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Question on moment arm for analysis/design
4

Question on moment arm for analysis/design

Question on moment arm for analysis/design

(OP)
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?

RE: Question on moment arm for analysis/design

I would check the tension in the anchors based on d+2g but the shear in the welds based on d.

Technically one of the lines of anchors will not be taking load and it will be a compressive force on the brick/concrete but I have found it is usually conservative to take loads at the anchors.

RE: Question on moment arm for analysis/design

(OP)
I appreciate the response, but can you explain why?  I understand the reasoning for a baseplate, but not for the detail shown for the reasons stated.

RE: Question on moment arm for analysis/design

The tension in the top anchor must be increased to allow for some prying action.  To be conservative, use the plastic moment in the steel angle and divide by about 85% of the distance between the anchor and the top of the angle.  Add this to the tension given by M/d.  

The real answer is somewhere in between.  Free body diagram is a vertical beam with a plastic pin at the bottom compression flange and a plastic pin at the tension anchorage with a tension load at the top of the beam.

RE: Question on moment arm for analysis/design

(OP)
Teguci-
I agree for a flexible angle, but you can make the angle stiff enough to neglect prying by following the procedures outlined in the steel manual of providing the stiffener.  I am more concerned with which moment arm to use.

RE: Question on moment arm for analysis/design

If the angle is 100% stiff, then tension in the anchorage is T x d/(d+g).  aka Moment arm is d+g.

Free body = pin support at the bottom of the compression flange; Tension force at the top of the tension flange; reaction at the wall anchorage

RE: Question on moment arm for analysis/design

I would probably use T = M/(d+g).

RE: Question on moment arm for analysis/design

StructuralEIT

As shown, I would base the tension on d+g.  The flange with the compressive force will transfer that force to the wall in compression through the angle leg parallel to the flange.  I suppose one could adjust for half the thickness of the flange.

I think you wanted to eliminate that bearing action by adding nuts between the angles and the wall.  Sort of,  like leveling nuts under column base plates without grout.  In that case, I would agree with you that the tension/compression forces in the bolts would be based on T/d.  The apparent difference in couples is taken by the moments in the welds connecting the angles to the beam.  If you draw a free body of each angle, you see that you need a moment in that weld equal to T*Δy.
 

RE: Question on moment arm for analysis/design

I agree with Jike as when the moment is applied, one of the bolts will be in tension, and the bottom or top of the beam will be the thrust point for the compression force as the beam is much stiffer than the angle. This leads to M/(d + g) for the force in the wall bolts.

Mike McCann
MMC Engineering

RE: Question on moment arm for analysis/design

(OP)
I appreciate the responses, but miecz is the only one who attempted to explain why.  

If you believe that the tension in the bolt is less than the shear in the weld where does that force go?
I guess I am picturing a FBD of this angle with a Tension force in the horizontal leg of T1=M/d (pointing to the left), and a horizontal force from the anchor of T2=M/(d+g) (pointing to the right).  T1 > T2 and the only other force that can be present is the top of the angle bearing on the wall as it tries to rotate, but this will only add to the forces pointing to the left.
I am failing to see, just by statics, how the force in the anchor can be less than that in the weld (i.e. how T2 applies).

RE: Question on moment arm for analysis/design

StructuralEIT:

The difference lies in the difference of the moment arms...

It is true that the moment arm for the welds is "D", the distance between the welds and the depth of the beam.

However, assuming that the thickness of the angle is enough to transfer the weld force to the anchor bolt without bending, then the effective moment arm of the connection to the bolts is increased by a distance "g", the gage dimension.  If the angle fails, then you have a connection failure.  As this is not what we want design for, the angle must be designed to transfer the force without failure.

Mike McCann
MMC Engineering

RE: Question on moment arm for analysis/design

The previous free-body is for a non-stiffened angle.  As Teguci mentioned above, there is prying-type action occurring that creates a different force in the bolts when compared to the welds on the flanges.

The attached sketch below is for a stiffened condition.  To achieve the "rigid" condition, the welds now must take not only shear but moment along the horizontal leg length.  This creates a force at the bolts = M/(d+2g) which is also the shear at the welds.  The shear at the welds is no longer M/d since there are new moments at the end of the beam that change the total moment to something less.

 

RE: Question on moment arm for analysis/design

"I am failing to see, just by statics, how the force in the anchor can be less than that in the weld "

That's because it isn't, shear remains constant, based on the (d+g) moment arm.
(Assuming there is no prying.)

