Estimations of the Influence of the Non-Linearity of the Aerodynamic Coefficients on the Skewness of the Loading. Vincent Denoël *, 1), Hervé Degée 1)

Transcription

1 Estimations of the Inflence of the Non-Linearity of the Aerodynamic oefficients on the Skewness of the Loading Vincent enoël *, 1), Heré egée 1) 1) epartment of Material mechanics and Strctres, Uniersity of Liège, Belgim ABSTRAT This paper is deoted 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 hae howeer shown that it is possible to accont for the non linearity of this loading. This non linearity can come (i) from the sqared elocity or (ii) from the shape of the aerodynamic coefficients (as fnctions of the wind angle of attack). In this paper, we show that this second origin can hae significant implications on the design of the strctre, particlarly when the non linearity of the aerodynamic coefficient is important (obios, isn t it?) or when the transerse 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 elocities. In order to simplify the representation of these loads, approached models are generally considered. Since a conenient linear approximation gies accrate reslts in many cases, sch a model has been widely sed dring the last decades (e.g. aenport 196, 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 respectiely the drag and lift forces and the pitching moment. This general definition of the loading inoles the well-known fltter deriaties and aerodynamic transfer fnctions (e.g. Simi, 1996). After haing measred these fnctions in wind tnnel experiments, the dynamic response and stability stdies of the whole strctre can be realized.

2 As a particlar case of this linear approximation model, the linear qasi-steady theory (Fig. 1) proides ery particlar approximations of the fltter deriaties and of the transfer fnctions in terms of the sal aerodynamic coefficients (, L, M ) and their deriaties ' ' ' with respect to the angle of attack (, L, M ). Een if this model is limited becase of its inability to represent transient freqency dependent forces, it can howeer 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. Een if it is also limited to low freqency motions, this theory presents howeer the adantage to gie a non linear model of 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. In this paper, we focs on some of these phenomena (mainly on the effects of nonlinearity of the aerodynamic coefficients with regard to the angle of attack of the incident wind) and on the way to accont for the loading terms related to them. Wind Loading Models Linear Approximation Model Qasi-steady Theory Linear Qasi-steady Theory Fig. 1. Schematic iew of wind loading models QUASI-STEAY FORMULATION OF THE WIN LOAING In order to clarify the context, we first present a short recall of the sal linear qasisteady theory. 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 respectiely the air density, the width of the deck and the constant wind elocity sed for the experiment. V i F L F M F Fig.. Aerodynamic forces (drag, lift, moment)

3 For the range of wind elocities considered in practical applications (high Reynolds nmber), these aerodynamic coefficients can be considered to be independent of the wind elocity. On the other hand these coefficients are ery dependent of the angle of attack i of the wind with respect to the bridge deck. This is illstrated at Fig. 3 which represents aerodynamic coefficients of seeral famos Eropean bridges. Any of these coefficients is indeed a non-linear fnction of the angle of the angle of attack. For a conenient comparison, dotted lines represent tangents at the origin.. L M Normandie Bridge (France) Viadc de Milla (France) Vasco de Gama Bridge (Portgal) i ( ) -1 1 i ( ) Fig. 3. Examples of aerodynamic coefficients i ( ) Proided the displacements of the strctre are slow or their amplitde remains small, Eqs. may be sed to estimate the bffeting forces acting on a bridge deck. For example, the drag force can be estimated by: 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 elocity is also time dependent since it depends of the mean wind speed (U ) and of the trblence components ( and ). In order to partially accont for a flid-strctre interaction, relatie ales mst be considered for both the wind angle of attack and the wind elocity. With notations of Fig. 4, these qantities can be expressed by: t () ht () i() t = ArcTan α() t U () t p() t + (4) V t = U + () t p() t + () t h() t () ( ) ( )

4 where the pper dot denotes time deriaties. Introdcing Eqs. 4 into Eq. 3 gies 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) (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, 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 properties of the bridge; The se of Monte arlo simlations, which consists in generating wind histories and soling seeral 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 (ariances), and therefore this response was generally considered as 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 behae linearly, the strctral response is Gassian too. After haing linearized Eqs. 4, introdced the reslting eqations into Eq. 3, replaced the exact expression of aerodynamic coefficient by a linear approximation, and finally remoed the sbseqent qadratic terms, the linear qasi-steady formlation can be obtained: 1 ( ) ' ( ) ' ( ) ' ( ) F () t = ρbu p t + + α + + h t ( ) t t 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 elocities) of the strctre or of the components of the

