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Measurements of wind-induced vibrations on a concrete chimney
, Journc, l o f Industrial Aerodynamics, I (1975/76) 239--247 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
MEASUREMENTS CHIMNEY
OF WIND-INDUCED
VIBRATIONS
239
ON A CONCRETE
F.P. MULLER and H. NIESER Institut ffir Beton und Stahlbeton, University o f Karlsruhe, Karlsruhe (W. Germany) (Received February 13, 1975)
Summary In the years 1971--1973, measurements of wind velocity and structure vibration were made on a concrete chimney of 180-m height in South Germany. All data were recorded continuously in an analogue form. From the results, the statistical properties of excitation and response, as well as the relationship between these two quantities, were determined. The report contains a short description of the problems that generally occur with measurements of this type and shows methods of solving them. Some of the results are given as mean values and functions of correlation and spectral density. They show reasonable agreement with the assumptions already known concerning the profile of mean wind velocity and gust spectra. The data of wind velocity at different levels were analysed to determine coherence and phase of cross spectral functions. The power spectra of structure response were calculated theoretically from the data of wind velocity and then compared with the experimentally derived spectra. The result of this study is that theory and experiment are in good agreement if cross correlation of wind velocity is taken into account.
1. I n t r o d u c t i o n In the last two years, measurements of wind velocity and structural vibrat i o n s w e r e c o n d u c t e d o n a c o n c r e t e c h i m n e y o f 1 8 0 - m h e i g h t in S o u t h e r n G e r m a n y . T h e b u i l d i n g is l o c a t e d in a c o m p a r a t i v e l y f l a t r e g i o n o f t h e u p p e r R h i n e V a l l e y w e s t o f t h e c i t y o f K a r l s r u h e . W i t h i n a r a d i u s o f s o m e 4 0 0 m it is s u r r o u n d e d b y r e f i n e r y s t r u c t u r e s u p t o 3 0 m in h e i g h t . T h e p r e v a i l i n g winds come from the south-west. T h e c h i m n e y d e s i g n is o f c i r c u l a r - r i n g c r o s s - s e c t i o n , t h e d i a m e t e r a n d w a l l thickness of which taper continuously from the base to the top. A sketch of t h e s t r u c t u r e is s h o w n in Fig. 1. T h e n a t u r a l f r e q u e n c i e s o f t h e c h i m n e y a r e as f o l l o w s : 1st mode: 2nd mode: 3rd mode:
no = 0 . 2 6 s -1 nl = 0.99 s -l n2 = 2 . 3 4 s -~
240
MASS
B~m] s[cm] ~+ 180_rnWD,WV,A L,,75 17,0
WO=WINODIR[CTION WV•WIND VELOCITY
y+130m WV
5,9g
17,0
A=A-4 CCELERATION~o S~SIR~SS LININGQ
,z_+80 m
_ -
7,Z,2 25,0-
j+35m WV
8,82 60,O
,__ -+0 $ ....
9,8~_ 60,_0_
~'~I /.._
STIFFN[SS DISTRIBUTION 9,Z~[ Mp/m] 6L, [ m~]
10,9
12,6
17,L, / ~
36,2
38,7
131.6
1&3,6
187,9
s j
Fig. 1. Dimensional sketch o£ the chimney and arrangement o£ the measuring points. Measurements of wind velocity were made at three different levels at 35 m, 130 m and 180 m above ground. These data were recorded continuously in analogue form as well as the acceleration at the top and strain at the base of the building (Fig. 1). 24 runs varying in length from 10 to 15 min were analysed by computer to estimate the mean value, variance, power spectral density and cross-correlation spectral density of each quantity measured. Analysis of records The instruments used for the measurements of wind velocity were threecup anemometers arranged circumferentially around the chimney in sets of 4 at each of the 3 levels. Because of the inertia of cup anemometers and the influence of the building on the airflow, the recorded data do not represent the true windspeed. Therefore a correction had to be applied to the data. The influence of the instrument's inertia was to a large extent eliminated by prior calibration in a wind tunnel at fluctuating velocities. In this way, a relationship was derived between the true velocity, v, on the one hand, and the recorded velocity, v*, and its differential dv*/dt on the other hand [ 1 ] v = f (v*, ~-TdV*] The corrected data were compared with that of a hot-wire anemometer. The comparison shows that for frequencies up to 1.5 s -1 the two were in good agreement. Wind tunnel studies were also conducted to determine the influence of the flow around the building on the wind velocity which demonstrated that the
241
flow can be described b y a potential equation in a definite zone in the approaching airstream (Fig. 2). At each level of the chimney, the instrument located in the most favourable position with regard to deviation from potential flow was recorded. Thus, the free-stream velocity could be calculated from the recorded data b y means of the potential equation. j~-~-
~ ----~.MEASUREDDATA>THEOREIICALDATA
........
