Numerical Simulation of the Wind Action on a Long-Span Bridge Deck V = W (I = 1, 2) on the Solid Boundary Ã (3) N+ 1 I I Vs 2 4) Calculate Vi With

A. L. Braun and A. M. Awruch Numerical Simulation of the Wind

A. L. Braun Action on a Long-Span Bridge Deck PPGEC/UFRGS A numerical model to study the aerodynamic and aeroelastic bridge deck behavior is Av. Osvaldo Aranha, 99 – 3o andar presented in this paper. The flow around a rigid fixed bridge cross-section, as well as the 90035-190 Porto Alegre, RS. Brazil flow around the same cross-section with torsional motion, are investigated to obtain the www.cpgec.ufrgs.br aerodynamic coefficients, the Strouhal number and to determine the critical wind speed [email protected] originating dynamic instability due to flutter. The two-dimensional flow is analyzed employing the pseudo-compressibility approach, with an Arbitrary Lagrangean-Eulerian (ALE) formulation and an explicit two-step Taylor-Galerkin method. The finite element method (FEM) is used for spatial discretization. The structure is considered as a rigid A. M. Awruch body with elastic restrains for the cross-section rotation and displacement components. PPGEC/UFRGS & PROMEC/UFRGS The fluid-structure interaction is accomplished applying the compatibility and equilibrium Av. Sarmento Leite, 425 conditions at the fluid-solid interface. The structural dynamic analysis is performed using 90050-170 Porto Alegre, RS. Brazil the classical Newmark’s method. www.mecanica.ufrgs.br/promec Keywords: Fluid-structure interaction, Finite Element Method (FEM), Large Eddy [email protected] Simulation (LES), aeroelasticity, aerodynamics

explicit two-step Taylor-Galerkin method with an Arbitrary Introduction Lagrangean-Eulerian (ALE) description. A similar Taylor-Galerkin formulation was used by Tabarrok & Su (1994) and by Rossa & Wind tunnel tests for assessment of aerodynamic and aeroelastic Awruch (2001), but with a semi-implicit scheme. The (ALE) scheme informations in the study of bridge girders performance are was first presented by Hirt et al. (1974) in a numerical work. Since numerically simulated in this work. The usual way to obtain these this first paper many other authors used this description with the informations is using representative models in a wind tunnel. same concepts. The classical Smagorinsky’s model, similar to that However, with the improvement in computers technology and presented by Kuroda (1997), was employed for the sub-grid scales computational fluid dynamics (CFD) algorithms, many of these simulation. The finite element method was used for spatial problems can also be analyzed by numerical simulation.1 discretization. The structure was considered as a rigid body with Long-span bridges, such as suspension bridges for example, elastic restrains for the cross-section rotation and displacement must be designed to support, from a static point of view, the mean components. The coupling between fluid and structure was wind forces (using the drag, lift and pitching moment coefficients). performed applying the compatibility and equilibrium equations at Besides, considering that such structures show low damping and the interface. The structural dynamic analysis was accomplished low stiffness, they are subjected to aeroelastic phenomena, such as using the classical Newmark’s method (Bathe, 1996). Examples are flutter, galloping and vortex shedding induced vibrations. Only the presented to illustrate the capability of the computational method. first case will be studied in the present work. The term aeroelasticity is used when the aerodynamic forces Governing Equations for the Flow Simulation produce some kind of structural instability as a consequence of the interaction between these forces and the structural motion. The The governing equations, considering the pseudo- flutter phenomenon is a type of aeroelastic instability that begins compressibility approach in an isothermic process, Large Eddy when the effective damping (structural + aerodynamic) becomes Simulation (LES) with Smagorinsky’s model for turbulent flows and negative. an Arbitrary Lagrangean-Eulerian (ALE) description, are: Kawahara & Hirano (1983) were one of the first authors to a) Momentum equations: analyze numerically the wind action on a bridge cross-section. They used the Finite Element Method (FEM) to obtain the aerodynamic   ∂v   ∂vi ∂vi 1 ∂p ∂  j ∂vi  ∂vk coefficients as functions of the angle of attack of the wind and the + ()v j − wj + δij − ()ν +ν t + + λ δij  = 0 ∂t ∂x ρ ∂x ∂x   ∂x ∂x  ∂x  Strouhal number. Kuroda (1997) employed two different numerical j j j   i j  k  procedures to study the approaching span of the Great Belt East (i, j, k = 1, 2) in Ù (1) Bridge: the Finite Element Method and the Finite Difference Method (FDM), and Large Eddy Simulation (LES) with the  ∂v  2 1 1 ∂vi j  Smagorinsky’s model for the turbulent flow. He also presented the being νt = ()CS ∆ ()2Sij Sij 2 with Sij = + and 2 ∂x ∂x  results referring to the aerodynamic coefficients for various angles  j i  1/2 of attack and the Strouhal number for both numerical methods Ä = (Äx Äy ) , where Äx and Äy are the element dimensions in the employed in his analysis. Larsen & Walther (1997) analyzed several global axis direction x and y, respectively. bridge decks observing their aeroelastic behaviour using a numerical b) Mass conservation equation: code based on Discrete Vortex Simulation (DVS), presenting the respective critical flutter velocity (using the flutter derivatives). ∂v Recently, (2002) applied a direct method for the flutter ∂p ∂p 2 j Selvam et al. + ()v j − wj + ρc = 0 (j = 1, 2) in Ù (2) analysis. They used (FEM) and (LES). ∂t ∂x j ∂x j In this work, the analysis of the flow of a slightly compressible fluid in a two-dimensional flow domain was carried out using an p 2 which is obtained considering ∂ = c . ∂ρ Paper accepted August, 2003. Technical Editor: Aristeu da Silveira Neto.. The boundary conditions of Eqs. (1) and (2) are the following:

