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Vortex induced vibration of reinforced concrete chimneys: in situ experimentation and numerical previsions
Journal of Wind Engineering and Industrial Aerodynamics 74—76 (1998) 765—776
Vortex induced vibration of reinforced concrete chimneys: in situ experimentation and numerical previsions P. D’Asdia, S. Noe`* Department of Civil Engineering, University of Trieste, Piazzale Europa 1, I-34127 Trieste, Italy
Abstract The paper presents the main aspects of a numerical model aimed at evaluating the response of cylindrical structures subjected to vortex shedding excitation. The model can asses displacement amplitude and stresses for wind velocities lower than/equal to/greater than the critical wind speed. In the lock-in range it can simulate the displacement limitation due both to damping or to aeroelastic force equilibrium. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Vortex shedding; Chimney; Experimental response; Numerical model
1. Introduction Among aeroelastic phenomena a typical one is the vibration of slender cylindrical structures excited by detachment of vortices. The critical fluid velocity corresponds, through the well known Strouhal relation, to the coincidence of the vortex shedding frequency with a natural structural one. Due to the wake—oscillator interaction, in a narrow range of velocities, close to the critical value, the vortex shedding, abandoning the Strouhal law, remains tuned on a structural natural frequency. The phenomenon, called lock-in, causes a highly nonlinear resonant response of the structure. The response is hysteretic and nonsymmetrical with respect to the structural frequency. Research on this phenomenon has been carried out both experimentally and theoretically. The first experiments in this field are ascribed to Strouhal [1] and until recently several authors have been performing wind tunnel tests [2—4]. Von Karman was the author of the first theoretical studies concerning the vortex wake caused by a still cylindrical body [5]. Later several researchers have tackled the problem either of the numerical or theoretical description of the interaction between the vortex
* Corresponding author. E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 6 9 - 5
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excitation and an elastically mounted cylindrical body. Theoretical models developed so far can be divided into two main classes [4]: single-degree-of-freedom (SDOF) models and coupled wake-oscillator models. Models of the former group are more widely used for design purposes and often allow only an assessment of the oscillation peak amplitude, while models of the latter group offer a tentative description of the interaction between vortex shedding and oscillating body. In turn SDOF models can be subdivided into models based on the negative damping concept [6,7] and models based on force-coefficient data [8,9]. This paper is focused on the main aspects of a new numerical model of the phenomenon, on its practical application to a reinforced concrete chimney, as well as on the comparison between theoretical previsions and experimental data collected in situ. Due to its characteristics the numerical model can be placed in an intermediate position between SDOF models and coupled wake-oscillator models. The wake—oscillator interaction is interpreted as a resonance phenomenon of a damped oscillator under the action of an alternating force in the cross wind direction. The frequency of the force is ruled by the oscillator motion and by Strouhal shedding frequency.
2. The numerical model The model is founded on the hypothesis that the response of an elastically suspended cylinder caused by the vortex shedding can be treated as a resonance phenomenon. A detailed description of the model is presented in Refs. [10,11]. Here the fundamental lines of its formulation are briefly reported. The model operates in the time domain. Its formulation schematically reproduces the load effect on the structure due to the variation in the surface pressure fields induced by the detachment of the vortices in the form of an alternating force per unit
Fig. 1. Cylinder cross section.
