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Modulus of Subgrade Reaction (k)

Modulus of Subgrade Reaction (k)

Modulus of Subgrade Reaction (k)

As a geotechnical engineer, I've always been dissatisfied with the way the topic of modulus of subgrade reaction is addressed in geotechnical textbooks and figured I would look to this forum to help clarify some concepts in my mind and hopefully the minds of others.

Firstly, after reading many threads, I now understand that this parameter is a second order problem with interdependence of both the structural system and geotechnical conditions. Asking the question, what is the k-value used for, I am guessing that it is used by the structural engineer to evaluate the deformed shape and resultant stresses in a concrete slab subject to a given load. The structural can then use this value to dimension reinforcement to keep stress levels/deflections within a safe/serviceable range. I'm also guessing that this analysis is somewhat iterative as the amount of reinforcement would influence the slab's structural stiffness and deflection under load. The soil's role in the problem is that depending on how the soil deforms in response to the stress (its stiffness represented by the parameter k) a different portion of the underside of the slab will be supported and influence the reaction force acting on the bottom of the slab.

For structural mats associated with large buildings, I've never seen anything in a geotech textbook showing how to correlate soil data with k. You would think that since the k parameter involves deflection, oedometer tests could be used in some way. I then realize however that depending on the stress level, a linear constant such as k would never adequately characterize the subgrade's reaction. Additionally, depending on the size of the loaded area, there will be deeper or shallower soil strata with varying compressibility characteristics involved. Anyone who has done a settlement calc should realize that you can't just use some USCS or CBR correlation to develop a single k parameter to relate stress and deflection in this instance. In some sense, it is scary to see people on this forum and elsewhere looking to use classification data and index tests to come up with this parameter. One can only hope that they are looking for it's use in a pavement or slab on grade application. My geotech textbook (Das) and at least the classes I took, IMHO, don't do a good job discussing the important boundary conditions that make the published correlations between CBR (and other geotech parameters)applicable.

My epiphany from the other day is that I realized that the available correlation charts generally come from the pavement engineering discipline and that this is an important distinction. In pavement engineering (which for concrete is looking at fatigue) and structural concrete slab on grade (limit state) the problem is relatively narrowly defined. The problem consists of "thin" slabs and lower (compared to buildings anyway) load levels. I've seen others on this site scoff or palm/face at people asking about the k value and correlations with soil parameters, saying it is not a pure geotech parameter. My counter argument however is that given a relatively narrow range of boundary conditions, I would expect there to be a correlation between CBR or R-value and k. In pavement and slab on grade loading scenarios, the stress level in the subgrade below the depth improved by compaction and captured in a geotech parameter such as the CBR is probably relatively unimportant. Therefore, using CBR as a proxy is appropriate considering the difficulty/rarity of performing a plate load test.

If you made it this far and are still reading, any comments refuting or confirming my statements based on your own experience would be much appreciated. I'd also love to know anything about a fundamental approach that could be used to evaluate the appropriate k-value for a large mat foundation.

RE: Modulus of Subgrade Reaction (k)

Mr. Shooter:

An interesting question you pose. However, with this quoted statement, perhaps you can explain why you are concerned. For instance, in your job are you seeing there is some failure, excessive construction cost, or other possibly practical reason involved so that you (or any of us) need a better way than using a k value estimated?.

it is scary to see people on this forum and elsewhere looking to use classification data and index tests to come up with this parameter.

As a reader of this forum can you cite examples of some poor engineering or other problem that has resulted from roughing out parameters for design. Guessing that structural designs may be scary, without documentation doesn't fly.

I've found numerous examples through the years where approximations are needed or justified for design and for holding down costs of the job and also the engineering part.

RE: Modulus of Subgrade Reaction (k)

Oldestguy, thanks for the response. The specific use of the word scary comes from my knowledge regarding the paper found at the website below (see page 7). Skimming this paper again, it may have seeded some of my recent thoughts on the topic and I had just forgot. I suppose this is the sole example to which I would apply this word.


I forget how I stumbled upon this particular website but as a sidebar I highly recommend it.

