Influence of the Non-Linearity of the Aerodynamic Coefficients on the Skewness of the Buffeting Drag Force. Vincent Denoël *, 1), Hervé Degée 1)

Transcription

1 Inflence of the Non-Linearity of the Aerodynamic oefficients on the Skewness of the Bffeting rag Force Vincent enoël *, 1), Hervé egée 1) 1) epartment of Material mechanics and Strctres, University of Liège, Belgim ABSTRAT This paper is devoted to the non linear qasi-steady aerodynamic loading. A linear approximation is often sed to compte the response of strctres to bffeting forces. Some researchers have however shown that it is possible to accont for the non linearity of this loading. This non linearity can come (i) from the sqared velocity or (ii) from the shape of the aerodynamic coefficients (as fnctions of the wind angle of attack). In this paper, it is shown that this second origin can have significant implications on the design of the strctre, particlarly when the non linearity of the aerodynamic coefficient is important or when the transverse trblence is important. INTROUTION Wind loads acting on blff bodies like bridge decks are complex fnctions of the components of the trblence and of the strctral displacements and velocities. In order to simplify the representation of these loads, approached models are generally considered. Since a convenient linear approximation gives accrate reslts in many cases, sch a model has been widely sed dring the last decades (e.g. avenport, 196, Simi and Scanlan, 1978). In its most general formlation, this linear model consists in decomposing the wind loads in three terms: (i) the static wind loading, (ii) the self-excited forces and (iii) the bffeting forces: F () t = F () t + F () t + F () t s se b F () t = F () t + F () t + F () t L Ls Lse Lb F () t = F () t + F () t + F () t M Ms Mse Mb (1) where F () t, FL ( t ) and FM ( t ) represent respectively the drag and lift forces and the pitching moment. This general definition of the loading involves the well-known fltter derivatives and aerodynamic transfer fnctions (e.g. hen and Kareem, ). After having

2 measred these fnctions in wind tnnel experiments, the dynamic response and stability stdies of the whole strctre can be realized. As a particlar case of this linear approximation model, the linear qasi-steady theory (Fig. 1) provides very particlar approximations of the fltter derivatives and of the transfer fnctions in terms of the sal aerodynamic coefficients (, L, M ) and their derivatives ' ' ' with respect to the angle of attack (, L, M ). Even if this model is limited becase of its inability to represent transient freqency dependent forces, it can however represent correctly the low freqency motions of the strctre. Frthermore this linear qasi-steady theory is also a particlar case of another more general model: the (non linear) qasi-steady theory. Even if it is also limited to low freqency motions, this theory presents however the advantage to give a non linear model for the wind loading. It is ths interesting in the sense that these non linear effects bring new physical phenomena that can t be enlightened with the sal linear models. This paper focses on some of these phenomena (mainly on the effects of non-linearity of the aerodynamic coefficients) and on the way to accont for them in nmerical simlations. Wind Loading Models Linear Approximation Model Qasi-steady Theory Linear Qasi-steady Theory Fig. 1. Schematic view of wind loading models QUASI-STEAY FORMULATION OF THE WIN LOAING In order to clarify the context, a short recall of the sal linear qasi-steady theory is presented first. The aerodynamic coefficients of a bridge deck are determined by measring the aerodynamic forces ( F, F L, F M ) acting on a fixed section placed in a wind tnnel: = F ; = F ; = F ρbv ρbv ρb V L M L M () where ρ, B and V represent respectively the air density, the width of the deck and the constant wind velocity sed for the experiment.

