AlllDO TTEfifiB

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A11 100992883

NBS BUILDING SCIENCE SERIES 74

c. a.
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OP STANDA&Oa

.NOV 2 6 1975 The Buffeting of Tall Structures by Strong Winds^

<

' ;

. lJUJ:r\,^^. n^,

Emil Simiu

Institute for Applied Technology

National Bureau of Standards

r,o-f axx • TR 4 3 5

Center for Building Technology OO '"^

Washington, D.C. 20234 c

"^

and

Daniel W. Lozier

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

Front Cover: Standard Oil of Indiana Building, Chicago.

Courtesy of Standard Oil of Indiana. (Edward D. Stone, Perkins and Will, Architects; Perkins and Will, Structural Engineers; E. Alfred Picardi, Chief Engineer )

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secrefary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued October 1975
Library of Congress Cataloging in Publication Data

Simiu, Emil. The Buffeting of Tall Sliuctures by Strong Winds.

(National Bureau of Standards Building Science Series; 74) Includes bibliographical references. Supt. of Docs. No.: C 13.29:2/74.

1. Wind-pressure. 2. Structural dynamics. 3. Tall buildings.

4. Buildings—Aerodynamics. I. Lozier, Daniel W., joint author.

II. Title. III. Series: United States. National Bureau of Standards. Building Science Series; 74. TA435.U.S8 No. 74 [TA654.5I 690'.08s [690'.2] 7.S-.30727

National Bureau of Standards Building Science Series 74

Nat. Bur. Stand. (U.S.), Bldg. Sci. Ser. 74, 90 pages (Oct. 1975) CODEN: BSSNBV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1975

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SO Catalog No. C13.29:2/74). Price $1.55 (Add 25 percent additional for other than U.S. mailing).
The Buffeting of Tall Structures by Strong Winds

Emil Simiu and Daniel W. Lozier

Certain shortcomings of current procedures for computing alongwind structural response have

been shown to result in unrealistic estimates of tall building behavior under the action of

strong winds. Differences between predictions of fluctuating response based on various such

procedures may be as high as 200%. In recent years, advances in the state of the art have

been made which provide a basis for significantly improved alongwind response predictions.

The purpose of the present work is to present a procedure for calculating alongwind response

which incorporates and utilizes these advances. The basic structural, meteorological and

aerodynamic models employed are described, and expressions for the alongwind deflections

and accelerations, consistent with those models, are derived. A computer program is presented for calculating the alongwind response of structures with unusual modal shapes or

for which the contribution of the higher modes to the response is significant. For more

common situations, a simple procedure is presented which makes use of graphs and on the

basis of which rapid manual calculations of the alongwind deflections and accelerations can

be performed. Numerical examples are given to illustrate the use of the computer program and

of the graphs. Results of numerical calculations are used to discuss some of the approximations and errors inherent in the models employed.

KEY WORDS: Accelerations; buffeting; building codes; buildings; deflections; dynamic

response; gust factors; structural engineering; wind engineering; wind loads.

ACKNOWLEDGMENTS

The writers wish to acknowledge useful comments and suggestions by R. D. Marshall,

R. A. Crist and R. A. Grot of the Center for Building Technology. They also wish to

express their appreciation to S. Haber of the Applied Mathematics Division for his advice

on the numerical techniques employed in the computer program.

iii

TABLE OF CONTENTS

Page

Abstract

Acknowledgments

List of Symbols

List of Figures -^^ List of Tables ^

1. Introduction ]^ 2. Structural, Meteorological and Aerodynamic Models. Alongwind Deflections and

Accelerations 2 2.1 Structural Behavior 2 2.2 The Atmospheric Boundary Layer 3 2.2.1 Horizontally Homogeneous Flow 3 2.2.2 Mean Wind Profiles 4 2.2.3 Relation between Wind .Speeds in Different Roughness Regimes .... 6 2.2.4 The Power Law Model 7 2.2.5 Longitudinal Velocity Fluctuations , 8 2.3 Mean and Fluctuating Pressures

2.3.1 Relation between Wind Pressures and Wind Speeds 10 2.3.2 Cross-Spectra of Fluctuating Pressures 12 2.3.3 Results of Pressure Measurements 14 2.4 Alongwind Deflections and Accelerations 16 2.4.1 Mean Response 18 2.4.2 Fluctuating Response: Deflections and Accelerations 19

3. Computer Program for Calculating Alongwind Deflections -and Accelerations. ... 20 3.1 Description of the Computer Program 20 3.2 Integration Procedure 23

4. Simplified Procedure for Calculating Alongwind Deflections and Accelerations. . 25 4.1 Introduction 25 4.2 Basic Assumptions 26 4.3 Total Fluctuating Response as a Sum of Background and Resonant

Contributions 26 4.4 Alongwind Deflections and Accelerations 29 4.4.1 Mean Response 29

4.4.2 Fluctuating Response: Deflections and Accelerations 30 4.5 Numerical Example 40

4.6 Comparison between Gust Response Factors Calculated Using Various

Current Procedures 45

5. Approximations and Errors in the Estimation of the Alongwind Response 54

5.1 Contribution of Higher Vibration Modes to the Response 54

5.2 Influence upon Calculated Response of the Deviation from a Straight

Line of Fundamental Modal Shape 56

5.3 Influence upon Calculated Response of Errors in the Estimation of the

Roughness Length 56

5.4 Spectra in the Lower Frequency Range and Alongwind Response 56

5.5 Acrosswind Correlation of the Pressures and Alongwind Response 58

6. Conclusions

_ ^-^ References ^2

APPENDIX: Computer Program Listing. Sample Input and Output 67

V
LIST OF SYMBOLS

a, a', a = Mean deflection, fluctuating deflection, acceleration of structure,

respectively.

a , a = Maximum probable deflection, maximum probable acceleration, respectively.

ni3. X m3. X A = Area

A^, Aj^ = Area of windward, of leeward side, respectively.

B = Building width

f) .= Function defined by Eqs. 4.19, 4.27.

C„ = Drag coefficient (C^ = C + C„)

D D w £ C = Mean pressure coefficient

P

C^,Cj^ = Mean pressure coefficient on windward, on leeward side, respectively.

C, ,C', C,„,C![ = Pressure coefficients defined by Eqs. 2.27a, 2.27b.

Iw w 1£ I

/ -I »

C ,C = Exponential decay coefficients (Eq. 2.18)

y z D - = Depth (alongwind dimension) of building

f,f^,fj^ = Similarity (Monin) coordinate (Eq. 2.12), value of f beyond which Eq.

2.11 holds, peak similarity coordinate, respectively.

= Nondimensionalized frequencies (Eq. 2.55 and 2.48)

Fp (t) = Force acting at point

1 2 Acceleration of gravity = 9.81 m/s

g (2),g..(z) = Peak deflection, peak acceleration factor, respectively (Eqs. 2.36,

^ ^ 2.40).

G(z), .= Functions defined by Eqs. 2.43, 2.45.

G.F. = Gust factor (ratio of maximum probable deflection to mean deflection).

H = Building height.

H(z,P^,n) = Mechanical admittance.

H {z,P^,n) = Complex conjugate of H(z,P^,n)

I , = Function defined by Eq. 2.54 rrL ^

J = Function defined by Eq. 4.17

= Function defined by Eq. 2.53

L = Turbulence scale

XM = Point of ordinate z

m = Mass of building per unit height

vi
= Generalized mass (Eq.'2.4)

Sl^ = Nondimensional generalized mass (Eq. 2.44)

n,n^ = Frequency, natural frequency in the rth mode, respectively

N^(Pj^,P2,n) = Alongwind velocity cross-correlation coefficient

N{P ^,P^,T\) = Alongwind pressure cross-correlation coefficient

p ,p ,p ' ,pj_ , p^ = Pressure, mean pressure, fluctuating pressure, fluctuating pressure

on the windward, on the leeward side, respectively

r ,r„ = Ratios defined by Eqs. 2.28a, 2.28b

r.,11r. = Ratios defined in Section 5.5

P. = Point of coordinates y.,z.

1 -^11

R ,R = Square root of acrosswind coherence function for longitudinal velo-

" ^ city fluctuations for pressure fluctuations, respectively.

jt = Function defined by Eqs. 4.20, 4.28

S ,S-.,Sp ,S ,S = Spectral density of fluctuating response, of acceleration, of force,

^ ^ P P " of pressure, of longitudinal velocity fluctuations, respectively

T,t = Time

u - Longitudinal velocity fluctuation

U = Mean wind speed

= Friction velocity

U = Nondimensionalized mean wind speed (Eq. 2.46)

= Fastest mile wind

w = Weight of building per unit volume

y = Horizontal coordinate in plane perpendicular to wind direction

Y = y/B (Nondimensionalized coordinate)

Y = Function defined by Eq. 2.57

rr

z = Height above ground

Z = z/H (Nondimensionalized coordinate)

= Roughness length

z^ = Zero plane displacement

a = Power law exponent (Eq. 2.9)

3 = Square of ratio between r.m.s of velocity fluctuation and friction

velocity (Eq. 2.13)

Y

= Constant defining modal shape in Eq. 5.2

vii
= Damping ratio in rth mode

Vi^(z) = Modal shape in r-th mode

u)^ = Circular frequency in r-th mode (Eq. 2.3)

p = Air density = 1.25 kg/m^

V ,v.. = Quantities defined by Eqs. 2.37, 2.41, 4.22, 4.24

= Quantities defined by Eq. 2.26b, Eq. 4.30, respectively.

viii
LIST OF FIGURES

2.1

2.2

2.3a,b

4. la,b,c

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

5.1

Page

Flow downwind of a surface roughness discontinuity (after Peterson [17]). . . 5

Ratios u^/u^j 6,31

Spectra of longitudinal velocity fluctuations and spectra of fluctuating

pressures at stagnation point [34 J

White noise spectrum, square of modulus of mechanical admittance, decaying

spectral curve

Function 3f

Function S>

Function ^,

Function

Function ^,

Function

Function

Function &

Function ^,

Function

Function

Function

Function &

Function

Function

Function ^,

Function ^

Modal Shapes

10, C 16

B/H = 0, C = 10, C

' z • y

16

B/H = 0.167, C = 10, C =16

z y B/H = 0.4, C = 10, C 16

z y B/H = 1.0, C = 10, C =16

z y B/H = 3,0, C = 10, C =16

z y

C = 6.3, C

B/H = 0, C

10

6.3, C = 10 y B/H = 0.4, C = 6.3, C = 10

z y B/H = 1.0,

B/H = 3.0, C

C = 4.0, C

6.3, C = 10

y 6.3, C = 10 y

6.4

B/H = 0, C = 4.0, C 6.4

' z y B/H = 0.4, C = 4.0, C = 6.4

' z y B/H = 1.0, C = 4.0, C = 6.4

z y

B/H = 3.0. C = 4.0, C = 6.4

z y

13

27

33

34

35

36

37

38

39

45

46

47

48

49

50

51

52

53

55

ix
LIST OF TABLES

Page

Suggested Values of for Various Types of Exposure 4

Approximate Ratios of Probable Maximum Speed Averaged over Period t

to that Averaged over One Hour (in Open Terrain) 5

Estimated Ratios of Design Pressures to Measured Pressures near the

Stagnation Point for Various Buildings

32 Values of , , ' *

'

^o ^d 'V 32 Values of —, — Corresponding to Various Y Curves

H H 11 54 Description of Buildings Selected as Case Studies '

"

Percent Contribution of Higher Modes of Vibration to Root Mean Square

of Fluctuating Response

Deflections and Accelerations for Various Roughness Lengths • • • -^^

Ratios [a 1^ / la r,7 max^f^ max^O.03

Ratios of Response for Various Values C^, C^ to Response Calculated

Using C = 10, C =16 60

^ z y

X
Chapter 1. INTRODUCTION

Following an increasing recognition of the importance in tall building design of the

dynamic effects due to the gustiness of wind, several procedures for computing alongwind

response have been proposed in the last decade [1, 2, 3], The purpose of these procedures

is to calculate equivalent static wind loads whose effect upon the structure is the same as that of the gusty wind.

As part of an effort aimed at evaluating and improving building code provisions on design for wind loads, an analysis of current procedures for computing alongwind structural

response - including the procedures described in the American National Standard A58.1 [4] and in the National Building Code of Canada [5, 6] - ha? been presented in Ref . 7. From

this analysis, the following points emerged. First, the alongwind cross-correlation of the

fluctuating pressures is represented in current procedures by models which are fundamentally

inadequate: in Ref. 6 is is assumed that pressures on the windward and leeward faces are perfectly correlated fo each other, whereas in Ref. 4 the reduction factor expressing the

alongwind correlation of these pressures, rather than being applied to their cross -spectrum

alone, is in effect applied to the entire response [8]. It is primarily for this reason that

the differences between the values of the fluctuating part of the response calculated in

accordance with the A58.1 Standard, on the one hand, and the National Building Code of

Canada, on the other hand, may in certain cases be as high as 200% [7] . Second, current procedures do not take into account the dependence of the longitudinal wind velocity

spectra upon height above ground. This was shown to result in significant overestimates

of the components of the fluctuating velocity which produce resonant excitation in windsensitive structures [7, 9]. Third, current procedures do not enable the analyst to estimate the contribution of the higher vibration modes to the structural response. Fourth,

most of the existing procedures do not provide an estimate of the alongwind accelerations

induced in structures by gusty winds

.

In the years following the development of the procedures described in Refs. 1, 2, 3, 4 and 6, developments of significance in the context of determining structural response to wind have taken place notably in the area of boundary layer meteorology. Progress has also

been made in understanding and evaluating the effect of the alongwind pressure cross- correlations upon building response. Finally, efficient numerical methods have been developed, from which economical computer programs, suitable for the analysis of structures

subjected to wind loads, could evolve. The purpose of this paper is to present a procedure

which incorporates and utilizes these advances and thus results in improved estimates of

alongwind response.

In the following chapters, the basic structural, meteorological and aerodynamic models

employed in this work are discussed, and expressions for the alongwind deflections and

accelerations, consistent with these models, are presented. An efficient procedure for

carrying out the required integrations, and the main features of the computer program

developed for the calculation of the alongwind response, are described. A complete listing

of the computer program, and a sample input and output for the program, are included in an Appendix. — Figures in brackets indicate the literature references at the end of the paper.

1
The use of the computer program is required in the case of structures with unusual

modal shapes or for which the contribution of the higher modes to the response is significant. For more common situations, in which the fundamental modal shape may be approximated

by a straight line and the contribution of the second and higher vibration modes may be

neglected, a simple procedure is presented, which makes use of graphs and on the basis of

which rapid manual calculations of the alongwind deflections and accelerations can be performed. Numerical examples are given to illustrate the use of the computer program and of

the graphs. Results of numerical calculations are then used to discuss some of the approximations and errors inherent in the models employed.

Chapter 2. STRUCTURAL, METEOROLOGICAL AND AERODYNAMIC MODELS.

ALONGWIND DEFLECTIONS AND ACCELERATIONS.

This chapter presents a description of the structural, meteorological and aerodynamic

models required for the calculation of the alongwina structural response. Questions related

to the definition of the wind climate for engineering design purposes are beyond the scope

of this work and will therefore not be discussed herein.

2 . 1 Structural Behavior

The structure is assumed to be 'linear and elastic. Its deformation may then be expressed as a sum of products of modal shapes and generalized coordinates. To each of the r vibration modes, where r = 1, 2, 3, ... there corresponds a modal shape y^(z), where z = height above ground, a natural frequency n^ and a mechanical damping ratio n^. The deforma

tion y(z,t) at point M (or ordinate z) of the structure under the action of a force F(_?^,t)

acting at point (or ordinate z^ can then be written as

y(z,t) = Ey^(z)y^(t) (2.1)

in which the generalized coordinates y^(t) are the solutions of the equations

y^(t) 2a)^n^y(t) + cojy^(t) = i_ (z^) F (P^ .t) (2.2) r

where O) = 2iTn (2 .3) r r ^ '

= (z)m(z)dz (2.4)

and m(z) = mass of structure per unit of height.

