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Experimental Study of The Dynamic Response of a Cable Under Wind Flow
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 12 (2019) 446–454
www.materialstoday.com/proceedings
DAS35
Experimental Study of The Dynamic Response of a Cable Under Wind Flow Vlad Daniel Urdareanu*, Ionut Radu Racanel, Mircea Degeratu, Costin Ioan Cosoiu, Cristian Lucian Ghindea, Radu Iuliu Cruciat Technical University of Civil Engineering of Bucharest, Lacul Tei bd, no 122, 72311, Bucharest, Romania
Abstract Nowadays, the high quality of available building materials, powerful machinery and high precision construction techniques allow us to build bridges with very long spans at efficient costs. Depending on the site and the spans, the most common solutions for these types of structures are arch bridges, cable stayed bridges, suspension bridges and even variations among the three. Because of their size and the nature of the crossed obstacles, these types of structures can be very difficult to access outside the carriageways or the pavements and so, inspection maintenance works are very difficult and costly. One critical structural element which is common for all these bridges are the cables. They are out in the open, exposed to a wide variety of phenomena which can cause them damage. Although they perform exceptionally well under tensile forces in longitudinal directions, they are very weak and prone to take damage under transverse loads. For example, it has been observed, that long cable stays tend to vibrate uncontrollably. If left uncheck, these vibrations can resonate and cause damage to the protective sheath, anchorages and even the cables themselves, eventually leading to their collapse. In order to prevent this, engineers must take into account the various mechanisms that can cause vibrations, determine their effect on the stays, and find ways to mitigate them in order to reduce the amplitudes under acceptable levels. Cable vibration simulation using numerical models is very difficult to do, due to the high degree of uncertainty, large volume of processing to be done and lack of validation methods for the results. This paper presents a scaled model analysis of a cable stay in a wind tunnel where vibrations are measured under different wind speeds. Finite element models of the cable are then done, using the measurements to correct certain design parameters in order to improve the precision of the calculated results. © 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of 35th Danubia Adria Symposium on Advances in Experimental Mechanics. Keywords: cable vibrations; vortex shedding; wind tunnel;
*Corresponding author. Tel: +40-726186453 E-mail address: [email protected] 2214-7853 © 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of 35th Danubia Adria Symposium on Advances in Experimental Mechanics.
V.D. Urdareanu et al. / Materials Today: Proceedings 12 (2019) 446–454
447
1. Introduction Cables are the key structural elements of long spanned bridges. This is especially true for cable stayed bridges, due to the fact that these types of structures are highly dependent on their purpose ability to take significant amounts of loads from the superstructure and transmit them to the pylons. This is done by high tensile forces in the cables and leads to an optimization of the distribution of stress throughout the structure. The cables are typically made up of several wires or strands of high strength steel put together in cylindrical, polyethylene sheaths. Evidently, the purpose of the steel elements is to take the loads of the superstructure, while the polyethylene sheaths are used for protection against the various weather phenomena such as wind, rain or icing. While it is true that the two systems should work very well together in obtaining long lasting, high strength and very cost-efficient structural elements, the combination between them can also lead to unforeseen and dangerous situations. Grouping the strands together and covering them with round sheaths ensures that they are protected against most water, ice or precipitation based phenomena. However, the external diameters of the stays are sized in the area in which vortex shedding can occur under different wind speeds. Reynolds numbers for typical sizes of cables suggest turbulent wind flows and stable frequencies of shedding. This in turn can lead to small repetitive oscillating forces acting on the cables. Stays are very strong and flexible structural elements, capable of resisting high tensile stresses. But because of their high slenderness compared to their large lengths, they are very weak to forces acting transversely to them and thus, susceptible to large displacements in the respective directions. On top of that, they are known to have small internal damping in the range of 0.5% to 1%. Taking all of the above into account, it can be said that long cable stays are prone to vibrate in real life conditions and that resonance phenomena is likely to occur. This can lead to damage of both the structural and protective components of the stays and even to their collapse if the vibration levels are left uncheck. It is common practice for engineers to take special measures against this by adopting vibration mitigation devices for the cables, at first in the design phase which will later be tested in wind tunnels. Designing these devices is not an easy task because of the very difficult analyses that have to be made. Wind tunnel tests are mandatory because of the high degree of uncertainty that finite element models give in cable vibration simulations under wind loads. The purpose of