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The measurement of cable tension by the vibratory method
Abstract
The need, in many cases, to know the tension of a cable in a structure led to seeking a simple and rapid method of measurement. This method is based on the theory of vibrating strings; it enables the tension to be obtained from the measurement of the frequency of vibration of the cable and a knowledge of its length and mass per unit length. The authors justify the application of the 'vibrating string' model to a cable, and specify the limits of its field of utilization. A simple rule has been established to determine whatever a given case lies within this field. The tension can be measured to an accuracy of better than 5%. Examples are given of the application to bridges incorporating prestressed cables. The authors also indicate the principal precautions to be taken for the correct use of this method.
The results of a recent experimental program of static and dynamic loads, carried out on the cable-stayed Tampico bridge are presented and analyzed. The bridge is located in the northern part of Mexico on a highway important for the commercial and industrial development of the area. Static and dynamic loads produced by trucks of known weight are used to measure strains induced along the stayed span. Ambient vibration testing of the superstructure and cables is carried out and dynamic parameters and modal shapes are derived. Vibrations produced by trucks crossing the bridge are used to calculate factors of dynamic amplification. A mathematical model was developed and calibrated with experimental results. It is concluded that the comparison of the numerical and experimental results of both testing programs did not reveal structural deficiencies in the bridge such as fatigue or deterioration.
From the frequency characteristic equation of a beam fixed and hinged at both ends, the frequency relationship between the two states was discovered, and a simple practical formula in the estimation of cable tension was proposed, which has definite physical meanings. The results from case studies have demonstrated that the formula can simulate the tension condition precisely for cables, so it can be used to identify the tension of cables with different slenderness ratios and flexural stiffness more easily and accurately. The influence of different cable boundary conditions on the cable tension calculation results was analyzed, and a cable tension formula which can take into account the rotational stiffness of the cable ends was proposed.
The cable tension test with the frequency method is widely used in construction control, assessment and long-term monitoring of cable-supported structures. Due to existence of intermediate supports, such as, absorbers and dampers, the calculation precision of cable tension is reduced. In order to acquire a cable tension calculation method considering influence of intermediate supports, an equivalent cable length calculation method based on vibration differential equation of a cable and energy principle was proposed here. This method could be applied in many formulas for calculatating cable tension under a fixed-fixed boundary condition. The results from case studies demonstrated that the method can simulate the tension condition of cables with intermediate supports precisely; meanwhile, compared with the finite element method, the method of equivalent cable length calculation is not complex, it can be easily used in practical engineering.
Modal analysis tests were conducted in laboratory conditions on a tensioned cable of 50 m length. Both low and high resonant frequencies of the cable were excited respectively by manual shaking and hammer strike. This paper first deals with resolution techniques of inverse problems which were used to improve the accuracy of the determination of the cable tension when the boundary conditions were identified as properly clamped. The simple estimate of the tension given by the vibrating string model was then corrected by about 4 %. Subsequently, this paper focuses on the influence of disturbances at the anchorages on the determination of the cable tension based on the measured resonant frequencies. The types of disturbances addressed were produced by a movable constraint close to the anchorages in order to simulate ill-defined end conditions. However, no significant degradation of the coherence function was observed for nonlinear vibrating conditions. Finally, the modal frequencies of the cable, which was progressively damaged, were measured showing evidence of the full tension recovery in broken wires. The practical outcomes of this study give an insight into the extended use of frequency monitoring of stay-cables.
Estimating the exact tension force of cables is very important to the construction and maintenance of the cable bridges. The cable tension changes continually from the first installation to the final construction stage. Moreover, in service status, this changes continuously as the vehicle loads change. Recently, many extradosed bridges are constructed and these bridges have the heavy and thick cables. But the existing formulas do not give the reliable estimations for these bridges because these formulas do not consider the tension force induced by self-weight (gravity effect). In this study, the gravity effect on the extradosed bridge cables is investigated and an improved estimation method considering the gravity effect is proposed. The relationship among the gravity effect, the natural frequency and the slenderness ratio is also clearly explained. The field test results show that the new method has the acceptable reliability compared with the existing methods for the extradosed bridge cables.
