**2. Aerodynamic phenomena**

Wind action on bridges can adversely influence their static and dynamic stability. Alongwind deflections are examples of static reactions. These are minor problems in the sense that the existing codes of practice cover with sufficient detail current geometries of decks allowing conservative calculations. The same is not true with regards to the dynamic stability of large and very large-span bridges, which is related with vibrations or oscillations of the deck as a result of fluctuations in the pressure distribution over its surface. For very large-span bridges, a customised approach is mandatory, both analytical and experimental. The collapse of the Tacoma Narrows suspension bridge in 1940 alerted the scientific community to the importance of aerodynamic studies of wind actions on structures and to the possible occurrence of aeroelastic phenomena.

#### **2.1. Classification**

Several types of aerodynamic phenomena can develop as a consequence of interaction of actual wind with bridge decks. The turbulence of natural wind will always induce a fluctuating force on structures. This is known as buffeting and the associated displacements are usually of small amplitude, in a way that does not change the topology of the flow around the deck. In other situations, called aeroelastic phenomena, the oscillation amplitude is large enough to interfere with the airflow. In this case, the flow pattern around a deck is affected by the structure's motion itself, giving rise to a very complex interaction between the deck's motion and aerodynamic forces.

The flexibility of long-span bridges renders them highly susceptible to aeroelastic phenomena. In this context, vortex-induced vibration (VIV), torsional flutter, coupled flutter, and galloping may arise. While in VIV, the amplitude is self-limited, in the other three phenomena, the amplitude of the deck's motion tends to increase continuously and, for this reason, they are categorised as aeroelastic instabilities. If damping is insufficient, the large amplitude of the motion may cause collapse of the structure. **Figure 3** systematises the classification of flow-induced vibrations on structures.

VIV may occur at low wind speeds, and when it does occur, it is for narrow ranges of wind speed specific for a given structure (**Figure 4**). Flutter-type instabilities arise at much higher wind speeds, above a critical value. In buffeting, turbulence in the incoming flow (a large band excitation) causes a response proportional to the dynamic pressure of the wind. The next sections address in more detail each type of flow-induced vibration.

#### *2.1.1. Buffeting*

higher depth-to-width ratio than one for a bridge with a double plane of cables, because the girder rigidity has to compensate the lack of torsional stiffness in the first case. This difference

In box-girder decks, the quest for high aerodynamic performance has resulted in the shift

Wind action on bridges can adversely influence their static and dynamic stability. Alongwind deflections are examples of static reactions. These are minor problems in the sense that the existing codes of practice cover with sufficient detail current geometries of decks allowing conservative calculations. The same is not true with regards to the dynamic stability of large and very large-span bridges, which is related with vibrations or oscillations of the deck as a result of fluctuations in the pressure distribution over its surface. For very large-span bridges, a customised approach is mandatory, both analytical and experimental. The collapse of the Tacoma Narrows suspension bridge in 1940 alerted the scientific community to the importance of aerodynamic studies of wind actions on structures and to the possible occurrence of

Several types of aerodynamic phenomena can develop as a consequence of interaction of actual wind with bridge decks. The turbulence of natural wind will always induce a fluctuating force

from thick rectangular decks to trapezoidal and streamlined decks (**Figure 2**).

in aspect ratio can lead to very distinct aerodynamic responses.

**Figure 2.** From general rectangular decks to trapezoidal and streamlined decks.

**2. Aerodynamic phenomena**

aeroelastic phenomena.

**2.1. Classification**

86 Bridge Engineering

As previously mentioned, buffeting refers to structural oscillations due to the turbulent fluctuations of oncoming wind. The structure can be considered absolutely rigid since the displacements are of very small amplitude. More specifically, the amplitude is much smaller than the thickness of the boundary layer1 , and thus the vibration of the structure does not disrupt the boundary layer, thus not changing the topology of the flow around the deck.

