2.2 Theoretical Analysis of Folsom Gate Failure and Its Quantitative Scenario
2.2.1 Summary
The Tainter gates installed at the Folsom Dam in California have circular-arc skinplates with a height
of 15.5 m and a radius of 14.33 m, as shown in Figure 2-9 (a).
The span is 12.8 m and the gross mass is 87
tons. One of these large-sized Tainter gates failed during operation on July 17, 1995.
failure occurred in Japan in July 1967, as documented by Ishii et al. (1980).
A similar Tainter gate
A self-excited vibration
mechanism which could have caused the gate failure was formulated and studied by Ishii & Imaichi (1977),
Ishii et al (1977) and Ishii & Naudascher (1984, 1992). Immediately after the Folsom Dam gate failure, Ishii
(1995a, 1995b, 1997) investigated the cause of the failure.
The results from these studies suggested that the
gate may have experienced a new type of violent self-excited vibration that was coupled with the flow rate
variation in the reservoir and that was different from the suggested mechanism for the gate failure in Japan.
In the Folsom Dam failure analysis, experimental modal analysis of a geometrically similar Tainter
gate was carried out primarily to establish the major vibration modes of the gate [see Ishii (1995); Anami &
Ishii (1998a)].
As a result, the rotational bending vibration mode of the skinplate, shown by the dashed lines
in Figure 2-9 (a) and by the red and black lines in Figure 2-10, and the whole gate vibration around the trunnion
pin, shown by the solid line in Figure 2-9 (a) and by the red and black lines in Figure 2-11, were identified.
Vibration in the former mode pushes and draws the water on the upstream reservoir, thus inducing an
extremely large hydrodynamic pressure in the reservoir. Such excessive so-called “push-and-draw pressures”
result in a significant added mass effect, which drastically lowers the vibration frequency of the skinplate from
that found in air [see Anami & Ishii (1998b)]. The coupling of this low frequency streamwise vibration of the
skinplate with the whole gate rotary mode about the trunnion pin can induce a violent self-excited vibration
with the aid of the so-called “flow-rate-variation pressure” due to the flow-rate changes under the gate [see
Anami & Ishii (1999, 2000, 2001, 2003)], and hence could destroy the gate, as detailed in Anami (2002).
In order to address the excessive push-and-draw pressure and the flow-rate-variation pressure, a
theoretical analysis was undertaken by Anami, Ishii & Yamasaki (2000), Anami (2002) and Ishii (1992) for a
reservoir flow field simplified to the extent that theoretical calculations could be carried out, where the
(a)
(b)
Figure 2-9: An 87-ton Tainter gate from the Folsom Dam in California, showing two predominant natural
vibration modes, over the curved dam crest: (a) Side view; (b) Simplified two-dimensional flow field for
theoretical calculations.
10
circular-arc skinplate was replaced by a vertical flat rigid weir plate undergoing streamwise rotational vibration,
and the curved dam crest against which the gate seals was modeled as a horizontal flat bed surface with a
vertical step down at the gate position to allow for the greater depth of the reservoir, as shown in Figure 2-9 (b).
Such modeling for the curved skinplate undergoing bending vibration and the curved dam crest is quite crude
but absolutely necessary to accomplish an empirical evaluation of the excessive push-and-draw hydrodynamic
pressure for the actual reservoir flow field of the Tainter gate.
As shown in Figure 2-9 (a), the circular-arc skinplate, inclined 9.5 degrees towards the downstream
side, is installed over the dam crest which is described by the Creager curve [see Creager & Justin (1927)] and
which is preceded by a vertical step 6.56 m-upstream of the skinplate. In order to evaluate the excessive
push-and-draw pressure and the flow-rate-variation pressure for such a practical reservoir flow field, the
empirical method using the theoretical analysis for the simplified flow field model was adopted.
An
approximately 1/38-scale model of Folsom Dam Tainter gate was installed over the same scaled model of the
curved dam crest, and the skinplate was forced to undergo streamwise rotational vibration and rotary vibration
around the trunnion pin. The push-and-draw hydrodynamic pressure and the flow-rate-variation pressure on
the skinplate were measured for various values of both the distance from the skinplate to the vertical step and
the downstream inclination angle. Measured data were compared with corresponding theoretical results
calculated for the simplified flow field model, to determine empirical values of two pressure correction factors,
the so-called “pressure coefficient”, “instantaneous flow-rate coefficient” and “reduced reservoir depth” [refer
to Anami (2002) and Anami, Ishii, Knisely & Oku (2005)].
Figure 2-10: Skinplate streamwise bending vibration mode of Folsom dam Tainter-gate.
