Von Misers criterion or Principle stress? 3

[george69]

Hi All

My question is in relation to Von Misers Yield stress criterion vs Principle stress limits.

If I have a simply supported beam with a point load in the middle, should the longitudinal bending stresses in the beam be checked to priciple stress limitations or to Von misers criteria.

The reason I ask is that I thought both methods were acceptable but were just differnet ways of looking at stresses. However, when you look at the Australian code provisions, one requirement is to have s1<=0.66Fy when checking priciple stresses, whereas the other requirement is to use VM equation; s1^2-s1s2+s2^2<= (Fy/1.1)^2. Note the 1.1 is the FOS on yield.

Clearly the VM equation will give you a better answer by allowing you to accept higher stresses.( i.e 1/1.1 = 0.9 compared to 0.66) So... why would you not ALWAYS use VM equation( and in my example, with s2 value zero)???

From what I have gathered from others, it seems as though both provisions must always be checked. But at the moment I can not see why. Is there a requirement that BOTH s1 and s2 must be present on an element such that one can use VM equation?


many thanks to all of you

 

[PEinc]

I've had the same question myself. A state transportation department near me requires the same VM analysis. I usually have bending in one direction. Therefore, no s2. I must be missing something.

Anyone have a good explanation?

http://www.PeirceEngineering.com

 

[Ron]

I typically only use von Mises stress analysis for plates and membranes. It can be useful for localized stress risers in other shapes, but awfully hard to model correctly in many shapes.

 

[abusementpark]

I'm not sure if I understand the question. Are you talking about the combined effect of longitudinal bending stress and shear stress as a failure criterion?

I feel that this is generally neglected in wide flange members because different portions of the beam do the majority of the work in resisting bending and stress stress, respectively. In other words, the flanges resist most of the bending, and the web resists most of the shear.

 

[george69]

Actually no. The case I am referring to is a simplistic case where the shear stress is 0. An example would be a simply supported beam with UDL on it. Thus, at midspan, the only stress is the longitudinal bending stress. The question is which code provision to use and is best representative of the stress condition. That is; s1<0.66Fy or VM criteria(where s2 = 0) or maybe both criteria?

Example: simple supported beam 5m long, udl 10Kn/m, Z=100e3mm3. Mmid = 31.25kNm. Say steel yield stress is Gr 450Mpa.

stress and midspan = M/Z = 312.5Mpa > 0.66Fy( which is the principle stress code limit and appears to be no good). But when using the VM criteria 312.5Mpa < Fy/1.1 (VM criteria with FOS of 1.1 on yield and appears to be OK!)

 

[Ron]

It is generally accepted to use only the 0.66Fy criterion. The failure predictability under simple bending is such that further characterization of the stress distribution is not necessary.

Besides, you are generally required to evaluate under the more stringent criterion.

 

[TERIO]

Here are my thoughts -

If using principal stress limits you would check the location of maximum bending against 0.66Fy and maximum shear against 0.4Fy.

If you are using von Mises then I think you would be obligated to examine all locations on the beam. Even if neither the bending or the shear is at a maximum the combined effect could be. Since you have done a somewhat more detailed analysis you could reasonably use a smaller factor of safety.

 

[EdR]

Going back to the basic definition of Von Mises stress you find that it related by a constant to the maximum shearing stress on the octahedral plane (sigvm = (3/sqrt(2))tauoct)) so basically a Von Mises criterion is a shearing stress criterion while a principal stress criterion is a normal stress criterion. As such both should be checked....

AS to what the applicable code requirement is I don't know....

Ed.R.

 

[Lion06]

I'm with Ron. I'd only use the VM stress for plates or shells. Even with plates, it's really only applicable because of the torsional shear stresses. If you have a simply supported plate (on two opposite sides, with the opposing two sides free) then you have a beam situation - though not as clearly defined because of the absence of flanges.

 

[omairzali]

First of all, to correct EdR statement, Von mises criteria extends from from the root criterion of Tresca condition (i.e. yeilding failure occurs when maximum shear stress within the material reaches the maximum shear stress sustainable under uniaxial stress conditions)....you got to understand this via mohr circle! The only difference in Von mises and tresca criterion is the yeilding surface...see this visual for better understanding

http://en.wikipedia.org/wiki/File:Yield_surfaces.svg

So the Von mises criterion is one of the YIELDING criteria, not Shearing stress criterion.

(Remember that for materials subjected to uniaxial stress conditions, material yeilds at a simple known Yeild stress.
However, for materials subjected to Bi-axial or Tri-axial stress conditions, material yeilds at a complicated range of stress. This complicated range of stress is defined and approximated by certain yeild criteria, which includes Von Mises.)

