- Email: [email protected]
Dynamic response of guyed masts
PII: S0141-0296(97)00206-X
Engineering Structures, Vol. 20, No. 12, pp. 1097–1101, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0141–0296/98 $19.00 + 0.00
Dynamic response of guyed masts Murty K. S. Madugula, Yohanna M. F. Wahba and Gerard R. Monforton Faculty of Engineering, College of Engineering and Science, University of Windsor, Windsor, Ontario, Canada, N9B 3P4
Natural frequencies of guyed masts were determined by modelling the mast as truss elements in one model and as beam-column elements in another model. Guys were modelled as cable elements in both cases. The natural frequencies from the two models agreed closely with each other. For the purpose of verification of the finite element modelling, two triangular steel latticed scale-modelled guyed masts were fabricated and tested on a shake table specially designed to test guyed masts. The experimental results were in good agreement with the results from the finite element analysis. The finite element modelling was then applied to six existing towers which are representative of the design practices in Canada. Based on the results of these analyses, the influence of ice accretion, guy initial tensions and torsion resistors on the dynamic characteristics of the guyed masts is discussed. 1998 Elsevier Science Ltd. All rights reserved Keywords: guyed masts, ice load, latticed towers, natural frequencies, wind load
accompanied with moderate winds can lead to galloping of guys, resulting in unacceptable stress levels throughout the structure. A guyed mast is an example of a structure with nonlinear behaviour even under working conditions. This nonlinear behaviour is due to the change in the stiffness of the guys that occurs with a change in the guy tension. Thus, the natural frequencies of the mast depend on the magnitude of the guy tension at the time of vibration. This paper investigates the natural frequencies of guyed masts. For the purpose of verification of the finite element models used, the analytically determined natural frequencies of small-scale model masts are compared to those obtained experimentally. Six existing masts (differing in height, number of guy levels, wind and ice loading) representing typical guyed masts in Canada were used in the investigation of the effect of icing, initial guy tension and torsion resistors on the natural frequencies of such structures.
1. Introduction Guyed masts are used extensively in the telecommunications industry and the determination of their natural frequencies is important for the understanding of their behaviour under different dynamic loads. These dynamic loads can arise from wind and earthquakes, sudden rupture of guys, galloping of guys, sudden ice shedding from iceladen guy wires, etc. In some cases these vibrations have led to the loss of signals for excessive periods of time due to the displacement and rotation of the antennas, and in other cases have resulted in permanent deformation or failure. Since 1959, there have been approximately 100 confirmed collapses of towers in the United States1. For selfsupporting lattice towers, the first few natural frequencies are important; however, for guyed masts, higher modes of vibration are also significant and can easily be excited by wind and ice loads. Determination of natural frequencies is the first step in determining the dynamic response of guyed masts. It has been reported by Saxena et al.2 that several incidents where heavy icing combined with moderate wind have resulted in serious misalignment of towers, and complete failure in some other cases. Novak et al.3 have shown that accumulation of ice on some parts of the guy wires
2. Modelling The two finite element approaches outlined below use an available commercial computer package ABAQUS4. 2.1. Truss model This model represents a detailed and accurate modelling of the guyed mast. Each member (legs, diagonals and horizontals) forming a three-legged latticed mast was mod-
*Corresponding author. Tel.: + (1) 519-253-4232, ext. 2559; fax: + (1)519-971-3686; e-mail: [email protected]
1097
1098
Dynamic response of guyed masts: M. K. S. Madugula et al.
