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(OP)
First: Yes, I searched and found 14 threads on the subject.  No, they did not answer the question.

We have been conversing at the office about laterally loaded piles in sand.  We normally provide a couple methods for the structural to use to calculate capacity (Lpile criteria and the one I am going to discuss).

From Poulos and Davis, they indicate for softer cohesive soils that K(sub h)=n(sub h)*z/d.  z=depth   d=pile diameter

They go on to say that “For piles in sand, assuming that the modulus of elasticity depends only on the overburden pressure and the density of the sand,  Terzaghi showed that n(sub h)=A(gamma)/1.35 tcf”  The then go on to provide typical values of N(sub h), such as 7 tcf for dry loose sand (after Terzaghi, 1955). (The way the book is laid out, it implies the use of the same equation as above.)

I have a Das book that provides the equation for sand as K(sub h)=n(sub h)*z    z=depth   Das provides values of 280-350 pcf for N(sub h), for dry loose sand.

As you can see the Poulos and Davis equation includes the diameter of the pile, while Das does not.  Also, the values for N(sub h) are quite different.

So, my question is, which appears to be more correct in your experiences with piles in sand?  Should the diameter be a factor in sand?

(OP)
Also noticed that the units would not be consistent between the two.

Das is presenting the same equations as Poulos and Davis, but the units for Das' K(sub h) would be tsf, not tcf. Another way to say it is that
Das K(sub h) = Poulos & Davis K(sub h)*d
Both sources give the same equation for T
T=fifth root of (EI/N(sub h))
which is the equation you use in the graphs.

Regarding the values for N(sub h), which version of Das do you have?  I have the 2nd edition, 1990.  It gives the values of N(sub h) in kN/m^3.  When I convert the values for loose dry sand, they are 6 to 7 tcf which agrees quite well with Terzaghi's values.

(OP)
I have the 3rd edition.  The value is offered in pcf and in Kn/m^3.  I converted the metric to English, and that seems to correlate, as you have said.  This must be a misprint on the English units.

The question still remains should the value be divided by the diameter of the pile?  Consistent with my OP, both sources also reference Vesic (1961) for cohesive soils.  In the initial portion of the equation, the “0.65” term is divided by the diameter in the Poulos and Davis version and not in the Das.

“Another way to say it is that Das K(sub h) = Poulos & Davis K(sub h)*d”  OK, but based on my last paragraph, Vesic “does not =” Vesic in the two books.

So, after looking in another reference, the subgrade modulus there is reported as psi/in.  I take it that the Das version is expecting that the value will be applied per unit length (divide by the diameter, as it is shown on the Poulos and Davis book).  Seem correct?  Also, do you normally see K(sub h) given in tsf or tcf (or metric equivalent)?

I don't have the Poulos and Davis book, but NAVFAC gives the same equation that you are quoting from Poulos and Davis, so I am basing my comments on that.  I do have another book, Prakash and Sharma (1990) "Pile Foundations in Engineering Practice" which I like because it goes into some detail on the theories.

Prakash and Sharma present the subgrade reaction approach similar to Das.  K is in Force/Length^2 and is equal to p/y where p is the soil reaction per unit pile length and y is the deflection.  Also, K=N(sub h)*depth, so N(sub h) has units of Force/Length^3.  Technically, N(sub H) is the "constant of modulus of horizontal subgrade reaction" or the "coefficient of variation of horizontal subgrade reaction".  K is the modulus of horizontal subgrade reaction.

Okay, you probably knew all that, but I wanted to make sure we're on the same page.  My point about both sources using the same equation for T is that it is irrelevant (at least for soils with increasing modulus with depth) how K is defined since the equation for T relies on N(sub h) which is in units of Force/Length^3.  N(sub h) should not be divided by the diameter.

For soils with modulus constant with depth, you calculate R, which is equal to (EI/K)^(0.25) in Das and Prakash & Sharma. If you check the equation for R in Poulos & Davis, I bet you will find R=(EI/Kd)^(0.25), so the modulus is multiplied by the pile diameter.  Prakash & Sharma report that the Canadian Foundation Engineering Manual approaches subgrade reaction this way, first dividing by the diameter and then multiplying it.

To be thorough, if you expect your clients to use the subgrade reaction approach for laterally loaded piles, you should recommend a value for K or N(sub h), and state how R or T is to be calculated with your recommended value.

I'm simply, at this time, will suggest another source to check out - M.J. Tomlinson's book on Pile Design.  It has an extensive section on subgrade reaction on piles and extensive worked out examples.

Panars, this thread touches the topic I have some confusion with when use the given K value from the geotechnical consultants in designing laterally loaded piles and check against an allowable capacity or displacement (say, 0.25”).

We have the exact words from the geotechnical report: ”For sands, it is assumed that the elastic modulus depends only on the effective overburden pressure and the relative density”. My confusion is

1.Typically, the geotechnical reports give the modulus of subgrade reaction K(sub h) in kcf. And we can further find some words like “the K(sub h) applies to a pile width (diameter) of 1ft only. For other pile diameters a correction must be applied as follows:

k(sub s)=K(sub h)*1ft/B      B=pile diameter.

In calculation of spring constant, K (in K/ft), for a pile diameter B and segment length of L (in the analysis model, we do this from time to time) we have then

K=K(sub s)*B*L = K(sub h)*1ft*L

Apparently this means the spring stiffness is independent of pile diameter, in another word, under a given horizontal load we’ll get the same lateral displacement no matter the pile is 48” diameter or 20” diameter. Am I right in the interpretion?

