2 way Post tensioned slab 5

In design a post tensioned flat plate, can the no. of tendons in the column strip be the same or less than those in the middle strip. What should the proportion be? For example if the majority of the load is located more in the middle strip, does the code (any code) allow more tendons to be located in the middle strip?

 

[Taro]

2-way p.t. flat plate slabs are typically reinforced using a banded-distributed tendon arrangement. In one direction, tendons are uniformly spaced across the slab width, much like a one-way slab. In the other direction, all tendons are bunched together at the column lines. The primary advantage of this arrangement is constructability (the tendons do not have to be woven in each direction to achieve the proper profiles). However, it is possible to arrange the tendons in a different configuration as long as you provide an adequate load path to the columns.

 

[Ingenuity]

Aimmee,

First, column and middle strips are used as a structural approximation, yet convenient method, of recognizing the varying structural actions across design sections. All c/m strips are doing is approximating the transverse distribution of moments across the panel width. You could have many more design strips and vary your reinforcement (rebar and P-T) for each and every strip but this would be laborious and difficult to construct.

It is common to use 75/25 in c/m strips for M-ve moments respectively, and maybe 60/40 for M+ve moments, but it depends on your panel aspect ratio. You can vary this ratio provided you satisfy static's!

Having said that, ACI 318 (USA code) permits the use of banded tendons in one direction (100/0 basically) and uniform tendons (50/50 basically) in the other direction. IMO, this has many shortcomings from a structural behavior and serviceability view point, but it is a common technique in the USA, and easy to do (easy to design and construct) - but NOT necessarily structurally correct!

As Taro also stated, banded/uniform tendons are easy to construct, for UNbonded P-T because it avoids the "weaving" of long, full-length tendons. For bonded PT (in a flat duct) it is relatively "easy" to "weave" the tendons because the ducts are usually in 20' lengths and can be placed and "weaved" first in short lengths then joined, then strands inserted after weaving is complete (but before concrete placement!).

To sort of answer your question...your distribution of reinforcement (rebar and/or P-T tendons) must follow the structural actions and you must have a load path back to your supporting structure in both directions - most often we use an elastic distribution of bending moments to determine the placement of reinforcement systems and P-T systems are no exception to this.

You mentioned you have more load in the middle strips - do you have large and numerous point loads in the mid-panel area/s?

You may wish to check out a recent thread sort of related to this subject:

"Flat plate with point loads"
thread507-20572

HTH

 

[Taro]

Ingenuity,

Can you expand on your statements that the banded-distributed tendon layout "has many shortcomings from a structural behavior and serviceability view point" and is "NOT necessarily structurally correct"? Millions of square feet of post-tensioned buildings have been built using this method in my area, and I have not heard of any horror stories so far that are attributed to the banded-distributed tendon layout. I would appreciate it if you could point me in the right direction for documentation of these problems.

 

[rapt]

Aimmee,

Inginuity's comments in his first paragraph are correct for distribution, just needs a little more detailed comment to try to simplify things.

If the span length of the column strip and middle strip and the concrete shape are the same then more tendons will be required in the column strip than the middle strip.

If the loads are uniform, an approximate logic for this assuming that the column strip and middle strip widths are equal and using load balancing logic, is

Middle strip, half of load carried in each direction to supporting column strips so 25% of total load carried in each middle strip in each direction

Column strip, must support itself plus load from transverse column strip so 75% of total load is carried in column strip.

If the tendons are laid out in a correct two way load balance distribution then there will be three times as many tendons in the column strip as in the middle strip.

This tendon layout very closely matches the elastic moment pattern in a two way slab so it very closely matches the way the slab wants to act without significant forced redistribution of moments. This has always been my design philosophy. Putting the reinforcement (including tendons) where the moments are sounds logical. All RC design is done this way and PT design in many parts of the world is as well

If the loading is not uniform as you have suggested, then the proportions may vary but the main premise remains that the middle strip load must be carried to the support lines to be carried to the columns by the column strips to the columns. The column strips must support themselves plus the load they receive from the middle strips.

The middle strip does not need to be carried 50/50 in each direction. This can be varied especially if span lengths vary but the total load carried in the 2 directions must be 100%. If it is varied, then the column strips must be designed to carry the proper proportion of the middle strip loads being distributed to them. Variations in this will often be needed if the span lengths vary if you want to match the actual moment pattern. In that case, the longer span distribution will probably be reduced in the column strip and increased in the middle strip and the short span direction the reverse. The limit of this is a continuous support in one direction with equally spaced tendons in the other direction.

