Plastic Moment of resistance of beam with welded tee at bottom
Plastic Moment of resistance of beam with welded tee at bottom
(OP)
Does anyone know of software that will calculate the plastic moment of resistance of a steel beam with welded Tee section at the bottom? I started writing a spreadsheet, but is a little laborious to cover all cases because the PNA can fall above the bottom flange of the beam, below the bottom flange of the beam or within the bottom flange of the beam.






RE: Plastic Moment of resistance of beam with welded tee at bottom
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
I am almost finished my spreadsheet, but any shortcut as a check on it would be of interest.
RE: Plastic Moment of resistance of beam with welded tee at bottom
Shortcut:
- treat tension portion of section as just the two lower flanges.
- treat compression portion as top flange and whatever web is needed to balance the areas.
It's an approximation but a sound one. UK codes contain a similar method as an option.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
I will try both methods and see how they compare with my spreadsheet answer. My spreadsheet still requires trial and error of the PNA location, but after that it does the calcs. I thought I could use the goal seek of Excel, but it does not always give me the right answer, so I did the trial and error manually.
They are both good ideas, BAretired's method really simplifies it the nth degree, and I suspect may give reasonably accurate answers in most cases. You guys are awesome.
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
I always wonder about fit up given that both members are likely to have natural camber and one is likely to have some deflection, even if jacked.
Perhaps:
1) make backer bar permanent structure and intermittent.
2) move backer bar to other side of stem.
3) connect backer to existing with fillet welds each side.
4) connect tee to backer with fillets one side.
5) throw in stiffeners at 1/4 points-ish
I've never done this but suspect that it might be rather field friendly.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
I would be inclined to do it as ajk1 suggests except that I would prefer a small angle rather than a plate or bar. That would tend to preserve the right angle a little better.
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
It might be a good idea to add nominal stiffeners between the flange of the beam and the tee, although it could be argued that they are not required.
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
1) It's kinda hard in the vertical position.
2) Depending on how much of a stickler you are for weld inspection, considerable inspection costs could be eliminated.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
What are the stiffeners doing, and what spacing are they at, or are they only at the ends. I don't see any mention of stiffeners in Newman's on-line notes.
RE: Plastic Moment of resistance of beam with welded tee at bottom
The angles would be shipped loose and I'm assuming that the heel side fillet weld to the existing beam is achievable. You'd need to do it anyhow for the backer bars although, admittedly, you'd be much less concerned about the weld quality in a true backer bar attachment. The tee would only be welded to the angles on the freely accessible side.
I was proposing welding the angles into place and then, subsequently, welding the tee into place. As I showed in my crappy sketch, the angles would be welded to the beam in two locations. However, it would be the heel weld that would be doing all of the work.
The stiffeners would be stabilizing the Tee. They would be particularly relevant with my proposal as the tee would only be welded on one side which would introduce a small amount of eccentricity to things.
See figure 3 for an example: Link
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
I have a feeling that goal seek did not work (usually, for other spreadsheets that I have used it on it worked fine)is that perhaps it was using too coarse an increment and missed the number that would have satisfied the iteration. Seems strange though.
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
If you are using Staadpro; Section wizard tool may find helpful to you.
can easily calculate elastic and plastic section properties of custom and built-up shapes.
Cheers! -VH
RE: Plastic Moment of resistance of beam with welded tee at bottom
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
It is not wasteful at all. If you are using the plastic moment of the built-up section, then the weld must develop As.Fy of the tee between the ends of the tee and the point where the plastic moment is required.
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
- If you'll be plastifying the reinforcing, then it needs to be developed for Fy between the point of max moment and the end of the member. For this, one can utilize the termination welds and all of the stitch welding in between. As such, this isn't usually too onerous to satisfy.
- If the reinforcement doesn't extend full length, then it needs to be developed past the theoretical cutoff point for the amount of force assumed in the reinforcement at that point. This is MQ/I if you're elastic, As*Fy if you're plastic, or something a little fancier if you're partially plastic and feeling ambitious. It's rare for the termination welds to require As*Fy development in my experience.
- I don't think that VQ/I is technically valid for locations where the section moment exceeds the first yield moment.
