WIND EFFECT ON STRUCTURES
In the design of long span suspension bridges, notably suspended bridges, the wind action is of primary concern. With the failure of the Tacoma Narrows suspension bridge in 1940 in Washington, a deep general realization of the potential aeroelastic nature of the wind phenomena was firmly established the world over. Therefore to understand the response of long span suspension bridges under wind excitation, the basic wind phenomena needs to be clearly understood. Hence this chapter focuses on and reviews a number of topics connected with the effect of the wind on long-span suspension bridges.
The aerodynamic effect of wind on bridges are primarily vortex shedding, galloping, torsional-divergence, flutter and buffeting. They are discussed below.
The criteria for the design of long spanned suspension bridges are concerned with the static and dynamic responses of the bridge under wind loading. A basic knowledge of the wind forces that are required to understand the issues involved in the design is explained in the following section.
The design of long span suspension bridges is often governed by aeroelastic instability. Aerodynamic design involves calculation of the critical velocity for the onset of flutter. It is to be ensured that the wind velocity does not exceed the predicted critical velocity to avoid failure due to flutter.
Arrol and Chatterjee (1981) report that frequencies other than the fundamental one should be considered in design. They mention that the designers should remember that the position of maximum stress would not always be at mid-span, or a support, and the stress value will depend upon the mode shape. In a simply supported span the second mode maximum stress is at the quarter points and will have a value four times that of the fundamental mode maximum stress, occurring at mid span.
Figure 2.1 Relative bending moment diagrams due to 1st and 2nd modes of vibration.
(Picture from Arrol and Chatterjee, 1981)
There are static and dynamic concerns to be considered for a safer design of bridges as discussed by Simiu and Scanlan (1986) and Larsen (1992). They are described below.
Here the considerations are overturning, excessive lateral deflection, divergence, and lateral buckling (Selvam et. al., 1998). Usually the static phenomena are not critical for the design of bridges. The issues related to static behavior can be checked by the aerodynamic force components like drag force, lift force and pitching moment. The static issues are taken care of by the plot of the coefficients of drag, lift and moment against the angle of incidence of wind. This is explained in chapter four.
From Newton’s second law, the motion of mass is described by the differential equation
Where is the time dependent load acting on the mass, is the stiffness coefficient and is the coefficient of damping. This equation can be rewritten in the form
Where and . Here is the natural circular frequency and 2 is the critical damping coefficient. Three cases arise based on being less than, equal to, or greater than unity resulting in under-damped, critically damped and over-damped response respectively.
Dynamic behavior includes the responses due to vortex shedding excitation, self-excited oscillations and buffeting by wind turbulence (Selvam 1998). Sachs (1978) states that suspension bridges could oscillate in two natural modes, vertical and torsional. In the vertical mode, all joints at any cross-section move the same distance in the vertical plane, while in the torsional mode every cross-section rotates about a longitudinal axis parallel to the roadway.
Unlike the static behavior, the dynamic behavior is critical and important to be considered during design.
2.3 Aerodynamic Instability
Aeroelasticity is the discipline concerned with the study of phenomena wherein the aerodynamic forces and structural motions interact significantly. When a structure is subjected to wind flow, it may vibrate or suddenly deflect in the airflow. This structural motion results in a change in the flow pattern around the structure. If the modification of wind pattern around the structure by aerodynamic forces is such that it increases rather than decreasing the vibration, thereby giving rise to succeeding deflections of oscillatory and/or divergent character, aeroelastic instability is said to occur (Simiu and Scanlan, 1986). The aeroelastic phenomena that are considered in wind engineering are vortex shedding, torsional divergence, galloping, flutter and buffeting.
