Method for determining tension in a rod

A method of determining the tension in an elongated rod is described. The method includes impacting the rod and measuring the vibration response of the rod. Further, the method entails generating a theoretical vibration response of the rod based in part at least on the structural properties of the rod. Thereafter, the method entails adjusting the theoretical vibration response to cause the theoretical vibration response to conform with at least a portion of the measured vibration response. Based on the adjusted theoretical vibration response, the tension of the rod is determined.

FIELD OF THE INVENTION

The present invention relates to evaluating the integrity of tension members in concrete structures. More particularly, the invention relates to determining the tension in tension rods or anchor tendons which secure trunnions for movable gate spillways in dams.

BACKGROUND

Movable gate spillway dams are used in controlling the water flow over dams, particularly power generation dams and flood control dams. These dams include movable portions that provide a means of to release large quantities of water. One particular version includes the so-called Tainter Gate Spillway on a concrete dam. The Tainter gate is basically a curved flood gate panel which forms an outer surface of a sector of a circle. Water upstream is held by the curved flood gate panel when the gate is raised. The curved panel is supported on a radial structure that forms a pivot opposite the downstream side of the curved panel. A side view of the gate is thus seen as a sector of a circle. The pivot engages a trunnion that is connected to and supported by the concrete dam structure. The gate can then pivot about the trunnion to raise or lower the gate.

The trunnion is secured to the concrete dam structure by tension rods or anchor tendons. Each rod or tendon is encased in a sheath, and the sheathed rod is embedded in the concrete. One end of the rod or tendon is anchored within the concrete structure to resist the rod being pulled out of the structure. The other end of the rod extends outwardly and is not embedded in concrete. The rod portion not embedded in concrete is connected to a trunnion and secured thereto generally by a nut threaded on the end portion of the rod and abutting an anchorage plate on the trunnion. This arrangement provides for the rod or tendon to be in tension and resist the force of water held back by the gate.

This construction approach is beneficial in terms of cost as compared to methods in which the anchorage structure for the trunnions is not encased in concrete. However, the construction described does no lend it self to visual inspection of the tension rods. The ability to periodically assess the structural integrity of the rods is important in maintaining safe and efficacious dam function. When tension rods or tendons are damaged, catastrophic failure can ensue or a gate may need to be permanently closed at a large cost and possible loss of the ability to control a maximum probable flood. Accordingly, there exists a need for a method of periodically assessing the integrity of the rods or tendons to determine whether they are in good condition.

SUMMARY OF THE INVENTION

The invention provides a method of determining the tension in a rod that anchors a movable gate spillway trunnion in a dam. The method comprises mounting one or more transverse motion sensors to the rod. The rod is impacted and transverse motion data is recorded from the one or more sensors over a period of time. The method further includes analyzing the transverse motion data and producing a measured frequency response of the rod. A theoretical model of the rod is also generated, and the theoretical frequency response of the model is calculated in order to derive a series of theoretical modal frequencies for the model. The method includes superimposing the series of modal frequencies for the model on the measured frequency response of the rod and adjusting a tension parameter in the model such that at least one modal frequency of the model is approximately equal to a resonance frequency in the measured frequency response.

The present invention also entails a method of determining the tension in an elongated rod. The method basically includes impacting the rod and measuring a vibration response of the rod. Further, the method entails generating a theoretical vibration response of the rod and comparing the measured and theoretical vibration responses. The theoretical vibration response is adjusted to cause at least a portion of the theoretical vibration response to conform with at least a portion of the measured theoretical vibration response. The tension in the rod is determined based on the adjusted theoretical vibration response.

DESCRIPTION OF THE INVENTION

The present invention relates to determining the tension in a rod. In one particular embodiment, the invention relates to assessing the structural integrity of tension rods or anchoring tendons for securing a movable flood gate trunnion to a concrete dam structure. More particularly, the invention relates to determining the tension in a rod in situ in a trunnion anchorage system. Before describing the invention in detail, a general description is in order of a dam and movable flood gate system in which the invention may be utilized.

Dams generally provide one or more fixed elevation spillways over which water may flow when a dammed reservoir water level reaches a certain stage. In some applications, however, it is desirable to permit the reservoir level to rise substantially above the fixed spillway level before permitting flow over the spillway. Power generation dams are examples of such applications. In order to provide this functionality, movable floodgates are employed. An example of such a movable flood gate is the so-called Tainter gate.

