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Serviceability criteria for wind-induced acceleration and damping uncertainties
Journal of Wind Engineering and Industrial Aerodynamics 74—76 (1998) 1067—1078
Serviceability criteria for wind-induced acceleration and damping uncertainties L.C. Pagnini*, G. Solari DISEG, Department of Structural and Geotechnical Engineering, University of Genova, Genova, Italy
Abstract Classical serviceability criteria for wind-induced accelerations are discussed pointing out their conventional character and unreliability due to the uncertain knowledge of structural damping. A theoretical frame of some advanced probabilistic criteria is given showing that, in this context, damping uncertainties can be easily taken into account together with wind velocity randomness. It is demonstrated that the joint use of damping models, closed-form solutions and uncertainty propagation techniques previously formulated by these authors leads to analytical formulae, the application of which is as simple as the use of classical conventional procedures. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Acceleration; Damping; Dynamic response; Reliability; Serviceability; Uncertainty; Wind velocity
1. Introduction It is well known that buildings should be designed so that wind-induced acceleration will not cause unacceptable discomfort to their occupants [1]. The literature is rich in methods aimed, on one side, at determining structural response to wind actions and, on the other, at formulating suitable serviceability criteria based on human response. The application of classical methods for determining acceleration [1—3] shows that this quantity is most affected by wind velocity and structural damping. Wind velocity is, by its own, a stochastic process. Structural damping, though recognized as the most uncertain parameter, is usually considered as deterministic.
* Corresponding author. E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 9 8 - 1
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Current serviceability criteria based on human response to wind-induced motion compare suitable acceleration parameters related to fixed mean return periods of the wind velocity [4—7] with deterministic conventional thresholds. Methods correctly inspired in rational probabilistic approaches according to classical reliability analyses are still rare and not much applied in engineering practice [8]. This paper discusses classical serviceability criteria pointing out their conventional character and unreliability due to damping uncertainties. A theoretical frame of some advanced probabilistic criteria is given showing that, in this context, damping uncertainties can be easily taken into account together with wind velocity randomness. It is demonstrated that the joint use of damping models [9,10], closed-form solutions [2,3] and uncertainty propagation techniques [11,12] leads to analytical formulae, the application of which makes advanced probabilistic serviceability criteria as simple as classical conventional procedures.
2. Wind-induced acceleration Consider a vertical structure referred to a Cartesian reference system x, y, z; z coincides with the structural axis and is directed upwards. The structure is subjected to a static force due to the mean wind velocity uN aligned with x and to three zero mean fluctuating actions due to the u@ longitudinal turbulence (parallel to x), to the v@ lateral turbulence (parallel to y) and to the vortex shedding s@, here treated as stochastic stationary Gaussian uncorrelated processes, assuming u@/uN @1, v@/uN @1. The structure has a linear visco-elastic behaviour. It possesses three uncoupled components of motion, the alongwind and crosswind displacements, directed towards x, y, and the h torsional rotation around z. Each a"x, y, h component of motion is a stochastic stationary Gaussian process, whose second temporal derivative is denoted by a( . It is assumed that a( only depends on the contribution of the related fundamental mode t . a1 Neglecting the dependence on z for simplicity of notation, the probability density function (pdf ) f, the mean value E[ ) ] and the variance »[ ) ] of the maximum generalized acceleration a( in the period *¹ over which wind velocity is averaged .!9 are given by [1,2]
A
B C
A
BD
a( a( 2 a( 2 f ( (a( )"2n *¹ .!9 exp ! .!9 exp !2n *¹exp ! .!9 a.!9 .!9 a1 a1 2p2( 2p2( p2( a a a E[a( ]"g ( p ( , .!9 a a
S
p2 p2( a, »[a( ]" .!9 6 s2 a
1 DI (u ), p ( "K u2 a * m a * a a1
0.5772 g "s # , a a s a
,
(1)
(2)
s "J2 ln(2n *¹), a a1
(3)
where n is the fundamental frequency in the a direction, g ( , p ( are the peak factor a1 a a and the root mean square (RMS) value of a( , m is the damping coefficient in the a1
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a direction, and u is the shear velocity, chosen here as the representative parameter of * the mean wind velocity uN . K is a quantity independent of m , u ; DI is a function of a a1 * a u independent of m . Limited to slenderline-like vertical structures, they are given by * a1 [3] K "t a a1
o*2(0.6h)bhj aC K , xu ax 2m a1
C DI "+ s2 DI , s "J (0.6h) ae , a ae ae ae e C xu e
d nJ p u ae C[nJ hI ] (e"u, v) , DI " ae ae ae 4 (1#1.5d nJ )5@3 u ae
G C
DH
(4)
(5)
2 nJ 1!nJ p as as exp ! C[hI ] , DI " as as 4 JpB (0.8h) B (0.8h) a a
(6)
k dC h n d ¸ (0.6h) hI " ae u ze , nJ " a1 e e (e"u, v) , ae d ¸ (0.6h) ae d u *(0.6h) e e u *
(7)
k h hI " as , as ¸b
n b a1 nJ " , as ¶ Su *(0.8h) a *
1 1 (1!e~2g), g'0, C[g]" ! g 2g2
(8)
C[0]"1,
(9)
where h and b are the height and the width of the structure; j "j "1, j "b; + is x y h e the sum over three terms with indices e"u, v, s, corresponding to the contributions of the longitudinal turbulence, of the lateral turbulence and of the vortex shedding; m is a1 the modal mass in the a generalized direction; o is the air density; *(z)"2.5 ln(z/z ), 0 z being the roughness length; J "2I , J "I , J "1, I , I being the u@, v@ turbu0 u u v v s u v lence intensities; ¸ , ¸ are the integral length scales of the u@, v@ turbulence compou v nents; d "6.868, d "9.434; C , C are the exponential decay factors of the u@, v@ u v zu zv turbulence components in the z direction; C "c , C "cl, C "c , c , cl, c being xu d yu hu m d m the drag, lift and torsional moment coefficients; C "c@ !cl, C "c #c@l, xv d yv d C "c@ , c@ , c@l c@ being the prime angular derivatives of c , cl, c , C "cJ , hv m d m d m xs ds C "cJ l , C "cJ , cJ , cJ l , cJ being the RMS drag, lift and torsional moment wake ys s hs ms ds s ms coefficients; ¶ "2, ¶ "1, ¶ depends on the structural shape; S is the Strouhal x y h number; B is the bandwidth parameter; ¸ is the vortex correlation length (in b@s); a KM , k are the mean modal wind factor and the equivalent correlation factor defined ax ae in Refs. [2,3]. Analysing three-dimensional buildings as slenderline-like usually leads to significant overestimations of the alongwind acceleration, to reliable estimates of the crosswind acceleration, to underestimate the torsional acceleration [13]. Formulae are given in Ref. [14] for reducing DI (Eq. (5)) in order to account for threexu dimensionality effects. Criteria are available in the literature aiming at obtaining non-directional reference a values of the acceleration as suitable combinations of their generalized directional
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components a( [5,7]. It is well recognized, above all for tall buildings, that the crosswind acceleration usually dominates the alongwind acceleration [8] therefore representing the main quantity to be verified for serviceability.
