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Statistical study for mean wind velocity in Shanghai area
Journal of Wind Engineering and Industrial Aerodynamics 90 (2002) 1585–1599
Statistical study for mean wind velocity in Shanghai area Yaojun Ge*, Haifan Xiang State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China
Abstract In order to predict mean wind velocity in the Shanghai area, the probability distribution model of extreme values has been used in the statistical analysis of joint distribution of wind speed and corresponding direction in this paper. The theoretical model of the joint probability distribution with directional independent coefficients is set up to describe joint distribution of wind speed and direction, and its application method is developed into three statistical steps, data processing of wind speed records, examination of joint distribution model and estimation of distribution model parameters. These application methods have been used to calculate extreme wind velocity in the surface measurement stations, Baoshan, Chuansha and Longhua, and at the deck level of the Yangpu Bridge in Shanghai. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Mean wind velocity; Wind direction; Statistical study; Joint distribution; Shanghai area
1. Introduction Wind loading is usually expressed by wind velocity, which can be divided into two components according to wind velocity records; long-period mean wind and shortterm fluctuation [1]. The mean wind velocity is statistically described by random variable model. If the statistical data of mean wind velocity records are large enough, the distribution of the data will appear in a fixed pattern, which is usually defined as the probability model of mean wind velocity distribution. Both because what we are concerned most is the extreme velocity record in the statistical analysis and the *Corresponding author. Tel.: +86-21-65982398; fax: +86-21-65984882. E-mail address: [email protected] (Y. Ge). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 2 7 2 - 6
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purpose of the statistical extrapolation is to determine the maximum velocity in a given return period, as far as statistical theory is concerned, it is relatively suitable to use extreme distribution models suggested by Gumbel, Frechet and Weibull [2–7]. The effect of mean wind on structures is related to not only wind velocity but also wind direction. Mean velocity is naturally uneven at different directions, while most engineering structures, especially long-span bridges, usually have significant difference in dimensions at different directions in space. It is necessary, therefore, to study joint action of wind velocity and wind direction on windinduced responses of structures. There are three typical methods to represent the joint effect of wind velocity and wind direction [8], the steady random process [9], the maximum coefficient method [10], and the joint distribution probability [11], which is the most popular way to consider the joint action of wind velocity and wind direction [12–15]. This paper describes a joint distribution model of wind velocity and wind direction, and carries out practical studies about data processing of wind records examination of joint distribution models, estimation of distribution parameters and so on. Three state meteorological stations, Baoshan, Chuansha and Longhua, have been taken as typical examples for describing the mean wind velocity in Shanghai area, and special attention has been paid on the estimation of the mean velocity at the site of the Yangpu Bridge across Huangpu River in Shanghai.
2. Joint distribution model From the views of meteorology, aerodynamics and dynamics of structure, the effect of wind on structures is relevant to wind direction besides wind velocity. Because of the existence of the different elementary forms of atmospheric circulation and local obstacles, the extreme wind climate at every place, which is normally described by a wind rose diagram, is uneven along each direction. When exposed to certain aerodynamic loads, the response of a structure, especially a long-span bridge, is relevant to wind direction too. This paper, therefore, will put forward a joint distribution model of wind velocity and direction with independent coefficients as follows. 2.1. Elementary hypothesis (a) The mean wind velocities at different directions of the same place follow the same distribution model, which can be optimally fitted with all samples of wind velocity records at all directions of a certain place. (b) The distribution parameters of the best fitted model at different directions of the same place are mutually independent and can be optimally estimated only with the sample of wind velocity records at the corresponding direction.
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2.2. Probability model The joint distribution probability model with independent parameters has three forms as follows: x ba Model A ðGumbelÞ: FGa ðxÞ ¼ exp exp ; ð1Þ aa x ba ga Model B ðFrechetÞ: FFa ðxÞ ¼ exp ; aa
ð2Þ
x ba g a Model C ðWeibullÞ: FWa ðxÞ ¼ exp ; aa
ð3Þ
in which the parameters, aa ; ba and ga ; are measures of dispersion, location and shape in compass direction a; respectively. 2.3. Statistical steps The joint distribution model of wind velocity and wind direction can be obtained in three steps as follows: (a) The original data of wind velocity record is collected and processed, which includes the selection of statistical stations, collection of daily records, and modification of standard height. (b) The joint distribution model of wind velocity and direction is examined. One of the three distribution types is optimally determined based on the extreme sample of wind record data at all directions. (c) The optimal estimation of independent parameters of joint distribution models is carried out. Two or three independent parameters of the model are estimated by using the extreme sample of wind velocity record data at the corresponding direction.
3. Data processing of wind record Because of the effect of many factors, the original wind velocity record usually lacks adequate representation and consistency. Data processing of original wind velocity record must be carried out before the statistical analysis and probability description of mean wind velocity. According to some field measurement and research results, there are several factors affecting the original wind velocity record, which can be summarized as follows.
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3.1. Selection of statistical stations Mean wind velocity randomly changes with spatial location and time sequence. In general, the statistical analysis model of mean wind velocity should reflect the randomness of both space and time so that if only described by random process models can the dual randomness of space and time be included. In order to evade the difficulty caused from random process, this paper reflects the space randomness of mean wind by adopting a sample average method, which is very similar to the time average method used to describe the randomness of time. The characteristics of mean wind velocity on the interested site can be approximately described by the weighted average of original wind velocity data collected at several statistical stations, which have the typical wind environment around the site. 3.2. Collection of daily record The original daily records of wind velocity in Shanghai area sometimes show a lack of consistency during the consecutive recording time because of the variation of such factors as apparatus type, recording method, sampling period, and so on. In order to utilize all original data over the recording period, the corresponding observation condition in which the wind velocity data were recorded must be comprehended so that the velocity data collected at different observation conditions can be adjusted to a standard condition. 3.3. Modification of standard height There are mainly two models to describe the velocity profile in the natural boundary layer, that is, the exponential model and the logarithm model. The velocity record modification of standard height can be performed according to these two models. An exponential model was adopted in this study.
4. Examination of distribution model It mainly depends on statistical examination as to whether the data of velocity record can be fit into some joint distribution model. The most widely used methods of examination are w2 examination, K–S examination and likelihood ratio examination. In the case of model examination of velocity distribution, however, the probability plot correlation coefficient (PPCC) examination has been widely used. 4.1. PPCC method The PPCC can be defined as [16] P % ðXi XÞ½M i ðDÞ MðDÞ gD ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; P P 2 % ðXi XÞ ½Mi ðDÞ MðDÞ2
ð4Þ
Y. Ge, H. Xiang / J. Wind Eng. Ind. Aerodyn. 90 (2002) 1585–1599
X% ¼
1X Xi ; n
MðDÞ ¼
1X Mi ðDÞ n
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ð5aÞ
ð5bÞ
in which n is the capacity of the sample, D is the probability distribution to be examined, and X1 ; X2 ; y; Xi are the smallest sample, the second smallest; y; the ith smallest, respectively. For a random variable X which follows the probability distribution D; the minimum distribution of X ; the next minimum distribution and even the ith minimum distribution can be defined, and the mean value of each distribution is supposed to Mi ðDÞ: If the sample data are produced by the distribution D; besides location measure (relevant to mean value) and dispersion measure (relevant to deviation value), Xi for all i is approximately equal to the theoretical value Mi ðDÞ: The Xi BMi ðDÞ plot (called probability plot) is almost linear. These linear characteristics result in gD D1:0: Consequently, the closer the distribution D is to the actual distribution of the sample data, the lesser the difference between gD and 1.0 will be. 4.2. Modified PPCC method It is easy to see that the PPCC method is strictly based on a specific maxim sample, which is taken from the parent distribution and should reflect all the characteristics of the parent sample. Because of the randomness of sampling, however, the PPCC values due to different maxim samples may be varied. Furthermore, the maxim sample is intimately relevant to the sample capacity. A modified PPCC method, called the extreme sample convergence check approach, has been suggested to examine which joint distribution model among three types is optimally fitted to represent the parent distribution [17]. This method can compare not only the PPCC coefficient gD of different distribution models so as to determine the optimal distribution model, but also the PPCC coefficient gD corresponding to different capacities of the extreme sub-samples so as to find the rule about the gradual change of coefficient gD with the capacity of a sample.
