Repost: Induced bending moment in a thick shim

[koopas]

This is a repost of my thread that was deleted yesterday.

I was reading through an old Douglas structural repair training class manual and found an example on shimming.

The problem involves DD6 rivets, each carrying a load of 1,000 lb (MS of 0.18), and the introduction of a 1/16-inch-thick shim in the joint. The calculated induced bending moment on the rivet shank is 1/16 * 1/2 * 1,000 = 31.3 in*lb.

The shank section modulus is given as 0.000674, and the bending stress is then computed as 31.3 / 0.000674 or 46,500 psi. A bending allowable of 70,000 psi is listed.

In the last step, a combined MS (shear & bending) is computed as follows:

1 / [ (46,500 / 70,000)^2 + (1/1.18)^2 ]^(1/2) - 1 = -0.075.

Here are my questions:

1. How is the shank section modulus computed for this fastener? Is the moment of inertia manually calculated, then divided by half the shank diameter (location of "upper" fiber of greatest stress)? Where can I find a list of shank section moduli for common aircraft fastners?

2. Where does the bending allowable of 70,000 psi come from? Is it simply Ftu of the rivet material (2024-T4?)

3. Since the "combined MS" is negative, the margin of safety is eliminated and it is concluded that thick shim induces excessive bending stresses. How can one start with a positive margin for shear (MS = 0.18), a positive margin for bending (tensile) stress (70,000/46,500 - 1 ~ +0.50), and end up with a NEGATIVE MS for the combined solution? This is related to question 4 below.

4. Where does that "combined MS" formula come from? Since I can see that the formula involves shear and tensile stresses, is Mohr circle somehow at the basis of this fiasco?

5. Niu's "Airframe Structural Design" book pp. 244-5 gives a example on how to calculate the acceptability of a shim. Essentially, you calculate the tensile sustained residual stress based on the deflection (gap), loading type, and fixity in the joint. This calculated stress is then compared to the maximum allowable tensile sustained stress found in Fig. 4.7.7 on pp 116 for different materials and grain directions. If the calculated tensile sustained residual stress is less than the tabulated maximum allowable tensile sustained stress, the gap does not need to be shimed.

How do you quantify the acceptability of a gap or the need for a shim?

Thanks for your continued support.
Alex
Alexandre_Ly@hawaiianair.com

 

[jetmaker]

koopas,

Q1) Section Modulus, z, is simply I/c. Therefore, for a rivet, I=pi*d^4/64, and c=d/2. So yes... you are correct in your statement. Do not know of any source for the data.

Q2) The bending allowable of 70000psi comes from the material properties of the fastener. Based on historical test data.

Q3) The equation 1 you stated is a Von Mises failure criteria. It basically takes the vector sum of the stresses. That's how you went from a +ve margin to a -ve one.

Q4) Nothing to do with Mohr's Circle. It is the Von Mises Failure Criteria.

Q5) Gaps in excess of 0.005" NEED to be shimmed. Only exception is if you are connecting very flexible material or materials that are lightly loaded in service. The residual stresses induced by closing a gap are particularly detrimental under cyclic/fatigue loading. Therefore, best practice is to ALWAYS SHIM A GAP. The real question is... "when can I use a free shim versus a structural shim?".

Have fun.

 

[koopas]

Jetmaker,

Again, thanks for taking the time to help.

Regarding answer (2), is the maximum bending allowable simply Ftu of the fastener material? Do you have a source for the historical data?

I'll do more research and reading on Von Mises stress.

Have a good weekend,
Alex

 

[jetmaker]

koopas,

Sorry, but I do not have the historical data that this was derived from. And as for using Ftu, that would be conservative. I checked allowables on 2024-T3 extruded rod, and found that Ftu=61 ksi.

Let me know what you find out.

Thanks.

 

[Sparweb]

Koopas, open Bruhn to chapter C1 (I think) and you can get an appreciation of combined stresses. Skip Mohr's circle, 'cause I don't think aircraft people use it, and jump to the octahedral stress theory at the end of the chapter. This doesn't plug anything into the Von Mises failure criteria that Jetmaker gave you, but you will get an idea of how the stresses can combine.

Tension and shear do not play independently, though often we treat them like they do. Always combine them in all fasteners.

The bending allowable comes from the fact that you're bending a driven DD rivet. 70 ksi sounds right around the Ftu of the DD rivet, but I don't have the numbers handy.

Mil-Hdbk-5 is one way to "guesstimate" a tensile allowable for a rivet, the material of the rivet, you can flip to that material and learn more about it. The "historical test data" is the test data that backs up every single number in Mil-Hdbk-5.

Mil-Hdbk-5E used to have a table (which has been taken out since revision G) of how to calculate margins of safety in round and tubular sections (chapter 1). I don't know why it was taken out, but if you can find an old copy, you'll see it.


STF

 

[koopas]

Sparweb,

I spent half of the day reading Bruhn, starting with C1. Whoa...I thought I was back in school for a moment.