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length, with changing direction, frequency and phase as a function of the relative wind velocity. The direction of the force is always perpendicular to the relative velocity. The lock-in phenomenon is reproduced by making the frequency of the force coincide with the oscillation frequency of the structure in a pre-fixed range of the frequency ratio. Outside such range the frequency of the pulsating force is calculated, following Strouhal’s relation, as a function of the relative velocity. As the frequency varies, the continuity of the exciting force is ensured by a suitable modification of its phase. The section of a unit length of the cylinder is represented in Fig. 1. The action of vortex shedding at time t is represented by the force F with 4)%(t) direction perpendicular to the relative velocity of time t!*t : V . 8 3%-(t~*t) The vector F is computed at time t as a function of the values at time t!*t 4)%(t) according to the following expressions: F "1oDC »2 sin(u t#u ), 4)%(t) 2 4)%(t) 4)%(t) L w 3%-
A
B
yR a "arctan (t~*t) (t) » w(t~*t)
(yR '0Na'0),
» u "2p St w 3%-(t) , S5(t) D u " (t) q
p , !q (y/0)n~1 (y/0)n
» "J»2 #yR 2 , w 3%-(t) w(t~*t) (t~*t) u "u if 4)%(t) S5(t)
u u S5(t))X or S5(t)*X , L U uJ uJ (t) (t)
u u "u if X ) S5(t))X , U 4)%(t) -0#,(t) L uJ (t) u "[u !u ]t#u , 4)%(t) 4)%(t~*t) 4)%(t) 4)%(t~*t)
(1) (2)
(3)
(4) (5) (6a)
(6b) (7)
where o is the density of air, D the cylinder outer diameter, C and C are the lift and L D the drag coefficients, respectively, St is the Strouhal number, the ratios X and X are L U the lower and upper limits of the pre-defined lock-in range and q ;q are the (y/0)n (y/0)n~1 instants of the last two zero cross wind displacements y. Four different formulations can be adopted depending upon the four options A to D in Table 1, where u is one (usually the first) natural structural circular frequency. n All the numerical results presented below have been obtained according to option A. Different choices do not influence the lock-in response while the behaviour for wind velocities close to the lock-in range may be different. From this point of view there is of course an interaction between the values fixed for X and X and the formulation adopted. L U
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Table 1 Possible options for parameters and uJ and u (t) -0#,(t)
uJ (t) u -0#,(t)
A
B
C
D
u (t) u (t)
u n u (t)
u (t) u n
u n u n
Fig. 2. Experimental (left) versus numerical (right) response for »/f D"4.294, m"0.15%. n
Fig. 3. Experimental (left) versus numerical (right) response for »/f "5.046, m"0.15%. nD
Fig. 4. Experimental (left) versus numerical (right) response for »/f D"5.904, m"0.15%. n
As to the drag forces, the components of the vector F are computed at time t as: d(t) F "1oDC »2 , D w(t~*t) $x(t) 2
(8)
A B
yR F "!1oDC yR 2 $y(t) 2 D DyR D
. (t~*t)
(9)
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Fig. 5. Numerical undamped response for »/f D"5.046. n
The numerical model has been implemented within a finite element code performing dynamic analysis with Newmark time step integration. The numerical simulations have been first compared with wind tunnel tests [3,4]. A few of the results of such analyses are briefly reported in Figs. 2—4, in which recorded and numerical displacements are compared at respectively 85%, 100% and 117% of the critical wind speed (Re"4900—6800; Sc"15). The following series of values have been adopted for the analyses: St"0.20, C "0.42, C "1.20, X "0.95, L D L X "1.05. U Figs. 2—4 show a good agreement between the experimental and the simulated responses: the absolute value of the displacements could closely match through a minimal adjustment of the (measured) damping. A valuable feature of the model is the self-limitation capability in critical undamped conditions. The undamped response of the same numerical model used for the comparison with the wind tunnel tests with wind speed »"5.046f D (close to the n critical value) is reported in Fig. 5. The model automatically self-limits the maximum response to values close to 0.25D.
3. In situ experimentation on the chimney Here follow some preliminary results of a measurement campaign in situ on a reinforced concrete chimney. The chimney, which is part of the new municipal waste disposal incinerator in Trieste, is perfectly cylindrical (see Fig. 6). The stack is 100.0 m high with an external diameter of 6.3 m. The experimental campaign is expected to last for approximately 12 months. The final instrumentation plan is shown in Fig. 7. Up to now the biaxial accelerometers (positions 2 and 4 in Fig. 7), the anemometer and the wind-vane (position 1) have been installed and a number of time series of the dynamic response of the structure under wind action and of the wind velocity and direction have been obtained. The dynamic behaviour has been measured for different wind velocities in a range next to the critical value. Examples of the experimental data obtained are the first two diagrams of Figs. 8 and 9, which report two wind speed recordings over 3000 seconds with the corresponding registered structural response in terms of cross wind
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Fig. 6. The chimney.
acceleration of the chimney top. The analysis of the available data allows the evaluation of the first natural frequencies. The spectra of the acceleration recordings are shown in Fig. 10. The peaks relevant to the first two natural frequencies are very close to one another and correspond to two different cantilever modes of vibration (the chimney shows a symmetry plane due to the presence of openings at the basis and of three internal flues).
