Skip to main content

The Nonlinear Theory of Beams

  • Chapter
  • First Online:
Nonlinear Structural Mechanics

Abstract

The nonlinear theory of beams undergoing planar motions is presented in its kinematic, dynamical, and constitutive aspects. The classical form of the equations of motion and the associated weak form are derived together with ad hoc approximate theories for planar weakly nonlinear motions such as the Mettler theory. Experimental results that corroborate the analytical predictions are presented for beams restricted to planar motion. The theory is then generalized to three-dimensional finite motions in the context of exact, intrinsic, and induced theories derived from three-dimensional theory. Different constrained versions of the theory, the linearized elastodynamic problem, the axial-torsional-shearing/flexural uncoupling, and the nonlinear coupling between different load-carrying mechanisms are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 349.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 449.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 449.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The base curve can be any convenient material curve or line. It is often taken to coincide with the centerline (i.e., the line that passes through the centers of masses of the cross sections, denoted here by \({C}^{\text{ o}}\)) so as to obtain the simplest form of the inertial forces. On the other hand, this choice does not necessarily lead to the simplest expressions of the applied generalized force and moment resultants.

  2. 2.

    When the unit vectors (\({\mathbf{\mathit{b}}{}_{2}}^{\text{ o}},\,{\mathbf{\mathit{b}}}_{3}^{\text{ o}}\)) are taken to be collinear with the principal axes of inertia of the cross section and \({\mathcal{C}}^{\text{ o}}\) is chosen coincident with the centerline, the intrinsic reference frame \(({C}^{\text{ o}},{\mathbf{\mathit{b}}{}_{2}}^{\text{ o}},\,{\mathbf{\mathit{b}}}_{3}^{\text{ o}})\) represents the principal inertia reference frame of the cross section at s with origin in the center of mass \({C}^{\text{ o}}.\)

  3. 3.

    The slenderness ratio is defined as the ratio h ∕ l between the beam thickness and span or the ratio r ∕ l between the radius of gyration \(r := \sqrt{J/A}\) of the cross section and the beam span given that A is the area and J the second area moment of the cross section.

  4. 4.

    Historically, the correct order would be Bernoulli–Euler as pointed out in [23]. However, the order Euler–Bernoulli has prevailed in the literature.

  5. 5.

    These expressions may be obtained directly by imposing the unshearability constraint in the linearized strain–displacement relationships (5.6).

  6. 6.

    The assumption \(\mathbf{\mathit{f }} \cdot {\mathbf{\mathit{b}}}_{1} = 0\) may be relaxed by considering longitudinal forces \(\mathbf{\mathit{f }} \cdot {\mathbf{\mathit{b}}}_{1}\neq 0\) so long as they are away from a resonance condition with the elastic axial modes of vibration.

References

  1. Abdel-Ghaffar AM (1980) Vertical vibration analysis of suspension bridges. ASCE J Struct Div 106:2053–2075

    Google Scholar 

  2. Abdel-Ghaffar AM (1982) Suspension bridge vibration: continuum formulation. J Eng Mech-ASCE 108:1215–1232

    Google Scholar 

  3. Abdel-Ghaffar AM, Rubin LI (1983) Nonlinear free vibrations of suspension bridges: theory. J Eng Mech-ASCE 109:313–345

    Google Scholar 

  4. Abdel-Ghaffar AM, Khalifa MA (1991) Importance of cable vibrations in dynamics of cable-stayed bridges. J Eng Mech-ASCE 117:2571–2589

    Google Scholar 

  5. Addessi D, Lacarbonara W, Paolone A (2005) On the linear normal modes of planar prestressed elastica arches. J Sound Vib 284:1075–1097

    Google Scholar 

  6. Addessi D, Lacarbonara W, Paolone A (2005) Free in-plane vibrations of highly pre-stressed curved beams. Acta Mech 180:133–156

    Google Scholar 

  7. Addessi D, Lacarbonara W, Paolone A (2005) Linear vibrations of planar pre-stressed arches undergoing static bifurcations. In: Proceedings of the EURODYN 2005, Paris, Sept 4–7, 2005

    Google Scholar 

  8. Agar TJA (1989) The analysis of aerodynamic flutter of suspension bridges. Comput Struct 30:593–600

    Google Scholar 

  9. Agar TJA (1989) Aerodynamic flutter analysis of suspension bridges by a modal technique. Eng Struct 11:75–82

    Google Scholar 

  10. Akhtar I, Marzouk OA, Nayfeh AH (2009) A van der Pol-Duffing oscillator model of hydrodynamic forces on canonical structures. J Comput Nonlin Dyn 4:041006-1-9

    Google Scholar 

  11. Akhtar I, Nayfeh AH, Ribbens CJ (2009) On the stability and extension of reduced-order Galerkin models in incompressible flows: a numerical study of vortex shedding. Theor Comp Fluid Dyn 23:213–237

    Google Scholar 

  12. Allan W (1874) Theory of arches. D. Van Nostrand, New York

    Google Scholar 

  13. Allgower EL, Georg K (1990) Numerical continuation methods: an introduction. Springer, Berlin

    Google Scholar 

  14. Allgower EL, Georg K (1997) Numerical path following. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol 5. North-Holland, Delft, NL, pp 3–207

    Google Scholar 

  15. Ames WF (1002) Numerical methods for partial differential equations, 3rd edn. Academic, New York

    Google Scholar 

  16. Anderson TJ, Nayfeh AH, Balachandran B (1996) Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam. J Vib Acoust 118:21–27

    Google Scholar 

  17. Andrade LG, Awruch AM, Morsch IB (2007) Geometrically nonlinear analysis of laminate composite plates and shells using the eight-node hexahedral element with one-point integration. Compos Struct 79(4):571–580

    Google Scholar 

  18. Anselone PM, Moore RH (1966) An extension of the Newton Kantorovic method for solving nonlinear equations with an application to elasticity. J Math Anal Appl13:475–501

    Google Scholar 

  19. Antman SS, Warner WH (1967) Dynamical theory of hyperelastic rods. Arch Rat Mech Anal 23:135–162

    MathSciNet  Google Scholar 

  20. Antman SS (1972) The theory of rods. In: Flügge S, Truesdell C (ed) Handbuch der Physik Via/2, pp 641–703

    Google Scholar 

  21. Antman SS (1990) Global properties of buckled states of plates that can suffer thickness changes. Arch Ration Mech Anal 110:103–117

    MathSciNet  Google Scholar 

  22. Antman SS (1998) The simple pendulum is not so simple. SIAM Rev 40:927–930

    MathSciNet  Google Scholar 

  23. Antman SS (2005) Problems of nonlinear elasticity. Springer, New York

    Google Scholar 

  24. Antman SS, Lacarbonara W (2009) Forced radial motions of nonlinearly viscoelastic shells. J Elast 96:155–190

    MathSciNet  Google Scholar 

  25. Arbabei F, Li F (1991) Buckling of variable cross-section columns. Integral–equation approach. J Struct Engng 117:2426–2441

    Google Scholar 

  26. Arena A (2008) Modellazione non lineare ed analisi della risposta dinamica di ponti sospesi. MS Thesis (in Italian). Sapienza University of Rome

    Google Scholar 

  27. Arena A, Formica G, Lacarbonara W, Dankowicz H (2011) Nonlinear finite element-based path following of periodic solutions. Paper no. DETC2011-48681, 2011 ASME IDETC, Washington DC USA, August 28–31, 2011

    Google Scholar 

  28. Arvin H (2012) Nonlinear modal analysis of a rotating composite Timoshenko beam with internal resonance. PhD Dissertation, Amirkabir University (Iran) and Sapienza University of Rome (Italy)

    Google Scholar 

  29. Arena A, Lacarbonara W, Marzocca P (2011) Nonlinear aeroelastic formulation for flexible high-aspect ratio wings via geometrically exact approach. Paper No. AIAA-11-937605, 52nd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics & Materials Conference, Denver, CO, April 4–7, 2011

    Google Scholar 

  30. Arena A, Lacarbonara W, Marzocca P (2011) Nonlinear dynamic stall flutter for flexible high-aspect ratio wings. ENOC 2011 7th European Nonlinear Dynamics Conference, Rome, July 24–29, 2011

    Google Scholar 

  31. Arena A, Lacarbonara W, Marzocca P (2012) Nonlinear post-flutter analysis for flexible high-aspect-ratio wings. 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, USA, April 23–27

    Google Scholar 

  32. Arena A, Lacarbonara W (2012) Nonlinear parametric modeling of suspension bridges under aeroelastic forces. Nonlinear Dynam, DOI: 10.1007/s11071-012-0636-3

    Google Scholar 

  33. Arena A, Lacarbonara W, Marzocca P (2012) Unsteady aerodynamic modeling and flutter analysis of long-span suspension bridges. Paper No. DETC2012/CIE-70289, ASME IDETC/CIE 2012, August 12–15, 2012, Chicago, IL

    Google Scholar 

  34. Argyris J (1982) An excursion into large rotations. Comput Meth Appl Mech Eng 32:85–155

    MathSciNet  Google Scholar 

  35. Asplund SO (1943) On the deflection theory of suspension bridges. Alqvist & Wiksells boktryckeri. Uppsala, Stockholm

    Google Scholar 

  36. Atluri S (1973) Nonlinear vibrations of a hinged beam including nonlinear inertia effects. J Appl Mech 40:121–126

    Google Scholar 

  37. Augusti G, Spinelli P, Borri C, Bartoli G, Giachi M, Giordano S (1995) The CRIACIV Atmospheric Boundary Layer Wind Tunnel. In: Wind engineering: retrospect and prospect, IAWE, International Association for Wind Engineering, vol. 5, Wiley Eastern Limited, New Delhi

    Google Scholar 

  38. Auricchio F, Taylor RL (1997) Shape-memory alloys: modelling and numerical simulations of the finite-strain superelastic behavior. Comput Method Appl M 143:175–194

    Google Scholar 

  39. Auricchio F, Taylor RL, Lubliner J (1997) Shape-memory alloys: macro-modelling and numerical simulations of the superelastic behavior. Comput Method Appl M 146:281–312

    Google Scholar 

  40. Auricchio F, Petrini L (2004) A three-dimensional model describing stress-temperature induced solid phase transformations. Part I: solution algorithm and boundary value problems. Int J Numer Methods Eng 61:807–836

    Google Scholar 

  41. Auricchio F, Petrini L (2004) A three-dimensional model describing stress-temperature induced solid phase transformations. Part II: thermomechanical coupling and hybrid composite applications. Int J Numer Methods Eng 61:716–737

    Google Scholar 

  42. Avramov KV, Pierre C, Shyriaieva NV (2008) Nonlinear equations of flexural-flexural-torsional oscillations of rotating beams with arbitrary cross-section. Int Appl Mech 44: 582–589

    MathSciNet  Google Scholar 

  43. Ball JM (1978) Finite-time blow-up in nonlinear problems. Nonlinear Evolution Equations. Academic, New York, pp 189–205

    Google Scholar 

  44. Balachandran B, Preidikman S (2004) Oscillations of piezoelectric micro-scale resonators. In: Topping BHV, Mota Soares CA (eds) Computational structures technology, progress in computational structures technology, pp 327–352

