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Correct Bending Moment in an Embedded Washer Plate (Anchor)

Correct Bending Moment in an Embedded Washer Plate (Anchor)

Correct Bending Moment in an Embedded Washer Plate (Anchor)

(OP)
I need to reduce the overall cost of a connection by as much as possible, and the only area where I see I could achieve that would be in the washer plate thickness. Right now I am not sure if my washer plate is too thick, so I am questioning whether I have the right bending moment.

The problem:

I need to fully develop a 1" threaded rod that has a 10" embedment depth in CIP concrete. In order to achieve the pullout and shear strength required, I need to attach a 4"x4" washer plate centered with a 1-1/8" diameter hole. The washer plate would be held in place by a nut threaded from the bottom. My question is what would be the most accurate way to determine the true moment in the washer plate?

I've seen some calculations (from an overly conservative engineer) where they considered M=PL/4, but that is ridiculous.

I believe considering a cantilever beam with a UDL (M=(wL^2)/2) is still too conservative. (half the moment as above)

Now I read about elastic theory of plates, and got that the max moment would be M=0.048wL^2
This number I like, but since the deflection is much greater than the thickness of the plate, it is not really applicable. However, since the plate is confined in the concrete, as in it is not going to deflect anywhere, would the moment I get from the elastic theory of plates be okay to use to design the washer plate thickness?

Any help or input on this would be greatly appreciated.

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

Why do you require a 4"x4" plate? If you require a uniform distribution of pressure, then the statics is pretty straightforward. M = PL/8 where L = 4".

Perhaps a circular plate would be more efficient. What is your source for M=0.048wL^2? Your comment about the plate being confined in the concrete and not going to deflect anywhere is totally without merit.

BA

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

(OP)
The 4"x4" plate is required to increase the projected breakout area, so as to get a higher pullout capacity. I cannot increase the embedment depth anymore, as the foundation is restricted in height.

That is what I had used (except (wL^2)/2 , but it comes out to the same moment). I am just thinking that it is a square plate and it would behave in bending similar to a two-way slab (with a point load in the center), thus reducing your max moment because of two way action.

I got the M=0.048wL^2 from some online course notes or an article on plate theory. I'll provide the link tomorrow.

Just wanted to see if I am out to lunch on this, as I am getting an ear full in the office for being too conservative in my designs. So I am questioning whether my analysis is indeed to conservative.

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

In the first site below, under Plates->Simple bending->Rectangular->4 corners, you can superpose the two loading conditions: Uniform and Conc.load (centered) to obtain the deflections and stresses in a square plate supported in the center with a uniform load (but without a hole). The result is M=0.043Pa where P is the total load, a is the side and M=1.41Mx=1.41My is the combined bending moment. In the form of a beam equation this would give, as w=P/a and L=a, M=0.043wL2, so you are not far from this.
However there is no account for the hole here: a more realistic approach would be by taking an annular plate with the same area as the square.
Also, depending on the thickness, shear in the plate at a diameter determined by nut size could be governing.

prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

If you take the load P and divide it by the entire plate area (P/16in^2) that is the uniform pressure over the whole plate. Then treating it as 4 cantilevers (assuming 2 orthogonal yield lines crossing at the center of the plate) would give M=(P/16)*(2^2)/2=0.125*P. That's not considering any moment restraint at the ends of the plate from clamping of the concrete, which I would ignore. Using what Prex posted the 0.043*wL^2 would give M=0.043*(P/4)*4^2=0.172*P. But I am not sure my assuming orthogonal yield lines occurring at the midspan of each side is correct in lieu of assuming diagonal yield lines.

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

Do you really need to do this? as a cost reducer it sounds dismally ineffective because you only save material cost, a small part of the total because the thinner piece has the same labor cost.

Draw the washer, including the hole, shade the area under the nut, that is the effective area. Divide the force by effective area above and below and check a series of free body pieces, probably starting at a corner and moving towards the center, then repeat starting at an edge. That should find your worst case.

Michael.
Timing has a lot to do with the outcome of a rain dance.

