Experimental and Computational Analysis of Large-Amplitude Flutter in the Tacoma Narrows Bridge: Wind Tunnel Testing and Finite Element Time-Domain Simulation

Abstract

Nonlinear wind-induced vibrations and coupled static–dynamic instabilities pose significant challenges for long-span suspension bridges, especially under large-amplitude and high-angle-of-attack conditions. However, existing studies have yet to fully capture the mechanisms behind large-amplitude torsional flutter. To address this, wind tunnel experiments were performed on H-shaped bluff sections and closed box girders using a high-precision five-component piezoelectric balance combined with a custom support system. Complementing these experiments, a finite element time-domain simulation framework was developed, incorporating experimentally derived nonlinear flutter derivatives. Validation was achieved through aeroelastic testing of a 1:110-scale model of the original Tacoma Narrows Bridge and corresponding numerical simulations. The results revealed Hopf bifurcation phenomena in H-shaped bluff sections, indicated by amplitude-dependent flutter derivatives and equivalent damping coefficients. The simulation results showed less than a 10% deviation from experimental and historical wind speed–amplitude data, confirming the model’s accuracy. Failure analysis identified suspenders as the critical failure components in the Tacoma collapse. This work develops a comprehensive performance-based design framework that improves the safety, robustness, and resilience of long-span suspension bridges against complex nonlinear aerodynamic effects while enabling cost-effective, targeted reinforcement strategies to advance modern bridge engineering.

1. Introduction

The catastrophic wind-induced collapse of the original Tacoma Narrows Bridge in 1940 [1] highlighted the destructive potential of aerodynamic instabilities and served as a catalyst for the emergence of bridge wind engineering as a distinct research discipline. In the 1970s, Scanlan pioneered the hybrid time-frequency self-excited force theory for flutter analysis [2,3,4,5,6,7], marking a shift toward quantitative research in bridge aerodynamics. However, this foundational theory is based on small-amplitude and linearity assumptions, which limit its applicability to real-world nonlinear behavior. Subsequent studies identified a distinct nonlinear flutter mechanism termed soft flutter [8,9,10,11,12,13,14], characterized by non-catastrophic limit cycle oscillations (LCOs) that occur after the critical flutter speed is exceeded. In this regime, structural vibrations stabilize into oscillations with amplitudes that increase gradually with wind speed, a key signature of soft flutter. Previous studies further documented associated Hopf bifurcation phenomena [12,15], providing deeper insight into the underlying nonlinear dynamics. Based on these insights, several nonlinear flutter self-excited force models were proposed for accurately reproducing both soft flutter and bifurcation dynamics. To address issues such as spurious numerical modes found in some frequency-domain formulations, time-domain self-excited force models were subsequently proposed [16,17,18,19]. These models have demonstrated high fidelity in reproducing experimentally observed vibration time histories, confirming their effectiveness for simulating complex nonlinear aeroelastic responses.

The Tacoma Narrows Bridge remains a landmark case study in bridge aerodynamics. Nevertheless, the precise mechanisms that triggered its flutter instability and the sequence leading to its wind-induced collapse continue to be debated. Scanlan’s classical flutter theory [2,4,20] reveals significant discrepancies between theoretical critical wind speeds and those obtained from experimental and historical data. Larsen et al. [20] attributed this deviation to actual damping ratios exceeding theoretical values, while Matsumoto et al. [13] suggested that variations in the angle of attack between the oncoming flow and the deck’s normal vector contributed to an elevated flutter threshold. Farquharson et al. [1] further highlighted the inherent uncertainties in estimating full-scale critical wind speeds. Although previous studies employed nonlinear theories to simulate the bridge’s post-critical behavior [21,22], the underlying failure mechanisms still remain inadequately understood.

Conventional spring-suspended free-vibration rigs are inherently constrained, limiting their application in experiments requiring large amplitudes and high angles of attack. Operational thresholds are defined as follows: large amplitude (torsional amplitude ≥ 15° or vertical amplitude ≥ L/50, where L denotes span length) and high angle of attack (initial α > 5° or post-static-deformation α > 8°). Although innovative devices [14,23] enable large-amplitude/high-angle-of-attack testing for replicating flutter phenomena in long-span bridges such as the Tacoma Narrows Bridge, significant design constraints persist. To address these limitations, this study developed a novel 2-DOF free-vibration apparatus capable of coupled large-amplitude vertical-bending and torsional oscillations. This system establishes a critical experimental platform for subsequent section model investigations.

To overcome these limitations, we developed a novel free-vibration wind tunnel experimental apparatus capable of conducting tests at large amplitudes and high angles of attack. Using this apparatus, we performed sectional model tests on the Tacoma Bridge, observing vibration amplitudes reaching 35° with Hopf bifurcation phenomena. Nonlinear self-excited force coefficients were subsequently derived from the experimental data. Full-bridge aeroelastic model tests were then conducted, revealing large-amplitude anti-symmetric torsional flutter. Nonlinear time-domain finite element analysis was subsequently performed using the ANSYS platform. This analysis successfully reproduced the large-amplitude flutter, with the steady-state flutter amplitude closely matching the full-bridge experimental results. The simulations identified the hanger at the quarter point of the main span as the initial failure location.

