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1、外文原文Dynamic behavior and stability of transmission line towers under wind forcesRonaldo C. Battistaa,b, Rosngela S. Rodriguesa, Michle S. PfeilaaCivil Engineering Program,COPPEUniversidade Federal do Rio de Janeiro,CP 68506,Rio de Janeiro, CEP 21945-970, BrazilbCOPPETEC Research, Consulting & Design
2、,CP 68506, Rio de Janeiro, CEP 21945-970, BrazilAbstractA 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 fr
3、om 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. I
4、n order to reduce the TLTs 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
5、 uncontrolled systems. 2003 Elsevier Ltd. All rights reserved.Keywords: Transmission line; Stability; Dampers; Wind force; Dynamics; Steel tower1. IntroductionA new analytical-numerical modeling has been applied to a chosen type of steel transmission line towers (TLT): a conventional 32.86 m-high se
6、lf-supporting tower. The structural modeling of the chosen TLT is based on observation of the systems 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 e
7、lectric cables have brought up the importance of fluid flowcablesstructure interaction when evaluating the towers behavior under the action of wind forces, leading to the new analytical-numerical modelling for the structural analysis of TLTs, as originally proposed by Rodrigues 1 and Rodrigues et al
8、. 2 and, almost simultaneously, by Yasui et al. 3 analyzing the differences in the behavior of power lines supported by tension- or suspension type transmission line towers. The overall results from the performed analyses were used to unveil the mechanism of collapse and envisage a remedial measure
9、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. De
10、scription of the structural modelFor 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 aeri
11、al 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
12、 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 t
13、o allow for 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.Fig. 1. 3D-FEM model of the str
14、uctural system.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 simulat
15、ed 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 insulators suspension rods are considered in a geometric non-linear static analysis.Following the static equili
16、brium state, the time history response of the structure under wind action is obtained for the superposition of the n significant modes as follows:where mj is the modal mass, j is the modal damping ratio, j the circular frequency, r(t); (t)and (t); are respectively, the displacement, velocity and acc
17、eleration at time t, j the vibration mode shape and j T 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 tak
18、en or not into account.3. Wind forcesThe 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:Referring to Eq. (2), (z) is the mean wind velocity in
19、the horizontal direction at z coordinate, i.e., (z) is 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 u(y,z,t)turbulenceis statistically determined as a function of the mean wind velocity (z); the
20、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 velocitydrag forcehas the expression:where is air density, A the effective area of the structure, CD() the drag coefficient corresponding
21、 to angle of attack and U(t) 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.The cross-spectral den
22、sity 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 velocitie
23、s (t) is performed by the autoregressive method, which consists of expressing the instantaneous value of (t) as a linear combination of some previous values of (t) plus a random impulse. The field of spatially correlated fluctuating wind velocities u(t) is obtained by pre-multiplying vt to a matrix
24、containing cross-correlation information between the generated signals given by the cross-spectral density function 8.3.1. Mean wind forcesThe 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 colla
25、psed. 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 where (z) is the design mean wind speed at reference height z =10 m, U0 =50 m/s
26、 the basic wind speed, S1 = 1.00 is the topographical factor, S2 = 0.69 is the combined exposure factor and S3 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 pow
27、er law (Eq. (5) using (zref) = (10) and p, exponent related to the terrain roughness, equal to 0.15 (farmland, scattered trees and low buildings):3.2. Turbulence numerical simulation along the transmission line axisIn the auto-regressive method, the turbulence u(y, z, t) simulation is a linear combi
28、nation of p values added to a zero-mean random impulse with variance 2Nu.where s are the auto-regressive parameters, p is the auto-regressive order and N(t) the zero-mean random process and variance equal to 1. According to Buchholdt et al. 8, the parameters s are to be determined with a solution of
29、 an algebraic system ofFig. 2. Mean wind velocityvertical profile.equations:where Ru is the autocorrelation function of u(t) process, determined by the inverseFourier transform of the energy spectrum Su(n). With u2 as the u(t) variance, Nu2 inEq. (6), is given byUsing the procedure described above i
30、n 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 d
31、etermined according to Eq. (3), considering three angles of attack: =0 (orthogonal to the transmission line axis), =45and =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 (); were t
32、hose given in the Brazilian design code 9.4. Self-supporting tower analysisThe 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 c
33、onductors(d =25.16 mm), two EHS lightning cables (d = 9.15 mm) and mean span equal to 450 m. The chains of glass pieces insulators are mounted on 2.90m length suspension-rods.4.1. Soilstructure interactionThe soilstructure interaction was performed taking into account two types of soil: medium sand
34、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 oscillati
35、on 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 an
36、d consequent very small settlements. 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
37、 soil and consequent very small settlements.Fig. 3. Delta towersilhouette and frontal view.4.2. Free vibration analysisThe result 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. The
38、se 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 towers dominant vibration modes. The fundamental period equal to 6.34 s (i.e., low frequency, f = 0.158 Hz) means tha
39、t, when exposed to the dynamic effects of the atmospheric turbulence, the fluctuating response of the low damped tower-cables coupled system in the along-wind and across-wind directions can be significant.Table 1Natural vibration periods and frequencies and modal shapes descriptionTw=Tower, EC=Elect
40、ric Conductor, LC=Lightning Conductor.Fig. 4. Mode shape 1lateral oscillation (T =6.34 s).Fig. 5. Mode shape 7lateral oscillation (T=2.08 s).4.3. Time domain analysisThe 3D-FEM model was analysed in the time domain (total time interval, Tmax, 840 s), considering the first 10 vibration modes in the r
41、esponse calculation. The wind forces transverse to the transmission line axis (=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 1.26m in
42、the along-wind direction, while in the vertical direction resulted in 1.34 m, both at time t =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 as
43、sumed 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, e and the standard deviation, e ; of the extreme values are given 11 by Eqs. (10a) and (10b):where
44、is the zero-crossing frequency, T is the time duration, is the standard deviation of the sample and =0.5772 is the Eulers 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 devi
45、ation 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 analysisHaving determined the displacements at nodal point 1 in the time domain (Fig. 7)
46、, 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 00.24 Hz, show the three peaks corresponding to the vibration mode shapes
47、 1, 2 and 3 (see Table 1).Fig. 6. Detail of the flexible cantilevered truss (nodal point 1).Fig. 7. Time historiesdisplacements 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
48、 response x(t),in the along-wind direction:where Sfwind() and Sx() denote, respectively, the modal power spectra density functions of the excitation force and displacement amplitude and H() 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 j , stiffness kj and damping ratio j ; subjected to a forcing excitation with frequency ; the complex frequency-response function takes the form:Fig. 8. Response spectradisplacements at node 1.The maxima of mo