# Nonlinear Aeroelastic in-Plane Behavior of Suspension Bridges under Steady Wind Flow

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Continuous Model

#### 2.1. Equations of Motion

#### 2.2. Damping Forces

#### 2.3. Aerodynamic Forces

#### 2.4. Dimensionless Equation of Motion

## 3. Linear Undamped Oscillations

#### 3.1. Anti-Symmetric Modes

#### 3.2. Symmetric Mode

## 4. Aeroelastic Stability Analysis

#### 4.1. Bifurcation Equation

#### 4.2. Linear Stability Analysis and Critical Wind Velocity

#### 4.3. Limit-Cycle Analysis

- when $u<{u}^{*}$, it is $\u25b5\left(u\right)<0$, so that no real solutions ${a}_{\pm}$ exist;
- when ${u}^{*}<u<{u}_{c}$, it is $\u25b5\left(u\right)>0$; however, due ${d}_{1}u+{d}_{0}<0$, it is $\sqrt{\u25b5\left(u\right)}<{d}_{3}$, so that both the ${a}_{+}$ and ${a}_{-}$ roots are real;
- when $u>{u}_{c}$, in addition to $\u25b5\left(u\right)>0$, it is also ${d}_{1}u+{d}_{0}>0$; consequently, $\sqrt{\u25b5\left(u\right)}>{d}_{3}$, which entails that only ${a}_{+}$ is real.

#### 4.4. Stability Analysis

- the lower non-trivial solution ${a}_{-}$ is unstable in its domain of existence ${u}^{*}<u<{u}_{c}$;
- the upper non-trivial solution ${a}_{+}$ is stable in its domain of existence $u>{u}^{*}$.

## 5. Finite-Dimensional Model

## 6. Numerical Results

#### 6.1. Linear Stability Analysis

#### 6.2. Galloping Response

#### 6.3. Validation by Finite-Dimensional Model

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MSM | Multiple Scale Method |

