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Wind-resistant design manual for highway bridges in Japan
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Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 1499–1509
Wind-resistant design manual for highway bridges in Japan Hiroshi Sato* Structures Research Group, Incorporated Administrative Agency, Public Works Research Institute, Minamihara 1-6, Tsukuba-shi, Ibaraki-ken 305-8516, Japan
Abstract In Japan, wind-resistant designs of highway bridges up to the span length of 200 m are conducted according to the Wind-Resistant Design Manual for highway bridges. This manual is also applicable to the highway bridges up to the span length of 300 m with minor modification of its provisions. In this paper, the manual is outlined first to illustrate windresistant design procedure in Japan. Then turbulence effects included in the manual are described with Dr. Davenport’s contributions to the manual. r 2003 Elsevier Ltd. All rights reserved. Keywords: Wind-resistant design; Manual; Highway bridges; Turbulence effects
1. Introduction Highway bridges up to the span length of 200 m in Japan are designed according to the specification for highway bridges. With regard to wind-resistant design, the specification prescribes design wind load of bridges and dimensions of pipe structures, however, there is no sufficient provision on wind-induced vibrations in the specification. The wind-resistant designs of long-span bridges in Japan were mostly based on the Wind-Resistant Design Criteria for Honshu–Shikoku bridges. The criteria was originally formulated in 1964 and revised in 1976 by the ad hoc committee in the Japan Society of Civil Engineers. The criteria prescribes the design wind load
*Corresponding author. Tel.: +81-29-879-6726; fax: +81-29-879-6739. E-mail address: [email protected] (H. Sato). 0167-6105/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2003.09.012
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including the effect of gust and the evaluation method of wind-induced vibrations based on wind tunnel testing. Since the revision of the criteria, wind engineering had made remarkable progress and a considerable amount of aerodynamic data on bridges had been acquired. It was thought that wind-resistant designs for limited types of bridges could be made without wind tunnel testing if reliable formulae for the estimation were provided. It was also thought that prediction of wind-induced vibration would become more reasonable and reliable if the effect of turbulence was incorporated appropriately in the design. From these reasons, the working group was organized in the Japan Road Association in 1984 to prepare the Wind-Resistant Design Manual for highway bridges. The working group was reformed into the Wind-Resistant Design Committee in 1986, and the manual was published in 1991. At present in Japan, wind-resistant designs of highway bridges up to span lengths of 200 m are conducted according to the Wind-Resistant Design Manual for highway bridges. This manual is also applicable to highway bridges up to span length of 300 m with minor modification of its provisions. If span length of a bridge becomes longer than 300 m; a specific design criteria will be established for the bridge. In this paper, Wind-Resistant Design Manual for Highway Bridges is outlined to illustrate wind-resistant design procedure in Japan.
2. Procedure of wind-resistant design using the Wind-Resistant Design Manual The procedure of wind-resistant design using the manual is as follows. The wind properties (basic wind speed, design wind speed, turbulence intensity, etc.) are decided first. The design wind load is calculated, and the static design of the bridge is made. At this stage, where major dimensions of the bridge are determined, wind-induced vibrations to be studied further are chosen considering the design wind speed, the deck width and the span length of the bridge. For a bridge with short span length, no study on wind-induced vibration is required. For long-span bridges, wind-induced vibrations specified above should be predicted using the formulae provided in the manual. Then the wind-induced vibrations are evaluated. The formulae were established considering past wind tunnel test results. Since the formulae are expressed by means of only a few parameters, the prediction error of the formulae is not negligibly small. Therefore, the formulae were established, so that their prediction may become safer. It means that evaluation ‘good’ based on the formulae is almost always ‘good’, however, evaluation ‘no good’ based on the formulae is not always ‘no good’ because of the safety margin of the formulae. In case that the evaluation based on the formulae turns out to be ‘no good’, the bridge engineer can modify the design or he can predict wind-induced vibrations more accurately by means of wind tunnel testing. The evaluation based on wind tunnel testing has priority over that based on the formulae.
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3. Wind properties used in design Wind properties are modeled fundamentally according to Davenport [1]. The terrains are classified into four, namely rough sea (Terrain 1), open farmland (Terrain 2), suburbs (Terrain 3) and city centers (Terrain 4). 3.1. Basic wind speed U10 The basic wind speed U10 is defined as the mean wind speed over open farmland (Terrain 2) at an elevation of 10 m; averaged over a period of 10 min: Using the meteorological data at weather stations in Japan, extreme wind speeds were estimated. The return period was 100 years. The basic wind speeds were classified into four categories, namely 30, 35, 40 and 45 m=s: 3.2. Design wind speed Ud The power law matched with the log profile at an elevation of 30 m was applied to the mean wind speed profile. The design wind speed for dynamic design can be obtained from the following formula: Ud ¼ U10 E1 ;
ð1Þ
where E1 is the correction factor for altitude and terrains. 3.3. Turbulence properties Typical values for turbulence properties such as turbulence intensities and power spectral density functions are provided in the manual. su ; r.m.s. of longitudinal velocity fluctuation, was assumed to be 2:5u; where u is the friction velocity.
