Hello everyone

While doing the project

"Rectangular Hollow Section beam calculator"
230724_ebbeam_rhs.html (-> 230720_ebbeam_rhs.php)
which I mentioned here as thread
"web-based beam calculator - RHS's only"

I needed a neat way to go from stress in a beam to its deflection.
So came by this unexpected derivation.

* have you seen it before?
* is it known to be useful?
* any thoughts???

Stating again - if you think in terms of stress not force, you get an
expression for beam deflection which has uses...

Regards,
Rich Smith

--------------------------------

Derivation - deflection of symmetrical section in relation to stress

This expression for elastic deflection of a beam would be very useful
"on-site" because everything you need to know can be obtained with a
tape-measure (and a length of string!) to calculate the stress on a
beam...

Symbols - all as familiarly used

"I" = Second Moment of Area
"Z" = Section Modulus
"M" = a Moment; a bending and/or turning force
"F" = a force (in Newtons; N)
"L" = length of the beam from supports to load
(length beyond supports is irrelevant)
"H" = height of symmetrical section in direction it is being bent
"sigma" (Greek symbol) = stress (in N/m^2)
"." == "*" meaning multiply

(1)
M=FL/4 ("simple" beam)

(2)
the massively significant equation which combines material property
and geometric property...
M=sigma.Z

noting "Z" = I/half-height for a *symmetrical* section
Z=2I/H
thus
M=2.sigma.I/H

(3)
and deflection
d=F.L^3/48EI ("simple" beam - supported each end; load in the middle)

Restating and taking the derivation a step further:
M=FL/4 => F=4M/L
M=2.sigma.I/H
d=FL^3/48EI

F=4M/L substituted into d=FL^3/48EI
d=(4M/L).L^3/48EI

Cancelling L^-1 and L^3 to L^2
4/48=12

d=ML^2/12EI

From "fundamental"
M=sigma.Z
for a symmetrical section
(eg. I-beam; circular, rectangular or square hollow section)
we derive
M=2.sigma.I/H

Substituting M=2.sigma.I/H into d=ML^2/12EI

(2.sigma.I/H)L^2/12EI=2.sigma.I.L^2/12EIH

"I" cross-cancels completely.
2/12=1/6

So...
d=sigma.L^2/6EH

If you had a beam and stretch a string between its ends, a tape
measure / rule in the middle between string and beam would measure the
deflection.
The tape measure also gives you the length of the beam and the height
of the beam (or "depth")

eg. a 200x100x12RHS used on its major axis would be measured as having
a height of 0.2m (200mm)...

as
d=sigma.L^2/6EH
then
sigma=6EHd/L^2

So there with L, d and H given by your tape (and d needing your
string).
Steel always has an elastic modulus E of around 210GPA (210e9Pa) - so
you know E already.

You can either deduce stress "sigma" from a measured deflection
sigma=6EHd/L^2

or calculate the deflection you would get at the yield stress and go
measure how much of the beam capacity is currently being used.
d=sigma_yield.L^2/6EH
You can take the nominal yield stress specification of the steel as
its yield stress - so for an "S355" steel (which would be typical of
an RHS) that would be 355MPa = 355e6Pa

As I said, this can only be used for symmetrical sections like I-beams
and RHS's and cylindrical tubes.
The "H" has that implication that the section must be symmetrical (you
could not use this for eg. angle-iron ("L-shape") which is not
symmetrical).

"Richard Smith" wrote in message news:lyjztvz84q.fsf@void.com...

Hello everyone

While doing the project

"Rectangular Hollow Section beam calculator"
230724_ebbeam_rhs.html (-> 230720_ebbeam_rhs.php)
which I mentioned here as thread
"web-based beam calculator - RHS's only"

I needed a neat way to go from stress in a beam to its deflection.
So came by this unexpected derivation.

---------------------

I had a slightly different approach. The deflection that matters most is
near or at the yield point, and can be figured or found with an on-line
calculator, so I placed a marker of that height at the beam center. Instead
of string I sighted down the beam from the end, standing out of the way in
case something went wrong. The idea was to relate deflection with a load to
without it, to know when to stop when straightening the slightly bent second
hand steel.

