DYNAMIC ANALYSIS OF WIND EFFECTS BY USING AN ARTIFICAL WIND FUNCTION

Transcription

1 2008/3 PAGES RECEIVED ACCEPTED J. GYÖRGYI, G. SZABÓ DYNAMIC ANALYSIS OF WIND EFFECTS BY USING AN ARTIFICAL WIND FUNCTION J. Györgyi, Prof Budapest University of Technology and Economics, Dept. of Structural mechanics, Arany János 96/b, H-1221 Budapest G. Szabó Civil engineer, PhD student Budapest University of Technology and Economics, Dept. of Structural mechanics, Arany János 96/b, H-1221 Budapest Research field: Dynamics of structures, moving forces of on structures, wind effect ABSTRACT KEY WORDS During the application of standards for calculating wind effects, we used the quasistatical analysis calculated with the dynamic factor, size factor and force coefficient. From the Eurocode we can get these factors only in the case of special structures. If we know the power spectral density function (calculated from the nondimensional power spectral density function given by the Eurocode), we can calculate the artificial wind velocity function and determine the dynamic calculation of the effect of dynamic and size factors. During the calculation of vortex shedding we calculated by the correlation length given in the Eurocode and used the modal analysis to calculate the displacement and initial forces from vortex shedding. Using this method we have got fewer values than in the case of the approximate statical calculation. In the standard we can find for the force coefficient, Strouhal numbers in the case of special cross section parameters only. Using the Fluent program system we can get the result for these parameters without standards and wind channel experiments. wind effect artificial wind function fluent program system 1 INTRODUCTION In engineering practice during the wind calculation of structures the standard (MSZ EN :2007) is very important. We have got the most important parameters for the calculation of different structures from the standard, but the parameters are only in the case of some kind of structure, and if we want to calculate special structures with a special form, we have to do wind channel experiments. The standard uses the quasi-statical calculations. It can be very conservative and gives larger displacements and internal forces. The first results of correct dynamic calculation are in Györgyi and Szabó (2007). The results of dynamic calculations depend upon the natural frequency of the structure. If we calculate the soil-structure interaction the value of the natural frequency decreases, and the wind effect changes. Calculation of soil flexibility, the natural frequencies of the system are decreasing, which has a large effect on the critical velocity in the case of vortex excitation. Therefore, the calculation of soil-structure interaction can be important for high buildings and chimneys, which we can see in Györgyi (2006). We analysed this problem in the case of a chimney on Fig CALCULATION OF WIND DIRECTION EFFECTS 2.1 Application of the MSZ EN :2007 In the MSZ EN :2007 the peak velocity pressure is. (1) 2008 SLOVAK UNIVERSITY OF TECHNOLOGY 21 Gyorgyi.indd :43:53

