Pile Group - Pile Spring Stiffness 4

[UP830]

Hello,

I am doing some independent learning of pile groups. Finding good resources for pile groups is hard in my opinion. There only ever seems to be a page or two on pile groups in most text books. If anyone has some good references for pile groups could they please comment.

Also, could someone tell me where the below formulae for pile spring stiffness comes from? I have searched and searched and searched and have not found it. Or, if you have another pile spring stiffness formulae, please let me know its reference.

Thank you.

 

[Guest262653]

Here are a few references:
- Single Piles and Pile Groups under Lateral Loading, Reese
- Pile Design and Construction Practice, Tomlinson
- Pile Foundation Analysis and Design, Poulos & Davis
- Piling Engineering, Fleming
- Recommendations on Piling (EA-Pfähle), German Geotechnical Society

 

[JAE]

A small section from the Foundation Handbook by Fang. Not sure if this will help or not:


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[dik]

OP:

Can you tell me what the variables are:

Ep, Eb, Er, As, B, and Ir? I assume E is Young's Mod for soil, and L is the pile length. Do you have a source for the formula?

Missing a value for Ab in the formula.

Dik

 

[EireChch]

DIK/OP,

The formulae is from pg 136 of Tomlinson - Pile Design and Construction 4th Edition.
Ab and As are area of base and shaft, respectively.

I have plugged the formulae into a calculator but getting a different result? I think the Author has used 80kN as Wb. The second part of the equations is correct which equals 1.86E-3, but i am not getting the same answer for the first part which should equal 1.38E-4?

I am not sure if there is a specific formulae for pile spring stiffness. Isnt a spring value just the displacement divided by the load. In your case the displacement is 2mm and the total applied load is 131kN. Hence spring stiffness is 131kN/2mm = 66kN/mm.

For every 66kN applied you will get 1mm of settlement.

Hope this helps.

 

[dik]

Thanks very much.

Dik

 

[SlideRuleEra]

Some of the confusion may be because the OP's example has either the pile's base is larger than the shaft (maybe a bell caisson is intended) or the math is wrong.
If the shaft and the base have the same diameter, B = 0.3 m, than A_s = 0.071 m², not 0.09 m².

Eriechch - If I use A_s = 0.09 m² in the equation's first term, get 1.38 E-4.

1) Finding good resources for pile groups is hard...
2) ...have another pile spring stiffness formulae, please let me know its reference.

1) Pile groups are complex and poorly understood, so it is really no consensus on how to handle pile group design. In the broadest terms, pile group efficiency applies to friction piling not point-bearing piling.

There are three common equations to calculate the "efficiency" of piling in a group. None of them are particularly accurate nor do they apply to all soils or pile group layouts. Efficiency compares the load that each pile in a group can carry, compared to how much load a single isolated pile can carry (in the same soil).

n₁ = number of piles in a row (plan view of pile layout)
n₂ = number of piles in a column (plan view of pile layout)
D = pile diameter (approximate diameter for "square" piling, like steel HP or concrete)
d = pile spacing (center to center)

Note that all three formulas are based ONLY on pile size and layout geometry, not soil properties. Also, the formulas apply ONLY to rectangular or square layouts.

There are other ways to calculate pile efficiency, I like "Feld's Rule", it is very simple:
Look at each pile in a group (any layout), one pile at a time, and reduce pile efficiency by 1/16 for each adjacent pile (in any direction, including diagonal). Average results for all piles to get efficiency of the group. Feld's rule does NOT consider pile spacing - I just keep spacing in the commonly accepted 3 x pile diameter range.

Once you have a handle on pile efficiency, have to look at "pile group block failure"... but not right now.

2) At the most fundamental level, Hooke's Law: k = f x e.
Consider a simplified point-bearing pile (no skin friction)

The "pile spring constant" (N/mm) can be calculated from the pile material's Modulus of Elasticity. Pile elastic shortening under load can be calculated.