352 / Vol. XXV, No. 4, October-December 2003 ABCM Numerical Simulation of the Wind Action on a Long-Span Bridge Deck v = w (i = 1, 2) on the solid boundary Ã (3) n+ 1 i i vS 2 4) Calculate vi with:

v = vˆ on the boundary Ãv or 1 a 1 1 2 n+ 2 n+ 2 ~ n+ 2 1 Ät ∂Äp p = ˆp on the boundary Ãp (4) vi = vi − (9) ñ 8 ∂xi

  ∂v   ó n n 1 n − p  ∂vi j  ∂vk ij j 5) Calculate v + v Äv with:  äij + ()í + í t + + ë n j = = Si i = i + i ñ  ∂x ∂x  ∂x ñ   j i  k  (i,j,k=1,2) in Ã (5)    ∂v  ó  ∂vi 1 ∂p ∂  ∂vi j  Ävi = Ät− rj − äij + í + + ∂x ñ ∂x ∂x  ∂x ∂x   j j j   j i  In these equations, vi and p (the velocity components and the (10) 1 n+ 2 pressure, respectively) are the unknowns. The viscosities í ì  = ñ ∂vk  ë äij  ∂xk  and ë ÷ , the specific mass ñ and the sound velocity c, are the  = ñ ì n+1 n fluid properties. The eddy viscosity í t depends of 6) Calculate p = p + Äp with: t = ñ derivatives of the filtered velocity components, of the element n+ 1 dimensions and of the Smagorinsky’s constant C . For a purely   ∂v  2 S  ∂p 2 j  Äp = Ät− rj − ñc  (11) Eulerian description, the mesh motion velocity w at each nodal ∂x  ∂x   j  j  point, with components wi , is equal to zero. Now, for a purely Lagrangean description, the mesh motion velocity at each nodal These expressions must be employed after applying the classical point is equal to the fluid velocity, i.e. vi = wi (i = 1, 2). Finally, in Galerkin technique into the finite element method (MEF) context. an Arbitrary Lagrangean-Eulerian formulation, w ≠ 0 and w ≠ v . As the scheme is explicit, the resulting system is conditionally On the boundaries Ã and Ã , prescribed values for velocity stable, with a stability condition given by: va p and pressure, vˆ and ˆp , respectively, must be specified, while on Äx Ät < á i (i = 1,..., NTE) (12) ˆ i c v Ãó the boundary force t must be in equilibrium with the stress + i tensor components ó . In Eq. (5), n is the direction cosine ij j where á (which is a real number less than one) is a safety between a vector perpendicular to Ã and the axis x . ó j coefficient, Äxi and vi are the i-th element characteristic Initial conditions for the pressure and the velocity components at dimension and the velocity, respectively, and NTE is the total t = 0 must be given. number of elements. Although variable time step could be adopted (Teixeira & a The Algorithm for the Flow Simulation Awruch , 2001), in this work an unique value of Ät will be used for the whole process, adopting the smallest one from those obtained by Expanding the governing equations in a Taylor’s series up to Eq. (12). second order terms, the algorithm for the flow simulation contains the following steps (Braun, 2002): 1 The Fluid-Structure Coupling ~ n+ 2 1) Calculate vi with: In the present work, the structure is idealized as a two- dimensional rigid body. Displacement and rotations take place on 1  the plane formed by the axis x1 and x2; the body is restricted by ~n+ 2 n Ät  ∂vi 1 ∂p vi = vi + − rj − äij + dampers and springs, as indicated in Fig. 1. 2 x ñ x  ∂ j ∂ j n (6)   ∂v   2 n  ∂  ∂vi j  ∂vk  Ät  ∂ vi  í + + ë äij  +  rjrk   ∂x  ∂x ∂x  ∂x 4 ∂x x j   j i  k    j k 

whererj = (v j − w j ) and v = (ν +ν t ) .

n+ 1 2) Calculate p 2 with:

 n 1  ∂v  2 n  n+ 2 n Ät  ∂p 2 j  Ät  ∂ p  (7) p = p + − rj − ñc  +  rirj   2  ∂xj ∂xj   4  ∂xj∂xi     Figure 1. Structure model, formed by a rigid body restricted by springs 1 1 and dampers. Structure degrees of freedom: u = displacement in the n+ 2 n+ 2 n 1 3) Calculate Äp = p − p . (8) direction of axis x1. u2 = displacement in the direction of axis x2.è = rotation around the axis x3 (perpendicular to the plane formed by the axes x1 and x2).

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2003 by ABCM October-December 2003, Vol. XXV, No. 4 / 353 A. L. Braun and A. M. Awruch

The structural dynamic equilibrium equation is given by the following matrix expression: Deriving Eq. (15) with respect to time, taking into account matrix L and equation (16), the following expression is obtained: c c ~ M U + C U + K U c = Qc (13) ~ S ~ S ~ S ~ S ~ S ~ S ~ S I I c c 0 0 − l1è  U = V = LU + L'( è )U , where L'( è ) =   (17) where is the mass matrix, the damping matrix, the ~ ~ ~ ~ ~ ~ ~ M C K S F S S 0 0 − l2è  ~ S ~ S ~ S c c stiffness matrix and U ,U and U c the acceleration, velocity and Equation (15) and Eq. (17) are applied to each node at the ~ S ~ S ~ S interface, where the equilibrium condition must be also satisfied, generalized displacements, respectively. Finally, Qc is the load that means that the load S acting on the structure at the interface, ~ ~ S must be equal to the load S given by Eq. (5), but with an opposite vector. ~ The subscript S means that these matrices belong to the structure signal (because here the fluid action on the structure is considered, and the superscript C indicates that these values correspond to the while Eq. (5) represents the boundary action on the fluid). S can be gravity center of the solid body. Equation (13) can be written as ~ follows: transported to the center of gravity of the body, obtaining:

