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A deterministic model for along-wind motion of buildings
A deterministic model for along-wind motion of buildings T. Balendra Department of Civil Engineering, National University of Singapore, Singapore 0511
G.K. Nathan and Kok Hin Kang Department of Mechanical and Production Engineering, National University of Singapore, Singapore 0511 (Received July 1988)
The along-wind response of a slender vertical structure in a turbulent atmospheric boundary layer is investigated by using a linearized timedomain technique. The methodology employs the classical flexural beam theory. The atmospheric wind turbulence is modelled by using cosinusoidal functions with closely spaced frequencies and with random phase angles uniformly distributed between 0 and 2n. The amplitudes of cosinusoidal functions are determined by using the power spectrum of the fluctuating wind velocity. The proposed model predicts the peak responses by using the predetermined quasi-static drag coefficients of a given geometric shape. The predicted peak responses are found to compare reasonably well with the published experimental results for a square building and a rectangular one. By referring to the proposed model, the influence of damping and shape of the building on alongwind response is discussed.
Keywords: wind turbulence, buildings, flexural beam theory
The along-wind response of a structure is determined by drag force in the direction of wind flow. Owing to spatial variation of wind velocities, the drag force is a function of time and space. Analytical methods to evaluate the response of slender str~tures to this drag force have been proposed by a number of researchers 1-3. However, these methods are based on a probabilistic approach, which is applicable for linear structures. For nonlinear behaviour of structures, a time-domain approach or deterministic analysis is required. The purpose of this investigation is to develop a simple and reliable deterministic model to predict the along-wind response of slender structures. As a first step, the proposed model is tested for linear structures. The drag force due to turbulence is expressed deterministically by using the power spectral density of the fluctuating component of the wind. The governing equation of motion is linearized and solved by using Galerkin's technique. The predicted peak displacements and accelerations by the proposed model are compared with published experimental results of a square building4 and a rectangular building s. 0141-0296/89/01016-07/$03.00 © 1989 Butterworth & Co (Publishers) Ltd
16 Eng. Struct. 1989, Vol. 11, January
Analytical model The slender structure is modelled as a uniform flexural beam of height H and width b. The wind is characterized by a mean flow with superimposed turbulence (gust) in the along-wind direction, as shown in Figure 1. The mean wind velocity, O(z), is assumed to vary with height z according to the power law
O(z) = Oo(Z/ZoP
(1)
where A is a constant that varies with the type of terrain and 0 0 the mean wind velocity at reference height Zo. The gust is a function of height and time. However, for simplicity, the gust is made independent of height by using a mean value obtained by integrating over the height of the vertical structure. The time-dependent gust is represented by cosinusoidal functions with closely spaced frequencies and with random phase angles. Such representation is classified as 'pseudo-turbulence', where there is a regular pattern with constant periodicity in time and uniform in space. With this assumption, the
Model for along- wind motion." T. Balendra et al. J
where p is the density of air and U~ = (0 2 + 2Uu + u 2) - 2h(O + u) + h 2
(5a)
UZ~ = D1 - Dz + D3
(5b)
or f J
f
J
f
f
I I
in which D1, D2 and D 3 are the main aerodynamic excitation term, main aerodynamic damping term and nonlinear damping term, respectively.
I
I I I I I I I I I I
Governing equations of motion and boundary conditions
If m is the mass per unit height and El is the flexural rigidity per unit height of the beam, then the governing equation for along-wind motion is m[9 + co + (Elp")" = FD(Z, t) x
/
I
(6)
in which c is the viscous damping coefficient associated with damping forces per unit height. In view of equations (3) and (5), equation (6) may be written as: mp + cp + lpbD2CD d- (EIp")" = ½pbCo(D1 + D3)
(7) The boundary conditions to be satisfied at z = 0, assuming a rigid foundation, are p(0) = 0
p'(0) = 0
(8a)
whereas the boundary conditions at z = H are p'"(H) = 0
p"(h) = 0 Figure 1 Wind profile and geometry of slender structure
Solution technique
instantaneous velocity U of incoming wind may be expressed as U(z, t) = O(z)i + u(t)i
(2a)
U(z, t) = U(z)i q- ~. Urn COS(~rn t q- ~m)] m=l
(2b)
where i is the unit vector in the along-wind direction. In equation (2b), u,, is the amplitude of the gust component with frequency o9,., which will be determined from the power spectrum of atmospheric turbulence in the alongwind direction; and q~,,the random phase angle uniformly distributed between 0 and 2n.
Accurate prediction of aerodynamic forces on an oscillating bluff body is a very difficult task. Thus, the simplified quasi-static representation of aerodynamic forces which assumes that the forces are the same as if the body is at rest in a uniform stream is considered. This is equivalent to assuming that the aerodynamic forces develop instantaneously. The unsteady drag force in the direction of the resultant wind velocity can be calculated by using the drag coefficient CD given by ESDU 6. If the velocity of the structure is denoted as ~z, t), then the velocity of wind relative to the structure is (3)
The drag force per unit height is correspondingly FD(Z, t) = ~pbCD 1 U,2
When the gust is represented by a periodic function given in equation (2b), the aerodynamic forcing function appearing on the right-hand side of equation (7) is periodic because of D1. Owing to the term D3, the forcing function is nonlinear. Furthermore, the coefficient of main aerodynamic damping term is periodic because of D 2. In this study the equations of motion are linearized by using the following assumptions: (a) the term D 3 =/~2 appearing in the aerodynamic forcing function is neglected; (b) the periodic aerodynamic damping coefficient involving D 2 is replaced by time-average values. Thus the aerodynamic damping in equation (7) is replaced by T
Aerodynamic forces
Ur = 0 -t- u --/9
(8b)
(4)
CD
Efo
1
(U + u) dt p
where T is the period of aerodynamic forcing function. Similar assumptions were also made by Patrickson and Friedmann 7 in a study on torsional response of buildings to wind, where the turbulence was described by a single sinusoidal function with an arbitrary amplitude. With these assumptions, the governing equations of motion become linear partial differential equations. The linearized periodic forcing function is now expanded in a complex Fourier series as follows: Fx(z, t) = ½pbCDD1 = ~ F=(z)eik~', k k = 0 , _+1, + 2 . . . . . _+oo
(9a)
Fx(z ) = ~1 foT Fx(z, t)e _ ik~ot dt
(9b)
Eng. Struct. 1989, Vol. 11, January
17
Model
for along-wind
1". B a l e n d r a
motion.