4. Design wind load The design wind load Pd for static design can be obtained from the following formula: Pd ¼ 0:5rUd2 Cd An G;
ð2Þ
where Cd is the drag coefficient, An the projected area, G the gust factor. Design wind speed Ud for design wind load is 40 m=s: Gust factor G was determined so that the shear force induced by the design wind load may become equivalent to those estimated by the gust response analysis proposed by Davenport [2]. 1.9 is used for gust factor G: In the manual, the typical values for Pd or Cd are provided according to the type of bridges and members.
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5. Wind-induced vibration 5.1. Prediction of wind-induced vibrations by the formulae The critical wind speed for flutter and galloping can be predicted by simple formulae as well as by wind tunnel testing. The formula for prediction of critical wind speed of flutter is as follows: Ucf ¼ 2:5fy B;
ð3Þ
where Ucf is the critical wind speed for flutter, fy the natural frequency of the 1st torsional mode, B the width of bridge section. The formula for prediction of critical wind speed of galloping is as follows: Ucg ¼ 8fh B : when angle of attack is almost 0 ;
ð4Þ
Ucg ¼ 4fh B : when angle of attack is positive;
ð5Þ
where Ucg is the critical wind speed for galloping, fh is the natural frequency of the 1st bending mode. The amplitude of vortex-induced vibrations can be predicted by simple formulae as well as by wind tunnel testing. The effects of turbulence on vortex-induced vibrations are incorporated. The formulae for prediction of critical wind speed and amplitude of vortex-induced vibrations are as follows: Ucv ¼ 2:0fh BFfor bending;
ð6Þ
Ucv ¼ 1:33fy BFfor torsion;
ð7Þ
Ac ¼ Ae Ems Et ;
ð8Þ
Ae ¼ 0:05bds ðd=BÞ=ðmr dh ÞFfor bending ðin h=BÞ;
ð9Þ
Ae ¼ 13:2bds ðd=BÞ3 =ðIpr dy ÞFfor torsion ðin degreeÞ;
ð10Þ
Et ¼ 1 15bt ðB=dÞ1=2 Iu2 X0Ffor bending;
ð11Þ
Et ¼ 1 20bt ðB=dÞ1=2 Iu2 X0Ffor torsion;
ð12Þ
where Ucv is the wind speed for the maximum amplitude of vortex-induced vibration, Ac the corrected maximum amplitude of vortex-induced vibration, Ae the maximum amplitude of vortex-induced vibration for rigid model in smooth flow, Ems the correction factor for vibrational mode (about 4=p), Et the correction factor for the effect of turbulence, bds the correction factor for sectional shape, d the depth of bridge section, mr ; Ipr the reduced mass or reduced mass moment of inertia ðmr ¼ m=ðrB2 Þ; m the mass per unit length of the bridge, Ipr ¼ Ip=ðrB4 Þ; Ip the mass moment of inertia per unit length of the bridge), dh ; dy the structural damping (logarithmic decrement), bt the correction factor for sectional shape, Iu the intensity of turbulence.
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5.2. Prediction of wind-induced vibrations by the wind tunnel testing Wind tunnel testing increases the reliability of wind-resistant design of bridges. Since the prediction of wind-induced vibrations based on the formulae may often provide safer value, there is a fair chance that wind tunnel study provides more economical design. Turbulence has significant effects on wind-induced vibrations. Therefore, standard wind tunnel testing methods in turbulent flow (full aeroelastic model test, taut-strip model test) were described in the manual as well as conventional testing methods (spring-mounted rigid model test, measurement of steady aerodynamic forces). 5.3. Evaluation of wind-induced vibrations The method of verification of flutter, galloping and vortex-induced vibrations are provided in the manual. For flutter and galloping, the following inequalities shall be satisfied: Ucf > Ur Ffor flutter;
ð13Þ
Ucg > Ur Ffor galloping;
ð14Þ
Ur ¼ Ud Er1 Er2 ;
ð15Þ
Ud ¼ U10 E1 ;
ð16Þ
where Ucf is the critical wind speed for flutter, Ucg the critical wind speed for galloping, Ur the reference wind speed for flutter and galloping, Ud the design wind speed, Er1 the correction factor for the effect of gust, Er2 the safety factor (1.2), U10 the basic wind speed, E1 the correction factor for altitude and terrains. It was found that onset velocity for negative damping, which caused galloping, increased remarkably in turbulent flow [3,4]. It suggested that Ucg would be high enough when turbulence intensity is high. Therefore, Erl ; correction factor for the effect of gust, was determined as 1 for galloping, while Er1 for flutter was 1.1–1.25 depending on wind turbulence. For vortex-induced vibrations, unless the following inequality (17) is satisfied, the maximum amplitude of the vortex-induced vibration shall be less than the allowable amplitude shown in Eqs. (18) or (19). Ucv > Ud ;
ð17Þ
ha ¼ 0:04=fh Ffor bending ðin mÞ;
ð18Þ
ya ¼ 2:28=ðbfy ÞFfor torsion ðin degreeÞ;
ð19Þ
where Ucv is the wind speed for the maximum amplitude of vortex-induced vibration, ha the allowable amplitude for bending vortex-induced vibration, fh the natural frequency of the 1st bending mode, ya the allowable amplitude for torsional
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vortex-induced vibration, b the distance between the deck center and the center of outmost lane, fy the natural frequency of the 1st torsional mode.
