Abstract
The two-edge box girder has been widely used as a stiffened girder in cable-stayed bridges. However, such girders have weakness in aerodynamic stability. To improve its aerodynamic stability, some previous researchers have given slope to the edge box instead of installing additional attachments to aerodynamically stabilize the bridge. For wind load design, an angle of attack (AOA) has to be considered. However, the effect of AOA has not been studied for sloped box girder yet. In the present study, the effect of AOA on the static wind load coefficient was investigated for 2-edge sloped box girder. A series of wind tunnel tests was performed by varying the box slope angle from 0° to 17° where AOA was set from −10° to 10°. Results showed that the lateral wind force is considerably reduced with the increase of the box slope angle except the case with the physical angle of 8°–11°. For practical AOA range, the box slope should be larger than 15° to minimize the aerodynamic static lateral force on the girder.
1. Introduction
Wind load is one of the most important design loads on bridges, especially for long-span bridges with low damping and high flexibility. 2-edge box girder section shown in Figure 1 has been widely used for various civil engineering practices including stiffened girder for long-span cable-stayed bridges since it has good constructability and cost effectiveness [1, 2].
However, 2-edge box girder section is a type of bluff bodies that is usually elongated with sharp corners. It is known that 2-edge box girder section has weak aerodynamic stability; it has not only dynamic stability with high vortex-induced vibration (VIV) and low flutter speed but also static stability with high aerodynamic static coefficient [3, 4].
To improve aerodynamic stability of the bridge section, several studies have been conducted by a number of researchers. For example, the effect of various attachments such as fairing, flap, edge plate, side plate, and baffle plate has been investigated to improve the aerodynamic stability of the edge girder section [5–8]. However, all these methods mentioned above have to install additional attachments onto a bridge. These can be effective and proper methods when existing bridges do not have enough aerodynamic stability.
To overcome the limitation of the need to install some additional attachments, Daito et al. [9] have designed a steel box with a slope. By providing a slope to the box section, the vibration force can be reduced compared to the ordinary steel box girder section, as shown in Figure 2. Suppose that θ (angle of attack, AOA) is 0° in Figure 2; when wind flows from left side to right side, the segregation point (vortex starting point) is located in p.1 for a rectangular box as shown in Figure 2(a). By decreasing the distance between boxes, the vibration force can be reduced as shown in Figure 2(b). However, this method is inappropriate since the torsional stiffness of the whole girder can be significantly affected by the distance between the two boxes. When a specific slope range is applied to the box as shown in Figure 2(c), the segregation point may move from p.1 to p.3 and the vibration force can be reduced compared to the case shown in Figure 2(a). This increased aerodynamic stability is the basic idea of achieving stabilized section. If the stabilized section is properly considered in the initial design of the bridge section, the aerodynamic stability can be increased without additional attachment after the construction of the bridge.
Daito [10] focused on the improvement of the flutter speed of 2-edge sloped box girder. The static aerodynamic response of such girder is also an important parameter that governs the wind resistance [11], and Lee et al. [12] have studied the effect of box slope on static aerodynamic response of 2-edge sloped box girder. In their study, box slopes were varied from 0° to 17° and static coefficients were obtained through a series of wind tunnel tests, where AOA was set as 0°. Results were compared with CFD (computational fluid dynamics) analysis, and it was found that the test results agree well with those from CFD analysis. Thus, their test methodologies can be used to evaluate the static aerodynamic coefficient of 2-edge sloped box girder. Their results showed that drag coefficients were generally decreased with increasing box slope angle except for box slope range from 8° to 11° when B/H ratio is 9.2.
Figure 3 shows the streamline for the different box slopes (9° and 21°) obtained from CFD analysis [13]. It can be seen that a localized vortex (air bubble) is generated near the left box when the box slope angle is 9°. Thus, air pressure is applied on the windward box (left box) first. Then, the secondary air pressure acts on the leeward box (right box). A similar streamline was observed for the case with the box slope angle of 8°–11°, and aerodynamic static coefficient is increased. On the other hand, for the box slope angle of 21°, a vortex is generated in the space between the windward and leeward box, as shown in Figure 3(b). Due to the interaction between the wind and deck shape, a recirculation zone is formed and the streamline around the bridge is quite similar to that of the single box girder [14–16]. A similar streamline was observed for the case without the box slope angle of 8°–11°. From this result, it can be found that sloped box girder does not give improved aerodynamic static coefficient for all sloped section, and the slope of the box girder should be carefully determined.
