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Dynamic behavior and stability of transmission line towers under wind forces
ARTICLE IN PRESS
Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 1051–1067
Dynamic behavior and stability of transmission line towers under wind forces Ronaldo C. Battistaa,b,*, Rosa# ngela S. Rodriguesa, Miche" le S. Pfeila a
Civil Engineering Program, COPPE–Universidade Federal do Rio de Janeiro, CP 68506, Rio de Janeiro, CEP 21945-970, Brazil b COPPETEC Research, Consulting & Design, CP 68506, Rio de Janeiro, CEP 21945-970, Brazil
Abstract A new analytical-numerical modelling for the structural analysis of transmission line towers (TLT) under wind action is presented and proposed as a rational procedure for stability assessment in a design stage. The numerical results obtained from a 3D finite element model are discussed in relation to the dynamic behavior and the mechanism of collapse of a typical TLT. A simplified two degree-of-freedom analytical model is also presented and shown to be a useful tool for evaluating the system fundamental frequency in early design stages. In order to reduce the TLT’s top horizontal along-wind displacements in the cross-line direction, nonlinear pendulum-like dampers (NLPD) installed on the towers are envisaged and their efficiency is demonstrated with the aid of comparisons between numerical results obtained from the controlled and the uncontrolled systems. r 2003 Elsevier Ltd. All rights reserved. Keywords: Transmission line; Stability; Dampers; Wind force; Dynamics; Steel tower
1. Introduction A new analytical-numerical modelling has been applied to a chosen type of steel transmission line towers (TLT): a conventional 32.86 m-high self-supporting tower. The structural modelling of the chosen TLT is based on observation of the system’s *Corresponding author. Civil Engineering Program, COPPE–Universidade Federal do Rio de Janeiro, CP 68506, Rio de Janeiro, CEP 21945-970, Brazil. Tel.: +55-21-2562-8477; fax: +55-21-2562-8484. E-mail addresses: [email protected] (R.C. Battista), [email protected] (R.S. Rodrigues), [email protected] (M.S. Pfeil). 0167-6105/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0167-6105(03)00052-7
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behavior and video images of some recent accidents in Brazil, when storm wind velocities reached values close to 100 km/h. The dynamic characteristics of the towers and the lateral movement of the electric cables have brought up the importance of fluid flow–cables–structure interaction when evaluating the towers behavior under the action of wind forces, leading to the new analytical-numerical modelling for the structural analysis of TLT’s, as originally proposed by Rodrigues [1] and Rodrigues et al. [2] and, almost simultaneously, by Yasui et al. [3] analysing the differences in the behavior of power lines supported by tension- or suspensiontype transmission line towers. The overall results from the performed analyses were used to unveil the mechanism of collapse and envisage a remedial measure to attenuate top horizontal displacements and overall stresses, which is the installation of non-linear pendulum-like dampers (NLPD) on the top of the TLT, similar to the ones that have been proposed by Pinheiro [4], Battista et al. [5] and Battista and Pinheiro [6] for other slender and tall towers.
2. Description of the structural model For simulating the actual behavior of the transmission lines and towers under wind action, the transmission line itself has to be included in the 3D finite element model (Fig. 1), which is composed of a central tower and adjacent spans of electric conductors and aerial wires for lightning protection. The tower structure and all cables were discretized with spatial frame elements. These elements instead of the most commonly adopted truss elements were used in the discretization of the tower structure to allow for the small bending stresses introduced by the rigid bolted connections which may be important in the evaluation of the ultimate structural strength. Although cables fundamental frequencies are not highly sensitive to the type of element used in their discretization, spatial frame elements with the actual bending stiffness of the cables were chosen to allow for
Fig. 1. 3D-FEM model of the structural system.
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numerical stability in situations where the cables experience very large displacement amplitudes and tension variations close to reversion of signal. This will be the case in the next step of this study when a non-linear dynamic analysis is to be performed. The chain of insulators and the linkage of the tower to the lightning conductors were modelled as double-hinged suspension-rods, allowing for the actual mechanical behaviour. The neighbouring towers and the transmission line continuity, indicated by dashed lines in Fig. 1, are simulated in the model through adequate boundary conditions, involving elastic, inertial and kinematical characteristics. The dead weight and pre-tension loadings in the catenary cables and in the insulator’s suspension rods are considered in a geometric non-linear static analysis. Following the static equilibrium state, the time history response of the structure under wind action is obtained for the superposition of the n significant modes as follows: mj r.j ðtÞ þ 2xj oj mj r’j ðtÞ þ o2j mj rj ðtÞ ¼ fTj Fwind
j ¼ 1; yn;
ð1Þ
where mj is the modal mass, xj is the modal damping ratio, oj the circular frequency, rðtÞ; r’ðtÞ and r.ðtÞ; are respectively, the displacement, velocity and acceleration at time t, fj the vibration mode shape and fTj Fwind is the generalized modal wind force. Mean wind forces were not considered for determining the frequencies and oscillation modes of the cables, as it can be shown [3] that frequencies have very close values independent if these forces are taken or not into account.
3. Wind forces The wind velocity field is expressed in Eq. (2) only in terms of its horizontal component U in a system of cartesian coordinates (x; y; z), where x is the along-wind direction and z is the vertical direction: % þ uðy; z; tÞ: U ¼ UðzÞ
ð2Þ
% is the mean wind velocity in the horizontal direction at z Referring to Eq. (2), UðzÞ % coordinate, i.e., UðzÞis constant in direction and magnitude, and is a function of the height z: The small fluctuation of the mean wind velocity in the longitudinal direction uðy; z; tÞ—turbulence—is statistically determined as a function of the mean % wind velocity UðzÞ; the roughness length and the altitude above the ground level. The global wind force time history Fwind ; defined in terms of its component in the direction of the mean velocity—drag force—has the expression: 1 FD ðtÞ ¼ rACD ðaÞ½UðtÞ2 ; 2
ð3Þ
where r is air density, A the effective area of the structure, CD ðaÞ the drag coefficient corresponding to a angle of attack and UðtÞ is the flow velocity time history. The power spectral density function Su used in this work to characterize the energy distribution of the longitudinal fluctuating component u of the wind velocity (Eq. (2)) is the one suggested by Simiu and Scanlan [7].
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The cross-spectral density between the fluctuating velocities u1 and u2 corresponding to two locations along the cable span is taken as the product of the spectrum Su and an exponential decaying function of the distance between the two location points [7]. The generation of a field of uncorrelated fluctuating wind velocities vðtÞ is performed by the autoregressive method, which consists of expressing the instantaneous value of vðtÞ as a linear combination of some previous values of vðtÞ plus a random impulse. The field of spatially correlated fluctuating wind velocities uðtÞ is obtained by pre-multiplying vðtÞ to a matrix containing cross-correlation information between the generated signals given by the cross-spectral density function [8]. 3.1. Mean wind forces The map of basic wind velocities U0 ; given in the Brazilian design code ABNT/ NBR6123 [9], indicates the value 50 m/s in the region in Brazil where the towers collapsed. This velocity is referred to a gust of 3 s time-duration, return period equal to 50 years, in open terrain at a height equal to 10 m. The design mean wind velocity (averaged over 10 min) was calculated according to % Uð10Þ ¼ U0 S1 S2 S3 ¼ 37:95 m=s;
ð4Þ
% is the design mean wind speed at reference height z ¼ 10 m, U0 ¼ 50 m/s where UðzÞ the basic wind speed, S1 ¼ 1:00 is the topographical factor, S2 ¼ 0:69 is the combined exposure factor and S3 ¼ 1:10 is the statistical factor (risk factor and service life required). The mean wind velocity profile along the height of the Delta tower, as depicted on Fig. 2, was constructed by the power law (Eq. (5)) using % ref Þ ¼ Uð10Þ % Uðz and p; exponent related to the terrain roughness, equal to 0.15 (farmland, scattered trees and low buildings): p z % % UðzÞ ¼ Uðzref Þ : ð5Þ zref
3.2. Turbulence numerical simulation along the transmission line axis In the auto-regressive method, the turbulence uðy; z; tÞ simulation is a linear combination of p values added to a zero-mean random impulse with variance s2Nu : uðtÞ ¼
p X
fs uðt s DtÞ þ sNu NðtÞ
ð6Þ
s¼1
where fs are the auto-regressive parameters, p is the auto-regressive order and NðtÞ the zero-mean random process and variance equal to 1. According to Buchholdt et al. [8], the parameters fs are to be determined with a solution of an algebraic system of
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Fig. 2. Mean wind velocity—vertical profile.