Turbulence has a stochastic nature; pressure and velocity are random. The turbulence energy of natural wind concentrates in the lower frequency of the velocity spectrum. That is to say that if the characteristic lateral length of the structure is comparatively small (up to about 10<sup>2</sup> m),


**Figure 3.** A classification of flow-induced vibrations, including aeroelastic phenomena.

<sup>1</sup> Boundary layer is the thin region near a solid surface where viscous effects are dominant and fluid velocity evolves from zero at the surface to nearly its free-stream value away from it. The depth of this layer is known as its thickness.

*2.1.2. Vortex-induced vibration*

amplitude.

deck, in addition to occurrences on the other side.

to a natural frequency of the bridge.

a certain wind speed.

*2.1.3. Torsional flutter*

*2.1.4. Galloping*

for galloping to happen whenever *CL*

Vortex-induced vibration (VIV) can appear as either vertical or torsional oscillations. It is caused when the flow separates at spots on the deck's contour so that vortices are shed periodically from one and the other sides of the bridge deck. Flow may separate at protuberances on the upper or lower surfaces of the deck, like traffic barriers or guard-rails. A detailed discussion of the flow physics of vortex-shedding excitation can be found in two recent reviews [7, 9]. When studying the aerodynamics of the Vila-Real Bridge, the authors concluded that VIV would not occur when vortices were shed from just one side of the deck [10]. The addition of road equipment to the top of the deck raised the possibility of vortex-shedding above the

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The nature of the resulting aerodynamic forces acting on the deck makes VIV self-limited in

Vortex-induced vibrations can cause discomfort to the users of a bridge or even its failure by fatigue in the long term. It should be pointed out that the amplitude can become of concern in situations for which the system has low damping and the vortex shedding frequency is close

Lock-in is an interesting phenomenon associated to VIV, in which the high amplitude of the oscillating motion is maintained over a somewhat wider range of wind speeds rather than for

Decks known to be particularly susceptible to VIV are plate-girder and deep box-girder decks.

Torsional flutter, sometimes called stall flutter, happens when the damping associated with torsional motion is very low. Here, damping is to be understood as the ensemble of structural and aerodynamic damping of the structure acted by wind. A critical velocity can be defined as the value for which the total damping becomes zero. In the onset of torsional flutter, lift causes the body to vary its pitch in a way that from its interaction with the wind a sustained torsional motion results. Once established, torsional flutter manifests as a rapid rotational oscillation. It should be said that static deformations induced by wind loadings at velocities near flutter

Torsional flutter can occur, in occasions, when propitious conditions are met, in flexible

Galloping, also known as bending flutter, is a single degree-of-freedom aeroelastic oscillation in the direction transverse to the wind, which can reach large amplitude. There is potential

total damping of a bluff body undergoing wind-excited oscillations consists of a structural

(*α*) exhibits an accentuated negative slope. Indeed, the

occurrence may possibly change the effective wind attack angle.

bridges with channel type or H-shaped plate girder cross-sections.

**Figure 4.** A generic graph of the relative amplitudes of the various flow-induced vibrations. Adapted from [4].

the aerodynamic force resulting from the turbulent flow acts as a spatially uniform fluctuating force; otherwise, the fluctuating wind can no longer be considered spatially uniform since the length scale of turbulence in the span-wise direction is comparable to, or smaller than, the span of the bridge. In this latter case, the wind peak speed effect is reduced. Indeed, the deck is not simultaneously excited over the whole of its length with the peak value of wind fluctuation; there are spatial and temporal variations.

Therefore, to optimise the dimensioning of the structure, instead of the wind peak speed, a reduced value should be used in the design. This entails the longitudinal cross-correlation of turbulent fluctuations. The problem is rather complex but the accumulated data concerning wind statistical properties and the development of analytical models associated with numerical techniques make it possible to tackle the influence of the three-dimensionality of natural wind fluctuations. From this, it is possible to derive reduction factors having in mind the relation between bridge span and turbulence length scale. If the spatial dimension of the correlation is considerably smaller than the bridge span, then the reduction factor will have a small value. Several examples of this procedure applied to bridges are available in the literature, and an interesting one concerns the Lion's Gate Bridge in Vancouver, Canada [5]. This analytical approach was developed to address the effect of gusts on aircraft wings and was transposed to civil engineering by Davenport [6].

