RETROSPECTIVE CONSIDERATION OF A PLAUSIBLE VIBRATION ... · 314 int. j. mech. eng. & rob. res. 2014...

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Int. J. Mech. Eng. & Rob. Res. 2014 Noriaki Ishii et al., 2014

RETROSPECTIVE CONSIDERATION OF APLAUSIBLE VIBRATION MECHANISM FOR THEFAILURE OF THE FOLSOM DAM TAINTER GATE

Noriaki Ishii1*, Keiko Anami2 and Charles W Knisely3

*Corresponding Author: Noriaki Ishii, [email protected]

A massive 87-ton Tainter gate installed at the Folsom Dam failed during operation on July 17,1995. To avoid similar future failures all potential causes of the accident must be considered.For this purpose, experimental modal analyses were undertaken on one of the similarly designedremaining Tainter gates to identify its in-air natural vibration characteristics. The experimentalmode shapes and frequencies are presented. Subsequently, a Finite-Element Method (FEM)model of the Folsom gate was constructed and validated by comparing the calculated naturalvibration characteristics with the corresponding measured values. The validated FEM modelwas then used to calculate the static structural deformation of the Folsom gate and correspondingstresses. From the methodical investigation presented in this paper, the static loading due toincreased trunnion friction on the Folsom Tainter gate was insufficient to produce failure. Finally,a conceptual model of a closed energy cycle for the self-excited vibration of the Folsom DamTainter gate is developed. This conceptual model was validated through tests on a two-dimensional 1/31-scale model of the failed Folsom gate.

Keywords: Folsom Dam gate, Tainter gate vibration, Radial gate, Failure investigation, Modeshape, Coupled-mode self-excited vibration

INTRODUCTIONA radial gate (also called a “Tainter gate”)installed at the Folsom Dam in California,having a height of 15.5 m, a width of 12.8 m, aradius of 14.33 m and a mass of 87.03 × 103

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1 Department of Mechanical Engineering, Osaka Electro-Communication University, 18-8, Hatsu-chou, Neyagawa, Osaka 572-8530,Japan.

2 Department of Mechanical Engineering, Ashikaga Institute of Technology, 268 Omae-cho, Ashikaga, Tochigi 326-8558, Japan.3 Department of Mechanical Engineering, Bucknell University, Lewisburg, PA 17837, USA.

kg, failed early in the morning of July 17, 1995,as shown in Figure 1a, unleashing a thunderingtorrent of white-water. Figure 2a shows the viewfrom downstream of the Folsom Dam Taintergate before failure, while Figure 2b shows the

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Figure 1: Two Tainter Gate Failures: (a) An 87-ton Gate at the Folsom Dam in the USAon July 17, 1995, and (b) A 37-ton Gate at the Wachi Dam in Japan on July 3, 1967

(Courtesy of the Asahi Newspaper Co. Ltd.)

(a) (b)

Figure 2: Two Views of Folsom Dam Tainter Gate: (a) Before Failure, and (b) AfterFailure

(a) (b)

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same gate after failure, opened like a hingeddoor and hanging on its massive chains.

Figure 3 shows a side view of the FolsomDam radial gate before failure. When the waterdepth over the dam crest was at 14.02 m, aroutine gate opening operation was initiated.In an interview, the gate operator explained thatwhen the gate opening reached 0.76 m, hewas standing on the catwalk just above thegate. He felt a small steady vibration, whichwas initially very light, but intensifying veryquickly. He pressed the stop button to stopraising the gate and, as he turned around, hesaw the gate slowly emerging from the bay inthe downstream direction like a large hingedgarage door (Ishii, 1995a).

A different gate failure incident is portrayedin Figure 1b. A radial gate at the Wachi Dam

in Japan, with a height of 12 m, a width of 9 m,a radius of 13 m and a mass of 37 tons, failedand was swept about 140 m downstream onJuly 2, 1967 (Ishii and Imaichi, 1980). About30 years ago, the review team for the gatefailure in Japan suggested that flow-inducedvibration was a possible cause of the failure.At that time, however, the detailed mechanismof such flow-induced vibration was unknownand the dynamic characteristics of radial gateswere not well-studied.

In the investigation of the Tainter gate failureat the Wachi Dam, several studies wereconducted by Ishii and Imaichi (1976, 1977a,1977b and 1980) and Ishii and Naudascher(1984 and 1992), where it was assumed thatthe center of the circular-arc skinplate did notcoincide, in general, with the trunnion center.

Figure 3: Side View Drawing of 87-ton Folsom Dam Tainter Gate

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Such a Tainter gate is called an “eccentricTainter gate”. When the skinplate center isshifted to a location beneath the trunnion pin,as shown in Figure 4, the fluid force acting onthe skinplate tends to close the gate opening.A gate with such a design is called a “press-shut device”. The gate is statically stable, sincethe gate is pressed closed by the hydrostaticpressure. Such a gate, however, is notnecessarily dynamically stable. In fact, in manycases, such a gate is dynamically unstable.When the skinplate moves downward, theapproach flow is decelerated, thus increasingthe fluid pressure. Supposing the skinplatecenter is below the trunnion pin, the increasingfluid force beneath the gate counters thehydrostatic load and raises the gate permitting

greater discharge. The increased dischargecreates a low pressure region beneath thegate, adding to the hydrodynamic load andcausing the gate to close. As this process isrepeated, energy is supplied to the gate andthe gate begins to undergo self-excitedvibration. A stability criterion for this type offlow-induced vibration of Tainter gates hasformulated by Ishii and Naudascher (1992).

The initial research after the Wachi failurefocused on this “eccentricity instability” whichrequires a displacement of the skinplate centerrelative to the trunnion pin. Since the failedWachi gate did not have any significanteccentricity, the consensus at the time was thatthe self-excited vibration mechanism identifiedby Ishii could not have actually occurred.

Figure 4: Side View Drawing of an Eccentric Tainter Gate Suspended by a Springand Damper

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More than 28 years after the Wachi gatefailure, the 87-ton Tainter gate failed at theFolsom Dam. The report of the failure on July19, 1995 in The Japan Times included thefollowing passage:

A 90-ton gate at Folsom Dam buckledMonday (July 17), unleashing athundering white-water torrent ... Theproblem started as operators werereopening the huge metal gate—one ofeight spillways—after the flow had beentemporarily shut down. “We wereopening the gate, and it started tovibrate. We don’t know why”, said TomAiken, dam manager for the US Bureauof Reclamation.

In this report, the dam manager for theUSBR clearly indicates that vibrations playeda role in the Folsom Dam gate failure. He hadalso previously mentioned the role of vibrationsin the following excerpt from a televised pressconference on July 17, 1995 (KXTV-10, 1995):

This morning at 8 o’clock we weremaking a routine gate change on our No.3 spillway gate and our operator said henoticed something bind up a little bit andthe gate vibrated and suddenly gave wayand right now we have about 40,000cubic ft/s being released.

