trunnion design

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INSPECTION OF TRUNNION RODS AT GREENUP DAM

Mark A Cesare1

J. Darrin Holt, P. E.2

ABSTRACT

Post-tensioned trunnion anchor rods are widely used as the structural support of Tainter

gates at many dams throughout the United States. There may be hundreds of these rodsthat are used for transferring the Tainter gate forces to the monolithic structure of the

dam. Previous failures have occurred in some rods whereby their post-tensioned stress

levels have unknowingly released. When this occurs a rod may no longer function asoriginally intended. Identifying rods that have experienced such failure is a major

problem for dam operators, a safety issue, and the subject of current research.

The Greenup Lock and Dam is located north of the town of Greenup, KY. Since its

original construction, about 950 trunnion rod lift-off tests have been preformed tomanually determine their in-situ states of tension. In 2010, a newly developed

nondestructive test (NDT) utilizing Dispersive Wave Propagation was applied to 104 of Greenup’s trunnion rods to collect data used for making preliminary computation of their

in-situ loads.

This paper describes the three-fold approach to nondestructively estimating tension in

trunnion rods. These approaches are based upon historical data, Dispersive Wave

Propagation and vibration studies, and the use of a limited number of lift-off tests for calibration and validation. Discussions will include a derivation of the mathematical

model, a description of the statistical approaches applied, and a discussion of howcomputational estimates are made for a rod’s tension along with the statistical

uncertainties.

INTRODUCTION

Construction of the dam section of the Greenup project was started in 1958 and the pool

reached normal elevation by 1962. The project has 9 Tainter style gates supported by 10 piers. Each trunnion anchorage is held in place by a trunnion beam with two sets of 102

anchor rods. The rods are 1 1/8” (nominal) in diameter and were tensioned to a load of

100,000 lb at the time of construction. A pier cross section is shown in Figure 1. Theanchor rods are about 75 feet long and at about a 20 degree angle to horizontal. The

embedded ends of the rods are, according to the as built drawing, in an open gallery. The

rods pass through tubes to hold protective grease and finally through a steel trunniongirder and a heavy steel plate. Approximately 24 inches of rod is left cantilevered past the

trunnion girder. This is used to grip the rod for tensioning and testing. In this design the

rods are locked off using a chock insert directly in the face plate, so there is no grip nut.

The rods were placed in steel tubes and the annular space filled with NO-OX-ID grease.

1 Member USSD, 2730 Rowland Road, Raleigh NC 27616, [email protected] Member USSD, 2730 Rowland Road, Raleigh NC 27616, [email protected].



The cantilevered exposed length was also coated in the same grease for protection from

the environment.

Figure 1. Pier Cross Section

HISTORICAL INSPECTION DATA

We are quite fortunate to have records of almost 1000 lift-off tests preformed at theGreenup project from 1968 to 1994. Each pier and rod group has had rods tested at some

point. However, there remain some rods that have never been tested, while others rods

were tested more than once over the years. Table 1 summarizes the lift-off data found.

Small batches of rods have been tested about every 6 years. While large batches of rodswere tested in 1981 and 1886. Out of the approximately 1000 lift-off tests only 4 actual

liftoffs are recorded. The inspection records also report qualitative assessment of the rods

such as “Rod Bent”, “Rod Pitted”, and “Rod Missing” but offer no further details.

Table 1. Historical Liftoff Data

Year Number of Rods

Tested

Target Test Loads Comments

1968 20 100 kips One lift-off at 82 kips

another at 104 kips

1969 20 100 kips No lift-offs


1976 60 84 – 88 kips No lift-offs

1981 275 87 kips One liftoff of 80 kips

Some rods reported bent


1986 483 86 kips No lift-offsSome rods reported bent

2 Innovative Dam and Levee Design and Construction



others reported pitted

1994 52 Visual only One rod missing

2010 104 Dispersive wave tests

2012? 20820

Dispersive wave testsLiftoff tests

0

100

200

300

400

500

600

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

Load (kips)

C o u n t

Tests from 1976, 1981, 1983, & 1986

Tests from 1986, 1969, 1970

l i f t o f f a t 8 0 k i p s , P e i r 5 , K y s i d e , 1 - 4

( 1 9 8 1 )

l i f t o f f a t 8 2 k i p s , P e i r 8 , K y s i d e , 3 - 1 3 ( 1 9 6 8 )

l i f t o f f a t 1 0 2 k i p s , P e i r 7 , K y s i d e , 2 - 1 ( 1 9 8 1 )

l i f t o f f a t 1 0 4 k i p s , P e i r 1 , K y s i d e , 4

- 1 ( 1 9 6 8 )

