[3] M. M. Shokrieh and M. Heidari-Rarani, “A comparative study
for beams on elastic foundation models to analysis of mode-
l delamination in DCB specimens,”Structural Engineering and
Mechanics,vol.37,no.2,pp.149–162,2011.
[4] E. Winkler,Die Lehre Von Der Und Festigkeit, Dominicus,
Prague, Czech Republic, 1867.
[5] K. Wieghardt, “U den balken auf nachgiebiger unterlage,” ̈
Zeitschrift fur Angewandte Mathematik und Mechanik ̈ ,vol.2,pp.
165–184, 1922.
[6] A. D. Kerr, “Elastic and viscoelastic foundation models,”Journal
Applied Mechanics Transaction,vol.31,no.4,pp.491–498,1964.
[7] M. M. Filonenko-Borodich, “Some approximate theories
of elastic foundation,”Uchenyie Zapiski Moskovskogo Gosu-
darstvennogo Universiteta Mekhanica,vol.46,pp.3–18,1940.
[8] S. N. Sokolov,Circular Plate on a Generalized Foundation,
Inzhinierny Sbornik, Academy of Sciences, Moscow, Russia,
1952.
[9]P.L.Pasternak,On a New Method of Analysis of an Elastic
Foundation by Means of Two Constants, Gosudarstvennoe
Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture, Moscow,
Russia, 1954.
[10] M. Het ́enyi, “A general solution for the bending of beams on
an elastic foundation of arbitrary continuity,”Journal of Applied
Physics,vol.21,no.1,pp.55–58,1950.
[11] A. D. Kerr, “A study of a new foundation model,”Acta Mechan-
ica,vol.1,no.2,pp.135–147,1965.
[12] A. Foppl,Vorlesungen Uber Technische Mechanik,B.B.Teubner,
Leipzig, Germany, 1909.
[13] E. Reissner, “A note on deflection of plates on a viscoelastic
foundation,”Journal Applied Mechanics,vol.25,pp.144–155,
1958.
[14] J. S. Horvath,A study of analytical methods for determining
the response of mat foundations to static loads [Ph.D. thesis],
Polytechnic Institute of New York, Brooklyn, NY, USA, 1979.
[15] J.J.Horvath,“Newsubgrademodelappliedtomatfoundations,”
Journal of Geotechnical Engineering,vol.109,no.12,pp.1567–
1587, 1983.
[16] A. Worku, “Calibrated analytical formulas for foundation
model parameters,”International Journal of Geomechanics,vol.
13,no.4,pp.340–347,2013.
[17] J. S. Horvath, “Discussion of modified Vlasov model for beams
on elastic foundations,”Journal of Geotechnical Engineering,vol.
118, no. 9, pp. 1482–1484, 1992.
[18] J. S. Horvath and R. J. Colasanti, “Practical subgrade model
for improved soil-structure interaction analysis: model devel-
opment,”International Journal of Geomechanics, vol. 11, no. 1,
pp.59–64,2010.
[19] I. E. Avramidis and K. Morfidis, “Bending of beams on three-
parameter elastic foundation,”International Journal of Solids
and Structures,vol.43,no.2,pp.357–375,2006.[20] K. Morfidis, “Exact matrices for beams on three-parameter
elastic foundation,”Computers and Structures,vol.85,no.15-16,
pp.1243–1256,2007.
[21] K. Morfidis, “Vibration of Timoshenko beams on three-
parameter elastic foundation,”Computers and Structures,vol.
88, no. 5-6, pp. 294–308, 2010.
[22] Y. Zhang, “Tensionless contact of a finite beam resting on Reiss-
ner foundation,”International Journal of Mechanical Sciences,
vol. 50, no. 6, pp. 1035–1041, 2008.
[23] E. J. Sapountzakis and A. E. Kampitsis, “Nonlinear analysis
of shear deformable beam-columns partially supported on
tensionless three-parameter foundation,”Archive of Applied
Mechanics,vol.81,no.12,pp.1833–1851,2011.
[24] J. Wang and S. Zeng, “Dynamics of piezoelectric smart beams
using a three-parameter elastic foundation model,”Journal of
Intelligent Material Systems and Structures,vol.22,no.1,pp.3–
16, 2011.
[25] R. J. Colasanti and J. S. Horvath, “Practical subgrade model
for improved soil-structure interaction analysis: software imple-
mentation,”Practice Periodical on Structural Design and Con-
struction,vol.15,no.4,pp.278–286,2010.
[26] J. S. Horvath and R. J. Colasanti, “New hybrid subgrade
model for soil-structure interaction analysis: foundation and
geosynthetics applications,” inProceedings of the Geo-Frontiers:
Advances in Geotechnical Engineering, pp. 4359–4368, Dallas,
Tex, USA, March 2011.
[27] S. Limkatanyu and E. Spacone, “Frame element with lateral
deformable supports: formulations and numerical validation,”
Computers and Structures,vol.84,no.13-14,pp.942–954,2006.
[28] J. H. Argyris,Energy Theorems and Structural Analysis,Butter-
worths & Co., London, UK, 1960.
[29] S. Limkatanyu, K. Kuntiyawichai, E. Spacone, and M. Kwon,
“Natural stiffness matrix for beams on Winkler founda-
tion: exact force-based derivation,”Structural Engineering and
Mechanics,vol.42,no.1,pp.39–53,2012.
[30] S. Limkatanyu, N. Damrongwiriyanupap, M. Kwon, and P.
Ponbunyanon, “Force-based derivation of exact stiffness matrix
for beams on Winkler-Pasternak Foundation,”Zeitschrift f ̈ur
Angewandte Mathematik und Mechanik,2013.
[31] S. Wolfram,Mathematica Reference Guide, Addison-Wesley,
RedwoodCity,Calif,USA,1992.
[32] R. L. Taylor, “FEAP: a finite element analysis program. User
manual: version 7.3,” Department of Civil and Environmental
Engineering, University of California-Berkeley, 2000.
[33] R. D. Cook,VisualFEA and General User Manual,JohnWiley
and Son, New York, NY, USA, 2001.
[34] K. Morfidis,Research and development of methods for the
modeling of foundation structural elements and soil [Ph.D.
thesis], Department of Civil Engineering, Aristotle University
of Thessaloniki, Greece, 2003.
[35] B. N. Alemdar and P. Gulkan, “Beams on generalized founda- ̈
tions: supplementary element matrices,”Engineering Structures,
vol.19,no.11,pp.910–920,1997.
[36] M. Eisenberger and J. Bielak, “Remarks on two papers dealing
with the response of beams on two-parameter elastic founda-
tions,”Earthquake Engineering and Structural Dynamics,vol.18,
no.7,pp.1077–1078,1989.
[37] C. V. G. Vallabhan and Y. C. Das, “Modified vlasov model for
beams on elastic foundations,”Journalofgeotechnicalengineer-
ing,vol.117,no.6,pp.956–966,1991.