Methods for calculating the modal shapes and the natural frequencies of structures are described, for example, in Ref. 10. Suggested values for the mechanical damping ratios for

steel frames and reinforced concrete frames are 0.01 and 0.02, respectively [4,6]. Lower values of the mechanical damping ratios may have to be used, for example in the case of

welded steel stacks or of certain prestressed concrete structures or of structures of the

2
framed tube type [6, 62]. In addition to the mechanical damping, the aerodynamic damping

may, in principle, also be taken into account. The aerodynamic damping is associated with

changes in the relative velocity of the air with respect to the building as the latter

oscillates about its mean deformed position (see Ref . 32) . It appears that it may be

unconservative to rely upon the effect of the aerodynamic damping; for this reason, the

latter is not taken into account in Refs. 4,6 and will also be neglected in this work.

2 . 2 The Atmospheric Boundary Layer

2.2.1 Horizontally Homogeneous Flow

For the purpose of estimating alongwind response it is useful and convenient to consider the model of , a horizontally homogenous boundary layer flow. Implicit in this model are the following assumptions:

1. At a sufficiently large height above ground, where the effects of the ground

friction become negligibly small (i.e., at the level of the so-called gradient

wind), the flow is horizontally homogeneous.

2. The terrain is horizontal.

3. The roughness of the terrain is uniform over a sufficiently large fetch.

The first of these assumptions may be applied in the case of large-scale, extratropical

storms, but does not, strictly speaking, hold in the case of mature hurricanes or severe

local storms such as thunderstorms. Indeed, in mature hurricanes, in the region of highest

winds the curvature of the isobars is relatively large and the wind speeds depend upon

distance from the center of the storm. It is presumably for these reasons that the hurricane mean wind profiles implicit in the Southern Building Code [11) (non-dimensionalized

with respect to the mean speed at 10m above ground) differ markedly - on the unconservative

side - from profiles associated with other types of storms. Recent research of a preliminary nature suggests, however, that in the lowest few hundred meters of the atmosphere,

the differences between profiles corresponding to mature hurricanes, on the one hand, and to extratropical storms (i.e., to horizontally homogeneous flow), on the other hand, are relatively small and may be neglected in structural engineering applications [12].

Thunderstorm winds are caused by strong, localized downdrafts which spread over the

ground in the manner of a wall jet [57] . The intensity of thunderstorm winds is therefore

highly dependent upon distance from the center of the downdraft. Since the model which

best describes the thunderstorm wind flow is of the wall jet, rather than of the parallel

boundary layer type, it is likely that differences exist between mean wind profiles typical

of thunderstorms, on the one hand, and of large-scale extratropical storms, on the other

hand. It is also noted in this connection that the notion of gradient wind has no meaning

in the case of thunderstorms. Nevertheless, in current building codes, it is tacitly

assumed that the wind structure may be considered to be the same in thunderstorms as in large-scale, extratropical storms. The question as to whether this assumption is acceptable

from an engineering viewpoint is one to which no clear answers, are currently available and

which merits careful research. Also, little information is available regarding the structure

of the flow in non-horizontal terrain. Research into this question has been recently re- ported by Sacr^ [14]

.

3
The effect upon the flow of a change in roughness of terrain has been investigated by

various writers, among whom Panofsky and Townsend [151, Taylor [161 and Peterson [171;

however, the results obtained so far do by no means appear to be definitive. A tentative

result derived by Peterson [17] will be presented subsequently herein.

2.2.2 Mean Wind Profiles

It can be shown, on the basis of both theoretical and experimental results, that for

high wind speeds the mean wind profile in the boundary layer of horizontally homogeneous

flows may be described throughout the height range of interest to the structural engineer,

by the relation [18,19,20,21,22]:

U(z) = 2.5 u. £n (2.5)

where U(z) = mean speed at height z above ground, z^ = zero plane displacement, z^ = roughness length and u^ = friction velocity, given by

U(z^)

2.5 S,n Vd (2.6)

in which z is any given reference height. In meteorological work, the standard reference

K height is z = 10m. Eq. 2.5 is commonly referred to as the logarithmic law. Suggested

R values of the roughness length z^ are given in Table 2.1 (see Refs. 20,22,24). For example,

at Sale, Australia, for terrain described as open grassland with few trees, at Cardington,

England, for open farmland broken by a few trees and hedgerows, and at Heathrow Airport in London, z^ - 0.08m [20,22,58,59]. At Cranfield, England, where the ground upwind of the

anemometer is open for a distance of half a mile across the corner of an airfield, and

where neighboring land is broken by small hedged fields, z^ = 0.095m [20,60].

Table 2.1 - Suggested Values of z^ for

Various Types of Exposure

Type

of Exposure Coastal^ Open

Outskirts of

Towns, Suburbs

Center of Towns Centers of

Large Cities

z^ (meters)

0.005-

0.01

0.03-

0.10

0.20-0.30 0.35-0. A5 0.60-0.80

Applicable to structures directly exposed to winds blowing from open water.

The zero plane displacement may in all cases be assumed to be zero, except that in centers of large cities the smaller of the values z^ = 20m and z^ = 0.75h, where h = average height

of buildings in the surrounding area, may be used.

It has been estimated [19,20] and verified experimentally [21] that the logarithmic law

is valid up to a height

z, - 0.03

^
where f is the Coriolis parameter. In terrain for which = 0.07m and U(10) = 26.8 m/s

(hourly mean), u. = 2.16 m/s and, with f - lO'^s"^, z, - 650 m. log . According to the National Building Code of Canada [6] the assumption that the exposure

is suburban or urban may not be used in design unless the appropriate terrain roughness

persists in the upwind direction for at least one mile. In the ANSI A58.1 Standard [4] it

is suggested that the type of exposure corresponding to. centers of large cities may be

assumed only if the terrain is heavily built-up for at least one-half mile upwind of the

structure, with at least 50% of the buildings being in excess of six stories. It can be seen that these criteria do not contain any provisions relating height above ground to length of fetch required for the assumption of horizontal homogeneity to hold. In this

connection, Peterson [17] suggests that the flow downwind of a discontinuity can be divided

into three zones, as shown in Fig. 2.1. In zone 1, the velocity is essentially equal to the velocity upwind from the discontinuity. In zone III it may be assumed that the flow is adjusted to the new roughness conditions, i.e., is determined by the same parameters z^^j

u^2 that would control the flow if the roughness length were everywhere z^^- I" zone II, the flow varies gradually from line AB, where it is presumably nearly the same as upwind of

the discontinuity, to line AC, where it may be described in terms of the parameters z^^j

u^2- Peterson's results were obtained on the basis of theoretical considerations alone and

have not been verified experimentally. Further research into the question of the flow

transition due to a change of terrain roughness is therefore believed to be necessary.

Fig. 2.1 - Flow downwind of a surface roughness discontinuity (after Peterson [17])

In designing tall buildings it is reasonable to use mean wind speeds averaged over a period of one hour [4,6]. If the mean winds are averaged over periods t different from

one hour, the mean winds U*^ averaged over one hour may be obtained using Table 2.2 [56].

Table 2.2. Approximate Ratios of Probable Maximum Speed

Averaged over Period t to that Averaged over One Hour (at 10m Above Ground in Open Terrain)

t 2 5 10 30 60 lOD 200. 500 1000 3600

(Seconds)

1 .53 1.47 1.42 1.28 1.24 1.18 1.13 1.07 1.03 1.00

5
For values of t not included in Table 2.2, linear interpolation is permissible. If the

wind speeds are given in terms of fastest miles, the averaging period t in seconds is given

by t = 3600/v^, where v^ is the fastest mile speed in miles per hour.

2.2.3 Relation between Wind Speeds in Different Roughness Regimes

Consider two adjacent terrains, each of sufficiently large fetch, the roughness lengths

of which are denoted z^^, z^, respectively, and over which the mean wind speeds may be

described, respectively, by Eqs . 2.7a and 2.7b:

= 2.5 u^^ in

"dl (2.7a)

"ol

U = 2.5 u^ £n (2.7b)

It can be shown that the relation between the friction velocities u^^, u^ as

2 *1 2 A +(£n —- - C)

"*1 .2 ..2

can be written

(2.8)

C)

in which f is the Coriolis parameter and A, C are meteorological constants determined

experimentally [18,20]. For the purpose of calculating the ratios u^^/u^^, it may be

assumed A = 4.5, C ='0; the error involved if these values are used can be shown to be

negligibly small [20]. Since the dependence of Eq. 2.8 upon u^^^ and f is very weak, the

ratio u^^/'J*]^ ""^X determined, for practical purposes, simply as a function of the roughness lengths z^^, z^^' shown in Fig. 2.2. The following numerical example, in which the

data are excerpted from Refs. 20, 22, illustrates the use of Fig. 2.2.

1.50

1.40

1:30

1.20

I.IO

1^0=0

Zq- 002m.

DOSmv^

.OIm>

008

0.20

12m

m

•t 40m 1 0.2 0.5 10 1.5 2.0 ROUGHNESS LENGTH IN METERS

Fig. 2.2 - Ratios u^/u^^

6
At Heathrow airport (near London) z^^ = 0.08m, z^^ = 0, and the measured wind speed at 10m above ground was 1)^^(10) = 11.7m/s. Using Eq. 2.7a,

"*i = ^^'^^n = ""/^ ^"08

At the Post Office Tower (in the center of London) z^ = 0.78m and z^ = 21m. From Fig.

2.2, u^/u^^ = 1.17, i.e., u^ = 1.13 m/s. The calculated value of the wind speed at z = 195m at the Post Office Tower is then,

195-21

U(195) = 2.5 X 1.13 in

q jl

= 15.3 m/s

which coincides with the measured wind speed reported in Ref. 22. It is also possible to obtain from Fig. 2.2 ratios u^/u^ for the case, z > z . Thus, let z , = 0.08m and z = 0.01m. Let z ^ = 0.20m. Then, for Fig. 2.2, u^^/u, = ol o o2 » o > *2' * 1.20, u^2/"*i = 0.06. It follows that u^/u^^ = 1.06/1.20 - 0.88.

2.2.4 The Power Law Model

Historically, the first representation of the mean wind profile in horizontally homogeneous terrain has been the so-called power law:

U(z) = U(z )(f-)'^ (2.9)

R z^

in which Zj^ is any given reference height and a is an exponent dependent upon the roughness

of the terrain.

To calculate the relation between wind speeds over terrains with different exposures,

an empirical model has been proposed according to which there corresponds to each of three

standard exposures a given exponent in Eq. 2.9 and a given gradient height, independent of

wind speed [1,2,3,4,6]. Criticisms of this model by various authors have been summarized

in Ref. 20. On the basis of recent experimental and theoretical atmospheric boundary layer research,

it is now widely recognized that for high winds the logarithmic law is a superior representation of mean wind profiles in the lowest few hundred meters of the atmosphere [19,20,21,

22,23,24]. It is therefore the logarithmic law that will be used in the calculations performed herein. The logarithmic law has, in addition, the advantage of being consistent

with the expression of the longitudinal velocity spectra. Indeed, as will be shown subsequently, the spectra are defined in terms of the friction velocity u^ and are thus implicit

functions of z^, z^. Thus, if the power law is used, it is necessary that some approximate

relation between the parameter, a, describing the mean wind profile, and the parameters z^,

z^, characterizing the wind spectra, be assumed, since no unique relation between a and

z^, z^, independent of z, z^^, exists. On the other hand, no such approximations are re- quired - and the errors inherent in them are thus eliminated - if the mean profiles are described by the logarithmic law.

7
2.2.5 Longitudinal Velocity Fluctuations

In the methods for determining gust factors described in the ANSI A58.1 Standard [4] and in the National Building Code of Canada [6], the validity of the following expression

for the wind spectrum is assumed:

nS (z,n) 2 —2 - \ ' 2,4/3 (2.10)

u^ (1 + X )

'

in which n is the frequency in Hz, S (z,n) = spectrum of longitudinal wind fluctuations, x = " 2 l,200n/U(10) , U(10) = mean speed at z = 10 m. (Some authors use the notation tcU (lO+z.) =

2 2 u^ , where k = [0 . 4/2-n (10/z^) ] and z^ is expressed in meters.) According to Eq. 2.10,

S^(z,n) is independent of height. On the basis of both theoretical and experimental results

it has been established, however, that Eq. 2.10 does not reproduce correctly the dependence

on z in the higher frequency part of the spectrum (see, for example, Ref. 25, p. 399), and

that an excellent representation of this dependence is given by the relation [24,26,27,28]:

nS rz,n)

= 0.26f (2.11)

in which

f = n(z-z^)/U(2) (2.12)

is known as the Monin, or similarity, coordinate. For engineering purposes it may be

assumed conservatively that Eq. 2.11 is valid for f > 0.2 [26].

In the lower frequency range, similarity breaks down and the spectra are not capable

of being described by a universal relation [24,25,27]. A useful description may, however,

be obtained, if it is remembered that (1) the one dimensional spectrum approaches a nonzero value as the frequency approaches zero; (2) the product nS^(n) reaches a maximum at some value 0 < f < f^ (in which f^ is the value of f beyond which Eq. 2.11 is valid),

i.e., the derivative of nS^(n) with respect to f vanishes at f = f^^J (-5) the relation

u^ = f^S^in) dn = B uf (2.13)

in which u = mean square value of longitudinal velocity fluctuations, with 3 - 6.0 [1,3,4,

29] , holds.

The above requirements are satisfied by the curve

u^ (l+SOf)^'"*

which also approximates very closely (from a designer's viewpoint, on the slightly conservative side) the spectrum in the higher frequency range (Eq. 2.11), and may therefore be

used as a representation of the, entire spectrum. Eq. 2.14 differs from a similar expression

proposed by Kaimal et al [28] in that it satisfies Eq. 2.13 with 3 - 6.0, rather than 3

=

4.75 as in Ref. 28, and yields therefore more conservative estimates of the dynamic response
In Eq. 2.14, f^^ = 0.03 . In reality, the spectra in the low frequency range, and

therefore the peak similarity coordinate f , appear to vary strongly [27], indeed quite

erratically [25], between sites and between atmosphere and laboratory. According to Owen

[25] , this unpredictable variation may be caused by meso-scale phenomena. Also, f^ varies

with height. According to available results of measurements [27,30], for neutral stratification--which prevails in strong winds--f^ - 0.02-0.03 at z = 3-6m, f^ = 0.025-0.04 at z = 15/20m, f^ - 0.04-0.08 at z = 30-60m, f ^ = 0 . 1 at z = 90 m. The question therefore arises

as to whether the assumption f^ = 0.03 , which is implicit in Eq. 2.14, might not be the source of significant errors in the estimates of structural response. To answer this question, it is necessary to study the influence upon response of the variation of f^^. For this purpose, an alternative expression of the longitudinal spectra was developed

which is consistent with the requirements previously described, but in which f^ is a parameter that may be varied, rather than being a given constant as in Eq. 2.14. It is con- venient, and as will be shown subsequently, sufficiently accurate for engineering purposes,

to represent the spectra in the low frequency range as,

nS (z,n) c, + a,f + b,f^ 0 < f < f, (2.15) u _ 1 1 1 — 1

ul + a^f + b^f^ f^ < f < f^ (2.16)

The coefficients in Eqs. 2.15 and 2.16 result from the three conditions stated just

before Eq. 2.13, and from the condition that Eqs. 2.15, 2.16, and 2.11 form a curve con- tinuous at f = f, and f = f . Thus,

1 s ^1 = 0 ^ = 2b^f^

-2 - '^2^1

c (f^-2fp (B-B^) - 2/3 B^(f^/2 - 2f^)

2 ? (3/2 + In^) (fg-2fj) - (f^/2-2fp

^ ^1 ' ^2 (for f ^ 2f ) °2 f (f -2fJ ^

s s 1

2/3B. (3/2 + In 2) - (B - BJ

b2 = 2 ^ i- (forf^ = 2f^)

s

in which ^ = ^2 - ''2/^1

0.39 f-2/^

s

9
To summarize: the spectra of the longitudinal velocity fluctuations may be described

by Eq. 2.14, in which case it is implied that f^^ = 0.03 , or , alternatively, by Eqs. 2.15,

2.16, 2.11, in which case the values of f^^ and f^ may be chosen as design parameters. It is

reasonable (i.e., slightly conservative) to assume f^ = 0.2. Then, to assess the influence

of the variation of f^ upon the structural response, calculations may be carried out based

on values of within the range 0 < f^ < 0.2.

