two way slab strong bands one direction

[structSU10]

I am looking at an existing two way concrete slab with drop caps from the mid 60s that has one direction very heavily reinforced - good for 2x the stated design live load - while the other direction has much less reinforcing and doesn't quite achieve the stated design live load. I am using RAM concept for this analysis right now. Is anyone aware of an different analysis method for these systems they may have used? I would be surprised if load can be redistributed the other way, or that treating one direction as a one way slab supported by the stronger bands would change things too much, but maybe I am off with that thought.

The odd thing is other floors in the same structure are reinforced as a more conventional two way system, with similar bars each way.

 

[KootK]

Do elaborate on that. I am sure that you are aware of what moments plate theory predicts, and that a slab is not a collection of "unit-width beams" due to the Poisson effect.

Sure. Some time ago now, I did a pretty exhaustive study of the theoretical underpinnings of the traditional methods including cover to cover reads of:

1) Park's book on reinforced concrete slabs and;

2) Hillerborg's first book on the strip method, before he dumbed it down for the masses in the blue version.

What I found was this:

3) The traditional methods, when used as their authors intended, will reliably produce safe designs.

4) The authors of the traditional methods did not just turn a blind eye to the many nuances of elastic plate theory. Rather, they considered those things in excruciating detail and then, usually, distilled and baked them into the very practical methods that we've come to know and love. That process of consideration and distillation to the simplest, safe procedures was very elegant in my opinion.

The strip method (a quasi-scientific method that ignores plate action (twisting resistance and Poisson´s effect)

In reference to the equivalent frame strip method that most everybody uses these days:

1) Poisson's effect is not ignored but, rather, is dealt with by acknowledging that Poisson's ratio is effectively zero for cracked concrete at ULS.

2) Twisting resistance is not ignored but, rather, is dealt with by ignoring that resistance and reassigning the associated demand to the more reliable, flexural resistance mechanisms. This is little different that what folks in modern times do with FEM models via Wood-Armer and the like.

 

[centondollar]

"It's not, except that if you have a finite set of rebar lengths you're no longer using your own principles. You no longer have a continuum, you no longer follow the BM, but choose a quasi-scientific method of roughly placing rebars... a mess really."
The continuum is guaranteed by the massive contribution of the concrete to the stiffness of the slab. Surely you know that the rebar does not significantly affect the stiffness of the slab. There is no "mess" to be found in using elastic analysis and a finite set of rebar sizes and laps?

"Your other comments are so absurd that I don't even want to comment them (such as denying the existence of a lower bound theorem of plasticity). I give up, you do you... I guess many buildings just miraculously stand up."
You must be confusing the combination of elastic analysis and plastic sectional design with plastic analysis. The nature of plastic frame or slab analysis (estimating a failure mechanism and attempting to find the lowest energy collapse load) is always an upper-bound theorem: the correct collapse load is approached "from above" (from a larger value), not from below. The same can be said for finite element analysis with conforming elements.

 

[centondollar]

Kootk:

"1) Poisson's effect is not ignored but, rather, is dealt with by acknowledging that Poisson's ratio is effectively zero for cracked concrete at ULS."
If you have zero poisson´s ratio in a slab, it has already collapsed into a million pieces. Any uncracked part of the section - which does exist in both SLS and ULS, as you know - will contribute to plate behavior and make the slab stiffer than adjacent, isolated beams.

"2) Twisting resistance is not ignored but, rather, is dealt with by ignoring that resistance and reassigning the associated demand to the more reliable, flexural resistance mechanisms. This is little different that what folks in modern times do with FEM models via Wood-Armer and the like."
The mechanism of twisting in the plate model is just as reliable (or unreliable) as the primary direction bending mechanisms; there is nothing special about the flexural components in a plate. I´m sure that some authors of simplified design methods use such justifications, but they are not based on structural mechanics.

As for Wood-Armer or other methods that incorporate the twisting effects, I would place more trust in them, because they are not based on more or less arbitrary assumptions. "Reassigning demand" is something I often read and hear about, but I´ve not yet experienced a sufficient explanation for why such assumptions are any more valid than a plate model. Recall that once cracking has affected the entire plate or larger structural assembly (e.g., frame), the internal forces are once again predicted by elastic theory (due to identical relative stiffnesses), while deflections, vibration characteristics and buckling capacity are changed; the "reassignment" of moment leads to the plate model, not to unit-width beams.

