncs an
I 1' n I M{ t!;t't il !tl( I tyl)t l( )l )vNn Mt( tli l3
I-IGURE 1.2.3. Apparcnt rnotion ol'an air particle due to
the earth's rotation.
t';rstcrly wind). In the Southern Hemisphere the reverse of these statements is
llllc.
ll' the Coriolis parameter is defined as
f:2llsind (r.2.3)
wlrc:rc d is the latitude of the point considered, it follows that the Coriolis force
;rcting per unit of mass in a plane (P) parallel to the surface of the earth (Fig.
I 2.4) on an element of air moving in such a plane with velocity v relative to
tlre carth will have the magnitude
F, : mfu
f lrc values of f are given in Table 1.2. 1 as functions of latitude.
(1.2.4)
fll(;lllll,l 1.2.,1. l 'r,ntlr,lr.trl:, ol tlrt'lrltrlion vccl()r (').
14 n lM()l;l'l ll lllc ()ll l(;t,l n ll()Nl;
TABLE 1.2.I. Corirllis l)ararnclcr.
Latitude
(deg)
f:2asinb
(s -')
l-ittitudo
(rlcg)
.f -- 2a sin $(r ')
0
5
10
r5
20
l5
t( )
l'r
,lo
.ll
0
0.1211 x l0 a
0.2533
0.3'175
0.49ttu
() (rI(r4
( ) /:l().1
O lJ l(r\
o (n/1
I Oll\
1.1172 x 10-4
t.1947
r.2630
r.3218
I .3705
1.4087
t.4363
1.4629
1.4584
-50
5.5
()( )
()-5
10
15
tt( )
1{-5
9o
1.2.3 The Frictionless Wind Balance
At sullicicrrtly grcat hcights, the eft'ects on the wind due to friction along the
ground bccome negligible, and the horizontal motion of air relative to the
surface of the earth is determined, in unaccelerated flow, by the balance among
the pressure gradient, the Coriolis, and the centrifugal force.
The effect of the forces acting on an elementary mass of air is shown in
Fig. 1.2.5 (the mass is assumed to be in the Northern Hemisphere). If the
particle started to move in the direction of the pressure gradient force (denoted
P), it would be deflected by the Coriolis force F.o @ig. I .2.5a). The particle
would then move in the direction of the resultant of P and F..o, shown as
direction II in Fig. 1.2.5b. The deflecting force would now become F,.6, to
which there would correspond a new direction of motion (direction III in Fig.
1.2.5b). When a steady state is reached, the wind flows along the isobars as
shown in Fig. 1.2.5c.
The isobars in Fig. 1.2.5 are depicted as straight, which means that in the
case shown there is no centrifugal force. However, in the more general case
of a curved isobar centrifugal forces will be involved. This case is taken up
below.
The steady velocity for which a balance between the pressure gradient force
and the Coriolis force alone obtains is called the geostrophic wind velociry G
and is related to the pressure gradient by the equation
do/dn2u:Csin6:P:-'
p
(1.2.s)
,, .,, 'Q!!'pl' (t.2.6)
I ;' n I M( ): il ,t lt ||t( il\ I rlt( )l ryNAMt(:l
High pressure
Low pressure
(b) (c)
FIGURE 1.2.5. Frictionless wind balance in geostrophic flow.
rvlrcrc P is the magnitude of the vector P,/is the Coriolis parameter, and p is
tlrt'lrir density.
I l' the isobars are curved (Fig. 1 .2 .6) , the force P as well as the centrifugal
I'r.cc C will act on the elementary mass of air in the direction normal to the
r:,.birrs, and the resulting steady wind will again flow along the isobars. Its
High pressure High pressure
ll'rr\:tion Of
wrrrrl (in the
N,rr tltorn
I l"rrst)here)
Direction of
wind (in the
Northern
Heqrisphere)
Low preisurc
(cycl<xric circulirlr0il)
(n)
Itl(llllll,l 1.2.(r. lrlictiorrlt.ss rvrrrrl lr;rl;rrri r
Low pressure
{anticyclonic circulation)
(h)
ttr ( y( l()ni(' lurtl lrrrlit'yr'lorrit' llow
16 n tM()t;l'l tt Ittc otti(;lll n ll()Nl;
velocity results fiom the relations
vn,f -r yl : ,"r _ dp/dnp (1.2.7)
where, if the mass of air is in the Northern Hemisphere, the positive or the
negative sign is used according as the circulation is cyclonic (around a center
6fl<tw prcssurc) or anticyclonic (around a center of high pressure), and where
r is tlrc nrrlirrs ol'crtrvalurc of thc airtra.icctory.* The velocity I/r,is calledthe
.tittttlit,ttt rt,itttl tt,lttt'ilv; it is ct;ttal ltl thc geostrophic wind velocity in the
p;rrtrt rrlrrr t:rst' irr wlrit'lr lltc ('tlrviltlllc tll' (hc isobars iS zerO. If the radiuS of
r'llr\';ll1!(' rs ltrvlt'. trr lllt' Nol'tltt'r-rt I lt:rrtisphcrc
'{ 'l',':: - ({)')'' (1.2.8)l';,,
lirr cyckrrric wirrtls, atrd
vr,: t{, -le)' -;#)" o 2s)
for anticyclonic winds. The sign of the radicals is given by the condition that
Vc,: Owhendp/dn: 0. It follows from the expressions fot Vr, that for the
same values of r, f and dp/dn, anticyclonic winds are weaker than cyclonic
winds [1-1, p. 121].
The foregoing discussion explains Buys-Ballot's /aw, which states: If, in the
Northem Hemisphere, a person stands with his back to the wind, the high
pressure will be on his right and the low pressure will be on his left. In the
Southern Hemisphere the reverse is true.
1.2.4 Effects of Friction
The surface of the earth exerts upon the moving air a horizontal drag force,
the effect of which is to retard the flow. The effect of this force upon the flow
decreases as the height above ground increases and, as indicated previously,
becomes negligible above a height 6 known as the height of the boundary layer
of the atmosphere. Above this height the frictionlesswind balance is estab-
lished, and the wind flows with the gradient wind velocity along the isobars.
The atmosphere abovc thc boundary layer is called the free atmosphere (Fig.
1.2."t).
*Strictly sllctrkirrl'., llrc r:rtlirrs ol ( lr v:r1lrt' ol lllc triricct()t] rtr:ry tlillt'r lirrltt lltc radius ol'curvaturc
ll llrc isglr:rr'.'llrc tlilltrclr,r.rrlrV lrc lreliletlctl, ltowrrvt'1. il tt r':rtt lx':tssttlttctl th:rl lhc wintl llrlw
is :rllptrrx irttitlt'ly slt':trly
A I Mil'.t'l il ltt( I li I rl to| ryNn Ml( :: l7
S (fricliorr lorr:r')
Irl',rlltoil!l!'t('
(;r,r(il,nt wrnrl luvtl
High pressure
Low l)ressure
(h)
,tt IItr':,1)lt('t i(. lroulrtlltt.y l:tyt.r
-layer depth)
FIGURE 1.2.7. The atmospheric boundary layer.
It is the wind regime within the boundary layer of the atmosphere that is of
'lrr.ct interest to the designer of civil enginlering structures. The questions oftlrt' boundary-layer height, of the variati,cn of wind speed and direction withlr.rrht above ground, and of the turbulence structure within the boundary layer.ut' therefbre discussed in more detail in Chapter 2.
It will be noted here that unrike the gradient wind velocity, the steady_stater'irrrl velocity within the boundary layer crosses the isobars. consider a geo_''t.rphic flow (i.e., a flow in which the isobars may be assumed to be straight)
'rrrtl the balance of the forces acting on particles A and, B, which move hori_z'rrrirlly within irs boundary rayer (Fig. t.z.s;. If ,4 (Fig. r.2.ga) is at a higherl('vcl than B (Fig. 1.2.8b), its speed u and (by virtue oithe relation F, : mfu)rts ('.riolis force will be largerthan those orb. tre deviation;;gr; " betweentlrt'wind direction and the isobars will therefore be smaller fir ttre highertl;rs(cr) particle. The angle o will be zero at the gradient level and will reachrts.rrraximum value os nearthe ground. In the Northern Hemisphere the windr.ltrcity in the boundary rayer may thus be represented by a spiral, as in Fig.l.).9.
High pressure
F" (Coriolis force)
l)[oction A
It()lt()rr
P {Jlressrrrc r;r;rlrr.rrt lorr r.)
Low l)t{:ssrlr(:
(r)
lf l( ll lltl,) l.2.ll. llrl:rrrt'c ol lor, {.., rr tlrr
18 ATMOSPHERTC ctRCUl AroNsi
FIGURE 1.2.9. Wind velocity spiral in the atmospheric
boundary layer.
In the case of a cyclonic storrn (or flow around a center of low pressure),
near the ground, the wind will cross the isobars toward the center. The air will
thus slowly converge and ascend. If the low-level convergence exceeds high-
level divergence, the mass and weight of the air column at the center of the
storm gradually increase and therefore the inward-directed pressure gradient
force decreases. As a result of such a decrease, the center of low pressure is
dissipated and filling is said to occur.
In the case of an anticyclone, the wind near the ground will cross the isobars
away from the center of high pressure. In the lower portions of a high, if low-
level divergence exceeds high-level convergence, the atmosphere will tend to
spread out and sink, and dissipation of the center will occur.
1.3 ATMOSPHERIC MOTIONS
Most atmospheric pmcesses can be described in terms of the quantities briefly
discussed in the prcccding sections: wind velocity (i.e., horizontal and vertical
wind), pressurc, tcmpcrature , density, and moisture. The behavior of these six
quantities is govcrncrl by six cquations: the equation of state, the first law of
thermodynanrics. tlrc ctluirtirtns <ll'continuity of mass and moisture, and the
horizontal antl vcrlicll ccprations ol'tnotion. Proviclccl that an adequate data
basc cxists, llrr:sc r:r;rrirlit)ns clur bc intcgratcd to yicltl it t;tutrttitative dcscription
ol itttttosphcric t'tttttltltrttts itl s()tnc shorl tittre irltcl llrc t.ollecli6rr el'llrr-r rlirlir.'l'hc calculatctl vitlttcs ol lhc six variablcs obllirrctl by irrtcgnrtiorr can (hcrr lrr.:
ttsctl as initial corttlitiotts lilr a l'urthcr inlcgnrtiorr;;11rp. 'l'his succcssivc ap-
Pfilxitnation pK)(:L!ss is lhc lrasis of numcrical wclllrcl procliction tcchniquoslllirl came into bcing lirllowing the increasecl availability ol- observations-
rncluding, more reccntly, observations obtained by satellites (Fig. 1.3.1)-and
tlrc: clevelopment of modern electronic computers.
Atrnospheric motions may be described as superpositions of interdependent
lkrws characterized by scales ranging from approximately one millimeter to
llr.usands of kilometers. To analyze such motions, it is convenient to classifz
llrr:rn according to their horizontal scale. In meteorology three main groups of
rrtrnospheric scales are commonly defined: microscale, mesoscale, and synoptic
st'irlc. According to the classification of Il-g], the synoptic scale includes mo_
tions with characteristic dimensions exceeding 500 kmor so and time scalesol two days or more. The microscale includes motions with characteristic di-
rrrc:nsions of less than 20 km or so and time scales of less than one hour. The
nrcsoscale is defined by dimensions and periods between those characteristic
ol' rnicroscale and synoptic scale.
lrlOtIRli 1.3.1. Srrtcllitc
( )ccanic ;rrrtl Alrrrosplrcric
vicw ol lrrrrrrt'irrrt' lrili
Ailtrrtrrisl rrrl rnrr )
r;r AIM()i;l't r nt(: M()lt()Nli 1g
Se pl . 113. 1974 (coLrnosy National
20 n lMosl'l ll lll(; (;lilc(ll All()N:;
Nortlr Polc
Polar easterlies
Polar front
,z .z .z
t/ tt t/ t/
l,'l(,llil{l,l 1..1.,1. lltt'llrrlrrr rrrt'ritliorrirl r'ircttluliott rnodel. After General Meterology
lry ll l{. llyt'r:. ('opyrill,lrl l()17, l(4,1 hy llrc Mc(iraw-Hill BookCompany, Inc. Used
wrllr 1rt'nnissiort ol Mt ( ilrrw I lill lltxrk ('olrtpittty.
1.3.1 The General Circulation
The combined effects of the earth's rotation and of friction break the thermal
circulation cell of Fig. l.l .2 into a pattern that consists basically of three
circulation cells as represented in Fig. 1.3.2 [l-2]. The theoretical pattem is
compatible with the existence (at sea level) of a high pressure belt at the horse
latitudes and of a low pressure belt at the polar front.
In reality, the tricellular meridional circulation model is complicated by
seasonal and by geographical effects. Seasonal effects consist of variation in
position and intensity of the pressure belts and are caused by the annual march
of the sun north and south of the equator. Geographical effects are caused by
the difference in physical properties and by the uneven distribution of water
and land over the globe.
In summer, because the ocean surface warns up more slowly than the land,
the air is colder over the ocean than over land. Just as in Fig. 1.1. I fluid flows
in tube 2 from the colder to the warmer tank, air near the surface will be driven
in summer by a pressure gradient force directed from the ocean toward the
land. On the other hand, in winter the air is colder over land and the oceans
become heat sources.
1.3.2 Thermally Direct Secondary Circulations: Monsoons and
Hurricanes
Secondary circulations are said to be of the thermally clircct typc if the centers
of high or low pressure (i.e., the highs or the lows) aroutttl which thcy develop
are formed by heating or cooling of the lower alttros;rhe rc.
Monsoons, Monsoor.rs itrc scirsorral winrls tlurt lirrrrr r'ells ol (lrc gcncral cir-
c:ulalion irntl tlcvr:krp rrrtrttntl llrclrrurlly prrxlttt'etl t'ottltttr'ttlitl Irighs irt wiltlcr
I:r AIM{}!,t ,tililt(;M()Il(iNl; 2l
;rtttl ltlws itt sttttttttct. ()wirrg lrl tltc vlrsl llrtttl rturss ol llrt.Asi;rrr r'onlirrt.rrl,
lll()llso()ll cllccts irl.r'(l('v('lolx'rl ttxrsl strorrgly irr Asitr. wlrcrc llrcy lurvt: tr t'olt
sitlcnrhlr: inl]rrcltct' orr lltc sr:lrsonal charrgcs ol welrllrt.r l)itllcnrsi.
Hurricanes. -lr<lpiclrl cycklncs are storms that dcrivc all thcir cnorgy I'r<lrn thc
llrtt:nl heat relcascd by tho condcnsation of watcr vapor and <lriginatc, gcncrally,
lrt'lwccn the 5 and 20 latitude circles. Their diamctcrs arc usually of the order
,rl' scvcral hundred kilometers. The depth of the atmosphere involved is of the
orrlcr of ten kilometers. Hurricanes are definedas tropical cyclones with surface
wirrd vefocities exceeding about l2o kmlhr. Spacecraft views of hurricanes are
slrown ln Figs. 1.3.1 and 1.3.3.
Hurricanes (known as typhoons in the Far East and cyclones in the region
ol Australia and the Indian Ocean) occur most frequently during the late sum-
;1*W*d,*
lrl(;llltli l.-f,1. Ilrrrlit'ilrc (illulvr ir, .,(.r.u l)v
( )t t':rrrit' lrntl A( rrlrsplrt'rit' Atlrr rirrr:,1 r ;rl inrr I
llrt' Apollo crcw (cotrrtcsy Nlrliorrirl
lrl( ll ll{l,l 1..1.:1. Mcrrrr tlircctions ol' hurricane motions [1-9].
lnor ancl oarly autumn months (August-September on the Northem Hemisphere,
February-March on the Southern Hemisphere), except in the Northern lndian
Ocean. Hurricanes normally travel as whole entities at speeds of 5 to 50 km/
hr. The mean directions of hurricane motions are shown in Fig. 1.3.4. It is
noted, however, that individual hurricanes may follow unusual, indccd erratic,
paths. World tropical cyclone statistics are presented in Fig. 1.3.5 ll-91. Data
on tropical cyclones reaching the United States coastline are presented in some
detail in Sect. 3.3. For detailed basic information on hurricanes, scc Il-10]
and [3-62].
As seen in a vertical plane section, the structure of a hurricanc in the mature
stage consists of five main regions, represented schematically in Fig. 1.3.6, in
which approximate dimensions are also shown. Region I consists of a nrughly
circular, relatively dry core of calm or light winds, calk:cl thc cyc, around
which the storm is centered. The air rises slowly near thc pcritnctcr of thc eye
and settles in its center. Region II consists of a vortox in which warm, moist
air is convected at high altitudes (by the thermodynantic Incchanism discussed
in Sect. 1.1) and forms tall convective clouds. Conrlcttsulion of water vapor
occurs as the moist air rises, and this results in intcnsc lirinlirll and thc rclease
of vast amounts of latent heat. It has been estimatctl lhitt lltc cttrrtlcnsation heat
energy released by a hurricanc in one hour may bc ctltrivltleltl lo thc clcctrical
energy used in the cntirc Unitcd States in onc wct:k ll t)1. 'l'lrc 1ir l'lows out
of region II into an outllow ltrycr (rcgion lll). ln lcgiorr lV llrt' llow is vortcxtlike
and settles vcry sl<lwly iltlo tllc lltlunclaty lltyct'tcgiolt V. llclow rcgion II,
where strong uptlrllts i1i. l)t('sc1l , scl)itfttti()rl ol'lltt'lrottttrl;rly l;rycl-lllily occur.
According to (illrlrlutr :rrrtl llrrtlson ll l ll :rrr r'xlttcssiott ol (lrc lirl'nr
n IM():ll'l il lil(; (;llr(;t,l n ll()Nli
I tllt
1t ,lt ( ','
1,, \ /{,,, ti", rI t'l) / I
(r.3.r)
(llllotlr t,xl,rl[,r.r^V
ai
(.)
o
o.
ra
(.)
14
p
f,i o@>@
o:-
_COc -O
oz
I S R; Pf tslS-r:HSits;i:l vs.!:Eeg5 3n-1i-.
-o C q V " h.:q
o -!biEao i-\i f o f,u9Qoo<j
o@'io
9 *-
E 3>f
a cc:
- rf i
-O s
l;\
,9:@
{ l,'i r
:U@L@o-z
i; oU.o oc:.g o?
I O a6
i di";i1alr
; oi@; uo
-O-
Ud
3=3>l'3P
6 :+c:o Y!r.. F:I
b ^ 
tn'j
'll
at
o
I
z
O
.l>
1:
t:
i:
l?
l=
l"i:
l;
l3l^
lo
l:
l>
;
z
o
:
;
qo
OO
crooo,o rj .cj v ;
":-ON
oc>o
ug.EaEaso o= 9=- oo vo
!'3:;t;I;6r-9f,9b'u<=<"'I I
Ohci-ri
>^ 6 O
3-g a b
X'6 o I qbs> € ii,;> o 6 a c^<i@ < >5lt;lri
\
)
24 n lM()..il,l ll lil(; (;llt(;(,1 n ll()Nli
h (km)
tt (krrr)
lil(lllltl,l 1..1.(r. Slrrrcturc ()l' a hurricane.
in which p6r is thc pressure that is approached as the radius r + @, p0 is the
pressure at the center of the hurricane eye, (pn - pdlp is assumed independent
of height, and Ra is twice the radius of maximum dp/dr, is fairly representative
of typical hurricane pressure fields. If this description of the pressure gradient
field is used, the gradient wind velocity field results from the expression of the
gradient wind (valid for cyclonic winds) derived in Sect. 1.2. There results
from this expression that the gradient velocity reaches a maximum at a radius
of the order of R.. From this radius the velocity decreases rapidly to zero at
the center of the eye, and more slowly to the relatively small values that obtain
at large distances from the center.
While the gradient wind velocity is directed along the isobars (see Fig.
1.2.5c), in the boundary layer the wind velocity has a radial component directed
toward the low pressures, as was shown in Sect. l.2.It is this component that
effects the inflow of the warm moist air at the ocean surface into region II,
thereby maintaining the supply of energy of the storm. Over land the dissipative
effect of friction increases, while the supply of energy in the form of warm
moist air tends to be cut off. As a result tropical storms over land usually fill
up within a few days at most.
The destructive effects of hurricanes are considerable and are due to the
direct action of the wind-which may reach peak surface velocities of 250 km/
hr or more-and, usually to an even larger extent, to the massive piling up of
water by the wind known as storm surge, together with flooding by heavy
rainfall (Fig. 1 .3.7).ln 1992 Hurricane Andrew alone caused damage estimated
at more than $20 billion I l-121.
Arctic Hurricanes. An'tic' lrrrrr.icltncs tlcsignulc polirr krws with a sylnlnctrical
cl6utl signittrrr.t'iurtl winrls ol ;rl lt';rst.lO rrt/s (5ll krrots), wlriclr rcgulitrly cxccccl
llrr: corrvenliorlrl llrrr'slroltl lol lrrrn'it':utc lirt'cc witttls. ;rrtrl irt wlriclr lltrxcs <ll
T-
6x-
I
l0
I
'l'
)ll
I r A I M{ ,:,t,t il ilt(; M( ) ll( )Nl; 25
.::a'!4
:fS-*;*.J..''r
Ed {*$sw
L*il
1ffi,j+i$iffi
l.'l(;uRE 1.3.7. Hurricane damage, Mississippi (courtesy National oceanic and At-
rr rospheric Administration).
lrr:at at the sea surface are largely responsible
l('nance of the storm Il-131.
1.3.3 The Extratropical Cyctone
srrch circulations are produccrr cirrrcr hy rhc mcchanical action of mountainlrltrricrs on large-scale atmospltt'r'ic crrrrcrrls, ol by the interaction of air masses
:rlong fionts. An examplc rll'tlltrttirgt't'rruserl by an cxtratropical storm is shown
rrr Iiig. 1.3.8.
Air tnasscs arc charitclct'izcrl lry rt'lrrtrvt'ly rrrrilirlrr physical propcrties overlrolizontal tlislirnccs colrrllrllrlrlt'ln 111,.,1,,,,,.,rsiorrs ol'.ra"un, rlr contincnts.'l'lle ir physicitl PtrrPt:t'tics lttc;tcr;ttttr'rl ur tlr(' :,our'( (. regiorr arrtl nr:ty 5c rr16rlilicrlrlttl'irtg strbscrlrrcrrl ll'irvt'l ol llrr.;rrr rrr;r.,r, Arr rrr:rsst,s lrury be: r.llrssilicrl, lrc
"lt
.t
l
f,ii "-e- q'
',lE-* k
kilr- .-- !t*;E; t
#Sr
r.i*si',,.: .'4&''.,
for the structure and the main-
26 n l Mo:itllil iltc (;ilrot,l A lloNl;
FIGURE 1.3.8. Damage caused by winter storm, Fire Island, New York, March 7,
1962 (courtesy National Oceanic and Atmospheric Administration).
cording to the source region, into three main groups: arctic, polar, and tropical;
each of these may in turn be divided into continental and maritime. Continental
polar air, for example, is dry and cold, whereas maritime tropical air is rnoist
and warm.
Transition zones between air masses are called frontal zones. The variation
of the physical properties of the atmosphere across frontal zones being fairly
rapid, the latter may be idealized as surfaces of discontinuity known as f-rontal
surfaces. The intersection of a frontal surlace with a surface of equal elevation
with respect to the sea level is called a front.
The equilibrium slope of the front between two air masses can be calculated
approximately on the basis of simple hydrostatic considerations and varies
normally between l/50 and l/400.
A front is rclcrrcd to as a <'old.f'ront or as a warm .frrn.l (Fig. 1 .3.9) according
as it movcs in thc tlircctiott <tl'thc wanlcr tlr coltlcr irir. Gcncrally, a warm
l-r<lnt rrrovcs slowly irntl is rro( rrssociir(crl wilh violclrl wt'itlhc:r conclitions. On
llrtr othcr hturtl, ir t'olrl lirrrrl ciur nrovc: rapirlly ltttrl r';tttsc s('v(:r'(: wcitlhcr. Irrc-
I.r AIM{r:il,lilIil( t\,4o111 ;11, 27
Wtttttt + -/Cold
Wrrrrr- C*""ll
Warnr lrcrrl slopr: Colrl lrorrl slo;tc
Ir'l(;tJlll,l 1.3.9. Warm and coltl lirrrrt skrps.
rltrcntly ahead of cold fronts squall lines developthat may be associated with
lruge thunderstorms and with tornadoes. The disturbance of the temperature,
vclocity, or pressure gradient field may cause wavelike perturbations on the
lhrnt that propagate as waves in a continuous medium. Major disturbances may
('iruse waves whose amplitudes increase with time and develop into intense
vortices. The formation and development of the most intense large-scale cir-
t'ulation in middle latitudes, the extratropical cyclones, is connected with such
rrrrstable waves occurring predominantly along a front. on the average, the
t'xtratropical cyclones move eastward with velocities of the order of 20 km/hr
irr summer and 50 km/hr in winter.
1.3.4 Local Winds
'l'hc influence of small-scale local winds on the general circulation is negligible.
ll.wever, their intensity may sometimes be considerable and in certain cases
f.ovcrn the design of buildings or structures.
Foehn winds. Air ffowing across a mountain ridge is forced by the mountain
skrpe to rise. If the air ascends to sufficiently great heights, condensation and
prccipitation due to adiabatic cooling will occur on the windward side. After
lurving thus lost most of its initial water-vapor content, the air passes over the
t |cst and is forced to descend. Consequent adiabatic compression results in
lrigh temperatures of the dry descending air. An example of a foehn wind is
srrggested in Fig. 1.3.10.
-5"C
_,f;-l_
l^ ( ,1 ' ) -.,. r-3,ooo m.L\- ,,r' ) V,"::i\,( ,/
ilt&" 
"',))_
l,'l(ll lltl'l t..1, ll). lror.lrrr rvirrtl.
+15"C
28 AIMosPHlBtcciltculAlloNS
In the United States intense and highly turbulent winds of the fbehn typc,
called chinook winds, develop on the slopes of the Rocky Mountains. In winter
chinook winds are notable for bringing sudden high temperature rises, with
unusually rapid dissipation of local snow.
The Bora. The adiabatic heating during the clescent of a very cold mass of
1ir that has passcrl ovcr il rn<lunlain hitrricr tlr it plitloittt tltily nol hc sullicient
to t'llrrrgc it irrlo it wiu'nr wintl rll'tltt'loclttt ly1rc. As tltc still crlltl air falls
grirvrl;rlror!irllv irrlo llrt'wrrlrrrcr rcgton ou lltt'lr'r sirlc, ils polr'rrlilrl cncrgy is
I ptiliFtlr-rl itrll kittr.lir. 11!lrlrgV Wttrrll nl r.rlrr.rrrt. rnlcrrsi(y tuit.y lhus bc pro-
'lu' e,l. r ltrltnr leti:+:rl lr1, g_ii,ilr t'l l5(l ,r(Xl ktrr/lrt sr'rntirlr'tl hy pcriotls ol'calm.
Wlttrl;q ol tlte !rrrtrt ty|e rrr r ur llt rttr-ilri rvltr'!t' tt slcr'p slopc scpitratcs a cold
Flitlt,:Il ltlrll !l tr.taal lr!!t!i! Airrnrrp- llrr-lreil kttowtt lrorit wittds arc thosc that
rrl r ur il ial:-;rlr- erirrl f'liiilrr- uir lllr: irrlllrlir:;l crursl ol tlrc Adriatic.
Jel FJtetI Wltlcl*. I lrt- ir'l rllrt'l t'urrsisls ol irrr int'rcasc in wind intcnsity due
ln tupngrirplrtr rrl r orrliplrirrlrorrs llrirl pirrtlttt't lr ('onvcrgcnce of streamlincs. The
Intrrti;rl rvirrrl ol llrr'lowct ltlrtittc Virllery irt soutltcrn France is a wcll-known
r'rirnrplt' ol ir lxrnr winrl inlcnsilictl by .yot cllbcts.
Thunderstorrns. A necessary condition for the occurrence of thundcrstoms
is thc fbrmation of tall convective clouds produced by the upward motion of
woffi, moist air. The motion may be started by thermal instability or by the
presence of mountain slopes or of a front. Thunderstorms are classified ac-
cordingly as thermally convective, orographic, and frontal.
If condensation of the water vapor contained in the ascending air produces
heavy precipitation, viscous drag forces exerted by the rain on the air through
which it falls contribute to the initiation of a strong downdraft. Part of the
falling water is evaporated in the underlying atmosphere that is thus cooled
and therefore sinks. The cold downdraft spreads over the ground in the manner
of a wall jet (i.e., a flow caused by a jet impinging on a wall) and produces
squally winds. This stage in the life cycle of a thunderstorm associated with
strong downdrafts usually lasts from 5 to 30 min and is called the moture stage
ll-141. As the energy supplied by the updraft is depleted, dissipation of the
thunderstorm occurs. A schematic vertical cross section through a thunderstorm
cell in the mature stage is shown in Fig. I .3 . I 1 . Characteristic of thunderstorms
is the sharp wind speed increase, known as Jirst gasl, which is associated with
the passage of the discontinuity zone between the cold downdraft and the
surrounding air.
Tornadoes. Tornadoes contain the most powerful of all winds t1-151. A tor-
nado consists of a vortex of air, typically of the order of 300 m in diameter,
that develops within a severe thunderstorm and moves with respect to the
ground with spceds of the order of 30-100 km/hr in a path, approximately 15
krl long, clircclcrl prcdorninanlly toward the northcast. Thc maximum tangen-
\ 
l)rroction ol movement
\-\
FIGURE 1.3.11. section through a thundersrorm in the marure stage Il-14].
tial speeds of tornadoes have been estimated to be of the orderof 350 km/hr,but the possibility that some may actuaily be considerably higher has not beennrlcd out.
Tornadoes are observed as funnel_shaped clouds (Fig. I .3.12). The tangen_(i.l speeds are probably highest at the iunnel edge uid d.op off toward theccnter and with increasing distance outside the funnel.
l .r n tM( )t;t,ilt iltc M()il()Nli 29
krn
2
l|l(;tjRlJ 1.3.12- 'lirrttitrlo ltttttt('l (utllrl('ry N:rtiorr:rl()t'crrrric irntl Atrrlrsphcric Arlrnin-islrilli()lt).
30 ATMOSPHERIC CIRCULATIONS
FIGURE 1.3.13. Balance of forces in tornado vorrex.
Slttt t' {ltt' t'r'trltilirliirl lott'cs itt llrt' lolrurtlo v()t1cx l:u' cxcccrl lhc Coriolis
lrtlrt=',. llrt'l;tllr'r ttt;tv lrt'ttt'1',lrt'lrrl;ur(l llr('1ir;rtlrrlrl wirrrl t.t;rurlir)n (scc Sect.I .'l rri:rt lrt- rr,t illr.rr :r',
I
tlt, t' l 1,. Ltlrrrit ili, lll{. r yr ll,,ltn1rllir
lirtitt llti r rillr-t nl ll!r. t Iilr=r, l, t', lltr,
IrLiIIIFIrI it|ilit!' IItr' t;trIttt',
tllt ,1,
lt
(1..r.2)
I rirrl | 11' I I I t, \\'lurlr r('l)r('r.t'rrl:; rlrc lon.t.s lrc(irtg on a l)lrrticlc in alilllt;lrlo \'iltlt'\, tl r:ttt lrt' rr'r'rr llritl llrt' ;rrt'sstn' ilt lt t<lrnad<l (locfcilr,ics lowardll5 ( ('lllt'l Iltr' rllllt'tt'trt't' lrt'lwt't'n llrt' plt'ssrrlc a( thc ccntcr iyxl at a f'ew
Irrrrrrlrt'rl lt't'r lr,rrr rlrt' t'cnrt:r' .l rlre v()11cx .ury bc as high as 0. I ol' one
irlrrrospltt'n', or lrlxrul J(XX) llsl'.'l'.rrrrtlrr's lr:rvc rrls. hccrr rcp.r1cd, although much less fiequcntly rhan in
thc Unitcrl Statcs, in Australia, wcstern Europe, India, and Japan. -l.ornadoes
that occur in Japan are known as totsumaki.s. Typical diameteri lilr {atsumakis
are of the order of 50 m. Their forward speeds are of the ordcr o1'40--50 km/
hr; the average length of their paths, which are directed gencrally toward the
northeast, is about 3 km and their maximum tangential .speecls arc probably
about 200 km/hr t1-161.
The destructive effects of tornadoes on buildings are illustrate<t by Fig.
1.3.14.
ADDENDUM: LOSSES DUE TO W|ND STORMS
wind storms are the largest single cause of economic and insured losses due
to natural disasters, well ahead ofearthquakes and floods [1-17, 1-lg]. In theunited States, between 1986 and 1993 hurricanes and tornadoes caused about
$41 billion in insured catastrophic losses, compared with $6.1g billion for all
othernatural hazards combined [1-18, p.4], hurricanes being the largest con-
tributor to the losses [1-19]. In Europe, in 1990 alone, four winter storms
caused $10 billion in insured losses, and an estimated $15 billion in economic
losses [ -20 to l-221.
ru'rrrrl vt'lot r(y, r'is tlrc rurliirl rlistance
;rll (l('rrsr{y, tntl tllthlr is l[c prcssure
ilrIIntNot: 31
I"IGURE 1.3.14. Tomado damage in Rochester, Indiana (courtesy prof.essor u. F.Koehler, Ball State University).
REFERENCES
l-l w' J' Humphreys, physics of the Air, McGraw-Hilr, New york, 1g40 (reprint,Dover. New york. I964).
l-2 H. R. Byers, General Meteorology, McGraw_Hill, New york, 1944.l-3 G' J' Haltinerand F. L. Martin, Dynamicnr and physicar Meteororogy,McGraw-Hill,New york, 19-57.
l-4 L' T' Matveev, Fhrt,st2s t..f rrtt'Atttr-,s1ilttrc,'167-513g0, u.s. Department ofcommerce, Nationar Tcc:rrrrit'rrr rr'rr.rrrrrri.' scrvicc, springfield, va.l--5 A. Miller. Mctutnil.1i.v, (,lurr.lr.s 1,, Mt.rril, (..1'rrrbus, OH, I971.l-6 M' Ncihurgcr,.t. (;. ririrr;icr.;uur W r) rr.rurcr., (/rttrcrsruntrirtlg tht A!n._s1;haric I.)nt,innncttl, W. ll. l;rt.r.trr;rrr. S;rrr l;l;urr.ist.o, 1973.
32 AtM()t;t't l nt(: (ilt(;ut n il()Nli
l-l H. Goldstcin, Chssit'ul Mcclrttttit',t, Atltlison Weslcy, Ncw Yolk, 1950.
l-8 F. Fiedlerand H. A. Panof.sky, "Atrrxrsphct'ic Scalcs and Spccrral Gaps," Ilull.
Am. Meteorol. Soc., 51 (Dec. 1970), I I 14,l I 19.
l-9 Hurricane, U.S. Department of Commercc, ESSA/PI 670009, 1969.
l-10 R. A. Anthes, Tropical Cyclones: Their Evolution, Structure and Effects,
Monograph No. 41, Am. Meteorol. Soc., Boston, 1982.
1-l I H. E. Graham and G. N. Hudson, Surface llinds Near the Center of Hurricanes
(and Other Cyclones), National Hurricane Research Project. Report No. 39,
U.S. Department of Commerce, Washington, DC, 1960.
l-12 R. D. Marshall, Wind Load Provisions of the Manufactured Home Construction
urul Safety Standards: A Review and Recommendations for Improvemer?/, NIS-'l'lR -5 189, National Institute of Standards and Technology, Gaithersburg, MD,
I 993.
I 1.1 S. Busingcr, "Arctic Hurricanes," Am. Scientist, 79 (1991), l8-33.
l-14 'l'ltuntle rstorm, Report of the Thunderstolm Project, U.S. Department of Com-
rncrcc, Washington, DC, 1949.
l-15 E. Kessler, "Tomadoes," Bull. Am. Meteorol. Soc., 5l (Oct. 1970), 926-936.
1-16 H. Ishizaki et al. "Disasters Caused by Severe Local Storms in Japan," Bull.
Diaster Prev. Res. lesf., Kyoto University, 20 (March l97l),227-243.
l-11 G. A. Berz, "Global Warming and the Insurance Industry," Interdisciplinary
Science Reviews, 18 (1993), 120-125.
1-18 D. D. Mclean, Chairman's Report to the Annual Meeting, First Annual Meeting
of Insurance Institute for Property Loss Reduction, Seattle, October 12, 1994.
1-19 A. C. Boissonade and S. K. Gunturi, "A Knowledge-based Computer System
for Financial Wind Risk Management," Proceedings, Computing in Civil En-
gineering (K. Khozeimeh, ed.), Am. Soc. Civil Engineers, New York, 1994.
l-20 G. Berz and K. Conrad, "Stormy Weather: The Mounting Windst<lrm Risk and
Consequences for the Insurance Industry," Ecodecision, April 1994, pp. 65-68.
I-21 Winter Storms in Europe-Analysis of 1990 Losst,s arul l;'utur( L()ss Potential,
Munich Reinsurance Company, D-80791 Munich, 1993.
l-22 Windstorm-New Loss Dimensions of a Natural llultnl, Munich llcinsurance
Company, D-80791 Munich, 1990.
CHAPTER 2
THE ATMOSPHERIC BOUNDARY
LAYER
As was indicated in chapter l, the Earth's surface exerts on the moving air ahorizontal drag force, whose effect is to retard the flow. This effect is diffusedby turbulent mixing throughout a region referred to as the ot_orp,lr"rf, bound_ary layer. The depth of the boundary rayer normally ranges in the case ofneutrally stratified flows from a few hundred meters to sl"ueJ kilometers,depending upon wind intensity, roughness of terrain, una ungt" of latitude.within the boundary rayer, the wind speed increases with elev?tion; its mag-nitude at the top of the boundary layeris often referred to as the gracrient speed.Outside the boundary layer, that is, in the f."" ut_orpt;;", ;# wind flowsapproximately with the gradient speed along the isobars.This chapter is devoted to the study of aspects of atmospheric boundary-layer flow that are of interest in structural design. The theoretical and experi_mental results presented include descriptions of irean wind profileq the relationbetween wind speeds in different roughn"r, regimes, and the structure of at-rnospheric turbulence. Since the structural engineer is concerned fimarlty wittrthe effect of strong winds, unress otherwise noted it will be assumed in thelbllowing that the flow is neutrally stratified. The justification of this assump-tion.is that' in strong winds, mechanical turbulence* dominates the heat con-vection by far, so that thorough turbulent mixing tends to p.odu." neutrarstratification, .iust as in a shailow raycr .f incompressible fluid mixing tends toproduce an isothermar state. Ars., sircc wincr speeds are considerably lowerthan the specd of sound, inc<lrrr1'l'cssirririry rrury bc assumed in the study of theclynamics of thc flow.
'r'A tltrirlilitlivc rk:sclipliolr ol-llrc rrrt.clrrrrtrr.:rl ltrrlrrrh,rrrr.plrt.rr6rrrt,rr6rr is prcsclrlul i1 Sccl.,1..l.
33
34 il[ AtMr)til'ilt ilt(: rr()uNt)nny rnyt tl
2.1 GOVERNING EQUATIONS
2.1.1 Equations of Mean Motion
The motion of the atmosphere is governed by the tundamental cquations of
continuum mechanics that include the equation of continuity-a consequence
of the principle of mass conseryation-and the equations of balance of mo-
menta, that is, Newton's second law. These equations must be supplemented
by phenomenological relations, that is, empirical relations that describe the
spcci(ic rcsponse to external effects of the continuous medium considered. (In
llrc cirsc ol'a lincarly elastic body, for example, the phenomenological relations
t'orrsist ol'lltc: so-citllccl Hookc's law.)
ll tlrc t'r;rrrrtiorr ol'conlinuity and the equation of balance of momenta are
irvcllrp.t'tl with rcsllcct to litttc:, ancl if'tcrms that can be shown to be negligible
;rlt'rlroplrctl ll 1.2 21" thi: lirlkrwing cquations describing the mean motion in
llrt' llrrrrrtllrty lrrye l ol tltt: rtlttursphcrc arQ obtained:
0u }tlU IV Idx 0v
AV AV dVt/- + v- t w- 0x dv 02,
ll) I 0nW+Lil; p 6x -fv*:*:o (2rr)
l0n l0r.r-+ t.fU- -;r:0pdy p dz (2.r.2)
l0o
;;,r*8:o (2'l'3)
AU AV AW
E* i,r+E:o (2.t.4)
where U, V, and W are the mean velocity components along the axes x, y, and
z of a Cartesian system of coordinates, whose z axis is vertical; p, p,f, and g
are the mean pressure, the air density, the Coriolis parameter, and the accel-
eration of gravity, respectively; and r, and r,, are shear stresses in the x and y
directions, respectively. The x axis is selected, for convenience, to coincide
with the direction of the shear stress at the surface, denoted rs (Fig. 2.1.1).
It can be seen, by differentiatingEq.2.l .3 with respect to,r or y, that the
\,-..Itl( il llll,l :.1. 1 " ( 'oorrlirrrrlt' rrrts
Irovl Itl.itNti |(Jt In Il()Ni; 35
vt'tlical vltriittiott ol lltr'ltot izotttrtl l)rL:ssur('1ll=lrrlrt'rrl tlt'1x.rrrls rr;xrrr llrc hori-
zorrtal dcnsity gr':rrlit'rrl. lior llrc l)urposc ol tlris it.xt, it will bc srrllicicnt kl
t'ottsidcrorrly lklws irr wlrit'lr llrc horizontal dc:nsily gllrtlicnl is rrogligiblc (c.g.,
l):rft)tft)pic l1ows; c.g., scc l2-2 1). In this casc rlrr: lrolizorrtul prcssurc gradient
tlocs not vary with lrciglrt irnd l.hus has, througlrotrl lhc lroundary layer, the
srrure magnitude as at tlro top of the boundary layor:
(2.t.s)
whcre z' is the gradient velocity, r is the radius of curvature of the isobars,
:urtl n is the direction of the gradient wind (see Eq. 1.2.7).If the geostrophic
rrlrproximation may be applied, it follows from Eq. 1.2.6 that
H: ,lrn,, *Y7
l0n-: : fv"pdx
wlrcre U, and Vr are the components of the geostrophic velocity G along the
r and y axes.
The boundary conditions may be stated as follows: at the ground surface
lcvel the velocity vanishes, while at an elevation from the ground equal to the
lroundary-layer thickness, the shear stresses vanish and the wind flows with the
liurdient velocity.
2.1.2 Mean Velocity Field Closure
'lir solve the equations of mean motion, it is necessary that phenomenological
t:lations (also referred to as closure relations) be assumed defining the stresses
t,, and r,. A well-known assumption [2-l] is that an eddy viscosity Kand a
rnixing length L may be defined such that
l0n-+:-fu"pdy
(2.1.6a)
(2.1.6b)
(2.1.7a)
(2.r.7b)
(2.1.8)
AUru : pK(x. !. z) 
ĉ,Z.
AVr,, : pK(x. j, z) 
d̂z.
K(x, y, z) :
'f 'lrc usc ol' ljc;s. 2.1.1 -2.1 .tl in t'orr jrrnt'ti'n with l,)qs. 2.1.1-2.1.4 is rcfcrred
Io its lhc tttcattvckrcity licltl t'krsrrrt' lrr lir;s. 2.1.7 cithcr lho ultly viscosity
ot'lltc ntixittg lcrrglh lir:kl rrrrrsl lrt. ,.tpr.t rlit.rl.
1.21,, v, ., l(ur:)' * (Yu,)'1"
36 ilil n lM():;t't il nt(; lt()t,Nt)nt ty
2.1.3 Mean Turbulent Field Closure
From the equations of balance of momenta
equation may be derived (e.g., see [2-31):
fbr the mean motion, thc lirllowing
.., a /q'\l l r, oI)* u,\:, )l- liA *lalu-l3x e).,h(9.
+ve.n).
r,,3V);al
(2.1.e)
(2.1.10)
(2.t .1t)
(2.t.t2)
(2.1.13)
c:0
rvlrt'rt. llrt'lr;rrs irrrlit'lrlt'irvcllrging willr rcspcct to time, u, tt, w are turbulent
lrlor rly llrrclrrrtliutrs irr llrr r, t,, ;: tlirct.tions, rtspcctively.
Iu I /' I r,'')"'
r:; llrr rt'srrll;rrrl llrrtlturlrrrft vt.lrxily, /,'is llrt.llrrctrurtirrg prssurc, and e is the
t;tlc of ctt('tf.v rltsstlt:tltott l)('r unil rrurss. lit;rr:rliorr 2.1.9 is rclbrrccl to as the
tttrltttlttt! /.ittt'tit' .'tt.'t,q.\' r'rlttrtliott rrrrtl c'xPlt'sscs llrt. blrllurcc ol'turbulcnt cnergy
ittlvcctiott (llle (rrttls itt lltc lit'st brlrt'kel ). pnrtlrrcliorr (thc lcrr.ns in thc second
brackct), clillirsion, antl tlissipirtiorr. 'l'hc rrsc ol lxl. 2.1.9 and attendant phe-
nomenological rclations-in c<lnjuncti<ln witlr llqs. 2. l.l-2.1.4 is rcf'crred to
as the mean turbulent field closurc. Phcnorncrrokrgical dcscriptions of the quan-
tities involved in Eq. 2.1 .9 havc becn attorrrptccl by various authors lz-4,2-5,
2-61 . Successful predictions of boundary-laycr characteristics based on Eq.
2.1.9 and various phenomenological descriptions have been reported in the
literature [2-7], although differences of opinion with regard to the relative merits
of these descriptions still exist.
In particular, the mean turbulent field closure appears to be advantageous in
the study of three-dimensional boundary-layer flows. Following [2-g] and
12-91, [2-10] proposed the relations
{rf,+ r2,lt'': porq,
/p's2\ t.1..) /v\' \; * z ) : q'n''^^.to'taz 1;1
(qr)r,,
L -- LaQ/6)
Tu : '''|Ul6z )Vl0z
in which ar = 0.16, 6 is the boundary-layer thickness, and e,. is the resultanl
velocity at the edgc of thc boundary layer (or thc gr:rtlicnt vclocity in atmo-
sphcric filrw).
:r:' tull ANvi llll llYl't r()r rr rlirNil()ril,/()NrAr ry il{rMr}rir Nr ,t,:,il()w 37
1.O0.5
v/6
FIGURE 2.1,2. Empirical functions. From J. F. Nash, .,The calculation of rhree_I)imensional rurbulent Boundary Layers in Incompressible Flow," J. Fluiel Mech.,37(1969), Cambridge University press, New york, p. 629.
In the case of the mean turbulent field closure in which Eqs. 2. r.g-2.r.r3
are used, the empirical functions that have to be specified are the diffusionl'unctions a2(yl6), and the dissipation length z7(y/6). Reference 2-r0 proposeslbr these functions the form represented infrg.'Z.t.Z.
2.1.4 Second-Order Closure
'fhe second-order closure consists in supplementing the equations of balancerf momenta and of continuity by the Riynolds 
"qirations, which govem thebehavior of the stress tensor components and are dlrived from first principles
12-111. Reynolds equations contain unknown terms, including triple velocitycorrelations, for which suitable phenomenological relation, ,rruit be sought. Torbtain such relations, the method of invariant modeling has been proposed,
which is based upon the following requirements. The -od"l"d terms must: (r)cxhibit the tensor and symmetry properties of the original terms in Reynolds
cquations, (2) be dimensionally correct, (3) be invariani under a Galilean trans-lbrmation, that is, a translation of the coordinate axes, (4) satis$z all the general
conservation laws [2-1 r, z-rz]. The second-order closure has been applied, forcxample, to the study of the flow structure in the boundary layer near a suddenchange of surface roughness 12-131.
2.2 MEAN VELOCITY PROFILES IN HORIZONTALLY
HOMOGENEOUS FLOW
It lnay bc assurnccl that in littgc st'lrlc sl()lnl:i, within a horizontal sitc ol unilirrm(lttghncss ovcr a sullicicrrlly lltrgt lt'lt lr ;r rr'p,ion cxisls ovcr w6ich rho ll.w is
3B ilil AtM()lipltilil(; t()t,Nt)Aily tAyl il
FIGURE 2.2.1. Growth of a two-dimensional boundary layer along a flat plate.
horizontally homogeneous. The existence of horizontally homogeneous atmo-
spheric flows is supported by observations and distinguishes atmospheric
boundary layers from two-dimensional boundary rayers such as occur along flat
plates. Indeed, it is known that in the latter case the flow in the boundary layer
is decelerated by the horizontal stresses, so that the boundary-layer thickness
grows as shown in Fig. 2.2.1 . rn atmospheric boundary layers, however, the
horizontal pressure gradient-which, below the gradient height, is only partly
balanced by the coriolis force (Fig. 1.2.8)-"re-energizes" the fluid and coun-
teracts boundaryJayer growth. Horizontal homogeneity of the flow is thus
maintained 12-141.
