Numerical simulation of the wind action on a long-span bridge deck

Numerical simulation of the wind action on a long-span bridge deck

A. L. Braun^I; A. M. Awruch^II

^IPPGEC/UFRGS Av. Osvaldo Aranha, 99 – 3o andar 90035-190 Porto Alegre, RS. Brazil www.cpgec.ufrgs.br, [e-mail]

^IIPPGEC/UFRGS & PROMEC/UFRGS Av. Sarmento Leite, 425 90050-170 Porto Alegre, RS. Brazil www.mecanica.ufrgs.br/promec, [e-mail]

ABSTRACT

A numerical model to study the aerodynamic and aeroelastic bridge deck behavior is presented in this paper. The flow around a rigid fixed bridge cross-section, as well as the flow around the same cross-section with torsional motion, are investigated to obtain the aerodynamic coefficients, the Strouhal number and to determine the critical wind speed 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 body with elastic restrains for the cross-section rotation and displacement components. The fluid-structure interaction is accomplished applying the compatibility and equilibrium conditions at the fluid-solid interface. The structural dynamic analysis is performed using the classical Newmark's method.

Keywords: Fluid-structure interaction, Finite Element Method (FEM), Large Eddy Simulation (LES), aeroelasticity, aerodynamics

Introduction

Wind tunnel tests for assessment of aerodynamic and aeroelastic informations in the study of bridge girders performance are numerically simulated in this work. The usual way to obtain these informations is using representative models in a wind tunnel. However, with the improvement in computers technology and computational fluid dynamics (CFD) algorithms, many of these problems can also be analyzed by numerical simulation.

Long-span bridges, such as suspension bridges for example, must be designed to support, from a static point of view, the mean wind forces (using the drag, lift and pitching moment coefficients). Besides, considering that such structures show low damping and low stiffness, they are subjected to aeroelastic phenomena, such as flutter, galloping and vortex shedding induced vibrations. Only the first case will be studied in the present work.

The term aeroelasticity is used when the aerodynamic forces produce some kind of structural instability as a consequence of the interaction between these forces and the structural motion. The flutter phenomenon is a type of aeroelastic instability that begins when the effective damping (structural + aerodynamic) becomes negative.

Kawahara & Hirano (1983) were one of the first authors to analyze numerically the wind action on a bridge cross-section. They used the Finite Element Method (FEM) to obtain the aerodynamic coefficients as functions of the angle of attack of the wind and the Strouhal number. Kuroda (1997) employed two different numerical procedures to study the approaching span of the Great Belt East Bridge: the Finite Element Method and the Finite Difference Method (FDM), and Large Eddy Simulation (LES) with the Smagorinsky's model for the turbulent flow. He also presented the results referring to the aerodynamic coefficients for various angles of attack and the Strouhal number for both numerical methods employed in his analysis. Larsen & Walther (1997) analyzed several bridge decks observing their aeroelastic behaviour using a numerical code based on Discrete Vortex Simulation (DVS), presenting the respective critical flutter velocity (using the flutter derivatives). Recently, Selvam et al. (2002) applied a direct method for the flutter analysis. They used (FEM) and (LES).

In this work, the analysis of the flow of a slightly compressible fluid in a two-dimensional flow domain was carried out using an explicit two-step Taylor-Galerkin method with an Arbitrary Lagrangean-Eulerian (ALE) description. A similar Taylor-Galerkin formulation was used by Tabarrok & Su (1994) and by Rossa & Awruch (2001), but with a semi-implicit scheme. The (ALE) scheme was first presented by Hirt et al. (1974) in a numerical work. Since this first paper many other authors used this description with the same concepts. The classical Smagorinsky's model, similar to that presented by Kuroda (1997), was employed for the sub-grid scales simulation. The finite element method was used for spatial discretization. The structure was considered as a rigid body with elastic restrains for the cross-section rotation and displacement components. The coupling between fluid and structure was performed applying the compatibility and equilibrium equations at the interface. The structural dynamic analysis was accomplished using the classical Newmark's method (Bathe, 1996). Examples are presented to illustrate the capability of the computational method.