equations: Ru ðk DtÞ ¼
p X
fs Ru ððk sÞ DtÞ;
k ¼ 1; 2; yp;
ð7Þ
s¼1
where Ru is the autocorrelation function of uðtÞ process, determined by the inverse Fourier transform of the energy spectrum Su ðnÞ: With s2u as the uðtÞ variance, s2Nu ; in Eq. (6), is given by s2Nu ¼ s2u
p X
fs Ru ðs DtÞ:
ð8Þ
s¼1
Using the procedure described above in a manner applied by Pfeil and Battista [10], 12 fluctuating wind velocity histories were generated associated to points along the transmission line axis, with longitudinal turbulence intensity equal to 0.14 and root mean square (RMS) value equal to 6.18 m/s. Then, the wind force time histories were determined according to Eq. (3), considering three angles of attack: a ¼ 0 (orthogonal to the transmission line axis), a ¼ 45 and 30 ; all in a horizontal plane. Equivalent nodal forces were applied according to influence lengths to the cables and chains of insulators and the drag coefficients, CD ðaÞ; were those given in the Brazilian design code [9].
4. Self-supporting tower analysis The self-supporting tower selected to be analysed is a Delta type (Fig. 3) with ASTM A36 and A572 steel angles, connected by bolts. It is part of a 230 kV transmission system designed for three simple Grosbeak type electric conductors
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Fig. 3. Delta tower—silhouette and frontal view.
(d ¼ 25:16 mm), two EHS lightning cables (d ¼ 9:15 mm) and mean span equal to 450 m. The chains of glass pieces insulators are mounted on 2.90 m length suspension-rods. 4.1. Soil–structure interaction The soil–structure interaction was performed taking into account two types of soil: medium sand and clay soil. Linear elastic springs and rigid elements were used to simulate, respectively, the soil reaction and the reinforced concrete footings. The study of the structural dynamic characteristics has shown that, whichever is the type of soil selected, the first 10 lower value natural oscillation frequencies do not change. This was an expected result since the relevant design factor of a transmission line tower foundation is the overturning moment arising from the action of wind. Footings designed for this type of tower and load result in low tension and compression stresses on the soil and consequent very small settlements. 4.2. Free vibration analysis The results from a free vibration analysis of the structural system under initial stresses is shown in Table 1, together with a few of the modal shapes depicted in Figs. 4 and 5. These results serve readily to give emphasis to the most important aspect of the structural system behavior: the electric cables lateral oscillation under the action of wind excites the tower’s dominant vibration modes. The fundamental period equal to 6.34 s (i.e., low frequency, f ¼ 0:158 Hz) means that, when exposed to the dynamic effects of the atmospheric turbulence, the fluctuating response of the lowdamped tower-cables coupled system in the along-wind and across-wind directions can be significant.
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Table 1 Natural vibration periods and frequencies and modal shapes description Mode
1 2 3 4 5 6 7 8 9 10
Period (s)
6.34 5.83 5.00 3.53 3.37 3.02 2.08 1.14 0.74 0.56
Frequency (Hz)
0.158 0.172 0.200 0.283 0.297 0.331 0.481 0.877 1.351 1.786
Mode shapes Involved components
Tower top displacement direction
Tw/EC EC Tw/LC LC/EC LC/EC LC/EC Tw/LC/EC LC/EC Tw/LC/EC Tw/LC/EC
Lateral Longitudinal Lateral Longitudinal Longitudinal Longitudinal Lateral Longitudinal Lateral Longitudinal
Tw=Tower, EC=Electric Conductor, LC=Lightning Conductor.
Fig. 4. Mode shape 1—lateral oscillation (T ¼ 6:34 s).
Fig. 5. Mode shape 7—lateral oscillation (T ¼ 2:08 s).
4.3. Time domain analysis The 3D-FEM model was analysed in the time domain (total time interval, Tmax ; 840 s), considering the first 10 vibration modes in the response calculation. The wind
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forces transverse to the transmission line axis (a ¼ 0 ), coinciding with the fundamental vibration mode direction, was the most unfavourable loadcase. The maximum horizontal displacement at the free end of the flexible cantilevered truss (Fig. 6) resulted equal to 1.26 m in the along-wind direction, while in the vertical direction resulted in 1.34 m, both at time t ¼ 408 s. It should be noticed that these large amplitude displacements are not expected from the conventional design calculations for this kind of structure. The structural response under wind action can be assumed to be a stationary Gaussian process. In that case, the probability density function for maxima converges to the Cartwright and Lonquet-Higgins probability density function (Eq. (9)) and the mean, Z% e ; and the standard deviation, sZe ; of the extreme values are given [11] by Eqs. (10a) and (10b): Z2 F ðZe Þ ¼ exp uT exp e2 ; ð9Þ 2s Z% e Dð2 ln uTÞ1=2 s þ
0:5772s ð2 ln uTÞ1=2
;
p s sZe ¼ pffiffiffi ; 1=2 6ð2 ln uTÞ
ð10aÞ
ð10bÞ
where u is the zero-crossing frequency, T is the time duration, s is the standard deviation of the sample and g ¼ 0:5772 is the Euler’s constant. Hence by using Eqs. (10) and taking into account just the fluctuating part of the displacements in the along-wind and vertical directions, the mean and the standard deviation of the extreme values of the displacements at node 1 (Table 2) are determined for each direction. The across-wind direction was omitted, since the related displacements are negligible. 4.4. Frequency domain analysis Having determined the displacements at nodal point 1 in the time domain (Fig. 7), the density spectra Sx and Sz are obtained with the application of the fast Fourier transform algorithm to the displacements time histories. The resultant response spectra displayed in detail in Fig. 8, for a frequency range 0–0.24 Hz, show the three peaks corresponding to the vibration mode shapes 1, 2 and 3 (see Table 1).
Fig. 6. Detail of the flexible cantilevered truss (nodal point 1).
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Table 2 Displacements at nodal point 1 (Fig. 6)—statistics of extremes Direction
Standard deviation
Most probable value (m)
Along-wind (X) Vertical (Z)
0.084 0.089
1.16 1.24
x[m]
Along-Wind Direction(X) 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 0
200
400
600
800
1000
Time [sec]
z[m]
Vertical Direction(Z) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2
0
200
400
600
800
1000
Time [sec] Fig. 7. Time histories—displacements at node 1.
Working strictly in the frequency domain, Eq. (11) expresses the transfer relation between power spectra density functions of the stationary random excitation and response xðtÞ; in the along-wind direction: Sx ðoÞ ¼ jHðoÞj2 Sfwind ðoÞ;
ð11Þ
where Sfwind ðoÞ and Sx ðoÞ denote, respectively, the modal power spectra density functions of the excitation force and displacement amplitude and HðoÞ is the modal complex-frequency-response function. Applying this formulation to a system with a single generalized degree of freedom (any of the lateral oscillation modal shapes) with frequency oj ; stiffness kj and damping ratio xj ; subjected to a forcing excitation with frequency o; % the complexfrequency-response function takes the form: HðoÞ % ¼
1 for xj > 0: kj ½1 þ 2ixj ðo=o % j Þ ðo=o % j Þ2
ð12Þ
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Along-Wind Direction (X) f1=0.158Hz f2=0.172Hz
2
S x [m s]
0.05
f3=0.200Hz
0.03
0.00 0.24
0.12
Frequency [Hz]
Vertical Direction (Z) f1=0.158Hz f2=0.172Hz f3=0.200Hz
2
S z [m s]
0.05
0.03
0.00 0.12
0.24
Frequency [Hz] Fig. 8. Response spectra—displacements at node 1.
