Question on moment arm for analysis/design 4

[Lion06]

I hope I don't come across foolish here, but I'm having a hard time understanding something. Let me explain the sketch I've posted before people start tearing it apart. This is not an actual detail, it is merely to help get across the point of my question.
The question is if you have a detail similar to this (whether it is a beam to column, or a column to ftg), would you use d for the moment arm to design the tension in the top anchor or (approximately) d+2g ( I know it would be from the top anchor to the centroid of bearing at the bottom angle, but just for argument's sake say d+2g)?
Does your answer change if you provide stiffeners such that the angles can be considered very stiff?

I'll give my opinion and explanation, then you can tear that apart.
I think you should use d as the moment arm in either case. I am differentiating this from a baseplate because a baseplate is a single (considered rigid) element that has a moment applied to it. This detail (whether stiffeners are present or not) has two individual angles with a tension and compression force applied seperately at a given location. While the baseplate is seeing 0 net force, moment only (assuming moment only and no axial load), these angles are each seeing a tension (or compression) force via the weld (not a moment only with 0 net force like a baseplate). I believe this applied whether the angles can be considered infinitely stiff or not because of the above reasons and the fact that they are so close to the end. If the angles were WT's and extended for some distance into the span of the beam such that the WT's had the opportunity to become fully engaged in helping to resist the moment, I would feel differently, but the angles (as they are currently shown) do not have that ability.

Any opinions?

 

[cms1]

Let's first address the situation where the angles are considered to be infinitely stiff.

The the reactions will then be M/(d + ~2g) -- I use approximately because of the compression reaction need not be located at the bolt centroid, but it is easiest to talk about it as if it does.

The shear at the welds will be equal to the reactions. Try drawing a V-diagram as though the angle-beam-angle were a single member loaded with two point loads at the beam flanges. These loads would be equal to M/d - with opposite signs. (Alternatively, you could load it with a single moment at the center, but I don't think it illustrates what is going on as well as the force couple)

 

[miecz]

StructuralEIT

I don't agree that the force in the weld is based on M/d. I merely referred to JAE's excellent sketch. It's true those forces are are represented as a vertical couple, but once in the web, that couple becomes a moment, and that moment adds force to the flange, so the flange force beyond the weld is greater than the weld shear, which is equal to the bolt force. This model, of course, ignores some secondary effects, such as the bending stresses in the web beyond the weld.

 

[Lion06]

apsix-
isn't d+g (or d+2g) only the lever arm at the section where the angles connect to the wall? It is not the moment arm 2" away from the wall where the angle leg is welded to the beam flange.

I guess I am having trouble visualizing the flow of force. I am picturing a short cant with a moment only applied (either a moment or a force couple at the flanges). Either way, the moment is flowing out of the beam, into the welds, through the angles and ultimately to the wall. It is not starting with the wall and working its way backwards. When the forces are transitioning from the beam to the angle (via the weld), it doesn't know whether the anchor to the wall is 1" up or 100' up. In the attached sketch, would you say that the weld is taking moment only (per JAE's second sketch) and virtually 0 tension/compression into those angles? I wouldn't.

 

[Lion06]

My sketch didn't make it on the last post. Here it is.

 

[rb1957]

forgive me for not reading all the posts. from a free body perspective, the shear is reacted by the web clip, and the moment is reacted over the base of the angles, ie d+2g.

 

[PMR06]

"it doesn't know whether the anchor to the wall is 1" up or 100' up"

True, but just like any other structure, you solve for external reactions based upon the stiffness of the structure, then cut free body diagrams to get internal forces. Another example, in a continuous beam a load on one bay doesn't "know" if the beam is simple supported or if there are several supports. The load "goes" where the global stiffness of the structure dictates.

BTW, how do you quote in a response?

 

[Lion06]

Ok, I've convinced myself using JAE's second sketch that the additional shear from the moment in the welds does make the total shear in the welds M/d even though the force in the anchor is M/(d+2g) I'm attaching JAE's sketch and my math.
Thank you all for the discussion and bearing with me with stuff like this. I really can't ever take anything for granted and I really NEED to understand the why and how.
Thanks a bunch!!!

 

[Lion06]

By the way, star for JAE for the sketch and star to miecz for the idea of the moments in the welds adding shear.

Also, if anyone has time to comment on the sketch I recently posted with the weld of the tip of angle to tip of flange, I would appreciate it. I don't believe there is an opportunity in that case for the weld to develop moment.

 

[msquared48]

I have thought about this a lot over the last few hours and havve to amend my original thoughts.

1. The angles, by design, will most likely be the same size for ease of construction. That being said, and also assuming that the angles will not be allowed to go into the plastic range, then the legs will not yield. Hence the design moment arm will be closer to the d + 2g, not the d +g that I previously mentioned.

2. The use of d + 2g is less conservative than the use of d + g for the tension in the bolts, and even less conservative than d.

3. The tension is seen in the bolts, and the compression in a compression block of varying stress on the flange of the opposing angle bracket. Tis is true if there is no back plate at the end of the beam. If there is a back plate, then there is a larger area triangular stress block that starts below the bolt in tensoion and proceeds to the limit of the lower angle flange.