NRT | Non Resonant Terms |

c.c. | complex conjugate |

## Appendix A. Calibration of the Damping Coefficients

## References

- Miyata, T. Historical view of long-span bridge aerodynamics. J. Wind Eng. Ind. Aerodyn.
**2003**, 91, 1393–1410. [Google Scholar] [CrossRef] - Larsen, A.; Larose, G.L. Dynamic wind effects on suspension and cable-stayed bridges. J. Sound Vib.
**2015**, 334, 2–28. [Google Scholar] [CrossRef] - Irvine, H.M.; Caughey, T.K. The linear theory of free vibrations of a suspended cable. Proc. R. Soc. Lond. A Math. Phys. Sci.
**1974**, 341, 299–315. [Google Scholar] - Irvine, H.M. Cable Structures; Dover Publications Inc.: New York, NY, USA, 1992. [Google Scholar]
- Hagedorn, P.; Schäfer, B. On non-linear free vibrations of an elastic cable. Int. J. Non-Linear Mech.
**1980**, 15, 333–340. [Google Scholar] [CrossRef] - Rega, G.; Luongo, A. Natural vibrations of suspended cables with flexible supports. Comput. Struct.
**1980**, 12, 65–75. [Google Scholar] [CrossRef] [Green Version] - Luongo, A.; Rega, G.; Vestroni, F. Planar non-linear free vibrations of an elastic cable. Int. J. Non-Linear Mech.
**1984**, 19, 39–52. [Google Scholar] [CrossRef] [Green Version] - Rega, G. Nonlinear vibrations of suspended cables-Part I: Modeling and analysis. Appl. Mech. Rev.
**2004**, 57, 443–478. [Google Scholar] [CrossRef] - Rega, G. Nonlinear vibrations of suspended cables-part II: deterministic phenomena. Appl. Mech. Rev.
**2004**, 57, 479–514. [Google Scholar] [CrossRef] - Luongo, A.; Piccardo, G. A continuous approach to the aeroelastic stability of suspended cables in 1: 2 internal resonance. J. Vib. Control
**2008**, 14, 135–157. [Google Scholar] [CrossRef] [Green Version] - Luongo, A.; Zulli, D. Dynamic instability of inclined cables under combined wind flow and support motion. Nonlinear Dyn.
**2012**, 67, 71–87. [Google Scholar] [CrossRef] - Foti, F.; Martinelli, L. A unified analytical model for the self-damping of stranded cables under aeolian vibrations. J. Wind Eng. Ind. Aerodyn.
**2018**, 176, 225–238. [Google Scholar] [CrossRef] - Bleich, F. The Mathematical Theory of Vibration in Suspension Bridges: A Contribution to the Work of the Advisory Board on the Investigation of Suspension Bridges; US Government Printing Office: Washington, DC, USA, 1950.
- Pugsley, A. The Theory of Suspension Bridges; Edward Arnold Publishers Limited: London, UK, 1968. [Google Scholar]
- Abdel-Ghaffar, A.M. Suspension bridge vibration: continuum formulation. J. Eng. Mech. Div.
**1982**, 108, 1215–1232. [Google Scholar] - Hayashikawa, T.; Watanabe, N. Vertical vibration in Timoshenko beam suspension bridges. J. Eng. Mech.
**1984**, 110, 341–356. [Google Scholar] [CrossRef] - Kim, M.Y.; Kwon, S.D.; Kim, N.I. Analytical and numerical study on free vertical vibration of shear-flexible suspension bridges. J. Sound Vib.
**2000**, 238, 65–84. [Google Scholar] [CrossRef] - Luco, J.E.; Turmo, J. Linear vertical vibrations of suspension bridges: A review of continuum models and some new results. Soil Dyn. Earthq. Eng.
**2010**, 30, 769–781. [Google Scholar] [CrossRef] - Farquharson, F. Aerodynamic stability of suspension bridges, University of Washington Experiment Station. Bull
**1952**, 116, 1949–1954. [Google Scholar] - Bleich, F. Dynamic instability of truss-stiffened suspension bridges under wind action. Proc. ASCE
**1948**, 74, 1269–1314. [Google Scholar] - Selberg, A. Aerodynamic effects on suspension bridges. Int. Conf. Wind Effects Build. Struct.
**1963**, 2, 462–479. [Google Scholar] - Scanlan, R.H.; Tomo, J. Air foil and bridge deck flutter derivatives. J. Soil Mech. Found. Div.
**1971**, 97, 1717–1737. [Google Scholar] - Larsen, A. Advances in aeroelastic analyses of suspension and cable-stayed bridges. J. Wind Eng. Ind. Aerodyn.
**1998**, 74, 73–90. [Google Scholar] [CrossRef] - Chen, X.; Kareem, A. Aeroelastic analysis of bridges: effects of turbulence and aerodynamic nonlinearities. J. Eng. Mech.
**2003**, 129, 885–895. [Google Scholar] [CrossRef] - Salvatori, L.; Borri, C. Frequency-and time-domain methods for the numerical modeling of full-bridge aeroelasticity. Comput. Struct.
**2007**, 85, 675–687. [Google Scholar] [CrossRef] - Arena, A.; Lacarbonara, W. Nonlinear parametric modeling of suspension bridges under aeroelastic forces: torsional divergence and flutter. Nonlinear Dyn.
**2012**, 70, 2487–2510. [Google Scholar] [CrossRef] - Wu, T.; Kareem, A.; Ge, Y. Linear and nonlinear aeroelastic analysis frameworks for cable-supported bridges. Nonlinear Dyn.
**2013**, 74, 487–516. [Google Scholar] [CrossRef] - Arena, A.; Lacarbonara, W.; Marzocca, P. Post-critical behavior of suspension bridges under nonlinear aerodynamic loading. J. Comput. Nonlinear Dyn.
**2016**, 11, 011005. [Google Scholar] [CrossRef] - Novak, M. Aeroelastic galloping of prismatic bodies. J. Eng. Mech. Div.