Now coming back to the question:
The case of the allowable stress comes from the fact that it is the result of the conservative approximation of
Plastic Section Modulus (abt x-axis)-> Zx , and Elastic Section Modulus (abt x-axis) -> Sx
i.e. Zx / Sx = 1.1....recall the allowable bending stress formula = 0.6*Mn/Sx = 0.6*(Mp)/Sx = 0.6*(Fy*Zx)/Sx
where, Mp = Plastic moment (due to full yeilding of the crossection)
Hence, you get allow. bend. stress= 0.6*Fy(1.1)= 0.66Fy

So the moral of the story is the codes are conservative to a reasonable degree (and they have to be!) and you gott to follow them.

 

[george69]

omairzali. Agree with your comments. So in short are you saying that both criteria need to be checked all the time regardless of stress condition. Remember that for a simply supported beam, as per my example above, has only one stress(bending in long direction).

I gather from your comments that the principle stress needs to be less than 0.66Fy, AND ALSO when substituted into VM equation the stress needs to satisfy <=Fy/1.1 (obviously with s2 being 0). That is both criteria need to be satisfied!

 

[omairzali]

george69, basically once you satisfy the check of 0.66Fy, you are automatically satisfying the Von mises (since 0.66 Fy gives you a lower allowable bending stress than the von-mises criterion). So, technically the code is helping you in a way to NOT check the von-mises criterion. (for the purpose of saving time and effort).

The other thing is...in your example, you just dont have only bending stress,you also have shear stresses acting too.

 

[george69]

Omairzali

I see your point and yes, satisfying 0.66Fy automatically satisfies VM..

BTW, in my example, it was a simplistic case looking at midspan, where shear is 0. Just to emphasis my case with the longitudinal bending stress

Thanks again for your input and help!!

 

[prex]

george69,
reread carefully the standard you are using. It is simply impossible that a structural standard would limit an actual (equivalent) stress to F[sub]y[/sub]/1.1 .
And whether the equivalent stress is calculated with VM or Tresca doesn't change much, as correctly noted above by omairzali.
The criterion F[sub]y[/sub]/1.1 is likely used by your standard (that I don't know of) for factored loads in the ultimate state of stress, in that case you would be mixing up oranges and apples.

prex

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[Lion06]

I'm just asking the question, but how would you even apply a VM stress criteria to a simply supported beam? That would require checking many different locations at many different sections. At the point of maximum longitudinal stress (max moment at the extreme fiber) there is 0 shear stress (so VM stress doesn't really apply, right?), and at the point of maximum shearing stress the longitudinal stress is 0 (so again, VM stress doesn't really apply, right?).

Even for the case of a beam continuous over a support where there is max moment and shear at the same section, the parts of the beam that take the respective forces are different. I don't imagine that an extremely localized yielding (at some arbitrary location between the neutral axis and the extreme fiber) is detrimental to the beam. That is the reason that shear yielding has a reduction factor of 1.0 in LRFD, right?

 

[BAretired]

At the midspan of a simple beam uniformly loaded, [&sigma;]2 = [&sigma;]3 = 0, so the von Mises yield criterion simplifies to [&sigma;]1 = [&sigma;]y. The material at any depth is subjected to pure axial stress. Failure occurs when the stress reaches yield point, so there is no difference between the two methods.

Check out this link:

http://en.wikipedia.org/wiki/Von_Mises_yield_criterion

BA

 

[george69]

Hi BAretired and thanks for your comments.
The issue is not so much the equations, or what the euqations reduce to. It;s really about the factors of safety. s1<0.66Fy(for Principle stresses) versus s1<Fy/1.1(for VM criteria) as a code requirement.

Thanks to all who answered this query - Big Help!!!

 

[Lion06]

george-

I'm still questioning the applicability of the VM stress. Does your code require you to check that? AISC doesn't (at least as far as I know). The max longitudinal stress is always as a location where there is 0 shear stress, and the max shear stress is always at a location where there is 0 longitudinal stress (for a beam element). Because of this, VM stresses really only apply at a location where the shear/longitudinal stresses are not a maximum. How can you possibly check these locations efficiently?

 

[apsix]

george69
Can you give an AS code reference on the stress limits you quote? I couldn't find it in AS4100 or AS3990.

If you have to satisfy both requirements then obviously the principal stress limit will govern for a SS beam, as noted above.

I see the real value of the VM limit check being for combined flexural, compression, shear and/or torsion stresses, as determined by FEA.

 

[george69]

hi apsix

Sure. AS1250 is the working stress code. I think it is clause 5 that stipulates the principle stress limit. However, the VM yield limit is referred to in the Source book for the AS1250. The Source book is a commentary on the 1250.