I-Beam Deflection Calculation Problem

[Motorhead]

I work at a community college and doing some bending beam experiments for student labs. When doing simply supported beam deflection calculations and measurements, I am seeing some inconsistent data that I cannot explain.

If I test a basic cross sectional shape (square, rectangle, tube), I get very good results with theory. When I try to do an I-Beam however, the data is way off from theory...on the order of 50% more measured deflection that what have been predicted. Even if the area moment of Inertia between basic shape and I-beam are about the same.

I know the general theory says that stiffness should be proportional to 1/(EI). So why should I see a difference? My gut feel says that there is something different on how the load is transferred between the web and the flange that is not present in the basic shape. In other words, the strain may not be constant laterally across the cross section as basic theory would imply.

My background is Mechanical and my basic structural books do not address anything like this. I am hoping this is something addressed and understood by structural engineers. If you could help with some insight, it would be much appreciated. It always a little frustrating when you cannot explain unknown results to students.

 

[JAE]

Post a sample calculation of your I-beam anticipated deflection - just for checking.

 

[Motorhead]

I don't have the exact numbers with me, but it was a simply supported beam (approx 3 feet in length) with point load applied at mid-span. Load is applied downward on the top part of the flange. The equation used for maximum deflection was ymax = -PL^3/(48EI). Material is A36. I have calc. max/min stress and was no where near yield strength.

 

[desertfox]

hi Motorhead

So what deflection did you measure? also you mention the moment area method so what figure of deflection have you calculated for that? and can you upload your working out.
Finally what size "I" beam is it as you have not stated it.

desertfox

 

[SteelPE]

More information is going to be required. The beam size, your calculations, support conditions and loading are definitely going to need to be stated.

However, since I like to jump to conclusions your deflection calculation is not technically correct, you need to add in shear deflection into your calculation. Typically this is ignored because the results are so small but maybe this is causing you some problems?

I have only run this calc a few times in school so I am not sure what the major influences are in the part of the calculation.

 

[JoshPlumSE]

What sort of span to depth ratio do you have for your beams? Wide flanges have such a skinny web, that it wouldn't surprise me if you're getting some shear deformation that you did not anticipate in your hand calculations.

After all, shear deformation usually just isn't on people's radar. I don't even know the formula for shear deformation of a beam with a point load. Anyone have tht handy?

I ran through some virtual work equations (remember those?!) to come up with the following. Though, I don't have 100% confidence in them. I just don't do this type of calc very often:

Delta = P*L / (4*A*G)

where w = distributed load, L = length of beam, A = shear area (approximately equal to the area of the web), G = shear modulus.

 

[nickky]

I agree with SteelPE. Please upload more info and your calculation for us to see.

By the way, sometimes even issues like lateral buckling, torsional buckling and web buckling may effect although they don't happen or seen by eyes. Although the sapn of 3 feet is short but everything here is based on proportions. Tell us if you have provided lateral support for the top flange which is in compression.

 

[BAretired]

Give us your calculations and your test results. Otherwise, we cannot offer any useful information.

BA

 

[weab]

I doubt this is the case, but confirm that you are not stressing the beam beyond the yield stress. Otherwise your deflections will be off.

 

[steve1]

According to Blodgett the following is the formula for calculating shear deflection of a simply supported beam with a concentrated load:

delta = (P)(L)(alpha)/(4)(A)(Es)

alpha is a form factor for which he provides an equation

This is in section 2.6-3 of my Blodgett book

 

[Lion06]

The shear deflection, using Castigliano's Theorem (I've always liked Castiglian's Theorem - both for deflection and indeterminate analysis), is the strain energy, U, as a result of the shear. This is equal to the integral of
(fs*V/(GA))*(partialV/partialP)dx, where fs is the form factor, V is the shear force function, G is the shear modulus, A is the area, and (partialV/partialP) is the partial derivative of the shear function with respect to the concentrated load P.

You can quickly see that V=0.5P, therefore, (partialV/partialP)=0.5.

This gives the shear deflection as the integral of (fs*0.5P/GA)*0.5dx from L to o.

Integrating gives (fs)PL/(4GA), as noted above, where fx=alpha=form factor

The form factor is A/I^2*(integral of ((Q^2)/(b^2))dA)

Shear deflections are usually insignificant, as noted above, but if you have a short span beam (3'), for a W8, you could get significant shear deformation.

 

[Motorhead]

Thank you very much for the insight provided already. I will confess to being ignorant of shear deformation. Must be something not taught in general mechanical engineering curriculum. We have some CE's around here and will consult there library of books for more information. Is there any book in particular any of you would recommend consulting for this type of calculations?

As to particulars to my setup, this is something we ran weeks ago and I do not have the exact setup numbers or calculations and have broken down the setup. I would like to investigate this further though for my own learning. I will see if I can re-setup the experiment and report back with exact numbers.

Thanks for your help so far.
Jeff

 

[Lion06]

Castigliano's Theorm is in Hibbeler's Structural Analysis text (I'm sure it's in other, too).

 

[desertfox]

Hi Motorhead

From Roarks stress and strain it says vertical deflection due to shear is is negligible under the conditions stated at the beginning of chapter 7 page 89 5th edition, in those conditions a metal beam of compact section should have a span to depth ratio of not less than 8, it also says that ratio increases to 15:1 for beams with relatively thin webs.
So when you dig up your beam have a look at the span to depth ratio it might give a clue.
I have also posted a site that gives info about shear deflections.

http://www.bth.se/tek/amt/mtd009.ns...0CE00529A64/$FILE/przemieniecki_p_201_208.pdf

Another article I posted here says unless the beam is very short or heavily loaded it is shear deflection is normally about 1% of the total deflection.(page 17)

http://courses.washington.edu/mengr354/jenkins/notes/chap3.pdf

desertfox

 

[rb1957]

i think steve's on the right road ... the difference between an I-beam and a basic (rectangular) section is deflection due to shear (particularly if the beams have the same I) ... the I-beam will have a much thinner web (duh) and concentrate the area away from the NA.

pix of the X-sections would be nice.

the cynic in me wonders if this could be a student post ??

 

[desertfox]

Hi rb1957

Its not a student post

desertfox

 

[csd72]

Check that the beam is not buckling.

 

[hokie66]

I wonder if the difference is not something to do with the test setup and measurement rather than shear deflection. Other than the W section, the others are torsionally stiff. So a slight eccentricity in loading, support conditions, or in positioning a dial gauge could give misleading deflection readings.

 

[civeng80]

I remember doing these kind of experiments in 2nd or 3rd year structural mechanics courses.

Results are always out quite a bit from theory, but in this case I dont think it would be because of shear deflection. I recall shear deflection is only significant in low span to depth ratios. I recall we always had to comment on our assumptions and reasons for variations. I believe Hokie66 is on the right track with his comment above.

 

[IJR]

Strange

When in college I was given an air table and fancy metering to measure g. My finding was about 1.8go where go is what Newton found out with no air table or a casio.

Now this makes me think hokie66 may be right.

And why did deflections by theory and practice match exactly when they built The Clinton Library?

respects
ijr