On the Dynamics of Suspension Bridge Decks with Wind-induced Second-order Effects

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MMPS 015 Convegno Modelli Matematici per Ponti Sospesi Politecnico di

Dynamics of Suspension Bridge Decks with Wind-induced Second-order

Engineering - Sapienza University, Rome (Italy) Department of

1 MMPS 015 Convegno Modelli Matematici per Ponti Sospesi Politecnico di Torino Dipartimento di Scienze Matematiche Settembre 015 On the Dynamics of Suspension Bridge Decks with Wind-induced Second-order Effects Gianfranco Piana Department of Structural and Geotechnical Engineering - Sapienza University, Rome (Italy) Department of Structural, Geotechnical and Building Engineering - Politecnico di Torino (Italy)

2 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS 015 / 48

3 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS / 48

4 Motivations The prediction of the critical wind speed for dynamic aeroelastic instability requires the knowledge of the natural frequencies and mode shapes of the structure. They are generally obtained by a modal analysis conducted with respect to the deformed configuration under permanent loads, and therefore used in both experimental tests and numerical computations. G. Piana MMPS / 48

5 Motivations The interaction between bridge vibration and wind is usually idealized as consisting of two kinds of forces: motion-dependent and motionindependent. According to this schematization, the equation of motion in presence of the aerodynamic forces can be expressed in the following general form: { } { } M + C + K = F + F [ ]{ δ} [ ]{ δ} [ ]{ δ} ( δ, δ) md mi [M], [C], and [K] are the generalized mass, damping, and stiffness matrices, respectively; {δ} is the displacement vector, {F} md is the motion-dependent aerodynamic force vector, and {F} mi is the motionindependent wind force vector. G. Piana MMPS / 48

6 Motivations Sometimes, the cable geometric stiffness matrix is added: { } { } [ ]{ } [ ]{ } ([ ] c δ δ ){ } (, ) g δ δ δ M + C + K + K = F + F Cause deformation! Causes dynamic instability! md mi { F} mi is usually neglected in dynamic stability analyses, even if it affects the bridge global stiffness. G. Piana MMPS / 48

7 Objectives 1 To investigate the effects of steady aerodynamic loads on stability and natural frequencies of long-span suspension bridges through a simplified analytical model. To put these effects in relation to the classic linear flutter analysis. Taking into account the second-order effects induced by a constant transverse wind in the bridge equations of motion, we will derive a generalized eigenvalue problem. We will show that the natural frequencies of a suspended deck-girder depend upon the average wind loading. G. Piana MMPS / 48

8 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS / 48

9 Reference model The single (central) span suspension bridge model is considered, and the linearized integro-differential equations describing the flexural-torsional deformations of the bridge deck-girder are adopted as starting point. Deck-girder: modeled as an elastic beam of constant cross-section, deformable in flexure and torsion, but inextensible and not deformable in shear; Main cables: modeled as purely extensible elastic elements; Single span suspension bridge model Hangers: assumed to be inextensible. G. Piana MMPS / 48

10 Reference model Flexural-torsional deformation of a suspended deck-girder subjected to a distributed vertical load, p(z), and a distributed torque moment, m(z) G. Piana MMPS / 48

11 Basic equations According to the linearized theory, the integro-differential equations governing the flexure and torsion of suspension bridges subjected to a vertical load p(z) and a torque moment m(z), distributed along the deck-girder, are: g Ec Ac 4 d v d v q l EI H = p vd, z 4 dz dz H l 0 g Ec Ac 4 d ϑ ( ) d ϑ q l EIω GI d. 4 t + Hb = m b z ϑ dz dz H l 0 v(z) is the vertical deflection and ϑ(z) is the torsion of the deck-girder; l is the deck-girder/main cables span; f is the main cables sag; q g is the bridge weight per unit length; μ g the bridge mass per unit length; H is the horizontal component of the tension in both cables due to the bridge weight q g (H = q g l /8f); E c A c is the extensional stiffness of both main cables; EI the flexural stiffness (about the x-axis) of the deck-girder; G I t and E I ω are the primary (St. Venant s) and warping torsion stiffness of the deck-girder, respectively. G. Piana MMPS / 48

12 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS / 48

13 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS / 48

14 Governing equations of motion We will study the free oscillations of a suspension bridge deck-girder subjected to a transverse constant wind (mean wind). We shall analyze the small flexural and torsional oscillations about the original (rectilinear) configuration, taking into account the second-order effects induced by the aerodynamic loads into the equations of motion of the suspended deck-girder. As is well known, the mean wind loading consists of the quasi-static load arising from the wind flow past the bridge, and is determined by its size and shape, the air density, the square of the mean wind speed, and its angle of inclination to the structure (angle of attack). G. Piana MMPS / 48

