- Email: [email protected]
Accelerations and comfort criteria for buildings undergoing complex motions
Journal of Wind Engineering and Industrial Aerodynamics, 41-44 (1992) 105-116 Elsevier
105
A c c e l e r a t i o n s a n d c o m f o r t c r i t e r i a for b u i l d i n g s u n d e r g o i n g c o m p l e x motions W. H. Melbourne and T. R. Palmer Department of Mechanical Engineering, Monash University, Wellington Road, Clayton, Victoria 3168, Australia
Abstract Acceleration criteria to achieve acceptable occupancy comfort in buildings have been developed in terms of peak accelerations as a function of motion frequency and return period. The derivation ofaccelerations felt ~y an occupant in a building undergoing complex motions with significant contributions from several modes is discussed. The way in which component accelerations at a point resolve into a record which is non-ergodic is illustrated with model measurements.
1. I N T R O D U C T I O N Acceleration criteria to achieve occupancy comfort in tall buildings has received somewhat varied attention over the past twenty years. The pioneering work of Chen and Robertson (1973) gave valuable information about human perception to sinusoidal excitation as a function offrequency, and they suggested using an "occupancy sensitivity quotient" which defines the ratio of tolerable ampfitudes ofmotions to tile threshold motion for halfthe population. ~ l e work by Reed (1971) gave the first full scale evaluation of occupants' response to accelerations on two buildings and gave the first criteria in term a of standard deviation of acceleration for a return period for the frequencies ofthose buildings. Irwin, in a series ofpapers, further studied the response ofh-mans to sinusoidal acceleration over. a range of frequencies and through. . a number of unpublished. . full scale studies of crane operators and braiding occupants was primarily responsible for the standard deviation acceleration criteria in ISO 6897, and he focused these on tall buildings in Irwin (1986). In North America some use appears to be made of an unreferenced peak acceleration criterion of 20 milli-g once in 10 years with no reference to frequency. Melbourne (1980) using the earlier work of Chen and Robertson, Reed and some full scale observations.developed criteria for tall .buildings undergoing normally distributed oscdlatmns, as d.istinct from the smusoidal oscillations used in the human perceptmn expertments. This development related the accelerations of the normally distributed and sinusoidal motions on the basis that it was the peak accelerations which were the most important. Chen and Robertson had suggested that even the third derivative of displacement (jerk as they termed it) could be significant. The criterion from this work was that structures should be designed such that peak horizontal accelerations do not 0167-6105/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
106 exceed 10 milli-g per year on average for frequencies in the range 0.2 to 0.3 Hz, with the implication that motions perceptible by most people would occur during one to two storms per annum. Later Melbourne and Cheung (1988) commenced the development of frequency dependent criteria for peak accelerations to use for buildings undergoing complex motions and for different distributions. This paper will complete the development of the peak acceleration criteria for occupancy comfort of tall buildings and then go on to show how accelerations from bui.~,mgs undergoing complex motions of sway and rotation in the lower modes and accelerations from combined modes can be determined and related to the peak acceleration criteria. 2. DEVELOPMENT OF PEAK ACCELERATION C R I T E R I A The development of acceleration criteria referenced to peak accelerations is based on the conclusions of Chen and Robertson (1973) that it is the second or even third derivative of displacement which is most relevant to human comtbrt, or rather discomfort. Criteria based on the standard deviation of acceleration ignore the probability distribution ofthe peak accelerations which varies greatly between a sine wave and a normally distributed process and very significantly, for example, between the cross-wind response of a building operating near the peak of the cross-wind force spectrum and the along-wind response. Hence the nee d to base acceleration criteria for occupancy comfort in tall buildings on peak acceleration rather than the standard deviation of acceleration. Horizontal acceleration criteria in terma of standard deviation of acceleration for the worst 10 consecutive minutes in a 5 year return period for buildings as a function of frequency have been given by Irwin's E2 Curve and Curve 1, Figure I in ISO 6897. This curve can be fitted by the equation o~ = exp(-3.65
-
0.411an)
(1)
where o~ is the standard deviation of acceleration in the horizontal plane n is the frequency of oscillation with an approximately normal distribution. This curve is not so much described as criteria per se by Irwin, but ISO 6897 describes it as a satisfactory magnitude and the level at which about 2% of the occupants will comment adversely, and Irwin notes that "in general the criterion for satisfactory magnitudes of vibration in each of a number of categories (which he describes) is based on a minimum adverse comment level of 2% of the opulation involved, qlds definition of acceptability was also suggested earlier Reed. It has been shown (Melbourne 1988) that Reed's standard dewation criterion of 5 milli-g for a 6 year return period, Melbourne's one year return period peak acceleration of 10 milli-g and'the North American 10 year return period peak acceleration of 20 milli-g in the frequency range 0.2 to 0.3 Hz are all in reasonable agreement with Equation 1. Hence it was concluded that with this degree of agreement it was reasonable to use Equation 1 as a base for developing frequency dependent peak acceleration criteria for horizontal motion in buildings for occupancy comfort. ~lle peak acceleration has been obtained from the standard deviation expression, Equation 1, on the assumption that it relates to a normally distributed process, using
107 A
x
=
ga~
(2)
where g is tile p e a k factor = ~]~ In nT for a n o r m a l l y distributed process (Melbourne 1977) n is tile frequency T is tile d u r a h o n in seconds (i.e. for 10 m ln T = 6 0 0 sec) wlfich gives for a 5 y e a r return period
-
100
5O
I
"
'
J
H e l b o u r n e ' s ( 1 9 8 8 ) maximum peak h o r i z o n t a l a c c e l e r a t i o n c r i t e r i a w i t h r e f e r e n c e t o I r w i n ( 1 9 8 6 ) , Reed ( 1 9 7 1 ) , Chen and R o b e r t s o n ( 1 9 7 3 ) and H e l b o u r n e ( 1 9 8 0 ) f o r T = 600 s e c o n d s , and r e t u r n p e r i o d R y e a r s . •~,
£n
x ffi 42£n noT ( 0 . 6 8 + e - ~ )
bO !
