Assessment of wind-induced structural fatigue based on joint probability density function of wind speed and direction

Transcription

1 The 1 World Congress on Advances in Civil, Environental, and Materials Research (ACEM 1) eoul, Korea, August 6-3, 1 Assessent of wind-induced structural fatigue based on oint probability density function of wind speed and direction *Tao-Tao Lin 1), Jun Chen ), Xiang Li 3) and Xiao-Qin Zhang 4) 1), ), 3), 4) Departent of Building Engineering, Tongi University, hanghai, China ) cechen@tongi.edu.cn ABTRACT This paper investigates assessent ethod for wind-induced structural fatigue daage with consideration of oint probability density function of wind speed and wind direction (JPDF). By introducing the JPDF into the traditional equivalent narrow-band ethod, a new forula for calculating fatigue life is derived. A double integral operation of JPDF is introduced into the calculation procedure to replace the original sectionalaccuulation of fatigue daage. Detailed derivation procedure for the forula is presented, followed by a brief description of the selection of JPDF. Applicability of the suggested ethod is verified by applying to a steel antenna, which is installed on the top of a high-rise building, to predict its fatigue life. 1. INTRODUCTION Higher, lighter and longer is the word-wide trend for construction of civil structures. As a result, wind-induced fatigue daage becoes a aor concern in the structural design of slender structures as antennas, TV towers, lap posts, asts etc. For assessent of wind-induced fatigue of slender steel structures, the effect of wind directionality, though very iportant, has either been copletely ignored or has been considered in a sectional anner that fatigue daage is first assessed in several wind direction sectors and accuulated daage is then coputed and taken as the final result. Methods for assessing wind-induced structural daage can be classified into two types as tie doain and frequency doain. Tie doain ethods usually obtain the structural stress tie history by dynaic analysis and count the nubers of stress cycles by rainflow technique. Frequency doain ethods transfor strain records into frequency-doain by the theory of rando vibration, which includes equivalent narrowband ethod (Wirsching 198) and upper and lower-bound solutions (Holes ). Equivalent narrow-band ethod assues the stress process as a narrow-band process and then corrects the result with an epirical factor. In addition, upper and lower-bound solutions introduce epirical expression of standard deviation of stress and then derive the closed-for solutions of upper-bound and lower-bound. Gu (1999) proposed the so- 1) Graduate tudent ) Professor 3) Engineer 4) Engineer

2 called tie-frequency doain ethods by cobining the higher accuracy of tie doain ethods with the higher coputational efficiency of frequency doain ethods. This ethod gains stress power spectru of the key point by the theory of rando vibration firstly. Then it siulates stress tie history by using Monte Carlo ethod and calculates daage by rainflow counting ethod. It is well known that wind directionality has a significant effect on the fatigue daage. One will have a very conservative assessent of fatigue life if the wind directionality hasn t been considered. On the other hand, researches on oint probability properties of wind speed and wind direction (JPDF) has achieved significant iproveent in the past decade. In this connection, we attept in this study to assess the structural fatigue life by introducing a JPDF odel into the traditional wind-sector-based ethod. The calculation procedure is based on equivalent narrow-band ethod which assues the stress process as a narrow-band process and then corrects the result with an epirical factor. A double integral operation of JPDF is introduced into the calculation procedure to replace the original sectional-accuulation of fatigue daage. Therefore, the fatigue assessent accuracy can be iproved in theory. Finally, the suggested ethod has been applied to a steel antenna with round-section to assess its along-wind fatigue life.. PROCEDURE FOR WIND-INDUCED FATIGUE AEMENT UING JPDF We derive in this section a new coputation procedure by replacing the sectional daage calculation with a continuous JPDF function in the equivalent narrow-band ethod. The structural stress process is assued as narrow-band process for which the daage D NB calculation forula can be established. Then, an epirical factor λ is introduced to account for the broad band effect and the final daage will be D NB * λ. Note the polar coordinates express of oint probability distribution of wind speed (radial direction) and wind direction (angular) in Fig.1. uppose the structural stress response due to the differential range ( v< V < v dv, <Θ< d ) in Fig.1 is a zeroean, stationary and narrow-band Gauss process. In sector-based approach, the stress response is assued due to a wind-speed-direction sector as shown in Fig.1b. Therefore, peaks of the stress process peaks, Pt (), will have a Rayleigh distribution as f ( p) p = exp Pv, σv, σv, p (1) Where f ( p) Pv, is probability density function of the stress peaks, σ v, is the standard deviation of the stress process.

