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# Fastener Bending Analysis Using Bruhn Supplement

## Fastener Bending Analysis Using Bruhn Supplement

(OP)

Hi all

I currently need to do a bolt bending calculation in order to choose an appropriate fastener.

The background is that I have a single shear joint between a 0.120" thick 7075-T6 rib foot and a 0.220" thick 7075-T7351 wing skin. Per drawing, the gap between the two items is 0.125", which is filled with a non-structural (floating) shim. In order to carry out a repair I need to insert an additional 0.150" shim into the joint, bringing the total thickness of non-structural shims in the joint to 0.275". The existing fastener is a TL200-4 (1/4" Taper-Lok with protruding shear head) and I propose to use a TL400-5 (5/16" Taper-Lok with protruding tension head) for the repaired configuration.

I intend to use the analysis method outlined in Article D3.5a in the Bruhn Supplement. In that analysis, the statement is made that "the interaction of shear and bending [in the fastener] is not critical", which seems fairly reasonable after some contemplation. Based on this, the shear and bending is treated separately. The shear is treated in the same way as usual (i.e. minimum of fastener shear allowable and bearing allowable in each thickness determines the allowable) and the remainder of the analysis focuses on the bending.

In essence, the bending analysis calculates the resulting bending moment in the fastener and compares it to an allowable moment which is obtained by multiplying the ultimate bending allowable of the fastener shank by a 0.88 knockdown factor to account for the tension induced by the bending. In Fig D3.19a it clearly illustrates that the maximum bending moment in the fastener occurs where the shank meets the head and nut. This is consistent with the assumed fastener FBD given in Bruhn Supp. Figure D1.43.

Based on this I would have argued that the critical section for bending would be in the thread minor diameter, since those threads between the unthreaded section of the shank and the nut would see the same maximum bending moment and tension. However, in the footnote to article D3.19a we are told to use the bolt shank bending allowable, and referred to Table A45 (in Appendix A of the supplement), where the allowables are based on the full shank diameter. The same is seen in Tables D1.1 & D1.2a (in the original book), where the tension allowables appear to be based on the thread minor diameter but the bending allowables are based on the full shank diameter. These tables may be intended for use in socket analyses or double shear lug/pin analyses, but I don't believe that the bending allowables in these tables are applicable to the case of fasteners in single shear.

Does anyone have any ideas about why this apparent inconsistency exists, or have I overlooked something?

Regards

Oz

### RE: Fastener Bending Analysis Using Bruhn Supplement

1st, congrations on an excellent post; a well researched, phrased question.

a comment on the FBD ... McCombs assumes that the end moment (at the head of the fastener) is reacted by an off-set prying load.  This is conservative, particularly in your case, as it ignores the preload in the fastener (which would make the problem very difficult to analyze).  But it does represent the ultimate condition where the head is gapping on one side so i guess it isn't that bad.

i'm not so sure that interpreting FigD3.19a as showing that the moment under the head is equal to the moment under the nut.  without doing all the calcs, i think that if you have an unsymmetric joint (different sheet thicknesses) that you'd probably have different end moments.  the point being that you might be able to put the nut on the lower loaded side.

recognise that the shear assumptions are somewhat conservative.  it looks to me as though McCombs allows uniform shear over the thickness of the sheets.  Another common assumption is that the load peaks nearer the parting plane, ie a triangular distribution, which will reduce the moment arm.

Recognise that 0.88 is a fudge factor, particularly when you can calc something more specific for your geometry.

IMHO, you can use the procedure as written, as i think there are plenty of conservatisms within it.  i'd have no problem using the min. dia. (rather than the full shank) if it passes ... but then if it did you probably wouldn't've asked.

### RE: Fastener Bending Analysis Using Bruhn Supplement

(OP)

I have had a think about the points you raised and I have a few comments as follows:

I agree that in the ultimate case, ignoring the preload in the fastener is probably reasonable.

As regards the moments, I will think out loud for a minute...the bearing loads on the fastener result in a net moment on the fastener. This moment must be reacted by opposing moments at the head and tail, each of these opposing moments being generated by a force couple composed of the tension in the fastener (ignoring preload) and a prying load on the head or nut. The proportion of moment attributable to the head and tail is a statically indeterminate problem, and the most obvious analogy is a beam with 100% fixity at each end, which would result in a 50-50 split. Although there will of course be less than 100% fixity, if the amount of fixity at the head and tail remains equal, then there should still be a 50-50 split. The only factors that would change the 50-50 proportion would be the stiffness of the fastener head and tail or the difference in material stiffnesses. I don't think a difference in sheet thickness would influence the stiffness significantly. Personally I think that the stiffnesses of the head and tail are similar enough that for practical purposes we can assume that the moment reacted by the head is equal to that reacted by the tail, and I interpret Fig. D3.19a to be confirming this.