5. Estimation of model parameters There are many coefficient estimation methods, for example, digital characteristics method, least-squares method, sequence statistics method, and maximum likelihood method. Among them, the maximum likelihood method has the highest estimation efficiency. If possible, the maximum likelihood method is always suggested. But when it fails sometimes, the least-squares method can be used instead.
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5.1. Estimation of shape parameter g The shape parameter g of joint distribution probability models B and C can be estimated by the maximum PPCC method in which an iterative computing procedure should be used to produce the optimum value of g corresponding to the maximum value of gD : 5.2. Regulation of distribution model If the value of the shape parameter g is in the range of 0ogo25; no regulation is required. But if gX25 or gp0; the original type of joint distribution probability model B or C can be adjusted to model A according to the statistical theory. 5.3. Estimation of parameters a and b (a) For the joint distribution probability model A, the parameters of dispersion and location, a and b; can be directly estimated by the maximum likelihood method as follows [15]: n n x
x
X X i i # xi exp exp ðx% aÞ ¼ 0; ð6aÞ # # a a i¼1 i¼1 " # n x
1X i # exp b ¼ a# ln : n i¼1 a#
ð6bÞ
(b) For the probability model B, the parameters of a and b can be estimated by the maximum likelihood method at first [15]: n X i¼1
# ðgþ1Þ g þ 1 ðxi bÞ ng
n X
# g ðxi bÞ
i¼1
" #1=g n 1X g # ðxi bÞ : a# ¼ n i¼1
n X
# 1 ¼ 0; ðxi bÞ
ð7aÞ
i¼1
ð7bÞ
If the maximum likelihood method fails to estimate the parameters, the least-square method can be adopted [15]: a# ¼
Pn
ðxi xÞ½M % i ðDÞ MðDÞ ; Pn 2 i¼1 ½Mi ðDÞ MðDÞ
i¼1
ð8aÞ
# b# ¼ x% aMðDÞ:
ð8bÞ
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(c) For the probability model C, the parameters of a and b can also be estimated by the maximum likelihood method at first [15]: n X
ðb# xi Þg1
i¼1
n n X g1 X ðb# xi Þg ðb# xi Þ1 ¼ 0; ng i¼1 i¼1
" #1=g n 1X g ðb# xi Þ : a# ¼ n i¼1 If this fails, the least-square method can be used [15]: Pn ðxi xÞ½M % i ðDÞ MðDÞ ; a# ¼ i¼1 Pn ½M ðDÞ MðDÞ2 i i¼1 # b# ¼ x% þ aMðDÞ:
ð9aÞ
ð9bÞ
ð10aÞ
ð10bÞ
6. Application in Shanghai area Shanghai lies in the southeast coast of China, and has the elementary characteristics of oceanic climate. Despite the fact that the area of Shanghai is not very large, in light of the extreme wind velocity mainly affected by typhoon, there are still some differences of the wind velocity records among 11 state meteorological stations in Shanghai. The present study, therefore, concentrates at three typical meteorological stations around Shanghai City, that is, the Baoshan meteorological station at the north of Shanghai, the Chuansha station at the east and the Longhua station at the southwest, shown in Fig. 1. 6.1. Data processing of maximum daily record The daily maximum values of wind velocity for each of the 16 standard compass directions have been collected over a period of 38 consecutive years at the Baoshan Station, 37 years at the Chuansha Station, and 35 years at the Longhua Station. The capacity of wind velocity record sample collected at Baoshan is 13 880, covering the maximum velocity values and corresponding directions on all 13 880 days from January 1, 1959 to December 31, 1996. The capacity of wind velocity record sample collected at Chuansha is 13 514 from January 1, 1959 to December 31, 1995. The capacity of wind velocity record sample collected at Longhua is 12 772 from January 1, 1956 to December 31, 1990 except for several holidays. All wind velocity data collected are properly processed into the 10-min average values at the 10-m height above the ground as shown in Tables 1–3 [18].
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Fig. 1. Brief map of Shanghai.
6.2. Examination of distribution model In order to carry out the modified PPCC method, two series of maxim samples from daily records at each station have been generated assuming that the observations are independent. The first series of maxim samples have been picked up by the period extreme value method with period time of 1 year, 6 months, 3 months, 1 month, 15 days and 8 days, respectively. Based on these six maxim samples, the result based on the modified PPCC method shows that the maximum wind velocities at the Baoshang station (BS) and the Chuansha station (C.S.) in Table 4 generally follow distribution model A, a Gumbel distribution. The PPCC method cannot be used at the Longhua Station since the data is given as frequencies in wind velocity periods, and not as time history. The second series of maxim samples have been produced by the peak-over-threshold method with the threshold values of 0.27%, 0.54%, 1.1%, 3.3%, 6.6% and 12.5%, respectively. According to the modified PPCC method, the best-fitted distribution of six maxim samples at each station shown in Table 5 has been confirmed to be a Gumbel distribution [15]. 6.3. Parameter estimation in measurement stations The maximum likelihood method has been chosen to estimate the parameters aa and ba for the wind velocity in azimuth direction a at each station. The
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Table 1 Frequency of daily maximum records at Baoshang station Comp. direct.