V LUUl,, ,,oo
'L,/', /
Fig. 2. Difference in percent between theoretica| atld measured data of wind velocity around a cylinder.
Generally, accelerometers, as arranged at the t o p of the chimney, are not suitable to respond to slowly varying movements. The sensitivity of the instruments employed permitted the investigators to record all vibrations with frequencies above 0.6 times the fundamental frequency of the building and with amplitudes greater than 1 mm. For computation of the displacements of the chimney t o p from measured acceleration data, double integration yields large errors because, comparatively, this method is very sensitive. Therefore, a mechanical model was chosen to determine the displacement in digital form (Fig. 3) [1]. It should be noted that the data calculated in this manner are high-pass filtered. Therefore, the vibrations can only be judged in the range of the natural frequencies of the building. Experimental results From inspection of the dynamic components of displacement in and normal to the wind direction (Fig. 4), it can be seen that the vibrations normal
242
~ +rt)
~ +x(t)
~
I
r it:
,t)
-6 It-~l ITI.e .sinu !t-z', d~"
=
o
I
~V co: CIRCULARFREQUENCYOF[HI
IF~'=&,5 AND £ ~ 2,Gw,WE OBTAIN r(tj ~ 0,985xit).
SPRANG-MASS SYSTEM
£: 4:
CIRCULARFREQUENCYOF THE EXCITATIONFUNCTION LOGARITHMrEDAMPINGDECREMENT
ASSUMING x { t } r(t), THE ERRORWILL BE :~ 1,5°/o.
Fig. 3. Mechanical model for determining the deflections x ( t ) from acceleration data ~(t).
[cm] 12
AMPLTuo NTHEWNOORE N /
10 -
--
- AMPLITUDE NORMAL TO THE W I N D /
8 O-
@
L, 2 0
I
I
&
I
I
8
I
I
12
[
[
16
[
I
20
I
I
2/,
I
J
I
28
I
32
I
(180) 36 ~m/sec]
Fig. 4. Oscillation amplitude versus wind velocity.
to th e wind reach a m a x i m u m value at relatively slow velocities. This effect is due to resonant vor t e x shedding excitation. A calculation of the Strouhal n u m b e r yields S ~ 0.19. The lift response is of the same order of magnitude as the dynamic drag response, however statical displacements must be added to the dynamic response in the wind direction. Thus, the loading in the drag direction determines the stability of the structure in this case. Measurements of wind velocity at different levels confirm the pow er law o f the mean wind velocities at least at the measuring points (Fig. 5). Nevertheless, th e power law e x p o n e n t was found to have a value of 0.35 which was greater than the values generally assumed for such a t y p e of surface. The results o f the spectral density calculations which are shown by s m o o t h curves on Fig. 6, are in good agreement with the assumptions of Davenport [2]. The
243 z =180 m
+~ z =130 m
SYM•L
V(180) RUN[m/sec]
L • l o L + + l ° ~124
oi
\O
11~ 2o~7 115
71,3
118 19,0 [72 37,3 1 ~3 3],~ 31,1
z
ko,3~1
z = 35m \\o o
0
0,5
Vlz) V(18Oi
1,1]
Fig. 5. Profile of the mean wind velocity, n.SvvInl T
3
6v
1
I
/
/
4"
k
/
~.,.-~ \ \
/ "~ \\\
/
/ /
/,/ /-
O'2F L
~
/
.~_k\\
\~\ \\\\\
2B,2
3
23 130
29,7
~ +
x mo +L, ++
+i,1 ++,+
18,1
THEORETI[ALRELATIONSHIP
\\\~\
11 10-1
22 130
~ ~3 Js
~k~'~
/
1
AOCORDN I GTODAVENPORI
5
n
~ ' ~ + ) 100
101
10?
