Frequency Monitoring of Stay-Cables

Transcription

1 Frequency Monitoring of Stay-Cables D. Siegert 1, L. Dieng 2, M. Goursat 3, F. Toutlemonde 1 1 LCPC, 58 boulevard Lefebvre Paris 2 LCPC, route de Bouaye Bouguenais 3 INRIA, domaine de Voluceau-Rocquencourt Le Chesnay Abstract 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. 1 INTRODUCTION 1.1 Background The vibrating string model is widely used to evaluate the tension of long stay-cables from the first measured modal frequencies [1] [2] [3] [4]. Since the stay-cable bending stiffness is not taken into account, the vibrating string model is only approximate. The tensioned beam model was thus considered in this investigation in order to improve the accuracy of the cable tension estimate. The partial differential equation of the transversal motion of a uniform beam submitted to a constant axial force T is EI 4 y x 4 T 2 y x 2 + µ 2 y t 2 = 0 where EI is the bending stiffness, y is the transversal displacement, x is the axial coordinate, t is the time and µ is the mass per unit length of the beam. A schematic drawing of the beam model is shown in Figure 1. The above equation is made dimensionless by scaling the variables x and y with the span length L of the beam and the time by one half of the fundamental period of vibration of the string model. Dimensionless variables are denoted by a prime, the dimensionless equation of vibration of the tensioned beam is then ɛ 4 y x 2 y 4 x + 2 y 2 t = 0 2 where ɛ = EI is a dimensionless parameter. The first term in the dimensionless partial differential equation T L 2 corresponds to a singular perturbation. When the perturbation parameter ɛ tends towards zero, the equation of motion becomes a second order differential equation related to the string model rather than a fourth order differential equation and then two boundary conditions are required instead of four. Therefore, for small enough values of the perturbation parameter, it is relevant to consider the effect of the bending stiffness as a small perturbation. A criterion derived from the asymptotic approximation to the first modal frequency of tensioned beam with clamped end conditions was ɛ < as proposed in reference [2]. This criterion ensures that the maximum model error

2 L T y l T x Movable constraint Figure 1: Model of the tensioned beam with clamped extremities. was less than 2% when the tension is estimated from the first measured frequency with the vibrating string model using the relation T = 4µL 2 f 2 n n 2 where f n is the frequency of the n th mode of vibration. To overcome the limitations of the vibrating string model, advanced methods were proposed to take into account the bending stiffness of the strand with clamped endconditions [5] [6]. However, these methods are based on approximate analytical estimates of the eigen-frequencies, the accuracy of which gets worse as the mode order increases, thus limiting the number of frequencies required to validate the model. The vibration analysis was also extended to the case of short cable lengths. The tensions of hangers within the range length of 6 to 20 m of a bow-string bridge were determined with the first eight measured frequencies using a finite element model taking into account the inertia and the elastically restrained conditions of the anchorage device [7] [8]. 1.2 Motivations This paper focuses on the effectiveness of the updating model procedures based on the measured resonant frequencies of stay-cables for improving their accuracy when the perturbation parameter ɛ is close to The paper also deals with the identification of the modification of some boundary conditions and damage detection. The bending stiffness of the cable and the tension were the two parameters involved in the model updating procedure based on the sensitivity of the resonant frequencies. The tensioned beam model with simple boundary conditions such as properly clampled end-conditions was considered in the proposed analysis. The issue of the validation of the finite element model was experimentally addressed regarding the measuring conditions, the modal frequencies identification methods and some details of the boundary conditions that might be encountered in field tests. Actually, it is not uncommon that the fixed ends of the cable are not precisely located or properly clamped when for example the cable is in contact with other parts of the structure. The perturbation of the boundary conditions was produced in the test by a rigid movable constraint that was either adjusted to remain in contact with the cable during the vibrations or in a position leading to multi-strikes which might be identified in the response as a non-linear behaviour of the system. Results of the vibration analysis of the progressively damaged strand with broken wires are also presented. 2 EXPERIMENTAL INVESTIGATIONS 2.1 Testing conditions The modal tests were carried out at the LCPC Centre de Nantes with a cable of 50 m length between the two fixed ends. The main characteristic values of the set-up are reported in Table 1. The cable was a locked coil strand made of 159 steel wires. The cross section of the strand is shown in Figure 2. The helical wires in the outer layers of a locked coil strand are shaped and designed to contact one another when the strand is axially loaded with a tension force. The shaped wires were 4.5 mm high with a cross-section area of 16.3 mm 2. The bending stiffness value of the strand is not accurately defined from the geometrical and elastic characteristics of the wires because it depends in a complex manner on the inter-wire contact conditions. However, an approximate upper bound estimate is given by considering that the strand behaves in flexion like a steel beam of the same diameter. The theoretical estimate of the bending stiffness of the strand is then 90 kn m 2.

