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An analytical study of the across-wind response of cylinders due to vortex shedding
An analytical study of the across-wind response of cylinders due to vortex shedding T. K. Datta and A. K. Jain Civil Engineering Department, Indian Institute of Technology, Hauz Khas, New Delhi, India (Received October 1985; revised February 1986)
The across-wind response of tapering cylinder-like structures, such as chimneys, due to periodic shedding of vortices is obtained under lock-in conditions. The method of analysis incorporates aerodynamic effects and describes the lift forces by a linear model monoharmonic in nature. Under lock-in conditions, the critical wind speed and the lift and aerodynamic coefficients are assumed to be uniform throughout the height of the chimney, although its outside diameter varies. The aerodynamic and lift coefficients are derived from a lift oscillator model which adequately describes the experimental results of the vortex-induced oscillation (at lock-in) of an elasticallysupported cylinder. With the help of an example problem, the paper shows how the response of the chimney is sensitive to the choice of the critical diameter used for the determination of critical wind velocity and the lift and aerodynamic coefficients. It also emphasizes the importance of the lock-in condition in the second mode in relation to bending stresses. Key words: across-wind response, vortex shedding, lock-in, chimneys, bending stresses Vortex-induced oscillation of tall chimneys due to wind has long been recognized and has assumed importance in design as a check against resonant aerodynamically induced vibrations, or excessive stresses developed due to resonant vibration. Literature on the subject indicates that this oscillation could be periodic or random, depending upon the flow regime, smoothness of the surface of the structure and proximity of other tall structures. For a fairly smooth surface and uninterrupted wind flow, periodic shedding of vortices is expected in subcritical (103 < Re < 105) and super-critical (Re > 3.5 x 106) flOW conditions. For velocity of wind exceeding the critical flow condition (0.5 × 106< Ro < 3.5 z 10~'), the vortex-induced forces tend to become random and par tially correlated along the height of the structure. 1 The oscillation across the main wind direction can become quite significant if either the vortex shedding frequencies coincide with the structure's frequency or the natural frequencies of the structure fall within the dominant 0141-0296/87/01027-o5/$03.00 © 1987Butterworth & Co (Publishers)Ltd
frequency range of the cross-wind force spectrum (for random excitation). The magnitude of the cross-wind oscillation is proportional to the lift forces which primarily depend upon the Strouhal number S, lift coefficient CL, aerodynamic and mass damping parameters and the correlation length. The values of the above parameters depend upon the flow condition and the geometric properties of the structure, and they should be determined from tests on actual structures. Basu and Vickery2developed a mathematical model for full-scale application to slender structures and validated it by comparing the predicted response with the experimentally measured response of a prototype 330m high TV tower. The model was based on laboratory experiment results and was developed within the framework of random vibration theory incorporating the nonlinear aerodynamic damping force. Details of the model and the results of other prototype tests are given in references 2-5. For the purposes of this paper, which is merely illustrative, Eng. Struct., 1987, Vo[. 9, J a n u a r y
27
A c r o s s - w i n d response o f cylinders: 7-. K. Datta a n d A. K. Jain (Not t o scale) -~
ki--10.5 m
- ~ x21
21 ~'60~ 20 v
-t~ x 2 0
D E
{ )
0 cN
t
i+1 Q 0
-~ xi+ -I~ x [
/ / = 12.5
1
D = 12.5m t =0.2m
,
3 2
I
-t~ x 2 LD," X 1
!0 m
a
b
c
d
Figure 1 Details of chimney: (a) chimney; (b) section A - A ; (c) lumped mass idealization; (d) d y n a m i c degrees of f r e e d o m Table I Data for c h i m n e y Distance from base (m)
Outside diameter (m)
Thickness (m)
0 12.5 25 37.5 50 62.5 75 87.5 100 112.5 125 137.5 150 162.5 175 187.5 200 212.5 225 237.5 250
20 20 19.5 19 18.5 18 17.5 17 16.5 16 15.5 15 14.5 14 13.5 13 12.5 12 11.5 11 10.5
0.85 0.85 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.24 0.23 0.22 0.21 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Frequencies: 0)1 = 2.1617 rad s 1; 0)2 = 8.3379 rad s-l; 0)3 : 19.6637 rad s -~
the values used in the analysis are obtained from tests on model chimneys and elastically-supported cylinders. Most of the available laboratory data generally refer to forces on static cylinders for which lift force is found to be harmonic 6 with a value of CL = 0.6. For elastically supported cylinders, experimental data are fitted to both linear and nonlinear models. The nonlinear model resembles the Van der Pol type equation, while the linear model is expressed by a monoharmonic function. 7-9 The excitation lift force can be approximated by a monoharmonic function, particularly in regions of Reynolds number where vortex shedding is predominantly periodic or at the lock-in condition.~° From various model tests on circular cylinders and chimneys, it is shown that the Strouhal number for circular chimneys may be taken to be about 0.2. Also, tests on model chimneys reveal that the resonant wind velocity is
28
E n g . S t r u c t . , 1 9 8 7 , V o l . 9, J a n u a r y
influenced primarily by the upper portion of the chimney. u These two observations are used in most analyses to determine the critical wind velocity under lock-in conditions. While the Strouhal number is generally considered as 0.2 for chimneys in subcritical and supercritical flows, no consistent value of CL is reported in the literature. Wootton 12used a value of CL for the response evaluation of large circular stacks in wind which was dependant upon the tip displacement of the tower and a nondimensional aerodynamic damping coefficient. This value was originally proposed by Vickery. ~3 Vickery et al. 14 showed that for random vortex excitation, the rms value of CL ranges between 0.05 and 0.15. For safe dynamic design of reinforced concrete (RC) chimneys, Maugh and Rumman 15 used a conservative value of CL of 0.66 and ignored the aerodynamic effects. Kwok et al. J6 proposed a sinusoidal lock-in excitation model to predict the across wind response of a circular tower tested in boundary layer flow in a wind tunnel. They predicted a maximum value of CL ~ 0.08. The work described so far only considers lock-in vibration of the chimney in the first mode. As far as the bending stress is concerned, however, the second mode lock-in could be more severe. In the present study, the response of a tapered chimney is investigated under periodic shedding of vortices. The equation of motion for across wind vibration considers the aerodynamic effects. The aerodynamic and lift coefficients used in the analysis are obtained by fitting the response of a lift oscillator model to the experimentally observed oscillation of an elastically supported cylinder under lock-in conditions. Using the present method of analysis, an example problem is solved to investigate the influence of the important parameters on the response of the chimney and the importance of the lock-in condition in the second mode of vibration.