c T M1 0 0 u1  C11 0 0 u1  Q = − L S dÃ (18)       ∫ÃS 0 0 0 0 ~ S ~ ~  M 2 u2  +  C22 u2  +  0 0 M    0 0 C    3è   33è  (14) where LT is the transpose matrix of L , given by Eq. (15), and S c ~ ~ ~ 0 0 QS1  K11 u1        c  contains the two components of the fluid boundary force acting on 0 K22 0 u2 = QS2    the structure at a point located on the structure surface ÃS (ÃS  0 0 K è   c   33  M x3 represents also the solid-fluid interface); these forces S are given   ~ by Eq. (5), but with an opposite signal. It must be noticed that the hypothesis of a rigid structure is proper when deformations of the cross-section are small compared to the rotation and displacement components. 2g At the solid-fluid interface, the compatibility condition must be A' satisfied, or in other words, the fluid velocity and the structure velocity must be the same at the common nodes of both fields. The l2(theta) compatibility condition and the translation of variables evaluated at l C' 1g the center of gravity of the body to a point located at the fluid- 1(theta) X structure interface may be written with the following expressions: l X2g I I c 1 0 − l2  U = V = LU with L =   (15) A ~ S ~ F ~ ~ S ~ 0 1 l1   x 2gA l 1g where S and F are referred to the structure and the fluid, 1gA X theta C respectively, and the superscript I is referred to the interface. It is x I important to notice that the both vectors U and V I have two l x A   1g   1 g  A ~ S ~ F l =   =   = x components that correspond to the global axis direction. However, ~ l x A ~ g  2 g   2 g  c U has three components, because it includes the rotation around Figure 2. Rigid body motion. The subscripts “g” and “l” are referred to ~ S quantities related to global and local axis, respectively. an axis perpendicular to the plane formed by x1 and x2. Values of c U can be transported to the solid-fluid interface (or to nodes To determine the coupling effects between the fluid and the ~ S structure, in the finite element method (FEM) context, consider an belonging to the structure boundary) through a translation matrix L , element belonging to the fluid domain in contact with the solid ~ body, as indicated in Fig. 3, where it can be observed that only as given by Eq. (15), being l and l the distance components 1 2 points 1 and 2 are in contact with the structure. between the gravity center of the body and the point under consideration, measured in the global system. Considering Fig. 2, it is observed that the distance components from a boundary point to the body gravity center are functions of è, and it may be written as follows:

xA  l1( è ) cosè − senè 1g  A l( è ) = = = R x (16)     A  ~ l2( è ) senè cosè x ~ ~ g     2g 

354 / Vol. XXV, No. 4, October-December 2003 ABCM Numerical Simulation of the Wind Action on a Long-Span Bridge Deck

(they were calculated previously in the flow analysis). Referring to Fig. 3, at nodes 3 and 4, the values V F and P F are known. All ~ ~ these considerations lead to the elimination of the second expression of Eq. (19) when the governing equations which describe the solid body motion are built, taking into account the solid-fluid coupling effect. The first expression of Eq. (19) can be re-written as:

I F 1 MM II V + AD II V I + MM IF V + AD IF V F − GP I = S I (20) ~ ~ ~ ~ ~ ~ ~ ~ ñ ~ ~

Equation (15) with matrix L and Eq. (17) with matrix L' ( è ) ~ ~ are considered for each node at the interface. Then, when an element Figure 3. Element of the fluid domain in contact with the solid body. side with two nodes and lying on the fluid-structure interface is considered, Eq (15) and Eq. (17) are written in the following form: The momentum equations in its matricial form, at element level I c I I c c (e), can be obtained by applying the Galerkin method to the Eq. (1), U = V I = T U ; U = V = T U + T' ( è )U (21) writting: ~ S ~ ~ ~ S ~ S ~ ~ ~ S ~ ~ S