4. Comparison between the experimental data and the numerical simulation The experimental response of the chimney has been compared with the numerical previsions. The recorded wind speed time histories have been used for the simulations. At first a simple model formed by two nodes and a beam element has been used, by imposing both the aeroelastic and the aerodynamic force relative to the upper half of the chimney to the top node. Both forces have been computed according to the rules and expressions described above as functions of the relative wind velocity and of the instantaneous value of the oscillation frequency computed at the top. For other analyses a 21 nodes discretization has been used on the whole height of the chimney. The increase in the number of degrees of freedom does not represent a problem for the wind load model. The evaluation of the forces is carried out for each node separately as a function of both the relative wind velocity and of the instantaneous oscillation frequency of the same node and of the corresponding stack length. Since the model oscillates according to the first mode only, in practice the instantaneous circular frequency turns out to be the same for all nodes. However,
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Fig. 7. Instrumentation plan.
from a theoretical point of view, there are no obstacles preventing the apparent instantaneous circular frequency to be different from node to node due to the superimposition of different modes. In the case of the 21-node model a logarithmic wind speed distribution along the whole height of the chimney has been assumed with speed value at the top of the chimney equal to that recorded experimentally. Therefore perfectly correlated wind histories at the various heights have been used. Since simultaneous wind speed recordings on several points of the chimney trunk were not available, this was the only possible way for proceeding. The increase in the dynamic response induced by the perfect correlation has been taken into account by slightly increasing the value of the structural damping ratio (fixed at n "2.5%). All comparisons have been performed by using the values St"0.20, C "0.42, C "1.20, L D X "0.95, X "1.30. L U The lower parts of Figs. 8 and 9 present the simulated (21 nodes model) cross wind acceleration histories. The two wind histories have speed values between 1.7 and 17.7 m/s. The critical value of the wind speed for the chimney is: 11.6 m/s (St "0.20; Re"4.9]106; Sc"89 for m"2.5%).
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Fig. 8. Recording n. 86. Wind time history (top), experimental (middle) and simulated (bottom) response for the chimney.
It is noteworthy that the results, both experimental and computed ones, are qualitatively similar to one another and that there is good agreement between the experimental maximum oscillation amplitudes and the amplitudes computed with the model (about 0.03 m). As expected, with the 2-node model the maximum acceleration amplitude (not reported) is over-estimated of a factor of about 2.5.
5. Other numerical results Analyses conducted with the 21-node chimney model, by maintaining the wind speed constant along the height with different values of the damping ratio highlight
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Fig. 9. Recording n. 93. Wind time history (top), experimental (middle) and simulated (bottom) response for the chimney.
some additional features of the vortex shedding numerical model. The self-limiting capability of the undamped critical response, already shown for the 2-node simulation of the wind tunnel tests, is illustrated by Fig. 11 in the case of the chimney. The model self-limits the maximum displacement amplitude to 3.4 m (0.54D). Moreover Fig. 12 shows the asymmetry of the numerical response with respect to the critical wind speed (» /f D"5), which is similar to the experimental evidence. The graph reports the 8 n maximum steady-state crosswind top displacement vs. the reduced wind speed for different values of the structural damping ratio. The maximum amplitude is always reached for wind speeds higher than the critical one. Some comparisons have been performed between theoretical previsions from Eurocode 1 [12] and numerical results. According to EC1, with a correlation length of 6D
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Fig. 10. Recordings n. 86 (left) and n. 93 (right). Experimental acceleration power spectrum.
Fig. 11. Undamped numerical response of the chimney.
(about 38% of the full length) and a logarithmic decrement d"0.03 (i.e. m"0.5%), the maximum estimated top displacement is about 0.2 m. The value is close to the numerical prevision for m"0.7%, adopting a logarithmic wind profile with critical wind speed at a 3D distance from the top of the chimney. It is worth noting that in this case all nodes lower than 6D from the top are outside the pre-defined lock-in range X !X . The assessments are in substantial agreement, even if EC1 adopts a more L U cautious value of the damping ratio.