3. Damping modelling and uncertainties A wide and homogeneous set of full-scale measurements of buildings and other structures was previously gathered. Based on the analysis of this data, limited to small displacements and flexural modes, the structural damping coefficient m was exa1 pressed by [9,10] b m " 1 #b n #e , a1 n 2 a1 m a1
(10)
where n is given in Hz and m is in % of critical damping; b and b are model a1 a1 1 2 parameters; e is the zero mean model error independent of b and b . m 1 2 The application of the first-order second-moment (FOSM) technique [11] to Eq. (10) leads to the relationships: E[b ] 1 #E[b ]n , E[m ]" 2 a1 a1 n a1 »[b ] 1 #»[b ]n2 #2C[b , b ]#»[e ], »[m ]" 1 2 m a1 2 a1 n2 a1
(11)
where C[., :] is the covariance. Statistical moments of b , b , e are given in Refs. 1 2 m [9,10] for four different classes of multi-storey buildings. It is usually accepted that damping pdf f is properly represented by a log-normal ma1 model.
4. Conventional serviceability criteria Current serviceability criteria based on human response to wind-induced acceleration can be framed into two classes: 1. RMS acceleration criteria [5,6]. The structure is reliable if p )p*, where p is the a a a RMS value of the acceleration a due to a mean wind velocity u "u* with mean * * return period RM "RM *; RM * is a reference value of RM , p* is an allowable limit of p . a a 2. Expected maximum acceleration criteria [4,7]. The structure is reliable if E[a ])a* , where E[a ] is the expected maximum value of a in *¹ due to .!9 .!9 .!9 a mean wind velocity u "u* with mean return period RM "RM *; a* is an allow* * .!9 able limit of E[a ]. .!9
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As a matter of fact, damping uncertainties deeply propagate on the estimation of the wind-induced acceleration. Referring to methods (1) and (2), p ( , E[a( ] become a .!9 random variables whose pdf are given by
A
B
1 K2u4 DI a * a , f ( (p ¨ )"2K2u4 DI f a a1 a * ap3( m p a p2( a a
(12)
A
B
1 g2( K2u4 DI a a * a . f f [a( ](E[a( ])"2g2( K2u4 DI a a * aE3[a( ] ma1 E2[a( ] E .!9 .!9 .!9 .!9
(13)
The first and second statistical moments of p ( , E[a( ] can be rigorously determined .!9 a by applying classical probability rules. Simple approximations of these quantities can be obtained by expanding p ( , E[a( ] in Taylor Series around m "m0 . a1 a .!9 a1 Assuming m0 "E[m ] and retaining up to first-order derivative terms, the FOSM a1 a1 technique gives rise to E[p ( ]"p0( , »[p ( ]"1(p0( )2X2[m ] , a a a 4 a a1
(14)
»[E[a( ]]"g2( »[p ( ], a .!9 a
E[E[a( ]]"g ¨ E[p ( ], a .!9 a
(15)
where the superscript 0 denotes quantities evaluated in the expansion point; p0( is a given by Eq. (3) for m "m0 , and X[ ) ] is the coefficient of variation. a1 a1 Retaining up to second-order derivative terms and using the SOSM technique [12], Eq. (14) becomes
A
B
A
B
E[p ( ]"p0( 1#3X2[m ] ; »[p ( ]"1(p0( )2X2[m ] 1# 9 X2[m ] , a a 8 4 a 16 a1 a a1 a1
(16)
Eq. (15) remains unchanged. It is verified that in most cases of engineering interest the pdf of p ( , E[a( ] spread a .!9 up to the point that traditional estimates based on nominal values of m are fully a1 unreliable. The numerical applications at the end of this paper fully confirm this situation.
5. Probabilistic serviceability criteria In accordance with methods stated in Ref. [8], probabilistic serviceability criteria based on human response to wind-induced acceleration can be framed into two classes: (1) Maximum acceleration criteria. The structure is reliable if PK )PK *, where PK is the probability that the maximum acceleration aL in a reference period ¹A*¹ exceeds a limit aL * defined herein as deterministic; PK * is an allowable limit of PK . (2) Acceleration persistence criteria. The structure is reliable if P)P*, where P is the probability, i.e. the fraction of time, that a suitable measure of the acceleration in *¹ exceeds a limit a* defined herein as deterministic; P* is an allowable limit of P.