The maxima of modal responses may be determined by considering the excitation as a sinusoidal force with the same magnitude as the modal generalized force Fmj in the natural frequency oj ; applied in direction X at the free end of the flexible cantilevered truss (Node 1 in Fig. 6). In the case of the structural system analysed herein, the first 10 natural frequencies are sufficiently close and the modal damping factors are low, leading to coupling of the vibration modes. It is then possible to apply the square root of the sum of the squares (SRSS) method to the modal responses presented in Table 3, which yields a displacement amplitude x ¼ 1:389 m in the along wind direction (X direction) at El. 31.00 m (Node 1 in Fig. 6): x ¼ ½x21 þ x23 þ x27 þ x29 1=2 :
ð13Þ
When the contributing modes have very close frequencies, the response amplitude may be also calculated by combining the modal responses (Table 3) in accordance to the complete quadratic combination method (CQC), in order to take into account the contribution of the other modes through the cross terms. The displacement amplitude at node 1 in the along wind direction (X) was then determined by the CQC method, resulting in x ¼ 1:392 m; a result which, as expected, is very close to that given by the SRSS method. In the CQC method, as the damping ratios are the same for all modes, xj ¼ 2%, the cross-correlation coefficients rnm between modes m and n were calculated according to Eq. (14); the
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Table 3 Modal responses in the frequency domain—along-wind direction Mode
Displacement (m) in X Direction, El. 31.00 m
1 3 7 9
1.26 0.08 0.21 0.54
modal damping ratios in this case were taken as nearly mass proportional: rnm ¼ rmn ¼
8x2 ð1 þ rÞr3=2 ; ð1 r2 Þ2 þ 4x2 rð1 þ rÞ2
0prnm p1;
ð14Þ
where r ¼ on =om and om > on : 4.5. Tower and cables strength analysis Checking the interaction equations for the load and resistance factor design criteria (LRFD), in accordance with the design codes for steel structures—as for example ABNT/NBR8800 (1986), or the ASCE or yet the EUROCODE recommendations—the angle sections of those members connected to the foundation have exceeded the prescribed limit values, whichever limit state was verified: safety or serviceability. Other members, around level 28.00 m, were also ill-proportioned. These calculations were made by taking the stress resultants at one instant of time where a peak in time history occurred for the axial force in these angle section members. As indicated by the envelope of tension forces in the cables (Table 4), the limit tension ratio for maximum wind velocity recommended by the Brazilian code for transmission lines was not surpassed. The analysis resulted in working ratios equal to 24% for the electric conductors and 20% for the lightning conductors; in both cases not greater than the recommended limit, 50% of the nominal cable strength. These results rule out the possibility that the collapse of these transmission line towers had been caused by rupture of cables under the action of wind. The existing damping devices (Stockbridge) were not included in the 3D-FEM model. They are useful to attenuate high frequency small amplitude movements that result from low speed winds that may lead to rupture caused by fatigue. Conversely, in the present analysis the movements are of another nature; they are of low frequency and considerable large amplitude. It is important to emphasize here that the linear dynamic analysis carried out with the tower–cables coupled model is just a preliminary step of a complete appraisal of this aerodynamic problem. Further steps should be taken with a non-linear dynamic analysis, in order to take into account adequately the large displacements of the cables that are allowed by large angular displacements of the suspension-rods.
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Table 4 Envelope of tension forces in the cables Cable Type
External conductor Central conductor Lightning conductor
Tension–T (kN) Minimum
Maximum
25.85 28.71 13.05
33.62 30.83 13.86
Nominal cable strength, Rn (kN)
Working ratio T/Rn
137.80 137.80 69.86
0,19–0,24 0,21–0,22 0,19–0,20
Insulators θ1 Wind
Cable Mass θ2
Double Pendulum
Fig. 9. Double pendulum behavior—comparison.
5. The important dynamic effect of the suspension-rods An important conclusion from this study is the fundamental role the suspensionrods play in the coupled tower–transmission line system; the height of the chain of insulators (or of the suspension-rod) defines the dynamic characteristics of the tower–cables coupled model. The system has the tendency to behave like a double pendulum in the orthogonal direction to the transmission line axis, and a preliminary evaluation of the system fundamental frequency can be made with a two-degree-offreedom model (Fig. 9). Taking the following transmission line characteristics as used in the 3D model: (a) medium span: L =450 m; (b) chain of insulators: height=2.65 m ) suspension-rod height L1=2.90 m; (c) electric conductors: Grosbeak (weight: m ¼ 13:0 N/m, maximum tension: Trmax ¼ 31500 N); (d) and the sag s of cables hanging in catenary shape under the action of their own weight as given by Eq. (15) (where Tr0 is equal to the tension at the middle of the span): Tr0 mL s¼ cosh 1 ; ð15Þ 2Tr0 m where s ¼ 10:45 m was obtained by considering that Tr0 ¼ Trmax ; when the suspension points are at the same level and thus the tension variation is very small.
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The fundamental frequency can be estimated with the double pendulum model as given by Eq. (16), where L1 is the suspension-rod height and g is the acceleration of gravity rffiffiffiffiffiffi 0:6 g fdp ¼ ð16Þ ðHzÞ; L2 =L1 B2:0 2p L1 The length L2 corresponds to the vertical distance between the straight line passing through the tips of two consecutive suspension-rods and the catenary centre of mass, i.e., L2 ¼ 2s=3: The model fundamental frequency based on the double pendulum linearized model (Fig. 10a) results in fdp ¼ 0:175 Hz which may be considered a simple and close estimation of f1 at a preliminary design stage, compared to the result from the 3D-FEM analysis, f1 ¼ 0:158 Hz. For the 3D-FEM Delta tower model with the tips of the suspension-rods restrained (Fig. 10b) each one of the swinging cables behaves like a simple pendulum and yields a fundamental frequency f1 ¼ 0:168 Hz, which may be approximated by the frequency given by the simple pendulum model, fp ¼ 0:189 Hz. The oscillation frequencies and mode shapes obtained for the 3D-FEM model considering the suspension-rods restrained are compared in Table 5 to those found with the new proposed model (free suspension-rods). As one can see, the values of frequencies are reasonably close for the two models. The main difference between these two models is that the tower is much more mobilized when the suspension-rods are free to swing and, as a consequence, it experiences larger amplitudes of the lateral top displacement. For restrained suspension-rods, the cables behavior begins to approach the one displayed by a simple pendulum with smaller oscillation amplitudes. This latter 3D model is close to that used in design practice, in which tower and cables are not coupled as in the 3D-FEM model, and the resultant of the
Insulators (L1)
Cable Mass
θ2
dp
0.6
g
2π
L1
θ2
Cable Mass
Bracing Bars (Mass = zero)
g = acceleration of gravity
=
Conductor Projection (L 2 )
Conductor Projection (L 2)
θ1
f
Insulators (L1)
Hz ; for L 2 /L1 ~ 2.0
(a) Double Pendulum (1)
fp =
1
g
2π
L2
Hz
(b) Simple Pendulum (2)
3D -F EM
f1 =0.158Hz
f 1 =0.168Hz
Pendulum
fdp =0.175Hz (1)
f p =0.189Hz (2)
Fig. 10. Pendulum formulation—results.