Governing Equations for the Flow Simulation

The governing equations, considering the pseudo-compressibility approach in an isothermic process, Large Eddy Simulation (LES) with Smagorinsky's model for turbulent flows and an Arbitrary Lagrangean-Eulerian (ALE) description, are:

a) Momentum equations:

being with and Ä=(Ä_xÄ_y)^1/2, where Ä_x and Ä_y are the element dimensions in the global axis direction x and y, respectively.

b) Mass conservation equation:

which is obtained considering .

The boundary conditions of Eqs. (1) and (2) are the following:

In these equations, v_i and p (the velocity components and the pressure, respectively) are the unknowns. The viscosities and , the specific mass ñ and the sound velocity c, are the fluid properties. The eddy viscosity depends of derivatives of the filtered velocity components, of the element dimensions and of the Smagorinsky's constant C_S. For a purely Eulerian description, the mesh motion velocity w at each nodal point, with components w_i, is equal to zero. Now, for a purely Lagrangean description, the mesh motion velocity at each nodal point is equal to the fluid velocity, i.e. v_i = w_i (i = 1, 2). Finally, in an Arbitrary Lagrangean-Eulerian formulation, w ¹ 0 and w ¹ v.

On the boundaries and Ã_p, prescribed values for velocity and pressure, and , respectively, must be specified, while on Ã_ó the boundary force must be in equilibrium with the stress tensor components ó_ij. In Eq. (5), n_j is the direction cosine between a vector perpendicular to Ã_ó and the axis x_j.

Initial conditions for the pressure and the velocity components at t = 0 must be given.

The Algorithm for the Flow Simulation

Expanding the governing equations in a Taylor's series up to second order terms, the algorithm for the flow simulation contains the following steps (Braun, 2002):

1) Calculate with:

where r_j=(n_j-w_j) and = (n+n_t).

2) Clculate p with:

3) Calculate

4) Calculate with:

5) Calculate with:

6) Calculate with:

These expressions must be employed after applying the classical Galerkin technique into the finite element method (MEF) context.

As the scheme is explicit, the resulting system is conditionally stable, with a stability condition given by:

where á (which is a real number less than one) is a safety coefficient, Äx_i and v_i are the i-th element characteristic dimension and the velocity, respectively, and NTE is the total number of elements.

Although variable time step could be adopted (Teixeira & Awruch^a, 2001), in this work an unique value of Ät will be used for the whole process, adopting the smallest one from those obtained by Eq. (12).

The Fluid-Structure Coupling

In the present work, the structure is idealized as a two-dimensional rigid body. Displacement and rotations take place on the plane formed by the axis x₁ and x₂; the body is restricted by dampers and springs, as indicated in Fig. 1.

The structural dynamic equilibrium equation is given by the following matrix expression:

where is the mass matrix, the damping matrix, the stiffness matrix and and the acceleration, velocity and generalized displacements, respectively. Finally, is the load vector.

The subscript S means that these matrices belong to the structure and the superscript C indicates that these values correspond to the gravity center of the solid body. Equation (13) can be written as follows:

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.

At the solid-fluid interface, the compatibility condition must be satisfied, or in other words, the fluid velocity and the structure velocity must be the same at the common nodes of both fields. The compatibility condition and the translation of variables evaluated at the center of gravity of the body to a point located at the fluid-structure interface may be written with the following expressions:

where S and F are referred to the structure and the fluid, respectively, and the superscript I is referred to the interface. It is important to notice that the both vectors and have two components that correspond to the global axis direction. However, has three components, because it includes the rotation around an axis perpendicular to the plane formed by x₁ and x₂. Values of can be transported to the solid-fluid interface (or to nodes belonging to the structure boundary) through a translation matrix , as given by Eq. (15), being l₁ and l₂ the distance components 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:

Deriving Eq. (15) with respect to time, taking into account matrix and equation (16), the following expression is obtained:

Equation (15) and Eq. (17) are applied to each node at the interface, where the equilibrium condition must be also satisfied, that means that the load acting on the structure at the interface, must be equal to the load given by Eq. (5), but with an opposite signal (because here the fluid action on the structure is considered, while Eq. (5) represents the boundary action on the fluid). can be transported to the center of gravity of the body, obtaining:

where is the transpose matrix of , given by Eq. (15), and contains the two components of the fluid boundary force acting on the structure at a point located on the structure surface Ã_s (Ã_s represents also the solid-fluid interface); these forces are given by Eq. (5), but with an opposite signal.