The cross-spectrum of the longitudinal fluctuating velocities at two points P^^, may

be expressed in the form:

S^(P^,P2,n) = Sy^(P^,n)Sy^(P2.n)R^(y^,y2,z^,Z2,n)N^(P^,P2,n) (2.17)

in which y^>^^ the coordinates of point P^ (i = 1 2) in a plane perpendicular to the

direction of the mean wind. The functions and are defined as the acrosswind and the

alongwind cross-correlation coefficient, respectively. By definition, if the points P^, P^ are in the same plane perpendicular to the mean wind direction, '^^ (Pj^ '"-^ ^ ^' Otherwise,

N^(Pj^,P2,n) < 1; its magnitude decreases as the alongwind separation between P^^, P^ increases

and as the frequency n increases. The acrosswind cross-correlation coefficient is a complex

quantity, the modulus of which is known as the square root of the acrosswind coherence

function [31] . However, its imaginary part has been determined by measurements to be

negligible for wind engineering purposes [1,2,3,32]. The following expression for the

acrosswind cross-correlation coefficient has been proposed by Davenport [3]

:

-2n[C^(z^-z^)^.C^(y^-yp^]l/^ (2.18)

R^(y^.z^,y2.Z2.n) = exp

u(z^) . U{z^)

On the basis of wind tunnel measurements, Vickery [3] suggested that it is reasonable to assume in engineering calculations C = 10, C =16. Expressions for the acrosswind crossz y correlation equivalent in practice to Eq. 2.18 with C = 10, C = 16 are presently used in

z y building codes. It appears, however, that the exponential decay coefficients depend on surface roughness conditions [13], on height above ground and, quite strongly, on wind speed

[63,64]. In open terrain, may decrease by a factor of three if U(10) decreases from

30m/s to lOm/s. C was also found to decrease as U(10) decreases, although less than C . 2 y also decreases as the height above ground increases. For example, for = 110m,

z- = 150m, measured values of C were about one half as much as for z, = 30m, z- = 80m

(see

2 Table 1 and Figs. 4 and 9 of

z Ref. 64). The dependence of the exponential

12

decay

coefficients upon terrain roughness, height above ground and wind speed is insufficiently

documented and therefore represents a source of uncertainty in engineering calculations.

Additional research in this area - both theoretical and experimental - is therefore believed

to be necessary.

2.3 Mean and Fluctuating Pressures

2.3.1 Relation between Wind Pressures and Wind Speeds

The pressure acting at a point P of elevation z on the surface of a building immersed

in a flow which has a steady velocity U(z) may be expressed as p(P) = l/2pCp(P) U^(z) (2.19)

10
In this expression U(z) = undisturbed mean velocity of the flow, i.e., its velocity at a sufficiently large distance upwind from the building, and is a dimensionless mean pressure coefficient. For blunt bodies in turbulent flow, C^CP) must be determined by experiment .

In the case of an unsteady velocity, the pressure may be expressed as follows:

p(P) = p(P) + p'(P) = l/2pCp(P)[U(z)+uCz)]^ + pC^(P)B(z)u(z,t) (2.20)

in which p(P) = mean pressure, p' (P) and u(z) = pressure and velocity fluctuations, C^{P) = added mass coefficient and B(z) = width of structure. Such an expression is reasonable

under the assumption that the transverse building dimension is small compared to the scale

of the energy containing eddies of the turbulence.

The question of the relative importance of the added mass term (the last term in Eq.

2.20) has been examined by Vickery and Kao [32]. On the basis of wind tunnel pressure

measurements [33], these writers showed that in determining pressures on bluff bodies in turbulent flow the adde4 mass term may be neglected. The same result was obtained in wind

tunnel tests reported by Bearman [34] and by Petty [35] . Neglecting the added mass term,

it follows from Eq. 2.20 and the definition of average values that

2 ~2

p' = 1/2 pC u2[2 ^ + " ' "

] (2.21)

, ^ U and

9 ~

p = l/2pC U^[l + ~ ] (2.22) P U -^—1/2

If u(z)/U(z) is small (i.e., u (z) /U(z) < 0.1) and the linear dimensions of the body

are small compared with the length characteristics of turbulence, the assumption that the

non-linear term in Eq. 2.21 may also be neglected is confirmed by experimental evidence [36],

1/2 In the atmosphere it may be assumed that u (z) /U(z) - 2 .45u^/2 . Su^ln [(z-z^)/z^], in

which u^ , z^, z^ = frictional velocity, zero plane displacement, roughness length, respectively (Eqs. 2.5 and 2.13). For tall buildings this ratio is of the order of 0.05 to 0.3,

depending upon building height and roughness of terrain. Also, except in the case of slender,

line-like structures, the ratio of building dimension to scale of turbulence is not neces- sarily very small. Questions may thus arise as to whether the non-linear term may be

neglected in the case of buildings with typical widths in atmospheric flow. Practical

difficulties have prevented so far the carrying out of simultaneous full-scale measurements

of p'(P), U(z) and u(z) . However, wind tunnel measurements have been performed [32,33] and

appear to confirm the assumption that Eq. 2.21 may be linearized. Also, the effect of the

non-linear term was analyzed in Ref. 37, according to which the contribution of this term to the fluctuating alongwind response of a 300m tall structure appears to be of the order of 5% (i.e., a contribution to the total alongwind response of about 3%). It thus appears that

the linearization of Eq. 2.21 is acceptable and that

11
p'(P) = pCpU(z)u(z)

p = 1/2 pC

(2.23a)

(2.23b)

This simplified model of the relation between alongwind fluctuating pressures and velocities

represented by Eq. 2.23a is used in all existing procedures for calculating dynamic alongwind

response [1, 2, 3, 4, 6]. Eq. 2.23a is applied in these procedures to express the pressures

on both the windward and the leeward sides of the building:

p^(P) = pC^U(z)u(z) (2.23c)

P^(P)

in which

C = w

C = (2.24b)

l/2pU^(z)

= pCj^U(z)u(z) (2.23d)

P (z) —W

-—~— (2.24a)

l/2pU^(z)

p^, p^ are the average values of the mean pressure at elevation z on the windward and the

leeward faces of the building, respectiv^ely . Attempts to verify by measurements the validity

of Eqs. 2.23, 2.24 are further discussed in Section 2.3.3.

It is implicit in Eq. 2.23d that the pressure fluctuations on the leeward side are proportional to the velocity fluctuations in the upstream flow. As noted in Ref. 8, in

reality the flow in the wake of the building is quite dissimilar from that in the oncoming

flow. Measurements suggest that the fluctuating pressures on the leeward side are small

compared to those on the windward side and, therefore, that Eq. 2.23d may be slightly

conservative [8,38,39].

2.3.2 Cross-Spectra of Fluctuating Pressures

If Eq. 2.23 is used,

Sp(P^,P2.n) = Cp(PpCp(P2)p^U(zpU(z^) S^(P^,P2,n) (2.25a)

in which the cross-spectral density of the velocity fluctuations can be written as

S (P.,P_,n) = S^'^^(P,,n) S^'^^(P^,n) R (P,,P„,n) N (P^,P„,n) (2,25b) u 1 2 u 1 u 2 u 1 2 u 1 2

and in which C (P.) = C or C„ according as P. (i = 1,2) is on the windward or leeward side,

p i w X, " 1

' Useful information regarding the magnitude of (Pj^ > ^2 provided by results of measurements of cross-spectra of pressures on the windward and leeward sides of structures. Such

measurements, carried out on full-scale buildings, have been reported by Lam Put [40] and by

12
Fig. 2.3 - Spectra of longitudinal velocity fluctuations and spectra of fluctuating pressures

at stagnation point: (a) L /B = 1.20; (b) L /B = 2.57 [34]

13
Van Koten [29] who found that except for very low frequencies,

^yj^^^iyj>^2l'^^

^ ^'^ ^^^^ subscripts w, i indicate that the points P^, are on the windward and leeward sides,

respectively; points p^^, P^^^^ and P^^, ^^'^^ coordinates y^, and y^ , z^, respectively).

From results of wind tunnel measurements reported by Kao (Figs. 5.1, 5.19, 5.21 of Ref. 33)

it also follows that, except for extremely low frequencies corresponding to eddies of negligible energy, N(P^^, ^21' nearly zero. More recently. Holmes [61] reported full-scale

measurements which suggest that it is reasonable to assume \^w' '2V =i-^ ^1 - ^"'^^ (2.26a)

where

5 = 3-85n Ax^2. 26b)

0 = 2.5 u^ (2-n ^-2- - 1) (2.26c)

In Eq. 25b, Ax is the smaller of the quantities 4H, 4B or 4D, where H, B, D are the height,

the width and the depth of the building, respectively [2].

The presence of a body in a turbulent flow produces, around that body, changes in the mean flow and boundary layer effects which result in a distortion of the oncoming turbulence

[34, 40, 42, 43, 44]. This distortion is not reflected by Eqs. 2.23, 2.25a, 2.25b. The

extent to which the turbulence distortion may affect the pressures on a body immersed in an isotropic turbulent flow has been investigated by Bearman for a two-dimensional bluff body

with a flat windward face [34], by Marshall for a circular disk [42] and by Petty for a circular cylinder [35]. These authors' results show that, at low frequencies, the pressure

spectra at the stagnation point are indeed proportional to the velocity spectra, i.e., may

be predicted by Bernoulli's equation, even for ratios L^/B of turbulence scale to transverse

body dimension considerably smaller than those typical of buildings in atmospheric flows (in

the atmosphere is of the order of a few hundred meters [31]). However, at higher frequencies, due to fluid strain effects, the pressure fluctuations are attenuated and the

pressure spectra drop off approximately 1.75 times as fast as the spectra of the oncoming

turbulence [34] (see Figure 2.3). It has also been observed that the intensity of high

frequency surface pressure fluctuations tends to increase with distance from the stagnation

point but that this increase is generally insufficient to offset the attenuation along the

stagnation streamline [42]. Thus, at higher frequencies, estimates of the pressure fluctuation components at the stagnation point based upon Bernoulli's equation appear to be con- servative. The differences between such estimates and the measured values decrease, however,

as the ration L^/B becomes larger. Bearman also suggests that the attenuated higher frequency components of the presssure fluctuations are better correlated than the corresponding

components of fluctuating velocities in the oncoming flow.

2.3.3 Results of Pressure Measurements

Measurements of pressures on the surface of a high-rise building model and of mean and

fluctuating velocities were carried out by Sadeh, Cermak and Hsi [44] in a shear flow simulating the atmospheric boundary layer. On the basis of these measurements (p. 50 and Fig.

4.8, p. 77, Ref. 33), the ratio

14
-1/2

-1/2

2p(P) uiz)

which is unity if Eq. 2.25a holds - was calculated along the centerline of the windward face

and found to have the values .97, 1.07, .85, 1.16, 1.16 at z/H = 0.45, 0.55, 0.65, 0.75 and

0.85, respectively, where z = height above ground and H = building height.

In the writers' opinion, research aimed at developing an improved model of the relation

between alongwind fluctuating pressures and velocities is clearly desirable. It appears,

however, that in the absence of such a model, Eqs. 2.23c, 2.23d may be used for engineering

purposes, the ensuing errors in estimating of the response being on the conservative side.

That this is the case is suggested by results of full-scale pressure measurements on buildings

reported in Refs. 46,47,48,49,50, as will be subsequently shown. For the windward and leeward sides of a tall building, the mean pressure coefficients

specified by the ASS. 1 Standard are = 0.8 and C^^ = -0.6, respectively. These values

appear to be confirmed by wind tunnel tests [44,51]. However, measurements suggest that

pressure coefficients for full-scale buildings are smaller than those obtained in wind

tunnel tests, i.e., are slightly decreasing functions of Reynolds number. It is therefore

of interest to compare pressures calculated using eqs. 2.23c, 2.23d (with = 0.8, =

-0.6) - which will be referred to herein as design pressures - to pressures estimated from

measurements reported in the literature. The measurement results are given in the form

[P (z)]

„ w ^meas , C = 2 27a)

l/2pU^(2^)

-1/2

2

- ^meas (2.27b)

w 2 l/2pU^(z ) d

where z is the anemometer height. It follows from Eq. 2.24a and the definitions of C^^, C^^

r

- ^Pw^'^^design ^ 0.8 ^2. 28a) w r

j-^-. U(z ) 2 '^w^ •^'measured C, ( ,. , -. ) Iw^U(z)

r ,2 — 1/2

r-

= ^design ^ 0. 8pU (z)u^ (z) 28b)

'^'^^w ^measured

with similar expressions for the quantities r , r' corresponding to the leeward side. To _ 1/2

estimate the ratios r', r , r' r„ it was assumed that u (z) /U(z) - l/£n[z-z,)/z ] and

w w £ )6 do

U(z )/U(z) = «.n[(z -z,)/z ]/2.n[(z-z )/z ] (see Eqs. 2.5 and 2.13). The roughness parameters

3. 3 d O Q O

15
used in the calculations were = 0.7 m, = 10 m for centers of large cities, z^ = 0.40m;

z^ = 0 for towns, z^ = 0.25 m, z^ = 0 for surburban areas. The results of the calculations,

which are given in Table 2.3, were quite insensitive to even large differences in the assumed

values of z^ and z^. To the extent that the measurements reported are reliable and that the

assumptions used in estimating the ratios r , r„ , r', r' are correct, the results of Table W X, w x. 2.3 suggest that the values of the pressures calculated in accordance with Eqs . 2.23c, 2.25d,

In which C = 0.8, = -0.6, are indeed conservative.

Table 2.3 - Estimated Ratios of Design Pressures to Measured Pressures near the Stagnation

Point for Various Buildings

Building

Height

Pressure

Transducer

Elevation

Anemometer

Elevation

Ref

.

Location Exposure meters Iw Cw

C

'

rw

r £

r'wr

I

50 London Large city 66 47 50 .42 . 14 .30 .08 1.80 4. 00 1 .25 3. 50

48 Kobe Town 107 75 120 .69 .50 .20 . 15 1.01 0. 84 1 .36 1. 22

49 Tokyo Large city 152 136 164 .80 . 13 .20 0.94 4. 20 1 .45

46 Tokyo Large city 120 95 124 .45 .40 .29 .14 1.57 1. 20 1 .00 1. 55

47 Monash U. Suburban 43 33 53 .74 .14 .40 1.57 1. 20 1 .00 1. 55

2 . 4 Alongwind Deflections and Accelerations

Consider a linearly elastic structure subjected to the action of a stationary random

force F (t) of known spectral density S (n) and applied at a point P . The spectral den1 P sity of the fluctuating deflection a' (M,t) at some point M (of ordinate z) of the structure

can be shoi-m to be [52] : S^(z,n) = H*(z,P^,n)H(z,P^,n)Sp (n) (2.29)

in which H(z,Pj^,n) is the mechanical admittance, i.e., the structural response divided by

i 2 Tin t

* e , and H (z,P^,n) is the complex conjugate of H(z,P^,n). From Eq. 2.2, in which F{P^,t)

_ ^i27Tnt^ 2.1, it follows,

y„(z)u (z ) H(z.P^,n) = E ^

(2.30)

,1 2,, n . n , 4iT M (1- —T-* in —

)

r 2 r n n r

r where is given by Eq. 2.4.