Theory The chimney is considered as an assemblage of 2D beam elements, each assumed to have constant diameter over
Across-wind response of cylinders: 7. K. Datta and A. K. Jain Table2
Maximum tip displacement and base moment
Assumed section
First mode lock-in
Second mode lock-in
hc (m)
Dc (m)
U (ms -1)
Tip displacement Base moment (m) (xl0SKNm)
U (ms -1)
Tip displacement Base moment (m) (xl05KNm)
232.5 200 175
11.5 12.5 13.5
19.439 21.503 23.223
0.286 0.382 0.463
74.976 82.939 89.574
0.106 0.174 0.123
3.925 4.606 5.318
the elemental length. The dynamic degrees of freedom are considered as the sway degrees of freedom only as shown in Figure l(d). The across-wind vibration due to the periodic shedding of vortices is described by the following equation of motion:
M X + C X + R X = P(t)
(1)
10.737 12.142 15.208
resonance curve. By fitting the response of the lift oscillator to this resonance curve, the following relationships are derived: 1 CL(K) = 4~ov-~l~o/pD 2 y2(K) = ~ ( K 2 - K~o)/pD 2 yI(K) = 2m(-qK - vKo)/p-D 2
where M is the diagonal lumped mass matrix; R is the condensed stiffness matrix corresponding to the sway degrees of freedom; C is the structural damping matrix which is not explicitly knpwn but.can be defined in terms of modal damping; X, X and X are, respectively, the displacement, velocity and acceleration in the structural coordinates; and P(t) is the time varying load vector given by:
P(t) = r y I ( K ) A X / U + ry2(K)AX/Dc + 0.5 rCe(K)A1 sin wt
(2)
r = 0.5 pUZ(2O~)
(3)
where D¢ is the diameter at a section in the upper part of the chimney used for the calculation of critical wind velocity; p is the mass density of air; A is a diagonal matrix whose ith element is given by 0.5(ai + ai_l)l, a~ being the ratio of D~/D¢ and l is the elemental length; I is a vector of unity; yl(K), y2(K) and Q ( K ) are the aerodynamic and lift coefficients, respectively, which depend on the nondimensional frequency K = D¢w/U; w is the lock-in circular frequency; U is the critical wind speed given by U = wD~/2crS; and S is the Strouhal number. The force vector P(t), as given by equation (2) is obtained with the assumptions that: lock-in has occurred with one of the structure's natural frequencies, to • at lock-in, there is a perfect correlation of lift and aerodynamic forces over the entire length of the chimney • aerodynamic coefficients Yl(K) and y2(K), and the lift coefficient CL(K), are constant over the entire length of the chimney • the magnitude of aerodynamic and lift forces at any point on the chimney depends on the critical wind velocity U (and not on the local mean wind velocity at that point)
(4) (5) (6)
In equations (4)-(5), D is the diameter of the vibrating cylinder; ~ is its mass; v is an equivalent damping ratio of the lift oscillator; "0 is the structural damping ratio; if0is the maximum nondimensional amplitude of acrosswind vibration recorded in the experiment; K0 is the nondimensional frequency of the lift oscillator. It is shown in reference 1 that, for a good fit, K0 = 1.05 and v = 0.052. Values of other parameters required to obtain CL, yl(g) and yz(K) from equations (4)(6) are taken from reference 17. It should be pointed out that these parameter values were obtained for a specific set of conditions in uniform flow and for pure translational motion of the elastically sprung body. The values of these parameters are, therefore, not universal. They are used in this work for illustrative purposes. For lock-in vibration in a particular mode characterized by the natural frequency w and mode shape function ~b, the equations of motion, (1) and (2), may be written in modal coordinate form as:
m(2 + 2rlo~±+ toZz) = r(alyl(K)~./U + aty2(K)z/Dc + 0.5a2CL(K) sin oJt)
(7)
where m = ~brMcb; al = ~brA~b; a2 = ~brAl; z, and 2 are the displacement, velocity and acceleration, respectively, in modal coordinates. Rearranging equation (7), the equation of motion in modal coordinates takes the form:
•
The aerodynamic coefficients yI(K) and y2(g), and the lift coefficient CL(K), are obtained from an empirical linear oscillator model 1 which adequately describes the response of an elastically-supported cylinder at lock-in as observed in experimental tests. 17 The experimental results which show the across-wind displacement of the cylinder against nondimensional frequency resemble a
+ (2rico - raxyl(K)/mU)2
+ (oo2 - raay2(K)/mDc)z = ra2CL/m sin wt
(8)
The solution of equation (8) gives the value of z used to obtain the displacement or moment response from the following relationships: X = ~z
(9) (10)
mb = q~bZ
where ~ is the mode shape coefficient for bending moment and Mb is the moment vector. It is interesting to study the role of aerodynamic effects in modifying equation of motion (8) which may be rewritten as:
+ 2rj-~i~ + ~oZz = ra2CL/mSinwt in which: = g~
-~ = ~/[g]" - ~
Eng. Struct., 1987, Vol. 9, J a n u a r y
(11)
(12)
29
Across-wind response of cylinders: T. K. Datta and A. K. Jain 1.5
-g 0.9
D c : 13.5 (case l)
Z
z 0.8
o
_
g %
0.7 D c = 13.5 m (case ])
× 0.6
v
0,75
o.5
/
12.5 m (case 1])
~ 0 . 4 ~
E
o
g
E .~ 0.3 -
£3
~ 0.2 O
0
0.1 0 0
1 25
I 50
I 75
I 100
I 125
I 150
I ~ 175 2 0 0
225
250
25
50
75 100 125 150 175 Distance from base (m)
200
225
250