+ :,,oo+,,o+
+'+++,I+
103
Fig. 6. Power spectra of horizontal wind velocity.
u p p e r limiting f r e q u e n c y is d e t e r m i n e d b y the limited f r e q u e n c y response o f the cup-anemometers. As can be seen b y cross-correlation f u n c t i o n s the gusts reach various levels o f the building with d i f f e r e n t t i m e lags. This is s h o w n b y a shifting o f t h e
244 peaks relative to the origin r = 0 (Fig. 7). This difference of phases produces a favourable effect cn the strain distribution of the building and may be taken into account at the design stage. The recognition of this phenomenon
,/RI,(O)R~,,(O~ ~ ~
+~.7
/e\/;2
~,5
RUN22 SYMBOLz[m] z'[m] ! Vm[m/sec] @ + 130 180 r 30,25
I\
[01 : ~30 351~3,00
0,4L:'~
I®1 o 1130i3sl 25,05
o,1
0
"" f
f
~'
..,
-L,O
-30
-20
-10
10
20
30
L,O
50 +r[sec]
Fig. 7. Cross-correlation functions of wind velocity.
is expressed in the cross-spectrum, generally stated as a coherence function and a phase relationship. The results of wind velocity measurements are in good agreement with the mathematical functions given in Figs. 8 and 9. The functional phase relationship, generally not specified in previous publications, may be described clearly by a linear function as shown in Fig. 9.
RUN 22
C~oh ,~
z'[m; Vm[m/sec][ __ 1130 _20,25 4 130 35 23,00/ 180 35--,--25,05
SYMBOLz[ml
~_~ + 13o
1,or+__t
•
0,8 ~,~
o
[
0,/-.* 0,2
+++o~ [
0
ntz-z'l I
0,1
o o o • o
n o
0,2
~
-e+ o+
0,3
~/m= ,~[V{z)*V(z':;]
++ oo • o
0,/.,
0,5
•
o
0,6
o
0,7
•
•
o
•
o
l • I"° O,B 0,9
°t _ n ]z-z'] ~m 1,0
Fig. 8. Coherence-function for two components of wind velocity.
245
RUN 22 SYMBOLI z [ml z'[mil Vm [m,,'sed]
+ '130 180! 30,25
2
18035 I
I
-0,~
I
-0,3
-0,2
I
I
~, 0,1
-0,1
I
0,2
I
I
0,3
0,~
"
25,05 I Z-Z'
n
)~
-11 :z-z'l~ I ~
u,"
_~7!z;
-3 o
Fig. 9. Relationship of phase between two components of wind velocity. Comparison with theoretically determined data The cross-spectrum Spp, of the wind forces was calculated from the experimentally derived cross-spectrum of the wind velocity, taking into account the relation between wind forces and wind velocity, according to McNown and Wolf [3]
p(t)
=
1 V(t) 2 ~pC D - - -2- +
pC M
Ao dV(t) d dt
Thus, the power spectrum of the displacements at the top of the chimney could be calculated from the fundamental relation as follows [ 4] h h 0
0
It should be noted that for the purpose of calculation a logarithmic damping decrement 0 = 0.045 was assumed. Power spectra of the displacements were computed with different values of the virtual mass coefficient cM. It may be seen that the influence of c M is exerted only for large values of the non-dimensional frequency ratio h n / V (h = total height of the building, V = mean wind velocity). In this frequency range the spectral densities are extremely small. Consequently, the value of CM may be neglected for further computations. The power spectra of the displacements, theoretically determined from the data of wind velocity measurements, were compared with the experimentally derived spectra. Theoretical and experimental results agreed in so far as only
246 vibrations of the fundamental mode are significant and that all higher modes are negligible. Hence, it may be assumed that the shape of the structural deflection at any time was identical with the shape of the first mode. Ttmrefore, a well-defined relationship exists between the displacements at the top and the bending moments at the base as well as between the spectral densities of these two quantities. Thus, the power spectra of the displacements could be computed from that of the base bending moments. This assumption cannot be automatically transferred to other systems. It holds true in the case of buildings, the section and stiffness of which are relatively constant over the entire height, or at least not changing abruptly. The power spectra of the displacements at the top of the chimney determined by different methods for a specific measurement are given in Fig. 10.
