Wind action according to the Brazilian Code: a case study
Transcrição
Wind action according to the Brazilian Code: a case study
EACWE 5 Florence, Italy 19th – 23rd July 2009 Flying Sphere image © Museo Ideale L. Da Vinci Wind action according to the Brazilian Code: a case study 1st A.M.Wahrhaftig, 2nd R.M.L.R.F. Brasil Department of Construction and Structure, Polytechnic School of Federal University of Bahia – [email protected] – Rua Aristides Novís, no 02, 5o andar – Federação, Salvador – BA, Brazil, CEP 40210-630. Department of Structural and Geotechnical Engineering, Polytechnic School of University of São Paulo – [email protected] – Av. Prof. Almeida Prado tv. 2, n. 83, Cidade Universitária São Paulo – SP, CEP 05508-900. Keywords: Wind action, Brazilian code, Dynamic of structures, Experimental investigation, Natural Frequencies. ABSTRACT Wind was not a problem for low-rise heavy structures of the past, but has come to be as the constructions have become more and more slender, using less material. The danger of the wind causing accidents is particularly significant for power lines structures, radio, TV and microwaves towers, radar antennas and other such structures, according to Blessmann (2001). Wind effects upon slender poles and towers are reported by Simui; Scalan (1996), Sachs (1972), Kolousek et al (1984) and Navara (1969). Accidents with power lines tower, one of them involving the fall of more than 10 consecutive towers in the state of São Paulo, Brazil, were reported by Blessmann (2001). Further, Blessmann (2001) reports a study on the effect of hurricanes in Miami at 1950 that completely destroyed 11 metal radio towers due to buckling of individual members. Occurrences of accidents with mobile phone antennas supporting poles in Brazil are reported by Brasil e Silva (2006). In this paper, we call poles long bar structures with circular or polygonal section. In the other hand, towers are metal frames, stayed or not. It is particularly important to investigate the effects of the wind on slender structures. A Contact person: 1rd A.M.Wahrhaftig, Department of Construction and Structure Of Polytechnic School of Federal University of Bahia, Rua Aristides Novís, no 02, 5o andar - Federação, CEP 40.210-630, (5511) 3283-9725. E-mail [email protected]. particularly interesting case are de mobile phone antennas supporting poles. In Brazil, a profound reform of the legal apparatus of telecommunications allowed for a new structure for that industry. Previously government owned Telebrás System was sold to private enterprise 29 of July of 1998 in 12 consecutive auctions carried out at the Rio de Janeiro Stock Exchange, the largest such an operation in the world to the time. Brasil and Silva (2006) report that during the implementation of the mobile phone system in Brazil, more than 10.000 support structures were designed, manufactured and installed. 2000 of those were reinforced concrete poles. In the early 1990 years, there were not enough experienced companies and personnel to supply the existing demand. Other products makers had to adapt their production lines to the telecommunications market. In the other hand, structural engineers specialized in other applications had to adapt their mathematical models to the analysis of such structures. Some of those models, according to Brasil and Silva (2006), consider the wind effects as static loads, neglecting the dynamic aspects. Nevertheless, for structures whose first natural frequency is under 1 Hz, the dynamic effects of Wind are important and to consider them static or deterministic in nature is too rough an approximation. About the importance of considering dynamic effects of wind, Durbey, C. & Hansen, O S. (1996) wrote that flexible structures could vibrate in different modes when excited by the Wind. Further they wrote that for slender structures, the dynamic effect of wind might cause resonance. The Brazilian code dealing with the analysis of structures excited by the wind is NBR 6123/88 – Loads due to wind upon constructions. The Code provides models to consider the effects of wind to design structures. All of them consider