Traditional flutter design criteria for bridges, rooted in Scanlan’s linear self-excited force theory, assume that once the critical wind speed is exceeded, rapid divergent flutter occurs, leading to structural failure. However, the present finding indicates that bluff-body sections often experience a different phenomenon known as soft flutter. In contrast to classical flutter, soft flutter is characterized by a gradual increase in torsional amplitude beyond the critical wind speed, forming limit cycle oscillations (LCOs) whose amplitudes scale proportionally with wind speed. This discovery challenges the adequacy of a single flutter-check wind speed in design practice. Instead, it suggests that long-span bridges can remain safe under soft flutter conditions, provided that oscillation amplitudes remain below damage thresholds for key structural components. To better reflect this behavior, we introduce a three-tiered performance-based flutter-check framework, accounting for both structural resilience and environmental conditions. This shifts the design paradigm from rigid wind speed limits to amplitude-controlled safety boundaries. It enables safer, more flexible design of long-span bridges without overly conservative constraints, thereby promoting innovation while maintaining aerodynamic stability. Figure 1 presents the technical approach diagram of this article.

2. Nonlinear Characteristics of Wind-Induced Vibration of a Large-Span Suspension Bridge

2.1. Project Overview

This subsection introduces a novel large-amplitude free-vibration wind tunnel apparatus. Using this facility, large-amplitude sectional model tests of the Tacoma Bridge were conducted. The experimental results were analyzed to develop a simplified self-excited force model based on Taylor-series expansion, and its accuracy was validated through back-calculation against the experimental results.

2.2. Chirp Nonlinear Self-Excited Force Properties

2.2.1. Measurement Technology and Test Equipment

High-Precision Five-Component Piezoelectric Force Balance

A high-precision five-component piezoelectric balance (Figure 2) was designed and fabricated to accurately quantify nonlinear self-excited forces during bridge section vibrations. The balance measures both dynamic and quasi-static forces (Fx, Fy) along two orthogonal directions, as well as moments (Mz, Mx, My) about three mutually perpendicular axes. The corresponding measurement ranges are ±0.3 kg for forces and ±3.0 kg·m, ±5.0 kg·m, ±10.0 kg·m, and ±3.0 kg·m for the respective moment components. This custom-designed system provides high metrological accuracy across a broad operational range, enabling reliable force characterization under large-amplitude and high static angle-of-attack conditions.

The model consists of a measurement segment and a compensation segment, integrated with a force balance system. The embedded balance exclusively captures aerodynamic forces acting on the outer surface of the measurement section. Notably, the mass and mass moment of inertia of this section account for only approximately 10% of the total model’s inertia properties. This configuration offers several advantages: it significantly reduces the contribution of inertial force components to the overall dynamic response; improves the accuracy of aerodynamic self-excited force quantification; and avoids the error amplification typically associated with traditional “large-number subtraction” methods during force extraction [12,15].

2.3. Support Device for Large-Amplitude and Large-Attack-Angle Free-Vibration Wind Tunnel Test

Figure 3 and Figure 4 illustrate the large-amplitude, high-angle-of-attack free-vibration wind tunnel apparatus. This system offers a wide tuning range for both torsional and vertical-bending natural frequencies and maintains high stiffness linearity across both degrees of freedom. The design achieves full decoupling between torsional and vertical stiffness systems. Vertical stiffness is generated through a cantilevered spring assembly constrained by linear guides that only allow for vertical translation, whereas torsional stiffness is independently controlled via a dedicated rotational stiffness applicator consisting of a spring-loaded hub equipped with a V-groove wire transmission system. Compared to conventional rigs, this apparatus supports torsional oscillations exceeding 30° while preserving linear stiffness characteristics under extreme amplitudes. The fully decoupled stiffness components also allow for independent adjustments of stiffness parameters through simple spring replacement, without any structural modifications. Furthermore, the system supports broad compatibility with various section models and offers cost-efficient operation, effectively overcoming the limitations of traditional free-vibration devices through its optimized mechanical configuration. Specifically, large torsional amplitude refers to angular displacements ≥ 15°, while large vertical amplitude corresponds to displacements ≥ 1/50 of the model span length. Additionally, a large angle of attack is defined as a static initial angle > 5°, or a dynamic equivalent angle > 8° during oscillation cycles.