16
If the structure is acted upon by two stationary random forces F (t) and F (t) applied

1 2 at points . P2 > respectively, the spectral density of the response a'(M,t) depends not only

upon the spectral densities, but also on the cross-spectral densities of the two forces, i.e.

[52]:

S^(z,n) = H*(z,Ppn)H(n,P^,n)Sp (n) + H* (z , P2 .n)H(z , P2 ,n) Sp (n) + H^^.P^.n)

^1 . ^2 H(z,P n)S (n) + H*(z,P.,n)H(z,P n)S (n) (2.31)

P P P P

1 2 ^1 *^2

If a distributed stationary random loading is applied to an area A, Eq. 2.31 may be

generalized:

S^(z,n) = /^/^H*(z,P^,n)H(z,P2,n)Sp(P^,P2,n)dA^dA2 (2.32)

in which P,,P^ are the centers of elemental areas dA, , dA^; S (P,,P^,n) = cross-spectral 12 12 p l 2 density of pressures at points ^j^>P2' It can be shovm that if the damping is small and the peaks are well separated,

the imaginary part of the product H*CM,Pj,n) H(M,P^,n) is negligible [52], and

y^(z)yjz)M^(z Jyjz J

H (z,P. ,n)H(z,P^,n) = Z E ^

1' ^ ^ ' 2' ^

,^ 4,, 2 2 r s 16Tr M M n n

r s r s

(1-n^/n^) {l-n^/nh + 4n n (n/n ) (n/n ) (2.33)

r 2, 2,2 , 2 2 , 2-, r,-, 2 , 2,2 ,22,2, [(1-n /n^) + 4n^n /n^][(l-n /n^) + 4r]^n /n^]

The cross-spectrum Sp(Pj^,P2,n) used in Eq. 2.32 is described by Eqs . 2.25a, 2.25b and 2.18.

It was indicated in Section 2.2.5 that, by definition, if P^>P2 both on either the windward or the leeward side of a building N(P^,P2,n) = 1. In view of the results reported in

Refs. 33, 38, 39, 40, 61 and quoted in Section 2.3.2, if P^ , P^ are points on opposite sides of

a building, it is reasonable and conservative to assume

N(P^,P2,n) =1 0 < n < n^ (2.34)

N(P^,P2.n) E N^(P^^.P^^,np n^ < n < » (2.35)

where Nu (P , P^„, n.) is given by Eqs. 2.25 with n = n , ana n is a sufficiently large

IW 2Ji I id

frequency which may be assumed to be equal to, say, 0.9n^ (see also Section 4.3).

The peak factor for the deflections, i.e., ratio between the maximum probable value and

the root mean square of the fluctuating deflections, may be written, approximately, as [1]:

g (z) = (2 Inv (z)T)^^^ + 0.577/(2 Inv (z)T)^''^ (2.36)

3 3- 3.

17
in which T = duration of wind loading, assumed to be one hour (3,600 sec) and

v^(z) = /^n\(z,n)/a'^z) (2.37)

The spectral density and the mean square value of the alongwind acceleration of the

building may be written as:

S..(z.n) = (2TM)\(z,n) (2.38)

d a

..2 a> a (z) = /^S^.(z,n)dn (2.39)

The peak factor for the acceleration may be written as:

g..(z) = (2 lnv..(z)T)^/^ + 0.577/(2 lnv..(z)T) (2.40)

a a a

in which.

v..(z) = /°°n^S..(z,n)/a^(z) (2-41)

a o a

If, as is typically the case for modern tall buildings, the intervals between natural

frequencies in successive vibration modes are sufficiently large and the damping is light,

the terms involving cross-products may be neglected [52], and the following relations are obtained

:

2.4.1 Mean Response

in which

a(z) = %(C^ + C^) pBH^G(z)/4TT^m (2.42)

% a, G(z) = |[y(z)G^/M^f^] (2.43)

'^12- M = / y (Z)dZ (2.44)

r o r

1^2

= rU^(Z)u^(Z)dZ (2.45)

U(Z) = U(Z)/u^ (2.46)

Z = z/H (2.47)

f = nrH/u^ (2.48)

r

*

18
19
Chapter 3. COMPUTER PROGRAM FOR CALCULATING ALONGWIND DEFLECTIONS AND ACCELERATIONS

3. 1 Description of the Computer Program

A Fortran program has been written to compute the integral

1^^.^^ (Eq. 2.54) and the mean deflection response (Eq. 2.42), the fluctuating deflection and acceleration (Eqs . 2.49 and

2.51), and the peak factors (Eqs. 2.36 and 2.40). . The program consists of a main program,

called MAIN, and eight subprograms, called INPUT, INIT, TRIPLE, F, UTILDA, XMU, STILDA and

FISTAR. Listings and a sample run appear in the Appendix. The program may be obtained on magnetic tape from the National Technical Information Service, Springfield, Va. 22151. MAIN contains the quadrature method for the integral I , . The integrand for I , , rrL rrL which is itself a triple integral, is computed by the subprogram TRIPLE. MAIN also contains

the coding for the mean and fluctuating responses, and the peak factors. MAIN calls the

subroutines INPUT and INIT, which control the reading of input parameters from cards and the

setting of parameters for the numerical integration, respectively. After INPUT and INIT have been called, MAIN completes the calculation with the aid of the other subroutines, and

the cycle is begun again with another call to INPUT. If there is no more input, a blank

card will cause a normal program exit.

The card deck read by INPUT has the following form:

Card 1, format (312)

IREP Selects one of two spectral representations of longitudinal wind fluctuations. Representation 1 corresponds to Eq. 2.14. Representation 2 is

given by Eqs. 2.15 and 2.16 and is devised to study dependence of the

solution on the peak similarity coordinate

IPRINT Selects one of two options for output printing by the main program. As the approximate integration of 1^,^,^^ progresses from ^ = 0 to ^' = some cutoff value, option 1 causes running information to be printed at each

sample value of ^' in the quadrature. Option 2 suppresses this printing.

Output formats will be described below.

RLIM The number of vibrational modes to use in the calculation, between 1 and

Card 2, format (3F6.0)

H Height of building, H, in meters.

BCON Width of building, B, in meters. XMASS Mass of building per unit height, m, in kilograms per meter.

20
Card 5, format (8F6.0)

EN(1) Natural frequencies n^.n^, . . . , of the building in modes 1, 2,. . in cycles per second.

Card 4, format (8F6.0)

ZETA(l) Damping ratios n^^, r\^, . . ., in modes 1, 2,...

Card 5, format (7F6.0)

ZO Roughness length, z^, in meters. CZ Exponential decay coefficient, C^. CY Exponential decay coefficient, C . y DCON Zero plane displacement, z^, in meters. BETACN Coefficient 3 in Eq. 2.13.

FX Peak similarity coordinate, f ^

, used only for spectral representation 2 (see input parameter IREP)

.

FS Value of similarity coordinate beyond which Eq. 2.11 is valid, used only

for spectral representation 2.

Card 6, format (2F6.0)

USTAR Friction velocity, u^, in meters per second (for calculation of u^ , see Section 4.4, steps 1 through 4). T Duration of storm in seconds.

Card 7, format (4F6.0)

CW Mean pressure coefficient on windward side, C . w CL Mean pressure coefficient on leeward side, C^^. XN Alongwind cross-correlation coefficient, N, for large values of the

frequency, n (Eq. 2.61).

RHO Air density, p

- 1.25 kg/m^.

All of these values are printed by subroutine INPUT, after being read in. For details on

the selection of input parameters, the reader is referred to Sections 2.2, 4.2, 4.4.

The subprograms for the computation of the integrals which appear in the expressions of

the alongwind response will now be described. Further details on the integration procedure

are given in Section 3.2.

21
The subprogram INIT determines the quadrature sample points and weights (program variables FTILDA and ATILDA) for the integration with respect to (Eq. 2.54) from NF^, NF^.'NF^

and NF^; these parameters specify the number "of subintervals to take in each of four special , subranges of the semi -infinite range (0, <=°) . In the r-th vibration mode, these subranges

are determined by the three points (^^ - 2)/30, ^'^ - 2 and + 2, and the points 0 and «>, where is given by Eq. 2.55. The program variable FPEAK stands for This scheme

allows smaller subintervals to be taken in the numerical integration for ^ close to zero and

which are points at which the integrand becomes large. Numbers NN^,NN2,NNj and NN^ are

also specified in INIT. These give the number of subdivisions to take on each edge of the

"unit cube" (after an appropriate transformation of the physical problem variables, see Sec.

3.2) in the evaluation of the triple integral that is involved in the integrand of I^.^.^- Subroutine INIT must be recompiled to change the values of the NF or NN. The subprogram TRIPLE computes the triple integral '"'^^.C^)* given by Eq. 3.4, that occurs in the integrand of I^^j^ This is the same as the quadruple integral of Eq. 2.57

after a change of variable. The integrand for the triple integral is computed by the FORTRAN

function F. Values of the functions iJ^(Z), lJ(Z) (see Eq. 2.46 and Eqs. 4.1, 4.2) and S(Z,n)

which occur in the integrand of the triple integral are computed by the Fortran functions

XMU, UTILDA and STILDA. These functions represent modal shapes, the non-dimensionalized

mean speed and the spectrum of longitudinal wind fluctuations, respectively. One of two different spectral representations is selected in STILDA according to the value of the input

parameter IREP, as has already been described. Finally, the function FISTAR computes the

function cj)*^(^) (Eq. 2.56) that also enters into the integrand of I^^^^- The printed output will now be described. The first step lists the input values read

from cards by subroutine INPUT. For each mode, R, in succession, the following information

is printed. First, the input parameters determined by subroutine INIT are listed. If the

output print parameter IPRINT is one, subsequent pages will contain, for each value of

used in the numerical quadrature of Eq. 2.56, the following:

A line containing the calculated value of (^) , (named "avg. of integrals") and an estimate of the error in calculating it. A table of four rows and nine columns, containing the following quantities (denote the

quantity in the i-th row and the j-th column by A^ ^):

= ^; A. = i = 2, 3, 4 1.1 '1,1

A. T = (j)* (?); A. ^ = A. ,(})* (?), i = 2, 3, 4 1.2 ^rr 1,2 i,l rs

A. = aA. , i = 1,..., 4; a is the coefficient used by the quadrature formula m

connection with the abscissa f.

A. . = A. ,

• Y (f) (The "term" in the quadrature sum).

1,4 1,3 rr^ ^ ^ „ A^ ^

= the sum of the current and all preceding (i.e., for preceding values of f)

values of A^ ^. (The "partial sum" of the quadrature sum; its value for the last ^ is the

calculated approximation to the integral I ,.)

22
A = an estimate of the variance of A. ,, due to the random factors in the calculation

1.6 1,4'

of

A. ^ = a similar estimate of the standard deviation of A. ^. 1.7 1,5

A. o = same as A. _ but with the factor unity in lieu of <t>* ft). 1.8 1,3 ^rr*--^

A. = same as A. but using A. instead of A. _. Its value for the last r is the i>y i>-' i>" i)-3 calculated approximation to the first term in the sum of Eq. 4.13 and corresponds to the so- called background response (see Sections 4.2 and 4.3).

If IPRINT is set equal to two the printout is modified to print the information about

Y^^(?), and the table, only for the last value of

Finally, the values at the top of the building are printed of the mean alongwind

deflection (in m) , the root mean square of the fluctuating deflections (in m) and acceler2 ations (in m/s ), and the peak factors for the deflections and accelerations as computed

from the modes so far encountered (i.e., from the first mode on the page with the output

for R,S = 1,1, from the first and second modes on the page with the output for R,S = 2,2, etc . ) .

3.2 Integration Procedure

The integrals being evaluated are:

^rsL - C^' ^rs^^'^^^rs^^^'^^ ^'''^ where

y (z/H)y (z/H)

r s r s

= [C^ + 2C C„N(u^^/H) + ch

•s ^ w w 2, * x,-"

[i-(^/^^)^][i-(^/?^)^] + [2n^(^/^^)] [2113(^/^3)]

([i-(^/^j']'^[2n (f/i)]')([i-(R)']'-t2n (^'/^ )]')

(3.3)

^s(^) = /W/X(Zi)y3(Z2)^f(Zi)?r(Z2)[?'(^.z^)?(^'.Z2)]

1/2

-2C ?[(Z -Z )^+(C B/C H)^Y -Y )2]^/

exp( 5 ^-4r ^ — )dY dY dZ dZ

lJ(Z ) + U(Z )

/VJ/^P3(Zpu3(Z2)Lf(ZpU(Z2[?(^,Zp?(^,Z2)] 1/2

-2C ?[(Z -Z )2+(C B/C H)\^]^/^

exp( 5 )dZ dZ dt (3.4)

U(Z^) + IJ(Z2)

?(^,Z) = (3.5)

23
The basic integration scheme has two parts. The integration with respect to ^ is done

a ? -2by a form of trapezoid rule, as follows: four numbers, called NF^, NF^, NF^ and NF^, are

specified, and a trapezoid rule using NF subintervals is applied on the interval [0 , ]

,

1 r-2

one using NF^ subintervals is applied on the interval [ , , one using NF^ subintervals is applied on the interval -2,^* +2], and one using NF^ subintervals on +2,

^r-2

^ r

~ NF

^' Firi3lly> ^ correction is made to compensate for discarding the rest of [0, <»J : the coefficient of the last integrand value is modified by adding to it,

3 „ ^ - 2

20'- r NF^

This corrects properly when L = 0, and of course also whenever the resulting last term in the

quadrature sum is negligible compared to the sum itself. That last term is one of the auxiliary quantities pointed out by the program. If it is not negligible, and L is not zero, a correction should be made by adding

6L

20 - 6L

times the last term (as printed out) to the sum. (The source of these corrections is a simple extrapolation of the integrand,

from the highest value of ^ used, to infinity. The extrapolation assumes that this function

can be represented, on that interval, by its asymptotic form,

CX (z)^2L-(23/3) —r , s

^

to all the accuracy needed.)

The procedure just outlined for integrating with respect to is applicable in the case

r = s. For the r / s case, where there is cross-mode coupling, the procedure must be suitably modified.

For each of the four intervals of ^' values mentioned above, an integer N is specified.