this study is to determine the dynamic response of an inclined cable under the effects of wind blowing perpendicular to it and extrapolating these results in order to get a more exact estimation of the behavior of real cable stays used in bridges. In order to do this, a scaled model of a cable stay was tested in a wind tunnel under different wind conditions. Numerical models of these tests were done and their results compared to the recordings. The design parameters of the finite element models were corrected in order to get more precise results between the actual responses and the simulated ones. 2. Test model A scaled model of a cable stay was done in order to simulate its vibrations and record the amplitudes and frequencies of the oscillations. The tests were done in the wind tunnel of the Hydrotechnics Faculty of the Technical University of Civil Engineering of Bucharest (Fig. 1). It is 25.00m long tunnel with 2.75m side, squared section. The cable was made up of a 3-wire strand, each with a diameter of 3mm and yield strength of 1860MPa, encased in a 50mm sheath filled with foam, leading to a weight of about 224 g/m. The cable was placed so that it would be perpendicular to the wind flow, diagonally at about 45° in section (Fig. 2 and 3). It was anchored to the corners of the tunnel, thus having a length of 2.404m, as seen in Fig. 2. Attached to the cable there were three-dimensional accelerometers placed at key points in the middle and quarters of the cable. These were used to measure accelerations in three orthogonal directions, which from now on will be mentioned as “X, Y and Z” according to the following: - “X” – along the cable - “Y” – perpendicular to both the cable and the wind flow - “Z” – perpendicular to the cable and parallel to the wind flow
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Fig. 1. (a) View of the wind tunnel
Fig. 2. Cable model in the wind tunnel
Corresponding measurements were taken from each device with a frequency of 1200Hz for 27 cases, considering 3 different internal stresses in the cable and varying wind speeds between 0 and 15m/s. The free vibration responses of the cable were also studied for each internal stress in order to get a baseline of the dynamic parameters (periods of vibration, internal damping, confirmation of the internal stress state). This was done by inducing a one time impulse on the cable in the form of a prescribed displacement.
Fig. 3. Cable sketch
V.D. Urdareanu et al. / Materials Today: Proceedings 12 (2019) 446–454
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In order to present better, more comprehensive references, the 3 accelerometers will be named according to their place on the cable as follows: - “Acc-Upper” – the upper most accelerometer - “Acc-Center” – the accelerometer in the middle of the cable - “Acc-Lower” – the lower most accelerometer The similitude between the cable model and an actual cable stay is presented in Table 1. The Reynolds criterion is the most important one, because suggests the type of flow. Table 1. Similitude of the test model
Structural mechanics
Characteristic Length
[m]
Section
-
Real cable stay
Cable model
Scale
200
2.404
1:83
37x7Ø5
3Ø3
-
Area
[mm2]
5143
21
1:243
Moment of inertia
[mm4]
7946
12
1:666
Axial stiffness
[N/mm]
5014
1720
1:3
Bending stiffness
[N/mm]
1.94E-07
1.67E-04
1:0.00116
Modulus of elasticity
[GPa]
195
195
1:1
Yield limit
[MPa]
1770
1860
1:0.95
Exterior diameter of sheath
[mm]
155
50
1:3
Weight per meter
[N/m]
500
2.2392
1:223
3
2701.14
116.25
1:23
[m/s]
12
3
1:4
Equivalent density Fluid mehanics
Measuring unit
Air speed
[kg/m ]
Reynolds criterion
-
117246.596
9455.371
1:12
Euler criterion
-
1395
28.125
1:50
Froude criterion
-
0.00144
1.671918721
1:0.00086
3. Numerical models Adequate numerical models of the scaled specimen would have to correctly simulate both the transient wind effects on the cable and the dynamic nonlinear behavior of the vibrating cable. Computer models containing both computational fluid dynamics and dynamic structural mechanics require great amounts of computational power and take up a lot of time to solve. Furthermore, the formulations of the boundary elements between the two sides have proven to be unstable in dynamic analyses. In order to simulate the dynamic response of the cable under wind flow with less computational power and more precise results, the calculation method consisted in separating the numerical approach in 2 different finite element models. The first one (Fig. 4,a) was a 2D model made considering computational fluid dynamics and was used to determine the effect of the wind on the cable, such as vortex shedding and variation of lift and drag coefficients in time. It made up of a rectangle with a circle inside, the first representing the air mass and the second one being the cable. The sides were modeled as boundary conditions with the following properties - left side – inlet - upper and bottom sides – no friction walls - cable – no friction wall - right side - outlet
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V.D. Urdareanu et al. / Materials Today: Proceedings 12 (2019) 446–454
The model consists of roughly 650000 quadratic elements with sizes varying from about 0.01mm on the side of the cable to about 2cm at the margins. Special care was taken in order to ensure at least 2.5 diameters in front and on the sides of the circle, and at least 5 meters in it’s wake. The turbulence model used was k-ω Shear Stress Transport, with a time step of 1e-5s. This was used to simulate the aeroelastic effects of the wind blowing on the cable perpendicular to it and to provide the input for the second model.
