Bridge Wind Load FSI Analysis
Bridge Wind Load FSI: Theoretical Foundations
Long-span bridges (suspension bridges, cable-stayed bridges) are flexible against wind, and their dynamic response under wind loads governs structural safety. The collapse of the Tacoma Narrows Bridge in 1940 is a classic example. It is necessary to accurately predict not only static wind pressure but also wind-structure interaction phenomena such as Vortex-Induced Vibration, Flutter, galloping, and buffeting.
How are they different?
| Phenomenon | Mechanism | Characteristics |
|---|---|---|
| Vortex-Induced Vibration (VIV) | Lock-in of Kármán vortices | Resonance within a limited wind speed range |
| Flutter | Aerodynamic negative damping | Divergence above critical wind speed |
| Galloping | Cross-section shape dependent instability | Occurs in rectangular sections |
| Buffeting | Turbulence in natural wind | Random response |
Governing Equations
What equations describe the wind response of a bridge?
Scanlan's flutter differential equation is fundamental. Lift $L$, drag $D$, and moment $M$ per unit length are expressed using self-excited aerodynamic coefficients (flutter derivatives).
Here, $K = B\omega/U$ is the reduced frequency, $H_i^*$, $A_i^*$ are flutter derivatives obtained from wind tunnel tests or CFD. $B$ is the deck width, $h$ is the vertical deflection, and $\alpha$ is the torsional angle.
Can't flutter derivatives be determined theoretically?
For thin airfoils, they can be derived analytically from Theodorsen's function, but bridge deck sections are bluff bodies, so we must rely on wind tunnel tests or CFD. The recent approach is to identify $H_i^*$, $A_i^*$ from CFD using the forced oscillation method.
Critical Flutter Wind Speed
How is the critical flutter wind speed determined?
Solve the eigenvalue problem for a 2-degree-of-freedom coupled system (vertical deflection $h$, torsion $\alpha$).
Substitute the flutter derivatives into the self-excited aerodynamic terms on the right side. The wind speed at which the system damping becomes zero is the critical flutter wind speed $U_{cr}$. For the Akashi Kaikyō Bridge (part of the Honshū-Shikoku Bridge system), the design requirement was $U_{cr} > 78$ m/s.
"10,000 Hours of Wind Tunnel Testing" for the Akashi Kaikyō Bridge – The Longest Suspension Bridge Protected by Bridge FSI Theory
The Akashi Kaikyō Bridge (completed in 1998), with a total length of 3,911m and a central span of 1,991m, is the world's longest suspension bridge. The biggest challenge in its design was "ensuring the flutter speed is sufficiently higher than the maximum wind speed of 80 m/s during typhoons." The design team conducted over 10,000 cumulative hours of wind tunnel testing using a 1:100 scale model, iteratively optimizing the cross-section shape of the stiffening girder between the main towers. The final slim box girder section adopted suppresses Kármán vortex shedding while ensuring a flutter speed more than 1.7 times the design wind speed. For FSI design of bridges of this scale, CFD is still used as "pre-screening" for wind tunnel tests; the standard workflow is to narrow down candidate cross-section shapes from dozens to a few using CFD calculations before proceeding to experiments.
Computational Methods for Bridge Wind Load FSI
Identification of Flutter Derivatives via CFD
Please explain the procedure for determining flutter derivatives using CFD.
The forced oscillation method is standard.
1. Create a 2D cross-section model (deck section only)
2. Perform CFD with prescribed sinusoidal vertical oscillation $h(t) = h_0 \sin(\omega t)$
3. Separate the in-phase component (stiffness term) and out-of-phase component (damping term) from the lift time history
4. Identify $H_1^*$~$H_4^*$, $A_1^*$~$A_4^*$
5. Perform similarly for torsional oscillation $\alpha(t) = \alpha_0 \sin(\omega t)$
Use 2D RANS or LES for CFD. The $k$-$\omega$ SST turbulence model has a proven track record for bridge deck sections.
Is 3D full-bridge model FSI not necessary?
