You may check the book "Wave motion in elastic solids, by karl Graff, Dover Publications 1975", which as several analytic solutions of beams on different types of foundations. Good luck

Many thanks Khulief, I will try it.

In general continous systems have only approximate series solutions.

See the following article:

M. Abu-Hilal 2006 Journal of Sound and Vibration 297, 477-491. Dynamic response of a double Euler-Bernoulli beam due to a moving constant load.

you can use this paper

فکر کنم این مقاله که پیوست شده، برای شما بسیار مفید واقع بشه

Please refer to textbooks  on Advanced Mechanics of Materials. Normally, these books have a chapter on Continuously Supported Beam on Elastic Foundation. Here are a few useful references:

1. Beams on Elastic Foundation by M. Hetenyi, University of Michigan Press, Ann Arbor, MI, 1979.

 2. Advanced Mechanics of Materials  by Robert D. Cook and Warren C. Young,  2nd Ed., Prentice-Hall, Nov. 1999 

3. Advanced Mechanics of Materials by A.P. Boresi and R.J. Schmidt, John Wiley & Sons, 2003.

4. Advanced Mechanics of Materials and Applied Elasticity by A.C. Ugural and S.K. Fenster, 5th Ed., , Prentice Hall, 2012

Hope what I have written above will be helpful. Thank you and wish you good luck.

Please refer to textbooks  on Advanced Mechanics of Materials. Normally, these books have a chapter on Continuously Supported Beam on Elastic Foundation. Here are a few useful references:

1. Beams on Elastic Foundation by M. Hetenyi, University of Michigan Press, Ann Arbor, MI, 1979.

 2. Advanced Mechanics of Materials  by Robert D. Cook and Warren C. Young,  2nd Ed., Prentice-Hall, Nov. 1999 

3. Advanced Mechanics of Materials by A.P. Boresi and R.J. Schmidt, John Wiley & Sons, 2003.

4. Advanced Mechanics of Materials and Applied Elasticity by A.C. Ugural and S.K. Fenster, 5th Ed., , Prentice Hall, 2012

Hope what I have written above will be helpful. Thank you and wish you good luck.

Thanks Ramesh for the useful references.

Yehia

Download Chapter 2(202).pdf from the link given below:

http://www.me.ust.hk/~meqpsun/Notes/Chapter4(202).PDF

please see this pdf

ANALYTICAL SOLUTION OF BEAM ON ELASTIC

FOUNDATION BY SINGULARITY FUNCTIONS

 By Dobromir Dinev

Many new papers in this area. Good to see. Thanks.

El-Sayed T and Farghaly S. (2016) Exact free vibration analysis of Timoshenko stepped shaft carrying elastically supported eccentric masses with application on SWRO mechanical system. Desalination 385: 194-206.

[1] S.H. Farghaly, K.M. Zeid, An exact frequency equation for an axially loaded beam-mass-spring system resting on a Winkler elastic foundation, Journal of Sound and Vibration, 185 (1995) 357-363.