RE: Modulus of Subgrade Reaction (k)

I didn't read the paper in detail, but I didn't see the settlement problem evaluations as related to using a k value for slabs, which is where we usually find it used. I gathered from your post that you also had that as the main usage for modulus of subgrade reaction. That deep material below a multi-storied building hardly is a subgrade.

RE: Modulus of Subgrade Reaction (k)

The modulus of subgrade reaction is a basis for structural engineers to evaluate deformation from point loads that may or may not be applied to the slab. Say, you know a racking system will be installed in the completed building. The rack load may be applied anywhere in the building, so the structural engineer needs to account for these point loads.

The modulus of subgrade reaction does not influence deeper-seated geotechnical issues.

So, knowing that we are interested in the subgrade support immediately below the slab. I always correlated k to soaked CBR value. We know the CBR sample is compacted to 95 or 100 percent relative compaction, we know the CBR is soaked (i.e., the value will be conservative) and correlations have been developed.

I will not comment on using k to address settlement. Structural engineers have been striving to get k to address settlement for years and I don't buy it at all!


ípapß gordo ainÆt no madre flaca!

RE: Modulus of Subgrade Reaction (k)

Recently I had some discussions with structural engineers who wanted to use k to check settlements but I am glad I could stop them for doing that as f-d mentioned...anyway, this is an interesting thread...

RE: Modulus of Subgrade Reaction (k)

Thanks fattdad, I think that between you, oldestguy, and mulling over the concept for a few days I have more sharply defined the issue in my head. Although not strictly a soil parameter, k can be correlated to soil parameters for slab on grade or concrete pavement design. I do think care needs to be taken that if using correlations, they were developed for the appropriate plate configuration. My original confusion/concern came about because I had always remembered seeing k also used for "mat/raft foundations" for large buildings in my geotech classes. Clearly the correlation between k from a 30" circular plate (pavements?) or 12" square plate (slab on grade?) load test and CBR would not be applicable for the mat foundation case. The soil strata involved in such a loading condition would be significantly deeper than could be captured by these small sizes. From what I gather, the k-value is only relevant for mat foundations using the approximate flexible method of analysis and seems more geared towards selecting the structural mat reinforcement than evaluating a geotechnical condition. Geotech conditions (total/differential settlement) which are going to control your foundation type selection, would be evaluated separately from k considering the entire mat, applicable loading cases, and be evaluated in the conventional way (shallow foundation analysis). I agree with f-d that using only k to evaluate settlement is not reasonable at all. The fact that settlement is proportional to the logarithm of change in stress as opposed to a linear relationship is evidence of that. I did find some equations to modify k from a 12" square plate to an arbitrarily sized mat (as long as the soil stressed by the larger mat is identical to the soil stressed by the plate) that do seem reasonable. I also found an equation relating the k value to the soil elastic modulus, poisson's ratio, and foundation width (Vesic 1961). This to me seems to be the most fundamental way to calculate a k-value for a arbitrary soil profile and load configuration.

All of this still leaves the open questions as to whether there is any guidance on the selection of a k-value for a large structural mat application? Also what would be the upper bound loading condition for which the values obtained from the plate load test and correlations with parameters such as CBR would be valid? I feel like trying to come up with a weighted soil subgrade elastic modulus and using that with the Vesic 1961 equation would, short of FEM, be the most reasonable analytical approach? I'm guessing that E values should be selected to represent pre-construction conditions (no consolidation, no strength gain through gradual load application)? Thanks again to any readers and commenters.

RE: Modulus of Subgrade Reaction (k)

Coduto's Foundation Design book has a good discussion on estimating k values for mats. Basically, it is the actual bearing pressure of the mat divided by the anticipated settlement. Since the settlement will be larger in the center of the mat (assuming the mat is somewhat flexible), the k value will be lower in the center, and higher at the edges/corners. Hence, the k value varies depending on the location with the mat.

RE: Modulus of Subgrade Reaction (k)

Thanks moe333, I will likely order the book. Forgive me if it is described in this reference but how would you structure a recommendation in this instance? Would you give one value representative of the center condition? An average value?

