Effects of guardrails on wind environment for vehicles and aerodynamic stability for bridges with box girders

Abstract

Normally strong winds in mountainous areas possess potential threats to the safety of vehicles travelling over the long-span bridges. Generally, decreasing the porosity of the guardrails could improve wind environment for vehicles, while the changed flow field around the bridge’s girder may weaken the structural aerodynamic stability simultaneously. To solve the two seemingly contradictory issues, such a long-span suspension bridge in mountainous areas is taken as the case study, and the guardrails are optimized with different schemes. The effects on wind environment for vehicles under normal traffic conditions are first studied by computational fluid dynamics (CFD) simulations. The further effects on the aerodynamic stability of the bridge under extreme winds are then determined by wind tunnel tests, and the observed non-divergent flutter is explainedbythe change in dynamic flow field. Results show that reducing the porosity of guardrails does improve the wind environment above the bridge deck, and the improvement on wind environment increases with the increase in angle of attack. After closing the guardrails completely, however, the girder appears non-divergent vibration different from the linear theoretical flutter when the critical wind speed is exceeded. The different post-flutter behaviors at different angles of attack are mainly related to the synchronization condition between the movement of vortex and the motion of the girder.

Keywords

box girder guardrails wind environment aerodynamic stability non-divergent flutter

Introduction

Box girders with streamlined cross sections show good aerodynamic performance and have been widely used for long-span cable-supported bridges. The single box girder is a common type. For instance, the Sutong Bridge with a main span of 1088 m (Ma et al., 2018), and the Russky Bridge with a main span of 1104 m (Syrkov and Krutikov, 2014) belonging to cable-stayed bridges, the Taizhou Yangtze River Bridge with two main spans of 2 × 1080 m (Tao et al., 2017), and the Great Belt Bridge with a main span of 1624 m (Larsen, 1993) belonging to suspension bridges are all composed of the single box girder.

However, the structural wind-resistance performance and the windproof precaution for vehicles are still two key issues for long-span cable-supported bridges. In general, when it displays more streamlined characteristics and show better aerodynamic stability, the box girder has fewer disturbances to the approaching flow. As a result, strong winds pass directly through the deck, placing vehicles in a dangerous wind environment. All the time, rolloveraccidents of vehicles travelling over bridges under crosswinds are common. In January 1990, the great storms in UK caused more than 400 wind-induced vehicle accidents of which rollover accidents are very common (Baker and Reynolds, 1992). On August 11, 2004, seven high-sided vehicles on the Humen Bridge in China were overturned by a sudden gale, leading to traffic disruption and direct economic losses (Wang et al., 2019). In September 2005, three trucks driving on Minjiang Bridge in Qingzhou were overturned by Typhoon Talim (Zhu et al., 2012). Because express ways have the characteristics of full closure, large traffic flow and high speed of vehicles, it may cause serious casualties and economic losses once an accident occurs.

Guardrails belong to secondary structures of the bridge, but they have obvious effects on the approaching flow. To improve the running safety of vehicles, decreasing the porosity of the guardrail or even setting windproof precautions like the wind barrieris an effective way by shielding the approaching flow (Laima et al., 2018) and reducing the lateral wind force on vehicles (Kozmar et al., 2014). On the contrary, when wind barriers are added, the box girder displays bluff characteristics more, leading to deterioration of the aerodynamic stability. Taylor et al. (2009) employed the discrete vortex method to analyze the aerodynamic stability of a footbridge, and found that reducing the height of pedestrian guardrail could improve the critical flutter wind speed. Buljac et al. (2017) studied the effects of wind barrier at the windward side on flutter performance of three typical bridge decks. They found that the windward wind barrier would aggravate the torsional instability, especially the bluff cross section of girder, and the critical flutter wind speed decreases with the its porosity decreases. Nagao et al. (1997) confirmed that the form and position of guardrail have significant effects on the vortex-induced vibration (VIV) of bridges. Zhu et al. (2015) carried out a series of wind tunnel tests and found that the porosity of pedestrian guardrail has a significant effect on the VIV response of stiffening girder at large angles of attack. The wind-resistance performance of the bridge and the windproof precaution for vehicles are two contradictive issues which were usually treated separately.

Nowadays, construction of highway is extending to mountainous areas with the rapid development of the transportation system in China, and an increasing number of cable-supported bridges span deep canyons. The single box girder still shows its advantages and has been adopted by some bridges, such as the Puli Bridge and the Longjiang Bridge. The maximum heights from their decks to the ground reach about 400 m and 285 m, respectively. Unlike coastal and plain areas, strong winds in mountainous areas are easy to block and deflect, showing large angles of attack (Li et al., 2017a, 2017b; Zhang et al., 2020) and affecting the aerodynamic stability of the bridges (Tang et al., 2020). The strong winds driven by local thermal effects could be recorded daily (Zhang et al., 2018), possessing potential threats to the running safety of vehicles. Due to the limited understanding of the complex wind characteristics and the insufficient prediction of the extreme climate, the aerodynamic stability for the bridges and the running safety for vehicles become more prominent, and it is necessary to solve the two issues simultaneously.