    Google Scholar 

  45. Ban RE, Chan TF (1986) PLTMGC: A multi-grid continuation program for parameterized nonlinear elliptic systems. SIAM J Sci Stat Comput 7:540–559

    Google Scholar 

  46. Bank RE (1998) PLTMG: A software package for solving elliptic partial differential equations, Users’ Guide 8.0. Software, Environments and Tools 5. J Soc Ind Appl Math

    Google Scholar 

  47. Bardin BS, Markeyev AP (1995) The stability of the equilibrium of a pendulum for vertical oscillations of the point of suspension. J Appl Math Mech 59:879–886

    MathSciNet  Google Scholar 

  48. Bartoli G, Righi M (2006) Flutter mechanism for rectangular prisms in smooth and turbulent flow. J Wind Eng Ind Aerodyn 94:275–291

    Google Scholar 

  49. Başar Y, Krätzig WB (1985) Mechanik der Flächentragwerke. Friedrich Vieweg & Sohn, Braunschweig/Wiesbaden

    Google Scholar 

  50. Başar Y (1987) A consistent theory of geometrically non-linear shells with an independent rotation vector. Int J Solids Struct 23:1401–1415

    Google Scholar 

  51. Başar Y (1993) Finite-rotation theories for arbitrary composite laminates. Acta Mech 98:159–176

    MathSciNet  Google Scholar 

  52. Başar Y, Montag U, Ding Y (1993) On a isoparametric finite-element for composite laminates with finite rotation. Comput Mech 12:329–348

    Google Scholar 

  53. Başar Y, Ding Y, Schultz R (1993) Refined shear-deformation models for composite laminates with finite rotations. Int J Solids Struct 30:2611–2638

    Google Scholar 

  54. Başar Y, Itskov M, Eckstein A (2000) Composite laminates: nonlinear interlaminar stress analysis by multi-layer shell elements. Comput Method Appl Mech Engrg 185:367–397

    Google Scholar 

  55. Batra RC (2006) Elements of continuum mechanics. AIAA Educational Series

    Google Scholar 

  56. Batra RC (2007) Higher-order shear and normal deformable theory for functionally graded incompressible linear elastic plates. Thin-Walled Struct 45:974–982

    Google Scholar 

  57. Batra RC, Porfiri M, Spinello D (2006) Electromechanical model of electrically actuated narrow microbeams. J Microelectromech S 15:1175–1189

    Google Scholar 

  58. Batra RC, Porfiri M, Spinello D (2008) Vibrations of narrow microbeams predeformed by an electric field. J Sound Vib 309:600–612

    Google Scholar 

  59. Bazant Z, Cedolin L (1991) Stability of structures. Oxford University Press, New York

    Google Scholar 

  60. Behal A, Marzocca P, Rao VM, Gnann A (2006) Nonlinear adaptive control of an aeroelastic two-dimensional lifting surface. J Guid Contr Dynam 29:382–390

    Google Scholar 

  61. Beletsky VV, Levin EM (1993) Dynamics of space tether systems. Advances in the astronautical sciences, vol 83. American Astronautical Society, San Diego

    Google Scholar 

  62. Belyayev NM (1924) Stability of prismatic rods subject to variable longitudinal forces (in Russian), in Engineering construction and structural mechanics, Leningrad

    Google Scholar 

  63. Benedetti D, Brebbia C, Cedolin, L (1972) Geometrical nonlinear analysis of structures by finite elements. Meccanica 7:1–10

    Google Scholar 

  64. Bernardini D, Pence TJ (2002) Shape-memory materials, modeling. In: Schwartz M (ed) The encyclopedia of smart materials, vol 2. Wiley, New York, pp 964–980

    Google Scholar 

  65. Bernardini D, Pence TJ (2009) Mathematical models for shape memory materials. In: Schwartz M (ed) Smart materials. CRC Press, Boca Raton, pp 20.17–20.28

    Google Scholar 

  66. Beran PS, Strganac, TW, Kim K, Nichkawde C (2004) Studies of store-induced Limit-Cycle Oscillations using a model with full system nonlinearities. Nonlinear Dynam 37:323–339

    Google Scholar 

  67. Biezeno CB, Koch, J (1923) Over een nieuwe methode ter berekening van vlokke platen met toepassing op enkele voor de techniek belangrijke belastingsgevallen. Ing Grav 38:25–36

    Google Scholar 

  68. Bigoni D (2012) Nonlinear solid mechanics. Bifurcation theory and material instability. Cambridge University Press, Cambridge.

    Google Scholar 

  69. Bleich F, McCullough CB, Rosecrans R, Vincent GS (1950) The mathematical theory of vibration in suspension bridges: A Contribution to the work of the Advisoy Board on the Investigation of Suspension Bridges. Department of Commerce, Bureau of Public Roads, USGPO, Washington, DC

    Google Scholar 

  70. Blekhman II (2000) Vibrational mechanics. Nonlinear dynamic effects, general approach, applications. World Scientific, Singapore

    Google Scholar 

  71. Blevins RD (1977) Flow-induced vibration. Van Nostrand Reinhold, New York

    Google Scholar 

  72. Boley BA, Weiner JH (1960) Theory of thermal stresses. Wiley, New York

    Google Scholar 

  73. Bolotin VV (1964) The dynamic stability of elastic systems. Holden-Day, San Francisco

    Google Scholar 

  74. Boonyapinyo V, Lauhatanon Y, Lukkunaprasit P (2006) Nonlinear aerostatic stability analysis of suspension bridges. Eng Struct28:793–803

    Google Scholar 

  75. Borri M, Mantegazza P (1985) Some contributions on structural and dynamic modeling of helicopter rotor blades. L’Aerotecnica Missili e Spazio 64(9):143–154

    Google Scholar 

  76. Bouc R (1967) Forced vibrations of mechanical systems with hysteresis. In: Preceedings of the 4th International Conference on Nonlinear Oscillations, Prague, Czechoslovakia

    Google Scholar 

  77. Bouc R (1971) Modele mathematique dhysteresis. Acustica 24:16–25

    Google Scholar 

  78. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, Berlin

    Google Scholar 

  79. Brokate M, Sprekels J (1996) Hysteresis and phase transitions. Springer, New York

    Google Scholar 

  80. Brownjohn JMW (1994) Observations on non-linear dynamic characteristics of suspension bridges. Earthquake Eng Struc 23:1351–1367

    Google Scholar 

  81. Brownjohn JMW, Dumanoglu AA, Taylor CA (1994) Vibration characteristics of a suspension footbridge. Eng Struct 16:395–406

    Google Scholar 

  82. Brownjohn JMW (1996) Vibration characteristics of a suspension footbridge. J Sound Vib 202:29–46

    Google Scholar 

  83. Buechner HF, Johnson MW, Moore RH (1965) The calculation of equilibrium states of elastic bodies by Newton’s method. Proceedings of the 9th Mid Western Mech Conf, Madison

    Google Scholar 

  84. Burgess JJ, Triantafyllou MS (1988) The elastic frequencies of cables. J Sound Vib 120: 153–165

    Google Scholar 

  85. Burgess JJ (1993) Bending stiffness in a simulation of undersea cable deployment. Int J Offshore Polar Eng 3:197–204

    Google Scholar 

  86. Caflisch R, Maddocks JH (1984) Nonlinear dynamical theory of the elastica. Proc R Soc Edin 99A:1–23

    MathSciNet  Google Scholar 

  87. Capecchi D, Vestroni F (1985) Steady-state dynamic analysis of hysteretic systems. J Eng Mech-ASCE 111:1515–1531

    Google Scholar 

  88. Capecchi D, Vestroni F (1990) Periodic response of a class of hysteretic oscillators. Int J Non Linear Mech 25:309–317

    Google Scholar 

  89. Carpineto N, Vestroni F, Lacarbonara W (2011) Vibration mitigation by means of hysteretic tuned mass dampers. In: Proceedings of EURODYN 2011, Leuven, July 4–5, 2011

    Google Scholar 

  90. Carpineto N (2011) Hysteretic tuned mass dampers for structural vibration mitigation. PhD Dissertation, Sapienza University of Rome

    Google Scholar 

  91. Carrera E (1999) Transverse normal stress effects in multilayered plates. J Appl Mech 66: 1004–1012

    Google Scholar 

  92. Carrera E, Parisch H (1997) An evaluation of geometrical nonlinear effects of thin and moderately thick multilayered composite shells. Compos Struct 40(1):11–24

    Google Scholar 

  93. Carrera E (2002) Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch Comput Method E 9:87–140

    MathSciNet  Google Scholar 

  94. Carrera E, Ciuffreda A (2005) A unified formulation to assess theories of multi-layered plates for various bending problems. Compos Struct 69:271–93

    Google Scholar 

  95. Cartmell M (1990) Introduction to linear, parametric and nonlinear vibrations. Chapman and Hall, London

    Google Scholar 

  96. Casciaro R (2005) Computational asymptotic post–buckling analysis of slender elastic structures, CISM Courses and Lectures NO. 470. Springer, New York

    Google Scholar 

  97. Castro FM (1991) Mechanical switches snap back. Mach Des 63:56–61

    Google Scholar 

  98. Caughey TK (1960) Sinusoidal excitation of a system with bilinear hysteresis. J Appl Mech 643:640–643

    MathSciNet  Google Scholar 

  99. Cesari L (1971) Asymptotic behavior and stability problems in ordinary differential equations. Springer, Berlin

    Google Scholar 

  100. Cevik M, Pakdemirli M (2005) Non-linear vibrations of suspension bridges with external excitation. Int J Non Linear Mech 40:901–923

    Google Scholar 

  101. Chan TF, Keller HB (1982) Arc-length continuation and multi-grid techniques for nonlinear eigenvalue problems. SIAM J Sci Statist Comput 3:173–194

    MathSciNet  Google Scholar 

  102. Chang WK, Pilipchuk V, Ibrahim RA (1997) Fluid flow-induced nonlinear vibration of suspended cables. Nonlinear Dynam 14:377–406

    MathSciNet  Google Scholar 

  103. Chelomeĭ V N (1939) The dynamic stability of elements of aircraft structures. Aeroflot, Moscow

    Google Scholar 

  104. Chen X, Matsumoto M, Kareem A (2000) Time domain flutter and buffeting response analysis of bridges. J Eng Mech-ASCE 126:7–16

    Google Scholar 

  105. Chen X, Kareem A (2000) Advances in modeling of aerodynamic forces on bridge decks. J Eng Mech 128:1193–1205

    Google Scholar 

  106. Cheng J, Jiang J-J, Xiao R-C, Xiang H-F (2003) Series method for analyzing 3D nonlinear torsional divergence of suspension bridges. Comput Struct 81:299–308

    Google Scholar 

  107. Cheng J, Jiang J-J, Xiao R-C (2003) Aerostatic stability analysis of suspension bridges under parametric uncertainty. Eng Struct 25:1675–1684

    Google Scholar 

  108. Cheng SH, Lau DT, Cheung MS (2003) Comparison of numerical techniques for 3D flutter analysis of cable-stayed bridges. Comput Struct 81:2811–2822