To determine the coupling effects between the fluid and the structure, in the finite element method (FEM) context, consider an element belonging to the fluid domain in contact with the solid body, as indicated in Fig. 3, where it can be observed that only points 1 and 2 are in contact with the structure.

The momentum equations in its matricial form, at element level (e), can be obtained by applying the Galerkin method to the Eq. (1), writting:

where contains the time derivative coefficients from the velocity components contains the coefficients of advective and diffusive terms, contains the coefficients of pressure derivative terms with respect to x₁ and x₂ and, finally, is a vector containing the boundary integrals resulting from the integration by parts of pressure and diffusive terms.

In Eq. (19), and contain, respectively, acceleration and velocity components corresponding to nodes 1 and 2 of Fig. 3, while and contain variables corresponding to nodes 3 and 4 of the same figure. A similar remark can be made with respect to the vectors of pressure gradients and boundary forces . Matrix contains elements coming from the connection of node 1 with itself and with node 2, and the connection of node 2 with itself and with node 1. Matrix reflects the connection between the nodes 1 with 4 and 2 with 3. Similar commentaries can be made with respect to matrices and .

Regarding the structural analysis, only the first matricial expression of Eq. (19) is necessary, because only this equation contributes to the assembling of the overall dynamic equilibrium equation. On the other hand, as the structural and the flow analysis 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 and at nodes that do not have any contact with the structure (they were calculated previously in the flow analysis). Referring to Fig. 3, at nodes 3 and 4, the values and 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:

Equation (15) with matrix and Eq. (17) with matrix are considered for each node at the interface. Then, when an element side with two nodes and lying on the fluid-structure interface is considered, Eq (15) and Eq. (17) are written in the following form:

Referring again to Fig. 3, the matrices and are given by:

The contribution from , on the side 1-2 of the element (e), to the total load acting at the gravity center of the body, can be calculated as:

Considering Eqs. (14), (15), (17) and (20), with the last one multiplied by ñ, the structural dynamic equilibrium equation, taking into account the solid-fluid coupling effect, is given by:

where NTL is the total number of fluid elements in contact with the structure, having at least one straight segment common to the solid body surface, forming the solid-fluid interface. The matricial Eq. (24) is re-written as:

As can be noticed, is a non-symmetric matrix, because it contains the advective terms and . This last term leads to the non-linearity of matrix .

In this work, a monolithic coupling between fluid and structure was not considered. The analysis for both fields is made in a sequential way. Firstly, Eq. (6) to Eq. (11) are solved, with the 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 time steps may be used for the fluid and the structure, here the same time step was adopted, because the computer time required by the structure analysis is negligible with respect to the processing time demanded by the flow analysis. Furthermore, compatibility and equilibrium conditions are more accurately imposed if the same time intervals are employed.

Strouhal Number and Aerodynamic Coefficients Calculation

The Strouhal number (St) can be calculated with , where V₀ is a reference velocity, L₀a reference dimension and f_v is the shedding frequency of a pair of vortices. It depends of the immersed prism cross-section, its oscillations, its superficial details, the Reynolds number and the flow characteristics. Formulation to calculate this number is presented in many publications and texts (for example, Schlichting, 1979). When the Strouhal number of a flow with a given immersed structure is known, it is possible to obtain the velocity , which will produce the resonant phenomenon on the vibrating body. It occurs when the shedding frequency of a pair of vortices is approximately equal to the structural natural frequency.

The drag coefficient C_D is related to the acting forces on the structure in the flow direction, while the lift coefficient C_Lis related to the acting forces on the structure in the transversal-to-flow direction. Finally, the pitching moment coefficient C_Mis related to the torsional moment acting at the gravity center of the immersed prism. The three coefficients can be calculated using the following expressions:

where and are the forces in the directions x₁ and x₂, respectively, acting on the structure at node i, located on the interface. and are the projections in the directions x₁ and x₂, respectively, of the distance between the gravity center and node i. NTN is the total number of nodes located on the solid-fluid interface. The forces and are the components of the force vector , given by Eq. (20). These forces are applied to the structure on each fluid element side belonging to the interface.

The pressure coefficient at a point i, , located on the interface, is related to the pressure acting at that point. This coefficient can be calculated using the following expression:

where p_i is the pressure at node i and P₀is a reference pressure (for example, the pressure in an undisturbed area of the flow). With the instantaneous values of pressure coefficients, the time history may be obtained and then the mean pressure distribution on the body surface, for a given time interval, may be calculated.