4. Considering comment 3, the moment arm value of d +2g is only an approximation of the true moment arm.

Mike McCann
MMC Engineering

 

[miecz]

StructuralEIT-

Agree with your analysis except for one major point: The "V" that you calculate is the force in the flange to the left of the weld. The force in the weld is M/(d+2g). It has to be, since, if you take a free body of the angle and sum forces in the x direction, the shear in the weld has to equal the force in the bolt.

 

[Lion06]

miecz-
I agree. I mis-spoke when I said the total shear in the weld. What I really meant to say was the total load in the flange (some from the direct shear in the weld and some from the moment in the weld).
The horizontal shear in the weld will be m/(d+2g), but there will be a vertical component required to add to it vectorially. So that being said, I would still probably just size the weld for M/d (to keep from having to calc the moment in the weld and add the shears vectorially), and size the anchors for M/(d+2g)

 

[swearingen]

This is seriously simplistic, but gets the point across somewhat clearer. It's the free-body of the beam that really sets the mind straight. There is another resisting moment provided vertically by the angle that helps reduce the force of the weld at the beam/angle interface that isn't immediately apparent when looking at the whole picture.

Note that I'm showing it to be M/(d+g) only because I assumed the lower angle to be flush against the concrete.



If you "heard" it on the internet, it's guilty until proven innocent. - DCS

 

[nutte]

I'm late to the discussion, but here's my two cents. For the sake of discussion, I've assumed the angles are stiff enough to keep prying considerations from adding additional load to the bolts.

The beam has a flange force equal to M/d (or more precisely M/(d-tf)). This will be the shear force resisted by the weld, and this is the tension I would use for the anchor. I see the flange force going through the weld into the angle, then through the angle (bending), to the anchor.

This beam has a moment, but let's say it had no moment, only tension on it. Let's make this total tension equal to 2*M/d. Assuming the flanges take all the tension, this gives a flange force of M/d, the same as my suggestion above for the moment.

So far, the top flange has virtually the same stress in each scenario. If the top flange is loaded to full stress, it doesn't really know if the bottom flange is at the same stress, or if it is at the reversal of that (T/C).

For the axial case, beam loaded in tension only, nobody would question that the anchor sees half of that tension force, or M/d.

I contend that these situations are the same, and the anchor tension would be M/d for the moment case.

 

[miecz]

Nutte-

The two cases are not the same. The moments due to T'd and C'd (see above) are additive in the case of an applied moment. These moments cancel each other in the case of a tensile force.

 

[haynewp]

The FBD's that JAE posted earlier have the right principles. The only conservative change I might make is to shorten the moment arm to d+g to get the shear in the weld (pivot about the bottom flange).

 

[rb1957]

FWIW, there are two choices about how to react the applied load. either ...

1) the moment due to P can be reacted due pin loads on the two angles ... moment arm = d+2g. this imples that there is a moment on the welds between the angles and the beam (consider a FBD of the angles, the offset in the forces ("g" apart) has to be reacted somehow).

or 2) the moment due to P is reacted by a couple at the welds to the angles ... moment arm = d. this implies that the angles have a small moment reaction to the rest-of-the-world, again due to the offset in the forces on the angle (offset = g).

or 3) some combination of the above and/or some fussing with the reactions (is the compression reaction at the fastener, at the tip of the angle, at the base of the angle, ...)

 

[nutte]

I disagree with part of JAE's second free body diagram. I apologize for not having time to sketch my own, so bear with this wordy description.

Cut a section just left of the angles. The flange force in each flange will be M/d. This assumes all the stress is in the flanges, with none in the web.

Now go right a little, at the angles. At the interface of the flange and angle, the horizontal force, called T1 in JAE's sketch, has to be equal to M/d for the horizontal forces to be in equilibrium. This is independent of the T' and C' moment, which I haven't fully reconciled in my mind yet. Regardless, for the summation of forces in the x direction to equal zero, T1 must equal M/d.

Now go to the angle. The bottom force, T1, will have to equal the anchor force, T2, for the summation of forces in the x direction to be zero.

Finally, I'm picturing another free body diagram with only the top angle removed (we have beam and bottom angle, no top angle), with the total compressive force in the bottom angle resolved as a point load at the anchor. The only external force is the moment. There will be an internal horizontal force at the top flange to angle interface, T1, which I propose above is M/d. The only other horizontal force is the anchor force C2, which has to equal T1.

T1=T2=C1=C2=M/d.

 

[rb1957]

nutte,
"Now go to the angle. The bottom force, T1, will have to equal the anchor force, T2, for the summation of forces in the x direction to be zero."

sure the force into (and out of) the angle can be M/d, but the angle isn't balanced for moments as the forces are offset (by "g"). to satisfy the FBD, there would need to be a moment on the vertical leg of the angle (as originally drawn) to react this moment. also, these moments restore the overall FB balance; consider the FB of the beam and angles.

 

[nutte]

Sure, there are moments to keep everything in equilibrium. But forces in the x direction still have to be balanced. And that was the original question, what is the anchor force.

 

[Lion06]

Just to get an outrageous example......... if you have a case where the g distance is much, much larger than the d distance (say by a factor of 100), would you size the weld (and anchor) for that much, much smaller force?