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Referring to method (1), it is assumed that the maximum acceleration aL in ¹ is the maximum acceleration a in the *¹ interval where the shear velocity u takes its .!9 * maximum value uL in ¹. It is easy to verify that this definition is correct and slightly * prudential in the alongwind direction where acceleration increases with increasing u . * On the other hand, it gives suitable approximations in the crosswind and torsional generalized directions, if the resonant shedding of vortices occurs, as it is typical of building structures, at high u values. Considering single generalized a directions: * = PK " f (L (a(L ) da(L , (17) a aL * == L (a( DuL , m ) f (uL ) f (m ) duL dm , (18) f( L f (L (a( )" a.!9@u*,ma1 .!9 * a1 uL * * ma1 a1 * a1 a 0 0 is the pdf of a( (Eq. (1)) conditioned by the where f (L is the pdf of a(L ; f ( L a.!9@u*, ma1 .!9 a occurrence of uL , m ; f L is the pdf of uL ; uL and m are statistically independent. * a1 u* * * a1 The first and second statistical moments of a(L can be rigorously determined by applying classical probability rules. Simple approximations can be obtained by expanding a( in double Taylor series around uL "uL 0 , m "m0 . .!9 * * a1 a1 Assuming uL 0 "E[uL ], m0 "E[m ] and retaining up to first order derivative terms, * * a1 a1 the FOSM technique leads to the formulae
P
PP
E[a(L ]"(E[a( ])0; .!9 »[a(L ]"(»[a( ])0#[(E[a( ])0]2(4b2X2[uL ]#1X2[m ]), .!9 .!9 * 4 a1 DI @ 0 b"1# a uL 0 , 4DI 0 * a
(19) (20)
where (E[a( ])0, (»[a( ) ])0 are given by Eq. (2) for u "uL 0 , m "m0 ; DI 0 is given by a1 a * a1 .!9 .!9 * Eqs. (4)—(6) for u "uL 0 ; DI @0 is the first derivative of DI 0 with respect to u , calculated a * * a * in uL "uL 0 . * * Retaining up to second-order derivative terms and using the SOSM technique, Eq. (19) becomes E[a(L ]"(E[a( ])0(1#cX2[uL ]#3X2[m ]), 8 .!9 * a1 »[a(L ]"(»[a( ])0#[(E[a( ])0]2](4b2X2[uL ]#1X2[m ] 4 .!9 .!9 * a1 #c2X2[uL ]# 9 X2[m ]#3c2X2[uL ]X2[m ]), 64 4 * a1 * a1 2DI 0DI A0!DI @ 0 DI @ 0 a (uL 0 )2, c"1# a uL 0 # a a * 8(DI 0)2 DI 0 * a a
(21) (22)
where DI A0 is the second derivative of DI 0 with respect to u , calculated in u "uL 0 . a a * * *
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It is to be noted that DI @ 0"DI A0 represent suitable first-order approximations. In a a such case, b"c"1 drastically simplify Eqs. (19) and (21). It is also relevant to point out that, despite the convolution integrals in Eq. (18), the pdf of a(L usually retains the typical shape of the type I extreme distribution of a( (Eq. (1)). Therefore, especially .!9 due to the randomness of u , m , a log-normal pdf should represent a suitable * a1 approximation of the actual distribution of a(L [8,12]. Referring to method (2), a suitable measure of the acceleration in *¹ is represented by its maximum value a . Considering also in this case single generalized a direc.!9 tions: = P"
P
f (J (a(J ) da(J , .!9 a.!9 .!9
(23)
a* == f (J (a(J " f( (a(J Du , m ) f (u ) f (m ) du dm , a.!9 .!9 a.!9@u*, ma1 .!9 * a1 u* * ma1 a1 * a1 0 0
PP
(24)
is the maximum value of a( in *¹ taking is the pdf of a(J ; a(J where f (J .!9 .!9 a.!9 u randomness and m uncertainties into account; f ( is the pdf of a( (Eq. (1)) * a1 a.!9@u*, ma1 .!9 conditioned by the occurrence of u , m ; f is the pdf of u ; and u and m are * a1 u* * * a1 statistically independent. The analogy of Eqs. (18) and (24) points out that FOSM and SOSM expressions of E[a(J ], »[a(J ] are given by Eq. (19) and (21) replacing uL by u . However, since .!9 * * .!9 X[u ] is much greater than X[uL ], FOSM and SOSM techniques often provide, in * * this case, unreliable estimates. Jointly referring to methods (1) and (2) above, it is finally noted that, due to the mathematical structure of Eqs. (18) and (24), the role of damping uncertainties reduces with respect to velocity randomness. Under this point of view, differently from methods (1) and (2), f (m )"d(m !m0 ) and X[m ]"0 often represent ma1 a1 a1 a1 a1 reasonable first-order approximations (d being the Dirac function). It follows that PK +1!F L (uL *), u* *
P+1!F (u*), u* *
(25)
where F L and F are the cumulative distribution functions (cdf) of uL , u ; and uL *, u* u* * * u* * * are the values of u in correspondence of which E[a ]"aL *, a* for m "m0 . It is a1 * .!9 a1 implicit that Eq. (25) applies if X[a ] is small and E[a ] is, as previously assumed, .!9 .!9 a monotonic function of u . *
6. Applications Serviceability criteria herein discussed are applied to calculate the crosswind acceleration at the top of a square plan multi-storey steel building with h"200 m,
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b"40 m; n "0.225 Hz, t (z)"z/h, m "21 333 333 kg, E[m ]"0.0159, X[m ]" yl yl yl yl yl 0.436. Damping is log-normally distributed. Aerodynamic properties are schematized by c "1.2, cl"0.0, c@ "0.0, c@l"!3.4, cJ "0.0, cJ l "0.5; S"0.1, ¸"1.0, $ $ $4 s B "0.25. y Wind is characterized by z "1 m, I (z)"0.4Jb /ln(z/z ), b "4.50, I (z)" 0 u v 0 u u 0.775I (z); ¸ (z)"300(z/300)e (¸ , z in m), e"0.46, ¸ (z)"0.24¸ (z); C "11.5, u u u v u zu C "0.7C ; o"1.25 kg/m3; *¹"600 s [3]. The pdf of u , uL (for ¹"1 yr) are zv zu * * modelled by Weibull and extreme type I distributions (Fig. 1, solid and dotted line):
AB
C A BD
C A BD
u k u k k u k~1 * f (u )" exp ! * , F (u )"1!exp ! * , * * u * u * c c c c
(26)
f L (uL "a expM!a exp(uL !u)!exp[!a(uL !u)]N, u* * * * (27) F L (uL )"expM!exp[!a(uL !u)]N, * u* * where k"1, c"0.333 m/s, E[u ]"0.333 m/s, X[u ]"1.00; a"3 s/m, u"2 m/s, * * E[uL ]"2.192 m/s, X[uL ]"0.195. * * Using the conventional serviceability criterion proposed in Ref. [5], R*"5 years implies u "2.5 m/s; p* "0.049 m/s2 corresponds to a* "0.172 m/s2. Assuming * .!9 .!9 m "m0 , Eq. (2) and (3) give E[y( ])0"0.1484 m/s2(a* , which means that the y1 y1 .!9 .!9 building is reliable. Fig. 2 shows the pdf of E[y( ] for RM *"5 yr. Solid lines correspond to rigorous .!9 probabilistic analyses (RPA); dotted and dashed lines correspond to log-normal models based on statistical moments evaluated through FOSM and SOSM techniques (Table 1). The probability that E[y( ]*a* is 0.315 (by RPA). It follows that .!9 .!9 the above criterion, as well as analogous criteria, is clearly unreliable due to damping uncertainty. Fig. 3 shows the pdf of y(L for ¹"1 yr (with uncertain damping) using the same symbols in Fig. 2. Table 2 gives the mean and RMS values of y(L corresponding to
Fig. 1. Pdf of the shear velocity.