Reasonable optimization on the aerodynamic shape of the bridge’s girder is a reliable way. As discussed above, the guardrails can significantly affect the wind environment for vehicles and aerodynamic stability for bridges, so it is necessary to optimize their form and improve the applicability of the box girder in mountainous areas. In this paper, such a long-span suspension bridge under construction is taken as the research object, and the content is organized into five more sections. In Section 2, the engineering case is described. In Section 3, the improvement of guardrails on wind environment for vehicles is studied by means of CFD numerical simulations. Then Section 4 is devoted to investigate the aerodynamic stability of bridges with different guardrails under extreme winds by wind tunnel tests. In Section 5, the corresponding aerodynamic mechanism is explained by the change in dynamic flow field. Finally, in Section 6 some conclusions are drawn.

Case descriptions

A suspension bridge with a main span of 780 m under construction is taken as the case study. The bridge is in mountainous areas in southwest China where the river turns sharply many times, as shown in Figure 1. The elevation at the mid-span deck is about 1570 m, and its distance to the bottom of the canyon exceeds 312 m. The highest elevation of the mountain peaks on the east side is 2260 m, while that on the west side is relatively low, but it is very steep with a maximum height difference of 760 m. Moreover, the river course 1.25 km away from the upstream of the bridge site is very tortuous, with the maximum bending angles of 75.6°. Due to the complex and special terrains, the bridge is frequently suffered from strong winds with large angles of wind attack. According to the results of numerical simulation, both positive and negative angles of attack may occur at the bridge site, and the angle of wind attack is far beyond the normal situation ranging from −3° to +3° (Wu et al., 2019). The Chinese code (Ministry of Transport of the People’s Republic of China, 2018) points out that the tested angles of wind attack in wind tunnels could be extended to ±5° for those bridges in mountainous areas.

Figure 1.

General arrangement of the target bridge: (a) terrains around the bridge site (excerpted from Google Earth), (b) elevation (unit: m), and (c) cross-section of the girder (unit: mm).

The bridge is designed to be a four-lane dual carriageway, and the lanes are marked with 1 to 4 according to the traffic flow direction, as shown in Figure 1(c). The girder of the bridge is a streamlined box of which the cross-sectionhas a width of 31.4 m and a height of 3 m. Two marginal guardrails for vehicles are installed on both sides of the girder and named as the windward and the leeward guardrails, respectively. Each guardrail has a height of 1.56 m and contains five bars. A central guardrail is also installed. As both ends of the bridge are connected with tunnels, vehicles may suddenly be attacked by crosswinds when travelling over the bridge. Under normal conditions, therefore, it is necessary to ensure their safety. Under extreme winds, the traffic may be closed, but the aerodynamic stability of the bridge must be satisfied.

Improvement on wind environment for vehicles

Forms of the guardrails

The windward guardrail can significantly affect the separation of the approaching flow, so its form has important influence on the flow field above the girder. Due to the uncertain wind direction in practical engineering, the marginal guardrails were optimized simultaneously, as shown in Figure 2. The porosity of the marginal guardrails, which is expressed as a non-dimensional parameter φ, was first changed. Specifically, four porosities with the same height of 1.56 m were considered. They were marked with cases 1 to 4 corresponding to φ = 100% (no marginal guardrails), φ = 67% (original scheme), φ = 20%, and φ = 0% (bluff scheme), respectively. As the height of the marginal guardrails is smaller than a lorry’s, three optimized schemes were then considered. Based on case 4, the solid part with the length of 1.56 m was raised by 0.69 m, 1.28 m, and 1.56 m, respectively, while the lower part kept the porosity of 67%. The three schemes were marked with cases 5 to 7. In addition, although other non-structural components such as handrails for staff and maintenance tracks may affect the aerodynamic stability, they were omitted to outstand the effects of different marginal guardrails more clearly.

Figure 2.

Schematic diagram of computational cases.