n.S. [crnZ] lO
'
:
z,OI,8
- -
FROMEASUREMEO NTFS 101.' . . . . . ri. . ~ ~.:_ ~ ' I ' !~ A EM LER A IIN H IR EG IEH O IE FBID OnTrF sT ~ . ~ ~ ~'|~ FC RC O EO A SA U M N SRESSATTHEBASE 100.-, "~.~ THEORETC IALD LY ETERMN IED |
i
10-1 /
,
-
'
I
[ } '
;~ ~"~
"!]il ..... ,~/~" ../L
:
~ ~,.-~,
/"
FUNCTI / IIONI
]ii~ Ji ~ RUN24 1o-026 ~]]! l[ I Vi(IBO] -31,1 [rnlsec]
10"210-3 2 10- Z 5 7 10" 2 n [l/sec ] Fig. 10. Power s )ectra of displacements at a height of 180 m. The results of the other measurements show the same tendency. The differences between Curve (2) and Curve (3) in the low-frequency range are due to the high-pass filtering action of the time-dependent acceleration function. Over the entire frequency range, the theoretically derived curve lies above the experimentally determined curves. Among others, the reason for this variation may be the assumption of fully-correlated wind forces in the horizontal direction. The probability distribution of the response shows a fairly good agreement with the normal distribution (Fig. 11). This confirms that the response of a structure to wind may be treated as a stationary Gaussian process, defined by the values of the mean and root-mean-square responses. The largest value recorded was about 3.5 times the standard deviation, ax.
247
o
p{×l ox - -0,5 o
.L o
NORMALDISTRIBUTION..... ~
o~f+
•
V~180)
0,~°o
SYMBOL RUN [m,"secj I
o~'. &
"
"+
*
22
+
2" J
"
32,3 33,& 31J
°
Fig. 11. Probability distribution of the displacements at the top. By means o f stochastic calculations considering the relationship and parameters derived f r om measurement data, gust loading factors were determined as a f u n c t i o n of the natural frequencies, of the structural height and of mean wind velocity {Fig. 12). The results are compared with t hat of a deterministic m e t h o d [5], which are shown by the br oke n line on Fig, 12. There are great differences, particularly in the low-frequency range. To confirm these values, further e x p e r i m e n t a t i o n and measurements are necessary.
) 1
~c0nst ---4maxx
__
/,,0
~KS!HLAICH IDETERMINIS~ICMEIHODI ,~,~_~~ ,/h=60m
. . . . . .
0
h
[-\ h-l!Oml
i i [ , , ; ' , l
+
!
i
.
_
. . . .
0,5 1,0 1,5 no [I/sec] Fig. 12. Oust loading factors for V(10) = 20 [m/s].
248
References 1 H. Nieser, Schwingungsberechnung turmartiger Bauwerke bei Belastung durch biSigen Wind, Dissertation, Universit~it Karlsruhe, 1974. 2 A.G. Davenport, The spectrum of horizontal gustiness near the ground in high winds, Q. J. R. Meteorol. Soc., 87 (1961) 194--211. 3 J.S. McNown and L.W. Wolf, Resistance to unsteady flow: I. Analysis of tests with flat plate, Engineering Research Institute, Univ. of Michigan 2446--I.P., Internal report to Sandia Corp., June 1956. 4 I.D. Robson, An Introduction to Random Vibration, Edinburgh University Press, 1964. 5 J. Schlaich, Beitrag zur Frage der Wirkung yon Windsti~ssen auf Bauwerke, Bauingenieur, 41 (1966) 102--106.