3 D L µ l (mm) (m) (kg m 1 ) (m) Table 1: Characteristics of the tensioned cable. Figure 2: Cross section of a locked coil strand. The axial tensile force was applied to the strand with a hydraulic jack. When the tensile force reached the test value of 900 kn, the system was held in position with nuts. The axial force was measured with a load cell of limited accuracy due to the thermal drift of the electronic conditioning system, leading to variations within a range of 5% in a day. The extremities of the cable were held by a specific system of jaws shown in Figure 3. Figure 3: Anchorage system of the strand. The movable constraint used to investigate the effect of permanent or non-permanent contacting conditions during the cable vibrations was located one meter from the clamped fixed end. The device is shown in Figure 4. 0bviously, when the movable constraint was adjusted to be far enough from the strand, the cable was not contacting the constraint during the transversal vibrations. Figure 4: Movable constraint device. 2.2 Experimental modal identification The modal test were carried out with two kinds of excitation, the manual shaking excitation and the impulse hammer, in order to extend the range of frequencies excited. The vibrating response of the cable was measured with one capacitive accelerometer operating within the frequency range of 0 to 500 Hz. The transient responses were filtered by a low pass filter with a cut-off frequency of 75 Hz and recorded with samples at a sample frequency of 600 Hz. Both the accelerometer and the manual shaking excitation were located approximatively at 15m from the fixed end of the cable. The location of the 320 g impact hammer was close to the fixed end. The impulse force pro-

4 duced during the hammer strike was recorded for calculating the frequency response functions and the coherence functions. However the following analysis will be mainly based on the results of the output-only analysis. The time histories of the acceleration responses of the cable to the hammer hit and to the manual shaking are respectively shown in the Figures 5 and 6. Since the first modes of vibration were low damped and easily excited, the duration of the free vibrations produced by manual shaking lasted a couple of minutes. 10 acceleration (m/s 2 ) time (t) Figure 5: Hammer hit transient response acceleration (m/s 2 ) time (t) Figure 6: Manual shaking free response. The resonant frequencies were first obtained from the Discrete Fourier Transform (DFT) of the acceleration time records. The modulus of the DFT of the times histories recorded for the shaken and struck cable are shown in Figure 7 for the test configuration in which there is no constraint between the fixed ends. As it was expected, only the first five frequencies were well excited with the manual shaking. Except the first resonant frequency at 2.34 Hz, the higher frequencies in a range up to 60 Hz were excited by the hammer strike. In the range of 0 to 60 Hz, all the first 24 resonant frequencies were identified. The signals were also processed with the COSMAD toolbox developped under the Scilab software to extract the resonant frequencies. The COSMAD toolbox uses a covariance driven subspace method which is decribed in full detail elsewhere [9]. Figure 8 shows the stabilisation diagram of frequencies related to the identification of the first frequency of vibration within the range of 0 to 5 Hz from the accelaration free response to the manual shake. The model order of the subspace processing required to successfully extract the first mode was lower than 80. Figure 9 displays the stabilisation diagram of frequencies obtained from the processing of the transient response to the ham-

5 Mag-DFT e-005 1e-006 1e-007 1e-008 1e-009 1e-010 1e-011 1e frequency (Hz) manual shake hammer hit Figure 7: Magnitude of the DFT of the cable response. mer strike. All the resonant frequencies in the range of 0 to 60 Hz were extracted using a model order above 300, except the first mode wich was not excited enough by the hammer strike. The results of the identification of the resonant frequencies are reported in Table 2. The differences between the frequency estimates from the peaks of the DFT and from COSMAD do not exceed 0.03 Hz. Stabilization diagram Frequency Model order Figure 8: Frequency stabilisation diagram in the range of 0 to 5 Hz.

6 Stabilization diagram Frequency Model order Figure 9: Frequency stabilisation diagram in the range of 0 to 60 Hz. n DFT estimate COSMAD estimate f n (Hz) f n (Hz) Table 2: Comparison of the resonant frequency estimates from the peak of the DFT and the subspace method (COSMAD).