As seen inFIG. 1, a dam100includes a spillway110defined between two piers120. Mounted between piers120is Tainter gate130comprised of a curved panel132supported on radial arms134. Radial arms134converge at pivot shaft136that is rotatably supported by trunnions140. Sides of curved panel132may slidably engage with curved leak-limiting slots122in side faces of piers120. It is to be appreciated that a water reservoir is on the convex or upstream side of curved panel132. When gate130is closed the panel is in the lower position as shown inFIG. 1and water is prevented from flowing downstream over the dam by curved panel132. When gate130is opened by rotating radial arm structure134about pivot shaft136, water from the reservoir is permitted to flow under panel132, over spillway110, and downstream from dam100.

Trunnions140are supported on ledges124that form a port of piers120. Pivot shafts136are journaled in trunnions140. Tension rods or anchorage tendons150extend in a generally upstream direction into piers120and are anchored in the piers by tendon anchorage plates or anchorage means152embedded in the concrete of the piers. Downstream ends of rods150are secured to trunnions140to prevent gate130from being pushed downstream by the water held back by panel132. Rods150typically extend through trunnions140to facilitate securing the rods to the trunnions. A typical means of securement is provided by dry-mounted trunnion anchor plates154positioned on the downstream side of the trunnions. Rods150extend through openings in anchor plates154and nuts156are tightened on the rods as shown in FIGS.2A &2B. In some cases, nuts156are threaded and engage threaded portions of rods150. In other cases nuts156are of the so-called Howlett grip nut type. In any case, nuts156provide a secure connection of rods150to trunnions140such that the rod portions between trunnions140and anchorage means152are tensioned. An untensioned or free end portion160of each rod150extends downstream of each nut156.

As shown inFIGS. 2A and 2B, at least between anchorage plates152and trunnion140, each rod150is encased in a liner158. Each liner158is typically a plastic sheath-tube within which rod150is free to move. Liners158prevent rods150from being in contact with the concrete of pier120along the tensioned portions of the rods.

When installed, rods or tendons150are post-tensioned for restraint of trunnions140and thus of gate130. That is, trunnions140are secured against piers120by, for example, tightening nuts156to produce a persistent tension in rods150. Obviously, the degree to which rods150are maintained in tension at all times is related to the safe function of gate130. Nut loosening, rod deterioration, or rod fracture are among factors that may result in a reduction or loss of tension in rods150and in consequent compromise of the safe and effective functioning of gate130. The instant invention is directed at measuring the tension in tendons or rods such as rods150in situ.

Turning now to a detailed description of the instant invention, a method is provided for measuring tension in rod150in dam100described above. This method is based on measuring the vibration response of rod150to an applied impact. In particular, the method provides for inferring the tension in rod150by system identification from an impact vibration response measured at a point on the untensioned end portion160of the rod. The impact vibration response may be initiated, for example, by striking untensioned portion160with an impacter or hammer12as illustrated inFIG. 2B.

Generally, the method exploits the dependence on rod tension of the transverse or flexural vibration response of rod150. Considering rod150to be modeled as a rod of uniform cross-section and material properties that is under axial tension, the equation of motion is:

u=transverse deflection of the rod at position x along the rod,

E=elastic modulus of the rod material,

I=area moment of inertia of the rod cross section,

m=mass per unit length of the rod.

Solution of the Equation 1 for a case where rod150is provided an initial disturbance, such as an impact for example, illustrates that the model rod will undergo periodic flexural deformation and exhibit an infinite number of resonance frequencies or modal frequencies. This flexural deformation entails transverse cyclic deformation of rod150. The modal frequencies are dependent, according to Equation 1, on the parameters E, I, T, and m. E is known from construction specifications. Area moment of inertia, I, can be computed from the known cross-sectional dimensions rod150. T is an assumed or initially estimated tension in the rod. The mass per unit length, m, can be determined from the known density of rod150, the rod's cross section and the rod's length.

