Deflection of Slab with step
Deflection of Slab with step
(OP)
Hi all
Me and a couple of engineering friends were arguing about the deflection
of a slab with a step in it compared to one without a step. (See attached sketch).
My gut feeling is that it will create extra stiffness and reduce the deflection if
it has a step. I will however never take this into consideration if I designed such a
slab.
What are your thoughts?
Me and a couple of engineering friends were arguing about the deflection
of a slab with a step in it compared to one without a step. (See attached sketch).
My gut feeling is that it will create extra stiffness and reduce the deflection if
it has a step. I will however never take this into consideration if I designed such a
slab.
What are your thoughts?






RE: Deflection of Slab with step
Deflection is a function of the moment of inertia. You're increasing the depth of the slab, and therefore the moment of inertia, over a very small width and I think that this would be negligible in the deflection calculations.
Because I`m assuming this is concrete, I lean towards increased deflection because of development issues for the bars going around the corners and cracking at the corners.
In practice, I've never relied on a step like this because of these concerns.
RE: Deflection of Slab with step
Try to run a few models in your analysis software and you'll see the difference. The best way to think of it is to reduce it to an absurdly large step and you'll see the deformation of that intermediate element.
RE: Deflection of Slab with step
1) Sketch A illustrates avscorreia's "extended length" concept. A lot of designers fail to capture this in the design of tall steps essentially because a) they don't recognized the issue and b) it's a tough/impossible thing to model directly in most of the common slab design software packages.
2) Sketch B illustrates Once's anchorage slip/softening concept (I think) using a strut and tie presentation which is how I've been thinking about this. In particular, the loss of efficiency in the upper, opening corner joint tends to introduce flexibility to the joint.
3) As an aside, I've previously attempted to assess slab steps for moment capacity using the strut and tie model shown below. I've found that the width of the slab step will generally need to be about twice the thickness of the slab coming into the opening joint. Interestingly, this nicely matches the standard details of many firms that show it just that way.
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: Deflection of Slab with step
(On a side note, I confess that I'm guilty of, before posting, running a couple of tests in SAP2000 just to be sure...)
I also try to use the 2h rule of thumb for proportioning the step width, combined with generous lengths for tie anchorage.
RE: Deflection of Slab with step
I've wasted a lot of time staring at these. I think it's very difficult to generalize. It depends on the height of step relative to the slab t but also where the step occurs. Is the step between supports to it's like a beam and slab is in neg bending or reverse? Or does it travel through both conditions? I've mostly resorted to trying to move them to supports and then specifying the remaining area to be filled after. Although in practice I haven't heard of any problems so like most things it seems to sort itself out.
Seems that the s&t may be complicated as well. Kootk - that st implies no horizontal shear across the joint, i.e. you could build this in pin truss? I don't think that's true unless you're exactly midspan uniform load. Maybe I'm wrong, would have to stare at it longer.
RE: Deflection of Slab with step
Yeah, nothing says "middle management bores the snot out of me" like a quality, mid-day, tri-colour eng-tips sketch.
Interesting. I've actually gone the opposite way.
For modest steps, I'll move the step as close to mid-span as possible. I don't like having to run two-way column top steel through the steps. The bars there are dense and often large making for difficult moment transfer through the joint. Additionally, I question the validity of punching shear provisions where a slab step is involved. I figure that, at mid-span positive moment locations, the bars are usually nice and small and the moment distribution across the design strip is relatively uniform. This makes for easier joint design.
If a step is so tall that it becomes a stiff beam essentially providing one-way support to the adjacent slab, then I will try to move the step to the supports. But, then, I probably won't also have concentrated top steel over the columns.
Yes! That's exactly right and a keen observation. It's a neat feature of the steps that throws me a little every time that I consider it. For a step of a meaningful height, I believe that there truly is no shear transverse to the leg of the concrete forming the step. This conclusion is also supported by the moment diagram that I provided above. By definition, where moment is constant, there is no shear...