On August 15, 1995, about a month after thegate failure, the first author had the opportunityto interview James Taylor, gate operator of thefailed gate, who was standing on the catwalkjust over the gate at the time of failure. He tooindicated that vibrations accompanied thegate failure as follows:

When spill gate No. 3 reached around2½ feet, I felt an unusual vibration. I

immediately pushed the stop button andstarted to check the gate. The vibrationintensified and as I turned to check thegate, I could see the right side (lookingdownstream) of the gate separating fromthe structure and water coming outaround the gate …

Based on these clear eyewitnesstestimonies, some sort of vibration wasevidently involved in the Folsom Dam Taintergate failure. The Forensic Report concerningthis failure assigns blame, however, toexcessive static loading (due to increasedtrunnion friction) of cross braces withinsufficient stiffness and strength. Thisconclusion is based on several runs of a finiteelement code that was not validated, but whichproduced results in agreement with the modeof failure—the failure of the bolts thatconnected Brace 3E-4D. The fact that thisconclusion was at odds with eyewitnessreports was essentially ignored.

Ishii (1995a) had an initial suspicion that“eccentricity instability” may have played arole in the Folsom Dam failure. The radialgate at the Folsom Dam, however, did notpossess such an eccentricity between theskinplate center and trunnion center. Theobservations of the gate operator and,subsequently, the validated FEM modelresults (to be discussed subsequently in thispaper) that were inconsistent with a staticfailure, lead naturally to consideration of a newmechanism for a “non-eccentricityhydrodynamic instability” to which non-eccentric radial gates may be susceptible.This new dynamic instability is a two degrees-of-freedom coupled-mode vibration resultingfrom the coupling of two vibration modes

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through inertia forces and hydrodynamicforces acting on the gate.

As a first step in the exploration of apotential vibration mechanism for the FolsomDam gate, in late September 1995, Ishii(1995b) conducted in-air experimental modalanalysis on one of the remaininggeometrically similar Folsom Dam gates(subsequently designated as the “originalgate”), employing an impact hammer andaccelerometers. From this modal analysis, arelatively precise assessment of the naturalvibration characteristics of the failed gate wasobtained. After this initial modal analysis, theremaining gates were reinforced.Subsequently, in the middle of November1996, a similar experimental modal analysiswas conducted on one of the reinforced FolsomDam gates in order to evaluate the dynamiceffectiveness of the reinforcement (Ishii, 1997).

The Finite-Element Method (FEM) has beendeveloped over several decades and is nowroutinely used as a tool to identify naturalfrequencies and mode shapes. However, themethod cannot provide information ondamping ratios. Further, any FEM analysiscontains many uncertainties, such as mass,stiffness and constraint conditions, which maysignificantly affect the results. Experimentalmodal analysis, on the other hand, can reliablyidentify the natural vibration mode shapes andfrequencies, as well as the damping ratios. Thedrawback to experimental modal analysis ofsuch large structures is the difficult, dangerous,and time intensive placement of expensiveaccelerometers. However, without suchexperimental input, the basic inherentuncertainties of the FEM analyses cannot bequantitatively assessed.

An FEM analysis was conducted on theoriginal Folsom Dam gate, using a highprecision finite element model with a largenumber of plate elements. The resultingcalculated frequencies and mode shapes arepresented and compared with thecorresponding experimental results. Theagreement between numerical andexperimental mode shapes and frequenciesallows the FEM uncertainties to be quantified.When the agreement is good, the FEM modelhas been validated. In addition, the staticdeformations and the static tensile loads werecalculated for a range of frictional forces at thetrunnion pins. These FEM deformations andloads contradict and call into question theconclusions of the USBR Forensic Report onthe failed Folsom Dam Tainter gate. The claimthat the gate failed due to excessive staticloading brought about by increased trunnionfriction and insufficient stiffness and strengthin critical structural members could not besubstantiated using the validated FEM model.The conclusions of the Forensic Report arealso at odds with eyewitness accounts ofvibration occurring just before failure.

In this study, the framework of themechanism behind a dynamic instability ofTainter gates, unrelated to eccentricity of thegate, is presented. Its potential applicability tothe failed Folsom Dam gate is detailed. Twoinherent modes of structural vibration of theoriginal Folsom Dam gate, as identified by in-air experimental modal analysis, aredocumented. In addition, similar in-air modalinformation is also presented for the reinforcedFolsom gates. Impact force inputs,acceleration responses, and transfer functionsare shown and used to identify the in-air natural

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frequencies, mode shapes and the dampingratios.

The proposed dynamic instabilitymechanism, unrelated to gate eccentricity,involves the coupling of the rigid-body gaterotational vibration around the trunnion pinswith a bending mode of the skinplate. Themechanism is formulated and presented interms of a closed energy feedback cycle.

Finally, the existence of this proposeddynamic instability mechanism isdemonstrated using a two-dimensional 1/31-scale model of the Folsom gate, where thelower-mode bending vibration of the skinplateis represented by a streamwise rotationalvibration of the rigid model skinplate. The in-water natural frequency of the streamwiserotational skinplate vibration is fixed at tworepresentative values, while the naturalfrequency of the whole gate vibration aroundthe trunnion pin is varied over a wide range ofvalues bracketing the in-water natural vibrationfrequency of the skinplate. The model gatewas placed in a channel and exposed to flowingwater. The gate vibration amplitudes wererecorded.

MATERIALS AND METHODSThe materials presented in this report havebeen drawn from three distinct, but relatedstudies. The materials and methods for eachof the individual studies are presentedseparately in the following sections.

Experimental Modal Analysis of theFolsom Dam Tainter GateIn late September 1995, an in-air experimentalmodal analysis was conducted on Tainter gateNo. 2, a gate that is geometrically similar to

the failed Tainter gate. Subsequently, in mid-November 1996, a similar vibration test wasundertaken on a reinforced Tainter gate.

As shown in Figure 3, the arm structure ofthe original gate had only two diagonalmembers between the lower two arms. Thewhole gate was hoisted by two chains with alength of 16.4 m, hooked to the gate near thebottom of the skinplate. A list of the mass ofmajor members of the Folsom Dam Taintergate is presented in Table 1, showing themass of the skinplate with horizontal girdersand leaf bracing was 59,732 kg, or about 68%of the whole gate mass of 87,033 kg.

To excite structural vibration of the gate, itwas raised (with no water behind the gate) byabout 0.3 m and struck with a 5.4 kg impacthammer (PCB GK-086B50) which wasinstrumented with a force transducer (PCBGK-206M06). The center of the lowerhorizontal girder was struck in both the radialand tangential directions, respectively, asshown by the arrows in Figure 5. The impactposition was chosen to excite as effectivelyas possible structural vibration modes

Table 1: Mass of Major Members ofOriginal Folsom Dam Gate

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associated with the bending flexibility of theskinplate and the radial arms. The radialimpact was delivered by striking the flangecenter of the horizontal girder in the upstreamdirection, along the web line. The tangentialimpact was delivered by striking the web nearthe skinplate of the horizontal girder in thedownward direction, perpendicular to the web.

The measurement locations of the gateacceleration responses to the impact forceare identified by small circles in Figure 5. Theacceleration responses were measured withaccelerometers (GK-308B piezo-type by PCBCo. for the original gate, and LS-20C servo-type by RION Co. for the reinforced gate). Inthese tests 12 accelerometers, mounted onmagnetic bases and positioned at variousmeasurement locations, were used for eachimpact. For the skinplate vibrations, theaccelerometers were systematically placed at72 positions along the centers of the verticalbeam flanges. For the radial arm vibrations,

the accelerometers were systematically placedat 64 positions along the center of the radialarm webs. In addition, the accelerationresponses of the chains were measured at 2positions on each chain to examine their lateralvibrations coupled with gate vibrations. Theacceleration responses of the hoist framessupporting the reduction gear for chain windingwere also measured to examine the effect ofthe chain elasticity on the gate vibrations. Thehoist frame acceleration responses wererecorded in the streamwise, vertical, andspanwise directions at 2 positions on eachhoist frame. Additionally, the accelerometerswere systematically placed at 18 positions onthe cross beam webs to measure the radialand tangential acceleration responses. In total,acceleration responses were measured at162 points.

