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Two way slabs
6

Two way slabs

Two way slabs

(OP)
I appreciate JAE's reply. It does seem ridiculous to design for full load in the longer direction when the shorter direction takes more of the load. I've read that slabs should be designed for equal deflection in each direction. That is where the deflection is the fourth power ratio of the length to the width, thus for both directions to act together, the load should be apportioned according to that ratio. That makes perfect sense to me. To design for the longer direction taking full load, that reinforcement would never be activated. Does the ridiculous actually rule and common sense become the ridiculous?

RE: Two way slabs

2
The British code, BS8110, shares the load between the two spans as follows

Msx=asx.n.lx^2

asx=(ly/lx)^4/(8.(1+(ly/lx)^4)

Msx = max moment per unit width
asx = moment coefficient
ly,lx = slab spans in x and y dirns
n = unit ultimnate load

Then a similar equation for the y-direction. This is for slabs with unrestrained corners. There is a more complex formula where corners are restrained.

Carl Bauer

RE: Two way slabs

3
A brief comment regarding the question of designing for full load in both directions.  

If a 2 way slab is bounded by supporting members, then the slab itself can be designed for partial load in each direction, and the support members are designed to transfer the load along all of the slab edges to the columns.

If you remove all edge support members to give a simple flat slab supported on columns, you have to replace the edge support somehow.  The design moments for the edge supports don't disappear when the edge members are removed - they have to be taken by strips of slab spanning between the columns instead.

In other words, if you design the slab for partial load in each direction, you provide a load path from the slab interior to the slab edges, but have not provided any load path to the columns.

In summary, the full load must always be used in both directions for the total floor structure - either the slab on its own (when the edge strips require more reinforcement than the interior) or for the combination of slab and edge members.

RE: Two way slabs

austim, I understand what you're saying, but "full load in both directions" violates equilibrium.  If I have 100 psf on a floor, and the slab is two-way, then the ACI Code requires me to take a full bay width of load in the north-south direction.  This puts ALL of the floor load into the supporting column and the reinforcing in the north south direction is designed to take ALL of the floor load.

Now, after that is complete, I have to take a full bay width of load in the east-west direction.  This puts the same amount of load, again, into the supporting columns.  This isn't a case of seperate load combinations as the reinforcing in the east-west direction (for the column and middle strips) takes ALL the load as well (just like the N-S rebar).

What actually happens, if you model the slab with finite elements, is you get a shared effect between the N-S and E-W slab strips, each taking a portion of the total floor load depending upon the relative stiffnesses of each slab strip, and also whether there are monolithic beams present that add to the stiffness.

With a FE analysis, you get much smaller moments and shears than with the ACI two-way action.  kronosconcrete's issue, I think, was how you distributed the total floor load to each orthogonal strip.  The ACI code just sidesteps that, as far as I can see, and offers a very conservative approach.

RE: Two way slabs

(OP)
I greatly appreciate these replies. Doesn't it boil down to that for a good design the orthogonal directions should act in concert. That would be reinforcement or in my particular concern, post tensioning, creating an equal deflection response. The longer span feels less load, as the shorter span takes the majority. If length to width is greater than two, then it's a one way slab. Why not make a one way slab be designed for full load its longer dimension too? If a load causes the short span to fail, the longer span won't save the day, unless it acts together with the short one. That can only be done by that inverse fourth power ratio distribution, as deflection is a fourth power phenomena.

The PCA Notes on ACI 318-99 in chapter 28 shows all tendons in one direction being within the column width. Post tensioning seems to make rebar virtually obsolete except for minor strengthening  when certain stresses are beyond the tendon's limits. The allowable span to thickness ratio of 45 beats the old rebar limit of 32.

RE: Two way slabs

JAE,

I strongly agree with austim.  A 2-way plate MUST carry 100% of the load in each direction in order to satisfy static equilibrium.  This is clearly documented in most any introductory text on reinforced concrete.  James MacGregor's text includes an excellent historical commentary on the subject.  I have investigated old plate slabs which were designed for less than 100% of the load in each direction and they are in very bad shape.

RE: Two way slabs

Hi Kronosconcrete, JAE, (and others ?)

In case anyone (other than Taro, whose agreement with me was very welcome) read a previous version of this posting, I should point out that this is Revision 1.  The original version included sketches which (due to a translation problem between input screen and final post) looked more than one of Picasso's works than an engineering sketch, and so I red flagged it as 'offensive'.  I have also slightly edited some of the accompanying text.

Since I have always practised well outside USA I have never been required to comply with any of the ACI standards!  (I may have been fortunate in this ?)   I accept that as a result I may have misunderstood the context of the original discussion.