0–2 (m/s)
2–4 (m/s)
4–6 (m/s)
6–8 (m/s)
8–10 (m/s)
10–12 (m/s)
12–14 (m/s)
14–16 (m/s)
16–18 (m/s)
18–20 (m/s)
20–24 (m/s)
S (%)
N NNE NE ENE
0.303 0.144 0.173 0.231
3.285 3.919 4.049 6.088
2.500 3.473 2.622 3.581
0.411 0.367 0.411 0.634
0.022 0.050 0.029 0.065
0.115 0.216 0.086 0.050
0.072 0.259 0.043 0.065
0 0.036 0.022 0.029
0.007 0.029 0.014 0
0 0.014 0.007 0
0 0 0 0
6.715 8.509 7.457 10.742
E ESE SE SSE
0.209 0.058 0.173 0.079
6.463 4.107 2.471 2.522
4.654 4.294 2.601 3.264
0.504 0.612 0.360 0.584
0.029 0.036 0.014 0.022
0.058 0.122 0.108 0.101
0.036 0.137 0.072 0.130
0.007 0.014 0.014 0.007
0 0.022 0.014 0.014
0 0.007 0 0
0 0.014 0 0
11.960 9.424 5.829 6.722
S SSW SW WSW
0.050 0.065 0.058 0.065
1.513 1.326 1.787 1.722
1.390 1.268 0.872 1.066
0.166 0.151 0.094 0.180
0.007 0 0 0.022
0.043 0.014 0.007 0.043
0.050 0.029 0.022 0.058
0 0 0.007 0.022
0.014 0.007 0 0.007
0 0 0 0
0 0 0 0
3.235 2.860 2.846 3.184
W WNW NW NNW
0.094 0.072 0.058 0.130
1.679 1.628 1.578 3.718
1.174 1.880 1.412 3.610
0.209 0.454 0.339 0.569
0.007 0.022 0.022 0.050
0.036 0.159 0.159 0.324
0.043 0.274 0.216 0.382
0.007 0.079 0.014 0
0.014 0.036 0.022 0.043
0 0 0.007 0
0 0 0 0
3.264 4.604 3.826 8.826
S (%)
1.960
47.85
39.66
6.045
0.396
1.643
1.888
0.259
0.245
0.036
0.014
100.0
Table 2 Frequency of daily maximum records at Chuansha station Comp. 0–2 2–4 4–6 direct. (m/s) (m/s) (m/s)
6–8 (m/s)
8–10 (m/s)
10–12 12–14 14–16 16–18 18–20 20–22 S (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (%)
N NNE NE ENE
0 0.007 0 0.007
0.488 0.851 0.614 0.814
1.813 2.901 3.115 3.382
2.768 4.129 4.166 3.589
1.110 1.924 1.665 0.851
0.333 0.548 0.333 0.222
0.037 0.074 0.074 0.022
0.007 0.052 0.044 0.037
0 0.015 0 0
0 0 0 0
0 0 0 0.007
E ESE SE SSE
0 0 0 0
0.844 0.444 0.281 0.215
3.507 2.412 1.532 1.628
3.300 4.018 2.923 2.960
0.577 1.021 1.384 1.399
0.126 0.274 0.274 0.444
0.030 0.030 0.022 0.074
0 0.015 0.037 0.007
0 0.007 0 0
0.007 0 0 0
0 0 0 0
8.391 8.221 6.453 6.726
S SSW SW WSW
0 0.007 0 0
0.363 0.289 0.192 0.311
1.783 1.125 0.814 0.947
3.167 1.672 0.895 0.895
1.095 0.377 0.096 0.126
0.244 0.074 0.007 0.059
0.044 0 0.007 0
0.007 0 0 0
0 0.007 0 0.007
0 0 0 0
0 0 0 0
6.704 3.552 2.013 2.346
W WNW NW NNW
0 0 0 0
0.244 0.311 0.252 0.207
1.006 1.317 1.376 1.332
0.992 1.746 2.153 2.494
0.363 0.777 1.413 1.273
0.215 0.548 0.636 0.481
0.015 0.096 0.133 0.059
0.022 0.022 0.074 0.015
0 0.015 0 0.007
0 0 0 0
0 0 0 0
2.856 4.832 6.038 5.868
S (%)
0.022 6.719 29.99
4.817
0.718
0.340
0.059
0.007
0.007
41.87
15.45
6.556 10.500 10.012 8.931
100.0
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Table 3 Frequency of daily maximum records at Longhua station Comp. 0–2 2–4 direct. (m/s) (m/s)
4–6 (m/s)
6–8 (m/s)
8–10 10–12 12–14 14–16 16–18 18–20 20–28 S (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (%)
N NNE NE ENE
0 0.023 0 0.023
0.876 1.040 0.900 1.557
2.378 2.980 3.199 5.890
1.369 1.400 2.245 2.871
0.305 0.329 0.579 0.712
0.055 0.055 0.110 0.141
0 0.031 0 0.031
0 0.008 0.008 0.008
0 0 0 0
0 0 0.031 0
0 0 0 0.008
4.983 5.867 7.071 11.241
E ESE SE SSE
0.008 0.008 0.008 0.008
2.323 1.259 1.283 1.392
6.821 4.380 3.833 4.161
2.159 2.159 2.057 2.409
0.469 0.493 0.735 0.798
0.110 0.141 0.141 0.110
0.031 0.031 0.031 0.031
0 0.008 0.008 0.008
0.008 0 0 0
0 0 0 0
0.008 0 0 0
11.937 8.479 8.096 8.917
S SSW SW WSW
0 0.008 0.008 0.031
0.821 0.602 0.438 0.516
1.916 0.900 0.571 0.712
0.790 0.250 0.086 0.219
0.164 0.055 0.055 0.055
0.031 0.008 0 0.031
0.031 0 0 0.008
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
3.755 1.823 1.158 1.572
W WNW NW NNW
0.023 0.008 0.008 0
0.743 1.502 1.259 1.095
1.369 2.879 3.364 2.558
0.602 2.159 2.057 1.643
0.164 1.072 0.954 0.438
0.110 0.438 0.274 0.055
0.055 0.110 0.086 0.031
0 0.031 0.008 0
0 0.008 0 0
0 0 0 0
0 0 0 0
3.066 8.206 8.010 5.820
S (%)
0.164 17.61
7.376 1.807
0.508
0.086
0.016
0.031
0.016
47.92
24.48
100.0
Table 4 Values of PPCC based on the period samples Station name
Distr. model
12 month
6 month
3 month
1 month
15 day
8 day
B.S.
A B C
0.97 0.92 0.93
0.95 0.92 0.97
0.94 0.90 0.97
0.95 0.91 0.96
0.95 0.90 0.91
0.95 0.92 0.90
C.S.
A B C
0.99 0.99 0.90
0.99 0.92 0.91
0.99 0.93 0.92
0.99 0.92 0.90
0.99 0.93 0.91
0.99 0.90 0.92
maximum-likelihood estimators of aa and ba at three stations can be derived from Eqs. (6a) and (6b). The parameters aa and ba of the 16 compass directions in these measurement stations can be calculated and unified to the time unit of year in Table 6. In order to obtain the parameters in the most unfavorable condition, three extreme samples, including the maxim samples of 1 year, 3 months and 1 month, are put into estimation at each station. In the case of the Baoshan station, the parameters of the joint distribution model A at almost all directions are controlled by the maxim sample of 1 year except the parameters at the direction of SSW. The distribution parameters at all 16 directions are controlled by the maxim sample of
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Table 5 Values of PPCC based on the POT samples Station name
Distr. model
0.27 (%)
0.54 (%)
1.1 (%)
3.3 (%)
6.6 (%)
12.5 (%)
B.S.
A B C
0.96 0.87 0.86
0.97 0.95 0.95
0.97 0.95 0.95
0.98 0.95 0.95
0.93 0.94 0.95
0.94 0.94 0.94
C.S.
A B C
0.99 0.97 0.97
0.99 0.98 0.98
0.97 0.95 0.94
0.97 0.94 0.93
0.98 0.96 0.95
0.99 0.96 0.95
L.H.