PS. The large safety factors for both material properties and loads are the most likely reason for the "success" of simple - yet incorrect - design methods which seem prevalent in reinforced concrete. On the other hand, modelling reinforced concrete mathematically is no easy feat, so perhaps ignorance is bliss and so-called simplified procedures are the best we´ll get in the near future.

 

[KootK]

If you have zero poisson´s ratio in a slab, it has already collapsed into a million pieces.

I disagree. I feel that it's about the rebar and the rebar in one reinforcing direction is unaffected by what's going on with the rebar in the orthogonal direction. It is in that sense that the Poisson's ration may be taken a zero.

The mechanism of twisting in the plate model is just as reliable (or unreliable) as the primary direction bending mechanisms

I disagree. I feel the mechanism of twisting resistance in a concrete slab is basically unreinforced concrete torsion in a cross section not well proportioned to resist torsion.

..there is nothing special about the flexural components in a plate.

What's special about the flexural resistance mechanism in a slab, relative to the torsional resistance mechanism, is that it's ductile.

As for Wood-Armer or other methods that incorporate the twisting effects, I would place more trust in them, because they are not based on more or less arbitrary assumptions.

There's nothing at all arbitrary about it. It's utterly strategic.

"Reassigning demand" is something I often read and hear about, but I´ve not yet experienced a sufficient explanation for why such assumptions are any more valid than a plate model.

Not more valid but simply valid in their own right. This reassignment is precisely what most folks are doing with FEM plate models by way of method such as the Wood-Armer method. That's what the Wood-Armer method does. I've not yet seen any, credible concrete slab design method proposed that relies on slab twist for primary resistance. If there's one kicking around out there someplace, I'd love to hear about it.

Recall that once cracking has affected the entire plate or larger structural assembly (e.g., frame), the internal forces are once again predicted by elastic theory (due to identical relative stiffnesses)

You mentioned that in the redistribution thread and here, as there, I disagree. This notion that, at first cracking, identical hairline cracks instantaneously occur everywhere such that relative stiffness is unaffected is hopelessly optimistic. Once appreciable cracking takes place within a concrete element, the elastic stress distribution becomes just one of many stress distributions that satisfy equilibrium. Our preference for the elastic stress distribution is really just a function of convenience and our expectation that designing to it will improve serviceability.

..the "reassignment" of moment leads to the plate model, not to unit-width beams.

If all you see when you look at the strip method is "unit-width beams", then I'm afraid that you don't understand the strip method. And I don't know how to help you with that other than to refer you to the publications that I mentioned earlier.

 

[KootK]

...most likely reason for the "success" of simple - yet incorrect - design methods which seem prevalent in reinforced concrete.

So you figure that the whole world, including the brilliant engineers of the past, are wrong about slab design and, rather, it is you alone who knows the truth of it? Perhaps you've studied probability at some point in your training?

 

[centondollar]

"I disagree. I feel that it's about the rebar and the rebar in one reinforcing direction is unaffected by what's going on with the rebar in the orthogonal direction. It is in that sense that the Poisson's ration may be taken a zero."
Rebar often has a negligible effect on the total stiffness of a cross-section (or fiber in a plate).

"I disagree. I feel the mechanism of twisting resistance in a concrete slab is basically unreinforced concrete torsion in a cross section not well proportioned to resist torsion."
The "cross-section" in the plate is a fiber, and looking at the problem from that perspective, it is essentially a solid rectangle subjected to twisting. Solid rectangles handle torsion well, and the twisting engages the rebar and compression block (orthogonal to the main flexural directions); I do not see a reason to disregard this part of the plate model. If you discard twisting, you discard a central part of the plate model. For certain types of loading, geometry and boundary conditions, twisting action can be significant (20%,...,50% of flexure in main directions).

"What's special about the flexural resistance mechanism in a slab, relative to the torsional resistance mechanism, is that it's ductile."
In practice, slab "torsion" is just bending along a plane that includes rebar and compressed concrete, so I don´t understand the ductility argument.

"There's nothing at all arbitrary about it. It's utterly strategic."
If by strategic, you mean strategic in the sense that it predicts a smaller requirement for rebar and thus saves money, I agree.

"I've not yet seen any, credible concrete slab design method proposed that relies on slab twist for primary resistance. If there's one kicking around out there someplace, I'd love to hear about it."
I did not say that twist is to be relied on for resistance, but rather that it must be accounted for in the analysis. The total strain energy includes the twisting energy; if one decides to use Wood-Armer to incorporate the twisting into two orthogonal reinforcement directions, one does not ignore twist. If one models plates as unit-width beams, one ignores twist, and thus makes an error (of varying proportion) in the analysis stage.