the real dynamic load as an equivalent static one (Blessmann, 1989). The choice among them is related to the frequency of the first vibration mode. In the first model, called static model, the influence of the fluctuating part of the Wind is taken into account by a Gust Factor in the calculation of the characteristic Wind velocity. It is considered that no resonance occurs.The model that deals specifically with the dynamic responses along the average Wind, called the discrete model, is described in Chapter 9 of NBR 6123/88 – Loads due to Wind upon constructions. It considers that the fluctuations of the wind occur in the band of the lower frequencies of the structure. The procedure starts with the computation of the natural frequencies that are used in order to obtain the corresponding dynamic amplification factors. Thus, the frequency computation process is fundamental to the procedure. To evaluate the procedures of NBR 6123/88 – Loads due to wind upon constructions., we apply them to a mobile phone antenna support pole. The dynamic computation carried up to the 5th vibration mode. We considered the geometric stiffness in the procedure, that is, the influence of the axial loads were taken into account. Mass discretization and modal shapes were obtained via Finite Element models. Frequencies were computed from the corresponding eigenvalue problem were the stiffness matrix included the geometric stiffness part, a nonlinear effect. The procedure is used to linearize second order effects. Structural data was collected in the field. Local survey indicated the presence of antennas and other accessories fixed to the structure, which are additional masses and forces. Field determination of the frequencies of the structure under environmental excitation was carried out using a piezoresistive accelerometer, with DC response, with high sensibility, capable of measuring low accelerations. This device was fixed to the top of the pole. Data acquisition was carried out for 40 hours. The fundamental frequency of the structure was obtained from the time series via FFT. The resultant value was 0.53 Hz. Applying the Brazilian Code to compute the resulting maximum bending moment the response of the so-called discrete model NL is 55.69 % larger the response of the static model. We observed little influence of higher modes than the fundamental one. These are responsible for 66% of total dynamic response of the structure. 1. WIND ACTION ACCORDING TO BRAZILIAN CODE The basic aim of NBR 6123/88 - Forces due to the wind in constructions (1988) is to define in the calculations imposed conditions for the forces due to the static and dynamic wind action. NBR 6123/88 presents two possible models of calculation for the wind action in structures, namely: static forces generated by the wind or static model and discrete dynamic model, that will be described, in this section. The consideration of the dynamic effect and extreme vibration of the structures due to the wind action is described in item 9 of NBR 6123/88. Blessmann (1989) clarifies that the Brazilian code presents a equivalent static action of the wind, based in the method of random vibration considered by Davenport. Differs from it in the parameters determination which define this action. The existing recommendations in NBR 6123/88 for the dynamic analysis take into account the variation in the module and in the orientation of the average wind speed. The average speed produces static effect in the structure, whereas the fluctuations or gusts produce important oscillations, “especially in high constructions”. This model of dynamic analysis of high structures is also commented by Simiu& Scalan (1996) who associates it with the necessity of the induced vibrations analysis for floating loads. NBR 6123/88 incorporates these concepts and says that constructions with basic period superior of 1 s, frequencies up to 1 Hz, can present important floating reply in the direction of the average wind. 1.1 Static forces developed by the wind or Static Model. The static forces due to the wind are determined as following. The basic speed of the wind, V0, is related to the