15 Governing equations of motion The mean wind, blowing with velocity U, induces the aerodynamic loads of lift, drag, and moment on the deck-girder (here the lift is neglected). We assume that the deck-girder is horizontal in its original configuration and that the wind, constant in space and time, invests the girder with zero angle of attack (i.e., it has horizontal direction). Single span suspension bridge subjected to steady aerodynamic drag,, and moment, m, uniformly distributed along the deck-girder z p x G. Piana MMPS / 48

16 Governing equations of motion We have the following expressions for the steady aerodynamic loads: 1 px = Ds( 0) = ρu BCD( 0 ), 1 mz = Ms( 0) = ρu B CM ( 0 ). drag moment ρ is the air density; U is the mean wind velocity; B is the deck width; and C D (0) and C M (0) are the aerodynamic coefficients of drag and moment, respectively, evaluated for zero angle of attack. G. Piana MMPS / 48

17 Governing equations of motion p x induces a bending of the deck-girder in the horizontal plane (xz-plane), and therefore produces the bending moment about the vertical axis: m y z m ( y = px l z). Considering a deformed configuration of the deck-girder characterized by v(z) and ϑ(z), the following destabilizing distributed actions arise: ( m y ϑ )" vertical load my v" torque moment G. Piana MMPS / 48

18 Governing equations of motion Moreover, the steady torsion ϑ(z) produces also an additional distributed aerodynamic moment. This can be expressed in the following form, linearized in a neighborhood of ϑ = 0: 1 ( ) Δ m ( 0) ( ) z z = ρu B C M ϑ z = = μϑ z, s ( ) being C M (0) = (dc M /dϑ) ϑ = 0, and where we have set 1 μs = ρ U B CM ( ) 0. G. Piana MMPS / 48

19 Governing equations of motion Now, with the aim of analyzing the free bending-torsional oscillations of the bridge, let us replace the distributed vertical load p and torque moment m with the forces of inertia: d v d ϑ p = μg, m= I, ϑ dt dt μ g g Ec Ac 4 d v d v q l EI H = p vd, z 4 dz dz H l 0 g Ec Ac 4 d ϑ ( ) d ϑ q l E Iω GI d. 4 t + Hb = m b z ϑ dz dz H l 0 where is the bridge mass per unit length and I ϑ denotes the bridge polar mass moment of inertia per unit length, about the z-axis. G. Piana MMPS / 48

20 Governing equations of motion Thus, taking into account the second-order effects induced by the aerodynamic drag and moment, the bending-torsional oscillations of the bridge are governed by the following partial differential equations: vertical load 4 ( ) v v myϑ v y EI H + = μ 4 g + h R t + hl t z z z t z () () 4 ϑ ϑ v ϑ y E Iω GI + Hb + m = I b h t h t 4 z z z t z torque moments ( ) t y μϑ s ϑ R() L() h R and h L being the additional horizontal components of the cables tension: ( ) ( ) Ec A q c g l Ec A q c g l hr () t = v( z, t) b ( z, t) d z, h ( ) (, ) (, ) d. 0 L t v z t b z t z l H ϑ = + ϑ l H 0,, G. Piana MMPS / 48

21 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS / 48

22 Antisymmetric modes In the case of antisymmetric oscillations of the deck-girder, the functions v(z,t) and ϑ(z,t) are such that the additional forces h R and h L are identically zero. The equations of motion are therefore reduced to the following coupled equations: EI ( m ) yϑ 4 v v v H + = μ 4 g z z z t 4 ϑ ( ) ϑ v ϑ EIω GI, 4 t + Hb + m y μϑ s = Iϑ z z z t, where z m ( y = px l z). G. Piana MMPS 015 / 48

23 Solution for antisymmetric modes m y Substituting the expression of in the previous equations, we obtain: 4 v v ϑ z ϑ v E1 ( z, t) : = EI H + p ( ) 0, 4 x ϑ+ l z + + μg = z z z z t 4 ϑ ( ) ( ) ϑ z v ϑ E z, t : = EIω GI ( ) 0. 4 t + Hb + p x l z μϑ s + Iϑ = z z z t The solution to the previous system can be found in the following variableseparable form: nπ z v z, t V t η z, ϑ z, t t ψ z η z = ψ z = sin, ( ) = ( ) ( ) ( ) =Θ( ) ( ) with ( ) ( ) so that the boundary conditions η(0)=η(l)=η"(0)=η"(l)=ψ(0)=ψ(l)=ψ"(0)= =ψ "(l)=0 are satisfied. l G. Piana MMPS / 48

