Time history response of a tall building with a tuned mass damper under wind force

Journal of Wind Engineering and Industrial Aerodynamics, 41.44 (1992) 1949-1960 Elsevier

1949

Time History Response of a Tall Building with a Tuned Mass Damper under Wind Force

A. Kawaguchi, A. Teramura and Y. Omote Technical Research Institute, OBAYASHICORPORATION, 4-840 Shimokiyoto Kiyose-shi, Tokyo 204, Japan Abstract Tall buildings and observation towers are occasionally vibrated under strong winds and become uncomfortable for occupants. Therefore, various types of dampers are being developed at present to reduce the vibration in those structures. However, there is no sure way to predict the wind-induced response of a structure with a damper system and to estimate the suppressing effects of dampers under wind loadings. Therefore, authors have tried to simulate a time history wind force, which was not really in the last, to predict the response of a building with a tuned mass damper (TMO), and to discuss the suppressing effect of dampers under wind force. In this paper, it is shown that the simulation of wind loadings in time history is developed practically from the frequency-domain datas of wind, and a computatinal method of response prediction is presented for the structure with the TMD. Using that method, the behaviors of the TMO with various mass ratios and the suppressible quantities can be easily estimated. I, Introduction Recently, a large number of high-rise residential blocks and office buildings have been constructed in urban area in Japan with the aim of intensifying the use of the available land. At the same time, construction of slender structures such as observation towers as local landmarks has been getting popular inprovincial areas. Because these are relatively light structural systems with low stiffness from the point of a seismic design, they have long natural periods and are easily excited by strong winds. Therefore, various types of dampers are being developed at present to reduce the vibration of buildings and to improve the comfort of occupants in these buildings. '>=>3> The City-Corp Center Building4~in the U.S.A.,CN TowerS>in Canada and Sydney Tower 6> in Australia are structures famous for equipping with dampers. KiureghianT>and LuffS>studied about equipments or optimal tuned mass dampers in structures theoretically. Experimental studies on their suppressing effects were carried out by Tanaka, 9> Modie'°>and others.

Under these circumstances, the prediction of wind-induced vibration of a building has become an important problem in ensuring the structural safety of a building and the comfort of occupants in the building. 0167-61~5/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

1950

[VelocityJ Gust Spectrum

JGeneraliTedl "lWind Force ] Aero

Force

Admittance Spec~='um

J"1Response Wechanical Response Admittance Spectrum

t

Time-Oomain Response

Time-Domain Wind Force Fig. 1 Outline of the Computational Procedure

Vibration~ Equation

The wind-induced vibrations of buildings are divided roughly into three categories according to excited forces as follows: (~) Buffeting vibration excited by gust wind, (~) Lateral vibration excited by vortex occuring behind the building. (~) Lateral vibration excited by galloping instability. Tall buildings have been desiged against wind loadings based on the results of detailed wind tunnel tests. To predict wind-induced vibrations without wind tunnel tests, the procedure implemented in frequency-dcmain proposed by Davenport TM has normally been used to estimate the along-wind response of a structure for a wind resistant design. This procedure is a very rational method which is based on a stochastic approach. However, i t is inadequate to handle the problem of plastic response and collapse due to the action of the wind load and of coupling with vibration control systems. Also, structural engineers require simulations of wind=induced vibration, like the time history responses analysis for a seismic design. Kanda, '=) Tamura'~)and Okuma~have conducted response analyses through simulation of fluctuating wind forces. The authors, considering the background above, have undertaken to simulate the time history wind force from the frequency-domain data for a t a i l building, on the assumption that ~ response analysis w i l l take the form of a modal analysis. And then response analyses for the building with or without a tuned mass damper were conducted using the modal data of the original building alone, and investigations have been made on the suppressing effect of ~ampers. The outline of the computational procedure is shown Fig. 1. 2. Vibration Equations Generally, a structure with a tuned mass demper(TMD) is modeled on a multidegree-of-freedom system as shown in Fig, 2, Considering the motion of a TMD equal to the control force, the vibration of an original structure wit}i a TMD can be expressed as equation (1). Moreover, i t is not necessary to calculate the dynamic property of an entire structure system inclued a damper, using the modal characteristics of an original structure.