L.C. Pagnini, G. Solari/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 1067–1078
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Fig. 2. Pdf of E[y( ] for RM *"5 yr. .!9
Table 1 Results of expected maximum acceleration criterion (y( in m/s2) .!9 Method
RPA
FOSM
SOSM
E[E[y( ]] .!9 S[E[y( ]] .!9
0.1588 0.0340
0.1484 0.0329
0.1594 0.0329
Fig. 3. Pdf of y(L for ¹"1 yr.
RPA, FOSM and SOSM techniques. It also gives the exceedence probability pL of a conventional perception threshold aL *"0.1 m/s2. It is apparent that the log-normal model of y(L , applied to Eq. (19) and especially to Eq. (21), gives rise to results with excellent approximation.
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Table 2 Results of maximum acceleration criterion (y(L in m/s2) Method
E[y(L ] S[y(L ] PK
Uncertain damping
Deterministic damping
RPA
FOSM
SOSM
RPA
FOSM
SOSM
0.1204 0.0768 0.51
0.1050 0.0601 0.43
0.1211 0.0694 0.47
0.1130 0.0694 0.47
0.1050 0.0555 0.44
0.1134 0.0561 0.51
Fig. 4. Pdf of y(J . .!9
Table 3 Results of acceleration persistence criterion (y(J in m/s2) Method
E[y(J ] .!9 S[y(JR ] .!9 P
Uncertain damping
Deterministic damping
RPA
FOSM
SOSM
RPA
FOSM
SOSM
0.0026 0.0102 0.0017
0.0006 0.0017 —
0.0021 0.0023 —
0.0025 0.0094 0.0014
0.0006 0.0017 —
0.0021 0.0022 —
Fig. 4 shows the pdf of y(J (with uncertain damping). Table 3 gives its mean .!9 and RMS values confirming that, in this case, FOSM and SOSM techniques lead to relevant errors. It also furnishes the exceedence probability P of the perception threshold a*"aL *. Consistent with the persistence properties of the selected wind velocity process, P"0.0017 means that a* is averagely exceeded 15 h per year.
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Calculating (E[y( ])0 (Eq. (2)) as a function of u and inverting it, for .!9 * aL *"a*"0.1 m/s2, uL *"u*"2.156 m/s. Using Eq. (25), PK "0.46, P"0.0015. The * * comparison between these values and those reported in Tables 2 and 3 points out the approximations involved by the application of methods of different level.
7. Conclusions and perspectives It is demonstrated that the uncertain knowledge of structural damping makes classical serviceability criteria for wind-induced acceleration purely conventional and often unreliable. The joint use of suitable damping models, closed-form solutions of building acceleration and uncertainty propagation techniques makes the application of probabilistic serviceability criteria as simple as the use of current conventional procedures. All formulae and criteria herein developed consider single generalized components of the structural acceleration assuming the structure as slenderline-like. First and second statistical moments of these quantities are calculated through Taylor series expansions around the mean values of random parameters. Acceleration limit states are assumed as deterministic. Perspectives for further advances are represented by the application of fully 3-D structural models, by evaluating the pdf and statistical moments of the reference value of the acceleration as suitable combinations of the pdf and statistical moments of its generalized directional components, by expanding Taylor series in a neighbourhood of the checking point [15] consistently with established second-order reliability approaches, by including probabilistic limit states in the formulation [16].
References [1] E. Simiu, R.H. Scanlan, Wind Effects on Structures, Wiley, New York, NY 1996. [2] G. Piccardo, G. Solari, 3-D wind-excited response of slender structures: basic formulation, closed form solution applications, J. Struct. Eng. ASCE. (1998) submitted. [3] G. Piccardo, G. Solari, Closed form prediction of 3-D wind-excited response of slender structures, J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 697—708. [4] P.W. Chen, L.E. Robertson, Human perception thresholds and horizontal motion, J. Struct. Div. ASCE 98 (1972) 1681—1695. [5] R.J. Hansen, J.W. Reed, E.H. Vanmarcke, Human response to wind-induced motion of buildings, J. Struct. Div. ASCE 99 (1973) 1589—1605. [6] A.W. Irwin, Perception, comfort and performance criteria for human beings exposed to whole body pure yaw vibration and vibration containing yaw and translational components, J. Sound Vib. 76 (1981) 481—497. [7] W.H. Melbourne, T.R. Palmer, Accelerations criteria for buildings undergoing complex motions, J. Wind Eng. Ind. Aerodyn. 41—44 (1992) 105—116. [8] J. Kanda, O. Nakamura, Y. Tamura, K. Uesu, Probabilistic criteria for serviceability limit of wind induced response, Proc. Int. Coll. on Structural Serviceability of Buildings, IABSE, AIPC, IVBH, Goteborg, 1993, pp. 59—66. [9] G. Lagomarsino, Forecast models for damping and vibration periods of buildings, J. Wind Eng. Ind. Aerodyn. 48 (1993) 221—239.
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[10] G. Lagomarsino, L.C. Pagnini, Criteria for modelling and predicting dynamic parameters of buildings, Report ISC-II, 1, Istituto di Scienza delle Costruzioni, University of Genova, Italy, 1995. [11] G. Solari, Evaluation and role of damping and periods for the calculation of structural response under wind loads, J. Wind Eng. Ind. Aerodyn. 59 (1996) 191—210. [12] G. Solari, Wind-excited response of structures with uncertain parameters, Prob. Eng. Mech. 12 (2) (1997) 75—87. [13] G. Solari, Mathematical model to predict 3-D wind loading on buildings, J. Eng. Mech. ASCE 111 (1985) 254—276. [14] G. Solari, Gust buffeting. II: dynamic alongwind response, J. Struct. Eng. ASCE 119 (1993) 383—398. [15] A.M. Hasofer, N.C. Lind, An exact and invariant first order reliability format, J. Eng. Mech. Div. ASCE 100 (1974) 111—121. [16] L.C. Pagnini, G. Solari, Probabilistic damping modeling and building reliability under wind loads, Proc. Structural Engineers World Congress, San Francisco, USA, 1998, in press.