MEASUREMENTS CHIMNEY
OF WIND-INDUCED
VIBRATIONS
239
ON A CONCRETE
F.P. MULLER and H. NIESER Institut ffir Beton und Stahlbeton, University o f Karlsruhe, Karlsruhe (W. Germany) (Received February 13, 1975)
Summary In the years 1971--1973, measurements of wind velocity and structure vibration were made on a concrete chimney of 180-m height in South Germany. All data were recorded continuously in an analogue form. From the results, the statistical properties of excitation and response, as well as the relationship between these two quantities, were determined. The report contains a short description of the problems that generally occur with measurements of this type and shows methods of solving them. Some of the results are given as mean values and functions of correlation and spectral density. They show reasonable agreement with the assumptions already known concerning the profile of mean wind velocity and gust spectra. The data of wind velocity at different levels were analysed to determine coherence and phase of cross spectral functions. The power spectra of structure response were calculated theoretically from the data of wind velocity and then compared with the experimentally derived spectra. The result of this study is that theory and experiment are in good agreement if cross correlation of wind velocity is taken into account.
1. I n t r o d u c t i o n In the last two years, measurements of wind velocity and structural vibrat i o n s w e r e c o n d u c t e d o n a c o n c r e t e c h i m n e y o f 1 8 0 - m h e i g h t in S o u t h e r n G e r m a n y . T h e b u i l d i n g is l o c a t e d in a c o m p a r a t i v e l y f l a t r e g i o n o f t h e u p p e r R h i n e V a l l e y w e s t o f t h e c i t y o f K a r l s r u h e . W i t h i n a r a d i u s o f s o m e 4 0 0 m it is s u r r o u n d e d b y r e f i n e r y s t r u c t u r e s u p t o 3 0 m in h e i g h t . T h e p r e v a i l i n g winds come from the south-west. T h e c h i m n e y d e s i g n is o f c i r c u l a r - r i n g c r o s s - s e c t i o n , t h e d i a m e t e r a n d w a l l thickness of which taper continuously from the base to the top. A sketch of t h e s t r u c t u r e is s h o w n in Fig. 1. T h e n a t u r a l f r e q u e n c i e s o f t h e c h i m n e y a r e as f o l l o w s : 1st mode: 2nd mode: 3rd mode:
no = 0 . 2 6 s -1 nl = 0.99 s -l n2 = 2 . 3 4 s -~
240
MASS
B~m] s[cm] ~+ 180_rnWD,WV,A L,,75 17,0
WO=WINODIR[CTION WV•WIND VELOCITY
y+130m WV
5,9g
17,0
A=A-4 CCELERATION~o S~SIR~SS LININGQ
,z_+80 m
_ -
7,Z,2 25,0-
j+35m WV
8,82 60,O
,__ -+0 $ ....
9,8~_ 60,_0_
~'~I /.._
STIFFN[SS DISTRIBUTION 9,Z~[ Mp/m] 6L, [ m~]
10,9
12,6
17,L, / ~
36,2
38,7
131.6
1&3,6
187,9
s j
Fig. 1. Dimensional sketch o£ the chimney and arrangement o£ the measuring points. Measurements of wind velocity were made at three different levels at 35 m, 130 m and 180 m above ground. These data were recorded continuously in analogue form as well as the acceleration at the top and strain at the base of the building (Fig. 1). 24 runs varying in length from 10 to 15 min were analysed by computer to estimate the mean value, variance, power spectral density and cross-correlation spectral density of each quantity measured. Analysis of records The instruments used for the measurements of wind velocity were threecup anemometers arranged circumferentially around the chimney in sets of 4 at each of the 3 levels. Because of the inertia of cup anemometers and the influence of the building on the airflow, the recorded data do not represent the true windspeed. Therefore a correction had to be applied to the data. The influence of the instrument's inertia was to a large extent eliminated by prior calibration in a wind tunnel at fluctuating velocities. In this way, a relationship was derived between the true velocity, v, on the one hand, and the recorded velocity, v*, and its differential dv*/dt on the other hand [ 1 ] v = f (v*, ~-TdV*] The corrected data were compared with that of a hot-wire anemometer. The comparison shows that for frequencies up to 1.5 s -1 the two were in good agreement. Wind tunnel studies were also conducted to determine the influence of the flow around the building on the wind velocity which demonstrated that the
241
flow can be described b y a potential equation in a definite zone in the approaching airstream (Fig. 2). At each level of the chimney, the instrument located in the most favourable position with regard to deviation from potential flow was recorded. Thus, the free-stream velocity could be calculated from the recorded data b y means of the potential equation. j~-~-
~ ----~.MEASUREDDATA>THEOREIICALDATA
........