The soil-pile spring constant can be determined by a pile load test:

In my sketch the red line (approximately) parallels the pile settlement graph. The slope of the red line is the soil-pile spring constant. Note that a typical pile load test is carried out to twice the pile design load... to make sure there is enough data to get line slope up to, and just beyond design load. Soil-pile settlement under load can now be calculated.

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[UP830]

SLIDERULEERA - thank you for the information. It has been very helpful.

I am looking at a pile group that comprises 2no. rows of 13 piles. Pile spacing is 1.75m. The piles are to be founded at 4.75m into dense sand and gravel. From my research I have decided that I do not need to consider pile group efficiency. I am conservatively taking the sum of the individual pile capacity as my total capacity. Even with ignoring efficiency, the capacity is still enough.

I am now looking pile group settlement. From my research, settlement of pile groups are larger than the settlement of a single pile due to the load shed from adjacent piles. I am using Piling Engineering by Flemming et al. as my reference. I have looked at two methods for assessing pile group settlement.

1 - Randolph and Wroth (1979) proposed the method using the α interaction factor, see page 189 and 190 of Piling Engineering. There is a αs factor which is the interaction factor for shaft and a αb factor which is the interaction factor for base.

The sum of αs and αb is then multiplied by the settlement of a single pile. Based on a pile length of 4m, 0.3m dia, and spacing of 1.75, I get a total α of 1.3. Therefore if my settlment for a single pile is 20mm then pile group settlement is 20x1.3 = 26mm. This method takes no consideration of the amount of piles in the group.
.
2 - Butterfield and Douglas (1981) , i think, proposed the decreasing stiffness method (see page 191) of each pile due to interaction effects is by means of an efficiency, ηw. You calculate ηw then multiply it by the calculated single pile settlement. ηw is essentially the number of piles to the power of 0.5. i.e. If you have 6 piles, ηw = 6^0.5 = 2.45. If you single pile settlement was 20mm your pile group settlement is 20*2.45 = 49mm.

As can be seen, there is some variation in the results. I would welcome anyone recommendations, thoughts. Maybe I am miss interpreting the methods. I can upload calculations if needed. I was also thinking of taking the average of the two methods?

 

[SlideRuleEra]

UP830 - Here are my calcs for the assumed 26 friction pile group you described. I get 28mm predicted settlement for the group, compared to 20mm for a single pile.

Have included a note on the calcs; would want to talk to a geotech about the soils since pile spacing d/D = 5.8 is pretty far apart for significant group action. Also, since the pile group is long and "thin" (13 piles long x 2 piles wide), would want an opinion on how that could affect group action.

IMHO, these two factors may?? make group predicted settlement a little less than 28mm.

Note: A readable .pdf of the calcs is attached.

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[LRJ]

UP830: to extend the interaction factor (α) approach to more than two piles, you need to do these calculations for each pile affecting every other pile (i.e. generate a matrix of α values). You might get a different (softened) response for each pile. If you have a pile cap or other pile head fixity, it is normal to take the average response of all piles as the pile group response.

SlideRuleEra (and UP830): I'm not familiar with the method(s) you are using, but it seems of that the efficiency factor (η) is being conflated with a softening factor. The efficiency factor affects the capacity of the pile group whereas the softening factor increases the settlement for the same applied load. A reduction in capacity is not necessarily equal to the reduction in stiffness.

 

[SlideRuleEra]

I'm not familiar with the method(s) you are using...

"Feld's Rule" is an acknowledged method of calculating approximate friction pile group efficiency. It was promoted by Poulos and Davis in 1980, but has flaws, as do all other pile group efficiency formulas that I know of. (I discussed this in my 13 June 2018 post in this thread).

A few references are:

"Tall Building Foundation Design", Poulos

"Geotechnical and Geoenvironmental Engineering Handbook", Rowe

"Geotechnical Engineering", Venkatramaiah

"Piles and Pile Foundations", Viggiani, Mandolini, Russo

I prefer Feld's Rule since it is both simple and works for any pile plan in addition to those that are square or rectangular.
Also, on anything other than an in-house project on an established site (electric generating station), we bring in a Geotech.