The method includes conducting a test to cause rod150to vibrate or exhibit a vibration response and recording that response. The method further includes developing a theoretical or expected vibration response of a model system including a rod of the same dimensions and properties as rod150. The development of the expected vibration response includes initially estimating the tension in rod150and entering this tension as the parameter T of the model rod system and Equation 1. The expected and measured vibration responses are then compared. If the expected and actual or measured vibration responses are not sufficiently congruent, the method includes adjusting T in the model system and generating a new theoretical vibration response. This process is repeated until the theoretical vibration response is sufficiently congruent with the measured or actual vibration response of rod150. The value of T at this point in the process is the measured tension in rod150.

Obtaining an actual or measured vibration response of rod150includes mounting a transverse motion sensor10on the untensioned portion160of the rod and connecting the sensor to an appropriate signal conditioning and recording system of well known design. In one embodiment sensor10is a piezoelectric accelerometer with a connection11to a signal conditioning and recording system (not shown) that includes a charge amplifier, and an amplifier, a filter, a digital data acquisition system, and a computer with display all of which are well known. Accelerometer10may be mounted to rod portion160variously including a threaded stud connection, a magnet, or an adhesive all of which are well known accelerometer mounting approaches. The method further includes applying a transversely-directed impact to untensioned end portion160and recording the resulting signal from accelerometer10for a period of time. In one embodiment, the impact is delivered by a manually wielded hammer or impacter12to produce a momentary impact force F as shown inFIG. 2B. The period of time for recording varies with the condition of the rod. The recording time period is typically between 1 and 10 seconds. The resulting recorded signal is known as the impact or transient response. Transient response20ofFIG. 3is typical of an impact or transient response obtained for a rod such as rod150. Transient response20is a typical “ringing” response illustrated by a damped cyclic or oscillatory pattern. In the example ofFIG. 3, the measured acceleration cycled between plus 3 and minus 3 acceleration units and gradually diminished over about 2 time units.

In some cases, when a measured transient or impact response is obtained as described above, the acceleration signal may be clipped during at least an early portion of the recorded response. Clipping is evidenced by the recorded transient response exhibiting oscillation between constant plus and minus values during an initial portion of the recording period. Clipping is generally due either to saturation or overloading of the accelerometer or one of the components in the signal conditioning and recording. In the event of clipping, the test may be repeated using a smaller impacter or hammer12, and such repeating may be continued until an unclipped response is obtained.

In some cases the recorded signal is too small or damps out too rapidly to be usable. The degree to which this can be a problem is dependent at least in part on the precision of sensor10and signal conditioning and recording system. Generally, however, a transient response that damps out in less than a few seconds is inadequate. When a test results in an inadequate recorded signal, a larger impact may be needed. The test may be repeated using a larger impacter12, for example.

Once an unclipped and adequately large transient response signal has been recorded, the resulting data are, in one embodiment, subjected to numerical Fourier transformation using commonly available software to produce the actual frequency response such as the frequency response30shown inFIG. 4. It is acknowledged that there are alternative approaches to obtaining vibration frequency response, including analog filtering. Frequency response30is a complex quantity expressed as a function of frequency. Frequency response30may be conveniently expressed as a magnitude32and a phase34, each represented as a function of frequency. The magnitude portion32exhibits a series of maxima36, each maximum occurring at typically a different frequency. The associated series of frequencies is sometimes referred to as the modal frequencies of rod150in situ. Having obtained measured frequency response30, the method includes generating a theoretical or expected vibration response and expressing this response as an expected spectral decomposition.

In one embodiment, the expected spectral decomposition for rod150is generated by finite element analysis (FEA). It is recognized that other well known approaches can be used to generate the expected spectral decomposition, including analytical solution of Equation 1. However, the finite element analysis may be straightforwardly accomplished by any of a variety of computer applications that are fast and easy to utilize.