2.4. Experimental Phenomena of the Large-Amplitude Nonlinear Vibration Segment Model of the Old Tacoma Bridge

2.4.1. Observations of the Segment Model Experiment with Large Amplitude

Section model tests were conducted in the TJ-1 Boundary Layer Wind Tunnel (1.8 m height × 1.8 m width × 12 m length), where the turbulence intensity was maintained below 1%. The experimental blockage ratio, calculated based on the width of the Tacoma Bridge model and the tested angle-of-attack range, was 4.6%, remaining within the recommended <5% limit for boundary-layer facilities. Tests were performed at a Reynolds number of 1.99 × 10⁵, corresponding to the subcritical flow regime. The free-vibration section model of the original Tacoma Narrows Bridge is shown in Figure 5 and Figure 6, and

Table 1. The parameters of the Tacoma Bridge segmental model.

Geometric Scale Ratio	Model Width	Model Height	Mass	Mass Moment of Inertia	Vertical Bending Frequency	Torsional Frequency
	m	m	kg	kg·m2	Hz	Hz
1/35	0.352	0.012	3.42	0.1608	1.51	2.62

presents the parameters of the wind tunnel sectional model for the Tacoma Narrows Bridge. Large-amplitude and high-angle-of-attack experiments revealed pronounced bifurcation phenomena, with two distinct dynamic behaviors observed. Bifurcation vibration refers specifically to Hopf bifurcation, a key concept in nonlinear dynamics. In the context of bridge aerodynamics, this phenomenon is observed as the convergence of flutter amplitude to stable limit cycles when excitation surpasses critical thresholds and as amplitude decay when excitation remains subcritical. Two distinct bifurcation behaviors emerged. First, a monostable system exhibited a single steady-state amplitude governed by excitation intensity: subcritical excitation resulted in amplitude decay, moderate excitation led to divergence toward a steady-state amplitude, and supercritical excitation caused attenuation back to the same steady state (Figure 7 and Figure 8). Second, a bistable regime was exhibited, featuring dual coexisting steady-state amplitudes. Small excitation converged to the lower-amplitude state, while moderate excitation triggered a transition to jump to the higher-amplitude state. All bifurcation phenomena observed in this study have been identified as Hopf bifurcations using phase-space representations (Figure 7d and Figure 8c).

2.4.2. Interpretation of the Segment Model Experiment with Large Amplitude

Figure 9 illustrates the schematic definition of two coordinate systems, including the wind axis and body axis for structural wind resistance analysis. The wind-axis system was adopted, and all subsequent displacements, vibration amplitudes, and self-excited forces are defined relative to the wind-axis coordinates.

Simultaneous force–vibration measurements were conducted on both an H-shaped bluff section (representative of the original Tacoma Bridge) and a closed box girder section under large-amplitude and high-angle-of-attack conditions. The nonlinear self-excited forces are modeled using polynomial formulations (Equations (1) and (2)), with vertical-motion parameters kept constant. The torsional motion parameters exhibited clear dependence on both amplitude and velocity. The highest-order coefficient (p) was adaptively determined for different flow conditions to capture the evolving aerodynamic behavior.

$$\begin{array}{l}{L}_{se}(\dot{h},\dot{\alpha },\alpha ,h)=\rho U^{2}B\left[H_{1000}{\displaystyle \frac{\dot{h}}{U}}+{\displaystyle \sum _{m=0,p=0}^{m=\infty ,p=\infty }{H}_{0mp0}{\left(\frac{B\dot{\alpha }}{U}\right)}^m{\alpha }^p+H_{0001}\frac{h}{B}}\right]\\ M_{se}(\dot{h},\dot{\alpha },\alpha ,h)=\rho U^2{B}^2\left[A_{1000}{\displaystyle \frac{\dot{h}}{U}}+{\displaystyle \sum _{m=0,p=0}^{m=\infty ,p=\infty }{A}_{0mp0}{\left(\frac{B\dot{\alpha }}{U}\right)}^m{\alpha }^p}+A_{0001}{\displaystyle \frac{h}{B}}\right]\end{array}$$

(1)

$$\begin{array}{l}{L}_{se}(\dot{h},\dot{\alpha },\alpha ,h)=\rho U^{2}B\left[H_{1000}{\displaystyle \frac{\dot{h}}{U}}+{\displaystyle \frac{B\dot{\alpha }}{U}}\left({\displaystyle \sum _{p=0}^{p=\infty }{H}_{01(2p)0}{\alpha }^{2p}}\right)+\alpha \left({\displaystyle \sum _{p=0}^{p=\infty }{H}_{00(2p+1)0}{\alpha }^{2p}}\right)+H_{0001}{\displaystyle \frac{h}{B}}\right]\\ M_{se}(\dot{h},\dot{\alpha },\alpha ,h)=\rho U^2{B}^2\left[A_{1000}{\displaystyle \frac{\dot{h}}{U}}+{\displaystyle \frac{B\dot{\alpha }}{U}}\left({\displaystyle \sum _{p=0}^{p=\infty }{A}_{01(2p)0}{\alpha }^{2p}}\right)+\alpha \left({\displaystyle \sum _{p=0}^{p=\infty }{A}_{00(2p+1)0}{\alpha }^{2p}}\right)+A_{0001}{\displaystyle \frac{h}{B}}\right]\end{array}$$