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Table 5 Natural frequencies and mode shapes considering suspension-rods effects (New model) suspension-rods not-restrained
Suspension-rods restrained
Mode Frequency (Hz)
Vibration mode
Tower’s top displacement direction
Frequency (Hz)
Vibration mode
Tower’s top displacement direction
1 2 3 4 5 6 7 8 9 10
Tw/EC EC Tw/LC LC/EC LC/EC LC/EC Tw/LC/EC LC/EC Tw/LC/EC Tw/LC/EC
Lateral Longitudinal Lateral Longitudinal Longitudinal Longitudinal Lateral Longitudinal Lateral Longitudinal
0.168 0.202 0.241 0.283 0.330 0.340 0.510 0.877 1.389 2.128
Tw/EC Tw/LC LC/EC LC/EC LC/EC LC/EC Tw/LC/EC LC/EC Tw/LC/EC Tw/LC/EC
Lateral Lateral Longitudinal Longitudinal Longitudinal Longitudinal Lateral Longitudinal Lateral Longitudinal
0.158 0.172 0.200 0.283 0.297 0.331 0.481 0.877 1.351 1.786
Tw=Tower, EC=Electric Conductor, LC=Lightning Conductor.
wind forces acting on electric and lightning conductors as well as on chains of insulators are applied to the tower’s linkage nodes. Table 6 summarizes and compares the results obtained with different models for the transverse displacements at the free end of the flexible cantilevered truss (Node 1, Fig. 6), considering the wind direction transverse to the transmission line axis. The models used for such comparison are the following: 1. design practice (static analysis): The electric cables are not discretized; the wind design dynamic pressure q ¼ 2:4 kN/m2 is uniformly distributed along electric cables and the height of the tower, considering the obstruction effective area. The resultant of the wind forces on the cables are applied to the tower’s linkage nodes. 2. equivalent cable-tower coupled model (static analysis): The new proposed model considering the mean wind velocity only, without turbulence. 3. simplified cable-tower coupled model (time-history analysis for turbulent wind): The new proposed model with the suspension-rods restrained. 4. cable-tower coupled model (time history analysis for turbulent wind): The new proposed model. 6. Non-linear pendulum-like dampers as a solution for the problem Since the kinetic mechanism that leads to structural collapse has been identified, the next step is to attenuate the amplitudes of the tower’s top horizontal cross-line displacements in the along-wind direction, by means of an auxiliary dynamic device. Based on the works by Pinheiro [4] and Battista and Pinheiro [6], on dynamic control of slender towers under environmental loadings, the proposed remedial measure is to install non-linear pendulum-like dampers (NLPD), as depicted in Fig. 11, to reduce the amplitudes of horizontal cross-lines displacements at the
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Table 6 Displacements response at the free end of the flexible cantilevered truss (node 1, Fig. 6) of the tower for a ¼ 0 wind direction Hypothesis
1. Design practice (static analysis) 2. Coupled model (static analysis) Mean wind velocity 3. Coupled model (time-history analysis) Suspension-rods restrained and turbulent wind 4. Coupled model (time-history analysis) Turbulent wind Probable value for the new proposed model 4 (standard deviationD0.08)
Displacements (m) Along-wind direction (X)
Vertical direction (Z)
Max./min. Min. Max. Min. Max.
70.51 0.60 0.60 0.73 0.002
70.51 0.61 0.61 0.003 0.67
Min. Max.
1.26 0.10 1.16
0.12 1.34 1.24
towers due to the sway motion of the transmission lines induced by wind. The dimensions proposed on Fig. 11 were determined with the solution of the non-linear two-degree-of-freedom system which formulation is summarized in the following section. 6.1. Equations of controlled motion It is possible to design NLPDs to attenuate the amplitudes of vibration in the dominant fundamental mode by applying the 2D modal equations (Eq. (17)) for the simple model illustrated in Fig. 12, where subscript p stands for the pendulum’s properties. For that, the related modal properties (modal mass M; force F ; stiffness K and/or frequency O) have to be extracted from the 3D-FEM model proposed herein. The Runge–Kutta method was used in the solution of the nonlinear system of equations: ( ðM þ mp Þx. þ C x’ þ Kx þ mp lðsin yÞ00 ¼ F0 sin ðoe t þ aÞ ð17Þ mp l 2 y. þ cp y’ þ kp y þ mp gl sin y þ mp l x. cos y ¼ 0 where ðsin yÞ00 ¼ y. cos y y’ 2 sin y: The comparisons between the controlled and uncontrolled systems as well as the angular amplitude of the NLPD are shown in Fig. 13, for NLPD having damping ratio xp ¼ 5% and stiffness kp ¼ 1 kN m/rad. The reduction of the horizontal cross-line displacements in the along-wind direction in the first mode of vibration reaches an efficiency rate over 90% when the NLPD system is designed to work almost in resonance with the first mode of the structural system under wind action. Because of the lack of space, the presentation and discussion of results from parametric and sensitivity analyses are herewith prohibitive.
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See Detail
X Body of mass mp equivalent to a steel sphere with D=0.28m. El. 20.73m
Perspective
Frontal View
Detail
Fig. 11. The delta tower and the non-linear pendulum-like dampers.
Fig. 12. Tower with NLPD and analogous simplified mechanical system.
Fig. 13. Controlled and uncontrolled structure responses for oe =o1 ¼ 0:9 and op =o1 ¼ 0:9:
7. Conclusions A 3D-FEM model was constructed for analysing the dynamic coupled behavior of transmission lines and towers under the action of wind. The distinguishing feature of
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this model is its ability to account for the inertia forces which arise in the towers dynamics with the wind induced sway motion of the electric cables. The suspensionrods formed by the chains of insulators were identified as the most important component of the system when it comes to the analysis of wind flow and tower-lines coupled model interactive dynamic behavior and response. To attenuate the towers’s top horizontal cross-line displacements in the along-wind direction, NLPDs were envisaged leading to an efficiency rate around 90% when the NLPD system is designed to work almost in resonance with the wind-induced motion in the first mode of oscillation of the coupled structural system under wind action. The authors realize that the presented results obtained with the proposed new modelling are just a preliminary step for the better understanding of the mechanical behavior of transmission line towers under the action of wind. The proposed model has yet to be augmented with appropriate non-linear dynamics to allow for very large angular displacements of the suspension-rods and cables to better describe what seems to be its actual interactive mechanical behavior. Very recent preliminary results, which have been obtained by the authors from a non-linear model, seem very promising and shall be reported in the near future.
References [1] R.S. Rodrigues, Collapse of transmission line towers under the action of wind, M.Sc. Thesis, COPPE/ UFRJ, Rio de Janeiro, April, 1999 (in Portuguese). [2] R.S. Rodrigues, R.C. Battista, M.S. Pfeil, Unveiling the dynamic mechanism of collapse of transmission line towers under wind forces, XXIX Jornadas Sudamericanas de Ingenieria Estructural—Jubileo Prof. Julio Ricaldoni, Punta del Este, 2000, (in Portuguese). [3] H. Yasui, H. Marukawa, Y. Momomura, T. Ohkuma, Analytical study on wind-induced vibration of power transmission towers, J. Wind Eng. Ind. Aerodyn. 83 (1999) 431–441. [4] M.A.S. Pinheiro, Non-linear pendulum absorber of lateral vibrations in slender towers, M.Sc. Thesis, COPPE/UFRJ, Rio de Janeiro February, 1997 (in Portuguese). [5] R.C. Battista, N.F.F. Ebecken, L. Bevilacqua, Dynamical analysis of an offshore platform with vibration absorbers, IUTAM-Symposium on Recent Developments in Nonlinear Oscillations of Mechanical Systems, Vietnan, 1999. [6] R.C. Battista, M.A.S. Pinheiro, Oscillations reduction of compliant structures of offshore platforms, Proceedings of the 21st Iberian Latin American Congress on Computational Methods in Engineering, Rio de Janeiro, 2000 (in Portuguese). [7] E. Simiu, R.H. Scanlan, Wind Effects on Buildings, Wiley, New York, 1996. [8] H.A. Buchholdt, S. Moossavinejad, A. Iannuzzi, Non-linear dynamic analysis of guyed masts subjected to wind and guy ruptures, Proc. Inst. Civ. Eng. September (Part 2) 81 (1986) 353–359. [9] ABNT. NBR 6123 Wind forces on buildings, ABNT, Rio de Janeiro (Brazilian Code) (in Portuguese). [10] M.S. Pfeil, R.C. Battista, Aerodynamics stability analysis of cable stayed bridges, J. Struct. Eng. 121(12) (1995) 1784–1788. [11] R.W. Clough, J. Penzien, Dynamics of Structures, McGraw-Hill Books Co, Singapore, 1993 (Chapter 2).
Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 1051–1067
Dynamic behavior and stability of transmission line towers under wind forces Ronaldo C. Battistaa,b,*, Rosa# ngela S. Rodriguesa, Miche" le S. Pfeila a
Civil Engineering Program, COPPE–Universidade Federal do Rio de Janeiro, CP 68506, Rio de Janeiro, CEP 21945-970, Brazil b COPPETEC Research, Consulting & Design, CP 68506, Rio de Janeiro, CEP 21945-970, Brazil
Abstract A new analytical-numerical modelling for the structural analysis of transmission line towers (TLT) under wind action is presented and proposed as a rational procedure for stability assessment in a design stage. The numerical results obtained from a 3D finite element model are discussed in relation to the dynamic behavior and the mechanism of collapse of a typical TLT. A simplified two degree-of-freedom analytical model is also presented and shown to be a useful tool for evaluating the system fundamental frequency in early design stages. In order to reduce the TLT’s top horizontal along-wind displacements in the cross-line direction, nonlinear pendulum-like dampers (NLPD) installed on the towers are envisaged and their efficiency is demonstrated with the aid of comparisons between numerical results obtained from the controlled and the uncontrolled systems. r 2003 Elsevier Ltd. All rights reserved. Keywords: Transmission line; Stability; Dampers; Wind force; Dynamics; Steel tower
1. Introduction A new analytical-numerical modelling has been applied to a chosen type of steel transmission line towers (TLT): a conventional 32.86 m-high self-supporting tower. The structural modelling of the chosen TLT is based on observation of the system’s *Corresponding author. Civil Engineering Program, COPPE–Universidade Federal do Rio de Janeiro, CP 68506, Rio de Janeiro, CEP 21945-970, Brazil. Tel.: +55-21-2562-8477; fax: +55-21-2562-8484. E-mail addresses: [email protected] (R.C. Battista), [email protected] (R.S. Rodrigues), [email protected] (M.S. Pfeil). 0167-6105/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0167-6105(03)00052-7
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behavior and video images of some recent accidents in Brazil, when storm wind velocities reached values close to 100 km/h. The dynamic characteristics of the towers and the lateral movement of the electric cables have brought up the importance of fluid flow–cables–structure interaction when evaluating the towers behavior under the action of wind forces, leading to the new analytical-numerical modelling for the structural analysis of TLT’s, as originally proposed by Rodrigues [1] and Rodrigues et al. [2] and, almost simultaneously, by Yasui et al. [3] analysing the differences in the behavior of power lines supported by tension- or suspensiontype transmission line towers. The overall results from the performed analyses were used to unveil the mechanism of collapse and envisage a remedial measure to attenuate top horizontal displacements and overall stresses, which is the installation of non-linear pendulum-like dampers (NLPD) on the top of the TLT, similar to the ones that have been proposed by Pinheiro [4], Battista et al. [5] and Battista and Pinheiro [6] for other slender and tall towers.
2. Description of the structural model For simulating the actual behavior of the transmission lines and towers under wind action, the transmission line itself has to be included in the 3D finite element model (Fig. 1), which is composed of a central tower and adjacent spans of electric conductors and aerial wires for lightning protection. The tower structure and all cables were discretized with spatial frame elements. These elements instead of the most commonly adopted truss elements were used in the discretization of the tower structure to allow for the small bending stresses introduced by the rigid bolted connections which may be important in the evaluation of the ultimate structural strength. Although cables fundamental frequencies are not highly sensitive to the type of element used in their discretization, spatial frame elements with the actual bending stiffness of the cables were chosen to allow for
Fig. 1. 3D-FEM model of the structural system.
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numerical stability in situations where the cables experience very large displacement amplitudes and tension variations close to reversion of signal. This will be the case in the next step of this study when a non-linear dynamic analysis is to be performed. The chain of insulators and the linkage of the tower to the lightning conductors were modelled as double-hinged suspension-rods, allowing for the actual mechanical behaviour. The neighbouring towers and the transmission line continuity, indicated by dashed lines in Fig. 1, are simulated in the model through adequate boundary conditions, involving elastic, inertial and kinematical characteristics. The dead weight and pre-tension loadings in the catenary cables and in the insulator’s suspension rods are considered in a geometric non-linear static analysis. Following the static equilibrium state, the time history response of the structure under wind action is obtained for the superposition of the n significant modes as follows: mj r.j ðtÞ þ 2xj oj mj r’j ðtÞ þ o2j mj rj ðtÞ ¼ fTj Fwind
j ¼ 1; yn;
ð1Þ
where mj is the modal mass, xj is the modal damping ratio, oj the circular frequency, rðtÞ; r’ðtÞ and r.ðtÞ; are respectively, the displacement, velocity and acceleration at time t, fj the vibration mode shape and fTj Fwind is the generalized modal wind force. Mean wind forces were not considered for determining the frequencies and oscillation modes of the cables, as it can be shown [3] that frequencies have very close values independent if these forces are taken or not into account.
3. Wind forces The wind velocity field is expressed in Eq. (2) only in terms of its horizontal component U in a system of cartesian coordinates (x; y; z), where x is the along-wind direction and z is the vertical direction: % þ uðy; z; tÞ: U ¼ UðzÞ
ð2Þ
% is the mean wind velocity in the horizontal direction at z Referring to Eq. (2), UðzÞ % coordinate, i.e., UðzÞis constant in direction and magnitude, and is a function of the height z: The small fluctuation of the mean wind velocity in the longitudinal direction uðy; z; tÞ—turbulence—is statistically determined as a function of the mean % wind velocity UðzÞ; the roughness length and the altitude above the ground level. The global wind force time history Fwind ; defined in terms of its component in the direction of the mean velocity—drag force—has the expression: 1 FD ðtÞ ¼ rACD ðaÞ½UðtÞ2 ; 2
ð3Þ
where r is air density, A the effective area of the structure, CD ðaÞ the drag coefficient corresponding to a angle of attack and UðtÞ is the flow velocity time history. The power spectral density function Su used in this work to characterize the energy distribution of the longitudinal fluctuating component u of the wind velocity (Eq. (2)) is the one suggested by Simiu and Scanlan [7].
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The cross-spectral density between the fluctuating velocities u1 and u2 corresponding to two locations along the cable span is taken as the product of the spectrum Su and an exponential decaying function of the distance between the two location points [7]. The generation of a field of uncorrelated fluctuating wind velocities vðtÞ is performed by the autoregressive method, which consists of expressing the instantaneous value of vðtÞ as a linear combination of some previous values of vðtÞ plus a random impulse. The field of spatially correlated fluctuating wind velocities uðtÞ is obtained by pre-multiplying vðtÞ to a matrix containing cross-correlation information between the generated signals given by the cross-spectral density function [8]. 3.1. Mean wind forces The map of basic wind velocities U0 ; given in the Brazilian design code ABNT/ NBR6123 [9], indicates the value 50 m/s in the region in Brazil where the towers collapsed. This velocity is referred to a gust of 3 s time-duration, return period equal to 50 years, in open terrain at a height equal to 10 m. The design mean wind velocity (averaged over 10 min) was calculated according to % Uð10Þ ¼ U0 S1 S2 S3 ¼ 37:95 m=s;
ð4Þ
% is the design mean wind speed at reference height z ¼ 10 m, U0 ¼ 50 m/s where UðzÞ the basic wind speed, S1 ¼ 1:00 is the topographical factor, S2 ¼ 0:69 is the combined exposure factor and S3 ¼ 1:10 is the statistical factor (risk factor and service life required). The mean wind velocity profile along the height of the Delta tower, as depicted on Fig. 2, was constructed by the power law (Eq. (5)) using % ref Þ ¼ Uð10Þ % Uðz and p; exponent related to the terrain roughness, equal to 0.15 (farmland, scattered trees and low buildings): p z % % UðzÞ ¼ Uðzref Þ : ð5Þ zref
3.2. Turbulence numerical simulation along the transmission line axis In the auto-regressive method, the turbulence uðy; z; tÞ simulation is a linear combination of p values added to a zero-mean random impulse with variance s2Nu : uðtÞ ¼
p X
fs uðt s DtÞ þ sNu NðtÞ
ð6Þ
s¼1
where fs are the auto-regressive parameters, p is the auto-regressive order and NðtÞ the zero-mean random process and variance equal to 1. According to Buchholdt et al. [8], the parameters fs are to be determined with a solution of an algebraic system of
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Fig. 2. Mean wind velocity—vertical profile.
