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The Eighth Asia-Pacific Conference on Wind Engineering,December 10–14, 2013, Chennai, India

Characteristics of Weibull coefficients of wind velocity by wind directions for evaluating safety of railway vehicles under strong wind

Saki Tanimoto1, Keiji Araki2, Takaaki Fukuhara3.

1Researcher of Disaster Prevention Technology Division, Railway Technical Research Institute, Tokyo, Japan, [email protected]

2Senior Researcher of Disaster Prevention Technology Division, Railway Technical Research Institute, Tokyo, Japan, [email protected]

3Assistant Senior Researcher of Disaster Prevention Technology Division, Railway Technical Research Institute, Tokyo, Japan, [email protected]

ABSTRACT In order to evaluate safety of railway vehicles under strong wind, we calculated Weibull coefficients cd and kd of 10-minute maximum instantaneous wind velocity by 16 wind directions. We used data of the wind velocity and the wind directions that have been obtained at 772 AMeDAS sites in Japan for calculations of cd and kd. In addition we estimated probabilities of wind velocity exceeding the critical wind speed of overturning by using cd and kd, the relative frequency of each wind direction. In addition, it is shown that the consideration of wind directions is important to evaluate safety of railway vehicles under strong wind.

Keywords: Weibull distribution, estimating safety of railway vehicles, wind direction

Introduction

Railway Technical Research Institute (RTRI) has been developing methods for evaluating safety of railway vehicles under strong wind [Fukuhara and Imai (2007) and Imai et al. (2009)]. These methods evaluate the probability of strong wind exceeding the critical wind speed of overturning. Indices for evaluating safety include the probability of strong wind as external force and the critical wind speed of overturning as resistance to wind for a railway vehicle. The critical wind speed of overturning is defined as the wind speed when the wheel load on the windward side of railway vehicles under strong wind becomes zero [Hibino and Ishida (2003)]. If wind speed is more than the critical wind speed of overturning, the railway vehicles are possible to overturn. The value of the critical wind speed of overturning depends on not only the shapes of railway vehicles and railway structures around the line, and running speeds of railway vehicles, but also the wind angle to the railway lines, as shown in Fig. 1. In Fig. 1, the value of the critical wind speed of overturning has a minimum value when the wind angle to the railway line is 80 degrees. The wind directions sometimes cannot be obtained although the value of the critical wind speed of overturning varies according to the change of the wind angle. In our previous studies for estimating safety, the critical wind speeds of overturning was supposed to be constant because of the assumption that only wind velocity can be obtained for the sake of simplification of the calculations. We used only the minimum value of the critical wind speeds of overturning to avoid overestimation of safety (the wind angle is 80 degrees in Fig. 1). But natural wind blows from various directions. Therefore, in this study we consider wind directions to estimate safety more accurately.

Proc. of the 8th Asia-Pacific Conference on Wind Engineering – Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds)Copyright c© 2013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: 978-981-07-8011-1doi:10.3850/978-981-07-8012-8 139 1065

Proc. of the 8th Asia-Pacific Conference on Wind Engineering (APCWE-VIII)

In order to evaluate safety of railway vehicles under strong wind in the case of considering wind directions, we calculated Weibull coefficients by the wind directions. In this paper, we consider that the value of the critical wind speeds of overturning depends only on the wind angle, that is, we consider that the shapes of railway vehicles and railway structures around the line, and running speeds of railway vehicles are constant. This paper introduces some examples of the Weibull coefficients by wind directions. In addition, the results of the safety evaluation made by using the Weibull coefficients by wind directions are shown.

The indices of evaluations of safety

It is generally known that the probability density function (PDF) f(u) of the wind

velocity u follows the Weibull distribution as shown in Eq. (1) [Shiraishi et al. (1984)].

( ) −=− kk

cu

cu

ck

uf exp1

(1)

where c and k are the Weibull coefficients. We defined an index Px for evaluating safety as the occurrence probability of wind velocity exceeding the critical wind velocity of overturning, ( )θcU . The index of Px is a function of the Weibull coefficients c and k, as shown in Eq. (2).