On the skinplate, the radial (streamwise)and tangential acceleration responses weremeasured at each measurement location,

Figure 5: Radial and Tangential Impact Position () and Measurement Points (o) forExperimental Modal Analysis of Folsom Dam Tainter Gate, Showing (a) Top View of the

Gate, (b) Side View of the Gate, and (c) Upstream View of the Skinplate

(a) (b) (c)

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The recorded data were monitored using FFTanalyzers (AD3525 by A&D Co.) to permitcareful adjustment of the input levels to the datarecorder for optimum recording.

Precision FEM Model DevelopmentBased on the blueprint of the original FolsomDam Tainter gate, a finite element model(frame structure) of the gate was created.Generally, in the Finite Element Method (FEM),simplified modeling is made in many cases.However, past experience suggests that for acomplicated structure like a Tainter gate, theprecision of the modeling has a significantinfluence on the accuracy of the subsequentoutput. For this reason, the model form wascreated faithfully in precise detail accordingto the blueprint, as discussed in Anami (2002)and Anami et al. (2006).

since vibration in the spanwise direction isinsignificant. For measurement locations onthe radial arms, however, vibrations in thetangential and spanwise directions weremeasured, since the vibration in thestreamwise direction was less significant.

A schematic diagram of the instrumentationis shown in Figure 6a and the subsequent flow-chart for the computer-based modal analysisis shown in Figure 6b. The circled “F”represents the impulsive force data, and thecircled “A” the acceleration response data. Theimpact force data and the accelerationresponse data were amplified throughdedicated exclusive amplifiers, and thenrecorded on tapes using a data recorder (XR-50H with VHS video tapes by TEAC Co. forthe original gate, and RD-135T by TEAC Co.with digital audio tapes for the reinforced gate).

Figure 6: (a) Instrumentation, and (b) Data Processing Flowchart for Modal Analysisof the Folsom Dam Gate

(b)

(a)

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The material properties of iron for all of thestructures of Folsom gate were given asfollows: 208.7 GPa for Young’s modulus, 78.89GPa for modulus of transverse elasticity, and7,870 kg/m3 for density. Correspondingly,Poisson’s ratio was taken as 0.323.

The FEM gate model was created usingstandard elements: the “plate element”, the“beam element” and the “DOF spring element”.The plate element is a 2-dimensional element,consisting of 3 or 4 nodes, which has theadvantage that the constraint conditions aregiven correctly, although the number of nodesincreases. Thus, in addition to the skinplateitself, the main horizontal girders of the I-beamwere also defined with 3 plate elements forthe flanges and web. Furthermore, the H-beams of the radial arm and cross beam werealso defined as 3 plate elements. The L-anglesteel struts between the radial arms weredefined by 2 plate elements.

The beam element can more simply modelthe structure as 2 nodes, but it is unsuitablefor representing complicated constraintconditions. Therefore, in the present modeling,only the T-beams reinforcing the skinplate andthe leaf-bracing L-angles were defined usingthe beam element, where each cross-sectionalform was specified. The massive chains, whichare attached to each spanwise side of theskinplate to hoist the gate, were defined as 1-DOF spring elements. The spring constant was0.718 × 108 N/m per chain, as calculated fromthe actual chain geometry.

In FEM analysis, the element division is avery significant aspect in determining theaccuracy of the analysis results. It is trivial thatthe greater the element division number is, themore accurate are the results. However, the

number of elements is restricted by theperformance of the computer. Thus, it isnecessary to make an effective analysis withthe limited number of elements, where theelement division was concentrated in thesignificant portions in the model. In the presentanalysis, the fine element divisions were madefor the skinplate and radial arms, since theexperimental modal analysis showed that theywere significant factors in determining thenatural vibrations of the whole gate. The totalelement number of the whole gate is 23,124,and the total nodes number is 23,967.

All structural elements were joined with bolts.Assuming sufficient rigidity, the junctions werespecified as rigid constraint conditions. In thismanner, high rigidity was given between oneindependent node and two or moresubordinate nodes, thus enabling thetransmission of the translation force andmoment produced in the independent node tothe subordinate nodes.

The trunnion pin is embedded in a concretebase, thus having only one degree of freedomto move around its rotation axis. Therefore, theflexibilities except for around its rotation axiswere constrained for the trunnion pin. The chainends have degrees of freedom in the rotationaland translational directions, while the degreeof freedom in the spanwise direction wasconstrained.

The high accuracy FEM model is shown inFigure 7, where the 3-dimensional views fromthe downstream and upstream sides areshown in Figures 7a and 7b, respectively. Thepoints of intersection in these views are theelement nodes. The elements surrounded bythe nodes on the skinplate or on the radial armsare plate elements, and the lines connecting

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Figure 7: Precise FEM Model Construction of the Original Folsom Dam Tainter Gate

the nodes of horizontal girders and skinplatereinforcement T-beams are beam elements.The numbers from 1 to 4 indicate the radialarms from the top. In order to identify theanalysis results, the junctions of the arms andthe trunnion pin hub on the right hand side aredenoted “C”, junctions of the arm struts andthe vertical struts are denoted “D”, “E” and “G”from the trunnion pin side, and junctions of thearm struts and the horizontal girders aredenoted “H”. The left hand junctions also aresimilarly denoted “T”, “S”, “R”, “P” and “O” asshown in Figure 7.

Design of 1/31-Scale ModelThe major purpose of model gate testing wasto obtain data concerning the coupled vibrationof the streamwise skinplate bending rotationand the whole gate rotation, in order to confirmthe coupled-mode self-excited vibrationmechanism postulated later in this paper. The

data can also be used subsequently to verifythe theoretical analysis of this mechanism. Oncethe theoretical analysis is verified, it can thenbe applied to calculate the hydrodynamicstability of the Folsom Dam gates, as well as ofother gates, under various conditions.Fundamentally, the model gate does notnecessarily need to be an exact scaledduplicate of Folsom Dam gate. Any combinationof skinplate streamwise flexibility, skinplaterotation center height, and the in-air dampingratios and so on, are all possible for the modelgate. Based on these model scale experiments,as well as on additional full-scale gate testing,Anami (2002) has developed a theoreticalanalysis that will be presented in a subsequentpaper (Anami et al., 2014).

The two-dimensional 1/31-scaled modelgate is shown in Figure 8. The rigid skinplateis supported by a horizontal coiled spring (B)

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just behind the skinplate, which represents thestreamwise skinplate bending elasticity in thefull-scale gate. The skinplate has its rotationcenter placed on the radial arm for easyadjustment, and performs streamwiserotational vibrations, denoted by. The validityof such crude modeling was confirmed by theexperimental modal analysis. The whole gateis suspended by a vertical coiled spring (A),which represents the chain elasticity in the full-scale gate. Thus, the gate performs rotationalrigid-body vibrations around the trunnion pin,denoted by . The position of both thehorizontal and vertical springs can be adjusted,thus permitting a fine adjustment of eachnatural vibration frequency. The model gatewas so adjusted at static equilibrium with all

upstream heads that the skinplate centercoincides exactly with the trunnion center, thatis, it is non-eccentric. The streamwise vibrationof the skinplate naturally shifts its center fromthe trunnion. However, assuming small-amplitude vibrations in this study, theeccentricity has a negligible effect in excitingthe whole gate around the trunnion, whencompared with that due to inertial torque of theskinplate itself.

