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]

I would add the option of carbon fiber reinforcing to Koots list if the load increase is not that large.

Is there a way to get around fireproofing that stuff on slab soffits these days? That's been an obstacle for me in the past. That said, I feel that it would be reasonable to attempt to talk AHJ into accepting un-fireproofed FRP in a situation like this. The odds of full load in conjunction with raging fire are surely pretty low. This strategy has certainly been employed with external post-tensioning in the past.

 

[hardbutmild]

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.

They literally say THIS IS LOWER BOUND SOLUTION: choose stress field... I will repeat it since you seem to have a reading problem.

It is convenient to reformulate the lower bound theorem as follows:
"In a plastic design [highlight #EF2929]a stress field is chosen[/highlight] 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."

You also brushed over the fact that I gave you another research paper where they talk about a lower bound solution, while you repeat that there is no such thing as a lower bound solution.

A plate twists and bends. A shear-deformable plate also deforms in shear.

You will have to excuse me since english is my second lanugage... isn't equilibrium a force thing? Isn't what you're talking about compatibility of deformation?

When I look at the definition of a lower bound theorem:
"If an equilibrium distribution of stress can be found which balances the applied load and nowhere violates the yield criterion, the body (or bodies) will not fail, or will be just at the point of failure."
it seems like a lot of words if it is supposed to be:
"If [highlight #EF2929]YOU MAKE AN ELASTIC ANALYSIS THAT[/highlight] nowhere violates the yield criterion, the body (or bodies) will not fail, or will be just at the point of failure."

what is even the point of that theorem if it only applies to the elastic analysis results? Furthermore, if only elastic analysis can satisfy the lower bound solution (and you're saying that only an elastic analysis produces equilibrium) why does it say at the end "will not fail, or will be at the point of failure"... elastic analysis is exact, how can it be either exact or conservative if it is strictly exact?

 

[hardbutmild]

Is there a way to get around fireproofing that stuff on slab soffits these days?

maybe near surface mount it and cover it with anything fire resistant from the bottom? might work

 

[KootK]

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.

Parallel axis theorem or no, this make no sense to me in the context of cracked concrete. It's a bit like suggesting that the stiffness of a truss is unaffected by the member chosen for one of the chords.

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.

Reinforced but not reinforced for torsion. Those are different things. Most beams have flexural reinforcement but that doesn't mean that they don't also need stirrups at times.

The aspect ratio is not relevant;

I think that it's utterly relevant. High aspect ratio things experience torsion quite differently from stocky members. It's a bit like the difference between torsion in an HSS steel member (predominantly St.Venant) and torsion in a wide flange (predominantly 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.

What you claimed was that, because of the flexural nature of the warping mechanism, you didn't understand my ductility concern. The point that I was trying to make was that line of reasoning was equivalent to saying "because I've used the strip method, I now have this warping mechanism available and, therefore, ductility is not a concern". As far as I know, the warping mechanism isn't available unless you've applied the strip method, either the old school way or via FEM applied strip method with Wood-Armer etc.

And that flexural stress is based on beam theory formulas.

And beam theory formulas are entirely valid for plate design within the context of the strip method. It's not a big secrete of any kind.

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.

Certainly, the equivalent frame method was, in part, developed for its computational expediency when things were done by hand. That said, what most people are still doing when they use SAFE and RAMConcept for slab design is just another version of strip design with the FEM software handling some of the tedious accounting. Consider:

1) FEM and hand-strip both integrate the design actions over user defined strips. This is the reason ConcePT an SAFE have the functionality of defining strips.

2) FEM and hand-strip methods both generally involve not relying on slab twist for primary resistance. The ACI doc shown below is basically all about how we manage that.

3) FEM and hand-strip methods both result in inflection points and bar extensions based on the integration of design actions rather than moment contours (the 1000 bar approach).

4) FEM and hand-strip done via equivalent frame both model a single story and the columns above and below.

It may not be grossly inaccurate or unsuitable, but it is nevertheless not the most accurate method available.