RE: Modulus of Subgrade Reaction (k)

See attached example

RE: Modulus of Subgrade Reaction (k)

Moe333, thanks again! I see what you mean about this text. I really appreciate the author's paragraph at the bottom of page 364. Even in school I've always understood geotechnical engineering to be one of the more experience based sub-disciplines within civil engineering. Practice has confirmed this notion thus far. Along those lines, if you are not fortunate enough to have a professor who can speak from practical experience or who knows how practicing engineers function, then the substantial bridge between theory and putting something on paper to be constructed is not effectively bridged. Nuggets like the one on the bottom of 364 is another way to partially bridge the gap. I've found similar statements in Bowles texts that I have borrowed from others. No such luck from Das :(

RE: Modulus of Subgrade Reaction (k)

the textbook example just irritates me! The soil properties to depths of 200 meters (160 ft) would play into the analyses of a loaded slab and resulting settlement. The, "k" value is being used in the wrong context by a structural engineer who has an equation. Just not my thing at all! Is there a subgrade modulus to help design the slab? Sure, there is! Unfortunately, the ground that's being stressed is not consistent with that value!


ípapß gordo ainÆt no madre flaca!

RE: Modulus of Subgrade Reaction (k)

I don't have a big problem with Coduto's method. It seems to take into account the long term bending of the slab, from which shears and moments in the mat can be estimated. A similar approach is presented in the following article. As Coduto discusses, settlement estimate would be provided based on conventional methods.


RE: Modulus of Subgrade Reaction (k)

This is an interesting subject, and has been discussed in detail on previous posts.
Search for "Modulus of Subgrade Reaction - Beating a Dead Horse Perhaps" among others.
k is not constant (of course), and depends on the foundation size and stiffness variation, in addition to the lower zone soil properties, just as SS has noted.
The k used for pavement design has nothing to do with the k used in settlement analysis, but some engineers are often confused and do not separate the two domains.

RE: Modulus of Subgrade Reaction (k)

To me, a structural engineer, k is a measure of the elastic property of a supporting material. No more, no less.
And as a structural engineer with some soils experience (and experience working with Geotechs), I know that k is highly variable and the way to deal with that is to allow for a reasonable range of k values. There are no absolutes below your feet.

RE: Modulus of Subgrade Reaction (k)

I respectfully disagree. k changes, even when the soil properties do not. That is, k reduces as the mat dimension increases, since the pressure bulb deepens.
k changes as load increases (re-compression vs virgin for clays),
k depends on both the structure's foundation and the soil.

RE: Modulus of Subgrade Reaction (k)

Notice the second to last paragraph in moe333's posted image - on the right side of the "book" near the bottom:

"Because it is so difficult to develop accurate k values, it may be appropriate to conduct a parametric studies [sic] to evaluate its effect on the mat design.
ACI 1999 suggests varying k from one-half the computed value to five or ten times the computed value, and basing the structural design on the worst case condition."

This is what is appropriate to do and what we typically do. We would use a range of k values in the analysis and design based on the soft values of k and then, using higher values of k can check soil stresses.

It is interesting in the structural world how sometimes, many times, the structure isn't all that much affected by variations in the k.

In some cases, with framed buildings, using a fixed support to replicate a footing in the model is a poor choice and using a spring support - even with widely varying values of k - doesn't affect the structural results all that much except when you compare it to an infinitely rigid support.

Now we do not use a range of k values for estimating settlements or deflections other than to get a "feel" for how the variation of k alters these numbers.

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RE: Modulus of Subgrade Reaction (k)

As a late response to the above discussions, I'll just cite the definition of the Winkler modulus, which is load (pressure)/settlement. This should clarify all doubts.
Of course we should specify the dominion. Are we dealing with the limited subset of immediate settlement (as Fattdad strongly implies) or are we dealing with total, time-dependent settlements and at which percentage of total consolidation. Are we dealing with dynamic or static settlements?
The by-the-definition procedure allows us to figure out any situation, at the desired load, once we decided the dominion (immediate or total settlements). Of course the consolidation parameters must be different in both cases.
If we have a slab, we just need to calculate settlements at various points (as described for example in Bowles, Das and other textbooks) by using the appropriate shape modificators. If the soil is layered, we should use appropriate methods relevant to such pattern.


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