The major specifications for the model gateare listed in Table 2. The skinplate has a radiusRa of 455 mm and a mass of 10.0 kg. Theskinplate has a spanwise length W0 of 290 mm,and inserted in a 300 mm-wide channel,leaving a gap of about 5 mm from the channel

Figure 8: Cross-Sectional View of a 1/31-Scaled Coupled-Mode Vibration Modelof the Folsom Dam Gate

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wall on each side of the skinplate. The wholegate has a mass of 15.1 kg. The skinplaterotation center height Rs from the skinplatelower end, shown in Figure 8 is 515 mm. Themoment of inertia of the skinplate around itsstreamwise rotation center, I, takes a valueof 0.944 kgm2, and the moment of inertia ofthe whole gate around trunnion pin, I, takeson a value of 2.220 kgm2.

It was significant in the model tests to adjustthe in-water streamwise vibration frequency ofthe skinplate to be near that of the Folsom DamTainter gate, because the instantaneouscoefficient of flow-rate-variation under the gateis strongly dependent upon the streamwisevibration frequency of the skinplate. From thefield vibration tests, the in-air streamwiseskinplate vibration frequency was found to beabout 27 Hz. This in-air frequency is reducedin flowing water, due to the added mass effectof water upon the skinplate. With a calculatedadded mass, the in-water vibration frequencywas estimated to be about 6 Hz (Anami andIshii, 1998). In order to realize this in-waterfrequency in the present model tests, the in-air streamwise vibration frequency of themodel skinplate, , was set at one of twovalues, 10.6 Hz or 12.3 Hz

As shown in Figure 8, the skinplate isinclined downstream at an angle of 22°, similar

to the inclination of the Folsom Dam Taintergate. The major purpose of the present modelgate testing is to demonstrate the self-excitedvibration due to the coupling of two inherentmodes of vibration. For this reason, the channelfloor just under the gate is inclined such thatthe press-open angles0 becomes zero, thuseliminating the press-open hydrodynamicpressure component. With such asimplification of the model test setup, an easyphysical understanding of the test results canbe obtained.

The model gate was exposed to a waterflow under the gate with an upstreamsubmergence depth d0 of 420 mm. The outflowis not submerged. With given values of thegate submergence depth d0 and the skinplatein-air streamwise vibration frequency, , thebasic Froude number F0 which determines thedynamic similarity of fluctuating flow field,defined by

agdF 0

0 2 ...(1)

takes on values of 13.9 and 16.0 in the presentmodel tests. The gate opening mean heightwas adjusted to 5 mm, which is 1.2% of thegate submergence depth d0. In the presentmodel gate tests, the in-air whole gate vibrationfrequency was varied from 2.6 Hz to 7.3Hz, so as to include the in-water naturalstreamwise vibration frequency of the skinplateat around 6 Hz.

RESULTS AND DISCUSSIONAs in the section on the Materials and Methods,the results of each of the three studies will bereported sequentially, each successive onebuilding on the results from the previous study.

Table 2: Major Specifications for FolsomDam Model Gate

B 5 mm s0 0° W0 290 mmd0 420 mm s 16° Rc 375 mmRa 455 mm a 0.003 RS 515 mmI 2.220 kgm2 0.105 10.6 HzI 0.944 kgm2 0.133 12.3 Hz

I 0.289 kgm2

aa

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Experimental Modal AnalysisCalculation of Inverse ModalIndicator Functions (IMIF)The recorded impulsive force and theacceleration response data were processedby the FFT analyzer (AD3525) to display thecorresponding real time waveforms and tocalculate their power spectra. As an example,the real time waves for the impulsive forcesand the accelerations measured at the centerof the skinplate are shown in Figure 9, whereFigure 9a is for the tangential direction andFigure 9b for the radial direction. Typically, the

impulsive forces during about 5 to 8 msreached a maximum level of about 15 to 17kN. Each acceleration power spectrum wasdivided by the respective input impulsive forcepower spectrum to calculate the transferfunctions. The impact tests were repeatedthree times for each acceleration responsemeasurement location to permit averaging ofthe calculated transfer functions. The bottomplots in Figure 9 show the averaged transferfunctions.

The calculated transfer functions at all 162measurement locations were input into a

Figure 9: Representative Impulsive Input, Acceleration Response and AveragedTransfer Functions of In-Air Vibration of Folsom Dam Tainter Gate in (a) The Tangential

(Up and Down) Direction, and (b) The Radial (Streamwise) Direction

(a) (b)

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software package for modal analysis (AD1461by A&D ZONIC Co.) to calculate the InverseModal Indicator Functions (IMIF) as an averageof all transfer functions, which is defined by

n

ii

n

iii

T

TTRealIMIF

1

2

1

)(0.1 ...(2)

where Ti represents the i-th transfer function,Real(Ti) represents the real part of i-th transferfunction, and |Ti| represents the amplitude ofthe transfer function. Therefore, the secondterm on right-hand side of Equation (2)represents cos in which is the phase-lagof the vibration relative to the impulsive force,taking on a value of 90° in the resonance state.Thus, the peak level of IMIF represents thestrength of the natural vibration of the gate. Thecalculated IMIF results are shown in Figure 10,where Figure 10a is for the tangential directionand Figure 10b is for the radial direction. Allthe peaks in the IMIF do not necessarily providea reasonable mode shape. Detailed

examination of the vibration mode shapes wasundertaken for each IMIF peak, permitting theidentification of several meaningful modeshapes which exhibited orderly regularity intheir amplitudes. The meaningful modes areidentif ied by mode names on theircorresponding IMIF peak in Figure 10. Asubscript “z” in the mode names denotes atangential (up-and-down, i.e., around thetrunnion pin) mode shape, while a subscript“x” denotes a radial (streamwise direction)mode. The mode designations for thesenatural vibrations are also included on thetransfer function plots in Figure 9.

Vibration Mode, VibrationFrequency and Damping RatioMajor vibration modes are shown in Figure 11and Figure 12, where Figures 11a and 12ashow side views and Figures 11b and 12bshow the 3D view of the gate. Figures 11c and12c show the spanwise vibration amplitudedistribution at labeled heights. The gray andblack lines show the instantaneous vibrationshapes with maximum amplitude. The

Figure 10: Inverse Modal Indicator Functions for In-Air Vibration of Folsom DamTainter Gate in (a) The Tangential (Up and Down) Direction, and (b) The Radial

(Streamwise) Direction

(b)(a)

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amplitudes are enlarged to provide easyrecognition of the mode shapes.

Figure 11 shows the most significant modeMz2, which corresponds to rotational vibrationof the whole gate around the trunnion pin. Theheavy chains and the hoist frames provide

elastic support. The natural vibration frequencyof this mode was 6.88 Hz, and the dampingratio was 0.012.