Figure 2-11: Whole gate vibration mode around the trunnion pin of Folsom dam Tainter-gate.
11
Applying the developed theoretical methods to evaluate the vibration-induced hydrodynamic
pressures, the failure analysis of Folsom dam Tainter-gate can be made. First, the excessive push-and-draw
pressure can be integrated over the wetted area of the skinplate to determine the resultant hydrodynamic load
on the failed Folsom Dam Tainter gate. The resultant force is non-dimensionalized by the static skinplate load.
Secondly, the dynamic instability of the failed Tainter gate can be theoretically examined. Thirdly, the
stability criteria for preventing such a hydrodynamic instability can be presented, and in addition the required
amplitude of the initial displacement trigger needed to overcome the damping effects due to mechanical friction
and thereby trigger violent self-excited vibration can be addressed. Finally, an FEM analysis of the structure
can be presented to determine the small amplitude vibration and the associated excessive hydrodynamic
pressure load on the gate at the moment of incipient failure. Refer to Anami (2002) for details.
2.2.2 Resultant Hydrodynamic Load on gate
The flow-rate variation pressure is about 1/10-smaller in amplitude, compared with the
push-and-draw pressure, and hence the resultant hydrodynamic load on gate can be calculated by the
push-and-draw pressure.
The skinplate of the failed Tainter gate at Folsom dam possessed a 1/2-wave length
bending vibration mode in the spanwise direction. Integration of the push-and-draw pressure over the wetted
area of the skinplate with this bending mode yields the hydrodynamic load on gate, WbL:
WbL 

2
0
 pW0 R S 0 gd 0  p b 0 dy
(2-1)
1
where pb0 is the dimensionless amplitude of push-and-draw pressure, given by expression (3-61) in Part II of
Anami (2002), which is a function of the reduced vertical axis-y along the skinplate, the Froude number F, the
reduced reservoir depth ratio * and the reduced skinplate-rotation-center height rs, refined by
y  Y / d0 ,
F
d0
 w ,
g
 *  dr / d0
, rs  RS / d 0
(2-2)
d0 represents the skinplate submergence depth and W0 the skinplate spanwise length. Rs0 represents the
skinplate streamwise vibration amplitude at the spanwise center along its bottom.
The static hydraulic load WS, given by WS=gW0d0/2, can be used to normalize the resultant
hydrodynamic load as follows:
WbL  Ws
  p RS 0
 1
Ws
4 d0

0
1
pb 0dy
(2-3)
The failed Tainter gate at the Folsom Dam had a skinplate inclination angle s=9.5o and a reduced
overhang distance (=L0/d0) of 0.5, where both of the pressure coefficient p and the reduced reservoir depth
ratio* take on an empirical value of 1.0. Therewith, the reduced resultant hydrodynamic load on the gate can
be calculated from equation (2-3), as shown by the dashed lines in Figure 2-12 as a function of the skinplate
centerline vibration amplitude. In this figure, the auxiliary parameter is the Froude number F, unknown for
the failed gate at this moment.
12
–3
0.9×10
4.0
3.0
2.0
(→Streamwise vib.
amplitude 11.9 mm)
Reduced resultant hydrodynamic load
(WbL+Ws)/Ws
5.0
F=80
Folsom gate
70
60
47.2
50
40
30
1.6
20
1.0
inducing failure load 333kN
on 4–bolts for a diagonal member
of Folsom dam Tainter–gate
0
1
2
3
4
5
–3
Reduced vib. amplitude R S0/d0［×10 ］
Figure 2-12: Resultant hydrodynamic load on Tainter gate at Folsom Dam, with the pressure coefficient
p=1.0 and the reduced reservoir depth *=1.0 for the skinplate inclination angle s=9.5o and the
overhang distance =0.5.
As shown in Figure 2-9(a), the Folsom Dam submergence depth d0 was 13.26 m. Hence, for
example, an abscissa value in Figure 2-12 of 1.0×10-3 would represents a skinplate vibration amplitude of 13.26
mm at the spanwise center along its bottom.
At F=50, for example, this small vibration would then induce a
large resultant hydrodynamic load about 1.7 times the static load.
2.2.3 Dynamic Instability of Folsom Dam Tainter-Gate and Its Trigger Displacement
Major specifications of the failed Tainter gate at Folsom Dam are shown in Table 2-4, in which
mechanical constants such as the in-air vibration frequencies a, in-air damping ratios a and rotation center
height RS have been determined by field vibration tests on a remaining gate of the same design as the failed
gate [see Ishii (1995a, 1995b) for details]. The gate was installed over a curved dam crest with a 9.5 o
inclination angle and a geometrical press-open angle of 8.5o.
crest tangent and the normal to the skinplate secant.)
