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Geometrically Exact Equations for Beams

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Encyclopedia of Continuum Mechanics

Synonyms

Bars; Beams; Canonical equations; Geometrically nonlinear equations; One-dimensional continua; Rods

Definitions

Beams are structures with one dimension that is much larger than the other two. They may be made from isotropic or anisotropic materials and may be homogeneous or inhomogeneous. They may be prismatic or initially curved and twisted. The term geometrically exact refers to there being no approximations in the geometry of either the reference line or the reference cross section. The geometrically exact equations of motion in their most fundamental form are first-order partial differential equations (PDEs), which may be written with or without displacement or rotational variables. Without these variables, they are referred to as intrinsic equations of motion and may be derived via vectorial or analytical mechanics. To be useful, they must be augmented by kinematical PDEs, often first order, defining generalized strains and generalized velocities in terms of partial...

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References

  • Antman SS, Warner WH (1966) Dynamical theory of hyperelastic rods. Arch Ration Mech Anal 23:135–162

    Google Scholar 

  • Bauchau OA, Kang NK (1993) A multibody formulation for helicopter structural dynamic analysis. J Am Helicopter Soc 38(2):3–14

    Google Scholar 

  • Berdichevsky VL (1976) Equations of the theory of anisotropic inhomogeneous rods. Dokl Akad Nauk SSR 228:558–561

    Google Scholar 

  • 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 

  • Borri M, Ghiringhelli GL, Merlini T (1992) Linear analysis of naturally curved and twisted anisotropic beams. Compos Eng 2(5–7):433–456

    Google Scholar 

  • Cesnik CES, Hodges DH (1997) VABS: a new concept for composite rotor blade cross-sectional modeling. J Am Helicopter Soc 42(1):27–38

    Google Scholar 

  • Chang CS, Hodges DH (2009a) Stability studies for curved beams. J Mech Mater Struct 4(7):1257–1270

    Google Scholar 

  • Chang CS, Hodges DH (2009b) Vibration characteristics of curved beams. J Mech Mater Struct 4(4):675–692

    Google Scholar 

  • Chang CS, Hodges DH, Patil MJ (2008) Flight dynamics of highly flexible aircraft. J Aircr 45(2):538–545

    Google Scholar 

  • Ghorashi M, Nitzsche F (2008) Steady state nonlinear dynamic response of a composite rotor blade using implicit integration of intrinsic equations of a beam. Int Rev Aerosp Eng 1:225–233

    Google Scholar 

  • Ghorashi M, Nitzsche F (2009) Nonlinear dynamic response of an accelerating composite rotor blade using perturbations. J Mech Mater Struct 4(4):693–718

    Google Scholar 

  • Hegemier GA, Nair S (1977) A nonlinear dynamical theory for heterogeneous, anisotropic, elastic rods. AIAA J 15(1):8–15

    Google Scholar 

  • Hodges DH (1990) A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. Int J Solids Struct 26(11):1253–1273

    Google Scholar 

  • Hodges DH (2003) Geometrically-exact, intrinsic theory for dynamics of curved and twisted anisotropic beams. AIAA J 41(6):1131–1137

    Google Scholar 

  • Hodges DH (2006) Nonlinear composite beam theory. AIAA, Reston

    Google Scholar 

  • Leamy M (2012) Intrinsic finite element modeling of nonlinear dynamic response in helical springs. J Comput Nonlinear Dyn 7:031,007

    Google Scholar 

  • Leamy MJ, Lee CY (2009) Dynamic response of intrinsic continua for use in biological and molecular modeling: explicit finite element formulation. Int J Numer Methods Eng 80:1171–1195

    Google Scholar 

  • Love AEH (1944) Mathematical theory of elasticity, 4th edn. Dover Publications, New York

    Google Scholar 

  • Masjedi PK, Ovesy HR (2015a) Chebyshev collocation method for static intrinsic equations of geometrically exact beams. Int J Solids Struct 54:183–191. https://doi.org/10.1016/j.ijsolstr.2014.10.016

  • Masjedi PK, Ovesy HR (2015b) Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations. Acta Mech 226:1689–1706. https://doi.org/10.1007/s00707-014-1281-3

  • Palacios R (2011) Nonlinear normal modes in an intrinsic theory of anisotropic beams. J Sound Vib 330(8): 1772–1792

    Google Scholar 

  • Patil MJ, Hodges DH (2006) Flight dynamics of highly flexible flying wings. J Aircr 43(6):1790–1799

    Google Scholar 

  • Patil MJ, Hodges DH (2011) Variable-order finite elements for nonlinear, intrinsic, mixed beam equations. J Mech Mater Struct 6(1):479–493

    Google Scholar 

  • Reissner E (1973) On one-dimensional large-displacement finite-strain beam theory. Stud Appl Math LII(2):87–95

    Google Scholar 

  • Simo JC, Vu-Quoc L (1988) On the dynamics in space of rods undergoing large motions – a geometrically exact approach. Comput Methods Appl Mech Eng 66: 125–161

    Google Scholar 

  • Sotoudeh Z, Hodges DH (2011) Modeling beams with various boundary conditions using fully intrinsic equations. J Appl Mech 78(3):article 031010

    Google Scholar 

  • Sotoudeh Z, Hodges DH (2013a) Structural dynamics analysis of rotating blades using fully intrinsic equations; Part I: theory and verification of single load path configuration. J Am Helicopter Soc 58(3):article 032003

    Google Scholar 

  • Sotoudeh Z, Hodges DH (2013b) Structural dynamics analysis of rotating blades using fully intrinsic equations; Part II: verification of dual load path configurations. J Am Helicopter Soc 58(3):article 032004

    Google Scholar 

  • Yu W (2013) VABS: cross sectional analysis tool for composite beams. AnalySwift Website. http://analyswift.com/. Accessed July 2013

  • Yu W, Hodges DH, Volovoi VV, Cesnik CES (2002) On Timoshenko-like modeling of initially curved and twisted composite beams. Int J Solids Struct 39(19):5101–5121

    Article  MATH  Google Scholar 

  • Yu W, Hodges DH, Ho JC (2012) Variational asymptotic beam sectional analysis – an updated version. Int J Eng Sci 59:40–64

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia

    Dewey H. Hodges

Authors
  1. Dewey H. Hodges
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Correspondence to Dewey H. Hodges .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Institut für Mechanik, Magdeburg, Germany

    Holm Altenbach

  2. Griffith University (Gold Coast Campus), Griffith School of Engineering Griffith University (Gold Coast Campus), Southport, Queensland, Australia

    Andreas Öchsner

Section Editor information

  1. Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, 30332-0150, Atlanta, Georgia, USA

    Dewey Hodges

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Hodges, D.H. (2018). Geometrically Exact Equations for Beams. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_53-1

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  • DOI: https://doi.org/10.1007/978-3-662-53605-6_53-1

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53605-6

  • Online ISBN: 978-3-662-53605-6

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

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