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bt = 1 or 0 (0 for hexagonal cross-section, 1 for others), correction factor for influence of turbulence Iu = intensity of turbulence m = mass of girder per unit length (kg fs2/m2) r = air density dh = logarithmic damping decrement of the structure in heaving vibration mode

2. Torsional vibration

where

0 = torsional mode amplitude (degree)

D = effective height of girder

Ip = polar inertia moment of mass of girder per unit length (kgfs2) d0 = logarithmic damping decrement of the structure in torsional vibration mode

3. Amplitude estimation from experimental data: As indicated in Equation 13.8, the occurrence of self-excited vibration is assessed as the sum of structural damping and aerodynamic damping. In wind tunnel tests, at specified wind velocity, aerodynamic during experiment from measured total damping. The aerodynamic damping usually is as shown in Figure 13.53. That is called the ''A-d curve.''

When A-d curve is obtained, the amplitude of the vibration at specified wind velocity can be estimated for the structure with any structural damping. In the case where the structural damping is ds, setting the value of — ds on the vertical axis and obtaining the intersection point on the curve, the corresponding amplitude on the horizontal axis is the estimated amplitude As of the steady-state vibration at the specified wind velocity.

Amplitude

U=specified wind velocity

FIGURE 13.53 Aerodynamic damping to amplitude at specified wind velocity (AS curve).

13.5 Examples of Real Bridges

In this section, some bridges that had aeroelastic vibration problems are introduced. It is considered that it is very important for bridge engineers to know the damage caused to the bridge by wind action, which would be helpful when they design new bridges.

13.5.1 Collapse of Tacoma Narrows Bridge [43]

The collapse of Tacoma Narrows Bridge in Figure 13.54 is very famous in the bridge engineering field as the accident was caused by a wind action. The accident occurred at a wind speed of 19 m/s in 1940. The Tacoma Narrows Bridge was constructed by utilizing the most advanced analytical method of suspension bridge in those days. Although the effect of wind load as the drag force of the girder was taken into account in the design of the bridge, the occurrence of aeroelastic vibration of the girder was not taken into consideration. The girder configuration was similar as shallow H-shape section shown in Figure 13.55. This type of girder section induces vortex-excited vibration in heaving and torsional modes and the flutter as the torsional self-excited vibration as explained in Section 13.4. As is well known, the bridge collapsed in the torsional vibration with large amplitude caused by the flutter. It was reported that the Tacoma Narrows Bridge was vibrating from the beginning of the erection and investigation was done to control the vibration by attaching fairings on the girder ends so as not to generate the separation flow from the leading edge of the girder as shown in Figure 13.56. Unfortunately, the bridge collapsed before the fairing could be attached. After the collapse, the cause was investigated from various points of view. The girder configuration chosen from various investigations was a truss-stiffened girder reducing the wind load and the possible occurrence of aeroelastic vibrations. The truss-stiffened girder developed in the investigation of the collapse of the Tacoma Narrows Bridge has been used mainly in United States and Japan as the girder of long-span suspension bridges.

13.5.2 Vortex-Excited Vibration of the Great-Belt Bridge [44]

The Great-Belt Bridge (Figure 13.57) was constructed as the second long-span suspension bridge in 1998. The aeroelastic performance was investigated in the wind tunnel by using a full span 1j0-scaled model. The occurrence of vortex-excited vibration in the smooth flow was predicted through the wind tunnel tests. The vibration, however, was not predicted in the turbulent flow. Therefore, at the beginning of erection, the girder was constructed without attaching a device to suppress the vortex-excited vibrations. Just before being opened to traffic, however, the bridge girder vibrated in the vortex-excited vibration. It was considered that the device to suppress the vortex-excited vibration should have been attached on the bridge girder, when the vortex-excited vibration was observed. After the observation of the vibration, a vane-type device was prepared and attached on the lower corners of the bridge girder as shown in Figure 13.58. As a result, the vortex-excited vibration was suppressed.

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