Now we're getting someplace.

 

[centondollar]

"They literally say THIS IS LOWER BOUND SOLUTION: choose stress field... I will repeat it since you seem to have a reading problem."
The "stress field to be chosen" is the solution to a boundary value problem that includes equilibrium equations and boundary conditions - it is not something that you can conjure at will! Perhaps you are the one with lacking reading comprehension, or perhaps you misunderstood what that text means, or what was taught in school.

"You also brushed over the fact that I gave you another research paper where they talk about a lower bound solution, while you repeat that there is no such thing as a lower bound solution."
I did not say that there is no lower bound solution. An elastic analysis in combination with plastic sectional design fulfills the lower bound theorem of plasticity.

"You will have to excuse me since english is my second lanugage... isn't equilibrium a force thing? Isn't what you're talking about compatibility of deformation?"
A boundary value problem must satisfy equilibrium (gradient of stress tensor = external forces), kinematic equations (compatibility of displacement and strain according to a dimension reduction (bar, beam, plate, shell) model or the full 3D problem) and boundary conditions (displacements, rotations and surface traction forces along the boundary of the solution domain). These conditions are fulfilled for elastic plate analysis, for example, but not by assuming that a plate is a sum of beams.

"Furthermore, if only elastic analysis can satisfy the lower bound solution (and you're saying that only an elastic analysis produces equilibrium) why does it say at the end "will not fail, or will be at the point of failure"... elastic analysis is exact, how can it be either exact or conservative if it is strictly exact?"
First off, I must clarify one thing. An elastic solution can be either linear or nonlinear, with the non-linearity arising from geometry (e.g., beam-column effect), material (e.g., cracking) or boundary conditions (e.g., compression-only supports). An iterative solution to such a problem (using a proper dimension reduction model, such as the bar/beam/plate model, or the full 3D model) also satisfies equilibrium as long as the convergence of the nonlinear analysis is ensured. Similarly, an elastic-plastic analysis - solved by iteration and ensuring convergence - also satisfies equilibrium. The important thing is that the solution is based on a proper mathematical model that provides EQUILIBRIUM, which the "slab made of unit width beams" does not.

Secondly, the conventional design using elastic analysis (or any other solution of the BVP that represents reasonably accurately EQUILIBRIUM and BOUNDARY CONDITIONS) is conservative according to the lower-bound theorem because the sectional resistance is calculated by assuming plastic (compression block, yielding rebar) failure.

 

[KootK]

If this is true, then I stand corrected. To my knowledge, strip methods were not derived from plate theory.

The clips below are from the table of contents of the Park & Gamble book that I mentioned previously. They spend two chapters on elastic pate theory and then the remaining ten chapters on how those results inform the various permutations of the strip design method.

 

[Brad805]

Koot, yes, that could be a problem that I did not immediately see. I have not tried using CF where we needed a fire rating. I am sure Sika has run into this, so I would be curious what they have said. The study below from 2007 suggests using insulation, but that will depend on the AHJ.

NRC Document

 

[centondollar]

"Parallel axis theorem or no, this make no sense to me in the context of cracked concrete. It's a bit like suggesting that the stiffness of a truss is unaffected by the member chosen for one of the chords."
It is not at all like that. Do the calculation and you will notice this.

"Reinforced but not reinforced for torsion. Those are different things. Most beams have flexural reinforcement but that doesn't mean that they don't also need stirrups at times."
Torsion in the plate model is bending along the edge surfaces of an infinitesimal cube, which essentially means that orthogonal reinforcement will pick up the twisting effect. The rebar will pick up the twisting if the designer knows the content of the finite element computer program output and designs the rebar accordingly.

"I think that it's utterly relevant. High aspect ratio things experience torsion quite differently from stocky members. It's a bit like the difference between torsion in an HSS steel member (predominantly St.Venant) and torsion in a wide flange (predominantly warping)."
The size of the twisting is affected by the aspect ratio, yes, but plates do not experience warping torsion or St Venant torsion in the sense that a beam does. The effect of small aspects ratios, irregular loads and irregular boundary conditions (e.g., columns instead of wall supports) do increase the relative size of the twisting moment, but it doesn´t mean that the plate acts like a beam in torsion.

