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Acta Mechanica Solida Sinica

, Volume 24, Issue 6, pp 510–518 | Cite as

Buckling Loads and Eigenfrequencies of a Braced Beam Resting on an Elastic Foundation

  • Yin Zhang
  • Yun Liu
  • Peng Chen
  • Kevin D. Murphy
Article

Abstract

The eigenvalue problems of the buckling loads and natural frequencies of a braced beam on an elastic foundation are investigated. The exact solutions for the eigenvalues are presented. The eigenvalues vary with the different parameters and are especially sensitive to the brace location. As the beam of a continuous system has infinite eigenvalues and these eigenvalues are influenced differently by a brace, the eigenvalues show rich variation patterns. Because these eigenvalues physically correspond to the structure buckling loads and natural frequencies, the study on the eigenvalues variation patterns can offer a design guidance of using a lateral brace of translation spring to strengthen the structure.

Key words

beam elastic foundation buckling eigenfrequency curve crossing curve veering 

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References

  1. [1]
    Suo, Z., Wrinkling of the oxide scale on an aluminum-containing alloy at high temperatures. Journal of the Mechanics and Physics of Solids, 1995, 43: 829–846.CrossRefGoogle Scholar
  2. [2]
    Sridhar, N., Srolovitz, D.J. and Suo, Z., Kinetics of buckling of a compressed film on a viscous substrate. Applied Physics Letters, 2001, 78: 2482–2484.CrossRefGoogle Scholar
  3. [3]
    Huang, R. and Suo, Z., Instability of a compressed elastic film on a viscous layer. International Journal of Solids and Structures, 2002, 39: 1791–1802.CrossRefGoogle Scholar
  4. [4]
    Huang, R. and Suo, Z., Wrinkling of a compressed elastic film on a viscous layer. Journal of Appllied Physics, 2002, 91: 1135–1142.CrossRefGoogle Scholar
  5. [5]
    Bowden, N., Brittain, S., Evans, A.G., Hutchinson, J.W. and Whitesides, G.M., Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer. Nature, 1998, 393: 146–149.CrossRefGoogle Scholar
  6. [6]
    Jiang, H., Sun, Y., Rogers, J.A. and Huang, Y., Post-buckling analysis for the precisely controlled buckling of thin film encapsulated by elastomeric substrates. International Journal of Solids and Structures, 2008, 45: 2014–2023.CrossRefGoogle Scholar
  7. [7]
    Zhang, Y. and Zhao, Y., Discussion on modeling shape memory alloy embedded in a composite laminate as axial force and elastic foundation. Materials and Design, 2007, 28: 1016–1020.CrossRefGoogle Scholar
  8. [8]
    Zhang, Y. and Liu, Y., Local bending of thin film on viscous layer. Acta Mechanica Solida Sinica, 2010, 23: 106–114.CrossRefGoogle Scholar
  9. [9]
    Zhang, Y. and Zhao, Y., A study of composite beam with shape memory alloy arbitrarily embedded under thermal and mechanical loadings. Materials and Design, 2007, 28: 1096–1115.CrossRefGoogle Scholar
  10. [10]
    Khang, D., Jiang, H., Huang, Y. and Rogers, J.A., A stretchable form of single crystal silicon for highperformance electronics on rubber substrates. Science, 2006, 311: 208–212.CrossRefGoogle Scholar
  11. [11]
    Huang, J., Juszkiewicz, M., de Jeu, W.H., Cerda, E., Emrick, T., Menon, N. and Russel, T., Capillary wrinkling of floating thin films. Science, 2007, 317: 650–653.CrossRefGoogle Scholar
  12. [12]
    Plaut, R.H. and Yang, J.G., Lateral bracing forces in columns with two unequal spans. Journal of Structural Engineering, 1992, 119: 2896–2912.CrossRefGoogle Scholar
  13. [13]
    Plaut, R.H., Requirements for lateral bracing of columns with two spans. Journal of Structural Engineering, 1992, 119: 2913–2931.CrossRefGoogle Scholar
  14. [14]