'  II IF  I   II IF  I  Referring again to Fig. 3, the matrices T and T are given by: MM MM V AD AD V ~ ~  ~ ~  ~   ~ ~  ~     +    = MM FI MM FF F ADFI ADFF V F  V      1   1   ~ ~  ~   ~ ~  ~  (19) 1 0 − l2 0 0 − l1   L   L' ( è )  I   I  0 1 l1    0 0 l1  GP S 1 ~  − 2 ~  1  ~  ~  T =   = ; T'( è ) =  è =     +   ~  2  L ~  2  ' ñ GPF S F 1 0 − l2   0 0 − l1 L ( è )       ~    ~   ~  ~   2   2  0 1 l1  0 0 − l2  where MM contains the time derivative coefficients from the (22) ~ velocity components V , AD contains the coefficients of advective The contribution from S I , on the side 1-2 of the element (e), to ~ ~ ~ and diffusive terms, GP contains the coefficients of pressure the total load acting at the gravity center of the body, can be ~ calculated as: derivative terms with respect to x1 and x2 and, finally, S is a vector ~ ˆ c T I containing the boundary integrals resulting from the integration by QS = −T S (23) ~ ~ parts of pressure and diffusive terms. I In Eq. (19), V and V I contain, respectively, acceleration and Considering Eqs. (14), (15), (17) and (20), with the last one ~ ~ multiplied by ñ, the structural dynamic equilibrium equation, taking velocity components corresponding to nodes 1 and 2 of Fig. 3, while into account the solid-fluid coupling effect, is given by: F V and V F contain variables corresponding to nodes 3 and 4 of ~ ~  NTL  T II   c M + ∑ T ρ MM T  U + the same figure. A similar remark can be made with respect to the  ~ S i=1  ~ ~ ~ i  ~ S vectors of pressure gradients GP and boundary forces S . Matrix NTL ~ ~   T II T II   c c (24) C + ∑ T ρ AD T + T ρ MM T' U + K U = MM II contains elements coming from the connection of node 1  ~ S i=1  ~ ~ ~ ~ ~ ~ i  ~ S ~ S ~ S ~ NTL  T IF F T IF F T I  c  with itself and with node 2, and the connection of node 2 with itself −  ∑ T ρ MM V + T ρ AD V − T GP  + Q   i=1  ~ ~ ~ ~ ~ ~ ~ ~ i ~  and with node 1. Matrix MM IF reflects the connection between the  S  ~ nodes 1 with 4 and 2 with 3. Similar commentaries can be made where NTL is the total number of fluid elements in contact with the structure, having at least one straight segment common to the solid with respect to matrices AD II and AD IF . ~ ~ body surface, forming the solid-fluid interface. The matricial Eq. Regarding the structural analysis, only the first matricial (24) is re-written as: expression of Eq. (19) is necessary, because only this equation c c c contributes to the assembling of the overall dynamic equilibrium M U + C U + K U c = Q (25) equation. On the other hand, as the structural and the flow analysis ~ S ~ S ~ S ~ S ~ S ~ S ~ S are performed in a sequential form in this work, the system constituted by the solid body and fluid elements with one or more sides common to the solid-fluid interface have prescribed values of V and P at nodes that do not have any contact with the structure ~ ~

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2003 by ABCM October-December 2003, Vol. XXV, No. 4 / 355 A. L. Braun and A. M. Awruch

As can be noticed, is a non-symmetric matrix, because it P − P C C i = i 0 (27) ~ S p 1 ñV 2 contains the advective terms and T T MM II T'( è ) . This last term 2 0  ~ ~ ~  leads to the non-linearity of matrix C . where P is the pressure at node i and P is a reference pressure (for ~ S i 0 example, the pressure in an undisturbed area of the flow). With the In this work, a monolithic coupling between fluid and structure instantaneous values of pressure coefficients, the time history may was not considered. The analysis for both fields is made in a be obtained and then the mean pressure distribution on the body sequential way. Firstly, Eq. (6) to Eq. (11) are solved, with the surface, for a given time interval, may be calculated. smallest Ät calculated with Eq. (12) and applying the boundary conditions given by Eq. (3) to Eq. (5). After, Eq. (25) is solved using the Newmark’s method (Bathe, 1996). Although different The Automatic Mesh Motion Scheme time steps may be used for the fluid and the structure, here the same Taking into account that the immersed body in the fluid can time step was adopted, because the computer time required by the move and rotate in its plane and that the flow is described by an structure analysis is negligible with respect to the processing time Arbitrary Lagrangean-Eulerian (ALE) formulation, a scheme for the demanded by the flow analysis. Furthermore, compatibility and mesh motion is necessary, establishing the velocity field in the equilibrium conditions are more accurately imposed if the same time w intervals are employed. fluid domain, such that the element distortion will be as smaller as possible, according to the following boundary conditions:

Strouhal Number and Aerodynamic Coefficients Calculation I I w int erface= V = U ; w external = 0 (28) f L ~ F ~ E boundaries ~ The Strouhal number (St) can be calculated with St = v 0 , V 0 In the present work, the mesh motion scheme is similar to that b where V0 is a reference velocity, L0 a reference dimension and fv is used by Teixeira & Awruch (2001). Considering that i is an inner the shedding frequency of a pair of vortices. It depends of the point in the fluid field and j is a boundary node, the mesh velocity immersed prism cross-section, its oscillations, its superficial details, components at node i , in the direction of the axis xk, are given by: the Reynolds number and the flow characteristics. Formulation to calculate this number is presented in many publications and texts NS a w j (for example, Schlichting, 1979). When the Strouhal number of a ∑ ij k flow with a given immersed structure is known, it is possible to wi j=1 (k = 1, 2) (29) k = NS obtain the velocity V0 R , which will produce the resonant ∑aij phenomenon on the vibrating body. It occurs when the shedding j=1 frequency of a pair of vortices is approximately equal to the structural natural frequency. where NS is the total number of nodes belonging to the boundary The drag coefficient CD is related to the acting forces on the lines and aij are the influence coefficients between the inner points structure in the flow direction, while the lift coefficient CL is related 1 and the boundary lines of the flow field, being a , where to the acting forces on the structure in the transversal-to-flow ij = n dij direction. Finally, the pitching moment coefficient CM is related to () the torsional moment acting at the gravity center of the immersed dij is the distance between i and j, and n ≥ 1 . The exponent n can be prism. The three coefficients can be calculated using the following adjusted by the user. Although regions with purely Eulerian and expressions: purely Lagrangean descriptions mat be used simultaneously with the ALE formulation, this alternative may result in more complex and NTN NTN NTN less efficient codes. It may also lead to more difficulties to control I I S I S I − S l + S l ∑ 1 i ∑ 2i ∑ ()1 i 2i 2i 1i mesh distortions. i=1 ; i=1 ; C i=1 CD = CL = M = 1 2 1 2 1 2 ñV0 L0 ñV0 L0 ρ()V0L0 Examples 2 2 2 (26)