24 Solution for antisymmetric modes We apply the Galerkin Method by imposing the following integral conditions: l 0 nπ z Ei ( z, t) sin dz = 0, i = 1,, l and we finally obtain the following differential system in matrix form: 4n π 4n π EI + H Θ l l μg 0 V l l V 0 I ϑ 4n π 4n π Θ 0 EI GI ω + t + H b 1 0 ( 3+ 4n π ) 1 V 0 0 V 0 px μs, = Θ Θ 0 ( 3+ 4n π ) 0 1 ( n ) G. Piana MMPS / 48

25 Linear eigenvalue problem Rewritten in symbolic form: [ ]{} [ ] p μ M q + K {} q p K { q} μ K { q} = { } ω Looking for a general solution in the form q = q 0 e t, where ω represents the angular frequency of free oscillation, we obtain: and therefore: x g s g ([ ] p μ K p [ ] x K g μ s K g ω M ){ q } = { } ([ ] p μ K p [ ]) x K g μ s K g ω M 0 0, det = 0, ω { } { } i 0. through which can be obtained as a function of p and. x μ s G. Piana MMPS / 48

26 Purely dynamic problem p x = μ = s 0 4n π 4n π EI + H ωμg 0 l l V 4n π 4n π Θ 0 EI GI ω + t + H b ω I ϑ l 0 = 0 l from which we obtain the natural frequencies of the unloaded structure: ω vn nπ 4n π EI H = + l l μ μ g g, ω ϑn nπ 1 4n π = EI GI ω + t + H b l Iϑ l. flexural torsional G. Piana MMPS / 48

27 Purely dynamic problem First antisymmetric torsional mode shape G. Piana MMPS / 48

28 Purely static problem μ = = g I ϑ 0 4n π 4n π p x EI + H ( 3+ 4n π ) 1 det l l = 0. px 4n π 4n π ( 3+ 4n π ) EI GI ω + t + Hb μ s 1 l l If p = 0: x If μ = 0 : s μ sc n 4n π 4n π = EI GI ω + t + H b l l xcn l( 3 4n π ) 4nπ 4n π p = EI + H μ + l torsional divergence lateral-torsional buckling scn G. Piana MMPS / 48

29 Purely static problem If p 0 and x μs 0 (interaction between torsional divergence and lateraltorsional buckling): μ μ s p x = 1 p sc1 xc1 Nondimensional moment versus nondimensional drag ( μ g = Iϑ = 0, n = 1 ) G. Piana MMPS / 48

30 Complete characteristic problem px μs μg 0, 0, 0, 0 I ϑ 4n π 4n π p x EI + H ωμg ( 3+ 4n π ) 1 det l l = 0 px 4n π 4n π ( 3+ 4n π ) EI GI ω + t + Hb μs ω I ϑ 1 l l For each n, we obtain two characteristic surfaces (one for the flexural and one for the torsional natural frequencies). G. Piana MMPS / 48

31 Characteristic surfaces Characteristic surface plotting the nondimensional flexural frequency squared in terms of the nondimensional drag and moment loads (n = 1) G. Piana MMPS / 48

32 Characteristic surfaces Characteristic surface plotting the nondimensional torsional frequency squared in terms of the nondimensional drag and moment loads (n = 1) G. Piana MMPS / 48

33 Limit cases flexural torsional Nondimensional flexural (left) and torsional (right) frequencies squared versus nondimensional drag for μ = 0 (n = 1) s G. Piana MMPS / 48

34 Limit cases flexural torsional Nondimensional flexural (left) and torsional (right) frequencies squared versus nondimensional drag for p = 0 (n = 1) x G. Piana MMPS / 48

35 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS / 48

36 Numerical example Case study: long-span suspension bridge (L main = 1400 m) All data are taken from: L. Salvatori, C. Borri / Computers and Structures 85 (007) G. Piana MMPS / 48

37 Numerical example G. Piana MMPS / 48

38 Numerical example Aerodynamic properties: streamlined cross-section with semicircular fairings and width-to-height ratio B/D = 14.3 C = 0.71, C = 5.59, C = 1.3 U, = 75.8 m s, f = 0.36 Hz D L M cr fl fl G. Piana MMPS / 48