2. How about pile in clay? How to deal with the soil in between cohesive and cohesionless?

If the statements given above are from the same geotech report, then I think they are being inconsistent.  The statement that "the elastic modulus depends only on the effective overburden pressure and the relatively density" implies that the parameter does not vary with loading nor pile size.  Yet in the next statement that "the K(sub h) applies to a pile width (diameter) of 1 ft only" they are saying that it does vary with pile size.  From what I have learned over the years, the experts (Terzaghi, Davisson, Matlock, and Reese) generally agree that for small strains, the soil stiffness response is independent of pile diameter. The theory is that as the pile diameter increases the zone of influence also increases.  Displacement for a given pressure is directly proportional to the length of the influence zone.  So if the diameter increases by N times, the pressure for a given load decreases by N times, and the length of the influence zone increases by N times.  The N values cancel, so the deflection for a given load per unit length of pile is independent of the diameter (I know this is confusing, but I've tried my best. This was my third attempt at writing it.)

My opinion is that for sands where the horizontal subgrade modulus increases linearly with depth, you need the coefficient of variation of horizontal subgrade reaction in units of Force/Length^3.  When multiplied by the depth, this gives you the horizontal subgrade modulus in units of F/L^2.  Multiply that by the force by unit length of the pile, and you get displacement.

For overconsolidated clays where the horizontal subgrade modulus is relatively constant with depth, you need the horizontal subgrade modulus in units of Force/Length^2.  This is commonly assumed to be 67 times the undrained shear strength.

From your description, I am assuming you are trying to analyze the pile similar to a vertical flexible beam on grade, perhaps using a strucutral finite element program with the soil modeled as springs.

If this is the case, for granular soils, the soil spring stiffness will increase with depth. From your example, it does not appear that you are accounting for this.  If k is the spring constant for the finite element model in units F/L, then k is determined by multiplying the coefficient of variation of horizontal subgrade reaction by the depth of the center of the pile segment and by the length of your pile segment.  The spring stiffness is independent of the pile diameter.  So for this particular pile segment, the displacement for a given force per pile length will be independent of the pile diameter.

HOWEVER, this does not mean that a 1-foot diameter pile and a 3-foot diameter pile will have the same displacement for the same load applied to the top of the pile.  The pile diameter has a large influence on the stiffness of the pile, EI (modulus of elasticity times the moment of inertia).  The moment of inertia is directly proportional to the diameter raised to the fourth power.

You can confirm this by using the subgrade reaction approach (presented in NAVFAC 7.2) or LPILE.  I considered two piles, one 1-foot in diameter and the other 3-foot in diameter.  All the other parameters are the same.  The piles are concrete (assumed to be uncracked for simplification) with E=3600 ksi. The sand has a coefficient of variation of horizontal subgrade reaction equal to 10 tons/ft^3 (11.5 pci), and the piles are 50 feet long.  Both piles are assumed to be free head piles and a 10 kip load is applied at the ground surface.  The 1-foot diameter pile will deflect 0.85 inches and the 3-foot diameter pile will deflect 0.145 inches (theoretically of course).

Panars, You are right, the lateral displacement of a pile depends not only on the the soil spring stiffness, also on the moment of inertia of the pile.

I can understand that the modulus of cohesionless soils increase with depth while in cohesive soils (another word: stiff over-consolidated soils) the value keeps the same (at least considered this way).  Accordingly, the modulus of horizontal subgrade reaction for sand is Ks(in k/ft^3)*depth (in ft), for clay is Ks (k/ft^2). To get the spring stiffness (in k/ft)of a pile segment of length L, K=Ks*L (k/ft^2*ft=k/ft). Naturally, there is no room for pile diameter, even for unit correction.

To some extent, I understood the reasoning of the independence of the pile diameter from the modulus. But it is a bit harder for me when think about this with regard to a concrete slab on elastic subgrade. Similarly, we can use the analogy of a series springs to represent the subgrade. The spring stiffness is the product of the modulus and a small bearing area. Therefore the modulus given should be in k/ft^3, then you have k/ft^3*ft^2=k/ft (no "diameter" here to be regardless). How to compare this to the laterally load pile?

There is an important distinction between the modulus of VERTICAL subgrade reaction for slabs on grade and the modulus of HORIZONTAL subgrade reaction for laterally loaded piles.  While the horizontal modulus is independent of pile diameter, the vertical modulus is DEPENDENT on the size of the slab.  For slabs on grade, the modulus of vertical subgrade reaction needs to be decreased for increasing slab dimensions.

Even though the basic theory is the same, slabs on grade and laterally loaded piles are treated differently, and you have to keep this in mind when looking at the subgrade reaction parameters.  Assuming the modulus of horizontal subgrade reaction is independent of pile diamter seems to work well for pile diameters normally encountered in practice.  For very large pile diameters, the pile no longer acts as flexible but more as a rigid body. A recent trend is to analyze very stiff laterally loaded piles using strain wedge theory, instead of a flexible member supported by elastic springs.

In retrospect, I would expect that the statement in the geotech report about adjusting the subgrade modulus for different pile diameters was the adjustment factor for slabs on grade wrongly being applied to piles.

I learnd "For slabs on grade, the modulus of vertical subgrade reaction needs to be decreased for increasing slab dimensions" in other documents.

Thanks Panars for your valuable input!

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