I I Analysis of the Flow Around a Rectangular Prism where S1 and S2 are the forces in the directions x1 and x2, respectively, acting on the structure at node i, located on the This example presents a prism with a rectangular cross-section free to oscillate in the transversal-to-flow and rotational directions. interface. l and l are the projections in the directions x and x , 1i 2i 1 2 Through this problem, the program performance for large and respectively, of the distance between the gravity center and node i. coupled motion is observed, even that this cross-section form is not NTN is the total number of nodes located on the solid-fluid usually employed in bridge structures. In this study, special attention I I is given to the structural dynamic response in the two degrees of interface. The forces S1 and S2 are the components of the force freedom of the cross-section and to the finite element mesh motion, I vector S , given by Eq. (20). These forces are applied to the remembering that a special scheme is employed for the non-linear ~ dependence with respect to the cross-section rotation in the structure on each fluid element side belonging to the interface. compatibility condition at the solid-fluid interface. i The pressure coefficient at a point i, C p , located on the In the present section, results obtained for a flow around a interface, is related to the pressure acting at that point. This rectangular prism with a Reynolds number equal to 1000 are coefficient can be calculated using the following expression: presented. The rectangular cross-section exhibits a height/width relation (h/B) of 0.2. The geometry and the boundary conditions, in

356 / Vol. XXV, No. 4, October-December 2003 ABCM Numerical Simulation of the Wind Action on a Long-Span Bridge Deck a non-dimensional form, are shown in Fig. 4. In addition, initial 0.30 velocity and pressure field for the fluid-structure interaction problem are those of a developed flow obtained with a fixed body. 0.20

0.10

0.00

angular displ. -0.10

-0.20

-0.30

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00 time

Figure 4. Flow around a rectangular prism: geometry and boundary 0.12 conditions. 0.10 0.08 The finite element mesh has 5865 nodes and 5700 quadrilateral 0.06 bi-linear isoparametric elements and is shown in Fig. 5. A non- 0.04 0.02 ∗ -4 dimensional time step Ät = 1.0x10 was adopted. The fluid and 0.00 structural data are presented in Table 1. -0.02

angular veloc. -0.04 -0.06 Table 1. Rectangular prism: dimensionless data for the fluid and the structure. -0.08 -0.10 Rectangular Prism - Reynolds 1000 -0.12 Specific mass (ñ) 1.0 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00 Volumetric viscosity (ë) 0.0 time Reynolds number (Re) 1000 0.04 Mach number (M) 0.06 0.03 Fluid data Reference/inflow velocity (V0) 1.0 Characteristic dimension (D) 1.0 0.02 * 4 Dimensionless longitudinal stiffness (K 11) 3x10 * 0.01 Dimensionless transversal stiffness (K 22) 0.7864 * Dimensionless torsional stiffness (K 33) 17.05 0.00 * Dimensionless longitudinal mass (M 1) 195.57 -0.01 * angular accel. Dimensionless transversal mass (M 2) 195.57 * -0.02 Dimensionless torsional mass (M 3) 105.94 * 7 Dimensionless longitudinal damping (C 11) 1x10 -0.03 Structural data * Dimensionless transversal damping (C 22) 0.0325 -0.04 * Dimensionless torsional damping (C 33) 0.0 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00 time 10 9 8 7 6

Y 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 24 X Figure 5. Rectangular prism: finite element mesh.

In Fig. 6 the time histories related to angular and vertical displacements, velocities and accelerations are presented. It is important to notice that the time used in these figures is dimensionless. These results are very similar to those obtained by Figure 6. Rectangular prism: time histories of vertical and rotational Sarrate et al. (2001), using a different method. displacements, velocities and accelerations.

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2003 by ABCM October-December 2003, Vol. XXV, No. 4 / 357 A. L. Braun and A. M. Awruch

0.04

8 0.03

0.02 7

0.01 6

0.00 5 Y -0.01 vertical veloc. 4 -0.02

3 -0.03

-0.04 2

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00 2 3 4 5 6 7 8 9 10 time X (II a) 0.015

8 0.010 -0.16

7 -0.23 -0.02 0.005 -0.02

6 0.02 -0.72 0.02 0.000 -1.63 -0.58 5 0.11

Y 0.16

vertical accel. -0.02 -0.005 -0.16 0.16 0.07 -0.05 4 -0.10 0.05 0.07 -0.010 -0.05 3 0.05 0.03 -0.02

-0.015 2 0.03 0.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00 time 2 3 4 5 6 7 8 9 10 X Figure 6. (ontinued). (I b)

The streamlines and the pressure field are shown in Fig. 7, in 8 three instants (t* = 439, t* = 442 and t* = 448). It can be observed 7 the presence of high pressure gradients and large vortices alternating between the lower and the higher surfaces. The streamlines show 6 that the cross-section orientation with respect to the free flow 5 direction modifies the boundary layer form. This conclusion is the Y same observed in bluff bodies, where the flux-ward dimension is 4 one of the parameters that determine the forms of the boundary layer and wake. In Fig. 8 it is verified the distortion of the mesh in an 3 instant where extreme structural rotation is reached. 2

2 3 4 5 6 7 8 9 10 X 8 0.06 0.15 (II b) 0.05 0.07 0.11 7 0.16 0.02 0.18 -0.02 8 -0.15 6 0.23 -0.10 -0.18 0.14 -0.15 -0.58 7 0.00 0.31 -0.10 -0.32 -0.32

Y 5 0.40 0.05 0.00 -0.58 0.11 -1.94 6 -0.05 -0.02 4 -0.02 -1.94 -0.58 0.11 -0.18 0.00

0.05 -0.32 -0.05 Y 5 0.40 0.07 0.31 3 0.04 0.02 0.23 -0.58 0.00 -0.05 -0.18 4 -0.10 2 -0.15 -0.02 0.18 -0.05 0.16 0.11 3 0.00 2 3 4 5 6 7 8 9 10 0.15 0.02 X 0.07 (I a) 2 0.04 0.14

Figure 7. Rectangular prism: (I) pressure contours and (II) streamlines 2 3 4 5 6 7 8 9 10 contours; (a) t* = 439; (b) t* = 442 and (c) t* = 448. (I c)X Figure 7. (Continued).