equations: Ru ðk DtÞ ¼
p X
fs Ru ððk sÞ DtÞ;
k ¼ 1; 2; yp;
ð7Þ
s¼1
where Ru is the autocorrelation function of uðtÞ process, determined by the inverse Fourier transform of the energy spectrum Su ðnÞ: With s2u as the uðtÞ variance, s2Nu ; in Eq. (6), is given by s2Nu ¼ s2u
p X
fs Ru ðs DtÞ:
ð8Þ
s¼1
Using the procedure described above in a manner applied by Pfeil and Battista [10], 12 fluctuating wind velocity histories were generated associated to points along the transmission line axis, with longitudinal turbulence intensity equal to 0.14 and root mean square (RMS) value equal to 6.18 m/s. Then, the wind force time histories were determined according to Eq. (3), considering three angles of attack: a ¼ 0 (orthogonal to the transmission line axis), a ¼ 45 and 30 ; all in a horizontal plane. Equivalent nodal forces were applied according to influence lengths to the cables and chains of insulators and the drag coefficients, CD ðaÞ; were those given in the Brazilian design code [9].
4. Self-supporting tower analysis The self-supporting tower selected to be analysed is a Delta type (Fig. 3) with ASTM A36 and A572 steel angles, connected by bolts. It is part of a 230 kV transmission system designed for three simple Grosbeak type electric conductors
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Fig. 3. Delta tower—silhouette and frontal view.
(d ¼ 25:16 mm), two EHS lightning cables (d ¼ 9:15 mm) and mean span equal to 450 m. The chains of glass pieces insulators are mounted on 2.90 m length suspension-rods. 4.1. Soil–structure interaction The soil–structure interaction was performed taking into account two types of soil: medium sand and clay soil. Linear elastic springs and rigid elements were used to simulate, respectively, the soil reaction and the reinforced concrete footings. The study of the structural dynamic characteristics has shown that, whichever is the type of soil selected, the first 10 lower value natural oscillation frequencies do not change. This was an expected result since the relevant design factor of a transmission line tower foundation is the overturning moment arising from the action of wind. Footings designed for this type of tower and load result in low tension and compression stresses on the soil and consequent very small settlements. 4.2. Free vibration analysis The results from a free vibration analysis of the structural system under initial stresses is shown in Table 1, together with a few of the modal shapes depicted in Figs. 4 and 5. These results serve readily to give emphasis to the most important aspect of the structural system behavior: the electric cables lateral oscillation under the action of wind excites the tower’s dominant vibration modes. The fundamental period equal to 6.34 s (i.e., low frequency, f ¼ 0:158 Hz) means that, when exposed to the dynamic effects of the atmospheric turbulence, the fluctuating response of the lowdamped tower-cables coupled system in the along-wind and across-wind directions can be significant.
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Table 1 Natural vibration periods and frequencies and modal shapes description Mode
1 2 3 4 5 6 7 8 9 10
Period (s)
6.34 5.83 5.00 3.53 3.37 3.02 2.08 1.14 0.74 0.56
Frequency (Hz)
0.158 0.172 0.200 0.283 0.297 0.331 0.481 0.877 1.351 1.786
Mode shapes Involved components
Tower top displacement direction
Tw/EC EC Tw/LC LC/EC LC/EC LC/EC Tw/LC/EC LC/EC Tw/LC/EC Tw/LC/EC
Lateral Longitudinal Lateral Longitudinal Longitudinal Longitudinal Lateral Longitudinal Lateral Longitudinal
Tw=Tower, EC=Electric Conductor, LC=Lightning Conductor.
Fig. 4. Mode shape 1—lateral oscillation (T ¼ 6:34 s).
Fig. 5. Mode shape 7—lateral oscillation (T ¼ 2:08 s).
4.3. Time domain analysis The 3D-FEM model was analysed in the time domain (total time interval, Tmax ; 840 s), considering the first 10 vibration modes in the response calculation. The wind
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forces transverse to the transmission line axis (a ¼ 0 ), coinciding with the fundamental vibration mode direction, was the most unfavourable loadcase. The maximum horizontal displacement at the free end of the flexible cantilevered truss (Fig. 6) resulted equal to 1.26 m in the along-wind direction, while in the vertical direction resulted in 1.34 m, both at time t ¼ 408 s. It should be noticed that these large amplitude displacements are not expected from the conventional design calculations for this kind of structure. The structural response under wind action can be assumed to be a stationary Gaussian process. In that case, the probability density function for maxima converges to the Cartwright and Lonquet-Higgins probability density function (Eq. (9)) and the mean, Z% e ; and the standard deviation, sZe ; of the extreme values are given [11] by Eqs. (10a) and (10b): Z2 F ðZe Þ ¼ exp uT exp e2 ; ð9Þ 2s Z% e Dð2 ln uTÞ1=2 s þ
0:5772s ð2 ln uTÞ1=2
;
p s sZe ¼ pffiffiffi ; 1=2 6ð2 ln uTÞ
ð10aÞ
ð10bÞ
where u is the zero-crossing frequency, T is the time duration, s is the standard deviation of the sample and g ¼ 0:5772 is the Euler’s constant. Hence by using Eqs. (10) and taking into account just the fluctuating part of the displacements in the along-wind and vertical directions, the mean and the standard deviation of the extreme values of the displacements at node 1 (Table 2) are determined for each direction. The across-wind direction was omitted, since the related displacements are negligible. 4.4. Frequency domain analysis Having determined the displacements at nodal point 1 in the time domain (Fig. 7), the density spectra Sx and Sz are obtained with the application of the fast Fourier transform algorithm to the displacements time histories. The resultant response spectra displayed in detail in Fig. 8, for a frequency range 0–0.24 Hz, show the three peaks corresponding to the vibration mode shapes 1, 2 and 3 (see Table 1).
Fig. 6. Detail of the flexible cantilevered truss (nodal point 1).
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Table 2 Displacements at nodal point 1 (Fig. 6)—statistics of extremes Direction
Standard deviation
Most probable value (m)
Along-wind (X) Vertical (Z)
0.084 0.089
1.16 1.24
x[m]
Along-Wind Direction(X) 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 0
200
400
600
800
1000
Time [sec]
z[m]
Vertical Direction(Z) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2
0
200
400
600
800
1000
Time [sec] Fig. 7. Time histories—displacements at node 1.
Working strictly in the frequency domain, Eq. (11) expresses the transfer relation between power spectra density functions of the stationary random excitation and response xðtÞ; in the along-wind direction: Sx ðoÞ ¼ jHðoÞj2 Sfwind ðoÞ;
ð11Þ
where Sfwind ðoÞ and Sx ðoÞ denote, respectively, the modal power spectra density functions of the excitation force and displacement amplitude and HðoÞ is the modal complex-frequency-response function. Applying this formulation to a system with a single generalized degree of freedom (any of the lateral oscillation modal shapes) with frequency oj ; stiffness kj and damping ratio xj ; subjected to a forcing excitation with frequency o; % the complexfrequency-response function takes the form: HðoÞ % ¼
1 for xj > 0: kj ½1 þ 2ixj ðo=o % j Þ ðo=o % j Þ2
ð12Þ
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Along-Wind Direction (X) f1=0.158Hz f2=0.172Hz
2
S x [m s]
0.05
f3=0.200Hz
0.03
0.00 0.24
0.12
Frequency [Hz]
Vertical Direction (Z) f1=0.158Hz f2=0.172Hz f3=0.200Hz
2
S z [m s]
0.05
0.03
0.00 0.12
0.24
Frequency [Hz] Fig. 8. Response spectra—displacements at node 1.