e t al.
in which T = (2r~/co) is the period of the forcing function. The partial differential equation is now reduced to ordinary differential form by eliminating the spatial variable, z, through Galerkin's method. An approximate solution to the problem may be obtained by assuming the solution to be a linear combination of simple polynomial shape functions that belong to admissible class, i.e. functions satisfying the kinematic or geometric boundary conditions only s. Thus for the periodic forcing function, the displacement of the flexural beam is expressed as
the turbulent intensity in the along-wind direction, and L,~ is the integral length scale of turbulence. When the fluctuating velocity component is expressed as
u(t) = ~
u,. cos(¢o,t + ~b,)
(14)
. = 1
the total energy per unit height present in the fluctuations is given by E, = ~
u, cos(~o.,t + 4).) m = l
p(z, t) = ~ pk(z)eik'°'
(lOa) j = 1, 2 . . . . h
)
1 2 . = 1
(15)
k = 0, _+ 1, + 2, _+3. . . . oo
k
pk(z) = ~ Hajkn ~
dt =
In view of equation (13),
(10b)
E. = ~
G.(n, z) dn dz =
0.(n) dn
(16)
0
in which the shape functions r/~ are determined from rlj = (z/n}j + a
j = 1, 2 . . . . h
(11)
The unknown coefficients ajk are determined by setting the error function, obtained by substituting the approximate solution into the differential equations, to be orthogonal to the assumed shape functions over the domain of the existence of the problem. Integrating by parts and substituting the boundary conditions yields a set of simultaneous complex algebraic equations in the form:
(12a)
EA] {x} = {v} in which [A] = [[K] + iko~[C]
(12b)
(kco)2EM]]
-
< X > = {X} T= (alka2k ... ahk)
(12c)
< F > = {F} T= (Q,,Qzk
(12d)
. . . Qh,)
Amplitudes of turbulent components The power spectrum of fluctuating along-wind velocities in the atmosphere at height z could be represented by the Von Karman equation 9, which takes the form nGu(n,a2~z ) - 4 n ( ~ ) / [ 1 - 7 0 . 8 ( n L u x ' ~52 ]/U 6]_] \
(13,
where G.(n, z) is the power spectral density at frequency n of the velocity component u (the corresponding power spectrum at height H is depicted in Figure 2), a,/U is
It should be noted that 0.(n)An is the measure of the energy associated with the frequency components in the range between n - ½An and n + ½An. Thus the amplitude u, is chosen so that its contribution to the total energy is equivalent to the area under the spectral density curve within the chosen frequency interval A n [ = ~ n . + l nm- 1)] at frequency n , ( = co,/2r0.
Computer implementation The infinite series in equation (2b) is truncated by choosing appropriate lower and upper limits on the frequency axis, so that the power contained within the frequency range represents adequately the total energy of the wind. With reference to the typical wind spectrum given in Figure 31° , it is observed that the peak of the wind spectrum occurs at about 0.04 Hz. The lower and upper limits of the frequency range should therefore be on either side of the spectral peak. Since the frequency of a tall building is generally greater than 0.05 Hz, the lower limit of the frequency is arbitrarily chosen to be 0.025 Hz, on the assumption that the components below this frequency will make marginal contribution to the maximum response. Likewise, the upper limit of the frequency is fixed at 4 Hz allowing for resonant effects due to higher harmonics of the building. With this choice, equation (2b) yields a gust period of 1/0.025 = 40 s. Determination of u The gust amplitude u. is calculated by using the procedure outlined in the preceding section for which the power
3.0 2.5 t~
2.0
>.
1.5
t~ 1.0 L
0.5 0 i0 -2
,
,
,
,
l,i,]
i
1
,
i
1,1,1
10-1
I
10 0
I
,
,
illl
101
10 -3
n Lux/U (HI Figure 2
18
Power s p e c t r u m of a l o n g - w i n d velocity
Eng. Struct. 1989, Vol. 11, January
10 -2
10 -1 Frequency
Figure 3
10 0
101
(Hz}
Distribution of w i n d energy w i t h f r e q u e n c y 1°
10 2
Model for along-wind motion: T. Balendra et al. spectrum within the selected frequency range is discretized by using an appropriate frequency spacing. The latter is determined as follows. Each harmonic term, co~in the series given by equation (2b), will primarily influence a system with a similar frequency and will have a small influence on other systems. The systems that will be excited significantly, say up to 100% of the resonance peak, by a particular harmonic eJ~ are determined as described below. If 4 is the ratio between the exciting frequency co= and the frequency of the system COo(=2nno), then the dynamic magnification for a damping ratio ( is given by
by Melbourne 5 on the response ofa CAARC rectangular building are used to verify the proposed analytical model. Example 1
The initial value of the frequency will be the lowest frequency chosen for the incoming wind. The total number of harmonics and the percentage of the resonance peak to be specified for a given percentage of damping ratio are determined through an iterative procedure so that the variance of the desired time history of the wind velocity given by equation (2b) represents adequately the energy contained under the power spectrum within the chosen frequency range. The phase angles required in the expression for u given by equation (2b) areobtained by using a random-number generator subroutine.