(3)
~21n nTexp(-3.65-O.411n n)
R
exp
for 0.06
(-3.65
- 0.4I £n no)
0.5
20 return period 0 "14 4J
g
.~ 10 y e a r
--
10
5
al u u
.-j U
0
$ year exp
0.2
Im
fl
It
0.05
•
It ~
I
,
0.5 I
II
( - 3 . 5 5 - 0 . 4 I tn n o )
I
!
*
1.0
O. 1 frequency,
FIG.1
0.5
~
IrwinVs E2 Curve and ISO 6897 ( 1 9 8 4 ) Curve 1, standard deviation hort zontal acceleration criteria f o r 10 m i n u t e s in 5 year return period for a building (i.e. approximately normally distributed response).
n o . Hz
Horizontal acceleration criteria for occupancy comfort in buildings
108 It must be noted t h a t to evaluate a peak acceleration from the earlier Chen and Robertson, and Irwin data based on experiments with sinusoidal motion g = ~ . To genervfise this acceleration criteria for other return periods it is necessary to chose relationships for wind speed vs return period and response vs wind speed. Fgr the former a Gmnbel line fit to daily data typical of t h a t for areas in which extremes are generated both by thunderstorm and extensive pressure systems has been used as follows: 12 =
30+31nR
(4)
where R is return period in years. For the latter a response based on t~2~ has been used as being a compromise between alongwind and crosswind respot.se. A fit to the factor generated by these assumptions is as follows: response for return period R years response for return period 5 years
=
0.68+
InR 5
(5)
Combining these gives the peak acceleration criteria for occupancy comfort in buildings of
Plots ofthe acceleration criteria are given as a function of frequency in Figure I ibr a period of 10 minutes of maximu m wind in a return period of R years. The period of 10 minutes has bee.n used both to fit in with the original curve of Irwin and .ISO 6897 and because it m typical of a period ofmaximum response in areas don~nated by thunderstorm activity, and where mean design wind speeds tend to be backworked artificially from peak wind speed data. For regions where maximum response may occur through longer periods, such as an hour, tile maximum hourly mean wind spced will be less than the maximum 10 minute mean wind speed, and the value of T in Equation 7 would increase ',~ 3600 seconds. 3. A C C E L E R A T I O N S F R O M C O M P L E X MOTIONS Motion which is significant to occupancy comfort in buildings can be identified, for analytical convenience, as arising from several sources one of which may be dominant or several of which may combine to give the resultant mohon felt by the occupant. These sources are identified as follows: (i)
(ii) (iii)
motion in the first two bending(or sway) modes about orthogonal axes (usually for the worst case these can be conveniently divided into along-wind and cross-wind motions), rotational motion about a vertical axis, often either a 2nd or 3rd mode but can be higher, and for buildings with stiffness and mass asymmetry, complex bending and rotational motion in one or both of the lower modes, a n d any one of these motions may not be normally distributed.
109 When the highest acceleration felt by an occupant is dominated by one of these motions the problem of evaluating either standard deviation or peak acceleration from analytical or wind tunnel studies in relatively straight forward. The problem arises when two or more motions may make a significant contribution and to determine how they combine to affect an occupant if there are significant differences in frequency of the contributing motions. The simplest way to combine the accelerations from two motions, such as resulting from the first two bending modes, is to take the square root of the sum of the squares, i.e.
It is easy to see the effect of one of these motions becoming dominant by noting that if one standard deviation acceleration is twice the other then the smaller contributes only 10% to the resultant acceleration. The difficulties which arise in determining accelerations to compare with the criteria may be summarised as follows:
(i) (ii) (rid
when the peak accelerations of a motion are not normally distributed, when the accelerations result from a complex mode, and when the acceleration (standard deviation) from one mode of motion is not at least twice any others.
3.1. Motions for w h i c h p e a k s a r e n o t n o r m a l l y d i s t r i b u t e d The probability distributions ofpeaks ofbuilding motions has been discussed in some detail by Melbourne (1977). In summary it can be concluded that response primarily due to along-wind excitation or to cross-wind excitation well away from the peak of the cross-wind force spectrnm can be regarded as having peaks which are approximately normally distributed. It is response due to cross-wind excitation occurring near the peak of the cross-wind fore spectrum for which the dmtribution of peaks departs significantly from being normally distributed and due to dependence between the motion and the excitation process (vortex shedding) results in peak factors considerably lower than for a normally dmtnbuted process. For sharp-edged buildings the peak of the cross-wind force spectrum occurs at a Reduced Velocity of approximately 10 (F, = v_~where is mean wind speed at the top of the building, n is frequency ofcross-wind mode, and b is width normal to the wind directlo.n). Any cross-wind, or rotahon about the vertical axis, resl)onse overating wtthm 20% of the Reduced Velocity at the ))eak ofthe cross-wincl force spectrnm could expect to have a distribution ofpeaks ~vhich deviate si2nificantlv l~om a normally distributed process. UnfortunateTy, for tall buildings, the maximum accelerations tend to come from the cross-wind response and they are most likely to exceed the comfort criteria when operating at Reduced Velocities n e ~ the peak of th e cross-w_ind force spectr-m. Hence in many, ifnot all, crittcal caf~es, it ts essential to establish the actual value of the peak factor from wind tunnel measurements on a full or simplified linear mode aeroelastic model. This is a situation where use of standard deviation acceleration data or data from a force balance test assuming a normally distributed response would result, in a significant overestimate of
110 tl~e acceleration effect on occupancy comfort which may result in the requirement to add a damping system which may not be necessary. The best way to illustrate this is by example. Evaluation from data obtained from a linear mode aeroelastic model wind tunnel test of a 240m tall, relatively slender, buildingpredicted a standard deviation acceleration response for a 5 year r e t u r n period of o5 =
millig
6.6
(8)
This response was dominated by a cross-wind motion in the first mode with a full scale frequency of 0.15 Hz. Retbrence to the standard deviation criteria in Figure I shows that this acceleration would fall significantly above the recommended criterion of 5.7 milli-g. The peak factor for 10 minutes of operation for the peaks normally distributed would be approximately g
=
~/21n 600
=
3.0
•
0.15 (9)
giving a peak acceleration of y
=
3.0
•
6.6 =
19.8
miili-g
(10)
similarly falling outside the criterion of 17 milli-g. However the operating condition was close to the peak of the cross-wind force spectrum and an upcrossing analysis to give the probability distribution of the peaks, showed that the peak factor for 10 minutes of operation was approximately 2.3. This gave a peak acceleration of y
=
2,5
.