However, I am distressed to think that my comments have been interpreted as a violation of equilibrium.  I obviously have not put my line of thought sufficiently clearly, since no violation of equilibrium is involved.

Some confusion may easily arise from a failure to differentiate between conditions in a slab supported along all edges and a slab supported by corner columns only.

In an edge supported rectangular slab, each edge provides a distributed reaction to a portion of the total load, the exact distribution of reactions being dependent on the plan geometry of the slab.  If one ignores any need for torsional restraint, the corners are generally very lightly loaded.

In a slab supported only by corner columns, the edge reactions (totalling 100% of the applied load) which would have been calculated in the previous case have to be transferred by beam action along the edges (either by beams or by edge strips with increased reinforcement).  But that is not adding 100% to 100%; it is only taking the original 100% and transferring it from the distributed edge supports to concentrated reactions at the columns (which still only total 100%).

I will try to show what I mean; please forgive any crudeness in my sketching.  (pencil and paper would be much easier than this).

Consider a single rectangular slab, with beams (or equivalent in more heavily reinforced strips) at the edges, and columns at all four corners, and subjected to a load W, uniformly distributed over the full slab area.

    
   P ____________________________Q
    |                                                 |
    |                                                 |
    |                                                 |B
    |                                                 |
    |                                                 |
    |____________________________|
   S                          L                       R


 
     _____________L____________                      
    |                                            |  
    |                                            |  
    |                                            |  
    |                                            |  
   P,S                                      Q,R



     __________B_________                       
    |                                  |  
    |                                  |  
    |                                  |  
    |                                  |  
   P,Q                             S,R

I hope that few would dispute that, for a uniformly distributed load of W, each column will carry a load of 0.25W (assuming, of course, equally stiff columns and no foundation settlement, etc, etc).

Whichever way you view the total structure in elevation,  you see a horizontal bending member, (consisting of either a slab alone, or slab plus two edge beams), spanning between double columns at each end, each individual column loaded to 0.25W.

Each elevation shows a bending member loaded with the full load of W, spanning between end reactions of 0.5W, and equilibrium is satisfied.  

This gives you total moments at midspan in the two directions of WL/8 or WB/8, which the total 'deck' structure must carry.  If anything less than the full W were to be used to calculate total moments at deck level in both directions, then equilibrium would not be satisfied, since the total upward reaction is clearly 1.0W in both elevations.

Suppose, for example, the proportions are such that the interior of the slab may be designed to take 70% of the total load W in the short direction and only 30% in the long direction.  Such a distribution of load internally only works if the edges are designed to carry the balance (0.3W on the short edge beams or strips, 0.7W on the long edges), giving 1.0W to be carried in both directions, and equilibrium satisfied.

I realise that all of the above may well totally contradict the detailed requirements of the ACI Code - if so, I'm sorry about that, but I'm afraid that I will lose little sleep over that .  I am confident that in principle none of the above will conflict with any valid FEA analysis.  

Any reader who has got this far and is now looking for a check box that says "Let 'Austim' know that this post was excessively verbose" has my fullest sympathy

RE: Two way slabs

(OP)
Thank you Austim. The supporting beams will be taken into account by the Equivalent Frame Analysis. The moment on the column becomes less as the beams' moment of inertia and "C", torsional factor, increase.
With post tensioning, there is no more column and middle strips, all is designed as a beam strip. Because the ACI 318-99 code allows spans of 45 the slab depth, definitely a greater efficiency is manifesting.
Wouldn't the moments each direction need to be equal? Since the L the longer direction is greater, the W needs to be less to make it the same. That's just a one degree adjustment. The deflection issue has yet to be addressed by anyone. That is the fourth degree thing. For the strain in tension in both directions to be proportionate to create equal vertical deflection, the strain in the long way will be short side to the fourth divided by long side to the fourth, which will have to express itself as discreet W's for each direction.
If this isn't true, I'd sure like to know why.
I respect the notion of slabs not being designed for full W being inferior, but if providing equal deflection and thus a balanced tension isn't calculated and implemented, the system would seem to be out of relationship. Thanks for hanging in with this.
I'm learning the Code from the Notes on ACI 318-99 by the PCA. The procedure for handling the perpendicular direction is never specifically addressed, at least in the Two Way Slab sections that I'm focussing on. Chapter 28 specifically addresses post tensioning and has great examples for a thorough analysis.
I feel nothing is truly understood until it appeals to common sense, as this is all a common sense mathematical construct to simulate results from experience.

RE: Two way slabs

Hi Kronosconcrete,

I feel yet another severe attack of verbosity coming on; you really shouldn't shouldn't have encouraged me to 'hang in there' .

I will refrain from making any comment about anything in the ACI Code and associated documents, in view of my total ignorance of them.