(2)

where

• $L_{se}$: Self-excited lift force
• $M_{se}$: Self-excited torsional moment
• $h$: Vertical displacement
• $\alpha$: Torsional displacement
• U: Undisturbed wind speed
• $\rho$: Air density
• B: Deck width
• $H_{abcd}$: Nonlinear self-excited force terms associated with vertical-bending motion
• $A_{abcd}$: Nonlinear self-excited force terms associated with torsional motion

The order (*p*) of the nonlinear self-excited force model is determined using an adaptive approach based on the procedure proposed by Qian et al. [21], which consists of three main steps: (1) The work and reactive components of the experimentally measured self-excited forces/moments are calculated. Then, an initial assumption is made by setting the polynomial order p to i (i = 1, 2, 3, …). Using the equivalence principle of work and reactive power, the corresponding nonlinear self-excited force parameters are estimated through the least-squares method. Subsequently, the residuals between the experimentally *measured work/reactive components and those predicted by the i-th order model are obtained. (2) p is incremented to i+1, and the same procedure is repeated to compute the residuals for the (i+1)-th order model. If the relative difference between the residuals of these two consecutive models is less than 10%, it indicates that their simulation accuracies are comparable, and, thus, the i-th order model is selected as the final result (p = i). (3) Conversely, if the residual difference exceeds 10%, p is further incremented to i+2, and the residuals for the (i+1)-th and (i+2)-th order models are compared. This iterative process continues until the residual difference between two consecutive models falls below 10%, at which point the lower-order model in the pair is adopted as the optimal solution for p. This adaptive approach is supported by a dedicated computational program developed to autonomously determine the optimal order p.

2.4.3. Model Validation of the Segment Model Experiment with Large Amplitude

The motion time–history was numerically reconstructed using the Runge–Kutta method (Figure 10). The simulated and experimentally measured trajectories exhibit strong convergence in both evolutionary trends and steady-state amplitudes across various wind speeds, validating the accuracy of the nonlinear model. For instance, at a 9° angle of attack and a wind speed of U = 3 m/s, the simulated vertical-bending displacement simulations remain within narrow error bounds throughout the time series. Similarly, the torsional displacement closely followed the experimental trend, with peak deviations remaining within acceptable limits. This level of agreement was consistently maintained at higher wind speeds, such as U = 6.4 m/s, confirming the robustness of the model under varying flow conditions. The steady-state amplitudes obtained from experimental measurements and those derived via the Runge–Kutta-based inversion method were compared and are shown in Figure 11. Across the wind speed range from the flutter onset to a maximum amplitude of 35°, the inverted and measured torsional and vertical responses align closely, with a maximum deviation of less than 5%. This close agreement confirms that the nonlinear self-excited force model accurately captures the amplitude evolution of flutter-induced vibrations across the full wind speed spectrum, thereby validating both its theoretical soundness and practical applicability.

Figure 12 and Figure 13 present three-dimensional surface plots and contour maps of the combined nonlinear flutter derivatives at an angle of −3°. The results reveal that these derivatives exhibit nonlinear variation with wind speed. At low wind speeds (<3 m/s), the derivatives show little to no amplitude dependence. However, beyond this threshold, amplitude sensitivity increases significantly. Conversely, the derivatives demonstrate a strong nonlinear correlation with wind speed across the entire operational range, while showing invariance for amplitude under all tested conditions.

2.4.4. Analysis of the Variation in Total Damping Coefficient with Amplitude and Bifurcation Mechanism of Flutter

The analysis of the evolution of the torsional-mode total damping ratio to amplitude (Figure 14 and Figure 15) provides insights into the vibrational development mechanism. This composite ratio, which includes both aerodynamic and structural damping components, governs the system’s energy dissipation and growth dynamics. The zero-crossing point on the damping curve corresponds to the stable equilibrium amplitude during steady-state vibration. In bifurcation scenarios featuring dual zero-crossings, the lower amplitude represents the Hopf bifurcation threshold. For example, at a 9° angle of attack and a wind speed of U = 6.4 m/s, the equivalent linearized torsional damping ratio is negative at small amplitudes. As the amplitude increases, the damping ratio progressively rises and reaches zero at the stable equilibrium point. Conversely, at 15° and U = 5.6 m/s, the damping curve shows two zero-crossings, with the lower-amplitude intercept identifying the critical bifurcation threshold. This damping evolution analysis offers a comprehensive understanding of the vibrational development mechanisms, accurately identifies equilibrium and bifurcation states, and reveals the aerodynamic origins of bifurcation phenomena.

