Confirmation about failure theories and Von Mises

[breaking_point]

I've been learning about failure theories of materials, namely the Von Mises and Tresca theories. I read that these theories are used because the accuracy of just comparing the yield stress to the maximum yield for a uni-axial test in insufficient in more complex stress states.

One thing I did notice is that when we design a beam by hand, say simply supported beam with a uniform load, we do not use these theories. Is this because we typically design and check the beam at maximum bending moment and shear for maximum normal and shear stress respectively, but these do not occur occur at the same point? i.e. at the maximum normal stress from bending there is no shear stress, and at the maximum shear stress there is no normal stress, so it is okay not to just use a basic check of normal stress against allowable yield stress and shear stress against the allowable shear stress?

I hope this makes sense, and someone can clear up my understanding!

 

[dik]

Partially that... and also, it's not common for steel beams to be loaded in shear to the point of failure... in 45 years, the number I've encountered could be counted on my fingers... not, that you shouldn't be careful.

Dik

 

[breaking_point]

Hi Dik,

I appreciate the reply! I read that ductile materials are typically governed by their shear strength, and in a uni-axial test of steel, the specimen normally ruptures along a 45 degree plane (the maximum shear plane). However, in the way typical structures are design normal stresses from bending are normally prominent, and therefore shear is less of a concern (but still important to check). Is this correct?

Cheers

 

[271828]

Two things:

von Mises is used to detect yielding. With the ultimate strength methods that are used today, yielding doesn't indicate failure, and is of little interest.

There is little or no interaction of shear and bending in steel beams. See Daley et al. that is cited in the 2016 AISC Commentary.

 

[dik]

Typical tension tests of mild steel usually fail by what is referred to as a cup and cone type of failure... the steel necks down locally and creates a triaxial stress state... the central portion fails in a 'brittle' tensile failure through the grain boundaries which caused the outer remaining portion to fail in a shear failure. This develops the 'classic' cup and conde failure observed.

Keep shear in the back of your mind, and, when you have heavy loads on short spans it is often prevalent. It's been several years since I was last involved with a project where shear was a significant factor.

putting the horse before the cart... modified slightly
Dik

 

[breaking_point]

271828 - Thank you for the reply! Please could you elaborate why we aren't interested in yielding? If we were to design a steel beam (assuming fully restrained and shear is not an issue), we could size it based on the sigma = MY/I equation, where we could calculating a resisting moment and the yield stress would be used as sigma. If we were to design a beam using an allowable stress beyond the yield stress, would this not make our elastic deflection calculations (based on a linear young's modulus) inaccurate?

Dik - thanks again for the information!

Thanks

 

[271828]

We typically check strength and serviceability limit states separately.

In North America (I assume the method is similar elsewhere.), when we're checking strength, we compare the factored load and the design strength. The latter is, roughly speaking, the strength (moment, shear, etc.) that is predicted to cause failure, multiplied by a resistance factor.

Take the example of a braced beam with stocky web and flanges in bending. As the moment increases, the outside corners of the compression flange tips yield first. (Compressive residual stresses are maximum near there.) However, additional moment can be applied, meaning it hasn't reached its actual failure moment. As the moment increases, yielding will spread until nearly the entire section has yielded. At that point, no further moment can be added. That's the moment we'll factor down a little to get the design strength and compare to the factored loads. Deflection is not checked at this step.

Next comes serviceability: When calculating deflection, we're not using factored loads. The load levels are much lower, so linearly elastic calculations are accurate enough.

 

[gravityandinertia]

First of all, failure theories are used in beam design for code based systems, however their application were done by the code writers in their investigation of what is safe. From there, along with testing they derived the limiting states and values that the codes use.

Additionally, I think most people don't take enough time to understand the background of these theories. Tresca criterion is built off the assumption that materials only fail in shear. This is where most people get lost. They think of a uniaxial tension test and say that there is no shear present in that load, and yet the sample still fails. Which means it's necessary to understand Mohr's circle and that a stress state at any point can be "reoriented." If you draw Mohr's circle for a uniaxial tensile test, you will find when the uniaxial stress reaches yield, the maximum shear for any orientation of that element is yield stress/2. This becomes the shear criteria for yield. The Tresca criteria now says that if the maximum shear in your design, {principal stress 1 - principal stress 3}/2 < yield stress/2 you haven't yielded. Of course, since you're dividing both sides by 2, those cancel and you are left with the criteria, {principal - principal 3} < yield stress.

von Mises is built off the assumption that each material has an ability to handle a certain amount of distortion or distortion energy. Sometimes, opposing stresses, cancel each other out. For example picture a square, if all sides of the square have equal stresses acting on them, the square will "shrink" uniformly and not be likely to fail. However flip the direction of only two opposing sides, now you have a condition where the material is being pinched in 1 direction then pulled in the direction 90 degrees to that. This will create significant distortion from the materials initial shape. This configuration under the same magnitudes of stress as the previous is much more likely to fail.

These theories are also applied in FEA as parts go past yield to determine the plasticity level, or permanent deformation as well as stiffness at specific levels of strain.

 

[rb1957]

and in beam stress analysis we're looking at details. The caps (extreme flanges) react tension or compression, so a simple stress and a simple failure criteria. Webs react shear (and usually as a stable non-buckling web) so again a simple stress and a simple failure criteria.

If you Really wanted to work the analysis to death you could consider points away from the maximum stress, where maybe the interaction creates a more critical location. This would be relevant if your beam cross-section was unusual (like maybe a web with stepped thickness.

another day in paradise, or is paradise one day closer ?

 

[Tomfh]

i.e. at the maximum normal stress from bending there is no shear stress, and at the maximum shear stress there is no normal stress, so it is okay not to just use a basic check of normal stress against allowable yield stress and shear stress against the allowable shear stress?

The codes simplify it in their own different ways.
E.g. in the Australian code you can assume the flanges alone carry bending and the web alone carries shear. Or more commonly you assume the whole cross section carries bending and provided the bending stress doesn’t exceed 75% of the bending yield capacity then you happily use the full web shear capacity. If you exceed 75% bending yield, then combined stresses can cause von mises/tresca failure like you are concerned about, and you need to limit your web shear stresses accordingly, so as to prevent these multiaxis yielding failure. Clauses AS4100 5.12.2 and 5.12.3

As you say, the max shear and max bending don't tend to occur together.