Fig. 4 (a) Computational fluid dynamics model (b) Structural mechanics model
The second model was a 3D model of the cable using 49 Timoshenko formulated beam elements with 2 nodes each and 6 degrees of freedom in one, considering the structural mechanics domain (Fig. 4,b). Direct integration time -history analysis was done using Hilbert-Hughes-Taylor Alpha method, with a time step size of 1e-5s. This was used to simulate the dynamic response of the cable under the effect of the forces predetermined from the previous model. It was expected that the main problems of this model would be the time step size and the damping model. The first one was solved by using a very small time step, about 50 times smaller with respect to the time step used in the recordings. For the second problem, the traditional Rayleigh damping method was used, where factors α and β were later corrected according to the results of the real case. 4. Results As expected, it was found that air blowing on the cables induces oscillations with different frequencies, depending on the wind speed, in all the studied cases. The amplitudes of vibration are larger in “Y” direction, perpendicular to the air flow as compared to the ones in the “X” direction. This was the case for both the numerical model approach and the actual situation. The main reason of the occurring vibration is vortex shedding in the wake of the cable. This can be clearly seen in the finite element models results, shown in Fig. 5. As the wind blows over the cable, there are high speed areas forming on the sides of the cylinder. Due to imperfections and turbulence considered in the model, these areas are unstable and tend to shift ever so slightly. This in turn, induces small pressure variations in the back of the cable. In time, the scale of this phenomenon increases and eventually leads to vortices forming and shedding. All of the above can be seen in the following pictures, depicting velocity, pressure and vorticity contours from the finite element models. The pressure changes induce force variations in time on the cable, in both “Y” and “Z” directions. It was found that these oscillations are sinusoidal in shape, with constant amplitudes. Both the amplitudes and shedding frequencies were higher and as the wind speed would increase. The first time steps of the computational fluid dynamics analysis were neglected, because of the inadequate convergence factors. This is usual for this time of analysis, as it takes some time for the fluid flow to stabilize. Both the neglected values and the considered ones can be seen in the following illustration. The lift force’s amplitudes were much greater then the drag force’s, while the maximum values of the force tend to be the same. It was also noted that the frequencies of the drag coefficient twice the ones of the drag coefficient.
V.D. Urdareanu et al. / Materials Today: Proceedings 12 (2019) 446–454
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The values of the lift and drag forces could not be verified based only on the results of this model. Instead, the frequencies of the vibration were compared and were found to match the recordings with errors under 5%.
Fig. 5. (a) Wind speed contour (b) Pressure contour (c) Vorticity contour
Fig. 6. Time variations of the forces acting on the cable
The forces and their variations (Fig. 6) were taken from the computational fluid dynamics model and introduced in the structural mechanics model as external excitations. It was found that, under the excitation provided by the wind, the cable would vibrate with more or less stable amplitudes, depending on the wind speed. The vibrations were composed of several interlaced oscillations, each corresponding to one of the cable’s own modes of vibration. The deformed shapes of the vibrations were consistent with one of the first 6 modes of the cable, but mostly with the first one. As the internal stress was increased, the
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presence of the upper modes was less and less present, eventually leading to only the first mode to be significant. This shows the important role of the internal forces in the cable’s response. The level of tensile stress in the cable can change the periods of the cable and so change it’s response under similar loads. This can be seen in Fig. 7 and Fig. 8, depicting the cable’s response under a 5m/s wind flow and different tensile stresses of 655N, respectively 288N.
Fig. 7. Cable vertical response under wind speed of 5m/s time domain (a) 655N Internal stress (b) 288N Internal stress
The results were processed and analyzed separated in two directions, “Y” (vertical) and “Z” (horizontal). Certain corrections and transformations had to be made in order to get a better understanding of the phenomena. In order to do this, two different approaches were considered, one with time-history variations of the acceleration levels in the three points in the time domain and one in the frequency domain for the same recordings.
