• Previous Article
    Harnack type inequalities for some doubly nonlinear singular parabolic equations
  • DCDS Home
  • This Issue
  • Next Article
    Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems
December  2015, 35(12): 5879-5908. doi: 10.3934/dcds.2015.35.5879

A partially hinged rectangular plate as a model for suspension bridges

1. 

Dipartimento di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale "Amedeo Avogadro", Viale Teresa Michel 11, 15121 Alessandria, Italy

2. 

Dipartimento di Matematica Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano

Received  July 2013 Published  May 2015

A plate model describing the statics and dynamics of a suspension bridge is suggested. A partially hinged plate subject to nonlinear restoring hangers is considered. The whole theory from linear problems, through nonlinear stationary equations, ending with the full hyperbolic evolution equation is studied. This paper aims to be the starting point for more refined models.
Citation: Alberto Ferrero, Filippo Gazzola. A partially hinged rectangular plate as a model for suspension bridges. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5879-5908. doi: 10.3934/dcds.2015.35.5879
References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[2]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I,, Comm. Pure Appl. Math., 12 (1959), 623. doi: 10.1002/cpa.3160120405. Google Scholar

[3]

B. Akesson, Understanding Bridges Collapses,, CRC Press, (2008). Google Scholar

[4]

O. H. Ammann, T. von Kármán and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge,, Federal Works Agency, (1941). Google Scholar

[5]

G. Arioli and F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge collapse,, Appl. Math. Modelling, 39 (2015), 901. doi: 10.1016/j.apm.2014.06.022. Google Scholar

[6]

E. Berchio, A. Ferrero, F. Gazzola and P. Karageorgis, Qualitative behavior of global solutions to some nonlinear fourth order differential equations,, J. Diff. Eq., 251 (2011), 2696. doi: 10.1016/j.jde.2011.05.036. Google Scholar

[7]

J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges,, Earthquake Engineering & Structural Dynamics, 23 (1994), 1351. doi: 10.1002/eqe.4290231206. Google Scholar

[8]

K. Friedrichs, Die randwert und eigenwertprobleme aus der theorie der elastischen platten (anwendung der direkten methoden der variationsrechnung),, Math. Ann., 98 (1928), 205. doi: 10.1007/BF01451590. Google Scholar

[9]

P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.051110. Google Scholar

[10]

F. Gazzola, Nonlinearity in oscillating bridges,, Electron. J. Diff. Equ., (2013), 1. Google Scholar

[11]

F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems,, Lecture Notes in Mathematics, (1991). doi: 10.1007/978-3-642-12245-3. Google Scholar

[12]

F. Gazzola and R. Pavani, Blow up oscillating solutions to some nonlinear fourth order differential equations,, Nonlinear Analysis, 74 (2011), 6696. doi: 10.1016/j.na.2011.06.049. Google Scholar

[13]

F. Gazzola and R. Pavani, Wide oscillations finite time blow up for solutions to nonlinear fourth order differential equations,, Arch. Rat. Mech. Anal., 207 (2013), 717. doi: 10.1007/s00205-012-0569-5. Google Scholar

[14]

D. Imhof, Risk Assessment of Existing Bridge Structure,, PhD Dissertation, (2004). Google Scholar

[15]

T. Kawada, History of the modern suspension bridge: Solving the dilemma between economy and stiffness,, ASCE Press, (2010). doi: 10.1061/9780784410189. Google Scholar

[16]

G. R. Kirchhoff, Über das gleichgewicht und die bewegung einer elastischen scheibe,, J. Reine Angew. Math., 1850 (2009), 51. doi: 10.1515/crll.1850.40.51. Google Scholar

[17]

W. Lacarbonara, Nonlinear Structural Mechanics,, Springer, (2013). doi: 10.1007/978-1-4419-1276-3. Google Scholar

[18]

R. S. Lakes, Foam structures with a negative Poisson's ratio,, Science, 235 (1987), 1038. doi: 10.1126/science.235.4792.1038. Google Scholar

[19]

A. C. Lazer and P. J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems,, Ann. Inst. H. Poincaré Anal. non Lin., 4 (1987), 243. Google Scholar

[20]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis,, SIAM Rev., 32 (1990), 537. doi: 10.1137/1032120. Google Scholar

[21]

M. Lévy, Sur l'équilibre élastique d'une plaque rectangulaire,, Comptes Rendus Acad. Sci. Paris, 129 (1899), 535. Google Scholar

[22]

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity,, Fourth edition, (1927). Google Scholar

[23]

E. H. Mansfield, The Bending and Stretching of Plates,, Second edition, (1989). doi: 10.1017/CBO9780511525193. Google Scholar

[24]

V. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains,, Mathematical Surveys and Monographs, (2010). doi: 10.1090/surv/162. Google Scholar

[25]

P. J. McKenna, Torsional oscillations in suspension bridges revisited: Fixing an old approximation,, Amer. Math. Monthly, 106 (1999), 1. doi: 10.2307/2589581. Google Scholar

[26]

P. J. McKenna and C. Ó. Tuama, Large torsional oscillations in suspension bridges visited again: Vertical forcing creates torsional response,, Amer. Math. Monthly, 108 (2001), 738. doi: 10.2307/2695617. Google Scholar

[27]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rat. Mech. Anal., 98 (1987), 167. doi: 10.1007/BF00251232. Google Scholar

[28]

P. J. McKenna and W. Walter, Travelling waves in a suspension bridge,, SIAM J. Appl. Math., 50 (1990), 703. doi: 10.1137/0150041. Google Scholar

[29]

A. Nadai, Die Elastischen Platten,, Springer-Verlag, (1968). doi: 10.1007/978-3-642-99170-7. Google Scholar

[30]

C. L. Navier, Extraits des recherches sur la flexion des plans élastiques,, Bulletin des Sciences de la Société Philomathique de Paris, (1823), 92. Google Scholar

[31]

R. H. Plaut and F. M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges,, J. Sound and Vibration, 307 (2007), 894. doi: 10.1016/j.jsv.2007.07.036. Google Scholar

[32]

R. Scott, In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability,, ASCE Press, (2001). doi: 10.1061/9780784405420. Google Scholar

[33]

E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications,, Marcel Dekker Inc., (2001). doi: 10.1201/9780203908723. Google Scholar

[34]

Tacoma Narrows Bridge Collapse, http://www.youtube.com/watch?v=3mclp9QmCGs,, 1940., (). Google Scholar

[35]

O. Zanaboni, Risoluzione, in serie semplice, della lastra rettangolare appoggiata, sottoposta all'azione di un carico concentrato comunque disposto,, Annali Mat. Pura Appl., 19 (1940), 107. doi: 10.1007/BF02410542. Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[2]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I,, Comm. Pure Appl. Math., 12 (1959), 623. doi: 10.1002/cpa.3160120405. Google Scholar

[3]

B. Akesson, Understanding Bridges Collapses,, CRC Press, (2008). Google Scholar

[4]

O. H. Ammann, T. von Kármán and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge,, Federal Works Agency, (1941). Google Scholar

[5]

G. Arioli and F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge collapse,, Appl. Math. Modelling, 39 (2015), 901. doi: 10.1016/j.apm.2014.06.022. Google Scholar

[6]

E. Berchio, A. Ferrero, F. Gazzola and P. Karageorgis, Qualitative behavior of global solutions to some nonlinear fourth order differential equations,, J. Diff. Eq., 251 (2011), 2696. doi: 10.1016/j.jde.2011.05.036. Google Scholar

[7]

J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges,, Earthquake Engineering & Structural Dynamics, 23 (1994), 1351. doi: 10.1002/eqe.4290231206. Google Scholar

[8]

K. Friedrichs, Die randwert und eigenwertprobleme aus der theorie der elastischen platten (anwendung der direkten methoden der variationsrechnung),, Math. Ann., 98 (1928), 205. doi: 10.1007/BF01451590. Google Scholar

[9]

P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.051110. Google Scholar

[10]

F. Gazzola, Nonlinearity in oscillating bridges,, Electron. J. Diff. Equ., (2013), 1. Google Scholar

[11]

F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems,, Lecture Notes in Mathematics, (1991). doi: 10.1007/978-3-642-12245-3. Google Scholar

[12]

F. Gazzola and R. Pavani, Blow up oscillating solutions to some nonlinear fourth order differential equations,, Nonlinear Analysis, 74 (2011), 6696. doi: 10.1016/j.na.2011.06.049. Google Scholar

[13]

F. Gazzola and R. Pavani, Wide oscillations finite time blow up for solutions to nonlinear fourth order differential equations,, Arch. Rat. Mech. Anal., 207 (2013), 717. doi: 10.1007/s00205-012-0569-5. Google Scholar

[14]

D. Imhof, Risk Assessment of Existing Bridge Structure,, PhD Dissertation, (2004). Google Scholar

[15]

T. Kawada, History of the modern suspension bridge: Solving the dilemma between economy and stiffness,, ASCE Press, (2010). doi: 10.1061/9780784410189. Google Scholar

[16]

G. R. Kirchhoff, Über das gleichgewicht und die bewegung einer elastischen scheibe,, J. Reine Angew. Math., 1850 (2009), 51. doi: 10.1515/crll.1850.40.51. Google Scholar

[17]