Static CFD model

The scale ratio of the CFD model is 1/45 which is consistent with that of the experimental model. As shown in Figure 3, the computational domain was assumed to be 24B in the mean-flow direction and 12B in the cross-flow direction where B is the width of the box girder. To improve the accuracy and efficiency of calculation, the element size progressively increased from the boundaries of the girder to the computational boundaries, and the grids number for each case was more than 160,000. The boundary conditions were set as follows: the windward boundary 6B away from the center of girder was set as the velocity-inlet; the leeward boundary 18B away from the center of girder was set as the pressure-outlet; the upper and lower boundaries depended on the direction of the inlet flow; and the girder was set as the smooth wall. Unsteady Reynolds-averaged Navier-Stokes (URANS) simulations were performed by using the k-ω SST model and the adopted time-step was 10⁻³ s. The discretized problem was numerically solved by utilizing a SIMPLE pressure-velocity coupling algorithm. Momentum equation, turbulent kinetic energy equation and turbulent dissipation rate equation were all solved by second order upwind schemes. The CFD software FLUENT was employed.

Figure 3.

Computational domain and boundary conditions.

Results and discussions

The wind velocity field above the box girder is mainly focused. The effect of the inlet wind speed U₀ on the wind field above the bridge deck is first investigated, taking 0° angle of attack as an example, Figure 4 shows the mean streamwise velocity profiles normalized by U₀ above the four lanes. It can be found that the effect of the inlet wind speed on wind environment for vehicles is very limited, so the inlet wind speed U₀ is taken as 10 m/s in the follow-up study. Then, 3° of attack α with −5°, 0°, and 5° were considered as discussed in Section 2. Figure 5 shows the mean streamwise velocity profiles. To better understand the effects of the guardrails, the contours of the mean streamwise velocity around the box girder are also shown.

Figure 4.

Profiles of the mean streamwise velocity under different U₀ at null angle of attack.

Figure 5.

Profiles and contours of the mean streamwise velocity: (a) α = −5°, (b) α = 0°, and (c) α = 5.

At −5° angle of attack, when the guardrails are not installed, the incoming flow is unimpeded above the bridge deck, so the mean wind speeds above the windward lanes, that is, Lane 1 and Lane 2, are approximately equal to the incoming wind speed, except the position near the bridge deck where the mean wind speed is 0 m/s. However, the streamwise wind speed in the 0 to 2 m range above the two leeward lanes, that is, Lane 3 and Lane 4, is significantly reduced due to the sheltering of the central guardrail. After the guardrails on both sides are set up, the mean wind speed in the range of 0 to 3 m above the four lanes decreases obviously, and the windproof effect enhances with the decrease of φ. It should be noted that there is velocity enlargement in Lane 1 and Lane 2 for the range of more than 3 m above the bridge deck, which is more obvious and more lanes are affected at 0° angle of attack. When the guardrails are closed completely, that is, φ = 0%, the box girder acts like a bluff body, a vortex is formed behind the windward guardrail, which makes the mean wind speed near the deck surface negative, that is, opposite to the direction of the incoming wind. When the distance of the solid part to the deck surface increases to 0.69 m, that is, Height 1, the vortex behind the windward guardrail moves up, so the negative mean wind speed area moves up, too. However, the incoming flow can partially pass the windward guardrail through the gap between the solid plate and the deck surface, so the mean wind speed near the deck surface increases. However, when the distance further increases to 1.28 m or 1.56 m, that is, Height 2 or Height 3, the vortex-shedding phenomenon is excited behind the windward guardrail. Figure 6 shows the contours of RMS turbulent streamwise wind speed. It can be learnt the wind speed fluctuates apparently above the windward lanes, so the wind loads on the vehicles are constantly changing, which has a bad impact on the stability and comfort of driving.

Figure 6.

Contours of RMS turbulent streamwise wind speeds (α = −5°): (a) case 5, (b) case 6, and (c) case 7.

Although the variation trends of streamwise wind speed with φ are similar at different angles of attack, there are great differences in the flow field around the girder and value of wind speed. With the increase in angle of attack, the windproof effect of the guardrails is enhanced and the streamwise wind speed above the deck decreases in general, which is due to the blunt characteristic of the cross section. When the guardrails are closed completely, the size of the vortex behind the windward guardrail is expanded, especially at 5° angle of attack, so negative wind speeds are observed in a larger scope. With the solid part of guardrails is moved upwards, the mean wind speed near the deck surface increases as the incoming flow can partially pass through the gap. Meanwhile, another small vortex behind the central guardrail is formed (see Figure 5), which causes a negative mean velocity in a certain height range above Lane 3. It should be noted that the vortex-shedding phenomenon is excited when the solid part is moved to Height 3 at 0° angle of attack. For this case, no vortex is observed above the leeward lanes, so the flow direction is nearly parallel to the deck surface.