The Automatic Mesh Motion Scheme

Taking into account that the immersed body in the fluid can move and rotate in its plane and that the flow is described by an Arbitrary Lagrangean-Eulerian (ALE) formulation, a scheme for the mesh motion is necessary, establishing the velocity field w in the fluid domain, such that the element distortion will be as smaller as possible, according to the following boundary conditions:

In the present work, the mesh motion scheme is similar to that used by Teixeira & Awruch^b (2001). Considering that i is an inner point in the fluid field and j is a boundary node, the mesh velocity components at node i , in the direction of the axis x_k, are given by:

where NS is the total number of nodes belonging to the boundary lines and a_ij are the influence coefficients between the inner points and the boundary lines of the flow field, being , where d_ij is the distance between i and j, and n > 1. The exponent n can be adjusted by the user. Although regions with purely Eulerian and purely Lagrangean descriptions mat be used simultaneously with the ALE formulation, this alternative may result in more complex and less efficient codes. It may also lead to more difficulties to control mesh distortions.

Examples

Analysis of the Flow Around a Rectangular Prism

This example presents a prism with a rectangular cross-section free to oscillate in the transversal-to-flow and rotational directions. Through this problem, the program performance for large and coupled motion is observed, even that this cross-section form is not usually employed in bridge structures. In this study, special attention is given to the structural dynamic response in the two degrees of freedom of the cross-section and to the finite element mesh motion, remembering that a special scheme is employed for the non-linear dependence with respect to the cross-section rotation in the compatibility condition at the solid-fluid interface.

In the present section, results obtained for a flow around a rectangular prism with a Reynolds number equal to 1000 are presented. The rectangular cross-section exhibits a height/width relation (h/B) of 0.2. The geometry and the boundary conditions, in a non-dimensional form, are shown in Fig. 4. In addition, initial velocity and pressure field for the fluid-structure interaction problem are those of a developed flow obtained with a fixed body.

The finite element mesh has 5865 nodes and 5700 quadrilateral bi-linear isoparametric elements and is shown in Fig. 5. A non-dimensional time step Ät* = 1.0x10^-4 was adopted. The fluid and structural data are presented in Table 1.

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 Sarrate et al. (2001), using a different method.

The streamlines and the pressure field are shown in Fig. 7, in three instants (t* = 439, t* = 442 and t* = 448). It can be observed the presence of high pressure gradients and large vortices alternating between the lower and the higher surfaces. The streamlines show that the cross-section orientation with respect to the free flow direction modifies the boundary layer form. This conclusion is the same observed in bluff bodies, where the flux-ward dimension is 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 instant where extreme structural rotation is reached.

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 & Awruch^a (2001), was applied here with the same success.

Numerical Study of the Great Belt East Bridge Cross-section

In this section, results of the numerical simulation of the wind action on a cross-section belonging to the Great Belt East Bridge are presented, including the aerodynamic and the aeroelastic behavior. The studies are accomplished by fixed and oscillating sectional models, according to the usual wind tunnel techniques.

The Great Belt East Bridge is located in Denmark, precisely in the Great Belt Channel, an important international shipping route. The design phase was initiated in 1989, being it opened to the traffic in 1998. It is a suspension bridge, with a superstructure constituted by two approaching spans of 535 m (each one) and a central span of 1624 m, which will be studied in this work. In Fig. 9, general aspects of the bridge are shown. The pictures were taken from Larsen & Walther (1997).

Firstly, the fixed cross-section was analyzed and the aerodynamic coefficients were obtained as functions of the angle of attack of the wind direction. The Strouhal number was also calculated. Finally, free oscillations of the cross-section in the vertical and the rotational degrees of freedom were allowed in order to carry out dynamic instability investigations.

Analysis of the Flow Around the Fixed Cross-Section

The computational domain and the boundary conditions used in this example, are illustrated in Fig. 10. As can be noticed, the inflow boundary conditions are functions of the angle of attack of the wind direction. Four different values of the angle of attack were studied: -10º, -5º, 0º e +5º. The initial pressure and velocity were assumed equal to zero.