    Google Scholar 

  109. Chetayev NG (1961) The stability of motion. Pergamon Press, New York

    Google Scholar 

  110. Cheung YK (1968) The finite strip method in the analysis of elastic plates with two opposite simply supported ends. Proc Instn Civ Engrg Lond 40:1–7

    Google Scholar 

  111. Cho KN, Bert CW, Striz AG (1991) Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory. J Sound Vib 145(3):429–442

    Google Scholar 

  112. Cho MH, Parmerter RR (1993) Efficient higher-order composite plate-theory for general lamination configurations. AIAA J 31(7):1299–1306

    Google Scholar 

  113. Ciarlet PG, Destuynder PA (1979) Justification of a nonlinear model in plate theory. Comp Method Appl Mech Engrg 17/18:227–258

    Google Scholar 

  114. Ciarlet PG (2002) The finite element method for elliptic problems. Society for Industrial and Applied Mathematics, Philadelphia, PA

    Google Scholar 

  115. Clark R, Cox D, Curtiss HCJ, Edwards JW, Hall KC, Peters DA, Scanlan RH, Simiu E, Sisto F, Strganac Th W (2004) A modern course in Aeroelasticity. Series: Solid mechanics and its applications 116, 4th edn. Kluwer Academic, New York

    Google Scholar 

  116. Coddington EA, Levinson N (1955) Theory of ordinary differential equations. McGraw-Hill Book, New York

    Google Scholar 

  117. COMSOL Multiphysics (2008) COMSOL Multiphysics/User’s Guide Version 3.5. COMSOL AB, Stokholm, Sweden

    Google Scholar 

  118. Connor J, Morin R (1970) Perturbation techniques in the analysis of geometrically nonlinear shells. In: Proc Symp Int Union of Theoretical and Applied Mechanics, Liege, vol 61, 683–705

    Google Scholar 

  119. Cosmo ML, Lorenzini EC (1997) Tethers in space handbook, 3rd edn. Smithsonian Astrophisical observatory. NASA Marshall Space Flight Center, Huntsville, Alabama

    Google Scholar 

  120. Costello GA (1997) Theory of wire rope. Springer, New York

    Google Scholar 

  121. Cosserat EF (1909) Theorie de corps deformables. Hermann, Paris

    Google Scholar 

  122. Crandall SH (1956) Engineering analysis. McGraw-Hill, New York

    Google Scholar 

  123. Crespo da Silva MRM, Hodges DH (1986) Nonlinear flexure and torsion of rotating beams with application to helicopter rotor blades-I. Formulation. Vertica 10:151–169

    Google Scholar 

  124. Crespo da Silva MRM, Glynn CVC (1978) Nonlinear flexural-flexural-torsional dynamics of inextensional beams I. Equations of motion. J Struct Mech 6:437–448

    Google Scholar 

  125. Crespo da Silva MRM (1988) Nonlinear flexural-flexural-torsional-extensional dynamics of beams-II. Response analysis. Int J Solids Struct 24:1235–1242

    Google Scholar 

  126. Crisfield MA (1991) Non-Linear Finite Element Analysis of Solids and Structures, vol 1. Wiley, New York

    Google Scholar 

  127. Crisfield MA (1997) Non-linear finite element analysis of solids and structures, vol. 2. Wiley, New York

    Google Scholar 

  128. Dankowicz H, Schilder F (2011) An extended continuation problem for bifurcation analysis in the presence of constraints. J Comput Nonlinear Dyn 6:031003

    Google Scholar 

  129. Davenport AG (1966) The action of wind on suspension bridges. In: Int Symp on Suspension Bridges, Lisbon, 79–100

    Google Scholar 

  130. Demasi L (2009) 6 Mixed plate theories based on the Generalized Unified Formulation. Part I: Governing equations. Compos Struct 87:1–11. Part V: Results. Compos Struct 88:1–16

    Google Scholar 

  131. De Miranda M (1998) Storebaelt East Bridge - Aspetti del montaggio e della realizzazione (in Italian). Costruzioni Metalliche 6

    Google Scholar 

  132. Den Hartog JP (1934) Mechanical vibrations. McGraw-Hill, New York

    Google Scholar 

  133. Depuis GA, Pfaffinger DD, Marcal PV (1970) Effective use of the incremental stiffness matrices in nonlinear geometric analysis. In: Proc Symp Int Union of Theoretical and Applied Mechanics, Liège, vol 61, 707–725

    Google Scholar 

  134. Diana, G, Bruni S, Collina A, Zasso A (1998) Aerodynamic challenges in super long span bridge design. In: Larsen A, Esdahl E (eds) Proceedings of the International Symposium on Advances in Bridge Aerodynamics, 10–13 May, Copenhagen. Balkema, Rotterdam

    Google Scholar 

  135. Ding Q, Chen A, Xiang H (2002) Coupled flutter analysis of long-span bridges by multimode and full-order approaches. J Wind Eng Ind Aerodyn 90:1981–1993

    Google Scholar 

  136. Dinnik AN (1929) Design of columns of varying cross section. Trans ASME 51:105–114

    Google Scholar 

  137. Dinnik AN (1932) Design of columns of varying cross section. Trans ASME 54:165–171

    Google Scholar 

  138. Di Egidio A, Luongo A, Paolone A (2007) Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams. Int J Non Linear Mech 42:88–98

    Google Scholar 

  139. Di Sciuva M, Icardi U (1995) Analysis of thick multilayered anisotropic plates by a higher-order plate element. AIAA J 33(12):2435–2437

    Google Scholar 

  140. Doedel EJ, Paffenroth RC, Champneys AR, Fairgrieve TF, Kuznetsov, Yu A, Sandstede B, Wang X (2001) AUTO 2000: Continuation and bifurcation software for ordinary differential equations (with HomCont), Technical Report, Caltech

    Google Scholar 

  141. Drozdov AD (1996) Finite elasticity and viscoelasticity. World Scientific, Singapore

    Google Scholar 

  142. Ecker H, Dohnal F, Springer H (2005) Enhanced damping of a beam structure by parametric excitation. In: Proceedings of European Nonlinear Oscillations Conf. (ENOC-2005) Eindhoven, NL

    Google Scholar 

  143. El-Bassiouny AF (2007) Parametric excitation of internally resonant double pendulum. Phys Scripta 76:173–186

    Google Scholar 

  144. Einaudi R (1936) Sulle configurazioni di equilibrio instabile di una piastra sollecitata da sforzi tangenziali pulsanti. Atti Accad Gioenia Catania 1 (serie 6), mem. XX:1–20

    Google Scholar 

  145. Eisley JG (1964) Nonlinear vibration of beams and rectangular plates. Z Angew Math Mech 15:167–175

    MathSciNet  Google Scholar 

  146. Elishakoff I (2000) Both static deflection and vibration mode of uniform beam can serve as a buckling mode of a non-uniform column. J Sound Vib 232:477–489

    MathSciNet  Google Scholar 

  147. Elishakoff I (2005) Eigenvalues of inhomogeneous structures. CRC Press, Boca Raton

    Google Scholar 

  148. Engesser F (1909) Ueber die Knickfestigkeit von Staeben veraenderlichen Traegheitsmomentes (in German). Zeitschrift der Oesterreichischer Ingenieur und Architekten Verein 34:506–508

    Google Scholar 

  149. Eringen AC (1976) Nonlocal field theories. In: Eringen AC (ed) Continuum physics, vol 4. Academic, New York

    Google Scholar 

  150. Euler L (1759) Sur la force des colonnes (in French). Memoires de L’Academie des Sciences et Belles-Lettres 13:252–282

    Google Scholar 

  151. Evensen JA, Evan-Iwanowski RM (1996) Effects of longitudinal inertia upon the parametric response of elastic columns. J Appl Mech 33:141–148

    Google Scholar 

  152. Faedo S (1949) Un nuovo metodo per l’analisi esistenziale e quantitativa dei problemi di propagazione. Ann Sc Norm Sup Pisa - Classe di Scienze, Ser. 3,1 no. 1–4:1–41

    Google Scholar 

  153. Faraday M (1831) On a peculiar class of acoustical figures; and on certain forms assumed by a group of particles upon vibrating elastic surfaces. Philos Tr R Soc S-A 121:299

    Google Scholar 

  154. Farquharson FB, Smith, FC, Vincent GS (1950) Aerodynamic stability of suspension bridges with special reference to the Tacoma Narrows Bridge. Part II: Mathematical analyses. Bulletin 116. University of Washington Press, Engineering Experimental Station, Seattle, WA

    Google Scholar 

  155. Ferreira AJM, Roque CMC, Martins PALS (2004) Radial basis functions and higher-order shear deformation theories in the analysis of laminated composite beams and plates. Compos Struct 66:287–293

    Google Scholar 

  156. Fiedler L, Lacarbonara W, Vestroni F (2007) Vibration behavior of thick composite laminated plates subject to in-plane pre-stress loading. In: Proceedings of the DECT’07–2007 ASME Engineering Technical Conferences, DECT2007–35532, Las Vegas, Nevada, 4–7 September 2007

    Google Scholar 

  157. Fiedler L, Lacarbonara W, Vestroni F (2009) A general higher-order theory for multi-layered, shear-deformable composite plates. Acta Mech209:85–98

    Google Scholar 

  158. Fiedler L., Lacarbonara W., Vestroni F (2010) A generalized higher-order theory for buckling of thick multi-layered composite plates with normal and transverse shear strains. Compos Struct 92:3011–3020

    Google Scholar 

  159. Finlayson BA (1972) The method of weighted residuals and variational principles. Academic, New York

    Google Scholar 

  160. Foltinek K (1994) The Hamilton theory of elastica. Am J Math 116:1479–1488

    MathSciNet  Google Scholar 

  161. Fonda A, Schneider Z, Zanolin F (1994) Periodic oscillations for a nonlinear suspension bridge model. J Comput Appl Math 52:113–140

    MathSciNet  Google Scholar 

  162. Föppl A (1907) Vorlesungen über technische Mechanik, B.G. Teubner, Bd. 5., Leipzig

    Google Scholar 

  163. Formica G, Lacarbonara W, Alessi R (2010) Vibrations of carbon nanotube-reinforced composites. J Sound Vib 329:1875–1889

    Google Scholar 

  164. Formica G, Arena A, Lacarbonara W, Dankowicz H (2013) Coupling FEM with parameter continuation for analysis and bifurcations of periodic responses in nonlinear structures. J Comput Nonlin Dyn 8, 021013

    Google Scholar 

  165. Frahm H (1911) Device for damping vibration of bodies, US Patent 989958

    Google Scholar 

  166. Fremond M (2002) Non-smooth thermomechanics. Springer, Berlin

    Google Scholar 

  167. Frisch-Fay R (1962) Flexible bars. Butterworths, Washington, D.C

    Google Scholar 

  168. Fung YC (1990) Biomechanics: motion, flow, stress, and growth. Springer, New York

    Google Scholar 

  169. Garcea G, Trunfio GA, Casciaro R (2002) Path-following analysis of thin-walled structures and comparison with asymptotic post-critical solutions. Int J Numer Methods Eng 55:73–100

