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End Plate with hole? how to size?

End Plate with hole? how to size?

End Plate with hole? how to size?

(OP)
Here is a crazy plate design I am trying to work out.

I have a square (10"x10") end/cap plate with a 2"dia hole at the center.  The plate will be welded on 4 sides to a steel "box". Through the hole is a bolt with a washer/nut on the back-side.  This bolt will be tensioned so that the 10x10 plate will want to bow outward.

How do I go about sizing this plate, and getting reactions at the perimeter for sizing the welds?

RE: End Plate with hole? how to size?

Look in Roark's book Formulas for Stress and Strain.  I'm pretty sure they have formulas for the situation you're describing.

RE: End Plate with hole? how to size?

I agreewith nutte - Roark is real handy.  You could also check out some of Timoshenko's works.

Mike McCann
McCann Engineering

RE: End Plate with hole? how to size?

(OP)
Ok, I got a Rourke book and will look through it. I've only ever used it for walls of concrete pits, or storage sheds.

RE: End Plate with hole? how to size?

Another option is to model the plate on, say, RISA-3D, and analyze for the point load.

DaveAtkins

RE: End Plate with hole? how to size?

Good idea Dave.  I would model it with the hole and with a ring load of the bearing width of the washer at the edge of the hole.  Should be interesting.

Mike McCann
McCann Engineering

RE: End Plate with hole? how to size?

(OP)
Well, I'm not finding anything in Rourke yet. I guess I might need to model it.

RE: End Plate with hole? how to size?

This is a pretty easy problem using yield line theory.  That's what I'd do for sure.

RE: End Plate with hole? how to size?

I doubt Roark has a square plate with round hole.  You could treat it as a round plate with round hole and that would likely be close enough.  Summing moments about a centerline will get you the average bending moment in it- perhaps that's the yield line theory.

RE: End Plate with hole? how to size?

I am not very savvy with finite element modeling but this seems like it would be a good application.

RE: End Plate with hole? how to size?

"Summing moments about a centerline will get you the average bending moment in it- perhaps that's the yield line theory."

Nope, not quite.  Most good concrete design books have a section on yield line theory.  This one should take about 15 min. max to figure out the load to cause the collapse mechanism.

RE: End Plate with hole? how to size?

Roarke doesn't have this.  I would model this up with finite elements.  Be sure to include some of the tube in your model, say 10 inches.  That way, you don't have to use pinned or fixed edges at the edge of your plate.  Also, I would use rigid links for the weld, so it's easy to extract the weld forces.

RE: End Plate with hole? how to size?

Obviously the yield lines should come in from each corner. Depending on the size of the washer/nut I might still do some kind of check in the immediate area of the hole for pull-through.

RE: End Plate with hole? how to size?

OK, using AISC LRFD, I get:

treq = sqrt( Pu / (2*phi*Fy) * (B-d) / (B-0.7071d) )

Pu = the factored load at the bolt hole.
phi = 0.9 for bending (if you want ASD, then put Omega=1.67 beside the Pu instead)
B = square plate width
d = bolt hole at the center of the plate

Use this only for comparison for whatever you end up with because it is NOT guaranteed!  

Better yet, use it for comparison with your own YLT result!

[Soapbox On]It took very little time to derive this, even symbolically.  Probably would've been <5 min. with numbers instead.  Serious steel design folks owe it to themselves to take that concrete book home one weekend and become fluent in yield line theory.  It is frequently useful and certainly faster and easier to interpret than a FE model.[Soapbox Off]

The only gotcha is that there's no way to get deflections from this.  If you have an application that you're concerned with deflection, then you'll have to use a FE model.  When I do something like this, I either approximate the deflections using some simplified model or make the FE model for deflections only.

RE: End Plate with hole? how to size?

haynewp, that's the way I see it to: the mechanism has YLs coming from the corners, terminating at the edge of the hole.

As for a local check, any suggestions for how to do such a thing?  I think I'd just use a bigger washer and forget that one.

RE: End Plate with hole? how to size?

I would try an FE model but I am not sure how to model in the stiffening around the hole from the attachment itself (again I am not real good with FE I was 'raised' around old-school guys) or might just put in an oversized washer.



RE: End Plate with hole? how to size?

I guess we're the same in that regard.  I wouldn't know how to model such a thing.  

A simplified model would be easy enough, but I think any realistic model must include difficult items like residual stresses around the hole from hole punching or drilling.

RE: End Plate with hole? how to size?