5 trblence. Since the ertical strctral elocity ( 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 ertical, horizontal and torsional motions cold exist. Higher degree polynomial approximation of the aerodynamic loading Een if the theoretical possibilities of stdying the dynamic response of strctres sbjected to non-gassian loading, i.e. to forces expressed by non-linear fnctions of the wind trblence, is qite old now (Ltes 1986, Soize 1978), recent researches hae deeloped new means of applying this theory in practical cases. Two main ways 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 1997, Kareem 1998); Another possibility consists in soling a particlar set of eqations (the moment eqations) deried from the Fokker-Planck-Kolmogoro eqations (i Paola, 199). The main disadantage of this method is that the force mst be represented by a Marko 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 soling a simple set of algebraic eqations. Both of these methods allow acconting for a non linear loading proided it is expressed as a polynomial approximation of the actal loading. So instead of linearizing Eq. 3, a higher order polynomial approximation can be sed : 1 l k p( t) ' h( t) ' ( t) ( t) F () t = ρbu ( ) α t + F kl l+ k U U l= k= U (6) Table 1: Vales of the parameters F kl k= k=1 k= k=3 k=4 l= '' ' ''' '' '''' ' l=1 ' ''' '' '''' ' l= ' ''' '' '''' l=3 ' ''' '' '''' l=4 ' ''' '' '''' This expression is howeer obtained by neglecting non linear terms of the strctral motions. This means that only non-linear components of bffeting forces can be acconted for F kl

6 in these approaches. The ales of the coefficients F kl in Eq. 6 are gien in Table 1 for the first ales of k and l.. These coefficients are expressed in terms of the first for deriaties of () i with respect to the angle of attack; these deriaties are defined by : 3 4 ' i '' i ''' i '''' () i = + i (7)! 3! 4! '' It can be seen for example that the cratre 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. RESPONSE OF STRUTURES TO NON GAUSSIAN LOAINGS Amongst both aailable methods to compte the response of a strctre to a non Gassian loading we follow the one proposed by Kareem and Spanos. The theoretical basics of this method won t be presented in the paper (see e.g. Grley 1997). We will jst present some interesting reslts that hae been obtained with it. Let s consider the eqation of motion goerning the dynamics of a single degree of freedom system sbjected to a particlar loading (Eq.8): ( ) x () t + ξϖ x () t + ϖ x () t = γ U + U() t + () t (8) 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 of which PS is gien by: S α σ ω = π α + ω ( ) (9) where σ = I U is the ariance of the process, I is the trblence intensity and α is a freqency-shaping parameter. Second order response Een 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 ariance of the displacement σ can be obtained. x qad As a comparison, we also compte the ariance of the response ( σ x lin ) obtained by neglecting the non-linear terms of the loading. The ratio between these two ariances is ths a measre of the inflence of the qadratic term of the loading on the second order characterization of the strctral motion. An

7 approached expression of this ratio is gien at Eq. 1, where Ψ represents the ratio between the qasi-static components of the PS and the total ariance of the response. ( ) σ x 1 / qad I + ϖ α Ψ 1 ( 1 ) 1 I σ + x + Ψ ( ϖ / α) lin (1) Fig. 5 illstrates this ratio of ariances in the most realistic case (which appears to be also the worst case), namely when ϖ / α is mch larger than nity x qad x lin =.5 1. = I Fig. 5. Inflence of the qadratic term of the loading on the ariance 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 ales of the wind intensity ( I 15% to I % ), the qadratic term of the loading doesn t affect significantly the ariance of the response. This obseration shold howeer be mitigated if the coefficient of () t in the loading wold hae been larger than the coefficients of U. This figre shows also that the ratio of the ariance 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 σ +, + Ψ + xqad Ψ ( Ψ ) = = = 1+ ( 1 Ψ) σ I x I qad, QS ( Ψ= 1) (11) This obseration is interesting since it shows that the exact ariance of the response nder qadratic loading σ can be estimated by mltiplying the qasi-static contribtion of the x qad ariance by a simple fnction of Ψ and I.