Figure 3 D e s i g n m o m e n t s f o r lock-in s e c o n d m o d e : w 2 = 8.3379 rad s - l ; r / = 5%
Distance f r o m base (m)
Figure 2 D e s i g n
moments
for lock-in first mode:
~o~ = 2.1617
rad s- ~; r / = 5%
g
Z c~
%
g = 1 - raly2(K)/wZmD~
X
= razyl (K)/2rn[gpUoJ
(13)
Equation ( 11) describes a single degree of freedom system having a frequency ~ and damping ratio ~, and acted upon by a harmonic force at frequency w. The system damping ~ incorporates aerodynamic damping and the system frequency ~ depends on the value of g which in turn depends on the aerodynamic coefficient y2(K).
1 0.8
. Case 1 ~ C a s e
0.6 ~ J
~
2 ~ Case 3
->'<"~
0.4 ~ E o E o.,
"
Maximum tip displacement (m)
Ca2se Case 1
0.82
0.50
&
0 E3
0
Figure 4
I 25
I 50
I 75
I I I ~"-'-?~ 100 125 150 175 Distance from base (m)
200
I 225
250
C o m p a r i s o n o f d e s i g n m o m e n t s f r o m cases 1, 2 a n d
3
Results Using the method of analysis described, the response of a chimney of height H = 250m (Figure l(a)) is obtained. The details of the chimney are shown in Table 1. For the analysis, the chimney is divided into 20 beam elements, as shown in Figure 1(c). The first three natural frequencies of the structure obtained from this lumped mass idealization are given in Table 1. Since the critical wind velocity for lock-in conditions is influenced primarily by the upper part of the chimney, ~1 three different sections in the upper one-third portion of the chimney are selected for evaluation of the critical wind velocity. The diameters of the chimney at these sections and their heights above the base are shown in Table 2. The responses of the chimney are computed for different critical wind velocities as obtained with the help of the diameters of the chimney at the three sections. These responses are compared in order to investigate how the choice of critical diameter used in the determination of critical wind velocity influences the response of the chimney. For the computation of the critical wind velocity, the Strouhal number is assumed to be 0.2. With this value of the Strouhal number, the values of CL(K), yl(K) and y2(K), as obtained from equations (4)-(6) are 0.266, 0.044 and 1.165, respectively. It is interesting to note that the rms lift coefficient CL obtained from laboratory tests, 4 varies between 0.15 and 0.23. The average measured CL for a RC tower I~ was found to be 0.173. Basu and Vickery4 used rms values of CL of 0.15 and 0.23 for predicting the response of a TV tower in the first and third modes which compared favourably with the experimentally measured values. Thus, the present empirical model, though valid for periodic shedding of
30
Eng. Struct., 1987, Vol. 9, J a n u a r y
vortices, uses a somewhat higher value of CL. The value of y1(K) in the present model provides an aerodynamic damping which is about one-third of the structural damping. The nonlinear aerodynamic damping model proposed by Basu and Vickery4 can incorporate a wide range of aerodynamic damping depending upon the flow condition. According to their model, the amount of aerodynamic damping, as observed in the present study, is typical of the low amplitude case for a linear system. For lock-in in the first mode, the maximum bending moments (design moments) at different points along the height of the chimney are shown in Figure 2. It is seen that the design moments at different sections of the chimney are quite sensitive to the choice of De. There may be as much as 40% change in base moment if hc is changed from 0.93 to 0.70 H. Figure3 shows the design moments along the height of a chimney when there is lock-in in the second mode. Here again, it is observed that a small change in the value of Dc may lead to considerable change in the design moments, especially at the base of the chimney. Table 2 compares the responses between the lock-in conditions at modes 1 and 2. For the displacement response, the first mode lock-in is critical. The second mode lock-in, if it occurs, may lead to a base moment about three times that for the first mode lock-in. Thus, the possibility of lock-in in the second mode must be investigated for safe design of the chimney. In Figure 4, design moments from cases 1, 2 and 3 are compared. In case 1, the aerodynamic effects are neglected and responses are obtained with a value of CL ---- 0.6 which is generally used for the stationary cylinders. In case 2, only the quadrature component
Across-wind response of cylinders: 1-. K. Datta and A. K. Jain of the aerodynamic effect is considered; the Strouhal number is so adjusted that the aerodynamic coefficient y2(K), corresponding to the displacement component, becomes zero as described in the theory. For this case, the Strouhal number becomes 0.167 and the corresponding values of CL and yl(K) are 0.266 and -0.0103, respectively. In case 3, the full aerodynamic effect is considered. The comparison shows that the response in case 1 is significantly higher than the other two cases and any design based on this analysis would be too conservative. The response in case 2 is higher than that in case 3. This increase in the response is partly due to the increased value of the critical wind velocity (Strouhal number being less) and partly due to the resonating effect as explained in the theory.
