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# Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

## Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

(OP)
All -

When designing a large mat foundation we (i.e. structural engineers) will typically use the modulus of subgrade reaction as the basis for the compression only soil springs we include in our analysis model.

Ideally, I get this value from the geotechnical engineer for the project. In addition, we usually n get formulas for adjusting this modulus of subgrade reaction based on the size of the mat. These formulas are usually pretty standard and I can find some variation of them in my foundation text books (Braja Das, Bowles, et cetera) However, the formulas that I recall seeing are all based on square or rectangular foundations.

Does anyone have any guidance for how to adjust these modulus of subgrade reaction for a circular or octagonal mat? I don't recall seeing these types of formulas before.

My tendency would be to use the same adjustment that I'd use for a square of the same contact area. However, I'd like to have a better reference for why I'm doing this other than just my own educated guess.

Thanks in advance for any assistance you can provide.

### RE: Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

For the wind turbines which we have worked on, we have been using the bulk shear modulus "G", as calculated from the initial bulk modulus "G0". From that we can calculate the foundation stiffness for a circular (or octagonal) footing over bedrock, or a half-space, for the four fundamental modes of motion.
Dave

Thaidavid

### RE: Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

(OP)
I can see that for estimating the dynamic properties of the soil stiffness. But, for static properties, the soil stiffness would usually (in my experience) be based on modulus of subgrade reaction instead of shear modulus.

### RE: Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

It was our understanding that, given the nature of a turbine installation, the dynamic stiffness effects would be the governing consideration. Beyond that, I am unable to comment.
Good luck,

Dave

Thaidavid

### RE: Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

I eschew the notion that the modulus of subgrade reaction is influenced by the size of the loaded area. Geotechnical performance certainly is, we know that! The size of the loaded area influencing the change of stresses and the likely magnitude of settlement/compression/consolidation, etc.

The modulus of subgrade reaction is not soil modulus and is not related to pore pressure dissipation. It's something else and unrelated (to me) to the magnitude of settlement.

I know folks will disagree and I'm reluctant to type this reply. Just floating out one man's perspective!

f-d

ípapß gordo ainÆt no madre flaca!

### RE: Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

Josh,
Not quite answering your direct question, but part of the fundamental behavior is to separate the short term "k" spring constant, (or bed of springs) from the long term behavior. That is, the mat foundation to saturated clay vertical response to applied load will be much stiffer for quickly applied and short duration seismic loading [undrained response, Eu] as compared with the 10 year nearly constant load consolidation [drained, Edr]. Potentially by a factor of 2 or 3.
That variable is stronger than the plan shape of the foundation for resulting differences in shear and moments.
Your only hope of reasonable foundation analysis results is a bounded solution on k. Your users probably don't want to hear this, but for foundation design, you need at least 4 sets of results. Static, low k. Static, high k. Short term, low k. Short term, high k. Now if you can automate that, that would be just one more reason to join your club.

### RE: Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

(OP)
Perhaps I need to bound the problem a bit more....
• Statics vs dynamics is really a different discussion. So, I'd prefer to just stick to the static solution for now....
• Also, I appreciate the fact that soils are certainly not linear and that soil properties have a lot of variation in them. Hence the wisdom of running your analysis with a range of values. I completely agree. Especially when the time dependent concepts that ATSE mentioned are thrown in as well.
• My belief is that the equations I'm starting with (and looking for) are based mostly on foundations on sandy soils.
• I have to admit ignorance as I don't know whether the equations are derived for initial stiffness or long term stiffness.
• Please forgive me if my terminology is not quite right. I use the term subgrade modulus somewhat interchangeably with soil modulus. I believe this is what Fatdad was pointing out.
What I'm really interested in is this general equation from Braja Das's Principles of Foundation Engineering for adjusting the spring stiffness used in an analysis based on the size of the foundation:

Right now, I'm really most interested in how to convert this equation into something that works for a circular or octagonal foundation.

This may develop into a bit more of a research project down the line... to better understand the basis and limitations of this equation and such. I know (mostly through 2nd hand stories) that it is sometimes difficult to get geotechnical engineers to give us structural types a modulus of subgrade reaction in the soils reports. Perhaps this discussion (and my further research) will help me to understand why.

### RE: Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

The structural design of a slab requires a value for the modulus of subgrade reaction. For the case of a point load appearing at some random location on the slab, the subgrade modulus and the point load work in conjunction to inform the design of slab reinforcement. Such point loads spread through the thickness of the slab, react against the soil and are quickly attenuated with depth.

Overall, the slab on grade may be designed for a uniform load of 1,000 psf, even though a random point load (think rack leg) applies 15 kips.

If we take a circular area of 6,000 sf (diameter of 155 ft) that's supporting an overall load of 1,000 psf, we can also look at how the 6M pounds reacts against the soil. That has nothing to do with some spring constant analogy. That has to do with elastic theory (the attenuation of load with depth), the soil's response to changing loads and the original soil pressure. If the soils are fine grained, we use consolidation theory to calculate such settlement. If the soils are coarse grained, we typically use elastic theory. That requires a soil modulus value, which is not the same as the modulus of subgrade reaction!

Structural engineers have tried to imagine some short term and long term subgrade modulus to help with this, but it's not consistent with geotechnical engineering practice.

Just use elastic theory to get the attenuation of loads and then determine how these load will lead to settlement. If it's coarse grained, I'd use Hook's law. If it's fine grained (and saturated), I'd use consolidation theory. I would never use subgrade modulus to guess performance when the loads are changing to depths of dozens of feet!

f-d

ípapß gordo ainÆt no madre flaca!

### RE: Adjustment for Modulus of Subgrade Reaction for Circular or Octagonal Foundations

You will find some equivalency between square and round footings in various texts - I've seen in Richard Hall and Woods. The following is from a Ph.D. thesis:

The circular footing is changed to an equivalent square footing with the size of B = R * π^0.5 .

http://www.eng.ox.ac.uk/civil/publications/theses/...

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