8 Third order analysis Seeral athors (Soize 1978, Grley 1997) hae shown that the extreme ales 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 = = (1) (3) (4) x x 3 ; γ σ x σ x n represents the nth centred moment of the process x. Grley and Kareem propose to compte the extreme ales 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. 6). This factor is expressed as a fnction of the mean crossing rate (ν) and of the dration of the obseration T. The correcting factor is of corse eqal to nity for a Gassian process, for which γ 3 = andγ = 4 3. e 4 Fig. 6. orrecting factor proposed by Grley and Kareem to accont for the effect of non Gassianity on extreme ales. If it is desired to estimate the effects of the non linearity of the loading on the extreme ales of the displacement, it is ths necessary to compte higher order statistical characteristics of the response. Exactly as the ariance 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 deelopments, 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 shold be neglected, the response wold be Gassian and the skewness coefficient wold be eqal to zero. The ratio γ 3 x γ qad 3 x, similar to the lin ratio sed in the preios paragraph to characterize the second order response, is ths meanless. It is then chosen to ealate, instead of this ratio, the importance of the non linearity on the skewness of the response, related to the skewness of the loading. Fig. 7

9 represents the ratio between these ales for seeral damping coefficients and eigen freqencies. 3x 3f =.5 =.1 =. =.5 = Fig. 7. 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 obsered 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 obserations 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 obios to obtain an intermediate skewness coefficient for the response, this coefficient being smaller for strctres haing an important dynamic contribtion, i.e. for lightly damped and soft strctres. The analytical relations sed to draw figre 7 can be sefl in seeral 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 α). Instead of establishing and integrating the bispectrm of the response, these analytical relations gie directly the ales of the skewness coefficient of the response, which can be sed to estimate the non Gassianity of the response and hence its extreme ales. NON LINEAR LOAING INLUING THE NON LINEARITY OF THE AEROYNAMI OEFFIIENTS Let s come back to the analysis of the bffeting response of strctres sbjected to a trblent wind flow. The polynomial approximation of Eq. 6 is here limited to the second order, leading to:

10 '' F () t () t ' () t () t ' ()() t t () t = ρbu U U U U + U (13) This means that the non linearity of the aerodynamic coefficient can be taken into accont p to its second order deriatie only. We also sppose, as it is often the case, that the components of the trblence t ( ) and t ( ) are independent Gassian processes: [ ] E () t = ; E () t = σ [ ] E [ t t ] E () t = ; E () t = σ ()() = ; (14) With these notations and assmptions, the statistical properties of the loading can be compted p to the third order: _ '' F () t σ σ = U + U ρbu 4 4 σ F σ ' σ σ ' σσ σ = U U U U U ρbu m 1 ρbu '' + + (3) 4 F 3 σ 3 4 U = U '' 4 σ 4 σσ U σ + ( + ) + U 4 ' ' '' σ ' σσ ' '' σσ ( ) U U U 6 3 '' σ + ( + ) 6 U (15) In the following these relations will be considered as reference ales. Fig. 8 illstrates these three first non-dimensional statistical moments for seeral ales of the wind intensities I = σ / U and I = σ / U. These are compted for the drag coefficient of the Viadct of Milla (see Fig. 1). The coefficients of the qadratic approximation are obtained by a least sqare fit with Gassian weight distribtion 1 : 1 It can be proed that the actal statistical distribtion of the angle of attack is almost Gassian when both wind intensities are almost the same (enoel, 5)

11 = = = (13) ' ''.83 ;. ; 3.4 I Mean I I Variance.5 5. I I Skewness I Fig. 8. Exact statistical characteristics of the loading as a fnction of the wind intensities From these reference ales of the statistical moments of the loading, two particlar approximations can be deried. 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 elocity are kept, the sbseqent expression of loading remains non-linear and hence non Gassian. Only the last term is remoed from Eq. 13. The approached statistical ales obtained nder this '' hypothesis are determined by imposing = in Eqs Mean Variance Skewness I I I I.5 5. I.5 5. I Fig. 9. Approached statistical characteristics of the loading as a fnction of wind intensities The first deelopments concerning the non Gassianity of the aerodynamic loading correspond to this hypothesis. Many athors (Grigori 1986, Benfratello 1996, Kareem 1998, Gsella, Floris ) hae stdied the effects of the wind intensity on the skewness of the loading. Their deelopments were mainly based on a one-dimensional trblence field ( I = ). In this case, the comparison of Figs. 8 and 9 shows that the exact and approached means, ariances ad skewnesses are exactly the same, as only the ales along the ertical axis hae to be compared. From a theoretical point of iew, this means that the aerodynamic coefficients can be linearized proided the transerse wind intensity is eqal to zero.