6. Dr. Davenport’s contributions One of the main features of the manual is incorporation of turbulence effects on wind-induced vibrations, where Dr. Davenport has made a great contribution. Therefore, the manual was influenced directly and indirectly by Dr. Davenport’s works. Some of the direct contributions of Dr. Davenport’s to the wind-resistant design methods described in the manual are as follows: (1) In the manual, simple formulae for predicting amplitude of vortex-induced vibrations are provided. The effects of turbulence on vortex-induced vibrations are incorporated. Wind turbulence properties can be predicted based on the model proposed by Dr. Davenport [1]. (2) Design wind load shall be calculated considering the effects of gust response, which was predicted according to the methods proposed by Dr. Davenport [2]. (3) Er1 ; correction factor for the reference wind speed for galloping, was determined as 1, because onset velocity for negative aerodynamic damping, which causes galloping, would increase remarkably in turbulent flow. The aerodynamic damping was measured by the author under the supervision of Dr. Davenport [3]. Since the gust response analysis methods are so well known, there may be no need for me to describe Dr. Davenport’s contribution here. Our research on the wind resistant design methods for vortex-induced vibrations and galloping and Dr. Davenport’s contributions are described in the following sections. 6.1. Design for vortex-induced vibrations Vortex-induced vibrations may take place to long-span bridges at wind speeds considerably lower than their design wind speed. For the design of long-span bridges, therefore, prediction of amplitude of vortex-induced vibrations becomes very important. The mechanism and countermeasures of the vortex-induced vibrations were studied in Japan in 1970s, but most of the studies were based on wind tunnel experiments in smooth flow. To understand the effects of turbulence on the vortex-induced vibrations, wind tunnel studies were conducted at the Public Works Research Institute in 1980s. In the study, vortex-induced vibrations of taut-strip models for 16 types of bridge cross section were measured in smooth flow and in three types of turbulent flow. Used bridge cross sections were rectangular section, trapezoidal section, hexagonal section, two-plate section and two-box section. B=d of the cross section ranged from 1 4 to 10. Scale ratio of the model was assumed as 200 ; and the model length was 1:2 m: Turbulent flow was generated by spires and floor roughness. Turbulence intensities for longitudinal wind speed ðIuÞ and vertical wind speed ðIwÞ were as follows: Iu ¼ 6:8%; Iw ¼ 5:1% for turbulent flow 1; Iu ¼ 11:5%; Iw ¼ 7:8% for turbulent flow 2; and Iu ¼ 22:4%; Iw ¼ 14:1% for turbulent flow 3. Integral scale of longitudinal wind
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1 speed (Lxu) was a little smaller than 200 of typical natural wind’s scale, and that of 1 vertical wind speed (Lxw) was about 200 of typical natural wind’s scale. In smooth flow, vortex-induced vibrations of vertical bending mode were observed for 14 types of cross sections. It was found that amplitude of the vortex-induced vibrations decreased in the turbulent flow. Relationship between the maximum amplitude and turbulence intensity Iw is shown in Fig. 1. In the figure, nondimensional amplitude ðh=BÞ was multiplied by reduced mass mr and structural damping dh : The ratios of the amplitude ðh=BÞmr dh to that in smooth flow are shown in Fig. 2. From the figures it was found that maximum amplitude decreases with turbulence intensity. The turbulence effects on vortex-induced vibrations were a little smaller for hexagonal sections than for other sections. It also seemed that the turbulence effects increased a little with B=d: Based on these results and others, the formulae for predicting amplitude of vortexinduced vibrations were developed as were shown in Eqs. (8)–(12). To utilize these formulae, turbulence intensities at the bridge site should be predicted. In the manual, wind properties are modeled fundamentally according to Dr. Davenport [1]. Dr. Davenport’s model was also applied to prediction of turbulence intensities as follows. In the manual, the power law matched with the log profile at an elevation of 30 m was applied to the mean wind speed profile. Intensity of turbulence Iu was calculated first at an elevation of 30 m assuming su ¼ 2:5u and the log profile. Then the effect of altitude on Iu was corrected by the power law profile. Therefore, Iu can be predicted by the following formula:
Iu ¼ ð30=zÞa =½lnð30=z0 Þ;
ð20Þ
where z is the altitude of the bridge (in m), a the exponent of the power law profile, z0 the roughness length (in m).
Fig. 1. Effects of turbulence on vortex-induced vibration amplitude ðh=BÞmr dh :
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Fig. 2. Ratios of the amplitudes ðh=BÞmr dh to those in smooth flow.
