Deflection of slab-on-grade 2

[ajk1]

Is there a way to approximately calculate the deflection of an existing 5" slab-on-grade, 6" granular stone base under it, when subject to a concentrated shoring post load of about 5000 pounds)?

The soil under the granular base is quite stiff (at least 200 pounds per cubic inch coefficient of subgrade reaction). The slab-on-grade has been in place for many decades.

I ask the question because we have to temporarily support a high pressure steam line that is currently suspended at 8 foot centres from the floor above, but the floor above is about to be repaired and the suspension hangers will of course be temporarily ineffective.

The plan is to temporarily support the steam line with steel A-frames at 8 foot cetres which are supported on the existing slab-on-grade.

When the suspended slab above is demolished, its load will be transferred to the shores and from them to the slab-on-grade, which may cause a bit of deflection of the slab-on-grade thereby causing the A-frame to move down a bit. My feeling is that the movement is tiny, but I would like to have a calculation to prove that. So the question is how to calculate is my question. An approximate type of calculation method will suffice.

 

[structSU10]

I would think for a rough number treating in like a spring supported beam, using a reasonable spring constant for the subbase, should give you a decent idea. If you have access to a finite element program that can do plates and springs, it would be fairly quick to check a few cases. hand calc may be cumbersome, and difficult to determine an 'effective' slab width.

 

[ajk1]

Thanks structSU10. I will look into that.

 

[Ron]

Use elastic layer analysis as if it were treated as a pavement.

 

[Hokie93]

Deflection for a concentrated load acting on an infinitely large plate on an elastic foundation is covered in Chapter 8 of Timoshenko's "Theory of Plates and Shells", second edition. While his assumptions undoubtedly do not match your conditions precisely, it should give a reasonable estimate of the deflection provided you are not loading the slab adjacent to a free edge or a joint.

w_max = PL²/8k, where w = deflection and L = lambda.

L² = SQRT (k/D), where k = modulus of subgrade reaction.

D = Eh³/12(1-nu²), where nu = Poisson's ratio for the plate/slab-on-grade and h = the plate/slab-on-grade thickness.

 

[Denial]

On my website (

http://rmniall.com)

you will find a pair of spreadsheets for analysing infinite and semi-infinite slabs supported on an elastic foundation.[ ] However it does not do the "layer analysis" referred to by Ron, so you would have to somehow adjust value of the modulus of subgrade reaction of your subgrade to reflect the slight(?) stiffening effect of the upper layer of the "granular base".

 

[ajk1]

Thanks Hokie93 and Denial -that information is precisely what I was looking for, given that I don't want to spend a lot of time on this. I will check both out and see how it goes. I don't think that it will matter if I over-estimate the deflection by neglecting any stiffening effect of the granular layer.

Thanks very much. Much appreciated.

 

[Denial]

ajk1

I have verified that Hokie93's formula will give the same answer as my (infinite) spreadsheet provided in the latter you use an adequately small loaded area.[ ] So provided you satisfy Hokie's warning about the load being sufficiently away from any slab edge you should use his formula.[ ] It will be much quicker.

If your load is close to a slab edge you could use my (semi-infinite) spreadsheet, with your load applied over a small area positioned exactly at the edge.

 

[TLHS]

Why are you particularly worried about deflection? It doesn't seem like any reasonable deflection that doesn't fail your slab would be significantly greater than the deflections you would have seen supporting from above. Also, your supports presumably would all deflect around the same amount. Additionally, over an 8ft span, your pipe should be able to deal with a bit of deflection.

Are there equipment nozzles or some other items that might make you worried about an exact deflection number?

Just as a rough idea of what we're talking about... Let's say you have a 6"x6" plate on your shoring. Let's project the load through the 5" thick concrete to the subgrade at a 45 degree angle, neglecting bending. Your load would be acting over a 16"x16" area of the subgrade. At 200lb/in^3 stiffness and a stress of 5000lb/256in^2 you get a deformation of about a tenth of an inch. This isn't right, but it's probably right enough. I don't think you're going to have to worry about this unless your system is really sensitive.

