The influence of the design methodology in the response of transmission towers to wind loading

Transcription

1 Journal of Wind Engineering and Industrial Aerodynamics 91 (23) The influence of the design methodology in the response of transmission towers to wind loading A.M. Loredo-Souza a, *, A.G. Davenport b a Departamento de Engenharia Civil, Laborat!orio de Aerodin#amica das Constru@*oes, Universidade Federal do Rio Grande do Sul, Av. Osvaldo Aranha, 99/35, Porto Alegre RS , Brazil b Boundary Layer Wind Tunnel Laboratory, University of Western Ontario, London, Ont., Canada N6A 5B9 Abstract From a theoretical approach, the design procedure for the establishment of wind loading on transmission towers was reviewed and current procedures, suchas Davenport s gust response factor (GRF), were compared with the statistical method using influence lines (SIL), which is considered more realistic. This latter approach can account for unbalanced loading effects, shear and axial loads and the effects of higher modes of vibration in the calculation of the response factors. Several responses due to certain assumed transverse wind characteristics were calculated for some typical transmission towers. The main findings were: (a) Peak loads calculated using SIL were larger than peak loads given by the GRF. (b) The dynamic response of transmission structures is strongly dependent on the turbulence intensity level and its spectrum. (c) For members in which there is reversal in the forces on the load position, the resonant response in the second mode of vibration was bigger, even by four to five times, than the corresponding one in the first mode. Although this effect is not as severe in terms of resulting stresses when all the components are computed in the peak responses, it can lead to fatigue problems. From the current results it can be concluded that the incorporation of the dynamic properties of transmission structures in the design methodologies is needed and that the statistical method using influence lines is a more correct approach since it allows for the inclusion of a larger number of factors in the design methodology. r 23 Elsevier Ltd. All rights reserved. Keywords: Wind; Towers; Transmission lines; Codes; Steel structures *Corresponding author. Tel./fax: address: lac@cpgec.ufrgs.br (A.M. Loredo-Souza) /3/$ - see front matter r 23 Elsevier Ltd. All rights reserved. doi:1.116/s (3)48-5

2 996 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Introduction Transmission line systems are very wind sensitive. The response of structures to wind action may envolve a wide range of structural actions including resultant forces, bending moments, cable tensions, as well as deflections and accelerations [1]. A typical form of response for structures subjected to wind loading is illustrated in Fig. 1. The total response is formed by the three components: (i) Mean ð%rþ: mean response in time; (ii) Background ð*r B Þ: the energy is spread over a broad range in the low frequency range; (iii) Resonant ð*r Rj Þ: consists of a series of highly concentrated peaks centred on the natural frequencies of the structure. An important tool in the analysis of the response is the use of influence lines or functions. Fig. 2 [2] shows the influence function for axial tension, F T, in a bracing member of a lattice tower of the Eiffel type, in which the lines of the main legs of the braced panel would intersect if produced upwards. The unit load at C, immediately above the braced panel produces the maximum tension in the member. The unit load B at the intersection point of the legs has no effect on the bracing member, since the load is resisted wholly by the main legs. The unit load A at the top of the tower puts the bracing member into compression. The influence line is therefore positive in sign below B and negative above it. Such towers are frequently shaped so that the main legs intersect at the center of pressure of the mean wind loading profile with the objective of minimizing the loads in the bracing members. Fig. 2 and respective explanation were extracted from Cook [2]. Wind loading is by its nature a dynamic force, which effect on a structure as a whole is to start it vibrating at its natural frequency and so inducing dynamic bending. This causes shear and bending stresses at all points, depending on the mass and acceleration of that point. Tower motion is dominated by structural damping, being also influenced by damping of aerodynamic origin. Typical values of structural damping for transmission structures are given in Table 1, which is extracted from ASCE [3]. The aerodynamic damping for the tower can be estimated by using modal analysis as [4]: z aj ¼ C j ¼ C j ; C crit 2o j M j ð1þ Fig. 1. Response of a structure to wind: (a) time history; and (b) power spectrum.