6.6 =
16,5
milli-g
(11)
and the conclusion that the building would satisfactorily meet occupancy comfort criteria. 3.2. M o t i o n s r e s u l t i n g f r o m c o m p l e x m o d e s Buildings with. asymmetry between the effective shear centre in rotation about a vertical axm and tile mass d i s t n b u h o n about such an axis have complex mode shapes and typically either the first or second mode may combine a h o r i z o n t a l displacement and a significant rotation about a vertical axis. Fortunately a numbe~ of modal analysispackages cope with this three-dimensional behaviour and describe the mode shapes in terms of a horizontal displacement from, and rotation about, a defined vertical axis, and there is a simple linear relationship between these two components. For the case of pure bending or sway motion the horizontal displacement at tile top of a building may be related to the base bending moment through tile inertial moment caused by the integration of maximum mass tlmes acceleration components up tile building. Hence for a g~ven peak of the fluctuating base moment under wind action tile peak horizontal displacement and hence acceleration at any level on the building can be obtained. For the situation where there is a complex mode tile same approach is used to obtain the peak acceleration due to horizontal displacement at a point on the defined vertical axis. If it is then required to obtain the horizontal acceleration at some point
111 away from this axis, such as in a room at the corner ofthe building, the horL-.ontal acceleration due to rotation for a given displacement at the defined vertical axis must be added vectorally to the horizontal displacement at the axis to give the resultant acceleration experienced at the corner of the building. 3.3. M o t i o n s r e s u l t i n g f r o m s i m i l a r c o n t r i b u t i o n s f r o m t w o o r m o r e modes To investigate this phenomenon directly several aeroelastic model studies of buildings were conducted in which horizontal accelerations were measured directly in two orthogonal directions, X and y (nominally along-wind and cross-wind) at the top of the building. This is the type of acceleration information which is readily available from combining model measurements of base moments with building modal information as described in §3.2. Using the discrete digitised accelerations a resolved acceleration response was computed from x'3, =
+
y2
(12)
Whilst this gives a trace of the magnitude of the acceleration which would be felt by an occupant it does not, in this simple form, provide any information about the direction of the acceleration felt and such a combined acceleration trace doubles the frequencies involved as acceleration values in opposite directions (+ve and -ve) appear as +ve with a +ve mean. Methods of resolving acceleration information with direction in reality proved not to be helpful in understanding what was happening. An example of the simply resolved acceleration trace from two acceleration components of similar magnitude with a frequency ratio 1.7 is given in Figure 2. Spectra of the original and resolved accelerations are given in Figure 3. From this example it can be seen why the problem is complex or at least very difficult to describe analytically. Starting with two discrete frequencies of 19.3 and 33.0 Hz, it can be seen how they resolve spectrally into two discrete frequencies of twice the value, 38.6 and 66 Hz. However on inspection ofthe trace ofthe resolved acceleration in Figure 2 it can be seen that one is not simply superimposed on the other, but that they occur in bursts dominated by one frequency and then the other. As one component becomes larger relative to the other so the longer it dominates in the resolved acceleration until only the frequency of the dominant signal can be observed. At the lower end of the frequency raho range, somewhere between 1.1 and 1.2 there is a similar process which is illustrated in spectra given in Figure 4. Here the acceleration components are similar in magnitude. At a frequency ratio of 1.18 the two components resolve into two dmcrete frequencies of double the value (again occurring in bursts ofone followed by the other). At a frequency ratio of 1.08 the two frequencies appear to merge with a broader band peak centred around a frequency told-way between the double values of both. However this is j,mt a problem of spectral resolution as the resolved traces still show bursts at one frequency and then the other. In fact what is happening becomes immediately obvious if one looks at a.resolved displacement response in the horizontal plane, which looks like a senes of irregular elliptic paths. Even with identical frequencies in two directions, the motion appears alternately (for a number of cycles) more in one direction than the other which is why, when the frequencies are different, the resolved response exhibits bursts of one frequency and then the other. The reason why there can be no simple analvtical method
112
3t
,
,
,
,
,
]
2
l
0
0
1
2
3
Time Is]
F i g u r e 2 Trace of resolved acceleration from two acceleration components of similar magnitude with a frequency ratio of 1.7
0,09
1
--I
L
0.08
'
0.008
I
J
-
i
0.007
0.07
~ '
1
m
0.008
0.00 0.005
0,05 0,004 lJ"J
0,04
0.003
0.03
0.02
0,002
0.01
0.001
0.00 0
lO0
0
l
I _
100
=.