place where the structure will be constructed. By definition it is the speed of a gust of 3 seconds, exceeded in average once in 50 years, measured 10 m above ground, in open and flat area. The Brazilian norm brings isopleths of the basic speed of Brazil. As general rule, one admits that the basic wind can blow of any horizontal direction. When it is calculated, the basic speed is multiplied by the factors S1, S2 and S3 to obtain the characteristic velocity of the wind Vk, for the considered part of the construction, so: Vk = V0 S1S2 S3 (1) The topographical factor S1 takes in account the variations of the relief of the land and the increase of the wind speed in the presence of mounts and slopes, but it doesn’t consider the reduction of the turbulence when the wind speed increase. The S2 factor considers the combined effect of the ground asperity, the variation of the wind speed with the height above ground and the dimensions of the construction or part of it in consideration. The NBR 6123/88 suggests that the land asperities should be divided in 5 categories. Regarding the dimensions, the constructions had been divided in 3 classes. To take in account the height of the land in the calculation of the S2 factor, the Brazilian Norm establishes formula (2). S2 ( z ) = bFr ( z /10) p (2) with p e b given on Table 1. For this model calculation, the wind dynamic action is taken in account by mean factor Fr specified for open ground in level or approximately in level, with few isolated obstacles, such as low trees and constructions (Category II). The time that defines the gust factor is a function of the construction class. It will be of 3 s, 5 s or 10 s, according to the construction class A, B or C, respectively. The characteristic velocity of the wind is then used to determine the wind pressure by q = 0, 613Vk 2 The component of the global force in the wind direction, drag force Fa, is given by (3) Fa = Ca qAe (4) where Ca denotes the drag coefficient and Ae the effective frontal area (area of the orthogonal projection of the construction, structure or structural element on a perpendicular plan to the wind direction). The drag coefficient is a function of the Reynolds number, of the dimensions and of the body forms, given by Re = 70000Vk L1 (5) where Vk is given in m/s in (1) and, L1, is the reference dimension. The drag coefficients possess values prescribed in tables or abacuses of NBR 6123/88 for diverse situations of calculation. Table 1: Exponent p and parameter b (NBR 6123/88). 1.2 Category of asperities I II III IV V p b 0.095 1.23 0.15 1 0.185 0.86 0.23 0.71 0.31 0.5 Discrete dynamic model If constructions possess variable properties along the height, as normally founded in telecommunications poles, it must be represented by a discrete model, as shown in Fig.1. z mn xn mn-1 xn-1 mi zi xi m1 x1 x Figure 1: Model for a discrete dynamic mode (NBR 6123/88). The NBR 6123/88 prescribes that the calculation of the total dynamic response must be considered as the superposition of the average and floating responses, as follows. The design speed must be gotten using the expression (6), V p = 0, 69V0 S1S3 (6) correspondent to the average speed of 10 minutes in 10 meters of height above the ground, in category II ground. When it is desired to determine the modal contributions in the dynamic response of the discretized model, for the degree of freedom i, the total load X i on the wind direction is the sum of the mean component Xi and the fluctuant component X̂ i , so: X i = X i + Xˆ i where the mean load Xi is: (7) ⎛ zj ⎞ X i = q0b C j A j ⎜ ⎟ ⎝ zr ⎠ 2p 2 (8) and the fluctuant component X̂ i is defined by: Xˆ i = FH ψ i xi (9) where: ψi = mi m0 (10) n ∑ βi xi i =1 n FH = q0b A0 2 ∑ ψi xi ξ (11) 2 i =1 βi = Cai Ai ⎛ zi ⎞ ⎜ ⎟ A0 ⎝ zr ⎠ q0 = 0, 613V p 2 p (12) (13) the parameters b and p are indicated in NBR 6123/88; z r is the reference level and again, Vp is the design velocity, q0 (in N/m2) is the dynamic pressure, zi , x i , mi , m0 , Ai , A 0 , ξ e Cai are, respectively, the height, the correspondent vibration mode in coordinate i; the concentrated mass in the degree of freedom i; the