roarks formula confusion 5

[greycloud]

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

 

[JLNJ]

Maybe units are buried in the constant(s).

 

[Structcon]

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

stress therefore is psi

 

[LonnieP]

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

LonnieP

 

[TehMightyEngineer]

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

Maine EIT, Civil/Structural.

 

[IDS]

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/

 

[JStephen]

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.

 

[JedClampett]

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.

 

[greycloud]

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.

 

[greycloud]

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.

 

[rb1957]

"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

 

[brut3]

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.

 

[IDS]

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/

 

[JedClampett]

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).

 

[greycloud]

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

 

[brut3]

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

 

[rb1957]

"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

 

[IDS]

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/

 

[IDS]

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/

 

[greycloud]

IDS:

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.