For the whole panel, the total load carried in each direction must be 100%. So, if extra load is carried in the middle strip in one direction compared to the other then the reverse is true for the column strips so that the total load carried in the 2 directions is 100%. If the spans in one direction are significantly longer than the other direction, then the column strip widths will not equal the middle strip widths in the shorter span direction so the distribution may need to vary.

The extreme of this is that the tendons are equally spaced in one direction and banded in the other. 100% of the load is still carried in each direction so statics is satisfied and a load path is supplied to the support. Unfortunately, the load path supplied is a One Way load path and the elastic load path is a Two Way load path. The result is that the tendon and reinforcement layout in no way matches the elastic moment pattern.
Large amounts of redistribution are required for the moments to change from the two way pattern in the uncracked state to the one way pattern at failure. Codes do not allow this amount of redistribution because of ductility issues but this has been forgotten in the case of prestressed two way slabs in USA. The redistribution required for the to happen induces extremely large curvatures in the slab which makes the slab non-ductile. Codes limit the amount of redistribution allowed to between 0 and 25-40% depending on the code and the level of reinforcement (reinforcement ratio). In this case the redistribution is 100%. Far too high.


This design method is very limited in its application. Stresses at service (based on the elastic moment pattern allowing for transverse distribution across the width, not average moments) must be kept very low to ensure that cracking does not occur at service. If cracking occurs at service, then the redistribution of moments will begin to at service and large unsightly unrestrained cracks will occur and deflections will increase considerably. Partial prestress design is not allowed for this case unless a two way pattern of bonded reinforcement is supplied. Slabs supporting large point loads and non uniform load patterns should not be designed this way. The proper moment distribution should be considered for these cases.

 

[Ingenuity]

Taro,

I will try and explain what I am referring to...it is a little difficult to put into words in a forum like this...and with no sketches...but I will try...

Just for argument sake, if we have an equal and multi-span flat plate structure in X and Y directions (i.e. square panels) and undertook an elastic analysis we would arrive at bending moments, shears and deflections, and the negative moments would peak over columns regions (in both directions) and be reduced in the middle strips (over the transverse column tendons), so we will see that the transverse distribution of the bending moments varies across the width of the slab panel. For positive moments, again we will see a varying distribution of moments across the panel width.

Now, if this was a RC flat plate we would provide rebar in column strips and middle strips in each direction, no questions asked.

For a PT flat plate, the elastic behavior is really one and the same as RC, and if anything, will behave more elastically as the onset of cracking is delayed due to the effects of the uplift and pre-compression by the prestress.

If we take a RC flat plate that we decide to only reinforce with rebar in the column strips (basically banded rebar, top and bottom) for 100% of the load, and in the orthogonal direction we provided top and bottom rebar that was uniformly spaced across the full panel width what would happen under load? (PS I am not advocating that this is a good method of reinforcing a structural slab! but I want to use it as an analogy). At service load levels it will crack, lots of cracking, and the type of cracking will be wide and somewhat UN-controlled and contrary to what we typically design and strive to achieve in a "good" design - why? Because there needs to be moment re-distribution (basically cracking) to take place so that the RC floor plate can "act" or behave as it was reinforced. Large areas of slab with significant moments have no reinforcement to resist those moments. For the slab to "work", these moments must redistribute to other areas where there is reinforcement. This effect can only happen if we provide a LOAD PATH to the supports, otherwise it will collapse. In this case, the elastic load path is 2-way but the reinforcement load path provided is one way, so the failure load path is one way. The elastic 2-way moments redistribute to a failure 1-way moment pattern. Unfortunately, for this to happen large cracks will develop and excessive deflections will occur to accommodate the redistribution of moments and actions. For "correctly" designed slabs (using column and middle strips both ways and adequate reinforcement) these cracks will be fine, well distributed and not full depth. Codes of practice control/limit the amount of moment re-distribution that can be assumed in design based upon the level of ductility of the section - an unreinforced section of concrete has no ductility! This is why we do not reinforce RC slabs in this way. At service, the redistribution has already commenced and excessive cracking and deflection will occur under normal service conditions. This is defined as structural serviceability failure. While the slab will not collapse (normally unless it is too non-ductile) it will have "failed" in terms of cracking and deflection.