358 / Vol. XXV, No. 4, October-December 2003 ABCM Numerical Simulation of the Wind Action on a Long-Span Bridge Deck

5 Y

4 (a) 3

2 3 4 5 6 7 8 9 10 X (II c) (b) Figure 7. (Continued). Figure 9. General characteristics of the Great Belt East Bridge: (a) cross- section; (b) elevation. 8 Firstly, the fixed cross-section was analyzed and the 7 aerodynamic coefficients were obtained as functions of the angle of attack of the wind direction. The Strouhal number was also 6 calculated. Finally, free oscillations of the cross-section in the vertical and the rotational degrees of freedom were allowed in order 5

Y to carry out dynamic instability investigations.

4 Analysis of the Flow Around the Fixed Cross-Section 3 The computational domain and the boundary conditions used in this example, are illustrated in Fig. 10. As can be noticed, the inflow 2 boundary conditions are functions of the angle of attack of the wind

2 3 4 5 6 7 8 9 10 direction. Four different values of the angle of attack were studied: - X 10º, -5º, 0º e +5º. The initial pressure and velocity were assumed equal to zero. Figure 8. Rectangular prism: finite element mesh at t* = 448.

It can be observed that a satisfactory performance has been obtained for this example. The ability of the code to study fluid- structure interaction problems, where immersed bodies move due to the flow action with large displacements and rotations was also confirmed. In addition, the main characteristics of flows around bluff bodies were well simulated. The analysis of this kind of problem is only possible if a special Arbitrary Lagrangean-Eulerian (ALE) description is used. Another important aspect that is the mesh motion model, used previously by Teixeira & Awrucha (2001), was applied here with the same success. Figure 10. Great Belt East Bridge: geometry and boundary conditions for the fixed cross-section. Numerical Study of the Great Belt East Bridge Cross-section In this section, results of the numerical simulation of the wind The finite element mesh employed in this problem has 8175 action on a cross-section belonging to the Great Belt East Bridge are bilinear isoparametric elements with 8400 nodes, and is shown in presented, including the aerodynamic and the aeroelastic behavior. Fig. 11. The studies are accomplished by fixed and oscillating sectional models, according to the usual wind tunnel techniques. 90 80

The Great Belt East Bridge is located in Denmark, precisely in 70 the Great Belt Channel, an important international shipping route. 60

The design phase was initiated in 1989, being it opened to the traffic Y 40 in 1998. It is a suspension bridge, with a superstructure constituted 30 by two approaching spans of 535 m (each one) and a central span of 20 1624 m, which will be studied in this work. In Fig. 9, general 10 0 0 50 100 150 200 250 aspects of the bridge are shown. The pictures were taken from X Larsen & Walther (1997). Figure 11. Great Belt East Bridge: finite element mesh.

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2003 by ABCM October-December 2003, Vol. XXV, No. 4 / 359 A. L. Braun and A. M. Awruch

5 The Reynolds number used in the four cases is 3.0x10 . The α=−10° other constants used in the analysis are presented in Table 2. From 60 the well-known Courant stability condition, the time step is Ät = 1.15x10-4 s.

50 Table 2. Great Belt East Bridge: data used to determine aerodynamic coefficients. Y Great Belt East Bridge - 40 Constants Reynolds 3x105 Specific mass (ñ) 1.32 Kg/m3 2 30 Volumetric viscosity (ë) 0.0 m /s 30 40 50 60 70 X Kinematic viscosity (í) 5.78x10-4 m2/s

Sound velocity (c) 337.0 m/s α=−5°

Reference/inflow velocity (V0) 40.0 m/s 60 Smagorinsky’s constant (CS) 0.2 Charact. dimension/cross-section(D) 4.40 m 50 The investigated mean coefficients, obtained from the time histories, are plotted in Fig. 12 as functions of the angle of attack, Y and compared with the experimental results given by Reinhold et al. 40 (1992) and the numerical results obtained by Kuroda (1997).

1.00 30 0.80 30 40 50 60 70 X CD 0.60 α=0° 0.40 CL 60

0.20 CM

0.00

50 -0.20 CD,CL and CM

-0.40 Y

-0.60 40

-0.80 Present work Reinhold et al. (1992) - (exp.) -1.00 Kuroda (1997) - (num.) 30 -1.20 30 40 50 60 70 X -10.00 -5.00 0.00 5.00 10.00 angle of attack(degrees) α=+5° Figure 12. Great Belt East Bridge: numerical and experimental results for aerodynamic coefficients as functions of the angle of attack. 60

The Strouhal number, obtained from the vertical velocity 50 component time history V2 at a point located a distance 0.2 B behind the cross-section (with zero angle of attack), is 0.18. Comparisons of Y some of the results obtained for Strouhal number of the referred bridge are shown in Table 3. 40

Table 3. Strouhal number for the Great Belt East Bridge. 30 30 40 50 60 70 Strouhal number X Reference - Reynolds 3x105 Figure 13. Great Belt East Bridge: streamlines contours for different Present work 0.180 angles of attack. Larsen et al. (1998) (numer.) 0.170 Wind tunnel tests (from: Larsen et al. (1998)) 0.160 Aeroelastic Analysis: Flutter The streamlines observed for the different angles of attack are In this work, Flutter analysis is accomplished in two different presented in Fig. 13 and are similar to those obtained by Kuroda ways: a) direct method, presented by Selvam et al. (2002), and b) (1997). * using the flutter derivative A2 , introduced by Scanlan & Tomko (1971), being this coefficient related to the aerodynamic damping due to torsional rotations. Both methods have the same experimental procedure and are based on the observation of the structural response to cross-section rotations for various wind velocity values.