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The wind in real field has an angle of attack (AOA). In case of bridges, aerodynamic static coefficient of bridge section can be substantially affected by AOA, and it must be considered in design procedure. To estimate static wind load, the range of AOA −3° to 3° is generally considered. However, in some cases, the range of wind AOA should be decided by wind environment assessment. When AOA is considered for the sloped box section, the physical angle will be changed. For example, when AOA is minus 3° (that is, 3° in counter-clockwise (CCW) direction) with box slope of 15°, the physical angle is equal to 12°, as shown in Figure 4 [13].
In summary, the sloped box girder section proposed by Daito et al. [4] was investigated in this study. They reported that the dynamic stability can be improved by using the proposed section. Also, Lee et al. [12] found that the static wind drag force is reduced by applying the sloped box girder, where AOA was set as 0°. For practical use of such section, the variation of AOA must be considered.
In this study, the effect of AOA on the aerodynamic static coefficient was investigated for the 2-edge sloped box girder through a series of wind tunnel tests for practical design purpose. For this, the verified wind tunnel test methodology conducted by Lee et al. [12] was applied. Box slopes were varied from 0° to 17° where AOA was considered from −10° to 10° by 1° step for each case. From the results, it can be found that the minimum box slope of 15° should be used for the test section to minimize the aerodynamic static lateral force on the girder.
2. Wind Tunnel Test
2.1. Test Section
Figure 5 shows target section considered in this study. It is a composite cross section of cable-stayed bridge which has 700 m span with 4 lanes. The width, B, and height, H, of the target section are 34.0 m and 3.7 m, respectively. Thus, the B/H ratio is approximately 9.2 which is the average B/H of the cable-stayed bridge [12]. The width of the box is 4.8 m, and outer height of the box is 1.9 m. The inner box height is 3.4 m. This results in a box slope of 17°. The wind speed was measured by heated wire anemometer which was installed in front of the model during the test, as shown in Figure 5(b).
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For the wind tunnel test, a 1 : 70 scale model was manufactured. Detailed dimensions of the model for the wind tunnel test are shown in Table 1. The surface of the test model was fabricated as smooth as possible so that the effect of surface roughness was negligible. The test model was also made to have sufficient rigidity so that no deformation of the model would occur during the wind tunnel test. To satisfy these characteristics, the test model was made by using acrylonitrile butadiene styrene (ABS) and Foamex materials. The scaled model was designed to have the same center of mass and moment as those of the original section. The experiment pathway blockage ratio, which is the ratio of wind tunnel cross section and projection area of the model, was 3.2%. It was controlled to be less than 5% [17]. The length and width ratio of the scaled model was designed about 2.06, which is more than 2. The production error of this model was less than 1 mm. Box slope was varied by adding or removing various attachment brackets, as shown in Figure 6(a). Figure 6(b) shows the example of the test section with 9° box slope angle.
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2.2. Test Setup, Method, and Parameters
The test was performed at Wind Tunnel Lab in Korea University. The dimension of the inlet part of the wind tunnel was 1.0 m × 0.8 m × 1.5 m (width × height × length). 3-component load cell (LMC-3501-5, Nissho) were used to obtain 3-dimensional aerodynamic forces. The load cell was securely fixed to each side of the model with bolts, as shown in Figure 7. The sign convention for static aerodynamic coefficients used in this study is shown in Figure 8.
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The drag direction is the same as wind direction. The lift direction is the orthogonal direction of the drag direction. Drag and lift directions are defined as global coordinate of wind direction. Aerodynamic static coefficients can be obtained from formulations. For example, the aerodynamic static coefficient can be obtained from the following equations:where Cd is the aerodynamic drag force coefficient, Cl is the aerodynamic lift force coefficient, Cm is the aerodynamic moment force coefficient, Fd is the drag force, Fl is the lift force, Fm is the moment force in the model, ρ is the density of air, U is the velocity, and B is the reference length.
In Figure 8, the x and z directions represent the local coordinate of the section. Between the local and global (wind direction) coordinate, the following relationships are satisfied:where Cx is the aerodynamic x-direction force coefficient, Cz is the aerodynamic z-direction force coefficient, and θ is an angle of attack (AOA).