Buffeting is proportional to the flow's dynamic pressure and thus the amplitude of buffetingrelated vibration exhibits a quadratic evolution with the flow velocity. In severe cases, it can destabilise vehicles or pedestrians. On the other hand, minor vibrations, which may go unnoticed, may cause in the long-term damage by fatigue to structural components.

Even bridges with a carefully designed cross-section of the deck can still be susceptible to buffeting. Buffeting may also stem from irregular flow separations along the contour of the bridge deck, which generate a new vortex field to be added to that of the oncoming turbulence [7]. Nevertheless, if the fluctuations associated with those separations lose their stochastic nature, and periodical vortical structures are formed in the wake, then this may turn out to be what is called vortex-shedding, to be dealt with in the next section. Buffeting refers also to a different effect: increased turbulence in the wake of a structure interfering with a nearby second one placed downwind (as in the l'Iroise Bridges), and the term wake-buffeting may be used in this case (Section 6.3.3 of [8]).

#### *2.1.2. Vortex-induced vibration*

Vortex-induced vibration (VIV) can appear as either vertical or torsional oscillations. It is caused when the flow separates at spots on the deck's contour so that vortices are shed periodically from one and the other sides of the bridge deck. Flow may separate at protuberances on the upper or lower surfaces of the deck, like traffic barriers or guard-rails. A detailed discussion of the flow physics of vortex-shedding excitation can be found in two recent reviews [7, 9].

When studying the aerodynamics of the Vila-Real Bridge, the authors concluded that VIV would not occur when vortices were shed from just one side of the deck [10]. The addition of road equipment to the top of the deck raised the possibility of vortex-shedding above the deck, in addition to occurrences on the other side.

The nature of the resulting aerodynamic forces acting on the deck makes VIV self-limited in amplitude.

Vortex-induced vibrations can cause discomfort to the users of a bridge or even its failure by fatigue in the long term. It should be pointed out that the amplitude can become of concern in situations for which the system has low damping and the vortex shedding frequency is close to a natural frequency of the bridge.

Lock-in is an interesting phenomenon associated to VIV, in which the high amplitude of the oscillating motion is maintained over a somewhat wider range of wind speeds rather than for a certain wind speed.

Decks known to be particularly susceptible to VIV are plate-girder and deep box-girder decks.

#### *2.1.3. Torsional flutter*

the aerodynamic force resulting from the turbulent flow acts as a spatially uniform fluctuating force; otherwise, the fluctuating wind can no longer be considered spatially uniform since the length scale of turbulence in the span-wise direction is comparable to, or smaller than, the span of the bridge. In this latter case, the wind peak speed effect is reduced. Indeed, the deck is not simultaneously excited over the whole of its length with the peak value of wind fluctuation;

**Figure 4.** A generic graph of the relative amplitudes of the various flow-induced vibrations. Adapted from [4].

Therefore, to optimise the dimensioning of the structure, instead of the wind peak speed, a reduced value should be used in the design. This entails the longitudinal cross-correlation of turbulent fluctuations. The problem is rather complex but the accumulated data concerning wind statistical properties and the development of analytical models associated with numerical techniques make it possible to tackle the influence of the three-dimensionality of natural wind fluctuations. From this, it is possible to derive reduction factors having in mind the relation between bridge span and turbulence length scale. If the spatial dimension of the correlation is considerably smaller than the bridge span, then the reduction factor will have a small value. Several examples of this procedure applied to bridges are available in the literature, and an interesting one concerns the Lion's Gate Bridge in Vancouver, Canada [5]. This analytical approach was developed to address the effect of

