I recently derived Beam deflection vs. stress - with applications . Which, unstated, is for the only case I have dealt with, a point centrally-loaded simply-supported-beam. Such as when you put a beam across a hold of a boat, strop a chain-block to its middle and hoist a load in the hold.
I was asked "Could you derive a similar expression for a simply-supported-beam uniformly-distributed-load?".
As best I can write here it is
y = 5 sigma L^2 / 24 E H
I have typeset it in this PDF: deflection vs. stress for a SSB-UDL.
Yes it is. The application is to help an engineer explain the basis of a design to a customer. When the need is to show in a single succinct formula the relationship between stress in the structure and the elastic deflection the structure will have.
This equation is very similar, with only the constant term different,
between the SSB-UDL case and the SSB-CL case.
This is expected. The only variables operating are the stress state
within the beam and the deflection of the beam that results in.
Everything is about what is, internally within the beam. Nothing
about the external load state, which does differ and needs describing
in different ways.
The different loading - distributed vs point-central - will
produce a different shape of the deformed beam. But given
linear-elastic proportionality, for the point of maximum deflection
which is at the mid-span of the beam in both cases, the only
difference is that constant term in the overall proportionalities.
Pleasingly neat...
(R. Smith, 16Oct2023)