    Google Scholar 

  170. Galerkin BG (1915) Series occurring in some problems of elastic stability of rods and plates. Eng Bull 19:897–908

    Google Scholar 

  171. Ganapathi M, Makhecha DP (2001) Free vibration analysis of multi-layered composite laminates based on an accurate higher-order theory. Compos Part B-Eng 32:535–543

    Google Scholar 

  172. Gattulli V, Lepidi M (2003) Nonlinear interactions in the planar dynamics of cable-stayed beam. Int J Solids Struct 40:4729–4748

    Google Scholar 

  173. Gattulli V, Lepidi M, Macdonald JHG, Taylor CA (2005) One-to-two global-local interaction in a cable-stayed beam observed through analytical, finite element and experimental models, Int J Non-Linear Mech 40:571–588

    Google Scholar 

  174. Gaudenzi P (1992) A general formulation of higher-order theories for the analysis of composite laminated plates. Compos Struct 20:103–112

    Google Scholar 

  175. Gaudenzi P, Barboni R, Mannini A (1995) A finite element evaluation of single-layer and multi-layer theories for the analysis of laminated plates. Compos Struct 30:427–440

    Google Scholar 

  176. Ge Z, Kruse HP, Marsden JE (1996) The limits of Hamiltonian structures in three dimensional elasticity, shells, and rods. J Nonlinear Sci 6:19–57

    MathSciNet  Google Scholar 

  177. Gimsing NJ (1997) Cable supported bridges: concept and design, 2nd edn. Wiley, New York

    Google Scholar 

  178. Lanzara G, Yoon Y, Liu H, Peng S, Lee W-I (2009) Carbon nanotube reservoirs for self-healing materials. Nanotechnology 20:335704

    Google Scholar 

  179. Glauert H (1947) The elements of Aerofoil and Airscrew theory, 2nd edn. Cambridge University Press, New York

    Google Scholar 

  180. Glover J, Lazer AC, McKenna PJ (1989) Existence and stability of large scale nonlinear oscillations in suspension bridges. ZAMP 40:172–200

    MathSciNet  Google Scholar 

  181. Goldstein H, Poole CP, Safko JL (2002) Classical mechanics, 3rd edn. Addison Wesley, Reading

    Google Scholar 

  182. Guckenheimer J, Holmes P (1985) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York

    Google Scholar 

  183. Hafka RT, Mallet RH, Nachbar W (1971) Adaptation of Koiter’s method to finite element analysis of snap-through buckling behaviour. Int J Solids Struct 7:1427–1447

    Google Scholar 

  184. Hale JK (1969) Ordinary differential equations. Wiley-Interscience, New York

    Google Scholar 

  185. Hall BD, Preidikman S, Mook DT, Nayfeh AH (2001) Novel strategy for suppressing the flutter oscillations of aircraft wings. AIAA J 39:1843–1850

    Google Scholar 

  186. Han S-C, Tabiei A, Park W-T (2008) Geometrically nonlinear analysis of laminated composite thin shells using a modified first-order shear deformable element-based Lagrangian shell element. Compos Struct 82(3):465–474

    Google Scholar 

  187. Handbook (1986) Tethers in space. in Proceedings of the first International Conference on Tethers in Space, Sept 17–19, Arlington, VA

    Google Scholar 

  188. Handbook (1988) Space Tethers for Science in the Space Station Era, Societá Italiana di Fisica, Conference Proceedings, 14, Bologna

    Google Scholar 

  189. Hansen MH, Gaunaa M, Madsen HAA (2004) A Beddoes-Leishman type dynamic stall model in state-space and indicial formulations, Report No. R -1354(EN), Risø National Laboratory

    Google Scholar 

  190. Hartlen R, Currie I (1970) Lift-oscillator model for vortex-induced vibration. Proc Am Soc Civ Eng 96:577–591

    Google Scholar 

  191. Hirai A, Okauchi I, Miyata T (1966) On the behaviour of suspension bridges under wind action. Paper No. 8. Int. Sypm. on Suspension Bridges, Lisbon, 240–256

    Google Scholar 

  192. Hodges DH, Dowell EH (1974) Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. NASA TN D-7818

    Google Scholar 

  193. Hodges DH, Atilgan AR, Danielson DA (1993) A geometrically nonlinear theory of elastic plates. J Appl Mech 60:109–1126

    Google Scholar 

  194. Hodges DH, Atilgan AR, Danielson DA (1993) A geometrically nonlinear theory of elastic plates. J Appl Mech 60:109–116

    Google Scholar 

  195. Hodges DH (1999) Non-linear in-plane deformation and buckling of rings and high arches. Int J Non Linear Mech 34:723–737

    Google Scholar 

  196. Hodges DH, Wenbin Y, Mayuresh JP (2009) Geometrically-exact, intrinsic theory for dynamics of moving composite plates. Int J Solids Struct 46:2036–2042

    Google Scholar 

  197. Holzapfel GA (2000) Nonlinear solid mechanics. Wiley, Chichester

    Google Scholar 

  198. Hsu CS (1963) On the parametric excitation of a dynamic system with multiple degrees of freedom. J Appl Mech 30:367–372

    Google Scholar 

  199. Hua XG, Chen ZQ (2008) Full-order and multimode flutter analysis using ANSYS. Finite Elem Anal Des 44:537–551

    Google Scholar 

  200. Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice Hall, Upper Saddle River

    Google Scholar 

  201. Ibrahim RA (2004) Nonlinear vibrations of suspended cables - Part III: Random excitation and interaction with fluid flow. Appl Mech Rev 57:515–549

    Google Scholar 

  202. Ibrahim RA (2005) Liquid sloshing dynamics. Theory and applications. Cambridge University Press, Cambridge

    Google Scholar 

  203. Ibrahim RA (2008) Parametric random vibration. New York, Dover

    Google Scholar 

  204. Iooss G, Adelmeyer M (1992) Topics in bifurcation theory and applications. World Scientific, Singapore

    Google Scholar 

  205. In-Soo S, Uchiyama Y, Yabuno H, Lacarbonara W (2008) Simply supported elastic beams under parametric excitation. Nonlinear Dynam 53:129–138

    MathSciNet  Google Scholar 

  206. Irvine HM, Caughey TK (1974) The linear theory of free vibrations of a suspended cable. Proc R Soc London, Ser A 341:299–315

    Google Scholar 

  207. Irvine HM (1984) Cable structures. Dover Publications, New York

    Google Scholar 

  208. Iwan WD (1965) The steady-state response of the double bilinear hysteretic oscillator. J Appl Mech 32:921–925

    MathSciNet  Google Scholar 

  209. Iwan WD, Blevins RD (1974) A model for vortex induced oscillation of structures. J Appl Mech Trans ASME 41:581–586

    Google Scholar 

  210. Jacobs EN, Ward KE, Pinkerton RM (1933) The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel. NACA Report No. 460

    Google Scholar 

  211. Jacover D, McKenna PJ (1994) Nonlinear torsional flexings in a periodically forced suspended beam. J Comput Appl Math 52:241–265

    MathSciNet  Google Scholar 

  212. Jensen JS (1998) Non-linear dynamics of the follower-loaded double pendulum with added support-excitation. J Sound Vib 215:125–142

    Google Scholar 

  213. Jones RM (1975) Mechanics of composite materials. McGraw-Hill Book Company, New York

    Google Scholar 

  214. Jones KF (1992) Coupled vertical and horizontal galloping. J Eng Mech-ASCE 118:92–106

    Google Scholar 

  215. Johnson MW Jr, Urbanik TJ (1984) A nonlinear theory for elastic plates with application to characterizing paper properties. J Appl Mech 51:146–152

    Google Scholar 

  216. Kantorovich LV, Krylov VI (1964) Approximate methods of higher analysis. Interscience Publishers, New York

    Google Scholar 

  217. Kapitza PL (1965) Collected Papers of P.L. Kapitza, Edited by D. TerHarr, Pergamon Press, NY

    Google Scholar 

  218. Krauskopf B, Osinga HM, Galan-Vioque J (eds) (2007) Numerical continuation methods for dynamical systems. Springer and Canopus Publishing Limited, New York

    Google Scholar 

  219. Krylov N, Bogoliubov N (1935) Influence of resonance in transverse vibrations of rods caused by periodic normal forces at one end. Ukrainian Sc. Res. Inst. of Armament, Recueil Kiev.

    Google Scholar 

  220. Kevorkian J, Cole JD (1996) Multiple scale and singular perturbation methods. Springer, New York

    Google Scholar 

  221. Kienzler R, Bose DK (2008) Material conservation laws established within a consistent plate theory. In: Jaiani G, Podio-Guidugli P (eds) Proc symp int union of theoretical and applied mechanics on relations of shell plate beam and 3D models, Tbilisi, Georgia, April 23–27, 2007

    Google Scholar 

  222. Ko JW, Strganac TW; Kurdila AJ (1998) Stability and control of a structurally nonlinear aeroelastic system. J Guid Control Dynam 21:718–725

    Google Scholar 

  223. Komatsu S, Sakimoto T (1977) Ultimate load carrying capacity of steel arches. J Struct Div-ASCE 103(12):2323–2336

    Google Scholar 

  224. Koiter WT (1945) On the stability of elastic equilibrium. PhD Thesis, Delft. English transl

    Google Scholar 

  225. Koiter WT (1970) On the stability of elastic equilibrium (Translation from Dutch). Tech. Rep. AFFDL-TR-70-25, Airforce Flight Dynamics Lab

    Google Scholar 

  226. Koiter WT (1970) Comment on: The linear and non-linear equilibrium equations for thin elastic shells according to the Kirchhoff–Love hypotheses. Int J Mech Sci 12:663–664

    Google Scholar 

  227. Kholostova OV (2009) On the motions of a double pendulum with vibrating suspension point. Mech Sol 44:184–197

    Google Scholar 

  228. Krupa M, Poth W, Schagerl M, Steindl A, Steiner W, Troger H, Wiedermann G (2006) Modelling, dynamics and control of tethered satellite systems. Nonlinear Dynam 43:73–96

    MathSciNet  Google Scholar 

  229. Kuhlmann G (2003) Ein hierarchisches inhomogenes Volumenelement zur Berechnung dickwandiger Faserverbunde. Ph.D. Thesis, Shaker Verlag, Aachen, Germany

    Google Scholar 

  230. Yuri A. Kuznetsov YA (2004) Elements of applied bifurcation theory. Springer, New-York

    Google Scholar 

  231. Hartman P (1982) Ordinary differential equations. Birkhaüser, Boston

    Google Scholar 

  232. Iyengar NGR (1998) Structural stability of columns and plates. Wiley, New York

    Google Scholar 

  233. Lacarbonara W, Chin CM, Nayfeh, AH (1997) Two-to-one internal resonances in parametrically excited buckled beams. AIAA Paper No. 97–1081, 38th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics & Materials Conf, Kissimmee, FL

    Google Scholar 

  234. Lacarbonara W, Nayfeh AH, Kreider W (1998) Experimental validation of reduction methods for nonlinear vibrations of distributed-parameter systems: analysis of a buckled beam. Nonlinear Dynam 17:95–117