( )( )

( )−=≡∞ k

c

Ux c

UduufP

c

θθ

exp (2)

where θ is the angle which the wind direction makes with the direction of train running. The value of θ in the direction in which the vehicles run is zero. The index of Px considering 16 wind directions is shown in Eq. (3).

( ) ( )=

×−≡16

1exp

d

k

d

cx dw

cU

Pdθ (3)

20

30

40

50

60

0 20 40 60 80 100 120 140 160 180

The minimum value of the critical wind velocity of overturning (31.9m/s, °= 80θ )

A general local train vehicle is assumed. The velocity that train runs is 100km/h. The lateral acceleration is 0m/s2.

Wind angle to the railway lines θ (degree)

The

criti

cal w

ind

velo

city

of

ove

rturn

ing

(m/s

)

Fig.1 Example of the critical wind velocity of overturning

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where cd and kd are the Weibull coefficients by 16 wind directions, d is the value of the wind directions expressed by 16-point compass (N, NNE, NE, …, NNW) and w(d) is the relative frequency of the wind direction d.

Eq. (2) and (3) show that the Px in the case of not considering the wind directions is obtained from c and k, and that the Px in the case of considering the wind directions is obtained from cd, kd and w(d). The value of Px decreases with safety levels increasing, because of the definition of the Px mentioned above. Weibull coefficients cd and kd

To calculate cd and kd, we used the 10-minute wind velocity and the 10-minute wind directions expressed by 16-point compass (N, NNE, NE, …, NNW) that the Japan Meteorological Agency (JMA) obtained at their observational sites (called AMeDAS) in Japan. The data of wind velocity and the wind directions are obtained at 772 sites of the AMeDAS where the JMA observed the maximum instantaneous wind velocity over a 10-minute period in March 2012. The total number of the data at each site is different from each other because the observation at each site started at different time.

From the data at the 772 sites of AMeDAS, the probability distributions of the maximum instantaneous wind velocity over a 10-minute period by the 16 averaged wind directions and of all wind directions over 10-minute period were provided. The probability distributions of the wind velocity were approximated to Weibull distributions by the least-squares method. As a result, cd and kd at the 772 sites were obtained. In addition, the values of w(d) were obtained from the frequency of the 16 averaged wind directions over 10-minute period.

Fig.2 shows examples of comparisons between the relative frequency distribution of observed values and that of the PDF approximated to the Weibull distributions. In Fig.2, the value of w(d) in the top four of 16 wind directions are indicated. Fig.2 also shows all wind directions. In this paper, site (A) is located in a coastal area, site (B) is located in an inland area, site (C) is located in an islands and site (D) is located near an urban area. At site (A) high frequency of strong wind is expected and site (B) low frequency of strong wind is expected. As shown in Fig. 2, the frequency of strong wind of all wind directions at site (A) is the highest of 4 sites (the value of the frequency is 0.001 when the wind velocity is more than 30m/s), and the frequency of strong wind in all wind directions at site (B) is the lowest of 4 sites (the value of the frequency is 0.001 when the wind velocity is more than 10m/s). The frequencies of wind velocity of 20m/s at site (C) and (D) are 0.001, but the peaks of the frequency distributions are different. Thus the distributions of the relative frequency and PDF of all wind directions vary depending on the sites. Additionally, the distributions of the relative frequency and PDF by 16 wind directions also vary depending on the sites. Furthermore, the distributions in each site are different by 16 wind directions. As shown in Fig. 2, the frequency of wind velocity of 15-20m/s in N and NNW directions of site (D) is 0.001 and the frequency of wind velocity of less than 15m/s in the other 2 wind directions of site (D) is 0.001. The values of cd, kd and w(d) at sites (A) - (D) are shown in Table 1 - 4, respectively. Table 1 - 4 show that cd, kd and w(d) are different by 16 wind directions at each site. The result that the relative frequency distributions of wind velocity are different by 16 wind directions at each site suggests that the result of the safety evaluation is different between the case of considering 16 wind directions (the case of estimations of Px using cd, kd and w(d)) and the case of not considering them (the case of estimations of Px using c and k).