(In the FORTRAN program, these 4 values of N are denoted by NN(1), NN(2) , NN(3) , NN(4),

3 respectively.) The unit cube is divided into N congruent subcubes by dividing the interval

[0,1] on each axis into N equal subintervals. In each subcube, two points R and R' are chosen at random, independently of each other. The point S that is opposite R - with respect

to the center of the subcube - is determined, as is the point S' that is opposite R'. The

3 values of the integrand at the points R and S, in all the N subcubes, are summed; and the

3 sum is divided by 2N , to form an average that we shall call "Y". The same is done with the

points R' and S', to form a second average, "Y"'. Y and Y' can both be regarded as approximations to the desired triple integral Y^^ (^) ; their average, (Y + Y')/2, is a better approximation and is used as the value of Y (f )

.

rs

24
The randomness in the selection of R and R', if nothing else, insures that there will b some error in Y, Y', and (Y + Y')/2. (There are, of course, other sources of error; the

randomness is introduced in order to decrease the total error. The theory behind this integration procedure is given in Ref. 53). The program calculates, as an estimate of the likely

size of the error of (Y + Y')/2, the quantity,

^ = ^

[kil(F(Rk)+F(S^)-F(Rp-F(S')3^]^/^

4N

3 wliere "£" denotes the integrand and the summation indicated is over the N subcubes - k bein

the number of the subcube for each particular summand. E is a positive quantity; the absolute value of the error of (Y + Y')/2 is likely to be about as big as E - only occasionally

greater than 2E and seldom greater than 3E. The random points used .„ thus calculating V (J) for one particular value of * should

not be the same as those used for other values of f . New random points should be generated

for each ^, in a manner as independent as possible of the points used for the other values

of Doing that generally results in some cancelling of the errors of the individual

Y^^ ('^) when they are combined in the integration with respect to ^. One remark must be made about the use of the term "random" in the above description.

The points R and R' are not found by using any "genuinely ;random" physical process - such as coin-tossing or observing radioactive decay - but are generated by a determinate mathematical process whose results are thought to mimic those of random processes. The numbers so generated should perhaps be called "pseudorandom" rather than "random"; but it is customary

to call them "random". The particular random numbers used in the present version of the

program are generated in the subroutine TRIPLE, under the names X(I) and XPRIME(I). The

mathematical "random number generator" used there is an unusual one; any standard one could

probably be used in its place.

Chapter 4. A SIMPLIFIED PROCEDURE FOR CALCULATING ALONGWIND DEFLECTIONS AND ACCELERATIONS

4 . 1 Introduction

The computer program described in Chapter 3 may be used to calculate the response of structures with any modal shapes and natural frequencies. The purpose of this chapter is to present a simplified procedure for calculating alongwind response, applicable to structures

for which it may be assumed, first, that the fundamental modal shape is linear and, second,

that the response to wind loading is dominated by the fundamental mode. The first of these

assumptions is acceptable in a large number of situations of practical interest, such as in

the case of typical multistory framed structures (see, for example, Ref. 10, p. 428 or Ref.

55, p. 60 and p. 242); moreover^ as will be shown in Chapter 5, even if it deviates rather

significantly from a straight line, the fundamental modal shape may be assumed to be linear

without introducing errors in the estimated^ ratio of fluctuating to mean reponse that exceed

a few percent. The second of these assumptions will generally hold if the ratios of natural

25
frequencies in the second and higher modes to the fundamental frequency are sufficiently

large (see Chapter 5)

4 . 2 Basic Assumptions

In developing the procedure presented in this chapter, the following assumptions were

used

1. The behavior of the structure is linear. Its fundamental mode of vibration is a linear function of height above ground, i.e., y, (z) = C —, where C is an arbitrary

1 H constant. The graphs developed herein are based upon the value C = 0.26.

2. Th,e contribution of the second and higher modes of vibration to the response is

negligible.

3. The mean velocity profile is described by the relations,

z-z , Ufz) = 2.5 u^ln

* z > z,+10 (4.1J

z — d

o

U(z) = 2.5 u,£n z < z^+lO (4.2) * z — d

o

where z, z , z, are expressed in meters,

o d

^ The use of the logarithmic profile above elevation (z+10) meters implies the

assumption of horizontal homogeneity of the flow (see Chapter 1). Eq. 4.2 is used, conservatively, on account of the uncertainty regarding the actual nature of

the flow near a building for z < or so. The mean velocity U(z) used in Eqs

4.1, 4.2 is averaged over a period of one hour.

4. The spectral density of the longitudinal wind speed fluctuations is described by

Eq. 2.14. The justification of this assumption was presented in Section 2.2.

5. The mean and fluctuating pressures are described by Eqs. 2.23c, 2.23d, 2.24a,

2.24b. (For tall buildings it is commonly assumed C = 0.8, C„ = 0.5, C„ = C +

^ ^ ^ w 2. D w

C, = 1.3.)

6. The spatial cross-correlation of the fluctuating pressures in the acrosswind and

alongwind directions may be described by Eq. 2.58 and 2.60,2.61, respectively.

For design purposes, the values recommended for the exponential decay coefficients

are C = 10, C = 16 [3,7]; comparable values are used in Refs. 1,2,4,6. However,

z y to permit the evaluation of possible errors due to uncertainties with regard to the actual values of the coefficients C^,C^, (see Sect. 2.2.5) calculations can also be carried out using different sets of values. Thus, graphs were developed

that correspond, in addition to the values C = 10, C = 16, to the values C =

z y z 6.4, C = 10 and C = 4.0, C = 6.4.

y z y

4 . 3 Total Fluctuating Response as a Sum of Background and Resonant Contributions

Consider a single degree of freedom linearly elastic system with natural frequency n^ and damping ratio r|. Let this system be subjected to the action of a forcing function with a 26
S(n)

- (a) White noise spectrum, (b) Square of modulus of mechanical admittance,

(c) Decaying spectral curve

27
spectrum S(n) such that (Fig. 4.1a)

where

S(n) = n > 0 (4.3)

where is a constant. The mean square value of the response can be written as

= /~ |H(n)|2dn (4.4)

The quantity |H(n)| , which is represented in Fig. 4.1b, is an analytic function; therefore,

the integral in Eq. 4.4 can be evaluated by means of the residue theorem to yield (see Ref.

10, p. 501):

"T TM1

X = -r^ S (4.6)

4ri 0

^

~2

If the damping ratio n is small, the bulk of the contributions to the total value x is An^

due to the forcing function components of frequencies n^-An^^ < n < n^^+Anj^, where -— is

a small number (hatched area in Fig. 4.1a). If, as in the case of atmospheric turbulence,

the spectral curve S(n) has the shape of a decaying curve (Fig. 4.1c), x may be written as the sum of three contributions:

Y = f V 1 ^ x^ = (x^), * (x^), + (x^), (4.7)

in which . (x^)^ = 's(n)|H(n)rdn (4.8)

h

= A S(n)|H(n)rdn (4.9)

r 1

(x^)3 = S(n)|H(n)rdn (4.10)

1 1

2 1 2 2 Assuming again that n is small, (x )^ -

^-pj— S(n^O); for 0 < n < n^- A , |H(n) 1 = 1; and (x ) is negligibly small. Thus,

or

-y n -A irn = /^^ S(n)dn + ^ S(np (4.11)

x^ = A(n)dn + S(nj) (4.12)

28
The first and the second term of the sum in Eq. 4.12 are usually referred to as the background part and the resonant part of the response, respectively.

The above relation can similarly be applied to Eq. 2.54, i.e..

(4.13)

where ? = n,H/u^

i i * The first and the second term of the sum in Eq. 4.13 correspond to the background and

resonant parts of the response, respectively.

To verify the extent to which the approximation involved in Eq. 4.13 is acceptable,

numerical computations were carried out - using the program described in Chapter 3 - for

about 120 cases corresponding to a wide range of typical buildings and terrain roughness

conditions. The calculations showed that the approximation is of the order of 1%. It was also verified that, for L = 1,2,3, the background term may be neglected and that

^llL^^^l'^l^^l^ (L= 1,2,3) (4.14)

4.4 Alongwind Deflections and Accelerations

4.4.1 Mean Response

From Eqs . 2.42, 2.43, 2.44, 2.45 it follows that the mean deflection may be written as

2 pu^ z z

a(z) = 0.238 — J(^, H) B ^ (4.15)

mn^

3 or, with p = 1.25 kg/m , = 1.3 and all quantities being expressed in SI units (m, kg, s)

2 i(z) = 0.387 —2 J(j^, ^, H) B I

(4.16)

mn^

where

z z J(-§, H) = ^—^ fl ZU^Z)dZ (4.17)

(2.5 u^)

z Zj

The function J(-j7, -rr, H) defined by Eq. 4.17 may be expressed as, H rl

^,H) = ^) . ^[Ki(^o^^^d'^2(^o^^^dS^^o^] ^'-''^ H

where all dimensions are in meters. can be obtained from Fig. 4.2 and k^,k^,k^ are given

in Table 4.1.

29
4.4.2 Fluctuating Response: Deflections and Accelerations

Let the quantities h and be defined as follows:

= ^ r Y^^(^)d^ (4.19)

n,H

where = jj^. It follows from Eqs. 2.49 through 2.54 and Eqs. 4.19, 4.20, that

—5

a(z)

-J—1/2 7 Cl/2

a (z) = 50 nj ^— £(z) (4.23)

v»(z) = n^ (4.24)

-1/2 -.5 1/2

2 "2

where a' (z) and a (z) are the root square mean of the fluctuating deflection and of

the acceleration, respectively. The peak factors associated with these quantities are

g (z),= [2iin 3,600 VJ^^^ + rjj (4.25a)

^ [2Jln 3, 600V 1^'^

* a-"

g..(z) = [2Jln 3,600 n ]^^^ + ry (4.25b)

^ [2£n 3,600 n^]^'

respectively, i.e., the maximum probable value of the fluctuating deflection and of the

-1/2 1/2

2 ..2 acceleration are g (z) a' (z) and g"(z) a (z) , respectively, and the gust response

£L 3. factor is —2 1/2 G.F. = 1 + g (z)5li^^ (4.26)

^ a(z)

The quantity fe may be obtained as follows:

in which the value of £> - corresponding to the values of the exponential decay coefficients

C = 10, C = 16, and calculated using the conservative assumption N E 1 - can be found in

z y

Fig. 4.3. For z^ ^ 0, Eq. 4.27 is approximate; the approximation was verified by numerical

calculations to be of the order of 1%.

30
Table 2.1 - Suggested Values of for

Various Types of Exposure

Type

of Exposure Coastal^ Open

Outskirts of

Towns, Suburbs

Center of Towns Centers of

Large Cities

(meters)

0.005-

0.01

0.03-

0. 10 0.20-0.30 0.35-0.45 0.60-0.80

Applicable to structures directly exposed to winds blowing from open water.

Table 2.2 - Approximate Ratios of Probable Maximum Speed

Averaged over Period t to that Averaged

over One Hour (at 10m Above Ground in Open Terrain)

t 2 5 10 30 60 100 200 500 1000 3600

(Seconds)

1.53 1.47 1 .42 1.28 1.24 1.18 1.13 1.07 1.03 1.00

Fig. 2.2 - Ratios u^/u^

31
Table 4.1 - Values of k . k* . k

z 0 (meters)

0.01 0.05 0. 07 0.10 0.20 0.30 0.40 0.60 0.80 1.00 1.20

740 ^ ^ j> 70/1 1 71 X O U 1 O J 1 1 ^ i. u i. Qfl OO

36 33 26 22

3.95 3.20 2.65 2.25

Table 4.2 - Values z^/H, z^/H Corresponding to

Various ^ ^ Curves

CURVE z /H

0 z^/H CURVE /H z^H

A 1 3 X 10

^ 0. J' 3 4 X -3 10 0.04

B 3 4 X 10"^ 0. K 5 4 X 10-3 0.

C 8 3 X 10-^ 0. K' 5 4 X 10-3 0. 15

D 1 1 X 10-^ 0. L

' 8 0 X -3

10 0.

E 1 9 X 10-4 0. L' 8 0 X 10-3 0.10

F 4 7 X 10-4 0. M 1 3 X 10-2 0.

G 1 0 X

10--^ 0. M' 1 3 X 10-2 0.20

H 1 6 X 10-3 0. N 1 8 X 10-2 0.

I 2 2 X 10-3 0. N' 1 8 X 10-2 0.45

I

'

2 2 X 10-3 0.06 0 2 7 X 10-2 0.

J 3 4 X .10-3 0. 0' 2 4 X 10-2 0.30

Note: If z,/H is of the order of 0.1 or less and z /H < 10 2 in determining '^^^ it may be

d o i i assumed z. = 0. -

32

34
Fig. 4.4 Function ^ . , B/H = 0, C = 10. C =16 ll Z y

35
36

38
1 5 10 15 20 25 30 35 40
The quantity K, may be obtained as follows

o

- ^^wS^ - ^^1 ^

,B ^o D ^ , ^

= z—rrr w- \i w h- ^i^ ^^-^s)

N = i - — (1 - e'^^1) (4.29) h 2,1

"l^^ (4.30)

H-z

0 = 2.5 (£n 2^ - 1) (4.31)

o

The values of Y^^ corresponding to the values of the exponential decay coefficients = 10,

= 16 can be found in Figs. 4.4 through 4.8. Linear interpolation in and between Figs. 4.3

through 4.8 is permissible.

To evaluate possible errors due to uncertainties with regard to the actual values of

C^,C^, the quantities S, Y^^^ corresponding to = 6.3, = 10 and = 4.0, = 6.4 are given in Figs. 4.9 through 4.13 and 4.14 through 4.18, respectively.

The Y^^ curves are identified by capital Roman letters to which there correspond

ratios z^/H, z^/H, listed in Table 4.2. For convenience. Tables 2.1, 2.2 and Fig. 2.2 have

also been included in this chapter.

4 . 5 Numerical Example

Consider a building for which the height H = 400m, the width B = 66m, the depth (i.e.,

the dimension in the alongwind direction) D = 45m, the damping ratio n = 0.01, the fundamen3 1 tal frequency n^^ = 0.09 Hz, the weight per unit volume w = 1,500 N/m (IN ^

^ ^

- ^ lb). The

basic wind speed (i.e. the fastest mile wind at 10 m above ground in open terrain) is v.f

" 75mph (1 mph = 0.4474 m/s) . The building is located in the center of a large city for which

it may be assumed z^ = 0.65m, = 20m (see Table 2.1). It is assumed, in addition, that

the air density p = 1.25 kg/m^, the acceleration of gravity g = 9,81 m/s^, the mean pressure

coefficients on the windward and leeward sides are C = 0.8, C. = 0.5 and thus the drag

w J, " coefficient C„ = C + C„ = 1.3, the acrosswind correlation coefficients C = 10, C = 16,

D w £ z y

[In view of the uncertainty with respect to the actual values of C^, C (see Sect. 2.2.5)

it is advisable that calculations be also carried out corresponding to lower values of

C , C (see Sect, 5.5)1.

z y

Step 1. Averaging time for the given fastest mile wind speed (see Section 1.2.2):

3600 3600

T = — = -^r^ = 48 s v^ 75

Step 2. Mean hourly speed at 10 m above ground in open terrain corresponding to the given

fastest mile wind (see Table 2,2),:

V U^(IO) = = 59.6 mph = 26.8 m/s

40
Step 3. Friction velocity in open terrain (for which it is assumed = 0.07m (See Eq.

2.6); 1

"l^^^^ 26.8 .

, "*i = 77;—To = 1274 = 2-^^ ""/^ 2. Sx-n ol

•Step 4. Ratio between friction velocities over terrains with z , = 0.07m and z = 0.65m, ^ ol o respectively (see Fig. 2.2):

= 1.17

"*1

i.e., = 1.17 X 2.16 = 2.54 m/s.