To better compare the improvement on wind environment, the container truck which is prone to rollover under crosswinds is selected as research object. The Chinese Code (Ministry of Transport of the People’s Republic of China, 2018) recommends that the wind speed within the driving height could be evaluated by the equivalent wind speed which is defined as

U_{eq} = \sqrt{(\frac{1}{z_{r}}) \int_{0}^{z_{r}} {\tilde{U}}^{2} (z) dz}

(1)

where $U_{eq}$ is the equivalent wind speed above bridge deck; $z_{r}$ is the height of vehicle, that is, 3.8 m; z is the distance from deck; and $\tilde{U} (z)$ is the streamwise wind speed z from bridge deck.

Compared with the original scheme (case 2), the percentage changes inequivalent wind speeds with different cases are listed in Table 1. It can be found that with the decrease of porosity, the equivalent wind speeds above the four lanes decrease in general. The improvement on wind environment also increases with the increase in angle of attack. When the marginal solid guardrails are installed, however, the windproof effect is enhanced at −5° angle of attack, but weakened at 5° angle of attack as the solid part is moved upwards.

Table 1.

Percentage changes in equivalent wind speeds for container truck.

Lane	α	Case 3 (φ = 20%)	Case 4 (φ = 0%)	Case 5 (Height 1)	Case 6 (Height 2)	Case 7 (Height 3)
1	−5°	−20.2%	−32.8%	−46.6%	−50.0%	−51.4%
	0°	−41.5%	−64.4%	−64.9%	−48.5%	−56.5%
	5°	−70.8%	−81.1%	−60.7%	−40.9%	−34.0%
2	−5°	−19.0%	−28.0%	−45.9%	−47.5%	−50.0%
	0°	−41.9%	−64.9%	−68.8%	−52.5%	−57.6%
	5°	−71.7%	−80.0%	−56.3%	−37.8%	−32.4%
4	−5°	−16.6%	−26.5%	−40.7%	−45.1%	−48.7%
	0°	−41.5%	−67.3%	−74.0%	−63.6%	−62.1%
	5°	−74.7%	−71.8%	−67.3%	−49.1%	−44.9%
4	−5°	−16.5%	−25.4%	−40.2%	−46.8%	−49.9%
	0°	−42.4%	−65.2%	−74.0%	−63.9%	−62.4%
	5°	−73.0%	−62.0%	−65.7%	−41.3%	−36.9%

Then, the effects of different inlet wind speeds are compared. Taking case 2 as an example, the variation of wind speed above bridge deck is evaluated by the influence coefficient which is defined as

λ_{s} = \frac{U_{eq}}{U_{0}}

(2)

where $λ_{s}$ is the influence coefficient of wind speed above bridge deck; and $U_{0}$ is the inlet wind speed.

Five inlet wind speeds U₀ with 5 m/s, 7.5 m/s, 10 m/s, 12.5 m/s, and 15 m/s were computed, Lanes 1 and 4 are taken as the examples. The influence coefficients of wind speed on bridge deck under different inlet velocities are listed in Table 2, indicating that the effect of the marginal guardrails on the wind environment for vehicles is little affected by the inlet wind speed.

Table 2.

Influence coefficients of wind speed on bridge deck for container truck.

Lane	α	$U_{0}$ = 5 m/s	$U_{0}$ = 7.5 m/s	$U_{0}$ = 10 m/s	$U_{0}$ = 12.5 m/s	$U_{0}$ = 15 m/s
1	−5°	0.80	0.80	0.80	0.80	0.80
	0°	0.82	0.81	0.81	0.81	0.81
	5°	0.74	0.74	0.75	0.75	0.75
4	−5°	0.79	0.79	0.79	0.79	0.79
	0°	0.72	0.72	0.72	0.72	0.72
	5°	0.46	0.47	0.46	0.47	0.47

In summary, reducing the porosity of guardrails can significantly improve the wind environment above the bridge deck, which is conducive to the driving safety under crosswinds, especially closing it. The improvement on wind environment or vehicles increases with the increase in angle of attack. However, too high guardrails can cause vortex shedding, leading to the increase in turbulent velocity, which is not good for driving safety.

Aerodynamic stability of the bridge

Test setup

The tests were carried out at XNJD-1 boundary layer wind tunnel. The model was suspended by eight stretching springs to form a two-degree-of-freedom vibration system in vertical and torsional directions, as shown in Figure 7. The experimental model has the scale ratio of 1/45 with a length of 2.1 m.

Figure 7.

Wind tunnel test model.

A three-dimensional (3D) finite element model was constructed using ANSYS software to obtain the dynamic characteristics of the suspension bridge. The first order symmetric vertical and first order symmetric torsional frequencies of the bridge are 0.2043 Hz and 0.4613 Hz, respectively. Based on the dynamic characteristics of the real bridge, the mass and mass moment of inertia of the experimental system were 11.34 kg and 0.737 kg·m² per meter, respectively. The vertical and torsional frequencies of the test model were 2.018 Hz and 4.439 Hz, and the damping ratios in vertical and torsional directions were 0.380% and 0.346%, respectively. In the tests, the incoming wind speed was continuously increased, and the displacement of girder was recorded by a computer acquisition device at each wind speed. The simulation of wind attack angle by rotating stiffening girder model.