2 Fig. 1 The dates of the analysed chimney 22 Gyorgyi.indd :44:00

3 Here the v m (z) is the mean wind velocity:. (2) The value of the basic wind velocity in Hungary is 20 m/s, c r (z) is the roughness factor in different terrain categories, and c 0 (z) is the orography factor. The I v (z) is the turbulence intensity function: if z min = z z max, if z < z min. (3) The z 0 and z min have different values in the case of different terrain categories; from these we can calculate the k I turbulence factor. The σ v is the standard deviation of the turbulence. The wind force is calculated from the. (4) The z e is the reference height for external wind action, and A ref is the reference area. The c s size factor takes into account the reduction effect on the wind action due to the nonsimultaneity of the occurrence of peak wind pressures on the surface:. (5) Here B 2 is the background factor, which depends on the height and width of a structure and the L (z s ) turbulent length scale at the reference height. z s is the reference height for calculation of c s and c d factors. The c d dynamic factor can be calculated from the power spectral density function from the Eurocode, and δ is the logarithmic decrement of the damping. If we calculate the Re(v(z s )) Reynolds number at the given roughness of the structure, we can calculate the c f,0 coefficient. Multiplying it with the end-effect factor, we can get the c f force coefficient in the formula (4). 2.2 Dynamic calculation for wind direction excitation In the standards there are some quasi-statical methods for calculation of the displacement from wind excitation. But the wind velocity is changing; therefore, the Eurocode applies the dynamic and size factors and calculates with the turbulence intensity function. If we know the wind velocity functions overthe height of a structure, we can calculate the dynamic forces. From the dynamic calculations we can get the time-dependent displacements and internal forces. The EN uses the S L (z, n) non-dimensional power spectral density function. From it we can calculate the S v (z, n) one-sided variance spectrum in the form:. (8) The σ v standard derivation can be calculated by multiplying the terrain factor by the basic wind velocity. In (7) we saw the redaction effect of R h (η h ) and R b (η b ) aerodynamic admittance functions; therefore, the v(z,t) was multiplied by. If we know the S v (z, n), we can calculate at the z height of the structure the artificial wind function of the dynamic part of the wind function - Bucholdt, H. A. (1999) - in the form: (9) formula. Here k p is the peak factor and the R 2 is the resonance response factor: (6). (7) R h (η h ) and R b (η b ) are the aerodynamic admittance functions, which depend on the size of the structure and from the f L (z s, n 1,x ) non-dimension frequency (n 1,x is the first frequency of the structure at a wind direction). The S L (z, n 1,x ) is a non-dimensional In the formula ϕ i changes randomly between 0 and 2π. In Fig. 2 we can see three different functions on the top of the structure using the EN at terrain category III. Calculating by the time dependent dynamic force: (10). The peak wind s velocity:. (11) The turbulent part depends on the turbulence intensity function, which changes during the height. The c s size factor takes into account the reduction effect on the wind s action due to the 23 Gyorgyi.indd :44:03

4 Fig. 2 The artificial wind function at the top in the case of different random v [m/s] Fig. 3 The artificial wind function at a different level 24 Gyorgyi.indd :45:07

5 nonsimultaneity of the occurrence of the peak wind pressures on the surface. The velocity of the turbulent part during the height:. (12) Fig. 3 shows the wind velocities at a different height. During the calculation we have to allow for the changing of the turbulence during the height and the correlation between the velocities of the neighbouring height points velocity. If we know the wind function, we can calculate the wind force using the force coefficient from the standard. On one part of the structure it will be: (13) If we calculate the Re(v(z s )) Reynolds number at the given roughness of the structure, we can calculate the c f,0 coefficient. Multiplying it with the end-effect factor, we can get the c f force coefficient in formula (13). The matrix differential equation of the system - if we want to calculate by structural damping - Györgyi (1996) is:. (14) Here M is the mass and K is the stiffness matrix of the structure; x is the displacement vector of the system and q is the force vector. If we know the M normalised eigenvectors of the system in the V matrix: V T MV = E,, (15) we can calculate the displacement vector in x(t) = Vy(t) form. Multiplying the equation from the left by V T :. (16) Because of the orthogonality, we have to solve some one degree of freedom system:. (17) For the solution of this differential equation we can use the numerical integration methods from Bathe and Wilson (1976). On Fig. 4 we can see the displacement at the top and on Fig. 5 the bending moment at the bottom from the dynamic calculation in the function of the time, the results of statical calculation and from the Eurocode. We can see that the Eurocode gave larger values. Fig. 6 shows the bending moment at the bottom from the dynamic calculation in the function of the time in the case of a different random. In the maximum of the moment, are not too large Fig. 4 The displacement at the top from the dynamic calculation and from the Eurocode 25 Gyorgyi.indd :45:43

6 Fig. 5 The bending moment at the button from the dynamic calculation and from the Eurocode Fig. 6 The bending moment at the button in the case of a different random 26 Gyorgyi.indd :45:47