"You mentioned that in the redistribution thread and here, as there, I disagree. This notion that, at first cracking, identical hairline cracks instantaneously occur everywhere such that relative stiffness is unaffected is hopelessness optimistic."
That is not what I wrote. What I wrote was that cracking continues, and that finally, the relative stiffness of vertical plate fiber is identical to what it was in the uncracked state.

"Once appreciable cracking takes place within a concrete element, the elastic stress distribution becomes just one of many stress distributions that satisfy equilibrium. Our preference for the elastic stress distribution is really just a function of convenience and our expectation that designing to it will improve serviceability."
This is incorrect. The reason for using the elastic stress distribution (in combination with plastic sectional design) is that it predicts the largest possible internal forces at the uncracked and fully cracked stages. For all possible stages inbetween those extremes, the slab will be able to resist the loading, because the stresses are continuously redistributed. Force follows stiffness.

"If all you see when you look at the strip method is "unit-width beams", then I'm afraid that you don't understand the strip method. And I don't know how to help you with that other than to refer you to the publications that I mentioned earlier. "
It is not all I see, but it is a main principle of the method, because it is not based upon elastic theory, plate theory or yield line theory.

 

[centondollar]

"So you figure that the whole world, including the brilliant engineers of the past, are wrong about slab design and, rather, it is you alone who knows the truth of it? Perhaps you've studied probability at some point in your training? "
Plenty of things have been done wrong in the past, wittingly or unwittingly, by both engineers and academics. Probability theory is not needed to recognize the evolution of the body of knowledge of engineering. Furthermore, I am certain that no "brilliant engineers" (high-rise top-shots?) use strip methods these days if their goal is to milk out all available capacity out of a slab, because the method ignores plate stiffness and therefore overestimates internal forces.

 

[KootK]

@centondallar: indulge me in a little experiment. I believe the square below to be black in color. What color do you think it is?

 

[centondollar]

The square is black. Was this an underhanded attempt at a "this person must be a bot"-joke? I expected civil discussion, but that seems to be too much to ask for on the internet.

 

[hardbutmild]

You must be confusing the combination of elastic analysis and plastic sectional design with plastic analysis. The nature of plastic frame or slab analysis (estimating a failure mechanism and attempting to find the lowest energy collapse load) is always an upper-bound theorem: the correct collapse load is approached "from above" (from a larger value), not from below. The same can be said for finite element analysis with conforming elements.

I guess that Attard and Base in this paper

https://www.sciencedirect.com/science/article/pii/B978008023256050036X

were wrong, because they open their introduction with this sentence "In what Hillerborg called his 'Strip Method' he attempted to formulate a practical and simple method of design for two-way spanning reinforced concrete slabs, based on the lower bound theorem of plasticity."

In this book

https://www.bookdepository.com/Desi...h-Stress-Fields-Aurelio-Muttoni/9783034898850

Muttoni, Schwartz and Thürlimann say near the beginning (page 15 and 16):

It is convenient to reformulate the lower bound theorem as follows:
"In a plastic design a stress field is chosen such that the equilibrium conditions and the statical boundary conditions are fulfilled. The dimensions of cross-section and the reinforcement have to be proportioned such that the resistances are everywhere greater than or equal to the
corresponding internal forces."
"In the design of framed structures instead of selecting a stress field one has to select a distribution of stress resultants, i.e. internal forces and moments. The admissible stress state corresponding to a system of loads determines the resultants in all parts of the structure. Thus all structural components and, especially important, details can be accurately proportioned and designed."

This is their book on stress fields (i.e. a fancy version of strut and tie) where it is quite clearly stated that stress field method is indeed a lower bound plastic solution from analysis to design. And these guys kind of know their stuff, they're a big deal.

 

[KootK]

The reinforced topping idea would work but but could get expensive, especially if the added weight triggered foundation upgrades.

Another alternative might be to introduce beams such that the system could be analyzed as being truly one way. The Achilles heel of such a scheme, however, is surely the existing services running below the slab. That said, if those services must remain, there might still be a way to pull it off:

1) Something akin to the stub girder setup.

2) External post tensioning.

3) Make the "beams" concrete curb-ish things on the topside hidden within a raised floor (also not cheap).

You'd want a pretty stiff beam to get this thing acting convincingly one-way.