Test results

The effects of guardrails with different porosities are first discussed. The angles of attack −5°, 0°, and 5° were tested. According to the test results, the vibration contains both the torsional and the vertical components, and the former is more obvious. Figure 8 shows the standard deviation (STD) of the torsional displacement versus the wind velocity corresponding to the full scale bridge. When the STD torsional displacement exceeds 0.5°(Ministry of Transport of the People’s Republic of China, 2018), the vibration is regarded as unstable, and the critical wind speed is alsolisted in Figure 8.

Figure 8.

STD torsional displacement and critical wind speed (different porosities): (a) α = −5°, (b) α = 0°, and (c) α = 5°.

The box girder without the marginal guardrails, that is, φ = 100%, shows good aerodynamic performance, because the unstable vibration is not observed at the 3° angles of attack within the tested wind velocities. With the original marginal guardrails, that is, φ = 67%, the box girder still keeps good aerodynamic performance at 0° and −5° angles of attack, but the unstable vibration occurs at 5° angle of attack though the critical wind speed is high. When φ further decreases to 20% or even 0%, the marginal guardrails become blunt, so the box girder shows bad aerodynamic performance at 0° and 5° angles of attack, especially at the large angle of attack. In the situation, although the wind environment for vehicles is improved, the aerodynamic stability of the bridge is weakened seriously. The lowest critical wind speed of the bridge is only 26.98 m/s when the angle of attack is 5° and the marginal guardrails are closed totally. For this case, the STD torsional displacement increases rapidly after the critical state, and reaches its peak at the wind speed of 49.70 m/s. After that the displacement decreases, and the vibration becomes stable again when the wind speed exceeds 65.66 m/s. This phenomenon will be further studied later.

Then, the effects of guardrails with different heights are discussed. At the 3° of attack, Figure 9 shows the STD torsional displacement of the girder and the critical wind speed. With the increase in height of the marginal guardrails, the distance from the upper solid part to the deck surface increases, and more winds could pass through this gap. At 0° and −5° angles of attack, the bad aerodynamic stability of the bridge gradually recovers. The higher the guardrail is, the higher the critical wind speed becomes. Although the aerodynamic stability could be largely improved, the critical wind speed of the bridge is still lower than that of the original scheme, that is, case 2.

Figure 9.

STD torsional displacement of the girder and critical wind speed (different heights): (a) α = −5°, (b) α = 0°, and(c) α = 5° (Note: “—” represents the flutter instability was not observed within the maximum tested wind speed.)

At −5° angle of attack, the box girder remains good aerodynamic stability no matter what the form of the marginal guardrails is. For cases 6 and 7, it should be noted that the vortex-shedding phenomenon behind the windward guardrail was observed in Section 3.3, but vortex-induced vibration of the girder did not occur in the wind tunnel tests. The probably reason is that the vortex is shed from the solid part and away from the deck surface, as shown in Figure 5(a). As a result, the fluctuation of the aerodynamic force acting on the girder is not strong enough to drive it to vibrate.

Further discussions

Unlike the linear theoretical flutter phenomenon, the vibration of the girder is not divergent when the critical wind speed is exceeded. In fact, the torsional amplitude of the girder increases rapidly near the critical state, but its trends to be stable or even decrease with the further increase in wind speed. To understand what type of the vibration is, the box girder with the solid marginal guardrails, that is, case 4, is focused. Figure 10 shows the STD heaving and the STD torsional displacements corresponding to the full scale bridge within the velocity range where the vibration occurs.

Figure 10.

STD displacements in the wind speed range where the vibration occurs: (a) α = 0° and (b) α = 5.

At null angle of attack, the heaving motion and the torsional motion have a strong coupling effect, and their displacements both increase with the increase in wind speed. Taking three typical wind speeds as examples (see Figure 10(a)), Figure 11(a) illustrates the time series of the displacements in the two directions and the corresponding spectrum characteristics. When the wind speed increases from 40.1 m/s to 77.5 m/s, the percentage changes in the STD heaving and the STD torsional displacements are 311.3% and 26.2%, respectively, indicating that the participation level of the heaving motion increases. As a result, the vibration frequency decreases. These characteristics indicate that the box girder acts like a streamlined body and the vibration is the coupled flutter.

Figure 11.

Time series of displacements and spectrum characteristics : (a) α = 0° and (b) α = 5.