y2(K) z ± p to ~'0 b'
4, Conclusions The vortex-induced oscillation of cylinder-like structures such as chimneys under lock-in conditions is investigated under a set of simplifying assumptions aimed at illustrating certain features of the response. Under periodic shedding of vortices, the response of the chimney is shown to be quite sensitive to the choice of the critical diameter of the chimney used for the determination of the critical wind speed, the lift coefficient and the aerodynamic coefficients. Furthermore, the lock-in conditions in the second mode are found to be more critical for bending response; the first mode lock-in is critical only for displacement response. Thus, for the safe design of the chimney, the possibility of lock-in conditions in the second mode must be investigated. Nomenclature A diagonal matrix described in equation (2) ratio of diameter of ith element to critical ai diameter parameters defined in equation (7) a l , a2 C structural damping matrix CL, C L ( K ) lift coefficient critical diameter De diameter of ith element D diameter of vibrating cylinder g parameter defined in equation (13) H height of chimney height of section corresponding to critical hc diameter I vector of unity K nondimensional frequency nondimensional frequency of lift oscillator Ko l element length M mass matrix vector of bending moment Mb m parameter defined in equation (7) m mass of vibrating cylinder P(t) load vector R stiffness matrix Reynolds number Re r parameter defined in equation (3) S Strouhal number U critical wind speed X structural displacement R structural velocity R structural acceleration yl(K) aerodynamic (velocity) coefficient
aerodynamic (displacement) coefficient displacement in modal coordinates velocity in modal coordinates acceleration in modal coordinates mass density of air lock-in circular frequency, structural frequency parameter defined in equation (12) maximum nondimensional amplitude of across-wind vibration of the circular cylinder equivalent damping ratio of lift oscillator structural damping ratio parameter defined in equation (12) parameter defined in equation (13) mode shape function mode shape coefficient for bending moment
References 1 Simiu, E, and Scanlan, R. H. "Wind effects on structures'. John Wiley and Sons, New York, 1978, pp 260-285 2 Basu, R. I. and Vickery, B. J. 'A comparison of model and full-scale behaviour in wind of towers and chimneys', Proc. Wind Tunnel Modelling for Civ. Eng., Cambridge University Press, New York, 1982, pp 94-106 3 Vickery, B. J. and Basu, R, I. 'Across-wind vibrations of structures of circular cross-section. Part I: development of a mathematical model for two-dimensional conditions', J. Wind Eng. Ind. Aerodyn,, 1983, 12, 49-73 4 Basu, R. I. and Vickery, B. J. 'Across-wind vibration of structures of circular cross-section. Part II: development of a mathematical model for full-scale applications', J. Wind Eng, Ind. Aerodyn., 1983, 12, 75-97 5 Vickery, B. J. and Basu, R. I. 'Simplified approaches to the evaluation of the across-wind response of chimneys', J. Wind Eng. Ind. Aerodyn., 1983, 14, 153-166 6 Bishop, R. E. D. and Hassan, A. Y. 'The lift and drag forces on a circular cylinder oscillating in a flowing fluid', Proc. Roy. Soc., London, 1964, Series A (277), 51-74 7 Skop, R. A. and Griffin, O. M. 'A model for the vortex excited response of bluff cylinders', J. Sound Vib., 1973, 27(2), 225-233 8 Griffin, O. M., Skop, R. A. and Koopmann, G. H. "The vortex excited resonant vibrations of circular cylinders', J. Sound Vib., 1973, 31(2), 235-249 9 Skop, R. A. and Griffin, O. M. 'On a theory for the vortex excited oscillations of flexible cylindrical structures', J. Sound Vib., 1975, 41(3), 263-274 10 Hallam, M. G., Heaf, N. J. and Wootton, R. L. 'Dynamics of marine structures', CIRIA Rep. UR 8, London, 1978, pp 55-58 I1 Scruton, C. 'On the wind excited oscillations of stacks, towers and masts', Syrnp. Wind Effects Buildings Struct., NPL, Teddington, 1963, 806-812 12 Wootton, L. R. 'The oscillations of large circular stacks in wind', Proc. Inst. Civ. Eng., 1969, 43, 573-598 13 Vickery, B. J. and Watkins, R. D. 'Flow induced vibrations of cylindrical structures', Proc. Ist Aust. Conf. HydrauL Fluid Mech., Pergamon, Oxford, 1963, pp 112-121 14 Vickery, B. J. and Clark, A. W. 'Lift or across-wind response of tapered stacks', J. Struct. Div., Proc. ASCE, 1972, 98, ST1, 1-20 15 Lawrence, C. M. and Rumman, W. S, 'Dynamic design of reinforced concrete chimneys', ACI J., September 1967, 64, 558-567 16 Kwok, K. C. S. and Melbourne, W. H. 'Wind induced lock-in excitation of tall structures', J. Struct. Div., Proc. ASCE, 1981, 107, ST1,57-71 17 Griffin, O. M. and Koopman, G. H. 'The vortex excited lift and reaction forces on resonant vibrating cylinders', J. Sound Vib., 1977, 54(3), 435--448 18 Ruscheweyh, H. 'Wind loading on the television tower, Hamburg, Germany', J. Ind. Aerodyn., 1975/1976, 1(3), 315-333
Eng. Struct., 1987,Vol. 9,January 31
The across-wind response of tapering cylinder-like structures, such as chimneys, due to periodic shedding of vortices is obtained under lock-in conditions. The method of analysis incorporates aerodynamic effects and describes the lift forces by a linear model monoharmonic in nature. Under lock-in conditions, the critical wind speed and the lift and aerodynamic coefficients are assumed to be uniform throughout the height of the chimney, although its outside diameter varies. The aerodynamic and lift coefficients are derived from a lift oscillator model which adequately describes the experimental results of the vortex-induced oscillation (at lock-in) of an elasticallysupported cylinder. With the help of an example problem, the paper shows how the response of the chimney is sensitive to the choice of the critical diameter used for the determination of critical wind velocity and the lift and aerodynamic coefficients. It also emphasizes the importance of the lock-in condition in the second mode in relation to bending stresses. Key