Serviceability criteria for wind-induced acceleration and damping uncertainties L.C. Pagnini*, G. Solari DISEG, Department of Structural and Geotechnical Engineering, University of Genova, Genova, Italy
Abstract Classical serviceability criteria for wind-induced accelerations are discussed pointing out their conventional character and unreliability due to the uncertain knowledge of structural damping. A theoretical frame of some advanced probabilistic criteria is given showing that, in this context, damping uncertainties can be easily taken into account together with wind velocity randomness. It is demonstrated that the joint use of damping models, closed-form solutions and uncertainty propagation techniques previously formulated by these authors leads to analytical formulae, the application of which is as simple as the use of classical conventional procedures. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Acceleration; Damping; Dynamic response; Reliability; Serviceability; Uncertainty; Wind velocity
1. Introduction It is well known that buildings should be designed so that wind-induced acceleration will not cause unacceptable discomfort to their occupants [1]. The literature is rich in methods aimed, on one side, at determining structural response to wind actions and, on the other, at formulating suitable serviceability criteria based on human response. The application of classical methods for determining acceleration [1—3] shows that this quantity is most affected by wind velocity and structural damping. Wind velocity is, by its own, a stochastic process. Structural damping, though recognized as the most uncertain parameter, is usually considered as deterministic.
* Corresponding author. E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 9 8 - 1
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Current serviceability criteria based on human response to wind-induced motion compare suitable acceleration parameters related to fixed mean return periods of the wind velocity [4—7] with deterministic conventional thresholds. Methods correctly inspired in rational probabilistic approaches according to classical reliability analyses are still rare and not much applied in engineering practice [8]. This paper discusses classical serviceability criteria pointing out their conventional character and unreliability due to damping uncertainties. A theoretical frame of some advanced probabilistic criteria is given showing that, in this context, damping uncertainties can be easily taken into account together with wind velocity randomness. It is demonstrated that the joint use of damping models [9,10], closed-form solutions [2,3] and uncertainty propagation techniques [11,12] leads to analytical formulae, the application of which makes advanced probabilistic serviceability criteria as simple as classical conventional procedures.
2. Wind-induced acceleration Consider a vertical structure referred to a Cartesian reference system x, y, z; z coincides with the structural axis and is directed upwards. The structure is subjected to a static force due to the mean wind velocity uN aligned with x and to three zero mean fluctuating actions due to the u@ longitudinal turbulence (parallel to x), to the v@ lateral turbulence (parallel to y) and to the vortex shedding s@, here treated as stochastic stationary Gaussian uncorrelated processes, assuming u@/uN @1, v@/uN @1. The structure has a linear visco-elastic behaviour. It possesses three uncoupled components of motion, the alongwind and crosswind displacements, directed towards x, y, and the h torsional rotation around z. Each a"x, y, h component of motion is a stochastic stationary Gaussian process, whose second temporal derivative is denoted by a( . It is assumed that a( only depends on the contribution of the related fundamental mode t . a1 Neglecting the dependence on z for simplicity of notation, the probability density function (pdf ) f, the mean value E[ ) ] and the variance »[ ) ] of the maximum generalized acceleration a( in the period *¹ over which wind velocity is averaged .!9 are given by [1,2]
A
B C
A
BD
a( a( 2 a( 2 f ( (a( )"2n *¹ .!9 exp ! .!9 exp !2n *¹exp ! .!9 a.!9 .!9 a1 a1 2p2( 2p2( p2( a a a E[a( ]"g ( p ( , .!9 a a
S
p2 p2( a, »[a( ]" .!9 6 s2 a
1 DI (u ), p ( "K u2 a * m a * a a1
0.5772 g "s # , a a s a
,
(1)
(2)
s "J2 ln(2n *¹), a a1
(3)
where n is the fundamental frequency in the a direction, g ( , p ( are the peak factor a1 a a and the root mean square (RMS) value of a( , m is the damping coefficient in the a1
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1069
a direction, and u is the shear velocity, chosen here as the representative parameter of * the mean wind velocity uN . K is a quantity independent of m , u ; DI is a function of a a1 * a u independent of m . Limited to slenderline-like vertical structures, they are given by * a1 [3] K "t a a1
o*2(0.6h)bhj aC K , xu ax 2m a1
C DI "+ s2 DI , s "J (0.6h) ae , a ae ae ae e C xu e
d nJ p u ae C[nJ hI ] (e"u, v) , DI " ae ae ae 4 (1#1.5d nJ )5@3 u ae
G C
DH
(4)
(5)
2 nJ 1!nJ p as as exp ! C[hI ] , DI " as as 4 JpB (0.8h) B (0.8h) a a
(6)
k dC h n d ¸ (0.6h) hI " ae u ze , nJ " a1 e e (e"u, v) , ae d ¸ (0.6h) ae d u *(0.6h) e e u *
(7)
k h hI " as , as ¸b
n b a1 nJ " , as ¶ Su *(0.8h) a *
1 1 (1!e~2g), g'0, C[g]" ! g 2g2
(8)
C[0]"1,
(9)
where h and b are the height and the width of the structure; j "j "1, j "b; + is x y h e the sum over three terms with indices e"u, v, s, corresponding to the contributions of the longitudinal turbulence, of the lateral turbulence and of the vortex shedding; m is a1 the modal mass in the a generalized direction; o is the air density; *(z)"2.5 ln(z/z ), 0 z being the roughness length; J "2I , J "I , J "1, I , I being the u@, v@ turbu0 u u v v s u v lence intensities; ¸ , ¸ are the integral length scales of the u@, v@ turbulence compou v nents; d "6.868, d "9.434; C , C are the exponential decay factors of the u@, v@ u v zu zv turbulence components in the z direction; C "c , C "cl, C "c , c , cl, c being xu d yu hu m d m the drag, lift and torsional moment coefficients; C "c@ !cl, C "c #c@l, xv d yv d C "c@ , c@ , c@l c@ being the prime angular derivatives of c , cl, c , C "cJ , hv m d m d m xs ds C "cJ l , C "cJ , cJ , cJ l , cJ being the RMS drag, lift and torsional moment wake ys s hs ms ds s ms coefficients; ¶ "2, ¶ "1, ¶ depends on the structural shape; S is the Strouhal x y h number; B is the bandwidth parameter; ¸ is the vortex correlation length (in b@s); a KM , k are the mean modal wind factor and the equivalent correlation factor defined ax ae in Refs. [2,3]. Analysing three-dimensional buildings as slenderline-like usually leads to significant overestimations of the alongwind acceleration, to reliable estimates of the crosswind acceleration, to underestimate the torsional acceleration [13]. Formulae are given in Ref. [14] for reducing DI (Eq. (5)) in order to account for threexu dimensionality effects. Criteria are available in the literature aiming at obtaining non-directional reference a values of the acceleration as suitable combinations of their generalized directional
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components a( [5,7]. It is well recognized, above all for tall buildings, that the crosswind acceleration usually dominates the alongwind acceleration [8] therefore representing the main quantity to be verified for serviceability.
