how you knew beam deflect vs stress

Jim - lost previous thread. You used that formulation for beam deflection in relation to beam

*stress*...

For a "simple beam" (like a plank bridging two points)

deflection = sigma.L^2 / 6.E.H

sigma = stress L = beam length between the two supports E = elastic modulus; Young's modulus H = the height of a *symmetrical* section (this only works for symmetrical sections)

How did you know of this formulation? Did you derive it yourself? Or is it known in engineering?

Broader explanation for a general audience:

The deflection vs. *stress* formulation enables you to estimate if a beam you find is bearing an acceptable load.

With the deflection vs *force* formation you need much more information

deflection = F.L^3 / 48.E.I I is the second moment of area and you need every linear dimension of beam's cross-sectional shape to calculate it. Then how would you know what the force is?

With the *stress* formulation you can take a reasonable guess at what the beam's yield stress is - any I-beam not otherwise marked a reasonable guess is 275MPa, and for any Rectangular Hollow Section (including Square Hollow Section) you can reasonably guess at 355MPa.

You can rearrange deflection = sigma.L^2 / 6.E.H to isolate "sigma" the stress, and given you know can easily measure with a tape-measure and rule the beam's length, height and deflection, and E is always around 210GPa for steel, you can get the stress. Then evaluate - is that acceptable?

--------------------------------- I took extreme fiber stress for my guesstimate of the surplus steel or wood beam's properties and found the maximum deflection for a center load from several on-line calculators. Likewise for columns I entered measured dimensions into on-line calculators and ensured the ends were pinned or could swivel to avoid cantilever stresses. The only load-induced damage to retired tripod columns has been slight egg-shaping of the pin holes, and it's about equal on both sides which supports my assumption that the asymmetrical hangers do self adjust to evenly distribute the load. The other damage was from accidents such as falling trees.

For the gantry beam's center splice I made the sum of its components moments of inertia greater than that of the C channels, and supported the channel flanges at the splice ends to compensate for the area lost to bolt holes. I tried calculating from basic principles for practice but didn't trust my result enough to stand under it.

Then I proof-tested the completed assembly. As far as possible the pinned-joint geometry allowed simple calculations, no Vierendeel trusses or dependence on torque and friction, only bolt shank shear and bearing pressure of a closely sized (match-drilled) hole. jsw

Hi there

I've done a typeset presentation of the derivation as a PDF at

All the formulae should be nice and easy to read and follow (?!).

Rich S

Hi there

I've done a typeset presentation of the derivation as a PDF at

All the formulae should be nice and easy to read and follow (?!).

Rich S

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How I knew was by taking the deflection from on-line beam calculators at working and assumed yield point (36KSI) stress levels, then comparing to the observed deflection at a measured loading. It was also a test for hidden damage in the former pallet rack shelf support beams, many of which needed straightening.

They are good and cheap enough for hobby use to avoid accidents. I bought that type after an industrial grade load cell like this from an auction cracked. My background taught me that you don't know when to stop what you're doing unless you can measure it, even if it's only digging a hole or splitting firewood.

Done a webpage

"Beam deflection vs. stress"

You get a mention you will recognise. Have I got you correctly?

Done a webpage

"Beam deflection vs. stress"

You get a mention you will recognise. Have I got you correctly?

------------------------------------- It's fine, I don't need to be identified, but I don't have any engineering degree, live in New Hampshire near the Atlantic and sighted a fixed height limit gauge from beyond the end instead of getting close enough to the center of the highly stressed and possibly unstable beam to compare string to a scale. Since the steel was painted scrap with unknown properties and history including possible cracks I went with direct measurement instead of calculation, though I did practice it with the averaged dimensions of my CAD model as a cross-check that I was running the calculators correctly. I used Ix to compare and nearly match the relative stiffnesses of components of the spliced beam, to distribute the stresses evenly among the bolts.

If a project at work justified it the structural design would be given to a real mechanical engineer so I didn't need to know more than the basics, to catch bad assumptions from electrical engineers. I chose to become a factotum who filled the gaps between the specialists and knew how to build and test their concepts, a position with little competition, what Arnold Wilkins did for Robert Watson-Watt. I had learned the graphical construction and algebraic model of wave interference that he used to demonstrate radar. As another tech rather rudely displayed on a bumper sticker, "Techs do what engineers can only dream of".

I had struggled with higher math at university once it ceased to be intuitive, beyond Differential Equations. Much later in night school it was easier when taught as a practical tool instead of an art form, but I couldn't handle the long late night commuting I'd need to complete an EE degree. Waking up when the wheels hit the roadside gravel was convincing.

In a few words you present a lot. Caused me some thinking.

I will leave that web-page as-is, if that's alright. It's a topic I want to leave well alone now.

Best wishes, Rich S

In a few words you present a lot. Caused me some thinking.

I will leave that web-page as-is, if that's alright. It's a topic I want to leave well alone now.

Best wishes, Rich S

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

It's fine as-is.

Was I correct to assume that matching Ix between the channel pairs and a splice plate bolted between the webs would evenly distribute the shear and bearing stresses on the bolts that joined them, regardless of the bolt pattern geometry? IOW would the channels and vertical splice plate both deflect identically where they overlapped?

The splice was more complicated, with horizontal plates joining the flanges at the discontinuity where the channels abut, but one vertical plate and a row of bolts is easier to describe and consider. I couldn't find an example of a splice in such narrow channel to analyze or copy.

the trolley wheels occupied almost all the space between flanges I lengthened the web plate(s) enough to let the shear and bearing at the center and outer end bolts support the moment at max load. Actually I grabbed a likely-looking piece of scrap 3/8" plate at a helpful steel constructor's shop and then tried to figure out its best use. It was larger than my rough estimate of the minimum size.

I learned long ago that if I have enough of the critical material to build something one way it's enough to build it several other ways, so I could order (or scrounge) material before finishing a detailed design, adapt to changing specs and deliver faster than the competition. At Mitre the engineers did the same, they ordered the critical components and then handed me the project to complete the design and build it.

Sorry haven't managed to look at your question. (Earthing/Grounding - made a "wander-lead" from the new Main Earthing Terminal and found visible Earth-bonding in the bathroom did not connect to Earth/Ground - equipotentially connected everything but somehow a connection to Earth hadn't happened - started diagnosing what was wrong).

I have been asked if I could derive an equation like

"y=sigma.L^2/6EH" for a simply-supported-beam centrally-loaded

for a simply-supported-beam uniformly-distributed-load.

So an engineer can show to the client in a single "sparse" equation the relationship between internal stress state and deflection.

Never done uniformly-distributed-load beam calculations. The formulae - yes I have looke them up, but lacking experience of their application - theory <-> practice - am a little wary of "just" juggling mathematical symbols.

Rich S

Sorry haven't managed to look at your question. (Earthing/Grounding - made a "wander-lead" from the new Main Earthing Terminal and found visible Earth-bonding in the bathroom did not connect to Earth/Ground - equipotentially connected everything but somehow a connection to Earth hadn't happened - started diagnosing what was wrong).

I have been asked if I could derive an equation like

"y=sigma.L^2/6EH" for a simply-supported-beam centrally-loaded

for a simply-supported-beam uniformly-distributed-load.

So an engineer can show to the client in a single "sparse" equation the relationship between internal stress state and deflection.

Never done uniformly-distributed-load beam calculations. The formulae - yes I have looke them up, but lacking experience of their application - theory <-> practice - am a little wary of "just" juggling mathematical symbols.

Rich S

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In the limiting case where the uniformly loaded, fixed end beam has zero stiffness it will hang in a "catenary".