Figure 2. Historical Lift-off data grouped by maximum/liftoff load

SAFETY AND ACCESSING THE RODS

There is no roadway on the Greenup Dam. All equipment must be hand carried across the

dam by a walkway and down a series of ladders on the downstream face of the pier. A

custom built steel work platform was provided. As shown in Figure 3, the platform provides an area below the rods for one crew member to access the rods. Other crew

members must stay on the lowest concrete landing just above the rods. The DispersiveWave Propagation test requires only one crew member near the rods to measure the rod’s

length and diameter, attach accelerometers using their magnetic bases and impact the sideof the rod with small hammer. The remainder of the crew can operate the data acquisition

system and record data from the landing above. Because of the large amount of strain

energy stored in the rods compared to their mass it is extremely important to minimizethe crew’s exposure to the rods. At no time were the crew permitted to stand directly in

front of a tensioned rod.




Figure 3. Work Platform and concrete landing

THEORETICAL MODEL OF VIBRATION

A symbolic diagram of the rod is shown in Figure 4.

Cantilever

No Tension Tension

Embedded

Figure 4. Beam model of Trunnion Rod

The support condition at the face of the trunnion girder is modeled as a pin with anadditional rotational spring. The spring accounts for the local deformation of the trunnion

girder’s flange plate and the gripping wedge. The dead end is show as a fixed connection

but could be modeled as a pin or stiffened pin. The preferred method of analysis is a

modal finite element model using beam elements. A model was developed that includesterms in the stiffness and mass matrices to account for the following effects: bending and




shear deformation, lateral and rotational inertia, and stiffening due to tension. The

formulation of the stiffness and mass matrices used here are given in the referenced works by Przemieniecki, and Weaver and Johnston. No damping was used in the model.

Approximately 400 elements were used to model the trunnion rod in two sections: the

cantilever and the embedded portion. The tension based stiffness modification was only

applied to the embedded portion of the rod since the cantilever is unstressed. Once theglobal stiffness matrix was assembled additional rotational stiffness was added at the pin

connections to its rotational degrees of freedom. The frequencies of the vibration modes

were found by solution of the generalized Eigen-value problem:

02 x M xK ff ff

Where,

ff K is the stiffness matrix reduced to the free degrees-of-freedom,

ff M is the mass matrix reduced to the free degrees-of-freedom,

x is a vector of modal displacements, and is the frequency of vibration.

The solution to the Eigen-value problem yield a set of ’s which stratify the equation.

These are the frequencies of vibration of the various modes. Each mode has a

corresponding x vector which is the shape of that mode. The mode shapes were

examined to determine if their motion is predominantly in the cantilever or embedded portion of the rod. We only expect to be able to observe the mode with dominant motion

the cantilever.

The model results for the broken rod are shown in Figure 5 where they are compared to

recorded spectral response. The broken rod model has no tension and no additionalstiffness at the pin connections. The calculated modes at 162 Hz and 1216 Hz are

dominated by motion of the interior portion of the rod and were expected to be observed in the exterior rod (the only part we can attach gages.) The calculated modes at 41 Hz,

309 Hz, 835 Hz and 1735 Hz can be seen to correspond to observed modes.




Figure 5. Comparison of calculated and observed vibration modes for a broken rod.

COLLECTING DATA FROM THE RODS

For the Greenup Dam 104 rods on Pier 2 were inspected. As expected, the analytic model

has shown that the vibration modes are quite sensitive to the length of the cantilever and to less extent to the diameter of the rod. Each of these quantities was carefully measured

by field crews. A histogram of the lengths of all rods is presented in Figure 6. A large

portion of the rods are between 24 ¼ and 25 inches and a few outlying lengths cover arange from 23 ¾ to 25 ¼. The rod diameters were measured; a histogram of the recorded

diameters is shown in Figure 7. Most of the rods are between 1.140 and 1.145 inches with

a total range of 1.130 to 1.155 inches.

.