The trick, I believe, is to recognize that vertical shear can still be transferred across the joint. It just presents itself as longitudinal shear in the vertical leg of the step rather than transverse shear. In that respect, it's actually a better mechanism of vertical shear transfer as it's both stiffer and stronger. It's also easily accommodated by the strut and tie model.
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: Deflection of Slab with step
RE: Deflection of Slab with step
I don't see it for conventionally considered slab loads. Any chance the non-STM presentations below swing you around to my way of thinking?
To the extent that any membrane forces would develop in the slab (shrinkage/caternary), those would definitely produce high shear stresses across a step. Interestingly, if the step failed in shear it may well still function in bending and, consequently, remain able to satisfy the original design intent.
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: Deflection of Slab with step
Still wondering if that moment diagram depends on a roller at one end though. If you have restraint from the diaphragm, so a restrained x translation (or spring) it seems like that diagram is no longer valid.
RE: Deflection of Slab with step
Absolutely it does. With restraint, you get those membrane forces that I mentioned above. Membrane forces presenting as axial slab loads beyond the step would present as shear and flexure -- perhaps self limiting -- across the step. I've really been sticking to the small deformation assumptions that we normally apply in slab design.
I've got a great wishy-washy answer prepared for you. I expect that you'll find it wholly unsatisfying.
1) Regardless of the detailing employed, which may involve closed ties, my intent is always to create opening and closing corner joints. There's really no such thing as torsion in the step in the general "slab fold" case. Just flexure and shear transfer. If I go closed ties, I'll try to go with lapped U-bars for constructability.
2) The typical details used by my firm, and many others it seems, have some technical detailing deficiencies in my opinion. It's usually a combination of Z-bars with several of the hooks facing the mechanically disadvantageous way.
3) Often, I'm creating a detail that's meant to deal with both sagging and hogging moments. That muddies the detailing waters a bit at often is what gets me back to something that looks like closed ties / torsion even thought that's not my real intent.
As a retirement project, I'd very much like to author an article on this (structuremag etc). With some of the advances that have been made in STM, and the utter free-ness of spreadsheet computing power, it seems to me that we could very easily be designing these things instead of just "detailing" them.
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: Deflection of Slab with step
This implies that I'm referring to non-linear analysis. In typical slab (or beam) design a roller or pin doesn't change the result as long as you looking at analysis in the undeformed configuration, i.e. 1st order. In this case you would get a different result with pin vs roller even with small deformation/undeformed analysis. I wasn't referring to any 2nd order catenary affect, just a straight statics analysis will give you axial. I haven't run any numbers and it's probably small, and this isn't really a truss so i'm sure the concrete is more than sufficient for that axial as shear.
Regardless of the detailing employed, which may involve closed ties, my intent is always to create opening and closing corner joints. There's really no such thing as torsion in the step in the general "slab fold" case. Just flexure and shear transfer. If I go closed ties, I'll try to go with lapped U-bars for constructability.
I didn't mean to imply torsion, just ties to simplify bending bars/construction. My typical detail is the same as yours (and everyone's) but on two recent jobs the detailer ignored the detail and provided closed ties, presumably because it's easier and he hoped I would ok it (which I did).
As a retirement project, I'd very much like to author an article on this (structuremag etc). With some of the advances that have been made in STM, and the utter free-ness of spreadsheet computing power, it seems to me that we could very easily be designing these things instead of just "detailing" them.
You could publish this in structuremag now, those articles are not very technical and they are very eager for articles. Off topic but I don't really understand that magazine - it's not technical enough to be very useful to engineers but it can't be a great marketing tool since the readership but must be entirely engineers.
RE: Deflection of Slab with step
If your target market is Engineers like structural analysis software or new structural products then it's a perfect valid marketing tool.