To perform the FEA, a model of rod150in situ is created. In one embodiment, a model of in situ rod150in dam100includes a rod or beam of uniform cross-section and density. The rod in the model is supported by a first support point at one end to represent anchorage plates152and a second support point to represent the connection of rod150to trunnion140and the support provided by the trunnion. The modeled rod is constrained by boundary conditions that represent the support arrangement at the first and second support points of rod150. The transverse displacement of rod150as well as the slope of the rod at the first support point is constrained to zero representing cantilever support of the upstream end of rod150by anchorage152. At the second support point the modeled rod is constrained to have zero transverse displacement representing vertically motionless trunnion140. Also at the second support point, bending of the model rod is constrained by a rotational spring representing the stiffness imposed by nut156on rod150. The nut stiffness is otherwise referred to as a torsional spring rate, Kr. The modeled rod is subject to a unit impact force applied transversely to a portion corresponding to untensioned portion160of rod150and representing force F applied to the rod as shown inFIG. 2B. A solution is obtained for Equation 1, subject to the above described boundary conditions, by means of the FEA and may be expressed as an expected transient or vibration response representing the expected vibration of rod150in response to impact F. In the FEA, the model tension, T, is fixed at zero in the portion of the modeled rod corresponding to untensioned portion160of rod130, while T is held at a constant value equal to an estimated or initial rod tension in the portion of the modeled rod between the first and second supports. For a given measurement of rod150, assumed or estimated values of the tension, T, and nut stiffness, Kr, are selected. Typically, the assumed or initially estimated value of T is the specified pre-tension obtainable from design and construction documents for dam100. An estimated value of nut stiffness, Kr, may be obtained from the known geometry of the nut by well known methods.

The FEA solution may produce an initial expected transient response similar to response20shown inFIG. 3, which is numerically transformed to an initial expected frequency response similar to frequency response30shown inFIG. 4. The expected frequency response magnitude may exhibit a series of maxima or resonances that define the initial expected modal frequencies for rod150in situ. The expected spectral decomposition is a series of generally increasing numbers, each number being the frequency in Hertz (Hz) at which an associated maximum in the frequency response occurs. The spectral decomposition can be expressed graphically as a horizontal frequency axis48from which vertical or spectral lines such as line48A and48B extend perpendicularly from points on the axis that represent the respective modal frequencies, as seen, for example, inFIG. 5.

FIG. 5also shows a portion of typical actual or measured frequency response40of rod150expressed as a graph of magnitude42versus frequency versus frequency. Comparing the locations of spectral lines48A and48B, for example, to the locations of the maxima42A and42B in magnitude42shows, in this example, that the spectral lines do not align with the maxima. In other words, the expected frequency response is incongruent with the actual or measured frequency response of rod150. Typically, especially in cases of the actual rod tension having reduced due to rod or other damage, the expected modal frequencies will be higher than the actually observed modal frequencies as is illustrated inFIG. 5. As can be appreciated fromFIG. 5, spectral lines48A and48B could be made to generally coincide with maxima42A and42B by horizontally shifting the lines along axis48.

This shifting of the expected spectral decomposition lines is the means of determining the tension of rod150. To shift the spectral decomposition, the tension, T, and second support point or nut stiffness, Kr, values are iteratively adjusted to bring the expected spectral decomposition into general congruence with the measured frequency response, at least as far as lines48A and48B are concerned. That is, the values of parameters T and Krare adjusted so as to shift expected spectral lines so that at least the first two spectral lines48A and48B generally coincide with the identified first two maxima42A and42B in the magnitude of the measured or actual frequency response. This is done making trial adjustments to T and Krand generating a new expected spectral decomposition by, for example, re-executing the FEA as described here above. This process is repeated until general congruence is obtained.

Experience shows that the value first or lowest frequency in the expected spectral decomposition, is dependent on the values of Krand T, while the value of the second frequency in the decomposition is dependent mainly only to the value of T. That is, for example, the frequency of spectral line48A inFIG. 5is dependent on the value of both Krand T. However, the frequency of the second expected spectral line, line48B for example, depends mainly on T. The method includes changing Krand T incremental amounts, generating another expected frequency response, and observing whether the first frequency of the expected spectral decomposition is closer in value to the maximum of the measured frequency response magnitude of rod150. This process is repeated until the expected and measured first expected spectral frequency generally coincides with the first maximum42A in magnitude response42shown inFIG. 6. During this process Kris generally adjusted first to determine if spectral line48A can be shifted sufficiently to obtain congruence with maximum42A. If not, the value of T may also be adjusted. Once congruence is obtained between line48A and maximum42A, the process is continued by adjusting only T until congruence of line48B with maximum42B is obtained. The final adjusted value of T is the measured tension in rod150. It is generally not necessary to assess the congruence of other, higher modes although the higher modes will shift in frequency due to the adjustments described above and will be congruent with higher observed modes.FIG. 6data was in ideal laboratory conditions and the first ten modes align with the observed modes.