    Google Scholar 

  235. Lacarbonara W (1999) Direct treatment and discretizations of non-linear spatially continuous systems. J Sound Vib 221:849–866

    MathSciNet  Google Scholar 

  236. Lacarbonara W, Vestroni F (2002) Feasibility of a vibration absorber based on hysteresis. In: Proceedings of Third World Congress on Structural Control, Como, April 7–12, 2002

    Google Scholar 

  237. Lacarbonara W, Rega G, Nayfeh AH (2003) Resonant nonlinear normal modes. Part I: analytical treatment for structural one-dimensional systems. Int J Non Linear Mech 38:851–872

    Google Scholar 

  238. Lacarbonara W, Rega G (2003) Resonant nonlinear normal modes. Part II: activation/orthogonality conditions for shallow structural systems. Int J Non Linear Mech 38: 873–887

    Google Scholar 

  239. Lacarbonara W, Chin CM, Soper RR (2002) Open-loop nonlinear vibration control of shallow arches via perturbation approach. J Appl Mech 69:325–334

    MathSciNet  Google Scholar 

  240. Lacarbonara W, Vestroni F (2003) Nonclassical responses of oscillators with hysteresis. Nonlinear Dynam 32:235–258

    Google Scholar 

  241. Lacarbonara W, Bernardini D, Vestroni, F (2004) Nonlinear thermomechanical oscillations of shape-memory devices. Int J Solids Struct 41:1209–1234

    Google Scholar 

  242. Lacarbonara W, Paolone A, Yabuno, H (2004) Modeling of planar nonshallow prestressed beams towards asymptotic solutions. Mech Res Commun 31:301–310

    Google Scholar 

  243. Lacarbonara W, Camillacci R (2004) Nonlinear normal modes of structural systems via asymptotic approach. Int J Solids Struct 41:5565–5594

    Google Scholar 

  244. Lacarbonara W, Arafat HN, Nayfeh AH (2005) Nonlinear interactions in imperfect beams at veering. Int J Non Linear Mech 40:987–1003

    Google Scholar 

  245. Lacarbonara W, Paolone A, Vestroni F (2005) Galloping instabilities in geometrically nonlinear cables under steady wind forces, Paper. No. 20th ASME Biennial Conference on Mechanical Vibration and Noise, Long Beach, CA, Sept 25–28

    Google Scholar 

  246. Lacarbonara W, Yabuno H (2006) Refined models of elastic beams undergoing large in-plane motions: theory and experiment. Int J Solids Struct 43:5066–5084

    Google Scholar 

  247. Lacarbonara W, Yabuno H, Hayashi K (2007) Nonlinear cancellation of the parametric resonance in elastic beams: theory and experiment. Int J Solids Struct 44:2209–2224

    Google Scholar 

  248. Lacarbonara W, Antman SS (2007) Parametric resonances of nonlinearly viscoelastic rings subject to a pulsating pressure. 21st ASME DETC Conf, No. DETC2007-35245, Las Vegas, USA

    Google Scholar 

  249. Lacarbonara W, Paolone A, Vestroni F (2007) Elastodynamics of nonshallow suspended cables: linear modal properties. J Vib Acoust 129:425–433

    Google Scholar 

  250. Lacarbonara W, Paolone A, Vestroni F (2007) Nonlinear modal properties of nonshallow cables. Int J Non Linear Mech 42:542–554

    Google Scholar 

  251. Lacarbonara W, Colone V (2007) Dynamic response of arch bridges traversed by high-speed trains. J Sound Vib 304:72–90

    Google Scholar 

  252. Lacarbonara W, Paolone A (2007) Solution strategies to Saint–Venant problem. J Comput Appl Math 206:473–497

    MathSciNet  Google Scholar 

  253. Lacarbonara W (2008) Buckling and post-buckling of non-uniform non-linearly elastic rods. Int J Mech Sci 50:1316–1325

    Google Scholar 

  254. Lacarbonara W, Pacitti A (2008) Nonlinear modeling of cables with flexural stiffness. Math Probl Eng, Article ID 370767, 21 pages, 2008. doi:10.1155/2008/370767

    Google Scholar 

  255. Lacarbonara W, Antman, SS (2012) Parametric instabilities of the radial motions of nonlinearly viscoelastic shells subject to pulsating pressures. Int J Non Linear Mech  47:461–472

    Google Scholar 

  256. Lacarbonara W, Ballerini S (2009) Vibration mitigation of a guyed mast via tuned pendulum dampers. Struct Eng Mech 32

    Google Scholar 

  257. Lacarbonara W, Arena A (2011) Flutter of an arch bridge via a fully nonlinear continuum formulation. J Aerospace Eng 24:112–123

    Google Scholar 

  258. Lacarbonara W, Pasquali M (2011) A geometrically exact formulation for thin multi-layered laminated composite plates. Compos Struct 93:1649–1663

    Google Scholar 

  259. Lacarbonara W, Antman SS (2007) Parametric resonances of nonlinearly viscoelastic rings subject to a pulsating pressure. Paper DETC 2007–35245, 21th ASME Biennial Conf. on Mechanical Vibration and Noise

    Google Scholar 

  260. Lacarbonara W, Antman SS (2008) What is parametric resonance in structural dynamics. Proceedings of the 6th Euromech Nonlinear Dynamics Conf., St. Petersburg, Russia

    Google Scholar 

  261. Lacarbonara W, Arvin, H, Bakhtiari-Nejad, F (2012) A geometrically exact approach to the overall dynamics of elastic rotating blades – part 1: linear modal properties. Nonlinear Dynam, 70:659–675

    Google Scholar 

  262. Lacarbonara W, Cetraro M (2011) Flutter control of a lifting surface via visco-hysteretic vibration absorbers. Int J Aeronautical Space Sci 12(4):331–345

    Google Scholar 

  263. Lagoudas DC (ed) (2010) Shape memory alloys: modeling and engineering applications. Springer, New York

    Google Scholar 

  264. Lanzo AD, Garcea G, Casciaro R (1995) Koiter post–buckling analysis of elastic plates. Int J Numer Methods Eng 38:2325–2345

    Google Scholar 

  265. Lau DT, Cheung MS, Cheng SH (2000) 3D flutter analysis of bridges by spline finite-strip method. J Struct Eng-ASCE 126:1246–1254

    Google Scholar 

  266. Lazer AC, McKenna PJ (1990) Large-amplitude periodic oscillations in suspension bridges: some new connection with nonlinear analysis. SIAM Rev 32:537–578

    MathSciNet  Google Scholar 

  267. Lee HK, Simunovic S (2001) A damage constitutive model of progressive debonding in aligned discontinuous fiber composites. Int J Solids Struct 38:875–895

    Google Scholar 

  268. Lee CL, Perkins NC (1995) Three-dimensional oscillations of suspended cables involving simultaneous internal resonances. Nonlinear Dynam 8:45–63

    MathSciNet  Google Scholar 

  269. Lee J (1997) Thermally induced buckling of laminated composites by a layer-wise theory. Compos Struct 65:917–922

    Google Scholar 

  270. Lee SY, Kuo YH (1991) Elastic stability of non-uniform columns. J Sound Vib 148:11–24

    Google Scholar 

  271. Leipholz H (1970) Stability theory. Academic, New York

    Google Scholar 

  272. Leissa AW (1969) Vibration of plates. NASA SP-160

    Google Scholar 

  273. Li H, Balachandran B (2006) Buckling and free oscillations of composite microresonators. J Microelectromech Syst 15:42–51

    Google Scholar 

  274. Li H, Balachandran B (2006) Buckling and free oscillations of composite microresonators. J Microelectromech S 15:42–51

    Google Scholar 

  275. Li H, Preidikman S, Balachandran B, Mote Jr. CD (2006) Nonlinear free and forced oscillations of piezoelectric microresonators. J Micromech Microeng 16:356–367

    Google Scholar 

  276. Li QS, Cao H, Li G (1994) Stability analysis of bars with multi-segments of varying cross-section. Comput Struct 53:1085–1089

    Google Scholar 

  277. Li QS, Cao H, Li G (1995) Stability analysis of bars with varying cross-section. Int J Solids Struct 32:3217–3228

    Google Scholar 

  278. Li QS, Cao H, Li G (1996) Static and dynamic analysis of straight bars with variable cross-section. Comput Struct 59:1185–1191

    Google Scholar 

  279. Li QS (2000) Buckling analysis of multi-step non-uniform beams. Adv Struct Engng 3: 139–144

    Google Scholar 

  280. Love AEH (1906) The mathematical theory of elasticity. Cambridge University Press, Cambridge

    Google Scholar 

  281. Lubarda VA (2004) Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics. Appl Mech Rev 57:95–108

    Google Scholar 

  282. Luongo A, Rega G, Vestroni F (1984) Planar non-linear free vibrations of an elastic cable. Int J Non Linear Mech 19:39–52

    Google Scholar 

  283. Luongo A, Rega G, Vestroni F (1986) On nonlinear dynamics of planar shear undeformable beams. J Appl Mech 108:619–624

    Google Scholar 

  284. Luongo A, Rega G, Vestroni F (1984) Planar non-linear free vibrations of an elastic cable. Int J Non Linear Mech 19:39–52

    Google Scholar 

  285. Luongo A, Paolone A, Piccardo G (1998) Postcritical behavior of cables undergoing two simultaneous galloping modes. Meccanica 33:229–242

    Google Scholar 

  286. Luongo A, Vestroni F (1994) Nonlinear free periodic oscillations of a tethered satellite system. J Sound Vib 175(3):299–315

    Google Scholar 

  287. Luongo A (1997) Appunti di Meccanica delle Strutture. Lecture Notes (in Italian), L’Aquila

    Google Scholar 

  288. Luongo A, Paolone A (2005) Scienza delle costruzioni, vol. 2: Il problema di de Saint Venant (in Italian). CEA, Milan

    Google Scholar 

  289. Ma C, Huang C (2004) Experimental whole-field interferometry for transverse vibration of plates. J Sound Vib 271:493–506

    Google Scholar 

  290. McComber P, Paradis A (1998) A cable galloping model for thin ice accretions. Atmos Res 46:13–25

    Google Scholar 

  291. McConnell KG, Chang CN (1986) A study of the axial-torsional coupling effect on a sagged transmission line. Exp Mech 26:324–328

    Google Scholar 

  292. Magnus W, Winkler DT (1966) Hill’s equation. Wiley-Interscience, New York

    Google Scholar 

  293. Mailybaev AA, Yabuno H, Kaneko H (2004) Optimal shapes of parametrically excited beams. Struct Multidisciplinary Optim 27(6):435–445

    MathSciNet  Google Scholar 

  294. Makhecha DP, Ganapathi M, Patel BP (2001) Dynamic analysis of laminated composite plates subjected to thermal/mechanical loads using an accurate theory. Compos Struct 51:221–236

