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roarks formula confusion
5

roarks formula confusion

roarks formula confusion

(OP)
Greatings everyone

I'm trying to get how roark in his book "formulas for stress and strain" got the stress equation for bending stresses in rectangular plates.

I noticed that the equation is composed of the moment formula divided by just T^2 so how did he arrive at this result.

I hope you can help me resolve this mystery.

Thanks in advance

RE: roarks formula confusion

Maybe units are buried in the constant(s).

RE: roarks formula confusion

beta is unitless.
q is in psi.
b is in inches.
t is in inches.

stress therefore is psi

RE: roarks formula confusion

A typo maybe? Section modulus should be B*T^2/6. Maybe the '/6/ got left out. Is his answer correct?

LonnieP

RE: roarks formula confusion

Maybe check to see if the book has any errata online somewhere?

Maine EIT, Civil/Structural.

RE: roarks formula confusion

Seventh Edition gives maximum stress in short (b) direction as Beta.q.b^2/t^2, and Beta for an infinitely long plate is 0.75.

Stress = M/Z = (q.b^2/8)/(t^2/6) = 0.75q.b^2/t^2

So it appears to be correct (at least in the 7th Edition, for an infinitely long plate).

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: roarks formula confusion

I'm not sure what formula you're looking at. But in a lot of the plate cases, the bending moment is bending moment per unit width, and the bending stress is then Mc/I = M(t/2)/(1/12*1"*t^3) = 6M/t^2. If the 6 is missing, it may be a typo. The units on M or in-lbs/in.

RE: roarks formula confusion

As I'm sure you all know, Roark is a compilation of calculations and work by others, with just the final formula given. So I find it's most useful to direct me to the original authors work, like Timoshenko's "Theory of Plates and Shells" and the Bureau of Reclamation's "Moment and Reactions of Rectangular Plates"
As far as errata, you probably could publish one and make money. There's mistakes. Sometimes the controlling case is not given. Take everything in there with a healthy dose of skepticism.

RE: roarks formula confusion

(OP)
First of all, thanks everyone for responding

LonnieP: as you said the 6 is somehow missing but so is a b for base length. B is used here as coefficient which i dont know is based on what.

IDS: beta seems to be part of the moment formula. you can see that if you take a look at timoshinko's "Theory of plates and shells" Beta is mentioned as a numerical factor in the bending moment formulas. so I dont think it relates to the missing terms of the section modulous.

RE: roarks formula confusion

(OP)
Jed: I'm indeed trying to compare the results with that in timoshinko to get what is beta and how he arrived at the stress formula so it would be helpful if u assist me in that.page 127 of timoshinks's mentions Beta as a numarical factor depending on abscissa of x point whatever that means.

RE: roarks formula confusion

"LonnieP: as you said the 6 is somehow missing but so is a b for base length. B is used here as coefficient which i dont know is based on what."

first off, it's moment per unit width (ie b = 1")
second, 6 is hidden within the other coefficients.
third, if you want to understand the equations, check the references, i'm sure you'll be lead to our favourite irish stressman ... Tim O'Shenko.

Quando Omni Flunkus Moritati

RE: roarks formula confusion

Greycloud,

Please see the attached derivation I came up with. I assumed that the infinitely long plate was acting as a simply supported beam across the short span and then followed through with a simple Fb = M/S calculation. I'm not 100% sure that is exactly how the equation was derived but the logic seems to make sense to me and produces the 0.75 Beta factor that you were looking for.

RE: roarks formula confusion

Quote:

IDS: beta seems to be part of the moment formula. you can see that if you take a look at timoshinko's "Theory of plates and shells" Beta is mentioned as a numerical factor in the bending moment formulas. so I dont think it relates to the missing terms of the section modulous.

If your version gives the same formula as the one I quoted then it is correct. Why do you think it is wrong?

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: roarks formula confusion

It says in Timoshenko that "beta is a numerical factor....". The formulas for moment are beta*q*a^2. So if q is a pressure and a is a length, the units are force or moment/length (in-lb/in). Roark (or more likely his grad students) converted the moment to a stress by multiplying by a 6/t^2 factor (the section modulus).

RE: roarks formula confusion

(OP)
brut3: can you tell me where u got the moment equation from? by the way i didn't read timoshinko's fully.

IDS: beta is a factor added to the moment equation not the stress formula meaning it does not incorporate terms from the section modulus. that is what i meant to say. so if you disagree please explai why

Jed: that is right and what I want to know is where did the 6 in the section modulus go in roark's. hope u can help with that

RE: roarks formula confusion

From a simply supported beam maximum moment formula wl^2/8 (w = pressure * tributary width)

RE: roarks formula confusion

"what I want to know is where did the 6 in the section modulus go in roark's." ...

why do you not think it is absorbed within the beta parameter ? the Roark expression is very clear (to my reading) ...
stress = beta*pressure*b^2/t^2, b is a plate dimension (short side of a rectangular plate), t is thickness, beta is a look-up. clearly beta is not dimensionless.

possibly you're confused with Timoshenko's beta not being the same as Roark's ? possibly Timoshenko absorbed the 6 into his beta (it's been a while since i cracked that book !)

