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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).

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

[34]

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

[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.

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

[34]

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

[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.

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