RE: Question on moment arm for analysis/design

(OP)
So you reduce the shear in the welds by making the angle rigid (from M/d to M/(d+2g) or M/(d+g) depending on your preference)?  I understand that the weld is now also taking moment, but the shear is reduced?
I understand the concept of prying and the additional force it exerts on the bolts.  I made a pretty slick spreadsheet using the prying equations for wind moment connections.

RE: Question on moment arm for analysis/design

The shear will never be M/d, the lever arm can't be less than (d+g).
The shear will only be M/(d+2g) if the bottom angle is very stiff, eg. by using stiffeners or the like. I would use (d+g) in my design unless there is a very good reason not to.
 

RE: Question on moment arm for analysis/design

(OP)
apsix-
csd, miecz, msquared, and JAE have all stated that the welds should be based on the "d" dimension.  JAE has added the qualification that this would only apply if the angle is not rigid.

RE: Question on moment arm for analysis/design

StructuralEIT-

I believe the force in the tension in the bolt is equal to the shear in the weld but the shear in the weld is less than the force in the flange beyond the weld.   Where does that extra force in the flange go? It goes into the web of the beam.  It gets there as a couple thru the weld.  Looking at JAE's second sketch, you see that moment shown as T'a, where T' is represented as perpendicular to the flanges.  Once in the web of the beam, picture the 2 couples in the web with forces parallel to the flanges.  Those forces are the difference between the force in the flange and the shear in the weld.   

RE: Question on moment arm for analysis/design

(OP)
but those are vertical forces.  Additionally, if you look at JAE's first sketch (the one assuming the angle is not rigid), he is using the "d" dimension for the shear in the welds.

RE: Question on moment arm for analysis/design

StructuralEIT
Referring to JAE's first sketch; obviously one of us is wrong, at this stage I don't believe that it is me.

I'll try to explain. The lever arm is the distance between the points of compression (C) and tension (T) reactions on the connection.
The location of C depends on the stiffness of the bottom angle; it's either at the bottom flange or near the bottom bolt.
The location of T is always at the top bolt, it makes no difference if the top angle is flexing or not.

RE: Question on moment arm for analysis/design

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)  



 

RE: Question on moment arm for analysis/design

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.

RE: Question on moment arm for analysis/design

(OP)
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.

RE: Question on moment arm for analysis/design

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.

RE: Question on moment arm for analysis/design

"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?

RE: Question on moment arm for analysis/design

(OP)
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!!!

RE: Question on moment arm for analysis/design

(OP)
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.

RE: Question on moment arm for analysis/design

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

RE: Question on moment arm for analysis/design

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.

RE: Question on moment arm for analysis/design

(OP)
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)

RE: Question on moment arm for analysis/design

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

RE: Question on moment arm for analysis/design

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.

RE: Question on moment arm for analysis/design

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.

RE: Question on moment arm for analysis/design

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).  

RE: Question on moment arm for analysis/design

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, ...)

RE: Question on moment arm for analysis/design

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.

RE: Question on moment arm for analysis/design

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.

RE: Question on moment arm for analysis/design

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.

RE: Question on moment arm for analysis/design

(OP)
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?

RE: Question on moment arm for analysis/design

Absolutely - my fbd shows why this is true, given in your hypothetical that the stiffnesses of the parts can be ignored.



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

RE: Question on moment arm for analysis/design

nutte,

One thing that I sensed in drawing out the second free-body diagram was that the moment in the beam isn't constant.

My first thoughts on this were that there is an "M" in the beam that must be simply resolved through to the end bolts.

However, with the rigid angles welded to the end of the beam, I believe that the "M" that sort of comes in from the left, actually changes across the angle leg widths.

So you can't just say that M/d is the force in the flange because the M varies as you move from the left of the beam towards the end of the beam.

This is tough to visualize because we usually look at beams as single line sticks.  This problem, to be correctly analyzed, would need an FEM analysis to really see what is going on since there are moments applied to the end of the beam at the left and at the top and bottom flange ends.
 

RE: Question on moment arm for analysis/design

Quote:

"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."
  OK

Quote:

"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"
It's not instantaneously all taken out of the flanges. It is taken out along the length of the angle's horizontal leg. The average shear force in the weld is as shown in JAE's angle FBD.