- For ajk1's situation, I see no demand based justification for full pen welding. Which isn't to say that we won't do it anyhow for other reasons.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
Using the relationship VQ/I for shear flow may be adequate for the present case. I don't know as I haven't checked it. But one cannot disagree with needing a total of As*Fy at each end of the tee. How else is the tension going to get into the tee other than by the weld under discussion?
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
Thank you for posting this canwest. While I'd never attempted to calculate the plastic version of VQ/I, I must confess that I was one of those folks who erroneously assumed that it would be greater than VQ/I. It's good to know the truth of the matter.
I agree with this as well although, for the sake of precision, I'd tweak it to "a total of As*Fy
at each end of the teeeither side of the point of maximum moment." All of the welds from M_max to the ends of the reinforcement can be counted towards this if one assumes a degree of ductility in the welds which, frankly, may not exist.@BA: just realized that you provide a more precise version of your statement previously. My bad. It was:
Try this on for size:
In these scenarios, we design our welds to satisfy two requirements:
1) We design our intermittent/continuous welds to deal with the increment of moment at each location. This is the VQ/I stuff for elastic situations and the analogous formulation for plastic situations.
2) We design our intermittent welds and termination welds such that, when taken together, they are capable of developing the entire force delivered to the reinforcing at the point of maximum moment.
Additional observations:
1) When reinforcement extends the full length of the original member, termination welds are technically not required.
2) When reinforcement extends only partial length, the VQ/I welds (set out precisely and varying) will not adequately resist the entire force delivered to the reinforcing at the point of maximum moment. Additional termination welds are required to make up the deficiency. These are the MQ/I welds or the analogous formulation for plastic situations (often conservatively taken as As x Fy).
I've attempted to summarize all this graphically below.
In the document that canwest supplied above, Mr. Muir presents the equation shown below for the shear flow in an elastic wide flange. I believe it to be in error and feel that the expression should be multiplied by d/2. That doesn't change any of the conclusions developed in the document however.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
I will print out Kootk's latest message and study it more. Looks quite comprehensive.
RE: Plastic Moment of resistance of beam with welded tee at bottom
Yeah, I think that it's just like cover plate theory with the a' stuff replaced by the tension/shear lag bit that we discussed above.
This strategy would require the more heavily loaded welds to yield and redistribute load to the more lightly loaded welds. How to we feel about that aspect of it? I've always thought of welds as not being very ductile. But then most welded connections require some degree of redistribution capacity in order to function as designed. AISC has a 1.25 factor that is applied to non-uniformly loaded welds of a different sort. I'd be tempted to apply it to this strategy as well.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
That's exactly right ajk1. Graphically and physically, that's what MQ/I "is". Think of it this way:
1) if you extended your reinforcing full length and specified a ridiculous weld spacing that perfectly matched shear flow demand, you would have just enough weld to develop the force in the reinforcing on either side of the peak moment location. The math just works out that way.
2) if you now cut the reinforcement short and leave the welds as they were, your total weld capacity would now fall short of being able to develop the required force in the reinforcing. And the shortfall would be exactly equal to the shear flow demand that would have otherwise been present between the end of the reinforcing and the end of the member.
3) The MQ/I-ish termination welds represent the reinstatement of the shortfall described in #2.
I'd like to revise my "what to do" recommendations to this:
1) Do VQ/I along the length.
2) Do MQ/I at the ends if partial length.
3) Do the tension lag extension that we discussed.
This strategy will be simple, reasonably efficient, and conservative for all cases where section plasticity is involved. And the requirement to fully develop the peak flange force will be automatically satisfied.
I've come to feel that developing As x Fy uniformly may not be the best way to go. That strategy would make the full, reinforced section available to you at the location of peak moment demand but, possibly, nowhere else. Presumably, you want the extra strength and stiffness available to you along much of the length of the member. For that assumption to be valid, I believe that the reinforcing section needs to be developed for MQ/I-ish forces beyond the theoretical cutoff points.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
A lot of piece of mind for a little extra weld.
"We shape our buildings, thereafter they shape us." -WSC
RE: Plastic Moment of resistance of beam with welded tee at bottom
Thanks MJB35. That is almost always my approach...be conservative. Then can sleep better at night.
RE: Plastic Moment of resistance of beam with welded tee at bottom
Oh no, doubt it. I figured out much of it during the course of this thread.