,~1~
0.000 0
_
!
100
Frequency [Hz]
F i g u r e 3 Spectra of tile two component accelerations and the resolved acceleration in Figure 2
113 0.0040
|
'/
I
0.0035
400E-6
m
350E-6
m
300E-6
0.0025
m
~OE-6
0.0020
m
200E-6
'
!
n
0.0030 m
Of)
0.0015
150E-6
m
O.OOlO 0.0005
0.0000
]OOE-6 50.0E-6
m
J L
,
L
I iO0
,
0
I
-~J-
O.OOEO
0
iO0
0
JO0
iO0
Frequency [Hz]
0.008
I
500E-6
m
0.005
400E-6
0,004 300E-6 .~ 0,003 ol
200E-9
u
0.002 IOOE-6
0.001 0,000
I iO0
•
0
e
I iO0
. O.OOEO
Frequency IHzl
F i g u r e 4 Spectra of two component accelerations for frequency ratios of 1.18 and 1.08 and the resolved accelerations
'
114
of combining two or more such acceleration components is because the resolved response, either as a displacement or modulus of acceleration, as a function of time, is not ergodic. The most coaservative approach to combining two or more accelerations is to take the square root of the s~lm of the squares of the standard deviations, i.e. %
= ~i
+ %
(13)
It remains therefore to establish the conservatism of such an approach for a range of ratios of component frequencies and magnitudes. This can be done with referenc~o to Table 1 which contains the summarised results of the experimeuts introduced at the beginning of this section, in particular the values of the peak accelerations determined from a knowledge of the component standard deviation accelerations with the peak accelerations determined from the record of continuously resolved accelerations. From the ratios of the peak accelerations determined by the two methods it can be seen that there is no obvious functional relationsl~p. For frequency ratios up to 1.18 with component amplitudes more than a factor of two apart the ratio of peak accelerations is approximately one (witldn +5%). Hence it can be concluded that with these restrictions the simple method of obtaining a resolved peak acceleration from the component standard deviation is reasonably acceptable, essentially because one component dominates as discussed in §3. A similar situation occurs where the component amplitudes are similar up to a frequency ratio of 1.08, but here a change is becoming detectable. For frequency ratios beyond 1.18 and component amplitudes similar, the peak accelerations from the continuously resolved accelerations are typically 80% of the peak accelerations derived from the component standard deviation accelerations. The reason is obvious when o~ieolOsOk~atth~(~een? ~ctmfSofrOmbth~ ~e~otrdoefl~obre~ionU~uslyresolved acceler' ; Y Y 'p • . . It is concluded, at this stage, that there is no simple way of getting peak accelerations from a knowledge of the component standard deviation accelerations, other than to note that if the components vary by more than a factor of two or the frequency ratio is below 1.1, in wlfich case the simple method of combining through the square root of the sum of the squares of the standard deviations is satisfactory for an accuracy of +5%. Outside this range it is necessary to obtain peak accelerations from a record of continuously resolved accelerations. Also, when the component frequencies are very different some aver at~ng of the acceleration criteria as a function of frequency would be reqmred. Q
REFERENCES
1. Melbourne, W.H. (1980). Notes and Recommendations on Acceleration Criteria for Occupancy Comfort in Tall Structures, Private Communications in Commercial Wind Tunnel Investigation Reports. 2. Reed, J.W. (1971). Wind Induced Mo~ion a n d H m n a n Comfort. Research Report 71-42, Massachusetts Institute of Technology. 3. C_hen, P.W. and Robertson, K.E. (1973). H - m a n Perception Thresholds to Horizontal Motion, Jnl. of Structural Division, ASCE, Vol.98, pp 1681-1695. 4. Irwin, A. (1986). Motion m Tall Buildings, Proc. of Conf. on Tall Buildings, Second Century of the Skyscraper, Chicago.
115 5. Melbourne, W.H. and Cheung, J.C.I~ (1988). Designing for Serviceable Accelerations in Tall Buildings, 4th Int. Conf. on Tall Buildings, Hong Kong and Shanghai, pp 148-155. 6. Melbourne, W.H. (1977). Probability distributions associated with the wind loading of structures, Civil Engineering Transaction of the Inst. of Engineers, Australia, Vol. CE19, No.l, pp 58-67.
2.96 0.41 2.33 2.99 0.82 0.96 0.95 0.96 0.84 0.73 0.87
1.00 1.00 1.08 1.18 1.00 1.08 1.18 1.23 1.33 1.50 1.70
ny
Standard deviation ratio a~
Frequency ratio
28.07 6.41 27.92 30.89 12.11 11.45 11.25 4.95 4.59 4.96 5.14 9.47 15.80 11.97 10.34 14.79 11.96 11.8,5 5.17 5.47 6.78 5.92
% average peak, ~eak accel factor go. 3.66 108.85 3.72 63.43 3.72 113.09 3.61 117.58 3.75 71.81 3.58 59.39 3.45 56.51 3.72 26.63 3.64 25.99 3.51 29.48 3.35 26.26
Peak acceleration from component standard deviations
3.72 3.64 3.72 3.54 3.95 3.36 2.46 2.87 2.90 2.84 2.89
peak factor g
=
110.63 62.06 113.09 115.29 75.64 55.74 40.30 20.55 20.71 23.86 22.66
peak accel xy = g ~
+
Pe~i¢ acceleration from resolved acceleration
1.02 0.98 1.00 0.98 1.05 0.94 0.71 0.77 0.80 0.81 0.86
z~ I~omcmns o's
f~mn~ved
Ratio
Table 1 Comparison of peak accelerations determined from component standard deviation accelerations and continuously resolved accelerations (accelerations measured at top of model building)
105
A c c e l e r a t i o n s a n d c o m f o r t c r i t e r i a for b u i l d i n g s u n d e r g o i n g c o m p l e x motions W. H. Melbourne and T. R. Palmer Department of Mechanical Engineering, Monash University, Wellington Road, Clayton, Victoria 3168, Australia
Abstract Acceleration criteria to achieve acceptable occupancy comfort in buildings have been developed in terms of peak accelerations as a function of motion frequency and return period. The derivation ofaccelerations felt ~y an occupant in a building undergoing complex motions with significant contributions from several modes is discussed. The way in which component accelerations at a point resolve into a record which is non-ergodic is illustrated with model measurements.