Under equilibrium conditions, in horizontally homogeneous flow Eqs. 2.1.1
and2.l .2, in which Eqs. 2.1.6 are used, become
v^ - v: !0". p.f az (2.2.ta)
(2.2.tb)
(2.2.2a)
(2.2.2b)
2.2.1 The Ekman Spirat
If in the above model the shear stresses are represented by Eqs. 2.1.7 andif,
in addition, it is assumed that the eddy viscosity is constant, thJmodel obtained
is called the Ekman spiral. Equations 2.2.1 thenbecome a system with constant
coefficients. With the boundary conditions U : V: 0 for z : O and U :U* V : Vrfor z: oo, the solution of the system is
u--u:-!0"" pfaz
IU : 6Gt1 - e-oz(cos az - sin az.)l
v:4ctt2
where a : (.l.l2K)v2
* e-"Z(cos az. * sin uz.)l
i':' Ml AN Vt t{'{ tiy t,il()l ll ll; lN ll()lll.1()NlAl ly ll(,Mrxil Nl ()(,li lt()w 39
Ucluations 2.2.2. wlrrclr tk:scribc tltc Ilkntittt s;rit'irl. irtl' rcl)lcric:tltctl sclter
rrratically in liig. 1.2.(). 'l'lrc ap,rocrnont ol' lhr:su ctlttitliotts witlt obscrvrrlitttts
lrirs bccn fountl to be rrnsirtislactory, howcvol'. liot cxruttplc, whilc acc<lrding
to Eqs. 2.2.2 lhc irngle rr,, hctwcen thc surlacc strc:ss 11 antl thc geostrophic
wind direction (trigs. 2.1.1 and 1.2.9) is 45o, obscrvations indicate that, in
blrotropic flows, dcpcnding chiefly upon roughncss ol'tcrrain, this angle may
rrrrrge between approximately 6" and 30'. The causc of the discrepancies is
thc assumption, mathematically convenient but physically incorrect, that the
t,tldy viscosity does not depend on height.
2.2,2 The Turbulent Ekman Layer
Mcteorologists have attempted to solve Eqs. 2.2.1 using assumptions on the
vlriation of eddy viscosity with height that are more plausible than the as-
surnption of constancy. A survey of corresponding solutions can be found in
l2 2l and 12-151.
A different type of approach was recently developed in l2-l4l in which,
lrrther than resorting to a mean velocity field, closure is based on similarity
t'onsiderations analogous to those used in the theory of two-dimensional bound-
irry layer flows. In this approach the boundary layer is divided into two regions,
ir surface layer and an outer layer. It is logical to assert that the surface shear
r,, rnust depend upon the flow velocity at some small distance z from the ground,
thc roughness ofthe terrain (i.e., a roughness length zo), and the density p of
thc air. Thus rs may be expressed as a function F of these quantities:
ro : F(Ui * Vi, z, zo, p) (2.2.3)
where i and j are unit vectors in the x and y directions, respectively. It is
r'onvenient to write Eq. 2.2.3 in nondimensional form as
ui+vj (2.2.4)
where the quantity
U*: (2.2.s)
is known as the shear velocity ol' lhc lkrw and.ll is some function of the ratio
;:/;1y. Equation 2.2.4 is a lirnrt 1vl' 1111: vve:ll-known "law of the wall" and de-
scribcs thc flow in thc surlitc:g litycr.
In thc outcr laycr it can hcr sirrrillrrly irsscl'lc(l that the reduction of velocity
l(t/,i + V*.)) * (Ui + lz.i)l rrl lu'rglrt ,'ttrtrsi tlcpond upon the surface shear
r1y, the: hr:ight to wlrich thrr cllt't'l ol llrc witll sttt:ss hits diffused in the flow,
U4.
: /, (:)\(0/
(?)"'
40 nil AtM()t;t'ilI tl(i tx)(,Nt)nt ty tAyt l
that is, thc boundary-laycr thickncss 6,arrrl thc tlursity p ol'thc air. 'l'hc cxpros-
sion of this dependence in nondimensional lirrrrr is known as thc '.vckrcity
defect law":
ui+vj _ u|i + v|i (2.2.6)u4 u4
where f2 is some function to be defined.Il it is postulated that a gradual change occurs from conditions near the
gnrrrnrl to contlilions in the outer layer, it may be assumed that a region of
ovcr'f ir1r e xists in which b<llh laws are valid. Let Eq. 2.2.4 be written in the
lorttt
(;) (*)l (2.2.7)
f irrrrrr tlrc lirnn ol' L<trs. 2.2.6 and 2.2.7, and the condition that their right-hand
sidcs bc cqual in the overlap region, it follows that a multiplying factor inside
the function 11 must be equivalent to an additive quantity outside the function
f2. rn the case of the analogous two-dimensional problem, it is well-known that
the two functions must be logarithms tz-16, 2-ril. The requirements of the
problem at hand will be satisfied if f1 and f2 are defined as [2-14]
., (;)
ui t v.i
-.1 tll *.
where B and k are constants. Substituting Eqs. 2.2.g anct 2.2.9 in Eqs. 2.2.7
and 2.2.6, respectively,
f{0 :1ln g'lkyi
fzG):gngr/!i+fj
i ('"; + r'.e) i
Usi + VRj , t /,-t--ttt't4 k \
u. 16^ : - ln -tt4 k 2.0
vu--4
llq: k
(2.2.8)
(2.2.e)
(2.2.10)
(2.2.tt)
(2.2.12)
(2.2.t3)
Ui+Vi
U4
ui+vi 'i) ' . f iU4
If Eqs. 2-2.10 and2.2.l 1 are now equated in the overlap region, there result
:':, Ml nN vt tr){ ili t,n()l ll Il; lN ll()lll,/()Nlnl ly ill )M(t(it Nt ()t |; |()w
lhrrrr wlriclt thert. lirlIrws
, " , t/.'(; (nt+rn'"\ ttt
\ ;,,,/ [
41
It can further be shown that the boundary-laycr thickness 6 may be expressed
lts
(2.2.14)
(2.2.1s)
(2.2.16a)
2.2.15 is established
whcre c is a constant. To prove this relation, let Eqs. 2.2.1a and 2.2.1b be
rrrultiplied by the unit vectors j and i, respectively. From the expressions thus
rrlrtained, and remembering that r, : r0, r, : O at the surface and that r, :
/,, : 0 at z : 6, it follows that
^Uxd -._ f
J trt * vi - (usi + vs:)r az : !t*i
[ ,.r, (#) o, : T [,u, ot
: const a-pl
'l'hat is, Eq. 2.2.16 is verified and the validity of Eq.
l),-141. Equation 2.2.14 may then be written as
(2.2.16)
where the integration is carried out over the boundary-layer depth. Since the
lrrrlk of the mass transport takes place in those parts of the boundary layer
whcre Eq. 2.2.6 holds-which include the overlap part of the surface layer
tlrrwn presumably to a very small height-the velocity profile in Eq. 2.2.16
rrrry be approximately described by Eq. 2.2.6. rf Eq.2.2.15 is now substituted
irrl<r Eq. 2.2.6 andEq.2.2.5 is used, the left-hand side of 8q.2.2.16 becomes
o:lu' . (^n- o)')''' T (2.2.17)
llquation 2.2.17 was oblairrctl irrrlt'pcrrtlcntly in [2-18] and [2-5]. The deri-
vrrtion o1'12-51 is bascd <lrr tlrt'lrrrlrrrlr.rrl crrcrgy equation and the assumption
rrl'ir rnixing lcngth pnrporlionlrl to .'. 'l'lrc tlrurntities A and B arc univcrsal
('()nslanls. lironr lho arr:rlysis ol olrst'rv;r(r('ns il wrrs liluncl that 4.3 < 1l < 5.3
irrrtl 0 < A < 2.tl12-14.215, I l!'1. .) l{). ). )O.221.2-22,2-2'3.2-241. On
(lre llitsis ol'cxpcritttt:ttls ilt lltt' u,urrl lrrrrrrr'l ;rrrtl irr lhc: irtrrroslthcn., llrc wt.ll
THE ATMOSPHERIC BOUNDAIIY LAYER
known von Kdrmin's constant is generally assumcd to bc ft = 0.4.r,
coefficient c in Eq. 2.2.15 is of the order of 0.25-0.3 \Z-ZO, 2-261.
2.2.3 The Logarithmic Law
Equation 2.2.1O may be written as
(2.2.18)
(l ll'l), rvlrt'rc .r rr llrt'lrt'ip.lrl rrlrovt'tlrc srrrlircc, z, is the roughness length,
rrtttl l/( I tr' lltt' t!rr';ttt lvttttl s1x't'rl. lir;rurlion l.l.lli is known as thc logarithmic
I it tr'
llr=r r=ttl lillr'i{rittt'l('ornluliir;rl tt'sr'irlt lr lurs cstlrblishccl thirl thc hcight above
gtrttttttl .'r ltp t{r t!lttr lr lttl ,r ,r tll nr;ry lrc:tsstrrrrt.tl lo lrr: irpproxirnatcly valid,
i:; rlr'Ilirrrl lrv lltr' Ir'lirlrrrrr
It(2.):!r*n1o ?.o
i1 ,"'.,' (2.2.19)
whcrc b is a corrstlrrt. lhc onlcr ol'rrragnitutlc ol'which is 0.015-0.03 12-26,2.271. As notcd in 12-261,l.c1.2.2.19 oxprcsscs the fact, well-kn.wn from
laboratory experiments-including cxpcriments conducted in rotating wind tun-
nels [2-28, p. 148, 2-29]-that the logarithmic layer extends to some fiaction
(of the order of lo%) of the boundary layer depth 6 (see Fig. 2.2.2). Figure
2.2.3 [2-30] represents averages of 14 mean wind profiles (average mean speed
at 9.1 m above ground u(9.1) : 5.3 m/s) measured in nearry neutral flow near
Dallas, Texas. It is seen that for the profiles of Fig. 2.2.3 the logarithmic law
provides a good description of the data up to at least 100 m elevation. This is
in agreement with Eq. 2.2.19.Indeed, for U(9.1) : 5.3 m/s, zs : 0.03 m,
f = 0.77 x 10-4 Qable 1.2.1), and b = 0.022, Eqs. 2.2.1g arnd'2.2.19 yield
zr = 100 m. Note in Figs. 2.2.2 and2.2.3 that the use of the logarithmic law
for heights exceeding z7 is conservative from a structural design viewpoint.
Equation 2.2.19 may also be shown to follow from the assumption that, in
the region 0 1 z 1 21, the shear stress r, differs little from the surface stress
rs (see, for example,I2-1, p. a89l), and the component Izof the velocity is
small. Integration of Eq. 2.2.la over the height z7 yields
ru : ro * ,f I', (vc - V) dz = ro * pf Vrzl (2.2.20a)
*The acttral valuc ol ,/< has in roconl ycars bccomc thc objcct ol sorlc ilchatc 12-251. Hgwcvcr,cllculalions ol inlcrcst in t:rrgirrccring upplications dcscrihcrl irr tlris lcx{ urt rxrt allcctctl signili-
t'lrrrlly by lht' irtlrr:rl vlrlrrt. ol (.
l,;' Mt AN Vl l{x.ily t,t t()t iltsi tN il()nt./()NtAt ty
r) Z.3ti crn
llttYl 11 ,; Nt ()ljti ll()W 43
4.O
3.O
2.O
1.0
_ o.B
I,9 0.6
6 o.+a
tt) o.2It
0.1
o.08
o.o6
o.04
0.5 0.75 1.0 1.25 1.5
VELOCITY (m/s)
FIGURE 2.2.2. Mean wind profile as
measured in a rotating wind tunnel
12-291. Copyright @ l9j5 by D. Reidel
Publishing Company.
10 1? t4 t(; 1r1 20 22 24
ll/rt.
l,'l(JIllll,l 2.2.J. Avcrugc ol l,l nt(.iut \\,lt(l ;rrolrlt. rr.t.otrlt.rl rrrlrr. l)irlllrs. ,l,cxlrs. Allcr.It II 'l'lrLrillcl.:ttttl II (). l,lrP1x" "Wttrrl ;u!11 l('nrlx'rrrulr. l,r'olilc ('hlrr.rrt.lt'r.islit.s liirrrr()lrst'r'vitliorrs olr rr l,:l(X) li 'lirr.vr'r." ./ 11t1tl At,.t .l (l()1y1 y, l()() .t0(r, Alilt.r.it.;rrrMr'lt'on rlolt it'lrl Sot, it'l y.
u*= O.147 m/s
ze= O.OO9 1 cm
-Ulu*= 
ltXlnlo; zo=3 cr
o 50F<(T9.1-T320)<60F
llll n lM()l;l'l ll lilo lr()llNl)nl rY lAYl lr
lpJ vrz,l : qru
where 4 is a small number. Using Eqs. 2.2.5 and2.2.13,
:,:, Ml nN vt trI tt\ I,t i()t It1; tN lt()l il./()Nlnt t\ ilirM{)(,t Nt ()t tl; tt()w
'l'Altl,lt 2.2.1. Vnlucs ol'Sur.ljrt, ltorrghlrcss l,t,rrglh (11) iurl ol.Srrll:rt.t. l)r:rgCrrclliciclrls lirr V:u.ious 'l,y;x.s ol' ,l.rrrains
45
lt is slrowrr in 12 2(rl irntl l2-3 ll that thc logarithmic law holds, for practical
l)urlr(rs('s. t'vt'rr lrt'yontl lrciglrts lrt wlrich r7 is of the order of 30%.
Il. lor r'r:rrrrplt'. / l() 'r st't' '. Il - 30 m/s at l0 m above ground, zo :
(l (f1 rrr (()lx'r l('rr:rirr). :rrul /r 0.02, il lillltlws then from Bqs.2.2.18 and
.'.' lt) tlrrt .:1 ,l(X) rrr. lrr llrr.'t'irsc ol'stnrng winds, the validity of the
Io1';1,r,1r"r'. 1;rw rrp 1o t'lt'v:rliorts rll'tltc ortlcr of 200 m has been confirmed by
nrr':rsur('nr('nls rt'por1t'tl irr l2 12l lrrrd l2-331, as well as by observations at Sale
f J .l-lf rrrrrl ('r'rrrrlit'kl l2 .i-5 1 rrnirlyz.ccl in 12-221.
()rr:rct'orrnt rll (hc lirrito hcight of the roughness elements, the following
cnrpirical urodilication of'Eq. 2.2.18 is required 12-361. The quantity z, rather
than dcnoting hcight above ground, is defined as
1-1R LJ (2.2.22)
where z, is the height above ground and za is a length known as the zero plane
displacement 12-371. The quantity z will be referred to as the effective height.
The flow parameters ze and z./ are determined empirically and are functions of
the nature, height, and distribution of the roughness elements [2-38]. The
roughness length z6 is a measure of the eddy size at the ground. It is suggested
in [2-33] that reasonable values of the zero plane displacement in cities may
be obtainedusing the formula
Type of Surlircc
Sand"
Snow surface
Mown grass (-0.01 m)
Low grass, steppe
Fallow field
High grass
Palmetto
Pine forest(mean height of trees: 15 m; one treeper l0 m2; Z+ = 12 m l2*4}l\
Sparsely builrup surburbsb
Densely built-up suburbs, townsb
Centers of large cities,
(2.2.20b)
\2.2.21)
(2.2.22a)
(t'nr)
0.ol o.I
0. I 0.6
0.l-l
t-4
2-3
4-to
l0-30
90-100
20-40
80-120
200-300
t.2,1.9
t.9-2.9
1.9-3.4
3.4-5.2
4.1-4.7
5.2-7 .6
7 .6-13.0
28.0-30.0
10.5-15.4
25.1-3s.6
61.8- 1 10.4
l0tr<
rlu2*zt: fvg
nk ,u4:.fB'*:Dj
I
il
il
{t
tl
:l
I
"Reference [2-38].
"Values of eo to be used in conjunction with the assumption 2,1 : 0 [2_42].
wake effects). For this reason values of zo in built_up terrain may differ con_siderably from experiment to experiment. ihe values listed in Table2.z.l areintended for use in structural engineering calculations in .on;un.tion with theassumption Za : 0- They are based on a careful analysis oi fuii-scare data,given in 12-421.
The surface drag coefficient r (Eq. 2.2.23) for windflow over water surjhces
1"pr9r upon wind speed. on the tasis of a large numbe, or -"urur.ments,the following empirical relations were proposed for the range 4 < u(r') <20 mls [2-431:
r : 5.1 x l0-4 [U(10)]046
r : lo-a [7.5 + 0.67U(10)]
(2.2.24a)
(2.2.24b)
where 11 is the general roof-top level.
Typical values ofzs forvarious types ofterrain, and the corresponding values
of the surface drag coelilcients (defined as
where u(10) is the mean wind speed in m/s at r0 m above the mean waterlevel. According to [2-44], tor UltO; ) 20 m/s or so l< is constanr.A more recent evaluation of existing measurements led to the expressionproposed in [2-45] for wind speeds U(10) up ro 40 m/s:
r:0.00r-s [r + exp ( tl0!!.5)l ' .0.00104 e.2.2s)I t..5. /l
For examplc. il utlO) : 20 nr/s. ir irrr.ws liirrrr l:qs. 2.2.25 ancr 2.2.23 thatx , 2.5 X l0 I ancl 7,, : 0.3-5 c.rrr. ll t.lrn lrt.vt.l.ilit:tl lhlrl crr<lrs irr the cstirnationo1'wincl spoctls cluc ttl tlnccr-lrtittlrt's :rss.t irrt'<l witlr tlillcrcrrccs lrr'.ng [Jqs.2'2'21t' 2'2'241't, antl 2.2.25 ittt' ittsil'trrlrt;rrt Atkliliorurl inlirnrrirli., .rr lhcsurlirct'tlrlrg lirr-wirrtr fl.w.vt.r'lrrr',,,.,',,,, ,. prr.r;t.rrrt.rr irr l?,r{rl :rrrtr lr rr.2 l
,r:H-?
.:I n I'
| ln 110/zo) I
(2.2.23)
in which z0 is expressed in meters) are given in Table 2.2.1 [2-39,2-40,2-41,
2-421. Table 2.2.I also incluclcs suggested values o1'2,' lirr built-up terrain. The
determination <ll'rcproscn(irtivc wintl pnrfilcs in brrill rrp tcrrain is gcncrally
rlillicLrlt on acc()unl ol lot'rrl llow irrlllrrxrgcncitics (c.9.,llrosc associlr(crl wilh
46 ltE AlMosplt tirc llouNunny InyFR
According to 12-1331, the influence on thc wavcs on thc wind prolilc appcars
to be restricted to elevations below three wave heights; in this zone wind speeds
are lower than indicated by the logarithmic profile.
2.2.4 The Power Law
Historically the first representation of the mean wind profile in horizontally
horrrogcncous lcrrain has been the powcr law, proposed in l9l6 by 12-471:
(2.2.26)
rrlrerr-,* irr;ur r'\lr(llt(=ttl rL'|t'tttlt'irl rrlxrrr rorrglrrrr.ss 0l lctrrain ancl ;*l and zsz
r lt=lir rlr. lir. tprltl,, irl rlt,t. f, rorrrrr I
Itt l.t .lHl il lii rrfiirltiit(=rl ( l) tlrirt lltr'powcr l:rw lroltls wilh consllrrrt cxponent
rr ttp ll lltr= gtiulir'ltl lrr'igltl ri;rttrl (,t)llrirl i rtst'll rs ir lrrrrcliolt ol'rv alonc. The
fi1ril nl llrt'rit' rri:iiiittlf tlttr; lrrrIltr.r llt:tl
/-.\"l/(.',,r) {/(;,,,1 ('"' }
\ r ,.'/
"l;" (;) (2.2.27)
'l-ho socott(l itssttttt;rliott lcl)r'cscnls rrr t'llt'r'l tur crrginccling sirrrplilicltion of the
boundary-laycr clcptlr clcscri;rtiorr givcn by l;.t1.2.2.1-5. Vllucs ol'D and a
recommendcd for dcsign l)url)oscs irr l2-4ttl ancl l2-491 arc slrown in Table
2.2.2. Yalues of 6 (in opcn tr:rrairt artrl ccntcrs ol' largc citics) sintillr to those
given in Table 2.2.2 were pK)poscd in 193,5 by Pagon [2-.5t), p. 7441. The
ASCE 7-95 Standard 12-1391 is bascd on the valucs ol' a arrtl d given in
12-491. However, [2-1391 uses 3-s gust speeds instead ol'fastosr-rnilc speeds,
and the power law exponents are adjusted accordingly-scc 'I'ablc 2.2.2.
Currently, the logarithmic law is regarded by meteor<lkrgists as a superior
representation of strong wind profiles in the lower atmosphcrc 12-26,2-51,
2-52, 2-53, 2-54, 2-551.
TABLE 2.2.2. Yalaes of 6 and c, Recommended in [2-48], 12-491, and [2-1391
Centers of
Large
Cities
Suburban
Coastal Areas Open Terrain Terrain
Reference
6
(m)
0
(m)
6
(m)
o
(m)
2-48
2-49
2-139*
l/ 10
l/il 5
0. r6
l7
t 19.5
275
274
274
0.28
I 14.5
U1
400
366
366
213
2ll
0.40 520U3 457U5 457
*lirr 3-s grrsls l,'l(lllltl,l 2.1..1. Wrrrrl lt.lot rty lrlrliles
Ml AN Vl lor 11y 1,11111 ll l:; tN ilOlll,/()NtAt ty ltilM0rit l.ll (,t |; Ilrw 47
2.2.5 Relation botwecn Wind Speeds in Dlfferent Roughness
Regimes
Considcr tw<l acliaccnl lcrrt'ltitts, cach of'unilirrrrr nrrrglrrrcss iurrl ol'srrllicirrrrtly
large fetch. Lct tlrc nlttgltttoss lcngths fbr thc lwo lcl'ruins bc rlcnolctl by 1111
and 20, and assume that z9r ( zs. The retardation ol'thc llow hy surl'acc l'riction
will be more effective over the rougher terrain; thcrclirrc, if the geostrophic
speed is the same over both sites, at equal elevations the mean wind speeds
will be lower over the rougher site. A schematic representation of the respective
wind profiles is shown inFig.2.2.4.
The profiles of Fig. 2.2.4 suggest the following procedure for relating wind
speeds in different roughness regimes. To calculate the wind speed U(z* z6)
over the rougher terrain if the speed U(24, zo) is known, Eq. 2.2.27 is applied
to each profile; then the quantity G is eliminated from the two relations thus
obtained, and
(2.2.28)
where cv(zo),6(ae) and ot(zo),6(261) correspond to the roughness lengths Zs and
Zs1, respectively. Equation2.2.28 was proposed in [2-48] and will be referred
to as the power law model.
Recently, an alternative procedure has been proposed that is based on results
of both theoretical and experimental studies I2-22l.If the speed U(zd, zs) is
known, it follows from Eq. 2.2.18
u(2,, zs): (6)*"'(?)" " (r(,,,,,0,)
48 lltt AIM()lil'ilt tit(; tr()t,Nt)nny tnyt n
where the notation of Eq. 2.2.22 is used. Applying now Eq. 2.2.29 to the two
profiles represented in Fig. 2.2.4 and eliminating G,
luprrrtiorr 2.2..10 tlt:tcrrrrincs tho valr.rc of the friction velocity a*. Then
[r'* ('".* - o)'),.:lu' * (," 
^^,- 
o)'l''' u*, (2.230)
U(2,,r,:,rr)
U*t:-'' 2.5 ln(zrlzor)
{/(;,,, rrr) 2.5u* lnk
4O
(2.2.2e)
(2.2.31)
l'irlrurtitrrrs l.l.19, l.l. 10, rrrrtl 2.2.31 will be ref'erred to as the similarity model.
As lrrrs bct'rr slrown in 12 22l, tl"rc unccrtainty with regard to the exact values
of llrc corrstiurts ,4 urrcl /i in tlq. 2.2.30 turns out to be of little consequence
insolirr as cstinratcs ol'wind speeds in the lower atmosphere are concemed.
With possible errors of the order of 3% or less, it may be assumed A : 1.4
and B : 4.7. Also, the dependence of the results on z* andl is insignificant
and may be neglected. For practical purposes, therefore, the ratios u*lu*1 may
be calculated simply as functions of the roughness lengths Z1y and zor. The
dependence of u*/u*1 upon Ze and zor can be represented by the relation [2-56]
(2.2.32)
However, subsequent research has shown that the similarity model must be
subjected to empirical adjustments in the case of terrain for which z, ) 0.30
m or so. Table 2.2.3 lists ratios u*/u*1 based on full-scale measurements,
corresponding to zs1 : 0.07 m and various values ze of practical interest
12-421.
The application of the similarity model will now be illustrated by a numerical
example. The data used in the example were obtained by measurernents in and
near London and were reported in [2-33]. At Heathrow, Z,t : 0.08 m, 2,,, =
0, and the measured mean wind at a height above ground zsr : l0 m is U(zrr,
zo) : 11.7 m/s. The mean wind U(z' Zo) at a height above ground z, : 195
m is sought at the Post OfficeTower in London, where zo : 2.5 m (2,7 : 0).
TABLE2.2.3. Ratios u*lu*rf<tr za1,: O.07 m and Various Values ar12-421
z r 0.0706u4 /Zo\
-:t-lu *1 \zor ,/
zo (m)
tt *l u a.r
0.(n5
o.83
0.07
I .(X)
0.30
I 15
r.00
1.33
2.50
1.46
:';' Ml ANVI lr)rllt l,lt{}l ll l:; lNll()lll./()Nlnl lyll{}il/(){il Nt ()t ,:; lt()W 49
litrrrrr l')t;. 2.2.)t), rt,t O.(Xrti rrr/s. liRilrr 'l'irblt. ).).1. 11 ,711 ,,r 1.4(l; that
is, r.r,,. - l.4l rrr/s. Usltli lit;s. )..2.31, {/(2,,,1,) l-5..1,1 rrr/s. lt is notcd that
this rcsult coincitlcs witlr tlrr. irctuirl rncasLrrctl spccrrl ll .l.ll.If thc rncan spcc:rl rrcrul rhc l)ost officc'l'owcr ut ;,, r95 rn is calculated
using the powcr law rrroclcl (hq.2.2.28) with thc paranrctcrs cv and 6 suggested
in 12-481, therc rcsults U(2.", z.t) : 13.4 m/s versus the measured 15.3 m/s
speed.
2.2.6 Effect of Thermal Convection on Mean Speed profiles in
Strong Winds
It is of interest to estimate the extent to which the effect of thermal convection
is significant in structural engineering and extreme wind climatological calcu-
latlons. To do this, we use the following expression, based on the work of
Monin and Obukhov f2-2, p. 282;2-51:
(2.2.33)
where u* : friction velocity, k : von Kdrmdn's constant, zs : roughness
lcngth, / : Monin-Obukhov function, and t : Monin-Obukhov length. If
the stratification is neutral, L: a, tl, : O, and Eq. 2.2.33 becomes the well-
known logarithmic law (Eq. 2.2.18).
The length L is defined by the following expression [I-4, p. 281]:
U(z):T1,":-r(;))
,-8 Qo
T cpp
where g : acceleration of gravity (S : 9.81 mls21, T: absolute temperature,
r;, : specific heat at constant pressure (co:240 callkg degree [l-4,p. 132]),p : air density (p = l.2kglm'|. and Q6 : eddy heat flux (usual orders of
rrragnitude for Qo are 10 to 60 callm2ls [l-4, p.276]).
unstable stratification. In unstable air the following expression will be used
|or tklL):
l, u* (2.2.34)
(2.2.3s)r (;): I:: , - - 160 t^rT
Equati.n 2.2.35 was pn)pos(:tl irr ltcl. I15. Acc'rding to Rcl,. 2-51 , it
providcs a vory gtxrcl lrt to cxllcrirrrcrrl;rl tllr(ir ovt'r'rrrrilirrrn tcrrain and lor0 >
:ll, > -2. (N<ltc thal /, is by tlt'lirtitiorr nt'|;11'ur,' rt lltc slrlrtilicati<lrr is unstirblc.)
litlrrittiorr 2.2.35 is lcprcscnlcrl irr lirp1. .t..t 5
50 t Ht A t M()lit,ilt nto ti()uNt)n ny t n yl il
FIGURE 2.2.5. Function {(z/L) for unstably stratified flow. From E. Simiu, ..Thermal
convection and Design wind Speeds," Journal of the structural Division, ASCE, l0g
(July 1982), 16l t-1615.
stable stratification. In the case of stable stratification it may be assumed
that
(2.2.36)
[2-25,2-51]. The length I is defined by Eq. 2.2.34; however, empirical studies
reported in [2-58] suggest that under stable stratification conditions it may be
assumed that
3.0
2.O 2.5 3.0 3.5 4.O 4.5 5.O
_z
L
L = l. I x l03a3x
where a* is expressed in m/s.
Table 2.2.4 lists estimated dcviations from
2.2.18) for three representativc c.lscs of interest
(2.2.37)
the logarithmic profile (Eq.
in structural engineering ap-
TABLE 2.2.4. Deviation of Mcan wirrd speeds from Logarithmic profile [2-59]
(':rsc l " Case 2t' Case 3'
Elevation 50 rrr
Unstable stratiliculion l'X,
Stable stratilicltioll l'X,
200 nr
_4%,
4'n,
l-5 m
-4%
l5m
- - LL /O
- t2%
"Hourly wirrtl s;xt'rl lrt/'llorrrly wirrtl spct,tl :rt
'llorrlly wirrrl sl)r'r'(l ;tt
lO nt r'lt'vlrliorr ovt'r ollt'rr lt:r':tilt
lll ilr r'li \:tlI)il rrv(.t rll)(.n l(.il;In
lll ltr r'lIr.rlrrrrr 'rv(.t r'l)(.il l(.il:ilil
l5 rrr/s
l.)rrr/s
1 rtr/',
p lr n t M( ll ;t 'l il iltc t t,nBl,t t NCL 51
Plicitlirlnsl.'l'ltc tcsrrlls ol 'l'irhlc 2.2.4 show llrirl srrt'lr tkrvirrliorrs rnay indeedbc ncgloctccl wltctr cslirrritlirrg wind prcssuros oll slrucluros (scc Case l) orwhen reducing to it c(tltlttt()ll clcvation largcsl rrr<lltllrly or ycarly wind speeds
rccorded at a wcathcl' sruti.n (scc case 2). Howcvcr, lbr wind speeds u(10).f the order of 5 m/s thc dcviations from a logarithmic profile are significant
(see case 3). The lattcr conclusion is of interest fbr the design of structures,
such as smoke stacks, that exhibit a significant across-wind response at lowwind speeds. This response is usually enhanced if, as in the casl in unstablystratified flow, the actual mean wind profile is closer to being uniform than8q.2.2.18.
2.3 ATMOSPHERIC TURBULENCE
Figure 2.3.1 shows that wind speeds vary randomly with time. This variationis due to the turbulence of the wind flow. Information on the features of
atmospheric turbulence is useful in structural engineering applications for three
main reasons. First, rigid structures and members are suu3e&eo to time-depen-
dent loads with fluctuations due in part to atmospheric turbulence. Second,
lThe estimates were based on the assumption that in unstably stratified flows e : 50 kcal/m2 sand 7': 290"-
*
e
Itl(,illl{l,l 1,.1. l= Wrrr'l .,lrr'r.rl r('(.()t(l
52 iltt AtMoril'ilt nt(; tr()t,Nt)At ty tAyt lt
flexible structures may cxhibit rcsorriurl irrrrlllilicltiorr c:llbcts inclucc:cl by voktc-
ity fluctuations. Third, the aerodynanric bchavi()r ol' structurcs-and, crlrrc-
spondingly, the results of tests conductcd in thc laboratory-rnay depend
strongly upon the turbulence in the air flow.
The following features of the atmospheric turbulence are of interest in var-
ious applications: the turbulence intensity; the integral scales of turbulence; the
spectra of turbulent velocity fluctuations; and the cross-spectra of turbulent
velocity fluctuations. Also of interest to structural designers is the dependence
of the largest wind speeds in a record upon averaging time.
2.3.1 Turbulence lntensity
'l'hc sirrrplcst clcscriptor of atmospheric turbulence is the turbulence intensity.
Lct u(z) dcnotc thc vclocity fluctuations parallel to the direction of the mean
spccd in a t.urbulcnt flow pussing a point with elevation z (Fig. 2.3.1). The
longitudinal turbulence intensity is defined as
ulottzI(71:
u(z)
(2.3.1)
where U(z) : mean wind speed at elevation z and rl r/2 - root mean square
value of z.J Vertical and lateral turbulence intensity may be similarly defined.
The longitudinal turbulence fluctuations can be written as
(2.3.2)
where z* : friction velocity (see Eq. 2.2.18).It is commonly assumed that B
does not vary with height.+ Values of B suggested for structural design purposes
on the basis of a large number of measurements are listed in Table 2.3.1
12-421.
The averaging time in Eqs. 2.3.1 and 2.3.2 should be equal to the duration
of the strong winds in a storrn. It is commonly assumed that this duration is
between 10 minutes and t hour.
tThe altemative notation o, : u2t/2 is also commonly used.iThis use of the notation 0 should not be confused with its use as the safety index (Appendix
A3).
TABLE 2.3.1. Values of p Corresponding to Various Roughness Lengths
u2 : Bu'*
zo (m)
a
0.005
6.5"
0.07
(r.0
0.30
5.25
r.00
4.1t5
2.50
4.00
"Based on mcasurcnrcnls rclxrrlctl irt l2 7t); irrxl rrsctl irr i'orrjrrrrcliorr witlr luls
Scc also 12-l'12l'.
2.2.23 itntl2.2.25
i, ll n I M( )l;t't lt ttll . tt,ntttJt I N( lt 53
lrrlr cxarrrlllc, il l: l(l rrr. :1y - 0.07 nr, irrrtl l/( lO) l0 rrr/s, il lolkrws
lirrnt Eqs.2.l.ltJ,2..1.1,1..1.2, and'l'ablc 2..1.1 tlrlr( thc trrlbrrlcrrcc intcnsity
is /(30) : 0.162.
2.3.2 lntegral Scales of Turbulence
'lhe velocity fluctuations in a flow passing a point (Fig. 2.3.1) may be consid-
cred to be caused by a superposition of conceptual eddies transported by the
rnean wind. Each eddy is viewed as causing at that point a periodic fluctuation
with circular frequency <,s : 2rn, where n is the frequency. By analogy with
lhe case of the traveling wave, we define the eddy wave length as )t : (Jln,
where U : wind speed, and the eddy wave number, K : 2rl)t. The wave
lcngth is a measure of eddy size.
Integral scales of turbulence are measures of the average size of the turbulent
cddies of the flow. There are altogether nine integral scales of turbulence,
corresponding to the three dimensionsof the eddies associated with the lon-
gitudinal, transverse, and vertical components of the fluctuating velocity, u, u,
rrnd w. For example, Ii, Ll,, and L'; are, respectively, measures of the average
longitudinal, transverse, and vertical size of the eddies associated with the
longitudinal velocity fluctuations (-r is the direction of the mean wind u and
of the longitudinal fluctuations z).
Mathematically, 1, is defined as
u: Ru,ur(x) dx (2.3.3)
where R,,rr(x) is the cross-covariance function of the longitudinal velocity compo-
rrents z1 = u(xr, !r, Zr, t) and u2 : u@r -l x, yr, 21, t), defined in a manner
rrnalogous to Eq. A2.29, / : time, and u2t/2 is the root mean square value
of u I (and a2). Note that in horizontally homogeneous flow, tj is independentolxl and y1 . Similar definitions apply to the other integral turbulence scales.
From their mathematical definition it follows that integral scales are small
il'the cross-covariance functions are rapidly decaying functions of distance,
rrnd conversely. Velocity fluctuations separated by a distance considerably larger
than the integral scales are uncorrelated, and will therefore act on a structural
c:lcment at cross-purposes. For example, values of r), and Li that are small
compared to the dimensions of a panel normal to the mean wind indicate that
thc effect of the longitudinal velocity fluctuations upon the overall wind loading
is small. However, if D" and Li, uc largc, the eddy will envelop the entire
pancl, and that effect will be signilicrrrrt.
Equation 2.3.3 can be translirrrrrcrtl il it is rrssumed that the flow disturbance
lrirvcls with thc vclocity L/(r) lrrrtl. tlrclt'lirlc. lhir( thc fluctuation u(x1, r -l t)
rrury hc irlcntiliccl with a(,r1 tlll, r). wlrcrt./ tintc (Taylor's hypothesis).
'l'hcn
#t:
54 THE AlMOSPI IFRIC R(}I'NI)NIIY IAYI II
Li, : Il,,1r) dr (2.3.4)
where R,(r) is the autocovariance function of the fluctuation a(x1, l). Thc lcngth
of the record from which R,(z) is estimated should be the same as that used to
estimate (J and u2 (i.e., about one hour; see Sect. 2.3.1).
Estimates of turbulence scales depend significantly upon the length and the
degree of stationarity of the record being analyzed, and usually vary widely
from experiment to experiment. For example, for open exposure, measured
values of Lj reported in [2-60] (Part 2, pp. 31 and 32) vary between 120 m
and 630 m at 150.8 m elevation (the average value being 400 m); between 110
rn and 690 m at 110.8 elevation (average value: 350 m); between 60 m and
(r50 rn at 80.8 m elevation (average value: 300 m); between 130 m and 450
rn at 50.8 m elevation (average value: 200 m); and between 60 m and 460 m
at 30.U rn elevation (average value: 200 m). Data reviewed in [2-61] suggest
that Li is a decreasing function of terrain roughness. For example, the following
data are listed in [2-61]:
Site z(m) zo(m) Ii@)
#\,:,
Cardington
Round Hill
Brookhaven
15 0.01 82
t7 0.04-0.10 5516 1.00 36
The following empirical expression was proposed in [2-6ll for the height
range z: 10-240 m:
Il, : Cz^ (2.3.s)
where C and m are given in Fig. 2.3.2 and z is the elevation (t) and z in
meters). The application of F,q.2.3.5 to the data just listcd yields, approxi-
mately, the values 1: l5O m (Cardington), 140-120 rrr (lbund Hill), 70 m
(Brookhaven), which are about twice as high as the mcasrrrctl values.
According to 12-611 the integral scales L) and I.i, lrcr, rcspcctively, about
one-third and one-half the integral scale Ij as givcn by lirg. 2.3.-5. However,
according to 12-621, a better estimate of L) is obtuinrrtl lirrrrr lhc cxprcssion
L', : 0'2Li, (2.3.6)
It has been suggested that
I \' = 1-"rt 5
(z in rr-rctcrs) l2 l.'l()1. 'l'lrt' cxplcssiorr
/,1, 0."[.:
(2.3.1)
(2..1.t|1
:, ll n lM(,:it't il ilt(. il/ntlt,l l N(;r 55
0.001 0.01 0.1 1.O 10
zq (meters)
l''l(;tlRE 2.3.2. values of c and m as funcrions of zo [2-611. Reprinted with permis-
',r.rr l'rom J. counihan, "Adiabatic Atmospheric Boundary Layers: A Review and
;\rr:rlysis of Data from the Period 1880-1972," Atmospheric Environment, 9 (1975),li/l 905, Pergamon Press.
\virs proposed in 12-641 and confirmed by subsequent measurements, as indi-
,;rlctl in 12-611.
2"3.3 Spectra of Longitudinal Velocity Fluctuations
fhe Energy cascade. It was mentioned in Sect. 2.3.2 that the turbulent
vchrcity fluctuations may be considered to be caused by a superposition of
.rltlics, each characteized by a periodic motion of circular frequency a : 2rn
t.r flt u wave number K :2rl\,, where x is the wave length). The total kinetic
('n('r'gy of the turbulent motion may, correspondingly, be regarded as a sum of
, 'rrtlibutions by each of the eddies of the flow. The function E(K) representingtlrt' tlcpcndence upon wave number of these energy contributions is defined as
tlrt' cncrgy spectrum of the turbulent motion.
ll'thc equations of motion of the turbulent flow are suitably transformed, it
,;rn bc shown that the inertial terms in these equations are associated with
tr;rrsl'cr of energy from larger eddies to smaller ones, while the viscous terms
,rrtount fbr energy dissipation 12-63]. The latter is effected mostly by the
',rrr:rllcst cddies in which the shear deformations, and therefore the viscous
',lr('srics, arc large. In the absencc of sources of energy, the kinetic energy of
tlrt' ltrr-bulcnt motion will decrclsc thirt is, the turbulence will decay-faster
rl tlrc viscosity eff'ccts are largc, rrr.rc skrwly if'these effects are small.
Molc prcc:iscly, in thc lallsl t'rrst' rlrt' tlt'r'lry tirr-rc is long if compared to the
lrr'riotls ol'thc cclclics irr llrc hi1',lr wlrvr. rrrrrrrlrt:r rangc. Thc energy of these
,,ltlit's rrriry thcrclirrc hc consitlt'tt'tl lo lrt' rrpploxirrrtrlcly stcacly. This can only
lrt'lltt't'trsc il thc: c:rrc:rgy le.tl inlo llrr.trr llrrorrl',lr ilrertiirl lr-utrsl'cr l'nrrrr lhc largcr
ltklit's is lrlrltrrrct:rl by lhc (.ncllly rlr:,r,1|;rlr.tl lltrorrglr visr'ous rlli'cts.'l'ltc srrurll
llll n lM()i'il'lll lll(; ll()(,Nl)nl lY lnYl ll
eddy motion is thcn dctcrrninccl solcly by lhc nrtc ol crrcrgy lrarrslcr'(or'. ctlrriv-
alently, by the rate of energy dissipation, dcnotcd t (scc [Jq. 2.1.9) antl by thc
viscosity. The assumption that this is the case is known as Kolmog,orov's.first
hypothesis. It follows from this assumption that, since small eddy motion is
dependent solely upon internal parameters of the flow, it is independent of
extemal conditions such as boundaries and that, therefore, local isotropy-the
absence of preferred directions of small eddy motion-obtains.
It may further be assumed that the energy dissipation is produced almost in
its crrtircly by (hc vcry smallest eddies of the flow. Thus, at the lower end of
llu' lrilihcl wrrvc rrrrrrrbcr rlngc t<l which Kolmogorov's first hypothesis applies,
tlrt' rrrllrrt'rrr't'ol (lrc viscosily is srrrirll. ln this subrange, known asthe inertial
tttl,t,rtt,t:r'.llrt't'rltly rnol iort ttury lrt'lrssuntctl to bc independent of viscosity,
;ur,l llrrr:, rlt'lt'rrrrint'tl solcly lry (lrt' r'trtc ol'cnerrjy transfer (which, in turn, is
t't1rr;rl lo llrr'r;rlt'()l ('lr('rl'.y rlrssiplrtiorr). linrrrr this assumption, known as Kol-
nttt,tltt,(tt".\.\('tttn(l ltvltttlltt',ti,t, il lollows tlrlrt lr rclirtion involving E(r$ and e
Itolrls lot :;ttlltt tt'nlly lrrglr A:
l'll',(K ). (. r | 0
whorc /j(/() is llrc cncrgy pcr ruril wavc nunll)or.
(2.3.9)
The clirncnsions ol'thc quantitics within brackots in Eq. 2.3.9 are[L3T 2f,
[L-r], and ILtT '1, respectively. From climensional consiclcrations (see Sect.
7.1) it follows immediately that
E(K) : ar62t3Y-s/3 (2.3. r0)
in which rz1 is a universal constant. On account of the isotropy, the expression
of the longitudinal velocity fluctuation spectrum* [which will be denoted S(K)]
is, to within a constant, similar to Eq. 2.3.10. Thus
S(rK) : aezt3 K-s/3 (2.3.t1)
in which it has been established by measurements that a : 0.5 12-211.
Spectra in the lnertial Subrange. Measurements carried out in the surface
layerof the atmosphere confirrn the assumption that in horizontally homoge-
neous, neutrally stratified flow the energy production (see Eq. 2.1.9) is ap-
proximately balanced by the energy dissipation [2-3]. The expression of this
balance may be written as
ro dU(z)
p dz.
'r'A tldrrilctl tlistrrssion 1)l :il)('( lrr is plt'st'rrlt'tl irr Allllcrrtlix A)
(2.3.12)
w I rt'rrr
l-{/(:)-.a,1, lrr'O lr)
ll lrqs. 2.2.5,2.3.12, and 2.3.13 are used.
"1*e : 
kz
Srrlrstituting Eq.2.3.14 into Eq. 2.3.1l, if it is assumed that
2rnK:-
u(z)
tlrcrc results
, .r n t Milr;t'ilt ilt{ , il,ttltt,il N(;t 57
(2.3.t3)
nS(z,n) ^^_^_)/,- --- : 0.28t--''u4 I
rvlrcrc the nondimensional quantityt
^nzt: uu
n known as the Monin (or similarity) coordinate, and
(2.3.14)
(2.3.ts)
(2.3.16)
(2.3.17)
S(2, n) dn : S(2, K) dK (2.3 . 1 8)
f 'tlrurtion 2.3.15 implies the validity of raylor's hypothesis (see Sect. 2.3.2).'l'hc left member of Eq. 2.3.16 is called the reducetl spectrum of the lon-
1'rtrrtlinal velocity fluctuations and is seen to be a function of height. Although
rrrtlividual samples may deviate considerably from the predicted values, Eq.' ]. 16 is, on the average, a very good representation of spectra in the high-
I r trlrrcncy range [2-5 1, 2-52, 2-53, 2-&, 2-65, 2-67] and may, for engineering
l)rrl)oscs, be conservatively assumed to be valid for f > O.Z [2_64, p.27,; (t] , 2-691. As in the case of thc logarithmic law, for high wind speedi such
;r:r iuc irssutlcd in structural dcsigrr (<ll'the order of 2o mls, say or more), it is
r,':rsorr:rblc t() apply F,q.2.3.1(r tlrRrrrglrout the height range of interest to the
'.1 nlt'lurll cnginccr.
Iltls ttst'ol lllt'st:ttttl:tttl ltolrrli()rt / slrrrrtlrl n.t lx tr)rlu.,r'(l rvillr its previous rrsc:rs llrt.(.1yri9lis
l ';il ;[ [('l('r.
58 ilil n tM()rit,t I nt(i il()t]Nt)nny tnyt il
Spectra in the LOwer-Frequency Range. 'l'lrc krwcr-li-c(luoncy ltttgcr is tlc-
fined between n : 0 and thc lowcr cnd ol'thc incrliul subrangc. As rtotccl in
[2-511, 12-52], and [2-65], in the lower-f-requency rangc sirrilarity brcaks down
and the spectra cannot be described by a universal relation. Howevcr, dcscrip-
tions that are useful for engineering purposes may be obtained by noting that:
l. The value of the spectra for n : 0 is
s(0) : +,ft], (2.3.te)
wlrt'r'e rr' is lhc nlcan squilrc valuc of the longitudinal fluctuations, U is
tlrt'rrrt';rrr vckrc'ily, untl /,) is thc longitudinal integral scale.f F;q.2.3.19
Iollows lirrrrr lir;s. 2..1.4 ancl A2.25.
l. 'l'he tlclivltivc ol'S(rr) wilh rcspect to n vanishes atn :0. (This follows
liirrrr lit;. 42.25.)
3. 'l'hc spcctrurn S(n) is lnonotonically decreasing.
4. 'l'hc spcctrum S(n) is continuous at the lower end of the inertial subrange
with the curve S(n) given by 8q.2.3.16.
5. The area under the spectral curve in the lower-frequency range is equal
to the mean square value of the longitudinal velocity fluctuations (Eq.
2.3.2) less the area under the spectral curve S(n) represented by Eq.
2.3.16. (This follows from Eq. A2.15.)
Two comments on lower frequency spectra are in order. First, as in the
case of the mean speed U, the mean square value u2, and the integral scale
Lj, estimates of spectra in the lower-frequency range depend upon the length
of record being used. For consistency, the length of the record from which S(n)
is estimated must be the same as that for (J, u2, and I). As indicated in Sects.