words: across-wind response, vortex shedding, lock-in, chimneys, bending stresses Vortex-induced oscillation of tall chimneys due to wind has long been recognized and has assumed importance in design as a check against resonant aerodynamically induced vibrations, or excessive stresses developed due to resonant vibration. Literature on the subject indicates that this oscillation could be periodic or random, depending upon the flow regime, smoothness of the surface of the structure and proximity of other tall structures. For a fairly smooth surface and uninterrupted wind flow, periodic shedding of vortices is expected in subcritical (103 < Re < 105) and super-critical (Re > 3.5 x 106) flOW conditions. For velocity of wind exceeding the critical flow condition (0.5 × 106< Ro < 3.5 z 10~'), the vortex-induced forces tend to become random and par tially correlated along the height of the structure. 1 The oscillation across the main wind direction can become quite significant if either the vortex shedding frequencies coincide with the structure's frequency or the natural frequencies of the structure fall within the dominant 0141-0296/87/01027-o5/$03.00 © 1987Butterworth & Co (Publishers)Ltd
frequency range of the cross-wind force spectrum (for random excitation). The magnitude of the cross-wind oscillation is proportional to the lift forces which primarily depend upon the Strouhal number S, lift coefficient CL, aerodynamic and mass damping parameters and the correlation length. The values of the above parameters depend upon the flow condition and the geometric properties of the structure, and they should be determined from tests on actual structures. Basu and Vickery2developed a mathematical model for full-scale application to slender structures and validated it by comparing the predicted response with the experimentally measured response of a prototype 330m high TV tower. The model was based on laboratory experiment results and was developed within the framework of random vibration theory incorporating the nonlinear aerodynamic damping force. Details of the model and the results of other prototype tests are given in references 2-5. For the purposes of this paper, which is merely illustrative, Eng. Struct., 1987, Vo[. 9, J a n u a r y
27
A c r o s s - w i n d response o f cylinders: 7-. K. Datta a n d A. K. Jain (Not t o scale) -~
ki--10.5 m
- ~ x21
21 ~'60~ 20 v
-t~ x 2 0
D E
{ )
0 cN
t
i+1 Q 0
-~ xi+ -I~ x [
/ / = 12.5
1
D = 12.5m t =0.2m
,
3 2
I
-t~ x 2 LD," X 1
!0 m
a
b
c
d
Figure 1 Details of chimney: (a) chimney; (b) section A - A ; (c) lumped mass idealization; (d) d y n a m i c degrees of f r e e d o m Table I Data for c h i m n e y Distance from base (m)
Outside diameter (m)
Thickness (m)
0 12.5 25 37.5 50 62.5 75 87.5 100 112.5 125 137.5 150 162.5 175 187.5 200 212.5 225 237.5 250
20 20 19.5 19 18.5 18 17.5 17 16.5 16 15.5 15 14.5 14 13.5 13 12.5 12 11.5 11 10.5
0.85 0.85 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.24 0.23 0.22 0.21 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Frequencies: 0)1 = 2.1617 rad s 1; 0)2 = 8.3379 rad s-l; 0)3 : 19.6637 rad s -~
the values used in the analysis are obtained from tests on model chimneys and elastically-supported cylinders. Most of the available laboratory data generally refer to forces on static cylinders for which lift force is found to be harmonic 6 with a value of CL = 0.6. For elastically supported cylinders, experimental data are fitted to both linear and nonlinear models. The nonlinear model resembles the Van der Pol type equation, while the linear model is expressed by a monoharmonic function. 7-9 The excitation lift force can be approximated by a monoharmonic function, particularly in regions of Reynolds number where vortex shedding is predominantly periodic or at the lock-in condition.~° From various model tests on circular cylinders and chimneys, it is shown that the Strouhal number for circular chimneys may be taken to be about 0.2. Also, tests on model chimneys reveal that the resonant wind velocity is
28
E n g . S t r u c t . , 1 9 8 7 , V o l . 9, J a n u a r y
influenced primarily by the upper portion of the chimney. u These two observations are used in most analyses to determine the critical wind velocity under lock-in conditions. While the Strouhal number is generally considered as 0.2 for chimneys in subcritical and supercritical flows, no consistent value of CL is reported in the literature. Wootton 12used a value of CL for the response evaluation of large circular stacks in wind which was dependant upon the tip displacement of the tower and a nondimensional aerodynamic damping coefficient. This value was originally proposed by Vickery. ~3 Vickery et al. 14 showed that for random vortex excitation, the rms value of CL ranges between 0.05 and 0.15. For safe dynamic design of reinforced concrete (RC) chimneys, Maugh and Rumman 15 used a conservative value of CL of 0.66 and ignored the aerodynamic effects. Kwok et al. J6 proposed a sinusoidal lock-in excitation model to predict the across wind response of a circular tower tested in boundary layer flow in a wind tunnel. They predicted a maximum value of CL ~ 0.08. The work described so far only considers lock-in vibration of the chimney in the first mode. As far as the bending stress is concerned, however, the second mode lock-in could be more severe. In the present study, the response of a tapered chimney is investigated under periodic shedding of vortices. The equation of motion for across wind vibration considers the aerodynamic effects. The aerodynamic and lift coefficients used in the analysis are obtained by fitting the response of a lift oscillator model to the experimentally observed oscillation of an elastically supported cylinder under lock-in conditions. Using the present method of analysis, an example problem is solved to investigate the influence of the important parameters on the response of the chimney and the importance of the lock-in condition in the second mode of vibration.