Step 5. Determine the 'following quantities:

B 66

H 400

2 o 0.65 , , ,„-3

1.6 X 10 H 400

^ = 20_ . 0 OS H 400 "'^

^ = V! . 0,09x400 ^ 2

1 u^ 2.53

= 111.5

= 35.25 \ (Table 4.1)

k^ = 3.76

3' = 17.1 (Fig. 4.2)

ft = 8.9 (Fig. 4.3)

= .8 x 10"^ (Table 4.2 and Fig. 4.5)

m = B X D X w/g = 445,000 kg/m (mass of building

per unit height)

Step 6. Determine the quantities ^ ^ 0 = 2.5u^(2.n i - 1) = 34.0 m/s (Eq. 4.31)

^1 3.85n Ax

C

= — = 1.83 (Eq. 4.30)

U

41
(In Eq. 4.30, Ax is the smaller of the quantities 4B, 4D and 4H) 11 -zF

N = i- - i— (i-e '^l) = 0.4

^1

(Eq. 4.29)

Step 7 Calculate quantity J (Eq. 4.18)

ri

= 17.1 + —-—J

[111. 5 + 20x35. 25+(20)^x3. 76] - 17.1

(400)

Step 8 Calculate quantity ^) (Eq. A. 21)

= (1 - ^) I - 8.45

Step 9 Calculate quantity ^ (Eq. 4.29)

Jl= .562 ^^^/n = 6.4

Step 10 Mean Response

a(z) = 0.387 —- J B ^ = 0.76 ^ (meters)

Z H H

where z is the height above ground of the point considered. The maximum mean deflection occurs at the top of the building and is a (H) = 0.76 m.

Step 11 Ratio of root mean square of fluctuating deflections to mean deflection (Eq. 4.21)

-1/2

i(z)

^ 0.278

Step 12 Peak factor for fluctuating deflections (eqs. 4.22 and 4.25a)

^a = "l

=

[2lx\ 3600 V ]^^^ = [2lvi (3600x0.059)]^"^^ = 3.28

3.

, TO 0.577 ^ 'a

= ^-^^ * 372^= ^-46

42
Step 13 Gust Response Factor (Eq. 4.26)

G.F. = 1 + g 5.!— =1.96

a(z)

The maximum probable deflection is:

a (H) = (G.F.) a(H) = 1.96 x 0.76 = 1.49m max

Step 14 Root mean square of accelerations (Eq. 4.23)

-J 1/2 c> 1/2

a (z) ^ 50 n: ~— a(z) = 0.045 ^ m/s" - 0.0047 ^ 1 J H H

Step 15 Peak factor for accelerations (Eq. 4.24, 4.25b):

(2£n 3,600 n^]^^^ = 3.40

_ .„ 0.577 , g" = 3.40 + ,

„

= 3.57

*a 3.40

•1/2 "2 Step 16 Maximum probable acceleration a = g- a (H) max a

-1/2

g.. a^(H) = 3,57x0.047 = 0,17 m/s^ - 0,017 g

3-

4 . 6 Comparison between Gust Response Factors Calculated Using Various Current Procedures

For the building just considered, the values of the gust response factors calculated in accordance with the procedures described in the ANSI A58.1 Standard [4], Ref. 6 and Ref, 3 are

1.53, 2.83 and 3.38, respectively, versus 1.96, as calculated herein. As noted in Ref. 7, the gust factors calculated using Refs. 3 and 6 are larger, first, because the pressures

on the windward and leeward sides are conservatively assumed to act in phase (i.e., to be

perfectly correlated) and second, because the variation with height of the spectrum of the

longitudinal velocity fluctuations is ignored. Gust factors are underestimated by the

procedure of Ref. 4 because the alongwind pressure correlation coefficient, rather than

being applied as a reduction factor to just the cross-spectra of pressures acting on opposite

sides of the building, is applied to the entire alongwind response, in violation of basic

random vibration theory (see Ref. 8).

43
44
45
46

48
r I

I

49

I
50
51
52
53
Chapter 5. APPROXIMATIONS AND ERRORS IN THE ESTIMATION OF THE ALONGWIND RESPONSE

In this chapter results of numerical calculations are presented on the basis of which

estimates can be made of the errors associated with uncertainties regarding certain features

and parameter values of the models employed. The calculations were carried out for three

typical buildings selected as case studies and described in Table 5.1. The wind speed at 10 m above ground in open terrain (z^ = 0.07m) was assumed to be v^ = 75 mph, where v^ = fastest mile of wind.

Table 5.1

Description of Buildings Selected as Case Studies

Building H B D

'^l w

meters Hertz Newton/meter^

1 365 60 45 0. 10 0.01 1,500

2 150 60 45 0.20 0.01 1,500

3 45 45 45 1.00 0.01 1,500

Note: 1 Newton - -.—^ lb

4. 25

5 . 1 Contribution of Higher Vibration Modes to the Response

The root mean square of the fluctuating deflection and of the accelerations were calculated for buildings 2 and 4 in open exposure (z^ = 0.07 m, z^ = 0) and city exposure (z^ =

0.80m, z^= 21m). The exponential decay coefficients assumed in the calculations were C^ =

10, Cy = 16. The assumed modal shapes in the first three modes are represented in Fig. 5.1.

The damping ratios were assumed to be = 1, = = 0.01. Calculations were carried out separately for the cases i^2^"i " '^3/'^! " ^^'^ '^2^"l

~ 2.5, i^j/n^ = 5. The con- tributions of the higher (i.e., of the second and third) modes of vibration to the response

are listed in Table 5.2. The contribution of the cross-mode product terms was also included in Table 5.2. This

contribution represented about one half of the amounts shown in columns (1) and (5) and was altogether negligible in all other cases.

It is seen that for the larger values of the ratios n^/n^, n^/n^ in Table 5.2 - which

are comparable to those encountered in practice in the case of typical tall buildings - the

contribution of the higher modes to the fluctuating deflections is negligible. However, the

contribution to the accelerations may be of the order of 10%.

54
.26 .32 .39

X

FIRST MODE SECOND MODE THIRD MODE !

i Fig. 5.1 - Modal Shapes !

Table 5.2 Percent Contribution of Higher Modes of Vibration to Root Mean Square of Fluctuating Response

Open Exposure Center of Large City

Building n2/n^=1.2; n^/n^=l 5 n2/n^=2. 5; n2/n^=l 2; n2/n^=1.5 n2/n^=2.5; "3/^1=^

Defl. Accel Defl. Accel

.

Defl. Accel

.

Defl

.

Accel

.

(1) (2) (3) (4) (5) (6) (7) (8)

1 5 14 0.1 10 7 8 0.1 7

2 8 20 0.1 10 14 9 2 8

55
5 . 2 Influence upon Calculated Response of the Deviation from a Straight Line of Funda - mental Modal Shape

A convenient means for estimating the influence upon response of the shape of the

fundamental mode of vibration is provided by the expression

—1/2 ^ - 2a

1 + Y + ct ^

a

derived by Vickery [3] on the basis of the assumptions that the power law (Eq. 2.9) holds,

that the spectra of the turbulent velocity fluctuations are independent of height and that

the fundamental modal shape is described as follows:

y^tz] = i^)'^ (5.2)

—1/2

where y is a constant. In Eq. 5.1, a' = r.m.s. of the fluctuating deflection, a = mean deflection, Q = function of geometrical, mechanical and environmental parameters,

independent of Y- It may be assumed, roughly, that a can vary between 0.10 for open expo- sure and 0.40 for centers of large cities. It follows from Eq. 5.1 that, for a = 0.10, the

ratios a' /a calculated assuming y

~ 0.5 and y = 1.5 differ by less than 1% from that

calculated assuming Y = 1 (i.e., a linear fundamental modal shape). For a = 0.4, the corres- ponding differences are of about 3%. It is thus seen that deviations from a straight line

of the fundamental modal shape have an insignificant effect upon the calculated ratio

—1/2 - a' /a.

5 . 3 Influence Upon Calculated Response of Errors in the Estimation of the Roughness

Length

To estimate the magnitude of the error associated with uncertainties regarding the

actual value of the roughness length, the response of buildings 1,2,3 was calculated for

coastal (z^ = 0 . 005-0 . 01m) , open (0. 03-0. 08m) , surburban (z^ = 0.20-0.30 m) , center of town

(z^ = 0.40 m) and center of large city (z^ = 0.60-0.80 m) exposures. The zero plane displacement was in all cases assumed to be zero. The results of the calculations are shown in

Table 5.3.

It is seen from Table 5.3 that the sensitivity of the results to even large errors in

the estimation of the roughness lengths is tolerably small. It is also noted that the

alongwind deflections and, consequently, the design wind loads are higher by about 15% in

the case of the coastal, than in the case of the open exposure.

5.4 Spectra in the Lower Frequency Range and Alongwind Response

It was shown in Section 2.2.5 that no universal relation exists that describes the

shape of the spectral curve in the lower frequency range and that the peak similarity

coordinate appears to vary strongly between sites and between atmosphere and laboratory.

To estimate the effect of this variation, the reponse of buildings 1,2,3 was calculated in

56
57
the cases z = 0.07m, z, = 0 and z = 0.80m, z, = 20m assuming values of the peak similarity

o d o d coordinate = 0.03 (Eq. 2.14) and = 0.01, = 0.10, = 0.19 (Eqs . 2.15,2.16). The

ratios fa 1 ^ / [a 1„ of the maximum probable response calculated using the values of max fj^ max-^ 0.033

the peak similarity coordinate f^ and 0.033, respectively, are given in Table 5.4.

Table 5.4

ios [a ]^ /[a ]„ max^f^ max^0.03

Building 1 Building 2 Building 3 Exposure Open Large City Open Large City Open Large City

f^ = 0.01 1 .00 1. 00 1.00 1.00 1.00 1.00

= 0.10 .98 .97 .97 .94 .95 .91

f^ = 0.19 .97 .96 .96 .93 .93 .87

The results of Table 5.4 suggest that Eq. 2. 14 ;to which there corresponds f^ = 0.03 ) is slightly conservative. As was mentioned in Section 2.2.5, according to available measurement results, between z = 3m and z = 60m, f^^ - 0.02-0.08; the occurrence, in the case of

building 3, of a fluctuating response corresponding to f^ = 0.10 or f^^ = 0.19 is therefore

believed to be unlikely. It is also noted that, in view of Eq. 4.14, the influence of the

spectral curve shape in the lower frequency range upon the value of the accelerations is negligible

.

5 . 5 Acrosswind Correlation of the Pressures and Alongwind Response

As indicated in Sect. 2.2.5, the exponential decay coefficients C , C^ appear to depend

upon terrain roughness, height above ground and wind speed. For example, in the case of

moderate wind speeds - such as usually occur during full-scale measurements of structural

response - the values of C^, appear to be considerably lower than those corresponding to high winds. Values as low as = 4 have been reported in the literature [50]. In view of the uncertainties with regard to the actual values of the coefficients C , C , y z it is of interest to estimate the errors in the calculated alongwind response that correspond

to possible errors in the assumed values of those coefficients. The alongwind response of

buildings 1, 2, 3 in open (z^ = 0.07m, = 0), and large city (z^ = 0.80m, = 0) exposures

was therefore calculated separately for C = 10, C =16 (case 1) , for C = 4, C = 6.4 fcase

z y z y

6) and for four intermediate cases in which C^, were assumed either constant throughout the

frequency range (case 4) or to have lower values at low frequencies and higher values near

and beyond the fundamental frequency n^ , (cases 2, 3, 5).

58
The ratios r. = fa l./fa 1,, r. = fa K in which cases 1 and i are denoted

1 max-*! max-"! i maxM

by the indices 1 and i, respectively (i = 2,3,4,5,6), are given in Table 5.5. It is seen that even a considerable change in the values of C , C in the lower frequency range affects

z y little the calculated response (cases 1,2,3). However, large decreases in these values near

the fundamental frequency of the structure do increase the total' response, by as much as almost 10-20% in the case of the deflection and 160% in the case of the acceleration.

59
CO 00 H f~M

OJDO LO O o O o o

nS -4 ,

1

, , <~)

H

O)

CQ

II CO H o o O o o

X * o

II oo

I-t C_j DOc o

=^ CM LO 00

(/) o o o o

;3 a 5-, r—

(

"O

0)

4_> a

to

fH

CQ GJ

(1> o hO

en o O —1 Co

Cl

ifi a> Dio H to rO LO H

{J)

<DDO rM u o O o o

(/I <u &0C

H > TJ

<M - CNJ LO o CQ •H

fH

ca o LO r\i ;> H o o o o

u

o

CD

t/) o

Oh

dJ q; —« 8 —' 8 ^ 8 > C V C V

M-i cn c

o 0 8 V V 8 V 8

+-t V V r-A V —

'

V V -—I V

(/I C c c c c c c C C c

o V V CTt V CTi V V V

•H o O o o o o

4J

CTj

1

LTi O o v£5 O \D

(Uo

o o vO O o

<D

t/l LO

05 U

60
Chapter 6. CONCLUSIONS

In this work practical procedures for calculating alongwind deflections and accelerations

have been presented, which incorporate recent advances in the state of the art and are therefore believed to provide improved estimates of alongwind response. The procedures presented

herein account for the variation of wind spectra with height and make allowance for the

imperfect correlation of pressures acting on the windward and leeward building faces.

Numerical examples are given, and the gust response factors obtained are compared to those

calculated using procedures described in current codes and standards.

For structures with unusual modal shapes or for which the influence of vibration modes

is significant the computer program presented in Chapter 3 and listed in the Appendix

should be employed. In the case of typical tall structures, for which the ratios of higher

to fundamental frequencies are not unusually low (e.g., for which n^/n^ > 2) and for which

the fundamental modal shape may be expressed as U-j^(z) - (z/H)^, where 0.5 < Y

< 1.5, the

simplified procedure presented in Chapter 4 may be used.

Results of numerical calculations have been presented which show that the sensitivity

of the results to even large errors in the estimation of the terrain roughness is tolerably

small. It was noted, however, that the alongwind deflections and, consequently, the design

wind loads may be higher by about 15% or so in the case of coastal, than in the case of open

exposure. It was found that the effect upon the response of possible variations of the

2 frequency, n, for which the reduced spectrum n S.{n)/u^ reaches a maximum is small. An investigation was also carried out of the effect upon the response of changes in the assumed

values of the exponential decay coefficients C , C^ in the expression for the acrosswind

crossrcorrelation of the fluctuating pressures. It was concluded that even a considerable

change in the values of C , C in the lower frequency range has a negligible effect on the

y z response. However, large decreases in these values near the fundamental frequency do significantly increase the calculated accelerations. The sensitivity of the calculated

accelerations to changes in the values of the exponential decay coefficients suggests that

measurements of accelerations, which are relatively simple and inexpensive, might offer

useful information on the actual magnitude of these coefficients.

It was noted that additional research is desirable to improve current aerodynamic models

describing the relation between wind speed fluctuations and the associated fluctuating

pressures. It was also noted that according to full-scale measurements reported in the

literature, the exponential decay coefficients C , C - which are a measure of the spatial

y z cross-correlation of the wind speed and of the pressure fluctuations - depend upon terrain

roughness, height above ground and wind intensity. In particular, the values of C^, C^,

under moderate wind conditions, appear to be considerably lower than those corresponding to

high winds. Therefore, full-scale measurements of alongwind structural response - particularly measurements of alongwind accelerations - can be useful for validating design procedures

only to the extent that information on the dependence of C , C on the aforementioned

y z parameters becomes available. Additional theoretical and experimental research aimed at

obtaining such information is therefore believed to be necessary.

61
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Sciences Laboratory, Nihon University, Furabashi, Chiba, Japan, 1967-1971.