A B C
0.99 0.91 0.90
0.99 0.89 0.88
1.00 0.90 0.88
0.99 0.93 0.92
0.99 0.93 0.92
0.99 0.96 0.95
Table 6 Parameters of the model A in measurement stations No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Comp. direct
N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW Non
Baoshang station
Chuansha station
Longhua station
a (m/s)
b (m/s)
a (m/s)
b (m/s)
a (m/s)
b (m/s)
2.4008 4.1856 2.9048 2.4427 2.0873 4.0492 2.5482 2.8722 2.1395 2.0054 1.9607 2.9108 3.3326 4.0748 4.1031 3.7250 3.7561
7.3020 9.9518 7.8151 8.5724 7.7565 9.4166 7.0570 8.3996 6.7052 6.5453 5.5037 7.2782 6.9781 8.8939 8.2240 10.1731 16.7761
2.8208 2.4911 2.2863 2.1117 2.0157 2.3685 2.8976 2.4384 2.7310 2.2348 2.3947 2.3947 2.5440 3.0293 3.1298 3.3317 2.1285
11.5768 12.6816 12.0952 11.2737 10.3525 11.3791 11.6564 11.6714 11.5979 9.6609 7.7686 8.7218 10.2042 12.3536 12.3661 11.7972 15.5242
0.6232 0.9872 1.5753 1.1926 1.4351 0.9372 0.9252 0.8718 1.1987 0.8218 1.4095 1.1965 1.2577 0.9447 0.8077 0.7713 1.7003
10.3439 10.8932 11.6096 11.8330 11.5892 11.6155 11.6306 11.4606 9.8680 8.0567 6.4235 8.2560 11.2305 13.6021 12.8571 10.9162 16.2174
3 months at the Chuansha station. Except the fact that the parameters at SW are controlled by the maxim sample of 3 months, the parameters at the other directions of the Longhua station are controlled by the maxim sample of 1 year. 6.4. Statistical estimation at specific site The above model can be used to obtain surface or upper-level description of the wind climates in the meteorological station or a specific site, for example, the site of
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the Yangpu Bridge across Huangpu River in Shanghai City. In the absence of actual deck-level data in the bridge site, the probability distribution of gradient wind speed and direction at the bridge site can be estimated by the Gumbel model derived from the surface anemometer records in the meteorological station, and this can be done by directly adjusting the parameters aba ¼ xasa ;
ð11aÞ
bba ¼ xbsa
ð11bÞ
in which aba and bba are the parameters for gradient or deck-level wind speeds at the bridge site, and asa and bsa are the parameters for surface wind speeds in the record station. For an anemometer located in approximately homogeneous terrain, x can be assumed to be independent of wind direction and is given by the power-law profile as ab zs db x¼ ð12Þ ds zb in which zs and ds are, respectively, the height and the atmospheric boundary thickness where the surface measurements were made; zb and db are, respectively, the height of the bridge deck and the boundary thickness at the bridge site, and as and ab are the exponents at the surface and bridge sites, respectively. After all parameters aba and bba and sample number na of the 16 compass directions in the three measurement stations are obtained, the weighted average parameters and the sample size for the deck level of the bridge site can be computed as follows: a% ba ¼
3 X
Zk akba ;
ð13aÞ
Zk bkba ;
ð13bÞ
k¼1
b%ba ¼
3 X k¼1
n% a ¼
3 X
Zk nka
ð13cÞ
k¼1
in which Zi ði ¼ 1; 2; 3Þ are the weight factors due to the distances di ði ¼ 1; 2; 3Þ between the bridge site and each measurement station, and can be calculated by Z1 ¼ d2 d3 =D;
ð14aÞ
Z2 ¼ d3 d1 =D;
ð14bÞ
Z3 ¼ d1 d2 =D;
ð14cÞ
D ¼ d1 d2 þ d2 d3 þ d3 d1 :
ð15Þ
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Table 7 Parameters of mean wind velocity No.
Dir.
a
b
m
s
U100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW Non
1.883 2.470 2.219 1.878 1.825 2.370 2.064 2.000 1.983 1.644 1.897 2.117 2.319 2.592 2.583 2.515 2.482
9.784 11.17 10.58 10.63 9.993 10.85 10.21 10.57 9.429 8.095 6.565 8.098 9.570 11.73 11.25 10.96 16.17
10.87 12.60 11086 11.72 11.05 12.22 11.40 11.72 10.57 9.04 7.66 9.32 10.91 13.22 12.74 12.42 17.60
2.41 3.17 2.85 2.41 2.34 3.04 2.65 2.56 2.54 2.11 2.43 2.71 2.97 3.32 3.31 3.23 3.18
18.44 22.53 20.78 19.27 18.39 21.75 19.70 19.77 18.55 15.66 15.26 17.84 20.24 23.65 23.13 22.54 27.59
Based on the above procedure, all weighted average values of the parameters a and b; the mean value m and mean square deviation s; and the expected wind velocity U100 for the returned period of 100 years in the 16 compass directions and the nondirection sample at the deck level of the Yangpu Bridge have been obtained in Table 7. Some conclusions can be drawn from Table 7 as follows: (a) The mean or expected values of wind velocity are uneven at different directions. The values at the direction of SSW, SW and WSW are relatively smaller, but the values at WNW, NW and NNW are larger. The ratios of the non-directional expected value in the 100-year return period to the maximum (WNW) and minimum (SW) expected values are about 1.17 and 1.81, respectively. (b) The standard deviations s=m of wind velocity are also different at each direction. The largest and smallest values of standard deviations are approximately equal to 0.32 at SW and 0.18 at the non-direction item, respectively. (c) As opposed to the numbers of expected velocities U100 ; the current code in China would have given Um ¼ 33:9 m=s for all directions. The expected velocities listed in Table 2 are much less than this figure, in particular, Um =U100 ¼ 1:23 at the non-direction item.
7. Conclusions Based on the statistical analysis of mean wind velocity in the Shanghai area, the joint probability distribution models of extreme values is applied to statistical
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analysis of joint action of wind speed and its direction in this paper. Three analytical models of the joint probability distribution with directional independent parameters are used to describe joint distribution, and its application method is developed into three steps, data processing of wind records, examination of distribution models and estimation of model parameters. These approaches have been used to estimate extreme wind velocity at the surface level of three meteorological stations in the Shanghai area and the deck level of the Yangpu Bridge across Huangpu River. It can be concluded that mean values and mean square deviations of wind velocities vary with directions and this should be considered in the definition of mean wind velocity.
Acknowledgements The research was financially supported by the National Science Foundation of China under Grant No. 59895410.
References [1] G.H. Li, Stability and Vibration of Bridge Structures, China Railway Press, China, 1992 (in Chinese). [2] E.J. Gumbel, Statistics of Extremes, Columbia University Press, New York, 1958. [3] E.J. Gumbel, Statistical theory of extreme values and some practical applications, NBS Applied Mathematics Series 33, US Department of Commerce, 1954. [4] H.C. Shellard, The estimation of design wind speed, Proceedings of the First International Conference on Wind Engineering, Teddington, 1963, pp. 29–52. [5] A.G. Davenport, The dependence of wind loads on meteorological parameters, Proceedings of the International Seminar of Wind Effects on Buildings and Structures, Ottawa, 1967. [6] A.G. Davenport, N. Isyumov, T. Jandali, A study of wind effects for the Sears Project, University of Western Ontario, Engineering Science Research Report, BLWTL-5-71, 1971. [7] J.R. Mayne, The estimation of extreme winds, J. Wind Eng. Ind. Aerodyn. 5 (1979) 109–137. [8] E. Simiu, R.H. Scanlan, Wind Effects on Structures, 3rd Edition, Wiley, New York, 1996. [9] P. Sparis, J. Autonogiannakis, P. Papadopulos, Markov matrix coupled approach to wind speed and direction simulation, Wind Eng. 19 (3) (1995) 121–123. [10] E. Simiu, J.J. Filliben, Wind direction effects on cladding and structural loads, Eng. Struct. 3 (1981) 181–186. [11] C.J. Baynes, The statistics of strong wind for engineering application, Ph.D. Thesis, University of Western Ontario, 1974. [12] L. Gomes, B.J. Vickery, On the prediction of extreme wind speeds from the parent distribution, Research Report No. R241, University of Sydney, 1974. [13] N.J. Cook, Towards better estimation of extreme winds, J. Wind Eng. Ind. Aerodyn. 9 (1982) 295–323. [14] R.I. Harris, Gumbel revisited: a new look at extreme value statistics applied to wind speed, J. Wind Eng. Ind. Aerodyn. 59 (1996) 1–22. [15] Y.J. Ge, Reliability theory and its applications to wind induced vibration of bridge structures, Ph.D. Thesis, Tongji University, 1997 (in Chinese). [16] E. Simiu, J.J. Filliber, Statistical analysis of extreme winds, Technical Note 868, National Bureau of Standards, Washington, DC, 1975.
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[17] Y.J. Ge, Z.X. Lin, H.F. Xiang, Extreme sample convergence check method for wind velocity distribution, Proceedings of the Fifth National Conference on Wind Engineering and Industrial Aerodynamics, Hunan Province, 1998 (in Chinese). [18] Y.J. Ge, Wind speed daily records of the Baoshang, Chuansha and Longhua meteorological stations, Technical Report of Tongji University, 1997 (in Chinese).
