Calculation of third order joint acceptance function for line-like structures

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1 Calculation of third order joint acceptance function for line-like structures N. Blaise, T. Canor & V. Denoël University of Liège (Belgium)

2 Introduction Second order analysis Third order analysis Fourth order analysis Conclusion

Introduction Second order analysis Third order analysis Fourth order analysis Conclusion Line-like structure Gorsexio bridge (16 kms west of

3 Introduction Second order analysis Third order analysis Fourth order analysis Conclusion Line-like structure Gorsexio bridge (16 kms west of Genoa, 144 meters span, 177 meters high) Wind pressure measurements to perform first a Gaussian and next a non-gaussian static analysis Annexe

4 Wind pressure measurements Refined grid Wind properties : turbulence intensity, correlation, etc... Topography, structure geometry, etc... Structural responses of interest : global or local

5 Wind pressure measurements Refined grid Wind properties : turbulence intensity, correlation, etc... Topography, structure geometry, etc... Structural responses of interest : global or local Coarse grid Inaccurate estimation of cumulants of the structural responses Scale effects (Central limit-theorem)

6 Scale effects Aerodynamic fluctuating resultant force f (t) = l 0 p(x, t)dx zero-mean fluctuating non-gaussian aerodynamic pressures p(x, t) (s.d. σ p) Cumulants of f (t) - level of correlation in the aerodynamic field High Low σ 2 f l 2 σ 2 p γ 3,f γ 3,p γ e,f γ e,p σ 2 f, γ 3,f, γ e,f 0

7 Scale effects Aerodynamic fluctuating resultant force f (t) = l 0 p(x, t)dx f (t) = N i p i (t) x i zero-mean fluctuating non-gaussian aerodynamic pressures p(x, t) (s.d. σ p) Cumulants of f (t) - level of correlation in the aerodynamic field High Low σ 2 f l 2 σ 2 p γ 3,f γ 3,p γ e,f γ e,p σ 2 f, γ 3,f, γ e,f 0 Evolutions of the cumulants of f (t) and of structural responses?

8 Introduction Second order analysis Third order analysis Fourth order analysis Conclusion

9 Model of the wind velocity field Random stationary 1-direction wind flow Fluctuating gaussian turbulence component u(ξ, t) with standard deviation σ u Wind velocities perfectly correlated in the small direction (line-like structure) Turbulence intensity [0.1 ; 0.5] I u = σ u U

10 Model of correlation : decaying exponential function 1 ρ ij = e l Lu ξ i ξ j Integral length scale (mean gust dimension) : L u correlation EC 1 : Reference integral length scale L u of 300m at the reference height of 200m Parameter φ = l and change of variable s ij = ξ i ξ j L u 1 Dyrbye, C. and Hansen, S.O. (1997). Wind Loads on Structures. Wiley & Sons, Ltd.

11 Bernoulli s equation p(ξ, t) = 1 ρdc (U + u(ξ, t))2 2 Gaussian aerodynamic pressures (quadratic term neglected) p(ξ, t) 1 2 ρdcu2 + ρdcu u(ξ, t) = a + b u(ξ, t) Cross-cumulants of order 2 (covariances) between aerodynamic pressures κ 2pij = b 2 σ 2 ρ ij

12 Simply supported line-like structure Structural response I (ξ) : Response influence function Second cumulant (variance) ˆ 1 R(t) = I (ξ)p(ξ, t)dξ 0 1 κ 2R = I (ξ i )I (ξ j )κ 2p ( ξ i ξ j )dξ i dξ j 0

13 Simply supported line-like structure Structural response Second cumulant (variance) ˆ 1 R(t) = I (ξ)p(ξ, t)dξ 0 1 κ 2R = I (ξ i )I (ξ j )κ 2p ( ξ i ξ j )dξ i dξ j 0 Considered response : reaction on the left Response influence function I (ξ) = 1 ξ

14 Dyrbye and Hansen 1 used (i) change of variable s ij = ξ i ξ j and (ii) interchange of the order of integration to replace 1 κ 2R = b 2 σ 2 u by two single integrals 2 nd order influence function 0 I (ξ i )I (ξ j )ρ(s)dξ i dξ j 2 nd order joint acceptance function ˆ 1 s k(s) = 2 I (ξ)i (ξ + s)dξ 0 κ 2R = b 2 σ 2 u ˆ 1 0 k(s)ρ (s) ds 1 Dyrbye, C. and Hansen, S. O. (1988). Calculation of joint acceptance function for line-like structures. Journal of Wind Engineering and Industrial Aerodynamics vol. 31-2,

15 Closed-form solutions κ 2R (φ) b 2 σ 2 u Full correlation : κ 2R(0) b 2 σ 2 u k(s) = 1 3 ( s 3 3s + 2 ) = 2 3φ 1 φ φ 4 e φ ( 2 φ φ 4 ) = 1 0 k(s)ds = 1/4