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0.001

0.01

0.1

1

0 10 20 30 40

E ESE W WNW

E ESE W WNW

Maximum instantaneous wind velocity (m/s)

Prob

abili

ty Approximated

Observed

Site (B)


0.001

0.01

0.1

1

0 10 20 30 40

All directionAll direction

Prob

abili

ty

Observed Approximated

(The dots are the relative frequency distribution of observed valuesand the lines are the PDF approximated the Weibull distributions.)

Fig. 2 Examples of the probability distributions of the wind velocity by wind directions

0.001

0.01

0.1

1

0 10 20 30 40

N WNW NW NNW

N WNW NW NNWApproximated


Prob

abili

ty

Observed

Site (D)

0.001

0.01

0.1

1

0 10 20 30 40


Prob

abili

ty



0.001

0.01

0.1

1

0 10 20 30 40

NE WSW W WNW

NE WSW W WNW


Prob

abili

ty Approximated

Observed

Site (A)

0.001

0.01

0.1

1

0 10 20 30 40

N NE ENE E

N NE ENE EApproximated


Prob

abili

ty

Observed

Site (C)

0.001

0.01

0.1

1

0 10 20 30 40


Prob

abili

ty



0.001

0.01

0.1

1

0 10 20 30 40


Prob

abili

ty



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Table 1: The Weibull coefficients cd and kd by the wind directions and the relative frequency of wind distributions w(d) at site (A)

N NNE NE ENE E ESE SE SSE S cd 6.07 9.63 12.68 7.90 6.66 6.37 5.17 6.82 7.41 kd 1.23 1.47 2.12 2.04 2.05 1.83 1.62 1.71 1.56

w(d) 0.01 0.08 0.18 0.03 0.02 0.01 0.01 0.02 0.02 SSW SW WSW W WNW NW NNW all

cd 7.58 9.07 11.40 14.64 14.64 7.44 5.25 12.13 kd 1.76 1.73 1.87 2.06 2.28 1.90 1.75 1.76

w(d) 0.02 0.03 0.11 0.28 0.15 0.02 0.01 1.00

Table 2: The Weibull coefficients cd and kd by the wind directions and the relative frequency of wind distributions w(d) at site (B)



cd 2.78 3.25 3.37 3.78 3.88 3.05 2.54 2.59 kd 1.94 1.83 1.92 2.40 2.33 1.85 1.72 1.53

w(d) 0.03 0.04 0.05 0.10 0.09 0.03 0.02 1.00

Table 3: The Weibull coefficients cd and kd by the wind directions and the relative frequency of wind distributions w(d) at site (C)


w(d) 0.10 0.08 0.09 0.09 0.09 0.08 0.06 0.06 0.07

SSW SW WSW W WNW NW NNW all cd 7.35 6.93 5.63 5.14 6.20 8.83 9.87 8.15 kd 2.71 2.35 1.77 1.59 1.41 2.13 2.91 2.52

w(d) 0.06 0.04 0.02 0.02 0.03 0.04 0.07 1.00

Table 4: The Weibull coefficients cd and kd by the wind directions and the relative frequency of wind distributions w(d) at site (D)



cd 5.23 3.68 2.52 2.61 2.92 3.22 4.39 4.87 kd 1.50 1.36 1.93 2.73 3.10 2.63 1.74 1.61

w(d) 0.02 0.03 0.02 0.05 0.10 0.12 0.10 1.00

(The values of cd, kd and w(d) are rounded off to the hundredth.)