Fig. 8. Cable vertical response under wind speed of 5m/s (a) 655N Internal stress (b) 288N Internal stress
The frequency analysis was found to give a much better picture of the dynamic response of the cable, as it illustrates the participation of several interlaced oscillations. The first one shows all 3 recorded points moving in the same direction with comparable values, corresponding to the first mode of vibration of the cable. The second and third modes are also present but with smaller participation factors, as can be seen by spikes in the Fourier spectra. The second picture shows that the third mode is more dominant, with small participations from the first, second and fourth modes. This can also be seen in the time domain, where the upper and lower points moved opposite from the center one, suggesting a 3 wave vibration. The cable’s response in the “Y” direction was consistent with the one in the “Z” direction but at lower amplitudes of acceleration. Accelerations in the “X” direction were very low, and so neglected because they could have been influenced by the vibrations of the tunnel itself. Since the recordings showed similar dynamic responses of the cable
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in the “Y” and “Z” directions, the study focuses only on the vertical accelerations for the second part. Fig. 9 shows a comparison between the “Y” and “Z” accelerations.
Fig. 9. Comparison between vertical and horizontal response of the cable
The recordings show that the internal damping of the cable was very low, ranging between 0.3% and 2%, depending on the internal tensile stress. Increasing the tension in the cable would lead to a decrease in damping and main period of vibration. Calculating the Rayleigh damping model’s α and β coefficients based on the recorded damping did not yield results with sufficient accuracy. It was found by trial that the best results correspond to a variation of 0.5% to 1% damping over the first three main periods of vibration. The time step was also a problem, since different time steps gave different results. This was fixed by decreasing the time step to a level after which further decreasing did not change the results. A time step of 10-5s was found adequate. After tuning all the design parameters, the simulated response and the recorded one were quite close, with errors around 5%, which were considered acceptable. This can be seen in Fig. 10.
Fig. 10. Comparison between measured and estimated cable response
5. Conclusions The vibrations due to wind blowing on the cable were significantly larger in the direction perpendicular to the air flow (vertical) as compared to the ones in the flow’s direction. In most cases, the fundamental mode of vibration was found to be dominant, but there were recordings in which the upper modes would dictate the response, especially the first three. The highest oscillation amplitudes were recorded for relatively low wind speeds of 3 to 5 m/s, for which the shedding frequency overlapped the cable’s frequency and so the resonance phenomena were stronger. It is surprising that although higher wind speeds result in greater forces, the strongest dynamic responses of the cable were recorded
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for relatively moderate wind speeds. Scaling the interval to full scale would suggest that the most dangerous wind speeds for vortex shedding induced vibrations are around 10 to 15 m/s. It must be noted that this study only takes into account vibrations due to vortex shedding. In reality, there are many other vibration inducing phenomena such as wake galloping, icing on the cable, rain-wind combination or even traffic loads. One of the goals of this study was to find a suitable design approach for accurate estimations of cable stays vibrations. Acceptable simulations of cable dynamic responses under wind conditions using finite element models are reliable but very difficult to do, due to the large number of parameters involved. The most important ones are related to the damping model, integration method and time step size. The key parameters needed for precise finite element method analyses of cable vibrations are the time step and the damping model considered. The time step issue can be solved by decreasing the time step size until acceptable results are obtained, thus increasing the computational power required to run the model. As for the damping model, the most common method used was Rayleigh damping, via mass and stiffness coefficients α and β. It was found that the coefficients calculated based on the recorded damping and periods of vibration did not give the most precise results. The most accurate estimation of the behaviour of the cables was obtained by considering α and β according to 1% damping for the first mode of vibration of the cable and 0.5% for the third and last mode of vibration. Precise simulations of cable dynamic responses under wind conditions using finite element models are reliable but very difficult to perform, due to the large number of parameters involved, which is why scaled model tests are recommended for any wind susceptible constructions. References [1] M. Rades, Mechanical vibrations, Printech, Bucharest, 2008 [2] E.L. Wilson, Three dimensional static and dynamic analysis of structures: A physical approach with emphasis on earthquake engineering, second ed., Computers and Structures Inc; Berkeley, California, 1998 [3] Y.A. Cengel, J.M. Cimbala, Fluid Mechanics: Fundamentals and Applications, 4th edition, McGraw Hill Education, New York, 2017 [4] M. Degeratu, Dimensional Analysis, Similarity and Modelling - Guidance for Applications in Fluid Mechanics, Romanian Academy of Scientists, Bucharest, Romania, 2015 [5] A.K. Chopra, Dynamics of structures: Theory and applications to earthquake engineering, Prentice Hall, Upper Saddle River, 2011.