Buffeting is proportional to the flow's dynamic pressure and thus the amplitude of buffetingrelated vibration exhibits a quadratic evolution with the flow velocity. In severe cases, it can destabilise vehicles or pedestrians. On the other hand, minor vibrations, which may go unno-

Even bridges with a carefully designed cross-section of the deck can still be susceptible to buffeting. Buffeting may also stem from irregular flow separations along the contour of the bridge deck, which generate a new vortex field to be added to that of the oncoming turbulence [7]. Nevertheless, if the fluctuations associated with those separations lose their stochastic nature, and periodical vortical structures are formed in the wake, then this may turn out to be what is called vortex-shedding, to be dealt with in the next section. Buffeting refers also to a different effect: increased turbulence in the wake of a structure interfering with a nearby second one placed downwind (as in the l'Iroise Bridges), and the term wake-buffeting may be

gusts on aircraft wings and was transposed to civil engineering by Davenport [6].

ticed, may cause in the long-term damage by fatigue to structural components.

there are spatial and temporal variations.

88 Bridge Engineering

used in this case (Section 6.3.3 of [8]).

Torsional flutter, sometimes called stall flutter, happens when the damping associated with torsional motion is very low. Here, damping is to be understood as the ensemble of structural and aerodynamic damping of the structure acted by wind. A critical velocity can be defined as the value for which the total damping becomes zero. In the onset of torsional flutter, lift causes the body to vary its pitch in a way that from its interaction with the wind a sustained torsional motion results. Once established, torsional flutter manifests as a rapid rotational oscillation.

It should be said that static deformations induced by wind loadings at velocities near flutter occurrence may possibly change the effective wind attack angle.

Torsional flutter can occur, in occasions, when propitious conditions are met, in flexible bridges with channel type or H-shaped plate girder cross-sections.

#### *2.1.4. Galloping*

Galloping, also known as bending flutter, is a single degree-of-freedom aeroelastic oscillation in the direction transverse to the wind, which can reach large amplitude. There is potential for galloping to happen whenever *CL* (*α*) exhibits an accentuated negative slope. Indeed, the total damping of a bluff body undergoing wind-excited oscillations consists of a structural damping (considered positive) and an aerodynamic damping, related to *d CL* /*d* + *CD*. The system can develop divergent oscillations if the total damping becomes negative. The condition for negative aerodynamic damping to occur is known as the Den Hartog criterion [11]. Galloping occurs above this critical wind speed.

Galloping can occur if Den Hartog's criterion is not observed. Vaz et al. [16] numerically studied two decks that were found stable. Nevertheless, galloping can be an important phenomenon in bridge cables, especially when the section's geometry may be changed by ice deposition.

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From what has been said, it is understood that, as far as bridges are concerned, VIV is the aeroelastic phenomena that deserves the most attention. It is a major concern since the 1940s, when large oscillations began to be observed in the original Tacoma Narrows Bridge, at rela-

The occurrence of aeroelastic phenomena on long-span bridges has been widely studied. Good starting points for further reading are the reviews of Miyata [17] and of Saito and Sakata [18]. In aerodynamics, minor details (from the point of view of design) in the body contour can lead to significant modifications of the flow pattern and aerodynamic response. While this means that the aerodynamic can be improved by introducing, still at the design stage, inexpensive and simple-to-implement modifications, it also means that it is important to consider in the aerodynamic study the various stages of the deck construction since the addition of equipment, such as median dividers, guard rails, and border beams, can noticeably change

The overall shape of the bridge deck's cross-section strongly influences the occurrence of aeroelastic phenomena [23–25]. A numerical study of four generic cross-section shapes developed from the well-known plate girder section of the First Tacoma Narrows Bridge, by adding horizontal plates and fairings, showed that the closed section with fairings displayed the best aerodynamic performance (in terms of drag loading and oscillating lift) and the parent