6.2. Design for galloping When a model of a long-span bridge with bluff box section is tested in smooth flow, divergent vibration called galloping is sometimes observed. In turbulent flow, the divergent amplitude vibration may turn to less divergent but more random vibration. In order to understand the turbulence effects on galloping, unsteady aerodynamic forces acting on the 1:2 retangular prism were measured in smooth and turbulent flow at the University of Western Ontario by the author under the supervision of Dr. Davenport [3]. The measurement of lift force on the model, whose longer side was set along the wind, is described here. Turbulence was generated by a coarse grid placed upstream of the model. The mesh size and the bar size were 0.61 and 0:15 m; respectively. The intensity of turbulence used in the experiment was slightly smaller than that usually found in the natural wind. The ratio of the integral scale of the turbulence to the width of the model was smaller than the ratio of the integral scale of natural wind to the width of a long-span bridge section. The model had a relatively large aspect ratio ð0:10 m 0:20 m 2:13 mÞ: End plates were not attached to the model. The lift force was measured by the pneumatic averaging method. This consists of two kinds of averaging methods, i.e. continuous averaging by porous material and discrete avaraging by a manifold. Power spectral density functions (PSDFs) of the fluctuating lift on the model at rest, SL ; are shown in Fig. 3. The turbulence broadens the peaks of the PSDF of the lift and also lowers the peak reduced frequency (from 0.16 in smooth flow to 0.12 in turbulent flow). Unsteady lift force was measured by the forced oscillation method. Then aerodynamic damping ratio for heaving motion zah was estimated. The results are
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fSL/ (0.5 U2B)2
1507
fB/U
0.05
Lift
Smooth flow
h
Turbulent flow
0.00
Wi nd
-0.05
Aerodynamic Damping ah
0.10
Fig. 3. Power spectral density functions of the fluctuating lift on the rectangular model at rest.
1.0E-03
1.0E-02
1.0E-01
1.0E+00
fB/U
Fig. 4. Aerodynamic damping for heaving motion of the rectangular model.
shown in Fig. 4. In smooth flow, sharp peak of positive aerodynamic damping appears at fB=U ¼ 0:17: The aerodynamic damping turns to negative at fB=Uo0:15: According to free vibration test of the model conducted in smooth flow, divergent vibration of heaving mode, which can be regarded as galloping, was observed in the corresponding wind speed range. Aerodynamic damping in the turbulent flow is also shown in Fig. 4. The pattern is similar to that of smooth flow, however, the positive damping peak shifts to the lower reduced frequency and the peak becomes broader in the turbulent flow. The turbulence decreased the critical reduced frequency to fB=U ¼ 0:10 (or increase the critical reduced wind speed to U=ðfBÞ ¼ 10), where the aerodynamic damping turns to negative. The wind-induced vibration observed in the free vibration test was less divergent in the turbulent flow than in smooth flow.
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Similar studies were conducted at the Public Works Research Institute for a continuous box girder bridge [4]. It was also concluded that turbulence decreases the reduced frequency (or increases the reduced wind speed) for the onset of negative damping the causes galloping. As was described in Eq. (14), critical wind speed for galloping should be higher than the reference wind speed that can be obtained from Eq. (15). So far Er1 for galloping in Eq. (15), correction factor for the effect of gust, had been the same as Er1 for flutter, which was larger than 1. Considering the turbulence effects on galloping, Er1 was determined as 1 for galloping. 7. Revision of the manual Although the formulae to predict wind-induced vibrations of bridge girders are provided in the manual, only the outlines are described for towers and cables. Detailed description of the design methods for towers and cables has been required. In the manual, assumed bridge types were suspension bridges, cable-stayed bridges and box girder bridges. Recently, plate girder bridges with very small torsional rigidity have been applied to relatively long span length in Japan. Noise barriers, which make girders bluff, are often attached to highway bridges in city area. In some cases, two bridges are constructed very closely to each other, which may cause buffeting problems. Wind-resistant design methods for these bridges have been also required. From these reasons, the committee for the Wind Resistant Design Manual was reorganized at the Japan Road Association in 2000 to revise the manual. The revised manual will hopefully be published in 2003. 8. Conclusions In this paper, the Wind-Resistant Design Manual for highway bridges was introduced. The main features of the manual are as follows: (1) The critical wind speed for flutter and galloping can be estimated by simple formulae as well as by wind tunnel testing. (2) The amplitude of vortex-induced vibrations can be estimated by simple formulae as well as by wind tunnel testing. The effects of turbulence on vortexinduced vibrations are incorporated. (3) Wind properties and design wind load are determined according to the models or methods proposed by Dr. Davenport.
Acknowledgements The author would like to express his gratitude to the members of Wind-Resistant Design Committee in Japan Road Association. The discussions at the Committee were essential for preparing the paper as well as the manual.
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References [1] A.G. Davenport, The Interaction of Wind and Structures, Engineering Meteorology, Elsevier, Amsterdam, 1982, pp. 527–572. [2] A.G. Davenport, Buffeting of a suspension bridge by storm winds, Proc. ASCE 88 (1962) ST3. [3] H. Sato, On the aerodynamic forces on a rectangular prism in smooth and turbulent flow, including motion-induced effects, Thesis of Master of Engineering Science, University of Western Ontario, 1983. [4] N. Narita, K. Yokoyama, H. Sato, Y. Nakagami, Aerodynamic characteristics of continuous box girder bridges relevant to their vibrations in wind, Seventh International Conference on Wind Engineering, Aachen, Vol. 4, 1987, pp. 283–29.
Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 1499–1509
Wind-resistant design manual for highway bridges in Japan Hiroshi Sato* Structures Research Group, Incorporated Administrative Agency, Public Works Research Institute, Minamihara 1-6, Tsukuba-shi, Ibaraki-ken 305-8516, Japan
Abstract In Japan, wind-resistant designs of highway bridges up to the span length of 200 m are conducted according to the Wind-Resistant Design Manual for highway bridges. This manual is also applicable to the highway bridges up to the span length of 300 m with minor modification of its provisions. In this paper, the manual is outlined first to illustrate windresistant design procedure in Japan. Then turbulence effects included in the manual are described with Dr. Davenport’s contributions to the manual. r 2003 Elsevier Ltd. All rights reserved. Keywords: Wind-resistant design; Manual; Highway bridges; Turbulence effects
1. Introduction Highway bridges up to the span length of 200 m in Japan are designed according to the specification for highway bridges. With regard to wind-resistant design, the specification prescribes design wind load of bridges and dimensions of pipe structures, however, there is no sufficient provision on wind-induced vibrations in the specification. The wind-resistant designs of long-span bridges in Japan were mostly based on the Wind-Resistant Design Criteria for Honshu–Shikoku bridges. The criteria was originally formulated in 1964 and revised in 1976 by the ad hoc committee in the Japan Society of Civil Engineers. The criteria prescribes the design wind load
*Corresponding author. Tel.: +81-29-879-6726; fax: +81-29-879-6739. E-mail address: [email protected] (H. Sato). 0167-6105/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2003.09.012
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including the effect of gust and the evaluation method of wind-induced vibrations based on wind tunnel testing. Since the revision of the criteria, wind engineering had made remarkable progress and a considerable amount of aerodynamic data on bridges had been acquired. It was thought that wind-resistant designs for limited types of bridges could be made without wind tunnel testing if reliable formulae for the estimation were provided. It was also thought that prediction of wind-induced vibration would become more reasonable and reliable if the effect of turbulence was incorporated appropriately in the design. From these reasons, the working group was organized in the Japan Road Association in 1984 to prepare the Wind-Resistant Design Manual for highway bridges. The working group was reformed into the Wind-Resistant Design Committee in 1986, and the manual was published in 1991. At present in Japan, wind-resistant designs of highway bridges up to span lengths of 200 m are conducted according to the Wind-Resistant Design Manual for highway bridges. This manual is also applicable to highway bridges up to span length of 300 m with minor modification of its provisions. If span length of a bridge becomes longer than 300 m; a specific design criteria will be established for the bridge. In this paper, Wind-Resistant Design Manual for Highway Bridges is outlined to illustrate wind-resistant design procedure in Japan.
2. Procedure of wind-resistant design using the Wind-Resistant Design Manual The procedure of wind-resistant design using the manual is as follows. The wind properties (basic wind speed, design wind speed, turbulence intensity, etc.) are decided first. The design wind load is calculated, and the static design of the bridge is made. At this stage, where major dimensions of the bridge are determined, wind-induced vibrations to be studied further are chosen considering the design wind speed, the deck width and the span length of the bridge. For a bridge with short span length, no study on wind-induced vibration is required. For long-span bridges, wind-induced vibrations specified above should be predicted using the formulae provided in the manual. Then the wind-induced vibrations are evaluated. The formulae were established considering past wind tunnel test results. Since the formulae are expressed by means of only a few parameters, the prediction error of the formulae is not negligibly small. Therefore, the formulae were established, so that their prediction may become safer. It means that evaluation ‘good’ based on the formulae is almost always ‘good’, however, evaluation ‘no good’ based on the formulae is not always ‘no good’ because of the safety margin of the formulae. In case that the evaluation based on the formulae turns out to be ‘no good’, the bridge engineer can modify the design or he can predict wind-induced vibrations more accurately by means of wind tunnel testing. The evaluation based on wind tunnel testing has priority over that based on the formulae.
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3. Wind properties used in design Wind properties are modeled fundamentally according to Davenport [1]. The terrains are classified into four, namely rough sea (Terrain 1), open farmland (Terrain 2), suburbs (Terrain 3) and city centers (Terrain 4). 3.1. Basic wind speed U10 The basic wind speed U10 is defined as the mean wind speed over open farmland (Terrain 2) at an elevation of 10 m; averaged over a period of 10 min: Using the meteorological data at weather stations in Japan, extreme wind speeds were estimated. The return period was 100 years. The basic wind speeds were classified into four categories, namely 30, 35, 40 and 45 m=s: 3.2. Design wind speed Ud The power law matched with the log profile at an elevation of 30 m was applied to the mean wind speed profile. The design wind speed for dynamic design can be obtained from the following formula: Ud ¼ U10 E1 ;
ð1Þ
where E1 is the correction factor for altitude and terrains. 3.3. Turbulence properties Typical values for turbulence properties such as turbulence intensities and power spectral density functions are provided in the manual. su ; r.m.s. of longitudinal velocity fluctuation, was assumed to be 2:5u; where u is the friction velocity.