3 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Fig. 2. Tension in shear bracing member of lattice tower (after Cook [2]). Table 1 Approximate dynamic properties for transmission structures [3] Type of structure Fundamental frequency, e T (Hz) Damping ratio, z s Lattice tower H-frame Pole where C j is the modal damping at mode j; C crit the critical modal damping and M j the modal mass at the jthmode of vibration. The modal damping is given by C j ¼ Z H Cm 2 j ðzþ dz ð2þ with C ¼ r a %VC D w: Recognizing that generally the mean velocity %V; the drag coefficient C D and the tower s width w vary with height, we have: R H r z aj ¼ a %VðzÞC D ðzþwðzþm 2 j ðzþ dz R 4pf H Tj mðzþm2 j ðzþ dz : ð3þ A fundamental concept is the solidity ratio f defined as the ratio between the total frontal area of all individual members (effective or solid area), A s ; and the frontal envelope area, A; i.e., the area of the limiting surface that includes all the individual members: f ¼ A s =A: ð4þ

4 998 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Theoretical approaches 2.1. Statistical method using influence lines Based on the above considerations and the statistical method described in Loredo- Souza [5], we can write the main equations assumed for the calculation of each of the three kinds of response of the tower. The load on small cross-section of structure will be: d %FðzÞ ¼ 1 2 r a % V 2 ðzþc D ðzþfðzþwðzþ dz ð5þ for the mean component, and dfðz; tþ ¼r a %VðzÞvðz; tþc D ðzþfðzþwðzþ dz ð6þ for the fluctuating component. The three kinds of reponse are then: (a) Mean response: The mean response is given by %r ¼ 1 Z H 2 r a V % 2 z 2afðzÞCD H ðzþwðzþiðzþ dz H Z H z 2afðzÞCD ¼ q H ðzþwðzþiðzþ dz: ð7þ H (b) Background response: The background response is given by: Z H Z H *r 2 B ¼ r 2 a C DðzÞC D ðz ÞfðzÞfðz Þ½ %VðzÞvðz; tþiðzþš½ %Vðz Þvðz ; tþiðz ÞŠ *r 2 B ¼ r2 a wðzþwðz Þdz dz ; Z H Z H C D ðzþc D ðz ÞfðzÞfðz Þ %VðzÞ %Vðz Þs v s v Rðv z ; v z ÞiðzÞiðz Þ wðzþwðz Þdz dz ; ð9þ where Rðv z ; v z Þ is the cross-correlation coefficient between v at the two heights z and z and can be expressed by: Rðv z ; v z Þ¼ vðz; tþvðz ; tþ De ð Dz=z L v Þ ; ð1þ s v s v where Dz ¼jz z j: Then: Z H Z H *r 2 B ¼ð2q HI v Þ 2 C D ðzþc D ðz ÞfðzÞfðz z a z a Þ H H e ðdz=z L v Þ iðzþiðz ÞwðzÞwðz Þdz dz ; ð11þ (c) Resonant response: The spectral density of the modal generalized force is S Qj ðf j Þ¼r 2 a w2 ðf Þ Z H Z H C D ðzþc D ðz ÞfðzÞfðz Þ %VðzÞ %Vðz ÞS v ðz; z ; f Þm j ðzþm j ðz ÞwðzÞwðz Þdz dz ; ð8þ ð12þ

5 where w 2 ðf Þ is the aerodynamic admittance, m j ðzþ is the mode shape, S v ðz; z ; f Þ is the cross spectral density of wind velocity. The latter is used in the calculation of the correlation of the individual frequency components of wind turbulence, or coherence. The square root of the coherence, when plotted against the reduced frequency, can be approximated by an exponential function of the form: C Dz gðdz; f ÞDe j jf =VðzÞ ; ð13þ where C is the exponential decay factor for narrow band correlation. The spectra of the generalized force may be simplified to f j S Qj ðf Þ¼ð2q H I v Þ 2 f Z js v ðf Þ H Z H w 2 ðf Þ C D ðzþc D ðz ÞfðzÞfðz z a z Þ H H s 2 v C Dz e j jf = %V H=2 m j ðzþm j ðz ÞwðzÞwðz Þdz dz : ð14þ The variance of the generalized modal co-ordinate, y j ; can be given approximately by s 2 R ðy jþe p 1 f j S Qj ðf j Þ ð15þ 4ðz aj þ z s Þ K 2 j which is obtained by integrating the spectral density for modal response (displacement) over a narrow band in the vicinity of the natural frequency corresponding to mode j. K j ¼ o 2 j M j is the modal stiffness, and M j the modal mass. The resonant response for local effects (shear, bending moment, etc.) is obtained by multiplying the square root of Eq. (14) by the response participation factor: *r Rj ¼ s R ðy j Þ Z H ARTICLE IN PRESS A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) mðzþo 2 j m jðzþiðzþ dz; where o j natural circular frequency of vibration for mode j; and therefore: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R H p f j S Qj ðf j Þ *r Rj ¼ mðzþm jðzþiðzþ dz R 4 ðz aj þ z s Þ H : ð17þ mðzþm2 j ðzþ dz (d) Total response: The total peak response #r is given by #r ¼ %r þ g s *r ð18þ withthe total fluctuating response *r (rms value) being: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *r ¼ *r 2 B þ X j *r2 Rj: ð19þ The statistical peak factor g s is given by p g S ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lnðuTÞþpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :577 ð2þ 2lnðuTÞ being the time T around s and the crossing rate u estimated by P f 2 u 2 j *r 2 Rj ¼ *r 2 B þ P *r 2 : ð21þ Rj a ð16þ

