In general, any mechanism that produces an unbalanced moment may also serve to initiate a rigid body rotation of a Tainter gate about the trunnion pin. From the modal analysis testing on an intact Folsom Dam Tainter gate, and an understanding of the concepts of flow-rate variation pressure and push-and-draw pressure presented in Chapters 4 and 5, respectively, a conceptual model of the vibration mechanism can be formulated. The whole gate rotation induces a flow-rate variation pressure and a coupled inertia torque on the skinplate, as presented in Chapter 4. Both the flow-rate-variation pressure and the inertia torque excite the skinplate to rotate in a bending mode shape about a horizontal nodal line. In the present chapter we will develop the theory behind such an instability mechanism, called the self-excited coupled-mode instability, culminating in the graphical representation of the Folsom Dam gate instability in terms of a dynamic stability criterion diagram under the conditions at which failure occurred.
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In this book Tainter gates references can be found in most every of chapter. One special type of instability, eccentricity instability was presented in Chapter 8. The unbalanced moment acting on the gate due to the eccentricity is an essential feature of the eccentricity instability mechanism. In general, any mechanism that produces an unbalanced moment may also serve to initiate a rigid body rotation of the Tainter gate about the trunnion pin. In the present chapter we will present the theory behind another instability mechanism, called the self-excited coupled-mode instability.
Tainter gates, or radial gates as they are also known, are frequently used as crest gates on dam spillways for water level regulation. The design of these Tainter gates is such that the resultant hydraulic load due to pressure exerted on the skinplate usually passes through the trunnion pin. In this manner, the hydraulic load is usually borne by the trunnion pin as shown in Figure 1. Mechanical friction in Tainter gates is much lower than in other types of gates, such as vertical lift gates. Further, a portion of the gate weight is also carried by the trunnion pin, permitting the use of a relatively a small capacity hoisting motor and smooth operation of the gate. For this reason, Tainter gates are well suited for larger installations, such as the crest gates on the Itaipu Dam in Brazil, as was shown in Figure 1 in Chapter 8, as well as many other sites world-wide. Tainter gates are also installed on the dam crest at the Folsom Dam on the American River near Sacramento, California. A drawing of a Tainter gate from the Folsom Dam is shown in Figure 1. The gate has a height of 15.5 m, a width of 12.8 m with a skinplate radius of 14.33 m. The whole gate mass is 87.03 × 103 kg.
Figure 1. Side view of the 87-ton Tainter gate at the Folsom Dam in California, showing two predominant modes of vibration
According to an eyewitness account, one of these massive Tainter gates experienced flow-induced vibrations and failed, early in the morning of July 17, l995. The gate operator was on the catwalk just above the gate, opening the gate. He said he felt a small steady vibration starting up, initially very light, but then quickly intensifying. Upon his initial observation, the operator pushed the stop button, thinking to close the gate. As he turned from the control panel, he saw one side of the gate moving slowly in the downstream direction, rotating about the other side of the gate, like a large hinged garage door, as detailed by Ishii (1995a).
In an earlier case, a Tainter gate, with a height of 12 m, a width of 9 m, a skinplate radius of 13 m and a mass of 37 × 103 kg at the Wachi Dam in Japan failed and was swept downstream about 140 m on July 2, 1967 (Ishii et al., 1980). The team reviewing this gate failure in Japan suggested more than 35 years ago that flow-induced vibrations may have been a possible cause. However, at that time there was no known vibration mechanism and the structural dynamics of such large gates had not been well studied.
In examining the Wachi Dam gate failure, the eccentricity instability presented in Chapter 8 was identified. This dangerous type of flow-induced vibration of radial gates results from the eccentricity of the curved skinplate center relative to the trunnion center. A careful examination of the Wachi gate drawing revealed that the gate eccentricity was only 72 mm, and this small eccentricity was later shown to be insufficient to excite the eccentricity vibration mechanism.
Ishii (1995a) had an initial suspicion that eccentricity instability may have played a role in the Folsom Dam failure. For the onset of the eccentricity instability, however, the eccentricity of the skinplate center from the trunnion pin center would have had to have been more than 0.42 m for the Folsom gate. The radial gate at the Folsom Dam did not possess such a large eccentricity.