Bridge Wind Load FSI Analysis
Theory and Physics
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.
Physical Meaning of Each Term
- Structure-Thermal Coupling Term: Thermal expansion due to temperature change induces structural deformation, and deformation affects the temperature field. $\sigma = D(\varepsilon - \alpha \Delta T)$. 【Everyday Example】Railroad tracks expand in summer, narrowing the gaps – a classic case of temperature rise → Thermal Expansion → stress generation. Warping of circuit boards after soldering is also due to differences in thermal expansion coefficients between materials. Engine cylinder blocks experience thermal stress from temperature differences between hot and cold parts, potentially leading to cracks.
- Fluid-Structure Interaction (FSI) Term: Bidirectional interaction where fluid pressure/shear forces deform the structure, and structural deformation changes the fluid domain. 【Everyday Example】Cables of a suspension bridge vibrating in strong wind (Vortex-Induced Vibration) – wind forces shake the structure, the shaking structure alters the airflow, further amplifying the vibration. Blood flow in the heart and elastic deformation of blood vessel walls, and aircraft wing flutter (aeroelastic instability) are also typical FSI problems. One-way coupling may suffice in some cases, but bidirectional coupling is essential for large deformations.
- Electromagnetic-Thermal Coupling Term: A feedback loop where Joule heating $Q = J^2/\sigma$ causes temperature rise, and temperature change alters electrical resistance. 【Everyday Example】The nichrome wire in an electric heater glows red when current flows, generating heat (Joule heat) – as temperature rises, resistance changes, altering current distribution. Eddy current heating in IH cooking heaters and increased sag of power lines due to temperature rise are also examples of this coupling.
- Data Transfer Term: Interpolation resolves mesh mismatch between different physical fields. 【Everyday Example】When calculating "feels-like" temperature by combining "air temperature data" and "wind data" in weather forecasting, interpolation is needed if observation points differ – similarly in CAE coupled analysis, structural and CFD meshes generally don't match, so the accuracy of data transfer (Interpolation) at the interface directly affects result reliability.
Assumptions and Applicability Limits
- Weak Coupling Assumption (One-way coupling): Valid when one physical field affects the other but the reverse is negligible.
- Cases Requiring Strong Coupling: Large deformations in FSI, strong temperature dependence in electromagnetic-thermal coupling.
- Time Scale Separation: Subcycling can improve efficiency when characteristic times of each physical field differ significantly.
- Interface Condition Consistency: Ensure energy and momentum conservation at the coupling interface is satisfied numerically.
- Non-applicable Cases: When three or more physical fields are strongly coupled simultaneously, monolithic methods may be necessary.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Thermal expansion coefficient $\alpha$ | 1/K | Steel: ~12×10⁻⁶, Aluminum: ~23×10⁻⁶ |
| Coupled interface force | N/m² (pressure) or N (concentrated force) | Check force balance between fluid and structure sides |
| Data transfer error | Dimensionless (%) | Interpolation accuracy depends on mesh density ratio. Below 5% is a guideline. |
Numerical Methods and Implementation
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."
Monolithic Method
Solves all physical fields simultaneously as one system of equations. Stable for strong coupling, but implementation is complex and memory consumption is high.
Partitioned Method (Partitioned Iterative Method)
Solves each physical field independently, exchanging data at the interface. Easy to implement and allows leveraging existing solvers. Suitable for weak coupling.
Interface Data Transfer
Nearest neighbor (simplest but low accuracy), projection (conservative), RBF interpolation (robust against mesh mismatch). Balance between conservation and accuracy is crucial.
Sub-iteration
Perform sufficient iterations within each coupling step to ensure interface condition consistency. Residual criteria are scaled based on typical values for each physical field.
Aitken Relaxation
Automatically adjusts the relaxation factor for coupling iterations. An adaptive technique that prevents divergence from over-relaxation and accelerates convergence.
Stability Condition
Beware of the added mass effect (in fluid-structure coupling when structural density ≈ fluid density). Apply Robin-type interface conditions or the IQN-ILS method if unstable.
Analogy for Aitken Relaxation
Aitken relaxation is like "balancing a seesaw." If one side pushes too hard, the other side flies up, and the reaction causes it to push too hard again – Aitken relaxation automatically adjusts the pushing force to suppress this oscillation. When coupling iterations oscillate and fail to converge, it's an adaptive technique that automatically adjusts the next correction based on the previous correction amount.
Practical Guide
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 |
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