Journal of Engineering Mathematics

, Volume 89, Issue 1, pp 163–176 | Cite as

Exact stiffness matrix for nonlocal bars embedded in elastic foundation media: the virtual-force approach

  • Suchart Limkatanyu
  • Woraphot Prachasaree
  • Nattapong Damrongwiriyanupap
  • Minho Kwon


This paper proposes a solution to the exact bar-foundation element that includes the bar nonlocal effect. The exact element stiffness matrix and fixed-end force vector are derived based on the exact element flexibility equation using the so-called natural approach. The virtual-force principle is employed to work out the governing differential compatibility equation as well as the associated end-boundary compatibility conditions. Exact force interpolation functions are used to derive the exact element flexibility equation and can be obtained as the analytical solution of the governing differential compatibility equation of the problem. A numerical example of a nanowire-elastic substrate system is used to verify the accuracy and efficiency of the natural nonlocal bar-foundation model and to demonstrate the superiority over its counterpart, a displacement-based model. The effects of material nonlocality on the system responses are also discussed in the example.


Bar elements Elastic substrate media Finite element Nanowires  Natural stiffness matrix Nonlocal elasticity Virtual-force principle 



This study was partially supported by the Thailand Research Fund (TRF) under Grants MRG4680109 and RSA5480001 and by the STREAM Research Group under Grant ENG-51-2-7-11-022-S, Faculty of Engineering, Prince of Songkla University. Any opinions expressed in this paper are those of the authors and do not reflect the views of the sponsoring agencies. Special thanks go to senior lecturer Mr Wiwat Sutiwipakorn for reviewing and correcting the English in the paper. In addition, the authors would like to thank two anonymous reviewers for their valuable and constructive comments.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Suchart Limkatanyu
    • 1
  • Woraphot Prachasaree
    • 1
  • Nattapong Damrongwiriyanupap
    • 2
  • Minho Kwon
    • 3
  1. 1.Department of Civil Engineering, Faculty of EngineeringPrince of Songkla UniversitySongkhlaThailand
  2. 2.Civil Engineering Program, School of EngineeringUniversity of PhayaoPhayaoThailand
  3. 3.Department of Civil Engineering, ERIGyeongsang National UniversityJinjuSouth Korea

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