1. I N T R O D U C T I O N Acceleration criteria to achieve occupancy comfort in tall buildings has received somewhat varied attention over the past twenty years. The pioneering work of Chen and Robertson (1973) gave valuable information about human perception to sinusoidal excitation as a function offrequency, and they suggested using an "occupancy sensitivity quotient" which defines the ratio of tolerable ampfitudes ofmotions to tile threshold motion for halfthe population. ~ l e work by Reed (1971) gave the first full scale evaluation of occupants' response to accelerations on two buildings and gave the first criteria in term a of standard deviation of acceleration for a return period for the frequencies ofthose buildings. Irwin, in a series ofpapers, further studied the response ofh-mans to sinusoidal acceleration over. a range of frequencies and through. . a number of unpublished. . full scale studies of crane operators and braiding occupants was primarily responsible for the standard deviation acceleration criteria in ISO 6897, and he focused these on tall buildings in Irwin (1986). In North America some use appears to be made of an unreferenced peak acceleration criterion of 20 milli-g once in 10 years with no reference to frequency. Melbourne (1980) using the earlier work of Chen and Robertson, Reed and some full scale observations.developed criteria for tall .buildings undergoing normally distributed oscdlatmns, as d.istinct from the smusoidal oscillations used in the human perceptmn expertments. This development related the accelerations of the normally distributed and sinusoidal motions on the basis that it was the peak accelerations which were the most important. Chen and Robertson had suggested that even the third derivative of displacement (jerk as they termed it) could be significant. The criterion from this work was that structures should be designed such that peak horizontal accelerations do not 0167-6105/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
106 exceed 10 milli-g per year on average for frequencies in the range 0.2 to 0.3 Hz, with the implication that motions perceptible by most people would occur during one to two storms per annum. Later Melbourne and Cheung (1988) commenced the development of frequency dependent criteria for peak accelerations to use for buildings undergoing complex motions and for different distributions. This paper will complete the development of the peak acceleration criteria for occupancy comfort of tall buildings and then go on to show how accelerations from bui.~,mgs undergoing complex motions of sway and rotation in the lower modes and accelerations from combined modes can be determined and related to the peak acceleration criteria. 2. DEVELOPMENT OF PEAK ACCELERATION C R I T E R I A The development of acceleration criteria referenced to peak accelerations is based on the conclusions of Chen and Robertson (1973) that it is the second or even third derivative of displacement which is most relevant to human comtbrt, or rather discomfort. Criteria based on the standard deviation of acceleration ignore the probability distribution ofthe peak accelerations which varies greatly between a sine wave and a normally distributed process and very significantly, for example, between the cross-wind response of a building operating near the peak of the cross-wind force spectrum and the along-wind response. Hence the nee d to base acceleration criteria for occupancy comfort in tall buildings on peak acceleration rather than the standard deviation of acceleration. Horizontal acceleration criteria in terma of standard deviation of acceleration for the worst 10 consecutive minutes in a 5 year return period for buildings as a function of frequency have been given by Irwin's E2 Curve and Curve 1, Figure I in ISO 6897. This curve can be fitted by the equation o~ = exp(-3.65
-
0.411an)
(1)
where o~ is the standard deviation of acceleration in the horizontal plane n is the frequency of oscillation with an approximately normal distribution. This curve is not so much described as criteria per se by Irwin, but ISO 6897 describes it as a satisfactory magnitude and the level at which about 2% of the occupants will comment adversely, and Irwin notes that "in general the criterion for satisfactory magnitudes of vibration in each of a number of categories (which he describes) is based on a minimum adverse comment level of 2% of the opulation involved, qlds definition of acceptability was also suggested earlier Reed. It has been shown (Melbourne 1988) that Reed's standard dewation criterion of 5 milli-g for a 6 year return period, Melbourne's one year return period peak acceleration of 10 milli-g and'the North American 10 year return period peak acceleration of 20 milli-g in the frequency range 0.2 to 0.3 Hz are all in reasonable agreement with Equation 1. Hence it was concluded that with this degree of agreement it was reasonable to use Equation 1 as a base for developing frequency dependent peak acceleration criteria for horizontal motion in buildings for occupancy comfort. ~lle peak acceleration has been obtained from the standard deviation expression, Equation 1, on the assumption that it relates to a normally distributed process, using
107 A
x
=
ga~
(2)
where g is tile p e a k factor = ~]~ In nT for a n o r m a l l y distributed process (Melbourne 1977) n is tile frequency T is tile d u r a h o n in seconds (i.e. for 10 m ln T = 6 0 0 sec) wlfich gives for a 5 y e a r return period
-
100
5O
I
"
'
J
H e l b o u r n e ' s ( 1 9 8 8 ) maximum peak h o r i z o n t a l a c c e l e r a t i o n c r i t e r i a w i t h r e f e r e n c e t o I r w i n ( 1 9 8 6 ) , Reed ( 1 9 7 1 ) , Chen and R o b e r t s o n ( 1 9 7 3 ) and H e l b o u r n e ( 1 9 8 0 ) f o r T = 600 s e c o n d s , and r e t u r n p e r i o d R y e a r s . •~,
£n
x ffi 42£n noT ( 0 . 6 8 + e - ~ )
bO !