2.3.1 and 2.3.2, for structural engineering purposes this lcngth should be equal
to the duration of the strong winds in a typical storm. Corrmonly this is as-
sumed to be I hour, although record lengths as low as l0 rninutes are used by
some workers. The l-hour period beyond which winds in a typical storm may
be assumed to become relatively weak is sometimes rcf'crrccl to as the "spectral
gap" (or quiescent period) in a conventional reprcscntation of wind activity
corresponding to a continuous range of periods, including daily, monthly, sea-
sonal, yearly, and secular periodicities [2-681. Spcctra ol' longitudinal wind
speed fluctuations for periods longer than about I lrtlttr ctlrrcspond to meso-
meteorological flow pattems. Thcy wcrc tcntativcly Ittotlclctl by Van der Hoven
rBy virluc ol (he tlelinitiorr ol (lrr's1x't'trirl tlt'rrsily, lit1. 2.1. l() rrrrplics ir vltrisltittgly snrall, rathcr
than it lirritc. t'orrlrilruliorrs ol llrrrlrr:rlirr1l totttlxrnt'nls willt zt'lr ltt'tlucrtty t() lllc tlrolln squatc
virlttc ol lltt' lltttltt;tlrotts
U
:r n tM()l ;t,t il ttlL ililillt,t I N(:l 59
ll l'l ll' wlttl ttolr'tl lltt' t'ris(t'rtcc ol's1rt:cllrrl lx.;rks irl pt'r'itxls ol'lrlrout 4 clays.lilut'lrr:rlions willr;x.r'irxls lorrgcr Lharr tlrtlsc tylricrrl ol'llrc: spcctral gop ir"tlrsrt:grrnlcd in slnrclru'irl crrgirrccring rnotlcls. 'l'lris irlkrws thc usc o1'Eq.2.3.19
:rrrrl ilcttts 2 to 5 trllrvcr as c()n.tponcnts 9l' a rcirs.,rlrflc rnicnrmeteoiological
rrrtxlcl all<lwing thc cslitttittion of longitudinal spcctra fbr periods shorter than
;rlxrut I hour.
A sccond comment pertains to the relation between the frequency zps11 &tr'lriclr the curve ns(n) reaches a maximum and the integral scale ri. As shown
rrr l2-(rll, the assumption has been used in the literature that
rxlu
" 2t flpeak
f lrrr.vsvsl, it was pointed out in 12-731 that the estimation of Il based on
rrrt';rsrrrcd values of u and npeal can be in error several fold, owing to the',('rsitivity of Il to the assumptions conceming the spectral shape between n
. 
o irnd_ n. : ,peak.This shape is in general unknown and, thereiore, so is therll;rtionship between npear and Il,.
Expressions for the Spectrum used for structurar Design pur-
,roses. The curve
t'l nS(2, n)
, u'*
2o0f
(2.3.20)
(2.3.21)
(2.3.22)
rrlrrrsc lirrm was proposed in [2-66], approximates very closely Eq.2.3.16 intlr. rrrcftial subrange (zis the height above ground, n isihe trequency inHertz,
rr , ;rrtl.f'are given by Eqs. 2.2.18 and 2.3.r7, respectively). Ii .un t" verified
rlr;r( lir1. 2.3.21implies that
,/ : 6u2*
rrlrtlr, lilr built-up terrain (zo > 0.30 m, see Table 2.3.1), may result in an,,\('l('stirnation of structural response of the order of 5%. Requirements pre_
rrrrrr5;fy listed pertaining to the value of .{n) and ds(n)ldn at-n :0 are not''.rrr:;lit'tl. However, this is inconsequential as far as the design of most land-lr.r:'t'rl structures is concerned, since their fundamental frequeicies of vibration;r(' rriuillly higher than the frequcncy corrcsponding to tire lower end of the
rrrt rtr:rl subrange. Therefore, pr<lvitlcrl that Eq. 2.3.22 is satisfied, the response,'l :;ut'll structurcs does not dcpcrrrl sig,rrilit.rrrrtly upon the shape of the spectrum
rrr llrt' lowor l-rcqucncy rangc (st.t' St.t.l. (). l.-l).
Ilrc tlcvcl.pr'cnt of'l'\. 2..1.)l ll 701 wrrs rrr.rivutccl by criticisrn of thelrrlllv*1rt* cxprcssion, pnr;rost.tl irr l.r'lll;rrrtl rrst,tl irr rhc Natignal Building('rrrlt' ol ('unutll l2-721:
60 Tt tE nTMOSt'Ht-llto tx)(,Nt)ntiy tnyt il
ns(z.. nt -r'l- :4.0ui ''" tl * "'ro"
(2.3.23)
in which x : l2OOnlU(l0); n is expressed in Hertz and U(10) is the mean
wind speed, in meters per second, at z : l0 m. Equation2.3.23 was obtained
by averaging results of measurements obtained at various heights above ground
and does not, therefore, reflect the dependence of spectra on height. In the
irbscncc ol' rnodcls capable of describing this dependence-such models were
rrrrf y tlr:vckrpcrl subscqucntly in the 1960s-Eq. 2.3.23 and similar expressions
Prolxrst'tl in llrt' litcr-lrlurc havc pnrviclcd useful first approximations of the
lonpiiiurlirr;rl trllrrrlt'lrct: r.il)(:ctrit in lhc atl'nospheric boundary layer. It is noted
llr:rt llrt' th'Pt'nrlt'rrt'r' ol's;rectllr rln hcight is clcarly suggested by data published
rrr lJ 7ll (lrip. l.1..lir).
As rrrt'rrliolrr'tl t'trlliu', llrc spccttrl tlistribution in the lower-frequency range
lrlrs littlt' irrllrrcrree on brriltlirrg tcsponsc; however, the magnitude of the tur-
lrrrlcrrl llrrclrrrtiorr r'orrrponcnts at licquencies cqual,or close, to the natural
l'r'cclucncics ol'u tall structurc rnay affect its response very significantly. It is
thcrolorc ol'intcrcst to comparc thc higher-frequency components inEq.2.3.23
to those of Eq. 2.3. l6 (or, equivalently , Eq. 2 .3 .21). Such a comparison shows
that Eq. 2.3.23 may overestimate the longitudinal spectra of turbulence in the
higher-frequency range by as much as 100-4fi)%, as can be seen in Table
2.3.2 and Fig. 2.3.3b.
It is also noted that Eq. 2.3.23 yields z2 : 6u2*, and that it implies S(0): 0, or U:0 (see Eq. 2.3.19), which is physically not possible [2-3].
The von Kiirmrin spectrum [2-1341
nS(n)
a-
U-*
[' 
_'
(2.3.24)
was proposed before the development of Eq. 2.3.16. Equation 2.3.24 satisfies
the conditions S(n) + O and dS(n)ldn : O for n : 0. However, for Eq. 2.3.24
to be consistent with Eq. 2.3.16, it can easily be shown that it would be
necessary to have Ii = 0.303t22, which does not appear to be the case in the
atmosphere. That Eq. 2.3.24 is, in general, not consistent with Eq. 2.3.16 can
be explained physically by the fact, discussed earlier in connection with Kol-
mogorov's hypotheses, that the higher-frequency spectrum is independent of
the large-scale features of the turbulence that determine Lj. Equation 2.3.24
is not used in applications where the magnitude of the higher-frequency com-
ponents of the longitudinal velocity fluctuations is of interest. However, it can
be used in applications in which the effect of the low-frcqucncy component
could bc irnporlant, suclt irs thc analysis of structurcs with vcry long nalural
48q'U
,,r(+)l
:1 N
-'l -6l-r5
' .1 A l lv'l( ', il,ill ttl( il,1il il,1 1 N(.1 6l
0.002 0.005 0.01
Wave number -lL- cycles/meteru(r2)
(a)
(b)
lfl(;UltE 2.3-3- (u) Longitudinal turbulcncc spcctra measured at Sale, Australia (based
rrrr 20 rccords)12-ill. Frorn A. G. I)lvcrrporl, "'t'hc Spectrum of Horizontal Gustiness
Ncrrr thc Gr<rund in High Winils,'' ettrrt. .l . lilt.1rrl Mcteontl. Soc., g7 (1961):202.
1/r; ('ornp:uison ol'spcctra givcrr by litls. J..1.21 rrrrtl 2.3.23. From E. simiu, ,,wind
Sltct'trir rrrxl I)ynurric Akrngwintl ltcsPorrst'," .l . ,\rr.rrt.l)ir,., ASCE 100 (1974): lg97_
t9 t0
Eq.2.3.21
U(10): 30 m/s, eo = 0.08 m
n (cycles/s)
62 lltE AtMOS|'l tLtilo t]()tjND^lty tAyt il
TABLE 2.3.2. Yafues of nS(n)lu?* lirr 7., : l).0t1 nr an<l {/(10) : 30 rn/s l2-7(ll
z:100m z:300mn
Cycles per
Second
(l)
Eq.2.3.16
or 2.3.21
(3)
Eq.2.3.16
or 2.3.21
(s)
All Values
of z,
8q.2.3.23
(6)
f
(4)
f
(2)
0.1
0.2
0.-5
1.0
0.255
0.450
1.125
2.250
0.70
o.43
0.24
0.15
0.586
1.172
2.930
5.860
0.37
0.23
0.13
0.08
t.4l
0.98
0.54
o.34
pclirxls ol' vibnrlion (c.g., corrrpliant off.shore platforms, which have motions
with lrcrirxls or irlrotrt 50 io 120 s). A modified form of the von Kdrmiin
ril)r:ctnur, bascrl orr lirs( principlcs and reflecting the variation of the spectrum
witlr lrc:ight irbovc gnrun(|, wils rcccntly proposed by Harris l2-l4}].
ljor thc purposc ol' studying thc sensitivity of tall building response to changes
in the valuc ol'various parameters determining spectral shape, an alternative
expression fbr the spectrum, consistent with Eq. 2.3.16, was proposed in
t2-lo). This expression depends upon the parameter 0 and an additional pa-
rameter allowing the modification of the shape of the lower-frequency part of
the spectrum, and is subject to the constraint imposed by Eq. 2.3.2. A similar
expression was developed in [2-74] to study the sensitivity of compliant struc-
tures to changes in the values of the parameters B and Ij, and to changes in
the shape of the lower-frequency portion of the spectrum consistent with Eq.
2.3.2. The expression of [2-74] is
.f < .f^ (2.3.25a)
f^<f<f, (2.3.25b)
f--f, (2.3.25c)
where a* and/are given by Eqs.2.2.18 and 2.3.17, n is expressed in Hertz,
z is the height above the surface (in the case of flow over the ocean, the height
above the mean water level), f, is the lower limit of the inerlial subrange (f,
- 0.2),f-is a parameter allowing changes in the shape of the spectral curve
forf < f,, and
ariQ)0
( o,f + b,72 + d,13
I
nS(2, n) j---- :\ cz * azf t brf,u'x 
I
I o.ze.s ''''
At:
z
8r : 0.26.f ,2tl
(2.3.zsd)
(2.3.25c)
!,,,,) '''5r
6lL,-.1',1' , 2(.1 ,,, .lil
tt2 : *2b2f,n
23 n lMr,!.t'ilt ilt(; il,lilitit tN(;t
i 2!",,(.1', -- .l',,,1 r .1,( 1., * zt,,,,l tnfr.
, -2lo,f^ o I,,, - I*l.f - Pt + b2(f^ -f ,'I
b,:t-t.sf.dl
t'z:0t-azf,-brf?
(2.3.2st)
(2.3.259)
(2.3.zsh)
(2.3.2s1)
(2.3.2sj)
llquations 2.3.25 are plotted in Fig. 2.3.4 for k : 0.4, zo : 0.001266 m,I : 35 m, U(35) :45 m/s (u*: 1.76 m/s), B : 6.0, f,:0.22, U: l8O
rn, andJ, : 0.07. Also plotted in Fig. 2.3.4 isF;q.2.3.23 (intemrpted lines).
Eqs.2.3.25
'..r*/-ur rtrt
r1llz)
l,'l(;tJl{lt 2.3.4. Spcctrl ol' hrrrgitrrrlirrrrl vt'locil.y lluctr.ralions (Eqs. 2.3.25).
I
I
o
0 0.150.0500
64 ilil n tM():;t'ilt nt(i lr()t,Nt)nl ty lAyt I
Unlikc l:q.2.3.24, Uqs. 2.3.25 atc cotrsis(ctrt with Ilt1. 2.3.1(r. lkrwt'vo', tlrc:y
do not satisfy the requirement dS(n)ldn : 0 lirr rr : 0. 'l'his rcquilotrtr:llt could
be satisfied by modifying Eq. 2.3.25a in thc imrncdiatc vicinity ol'a : 0.
However, such a modification is not necessary in practice since its efl'ect on
results of engineering calculations would be negligible. Finally, it is seen in
Fig. 2.3.4 that Eq. 2.3.23 significantly underestimates the spectral ordinates
at very low frequencies. This is due to the fact, noted earlier, that Eq. 2.2.23
implies that Il : g.
Finally, we mention the spectrum proposed by Hanis in 1968 12-1371:
(2.3.26)
wlrt'rt r I.lt(X)rrl{/(10). Likc F.q. 2.3.23, Eq. 2.3.26 does not reflect the
vlrrilrtion ol'thcr spcrclnurr with height above ground. However, it has over Eq.
2.3.23 tlte: rrtlvirrrtagc that it irnplies a nonzero integral scale of turbulence lf: 10001 U(r.)l IJ(I0)l/0 (in rneters).
2.3.4 Cross-Spectra of Longitudinal Velocity Fluctuations
The cross-spectruml' of two continuous records is a measure of the degree to
which the two records are correlated and is defined as
S'j,,r(r, n) : lf,,,r(r, n) + iSf,,r(r, n) (2.3.21a)
in which i: Jl. The real and imaginary pafts in Eq. 2.3.21a are known
as the co-spectrum and the quadrature spectrum, respectively. The subscripts
u1 and u2 indicate that the two records are taken a.t points M1 and M2, the
distance between which is denoted by r.
The coherence function is defined as [2-751
Q(r, n): [Coh(r, n)]2 : cf,,ur{r, n) + ql,,,r(r, n) (2.3.27b)
where
nS(n\ x.-:4.0-u'* -'" (2 + x2)s'6
) 15f,,,,{r, n\12c;tu2tr. nt : se:;)se, n)
) 1sf,,1r. r)12Qituttr.r) : ik:;)Nru n)
(2.3.27c)
(2.3.27d)
In Eqs. 2.3.27c and d, S(21, n) and S(22, n) are the spectra of the longitudinal
velocity fluctuations at points M1 and M2.
lA tlctailctl rlistrrssion ol tlrss spt'tlt:r is ptcst:ttlttl irr Allpt:rulix A,?.
:| n I M{ }t;t't ti til( I t,ililt,t I N(:t 65
'l'lrc lirlklwirlg ('xl)t('\\t('rr lot llrt's(luiur txrl ol llrr't'olrt,n'nt'c lirrrcli6rr (also
kltowtt its tlltn'()w lt:urtl t.trrss t'orr-crlirliorr) wiri l)ll)l)()ri(,rl irr lJ-7(rl:
Clrrh(r, n) : (, t (2.3.28)
wlrcrc
I
lil
ll
i:
r r'. altemativ ely [2-7 6],
nlc1,k, - zr)2 + C1Jy, - yr)tlt,,
u(t0) (2.3.2e)
.f: (2.3.30)
Irr llqs. 2.3.29 and 2.3.3O, !r, !2, and 21, Z2 are the coordinates of points M,,
A / 
' . the line M 1, M2 is assumed to be perpendicular to the direction of the meanrvirrcl, u(10) is the wind velocity at 7 : 10 m, and the exponential decay
t otrllicients C, C, (or C1r, Cy) are determined experimentally.
ln homogeneous turbulence the quadrature spectrum vanishls [2-64]. rnthe;rlrrursphere it appears that the ratio of quadrature spectrum to co-spectrum is
:.rrurll and that the square root of the coherence function may therefore be
:rss.rned, for engineering purposes, to be approximately equal to the reduced
t rr spectrum cu,ur. on the basis of wind tunnel measurements, it has been
:;rrggcsted in 12-771 that it is reasonable to assume in engineering calculations
S'u,ur(r, n) : Stt2(zr, n)Stt22r, n1"-i (2.3.31)rvlrc,rcf is defined by Eq. 2.3.30 and Cr: 10, C, : 16.* It appears, however,tlr:rl the exponential decay coefficients C, C, (or Cv, Cr), iatherthan being
rrrtlcpendent of roughness, are generally larger for iougir surface conditions
.rrt'lr as urban areas than for smooth surfaces l2-el. Moieover full-scale mea_',rrrr:rnents indicate that the exponential decay coefficients depend on height
;rlxrvc ground and, quite strongly, on wind speed, as shown in f,igs. 2.3.5 ind; \.6 12-60, 2-781. The dependence of the exponential decay coeftcients upon
rvirrtl speed is illustrated in Figs. 2.3.7aand2.3.7b, which represent Eq. z.i.zs
'A rrrtrtlified modcl of the spatial structure of turbulence proposed in t2-147) eliminates thel'lhrwirrg two drawbacks of Eq. 2.3.31. First, Eq. 2.3.31 does not allow for negative values oftlrt'lttl'lrttloncc co-spcctrum, regardless of spatial separation. For homogeneous turbulence thisrIr|lics lltat' contrary to its definition, thc mcrn ol'lhc lluctuating longitudinal velocity component
'hrs ttttl vitnish. Sccond' 8q.2.3.31 inrplics lurgc colrclalionsirlthe low-frequency components
' r'r'tr il lltc scpitlitlion is largc. Thc rrtort rt':rlrstit rrlrxlcl pnrposcd in 12-All may result in atr'(lrr( li()f) ol lltc calcttlirtctl rcsoniutl ltslxrrs(' ol slt'lrrlt'r slntclrtros by as rnuch 1ts 25%. Sce also
| .' t,lr{, 2 t491.
66 THE ATMOSPHFIIIC f]Ot]NI)AIIY IAYI II
0102030405060
u(10) (m/s)
Irl(;llltl,l 2.J.-5. Virliation ol-cxponential decay coefficient C,,, with wind speed (open
Irllrrirr) l.l 781.
and nlcasurctl valucs ol'thc square root of the coherence function for records
(takcn at points of cqual clevation) with U(10) : 20.8 m/s (Cr:u : 3.5) and
U(10) : 35.2 mls (Cr, : 8.8) t2-601. The dependence of the exponential
decay coeflicients upon terrain roughness, height above ground, and wind speed
is insufficiently documented and therefore represents a source of uncerlainty in
structural engineering calculations.
It was pointed out in Sect. 2.3.2 that relatively large uncertainties remain
concerning the integral scales of turbulence. In view of the close physical
r 0.0
8.0
6.0
4.0
C,,
2.O
0
F'I(;URI,l 2.3.6. Virrirrtiorr ol (', with wincl spcctl anrl hciglrl (opcrr lclr:rirr) l2 Tlll
o
(:ry 3.5
t/(10) 20.8 m/s
ao
;,:t n lM(lt;t,t il lll(. lUl ilIlt tN(:t 67
0.1 0.2 0.3 0.4
nll yr-!ztl/ul'tol
i -.'C,lt, - ',1I - (/t:\ (2.3.32)
with cr : 3.0 ovcr warcr.rrtl (', (r.0 rvr:r r.nd. A thcorctical approachtcptlrlccl in l2-t'tt)l suggosts thirt tlrr krrrliitrrrlirrirl cohcrcrrcc clcpcntls up1;n thc
-. 0.6
5
€ 0.4O
-* o.o|,
I
IE o.aoO
o.20.1
nlly*y2ll/u11o)
Run 'l 1 7l
lv,I v,]
l(meters) |
I o I o I rz Il"lolssl
l:l;l:: I
(b)
FIGURE 2.3.7. Measured values of Coh(l lr _ !zl, n) t2_6O1.
rclationship between turbulence cross-spectra and integral scales, similar un-certainties can be expected concerning the exponential decay coefficients.
Nevertheless, results of recent research quoted in lz-sol ,ugg".i that the valueL. = t0 is acceptable or even conservative from a structural design viewpoint.A similar conclusion regarding the value Cy = 16 follows froil 1Z_St1, ac_
:":d:g^,g,ylich. C, is a funcion of the raiio ly, - yrltz, as shown in Fig.2.3.8. Additional research into the vertical and lateral coherence of the lon_gitudinal velocity fluctuations is reported in [2-62] and, [2-g2,2-g3,2-g4,2-85, 2-86, 2-871.
In some applications the longitudinal (along-wind) coherence of the longi-tudinal velocity fluctuations is of interest. According to [2-gg], the longitudinalcoherence between the fluctuations at two points M,1x, , y, zj una M2(x2, y, z)can be expressed by Eq. 2.3.28, where
9oo
Run 1 18
68 lHf n tMosit,ilt llt(; tlotJNt)ntly tAyr lt
50
40
cy 30
20
0.t2345
lv1-vrltz
l"l(,lllltf,l 2..1.t1. l)clrcntlcncc 1rl'(i,. upon ly, - yrllz according to [2-g01. copyrightr, ) l()t{ l by l). ltcirlcl l,rrlrlislring ('ornpany.
Irrrlrrrlcrrcc ilrtcnsity /(1), thc distance lr, - rrl, and integral scale He), asslrowrr in lrig" 2.3.9.
2.3-5 spectra and cross-spectra of vertical and Lateral velocity
Fluctuations
It is shown in [2-31 that the spectra of vertical fluctuations up to about 50 m
may be estimated by the formula
10
o
c
J
N
I
X
co()
o
FIGURE 2.3.9. t,ongiruclinal
a : l(2.)lx, - .r.,11.;(r) 12 8el.
12
nllru
cohcrcncc as a lunction o| nl.i,l IJ f ur thrcc valucs o1.('opyrigh( () 1979 hy I). ltcitlcl l)rrhlishing (-orrrpirny.
:) ll A lM(l:;l,t il iltl . il,t il tut t N( il 69
rr,\,,,(l:, rr) _ .1..1(r/'
u)r, I I l(ll"' (2..j.-1.j1
(2.3.3s)
According to n)casurcnrcnts rcportccl in [2-60, 2 ttOl, thc cn)ss-spectrum of
vcrlical fluctuations ilt lwo points M1 and M2, ol'clcvution z may be expressed
ils
S.,-(Ay, n) : S.(2, 11\s-8navtu171 e3.34)
in which Ay is the horizontal distance between the points M1 and. M2.
The spectrum of the lateral velocity fluctuations may be written as
nS,(n) l5f-;T:(r+riJ-s'
where c(r) is a coeflicient that depends on t and z, is the longitudinal turbulent
lluctuation. If Eqs. 2.2.18 and 2.3.2 are substituted into nq. Z.Z.Ze .
0t '' r(tl \tt2.5 tnetzu) /
'l'he form of Eq. 2.3 .35 was proposed in [2-661 . Equations 2.3 .33 and 2.3 .35 ,irr which the parameter/is given by Eq. 2.3.17, are consistent with the re-
tluirement that, in the higher frequency range, the ratio of the vertical and
lrrteral to the longitudinal spectra is equal to 413 [2-651.
Cross-spectra of lateral velocity fluctuations can tentatively be assumed to
bc given by an expression similar toEq.2.3.3l , with exponential decay coef-
licients lower by about 33% rhan those used in Eq. 2.3.31 [z-go, 2-go]. Al-lcrnative expressions for the spectra of vertical and lateral velocity fluctuaiions,
based on a modified von Kdrmdn formulation which takes into account the
variation of spectra with height, were proposed in [2-140].
2.3.6 Dependence of Wind Speeds on Averaging Time
It fbllows from the definition of the mean value that mean wind speeds depend
upon the averaging time. As the length of the averaging interval decreases, the
rnaximum mean speed corresponding to that length increases. The relation
bctween the wind speed averaged over / seconds, u,(z), and the hourly speed,
L/3u1p(z), may be written as
U,(z) : Uzr,cnk) + cG)71n (2.3.36)
(2.3.37)
'l'hc cocllii:icnt ('(1) is clclcrtttittetl on llrc lrlrsis <ll's(atistical stuclics of wind
sPcctl rccorcls. llcsrrlts tll'sur'lt slu(licH w('t(' rclx)tlcrtl l-ly I)urst l2-9 ll anrl arc
U,(2,) : U,,,,n,(.) (l
70 ilil n tMofit't il til(: n()t,Nt)nily tAyt il
100
r (s)
1 0,000
l"l(illl{l'l 2.-1,10. It:rrio o| pnrbablc maximum speed averaged over period r to that
luvcragctl ovct' ()llc lxrur l2-921.
plotted in Fig. 2.3.10, which corresponds to open terrain conditions (zo = 0.05m) and an elevation z : l0 m. values of c(t) consistent with Fig. 2.3.10 arelisted in Table 2.3.3.
Experimental results presented in 12-931suggest that Eq. 2.3.36 is appli-
cable, with the values of the coefficient c(t) of rable 2.3.3, to wind ,p""0,
over terrains with roughness lengths of up to Zs : 2.50 m.
Mean speeds used in the design of tall buildings are hourly averages, while
information on wind intensities is currently provided in terms of fastest milewind speeds at about l0 m above ground in open terrain. Fastest mile wind
speeds are averaged over the time required for thi passage over the anemometer
of a volume of air with a horizontal length of one mile. From this definition itfollows that for the fastest mile u|the averaging time in seconds is r : 3600/
UJ, w^here Uyis given in miles per hour. For eiample, if UJ.: 90 mph, thent :40 s and the corresponding hourly mean is, from Fig. i.z.to,9ofl.2g =70 mph (31 m/s). A recent study 12-1441 essentially.o-nfi.-, the validity ofFig. 2.3.10. For hurricane winds see Sect. 2.4.3.
2.4 HORIZONTALLY NONHOMOGENEOUS FLOWS
Horizontal nonhomogeneities of atmospheric flows may be ascribed either toconditions at the Earth's surface (e.g., changes in surface roughness, topo-
TABLE 2.3.3.Coefficient c(t)
ll
t:(r) 3.00
l0 20
2.32 2 (X)
30 -50
r .71 r .35
100 200 3(x)
1.02 0.70 0.54
600 1000 3600
0.3(r 0. l(r 0.(X)
1.5
81.4
5
] t'.
1.2
l_1
1.0
:,,1 ll()1il/()NtAt lY N()Nll{}M()(it Nt ()U:i lt{)w:; 71
glirllhic lcirttttcs ol llrc tt'rr;rirr) or'1o tlrc nrclcorrlogit'lrl rurlrrr.t'ol tlrc llow (as
irr tltc cuso ol'ttrrpit'irl tyt'krrrcs rlr ol'thurtrlr-:rstorrrrs). Wlrilc thc structure of
llrrizonlally honrogcrrr:orrs llows is basically wcll rrrrtlclsltxrcl, rcsults obtained
in thc study ol'horizorrtully rxrrrhonrogcncous llows urc lo a large extent still
irrcornplctc or tcntalivc. Sornc of these rcsults arc, ncvortheless, of interest to
lhc designer and will thcrcfbre be discussed hcrcin.
2.4.1 Flow near a Change in Surface Roughness
ln the case dealt with in the preceding sections, of a horizontally homogeneous
lkrw, it is assumed that the surface roughness is uniform over an infinite plane.
In reality, a site is limited in size; the flow near its boundaries is therefore
rrll'ccted by the surface roughness of adjoining sites.
Useful information on the flow structure in the transition zones may be
ohtained by considering the simple case of an abrupt roughness change along
:r line perpendicular to the direction of the mean flow [2-13, 2-94,2-95, 2-96,
) 9l ,2-981 (Fig. 2.4.1). Upwind of the discontinuity, the flow is horizontally
lromogeneous and, near the ground, governed by the parameters Zs1 &\d u.a1.
l)ownwind of the discontinuity, the flow will be disturbed over a height h(x).
'f 'his height, known as the depth of the internal boundary layer, increases with
(ho distance x until the entire flow adjusts to the roughness length zs2 of the
lcrrain downwind of the discontinuity.
If the investigation is limited to the lower portion of the boundary layer, it
rrray be assumed that the flow is two-dimensional. For steady flow, and ne-
glccting the pressure gradient force-the effect of which was shown to be
insignificant [2-98]-the equations of continuity and of balance of momenta
nray be written as
ua+w9!:taJ0x 0z pAz
AU AW
-+-:00x 0z
(2.4.1)
(2.4.2)
Since Eqs. 2.4.1 and2.4.2 contain three unknowns, a third equation is required
Io close the system. In the solution of [2-96] the mean turbulent field closure
was used (F,q.2.1 .9), which, for two-dimensional flow and with phenome-
rrolrrgical relations similar to those proposed in 12-91 and [2-10] (see Eqs.
l. I . l0-2. l. 13) takes the form
II
---J1
125
--{ " fii,{c
zo1, u 11
l,'l(Jtll{lt 2.4.1. l;low zorrt's rlowrrwrrrl ol rr t'lurrrgc irr loLrghncss ol'lcr-rlin.
llll n tM()lit'l il til(: tl()t,Nt)nny tn yl n
in which l, is the mixing length.
In horizontally homogeneous flow, the validity in the surface layer of the
Iogarithmic law implies the following expression for the mixing length [2-l]:
L : kz e.4.4)
([ o l) Ii,ll.wirrg M.nin 12-991, ir is assumed in [2-96] that in Eq. 2.4.3
rlrt' srunt' cxPressiorr lirr /. holils ncar the ground throughout the flow, inituding
(lrc rlistrrrlrt'tl lkrw tlowrrwind ol' thc discontinuity.
'l'lrc lrorrntliuy corrtliliorrs lilr Eqs. 2.4.1-2.4.3 are
;, ,t ll( )lit,/( )N tAl I y N( )Nt t( )M( )( ,t Nt ( )llt; I I ( )wl; 73
describccl irr lclrrrs ol llrt'I;u;rnr(:l(:r's io2, u*2. For practical purposes it may be
assumed that ( l) tlrc Plrlilt' t'ollt'sporrcling to these parameters is completely
established at distanccs ol rrrorc llrrrn -5 krn downward from the roughness
change, (2) for a distancc dowrrwirrtl ol' thc roughness change of less than 500
m the profile is the same as upwind ol'the discontinuity, and (3) in the interval
500 m < "r < 5 km the profile is logarithmic below line AB, with zero speed
at the ground surface, and a speed at elevation x/12.5 equal to the speed at
that elevation upwind of the roughness change t2-421.
A more "exact" model of the internal boundary layer growth is
(2.4.10)
where zs, is the larger of zs, and zo, [2-100]. Equation z.4.lo was based on the
analysis of a considerable number of data and holds for both smooth-to-rough
and rough-to-smooth transition. It is approximately valid for values h(x) <
0.26 where 6 is the boundary layer depth. For additional references on flows
near a change in surface roughness, see [2-100] and [2-138].
2.4.2 Wind Flow over Hills
wind tunnel investigations of simulated flows over ramps and escarpments are
reported in [2-101, 2-102,2-1031. For open terrain conditions, ratios (u2lu)2
at various stations given in t2-1011 are represented in Figs. 2.4.2 and,2.4.3.
(uz and u1 denote wind speeds at height z above ground downwind and upwind
of the ramp, respectively.) Measurements of l2-lo2l tend to corroborate these
results. The results of [2-101] and 12-1021also suggest that for ramps with
slopes of about 2O% to 35%, the ratios (J2l()1 are, for practical purposes,
independent of slope. However, for a ramp with a l0% slope, the ratios (U2
- U)lUl are only about one-half as large as in the case of a 2O% slope
[2-101]. More detailed wind tunnel measurements of ratios u2lu1 for escarp-
U A(rlpt w dtrlpl
0.t6 0x -o.to a, _ritll , 
I (rtp il(rtptfitlt\
p oz az \o.ro az I az )
(rlol3/2
- ', -n- L :0 (2'4'3)
h(x) : o.28zo,(*)'-
I
I
'l
i
i
ii
ii
l
i
L
I
I
I
I
I
l
L
l
ri
IJ : 2.5u*t ln L
Zot
w:0
r : puz*t
U:O
w:o
1,"-,
(2.4.s)
(2.4.6)
(2.4.7)
(2.4.8)
(2.4.9)
l., : olo o,. (#),,,1'
)0;z:zozi)
(see Eqs. 2.1.7 -2.1.8).
Equations 2.4.1-2.4.3 with the boundary conditions, Eqs.2.4.5_2.4.9, werc
solved numerically in [2-96) for various values of the parameter m : ln(261/
zoz). In the case of the smooth-to-rough transition, the calculations indicate that
three regions may be distinguished downwind of the discontinuity (Fig. 2.a.D.
In region I (above line AB, approximately defined by a slope or t : iz.5;, the
velocity is essentially equal to the velocity upwind of the discontinuity. This
result is consistent with conclusions reached independently by other authors
12-941 and 12-95). In region III (below line AC, defined by a slope of aboutl: 100) it may be assumed, at least very roughly, that the flow is adjusted to
the new roughness conditions, that is, is determined by the same parameters
Zoz, u*z that would control the flow if the roughness length were everywhere
Zs2. rn region II, as the distance downwind from the discontinuity increases,
the velocity profiles deviate increasingly from the profile given by Eq.2.4.5
and the turbulcnc:c cncrgy varics graclually l'rom linc AB, whtre it is prcsunrably
ncarly lhc sillllc:ts tlpwitttl ol'lhc tliscontirrrriiy, lo lirrc,4(1, wlrcrc it lrrtry bc l'l(;tll{l,l 2.4.2. Wirrrl lrtoltlt:, r)\'( r ;ul (':i( iul)nr('nl ll lOll
74 iltf ArM(xil,ilf nto t(tt,Nt)Any tAyt tt
FIGURE 2.4.3. Wind profiles over an escarpment t2_l0ll.
ments with 25%, 50%, and loo% slopes and for a cliff, as well as measure-
ments of the root mean square of the longitudinal turbulence fluctuations, are
reported in [2-103]. The ratios urlu, of t2-1031 are similar to those of Fig.
2.4.2, except at low elevations (about 5 m above ground) where they are larger
by about20%.
Results of theoretical and numerical studies of wind flows over hills have
been reported in [2-lM, 2-105,2-106,2-lo1 ,2-108]. For a hill with maximum
height ft, a longitudinal scale L(L >> /r) and a profile hf(xlL), wheref(xlL)< 1 (Fig. 2.4.4), the following resulr was obtained in [2-1041:
U,_-_:: .l *ul
ho ln2(Llz$tr(o)(", z) (2.4.11)
in which U2 is the wind speed at (x, a), U1 is the wind speed at (x : -@, z),
ze is the roughness length, x is the horizontal distance (see Fig. 2.4.4), z is the
height above surface of the hill at the point considered, il(0) is the approximate
value of a dimensionless quantity representing the perturbatlon to the upwind
velocity due to the presence of the hill,
Lln(llz) rLn(zll) + ln(//zo)l
li'l(Jllltl,l 2,4.4. l,rolilc: ol l low lrill.
'4 ll()|il,/oNlnl lY N()Nl t()Ml ,{it Nt (}(,li |()wl;
lnd
(2.4.t2)
(2.4.13)
(The quantity / is the thickness of the intemal boundary layer createdby the
change in surface shear stress as the air flows over the hill. This internal
boundary layer is similar to that caused by changes in terrain roughness.) For
any hill symmetric about x : Q, ;(0) can be expressed in terms of Kelvin
lunctions as shown in [2-104]. In the particular case
t f" .l 't.ttt,\lt.\lt ,l
,, J ,, elt,l
/x\ Ir\z) : T +1ld
I /L\""//.r : g \;/
(2.4.t4)
in which L is the horizontal distance from the top of the hill to the point at
which the height is half the maximum height ft, the quantity o : l. Values of
l?(0) corresponding to the profile 2.4.14 are represented in Fig. 2.4.5 atxlL:
0 (top of the hill), xlL: -0.5 and xlL:0.5, for llzo: 1gz, Llzo:2.1 t
lOa (curves A), llz(): 10a, Llzo : 3.2 x 105 (curves B), and llzo : 1gs,
Llzo:3.6 x 106 (curves C). Values of t(U2 * U)lUl (L/ft) calculated in
l2-l04l are listed in Table 2.4.1. The analysis and results of [2-104] are valid
- 01l
L
l,'l(;tll{U 2.4.5. Vulucs ol r?("'
f ;low Ovcr a Low llill." Qrtrtrt
0 o4 08;(.))_>. -0.2-0.1 6 iio)e
, " )=ou
l;rortr l' S .l;rt'ksotr:rtttl .1 . ('. ll. lltltl. "'l'urbulcrrrt
.lt,ut llrtvtl ll,'1,'r,trtl . ,lttt.. ll)l ( 197.5). 929 ().55,
76 llt n tM()sl,lFilto n()(,Nt)Ally tAyt n
TAIII-|t 2.4.1. Values of l(U, - Il)lIltl(l,lltl il 'l'op ol'lliil
zll zull : 10 ' ,.t/l - ft) 4 ;,,// - l0 5
0.0
0.1
0.3
0.6
1.0
t.5
2.1
2.09
2.46
2.33
2.20
2.O8
1.9'7
l .88
1.87
2.13
2.07
1.97
1.87
1.79
1.73
t.72
1.92
l .85
1.78
t.72
1.66
t.62
lirl hills in rural tcrrain (zo : 0.03 m) with 0.1 < Z < l0 km and with ratioshll, > l(1,,/1,)0 '. For cxample, if a6 : 0.025 rr7, L :500 m, and h : 25 m,
tlrcrr, lirrrrr |tq.2.4.13,2.111 l: 1.0 x l0 3, to which there corresponds, from'l'ablc2.4.l, UzlUt: l.l2atzll:0.1 (or z = 2.5 m). Thetheorybecomes
lcss accuratc in rough terrain (zo : 0.5 m), the actual speeds U2 being lower
than thosc givcn by Table 2.4.1.
For flow over escarpments (Fig. 2.4.6), the following relation is derived in
[2- lOe]:
!:=r+!!1n(LtzdhUt L 4r ln(zlzi "'
(zlLi)2+U+(x/L)12
klL\2+.1-(xtL)12 (2.4.rs)
in which notations similar to those of Eq. 2.4.1r are used. It is suggested in
12-1091that Eq. 2-4.15 may be applied to flow over escarpments with t <<
5 km and with slopes as large as 202" or so. For example, if L : 250 m,h:50 m, and zo:0.025 m, for x: L andz: l0mthe ratio U2lU,:
1.19. According to [2-1031, Eq. 2.4.15 provides useful indications of the trendsof the variation of u2lu1 with x and z, rather than dependable quantitative
results.
Full-scale and wind tunnel measurements of flows over two- and three-
dimensional hills and over embankments are reported in [2-l0g] (which extends
the analytical approach of [2-1041ro three-dimensional hills), and [2-tlo,2-lll,2-112, 2-113,2-114, 2-115,2-l4zl. As noted in [2-ll0], estimates obtained
independently in l2-lo4l, [2-105], and [2-106] agree well with each other and
with the full-scale measurements of [2-110].
l,'l(Jlllll,) 2.4.(r. lrlow ovL:t' cscitrpnlcllts noltrlions l2 1091.
\l IYN()Nl l()M()(il Nt t)t,i; lt()Wl; 77
A sirtrplc: ttterlltrxl lot r'itlt'ulrrlirrg wirrtl spr:ctl irrt.r't.:rscs ( "spectl-ups") lirrbuildings l<lcalctl ()ll lw() tlitttt'ttsiotral ridgcs or cscarl)nrcnls or orr axysinrrnctrichills is includccl irr llrc AS('lj 7-9-5 Stancl:rrcl 12-l'3gl iintl, in corrrputcrized fbrm,as parl of thc cliskcilc "l)cvckrpmcntal cornpu(cr-llascrl Vcision of ASCE7-95 Standard Provisi'ns krr wind Loads" [17--51 appcnded to this book.
2.4.3 The Hurricane Boundary Layer
The horizontal inhomogeneity of a hurricane wind flow over a uniform, hori-
zontal surface is associated with the variation of the pressure gradient with
distance from the centerof the storm (see Eq. 1.3.1). tn aerivinfthe logarith-mic description of the mean velocity profilei near the ground ftq. Z.Z.fS; itwas assumed that the flow in the free atmosphere is geostrophic lSect. 2.2).This assumption does not hold in the region of highest winds of the mature
hurricane; the question therefore arises as to wheiher or not Eq. 2.2.1g isapplicable in this region.
Several analytical solutions of the hurricane boundary{ayer problem have
been atempted so far [2-116,2-117,2-llg,2-llg, z-l2iJ], att oi wnicn appty
to steady, axisymmetric mean flows. The solutions of [2-116] through p lirjl
are based on the assumption that the eddy viscosity is constant, and they .unnoi
therefore provide a reliable detailed description of the flow near the ground. Aconsiderably more realistic modeling of the turbulence effects is used in
l2-l2ol, in which the equations of motion and continuity are supplemented by
the turbulence closure relations discussed in Sect. 2.1 (Eqs. 2.1.g-2.1. 13). Th;
system of equations thus obtained-in which the expre.ssion for the pressure
gradient field given by Eq. 1.3.1 was used-was solved numerically assuming
values of the surface roughness of 0.002 m to 0.90 m, differences between thehigh pressure in the far field and the low pressure at the storm center of 60 mbto 140 mb, and radii at which the gradient wind has a maximum value of 30km to 50 km. According to 12-1201, in the lowest 400 m of the boundary layer
the mean wind profiles differ only insignificantly from the logarithmic profiles
described by Eq. 2.2.18.
As Table 2.4.2 shows, for decaying hurricanes the increase of mean wind
TABLE 2.4.2. Yariation of wind speeds with Height in Hurricanes caror andEdna
Height
above Ground
(m)
Carol Edna
Mean Max. l-rnin Mean Max. 1-min
I 1.3
22.9
45.7
r08.2
t25.(l
t4.s
18.1
24.1
29.1
22.8
l(). I
t58
l.t tl
r t.8
20.3
259
17.0
25.9
30 rJ
78 IltF ATMospHEntc frot,ND^ny iAyt n
spceds with hcight in appnrxintalc ircconlirrrcc wilh tho logarithnric l1w w1s
documented in 1954 following thc passagc ovor lJnxrkhaven National Labo-
ratory of hurricanes Carol and Edna 12-121, p. 461.
More recently [2-122] reported extensive observations of mean wind speeds
recorded at elevations from 9. I m to 390 m during the passage of four decaying
tropical cyclones over northwestern Australia. The mean wind profiles were in
most cases irregular, and as noted in [2-1221, a wind speed maximum was
often observed at 60-200 m. Nevertheless, the profiles corresponding to the
largest l0-min wind speed observed during each storm at 9.1 m were by and
large consistent with the logarithmic law and a roughness length of 1 to 4 cm,
as can be seen in Table 2.4.3, in which the only significant anomaly is the
speed observed during cyclone Karen at 59.7 m elevation.
whether the logarithmic profile holds in the case of mature hurricanes re-
mains an open question. Implicit in the provisions of the 1975 Southern Build-
ing code 12-1231is the assumption that hurricane wind profiles are considerably
flatter than would be indicated by the logarithmic law. To date there is no
conclusive evidence that this is the case. Since design wind speeds specified
in building codes correspond to an elevation of l0 m or so, the use of this
assumption in the design of tall structures might be imprudent.
According to a study of tropical storm and hurricane records peak gust
factors for hurricane speeds are about lo% higher than indicated in Fig. z.3.lo
for extratropical storms t2-1351. The conclusions of 12-1351(see also Iz-1391,p. 155) were based on the analysis of about 12 records.
Reference !2-l4ll contains information on gust factors, longitudinal turbu-
lence intensities, scales and spectra in typhoon Mireille, that traveled over
omura bay (Nagasaki) and passed directly over the anemometer placed at
100-m elevation on a tall building at the shoreline. The gust factors were found
to decrease as the mean speed increased. The turbulence intensity during the
period of the strongest 10-min wind (about 25 mls) was over 25%. Thetur-
bulence scale was estimated to be 780 m during that period and 2g0 m during
the 10-min period preceding it. The von Kdrm6n spectrum (Eq.2.3.24), with
TABLE 2.4.3. l0-min Speeds at Various Elevations Corresponding to
Maximum 10-min Speed at 9.1 m during Four Tropical Cyclones
WindSpeed (m/s)Height
above Ground
(m)
Beryl
(12173 12:00)
Trixie Beverly Karen
(2175; 18:30) (3175:2r:O0) (3177; t90O)
9.1
59.7
19t.4
279.2
390. r
32.5
39.5
41
.57..5
30.5
5l
43.5
48.s
4ti..5
21
32
36
22
28
31
34
Noto: Nurrrbcrs ilr pirrcrrlltcscs irulit'irtc tltc rrronth, ycar, trrrrl lrorrr ((iM'l') f,'l(;tlRl,l 2.4.7 , I lrrrricrrrrr. wirul sllcctl reconl.
HoFtzoNTAt I y N( )Nt t( )M( x lt Nt ()t,s I I ow$ 79
thc lurbulonco scitlc:s.ittsl irrtlicatotl, rnatchctl thc rncirsul'ctl spcctra wcll, cxcept
lirr thc rangc ol'abrlut 0.o25 to 0. l5 Hz, whcrc it undcrcstimate<l the measured
spectra by as much as l0]ol, fbr certain frequencies. Finally, according to
l2-l4ll, surface wincl spccds in the eye can be comparable with or higher than
the estimated speeds at the gradient height level; see also comments following
Eq. 3.3.7 and Ref. 13-791.
Two more notes on hurricane winds are in order. First, in the immediate
proximity of the eye, flow separation occurs and the boundaryJayer assump-
tions break down (see Sect. 1.3). The implications of this phenomenon to the
designer are not yet well understood. Second, as the hurricane moves inland,
filling occurs (see Sect. 1.3) and the maximum winds tend to decrease. Em-
pirical descriptions of the wind intensity reduction as a function of distance
from the coastline were proposed in 12-1241, [2-1251, and [2-l5l]. According
to [2-1251, the ratios of peak gusts at 50 km, 100 km, and 150 km inland to
peak gusts at the coastline are, approximately, 0.90, 0.80, and 0.70, respec-
tively. See also [3-57] to t3-601.
A hurricane wind speed record, which clearly indicates the passage of the
eye, is shown in Fig. 2.4.7. The nonstationary character of the record of Fig.
2.4.7 is noteworthy, as is the contrast to Fig. 2.3.1. For techniques to char-
acteize turbulent fluctuations for nonstationary records, which are typical of
hurricanes but characterize other storms as well, see [A2-14] to [AZ-241.
2.4.4 Thunderstorm Winds
The cold air flow which, in a thunderstorm, spreads horizontally over the
ground was compared in Sect. 1.3 to a wall jet. Just as in the case of the wall
jet, the surface friction retards the spreading flow, which may thus be expected
to be similar, near the ground, to an ordinary boundary layer [2-126, 2-127,
2-1281.
of particular interest to the designer is the so-calledfrst gust (or gustfront),
that is, the wind occurring in a thunderstorm that exhibits a considerable and
relatively rapid change of speed and direction (Fig. 2.a.8). Following l}-l}9l
and [2-130], the wind speed increase and the time interval during which this
MDIIT l0 Plrl
I lll n I M( ): it ,l ll lll( I ll( )l,Nl )/\t ty I n., I I I
c. l77m f. 444m
b. 90m e. 355m
a. 45m d.266m
FIGURE 2.4.8. Thunderstom wind speeds recorded simultaneously at six elevations
from 45 m to 444 m above ground near Oklahoma City (courtesy of National Severe
Storms Laboratory, National Oceanic and Atmospheric Administration).
increase takes place will be referred to as the gust size A,V and the gust length
At, respcctively. Depending upon thunclerstorm intcnsity, thc gr-rst sizc rnuy
vary appnrxirtr:t{cly l'roltt S ltt/s to 30 rtr/s, whilc lhc gtrst lengtlr nuly rangc
ll'otn ir l'cw rrrirrulcs (o ?0 rrrirrrrlt's ()r-s().
'l'lrc llrtlllrlt'r'slotttt wiltrl tt'trtllls n';xrr'lt'rl irr l.) | lOl r;rr1'1'r':;t ilrrrl rlrlrirrl'tltt'
n lMr '"t ilI t{tl . tt()t ,Nt)nl ty lnyl it tllt( t:;{)il ()|t nlJ lt()w Bl
itttc:rvitl A/: (l) rr;t lo l(X) rrr :lltovt' gtrrrrrrtl wirrtls slrt't'tls v:rry wrllr lrt:iglrt irr
accortlancc witlr llrt'1o1,:rrrlltrrric lrrw, urrtl (l)) irlxrvr' l(X) lrr llrt'vtrr-ilrti1ll gl
wincl spccds witlr hc:iF,lrt is Ircgligiblc. ('t'his is rerrsolrirlrl-y t'orrrplrliblc with the
rccords of Fig. 2.4.8.) Ntl rolation betwccn wirrtl spcctls in tlillorcnt rclughness
regimes, based on a rational model of the thunclcrstorrrr wind lklw, has been
derived so far. To convert thunderstorm wind spccds rccordcd over open terrain
into wind speeds over built-up terrain, the samc procedure is used in current
practice that is applied to extratropical cyclone winds (Eq. 2.2.26), even though
the notions of gradient height and gradient speed have no meaning in the case
of thunderstorms. whether or not this practice is acceptable for structural en-
gineering purposes is a question that merits investigation, particularly if it is
recalled that, according to [2-131], about one-third of the extreme wind speeds
recorded in the United States are associated with thunderstoms.