Theory The chimney is considered as an assemblage of 2D beam elements, each assumed to have constant diameter over
Across-wind response of cylinders: 7. K. Datta and A. K. Jain Table2
Maximum tip displacement and base moment
Assumed section
First mode lock-in
Second mode lock-in
hc (m)
Dc (m)
U (ms -1)
Tip displacement Base moment (m) (xl0SKNm)
U (ms -1)
Tip displacement Base moment (m) (xl05KNm)
232.5 200 175
11.5 12.5 13.5
19.439 21.503 23.223
0.286 0.382 0.463
74.976 82.939 89.574
0.106 0.174 0.123
3.925 4.606 5.318
the elemental length. The dynamic degrees of freedom are considered as the sway degrees of freedom only as shown in Figure l(d). The across-wind vibration due to the periodic shedding of vortices is described by the following equation of motion:
M X + C X + R X = P(t)
(1)
10.737 12.142 15.208
resonance curve. By fitting the response of the lift oscillator to this resonance curve, the following relationships are derived: 1 CL(K) = 4~ov-~l~o/pD 2 y2(K) = ~ ( K 2 - K~o)/pD 2 yI(K) = 2m(-qK - vKo)/p-D 2
where M is the diagonal lumped mass matrix; R is the condensed stiffness matrix corresponding to the sway degrees of freedom; C is the structural damping matrix which is not explicitly knpwn but.can be defined in terms of modal damping; X, X and X are, respectively, the displacement, velocity and acceleration in the structural coordinates; and P(t) is the time varying load vector given by:
P(t) = r y I ( K ) A X / U + ry2(K)AX/Dc + 0.5 rCe(K)A1 sin wt
(2)
r = 0.5 pUZ(2O~)
(3)
where D¢ is the diameter at a section in the upper part of the chimney used for the calculation of critical wind velocity; p is the mass density of air; A is a diagonal matrix whose ith element is given by 0.5(ai + ai_l)l, a~ being the ratio of D~/D¢ and l is the elemental length; I is a vector of unity; yl(K), y2(K) and Q ( K ) are the aerodynamic and lift coefficients, respectively, which depend on the nondimensional frequency K = D¢w/U; w is the lock-in circular frequency; U is the critical wind speed given by U = wD~/2crS; and S is the Strouhal number. The force vector P(t), as given by equation (2) is obtained with the assumptions that: lock-in has occurred with one of the structure's natural frequencies, to • at lock-in, there is a perfect correlation of lift and aerodynamic forces over the entire length of the chimney • aerodynamic coefficients Yl(K) and y2(K), and the lift coefficient CL(K), are constant over the entire length of the chimney • the magnitude of aerodynamic and lift forces at any point on the chimney depends on the critical wind velocity U (and not on the local mean wind velocity at that point)
(4) (5) (6)
In equations (4)-(5), D is the diameter of the vibrating cylinder; ~ is its mass; v is an equivalent damping ratio of the lift oscillator; "0 is the structural damping ratio; if0is the maximum nondimensional amplitude of acrosswind vibration recorded in the experiment; K0 is the nondimensional frequency of the lift oscillator. It is shown in reference 1 that, for a good fit, K0 = 1.05 and v = 0.052. Values of other parameters required to obtain CL, yl(g) and yz(K) from equations (4)(6) are taken from reference 17. It should be pointed out that these parameter values were obtained for a specific set of conditions in uniform flow and for pure translational motion of the elastically sprung body. The values of these parameters are, therefore, not universal. They are used in this work for illustrative purposes. For lock-in vibration in a particular mode characterized by the natural frequency w and mode shape function ~b, the equations of motion, (1) and (2), may be written in modal coordinate form as:
m(2 + 2rlo~±+ toZz) = r(alyl(K)~./U + aty2(K)z/Dc + 0.5a2CL(K) sin oJt)
(7)
where m = ~brMcb; al = ~brA~b; a2 = ~brAl; z, and 2 are the displacement, velocity and acceleration, respectively, in modal coordinates. Rearranging equation (7), the equation of motion in modal coordinates takes the form:
•
The aerodynamic coefficients yI(K) and y2(g), and the lift coefficient CL(K), are obtained from an empirical linear oscillator model 1 which adequately describes the response of an elastically-supported cylinder at lock-in as observed in experimental tests. 17 The experimental results which show the across-wind displacement of the cylinder against nondimensional frequency resemble a
+ (2rico - raxyl(K)/mU)2
+ (oo2 - raay2(K)/mDc)z = ra2CL/m sin wt
(8)
The solution of equation (8) gives the value of z used to obtain the displacement or moment response from the following relationships: X = ~z
(9) (10)
mb = q~bZ
where ~ is the mode shape coefficient for bending moment and Mb is the moment vector. It is interesting to study the role of aerodynamic effects in modifying equation of motion (8) which may be rewritten as:
+ 2rj-~i~ + ~oZz = ra2CL/mSinwt in which: = g~
-~ = ~/[g]" - ~
Eng. Struct., 1987, Vol. 9, J a n u a r y
(11)
(12)
29
Across-wind response of cylinders: T. K. Datta and A. K. Jain 1.5
-g 0.9
D c : 13.5 (case l)
Z
z 0.8
o
_
g %
0.7 D c = 13.5 m (case ])
× 0.6
v
0,75
o.5
/
12.5 m (case 1])
~ 0 . 4 ~
E
o
g
E .~ 0.3 -
£3
~ 0.2 O
0
0.1 0 0
1 25
I 50
I 75
I 100
I 125
I 150
I ~ 175 2 0 0
225
250
25
50
75 100 125 150 175 Distance from base (m)
200
225
250