3 V i F L F M F Fig.. Aerodynamic forces (drag, lift, moment) For the range of wind velocities considered in practical applications (high Reynolds nmber), these aerodynamic coefficients can be considered to be independent of the wind velocity. On the other hand these coefficients are very dependent on the angle of attack i of the wind with respect to the bridge deck. This is illstrated in Fig. 3 which represents drag coefficients of some famos Eropean bridges. Any of these coefficients is indeed a nonlinear fnction of the angle of attack. For a convenient comparison, dotted lines represent tangents at the origin.. Normandie Bridge (France) Viadct of Milla (France) i ( ) Vasco da Gama Bridge (Portgal) Fig. 3. Examples of drag coefficients (remona, ) Provided the displacements of the strctre are slow and their amplitdes remain small, Eq. () may be sed to estimate the bffeting forces acting on a bridge deck. The sbseqent developments can be done similarly for the drag and lift forces as well as the pitching moment. For the sake of brevity in the notations, drag forces only are stdied in the following. The developments remain however also valid for the other aerodynamic coefficients. The time varying formlation for the drag force is: 1 F () t = [ i() t ] ρbv () t (3) where the aerodynamic coefficient is now time dependent (trogh the angle of attack i ) and where the wind velocity is also time dependent since it depends on the mean wind speed U and on the trblence components t ( ) and vt ( ) (Fig. 4).

4 In order to partially accont for a flid-strctre interaction, relative vales mst be considered for the wind angle of attack and the wind velocity. With notations of Fig. 4, these qantities can be expressed by: vt () ht () i() t = ArcTan α() t U () t p() t + (4) V t = U + () t p() t + v() t h() t () ( ) ( ) where the pper dot denotes a time derivative. Introdcing Eqs. (4) into Eq. (3) gives the non linear qasi-steady expression of the wind loading. As introdced before, it can be seen that this expression is a complex fnction of the components of the trblence and the motion of the strctre. U (t) V(t) v(t) i(t) h(t) (t) p(t) Fig. 4. isplacements of the strctre The components of trblence are known in a probabilistic way only (Simi, 1974). The most common methods that can be sed to compte the dynamic response of a bridge are therefore: A stochastic analysis procedre (see e.g. logh and Penzien, 1993), which is based, in its most basic formlation, on the comptation of the power spectral density (PS) of the response of the bridge as a fnction of the PS of the trblence components and of the mechanical and aerodynamic properties of the bridge; To se Monte arlo simlations; this consists in generating wind histories and solving several times a deterministic problem by means of step-by-step analyses. This method allows acconting for the complete non linear expression bt is rather slow since many generations are needed for a good accracy. Ths this method shold be sed essentially to check reslts obtained by a stochastic analysis. Linearization of the aerodynamic loading A cople of decades ago, the field of application of stochastic procedres was limited by the abilities of comptation means: probabilistic properties of strctral response cold ths be determined p to the second order only (variances). Therefore the response was generally considered to be Gassian. As the components of the trblence can be considered as Gassian, the classical procedre consists in simplifying the expression of the loading to a linear fnction of the trblence. The loading is then Gassian, and hence, if the strctre can be assmed to behave linearly, the strctral response is Gassian too. After having linearized Eqs. (4), introdced the reslting eqations into Eq. (3), replaced the exact expression of aerodynamic coefficient by a linear approximation, and finally

5 removed the sbseqent qadratic terms, the linear qasi-steady formlation can be obtained (remona, ): 1 ( ) ' ( ) ' ( ) ' ( ) F () t = ρbu p t + + α + + h t ( ) t t v t U U U U (5) It can be seen that this expression is a particlar case of Eqs. (1) since each term is a linear expression of the displacements (or velocities) of the strctre or of the components of the trblence. Since the vertical strctral velocity ( ht ( ) ) and the rotation of the deck ( α () t ) are present in this expression of the drag force, Eq. (5) shows that a copling between vertical, horizontal and torsional motions cold exist. Higher degree polynomial approximation of the aerodynamic loading Nowadays more complex theories allow acconting for a non linear loading provided it is expressed as a polynomial approximation of the actal loading. Instead of linearizing Eq. (3), a higher order polynomial approximation can be sed: 1 l k p( t) ' h( t) ' ( t) v ( t) F () t = ρbu ( ) α t + F kl l+ k U U l= k= U (6) Table 1: Vales of the parameters F kl (enoel, 5) F kl K= k=1 k= k=3 k=4 L= ' ' ' L=1 ' ' ' l= ' ' l=3 ' ' l=4 ' ' This expression is however obtained by neglecting non linear terms of the strctral motions. Non-linear components of bffeting forces only are kept in this approach. The vales of the coefficients F kl in Eq. (6) are given in Table 1 for the first vales of k and l. These coefficients are expressed in terms of the first for derivatives of ( i ) with respect to the angle of attack; these derivatives are defined by:

6 3 4 ' i i ' i () i = + i (7)! 3! 4! It can be seen that the crvatre of the aerodynamic coefficient is present in a second order term of the loading (k=, l=). It can be also checked that Eq. (5) is a particlar case of Eq. (6). In Eq. (6) the terms containing the derivatives of the bridge deck are related to the aerodynamic damping. Becase they have already been widely stdied and since the main aim of the paper concerns the effects of the non linear bffeting forces, these terms will be neglected in the following. Since it is able to inclde any order of non linearity, the non linear model represented by Eq. (6) is very general. In order to enlighten in closed form the most important inflences of the non linear loading, this paper focses on the second order approximation of Eq. (6) only: F () t t () ' vt () () t ' tvt ()() v() t = ρbu U U U U + U (8) The non linearity of the aerodynamic coefficient can therefore be taken into accont p to its second order derivative only. Let s also consider the habital hypothesis of Gassian ncorrelated components of the trblence t ( ) and vt ( ) : [ ] E () t = ; E () t = [ ] E [ t v t ] E v() t = ; E v () t = ()() = ; v (9) With these notations and assmptions, the statistical properties of the loading can be compted p to the third order:

7 _ F () t v = U + U ρbu 4 4 F ' v ' v v = U U U U U ρbu m 1 ρbu + + (3) 4 F U = U 4 v 4 U + ( + ) + U 4 ' v ' v ' v ' v ( ) U U U 6 3 v + ( + ) 6 U (1) In the following these relations will be considered as reference vales. Fig. 5 illstrates these first three non-dimensional statistical moments for several vales of the wind intensities I = / U and I v = v / U. These are compted for the drag coefficients of the bridges given in Fig. 3. The coefficients of the qadratic approximation are obtained by a least sqare fit with Gassian weight distribtion: ' Normandy Bridge : =.748 ; =.476 ; =.66 ' Milla V iadct : =.83 ; =. ; = 3.4 ' Vasco da Gama Bridge : = 337 ; = 3 ; = 1.54 (11) It can be proved that the actal statistical distribtion of the angle of attack is almost Gassian when both wind intensities are almost the same (enoel, 5), which jstifies the se of a Gassian weighting fnction.

8 Mean.5 Variance.5 Skewness Va sc o d a Gama Bridge Viadct of Milla Normandy Bridge I I I...5 Mean Mean..5 I I I...5 Variance Variance I I I...5 Skewness Skewness..5 Fig. 5. Exact statistical characteristics of the loading as a fnction of the wind intensities From these reference vales of the statistical moments of the loading, two particlar approximations can be derived. First approximation : linearization of the aerodynamic coefficient ( = ) As a first approximation, it cold be spposed that the aerodynamic coefficient is linear ( = ). Since non linear terms coming from the expression of the sqared velocity are kept, the sbseqent expression of loading remains non-linear and hence non Gassian (see Eq. (8), in which = ). The approached statistical vales obtained nder this hypothesis are determined by imposing = in Eqs. (1). They are represented in Fig. 6.