2.5 ATMOSPHERIC BOUNDARY LAYER EFFECTS ON OCEAN
FLOW
If suflicient data on wind flows over the ocean are available, it is possible to
model the ocean waves induced by those flows. Such modeling is referred to
as hindcasting. A vast literature on this topic is available (e.g., see [14-34,p.
5201).
Mechanisms by which kinetic energy is transmitted from the atmospheric
boundary layer to the ocean water are exceedingly complex. In some appli-
cations one may assume, however, the existence of uniform surface shears at
a hypothetical horizontal ocean/atmosphere interface (see Sects. 2.2.2 and
2.2.3). one among many instances were this assumption is used is the recent
modeling of wind-induced along-shore ocean currents over bottom topography
characterized by comrgations normal to the shore. It was shown in [2-145] that
the equations of motion of the wind-induced ocean motions can be represented
approximately by the equations
a* : AyH(x, y, z) -t eg{x, t)
di : - u* H(x. y.;) * cg2())
2 : eg(x, z, t) (2.s.D
where e is small, x is a basic along-shorc specd, y is proportional to the out-
ol-phase component of a strcarn lirnction lilr rnoli<tn due to the topography, z
is thc cnergy-cnstropy, ancl
,(r = ,:r t r1y I t\tl.
,tj, ,.\'. ,(r ,. 1,, I {r l)1r,, I tltll (l-5 l)
82 lltt AIM(xitlll tito tlot,Nt)nny tAvt n
H(x, y, z) : j.y" + zx t ,l(r,lil r:).*' 1t'r1l;xn, ,1; = I tD'l
(2.s.3)
and where 6 is the amplitude of the bottom topography comrgations, er is a
friction coefficient related to the eddy viscosity of the ocean flow, and €ze and
er(t) arc, respectively, the steady and fluctuating wind stress at the ocean
surface. The wind stress fluctuations of interest in this problem correspond to
the very low frequencies studied in [2-1431. Reflecting the effect of the bottom
comrgations, Eqs. 2.5.1 form a bistable system capable of chaotic behavior
even if the fluctuating wind stress is assumed to be harmonic, as was done in
12-t451.
The model used in [2-145] becomes more realistic if it is assumed that the
wind speed fluctuations are random, rather than harmonic. Using chaotic dy-
namics techniques developed in [6-101], and van der Hoven's results on the
spectra of low-frequency wind speed fluctuations [2-143], the case of random
wind excitation was studied in [2-146]. Among other results, [2-146] provides
estimates of lower bounds for the probability that during a specified time in-
terval, the amplitudes of the wind-induced fluctuating currents do not exceed
a safe threshold associated with the barrier of the unforced system's double
potential well.
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,Slr:r1rcs," liltruttl. ltt\\'t i'lt'tt't'rr,l l.t, tlt, l/1. .t"t1 .t71,
2-81
BB ilil ntMo: ;t,t lll(:l()uNt)Altyl/\\ttl
'2-104 I'. S..lltt'ksott ltrltl .l ('. ll. llrrrrt. "'lurlrrilcrrl l;low ovcr it l.()w llil. .l lirttttl
Mclrontl. Jar'., l0l (1975), ().)t) ()\'l
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2-106 P' A. Taylor, "Numerical Studics ol'Ncutrally Stratitied Planetary Bounclary
Layer Flow Above Gentle Topography," Bountl. Layer Meteorol., 12 (1977),
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23 (1982) rt5-124.
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Fourth International Conference on Wind Effects on Buildings aml Structures,
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Meteorol. Soc., lO4 (1978), jl9-j28.
2-lll F. Bradley, "An Experimental Study of the profiles of wind Speed, Shearing
Stress and rurbulence at the crest of a Large Hil1," "/. Royal Meteorol. soc.,
106 (1980), 101-123.
2-ll2 R. E. Britter, J. C. R. Hunt, and K. J. Richard, ..Air Flow Over a Two_
Dimensional Hill. studies of Velocity Speed-up, Roughness Ell'ects and rur-
bulence," J. Royal Meteorol. Soc., l07, (1981), 9l-110.
2-ll3 G. J. Jenkins et al., "Measurements of the Flow Structure arouncl Ailsa Craig,
a Steep, Three-Dimensional Isolated Hill," "/. Royal Meteorol. soc., r07 (l9gl),
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2-ll4 c. Sacr6, "An Experimental study of the Air Flow over a Hill in the Atmo-
spheric Boundary Layer," Bound. Loyer Meteorot., 17 (lgjg),3g1_401.
2-ll5 T. Hauf and G. Neumann-Hauf, "Turbulent wind Fkrw over an Embank-
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2-116 S. L. Rosenthal, A Theoretical Analysis of the Fie kl ol'Motiptt in the Hurricune
Boundary zrzyer, National Hurricane Research pro.iccr, Ilcp.rl No. 56, u.s.
Department of Commerce, Washington, DC, 1962.
2-ll7 R. K. smith, "The surface Boundary Layer of a Hurricanc,' 'l't,llus,20 (196g),
413-484.
2-ll8 G. F. carrier, A. L. Hammoncl and o. D. George, "A M.dcl of the Mature
Hurricane," J. Fluid Mech., 47 (1971), 145-1j0.
2-ll9 E. Simiu, "variation of Mean winds in Hurricanes," J. Eng. Mech. Div.,
ASCE, 100, No. EM4, Proc. paper 10692 (Aug. 1974) 833_837.
2-120 E. Simiu, V. c. Patel, and J. F. Nash, "Mean wind proliles in Hurricanes.,'
J. Eng. Mech. Div., ASCE, 102, No. EM2, proc. papcr 12044 (April 1976),
265-213.
2-l2l Survey of Meleorologicttl Frtttors Partincnt to Rt'tlut,tirn of Lpss sf'Lifc und
Propcrty in Hurricutrt' Siluttlitn.r, Nutionul Flurricanc Rcscarch prgjcct Rcp6r1No' 5, tJ.S. l)cP:trltttt'ttl ol ('onttttcrcc. Wclrlhcr Ilurcur.r, WasIirrgl1trr, IX',
March l()57
r.r( .t r, Bg
I lll l(..1. Wilsorr. "( lr;rr:rtlt.risl i(.s ()l lll(.srrlrr.lorrtl l.:ryt.l Wrrrrl Slrut.lrue irr'l'r.op
lclrl ('.yt lorr,'s,' llrrrt'lrrr ol Mclt'onrhrl.ly" l)r.lrt . ol St.it.lrt.t. irlrtl 'l'cchnology,
Mcllrottlttc, I'lt'Jr:tli'tl lirr Ittlcrttationirl ('orrli'rr'rrr't'on'llo;lrcul (lyclones, Perth,
Austlllilr. Nov- l()7().
2-123 Srnthcn Stt,tthtrd Buildittg Ci,rlr, llirrrrirrgltrrrr, Ala., 1965, p. l2_5.
2 124 w. Malkin, Filling ond Intensity Changt:; irr Hurricanes over Lanj, National
Hurricane Research Pnrject, Report No. 34, u.s. Department of Commerce,
Washington, DC, 1959.
2-125 J. L. Goldman and r. Ushiyima, "Decrease in Maximum Hurricane winds
afterLandfall ,".1. Struct. Div., ASCE, 100, No. STI, proc. paper 10295 (Jan.
t974), t29 t4l.
2-126 M. B. Glauert, "The Wall Jet," J. Fluid Mech., f (1956) 625.
2-127 P. Bakke, "An Experimenral Investigarion of a wall |et, J. Fluid Mech.,2
(.t957) 467.
2-128 J. Bumham and M. J. Colmer, On lnrge Rapid Wintl Ftuctuations Wich Occur
when the wind Had Previously Been Light , Technical Report No. 69261 , Royal
Aircraft Establishment, Famborough, U.K., 1969.
2-129 M. J. colmer, "on the Character of rhunclerstorm Gust-Fronts," Technical
Report Aero 1316, Royal Aircraft Establishment, Famborough, U.K., 1971.
2-l3o R. w. Sinclair, R. A. Anthes, andH. A. panofsky, variationof the LowLevel
winds During the Passing of a Thunderstorm Gust Franr, NASA Contractor
Report No. CR-2289, 1913.
2-l3l H. c. s. Thom, "New Distributions of Extreme wind speeds in the United
States, .r. Struct. Div., ASCE, No. ST7, proc paper 603g (July 196g), l7g7_
1801 .
2-132 S. D. Smith, "wind Stress and Heat Flux over the ocean in Gale Force
Winds," J. Phys. Oceanography, l0 (May l98O),'709-726.
2-133 L. Knigermeyer, M. Gninewald, and M. Dunckel, "The Influence of Sea
Waves on the Wind Profile," Bound. lnyer Meteorot.,14(lg7g),403 414.
2-134 T. von Kiirmein, "Progress in the Statistical rheory of rurbulence ," proc. Nat.
Acad. Sci., Washington, DC (1948), 530-539.
2-135 w. R. Krayer and R. D. Marshall, "Gust Factors Applied to Hurricane winds,"
Bull. Am. Meteorol. Soc., 73 (1992), 613-6lj.
2-136 J. Bidtry, Personal communication. 1981.
2-137 R. I. Harris, "The Nature of wind, in The Modern Design of wirul-sensitive
Structures, Construction Industry Research ancl Information Association. Lon-
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2-138 D. M. Deaves, "computations of wind Flow over changes in surface Rough-
ness," J. Wind Eng. Ind. Aerdyn., 7 (1981), 65-94.
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,ure.s, American Socicty o1'Civil E,nginccrs, New york, 1995.
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Winds," J. Wind l,)n,q. ltttl. .'lt,trnlltr. ll. (1990). 461 461 .
2-l4l Y. Tamura, K. Shilrrlulrr, :rrrrl lr.. llilri, "Wirrtl l{csponsc o{'a 'lirwcr ('l'yphtxrn
Obscrvation al thc N:tglrs:rki llrrr:, li'rr llrsr'lr l)orrrlorcrr)," .1. Witrtl lin.4. ltrt!.
Acnrl. . -50 ( l9().1). .1O() l lti
90 lltf n tMOSI,I l nt(; nouNl)nny tAyt n
2-142 A. I). Pcrcira, M. ('. (i. Silvrr, l). X. Vtr'1,',irs, ruttl A. (i. I-opcs, "Wilul 'l'Lrnncl
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CHAPTER 3
EXTREME WIND CLIMATOLOGY
Climatology may be defined as a set of probabilistic statements on long-term
weather conditions. The branch of climatology that specializes in the study of
winds is referred to as wind climatology. Wind climatology provides the de-
signer and the code writer with information on the extreme winds that might
affect a structure during its lifetime.* Such information is required for making
rational decisions on the magnitude of the wind loads to be used in design.
This chapter is devoted to a review of problems involved in the description
of the wind climate for structural design purposes and in the development of
criteria for the definition of design wind speeds. Procedures for estimating
extreme winds are presented, and the uncertainties inherent in these procedures
are discussed. Someof the material included herein is heavily dependent upon
probabilistic and statistical notions and tools. These are presented in some detail
in Appendix A1.
The reliability of climatological statements based on the analysis of extreme
wind speed data is clearly dependent upon the quality of the data. This topic
is discussed in Sect. 3.1. The question of the prediction of extreme wind speeds
in well-behaved wind climates and in hurricane-prone regions is dealt with in
Sects. 3.2 and 3.3, respectively. The dependence of extreme wind speeds upon
direction is discussed in Sect. 3.4. Information on the frequency of occurrence
of tornado winds of various intensities in the United States is presented in Sect.
3.5.
In the United States surfacc wirxl spoocls rcpofted by the Weather Service
have traditionally been exprossc(l itt tttilcs pcr hour (l mph : O.447 m/s). In
*Wincls othcr than thoso ol'inlcrcsl lirrn lr slrr( lurrl sirlr'ly vicwpoittt will bc rlcalt wilh in Chaptor
t5.
91
I X llil Ml wlNI) ( il tMn t( )l ( x iY
Itrtrriclrttc-rollr(ctl wot'k, lt:rrgllts ltt' lrt'rlrrrrrlly t'xplt'ssul irt rlrrrlrt'rrl rrrlt's (I
nuli - l.l5 rnilc). Iior corrve:rricrrcc, wlrt'r't'lrpPnrplirrtc:, lhcsc trnils will rrlso
be used herein.
3.1 WIND SPEED DATA
To provide useful information on the wind
speed data recorded at that location must
micrometeorologically homogeneous set.
climate at a given location, wind
be reliable and must constitute a
3.1.1 Reliability of Wind Speed Data
Wind speed data may be considered to be reliable if:
I . The instrumentation used for obtaining the data (i.e. , the sensor and the
recording system) may be assumed to have performed adequately and was
properly calibrated. If it can be determined that the calibration was not ade-
quate, the data must be adjusted-whenever the information needed for that
purpose is available.
Example The following information is excerpted from [3-l] regarding the
5-minute winds given in the original U.S. Weather Bureau records taken before
1932 "Up to 31 December 1921 , all recorded wind speeds were the uncor-
rected readings of 4-cup anemometers. From 1928 through 193 1, all speeds
from the older 4-cup anemometers were corrected to agree with the readings
of the 3-cup instruments, then being introduced, readings from which were not
corrected to true speeds. From I January 1932 onward all readings, whether
from 3- or 4-cup anemometers, were already corrected to true speed in the
original records." Official U.S. Weather Bureau instructions fbr the correction
of 3- and 4-cup anemometer readings are given in Table 3.1.1, which is ex-
cerpted from [3-2], and whose use will now be illustrated. At Williston, N.D.,
the original readings of the maximum 5-minute wind in 1922 and 1930 on
record at the National Oceanic and Atmospheric Administration are 56 mph
and 37 mph, respectively. Using the corrections of Table 3 . I . I , the true speeds
(according to U.S. Weather Bureau calibrations) are 56 - l2 - 44 mph and
37 - 2 : 35 mph, respectively.
2. The sensor was exposed in such a way that it was not influenced by local
flow effects due to the proximity of an obstruction (e.g., building top, or
instrument support). For most U.S. weather stations, the existence of such an
obstruction during the period of record is noted, in principle, in Local Cli-
matological Data Summary sheets (LCD Summaries) issued by the Environ-
mental Data Scrvicc ol'thc National C)ccanic and Atmospheric Admlnistration
l3-31.
3. 'l'hc: lrlrrxrsplrt'r'ic slrlrtilit'irtion lrray bc: assurrretl lo havc hccn rrcrrtr:rl. 'l'his
:t I wlNt):it ,l lt) l)n tn
'l'Alll,lt J.l.l. ('orn.t.liorrs lo lrrrlie:rlc<t Wirrrl Sllctrls l-l-21
Spt't'tls lrrtliclrlc:tl
By 3-Cup "S" 1'ypc
Anemometer, mph
[1928-l93l"l
By 4-Cup Anemometer,
mph [Up to 31 Dec,
1g2l"l
Corrections in
Whole Miles
per Hour
ob-rc
t7-26
27-35
36,44
45-52
53-61
62-70
7 t-79
80-87
88-96
97-t05
106-1 14
tt5-122
123-132
r33-139
140-149
150-r57
158-166
t6t 174
175-184
r85-192
193-2(n
ob-g
9-12
l3-16
t7-20
2t-24
25-28
29-32
33-36
37-39
40-43
44-47
48-5 I
52-54
55-58
59-62
63-65
66-69
70-73
74-77
78-80
8l-84
85-88
89-9r
92-95
96-99
100- 103
104-106
107-t l0
I I l-114
I l5-1 l7
1 l8-l2l
122-125
126-128
129-132
r 33- 136
t37 -140
l4t-t43
+l
0
-l
-J
-4
-5
-6
-l
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-t9
-20
-21
1a
-23
-24
-25
-26
a1
-28
-29*30
-31*32
- J-t
-34
- 3-5
"Reference [3- l].
'Movement of anemometer cups obst.rvt.rl
94 I x tltl Ml wlNt) ( il tM/\ t( )l ( )( iy
llsstlllll)ti()ll is itcccp(itblc lirl witrtl spcctls;rt lO rrr;rlrovt'glorrrrtl irr opcll lcll-irirr
in cxccss ol'2-5 rnph rlr so (scc Scct. 2.2.-5).
3.1.2 Micrometeorological Homogeneity of Wind Speed Data
A set of wind speed data is referred to herein as micrometeorologically ho-
mogeneous if all the data belonging to the set may be considered to have been
obtained under identical or equivalent micrometeorological conditions.
These conditions are determined by the following factors, which wiil be
brictly cliscussed below:
. Avcrirgirrg tirnc (i.c., whcther highest gust, fastest mile, one-minute av-
t'r:rgc, livc nlinulc avcragc, etc., was recorded).
o Ilciglrl irl'rovc gnrurrtl.
. l{orrglrrrc:ss ol' surnlunding tcrrain (exposure).
I . Avantging 'l'ime. If various averaging times have been used during the
pcriod o1'rccord, the data must be adjusted to a common averaging time. This
can be done by using Eq. 2.3.31 and Tables 2.2.1 and2.3.3, or Fig. 2.3.10.
(For hurricane wind speeds, see also Sect. 2.4.3.)
Data averaged over short time intervals, such as highest gusts or fastest
miles, may in certain cases be affected by stronger than usual local turbulence
effects, and thus provide a somewhat distorted picture of the intensity of the
mean winds. In principle, it is desirable, therefore, that the data used for the
description of the wind climate be averages over relatively long periods, say
five minutes or so. However, owing to the current data collection policy of the
u.S. National weather Service and the availability of 3-s gust speed data at a
large number of stations in the United States, the ASCE 7-95 Standard lz-1391
uses 3-s gust speeds at lO-m elevation as basic wind speeds.*
2. Height above Ground. If during the period of record thc erevation of the
anemometer has been changed, the data must be adjusted to a common elevation
as follows: Let the roughness length and the zero plane displaccment be denoted
by z6 and 27, respectively (zo and za are parameters that clefinc the roughness
of terrain; see Sect. 2.2). For strong winds (i.e., with speecls cxceeding l0
m/s or so), the relation between the mean speeds u(ar) and u(z) over horizontal
terrain of uniform roughness at elevation z1 and Z2 above ground, respectively,
can be written as
xThe National Weather Service and the Federal Aviation Administration are currently imple-
menting the Automated Surface Observing System (ASOS). It is anticipated that by the year 2000
there will be 1700 ASOS units in operation. The ASOS anemomctcr reading is sensed once a
second. Every fivc sccttnds it rlnning ilvcnlgL- is eomputcd, which is rclorrctl kr as a (igsl. A
2-rninute running avcrilgc ol thc 5 s irvcrllgcs is irlso conrprrlctl lrrrtl is rrst'il rrs rr rrrcirsrrrc ol thc
prcvailirrg winrl.'l'ltt'st'tltoiccs rtl:rvt'nrgirrg lirrrc wcle rlct'lrrcrl t() l)('nr()sl rrsr.lrrl lor:rvilrlion
purJxrscs l.l (rl{ 1 .
Equation 3. l. I lirllows clircctly from Eqs. 2.2.18 ;ttd 2.2.22. For open terrain
Za : O, and the values ol' the roughness lcngth 2,, can bc taken from Table
2.2.1. The power law (Eq. 2.2.26 and Table 2.2.2) may be used in lieu of
Eq. 3.1.1. As noted in sect.2.2.3, considerable uncertainties subsist with
regard to the values of the roughness parameters in built-up terrain. Good
judgment and experience are required to keep the errors inherent in the sub-
jective estimation of the roughness parameters within reasonable bounds. It is
clearly advisable to investigate in individual cases the effectof such possible
errors upon the predictions of extreme wind speeds.
3. Roughness of surrounding Terrain In many cases anemometer locations
have been changed during the period of record, for example, from a town to
a neighboring airport station. The corresponding records can, in principle, be
adjusted to a common terrain roughness by using the similarity model (Eqs.
2.2.29 and 2.2.31 and rable 2.2.3) described in Secr. 2.2. As indicated in
sect.2.4.1 , this model may be assurned to be applicable in horizontal terrain
if at each station the terrain roughness is reasonably uniform over a distance
from the anemometer of about 100 times the anemometer elevation. In terrain
in which sheltering effects by small-scale obstacles are present, the data may
be adjusted by using a procedure presented in [3-4].
A situation commonly encountered in practice is one in which, while the
anemometer may not have been moved, the roughness of the terrain surrounding
the anemometer has changed significantly over the years as a result of extensive
land development. In such situations the adjustment of the data to a common
roughness may pose insurmountable problems, unless detailed information on
the phases of the land development is available.
Anemometer elevation and location changes are listed for most u.S. weather
stations in Local Climatological Data Summaries t3-31. wind climatological
information for various locations around the world is available in [3-77 , 3-78].
3.2 ESTIMATION OF EXTREME WIND SPEEDS IN
WELL-BEHAVED CLIMATES
Infrequent winds (e.g., hurricanes) that are meteorologically distinct from and
considerably stronger than the usual annual extremes are referred to herein as
extraordinary winds. climates in which extraordinary winds may not be ex-
pected to occur are ref'erred to us wr'l/ ltehuvctl. In such climates it is reasonable
to assume that each of thc tlrrt:r in rr st'r.it's ol'thc largest annual wind speeds
contributcs to the dcscripliott ol'llrc plrb;rbilis( it'l'rr.rhlrvirlrof the extreme winds.
A statistical analysis ol'sttclt rt st'tir's r'lrrr llrt'rt'lir.t' bc cxpcclcd t<l yicld uscful
prctlictions ol' l<lng*lorrrr wirrtl t.xllt.rrrr.:,
'l'htrs, in tt wcll-hchltvtrtl t'lirrrrrlt',:r( :nrv 1'rvcrr st;rliorr ir rirrrtkrn virrilrblt'rrlry
;llMAll()ll rrl r 'ilttl Ml
I l(.'.tl
( /(;'. ,)
wlNl r:;t,t tll:; tN wt lt ttt ltnvt t) (,t lMn il:
lnf(;, ;',,111:ul
lrrl(i, :'.,111;'.qyl
r95
(3. r. r)
96 I x tilt N/lt wtNl) (.1 lMn l( )l { )( 'Y
bc tlclirrr,tl, wlticlr corrsisls ol tlrc l:rrIr':;l yt':rrly wtrrtl sllcctl. ll llrr'sllr(rorr ts
9pc lirr wIich wiltrl rccortls ()vcl lt rrrrtttlrt't ol tttttset'tllit'c yoilrs ittt'lrvltil:rble:,
thcn thc cuntulativc dislributiolr lirrrclion (('ll1'1 ol'tlris ritrlclottt vllriitlllr-: tttay
be estimated to charactcrizc thr: pnrblbilistit' bcltitvi<lr of thc largcst annual
wind speeds. The basic clesign wintl spectl is thcn dclincd as the speed corre-
sponding to a specified value p ol' thc Cl)lr or, cquivalently, to a specified
mean recurTence interval N.* R wind cttrrcsponding to an /V-year mean re-
currence interval is commonly referred to as the N-year wind.
This section is devoted to the question of estimating (1) the CDF of the
largest annual speeds and (2) errors inherent in the wind speed predictions.
Such errors include, in addition to those associated with the quality of the data
(sec Sect. 3.1), modeling errors and sampling errors. Modeling errors are due
to an inadequate choice of the probabilistic model itself. Sampling errors are
a consequence of the limited size of the samples from which the distribution
parametcrs are estimated and become, in theory, vanishingly small as the sam-
ple size increases indefinitely.
3.2.1 Probabilistic Modeling of Largest Yearly Wind Speeds
Extreme wind speeds inferred from any given sample of wind speed data de-
pend on the type of distribution on which the inferences are based. For large
mean recurrence intervals (".g., N > 50 years) estimates based on the as-
sumption that a Type II distribution is valid are higher than corresponding
estimates obtained by using a Type I distribution, while estimates based on a
reverse Weibull distribution wittL tail length parameter ^Y < 15 , say, are lower. f
According to [3-5], extreme winds in well-behaved climates may be assumed
to be best modeled by a Type II distribution with p : 0 and 7 : 9. However,
subsequent research has shown that this assumption is not borne out by analyses
of extreme wind speed data f3-6, 3-1 , 3-91. In [3-6] , 37 year-series of 5-minute
largest yearly speeds measured at stations with well-behaved climates were
ruUi""t"A to the probability plot correlation coefficient test (see Sect. A1.6) to
determine the tail length parameter of the best fitting distribution of the largest
values. Of these series, 72% were best fit by Type I distributions or by Type
II distributions with "v : 13 (which differ insignificantly from the Type I
distribution); ll% by Type II distributions with 7 < ^Y < 13; and 17% by
Type II distribution with 2 = I I 7. Virtually the same percentages were
obtained in [3-7] from the analysis of sets of 3'7 data generated by the Monte
Carlo simulation from a population with a Type I distribution. On the other
hand, the analysis of sets generated by Monte Carlo simulation from a Type
II distribution with tail length parameter 7 : 9 led to percentages differing
*Rccall that lV: tlf f ' 7r) (scc Appcntlix Al,Irq. Al-).
lDifl'crcnccr hctwccrr spet:tls cslirrrirlctl ott lltc basis ol T'ypc II tlistributions rntl lhc -l'ypc I
tlisll'ibrrlion ilttrt'rrst.lrs.y tlcclclrsr's. l)illi'rcrrt'cs hclwcctt s;rt:ctls lt:tsctl on lhc'l'ypt I tlisllilrtlliott
:uxl rcvt'rse Wt'ilrrrll rlislrilrttliotrs rtlso irtt tt':tse its 1 tlt't'rrl:tst's
l:lllMnllillt ot t"lltt Mt WtNt):;t,t tlrl ; lNt Wt lttililnvt tr(.t tMAilti 97
sigrrilicirnlly ltttltt lltost' r onr'sporrtlirrg to tlrc irclrrirl wirrtl spcc:tl tltrlir. ()rr thc
basis ol'thcsc t'csrllls il r'rrrr lrt'conliclcntly statcd that in woll-bchavccl clirlates
cxtrcmc wintl spr:ctls lrlt' rrrotle lctl rn<lre realistically by thc Type I than by the
Typc II distributiorr with ^y ,., 9. This conclusion was reinforced by studies
reported in [3-9], in which tcchniques similar to those of [3-7] were used in
conjunction with wind speed data at one hundred U.s. weather stations listedin [3-9].
As indicated earlier, the Type I distribution results in lower estimates of the
extreme wind speeds than the Type II distribution with 7 : 9. An interesting
result obtained in [,{1-36] is that at most stations in the United States even the
Type I distribution appears to be an unduly severe model of the wind speeds
corresponding to large mean recurrence intervals; at these stations a better fit
to the data is obtained by reverse weibull distributions (see also end of Ap-
pendix Al). Thus, structural reliability calculations based on the assumption
that the Type I distribution holds are in most cases conservative [Al-36]. For
this reason we will assume in this section that the Type I distribution model
holds. The degree of conservation inherent in this assumption is generally
modest for basic (5O-year) speeds, but it can be very significant for wind speeds
corresponding to nominal ultimate wind loads, i.e., 5O-year wind loads mul-
tiplied by a wind load factor (see Sect. A.3.3).
3.2.2 Estimation of and Confidence lntervals for the N-year Wind:
Numerical Example
It is shown in Sect. Al.7 that, given a set of data with a Type I extreme value
underlying distribution, several techniques can be used to estimate the param-
eters of the distribution and, hence, the value of the variate corresponding to
a given mean recurrence interval.* However, inherent in these estimates are
sampling errors. A measure of the magnitude of the latter can be obtained by
calculating confidence intervals for the quantity being estimated, that is, inter-
vals of which it can be stated-with a specified confidence that the statementis correct-that they contain the true, unknown value of that quantity. Tech-
niques that can be used to estimate the N-year wind, and confidence intervals
for the N-year wind, are discussed in some detail in Sect. A1.7. one of these
techniques is presented and illustrated below.
Using the approximation -ln[-ln(1 - l/N)] = ln N, it follows from Eq.Al .74 (which is based on the method of moments) that the estimated value
Dp of the N-year wind u1y is
0N = X I 0.78(lnN - 0.577)s (3.2.1)
where X and s are, respcctivcly, rlrrr sirrrrplc rncarn and the sample standard
deviation of the largest ycarly wilrtl spt'erls lirr (hc pcriod of recorcl.
*InAppendixAlthisvalucistlcrxrh.rl lry(i,q711,rllrr.rr./r I l/N:rrxl Nisllrt, 1rt.ir1 t1.(.rtrroncc
inlcrval, In this chaptcr lhc trottrliolr (i,( I l/N I r,. r,, rrr,r.rl
IXIlilMl wlNl)(.1 lMnl()l ()(iY
As Prt.vipttsly rurlctl, irtltr.:tcrr( iil lltt't':-lrlrItlt':; ol l'1u 11t'q s:tttlpllttl', t'ltot:.. ll
lirllows l.nrrrr I')t1s' n I 76 lrrrtl Al'/o (wlritlr:tlt'lrltst'tl ott lltc: tttcllrotl ol ttttr
llcllls) llrirt thc staurlanl tlovialiorr ol tltt'srtttplirtg ert'tlt's in thc cslilrr:rtiolt ol
r',y cln hc writtcn as
.\1)0r^) -- 0.781 l.& + l.46tln ru - O.SZtl I l.l(ln rV - O'S;Zt'l'' ;,/n
(3.2.2)
wlrt't'c rt is thc samPle size.
Example At Great Falls, Montana, the largest yearly fastest-mile wind speeds
rrt l0 rrr above ground during the period 1944-1971 (sample sizen :34) were
l3-t)l:
51, 65, 62, 58, &, 65, 59, 65, 59, 60, 64, 65,'73, 60, 61, 50, 74
60, 66, 55, 51, 60, 55, 60, 51,51,62,51,54,52,59,56,52' 49
(mph). The sample mean and the sample standard deviation for these data are
X:'5g mph and s : 6.41mph. From Eqs. 3.2.1 and3.2.2 it follows that
for N : 50 years and N : 1000 Years'
lso = 76 mph SD(i5o) : 3'7 mph
|wn = 91 mPh SD(irooo) = 6'4 mph
As shown in Sect. A1.7, the probabilities that u vl is contained in the intervals
0v1 + SD(01y), 0n t zsD(|il, and 0p + 3SD(0,'v) are approximately 68%'
gi%, ana 99%, respectively. These intervals are referred to as the 68%, 95%,
and 99% confidenci intervals for u7, and are shown fbr the 34-year Great
Falls sample in row 1 of Table 3.2.1.
It is also shown in Sect. A1.7 that the width of the confidence intervals can
TABLE 3.2.1. confidence Intervals for the N-year wind at Great Falls
I i;llMn ll()t\t ()t I i: ilil [It wtNt) :;l'l I t)t; lN wt il ilt ilAV' t) (]t lMn il ti gg
bc rctlLtcctl il it lttott't'lltr'rt'ttl t'slitturtor is uscd; lurwcvrrr', llrc intcrvals cannol
bc trarnrwcr lltirtt tlrrst' ohlrrirrr'tl by using the Crarrcr,llacl (C.R.) lower bouncl
(Eq.Al.77). tt<tr tlrr: (irt:;rt Irlrlls sltnple, the confidence intervals based on the
latte r are shown in linc (2) ol 'l'ablc 3.2.1. lt is seen that the differences between
the results of lines (l) and (2) ol"rable 3.2.1are small.'rhis is consistent with
the conclusion of Sect. Al.7 that the e{iiciency of the method of moments (Eq.
3.2.1) is generally adequate for structural design purposes.
In Table 3.2.1 the errors in the estimation of the 50-year wind are of the
order of lo% at the 95% confidence level. Since the wind pressures are pro-
portional to the wind speeds (see Chapter 4), the corresponding errors in the
estimation of the pressures are of the order of 2O%. An altemative approach
to accounting for sampling errors, which applies the theorem of total prob-
ability, is suggested in [3-51].
To reduce sampling errors, [3-8] resorted to the consolidation of records
from different stations, thereby creating "superstations" with large sample
sizes. This approach, if valid, would be quite attractive: for example, the
information yielded by a "superstation" consolidating 2o-year records taken
at 100 stations would be equivalent to information yielded by a 2000-year
record. However, according to [A1-15], statistical tests did not validate the
"superstation" concept for extreme wind analyses based on peak gust speeds.
For that concept to be valid, the population distributions for the station
records being consolidated would have to be identical, and in addition those
records would have to be mutually uncorrelated. In general, the first of these
conditions cannot be assumed a priori to be true. Second, if the records consist
of peak gust speeds, the observed lack of correlation between records taken at
different stations may be spurious; that is, it would be likely to occur even if
the corresponding mean wind speed records were well correlated. The apparent
lack of correlation would be an artifact due to the strong random variability of
the ratios between gust speeds and mean speeds. For these reasons the "su-
perstation" concept may yield inadequate results. In our opinion this is likely
to be reflected in the quality of the wind speed map specified by the ASCE
7-95 Standard [9-5], which is based largely on the "superstation" concept
applied to peak gust speed records.
3-2.3 Methods for Estimating the Extreme speeds at Locations with
lnsufficient Largest Yearly Wind Speed Data
There are about one hundred u.S. weather stations for which reliable and
relatively long wind speed records arc available (e.g., records overperiods of
20 years or more). Some of thcsc sl:rlions c()vcr arcas of tens of thousands of
square miles, over which-lilr rrrc(t:onrkrgical rcasons or owing to topographic
effects-the extreme wincl clirrrrrtt' is rrol rrt't'r,ssirrily uniform. Thcre arises
therefore in practice thc prohlcrrr ol t'slirrrrrlirrl'('xll('tn(: wintl spcccls at various
locations where long-tcntt t'ccottls ol llrc l:rr1'r':rl yt'lrll_y witrtl spcctl tlutl tlo 191
exisf.
Confidencc level 687 95%
Mean recunence
interval, N (years) s0 1000 50 1000 50 1000
(1) Estimated by
method of
moments 76 + 3.1 9l + 6.4 16 + 1.4 9l + 12.8 76 + I l.l 9l + 19.2
(2) Estinratcd using
C.R. lowcr bounil 76 I .l.l ()l Il5.o9l I5.0 '76 r 6.2 9l I lO.O 7(r I (). \
lo0 txiltt Mt wtNt)ct tMnt()t ((iy
Estimates Of Extreme Wind Speerls lrr a Marirrc EnvirOnmenl. l{t'lcrt'rtt't'
1.1-l ll lists ilrrcer rrrcllrorls tlrir( ,iu1', ur lrrrrrt rplt', :rvltilitblc (o clt'l'y ottt sttt'lt
cstitnatcs lilr lrrarinc cnvirorrrttcrtls wltt'tt' lltt' cxltctttc spcccls itte itssot'iltlctl
with extratnrpical strlrtrrs. 'l'hc lirst rrrt'llrrxl rnlrkcs usc ttl'climattlltlgical inlor-
mation on various paramctcrs ol' thc s(or.rrr iuttl ol' physical modcls rclating
those parameters to the surface wincl spcctls. lt is shown in Sect. 3.3 that such
a method can be applied to estimatc oxlt'cttlc wind speeds in hurricane-prone
regions. However, as noted in [3-lll, owing to the complexity of the surface
wind patterns in extratropical storms, the usefulness of this method appears to
be uncertain in regions where such storms are dominant.
A second method listed in [3-11] is the use of objective analysis schemes.
These consist of (1) an initial guess at the surface wind on a regular grid, (2)
an automated procedure for screening wind reports from ships to eliminate
erroneous readings, and (3) a procedure for correcting the initial guess on the
basis of the usable set of ship reports, which involves relations among the
surface wind speeds, sea-level pressures, and air and sea temperatures. Details
on objective analysis schemes and of errors culrently inherent in such schemes
(which may range from lO% to 30%) are given in [3-11].
The third method listed in [3-11] is referred to as direct kinematic analysis.
The method, which involves subjective judgment by experienced analysts, con-
sists in synthesizing discrete meteorological observations to obtain a continuous
field represented in terms of streamlines and isotachs. Objective or kinematic
analyses applied to a sufficient number of strong storms make it possible to
provide estimates of extreme winds that may occur at any one location. As
indicated in [3-11], one of the major diliculties in conducting such analyses
is that much of the vast store of existing data is currently not accessible in
readily usable form.
Estimation of Extreme Wind Speeds fromShort-Term Records. A prac-
tical procedure for estimating extreme wind speeds at locations where long-
term data are not available is described in [3-12]. The method, whose appli-
cability was tested for a large number of U.S. weathcr stations, makes it
possible to infer the probabilistic behavior of extreme winds from data con-
sisting of the largest monthly wind speeds recorded over a period of three years
or longer. Estimates based on the monthly speeds, denoted by 0n.., are ob-
tained by rewriting Eq. A1.74 as follows:
0N.^ = x. + o.zAPn(l2N) - 0.5771s* (3.2.3)
where X. and sa are, respectively, the sample mean and the sample standard
deviation of the largest monthly wind speed data, and N : mean recurrence
interval in years.
The standard dcviatitln tll'thc sanlpling error in thc cstirlation tll'I'x7.,,, is
obtaincd from l'it1s" Al .76 irrrtl A I .70 as
r:' Ir;ilMAil()N ()t I . |ltt N/,1t wtNt) l;t 'l It)ti IN wt |t tlt ltAVt t) (.t tMn il:
,S/)( /',,r,.,,, ) o /ttl l.(4 I 1.4()llrr( llN) O..5771
I r.rlrn(t2N) - o.sztl'l',, h:, (3.2.4)
i
ir
where n-: sample size.
Example At Great Falls, the sample mean and the sample standard deviation
of the largest monthly fastest-mile wind speeds at l0 m above ground for the
period September 1968 through August l97lx (sample size n*:36) aret^:42mph, s^:6.96 mph. From Eqs.3.2.3 and3.2.4, the estimates forN: 50 years and N : 1000 years are
0s0.. : 74 mph SD(0s0,) = 6.23 mph
iuxn,^: 90 mph SD(irooo..) : 8.85 mph
It is seen that the estimated speeds based on the set of 36 largest monthly data
are only slightly lower than those obtained from the set of 34 largest yearly
speeds (|so:76 mph and 01es0 : 91 mph; see Sect. 3.2.2); however, the
sampling errors are larger.
Similar calculations carried out for 67 sets of records taken at 36 stations
are reported in [3-12], where it was found that the differences iso,^ - lso,where 056 is the 50 year wind speed estimated from long-term largest yearly
data, were less than sD(0s0.) in 66% of the cases and less than twice the value
of sD(05s,-) in 95% of the cases. This remarkable result, confirmed by addi-
tional calculations reported in [3-13], indicates that the estimates based on
largest monthly wind speeds recorded over three years or more provide a useful
description of the extreme wind speeds in regions with a well-behaved wind
climate.
Inferences concerning the probabilistic model of the extreme wind climate
have also been attempted from data consisting of largest daily wind speeds
13-121, or of wind speeds measured at one-hour intervals 13-141. one problem
that arises in this respect is that data recorded on two successive duys u..
generally strongly correlated. A second and more serious problem is that the
daily (or hourly) data reflect a large number of events (e.g., moming breezes)
that are altogether unrelated meteorologically to the storms associated with the
extreme winds. These events can be viewed as noise that obscures the infor-
mation relevant to the description ol'thc cxtreme wind climate. Indeed, it was
verified in [3-12] that estimatcs ol' cxtr-cnrc winds based on daily data differ
significantly from estimatcs obllrirrt:tl lor krrrg,lcrrrt rccords of largest yearly
speeds. This conclusion is a.litrtittri ltut'lirl irrlc'r'ctrc:cs based 9n hourly data.
*For the actual data. scc thc l,octrl ('liru:rlolo1,r, .rl I ):rt.r ',rrrrrrr;rrit.s li1. tltr. ycitrs l()()ll l.)7.1: "'--=\,;' \
T
'l
i
I
nunomartins
Oval
1O2 I x I ill Mt wtNl) (;t tMn t( )t ( x iy
l,lstirrlrlcs ()l'L:xirL:nlL: wirttl spct'tls lr;r:;t'rl orr st'ls ol tllrllr itt cxccss ol s1r't'rltcrl
{hrcshokls (scc cncl ol'Sccl . A I.7) lrrt' rt'llrlcrl irr l.t l0l lirt' sltorl r('('()11ls.
It was shown in Scct. 2.4.4 tltal lhrrrttlt'r.slot'rrr wirttls havo I'caiures tlrirt rlillcr
markedly fiom thosc ol'othcr typcs ol'wirrtl. (ic:ncrally, extremc wind spocds
are analyzed without separating thundcrslonr tlata lrcm the other extremes.
Whether it would be useful to extract thcsc tlata l'rorn the mixed sets and analyze
them separately-despite the difficulties this would entail-is still being debated
t3-s01.
3.3 ESTIMATION OF EXTREME WIND SPEEDS IN
HURRICANE-PRONE REGIONS
We now consider the prediction of extreme winds in climates characterized by
the occurrence of hurricanes. It was suggested in Sect. 3.2 that in a well-
behaved wind climate each of the data in a series of the largest yearly speeds
contributes to the description of the probabilistic behavior of the extreme winds.
However, in a hurricane-prone region most of the speeds in a series of the
largest yearly winds are considerably lower than the extreme speeds associated
with hurricanes; they may therefore be irrelevant from a structural safety point
of view. This situation is illustrated by the plot of Fig. 3.3.1, which shows the
S-minute largest speeds recorded at Corpus Christi, Texas, between l9l2 and
1948 [3-6]. It may then be argued that in hurricane-prone regions the series of
the largest yearly speeds cannot provide useful statistical information on winds
of interest to the structural designer, much in the same way as the population
of a first-grade classroom-which might include a teacher-is of little use in a
statistical study of the height of adults. That this is the case is suggested below.
The abscissa in Fig. 3.3.1 represents the reduced variate
t / l\lv:-tnl-tnlr-:ll" | \ N/l
where N is the mean recurrence interval. In virtue of Eqs. Al .43 and A1.45,
a Type I extreme value cumulative distribution function would be represented
in Fig. 3.3.1 by a straight line, whose intercept and slope would be equal to
the distribution parameters p and o, respectively. To the extent that the pop-
ulation of largest yearly speeds would be described by a Type I distribution,
the actual data would then approximately fit a straight line. In Fig. 3.3.1 this
is roughly the case as far as the winds of less than hurricane force are concemed.
However, if-as in Fig. 3.3.l-the hurricane-force winds are included in the
set being analyzed, clearly the fit of a Type I distribution to the data is extremely
poor.
A bettcr fit can bc obtainccl il-a Typc II distribution with a snrall valuc ol'
thc tail lcngth pirrrrrrtclcr is usctl. Howcvcr, As slrowrr in 1.1 (rl. 1ln:1lir'1i1;11r,i,r1'
cxlrclnc wintls irr ltttl-rit'lrltt'plottc tcgiotts b:rscrl ort'l'ypt'll tlisl r.ilrrrliorrs lrlt'irr
t--Ft--FatsHia-!!txxx,FtsFa__xt FFrxF!a
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ooooooooooooCOoooooooooooooo
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l0,l r,(rIilMt wtNt){]tl\/nt()tI)(iY
1t,,.r , .i',( s tlnloalisiic. l'ot cxlttttpl,', ltltrtrl' r'rr{ lr ;l tlistrillrrtiorl trt tllt' l()l'l
lir!r, r,,ol.tl ol'thc llrrgcst yclu-ly slx'('(l:,;rl ('orlrrr:, ('lrlisti wotrltl yicltl, lirr'(lrc
r trr':rtr'rl 1000-ycar wind, a valu('()l l()fr0 rrrplr rr lirlicttlous rcsult l.l ()1.
',{ uous difficulties als() arisc il rrrixr"rl lirt't lrt'1 pnrbirbility distributions arr'
ii ,,1 lt 51. lndeed, sincc hurriciltlcs ill(' l:rl('('v('ll(s, thc number of hurricane
r,,r rr'plcrl cyclone) wind spccd cllttlt irr lt t'e,.'ttltl ol'thc largest yearly winds
,,i,.1 1v1'l at any one stati()n is small (t:.g., irr lrig. 3.3.1 only two of the data
r,l'r(':iL:nt hurricane wincl spccds). 'l'hcrclirrc thc confidence intervals for the
, tr!('nrc wind preclictions arc, in gcncral, unacccptably wide (e.g., for N :
l{x) years, of the ordcrof irgo(l + 0.6) at the 68% confidence level; see
1r l5l). It is for this rcason that the 50-year fastest-mile wind estimated in
1l 5l for Corpus Christi on the basis of a mixed Fr6chet distribution is only
/(r mph at 30 ft above ground in open terrain. This value appears to be severely
Iow; indeed, in the pcriod 1916-1970 Corpus Christi was hit by three devas-
tating hurricanes 13- l6l with fastest-mile winds of up to 120 mph at 23 ft above
ground in open terrain (see Corpus Christi 1970 Local Climatological Data
Annual Summary).
Because this series of the largest yearly winds does not appear to provide a
suitable basis fbr predicting hurricane wind speeds, alternative bases for such
predictions have been proposed in the literature, which are now briefly dis-
cussed.
3.3.1 Procedure Based on the Maximum Average Monthly Speed
In this procedure, proposed in [3-17], it is assumed that the behavior of the
extreme winds is described by the cumulative distribution function
F(u) :p,"^p[-(;) "] * (1 - p7)exp[ (;) '] (3 3 1)
where zr is the wind speed, p7 is the probability of an annual extreme wind
bcing produced by a tropical storm, and o is a scale parametcr. The parameter
,rr, determined in [3-17] as an empirical function of the mean number of
tropical storm passages per year through a five-degree longitude-latitude square,
is represented in Fig. 3.3.2.The parameter o is given in Fig. 3.3.3 as a function
ol' the maximum of the average monthly wind speeds recorded at the station
concerned over a reasonably long period (e.g., ten years or so).
The application of this procedure is illustrated in three cases: West Palm
llcach (Florida), Boston (Massachusetts), and Columbia (Missouri), for which
pt = 0.43, Pt': 0.72, andPr: O, respectively (Fig. 3'3.2). At West Palm
llcach, thc maximum of the average monthly speeds in the period 1952-1974
(olrtainccl lnrrn thc Local Climatological Data Summaries) was 13.9 mph at 30
It lrhovc gnlrntl. linrnr lrig.3.3.3, o - 5l mph. Thcrcfbre
o n ,.xll (;;) 
-' 
| , ,,r, ,.-,,1 (; )
Lr l :'llMn ll()N ()l I 1lilt Ailt wtNt, :,t 't t lr:, ll! lIrlilili t\|t t,lr{ r,il r ttMAll .. l(}1r
FIGURE 3.3.2. Probability p, of an annual extreme wind being produced by a tropical
storm. From H. C. S. Thom, "Toward a Universal Climatological Extreme Wind
Distribution," in Proceedings, lnternational Research Seminar on Wind ElTects on
Buildings and Structures, Vol. I, p.682. Copyright, Canada, 1968, University of
Toronto Press.
o 2 4 6 8 t0 12 14 l(i l8 20 22 24 26 2830 32 34
Maxirnrrrrr nr()nlllly,rvcr,r;r. wirrtl speed (mph)
FIGURE 3.3.3. Scalc panunet('r'o. l;rrrr ll (' S.'l'lrolrr, "Toward a Universal
Climablogical Extrcnrc Wirrtl l)islrilrrrlr.rr." nt l'trtt t't'tlirr,q.t. lntcrnational Research
Sctninar on Wintl lillccls otr llriltlirrl':, ;urtl Strrr, trrcs. Vol. l, p. 6112. Copyright,
('rulttllt. l9(113, Univcrsily ol lolonlo l'r, ,.',
-+-
0.17 o:11
lt( t'\
'| (r i2)
106 I x I nl Ml wlNl r ( rl lMn l( )l ( )(;Y
Recalling that N llll /,ilrv)1" i( lolkrws lrrrrr Ilt1. .1.-1.2 tlrlrl tlrt't'sltrrrtt(t'tl
N-year wincls lirr N 50" l(x), irrrrl l(x)o y('iilri itrt: /'5{) , 102 rrrPh, t'1,x,
120 mph, and t/l(xx) : l9ti trph, rcsllc:t'livt:ly.
At Boston, the highcst of thc avcrirgc rttrtrrlhly spccds rccordcd bctwccn
1950,1914 was 18.8 mph at 30 fi abovc grouncl in opcn terrain. To this valuc
thcrc corresponds o :63.6 mph. With Pr - 0.12, it follows from Eq. 3.3.1
that thc extreme wind estimates are zr5s : 106 mph and z/rur : I 19 mph. It is
norcrl that the estimates presented in [3-5| otc l/5s : 88 mph and ales : 93
rrrph, that is, considerably lower than those based on Eq. 3.3.1.