6. Conclusions The proposed model can simulate adequately both the lock-in response and the structural behaviour for wind velocities higher and lower than the lock-in range. In lock-in conditions the model is able to assess correctly the maximum displacement amplitude both when the phenomenon is ruled by structural damping
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Fig. 12. Numerical response: maximum amplitude.
(typical displacement values under some percent of the diameter) and when it is governed by the aeroelastic force equilibrium (with maximum displacements over 10—20% of the diameter). It is worth of note that contributions to such limitation come from the varying direction of the cross wind alternating force and from the increment of the drag force. Four different similar formulations of the same model have been presented. The comparison with different kinds of experimental data performed with the first of them are satisfactory. The research will be continued including tests of the other possible options. At the same time a wind tunnel experimental program is in progress to provide the necessary experimental verification.
Acknowledgements The authors are indebted to the Trieste Town Council for making the experimental campaign possible. This work is partially supported by a MURST (Italian Ministry of University and Scientific Research) grant.
References [1] V. Strouhal, U®ber eine besondere art der Tonerregung, Annalen der Physik, Band V (1878) 216—250. [2] C.C. Feng, The measurement of vortex-induced effects in a flow past stationary and oscillating circular and D section cylinders, Master’s Thesis, University of British Columbia, Vancouver, British Columbia, Canada, 1968. [3] I. Goswami, R.H. Scanlan, N.P. Jones, Vortex-induced vibrations of circular cylinders I: experimental data, J. Eng. Mech. ASCE 119 (11) (1991) 2270—2287.
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[4] I. Goswami, Vortex-induced vibrations of circular cylinders, Ph.D. dissertation, The John Hopkins University, Baltimore, MD, 1991. [5] T. von Karman, U®ber den Mechanismus des Widerstands den ein bewester Ko¨rper in einer Flu¨ssigkeit erfa¨hrt, Go¨ttingen Nachtrichten Math. Phis. K. (1911) 509—517. [6] R.I. Basu, B.J. Vickery, Across-wind vibrations of structures of circular cross-section. Part II. Development of a mathematical model for full-scale applications, J. Wind Eng. Ind. Aerodyn. 12 (1983) 75—97. [7] E. Simiu, R.H. Scanlan, Wind effects on structures, Wiley, New York, 1986. [8] T. Sarpkaya, Fluid forces on oscillating cylinders, J. Wtrwy. Port, Coast. Ocean Div. 104 (1) (1978) 19—24. [9] T. Staubli, Calculation of the vibration of an elastically mounted cylinder using experimental data from a forced oscillation, J. Fluids Eng. 105 (1983) 225—229. [10] P. D’Asdia, A. Gruden, S. Noe`, Distacco dei vortici su ciminiere: confronto tra modelli numerici, Proc. IV Italian National Conf. on Wind Engineering — IN-VENTO-96 — Trieste, 1996 (in Italian). [11] P. D’Asdia, S. Noe`, Un modello numerico per la valutazione del carico da vento su strutture snelle dovuto al distacco dei vortici (in Italian), in press. [12] Commission of the European Communities, Eurocode No.1: Basis of design and action on structures, Part 2—4: Actions on structures — Wind Actions, 1997.