reference mass; the equivalent mass for the degree of freedom i; the reference mass; the amplification dynamic coefficient; and the drag coefficient corresponding to the coordinate i. The dynamic amplification coefficients was computed by Galindez (1979) admitting a potential modal form, and has been transformed into abacuses and included in NBR 6123/88 for the five ground categories. To use them it is necessary to determine the width of the construction by n L1 = ∑ Ai i =1 (14) h where h is the edification height. The calculation process is finished by a similar form of the static one, through the superposition of the intervening variable effect. When more than one vibration mode is used in the solution, NBR 6123/88 establishes that the joint effect can be calculated by the criterion of the square roots. Let Q̂i be any static variable (force, bending moment, tension, etc) or a geometric variable (deformation, displacement, and rotation), correspondent to a mode i, the overlapping of effects is calculated by: 1/2 ⎡n ⎤ Qˆ = ⎢ ∑ Qˆ i 2 ⎥ ⎢⎣ j =1 ⎥⎦ (15) Beyond inducing longitudinal vibrations, the random fluctuations of the instantaneous speed regarding the average wind speed are responsible for structural vibrations in the perpendicular direction to the average flow direction. NBR 6123/88 prescribes that the resultant in the perpendicular direction to the wind direction can be calculated computing 1/3 of the effective forces in the wind direction. Thus, the final response of the structure developed by the wind actions must obey the rules of the vectorial calculation. Note that no vortex shedding consideration has being made. 2. DETAILS ABOUT THE INVESTIGATED STRUCTURE The investigated structure is a truncated cone metallic pole with 52 cm and 82 cm top and bottom diameters respectively. It is intended for the sustaining of the mobile phone broadcasting system. It is 30 meters high, hollow section. The external diameter (φext) and thickness (t) vary along the height. It is installed in the city of Aracaju, Sergipe, Brazil. The structure data were acquired in the field. The diameters were measured with a metallic tape measure and the thickness with an ultrasound equipment. For a same vertical line, they were carried out several thickness measurements, obtaining a relative average of the band. The union of the pole segments is formed by the sussessive fittings, by placing and screw-fastening of the metallic parts. Each superpositioning band has 20 cm of length. In these joints areas, the thickness of the transverse section correspond to the sum of the superpositioning bands measures. In Table 2 it can be found the properties and the discretization used to model the structure. The assessed slenderness of the structure is 256. The metallic pole sustains two working platforms, one situated at 20 m high and the other, in the superior extremity. There is still a set of antennas located at 27 m from the base and attached to the body of the pole through metallic devices. The platforms and the supporting devices follow the composition presented in Table 3, where φ designate the diameter of the platform. The local assessment revealed the presence of microwave (MW) antennas and of radio frequency (RF), that are listed with the rest of the structure accessories in Table 4. The data related to the antennas were obtained from the catalogue of the manufacturer. All the before mentioned devices represent additional masses and concentrated forces to the structure, as it is showed in Table 5. Table 5 presents the structural parameters and the parameters of the existing devices, the specific weight adopted for the material of the structure, the localized and distributed axial load. In Fig 2(a) they are presented photographic image of the pole. The geometry of the structure and the existing devices are schematically represented in Fig. 2(b). Height (m) 30.00 29.00 28.00 27.00 26.00 25.00 24.10 23.90 23.00 22.00 21.00 φext (cm) 52.00 53.00 54.00 55.00 56.00 57.00 57.90 58.10 59.00 60.00 61.00 Table 2 – Structure data and discretization of the model. t Height φext t Height φext (cm) 0.60 0.60 0.60 0.60 