9 Mean.5 Variance.5 Skewness Va sc o d a Gama Bridge Viadct of Milla Normandy Bridge I I I Mean Mean..5 I I I Variance Variance I I I Skewness Skewness..5 Fig. 6. Approached statistical characteristics of the loading as a fnction of wind intensities The first developments concerning the non Gassianity of the aerodynamic loading correspond to this hypothesis. Many athors (Grigori, 1986, Benfratello, 1996, Kareem, 1998, Gsella,, Floris, ) have stdied the effects of the wind intensity on the skewness of the loading. These early developments were based on a 1- trblence field ( I v = ). In this case, the comparison of Figs. 5 and 6 shows that the exact and approached means, variances and skewnesses are exactly the same, as the vales along the vertical axis only have to be compared. From a theoretical point of view, this indicates that the aerodynamic coefficients can be linearized provided the transverse wind intensity is eqal to zero. In practical applications, the transverse intensity I v is however generally not eqal to zero and the vertical axis of Figs. 5 and 6 is not sfficient anymore to represent the actal statistical moments. The comparison of both figres shows that a small transverse intensity can modify drastically the vales of the statistical moments. For example, for I = 1% and I v = 1%, the exact and approached non dimensional variances of the drag force on the Viadct of Milla are respectively eqal to 1.1E 4 and 6.9E 4. The originality of the paper lies in this observation and an important contribtion is to provide statistical moments p to the third order, in a - flow and acconting for the non linearity of the aerodynamic coefficients. Regarding the skewness coefficient, the significantly different patterns depicted in Figs. 5 and 6 show that it is considerably affected by the non linearity of the aerodynamic coefficient.

10 Frthermore, Fig. 5 exhibits negative skewness zones (for large ) that can not be explained with the approached model (Fig. 6). onseqently, the non linearity of the aerodynamic coefficient ( ) can trn the positively-skewed probability density fnction into a negatively-skewed one. In terms of extreme vales, this may have very important conseqences. It is interesting to notice that the skewness coefficient associated to Normandy Bridge is significant despite the apparent linearity of its drag coefficient (Fig. 3). Second approximation : linearization of the aerodynamic loading As a second approximation, the eqations of the linear qasi-steady loading cold be derived. In this method, linear terms of the loading only mst be considered: the first three terms of Eq. (8) only are ths kept. The non dimensional statistical characteristics of the loading are now mch simpler than Eqs. (1): _ (3) () t F ' m v F 3 U U F = ; = 4 + ; = 1 ρbu 1 1 ρbu ρbu (1) Exactly as for the statistical moments given in Eq. (1), these vales are obtained thanks to the theory of probabilities. Va sc o da Gama Bridge Viadct of Milla Normandy Bridge.5. Mean onstant Vale : Mean = Mean onstant Vale : Mean =.83 Mean onstant Vale : Mean = I I I Variance Variance Variance Skewness onstant Vale : Skewness = Skewness onstant Vale : Skewness = Skewness onstant Vale : Skewness =..5

11 Fig. 7. Approached statistical characteristics of the loading as a fnction of wind intensities They are plotted in Fig. 7. It can be checked that this method provides inaccrate reslts, even on the vertical axis. Frthermore since the loading is now Gassian, this method gives a skewness coefficient obviosly eqal to zero. RESPONSE OF STRUTURES TO NON GAUSSIAN LOAINGS Even if the theoretical backgrond concerning the stdy of the dynamic response of strctres sbjected to non-gassian loading is qite old (Ltes and H, 1986, Soize, 1978), recent researches have led to new ways of applying this theory in practical applications. Two major philosophies can be distingished: Third and forth order characteristics can be represented by bispectra and trispectra, similarly as second order characteristics can be represented by PSs (Grley et al., 1997, Kareem, 1998, Spanos, 1983); Another possibility consists in solving a particlar set of eqations (the moment eqations) derived from the Fokker-Planck-Kolmogorov eqations (i Paola, 199). The main disadvantage of this method is that the force mst be represented by a Markov chain; bt when this condition is flfilled, this second method is mch faster that the first one since probabilistic characteristics of the response can be estimated by solving a simple set of algebraic eqations. Amongst both of them, this paper is based on the first one. The theoretical basis of this method is not presented in the paper bt can be fond in the literatre (see e.g. Grley et al., 1997). The method is here applied to an interesting academic application: the stdy of a single degree of freedom system sbjected to the simplest qadratic loading: ( ) x () t + ξϖ x () t + ϖ x () t = γ U + U() t + () t (13) where ξ and ϖ represent the strctral characteristics, γ = ρb / is spposed to be constant and t ( ) represents a zero-mean Gassian and Ornstein-Uhlenbeck process whose PS is given by: S a ( ω) = π a + ω (14) where = I U is the variance of the process, I is the trblence intensity and a is a freqency-shaping parameter. This academic application is worthy of note becase it can be sed to estimate the effects of the non linearity of the qadratic loading in a - flow (Eq. (8)). This is discssed in the next section.