Vortex induced vibration of reinforced concrete chimneys: in situ experimentation and numerical previsions P. D’Asdia, S. Noe`* Department of Civil Engineering, University of Trieste, Piazzale Europa 1, I-34127 Trieste, Italy
Abstract The paper presents the main aspects of a numerical model aimed at evaluating the response of cylindrical structures subjected to vortex shedding excitation. The model can asses displacement amplitude and stresses for wind velocities lower than/equal to/greater than the critical wind speed. In the lock-in range it can simulate the displacement limitation due both to damping or to aeroelastic force equilibrium. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Vortex shedding; Chimney; Experimental response; Numerical model
1. Introduction Among aeroelastic phenomena a typical one is the vibration of slender cylindrical structures excited by detachment of vortices. The critical fluid velocity corresponds, through the well known Strouhal relation, to the coincidence of the vortex shedding frequency with a natural structural one. Due to the wake—oscillator interaction, in a narrow range of velocities, close to the critical value, the vortex shedding, abandoning the Strouhal law, remains tuned on a structural natural frequency. The phenomenon, called lock-in, causes a highly nonlinear resonant response of the structure. The response is hysteretic and nonsymmetrical with respect to the structural frequency. Research on this phenomenon has been carried out both experimentally and theoretically. The first experiments in this field are ascribed to Strouhal [1] and until recently several authors have been performing wind tunnel tests [2—4]. Von Karman was the author of the first theoretical studies concerning the vortex wake caused by a still cylindrical body [5]. Later several researchers have tackled the problem either of the numerical or theoretical description of the interaction between the vortex
* Corresponding author. E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 6 9 - 5
766
P. D+Asdia, S. Noe` / J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 765–776
excitation and an elastically mounted cylindrical body. Theoretical models developed so far can be divided into two main classes [4]: single-degree-of-freedom (SDOF) models and coupled wake-oscillator models. Models of the former group are more widely used for design purposes and often allow only an assessment of the oscillation peak amplitude, while models of the latter group offer a tentative description of the interaction between vortex shedding and oscillating body. In turn SDOF models can be subdivided into models based on the negative damping concept [6,7] and models based on force-coefficient data [8,9]. This paper is focused on the main aspects of a new numerical model of the phenomenon, on its practical application to a reinforced concrete chimney, as well as on the comparison between theoretical previsions and experimental data collected in situ. Due to its characteristics the numerical model can be placed in an intermediate position between SDOF models and coupled wake-oscillator models. The wake—oscillator interaction is interpreted as a resonance phenomenon of a damped oscillator under the action of an alternating force in the cross wind direction. The frequency of the force is ruled by the oscillator motion and by Strouhal shedding frequency.
2. The numerical model The model is founded on the hypothesis that the response of an elastically suspended cylinder caused by the vortex shedding can be treated as a resonance phenomenon. A detailed description of the model is presented in Refs. [10,11]. Here the fundamental lines of its formulation are briefly reported. The model operates in the time domain. Its formulation schematically reproduces the load effect on the structure due to the variation in the surface pressure fields induced by the detachment of the vortices in the form of an alternating force per unit
Fig. 1. Cylinder cross section.
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length, with changing direction, frequency and phase as a function of the relative wind velocity. The direction of the force is always perpendicular to the relative velocity. The lock-in phenomenon is reproduced by making the frequency of the force coincide with the oscillation frequency of the structure in a pre-fixed range of the frequency ratio. Outside such range the frequency of the pulsating force is calculated, following Strouhal’s relation, as a function of the relative velocity. As the frequency varies, the continuity of the exciting force is ensured by a suitable modification of its phase. The section of a unit length of the cylinder is represented in Fig. 1. The action of vortex shedding at time t is represented by the force F with 4)%(t) direction perpendicular to the relative velocity of time t!*t : V . 8 3%-(t~*t) The vector F is computed at time t as a function of the values at time t!*t 4)%(t) according to the following expressions: F "1oDC »2 sin(u t#u ), 4)%(t) 2 4)%(t) 4)%(t) L w 3%-
A
B
yR a "arctan (t~*t) (t) » w(t~*t)
(yR '0Na'0),
» u "2p St w 3%-(t) , S5(t) D u " (t) q
p , !q (y/0)n~1 (y/0)n
» "J»2 #yR 2 , w 3%-(t) w(t~*t) (t~*t) u "u if 4)%(t) S5(t)
u u S5(t))X or S5(t)*X , L U uJ uJ (t) (t)
u u "u if X ) S5(t))X , U 4)%(t) -0#,(t) L uJ (t) u "[u !u ]t#u , 4)%(t) 4)%(t~*t) 4)%(t) 4)%(t~*t)
(1) (2)
(3)
(4) (5) (6a)
(6b) (7)