7 Fig. 7 The effect of reduction of the turbulent part s velocity during the height differences, but during the correct analysis we have to calculate more than one wind function. Fig. 7 shows the effect of the reduction of the turbulent part s velocity during the height. The reduced values are equivalent to the effect of the c s size factor on the structure in Fig. 1. of the structure the 31.2 m length force and calculates the critical velocity in the middle of the 31.2 m. The mean wind velocity at the centres of the correlation length is v m,l1 = 25,00 m/s In the EN the value of the c lat force coefficient changes between 0.2 and 0.7 depending on the Reynolds number, which is: 3 CALCULATION OF THE VORTEX EXITATION 3.1 Application of the MSZ EN :2007 If we know the natural frequency s cross direction (n i,y ), we can calculate the critical velocity:. (18) In the EN it is 0.18 and from (18), the critical velocity is m/s. In the case of vortex excitation, there is a look-in effect. In this case the frequency of the vortex shedding is the same. In the Eurocode the correlation length is 6b, where the b is the diameter of the cylinder. Under the above correlation length there is excitation too, but here the frequency of the vortex shedding is different from the natural frequency of the system; therefore, during this part of the chimney, there is no resonance. The Eurocode puts from the top In our case the c lat force coefficient is 0.2. The logarithmic decrement of the structure in the EN is The Eurocode gives the procedure for calculating the displacement of the top: (19) Here K w is the correlation length factor, K the mode shape factor and Sc is the Scruton number from the standard. Using the φ 1 (z) vibration form, the accelerations are:. (20) If we know the accelerations, we can calculate the inertial forces and from them the internal forces of the structure. 27 Gyorgyi.indd :45:53

8 Tab. 1 Maximum of the displacement, moment and shear force at the dynamic calculation of the vortex shedding H 40 H 60 H 80 H 100 H 85 EN 2007 Displacement at top [cm] 9,18 17,39 26,91 15,41 26,48 29,70 Moment at bottom [knm] Shear force at bottom [kn] Dynamic calculation of vortex excitation If we want to get the dynamic calculation, we can calculate the force during all the chimneys, but the frequency will change. In Tab. 1 in the last column there is the displacement, moment and shear force from the quasi-statical calculation using the EN procedure, when there is a statical distributed force from the top to the correlation length. In the other column there are the results of the dynamic calculation at different positions of the correlation length. We can see that the larger values are from the application of the standard. In Fig. 8 we can see the moments at the button from different positions of critical velocity, the dynamic calculation, the correlation length excitation from theeurocode and the statical calculation by the Eurocode. We can see that the statical calculation by the Eurocode gives the largest values. 4 APPLICATION OF THE FLUENT PROGRAM SYSTEM 4.1 Calculation model In the case of simple shapes we can obtain the needed force coefficients and the Strouhal numbers from the standards or the literature. If we have a complex shaped cross section, we can apply the computational fluid dynamic (CFD) to calculate the flow pattern around the structure and the aerodynamic forces. There is one application of this software in the work of Lajos, et al. (2006). For the calculations we used Fluent commercial CFD software too. This code solves the Navier-Stokes differential-equation system (21) numerically. Fig. 8 The moments at the button from different positions of critical velocity 28 Gyorgyi.indd :46:12

9 2008/3 PAGES Fig. 9 Computational grid,, an unsteady solution to model the vortex shedding phenomena. The unsteady solution is solved using a second order difference scheme over time. We considered two boundary conditions with 1m/s and 10m/s velocity inlet boundary conditions. For these two velocities the time stepping was and s respectively. The turbulence was modelled with the k-ε model., 4.2 Results. (21) Here v is the kinematic viscosity, ρ is the air density, v is the wind speed, p is the pressure and g is the gravity. The first step during the modelling is to determine the contour of the structure and the computational domain. Secondly, the flow field must be divided into a certain number of cells. The two-dimensional mesh applied can be seen in Fig. 9. If the purpose is to determine the force coefficient in the wind s direction, a steady state solution can be enough, but we must apply The unsteady solutions provide the velocity field around the cylinder. The contour of the velocity field can be seen on Fig. 10. The velocity vector plot around the wall is shown on Fig. 11. The Fluent software shows us the forces acting on the body immersed in the fluid flow. From the force s time histories, the force coefficients can be visualized (Figs. 12 and Fig. 13). As we can see, the lateral coefficients (cy) oscillate around zero value, giving a periodical force to the structure. The force coefficient in the wind direction (cx) oscillates too, but around a well-determined value with very small amplitudes. From the Gyorgyi.indd :46:36

10 2008/3 PAGES Fig. 10 The contour of the velocity field Fig. 11 Velocity vector plot Fig. 12 Force coefficients at 1m/s Fig. 13 Force coefficients at 10 m/s forces, the coefficients and the Strouhal number were calculated using the following expression:, (22) In the above expression ρ is the air density, v is the wind speed, D is the chimney diameter and nx the frequency of the vortex shedding. As the calculated case is simple, we have the possibility of checking our results as shown on the following figures. On Fig. 14 from Zuranski (1986) and on Fig. 15 from the Eurocode, the calculated force coefficients are compared with the measured results (the black dots symbolize the calculated results at the wind velocities of 1 and 10 m/s). On Fig 16 from Zuranski (1986), the calculated Strouhal numbers are shown. Fig. 14 Force coefficient in the wind direction 30 Gyorgyi.indd :46:46