When you say "shear" assumptions, did you mean "bearing" assumptions? McCombs analysis does assume a uniform bearing stress of Fbru acting over the portion of the thickness necessary to react the load. If a triangular distribution was used, then the peak stress would also have to be limited to Fbru and this would have the effect of increasing the moments. Therefore, I think the triangular distribution may be of some use in an elastic analysis, but I don't think it is helpful in an ultimate analysis.

As regards my particular problem, I can show a positive margin either way, but the reason for this is that I am reverse engineering the loads. I am using the fastener bending analysis to calculate the maximum load that might be applied in the baseline configuration, and then applying this load to the new configuration. Thus, whichever diameter (min or full) I use for both analyses, the margin is not significantly affected. Some may argue that I should be happy that I have a positive margin and leave it at that...however I have this strange engineering trait that makes me want to understand how things work so that I can get the analysis right...some would call me crazy!

Regards

Oz

### RE: Fastener Bending Analysis Using Bruhn Supplement

g'day,

(and that's not taking the pi, i'm a fellow digger)

my end-moment ramble started 'cause i thought McCombs was conservative in reacting the moment (caused by the offset shear loads) as a couple between the bolt CL and the edge of the head; i was thinking that the fastener preload would allow the moment to be taken out on opposite sides of the head.  but it is reasonable to assume that under ultimate conditions the head is gapping on one side.  you've noted that the nut end of the fastener can react a smaller moment than the head, thus it's reasonable under ultimate conditions to allow the nut to off-load moment to the head, but that should be about a 1/4 of a gnat.

yeah, shear reaction = bearing.  it is a common assumption to allow higher shear reaction closer to the parting plane of the joint under ultimate conditions.  and you're right this should be checked against the bearing allowable, and would provide a rationale for determining the maximum peak shear reaction (= the bearing allowable).

but fasteners are usually critical in shear or bearing, and usually it's a significant difference.  and the tables for single shear allowables include all these effects.

### RE: Fastener Bending Analysis Using Bruhn Supplement

(OP)
G'day rb1957

Another Aussie huh? Are you in Oz or are you off contracting overseas somewhere?

I agree that before gapping occurs, the fastener preload would allow the moment to be taken on opposite sides of the head or tail, which would result in a pretty much constant tension in the bolt, which would be equal to the preload. As the shear load increases then the moment would start to create gapping on one side of the head or tail, and then you would calculate the bolt tension to be that which was necessary to supply the resisting moment via a couple composed of a tension load through the bolt centreline and a prying load between the head or tail and the sheet.

I think we are on different pages regarding the bearing distribution. McCombs assumes that the bearing stresses are already equal to Fbru, and then assumes that this bearing is applied over only part of the thickness. Therefore it is not possible to get a bearing distribution that would reduce the moment any further.

I think I may have been too conservative in my assumption of a 50-50 split between the head and tail. I think that this would apply in the elastic range, but I think you are correct in saying that the head could still continue to resist additional moment after the tail has reached its ultimate allowable. To use this approach you might add together the total bending moment allowables at the head and tail side and compare it to the total applied moment. An equivalent approach would be to take the average of the head and tail bending moment allowables and compare it to the Mo from D3.5a, which is simply half of the total applied moment. This would provide a not-insignificant improvement.

I have a spreadsheet with all the calculations in it so I decided that the easiest way to check the validity of the analysis was to set the shim thickness to zero, which would give an ordinary single shear joint. For the TL200-4 fastener the single shear allowable is 5668 lb, and for the three methods of fastener bending analysis the resulting allowables for zero thickness shim are as follows:

Using full shank diameter: 5558 lb
Using thread minor diameter: 3968 lb
Using average bending allowable (not average diameter): 4829 lb

For zero shim thickness we should expect a bending allowable close to the single shear allowable, since these allowables are verified by test, so it appears that using the full shank diameter gives an answer closer to reality. I still think that theoretically, the average bending allowable would be the one to use, but perhaps there is enough conservatism in the analysis already, that the use of the full shank diameter brings the analysis back in line with the test data.

Oz

### RE: Fastener Bending Analysis Using Bruhn Supplement

g'day

i've been OS for about 25 years, straight after uni.

your post sounds reasonable, i hadn't noticed McCombs used thickness as the variable and stress as the constant in calculating the sheet reaction.

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