RE: Question on moment arm for analysis/design

JAE, you just beat me to it.

RE: Question on moment arm for analysis/design

StructuralEIT-

I would design the anchor for that much smaller force.  The weld would have a small shear, but a large moment.  The weld would be designed per table 8-4, page 8-66 of the AISC Code, the lower sketch, with a large "a".

RE: Question on moment arm for analysis/design

JAE and haynewp have it - and again you can see it on the free-body diagram on the bottom of my sketch.  The angle redistributes the total moment.  My sketch is crude and is discrete, but I did it that way on purpose so it's easier to see.   



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

RE: Question on moment arm for analysis/design

What if you model the free body of angles and beam as another beam?  See attached.  Just load the beam with a couple equal to M/d and solve the rest using statics.

The the shear and moment diagrams explain what forces the parts are designed for.  The extra force not accounted for in a free body of just the angle is the constant shear thru the beam web.

Is this model valid?

RE: Question on moment arm for analysis/design

(OP)
If you had a case where the top angle were turned down and the bottom angle turned up, would you then design for a much larger anchor force for a rigid angle (say M/(d-2g)) and design for a smaller force if the angles were not rigid (M/d) as shown in JAE's first sketch?
I suppose so, it's just not so intuitive.

RE: Question on moment arm for analysis/design

(OP)
Now that it seems most are in agreement that the weld should be designed for a shear of M/(d+2g) and the additional moment, Can I please get an honest answer as to how many people would have sized the weld for M/d or M/(d+2g) without giving it a second thought prior to this discussion?

RE: Question on moment arm for analysis/design

I would have used M/d, and it would have been conservative.

Mike McCann
MMC Engineering

RE: Question on moment arm for analysis/design

M/(d+2g)

RE: Question on moment arm for analysis/design

Actually M/(d+2g) and if it were 5% overstressed, I would say OK.  Haha.

RE: Question on moment arm for analysis/design

to answer your previous post, i think most would use d-g, assume the compression reaction was applied at the heel of the angle.

RE: Question on moment arm for analysis/design

Hmm, I'm studying it more, free body diagrams of each piece, and I'm ready to concede the point.  I sure would have thought the anchor force would be M/d.

As for design, I'm with Mike, I would have used M/d and not thought twice about it.

RE: Question on moment arm for analysis/design

nutte,

I tried to make what you are suggesting theoretically happen and I can't. If the angles were only attached at the very toe to the beam with welds, and likewise at the very toe with a weld at the wall,  then the shear at the angle's toe would be as you say (M/d). But the assumptions don't work with statics. There has to be a different force transfer between the angle and the beam when using the rigid angle assumption or the angle would be unstable.     

RE: Question on moment arm for analysis/design

I just think it is amazing that "simple" statics problems like this one can throw us all for a loop.  

I'm also impressed by how interested so many of us are in a topic as basic (or essential) as this.  

 

RE: Question on moment arm for analysis/design

These are my favorite kinds of questions, ones that kind of combine strengths of material and statics. They are not based on one's experience and I think they really test your intuitive engineering ability.

RE: Question on moment arm for analysis/design

Yes, just when you think you have a good talent for visualizing load paths....boom, StructuralEIT lowers the boom on us.

 

RE: Question on moment arm for analysis/design

I probably would have used M/(d) for the weld and designed the bolt for M/(d+g), assuming there is no prying.

I still believe that M/(d+2g) is potentially unconservative and therefore wouldn't use it.

RE: Question on moment arm for analysis/design

Well, I would say you are wrong to use M/d for the weld, unless you are using that just to be conservative. I wouldn't use M/(d+2g) either.

RE: Question on moment arm for analysis/design

(OP)
my last question wasn't a right or wrong, just a "who would have done what without giving it a second thought before this thread".

RE: Question on moment arm for analysis/design

I agree with JAE.  We should never be embarrassed to present "foolish" questions here as Structural EIT was initially concerned.  

A star to the originator.

Mike McCann
MMC Engineering

RE: Question on moment arm for analysis/design

I would have designed the top bolt for M/(d+g), then added 30% for prying.

I would have designed the weld for a shear of M/(d+g), plus a moment of (M*g)/(d+g).

RE: Question on moment arm for analysis/design

I honestly don't know what I would have done.  The answer certainly wasn't obvious.  In fact, it was somewhat counterintuitive (is that a word?).  I love these little excercises.

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