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# Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

## Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

(OP)
From the geotech report for Mat slab:
A modulus of subrade reaction is given for a 30" diameter load plate = 200pci. However it is then stated that "depending on how the slab loads are applied, the value will have to be geometrically standard. The value should be adjusted for larger areas using the following expression for cohesionless soil:"

ks = k* [(B+1) / (2*B)]^2

Where
B = width of the loaded area, in feet
k = coefficient of vertical subgrade reaction for 707 in^2 loaded area

Questions:

1. Can someone help me understand what this is for or how it is applied?
2. What direction is the loaded area intended to be (i.e. the larger of the 2 directions if you have a rectangle)?
3. why is it being referred to as a coefficient now?

Say I have a 50' x 80' rectangular mat should the subgrade modulus be
k = 200*[(80+1)/(2*80)]^2 = 51.25 pci ?

Also why not provide a subgrade modulus that varies based location in the mat (i.e. higher at edges than at center or vice versa I can't remember which is correct).

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

RF;

This topic is too involved to cover it here in detail. The following two books will help you get started:

Foundation Analysis by R. F. Scott, 1981
Design & Performance of Mat Foundations, SP-152, ACI, 1995

In general, this vertical subgrade modulus method has many limitations and can be on the unconservative side if you have 3 or more column lines. Changing the Kv value by 300% will not affect your induced bearing pressure, shears and moments by more than 5%. Also as you have noticed, you have to assume one Kv value under the entire beam strip length. So what happens when the soil is stiff clay one the east side and loose silty sand on the west side of the building - is it rational to use the same Kv value? Further, how can you depend on a method that, in general, when input parameters are tripled, the output values are nearly unaffected?

That is why back in 1975, researched Lee, discouraged the use of this method. Dr. Poulos emphasized the same thing in his 2000 Buchanan lecture - see attached excerpt.

So where does this leave you? Try to get books that explain the Continuum method of raft foundation design.

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

RFreund,

It's an equation from Plate Bearing Test by Terzaghi. Modulus subgrade reaction ks is conceptual only, not soil parameter.
The story began when Terzaghi tried to relate the vertical deflection (d) of a shallow foundation to the load that applied on that foundation (q). Then he defined the relation as ks = load q / deflection d.
The test was performed on a certain size area of plate (on you case the plate is circular plate of 2.5 feet diameter, British standard), so a conversion is needed to use this ks for different sizes of plates / foundations.

The equation ks = k* [(B+1) / (2*B)]^2 from your Geotech report is for squared area B (width) = L (length)
For rectangle area the equation will be : ks = k* [ (m + 0.5) / 1.5 m ] where m = L / B and L > B

Hope that would help.

Cheers.

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

(OP)
@FixedEarth -

Thanks for the info. I would like to look into the Continuum method of raft foundation design, any references in mind?
I read through Scott and Bowels and was left thinking that they would have us vary the Subgrade Modulus based on the location underneath the mat (i.e. near center vs edge). Either Scott or Bowels mentions this. This is where most of my confusion was.

@Andre -

Thanks for the info. That makes sense, as I mentioned I was expecting some sort of equation to vary the Subgrade based on location. But it seems as though I will have a constant subgrade below the entire mat. Which I guess makes it easier for me. As all the springs are now the same (if I have a regular sized mesh in my FE model).

Thanks again!

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

Look at the two books by Hemsley: "Design Applications of Raft Foundations" & "Elastic Analysis of Raft Foundations". Also look at ACI 336.2R, "Suggested Analysis & Design Procedures for Combined Footings and Mats". You will need to work closely with the geotechnical engineer on having uniform subgrade and limiting total & differential settlements. If it is a simple mat foundation, you will want to have a "semi-rgid" mat thickness instead of "flexible" mat thickness.

The continuum method use the soil's "E" and "v" parameters. Since predicting the accuracy of "E" value for soil can be dificult, you need to check the reasonableness of your settlement and moment output. On your first few jobs, consult with experienced mat foundation designers.

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

It is a soil-structure-interaction problem. Read 'Basic SSI Concepts and Applications Overview', Report No. CGT-2002-2 by John S. Horvath, Ph.D, P.E., Manhattan College, August 2002

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

The series of articles and reports on the modulus of subgrade written by Horvath is particularly exhaustive.
He also describes a rigorous method to apply in mat foundations, which once I suggested to the structurist but apparently it is not immediately applicable in the commercial structural softwares, requiring separate FEM software.

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

(OP)
Thanks for the info guys.

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

Watch this video:
Clearly describes how the "k" value should be determined.

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

This is the link to Horvath's paper on the MK-R method which, according to the authors, overcomes the lack of coupling in the springs.

http://jshce.com/files/ceenge-2011-1.pdf
As soon as I read it I sent it to a structural engineer who was willing to try it out. He told me that his commercial structural software (and most likely all Others) couldn't apply the method although he could have used separate standalone applications.
Which wasn't good news, since structural guys ask for winkler data immediately applicable in their structural softwares.

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

(OP)
Interesting stuff, thanks guys!

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

From all research papers, books with mathematics that look like "ET" writen, only logical sense that explaind to me how to determine K modulus was looking at the video posted previously.
Althou I have to say that I have also truble understanding when the talk aboth interation proce between geotehnical and strucural engineer (around time 28:50 in the video).
Can anyone comment?

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

mar2085, the presentation is very interesting, although the skyscrapers projects are a tad above the average jobs you get (those I get for sure, alas).
The iteration procedure simulates the ridistribution of stresses in function of different loading configurations and time elapsed if I'm not wrong here.

### RE: Modulus of Subgrade Reaction - Geometrically standard - Reduction for larger areas

I think that they are trying to equal setlements that goetehnical model gives and those arising from winkler model wich in return has to do with twaeking of "k" modulus under the raft wich in return changes forces in the superstructure members.
Agree that the presentations is impressive and also agree that my skyscraper design hasnt yet been put to test , but if you can use this on BIG projects why not on smaller ones

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