At 5° angle of attack, the coupling effect between the heaving motion and the torsional motion becomes weak, as the heaving displacement is always small. The STD torsional displacement increases first with the increase in wind speed and reaches the peak when the wind speed is 46.1 m/s. After that it becomes to decrease. The STD heaving displacement shows a similar trend, but the wind speed corresponding to the peak is higher. It seems that the vibration is not the vortex-induced vibration, or else the lock-in region ranging from 27.6 m/s to 64.3 m/s would be so wide. Another important reason is that no vortex-shedding phenomenon for this case was observed in the CFD simulations. Now that it is not VIV, the vibration may be flutter. Taking three typical wind speeds as examples (see Figure10(b)), Figure 11(b) illustrates the time series of the displacements in the two directions and the corresponding spectrum characteristics. At the three wind speeds, the vibration frequency keeps the same value of 4.375 Hz which is slightly smaller than the torsional frequency of the model, indicating that the vibration is dominated by the torsional motion. As the box girder acts like a bluff body at 5° angle of attack, the above characteristics are reasonable. The aerodynamic mechanism of the non-divergent flutter at higher wind speeds will be investigated in the next section.

Aerodynamic mechanism of thenon-divergent flutter

Flutter and galloping of bridge are two main aeroelastic instability phenomena, which are closely related to self-excited forces. Aerodynamic instability means that the structure absorb energy from the outside continuously, so it is in an unstable state of divergence (Buljac et al., 2017; Chen et al., 2020). To explain the non-divergent flutter observed in the wind tunnel tests and make sure that the optimization on the guardrails is reasonable, dynamic CFD simulations are used to study the aerodynamic behavior.

Flutter derivatives

The Scanlan’s theory of flutter derivatives is widely adopted to compute critical flutter wind speeds. The self-excited forces on a bridge’s girder per unit length are defined as (Scanlan and Tomko, 1971)

\begin{matrix} L_{se} = \frac{1}{2} ρ U_{0}^{2} (2 B) \\ {{KH}_{1}^{*} \frac{\overset{\cdot}{h}}{U} + {KH}_{2}^{*} \frac{B \overset{\cdot}{α}}{U} + K^{2} H_{3}^{*} α + K^{2} H_{4}^{*} \frac{h}{B}} \end{matrix}

(3)

\begin{matrix} M_{se} = \frac{1}{2} ρ U_{0}^{2} (2 B^{2}) \\ {{KA}_{1}^{*} \frac{\overset{\cdot}{h}}{U} + {KA}_{2}^{*} \frac{B \overset{\cdot}{α}}{U} + K^{2} A_{3}^{*} α + K^{2} A_{4}^{*} \frac{h}{B}} \end{matrix}

(4)

where L_se and M_se are the self-excited lift and pitching moment; K = ωB/U₀ is the reduced frequency and ω is the circular frequency; h and α are the heaving and the torsional displacements, respectively; $H_{i}^{*}$ , $A_{i}^{*}$ (i = 1∼4) are the flutter derivatives related to the reduced velocity U₀/(fB), and f is the vibration frequency.

To realize the motion of the girder, the computation domain is re-divided into three parts, the rigid zone, the deforming zone, and the stationary zone, as shown in Figure 12(a). The rigid zone moves along with the girder to ensure the quality of the mesh around the girder. The deforming zone is discretized by triangular unstructured girds, and the stationary zone is discretized by quadrilateral structured girds. Different porosities of the guardrails are computed, and the total cell numbers for them are between 239,000 and 252,000. SDOF heaving and torsional vibrations are imposed respectively by user-defined functions (UDF), as shown in Figure 12(b). The amplitude of single peak is set as 0.025B for heaving vibration, and 3° for torsional vibration. The frequencies of heaving and torsional vibration are both 2 Hz. Turbulence model and computational method are the same as those in Section 3.2.

Figure 12.

Dynamic mesh model: (a) local computational mesh and (b) diagram for forced vibrations.

Based on the computed self-excited forces, the flutter derivatives under different reduced wind velocities can be obtained by the least square method. Figure 13 shows the coupled item, $A_{1}^{*} H_{3}^{*}$ , and the direct flutter derivatives, $H_{1}^{*}$ and $A_{2}^{*}$ , which are the most important items for the flutter stability (Matsumoto et al., 1999). The critical flutter state of the bridge is computed via a two degree-of-freedom flutter analysis method (Simiu and Scanlan, 1996), and the results are shown in Figure 14.

Figure 13.

Flutter derivatives of the girder: (a) α = −5°, (b) α = 0°, and (c) α = 5.

Figure 14.

Flutter conditions of the bridge: (a) critical flutter speed and (b) flutter frequency.