The maxima of modal responses may be determined by considering the excitation as a sinusoidal force with the same magnitude as the modal generalized force Fmj in the natural frequency oj ; applied in direction X at the free end of the flexible cantilevered truss (Node 1 in Fig. 6). In the case of the structural system analysed herein, the first 10 natural frequencies are sufficiently close and the modal damping factors are low, leading to coupling of the vibration modes. It is then possible to apply the square root of the sum of the squares (SRSS) method to the modal responses presented in Table 3, which yields a displacement amplitude x ¼ 1:389 m in the along wind direction (X direction) at El. 31.00 m (Node 1 in Fig. 6): x ¼ ½x21 þ x23 þ x27 þ x29 1=2 :
ð13Þ
When the contributing modes have very close frequencies, the response amplitude may be also calculated by combining the modal responses (Table 3) in accordance to the complete quadratic combination method (CQC), in order to take into account the contribution of the other modes through the cross terms. The displacement amplitude at node 1 in the along wind direction (X) was then determined by the CQC method, resulting in x ¼ 1:392 m; a result which, as expected, is very close to that given by the SRSS method. In the CQC method, as the damping ratios are the same for all modes, xj ¼ 2%, the cross-correlation coefficients rnm between modes m and n were calculated according to Eq. (14); the
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Table 3 Modal responses in the frequency domain—along-wind direction Mode
Displacement (m) in X Direction, El. 31.00 m
1 3 7 9
1.26 0.08 0.21 0.54
modal damping ratios in this case were taken as nearly mass proportional: rnm ¼ rmn ¼
8x2 ð1 þ rÞr3=2 ; ð1 r2 Þ2 þ 4x2 rð1 þ rÞ2
0prnm p1;
ð14Þ
where r ¼ on =om and om > on : 4.5. Tower and cables strength analysis Checking the interaction equations for the load and resistance factor design criteria (LRFD), in accordance with the design codes for steel structures—as for example ABNT/NBR8800 (1986), or the ASCE or yet the EUROCODE recommendations—the angle sections of those members connected to the foundation have exceeded the prescribed limit values, whichever limit state was verified: safety or serviceability. Other members, around level 28.00 m, were also ill-proportioned. These calculations were made by taking the stress resultants at one instant of time where a peak in time history occurred for the axial force in these angle section members. As indicated by the envelope of tension forces in the cables (Table 4), the limit tension ratio for maximum wind velocity recommended by the Brazilian code for transmission lines was not surpassed. The analysis resulted in working ratios equal to 24% for the electric conductors and 20% for the lightning conductors; in both cases not greater than the recommended limit, 50% of the nominal cable strength. These results rule out the possibility that the collapse of these transmission line towers had been caused by rupture of cables under the action of wind. The existing damping devices (Stockbridge) were not included in the 3D-FEM model. They are useful to attenuate high frequency small amplitude movements that result from low speed winds that may lead to rupture caused by fatigue. Conversely, in the present analysis the movements are of another nature; they are of low frequency and considerable large amplitude. It is important to emphasize here that the linear dynamic analysis carried out with the tower–cables coupled model is just a preliminary step of a complete appraisal of this aerodynamic problem. Further steps should be taken with a non-linear dynamic analysis, in order to take into account adequately the large displacements of the cables that are allowed by large angular displacements of the suspension-rods.
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Table 4 Envelope of tension forces in the cables Cable Type
External conductor Central conductor Lightning conductor
Tension–T (kN) Minimum
Maximum
25.85 28.71 13.05
33.62 30.83 13.86
Nominal cable strength, Rn (kN)
Working ratio T/Rn
137.80 137.80 69.86
0,19–0,24 0,21–0,22 0,19–0,20
Insulators θ1 Wind
Cable Mass θ2
Double Pendulum
Fig. 9. Double pendulum behavior—comparison.
5. The important dynamic effect of the suspension-rods An important conclusion from this study is the fundamental role the suspensionrods play in the coupled tower–transmission line system; the height of the chain of insulators (or of the suspension-rod) defines the dynamic characteristics of the tower–cables coupled model. The system has the tendency to behave like a double pendulum in the orthogonal direction to the transmission line axis, and a preliminary evaluation of the system fundamental frequency can be made with a two-degree-offreedom model (Fig. 9). Taking the following transmission line characteristics as used in the 3D model: (a) medium span: L =450 m; (b) chain of insulators: height=2.65 m ) suspension-rod height L1=2.90 m; (c) electric conductors: Grosbeak (weight: m ¼ 13:0 N/m, maximum tension: Trmax ¼ 31500 N); (d) and the sag s of cables hanging in catenary shape under the action of their own weight as given by Eq. (15) (where Tr0 is equal to the tension at the middle of the span): Tr0 mL s¼ cosh 1 ; ð15Þ 2Tr0 m where s ¼ 10:45 m was obtained by considering that Tr0 ¼ Trmax ; when the suspension points are at the same level and thus the tension variation is very small.
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The fundamental frequency can be estimated with the double pendulum model as given by Eq. (16), where L1 is the suspension-rod height and g is the acceleration of gravity rffiffiffiffiffiffi 0:6 g fdp ¼ ð16Þ ðHzÞ; L2 =L1 B2:0 2p L1 The length L2 corresponds to the vertical distance between the straight line passing through the tips of two consecutive suspension-rods and the catenary centre of mass, i.e., L2 ¼ 2s=3: The model fundamental frequency based on the double pendulum linearized model (Fig. 10a) results in fdp ¼ 0:175 Hz which may be considered a simple and close estimation of f1 at a preliminary design stage, compared to the result from the 3D-FEM analysis, f1 ¼ 0:158 Hz. For the 3D-FEM Delta tower model with the tips of the suspension-rods restrained (Fig. 10b) each one of the swinging cables behaves like a simple pendulum and yields a fundamental frequency f1 ¼ 0:168 Hz, which may be approximated by the frequency given by the simple pendulum model, fp ¼ 0:189 Hz. The oscillation frequencies and mode shapes obtained for the 3D-FEM model considering the suspension-rods restrained are compared in Table 5 to those found with the new proposed model (free suspension-rods). As one can see, the values of frequencies are reasonably close for the two models. The main difference between these two models is that the tower is much more mobilized when the suspension-rods are free to swing and, as a consequence, it experiences larger amplitudes of the lateral top displacement. For restrained suspension-rods, the cables behavior begins to approach the one displayed by a simple pendulum with smaller oscillation amplitudes. This latter 3D model is close to that used in design practice, in which tower and cables are not coupled as in the 3D-FEM model, and the resultant of the
Insulators (L1)
Cable Mass
θ2
dp
0.6
g
2π
L1
θ2
Cable Mass
Bracing Bars (Mass = zero)
g = acceleration of gravity
=
Conductor Projection (L 2 )
Conductor Projection (L 2)
θ1
f
Insulators (L1)
Hz ; for L 2 /L1 ~ 2.0
(a) Double Pendulum (1)
fp =
1
g
2π
L2
Hz
(b) Simple Pendulum (2)
3D -F EM
f1 =0.158Hz
f 1 =0.168Hz
Pendulum
fdp =0.175Hz (1)
f p =0.189Hz (2)
Fig. 10. Pendulum formulation—results.