12 Second order response Even if the loading is non linear and hence non Gassian, its PS ( S F ( ω ) ) can be expressed in terms of the PS of the trblence ( S ( ω ) ) (e.g. Floris, ). After mltiplication by the transfer fnction of the system, the PS of the response ( S ( ω ) ) can be obtained and finally, after integration along the freqencies, the variance of the displacement can be obtained. This is the exact variance. As a comparison, the variance of the x qad response ( x lin ) obtained by neglecting the non-linear terms of the loading can also be compted. The ratio between these two variances is a measre of the inflence of the qadratic term of the loading on the second order characterization of the strctral motion. An approached (becase it is valid for small trblence intensities, namely smaller than %) expression of this ratio is given by (Appendix A): ( ) I 1 / + ϖ a Ψ 1+ ( 1 ) 1 I + Ψ ( ϖ / a ) x qad x lin (15) where Ψ represents the ratio between the qasi-static component and the total variance of the response. Fig. 8 illstrates this ratio of variances in the most realistic case (which appears to be also the worst case), namely when ϖ / a is mch larger than nity x qad x lin =.5 1. = I Fig. 8. Inflence of the qadratic term of the loading on the variance of the displacement (as a fnction of the dispatching of energy between qasi-static and dynamic contribtions) This figre shows that the non linear term of the loading prodces a more significant inflence when the response of the strctre is essentially dynamic and not qasi-static (i.e. when Ψ= ). It can be seen that for sal vales of the wind intensity ( I 15% to I % ), the qadratic term of the loading doesn t affect significantly the variance of the response. This observation shold however be mitigated if the coefficient of () t in the loading was larger than the coefficient of U.

13 This figre shows also that the ratio of the variance of the response for Ψ= and Ψ = 1 1+ I 1+ I. Frthermore, for any Ψ this ratio cold be estimated by: is eqal to ( ) ( ) f I Ψ 1 ( 1 )( 1 I ) I +, + Ψ + xqad Ψ ( Ψ ) = = = 1+ ( 1 Ψ) I x I qad, QS ( Ψ= 1) (16) This shows that the exact variance of the response nder qadratic loading x qad can be estimated by mltiplying the qasi-static contribtion of the variance by a simple fnction of Ψ and I. Note that the fnction given in Eq. (16) is valid for an Orstein-Uhlenbeck loading process only. In practical applications, the trblence can very rarely be considered as an Orstein- Uhlenbeck process; bt let s sppose that the statistical moments of the loading have been compted following the developments of this section. This comptation implies simple algebraic operations only. If the strctre is very stiff, its response is mainly qasi-static and, in this case, the variance of the strctre is obtained by dividing the variance of the loading by the sqared stiffness. In practical applications, the response is divided into qasi-static and dynamic components. In this more general case, Eq. (16) can be sed to give an estimation of the complete variance. Indeed, the response inclding these two components can be estimated from the knowledge of the qasi-static conterpart only (nder qadratic loading). Althogh the fnction given in Eq. 16 is valid for an Ornstein-Uhlenbeck process, it cold be applied in good approximation to most sal PS fnction of wind trblence (Harris, Scanlan, avenport, Erocode 1),(enoel, 5). This approximation presents the advantage to give very fast reslts. Third order analysis Several athors (Soize, 1978, Grley et al., 1997) have shown that the extreme vales of non Gassian processes depend of their higher order statistical characteristics, namely their skewness and krtosis defined by: ( n ) _ where mx = E ( x x) m m γ = = (17) (3) (4) x x 3 ; γ x x n represents the nth centred moment of the process x. Grley et al. propose to compte the extreme vales as if the process was Gassian and then to mltiply this first estimation by a correcting factor that acconts for the non Gassianity (see Fig. 9). This factor is expressed as a fnction of the mean crossing rate (ν) and of the dration of the observation T. The correcting factor is of corse eqal to nity for a Gassian process, for which γ 3 = andγ = 4 3.