Interesting observation. I would argue that for both the steel and concrete scenarios, the following is true:
1) Where the extra reinforcing is first introduced (theoretical cutoff point), you need to quickly develop it for the portion of the flexural tensile demand that the reinforcing is assumed to be dealing with at that location.
2) The demand in #1 is based on the moment demand at the theoretical cutoff point.
3) The moment demand at the theoretical cutoff point is the integral of [applied shear x lever arm] up to the theoretical cutoff point.
4) If you connect the dots (1->2->3), I think it reasonable that reinforcing termination development is a function of the shear from the end of the reinforcing to the end of the reinforced member.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
I stand firm on this one. Try another version on for size:
1) Required bond is a function of required flexural tension demand at the point of theoretical cutoff (B below).
2) Required flexural tension is a function of the moment demand at the section associated with theoretical cutoff.
3) The moment demand associated with the theoretical cutoff location is a function of the aggregate shear from A to B below. Literally: M = INT(V(x))|A-->B.
If one agrees with points one through three, then one would have to conclude that required rebar development is a function of the summed shear demand from A to B which is analogous to MQ/I in our steel reinforcement problem.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
Yes, for partial length reinforcing scenarios, I believe that there would be something wrong with it.
If the reinforcing extended full length and one provided exactly VQ/I the whole way, then the aggregate horizontal shear transfer capacity would exactly balance the force demand in the reinforcement at the location of peak moment.
If you cut the reinforcement short, and the horizontal shear attachment along with it, then you would not have enough aggregate shear transfer capacity to balance the force demand in the reinforcement at the location of peak moment. The shortfall would be MQ/I.
The above pertains to an elastic system but the same logic would hold were plastic section capacity required.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
The moment at any point x is equal to the area under the shear force diagram between the support and point x. That is not in question, but the amount of additional reinforcement is a function of the difference between maximum moment and the moment which the main reinforcement can resist on its own.
I don't think that is quite the same problem as we are discussing in this thread although there are similarities.
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
But Mc - Mb = Mc - INT(V(x))|A->B. So still dependent on the aggregate shear beyond the extent of the reinforcement.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
The term "aggregate shear" is unfamiliar to me. What you are calling aggregate shear is in fact not shear at all. It is shear times distance with units of foot-pounds or kilo-Newton-meters. But if that is how you define aggregate shear, the additional reinforcement is carrying the aggregate shear defined by the area of the triangle between point B and midspan on your sketch and has nothing to do with the shaded area between points A and B.
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
t.L.Fy/2 = As.Fy (this assumes the full pen weld strength is Fy/2)
so L = 2As/t
where t is the stem thickness of the tee and L is the total length of weld required
For a WT180x32
As = 4070 mm2
t = 7.7 mm
L = 2(4070)/7.7 = 1057 mm
If the length of tee is 8m, the half length is 4m and the average length of weld required is 264 mm/m. If we assume the same provisions as CSA S16 specifies for composite action (shear studs may be placed uniformly between the point of zero moment and the point of maximum moment) then we could use 110mm of full pen weld at 400 mm o/c.
Alternatively, recognizing that full pen welds are not as flexible as shear studs, if we want to ensure that the shear flow is adequate at every section, we should space our welds at 400 mm o/c and use a length of weld varying from 220 at the end down to 0 at midspan. Of course, some reasonable minimum would apply, say 50 or 75 mm.
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
By aggregate shear, I simply meant the area under the shear diagram. It was not the greatest choice of phrasing in retrospect. However, you seem to have interpreted my meaning perfectly and one gets a little tired or repeating the phrase area under the shear diagram over and over again.
The area under the triangle from B to C does define:
1) the felexural tension demand in the cutoff bars at the location of peak demand (C) and;
2) the reinforcement development/anchorage required beyond that point.
For this analogy, however, I would argue that these things are not the parameters of interest. Rather, the parameter of interest is the demand for flexural tension development/anchorage past the theoretical bar cutoff points (B). And that is based on the portion of the moment at location B resisted by the cutoff bars and, by extension, the area under the shear diagram between points A and B.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
I believe this "check" to exist in CSA A23.3 in the code clause shown below where Mf would be the moment at the bar cutoff location. It's really a tough analogy to make in concrete because development in modern codes is complicated by all manner of extra stuff (tension lag, shear effects, etc). The parallels would have been easier to draw back in the bond stress days. Anyhow, here's how I see MQ/I playing out in concrete using the quoted code clause:
1) Sum the area under the shear diagram from the end of the member to the theoretical cutoff point to get Mf.