4. Design wind load The design wind load Pd for static design can be obtained from the following formula: Pd ¼ 0:5rUd2 Cd An G;
ð2Þ
where Cd is the drag coefficient, An the projected area, G the gust factor. Design wind speed Ud for design wind load is 40 m=s: Gust factor G was determined so that the shear force induced by the design wind load may become equivalent to those estimated by the gust response analysis proposed by Davenport [2]. 1.9 is used for gust factor G: In the manual, the typical values for Pd or Cd are provided according to the type of bridges and members.
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5. Wind-induced vibration 5.1. Prediction of wind-induced vibrations by the formulae The critical wind speed for flutter and galloping can be predicted by simple formulae as well as by wind tunnel testing. The formula for prediction of critical wind speed of flutter is as follows: Ucf ¼ 2:5fy B;
ð3Þ
where Ucf is the critical wind speed for flutter, fy the natural frequency of the 1st torsional mode, B the width of bridge section. The formula for prediction of critical wind speed of galloping is as follows: Ucg ¼ 8fh B : when angle of attack is almost 0 ;
ð4Þ
Ucg ¼ 4fh B : when angle of attack is positive;
ð5Þ
where Ucg is the critical wind speed for galloping, fh is the natural frequency of the 1st bending mode. The amplitude of vortex-induced vibrations can be predicted by simple formulae as well as by wind tunnel testing. The effects of turbulence on vortex-induced vibrations are incorporated. The formulae for prediction of critical wind speed and amplitude of vortex-induced vibrations are as follows: Ucv ¼ 2:0fh BFfor bending;
ð6Þ
Ucv ¼ 1:33fy BFfor torsion;
ð7Þ
Ac ¼ Ae Ems Et ;
ð8Þ
Ae ¼ 0:05bds ðd=BÞ=ðmr dh ÞFfor bending ðin h=BÞ;
ð9Þ
Ae ¼ 13:2bds ðd=BÞ3 =ðIpr dy ÞFfor torsion ðin degreeÞ;
ð10Þ
Et ¼ 1 15bt ðB=dÞ1=2 Iu2 X0Ffor bending;
ð11Þ
Et ¼ 1 20bt ðB=dÞ1=2 Iu2 X0Ffor torsion;
ð12Þ
where Ucv is the wind speed for the maximum amplitude of vortex-induced vibration, Ac the corrected maximum amplitude of vortex-induced vibration, Ae the maximum amplitude of vortex-induced vibration for rigid model in smooth flow, Ems the correction factor for vibrational mode (about 4=p), Et the correction factor for the effect of turbulence, bds the correction factor for sectional shape, d the depth of bridge section, mr ; Ipr the reduced mass or reduced mass moment of inertia ðmr ¼ m=ðrB2 Þ; m the mass per unit length of the bridge, Ipr ¼ Ip=ðrB4 Þ; Ip the mass moment of inertia per unit length of the bridge), dh ; dy the structural damping (logarithmic decrement), bt the correction factor for sectional shape, Iu the intensity of turbulence.
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5.2. Prediction of wind-induced vibrations by the wind tunnel testing Wind tunnel testing increases the reliability of wind-resistant design of bridges. Since the prediction of wind-induced vibrations based on the formulae may often provide safer value, there is a fair chance that wind tunnel study provides more economical design. Turbulence has significant effects on wind-induced vibrations. Therefore, standard wind tunnel testing methods in turbulent flow (full aeroelastic model test, taut-strip model test) were described in the manual as well as conventional testing methods (spring-mounted rigid model test, measurement of steady aerodynamic forces). 5.3. Evaluation of wind-induced vibrations The method of verification of flutter, galloping and vortex-induced vibrations are provided in the manual. For flutter and galloping, the following inequalities shall be satisfied: Ucf > Ur Ffor flutter;
ð13Þ
Ucg > Ur Ffor galloping;
ð14Þ
Ur ¼ Ud Er1 Er2 ;
ð15Þ
Ud ¼ U10 E1 ;
ð16Þ
where Ucf is the critical wind speed for flutter, Ucg the critical wind speed for galloping, Ur the reference wind speed for flutter and galloping, Ud the design wind speed, Er1 the correction factor for the effect of gust, Er2 the safety factor (1.2), U10 the basic wind speed, E1 the correction factor for altitude and terrains. It was found that onset velocity for negative damping, which caused galloping, increased remarkably in turbulent flow [3,4]. It suggested that Ucg would be high enough when turbulence intensity is high. Therefore, Erl ; correction factor for the effect of gust, was determined as 1 for galloping, while Er1 for flutter was 1.1–1.25 depending on wind turbulence. For vortex-induced vibrations, unless the following inequality (17) is satisfied, the maximum amplitude of the vortex-induced vibration shall be less than the allowable amplitude shown in Eqs. (18) or (19). Ucv > Ud ;
ð17Þ
ha ¼ 0:04=fh Ffor bending ðin mÞ;
ð18Þ
ya ¼ 2:28=ðbfy ÞFfor torsion ðin degreeÞ;
ð19Þ
where Ucv is the wind speed for the maximum amplitude of vortex-induced vibration, ha the allowable amplitude for bending vortex-induced vibration, fh the natural frequency of the 1st bending mode, ya the allowable amplitude for torsional
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vortex-induced vibration, b the distance between the deck center and the center of outmost lane, fy the natural frequency of the 1st torsional mode.
