I am looking for the resolution of infinite beam on elastic foudation for a "local load"
I don't need the formula but the solution of differential equation (order 4 )
(solution: zimmermann)
Not sure what you are getting at - but if you check out Poulos and Davis' book of "Elastic Solutions for Soil and Rock Mechanics," you might find what you are looking for. This is a pretty well complete compilation of elastic solutions.
AREMA publishes the Talbot reports where they explored track reactions to loading. A quick flip through the book did not give any high level math. What's the context of your problem?