Figure 12 shows the fundamental modeMx2, which corresponds to the streamwisebending vibration of the skinplate, associated

Figure 11: Fundamental Mode Shape of the Rigid Body (Whole Gate) Vibration Aroundthe Trunnion Pin for the Original Folsom Dam Tainter Gate Designated as Mode Mz2 (a =

6.88 Hz; a = 0.012) Showing (a) The Side View, (b) A 3D-View, and (c) The SpanwiseDistributions of Deflection Amplitudes

(a) (b) (c)

Figure 12: Fundamental Mode Shape of Skinplate Streamwise Bending Vibrationfor the Original Folsom Dam Tainter Gate Designated as Mode Mx2 (a = 26.9 Hz; a =0.002) Showing (a) The Side View, (b) A 3D-View, and (c) The Spanwise Distributions

of Deflection Amplitudes

(a) (b) (c)

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with the deformation of the radial armstructures. In the vertical direction, the skinplateamplitude corresponds to about a 3/4-wavelength bending mode, as shown in Figure12a. In the spanwise direction, there is alsonon-uniformity in the vibration amplitude,approximating a half wavelength mode, withnodes at the spanwise ends, as shown inFigures 12b and 12c. The natural vibrationfrequency of this mode was 26.9 Hz, and thedamping ratio was very small, with a value of0.002.

After the failure, the remaining FolsomTainter gates were reinforced. Additional plateswere welded onto the radial arms, additionaldiagonal members were attached betweenradial arms, additional diagonal braces wereattached between cross beams, additionalgirder bracings were added, and all jointswere welded. No changes were made to theskinplate. Additional correspondingexperimental modal analyses were made on

a reinforced gate. Results from the modalanalyses, given in Figure 13, show thefundamental mode Nx2, involving a streamwisebending vibration of the skinplate. Figure 13ashows a side view and Figure 13b shows the3D view of the gate. Figure 13c shows thespanwise vibration amplitude distribution atlabeled heights. As shown in Figures 13b and13c, the skinplate vibrated in a bending mode,similar to the mode Mx2. The nodal line islocated just a bit higher than that for the originalgate, between i = 3 and 4 (compare with ModeMx2 in Figure 12). The skinplate still exhibiteda half wavelength spanwise mode shape, withnodes at the spanwise ends. The naturalvibration frequency of this mode was 27.9 Hz,and the damping ratio was 0.004. Both thenatural vibration frequency and damping ratioare slightly larger, relative to the original gate.

Measured natural vibration characteristicsof the original and reinforced gates aresummarized in Table 3, including sketches of

Figure 13: Fundamental Mode Shape of Skinplate Streamwise Bending Vibration for theReinforced Folsom Dam Tainter Gate Designated as Mode Nx2 (a = 27.9Hz; a = 0.004)

Showing (a) The Side View, (b) A 3D-View, and (c) The Spanwise Distributionsof Deflection Amplitudes

(c)(b)(a)

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the vibration mode-shapes. There are threeapparent natural vibration modes of the wholegate in rotation around the trunnion pin, whichare grouped into mode group Mz2 and Nz2 forthe original and reinforced gate, respectively.

Since the gate mass was increased byreinforcement, the natural vibration frequenciesafter reinforced were reduced, except modeNz2 which showed an anomalous slightincrease, possibly due to increased stiffnessfor this mode. In the streamwise direction, thereare also three apparent bending modes for theskinplate for the original gate, but only two forthe reinforced gate. These streamwiseskinplate bending modes are grouped intomode groups Mx2 and Nx, for the original andreinforced gates, respectively. The naturalvibration frequencies after reinforcementincreased about 1 to 2 Hz. This increase isdue to the increased bending rigidity of theskinplate resulting from the reinforcement ofthe radial arm structures. The damping ratiosare quite small, even after reinforcement, withvalues of 0.008 to 0.021 for the whole gate

rotational vibration around the trunnion pin, and0.002 to 0.008 for the skinplate streamwisebending vibration.

The spring constant due to the elasticchains and hoist frames and the moment ofinertia of the whole gate around the trunnionpin determine the natural vibration frequencyof 6.88 Hz, measured for the vibration modeMz21. As a check, calculations for the springconstant due to the chains and the hoistframes were made to obtain a value of 1.18× 108 N/m. Additionally, calculations for themoment of inertia of the whole gate aroundthe trunnion pin also were made to obtain avalue of 1.309 × 107 kgm2. A natural vibrationfrequency of 6.85 Hz for the whole gaterotation around the trunnion pin was found,which is quite close to the measured value of6.88 Hz. One may therefore have confidencethat the vibration mode Mz2 is indeed that ofthe whole gate undergoing rigid-bodyrotational vibration around the trunnion pin,due to the elastic support from the chains andhoist frames.

Table 3: Mode Shapes, Modal Frequencies, and Modal Damping from ExperimentalModal Analysis of Folsom Dam Tainter Gate In Air

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Vibration Characteristics from FEMAnalysis of the Folsom Dam TainterGateVibration Analysis

The natural vibration frequencies and vibrationmodes were analyzed using FEM analysissoftware (MSC/N4W, NASTRAN for WindowsV3 by the MacNeal-Schwendler Co.). Theanalyzed frequency range was 0 to 50 Hz, thesame range as the experimental modalanalysis. The vibration modes obtained bycalculation are shown by animation on acomputer screen. The natural vibration modeswere identified as those modes for which thewhole gate was carrying out orderly vibration.The significant results are shown in Figures14 and 15. Figures 14a and 15a are sideviews, while Figures 14b and 15b are 3-dimensional downstream views. The red andblack lines show the instantaneous vibrationshapes with maximum amplitude. In order tohelp with visualization of the modes, theamplitudes are magnified.

Figure 14 shows the whole gate rotationalvibration mode around the trunnion pin with thefrequency of 7.3 Hz. This vibration mode is ingood agreement with the experimental modalanalysis result shown in Figure 11.Deformations are not apparent for any beamsof the gate. It appears that this vibration iscaused by the elastic support of the chains,which raise and lower the gate.

Figure 15 shows the skinplate streamwisevibration mode. In the streamwise verticalplane, the skinplate is performing the lowestfrequency bending mode vibration with thenode near the second horizontal girder. In thehorizontal plane, the skinplate is performing ahalf wavelength vibration with nodes near eachside of the skinplate. The natural vibrationfrequency is 25.4 Hz. This vibration mode is ingood agreement with the experimental modalanalysis result shown in Figure 12.

The vibration frequencies and mode shapesobtained by FEM analysis are listed in Table

Figure 14: Whole Gate Vibration Mode of a = 7.3 Hz from FEM Analysis

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4, where the results from the experimentalmodal analysis are also shown for comparison.The FEM analysis results for natural vibrationfrequencies are in good agreement with theexperimental modal analysis results. Thesimulated FEM frequencies have relative errorof 5.8% for the whole gate vibration, and 5.6%for the skinplate streamwise vibration.

As a result, one may conclude that thepresent FEM simulation results showed goodcorrelation for natural vibration frequencies and

corresponding mode shapes with the field testresults. Therefore, the finite element model haseffectively captured the essential physics evenfor the large-scaled Folsom Dam Tainter gate.

Static Deformation CharacteristicsStructural static deformation of the Taintergate, induced by the hydrostatic pressure,was calculated for the same FEM model thatwas validated by comparison with theexperimental modal analysis results. It is veryuseful to know what level of deformation the

Figure 15: Skinplate Streamwise Vibration Mode of a= 25.4 Hz from FEM Analysis

Table 4: Comparison of FEM Analysis Prediction of Modal Frequencies for FundamentalMode Vibrations of Folsom Dam Tainter Gate with Frequencies from Experimental

Modal Analysis

RelativeMode Vibration Mode shape Frequency Damping Frequency Damping Error

[Hz] Ratio [Hz] Ratio of Freq. [%]

Mx2

7.28 ― 5.8

25.4 ― 5.626.9 0.002

FEM analysisExp. Modal analysis

Mz2 6.88 0.012

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massive skinplate experiences due to thehydrostatic pressure.

