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# Von Mises in Hand Calc?2

## Von Mises in Hand Calc?

(OP)
Hello everyone,

I'm contributing a small section to a colleague's report that is showing compliance with structural aspects of FAR 23 for a fixed wing aircraft's control system. The control system is fairly agricultural, i.e. cables, bellcranks, pushrods and the like.

One component subject to analysis is a torque tube that features drive arms welded to it, with the drive arms being simple 4130N square tube. My colleague has analysed these arms as a cantilever beam.

The colleague remarked that he "had to go to Von Mises" to show the arms good, which got my attention. I've been around the proverbial block a couple of times, and have seen a reasonable number of hand stress calculations from various sources including 'heavy metal' manufacturers, however I cannot immediately recall a time where I've seen Von Mises stresses determined in said hand calcs. I have, of course, seen VM stresses and their related brethren used when reporting FEM results.

Is my sample size too small? Would anyone like to comment regarding the use of Von Mises stresses in hand calculations?

Thank you,
Greg

### RE: Von Mises in Hand Calc?

Did this engineer also show you the calculation performed? The Von Mises effective stress can be calculated by hand with relatively simple formulas. This is usually covered in a college Machine Design course. My textbook gives formulas for Von Mises effective stress in terms of both principle stresses and applied tensile and shear stresses.

The Von Mises technique is just a different, slightly less conservative failure criterion.

Basically, using a stress vs strain curve with some yield stress from uniaxial test data is very conservative because the area below said curve represents the total strain energy per unit volume stored in the material. Basing the notion of failure on this is not entirely accurate (although as mentioned it is conservative) because there are two components to this total strain energy. That is, the hydrostatic, and distortion components. The distortion component comes from the intergranular shear and is what is key to failure.

The Von Mises effective stress is the equivalent uniaxial tensile stress on a member which would create the same distortion energy as created by the actual applied stresses. But it's calculation may be formulaic if you have determined the critical stress location and critical stress element of the part, depending on the situation (again, we have no details of the actual analysis this person performed).

Most design books should develop the VM stress from first principles based on strain energy of a stress element.

Keep em' Flying
//Fight Corrosion!

### RE: Von Mises in Hand Calc?

Bruhn covers this subject quite well and includes a method that is simple to use as a hand calculation.

STF

### RE: Von Mises in Hand Calc?

difficult to see where a von Mises stress would apply in a part 23 control system ? a torque tube with cantilevered arms ??

but as a calc, sure; I guess the max principal was too high, and he was "smart" enough to see a tensile (or same sign, whatever) min principal would give a lower von Mises. So he's looking at a web ?? still don't see it ... on a cantilever arm, the caps take the bending (as a couple) and the web takes the direct shear. If you have too you've got bending and shear interaction ... all simple calcs, not seeing it still !?

### RE: Von Mises in Hand Calc?

Von Mises yield criterion was used by major oil companies for casing and tubular design way before FEM was even thought about.This question really needs a sketch,and a FBD to appreciate the situation.Feel like the analysis is being made difficult,when in fact a torque driving several lever arms is almost a classic strength of materials problem.It's solution is clearly seen as a sample problem in most strength of materials texts.Yeah, the FEB methods available today make it more an exact science,but the estimation of long ago using strength was good enough,it got us to the moon and back.

### RE: Von Mises in Hand Calc?

What caught my attention in the OPs post was the comment ""had to go to Von Mises" to show the arms good". I would dig in a little deeper as to why he had to go to Von Mises, is there valid rational why the Von Mises is acceptable when other methods did not show the component acceptable?

Probably a valid reason, but I would want to understand the rational.

### RE: Von Mises in Hand Calc?

von Mises is lower than max principal if the other principal stresses are the same sign, and larger when there are different signs. But as you say, where is a simple control system would you find multiple principal stresses ... a really heavy fitting, a plate-like web ? The closest I could visualise is on an arm fttg, using the full section (including the web) for bending and the web also for shear ... but that looks more like simple bending shear interaction, maybe make a principal stress from bending stress and shear stress, but von Mises (with multiple principals) ?

### RE: Von Mises in Hand Calc?

(OP)
Thank you to everyone for their input.
The VM yield criterion itself is not being questioned, nor how to calculate it - my interest is whether it regularly appears in the typical aircraft hand calcs that many of us are familiar with.

My reaction upon first reading was not unlike these:

#### Quote (rb1957)

difficult to see where a von Mises stress would apply in a part 23 control system ? a torque tube with cantilevered arms ??

#### Quote (MOHR1951)

Feel like the analysis is being made difficult,when in fact a torque driving several lever arms is almost a classic strength of materials problem.It's solution is clearly seen as a sample problem in most strength of materials texts.

#### Quote (rb1957)

The closest I could visualise is on an arm fttg, using the full section (including the web) for bending and the web also for shear ... but that looks more like simple bending shear interaction, maybe make a principal stress from bending stress and shear stress, but von Mises (with multiple principals) ?
which lead me to posting here to see whether I required calibration, or perhaps the use of VM in this type of calculation was slightly amiss.