(The press-open angle is the angle between the
The average gate opening B was 0.762m, or 5.7% of the
gate submergence d0 of 13.26m. Therefore, the assumption of a small gate opening is justified. Moments of
inertia of the whole gate and skinplate, I, were calculated numerically. Nondimensional parameters necessary
for theoretical calculations of the dynamic instability are presented in Table 2-5. The reduced height of the
rotation center, rs, defined by Equation (2-2), takes on a value of 0.72. The moment-of-inertia ratio of the
Table 2-4. Major specifications of the failed Tainter gate.
Flow
field
－
B
d0
s
14.02 m
0.762 m
13.26 m
9.5°
aq
Iq 1.31×107 kgm2
6
2
I 1.61×10 kgm Mechanical a
constants aq
RS
9.6 m
s0
a
8.5°
6.88 Hz
26.9 Hz
0.012
0.002
Table 2-5. Nondimensional parameters for theoretical calculations.
rs
a
Mechanical
I
constants a
F0
0.72
0.123
129
196
Correction
factors
for flow
13
cf
p
*
0.72
1.0
1.0
: Caluculated
: Measured
Model gate
Tainter gate "A"
Tainter gate "B"
0.56 (a=26.6; r s=0.43)
0.46 (a=3.2; r s=0.18)
0.5
Folsom
gate
0
0
Folsom gate
50
100
150
Basic Froude number F0
185
196
0.24 (a=129; rs=0.72)
133
137
0.31 (a=101; rs=0.55)
16
Frequency ratio wa
1
200
Figure 2-13: In-water to in-air vibration frequency ratio of the skinplate
streamwise vibration of Tainter gates.
skinplate to the whole gate, aI(≡I/Iq), is comparatively small at 0.123, which affects the coupling level of the
streamwise vibration of the skinplate with the rotational vibration about the trunnion pin. The water-to-gate
mass ratio a (≡d02W0/(I/RS2)) representative of a lightweight design, takes on a large value of 129, which
results in a significantly large in-water reduction of the streamwise vibration frequency relative to the skinplate
in-air value. The basic Froude number F0 defined by F0  d 0 / g  a takes on a large value of 196. The
instantaneous flow-rate coefficient cf was measured by model experiments to be 0.72 [see Anami, Ishii &
Takano (2003) for details].
The dynamic instability of the failed Tainter gate at Folsom Dam can be theoretically calculated [see
Anami(2002) and Anami & Ishii(2003) for details]. As part of such an analysis, the in-water to in-air frequency
ratio of the skinplate streamwise vibration, w/a (=F/F0) must be determined and the result is shown in
Figure 2-13. In this figure, the abscissa is the basic Froude number F0 and calculated results are shown by
solid and dotted lines. Small ordinate values in this figure represent significant added mass effects for the
in-water gate vibration.
Experimental test results are also plotted, showing good agreement with the
corresponding theoretical results [see Anami, Ishii & Takano (2004) for model gate tests, Anami, Ishii &
Knisely (2004b) for full-scaled Tainter-gate “A”, and Anami, Ishii & Knisely (2004a) for full-scaled
Tainter-gate “B”]. The frequency ratio w/a of the Folsom gate is 0.24 at F0=196. This result suggests
F=47.2 for the failed Tainter gate at Folsom dam.
Subsequently, the dynamic stability criterion curve for F=47.2 of the failed gate can be calculated and
is given by the solid line in Figure 2-14 (a), where the ordinate represents the required damping ratios c and
cq, necessary for complete dynamic stability against the fluid excitation. The abscissa is the frequency ratio
nw, which is defined by the ratio of the skinplate in-water inherent vibration frequency  nw to the whole gate
in-air rotational vibration frequency,aq.
nw=1.0 represents the resonance state of the coupled vibration. If
nw is smaller than 1.0, the coupled vibration is synchronized with the skinplate streamwise vibration with a
frequency of nw = 6.46 Hz (a = 26.9 Hz times 0.24), and an intense dynamic instability is expected since
the required stability damping ratio c shows a large peak with values beyond the expected structural damping.
For the failed Folsom dam gate, the whole gate vibration frequency about the trunnion pin is aq = 6.88 Hz (see
Table 2-4), with a corresponding value of nw = 0.94. For this frequency ratio the streamwise vibration damping
ratio a=0.002 (see Table 2-4) is plotted by the black dot. It is far smaller than the required damping ratio c
14
=0.051 for stability, resulting in an intense dynamic instability.