"What you claimed was that, because of the flexural nature of the warping mechanism, you didn't understand my ductility concern. The point that I was trying to make was that line of reasoning was equivalent to saying "because I've used the strip method, I now have this warping mechanism available and, therefore, ductility is not a concern". As far as I know, the warping mechanism isn't available unless you've applied the strip method, either the old school way or via FEM applied strip method with Wood-Armer etc."
The "warping mechanism" (to be clear: I refer to the plate twisting moment "Mxy" when you mention "warping mechanism") is not available at all in a strip method, since it is not based on the plate theory from which such moments arise. This is my understanding of the strip method.

"And beam theory formulas are entirely valid for plate design within the context of the strip method. It's not a big secrete of any kind."
If one makes assumptions that something behaves like a beam and applies beam formulas, the method obviously validates and justifies itself. It is no secret to me.


" That said, what most people are still doing when they use SAFE and RAMConcept for slab design is just another version of strip design with the FEM software handling some of the tedious accounting."
I sure hope that this does not happen in my country. Using finite element analysis and (presumably) plate elements, and then ignoring the plate action and designing as if the computer program output were a frame analysis output, is bad engineering.


"1) FEM and hand-strip both integrate the design actions over user defined strips. This is the reason ConcePT an SAFE have the functionality of defining strips."
A commercial company will do what its customers ask for and absolve itself of any responsibility. Thus, I do not think it is fruitful to quote what some FEA program providers offer.

"2) FEM and hand-strip methods both generally involve not relying on slab twist for primary resistance. The ACI doc shown below is basically all about how we manage that."
As far as I know, the latest ACI does not actually say that twisting action can be ignored.

"3) FEM and hand-strip methods both result in inflection points and bar extensions based on the integration of design actions rather than moment contours (the 1000 bar approach)."
I sure do hope that those integrations are done without ignoring plate action.

"4) FEM and hand-strip done via equivalent frame both model a single story and the columns above and below."
Well, yes, but they do not do it in the same manner, and the results are not the same. As soon as you put a plate into a FEM model, it will stiffen the structure and cause some energy to be absorbed by the twisting component.


I have read 447R-18, and it was actually one of the documents that brought this question to my mind and reinforced my views on the inadequacy of equivalent frame methods for anything else than extremely regular slab layouts, preferably supported on continuous members (shear walls) and without any holes or irregularities.

PS. If the computer spits out a solution that accounts for plate action, I believe it reasonable to also design according to those actions, and not to arbitrarily reduce the answers to some strips of "beams in plates".

 

[KootK]

tt is not at all like that. Do the calculation and you will notice this.

Sure, show me the particular calculation that you've got in mind and I'll give it a whirl. I practice in North America so my familiarity of the Eurocodes is limited.

Torsion in the plate model is bending along the edge surfaces of an infinitesimal cube, which essentially means that orthogonal reinforcement will pick up the twisting effect. The rebar will pick up the twisting if the designer knows the content of the finite element computer program output and designs the rebar accordingly.

That strain model that you've described, envisioned on a cube, implies the formation of a concrete strut within the cube. And that's essentially "concrete speak" for the need for diagonal tension shear resistance. In my opinion, the flexural steel needed for a warping mechanism is in addition to the concrete shear resistance required to support that mechanism.

...but it doesn´t mean that the plate acts like a beam in torsion.

There will be significant commonalities. We flexurally reinforce slabs much as we reinforce beams. We shear reinforce slabs much as we shear reinforce beams (stud rails etc). Torsion is somehow the lone action that is completely unlike beams? That's hard to swallow, especially given that, in many ways, torsion is just a gradient shear field.