An example of how said colleague has attempted to use VM in this situation is attached in the file below - unfortunately the report is bereft of FBDs. Please note during reading that I am not the preparer, checker nor approver of said information; I am just an interested observer.

### RE: Von Mises in Hand Calc?

I know you said you did not prepare this, but as an observer as you mentioned, I would be curious to see some diagrams. At a cursory glance, it is hard to tell what the numbers apply to without illustration.

If I were preparing I would
1. Show the component overview with free body diagram
2. Find and show the critical cross section
3. Find and show the critical point in that cross section (where stresses are highest). This is where your stress element of interest will be.

Any time I've ever found a principle or Von Mises stress, it is related to a stress element with applied axial and shear.

Also, it would be helpful to get some context on the shear stress. In the notes, the calc is simply (Vmax / Ashear) but this is only true for direct (transverse) shear, so it depends on what we are talking about here.

For shear due to bending you would have Tau = (V*q)/(I*b)
And for torsional shear, you would have Tau = (T*r)/J

Keep em' Flying
//Fight Corrosion!

### RE: Von Mises in Hand Calc?

at a glance ... combining bending stress with shear stress in a beam, really ?

bending stress is calc'd at max fiber, shear stress at NA ... why combine them ?? ok, at 2nd glance, for a square tube not unreasonable.

uniform shear stress ? with such a small margin, a better shear stress calc is worth it, and uniform is probably unconservative.
what about plastic bending ? (much higher allowable)

instead of principal (note the spelling) stress I'd use bending/shear interaction ...
Rt^2+Rs^2 = (83.7/85)^2+(4.4/50)^2 = 0.98 ... MS = 0.02

note his von Mises stress is higher than max principal (so why mention it ?)
I wouldn't've considered a 0.75" tube as a biaxial member.

### RE: Von Mises in Hand Calc?

I would write some too, but looks like rb1957 wrote it earlier:
"von Mises is lower than max principal if the other principal stresses are the same sign, and larger when there are different signs. But as you say, where is a simple control system would you find multiple principal stresses.."

Von mises would be applicable for an arm for simple strength/margin check.
However, if it were to go beyond this and was to be checked for fatigue, then max/min principle stresses might have been needed to be calculated.

But again, it really depends on the type of the part: (casting, sheet metal, machined part?)

If it is a simple strength check as SWComposites had mentioned it before, without "beating every stinking little fitting to death with a complicated FEM", we might apply this rule of thumb on this as well :)
In automotive, they'll mostly use von mises stresses to calculate fatigue life on parts. And it gives reliable results. In aerospace they will extract every element load from a "good" quality shell element and calculate every stress in detail and check it accordingly. So, it really is up to the complexity of the FEA model and the assumptions. But with rb1957's explanation and von mises checks in most cases where the structure isn't a "primary/secondary" structure, they might be taking the shortcut. They might have been good with Von Mises so far. If they had no problem with it, then it would be experimentally approved as well.

You might wanna check it with other stress engineers there to see if that's the general practice.

I really think every stress department needs weekly meetings of 1 hour to discuss their know-how and to agree on their methods. That's what we used to do when I worked for the wing-box structure. It helps not only at an individual level but at a team level.....

Spaceship!!
Aerospace Engineer, M.Sc. / Aircraft Stress Engineer

### RE: Von Mises in Hand Calc?

The VM criterion only predicts the onset of yielding. Many aircraft structures have a 1.5 factor of safety for the ultimate condition. Because of the relationship between Fty and Ftu (relatively close), this usually means the ultimate condition is critical (though plastic bending can shift that sometimes). So because yielding is not usually the critical scenario for static loading, and VM is only applicable to yielding, you don't see it used too often for aircraft structures.

### RE: Von Mises in Hand Calc?

with all of this "how many angels can dance on the tip of a needle" stuff (being very picky about failure mode/allowable) I'd want to see a FF in the calc, particularly if control system.

### RE: Von Mises in Hand Calc?

If I imagined this right, you have a control rod that undergoes torque and bending together while loaded. Both the peak shear from the torsion and peak bending interact at the extreme fiber which must be accounted for. Assuming elastic behavior up to ultimate, writing your margin of safety using the VM stress against FTU is acceptable and conservative. I should note that this also assumes the material is a non-brittle metal, i.e. it has some plasticity.

Interaction Equations are normally used when combining stresses in a hand calculation, so that's probably why you don't recall seeing von mises in hand written analysis. The interaction equation for a beam stressed in torsion + bending is:

Rb2 + Rst2 = 1

Some notes:
If your control rod has a hollow tubular cross section, don't apply plastic bending. tube walls collapse well before you can develop the plastic bending allowable. Use elastic bending, or look it up on a test derived curve. there are curves in Bruhn and Mil-hdbk-5 for limited materials.

If your control rod is also in compression, VM is NOT a good assumption, since the tube can buckle before rupturing. In that case use the appropriate interaction equation.

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