The static deformation was calculated forthe gate conditions at the time of failure, thatis, at a discharge opening of 0.76 m, and agate submergence depth was 13.26 m. Thewhole hydraulic load is 11.2 × 106 N. Eachelemental hydrostatic load acted on eachskinplate element submerged in water, in thedirection normal to the skinplate. In addition,the gravity force acted downward on eachelement of the whole gate.

In Figure 16a, the calculated loading isshown in a side view overlapped with the gatebefore deformation. Figure 16b is the 3-dimensional view from the upstream side. Theskinplate does not show any meaningfuldeformation in its upper portion, and showsits maximum static deformation at the bottomcenter portion. Deformation at each side of theskinplate is comparatively small. This limiteddeflection is because the sides of skinplate

are supported by the radial arms. The trunnionbeams show large deformations, but they donot become a problem.

The most significant result is that, as shownin Figure 16a, the bottom center of theskinplate shows a deformation of 19 mm inthe downstream direction. The downwarddeformation of 4 mm is due to the structuraldeformation of the skinplate itself, in additionto the elastic elongation of the chains.Relatively slender and weak members wereused in the Folsom Dam Tainter gate,especially relative to its massive size. Theseundersized components resulted in the largestreamwise deformation of 19 mm. Detailedcalculation of the hydrodynamic load due tostreamwise vibration (Anami, 2002; Anamiet al., 2002, 2005 and 2012) suggest that thehydrodynamic load reaches the same level ashydrostatic load with a vibration amplitude of8.4 mm along the skinplate spanwisecenterline, which is close to half of 19 mm static

Figure 16: Static Structural Deformation of Folsom Dam Tainter Gate with GateSubmergence of 13.26 m at Failure

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deformation. Therefore, the vibratorydeformation of the skinplate cannot bedisregarded.

Static Tensile Loads and Influenceof Frictional ForcesThe USBR concluded that the increasedtrunnion friction was the main factor in gatefailure, that is, an excessive static load arosesince the gate was raised against the trunnionfriction, thus inducing the incipient gate failure.In the USBR analysis of the failure, an FEManalysis was carried out for the givenhydrostatic load and trunnion friction tocalculate the static stress. Their results suggestthat excessive tensile force developed on thediagonal member between 3rd and 4th arms,exceeding the allowable load of the bolts. Sincesome doubt remains concerning the validityof FEM model by USBR, it was deemedprudent to re-examine the USBR conclusionsby using our previously validated FEM model.

As a result of these considerations, thetensile forces acting on each member inFolsom gate exposed to the loadingconditions at the time of failure were analyzedfor a gate discharge opening of 0.76 m and agate submergence depth of 13.26 m, wherethe trunnion friction coefficient was varied overa range of values, including the USBR-estimated maximum possible value of 0.3 atthe moment of gate failure. One of thecalculated results, the stress using a trunnionfriction coefficient of 0.3, is shown in Figure17. The significant aspect of this figure is thatthe maximum stress appears in the diagonalmember 3E-4D, that is, the member between3rd and 4th arm, as indicated by the orangecolor. From this load, one can determine thatthe tensile force acting on the bolted joint of

the diagonal member 3E-4D reached amaximum value of 240 kN.

The USBR Forensic Team concluded thatthe Folsom gate failure started from thebreaking of 4 bolts connecting the diagonalmember 3E-4D to the 3rd and 4th radial arms.The Forensic Team recovered similar 4-boltedmembers and attachment plate and undertooka careful tensile-loading test. As a result, it wasconfirmed that the range of the tensile failureload was 333 kN to 363 kN. The calculatedmaximum tensile force of 240 kN from thepresent FEM analysis, which is still sufficientlyless than the failure load of 333 kN, leads tothe conclusion that the static load alone didnot produce the observed failure.

The maximum tensile force on the diagonalmembers between arms was carefullycalculated with an increasing trunnion frictioncoefficient up to 0.7, as shown by the solid linein Figure 18. As shown by the solid symbols,

Figure 17: Static Stress and Tensile Forceon Folsom Dam Tainter Gate with GateSubmergence of 13.26 m and TrunnionFrictional Coefficient of 0.3 at Failure

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the calculated data increased linearly withincreasing trunnion friction coefficient, andhence the data are connected with the solidlines. If the trunnion friction coefficient is smallerthan about 0.1, the maximum tensile forceappears in the center diagonal member 3G-4E, while if it exceeds about 0.1, the maximumtensile force appears in the diagonal member3E-4D near the trunnion pin. The maximumtensile force in member 3E-4D exceeds thefailure load of 333 kN, only when the trunnionfriction coefficient reaches 0.42. This value isfar larger than the range of 0.22 to 0.28, whichwas estimated by USBR to be reasonablevalues for failure. In addition, this value of 0.42is still larger than the maximum possible frictioncoefficient 0.3, which was obtained by USBRfrom friction tests on the trunnion pin of thefailed gate.

The validated FEM model results indicatethat it was not possible for the Folsom Tainter

gate to fail only due to the excessive static loadresulting from the increased trunnion friction,as the Forensic Team concluded.

Supposing nonetheless that the trunnionfriction was the main cause of the Folsom gatefailure, logic then dictates that failure shouldhave occurred immediately upon opening thegate, that is, when the gate submergence depthwas near its maximum value and larger than13.26 m recorded at the time of the failure.The gate submergence depth for the closedgate was 14.02 m, since the Folsom gatefailed at the gate submergence depth of 13.26m and the discharge opening of 0.76 m.Therefore, upon incipient gate opening, thegate submergence depth was about 6% largerthan that at the moment of failure, thus resultingin a correspondingly larger hydrostaticpressure load and in a correspondingly largerinfluence of trunnion friction. Supposing thetrunnion friction to be the cause of gate failure,one must then accept that the accident shouldhave occurred at the beginning of the gateopening process.

Based on this reasoning, FEM calculationswere made again at the condition (the gatesubmergence depth of 14.02 m) when the gatestarted to open, to re-examine the maximumtensile forces on the diagonal members. Thecalculated results for the gate closed areshown by the open symbols and broken linesin Figure 17. The tensile forces are about 12%larger than those at the gate opening of 0.76m where failure occurred. However, the tensileforce for values of trunnion friction coefficientin the range of 0.22 to 0.28 is still significantlysmaller than the minimum failure load of 333kN. Moreover, not until a frictional coefficientvalue of 0.37 (which is still larger than the

Figure 18: Tensile Loads on Strut Bracewith Increasing Trunnion Friction

Coefficient Up to 0.7

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maximum frictional coefficient of 0.3 estimatedat the time of gate failure) does the maximumtensile force exceed the failure load of 333 kN.

The above results, in summary, suggest thatincreased trunnion friction alone could not havebeen the only contributing mechanism of theFolsom gate failure.

