Finite Elements on Generalized Elastic Foundation in Timoshenko Beam Theory
Publication: Journal of Engineering Mechanics
Volume 134, Issue 9
Abstract
Using the Vlasov foundation model, a modified approach of the continuous beam on elastic supports, leading to both a mechanical model and the proper foundation parameters of the generalized foundation is shown. Two formulations of the beam finite-element with shear deformation effect, resting on a two-parameter elastic foundation, characterized by distinct contributions of normal and rotary reactions are presented. The behavior of the second foundation parameter in the two formulations is governed by the bending cross section rotation of a beam. The first formulation, yielding a free-of-meshing stiffness matrix and equivalent nodal load vector, is based on the transcendental or “exact” solution of the governing differential equation of the beam resting on the elastic layer of constant thickness. Considering a linear variation of the layer thickness along the beam, the second formulation is based on the assumed polynomial displacement field. Numerical comparisons with the exact approach show that the cubic formulation leads to better results when the foundation parameters are variables. The practical utility of the analogy between a tensile axial force and the second foundation parameter is exemplified, too.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writer is indebted to Professor Henryk Stolarski from the University of Minnesota for his stimulating e-mails and to his colleagues Dipl-Engrs Anca Radulescu, Alecu Cipi, and Dan Maiorean from IPTANA Bucharest for their graphical assistance. Useful comments by the reviewers are also gratefully acknowledged.
References
Areiza-Hurtado, M., Vega-Posada, C., and Aristizabal-Ochoa, J. D. (2005). “Second-order stiffness matrix and loading vector of a beam-column with semirigid connections on an elastic foundation.” J. Eng. Mech., 131(7), 752–762.
Aydogan, M. (1995). “Stiffness-matrix formulation of beams with shear effect on elastic foundation.” J. Struct. Eng., 121(9), 1265–1270.
Bowles, J. E. (1996). Foundation analysis and design, 5th Ed., McGraw-Hill, New York.
Carpenter, C. W. (1973). “Ánalysis of plates on elastic foundation.” Int. J. Numer. Methods Eng., 7(3), 408–410.
Chiwanga, M., and Valsangkar, A. J. (1988). “Generalized beam element on two-parameter elastic foundation.” J. Struct. Eng., 114(6), 1414–1430.
Desai, C. S., and Kuppusamy, T. (1978). “Procedure for a soil-structure interaction problem.” Proc., Computing in Civil Engineering, ASCE, New York, 200–218.
Eisenberger, M., and Clastornik, J. (1987). “Beams on variable two-parameter elastic foundation.” J. Eng. Mech., 113(10), 1454–1466.
Eisenberger, M., and Yankelevsky, D. Z. (1985). “Exact stiffness matrix for beams on elastic foundation.” Comput. Struct., 21(8), 1355–1359.
Filonenko-Borodich, M. M. (1940). “Some approximate theories of the elastic foundation.” Uchenyic Zapiski Moskovskogo Gosudarstvennogo Universiteta, Mekhanika, 46, 3–18 (in Russian).
Gallagher, R. H. (1975). Finite-element analysis—Fundamentals, Prentice-Hall, Englewood Cliffs, N.J.
Hetenyi, M. (1946). Beams on elastic foundation, University of Michigan Press, Ann Arbor, Mich.
Horvath, J. S. (1989). “Subgrade models for soil-structure analysis.” Foundation engineering: Current principles and practices, ASCE, New York, 599–612.
Horvath, J. S. (1993). “Beam-column analogy model for soil-structure interaction analysis.” J. Geotech. Engrg., 119(2), 358–364.
Jones, R., and Xenophontos, J. (1977). “The Vlasov foundation model.” Int. J. Mech. Sci., 19, 317–323.
Kerr, A. D. (1964). “Elastic and viscoelastic foundation models.” J. Appl. Mech., 31, 491–498.
Kerr, A. D. (1965). “A study of a new foundation model.” Acta Mech., 1(2), 135–147.