In this paper, a finite difference formulation for vibration analysis of structural cables is introduced. This formulation incorporates into a unified solution the effects of the bending stiffness of cable and its sag-extensibility characteristics and provides a tool for accurate determination of vibration mode shapes and frequencies. Various cable-end conditions, variable cross sections, and intermediate springs and/or dampers are taken into account. Using a nondimensional form of this formulation, a parametric study was conducted an the effects of sag-extensibility and bending stiffness. The formulation is verified with available theoretical solutions and compared with finite-element analyses. Capabilities of this formulation are demonstrated through examples. A simple relationship among nondimensional cable parameters is also introduced for the range of parameters applicable to stay cables in cable-stayed bridges. This simple relationship provides an accurate tool for measurement of tension forces in stay cables using the vibration method.
Field load testing is an effective method to evaluate bridge performance and to calibrate structural models. This paper presents the load tests of the cable-stayed Rainha Santa Isabel Bridge which crosses the Mondego River near Coimbra, in Portugal. During these tests, several parameters were measured, like vertical displacements of the deck, horizontal displacements of the mast, rotations, strains and stay forces. Different types of equipment were used in order to get more accurate measurements. The experimental results are compared with the analytical values computed with a finite element model of the bridge.
Vibrating strings help in measuring relative displacements in a mechanical system. Since the ground natural frequency of a string increases when it is stretched, monitoring the ground frequency yields the current length of the string. Therefore a wire able to vibrate between two anchor points of a system acts as a relative displacement sensor. Excitation is usually achieved by means of an active coil, which is very close to the vibrating iron wire. Vibrating wire sensors (VWS) based on this excitation may prove obtrusive and one is limited to wires of small length.
The new VWS takes advantage of distributed passive magnets, which force the wire to vibrate mainly in its fundamental mode. The sensor proves scalable and much less obtrusive when fully embedded, since it can be made flat and very flexible.
On the basis of a simplified electromechanical modelling of the measurement process, a suitable distribution of magnets is proposed, which is proved numerically and experimentally to make the measurement robust with respect to mechanical uncertainties. Moreover, numerical simulations suggest measuring not the voltage in the vibrating wire but the current in an auxiliary circuit.
Extradosed bridges have loose and sagged cables that are different from those used in cable-stayed bridges. The total tension
force of an extradosed bridge cable estimated using the existing formulas is not accurate. Although the existing formulas
are widely used in the field, they do not consider the exact deflection curve, features of nonlinear oscillation, and stretching
force due to self-weight. In this study, a new method was developed to calculate the total tension force of the cable. In
the new method, the total tension force of the cable includes the applied axial load and the stretching force due to self-weight.
For the exact estimation of the applied axial load to the cable, the vibration of the cable was treated as a conservative
nonlinear oscillation system, and a new formula was derived using the perturbation technique. The new formula was used to
clearly explain the hardening effect due to the initial amplitude of the initial excitation, the presence of imaginary number
of frequencies, the correlation with the stretching force due to self-weight, and the softening effect of the slenderness
ratio on nonlinear natural frequency. For calculating the stretching force due to self-weight, the stretching of the cable
for the static deformation was considered. A field test showed that it is necessary to add the stretching force due to self-weight
to the estimated axial load for extradosed bridge cables and that the new method can be used to calculate the total tension
force of loose and sagged cables.
Keywordsextradosed bridge–cable tension–natural frequency–gravity effect–nonlinear oscillation
Proper identification of the cable’s resonant frequencies is critical to provide an accurate estimate of cable force. The
MUltiple SIgnal Classification (MUSIC) algorithm is implemented to estimate cable-stayed bridge cable tensions noninvasively from measured
cable motion. This algorithm performs eigenanalysis on the data sequence to estimate and eliminate noise contributions before
creating its frequency spectrum, providing a more robust estimation approach than traditional Fourier based frequency spectrums.
To aid in the selection of cable frequencies, a comprehensive finite difference cable model is simulated and compared to the
estimated MUSIC spectrums.
KeywordsCable-stayed bridges-Cable force estimation-Frequency spectrum estimation-Subspace methods
The microwave interferometry has recently emerged as an innovative technology, suitable to the non-contact vibration monitoring of large structures. The paper addresses the application of microwave remote sensing to the measurement of the vibration response in the longer cables of two cable-stayed bridges. In order to highlight the reliability and accuracy of the radar technique, the natural frequencies (and the cable tensions predicted from natural frequencies) identified from radar data were compared to the corresponding quantities obtained by using more conventional techniques. The results of the investigation highlight the accuracy and the simplicity of use provided by the microwave remote sensing, as well as its effectiveness to simultaneously measuring the dynamic response of all the stay cables of an array.
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