RE: Deflection of Slab with step
RE: Deflection of Slab with step
RE: Deflection of Slab with step
RE: Deflection of Slab with step
My choice of words was poor when I mentioned "small deformation". It was never my intent to imply that second order / non-linear analyses would be required for the axial loads to present themselves. Rather than "sticking to small deformation", it would have been more accurate of me to have simply said "sticking to ignoring membrane effects". Although even that gets murky with a step in play as membrane effects become primary/normal effects across the step. Anyhow, my bad.
For me, it would depend on the purpose of analysis:
1) For something like the OP's hypothetical test case -- an isolated simple span -- assuming pin-roller makes sense to me.
2) For evaluating the demands on the step, assuming pin-roller no longer makes sense to me as a result of your insights here. Most slabs will experience membrane forces as a result of the incidental restraint provided by shear walls, adjacent framing etc.
3) For slab flexural and shear design in general, I would definitely not want to assume pin-pin support. The membrane load path is difficult to assess in my opinion so I wouldn't want it entering into the load carrying capacity determination for conventional slab designs.
4) In my experience, most slab steps are not explicitly modeled when FEM is used. Rather, the slabs are analyzed/designed as though the steps were non-existent and then, at a later stage, the steps are "detailed". In this process, I don't think that it would make sense to attempt to model slab axial restraint since, in a slab modeled as though it were flat, membrane action isn't going to generate the interesting results at the slab step anyhow. This is, of course, more a commentary on how things are rather than how they ought to be.
These are interesting results. Kudos for taking the time to generate them. The difference in the moment diagrams is surprising to me. Apparently, you possess an intuition for the situation that I lack. I`ll try to find some time on the weekend to run a couple of studies myself. In the interest of keeping things simple and easily reproducible for anyone who may be following along, I think that I`ll do it like this:
- Steel.
- Weightless structure
- Point load applied at step
- Step at span 1/3 point
- Step height equal to 1/4 span
I may also tinker with changing the pin supports to springs to model some flexibility. I'm wondering if this might one of those situations where the effect is highly sensitive to the difference between full fixity and partial fixity. If anyone has recommendations regarding logical support spring constants, I'd welcome the advice.
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: Deflection of Slab with step
RE: Deflection of Slab with step
With a pinned end on one end and a roller on the other.
Although there is a difference, it rarely differs more than 10%.
It actually makes sense now that we ignore the step when designing slabs.
RE: Deflection of Slab with step
RE: Deflection of Slab with step
RE: Deflection of Slab with step
I messed around with some models over the weekend and came up
with some qualitative conclusions that I find pretty interesting. I wanted to make a comprehensive spreadsheet to study individual parameters in more detail but, as it turns out, that's hard. Here's what I've got:
1) A first order elastic analysis with lateral restraint at the supports will indeed yield two mechanisms of load resistance, one of which will involve axial load in the slab and shear across the step (henceforth known as the Bookowski effect). See the crude sketch below.
2) with the step at midspan, the Bookowski effect yields upper and lower slab moments that sort of oppose one another and produce no net vertical displacement at midspan.
3) we've been contemplating a step shifted towards the higher slab which produces tension reaction on the slab. If you shift the step towards the low slab (right of centre here), I think that you actually get compression reactions acting on the slab. This surprised me. In both cases, the lateral reactions on the slab produce an upwards deflection at midspan that will oppose the displacement induced by the imposed load.
4) in cases where the slab height is small in proportion to the span, I expect that the Bookowsi effect would be small. With deep steps, it seems to have a pronounced effect on bending moments.
5) since the two load paths share in resisting the applied loads, I believe that it would remain safe, in all cases, to reinforce stepped slabs ignoring the Bookowski effect and designing to the conventional moment diagram proposed at the top of this thread. In this respect, a shear failure across the step may actually serve to hasten the redistribution of moments to match the conventional load path. That said, I'm now considering using at least minimum shear reinforcement across my deeper steps to keep any shear cracks reasonably small.
Item number five involves quite a bit of judgement/conjecture on my part. I welcome any debate and/or dissenting opinions.
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.