The building is 38.1 m by 38.1 m in plan and 317.5 m in height. It has a natural frequency of 0.1 Hz, a damping ratio of 1% and an average density of 192 kg m -3. The intensity of the along-wind turbulence component, tr,/~/, and the corresponding integral length scale, L,x, are found to be 7.5% and 115 m respectively, at the top of the building. The frequency spacing, Aco(= ogiAx), required to determine u is obtained by setting the dynamic magnification factor to be 75 % of the resonance peak, at 1% damping. With the starting value of a~ fixed at 0.025 Hz, the value of AZ for the determination of u is found to be 0.0176. Reinhold and Sparks have made wind-tunnel measurements on a square-sectioned rigid model with an aspect ratio of 8.33:1 (height = 635 mm, breadth = width = 76.2 mm). The turbulent shear layer was developed in a low-speed wind tunnel. The mean velocity profile was found to follow a power law with a power index 0.37, while the measured velocity spectrum of the turbulence component was in good agreement with the empirical Von Karman spectrum function given in equation (13). Fluctuating wind pressures on the surface of the model were measured by using transducers placed at six elevations throughout the height; and modal force spectrum was then determined for the fundamental mode of the building, assumed to be linear. Using this modal force spectrum, the peak responses of the full scale building (317.5 m in height and 38.1 m x 38.1 m in plan) were then estimated. The estimated peak displacements and peak accelerations are tabulated in Tables I and 2 respectively. The predicted responses by the proposed analytical model using a value of 1.52 for CD are also presented in Tables 1 and 2. The results are obtained using nine harmonics ( k = - 4 to 4 in equation (10)); for each harmonic, five shape functions are used. The computed peak displacements agree well with the values estimated by Reinhold and Sparks to within 15.7%; they also compare well with the values predicted by using the method proposed by Simiu 2 to within 12.8 %. The peak accelerations compare reasonably well with the values estimated by Reinhold and Sparks to within 18.2%. When compared to Simiu's method, the peak acceleration of the along wind motion falls within 18.2%.
Number of harmonics and shape functions
Example 2
The Fourier series expansion of the aerodynamic forcing functions (with a period T = 40 s) may be approximated with sufficient accuracy by a relatively small number of harmonic components (values of k in equation (9a)) through a convergence study. For each harmonic k, the number of shape functions (values of h) required in equation (11) is similarly obtained by a convergence study.
The CAARC standard tall rectangular building is 31 m by 46 m in plan and rises to a height of 183.0 m above the ground level. It has a natural frequency of 0.2 Hz, a damping ratio of 1% and an average mass density of 157 kg/m 3. The intensity of the along-wind turbulence component, a,/gJ, and the corresponding integral length scale L,x, are respectively 9.5 % and 142 m at the top of the building ~1. The peak displacement responses in the along-wind direction are predicted separately for the angles of incidence of 0 ° and 90 °. At 0 °, the mean wind blows normal to the longer side of the building (46 m); and at 90 °, the wind blows normal to the shorter size of the building, (31 m). The drag coefficients obtained from ESDU 6 with the correction for the effect of turbulence
D = [(1 - X2)2 + (2(42) 2] -½
(17)
The frequencies of the system are then obtained from (18)
D = RD*
where D* is evaluated from equation (17) at Z = 1.0, namely D* = 1/2(. Solving the above second-order equation yields 412= [1 + (1 - (1 + 4(2)(1 - 4(2/RZ))½]/(1 + 4~2) (19a) and X~= [1 -- (1 - (1 + 4(2)(1 -- 4~2/R2))½]/(1 + 4(2) (19b) The required frequency spacing Aco is then given by Ao~ =
toiAz
"" ogiAx
(20)
ZIZ2 where AZ = Z 2 - Zt; thus starting with a known initial value, to t , leads to o9i+1 = o9i + Ao9 ~- 09i(1 + AZ) = O~z(1 +Az) ~
(21)
Comparison with experimental results The experimental results-furnished by Reinhold and Sparks 4 on a square building and the results furnished
Eng. Struct. 1989, Vol. 11, January
19
M o d e l for a l o n g - w i n d motion," T. Balendra et al. Table 1 Peak displacement in along-wind direction for prototype building Mean wind speed
Reduced velocity
U(H)
O(H)
(m/s)
no-b-
40 35 29 26
10.5 9.2 7.6 6.8
Table 2
Peak tip displacement in along-wind direction (m)
Predicted response
Reinhold & Sparks 4
Simiu 2 (c)
(b)
% Deviation (b) (a) - x 100% (b)
% Deviation (c) (a) . . . . . x 100% (c)
(a) 1.36 1.05 0.73 0.59
1.40 1.06 0,69 0,51
2.9 0.9 5.8 15.7
1.56
12.8 6.4 3.3
Peak acceleration in along-wind direction for prototype building
Mean wind speed
Reduced velocity
O(H)
LI(H) nob
(m/s)
40 35 29 26
10.5 9.2 7.6 6.8
Peak tip acceleration in along-wind direction ( P % ) g Predicted response
Reinhold & Sparks 4
(a)
(b)
2.9 2.3 1,6 1.3
3.3 1.6 1.1
Table 3 Peak displacement in along-wind direction normal to the longer side of the CAARC building Mean wind speed
Reduced velocity
O(H)
U(H)
(m/s)
n°a
46
7.4
Peak tip displacement (m)
Predicted response (a)
Tanaka & Lawen 11 (b)
% Deviation (b) - (a) ---x 100% (b)
0.447
0.591
24.4
Peak displacement in along-wind direction normal to the shorter side of the CAARC building Mean wind speed
Reduced velocity
O(H) (m/s)
LI(H) nob
Predicted response (a)
Tanaka & Lawen 11 (b)
% Deviation (b) - (a) - x 100% (b)
46
5.0
0.291
0.247
17,8
Peak tip displacement (m)
intensity are 1.204 and 1.114 for aspect ratios 3.98 and 5.98 respectively. The frequency spacing, Atn, required to determine u is obtained by setting the dynamic magnification factor to be 75 % of the resonance peak, at 1%0 damping. With the starting value of tn i fixed at 0.025 Hz, the value of Ag for the determination of u is found to be 0.0176.