Predicted Closed Energy CycleAs a result of the field vibration tests and theFEM analysis, the gate has been shown topossess two typical and relevant naturalvibration modes, as illustrated in Figure 19.One fundamental vibration mode is the rigid-body rotational lifting vibration of the whole gateabout the trunnion pin, denoted by . Thesecond significant mode of vibration waslowest frequency streamwise bending of theskinplate (see the dotted lines), denoted by.This lowest frequency bending mode of theskinplate is essentially a streamwise rotationof the skinplate about a point in the upper halfof the skinplate. When the massive skinplate,

which constitutes 68% of the whole gate mass,performs a rotational vibration, a large inertiatorque due to this rotational vibration arises,which can then excite the whole gate to vibratearound the trunnion pin. When the whole gatevibrates around the trunnion pin, an additionalinertia torque also arises, which can produceadditional skinplate rotation. If the naturalfrequencies of these two vibration modescoincide, the streamwise rotational vibrationof the skinplate and the whole gate rotationalvibration around the trunnion pin can couplewith each other through the inertia torques,resulting in “coupled-mode vibration”.

Such a coupled-mode vibration will beaccompanied by a flow-rate variation beneaththe gate. The flow-rate variation producespressure fluctuations, which, under certainconditions, can lead to a violent self-excitedvibration, which is then called “coupled-modeself-excited vibration”. The closed-loop energycycle for coupled-mode self-excited vibrationis shown in Figure 20. Also shown in Figure20 is the “push-and-draw hydrodynamicpressure,” which is a significant factor in gatefailure. Details of this closed energy loop arediscussed in the following paragraphs.

If the streamwise vibration of the skinplateis triggered by some random excitation, theskinplate inertia torque excites the whole gateto vibrate around the trunnion pin. Thestreamwise skinplate vibration and the wholegate rotation produce a change in the gateopening, resulting in a flow-rate variationbeneath the gate, inducing a “flow-ratevariation pressure”. If the lower end of theskinplate behaves as a press-shut device (seeNaudascher and Rockwell, 1993 for adiscussion of press-open and press-shut

Figure 19: Two Predominant NaturalVibration Modes of Folsom Dam

Tainter Gate

338


devices), the flow-rate-variation pressure feedsenergy back to the skinplate streamwisevibration, thus amplifying its vibrationamplitude, as shown near the bottom of Figure20. As a result, when the lower end of theskinplate exhibits press-shut behavior, violentcoupled-mode self-excited vibration mayoccur.

The coupled inertia torque acting on theskinplate due to the whole gate vibrationaround the trunnion pin actually serves tostabilize the gate due to its phasing relative to

the skinplate motion. The negative excitationof this inertia torque is shown by the dashedline in Figure 20. However, the stabilizing effectof the inertia torque from the whole gatevibration is comparatively small. It is significantto note that this type of coupled-mode self-excited vibration can occur, even when theskinplate center is concentric with the trunnionpin.

Of special note in this closed energy-cycleis that the streamwise skinplate vibrationinduces a push-and-draw pressure. The

Figure 20: Closed Energy Cycle for Coupled-Mode Flow-Induced Vibration, PossibleTainter Gate Susceptibility

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amplitude of this push-and-draw pressurecan be more than 10 times that of the flow-rate variation pressure, and it could havepotentially caused the Folsom Dam gatefailure. The gate fails when the loading dueto the push-and-draw pressure exceeds thematerial strength.

It is also significant that the push-and-drawpressure does not consume energy from thevibrating gate. The push-and-draw pressurehas associated with it a large added masseffect. This large added mass of water mustbe accelerated with the gate as it undergoesits streamwise vibration. The added masslowers the relatively high in-air frequency of thestreamwise skinplate vibration. As a result,when the in-water streamwise skinplatevibration frequency approaches the frequencyof the in-water whole gate rigid body rotationalvibration, the system approaches resonance.Near resonance, the comparatively smallamplitude flow-rate variation pressure canspontaneously amplify the amplitude ofcoupled-mode vibration.

Model Investigations of Coupled-Mode Self-Excited VibrationThe model gate was constructed to confirmthe coupled-mode self-excited vibrationmechanism just postulated in the precedingsection. In addition, the data obtainedconcerning the coupled-mode vibration of thestreamwise skinplate bending rotation and thewhole gate rotation will be used subsequentlyto verify the theoretical analysis of thismechanism.

Figure 21 shows typical records of the self-excited vibration waveforms at F0 = 16.0,which were measured at the lower end of the

gate. The upper waveforms show thestreamwise vibration , and the lower showthe up-and-down vibration. It was previouslyaccepted that a gate with the skinplateconcentric with the trunnion center could neverundergo any self-excited vibrations. However,the model gate clearly undergoesspontaneous vibrations, exponentiallyincreasing in amplitude, and approaching asteady vibration with a constant amplitude.These test results clearly demonstrate thatTainter gates essentially possess an inherentdynamic instability, even if the skinplate is non-eccentric to the trunnion pin.

Diagrams on the right side in Figure 21show the vibration trajectories of the lower endof the skinplate. In all cases, the vibrationtrajectory exhibits the characteristiccounterclockwise behavior of a press-shutdevice. The gate opening is reduced when theskinplate moves in the downstream direction.In the case of Figure 21a, where the in-airwhole gate vibration takes the frequency a

of 4.42 Hz and the damping ratio a of 0.0020,the trajectory has a steep slope. Both theexcitation ratios, and , measured from theexponentially increasing vibration-waveforms,were 0.0052. As a increases to 4.96 Hz asin Figure 21b and to 5.29 Hz as in Figure 21c,the streamwise vibration increases inamplitude, and as a result, the trajectoryinclination becomes smaller. Accordingly, theexcitation ratio also increases to 0.007 inFigure 21b and to 0.0091 in Figure 21c.Furthermore, whena is 5.85 Hz, as in Figure21d, the up-and-down and streamwisevibrations have nearly equal amplitudes,resulting in a trajectory inclination of 45°. In thisconfiguration, the gate motion can most

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Figure 21: Self-Excited Vibration Waveforms and Trajectories for 1/31-Scaled Modelof the Folsom Dam Gate at F0 =16.0 (d0 = 420 mm, a = 12.3 Hz) for Four Different

In-Air Whole Gate Vibration Frequencies

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effectively extract energy from the fluctuatingflow, resulting in a relatively large excitationratio of 0.0066 in spite of the fairly large in-airdamping ratio of 0.0127.

Measured data for in-water vibrationfrequency are plotted in Figure 22a, wheredata from two test results for the basic Froudenumber F0 = 13.9 and 16.0 are shown. Theabscissa is the frequency ratio nw(=nw/a)of the streamwise in-water natural vibrationfrequencynw to the in-air natural vibrationfrequency around the trunnion,a, wherea

was varied from 2.6 Hz to 7.3 Hz, and nw

took a constant value of 6.14 Hz for a =10.6 Hz and 6.10 Hz for a = 12.3 Hz,respectively. When the frequency ratio nw islarger than 1.0, the in-water to in-air frequencyratio, w(≡ w/a), of the whole gatevibration around the trunnion becomes 1.0,and the frequency ratio w(≡ w/a) of thestreamwise in-water vibration to the in-airvibration around the trunnion also becomes1.0. Thus, one may conclude that the vibrationis synchronized with the whole gate naturalvibration (the lower of the two frequencies).In contrast, when nw is slightly less than 1.0,both frequency ratios, w and w, take onthe same value as the abscissa, nw. Thus,one may conclude that the vibration issynchronized with the skinplate streamwisenatural vibration (again, the lower of the twofrequencies). When nw becomes still smaller,the vibrations do not synchronize.