3 N N d v W v v dv E W v v E (a) Differential range (b) sectional range Fig. 1 Differential range of wind speed and direction plane In order to calculate the probability density of stress process, we need to adopt the following two assuptions: (1) In stationary Gauss process, for any peak point, a iniu point which has the sae absolute value with the axiu point (peak) point can be found, regardless of the position where the iniu point appear. () We believe only the positive stress range contributes to fatigue daage. However, the negative stress range should be neglected. Consequently, the relation of the peak rando variable Pv, and the stress range rando variable v, will be = P () v, v, According to the basic knowledge of probability theory, the probability density f, can be expressed: function of stress range, ( ) v, dp 1 f f f exp( ) v, v, ( ) = Pv, = Pv, = dv, 4σv, 8σv, (3) In the differential range ( v< V < v dv, <Θ< d ), the nuber of cycles within the stress range to d is where T is tie duration, v, by f ( ) ( ) n = f Tf d (4) v,, v, v, is the rate of crossing in the range and can be obtained

4 ( ) 1 α, 1 ω v v, ω dω v, = = π α v, π v, ( ω) dω f (5) where v, ( ω ) is the power spectru of the stress process. Introducing Eq. (4) into the -N curve function N = K (where, K are aterial constants), we can calculate the fatigue daage in the differential wind-speed-direction range( v< V < v dv, <Θ< d ) and stress differential range ( to d ) as D NB, v,, ( ) fv, T fv, d = (6) K Therefore, the daage of narrow-band process within the total stress range will be D NB, v, Introducing Eq. (3) into Eq.(7), we get v, v, ( ) f T f d = (7) K D ( σ v ) Tf = Γ 1 K v, NB, v,, (8) where Γ ( ) is the Gaa function. An epirical factor λ was suggested by Wirsching (198) to count for the effect of wide-band process. The factor is defined as λ ( 1 a)( 1 εv ) v,, b = a (9) where aterial paraeter a and b are deterined by epirical function Eq. (1); ε v, is paraeter of spectral width and can be deterined by Eq. (11) a =.96.33, b= (1) ε v, = α v, 1 α α (11) v, 4 v, In Eq. (11), α v,, α v, and α 4 v, are the th, th and 4th oent of stress power spectru.

5 Thus, the daage of wide-band process, D,, can be obtained by WB v D ( σ v ) λ Tf = Γ 1 K v, v, WB, v,, (1) The probability of wind occurrence in the range ( v< V < v dv, <Θ< d ) can be calculated fro the JPDF f ( v ) V,, Θ by ( ) P = f v dvd (13) v, V, Θ, Based on all the above equations, the total fatigue daage D in the entire range ( < V <,<Θ< π ) will be π π T D= P D = Γ 1 f v, λ f σ dvd ( ) ( ) (14) v, WB, v, V, Θ v, v, v, K Let D=1, the fatigue life of wind-induced with a double integral operation of JPDF is T = Γ (15) π 1 fv, Θ ( v, ) λv, fv, ( σv, ) dvd K Eq. (14) and (15) are the forula for calculating wind-induced fatigue daage with consideration of oint probability density function of wind speed and direction. 3. ELECTION OF COMPUTATIONAL PARAMETER 3.1 Calculation of stress power spectru Consider low daping case and ignore the coupling effect of various vibration odes of a slender structure with round section, the stress power spectru of the key point can be expressed as (Wang 8)

6 ( ω) s n= k = = k 1 H rcosαω ( z) φ ( z)( z z1 ) dz z1 I H rsinαωn ( z) φn( z)( z z1 ) dz z1 I hn ( ω) h ( ω) (16) where, =k-1 is the ode of X-direction and n=k is the ode of Y-direction; oreover, the ax is the nubers of the ode of X-direction and the ax k is the nubers of the ode of Y-direction. r is outer radius of round-section. α is an angle with thdirection. ω is th frequency. (z) is ass per unit length. φ ( z) is th ode. z 1 is height of key point. I is oent of inertia. h (ω) is generalized displaceent spectru of th ode. 3. JPDF For JPDF odel, this paper uses the Farlie-Gubel-Morgensternoxign (FGM) odel suggested by Erde (11). ( ) ( ) ( ) ( ) ( )( ( )) FV, Θ v, = FV v FΘ 1 δ 1 FV v 1 FΘ (17) where FV, ( v, ) (JCDF); FV ( v ) is the arginal cuulative distribution function of wind speed; F ( ) Θ is oint cuulative distribution function of wind speed and direction Θ is arginal cuulative distribution function of wind direction, and 1 δ 1 is a correlation coefficient. Fro Eq. (17), we can easily get the JPDF as ( ) ( ) ( ) ( ) ( )( ( )) fv, Θ v, = fv v fθ 1 δ 1 FV v 1 FΘ (18) and the arginal PDF of wind speed and direction fv ( v ) and f ( ) Θ can be further expressed as Noral Weibull distribution and ixture von Mises odel of order 5 (Zhang 11; Erde 11). The coefficient δ can be obtained by the fitting the Eq.(18) to the easured JPDF. In this case, we use MATHCAD as a platfor for the fitting process (Zhang 11; Li 5). = α 1 α ω α v v fv ( v) z( v, φ1, φ) ( 1 ω) exp ; v I ( φ1, φ) φ β β β (19)