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Table 5 Natural frequencies and mode shapes considering suspension-rods effects (New model) suspension-rods not-restrained
Suspension-rods restrained
Mode Frequency (Hz)
Vibration mode
Tower’s top displacement direction
Frequency (Hz)
Vibration mode
Tower’s top displacement direction
1 2 3 4 5 6 7 8 9 10
Tw/EC EC Tw/LC LC/EC LC/EC LC/EC Tw/LC/EC LC/EC Tw/LC/EC Tw/LC/EC
Lateral Longitudinal Lateral Longitudinal Longitudinal Longitudinal Lateral Longitudinal Lateral Longitudinal
0.168 0.202 0.241 0.283 0.330 0.340 0.510 0.877 1.389 2.128
Tw/EC Tw/LC LC/EC LC/EC LC/EC LC/EC Tw/LC/EC LC/EC Tw/LC/EC Tw/LC/EC
Lateral Lateral Longitudinal Longitudinal Longitudinal Longitudinal Lateral Longitudinal Lateral Longitudinal
0.158 0.172 0.200 0.283 0.297 0.331 0.481 0.877 1.351 1.786
Tw=Tower, EC=Electric Conductor, LC=Lightning Conductor.
wind forces acting on electric and lightning conductors as well as on chains of insulators are applied to the tower’s linkage nodes. Table 6 summarizes and compares the results obtained with different models for the transverse displacements at the free end of the flexible cantilevered truss (Node 1, Fig. 6), considering the wind direction transverse to the transmission line axis. The models used for such comparison are the following: 1. design practice (static analysis): The electric cables are not discretized; the wind design dynamic pressure q ¼ 2:4 kN/m2 is uniformly distributed along electric cables and the height of the tower, considering the obstruction effective area. The resultant of the wind forces on the cables are applied to the tower’s linkage nodes. 2. equivalent cable-tower coupled model (static analysis): The new proposed model considering the mean wind velocity only, without turbulence. 3. simplified cable-tower coupled model (time-history analysis for turbulent wind): The new proposed model with the suspension-rods restrained. 4. cable-tower coupled model (time history analysis for turbulent wind): The new proposed model. 6. Non-linear pendulum-like dampers as a solution for the problem Since the kinetic mechanism that leads to structural collapse has been identified, the next step is to attenuate the amplitudes of the tower’s top horizontal cross-line displacements in the along-wind direction, by means of an auxiliary dynamic device. Based on the works by Pinheiro [4] and Battista and Pinheiro [6], on dynamic control of slender towers under environmental loadings, the proposed remedial measure is to install non-linear pendulum-like dampers (NLPD), as depicted in Fig. 11, to reduce the amplitudes of horizontal cross-lines displacements at the
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Table 6 Displacements response at the free end of the flexible cantilevered truss (node 1, Fig. 6) of the tower for a ¼ 0 wind direction Hypothesis
1. Design practice (static analysis) 2. Coupled model (static analysis) Mean wind velocity 3. Coupled model (time-history analysis) Suspension-rods restrained and turbulent wind 4. Coupled model (time-history analysis) Turbulent wind Probable value for the new proposed model 4 (standard deviationD0.08)
Displacements (m) Along-wind direction (X)
Vertical direction (Z)
Max./min. Min. Max. Min. Max.
70.51 0.60 0.60 0.73 0.002
70.51 0.61 0.61 0.003 0.67
Min. Max.
1.26 0.10 1.16
0.12 1.34 1.24
towers due to the sway motion of the transmission lines induced by wind. The dimensions proposed on Fig. 11 were determined with the solution of the non-linear two-degree-of-freedom system which formulation is summarized in the following section. 6.1. Equations of controlled motion It is possible to design NLPDs to attenuate the amplitudes of vibration in the dominant fundamental mode by applying the 2D modal equations (Eq. (17)) for the simple model illustrated in Fig. 12, where subscript p stands for the pendulum’s properties. For that, the related modal properties (modal mass M; force F ; stiffness K and/or frequency O) have to be extracted from the 3D-FEM model proposed herein. The Runge–Kutta method was used in the solution of the nonlinear system of equations: ( ðM þ mp Þx. þ C x’ þ Kx þ mp lðsin yÞ00 ¼ F0 sin ðoe t þ aÞ ð17Þ mp l 2 y. þ cp y’ þ kp y þ mp gl sin y þ mp l x. cos y ¼ 0 where ðsin yÞ00 ¼ y. cos y y’ 2 sin y: The comparisons between the controlled and uncontrolled systems as well as the angular amplitude of the NLPD are shown in Fig. 13, for NLPD having damping ratio xp ¼ 5% and stiffness kp ¼ 1 kN m/rad. The reduction of the horizontal cross-line displacements in the along-wind direction in the first mode of vibration reaches an efficiency rate over 90% when the NLPD system is designed to work almost in resonance with the first mode of the structural system under wind action. Because of the lack of space, the presentation and discussion of results from parametric and sensitivity analyses are herewith prohibitive.
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See Detail
X Body of mass mp equivalent to a steel sphere with D=0.28m. El. 20.73m
Perspective
Frontal View
Detail
Fig. 11. The delta tower and the non-linear pendulum-like dampers.
Fig. 12. Tower with NLPD and analogous simplified mechanical system.
Fig. 13. Controlled and uncontrolled structure responses for oe =o1 ¼ 0:9 and op =o1 ¼ 0:9:
7. Conclusions A 3D-FEM model was constructed for analysing the dynamic coupled behavior of transmission lines and towers under the action of wind. The distinguishing feature of
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this model is its ability to account for the inertia forces which arise in the towers dynamics with the wind induced sway motion of the electric cables. The suspensionrods formed by the chains of insulators were identified as the most important component of the system when it comes to the analysis of wind flow and tower-lines coupled model interactive dynamic behavior and response. To attenuate the towers’s top horizontal cross-line displacements in the along-wind direction, NLPDs were envisaged leading to an efficiency rate around 90% when the NLPD system is designed to work almost in resonance with the wind-induced motion in the first mode of oscillation of the coupled structural system under wind action. The authors realize that the presented results obtained with the proposed new modelling are just a preliminary step for the better understanding of the mechanical behavior of transmission line towers under the action of wind. The proposed model has yet to be augmented with appropriate non-linear dynamics to allow for very large angular displacements of the suspension-rods and cables to better describe what seems to be its actual interactive mechanical behavior. Very recent preliminary results, which have been obtained by the authors from a non-linear model, seem very promising and shall be reported in the near future.
References [1] R.S. Rodrigues, Collapse of transmission line towers under the action of wind, M.Sc. Thesis, COPPE/ UFRJ, Rio de Janeiro, April, 1999 (in Portuguese). [2] R.S. Rodrigues, R.C. Battista, M.S. Pfeil, Unveiling the dynamic mechanism of collapse of transmission line towers under wind forces, XXIX Jornadas Sudamericanas de Ingenieria Estructural—Jubileo Prof. Julio Ricaldoni, Punta del Este, 2000, (in Portuguese). [3] H. Yasui, H. Marukawa, Y. Momomura, T. Ohkuma, Analytical study on wind-induced vibration of power transmission towers, J. Wind Eng. Ind. Aerodyn. 83 (1999) 431–441. [4] M.A.S. Pinheiro, Non-linear pendulum absorber of lateral vibrations in slender towers, M.Sc. Thesis, COPPE/UFRJ, Rio de Janeiro February, 1997 (in Portuguese). [5] R.C. Battista, N.F.F. Ebecken, L. Bevilacqua, Dynamical analysis of an offshore platform with vibration absorbers, IUTAM-Symposium on Recent Developments in Nonlinear Oscillations of Mechanical Systems, Vietnan, 1999. [6] R.C. Battista, M.A.S. Pinheiro, Oscillations reduction of compliant structures of offshore platforms, Proceedings of the 21st Iberian Latin American Congress on Computational Methods in Engineering, Rio de Janeiro, 2000 (in Portuguese). [7] E. Simiu, R.H. Scanlan, Wind Effects on Buildings, Wiley, New York, 1996. [8] H.A. Buchholdt, S. Moossavinejad, A. Iannuzzi, Non-linear dynamic analysis of guyed masts subjected to wind and guy ruptures, Proc. Inst. Civ. Eng. September (Part 2) 81 (1986) 353–359. [9] ABNT. NBR 6123 Wind forces on buildings, ABNT, Rio de Janeiro (Brazilian Code) (in Portuguese). [10] M.S. Pfeil, R.C. Battista, Aerodynamics stability analysis of cable stayed bridges, J. Struct. Eng. 121(12) (1995) 1784–1788. [11] R.W. Clough, J. Penzien, Dynamics of Structures, McGraw-Hill Books Co, Singapore, 1993 (Chapter 2).