Fatigue of stay cables inside end fittings high frequencies of wind induced vibrations

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1 D. Siegert, P. Brevet Laboratoire Central des Ponts et Chaussées, France Fatigue of stay cables inside end fittings high frequencies of wind induced vibrations Summary A twenty year old stay cable was dismantled because it showed many ruptures at one of its terminations. The examination of the wire breakages showed evidence of fatigue. Analysis of the dynamic behaviour of the stay cable was carried out. It showed that the combination of flexural bending stresses with fretting stresses between the outside layer and wire serving lead to a stress range beyond the endurance threshold. Recorded measurements of transversal vibration displacements near the sockets show that high modes of wind induced vibrations (Karman Vortex) were involved in the fatigue occurrence. Examination of the internal state of three other cables confirms the analysis of displacement records in relation with the fatigue limit. 1. Introduction This paper deals with five years of investigations on a cable stayed bridge built on the sea-side over the river Loire in Western France. This bridge which had one of the longest suspended spans in the 1970 s is mainly characterised by two towers of 130 metres in height, seventy-two stay-cables and a metallic deck that is 720 metres long. One of the locked coil cables (72 mm diameter) in service for about twenty years appeared to be severely damaged with lots of wire breakages in the external layer as shown in the figure 1.

Figure 1 : Damaged stay cable at the lower anchorage After the removal of the cable, the internal state of the stay cable at the sockets was investigated.

2 Figure 1 : Damaged stay cable at the lower anchorage After the removal of the cable, the internal state of the stay cable at the sockets was investigated. The strand was made of 225 wires and 116 of them were found broken at the lower anchorage and only four at the upper one. The repartition of the wire breaks in the layers is given table 1. Layer n Wire shape Number of wires Number of broken wires 1 Z Z Z O O O O O 6 0 Centre wire O 1 0 Table 1 : Composition and repartition of broken wires in the locked coil strand All of the broken wires were located in a section between one and three centimetres inside the socket. Figure 2 shows the angular positions of the broken wires in the section and suggests the characteristic symmetry of the in-plane bending condition of stress. Observation of the fracture surfaces led to the conclusion that fatigue cracks were initiated at the inter-wire contacts or at the contact between the external layer and the serving wire. Figure 3 shows the surface fracture of a Z shaped wire. The surface fracture is fresh enough to enable us to recognize the fatigue crack propagation area initiated at the inter-wire contact.

$Figure 2 : Broken wire locations in the section of the strand : 0 shape wires, O cylindrical wires, broken wires Figure 3 : Surface fracture of an inter-wire fretting fatigue wire breakage.$ deformations of stay cables.

3 Figure 2 : Broken wire locations in the section of the strand : 0 shape wires, O cylindrical wires, broken wires Figure 3 : Surface fracture of an inter-wire fretting fatigue wire breakage. An analysis of the dynamic behaviour of the bridge was undertaken after the replacement of the damaged stay cable in order to look for correlations between traffic and wind actions and bending deformations of stay cables. In addition, the bending deformations of some stay cables at the anchorages were measured over a period of 4 months in order to get a first assessment of the fatigue stresses. The fatigue limit of stay cables submitted to lateral vibrations has been previously addressed in the literature (Hobbs and Ghavami (1982) and at the time of our investigation some collaborative research work on the fretting fatigue of steel wires was in progress at the Ecole Centrale de Nantes and Joseph Fourier University in Grenoble (Siegert (1997), Couroneau (1998), Delclos (2000), Olivella (2000)).

4 2. Monitoring of the bridge 2.1 Global monitoring The regular maintenance team has reported that the dismantled stay cable was undergoing severe vibrations compared to the others, with flutter characteristics and they had noticed six month previously that the painting of the strand was damaged close to the anchorage. The monitoring set up shown in figure 4 consists in measuring the movements of the main structural elements (deck, towers, stay cables) in order to find correlations between the different structural movements and between traffic and wind actions. The monitoring devices consisted of 25 accelerometers, three anemometers and an equipment for traffic analysis which had been set up after the replacement of the damaged stay cable with an identical one. Figure 4 : Global monitoring of the bridge : anemometers, accelerometers, traffic recorder. Figure 5 shows the displacement of the replaced cable recorded in the vicinity of the lower anchorage during a period of heavy vibrations and the corresponding power spectral density. The power spectral density representation of the recorded signal indicates that the energy of vibration is mainly located in the frequency range which is roughly between 30 and 35 Hz. For comparison, figure 6 deals with the state of vibrations of an another stay cable recorded at the same time which shows lower amplitudes of displacement and frequency range. The fundamental frequencies of vibration of the stay cables which depend on the mass per unit length, the tension and the length, are in the range of 0.6 to 1.3 Hz.