2) Use Mf/dv to get the flexural tension demand (Tf_tot) in all of the bars present at the theoretical cutoff point.
3) By ratio of bar areas, determine the flexural tension demand in just the bars being cutoff (Tf_co).
4) Ensure that the bars that are cutoff are developed/anchored for Tf_co beyond the theoretical cutoff point.
I believe that this establishes a clear chain connecting the required development of reinforcing bars beyond the theoretical cutoff point (#4) to the area under the shear diagram beyond the theoretical cutoff point (#1). Steps two and three, taken together, essentially do the job of Q/I which is to isolate the flexural stress in the connected "reinforcement" from that in the rest of the section.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
Perhaps we are defining the cutoff point in a different way. I say the cutoff point is the location where the full length reinforcement is just able to resist the moment by itself. Beyond that point, additional reinforcement is needed; thus there is no demand on the additional reinforcement at the cutoff point, but normal practice would be to extend the bars beyond the cutoff point for full or partial development.
If concentrated loads were added each end of the beam between points A and B, keeping the uniform load the same throughout, Mb would increase, creating greater demand on the full length reinforcement but the additional reinforcement would be unaffected by it because Mc - Mb would not change. There would still be no demand on the additional reinforcement at the theoretical cutoff point.
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
No, we've been thinking of the same theoretical cutoff point BA.
Based on the above statements, I believe that our difference of opinion stems from our telling two different stories regarding the flexural tension demand in the rebar at the cutoff locations. See my sketch below and let me know if it fails to accurately reflect your thinking.
With steel reinforcing of steel beams, the MQ/I story is saying that:
1) Immediately to the left of the theoretical cutoff point, all stresses are distributed in MC/I fashion throughout the unreinforced section and;
2) Immediately to the right of the theoretical cutoff point, all stresses are distributed in MC/I fashion throughout the reinforced section.
Making that pseudo-instantaneous jump is what creates the MQ/I and tension lag demands immediately beyond the cutoff point. My story of cutoff rebar flexural tension demand is consistent with the MQ/I philosophy. Your story is not. And that's not to say that my story is more correct than yours or, indeed, that my story is correct at all. Mine simply parallels MQ/I methodology. I also feel that my story is more compatible with strain compatibility (pun intended). But then, we assume strain compatibility in concrete in many instances where it clearly doesn't exist.
I've reviewed a few of the concrete textbooks that I have on hand with regard to this issue. All simply state that, for a number of reasons, bars should be extended beyond the theoretical cutoff point. And I don't dispute that will produce safe designs regardless of which of our stories more closely reflects the truth.
I was hoping that I could find something in print that would make one of our flexural tension demand stories the clear winner but, so far, I've come up empty handed. I've included MacGregor's version below for reference. It deals with overall, mobilizeable code capacity but doesn't say much regarding the expected stress in individual bars.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom
In the case of the WF reinforced with a WT, there is a similar problem. The stem and flange of the tee are by necessity, unstressed at the physical end of the tee. The flange is unstressed for some distance beyond that because of shear lag in the stem, so the prudent designer would take that into account when determining the length of the WT and would likely call for continuous weld in the portion beyond the cutoff point. That would be a matter of engineering judgment.
So far as the full penetration weld along the stem of the tee is concerned, I tend to agree with one of your earlier statements that to distribute the weld uniformly between the end of the tee and the point of maximum moment as we do for example with shear studs in composite beams, is not a prudent course of action because the welds are not as flexible as shear studs. For that reason, the weld metal should be distributed in a manner compatible with the shear flow along the stem of the tee.
BA
RE: Plastic Moment of resistance of beam with welded tee at bottom
RE: Plastic Moment of resistance of beam with welded tee at bottom
Oh no, in my estimation, this should never be necessary (nor advisable). If you're essentially running the reinforcement full length, I believe that you can just use intermittent VQ/I welds with no need for a true termination weld or tee extension (area under left over VQ/I diagram = zero in KootK speak). I'd just double the last weld length for good measure as BA recommended a ways back.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Plastic Moment of resistance of beam with welded tee at bottom