2.4 Vortex Shedding
Simiu and Scanlan (1986) states that when a body is subjected to wind flow, the separation of flow occurs around the body. This produces force on the body, a pressure force on the windward side and a suction force on the leeward side. The pressure and suction forces result in the formation of vortices in the wake region causing structural deflections on the body. The shedding of vorticity balances the change of fluid momentum along the entire body surface (Larsen and Walther, 1997). The shed vortices are convected downwind by local mean wind speed and viscous diffusion but will also interact to form large-scale coherent structures. The frequency in which the vortices are shed dictates the structural response. The structural member acts as if rigidly fixed, when the frequency of vortex shedding (also called wake frequency) is not close to the natural frequency of the member. On the other hand, when the vortex-induced and the natural-frequencies coincide, the resulting condition is called lock-in. During lock-in condition, the structural member oscillates with increased amplitude but rarely exceeding half of the across wind dimension of the body (Simiu and Scanlan, 1986). The lock-in condition is illustrated in Figure 2.2.
In the Figure 2.2, we see that the wake frequency remains locked to that of natural frequency for a range of wind velocities. As the velocity further increases, the wake frequency will again break away from the natural frequency. The extent of the shedding depends on the Reynolds number, which is defined as
Figure 2.2 Qualitative trend of vortex shedding frequency with wind velocity during lock-in (Simiu and Scanlan, 1986)
Simiu and Scanlan (1986) explain and give an insight into the understanding of the nature and extent of the vortex shedding phenomenon for different ranges of Reynolds number for two different cross-sections, a plate and a cylinder as shown in Fig 2.3 and 2.4. They also report that, the vortex-shedding phenomena is describable in terms of a non-dimensional number S, which is defined as
mean flow velocity
The number S takes on different characteristic constant values depending upon the cross-sectional shape of the prism being enveloped by the flow.
Figure2.3 (a) Re =0.3
Figure 2.3 (b) Re = 10
Figure 2.3 (a)-(d) Flow past a sharp edged plate showing the vortex shedding
(Pictures taken from Simiu and Scanlan, 1986)
Figure 2.4 (a) Re@ 1
Figure 2.4 (b) Re@
Figure 2.4 (c) 30 £ Re£ 5000
Figure 2.4 (d) 5000 £ Re£ 20000
Fig 2.4 (e) Re ³200000
Figure 2.4 (a)-(e) Flow past a Circular cylinder
(Pictures taken from Simiu and Scanlan, 1986)
From the figures, as illustrated by Simiu and Scanlan (1986), it is seen that for a very low Reynolds number, the flow remains the same, just circumventing the obstruction on its way. For higher Reynolds numbers, the flow starts to separate around the edges of the obstruction and vortices are generated in the immediate wake of the obstruction. Thereafter further increase in the Reynolds number causes the creation of cyclically alternating vortices and they are carried over with the flow downstream. From there on, the inertial effects become dominant over the viscous effects and turbulence sets in, resulting in shear of the flow. So this reasonably illustrates the vorticity phenomenon starting from a smooth and low speed flow to a turbulent and high-speed flow.
Simiu and Scanlan (1986) state that galloping is an instability typical of slender structures. This is a relatively low-frequency oscillatory phenomenon of elongated, bluff bodies acted upon by a wind stream. The natural structural frequency at which the bluff object responds is much lower than the frequency of vortex shedding. It is in this sense that galloping may be considered a low-frequency phenomenon. There are two types of galloping: Wake and Across-wind.
Wake galloping: It is considered of two cylinders one windward, producing a wake, and one leeward, within that wake separated at a few diameters distance away from each other. In wake galloping the downstream cylinder is subjected to galloping oscillations induced by the turbulent wake of the upstream cylinder. Due to this, the upstream cylinder tends to rotate clockwise and the downstream cylinder, anti-clockwise thus inducing torsional oscillations.
Across wind galloping: Across wind galloping in a bridge, is an instability that is initiated by a turbulent wind blowing transversely across the deck. Across-wind galloping causes a crosswise vibration in the bridge deck (Liu, 1991). As the section vibrates crosswise in a steady wind velocity U, the relative velocity changes, thereby changing the angle of attack (a ). Due to the change in a , an increase or decrease on the lift force of the cylinder occurs. If an increase of a causes an increase in the lift force in the opposite direction of motion, the situation is stable. But on the other hand if the vice versa occurs, i. e., an increase of a causes a decrease in lift force, then the situation is unstable and galloping occurs. Fig 2.6 gives an illustration of this process.