64. Shiotani, M., and Iwatani, Y., "Correlations of Wind Velocities in Relation to the

Gust Loading", Proceedings , Third International Conference on Wind Effects on Buildings

and Structures, Tokyo, 1971. »

65

APPENDIX

COMPUTER PROGRAM LISTING

SAMPLE INPUT AND OUTPUT

67

C/»/WlNDLOAD

INTEGER R,RVAL,SVAL,PLIM

DIMENSION PSUM(^) ,WP(4) ,FACTOR(^) ,WEIGHT(^) ,TERMC4) ,TERMV<4)

,

ISIGMA(A) ,PSUU(4) ,WEIGHU(A) ,TERU<A) ,XJH(4)

C0MM0N/BL0CK2/H ,BCON ,DCON,XMASS ,EN<8) ,ZETA(8) , Z 0 , CZ s CY , BETACN

,

» Fl ,FS ,USTAR,T ,CW,CL,XN,RHO , I REP, IPRINT ,RLIK

COMMON/ BLOCKl/KOUNT

C0MM0N/BL0CK3/FTILDA( 20 0 ) , AT I LDAC 20 0 ) , M , NF ( 4 ) ,NN<4)

COMMON/ BLOCKS/ RVAL.SVAL

COMMON/ BL0CK6/XMU INT < 8 ) , G I NT C 8

)

200 CONTINUE

KOUNT = 0 CALL INPUT

TFACl=CW*CW+2. 0»CW«CL+CL*CL

TFAC2=CW«CW+XN*2. 0«CW«CL+CL*CL

C SEE EQ.2.59 OF THIS REPORT FOR OCCURRENCE OF THIS FACTOR.

TCRIT=0.9«EN( 1 )*H/USTAR

DO 202 J=l ,4

XJHC J)=0 .

0

202 CONTINUE

GH=0.0

DO 250 R=l,R"LIM

RVAL=R

SVAL=RVAL

CALL INIT

C PERFORM FOUR-DIMENSIONAL INTEGRATION.

DO 205 J=l ,U

PSUU<J)=0.0

SIGMAC J)=0.

0

205 PSUM<J)=0.0

N = NN( 1 ) C M IS NUMBER OF STEPS IN FTILDA INTEGRATION

DO 230 1=1,

M

IF (I.GT.NF(1>) N=NNC2)

IF ( I .GT.NF( 1 )+NF( 2) ) N=NN(3)

IF (I.GT.NFC 1 )+NF<2)+NF<3)) N=NN<4)

W=FTILDA< I

)

CALL TRIPLE(N,W,Z,E)

IF C IPRINT. EQ. 2) GO TO 212

IF (MOD( 1-1 ,5) .EQ. 0 ) WRITE (6,210) RVAL,SVAL

WRITE (6,211) Z,E

210 FORMAT ( 19H1 INTEGRAL FOR R,S =,I2,1H,,I2)

211 FORMAT(//26H0AVG OF TRIPLE INTEGRALS = , E 1 2 . 5 , 4X , 9HERR EST =,E12.5)

GO TO 21A

212 IF (I.LT.M) GO TO 21A

WRITE (6,210) RVAL,SVAL

WRITE (6,211) Z,E

214 CONTINUE

WP( 1 )=1 .

0

WP(2)=W»W

WP(3)=WP(2)*WP(2)

WP(4)=WP( 3)»WP( 2)

PHISTR=FISTAR(W)

TFAC=TFAC1

IF (W.GE.TCRIT) TFAC=TFAC2

69
DO 215 J= 1,4

FACTORC J)-WPC J)»PHISTR

WEIGHTC J)=ATILDAC I )»FACTORC J)

TERM( J ) =Z«WE I GHT ( J

)

PSUMC J)«PSUM{ J)+TFAC«TERM< J)

C PSUM IS I SUB RRL (EQ. 2.54 OF THIS REPORT).

TERMVC J)»(WEIGHT< J)»E)»<WEIGHT< J)»E)

SIGMA(J)=SQRT( S1GMA<J)«SIGMA(J)+TERMV< J)

)

WEIGHU< J)=AT1LDA< 1 >*WP(J>

TERU( J)=Z»WEIGHU< J)

PSUU( J)=PSUU< J)-»-TERU< J)

C PSUU IS I SUB RRL ( EQ . 2.54) WITH FISTAR AND TFAC2 EQUAL UNITY.

215 CONTINUE

IF <1PRINT.EQ.2 .AND. I.LT.M) GO TO 230

WRITE' (6,220

)

220 FORMAT (// 1 9X , 1 2HMECH. ADMITT , 9X , 6HWEI GHT , I IX , 4HTERM, 4X , 1 IMPARTIAL

1SUM,6X,9HTERM VAR . , 1 OX , 5HSI GMA , 9X , 4HTERU , IX , 1 2HPART I AL SUMU ) WP( 1 )sFTILDA( I

)

DO 225 J=l ,4

WRITE (6,221 ) WP( J) ,FACTOR( J) ,WEIGHT( J) ,TERM( J) ,PSUM( J) ,TERMV< J)

,

ISIGMA( J) ,TERU( J) ,PSUU( J)

221 FORMAT( 1X,7E15.8,2E13.8)

225 CONTINUE

230 CONTINUE

DO 240 J=l ,4

XJH ( J ) =XJH ( J ) 'XJH { J

)

XJH( J>«XJH( J)*XMU(RVAL, 1 . 0 )»XMU(SVAL, 1 .0 )*PSUM( J)/

1 (XMUINT(RVAL)»XMUINT(SVAL)*(EN(RVAL)»EN(SVAL)*(H/USTAR>*»2)»«2>

XJH( J)=SQRT(XJH( J)

)

240 CONTINUE

GH=GH+XMU(R, I . 0

) •G 1 NT ( R) / ( XMU I NT ( R) ( EN( R) "H/USTAR ) *»2

)

DEF=G .5»(CW+CL)«RH0»BC0N»H»H«GH/(39.478418»XMASS)

RMSDEF=RHO»BCON»H*H»XJH( 1 )/ ( 39. 4784 1 8»XMASS

)

RMSACC=RH0*BC0N»USTAR»USTAR»XJH(3)/XMASS

XX=SQRT( 2. 0»ALOG(USTAR»XJH(2)»T/(H»XJH( 1 ) ) )

)

PEAKDF=XX+0 .577/XX

XXX=SQRT(2. 0«ALOG(USTAR*XJH(4)»T/(H»XJH(3) ) )

)

PEAKAC»XXX+0 .577/XXX

WRITE (6,280) DEF , RMSDEF , RMSACC , PEAKDF , PEAKAC

280 FORMAT (29H1 MEAN ALONGWIND DEFLECT ION . . . , E 1 6 . 8/

134H0RMS OF FLUCTUATING DEFLECT I ONS . . . , E 1 6 . 8/

224H0RMS OF ACCELERAT I ONS . . . , E I 6 . 8/

330H0PEAK FACTOR FOR DEFLECT I ON . . . , E 1 6 . 8/

432H0PEAK FACTOR FOR ACCELERAT 1 ON E 1 6 . 8

)

250 CONTINUE

GO TO 200

END

C/»/ INPUT

SUBROUTINE INPUT

C INPUT OF PARAMETERS FROM CARDS, SEE CHAP. 3 OF THIS REPORT.

INTEGER RLIM

COMMON/ BL0CK2/H , BCON , DCON , XMASS , EN( 8 ) , ZETA( 8 ) , Z 0 , CZ , CY, BETACN,

IFl ,FS,USTAR,T,CW,CL,XN,RHO, IREP, IPRINT,RL1M

READ (5,1) IREP, IPRINT , RLIM

IF (IREP.EQ.O) GO TO 30

70
1 F0RMAT<3I2)

READ (5,2) H,BCON,XMASS

READ (5,2) (EN( I ) , I-l ,RLIM>

READ <5,2) <ZETA< I ) , I»l ,RLIM)

, IF (IREP.EQ.l) READ (5.2) Z 0 . CZ , CY , DCON , BETACN

IF (IREP.EQ.2> READ (5,2) ZO , CZ • Y, DCON , BETACN , F 1 , FS

READ (5,2) USTAR,T

READ (5,2) CW,CL,XN,RHO

2 FORMAT ( 8F6. 0

)

WRITE (6,il) IREP,IPRINT,RL1M

11 FORMAT ( IHl ,35X,26HVALUES OF INPUT PARAMETERS//

li9H0CONTROL PARAMETERS/

114H0 I REP -,13/

214H IPRINT »,I3/

314H RLIM -,I3)

WRITE (6,12) H,BCON,XMASS

12 FORMAT(/22H0STRUCTURAL PARAMETERS/

IMHO H =,F12.3/

114H BCON =,F12.3/

314H XMASS »,F12.3)

WRITE (6,13) (EN( 1 ) , I«l ,RLIM)

13 F0RMAT(22H ( EN ( I ) , I « 1 , RLI M ) . . . ,8F I 0 . 3

)

WRITE (6,IA) (ZETA(I),I«1 ,RLIM)

14 F0RMAT(22H ( ZETA( I ) , I « 1 , RLI M ) . . . , 8F 1 0 . 3

)

WRITE (6,15) ZO ,CZ,CY, DCON, BETACN

15 FORMAT(/31H0MICROMETEOROLOGICAL PARAMETERS/

114H0 ZO »,F12.3/

114H CZ »,F12.3/

214H CY «,F12.3/

214H DCON =,F12.3/

314H BETACN »,F12.3)

IF (IREP.EQ.2) WRITE (6,16) Fl.FS

16 F0RMAT(14H Fl "•,F12.3/

114H FS =,F12.3)

WRITE (6,17) USTAR,T

17 FORMAT(/26H0CLIMATOLOGICAL PARAMETERS/

114H0 USTAR =,F12.3/

114H T =,F12.3)

WRITE (6,18) CW,CL,XN,RHO

18 FORMAT (/23H0AERODYNAMIC PARAMETERS/

114H0 CW *,F12.3/

114H CL =,F12.3/

214H XN »,F12.3/

314H RHO =,F12.3)

RETURN

30 WRITE (6,31)

31 FORMAT(///47H0NORMAL EXIT, END-OF-FILE ENCOUNTERED ON INPUT.)

STOP

END

C/»/INIT

SUBROUTINE INIT

C THIS SUBROUTINE SETS UP THE INTEGRATION PARAMETERS FOR THE CURRENT

C VALUE OF RVAL. NOTE THIS INTEGRATION SCHEME IS VALID ONLY FOR THE

C CASE RVAL«SVAL. SEE CHAP. 3 OF THIS REPORT FOR DESCRIPTION

C OF HOW THE PARAMETERS NF AND NN MAY BE CHOSEN.

71
INTEGER RLIM.RVAL.SVAL

COMMON/ BL0CK2/H ,BCON,DCON,XMASS ,EN( 8) , ZETA< 8 ) , Z 0 , CZ , CY , BETACN

,

IF I ,FS,USTAR,T,CW,CL,XN,RHO, I REP , IPR INT , RLI

M

COMMON/BLOCK3/FTILDA(20 0 ) .ATILDAC 20 0 ) , M, NF< 4 ) , NN( 4

>

COMMON/ BLOC K5/RVAL,SVAL

NF(l)alO

NF < 2 ) = 1

0

NF(3)=20

NF ( 4 > = i 0 NN(I)=5

NN( 2)=6

NN(3)=6

NN(4)=6

FPEAK=EN C RVAL) *H/USTAR

M=l

STEPOL=0

.

DO 110 1=1,4

IF (I.EQ.l) FTILDA(M)=0.

IF <I.EQ.2) FTILDA(M) = ( FPEAK-2 . >/ 3 0

.

IF <I.EQ.3) FTILDA(M)=FPEAK-2.

IF (I.EQ.4) FTILDA(M)=FPEAK+2.

IF (I.EQl) STEPNU=< <FPEAK-2. )/30. >/FLOAT(NF< 1 )

>

IF <I.EQ.2) STEPNU=( ( 29. •FPEAK-58. )/30. )/FL0AT<NF<2)

)

IF (I.EQ.3) STEPNU=4./FL0AT<NF(3)

>

IF (I.EQ.4) STEPNU=(2.»FPEAK ) /FLOAT < NF < 4 )

)

ATILDA(M)=( STEP0L+STEPNU)/2.

MM=NF( I )-l

DO I 0 1 J=l ,MM

MJ=M+J

FT I LDA( MJ ) =FT I LDA( M ) FLOAT ( J > "STEPNU

101 ATILDA<MJ>=STEPNU

M=M+MM+1

110 STEPOL=STEPNU

M = M-1

ATILDA(M>=ATILDA(M>+<2.*FPEAK-STEPOL>*3./20.

WRITE (6,115) RVAL,SVAL,M,NF,NN,FPEAK

115 F0RMAT<44H IVALUES OF INTEGRATION PARAMETERS FOR R,S = ,I2,1H,,I2/

19H0 M =,14/

217H0<NF( I ) , 1 = 1 ,4) . . . ,414/

317H0(NN(I),I=1,4)...,4I4/

49H0 FPEAK =,F12.3)

WRITE (6,116) (FTILDA( I ) , 1=1 ,M)

116 FORMAT(21H0(FTILDA( I ) , 1= 1 ,M) . . . , 1 0F8.3/( 21X, 1 0F8. 3)

>

WRITE (6,117) (ATILDA( I ) , 1=1 ,M)

1 17 F0RMAT(21HQ(ATILDA( I ) , I = I , M) . . . , 1 0 F8 . 3/ ( 2 IX , 1 0 F8 . 3 )

>

RETURN

END

C/*/TRIPLE

SUBROUTINE TRIPLE(N,W,Z,E)

C TRIPLE INTEGRAL EVALUATED BY A MONTE CARLO METHOD.

C INTEGRAL EVALUATED IS I SUB R,S OF FTILDA. (EQ. 2.57 OF THIS REPORT.

C SEE CHAP. 3 IN WHICH THE QUADRUPLE INTEGRAL IS TRANSFORMED INTO

C A TRIPLE INTEGRAL.)

C INPUT...W, THE CURRENT VALUE OF FTILDA

C N, THE NUMBER OF SUBDIVISIONS TO TAKE ON EACH EDGE OF

72
C THE UNIT CUBE FOR THE MONTE CARLO QUADRATURE

C R AND S ARE IN LABELLED COMMON AND ARE NEEDED ONLY IN

C THE FUNCTION SUBROUTINE F THAT EVALUATES THE INTEGRAND

C OF THE TRIPLE INTEGRAL

C OUTPUT... Z, THE APPROXIMATION TO THE TRIPLE INTEGRAL

C E, AN ERROR ESTIMATE FOR Z DOUBLE PRECISION D, DFRA , DK , DKSQ , DPR IME

COMMON/ BLOCK i/KOUNT

DIMENSION D<3) ,DPRIMEC3) , Q < 3 ) ,X ( 3 > ,XPR IME( 3 > , P< 3 > ,R< 3 ) , RPR IME< 3 )

f

1S<3) ,SPRIME<3)

C

C SIX SEQUENCES OF PSEUDO-RANDOM NUMBERS ARE GENERATED

C FOR THE MONTE CARLO QUADRATURE. THEY ARE OF THE FORM

CC X SUB K * FRACT. PART OF (K»«2)*SQRT<P)

C

C WHERE P TAKES THE PRIME VALUES 2,3,5,7,11 AND 13. C VALUES OF K AND K»»2 ARE STORED IN DOUBLE PRECISION

C TO ALLOW THE LARGEST POSSIBLE VALUE OF K**2. THE

C FOLLOWING ARITHMETIC STATEMENT FUNCTION CONTAINS

C A NON-STANDARD INTRINSIC FUNCTION, DMOD, TO PERMIT

C THE DOUBLE PRECISION FRACTIONAL PART TO BE TAKEN.

DFRA<D)=DMOD<D, l.ODO)

IF (KOUNT.GT.O) GO TO 10 DK=0. ODO

D( 1 )=DFRA(DSQRT(2. ODO )

)

D<2>=DFRA<DSQRT(3.0D0)

D(3)=DFRA<DSQRT(5. ODO)

DPRIMEC 1 )=DFRA(DSQRT<7. ODO)

)

DPRIME<2)=DFRA<DSQRT( 1 1 . ODO)

)

DPRIME(3)=DFRA(DSQRT( I . ODO )

)

10 CONTINUE

XN=N

RXN=1 . 0/XN

RXNH=0 .5«RXN

N3«N*N«N

XN3=N3

RXN3»l . 0/XN3

RXN3H=0.5»RXN3

Y»0.0

YPRIME=0.