1951

~,

Structure

Vibration

~:dergaSs

c=~k= XD[t)~TMD. Wind .. Generalized Genaral ized .~___~-P,|tJ Control Force lind Forcc/ 7 o, (t) ~ m, ) X,(t Structure

,

42 s

M

'1111

I I I I I I,

'

~///#//#///////////

Fig. 2 Malutj-0egree-of Freedom System for a Structure with a TMD

Fig. 4 Dimensions of

Fig. 3 Modal System fo r-th Mode

a Model Building

(i)

[m] {~(t)}+[c] {X(t)i+[k] {X(t)} = {Q(t) / + {F (t) }

where, {X(t)} • displacement. {X(t)} • velocity, {X(t)} • acceleration, [m]" mass. [c] : damping. [k] : stiffness, {Q(t)} : wind force. {F(t)} : control force of the TMD, t : time. In the wind resistant design, structural members of structurs should not be in the plastic region through a long duration of strong wind. Therefore, the structure is assumed to respond within the elastic region and the solution apply the mode superposition method. Expressing the displacement of the original sutructure by equation (2), the vibration for the r-th mode can be expressed as equation (3). IX(t)} = :E X,(t)

r'-'I

{~,}

(2)

X, (t)+2h, =,)~, (t) +o~,=X, (t)={e, (t)+F, (t)i/m,

(3)

where. X,(t) : generalized displacement, {$,} : natural vibration mode, mr : generalized mass (=~m,~ri=), Q, (t) : generalized wind force (=2Oi(t) ~r,). F,(t] : generalized control force (=l~F=(t) ¢,t), h, : damping coefficient, (o, : natural circular frequency, and subcription i : node number of mass (i=1,2, ..n), r : number of modal mode (r=1,2,..n,). Moreover, the effective mass of the r-th mode is expressed as the following equation, me=(Tml¢,~)=/()'ml¢,t=), for the original structure.

When a TMD is installed only at node point s vibration equation of the TMD w i l l be as follows,

in

mo~o (t)+co [~(o(t)-X, (t) } ÷ko {Xo (t)-Xs (t)} =0 And. equation (4) can be transformed into equation (5).

the structure,

the (4)

1952

(5)

~o (t) +2ho~ o {Xo (t) -Xs (t) } + ~oD={Xo (t)-×s (t)} =0

Here, mo, co and ko are the mass, damping and stiffness of the TMD, respectively, while Xo(t), ho and ~o are displacement, damping coefficient and natural circular frequency of the TMD. Xs is the displacement of the node s of the original strucLure and is estimated from equation (2). The generalized control force occuring from TMD motion is calculated by the following equation (6). oo

F,(t) = Fo(t) ¢s, = -moXo(t) Cs,

(6)

If, therefore, the item for the external force Q,(t) can be assumed, the time history response of the coupled vibration of the original structure and the TMD can be simulated by coupling and solving equations (3) and (5). Actually, it is necessary to use a shorter incriment time ~t) for time history response because the response of the structure was caluculated using the previous response values of the TMD. ~. A Model Building and a Tuned Mass Damper The particulars of the original model structure, which is a twelve stories building with a rectangular section, are given in Fig. 4 and Table 1, and those of the tuned mass damper (TMD), which is installed on the top of the model building, in Table 2. The mass of the TMD was set varying the mass ratio (u) to the effective mass of the 1-st mode of the original model from 0.5 to 1.0, 2.0 and 5,0%. These values are corresponded to the absolute mass weight ratio of 0.32, 0.64, 1.28 and 3.20 % of the total building. The optimum natural frequency and damping coefficient were selected for TMD using equations (7) and (8). ') Frequency Ratio Damping coefficient

: :

1/(1 + U) [3u/{8(1+u)3}] ','=

(7) (8)

These optimum dynamic properties of the TMD make a significant difference in the situations of loadings and the expectation of reducing any responses. These optimum equations were established to reduce the displacement of harmonic responses for a structure. Actually, in order to i n s t a l l a damper system in a structure, i t w i l l be necessary to adjust the dynamic properties

Table I .,,

Characteristics of the Model Building

Mode Effective

Mass

t-st

(t' s'/m)

Natural Frequency Damping Coefficient

(Hz) (~)

211

2-nd 3-rd 42

2

0,71 1,82 2,93 1.0

2,5

4,1

Table 2

Characteristics of the Tuned Mass Damper

Case No, Mass Rat U io

(~)

T-I

T-2

T-3

7-4

0,5

1,0

2,0

5,0

Natural Frequency (Hz) 0,71 0,70 0,70 0,67 Da~in@ Coefficient (~;) ,r,

,.