A classical example of this phenomenon is observed in ice covered power transmission lines. Galloping is reduced in these lines by decreasing the distance between spacing of the supports and increasing the tension of the lines.
Figure 2.5 Wake galloping
Picture from Simiu and Scanlan (1986)
Figure 2.6 Across wind galloping: Wind and motion components, with resultant lift and drag, on a bluff cross section. (Picture from Simiu and Scanlan, 1986)
In the figure,
2.6 Torsional divergence
Torsional divergence is an instance of a static response of a structure. Torsional divergence was at first associated with aircraft wings due to their susceptibility to twisting off at excessive air speeds (Simiu and Scanlan, 1986). Liu, 1991, reports that when the wind flow occurs, drag, lift, and moment are produced on the structure. This moment induces a twist on the structure and causes the angle of incidence to increase. The increase in results in higher torsional moment as the wind velocity increases. If the structure does not have sufficient torsional stiffness to resist this increasing moment, the structure becomes unstable and will be twisted to failure. Simiu and Scanlan, 1986, report that the phenomenon depends upon structural flexibility and the manner in which the aerodynamic moments develop with twist; it does not depend upon ultimate strength. They say that in most cases the critical divergence velocities are extremely high, well beyond the range of velocities normally considered in design.
Figure 2.7 a. Torsional divergence of an airfoil
b. Torsional divergence of Bridge deck (Picture from Liu, 1991)
The aerodynamic moment per unit span is given by
Where is density, is the mean wind velocity, is the deck width, is the angle of twist and is the aerodynamic moment coefficient about the twisting axis.
At zero angle of attack the value of this moment is
For a small change in away from , is approximated as given by
Now equating the aerodynamic moment to the structural resisting moment gives
Setting , in the above equation, we get
Divergence occurs when approaches infinity
Thus the critical divergence velocity is given as
The phenomenon of flutter is a very serious concern in the design of bridges. The failure of the Tacoma’s narrows bridge was due to the flutter. In the later part of this chapter, a review of the Tacoma’s Narrows bridge failure is reported to give a better insight into the flutter-induced instability that resulted in failure. The term flutter has been variously used to describe different types of wind-induced behavior. Flutter can be defined as a condition of negative aerodynamic damping wherein the deflection in the structure grows to enormous levels till failure once started. It is also known as classical flutter. The other types of flutter reported by Simiu and Scanlan (1986) are stall flutter and panel flutter.
Stall flutter is a single-degree-of-freedom oscillation of airfoils in torsion due to the nonlinear characteristics of the lift (Simiu and Scanlan, 1986). The stall flutter phenomenon can also occur with structures having broad surfaces depending on the angle of approaching wind. The torsional oscillation of a traffic stop sign about its post is an example of this phenomenon.
Panel flutter is a sustained oscillation of panels typically the sides of large rockets, caused by the high-speed passage of air along the panel. The most prominent cases have been in supersonic flow regimes and so have not appeared in the wind engineering context. Flag flutter is closely related to panel flutter.
The motion that is caused by the wind flow will either be damped out or will grow indefinitely until failure. The theoretical dividing line between these two states is the critical flutter condition and the wind speed at this condition is called critical wind speed.
2.8 Methods adopted to study critical velocity for flutter
The methods available for studying the aeroleastic instability are the free oscillation method and the forced oscillation method.
This method was used in this work for the study of flutter stability analysis of the structure during motion. In this method the structure is elastically suspended and is given an initial perturbation in terms of heave or pitch and the structure is left to oscillate freely. The lift, drag and moment generated due to the applied displacement is then measured and thus a time history data is generated. The governing equations of motion for translation and rotation are
Here and represents mass, moment of Inertia, heave, pitch, lift and moment respectively. and represents damping and stiffness coefficients with the subscripts and meaning heave and rotation respectively.