0,60 0.60 0.60 0.60 0.60 0.60 0.60 (m) 20.00 19.00 18.10 17.90 17.00 16.00 15.00 14.00 13.00 12.10 11.90 (cm) 62.00 63.00 63.90 64.10 65.00 66.00 67.00 68.00 69.00 69.90 70.10 (cm) 0.60 0,60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.76 (m) 10.00 9.00 8.00 7.00 6.10 5.90 5.00 4.00 3.00 2.00 1.00 0.00 (cm) 72.00 73.00 74.00 75.00 75.90 76.10 77.00 78.00 79.00 80.00 81.00 82.00 t (cm) 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Table 3 - Composition of the platform and suport. Platform φ = 2.5 m Floor sheet Lateral floor sheet Channel (U) 150 × 12.2 mm – Banister Angle (L) 102 × 76 × 6.4 mm – Banister Angle (L) 102 × 76 × 6.4 mm - Banister Angle (L) 102 × 76 × 6.4 mm – Floor suport Platform lower ring Joints Banister bolts Angles (L) 152 × 102 × 9.5 mm – Platform lower suport Total = Support set for antenna Tube φ = 1´ (25.4 mm) Angle (L) 203x152×19 mm Staples U (φ =1´) Top plate Total = Mass (kg) 116 46 96 68 77 43 14 3 5 33 500 Mass (kg) 6 50 1 1 58 Table 4 – Composition of the localized nodal masses. Device Antenna RF 2.6 m Antenna RF 1.23 m Antenna MW Platform Support for antennas Pipe φ = 1´ (25.4 mm) (Guide) Pipe φ = 3/4´ (19 mm) (LC) Total (kg) = Mass (kg/unit) 19 4 19 500 58 6 6 1st Plat (20 m) Quant. (kg) 2 37 1 4 2 38 1 500 6 345 0 0 0 0 924 Support (27 m) Quant. (kg) 3 56 0 0 0 0 0 0 3 173 0 0 0 0 228 2nd Plat (30 m) Quant. (kg) 1 19 1 4 0 0 1 500 6 345 1 6 1 6 880 (LC = Lightning Conductor, MW = Microwave, RF = Radio Frequency, Plat = Platform) Table 5 –Localized axial load and characteristics of the devices. Device Pole Ladder Cables 1st Platform Antenna of the 1ª platform Intermediate antennas Intermediate supports 2nd Platform Antennas of the 2ª platform Frontal area Variable 0.05 m2/m 0.15 m2/m 2.60 m2 1.99 m2 2.11 m2 0.56 m2 2.36 m2 0.90 m2 Height from 0 at 30 m from 0 at 30 m from 0 at 30 m Weight, distributed weight 77 kN/m3 0.15 kN/m 0.25 kN/m 20 m 9.06 kN 27 m 2.24 kN 30 m 8.63 kN (a) (b) Figure 2: the mobile phone antennas support pole: (a) photographic image of the pole, (b) geometric scheme of the structure (centimeters). 3. MEF MODELLING OF THE STRUCTURE For a bending bar, the deformation matrix is formed by the curvature of the bar axis to which the bending moment is related. In the dynamic solution for eigenvalues and eigenvectors, the nonlinear geometrical approach is given by the introduction of the geometrical rigidity matrix. The development of this matrix can be found in Bathe (1996), Clough (1993), Cook (1974), Cook et al. (2002) and Venâncio Filho (1975). It’s opportune to mention that the introduction of the geometrical rigidity matrix in the dynamic or static calculations, is recommended by Rutemberg (1982) and Wilson (1987) to linearize second order effects. Therefore, the rigidity matrix presented in Eq. (16) , for a bending bar with three degrees of freedom as indicated in Fig 3, becomes composed of two parts, one elastic, and the other, geometric, in which E is the elasticity module of the material, A is the area of the transverse section, L is the length of the bar, I is the section’s moment of inertia related to the perpendicular axis to the plan that contains the figure and F is the normal force of compression taken as positive. The elastic and geometric matrix showed in (16) were developed in the local system of the element. When dealing with structure with discrete system of generalized coordinates, it’s necessary to refer them to the system of global reference, following the existing correlation between the displacements in the two systems. The techniques of this procedure can be found in Cook (2002) and Bathe (1996). The structure presented in this work was modelled using frame elements with transverse sections, constant and variable, depending on the case. To the model were attributed the forces described in Table 5 with the corresponding masses. The connecting regions were treated as variable section frame elements of 0,2 m length and thickness corresponding to the sum of the thickness of the sections immediately above and below the jointzone. In Fig. 4 one finds the model in Finite Elements with a three-dimensional