(3)
~21n nTexp(-3.65-O.411n n)
R
exp
for 0.06
(-3.65
- 0.4I £n no)
0.5
20 return period 0 "14 4J
g
.~ 10 y e a r
--
10
5
al u u
.-j U
0
$ year exp
0.2
Im
fl
It
0.05
•
It ~
I
,
0.5 I
II
( - 3 . 5 5 - 0 . 4 I tn n o )
I
!
*
1.0
O. 1 frequency,
FIG.1
0.5
~
IrwinVs E2 Curve and ISO 6897 ( 1 9 8 4 ) Curve 1, standard deviation hort zontal acceleration criteria f o r 10 m i n u t e s in 5 year return period for a building (i.e. approximately normally distributed response).
n o . Hz
Horizontal acceleration criteria for occupancy comfort in buildings
108 It must be noted t h a t to evaluate a peak acceleration from the earlier Chen and Robertson, and Irwin data based on experiments with sinusoidal motion g = ~ . To genervfise this acceleration criteria for other return periods it is necessary to chose relationships for wind speed vs return period and response vs wind speed. Fgr the former a Gmnbel line fit to daily data typical of t h a t for areas in which extremes are generated both by thunderstorm and extensive pressure systems has been used as follows: 12 =
30+31nR
(4)
where R is return period in years. For the latter a response based on t~2~ has been used as being a compromise between alongwind and crosswind respot.se. A fit to the factor generated by these assumptions is as follows: response for return period R years response for return period 5 years
=
0.68+
InR 5
(5)
Combining these gives the peak acceleration criteria for occupancy comfort in buildings of
Plots ofthe acceleration criteria are given as a function of frequency in Figure I ibr a period of 10 minutes of maximu m wind in a return period of R years. The period of 10 minutes has bee.n used both to fit in with the original curve of Irwin and .ISO 6897 and because it m typical of a period ofmaximum response in areas don~nated by thunderstorm activity, and where mean design wind speeds tend to be backworked artificially from peak wind speed data. For regions where maximum response may occur through longer periods, such as an hour, tile maximum hourly mean wind spced will be less than the maximum 10 minute mean wind speed, and the value of T in Equation 7 would increase ',~ 3600 seconds. 3. A C C E L E R A T I O N S F R O M C O M P L E X MOTIONS Motion which is significant to occupancy comfort in buildings can be identified, for analytical convenience, as arising from several sources one of which may be dominant or several of which may combine to give the resultant mohon felt by the occupant. These sources are identified as follows: (i)
(ii) (iii)
motion in the first two bending(or sway) modes about orthogonal axes (usually for the worst case these can be conveniently divided into along-wind and cross-wind motions), rotational motion about a vertical axis, often either a 2nd or 3rd mode but can be higher, and for buildings with stiffness and mass asymmetry, complex bending and rotational motion in one or both of the lower modes, a n d any one of these motions may not be normally distributed.
109 When the highest acceleration felt by an occupant is dominated by one of these motions the problem of evaluating either standard deviation or peak acceleration from analytical or wind tunnel studies in relatively straight forward. The problem arises when two or more motions may make a significant contribution and to determine how they combine to affect an occupant if there are significant differences in frequency of the contributing motions. The simplest way to combine the accelerations from two motions, such as resulting from the first two bending modes, is to take the square root of the sum of the squares, i.e.
It is easy to see the effect of one of these motions becoming dominant by noting that if one standard deviation acceleration is twice the other then the smaller contributes only 10% to the resultant acceleration. The difficulties which arise in determining accelerations to compare with the criteria may be summarised as follows:
(i) (ii) (rid
when the peak accelerations of a motion are not normally distributed, when the accelerations result from a complex mode, and when the acceleration (standard deviation) from one mode of motion is not at least twice any others.
3.1. Motions for w h i c h p e a k s a r e n o t n o r m a l l y d i s t r i b u t e d The probability distributions ofpeaks ofbuilding motions has been discussed in some detail by Melbourne (1977). In summary it can be concluded that response primarily due to along-wind excitation or to cross-wind excitation well away from the peak of the cross-wind force spectrnm can be regarded as having peaks which are approximately normally distributed. It is response due to cross-wind excitation occurring near the peak of the cross-wind fore spectrum for which the dmtribution of peaks departs significantly from being normally distributed and due to dependence between the motion and the excitation process (vortex shedding) results in peak factors considerably lower than for a normally dmtnbuted process. For sharp-edged buildings the peak of the cross-wind force spectrum occurs at a Reduced Velocity of approximately 10 (F, = v_~where is mean wind speed at the top of the building, n is frequency ofcross-wind mode, and b is width normal to the wind directlo.n). Any cross-wind, or rotahon about the vertical axis, resl)onse overating wtthm 20% of the Reduced Velocity at the ))eak ofthe cross-wincl force spectrnm could expect to have a distribution ofpeaks ~vhich deviate si2nificantlv l~om a normally distributed process. UnfortunateTy, for tall buildings, the maximum accelerations tend to come from the cross-wind response and they are most likely to exceed the comfort criteria when operating at Reduced Velocities n e ~ the peak of th e cross-w_ind force spectr-m. Hence in many, ifnot all, crittcal caf~es, it ts essential to establish the actual value of the peak factor from wind tunnel measurements on a full or simplified linear mode aeroelastic model. This is a situation where use of standard deviation acceleration data or data from a force balance test assuming a normally distributed response would result, in a significant overestimate of
110 tl~e acceleration effect on occupancy comfort which may result in the requirement to add a damping system which may not be necessary. The best way to illustrate this is by example. Evaluation from data obtained from a linear mode aeroelastic model wind tunnel test of a 240m tall, relatively slender, buildingpredicted a standard deviation acceleration response for a 5 year r e t u r n period of o5 =
millig
6.6
(8)
This response was dominated by a cross-wind motion in the first mode with a full scale frequency of 0.15 Hz. Retbrence to the standard deviation criteria in Figure I shows that this acceleration would fall significantly above the recommended criterion of 5.7 milli-g. The peak factor for 10 minutes of operation for the peaks normally distributed would be approximately g
=
~/21n 600
=
3.0
•
0.15 (9)
giving a peak acceleration of y
=
3.0
•
6.6 =
19.8
miili-g
(10)
similarly falling outside the criterion of 17 milli-g. However the operating condition was close to the peak of the cross-wind force spectrum and an upcrossing analysis to give the probability distribution of the peaks, showed that the peak factor for 10 minutes of operation was approximately 2.3. This gave a peak acceleration of y
=
2,5
.