W. Lacarbonara, Nonlinear Structural Mechanics,, Springer, (2013). doi: 10.1007/978-1-4419-1276-3. Google Scholar

[18]

R. S. Lakes, Foam structures with a negative Poisson's ratio,, Science, 235 (1987), 1038. doi: 10.1126/science.235.4792.1038. Google Scholar

[19]

A. C. Lazer and P. J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems,, Ann. Inst. H. Poincaré Anal. non Lin., 4 (1987), 243. Google Scholar

[20]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis,, SIAM Rev., 32 (1990), 537. doi: 10.1137/1032120. Google Scholar

[21]

M. Lévy, Sur l'équilibre élastique d'une plaque rectangulaire,, Comptes Rendus Acad. Sci. Paris, 129 (1899), 535. Google Scholar

[22]

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity,, Fourth edition, (1927). Google Scholar

[23]

E. H. Mansfield, The Bending and Stretching of Plates,, Second edition, (1989). doi: 10.1017/CBO9780511525193. Google Scholar

[24]

V. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains,, Mathematical Surveys and Monographs, (2010). doi: 10.1090/surv/162. Google Scholar

[25]

P. J. McKenna, Torsional oscillations in suspension bridges revisited: Fixing an old approximation,, Amer. Math. Monthly, 106 (1999), 1. doi: 10.2307/2589581. Google Scholar

[26]

P. J. McKenna and C. Ó. Tuama, Large torsional oscillations in suspension bridges visited again: Vertical forcing creates torsional response,, Amer. Math. Monthly, 108 (2001), 738. doi: 10.2307/2695617. Google Scholar

[27]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rat. Mech. Anal., 98 (1987), 167. doi: 10.1007/BF00251232. Google Scholar

[28]

P. J. McKenna and W. Walter, Travelling waves in a suspension bridge,, SIAM J. Appl. Math., 50 (1990), 703. doi: 10.1137/0150041. Google Scholar

[29]

A. Nadai, Die Elastischen Platten,, Springer-Verlag, (1968). doi: 10.1007/978-3-642-99170-7. Google Scholar

[30]

C. L. Navier, Extraits des recherches sur la flexion des plans élastiques,, Bulletin des Sciences de la Société Philomathique de Paris, (1823), 92. Google Scholar

[31]

R. H. Plaut and F. M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges,, J. Sound and Vibration, 307 (2007), 894. doi: 10.1016/j.jsv.2007.07.036. Google Scholar

[32]

R. Scott, In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability,, ASCE Press, (2001). doi: 10.1061/9780784405420. Google Scholar

[33]

E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications,, Marcel Dekker Inc., (2001). doi: 10.1201/9780203908723. Google Scholar

[34]

Tacoma Narrows Bridge Collapse, http://www.youtube.com/watch?v=3mclp9QmCGs,, 1940., (). Google Scholar

[35]

O. Zanaboni, Risoluzione, in serie semplice, della lastra rettangolare appoggiata, sottoposta all'azione di un carico concentrato comunque disposto,, Annali Mat. Pura Appl., 19 (1940), 107. doi: 10.1007/BF02410542. Google Scholar

[1]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851

[2]

Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden. Higher--order implicit function theorems and degenerate nonlinear boundary-value problems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 293-315. doi: 10.3934/cpaa.2008.7.293

[3]

John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276

[4]

Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595

[5]

Dariusz Bugajewski, Piotr Kasprzak. On mappings of higher order and their applications to nonlinear equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 627-647. doi: 10.3934/cpaa.2012.11.627

[6]

Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331

[7]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061

[8]

Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920

[9]

Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358

[10]

John Baxley, Mary E. Cunningham, M. Kathryn McKinnon. Higher order boundary value problems with multiple solutions: examples and techniques. Conference Publications, 2005, 2005 (Special) : 84-90. doi: 10.3934/proc.2005.2005.84

[11]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127

[12]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[13]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83

[14]

Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596

[15]

Xiao-Yu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear two-point boundary value problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 31-43. doi: 10.3934/naco.2012.2.31

[16]

Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang, Jihong Shen. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244

[17]

Yanfei Lu, Qingfei Yin, Hongyi Li, Hongli Sun, Yunlei Yang, Muzhou Hou. Solving higher order nonlinear ordinary differential equations with least squares support vector machines. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019012

[18]

Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020009

[19]

Risei Kano, Yusuke Murase. Solvability of nonlinear evolution equations generated by subdifferentials and perturbations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 75-93. doi: 10.3934/dcdss.2014.7.75

[20]

Akisato Kubo. Nonlinear evolution equations associated with mathematical models. Conference Publications, 2011, 2011 (Special) : 881-890. doi: 10.3934/proc.2011.2011.881

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]