Miyahara, F., and Ergatoudis, J. G. (1976). “Matrix analysis of structural-foundation interaction.” J. Struct. Div., 102(1), 251–265.
Morfidis, K., and Avramidis, I. E. (2002). “Formulation of a generalized beam element on a two-parameter elastic foundation with semirigid connections and rigid offsets.” Comput. Struct., 80, 1919–1934.
Morfidis, K., and Avramidis, I. E. (2006). “Bending of beams on three-parameter elastic foundation.” Int. J. Solids Struct., 43(2), 357–375.
Narayanaswami, R., and Adelman, H. M. (1974). “Inclusion of transverse shear deformation in finite-element displacement formulation.” AIAA J., 12(11), 1613–1614.
Onu, G. (1983). “Shear effect in beam stiffness matrix.” J. Struct. Eng., 109(9), 2216–2221.
Onu, G. (1996). “Equivalences in soil-structure interaction.” Comput. Struct., 58(2), 367–380.
Onu, G. (2000). “Shear effect in beam finite-element on two-parameter elastic foundation.” J. Struct. Eng., 126(9), 1104–1107.
Pasternak, P. L. (1954). “On a new method of analysis of an elastic foundation by means of two foundation constants.” Gosudarstvennoc Izdatelstvo Literaturi po Stroitelstvu i Arkhiteture, Moscow, USSR (in Russian).
Pilkey, W. D. (2002). Analysis and design of elastic beams—Computational methods, Wiley, New York.
Przemieniecki, J. S. (1968). Theory of matrix structural analysis, McGraw-Hill, New York.
Selvadurai, A. P. S. (1979). Elastic analysis of soil-foundation interaction, Elsevier, Amsterdam.
Shirima, L. M., and Giger, M. W. (1992). “Timoshenko beam element resting on two-parameter elastic foundation.” J. Eng. Mech., 118(2), 280–295.
Soare, M. V. (1986). Discrete structures and equivalent continua in the mechanics of solids, Editura Academiei, Bucharest (in Romanian).
Sogge, R. L. (1981). “Laterally loaded pile design.” J. Geotech. Engrg. Div., 107(9), 1179–1199.
Sogge, R. L. (1984). “Microcomputer analysis of laterally loaded pile.” Laterally loaded deep foundations, STP835, ASTM, Philadelphia, 35–48.
Timoshenko, S. S. (1953). Collected papers, McGraw-Hill, New York.
Ting, B. Y., and Mockry, E. F. (1984). “Beam on elastic foundation finite-element.” J. Struct. Eng., 110(10), 2324–2339.
Vallabhan, C. V. G., and Das, Y. C. (1988). “Parametric study of beams on elastic foundations.” J. Eng. Mech., 114(12), 2072–2082.
Vallabhan, C. V. G., and Das, Y. C. (1991a). “Modified Vlasov model for beams on elastic foundations.” J. Geotech. Engrg., 117(6), 956–966.
Vallabhan, C. V. G., and Das, Y. C. (1991b). “A refined model for beams on elastic foundations.” Int. J. Solids Struct., 27(5), 629–637.
Vlasov, V. Z., and Leontiev, N. N. (1960). “Beams, plates, and shells on an elastic foundation.” Fizmatgiz, Moscow, USSR (in Russian).
Wilson, E. L. (1974). “The static condensation algorithm.” Int. J. Numer. Methods Eng., 8(1), 198–203.
Yokoyama, Y. (1991). “Vibrations of Timoshenko beam-columns on two-parameter elastic foundations.” Earthquake Eng. Struct. Dyn., 20, 355–370.
Zhaohua, F., and Cook, R. D. (1983). “Beam elements on two-parameter elastic foundations.” J. Eng. Mech., 109(6), 1390–1402.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 134 • Issue 9 • September 2008
Pages: 763 - 776
Copyright
© 2008 ASCE.
History
Received: Oct 3, 2006
Accepted: Mar 27, 2008
Published online: Sep 1, 2008
Published in print: Sep 2008
Notes
Note. Associate Editor: Jiun-Shyan Chen
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.