6.6 =
16,5
milli-g
(11)
and the conclusion that the building would satisfactorily meet occupancy comfort criteria. 3.2. M o t i o n s r e s u l t i n g f r o m c o m p l e x m o d e s Buildings with. asymmetry between the effective shear centre in rotation about a vertical axm and tile mass d i s t n b u h o n about such an axis have complex mode shapes and typically either the first or second mode may combine a h o r i z o n t a l displacement and a significant rotation about a vertical axis. Fortunately a numbe~ of modal analysispackages cope with this three-dimensional behaviour and describe the mode shapes in terms of a horizontal displacement from, and rotation about, a defined vertical axis, and there is a simple linear relationship between these two components. For the case of pure bending or sway motion the horizontal displacement at tile top of a building may be related to the base bending moment through tile inertial moment caused by the integration of maximum mass tlmes acceleration components up tile building. Hence for a g~ven peak of the fluctuating base moment under wind action tile peak horizontal displacement and hence acceleration at any level on the building can be obtained. For the situation where there is a complex mode tile same approach is used to obtain the peak acceleration due to horizontal displacement at a point on the defined vertical axis. If it is then required to obtain the horizontal acceleration at some point
111 away from this axis, such as in a room at the corner ofthe building, the horL-.ontal acceleration due to rotation for a given displacement at the defined vertical axis must be added vectorally to the horizontal displacement at the axis to give the resultant acceleration experienced at the corner of the building. 3.3. M o t i o n s r e s u l t i n g f r o m s i m i l a r c o n t r i b u t i o n s f r o m t w o o r m o r e modes To investigate this phenomenon directly several aeroelastic model studies of buildings were conducted in which horizontal accelerations were measured directly in two orthogonal directions, X and y (nominally along-wind and cross-wind) at the top of the building. This is the type of acceleration information which is readily available from combining model measurements of base moments with building modal information as described in §3.2. Using the discrete digitised accelerations a resolved acceleration response was computed from x'3, =
+
y2
(12)
Whilst this gives a trace of the magnitude of the acceleration which would be felt by an occupant it does not, in this simple form, provide any information about the direction of the acceleration felt and such a combined acceleration trace doubles the frequencies involved as acceleration values in opposite directions (+ve and -ve) appear as +ve with a +ve mean. Methods of resolving acceleration information with direction in reality proved not to be helpful in understanding what was happening. An example of the simply resolved acceleration trace from two acceleration components of similar magnitude with a frequency ratio 1.7 is given in Figure 2. Spectra of the original and resolved accelerations are given in Figure 3. From this example it can be seen why the problem is complex or at least very difficult to describe analytically. Starting with two discrete frequencies of 19.3 and 33.0 Hz, it can be seen how they resolve spectrally into two discrete frequencies of twice the value, 38.6 and 66 Hz. However on inspection ofthe trace ofthe resolved acceleration in Figure 2 it can be seen that one is not simply superimposed on the other, but that they occur in bursts dominated by one frequency and then the other. As one component becomes larger relative to the other so the longer it dominates in the resolved acceleration until only the frequency of the dominant signal can be observed. At the lower end of the frequency raho range, somewhere between 1.1 and 1.2 there is a similar process which is illustrated in spectra given in Figure 4. Here the acceleration components are similar in magnitude. At a frequency ratio of 1.18 the two components resolve into two dmcrete frequencies of double the value (again occurring in bursts ofone followed by the other). At a frequency ratio of 1.08 the two frequencies appear to merge with a broader band peak centred around a frequency told-way between the double values of both. However this is j,mt a problem of spectral resolution as the resolved traces still show bursts at one frequency and then the other. In fact what is happening becomes immediately obvious if one looks at a.resolved displacement response in the horizontal plane, which looks like a senes of irregular elliptic paths. Even with identical frequencies in two directions, the motion appears alternately (for a number of cycles) more in one direction than the other which is why, when the frequencies are different, the resolved response exhibits bursts of one frequency and then the other. The reason why there can be no simple analvtical method
112
3t
,
,
,
,
,
]
2
l
0
0
1
2
3
Time Is]
F i g u r e 2 Trace of resolved acceleration from two acceleration components of similar magnitude with a frequency ratio of 1.7
0,09
1
--I
L
0.08
'
0.008
I
J
-
i
0.007
0.07
~ '
1
m
0.008
0.00 0.005
0,05 0,004 lJ"J
0,04
0.003
0.03
0.02
0,002
0.01
0.001
0.00 0
lO0
0
l
I _
100
=.