V LUUl,, ,,oo
'L,/', /
Fig. 2. Difference in percent between theoretica| atld measured data of wind velocity around a cylinder.
Generally, accelerometers, as arranged at the t o p of the chimney, are not suitable to respond to slowly varying movements. The sensitivity of the instruments employed permitted the investigators to record all vibrations with frequencies above 0.6 times the fundamental frequency of the building and with amplitudes greater than 1 mm. For computation of the displacements of the chimney t o p from measured acceleration data, double integration yields large errors because, comparatively, this method is very sensitive. Therefore, a mechanical model was chosen to determine the displacement in digital form (Fig. 3) [1]. It should be noted that the data calculated in this manner are high-pass filtered. Therefore, the vibrations can only be judged in the range of the natural frequencies of the building. Experimental results From inspection of the dynamic components of displacement in and normal to the wind direction (Fig. 4), it can be seen that the vibrations normal
242
~ +rt)
~ +x(t)
~
I
r it:
,t)
-6 It-~l ITI.e .sinu !t-z', d~"
=
o
I
~V co: CIRCULARFREQUENCYOF[HI
IF~'=&,5 AND £ ~ 2,Gw,WE OBTAIN r(tj ~ 0,985xit).
SPRANG-MASS SYSTEM
£: 4:
CIRCULARFREQUENCYOF THE EXCITATIONFUNCTION LOGARITHMrEDAMPINGDECREMENT
ASSUMING x { t } r(t), THE ERRORWILL BE :~ 1,5°/o.
Fig. 3. Mechanical model for determining the deflections x ( t ) from acceleration data ~(t).
[cm] 12
AMPLTuo NTHEWNOORE N /
10 -
--
- AMPLITUDE NORMAL TO THE W I N D /
8 O-
@
L, 2 0
I
I
&
I
I
8
I
I
12
[
[
16
[
I
20
I
I
2/,
I
J
I
28
I
32
I
(180) 36 ~m/sec]
Fig. 4. Oscillation amplitude versus wind velocity.
to th e wind reach a m a x i m u m value at relatively slow velocities. This effect is due to resonant vor t e x shedding excitation. A calculation of the Strouhal n u m b e r yields S ~ 0.19. The lift response is of the same order of magnitude as the dynamic drag response, however statical displacements must be added to the dynamic response in the wind direction. Thus, the loading in the drag direction determines the stability of the structure in this case. Measurements of wind velocity at different levels confirm the pow er law o f the mean wind velocities at least at the measuring points (Fig. 5). Nevertheless, th e power law e x p o n e n t was found to have a value of 0.35 which was greater than the values generally assumed for such a t y p e of surface. The results o f the spectral density calculations which are shown by s m o o t h curves on Fig. 6, are in good agreement with the assumptions of Davenport [2]. The
243 z =180 m
+~ z =130 m
SYM•L
V(180) RUN[m/sec]
L • l o L + + l ° ~124
oi
\O
11~ 2o~7 115
71,3
118 19,0 [72 37,3 1 ~3 3],~ 31,1
z
ko,3~1
z = 35m \\o o
0
0,5
Vlz) V(18Oi
1,1]
Fig. 5. Profile of the mean wind velocity, n.SvvInl T
3
6v
1
I
/
/
4"
k
/
~.,.-~ \ \
/ "~ \\\
/
/ /
/,/ /-
O'2F L
~
/
.~_k\\
\~\ \\\\\
2B,2
3
23 130
29,7
~ +
x mo +L, ++
+i,1 ++,+
18,1
THEORETI[ALRELATIONSHIP
\\\~\
11 10-1
22 130
~ ~3 Js
~k~'~
/
1
AOCORDN I GTODAVENPORI
5
n
~ ' ~ + ) 100
101
10?