Typically, the modal frequencies expected based on the FEA are greater than the corresponding modal frequencies observed in the measured response of rod150. In such case, the adjustments to Krand T made in the above-described process amount generally to reducing the values of the Krand T. However, this may not always be the case and experimentation may be required in some cases.

To further illustrate the method, the actual values obtained in an example may be considered with the assistance ofFIGS. 5 and 6. The figures display a portion of measured frequency response40presented as a magnitude42and a phase44versus frequency as has been noted above. Magnitude response42exhibits first and second identified resonances or maxima42A and42B at about 65.76 and 85.80 Hz, respectively. For the rod150under study, the first two spectral lines48A and48B in the expected spectral decomposition occurred at about 49 and 92 Hz, respectively. The initial estimates of T and Krutilized to generate the initial expected spectral decomposition for rod150were 5000 lb and 10000 ft-lb/radian, respectively. By successive adjustments of Krand T, as here above described, the expected frequency response was modified such that spectral lines48A and48B shifted to approximately 49.3 and 91.1 Hz, respectively illustrated inFIG. 6. The value of Krobtained was 3×106ft-lb/radian. The final value of T obtained was 1223 lb, the measured actual tension in rod150.

A testing protocol provided by the invention is illustrated by flow chart200inFIG. 7. The protocol includes exposing the untensioned portion160of rod150and affixing one or more motion sensors10to the rod as indicated by block202and illustrated inFIG. 2B. Block204calls for obtaining the structural and material properties of rod150for use in generating the expected frequency response of the rod. Per block206, an impacter12is selected and utilized to strike untensioned end portion160and apply an impact force F as illustrated inFIG. 2B. The transient response is collected by the data acquisition system as provided in block208. Logic blocks210,212,214, and216describe the process of assuring a sufficiently large, unclipped transient response is obtained as described above. According to block218, the initial expected spectral decomposition for rod150is generated using a computer and FEA, for example, based on Equation 1 with known structural and material properties of the rod150and estimated values of Krand T as has been described before. As called for in block220the expected spectral decomposition for rod150is compared to the actual or measured frequency response of rod150. An assessment of congruency is made (Block222), and if congruency does not exist, the tension value, T, is adjusted (Block224) and a new or adjusted spectral decomposition is generated (Block218). As indicated by Block226, when congruency is obtained the value to which the tension, T, has been adjusted to obtain congruency is recorded as the measurement of the actual tension in rod130.

As has been noted here above, the invention is not limited to use of a particular kind of motion sensor10. There are many different types of such sensors including velocity and displacement sensors. Also, in some embodiments, more than one sensor10may be used in order to sense transverse motion of rod extension160in radially different directions. Additionally, the manner of exciting rod150in order to obtain a measured frequency response is not limited to impacting untensioned end160. There are numerous well known ways to excite a mechanical system and determine its frequency response. For example, untensioned end160could be exposed to a continuous cyclical driving displacement that is varied over a range of frequencies to generate a mechanical impedance frequency response from which modal frequencies can be determined. Additionally, operation of the gate or action of water on the spillway may be used to excite the rods.

Above, much of the discussion is focused on an example of an elongated rod used in a trunnion anchorage system. It is appreciated that the method of determining the tension of a rod, as disclosed herein, is applicable to a wide variety of rods used in a variety of applications. For example, the method of the present invention is applicable to tie-back rods. These are rods used for holding back retaining walls and other masonry or concrete walls such as sea walls. In addition, the present method of determining the tension in a rod can be used in the case of earth anchors used for what is known as soil or rock anchor nailing. Further, and in the way of another example, the method of determining the tension in a rod can be used in the case of tensioned rods that are used for holding guy cables in tension for tall guy towers.

The present invention may, of course, be carried out in other specific ways than those herein set forth without departing from the scope and the essential characteristics of the invention. The present embodiments are therefore to be construed in all aspects as illustrative and not restrictive and all changes coming within the meaning and equivalency range of the appended claims are intended to be embraced therein.