The difference (c -a) = 0.049 shows the excitation ratio representing the level of dynamic
instability, which is plotted over the reduced gate opening b (≡ B/d0) = 0.057 in Figure 2-14(b). If the gate
opening B decreases, the submergence depth d0 increases, because the upstream water depth is maintained at
14.02m. As a result, the skinplate streamwise vibration frequency nw decreases moving the operating point in
Figure 2-14a toward the left, while the stability damping ratio curve is essentially unaffected. Therefore, the
level of dynamic instability decreases with decreasing the gate opening, as shown in Figure 2-14 (b). In other
words, the excitation ratio was 0.009 with a nearly closed gate, but increased about five fold with increasing the
gate opening.
The large peak in c for nw <1.0 was calculated for aq= 0.012. This in-air damping ratio is due only
to the structural viscous damping for the whole gate vibration around the trunnion pin. It does not include the
expected significant damping effect due to Coulomb friction. Significant Coulomb friction is expected at the
trunnion pin and also at the skinplate rubber seals. Timoshenko, et al. (1974), show the damping effect due to
the Coulomb friction can be represented by an equivalent viscous damping ratio e, given by
e 
2 A0
,

 A
(2-4)
where A0 represents the static displacement of the hoisting chains and A the vibration amplitude of the skinplate
due to the whole gate rotary mode around the trunnion pin. As is evident form Equation (2-4), if the gate does
not oscillate, e becomes infinitely large and then no vibration is possible. On the other hand, if some
triggering level of displacement is initially given to the gate, the vibration can start, because e decreases with
increasing vibration amplitude. Therefore, according to this model, no Tainter gate is expected to undergo
spontaneous self-sustained vibrations from rest, even if it is dynamically unstable. However, any Tainter gate
with a dynamic instability can begin vibrating as soon as some threshold level of triggering displacement is
exceeded. Once vibration arises, e further decreases as the amplitude increases due to fluid excitation. As a
result, the excitation ratio increases monotonically, approaching its maximum value of 0.049 given in Figure
●
0.06
Synchronized with
whole gate vib.
Excitation ratio ( c–a)
Synchronized with
streamwise vib.
: Folsom gate datum
on its incipient failure
Stable
Stable
0.05
0.051
Unstable
aq
cq
aq
aqe
0
0
0.002
a=0.002
0.05
0.049
0.04
0.03
0.02
0.01
0
0
0.009
0.057 on failure
c
0.1
0.05
Reduce gate opening b( ≡B/d0)
0.5
1
1.5
2
Vibration frequency ratio nw≡nw/aq)
(a) Dynamic stability criterion curve
(b) Possible dynamic instability level
(B=0.762m; d0=13.26m)
vs. gate opening
Figure 2-14: Dynamic instability at F=47.2 for the failed Tainter gate at the Folsom Dam.
0.94
Stability damping ratio c, cq
2-14b.
15
Initial displacement trigger A [mm]
15
Unstable
10
t=0.3
8.5
t=0.15
5
5.0
Stable
0
0
0.5
1
Frictional coefficient at skinplate side seal, s
Figure 2-15: Threshold displacement trigger inducing self-excited vibration of Folsom gate,
supposing t=0.15 and 0.30 at the trunnion pin.
The task at hand now is how to estimate the threshold level of the initial displacement trigger Ai. If
the value of parameter aq is increased from it measured value of 0.012 by e, the stability damping ratio curve
c for nw < 1.0 in Figure 2-14 (a) would decrease. Trial and error calculations of dynamic instability were
made for various values of aq (= 0.012+e). With aq = 0.08, the neutral stability curve passes through the
Folsom gate data point, as shown by dotted curve in Figure 2-14a. Thus, for neutral stability, e takes on a
value of 0.068. If e is smaller than this value, or in terms of Ai, if
Ai 
2 A0 2 A0

 
  e  0.068
(2-5)
then the Folsom gate is unstable and subject to intense self-sustained vibration. The threshold value of the
initial displacement trigger Ai was calculated for the trunnion frictional coefficient t of 0.3 [possible maximum
value, suggested by USBR (1996)] and 0.15 (a value for a new trunnion), as shown in Figure 2-15. Here the
abscissa is the frictional coefficient at the skinplate side seals, s. For example, if s is assumed to be 0.5, a
small upward skinplate displacement of 8.5mm would be sufficient to trigger an intense dynamic instability of
the Folsom gate.