,~1~
0.000 0
_
!
100
Frequency [Hz]
F i g u r e 3 Spectra of tile two component accelerations and the resolved acceleration in Figure 2
113 0.0040
|
'/
I
0.0035
400E-6
m
350E-6
m
300E-6
0.0025
m
~OE-6
0.0020
m
200E-6
'
!
n
0.0030 m
Of)
0.0015
150E-6
m
O.OOlO 0.0005
0.0000
]OOE-6 50.0E-6
m
J L
,
L
I iO0
,
0
I
-~J-
O.OOEO
0
iO0
0
JO0
iO0
Frequency [Hz]
0.008
I
500E-6
m
0.005
400E-6
0,004 300E-6 .~ 0,003 ol
200E-9
u
0.002 IOOE-6
0.001 0,000
I iO0
•
0
e
I iO0
. O.OOEO
Frequency IHzl
F i g u r e 4 Spectra of two component accelerations for frequency ratios of 1.18 and 1.08 and the resolved accelerations
'
114
of combining two or more such acceleration components is because the resolved response, either as a displacement or modulus of acceleration, as a function of time, is not ergodic. The most coaservative approach to combining two or more accelerations is to take the square root of the s~lm of the squares of the standard deviations, i.e. %
= ~i
+ %
(13)
It remains therefore to establish the conservatism of such an approach for a range of ratios of component frequencies and magnitudes. This can be done with referenc~o to Table 1 which contains the summarised results of the experimeuts introduced at the beginning of this section, in particular the values of the peak accelerations determined from a knowledge of the component standard deviation accelerations with the peak accelerations determined from the record of continuously resolved accelerations. From the ratios of the peak accelerations determined by the two methods it can be seen that there is no obvious functional relationsl~p. For frequency ratios up to 1.18 with component amplitudes more than a factor of two apart the ratio of peak accelerations is approximately one (witldn +5%). Hence it can be concluded that with these restrictions the simple method of obtaining a resolved peak acceleration from the component standard deviation is reasonably acceptable, essentially because one component dominates as discussed in §3. A similar situation occurs where the component amplitudes are similar up to a frequency ratio of 1.08, but here a change is becoming detectable. For frequency ratios beyond 1.18 and component amplitudes similar, the peak accelerations from the continuously resolved accelerations are typically 80% of the peak accelerations derived from the component standard deviation accelerations. The reason is obvious when o~ieolOsOk~atth~(~een? ~ctmfSofrOmbth~ ~e~otrdoefl~obre~ionU~uslyresolved acceler' ; Y Y 'p • . . It is concluded, at this stage, that there is no simple way of getting peak accelerations from a knowledge of the component standard deviation accelerations, other than to note that if the components vary by more than a factor of two or the frequency ratio is below 1.1, in wlfich case the simple method of combining through the square root of the sum of the squares of the standard deviations is satisfactory for an accuracy of +5%. Outside this range it is necessary to obtain peak accelerations from a record of continuously resolved accelerations. Also, when the component frequencies are very different some aver at~ng of the acceleration criteria as a function of frequency would be reqmred. Q
REFERENCES
1. Melbourne, W.H. (1980). Notes and Recommendations on Acceleration Criteria for Occupancy Comfort in Tall Structures, Private Communications in Commercial Wind Tunnel Investigation Reports. 2. Reed, J.W. (1971). Wind Induced Mo~ion a n d H m n a n Comfort. Research Report 71-42, Massachusetts Institute of Technology. 3. C_hen, P.W. and Robertson, K.E. (1973). H - m a n Perception Thresholds to Horizontal Motion, Jnl. of Structural Division, ASCE, Vol.98, pp 1681-1695. 4. Irwin, A. (1986). Motion m Tall Buildings, Proc. of Conf. on Tall Buildings, Second Century of the Skyscraper, Chicago.
115 5. Melbourne, W.H. and Cheung, J.C.I~ (1988). Designing for Serviceable Accelerations in Tall Buildings, 4th Int. Conf. on Tall Buildings, Hong Kong and Shanghai, pp 148-155. 6. Melbourne, W.H. (1977). Probability distributions associated with the wind loading of structures, Civil Engineering Transaction of the Inst. of Engineers, Australia, Vol. CE19, No.l, pp 58-67.
2.96 0.41 2.33 2.99 0.82 0.96 0.95 0.96 0.84 0.73 0.87
1.00 1.00 1.08 1.18 1.00 1.08 1.18 1.23 1.33 1.50 1.70
ny
Standard deviation ratio a~
Frequency ratio
28.07 6.41 27.92 30.89 12.11 11.45 11.25 4.95 4.59 4.96 5.14 9.47 15.80 11.97 10.34 14.79 11.96 11.8,5 5.17 5.47 6.78 5.92
% average peak, ~eak accel factor go. 3.66 108.85 3.72 63.43 3.72 113.09 3.61 117.58 3.75 71.81 3.58 59.39 3.45 56.51 3.72 26.63 3.64 25.99 3.51 29.48 3.35 26.26
Peak acceleration from component standard deviations
3.72 3.64 3.72 3.54 3.95 3.36 2.46 2.87 2.90 2.84 2.89
peak factor g
=
110.63 62.06 113.09 115.29 75.64 55.74 40.30 20.55 20.71 23.86 22.66
peak accel xy = g ~
+
Pe~i¢ acceleration from resolved acceleration
1.02 0.98 1.00 0.98 1.05 0.94 0.71 0.77 0.80 0.81 0.86
z~ I~omcmns o's
f~mn~ved
Ratio
Table 1 Comparison of peak accelerations determined from component standard deviation accelerations and continuously resolved accelerations (accelerations measured at top of model building)