Now if we take a similar flat plate but use P-T in the same banded/uniform distribution we would expect to see a similar behavior - however, depending on the level of balanced load and pre-compression, we will see a delay in the onset of cracking. Again, an unreinforced section of concrete with some P/A does not have much ductility.

But in P-T structures we use load-balancing? Load balancing is a very powerful tool for the engineer. Static's and equilibrium will tell us that we have to "balance" the full load in BOTH directions. From load-balancing view point, the panel tendons may be distributed between the X and Y directions arbitrarily provided the column line tendons satisfy static's and equilibrium. Load balancing can be achieved by band/uniform arrangements, or column/middle strip each way and at the BALANCED load level they will behave somewhat similarly. BUT, at loads other than the balanced load, and especially at overload, the banded/uniform tendons do not perform as well as those with column strips in each direction with corresponding middle strips.

If you look at the crack patterns in a banded/uniform slab at ultimate flexural limit state, after all cracking has occurred, moments have redistributed to where there are resisting moments etc, plastic hinges have formed, you will notice that the loads are all being carried by the equally spaced tendons to the banded tendons over the entire length and then to the support by the banded tendons. In other words, the load path to the support is a 1-way load path and is exactly the same as the load balancing load path. A 2-way load path requires middle strip reinforcement (of one type or another) and column strip reinforcement in both directions and this is not being provided, especially for the positive middle strip moments (between the banded tendons), in banded/uniform tendon design.

Also, when we consider a section's negative or positive moment capacity under ultimate (limit) strength state, the flat plate with banded tendons concentrated in a narrow width over the columns, is considered with the FULL panel width and no treatment of the "shear lag" effects are considered. Basically, it is permitting the use of mobilizing the full panel width, yet only reinforce (PT and/or rebar) in a narrow localized area. This, IMO, is wrong.

The "average" moment concept of banded/uniform tendons is just "assuming" the moment is uniform over the full panel width, and is very much simplifying the real situation.

Contrary to the PTI Manual statement "...forget about arbitrary column strip, middle strip and moment percentage tables which have been long familiar to the designer of the RC flat plates" P-T and RC plates are very similar in behavior. ACI has been slow in recognizing TRUE partial prestress, whilst other international codes have long permitted prestressed concrete as a “unified” material - "structural concrete", ranging from reinforced, to “partial" prestress, to “full” prestress.

The column/middle strip concept still uses average moments, but it is an average moment across the column or middle strips, respectively, NOT over the full panel width. Actual peak negative moments that over at the columns will be appreciably larger than the average column strip moments, and accordingly, the peak negative moments over the columns will be VERY significantly larger than the average of the full panel moments used in banded/uniform arrangements.

For banded/uniform tendon design, ACI 318 assumes the slab to be uncracked so its flexural tension levels are "hypothetical" stress levels - actual stresses are significantly different to these “hypothetical” values. The distribution of flexural stresses will approximately follow those of the elastic bending moments - bending moments that are based upon an assumed elastic material with a varying distribution across the panel width. All working stress calculations should be based on the actual stress distribution. Based on these correctly calculated stresses, banded/distributed tendon arrangements should not be allowed is the stresses are greater than in tension. The stress levels allowed in ACI should be compared to properly calculated stresses, not average moments/stresses, and the levels are far too high to provide correct crack control.

I have reviewed unbonded P-T flat plates, and often see cracks that are wide and not well distributed (sort of opposite to what we try and achieve in a "good" design) and these cracks are often full depth (unreinforced section cracks plus restraint effects can add to the net cracking situation). When I design partially prestressed slabs I ensure that there is sufficient mild steel rebar to control cracking in locations where flexural tensile stresses are exceeded, based upon a more rational (non-uniform) distribution of moments. For UN-bonded P-T flat plates with banded/uniform tendons where the design has assumed average moments across a full panel width the flexural stresses will be incorrect, there will be cracking, and the sections do not really have the ductility to be assuming this level of re-distribution that must occur. The result, if cracking does occur at service load levels, will be similar to the RC slab discussed above, large unrestrained cracks and excessive deflections compared to the calculated values.

Granted, many do not fail in a structural sense (secondary actions come into play), but in terms of serviceability criteria they will be cracked and will have deflected, and when one of the objective was to have controlled or no cracking they have "failed".