    Plaut, R.H., Murphy, K.D. and Virgin, L.N., Curve and surface veering for a braced column. Journal of Sound and Vibration, 1995, 187: 879–885.CrossRefGoogle Scholar
  15. [15]
    Chen, P.T. and Ginsberg, J.H., On the relationship between veering of eigenvalue loci and parameter sensitivity of eigenfunctions. Journal of Vibration and Acoustics, 1992, 114: 141–148.CrossRefGoogle Scholar
  16. [16]
    von Kármán, T., Dunn, L.G. and Tsien, H., The influence of curvature on the buckling characteristics of structures. Journal of the Aeronautical Sciences, 1940, 7: 276–289.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Tsien, H., A theory of the buckling of thin shells. Journal of the Aeronautical Sciences, 1942, 9: 373–384.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Hutchinson, J.W. and Budiansky, B., Dynamic buckling estimates. AIAA Journal, 1966, 4: 525–530.CrossRefGoogle Scholar
  19. [19]
    Pierre, C., Mode localization and eigenvalue loci veering phenomena in disordered structures. Journal of Sound and Vibration, 1988, 126: 485–502.CrossRefGoogle Scholar
  20. [20]
    Perkins, N.C. and Mote, C.D., Comments on curve veering in eigenvalue problems. Journal of Sound and Vibration, 1986, 106: 451–463.CrossRefGoogle Scholar
  21. [21]
    Timoshenko, S. and Gere, J.M., Theory of Elastic Stability. McGraw-Hill Book Company, Inc., 1981.Google Scholar
  22. [22]
    Jei, Y.G. and Lee, C.W., Does curve veering occur in the eigenvalue problem of rotors? Journal of Vibration and Acoustics, 1992, 114: 32–36.CrossRefGoogle Scholar
  23. [23]
    Leissa, A.W., On a curve veering aberration. Zeitschrift für Angewandte Mathematik und Physik(ZAMP), 1974, 25: 99–111.CrossRefGoogle Scholar
  24. [24]
    Kutter, J.R. and Sigillito, V.G., On curve veering. Journal of Sound and Vibration, 1981, 75: 585–588.CrossRefGoogle Scholar
  25. [25]
    Chen, P.T. and Ginsberg, J.H., Modal properties and eigenvalue veering phenomena in axisymmetric vibration of spheroidal shells. Journal of the Acoustic Society of America, 1992, 92: 1499–1508.CrossRefGoogle Scholar
  26. [26]
    Pierre, C. and Plaut, R.H., Curve veering and mode localization in a buckling problem. Zeitschrift für Angewandte Mathematik und Physik(ZAMP), 1989, 40: 758–761.CrossRefGoogle Scholar
  27. [27]
    Wang, J. and Mote, C.D., On the divergence buckling of a wide bandsaw plate by roll-tensioning. Journal of Sound and Vibration, 1986, 175: 661–675.CrossRefGoogle Scholar
  28. [28]
    Castanier, M.P. and Pierre, C., Predicting localization via Lyaponov exponent statistics. Journal of Sound and Vibration, 1997, 203, 151–157.CrossRefGoogle Scholar
  29. [29]
    Zhang, Y. and Murphy, K.D., Secondary buckling and tertiary states of a beam on a non-linear elastic foundation. International Journal of Nonlinear Mechanics, 2005, 40: 795–805.CrossRefGoogle Scholar
  30. [30]
    Graff, K.F., Wave Motion in Elastic Solids, Ohio State University Press, 1975.Google Scholar
  31. [31]
    Zhang, Y. and Murphy, K.D., Crack propagation in structures subjected to periodic excitation. Acta Mechanica Solida Sinica, 2007, 20: 230–240.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2011

Authors and Affiliations

  • Yin Zhang
    • 1
  • Yun Liu
    • 2
  • Peng Chen
    • 3
  • Kevin D. Murphy
    • 4
  1. 1.State Key Laboratory of Nonlinear Mechanics (LNM)Institute of Mechanics, Chinese Academy of SciencesBeijingChina
  2. 2.Faculty of Information and AutomationKunming University of Science and TechnologyKunmingChina
  3. 3.Department of Civil EngineerningGuangdong Technical College of Water Resources and Electric EngineeringGuangzhouChina
  4. 4.Department of Mechanical EngineeringUniversity of ConnecticutStorrsUSA

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