Al Cblumbia, Missouri, the probability of occurrence of hurricanes is nil
irntl lit1. 3.3.1 becomes
F(u) :".0[ (;) '] (3.3.3)
'l'hc maximum of the monthly wind speeds recorded between 195 l-1974 was
l-5.7 mph at 30 ft above ground in open terrain so that o : 51 .O mph and zr.e: 88 mph, 2,100 : 95 mph and z1ooo : 123 mph. It is noted that the estimated
extremes of [3-51 are lower, that is, u5o : 70 mph and 1]roo : 85 mph. The
extreme speeds at Columbia were also estimated assuming the validity of the
Type I distribution, with parameters inferred from the l95l 1974 series of the
largest yearly speeds at 30 ft above ground in open terrain. The results thus
obtained were t/56 : 66 mph, zroo : 69 mph, and ?1s00 : 8l mph, versus z'so: 88 mph, urut : 95 mph, and zr1ee,l : 123 mph, as estimated on the basis
of Eq. 3.3.1 with the attendant assumptions of [3-17].
Among these assumptions is the relation implicit in Eq. 3.3.1 and Figs.
3.3.2 and 3.3.3 between maximum average monthly speed and the extreme
wind speeds. No fundamental meteorological grounds are offered in [3-17] or
elsewhere in the literature for this relation which, frorn the evidence available
so far, does not appear to be justified.
3.3.2 Procedure Based on Climatological and Physical Models of
Hurricanes
To illustrate the principle of this procedure, an estimate will be made of the
probability that hurricane winds in excess of 155 mph will occur at any one
specific site on the Texas coast. The following information will be used in the
estimation:
o Average number per year of hurricanes with-wind speeds in excess of 155
mph moving inland in the United States, t iss. According to the National
Weather Service, there have been two such hurricancs in thc past 75 ycars
or so, the Labor Day Florida Kcys hurricanc in 1935 ittrtl ltttrricittlc Cltln-
illc in 1969 l3-181. A rcasonablc ostirnittc is tlrt'rr ll,tt - 2ltrrlricttrros/
(75 ycirrs) 0.027 htrrr/yc:irr.
l:l ll:llMn ll()N l)l I "llrl [Il Wllil):,1'l ll)i; lN llllllltl{ nl]l l'lt{rl.ll r |M/\il:, llll
r Avt'r'rrl.:t'rrrrrrlx'r lrr v('irr ol lrutt'it':rttt's nr()vlrl' rttl;rrrrl trr tlrt'I lrrilctl Sllttc:s,
1r,,,.'l'lris(ilr:ur{r(ytrrrlrt't'slitturtcrlI'nrruliil'.. t.t..[:r,, ll(rlrtrtricuttcs/
(63 ycrrrs) I .i"i,l lrrrr r'lyeltr.
o Avcragc rrrrrrrlrt'r'lx'r'y(:ll ol'all hun'icattcs trrovittg irrllrrrtl irr 'l-cxas, a7.
From Fig.'J.J.4. ttr = 2J hurricancs/((r-1 yclr's) = O.43 hurr/year.
o Average width ol'area swept by winds in cxccss ol'155 mph in one
hurricane, I'lz. According to [3-20], thc path of destruction of the Labor
Day Florida Keys hurricane was 35 40 miles wide. It will be assumed
conservatively that winds in excess of 155 mph affected a width W : 30
miles of that path. In the case of huricane Camille, it appears that it may
be assumed conservatively W : 20 miles [3-211. A reasonable value to
be used in the calculations is then W : (30 + 20)12 : 25 miles.
It will be assumed that the average number per year of hurricanes with speeds
in excess of 155 mph moving inland in Texas is
'-t ,t;tt
U1
(Implicit in Eq. 3.3.4 is the assumption that the probability distribution of the
hurricane intensities, given that a hurricane has occurred, is the same through-
out the U.S. Gulf and Atlantic coasts.) The length of the (smoothed) Texas
coast being about 375 miles, the probability sought is
t55U7
P(u > 155 mphl : ,'r" : :0.00042' 375
(3.3.4)
(3.3. s)
that is, approximately ll25OO per year.
The estimate just presented has several significant weaknesses. First, the
errors in the estimate of utfs could conceivably be large, the estimate being
based on a 7S-year-long record containing just two relevant data. Second, the
assumption that the rate of arrival of hurricanes is uniformly distributed over
the length of the Texas coastline overestimates the probability of hurricane
strikes over the coastline segment adjacent to the Mexican border (by about
25%), and underestimates that probability (by about25%) nearerthe Louisiana
border (Figs. 3.3.5 and 3.3.6). Third, the reliability of the estimate of er7 is
difficult to ascertain. Indeed, according to Fig. 3.3.6, u7: 1.6/(100 x l0)
entries/yearlnmi of coast x 0.53 huricanes/entry x 330 nmi of coast :0.28
hurr/year, versus 0.43 hurr/ycar, as obtained from the data of Fig. 3.3.4. (This
discrepancy is possibly thrc to lltc courtlirrgol'ccrtain tropical cyclones as
hurricanes in [3-l9lr';. lirrullr. tlrc esl inurlcs ol'14/:rrc largcly subjccfive, since
'r'Rcl'crcncc l3 l9l w:rs lcvisctl irr l()/li.rrrrl r', ul)(l.rl(,l,rnrrr.rllr lry llrt'N:rtionlrl llrrlritlrrrr'('crrlcr
l3 -521. Arlditiorlrl ittlirttt:rliott otr Norllr r\ll,rrrlr, lrrrrrr,,ur". r', ,r\r;rilirl)lc irr l1 -5 11 lrrtrl. on llrpt',
irr I I 5'1 1. lirr irtlirtrrtrliort tttt Wr':,lcttt Norllr l'.r, rlr, lr{rlr! .rl ( \r lor( s. rt't' I I 551
lrirrlrrrrl I
i
L
I
d>
ao\--g 5".'=o-.: -. cn, g2\o
9 -P
E!a
dgt
ool E.c.F !
^ilo =t
:(/oU(E>, = ,-j
dotr
.:!d
trE -
-^O
o9"i
LF*dJ9Yrc H
=o 5;h 5
.r-.i " O
_* boU
<. aJ -C
.o!s?e
e4A__ a0;llE@
rl o-'x
lrrrtl lrtrtl lrrrrl 'ir r r r'l'
z
J4zU_Eg
o>UOk =o(J(rU
=or2 -<-Ezo&-rrllo(/)r':tz
z 64- trlo zzuJ<
9U>E
KE.UU
FFzzUUOOtlc
i
tEiP
I:q
c.S;:\ (t)TE.\(r\908'r Eiin9()trl
tdErIOr?rHt%n
Ao,--^r=jfA^
t:11\\
qm
t'aJ.\
&\'p
^,/e )try iJ/ls{ t1| _-*1 )-V
lrrlnO
.' u/ 't !' '.1 
; l'j
l!t,l ,1,":,:- -1,, '.'i;:'l -,
109108
tll
Nurrrlr:r ol crrtrrr..;/l(X) yrr;rrt/lO ntnt of coasl
ONN
to rr rt, O A
9s)99 5qo!
Ratio (hurricanes/total srorms)
N
bo
su,4 s.)tlsr.rJl.lr:.il:rl.t.lu0l.)A.l lu.)t(l().tl U() Uotll:tU.t()llil.l() slsll(1 .)(ll u() Jlls ll lll
sJrlrlr(lr:(l().r(l l)Ur/r\.rlt.).t lxJ.l() ifuillolx)rU or.ll ()l r-lrr:().r(l(lu,)All.).).llJ l)lilr JArsuitlr).r(l
-rrr()J v 'spaads pully'r ouea!ilnH 6u//|€tu//ls3 Jo, arnpacotd opec auow
'uorleturlse eql ur pe^Io^ur serlureuecun eql lo cillJrd
InJ8urueew ,{1pcrs,(qd 'Juelo u JeJo feqt 'peeds flqtuoru eSeJa^e rununxuru
eql uo pesEq lepotu eql e{rlun'1eq1 sr s'€'€ puE ?'€'€ 'sbgJo ernlBeJ Injosrr rl
'oslv 'seleJ esuBJnsur Jo suorluurrxoJdde lsJg JoJ 'eldl'uexe JoJ 'suorSer ouo.l(l
-auBclunq ur seJnlcruls Jo eJnlreJ Jo ,{lrlrquqoJd eqt 1o seleturlse ssoJS Surluilr
JoJ esn etuos Jo eq ,{?lu peurllno lsnl qrBoJddu oql sesselr)lee,$ eseql etrdso( I
'lle^\ se l4 Jo uorleurlsc crll ul
lu€JuruSrs eq eJoJeJoql ,{uu ro.ue aql :plou aql ur uo{Bl aJo,r s}uetueJnsuour oU
'lZZ-fl (rtuu) pelteu sll?^rctul ocu€lslp IElspoc qlr.iv\ dutu rotscol .S.t.g .du()i)1.,1
t
JFts
I {A
_+(, o5i-
-At-
5 tD'O
3 --i qlnx
!iH='
1a-- ',o o
aD+
aa^
0a.n'3
= 
(-a d
9=io v;--
Fot! l a5 A;l f
- ?ft - u+!r t6='^
=yd 
5
=! 
ir5
J.9 - oxe
.DD@
-o 3
6 l./
YoI
\o+a; -J--
ur!) 6
-3b r*^
' a6 ry=-J
:'m r
*-/(-
F-,- )oz.-\\
\,1*
I@ I@
lt
il rrr dl
^o_o
=.? o!1
fii
3ao
ac
3g
o
o
o
3
Lake Charles, La.+
St. Marks, Fla.
Ft. Myers, Fla.
Miami, Fla.
Cape Hatteras, N.C
0rr Alx)t()lvv\It:) (tNtM tv\t IIIx I
112 I x tttl Ml wlNl) (
tlcvcl6pol in l-3,2-1 l.'l'his ir1-rprxrclr wrrs srrbsctlrrcrr(ly irplllit'tl irr l-l )'ll' wllerc
extreme wind speecls associatccl witlr lrtrrricrrncs wLlro cstirrratctl ott tltc bltsis ol'
the climatological and physical modcls dcscribcd bclow'
Climatological Models
1. The hurricane frequency of occurrence is modeled by a Poisson process
with a constant rate.
2. The probability distribution of the pressure difference between center and
pcriphcry of the storm, Ap,,"*, is lognormal. To eliminate values of Ap.,*
jrulgctl, in thc light of historical data, to be unrealistically high, the
tlislribrrtion is ccnsorcd stl that AP,."* ( 101.6 mm (4'00 in) of mercury
l.\2.]1.(Ntl(r:tlrirtA/).''.*:l0l.6mmcor.respondstothelowestatmo_
sgrlrcle l)rL:ssurc cvcr rccorclcd worldwide t3-25].) Theoretical studies ap-
pcru. (o cottlirttl this bound l3-61].
l. 'l'hc p(rbability distribution of the radius of maximum wind speeds, R,
is krgnorrnal. This clistribution is censored so that 8 km < R < 100 km
to avoid unrealistically "tight" or "broad" storms 13-231'
The average correlation coefficient between R and Ap-,^ is about -0'3'
(see 13-221, pp. 68 and 69.) All other climatological characteristics of
hurricanes are statistically independent.
The probability distribution of the speed of translation, s, is normal. This
distribution is censored so that 2kmlhr < r < 65 km/hr 13-231'
The probability distributions of the distance between any specified point
on the coast and the hurricane crossing point along the coast (or on a
line normal to the coast) are curves matching the historical data. Separate
curves are defined for entering, exiting, upcoast heading, and downcoast
heading storms.
For entering storms the probability distribution of the direction of storm
translation is a curve matching the historical data. For exiting upcoast
heading and downcoast heading curyes the distributions are uniform be-
tween 130" of the mean directions of storm translation. In all cases the
storm path is assumed to be a straight line.
Physical Models
l. The maximum gradient speed is given by Eq' 1'2'8 in which r : R' and
in which it is assumed that
- dAP,,t,,,
13-261. This rclation corresponcls to thc avcrage of data observed during
the 1949 hurricane that crossed Lake okecchobee, Florida [3-27, 3-2g].whether Eq- 3.3.7 can be assumed to be generally valid is uncertain.
For example, according to [2-l4ll, during typhoon Mireille observed
surface wind speeds over ground (which are lower than over the ocean)
were comparable in the region of the eye with estimated winds at the
gradient height. However, according to [3-79], measurements indicated
that the 10-m level sustained surface winds over water were generally
within 55% to 85% of the winds measured by reconnaissance aircraft at
500 m to 1800 m. Reference 13-791also suggests that the logarithmic
law appears to be valid up to about 1g0o m but that at aboul 3000-m
elevation winds may be less than at the surface.
Let the center of the storm be denoted by o, and consider a line oM
that makes an angle of ll5'clockwise with the direction of motion of
the storm. The l0-minute wind speed at l0 m above the ocean surface
at a distance r from o along line oM is denoted by u(10, r). The ratio
U(10, r)lU(10, R) is assumed to depend on r as shown in Fig. 3.3.7
13-261. Let the angle between a line oN and line oM be denoted by 0.
I li;llN,4nll()N()l l!llli fint WtNt):it't tt)l; tNil( ,tilil(nfJt I.t t.,rJt 1.t tMnil:, ll3
'l'ltc tttltxilttrrrrr rvrrrtl sl)(.(.(l:tl l0 r1 lrlrpvt'(lrt.6t.t.:rrr srrr.lltt.c, itvctirgctl
ovcr l0 lrrirrrrlt's, rr p,rv(.n by lhc clnpiriclrl rcllrliorr
t/( 10, R) : 0.865/",(/t) I 0..5,r (3.3.7)
t,l I 0,,)
rrr(;rjrr,r-r..1.7. rr;rrr',, lil';li , ttrt1111li) |r .r(,1
')
-1_
4.
6.
7.
50
40
30
E
E
d
l0
I
1
6
5
4
dp
dn
whr-'l'c rv is ohl:tinctl lirr crrrpiricirl tllrtrr l3-2(r. .1 2ltl
(3.3.6)
I l4 I X i lll Ml wlNl) ( ,l l[/n l( )l ( )( '|Y
l'lrc lO tttittrttc wirrtl sllectl l/1 lO,
(.rN is givcn by Lhc cxPlcssion l'l
U(10, r,0) : {/(lo' r') I ,' cos (/)
. ,l,lrt. witul vckrcity vector has a cornptlncnl. t.lirccted toward thc center of
lltt.s{tll'llt'().Theanglebetweenthatvcctorandthetangenttothecircle
tt'rr(r'tt'rl rrt O varies'iinearly between 0o and 10" in the region O I r
' /i :rrrtl l)ctween 10" and 25" in the region R < r < l'zR' and is
r'r1n;rl l,r .15" in the region r > l'2R13-261'
' llr,' :,lorttt clccaj results from a decrease with time of the difference
lrt lrl,r'r'il l)tcssure at the center and pressure at the periphery of the 
storm
rr ,r( ( ()r(llrtlce with the relation
r'. //) trl :t tlis(lttlt't' r' llrttt () irlolrg lirrc
2(rl:
(3.3.tt)
Lp(t) : APn,"* - 0.02[1 + sin {lr (3.3.e)
ir:r r:iilMn il()u ()r rrrr I\,4r wll.il,:ir ,r rr): ; rN ilr,rrrr{ Ar]r ll(,r!r (1Mn 1:, r15
l,cl llrt'lllrlxrlrilily llr:tl llrt'wttttl spt't'rl irr rrrry orrt.slorrrr is lcss tlriyr sgl.rrc
vttltlc' tr, bc tlt:ltolt'tl lry /',, l'lrt'pnrblrbility llr:rl llrt. lrililrest wirrtl {/ irr rr st1;rnrs
is lcss lltart ll t':rrr lrt.wttltr.n:rs
(3.3.11)
(3.3.rZa)
(3.3.tzb)
(3.3.t2c)
I''(U < uln) : 1"',', (3.3.10)
The probability that U < u in z years is denored by F(U I u, r). The totalprobability theorem (Eq. Al.5) yields
F(U < u, r) : i^ ofu < uln)p(n, r)n:0
where p(n, r) denotes the probability that n storrns will occur in r years,
Assuming thatp(n, r) is a Poisson process (Eq. A1.34), Eq. 3.3.11becomes
F(U < u, r:) : ! o' (\t)'" ^"n:O ' nl
- ')"3txt41'-(4- r:0 nt
_ --)tnl - 1,1-(
where \ is the annual rate of occurrence of hurricanes in the area of interest
for the site being considered. For z : r, F(u 1 u, r) is the probability of
occurrence of wind speeds less than u in any one year.
consider now the wind speed, ui.The probability that u I u,in any one
storm is
rrlrr rr. / . travel time in hours, Ap(f) and Ap"'o* are given in inches of
rrr'r{ rrry. and @ : angle between coast and storm track (0 < d < 180')'
llrr', rrttrtlcl is consistent with measurements repotled in 12-1241'
(rllrr'rt'tluctionofwindspeedsduetoincreasedsurfacefrictionoverland
r" r'rv('n by the t u'irollu'(10) : 0'85' where u/(10) and u'(10)
.rtr.lltcl0-minutespeedsatl0-melevationoverlandandoverwater,
rr':,lrcetively.ltcanbeverifiedthatthemodeldevelopedforextratropical
'.r,,,,rrs (Flqs.2.2.29 ancl 2'2'31 ' and Table 2'2'3) would yield a some-
rvlurt smaller ratio Ul( l0)/U''(10)'
/ llrt: clependence of wind speeds upon averaging time is modeled as in
Sr'ct.2.3.6.
Notc that physical models proposed in [3-28] are in some cases slightly
rir,rtliliccl wittr respect to the corresponding models o1' l3-26].
lrstirnates of the probabilities of occurrence of hurricane wind speeds were
rrtrrrrirrcd inl3-24lby assuming each of the areas adjoining 56 mileposts (Fig'
r 1.5) to be hit by m : 1000"hurricanes. The climatological characteristics of
llrc hurricanes were cletermined by Monte Carlo simulation from the respective
lrKrbabilistic models as fitted to historical data. For each 
of the rz hurricanes'
thc climatological characteristics used in conjunction with the physical models
dcscribedearlierdefineawindfieldwhichdependsuponthepositionofthe
hurricane.Toeachpositionofthehurricanewithrespecttothesiteofinterest
therecorrespondsawindspeedatthatsite.Windspeedscausedbyahurricane
at the site are calcurateJ ro, u sufficiently large number of such po-sitions. The
largcst among these speeds is the maximum wind speed caused by the hurricane
atthcsitc.Asctill.mspccdsisthusobtained,whichisusedasthcbasicset
ol'rllrlu lirr tlrt: cslirrurtion ol'tlrc prtlbilbility ol'()cctrffcllcc ol'ltttrricrrrtc wind
sPt't.tls, 'l'lrt'st' sPr'ctls :tR' tltttkt'tl lry rtrilgtriltttlc 'l'lrc i (lr srrt:rllt'sl spt't'tl ilt lt
s..'t ,,1 lr wiltrl spt't'rl:; is tlt'ttolt'tl lry tt,.
(3.3.13)
F(.U < ui, l) : e ^(t i/m+t) (3.3.t4)
For each of the mileposts in Fig. 3.3.5, estimates of hurricane wind speeds
corresponding to various probabilities of occurrence (or mean recurrence in-
tervals) were obtained in t3-24l both at the coastline and at various distances
inland lrom rhe coastlinc.
Results of a rccent stucly inrlicrrlt' llr:rt lrrrrr.it'irrrc wincl speed data obtained
by simulati<ln and ftrrnring lhc brrsis ol llrt't'stirrr:rtr.s ol'13-241 rnay be clcscribecl
by thc rcvcrsc Wcitrull rlistt'ibrrtiorr l l /ll 'llr;rr tlistr.ilrution lurs lirnitcrl uppcrtail and is consistcnl willr llrt':rsslrnlll rln tlr;rt lrrrrlt.lrrre witttl sPct:tls irrchrltltttlltl. Ilcl'crt:ncc 1,1 7l l lllrvi(l(':i 1rl()llrjrtrr)!r {)1 :rt r'r'ssirr1,, llr1;st' tl:rllr (llrt'
rllt(lr is ltlso ltvirilltltlt'irr ll.i ()l). lr:; rr,,.ll tr.. ( r)!nIrttr.r l)r1)t,tiuns lor llrt.(.sllu:tlt()lt
_tA -_"' m*1
Thus
I l6 I X llll Ml wlNl) { )l lMn l( )l (l( tY
()l tllc t(lvcrst: Wcilrttll tlislt'ilrrrtiorr l)ill;lllrt'l("ls:tttrl ol ltttttit'ltttt'witxl 
sPtetls
witlr vltt'iotts lllcan rcctlrt-ctlcc irttcl-vltls'
llsti.urtctl hurricanc win..l ,gr.,,1, tlillr'r lnrrrr tlrc 
"tt'rtc" s1.rc:ctls owittg ltt
.5se'vuti.., pnlhabilistil ltx)clolillg, lllysiclrl lrrtxlclittg' irrttl sarrtllliug 
crrors
(i.t'.'clrrtrscluctothelimitcclsiz,ctll.(Itctllttlrsirttrplc.sbr:ins.u'"..])ljtlrwind
spt'etls with lltcan .""u""nt" intervals tll'thc ordcr rll'50 ycars' it 
was shown
rrr I I ]01 rhirt thc *tunJura deviation ()f thc sanrpling errors, 
duc t<l thc limited
:;t.r .l t.lirrrrt.r.gical Jata availaule (about 100 years), is about 
l0% of the
,1.,1 ,,r':rr,'.1 s1,".,1*' it is about 15% fot 2000-year speeds'
I'r.t.t.tlrrrt.s ti,' "*ti,niting-r,u.ri.un" 
wind speeds-that-are similar conceptually
r. lt .)\l rvt'rt' ,r.,u"r.rf"J-in i: i1]^ana tS-:Zl for the purpose of studying
lnil | r, ;ril(' 1't'ttt'trtlccl *uu"'' A simplified methoJ for estimating hurricane 
wind
..1,,,,1', \\'it:' l)l()l)()scd in [3-56]'
Mrrrlr.' (--irrlo Simulations Based on the Shapiro Hurricane' Boundary'
I:ryt.t Mtt<Ial- 'l'hc #;;il; boundary-laver flow in a translating hurricane
\\';,'. .,l,r,r('\trrtrtlocl i"i;-;ii Lt uling empirical physical models^based on his-
r.rr,;rl ,lrrr:rt.l.gicat Jata,-as lndicated earlier in ittit section. Shapiro [3-57]
1|1'r,r.lr11ir'tI ,..' ..pp.o*i*ut" model of the hurricane boundary-layer 
flow based
orr ,r ,,rrrr1,lrlr,..t ,olution of the Navier-stokes equations' complemented 
by the
l()ll{,\\rnr' :rssrtttrptions: the boundary-layer height 6 and the 
eddy viscosity are
(..r.,r,,rr (;, I m; f ='i;1dt;t; '), uid th" frictional drag due to the
Irrrr',1;rltott vt'ltrcity i, "qtluf to the 
square of the flow velocity times an empirical
trrt t'rr , r,r.llicient rrr"il"J* rin"u.iy with flow velocity. Rather than obtaining
tlrr' lrrllv rrottlinear solution of the equations of motion' a simplified 
approach
w;1.'tt''t..lwltcreinthevelocitywaswrittenaSthesumofanaxisymmetrictetm
lttt,lllrt.lllsttwot",m,ofuseriesreflectingtheflowasymmetryduetohurricane
lriilr:,l.rlr()il. The facf that the series is truncated af'ter the first 
two terms is
t':,trrrr,rlt'tl by Shapiro to result in an approximation of the fully 
nonlinear so-
Irrrrrrrr lrr within uAoiZS% t3-57, p' f q96t' This errorcstirnatc doesnot 
include
nr,r,l, ltttg crrors; "u"" if 'f'L 
fully nonlinear solutions wcrc available' the Sha-
1,rr,, rrurtlcl would ";;;; uut" to iescribe 
the detailed s(ructure of the boundary
!,rv.r. cspecially near the eye wall [3-57' p' 19951'
liesrrlts of ri-ufution' Uu'"d on ih" Shapiro modcl wcrc reported in [3-58]'
rvlrrth used Shapiro's truncated series approach' The rcsults 
of the simulations
tlrllt:r from those of 13-241 inone main respect 13-591' Owing to a new and'
ilr ()ur opinion, *"diil. nuing rate model that is less conservative than its
toLrnterpart in Ref. t'iOl,the! yield.lower wind speeds inland' Otherwise' in
spitc of the use "t ln" Sit"pit 
mode-l'-they yield wind speeds-comparable to
those of 13-241. Although Ref' t3-581 rnual use of data and models 
from
13-601, whereas f3;4f ;?; basecl tn [3122] (i'e'' an earlier version 
of [3-60])'
this is not a Source of significant dill.erences. E,stimated wind 
speeds obtained
in 13_581 are lisrccl;; i;;i;3.3.1, which atso shows cstimatcs 
fronr [3-24],
t3-29l.arrtl|3-71|.Ntltc(Iur{(lrr:clllllparistlnstll'|3_5ttIllclwcclrtltt:cstitrratcs (,
i&
tr
tr
9'l
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z
tr6q)
I
al
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ro
Eq
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liE
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-:
69
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r60o
O€-O
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- ,:,,,&c<rl.l
Qr9n
r$€r
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N-a$
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ca
]fo
-o
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o
o
-:
!oo.o.
!o
E
io
.o
!
3.o
o
o
I
8r
!oabO
^re8
F2|=
i- -t'i9-'=!
-'E LE E!.. =!'. E'C
t' t'i .1
@9-$
9N6@
rr@o
r€r-
cno$
+-CO
:"." '-:" !,
ll7
I I B I x tltl Ml wllll) (,1 IMA l{ )l { )( iY
<ll'13 5ttlirrrtl 1.1 241irrc irrvirlitl 0wirr1', lo lrrr irrtorrsislt'rtt'y itt lltt'll:tltslirtlttlttiotts
of hourly spccds iukr lirstc:st rrrilc spt'ctls. 'l'ltcse Ir'ltttslitt'tttttlirttts lttt brtsctl itt
[3-58] on the moclcl ol'12-1351 lirr (lre e:slirrur(t's ol l.] -5ltl, bLt( ott lht: tlill'crcnt
model of [2-91] for the cstimatcs ol'1.1-241. lior 50-yc1r spcctls, tlillcronccs
between [3-24] and [3-58] excecd 10"1, lol trrilcposts 2(n. 3(X), ancl l3(X). For
2000-year winds the differences excccd lO'/,, l'<tr milcposts 700 and 1700. At
l0 m above open teffain, the hourly spced corrcsponding to the largcst 2000-
year estimate of [3-71] is about 47 mls at l0 m over water the cstimate of the
largest 2000-year hourly speedwould be about 41 x 1.2 : 56.4 mls'
Load Factors. It is shown in [3-71] that the wind load factor $,u : 1.3
spccified in the ASCE, 7-95 Standard would in most cases coffespond for wind-
scnsitive structures to nominal ultimate wind loads with mean recurrence in-
tcrvals of, roughly, 500 years or less. For the other sets of estimates of Table
3.3.1, the load factor d' : 1.3 would in many cases coffespond to nominal
ultimate wind loads with even shorter mean recuffence intervals' These results
are reflected in the average estimated ratios of 2000-year speeds to 5o-year
speeds, which are about I .3, 1.4,1.45, and 1.46 for the sets based on t3-7ll'
13-241, I3-2g1, and [3-58], respectively, so the squares of these values are
uUo"f t.Z, 1.g5,2.1, and 2.15, respectively. The results of [3-71] and Table
3.3.1 therefore indicate that for wind-sensitive structures, the wind load factor
for hurricane wind speeds should be larger than 1.3, even if hurricane design
wind speeds are multiplied by a factor of 1.05, as is done in the ASCE 7-95
Standaid t}-t3gl. For additional details, see [3-80], tA3-3ll and Sect. A3.3'
Estimates of Hurricane/Tropicat Cyclone Wind Speeds for Various Lo'
cations Outside the U.S. Estimates of hurricane/tropical cyclone wind speeds
based on models similar to those of 13-241 were repofted lor French overseas
departments and territories in [3-63] and are summarizcd in Table 3.3.2.
Estimated standard deviations of sampling errors in m/s were 2.2 (3.4),2.5
(4.2),2.0 (4.0), 1.6 (2.5) and 3.9 (12-3) for 50-yr (1000-vr) speeds at Gua-
daloupe, Martinique, R6union, New Caledonia, and Tahiti, respectively'
Estimates of hurricane speeds are reported in [3-64] for the Eastem Car-
ribean, Jamaica, and Belize and in [3-65] for the Northern Australia Coast.
For information on westem Norlh Pacific tropical cyclones, see
t3-ssl.
Saffir-Simpson Scare. The National Hurricane Center, the National Weather
Service, and emergency management departments use a classification of hur-
ricanes into five categories (Table 3.3.3). The central pressure and wind speed
portion of the classification was proposed by H. Saffir in 1970, while the storm
surge portion was added subsequently by R. Simpson. Thc avcraging timc,
height above grouncl, and surfacc cxposurc (i.c., whcllrcr ()l)cll lcll'llilt or wlttcr)
lil l:;,Mn 
'()N 
{)t I r.,llA,lt WtNt) :it,t tt)ti lN ,'t.'( A,t t,t tr', (.t tMn 
' 
:, l1g
'l'Alll,l,l -1.-].2- l'lslirrrrlrrl wirrl s;x'r'rrs (r(!r'irr s;x.r.rr irr r0 rrr ()vcr.irrt.()rcarr)
!.r-6rl
Mcan
rcturn
period
(yrs) Martiniquc
sr.
Barthilcrrry (iuatlcloupc NewIldunion Caledonia Tahiti
30
34
39
47
52
25
50
100
500
r000
32
37
4t
49
52
32
40
47
50
29
35
39
48
51
38
40
43
48
50
38
40
43
48
49
for the wind speeds are not specified in the classification. For this reason storrneffects and evacuation requirements for the various categories are described forthe use of, among others, emergency management personner 13-661, rather thanstructural engineers or building code officiils.
Mixed Distributions. Hurricane-prone regions are also subjected to winds notassociated with hurricanes (or tropical cycrones), whose effects can be ac-counted for by developing mixed distributions of hurricane and nonhurricanewind speeds. Since the occurrence of hurricane winds and the occurrence ofnonhurricane winds are independent events, it is possible to write
F(.U < u) : F11(J <. u)F7s11(J I u) (3.3.1s)
where F(u ( rz) is the probability that the wind speeds u associated with anystorm are less than u in any one year, and Fs(U ( z) ancl Fuu(J I u) arethe probabilities that hurricane speeds and nonhurricane wincl'ri""a, are less
thal a in any one year.-The probability Fs is determined as shown previouslyin this section' The probabitity F,vi7 is determined as shown in sect. 3.2.
TABLE 3.3.3. The Saffir-Simpson Scale
Category Description Mean WindSpeeds (m/s) Storm Surge(m) North AtlanticExamples
1
2
3
4
5
Minimal
Moderate
Extensive
Extreme
Catastrophic
33-42
43-49
50-.5f1
.59 #)
( ilctrlcr
lltlur (r()
t.2-1.6
t.7-2.5
2.6 3.8
3.9- -5.-5
( i r clrlcr
llr:rrr.5.-5
Agnes 1972
Cleo 1964
Betsy 1965
David 1979
Camille 1969
l2O {xllll Ml wlNlr (.
('irlculirliorrs tcptttlt:tl irr 1.1 l':l lsll(rw thlr( lltt'Prrrlrirbility /'lll 'r) r\ vir'
lulrlly tltc rurn" r,li I,'tr(/ < tt) lirl'lttcltlt lc('llll('ll('r'ilt(t'l'vltls N ' 5O yt:1r's'
Fo. N : 20 ycars, cstiptl(ccl wirrtl spce:cls tlr:rt irrcltttlc: tllt: cllcc( ttl'Itotllttlrricltltc
wincls cxceed the esl"imatccl hurricailc wiltcl spcctls lly al-tout 5%' Nrlto that
these conclusions are not neccssarily appliclrblc ntlrth tll'Capc Hattcras' whcrc
nonhurricanewindsmaycontroltheclcsigna(coflainlocatitlnsl3_33|.
3.4 WIND DIRECTIONALITY
Witrtl cllbcts on various structures and components depend not only on the
nrilgrritudo of the wind speeds but on the associated wind directions as well'
lr.i this rcason, knowledge of continuous joint probability distributions of ex-
trcrne wind speeds and directions would be useful for design and code devel-
0pmentpu,po,",.However,sofarnocrediblemodelsforsuchdistributions
have been proposed in the literature'
In the abience of such models, wind effects and their probability distributions
may be estimated in well-behaved wind climates on the basis of information
consisting of largest yearly wind speed data recorded for each octant over
periods oT ZO y"uit, tuy, oilong"r (see Sects' 8' l '2 and 8' l '3)' Such data have
teen published for a number of U.S. weather stations in [3-341. Summary
statistics of largest yearly wind data recorded at Sheridan, Wyoming, in the
period lg58-1l:i'7 (iee 1'aUte 8.1.2) are shown in Fig. 3.4.1. h is seen that in
ihi, "u*" 
wincls blowing from the noftheast are considerably weaker than north-
west or southwest winds.
As shown in Sect. 8.1.3, there are important practical applications in which
information is needed on the univariate probability distributions of the largest
yearly wind speeds associated with each of the principal compass directions,
una on the correlation coemcients for the largest yearly winds blowing from
any two directions. In well-behaved climates the largcst yearly wind speeds
ioi ony given direction are in most cases-though not always-adequately fitted
UV fVpJI distributions of the largest values. As indicated in [8-141, the cor-
retatlon between wind speeds occurring in any two of the eight principal com-
pass directions is in -ort .ur"r weak. For example, the estimated correlation
coeflrcients between wind speeds from directions i and i(i,i : 1,2, " ' ' 8)
are shown for Sheridan, wyoming, in Table 3.4.1. These values are fairly
typical. However, there are stations where the values of the correlation coef-
dci"nt, are higher (e.g., Detroit, Michigan, where 8 of the 28 estimated values
are larger than 0.45, although none exceeds about 0'6)'
An important practical p-ut"- faced by the designer is obtaining the largest
yearly wind ,p""d dutu for each of the eight principal compass directions at
io.ution, not covered in [3-341 . There are two such sources of data- One sourcc
consists of the original LrnpLrblishecl rccorcls storecl hy thc Nlr(irlnlrl Ot:canic and
Atrn0sphcric Arlrilinisf nrtiorr (NOAA). Ob(aining ittttl e xlrltt'lilrg thc rrcctlctl
rl:lltr lhltil tlrttst.rct'()ttls is lrollr irtcortvcrricnl rriltl littlr'('()llsllllllllI A st't'otttl
r ,t wllll, i,il lt I :ilr )Nn I il v
FIGURE 3.4.1. Summary statistics of largest yearly wind speeds by direction at Sher-
idan, Wyomine (1958,19j1).
source consists of published Local Climatological Data summaries issuedmonthly by NoAA. Directionar largest yearly speeds in the published datadiffer in a few cases from the corresponding speeds in the original records.
The reason for these differences is that the published data consists of (l) the
largest daily speed for every day ofthe year and (2) the direction for that speed.
TABLE 3.4.1. Estimated correlation Coefficients for Directional wind speedsin Sheridan, Wyoming
DirectionI I -0.05 0.352 1 0.01
3t
4
5 symmetrrc
6
1
lt
0.12 0.16
-0.1-5 -0.340.04 0.34
| 0 17
I
*0.031 -0.22 0.070.03 0.07 -0.010.10 -o.12 -0.430.03 -0.16 0.400.01 -0.16 -0.41t 0.20 0.01
| 0.32
I
20 30 40 50 50
122 IXiltt Mt WtNt )(jtMnt{)l (x;Y
Cgnsidcr, lirr cxartrplc, lIc ctrse wlrr'tt'in lr givr'tt ycltl lltt'lrrr'g.est prrlrlisllt'tl
speeds fbr wincls bckrwing l'nrrrr lhc r)()11lr lrxl lltc c:ltst arc 70 tttplt ittttl (r-5 rltPlr,
respectively. It is conceivablc that on thc salnc clay (ha( thc n<lrtlr witttl occurrccl'
the winds blowing from the east were 69 rnph. Thc highcst wind spccd f'rom
the east would not be reflected in the published data'
An exhaustive study of original and published data listed in 13-341 for 24
stations showed conclusively that the extreme wind speed estimates based on
published data differ insignificantly (by about 3% or less) from those based on
ihe original clata. It is, therefore, appropriate to base structural engineering
calculat"ions on the largest yearly directional fastest-mile wind speeds obtained
l'rorn l-ocal Climatological Data summaries t8-14]. A novel probabilistic ap-
pnxrch to thc modeling of directional extreme wind speeds, in which the ex-
trcnrc valuc distribution parameters are functions of direction, and which ac-
counts lirr the correlation among extremes across directions, was reported in
I 3-67 l.
In hurricane-prone regions estimates of hurricane wind effects can be carried
out on the basis of hurricane wind speed data generated by Monte Carlo sim-
ulation for each of 16 directions, as shown in sects. 3.3 and 8. 1 .3 (Eqs. 8. 1 .21-
8.1.23). Such data-used in 13-241 fot estimating extreme hurricane winds
blowing from any direction-are listed on tape in [8-9] (see also t3-7ll) for 56
mileposts (Fig. 3.3.5).
3.5 PROBABILITIES OF OCCURRENCE OF TORNADO WINDS
consider an area ,40, say, a one-degree longitude-latitude square, and let the
tornado frequency in that area (i.e., the average number of tornado occulTences
per year) be- denoted by D. The probability that a tornado will strike a particular
location during one year is assumed to be
aP(S):t-
Ao
(3.s.1)
where c is the average individual tomado area. In certain applications, for
example, the design of nuclear power plants, rather than the probability P(S)'
it is oi interest to esrimate the probability P(S, Z0) that a tornado with maximum
wind speeds higher than some specified value tr/' will strike a location in any
one year. This probability can be written as
P(5, vi : P(viP(s) (3.s.2)
where P(Z) is the probability that the maximum wind spccd in a tornaclo will
bc higher than 2,,.
tlsiirlu(cs 6l'p1rl'rlrbililics /'(S) in thc Urtitc:tl Slrrlt's rttt' sllowtt irl lril', 1.5' l.
r,, l,ll()ltnllll llll :; ()t ()(;(:tlilt il t.t(.t o; 11rt thlnt )() wtNt): ; t23
FIGURE 3.5.1. Tornado strike probability within 5-degree squares in thc contiguous
United States (units are l0 5 probability per year) [3-351.
which is taken from 13-351. Figure 3.5.1 is based on Eq. 3.5.1 in which D was
estimated from l3-year frequency data, a : 2.82 sq. miles (as estimated in
[3-36] for the state of Iowa), and,46 : 4780 cos S, where @ is the latitude at
the center of the one-degree square considered. Estimated probabilities p(zn)
are shown in Fig. 3.5.2, also taken from 13-351. These estimates are based
upon observations of 1612 tornadoes during l97l and 1972, and the rating
(largely subjective) of these tornadoes according to an intensity scale proposed
in[3-371.* It is noted that in estimating the probabilities of Fig. 3.5.2 it was
assumed that tornado path areas are the same throughout the contiguous United
States.
The maximum speed of the tornado corresponding to a specified probability
of occurrence can be estimated using Figs. 3.5.1 and 3.5.2. According to
t3-351, "in order to adequately prorect public health and safety, the determi-
nation of the design basis tomado is based on the premise that the probability
of occurence of a tornado that exceeds the Design Basis Tornado (DBT) should
be on the order of 10-7 per year per nuclear power plant." The required
probability P(Ze) is then determined from the relation
P(r/o)P(S) : t0-7 (3.s.3)
*According to this scale tomadoes may bc dividcd into the following classes: F0 (maximum wind
speed <72 mph), Fl ('73-112 mph). F2 (lll 157 rnph). F3 (l5tt 206 rnph), F4 (207-260 rnph).
F-5 (261-318 mph), and F6 (3 19 3ll0 rnphy
124 I X llll Ml wtNl) ( ;t tMn l( )l ( x iY
It too
=9flat73oz
;5
4
o.l o.2 0.5 I 2 5 lo 20 3040506070 80 90 95 9899
PERCENT PROBABILITY
l'l(;llltl'l -1.5.2. Percent probability of exceeding ordinate value ol'the wind speed [3-lsl
wlrt't' lhc value of P(S) for the location considered is taken f'rom Fig. 3.5.1.'l'lrc wincl speed corresponding to the probability P(lze) so determined is then
rrlrtrrirrccl l'rom Fig. 3.5.2. The average tornado intensity with a l0-7 probability
P('r ycar for each 5-degree square in the contiguous United States, based on
litl. 3.-5.3 and Figs. 3.5.1 and 3.5.2, is shown in Fig. 3.5.3 13-35].
lior nuclear power plant design purposes, the contiguous United States are
tlivided, in [3-35], into three tornado intensity regions shown in Fig. 3.5.4.'l'hc corresponding tornado winds are given in Table 3.5.1.
Thc pressure drop due to the passage of tomadoes can be estimated from
tlrc ccluation f<rr thc cyclostrophic wind. Using the relation Vr, : drldt, Eq.
I ..1.2 crrn hc wrillcrr lrs
1,, t,t t(rt rnilll llll :; ()t (xt(;lItl il N(.t ()t t()t tN/\t )() wtNt]:i r25
FIGURE 3.5.3. Calculated tomado wind speed by 5-clegree squares for l0 7 proba-
bility per year [3-351.
wherep is the pressure, / is the time, 2,, is the translationar speed, p is the air
density, R- is the radius of maximum rotational wind speed, and z, is the
maximum tangential wind speed* t3-351. Assuming R- is typically 150 ft for
intense tornadoes and that Vt = Vrur, Eq.3.5.4, in which the parameters of
Table 3.5.1 are used, yields approximately the values of rable z.s.z 1z-2s1.Following the development in [3-35] of the estimates summarized in Tables
3.5.1 and 3.5.2, vaious attempts to improve the probabilistic and physical
description of tomado winds have been reported [3-3g, 3-3g, 3-40, 3-41 ,3-42, 3-43,3-44,3-45, 3-46, 3-4i1. Using as a point of departure tornado risk
maps presented in [3-46], a regionalization of tomado risks that divides the
contiguous United states into four areas was proposed in [3-45] (see also [3-44,p. 4801. Regional tomado occurrence rate (per mi2 per year) were estimated in
[3-45] from a29-year (1950-1978) data bank maintained by the National Severe
Storms Forecast Center and comprising about 20,000 reported tornadoes. These
regional occurrence rates are corrected in [3-43] and [3-45] to account for:
l. Failure to record tomado intensity, which affects about lo% of the total
number of reported tornadocs. J'his corrcction is based on the assumption
that unrated tornadocs ntay bc apporlionccl anrong the various intensity
categories according lo lltc tcporlctl tor-rriukr licquencies lor those cate-
gories.
r'llrt'nrlational spcctl (,,, is tltc tlsttll:rtl ol tlrc tirrrllr'rrtrrl;rrrrl rirrli:rl vt'ftx,itit.s.
FOR ENTIRE CONTIGUOUS
FoR ALL STATES wEsT oF 1O5O w LoNGITUDE
( 1.5 -l)
o
o
oz
\'$rr
Not
N€
iN
:t,, I'il{ )ilnltil ilil :i ()t
l'Alll,lj J.5.1. llcgirlu:rl'lirrrratkr Wirrds
( x;cunt u N( .t ( )t t( )nNn t ,( ) wtNt)ti 127
Maxinturtt
Speed (,,,,*
Region (mph)
Rotational
Speed tr/,.,
(mph)
Translational
Speed 2,,
(mph)
Radius of
Maximum
Rotational
Wind Speed R,"
(f0
I
il
m
290
240
190
150
150
150
360
300
240
70
60
50
Temporal variations in tomado reporting efficiency. The number of re-
ported annual tomado occurrences in the United States has increased from
about 250 in 1950 to 850 rn 1979. The growing trend in the number of
reported tornadoes during this period has been ascribed to a correspond-
ing increase in population density. An explicit relation to this effect has
been proposed in [3-47].Corrections accounting for tornado reporting
efficiencies were effected in [3-45] by averaging the 1971-1978, 1970-
1918, 1969-1978, and 1950-1978 data and assuming that the true oc-
currence rates are equal to the largest of these estimates.
Possible errors in the rating of tornado intensities on the basis of observed
damage. The reason for the occunence of such errors is that maximum
tornado winds are in practice not measured, but inferred, largely on the
basis of professional judgment, from observations of damage to build-
ings, signs, and so forth [3-42].
Inhomogeneous distribution along the tornado path of buildings and var-
ious other objects susceptible of being damaged. In the possible absence
of such objects over the portions of the tornado path where the winds
are highest-or even over the entire tornado path-the rating of the tor-
nado is bound to be in error. The effect of corrections for such errors is
to increase the estimated probability of occurrence of tomadoes with
higher intensities.
Variation of tornado intensity along the tornado path. Accounting to this
factor results in smaller estimated risks of high tornado winds than would
be the case if the maximum tornado winds (by which tomado intensities
TABLE 3.5.2. Regional Pressure Drops and Pressure Drop Rate
Total Pressure Drop Rate of Pressure Drop
(psi/s)Region (psi)
I
ilu
2.
1_
4.
c.t
c.)
bo
l.)
o
cn
oF
$
la
ar)
F
trr
5.
3.0
2.25
r.5
2.O
1.2
0.6
124 I x I tll Ml wlNl) (.1 lMn l( )l ( x iY
arc ftrlc(l) wu'c unili)r'rtt lrlortpl lltc t'rtlttt'p:rllt. ('ot't'ccl itttrs clli'c'lctl in
l3-451, bascd upon lhc irrtalysis ol rhrt'rrrrcn(ctl lorrrarkrcs, lctl to risk
reductions by a llctor ol'abou( livc lirr li4 (orrratlocs antl about tcn lirr
F6 tornadoes.
The corrections for the factors listed involvc subjective judgments that may
be formalized by Bayesian techniques (see Eq. Al.6). In [3-45] the corected
rates of occurrence differ insignificantly from the uncorrected (prior) rates, with
the following exceptions. For the three areas of the regionalization map pro-
posed in [3-451 in which the most intense tomadoes recorded in the period
1950 1978 were F5, it was estimated in [3-45] that rates of occurrence of F6
tornadoes, rather than being zero, are about 1/20 times the rate of occurrence
of F5 tomadoes. For the fourth area of that map, in which the most intense
tornadoes recorded in the same period were F4, it was estimated that the
corrected rate of occurrence of F4 tomadoes is about six times the uncorrected
rate, and that the rate of occuffence of F5 tornadoes, rather than being zero,
is 1l2O times the corrected rate of occurrence of F4 tomadoes.
Reference [3-43] suggests that the velocity ranges associated in [3-37] with
the tornado ratings F 1 through F6 (see p. I I 1) are overconservative by amounts
varying from about 5% for Fl tornadoes to about 2O% or more for F6 torna-
does. The wind speed reductions proposed in [3-43] are used in [3-45] as a
basis for suggesting a reduction of the 360 mph, 300 mph, and 240 mph wind
speeds, specified in [3-35] for regions I, II, and III of Fig. 3.5.4, to 300 mph,
225 mph, and 200 mph, respectively. In the authors' opinion, the arguments
adduced in 13-431 in favor of such reduclions are tentative, in some instances
at least. For example, to support the contention that the maximum wind speeds
in a tornado classified as F3 are lower than the values proposed in [3-37],
[3-43] interprets a 133-167 mph estimate of the velocity causing the collapse
of a chimney during the Xenia, Ohio, tornado of 3 April 1974 13-42, p. 17151
simply as a 133 mph estimate 13-43, p. 16251. On the other hand, it should be
noted that the estimates of [3-35] and [3-37] are also tentative.
A position that is to some extent a compromise between [3-35] and [3-45]
was adopted in the American National Standard ANSI/ANS-2.3-1983 [3-481,
which divides the contiguous United States into three zones, denoted as 1,2,
and 3. Zones I and 2 cover, approximately, region I of Fig. 3.5.4, while zone
3 covers approximately regions II and III. Table 3.5.3 lists the maximum
tomado wind speeds Z-o^, the translational wind speeds Vr,, the radius of the
maximum wind speed R-, and the maximum atmospheric pressure drop po,
given in [3-48] for tomadoes corresponding to various probabilities of exceed-
ance.