Statistical study for mean wind velocity in Shanghai area Yaojun Ge*, Haifan Xiang State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China
Abstract In order to predict mean wind velocity in the Shanghai area, the probability distribution model of extreme values has been used in the statistical analysis of joint distribution of wind speed and corresponding direction in this paper. The theoretical model of the joint probability distribution with directional independent coefficients is set up to describe joint distribution of wind speed and direction, and its application method is developed into three statistical steps, data processing of wind speed records, examination of joint distribution model and estimation of distribution model parameters. These application methods have been used to calculate extreme wind velocity in the surface measurement stations, Baoshan, Chuansha and Longhua, and at the deck level of the Yangpu Bridge in Shanghai. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Mean wind velocity; Wind direction; Statistical study; Joint distribution; Shanghai area
1. Introduction Wind loading is usually expressed by wind velocity, which can be divided into two components according to wind velocity records; long-period mean wind and shortterm fluctuation [1]. The mean wind velocity is statistically described by random variable model. If the statistical data of mean wind velocity records are large enough, the distribution of the data will appear in a fixed pattern, which is usually defined as the probability model of mean wind velocity distribution. Both because what we are concerned most is the extreme velocity record in the statistical analysis and the *Corresponding author. Tel.: +86-21-65982398; fax: +86-21-65984882. E-mail address: [email protected] (Y. Ge). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 2 7 2 - 6
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purpose of the statistical extrapolation is to determine the maximum velocity in a given return period, as far as statistical theory is concerned, it is relatively suitable to use extreme distribution models suggested by Gumbel, Frechet and Weibull [2–7]. The effect of mean wind on structures is related to not only wind velocity but also wind direction. Mean velocity is naturally uneven at different directions, while most engineering structures, especially long-span bridges, usually have significant difference in dimensions at different directions in space. It is necessary, therefore, to study joint action of wind velocity and wind direction on windinduced responses of structures. There are three typical methods to represent the joint effect of wind velocity and wind direction [8], the steady random process [9], the maximum coefficient method [10], and the joint distribution probability [11], which is the most popular way to consider the joint action of wind velocity and wind direction [12–15]. This paper describes a joint distribution model of wind velocity and wind direction, and carries out practical studies about data processing of wind records examination of joint distribution models, estimation of distribution parameters and so on. Three state meteorological stations, Baoshan, Chuansha and Longhua, have been taken as typical examples for describing the mean wind velocity in Shanghai area, and special attention has been paid on the estimation of the mean velocity at the site of the Yangpu Bridge across Huangpu River in Shanghai.
2. Joint distribution model From the views of meteorology, aerodynamics and dynamics of structure, the effect of wind on structures is relevant to wind direction besides wind velocity. Because of the existence of the different elementary forms of atmospheric circulation and local obstacles, the extreme wind climate at every place, which is normally described by a wind rose diagram, is uneven along each direction. When exposed to certain aerodynamic loads, the response of a structure, especially a long-span bridge, is relevant to wind direction too. This paper, therefore, will put forward a joint distribution model of wind velocity and direction with independent coefficients as follows. 2.1. Elementary hypothesis (a) The mean wind velocities at different directions of the same place follow the same distribution model, which can be optimally fitted with all samples of wind velocity records at all directions of a certain place. (b) The distribution parameters of the best fitted model at different directions of the same place are mutually independent and can be optimally estimated only with the sample of wind velocity records at the corresponding direction.
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2.2. Probability model The joint distribution probability model with independent parameters has three forms as follows: x ba Model A ðGumbelÞ: FGa ðxÞ ¼ exp exp ; ð1Þ aa x ba ga Model B ðFrechetÞ: FFa ðxÞ ¼ exp ; aa
ð2Þ
x ba g a Model C ðWeibullÞ: FWa ðxÞ ¼ exp ; aa
ð3Þ
in which the parameters, aa ; ba and ga ; are measures of dispersion, location and shape in compass direction a; respectively. 2.3. Statistical steps The joint distribution model of wind velocity and wind direction can be obtained in three steps as follows: (a) The original data of wind velocity record is collected and processed, which includes the selection of statistical stations, collection of daily records, and modification of standard height. (b) The joint distribution model of wind velocity and direction is examined. One of the three distribution types is optimally determined based on the extreme sample of wind record data at all directions. (c) The optimal estimation of independent parameters of joint distribution models is carried out. Two or three independent parameters of the model are estimated by using the extreme sample of wind velocity record data at the corresponding direction.
3. Data processing of wind record Because of the effect of many factors, the original wind velocity record usually lacks adequate representation and consistency. Data processing of original wind velocity record must be carried out before the statistical analysis and probability description of mean wind velocity. According to some field measurement and research results, there are several factors affecting the original wind velocity record, which can be summarized as follows.
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3.1. Selection of statistical stations Mean wind velocity randomly changes with spatial location and time sequence. In general, the statistical analysis model of mean wind velocity should reflect the randomness of both space and time so that if only described by random process models can the dual randomness of space and time be included. In order to evade the difficulty caused from random process, this paper reflects the space randomness of mean wind by adopting a sample average method, which is very similar to the time average method used to describe the randomness of time. The characteristics of mean wind velocity on the interested site can be approximately described by the weighted average of original wind velocity data collected at several statistical stations, which have the typical wind environment around the site. 3.2. Collection of daily record The original daily records of wind velocity in Shanghai area sometimes show a lack of consistency during the consecutive recording time because of the variation of such factors as apparatus type, recording method, sampling period, and so on. In order to utilize all original data over the recording period, the corresponding observation condition in which the wind velocity data were recorded must be comprehended so that the velocity data collected at different observation conditions can be adjusted to a standard condition. 3.3. Modification of standard height There are mainly two models to describe the velocity profile in the natural boundary layer, that is, the exponential model and the logarithm model. The velocity record modification of standard height can be performed according to these two models. An exponential model was adopted in this study.
4. Examination of distribution model It mainly depends on statistical examination as to whether the data of velocity record can be fit into some joint distribution model. The most widely used methods of examination are w2 examination, K–S examination and likelihood ratio examination. In the case of model examination of velocity distribution, however, the probability plot correlation coefficient (PPCC) examination has been widely used. 4.1. PPCC method The PPCC can be defined as [16] P % ðXi XÞ½M i ðDÞ MðDÞ gD ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; P P 2 % ðXi XÞ ½Mi ðDÞ MðDÞ2
ð4Þ
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X% ¼
1X Xi ; n
MðDÞ ¼
1X Mi ðDÞ n
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ð5aÞ
ð5bÞ
in which n is the capacity of the sample, D is the probability distribution to be examined, and X1 ; X2 ; y; Xi are the smallest sample, the second smallest; y; the ith smallest, respectively. For a random variable X which follows the probability distribution D; the minimum distribution of X ; the next minimum distribution and even the ith minimum distribution can be defined, and the mean value of each distribution is supposed to Mi ðDÞ: If the sample data are produced by the distribution D; besides location measure (relevant to mean value) and dispersion measure (relevant to deviation value), Xi for all i is approximately equal to the theoretical value Mi ðDÞ: The Xi BMi ðDÞ plot (called probability plot) is almost linear. These linear characteristics result in gD D1:0: Consequently, the closer the distribution D is to the actual distribution of the sample data, the lesser the difference between gD and 1.0 will be. 4.2. Modified PPCC method It is easy to see that the PPCC method is strictly based on a specific maxim sample, which is taken from the parent distribution and should reflect all the characteristics of the parent sample. Because of the randomness of sampling, however, the PPCC values due to different maxim samples may be varied. Furthermore, the maxim sample is intimately relevant to the sample capacity. A modified PPCC method, called the extreme sample convergence check approach, has been suggested to examine which joint distribution model among three types is optimally fitted to represent the parent distribution [17]. This method can compare not only the PPCC coefficient gD of different distribution models so as to determine the optimal distribution model, but also the PPCC coefficient gD corresponding to different capacities of the extreme sub-samples so as to find the rule about the gradual change of coefficient gD with the capacity of a sample.