**1969**, 95, 115–142. [Google Scholar] - Novak, M. Galloping oscillations of prismatic structures. J. Eng. Mech. Div.
**1972**, 98, 27–46. [Google Scholar] - Robertson, I.; Li, L.; Sherwin, S.; Bearman, P. A numerical study of rotational and transverse galloping rectangular bodies. J. Fluids Struct.
**2003**, 17, 681–699. [Google Scholar] [CrossRef] - Chen, Z.; Tse, K.T.; Kwok, K.C.; Kareem, A. Aerodynamic damping of inclined slender prisms. J. Wind Eng. Ind. Aerodyn.
**2018**, 177, 79–91. [Google Scholar] [CrossRef] - Chen, Z.; Tse, K.T.; Kwok, K.; Kim, B.; Kareem, A. Modelling unsteady self-excited wind force on slender prisms in a turbulent flow. Eng. Struct.
**2020**, 202, 109855. [Google Scholar] [CrossRef] - Luongo, A.; Paolone, A.; Piccardo, G. Postcritical behavior of cables undergoing two simultaneous galloping modes. Meccanica
**1998**, 33, 229–242. [Google Scholar] [CrossRef] - Luongo, A.; Zulli, D.; Piccardo, G. Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables. J. Sound Vib.
**2008**, 315, 375–393. [Google Scholar] [CrossRef] [Green Version] - Macdonald, J.H.; Larose, G.L. Two-degree-of-freedom inclined cable galloping—Part 1: General formulation and solution for perfectly tuned system. J. Wind Eng. Ind. Aerodyn.
**2008**, 96, 291–307. [Google Scholar] [CrossRef] - Ferretti, M.; Zulli, D.; Luongo, A. A Continuum Approach to the Nonlinear In-Plane Galloping of Shallow Flexible Cables. Adv. Math. Phys.
**2019**, 2019, 1–12. [Google Scholar] [CrossRef] [Green Version] - Abdel-Rohman, M.; John, M.J. Control of wind-induced nonlinear oscillations in suspension bridges using a semi-active tuned mass damper. J. Vib. Control
**2006**, 12, 1049–1080. [Google Scholar] [CrossRef] - Abdel-Rohman, M.; John, M.J.; Hassan, M.F. Compensation of time delay effect in semi-active controlled suspension bridges. J. Vibr. Control
**2010**, 16, 1527–1558. [Google Scholar] [CrossRef] - Almutairi, N.B.; Zribi, M.; Abdel-Rohman, M. Lyapunov-based control for suppression of wind-induced galloping in suspension bridges. Math. Probl. Eng.
**2011**, 2011. [Google Scholar] [CrossRef] - Di Nino, S.; Luongo, A. Nonlinear aeroelastic behavior of a base-isolated beam under steady wind flow. Int. J. Non-Linear Mech.
**2019**, 119, 103340. [Google Scholar] [CrossRef] - Piccardo, G.; Pagnini, L.; Tubino, F. Some research perspectives in galloping phenomena: critical conditions and post-critical behavior. Contin. Mech. Thermodyn.
**2015**, 27, 261–285. [Google Scholar] [CrossRef] - Chen, X.; Kareem, A. Advances in modeling of aerodynamic forces on bridge decks. J. Eng. Mech.
**2002**, 128, 1193–1205. [Google Scholar] [CrossRef] - Piccardo, G.; Tubino, F.; Luongo, A. On the effect of mechanical non-linearities on vortex-induced lock-in vibrations. Math. Mech. Solids
**2017**, 22, 1922–1935. [Google Scholar] [CrossRef] - Simiu, E.; Scanlan, R.H. Wind Effects on Structures: Fundamentals and Applications to Design; John Wiley: New York, NY, USA, 1996. [Google Scholar]
- Ehsan, F.; Scanlan, R.H. Vortex-induced vibrations of flexible bridges. J. Eng. Mech.
**1990**, 116, 1392–1411. [Google Scholar] [CrossRef] - Reissner, H. Oscillations of suspension bridges. J. Appl. Mech.
**1943**, 10, 23–32. [Google Scholar] - Del Arco, D.C.; Aparicio, A.C. Preliminary static analysis of suspension bridges. Eng. Struct.
**2001**, 23, 1096–1103. [Google Scholar] [CrossRef] - Luongo, A.; Di Egidio, A.; Paolone, A. Multiple scale bifurcation analysis for finite-dimensional autonomous systems. Recent Res. Dev. Sound Vib.
**2002**, 1, 161–201. [Google Scholar] - Luongo, A. On the use of the multiple scale method in solving ’difficult’ bifurcation problems. Math. Mech. Solids
**2017**, 22, 988–1004. [Google Scholar] [CrossRef] [Green Version] - Nayfeh, A. Introduction to Perturbation Techniques; John Wiley & Sons: Hoboken, NJ, USA, 2011. [Google Scholar]
- Nayfeh, A. Problems in Perturbation; A Wiley Interscience Publicationp, Wiley: Hoboken, NJ, USA, 1985. [Google Scholar]
- Luongo, A.; D’Annibale, F.; Ferretti, M. Hard loss of stability of Ziegler’s column with nonlinear damping. Meccanica
**2016**, 51, 2647–2663. [Google Scholar] [CrossRef] - CNR. Guide for the Assessment of Wind Actions and Effects on Structures–CNR-DT 207/2008. 2008. Available online: http://cae-cube.ru/literatura/literatura-prochiye/obshcheye/CNR-DT_207.2008.pdf (accessed on 2 March 2020).
- Mannini, C.; Massai, T.; Marra, A.M.; Bartoli, G. Interference of vortex-induced vibration and galloping: Experiments and mathematical modelling. Procedia Eng.
**2017**, 199, 3133–3138. [Google Scholar] [CrossRef] - Orban, F. Damping of materials and members in structures. J. Proc. Conf. Ser.
**2011**, 268, 012022. [Google Scholar] [CrossRef] - Brownjohn, J. Estimation of damping in suspension bridges. Proc. Inst. Civ. Eng.-Struct. Build.
**1994**, 104, 401–415. [Google Scholar] [CrossRef]