0

5

10

15

20

25

30

35

23.5 23.75 24 24.25 24.5 24.75 25 25.25 25.5

Length (in)

C o u n t

Figure 6. Histogram of Rod Length

0

5

10

15

20

25

30

35

40

45

50

1.1200 1.1250 1.1300 1.1350 1.1400 1.1450 1.1500 1.1550 1.1600

Diameter (in)

C o u n t

Figure 7. Histogram of Rod Diameter




An accelerometer was attached to each rod and data was collected as the rods were struck

from the side with several different hammers. A total of 12 data sets were recorded for each rod. Each time history recorded 21,000 samples at 25,000 Hz for about 1 second

samples. For each rod a set of frequencies was observed. The lowest or fundamental

cantilever modes were observed between 48 Hz and 54 Hz for the apparently intact rods

and 43 Hz for the broken rod.

ANALSIS OF THE DATA

Since the length of the rods has the greatest effect on the frequency the first step is to plot

the length vs. lowest observed frequencies (f1). Figure 8 shows the general trend of

decreasing frequency with increasing length. The failed rod that was left in place is aclear outlier with frequency much lower than other rods of similar length. There is also

some scatter in the remaining points. All the rods are the same length internally and have

the same fasteners at each end. Therefore, we attribute these differences in frequency to

differences in tension either directly or as a secondary effect were the tension make the

griping system stiffer and increasing the frequencies. Similar patterns are seen in thesecond and third lowest observed frequencies.

40

42

44

46

48

50

52

54

56

23.60 23.80 24.00 24.20 24.40 24.60 24.80 25.00 25.20 25.40

Length (in)

F r e q u e n c y

( H z )

Figure 8. Comparing length and Frequency

The solid line in Figure 8 was found by the theoretical model. It clearly shows that

although there is some systematic bias in the frequency result the effects of length arewell modeled. For this line only the length was varied; the diameter was held constant at




the average and the tension held at 100K. The lower observed frequency is partially due

the mass of the gauge which is slowing the vibration.

The finite element model was then used to generate Figure 9 which plots tension in the

rod with frequency at the average length and diameter. At zero tension the frequency

does not reach zero as it would for a string since the rod is acting purely as a beam withsome bending stiffness. Notice that the curve is not as smooth as might be expected. The

slight undulations in this curve are due to the interaction of the vibration mode of the

interior and cantilever part of the rod.

Theoretical f1

44.0000

45.0000

46.0000

47.0000

48.0000

49.0000

50.0000

51.0000

52.0000

53.0000

54.0000

0 20000 40000 60000 80000 100000 120000

Tension (lbs)

F r e q u e n c y ( H z )

f1

Figure 9. Comparing Tension and Frequency.

Finally, we combine the 11 points of Figure 9 with the frequencies of Figure 8 and we

can interpolate the length and tension to effectively create contours of tension in thelength vs. frequency space. Two sets of finite element analysis were done to create the

chart in Figure 12. One obtained by varying the length create the previously mentioned

line in Figure 8 and the other varying the tension to create the points in Figure 9. It is

assumed that contour lines for other tensions remain parallel to the 100 kips line. Thisallows simple piecewise but linear interpolation to estimate the tension.




Tension/Frequency Model

40

42

44

46

48

50

52

54

56

58

23.40 23.60 23.80 24.00 24.20 24.40 24.60 24.80 25.00 25.20 25.40 25.60

Length (in)

F r e q u e n c y ( H z )

T =0 kips

T =10 kips

T =20 kips

T =30 kips

T =40 kips

T =50 kips T =60 kips

T =70 kips T =80 kips

T =90 kips

T =100 kips

Figure 10. Interpolating Tension from Length and Frequency

The effects of the diameter were found to be statistically not significant. It was found thatthe variance in the response due to the variance in diameter is small and of the same order

of magnitude as the underlying error.

Ideally at this point liftoff tests would be done to calibrate the theoretical model to actualdata from this particular dam. The Corps of Engineers is planning on liftoff testing about

20 rods and two different piers at Greenup. At that point we will be able to create acalibrated model. As a preliminary calibration of the data we adjusted the 100K solid line

to sit just at the top of the cloud of points in the length versus frequency diagram. We

believe this to be reasonable since historically only 2 of the 1000 previous lift-off tests

showed load greater than 100K and this was the initial tension of the rods.Figure 10 shows the estimated rod tensions grouped by load in a histogram. These

estimates may be slightly conservative (under estimating tension) since we would expect

more lift-offs than the 2 that were observed at the 86 kip load.





Weaver, W., P. Johnston, “Finite Element for Structural Analysis,” Prentice-Hall, New

Jersey

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