Figure 3 D e s i g n m o m e n t s f o r lock-in s e c o n d m o d e : w 2 = 8.3379 rad s - l ; r / = 5%
Distance f r o m base (m)
Figure 2 D e s i g n
moments
for lock-in first mode:
~o~ = 2.1617
rad s- ~; r / = 5%
g
Z c~
%
g = 1 - raly2(K)/wZmD~
X
= razyl (K)/2rn[gpUoJ
(13)
Equation ( 11) describes a single degree of freedom system having a frequency ~ and damping ratio ~, and acted upon by a harmonic force at frequency w. The system damping ~ incorporates aerodynamic damping and the system frequency ~ depends on the value of g which in turn depends on the aerodynamic coefficient y2(K).
1 0.8
. Case 1 ~ C a s e
0.6 ~ J
~
2 ~ Case 3
->'<"~
0.4 ~ E o E o.,
"
Maximum tip displacement (m)
Ca2se Case 1
0.82
0.50
&
0 E3
0
Figure 4
I 25
I 50
I 75
I I I ~"-'-?~ 100 125 150 175 Distance from base (m)
200
I 225
250
C o m p a r i s o n o f d e s i g n m o m e n t s f r o m cases 1, 2 a n d
3
Results Using the method of analysis described, the response of a chimney of height H = 250m (Figure l(a)) is obtained. The details of the chimney are shown in Table 1. For the analysis, the chimney is divided into 20 beam elements, as shown in Figure 1(c). The first three natural frequencies of the structure obtained from this lumped mass idealization are given in Table 1. Since the critical wind velocity for lock-in conditions is influenced primarily by the upper part of the chimney, ~1 three different sections in the upper one-third portion of the chimney are selected for evaluation of the critical wind velocity. The diameters of the chimney at these sections and their heights above the base are shown in Table 2. The responses of the chimney are computed for different critical wind velocities as obtained with the help of the diameters of the chimney at the three sections. These responses are compared in order to investigate how the choice of critical diameter used in the determination of critical wind velocity influences the response of the chimney. For the computation of the critical wind velocity, the Strouhal number is assumed to be 0.2. With this value of the Strouhal number, the values of CL(K), yl(K) and y2(K), as obtained from equations (4)-(6) are 0.266, 0.044 and 1.165, respectively. It is interesting to note that the rms lift coefficient CL obtained from laboratory tests, 4 varies between 0.15 and 0.23. The average measured CL for a RC tower I~ was found to be 0.173. Basu and Vickery4 used rms values of CL of 0.15 and 0.23 for predicting the response of a TV tower in the first and third modes which compared favourably with the experimentally measured values. Thus, the present empirical model, though valid for periodic shedding of
30
Eng. Struct., 1987, Vol. 9, J a n u a r y
vortices, uses a somewhat higher value of CL. The value of y1(K) in the present model provides an aerodynamic damping which is about one-third of the structural damping. The nonlinear aerodynamic damping model proposed by Basu and Vickery4 can incorporate a wide range of aerodynamic damping depending upon the flow condition. According to their model, the amount of aerodynamic damping, as observed in the present study, is typical of the low amplitude case for a linear system. For lock-in in the first mode, the maximum bending moments (design moments) at different points along the height of the chimney are shown in Figure 2. It is seen that the design moments at different sections of the chimney are quite sensitive to the choice of De. There may be as much as 40% change in base moment if hc is changed from 0.93 to 0.70 H. Figure3 shows the design moments along the height of a chimney when there is lock-in in the second mode. Here again, it is observed that a small change in the value of Dc may lead to considerable change in the design moments, especially at the base of the chimney. Table 2 compares the responses between the lock-in conditions at modes 1 and 2. For the displacement response, the first mode lock-in is critical. The second mode lock-in, if it occurs, may lead to a base moment about three times that for the first mode lock-in. Thus, the possibility of lock-in in the second mode must be investigated for safe design of the chimney. In Figure 4, design moments from cases 1, 2 and 3 are compared. In case 1, the aerodynamic effects are neglected and responses are obtained with a value of CL ---- 0.6 which is generally used for the stationary cylinders. In case 2, only the quadrature component
Across-wind response of cylinders: 1-. K. Datta and A. K. Jain of the aerodynamic effect is considered; the Strouhal number is so adjusted that the aerodynamic coefficient y2(K), corresponding to the displacement component, becomes zero as described in the theory. For this case, the Strouhal number becomes 0.167 and the corresponding values of CL and yl(K) are 0.266 and -0.0103, respectively. In case 3, the full aerodynamic effect is considered. The comparison shows that the response in case 1 is significantly higher than the other two cases and any design based on this analysis would be too conservative. The response in case 2 is higher than that in case 3. This increase in the response is partly due to the increased value of the critical wind velocity (Strouhal number being less) and partly due to the resonating effect as explained in the theory.