ScienceDirect Materials Today: Proceedings 12 (2019) 446–454
www.materialstoday.com/proceedings
DAS35
Experimental Study of The Dynamic Response of a Cable Under Wind Flow Vlad Daniel Urdareanu*, Ionut Radu Racanel, Mircea Degeratu, Costin Ioan Cosoiu, Cristian Lucian Ghindea, Radu Iuliu Cruciat Technical University of Civil Engineering of Bucharest, Lacul Tei bd, no 122, 72311, Bucharest, Romania
Abstract Nowadays, the high quality of available building materials, powerful machinery and high precision construction techniques allow us to build bridges with very long spans at efficient costs. Depending on the site and the spans, the most common solutions for these types of structures are arch bridges, cable stayed bridges, suspension bridges and even variations among the three. Because of their size and the nature of the crossed obstacles, these types of structures can be very difficult to access outside the carriageways or the pavements and so, inspection maintenance works are very difficult and costly. One critical structural element which is common for all these bridges are the cables. They are out in the open, exposed to a wide variety of phenomena which can cause them damage. Although they perform exceptionally well under tensile forces in longitudinal directions, they are very weak and prone to take damage under transverse loads. For example, it has been observed, that long cable stays tend to vibrate uncontrollably. If left uncheck, these vibrations can resonate and cause damage to the protective sheath, anchorages and even the cables themselves, eventually leading to their collapse. In order to prevent this, engineers must take into account the various mechanisms that can cause vibrations, determine their effect on the stays, and find ways to mitigate them in order to reduce the amplitudes under acceptable levels. Cable vibration simulation using numerical models is very difficult to do, due to the high degree of uncertainty, large volume of processing to be done and lack of validation methods for the results. This paper presents a scaled model analysis of a cable stay in a wind tunnel where vibrations are measured under different wind speeds. Finite element models of the cable are then done, using the measurements to correct certain design parameters in order to improve the precision of the calculated results. © 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of 35th Danubia Adria Symposium on Advances in Experimental Mechanics. Keywords: cable vibrations; vortex shedding; wind tunnel;
*Corresponding author. Tel: +40-726186453 E-mail address: [email protected] 2214-7853 © 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of 35th Danubia Adria Symposium on Advances in Experimental Mechanics.
V.D. Urdareanu et al. / Materials Today: Proceedings 12 (2019) 446–454
447
1. Introduction Cables are the key structural elements of long spanned bridges. This is especially true for cable stayed bridges, due to the fact that these types of structures are highly dependent on their purpose ability to take significant amounts of loads from the superstructure and transmit them to the pylons. This is done by high tensile forces in the cables and leads to an optimization of the distribution of stress throughout the structure. The cables are typically made up of several wires or strands of high strength steel put together in cylindrical, polyethylene sheaths. Evidently, the purpose of the steel elements is to take the loads of the superstructure, while the polyethylene sheaths are used for protection against the various weather phenomena such as wind, rain or icing. While it is true that the two systems should work very well together in obtaining long lasting, high strength and very cost-efficient structural elements, the combination between them can also lead to unforeseen and dangerous situations. Grouping the strands together and covering them with round sheaths ensures that they are protected against most water, ice or precipitation based phenomena. However, the external diameters of the stays are sized in the area in which vortex shedding can occur under different wind speeds. Reynolds numbers for typical sizes of cables suggest turbulent wind flows and stable frequencies of shedding. This in turn can lead to small repetitive oscillating forces acting on the cables. Stays are very strong and flexible structural elements, capable of resisting high tensile stresses. But because of their high slenderness compared to their large lengths, they are very weak to forces acting transversely to them and thus, susceptible to large displacements in the respective directions. On top of that, they are known to have small internal damping in the range of 0.5% to 1%. Taking all of the above into account, it can be said that long cable stays are prone to vibrate in real life conditions and that resonance phenomena is likely to occur. This can lead to damage of both the structural and protective components of the stays and even to their collapse if the vibration levels are left uncheck. It is common practice for engineers to take special measures against this by adopting vibration mitigation devices for the cables, at first in the design phase which will later be tested in wind tunnels. Designing these devices is not an easy task because of the very difficult analyses that have to be made. Wind tunnel tests are mandatory because of the high degree of uncertainty that finite element models give in cable vibration simulations under wind loads. The purpose of this study is to determine the dynamic response of an inclined cable under the effects of wind blowing perpendicular to it and extrapolating these results in order to get a more exact estimation of the behavior of real cable stays used in bridges. In order to do this, a scaled model of a cable stay was tested in a wind tunnel under different wind conditions. Numerical models of these tests were done and their results compared to the recordings. The design parameters of the finite element models were corrected in order to get more precise results between the actual responses and the simulated ones. 2. Test model A scaled model of a cable stay was done in order to simulate its vibrations and record the amplitudes and frequencies of the oscillations. The tests were done in the wind tunnel of the Hydrotechnics Faculty of the Technical University of Civil Engineering of Bucharest (Fig. 1). It is 25.00m long tunnel with 2.75m side, squared section. The cable was made up of a 3-wire strand, each with a diameter of 3mm and yield strength of 1860MPa, encased in a 50mm sheath filled with foam, leading to a weight of about 224 g/m. The cable was placed so that it would be perpendicular to the wind flow, diagonally at about 45° in section (Fig. 2 and 3). It was anchored to the corners of the tunnel, thus having a length of 2.404m, as seen in Fig. 2. Attached to the cable there were three-dimensional accelerometers placed at key points in the middle and quarters of the cable. These were used to measure accelerations in three orthogonal directions, which from now on will be mentioned as “X, Y and Z” according to the following: - “X” – along the cable - “Y” – perpendicular to both the cable and the wind flow - “Z” – perpendicular to the cable and parallel to the wind flow
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Fig. 1. (a) View of the wind tunnel
Fig. 2. Cable model in the wind tunnel
Corresponding measurements were taken from each device with a frequency of 1200Hz for 27 cases, considering 3 different internal stresses in the cable and varying wind speeds between 0 and 15m/s. The free vibration responses of the cable were also studied for each internal stress in order to get a baseline of the dynamic parameters (periods of vibration, internal damping, confirmation of the internal stress state). This was done by inducing a one time impulse on the cable in the form of a prescribed displacement.
Fig. 3. Cable sketch
V.D. Urdareanu et al. / Materials Today: Proceedings 12 (2019) 446–454
449
In order to present better, more comprehensive references, the 3 accelerometers will be named according to their place on the cable as follows: - “Acc-Upper” – the upper most accelerometer - “Acc-Center” – the accelerometer in the middle of the cable - “Acc-Lower” – the lower most accelerometer The similitude between the cable model and an actual cable stay is presented in Table 1. The Reynolds criterion is the most important one, because suggests the type of flow. Table 1. Similitude of the test model
Structural mechanics
Characteristic Length
[m]
Section
-
Real cable stay
Cable model
Scale
200
2.404
1:83
37x7Ø5
3Ø3
-
Area
[mm2]
5143
21
1:243
Moment of inertia
[mm4]
7946
12
1:666
Axial stiffness
[N/mm]
5014
1720
1:3
Bending stiffness
[N/mm]
1.94E-07
1.67E-04
1:0.00116
Modulus of elasticity
[GPa]
195
195
1:1
Yield limit
[MPa]
1770
1860
1:0.95
Exterior diameter of sheath
[mm]
155
50
1:3
Weight per meter
[N/m]
500
2.2392
1:223
3
2701.14
116.25
1:23
[m/s]
12
3
1:4
Equivalent density Fluid mehanics
Measuring unit
Air speed
[kg/m ]
Reynolds criterion
-
117246.596
9455.371
1:12
Euler criterion
-
1395
28.125
1:50
Froude criterion
-
0.00144
1.671918721
1:0.00086
3. Numerical models Adequate numerical models of the scaled specimen would have to correctly simulate both the transient wind effects on the cable and the dynamic nonlinear behavior of the vibrating cable. Computer models containing both computational fluid dynamics and dynamic structural mechanics require great amounts of computational power and take up a lot of time to solve. Furthermore, the formulations of the boundary elements between the two sides have proven to be unstable in dynamic analyses. In order to simulate the dynamic response of the cable under wind flow with less computational power and more precise results, the calculation method consisted in separating the numerical approach in 2 different finite element models. The first one (Fig. 4,a) was a 2D model made considering computational fluid dynamics and was used to determine the effect of the wind on the cable, such as vortex shedding and variation of lift and drag coefficients in time. It made up of a rectangle with a circle inside, the first representing the air mass and the second one being the cable. The sides were modeled as boundary conditions with the following properties - left side – inlet - upper and bottom sides – no friction walls - cable – no friction wall - right side - outlet
450
V.D. Urdareanu et al. / Materials Today: Proceedings 12 (2019) 446–454
The model consists of roughly 650000 quadratic elements with sizes varying from about 0.01mm on the side of the cable to about 2cm at the margins. Special care was taken in order to ensure at least 2.5 diameters in front and on the sides of the circle, and at least 5 meters in it’s wake. The turbulence model used was k-ω Shear Stress Transport, with a time step of 1e-5s. This was used to simulate the aeroelastic effects of the wind blowing on the cable perpendicular to it and to provide the input for the second model.
