Analysis of a nonlinear fish-bone model for suspension bridges with rigid hangers in the presence of flow effects

Alessio Falocchi; Justin T. Webster

doi:10.3934/dcds.2024164

Article Contents

2025, Volume 45, Issue 7: 2241-2280. Doi: 10.3934/dcds.2024164

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Analysis of a nonlinear fish-bone model for suspension bridges with rigid hangers in the presence of flow effects

Alessio Falocchi^1, and
Justin T. Webster^2, ,

1.
Dipartimento di Matematica, Dip. di Eccellenza 2023-2027, Politecnico di Milano, Italy
2.
University of Maryland, Baltimore County, Baltimore, MD, USA

^*Corresponding author: Justin T. Webster
^*Corresponding author: Justin T. Webster

Received: July 2024

Revised: November 2024

Early access: December 2024

Published: July 2025

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Abstract

We consider a dynamical system of nonlinear partial differential equations modeling the motions of a suspension bridge. This fish-bone model captures the flexural displacements of the bridge deck's mid-line, and each chordal filament's rotation angle from the centerline. These two dynamics are strongly coupled through the effect of cable-hanger, appearing through a sublinear function. Additionally, a structural nonlinearity of Woinowsky-Krieger type is included, allowing for large displacements. Well-posedness of weak solutions is shown and long-time dynamics are studied. In particular, to force the dynamics, we invoke a non-conservative potential flow approximation which, although greatly simplified from the full multi-physics fluid-structure interaction, provides a driver for non-trivial end behaviors. We describe the conditions under which the dynamics are uniformly stable, as well as demonstrate the existence of a compact global attractor under all nonlinear and non-conservative effects. To do so, we invoke the theory of quasi-stability, first explicitly constructing an absorbing ball via stability estimates and, subsequently, demonstrating a stabilizability estimate on trajectory differences applied to the aforesaid absorbing ball. Finally, numerical simulations are performed to examine the possible end behaviors of the dynamics.

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Mathematics Subject Classification: 35B41, 35G31, 35Q74, 74K20, 74H40, 70J10.

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Figure 1. Fish bone model on the left and a cross section for fixed $ x $ on the right. Dotted lines are the deck and cables section in rest position, $ s(x) $ is the cable initial shape, see Section 2.2

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Figure 2. Sketch of the side view of the suspension bridge with the quotes assumed positive; $ s(x) $ is the cable initial shape, see Section 2.2

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Figure 3. Plots $ \overline \theta_j(t) $ ($ j = 1, \dots, 4 $) on $ [0, 120s] $ with $ \delta = \zeta = 0 $, $ P = S = 0 $ and $ \beta = 0 $

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Figure 4. Plots $ \overline\theta_j(t) $ ($ j = 1, \dots, 4 $) on $ [0, 120s] $ with $ \delta = \zeta = 0 $, $ P = S = 0 $ and $ \beta = 10^{-2} $, $ \mathcal U = 30m/s $

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Figure 5. Plots $ \overline\theta_j(t) $ ($ j = 1, \dots, 4 $) on $ [0, 120s] $ with $ \delta = \zeta = 0 $, $ P = 0 $, $ S = \frac{EA}{2L} $ and $ \beta = 10^{-2} $, $ \mathcal U = 30m/s $

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Figure 6. Plots $ \overline\theta_j(t) $ ($ j = 1, \dots, 4 $) on $ [0, 120s] $ with $ \delta = \zeta = 0.01 $, $ P = 0 $, $ S = \frac{EA}{2L} $ and $ \beta = 10^{-2} $, $ \mathcal U = 30m/s $

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Table 1. TNB mechanical features

$ E $	$ 210 \quad 000MPa $	Young modulus of the deck (steel)
$ E_c $	$ 185 \quad 000MPa $	Young modulus of the cables (steel)
$ G $	$ 81 \quad 000MPa $	Shear modulus of the deck (steel)
$ L $	$ 853.44m $	Length of the main span
$ \ell $	$ 6m $	Half width of the deck
$ f $	$ 70.71m $	Sag of the cable, see Fig. 2
$ \mathcal{I} $	$ 0.154m^4 $	Moment of inertia of the deck cross section
$ K $	$ 6.07\cdot10^{-6}m^4 $	Torsional constant of the deck
$ J $	$ 5.44m^6 $	Warping constant of the deck
$ A $	$ \approx 1.85m^2 $	Area of the deck cross section
$ A_c $	$ 0.1228m^2 $	Area of the cables section
$ M $	$ 7198kg/m $	Mass linear density of the deck
$ H $	$ 45 \quad 413kN $	Horizontal tension in the cables, $ H=\frac{MgL^2}{16f} $
$ \mathcal{L}_0 $	$ 868.815m $	Initial length of the cables, see (5)

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