12 In practical applications, this transerse intensity I is howeer generally not eqal to zero and the ertical axis of Figs. 8 and 9 is no more sfficient to represent the actal statistical moments. Indeed, the comparison of these two figres shows that a small transerse intensity modifies drastically the ales of the statistical moments. For example, for I = 1% and I = 1%, the exact and approached non dimensional ariances are respectiely eqal to 1.1E 4 and 6.9E 4. Reagrding the skewness coefficient, it can be significantly affected by the non linearity of the aerodynamic coefficient. Indeed, Fig. 8 exposes a negatie skewness zone (for large I ) that can not be explained with the approached model. onseqently, the non linearity of the aerodynamic coefficient can trn the positiely-skewed probability density fnction into a negatiely-skewed one. In terms of extreme ales, this may hae ery important conseqences. Second approximation : linearization of the aerodynamic loading As a second approximation, we consider the eqations of the linear qasi-steady loading. In this method, only linear terms of the loading mst be considered: the first three terms of Eq. 13 are ths only acconted for. The non dimensional statistical characteristics of the loading are now mch simpler: _ (3) F () t σ ' ; F σ σ m = 4 ; F = + 3 = 1 ρbu 1 U U 1 ρbu ρbu (16) Fig. 1. Approached statistical characteristics of the loading as a fnction of wind intensities These ales are plotted on Fig. 1. It can be checked that this method proides inaccrate reslts, een on the ertical axis. Frthermore since the loading is now Gassian, this method gies a skewness coefficient obiosly eqal to zero. ESTIMATION OF THE INFLUENE OF NON LINEARITY OF THE LOAING ON THE RESPONSE OF THE STRUTURE Seeral famos researchers hae presented adanced methods to compte exactly the statistical moments of the response of a strctre sbjected to a non Gassian loading (see

13 preios paragraph). In this paper, it is desired to gie an estimation of the skewness of the response with a minimm comptation effort. Let s sppose that the reference statistical ales of the loading hae been compted following the deelopments of the preios paragraph. Note that this comptation implies simple algebraic operations only. If the strctre wold be ery stiff, its response wold be mainly qasi-static and, in this case, the ariance of the strctre cold be obtained by diiding the ariance of the loading by the sqared stiffness. In practical applications, the ratio Ψ between the qasi-static and the dynamic components of the PS is howeer not eqal to zero. In this more general case, Eq. 11 can be sed to gie an estimation of the complete ariance, i.e. the sm of the qasi-static and dynamic contribtions. This reslt is not rigoros since the fnction f ( Ψ ) was compted for an Ornstein-Uhlenbeck process and not for the actal PS of the wind loading. This approximation presents howeer the adantage to gie ery fast reslts. oncerning the skewness coefficients, we hae seen (Fig.7) that the skewness of the response is smaller than the skewness of the loading. Hence, the skewness coefficient of the loading cold be sed as bondary ale for the design of the strctre. If the dynamic component of the response is more important, its skewness coefficient can be significantly redced. Fig. 7 can ths help giing a better approximation of the skewness coefficient of the response. This ale can be estimated by the knowledge of: the skewness coefficient of the loading, the damping coefficient of the strctre, the eigen freqency of the strctre, an eqialent shaping factor α This eqialent shaping factor α mst be determined in accordance with the actal PS of the loading. Seeral athors (Mscollino 1995, Floris ) hae proposed some formlations for this eqialence which consists in replacing the actal PS by an Ornstein-Uhlenbeck one. 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. We hae seen that this linearization can lead to a significant inaccracy on the statistical characteristics. The deelopments presented in this paper were limited to the second order deriatie 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, we hae seen how the statistical characteristics of the loading cold be sed to determine estimations of the statistical characteristics of the response. In this part, the heay rigoros analysis method has been aoided and a simpler approach, based on simple and fast comptations, has been retained. This ery simple method can gie estimations of the response that cold be sed, for instance, for a pre-design of a strctre.

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