The static aerodynamic coefficients (Cd, Cl, and Cm) were evaluated by the test results with equation (1). Then, Cx and Cz values were obtained by using equation (2). These values are the final drag and lift coefficient including the effect of AOA. In the case of Cm, Cm value is not dependent for the coordinate system [18]. During the test, the air flow was assumed as uniform and wind speed was set as 8 m/s. Reynolds numbers were 2.57 × 105. This value was selected, since the aerodynamic coefficients are stabilized at this Reynolds number in previous study [12]. Considered box slope angles were 0°, 3°, 6°, 9°, 12°, 15°, and 17° (7 cases). AOA was considered from −10° to 10° by 1° step for each case. Thus, a total of 147 test cases were conducted as shown in Table 2. For accurate setup of AOA during the test, an AOA controller with protractor having 0.05° accuracy was used, as shown in Figure 9.
3. Test Results
3.1. Variation of Cx Coefficients with AOA
Figure 10 shows the relationship between the aerodynamic x-direction force coefficients, Cx, and AOA for various box slope angles. Cx is related to lateral force of the girder, and it should be close to zero to minimize the lateral forces. From Figure 10, it can be seen that Cx values are generally decreased with the increase of the box slope angle. However, Cx values with box slope angle of 9° (filled circles in Figure 8) are fairly close to or slightly higher than Cx values with box slope angle of 6°. Lee et al. [12] reported that the drag coefficient is increased due to secondary impact for the case with the box slope angle of 8°–11° for 0° of AOA. The case with box slope of 9° is in this range and shows incased Cx similarly with previous study.
Figure 11 shows the variation of Cx with box slope angle for various AOA. It can be seen that Cx is rapidly decreased with increasing box slope angle for AOA ranging from −10° to −6°. For AOA larger than −6°, complex behavior of Cx is observed. To investigate this complex behavior of Cx, the test results are divided into three groups as shown in Figures 12(a) and 12(b). Group A represents the test data of AOA from −10° to −6°. Group B has AOA from −5° to 5°. The rest of test data was categorized as group C. Group B includes values from −5° since Cx is slightly increased at the box slope of 9°.
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From Figure 12(a), for group A, it can be seen that Cx is decreased with the decrease of the absolute value of AOA, and Cx is sharply decreased with increasing box slope. Similarly with group A, Cx is also decreased with the decrease of the absolute value of AOA for group C. For low and high AOA, such as group A or C, the aerodynamic static coefficient can be directly affected by the wind force which acts in upper and lower slabs rather than wind force on the girder side [19]. This phenomenon can be explained from Figure 13. As a result, for groups A and C, Cx is increased with the increase of the absolute value of AOA due to increased windward wall.
The variation of Cx in group B shows clearly different behaviors compared to groups A and C, as shown in Figure 12(b). The difference of Cx for adjacent AOA for the same box angle is quietly nonuniform for group B. This means that Cx is suddenly changed at specific box slope angle. The regions where Cx is suddenly changed are marked in Figure 12(b) by using rectangular box. For example, when the box slope angle is equal to 12°, Cx is jumped for −1°, −2°, and −3° of AOA. For these cases, the physical angles are 9°, 10°, and 11°, respectively. These physical angles are in the range of 8°–11° and the secondary impact occurs as reported by Lee et al. [12]. The effect of secondary impact is described in Figure 3(a). The similar phenomenon was observed when the box slope angle ranged from 3° to 15° as presented in Figure 12(b). However, this phenomenon did not occur for the box slope angle of 0° and 17°, since the physical angle is not in the range of 8°–11°. Except the cases having physical angles of 8°–11°, Cx is decreased with the increase of the box slope angle (up to 64.8%). In summary, Cx is decreased up to 64.8% with the increase of the box slope angle except group C (high positive AOA region) and the cases having the physical angle of 8°–11°.
3.2. Variation of Cz Coefficients with AOA
Figure 14 shows the variation of Cz with AOA for various box slope angles. It can be seen that Cz is increased with increasing AOA for all box slope angles considered in this study. Cz is related to vertical force of the girder, and it should be close to zero to minimize the vertical forces. From the test results, it can be found that Cz is increased with the increase of the value of AOA. This phenomenon is also related with Figure 13. By increasing the value of AOA, forces acting on the bottom of the girder are increased, as shown in Figure 13.