0

E=0.0

DO 30 K3sl,N

DO 30 K2=1,N

DO 30 Klsl,N

DKsDK-i-1 . ODO

DKSQ=DK»DK

Q< 1 )«K1-1

Q<2)=K2-l

Q( 3)»K3-1

DO 15 I « 1 , 3 X( I )=DFRA(DKSQ»D( I )

)

XPRIMEC I )=DFRA(DKSQ»DPRIME< I >

)

15 CONTINUE

DO 20 I>1,3

73
Q(I)«^<I>«RXN

P< I )«Q(I)-»-RXNH

R( I )»Q( I )+RXN*X< 1

)

RPRIME( I )«Q< I )+RXN»XPRIME( I

>

PI2«2.0»P(1)

S< I )»P12-R< 1

)

SPRIME< I )=PI2-RPRIME< I

)

20 CONTINUE > FR=F(R,W)

FS=F<S,W>

G=FR+FS

FRPR1M=F(RPRIME,V)

FSPR1M=F( SPRIME.W)

GPRIME»FRPRIM+FSPRIM

Y»Y+G

YPR1ME"YPRIME+GPRIME

E«=E+(G-GPRIME)»»2

30 CONTINUE

Y=Y»fiXN3H

YPRIME=YPRIME»RXN3H

Z=0 .5«<Y+YPRIME)

E=0 .25«RXN3«SQRT<E)

RETURN

END

C/VF

FUNCTION F<V,W>

C CALCULATION OF INTEGRAND FOR TRIPLE INTEGRAL

C Y SUB R,S OF FTILDA

C <EQ. 2.57 OF THIS REPORT, WHICH IS TRANSFORMED INTO A TRIPLE

C INTEGAL BY A CHANGE OF VARIABLE. SEE CHAP. 3.)

C THE PROGRAM VARIABLE W STANDS FOR FTILDA. THE VALUES

C OF R AND S ARE STORED IN COMMON BLOCK /BLOCKS/.

C THE VARIABLE KOUNT IN /BLOCKl/ IS SET TO ZERO WHEN

C NEW INPUT PARAMETERS ARE READ BY SUBROUTINE INPUT. IT IS INCREASED

C BY ONE EACH TIME THIS SUBROUTINE IS EXECUTED, SO AT THE END OF THE

C WHOLE FOUR-DIMENSIONAL INTEGRATION, IT CONTAINS THE NUMBER OF TIMES

C THE INTEGRAND FOR THE TRIPLE INTEGRAL WAS CALCULATED.

INTEGER RLIM,RVAL,SVAL

DIMENSION V(3)

CO MMO N/ BLO C K 1 / KOUNT

COMMON/ BLO CK2/H , BCON , DCON , XMASS , EN < 8 ) , ZETA< 8 ) , Z 0 , CZ , CY , BETACN

,

IFl ,FS,USTAR,T,CW,CL,XN,RHO,IREP,IPRINT,RLIM

COMMON/ BLOCKS/ RVAL, SVAL

IF (KOUNT. GT.O) GO TO 2000

C INITIALIZATION.

CONl=2.0«CZ

C0N2=CY»BC0N/ (CZ«H)

2000 CONTINUE

Zl =V( I

)

Z2=V<2>

TT=V<3)

UTI«UTILDA(Z1

)

UT2bUTILDA<Z2)

ALPHA=< 1 . 0-TT)»XMU<RVAL,Zl ) XMU < SVAL , Z2 > 'UT I •UT2

ALPHA«ALPHA»SQRT<STILDA<W,Z1 ,UTl ) "ST I LDA< W , Z 2 ,UT2 >

)

74
BETA«CON i «SQRT ( < Z 1 -Z2)**2+< C0N2»TT)»*2)/ <UT1 UT2>

F«ALPHA»EXP(-BETA»W)

K0UNT«KDUNT*1

RETURN

END

C/«/XMU

FUNCTION XMU< I ,Z)

C COMPUTATION OF MODAL SHAPES.

COMMON/ BLOCKl/KOUNT

COMMON/ BL0CK6/XMU INT <8 ) ,GINTC 6)

DIMENSION XMUTAB<15,8>

C FIRST MODAL SHAPE

DATA XMUTAB< 1,1) ,XMUTAB< 2,1) ,XMUTAB< 3,1) ,XMUTAB( 4,1) ,XMUT AB< 5,1),

1 XMUTAB<6, 1 ) ,XMUTAB<7, 1 ) ,XMUTAB<8, 1 ) ,XMUTAB(9, 1 ) ,XMUTAB( 10,1),

2 XMUTAB( 11,1) ,XMUTAB< 12,1) ,XMUTABC 13,1) ,XMUTAB< 1 4 , 1 ) ,XMUT AB< 15,1)/

3 .00000, .01857, .03714, .05571, .07429,

4 .09286, .11143, .13000, .14857, .16714,

5 .18571, .20429, .22286, .24143, .26000/

C SECOND MODAL SHAPE

DATA XMUTABC 1 , 2 ) ,XMUTAB( 2 , 2 ) ,XMUTAB<3,2) ,XMUTAB<4, 2) ,XMUT ABC 5 , 2 )

,

1 XMUTAB<6,2) ,XMUTAB<7 ,2) ,XMUTAB(8,2) ,XMUTAB<9,2) ,XMUTAB( 10,2),

2 XMUTAB< 11,2) ,XMUTAB( 1 2 , 2 ) ,XMUTAB( 13 ,2) ,XMUTAB< 14,2) ,XMUTAB< 15,2)/

3 .00000, -.05377,-. 10620,-. 14350,-. 17210,

4 -.18310,-. 16310,-. 12920 ,-. 08235,-. 0 2325,

5 .05252, .12850, .22520, .29870, .32350/

C THIRD MODAL SHAPE

DATA XMUTAB( 1 ,3) ,XMUTAB(2,3) ,XMUTAB<3,3) ,XMUTAB(4,3) ,XMUT AB( 5 , 3 )

,

1 XMUTAB<6,3) ,XMUTAB( 7 , 3 ) ,XMUTAB< 8 , 3 ) ,XMUTAB< 9 , 3 ) ,XMUT AB< 10,3),

2 XMUTB( 1 I ,3) ,XMUTAB< 1 2 , 3 ) ,XMUTAB< 13,3) ,XMUTAB( 14,3) ,XMUTAB{ 15,3)/

3 .0000 0,-. 07302,-. 1336 0,-. 16530,-. 16940,

4 -.11011, .02515, .10890, .16540, .18120,

5 . 101 10, -.04585,-. 1928 0,-. 308 0 0 ,-. 3766 0/ C HIGHER MODAL SHAPES NOT USED IN PRESENT SUBROUTINE.

IF (KOUNT.GT.O) GO TO 5 C INITIALIZATION.

DO 10 Js4,8

DO 10 K« 1 , 1

5

10 XMUTABCK, J)=0 .

0

C COMPUTE INTEGRALS OF XMU(I,Z)»»2 AND OF XMU< I , Z ) 'UT I LDAC Z ) 2

.

C THESE ARE DENOTED BY XMUINT AND GINT, RESPECTIVELY, AND THE

C VALUES ARE RETURNED IN COMMON BLOCK /BL0CK6/.

DO 2 J*l ,3

S = 0. 0 T»0.0

DO 1 K«2,14

S=S+XMUTAB(K, J)*UTILDA(FL0AT<K-1 )/ 14. 0 )*»2

1 T=T+XMUTAB<K,J)»»2

S=S+0.5»XMUTAB< 15 , J)«UTILDA( 1 . 0 )»*2

T=T+0.5»XMUTAB( 15 , J)»»2

GINT< J)«S/ 14.

0

2 XMUINT< J)=T/ 14.

0

DO 3 J«4,8

GINT(J)«0.0

3 XMUINT<J)=>0.0

5 CONTINUE

75
Y"M.O*Z

J«Y

IF < J.EQ. 14) GO TO 30

XMU=XMUTAB( J+1 , I) + ( Y-FLOAT ( J ) ) * < XMUT ABC J+2 , I > -XMUT AB( J+ I , I )

)

RETURN

30 XMU=XMUTAB(J+i ,1)

RETURN

END

C/VUTILDA

FUNCTION UTILDACZ)

C NONDIMENSIONALIZED MEAN WIND SPEED AT ELEVATION Z. INTEGER RLIM

COMMON/ BLOCKl/KOU NT

COMMON/ BL0CK2/H , BCON , DCON , XMASS , EN ( 8 ) , ZETAC 8 ) , Z 0 , CZ , CY , BETACN

,

IFl ,FS,USTAR,T,CW,CL,XN,RHO, IREP, IPRINT,RLIM

IF (KOUNT.GT.O) GO TO 1 C INITIALIZATION.

ZA=( 10.0+DCON)/H

UC0NST=2.5»AL0G( 10.0/ZO)

DTILDA=DCON/H

HOZ0=H/Z0

I CONTINUE

UTILDA=UCONST

IF (Z.GT.ZA) UTILDA=2.5*AL0GC <Z-DTILDA)»HOZ0

)

RETURN

END

C/»/STlLDA

FUNCTION STILDA(V,Z,UT)

C THIS FUNCTION IS RELATED TO THE ASSUMED REPRESENTATION

C OF THE SPECTRUM OF LONGITUDINAL WIND FLUCTUATIONS. STILDA

C IS THE EXPRESSION

C N*S(Z,N)/ <FTILDA»USTAR»»2)

C THAT APPEARS IN EQ. 2.57 OF THIS REPORT.

C

C TWO REPRESENTATIONS OF STILDA ARE CODED HERE, ONE A

C MODIFICATION OF A FORMULA (EQ. 2.1A OF THIS REPORT)

C OF KAIMAL, THE OTHER A FORMULA (EQS. 2.15, 2.16 AND 2.11 OF

C THIS REPORT) DEVISED TO STUDY THE DEPENDENCE ON

C THE PARAMETER Fl, THE PEAK SIMILARITY COORDINATE. THE FIRST

C OR SECOND REPRESENTATION WILL BE SELECTED ACCORDING AS

C IREP IS 1 OR 2. INTEGER RLIM

COMMO N/ BLO C K 1 / KOUNT

C0MM0N/BL0CK2/H, BCON, DCON, XMASS, EN<8) ,ZETA(8) ,Z0 , CZ , CY , BETACN,

IF I ,FS,USTAR,T,CW,CL,XN,RHO, IREP, IPRINT.RLIM

IF (KOUNT.GT.O) GO TO 20

GO TO (11,12), IREP

CC INITIALIZATION FOR MODIFIED KAIMAL FORMULA.

II ZA=(DCON+10. 0)/H

DTILDA=DCON/H

GO TO 20

C

C INITIALIZATION FOR FORMULA DEPENDENT ON THE PARAMETER Fl.

12 BETAl =0. 39«FS*»( -2. 0/3. 0 )

76
BETA2=(2. 0/3.0)«BETAl

BETA3=BETACN-BETA1

BETA4=FS-2. 0»FI

BETA5=0 .5»FS-2. 0*F

1

C2 = (BETA4»BETA3-BETA2»BETA5>/< < 1 . 5+ALOG ( FS/F I ) ) BETA4-BETA5

)

IF (BETA^) 16,15,16

15 B2=2. 0»(BETA2*< 1 . 5+AL0G<2. 0 ) ) -BETA3 )/ < FS»FS

)

GO TO 17 16 B2=<BETA2-C2)/ (FS*BETA4)

17 B1=B2-C2/(F1»F1

)

DTILDA=DCON/H

ZA=( 1 0 . 0+DCON)/H

2 0 GO TO ( 1 00 ,200 ) , IREP

C

C MODIFIED KAIMAL FORMULA

100 CONTINUE

ZZ=AMAX1 (Z,ZA)

ZZF=(ZZ-DTILDA>/UT

STILDA=20 0. 0»ZZF/<C 1 . +5 0 . 0 'W^ZZF ) «• 1 .67)

RETURN

C

C FORMULA DEPENDENT ON THE PARAMETER Fl.

200 CONTINUE

ZZ=AMAX1 (Z,ZA)

ZUT=( ZZ-DTILDA)/UT

A=W«ZUT

IF (A.GE.FS) GO TO 202

IF (A.GE.Fl) GO TO 201

STILDA=Bl*ZUT»<A-2. 0»F1

)

GO TO 205

201 STILDA=B2*ZUT»(A-2. 0»F1 )+C2/W

GO TO 205

202 STILDA=0.26/<W»A*»0. 666667)

205 RETURN

END

C/»/FISTAR

FUNCTION FISTARCW)

C CALCULATION OF THE FUNCTION PHI STAR OF FTILDA.

C SEE EQ.2.56 OF THIS REPORT.

C W STANDS FOR FTILDA.

INTEGER RLIM,RVAL,SVAL

C0MM0N/BL0CK2/H,BC0N,DC0N,XMASS ,ENC8) ,ZETA(8 ) , Z 0 , CZ , CY , BET ACN

,

IFl ,FS,USTAR,T,CW,CL,XN,RHO, IREP, IPRINT,RLIM

CO MMON/ BLO CK5/ RVAL , SVAL

A=(W/EN<RVAL) )»<USTAR/H)

E=( W/EN( SVAL) )«CU5TAR/H)

G = 2. "ZETACRVAD'A

D=2.«ZETA(SVAL)*E

A=l . -A»A

E=l . -E«E

FISTAR = (A«E + G*D)/ < < A*A+G»G ) • < E*E+D*D)

)

RETURN

END

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cc a. a. 81

NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

K PUBLICATION OR RI-PORT NO. NBS BSS-74

2. (.ov't Ac( CSS 1 on

N,,.

3. R c ( 1 p icni ' s Accession N'

4. TITLE AND SUBTITLE

The Buffeting of Tall Structures by Strong Winds

5. Publication Dat<- October 1975

6. Performing Organization Code

7. AUTHOR(S)

Emil Simiu and Daniel W. Lozier

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as No. 9

13. Type of Report & Period Covered

BSS

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15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 75-30727

16. ABSTRACT (A 200-word or less factual summary of most significant information. It document includes a significant

bibliography or literature survey, mention it here.)

Certain shortcomings of current procedures for computing alongwind structural response

have been shown to result in unrealistic estimates of tall building behavior under

the action of strong winds. Differences between predictions of fluctuating response

based on various such procedures may be as high as 200%. In recent years, advances

in the state of the art have been made which provide a basis for significantly improved

alongwind response predictions. The purpose of the present work is to present a procedure

for calculating alongwind response which incorporates and utilizes these advances.

The basic structural, meteorological and aerodynamic models employed are described,

and expressions for the alongwind deflections and accelerations, consistent with those

models, are derived. A computer program is presented for calculating the alongwind

response of structures with unusual modal shapes or for which the contribution of

the higher modes to the response is significant. For more common situations, a simple

procedure is presented which makes use of graphs and on the basis of which rapid manual

calculations of the alongwind deflections and accelerations can be performed. Numerical

examples are given to illustrate the use of the computer program and of the graphs.

Results of numerical calculations are used to discuss some of the approximations and errors inherent in the models employed.

17. KEY WORDS (six to twelve entries; alphabetical order; capital! ze only the first l-etter of the first key word unless a proper name; separated by semicolons ) . . ^ . , Accelerations; buffeting; building codes; buildings;

deflections; dynamic response; gust factors; structural engineering; wind engineering;

wind loads.

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