elled as a two-node, three-dimensional truss element with three degrees of freedom at each node. Cables were modelled as 3-D cable elements (which are truss elements with no compression capability). Two-node and three-node cable elements were used. Each cable was divided into 12 or 20 cable elements depending on the length of the cable. All other components of the mast such as the outriggers (torsion resistors) were also modelled as 3-D truss elements. For the boundary conditions, the base of the mast was modelled as a hinge preventing the three displacements and the torsional rotation. The masses of the antennas and microwave dishes were modelled as additional lumped masses located at their mounting positions on the mast. Because of the inherent non-linear behaviour of tall guyed masts, geometric non-linearity was included in the finite element analysis procedure. The analysis was performed in two steps; in the first step the gravity loads resulting from the self weight of the tower, in addition to the forces due to the initial prestressing in the cables, were applied and equilibrium was achieved. In the second step, the natural frequencies of the structure were extracted. It should be noted here that the resulting model had a large number of degrees of freedom and was considered as a reference for the purpose of comparison with the beam-column model. 2.2. Beam-column model In this model, two-node elastic beam-column elements with six degrees of freedom at each node were used to model the mast. One element was used for every panel in the mast. A rigid link was used to model the outriggers (torsion resistors). Guy modelling and boundary conditions were identical to that of the truss model. In order to determine the equivalent bending, shear and torsional properties of the latticed mast, a typical part of the tower, consisting of four panels as shown in Figure 1, was modelled as a 3-D truss fixed at the base and free at the other end. A moment was applied at the free end and the resulting deflection was calculated, from which the flexural stiffness (EI) was estimated. A lateral force was applied at the free end from which the shear stiffness (GA) was estimated. Then, from the angle of twist due to a torsional moment applied at the top, the torsional stiffness (GJ) was determined. This process was repeated for every section of the mast where the element properties changed. This model resulted in a significant reduction in the number of elements, nodes and solution time.
3. Experimental verification For the purpose of verification of finite element analysis results, a shake table was built and two triangular steel latticed guyed mast models were tested in the Structural Laboratory of the University of Windsor. There are several difficulties in the dynamic testing of guyed masts. Some of the difficulties that had to be addressed for the development of this shake table were: (1) Size of the structure: guyed masts are very tall; their heights may vary anywhere from 40 to 600 m. Considering that the anchor radius for a typical tower is about 0.6–0.7 of the height of the tower, a relatively large shake table is required to model the tower with the anchoring guys. (2) Natural frequency of towers: the structure is flexible
and lightweight and has relatively small natural frequencies. The shake table should have optimal stiffness-to-mass ratio to avoid frequency interference between the shake table and the model masts. (3) Member sizes: typically, a 100 m guyed mast may have solid round legs of 51 mm diameter. In order to use a 1:20 scale model, a mast of 5 m height would be required, with an anchor radius of 3 m. The diameter of the legs will be only 2.5 mm and the bracing members will be still smaller, making the fabrication of the model difficult. The resulting design consisted of a rectangular table 3.5 m × 2.5 m, which allows testing of guyed mast models with a height of 3.6 m. The details of the design are given in Ref. 5. Model guyed mast #1 had a face width of 30.5 mm and was 2210 mm high. It was hinged at the bottom and laterally supported by cables anchored at 1400 mm radius from the base of the mast (Figure 1). Wires of 0.3 and 0.45 mm diameter were used to model the guys. Rods of 3.13 and 1.56 mm diameter, respectively, were used for the legs and bracing members of the tower. Model guyed mast #2 was 2450 mm high with levels at 571, 1334 and 2286 mm, and had an anchorage radius of 1550 mm. The guys were tensioned by turnbuckles and initial guy tensions were measured by custom built load cells. Accelerators were mounted at different locations on the mast to measure horizontal accelerations due to impact loads applied to the guyed mast. From the frequency domain inspection of the response, the natural frequencies were extracted and compared with analytical solutions obtained from the finite element models. Table 1 compares the natural frequencies obtained from the experimental scale-model guyed masts to those from the finite element models. From Table 1 it is seen that there is good agreement between experimental and numerical models and very close agreement between the two numerical models (truss and beam-column models).
4. Determination of natural frequencies of existing guyed masts Six guyed masts were chosen for the purpose of this study. These are existing masts that are representative of the design trends in Canada. The data for these guyed masts were obtained from different tower designers and care had been taken to ensure confidentiality of all the information regarding the designer, fabricator, owner and location of the guyed mast. Table 2 shows the details of these masts with respect to their height, number of guy levels, specified wind and ice loads, and location of torsion resistors. A computer program was developed to create the input file in the form of the finite element package ABAQUS, thus generating nodes, elements and boundary conditions depending on the type of model required (truss or beamcolumn model). This finite element program allowed for modelling of all geometrical and structural details of the mast (e.g. changes in cross-sectional properties, torsion resistors, eccentric locations of the antennas, tapering of the mast at the base, etc.). There are many parameters involved in the design of guyed masts. Usually, the end user would specify the required height and the maximum allowable degree of tilt/twist at the microwave dishes that would satisfy the
Dynamic response of guyed masts: M. K. S. Madugula et al.