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

(OP)
@prex Thank you for providing me a link to that site! I will try out the 4 sided plate calculation that you have guided me to. I'll also take a good look through the site as it seems to have a lot of useful information. Thanks again for making me aware of it. Good to see that the moment equation you provided in your post is similar to what I had found, but as you mentioned it does not account for the hole. The suggestion about using an annular plate may just be the way to go. I will see what I get using that approach and thanks for bringing up the shearing of the plate around the nut. I will have to check that as well.

@haynewp Interesting take. The only thing I would do differently would be to take out the hole area (~1 in^2), to get a plate area of 15 in^2. This would give M=0.133*P. However this topic is new to me, so I would have to read up on orthogonal yield lines and diagonal yield lines to get a better understanding of its application. Thanks for your suggestion.

@paddingtongreen Well according to my boss, a few years ago when our company was super busy I don't think anybody cared nearly as much about money as they do today. It is all about budget, being more competitive and saving every penny where possible. Right now I have the plate sized at 1" thick, but with the above suggestions I think a 1/2" thick plate could work. Factor in over 700 of these connections and the savings add up. I do agree with you on the labor being much more expensive, and in the grand scheme of things the savings would be tiny compared to the overall project cost, but it just goes back to the points of being more competitive on bids and every penny counts. No work...no job.

Regarding your suggestion, I do not understand the second part. Anywhere I could find an example of this or some reading on this topic?

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

A coworker of mine just went to a seminar about anchor rod design. They talked about adding plate washers to increase the breakout capacity. The presenter was not fond of plate washers. He mentioned that plate washers will transfer load at 45 degrees. This means to get the full use out of a 4 inch plate washer it'd need to be about 2 inches thick. Otherwise it will just bend excessively under load.

I recall reading elsewhere that plate washers are rather ineffective. I think it may have been in AISC Design Guide 1.

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

Another yield line could be a circle forming adjacent to and around the anchor hole. I haven't often relied on plate washers either.

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

I don't believe that yield lines form on circular curves, so I don't agree with haynewp.

With a 10" embedment, a standard nut and washer will satisfy the bearing requirements and the factored pull-out requirements of a threaded rod provided its yield strength does not exceed 400 MPa (58 ksi) and f'c = 25 MPa (3600 psi) or better.

Getting rid of the square plate entirely is the ultimate in economy.

BA

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

Found it! From AISC Design Guide 1:

"The addition of plate washers or other similar devices
does not increase the pullout strength of the anchor rod,
and can create construction problems by interfering with reinforcing
steel placement or concrete consolidation under the
plate. Thus, it is recommended that the anchorage device be
limited to either a heavy hex nut or a head on the rod. As an
exception, the addition of plate washers may be of use when
high strength anchor rods are used, or when concrete blowout
could occur."

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

BA, why not? I believe circular yield lines make up part of what is called fan mechanisms in slabs.

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

haynewp, I can't see how you could generate a circular yield line around the nut without creating radial yield lines extending to the edges of the plate. And if you are going to do that, one straight yield line at the edge of the nut across the 4" width of plate would seem to me to be a more critical yield line pattern.

For a 1" dia. rod, the nut size is 1.5" across the flats, so the cantilever is (4 - 1.5)/2 = 1.25". The uniform pressure is P/16, so the one way bending moment per inch of plate is (P/16)*L2/2 = (P/16)*1.252/2 = 0.0488P or a total of 0.195P across the full 4" plate. I do not see a more critical yield line than this.

BA

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

I didn't say what the most critical one was and I already brought up the one you just mentioned. You said you didn't believe yield lines formed on circular curves.

See page 141.
http://books.google.com/books?id=yTFOLdUaG9EC&...

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

They don't. They form in straight lines, but there can be a number of straight lines which may appear to be circumferential. When they form on circumferential lines, they require radial yield lines in order to create a mechanism.

BA

RE: Correct Bending Moment in an Embedded Washer Plate (Anchor)

OK, so a yield line that is "circularish" with radial lines is another possible but unlikely mechanism.

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