 

[KootK]

Was this an underhanded attempt at a "this person must be a bot"-joke? I expected civil discussion, but that seems to be too much to ask for on the internet.

It was just an attempt to lighten the mood a bit. You know, "if I say it's sunny out, you'll probably insist that it's raining...". Now I am starting to wonder if you're a bot though.

I expected civil discussion, but that seems to be too much to ask for on the internet.

Pfft. If this is too rough for your sensibilities, the internet may not be the place for you:

Retaining Wall - Flexural Reinforcement from Stem Into Footing

Rafter without fly brace?

 

[centondollar]

"It is convenient to reformulate the lower bound theorem as follows:
"In a plastic design a stress field is chosen such that the equilibrium conditions and the statical boundary conditions are fulfilled. The dimensions of cross-section and the reinforcement have to be proportioned such that the resistances are everywhere greater than or equal to the
corresponding internal forces.""
That theorem is valid if you use elastic (i.e. plate!) analysis. Equilibrium and boundary conditions are not satisfied for collections of unit-width beams spanning two directions. Unless Hillerborg used plate theory, his solution cannot possibly fulfil equilibrium and boundary conditions of a plate!

"This is their book on stress fields (i.e. a fancy version of strut and tie) where it is quite clearly stated that stress field method is indeed a lower bound plastic solution from analysis to design. And these guys kind of know their stuff, they're a big deal."
The combination of elastic analysis (equilibrium) and sectional design based on plasticity (lever arm and forces from concrete and steel) fulfils the lower bound theorem of plasticity, yes. However, the issue at hand was modeling of slabs using assumptions that are not based on equilibrium (plate theory), but rather on ad-hoc assumptions that boil down to assuming the existence of unit-width beams in a slab.

 

[centondollar]

"It was just an attempt to lighten the mood a bit. You know, "if I say it's sunny out, you'll probably insist that it's raining...". Now I am starting to wonder if you're a bot though."
Artificial intelligence is just a fancy word for applied mathematics, not magic - yet. =)

 

[hardbutmild]

What part of the method does not satisfy the equilibrium? All of the forces are accounted for, I don't understand... If what you suggest is true (that only elastic analysis methods can satisfy equilibrium and statical boundary conditions) wouldn't it make more sense to write "if a stress field is chosen to EXACTLY match the elastic solution"?

But the following problem makes my mind just explode, PLEASE ANSWER ME THIS:
If only the elastic solution is valid and plastic one is not, how can one CHOOSE a stress field? If what you say is true, there is no possibility of choosing, after all you mention quite clearly that elastic analysis is exact and has no room for artistic mumbo-jumbo such as choosing.

 

[Brad805]

Assuming the rebar layout and concrete is relatively well known, and the project requires only a nominal load increase this seems an excellent case to retain someone like Vector Analysis Group to a detailed NLFEA analysis to understand the real slab capacity. Analyzing 1/4 or 1/2 of the slab shown using NLFEA is not complicated if you know how to do it, and have the software.

I would add the option of carbon fiber reinforcing to Koots list if the load increase is not that large. In my experience the added thickness option leads to architectural problems at stairs.

How has the slab performed to date?

 

[KootK]

Rebar often has a negligible effect on the total stiffness of a cross-section (or fiber in a plate).

I feel that rebar has a pretty significant effect on the stiffness of a cracked concrete member, which is what ULS strip design speaks to. How could it not, it's the tension bit?

Solid rectangles handle torsion well

Solid, unreinforced, concrete rectangles do not handle torsion well at all, particularly when aspect rations are high as they will be with a concrete slab. With high aspect ratios, it basically just becomes a form of warping.

In practice, slab "torsion" is just bending along a plane that includes rebar and compressed concrete, so I don´t understand the ductility argument.

What you've described is precisely what we accomplish with Wood-Armer and the like: converting cross sectional torsion to a flexural mode of resistance via warping.

If by strategic, you mean strategic in the sense that it predicts a smaller requirement for rebar and thus saves money, I agree.

I see. So you feel that, not only is everybody but you designing their slabs incorrectly but, in addition, we're all doing that for the nefariously unethical purpose of saving contractors money at the expense of public safety? And you're worried about me not being able to conduct civilized discourse?

If one models plates as unit-width beams, one ignores twist, and thus makes an error (of varying proportion) in the analysis stage.

1) Generally, the strip method does not use unit-width beams but, rather, column and middle strips strategically defined to reflect the lateral distribution of flexural stress.