1099
Figure 1 Geometry of the experimental model tower Table 1 Comparison of the experimental and analytical results for the model guyed masts Mode
Natural frequency (Hz) Model guyed mast #1
Torsional 1st bending ( xplane) Top cable 2nd bending ( xplane) 3rd bending ( xplane)
Model guyed mast #2
Experimental
Beam model
Truss model
Experimental
Truss model
17.0 24.0
17.1 25.8
17.1 25.9
16.8 20.0
16.8 21.0
32.0 42.0
30.1 42.5
30.2 42.5
30.0 31.0
30.1 33.3
—
51.8
51.9
63.0
66.4
Height (mm)
# of guy levels
Wind pressure (Pa)
Ice thickness (mm)
Location of torsion resistors
46 61 76 91.5 116 122
3 3 4 4 5 6
450 500 306 1002 500 850
25 25 10 50 50 40
Level 3 No torsion resistors Level 4 Levels 2, 3 and 4 No torsion resistors Levels 4 and 5
Table 2 Details of towers used in the analysis Guyed mast #
I II III IV V VI
serviceability requirements of the communications network. However, the designer has the freedom to change other key parameters such as the face width of the mast, number and location of guy levels, radius of the anchors, torsion
resistors (if any), initial guy tensions, etc. Based on the designer’s choice for the above parameters, the sizing of the different members and the guys will be determined. In current practice, this is based on the static analysis of the masts.
Dynamic response of guyed masts: M. K. S. Madugula et al.
1100
usually specified to be 10%. The initial guy tensions of the six guyed masts included in this study were varied from 8% to 15% and the effect on the natural frequencies is shown in Table 4. As can be seen from Table 4, a change in the initial tension of the guys affects the natural frequencies of such masts by as much as 35%. It is also clear that the height of the mast is the parameter that has the largest effect on the fundamental natural frequency of these masts.
5. Effect of icing In this study, the natural frequencies and mode shapes were determined for the bare condition and for four different ice thickness case (10, 25, 40 and 50 mm) specified in CSA Standard S37-946. Mode shapes and natural frequencies of bare and iced guyed masts were compared. The analysis was performed using the truss and beam-column models. The first 100 natural frequencies and mode shapes were obtained. Then all ‘superfluous guy modes’ (i.e. guy modes that did not interact with the mast) were excluded. The remaining modes involved either ‘pure mast’ modes or modes involving coupling between the guy and mast motion; these are the ‘mast modes’ considered in the results. Guyed mast VI which represents an average height (122 m) mast with six guy levels subjected to high wind (850 Pa) and relatively high ice loading (40 mm radial ice thickness) is chosen to demonstrate the typical results of this study. The top two guy levels have six guys at each level and are connected to the mast through torsion resistors. The guyed mast has three microwave dishes that are modelled as concentrated mass elements that are rigidly connected to the mast. Table 3 compares the natural frequencies of guyed mast VI for radial ice accretions of zero, 10, 25 and 40 mm thickness. From Table 3, it is clear that these types of structures exhibit very low natural frequencies thus making them vulnerable to dynamic effects of wind even at low wind speeds. For an ice accretion of 10 mm, a 19% reduction in natural frequency results and this reduction reaches 45% for 40 mm ice accretion. Furthermore, for the natural frequencies obtained under iced cases, icing caused a considerable increase in the coupling between the mast modes and the guy modes, thus making these structures more vulnerable to galloping of the guys. In the absence of ice, the motion in the mast was negligible compared to that of the guys. However, for the iced conditions, not only did the number of coupled mast–guy modes increase, but also the effect of the guy motion on the mast was greatly magnified. Although it is not anticipated that uniform ice accretion of this magnitude would be formed throughout the guyed mast, smaller ice accretions would still produce appreciable reduction of natural frequencies.