I thought there would be a decent amount of info on pull through of bolts but I am not finding much so far. I think it would also be a function of the distortion of the remainder of the plate and not just in the local area of the washer.

I only found this in Design Guide 10 regarding anchor bolts:

"4.2.5 Anchor Rod Pull or Push Through
The nuts on the anchor rods can pull through the
base plate holes, or when leveling nuts are used and the
column is not grouted, the base plate can be pushed
through the leveling nuts. Both failures occur when a
washer of insufficient size (diameter, thickness) is used
to cover the base plate holes. No formal treatise is presented
herein regarding the proper sizing of the washers;
however, as a rule of thumb, it is suggested that the
thickness of the washers be a minimum of one third the
diameter of the anchor rod, and that the length and width
of the washers equal the base plate hole diameter plus
one inch."

RE: End Plate with hole? how to size?

271828,

Pullthough can be checked with YL too. It will probably be a small fan, but you could cheat and do it as a small square, much the original problem.

RE: End Plate with hole? how to size?

Tom I was thinking that too but it would seem to me that checking the two yield surfaces (the overall square versus a fan around the hole) independently might not be conservative.


I am trying to imagine how the deflection on the overall square plate influences the angles toward the middle of the plate, possibly a pull-through might occur earlier than what a fan yield line predicts. But I may be wrong.

RE: End Plate with hole? how to size?

So, AISC says the washer thickness should be 1/3 the rod diameter?  1/4" thick washers for 3/4" anchor rods?  I doubt that will happen!!!

DaveAtkins

RE: End Plate with hole? how to size?

Dave, why is that a strange washer size?  Of course, they're thinking of plate washers at the huge oversized holes.

haynewp, say there's a plate washer there.  I'm having a hard time envisioning a pull-through failure.  He might also have a leveling nut and washer underneath, sandwiching the end plate.  With a regular hardened washer, I can envision pull-through.

I think the most troublesome issue is overall deformation because that doesn't come out of the YL analysis and because treq might be pretty thin.  Elastic deflection is the only thing that IS easy to get from a FE model.  I really like using YLA for strength and FE for deflection.

RE: End Plate with hole? how to size?

If you really want to make this problem simple you can halve the load and solve strength and deflections for a one-way case.

RE: End Plate with hole? how to size?

Use a strip method.

Treat it as a square hole and apply load along four sides.

Take this load onto a 2" wide strip and analyse this as a beam. 4 beams total but all are identical.

This will be conservative, but I would think that your time to put this into an analysis would not be justified by the cost of the plate.

csd

RE: End Plate with hole? how to size?

csd, but that's no fun at all!  After all, that's why we do this, right?!

RE: End Plate with hole? how to size?

271828-

Using the yield line theory formula presented above, and neglecting local effects, I get a required plate thickness of .175 inches for a 3 kip load.  Is that what the formula says?  Seems a bit thin.

RE: End Plate with hole? how to size?

I get 0.208" for a 3 kip load and 36 ksi.  Not sure if that seems thin to me for 3 kips.

Want to re-emphasize that the formula is NOT guaranteed for design use and was provided for verification purposes only.  I feel pretty good about it, though.

RE: End Plate with hole? how to size?

271828,

Yes, it is certainly not because it pays well!

RE: End Plate with hole? how to size?

271828-

I'm not designing this.  I'm just trying to get a handle on yield line theory, as it looks like a valuable tool.  I was using 50 ksi.  For comparison, I modeled this up with a crude elastic finite element model.  With a .208 thick plate, and a 3 kip load, I get stresses near 90 ksi.  Of course, these are near the hole, where finite elements tend to find stress risers.  To get stresses below yield, I would need a 3/8" plate.  Just from a gut level though, 3/8" feels a lot better than 3/16" for a 3 kip factored load.

RE: End Plate with hole? how to size?

jmiec, I would expect large stresses like that from the FE model.  In reality, they wouldn't exist and load would distribute, eventually ending up with a collapse mechanism like assumed in the yield line analysis.  We're definitely focusing on "the end of the line" strength, excluding strain hardening that is.

3/8" just *feels* better because we use them more in applications like this.  In reality, I'd probably bump up the plate size to that anyway in this case, not because of 3 kips because that's small.

For one thing, perhaps there are other loads, gravity loads for example, that might push the plate size up.  

Deflection is still an issue too.  Your FE model is perfect for checking that.

There could also be constructability issues, like moments on the BP during construction.

Definitely some issues to think about.  I just like YLA because it gives a rational way to check the strength limit state without some hokie approximation.  Then I can go figure out how to deal with the others.

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