2) The strip methods do account for twist. If they did not, then then the moments that we design for with that method would be less that the static moments. Our designing our strips to the static moments is us capturing the component of resistance that would otherwise be resisted by the twisting mechanism.

It is not all I see, but it is a main principle of the method, because it is not based upon elastic theory, plate theory or yield line theory.

I disagree. The equivalent frame method that is at the heart of modern strip design is entirely based on elastic plate theory. The sometimes 2D nature of it doesn't change that. You'll also find homage to elastic plate theory in the strip method in:

a) Punching shear checks.
b) The definitions of column and middle strips that reflect the lateral distribution of moments as predicted by elastic analysis.

 

[centondollar]

"What part of the method does not satisfy the equilibrium? All of the forces are accounted for, I don't understand..."
A plate twists and bends. A shear-deformable plate also deforms in shear. A plate has non-zero Poisson ratio. Thus, a "strip method" cannot satisfy equilibrium, since it ignores the actual behavior of a plate.

"If only the elastic solution is valid and plastic one is not, how can one CHOOSE a stress field?"
You cannot "choose" the stress field and apply the lower-bound statement. That´s correct. You can, however, use the upper bound theorem of plasticity (choose a mechanism that produces minimal strain energy and collapse load) and ad-hoc assumptions about collapse mechanism, but it is not a lower-bound method.

"If what you say is true, there is no possibility of choosing, after all you mention quite clearly that elastic analysis is exact and has no room for artistic mumbo-jumbo such as choosing."
Elastic analysis is based on structural mechanics. Dimension reduction, equilibrium of stress and external load, constitutive law and kinematic assumptions. Plane stress or plane strain problems, stress function problems and so on - all based on structural mechanics, and not on ad-hoc assumptions about beams of unit width inside a plate.

 

[KootK]

Artificial intelligence is just a fancy word for applied mathematics, not magic - yet. =)

Supposedly, AI's make better therapists than humans do sometimes. It seems that all we really want, when it comes right down to it, is a receptive ear/mic.

 

[centondollar]

"I feel that rebar has a pretty significant effect on the stiffness of a cracked concrete member, which is what ULS strip design speaks to. How could it not, it's the tension bit?"
Use Steiner´s rule to calculate the 2nd moment of area by first ignoring the rebar and then accounting for it. You will notice that the difference is not very large for an ordinary slab.

"Solid, unreinforced, concrete rectangles do not handle torsion well at all, particularly when aspect rations are high as they will be with a concrete slab. With high aspect ratios, it basically just becomes a form of warping."
The slab is obviously reinforced and concrete is thus confined by both rebar and by the fact that the slab extends in all directions. We are not talking about plain concrete.

The aspect ratio is not relevant; the twisting effect happens in each plate fiber (or "in every cross-section in two directions" or "on the face of an infinitesimal plate element", if you will) and is resisted by the plate fiber.

"What you've described is precisely what we accomplish with Wood-Armer and the like: converting cross sectional torsion to a flexural mode of resistance via warping."
Wood-Armer is not strictly speaking based on warping (but I think I understand why you used that word), but yes, you are correct, and I never made claims to the contrary.

"I see. So you feel that, not only is everybody but you designing their slabs incorrectly but, in addition, we're all doing that for the nefariously unethical purpose of saving contractors money at the expense of public safety?"
I made no reference to who considers what strategic, so no, I do not feel what you wrote. My reply was the first thought that sprung into my mind when faced with the word "strategic" in combination with ad-hoc reinforced concrete theories that have served their purpose in the past.

"1) Generally, the strip method does not use unit-width beams but, rather, column and middle strips strategically defined to reflect the lateral distribution of flexural stress."
And that flexural stress is based on beam theory formulas.

"2) The strip methods do account for twist. If they did not, then then the moments that we design for with that method would be less that the static moments. Our designing our strips to the static moments is us capturing the component of resistance that would otherwise be resisted by the twisting mechanism."
If this is true, then I stand corrected. To my knowledge, strip methods were not derived from plate theory.

"I disagree. The equivalent frame method that is at the heart of modern strip design is entirely based on elastic plate theory. The sometimes 2D nature of it doesn't change that. "
I am not convinced of this, since "equivalent frames" are simply a remnant of a past in which computing power was limited and need for simple tools was great. An "equivalent frame" is an attempt at modelling something (usually columns and stiffened or unstiffened plates) as a beam precisely to avoid the difficulty of manual calculations with plate theory. It may not be grossly inaccurate or unsuitable, but it is nevertheless not the most accurate method available.