3. Damping modelling and uncertainties A wide and homogeneous set of full-scale measurements of buildings and other structures was previously gathered. Based on the analysis of this data, limited to small displacements and flexural modes, the structural damping coefficient m was exa1 pressed by [9,10] b m " 1 #b n #e , a1 n 2 a1 m a1
(10)
where n is given in Hz and m is in % of critical damping; b and b are model a1 a1 1 2 parameters; e is the zero mean model error independent of b and b . m 1 2 The application of the first-order second-moment (FOSM) technique [11] to Eq. (10) leads to the relationships: E[b ] 1 #E[b ]n , E[m ]" 2 a1 a1 n a1 »[b ] 1 #»[b ]n2 #2C[b , b ]#»[e ], »[m ]" 1 2 m a1 2 a1 n2 a1
(11)
where C[., :] is the covariance. Statistical moments of b , b , e are given in Refs. 1 2 m [9,10] for four different classes of multi-storey buildings. It is usually accepted that damping pdf f is properly represented by a log-normal ma1 model.
4. Conventional serviceability criteria Current serviceability criteria based on human response to wind-induced acceleration can be framed into two classes: 1. RMS acceleration criteria [5,6]. The structure is reliable if p )p*, where p is the a a a RMS value of the acceleration a due to a mean wind velocity u "u* with mean * * return period RM "RM *; RM * is a reference value of RM , p* is an allowable limit of p . a a 2. Expected maximum acceleration criteria [4,7]. The structure is reliable if E[a ])a* , where E[a ] is the expected maximum value of a in *¹ due to .!9 .!9 .!9 a mean wind velocity u "u* with mean return period RM "RM *; a* is an allow* * .!9 able limit of E[a ]. .!9
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As a matter of fact, damping uncertainties deeply propagate on the estimation of the wind-induced acceleration. Referring to methods (1) and (2), p ( , E[a( ] become a .!9 random variables whose pdf are given by
A
B
1 K2u4 DI a * a , f ( (p ¨ )"2K2u4 DI f a a1 a * ap3( m p a p2( a a
(12)
A
B
1 g2( K2u4 DI a a * a . f f [a( ](E[a( ])"2g2( K2u4 DI a a * aE3[a( ] ma1 E2[a( ] E .!9 .!9 .!9 .!9
(13)
The first and second statistical moments of p ( , E[a( ] can be rigorously determined .!9 a by applying classical probability rules. Simple approximations of these quantities can be obtained by expanding p ( , E[a( ] in Taylor Series around m "m0 . a1 a .!9 a1 Assuming m0 "E[m ] and retaining up to first-order derivative terms, the FOSM a1 a1 technique gives rise to E[p ( ]"p0( , »[p ( ]"1(p0( )2X2[m ] , a a a 4 a a1
(14)
»[E[a( ]]"g2( »[p ( ], a .!9 a
E[E[a( ]]"g ¨ E[p ( ], a .!9 a
(15)
where the superscript 0 denotes quantities evaluated in the expansion point; p0( is a given by Eq. (3) for m "m0 , and X[ ) ] is the coefficient of variation. a1 a1 Retaining up to second-order derivative terms and using the SOSM technique [12], Eq. (14) becomes
A
B
A
B
E[p ( ]"p0( 1#3X2[m ] ; »[p ( ]"1(p0( )2X2[m ] 1# 9 X2[m ] , a a 8 4 a 16 a1 a a1 a1
(16)
Eq. (15) remains unchanged. It is verified that in most cases of engineering interest the pdf of p ( , E[a( ] spread a .!9 up to the point that traditional estimates based on nominal values of m are fully a1 unreliable. The numerical applications at the end of this paper fully confirm this situation.
5. Probabilistic serviceability criteria In accordance with methods stated in Ref. [8], probabilistic serviceability criteria based on human response to wind-induced acceleration can be framed into two classes: (1) Maximum acceleration criteria. The structure is reliable if PK )PK *, where PK is the probability that the maximum acceleration aL in a reference period ¹A*¹ exceeds a limit aL * defined herein as deterministic; PK * is an allowable limit of PK . (2) Acceleration persistence criteria. The structure is reliable if P)P*, where P is the probability, i.e. the fraction of time, that a suitable measure of the acceleration in *¹ exceeds a limit a* defined herein as deterministic; P* is an allowable limit of P.