According to [3-72, p. D-ll, the ANSI/ANS-2.3.1983 Standard was not
approved by the Nuclear Regulatory Commission. Efforts to develop an im-
proved standard are under way. Reference [3-73] is an overview of recent
developments conccrning dcsign critcria fbr tomadoes. It notes that ncw Nu-
clcar Rcgrrlat<lry Corrrrrrission c:ri(cria ckr not clclinc tornaclo clcsign crileri:r on
trt l lt N(;t l; 129
'l'Alll,ll -1.5.J. Slrrrrrlitltl 'l!rt'rtirtlo ('har:rclerislics (llxlraclcrl li-6r1 ArlcricanNali'nal slanrl.rrl ANsl/ANs-2.J-l9tt-] with per'rissirrr 'l'thc publisher, ilreAmerican Nucletr Socill.y)
Probability ol'
Exceedance, per Ycar Zone
V rrru
(mph)
V,,
(mph)
Rnro*
(f0 (psi)
107
106
l0-5
320
250
180
260
200
140
200
150
100
70
55
40
57
45
32
45
JJ
25
540
435
320
453
355
253
355
270
185
t.96
1.35
0.70
1.46
0.85
o.4l
0.85
0.47
o.20
I
2
J
I
2
3
1
2
3
a probabilistic basis, although_the design parameters it accepts for new nuclear
reactordesigns are in the 10 6 range 13-74,3-151. Reference [3-76] is a studyof tornado climatology in the contiguous United States based on the National
Severe Storms Forcast Center's tomado data base for the period January l,
1954, through December 31, 1983. Strike probabilities were estimated in
[3-76] on the basis of expected tomado areas, conditional probabilities of tor-
nado intensities were based on affected area, rather than on number of occur-
rences, the intensity distribution was based on a weibull model, and design
wind speeds were based on regional intensity distributions. wind speed io
obtained were 50 to 100 mph lower than the estimates of [3-35], and tornado
design basis wind speeds suggested in [3-76] are 200 mph and 330 mph,
respectively, for the United States west and east of the Rocky Mountains. For
additional information on tornadoes, see also [3-69] and t3_701.It was noted in [3-40] that probabilities of a target being hit by a tornado
wind in excess of any specified threshold depend upon the iize olthat target.
This topic is analyzed in detail inf3-44,3-451, where the estimates are based
upon statistics of tomado intensities, path lengths, and path widths on the one
hand, and on the geometric characteristics of the target on the other. It is
suggested in [3-49] that tornado wind loads dominate the design of most trans-
mission lines over 10 miles in length over wide areas of the united States.
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PART B
WIND LOADS AND THEIR EFFECTS
ON STRUCTURES
I FUNDAMENTALS
CHAPTER 4
BLUFF.BODY AERODYNAMICS
The subject of aerodynamics covers a very wide range. of necessity, therefore,
only a few highlights can be emphasized in the present chapter. The field
received its great initial impulse from the efforts in the early twentieth century
to achieve heavier-than-air flight. Since that time it has continually received
strong contributions from a great variety of aerospace studies, and from the
sustained, intensive development of machines with internal flows, such as jet
engines, pumps, and turbines.
In addition, interesting new advances in applications of aerodynamics to
civil engineering structures have occurred in the last three decades. Dealing as
they do with the natural wind, these applications of aerodynamics are limited
mainly to relatively low-speed, incompressible flow phenomena. In this appli-
cation, aerodynamics is also closely associated with meteorology and concemed
in particular with turbulent flows in the boundary layer of the earth's atmo-
sphere.
Besides a primary concem with the mean velocity of the wind, two aspects
of these turbulent flows are of interest to the structural engineer: the state of
turbulence of the natural wind approaching a structure and the local or "sig-
nature" turbulence provoked in the wind by the strucrure itsell. Since moir
structures in civil engineering present bluff forms to the wind, emphasis is
placed, in wind engineering, upon bluff-body aerodynamics. This fact, char-
acteristic of a new situation not emphasized as strongly in aeronautical and
other previous studies, has occasioned new research on the details of, flclw
effects around bluff forms typical of such structures as buildings, towcrs, ancl
bridges. In this context, interest centers particularly on details ol'thc clcvcl-
opment of body pressures by the givcn flows.
As pointed out by Roshk<l in a rcccnt rcvic:w, "llrc: pnlblcrrr ol' blrrll lxxly
r35
136 nt lI I il()l )Y nl ll()l)YNnMl(ll
ll1;w rclrpritrs irlrrrosI errlilcly irr llrt't'trpttit';rl. rlt'st'r'iptive tt'ltltlt ol kttowlt'tlgc"
l4 tttil. Altlxrr.rgh ctx)rnl()us lrtlvlrnt'r:s lurve lrt't'tt lttttl lttc bcittg rrtitrlc itl coltt-
ltutali6nal lluicl clynarrrics (C-Irl)), so llrt'resrtlts ltitvc boctr Itltltle:sl irr tllc sllc-
cializ,cd branch ol-CFD known ils ('onrl)ulirliontrl wind cnginccrirrg (CWts)-
Lllrl)/CWE can proviclc qualitativc rcsrrlts ott wittcl llow lor pcdcstrian comlbrt
pufposcs (Chapter l5), although cvcn irr this casc no delinitive validations
itppcar to be available [4-89]. Howcvcr, lilr structural engineering purposes,
0wing t() the computational problcms arising in large Reynolds number, tur-
lrulcn(, separated flows (Sect. 4.3), current methods are inadequate and/or
lrnrhihitivcly expensive. For details on the current status of CW-E, see [4-90
ro 4 92, 4-951.
ln this chapter a few basic theoretical principles and experimental facts are
r-r:vicwed that lay a foundation for the study of wind engineering'
4.1 GOVERNINGEQUATIONS
4.1.1 Equations of Motion and Continuity
Consider a fixed elemental volume dV in a fluid. The vector velocity* of the
fluid is commonly expressed bY
u:ui+/j+wk (4.1.1)
where i, j, k are unit vector components along the usual three fixed rectangular
coordinate axes x, y, 3. For compactness of notation let x, y, Z be replaced
respectively by x1, x2, x3, I't, u, wby u1, tt2, tt3, and the unit vectors i, j, k by
i1, i2, ij so that Eq. 4.1.1 may be rewritten
3t : 
'?' ''i" 
(4'l'2)
i
The force acting on the fluid contained irt the volume dll consists of two
parts. The first part, referred to as the body lforce and caused by some force
held, such as gravity, will be denoted byFp dV, where p is the fluid density.
The second part is due to the net action on the fluid of the internal stresses
oa(i, i : 7,2,3). For example, the contribution to this action of the normal
stress 01 (see Fig.4.1.1) is
-otr dxz ar, + (o,, + # *,) *, dr, -- 
ai dxl cLrt dvl
dxt
FIGURE 4.1.1. Forces on an elementary volume of fluid
It can similarly be shown that the net force component in the i direction due
to the action of all the stresses o,; is
a,, dxadx.
,t I (i{'vt iltltN(, t(Jl,Ail()Nl; 137
t.,,* ff4,,tdxrdx.
), uo o,1:r 0x1
Du, I a",,
A p dV : Fip dV * ,",i,0,
Da:a+ Lu-Dt 0t i:r '0*i
(4.1.4)
Denoting the components of F by F,(i : 1,2,3), the force balance equations,
given by Newton's second law, are
where the operator DlDt, known as the substantial or the material deivative,
is defined as follows:
(t : 1 ,2,3) (4. r.s)
(4.1.6)
Since Eq. 4.1.5 is true for all volume elements, the factor dV may be divided
out of Eq. 4.1.5 and the equations of motion, in component form, of a fluid
particle can be written as
(4.1.1)
Various forms of this basic equation can be derived depending upon the nature
of the forces d and stresses o4 acting upon the fluid particle.
Before examining these particular cases, it will be useful to recall the prin-
ciple of mass conservation. This principle states that the rate of increase of the
fluid mass contained within a fixed closed surface must be equal to the differ-
ence between the rates of influx to and effiux from the volume enc.loscd by that
suriace. The equations of continuity can then be shown to be [4- | . 4-21:
Du, .l a",i,' : rF, *,1,#, (i : 1.2.3)
: oot' ,rvdr' (4. 1.3)
tlrr :rpltlicirliorrs wlrr:rt' llrclt.t'xisls lt sirtglt' inllx)t'lirtll tttt':tlt llttw vt'lot ily lti lotttltltttir'rl lry vlrri:rblc
c()n)l)on('nls. lltt'nrt':rrr llow is ollctt lirkt'tt:ts ltt'irr1g irt lltt" t rlit't'tliott. tltllt vclotrlv (l(:,il'lllll((l
rrs l/. fltc tcslx'r'livc r. \',.'(()llll)()ll('ttl:i lltt'tt lx'irt11 rlcsrlltt:rlr'rl ltr l/ I tt I' tt
II
rl
i)(Nri\
/lr
op
i)r
(-1.I n)
*138 Bl Ut I tr()t )y At n()t)yN^t\,4t(;r;
l'or lttt ittcottt;ltcssiltltr lltritl whe tt'irr rro r'lr;rrrg,r. irr tlt'lrsily /) ('(:(:ut.s, llris rr.rlrrt.t's
Itr
.', arr.L..', 0 (4.1.9)i r 0.r,
4.1.2 The Navier-Stokes Equations
tjnlikc a solid, a fluid under static conditions is incapable of suppor-ting any
slcady-state stresses other than normal pressure. In dynamic situations, on the
othcr hand, it may support shear in a time-dependent manner. Most often, in
lltricl-mechanical applications, it has been adequate to assume then that the
strcsscs involved are either normal pressures or ascribable to viscosity only.
lrluicls with internal shear stress proportional to the rate of change of velocity
with distance normal to that velocity are termed viscous or Newtonian. For
cxurnple the shear stress ol2 in the simple two-dimensional flow pictured in
lrig. 4.1.2 is expressed as
outotz: lt *
whcrc the proportionality factor is defined as the fluid viscosity.*
rl
FiGURE 4.1.2. Linear velocity increase with distance from a wall.
r'l'hc units ol viscosity arc
liler. lcngth lorcc timc
' ilr'('it vt'kre ily lcngth r
'l'ypitirl v:rlrrcs ol lr lirl t'x;rnrplt' lirr:rir:uttl wlt(cr irl 20,, :rrc
ltuu 0 ()(X)ltt p/t.rrr s, 111,,, O.Ol 1ilt,rtr s
(irttttttllt ttttil: trtr'/r,rrr'r, r!lrr,lr. I lrrisr. I 11lt.rtr s (].()(]lO') llrl r/lt
'l I (i()Vl llNlN(i l(Jlln ll()Nri 
-l 39
lirrllhcr, by rlivitlirrg tltc wlrolc s(rtss /r'rt,srrl o,, lt( it lltlitl ptlint itt(o prcssttrc
strcss (6r'sirrrply l)ft,,t,\ur(, tltirl is, rtorttutl sltcss) 1t ltttd tltvitrtorll' s(tcss, dclincd
as
(i, j : l, 2, 3) (4.1.1 l)
where
(4.1.t2)
du : 2r' (r, _ iu, -i, ,--)
I ( 6u, 8ri\eii:i\a*-i,)
(4.1. l0)
" fr. i:jur:Lo. i+j
the following breakdown of stress oii can be obtained:
(4.1.13)
@.r.4)oij : -pbij + 2r, ("i- +r, -i, "--)
Using this form of stress for a Newtonian fluid results in the equations of
motionx 
,
Dui _,_op.ig\r*/ r : \). 0"fi : oF, 0x, 7:r dx; ( (", - 16,i ^4,'^r)\ 
(4'l'15)
Equations 4.1.15 (i, j : 1,2,3) are the well-known NQvier-Stokes equa-
tions. If Flq.4.1 .12 is used, and if the viscosity p may be considered to be
ionstant throughout the fluid, then Eqs. 4.1.15 become
3
Z @uol\xo)k:1
0*,
Further simplification occurs in the case of an incompressible fluid, that is, one
for which Eq. 4.1 .9 holds. Equations 4.1 .16 can then be written in the vector
form
,o#: oF,- y-, (,; W. ) (4 116)
: pF - ; P ii -r tt > f it I tlX; i I (txt
aI.
-1
lcngllr tirrrc
Du
PDt
ilior :r rrrort'tlt'liriletl ir(('()tlrrl s('('. lirt t'rittttltle, l'l ll. l'l .ll, l'l ll' or l'l'll
A.t.t7t
*140 ilr Ur I il()t)y nt il()t)yNnMtcl;
4.1.3 Bernoulli's Equation
lJor a lluirl that, in aclditi<ln to hcirrg irrerrrrprcssiblc, is irryl,rr.irl (/r 0) irrrtl
is actcd upon by ncgligiblc bocly lirlccs, llt1. 4.1.17 rctluccs to
(4. l. l8)
ll'the coordinate axes are so oriented thatJl corresponds to the direction of
rnotion, and if the flow is steady, it follows immediately from the integration
ol' [lqs. 4. I . 18 that
;r :, llowlNA(:llllvl l)l'Alll v(|l lllxll()w t4l
Du i, al, .e Dr : -,?,; ''
jl"l'*4:consr
Ja//I/
srnenulrrurs \ \,,/
-('
dr
-\-
u
ll ovory point of a streamline. Equation 4.1.19 is a special
tltt'rtrcm and is most commonly written as
(4.1.19)
form of Bernoulli's
lpu' + p: const (4.1.20)
wlrcrc a is the flow velocity along a streamline. The quantity |puz has the
rlirncnsions of pressure and is referred to as lhe dynamic pressure.'l'his important equation is widely used to interpret the ie-lation between
i)rcssurc and velocity in atmospheric and wind tunnel flows. Detailed comments
on llcrnoulli's equation and its applicability in fluid flows-including flows in
which viscosity is present-are provided in Sect. 3.5 of [4-3].
4..2 FLOW IN A CURVED PATH..VORTEX FLOW
('orrsicler a two-dimensional flow u"rr")lio locally concentric streamlines a
tlistance dr apart and having radius of curvature r (Fig. 4.2.1). For the flow
to nraintain its curved path, it must experience an acceleration toward the center
trl'curvature of the streamlines of amountu2lr, where z is here used to designate
tlrc local tangential velocity of the flow. Let the pressure acting on the fluid
e:lcrrrcnt under consideration be denoted by p. The pressure differential from
onc strcamline to the next along r, which is responsible for this acceleration,
is r/7r. The equation of motion for the fluid element is then
rlp dA : p d.r dl llr
wltcrc trr is tltc llrritl tlcnsily lu'rd dA is thc arca <lf thc clcntcrrt in ir pllrrrt: rrtlrrrral
to llter plitltc ol'llrt'ligrrlc. 'l'lris rclulion inrlicirlcs lhlrt lltc l)rrsriur'(. t.lurrrge rror.rnirl
l() lllc sll'ctttttlitres ol'lt r'tttvt'tl llrtw irr tlrc: :rbscrnt'e ol irrry ollrt.r' lort.r.s rs
Bernoulli's equation (4.1.20) then permits calculation of the pressure along a
curved path of such a streamline flow.
In particular, one may consider the case wherein the flow is completely
circular and the value of p6 in Eq. 4.1.20 is the same on all streamlines. This
is the case of vortex.flow. Differentiation of Eq. 4.1.20 yields
.drdp : pu';
du doPui+,rr:o
which, when combined with Eq. 4.2.1, yields
FICURE 4.2.1. Flow in a curved Path
(4.2.1)
(4.2.2)
(4.2.3)
Equation 4.2.3 canthen be integrated to yield
ur:C:const (4.2.4)
'I'his simplc law sta(cs lirr irrr inconrprcssihle, inviscid fluicl thc thcrllcticlrl
(hypcrb1;lic) rclation bclwct'rr positiorurl t'rrtlius r antl tangcnliltl vclrtcily rr irt lt
.lit't'rrtrtc.r.
du dr
ur
142 lil ul I il( )t )y n t tt( )t )vNn Mtcii
Itt lttt ;tcltt:rl lit't' votlcx, ltowcvt'r. llrt' r'llct l:r ol vrscosily il'(. l)t(',.r(.nl ilswt'll.'l'llcy llltvc ttol llcctt ittt'ltttlt'rl rtr llrt'srrrrplt tlt'r'iv:rlion irlrovc.'l'lrcst.will
Ititvt', itt l)ilrt, lllc cll'cc( ol'"lockirrg," s()rn('l)()rtr()rr ol lhc lluitl (ltclrr llrt.t'r'lrlg.)
logclltcr alttl citusittg il to ft)lllc lrs lr liliitl lrtxl-y irrslt:lrtl ol'as thc pclli.t.l llrritl
rlcscrilrtxl by l-q. 4.2.4. T'hus locirlly, neru'thc t'crrlcr ol'a I'rcc vgrtcx. llrc
vclocity u intrtu:sc,t with radius, whcrclrs ucconling to Eq.4.2.4 il tlt,crttt,scs
witlr irrcrctrsing r. This latter condition lctually hokls outwarcl fnrrn a tnu1sitiln
tr',t\i.rt irr which rz attains its maximum valuc. 'l'hc value of a in such a region
rs rlt'Perrtlcnt on the values of the fluid viscosity and of the total angular mo-
nrcnlunr rrl'thc vortex. Figure 4.2.2 lllustrates qualitatively the pressure ancl
vt'krt'ity rclrrti<lns that hold in a free vortex occurring in a real fluid. It should
lrt'rrrtt'rl tlrirt thc free vortex here described differs from the forced or con-
ttttttttt'rl rrlTr'.r that may develop in a fluid held in a rotating container.'l'lrr' llcc vortex is of interest in many flows that occur in engineering ap-
lrlrt';rtirrrs. lior cxample, atmospheric flows along the curved isobars of the
rvt'rrf lrcr nrrl) ilrc described by generalizations of Eq.4.2.1 . These have been
rlt'strilrt'rl irr scct. 1.2, where additional Coriolis forces have been included.
.l :l lt{ )l lNl )n I lY I AYI I l" n lll ) :;l l'l\l tn I l( )ll 143
4.3 BOUNDARY I AYERS AND SEPARATION
'l'lrc lrngc: ol vist'o:;rly v;rlrrt's lo bc lirtrtttl itrrl()rt1l vrtriotts lluids is vory grcat.
'l'ltc viscosity ol lrir lrt rurlrrrirl rrrclorlrolrlgicitl 1tl'cssrtl't:s ancl {ctnpcraturcs how-
t'vcr has a r-clltively srrlrll virluc. Noncthclcss, in stlrrtc circumstances this small
viscosity plays irrr irrrprlllnt nrlc. An imporlanL rnanif'estation of the viscous
cll'ccts of air occut's in tlro lbrmation of boundary layers.
Consider an air llow over and along a stationary smooth surface. It is an
cxpcrimental fact that the air in contact with the surface adheres to it. This
cituses a retardation of the air motion in a layer near the Surface referred to aS
tlrc boundary layer. Within the boundary layer the velocity of the air increases
l'rom zero at the surface (no slip) to its full value, which corresponds to the
cxtemal (as opposed to boundary layer) flow t2-11. A boundary-layer velocity
profile is depicted in Fig. 4.3.1.
Air, since it has mass, evidences inertial effects according to Newton's
sccond law (or, more specifically, the Navier-Stokes equations). The two most
influential effects in an air flow are then viscous and inertial, and the relation
ol'these to each other becomes an index of the type of flow characteristics or
Phenomena that may be expected to occur. This index can be expressed as a
nondimensional parameter G", the. Reynolds number, which is a measure of
thc ratio of inertial to viscous forces. For example, consider a volume of fluid
with a typical surface dimension L. Then, by Bernoulli's theorem, the net
pressure p - po caused by fluid flow at velocity U, which is of the order of
t, p[J2, creates inertial forces on the fluid element enclosed by that volume which
I
p
HEIGHT
VELOCITY
.1.,!. L I \ lr( .rl lrrunrl;trv l;tyt't vt'lot'ily lllirlilt'.
u =C/r
lll(illlll'l 'l.l.f- I'r.ssrrrt' :rrrtl vt'krt iry rlisrrilruri.rr rr ;r 'rirrr'\ lr.r' Itl(llllll,l
144 ilt rJr I il( )t )y l\t il( )t )yNt\Mt(
irr'('()l llr('()r(lt't ()l /){/ /. . ()rr lltt'ollrr't lr;rrr,l. llrt'visr'otts stt't'sst's otr llrr"t'lr'tnt'rtl
Irr.t'ol lltt'onlcl ol lrllll,, sr> visto:;rly rt'l:rlt'rl l()r'('(':j itr(r ol tltt: otrlt'r rtl 1r.llll,' /,'. 'l'lre lirlio ol'irtctlilrl (o vistorrs lolt t s rs tlrt'rr ol'lho rlttlcr ol'
,l :l li()t ,Nt)nnY tnYt ilr; ANt) ::t I'nt tn |()N 145
FIGURE 4.3.2. Flow separation at corner of obstacle.
l)oint,* the wake will contain the effects of vortex formation. Depending upon
tlrc magnitude of the Reynolds number, the flow willbe turbulent to a greater
or lesser extent. Many turbulent flows may thus be typically viewed as wake
lkrws in which upstream objects have already "stirred" the flow in some such
rnanner as has been described. Turbulence can bc caused by means other than
thc stirring mechanisms mentioned above (e.g., by thermally induced convec-
tion), but for the majority of flows of importance to wind engineering, turbu-
lcnce can be considered to be initiated mechanically, as described. Thus, for
r:xample, trees, buildings, or telrain upstream of a given point play an important
role in developing the turbulence of the wind observed in the atmospheric
lroundary layer at that point. Descriptions of turbulence in the natural wind are
given in Sect. 2.3.
When turbulence is present, one turbulent layer of the fluid tends to produce
Iurbulent motions in adjacent layers, as, for example, in a wake or boundary
ltyer. This takes place through transfer of momentum from one layer to an-
other. A similar phenomenon occurs in the absence of turbulence when a
lirrninar, as opposed to turbulent, boundary layer is created. The difference
bctween a laminar and a turbulent boundary layer is that, in the former, the
transfer of momentum occurs at the molecular rather than the macroscopic
scale. The fluid viscosity p is in fact the result of such molecular transfers of
nx)rnentum. As noted in Sect. 2. I in the context of atmospheric flows, turbulent
lxrundary layers may be viewed as being governed by an equivalent kinematic
viscosity callcd eddy viscosity, whose value reflects the large momentum trans-
lcrs induced by turbulcncc.
(4.3.1)
wlre:rc: y : pl p is called the kinematic yi^r<'o,ril_y.* (See also Sect. 7.1.) Thus,
wlrcrr 61. is large, inertial effects preilominate; when it is small, the viscous
t'llccts are the stronger ones. It is noted that the concept of Reynolds number
rs. irr rclaticln to the boundaries affecting a flow, a very local thing; that is, the
st'lt'c(ion of the representative length Z for the calculation of G" depends upon
tlrr' irrtcrcst ol'the investigator in local details. Thus a flow over a given object
rrr;ry tlcvckrp a wide variety of Reynolds numbers, depending upon the partic-
rrl;rr rcgion focused on for study. When discussing the whole flow that envelops
rr llivcrr lxily, it is usual to select for the length -L some overall representative
tlirrrrnsion ol that body.
Iirrrrrtlrrry-layer separation occurs if fluid particles in the boundary layer are
srrllicit:rrtly dccelerated by inertial forces that the flow near the surface becomes
n'vt'r'sctl. 'fhcse deceleration effects occur as a result of the presence in the
llow ol'arlvcrse pressure gradients. Such severe adverse pressure gradients as
t'rrrr lrr: llrocluced, for example, by the flow over the comer of a bluff body
I't'nt'r'irlly cause flow separation. Through processes that are not well under-
strxxl, thc scparation layers generate discrete vofiices, which are shed into the
wrrkc llow bchind the bluffbody (Fig. 4.3.2). Such vortices can cause extremely
lril',lr srrctions ncar separation points such as comers or eaves.
Iilows ol-practical interest have Reynolds numbers ranging from nearly zero
to irs lrigh as 108 or lOe. Steadily increasing the Reynolds number of the flow
()v('r irr) obstacle generally produces a widely varying sequence of flow phe-
n()rncnlr lirr which the Reynolds number provides a convenient index, as is
st't'rr. lirr cxample, in Sect. 4.4.
ll'" as is true in most cases, the flow over a body has separated at some
''l vpical valucs of kinenratic viscosity for air and watcr are. respectively:/,i, : 0. 150 crn2/s at 20'C
r,".u : 0.01 crn2/s at 20'C
A t orrrnror rrrril lirr kincrrratic viscosity is thc .stote:
I stokc - I crrr/s - 0.001764 ltrls
A rrrclrrl :rplrlrlirrt:rlc lorrrttrl:r lor lltL'l{t:ynoltls rtrrrrrbcr in:rit'lrl lrllrul lO"(':urtl ;rlrrrosplrcrit'
lr( r;sllr( rs (r/ (X)() l// . r|lrt'ri {/ is in Ittt'lr'r's l)r'r s('('on(l :rrrtl /. in nr('l('rs l lrs lrr'r'rrrrrt s (r.) }O l//.
lor l/ rrr lt/:, :lrrl / rr l1-( l
olJ L(lle : '
pUL ll
p( )l , I!:
lLv
llrt lltt' t'rrsr' ol lrirlirrls (x ( lrrr( r( ('
tlrc lrtxly. in ;rttottlrrtttt rvillr llrr'
tr'tltttr' tlt;t1', lry ttttlttts ol llcrttttt'lttr
ol llrl,.,r'lrrr.rlrorr r:, rr:rr;rlly tk'sirt'tl lts l:rlc:rs possilrlt':rlorrl',
,rlnr "l , t'nlr'rllrr,' Irr':,r,lr( rlisttilrttlions 1o iltt li':rsr' lill :rrrrl
Iorrr
146 n UI I Ir( )t )y n t tr( )t ,yNn Mt( il
4.4 WAKE AND VORTEX FORMATIONS IN TWO-DIMENSIONAL
FLOW
lrt thc lilllowing discussion, lho lklw is assrrrrrctl (o bc smooth (laur"inlr) arrd
two-rlirncnsional, that is, indcpcntlcnl ol'lhc c<xrrclinatc normal to tlrc planc ol
vicwing. Consider a two-dimensional llow around the sharp-edgccl flat platc
shrrwn in Fig. 4.4.1 . At a very low Rcynolds number (e.g., ULlu: 0.3,
whcrc L is the dimension of the plate across the flow), the flow turns the sharp
corncr and follows both front and rear contours of the plate (Fig.4.4.la). At
rr slightly higher Reynolds number (&" = l0) obtained by merely increasing
tlrc llow velocity over the same plate, the flow separates at the corners ancl
t'rt:alcs two large, symmetric vortices behind the plate that remain attached t<l
(lrt: back of the plate (Fig. 4.4.1b). At increased Reynolds number (G" = 250)thc syrnmetrical vortices are broken and replaced by cyclically altemating vor-
tit'cs that form by tums at the top and bottom edges and are swept downstream
( l;ig. 4.4. I c). A full cycle of this phenomenon is defined as the activity betwecn
llrc occurrence of some instantaneous flow configuration about the body and
llrc ncxt identical configuration. At still higher Reynolds numbers, say Ge 2
It){)t) (Fig. 4.4.1d), the inertia forces predominate; large distinct vortices have
littlc possibility of forming and, instead, a generally turbulent wake is formed
Ilchintl thc plate, its two outer defining edges forming a "shear layer" con-
sisting ol'a long series of smaller vortices that accommodate the wake region
lo thc udjacent smooth flow region. Overall, these results dramatically illustratc
rDe e Q.3
(o)
lll(;llltl,l.l.,l.la. lihrw pirsl lr slt:u1r t'tllicrl pl;rtt.(11,. -- O I
.l 'l Wnlll nl.Jl'V()l lllXl0liMnll()Nl; tl.l tW(llrlMl il:;l{)t!At |()W 147
(b)
FIGURE 4.4.1b. Flow past a sharp-edged plate Ge = 10.
(, )
l,'l(illllll,l 4..1.lr'. l;kru |1r;l ir "lrirrI r'rl1'r'r; ,t1,,,. ,t,, _) 5()
t4B lll l,l I lt()t)Y At n()t )YNnMt|1
FIGURE 4.4.1d. Flow past a sharp-edged plate Ge > 1000.
thc changes in the flow with Reynolds number, proceeding from predominantly
viscous effects to predominantly inertial effects.
Next the renowned case of two-dimensional flow about a circular cylinder(rig. 4.4.2) is briefly examined. A number of flow situations can be created
by increasing the flow velocity, each situation being identified by a specific
llcynolds number range. At extremely low values of Reynolds number (G" =l) thc flow (assumed laminar as it approaches) remains attached to the cylinder
thrrrrgh<rut its complete periphery, as shown in Fig. 4.4.2a. At G" = 20, thelklw lirnn rcnrains symmctrical but flow separation occurs and large wake
ctklics urc lirnnctl which rcsidc ncar the downstream sudace of thc cylinclcr,
rrs srrggcstt:tl irr Iiig. 4.4.2b. lror 30 < (R" < -5000, al(crnuting vor.ticcs arc
slretl liirrll tltt't'ylilrrlcr rttrtl lirt'nr ir clcar "vorlL:x lr;ril.' rlowrrslreirrrr.'l'his
ltltt'ttolttt'ttott wirs litsl rt'1xrt1ctl by llt<rlrltl l4-.5 1 irtrtl vorr Klrrrrr;ur l.l O; 11r;*.
Qe=l
(o)
,l .t w^l,.t nt.lt I v(,t illx l()l tMnll()N:; lN tw()trlMl Nt;l{)Nnt ll()w 149
qe =20
(b)
@
VON KARMAN VORTEX TRAIL
30 39" S 5OOO
cuuuS.z{S ZU(JUUU 4"2ZOOAOO
(d) (e)
l|l(;uRE 4.4.2. (a) Flow past circular cylinder (Re = l. (b) Flow past circular cylinderill" = 20. (c) Flow past circular 30 < G" < 5000. (d) Flow past circurar cylinder
.5(XX) < G" < 200,000. (e) Flow past circular cylinder G" > 200,000.
1.1 .2c). The finer details ol this striking occurrence are still not fully under-
slrxrtl, and the process cor.rlirrrrcs to be the focus of many studies, both exper-
irncrrtrrl and thcoretical 14-21 l. llchintl thc cylinder there is establishcd a stag
gcrctl, stablc arrangctllcltl ol'vorlit'cs lhlrt rrrovcs <tffdownstream at a vcltrity
stttltcwltitt lcss thltrr tlurl ol lltr'sttt.lirrrrtrlirrg llrritl . In this rangc ol'llc:yrroltls
tttuttbcr lltc w:rkc llow is lr.litlrvcly snrtxrllr irnrl rcgrrlirr aplttl I'r-onr lltc vorlit'csr
lllclttsclvt:s. liigrrrt:.tr.,4..1 tlt'1rrr'ts llrt'slrt':rrrrlirrcs ol tlrc wlrkc llow lx'ltiltl ir
(c)
WAKE
5OOOs.4.3 2OOOOO
150 ilr ur r tt()t)Y AI lt()l)YNAMlcl;
l,'l(;tjRli 4.4.3. Vortex trail in water tunnel. Courtesy of the National Aeronautical
lislablishrncnt, National Research Council of Canada.
circular cylinder in a water tunnel [4-7] within the above-mentioned Ge range.
'l'hc lkrw in this photograph was made visible by the emission of dye from the
t'y I irttlcr.
As ltcynolds number further increases into the range 5000 < G" < 200 000,
(hc irttlchcd flow upstream of the separation point is laminar. In the separated
lLrw (hrcc-dirncnsional patterns are observed, and transition to turbulent flow
(x'('uls irr thc wake-farther downstream from the cylinder for the lower Rey-
rxrltls rrrrnrbcrs and nearer the cylinder surface as the Reynolds numbers increase
l.l l()l l,or thc largest Reynolds numbers in this range, the cylinder wake
rrrrtlcr-gocs transition to turbulence immediately after separation, and a turbulent
wrrkc is pnrtluced between the separated shear layers (Fig. 4.4.2d).
Ifcyonrl 61" = 2OO 000 (Fig. 4.4.2e) the wake narrows appreciably (giving
lisc lo lcss drag; see p. 158).
( )thcl bluff bodies, notably triangles, squares, rectangles, and other regular
;rrrrl ir.rcgrrlar prisms, give rise to analogous vortex-shedding phenomena.
'l'lrc prorllrnccd regularity of such wake effects was firQt"rgp-pged,by,$trquhal
l.l ttl wlxr pointccl oul that the vortex-shedding phenomenon is describable in
tcrrrrs ol':r rror.trlirncnsional number (the Strouhal number):
NDs-?
i
wlrerc N, is tlrr. lit't1rrt'rrr'y ol lirll cyclcs ol'votlcx slterrltlilrg. /) is it t'ltltlrtt:lct'islic
tlirrrt'nsiort ol llrc lrtxly lrlojt't'terl ott lr pltrttrr tttll'ttnl ltt lltr'tttr'lttt llow vclocily,
't .l w^t,t Ailll V(lt iltx I()llMn il()Ni; tN tw(,t)tMt N:;t()NAt il()w r5t
r-ffi--r-.-
, o.2
ll
I
o.1
107106
. "eff@f 3 3 ""etrt e R"6 STxf t ^3
O Smootho k/D=0.0003
^ k/D=o.oo12o k/D:0.0101
105
REYNOLDS NUMBER
l"l(;uRE 4.4.4. Relation between the Strouhal number and Reynolds number for cir-
t'trlar cylinder. From W. C. L. Shih, C. Wang, D. Coles, and A. Roshko, ,.Experi-
nrcnts on Flow Past Rough circular cylinders at Large Reynolds Numbers," J. wrut
I,)rg. Ind. Aerod., 49 (1993), 351-348.
irnd u is the velocity of the oncoming flow, assumed laminar. The number S
takes on different characteristic constant values depending upon the cross-sec-
tional shape of the prism being enveloped by the flow. Figure 4.4.4 t4-g6l
shows the relation of 3 to G" for a circular cylinder in the range 10s < G" <
107. The values of Fig. 4.4.4 were inferred from the unsteady pressure mea-
sured in smooth flow at about 90 degrees from the front stagnation point.
('oherent vortex shedding was noted to disappear at Reynolds numbers beyond
4 x l0s, and contrary to results reported by some observers and summarized
in l4-9], there was no increase of the Strouhal number to values near 0.5. Table
4-4.1 [4-l0l also lists a number of values of s for different cross-sectional
shapes for Reynolds numbers in the clear vortex-sheddingrange, the approach-
ing flow being laminar.
A certain amount of debate continues on the question of whether or not
lrcriodic vortex shedding gan still be exhibited at extremely large Reynolds
rrurnbers, say, G" >> 108. If one substitutes an effective eddy viscosity (see
scct. 2.2) for the actual kinematic viscosity of the fluid, it is conceivable that
ir ncw Reynolds number range can be calculated in which altemating vortex
shcdding from extremely largc bluff ob.jects can once more be forecast. In this
wly thc occasionally ohscrvul lrttgc vorlcx trails in ocean currents downstream
ol' islands may possibly lrc tt't'ottt'ilt'tl with srn:rllcr-scalc expcrimcntal <lbscr-
vlttiotts. Irigttrc 4.4..5, lttl ittslltttt'c, is lr lcpnltlrrcl ion ol'a satcllitc photogrlph
l4 Ill ol'tt voflox tllril irt tltt' itlrrroslrltt'tt' rn;rtlc visiblc by ckrutl Jlrc:sctrctr irr,
tltc votliccrs slttxl lhrrrt lltt' nrrtttttl;rrn l)('irh ()l ( irrrrrltrlrrpc lsltrrrtl ovr'r' 12(X) rrr
152 ill liltlt()t )Y nl ll()l )YNnMl(.1,
I'Altl,l,l 4.4.1. Slroulral Nurrrlx'r' lirr rr Vrrrit'l.y ol' Slritpr:s
Profile dimensrons, Value 0l Prolile drmensions,
in mm
Val0e ol:/
t= 2.O
0.120
r= 1.0
r25{l_l___T
{rz.s{
l*so--l
o.147
0.1 37
_-_>
,=0.5
ilH
['r]
0.120
r= 1.0
r2s{[-l-
+-t2.5c<-T
1..-so-t
0.150
1.0
f.-.-l
0.144
0.145
0.t42
/ 0.147
r= 1.5
rzs{f-.1
[-*-]
0.145 $-t_
[ru!
0.13 1
I
I
0.134
/ 0.137
r= 1.0
L-uo-t
o 140 E_l t_
l*rs*zs.l.zs!
0.12 I
0.153 0.143
125{l-J
L_*-J
0.145
r= l.O
0.135
0.168
+
G
0.156
0.160
I
,t
0.145
Cylinder 1l8OO..*r. tttOO 
Q
0.200
T-lLl
l-. .'l
o.1 l4
t 0.145
\i1,r.r,.. liRrril .'wirrrl lirr.t.t's orr Stlrrt.lrrR's," '/izrrr.r. AS('li, l2(r (l(x'l), ll.t,l ll()11 (r - lo IIIZ
.l .t w^t, I /\lillv{)l illxl(}ltMnll()Nt,tt.t tw(rtrtfi,,iltl.,t{rlll\l ttow
.r..\li<{ffCi .,t: *a\\."{f\ q "
. ,r*' .,* };1* ,, ,r"
.*
;
,,} P
*;w
^"htffi
i$:i. ..
&d'
i;:-le . ..,
..,,.s, 
'
.-ig
" q&:
:.l i!,i,,i.1;*6,'.:i,:-' .,,-'all$::,"i.'
I''IGURE 4.4.5. Satellite photo of cloud vortices downstream of Guadalupe Island (off
Itlla Califomia) [4-l l]. Courtesy of the National Aeronautics and Space Administra-
lion.
high off the Pacific coast of Mexico. The photograph spans some 250 km.
Assuming, as in [2-1171, an effective value of (kinematic) eddy viscosity an
= -50 m2ls, a full-scale Reynolds number of the order of 1010 for the phenom-
cnon (based on u = 1.5 x l0 5 m'ls; would be reduced to an effective valueol'((R")"s = 3000, which falls well within the laminar vortex-shedding range.
Assuming the island to be about 20 km long, the distance between successive
pcriodic voftex centers is roughly 55 km. Further, assuming a Strouhal number
lirr thc island peak as S : 0.12, a mean wind velocity of U :30 m/s, and
rrrr cfl'cctive island clirnrrrrsion ol'/) = 6000 m yields the vortex-shedding frc-(lucncy
0 l.)( t())
(il()()
I fi4 ill t,t I il( )l)Y n t il( )l )YNn Ml( ii
l;.
"';ffit; *:.."s,ffit
lrl(ltll{lt 4.4.6. Satcllite photo of Jan Mayen Island (Arctic Ocean). From Weather,
1I. l0 (Oct. 1916), 346.
wlrich in turn gives a shedding period of Z : 1/N" : 166l s. Employing S :
l/7'yiclcls a calculated vortex separation of
S : 30 x 166l : 50.000 m : 50km
rr rlislurrcc consistcnt with rough fireasurement of the photograph. Anothcr in-
tcrcs(ing photograph ol'largc-scalc vol'tcx shedding is prcscntcd in Fig. 4.4.6
l.l t2;.''
WIrcrr contlilions:rlc srrclr thirl ir rlistinct voflcx triril is l)rescn( in lhc wrrkc,
lt llow ct1)sli()v('r-:rll ol (lrc llrtly occttrs llltl hrts it ('()nllx)n('rrl rrolrnirl lo lltc
lll lurs lx'trr lrroul'lrl lo llrr' ,rlllrtlrott ol lltt :tttllllrs llltl lt sitttll;r |rrrlrlr rtr r', ltr';rlr'rl rrr l.l i{)l
.l ', llll(.1:;()t'l lW()l)tMl tl',lol lnl .lllllr lrlll/\l lrrl lM
t\
I
(-.._-- - -\)
- 
-_*7
\-'
I l(.llltl,l 4.4.7. lillcct ol'splittcr plate on flow behind a circular cylinder [4-13'
I r' I I
,r1,rr,r;11 11i11g llow dircction. Thus it becomes possible to inhibit the establish-
nr rr ol ir v()flcx trail by placing a "splitter plate" in the near wake of the
,.i rir'l;rlurtt l'rocly, as first pointed out in [4-13]. (See Fig. 4.4.7.) The action of
rlrr.. pl:rtt'is to prevent the flow crossover and thus to quiet the entire wake
ll,,s' t.)tr:rlitativcly, the presence of the plate has the same type of effect as
i, rr1'tlrt'rrirrg thc body in the stream direction and causing it to approach, to
.,,irr, ;rppnrxinration, the form of a symmetrical airfoil. Following this type of
,rtri,r();r('lr it can bc seen that elongated bodies, oriented with theirlong dimen-
r,,rr l';urrllcl to the main flow, tend to elicit relatively narrow wakes, many
r.. rtlr,rrt lrpprcciablc voftex production.
ll llows irbout square and rectangular prisms are compared (Fig. 4.4.8), the
,|il:il(' rs sccn (at reasonably high G") to produce flow separation followed by
,r n'rrlt'. (rrrbulcnt wake, whereas the more elongated rectangular form (de-
1,, r.lrr11, orr lcngth-to-width ratio) may exhibit separation at leading comers that
r. l,rllrrwt'rl rl<lwnstream by flow reattachment and finally, once more, by flow
,p,u;rtr()rr rrt thc trailing edge. Thus it is seen that not only does the bluffface
,,1 rlrr'lrotly prcscnted to the fluid affect the resulting wake, but the streamwise
i, rrlllr rrrrtl gcncral form of the body also play important roles in the wake form.
irr',lr.rr1r tlis(inction to the casc of Fig. 4.4.8b, if the rectangle is placed with
rr. l{)rr)'. tlirrrcnsion normal (o thc llow, the wake exhibits a strong voftex-
i,,,l,lrrr1, i'lnractcristic, lirlkrwctl at highcr G" by a turbulent wake not unlike
rl,.rr ;rr11f111'gtl by thc sltiu'1'r ctlgctl llirt platc (see Figs. 4.4.|c and4.4.ld).
,I II PRESSURE, LIFT, DRAG, AND MOMENT EFFECTS ON
! WO I)IMENSIONAL STRUCTURN I FORMS
! rlrrrt" ,l.5.l srrl',1qesls l s('( l11 )rl ol .r lrlrrll lrotly ltttttr'tst'tl itt lt llrlw ol vclot'ily
it llrt. ll6w will tlt'vt.l91r lpt:rl ;'t,..,.,rrr{.', /, r'\,('l llrr' lrrxly irr ttcr'olltrilt'c willr
!l, r il{}illli':; t't;tIt(iottl
156 tlt t.I I il( )t)Y At il( )t )YNAM|(:ii
.=.t.t^"t
(b)
lr'l(;Ljllli 4.4.8. Flow separation and wake regions of square and rectangular cylinders.
ATTACHMENT
t,'l{;llltl,l 4.5.1. l.ill irntl tlritg rtn irn itrhil!rt!y ltltt!l hrtly (4.5..s )
,t h il | c ri oN two DtMl NtiloNAl riffflll:lllll^l | ()llMli 157
wltetc tlrc cottslirttl lroltls irkrrrg a strcarnline irrrtl l/ rrl)rrsonts thc volocity on
tltc strctttrline itt thc itnrttcrcliatc vicinity ol'tlre hxly (i.c., irrrrrrccliatcly outside
llrc llrundary llyer tltal lirrrrrs on its surlacc). 'l'hcr intcgration of the pressures
ovcr thc body surlitce rcsulls in a nct fbrcc and a lnolnont. The components of
tlrc lirrcc in thc along-llow and across-flow dircctions are referred to as drag
ilnd li.li, respectivcly. 'l'hc drag, lift, and moment are quite obviously affected
lry llrth the shapc ol'thc body and the Reynolds number.
'l'hc body may, lirr cxample, be contoured with the express purpose of
rrrininrizi'ng drag and rnaximizing lift, resulting in an airfoil-like shape. Again,
;rs in rnany civil engineering applications, the shape of the body may not be
;rrucnable to such special adjustment; its form will most likely have been fixed
lry other design objectives than purely aerodynamic ones. Nevertheless, the
lill, clrag, and moment developed by the fluid flows about the structure will
rt'rrrlin of strong interest because these are effects that must be designed against.
ll is usual to refer all pressures measured at a structural surface to the mean
rlynrunic pressure )pU2 of the far upstream wind or the free-stream wind at
rorrrc distance from the structure (e.g., at a point well above it out of the
lrrrrrrrrlary layer). Thus nondimensional pressure cofficients Co are defined by
t pl l) t p r.'orrsl
P-PoLr: Ei,
FL
L, --' iPu'B
Fnt t, : t_pu)n
ttt( 'tt : 
1,,, ,: ,r:
(4.5.l)
(4.s.2)
(4.s.3)
(4.s.4)
wlrcrc U is the mean value of the reference wind and p - ps represents the
prl'ssuredifference between local and far upstream pressure p6. Such nondi-
rrrcrrsional forms enable the transfer of model experimental results to full scale,
irrrtl the establishment of reference values for cataloguing the aerodynamic
;rro;rcflies of given geometric forms.
Arralogously, the net wind-pressure forces (per unit of span) F1 and Fp in
tlrc lili and drag direction, respectively, can be rendered dimensionless and
r'rlrrcssed in terms of lift and drag cofficients Cy and Cp as
rvlte tc /J is somc typic:al t'cli't't'ttt'c tlittrcrtsiorr ol'lhc s(ructure. For the net flow-
tttrlttt'e:cl lnolncnt M lltcr crtrlcsPotttlittg cot'llir'icrrl is
Su br; rit ical
158 tlt Ut I tK)t)Y nt ll()l)YNnMl(
106
Reynolds number,4e
FIGURE 4.5.2. Evolution of mean drag coemcient with Reynolds number for a cir-
cular cylinder. After L. R. Wooton and C. Scruton, "Aerodynamic Stability," in The
Modern Design of Wind-Sensitive Structures, Construction Industry Research and In-
firrmaticrn Association, London, 1971, pp.65-81 and 14-221.*
When the flow is fluctuating as a consequence of oncoming turbulence,
vortex-associated flow changes, or signature (body-induced) turbulence, the
above quantities become time dependent. In such cases, when time-varying
litrccs ancl moments occur, mean values of force coefficients aS well aS spectral
clcnsity clistributions of these quantities are required for their fuller description.'
(Note that in two-dimensional flow L1 , Fp, ?fid M represent corresponding
valucs per unit of dimension normal to the plane of observation. In three-
dimensional cases, correct dimensionality is preserved by including an addi-
tional factor B in the denominator of each expression.)
Retuming to the prism of circular cross section in smooth flow, the variation
ol' its mean drag coefficient Cpmay be represented as in Fig. 4.5.2, where the
clcpendence on Reynolds number is shown. Note particularly how Cp drops
sharply in the rang6 of about 2 x l}s S Ge < 5 x 10s. This region of sharp
clrop is called the critical region and corresponds to a condition wherein the
f ransition from laminar to turbulent flow occurs in the boundary layer that forms
on the surface of the cylinder. The turbulent mixing that thus takes place in
tlrc boundary layer helps transport fluid with higher momentum toward thc
surlace of the cylinder. Separation then occurs much farther back and the wake
c()nscquently narrows, finally producing a value of the time-averaged C, that
is only about { of its highest value. As G" increases into the supercritical and
thcn ihe transcritical range (G" = 4 X 101. CD increascs once more but
rcnrains much l<lwcr than its subcritical values.
l,llcccrrl tl:rtu lrl ll(rl slrow llr:rl tlrt.tlrirg c0cllicicttls itt thc lt'giorr 5 l{)' (ll, ' lO/ :rtt'stttltllct'
lly ltllrttl l5%' llllrrr lltost'tttrlitltltrl irr lrig 4'5 2 rtlt lltt: hitsis ol t';ttllr': tltlrttttt;tltrttt
Sct' Appr'rrtlir A.)
aUt1
.o
O+
oo
Obrr
N'
o
107
n,, til l(,t:i ()N lw()l)lMt t!!;l(lt,tAt !iililt( illt t^t t()t tMl; l59
o" "o" , ,oaorra' 
l2o" l8o"
l,'l(;tlRE 4.5.3. Influence of Reynolds number on pressure distribution over a circular
, ylirrtlcr (after [4-22]).
lrigure 4.5.3 depicts a typical distribution of the mean pressure coefficient
;rlrorrt the circular cylinder in smooth flow as a function of angular position.