To investigate the effect of box slope angle for specific AOA, the variation of Cz with box slope angle for various AOA is plotted as shown in Figure 15. Also, data in Figure 15 are divided into three different group similarly with the case of Cx, as shown in Figures 16(a) and 16(b). Groups A, B, and C in Figures 16(a) and 16(b) are the same as previous Cx case. In the case of group A, Cz is slightly decreased with the increase of the box slope angle. The effect of AOA is negligible when the box slope angle is smaller than 9°. For group C, Cz is decreased with increasing box slope angle except box slope angles of 9 and 17°. However, the reduction is not large, and the variation of Cz due to AOA change is much significant. This is because the vertical force is more sensitive for AOA rather than the box slope angle as mentioned before (refer Figure 13). With increasing AOA value, the lift force is increased due to increased lift force on the lower slab of the girder, as shown in Figure 13.
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The variation of Cz with box slope angle in group B is more complex than that in groups A and C, as shown in Figure 16(b). Some unexpected changes (increases) of Cz were observed, and those are marked by using rectangular boxes in Figure 16(b). If these points in the rectangular box are removed, the variation curves are more smooth. Also, Cz has the maximum value at these points. For example, Cz has the maximum value when the box slope angle is 12° for −1° and −2° of AOA. For these AOA angles, the physical angle can be calculated as 10° and 11°, respectively, and physical angles are in the range of 8°–11°, which is the secondary impact range when B/H is equal to 9.2. When the secondary impact occurs, the vertical force can be increased, as shown in previous Figure 3(a) [20]. Thus, Cz is increased in this region. However, as shown in Figure 16(b), the effect of AOA is more significant than that of the box slope angle. For group B, the value of Cz is increased with the increase of the value of AOA. In summary, Cz, is sensitive to AOA, and the effect of box slope angle on Cz, is not large.
3.3. Variation of Cm Coefficients with AOA
Figure 17 represents relationship between Cm and AOA for various box slope angles. It can be seen that Cm is mostly increased with increasing AOA for all box slope angles considered in this study. The general variation shape of Cm curve is similar to Cz curve. It should be noted that Cm has positive sign for clockwise direction. Thus, Cm in positive AOA has larger value than that in negative AOA, since the wind load acts in the lower slab part of the girder and it makes the moment in clockwise direction for positive AOA, as shown in Figure 13. This means that Cm is largely governed by AOA rather than box slope angle, similar to the case of Cz.
To investigate the effect of the box slope angle on Cm, the variation of Cm with the box slope angle is plotted for three groups as shown in Figures 18(a) and 18(b), where groups A, B, and C are the same as the cases used for Figures 12 and 16.
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From Figure 18(a), it can be found that the absolute value of Cm is increasing with increasing box slope angle for groups A and C. The relationship between Cm and the box slope angle is almost linear, and the absolute minimum values of Cm are observed at the box slope of 0° for all cases in groups A and C. However, for group B, the box slope angle that has minimum value of Cm is not simply equal to 0° or 17°, and it varies depending on AOA. The minimum values of Cm are marked in Figure 18(b) as rectangular boxes. As mentioned before, these points fall into the physical angle of 8°–11°, which is the secondary impact range when B/H is 9.2. In the case of Cm, the secondary impact makes counter-clockwise moments and it reduces the Cm value.
The value of Cm is affected by both AOA and the box slope angle for group B, as shown in Figure 18(b). With increasing AOA, Cm is increased for group B. In the case of groups A and C, the effect of box slope angle is more significant than that of AOA, as shown in Figure 18(a).
In previous research [12], B/H of 6.7, 9.2, and 11.7 was considered. From the results, it can be seen that the effect of the secondary impact is increased with increasing B/H ratio. Thus, for larger B/H ratio than 9.2, it is expected that unexpected fluctuation of Cx may be increased and vice versa. In the case of Cz and Cm, the effect of AOA is larger than the box slope angle. Therefore, for other B/H ratios, the similar behavior with the case of B/H = 9.2 is expected for Cz and Cm.