IDS's q*b^2/8 is maximum moment in a beam, span b, with a distributed load q. as he shows, 0.75 = 6/8, and the "missing" 6 is found.

no?

Quando Omni Flunkus Moritati

RE: roarks formula confusion

Quote:

IDS: beta is a factor added to the moment equation not the stress formula meaning it does not incorporate terms from the section modulus. that is what i meant to say. so if you disagree please explai why

I'm not sure what you mean. Beta is the factor in the stress equation, not the factor for bending moment. If you want to use the factor for bending moment just divide Roarke's factors by 6, and then you can use Z per unit width, instead of t^2 in the equation.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: roarks formula confusion

Quote:

stress = beta*pressure*b^2/t^2, b is a plate dimension (short side of a rectangular plate), t is thickness, beta is a look-up. clearly beta is not dimensionless.

?

Stress = Beta * pressure * length^2/length^2, so beta is dimensionless (as said by Structcon in the 3rd post)

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: roarks formula confusion

(OP)
IDS:

Quote (I'm not sure what you mean. Beta is the factor in the stress equation, not the factor for bending moment. If you want to use the factor for bending moment just divide Roarke's factors by 6, and then you can use Z per unit width, instead of t^2 in the equation. )


timoshinko includes the beta as a factor in the bending moment equation but do u mean that roark altered this factor by multiplying it with 6?

the derivation u all made makes since but here is the problem, beta is a function of the aspect ratio but from your derivation it doesn't seem to be affected by the aspect ratio.

again thank you all for responding and for beering up with me.

RE: roarks formula confusion

Well, β in the above discussions depends indeed on the aspect ratio, but the case with b/a=∞ has been taken for direct comparison with the beam equation.
And if you look at Timoshenko (you should, as you cite it so often) you'll see that the factor β for the center moment in a simply supported rectangular plate equals 0.125 (b/a=∞), and you see that this is 0.75/6 or 1/8!
Also, if you compare β values in table 8 page 120 of Timoshenko with β values by Roark, you'll see that they exactly differ by a factor of 6.
So what are we discussing about? β is not a physical quantity, it is just a numerical coefficient, and any author can use its own definition of it without being in contrast with any theory.

prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads

RE: roarks formula confusion

Quote:

beta is a function of the aspect ratio but from your derivation it doesn't seem to be affected by the aspect ratio.

I hope the additional details given by prex above will clear this up. I just looked at the beta factor for a very long plate (where the bending moment/m width is the same as for a beam 1 metre wide), because this is the easiest to check, but you can see in Roark that beta varies depending on the configuration of the plate. Anyway, it's good to get confirmation that Roark's values are indeed equal to Timoshenko's multiplied by 6.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: roarks formula confusion

(OP)
well thank you indeed for the clarification prex u are correct indeed and thanks Doug for your help as well and to everyone else now it makes since.

since we are on this does anyone know what is the moment formula for a rectangular plate subjected to a uniform moment at middle of longer side.

RE: roarks formula confusion

"uniform moment at middle of longer side" ? maybe a point (discrete) moment at the middle of the long side ??

how is the plate supported ? corners only ? pinned along all edges ?? fixed ??

Quando Omni Flunkus Moritati

RE: roarks formula confusion

(OP)
it is the same case as moments around edges but instead applied in middle of the plate.

the plate is clamped from all sides

RE: roarks formula confusion

Quote (prex)

Also, if you compare β values in table 8 page 120 of Timoshenko with β values by Roark, you'll see that they exactly differ by a factor of 6.

They should not differ by a factor of exactly 6 because, for a plate, the section modulus includes a factor of (1 - ν2) where ν is Poisson's Ratio. When ν is 0.3 in the case of steel, S = bt2/5.46.

BA

RE: roarks formula confusion

if the plate has fixed edges, and the moment is applied on the edge ... then the plate will see nothing !?

Quando Omni Flunkus Moritati

RE: roarks formula confusion

rb, I'm guessing that the moment is applied at midspan in both directions but it isn't clear. In any case, it belongs in a separate thread.

BA

RE: roarks formula confusion

(OP)
BA.: stress equations for plates in timoshinko's use 6 always. the usage of the factor u are talking about was included in buckling strength as I remember.

rb:Regarding the bending case as BA said the moment is applied midspan of the plate along the shorter length with all edges clamped.it needs a separate thread indeed.