To better understand the contributions of individual self-excited force components to vibrational development, Figure 16 presents amplitude-dependent damping coefficient curves for the components: ${\mathrm{A}}_1^{\mathrm{*}}, {\mathrm{A}}_2^{\mathrm{*}}, {\mathrm{A}}_3^{\mathrm{*}}, {\mathrm{A}}_4^{\mathrm{*}}$. The analysis reveals that mechanical damping exhibits amplitude-dependent amplification, which is primarily attributed to nonlinear structural torsional damping at higher amplitudes. The damping coefficients of A₁* and A₄* remain invariant with amplitude, while A₃* (representing torsional stiffness) shows near-zero damping due to minimal energy dissipation. In contrast, A₂* displays significant amplitude-dependent variation, making it the primary contributor to the overall damping behavior.

3. Static and Dynamic Vibration Instability Analysis of the Old Tacoma Bridge

3.1. Theoretical Basis

This study establishes an integrated time-domain framework to analyze wind-induced static–dynamic nonlinear instabilities, combining structural nonlinearities (both geometric and material) with aerodynamic nonlinearities. The governing equations, derived from Lagrange’s equations or Hamilton’s principle, account for structural geometric nonlinearities (large displacements and rotations) as well as material nonlinearities (elastoplastic constitutive relationships). Nonlinear self-excited forces are rigorously incorporated into the structural equations through aerodynamic–structural interaction modeling. By directly solving these coupled equations in the time domain, the framework effectively captures the non-stationarity of wind loads, the nonlinear structural response, and aeroelastic effects, enabling a comprehensive analysis of the static–dynamic behavior of long-span suspension bridges under wind excitation. For comprehensive details regarding the computational methodology and governing equations, readers are referred to the pertinent studies by Qian Cheng et al. [21,24].

3.2. Test Verification

3.2.1. Wind Tunnel Test of Tacoma Bridge Full-Bridge Aeroelastic Model

Purpose of Test Condition Design

Experiments were conducted in the TJ-3 Wind Tunnel (2 m height × 15 m width × 14 m length), which is characterized by a turbulence intensity below 2%. Blockage corrections were applied using the fundamental solid blockage criterion, with the blockage ratio defined as the ratio of the model’s maximum frontal projected area to the test section’s cross-sectional area. The calculated blockage ratio of 1.6% is well below the standard 5% threshold for such facilities. Testing was conducted at a maximum Reynolds number of Re = 2.34 × 10⁵. The full-scale aeroelastic model of the Tacoma Bridge (Figure 17) was tested at a 1:110 scale. Experimental configurations included yaw angles of 0° and 20°, uniform and turbulent flow conditions, and enhanced damping conditions to simulate various operational scenarios.

Analysis of Test Results

Experimental measurements recorded maximum torsional amplitudes approaching 35°, with simultaneous observation of first-order symmetric and antisymmetric flutter modes consistent with full-scale behavior. Analysis of wind speed–amplitude relationships (Figure 18) reveals four key findings: (1) Increased damping effectively suppresses torsional amplitudes. At a wind speed of 2.4 m/s, the vibration amplitude with Damp 1 is 30% lower than the undamped case. Within the 1.5–2.3 m/s range, the amplitude under Damp 2 configuration measures approximately 20% of the undamped baseline. (2) Turbulent flow conditions generate distinct response characteristics compared to uniform flow. Within the 1.5–2.0 m/s wind speed range, the vibration amplitude under turbulent flow conditions measures 50% to 80% of that observed in uniform flow. (3) Tower fixation significantly influences structural vibration modes and critical wind speed. (4) Variations in deflection angles lead to notable modulation of torsional amplitudes.

Spatially resolved measurements of torsional, vertical-bending, and horizontal-bending amplitudes along stiffening girders (Figure 19, Figure 20, Figure 21 and Figure 22) reveal significantly greater horizontal displacements compared to vertical displacements, confirming the reduced horizontal bending stiffness of the Tacoma Bridge structure. Under 0° yaw uniform flow conditions, elevated amplitudes at 1/4, 3/4, and mid-span points demonstrate concurrent first-order symmetric and antisymmetric flutter modes, consistent with full-scale observations. Under turbulent flow conditions, distinct sectional responses are observed: torsional amplitudes diverge between 1/4 and 3/4 spans; fixed-pier configurations show a flutter initiation peak at 1.8 m/s, followed by the sequential development of symmetric–antisymmetric torsional flutter; and under 20° yaw conditions, near-linear torsional amplitude growth is observed, accompanied by pre-flutter multi-order vortex-induced vibrations, replicating historical Tacoma Bridge behavior.

3.2.2. Verification of Finite Element Model and Time-Domain Static Dynamic Analysis

Establishment and Accuracy Verification of Finite Element Model

A three-dimensional finite element model of the Tacoma Bridge was developed in ANSYS (Figure 23). The stiffening beams were modeled using Beam188 elements, and the cross section of the Beam188 element is shown in Figure 24, with nonlinear material properties through TB and TBPT commands in the element table (Figure 25). Modal validation demonstrated excellent agreement between the simulated and empirical natural frequencies, confirming the model’s accuracy (

Table 2. Comparative frequency table of the finite element model.