On a side note...(although it has little to do with P-T)… in early 1900's C.A.P Turner and his "mushroom" flat slabs were very controversial amongst engineers - there was significant disagreement about simply designing for 100% of static's - and even in ACI 1963 it was still permissible to design a RC flat slab for approximately 70% of static's!!!!

Sorry the above is so verbose.

 

[Taro]

Ingenuity,

Thank you for your in-depth reply. In the U.S., post-tensioned concrete has traditionally been designed for "full" prestressing such that service level stresses are limited in an attempt to preclude cracking. The load-balancing approach combined with the stress limits are the rationale for allowing the banded-distributed layout and it seems to work acceptably. However, I agree that it is not as rational and for ultimate strength conditions significant load redistribution must occur.

U.S. codes do include provisions allowing "partial" prestress, but many designers are not yet familiar with this approach and many design software programs do not accomodate it (insert advertisement for RAPT here). Also, there is the constructability issue. There would need to be compelling evidence that the banded-distributed layout is either unsafe or uneconomical in order for many to change their ways.

 

[rapt]

Taro,

I agree that the use of full prestressing combined with banded/distributed tendon arrangements will work. Unfortunately, the definition of full prestressing and the way it is calculated has changed over time.

In doing the calculations to determine the stress levels, the logic has to be correct to determine whether or not the slab is cracked. Effects of restraints of shortening due to supports or connection to stiff members need to be considered. The actual elastic moment distribution in the slab needs to be considered, at least in column and middle strips (yes they still are logically applied in working with the elastic moment patterns), so that the large negative moments near supports are not diluted too much.

Full prestressing used to mean no tension under the real loadings and moments applied to the concrete. Under this condition, banded/distributed tendon arrangements will work satisfactorily.

Then the stress level for uncracked members was raised to 6 root fc. This stress level is too high and definitely does not guarantee an uncracked member. The 6 root fc figure is simply a level at which, as long as there is reinforcement near the tension face, that crack control will be adequate and that complicated checks do not need to be carried out as the results of the complex checks will be ok so save time doing the calculations. The operative term here is "reinforcement near the tension face". In a banded/distributed tendon arrangement there is not always reinforcement at the tensile face where the elastic moment is applied and the stress in the concrete occurs. This limit does not mean that the slab is guaranteed to be uncracked at this level.

The stresses then were begun to be calculated based on "average moments" and compared to these same allowable stress levels somehow relating the moments to the load balancing logic rather than the elastic moment patterns. The actual column strip stresses will be 50% or more higher than the average and the very peak stresses will be far higher again. So, while designers think they are designing uncracked slabs they are not. The slabs will definitely be cracked in negative moment regions and may also be in positive moment regions where the stresses could be 20% higher or so than the average. Intyerestingly, the British Concrete Society Report TR43 on Prestressed Slabs limits the stress in the concrete to zero tension if the stresses are calculated on average moments, recognizing the inaccuracy and unconservative nature of the calculation method being used.

Now, the 2002 version of ACI has raised the bar again and the allowable level is 7.5 root fc based on average moments. These slabs are very cracked and assumptions of uncracked behaviour for crack control, redistribution and deflection are grossly unconservative and just plain wrong!!!

ACI also limits two way slabs based on average moments to an absolute maximum stress of 6 root fc (now 7.5). This limit was in recognition that the average moments are not the real figures. True partially prestressed slabs were not allowed by this clause. This limit was far too high when average moments are considered and the banded/distributed slabs are really partially prestressed without the designer knowing and allowing for it. Also, some designers were starting to ignore this limit and go even higher. Because of this, deflections and cracking are far worse than they should be.

It has gotten to the stage where many USA designers think that the average moments are the real moments and that column/middle strip logic really does not apply (they even had this added to ACI) and the banded/distributed arrangements are the way slabs actually want to act. They are extending the method outside the bounds for which it was developed and where it should stay, uncracked (zero tension under real moments), lightly loaded, uniformly loaded slabs with low steel ratios to remain ductile.

It is not so much that dersigners need to change their ways, rather, they must recognise that the method being used is an approximation at best and that the design must be kept within logical limits if the approximate design method is to be used. If going outside these limits then more accurate design methods are needed. The real method needs to be taught and understood so that designers can then understand why there are limits and apply them logically.