20
0.78 0.61
Eng. Struct. 1 9 8 9 , Vol. 1 1 , J a n u a r y
% Deviation (b) - (a) - × 100% (b)
Simiu 2
12.1 -0.0 18.2
3.2
9.4
1.5 1.1
6.7 18.2
(c)
% Deviation (c) - (a) ..... x 100% (c)
Tanaka and Lawen ~~ have made wind-tunnel measurements on a rectangular rigid model of 31 mm by 46 mm wide and 183 mm high with a generalized mass of 13.6 × 10 -3 kg. Thus the model simulates the CAARC building to a scale 1:1000. The mean velocity profile was found to follow a power law with a power index 0.28, while the measured longitudinal velocity spectrum of the turbulence component was in good agreement with the Von Karman spectrum. The dynamic response of the model was detected by the bending of the supported leaf spring and converted to the tip displacement of the building through static calibration. The measured root mean square displacements in the along-wind direction at two different orientations of the building are converted to peak values using the following peak factor given in ESDU: gD = (2 ln(n o To)~ + 0.577(2 loge(no To))- ~
(22)
where TO= 3600 s. At n o = 0.2 Hz, 9D is found to be 3.79. The peak displacements using this peak factor are presented in Table 3. The predicted responses by the proposed analytical model for the along-wind motion are also presented in Table 3. The results are obtained by using 19 harmonics (k = - 9 to +9); for each harmonic, the number of shape functions used for translational motion is 5. The computed peak displacement agrees well with the values measured by Tanaka and Lawen to within 17.8 %0 when the wind is normal to the shorter side of the building. When the longer side of the building is normal to the wind direction, the deviation between predicted and measured responses increases to 24.4%. This could be due to the peak factor used in converting the measured root mean square values to peak values. According to Tanaka and Lawen the peak factor at
Model for along-wind motion. T. Balendra et aL positive pressure was found to be in the range of 3.6 to 4.1.
3.0
Effects of damping and structural shapes
2.0
The effect of damping on the along-wind responses of the square building considered before is shown in Figures 4 and 5. It is seen that for any given reduced velocity, the peak displacement and peak acceleration at the top of the building decrease linearly with increase in damping ratio, on a log-log scale. The rate of decrease is found to be independent of the reduced velocity. The proposed analytical model is also used to study the effect of different structural shapes on the along-wind response. Three identical structures, but of different shapes, namely square, rectangle (breadth: width = 2:1), and equilateral triangle are chosen. The characteristic length, length of the side normal to wind, b, for each shape is determined on the basis of equal plan area. Other properties such as height, mass per unit height, fundamental frequency and damping are assumed to be the same as for the square structure considered before. The variation of normalized peak displacement at the top of the building with reduced velocity is shown in Figure 6. As seen from this figure, the displacement varies linearly with the reduced velocity on a log-log scale. Thus Pmax
1.0 : E
0.8
~o
0.6
-~
0.5
7 e-
o -~ 0.3 o.
. . . .
)
o 2 -
where k is the slope of the lines plotted in Figure 6. For
1
.
5
~
0.1
I
I
I
=
2
3
4
5
Damping ratio, ~ (~) 1.0
Figure 5
~" 0.8
Variation of peak acceleration with damping ratio
20I- . . . / a Triangle °
0
.
6
|
~
~
i
10~ 81-
"
.
.
.
ectan ,e
Square
"~ ~ •
;
1 0.2Q. Figure 6
6
l I 7 8 9 Reduced velocity, U(H)/n0b
I 10
I 11
Along-wind peak displacements for different shapes
a given reduced velocity, the triangular shape has the highest response, whereas the square shape has the least. 0.1
Figure 4
I
2
I
3
I
4
I
5
Damping ratio, ~ (~) Variation of peak displacement with damping ratio
Conclusions
The linearized deterministic model presented herein to predict the peak displacement and acceleration of slender
Eng. Struct. 1989, Vol. 11, January
21
Model for a l o n g - w i n d motion." T. Balendra et al. vertical structures due to a l o n g - w i n d m o t i o n s is f o u n d to give a c c u r a t e results. Thus, representing the gust by ' p s e u d o - t u r b u l e n c e ' seems to be reasonable. By using the p r o p o s e d model, the following conclusions are m a d e on the a l o n g - w i n d response of buildings. The p e a k a l o n g - w i n d d i s p l a c e m e n t a n d a c c e l e r a t i o n are found to decrease linearly with increasing d a m p i n g on a l o g a r i t h m i c scale, a n d the rate of decrease is i n d e p e n d e n t of the r e d u c e d velocity. W h e n three buildings of equal height a n d p l a n a r e a a n d differing only in their shape (square, rectangle, equilateral triangle) are c o m p a r e d for a l o n g - w i n d response, then for a given r e d u c e d velocity, a t r i a n g u l a r b u i l d i n g will have the highest response, whereas the square b u i l d i n g will have the least. The o r i e n t a t i o n of the structures is such that the characteristic length is n o r m a l to the d i r e c t i o n of wind.
References 1 Davenport, A. G. 'Gust loading factors' J Struct Div ASCE 1975, 97 9-38
22
Eng. Struct. 1989, Vol. 11, January
2 Simiu, E. 'Revised procedure for estimating along wind response' Struct Div ASCE 1980, 106, 1-10 3 Solari, G. 'Analytical estimation of the along wind response of structures' Wind En 9 Ind Aerodynam 1983, 14, 467478 4 Reinhold, T. A. and Sparks, P. R. 'The influence of wind direction on the response of a square section tall building' Proc Fifth Int Conf Wind Engng, Fort Collins, Colorado, 1979, 5 Melbourne, W. H. 'Comparison of measurements on the CAARC standard tall building model in simulated wind flows' J Wind Eng Ind Aerodynam 1980, 6, 73-88 6 Engineering Sciences Data Unit 'The response of flexiblestructures to atmospheric turbulence', ESDU Item no. 76001, Engineering Sciences Data Unit. London, UK, September 1976, pp 1 49 7 Patrickson, C. P. and Friedmann, P. P. 'Deterministic torsional building response to winds' J Struct Die ASCE 1979, liP3, 2621-2637 8 Balendra, T., Swaddiwuhipong, S., Quek S. T. and Lee S. L. 'Free vibration of asymmetric shear wall-frame buildings', Earthquake Eng Struct Dynam 1984, 12, 629-650 9 Reinhold, T. A. 'Measurements of simultaneous fluctuating loads at multiple levels on a model of tall building in a simulated urban boundary layer', Ph.D Thesis, Virginia Polytechnic Institute and State University, USA, 1977 10 Waller, R.A. Buildings on springs International Series of Monographs in Civil Engineering, Vol. 2, Pergamon Press, 1969 11 Tanaka, H. and Lawen, N. 'Test on the CAARC standard tall building model with a length scale of 1:1000', J Wind Engn9 Aerodynam 1986, 17, 15 29
G.K. Nathan and Kok Hin Kang Department of Mechanical and Production Engineering, National University of Singapore, Singapore 0511 (Received July 1988)
The along-wind response of a slender vertical structure in a turbulent atmospheric boundary layer is investigated by using a linearized timedomain technique. The methodology employs the classical flexural beam theory. The atmospheric wind turbulence is modelled by using cosinusoidal functions with closely spaced frequencies and with random phase angles uniformly distributed between 0 and 2n. The amplitudes of cosinusoidal functions are determined by using the power spectrum of the fluctuating wind velocity. The proposed model predicts the peak responses by using the predetermined quasi-static drag coefficients of a given geometric shape. The predicted peak responses are found to compare reasonably well with the published experimental results for a square building and a rectangular one. By referring to the proposed model, the influence of damping and shape of the building on alongwind response is discussed.