4,3

6,0

8,4 12.7

1953 of the damper, because the various characteristics of the structure used for the prediction remain uncertain.

In the analysis here, only the vibration along the weak axis in terms of stiffness of the original building was taken into account. In the prediction of the vibration of the model building, vibrations from the first to third modal mode were taken into account according to the modal analysis, and the suppressing effect due to the actions of the TUD was investigated. 4. Time History Wind Force Generally. on the assumption that a time history data w i l l have an average value of zero and w i l l conform to the stationary Gaussian process, the relationship between the spectra and the time history data can be expressed by a trigonometrical series. 's) According to that relationship, the time history data of the generalized wind force for the r - t h mode is transfomed a r t i f i c i a l l y from a spectrum as equation (9). kn

Q.(t) = Z[{2T-'SQ,(n,)}'I= cos(2 n n , t + ~ , ) ]

(9)

k=l

where, Q,(t) :time history data, T :duration time, So,(n,) "power spectrum, n, • frequency, qJ, : phase difference, subscript k • number of divisions of frequency. In this paper, while the model building was being vibrated by gust wind, fluctuating wind force was simulated in the along-wind direction on the wide wall surface of the model. On the assumption that the model building was to be b u i l t in an urban area, various items were set as follows. ' Profile of average wind velocity: ~z, = ~o(Zi/Zo),t4 9o:reference wind velocity (=23.6m/s, Zo=42m) 10'

I Presented by Davenport ,e~

Ov

I

10 0

10 .I

10-2 - 3 10

Table 3 Root meansquare of Genaralised Wind Force

/

Mode 1-st 2-nd Time Domain s. 69 2.44 Freq. Domain 5.71 2.44 1.01 1.00 Ratio 10 "=

10-'

Frequency

100

n (Hz)

Fig. 5 Spectrum of Wind Velocity

10'

3-rd 1.59 1.59 1.00

1954 ,==.

• Turbulence

• • • •

intensity

• o v / V , o = (6 K ) t / 2

K :a drag c o e f f i c i e n t of t e r r a i n (=0• 009) V,o:average wind v e l o c i t y at 10 ,! above ground Power spectrum of gust wind'nSv i (n) / o v2=2~2/[3 (1+~2) ,/3] n=nLzi/Vzi. Lzi=1200(Zi/10),/4 .shown in Fig. 516) Power spectrum of wind force" Sa,(n)=4P,2/Vi21X(n)l 2 Sv~(n) P~:mean wind pressure at node i (=1/2 pCiVi=A~) Wind force/Wind v e l o c i t y admittance" assumed equal to u n i t y xCn) 2=1 Power spectrum of generalized wind force for the r - t h mode: n

• • • •

n

So, Cn)= T T S a i j ( n ) $ i ~ $ j ~ I-1 j-I Cross spectrum of wind force" Soij (n)=Roij (n) {Soi (n)Soj (n)} '/= Cross-correlation for wind force: Ro~j (n)= exp(kznaz/V) • exp(-kln&l/V) , (kz=kl=8) Wind force c o e f f i c i e n t " C~= 1.4 Phase q~,' random in frequency, but i t is same phase between each mode.

The f l u c t u a t i n g wave form and the power spectrum of generalized wi.Jd force simulated by equation (9) are given in Figs. 6 and 7. respectively• The 0 10

. I-st

50 ,

100 time 150(seconds)200

250

300

-io

10~ 3=rd

Fig. 6 Time History simulated 10ooi I0"I,

10°

,-°,

for Generalized

Wind Forces

100

'

,,/

10"

0-~

I0~ . . . . ~,lJl

I0"~,

i~.t

0"~

3-rd

,-o,,

/

10"1

~ t

10": 10=4

10 "a

10"=

10"

100

I0'

10 "~

I0 "~

10 "1

10°

10'

Frequency n ( H z ) Frequency n (Hz) Fig, 7 Spectra of Generalized Wind Forces •-------: Simulated,- . . . .

10"~

10": 10" 10° I0' Frequency n (Hz)

:Original

1955

Table 4

List of Statistical Response for the Top of the Building

Freq. Oomain Time Oomain Analyses without TMDi without TUD TMD u:0.5~( TUG ~:1.0~ TUD u:2.0%JTUO u:5.0% o peak: 0 peak o peak o .oeek o peak t o peak I st 1.78 6.12 1.73 5.97 !.05 3.61 0.93 3.20 0.83 2.83 0.72 2.44 D~eplece2 nd 0.08 0.29 0.06 0.23 0.08 0.23 0.06 0.23 0.00 0.23 0.06 0.21 merit 3 rd 0.02 0.09 0.01 0.04 0.01 0.04 0.01 0.04 0.01 0.04 10,01 0.0~ (mm) Total 3.19 10.50 3.32 10.81 3.05 8.42 3.01 8.15 2.98 8.00 2.94 7.80 Response •

,,

,,,,,

! st Accelere o 2 nd tion 3 rd (9al} Tote1

3.50 12.10 !.00 3.72 0.57 2.19 3.69 13.10

3.43 11.69 1.95 0.82 3.06 0.82 0.39 1.52 0.39 3.56 12.35 2.22

6.73 3.06 h 52 8.40

1.68 5.99 0.82 3.05 0.39 1.52 1.98 7.92

1.41 0.81 0.39 1.75

4.89 1.11 3.82 3.00 0.75 2.02 1.52 0.39 1.52 7.10 !.48 5.54

Frequency Range (161:0.40 "~,1.42Hz, 2~d:t.42,,-3.64Hz, 3rd:3.64,~,5Hz, Total:O. 0033-,-5Hz) 1~0

50

-10 !0

;

-lOP

/

I00 time

150(seconds)200

I' I ~cceleration (gel)

I

/

250

300

I

~

I"

"

'

Fig. 8(1) T~e HisS;ely response computed for the Top of the Building wtTrtouT /MU

Displacemenl

tO":

Accelerat ion

I0'

" i

Ox

I0'

10° -:

I0 =

10"

0"

10".

0-=

,o...

I!tl,

IVII

tO-=. -. 10-=

I0-=

|O"t

10o

I0'

0"' 10"=

Frequency n (Hz)

Fig. 8(2)

/i!,

/

0-1

10"

100

10'

Frequency n (Hz) Response Spectra of the Top of the Building without TMD : Time-domain,. . . . .

: Frequency-domain

frequency range evaluated is between 0.0033 and 5 Hz, the frequency being divided into 1,500 divisions at a interval of 1/300 Hz. The time duration was 5 minutes. There is close agreement between the shapes of the spectra obtained through simulation and those computed in the frequency region. Also, the root mean square of the wind force agree well in the both analyses as shown in Table 3. I t is indicated that the v a l i d i t y of the simulation method of time history wind force using equation (9) was proved.

1956 50 0 I0~ Displacement (~m)

100 t~e

-I0~ I i0 L Acceleration (901)

|

5~k. J..,, ....... L-~.~- .L..I¢.~I~,. J= . . ,

-io~ Fig. 9(l)

150(seconds)200

.......

250

|

oj

300

|

1

V"'7 7

I

Time History response computed for the Top of the Building with TMD ( C~_~of T-3, IJ=2g ) 10 °

Displacement

"!

10"".

/

Accelerat ion

"IA[

N'

10".

I0".

I0-=.

tO'e

10"=

/ t111t

1

10".

I0" 10" Frequency

Itl, I!tl lVl/

tOo n (Hz)

/ I0"

| I"|

I I]W'

I0" I0 ° 10' FreQuency n (Hz)

Fig. 9(2) ResponseSpectra of the Top of the Building with TMD ( Case of T-3, u=2~) 5, Response Analysis The vibration equation was solved by the Runge-Kutta method, reducing the time interval to 1/5. The l i s t of the response at the top of the building without or with a TMO are given in Table 4. The time history waves and spectra obtained by analyses are shown in Figs. 8, 9 and 10. In the table, the statistical values of each mode were calculated from the integration of the spectrum, while the value of the time history response over the entire range of frequency was read directly from waves, In this paper, a report is made only on the fluctuating component, ignoring the average portions. 5.1 Comparison between the two methods for predicting the response Comparing the result calculated by the time-domain analysis and that by the frequency-domain analysis for the case of non=attached TMD, the results are summarized as follows : (D The tendencies of the spectrum at the natural frequencies and in the lower frequency region agree closely with each other. (~) Although there is acertain amount of dispersion in the responses in th~ high mode vibrations, the acceleration response is large at several tens of

1957 0 50 I00 tinm 10~ Relative Displacement (ram)

20~ Acceleration (gel)

I

15O[seconds) 200 I L

/

L

250 I

300 1

I

J

'o

-=o~ -,oF'"' r"r'"~lF / ,~lr"" iI" """'~''ll~F '"'r~"|,,r lr ~PI~"~' / ~ "w~''14 500k Control Force (kgf)

I

I

|

"-5ooF25G I~' '"~ ...." /?""'qr "- Ir""''""*"~r'ltll"'"/''p'.....VTrT........ ~,lt,'~ ~/~v ~'rl '4~1

Fig. I 0 ( 1 )

Time H i s t o r y

response o f t h e TMD

(

Relat ive DLsplacement

I0'-.

IO o .

,

tO = |.

,,

....

g=2~ )

Accelerat ion

11)°..

,o-,-

II.

I0 "t . . . .

Case o f T-3,

!

,o.,

/lt

tO-i,

"":

,o.,

I0"=

tO"

IO"

lYll

I0 °

Frequency n (Hz)

Fig. I0(2)

/Lllli

I0"*~

lO'*~

IO'

,o.,. tO"

/

10"

IVIA_ '"

Frequency n (Hz)

Response$~ectra of TMD ( Case of T-3, g =24)

percent and the displacement response remains at the level of several percent. (~)Thedifference is negligible in the primary vibration component. From the engineering point of view, i t can be said that there is satisfactory agreement between the time-domain and the frequency-domain analyses. 5.2 The behavior of the TMD The behavior of the TMD is given in Fig. 11. The control force increases with the increase of the effective mass ratio of TMD, while the motion of the TMD mass decreases according to it. Namely, when the mass ratio is increased from 0.5 ;~ to 5 ~,the control force increases by a factor of about 2, while the relative movement and acceleration of the TMD itself is reduced

1958 50"

.-. 4 0

5

0

"

Relative

x

isplacement

I Q

.~ 30

x

5.

~

Control Force -]600

Acceleration \~

c

/

"/'"~

| w

20

•-~ !0

0

¢;s

.p=

-

!

""

4o0 -2o

-"-:i

Power .....

-

o

,,

4-

¢

0

""

0.

_I 0.5

I

I

I I I

1

2

3

Mass Ratio

o

.o~

4 5

u (~)

Fig,, 11 Behavior of the Tuned Mass Damper to RelativeMass Ratio

100I

I- ....

tl- ....

l- ......

Tote[ Peek o

I-st Peak

II

75

X l X 50 o ,f,=l 4~

-o

25-

4iJ

Pio~, - - 0 - J,¢¢o, i o n

o.

~.

Symbol

o--

. . . . . 0,5

lAocle o

- - I - - - - 0 - - --1)-- - 0 - - --G--- - ~ ' 1

Mass Ratio

I 2

I I,I 3 4 5

u

(~)

Fig. 12 Suppression Effect of the Tuned Mass Oa~oer to Mass Ratio

to approximately 1/4. 5. While the spectrum is remarkably prominent in the f i r s t mode of the original building, the movement of the TMO is affected only a l i t t l e from the high-mode of the original building, 5.3 Suppression Effect of the TMO The suppression effect of the TMD can easily be observed in the wave form and the spectrum, Particularly, the spectral peak level of the responses are relatively lower compared with those without the TBD in the f i r s t mode of

1959

the model and the shape of the spectra in the primary natural frequency is s i g n i f i c u n t l y separated with a shallow valley. Although the effect of the TMD is to reduce the vibration in the tuned modal frequency, the motion of the model in other frequencies is almost not affected by the TMD. Therefore, estimating the suppression effect of the TMD in respect of quantity, the ratio of response at the top of the building with that without the TaD was calculated and shown in Fig. 12. An increase in the effective mass of the TMD results clearly in a marked reduction in the primary vibration component, to which the frequency of the TMD is tuned;the response is reduced to 60 ~when the mass ratio ~=0.5 ~, and to between 30 and 40 %when ~=5%. However, the reduction in the displacement response over the entire range frequency is small and only a minimal amount of the effect of the TMD is seen, especially i n t h e root mean square of the displacement, o. Since i t s effect is due to the dynamic characteristics of the external force, for which the predominant frequency is lower than the natural frequencies of the building and is over a wide range of frequency, the TMD makes i t d i f f i c u l t to control the long-period vibration component. On the other hand, i f the vibration of the building is restricted to the narrow range of frequency near or equal to that of the TBD, i t w i l l more effectively reduce the displacement response of the building satisfactorily.

6. Conclusion The following conclusions can be drawn from the investigation above, which was conducted through numerical experiments : It was confirmed that the simulation of time history wind force occured by gust wind, presuming modal analysis, can be used as an extension of the traditional frequency-domain analysis, and is effectively useful in implementing a time history response analysis. Therefore, this simulation method will be applied to wind force excited by vortex and excited galloping instability. ~) The response of the building with the tuned mass damper was analyzed, considering to the interaction of the original building and the damper, based on the modal analysis for the original building. Since this analysis is a simple method, which is not requred to establish the inherent characteristics of an entire structural system included a damper, it will be useful in the future. (~) The suppressing effect of a tuned mass damper was investigated by varying i t s mass ratio for a model building. I t was discovered that the damper can reduce the vibration of primary mode tuned to around 60% when the mass ratio to the primary effective modal mass is 0.5 ~, which is equal to 0.32 % of the absolute total mass of the building, and to around 45% when the effective mass ratio is raised to 2 %, which is 1.46 ~ of the absolute total mass. (~) The TMD, however, has v i r t u a l l y no effect on vibrations with a longer period than the primary vibration or on high-mode vibration of the building. I f i t is necessary to reduce vibration of a structure included high-modes, an active vibration control system w i l l be required.

1960

7. References 1) 7akeda [. : Base Isolation, Vibration Isolation and Vibration Controi of Structures, Gihodo, 1988, (in Japanese) 2) Architectural Institute of Japan:Vibration Control for Tall Building, 1989 (in Japanese) 3) Tsuji M.: Review of Various Mechanical Means for Suppressing Wind-Induced Vibration of Structures, Journal of Wind Engineering of Japan, No. 20,1984 (in Japanese) 4) Wiesner K.B.: Tuned Mass Dampers to Reduce Building Wind Motions, ASGE, 1979 5) Isymov N.,Oavenport A.G. and Monbaliu J.: CN Tower,Toronto, Model and Full Scale Response to Wind, Final Report, 12th Congress, International Associate of Bridge and Structure Engineering, 1984 6) Kowk K.C.S.: Full-Scale Measurements of Wind-Induced Response of Sydney Tower, Journal of W.E.I.A.,Vol. 13, No. 1983 T) Kiureghian A.D., Sackman J.L. and Nour-Omid B.: Dynamic Response of Light Equipment in Structures, Earthquake Engineering Research Center, Report No. UCB/EERC-81/05, April 1981 8) Luft R.W.: Optimal Tuned Mass Dampers for Buildings, ASCE,Vol. lO5, No. ST12, 1979 9) Tanaka H. and Mak C.Y.: Effect of ]uned Mass Damper on Wind Induced Response of Tall Builings, Journal of W.E.I.A.,vol. 4, P. 357-368,1983 lO)Modi V.J.:On the Suppression of Vibrations Using Nutation Dampers,Journal of Wind Engineering of Japan, No. 37,1988, P. 547-556 11)Davenport A.G.: Gust Loading Factors, ASCE,Vol. 93, No. ST3,1967 12)Kanda J. : Effects of Inelastic Behaviour on Gust Responses, Journal of Wind Engineering of Japan, No. 32,1987 (in Japanese) 13)Tamura Y.,et el. :Simulation of Wind-Induced Vibrations of Tall Buildings, Summaries of Technical Papers of Annual Meeting Architectural Institute of Japan, 1988 (in Japanese) 14)Okuma T.,et al.:Numerical Experiment on Wind-Induced Vibration of Chiba Port Tower,Journal of Wind Engineering, No, 41,1989 (in Japanese) 15)Hosiya M.: Vibration Analysis based on the Probablistic Approach, Kajima Publisher, 1981 (in Japanese) 16)Davenport A.G.:The Dependence of Wind Loads on Meteorological Parameters, Wind Effects on Buildings end Structures, 1967