Fig. 4 (a) Computational fluid dynamics model (b) Structural mechanics model
The second model was a 3D model of the cable using 49 Timoshenko formulated beam elements with 2 nodes each and 6 degrees of freedom in one, considering the structural mechanics domain (Fig. 4,b). Direct integration time -history analysis was done using Hilbert-Hughes-Taylor Alpha method, with a time step size of 1e-5s. This was used to simulate the dynamic response of the cable under the effect of the forces predetermined from the previous model. It was expected that the main problems of this model would be the time step size and the damping model. The first one was solved by using a very small time step, about 50 times smaller with respect to the time step used in the recordings. For the second problem, the traditional Rayleigh damping method was used, where factors α and β were later corrected according to the results of the real case. 4. Results As expected, it was found that air blowing on the cables induces oscillations with different frequencies, depending on the wind speed, in all the studied cases. The amplitudes of vibration are larger in “Y” direction, perpendicular to the air flow as compared to the ones in the “X” direction. This was the case for both the numerical model approach and the actual situation. The main reason of the occurring vibration is vortex shedding in the wake of the cable. This can be clearly seen in the finite element models results, shown in Fig. 5. As the wind blows over the cable, there are high speed areas forming on the sides of the cylinder. Due to imperfections and turbulence considered in the model, these areas are unstable and tend to shift ever so slightly. This in turn, induces small pressure variations in the back of the cable. In time, the scale of this phenomenon increases and eventually leads to vortices forming and shedding. All of the above can be seen in the following pictures, depicting velocity, pressure and vorticity contours from the finite element models. The pressure changes induce force variations in time on the cable, in both “Y” and “Z” directions. It was found that these oscillations are sinusoidal in shape, with constant amplitudes. Both the amplitudes and shedding frequencies were higher and as the wind speed would increase. The first time steps of the computational fluid dynamics analysis were neglected, because of the inadequate convergence factors. This is usual for this time of analysis, as it takes some time for the fluid flow to stabilize. Both the neglected values and the considered ones can be seen in the following illustration. The lift force’s amplitudes were much greater then the drag force’s, while the maximum values of the force tend to be the same. It was also noted that the frequencies of the drag coefficient twice the ones of the drag coefficient.
V.D. Urdareanu et al. / Materials Today: Proceedings 12 (2019) 446–454
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The values of the lift and drag forces could not be verified based only on the results of this model. Instead, the frequencies of the vibration were compared and were found to match the recordings with errors under 5%.
Fig. 5. (a) Wind speed contour (b) Pressure contour (c) Vorticity contour
Fig. 6. Time variations of the forces acting on the cable
The forces and their variations (Fig. 6) were taken from the computational fluid dynamics model and introduced in the structural mechanics model as external excitations. It was found that, under the excitation provided by the wind, the cable would vibrate with more or less stable amplitudes, depending on the wind speed. The vibrations were composed of several interlaced oscillations, each corresponding to one of the cable’s own modes of vibration. The deformed shapes of the vibrations were consistent with one of the first 6 modes of the cable, but mostly with the first one. As the internal stress was increased, the
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presence of the upper modes was less and less present, eventually leading to only the first mode to be significant. This shows the important role of the internal forces in the cable’s response. The level of tensile stress in the cable can change the periods of the cable and so change it’s response under similar loads. This can be seen in Fig. 7 and Fig. 8, depicting the cable’s response under a 5m/s wind flow and different tensile stresses of 655N, respectively 288N.
Fig. 7. Cable vertical response under wind speed of 5m/s time domain (a) 655N Internal stress (b) 288N Internal stress
The results were processed and analyzed separated in two directions, “Y” (vertical) and “Z” (horizontal). Certain corrections and transformations had to be made in order to get a better understanding of the phenomena. In order to do this, two different approaches were considered, one with time-history variations of the acceleration levels in the three points in the time domain and one in the frequency domain for the same recordings.