In streamlined box sections, VIV can likely be avoided or kept at insignificant levels if the angle between the horizontal extension of the deck's bottom plate and the lower inclined panels is kept smaller than 15°, in order to avoid flow separation [25]. Streamlining of the deck can possibly result in improved aerodynamic performance, with an increase in the critical wind speed for torsional flutter and decrease in the vortex-induced response [19, 26]. Even though streamlined decks have lower drag and postpone the onset of possible divergent aeroelastic instabilities to higher wind speeds, they may nevertheless exhibit oscillations at low

It should be said that trapezoidal decks are far from being tractable as thin airfoils and analytical methods, such as Theodersen's thin airfoil theory for the study of flutter, are not successful if applied without appropriate modifications. Bridge decks have to be considered as bluff bodies. The aerodynamic stability of super-long-span bridges can be effectively improved by introducing a slot at the centre of the girder [24]. This is the design being adopted for the proposed

tively low wind speeds.

**2.2. Influence of bridge features**

the deck's aerodynamic response [17, 19–22].

H-shaped cross-section, the worst [23].

wind speed due to vortex shedding excitation.

Messina Strait (**Figure 2**) and Gibraltar Strait crossings [27, 28].

*2.2.1. Deck cross-section shape*

Non-streamlined box-girders with relatively small width-to-depth ratio are particularly susceptible to galloping, since the flexural stiffness is much smaller than the torsional.

Galloping can occur at low, either steady or unsteady, wind speeds, for structures with low structural damping associated with bending, which is the case of suspension bridges, although the occurrence of galloping is more typical of overhead power lines subjected to ice deposition [11].

#### *2.1.5. Coupled flutter*

Coupled flutter, also named classic flutter, encompasses motion in both bending and torsion. It occurs in cases for which the structure has a pair of neighbouring frequencies in bending and in torsion.

Bridge decks that will exhibit coupled flutter if the wind speed exceeds a critical value are those with a streamlined trapezoidal cross-section, with large width-to-depth ratio, so that there are no large vortices shedding from the leading edge. Examples are the Great Belt and the Izmit Suspension Bridges, as pointed out by [7].

Classic flutter was first perceived in aircraft wings and Theodersen [12] developed a thin airfoil theory aimed at describing analytically the flutter phenomenon. Since the aerodynamics of trapezoidal decks is far from being tractable as that of a thin airfoil, this theory cannot be directly applied to bridge decks without corrections [13]. Scanlan and Sabzevari [14] proposed a formulation in which the aeroelastic forces (lift and torque) are expressed as functions involving coefficients known as flutter derivatives. These derivatives (which cannot be obtained analytically) have to be obtained through wind tunnel testing. The resulting equations can be solved for the critical wind speed for the onset of flutter.

In certain cases, designers may resort to simpler ways of estimating this critical speed. In the beginning of the 1960s, A. Selberg proposed a formula for thin plates. Under certain conditions, it can be adapted to bridge decks through the use of a factor that depends on the geometry of the cross-section. More recently, Bartoli and Mannini [15] proposed simple expressions for critically reduced wind speed and coupling frequency based on a reduced number of flutter derivatives.

#### *2.1.6. Comments*

Current design practice avoids coupled flutter, alias classic flutter, by having in mind from the beginning of the design sufficient separation between the lowest torsional natural frequency of the bridge and the fundamental bending frequency. For example, Section E.4.2 of the Eurocode 1 [8] establishes a minimum ratio of 2:1.

In the same way, torsional flutter is not a significant phenomenon in modern bridges due to the necessary stiffness imposed by practical reasons.

Galloping can occur if Den Hartog's criterion is not observed. Vaz et al. [16] numerically studied two decks that were found stable. Nevertheless, galloping can be an important phenomenon in bridge cables, especially when the section's geometry may be changed by ice deposition.

From what has been said, it is understood that, as far as bridges are concerned, VIV is the aeroelastic phenomena that deserves the most attention. It is a major concern since the 1940s, when large oscillations began to be observed in the original Tacoma Narrows Bridge, at relatively low wind speeds.