5 Time (s) amplitude (mm² / Hz) Frequency Hz Figure 5 : Time record of the vibrations of the replaced cable and related power spectral density. Time (s) Amplitude (mm² / Hz) Frequency Hz Figure 6 : Time record of the vibrations of a stay cable and related power spectral density which is not submitted to heavy vibrations. A correlation has been found between the induced vibrations and winds in the east west direction (nearly perpendicular to the plane of the stay cables) with a velocity

6 above 10m/s. No correlation has been found between heavy traffic or tower and deck vibrations and the high amplitudes of vibrations of the cables. An extensive literature review about wind induced vibrations of stay cables has been written by Matsumoto (1995). The frequency range of the Karman Vortex induced vibrations is calculated using the following expression : V f k = D where V is the velocity of the wind, D the diameter of the stay and S is the non dimensional Strouhal constant which takes a value in the range of 0.18 to 0.2 for a cylinder vibrating in the air. The application of the above formula with a wind velocity in the range of 10 m/s to 15 m/s gives frequencies in the range of 28 Hz to 41 Hz which correspond to the frequencies shown in the figure 5. S 2.3 Local monitoring In order to evaluate the bending deformations undergone by the stay cables in service conditions, the device shown in figure 7 was set up on selected stay cables : the cable which has been replaced (cable 1), the symmetric opposite on the same side (cable 2) and the symmetric cable on the opposite side (cable 3). The principle of the determination of the bending strain based on the measure of the displacement has been studied in great depth by Poffenberger and Swart (1965). The amplitude of displacement has been measured in four sections of the cable with reference to the axis of the anchorage socket which is articulated using eight optical sensors of displacement to carry out measurement in two perpendicular directions. The asymptotic deflection close to the anchorage is given by : β Y( x ) = [exp ( Ωx ) + ( Ωx ) 1] Ω where β is the angular deflection with reference to the socket axis as shown in figure 7 and Ω = Τ with T the tension in the cable, and EI the modulus of bending ΕΙ stiffness. The maximal bending stress evaluated at the embedded section is then deduced : σ = D 2 2 E cos ( α )Y,xx (0 ) socket x β Y Socket s axis 635 mm cable Figure 7 : scheme of the deflection measurements at the cable end.

7 The results of a rainflow analysis of the local monitoring of three stay cables over a period of about four months are given in table 2 respectively for the cables 1, 2 and 3. The calculation of the stresses are based on the following assumptions : EI = N.m² (no inter-wire slip) X = 685 millimetres (validity of the asymptotic development with respect to the mode of vibration) Cable 1, recently replaced, is the mot mechanically effected, about cycles and 2800 cycles were above 80 Mpa and 100 Mpa respectively. Fretting fatigue tests carried out in our laboratory are shown in figure 8. The value of the experimental fretting fatigue limit obtained is about 100 Mpa. This threshold is overcome, indeed the conditions of the laboratory tests are not actually completely representative of the complexity of the contact conditions involved at the socket of the cable. Locked cable n 1-72 Displacement range mm Stress range MPa Vertical motion Horizontal motion Locked cable n 2-72 Displacement range mm Stress range MPa Vertical motion Horizontal motion Displacement range mm Locked cable n 3-72 Stress range MPa Vertical motion Table 2 : Results of the rainflow processing of the local monitoring of cables 1, 2 and 3. Horizontal motion

300 250 200 σ (MPa) 150 100 50 0 100 000 1 000 000 10 000 000 100 000 000 Nb cycles Figure 8 : Experimental results of fretting fatigue behaviour of un-lubricated wires. 2. Evaluation of three other cables removed Stay cable 2 was removed for further inspection.

$The fatigue cracks have been initiated at the contacts with the serving wire. The figure 10 shows the surface fracture.$

8 σ (MPa) Nb cycles Figure 8 : Experimental results of fretting fatigue behaviour of un-lubricated wires. 2. Evaluation of three other cables removed Stay cable 2 was removed for further inspection. Two broken wires were found in the outer layer as shown in figure 9. These wire breakages were in a section located 10 cm inside the socket. The fatigue cracks have been initiated at the contacts with the serving wire. The figure 10 shows the surface fracture. It is worth to notice that the application of the ultrasonic technique failed to detect the two wire breaks because the location was too far inside the socket. Successful applications of the ultrasonic technique of inspection have been reported [Gronau (2001)]. Figure 9 : Broken wires found in cable 2. Figure 10 : Surface fracture of a broken wire in cable 2.

9 Two other cables were removed because of the damaged aspect of the painting but no rupture was found. 4. Conclusions The inspection of the removed stay cable 1 has shown evidence of fretting fatigue at the anchorage. This stay cable is expected to have undergone a high number of cycles of wind induced vibrations (> in 20 years) related to the von Karman vortex phenomena with a stress range above 80 Mpa. It is worth to notice that this stay cable seems to be prone to be submitted to a range of flexural stresses above the fretting fatigue limit at high frequencies. Moreover a decrease of about 50 Mpa of the maximal flexural stress range at the anchorage should be enough to restore endurance conditions. 5. References Couronneau, N. (1998) Propagation par fatigue d une fissure dans un fil constitutive d un câble, thèse N ED , Université de Nantes. Delclos, A. (2000) Comportement élastoplastique du contact entre fils d un câble Thesis ED Université de Nantes F Gronau, O., Klein, J. & Lobert, H. (2001) Experiences with non-destructive examination on rope terminations and fittings on selected exemples OIPEEC Round Table Conference : Ropes terminations and fittings, Bethlehem USA, Hobbs, R.E., Ghavami, K. (1982) The fatigue of structural wire strands, Int. J. of Fatigue, April 1982, pp Matsumoto, M., Ishizaki, H., Kitazawa, J., Aoki, J. & Fujii, D. (1995) Cable aerodynamics and its stabilization Int. Symp. On Cable Dynamics, Liege, Belgium. Olivella, S. (2000) Contribution à l étude du comportement et de la fatigue des câbles métalliques, Thesis N F Université Joseph Fourier - Grenoble. Poffenberger, J.C. & Swart, R.L. (1965) Differential displacement and dynamic conductor strain IEEE transactions, PAS-84, Siegert, D. (1997) Mécanismes de fatigue de contact dans les câbles de haubanage du génie civil Thesis ED Université de Nantes F

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