7 z( v, φ, φ ) = 1 exp ( v φ ) 1 1 π φ () 1 I = z v dv (1) ( φ, φ ) (, φ, φ ) 1 1 φ where, β is scale paraeter. α is shape paraeter. ω is weight. φ 1 and φ are paraeters which have the sae units with v. N ω fθ ( ) = exp kcos ( u) ; < π π I () = 1 ( k ) π 1 1 k I ( k) = exp k cos d π = (3) k =! where, N represents nubers of the odel peaks. ω is a weight, ω 1, ω = 1, (=1, N). k and u π function of the first kind. ( k ) < are paraeters. ( ) k I k is the odified Bessel N 4. NUMERICAL EXAMPLE 4.1 teel antenna The exaple used is a 87 high steel antenna with round-section that is installed on the top (the height is 46) of a high-rise building. For nuerical calculation purpose, the antenna is siplified as three sections whose lengths are 34, 7 and 6 respectively. The outer diaeter /thickness of each section is.4/3c, 1.75/.5c and 1./.c, respectively, see Fig. For dynaic response calculation, the ass density is set as kg/ 3, and a luped ass odel is used (see Fig. ) to calculated the natural frequency and vibrational odels. A Rayleigh daping odel is used with the daping ratio being.5. Kaial spectru odel is selected to siulate the along-wind turbulent coponent, and hiotani odel is used for the spatial correlation. The natural frequencies of the first six odes of the antenna are suarized in Table 1 and the ode shapes are presented in Fig. 3.

8 t 3 = c D 3 =1 N 8 t =.5 c D =1.75 Y Key point 3 W Key point E Key point 1 46 t 1 =3 c D 1 =.4 X Fig. A steel antenna Height of antenna () 5 4 Height of antenna () 5 4 Height of antenna () th and th ode th and 4th ode th and 6th ode Fig. 3 Vibration odes of the antenna Table 1 The first six natural frequencies of the antenna th Frequency(Hz)

9 4. JPDF Field easured wind data (Yang ) fro weather station near the antenna have been analyzed and the paraeters for the JPDF are calculated and suarized in Table and 3. The correlation coefficient is calculated as.115. Fig. 3 and Fig4 show the easured JPDF and the calculated JPDF by the odel defined by Eq. (18). Table Fitting results of wind-speed-pdf (Noral Weibull odel) Paraeters ω φ 1 φ α β R Fitting results Table 3 Fitting results of wind-direction-pdf (ixture von Mises odel (N=5)) (N=5) ω k u Fig. 4 Measured JPDF Fig. 5 Calculated JPDF 4.3 Assessent of wind-induced fatigue life The three key points selected for the fatigue life assessent is deonstrated in Fig.. The points are on the botto section with an angle of sixtieth, ninety and 1th degrees respectively. The wind-induced fatigue life under along-wind loading calculated as 536., and years respectively. 5. CONCLUDING REMARK A new wind-induced fatigue assessent approach has been suggested in this paper that has considered the oint probability density function of wind speed and direction (JPDF). This ethod uses a continuous JPDF rather than a sector based odel and therefore is ore accurate in theory copared with sectional-accuulation. Moreover,

10 it is ore flexible on getting the satisfying accuracy with saller integral intervals of wind speed and direction. Fatigue assessent of a steel antenna is used to verify the application of this new ethod. ACKNOWLEDGEMENT The finical support fro The Proect of National Key Technology R&D Progra in 1th Five Year Plan of China (1BAJ11B) to the second author is appreciated. REFERENCE Erde, E. and hi, J. (11), Coparison of bivariate distribution construction approaches for analysing wind speed and direction data, Wind Energy, 14(1), Gu, M., Xu, Y.L., Chen, L.Z. and Xiang, H.F. (1999), Fatigue life estiation of steel girder of Yangpu cable-stayed Bridge due to buffeting, J. Wind Eng. Ind. Aerodyn., 8, Holes, J.D. (), Fatigue life under along-wind loading - closed-for solutions, Eng. truct., 4(1), Li, M. and Li, X. (5), MEP-type distribution function: a better alternative to Weibull function for wind speed distributions, Ren. Energy, 3(8), Wang, Q.H. and Gu, M. (8), Directional wind-induced fatigue of a real steel antenna, J. of Tongi University (Natural cience), 36(8), (In Chinese) Wirsching, P.H. and Light, M.C. (198), Fatigue under wide band rando stress, J. truct. Div., 16(7), Yang, Y.X., Ge, Y.J. and Xiang, H.F. (), tatistic analysis of wind speed based on the oint distribution of wind speed and wind direction, truct. Eng., (3), 9-36(46). (In Chinese) Zhang, X.Q. (11), Modeling and application of oint probability density function of wind speed and direction, Master Thesis, Tongi University. (In Chinese)