The equations 2.1 and 2.2 can be rewritten as
The derivation of 2.3 and 2.4 will be explained in section 4.5 of chapter 4. Thus with the knowledge of the lift and moment forces from CFD calculations, for each instant of time the equations 2.3 and 2.4 are solved incorporating the fluid structure interaction to get the heave and pitch displacements. The pitch angle is then plotted against time. When the pitch angle dies down gradually with the passage of time, it means that the critical flutter velocity is not reached. When the pitch angle keeps growing it means that the critical flutter velocity is reached. Based on these plots the critical flutter velocity is calculated. This process is discussed in detail in chapter 4.
2.8.2 Forced Oscillation Procedure
In this method, the structure is forced in a torsional or heave sinusoidal motion relative to the flow with a prescribed frequency and amplitude (Hansen et. al., 1999). The lift and moment generated due to this applied force is measured and used for the calculation of the aerodynamic derivativesThe calculated aerodynamic derivatives are then used for the computation of the critical velocity for flutter. This process is described below.
The lift and moment loads exerted on an oscillating bridge section with 2 degrees of freedom namely the vertical or heave motion () and rotational or pitch motion () are given by the following equations (Simiu and Scanlan, 1986).
, is the reduced non-dimensional frequency
For the pure heave motion, the equations 2.5 and 2.6 become
For samples, (), =1,2,… , the equations 2.7 and 2.8 constitute two sets of over determined equations, which can be solved in the least squares sense as reported by Walther (1994) as follows.
The least square formulation for solving the equations 2.7 and 2.8 is
where and and are (x 2) matrices
and . The right hand side vectors, and are the lift and moment vectors
In a similar fashion, for pure pitch motion, equations 2.5 and 2.6 becomes 2.9
and the matrices and are
and again as before
Thus the equations 2.7 through 2.10 are solved using the least square principle to obtain the eight aerodynamic derivatives. The aerodynamic coefficients as defined by Larsen and Walther (1996) are as follows.
, and , are obtained from time dependent lift and moment coefficients by a pure torsional oscillatory motion described by .
Oscillatory one degree of freedom excitation in a pure vertical motion described by , yields the flutter derivatives , and ,
Here, is the phase shift of the aerodynamic forces with respect to the imposed motion of pure heave or oscillation.
Once the aerodynamic derivatives are computed, they are plugged into the equations of motion for heave and rotation (2.1 and 2.2) as shown below.
Rewriting the above heave equation (2.11), we get
Substituting angular frequency in heave, and damping ratio in heave,
In a similar fashion, the equation for rotational motion follows as
These two differential equations (2.13 & 2.14) are now based on the observation, that and are harmonic in time with a common frequency at the critical wind speed for the onset of flutter. The representation of heave and pitch in the complex notation is used in solving the flutter deterministic equations 2.13 and 2.14. The critical velocity for flutter is then calculated by plotting curves corresponding to the roots of the real and imaginary parts of the flutter determinant equation against the non-dimensional wind velocity as illustrated by Walther, 1994 and Larsen, 1995. The intersection point between the real and imaginary root curves defines the critical wind speed for flutter as
where is the critical flutter velocity and is the ordinate of the point of intersection in the plot.
When the critical wind speed for flutter is exceeded, the structure will become unstable and experience excessive deflections. Hence it is an important factor to be considered in design. Arrol and Chatterjee (1981) mention the following guidelines.
Vortex shedding: With respect to vortex shedding, if the critical wind speed for resonance in vertical and torsional modes (vertical modes only for trusses) is greater than the reference wind speed, the static and fatigue stress effects need to be checked from amplitude calculations appropriate to the mode shape.
Turbulence Response: If the natural frequency in first mode for vertical or torsional deflection is greater than 1 Hz, a dynamic analysis for stress effects need to be carried out to account for it.