At null angle of attack, the girder without the marginal guardrails, that is, φ = 100%, is a streamlined body. It can be seen that $H_{1}^{*}$ , $A_{2}^{*}$ , and $A_{1}^{*} H_{3}^{*}$ are negative and decrease with the increase in U₀/(f·B). In this situation, the uncoupled self-excited items always contribute to positive damping, while the coupled item always generates negative damping after considering the effect of the phase difference between the torsional and the heaving motion, which drives the bridge to coupled flutter at higher wind speeds. When the porosity of the guardrails decreases to 0%, the girder shows the characteristics of a bluff body. $A_{2}^{*}$ changes its sign from negative to positive as the increase of U₀/(f·B), and the uncoupled self-excited pitching moment also generates negative damping, leading to reduction of the critical flutter wind speed. Meanwhile, the coupling effect is weakened and the flutter frequency is closer to the torsional frequency. After $A_{2}^{*}$ becomes positive, however, it is relatively stable and increases slightly with the increase in wind speed. This may be the reason why the torsional flutter ended to be stable in the wind tunnel tests. It should be noted that $H_{1}^{*}$ becomes positive when U₀/(f·B) is about 2. Although $H_{1}^{*}$ soon becomes negative again, this implies that the girder has the possibility of the heaving instability at lower velocities.

With the increase in angle of attack, the major difference of flutter derivatives lies in $A_{2}^{*}$ , which reflects the torsional flutter of the bridge. At −5° angle of attack, when the marginal guardrails are closed completely, $A_{2}^{*}$ decreases first, followed by a quick increase, and then shows a downward trend with the increase of the reduced wind speed. In the process, the uncoupled term $A_{2}^{*}$ is always less than 0, which explains the phenomenon that no obvious vibration was observed in the wind tunnel tests. At 5° angle of attack, when the marginal guardrails are not set, $A_{2}^{*}$ still shows a decreasing trend with the increase of reduced wind speed, but its absolute value is smaller than those at the other 2° angles of attack. After installing the guardrails, $A_{2}^{*}$ with three porosities changes its sign from negative to positive, but they are different with the further increase in wind speed. When the porosity is 67%, $A_{2}^{*}$ always hovers around 0 and slightly less than 0, so torsional vibration occurs at high wind speed in the tests. When the porosity is 20%, $A_{2}^{*}$ continues to increase and tends to be stable in the range of 2.87 to 5.73, which also confirms the phenomenon that the torsional amplitude increases continuously first and then tends to be stable in the wind tunnel tests. When the guardrails are completely closed, $A_{2}^{*}$ does not increase blindly with the increase in the reduced wind speed, while then changes back to negative. As a result, the torsional amplitude not only does not diverge, but also attenuates in the tests.

In conclusion, as the decrease of porosity, the critical flutter wind speed decreases and the flutter frequency tends to torsional frequency, which is consistent with the experimental results. The different post-flutter behaviors at different angles of attack can be explained by the flutter derivative $A_{2}^{*}$ .

Flow field characteristics

To further explore the aerodynamic mechanism of flutter, especially at low porosity of marginal guardrails, the change in flow field characteristics and the input energy by aerodynamic forces are investigated. Since the flutter type is dominated by torsional motion when φ ≤ 20%, the flutter stability can be well evaluated by analyzing SDOF torsional vibration. For the SDOF torsional vibration, and the input energy by aerodynamic forces acting on the cross-section of the girder over a period can be computed by equation (5).

W_{T} = \int_{t 1}^{t 2} M (t) \overset{\cdot}{α} (t) dt

(5)

where $t 1$ and $t 2$ are start and end times, and $α (t)$ is the displacement of torsional motion

Figure 15 gives the input energy by pitching moment during a torsional cycle. The variation of input energy is similar to that of $A_{2}^{*}$ . At −5° angle of attack, $W_{T}$ is always less than 0 regardless of the porosity of the guardrails, which indicates that the girder cannot maintain the current vibration and its amplitude will decrease, which is consistent with the experimental results. At 0° and 5° angles of attack, when the porosity of guardrails is less than or equal to 20%, $W_{T}$ is greater than 0 as the increase of the wind speed, which means that the torsional flutter occurs. However, it should be noted that when the guardrails are closed completely, $W_{T}$ decreases again to 0 as the further increase of the reduced wind speed at 5° angle of attack, which confirms the phenomenon of amplitude attenuation in the wind tunnel tests.

Figure 15.

Input energy at different angles of attack: (a) α = −5°, (b) α = 0°, and (c) α = 5.

The flutter behavior of the bridge is different at the 3° angles of attack. To better understand how the angle of attack affects the flutter behavior, taking case 4 as an example, the contours of the static pressure coefficient in a torsional cycle are given in Figure 16 where the static pressure is normalized by $0.5 ρ U_{0}^{2}$ . The stream lines and the input energy per quarter cycle are also shown in the figure.