	Frequency (Hz)
Mode Shape	In Situ Measured Values [1]	Hu Calculated Values [25]	Calculated Values In This Article
1st S-V-B	0.1333	0.1321	0.1329
2nd S-V-B	0.2	0.201	0.1959
3rd S-V-B	0.35	0.3497	0.3458
4th S-V-B	0.45	0.5014	0.4922
5th S-V-B	0.6333	0.6744	0.4923
1st A-V-B	0.145	0.1385	0.1374
2nd A-V-B	0.275	0.2713	0.2677
3rd A-V-B	0.4	0.4201	0.4136
4th A-V-B	0.5667	0.5841	0.5697
1st S-T	-	0.2139	0.2435
1st A-T	0.2333	0.2129	0.2343

Note: S-V-B is symmetric vertical bending mode; A-V-B is antisymmetric vertical bending mode; S-T refers to symmetric torsion mode; A-T represents antisymmetric torsion mode.

). As summarized in Table 2, the natural modal frequencies derived from field measurements, Hu’s analytical results, and the proposed FE model demonstrate close agreement in fundamental modes but diverge at higher frequencies. For the first symmetric vertical bending mode, the empirical frequency was 0.1333 Hz, while the simulated frequency was 0.1329 Hz, resulting in a 0.3% deviation. For the second symmetric vertical bending mode, the empirical frequency was 0.20 Hz, compared to the simulated frequency of 0.1959 Hz, yielding a 2.05% error.

Time-Domain Self-Excited Force Mathematical Model and Mechanism

Nonlinear amplitude-dependent flutter derivatives, obtained from large-amplitude/high-angle-of-attack Tacoma Bridge section models, are expressed in the time domain. Using an aerodynamic model based on impulse response functions, the time-domain self-excited forces are decomposed into mean and fundamental frequency components, with each parameter calculated separately. Analysis of representative components (e.g., ${\mathrm{A}}_2^{\mathrm{*}},{\mathrm{A}}_3^{\mathrm{*}}$) reveals distinct amplitude-dependent trends. Specifically, parameter C2 increases significantly from 0.011 to 0.475 as the amplitude rises from 0 to 0.5 rad, demonstrating pronounced amplitude sensitivity and a heightened influence on self-excited forces at larger amplitudes.

Table 3. Self-excited force parameters under different amplitudes at a 0-degree attack angle.

Parameter	0 (rad)	0.1 (rad)	0.2 (rad)	0.3 (rad)	0.4 (rad)	0.5 (rad)
C1	0.000	0.001	0.001	0.016	−0.018	−0.055
C2	0.011	0.013	0.018	0.024	0.076	0.475
C3	0.203	−0.035	0.029	4.62E−04	−0.148	0.339
C4	−0.205	0.032	−0.031	−0.017	0.179	4.063
d3	0.098	0.101	0.128	0.087	−0.575	−1.818
d4	0.098	0.100	0.128	0.004	−0.584	−109.853

shows the wind-induced instability types and the corresponding critical wind speeds for the full-bridge aeroelastic model at various different wind attack angles. In ANSYS software (version ANSYS 17.0), the solver parameters were configured using the DSP excitation method (Figure 26). Given the Tacoma Bridge’s characteristic flutter behavior in the first-order antisymmetric torsional mode, a torsional displacement corresponding to this mode was applied at each node of the main girder. Three different time step lengths—0.05 s, 0.025 s, and 0.1 s—were tested. After evaluating the convergence of the calculation results, a time step length of 0.05 s was selected as the optimal choice. The cnvtol command was employed to control the convergence of displacement (u) and unbalanced force (f), with a convergence criterion of 0.5.

Our computational framework accounts for geometric, material, and aerodynamic nonlinearities and incorporates them into the computational model. Computational results indicate the presence of soft flutter behavior within the −3° to 3° angle-of-attack range, with amplitudes increasing as wind speed rises (Figure 27), consistent with the full-scale and sectional model experiments. At 0° angle of attack, the mid-span quarter point experiences maximum torsional amplitudes of 20° at 15 m/s, which increase to 24° at 18.6 m/s, confirming amplitude–wind speed correlation. The simulation accurately replicates the suspension rod failure progression observed in the prototype structure. Both experimental and finite element wind speed–amplitude curves show strong agreement, with the maximum deviations remaining below 10%. At 0° angle of attack, amplitude discrepancies remain within acceptable limits across all wind speeds, validating both the theoretical framework and the finite element methodology. This confirms that the finite element model, incorporating time-domain self-excited forces and material nonlinearity, can accurately simulate large-amplitude bridge flutter.