2.2.4 Impossible Failure by Static Loads
A precise FEM model of the failed Folsom gate was constructed on a computer (Epson VA-639E) by
a powerful pre- and post-processing tool (FEMAP Ver.5.00), where the gate was divided into 23,124 elements.
A FEM analysis software (MSC/N4W NASTRAN for Windows V3, the MacNeal-Schwendler Corporation)
computed the static stress and deformation at d0=13.26m, as shown in Figure 2-16(a), where t was given by its
possible maximum value of 0.3. The maximum tensile force of 240 kN appeared at the brace strut connecting
Points 3E and 4D, but this value is smaller than the minimal failure load 333 kN for the 4 bolts connecting the
brace to the strut. This result clearly suggests that the gate could not possibly fail due only to the static
hydraulic load and the increased trunnion friction.
More detailed calculations of the tensile loads with increasing t are presented in Figure 2-16 (b),
where the solid lines are for d0=13.26 m on failure (B=0.76m) and the dotted ones for d0=14.02 m at closed
16
(a) Static stress and deformation at t =0.3.
(b) Tensile load on strut braces, with increasing
trunnion frictional coefficient t up to 0.7.
Figure 2-16: Results of FEM analysis for the failed Tainter gate at Folsom dam, with gate
submergence depth d0 of 13.26m (B=0.76m) and 14.02m (B=0m).
gate opening (B=0 m). The ordinate represents the reduced tensile force relative to the failure load 333 kN of
the 4 bolts. Each dotted line (closed condition) is consistently about 10% larger than the corresponding solid
line (condition at failure). Therefore, if as has been suggested by some reports [USBR (1996)], the gate failure
was caused by the static hydraulic load coupled with increased trunnion friction, the gate should have failed not
at a gate opening B of 0.76 m, but rather immediately after the gate opening was initiated. Moreover, none of
calculated data ever exceeded the value of 1.0, even with the maximum possible value of 0.3 for t. Hence,
the gate would not have failed, not even upon incipient gate opening.
These FEM analysis results are significant because they suggest that the dynamic instability in this
present study must be seriously considered as the source of the Folsom gate failure.
2.2.5 Estimated Amplitude of Vibration at Incipient Failure
The present theoretical analysis, with the aid of an FEM analysis, can estimate the amplitude of the
skinplate streamwise vibration with incipient failure. The resultant hydrodynamic load induces a tensile force
acting on the most dangerous strut brace (3E-4D), as shown in Figure 2-17. Assuming the hydrodynamic
force to be an equivalent static force, an FEM analysis was undertaken. As a result, the minimal tensile failure
load of 333 kN for the 4 bolts was exceeded if the reduced resultant hydrodynamic load exceeded 1.6.
Following the solid line for F=42.7 in Figure 2-12, one sees that the reduced load of 1.6 is induced by a reduced
vibration amplitude of 0.9  10-3 which corresponds to 11.9 mm.
It is concluded here that the Folsom gate was initiated when the streamwise vibration of the
bottom end of the skinplate at its spanwise center reached only 11.9 mm, smaller than the static deformation of
19mm due to the static hydraulic load.
17
Figure 2-17: Tensile force acting on the strut brace 3E-4D vs. reduced resultant hydrodynamic force
acting on skinplate of Folsom dam Tainter-gate.
2.2.6 Quantitative Scenario of Failure
Tom Aiken, dam manager for the USBR, mentioned in a TV press conference “James Taylor, gate
operator, noticed something bind up a little bit and the gate vibrated and suddenly gave way…” [see Anami &
Ishii (2003) for detail]. The “Something bind[ing] up” must have triggered the displacement of the failed gate.
A likely scenario is as follows:
Step 1: The gate was cocked in the gate bay, because of the increased rust of the right-hand side trunnion
pin.
Step 2: The “binding up” was suddenly released as the hoist chain exerted increased upward force
resulting in a sudden upward jerking motion of the skinplate.
Step 3: This upward displacement exceeded 8.5 mm, and with the gate opening of 0.76 m, and the intense
dynamic instability was initiated, leading to violent coupled-mode self-excited vibration.
Step 4: The vibration ended when the streamwise vibration of the bottom center of the skinplate reached
11.9 mm in amplitude. The gate started to fail.
As deduced here, gates with essential dynamic instability due to fluid excitation do not
necessarily experience spontaneous vibrations.
They are, however, dangerous to use without requisite
attention to potential displacement triggers, because the gate’s essential instability can be suddenly initiated by
any displacement above the threshold value, resulting in potentially fatal failure, in the same manner as the
Folsom gate.
Field tests should be made to identify any gate with an essential instability, and promote its
completely stable long-term operation, before another gate failure occurs.
18