Rao, "Large amplitude vibrations of slender, uniform beams on elastic foundation
," Indian Journal of Engineering & Materials Sciences, vol.
Hetenyi, Beams on Elastic Foundation
: Theory with Applications in the Fields of Civil and Mechanical Engineering, The University of Michigan Press, Ann Arbor, Mich, USA, 1971.
The elastic soil model, which is used beams on elastic foundation, is first introduced by Winkler (1867).
Deflection line equation of beams on elastic foundation is including four constant of integration.
Although vibration analysis of beams on elastic foundation is a widely studied topic, there are only few papers that exist in the literature pertaining to the analysis of beams with elastically restrained ends.
Becker, "Dynamic response of beams on elastic foundation," Journal of Structural Engineering ACSE, vol.
Hetenyi, Beams on Elastic Foundation, The University of Michigan Press, Ann Arbor, Mich, USA, 1946.
Ting, "Finite beams on elastic foundation with restraints," Journal of the Structural Division, vol.
Some unexpected results in vibration of non-homogeneous beams on elastic foundation
. Chaos, Solitons Fractals, 2001, 12, 2177-2218.
The Winkler foundation model  is the most rudimentary mechanical subgrade model and has been widely adopted in studying the problem of beams on elastic foundation. In the Winkler foundation model, a set of 1D independent springs is attached along the beam to form the beam-foundation system.
Heidari-Rarani, "A comparative study for beams on elastic foundation models to analysis of model delamination in DCB specimens," Structural Engineering and Mechanics, vol.
Melnikov explains the mathematics behind the influence function methods (including an introduction to Fourier series), Green's functions, including construction based on defining properties, symmetry of the function, alternate construction of Green's functions and boundary-contact value problems, Kirchoff beam problems, including the bending of beams of uniform rigidity and beams on elastic foundations
, other beam problems, including Euler buckling problems, and bending of plates and shells.