Keywords: wind turbulence, buildings, flexural beam theory
The along-wind response of a structure is determined by drag force in the direction of wind flow. Owing to spatial variation of wind velocities, the drag force is a function of time and space. Analytical methods to evaluate the response of slender str~tures to this drag force have been proposed by a number of researchers 1-3. However, these methods are based on a probabilistic approach, which is applicable for linear structures. For nonlinear behaviour of structures, a time-domain approach or deterministic analysis is required. The purpose of this investigation is to develop a simple and reliable deterministic model to predict the along-wind response of slender structures. As a first step, the proposed model is tested for linear structures. The drag force due to turbulence is expressed deterministically by using the power spectral density of the fluctuating component of the wind. The governing equation of motion is linearized and solved by using Galerkin's technique. The predicted peak displacements and accelerations by the proposed model are compared with published experimental results of a square building4 and a rectangular building s. 0141-0296/89/01016-07/$03.00 © 1989 Butterworth & Co (Publishers) Ltd
16 Eng. Struct. 1989, Vol. 11, January
Analytical model The slender structure is modelled as a uniform flexural beam of height H and width b. The wind is characterized by a mean flow with superimposed turbulence (gust) in the along-wind direction, as shown in Figure 1. The mean wind velocity, O(z), is assumed to vary with height z according to the power law
O(z) = Oo(Z/ZoP
(1)
where A is a constant that varies with the type of terrain and 0 0 the mean wind velocity at reference height Zo. The gust is a function of height and time. However, for simplicity, the gust is made independent of height by using a mean value obtained by integrating over the height of the vertical structure. The time-dependent gust is represented by cosinusoidal functions with closely spaced frequencies and with random phase angles. Such representation is classified as 'pseudo-turbulence', where there is a regular pattern with constant periodicity in time and uniform in space. With this assumption, the
Model for along- wind motion." T. Balendra et al. J
where p is the density of air and U~ = (0 2 + 2Uu + u 2) - 2h(O + u) + h 2
(5a)
UZ~ = D1 - Dz + D3
(5b)
or f J
f
J
f
f
I I
in which D1, D2 and D 3 are the main aerodynamic excitation term, main aerodynamic damping term and nonlinear damping term, respectively.
I
I I I I I I I I I I
Governing equations of motion and boundary conditions
If m is the mass per unit height and El is the flexural rigidity per unit height of the beam, then the governing equation for along-wind motion is m[9 + co + (Elp")" = FD(Z, t) x
/
I
(6)
in which c is the viscous damping coefficient associated with damping forces per unit height. In view of equations (3) and (5), equation (6) may be written as: mp + cp + lpbD2CD d- (EIp")" = ½pbCo(D1 + D3)
(7) The boundary conditions to be satisfied at z = 0, assuming a rigid foundation, are p(0) = 0
p'(0) = 0
(8a)
whereas the boundary conditions at z = H are p'"(H) = 0
p"(h) = 0 Figure 1 Wind profile and geometry of slender structure
Solution technique
instantaneous velocity U of incoming wind may be expressed as U(z, t) = O(z)i + u(t)i
(2a)
U(z, t) = U(z)i q- ~. Urn COS(~rn t q- ~m)] m=l
(2b)
where i is the unit vector in the along-wind direction. In equation (2b), u,, is the amplitude of the gust component with frequency o9,., which will be determined from the power spectrum of atmospheric turbulence in the alongwind direction; and q~,,the random phase angle uniformly distributed between 0 and 2n.
Accurate prediction of aerodynamic forces on an oscillating bluff body is a very difficult task. Thus, the simplified quasi-static representation of aerodynamic forces which assumes that the forces are the same as if the body is at rest in a uniform stream is considered. This is equivalent to assuming that the aerodynamic forces develop instantaneously. The unsteady drag force in the direction of the resultant wind velocity can be calculated by using the drag coefficient CD given by ESDU 6. If the velocity of the structure is denoted as ~z, t), then the velocity of wind relative to the structure is (3)
The drag force per unit height is correspondingly FD(Z, t) = ~pbCD 1 U,2
When the gust is represented by a periodic function given in equation (2b), the aerodynamic forcing function appearing on the right-hand side of equation (7) is periodic because of D1. Owing to the term D3, the forcing function is nonlinear. Furthermore, the coefficient of main aerodynamic damping term is periodic because of D 2. In this study the equations of motion are linearized by using the following assumptions: (a) the term D 3 =/~2 appearing in the aerodynamic forcing function is neglected; (b) the periodic aerodynamic damping coefficient involving D 2 is replaced by time-average values. Thus the aerodynamic damping in equation (7) is replaced by T
Aerodynamic forces
Ur = 0 -t- u --/9
(8b)
(4)
CD
Efo
1
(U + u) dt p
where T is the period of aerodynamic forcing function. Similar assumptions were also made by Patrickson and Friedmann 7 in a study on torsional response of buildings to wind, where the turbulence was described by a single sinusoidal function with an arbitrary amplitude. With these assumptions, the governing equations of motion become linear partial differential equations. The linearized periodic forcing function is now expanded in a complex Fourier series as follows: Fx(z, t) = ½pbCDD1 = ~ F=(z)eik~', k k = 0 , _+1, + 2 . . . . . _+oo
(9a)
Fx(z ) = ~1 foT Fx(z, t)e _ ik~ot dt
(9b)
Eng. Struct. 1989, Vol. 11, January
17
Model
for along-wind
1". B a l e n d r a
motion.