I'lrt' rcsults are evidently sensitive to Reynolds number.*
'l'lrc drag coefficient of an elongated rectangular-section body in smooth flow
tlrrg. 4.5.4) [4-14,4-231 is also a function of the narrowness of its wake, but
rlrc krwer limit of wake width is approximately the full width of the body. The
n,rrkc width at somewhat lower G." is much greater than the body width, and
tlrrs is accompanied by higher cp;then, when flow reattachment to the body
lx'1iins to occur, the drag cocfficient drops. This is a function mainly of the
r'lrrngirtion blh of the brxly, irs shown in the figure. Flow in the critical region
l:' ilccompanied by turhulcrrcer, irncl thcrcfirrc this region is shown as a shaded
lr;urtl ol'possible valucs in lrig. 4..5.4.
liigrrrc 4.5.5 14-1.5 1 illrrsrrrrrcs tlrt' t'volrrliorr with Rcynolds number of the
rrrt':ttt tlrag cocflicicttt ol'it stlttirtt irt srrroollr llow tlrrrirrg successivc rlodifica-
Itotts ol'ils corncrs. Nolrr llrirt otrly llrt.slttlP t'orrrt'r't'tl stlrrar-o cxhihits practicirlly
rllrt'lrcsstttr:s ('on('sl)on(ling k) /l ll" irrrrl ll ll"i{1" ;rrt lt.lt.trt.tl l() itri llt(.plt.ssrrrt.trl llrt.
',lrr;',rlrliorr lloirrl :riltl lltt lltsc l)ti'r{ilr', rt rlx r livrll'
4r.61 xlOs
.4c=l.lxlO'
160 nt t,t I lt()t)y At n()l)yNAMt(]ri
0L
0
FIGURE 4.5.4. Effect
14-231.
24
of afterbody upon
6
drag of a
o
rectangular
b/h
cylinder 14-14),
unchanging drag with change of Reynolds number. This is simply accounted
for by the early separation ofthe flow at the upstream corners and the shortness
of the afterbody that practically precludes the possibility of flow reattachment,
whereas squares with rounded corners tend to possess the same kind of critical
region for the drag coefficient as seen earlier for the circular cylinder. Note
also, in the case of the circular section, the dependence of the drag upon the
roughness of the cylinder surface. This dependence was studied in detail in
14-241. (See also Sect. 11.1.1.)
Because of such effects, certain features of the flow in tests over wind tunnel
models can be expected to be independent of the Reynolds number, while others
may be quite sensitive to it. Thus it can be argued that cerlain Reynolds-
number-insensitive flow phenomena may be encountered in tests in which the
llow will always break cleanly away at the same identifiable points. certain
types of bodies such as the circular cylinder offer extended regions of possible
llow separation in which the location of the actual separation points depends
rupon Reynolds number. with such bodies the entire structure of the flow will
hc highly Reynolds-number-sensitive (see Secr. 7.3.2).
l'or cxtremely low Reynolds numbers the drag coefficient increases greatly
irs ir rcsult of viscous effects. This is illustrated in Fig. 4.5.6 t4-I41, which
1fL:;)icts Cpfor circularand square flat plates for 10-2 < G., < 107. (Analogous
cllccts on lift and moment do not necessarily follow, though some distortion
is vcry likcly.)
sincc thc prcssurc dil'l'crcnces across a sharp-cornered square vary with time,
llte soctional lili crrcllicicnt will also be a function of time: C1.: C1,Q). Figure
4.5.1 14- l6l illustra(cs tl.rc spcctral density of c7. plottcrl as a lirnction of
rrll/{/, whcrc rr is l'r'c:t;ucrrcy in Hz, B is thc dimcnsion ol'rhc: sitkr ol'tlrc scprarc,
ittttl l/ is ttlcitll (tttr"()trring vt:krcily (irssurnccl to bcconslirrrl llrlorrg.lrorrl tlte lc:gion
4h rIil(lii ()N twot)tMl Nl;l()Nnt 1;ililt(;ililtnt t()ilMli 161
2.2a--r-- il r --r--Tt-T- r --l
t ----'!- [-1r,4 f , t, =oozt' , , ,, , l'"_lI8J- r r rr r ___r rr r
1.2
0.8
0.4
---) u--D]n
(h)
r/h = 0.167
r/h:0.5
(circular section)
8105 2 8106 2 8l 07
Ee
- 
sanded surface k)
--- Smooth surface
FIGURE 4.5.5. Influence of Reynolds number, comer radius, and surface roughness
on drag coefficient, square to circular cylinders (r is the corner radius; k is the grain
size of sand). After [4-t51.x
o.t
to-2 lo-l
9tt
FIGURE 4.5.6. Typical rlr:rg coc{licicnt as a lirnction o1'Reynolds number [4-141.
*Motc tcccnl tllta lor circtrlitr cylittrlcts l,l tl(rl rrrt in gcncrirl tprrirliltrlivc itgrcclllcnt wilh tllrsc gl
IriP,.4.5.5 btrl intlicatc lltirl lirl l;tti,ic l{r'ynrltlr rurrrlx'rs thc cylirxk:r rrurglrrrr":ss brings irlxlrt ir
sonrcwhitl slr'ongcr irrcrcitsc irt tllrg
co
l07to6t05t04t03t02
k1h = 0.007
I'Fk/h: O.OO2
h:0.001 \--'
t62 lil ul I Ir lt)Y n t il( ltlYNAMtcl
50
20
10
2
1
0.5
o.2
0.05
0.02
0.01
0.005
0.002
0.02 0.05 0.100.20 0.50 1.00 2.00
NB/U
l"lGURE 4.5.7. Spectrum of lift fluctuations on a square-section cylinder for flow
nonnal to a face (G" : l0s). From B. J. Vickery, "Fluctuating Lift and Drag ona
l,.ng Cylinder of Square cross-Section in a Smooth and in a Turbulent Flow," ,/.
lluid Mech.,25 (1966), Cambridge Univ. Press, New York, pp. 481-494.
ol'lkrw under consideration). In both smooth and turbulent flow, a high spectral
1rt'rrk occurs at the Strouhal number nBlU : 0.12.'l'his is clcar evidence of periodic voftex shedding. For any given bluff body,
tlris shodding is not a purely sinusoidal phenomenon, as seen from the spread
t. rrtlrcr I'rcqucncics ol'thc spcctral peak in Fig. 4.5.7; however, a good first
lrprpIrrxirna(ion lo tho lili lirrcc pcr unit span occurring at the peak Strouhal
rrrrrrrbcr is givcn by
N
aN
p
N
d!
f,l
cn
lr, t,pU)B(-, sin <,:t
I
J
llo
li
s
il
llltblr
/lirii
/'f tr./,t \'\
.rt"# \:t,,ff tpj'dtTurbulent (
stream
Smooth -+_o+stream
('1..5.6)
1,, tt|(.11; ()N lw()t)lMl N:it{)l.JAl f;lltll(jillilnt t()t tM:i t63
t4
E
ao
k;OtoL
J
o(o
o
F
l
FO
lJL
Lo
]-
rrl
a
hUoO
a
(r
SMOOTH STREAM
U+
o5"1o"15o20"25"30"35"40"45"
ANGLE OF ATTACK, o
lfl(;URE 4.5.8. Variation of the coefficient of fluating normal force, C1y_.. with angle
ol attack for a rectangular prism. From B. J. Vickery, "Fluctuating Lift and Drag on
rr Long Cylinder of Square Cross-Section in a Smooth and in a Turbulent Flow," "/.Iluid Mech.,25 (1966), Cambridge Univ. Press, New York, pp. 481-494.
where Cl is a mean lift coefficient that depends on the particular cross section
shape and a : 2rn, r? satisfying the Strouhal relation.
The root mean square (rms) value of the fluctuating normal force coefficient
(ry,,,,. on the square section is shown in Fig. 4.5.8 t4-161 as a function of angle
ol'attack a with respect to the mean wind direction. Here the turbulencex is
sccn to lower the highest normal force below, and to raise the lowest normal
lirrce slightly above, the respective laminar values.
Figure 4.5.9 l4-ll presents two photographs of flow over proposed bridge
tlcck sectional forms as visualized in a water tunnel flow containing fine alu-
rrrinum particles. Figure 4.5.9a shows a section that produces severe flow
scparation; Fi9.4.5.9b portrays the flow-smoothing effect of a modified section
providing lower lift and drag.
Rcf'erence [4-10] prescnts mcan values of cp and c. obtained under laminar
llow conditions for a largc rrurnbcr ol-scctional shapes common in construction,
irs takcn lnrm il2-21ntl 14 lltl; scc'l'ablc 4.5.1.l4-171, an<l 14-621.
r'lltc ltttlrttlcttec clt;tt;tt'lt'rislits itt lltt trlr'rrrrrrrt ol lir1l .l 'r lJ wcrc tlrc lirllowirrli: klrpitrrtlil:rl
sr';rlt' l.:l/J. lirlt'r'lrl st:rlt' O.,l/1. lrrrlrrrlt nr ! ltl( u.,tl\, lll'i,
CNr..{r/2pua)u
164 ilt t,t I 80t)Y nt tt()t)YNnMt(;1;
FIGURE 4.5.9a. Visualization of water flow over a model bridge deck section. Cour-
tesy of the National Aeronautical Establishment, National Research Council of Canada.
FIGURE 4.5.9h. Visualization ol'watcr flow ovcr a paI1itlly slrcirrrrlirrctl rrrtxlcl lrlitlgc
dcck scction. (-ottrlcsy ol (lrt' Nrtlionul Acntnitulicitl lislitblislrrrrt'ttl , Nitliorrrrl l(cscirrclr
('outrt'il ol ('lrturtLt.
Prtlth arrrl w[r(l rlilFr lh)il
*M-r.
cD cL
2.O3 0
-[
r,96 -
2.01 o
-I 2.O4 0
-D#l
1.81 0
------- L- 2.O 0.3
-l 1.83 2.O7
_L r.99 -0.09
+lJ 1.62 - 0,48
-lF 2.O1 o
nllr+ll
ilhlll
Iit(;t:; ()N tw()l)lMl Nlit()Nnt :;ilIt,(;lUllAl l()ltMt; 1€5
'l'Alll,ltl 4.5.1. 'l'wrl l)irtttttsiottul l)rug urttl l,ilI ('rnllicicrrls lirr Slructrlrul
Slurpes
\rrlrrr'. Iilorrr .'Wirrtl lirrtt's ott Sltttr lttrr': /irrrrr A5('lr. ll(r ( l()(rl ), I l.l.l I l()t{ rrrrtl I ll Jl
166 Bt Ut I not)y Al tr()t)yNAMt(;ri
'l'hc rosults ol"l'ablc4.5.1 are irppliclrblc lo nrcnrbors with luryc lrspr:cl urtio
(ratio of length to width) \, or lo rrtcntbcrs with ond platcs (abutrncnts). For
members with small aspcct ratio (c.g., 
^ 
< l0) and no cnd platcs (abutmcnts),
end flow effects are significant, and thc drag cocflicients are smaller than in
Table 4.5.1 (see Sect. 4.6.2). The drag coeflicients are also modified by the
presence of turbulence in the oncoming flow. Experiments have shown that in
most cases of interest in practice these modifications are small Il2-2, 12-51.
For this reason wind tunnel tests aimed at measuring aerodynamic forces or
trussed frameworks with sharp-edged members are to this day conducted in
smooth flow [12-1, 12-6]. Note, however, that in some cases the effect of
turbulence on the drag force can be significant. For members with rectangular
cross section, this effect depends upon (l) the ratio blhbetween the sides of
the cross section and (2) the turbulence in the oncoming flow. If the ratio blh
is small, no flow reattachment occurs following separation at the front corners.
5
(b)
FIGURE 4.5.10. Separation layers in smooth flow (solid line) and in turbulent flow
(intemrpted line). After A. Laneville, I. S. Gatshore, and G. V. Parkinson, "An
Explanation of Sonrc Ell'ccts ol'Turbulence on Blufl'Bodics," /)rrcclrlirg.r, l,ourth
International Conl'crcttcc, Wirrtl llllccts on Buildings antl Slnrclrrn's, ('irrrrbritlgc t)niv.
Prcss, Carnhritlgc, l()77.
!--
Hrgher ,/
drae f
tf, tlt(;l;()N twot)tMt Nlit()NAt l;ltlt t(;lt,ttnt t('ttM!; 167
l)cpcntlirrg ttlxrtt ils tttlt'ttsily, thc) turbulcnccr cirrr crrlrirrrcc llrc llow (,ulti1n1r(.nl
in thc wlrkc ittttl, lltcrcliut. (:itusc stK)n8,cr sucliorrs lrrrtl lirrgcl tlrirtl (1,'ig.
4.5.10a). ll'thc ntlio /r//r is sullicicntly largc, tlrer turhulcrrcrr ciur t'rrrrst. lLrw
rcattachmcnt which wottltl rrot have occurrcrl irr srrrtxrlh llow arrtl llrrrs nsrrll
in reduced drag (lrig. 4.5.lob) 14-25,4-26|. A bcaurilul visualizltion o| r|c
llow around a body with rectangular cross scction (blh :0.4; srn<xrtlr lkrw,
Re : 200) is shown in Fig. 4.5.n [4-87] and may be compared, qualitativcly,
with the smooth flow case depicted in Fig. 4.5.100-see also [4-94]. -the
dependence of the drag coefficient upon turbulence intensity is shown for two
ratios blh in Fig. 4.5.12* 14-261. Additional studies on turbulence effects on
drag and lift of sharp-edged bodies are reported, in [4-271, 14-281, and [4-85].
The effect of turbulence in the case of bodies with rounded shapes is, essen-
tially, to reduce the Reynolds numberat which the critical region (Fig.4.5.2)
sets in. This is shown in [4-291, which includes, in addition, information on
the fluctuating lift and drag forces on a rigid cylinder due to vortex shedding
and to turbulence in the oncoming flow (see also t4-301).
For a recent, wide-ranging review of turbulence effects on bluff-body aero-
dynamics, see [4-87]. Reference [4-14] is compendium of drag effects that
contains limited data obtained in smooth flow on models of buildings and
structures.
|I'IGURE 4.5.11. Flow around rectangular cylinder (b/h - O.4, G." : 200). From Y.
Nakamura, "Bluff-Body Aerodynamics and Turbulence," ./. Wind Eng. Ind. Aerod.,
49 (1993). 6s-18.
t'Notc llral lir Itlh - l, (), irs ohl:tittcrl in l,l 2{rl lol srrrrxrllr lkrw <lillors by ll-xlrt l0%, lirrrrr tlrc
vitlttc lislcxl in 'l'irhlc 4..5.1. l)illctt'rr('cs ol llri$ ottlcr or l:rlgt'r' irrc c()null()n t:vt:rr lirr rr.strlls ol
sinrplt: wirrtl lunnrl {(:sls.
168 lrl t,l I lr()l)Y nl lt()l)YNAMI{;l
CD
048121620 _y,u., enl
FIGURE 4.5.12. Dependence of drag coefficient upon turbulence intensity. After A.
Laneville, I. S. Gartshore, and G. V. Parkinson, "An Explanation of Some Effects of
Turbulence on Bluff Bodies," Proceedings, Fourth Intemational Conference, Wind
Effects on Buildings and Structures, Cambridge Univ. Press, Cambridge, 1977.
4.6 REPRESENTATIVE FLOW EFFECTS IN THREE DIMENSIONS
Most flows have a three-dimensional character, principally as a result of their
contact with boundaries. For example, if a hypothetical laminar flow consisting
of an air mass displaced uniformly as a single unit encounters an object, it will
be diverted in several directions. Also the passage of such a flow along a surface
sets up boundary-layer velocity gradients. Three-dimensionality is clearly in-
herent in turbulent flows.
Although the general equations for fluid flow remain availablefor applica-
tion, few flow problems in three dimensions have been satisfactorily solved in
a purely analytical fashion because of the considerable complexities involved.
As a result, most three-dimensional studies rely partially or wholly upon ex-
periment. Therefore, this section is mainly concerned with broad aspects of
three-dimensional flows, with conditions of testing, and with some represen-
tative results obtained by test.
4.6.1 Cases Retaining Two-Dimensional Flow Features
The success of the two-dimensional flow models discussed in the previous
section has in a few cases been considerable because sorlc actual flows retain
certain two-dimcnsional t'catures, at least to a first approxinrllion. Consiclcr,
forexample, lhc casc ol'a long nld ol'squarc cnlss scclirttt itt:rtt lrit llow with
unilorm nroarr vcrkrcily norrrurl (o onc lircc:. lixccgrl rtt'rrr llrr't'rrrls ol llrc nxl,
40
o
(E
Fz
UJo
tr
Lr-
UJoo
zot-
J
UJ(E
(Eo(J
o.
o.4
't F nt t,nt 1;t Nln ltvt I l()w I lil (;t:; ||J ilillt I t)tMt N:;t()Nli 169
r1"/D
I,'IGURE 4.6.1. Spanwise correlation of the fluctuating pressure difference across the
t'cnter line of a long square-section cylinder for flow normal to a face (G" : 105;.
lirrm B. J. Vickery, "Fluctuating Lift and Drag on a Long Cylinder of Square Cross-
Scction in a Smooth and in a Turbulent Flow," J. Fluid Mech., 25 (1966), Cambridge
Oniv. Press, New York, pp. 481-494.
llrc mean flow may, in this case, be considered for practical purposes as two-
tlimensional. However, the effects associated with flow fluctuations are not
itlcntical in different strips, the differences between events that take place at
rrny given time increasing with separation distance. This is shown in Fig.4.6.1
l4-16] for the pressure difference between centerlines of top and bottom faces
ol'the rod under both laminar and turbulent approaching flow.* It is observed
tlrat the three-dimensionality of the flow manifests itself through spanwise loss
ol'correlation R7s between pressure differences (measured respectively between
;xrints,4 and A' at section.4 and points B and B' at section B), this correlation
krss being strongly accentuated when turbulence is present in the oncoming
llow. From this example one may infer that fluctuating phenomena, including
vortcx shedding, cannot nonnally be expected to be altogether uniform along
lhc cntire length of a cylinclrical botly, cvcn if the flow has uniform mean speed
:rntl thc body is gcrlrnctricirlly trrtilirrrn.
ln practicc, rncan llow t'orrtliliorrs rrpwirrtl ol'tall slcndcr structurcs arc usu-
rrlly no( unilorrrr, ls trssrrrrrctl irr (lrt'sirrrplr'r.('ilri('r tliscrrsscrl ahovo; inrlcctl, in
r'l'lrt'lrrrlrrrlt'rttt't'lr;uittlt'lislir's lvr'tr'lltr'',;trrrr';r" ttt llrr'r'r1x'tttttr'rrt ol liil',.'1.5.11.
170 ilt t,t I il()l)y nt il()t )yNnMr(:l
thc atlttospltcric ltrlLtlttl:rty lltyt:t'tlrt'rrrt'rrrr llow vclocily itrcreirscs witlr lrciglrr.
Also certain tall structurcs (c.g., sllreks):ur lt()t gcorrrctrically unilirrrrr.'l'lrcsc
important features-in addition to thc incitlcnl lurbLrlcncc-furthcr dccrcasc the
coherence of vortices shed in thc wako ol'structurcs.
4.6.2 Structures in Three-Dimensional Flows: Case Studies
The complexities of wind flow introduced by the geometries of typical struc-
tures and by the characteristics of the terrain and obstacles upstream emphasize
the need to carry out detailed studies of wind pressures experimentally using
wind tunnel models and simulation. In order to give some idea of the type of
results so obtained and to emphasize the important roles of the boundary layer
velocity profile and of the turbulence in such results, a few examples are cited
bclow.
wind flows about buildings are prime examples of three-dimensional flows
that cannot be described acceptably by two-dimensional models. Ftgure 4.6.2
il5-l ll suggests such a situation. Here a tall model building in a wind flow is
preccded by a lower building. This latter trips off a vortex in the space between
buildings. Air descending close to the windward wall flows through openings
beneath the building at ground level. Regions ofaccelerated flow are produced
around vertical and horizontal corners of the building. In the areas of vortex-
flow, through-flow, and corner streams, many design problems are presented
by the special characteristics of the locally accelerated flow. (See Sect. 15.3
and p. 188.)
A few examples are now shown of differences between drag or pressure
coefficients measured in a uniform and in a boundary layer flow. The existence
of such differences was first pointed out by Flachsbart in 1932 t4-311.we consider first the case of a rectangular plate normal to the wind in a
l"l(;tJltlt 4.(r.2. Mrrirr Icit(tttcs ol lhe llow rrnrrrn<l :r lrrll lrrriLlirrll rrrrrlt.l Il5 III
,l ri lll l'lll ',1 Nlnllvl ll()Wllll{il:;llJ llllll I l)l[/l N:;l{)Nl; lll
'l'Altl,l,l 4.(r.1. l)rrg ('rx'llil'irrrls l'rrr a llccllrrgrrlul l'lrrlt'Nolrrrirl lo Wirrrl irr
Srrrrxrlh l,'low l.l-lll, l! 2l
l{cctarrgular Platc in Nonnal Wind"
Itrrelurtgttllrr l)lirtc
on (inrrrnrl
(Standing on I-ong
Sidc)
Aspcct ratio I .0 2.O -5.0(',, 1.18 1.19 l.2O
10. 20. 40. oo
1.23 1.48 I .66 I .98
1.0 10. oo
l.l0 1.20 1.20
I'l'hc values listed in [4-10] were taken from [2-2]. Some of these values were incorrectly
trrrnscribed in [4-10] and therefore differ from those shown in this table.
snrooth flow. The drag coefficients depend strongly upon aspect ratio and upon
whether the plate is held in midair, as in the case of a tralfic sign, or stands
on the ground, as in the case of a free-standing wall; see Table 4.6.1. For
rcctangular plates on the ground, the drag coefficients of Table 4.6.1 are rea-
sonably consistent with mean drag coefficients obtained in boundary layer flow
14-931. Reference [4-93] contains additional results on free-standing walls,
rncluding pressures in the presence of a building upwind or downwind from
tlrc wall.
Note that the aerodynamic force normal to the plate is not necessarily largest
when the yaw angle a (Fig. 4.6.3) is zero. For a plate with aspect ratio X :
5, the dependence of the aerodynamic force normal to the face of the plate
upon cv is shown in Fig. 4.6.3. It is seen that for ot : 4Oo the aerodynamic
lirrce is larger by about 15% than in the case cv : 0o. A similar, though
sornewhat smaller, increase was reported in ll2-21for a plate girder with aspect
rrrtio X = 10.
The effect of turbulence on a square plate normal to the flow was studied in
14 251, where drag coefficients were measured for both smooth flow and tur-
bulent flow with 8.3% turbulence intensity and 7 .6 cm longitudinal turbulence
scale; see Table 4.6.2.
Note that the drag coefficients measured in smooth flow differ slightly among
1.2
08
(',, 0 4
0
o4
lr'l(;llltl,l 4.6.J. lX!l)(:nrlt'rrt't'ol tlt:r1'. tocllrr'rctrl lot pl;tlt'willt itspccl ntlio \ 5 ttpott
tlirccliorr ol lrolizon(rrl wirul I l.) .'l
172 lll Ul I ll()l)Y nl lr()l)YNAMI(:i;
'l'Alll,lil 4.(r.2. lh'ug ('rx'llicitrrls l'rrr
Sr;uarr: l)latc Nrlrrttrtl lo lhe Mr.un lr'krw
l4-2sl
Plate Sizc
(cm) Sntrxrlh 'l'urbulcnt
(',,
5.08 x 5.08
10.16 x 10.16
15.24 x 15.24
20.32 x 20.32
t.12
1.09
l.1l
l. l5
t.26
L22
t.20
I.t8
themselves and from the value of Table 4.6.1 (CD : l.l8). Note also that as
the ratio between the longitudinal scale of turbulence and the dimension of thc
plate decreases, the influence of the turbulence on the magnitude of the drag
coemcient becomes smaller. These results are further discussed in Sect. 7.3.3.
Figure 4.6.4a shows a model used for measurements reported by Flachsbart
in 1932 t4-311. The measurements were conducted in both smooth and shear
(boundary-layer) flow (Figs. 4.6.4b and c). The measured mean pressure coef'-
ficients Q, referred to the free-stream velocity, are shown in Fig. 4.6.4d for
smooth flow and Fig. 4.6.4e for boundary-layer flow (interrupted and solid
lines represent pressures and suctions, respectively). It is seen that the differ-
ences between the results obtainedin the two types of flow are significant.
Similar results were subsequently obtained in 14-321 and [4-33].
Figure 4.6.5a depicts mean flow patterns around a vertical wall of height-
to-width ratio I : 1 with uniform approaching flow. Figure 4.6.5b depicts the
same situation in boundary{ayer flow. Figures 4.6.6a and 4.6.6b display the
pressure coefficients developed on the faces of a cube resting on a horizontal
surface (due to flow normal to one face) first in uniform flow, then in a bound-
aryJayer flow. Figures 4.6.7a and 4.6.7b present similar results for a tall
building. It is noted that in Figs. 4.6.5b,4.6.6b and4.6.7b the pressure coef-
ficients are referred to the free stream velocity t4-201.
Loads on structural parts (e.g., cladding) are determined by the algebraic
sum of the extemal and intemal pressures acting on these parts. In the ideal
case of a hermetically sealed building, the internal pressure is not affected by
the external wind flow (Fig. 4.6.8a). If the building has an opening on thc
windward (leeward) side and is otherwise sealed, the wind flow will create a
positive (negative) internal pressure, as shown in Fig. 4.6.8b (Fig. 4.6.8c).
In most cases the opening or porosity distribution over the building envelopc
is not known, and intemal pressures could be either positive or negative (Fig.
4.6.8d). Building standards (e.g., [2-491) specify intemal prcssurc cocflicicnts
generally believed to be conseryative fbr use in design. lrrvcstiglrtiorrs into thc
magnitude of intcrnal prcssurcs and of thcir dcpcntlurt'c orr tirrrt' lrrt' rc;xrr1e:tl
in 14-521 to l4--571, which contuins adclitional rclcrctrccs.
,l ri lll l'lll iil Nln llvl ll()W lllll lri ll! llllll I lrlMl t..l!;l()l.l!i 113
Wrrrrl
---d>
Wtrrrl
('
lllo Io 2.O
l, llJlllll,,.l.(r.4. Srrrnnurry ol rrrrxh'l l(':,1:, rr :;rrlxrllr rrtrl lrottntiary-laycr llow. F'nrtl
\\'rrrlrlrtrt'k lrrrl gcsclrloss('nr' llr(l nllr'rrt ( i('lr;ru(l('." lry ( ). liluchsbafl, in Iirgtltri,t.tt
,1, t lt't!\l\',tttrtti.st'lrttr Vt'r,tttt'lrttit.tt,tlt .it (;tttttut:('tr, lV l,it'li'rrrrrg, 1,. l'rrrrllll, rttttl A.
llt t., (r'rls. ), Vcllirg vott ll . ( )lrlt'trlrotttp, l\ltttttr lt irrr,l ll,'rlirt, l().J2.
z
E
174 Bt t,t I B()l)Y nl lr()l)YNnMl(i:;
r.0 0
AP"p-ifr"
FIGURE 4.6.5a. Flow pattern and center line pressure distribution of a wall of height-
to-width ratio I : 1 in a constant velocity field. From W. D. Baines, "Effects of Velocity
Distribution on Wind Loads and Flow Patterns on Buildings," Proceedings, Sympos-
ium No. 16, Wind Effects on Buildings and structures, held at the National Physical
Laboratory, England, in 1963, published by HMSO London in 1965.
4.7 THE RELATION OF TIME-VARYING FORCES TO WIND
VELOCITY IN TURBULENT FLOW
For a given body immcrscd in a wind flow it is of intcrcsl (o crtttvc:tl irrlilrntation
on vclocity lluctualions into inlirnnation on prcssttrcs ovt't lltt'lrotly or on
rcsullanl lirrcc:s iurtl nl()nrc:nls. Sinc:c: tttrtst rr:itl lltlws lttr' sttlltt it'lttly t'oltllllcx
!-
H
r'0
4l llll lillAlloll()l llMl v^l tYlN(it()t t(:t t{ r wt[]D vt I { 
'( 
.ltY 175
lrlGURE 4.6.5b. Flow pattern and center line pressure distribution over a wall of
hcight-to-width ratio 1 : I in a boundary-layer velocity field. From W. D. Baines, "Ef-
lccts of Velocity Distribution on Wind Loads and Flow Pattems on Buildings," Pro-
,'rulings, Symposium No. 16, Wind Effects on Buildings and Structures, held at the
Nirtional Physical Laboratory, England, in 1963, published by HMSO London in 1965.
tlurt analytical calculation ol'such rcsults is not possible, it is usual to employ
lirrrnulas l'eaturing unknown cocllicicnls that may be evaluated by experiment.
4.7.1 Drag Forces
'l'lrc nct tlt'itg lirrcc c()nsisls ol lltr rt'sttll;url ()v('r'ir givcrrt botly surlircc ol'trll
t'orrr1'roncnls ol'clcrrtcntlrl lirtcr':; llurl ;rrr';rlrlqnr'tl witlr llrt'tlnrg, orirlorrg wintl,
176 tJttJt I tlot)Y AFllol)yNAMtct
--.5 5
-.6 0
)
--_ =-.-.80--......-...- 
-.2=-_.70- )
__--s__--
\--__
--.65 -
- 
-.60 --/
-\--.80-
at---
--., 
J
_-.70r'(_
'.(.
rilI,, 99
l/rl
f*'no
(a)
FIGURE 4.6.6a. Pressure distributions on the faces of a cube in a constant velocity
field. From w. D. Baines, "Effects of velocity Distribution on wind Loads and Flow
Patterns on Buildings," Proceedings, Symposium No. 16, Wind Effects on Buildings
and Structures, held at the National Physical Laboratory, England, in 1963, published
by HMSO London in 1965.
direction. The time-varying drag FoQ) on a body completely enveloped by a
flow is conventionally given by the formula
Fugy : [pu2gynzc,, (4.7.t)
whcre B is l lypicirl lrorly rlirncnsion irnrl (i7, is lhr.: rrsrrirl tlrirll t.rx'llicicrrl
ilttAil()N ()t ltMl
0.2 0
vnllYlN(i l()t t(;l :; t() 11y151; vt t(x;t ly 177
-0 ,0 0./0
(b)
lfl(luRE 4.6.6b. Pressure distributions on the faces of a cube in a boundary-layer
't'l.city field. From w. D. Baines, "Effects of velocity Distribution on wind Loadsrrrrtl lrlow Pattems on Buildings," Proceedings, Symposium No. r6, wind Effects on
lhrildings and structures, held at the National physical Laboratory, England, in 1963,
grrrblished by HMSO London in 1965.
Irr Eq. 4.7.1 a seconcl tcrrrr .l' rlrc lirrrrr pn\au1rlldtlc. is often included,
1r;ulicularly if the lluitl irr tprcstiorr is rclirrivcly rlcnsc, for examplc, as in thc
trtse: ol'walcr; (),,, is an cnrpitit':rl "virlrlrl nritss" erx'llicicnt intcn(lcd lo ucc()111
Ior cll'ccls linkctl trl thc lltrirl itt t t'lt'tirlron. At lrrrrlly llrc cocflicicrrl (1,, trppr:trr.s
Io bc ttsr.:l'trl itt citsc:s wlrt'n'itt lltr'llttirl ruirrr urvolvt'rl is tr;.r1'rrcciirblc rclir(ivc to
4 / lilt
{
l*,no
178
5
llr t,t I l1()l )Y nl ll()l)YNnMl(;i
. -0.9
z
f 
*'"0
Fro nt Bock
(a)
FIGURE 4.6.7a. Pressure distributions over the sides and top of a tall building model
in a constant velocity field. From W. D. Baines, "Effects of Velocity Distribution on
Wind Loads and Flow Pattems on Buildings," Proceedings, Symposium No. 16, Wind
Effects on Buildings and Structures, held at the National Physical Laboratory, England,
in 1963, published by HMSO London in 1965.
std?
4 / llll llt tn it()N ()t ItMt v^l tytN{i t()t t(jt j; t() wtNt) vt t(x;ilv l7g
(b)
lrlGURE 4.6.7b. Pressure distributions over the sides and the top of a tall building
rrurdel in boundary-layer vclocity ficld. From w. D. Baines, "Effects of velocity
l)istributi<rn on Wind Loatls irrrtl lilow P:rltcrns on Buildings," Proceedings, Sympos-
ittttt No. 16, Wind E,ll'ccts orr lhriltlings irntl Stnrcturcs, held at the National Physical
l.rrboratory, Iinglancl, in l()(r.1. prrlrlislrt.tl lry IIMS() Lonckrn in 1965.
f *'"0
.6
S idc
-0',1 to
-0.49
Bock
180 Bl tJt I lr( )l)Y n I ll()l)YNn Mloii
(o) HERMETIC BUILDING (b) WINDWARD OPENING
WIND
(c) sucrroN oPENTNG,r, 
?t_.il't:iro:,X:*.
FIGURE 4.6.8. Mean internal pressures in buildings with various opening distribu
tions. From H. Liu and P. J. Saathoff, "Intemal Pressure and Building Safety," ./.
Struct. Div., ASCE, f08 (1982), 223*2234.
the body mass. One can then visualize it as specifying a hypothetical mass
which, given the acceleration dUldt, accounts for the net force due to all thc
variously accelerated fluid elements in the entire flow around the body' In most
flows of interest in wind engineering. however, the entire term containing C.',,,
contributes only a negligible part to Fp. For this reason it is usually neglectctl
in this context, and it is not retained in what follows.
A three-dimensional flow will have three components, U(t)' V(t), and W(l),
in three mutually perpendicular directions. In the neutrally stratified flows* ol'
strong interest to wind engineering the mean wind velocity -U is horizontal,
and the wind then can be represented as the sum of mean and fluctuating
components.
U(t):D + uO)
V(t) : u(t)
W(t) : w(t)
(4.7.2t
the means of u, u, and w being zero.
One may then express drag in the horizontal direction by means of Fq.4'7 'l
with U(r) as in Eq. 4.7.2. In general,when time-varying vclocities arc llttts
introduced, the imperfect spatial correlation of thc vckrcily llucttutti(tns Itrtlsl
also be considered. Howcvcr, hcrc it is first assutnctl lhtrt lhc lrtxly in tlttcsliolt
*Sco Clt:rptcr l, 11.(), irtttl ('hitplt'l 2' 1l .l.l
4 / illl ilt tAil()N ()t ltMt VAnytN(i tr)lu F.i trr 6111.11, vt t(x.ilv 1Bl
r'':'rrllrt'icttlly stttitll t'ottt;trtretl lo llrc crlntlttliort rlrslrrrrt'e.s ol llre lltrgllirlrotrs 1,
r'.;ttttl tt', srl lltitl, lirt tlrt'l)ttrl)()scs ol'lhc: lllrrblt'trr rrl lr;rrrrl, llrcsr. lirller rrriry lte,,rttrltlt'lctl kr bc ptrr'lirt'll-y corlclulcrl. Sirrec irr llrt. lupilr wilrtls rrsulrlly ol'grllrlcsl
rrl{'r('st lo wintl crrgirrcclirrg u(tllU ftrrcly cxr.t't.tls o.l, r' nriry gcnuirlly bc
nt';rl,'r.',r.,,, witlr srrritll c:r'nrr yiclding
I"n(t) - F,, 1 plJulrTll(',,
rllrt'rt' thc stoarly arrcl tlro lluctuating parts of thc clrag lirrce are, respectively,
F,,: lpo2cotu, t u\Ol (4.7.4a)
:l!ltl
Fo: pUu(t)yz'Co @.7.4b)
I'rrrrrr litl. 4.1 .4b it is seen that Fp(t) varies directly as u(t). This is true ro a
lrrr,r ;rpPtrxirnation only, since observation of physical flows reveals that cp
irrirv ilscll'also vary as a function of the frequency components of ,r(/).
Irr orrlcr to cxamine the statistical characteristics of Fb@, it is useful to
, ,n"r(l('r' ils spcctral density Sro@).one first calculates its autocovariance func-
tirn (s('(' Appcndix A2., Eq. A2.21):
(4.1.3\
(4.1.s)
(4.7.6)
(4.7 .7)
(4.1 .8)
1fr/2
111,,(r) -' ,l'i f ) ,,rnb{,lnptr + 11 dt
I: prurFilcrpe4t + n
: FbG)Fb(t + r)
rrlllr t'
S6,(n) : t J* Rp,{r;cos 2trnr dr
r.'\1r1rt'rrrlix A2, Eq. A2.20), it follows that
.\'1,;,(rr) : p2O2 F C2rS,1n1
f trr rrlrrrlr rlris through hy (t,p(/; lll)'' yie:ltls rhc spectral density sc, of the fluc-
tn,rtrrl' rlr':tg cocllicicnl :
,!,,,{rr) 't. ;, 't;l'l'
llrr" t'rlttitliott will l.ltr vitlitl ovr't lltl t;rn1:t',r! lrt'rlrrt.rrt'it's ol',\',,(l) prrrvitletl lrll
r llr'r l\ rr'ntitin ;x'rli.ctl.y cot!rlirlr.rl in .ui!iuuri.{l ;rlrovt'_ llowcver, ltt.r.ttttst. in
182 ttt ut I tt()l)Y Al li()l )YNAMlcl
practice this assurrrptiorr tkre:s rrol lroltl, il is rrsturl (o ittcltttlr-: itrt lttlitrsllttcttt
factorto preservc thc validity ol'l')t1. 4.7.t'|.'l'his is tlonc by writirrg (4.7.11) as
(4.1.e)
where the newly introduced factor y21r1 is termed the aerodynamic admittance*
of the body in question and represents a modifying adjustment (fbr an actual
body) of the ideal case of a body enveloped by turbulence with full spatial
correlation. This modification brings the drag coefficient spectrum into align-
ment with actual conditions.
Thc aerodynamic admittance is a function of body shape and dimensions
ancl of the characteristics of the turbulence. For a given body it is thus a
tiequency-dependent function. Figure 4.7 .l l4-3slsuggests the manner in which
12(n) varies for a square flat plate placed normal to a turbulent flow with
uniform mean speed. The decrease of the aerodynamic admittance with in-
creasing frequency corresponds to the fact that the smaller turbulent eddies
have shorter wavelengths; thus those eddies with higher frequencies will suffer
loss of coherence more rapidly than do the large eddies. References [4-36] and
[4-37] appear to be among the earliest to have introduced and used aerodynamic
admittance concepts in buffeting problems.
4.7.2 Relation of Wind Pressures over Slender Buildings to Wind
Velocities
The type of arguments employed in Sect. 4.7.1 in relation to total drag forces
is now applied to the case of a high-rise building of rectangular plan form,
with the horizontal wind blowing normal to one face. In this instance, the
along-wind structural motion is dependent on the windward and leeward pres-
sure distribution in a manner that is conceptually simple.
The pressure acting at a point Q of elevation z on the surface of such a body
in a steady flow of velocity U(z) may be expressed as
p(Q) : ip(I2tz'tCo<Qt (4.7.10)
where p is fluid density and Co is the appropriate pressure coefficient at this
point.
In the case of unsteady flow U(z) : U(<) + u(2, r) the pressure may be
approximated by
p(Q, r) - pQ) + p',(Q, r) (4.1.1t)
*Thc ttso ol'tltis letttt irr wirrtl trttgitttrctittg is ltlt cxtcttsitllt ol lls otif itt:tl ttsc itt :rt'trtttltttlicitl
conlcxts l4 .141.
- ,\..(rr )56r(r) - 4Ci, :';',;' \ t(r)
4I llll ilt lAil{)| (}t ilMt vnl tylN(i l()l t( t!, lil wtt.Jtt vt t(t{.ilv
to-3 lo-2 to-r I to
^B/U
l.'l(lllRIl 4.7.1. Aerodynamic admittance of a square plate in turbulent flow. After p.
W llcarman, "Wind Loads on Structures in Turbulent Flow," in The Modern Design
,'1 ll'irul-Sensitive Stuctures, Construction Industry Research and Information Associ-
,rrrrrrr, l-ondon, 1971, pp. 42-48. By permission of the Director of the National physical
l;rl)orirtory, U.K., and the Director of the Construction Industry Research and Infor-
rrr.rtiorr Association, U.K.
rvlrt'rc 2 and p' have the following values:
(4.7.12)
UJoz
FF
=o
o
=z
oo(E
UJ
I -rrlr+-s!lpe) : rpcptetu'k I It_(z)l
p'(Q) : ) oc,Q)u'alz 4! . okfi{n) ro, ,r,,
rvlrcre overbars indicate the mean values.
A bricf numerical example is in order here. In the atmosphere u21z1t/2 =
' 'jrr,r, rrnd U(z) :2.5u*lnl(z - 2.,1)lz1l (see Eqs. 2.2.18 and.2.3.2 and Table
,,,' ', 
Forcxample, il'1,, - 0.03 rrr, ?.,1 = O, and U1 l0) :40 m/s atz : 50
rl'(':)
rt't.t () ()llt
',o lltitl lltc cltrrt ilr rrr-:glct'lirrli lltt' trorrl!rrr':rr lt'rrrr rrr litl .1.7. ll is lcss lhln ),X,.
184 lll ut I u()t)y nt n()t)YNnMt(;li
Morc gcncrally ancl anuklgotrsly lo llrr'rllrrg tcsttlls lrlrcatly tliscttssetl irt Scct.
4.'7.1, calc'J,lations reportcd in l4-.ltil itrtlit'irtt: tlrirt (hc lirlkrwing tclations arc
satisfactory, with insignificant erK)r, lirr 7r lttttl 7r':
If, as in the case of many buildings, the horizontal dimensions of the body
are small compared to the scale of turbulence, it is reasonable to assume that
thc fluctuating pressures affecting along-wind response-which consist entirely
ol'thosc on the windward and leeward faces-may be given by
pQ) : )pD'Q)c,,(Q)
p'(Q, t) : pOQ)u(Z, t)Cp(Q)
p,(e) : pU(z)u(2, t)C,(e*)
p,(e) : pry1a1u(2, t)C{e)
(4.1.t4)
(4.1.1s)
(4.7.16)
(4.',l .r1)
(4.1 .18)
(4.7.1e)
where Q", and Q1 are points on the windward and leeward faces, respectively,
and where
c,(Q*): #+
C/Q):ffi
where z is the elevation of point Q, or Q7. As discussed in Chapter 9, it is
usual in current procedures for estimating along-wind building response to
assume that Eqs. 4.7 .16 and 4.7 .17 are valid regardless of the ratio of building
transverse dimensions to the scale of turbulence. This point is brought up again
later in Sects. 4.7.3 and 4.7.4.
In calculating along-wind structural response (see Chapter 5) information is
required on the spatial correlation of pressures applied at any two points Q1
and Q2. Such information is supplied by the co-spectra of fluctuating pressures
(quadrature spectra being assumed negligible). Assuming the validity of Eqs.
4.1 .16 and 4 .7 .17 , the co-spectra take the form
S,',i,(Q,, Qz, n) : co(Q)cp(Qr)p'DQ)u(z)sc,,,,(Qr, Qz, n) (4'7 '20)
That is, the co-spectra of the pressures are proportional to the co-spectra of the
fluctuating longitudinal wind components in the undisturbed oncoming flow at
the elevations of the two points. The pressure coefficient C,IQi) rcpresents
windward or lccwarcl vatlucs dcpcnding upon whcthcr lhc poirrl Q, is on the
windwarcl <lr lccwitnl sitlc.
4/ lllr nt tAill )ft()t ilMt vnttytN(it()t ll:t t; t(lwll.lt )vt t()(]ily tB5
'l'ltc co spt't'tnrrrr,\f,,,,, rrury lrc cxllrcssctl irr llrc lollowilrl' 1,,t"t'
,\'j,,,,,t1t,. Q,. rr) ,tf,,,,tr'. tttN(ttl (4.1.21t
whcrc S,f,,,r(r, n) is lltr: ('() sl)octrum <lf thc lorrgitutlirrll vckrcity lluctuati()ns .lt
lxrints Q1 and Q2 (Qj bcing thc pro.jection ol'Q, orr u planc. nonnal to the mcan
wind direction, that contains Q), and r is thc clistancc bctween Q1 and Qi.'l'hc f'unction N(n) is ref'crrcd to as the along-wind cnrss-correlationcoefficient.
11' B1 and Qz are contained in the same vertical plane normal to the mean
wind (i.e., if their along-wind separation is zero), then N(n) = l. For nonzero
rrlong-wind separation, an expression of N(n) is given in Sect. 4.7 .4. In the
t'rtsc Q1 = Qz,
so,(Qr, d : c3(Q)p',O2k)s,(2,) (4.7.22)
4.7.3 Pressure Fluctuations on the Windward Face of Bluff Bodies
n theory of turbulent flow around two-dimensional bluff bodies has been de-
vclcrped in 14-391, which has subsequently been applied in [4-40] and [4-4ll
to the study of surface pressures generated by turbulent velocity fluctuations.
'l'hc theory is based, essentially, on the following assumptions: (l) the turbu-
Icnce intensity is of the same order of magnitude as, or lower than, the tur-
lrrrlence intensity typical of atmospheric flows, (2) the body is sufficiently long
{lrat end effects may be neglected, (3) in the flow region upwind of the body
:rrry velocity fluctuations induced by wake flow are statistically independent of
thc velocity fluctuations caused by the oncoming turbulence and so the latter
t'irn be studied separately. Fundamental to the approach of [4-391 is a gener-
rrlization of "rapid-distortion theory" which allows the linearization of the
er;uations describing the turbulent motion near the upstream face of the body.'I'his linearization follows from the assumption that the changes in the mean
llow associated with the presence of the body distort the turbulence sufficiently
r:rpidly so that, during the distortion process, the nonlinear inertial transfer of
cncrgy between eddies of different sizes is negligible.
The equations for the turbulent nondimensionalized vorticity vector a,)i G :
| . 2, 3) are then [4-39]
D6, +
Dt .i :t
\fi,cdr^ (i:1,2,3)" dxj (4.7.23)
wlrcrc l; (i : 1,2,3) is tlrc norrtlirncnsionalizcd velocity fluctuation vector'I'hc nondimensionalizccl prcssiur'('7)' is givcn by
lF' ittr,
=ldl; ilt )i / rr 
t"" , ,, t"4\
r r \ I t)\t ' ilt,/
(i r. 2. 3) (4.1.24\
186 lil t,t I tt()t)y nt n()t )yNnMtcli
wherc Oi (i : l,2, 3) is lhc trotttlitttt'rtsionirlizcrl rrrcirn vclocily vc(.[or, wlrosc
field is approximately irrotatiotrirl.* 'l'lrt' lrorrrrtlirry conclitions arc csscntially
the following: (l) at large distanccs lirrrrr (hc b<ily, thc velocitics approach
their values in the undisturbed flow ancl (2) in thc immediate vicinity of the
upwind surface of the body, the velocitics at, cach point are perpendicular to
the outward normal from that surface.
Calculations carried out on the basis of the above equations and boundary
conditions suggest, for example, that whenever alL" < I (where a is the typical
horizontal dimension of body and I is the longitudinal turbulence scale), Eq.
4.7.22 is applicable, on the windward face, up to frequencies r, = 0.15 Ula,where U is the mean speed of the undisturbed flow 14-411. For higher fre-
quencies the pressure spectra decay more rapidly than the velocity spectra so
that, for structural design purposes, Eq. 4.7.22 is conservative. That this is the
case appears to be confirmed by experimental results reported in t4-421,
t4-431, and [4-44].r Calculations also suggest that the smallei eddies are "piled
up" against the upward face of the body and therefore that the coherence of
the high-frequency pressure fluctuations is somewhat greater than the coherence
of the high frequency velocity fluctuations in the undisturbed flow. In structural
engineering computations this "piling-up" effect can be taken into account by
choosing appropriately small values of the exponential decay coefficients in
Eq.2.3.29 or 2.3.30.
4.7.4 Pressure Fluctuations on the Leeward Face of Bluff Bodies
According to Eqs. 4.7.16 through 4.7.19, the ratios of the pressures on the
leeward face to the pressures on the windward face are the same for both
fluctuating and mean pressures. Results of full-scale measurements suggest,
however, that the pressure fluctuations on the leeward side are less strong than
indicated by Eq. 4.7 .17 (e.g. , see Fig. 4 .7 .2 taken from t4-501). It is reasonable
to assume, therefore, that the use in design of Eq. 4.7.17 is conservative from
a structural safety viewpoint.
Also of interest for design purposes is the question of the extent to which
pressures on the windward side of a building are correlated to the pressures on
the leeward side. It is intuitively clear that this correlation cannot be perfect.