    Google Scholar 

  295. Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Ciffs

    Google Scholar 

  296. Mannini C, Bartoli G, Borri C, Borsani A, Ferrucci M, Procino L (2009) Recent developments in measurement and identification of bridge deck flutter derivatives. U. Peil, ed., WtG Berichte Nr. 11 - Windingenieurwesen in Forschung und Praxis, Dreiländertagung D-A-CH, Braunschweig, Germany, 1–15

    Google Scholar 

  297. Marzocca P, Librescu L, Silva WA (2002) Flutter, postflutter, and control of a supersonic wing section. J Guid Contr Dynam 25:962–970

    Google Scholar 

  298. Marzouk OA, Nayfeh AH, Arafat HN, Akhtar I (2007) Modeling steady-state and transient forces on a cylinder. J Vib Control 13:1065–1091

    Google Scholar 

  299. Marzouk OA, Nayfeh AH (2009) Reduction of the loads on a cylinder undergoing harmonic in-line motion. Phys Fluids 21:083103-13

    Google Scholar 

  300. Marzouk OA, Nayfeh AH (2010) Characterization of the flow over a cylinder moving harmonically in the cross-flow direction. Int J Non Linear Mech 45:821–833

    Google Scholar 

  301. Masri SF (1975) Forced vibration of the damped bilinear hysteretic oscillator. J Acoust Soc Am 57:106–111

    Google Scholar 

  302. Matsunaga H (1994) Free vibration and stability of thick elastic plates subjected to in-plane forces. Int J Solids Struct 31(22):3113–3124

    Google Scholar 

  303. Matsunaga H (1997) Buckling instabilities of thick elastic plates subjected to in-plane stresses. Compos Struct 62(1):205–214

    MathSciNet  Google Scholar 

  304. Matsunaga H (2000) Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Compos Struct 48(4):231–244

    Google Scholar 

  305. Matsunaga H (2001) Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses. Int J Mech Sci 43:1925–1944

    Google Scholar 

  306. Matsunaga, H (2002) Vibration of cross-ply laminated composite plates subjected to initial in-plane stresses. Thin Wall Struct 40:557–571

    Google Scholar 

  307. Matsunaga H (2006) Thermal buckling of angle-ply laminated composite and sandwich plates according to a global higher-order deformation theory. Compos Struct 72:177–192

    Google Scholar 

  308. Matsunaga H (2007) Free vibration and stability of angle-ply laminated composite and sandwich plates under thermal loading. Compos Struct 77:249–262

    Google Scholar 

  309. MATHEMATICA (2007) Wolfram Research Inc. Urbana Champaign, IL

    Google Scholar 

  310. McKenna PJ, Walter W (1987) Nonlinear oscillations in a suspension bridge. Arch Rat Mech Anal 98:167–177

    MathSciNet  Google Scholar 

  311. McLachlan NW (1962) Theory and application of mathieu functions. Dover, New York

    Google Scholar 

  312. Meirovitch L (1970) Methods of analytical dynamics. Mc-Graw-Hill, New York

    Google Scholar 

  313. Melan J (1853) Theorie der eisernen bogenbrücken und der hangebrücken. Leipzig. (1913) Theory of arches and suspension bridges. Translated by D B Steinman, Myron C. Clark, Chicago

    Google Scholar 

  314. Melde W (1859) Über Erregung stehender Wellen eines fadenförmigen Körpers. Ann Phys Chem 109:193–215

    Google Scholar 

  315. Mathieu E (1868) Mémoire sur le movement vibratoire d’une membrane de forme elliptique. J Math Pures Appl 137–203

    Google Scholar 

  316. Mettler E (1962) Dynamic buckling. In: Flugge (ed) Handbook of engineering mechanics. McGraw-Hill, New York

    Google Scholar 

  317. Mikhlin SG (1964) Variational methods in mathematical physics. Pergamon, Oxford

    Google Scholar 

  318. Miles J (1985) Parametric excitation of an internally resonant double pendulum. Z Angew Math Phys 36:337–345

    MathSciNet  Google Scholar 

  319. Mindlin RD (1951) Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. NASA Technical Paper 1903, Hampton, VA

    Google Scholar 

  320. Mindlin RD (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 38:31–38

    Google Scholar 

  321. Mishra SS, Kumar K, Krishna P (2008) Multimode flutter of long-span cable-stayed bridge based on 18 experimental aeroelastic derivatives. J Wind Eng Ind Aerodyn 96:83–102

    Google Scholar 

  322. Mittelmann HD, Roose D (eds) (1990) Continuation techniques and bifurcation problems, vol 92. ISNM, Birkhäuser

    Google Scholar 

  323. Miyata T (2003) Historical view of long-span bridge aerodynamics. J Wind Eng Ind Aerodyn 91:1393–1410

    Google Scholar 

  324. Mohr GA (1992) Finite elements for solids, fluids and optimization. Oxford University Press, Oxford

    Google Scholar 

  325. Moisseiff LS, Leinhard, F (1933) Suspension bridges under the action of lateral forces, with discussion. Trans Am Soc Civ Eng 98:1080–1141

    Google Scholar 

  326. Moon FC (2004) Chaotic vibrations. Wiley, New York

    Google Scholar 

  327. Morley A (1917) Critical loads for long tapering struts. Engineering 104:295

    Google Scholar 

  328. Murdock JA (1991) Perturbations. Wiley, New York

    Google Scholar 

  329. Naghdi PM (1972) The theory of shells and plates. In Truesdell S (ed) Flügges Encyclopedia of Physics, vol VI a/2. Springer, New York, pp 425–640

    Google Scholar 

  330. Nayfeh AH (1973) Perturbation methods. Wiley, New York

    Google Scholar 

  331. Nayfeh AH, Mook DT, Lobitz DW (1974) Numerical-perturbation method for the nonlinear analysis of structural vibrations. AIAA J 12:1222–1228

    Google Scholar 

  332. Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New York

    Google Scholar 

  333. Nayfeh AH (1981) Introduction to perturbation techniques. Wiley, New York

    Google Scholar 

  334. Nayfeh AH, Pai PF (1989) Non-linear non-planar parametric responses of an inextensional beam. Int J Non Lin Mech 24(2):139–158

    Google Scholar 

  335. Nayfeh AH, Balachandran B (1995) Applied nonlinear dynamics. Wiley-Interscience, New York

    Google Scholar 

  336. Nayfeh AH, Lacarbonara W (1998) On the discretization of spatially continuous systems with quadratic and cubic nonlinearities. JSME Int J C-Dyn Con 41:510–531

    Google Scholar 

  337. Nayfeh AH (2000) Nonlinear interactions. analytical, computational, and experimental methods. Wiley-Interscience, New York

    Google Scholar 

  338. Nayfeh AH, Arafat H, Chin, CM, Lacarbonara W (2002) Multimode interactions in suspended cables. J Vib Control 8:337–387

    MathSciNet  Google Scholar 

  339. Nayfeh AH, Pai PF (2004) Linear and nonlinear structural mechanics. Wiley, New York

    Google Scholar 

  340. Nayfeh AH, Arafat HN (2005) Nonlinear dynamics of closed spherical shells, Paper. No. DETC2005-85409. In: Proceedings of the 20th ASME Biennial Conference on Mechanical Vibration and Noise, Long Beach, CA, Sept. 25–28

    Google Scholar 

  341. Navier CLMHL (1823) Rapport et mémoire sur le ponts suspendus. Paris, Imprimerie Royale

    Google Scholar 

  342. Nazmy AS (1997) Stability and load-carrying capacity of three-dimensional long-span steel arch bridges. Comput Struct 65(6):857–868

    Google Scholar 

  343. Nemat-Nasser S, Shatoff HD (1973) Numerical analysis of pre- and post-critical response of elastic continua at finite strains. Comput Struct 3:983–999

    Google Scholar 

  344. Ng L, Rand RH (2002) Bifurcations in a Mathieu equation with cubic nonlinearities. Chaos Soliton Fract 14:173–181

    MathSciNet  Google Scholar 

  345. Noda N, Hetnarski RB, Tanigawa Y (2003) Thermal stresses, 2nd edn. Taylor & Francis, New York

    Google Scholar 

  346. Noor, AK, Burton, WS (1989) Assessment of shear deformation theories for multilayered composite plates. Appl Mech Rev 42(1):1–13

    Google Scholar 

  347. Nosier A, Kapania RK, Reddy JN (1993) Free vibration analysis of laminated plates using a layer-wise theory. AIAA J 31(12):2335–2346

    Google Scholar 

  348. Pagano NJ (1969) Exact solutions for composite laminates in cylindrical bending. J Compos Mater 3:398–411

    Google Scholar 

  349. Pagano NJ (1970) Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater 4:20–34

    Google Scholar 

  350. Pagano NJ, Hatfield SJ (1972) Elastic behavior of multi-layered bidirectional composites. AIAA J 10:931–933

    Google Scholar 

  351. Paolone A, Vasta M, Luongo A (2006) Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I. Non-linear model and stability analysis. Int J Non Linear Mech 41:586–594

    Google Scholar 

  352. Pandya BN, Kant T (1988) Flexural analysis of laminated composites using refined higher-order C 0 plate bending elements. Comput Method Appl M 66:173–198

    Google Scholar 

  353. Pasca M, Pignataro M, Luongo A (1991) Three-dimensional vibrations of tethered satellite system. J Contr Guid 14(2):312–320

    Google Scholar 

  354. Pasca M, Vestroni F, Luongo A (1996) Stability and bifurcations of transversal motions of an orbiting string with a longitudinal force. Appl Math Mech ZAMM 76(4):337–340

    Google Scholar 

  355. Pasquali M (2010) Geometrically exact models of thin plates towards nonlinear dynamic system identification via higher-order spectral approach. MS Thesis. Sapienza University of Rome

    Google Scholar 

  356. Pasquali M, Lacarbonara W, Marzocca P (2011) System identification of plates using higher-order spectra: numerical and experimental investigations. Paper No. 945175, 52nd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics & Materials Conference, Denver, CO, April 4–7

    Google Scholar 

  357. Pasquali M, Lacarbonara W, Marzocca P (2011) Advanced system identification of plates using a higher-order spectral approach: theory and experiment. Paper no. DETC2011-47975, 2011 ASME DETC, Washington DC, August 28–31

    Google Scholar 

  358. Patil MJ, Hodges D (2004) On the importance of aerodynamic and structural geometrical nonlinearities in aeroelastic behavior of high-aspect-ratio wings. J Fluid Struct 19:905–915

    Google Scholar 

  359. Pellicano F, Amabili M (2006) Dynamic instability and chaos of empty and fluid-filled circular cylindrical shells under periodic axial loads. J Sound Vib 293:227–252

    Google Scholar 

  360. Petrangeli MP and Associates (2008) Ponte della Musica: Verifica delle strutture in acciaio dell’arco, dell’impalcato e della soletta. codifica E281004300SXA, Rome

    Google Scholar 

  361. Petrolito J (1998) Approximate solutions of differential equations using Galerkin’s method and weighted residuals. Int J Mech Eng Educ 28:14–26

    Google Scholar 

  362. Picone M (1928) Sul metodo delle minime potenze ponderate e sul metodo di Ritz per il calcolo approssimato nei problemi della fisica-matematica. Rend Circ Mat Palermo 52: 225–253