6 1 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Velocity gust factor The current procedure for the design of transmission structures is largely based on the assumption of a static behaviour of the structure. A certain pattern of wind loading is assumed, generally a power law profile for velocities, the aerodynamic force coefficients determined and the corresponding pressures calculated. The peak wind velocity used can be estimated by #V ¼ %V þ g s s v ð22þ or s v #V ¼ %V 1 þ g s ¼ %Vð1 þ g s I v Þ: %V ð23þ The hourly mean wind speed is multiplied by a gust factor and, therefore, the corresponding mean wind pressure by the square of it Gust response factor An attempt to consider dynamic effects on the response of these structures is made through the gust response factor (GRF) method suggested by Davenport [6]. This is incorporated in the ASCE [3] guidelines for the loading of transmission structures, but with the resonant response component neglected. The approach is based on statistical methods which take account of the spatial correlation and energy spectrum of wind speed and the dynamic response of the transmission line system. The peak response is given by #r ¼ %F D i r G t ; ð24þ where %F D is the mean wind drag force, i r an influence coefficient and G t the gust response factor for the tower and is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G t ¼ 1 þ :75g s E x B t þ R t ; ð25þ withthe exposure factor E x being: p E x ¼ ffiffiffiffiffiffiffiffi z a ref 24k ; ð26þ h o where z ref is the reference height, h o the effective height (at approximate center of pressure of structure), a the power-law exponent and k the surface drag coefficient for which typical values are given in Table 2. The dimensionless resonant term R t is: R t ¼ :123 f 5=3 Th o 1 ð27þ %V o z being f T the tower s natural frequency, %V o the mean wind speed at effective height h o and z the tower s damping. The dimensionless response term corresponding to the quasi-static background wind loading on the tower, B t is given by (H the tower s height and L v the transverse

7 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Table 2 Typical values of the surface drag coefficient k [6] Type of terrain Power law exponent, a Surface drag coefficient, k (z ref ¼ 1 m) Open country, flat shorelines.1.15 Farmland, scattered trees and buildings.16.5 Woodland, suburbs integral scale of turbulence): 1 B t ¼ : ð28þ 1 þ :375H=L v There are, however, some simplifications in the GRF method. It does not account for unbalanced loading effects, shear and axial loads, nor the effect of higher modes of vibration in the calculation of the response factors. A unified approach for a variety of structural responses is possible through the statistical method using influence lines, which can easily incorporate all those factors. 3. Calculation of the responses Several responses due to certain assumed wind characteristics were calculated in a typical transmission tower. These values were obtained considering transverse wind on the tower only. The consideration of the conductors masses will, in general, decrease the natural frequencies of the towers, therefore increasing the resonant response. The aerodynamic admittance function was assumed to be equal to unit in the analyses. The tower studied is shown in Fig. 3 and it is an example of a suspension tower. The mass distribution, mðzþ; natural frequencies and mode shapes, are also shown. The solidity ratio, fðzþ; and drag coefficient distribution, C D ðzþ; are shown in Fig. 4, being the influence lines for tension in certain members shown in Fig. 5. Initially the response (tension) due to wind action was calculated in the four members indicated in Fig. 3. The mean, background and resonant components of the response were obtained following the formulation presented above. The response in the member was calculated for two kinds of terrain: open country, a ¼ :1; and suburban exposure, a ¼ :25: It was assumed an open country wind speed of 5 m/s at the tower s top, H ¼ 43:9 m, and a corresponding velocity for the suburban exposure. A third velocity value was included (V ¼ 42:4 m/s and a ¼ :143) because that is the value adopted by the Utility Company from which the tower was obtained and the responses given by different methods will be compared for this value. The results are shown in Table 1 withthe assumed parameters: turbulence lengthscale, L v ¼ 5 m, exponential decay factor for narrow band correlation, C ¼ 8; tower damping z ¼ :1 and statistical peak factor g s ¼ 3:6:

8 12 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Fig. 3. Transmission tower, natural frequencies and mode shapes and mass distribution. Fig. 4. Solidity ratio and drag coefficient for the tower. From the analysis of Tables 3 and 4 it can be observed that, for the members in which there is reversal in the efforts depending on the load position, the resonant response in the second mode of vibration was bigger, by four to five times, than the corresponding one in the first mode. Although this effect is not as severe in terms of resulting stresses when all the components are computed in the peak response, it can