360 / Vol. XXV, No. 4, October-December 2003 ABCM Numerical Simulation of the Wind Action on a Long-Span Bridge Deck

In Selvam et al. (2002), it is shown a method in which the Table 4. Great Belt East Bridge: structural data used in the flutter analysis. growth/decay rate is determined from the structural response, Great Belt East Bridge – Reynolds 105 – Structural data V observed in several reduced wind velocities given by V ∗ = 0 , 9 f B Longit. and transv. stiffness (K11, K 22) 3x10 N/m.m 6 Torsional stiffness (K 33) 7.21x10 N.m/rad.m where V0 is the inflow velocity, B is the bridge deck width and f is 4 2 the natural structural frequency. These values of the growth/decay Longit. and transv. mass (M1, M 2) 2.27x10 N.s /m.m 6 2 k k +1 k k k+1 Torsional mass (M 3) 2.47x10 N.m.s /rad.m rate are calculated with ãg / d = ( y − y ) y , where y and y 4 Longit. and transv. damping (C11, C 22) 3x10 N.s/m.m are the peak values in the same oscillation period. After, they are Torsional damping (C ) 0.00 N.m.s/rad.m transported to a chart in function of the reduced wind velocity, and 33 Vertical natural frequency (f ) 0.099 Hz the critical velocity corresponds to the point where the curve crosses h the velocity axis (growth/decay rate = 0). Angular natural frequency (fè) 0.272 Hz In the flutter derivatives method (Scanlan & Tomko, 1971), the Critical damping (æ) 0.002 exp exp experimental damping æè and the natural frequency ù è for In Fig. 14, time histories related to the angular displacement are each reduced wind velocity are obtained from the structural presented for each reduced wind velocity. From the rotation time response. These values are introduced into an expression, histories, the growth/decay rate as well as the logarithmic decrement representing the aerodynamic damping and given by: for each reduced velocity were obtained. In Table 5 all these values are presented. 4I  ù  A* V* æ è æexp (30) 0.04 2 ()= 4  è exp − è  V* = 2 ñB 0.03  ù è  0.02 where I is the mass moment of inertia, ñ is the specific mass of the 0.01 fluid, B is the bridge deck width, æè is the structural critical 0.00 damping and ù è the structural natural frequency. Eq. (30) may be -0.01 angular displ.(rad) also written, by experimental considerations, in a reduced -0.02 exp exp expression in terms of the logarithmic decrement ä ≅ 2ðæ as -0.03 follows: -0.04

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 time(s) I 2äexp A* V* è (31) 2 ()= − 4 0.05 ñB ð V* = 4 0.04

0.03

* 0.02 Thus, a curve of this coefficient A2 in function of the reduced wind velocities is built. The critical flutter velocity is obtained by a 0.01 critical condition expressed by: 0.00 -0.01

angular displ.(rad) -0.02 * 4Iæè A2 = (32) -0.03 ñB -0.04

-0.05

* 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 So, when A2 > 4Iæè ñB the aerodynamic damping is greater time(s) than the structural damping, originating negative damping and 0.04 oscillations with growing amplitudes. V* = 6 0.03

The geometry as well as the finite element mesh employed in 0.02 the determination of the critical velocity of flutter is the same that 0.01 was used previously (Fig. 10 and Fig. 11). Initially, a fixed cross- 0.00 section with an inclination of 1.8º was taken. After 30000 time -0.0 1 steps, the load boundary conditions at the body surface were -0.0 2

computed, and then, the body motion was allowed. The outflow angular displ.(rad) -0.0 3 boundary conditions were kept identical to the case where the body -0.0 4 remains fixed with zero angle of attack, with exception to the inflow -0.0 5 velocity, which changes in order to obtain the desired curves. -0.0 6

The physical properties and design values of the structure, 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 employed in the experiments, are found in Table 4. The structure is time(s) idealized such that only torsional rotations are allowed (because it Figure 14. Great Belt East Bridge: angular displacement time histories for was verified that coupling vertical displacements and rotations will the studied reduced wind velocities. not modify significativelly the critical velocity of flutter). The problem was analyzed for four reduced velocities: 2, 4, 6 and 10. These values correspond to the following inflow velocities: 16.86 m/s, 33.73 m/s, 50.59 m/s and 84.32 m/s, respectively. The flow was analyzed with Re = 105.

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2003 by ABCM October-December 2003, Vol. XXV, No. 4 / 361 A. L. Braun and A. M. Awruch

0.50 * V* = 10 derivative method, considering the critical condition as A 2 ≥ 0.40 1.62x10-2, yields a reduced velocity equal to 8.66 which corresponds 0.30

0.20 to a critical velocity equal to 73 m/s. In Table 6 below, comparisons

0.10 of the critical velocity obtained by this work and by other authors 0.00 (through numerical and experimental works) are shown. -0.1 0

-0.2 0 Table 6. Flutter velocity for the Great Belt East Bridge. angular displ.(rad) -0.3 0 -0.4 0 Great Belt East Bridge – Flutter velocity -0.5 0 Reference V (m/s) -0.6 0 crit