14 e 4 Fig. 9. orrecting factor proposed by Grley et al. to accont for the effect of non Gassianity on extreme vales. If it is desired to estimate the effects of the non linearity of the loading on the extreme vales of the displacement, it is ths necessary to compte higher order statistical characteristics of the response. Exactly as the variance can be obtained by integration of the PS, the third centred moment can be estimated by integration of another mathematical qantity: the bispectrm. Since, in or developments, the loading (Eq. (8)) is considered as a polynomial form of a Gassian process (t), the bispectra of the loading and of the response can be determined in an analytical way. If the non linear term of the loading was neglected, the response wold be Gassian and the skewness coefficient wold be eqal to zero. The ratio γ 3x γ qad 3x similar to the ratio sed lin in the previos paragraph is ths meaningless. Instead of this ratio it is ths chosen to evalate the ratio of the skewness coefficient of the response to the skewness coefficient of the loading. Fig. 1 represents this ratio as a fnction of the damping coefficient, the circlar freqencies and the parameter a. These crves have been obtained in closed form (by analytical integration of the bispectrm of the response) withot any other hypothesis. The reslting (heavy) relations are not provided in this paper bt can be fond in the literatre (enoel, 5). This figre points ot the convenient possibility to determine the skewness of the response as a fnction of the skewness of the loading.

15 3x 3f =.5 =.1 =. = = a Fig. 1. Importance of the skewness of the response as a fnction of the skewness of the loading, the damping coefficient and the eigen freqency (closed form) It can be observed that: the skewness of the response is always smaller than the skewness of the loading; the skewness of the response is small for lightly damped or soft strctres; These observations can be jstified by considering that the response of the strctre is composed of two contribtions: (i) the qasi-static contribtion, shaped like the applied force and ths characterized by the same skewness coefficient, and (ii) the dynamic component, rather shaped like a Gassian process. It seems ths obvios to obtain an intermediate skewness coefficient for the response, this coefficient being smaller for strctres having an important dynamic contribtion, i.e. for lightly damped and soft strctres. The analytical relations sed to draw Fig. 1 can be sefl in several applications. Let s imagine for instance that the actal PS of the trblence cold be approached by an Ornstein-Uhlenbeck process (e.g. by fitting parameter a). Several athors (Mscolino, 1995, Floris, ) have proposed some formlations for this eqivalence which consists in replacing the actal PS by an Ornstein-Uhlenbeck one. In the context of a fast estimation of the non Gassianty of the response, Fig. 1 shows that the skewness coefficient of the loading cold be sed as a bondary vale for the design of the strctre. If the dynamic component of the response is important, its skewness coefficient can be significantly redced and Fig. 1 can help giving a better approximation of it. This vale can be simply estimated by the knowledge of: the skewness coefficient of the loading, the damping coefficient of the strctre, the eigen freqency of the strctre, an eqivalent shaping factor a which is clearly mch simpler than compting and integrating the exact bispectrm.