7 3 PARAMETER IDENTIFICATION The bending stiffness of the strand and the tension were identified from the experimental estimates of the resonant frequencies using the tensioned beam model with properly clamped end conditions. The basic model updating procedure was performed on a finite element model of the beam made of 150 elements to ensure the accuracy better than 0.01 Hz of the calculated frequencies up to the 20 th mode order. The calculated frequencies are derived from the generalised eigenvalue problem of the finite element formulation (K 4π 2 f 2 i M)Φ i = 0 where K is the stiffness matrix, M is the mass matrix, f i is the i th eigenfrequency and Φ i is the related eigenvector. The updating procedure consists in minimising through an iterative procedure the following objective function J(T, EI) = nx i=1 (f i ˆf i) 2 = nf i ˆf i o t nf i ˆf i o where ˆf i is the i th experimental frequency, n is the number of frequencies considered in the model updating procedure and the superscript t denotes the transpose. The numerical solution consists in iterating the mean square solution of the over-determinated linear system of equations n o [S] k {δp } k+1 + ˆf 2 i (fi 2 ) k = 0 where k is the iteration step, δp is the correction vector of the updating parameters and [S] is the sensitivity matrix of the squared frequencies to the parameters which is derived from the partial derivatives of the stiffness matrix and the eigenvectors of the modes [S ij] k = f 2 i P j = {Φ i} t k K P j {Φ i} k The results of the identification of the tension and the bending stiffness of the strand are reported in Table 3 for numbers n of resonant frequencies in the range of 7 to 20. The initial value of the tension for starting the iterative procedure was the estimate given by the vibrating string model. Only three iterations were required to achieve the converge. The value of the updated tension is 877 kn, which is 3.85 % below the estimate of the vibrating string model. The updated estimate of the bending stiffness is 61 kn m 2, which is far below the theoretical estimate of 90 kn m 2. The value of the perturbation parameter ɛ is equal to The calculated eigenfrequencies with the updated parameters match the first 20 measured resonant frequencies, the maximun difference is 0.01 Hz. The condition numbers of the matrix SS t used to diagnose the ill-conditioned matrices of systems of linear equations are reported in the last column of Table 3. The condition number of the matrix SS t decreased with increasing the number of experimental frequencies taken into account in the updating procedure, leading to improved numerical conditions for solving the system of linear equations. n T EI condition (kn) (kn m 2 ) number Table 3: Parameter estimates of the tensioned beam model for different numbers of measured frequencies. 4 EFFECTS OF THE MOVABLE CONSTRAINT The movable constraint was first adjusted not to be in contact when the cable was at rest but close enough to be hit by the cable as it was vibrating. The resonant frequencies identified are shown in Figure 10. Both the low and the high frequencies were excited by the manual shaking and the effect of the strikes during the vibrations produced by the manual shaking excitation can be noticed. The identified frequencies from the manual shake excitation matched the frequencies extracted from the hammer strike up to the 7 th mode number as can be seen in Figure 10. For the higher modes, the differences between the frequencies increased abruptly. This result indicates that the identified modal properties depended on the conditions of excitation as expected for a nonlinear behaviour of the system. Input-output analysis were carried out with the impulse hammer to detect nonlinearities in the measured response of the cable from the coherence functions. The approach presented here is inspired by the analysis of nonlinear

8 Mag-DFT e-005 1e-006 1e-007 1e-008 1e-009 1e-010 1e-011 1e frequency (Hz) manual shaking hammer strike Figure 10: Magnitude of the DFT of the response of the cable striking the movable constraint. effects in a clamped-free beam with a rigid constraint as given by B. H. Tongue [10]. The coherence function which takes values between 0 and 1, was calculated in the case of the free vibration between the fixed ends of the cable and for the movable constraint either partially or fully in contact with the strand during the modal tests. Even when the cable never touched the constraint while it was vibrating, the coherence function was lower than one for the frequencies in the range of 0 to 20 Hz because of the signal to noise ratio. The coherence took values close to 1 around the resonant frequencies higher than 20 Hz, indicating that the behaviour is linear. The coherence function is shown in Figure 11 for the three different conditions tested with the movable constraint. The degradation of the coherence for the nonlinear case in the range of the higher frequencies is however very weak coherence frequency (Hz) Figure 11: Coherence functions for the different conditions labelled 1: without constraint, 2: non linear constraint, 3: permanent constraint. The frequencies identified for the cases of the non-permanent and permanent conctact are reported in Table 4 and compared with the case without the movable contraint device. The frequency estimates for the testing configuration of permanent contact between the cable and the movable constraint are also compared with the theoretical estimates based on the previous estimates of the tension and the bending stiffness. In the theoretical model the transversal displacement degree of freedom corresponding to the movable constraint was removed. The maximal