6. Dr. Davenport’s contributions One of the main features of the manual is incorporation of turbulence effects on wind-induced vibrations, where Dr. Davenport has made a great contribution. Therefore, the manual was influenced directly and indirectly by Dr. Davenport’s works. Some of the direct contributions of Dr. Davenport’s to the wind-resistant design methods described in the manual are as follows: (1) In the manual, simple formulae for predicting amplitude of vortex-induced vibrations are provided. The effects of turbulence on vortex-induced vibrations are incorporated. Wind turbulence properties can be predicted based on the model proposed by Dr. Davenport [1]. (2) Design wind load shall be calculated considering the effects of gust response, which was predicted according to the methods proposed by Dr. Davenport [2]. (3) Er1 ; correction factor for the reference wind speed for galloping, was determined as 1, because onset velocity for negative aerodynamic damping, which causes galloping, would increase remarkably in turbulent flow. The aerodynamic damping was measured by the author under the supervision of Dr. Davenport [3]. Since the gust response analysis methods are so well known, there may be no need for me to describe Dr. Davenport’s contribution here. Our research on the wind resistant design methods for vortex-induced vibrations and galloping and Dr. Davenport’s contributions are described in the following sections. 6.1. Design for vortex-induced vibrations Vortex-induced vibrations may take place to long-span bridges at wind speeds considerably lower than their design wind speed. For the design of long-span bridges, therefore, prediction of amplitude of vortex-induced vibrations becomes very important. The mechanism and countermeasures of the vortex-induced vibrations were studied in Japan in 1970s, but most of the studies were based on wind tunnel experiments in smooth flow. To understand the effects of turbulence on the vortex-induced vibrations, wind tunnel studies were conducted at the Public Works Research Institute in 1980s. In the study, vortex-induced vibrations of taut-strip models for 16 types of bridge cross section were measured in smooth flow and in three types of turbulent flow. Used bridge cross sections were rectangular section, trapezoidal section, hexagonal section, two-plate section and two-box section. B=d of the cross section ranged from 1 4 to 10. Scale ratio of the model was assumed as 200 ; and the model length was 1:2 m: Turbulent flow was generated by spires and floor roughness. Turbulence intensities for longitudinal wind speed ðIuÞ and vertical wind speed ðIwÞ were as follows: Iu ¼ 6:8%; Iw ¼ 5:1% for turbulent flow 1; Iu ¼ 11:5%; Iw ¼ 7:8% for turbulent flow 2; and Iu ¼ 22:4%; Iw ¼ 14:1% for turbulent flow 3. Integral scale of longitudinal wind
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1 speed (Lxu) was a little smaller than 200 of typical natural wind’s scale, and that of 1 vertical wind speed (Lxw) was about 200 of typical natural wind’s scale. In smooth flow, vortex-induced vibrations of vertical bending mode were observed for 14 types of cross sections. It was found that amplitude of the vortex-induced vibrations decreased in the turbulent flow. Relationship between the maximum amplitude and turbulence intensity Iw is shown in Fig. 1. In the figure, nondimensional amplitude ðh=BÞ was multiplied by reduced mass mr and structural damping dh : The ratios of the amplitude ðh=BÞmr dh to that in smooth flow are shown in Fig. 2. From the figures it was found that maximum amplitude decreases with turbulence intensity. The turbulence effects on vortex-induced vibrations were a little smaller for hexagonal sections than for other sections. It also seemed that the turbulence effects increased a little with B=d: Based on these results and others, the formulae for predicting amplitude of vortexinduced vibrations were developed as were shown in Eqs. (8)–(12). To utilize these formulae, turbulence intensities at the bridge site should be predicted. In the manual, wind properties are modeled fundamentally according to Dr. Davenport [1]. Dr. Davenport’s model was also applied to prediction of turbulence intensities as follows. In the manual, the power law matched with the log profile at an elevation of 30 m was applied to the mean wind speed profile. Intensity of turbulence Iu was calculated first at an elevation of 30 m assuming su ¼ 2:5u and the log profile. Then the effect of altitude on Iu was corrected by the power law profile. Therefore, Iu can be predicted by the following formula:
Iu ¼ ð30=zÞa =½lnð30=z0 Þ;
ð20Þ
where z is the altitude of the bridge (in m), a the exponent of the power law profile, z0 the roughness length (in m).
Fig. 1. Effects of turbulence on vortex-induced vibration amplitude ðh=BÞmr dh :
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Fig. 2. Ratios of the amplitudes ðh=BÞmr dh to those in smooth flow.