Application of Time-Domainized Self-Excitation Model and Analysis of Computational Results

At 0° angle of attack and a wind speed of 20.4 m/s, the maximum torsional amplitude reaches 32.03°, closely matching the large-amplitude flutter deformation observed in the Tacoma Bridge. Figure 28 presents the deformation of the ANSYS model at this critical amplitude. The visualization highlights significant antisymmetric torsional vibration in the main girder, with maximum displacements occurring at the quarter-span (L/4) and three-quarter-span (3 L/4) locations. While secondary torsion is observable in the side spans, the corresponding deformations are minimal in magnitude.

As the wind speed increased to 20.5 m/s at a 0° angle of attack, the hanger fracture was initiated. Figure 29 shows the stress distribution across hangers at a maximum torsional displacement of 32.03°. Critical stress concentrations were observed at the windward quarter-span location, reaching 1403 MPa, which exceeds the 1250 MPa tensile strength of the ASTM No. 233 high-strength cables historically used. This stress state led to the complete relaxation of significant hanger sections and fracture initiation at the quarter-span position (Figure 30). Notably, the main span stiffening girders remained elastic at the point of failure initiation. Subsequent computational analysis, including element removal, revealed adjacent hanger stresses of 1296 MPa and 1530 MPa in the subsequent timestep, resulting in sequential fracture propagation consistent with the failure progression documented in the Tacoma Bridge collapse.

Hanger fracture occurred at 20.5 m/s wind speed and a 0° angle of attack. Figure 31 shows the stress distributions across hangers at maximum torsional displacement under these conditions. At the model’s peak torsion angle (32.03°), windward hangers at the three quarters reached stresses of 1403 MPa (Figure 32), which induced significant stress relaxation. According to period-applicable ASTM specifications for welded bridges, the No. 233 high-strength cable material (ultimate tensile strength ≈ 1250 MPa) explains fracture initiation at quarter-span hangers, consistent with Figure 33 and Figure 34, showing stiffening girders remained elastic. Subsequent computational progression via element removal revealed adjacent hanger stresses of 1296 MPa and 1530 MPa in the following timestep (Figure 35), confirming sequential fracture propagation consistent with the Tacoma Bridge’s documented failure sequence.

Figure 36 compares experimental and simulated wind speed–amplitude relationships for the full bridge at 0° angle of attack (pre-failure conditions). The steady-state amplitudes calculated using ANSYS show strong agreement with the experimental data (Figure 34), with the maximum deviations remaining below 10%. Notably, the simulations slightly overestimate the amplitudes at lower wind speeds while underestimating them at higher velocities. These minor discrepancies may arise from parameter uncertainties in the time-domain implementation or from differences in amplitude-dependent damping between the experimental and computational models. Nonetheless, the observed correlation validates the integrated finite element methodology—incorporating nonlinear time-domain self-excited forces and material nonlinearity—as an accurate tool for simulating large-amplitude bridge flutter phenomena.

Figure 37 presents wind speed versus steady-state amplitude relationships from fully coupled time-domain ANSYS aeroelastic flutter analyses at −3° to 3° angles of attack (pre-failure conditions). The results show progressively increasing torsional amplitudes with wind speed, exceeding 30°, which is consistent with the soft flutter characteristics and validates previous large-amplitude free-vibration experiments on the Tacoma Bridge. Comparative analysis reveals distinct aerodynamic nonlinearities: at lower wind speeds, a 0° attack angle produces smaller amplitudes than ±3° conditions, while it results in larger amplitudes at higher wind speeds. Moreover, all configurations demonstrate a decrease in soft flutter amplitudes as wind speed exceeds critical thresholds.

Bifurcation phenomena were identified in the ANSYS full-bridge time-domain analysis of the Tacoma Bridge (Figure 38). At 0° angle of attack with U = 13.2 m/s (equivalent to 4.4 m/s for sectional models), a 3° initial excitation produces convergence to steady-state amplitude, while 2° excitation results in amplitude decay to zero. Notably, the critical excitation threshold for sectional models (4.6°) differs significantly from the full-bridge response, likely attributable to three-dimensional effects in long-span structures.

3.2.3. Implications for Bridge Design Enhancement and Recommendations for Wind-Resistant Codes

Currently, long-span suspension bridges often exhibit low flutter critical wind speeds, resulting in limited aerodynamic design margins and a high probability of wind speeds exceeding this critical threshold within their 100-year design lifespan. This poses significant challenges for the design and safety assessment of such bridges. When wind speeds surpass the flutter critical wind speed, linear flutter theory fails to reliably evaluate the structural safety of long-span suspension bridges. The findings of this study reveal that exceeding the flutter critical wind speed does not necessarily result in structural damage. As long as the flutter amplitude remains below a defined performance threshold and the stress in individual components does not exceed their load-bearing capacities, the structural integrity of the bridge can be maintained, effectively increasing the permissible flutter-limited wind speed. From a design perspective, strengthening only the vulnerable hangers—for example, by increasing their diameters or using higher-strength materials—can ensure that these components remain intact when the vibration amplitude reaches the threshold. This approach can significantly raise the allowable flutter wind speed with minimal additional cost.