11 2008/3 PAGES Fig. 15 Force coefficient laterally Fig. 17 Computational grids of the considered sections Fig. 16 Strouhal number as a function of the force coefficient 4.3 Results of the other type of cross section Besides the circle we tried other types of sections. We considered a rectangular section with two different side ratios: d/b=1 and d/ b=2.5. On Fig. 17 the computational grids can be seen. On Fig. 18 we showed the velocity field of the solution. During the solution we recorded the forces acting on the body. On Fig. 19 we compared the calculated force coefficients in the wind direction with the standard values. As in the case of the circle the frequencies of the force fluctuations can be calculated, and the Strouhal numbers can be obtained. On Fig. 20 we compared the Strouhal numbers with the standard values. The inlet velocity was 10 m/s, we used t= sec for a time step. We applied a k-ε model for modelling turbulence. 5 CONCLUSIONS The calculations with the artificial wind function in the case of a structure, which does have not the formulas in the standard, is a real solution method. If we do not have information about the force coefficients or the Strouhal number, we can use CFD software for simulations. We applied the Fluent commercial code for our analysis. A simple circular shape was considered. We obtained the needed coefficients and Strouhal number from the calculations. Good agreements were Gyorgyi.indd :46:58

12 Fig. 18 The contour plot of the velocity field found with the values from the measurements and standards. We tried other types of cross sections. We investigated rectangular Fig. 20 Comparison of the force coefficients in the wind direction sections with different side ratios. We got acceptable differences in the case of the force coefficients in the wind direction. In case of the d/b=1, ratio the Strouhal number matches with the standard value, but in the other case we overestimated it. Fig. 19 Comparison of the Strouhal numbers ACKNOWLEDGEMENT The authors are grateful for the support of the Department of Fluid Mechanics (University of Technology and Economics, Budapest) 32 Gyorgyi.indd :47:04

13 REFERENCES Bathe, K.J. - Wilson, E.L. (1976) Numerical methods in finite element analysis. Englewood Cliffs, New Jersey: Prentice-Hall, Inc.,528 pp, ISBN Bucholdt, H. A. (1999) An introduction to cable roof structures. Second edition, Thomas Telford Publications Ltd., London Györgyi, J. (1996) Application of direct integration in the case of external and internal damping. Proceedings of the Estonian Academy of Sciences Engineering, 2, 2, 1996, pp Györgyi, J. (2006) Effect of soil-structure interaction in case of earthquake and wind calculation of towers. In F. Darne, I. Doghri, R. El Fatmi, H. Hassis & H. Zenzri eds., Advances in Geomaterials and Structures, pp , The First Euromediterranean Symposium on Advances in Geomaterials and Structures, Hammamet, Tunisia Györgyi, J.- Szabó, G. (2007) Calculation of wind effect by dynamic analysis using the artificial wind function Dynamic train-bridge interaction at arc bridge, In: Jendželovský, N. (ed.) Proceedings of 6 th International Conference on New Trends in Statics and Dynamics of Buildings. 6 th International Conference on New Trends in Statics and Dynamics of Buildings, Bratislava, Slovakia, October 18-19, 2007 Bratislava: Slovak University of Technology in Bratislava, pp. 9-12, CD Author index 47, pp.1-14, ISBN Lajos, T., Balczó, M., Goricsán, I., Kovács, T., Régent, P., Sebestyén, P. (2006) Prediction of wind load acting on telecommunication masts, paper, No. A-0206, pp.1-8, IABSE Symposium on Responding to Tomorrow s Challenges in Structural Engineering, Budapest MSZ EN :2007, Eurocode 1: A tartószerkezetet érő hatások. 1-4, rész: Általános hatások. Szélhatás. Magyar Szabványügyi Testület, Budapest Zuranski, J. A. (1986) A szél hatása az építményekre. Műszaki Könyvkiadó, Budapest 33 Gyorgyi.indd :47:07