After reaching the yield point I counted hydraulic pump strokes and
increased the count until the beam (3" and 4" C channel) relaxed straight,
since permanent deflection isn't predictably linear in the strain hardening
range above yield.
https://www.autodesk.com/support/technical/article/caas/tsarticles/ts/2iMjLsg9VOc7z2KALCjdfR.html

I could have measured deflection with string but the chained setup wasn't
stable enough to risk getting that close. I hadn't yet determined where to
drill the bolt holes that would have permitted more stable connections.

One of the four channel sections clearly was higher strength steel and
required much more deflection to straighten. It also had slightly smaller
dimensions and bent more below the yield point, allowing the trolley track
to twist and deflect sideways instead of sagging at max proof loading,
though it held my 2100 Lb log OK. The scrap dealer had no more to replace
it.

In the calculators max deflection is given as a fraction of length, such as
L/360 for floors that support plastered or sheet rock walls, to avoid
cracking. They give it for the load centered or at two other points. It's
for the live load, the dead load deflection occurs before finishing the
walls. For unfinished areas I've seen L/180 suggested and it's what I use.
IIRC the (unmeasured) deflection to straighten the high strength channel was
roughly L/20.
jsw

Very appreciating considered engaged informed response.
How did I not guess that someone here would relate to some very like
experience! :-)
Will read again tomorrow and hopefully respond more.
Regards, Rich S

"Richard Smith" wrote in message news:lyleear9w2.fsf@void.com...

Very appreciating considered engaged informed response.
How did I not guess that someone here would relate to some very like
experience! :-)
Will read again tomorrow and hopefully respond more.
Regards, Rich S

------------------

While most of what I've been asked to design and build was electronic there
has been some necessary mechanical content, and more for my home projects,
the bucket loader, sawmill and gantry hoists, so I've tried to learn enough
engineering theory and practice that they don't break or weigh and cost 10x
more than needed. Some electrical engineers I worked for knew little outside
their specialty and couldn't solder.

I had almost no engineering perception - all my knowledge was
fundamental science.

When I realised my scientific career dreams were simply not going to
happen, working in construction for an hourly wage just to pay for a
meagre existence while rehabilitating into society :-) my engineering
knowledge lit-up. Going from almost zero to "these are somewhat
advanced topics we are talking about here" (they weren't, but it takes
a new part of you head with a new region of "landscape" perceiving
and analysing structural things in what becomes a familiar way).

I remember as if it were yesterday just over 15 years ago waiting for
the train to work to leave London Waterloo station and looking up at
the roof-trusses and I started to "trace" their design and see the
forces flowing.

Euler-Bernoulli beam and Finite Element Analysis modelling came when I
came back from Turkey only 8 years ago. Having realised I could have
so speeded-up my work as a welding engineer if I could independently
do at-least-approximate analyses of structures and challenge things I
knew were wrong but had not the means to estimate.

I "got" Finite Element Analysis modelling first - but realised it
didn't suggest how to design long structural elements - beams and
columns.
I learned (Euler-Bernoulli) beam while working as the gateperson at a
bricks-and-mortar residential construction site.

I had known for years into decades that I had no engineering
perception and needed to do something about it, but had no idea how.
Funnily enough (?), where as a welder and steel-erector ("ironworker")
was where it happened...

Regards,
Rich S

"Richard Smith" wrote in message news:lya5uqgk3n.fsf@void.com...

I remember as if it were yesterday just over 15 years ago waiting for
the train to work to leave London Waterloo station and looking up at
the roof-trusses and I started to "trace" their design and see the
forces flowing.

-----------------------

I do that too, most recently to a circus-type tent.

The knowledge was gained by trial and costly error:
https://en.wikipedia.org/wiki/Dee_Bridge_disaster

We still aren't immune to error:
https://en.wikipedia.org/wiki/Florida_International_University_pedestrian_bridge_collapse

FIU was supposedly the expert in that construction method.
"Unlike most bridges in Florida, the design for this project was overseen by
the university, not the Florida Department of Transportation (FDOT), in a
program known as the Local Agency Program (LAP)."

"Florida International University is known for its expertise in accelerated
bridge construction (ABC) and has attracted international scholars as PhD
students."