e t al.
in which T = (2r~/co) is the period of the forcing function. The partial differential equation is now reduced to ordinary differential form by eliminating the spatial variable, z, through Galerkin's method. An approximate solution to the problem may be obtained by assuming the solution to be a linear combination of simple polynomial shape functions that belong to admissible class, i.e. functions satisfying the kinematic or geometric boundary conditions only s. Thus for the periodic forcing function, the displacement of the flexural beam is expressed as
the turbulent intensity in the along-wind direction, and L,~ is the integral length scale of turbulence. When the fluctuating velocity component is expressed as
u(t) = ~
u,. cos(¢o,t + ~b,)
(14)
. = 1
the total energy per unit height present in the fluctuations is given by E, = ~
u, cos(~o.,t + 4).) m = l
p(z, t) = ~ pk(z)eik'°'
(lOa) j = 1, 2 . . . . h
)
1 2 . = 1
(15)
k = 0, _+ 1, + 2, _+3. . . . oo
k
pk(z) = ~ Hajkn ~
dt =
In view of equation (13),
(10b)
E. = ~
G.(n, z) dn dz =
0.(n) dn
(16)
0
in which the shape functions r/~ are determined from rlj = (z/n}j + a
j = 1, 2 . . . . h
(11)
The unknown coefficients ajk are determined by setting the error function, obtained by substituting the approximate solution into the differential equations, to be orthogonal to the assumed shape functions over the domain of the existence of the problem. Integrating by parts and substituting the boundary conditions yields a set of simultaneous complex algebraic equations in the form:
(12a)
EA] {x} = {v} in which [A] = [[K] + iko~[C]
(12b)
(kco)2EM]]
-
< X > = {X} T= (alka2k ... ahk)
(12c)
< F > = {F} T= (Q,,Qzk
(12d)
. . . Qh,)
Amplitudes of turbulent components The power spectrum of fluctuating along-wind velocities in the atmosphere at height z could be represented by the Von Karman equation 9, which takes the form nGu(n,a2~z ) - 4 n ( ~ ) / [ 1 - 7 0 . 8 ( n L u x ' ~52 ]/U 6]_] \
(13,
where G.(n, z) is the power spectral density at frequency n of the velocity component u (the corresponding power spectrum at height H is depicted in Figure 2), a,/U is
It should be noted that 0.(n)An is the measure of the energy associated with the frequency components in the range between n - ½An and n + ½An. Thus the amplitude u, is chosen so that its contribution to the total energy is equivalent to the area under the spectral density curve within the chosen frequency interval A n [ = ~ n . + l nm- 1)] at frequency n , ( = co,/2r0.
Computer implementation The infinite series in equation (2b) is truncated by choosing appropriate lower and upper limits on the frequency axis, so that the power contained within the frequency range represents adequately the total energy of the wind. With reference to the typical wind spectrum given in Figure 31° , it is observed that the peak of the wind spectrum occurs at about 0.04 Hz. The lower and upper limits of the frequency range should therefore be on either side of the spectral peak. Since the frequency of a tall building is generally greater than 0.05 Hz, the lower limit of the frequency is arbitrarily chosen to be 0.025 Hz, on the assumption that the components below this frequency will make marginal contribution to the maximum response. Likewise, the upper limit of the frequency is fixed at 4 Hz allowing for resonant effects due to higher harmonics of the building. With this choice, equation (2b) yields a gust period of 1/0.025 = 40 s. Determination of u The gust amplitude u. is calculated by using the procedure outlined in the preceding section for which the power
3.0 2.5 t~
2.0
>.
1.5
t~ 1.0 L
0.5 0 i0 -2
,
,
,
,
l,i,]
i
1
,
i
1,1,1
10-1
I
10 0
I
,
,
illl
101
10 -3
n Lux/U (HI Figure 2
18
Power s p e c t r u m of a l o n g - w i n d velocity
Eng. Struct. 1989, Vol. 11, January
10 -2
10 -1 Frequency
Figure 3
10 0
101
(Hz}
Distribution of w i n d energy w i t h f r e q u e n c y 1°
10 2
Model for along-wind motion: T. Balendra et al. spectrum within the selected frequency range is discretized by using an appropriate frequency spacing. The latter is determined as follows. Each harmonic term, co~in the series given by equation (2b), will primarily influence a system with a similar frequency and will have a small influence on other systems. The systems that will be excited significantly, say up to 100% of the resonance peak, by a particular harmonic eJ~ are determined as described below. If 4 is the ratio between the exciting frequency co= and the frequency of the system COo(=2nno), then the dynamic magnification for a damping ratio ( is given by
by Melbourne 5 on the response ofa CAARC rectangular building are used to verify the proposed analytical model. Example 1
The initial value of the frequency will be the lowest frequency chosen for the incoming wind. The total number of harmonics and the percentage of the resonance peak to be specified for a given percentage of damping ratio are determined through an iterative procedure so that the variance of the desired time history of the wind velocity given by equation (2b) represents adequately the energy contained under the power spectrum within the chosen frequency range. The phase angles required in the expression for u given by equation (2b) areobtained by using a random-number generator subroutine.