39 Numerical example Natural frequencies of the unloaded bridge (present model): f v 1,0 = Hz f ϑ = Hz 1,0 (1 st vertical antisymmetric) (1 st torsional antisymmetric) Critical wind speeds: U U cr, div cr, lat = m s = 86.4 m s (present model) (present model) Static instability U cr, fl = 75.8m s, f = 0.36 Hz fl (Salvatori & Borri) Dynamic instability G. Piana MMPS / 48

40 Numerical example Natural frequencies under steady wind (present model): Flexural U = 0 U = 0.5 U cr,fl U = 0.6 U cr,fl U = 0.7 U cr,fl U = 0.8 U cr,fl U = 0.9 U cr,fl f v1, Hz Diff., % Torsional U = 0 U = 0.5 U cr,fl U = 0.6 U cr,fl U = 0.7 U cr,fl U = 0.8 U cr,fl U = 0.9 U cr,fl f θ1, Hz Diff., % G. Piana MMPS / 48

41 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS / 48

42 Implications in flutter analysis In linear finite element analyses, the general aeroelastic motion equations of bridge systems are usually expressed in terms of the generalized modal coordinate vector {δ}: 1 1 [ ]{ } [ ] { } [ ] M δ + C ρu C δ + K ρu K { δ} = { } 0. (*) [M], [C], and [K] are the generalized mass, damping, and stiffness matrices, respectively; [C * ] and [K * ] are the generalized aerodynamic damping and aerodynamic stiffness matrices, respectively, that are functions of the flutter derivatives. By assuming harmonic oscillations in the form { δ} = { δ } i 0 e ωt, the flutter speed, U CF, and the flutter frequency, ω F, are obtained from the nontrivial solution to Eq. (*). G. Piana MMPS / 48

43 Implications in flutter analysis Based on what we have shown, Eq. (*) can be modified as follows: g [ ]{ } [ ] { } M δ + C ρu C δ + [ K] ρu K ρu K { δ} = { } 0, (**) where [K g ] is the generalized geometric stiffness matrix. Therefore, the flutter speed, U CF, and the flutter frequency, ω F, can be obtained from the nontrivial solution to Eq. (**), which is given by the following complex eigenproblem: det ω [ M] + ω [ C] ρu C i + [ K] ρu K g ρu K = 0. G. Piana MMPS / 48

44 Outline Motivations and objectives Reference model and basic equations Free vibrations and stability under steady aerodynamic loads Governing equations of motion Antisymmetric modes Numerical example Implications in flutter analysis Final remarks G. Piana MMPS / 48

45 Final remarks We presented a simplified analytical model through which we showed that the natural frequencies of a suspension bridge deck-girder are affected by the mean wind loading. We focused our analysis on the antisymmetric oscillations and we obtained the corresponding characteristic surfaces by solving a generalized eigenvalue problem in which the geometric stiffness matrix modifies the global stiffness of the system. We presented some numerical results and we suggested the possibility of modifying the classic linear equations used in flutter analysis, in order to take into account the effect of motion-independent wind loads on the bridge global stiffness. G. Piana MMPS / 48

46 Final remarks Future developments can regard: adding of self-excited forces implementation in a finite element code comparison to more sophisticated nonlinear models ( * ) (*) Abdel-Ghaffar, A.M. Suspension bridge vibration: continuum formulation. Journal of Engineering Mechanics- ASCE, 108: (198). Abdel-Ghaffar, A.M., and Rubin, L.I. Nonlinear free vibrations of suspension bridges: theory. Journal of Engineering Mechanics-ASCE, 109: (1983). Lacarbonara, W. Nonlinear structural mechanics: theory, dynamical phenomena and modeling, Springer, New York (013). G. Piana MMPS / 48

47 Final remarks G. Piana, A. Manuello, R. Malvano, A. Carpinteri, Natural Frequencies of Long-Span Suspension Bridges Subjected to Aerodynamic Loads, F.N. Catbas (ed.), Dynamics of Civil Structures, Volume 4: Proceedings of the 3nd IMAC, A Conference and Exposition on Structural Dynamics, Conference Proceedings of the Society for Experimental Mechanics Series, 014. G. Piana MMPS / 48

48 Thanks for your attention G. Piana MMPS / 48

Advanced aerostatic analysis of long-span suspension bridges *

44 Zhang / J Zhejiang Univ SCIENCE A 6 7(3):44-49 Journal of Zhejiang University SCIENCE A ISSN 9-395 http://www.zju.edu.cn/jzus E-mail: jzus@zju.edu.cn Advanced aerostatic analysis of long-span suspension