16 Discrete aerodynamic pressure field N N κ 2R I (ξ i )I (ξ j )κ 2p ( ξ i ξ j ) ξ i ξ j i=1 j=1

17 Discrete aerodynamic pressure field N N κ 2R I (ξ i )I (ξ j )κ 2p ( ξ i ξ j ) ξ i ξ j i=1 j=1

18 Discrete aerodynamic pressure field N N κ 2R I (ξ i )I (ξ j )κ 2p ( ξ i ξ j ) ξ i ξ j i=1 j=1

19 Introduction Second order analysis Third order analysis Fourth order analysis Conclusion

20 Bernoulli s equation : p(ξ, t) = 1 ρdc (U + u(ξ, t))2 2 Non-Gaussian aerodynamic pressures p(ξ, t) = a + b u(ξ, t) + c u 2 (ξ, t) a = 1 2 ρdcu2, b = ρdcu and c = 1 2 ρdc quadratic term taken into account γ 3,p 3I u and γ e,p 12I 2 u γ e,p = 4 3 γ2 3,p Cross-cumulants of order 3 between aerodynamic pressures κ 3pijk 2b 2 cσ 4 u (ρ ij ρ ik + ρ ij ρ jk + ρ ik ρ jk )

21 Analytical approach Third cumulant κ 3R = 1 0 6b 2 cσ 3 u I (ξ i )I (ξ j )I (ξ k )κ 3p (s ij, s ik ) dξ i dξ j dξ k 1 0 I (ξ i )I (ξ j )I (ξ k )ρ (s ij ) ρ (s ik ) dξ i dξ j dξ k

22 Analytical approach Replace triple integral 1 κ 3R = 6b 2 cσu 4 I (ξ i )I (ξ j )I (ξ k )ρ ( ) s ij ρ (sik ) dξ i dξ j dξ k 0 by three 3 rd order influence functions (single integrals) k 1(s 1, s 2) =... k 2(s 1, s 2) =... k 3(s 1, s 2) =... and two 3 rd order joint acceptance functions (double integrals) κ 3R 6b 2 cσ 4 u = 2 ˆ 1 ˆ s ˆ 1 ˆ 1 s1 0 0 [(k 1(s 1, s 2) + k 2(s 1, s 2))ρ(s 1)ρ(s 2)] ds 2ds 1 k 3(s 1, s 2)ρ(s 1)ρ(s 2)ds 2ds 1

23 Analytical approach Closed-form solutions κ 3R (φ) 6b 2 cσu 4 ( = e 2φ 4e φ (φ + 5) + e 2φ (2φ(2φ 7) + 19) + 1 ) 4φ 4 κ 3R (0) Full correlation : 6b 2 cσu 4 = 1 8

24 Discrete aerodynamic pressure field N N N κ 3R 6b 2 cσu 4 I (ξ i )I (ξ j )I (ξ k )ρ ( ) s ij ρ (sik ) ξ i ξ j ξ k i=1 j=1 k=1

25 Introduction Second order analysis Third order analysis Fourth order analysis Conclusion

26 Cross-cumulants of order 4 between aerodynamic pressures κ 4pijkl 4b 2 c 2 σ 6 (12ρ ij ρ ik ρ jl ) Fourth cumulant of the reaction 1 κ 4R 48b 2 c 2 σ 6 I (ξ i )I (ξ j )I (ξ k )I (ξ l )ρ ( ) s ij ρ (sik ) ρ ( ) s jl dξi dξ j dξ k dξ l 0

27 Quadruple integral κ 4R 48b 2 c 2 σ 6 1 I (ξ i )I (ξ j )I (ξ k )I (ξ l )ρ ( ξ i ξ j ) ρ ( ξ i ξ k ) ρ ( ξ j ξ l ) dξ i dξ j dξ k dξ l 0 Interchange of order of integration is challenging for a 4-dimensional domain Alternative : 8 quadruple integrals 1 ξi ξi ξj ρ (( )) ξ i ξ j ρ ((ξi ξ k )) ρ (( )) ξ j ξ l dξi dξ j dξ k dξ l 1 ξi 1 ξj 0 0 ξ i 0... ρ (( )) ξ i ξ j ρ ( (ξi ξ k )) ρ (( )) ξ j ξ l dξi dξ j dξ k dξ l 1 ξi ξi ξ... ρ (( )) ξ j i ξ j ρ ((ξi ξ k )) ρ ( ( )) ξ j ξ l dξi dξ j dξ k dξ l 1 ξi ξ i ξ... ρ (( )) ξ j i ξ j ρ ( (ξi ξ k )) ρ ( ( )) ξ j ξ l dξi dξ j dξ k dξ l 1 1 ξi ξj 0 ξ i ρ ( ( )) ξ i ξ j ρ ((ξi ξ k )) ρ (( )) ξ j ξ l dξi dξ j dξ k dξ l ξj 0 ξ i ξ i 0... ρ ( ( )) ξ i ξ j ρ ( (ξi ξ k )) ρ (( )) ξ j ξ l dξi dξ j dξ k dξ l 1 1 ξi 1 0 ξ i 0 ξ... ρ ( ( )) ξ j i ξ j ρ ((ξi ξ k )) ρ ( ( )) ξ j ξ l dξi dξ j dξ k dξ l ξ i ξ i ξ... ρ ( ( )) ξ j i ξ j ρ ( (ξi ξ k )) ρ ( ( )) ξ j ξ l dξi dξ j dξ k dξ l

