Flat plate with point loads 2

How do you design RC/PT flat plate with several point loads (column over) acting over the design strip? Obviously, the whole design strip doesn't contribute to the load carrying capacity of a single point load! In some cases, point load may be located closer to the middle strip but in other cases, it may also be located closer to the column strip.

Would anyone be able to advise?

 

[redhead]

A yield line analysis for this portion of the structure may be the best approach.

 

[ishvaaag]

Yet if you want to use prestress to sustain point loads on a slab better ensure there are enough strands under it to take the load to hanging points, or a reinforced concrete substructure/beam able to pass it to where the main supporting strands are. A load path issue.

 

[rapt]

Verada

You will have to use properly distributed tendons in each direction to carry the load to support lines and then to the supports. An FE analysis will show the distribution of moments across the spans in each direction. From this distribution you could attempt to determine logical widths of design strips both under the columns above and over the columns below in each direction. If the support layoput below is random and complex then this is your only real solution. You will have to divide the whole slab into design strips in each direction and place and profile tendons and extra untensioned reinforcement for each strip. These design strips could be as thin as 3 to 5 ft wide depending on the moment distribution.

Failing that, if the support layout below is relatively regular, you could determine strip widths based on the point load locations in the spans. I would spread the effect of a point load located at midspan over a width of about .2 of the span length in each direction (half each way). If the point load is applied closer to a support this width would reduce and the amount carried in each direction could vary depending on the relative locations in the 2 directions. The point load would be carried in 2 directions to the support lines where another design strip (column strip)carries them to the supports. These support design strips could be as wide as a standard column strip (yes real columns strips are logical in a properly distributed and designed PT flat plate, contrary to the opinion of the PTI). The remainder of the slab width between these design strips then needs to be designed as separate strips carrying slab load and normal live load only (half in each direction). These are also carried by the support (column)strips to the columns.

So you end up with a conventionally layed out flat plate with column strips and middle strips in each direction with some special middle strip areas which carry the point loads as well. All of the load carried in both directions but with tendons properly disrtributed in the slab to match the load and moment patterns.

You cannot use the very simplified, unconservative, banded/distributed tendon model used in USA for normal flat plates. Your tendons and reinforcement must be placed where the actual moments occur in the slab in each direction. This does not happen in a standard banded/distributed tendon arrangement in which areas experiencing significant moment have no reinforcement and therefore require redistribution (via uncontrolled cracking) of these moments to areas where there is reinforcement to provide a moment of resistance to support the applied moment.

 

[Ingenuity]

verada,

I concur with rapt's comments and will add to the banded/distributed comments - if you are going PT and designing to US standards then ACI 318 18.12.4 on tendon spacing is for "normal live loads and uniform loads" and it states that special consideration shall be provided for slabs with point loads. I would consider that these "special condiderations" are basically design for the actual actions that you have, do not use average panel moments, and go with a column/middle strip approach each way, and still consider your local effects within that middle or column strip for the point loading.

Regardless of whether it is a RC or PT flat plate the methods are the same - i tend NOT to differentiate between the two - prestressing is just another form of reinforcement system when it all comes down to it and one day I trust that the codes of practice will unify this correctly.

The PTI has a recent text called Design Fundamentals of PT Concrete Floors - 1999 and a section is on a case study of a concentratd loading to a flat plate. It does a comparison between a 2D and 3D analysis and tabulates the % differences between -ve and +ve moments and the differences are small based upon FULL PANEL WIDTH - unfortunately the readers of this section are left with the impression that you can analyse a flat plate structure with a significant point load and by averaging the moments across the full panel that you will be okay - this is incorrect. So be careful.

HTH

 

[rapt]

Expanding on Ingenuity's last paragraph,

1 A 2D and a 3D analysis will yield the same total moment in a panel and the same total negative and positive moments at each cross-section. The 3D model will give the extra benefit of defining a very accurate transverse distribution of these moments based on an elastic material. To do a 3D analysis and then take an average moment over the whole panel width is a waste of effort as the 2D model will give exactly the same result in a fraction of the time.
Unfortunately, the transverse distribution of moments in a 3D model is too accurate and we end up with a different moment at each mm or inch across the width, an impossible design situation. Appropriate design widths need to be selected. We have found that the use of the RC 2way division into column and middle strips is a very logical method of dividing up the moments in a 3D analysis. This allows some averaging, especially of the very sharp negative peak moment at the column which is really an anomoly of the analysis method anyway, but is nowhere near as illogical as the Full Panel Width approach. If a 2D analysis is done, a good approximation of the 3D result is achieved by using the RC transverse distribution factors between column and middle strips.

2 The stresses in a slab relate to the moment in the slab. If there is significant variation in moment over a certain width of slab, then the stresses will vary accordingly. For a flat plate with uniform loading, the assumption of Full Panel Width as the design width means that these stresses are averaged over the whole width of the panel. In actual fact, these stresses vary by several orders of magnitude in a maximum negative moment region at a support and by about 25 to 50% in a maximum positive moment region. In the negative moment region even the assumption of averaging over a column strip width and a middle strip width results in a difference in stress of approximately 3 times between the 2 strips and a stress 1.5 times the Full Panel Width figure.

The use of averaged moments is very misleading and does not represent the true stress conditions in the slab. The result of this is that the designer may decide that a slab is uncracked based on averaged moments, but, in actual fact, the slab is highly stressed in some areas and will crack causing unexpected serviceability problems with respect to wide unrestrained cracks and significantly increased deflections.

This calculation is completely unrelated to the tendon layout approach used, either banded/distributed or conventional column/middle strips. The elastic stresses in the slab are generally only slightly affected by the tendon layout as long as a logical load balancing layout is used.
The tendon layout and reinforcement pattern only come into the equation after cracking has occurred. If the bonded reinforcement is not placed where the higher elastic stresses occur in the slab, then the cracks will not be restrained and the effects mentioned in the previous paragraph will be worse.

For slabs with large randomly located point loadings applied on the slab, this is even more critical and placement of tendons and calculation of stresses/capacities must be directly related to the elastic moment pattern in the slab.