The dynamic similarity of the fluctuating flowdue to skinplate streamwise vibration issubject to the Froude number F, (different fromthe basic Froude number F0) defined by

wgdF 02 ...(3)

In the present model tests, the Froudenumber F ranged from 3.4 to 7.9 for F0 = 13.9and from 3.5 to 7.7 for F0 = 16.0.

The exponentially increasing or decreasingvibration waveforms suggest a significantcharacteristic of the self-excitation due to fluidmotion called the “fluid-excitation ratio”. If thein-air damping ratios a and a, due tomechanical and additional damping, weremeasured, one may calculate from anyspontaneous vibration waveforms, the fluid-excitation ratio for the whole gate vibration, f,and that for the streamwise vibration, f, bythe following expressions, respectively:

awa

cf I

C

2 ...(4a)

a

wa

cf I

C

2 ...(4b)

where and are the resultant excitationratios for the streamwise and whole gatevibrations.

In the present model gate, the mean valueof a was 0.003 from in-air free vibration testsof the whole gate around the trunnion. In thecase of model experiments, a fairly largedamping effect appears for the skinplatestreamwise vibration, due to leakage flowsfrom comparatively large clearances kept atboth sides of the skinplate. This kind ofadditional damping is intrinsic to a modelexperiment and cannot be avoided. For ameasure of the effect of these leakage flows,in-water free vibration tests of the skinplatewere made in detail to measure the dampingratio including this additional damping effect.As a result, the damping ratio of the skinplatestreamwise vibration, a, was 0.13 for F0 =16.0 and 0.11 for F0 = 13.9.

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Calculated values of the fluid-excitationratios based on experimental data areplotted in Figure 22b. Over the frequencyrange of nw, larger than 1.0, an instability thatis synchronized with the whole gate vibrationappears, and the maximum value of the fluid-excitation ratio f is 0.013. If nw becomesslightly smaller than 1.0, the instability issynchronized with the skinplate vibration,and the fluid-excitation ratio f rapidlyincreases, showing a maximum value of0.095 and then rapidly decreases with afurther decrease in nw.

CONCLUSIONBased on eyewitness accounts and on avalidated FEM analysis, it would appear verylikely that flow-induced vibrations must haveplayed a role in the failure of the Folsom DamTainter gate in 1995. At the time when theUSBR Forensics Team completed its report,the mechanism of coupled-mode self-excitedvibration of Tainter gates had not beenformulated. Other than the “eccentricity-instability” that had been developed as a resultof the Wachi Dam gate failure, there was noknown mechanism for the vibration of such amassive Tainter gate. With no significanteccentricity between the skinplate center andthe trunnion center, the Forensics Team ruledout the possibility of the “eccentricity instability,”and chose not to consider any other vibration-inducing mechanism. They ignored thetestimony of the operator who was on thecatwalk above the gate at the time of its failure,when based on an unvalidated FEM model,they concluded that the gate failed due to staticloading resulting from corrosion and increasetrunnion pin friction. The fallacy in thisconclusion, in addition to ignoring first handobservation, is that the failure should haveoccurred immediately upon opening the gate,not after having opened the gate 0.76 m.

The present study provides measuredmode shapes, frequencies and damping ratiosfor an original Folsom Dam gate, similar tothe failed gate, and for the a Folsom Dam gateafter a retrofitted reinforcement. A highprecision FEM model of the original FolsomDam gate yielded frequencies and modeshapes consistent with the measureddynamics of the original gate. This same FEMmodel showed that the loading on the diagonal

Figure 22: Vibration Frequency Ratio andFluid-Excitation Ratio for 1/31-Scaled

Model

0 0.5 1 1.5 2 2.50.00

0.05

0.10

f, f

f

f

■ ,□ : F0=13.9●,○ : F0=16.0

nw(≡nw/a)

0.095

0.013

■,● : f□,○ : f

0 0.5 1 1.5 2 2.50.0

0.5

1.0

1.5

2.0

2.5

w, w

□,○ : w (≡w/a)■,● : w (≡w/a)

■,□ : F0=13.9●,○ : F0=16.0

Synchronized toskinplate vib.

Synchronized towhole gate vib.

(a)

(b)

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member 3E-4D near the trunnion pin,identified as the point of failure by the USBR,did not exceed the minimum failure load asmeasured by the USBR. Even upon incipientgate opening, the loading, which was morethan 12% higher than with the gate open at0.76 m, did not exceed the minimum loadingfor failure. There is strong evidence that thegate did not fail due to static loading as wasconcluded by the USBR Forensics Team.

On the basis of the eyewitness testimonyand through a persistent fifteen-year researchprogram the mechanism of the coupled-modeself-excited dynamic instability of Tainter gateshas been formulated. The mechanism involvesthe coupling of two modes, in this case therigid-body rotation about the trunnion pin anda low frequency bending mode of the skinplate.When the two modal frequencies coalesce dueto the added mass effect of the waterenvironment, the inertial and hydrodynamicloadings create a situation whereby the motionof one mode serves as the forcing function forthe other mode. A 1/31 scaled model of theFolsom Dam gate was used to verify theexistence of this instability mechanism. Furthertesting has been undertaken and a theoreticalmodel of the mechanism has been developed.Future papers by the present authors will reporton these studies.

It is hoped that through this presentation ofthe data in this paper, an awareness of thecoupled-mode self-excited instabilitymechanism may be fostered among gatedesigners and in the general gate engineeringcommunity. The coupled-mode self-excitedinstability mechanism can lie dormant untilsome hard input causes an initiation of thevibration, which then quickly leads to gate

failure. An understanding of the mechanism isthe first step in designing gates that are notsusceptible to this instability mechanism.

Retrospectively, after identification of thecoupled-mode self-excited dynamic instability,reexamination of the Wachi gate failure showedthat the coupled-mode self-excitationmechanism is consistent with the gate vibrationin the Wachi failure. The unknown mechanismof gate vibration that was suspected at the timeof the Wachi failure (Ishii et al., 2008) has nowbeen plausibly identified. In the authors’ opinion,this dynamic instability mechanism also playeda major role in the Folsom Dam gate failure.The information contained in this paper has beenpreviously presented to the USBR. Theexistence of this coupled-mode dynamicinstability is significant and must be understoodby gate designers, owners, and operators toavoid future gate failures. In an effort to educatethe gate engineering community, the researchleading to the identification of this instabilitymechanism is documented in this paper. Theauthors’ hope is that others can understand thismechanism and the role it most likely played inthe Folsom Dam gate failure.

ACKNOWLEDGMENTFinancial support was provided by the Bureauof Reclamation, US Department of the Interior,and through a Science and Research Grantfrom the Japanese Ministry of Education. Theauthors would like to express their sincere andgreat thanks to Mr. Tom Aiken, Mr. CharlesHoward, Mr. Steve Herbst, Mr. John White, Mr.Steve Sherer and Mr. Bill Joy, for their kindsupport in undertaking this study of the FolsomDam gate failure. The authors would like toexpress their sincere thanks to Mr. Keiji

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Sakamoto and Mr. Shigeharu Mima, for theirkind support in conducting the impact hammerfield tests and also performing computercalculations for the modal analysis.

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RETROSPECTIVE CONSIDERATION OF A PLAUSIBLE VIBRATION ... · 314 int. j. mech. eng. & rob. res. 2014...

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