5. Estimation of model parameters There are many coefficient estimation methods, for example, digital characteristics method, least-squares method, sequence statistics method, and maximum likelihood method. Among them, the maximum likelihood method has the highest estimation efficiency. If possible, the maximum likelihood method is always suggested. But when it fails sometimes, the least-squares method can be used instead.
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5.1. Estimation of shape parameter g The shape parameter g of joint distribution probability models B and C can be estimated by the maximum PPCC method in which an iterative computing procedure should be used to produce the optimum value of g corresponding to the maximum value of gD : 5.2. Regulation of distribution model If the value of the shape parameter g is in the range of 0ogo25; no regulation is required. But if gX25 or gp0; the original type of joint distribution probability model B or C can be adjusted to model A according to the statistical theory. 5.3. Estimation of parameters a and b (a) For the joint distribution probability model A, the parameters of dispersion and location, a and b; can be directly estimated by the maximum likelihood method as follows [15]: n n x
x
X X i i # xi exp exp ðx% aÞ ¼ 0; ð6aÞ # # a a i¼1 i¼1 " # n x
1X i # exp b ¼ a# ln : n i¼1 a#
ð6bÞ
(b) For the probability model B, the parameters of a and b can be estimated by the maximum likelihood method at first [15]: n X i¼1
# ðgþ1Þ g þ 1 ðxi bÞ ng
n X
# g ðxi bÞ
i¼1
" #1=g n 1X g # ðxi bÞ : a# ¼ n i¼1
n X
# 1 ¼ 0; ðxi bÞ
ð7aÞ
i¼1
ð7bÞ
If the maximum likelihood method fails to estimate the parameters, the least-square method can be adopted [15]: a# ¼
Pn
ðxi xÞ½M % i ðDÞ MðDÞ ; Pn 2 i¼1 ½Mi ðDÞ MðDÞ
i¼1
ð8aÞ
# b# ¼ x% aMðDÞ:
ð8bÞ
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(c) For the probability model C, the parameters of a and b can also be estimated by the maximum likelihood method at first [15]: n X
ðb# xi Þg1
i¼1
n n X g1 X ðb# xi Þg ðb# xi Þ1 ¼ 0; ng i¼1 i¼1
" #1=g n 1X g ðb# xi Þ : a# ¼ n i¼1 If this fails, the least-square method can be used [15]: Pn ðxi xÞ½M % i ðDÞ MðDÞ ; a# ¼ i¼1 Pn ½M ðDÞ MðDÞ2 i i¼1 # b# ¼ x% þ aMðDÞ:
ð9aÞ
ð9bÞ
ð10aÞ
ð10bÞ
6. Application in Shanghai area Shanghai lies in the southeast coast of China, and has the elementary characteristics of oceanic climate. Despite the fact that the area of Shanghai is not very large, in light of the extreme wind velocity mainly affected by typhoon, there are still some differences of the wind velocity records among 11 state meteorological stations in Shanghai. The present study, therefore, concentrates at three typical meteorological stations around Shanghai City, that is, the Baoshan meteorological station at the north of Shanghai, the Chuansha station at the east and the Longhua station at the southwest, shown in Fig. 1. 6.1. Data processing of maximum daily record The daily maximum values of wind velocity for each of the 16 standard compass directions have been collected over a period of 38 consecutive years at the Baoshan Station, 37 years at the Chuansha Station, and 35 years at the Longhua Station. The capacity of wind velocity record sample collected at Baoshan is 13 880, covering the maximum velocity values and corresponding directions on all 13 880 days from January 1, 1959 to December 31, 1996. The capacity of wind velocity record sample collected at Chuansha is 13 514 from January 1, 1959 to December 31, 1995. The capacity of wind velocity record sample collected at Longhua is 12 772 from January 1, 1956 to December 31, 1990 except for several holidays. All wind velocity data collected are properly processed into the 10-min average values at the 10-m height above the ground as shown in Tables 1–3 [18].
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Fig. 1. Brief map of Shanghai.
6.2. Examination of distribution model In order to carry out the modified PPCC method, two series of maxim samples from daily records at each station have been generated assuming that the observations are independent. The first series of maxim samples have been picked up by the period extreme value method with period time of 1 year, 6 months, 3 months, 1 month, 15 days and 8 days, respectively. Based on these six maxim samples, the result based on the modified PPCC method shows that the maximum wind velocities at the Baoshang station (BS) and the Chuansha station (C.S.) in Table 4 generally follow distribution model A, a Gumbel distribution. The PPCC method cannot be used at the Longhua Station since the data is given as frequencies in wind velocity periods, and not as time history. The second series of maxim samples have been produced by the peak-over-threshold method with the threshold values of 0.27%, 0.54%, 1.1%, 3.3%, 6.6% and 12.5%, respectively. According to the modified PPCC method, the best-fitted distribution of six maxim samples at each station shown in Table 5 has been confirmed to be a Gumbel distribution [15]. 6.3. Parameter estimation in measurement stations The maximum likelihood method has been chosen to estimate the parameters aa and ba for the wind velocity in azimuth direction a at each station. The
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Table 1 Frequency of daily maximum records at Baoshang station Comp. direct.