The building is 38.1 m by 38.1 m in plan and 317.5 m in height. It has a natural frequency of 0.1 Hz, a damping ratio of 1% and an average density of 192 kg m -3. The intensity of the along-wind turbulence component, tr,/~/, and the corresponding integral length scale, L,x, are found to be 7.5% and 115 m respectively, at the top of the building. The frequency spacing, Aco(= ogiAx), required to determine u is obtained by setting the dynamic magnification factor to be 75 % of the resonance peak, at 1% damping. With the starting value of a~ fixed at 0.025 Hz, the value of AZ for the determination of u is found to be 0.0176. Reinhold and Sparks have made wind-tunnel measurements on a square-sectioned rigid model with an aspect ratio of 8.33:1 (height = 635 mm, breadth = width = 76.2 mm). The turbulent shear layer was developed in a low-speed wind tunnel. The mean velocity profile was found to follow a power law with a power index 0.37, while the measured velocity spectrum of the turbulence component was in good agreement with the empirical Von Karman spectrum function given in equation (13). Fluctuating wind pressures on the surface of the model were measured by using transducers placed at six elevations throughout the height; and modal force spectrum was then determined for the fundamental mode of the building, assumed to be linear. Using this modal force spectrum, the peak responses of the full scale building (317.5 m in height and 38.1 m x 38.1 m in plan) were then estimated. The estimated peak displacements and peak accelerations are tabulated in Tables I and 2 respectively. The predicted responses by the proposed analytical model using a value of 1.52 for CD are also presented in Tables 1 and 2. The results are obtained using nine harmonics ( k = - 4 to 4 in equation (10)); for each harmonic, five shape functions are used. The computed peak displacements agree well with the values estimated by Reinhold and Sparks to within 15.7%; they also compare well with the values predicted by using the method proposed by Simiu 2 to within 12.8 %. The peak accelerations compare reasonably well with the values estimated by Reinhold and Sparks to within 18.2%. When compared to Simiu's method, the peak acceleration of the along wind motion falls within 18.2%.
Number of harmonics and shape functions
Example 2
The Fourier series expansion of the aerodynamic forcing functions (with a period T = 40 s) may be approximated with sufficient accuracy by a relatively small number of harmonic components (values of k in equation (9a)) through a convergence study. For each harmonic k, the number of shape functions (values of h) required in equation (11) is similarly obtained by a convergence study.
The CAARC standard tall rectangular building is 31 m by 46 m in plan and rises to a height of 183.0 m above the ground level. It has a natural frequency of 0.2 Hz, a damping ratio of 1% and an average mass density of 157 kg/m 3. The intensity of the along-wind turbulence component, a,/gJ, and the corresponding integral length scale L,x, are respectively 9.5 % and 142 m at the top of the building ~1. The peak displacement responses in the along-wind direction are predicted separately for the angles of incidence of 0 ° and 90 °. At 0 °, the mean wind blows normal to the longer side of the building (46 m); and at 90 °, the wind blows normal to the shorter size of the building, (31 m). The drag coefficients obtained from ESDU 6 with the correction for the effect of turbulence
D = [(1 - X2)2 + (2(42) 2] -½
(17)
The frequencies of the system are then obtained from (18)
D = RD*
where D* is evaluated from equation (17) at Z = 1.0, namely D* = 1/2(. Solving the above second-order equation yields 412= [1 + (1 - (1 + 4(2)(1 - 4(2/RZ))½]/(1 + 4~2) (19a) and X~= [1 -- (1 - (1 + 4(2)(1 -- 4~2/R2))½]/(1 + 4(2) (19b) The required frequency spacing Aco is then given by Ao~ =
toiAz
"" ogiAx
(20)
ZIZ2 where AZ = Z 2 - Zt; thus starting with a known initial value, to t , leads to o9i+1 = o9i + Ao9 ~- 09i(1 + AZ) = O~z(1 +Az) ~
(21)
Comparison with experimental results The experimental results-furnished by Reinhold and Sparks 4 on a square building and the results furnished
Eng. Struct. 1989, Vol. 11, January
19
M o d e l for a l o n g - w i n d motion," T. Balendra et al. Table 1 Peak displacement in along-wind direction for prototype building Mean wind speed
Reduced velocity
U(H)
O(H)
(m/s)
no-b-
40 35 29 26
10.5 9.2 7.6 6.8
Table 2
Peak tip displacement in along-wind direction (m)
Predicted response
Reinhold & Sparks 4
Simiu 2 (c)
(b)
% Deviation (b) (a) - x 100% (b)
% Deviation (c) (a) . . . . . x 100% (c)
(a) 1.36 1.05 0.73 0.59
1.40 1.06 0,69 0,51
2.9 0.9 5.8 15.7
1.56
12.8 6.4 3.3
Peak acceleration in along-wind direction for prototype building
Mean wind speed
Reduced velocity
O(H)
LI(H) nob
(m/s)
40 35 29 26
10.5 9.2 7.6 6.8
Peak tip acceleration in along-wind direction ( P % ) g Predicted response
Reinhold & Sparks 4
(a)
(b)
2.9 2.3 1,6 1.3
3.3 1.6 1.1
Table 3 Peak displacement in along-wind direction normal to the longer side of the CAARC building Mean wind speed
Reduced velocity
O(H)
U(H)
(m/s)
n°a
46
7.4
Peak tip displacement (m)
Predicted response (a)
Tanaka & Lawen 11 (b)
% Deviation (b) - (a) ---x 100% (b)
0.447
0.591
24.4
Peak displacement in along-wind direction normal to the shorter side of the CAARC building Mean wind speed
Reduced velocity
O(H) (m/s)
LI(H) nob
Predicted response (a)
Tanaka & Lawen 11 (b)
% Deviation (b) - (a) - x 100% (b)
46
5.0
0.291
0.247
17,8
Peak tip displacement (m)
intensity are 1.204 and 1.114 for aspect ratios 3.98 and 5.98 respectively. The frequency spacing, Atn, required to determine u is obtained by setting the dynamic magnification factor to be 75 % of the resonance peak, at 1%0 damping. With the starting value of tn i fixed at 0.025 Hz, the value of Ag for the determination of u is found to be 0.0176.