**Figure 2.**Qualitative galloping response to smooth flow, and hysteresis loop, for prismatic bodies with a 2:1 side ratio.

**Figure 3.**Natural modes of the undamped model: First antisymmetric modes ${\varphi}_{A}\left(s\right)$ (black solid line); Second symmetric mode ${\varphi}_{S}\left(s\right)$ (gray solid line).

**Figure 4.**Bifurcation diagram for the case study: steady amplitude of the limit-cycle $vs$ the wind velocity.

**Figure 5.**Comparison between exact (finite difference and numerical integration) and analytical (perturbative) results. Time-histories (filled area) vs numerical integration of the bifurcation equation (black line), when: (

**a**) $u<{u}^{*}$, (

**b**) ${u}^{*}<u<{u}_{c}$, and (

**c**) $u>{u}_{c}$; (

**d**) bifurcation diagram, exact (red point bullets) vs analytical results (black solid line).

Mode | ${\tilde{\mathit{\omega}}}_{\mathit{c}}$ | ${\mathit{u}}_{\mathit{c}}$ | ${\mathit{\omega}}_{\mathit{c}}$ | ${\mathit{U}}_{\mathit{c}}$ |
---|---|---|---|---|

[rad/s] | [m/s] | |||

A | $8.77$ | $0.7$ | $2.12$ | $34.13$ |

S | $15.61$ | $1.05$ | $3.91$ | $51.3$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Di Nino, S.; Luongo, A.
Nonlinear Aeroelastic in-Plane Behavior of Suspension Bridges under Steady Wind Flow. *Appl. Sci.* **2020**, *10*, 1689.
https://doi.org/10.3390/app10051689

**AMA Style**

Di Nino S, Luongo A.
Nonlinear Aeroelastic in-Plane Behavior of Suspension Bridges under Steady Wind Flow. *Applied Sciences*. 2020; 10(5):1689.
https://doi.org/10.3390/app10051689

**Chicago/Turabian Style**

Di Nino, Simona, and Angelo Luongo.
2020. "Nonlinear Aeroelastic in-Plane Behavior of Suspension Bridges under Steady Wind Flow" *Applied Sciences* 10, no. 5: 1689.
https://doi.org/10.3390/app10051689