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Referring to method (1), it is assumed that the maximum acceleration aL in ¹ is the maximum acceleration a in the *¹ interval where the shear velocity u takes its .!9 * maximum value uL in ¹. It is easy to verify that this definition is correct and slightly * prudential in the alongwind direction where acceleration increases with increasing u . * On the other hand, it gives suitable approximations in the crosswind and torsional generalized directions, if the resonant shedding of vortices occurs, as it is typical of building structures, at high u values. Considering single generalized a directions: * = PK " f (L (a(L ) da(L , (17) a aL * == L (a( DuL , m ) f (uL ) f (m ) duL dm , (18) f( L f (L (a( )" a.!9@u*,ma1 .!9 * a1 uL * * ma1 a1 * a1 a 0 0 is the pdf of a( (Eq. (1)) conditioned by the where f (L is the pdf of a(L ; f ( L a.!9@u*, ma1 .!9 a occurrence of uL , m ; f L is the pdf of uL ; uL and m are statistically independent. * a1 u* * * a1 The first and second statistical moments of a(L can be rigorously determined by applying classical probability rules. Simple approximations can be obtained by expanding a( in double Taylor series around uL "uL 0 , m "m0 . .!9 * * a1 a1 Assuming uL 0 "E[uL ], m0 "E[m ] and retaining up to first order derivative terms, * * a1 a1 the FOSM technique leads to the formulae
P
PP
E[a(L ]"(E[a( ])0; .!9 »[a(L ]"(»[a( ])0#[(E[a( ])0]2(4b2X2[uL ]#1X2[m ]), .!9 .!9 * 4 a1 DI @ 0 b"1# a uL 0 , 4DI 0 * a
(19) (20)
where (E[a( ])0, (»[a( ) ])0 are given by Eq. (2) for u "uL 0 , m "m0 ; DI 0 is given by a1 a * a1 .!9 .!9 * Eqs. (4)—(6) for u "uL 0 ; DI @0 is the first derivative of DI 0 with respect to u , calculated a * * a * in uL "uL 0 . * * Retaining up to second-order derivative terms and using the SOSM technique, Eq. (19) becomes E[a(L ]"(E[a( ])0(1#cX2[uL ]#3X2[m ]), 8 .!9 * a1 »[a(L ]"(»[a( ])0#[(E[a( ])0]2](4b2X2[uL ]#1X2[m ] 4 .!9 .!9 * a1 #c2X2[uL ]# 9 X2[m ]#3c2X2[uL ]X2[m ]), 64 4 * a1 * a1 2DI 0DI A0!DI @ 0 DI @ 0 a (uL 0 )2, c"1# a uL 0 # a a * 8(DI 0)2 DI 0 * a a
(21) (22)
where DI A0 is the second derivative of DI 0 with respect to u , calculated in u "uL 0 . a a * * *
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It is to be noted that DI @ 0"DI A0 represent suitable first-order approximations. In a a such case, b"c"1 drastically simplify Eqs. (19) and (21). It is also relevant to point out that, despite the convolution integrals in Eq. (18), the pdf of a(L usually retains the typical shape of the type I extreme distribution of a( (Eq. (1)). Therefore, especially .!9 due to the randomness of u , m , a log-normal pdf should represent a suitable * a1 approximation of the actual distribution of a(L [8,12]. Referring to method (2), a suitable measure of the acceleration in *¹ is represented by its maximum value a . Considering also in this case single generalized a direc.!9 tions: = P"
P
f (J (a(J ) da(J , .!9 a.!9 .!9
(23)
a* == f (J (a(J " f( (a(J Du , m ) f (u ) f (m ) du dm , a.!9 .!9 a.!9@u*, ma1 .!9 * a1 u* * ma1 a1 * a1 0 0
PP
(24)
is the maximum value of a( in *¹ taking is the pdf of a(J ; a(J where f (J .!9 .!9 a.!9 u randomness and m uncertainties into account; f ( is the pdf of a( (Eq. (1)) * a1 a.!9@u*, ma1 .!9 conditioned by the occurrence of u , m ; f is the pdf of u ; and u and m are * a1 u* * * a1 statistically independent. The analogy of Eqs. (18) and (24) points out that FOSM and SOSM expressions of E[a(J ], »[a(J ] are given by Eq. (19) and (21) replacing uL by u . However, since .!9 * * .!9 X[u ] is much greater than X[uL ], FOSM and SOSM techniques often provide, in * * this case, unreliable estimates. Jointly referring to methods (1) and (2) above, it is finally noted that, due to the mathematical structure of Eqs. (18) and (24), the role of damping uncertainties reduces with respect to velocity randomness. Under this point of view, differently from methods (1) and (2), f (m )"d(m !m0 ) and X[m ]"0 often represent ma1 a1 a1 a1 a1 reasonable first-order approximations (d being the Dirac function). It follows that PK +1!F L (uL *), u* *
P+1!F (u*), u* *
(25)
where F L and F are the cumulative distribution functions (cdf) of uL , u ; and uL *, u* u* * * u* * * are the values of u in correspondence of which E[a ]"aL *, a* for m "m0 . It is a1 * .!9 a1 implicit that Eq. (25) applies if X[a ] is small and E[a ] is, as previously assumed, .!9 .!9 a monotonic function of u . *
6. Applications Serviceability criteria herein discussed are applied to calculate the crosswind acceleration at the top of a square plan multi-storey steel building with h"200 m,
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b"40 m; n "0.225 Hz, t (z)"z/h, m "21 333 333 kg, E[m ]"0.0159, X[m ]" yl yl yl yl yl 0.436. Damping is log-normally distributed. Aerodynamic properties are schematized by c "1.2, cl"0.0, c@ "0.0, c@l"!3.4, cJ "0.0, cJ l "0.5; S"0.1, ¸"1.0, $ $ $4 s B "0.25. y Wind is characterized by z "1 m, I (z)"0.4Jb /ln(z/z ), b "4.50, I (z)" 0 u v 0 u u 0.775I (z); ¸ (z)"300(z/300)e (¸ , z in m), e"0.46, ¸ (z)"0.24¸ (z); C "11.5, u u u v u zu C "0.7C ; o"1.25 kg/m3; *¹"600 s [3]. The pdf of u , uL (for ¹"1 yr) are zv zu * * modelled by Weibull and extreme type I distributions (Fig. 1, solid and dotted line):
AB
C A BD
C A BD
u k u k k u k~1 * f (u )" exp ! * , F (u )"1!exp ! * , * * u * u * c c c c
(26)
f L (uL "a expM!a exp(uL !u)!exp[!a(uL !u)]N, u* * * * (27) F L (uL )"expM!exp[!a(uL !u)]N, * u* * where k"1, c"0.333 m/s, E[u ]"0.333 m/s, X[u ]"1.00; a"3 s/m, u"2 m/s, * * E[uL ]"2.192 m/s, X[uL ]"0.195. * * Using the conventional serviceability criterion proposed in Ref. [5], R*"5 years implies u "2.5 m/s; p* "0.049 m/s2 corresponds to a* "0.172 m/s2. Assuming * .!9 .!9 m "m0 , Eq. (2) and (3) give E[y( ])0"0.1484 m/s2(a* , which means that the y1 y1 .!9 .!9 building is reliable. Fig. 2 shows the pdf of E[y( ] for RM *"5 yr. Solid lines correspond to rigorous .!9 probabilistic analyses (RPA); dotted and dashed lines correspond to log-normal models based on statistical moments evaluated through FOSM and SOSM techniques (Table 1). The probability that E[y( ]*a* is 0.315 (by RPA). It follows that .!9 .!9 the above criterion, as well as analogous criteria, is clearly unreliable due to damping uncertainty. Fig. 3 shows the pdf of y(L for ¹"1 yr (with uncertain damping) using the same symbols in Fig. 2. Table 2 gives the mean and RMS values of y(L corresponding to
Fig. 1. Pdf of the shear velocity.