28 Closed-form solution κ 4R 48b 2 c 2 σ 6 = 3e φ 40 ( e φ ((2φ 3)φ 2 + 6) 6(φ + 1) ) 2 [ 15e φ ( 2φ ) (φ(φ + 4) + 2) +e 3φ ( (2φ(6φ(16φ 95) ) 4275)φ ) + 15(φ + 1)(2φ + 1) ] 5e 2φ (φ(2φ(φ + 3)(φ(3φ + 22) + 31) ) ) Solution may seem unhandy but approximations available (e.g., Taylor series) κ 4R (0) Full correlation : 48b 2 c 2 σ 6 = 1 16

29 Introduction Second order analysis Third order analysis Fourth order analysis Conclusion

30 Skewness coefficient γ 3R (φ) = κ 3,R (φ)/κ 3/2 2R (φ) =... Full correlation : γ 3R (0) = 3 σu U = 3Iu = γ 3p Excess coefficient γ er (φ) = κ 4R (φ)/κ 2 2R (φ) =... ( σu ) 2 Full correlation : γ er (0) = 12 = 12I 2 U u = γ ep

31 Evolution of the cumulants 1.0 Linear: g x Κ 2 R Φ Κ 2 R 0 Κ 3 R Φ Κ 3 R 0 Κ 4 R Φ Κ 4 R 0 Γ 3 R Φ 3I u Γ e,r Φ 12I 2 u Φ

32 γ e,r = f (γ 3,R )

33 γ e,r = f (γ 3,R ) Could use the approximation γ er (φ) 4 3 γ 3R(φ) 2 for this type of structural response

34 Discrete aerodynamic pressure field

35 Discrete aerodynamic pressure field Improvements of the approximations for a same number of sensors?

36 Perspective Numerical admittance 1 : correction terms on cumulants and cross-cumulants of order 2, 3 and 4 for discrete aerodynamic pressures field 1 Denoël, V. and Maquoi, R. (2012). The concept of numerical admittance. Archive of Applied Mechanics vol. 82, 10-11, pages

37 Perspective Numerical admittance 1 : correction terms on cumulants and cross-cumulants of order 2, 3 and 4 for discrete aerodynamic pressures field May be used as is for simple structures : closed-form solutions 1 Denoël, V. and Maquoi, R. (2012). The concept of numerical admittance. Archive of Applied Mechanics vol. 82, 10-11, pages

38 The team......thanks you for your kind attention Read out more about us on : Contact me at : N.Blaise@ulg.ac.be

39 Questions?

40 Other type of structural responses : 2ξ 1 γ 3R (φ) = 0 while γ er (φ) Κ 2 R Φ Κ 2 R 0 Κ 3 R Φ Κ 3 R 0 Κ 4 R Φ Κ 4 R 0 Γ 3 R Φ 3I u Γ e,r Φ 12I 2 u Φ

41 Spectral analysis S ui u j (ω, s ij ) = Γ u(ω, s ij )S u(ω) Structural response Γ u(ω, s ij ) = e C ω s ij 2U Power spectral density ˆ 1 R(t) = I (ξ)p(ξ, t)dξ 0 S pi p j (ω) b i b j S ui u j (ω, s ij ) ˆ 1 ˆ 1 S R (ω) = I (ξ i )I (ξ j )S p(ω, ξ i, ξ j )dξ i dξ j 0 0

42 Spectral analysis S ui u j (ω, s ij ) = Γ u(ω, s ij )S u(ω) Structural response Γ u(ω, s ij ) = e C ω s ij 2U Bi-spectrum ˆ 1 R(t) = I (ξ)p(ξ, t)dξ 0 D pi p j p k (ω 1, ω 2 ) 2c i b j b k S ui u j (ω 1 )S ui u k (ω 2 ) + 2b i c j b k S ui u j (ω 1 )S uj u k (ω 1 + ω 2 ) + 2b i b j c k S ui u k (ω 2 )S uj u k (ω 1 + ω 2 ) ˆ 1 ˆ 1 ˆ 1 D R (ω 1, ω 2 ) = I (ξ i )I (ξ j )I (ξ k )D p(ω 1, ω 2, ξ i, ξ j, ξ k )dξ i dξ j dξ k 0 0 0

43 Spectral analysis S ui u j (ω, s ij ) = Γ u(ω, s ij )S u(ω) Structural response Γ u(ω, s ij ) = e C ω s ij 2U Tri-spectrum ˆ 1 R(t) = I (ξ)p(ξ, t)dξ 0 T pi p j p k p l (ω 1, ω 2, ω 3 ) =... ˆ 1 ˆ 1 ˆ 1 ˆ 1 T R (ω 1, ω 2, ω 3 ) = I (ξ i )I (ξ j )I (ξ k )I (ξ l )T p(ω 1, ω 2, ω 3, ξ i, ξ j, ξ k, ξ l )dξ i dξ j dξ k dξ l

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