16 ONLUSIONS This paper has presented two important featres in the analysis of strctres sbjected to non Gassian loadings. The first one consists in the determination of the statistical characteristics of the loading. They are commonly established by spposing that the aerodynamic coefficients can be linearized. This linearization can lead to a significant inaccracy on the statistical characteristics. The developments presented in this paper were limited to the second order derivative of the aerodynamic coefficients with respect to the angle of attack bt the same reasoning cold be easily extended to higher order polynomial approximations. As a second step, it has been shown how the statistical characteristics of the loading cold be sed to estimate the statistical characteristics of the response. In this part, the heavy rigoros analysis method has been avoided and a simpler approach, based on simple and fast comptations, has been retained. This effortless method can give estimations of the response that cold be sed, for instance, for a pre-design of a strctre. AKNOWLEGMENTS The athors wold like to acknowledge the Belgian National Fnd for Scientific Research. APPENIX Under the loading conditions of Eqs. (13) and (14) the PS of the non linear loading force is given by: S F 4 4a I I = + π + ω + ω a 4a ( ω) ( γu ) (A.1) where the first and second terms in the brackets respectively come from the linear and the non linear conterparts of the loading. The traditional white noise approximation (avenport, 196) prescribes the estimation of the response as the sm of the backgrond and the resonant components. This method can be applied to determine the response to the linear loading on the one hand and to the non linear one on the other hand. Their ratio is given by: x + qad xqad, qs xqad, dy xqad, qs x,, lin qs xqad dy xlin, dy = = + x,, lin xlin xlin qs xlin xlin dy xlin { 13 Ψ 1 Ψ (A.) which is now a fnction of the qasi-static and dynamic contribtions nder the linear and non linear loading. Recalling that the qasi-static response is obtained by dividing the variance of the loading by the stiffness and that the dynamic response is expressed as a fnction of the PS of the loading at the natral freqency, it is possible to write:

17 qad 4I + I I = = = 1 + 4I x, 4 qad qs F x, qs F lin lin (A.3) 4 I I S + qad = = a + ϖ 4a + ϖ = 1 + I S I lin a + ϖ ( a ) ( a ) ( ϖ ) 1 ϖ / x, + qad dy F x, + lin dy F ( ϖ) 4 ϖ / (A.4) The introdction of Eqs. (A.3) and (A.4) into (A.) is eqivalent to Eq. (15). This relation is approached becase it is based on the white noise approximation. A deeper investigation into this problem (enoel, 5) shows that this approximation is however sitable if the trblence intensity is smaller than %, which reslts in discrepancies smaller than 5%. REFERENES Benfratello S., Falsone G., Mscolino G. (1996), Inflence of the qadratic term in the along wind stochastic response of SOF strctres. Engineering Strctres, Vol. 18, hen X., Kareem A. (), Advances in modelling of aerodynamic forces on bridge decks, Jornal of Engineering Mechanics, Vol 18(11), logh R.W., Penzien J. (1993), ynamics of Strctres, Mc Graw-Hill : ivil Engineering series (second edition). remona., Focriat J.. (), omportement a vent des ponts, Presses de l Ecole Nationale des Ponts et hassées, Paris (in french). avenport A.G. (196), The response of slender line-like strctres to a gsty wind, Proceedings of the Institte of ivil Engineers, Vol 3, enoël V. (5), Application des méthodes d analyse stochastiqe à l étde des effets d vent sr les strctres d génie civil, Ph thesis, University of Liège, Belgim (in french). i Paola M., Mscolino G. (199), ifferential moment eqations for FE modelled strctres with geometrical non-linearities, International Jornal of Non Linear Mechanics, Vol 5(4), Floris., Valloni (), Non Gassian response to random wind pressres : a comparison of different approaches, Eropean onference on Strctral ynamics, Erodyn, Mnich (pp ). Grigori M. (1986), Response of linear systems to qadratic gassian excitation, Jornal of Engineering Mechanics ASE Vol 11(6) Grley K., Tognarelli A., Kareem A. (1997) Analysis and simlation tools for wind engineering, Probabilistic Engineering Mechanics, Vol 1(1), Gsella V., Materazzi A.L. (), Non Gassian along wind response analysis in time and freqency, Engineering Strctres Vol,

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