At 10kips/8ft of pipe, though, this is a pretty serious install.

Also, here's a fun article for structural point loads on slabs on grade:

http://www.structuremag.org/wp-content/uploads/2014/08/C-StructuralDesign-Azzi-Apr081.pdf

It's about strength rather than serviceability, but it still might be useful.

 

[Denial]

The simplicity of the Timoshenko formula given by Hokie93 got me thinking about whether there might be a comparable formula for when the point load is located on the very edge of a semi-infinite slab.[ ] Based on some of the work I did when I was developing my spreadsheet, I was able to work out that the deflection directly under a single point load on the edge is given by
[ ] [ ] w_max = c * PL²/k
using the same symbols as Hokie93 used.

In this formula c is a numerically determined coefficient whose value varies slightly with Poisson's ratio as follows:[tt]
nu [ ][ ][ ][ ] c
0.0 [ ][ ] 0.404
0.15 [ ] 0.426
0.30 [ ] 0.461
0.50 [ ] 0.521[/tt]

I no longer have access to Timoshenko's book.[ ] The fact that "my" c varies with nu adds raises a suspicion that the Timoshenko formula should also vary slightly with nu. If so, then presumably the given formula applies only to some particular value of nu.[ ] But then the exactness of Timoshenko's coefficient (1/8) hints otherwise.

 

[Teguci]

If the pipe is sensitive and the slab is flexible, I'd think lifting (ie preloading the supports) the pipe with spring cans attached to your temporary frames prior to demo, would take the movement out of the pipe completely. It's not a 3 ft diameter natural gas line, but if you're concerned about a 1/10th of an inch deflection, this would be a way to remove that variable and not concern yourself with the soil pumping area that was neglected when the slab was poured.

 

[Ron]

While both approaches vary the result with respect to Poisson's ratio, the variation in Poisson's ratio for concrete is quite small.....the limits usually being somewhere between 0.10 and 0.20, with 0.15 usually taken as the default value for pavement analysis.

Some polymer modified concretes can have a higher Poisson's ratio. Dynamic loading will also increase Poisson's ratio as compared to static loading.

 

[ajk1]

Thank you all for your tremendous help. As to your point TLHS about the current installation being hung from a floor that deflects, that is a good point too. About your comment that "At 10 kips/8ft of pipe, though, this is a pretty serious install", I am not sure where you are getting the 10 kips. The structural slab will be shored at 4 foot centres and the load on the shores will be 5 kips. The pipe will have its own independent "A" fare steel temporary supports, also by coincidence at 8 foot centres, but they will carry only a tiny load, namely the weight of pipe and contents.

Thanks again everyone for the help.

 

[TLHS]

Okay, I don't think I understand the layout you're talking about then. I was understanding that you were demoing the floor above and were installing A-Frame shoring just to support the pipe. With a 5,000 pound load at each A-Frame leg, that would mean a 10,000 pound vertical force total per A-Frame.

The A-Frames holds up part of the floor too?

I'm significantly less worried if it's a small-ish steam line.

 

[ajk1]

To TLHS: No; the "A" frames hold up only the pipes. The load on the "A" frames is tiny. The issue is the load on the nearby shoring posts that temporarily hold up the slab above; these shoring posts will bring load onto the slab-on-grade, causing it to deflect; the "A" frames, being supported by the slab-on-grade will go along for the ride. I probably did not make this clear in my original description, so thank you TLHS for giving me the opportunity to clarify.

If the shoring posts (not the "A" frames) are placed at 4 feet on centres around the perimeter of the new temporary openings that will be made to get at the corroded rebar and repair it, then the load per shore (not the "A" frame) may be in the range of perhaps 5,000 pounds per shore, although that is probably grossly overestimating the load, but I don't know how to calculate that without doing a computer model, but I am only just trying to get a "rough feel" for the deflection. I think the procedures that you and Hokie93 and Denial have provided to me should suffice.

(Complicating it all is that we have now found by GPR, and confirmed by cores, that there are locations of significant voids (up to 4") under the slab-on-grade in some locations. So of course my question relates to those locations where there is no void under the slab-on-grade).