1- Valsangkar, A. J. and Pradhanang, R., 1988, Vibrations of beam‐columns on two‐parameter elastic foundations, Earthquake engineering & structural dynamics, 16, 2, 217-225.
2-De Rosa, M. A., 1995, Free vibrations of Timoshenko beams on two-parameter elastic foundation, Computers & Structures, 57, 1, 151-156.
3-Yokoyama, T., 1996, Vibration analysis of Timoshenko beam-columns on two-parameter elastic foundations, Computers & Structures, 61, 6, 995-1007.
4- Eisenberger, M. and Clastornik, J., 1987, Beams on variable two-parameter elastic foundation, Journal of Engineering Mechanics, 113, 10, 1454-1466.
5- Zhaohua, F. and Cook, R. D., 1983, Beam elements on two-parameter elastic foundations, Journal of Engineering Mechanics, 109, 6, 1390-1402.
6-Naidu, N. R. and Rao, G. V., 1995, Vibrations of initially stressed uniform beams on a two-parameter elastic foundation, Computers & structures, 57, 5, 941-943.
7- Timoshenko, S. P., Goodier, J. N., 1970, Theory of Elasticity, Third Edition, McGrawHill Book Company, Singapore, 567 pp
8- Widad, B., Salah, K. and Souad, B., 2019, Soil non-homogeneity and soil-structure interaction effects on beam vibrations, Proceedings of the Institution of Civil Engineers-Structures and Buildings, 1-10.
9- Boudaa, S., Khalfallah, S. and Bilotta, E., 2019, Static interaction analysis between beam and layered soil using a two-parameter elastic foundation, International Journal of Advanced Structural Engineering, 11, 1, 21-30.
10- Mullapudi, T. R. S., 2010, Nonlinear finite element formulation of the soil structure interaction through two-parameter foundation model.
11- Avcar, M., 2016, Effect of material non-homogeneity and two-parameter elastic foundation on fundamental frequency parameters of Timoshenko beam, Acta Phys Pol A , 130, 375–378.
12-Froio, D. and Rizzi, E., 2017, Analytical solution for the elastic bending of beams lying on a linearly variable Winkler support, International Journal of Mechanical Sciences, 128, 680-694.
13- Mullapudi, R. and Ayoub, A., 2010, Nonlinear finite element modeling of beams on two-parameter foundations, Computers and Geotechnics, 37, 3, 334-342.
14- Obara, P., 2014, Vibrations and stability of Bernoulli-Euler and Timoshenko beams on two-parameter elastic foundation, Archives of Civil Engineering, 60, 4, 421-440.
15- Chen, J., Feng, Y. and Shu, W., 2016, An improved solution for beam on elastic foundation using quintic displacement functions, KSCE Journal of Civil Engineering, 20, 2, 792-802.
16- Ayvaz, Y. and Özgan, K., 2002, Application of modified Vlasov model to free vibration analysis of beams resting on elastic foundations, Journal of Sound and vibration, 255, 1, 111-127.
17- Basudhar, P. K., Yadav, S. K. and Basudhar, A., 2018, Treatise on Winkler modulus of subgrade reaction and its estimation for improved soil–structure interaction analysis, Geotechnical and Geological Engineering, 36, 5, 3091-3109.
18- Ma, J., Liu, F., Nie, M. and Wang, J., 2018, Nonlinear free vibration of a beam on Winkler foundation with consideration of soil mass motion of finite depth, Nonlinear Dynamics, 92, 2, 429-441.
19- Winkler, E., 1867, Die Lehre von der Elasticitaet und Festigkeit: mit besonderer Rücksicht auf ihre Anwendung in der Technik für polytechnische Schulen, Bauakademien, Ingenieue, Maschinenbauer, Architecten, etc (Vol. 1). Dominicus.
20- Chen, Y. H., 1987, General dynamic-stiffness matrix of a timoshenko beam for transverse vibrations, Earthquake engineering & structural dynamics, 15, 3, 391-402.
21- Paz, M. and Kim, Y. H., 1991, Structural dynamics, Springer.
22- Paz, M. and Dung, L., 1975, Power series expansion of the general stiffness matrix for beam elements, International Journal for Numerical Methods in Engineering, 9, 2, 449-459.
23- Paz, M., 1973, Mathematical observations in structural dynamics, Computers & Structures, 3, 2, 385-396.
24- Knopp, K., 1990, Theory and application of infinite series, Courier Corporation.
)