Fig. 8. Cable vertical response under wind speed of 5m/s (a) 655N Internal stress (b) 288N Internal stress
The frequency analysis was found to give a much better picture of the dynamic response of the cable, as it illustrates the participation of several interlaced oscillations. The first one shows all 3 recorded points moving in the same direction with comparable values, corresponding to the first mode of vibration of the cable. The second and third modes are also present but with smaller participation factors, as can be seen by spikes in the Fourier spectra. The second picture shows that the third mode is more dominant, with small participations from the first, second and fourth modes. This can also be seen in the time domain, where the upper and lower points moved opposite from the center one, suggesting a 3 wave vibration. The cable’s response in the “Y” direction was consistent with the one in the “Z” direction but at lower amplitudes of acceleration. Accelerations in the “X” direction were very low, and so neglected because they could have been influenced by the vibrations of the tunnel itself. Since the recordings showed similar dynamic responses of the cable
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in the “Y” and “Z” directions, the study focuses only on the vertical accelerations for the second part. Fig. 9 shows a comparison between the “Y” and “Z” accelerations.
Fig. 9. Comparison between vertical and horizontal response of the cable
The recordings show that the internal damping of the cable was very low, ranging between 0.3% and 2%, depending on the internal tensile stress. Increasing the tension in the cable would lead to a decrease in damping and main period of vibration. Calculating the Rayleigh damping model’s α and β coefficients based on the recorded damping did not yield results with sufficient accuracy. It was found by trial that the best results correspond to a variation of 0.5% to 1% damping over the first three main periods of vibration. The time step was also a problem, since different time steps gave different results. This was fixed by decreasing the time step to a level after which further decreasing did not change the results. A time step of 10-5s was found adequate. After tuning all the design parameters, the simulated response and the recorded one were quite close, with errors around 5%, which were considered acceptable. This can be seen in Fig. 10.
Fig. 10. Comparison between measured and estimated cable response
5. Conclusions The vibrations due to wind blowing on the cable were significantly larger in the direction perpendicular to the air flow (vertical) as compared to the ones in the flow’s direction. In most cases, the fundamental mode of vibration was found to be dominant, but there were recordings in which the upper modes would dictate the response, especially the first three. The highest oscillation amplitudes were recorded for relatively low wind speeds of 3 to 5 m/s, for which the shedding frequency overlapped the cable’s frequency and so the resonance phenomena were stronger. It is surprising that although higher wind speeds result in greater forces, the strongest dynamic responses of the cable were recorded
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for relatively moderate wind speeds. Scaling the interval to full scale would suggest that the most dangerous wind speeds for vortex shedding induced vibrations are around 10 to 15 m/s. It must be noted that this study only takes into account vibrations due to vortex shedding. In reality, there are many other vibration inducing phenomena such as wake galloping, icing on the cable, rain-wind combination or even traffic loads. One of the goals of this study was to find a suitable design approach for accurate estimations of cable stays vibrations. Acceptable simulations of cable dynamic responses under wind conditions using finite element models are reliable but very difficult to do, due to the large number of parameters involved. The most important ones are related to the damping model, integration method and time step size. The key parameters needed for precise finite element method analyses of cable vibrations are the time step and the damping model considered. The time step issue can be solved by decreasing the time step size until acceptable results are obtained, thus increasing the computational power required to run the model. As for the damping model, the most common method used was Rayleigh damping, via mass and stiffness coefficients α and β. It was found that the coefficients calculated based on the recorded damping and periods of vibration did not give the most precise results. The most accurate estimation of the behaviour of the cables was obtained by considering α and β according to 1% damping for the first mode of vibration of the cable and 0.5% for the third and last mode of vibration. Precise simulations of cable dynamic responses under wind conditions using finite element models are reliable but very difficult to perform, due to the large number of parameters involved, which is why scaled model tests are recommended for any wind susceptible constructions. References [1] M. Rades, Mechanical vibrations, Printech, Bucharest, 2008 [2] E.L. Wilson, Three dimensional static and dynamic analysis of structures: A physical approach with emphasis on earthquake engineering, second ed., Computers and Structures Inc; Berkeley, California, 1998 [3] Y.A. Cengel, J.M. Cimbala, Fluid Mechanics: Fundamentals and Applications, 4th edition, McGraw Hill Education, New York, 2017 [4] M. Degeratu, Dimensional Analysis, Similarity and Modelling - Guidance for Applications in Fluid Mechanics, Romanian Academy of Scientists, Bucharest, Romania, 2015 [5] A.K. Chopra, Dynamics of structures: Theory and applications to earthquake engineering, Prentice Hall, Upper Saddle River, 2011.