Beam deflection vs. stress - with applications

Qualification : this is the case of an SSB-CL - a simply-supported-beam centrally-loaded

The derivation

The familiar beam deflection equation for a "simple beam" is
y=FL^3/48EI
This is beam deflection vs. force.

While programming Rectangular Hollow Section beam calculator I "followed a hunch" and derived this alternative expression
y=sigma.L^2/6EH
which is beam deflection vs. stress.
The Second Moment of Area I cross-cancels and disappears from the derived equation, leaving beam height as the only section geometric characteristic remaining.

This formula is very useful for the specific situation of analysing if loading is acceptable on an existing beam in-service - without needing the design information.

Here is
derivation of deflection vs. stress
as a typeset PDF document with mathematic formulae well presented.

Known already?

Is this previously known?
It must be because it is "obvious". Including that when you are going through beam calculations, going from beam stress to beam deflection, you use "I" twice and that would lead to the same instinct which I had, in suspecting the two "I"'s cross-cancel.
However, coming from a scientific training and developing as a welder, self-taught in engineering, I have no idea where to look for what must be existing knowledge.

Is the equation used?
Asking around, an engineer now in Alaska is familiar with the technique of stretching a string from end to end of a beam and measuring deflection of the beam in the middle, as the distance from the string to the beam at that point. Then deducing the internal stress within the beam.
He didn't know of the "sigma=6EHy/L^2" derivation, and used online beam calculators in a mix of operations to get to the stress from the deflection.

Application - surprisingly readily done

The crux of this is that for structural steel beams, that you likely do not have the steel specification for an existing beam in service does not much obstruct applying the analysis.

"sigma=6EHy/L^2" gives the stress in that beam. Is that stress acceptable?

The yield stress of the metal of an in-service beam might be unknown. For structural steels this is much less of an impediment than might be assumed.

most "I" beams (and "H" columns) are S275 steel, which can be taken as having the specification minimum yield stress of 275MPa - where S355 the error is conservative ("on the safe side")
most Rectangular and Square hollow sections are from S355 steel with a yield stress of 355MPa
the lowest yield strength specification structural steel is 235MPa ("S235"), which is a rarely encountered - and is the lowest strength a cheap mass-produced steel can have - so there is a lower bound which is not much less than the assumed common "S275" strength
were a beam a special higher yield strength steel the error would be conservative

The purely geometrical component of beam performance ("open section", "closed section") so channels the steel specification which can optimally be used that the generalisation above holds very true for structural steel beams.

If the service stress determined for a structural steel beam is significantly less than 235MPa no concern remains. Which should apply to the majority of beams so evaluated. As services stresses approach or exceed any of these yield stresses that beam would be noted for further scrutiny.
So this is conserving resources - only beams bearing high stresses get detailed attention.

Ignore that centre-popping (hardness test!) or other indicators are that the beam has a higher yield stress. High-quality structural steels have yield stresses right down only just above the specification minimum (this gives a lovely to process (saw, hole-punch, bend, weld, etc) steel, and you cannot use above specification strength anyway). Only low quality beams will have a "wild" high hardness - and you do not want to be loading such beams significantly...
Only use higher yield stresses if you do have the specification information and the beam is from a conformant higher-yield steel.

Finishing note

Finding this was "a little delight" for me and I hope you enjoy sharing this interest. I taught myself "Euler-Bernoulli beam" while working as the gateperson at a construction site. Where I sat in the gatehouse otherwise "keeping an eye on things" for hundreds of hours immersing myself in this study.
I have used "Euler-Bernoulli beam" when making real structures and seen the amazing correlation between theory and practice in situations where lives depend on it and got excellent results. Hence sharing this happy topic.

(R. Smith, 05Oct2023, 07Oct2023 (conservative, etc.), 06Nov2023 (qual. ssb-cl))