9 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Fig. 5. Influence line for tension in members F1T, F1, F3AT and P9T for the tower. Table 3 Member responses for: L v ¼ 5 m, C¼ 8, z ¼ :1 (a) Member F1 F3AT V H (m/s) a R (N) 27,119 84,73 58, R B (N) 13,499 15,865 16, R R1 (N) ,651 12, r R2 (N) r (N) 83,125 16,12 131, , G (b) Member F1T P9T V H (m/s) a r (N) 278 1, ,424 15,725 r B (N) r R1 (N) r R2 (N) r (N) ,3 14,668 2,88 43,787 34,383 G

10 14 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Table 4 Member responses for: L v ¼ 5 m, C¼8, z ¼ :2 (a) Member F1 F3AT V H (m/s) a r (N) 27,119 84,73 58, r B (N) 13,499 15,865 16, r R1 (N) r R2 (N) r (N) 1st & 2nd modes 79, , , , r (N)1st mode 79,437 15, , , G 1st & 2nd modes G 1st mode only (b) Member F1T P9T V (m/s) a r (N) 278 1, ,424 15,725 r B (N) r R1 (N) r R2 (N) r (N) 1st & 2nd modes ,67 14,492 19,97 43,439 34,12 r (N) 1st mode ,836 14,322 19,856 43,95 33,861 G 1st & 2nd modes G 1st mode only lead to fatigue problems and therefore decrease the nominal strength of the members in a shorter time than expected. Table 5 presents response values (force on members) calculated from different methodologies. The design values (do to wind only) used by the Utility Company are compared with the gust response factor method and the statistical method using influence lines (SIL). The parameters adopted in the GRF method are: a ¼ :143; H ¼ 43:9m, V H ¼ 42:4 m/s, z ref ¼ 1 m, V 1 ¼ 34:3 m/s, k ¼ :4; h o ¼ 29:3m, V o ¼ 4 m/s, g s ¼ 3:6; L v ¼ 5 m, f T ¼ 1:7Hz and z ¼ :2: The value of the gust response factor obtained was G t ¼ 1:8; which was multiplied by the mean response to obtain the design value. The Utility method does not consider any dynamic effects in its procedure. From the analysis of Table 5 it can be seen that the SIL gave always higher response values than the GRF method and also the Utility method. The results show that there can be an increase of more than 3% in the members stresses by using SIL, in relation to the usual procedure. For the ratio SIL/GRF the difference was up to 2%. This figure can vary a great deal according to the properties and characteristics assumed for the wind and structure.

11 A.M. Loredo-Souza, A.G. Davenport / J. Wind Eng. Ind. Aerodyn. 91 (23) Table 5 Forces (N) on members obtained from different methodologies Member Utility method GRF method SIL method z ¼ :2 z ¼ :1 1st & 2nd modes z ¼ :2 1st & 2nd modes z ¼ :2 1st mode only F1T 12,635 12,163 14,668 14,492 14,322 F3AT F1 19,96 15, , , ,6 P9T 33,7 28,35 34,383 34,12 33, Conclusions The main findings in this study were: (a) Peak loads calculated using SIL were larger than peak loads given by the GRF. (b) The dynamic response of transmission structures is strongly dependent on the turbulence intensity level and its spectrum. (c) For members in which there is reversal in the forces on the load position, the resonant response in the second mode of vibration was bigger, even by four to five times, than the corresponding one in the first mode. Although this effect is not as severe in terms of resulting stresses when all the components are computed in the peak responses, it can lead to fatigue problems. From the current results it can be concluded that the incorporation of the dynamic properties of transmission structures in the design methodologies is needed and that the statistical method using influence lines is a more correct approachsince it allows for the inclusion of a larger number of factors in the design methodology. References [1] A.G. Davenport, The response of slender structures to wind, in: The application of wind Engineering Principles to the Design of Structures, Lausanne, Switzerland, February 23 27, [2] N.J. Cook, The Designer s Guide to Wind Loading of Building Structures. Part 1, Building Research Establishment, London, UK, [3] American Society of Civil Engineers, Guidelines for electrical transmission line structural loading. ASCE Manuals and Reports on Engineering Practice No. 74. New York, [4] A.G. Davenport, The response of tension structures to turbulent wind: the role of aerodynamic damping, in: Proceedings of the First International Oleg Kerensky Memorial Conference on Tension Structures, London, England, June 2 22, [5] A.M. Loredo-Souza, The behaviour of transmission lines under high winds. Ph.D.Thesis, University of Western Ontario, London, Ont., Canada, [6] A.G. Davenport, Gust response factors for transmission line loading, Proceedings of the Fifth International Conference on Wind Engineering, Colorado State University, Pergamon Press, Oxford, 1979.