Chapter C1.17's octahedral shear stress theory essentially tells you how to find combined M.S. for plane stress conditions, with and without shear stresses. Useful for fuselage stuff (thin plates), I guess. Good background on stress ratio's in C1.15.

Since I was interested in a rivet shank, I was looking for combined MS for a simple 1D beam in bending and shear. For some odd reason, I started reading chapter C4 and got "taken to school" in the round tubes sections. I was amused to find myself so overwhelmed by figures, graphs, empirical data, etc. There were also some typos referring you to the wrong figures so that also threw me for a loop. Overall, after a few hours of reading, my brain was fried and I still hadn't found what I was looking for.

I then flipped back to Chapter C3 and voila! It wasn't apparent that C3 covers "solid" shapes.

Under C3.12 "Strength under combined bending and flexural shear", the interaction equation is given as Rb^2 + Rs^2 = 1 and the MS as 1/ [Rb^2 + Rs^2]^0.5 - 1.

Now, my question is the following: for the ratio Rb, Bruhn uses a ratio of moments (in*lb) whereas ratio Rs is based on a ratio of shear stresses (psi).

Why doesn't Rb use a ratio of bending stresses in psi instead of a ratio of the moments? Does it matter? I guess not, if (C/I) is constant.

Thanks,
Alex

 

[Sparweb]

Keep your bug detectors turned on while reading Bruhn, it's chock full of them. Also: making sure you're looking at the right graph. In some cases, figure numbers are duplicated!

It's hard to appreciate in these days of spell checkers and graphics software how a book like Bruhn's could become so widely used, but the fact of the matter is that it brings together a heck of a lot of data.

Combined bending and shear stress interaction formulae still have some mysteries for me, because if you were to flip to D1, you would quickly come across the interaction formula for a bolt, in which Rb^2 + Rs^3 = 1 !

In a case where a linear increase in load generates a proportionally linear increase in stress, then it doesn't matter if you use stress or load for the ratio. I'm less comfortable with the truth of that in a case such as the bending moment in a column as it deflects under compression, where I don't think the relation of load/stress is linear.


STF

 

[Sparweb]

An acquaintance of mine has pointed out that stress ratios are used especially in cases of non-linear stress, such as in the plastic deformation range. Back to the books, for me, too!

STF

 

[SuperStress]

Keep in mind too, that margins of safety calculated for non-linear structure (such as beam-columns) are only "apparent" values since bending moments (or bending stresses) increase faster than the applied axial load increase.

The "true" margin of safety will always be closer to zero than the apparent value.

The true margin of safety is found by successive trials whereby some factor "A" is found by which ALL of the applied loads must be multiplied to give an MS=0. The true margin of safety is then A-1.

SuperStress

 

[darrenm]

Regarding shims, pretty much all you need to know is found in the following two references

[1] “Airframe Stress Analysis and Sizing” M.C.Y Niu 2nd ed. Section 9.11 and page 685.
[2] “Analysis and Design of Flight Vehicle Structures”, E.F.Bruhn. Section D3.5.

These include analysis techniques, allowable gaps, allowable preload stress etc etc.

With regards to the allowable moment and shear in the original RF/MS interaction formula. The shear allowable is just that, the allowable shear load for the fastener. The allowable bending moment is the moment associated with the fully plastic bending capability of the bolt as calculated by cozzones approximate method (according to the Boeing SRM copy I have).

I am writing an article on this exact topic for the information of my colleagues and have a list of 4 outstanding questions which I think will be impossible to answer but if you never ask you'll never know. Here goes.......

1. Are there any commonly used, published methods, verified by test, that allow the reduction in bolt strength to be calculated when a shim is installed? (in addition to the one mentioned at the start of this discussion thread!)

2 Regarding the formula mentioned at the start of this thread, does anyone know the accuracy and applicability of this formula ie tension / shear head csk or prot head fasteners. I have one vague page of data by anonomous, although the page has a "BAe Military" header on it, "unconfirmed" test results referred to on my data page show results similar to those that one would obtain from the 1/sqrt(Rs^2+Rb^2) formula although I do not have the material data to do a direct comparison.


3. Frequently in the aircraft industry we have large joints such as wing to fuselage joints that have shims installed over part of the joint. Are there any published methods that calculate the reduced flexibility of shimmed fasteners, this will be important when determining the load distribution amongst the fasteners. I would need this to enable limit and ultimate load distributions to be calculated. I have proposed a simple method in my article but would like something with proven accuracy because there are too many half truths out there already and I don't want to add to their number!

4. For a joint with shimmed and unshimmed fasteners, is it acceptable to assume that all of the fasteners can achieve their ultimate capability at the same time such that the joint allowable is the sum total of the allowable of all of the fasteners - normal aircraft industry practice for unshimmed bolts? My feeling is that shimmed and unshimmed bolts will fail at different strains (depending on the thickness of the shim) which will mean one type of bolt will be reacting a load less than its ultimate when the other type fails, hence, summing the allowables of all of the bolts will overestimate the joint capability.