where o is the density of air, D the cylinder outer diameter, C and C are the lift and L D the drag coefficients, respectively, St is the Strouhal number, the ratios X and X are L U the lower and upper limits of the pre-defined lock-in range and q ;q are the (y/0)n (y/0)n~1 instants of the last two zero cross wind displacements y. Four different formulations can be adopted depending upon the four options A to D in Table 1, where u is one (usually the first) natural structural circular frequency. n All the numerical results presented below have been obtained according to option A. Different choices do not influence the lock-in response while the behaviour for wind velocities close to the lock-in range may be different. From this point of view there is of course an interaction between the values fixed for X and X and the formulation adopted. L U
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Table 1 Possible options for parameters and uJ and u (t) -0#,(t)
uJ (t) u -0#,(t)
A
B
C
D
u (t) u (t)
u n u (t)
u (t) u n
u n u n
Fig. 2. Experimental (left) versus numerical (right) response for »/f D"4.294, m"0.15%. n
Fig. 3. Experimental (left) versus numerical (right) response for »/f "5.046, m"0.15%. nD
Fig. 4. Experimental (left) versus numerical (right) response for »/f D"5.904, m"0.15%. n
As to the drag forces, the components of the vector F are computed at time t as: d(t) F "1oDC »2 , D w(t~*t) $x(t) 2
(8)
A B
yR F "!1oDC yR 2 $y(t) 2 D DyR D
. (t~*t)
(9)
P. D+Asdia, S. Noe` / J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 765–776
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Fig. 5. Numerical undamped response for »/f D"5.046. n
The numerical model has been implemented within a finite element code performing dynamic analysis with Newmark time step integration. The numerical simulations have been first compared with wind tunnel tests [3,4]. A few of the results of such analyses are briefly reported in Figs. 2—4, in which recorded and numerical displacements are compared at respectively 85%, 100% and 117% of the critical wind speed (Re"4900—6800; Sc"15). The following series of values have been adopted for the analyses: St"0.20, C "0.42, C "1.20, X "0.95, L D L X "1.05. U Figs. 2—4 show a good agreement between the experimental and the simulated responses: the absolute value of the displacements could closely match through a minimal adjustment of the (measured) damping. A valuable feature of the model is the self-limitation capability in critical undamped conditions. The undamped response of the same numerical model used for the comparison with the wind tunnel tests with wind speed »"5.046f D (close to the n critical value) is reported in Fig. 5. The model automatically self-limits the maximum response to values close to 0.25D.
3. In situ experimentation on the chimney Here follow some preliminary results of a measurement campaign in situ on a reinforced concrete chimney. The chimney, which is part of the new municipal waste disposal incinerator in Trieste, is perfectly cylindrical (see Fig. 6). The stack is 100.0 m high with an external diameter of 6.3 m. The experimental campaign is expected to last for approximately 12 months. The final instrumentation plan is shown in Fig. 7. Up to now the biaxial accelerometers (positions 2 and 4 in Fig. 7), the anemometer and the wind-vane (position 1) have been installed and a number of time series of the dynamic response of the structure under wind action and of the wind velocity and direction have been obtained. The dynamic behaviour has been measured for different wind velocities in a range next to the critical value. Examples of the experimental data obtained are the first two diagrams of Figs. 8 and 9, which report two wind speed recordings over 3000 seconds with the corresponding registered structural response in terms of cross wind
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Fig. 6. The chimney.
acceleration of the chimney top. The analysis of the available data allows the evaluation of the first natural frequencies. The spectra of the acceleration recordings are shown in Fig. 10. The peaks relevant to the first two natural frequencies are very close to one another and correspond to two different cantilever modes of vibration (the chimney shows a symmetry plane due to the presence of openings at the basis and of three internal flues).
4. Comparison between the experimental data and the numerical simulation The experimental response of the chimney has been compared with the numerical previsions. The recorded wind speed time histories have been used for the simulations. At first a simple model formed by two nodes and a beam element has been used, by imposing both the aeroelastic and the aerodynamic force relative to the upper half of the chimney to the top node. Both forces have been computed according to the rules and expressions described above as functions of the relative wind velocity and of the instantaneous value of the oscillation frequency computed at the top. For other analyses a 21 nodes discretization has been used on the whole height of the chimney. The increase in the number of degrees of freedom does not represent a problem for the wind load model. The evaluation of the forces is carried out for each node separately as a function of both the relative wind velocity and of the instantaneous oscillation frequency of the same node and of the corresponding stack length. Since the model oscillates according to the first mode only, in practice the instantaneous circular frequency turns out to be the same for all nodes. However,
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Fig. 7. Instrumentation plan.