Figure 16.

Contours of pressure coefficient in a torsional cycle: (a) α = 0° and (b) α = 5.

As previous studies have described (Guo et al., 2020; Tang et al., 2018), the movement of vortex formed at the windward side toward the leeward side is the main cause of flutter. When the box girder moves clockwise, the vortex exists on the windward side of the box girder, and when the box girder moves counterclockwise, the vortex just moves to the leeward side of the box girder, and the vortex will promote the rotation of the box girder. At this time, the synchronization condition of the vortex movement and the girder motion is considered to be achieved. At low wind speeds, the vortex intensity is weak, and the inflow cannot transfer the vortex to the leeward side. Therefore, the movement of the vortex cannot be synchronized with the motion of the girder, leading to negative input energy provided by pitching moment, and the girder is stable. With the increase in wind speed, the vortex intensity strengthens, and the inflow with high velocity can make the vortex move to the leeward side. When the vortex moves faster, the synchronization condition is easier to achieve, so the pitching moment produces positive work and the aerodynamic instability occurs.

When the solid guardrails are installed on both sidesof the girder, the windward one strengthens the vortex, and the leeward one prevents the vortex from dissipating. At low wind speeds, the vortex at leeward side cannot dissipate, which obstructs the formation of vortex at windward side. As a result, the synchronization condition is not achieved, leading to negative input energy provided by the pitching moment, so the girder is stable. When the wind speed exceeds the critical wind speed, the leeward guardrail can no longer block the vortex due to its large size, and the synchronization condition is satisfied. Hence, the pitching moment generates positive work and increases rapidly, as shown in Figure 16, and the girder becomes unstable. However, the post-instability behaviors of the girder are different at different angles of attack. At null angle of attack, the existence of solid guardrails on both sides makes the separation point and the re-attachment point of approaching flow stable, during which the formation and movement of the vortex no longer change with the wind speed (see Figure 16a). With the increase in wind speed, although the synchronization condition can still be achieved and the input energy is positive, but it increases slowly, which explains why the torsional displacement in the tests does not diverge. At 5° angle of attack, the large angle of attack promotes the separation of approaching flow at high wind speeds, and a big vortex is formed over the over-blunt bridge deck (see Figure 16b). With the further increase of wind speed, the synchronization condition is gradually destroyed, and the input energy decreases or even becomes negative. This can well explain the phenomenon that the torsional amplitude increases first and then decreases in the wind tunnel tests at 5° angle of attack.

Conclusions

In this paper, the effects of guardrails on wind environment for vehicles and aerodynamic stability for bridges were studied by CFD simulations and wind tunnel tests. Some conclusions can be made.

Reducing the porosity of marginal guardrails can significantly improve the wind environment above the bridge deck, which is conducive to the driving safety, especially closing it. The improvement on wind environment increases with the increase in angle of attack. When the height of guardrails increases, the windproof effect enhances at −5° angle of attack, but weakens at 5° angle of attack. However, too high guardrails can cause vortex shedding, leading to the increase in turbulent velocity.

With the decrease in the porosity of guardrails, however, the aerodynamic stability of the bridge is weakened at the same time, especially at null and positive angles of attack. Unlike the linear theoretical flutter phenomenon, the torsional amplitude first increases rapidly but then trends to be stable or even decreases with the increase in wind speed. The different post-flutter behaviors at different angles of attack are mainly related to the synchronization condition between the movement of vortex and the motion of the girder, which can be explained the flutter derivative $A_{2}^{*}$ .

The movement of vortex formed at the windward side toward the leeward side is the main cause of flutter. When the guardrails are closed completely, the windward guardrail strengthens the vortex, and the leeward guardrail prevents the vortex from dissipating. When the wind speed exceeds the critical wind speed, the leeward guardrail can no longer block the vortex due to its large size, and the synchronization condition is achieved, so the girder becomes unstable. At null angle of attack, the solid guardrails make the separation point and the re-attachment point of approaching flow stable, during which the formation and movement of the vortex no longer change with the wind speed. The input energy by pitching moment increases slowly with the further increase in wind speed, so the vibration of the girder does not diverge. At 5° angle of attack, the large angle of attack promotes the separation of approaching flow at high wind speeds, and a big vortex is formed over the over-blunt bridge deck. The synchronization condition is gradually destroyed as the further increase of wind speed, and the girder becomes stable again.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: the authors are grateful for the financial supports from the National Natural Science Foundation of China (Grants 51708463, 51525804) and the Fundamental Research Funds for the Central Universities (2682019CX04).

ORCID iD

Haojun Tang

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