Nonlinear parametric modeling of suspension bridges under aeroelastic forces: torsional divergence and flutter

Abstract

A fully nonlinear model of suspension bridges parameterized by one single space coordinate is proposed to describe overall three-dimensional motions. The nonlinear equations of motion are obtained via a direct total Lagrangian formulation and the kinematics, for the deck-girder and the suspension cables, feature the finite displacements of the associated base lines and the flexural and torsional rotations of the deck cross-sections assumed rigid in their own planes. The strain-displacement relationships for the generalized strain parameters, the elongations in the cables, the deck elongation, and the three curvatures, retain the full geometric nonlinearities. The proposed nonlinear model with its full extensional-flexural-torsional coupling is employed to study the torsional divergence caused by the static part of the wind-induced forces. Two suspension bridges are considered as case studies: the Runyang bridge (main span 1,490 m) and the Hu Men bridge (main span 888 m) in China. The evaluation of the onset of the static instability and the post-critical behavior takes into account the prestressed condition of the bridge subject to dead loads. The dynamic bifurcation that occurs at the onset of flutter is also studied accounting for the prestressed equilibrium state about which the equations of motion are obtained via an updated Lagrangian formulation. Such a bifurcation is investigated in the context of the parametric nonlinear model considering the model parameters of the Runyang Suspension Bridge together with its aeroelastic derivatives. The calculated critical wind speeds for the onset of the static and dynamic bifurcations are compared with the results obtained via linear analysis and the main differences are highlighted. Parametric sensitivity studies are carried out to assess the influence of the design parameters on the instabilities associated with the bridge aeroelastic response.

Appendix

The nine components of the rotation matrix R ^o are given by

$$ \everymath{\displaystyle} \begin{array}{l} R_{11}^{\mathrm{o}}=\cos\phi_2^{\mathrm{o}} \cos \phi_3^{\mathrm{o}}, \\[5pt] R_{12}^{\mathrm{o}}=-\cos\phi_2^{\mathrm{o}} \sin \phi_3^{\mathrm{o}}, \\[5pt] R_{13}^{\mathrm{o}}=\sin \phi_2^{\mathrm{o}}, \\[5pt] R_{21}^{\mathrm{o}}=\sin \phi_1^{\mathrm{o}} \sin \phi_2^{\mathrm{o}} \cos \phi_3^{\mathrm{o}} +\cos \phi_1^{\mathrm{o}} \sin \phi_3^{\mathrm{o}}, \\[5pt] R_{22}^{\mathrm{o}}=\cos \phi_1^{\mathrm{o}} \cos \phi_3^{\mathrm{o}} - \sin \phi_1^{\mathrm{o}} \sin \phi_2^{\mathrm{o}} \sin \phi_3^{\mathrm{o}}, \\[5pt] R_{23}^{\mathrm{o}}=-\sin \phi_1^{\mathrm{o}} \cos \phi_2^{\mathrm{o}}, \\[5pt] R_{31}^{\mathrm{o}}=\sin \phi_1^{\mathrm{o}} \sin \phi_3^{\mathrm{o}} - \cos \phi_1^{\mathrm{o}} \sin \phi_2^{\mathrm{o}} \cos \phi_3^{\mathrm{o}}, \\[5pt] R_{32}^{\mathrm{o}}= \cos \phi_1^{\mathrm{o}} \sin \phi_2^{\mathrm{o}} \sin \phi_3^{\mathrm{o}} +\sin \phi_1^{\mathrm{o}} \cos\phi_3^{\mathrm{o}}, \\[5pt] R_{33}^{\mathrm{o}}=\cos \phi_1^{\mathrm{o}} \cos \phi_2^{\mathrm{o}}. \end{array} $$

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The vectors representing the rotations of the deck cross-section projected into the global frame $\{\mbox {$\mbox {\boldmath $e$}$}_{1},\mbox {$\mbox {\boldmath $e$}$}_{2},\mbox {$\mbox {\boldmath $e$}$}_{3}\}$ can be defined as:

$$ \everymath{\displaystyle} \begin{array}{l} \mbox {$\mbox {\boldmath $\Psi $}$}^{\mathrm{o}} = \bigl[\psi^{\mathrm{o}}_1\ \psi^{\mathrm{o}}_2\ \psi^{\mathrm{o}}_3\bigr]^\top,\qquad \breve{\mbox {$\mbox {\boldmath $\Psi $}$}} = \bigl[\breve{\psi}_1, \breve{\psi}_2, \breve{\psi}_3\bigr]^\top, \\[5pt] \psi^{\mathrm{o}}_1 = \phi^{\mathrm{o}}_1 +\phi^{\mathrm{o}}_3 \sin\phi^{\mathrm{o}}_2, \\[5pt] \psi^{\mathrm{o}}_2 = \phi^{\mathrm{o}}_2 \cos\phi^{\mathrm{o}}_1 - \phi^{\mathrm{o}}_3 \sin\phi^{\mathrm{o}}_1 \cos\phi^{\mathrm{o}}_2, \\[5pt] \psi^{\mathrm{o}}_3 =\phi^{\mathrm{o}}_2 \sin\phi^{\mathrm{o}}_1 + \phi^{\mathrm{o}}_3 \cos\phi^{\mathrm{o}}_1 \cos\phi^{\mathrm{o}}_2, \end{array} $$

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$$ \everymath{\displaystyle} \begin{array}{rll} \breve{\psi}_1 &=& \phi_1 \cos\phi^{\mathrm{o}}_2 \cos\phi^{\mathrm{o}}_3 + \phi_2 \bigl(\sin\phi^{\mathrm{o}}_2 \sin\phi_1 \\[5pt] && {} - \cos\phi^{\mathrm{o}}_2 \sin\phi^{\mathrm{o}}_3 \cos\phi_1\bigr) \\[5pt] && {} + \phi_3 \bigl(\sin\phi^{\mathrm{o}}_2 \cos\phi_1 \cos \phi_2 \\[5pt] && {} + \cos\phi^{\mathrm{o}}_2 \sin\phi^{\mathrm{o}}_3 \sin\phi_1 \cos\phi_2 \\[5pt] && {} + \cos\phi^{\mathrm{o}}_2 \cos \phi^{\mathrm{o}}_3 \sin\phi_2\bigr), \\[5pt] \breve{\psi}_2 &=& \phi_1\bigl(\sin\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_2 \cos\phi^{\mathrm{o}}_3 + \cos\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_3\bigr) \\[5pt] && {} + \phi_2 \bigl[\cos\phi_1 \bigl(\cos\phi^{\mathrm{o}}_1\cos\phi^{\mathrm{o}}_3 \\[5pt] && {} - \sin\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_2 \sin\phi^{\mathrm{o}}_3\bigr) - \sin\phi^{\mathrm{o}}_1 \cos\phi^{\mathrm{o}}_2\sin\phi_1 \bigr] \\[5pt] && {} + \phi_3 \bigl[-\sin\phi^{\mathrm{o}}_1 \cos\phi^{\mathrm{o}}_2 \cos\phi_1 \cos\phi_2 \\[5pt] && {} - \sin \phi_1 \cos\phi_2 \bigl(\cos\phi^{\mathrm{o}}_1 \cos \phi^{\mathrm{o}}_3 \\[5pt] && {} - \sin\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_2 \sin\phi^{\mathrm{o}}_3\bigr) \\[5pt] && {} + \sin\phi_2 \bigl(\sin\phi^{\mathrm{o}}_1 \sin \phi^{\mathrm{o}}_2 \cos\phi^{\mathrm{o}}_3 +\cos\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_3 \bigr) \bigr], \\[5pt] \breve{\psi}_3 &=&\phi_1 \bigl(\sin\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_3 - \cos\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_2 \cos\phi^{\mathrm{o}}_3 \bigr) \\[5pt] && {} + \phi_2 \bigl[\cos \phi_1\bigl(\sin\phi^{\mathrm{o}}_1 \cos\phi^{\mathrm{o}}_3 \\[5pt] && {} + \cos\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_2 \sin\phi^{\mathrm{o}}_3\bigr) + \cos\phi^{\mathrm{o}}_1 \cos\phi^{\mathrm{o}}_2\sin\phi_1 \bigr] \\[5pt] && {} + \phi_3 \bigl[\cos\phi^{\mathrm{o}}_1 \cos\phi^{\mathrm{o}}_2 \cos\phi_1 \cos\phi_2 \\[5pt] && {} - \sin \phi_1 \cos\phi_2 \bigl(\sin\phi^{\mathrm{o}}_1 \cos \phi^{\mathrm{o}}_3 \\[5pt] && {} + \cos\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_2 \sin\phi^{\mathrm{o}}_3\bigr) + \sin\phi_2 \bigl(\sin\phi^{\mathrm{o}}_1 \sin \phi^{\mathrm{o}}_3 \\[5pt] && {} -\cos\phi^{\mathrm{o}}_1 \sin\phi^{\mathrm{o}}_2 \cos\phi^{\mathrm{o}}_3\bigr)\bigr]. \end{array} $$

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