from a theoretical point of view, there are no obstacles preventing the apparent instantaneous circular frequency to be different from node to node due to the superimposition of different modes. In the case of the 21-node model a logarithmic wind speed distribution along the whole height of the chimney has been assumed with speed value at the top of the chimney equal to that recorded experimentally. Therefore perfectly correlated wind histories at the various heights have been used. Since simultaneous wind speed recordings on several points of the chimney trunk were not available, this was the only possible way for proceeding. The increase in the dynamic response induced by the perfect correlation has been taken into account by slightly increasing the value of the structural damping ratio (fixed at n "2.5%). All comparisons have been performed by using the values St"0.20, C "0.42, C "1.20, L D X "0.95, X "1.30. L U The lower parts of Figs. 8 and 9 present the simulated (21 nodes model) cross wind acceleration histories. The two wind histories have speed values between 1.7 and 17.7 m/s. The critical value of the wind speed for the chimney is: 11.6 m/s (St "0.20; Re"4.9]106; Sc"89 for m"2.5%).
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Fig. 8. Recording n. 86. Wind time history (top), experimental (middle) and simulated (bottom) response for the chimney.
It is noteworthy that the results, both experimental and computed ones, are qualitatively similar to one another and that there is good agreement between the experimental maximum oscillation amplitudes and the amplitudes computed with the model (about 0.03 m). As expected, with the 2-node model the maximum acceleration amplitude (not reported) is over-estimated of a factor of about 2.5.
5. Other numerical results Analyses conducted with the 21-node chimney model, by maintaining the wind speed constant along the height with different values of the damping ratio highlight
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Fig. 9. Recording n. 93. Wind time history (top), experimental (middle) and simulated (bottom) response for the chimney.
some additional features of the vortex shedding numerical model. The self-limiting capability of the undamped critical response, already shown for the 2-node simulation of the wind tunnel tests, is illustrated by Fig. 11 in the case of the chimney. The model self-limits the maximum displacement amplitude to 3.4 m (0.54D). Moreover Fig. 12 shows the asymmetry of the numerical response with respect to the critical wind speed (» /f D"5), which is similar to the experimental evidence. The graph reports the 8 n maximum steady-state crosswind top displacement vs. the reduced wind speed for different values of the structural damping ratio. The maximum amplitude is always reached for wind speeds higher than the critical one. Some comparisons have been performed between theoretical previsions from Eurocode 1 [12] and numerical results. According to EC1, with a correlation length of 6D
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Fig. 10. Recordings n. 86 (left) and n. 93 (right). Experimental acceleration power spectrum.
Fig. 11. Undamped numerical response of the chimney.
(about 38% of the full length) and a logarithmic decrement d"0.03 (i.e. m"0.5%), the maximum estimated top displacement is about 0.2 m. The value is close to the numerical prevision for m"0.7%, adopting a logarithmic wind profile with critical wind speed at a 3D distance from the top of the chimney. It is worth noting that in this case all nodes lower than 6D from the top are outside the pre-defined lock-in range X !X . The assessments are in substantial agreement, even if EC1 adopts a more L U cautious value of the damping ratio.
6. Conclusions The proposed model can simulate adequately both the lock-in response and the structural behaviour for wind velocities higher and lower than the lock-in range. In lock-in conditions the model is able to assess correctly the maximum displacement amplitude both when the phenomenon is ruled by structural damping
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Fig. 12. Numerical response: maximum amplitude.
(typical displacement values under some percent of the diameter) and when it is governed by the aeroelastic force equilibrium (with maximum displacements over 10—20% of the diameter). It is worth of note that contributions to such limitation come from the varying direction of the cross wind alternating force and from the increment of the drag force. Four different similar formulations of the same model have been presented. The comparison with different kinds of experimental data performed with the first of them are satisfactory. The research will be continued including tests of the other possible options. At the same time a wind tunnel experimental program is in progress to provide the necessary experimental verification.
Acknowledgements The authors are indebted to the Trieste Town Council for making the experimental campaign possible. This work is partially supported by a MURST (Italian Ministry of University and Scientific Research) grant.
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