y2(K) z ± p to ~'0 b'
4, Conclusions The vortex-induced oscillation of cylinder-like structures such as chimneys under lock-in conditions is investigated under a set of simplifying assumptions aimed at illustrating certain features of the response. Under periodic shedding of vortices, the response of the chimney is shown to be quite sensitive to the choice of the critical diameter of the chimney used for the determination of the critical wind speed, the lift coefficient and the aerodynamic coefficients. Furthermore, the lock-in conditions in the second mode are found to be more critical for bending response; the first mode lock-in is critical only for displacement response. Thus, for the safe design of the chimney, the possibility of lock-in conditions in the second mode must be investigated. Nomenclature A diagonal matrix described in equation (2) ratio of diameter of ith element to critical ai diameter parameters defined in equation (7) a l , a2 C structural damping matrix CL, C L ( K ) lift coefficient critical diameter De diameter of ith element D diameter of vibrating cylinder g parameter defined in equation (13) H height of chimney height of section corresponding to critical hc diameter I vector of unity K nondimensional frequency nondimensional frequency of lift oscillator Ko l element length M mass matrix vector of bending moment Mb m parameter defined in equation (7) m mass of vibrating cylinder P(t) load vector R stiffness matrix Reynolds number Re r parameter defined in equation (3) S Strouhal number U critical wind speed X structural displacement R structural velocity R structural acceleration yl(K) aerodynamic (velocity) coefficient
aerodynamic (displacement) coefficient displacement in modal coordinates velocity in modal coordinates acceleration in modal coordinates mass density of air lock-in circular frequency, structural frequency parameter defined in equation (12) maximum nondimensional amplitude of across-wind vibration of the circular cylinder equivalent damping ratio of lift oscillator structural damping ratio parameter defined in equation (12) parameter defined in equation (13) mode shape function mode shape coefficient for bending moment
References 1 Simiu, E, and Scanlan, R. H. "Wind effects on structures'. John Wiley and Sons, New York, 1978, pp 260-285 2 Basu, R. I. and Vickery, B. J. 'A comparison of model and full-scale behaviour in wind of towers and chimneys', Proc. Wind Tunnel Modelling for Civ. Eng., Cambridge University Press, New York, 1982, pp 94-106 3 Vickery, B. J. and Basu, R, I. 'Across-wind vibrations of structures of circular cross-section. Part I: development of a mathematical model for two-dimensional conditions', J. Wind Eng. Ind. Aerodyn,, 1983, 12, 49-73 4 Basu, R. I. and Vickery, B. J. 'Across-wind vibration of structures of circular cross-section. Part II: development of a mathematical model for full-scale applications', J. Wind Eng, Ind. Aerodyn., 1983, 12, 75-97 5 Vickery, B. J. and Basu, R. I. 'Simplified approaches to the evaluation of the across-wind response of chimneys', J. Wind Eng. Ind. Aerodyn., 1983, 14, 153-166 6 Bishop, R. E. D. and Hassan, A. Y. 'The lift and drag forces on a circular cylinder oscillating in a flowing fluid', Proc. Roy. Soc., London, 1964, Series A (277), 51-74 7 Skop, R. A. and Griffin, O. M. 'A model for the vortex excited response of bluff cylinders', J. Sound Vib., 1973, 27(2), 225-233 8 Griffin, O. M., Skop, R. A. and Koopmann, G. H. "The vortex excited resonant vibrations of circular cylinders', J. Sound Vib., 1973, 31(2), 235-249 9 Skop, R. A. and Griffin, O. M. 'On a theory for the vortex excited oscillations of flexible cylindrical structures', J. Sound Vib., 1975, 41(3), 263-274 10 Hallam, M. G., Heaf, N. J. and Wootton, R. L. 'Dynamics of marine structures', CIRIA Rep. UR 8, London, 1978, pp 55-58 I1 Scruton, C. 'On the wind excited oscillations of stacks, towers and masts', Syrnp. Wind Effects Buildings Struct., NPL, Teddington, 1963, 806-812 12 Wootton, L. R. 'The oscillations of large circular stacks in wind', Proc. Inst. Civ. Eng., 1969, 43, 573-598 13 Vickery, B. J. and Watkins, R. D. 'Flow induced vibrations of cylindrical structures', Proc. Ist Aust. Conf. HydrauL Fluid Mech., Pergamon, Oxford, 1963, pp 112-121 14 Vickery, B. J. and Clark, A. W. 'Lift or across-wind response of tapered stacks', J. Struct. Div., Proc. ASCE, 1972, 98, ST1, 1-20 15 Lawrence, C. M. and Rumman, W. S, 'Dynamic design of reinforced concrete chimneys', ACI J., September 1967, 64, 558-567 16 Kwok, K. C. S. and Melbourne, W. H. 'Wind induced lock-in excitation of tall structures', J. Struct. Div., Proc. ASCE, 1981, 107, ST1,57-71 17 Griffin, O. M. and Koopman, G. H. 'The vortex excited lift and reaction forces on resonant vibrating cylinders', J. Sound Vib., 1977, 54(3), 435--448 18 Ruscheweyh, H. 'Wind loading on the television tower, Hamburg, Germany', J. Ind. Aerodyn., 1975/1976, 1(3), 315-333
Eng. Struct., 1987,Vol. 9,January 31