The finite element mesh employed in this problem has 8175 bilinear isoparametric elements with 8400 nodes, and is shown in Fig. 11.

The Reynolds number used in the four cases is 3.0x10⁵. The other constants used in the analysis are presented in Table 2. From the well-known Courant stability condition, the time step is Ät= 1.15x10^-4 s.

The investigated mean coefficients, obtained from the time histories, are plotted in Fig. 12 as functions of the angle of attack, and compared with the experimental results given by Reinhold et al. (1992) and the numerical results obtained by Kuroda (1997).

The Strouhal number, obtained from the vertical velocity 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 some of the results obtained for Strouhal number of the referred bridge are shown in Table 3.

The streamlines observed for the different angles of attack are presented in Fig. 13 and are similar to those obtained by Kuroda (1997).

Aeroelastic Analysis: Flutter

In this work, Flutter analysis is accomplished in two different ways: a) direct method, presented by Selvam et al. (2002), and b) using the flutter derivative , 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.

In Selvam et al. (2002), it is shown a method in which the growth/decay rate is determined from the structural response, observed in several reduced wind velocities given by , where V₀ is the inflow velocity, B is the bridge deck width and f is the natural structural frequency. These values of the growth/decay rate are calculated with ã_g/d=(y^k - y^k+1)/y^k, where y^k and y^k+1k+1 are the peak values in the same oscillation period. After, they are transported to a chart in function of the reduced wind velocity, and the critical velocity corresponds to the point where the curve crosses the velocity axis (growth/decay rate = 0).

In the flutter derivatives method (Scanlan & Tomko, 1971), the experimental damping and the natural frequency for each reduced wind velocity are obtained from the structural response. These values are introduced into an expression, representing the aerodynamic damping and given by:

where I is the mass moment of inertia, ñ is the specific mass of the fluid, B is the bridge deck width, is the structural critical damping and ù_è the structural natural frequency. Eq. (30) may be also written, by experimental considerations, in a reduced expression in terms of the logarithmic decrement as follows:

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

So, when the aerodynamic damping is greater than the structural damping, originating negative damping and oscillations with growing amplitudes.

The geometry as well as the finite element mesh employed in the determination of the critical velocity of flutter is the same that was used previously (Fig. 10 and Fig. 11). Initially, a fixed cross-section with an inclination of 1.8º was taken. After 30000 time steps, the load boundary conditions at the body surface were computed, and then, the body motion was allowed. The outflow boundary conditions were kept identical to the case where the body remains fixed with zero angle of attack, with exception to the inflow velocity, which changes in order to obtain the desired curves.

The physical properties and design values of the structure, employed in the experiments, are found in Table 4. The structure is idealized such that only torsional rotations are allowed (because it was verified that coupling vertical displacements and rotations will 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.

In Fig. 14, time histories related to the angular displacement are presented for each reduced wind velocity. From the rotation time histories, the growth/decay rate as well as the logarithmic decrement for each reduced velocity were obtained. In Table 5 all these values are presented.

In Fig. 15 the curves to obtain the critical velocity of flutter by the direct method of Selvam et al. (2002) and by the flutter derivative (Scanlan & Tomko, 1971), respectively, are presented.

From Fig. 15, the reduced critical velocity obtained by the direct method is 8.18, corresponding to a velocity of 69 m/s. By the flutter derivative method, considering the critical condition as A^*₂> 1.62x10^-2, yields a reduced velocity equal to 8.66 which corresponds to a critical velocity equal to 73 m/s. In Table 6 below, comparisons of the critical velocity obtained by this work and by other authors (through numerical and experimental works) are shown.

Conclusions

A model for the numerical simulation of the wind action on bridges was described. The computational code was validated through the analysis of a rectangular cross-section and studies of the aerodynamic and aeroelastic behavior of the Great Belt East Bridge cross-section. The results show good agreement with those obtained by other authors. In future works, it is expected to explore other bridge decks, considering details such as cables, guardrails and aerodynamic appendages. It is also expected improvements in the efficiency in processing time, mainly in turbulent flows. An alternative is to use time integration with subcycles (Teixeira & Awruch^a, 2001), optimizing the time step on the computational domain. Another possibility is to use semi-implicit schemes for the flow analysis.

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Publication Dates

• Publication in this collection
  18 Mar 2004
• Date of issue
  Dec 2003