6.2. Design for galloping When a model of a long-span bridge with bluff box section is tested in smooth flow, divergent vibration called galloping is sometimes observed. In turbulent flow, the divergent amplitude vibration may turn to less divergent but more random vibration. In order to understand the turbulence effects on galloping, unsteady aerodynamic forces acting on the 1:2 retangular prism were measured in smooth and turbulent flow at the University of Western Ontario by the author under the supervision of Dr. Davenport [3]. The measurement of lift force on the model, whose longer side was set along the wind, is described here. Turbulence was generated by a coarse grid placed upstream of the model. The mesh size and the bar size were 0.61 and 0:15 m; respectively. The intensity of turbulence used in the experiment was slightly smaller than that usually found in the natural wind. The ratio of the integral scale of the turbulence to the width of the model was smaller than the ratio of the integral scale of natural wind to the width of a long-span bridge section. The model had a relatively large aspect ratio ð0:10 m 0:20 m 2:13 mÞ: End plates were not attached to the model. The lift force was measured by the pneumatic averaging method. This consists of two kinds of averaging methods, i.e. continuous averaging by porous material and discrete avaraging by a manifold. Power spectral density functions (PSDFs) of the fluctuating lift on the model at rest, SL ; are shown in Fig. 3. The turbulence broadens the peaks of the PSDF of the lift and also lowers the peak reduced frequency (from 0.16 in smooth flow to 0.12 in turbulent flow). Unsteady lift force was measured by the forced oscillation method. Then aerodynamic damping ratio for heaving motion zah was estimated. The results are
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fSL/ (0.5 U2B)2
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fB/U
0.05
Lift
Smooth flow
h
Turbulent flow
0.00
Wi nd
-0.05
Aerodynamic Damping ah
0.10
Fig. 3. Power spectral density functions of the fluctuating lift on the rectangular model at rest.
1.0E-03
1.0E-02
1.0E-01
1.0E+00
fB/U
Fig. 4. Aerodynamic damping for heaving motion of the rectangular model.
shown in Fig. 4. In smooth flow, sharp peak of positive aerodynamic damping appears at fB=U ¼ 0:17: The aerodynamic damping turns to negative at fB=Uo0:15: According to free vibration test of the model conducted in smooth flow, divergent vibration of heaving mode, which can be regarded as galloping, was observed in the corresponding wind speed range. Aerodynamic damping in the turbulent flow is also shown in Fig. 4. The pattern is similar to that of smooth flow, however, the positive damping peak shifts to the lower reduced frequency and the peak becomes broader in the turbulent flow. The turbulence decreased the critical reduced frequency to fB=U ¼ 0:10 (or increase the critical reduced wind speed to U=ðfBÞ ¼ 10), where the aerodynamic damping turns to negative. The wind-induced vibration observed in the free vibration test was less divergent in the turbulent flow than in smooth flow.
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Similar studies were conducted at the Public Works Research Institute for a continuous box girder bridge [4]. It was also concluded that turbulence decreases the reduced frequency (or increases the reduced wind speed) for the onset of negative damping the causes galloping. As was described in Eq. (14), critical wind speed for galloping should be higher than the reference wind speed that can be obtained from Eq. (15). So far Er1 for galloping in Eq. (15), correction factor for the effect of gust, had been the same as Er1 for flutter, which was larger than 1. Considering the turbulence effects on galloping, Er1 was determined as 1 for galloping. 7. Revision of the manual Although the formulae to predict wind-induced vibrations of bridge girders are provided in the manual, only the outlines are described for towers and cables. Detailed description of the design methods for towers and cables has been required. In the manual, assumed bridge types were suspension bridges, cable-stayed bridges and box girder bridges. Recently, plate girder bridges with very small torsional rigidity have been applied to relatively long span length in Japan. Noise barriers, which make girders bluff, are often attached to highway bridges in city area. In some cases, two bridges are constructed very closely to each other, which may cause buffeting problems. Wind-resistant design methods for these bridges have been also required. From these reasons, the committee for the Wind Resistant Design Manual was reorganized at the Japan Road Association in 2000 to revise the manual. The revised manual will hopefully be published in 2003. 8. Conclusions In this paper, the Wind-Resistant Design Manual for highway bridges was introduced. The main features of the manual are as follows: (1) The critical wind speed for flutter and galloping can be estimated by simple formulae as well as by wind tunnel testing. (2) The amplitude of vortex-induced vibrations can be estimated by simple formulae as well as by wind tunnel testing. The effects of turbulence on vortexinduced vibrations are incorporated. (3) Wind properties and design wind load are determined according to the models or methods proposed by Dr. Davenport.
Acknowledgements The author would like to express his gratitude to the members of Wind-Resistant Design Committee in Japan Road Association. The discussions at the Committee were essential for preparing the paper as well as the manual.
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References [1] A.G. Davenport, The Interaction of Wind and Structures, Engineering Meteorology, Elsevier, Amsterdam, 1982, pp. 527–572. [2] A.G. Davenport, Buffeting of a suspension bridge by storm winds, Proc. ASCE 88 (1962) ST3. [3] H. Sato, On the aerodynamic forces on a rectangular prism in smooth and turbulent flow, including motion-induced effects, Thesis of Master of Engineering Science, University of Western Ontario, 1983. [4] N. Narita, K. Yokoyama, H. Sato, Y. Nakagami, Aerodynamic characteristics of continuous box girder bridges relevant to their vibrations in wind, Seventh International Conference on Wind Engineering, Aachen, Vol. 4, 1987, pp. 283–29.