The aerodynamic design margins of long-span bridges remain limited, and existing wind-resistant design codes become inadequate when wind speeds exceed the flutter critical velocity, particularly for ultra-long-span suspension bridges. By contrast, adopting a performance-based design approach is technically feasible, with the following implementation process: (1) conducting large-amplitude flutter sectional model tests; (2) deriving nonlinear flutter derivatives from the experiments and calculating the corresponding time-domain self-excited force parameters; (3) performing time-domain simulations using finite element software such as ANSYS to identify vulnerable components and propose targeted reinforcement measures. This approach is viable because large-amplitude flutter testing equipment has already been developed domestically, and subsequent numerical modeling and simulation, such as ANSYS-based analyses, are well established within the wind-resistant structural engineering community. However, advancing this methodology requires a paradigm shift among engineering practitioners and regulatory bodies. To this end, it is necessary to promote and disseminate the concept of allowing controlled flutter while limiting its amplitude. The authors are actively contributing to this transition by completing research projects grounded in this theory and advocating its adoption through academic conferences and industry review panels, with the ultimate goal of integrating this method into future wind-resistant design codes.

4. Conclusions

This study investigates the nonlinear wind-induced vibration mechanisms of long-span suspension bridges through integrated experimental and numerical methods. Wind tunnel tests were conducted on H-shaped bluff sections and closed box girders using a high-precision five-component piezoelectric balance with a custom support system. In parallel, a finite element time-domain simulation framework was developed based on experimentally derived nonlinear flutter derivatives. Validation was performed through aeroelastic tests on a 1:110-scale model of the original Tacoma Narrows Bridge and corresponding simulations, which showed less than 10% deviation from experimental and historical wind speed–amplitude data. Failure analysis identified the suspenders as the critical failure components during the Tacoma collapse. The results revealed Hopf bifurcation in H-shaped sections—characterized by amplitude-dependent flutter derivatives and equivalent damping coefficients—and distinct angle-dependent nonlinear aerodynamic behavior in closed box girders. Based on these findings, a novel performance-based design framework is proposed to improve the accuracy of aeroelastic modeling and guide targeted structural reinforcement. This work provides a practical and cost-effective methodology to enhance the safety, aerodynamic stability, and resilience of long-span suspension bridges under complex wind loading conditions.

The principal findings are summarized as follows:

(1)

This study employs a novel large-amplitude/high-angle-of-attack free-vibration apparatus for long-span suspension bridge testing. Nonlinear polynomial modeling of self-excited forces/moments reveals bifurcation phenomena in H-shaped bluff sections, and torsional mode damping demonstrates pronounced amplitude dependence in these sections. Furthermore, closed box girders exhibit distinct flutter displacement responses and self-excited force spectral evolution patterns across varying angles of attack.

(2)

The full-bridge aeroelastic model successfully reproduced large-amplitude torsional flutter in the Tacoma Bridge. Notably, alternating occurrences of antisymmetric and symmetric torsional flutter modes were observed under specific wind conditions. Furthermore, hanger fractures initiated at quarter-span locations during high wind speeds, thereby demonstrating consistency with the failure pattern of the prototype bridge.

(3)

This study employs finite element analysis integrated with a time-domain self-excited force model, incorporating material nonlinearity to characterize the post-flutter behavior and collapse sequence of the Tacoma Narrows Bridge. Computed post-flutter wind speed–amplitude curves exhibited < 10% deviation from full-bridge aeroelastic model results. Crucially, the first-to-fail structural components were identified, enabling targeted reinforcement of vulnerable members. This demonstrates a cost-effective enhancement in flutter critical wind speeds in long-span suspension bridges.

The key contributions of this study are as follows: achievement of large-amplitude torsional flutter in free-vibration wind tunnel testing; reconstruction of Tacoma Bridge’s post-flutter response and identification of initial failure mechanisms via ANSYS simulations; development of an expanded wind safety framework permitting limited-amplitude flutter oscillations at elevated wind speeds, based on observed soft flutter phenomena with progressive amplitude growth. This integrated methodology provides novel insights for bridge aerodynamics research and wind-resistant design code development.

The core academic contributions of this study are as follows: It achieved ultra-large-amplitude testing (>30°) in free-vibration section model experiments, demonstrating significant methodological innovation. It utilized the ANSYS finite element platform to reproduce the large-amplitude torsional flutter of the Tacoma Narrows Bridge, precisely identifying the initial failure locations (components and positions). This addresses a critical gap in the existing literature by successfully simulating the Tacoma flutter failure sequence within a finite element framework. Subsequently, vulnerable components can be identified through finite element analysis. Implementing targeted reinforcement to these bridge structural elements effectively enhances wind resistance performance. This represents a novel performance-based theoretical approach to wind-resistant design.