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The estimation of Px

The value of Px can be estimated by using cd, kd and w(d) which were shown above. As an example, the estimations of Px in a virtual railway section are shown by using the values of cd, kd, c, k and w(d) at site (A) - (D). The frequency distributions of the 16 wind directions at site (A) - (D) are shown in Fig.3. We assumed a railway section as follows:

1. It is a virtual section from station I to station II as shown in Fig. 4. 2. The value of ( )θcU follows the distribution of Fig. 1. 3. The wind velocity and the wind directions are constant. In these assumptions, the value of Px were estimated regarding the following two

cases; a case of considering 16 wind directions and a case of not considering them. In the case of not considering 16 wind directions, it is assumed that the wind angle θ is 80 degrees with

( )θcU minimum. The values of Px in the case of considering 16 wind directions and in the case of not

considering them are shown in Table 5. The values of Px using cd, kd and w(d) at sites (A) - (D) in the case of considering 16 wind directions are different from the values of Px using c

Fig. 3 The wind rose of w(d) at site (A) – (D) in Table 1 - 4

Site (A) (The total number of the data is 184,372)

Site (B) (The total number of the data is 79,541)

Site (C) (The total number of the data is 165,173)

Site (D)(The total number of the data is 209,619)

0%

10%

20%

30%N

NNE

NE

ENE

E

ESE

SE

SSES

SSW

SW

WSW

W

WNW

NW

NNW

0%

10%

20%

30%N

NNE

NE

ENE

E

ESE

SE

SSES

SSW

SW

WSW

W

WNW

NW

NNW

0%

10%

20%

30%N

NNE

NE

ENE

E

ESE

SE

SSES

SSW

SW

WSW

W

WNW

NW

NNW

0%

10%

20%

30%N

NNE

NE

ENE

E

ESE

SE

SSES

SSW

SW

WSW

W

WNW

NW

NNW

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and k at sites (A) - (D) in the case of not considering them. The value of Px at site (A) in the case of considering 16 wind directions is approximately 1/10 to the value of Px in the case of not considering them, the value of Px at site (B) is approximately 1/100, the value of Px at site (C) is approximately 10,000 times, and the value of Px at site (D) is the same level, respectively. As a result, the value of the index Px varies depending on whether or not the wind directions are considered.

Conclusions

We obtained the values of cd and kd by 16 wind directions and w(d) at 772 sites in

Japan. The values of cd, kd and w(d) have enabled us to estimate the index of Px at 772 sites considering the 16 wind directions. In this paper, the values of Px at 4 of 772 sites were indicated as an example. In addition, it is shown that the consideration of wind directions is important in evaluating safety of railway vehicles under strong wind more accurately.

Fig. 4 The image of the assumed section

Station I Station II

Approximately 16km

θ

Wind direction

The direction that train runs

The Px NOT considering 16 wind directions

The Px considering 16 wind directions

(A) 3.89 10× -03 2.04 10× -04 (B) 2.60 10× -21 3.13 10× -23 (C) 1.40 10× -14 2.56 10× -10 (D) 6.80 10× -10 7.16 10× -10

Table 5: The Px at the assumed section estimated using the values of c, k, cd, kd and w(d) at site (A) – (D).

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References Fukuhara, T. and Imai, T. (2007), “Evaluation method of train operation control under strong wind considering

temporal fluctuations of wind velocity,” QR of RTRI, Vol. 48, No. 3, 148-152.

Imai, T. Araki, K., and Fukuhara, T. (2009), “A Method to estimate the probability of a train encountering gusts in an operating section,” QR of RTRI, Vol. 50, No. 1, 50-55.

Hibino, Y. and Ishida, H. (2003), “Static analysis on railway vehicle overturning under crosswind,” RTRI Report, Vol. 17, No. 4, 39-44. (in Japanese)

Shiraishi, N., Matsumoto, M., Shirato, H., and Yasuda, K. (1984), “Stochastic analysis of sixteen directional wind velocity in Japan – Identification of Weibull parameters –.” Journal of Wind Engineering, Vol.22, 13-16. (in Japanese)

Japan Meteorological Agency, Web site of Japan Meteorological Agency, http://www.jma.go.jp/jma/index.html (accessed 2012).

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