0–2 (m/s)
2–4 (m/s)
4–6 (m/s)
6–8 (m/s)
8–10 (m/s)
10–12 (m/s)
12–14 (m/s)
14–16 (m/s)
16–18 (m/s)
18–20 (m/s)
20–24 (m/s)
S (%)
N NNE NE ENE
0.303 0.144 0.173 0.231
3.285 3.919 4.049 6.088
2.500 3.473 2.622 3.581
0.411 0.367 0.411 0.634
0.022 0.050 0.029 0.065
0.115 0.216 0.086 0.050
0.072 0.259 0.043 0.065
0 0.036 0.022 0.029
0.007 0.029 0.014 0
0 0.014 0.007 0
0 0 0 0
6.715 8.509 7.457 10.742
E ESE SE SSE
0.209 0.058 0.173 0.079
6.463 4.107 2.471 2.522
4.654 4.294 2.601 3.264
0.504 0.612 0.360 0.584
0.029 0.036 0.014 0.022
0.058 0.122 0.108 0.101
0.036 0.137 0.072 0.130
0.007 0.014 0.014 0.007
0 0.022 0.014 0.014
0 0.007 0 0
0 0.014 0 0
11.960 9.424 5.829 6.722
S SSW SW WSW
0.050 0.065 0.058 0.065
1.513 1.326 1.787 1.722
1.390 1.268 0.872 1.066
0.166 0.151 0.094 0.180
0.007 0 0 0.022
0.043 0.014 0.007 0.043
0.050 0.029 0.022 0.058
0 0 0.007 0.022
0.014 0.007 0 0.007
0 0 0 0
0 0 0 0
3.235 2.860 2.846 3.184
W WNW NW NNW
0.094 0.072 0.058 0.130
1.679 1.628 1.578 3.718
1.174 1.880 1.412 3.610
0.209 0.454 0.339 0.569
0.007 0.022 0.022 0.050
0.036 0.159 0.159 0.324
0.043 0.274 0.216 0.382
0.007 0.079 0.014 0
0.014 0.036 0.022 0.043
0 0 0.007 0
0 0 0 0
3.264 4.604 3.826 8.826
S (%)
1.960
47.85
39.66
6.045
0.396
1.643
1.888
0.259
0.245
0.036
0.014
100.0
Table 2 Frequency of daily maximum records at Chuansha station Comp. 0–2 2–4 4–6 direct. (m/s) (m/s) (m/s)
6–8 (m/s)
8–10 (m/s)
10–12 12–14 14–16 16–18 18–20 20–22 S (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (%)
N NNE NE ENE
0 0.007 0 0.007
0.488 0.851 0.614 0.814
1.813 2.901 3.115 3.382
2.768 4.129 4.166 3.589
1.110 1.924 1.665 0.851
0.333 0.548 0.333 0.222
0.037 0.074 0.074 0.022
0.007 0.052 0.044 0.037
0 0.015 0 0
0 0 0 0
0 0 0 0.007
E ESE SE SSE
0 0 0 0
0.844 0.444 0.281 0.215
3.507 2.412 1.532 1.628
3.300 4.018 2.923 2.960
0.577 1.021 1.384 1.399
0.126 0.274 0.274 0.444
0.030 0.030 0.022 0.074
0 0.015 0.037 0.007
0 0.007 0 0
0.007 0 0 0
0 0 0 0
8.391 8.221 6.453 6.726
S SSW SW WSW
0 0.007 0 0
0.363 0.289 0.192 0.311
1.783 1.125 0.814 0.947
3.167 1.672 0.895 0.895
1.095 0.377 0.096 0.126
0.244 0.074 0.007 0.059
0.044 0 0.007 0
0.007 0 0 0
0 0.007 0 0.007
0 0 0 0
0 0 0 0
6.704 3.552 2.013 2.346
W WNW NW NNW
0 0 0 0
0.244 0.311 0.252 0.207
1.006 1.317 1.376 1.332
0.992 1.746 2.153 2.494
0.363 0.777 1.413 1.273
0.215 0.548 0.636 0.481
0.015 0.096 0.133 0.059
0.022 0.022 0.074 0.015
0 0.015 0 0.007
0 0 0 0
0 0 0 0
2.856 4.832 6.038 5.868
S (%)
0.022 6.719 29.99
4.817
0.718
0.340
0.059
0.007
0.007
41.87
15.45
6.556 10.500 10.012 8.931
100.0
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Table 3 Frequency of daily maximum records at Longhua station Comp. 0–2 2–4 direct. (m/s) (m/s)
4–6 (m/s)
6–8 (m/s)
8–10 10–12 12–14 14–16 16–18 18–20 20–28 S (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (%)
N NNE NE ENE
0 0.023 0 0.023
0.876 1.040 0.900 1.557
2.378 2.980 3.199 5.890
1.369 1.400 2.245 2.871
0.305 0.329 0.579 0.712
0.055 0.055 0.110 0.141
0 0.031 0 0.031
0 0.008 0.008 0.008
0 0 0 0
0 0 0.031 0
0 0 0 0.008
4.983 5.867 7.071 11.241
E ESE SE SSE
0.008 0.008 0.008 0.008
2.323 1.259 1.283 1.392
6.821 4.380 3.833 4.161
2.159 2.159 2.057 2.409
0.469 0.493 0.735 0.798
0.110 0.141 0.141 0.110
0.031 0.031 0.031 0.031
0 0.008 0.008 0.008
0.008 0 0 0
0 0 0 0
0.008 0 0 0
11.937 8.479 8.096 8.917
S SSW SW WSW
0 0.008 0.008 0.031
0.821 0.602 0.438 0.516
1.916 0.900 0.571 0.712
0.790 0.250 0.086 0.219
0.164 0.055 0.055 0.055
0.031 0.008 0 0.031
0.031 0 0 0.008
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
3.755 1.823 1.158 1.572
W WNW NW NNW
0.023 0.008 0.008 0
0.743 1.502 1.259 1.095
1.369 2.879 3.364 2.558
0.602 2.159 2.057 1.643
0.164 1.072 0.954 0.438
0.110 0.438 0.274 0.055
0.055 0.110 0.086 0.031
0 0.031 0.008 0
0 0.008 0 0
0 0 0 0
0 0 0 0
3.066 8.206 8.010 5.820
S (%)
0.164 17.61
7.376 1.807
0.508
0.086
0.016
0.031
0.016
47.92
24.48
100.0
Table 4 Values of PPCC based on the period samples Station name
Distr. model
12 month
6 month
3 month
1 month
15 day
8 day
B.S.
A B C
0.97 0.92 0.93
0.95 0.92 0.97
0.94 0.90 0.97
0.95 0.91 0.96
0.95 0.90 0.91
0.95 0.92 0.90
C.S.
A B C
0.99 0.99 0.90
0.99 0.92 0.91
0.99 0.93 0.92
0.99 0.92 0.90
0.99 0.93 0.91
0.99 0.90 0.92
maximum-likelihood estimators of aa and ba at three stations can be derived from Eqs. (6a) and (6b). The parameters aa and ba of the 16 compass directions in these measurement stations can be calculated and unified to the time unit of year in Table 6. In order to obtain the parameters in the most unfavorable condition, three extreme samples, including the maxim samples of 1 year, 3 months and 1 month, are put into estimation at each station. In the case of the Baoshan station, the parameters of the joint distribution model A at almost all directions are controlled by the maxim sample of 1 year except the parameters at the direction of SSW. The distribution parameters at all 16 directions are controlled by the maxim sample of
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Table 5 Values of PPCC based on the POT samples Station name
Distr. model
0.27 (%)
0.54 (%)
1.1 (%)
3.3 (%)
6.6 (%)
12.5 (%)
B.S.
A B C
0.96 0.87 0.86
0.97 0.95 0.95
0.97 0.95 0.95
0.98 0.95 0.95
0.93 0.94 0.95
0.94 0.94 0.94
C.S.
A B C
0.99 0.97 0.97
0.99 0.98 0.98
0.97 0.95 0.94
0.97 0.94 0.93
0.98 0.96 0.95
0.99 0.96 0.95
L.H.