The "strip theory" approach, where flutter derivatives obtained from 2D cross-section analysis are applied to a full-bridge structural model using modal methods, is mainstream in practice. However, 3D CFD-CSD coupling is sometimes performed when 3D effects (end vortices, spanwise phase differences) are significant.
CFD Analysis of Vortex-Induced Vibration
How do you simulate vortex-induced vibration?
Perform 2D cross-section CFD-CSD coupling. Solve the flow around the deck section using LES (or DES) and couple it with a spring-mass model representing the structure.
The lock-in phenomenon occurs when the vortex shedding frequency $f_s = St \cdot U/D$ (Strouhal number $St \approx 0.1$~$0.2$) approaches the structural natural frequency. To correctly predict the lock-in range and amplitude with CFD, sufficient time integration (over 100 vibration cycles) is necessary.
Buffeting Analysis
How is the randomness of natural wind handled?
The frequency domain method is efficient. Using Davenport's wind speed spectrum as input, combine the aerodynamic admittance function and the structural transfer function to obtain the response spectrum.
$S_u$ is the fluctuating wind speed spectrum, $\chi$ is the aerodynamic admittance, and $H$ is the structural frequency response function. In the time domain, there's also a method of generating an artificial fluctuating wind field (von Karman type) and applying it as the inlet condition for CFD.
Cross-validation of Wind Tunnel Tests and CFD – On-site Realities in Bridge Design
Wind tunnel tests still play the leading role in wind-resistant bridge design, but opportunities for CFD and FSI are increasing. Since wind tunnels use 1/100~1/200 scale models, there's a significant problem with Reynolds number mismatch compared to the real structure. CFD can, in principle, perform full-scale calculations and reproduce 3D effects and turbulence from bridge deck traffic that are difficult to replicate in wind tunnels. On-site, the standard practice is "grasping general trends with wind tunnels and refining details with CFD."
Bridge Wind Load FSI in Practice
The flow based on Japan's "Wind-Resistant Design Manual for Highway Bridges" is as follows.
1. Evaluation of Aerodynamic Characteristics of Cross-section: Obtain aerodynamic force coefficients ($C_D$, $C_L$, $C_M$) via wind tunnel test or CFD.
2. Static Stability Check: Verify divergence wind speed.
3. Vortex-Induced Vibration Check: Predict lock-in wind speed range and response amplitude.
4. Flutter Check: Ensure critical flutter wind speed is at least 1.2 times the design wind speed.
5. Buffeting Response: Maximum displacement/stress at design wind speed.
6. Consideration of Vibration Control Measures: TMD, flaps, fairings, etc.
How do you decide between CFD and wind tunnel tests?
In practice, wind tunnel tests are primary, with CFD used complementarily. However, recently CFD has been active in parametric studies of cross-section shapes. Wind tunnel tests cost several million yen per cross-section, but CFD allows rapid evaluation of design change effects.
CFD Mesh Design
What should I be careful about with CFD meshing for bridge deck sections?
| Parameter | Recommended Value | Reason |
|---|---|---|
| Computational Domain (Upstream) | 10B | Avoid upstream blockage |
| Computational Domain (Downstream) | 20B | Wake development |
| Computational Domain (Top/Bottom) | 10B | Avoid wall effects |
| First layer thickness on deck surface | $y^+ < 1$(LES),$y^+ \approx 30$(RANS) | Wall resolution |
Spanwise direction (3D LESRelated TopicsCoupled AnalysisAeroelastic Flutter AnalysisCoupled AnalysisWind Load-Structure Coupled AnalysisFluidBuilding Wind Load AnalysisFluid Analysis (CFD)Aeroelastic Analysis — Flutter Theory and Governing EquationsGlossaryFlutter — Explanation of CAE TerminologyFluid Analysis (CFD)Vortex-Induced Vibration (VIV) Related Simulators Experience the theory firsthand with the interactive simulator for this field All SimulatorsRate this article Thank you for your feedback! Helpful More details Report error |