4. Proposal of 2-Edge Sloped Box Girder Section for Wind Load Design
The static wind load per unit length acting on bridges in wind direction can be obtained using the following equation:where is the static wind load in wind direction, ρ is the air density (1.225 kg/m3), Vd is the design criteria wind velocity, Cx is the x-direction coefficient, G is the gust factor, and Br is the reference length (width).
should be minimized for efficient wind load design of the 2-edge sloped box girder to reduce the x-direction forces acting on the girder. It means that Cx should be small as possible considering the practical AOA range. In practical design, the range of AOA at −3° to 3° is generally used [21, 22]. Also, Cz and Cm should be close to zero to minimize the vertical forces and rotational moments in the girder.
Figure 19 shows the variations of aerodynamic static coefficients with box slope angle for the practical AOA range (from −3° to 3°). The shaded area in Figures 19(a)–19(c) represents the test results of Cx, Cz, and Cm, respectively. When the box slope is larger than 15°, this range is adopted from the results of Daito [10]. Daito [10] reported that flutter resistance of 2-edge sloped box girder is considerably improved when the box slope is larger than 15°. From Figure 19(a), it can be found that Cx is rapidly decreased with the increase of the box slope angle except the box slope angle of 9°. Furthermore, Cx values are almost converged to approximately 0.1 when the box slope angle is larger than 15° and has the minimum values. When the box slope is larger than 15°, the physical angle is larger than 12°, since the minimum AOA is −3°, and this physical angle is not in the secondary impact region (from 8° to 11° for B/H = 9.2). Thus, it can be concluded that the x-direction force can be minimized and flutter resistance can be improved when the box slope is larger than 15° based on the test results performed in this study and Daito [10].
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The variation of Cz with box slope angle for practical AOA range is shown in Figure 19(b). It can be seen that the variation of Cz is not large compared to Cx. For example, Cx is approximately 0.16 and 0.1 for box slope angles of 0° and 17°, respectively, when AOA is −3°. There are 60% changes in Cx. For the same AOA, Cz is approximately −0.46 and −0.40 for the box slope angle of 0° and 17°, respectively, and there is a little difference compared to Cx. Thus, the effect of the box slope angle on Cz is not large. Also, Cz is more close to zero when the box slope is larger than 15° compared to the case with box slope angle of 0°.
Figure 19(c) shows the variation of Cm with the box slope angle for the AOA ranging from −3° to 3°. In the case of Cx and Cz, the aerodynamic static coefficients are improved when the box slope is larger than 15° compared to 0° box slope case. However, in the case of Cm, the absolute values in shaded area are larger than 0° box slope case. This means that the larger torsional moment can occur when the sloped box section is used for 2-edge girder, especially for −3° and 3° of AOA. Thus, special care for torsional behavior is needed when using 2-edge sloped box girders.
5. Conclusions
Minimizing the static wind force is important for the design factor of long-span bridges. In this study, the aerodynamic static coefficients of a 2-edge sloped box girder were evaluated considering the angle of attack (AOA) through a series of wind tunnel tests. The test results are analyzed by introducing the physical angle concept to consider the effect of AOA. Finally, the effective box slope was determined based on the test results. The box slope should be larger than 15° to reduce the wind drag force and avoid the unexpected fluctuation of the wind drag force. The major findings of this study are as follows:(1)From wind tunnel test results, it can be found that Cx is rapidly decreased with the increase of the box slope angle except the cases having the physical angle of 8°–11° when AOA ranged from −10° to 5° (groups A and B). Also, Cx value is converged to 0.1 for practical range of AOA (−3° to 3°) when the box slope is larger than 15°.(2)Cz is increased with increasing AOA. This is because forces acting on the bottom of the girder are increased with increasing AOA. Similar phenomenon was observed for Cm. With increasing AOA, the forces that act on the girder slab make the rotational moments to the center of the girder. The effect of AOA on Cz and Cm is comparatively larger than that of the box slope angle.(3)The efficient box slope angle was determined based on the wind tunnel test results. It can be found that Cx tends to be minimized when the box slope is larger than 15°. When the box slope is larger than 15°, Cz is also smaller than the case of 0° box slope. However, larger torsional moment can occur with the increase of the box slope. Especially, for −3° and −3° of AOA, considerable increase in the torsional moments is expected and special care is necessary for these cases.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by a research grant from Kangwon National University.