The "warping mechanism" (to be clear: I refer to the plate twisting moment "Mxy" when you mention "warping mechanism") is not available at all in a strip method, since it is not based on the plate theory from which such moments arise. This is my understanding of the strip method.

Seriously, check out the Park book and then tell me that the strip method is not based on plate theory. The plate theory is pretty much all that it's based upon. On this, your understanding is demonstrably incorrect. Prove me otherwise. If the equivalent frame strip method is not based on elastic plate theory, please show me what it is based on.

A commercial company will do what its customers ask for and absolve itself of any responsibility. Thus, I do not think it is fruitful to quote what some FEA program providers offer.

My point was simply that, if software vendors go to the trouble of coding in strip functionality, it seems reasonable to me to assume that a lot of their customers use that strip functionality and, thus, probably ascribe to some version of the strip method. The hive mind isn't always correct but, most of the time, it is. Certainly, it would give me pause for thought if I found myself questioning the validity of a method that was highly supported by my peers.

PS. If the computer spits out a solution that accounts for plate action, I believe it reasonable to also design according to those actions, and not to arbitrarily reduce the answers to some strips of "beams in plates".

Truly, after all that we've discussed this far, it baffles me that you still feel entitled to claim that the strip method is "arbitrary". The Park & Gamble book is basically 700+p of just how not arbitrary it is. Is it perfect? No, no method is. But arbitrary?!?

Using finite element analysis and (presumably) plate elements, and then ignoring the plate action and designing as if the computer program output were a frame analysis output, is bad engineering.

As I've said repeatedly, the strip method does not ignore plate action.

As far as I know, the latest ACI does not actually say that twisting action can be ignored.

As I've said repeatedly, the strip method does not ignore plate action.

I sure do hope that those integrations are done without ignoring plate action.

As I've said repeatedly, the strip method does not ignore plate action.

 

[rapt]

Centondollar,

If you are talking about the ACI one way strip methods for PT flat slabs I wholly agree.

If we are talking about 2D frames in each direction with distribution to column/middle strips, I am afraid you have been badly misled.

2D frames on a regular rectangular grid of columns, as in this post, will give almost exactly the same results as an FEM plate analysis for the design moments where the Mxy moments are included in the Mx and My design moments. The only difference is that the FEM analysis defines the distribution across the width. Plate action does not change the total moment in each direction. There may be a small variation depending on how the column stiffnesses are modeled, especially end columns, but the total panel moment will still be the same either way.

You have obviously never done the comparison. If you had, like I have and I think Kootk has also, you would find that we are correct.

My point which you do not seem to understand is that in designing a slab analysed by FEM, the designer will still define logical strips over which to distribute the reinforcement, so the reinforcement pattern is averaged over a series of nominal widths. The normal solution to make this not too complex would be to use the code defined column and middle strips except in cases with complex loading/support arrangements. So your "exact mathematical solution" using elastic FEM is reduced to the same level of accuracy as our 2D equivalent frame solution as long as logical transverse distribution factors are used in the 2D method and as long as the designer has included the Mxy moments in the design moments in the FEM solution.

I have not read the latest version of the ACI document, but the first release of that document suggested that designers could ignore Mxy moments (treating them as Compatibility Torsion). That is obviously wrong and I hope they have fixed it in later versions as it is not compatibility torsion, it is some extra stress left over in converting principal stresses to X and Y design direction stresses. Several USA FEM concrete packages (and one Australian) do ignore Mxy by default and some have for a long time. Others include it as they should.

 

[dik]

Good summary, rapt...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Do you feel any better?

-Dik

 

[bugbus]

I just wanted to reiterate my earlier point that the stiffness of the cracked concrete in a particular direction, particularly as the slab approaches the ultimate condition and the tension stiffening effect becomes small, would basically be proportional to the quantity of reinforcement in that direction.

As far as I'm concerned this isn't really up for debate.

 

[KootK]

@centondollar: I deleted that thread with the PT optimization paper. I couldn't get the link set up such that anybody could download it (other than you apparently). I gave up in frustration.