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[LRJ]

Okay, that confirms it: you are applying an efficiency factor (which should be used to calculate the reduction of single pile capacity in a pile group) to the load-displacement (stiffness) response of a single pile in a pile group. This is incorrect: you cannot use these efficiency factors to predict pile group settlement. Capacity and stiffness are different and pile group effects affect them differently.

As an aside, but related to the above, if you have t-z or p-y springs in a structural model and you want to apply pile group effects, you need to be careful not to compound the spring softening. Moreover, sometimes if a t- or p-modifier are applied to a spring, that will also soften it; this should be avoided, with the pile group capacity reduction essentially 'cutting' the spring at the calculated value, and the spring softening handled separately using z- or y-modifiers.

 

[SlideRuleEra]

LRJ - Thanks, in the your first paragraph I see your point.
How large an error will combining combining stiffness and capacity introduce?
Using my understanding of the OP's example, I got 28 mm of settlement for the group compared to 20 mm for a single pile.
Would a better answer be, maybe... 24 mm (half my estimated differential settlement)?

Edit:

...the efficiency factor (η) is being conflated with a softening factor. The efficiency factor affects the capacity of the pile group whereas the softening factor increases the settlement for the same applied load.

Just realized my calcs are for friction piling (and are clearly labeled as such). There is no point bearing. How does a softening factor affect pile skin friction? For driven displacement piling in a group, soils in the vicinity are densified.

For cast-in-place piling (which may not be practical in soil where friction piling are needed) how does softening even apply?

Note: Not arguing... trying to learn something.

End of Edit.

I usually try to address an OP's question. In his latest post, the OP calculated settlement of 26 mm and 49 mm using two different methods. He has no idea if either one of the answers is reasonable. I presented my (flawed) calcs to give him a clue that 26 mm is more likely "better" than 49 mm.

I'm afraid your second paragraph goes beyond my limited knowledge of modern methods.

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[LRJ]

The error of combining stiffness and capacity reductions will depend on the magnitude of reductions as well as the formulation of your load-displacement response (e.g. type of t-z/p-y curves, etc.).

The 28 mm you got was from using an efficiency factor. To reiterate: you cannot use this approach to predict pile group displacement; it is a pile group capacity reduction method. Also, I have not said that the 28 mm is incorrect due to doubling up on the efficiency/softening factors: you've only used the efficiency factor and that is not a valid approach for predicting settlement.

Pile group effects are still applicable in the same way to driven piles. The precise formulation may differ somewhat to account for interface effects caused by the installation technique (this is perhaps more relevant for axial rather than lateral pile group effects), but the same underlying theory will still be applicable. You still shouldn't use an efficiency factor to predict the additional settlement of the pile group.

On densification: this is an argument for not reducing the capacity of pile groups in cohesionless soils. However, the softening effects are still possible. Whether or not pile capacity will be increased or decreased also depends on whether the cohesionless soil is likely to contract or dilate upon shearing.

 

[SlideRuleEra]

LRJ - Thank you.

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[UP830]

Apologies, I have been away. LRJ and SRE, thank you for your inputs and discsussion. They have been very helpful.

LRJ - please excuse me, but I am struggling to get my head around the matrix of α. I understand the idea of it, but calculating a value based on a number of adjacent piles is confusing me.

See below, I have calculated the α based on Pile 2 being affected by Pile 1. Therefore the α factor at Pile 2 is 1.41.

Pile 2 then affects Pile 3 so α at Pile 3 will be 1.41 + 0.41 = 1.82.

So on and so on, is this correct?

 

[LRJ]

For a start, it's unclear how you have calculated an α of 1.41. There appears to be a missing term based on the equations you provided previously.

When calculating the influence on other piles, it's not just a case of adding the same amount over and over. The influence of pile 1 on pile 3 should be calculated using the correct spacing between those piles. Pile 3 would also be affected by pile 2 (and possibly piles in the row above) based on the loading direction depicted in your drawing.

Also, for the depicted lateral loading, it is typical to specify a spacing beyond which there are no interaction effects (this is typically 8 diameters, centre-to-centre spacing).

The matrix of α values should be 10x10 in your case (because you have 10 piles).