A B C
0.99 0.91 0.90
0.99 0.89 0.88
1.00 0.90 0.88
0.99 0.93 0.92
0.99 0.93 0.92
0.99 0.96 0.95
Table 6 Parameters of the model A in measurement stations No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Comp. direct
N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW Non
Baoshang station
Chuansha station
Longhua station
a (m/s)
b (m/s)
a (m/s)
b (m/s)
a (m/s)
b (m/s)
2.4008 4.1856 2.9048 2.4427 2.0873 4.0492 2.5482 2.8722 2.1395 2.0054 1.9607 2.9108 3.3326 4.0748 4.1031 3.7250 3.7561
7.3020 9.9518 7.8151 8.5724 7.7565 9.4166 7.0570 8.3996 6.7052 6.5453 5.5037 7.2782 6.9781 8.8939 8.2240 10.1731 16.7761
2.8208 2.4911 2.2863 2.1117 2.0157 2.3685 2.8976 2.4384 2.7310 2.2348 2.3947 2.3947 2.5440 3.0293 3.1298 3.3317 2.1285
11.5768 12.6816 12.0952 11.2737 10.3525 11.3791 11.6564 11.6714 11.5979 9.6609 7.7686 8.7218 10.2042 12.3536 12.3661 11.7972 15.5242
0.6232 0.9872 1.5753 1.1926 1.4351 0.9372 0.9252 0.8718 1.1987 0.8218 1.4095 1.1965 1.2577 0.9447 0.8077 0.7713 1.7003
10.3439 10.8932 11.6096 11.8330 11.5892 11.6155 11.6306 11.4606 9.8680 8.0567 6.4235 8.2560 11.2305 13.6021 12.8571 10.9162 16.2174
3 months at the Chuansha station. Except the fact that the parameters at SW are controlled by the maxim sample of 3 months, the parameters at the other directions of the Longhua station are controlled by the maxim sample of 1 year. 6.4. Statistical estimation at specific site The above model can be used to obtain surface or upper-level description of the wind climates in the meteorological station or a specific site, for example, the site of
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the Yangpu Bridge across Huangpu River in Shanghai City. In the absence of actual deck-level data in the bridge site, the probability distribution of gradient wind speed and direction at the bridge site can be estimated by the Gumbel model derived from the surface anemometer records in the meteorological station, and this can be done by directly adjusting the parameters aba ¼ xasa ;
ð11aÞ
bba ¼ xbsa
ð11bÞ
in which aba and bba are the parameters for gradient or deck-level wind speeds at the bridge site, and asa and bsa are the parameters for surface wind speeds in the record station. For an anemometer located in approximately homogeneous terrain, x can be assumed to be independent of wind direction and is given by the power-law profile as ab zs db x¼ ð12Þ ds zb in which zs and ds are, respectively, the height and the atmospheric boundary thickness where the surface measurements were made; zb and db are, respectively, the height of the bridge deck and the boundary thickness at the bridge site, and as and ab are the exponents at the surface and bridge sites, respectively. After all parameters aba and bba and sample number na of the 16 compass directions in the three measurement stations are obtained, the weighted average parameters and the sample size for the deck level of the bridge site can be computed as follows: a% ba ¼
3 X
Zk akba ;
ð13aÞ
Zk bkba ;
ð13bÞ
k¼1
b%ba ¼
3 X k¼1
n% a ¼
3 X
Zk nka
ð13cÞ
k¼1
in which Zi ði ¼ 1; 2; 3Þ are the weight factors due to the distances di ði ¼ 1; 2; 3Þ between the bridge site and each measurement station, and can be calculated by Z1 ¼ d2 d3 =D;
ð14aÞ
Z2 ¼ d3 d1 =D;
ð14bÞ
Z3 ¼ d1 d2 =D;
ð14cÞ
D ¼ d1 d2 þ d2 d3 þ d3 d1 :
ð15Þ
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Table 7 Parameters of mean wind velocity No.
Dir.
a
b
m
s
U100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW Non
1.883 2.470 2.219 1.878 1.825 2.370 2.064 2.000 1.983 1.644 1.897 2.117 2.319 2.592 2.583 2.515 2.482
9.784 11.17 10.58 10.63 9.993 10.85 10.21 10.57 9.429 8.095 6.565 8.098 9.570 11.73 11.25 10.96 16.17
10.87 12.60 11086 11.72 11.05 12.22 11.40 11.72 10.57 9.04 7.66 9.32 10.91 13.22 12.74 12.42 17.60
2.41 3.17 2.85 2.41 2.34 3.04 2.65 2.56 2.54 2.11 2.43 2.71 2.97 3.32 3.31 3.23 3.18
18.44 22.53 20.78 19.27 18.39 21.75 19.70 19.77 18.55 15.66 15.26 17.84 20.24 23.65 23.13 22.54 27.59
Based on the above procedure, all weighted average values of the parameters a and b; the mean value m and mean square deviation s; and the expected wind velocity U100 for the returned period of 100 years in the 16 compass directions and the nondirection sample at the deck level of the Yangpu Bridge have been obtained in Table 7. Some conclusions can be drawn from Table 7 as follows: (a) The mean or expected values of wind velocity are uneven at different directions. The values at the direction of SSW, SW and WSW are relatively smaller, but the values at WNW, NW and NNW are larger. The ratios of the non-directional expected value in the 100-year return period to the maximum (WNW) and minimum (SW) expected values are about 1.17 and 1.81, respectively. (b) The standard deviations s=m of wind velocity are also different at each direction. The largest and smallest values of standard deviations are approximately equal to 0.32 at SW and 0.18 at the non-direction item, respectively. (c) As opposed to the numbers of expected velocities U100 ; the current code in China would have given Um ¼ 33:9 m=s for all directions. The expected velocities listed in Table 2 are much less than this figure, in particular, Um =U100 ¼ 1:23 at the non-direction item.
7. Conclusions Based on the statistical analysis of mean wind velocity in the Shanghai area, the joint probability distribution models of extreme values is applied to statistical
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analysis of joint action of wind speed and its direction in this paper. Three analytical models of the joint probability distribution with directional independent parameters are used to describe joint distribution, and its application method is developed into three steps, data processing of wind records, examination of distribution models and estimation of model parameters. These approaches have been used to estimate extreme wind velocity at the surface level of three meteorological stations in the Shanghai area and the deck level of the Yangpu Bridge across Huangpu River. It can be concluded that mean values and mean square deviations of wind velocities vary with directions and this should be considered in the definition of mean wind velocity.
Acknowledgements The research was financially supported by the National Science Foundation of China under Grant No. 59895410.
References [1] G.H. Li, Stability and Vibration of Bridge Structures, China Railway Press, China, 1992 (in Chinese). [2] E.J. Gumbel, Statistics of Extremes, Columbia University Press, New York, 1958. [3] E.J. Gumbel, Statistical theory of extreme values and some practical applications, NBS Applied Mathematics Series 33, US Department of Commerce, 1954. [4] H.C. Shellard, The estimation of design wind speed, Proceedings of the First International Conference on Wind Engineering, Teddington, 1963, pp. 29–52. [5] A.G. Davenport, The dependence of wind loads on meteorological parameters, Proceedings of the International Seminar of Wind Effects on Buildings and Structures, Ottawa, 1967. [6] A.G. Davenport, N. Isyumov, T. Jandali, A study of wind effects for the Sears Project, University of Western Ontario, Engineering Science Research Report, BLWTL-5-71, 1971. [7] J.R. Mayne, The estimation of extreme winds, J. Wind Eng. Ind. Aerodyn. 5 (1979) 109–137. [8] E. Simiu, R.H. Scanlan, Wind Effects on Structures, 3rd Edition, Wiley, New York, 1996. [9] P. Sparis, J. Autonogiannakis, P. Papadopulos, Markov matrix coupled approach to wind speed and direction simulation, Wind Eng. 19 (3) (1995) 121–123. [10] E. Simiu, J.J. Filliben, Wind direction effects on cladding and structural loads, Eng. Struct. 3 (1981) 181–186. [11] C.J. Baynes, The statistics of strong wind for engineering application, Ph.D. Thesis, University of Western Ontario, 1974. [12] L. Gomes, B.J. Vickery, On the prediction of extreme wind speeds from the parent distribution, Research Report No. R241, University of Sydney, 1974. [13] N.J. Cook, Towards better estimation of extreme winds, J. Wind Eng. Ind. Aerodyn. 9 (1982) 295–323. [14] R.I. Harris, Gumbel revisited: a new look at extreme value statistics applied to wind speed, J. Wind Eng. Ind. Aerodyn. 59 (1996) 1–22. [15] Y.J. Ge, Reliability theory and its applications to wind induced vibration of bridge structures, Ph.D. Thesis, Tongji University, 1997 (in Chinese). [16] E. Simiu, J.J. Filliber, Statistical analysis of extreme winds, Technical Note 868, National Bureau of Standards, Washington, DC, 1975.
Y. Ge, H. Xiang / J. Wind Eng. Ind. Aerodyn. 90 (2002) 1585–1599
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[17] Y.J. Ge, Z.X. Lin, H.F. Xiang, Extreme sample convergence check method for wind velocity distribution, Proceedings of the Fifth National Conference on Wind Engineering and Industrial Aerodynamics, Hunan Province, 1998 (in Chinese). [18] Y.J. Ge, Wind speed daily records of the Baoshang, Chuansha and Longhua meteorological stations, Technical Report of Tongji University, 1997 (in Chinese).