Classical and Stall flutter: For prevention of this type of instability, the critical wind speed is to be greater than 1.3 times reference speed. The designer must ensure one of the following. The critical wind speed exceeds the practical limiting value for the given site or the resulting amplitudes are of allowable levels. Criteria for acceptability may include considerations of fatigue or of user reaction as well as of ultimate strength.
Buffeting is defined as the unsteady loading of a structure by velocity fluctuations in the incoming flow and not self-induced (Simiu and Scanlan, 1986). Buffeting vibration is the vibration produced by turbulence. There are two types of buffeting. One type is caused by turbulence in the airflow, and the other type is caused by disturbances generated by an upwind neighboring structure or obstacle. The first type of buffeting can produce significant vertical and torsional motions of a bridge even at low speeds. This buffeting induced motion results in a gradual transition to large amplitude torsional oscillations, which could lead to the failure of a bridge. If the velocity fluctuations are clearly associated with the turbulence shed in the wake of an upstream body, the unsteady loading is referred to as wake buffeting. Wake buffeting is common in urban areas with many tall structures.
2.11 Tacoma Narrows Bridge Failure
The Tacoma Narrows Bridge failure in 1940, at Washington, USA is a classical example of the aerodynamic instability failure. Wind wrecked the 2800-ft. main span of the bridge on Nov 7, 1940. A wind of 42 mph was responsible for the accident, though higher winds had been experienced previously without damage (Bowers 1940). He reports that this wind caused a vertical wave motion that developed a lag or phase difference between opposite sides of the bridge giving the deck a cumulative rocking or side-to-side rolling motion. Failure appeared to begin at mid-span with buckling of the stiffening girders. The suspenders snapped and their ends jerked high in the air above the main cables, while sections of the floor system several hundred feet in length fell out successively breaking up the roadway toward the towers. Almost the entire suspended structure between the towers was ripped away and fell into the waters below, but the 1100-ft. side spans remained intact. Cables and towers survived and held up the weight of the side spans, though the latter sagged about 30-ft. as the towers went back sharply by the unbalanced pull of the side-span cables. Bowers reports that about five weeks before the failure, on the chance that an aerodynamic failure might be found, a 1:20 model of the deck was built and was tested in the wind tunnel. The tests showed the model to be aerodynamically unstable in certain winds and this condition was concluded to be the source of the oscillations of the bridge itself. The wind tunnel test was followed by studies towards remedial measures aimed at modifications that would improve the characteristics of the bridge. A contract for the installation of the deflector vanes was under negotiation when the collapse occurred.
The following pictures, Figures 2.8 –2.13 give an understanding of the instabilities the bridge suffered and also shows the mode of collapse during failure.
Figure 2.8: This photograph shows the twisting motion of the center span just prior to
Figure 2.9: The nature and severity of the torsional movement is revealed in this picture taken from the Tacoma end of the suspension span. When the twisting motion was at the maximum, elevation of the sidewalk at the right was 28 feet (8.5m) higher than the sidewalk at the left.
Figure 2.10: This photograph actually caught the first failure shortly before 11 o'clock as the first concrete dropped out of the roadway. Also note bulges in the stiffening girder near the far tower and also in the immediate foreground.
Figure 2.11: A few minutes after the first piece of concrete fell, this 600 foot section broke out of the suspension span, turning upside down as it crashed in Puget Sound. Note how the floor assembly and the solid girders have been twisted and warped. The square object in mid air (near the center of the photograph) is a 25-foot (7.6m) section of concrete pavement. Notice the car in the top right corner.
Figure 2.12: This photograph shows the sag in the east span after the failure. With the center span gone there was nothing to counter balance the weight of the side spans. The sag was 45 feet (13.7m). Also the immense size of the anchorage is illustrated.
Figure 2.13: This picture was taken shortly after the failure. Note the nature of the twists in the dangling remainder of the south stiffening girder and the tangled remains of the north stiffening girder.