view, a lateral and the discretization of the structure that count on 40 elements of bar. The vibration modes and frequencies, obtained by the Finite Elements Method, are shown in Fig. 5. q3 q2 q1 Figure 3: structural model. EA ⎡ 0 ⎢ L ⎢ 12 EI [ K ] = ⎢⎢ symmetric L3 ⎢ ⎢ ⎣⎢ ⎤ ⎥ ⎥ 6 EI ⎥ F − 2 − L ⎥ L ⎥ 4 EI ⎥ L ⎦⎥ 0 (a) 3D Figure 4: MEF modelling. ⎡ ⎢ 0 ⎢ ⎢ ⎢ symmetric ⎢ ⎢ ⎢⎣ 0 6 5 ⎤ 0 ⎥ ⎥ L⎥ − ⎥ 10 ⎥ 2 L2 ⎥ 15 ⎥⎦ (b) Lateral view (16) (c) Discretization Natural vibration modes Frequencies 1st mode 2nd mode 3rd mode 4th mode 5th mode 0.532 Hz 2.884 Hz 8.654 Hz 16.985 Hz 26.174 Hz Figure 5: natural modes and frequencies by FEM. 4. EXPERIMENTAL INVESTIGATION OF THE STRUCTURAL FREQUENCIES The investigation of the natural frequency of the structure under ambient excitement was carried out using piezoresistive accelerometers, manufactured by Bruel & Kjaer, with DC response and sensitivity of 1021 mV/g, provided with integrated cable, suitable to measure accelerations between ± 2g. This device was fixed to the superior extremity of the pole, as seen in Fig. 6. Figure 6: Structure’s instrumentation. The acquisition of data was carried out by the system ADS2000, AqDados (2003), from Lynx Computers, that was connected to a portable computer for the recording of the signals. The equipments were taken to the pole top, where they were placed on the surface of the working platform, and protected from the bad weather. The electric energy system of the station was used as source to feed the electronic equipments. The signals acquisition was carried out at the rate of 50 Hz and its elapsed time was 40h, 33 min, 22 s, beginning in December 11th , 2007, at 18 h 30 min 23 s. The acceleration time series can be seen in Fig. 7. One realized that the structure was under sufficient excitement of wind, having still occurred rain and strong winds during this period. The fundamental frequency of the structure was obtained from the time series of the signals acquisition by Fourier Transform (FFT) in the program AqDAnalysis 7.02 (2004). The obtained result was 0,53 Hz. In Fig. 8 one finds the analysis of the signal in the frequency domain. Figure 7: Temporal acceleration series. Figure 8: Signal in the frequency domain 5. CONCLUSIONS The experimental field investigation and the numerical computer modelling carried out with the Finite Element Method, performed to know the fundamental frequency of a telecommunication pole, allows to conclude the following: 1 the experimentally obtained frequency of the structure is 0.53 Hz below of 1 Hz, therefore, band where the modelling indicates the resonant action of the wind; 2 the numerical modelling using the matrix of geometrical rigidity and the discretization used to represent the structure were efficient to capture the geometric nonlinear effect of the system; 3 Applying the Brazilian Code to compute the resulting maximum bending moment (Fig.9), the response of the so-called discrete model NL is 55.69 % larger the response of the static model. We observed little influence of higher modes than the fundamental one. These are responsible for 66% of total dynamic response of the structure. Moment (kNm) 800 Static Model NL Dynamic Discret NL Dynamic Discret NL Dynamic Discret NL Dynamic Discret NL Dynamic Discret 600 Model Model Model Model Model - Mode 1 - Modes 1 and 2 - Modes 1 to 3 - Modes 1 to 4 - Modes 1 to 5 400 200 0 0 5 10 15 20 25 30 Height (m) Figure 9: wind action. REFERENCES Blessmann J. (2001). “Acidentes causados pelo vento”. (Accidents Caused by Wind), 4 ed, Ed by Federal University of Rio Grande do Sul. Porto Alegre.(in Portuguese). Simiu E., Scanlan R. (1996). “Wind effects on Structures”. John Wiley & Sons, New York. Sachs P. (1972 ).“Wind Forces in Engineering”, Pergamon Press. New York. Koulousek V., Fischer O., Naprstek J. (1984).“Wind Effects on Civil Engineering Structures”, Elsevier. New York. Navara A. (1969). “Dinâmica de Estruturas”, (Dynamic of Structures), LNEC – Laboratório Nacional de Engenharia Civil (National Laboratory of Civil Engineering). Lisboa.(in Portuguese). Brasil R.M.L.R.F. 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