    Google Scholar 

  363. Pignataro M, Rizzi N, Luongo A (1990) Bifurcation, stability and postcritical behaviour of elastic structures. Elsevier Science Publishers, Amsterdam

    Google Scholar 

  364. Pilipchuk VN, Ibrahim RA (1999) Non-linear modal interactions in shallow suspended cables. J Sound Vib 227:1–28

    Google Scholar 

  365. Pfeil MS, Batista RC (1995) Aerodynamic stability analysis of cable-stayed bridges. J Struct Eng-ASCE 121:1748–1788

    Google Scholar 

  366. Podio-Guidugli P, Virga EG (1987) Transversely isotropic elasticity tensors. Proc R Soc London, Ser A 411:85–93

    MathSciNet  Google Scholar 

  367. Podio-Guidugli P (1989) An exact derivation of the thin plate equation. J Elast 22:121–133

    MathSciNet  Google Scholar 

  368. Preidikman S, Mook DT (1997) A new method for actively suppressing flutter of suspension bridges J Wind Eng Ind Aerodyn 69/71:955–974

    Google Scholar 

  369. Preidikman S, Mook DT (1998) On the development of a passive-damping system for wind-excited oscillations of long span bridges J Wind Eng Ind Aerodyn 77/78:443–456

    Google Scholar 

  370. Proceedings (1995) of the 4th International Conference on Tethers in Space, April 10–14, Washington DC

    Google Scholar 

  371. Pugsley A (1968) The theory of suspension bridges, 2d edn. Edward Arnold, London

    Google Scholar 

  372. Pugno N, Schwarzbart M, Steindl A, Troger H (2009) On the stability of the track of the space elevator. Acta Astronautica 64:524–537

    Google Scholar 

  373. Quarteroni A, Sacco, R, Saleri, F (2007) Numerical mathematics. Springer, Berlin

    Google Scholar 

  374. Ramania DV, Keitha WL, Rand RH (2004) Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation. Int J Non Lin Mech 39:491–502

    Google Scholar 

  375. Rand RH (1996) Dynamics of a nonlinear parametrically-excited PDE: 2-term truncation. Mech Res Commun 23:283–289

    MathSciNet  Google Scholar 

  376. Rand RH, Armbruster D (1987) Perturbation methods, bifurcation theory, and computer algebra. Springer, New York

    Google Scholar 

  377. Reddy JN (1984) A simple higher-order theory for laminated composite plates. Trans ASME J Appl Mech 51:745–752

    Google Scholar 

  378. Reddy JN (2004) Mechanics of laminated composite plates and shells, 2nd edn. CRC Press, Boca Raton, FL

    Google Scholar 

  379. Rega G, Lacarbonara W, Nayfeh AH (2000) Reduction methods for nonlinear vibrations of spatially continuous systems with initial curvature. Solid mechanics and its applications vol 77. Kluwer, Dordrecht, p 235

    Google Scholar 

  380. Rega G (2004) Nonlinear vibrations of suspended cables - Part I: Modeling and analysis. Part II: Deterministic phenomena. Appl Mech Rev 57:443–479

    Google Scholar 

  381. Rega G, Lacarbonara W, Nayfeh AH, Chin CM (1999) Multiple resonances in suspended cables: direct versus reduced-order models. Int J Non Linear Mech 34:901–924

    Google Scholar 

  382. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12:69–77

    MathSciNet  Google Scholar 

  383. Ricks E (1979) An incremental approach to the solution of snapping and buckling problems. Int J Solids Struct 15:529–551

    Google Scholar 

  384. Rodrigues O (1840) Des lois géometriques qui régissent les déplacements d’un systeme solide dans l’espace. J de Math (Liouville) 5:380–440

    Google Scholar 

  385. Sabzevari A, Scanlan RH (1968) Aerodynamic instability of suspension bridges. J Eng Mech-ASCE 94:489–517

    Google Scholar 

  386. Saito H, Sato K, Otomi K (1976) Nonlinear forced vibrations of a beam carrying concentrated mass under gravity. J Sound Vib 46(4):515–525

    Google Scholar 

  387. Salinger AG, Burroughs EA, Pawlowski RP, Phipps ET, Romero LA (2005) Bifurcation tracking algorithms and software for large scale applications. J Bifur Chaos Appl Sci Engrg 15(3):1015–1032

    MathSciNet  Google Scholar 

  388. Salvatori L, Borri C (2007) Frequency- and time-domain methods for the numerical modeling of full-bridge aeroelasticity. Comput Struct 85:675–687

    Google Scholar 

  389. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems, 2nd edn. Springer, New York

    Google Scholar 

  390. Sanjuán MAF (1998) Using nonharmonic forcing to switch the periodicity in nonlinear systems. Phys Rev E 58:4377–4382

    MathSciNet  Google Scholar 

  391. Sarkar PP, Caracoglia L, Haan FL, Sato H, Murakoshid J (2009) Comparative and sensitivity study of flutter derivatives of selected bridge deck sections. Part 1: Analysis of inter-laboratory experimental data. Eng Struct 31:158–169

    Google Scholar 

  392. Sartorelli JC, Lacarbonara W (2012) Parametric resonances in a base-excited double pendulum, Nonlinear Dynam 69:1679–1692

    MathSciNet  Google Scholar 

  393. Scanlan RH (1987) Interpreting aeroelastic models of cable-stayed bridges. J Eng Mech-ASCE 113:555–575

    Google Scholar 

  394. Sears A, Batra RC (2004) Macroscopic properties of carbon nanotubes from molecular-mechanics simulations. Phys Rev B 69:235406-10

    Google Scholar 

  395. Seyranian AP, Yabuno H, Tsumoto K (2005) Instability and periodic motion of a physical pendulum with a vibrating suspension point (theoretical and experimental approach). Dokl Phys 50(9):467–472

    MathSciNet  Google Scholar 

  396. Somnay R, Ibrahim RA, Banasik RC (2006) Nonlinear dynamics of a sliding beam on two isolators. J Vib Control 12:685–712

    Google Scholar 

  397. Strømmen E, Hjoorth-Hansen E (1995) The buffeting wind loading of structural members at an arbitrary attitude in the flow. J Wind Eng Ind Aerodyn 56:267–290

    Google Scholar 

  398. Scanlan RH, Tomko JJ (1971) Airfoil and bridge deck flutter derivates. J Eng Mech-ASCE 97:1717–1737

    Google Scholar 

  399. Selberg A (1961) Oscillation and aerodynamic stability of suspension bridges. Acta Polytechnica Scandinavia 308

    Google Scholar 

  400. Seydel R (1994) Practical bifurcation and stability analysis. From equilibrium to chaos, 2nd edn. Springer, New York

    Google Scholar 

  401. Seyranian AP (2001) Regions of resonance for Hill’s equation with damping. Dokl Ross Akad Nauk 376:44–47

    Google Scholar 

  402. Shilov GE, Gurevich BL (1977) Integral, measure and derivative: a unified approach. In: Silverman RA (ed) Dover books on advanced mathematics. Dover Publications, New York

    Google Scholar 

  403. Shufrin I, Rabinovitch O, Eisenberger M (2009) Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach. Int J Solids Struct 46:2075–2092

    Google Scholar 

  404. Simo JC (1985) A finite strain beam formulation. The three-dimensional dynamical problem. Part I. Comput Method Appl M 49:55–70

    Google Scholar 

  405. Simo JC, Marsden JE, Krishnaprasad PS (1988) The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods and plates. Arch Ration Mech Anal 104:125–183

    MathSciNet  Google Scholar 

  406. Simiu E, Scanlan R (1996) Wind effects on structures. Fundamentals and applications to design, 3rd edn. Wiley-Interscience Publication, New York

    Google Scholar 

  407. Skeldon AC (1994) Dynamics of a parametrically excited double pendulum. Phys D 75: 541–558

    MathSciNet  Google Scholar 

  408. Skop RA, Griffin OM (1973) A model for the vortex-excited resonant response of bluff cylinders. J Sound Vib 27:225–233

    Google Scholar 

  409. Smith HJ, Blackburn JA, Grnbech-Jensen N (1992) Stability and Hopf bifurcations in an inverted pendulum. Am J Phys 60:903–908

    Google Scholar 

  410. Stachowiak T, Okada T (2006) A numerical analysis of chaos in the double pendulum. Chaos Soliton Fract 29:417–422

    Google Scholar 

  411. Stephenson A (1906) On a class of forced oscillations. Q J Math 37:353–360

    Google Scholar 

  412. Stephenson A (1908) On a new type of dynamic stability. Mem Proc Manch Lit Phil Soc 52: 1–10

    Google Scholar 

  413. Steinman DB (1934) A generalized deflection theory for suspension bridges. Trans Am Soc Civ Eng March:1133–1170

    Google Scholar 

  414. Steinman DB (1946) Design of bridges against wind: V. Criteria for assuring aerodynamic stability. Civil Eng ASCE February:68–76

    Google Scholar 

  415. Stevens KK (1966) On linear ordinary differential equations with periodic coefficients. SIAM J Appl Math 14:782–795

    MathSciNet  Google Scholar 

  416. Stojanovic R (1972) Nonlinear thermoelasticity. Springer, Wien

    Google Scholar 

  417. Structural Engineers Association of California (1995) Performance-based seismic engineering of buildings. Vision 2000 Report. SEAOC Publications, Sacramento

    Google Scholar 

  418. Strutt JWS (Lord Rayleigh) (1883) On maintained vibrations. Phil Mag 15:229–235

    Google Scholar 

  419. Strutt JWS (Lord Rayleigh) (1887) On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure. Phil Mag 24:145–159

    Google Scholar 

  420. Struble RA (1962) Nonlinear differential equations. McGraw-Hill, New York

    Google Scholar 

  421. Sugimoto N (1981) Nonlinear theory for flexural motions of this elastic plate. J Appl Mech 48:377–382

    Google Scholar 

  422. Synge JL, Chien WZ (1941) The intrinsic theory of elastic shells and plates. Theodore von Karman Anniversary Volume, California Institute of Technology, 103–120

    Google Scholar 

  423. Tang DM, Dowell EH (2001) Experimental and theoretical study on aeroelastic response of high-aspect-ratio wings. AIAA J 39(8):1430–1441

    Google Scholar 

  424. Tang DM, Dowell EH (2002) Experimental and theoretical study of gust response for high-aspect-ratio wing. AIAA J 40(3):419–429

    Google Scholar 

  425. Tang DM, Dowell EH (2004) Effects of geometric structural nonlinearity on flutter and limit cycle oscillations of high-aspect-ratio wings. J Fluid Struct 19:291–306

    Google Scholar 

  426. Task Committee on Cable-Suspended Structures (1977) Commentary on the tentative recommendations for cable-stayed bridge structures. J Struct Div Proc ASCE 103:941–959

    Google Scholar 

  427. Theodorsen T (1931) On the theory of wing section with particular reference to the lift distribution. JNACA REPORT No. 383