7. Effect of torsion resistors These fixtures are usually used at guy levels just below the parabolic microwave antennas in order to minimize the twist at these locations. Six guys, instead of three, are connected to the mast (eight in the case of the four-sided mast) at those locations. These torsion resistors help reduce the twist of the mast. It is found that their effect on the overall dynamic behaviour of the structure is insignificant (less than 5%).
8. Conclusions Based on the finite element analysis and experimental investigation, the following conclusions can be drawn. (1) Both the truss model and the beam-column model are accurate and can account for the inherent complexities involved in the guyed masts. However, the beam-column model results in appreciable savings in solution time. (2) Ice on guyed masts results in significant reduction in natural frequencies. (3) Compared to bare masts, the mode shapes of iced masts involve more coupling between the motion of the guys and that of the mast. (4) Iced guyed masts are vulnerable to dynamic wind effects, including those due to galloping of the guys. (5) The height of the mast has greatest effect on the lowest natural frequency of the guyed masts. (6) An increase in the initial tension of the guys results in a significant increase in its natural frequencies. (7) The effect of torsion resistors on the overall dynamic behaviour of the masts is insignificant (less than 5%).
6. Effect of initial guy tensions Acknowledgements
The initial guy tension has an effect on both the serviceability and safety of the masts. In the current Canadian Standard S37-946, the initial guy tension may vary from 8% to 15% of the breaking strength of the guy, but it is
The authors wish to acknowledge the members of the Canadian Standards Association Technical Committee on Antenna Towers (Committee S37) for supplying the data
Table 3 Natural frequencies for guyed mast VI for different ice accretions Mode number
1 3 7 9 15 17 19 21
Natural frequency (Hz) Bare condition
10 mm radial ice
25 mm radial ice
40 mm radial ice
0.448 0.451 0.465 0.473 0.503 0.504 0.526 0.526
0.373 0.377 0.394 0.406 0.413 0.439 0.453 0.458
0.296 0.302 0.319 0.326 0.343 0.357 0.372 0.384
0.253 0.261 0.276 0.281 0.306 0.311 0.323 0.351
Dynamic response of guyed masts: M. K. S. Madugula et al.
1101
Table 4 Effect of initial guy tensions on the natural frequencies of guyed masts Guyed mast #
Fundamental natural frequency (Hz) Initial guy tension as % of breaking strength
I II III IV V VI
8%
10%
12%
15%
1.00 0.75 0.64 0.56 0.47 0.42
1.11 0.81 0.71 0.61 0.51 0.45
1.22 0.89 0.77 0.66 0.56 0.48
1.36 0.99 0.86 0.73 0.62 0.51
for the guyed masts. Also, the financial support provided by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
3 4
References 1
2
Mulherin, N. D. ‘Atmospheric icing and tower collapse in the U.S., preliminary results,’ in Proceedings of the Seventh International Workshop on Atmospheric Icing of Structures, University of Quebec, Chicoutimi, Quebec, Canada, 3–7 June 1996, p. 457 (abstract only) Saxena, R., Popplewell, N., Trainor, P. G. S. and Shah, A. H. ‘Vibrations of complex guyed towers,’ in Proceedings of the 12th
5
6
Biennial Conference on Mechanical Vibration and Noise Control, Montreal, Quebec, Canada, 1989, pp. 1–7 Novak, M., Davenport, A. G. and Tanaka, H. Vibration of towers due to galloping of iced cables, Journal of Engineering Mechanics Division, ASCE 1978, 104, 457–473 Hibbitt, H. D., Karlsson, B. I. and Sorenson, J. ‘ABAQUS General Purpose Finite Element Code,’ Hibbitt, Karlsson and Sorenson, Inc., Providence, RI, 1995 Wahba, Y. M. F., Madugula, M. K. S. and Monforton, G. R. ‘Shake table for the dynamic testing of guyed towers,’ in Proceedings of the ASCE Structures Congress XV, Portland, Oregon, USA, Vol. 1, 13– 16 April, 1997, pp. 353–357 Canadian Standards Association ‘Antennas, towers and antenna-supporting structures,’ CSA S37-94, Etobicoke, Ontario, Canada, 1994
Engineering Structures, Vol. 20, No. 12, pp. 1097–1101, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0141–0296/98 $19.00 + 0.00
Dynamic response of guyed masts Murty K. S. Madugula, Yohanna M. F. Wahba and Gerard R. Monforton Faculty of Engineering, College of Engineering and Science, University of Windsor, Windsor, Ontario, Canada, N9B 3P4
Natural frequencies of guyed masts were determined by modelling the mast as truss elements in one model and as beam-column elements in another model. Guys were modelled as cable elements in both cases. The natural frequencies from the two models agreed closely with each other. For the purpose of verification of the finite element modelling, two triangular steel latticed scale-modelled guyed masts were fabricated and tested on a shake table specially designed to test guyed masts. The experimental results were in good agreement with the results from the finite element analysis. The finite element modelling was then applied to six existing towers which are representative of the design practices in Canada. Based on the results of these analyses, the influence of ice accretion, guy initial tensions and torsion resistors on the dynamic characteristics of the guyed masts is discussed. 1998 Elsevier Science Ltd. All rights reserved Keywords: guyed masts, ice load, latticed towers, natural frequencies, wind load
accompanied with moderate winds can lead to galloping of guys, resulting in unacceptable stress levels throughout the structure. A guyed mast is an example of a structure with nonlinear behaviour even under working conditions. This nonlinear behaviour is due to the change in the stiffness of the guys that occurs with a change in the guy tension. Thus, the natural frequencies of the mast depend on the magnitude of the guy tension at the time of vibration. This paper investigates the natural frequencies of guyed masts. For the purpose of verification of the finite element models used, the analytically determined natural frequencies of small-scale model masts are compared to those obtained experimentally. Six existing masts (differing in height, number of guy levels, wind and ice loading) representing typical guyed masts in Canada were used in the investigation of the effect of icing, initial guy tension and torsion resistors on the natural frequencies of such structures.
1. Introduction Guyed masts are used extensively in the telecommunications industry and the determination of their natural frequencies is important for the understanding of their behaviour under different dynamic loads. These dynamic loads can arise from wind and earthquakes, sudden rupture of guys, galloping of guys, sudden ice shedding from iceladen guy wires, etc. In some cases these vibrations have led to the loss of signals for excessive periods of time due to the displacement and rotation of the antennas, and in other cases have resulted in permanent deformation or failure. Since 1959, there have been approximately 100 confirmed collapses of towers in the United States1. For selfsupporting lattice towers, the first few natural frequencies are important; however, for guyed masts, higher modes of vibration are also significant and can easily be excited by wind and ice loads. Determination of natural frequencies is the first step in determining the dynamic response of guyed masts. It has been reported by Saxena et al.2 that several incidents where heavy icing combined with moderate wind have resulted in serious misalignment of towers, and complete failure in some other cases. Novak et al.3 have shown that accumulation of ice on some parts of the guy wires
2. Modelling The two finite element approaches outlined below use an available commercial computer package ABAQUS4. 2.1. Truss model This model represents a detailed and accurate modelling of the guyed mast. Each member (legs, diagonals and horizontals) forming a three-legged latticed mast was mod-
*Corresponding author. Tel.: + (1) 519-253-4232, ext. 2559; fax: + (1)519-971-3686; e-mail: [email protected]
1097
1098
Dynamic response of guyed masts: M. K. S. Madugula et al.
elled as a two-node, three-dimensional truss element with three degrees of freedom at each node. Cables were modelled as 3-D cable elements (which are truss elements with no compression capability). Two-node and three-node cable elements were used. Each cable was divided into 12 or 20 cable elements depending on the length of the cable. All other components of the mast such as the outriggers (torsion resistors) were also modelled as 3-D truss elements. For the boundary conditions, the base of the mast was modelled as a hinge preventing the three displacements and the torsional rotation. The masses of the antennas and microwave dishes were modelled as additional lumped masses located at their mounting positions on the mast. Because of the inherent non-linear behaviour of tall guyed masts, geometric non-linearity was included in the finite element analysis procedure. The analysis was performed in two steps; in the first step the gravity loads resulting from the self weight of the tower, in addition to the forces due to the initial prestressing in the cables, were applied and equilibrium was achieved. In the second step, the natural frequencies of the structure were extracted. It should be noted here that the resulting model had a large number of degrees of freedom and was considered as a reference for the purpose of comparison with the beam-column model. 2.2. Beam-column model In this model, two-node elastic beam-column elements with six degrees of freedom at each node were used to model the mast. One element was used for every panel in the mast. A rigid link was used to model the outriggers (torsion resistors). Guy modelling and boundary conditions were identical to that of the truss model. In order to determine the equivalent bending, shear and torsional properties of the latticed mast, a typical part of the tower, consisting of four panels as shown in Figure 1, was modelled as a 3-D truss fixed at the base and free at the other end. A moment was applied at the free end and the resulting deflection was calculated, from which the flexural stiffness (EI) was estimated. A lateral force was applied at the free end from which the shear stiffness (GA) was estimated. Then, from the angle of twist due to a torsional moment applied at the top, the torsional stiffness (GJ) was determined. This process was repeated for every section of the mast where the element properties changed. This model resulted in a significant reduction in the number of elements, nodes and solution time.
3. Experimental verification For the purpose of verification of finite element analysis results, a shake table was built and two triangular steel latticed guyed mast models were tested in the Structural Laboratory of the University of Windsor. There are several difficulties in the dynamic testing of guyed masts. Some of the difficulties that had to be addressed for the development of this shake table were: (1) Size of the structure: guyed masts are very tall; their heights may vary anywhere from 40 to 600 m. Considering that the anchor radius for a typical tower is about 0.6–0.7 of the height of the tower, a relatively large shake table is required to model the tower with the anchoring guys. (2) Natural frequency of towers: the structure is flexible
and lightweight and has relatively small natural frequencies. The shake table should have optimal stiffness-to-mass ratio to avoid frequency interference between the shake table and the model masts. (3) Member sizes: typically, a 100 m guyed mast may have solid round legs of 51 mm diameter. In order to use a 1:20 scale model, a mast of 5 m height would be required, with an anchor radius of 3 m. The diameter of the legs will be only 2.5 mm and the bracing members will be still smaller, making the fabrication of the model difficult. The resulting design consisted of a rectangular table 3.5 m × 2.5 m, which allows testing of guyed mast models with a height of 3.6 m. The details of the design are given in Ref. 5. Model guyed mast #1 had a face width of 30.5 mm and was 2210 mm high. It was hinged at the bottom and laterally supported by cables anchored at 1400 mm radius from the base of the mast (Figure 1). Wires of 0.3 and 0.45 mm diameter were used to model the guys. Rods of 3.13 and 1.56 mm diameter, respectively, were used for the legs and bracing members of the tower. Model guyed mast #2 was 2450 mm high with levels at 571, 1334 and 2286 mm, and had an anchorage radius of 1550 mm. The guys were tensioned by turnbuckles and initial guy tensions were measured by custom built load cells. Accelerators were mounted at different locations on the mast to measure horizontal accelerations due to impact loads applied to the guyed mast. From the frequency domain inspection of the response, the natural frequencies were extracted and compared with analytical solutions obtained from the finite element models. Table 1 compares the natural frequencies obtained from the experimental scale-model guyed masts to those from the finite element models. From Table 1 it is seen that there is good agreement between experimental and numerical models and very close agreement between the two numerical models (truss and beam-column models).
4. Determination of natural frequencies of existing guyed masts Six guyed masts were chosen for the purpose of this study. These are existing masts that are representative of the design trends in Canada. The data for these guyed masts were obtained from different tower designers and care had been taken to ensure confidentiality of all the information regarding the designer, fabricator, owner and location of the guyed mast. Table 2 shows the details of these masts with respect to their height, number of guy levels, specified wind and ice loads, and location of torsion resistors. A computer program was developed to create the input file in the form of the finite element package ABAQUS, thus generating nodes, elements and boundary conditions depending on the type of model required (truss or beamcolumn model). This finite element program allowed for modelling of all geometrical and structural details of the mast (e.g. changes in cross-sectional properties, torsion resistors, eccentric locations of the antennas, tapering of the mast at the base, etc.). There are many parameters involved in the design of guyed masts. Usually, the end user would specify the required height and the maximum allowable degree of tilt/twist at the microwave dishes that would satisfy the
Dynamic response of guyed masts: M. K. S. Madugula et al.