I like your suggestion that you should be able to distribute loads in proportion to the fourth power of the sides.  At first glance it sounds perfectly reasonable, and has the great merit of being extremely simple.

It is just a shame that it doesn't fit the real behaviour of plates very well. Until I (fortunately) checked back to Roark (and through him to Timoshenko and Woinowsky-Krieger 'Theory of Plates and Shells') I was all ready to agree with you that such distribution was a fair approximation for plates supported continuously on all sides.  But even for that case it is a surprisingly bad approximation.  For plates supported at corners only it is of no real value at all.

Firstly, why does your suggestion sound reasonable?  If the bending of a rectangular plate were really analogous to two intersecting narrow strips along the long and short centrelines (spanning between rigid supports), then your formula would be great.

Then, for example, in a square plate the bending moment at the plate centre would be exactly 50% of the comparable bending moment in a one way plate (with continuous support along two opposite edges only).  Similarly, the maximum local reaction at the middle of the edge supports would also be exactly 50%.

But what do Roark, Timoshenko and all the other clever theoreticians say of that?  They say that in a square plate, the central bending moment is 0.0479ql^2 (only 38% of the one-way moment of 0.125ql^2), and the maximum local reaction is 0.042ql (84% of the one way value of 0.5ql).  I don't much like the comparison.

In any case, the simple distribution that you propose only matches the strip deflections at the plate centre - you need a very different proportion to match the strip deflections, for example, at the quarter point of the long centreline.

Matters get far worse when you consider what happens if the plate is supported at the corners only.  If you now consider the two elementary strips to be supported on beams between the corner columns, you have to add the beam deflections to the plate bending deflections to get the total central deflections.  Whichever direction of bending you look at, the 'simple' formula becomes the sum of two terms, that is central deflection = K1*(short dimension)^4 + K2*(long dimension)^4, where K1 and K2 are pretty complex functions of the plate geometry and the edge beam stiffnesses. Were I to try to take that any further I would very quickly be right out of my depth.

Now may be the right time to bring up another view of the whole matter altogether.

I don't know what the PCA reference has to say about concrete plate behaviour, but I do know what the comparable Australian documentation has to say.  Although you may find it difficult to get hold of, I would strongly recommend that you try to find a copy of "AS 3600 Supplement 1-1994 Concrete structures - Commentary". (AS 3600 is a Limit State design code). I am not trying to suggest in any way that the state of knowledge of concrete structures is better in Australia than it is in USA, just that you may get some benefit from a different view of all this.

In an introduction to 'Two way slab systems' the Commentary includes the following statements (in the interest of some forlorn attempt at brevity I will only quote the best bits):

"Two-way slab systems are statically indeterminate to a large degree and can exhibit considerable variation in redistribution of moments from the uncracked state to final maximum capacity.  Recent tests ... have not only confirmed this but have indicated that when approaching maximum load capacity, the distribution of moments is controlled largely by the distribution of steel in the slab"

"Thus in the analysis stage, there is no unique moment field which the designer needs to determine.  Within wide limits, whatever moment the designer adopts should be acceptable for determining the flexural strength of the slab, provided that equilibrium is satisfied".  (Austim comment - isn't that a tremendous thing for them to say - just forget all of the fancy maths, assume a reasonable set of load paths that genuinely satisfy equilibrium, and your load capacity will be OK.  That, of course, doesn't say that cracking at working loads will also necessarily be OK.)

"Furthermore, the flexural strength of the slab is enhanced significantly by the development of very large in-plane forces (membrane action) as the slab approaches failure"

"All these facts suggest that in the design process, any analysis involving a high degree of refinement is quite unnecessary and bears no relation to reality"  (Austim comment - maybe those words should be displayed prominently in every structural design office).

Anything further from me would be totally superfluous.



RE: Two way slabs

(OP)
I agree that high degrees of refinement are unnecessary. These Code people need to create these mathematical approximations to have something to do. But, credit must be given that ever refining nuances gradually contribute to the elimination of that freak occurrence when unbelievably bad and uncanny luck invades.
I wonder how many structural or concrete engineers really keep up and thoroughly understand Code revisions and evolvement. This stuff can seem hopelessly obtuse material at the first few glances. Hanging in there and getting incremental assimilation eventually creates a common sense understanding of the intent.
I'm working on that. Because I have a building system that is perfect for post tensioning, I feel I need to understand this. Math was always my strong suit but teaching one's self from a book is a little tough for me.
It seems that where ever feasible, suspended slabs should have deep perimeter beams. Post tensioning really asks for that with it's thinner slabs and longer spans.
Back to the drawing board.
Thanks all. Positive results are not always immediately discernible.

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