20
0.78 0.61
Eng. Struct. 1 9 8 9 , Vol. 1 1 , J a n u a r y
% Deviation (b) - (a) - × 100% (b)
Simiu 2
12.1 -0.0 18.2
3.2
9.4
1.5 1.1
6.7 18.2
(c)
% Deviation (c) - (a) ..... x 100% (c)
Tanaka and Lawen ~~ have made wind-tunnel measurements on a rectangular rigid model of 31 mm by 46 mm wide and 183 mm high with a generalized mass of 13.6 × 10 -3 kg. Thus the model simulates the CAARC building to a scale 1:1000. The mean velocity profile was found to follow a power law with a power index 0.28, while the measured longitudinal velocity spectrum of the turbulence component was in good agreement with the Von Karman spectrum. The dynamic response of the model was detected by the bending of the supported leaf spring and converted to the tip displacement of the building through static calibration. The measured root mean square displacements in the along-wind direction at two different orientations of the building are converted to peak values using the following peak factor given in ESDU: gD = (2 ln(n o To)~ + 0.577(2 loge(no To))- ~
(22)
where TO= 3600 s. At n o = 0.2 Hz, 9D is found to be 3.79. The peak displacements using this peak factor are presented in Table 3. The predicted responses by the proposed analytical model for the along-wind motion are also presented in Table 3. The results are obtained by using 19 harmonics (k = - 9 to +9); for each harmonic, the number of shape functions used for translational motion is 5. The computed peak displacement agrees well with the values measured by Tanaka and Lawen to within 17.8 %0 when the wind is normal to the shorter side of the building. When the longer side of the building is normal to the wind direction, the deviation between predicted and measured responses increases to 24.4%. This could be due to the peak factor used in converting the measured root mean square values to peak values. According to Tanaka and Lawen the peak factor at
Model for along-wind motion. T. Balendra et aL positive pressure was found to be in the range of 3.6 to 4.1.
3.0
Effects of damping and structural shapes
2.0
The effect of damping on the along-wind responses of the square building considered before is shown in Figures 4 and 5. It is seen that for any given reduced velocity, the peak displacement and peak acceleration at the top of the building decrease linearly with increase in damping ratio, on a log-log scale. The rate of decrease is found to be independent of the reduced velocity. The proposed analytical model is also used to study the effect of different structural shapes on the along-wind response. Three identical structures, but of different shapes, namely square, rectangle (breadth: width = 2:1), and equilateral triangle are chosen. The characteristic length, length of the side normal to wind, b, for each shape is determined on the basis of equal plan area. Other properties such as height, mass per unit height, fundamental frequency and damping are assumed to be the same as for the square structure considered before. The variation of normalized peak displacement at the top of the building with reduced velocity is shown in Figure 6. As seen from this figure, the displacement varies linearly with the reduced velocity on a log-log scale. Thus Pmax
1.0 : E
0.8
~o
0.6
-~
0.5
7 e-
o -~ 0.3 o.
. . . .
)
o 2 -
where k is the slope of the lines plotted in Figure 6. For
1
.
5
~
0.1
I
I
I
=
2
3
4
5
Damping ratio, ~ (~) 1.0
Figure 5
~" 0.8
Variation of peak acceleration with damping ratio
20I- . . . / a Triangle °
0
.
6
|
~
~
i
10~ 81-
"
.
.
.
ectan ,e
Square
"~ ~ •
;
1 0.2Q. Figure 6
6
l I 7 8 9 Reduced velocity, U(H)/n0b
I 10
I 11
Along-wind peak displacements for different shapes
a given reduced velocity, the triangular shape has the highest response, whereas the square shape has the least. 0.1
Figure 4
I
2
I
3
I
4
I
5
Damping ratio, ~ (~) Variation of peak displacement with damping ratio
Conclusions
The linearized deterministic model presented herein to predict the peak displacement and acceleration of slender
Eng. Struct. 1989, Vol. 11, January
21
Model for a l o n g - w i n d motion." T. Balendra et al. vertical structures due to a l o n g - w i n d m o t i o n s is f o u n d to give a c c u r a t e results. Thus, representing the gust by ' p s e u d o - t u r b u l e n c e ' seems to be reasonable. By using the p r o p o s e d model, the following conclusions are m a d e on the a l o n g - w i n d response of buildings. The p e a k a l o n g - w i n d d i s p l a c e m e n t a n d a c c e l e r a t i o n are found to decrease linearly with increasing d a m p i n g on a l o g a r i t h m i c scale, a n d the rate of decrease is i n d e p e n d e n t of the r e d u c e d velocity. W h e n three buildings of equal height a n d p l a n a r e a a n d differing only in their shape (square, rectangle, equilateral triangle) are c o m p a r e d for a l o n g - w i n d response, then for a given r e d u c e d velocity, a t r i a n g u l a r b u i l d i n g will have the highest response, whereas the square b u i l d i n g will have the least. The o r i e n t a t i o n of the structures is such that the characteristic length is n o r m a l to the d i r e c t i o n of wind.
References 1 Davenport, A. G. 'Gust loading factors' J Struct Div ASCE 1975, 97 9-38
22
Eng. Struct. 1989, Vol. 11, January
2 Simiu, E. 'Revised procedure for estimating along wind response' Struct Div ASCE 1980, 106, 1-10 3 Solari, G. 'Analytical estimation of the along wind response of structures' Wind En 9 Ind Aerodynam 1983, 14, 467478 4 Reinhold, T. A. and Sparks, P. R. 'The influence of wind direction on the response of a square section tall building' Proc Fifth Int Conf Wind Engng, Fort Collins, Colorado, 1979, 5 Melbourne, W. H. 'Comparison of measurements on the CAARC standard tall building model in simulated wind flows' J Wind Eng Ind Aerodynam 1980, 6, 73-88 6 Engineering Sciences Data Unit 'The response of flexiblestructures to atmospheric turbulence', ESDU Item no. 76001, Engineering Sciences Data Unit. London, UK, September 1976, pp 1 49 7 Patrickson, C. P. and Friedmann, P. P. 'Deterministic torsional building response to winds' J Struct Die ASCE 1979, liP3, 2621-2637 8 Balendra, T., Swaddiwuhipong, S., Quek S. T. and Lee S. L. 'Free vibration of asymmetric shear wall-frame buildings', Earthquake Eng Struct Dynam 1984, 12, 629-650 9 Reinhold, T. A. 'Measurements of simultaneous fluctuating loads at multiple levels on a model of tall building in a simulated urban boundary layer', Ph.D Thesis, Virginia Polytechnic Institute and State University, USA, 1977 10 Waller, R.A. Buildings on springs International Series of Monographs in Civil Engineering, Vol. 2, Pergamon Press, 1969 11 Tanaka, H. and Lawen, N. 'Test on the CAARC standard tall building model with a length scale of 1:1000', J Wind Engn9 Aerodynam 1986, 17, 15 29