1099
Figure 1 Geometry of the experimental model tower Table 1 Comparison of the experimental and analytical results for the model guyed masts Mode
Natural frequency (Hz) Model guyed mast #1
Torsional 1st bending ( xplane) Top cable 2nd bending ( xplane) 3rd bending ( xplane)
Model guyed mast #2
Experimental
Beam model
Truss model
Experimental
Truss model
17.0 24.0
17.1 25.8
17.1 25.9
16.8 20.0
16.8 21.0
32.0 42.0
30.1 42.5
30.2 42.5
30.0 31.0
30.1 33.3
—
51.8
51.9
63.0
66.4
Height (mm)
# of guy levels
Wind pressure (Pa)
Ice thickness (mm)
Location of torsion resistors
46 61 76 91.5 116 122
3 3 4 4 5 6
450 500 306 1002 500 850
25 25 10 50 50 40
Level 3 No torsion resistors Level 4 Levels 2, 3 and 4 No torsion resistors Levels 4 and 5
Table 2 Details of towers used in the analysis Guyed mast #
I II III IV V VI
serviceability requirements of the communications network. However, the designer has the freedom to change other key parameters such as the face width of the mast, number and location of guy levels, radius of the anchors, torsion
resistors (if any), initial guy tensions, etc. Based on the designer’s choice for the above parameters, the sizing of the different members and the guys will be determined. In current practice, this is based on the static analysis of the masts.
Dynamic response of guyed masts: M. K. S. Madugula et al.
1100
usually specified to be 10%. The initial guy tensions of the six guyed masts included in this study were varied from 8% to 15% and the effect on the natural frequencies is shown in Table 4. As can be seen from Table 4, a change in the initial tension of the guys affects the natural frequencies of such masts by as much as 35%. It is also clear that the height of the mast is the parameter that has the largest effect on the fundamental natural frequency of these masts.
5. Effect of icing In this study, the natural frequencies and mode shapes were determined for the bare condition and for four different ice thickness case (10, 25, 40 and 50 mm) specified in CSA Standard S37-946. Mode shapes and natural frequencies of bare and iced guyed masts were compared. The analysis was performed using the truss and beam-column models. The first 100 natural frequencies and mode shapes were obtained. Then all ‘superfluous guy modes’ (i.e. guy modes that did not interact with the mast) were excluded. The remaining modes involved either ‘pure mast’ modes or modes involving coupling between the guy and mast motion; these are the ‘mast modes’ considered in the results. Guyed mast VI which represents an average height (122 m) mast with six guy levels subjected to high wind (850 Pa) and relatively high ice loading (40 mm radial ice thickness) is chosen to demonstrate the typical results of this study. The top two guy levels have six guys at each level and are connected to the mast through torsion resistors. The guyed mast has three microwave dishes that are modelled as concentrated mass elements that are rigidly connected to the mast. Table 3 compares the natural frequencies of guyed mast VI for radial ice accretions of zero, 10, 25 and 40 mm thickness. From Table 3, it is clear that these types of structures exhibit very low natural frequencies thus making them vulnerable to dynamic effects of wind even at low wind speeds. For an ice accretion of 10 mm, a 19% reduction in natural frequency results and this reduction reaches 45% for 40 mm ice accretion. Furthermore, for the natural frequencies obtained under iced cases, icing caused a considerable increase in the coupling between the mast modes and the guy modes, thus making these structures more vulnerable to galloping of the guys. In the absence of ice, the motion in the mast was negligible compared to that of the guys. However, for the iced conditions, not only did the number of coupled mast–guy modes increase, but also the effect of the guy motion on the mast was greatly magnified. Although it is not anticipated that uniform ice accretion of this magnitude would be formed throughout the guyed mast, smaller ice accretions would still produce appreciable reduction of natural frequencies.