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 Present work – direct method 69 time(s) Present work – flutter derivative 73 Figure 14. (Continued). Selvam et al. (2002) (num.) 65-72 Larsen et al. (1997) (num.) 74 Enevoldsen et al. (1999) (num.) 70-80 Table 5. Great Belt East Bridge: numerical results for flutter analysis. Wind tunnel tests (from: Larsen et al. (1998)) 73 Great Belt East Bridge - Reynolds 5 Results 10 Conclusions V* = 2 V* = 4 V* = 6 V* = 10 Growth/decay rate 0.131 0.270 0.311 -0.500 A model for the numerical simulation of the wind action on bridges was described. The computational code was validated Logarithmic decrement 0.176 0.205 0.291 -0.403 through the analysis of a rectangular cross-section and studies of the aerodynamic and aeroelastic behavior of the Great Belt East Bridge In Fig. 15 the curves to obtain the critical velocity of flutter by cross-section. The results show good agreement with those obtained the direct method of Selvam et al. (2002) and by the flutter by other authors. In future works, it is expected to explore other * derivative A2 (Scanlan & Tomko, 1971), respectively, are bridge decks, considering details such as cables, guardrails and presented. aerodynamic appendages. It is also expected improvements in the efficiency in processing time, mainly in turbulent flows. An 0.20 alternative is to use time integration with subcycles (Teixeira & a 0.15 Awruch , 2001), optimizing the time step on the computational domain. Another possibility is to use semi-implicit schemes for the 0.10 flow analysis.

0.05 A*2 = 0.0162 References 0.00 A*2 Bathe, K. J. Finite Element Procedures. Prentice Hall, Englewood -0 .0 5 Cliffs, NJ, 1996. Braun, A. L. “A Numerical Model for the Simulation of the Wind -0 .1 0 Action on Bridge Cross-Sections” (In Portuguese), Msc. Thesis, Federal University of Rio Grande do Sul, Porto Alegre, R.S., Brazil, 2002, 139p. -0 .1 5 Enevoldsen, I.; Pederson, C.; Hansen, S. O.; Thorbek, L. T. &

-0 .2 0 Kvamsdal, T. Computational wind simulations for cable-supported bridges. st 0.00 2.00 4.00 6.00 8.00 10.00 Wind Engineering into the 21 Century, Vol.2, pp. 1265-1270, 1999. V*=V/fB Hirt, C. W.; Amsden, A. A. & Cook, J. L. An arbitrary Lagrangean- (a) Eulerian computing method for all flow speeds. Journal of Computational 2.00 1.80 Physics, V.14, pp. 227-253, 1974. 1.60 Kawahara, M. & Hirano, H. A finite element method for high Reynolds 1.40 number viscous fluid flow using two step explicit scheme. International 1.20 1.00 Journal for Numerical Methods in Fluids, V.3, pp. 137-163, 1983. 0.80 Kuroda, S. Numerical simulation of flow around a box girder of a long 0.60 span suspension bridge. Journal of Wind Engineering and Industrial 0.40 0.20 Aerodynamics, V.67&68, pp. 239-252, 1997. 0.00 Larsen, A. & Walther, J. H. Aeroelastic analysis of bridge girder -0 .2 0 sections based on discrete vortex simulation. Journal of Wind Engineering -0 .4 0 Growth/decay rate -0 .6 0 and Industrial Aerodynamics, V.67&68, pp. 253-265, 1997. -0 .8 0 Larsen, A. & Walther, J. H. Discrete vortex simulation of flow around -1 .0 0 -1 .2 0 five generic bridge deck sections. Journal of Wind Engineering and -1 .4 0 Industrial Aerodynamics, V.77&78, pp. 591-602, 1998. -1 .6 0 Rossa, A. L. & Awruch, A. M. 3-D finite element analysis of -1 .8 0 nd -2 .0 0 incompressible flows with heat transfer. Proceedings of the 2 International

0.00 2.00 4.00 6.00 8.00 10.00 Conference on Computational Heat and Mass Transfer, COPPE/UFRJ,, Rio V*=V/fB de Janeiro, Brazil, pp. 22-26, October, 2001. (b) Reinhold, T. A.; Brinch, M. & Damsgaard, A. Wind tunnel tests for the Great Belt link. In: Proceedings International Symposium on Aerodynamics Figure 15. Flutter analysis: (a) by the flutter derivative * and (b) by the A2 of Large Bridge, pp. 255-267, 1992. direct method for the Great Belt East Bridge. Sarrate, J.; Huerta, A. & Donea, J. Arbitrary Lagrangean-Eulerian formulation for fluid-rigid body interaction. Computer Methods in Applied Mechanics and Engineering, V.190, pp. 3171-3188, 2001. From Fig. 15, the reduced critical velocity obtained by the direct Scanlan, R. & Tomko, J. J. Airfoil and bridge deck flutter derivatives. method is 8.18, corresponding to a velocity of 69 m/s. By the flutter Journal of Engineering Mechanics Division, EM6, pp. 1717-1737, 1971.

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Schlichting, H. Boundary-Layer Theory. McGraw-Hill Inc., New York, aTeixeira, P. R. F. & Awruch, A. M. Three dimensional simulation of 2nd ed., 1978, 815 p. high compressible flows using a multi-time-step integration technique with Selvam, R. P.; Govindaswamy, S. & Bosch, H. Aeroelastic analysis of subcycles. Applied Mathematic Modeling, V.25, pp. 613-627, 2001. bridges using FEM and moving grids. Wind and Structures, V.5, pp. 25-266, bTeixeira, P. R. F. & Awruch, A. M. Analysis of compressible fluids 2002. and elastic structures interaction by the finite element method. Proceedings Tabarrok, B. & Su, J. Semi-implicit Taylor-Galerkin finite element of the 16th Brazilian Congress of Mechanical Engineering, COBEM 2001, method for incompressible viscous flows. Computer Methods in Applied Uberlândia (MG – Brazil), pp. 26-30, November, 2001.. Mechanics and Engineering, V.117, pp. 391-410, 1994.

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