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The 15th International Conference on Wind Engineering
Beijing, China; September 1-6, 2019
Codified procedure for buffeting response of buildings and bridges
Kristoffer Hoffmann a, Marie Louise Pedersen a, Svend Ole Hansen a,b,c
where πb(π¦, π§) = ππ£m(π¦, π§)πf(π¦, π§)πΌR(π¦, π§)ππ’(π¦, π§). The function ππ’(ππ¦, ππ§) is the correlation
coefficient for the along-wind turbulence component separated by ππ¦ = |π¦2 β π¦1| and ππ§ =|π§2 β π§1|.
The dynamic part of the deflection may as an approximation be expressed as π(π‘)π(π₯, π¦),
where π(π‘) is a stochastic amplitude function. The spectral density of π(π‘) is then given by
ππ(π) = |π»π(π)|2ππ(π),
where π»π(π) is the frequency response function associated with mode π and the natural fre-
quency ππ.
Let the power spectrum of the along-wind turbulence be denoted by ππ’. The generalized load
where πr(π¦, π§) = ππ£m(π¦, π§)πf(π¦, π§)π(π¦, π§)βππ’(π¦, π§, π). The function ππ’(ππ¦, ππ§ , π, π£m) is the
normalized co-spectrum for the along-wind turbulence components separated by ππ¦ and ππ§. The
variance of the stochastic amplitude function π(π‘) is then found by an integration of the spectral
density, i.e.
ππ2 = β« |π»π(π)|2ππ(π)
β
0
ππ.
The damping consists of both aerodynamic and structural damping, and is often relatively low,
meaning ππ βͺ 1, and ππ(π) usually has most of its values at frequencies below ππ. In this case
the so-called white noise approximation may be applied3
Figure 3 illustrates that the proposed product format is a very good approximation of the quadruple
integral for all values of ππ§. The admittance combination factor of π/2 is clearly necessary to
π π β πΎπ
πU ππ
include to ensure a correct asymptotic behavior when both ππ¦ and ππ§ approach infinity. When
both ππ¦ and ππ§ approach zero, the product of double integrals (Eq. (4)) approaches π/2.
4.2 Resonant response factor
Let the mode shape along the π§ dimension be parabolic, i.e. ππ§(π§) = (π§/π)2. The resonant re-
sponse factor π 2 determined using the quadruple integral (Eq. (3)), using the product of two dou-
ble integrals (Eq. (5)), and using the proposed product format (Eq. (7)) are all presented in Figure 4,
using the expression π 2/(π2
2πΏππN,π’(ππ)) , for ππ¦ = 0 and ππ¦ = 0.5 β ππ§. The plots illustrate that
the proposed product format is a very good approximation of the quadruple integral for ππ§ β₯ 10.
The admittance combination factor of π/2 is clearly necessary to include to ensure a correct as-
ymptotic behavior when both ππ¦ and ππ§ approach infinity. For ππ§ < 10 the proposed format
is conservative.
Figure 4. The expression π 2/(π2
2πΏππN,π’(ππ)) evaluated for a parabolic mode shape determined by numerical integra-
tion of the quadruple integral (Eq. (3)), using the product of two double integrals (Eq. (5)), and using the proposed
product format (Eq. (7)). The plots consider ππ¦ = 0 (left figure) and ππ¦ = 0.5 β ππ§ (right figure).
For a typical type of structure covered by the present Eurocode, the natural frequency in Hertz of
may be approximated by ππ β 50/π, and adopting typical values of the decay constant, ππ§ = 10,
and the characteristic mean wind velocity, π£m = 25 m/s, implies that ππ§ β 20. This underlines
that typical structures correspond to situations where the proposed product format is a very good
approximation.
As seen in Figure 4, the three different formula has different asymptotic limits for ππ§ β 0, see
also Section 3.2. The quadruple integral approaches (5/6)2, the product of double integrals ap-
proaches π 2β β (5 6β )2, while the proposed product format approaches 1.