9 difference between the experimental and the theoretical estimates does not exceed 0.06 Hz. n without constraint stikes on the constraint permanent constraint experimental experimental experimental theoretical (Hz) (Hz) (Hz) (Hz) Table 4: Frequency estimates for the movable constraint effect, corresponds to the impulse hammer excitation in the nonlinear testing configuration. 5 DAMAGE DETECTION The strand was damaged in a section located at about 10 m from the fixed end. The damages were produced in the outer layer of the tensioned strand by sawing Z shaped wires until they were broken into two parts. The damage was progressively extended in steps of two broken wires up to six broken wires. Figure 12 shows the final damaged section. No change in the resonant frequencies was detected up to the 8 th frequency. This indicates that the global tension force in the strand was not modified after 6 wires were broken in the outer layer, corresponding to a reduction of 5 % of the cross-section. Assuming the full release of tension in the broken wires, the expected reduction of the first frequency related to this amount of damage is 2.5 % which might be detectable in our experimental conditions. The lack of loss of the tension force in the strand showed evidence that the tension in the broken wires was fully recovered due to the friction forces with the contacting wires in the outer layer. The tension and the bending stiffness modulus were calculated from the first ten measured frequencies of the damaged strand with 6 broken wires in the outer layer using the model updating procedure presented in section 3. The calculated values of the tension and the bending stiffness were respectively 862 kn and 59 kn m 2. The reduction of 3 % of the damaged strand bending stiffness is not significant. It is in the scatter range of the estimates from the measured frequencies. As can be seen in table 5, the measured frequencies up to the 22 nd mode match the calculated frequencies after updating the finite element model. The maximum difference does not exceed 0.01 Hz below the 15 th mode and 0.05 Hz between the 15 th mode and the 22 nd mode. Figure 12: Broken wires in the outer layer of the strand.

10 n measured calculated - updated model f n (Hz) f n (Hz) Table 5: Resonant frequency estimates measured and calculated after updating for the damaged strand. 6 CONCLUDING REMARKS The experimental results presented in this paper have shown that the updating procedure of the finite element model of a strand properly clamped at its extremities was effective in improving the accuracy of the tension determination in estimating the bending stiffness of the strand. The simple estimate of the tension given by the vibrating string model with the first measured resonant frequency was then corrected by about 4 %. The experimental conditions investigated were related to a value of the perturbation parameter ɛ close to which corresponds to intermediate stay-cable lengths. The first 10 measured frequencies were required to update the finite element model. However the measured and calculated frequencies match up to the 20 th mode with a precision better than 0.05 Hz. Nonlinear behaviour produced by a rigid constraint close to the fixed end is difficult to detect by analysing the coherence of input-output measures. However, significant shifts of the measured frequencies were noticed above the 8 th mode which were dependent on the conditions of excitation. The vibration analysis of the damaged strand with 6 Z shaped wires broken in the outer layer corresponding to a 5 % reduction of the section has shown evidence that there was no loss of tension in the strand. In fact the tension in the broken wires was recovered by friction actions with the neighbouring wires. In addition, no significant reduction of the bending stiffness was observed. However, the resonant frequencies are less sensitive to a local reduction of the bending stiffness when this reduction is located far from the clamped end than close to it. Moreover, the clamped ends are the more likely sites of the fatigue damages of stay-cables. ACKNOWLEDGMENTS Thanks to Daniel Bruhat, Richard Michel and Sébastien Noblet for their participation in the experimental set-up and the measurements. References [1] P. Mars, D. Hardy, Mesure des efforts dans les structures à câbles, Annales T.P. Belgique, n o 6, p , [2] J.L. Robert, D. Bruhat, J.P. Gervais, Mesure de la tension des câbles par méthode vibratoire, Bulletin de Liaison des Ponts et Chaussées, n o 173, p , mai-juin 1991.

11 [3] S.E. Chen, S. Petro, Nondestructive Bridge Cable Tension Assessment using Laser Vibrometry, Experimental Techniques, pp , March/April [4] M. Rak, J. Krolo, Z. Savor, Lj. Herceg, D. Damjanovic, Modal analysis of the Dubrovnok bridge, Proceedings EVACES 05, Bordeaux, pp , octobre [5] R. Geier, G. De Roeck, J. Petz, Cable Force Determination for the Danube Channel Bridge in Vienna, Structural Engineering international, 3, pp , [6] I. Yamagiwa, H. Utsuno, K. Endo, K. Sugii, Application of Simultaneous Identification of Tension and Flexural Rigidity at once to the Bridge Cables, IABSE Reports n o 82, [7] D. Siegert, M.M. Choinska, P. Brevet, D. Bruhat, F. Toutlemonde, Détermination de la tension des câbles à partir des fréquences de vibration, Revue française de génie civil, Vol. 8 n o 9-10, p , [8] D. Siegert, L. Dieng, P. Brevet, F. Toutlemonde, Parameter identification of a hanger or stay-cable model based on measured resonant frequencies, Proceedings EVACES 05, Bordeaux, pp , octobre [9] L. Mevel, M. Basseville, A. Benveniste, M. Goursat, Merging sensor data from multiple measurement set-ups for non-stationary subspace-based modal analysis, Journal of Sound and Vibration 249 (4), pp , [10] B. H. Tongue, Principles of Vibration, Oxford University Press, 1996.