7. Effect of torsion resistors These fixtures are usually used at guy levels just below the parabolic microwave antennas in order to minimize the twist at these locations. Six guys, instead of three, are connected to the mast (eight in the case of the four-sided mast) at those locations. These torsion resistors help reduce the twist of the mast. It is found that their effect on the overall dynamic behaviour of the structure is insignificant (less than 5%).
8. Conclusions Based on the finite element analysis and experimental investigation, the following conclusions can be drawn. (1) Both the truss model and the beam-column model are accurate and can account for the inherent complexities involved in the guyed masts. However, the beam-column model results in appreciable savings in solution time. (2) Ice on guyed masts results in significant reduction in natural frequencies. (3) Compared to bare masts, the mode shapes of iced masts involve more coupling between the motion of the guys and that of the mast. (4) Iced guyed masts are vulnerable to dynamic wind effects, including those due to galloping of the guys. (5) The height of the mast has greatest effect on the lowest natural frequency of the guyed masts. (6) An increase in the initial tension of the guys results in a significant increase in its natural frequencies. (7) The effect of torsion resistors on the overall dynamic behaviour of the masts is insignificant (less than 5%).
6. Effect of initial guy tensions Acknowledgements
The initial guy tension has an effect on both the serviceability and safety of the masts. In the current Canadian Standard S37-946, the initial guy tension may vary from 8% to 15% of the breaking strength of the guy, but it is
The authors wish to acknowledge the members of the Canadian Standards Association Technical Committee on Antenna Towers (Committee S37) for supplying the data
Table 3 Natural frequencies for guyed mast VI for different ice accretions Mode number
1 3 7 9 15 17 19 21
Natural frequency (Hz) Bare condition
10 mm radial ice
25 mm radial ice
40 mm radial ice
0.448 0.451 0.465 0.473 0.503 0.504 0.526 0.526
0.373 0.377 0.394 0.406 0.413 0.439 0.453 0.458
0.296 0.302 0.319 0.326 0.343 0.357 0.372 0.384
0.253 0.261 0.276 0.281 0.306 0.311 0.323 0.351
Dynamic response of guyed masts: M. K. S. Madugula et al.
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Table 4 Effect of initial guy tensions on the natural frequencies of guyed masts Guyed mast #
Fundamental natural frequency (Hz) Initial guy tension as % of breaking strength
I II III IV V VI
8%
10%
12%
15%
1.00 0.75 0.64 0.56 0.47 0.42
1.11 0.81 0.71 0.61 0.51 0.45
1.22 0.89 0.77 0.66 0.56 0.48
1.36 0.99 0.86 0.73 0.62 0.51
for the guyed masts. Also, the financial support provided by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
3 4
References 1
2
Mulherin, N. D. ‘Atmospheric icing and tower collapse in the U.S., preliminary results,’ in Proceedings of the Seventh International Workshop on Atmospheric Icing of Structures, University of Quebec, Chicoutimi, Quebec, Canada, 3–7 June 1996, p. 457 (abstract only) Saxena, R., Popplewell, N., Trainor, P. G. S. and Shah, A. H. ‘Vibrations of complex guyed towers,’ in Proceedings of the 12th
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Biennial Conference on Mechanical Vibration and Noise Control, Montreal, Quebec, Canada, 1989, pp. 1–7 Novak, M., Davenport, A. G. and Tanaka, H. Vibration of towers due to galloping of iced cables, Journal of Engineering Mechanics Division, ASCE 1978, 104, 457–473 Hibbitt, H. D., Karlsson, B. I. and Sorenson, J. ‘ABAQUS General Purpose Finite Element Code,’ Hibbitt, Karlsson and Sorenson, Inc., Providence, RI, 1995 Wahba, Y. M. F., Madugula, M. K. S. and Monforton, G. R. ‘Shake table for the dynamic testing of guyed towers,’ in Proceedings of the ASCE Structures Congress XV, Portland, Oregon, USA, Vol. 1, 13– 16 April, 1997, pp. 353–357 Canadian Standards Association ‘Antennas, towers and antenna-supporting structures,’ CSA S37-94, Etobicoke, Ontario, Canada, 1994