The correlation will be greater for eddies with large wave lengths-which can
be thought of as enveloping the body in the same manner as the mean flow-
and will decay as the wave lengths decrease. This dependence can be expressed
*An irrotational flow is one in which the components
,,:# y,,,,_y,_ _*, ".:y,_y,
are all zer<r.
lAccording to l4-(r71, ltowt:vcr, ul any givon Ircqucncy thc prcssun':rrrrl vchx rlv rpt.r'trr lurve lhc
sanrc skrpc.
4t lilt ilt tnit()N()t ilMt vnnylN(il()l t(:l :, t()wtNttvt t(x]ty t87
0.0 25.o 50.0 7s.0 100.0 125.0 1s0.0
Time (s)
lfl(;uRE 4.7.2. Yaiations with time of wind pressure on the windward and on the
k'cward wall of a building. After w. A. Dalgliesh, "statistical rreatment of peak Gust
,rrr Oladding." J. Struct. Div., ASCE, 97 (1971),2173-218j.
lr.y choosing an appropriate expression forthe function N(n) in Eq.4.7.2l.In
l.l-45| an expression for this function has been proposed of the form
o
d
c;
N(n)
t
ll: E- '1(r 
- e-'El
15.4nA,x
(4.1.2s)
(4.',|.26)
U
wlrcre u is the mean wind speed at elevation (213)H; Ar is the smallest of the
tlirrrensions B, H, and D; B is the width; F1 is the height, and D is the depth
.l'the prismatic body. Full-scale and wind tunnel measurements reported in
l,t-461, [4-471, t4-48], and [4-49] suggest that this expression is adequate for
pnrctical use.
4.7.5 Peak Local Wind Loads
'I'hc adequate design of roof members, roofing, cladding, and other elements
srrsccptible to failure due to the local action of wind (e.g., solar collectors
l'1 601) is of foremost impoftance for reasons of both safety and economy. It
is thcrcfbre desirable that wincl-inch.rccil loads on such elements be ascertained
:rs roalistically as possiblc l4-591.'l'hc clcrncnts potcntially ittvolvctl irt lirilurcs duc to local wind loads are
trstttrlly rclativcly rigitl so llurl llrt'tlyrr:urric lrrrrplilication ol'thc rcsponsc is
ltcgligiblc. Thc witrcl krlrtl rrclirrl'. ort lrrt clt'rrrt'nl is tlrr:rr cqr.ral to llrc sulrr, ovcr
lltt'crttirc itrcit ol'thc e:lctttcttl. ol llrc inslrrrrllrrrt'orrs l)11:ssutcsi intlrrccrrl by wirrtl.
I)trring cvcly sl()rttt llris loittl tt':tr'ltr's;r pritlr. lltt'clcrrrr:nl coltccrrrrctl, tirrtl its
- - .ti;q fP- -' - ^ -' -/zv-- -
188 uttl I B()DY nl not)YNAM|(;li
connections, must bc closignccl lo srrsllirr lhe llcuk wind load attaincd cluring
the N-year storm, whcrc N is thc lttcan rccut'rotlcc intcrval of the dcsign wind
speed specified for that element.
The total wind force acting on an element such as a roof member or a curtain
wall could, in principle, be measured directly. However, the experimental set-
ups required for such measurements are prohibitively expensive and impracti-
cal. For this reason forces acting over an element have recently been measured
by devices that automatically add pressures occurring simultaneously at several
points of the element, weighted by the respective tributary areas. In particular,
such techniques have been used at the University of Westem Ontario to measure
wincl l<rads <rn models of low-rise structures 14-71, 4-72, 4-'13, 4-74, 4-75,
4-16,4-771. These measurements, as well as results of full scale tests, [4-51,
4-1t1,4-79,4-801 have been used to develop new design load provisions for
rnain f-rames and for parts and portions of low-rise buildings that have been
recently incorporated in various standards, including [2-1391; see also Sect.
9.5. Local pressures can have strongly non-Gaussian distributions, especially
at comers and edges; see [,4.2-13].
4.8 SECONDARY WIND FLOW EFFECTS
In addition to the wind loads caused by the direct action upon thestructure of
the wind flow, it is of interest in certain situations to examine secondary effects
produced by wind, such as the blowing of roofing gravel [4-58,9-63], and the
drifting of snow. Systematic studies of these effects have been reported in
14-63,4-&, 4-65,4-66, 4-81, 4-82, 4-83, 4-841.
Mention is also made of wind action as a factor that influences the energy
consumption of buildings by increasing air infiltration. It is shown in [a-68]
that energy losses due to wind-induced air infiltration can be reduced signifi-
cantly by the sheltering effect oftrees acting as wind breaks; the energy savings
thus achieved may in certain cases be as high as l5%. The results of [4-68]
were obtained in wind tunnel tests and were subsequently confirmed by full-
scale measurements [4-69].
ADDENDUM
For the sake of its historical interest, we reproduce here a note by Count Buffon
describing the flow changes occurring upwind of a tower, for which it offers a
charming (if scientifically no longer tenable) explanation. A translation of the
note follows.
On Reflected Wind
I must rcporl hcrc lrrr obsr:rvlliorr which it sccnls to Ittt'ltits t'st'rt1tt'tl lltt'ltllcttiion
ol'plrysic'ists, cvt'rr llrorrglr cvt'r'yorrc is in ir posiliott lo vt'ttly rl. ll st't'rrts llutt
Atlt)t Nt){ tM l Bo
rcllcctctl witttl is slroufl('r llrrrrr rlit'ccl wirrtl, irrrrl llrt'rrron'ri() lrs ()lrc is cklscr to
tlrc obslaclc lhrrl rclle't'ts rl. I lravc cxpc:rir-:rrcctl llris ir rrrrrrrhcr-ol'tirncs by
irppnlaching l l()wcl'tlurt is alrrxrst l(X) lbcl lrigh irrrtl is situatccl at lhc nortlr
t'nrl ol'tny ganlcn in Monlb:rrtl. Whcn a stnng wintl l'rkrws lhrrrr tlro s<lutlr, up
Io thirty stcps lirrur llrc towcl onc fbcls stnrngly pushetl, alicr which thcrc is
:rn interval ol' livc ol six stcps where one coascs to bo pushcd and where thc
wind, which is rcllcctcrl by thc tower is, so to spcak, in equilibrium with the
tlircct wind. Aficr this thc closcr one approachcs tho tower, the more the wind
re llccted by it is violcnt. lt pushes you back much more strongly than the direct
wind pushed you forward. The cause of this effect, which is a general one and
t'rrrr be experienced against all large buildings, against sheer cliffs, and so forth,
rs not difficult to find. The air in the direct wind acts only with its ordinary
s1rccd and mass; in the reflected wind, the speed is slightly lower, but the mass
is considerably increased by the compression that the air suffers against the
obstacle that reflects it, and as the momentum of any motion is composed of
thc speed multiplied by the mass, this momentum is considerably larger after
I I rc compression than before. It is a mass of ordinary air that pushes you in the
lilst case, and it is a mass of air that is once or twice as dense that pushes you
b;rck in the second case.
d I'Hitloire Nanrelle. t t t6 Suppldment
ADDITIONS
A l'Article gui a pour titre: Des
Vents riglis, page zz4.
I.
Sur Ie Vent riflichi, pege 2+2.
TJ r': oors rapporter ici une obfervation
.;,ri rne paroir avoit 6chappC I l'attcncion
'l<'s Phyficiens , guoique tour le rnondeli'it en drat de la viLifer ; c'eft que le
vc'rr riflichi eft plus violent que Ie vent
rlrrcdt, & d'autant plus qu'on eft plus
1,,rt's de I'obftacle qui le rerrvoie. J'enri llir nonrbre de fois I'expirience , en
.rl,prochant d'une tour qui a prls de
, cnr pieds de hauteur & qt-ri fe trouve
lirtrdc au nord , ) l'extrCrnirC de rnon
i.rrrlin, I Montbard ', lbrfqrr'il lirrrfflc
rrrr grand venr du nridi, on lc lcrrr lirr-
r('nrent pouflc iulqu') trcnte p:rs rlt' l:rr,"'r i ry,ris <1uoi , il y a trn irrrcrv.rllc ,h.
tirrtg t-ru lix pas I oi l'on cclli rl'trre
ooufG & or) Ie veur, qtri eft r6flichi par
la tour, fair , pour ainfi dire , iquilibre
avec Ie vent direCt ; aprls cela, plus on
approche de Ia tour & plus Ie vent qui
en eft riflechieft violenr, il vous repou(e
en arriire avec beaucoup plus de force
que le venr direC! ne vous poufibit en
avant, La caufe de cer efter qui eft gi-
n6ral , & dorrt orr peut faire l'ipreuve
contre tous les grands bntinrens, conrre
les collines coupies ) plornb , &c. n'eft
pas difficile I rrouver, L'air dans le vent
dire& n'asir qu€ par fa vireffe & [a
nra(fe ordirraire ; dans Ie vent rdfichi,
la vite(G eft un peu dirninude , nrais Ia
rna{Ie eft confiddrablemenr augmerrrie
par la cornprefiion que I'air fo'rftie
contre l'obftacle qui Ie rdflCchir ; &
comnre Ia guantird de tout rrrouvelrlent
eft compolie de Ia vire(G nrulripli6e
par Ia nraflb , cette quantird eft bien plus
grande apris Ia courprefiion gu'supara-
vanr. C'eft unc nra(G d'air ordinaire, qui
vous pouffb dans le prenrier cas, & c'eft
ttrrc rnafG d'air une ou deux fois plus
lt:;:t" 
, U"t vous repcu(G dans le lccond
I rrr'silrrilc ()l n()le {)rl rcllcclcrl wiilrl l,trtttt llt\lt,ilr Nttlutrllt', (ir,ttr,tttlt tt l\trticuliin,, ('(ilttriltutt l(.\
lilrtlttt'.t tlr Itt Nttlurt, Itrrr M. lc lrtttlr rlr llrllltt, Vol I l. lttlcrtrlirrtl rlrr .l:rrrlirt t^l tlrr (':rltitrel rlu l{oi,
rk' l'Atrtrlcrilir liritrrr,'irist', rk' r'clle rli'r :ir t.ilr.r r=lr lo[rr' lrIrrrt'rrrr'. A l'irris, l)r' l,'lrrrPrrrrrrre
Itoyrrlc, I //ll
190 tJt t,r I tKltlY nt ti()t )YNnMt(;:;
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4-22 A. Iloslrko. "lixpt'rirrrcrrls orr lltc likrw l)lrst lr ('ttt'ttl;rt ('vlttttlt't :rl Vt'r'y lliglt
Itcynoltls Ntttttlrct." .l . I"lrtirl Mt'rlr. lll ( l(X)l), .\'15 lt(r.
ilt IItil tJ( I,; lgI
4 l.| .1 . ('ottlt'ltt'sttt' ;ttrrl A l ;rrrt'villc,"A ('otttplrisorr ol ('olt'r'liolr Mcllrrxls Iist.rl
itt lltt livitlrrrrlrott ol l)rr1i ('ocllicicrtl Mcirsurt'rrrr'rrl: lor I wo l)rrrrt.rrsrorurl ltt.t.l
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luntl Viltnttitnt,r, l(lrtlsrttltr'. Wr'r,l ( it'rrrr;rrrv, l()'/1. li. N:rrrrlirsclrct (c(1.). ,Sl)t.ill
gt'r' Vclllrg, lle llirr, l()/,1. pp lrr() -'O l
192 nt tJil D()t )y At n()l )yNnMt(;l
4-41 J. C. lt. Hurrt, "'l'ulhulcrrt Velocities Nr:r'llttl lilttcluirlittg Srrrllrce I'rcssrrrcs
on Structures in'l'ulhulcnl Wintls," irr I'nxvnlitrg.s tl tlrt Iirurtlr lnttnuttiorrttl
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4-42 B. J. Vickery and K. H. Kao, "Drag or Along-Wind Rcsponsc ol' Slcnder
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4-43 P. W. Bearman, "Some Measurements of the Distortion of Turbulence Ap-
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4-44 R. D. Marshall, Surface Pressure Fluctuations Near an Axisymmetic Stagnation
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94, No. 5T6, Proc. Paper 5980 (1968), 1295-1313.
4-46 R. Lam Put, "Dynamic Response of a Tall Building to Random Wind Loads,"
Pnx'ccdings of the Third International Conference on Wind Effects on Buildings
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4-47 H. van Koten, "The Fluctuating Wind Pressures on the Cladding and Inside a
Building," in Symposium on Full-Scale Measurements of Wind Effects on Tall
Buildings, University of Western Ontario, London, Canada, 1973.
4-48 J. D. Holmes, "Pressure Fluctuations on a Large Building and Along-Wind
Structural Loading," J. Ind. Aerodyn., l, | (1975), 249-278.
4-49 K. H. Kao, Measurements of Pressure-Velocity Correlation on a Rectangular
Prism in Turbulent F/ow, Report No. BLWT-20, University of Westem Ontario,
London, Canada, 1970.
4-50 W. A. Dalgliesh, "Statistical Treatment of Peak Gust on Cladding," J. Struct.
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4-51 R. D. Marshall, "A Study of Wind Pressures on a Single-Family Dwelling in
Model and Full-Scale," J. Ind. Aerodyn., 1,2 (1975), 177-199.
4-52 G. A. Euteneuer, "Druckansteig im Inneren von Gebduden bei Windeinfall,"
Der Bauingenieur, 45 (19'70), 214-216.
4-53 J. D. Holmes, "Mean and Fluctuating Internal Pressures Induced by Wind," in
Wind Engineering, Proceedings of the Fifth International Conference, Fort Col-
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4-54 H. Liu and P. J. Saathoff, "Internal Pressure and Building Safety," J. Struct.
Div., ASCE, 108 (19822), 2223-2234.
4-55 P. J. Saathoff and H. Liu, "Intemal Pressure of Multi-Room Buildings," -/.
Eng. Mechs. Dlv., ASCE, f09 (1983), 908-919.
4-56 R. L Harris, "The Propagation of Internal Pressures in Buildings," J. Wind
Eng. Ind. Aerodyn.,34 (1990), 169-184.
4-57 B. J. Vickery, "Intemal Pressures and Interactions with the Building Envelope,"
J. Wind Eng. Ind. Aerodyn., 53 (1994), 125-144.
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Struct. Div., ASCE,, 102, No. STl, Proc. Papcr lll3.l4 (l()7()). 1,17 l6O.
4-59 "Hanc<rc:k (illrss l}clrkagc: A ('ornbination ol' littrrs'f " l'.rr.t,ittct't itt,q Ncu.,l
Rtutnl (Miry l()76), ().
lll lllll l\l(,1 :, 103
,l 60 ll. |. Mclk'rur. "Wrrrrl l,orrrl lilli'c(s on l;lll l'irrlr Sol;rt ('ollt't lots." .l ,\trttr'1.
l,.lttg.,lll{lttxry. l,l t l.5l
,1 .61 "Ncw Appnxrtlrt's lo l)t'silr,rr Against Wirxl At'lion." irr A. (i. l)rrvcrrlxrr( (lil.),
Ctnrsc No/r,.r', 'l'lrr' llorrrrtluly Laycr Wirrtl 'l'rrrrrcl l.lrlxrlrrloty. tlnivclsily ol'
Wcstorn Orrtirrio, l,oruLrt, ('anacla, 197L
1-62 J. Blcssllun. Atnnlintinrirt dus Con,slrucor),s, Iililorit tlrr tJnivcrsiditdc, Pontr
Alegre, Brasil, l9tl.l.
,t-63 R. J. Kind arrtl l{. 1,. Wardlaw, Design o.l'Iilxtlir4ts A14,uinst Gravel Bktw-ofJ',
Repoft No. 1.5544, National Research Council ol'Canada, Ottawa, 1976.
4-64 R. J. Kind, "A Critical Examination of the Requirements fbr Model Simulation
of Wind-Induced Erosion/Deposition Phenomena Such as Snow Drifting," ,4r-
mos. Environ, f0 (1976), 219-227.
4-65 C. Mateescu and H. Popescu, Accumulations de neige sur les constructions,
Etude explrimentale sur mod?les, Annales de 1'Institut Technique du Bdtiment
et des Travaux Publics, S6rie EM, Pais, 1974.
4-66 J. Wianecki, Banc d'essais d'accumulation de la neige due au vent, Annales de
I'Institut Technique du B6timent et des Travaux Publics, Sdrie EM, Paris,
1976.
4-61 W. Z. Sadeh and J. E. Cermak, 'lTurbulence Effect on Wall Pressure Fluctua-
tions," J. Eng. Mech. Div., ASCE, 98, No. EM6, Proc. Paper 9445 (1912),
t356*1319.
4-68 G. E. Mattingly and E. F. Peten, "Wind and Trees: Air lnfiltration Effects on
Energy and Housing," J. Ind. Aerodyn.,2 (1977), l-9.
G. E. Mattingly (personal communication, April 1977).
H. P. Pao and T. W. Kao, "On Vortex Trails Over Oceans," Atmos. Sci.,
Meteorological Society of the Republic of China,Taiwan, 3 (1976), 28-38.
D. Surry and T. Stathopoulos, "An Experimental Approach to the Economical
Measurement of Spatially Averaged Wind Loads," J. Ind. Aerodyn., 2, 4 (Jan.
1978), pp. 385-397.
4-72 A. G. Davenport, D. Surry, and T. Stathopoulos, 'Wind Loads on Low Rise
Buildings: Final Report of Phases I and II-Parts 1 and 2," BWLT Report SS8-
1977, The Univenity of Westem Ontario, London, Ontario, Canada, Nov.,
t917.
4-73 A. G. Davenport, D. Surry, and T. Stathopoulos, "Wind Loads on Low Rise
Buildings: Final Report of Phase III-Parts I and 2," BWLT-SS4-1978. The
University of Western Ontario, London, Ontario, Canada, July, 1978.
4-74 L. Apperley, D. Surry, T. Stathopoulos, and A. G. Davenporl, "Comparative
Measurements of Wind Pressure on a Model of a Full-Scale Experimental House
at Aylesbury, England," .1. Ind. Aerodyn., 4 (1979),207-228.
4-15 T. Stathopoulos, "PDIi ol'Wind Pressures on Low-Rise Buildings," J. Struct.
Dlv., ASCE, 106 (l9lJ0). ()7.1 990.
4-76 T. Stathopoulos, l). Strrr.y,:rtttl A. (i. l)avcnport, "E,fl'ective Wind Loads on
Flat R<xrl.s." .l . Strutl. /)ir', AS('lr..l 107 (l9ltl),281 298.
4-11 'f'. Statl-roptrukrs, "Witul l,o;rrls ort lrrtvt's ol l.ow lltriltlings," .l . Strul. l)ir'.
AS('li, 107 (l9t'l l). l1).! I l()t'l
tl 18 K. .l . liutorr trrrtl .l . l{. M;rVtr,'. Iltr' Mr';r:.rut'rnr'nl ()l Wintl l)ttssutt' ott 'l'wrt
Story llottscs :rt Aylrslrrrty." .l lrnl '1'.'tt\lttt . | (l()75), (r7 l0().
4-69
4-70
4-7 |
194 ut ut I l()t )y nt ll()t )yNnMt(it;
4-19 J. l). llolrncs, Witttl ltttrl.s ()n 1,.,lr'lli,st'Iiltiltlirt.q,t A llryit'w, ('Sll((), l)ivisiorr
of Building Rcscarch, Iliglrctt, Viclorirr, Austlrrliir, I91i3.
4-80 T. Stathopoulos, "Wind Loacls orr l.ow ltisc lluildings: A Rcvicw ol'thc Statc
of the Art," Eng. Struct., 6 (l9tJ4). ll9 l3-5.
4-81 J. T. Templin and W. R. Schricvcr, "l.oarls duc to Drified Snow," J. Struct.
Div., ASCE, r08 (1982), t9t6-t925
4-82 J. D. Iversen, "Small-Scale Modeling of Snow-Drift Phenomena," in Wind
Tunnel Modeling for Civil Engineering Applications, T. A. Reinhold (Ed.),
Cambridge Univ. Press, Cambridge, 1982.
4-83 R. J. Kind and R. L. Wardlaw, "Failure Mechanisms of Loose-Laid Roof-
Insulation Systems (High-Rise Buildings)," J. Wind Eng. Ind. Aerodyn., 9
(1982), 32s-341.
4-84 R. J. Kind and R. L. Wardlaw, "Wind Tunnel Tests on Loose-Laid Roofing
Systems for Flat Roofs, " Proceedings , Second International Symposium on Roof-
ing Technology, National Bureau of Standards, Gaithersburg, MD, Sept. 1985.
4-85 I. S. Gartshore, "Some Effects of Upstream Turbulence on the Unsteady Lift
Forces Imposed on Prismatic Two Dimensional Bodies," J. Fluids Eng.,106
(1984), 418,424.
4-86 W. C. L. Shih, C. Wang, D. Coles, and A. Roshko, "Experiments on Flow
Past Rough Circular Cylinders at Large Reynolds Numbers," J. Wind Eng. Ind.
Aerodyn., 49 (1993), 351-368.
4-87 Y. Nakamura, "Bluff-Body Aerodynamics and Turbulence," -/. Wind Eng. Ind.
Aerodyn., 49 (1993), 65-78.
4-88 A. Roshko, "Perspectives on Bluff Body Aerodynamics," J. Wind Eng. Ind.
Aerodyn., 49 (1993), 79-100.
4-89 A. Gadilhe, L. Janvier, and G. Barnaud, "Numerical and experimental modeling
of the three-dimensional turbulent wind flow through an urban square, ' ' J . Wind
Eng. Ind. Aerodyn., 46-47 (1993),755-763.
4-90 S. Murakami (ed.), "Cunent Status of Computational Wind Engineering," -/.
Wind Eng. Ind. Aerodyn, 35 (1990), l-318.
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Ind. Aerodyn., 46-47 (1993), l-912.
4-92 D. Laurence and J.-D. Mattei, "Current State of Computational Bluff Body
Aerodynamics," J. Wind Eng. Ind. Aerodyn.,49 (1993),23,44.
4-93 C. W. Letchford and J. D. Holmes, "Wind Loads on Free-standing Walls in
Turbulent Boundary Layers," J. Wind Eng. Ind. Aerodyn., 5l (1994), l-21.
4-94 Y. Nakamura and Y. Ohya, "Vortex Shedding from Square Prisms in Smooth
and Turbulent Flows," J. Fluid Mech., f64 (1986), 77-89.
4-95 A. Baskaran and T. Stathopoulos, "Prediction of Wind Effects on Buildings
Using Computational Methods-Review of the State of the Art," Canadian J.
Civil Eng., 2l (1994),805-822.
CHAPTER 5
STRUCTURAL DYNAMICS
Structural dynamics is the discipline concemed with the study of structural
rcsponse to time-dependent loads. This chapter reviews certain elementary re-
sults of structural dynamics theory and derives expressions for the response of
structures subiected to distributed stationary random loads. These results are
then applied in the particular case where the loads are induced by wind to
obtain expressions for the along-wind response, including deflections and ac-
cclerations. Several of the results obtained will also be useful in other appli-
cations occurring throughout the text.
5.1 THE SINGLE.DEGREE-OF-FREEDOM LINEAR SYSTEM
('onsider the system represented in Fig. 5.1.1 consisting of a single mass ttt
('oncentrated at point B and of the member /8 assumed to have negligible
nrass. The displacement x(r) of the mass nr is opposed by (1) a restoring force
sLrpplied by the member /B and (2) a damping force due to the internal friction
that develops within the system during its motion. It is assumed that the re-
storing force is linear, that is, proportional to the displacement x(r), and that
rlrc damping is viscous, that is, proportional to the velocity dxldt. It follows
tltcn from Newton's secon(l law lhat the motion of the mass is described by
tlrc cquation
,,ri'I (\ I A.r I'll) (s. r. r)
wlrcre: /'i/) is tlrc lirtrc tk'gx'ttr['rtl l(]jr(l :r('ltnl', ott (ltt: tttttss, I is thc sllrirrg
t.6rrslllrl (11r lIc slilllrcss) ol llrt'rrrt'rrrlrrr ,'lll, r'ts kttowlt its lltt: cocllit'it'ttl ol
r95
196
FIGURE 5.1.1. Schematic of a single-degree-of-freedom system.
viscous damping, and the dot denotes differentiation with respect to time. It is
common to write Eq. 5.1.1 in the form
X -f 2(,(2zrn)* -l (2rn1)2x : F(t)
m
(s.1.2)
where
-l.trr
rurt I
,l'
[-tm
-I Ik' 2r\lm
.C
Sr * ,
2,,/km
(s.1.3)
(s.1.4)
are known as the natural frequency and the damping ratio of the system, re-
spectively.* The quantity zJtcm is known as the critical damping coefficient
and can be shown to be the value of the damping coefficient beyond which the
free motion of the system is nonoscillatory. The damping ratio is expressed as
a percentage of the critical damping.
5.1.1 Response to a Harmonic Load
It can be easily verified t5-ll that if
F(/) : Fo cos 2Tnt (s. r .s)
xThe quantity 22"11, is tclcrlctl to as lltc natural circular I'rcquoncy rrrrtl is t.orrrnrrrly rlcrrolctl hy
@1.
!i r llll lilN(il l l)t (int t ()t 1ilil t,{)M ltNt Ail r;yl;ltM 197
llrc slcatly-stlrle solrrliorr ol lit1. .5. 1 .2 is
r(/) ItuItQt) c<ts(2rnt $) (.s.1.6)
wlrcrc
d:tan , 2(t(nln)| -hk$ (s.t .7)
(s.1.8)
(s.1.10)
the action of a load
0, that is, to a load
H(n) :
'f'lrc quantity F(n) is known as the mechanical magnification factor or me-
t'lrttnical admittance function of the system with parameters m, n1, and (r.
Similarly the steady-state response to the load
F(t) : Fs sin ZTnt (s.1.9)
rrray be written as
4r2 nlm{lt - (nl n,12fiJt1r;7y
x(t) : FoII(n) sin(2rnt - g)
5.1.2 Response to an Arbitrary Load
Lct the system described by Eq. 5.1.2 be subjected to
ct;ual to the unit impulse function 6(r) acting at time / :
rlcfined as follows (see Fig. 5.1.2):
6(o:0 forr + 0)
FLI t
lllJ, bl)dt:t )
(5.1.1r)
Il(;lll{1,: 5.1.2. tlnil irrrprrlsc lillctiorr.
t98 st n[,o I ulln L t)YNAMloti
FIGURE 5.1.3. Load F(r).
The response of the system to the load 6(r) is a function of time and is denoted
G(/).
An arbitrary load F(r) (Fig. 5.1.3) may be described as a sum of elemental
impulses of magnitude F(r') dr' each acting at time z'. By virtue of the linearity
of the system, the response at time / to each such impulse is G(l - r')F(r')
dr' and the total response at time / is
(5.t.12)
where the limits of the integral indicate that all the elemental impulses that
have acted before time / have been taken into account. With the change of
variabler: t - r',F,9.5.1.12 becomes
LetF(t) - F6 cos 2rnt.It follows then from Eqs. 5.1.6 and 5.1.13 that
x(ty : J' * o,, - r')F(r') dr'
x(t) : J- *rou - r) dr
H(n) cost : f Qz) cos Ztrnr dr
and
(s. 1. 13)
(s.1. 14)
/y'(rr) sin q5 : I G(r) sin 2rtt'r tlrI
ir
()
Using now Iit1s.5.l.l4 rrntl 5.1.15, it is possible lo wlilc
(s. I . ls)
II(n) and G(r):
H21n1 :
,, I il[ litN(il t I)t (iltil ()l Iltl Il)r]il/ lll.ll All :;Yr;lltrrt 100
r"["
/1r(rr) crrs'rl J,, .\,, ,t,t1)c.s 2zutr1(i( r,) t'.,s )trttt, tlt, tlt; (.5.1.1(r)
l"'r"''
H)tr) sinro - ],, J,, ,t, r,\ sin 2rnr,G( r.,) sirr 2rttr; tlr, dr, (5.1.l7)
I'ltc addition of Eqs. 5.1.16 and 5.1.17 yields thc firllowing relation between
f-
Jn Gr')G(ru) cos 2rn(r1 - 12) dr1dr2 (5.1.18)
5.1.3 Response to a Stationary Random Load
'l'hc case is now examined in which the load ,F(r) is generated by a stationary
l)()cess with spectral density Sflrz). The expression for the spectral density of
tlrc response S"(n) can be derived using Eqs. A2.20, A2.21, and 5.1.13:
f-
,S,(n) : t ) _&(r)cos 2rnr dr
:2[- 1,,, tf'/2 'lJ - I .* V ) ,'rx(tlx1 -t r) dt lcos 2rnr dr
: , I f ',- 1- ['" ,, I I er1)F(t - 11) dr1J--(r,*T J-r,z lJo
x I tr,,r,, * r - rt\ clrtl]ro, znm a,Jo -l)
: t J; *',, [j, *", [J-- ^",' * 11 * 12)
' cos 2trnr arlarrlar,
: 2 [ I cO)c(rr)cos 2trn(r1 - 12) dr1 dr2-Jo Jo t
J;
r-
J _ *o,r I rt - r2lcos2rn(r I rr - r) d(r * 11 - 12)
, J,, j,, (i(rr)(i(r,)si n 2ntt(r1 - r.) dr1 dr2
J ,,r,r,t, ,, r,)srrrlrrr(r I rr r;)tl(r I 11 r;)
(5.l.lt)l
200 silrt,(;ttlnAl t)YNnMlcli
where, in thc last stcp, thc itlcrttity
cos2rnr = cos Zrnl(r I rt r:)-(r1 -r)l (s. r.20)
is used.
From Eqs. A2.20, A2.23 and 5.1.18, there follows
Sln) : nzg1so1nS (s.1.21)
This result is extremely useful in applications. See also [5-l] to [5-4].
5.2 THE MULTI.DEGREE OF FREEDOM LINEAR SYSTEM
5.2.1 Natural Modes and Frequencies of a Continuously Distributed
Structure
It may be regarded as an experimental fact that a continuously distributed elastic
structure with low damping, when excited by a sinusoidal force, will vibrate
in resonance at certain sharply defined characteristic frequencies. Associated
with each such resonant or natural frequency, there will also be a characteristic,
or modal, form of vibration amplitude distributed throughout the structure. Such
forms are called the normal modes of the structure. For example, Fig. 5.2.1
depicts the first four normal modes of a vertical cantilever beam.
These characteristic deflection modes and associated frequencies are prop-
erties ofthe structure, independent ofthe loads, and represent very fundamental
dynamical evidences of its internally distributed inertial and stiffness proper-
ties.* In fact, the set of normal modes may be regarded as a fundamental set
of special deflection forms by means of which any general deflection of the
structure may be expressed.
Thus, if z is a running coordinate (e.g., height) of a structure, the modal
deflection forms of lateral ("{ direction) vibration may be written as x; (z), where
FIGUR Fint fbur normal modes of a cantilcvcr bcam
*Dctails <ln pnlccrlrrtt:s lirrtlclt:r'rrrirring, rtortttirl ntotlcs antl ttitlttt:tl ltt'r1ttr'tttit'r tttrty lrt' lirtttltl. lilr
cx:rtnplc, irr l5 ll ot l5 11.
T
E 5.2-1.
11 Mt 1 l lll (i1l I ()l llg ltlr)M ttfJt At t r:yr;ltM 2Ol
r. 2" l. 'l'ltcrr :ttry tlclle:ctiott r(:, /) lllily lrt'exlttt'sst'tl :ts lltt' sttttt
whcrc the coeflicients t,(r) inclicate what fructiort ol'caclt ntotlc r,(z) ctltcrs (ltc
p,ivcn deflection pattern. The coefficients {r(l) arc callod Lhc gcntruliT.ed ut'
ttttlinat€s of the system.
An important property of the normal modes xi(z) is their mutual orthogonality
with respect to mass weighting, by which is meant that
I,r.,.- t'(t)"' (z)m(z) dz 
: o (i + i)
r
KE : ; J.r.,., .,.,[*k, t)]2 dz 6-2.3)
ll thc system is vibrating in the single resonance modex;(z), then
r(2, /) : )",t.)t,t,l
*(2, t) : xiQ)EiQ)
:.o thaf the kinetic energy becomes
KE - +Mtt?
wlrt'rc
M,
A./, is kttowtr lts lltr.: ,qlrtr'rrr/t
15.2. I )
(s.2.4)
(s.2.s)
(5.2.2)
wlrore rn(z) is the mass of the structure per unit length.
Since the system is actually continuously distributed but responds at each
rrl its resonant frequencies like a single vibrating entity (or single degree of
ln:cdom), it becomes very useful and convenient to liken continuous systems
to single-degree-of-freedom systems. It is helpful in this context to use the
('rrorgy approach. The kinetic energy of a single mass M is )M*', where i is
rrs vclocity of displacement. We now seek the corresponding energy for the
rlistributed system.
'f'he lateral displacement being x(2, t), the elemental kinetic energy at point
.is
im(z)lx(2, t)21 dz
'l'lrc kinetic energy (KE) of the whole system is therefore
I
\ t r,(: )l'rrr(.:) r/r (5.2.6)
.l..1 .r, r,,
,',1 trnttt ol lltt' rvslt'trr ilt lllt: itlt ttorlttltl lttrxlt'.
ii
l,!
2O2 liltr,otunnr t)yNAMtoli
ln this scnsc a cttntinttous sys(qnr vibrlrtirrg irr irny orrc ol'ils rlrnrrirl nroclcs
may be viewed as thclugh it worc sintply ir sirrglc{ogrcc<rl-l'rccdr)tl systotn
with a mass M; and velocity {,.
5.2.2 General Expression of the Response
Consider a structure for which it may be assumed that the displacement in the
direction x is the same for all points in the structure that have the same coor-
dinate z (Fig. 5.2.2).It can be shown [5-2] that if the damping ratio is small
the generalized coordinates €i(r) satisfy the equations
whcrc (,, n,, and M, are the damping ratio, the natural frequency, and the
gcncralizcd mass (Eq. 5.2.6) in the ith mode. The quantity QiG) is known as
Lhc gcncruliz.ed force in the ith mode and has the expression
{,(r) t 2(,(2rrt,)fr(/) + 12rni)2tilt):T
ei(t) : \i ,u,t)x{z) dz (s.2.8)
where .F1 is the height of the structure, and pk, t) is the time-dependent load
per unit of length acting on the system. It is seen that each of the equations
5.2.7 is of exactly the same form as the equation of motion of the single-
degree-of-freedom system Eq. 5. 1.2.
If the load nQ, t) is such that
nk, t) : F(t)6(:z * z) (s.2.9)
where 6(z - zt) is defined in a manner similar to Eqs. 5.l.ll, that is, if the
structure is subjected to a concentrated force F(r) acting at a point of coordinate
FIGURE 5.2.2. Schcrnatic ol'a slcndcr
structure.
H
.11. llrt: gcrrcrirliztrl krrt't' (),(l) will
Q,u) 
o'1"],,
=. xik.)
It lirllows from
t txrrdinate z is
(illt I ()l lnt I l,()M ltr.ll Alr !;Yr;llM 203
I 4,,
lt(:,, l).t,(:,1 tl:.
bcr
I.
F(t) (5.2. r0)
5.2.3 Response to a Harmonic Load
ll tr concentrated load
F(t) : Fs cos 2rnt (s.2. 1 1)
rs rrcting on the structure at a point of coordinate z1,by virtue of Eq. 5.2.10
tlrt: generalized force in the ith mode will be
Q,(t) : F11xi(z)cos 2rnt (s.2.12)
rrrrtl the steady-state solutions of Eqs. 5.2.7 wlll be similar to the solution 5.1.6
ol liq. 5.1.5:
t,(t) : Foxik)Hi@)cos (Zrnt - $') (5.2.13)
wlrcrc
Hi(n) :
+rznl u,{11 - (nlni1212 + 4y,2tnl n,\'\"'
, 2f i@lni)o, : tan 
-_ 
r| - (nln;)'
F,q. 5.2.1 that the response of the structure
x(2, t): Fo I xik)xik)Hi(n)cos(2rnt - $)
lt is convenient to write Eq. 5.2.16 in the form
x(2, t) : FoH(2,, 21, n)cosf2rnt - 6(2, zt, n)l
rvlrcrc, as fbllows immcrliittcly l'rrrn lrqs. A2.4t and N:2.4b,
(s.2.t4)
(s.2.ts)
at a point of
(s.2.t6)
(s.2.11)
I r,(:)r;(z1rff,f,,l*in,l, | |
(.s.2.
tt(t..7.r, lr) : Il ] r.,,.,,,,::r)r/,(rr)t.,s,/,, I I
ItJ)
. r L;.t;(z).r,(11)//,{rr)silt <15,o(z' z1' n) : lan 
{x1k.'lx1k.1)H,(rr)cos <},
l5.2.lt);
(s.2.22)
Similarly, the steady-state response at a point of coordinate z to a concen-
trated load
F(t): Fssin2rnt (5.2.20)
acting on the structure at a point of coordinate Z1 elr' be written as
x(2., t) : FyII(2, 21, n)sinl2rnt - 6(2, \, n)l 6.2.21)
5.2.4 Response to a Concentrated Stationary Random Load
[,ct thc rcsponse at a point of coordinate z to a concentrated unit impulsive
load 6(t) acting at time / : 0 at a point of coordinate z1 be denoted G{2, 4,
r). Following the same reasoningthat led to Eq. 5.1.13, the response x(2, t)
of the structure at a point of coordinate z to an arbitrary load F(r) acting at a
point of coordinate Z1 czn be expressed as
x(:2, t) : I, ou, zt, t)F(t - r) dr
Note the complete similarity of Eqs. 5 .2.11 , 5 .2.21 , and 5 .2.22 to Eqs. 5. 1 .6,
5.1.10, and 5.1.13, respectively. Therefore, by following the same steps that
led to Eq. 5.1.21, there results
S,(2, 21, n) : H2(2, 21, n)Sp(n) (s.2.23)
where S,(2, zr, n) is the spectral density of the displacement x(2, t), the me-
chanical admittance function H(2, q, n) is given by Eq. 5.2.18, and Sln) is
the spectral density of the force F(r).
5.2.5 Response to Two Concentrated Stationary Random Loads
Let x(2, /) now denote the response of the structure at a point of coordinate z
to the action of two stationary random loads F1(l) and Fr(t) acting at points of
coordinates Z1 and 22, respectively. The autocovariance of the response can be
written as
t l'''
R,(2, z) - lim .,. \ .r(2, r)x(z. t -t r) dtt'ql J t':
lt : il il Mt,l ll l)l (illl I (ll
l':',: I I";, I J,, (i(:.,r. r,\t,1u
+ J," r;t., ?.2.r1)[,2(t - r,ldr,l
Iilt I t)()M ItNt At I f ;Yt;il M
t r\ tlt r
I {'-x I I GQ, q, r)F{t -t r - r) dr2IJ0
('- 
|+ I G\2. zz, r)F2Q -t r - r) dr2 | drJo ' 'l
f- I r- I: I Gz,21.zlt| \ G{,z.zr,r)Rp,(r * rr - r2'sdr2ldr1Jo lJo l
r- I f- r+ Jo ctz. zz.rzllJo *.. z2'r)Rp,(r t rr - r27dr-,ldr,I
f- [l- |+ I CAz.z1.r'll \ G1z.zz.r2)Rp,p,(r I rr - r2ldr2ldr,Jo 'lJo 'l
("- I f- |+ I G(2. 22. ryt | \ Cl,z.21. r2lRprp,(r * rr - 'r2\ dr2ldr,Jo LJo l
(s.2.24)
wlrcre the definition of the cross-covariance function (Eq. A2.29) was used.
'l'he spectral density of the displacement x(2, t) is
l"-
S,(2, n) : 2 \ &(2, r)cos 2zrnr dr''J-
: 2 [ &(2, r)cos 2trnl(r i rr - rz) * (rt - r)l- J_-
' d(r + rr - rz) (5.2.25)
l.t'r Eq. 5.2.24 be substituted into Eq. 5.2.25. Using the relations
H(2., z,i, n)cos 4,k..:.i. trt J,] ",..Z;.7)coS 
2trnr dr (5.2.26)
II(2., z.i. rr)sirr y'r(.l, .1,. rt) [ 
' 
,r, ,'. .:,, r)sin 2trnr tlr (5.2.21\
.t ,,
206 sillt,ott,nnl r)yNnMtori
(which arc siurilar to llqs. 5.1.14 irrrtl 5.l.l5) rrrrtl
H(2, zr, n)H(2, 22, n)coslg(2, z.t, n) - 6Q., zz, n)l
: f f G{z' zt' r)G(z' z2' r2)cos Ztrn(rv - 12) dt1 dt2
(s.2.28)
H(2, zr, n)H(2, 22, n)sinfg(a, Zr, ft) - Qk, zz, n)f
- [- [- /:t- ,: .1,, J,, ,,., 21, r)G{2, Zz, rz) sin2rn(r1 - :,2) dr1 dr2
(s.2.29)
wlrich carr bc derived immediately fiom Eqs. 5.2.26 and 5.2.21 , and following
thc stcps that led to Eq. 5.1 .Zl, there results
S*(2, n) : H2(2, 4, n)Sp,(n) r H'(2, 22, n)Spr(n)
+ 2H(2, 21, n)H(2, 22, n)
' {S$,r,(nlcosl6k, q, n) - 6k, zz, n)l
+ Sf;,o,(n)sinf6(2, zr, n) - 6k, zz, n)l\ (5.2.30)
where S"(2, n) is the spectral density of the displacement at a point of coordinate
z, H(2, Z;, n) are the mechanical admittance functions defined as in Eq. 5.2.18,
Sa(n) is the spectral density of the force F,(r), and Scr,rr(n), Sf,p, are the co-
spectrum and the quadrature spectrum of the forces F1(r) and Fr(r) defined as
in Eqs. A2.33 and A2.34, respectively.
5.2.6 Effect of the Cross-Correlation of the Loads upon the
Magnitude of the Response
Consider two random stationary loads F1(r) and Fz(r) acting at points of coor-
dinates Zr and 22, respectively, and such that Ft(t) : Ft(t) at all times. By
definition, in this case the cross-correlation equals the autocorrelation, SF,r, :
S",(,e), and Sp,p, : 0 (Eqs. A2.21 and A2.29, A2.20 and A2.33, A2.23 and
A2.34). The loads F1(r) and Fr(t) are said to be perfectly correlated. The
spectral density of the response to the two loads can then be written as (Eq.
s.2.30)
S,(2, n) : {U'(r, zt, n) + H2(2, zz, n) + 2H(2.,24, n)H(7.,7.t, 11)
' cosld(2, 2.1, tt) - $k, z.z, n)llS7,,(rr)
In thc parlicullrr casc wltctt ;' r.,,
(s .2.3 r )
nI Mll il l)l {illl I ()l l lll l l,r )M IllJl All :iY:;ll [/l 2O1
(5.1..]l )
('tllsitlcr rrpw llre t'rrst'wlrclt'tlrc loatls /"r(l) itrrtl /',(l) irrc strch thlrt llrc:ir
, nrss covariipllce: /11,,1,,( r ) 0. 'l'hcn, by Llqs- A2.'1.1 irrrtl A2'14'
sj.,,,'{n) : sf,,,u(rr) - tt (s.2.33)
;rrrrl, ilthe statistical pnrpcrlics of the loads arc thc same, that is' if Sp,(n) :
'\1,(rr),
S,(2, n) - [H'(2, zt, n) + H2(2, zz, n)]Sp,(n) (s.2.34)
ttr. tl' zt : Zz,
S,(2, n) : 2H2(2, 21, n)Se,(n) (5.2.3s)
I'lr(' spectrum of the structural response to the action of the uncorrelated loads
r:, tlrus seen to be only one half as large as in the case of the perfectly correlated
lo;s115.
5.2.7 Distributed Stationary Random Loads
I lr(' spectral density of the response to a distributed stationary random load
rrury be obtained by generalizing Eq. 5.2.30 to the case where an infinite
rrrrrrrbcr of elemental loads rather than two concentrated loads are acting on the
.,tnrcture. Thus, if the load is distributed overan areaA, and if it is noted that
rrr lhc absence of torsion the mechanical admittance functions are independent
ll tlro across-wind coordinate y, the spectral density of the along-wind fluc-
trrrrting deflection may be written as
tt
S,(2. n) : I I Il(2. 21. n)H(z.72. 1)' J,t Jt
x {Sfio;@)cos[d(2, Zr, n) - Q(2, zz, n)f
+ Sf;;oi(n)sinI6k, zr, n) - 6Q, zz, n)l\ dAt dAz
(s.2.36)
rVlrt'r'er /)i anJ pi denotc prL:sstlr(rs ircting at points of coordinates 11, Z1 and y2,
,, n'sllcctivcly.
It r':ut bc vcriliccl that llirrlr lir1. 5.2.-l(r lltc:rc lirlltlws*
rll1,rrsitr1,, li1s.5.2.lllillxl 5.1.1(r.'1 ,' l.l :tllrl 'r'lt.A.r,lrr;rrrtl A].4/r. lrot':rtlclivlrliolrol lil
', .t l/ irr l('nns ()l cotttplt'x v;tti:tlrlt'r, r,r'1 1'r 't. l t)ll
208 l-;lru(;l(,nAt t)yNAMtoti
s,(2.r) : -L ) > rr(r)'\/(:)vr\(r "/ l6ra 7 7 ,,i ,,i ltt,u,
r_" {lr - (ntn,)zl2 + 4yl@tn;2\{t - (ntn)212 + +yllntn)2}
" ll ' - (;)'l l' - (;)'l + 4rci::,JI^ln*,,,,,*/,,,
x sfi,;1n1 dA, dA2 * 
fr r, :,1, - (;)' I - z r, I,[, - (;)' ]]
' J, J, x1Q)x1Q,)sfi,,{n)dA1dA,) 6.2.3i)
If the damping is small and the resonant peaks are well separated, the cross-
terms in Eq. 5.2.37 become negligible and
*? tO \ ^ I n 
*,rr,rr,,rr) s'oio;(r) dA I dAz
S,(z, r) : I (s.2.38)I6ranlul {11 - (ntn)212 + +y!1zntn,12l
5.3 EXAMPLE: ALONG-WIND RESPONSE
To illustrate the application of the material presented in this chapter, the case
of the along-wind response of tall buildings will be dealt with below.
5.3.1 Mean Response
If in Eq. 5.2.8 the load p per unit of length is independent of time, the cor-
responding along-wind deflection, which will be denoted by x(z), results im-
mediately from Eqs. 5.2.1 and 5.2.7:
:'l{ Pktx'(zl dzXlZr:./t 
- 
X[Z)i r'niM;
, : \: x?k)m(z) az
(5.3.1)
where
(s.3.2)
and p denotes thc linro-irrvarian( loacl.
As indicatctl in ('hirlttcl'4, lhc rttcan wincl loatl aclirrg orr ir lrrriltlilrg ol'wiclth
l,:| lXnMl'l I nl ('lltiWlrllr lltril,r,Nl;l
,4',z- |//l
t"I
FIGURE 5.3.1. Schematic view of a building.
/J tliig. 5.3.1) may be written as
p(z):ipG.+ C)BU2(Z) (5.3.3)
rvlrt'rc p is the air density, c, and c7 are the values, averaged over the building
rvrtlth, of the mean pressure coefficient on the windward face and suction
, .t'llicient on the leeward face, respectively, and U(z) is the mean speed at
rlt'vir(it>n z in the undisturbed oncoming flow. Equation 5.3.1 then becomes
I
I
ll
I
r-.-
(s.3.4)
5"3-2 Fluctuating Response to Wind: Deflections and Accelerations
A:; irrtlicatcd in Chapter4, thc co-spcctrum of the pressures at point Mr, M2of
, ootlinatcs (yr, zr), (.y:, :.,), rcsl'rcctivcly, may be written as
Sl,ii,,t(n) ='S,11'(::,' rrl,f/,1'1;',, rt)('olt(.1',,.v.r, zr . z.z, n)N(n) (-5.3.-5)
,tlr,.'r'c ,S,1/'(l;, rr) is thc slrr't'ltirl rlt'rrrrly ol llrr' plt'ssrrrcs irl poinl M,(i - 1,2)
:rrrtl ('olt(.)r1,.)r-r, 11, 1.,. /l) :ttrrl N( l) ;ttr'lltr':rr'r'oss wintl lrrrtl llrc irlolrg-wintl
. r.rss coltclirliorr cocllit'it'rrl, rcrpr't lrvt'ly lly rlt'lrrrrrorr, il hotlr M, ir,td M, irrt'
_ I ^ _ \- l{ IJ2<zlx,(zt dzx(z) : ,p(c, + ct)B l\j4*; *,1r1
p(y,z,t) dA
210 stttu(;tt,liAl l)YNnMlo:i
on the same-windward or lccwatd--lirco ol' tlrc stl-ucrlutc, N(rr) = I . 'l'hc
expression for Sr(z; , n)