    Google Scholar 

  428. Theodorsen T (1931) Theory of wing section of arbitrary shape. JNACA REPORT No. 411

    Google Scholar 

  429. Theodorsen T (1935) General theory of aerodynamic instability and the mechanism of flutter. JNACA REPORT No. 496

    Google Scholar 

  430. Thompson JMT, Walker AC (1969) A general theory for the branching analysis of discrete structural systems. Int J Solids Struct 5:281–288

    Google Scholar 

  431. Thompson JMT (1989) Chaotic phenomena triggering the escape from a potential well. Proc R Soc London, Ser A 421:195–225

    Google Scholar 

  432. Timoshenko SP (1908) Buckling of bars of variable cross section. Bulletin of the Polytechnic Institute, Kiev, USSR

    Google Scholar 

  433. Timoshenko SP, Gere JM (1961) Theory of elastic stability. McGraw-Hill, New York

    Google Scholar 

  434. Timoshenko SP, Young DH (1965) Theory of structures, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  435. Troger H, Steindl A (1991) Nonlinear stability and bifurcation theory. Springer, Wien

    Google Scholar 

  436. Triantafyllou MS, Howell CT (1994) Dynamic response of cables under negative tension: an ill-posed problem. J Sound Vib 173:433–447

    Google Scholar 

  437. Truesdell C (1954) A new chapter in the theory of the elastica. In Proc. 1st Midwestern Conf. Solid Mech. 52–54

    Google Scholar 

  438. Trusdell C, Toupin RA (1960) The classical field theories. In: Flugge S (ed) Encyclopedia of physics, vol III/1. Springer, Berlin, pp 226–793

    Google Scholar 

  439. Truesdell C, Noll N (1965) The nonlinear field theories of mechanics. In: Flügge S (ed) Handbuch der Physik, Band vol III/3. Springer, Berlin

    Google Scholar 

  440. Tuc̆ková M, Tuc̆ek J,, Tuc̆ek P, Kubác̆ek L (2011) Experimental design of hysteresis loop measurements of nanosized ε-Fe2O3/SiO3 A statistically-based approach towards precise evaluation of ε-Fe2O3/SiO3 hysteresis loop parameters. In: NanoCon 2011, Sept 21–23, Brno, Czech Republic

    Google Scholar 

  441. Ukeguchi N, Sakata H, Nishitani H (1966) An investigation of aeroelastic instability of suspension bridges. Int. Symp. on Suspension Bridges, Lisbon, Paper No. 11, 79–100

    Google Scholar 

  442. UNI EN 1991-1-7: Part 1-7: Azioni in generale - Azioni eccezionali (2006)

    Google Scholar 

  443. UNI EN 1991-2: Part 2: Carichi da traffico sui ponti (2005)

    Google Scholar 

  444. van der Pol B (1927) On relaxation-oscillations. London Edinburgh Dublin Phil Mag J Sci 2(7):978–992

    Google Scholar 

  445. Vaziri HH, Xie J (1992) Buckling of columns under variably distributed asial loads. Comput Struct 45:505–509

    Google Scholar 

  446. Verhulst F (1990) Nonlinear differential equations and dynamical systems. Springer, Berlin

    Google Scholar 

  447. Vestroni F, Luongo A, Pasca M (1995) Stability and control of transversal oscillations of a tethered satellite system. Appl Math Comp 70(2):343–360

    MathSciNet  Google Scholar 

  448. Vestroni F, Lacarbonara W, Carpineto N (2011) Hysteretic tuned-mass damper device (TMD) for passive control of mechanical vibrations, Italian Patent

    Google Scholar 

  449. Vijayaraghavan A, Evan-Iwanowski RM (1967) Parametric instability of circular cylindrical shells. J Appl Mech 985–990

    Google Scholar 

  450. Villaggio P (1997) Mathematical models for elastic structures. Cambridge University Press, Cambridge

    Google Scholar 

  451. Visintin A (1994) Differential models of hysteresis. Springer, Berlin

    Google Scholar 

  452. Vlasov VZ (1959) Thin-walled elastic bars (in Russian), 2nd edn. Fizmatgiz, Moscow

    Google Scholar 

  453. von Kármán T (1910) Festigkeitsproblem im Maschinenbau. Encyk D Math Wiss IV:311–385

    Google Scholar 

  454. Waisman H, Montoya A, Betti R, Noyan IC (2011) Load transfer and recovery length in parallel wires of suspension bridge cables. J Eng Mech-ASCE 137:227–237

    Google Scholar 

  455. Walker AC (1969) A method of solution for nonlinear simultaneous algebraic equations. Int J Numer Methods Eng 1:197–180

    Google Scholar 

  456. Walker AC (1969) A nonlinear finite ekment analysis of shallow circular arches. Int J Solids Struct 5:97–107

    Google Scholar 

  457. Wang CM, Wang CY, Reddy JN (2005) Exact solutions for buckling of structural members. CRC Press, Boca Raton

    Google Scholar 

  458. Weiyi C (1999) Derivation of the general form of elasticity tensor of the transverse isotropic material by tensor derivate. Appl Math Mech 20(3):309–314

    MathSciNet  Google Scholar 

  459. Wen RK, Medallah K (1987) Elastic stability of deck-type arch bridges. J Struct Eng ASCE 113(4):757–768

    Google Scholar 

  460. Wenbin YuW, Kimb JS, Hodges DH, Chod M (2008) A critical evaluation of two Reissner–Mindlin type models for composite laminated plates. Aerospace Sci Technol 12(5):408–417

    Google Scholar 

  461. Whitney JM, Pagano NJ (1970) Shear deformation in heterogeneous anisotropic plates. J Appl Mech 37:1031–1036

    Google Scholar 

  462. Wilcox B, Dankowicz H (2009) Design of limit-switch sensors based on discontinuity-induced nonlinearities. In: Proceedings of IDETC/CIE 2009, San Diego, CA

    Google Scholar 

  463. Wilcox B, Dankowicz H, Lacarbonara W (2009) Response of electrostatically actuated flexible MEMS structures to the onset of low-velocity contact. In: Proceedings of IDETC/CIE 2009, San Diego, CA

    Google Scholar 

  464. Wu Q, Takahashi K, Nakamura S (2003) The effect of cable loosening on seismic response of a prestressed concrete cable-stayed bridge. J Sound Vib 268:71–84

    Google Scholar 

  465. Wu W, Takahashi K, Nakamura S (2003) Non-linear vibrations of cables considering loosening. J Sound Vib 261:385–402

    Google Scholar 

  466. Wu Q, Takahashi K, Nakamura S (2004) Non-linear response of cables subject to periodic support excitation considering cable loosening. J Sound Vib 271:453–463

    Google Scholar 

  467. Wu Q, Takahashi K, Nakamura S (2007) Influence of cable loosening on nonlinear parametric vibrations of inclined cables. Struct Eng Mech 25

    Google Scholar 

  468. Yabuno H (1994) Nonlinear stability analysis for summed-type combination resonance under parametrical excitation (application of center manifold theory and Grobner basis with computer algebra). Nippon Kikai Gakkai Ronbunshu C Hen/Trans Jpn Soc Mech Eng C 60(572): 1151–1158

    Google Scholar 

  469. Yabuno H (1996) Buckling of a beam subjected to electromagnetic force and its stabilization by controlling the perturbation of the bifurcation. Nonlinear Dynam 10(3):271–285

    Google Scholar 

  470. Yabuno H, Ide Y, Aoshima N (1998) Nonlinear analysis of a parametrically excited cantilever beam: (Effect of the tip mass on stationary response). JSME International Journal Series C: Dynamics, Control, Robotics, Design and Manufacturing 41(3):555–562

    Google Scholar 

  471. Yabuno H, Nayfeh AH (2001) Nonlinear normal modes of a parametrically excited cantilever beam. Nonlin Dyn 25:65–77

    MathSciNet  Google Scholar 

  472. Yabuno H, Saigusa S, Aoshima N (2001) Stabilization of the parametric resonance of a cantilever beam by bifurcation control with a piezoelectric actuator. Nonlinear Dynam 26(2):143–161

    Google Scholar 

  473. Yabuno H, Okhuma M, Lacarbonara W (2003) An experimental investigation of the parametric resonance in a buckled beam, Paper VIB-48615, 19th ASME Biennial Conf. on Mechanical Vibration and Noise

    Google Scholar 

  474. Yabuno H, Kanda R, Lacarbonara W, Aoshima N (2004) Nonlinear active cancellation of the parametric resonance in a magnetically levitated body. J Dyn Syst Meas Contr Tran ASME 126(3):433–442

    Google Scholar 

  475. Yabuno H, Murakami T, Kawazoe J, Aoshima N (2004) Suppression of parametric resonance in cantilever beam with a pendulum (Effect of static friction at the supporting point of the pendulum). J Vib Acoust 126(1):149–162

    Google Scholar 

  476. Yakubovich VA, Starzhinskii VM (1975) Linear differential equations with periodic coefficients, vol 2. Wiley, New York

    Google Scholar 

  477. Yu P, Desai YM, Shah AH, Popplewell N (1992) Three-degree-of-freedom model for galloping. Part I: Formulation. J Eng Mech-ASCE 119:2404–2424

    Google Scholar 

  478. Yu P, Desai YM, Popplewell N, Shah AH (1992) Three-degree-of-freedom model for galloping. Part II: Solutions. J Eng Mech-ASCE 119:2426–2446

    Google Scholar 

  479. Yu P, Bi Q (1998) Analysis of non-linear dynamics and bifurcations of a double pendulum. J Sound Vib 217:691–736

    MathSciNet  Google Scholar 

  480. Yu W (2005) Mathematical construction of a Reissner–Mindlin plate theory for composite laminates. Int J Solids Struct 42:6680–6699

    Google Scholar 

  481. Zavodney LD, Nayfeh AH (1989) The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment. Int J Non Lin Mech 24:105–125

    Google Scholar 

  482. Zhang X, Sun B, Peng W (2003) Study on flutter characteristics of cable-supported bridges. J Wind Eng Ind Aerodyn 91:841–854

    Google Scholar 

  483. Zhang X, Sun B (2004) Parametric study on the aerodynamic stability of a long-span suspension bridge. J Wind Eng Ind Aerodyn 92:431–439

    Google Scholar 

  484. Zhang X (2007) Influence of some factors on the aerodynamic behavior of long-span suspension bridges. J Wind Eng Ind Aerodyn 95:149–164

    Google Scholar 

  485. Zhen W, Wanji C (2007) Buckling analysis of angle-ply composite and sandwich plates by combination of geometric stiffness matrix. Comput Mech 39:839–848

    Google Scholar 

  486. Zhen W, Wanji C (2006) Free vibration of laminated composite and sandwich plates using global-local higher-order theory. J Sound Vib 298:333–349

    Google Scholar 

  487. Ziegler SW, Cartmell MP (2001) Using motorized tethers for payload orbital transfer. J Spacecraft Rockets 38:904–913

    Google Scholar 

  488. Zienkiewicz OC, Morgan K (1983) Finite elements and approximations. Wiley-Interscience, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Lacarbonara, W. (2013). The Nonlinear Theory of Beams. In: Nonlinear Structural Mechanics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1276-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4419-1276-3_5

  • Published:

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-1275-6

  • Online ISBN: 978-1-4419-1276-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics