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Two Methods to Obtain Analysis Solution of Beam on Winkler Foundation

[+] Author Affiliations
Shun-qun Li, Shou-xi Chai

Tianjin Institute of Urban Construction, Tianjin, China

Ling-xia Gao

Dalian Nationalities University, Dalian, China

Paper No. OMAE2010-20516, pp. 377-384; 8 pages
doi:10.1115/OMAE2010-20516
From:
  • ASME 2010 29th International Conference on Ocean, Offshore and Arctic Engineering
  • 29th International Conference on Ocean, Offshore and Arctic Engineering: Volume 2
  • Shanghai, China, June 6–11, 2010
  • Conference Sponsors: Ocean, Offshore and Arctic Engineering Division
  • ISBN: 978-0-7918-4910-1 | eISBN: 978-0-7918-3873-0
  • Copyright © 2010 by ASME

abstract

Based on the differential equation of beam section, two methods for beam on Winkler foundation in complex conditions are presented with displacement, angular rotation, moment and shear compatibility conditions of the two adjacent beam elements utilized. Together with the two moment boundary conditions at beam ends, (4n+2) equations can be got. In order to obtain analytic solution, two additional equations are required. The first method to obtain additional equations is depended on the principles of minimum potential energy and virtual work, and the second one makes advantage of boundary conditions of shear at two beam ends. The proposed methods in this paper can be used to compute beam on Winkler foundation in complex conditions which include complex beam conditions, such as variable Young’s modules and section, complex foundation conditions, namely variable coefficient of subgrade reaction, and complex loads consisting of concentrated force, moment and random distributed load. The displacement equations both for beam elements and for the whole beam, which have same structure as the classic Hetenyi solution, can be derived from the proposed method. Therefore, the equations for angular rotation, moment and shear of the beam can also be obtained from their differential relationship with displacement equation. The computation of an example shows that the results by the proposed method are consistent with that by the finite element method when the elements are small enough. Furthermore, element partition in the proposed method is thoroughly different from that in the classic finite element method even the former is belong to the later in the overall meaning.

Copyright © 2010 by ASME

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