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Journal of Applied Mathematics
Volume 2013, Article ID 626287, 13 pages
Research Article

Exact Stiffness for Beams on Kerr-Type Foundation: The Virtual Force Approach

1Department of Civil Engineering, Faculty of Engineering, Prince of Songkla University, Songkhla 90112, Thailand
2Civil Engineering Program, School of Engineering, University of Phayao, Phayao 5600, Thailand
3Department of Civil Engineering, ERI, Gyeongsang National University, Jinju 660-701, Republic of Korea
4Department of Civil Engineering, Gangneung-Wonju National University, Gangneung 210-720, Republic of Korea

Received 12 May 2013; Revised 17 July 2013; Accepted 6 August 2013

Academic Editor: Fayun Liang

Copyright © 2013 Suchart Limkatanyu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper alternatively derives the exact element stiffness equation for a beam on Kerr-type foundation. The shear coupling between the individual Winkler-spring components and the peripheral discontinuity at the boundaries between the loaded and the unloaded soil surfaces are taken into account in this proposed model. The element flexibility matrix is derived based on the virtual force principle and forms the core of the exact element stiffness matrix. The sixth-order governing differential compatibility of the problem is revealed using the virtual force principle and solved analytically to obtain the exact force interpolation functions. The matrix virtual force equation is employed to obtain the exact element flexibility matrix based on the exact force interpolation functions. The so-called “natural” element stiffness matrix is obtained by inverting the exact element flexibility matrix. One numerical example is utilized to confirm the accuracy and the efficiency of the proposed beam element on Kerr-type foundation and to show a more realistic distribution of interactive foundation force.