+ :,,oo+,,o+
+'+++,I+
103
Fig. 6. Power spectra of horizontal wind velocity.
u p p e r limiting f r e q u e n c y is d e t e r m i n e d b y the limited f r e q u e n c y response o f the cup-anemometers. As can be seen b y cross-correlation f u n c t i o n s the gusts reach various levels o f the building with d i f f e r e n t t i m e lags. This is s h o w n b y a shifting o f t h e
244 peaks relative to the origin r = 0 (Fig. 7). This difference of phases produces a favourable effect cn the strain distribution of the building and may be taken into account at the design stage. The recognition of this phenomenon
,/RI,(O)R~,,(O~ ~ ~
+~.7
/e\/;2
~,5
RUN22 SYMBOLz[m] z'[m] ! Vm[m/sec] @ + 130 180 r 30,25
I\
[01 : ~30 351~3,00
0,4L:'~
I®1 o 1130i3sl 25,05
o,1
0
"" f
f
~'
..,
-L,O
-30
-20
-10
10
20
30
L,O
50 +r[sec]
Fig. 7. Cross-correlation functions of wind velocity.
is expressed in the cross-spectrum, generally stated as a coherence function and a phase relationship. The results of wind velocity measurements are in good agreement with the mathematical functions given in Figs. 8 and 9. The functional phase relationship, generally not specified in previous publications, may be described clearly by a linear function as shown in Fig. 9.
RUN 22
C~oh ,~
z'[m; Vm[m/sec][ __ 1130 _20,25 4 130 35 23,00/ 180 35--,--25,05
SYMBOLz[ml
~_~ + 13o
1,or+__t
•
0,8 ~,~
o
[
0,/-.* 0,2
+++o~ [
0
ntz-z'l I
0,1
o o o • o
n o
0,2
~
-e+ o+
0,3
~/m= ,~[V{z)*V(z':;]
++ oo • o
0,/.,
0,5
•
o
0,6
o
0,7
•
•
o
•
o
l • I"° O,B 0,9
°t _ n ]z-z'] ~m 1,0
Fig. 8. Coherence-function for two components of wind velocity.
245
RUN 22 SYMBOLI z [ml z'[mil Vm [m,,'sed]
+ '130 180! 30,25
2
18035 I
I
-0,~
I
-0,3
-0,2
I
I
~, 0,1
-0,1
I
0,2
I
I
0,3
0,~
"
25,05 I Z-Z'
n
)~
-11 :z-z'l~ I ~
u,"
_~7!z;
-3 o
Fig. 9. Relationship of phase between two components of wind velocity. Comparison with theoretically determined data The cross-spectrum Spp, of the wind forces was calculated from the experimentally derived cross-spectrum of the wind velocity, taking into account the relation between wind forces and wind velocity, according to McNown and Wolf [3]
p(t)
=
1 V(t) 2 ~pC D - - -2- +
pC M
Ao dV(t) d dt
Thus, the power spectrum of the displacements at the top of the chimney could be calculated from the fundamental relation as follows [ 4] h h 0
0
It should be noted that for the purpose of calculation a logarithmic damping decrement 0 = 0.045 was assumed. Power spectra of the displacements were computed with different values of the virtual mass coefficient cM. It may be seen that the influence of c M is exerted only for large values of the non-dimensional frequency ratio h n / V (h = total height of the building, V = mean wind velocity). In this frequency range the spectral densities are extremely small. Consequently, the value of CM may be neglected for further computations. The power spectra of the displacements, theoretically determined from the data of wind velocity measurements, were compared with the experimentally derived spectra. Theoretical and experimental results agreed in so far as only
246 vibrations of the fundamental mode are significant and that all higher modes are negligible. Hence, it may be assumed that the shape of the structural deflection at any time was identical with the shape of the first mode. Ttmrefore, a well-defined relationship exists between the displacements at the top and the bending moments at the base as well as between the spectral densities of these two quantities. Thus, the power spectra of the displacements could be computed from that of the base bending moments. This assumption cannot be automatically transferred to other systems. It holds true in the case of buildings, the section and stiffness of which are relatively constant over the entire height, or at least not changing abruptly. The power spectra of the displacements at the top of the chimney determined by different methods for a specific measurement are given in Fig. 10.
n.S. [crnZ] lO
'
:
z,OI,8
- -
FROMEASUREMEO NTFS 101.' . . . . . ri. . ~ ~.:_ ~ ' I ' !~ A EM LER A IIN H IR EG IEH O IE FBID OnTrF sT ~ . ~ ~ ~'|~ FC RC O EO A SA U M N SRESSATTHEBASE 100.-, "~.~ THEORETC IALD LY ETERMN IED |
i
10-1 /
,
-
'
I
[ } '
;~ ~"~
"!]il ..... ,~/~" ../L
:
~ ~,.-~,
/"
FUNCTI / IIONI
]ii~ Ji ~ RUN24 1o-026 ~]]! l[ I Vi(IBO] -31,1 [rnlsec]
10"210-3 2 10- Z 5 7 10" 2 n [l/sec ] Fig. 10. Power s )ectra of displacements at a height of 180 m. The results of the other measurements show the same tendency. The differences between Curve (2) and Curve (3) in the low-frequency range are due to the high-pass filtering action of the time-dependent acceleration function. Over the entire frequency range, the theoretically derived curve lies above the experimentally determined curves. Among others, the reason for this variation may be the assumption of fully-correlated wind forces in the horizontal direction. The probability distribution of the response shows a fairly good agreement with the normal distribution (Fig. 11). This confirms that the response of a structure to wind may be treated as a stationary Gaussian process, defined by the values of the mean and root-mean-square responses. The largest value recorded was about 3.5 times the standard deviation, ax.
247
o
p{×l ox - -0,5 o
.L o
NORMALDISTRIBUTION..... ~
o~f+
•
V~180)
0,~°o
SYMBOL RUN [m,"secj I
o~'. &
"
"+
*
22
+
2" J
"
32,3 33,& 31J
°
Fig. 11. Probability distribution of the displacements at the top. By means o f stochastic calculations considering the relationship and parameters derived f r om measurement data, gust loading factors were determined as a f u n c t i o n of the natural frequencies, of the structural height and of mean wind velocity {Fig. 12). The results are compared with t hat of a deterministic m e t h o d [5], which are shown by the br oke n line on Fig, 12. There are great differences, particularly in the low-frequency range. To confirm these values, further e x p e r i m e n t a t i o n and measurements are necessary.
) 1
~c0nst ---4maxx
__
/,,0
~KS!HLAICH IDETERMINIS~ICMEIHODI ,~,~_~~ ,/h=60m
. . . . . .
0
h
[-\ h-l!Oml
i i [ , , ; ' , l
+
!
i
.
_
. . . .
0,5 1,0 1,5 no [I/sec] Fig. 12. Oust loading factors for V(10) = 20 [m/s].
248
References 1 H. Nieser, Schwingungsberechnung turmartiger Bauwerke bei Belastung durch biSigen Wind, Dissertation, Universit~it Karlsruhe, 1974. 2 A.G. Davenport, The spectrum of horizontal gustiness near the ground in high winds, Q. J. R. Meteorol. Soc., 87 (1961) 194--211. 3 J.S. McNown and L.W. Wolf, Resistance to unsteady flow: I. Analysis of tests with flat plate, Engineering Research Institute, Univ. of Michigan 2446--I.P., Internal report to Sandia Corp., June 1956. 4 I.D. Robson, An Introduction to Random Vibration, Edinburgh University Press, 1964. 5 J. Schlaich, Beitrag zur Frage der Wirkung yon Windsti~ssen auf Bauwerke, Bauingenieur, 41 (1966) 102--106.