Note that the comparison of the resonant response factor for a linear mode shape, i.e. ππ§(π§) =π§/π, is equivalent to the results presented for the background response factor in the previous sub-
section, since the response influence function is assumed linear.
5 REPONSE INFLUENCE FUNCTIONS AND MODE SHAPES WITH CHANGING SIGNS
For non-constant sign response influence functions or mode shapes, it is not possible to express
the fluctuating part of the structural response in terms of the mean response. Instead, the fluctuating
The 15th International Conference on Wind Engineering
Beijing, China; September 1-6, 2019
part of the structural response may be expressed using a non-uniform reference load distribution
for the background response and by equivalent static loads for the resonant response. The aerody-
namic admittance functions may then be expressed using their analytic expressions, which exist
for several fundamental structure types, such as cantilever and simply supported structures. The
mode shapes associated with these fundamental structures are illustrated in Figure 5.
Cantilever Simply supported
Figure 5. Mode shapes associated with a cantilever and a simply supported structure with three supports.
For simplicity, only the resonant response factor is considered in the following, but the principles
are equivalent for the background response factor.
Let the model structure have its main direction along π§, and a uniform response influence func-
tion and mode shape along π¦. The characteristic response due to resonant turbulence alone is then π max,π = πp β 2 β πf β πΌπ’ β πm β π β π β πΎm β π ,
where the resonant response factor is
π 2 =π2
2πΏππN,π’(ππ) β πU
2 (2
πβ ππ¦) β πL
2(ππ§).
The analytic expressions of the one-dimensional admittance function for the cantilever structure is
πL2(π) =
2
3 β πβ
2
π2+
8
π4β πβπ (
2
π2+
8
π3+
8
π4),
and for a simply supported structure with three supports it is
The maximum amplitude of the fluctuating wind load per unit length may then be evaluated as πΉw,f = 2 β πp β πf β πΌπ’ β πm β π β πΎπ ,L β π ,
where
πΎπ ,L =πΎm
1π β« ππ§(π§)πΌR(π§)
π
0ππ§
=1
1π β« ππ§
2(π§)π
0ππ§
.
Note that this expression for the load distribution factor πΎπ ,L is not identical to the expression
defined in Section 3.4 for a constant sign mode shape. For the cantilever model structure the load
distribution factor becomes πΎπ ,L = 3 and for the simply supported model structure πΎπ ,L = 2.
Since the fluctuating part of the structural response due to resonant turbulence is expressed rel-
ative to the mode shape, the fluctuating wind load determined above is in principle an equivalent
static load. This also implies that the maximum amplitude of the fluctuating wind load per unit
length πΉw,f does not depend on the response influence function.
6 PERSPECTIVE
One of the advantages of the new procedure is that the actual correlation of pressures and forces
is taken into account via the cross-sectional admittance function. The current assumption of equiv-
alence between wind pressure correlation and velocity correlation does not provide consistent re-
sults, and this non-consistency could be removed by the new approach proposed. The approach
also facilitates the use of structure-specific wind load characteristics, such as aerodynamic admit-
tance functions, determined directly from wind tunnel experiments.
The new procedure will relatively easily accommodate codified extensions in form of mode shapes
with changing sign, across-wind buffeting response, and torsional buffeting response. The proce-
dure described above has been applied successfully in buffeting response analyses for long-span
cable-supported bridges. It is believed that the same approach may turn out to give a consistent
and operational description of buffeting wind actions on buildings.
7 REFERENCES
1 EN1991-1-4, Eurocode 1: Actions on structures β General actions β Part 1-4: Wind actions, CEN, 2007. 2 A. G. Davenport, The response of slender, line-like structures to a gusty wind. Proceedings of the Institution of Civil Engineers, 23(3), 389β408, 1962. 3 C. Dyrbye and S.O. Hansen, Wind Loads on Structures, John Wiley & Sons Ltd, 1997. 4 S.O. Hansen and S. Krenk, Dynamic along-wind response of simple structures, Journal of Wind Engineering and