JOORNAL OF

Journal of Wind Engineering ~ ( ~ [ ~ i ~ and Industrial Aerodynamics 72 (1997) 335-344 ELSEVIER

Elastoplastic response of steel damper subjected to wind by the typhoon model

M. Yo s h i da a'*, T. H o n g o a, K. K o n d o a, T. Fujii b

~Kajima Technical Research Institute, 19-1 Tobitakyu 2-chome, Chofit-shi, To@o 182, Japan b College of General Arts and Natural Science Education, Kyoto Sangyo University,

36 Kamikamo Honzan, Kita-ku, Kyoto 603, Japan

Abstract

The object of this study is to evaluate the wind-induced fatigue damage of a steel damper installed at an H-type steel beam for earthquakes. A typhoon model is proposed to evaluate the extreme wind speed and the duration time of a strong wind. The applicability of the two methods, Methods 1 and 2, approximating the elastoplastic response is examined, by compar- ing the approximated response with that calculated by an exact analysis. The typhoon model simulates the observed typhoon conditions well. The wind speeds and accumulated duration times by typhoons are obtained. The difference between approximated and exact responses is small by about 20% even when the deformation of damper becomes nine times of that at the yield point. Overall, it became evident that the across wind response affects the fatigue damage more than the along-wind.

Keywords: Typhoon model; Wind speed; Elastoplastic response; Damper; Fatigue damage

1. Introduction

Earthquake usually is the deciding factor in structural design load in Japan. However, the wind load is also an important factor because strong wind occurrs often and the duration time is long. Therefore, vibration-controlling systems have been adopted in some buildings to cope against large vibrations.

The objective damper is made of a steel plate and applied as the joint element of structural beam, which absorbs the energy by the plastic deformation of steel. It is designed to yield at the earthquake velocity of 10-17cm/s less than the design value(25 cm/s). Meanwhile, the stress by the design wind load is close to but a little

* Corresponding author.

0167-6105/97/$17.00 :t'i: 1997 Elsevier Science B.V. All rights reserved. PII S0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 2 6 4 - X

336 ,tl. )'u~hi~la ul a~ . / l l i m t l:n£, lint. h'roJrn. 72 (1997) 335 344

over the yield stress. The design wind speed is 37 m/s at 10 m high, of which return period is 100 yr in Tokyo. As described, wind load is smaller than earthquake load, but strong wind occurs often and the duration is extensive that it becomes necessary to consider the wind-induced fatigue damage.

The important items or terms to ev;_lluale the wind-induced fatigue darnage are as Ibllows:

( I ) Wiml spued: f h e extem of wind speed and its accumulated duration time must be delined.

(21 Responsu: To cwtluale lhe fatigue damage of damper by wind, the elastoplastic response analysis is necessary.

(3) Fali qm,: The random rcsponsc induced by the wind loads has to be taken into consideration.

In this paper, ~11 and 12) are mainh discussed. Fhe details of(3) will be discussed at othcr opport unities [ I ].

2. Wind speed of typhoon

Limiting to typhoon wind, the maximum wind speed and the accumulated duration time will be analytically cstimated. The main parameters regulating typhoon property are as follows: (1) Coriolis parameter f (2) pressure difference between center and periphery AP, (3) forward speed of lyphoon C, (41 radius of maximum cyclostrophic wind rM, (5) direction of typhoon progress % (61 pressure gradient ~ ~v,, '~ ,, (7) atmo- spheric circulation and fiiction by' ground surtilce.

Where. parameter (1) is delined as the function of only the latitude at a site, the procedures to decide (2)(5) are presented by' mitsuta and Fu,jii [2] and (6) is approximated by using the following Schloemer's equation for the atmospheric pressure P(r):

P(r) =/L, + Ap exp( rxl r). (1)

where/L, is the pressure al the typhoon center and r the radial distance from the typhoon cemer.

The unsettled parameters are in ~7). lhe parameters can be included in Eq. (31 below: however, it becomes complicated. Therefore, simple equations are proposed here in order to reduce computation lime to calculate the maximum wind speed for extremely long period and the accumulated duration time needed for fatigue damage evaluation.

The approxhnalion of typhoon wind speed IL,-,r) at height = and distance r from typhoon center is proposed by Mitsuta and Fujii (2j and is given by

l j-_.rl = i i(r)" O(-,r) (2)

where l l . ( r ) is tile friction-free wind speed. (;(--.rt is wind speed ratio ( V(-j-): I,'l:(r)). The coefficients in (;I---.rt should be changed when calculating wind speeds at a given height and place.

M. Yoshida et al./J. Wind Eng. Ind. Aerodyn. 72 (1997) 335-344 337

The friction free wind speed Vv(r) is expressed as in Eq. (3) and the ratio G(z,r) of V(z,r) to Vv(r) is approximated by Eq. (4) [2]:

V2(r)/rT + i V y ( r ) = (@/(?r)/p(r), 1/rT = (l/r){1 + [C/Vv(r)]sin q~} (3)

G(z,r) = G(z, oc ) + {G(z,rp) - G(z, oc )})(r/rp) k-1 exp{(1 - l/k)[1 - (r/rp)k]}, (4)

where p is the air density, rw the radius of curvature of air flow, ~b the angle of typhoon center to object site, rp the distance from typhoon center where G(z,r) is maximum, oo the distance that is far enough from the t yphoon center and k defines the wind

speed distribution in radial direction. When z is near the sea surface, G(z, rp) = 1.1, G(z, oo ) = 2, rp = rM/2 and k = 2.5,

2.1. Proposed equation of wind speed on land

Approximat ing @/3r as the differential of Eq. (l), the wind speed of friction-free wind Vv(r) defined by Eq. (3) is as follows:

Vv(r) = {(V '2 + 4V2) '/2 - V'}/2, (5)

where V 2 = Ap (rM/r)exp( -- rM/r)/p(r), and V' = C sin 4~ + f" r, To obtain the wind speed at any height on land, three coefficients rp, G(z,rp),

G (z, ,m ) in Eq. (4) are changed, according to the following assumptions: (A. 1) The effect of g round roughness is expressed by the coefficient E -- (ZB/ZG) ~ in

Table 1. Here, ZB and Z c mean the heights varying the wind speed and ~ defines the changing rate,

(A.2) The changes with height of three coefficients, G(z, rv), G(z, oo ) rp in Eq. (4), are assumed to be expressed by a power-law-like, as usually observed wind speed profile.

(A.3) The coefficients for wind near the sea surface are as given after Eq. (4) are assumed as the values for those of Roughness Category I in Table 1 and z = ZB - 5 m. However, G(ZB, oo ) is changed from 2 to 0.45 by simulation of the observed typhoon wind speed on land.

The coefficients on land are now decided as follows:

rp = rM0.5 ~, G(z,rp) = [1.1(E/E,)] t~, G(z, oo ) = [O.45(E/E.)] ~, (6)

where fl = ln(z/ZG)/ln(ZB/ZG) and the suffix I means the Roughness Category 1 in Table 1.

Table 1 Roughness category and wind speed ratio E (E = (ZB/Zo) ~) [3]

I II III IV Seashore Farm Small city Large city

Power value ~ 0.1 0.15 0.25 0.3 Lower limit Zarn 5 10 15 20 Upper limit Z~m 200 250 350 450 Wind speed ratio E 0.691 0.617 0.537 0.444

Note: ZB and Zc are assumed lower and upper height boundaries varying the wind speed,

33~ M }bshi&l ul .l, .I. llTm/t:'n,v. I;M .h'rodl'n. 72 (1997) 335 344

The wind speed of each typhoon is calculated by Eqs. (1) (6), and the accumulated duration lime of each wind speed is obtained by' repetition of the calculation, changing the typhoon parameters defined by' Milsuta and Fujii [2] with Monte-Carlo ran- domize method.

2.2. I'uriliualion ,~/ o'phoon mode/

Fig. 1 shows the wind speeds and their dircclions of the typhoon 9119 observed at Matsue Meteorological Observatory (M M O) and lzumo Airport (IA) near the Japan Sea. The typhoon center approached lo about 55 km from the two sites and passed along the westsidc of ,lapan Archipelago. Symbols [_30 • ',, A in Fig. 1 show the observed data, broken lines arc the almosphcric pressures used for the calculation and smooth full lines arc results of lhe calculation.

The typhoon parameters used for the calculation arc the observed data. The equations to evaluate wind direction, that is followed by force balances of wind forces, atmospheric pressure, friction three and coriolis force, are presented by Yoshida et al. [4]. From the results, the wind speed and direclion calculated by the proposed equations simulate well the observed ones. The Roughness Category of MMO is assumed as III and thai e l l a is the middle of Roughness Category I and 11 in Table 1 similar to those assumed by the observations of site.

2.3. Accmnula led duration lime and maximum wind speed

The 111axinlum wind speed and the accunlulated duration time are calculated disregarding the change of wind direction. The objective cites are Tokyo (35.78'N,139.5 E), Nagoya (35.16 N.136.8 El. Osaka (34.70:N,135.5' E) and Fukuoka {33.60 N,130.4'EI. The Roughness Category is assumed as 1I and the height is 10 m above the ground. The typhoon parameters are defined by the Monte-Carlo ran- domize method. The period of calculation is 10 000 yr and the landing frequency of

(m/sec) 3 0 r- wind speed ~ 1 w,na s e a a p

h hpa) ' / ~ N~IO observed [1000F ' ~ ~ pressure .~

- 0 - " . ~ 8 b~ "kq.,.-" center IA calculated ~ T J _ _~_~% , , wind speed " ~ - ' - k " ~ ~essure

. . . .

.% I i * ._.._,Z.--~ T- " ,k, LMO calculated

-4 -3 -2 -1 0 1 2 3 4 (hour)

Fig. I. Comparison of observed and calculated wind speeds and directions during Typhoon 9119,

M. Yoshida et al./J. Wind Eng. Ind. Aerodyn. 72 (1997) 335-344 339

typhoon of which center pressure is less than 980 hPa is assumed as the mean of 2.21 times/y when the landing area is from 127°E to 143°E [2].

It is well known that the statistically accumulated duration time of each observed wind speed is expressed by the Weibull distribution function. Therefore, the accumu- lated duration time of each wind speed V that is limited only to typhoon is also approximated by the following Weibull function, where the shape parameter of Weibull function is fixed as 1.0:

ln[t(V)] = l n ( T / 6 ) - (l/f)V, (7)

where t is the accumulated duration time, 6 the scale parameter and T the time parameter.

Fig. 2 shows the results for every city (Tokyo 0 , Nagoya A, Osaka [], Fukuoka x ). The calculated data are approximated by a bent line for every city by Eq. (7). Each bent line intersects at the wind speed of 35-38 m/s.

The maximum wind speeds by typhoons during 10 000 yr are 58 m/s(SW) in Tokyo, 63 m/s(SSW) in Nagoya, 64 m/s(SW) in Osaka and 59 m/s(SE) in Fukuoka. The maximum typhoon wind speeds of 100 yr return period are assumed as 36 m/s in Tokyo and 39 m/s in other cities, which are almost the same as the empirical values shown in Ref.[3].

3. E l a s t o p l a s t i c r e s p o n s e

To approximate elastoplastic responses from elastic responses, two methods (Method 1 and Method 2) are selected. The applicability of each approximation is examined by the comparison with the exact elastoplastic response which is analyzed

= i

10 3 ..-

= 1 0

= 1 0 : 5

-~ s3-- N -~ lO E 35

1 ' 10

wind

- • ~ Fukuoka

~ ~ / Osaka -

~ Nagoya " ~ k ~ t ,

Tokyo

I

20 30 40 50 speed (m/sec) at 10m above ground

Fig. 2. Accumulated duration time caused by typhoons during 10 000 yr.

340 M. ~'oshida el al . .I. l~/'tml Eng. hid. Aerodvn. 72 (1997) 335 344

step by step in the time domain. Hereafter, the two kinds of approximated responses are also called approx, response and the exact elastoplastic response of step-by-step analysis is called exact respoHse.

The method of approximation by Method 1 is shown m Fig. 3. The wind load is assumed to be not affected by the yielding of members [5]. However, the deformation of yielded member is restrained by the other surrounding members. Therefore, the load P shown in Fig. 3 includes the restrain force, i.e. the load is larger than the actual charging load, because the charging load of a yielded member is assumed not to be affected by yielding and also that the load is endured by the small deformation. Therefore, the apparent load deflection curve should be obtained by static frame analysis under the incremental pseudo-static loading.

Method 2 is based on energy balances shown as hatched areas in Fig. 4. It is usually used in the earthquake response analysis as in Ref. I-6]. However, for wind-induced response, the subtraction of mean Pm~ and P~,~ is necessary to obtain the virtual energy. Then the virtual anaplitude is APD or z~Pp because the vibration is oriented from the mean [7].

load (P) l e l a s ~ ," elasto-plast ic

i . , ,

p . _ - - ~

] / XD:elastic Xa,:elasto-plastic

XD Xo def lec t ion

Fig. 3. lllustratitm oldamperdellection by Method 1.

load (P) load (P)

A. I Pc, ppF~ ~

l l -

¢al deflection X ~6) deflection X

Fig. 4. Illustration of energy bahmce considered by Method 2.

M. Yoshida et al./.Z Wind Eng. Ind. Aerodyn. 72 (1997) 335-344 341

3.1. Elastoplastic response analysis in time domain

Fig. 5 shows the plan of 112 m high (25 story) structural steel building and the positions of objective steel dampers (8 in total per floor). Each damper is shaped like a butterfly and has been installed in the mid-length of an H-type steel beam as a joint from fifth floor to 23 floor. The shape is designed to hasten the plastic deformation at the nar row center part as shown in the sketch view of the joint. The detail results of deflection and fatigue shown in this paper are limited to the damper at fifth floor where the fatigue damage is the largest.

Fig. 6 shows the structural model that the members in frame including the damper are evaluated separately and that the other frames are modeled as shear bars. The natural period of the fundamental mode is 3.03 s and the critical damping ratio is 0.02 for the elastic vibration.

.44m

32m_ +

Y wind load A response direction ~ ~ ~irection

" ~ X

[] column • steel damper

i l at 5--23 floor

~ H-type steel beam

i

i

i

~ ~ detail of damper joint

Fig. 5. Plan of building and damper position.

C o l u m n

amper

(p,x)

[ am

Frame

rigid floor

i'

,,,, i i

',', modeled shear bar

Fig. 6. Structural model.

342 3I. }.shick~ et ~t/. ./ Ibiml Eng. DM. .lerodvn. 72 (1997) 335 344

The following two cases of wind conditions are considered for a building sited near the Tokyo Bay (Roughness Category ill

(1) Evaluating the fatigue damage during the lifespan o[" the building, the wind speeds are interwtls of 1 from 17 to 53 m:s.

(2) Estimating the change of response during an extreme typhoon, the wind speed is varied with time. The duration is 2 h and the wind speed is 30 53 m/s I-4].

For an easy comparison between the responses in the along-wind and across-wind directions, the deflection is fixed in lhe Y direction as shown in Fig. 5 and also the wind direction is fixed in the X or Y directions for each response. The wind load is adopted by the results of a wind-tunnel test.

Fig. 7 shows the response of a damper during an extreme typhoon of case (2). The maximum deformation of the damper in the along-wind and across-wind directions are nearly equal. The ratio of deformation to the yield point of steel, i.e. the first bend poinl of hysteresis curve, is nine.

3.2. Resulls ol aPtwoximalion Ol elaxloplastic dcJteclion

The deflections to evaluate the applicability of approx, responses arc the mean of along-wind and RMS deflection of across-wind because each of these is the dominant componenl.

Fig. 8 shows the ratios of approx, deflections to exact ones in along-wmd and across-wind directions and also the ratio of fatigue damages calculated by using these deflections. The fatigue damage evaluation is followed by Miner's law, when consider- ing the random elasto-plastic response [1].

The ratios shown in the figures are defined as follows: Merci ratio: Along-wind lnean deflection of approx, response to lhat of exact

response. RMS ratio: Root mean square deflection of across-wind approx, response to that of

exact one. Fatigue ratio: Damper 's fatigue damage by approx, response to that by exact

response.

load (tonf)

,o i @ -60

-1.5-1.0-0.5 0 0.5 1.0 1.5 lal deflectoin (cm)

load (tonf)

30 . . . . . . . . . .

-30 - [ ] ~ ~ 1 ~

_6ol ~ -1.5-i.0-0.5 () 0.5 1.0 1.5

ib, dcflcctoin (cm)

I~ig. 7. Hysteresis curve of damper deflection at Iiflh lloor during extreme typhoon. (a) Along wind response. (b) Across wind response.

M. Yoshida et al./J. Wind Eng. Ind. Aerodyn. 72 (1997) 335--344 343

Fatigue ratio Ratio by Method 1

1.3 I- Mean ratio \ - - , 1 2 / byMethod 1 , , / ~ ' ~ ' ~ - x

1"0F~738 40 42 4~,'46-48"-~-5 ~ ' ~ " ' 22-54 0.9 [ - " ~ _ . . " ~ - - ~ / \ (m/see)

oL ', ,: Fatigue ratio u.o [ , , , / by Method 2 0.7]- "~' l 0.6]- 'k ;'~-, Mean ratio 0 5 L '.~' by Method 2

Rms ratio Ratio by Method I f . .~ [ Fatigue ratio \ _ /

1.2 t by Method 1 " - . ~

24. 052, 4 l . O l . a ~ ~ Q ~ '- , ' , ~ ' , '

i~v .,o "--~', Rms ratio ](m/sec) 0"91 - 'k, "..,ba~ Method 2 J , 0.8 ]'- 4", ~ - ' ~ ' ~ . . / " 0 7 L Fatigue ratio " ' - ' - - - . - ' "

by Method 2

(a) (b)

Fig. 8. Ratios of approximated mean deflection in along-wind direction, root mean square (RMS) deflection in across-wind direction and fatigue damage. (a) Along wind response. (b) Across wind response.

For deflections, the along-wind mean ratio by Method 2 in Fig. 8a, the approx. responses (mean ratio and RMS ratio) by Method 1 and Method 2 in Fig. 8a and Fig. 8b simulate exact responses well up to the wind speed 53 m/s. According to Fig. 8b, both RMS ratios are nearly 1.0 until wind speed 40 m/s, Above 40 m/s, the RMS ratio by Method 1 increases linearly and the maximum is 1.2 at 53 m/s. Meanwhile, the shape of RMS ratio by Method 2 is concave and the minimum is 0.8 at wind speed 46 m/s.

The discrepancy of mean response by Method 1 shown in Fig. 8a is large, however, it does not weigh with the fatigue damage evaluation, considering the reasons described below.

Comparing the mean ratio and the fatigue ratio of Fig. 8a, the fatigue ratio is closet" to 1.0 than the mean ratio. Also comparing Fig. 8a and Fig. 8b, the along-wind fatigue ratio of 0.85-1.05 is closer to 1.0 than that of 0.7-1.3 for across wind. According to Fig. 7, these impley that the fatigue damage is mainly caused by the vibration and the mean deformation is not sensitive for the fatigue damage. Furthermore, according to the figures by Yoshida et al. [1], the total fatigue damage by along-wind response is about 0.5 times of that by across-wind response. The total value is calculated by the integration of fatigue damage at every wind speed considering the duration time of Fig. 2, disregarding the change of wind direction. Also according to Yoshida et al. [1], even if the change of wind direction is disregarded and estimated conservatively, the statistical total fatigue damage ratio of the damper by across-wind response to the ultimate strength is less than several percents during 100 yr. Here, the ultimate strength means the fatigue damage when the damper is cracked or destroyed.

4. Conclusion

The typhoon model and the approximating method to evaluate the elastoplastic response are proposed. The main object is the fatigue damage evaluation of a steel

344 ,,11 Yoshida e* a/ ,L f17~l~/Epzg. Ind. " r~ ~ ' l ' O ~ ' ~ n 72 (1997) 335 344

damper and its practical use. Studies on the typhoon model and response analysis are being promoted and the analytical values are very well. However, the preciseness of fatigue damage is not adequate, as the fatigue test results with the elastoplastic random vibration that the ultimate strength scatters between 0.6 and 1.6 times of that by a constant amplitude vibration. Therefore, the approximated evaluations of wind condition and elastoplastic response are preferable and sufficient for the practical evaluation of fatigue damage. Thc results are as follows:

(ll By introducing parameters of ground surface roughness as proposed in this paper, it becomes possible to calculate the wind speed of typhoon at any place and any height.

12) To ewtluatc the fatigue damage of steel damper briefly and to.judge the necessity of a laborious exact calculation of elastoplastic response, it is useful to evaluate the elastoplaslic response from elastic response by using the balances of loads or energies. The difference between the approximated and exact responses is small by about 20°A,.

Acknowledgemenls

The authors are grateful to professor Y. Mitsuta of Kyoto University, professor T. Ohkuma of Kanagawa University, and professor M. Yamada and associate professor Y. Uematsu of Tohoku University for their advice and kind guidance.

References

[1] M. Yoshida. T. Kobayashi. T. HukunlotO. S. Hanyuda. Y. Matuzaki, Estimation of wind-induced fatigue damage ratio of steed damper controlling slrnclural vibration, Steel Construction Eng., JSSC I (2) 11994) 103 118 (in Japanese}.

[2] Y. Mitsuta, K. Ft0iL Synthesis of typhoon wind patterns by numerical simulation. Natural Disaster Sci. 8(2}(1986) 49 61.

[3] AIJ, Recommendations for loads on buildings, Arch. Inst. of Japan. 1986 (in Japanese). [4] M. Yoshida, 3". Hongo, K. Kondo. T. F u.jii, Elastoplastic response of frame with dampers subjected to

wind by typhoon Model, 9th ICWE, I995, pp. 1585 1596. [5] J. Kanda, Effects of inelastic behavior on gust responses, J. Wind gng 32 11987) 63 64 (in JapanescJ. [6] AIJ. Ultimate strength and deformation capacity of Buildings in seismic design, Arch. Inst. of Japan,

1990 (in Japanese). [7] BJ. Vickery, Wind action on simple yielding structures..I. Eng. Mech. Div. Proc. American Society of

Civil Engineers, 1970, pp. 107 120.