RE: roarks formula confusion

Quote (greycloud)

BA.: stress equations for plates in timoshinko's use 6 always. the usage of the factor u are talking about was included in buckling strength as I remember.

On p. 127 of "Plates and Shells" which you quoted earlier, Timoshenko uses the factors β and β1 to calculate Mx and My for Hydrostatic Loading on a simply supported rectangular plate. He does not provide stress equations.

If Roark (I don't have his book) has interpreted the stress in the plate from Timoshenko's work, he should not be using a section modulus of bt2/6. He should be using the plate section modulus of bt2/6(1 - ν2).

BA

RE: roarks formula confusion

2
BA, the numerical factor we were speaking about is exactly 6.
The factor (1-ν2) is included in the values for β, as you can see from the fact that Timoshenko (p.120 as well as p.127) specifies ν=0.3 in the tables for β. Those tables serve to calculate plate moments, and the stress is simply 6M/t2 (M=moment per unit length).

prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads

RE: roarks formula confusion

I was going to say what prex just said.

I would add that my earlier statement that the bending moment per unit width in a very long slab is equal to that in a beam under the same (unit width) load was overly simplistic. Nonetheless, that's the way it works out.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: roarks formula confusion

prex and IDS,

Yes, you are both right. Sorry, greycloud for rocking the boat.

BA

RE: roarks formula confusion

(OP)
no problem man

RE: roarks formula confusion

Boat-rocking is good!

It prompted me to do a bit of research on the theory, confirm that the different looking formulas in Wikipedia, Pilkey and a couple of other sources all gave the same answer, which was the same as the Roarke values, and compare that with some quick FE analysis.

One thing I did discover is that although the formulas converge quite quickly for plates with an aspect ratio of up to about 5:1, for very long plates they converge quite slowly. I couldn't work out why the moment was increasing above wl^2/8 as the aspect ratio increased; it turned out I was just doing too few iterations.

Amazing that all this was worked out in the early 1800s.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: roarks formula confusion

(OP)
good for you man, you mentioned a couple of sources other than pelky and roark so can u give me their names please

RE: roarks formula confusion

(OP)
thanks

RE: roarks formula confusion

I still have a problem understanding how the stress in a plate can be identical to the stress in a rectangular beam when the moment per unit width is identical.

In the case of a beam, the top fibers compress and are free to expand sideways due to Poisson's Ratio. The bottom fibers stretch and are free to shrink sideways due to Poisson's Ratio. Under load, the beam's cross section is not exactly rectangular, but trapezoidal.

In the case of a plate, the top fibers are compressed but are not free to expand sideways. The bottom fibers stretch but are not free to shrink sideways. Under load, the cross section remains rectangular. If the applied moment is Mx, then My is required to prevent each little section of plate from becoming a trapezoid. This requires strain energy in the Y direction.

My current quandary is why that does not decrease the stress in the X direction. A uniformly loaded simply supported plate of span L and infinite width has a maximum moment of wL2/8. Does it have precisely the same stress as a square beam of the same depth?

For me, it is not intuitive.







BA

RE: roarks formula confusion

Quote:


For me, it is not intuitive.

It wasn't for me either, which is why I wasted* half a day looking into it.

For me the non-intuitive part was that the longitudinal curvature at the mid-span of a very long slab must be close to zero, so how can there be a longitudinal moment?

I re-intuited it by imagining a very long slab (if we build it round the Equator it will be effectively infinite) simply supported on the two long sides. If we now cut a unit length slice of the slab with frictionless cuts of zero width there will be no change in the state of the cut slice, because there is zero shear transfer across any transverse section, and the slice is still restrained longitudinally.

If we now increase the width of the cuts to allow the two cut faces of the slab to rotate; the base of the slab will expand in the longitudinal direction, and the top will contract. This will tend to reduce the transverse stresses, due to the Poisson's Ratio effect, so the slab will deflect downwards to maintain moment equilibrium. The transverse bending moment at mid-span will always be wL^2/8, and the maximum transverse stresses will always be wL^2/8/(d^2/6), to maintain moment equilibrium, but the deflection will increase (to 5wL^4/384EI), depending on the Poisson's Ratio. For a Poisson's Ratio of zero the longitudinal moment would have been zero before the cut, so the deflection would already have been 5wL^4/384EI; for any positive Poisson's Ratio the deflection (before the cut) would have been less.

The Poisson's Ratio effect provides an effective prestress in the longitudinal direction. This does not reduce the transverse bending moment (which is controlled by the applied loads), and hence does not change the transverse stresses, but it does reduce the vertical deflection.


* Not really wasted.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: roarks formula confusion

Thanks Doug, I think I'm beginning to see it more clearly now.

BA

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