L.C. Pagnini, G. Solari/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 1067–1078
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Fig. 2. Pdf of E[y( ] for RM *"5 yr. .!9
Table 1 Results of expected maximum acceleration criterion (y( in m/s2) .!9 Method
RPA
FOSM
SOSM
E[E[y( ]] .!9 S[E[y( ]] .!9
0.1588 0.0340
0.1484 0.0329
0.1594 0.0329
Fig. 3. Pdf of y(L for ¹"1 yr.
RPA, FOSM and SOSM techniques. It also gives the exceedence probability pL of a conventional perception threshold aL *"0.1 m/s2. It is apparent that the log-normal model of y(L , applied to Eq. (19) and especially to Eq. (21), gives rise to results with excellent approximation.
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Table 2 Results of maximum acceleration criterion (y(L in m/s2) Method
E[y(L ] S[y(L ] PK
Uncertain damping
Deterministic damping
RPA
FOSM
SOSM
RPA
FOSM
SOSM
0.1204 0.0768 0.51
0.1050 0.0601 0.43
0.1211 0.0694 0.47
0.1130 0.0694 0.47
0.1050 0.0555 0.44
0.1134 0.0561 0.51
Fig. 4. Pdf of y(J . .!9
Table 3 Results of acceleration persistence criterion (y(J in m/s2) Method
E[y(J ] .!9 S[y(JR ] .!9 P
Uncertain damping
Deterministic damping
RPA
FOSM
SOSM
RPA
FOSM
SOSM
0.0026 0.0102 0.0017
0.0006 0.0017 —
0.0021 0.0023 —
0.0025 0.0094 0.0014
0.0006 0.0017 —
0.0021 0.0022 —
Fig. 4 shows the pdf of y(J (with uncertain damping). Table 3 gives its mean .!9 and RMS values confirming that, in this case, FOSM and SOSM techniques lead to relevant errors. It also furnishes the exceedence probability P of the perception threshold a*"aL *. Consistent with the persistence properties of the selected wind velocity process, P"0.0017 means that a* is averagely exceeded 15 h per year.
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1077
Calculating (E[y( ])0 (Eq. (2)) as a function of u and inverting it, for .!9 * aL *"a*"0.1 m/s2, uL *"u*"2.156 m/s. Using Eq. (25), PK "0.46, P"0.0015. The * * comparison between these values and those reported in Tables 2 and 3 points out the approximations involved by the application of methods of different level.
7. Conclusions and perspectives It is demonstrated that the uncertain knowledge of structural damping makes classical serviceability criteria for wind-induced acceleration purely conventional and often unreliable. The joint use of suitable damping models, closed-form solutions of building acceleration and uncertainty propagation techniques makes the application of probabilistic serviceability criteria as simple as the use of current conventional procedures. All formulae and criteria herein developed consider single generalized components of the structural acceleration assuming the structure as slenderline-like. First and second statistical moments of these quantities are calculated through Taylor series expansions around the mean values of random parameters. Acceleration limit states are assumed as deterministic. Perspectives for further advances are represented by the application of fully 3-D structural models, by evaluating the pdf and statistical moments of the reference value of the acceleration as suitable combinations of the pdf and statistical moments of its generalized directional components, by expanding Taylor series in a neighbourhood of the checking point [15] consistently with established second-order reliability approaches, by including probabilistic limit states in the formulation [16].
References [1] E. Simiu, R.H. Scanlan, Wind Effects on Structures, Wiley, New York, NY 1996. [2] G. Piccardo, G. Solari, 3-D wind-excited response of slender structures: basic formulation, closed form solution applications, J. Struct. Eng. ASCE. (1998) submitted. [3] G. Piccardo, G. Solari, Closed form prediction of 3-D wind-excited response of slender structures, J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 697—708. [4] P.W. Chen, L.E. Robertson, Human perception thresholds and horizontal motion, J. Struct. Div. ASCE 98 (1972) 1681—1695. [5] R.J. Hansen, J.W. Reed, E.H. Vanmarcke, Human response to wind-induced motion of buildings, J. Struct. Div. ASCE 99 (1973) 1589—1605. [6] A.W. Irwin, Perception, comfort and performance criteria for human beings exposed to whole body pure yaw vibration and vibration containing yaw and translational components, J. Sound Vib. 76 (1981) 481—497. [7] W.H. Melbourne, T.R. Palmer, Accelerations criteria for buildings undergoing complex motions, J. Wind Eng. Ind. Aerodyn. 41—44 (1992) 105—116. [8] J. Kanda, O. Nakamura, Y. Tamura, K. Uesu, Probabilistic criteria for serviceability limit of wind induced response, Proc. Int. Coll. on Structural Serviceability of Buildings, IABSE, AIPC, IVBH, Goteborg, 1993, pp. 59—66. [9] G. Lagomarsino, Forecast models for damping and vibration periods of buildings, J. Wind Eng. Ind. Aerodyn. 48 (1993) 221—239.
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[10] G. Lagomarsino, L.C. Pagnini, Criteria for modelling and predicting dynamic parameters of buildings, Report ISC-II, 1, Istituto di Scienza delle Costruzioni, University of Genova, Italy, 1995. [11] G. Solari, Evaluation and role of damping and periods for the calculation of structural response under wind loads, J. Wind Eng. Ind. Aerodyn. 59 (1996) 191—210. [12] G. Solari, Wind-excited response of structures with uncertain parameters, Prob. Eng. Mech. 12 (2) (1997) 75—87. [13] G. Solari, Mathematical model to predict 3-D wind loading on buildings, J. Eng. Mech. ASCE 111 (1985) 254—276. [14] G. Solari, Gust buffeting. II: dynamic alongwind response, J. Struct. Eng. ASCE 119 (1993) 383—398. [15] A.M. Hasofer, N.C. Lind, An exact and invariant first order reliability format, J. Eng. Mech. Div. ASCE 100 (1974) 111—121. [16] L.C. Pagnini, G. Solari, Probabilistic damping modeling and building reliability under wind loads, Proc. Structural Engineers World Congress, San Francisco, USA, 1998, in press.