Vertical and Lateral Soil Springs 2

[James B]

Hi All,

I have recently come across a new request in my practice. A structural engineer has requested a lateral soil spring for a shallow (both concrete pad [not mat or raft] and steel grillage) foundation. I have provided vertical soil springs for raft/mat foundations and lateral soil springs for deep foundations many times but have never been asked for either for a small spread footing. When I query the structural on why this is required he needs it as a "program input" - which is my first red flag - he does not understand what the parameters are or how they are used.

Regardless, I am confident in providing a vertical subgrade modulus using Winkler and Terzhagi methods for a spread footing by the equation of Kz = Qa/S. In my case we have about firm soils (silty clay) with an undrained shear strength of 50 kPa, an allowable bearing capacity of 50 kPa and the structure's allowable settlement is 25 mm = a very simple result of Kz = 50 / 0.025 = 2 MN/m3.

For the lateral modulus I am at a loss of how to interpret this by hand or other methods. I have tried finding papers and only come across an ASCE paper that indicates the lateral modulus may be obtained by Kh = 2Es/b. In my case I have assumed an Es = 10 MPa (firm silty clay) and b = 1.5 m (foundation dimension) which results in Kh = 13.3 MN/m3. This is 6x the vertical modulus and clearly something is missing in this equation, unless I have severely overestimated the Es which should be 3 MPa and this seems unreasonably low for a material with an Su = 50 kPa. Is there any guidance on lateral modulus for shallow foundations?

I also understand that typically lateral modulus is 1.5 - 2x higher than vertical, I just can't seem to find a paper that can justify this to provide to the structural engineer as backup documentation.

 

[Settingsun]

50 kPa allowable for Cu = 50 kPa seems low. Is this local knowledge that the ground is particularly subject to settlement? If so, typical equations for lateral stiffness won't be applicable for the same reason they wouldn't be for vertical stiffness.

I think part of the difference is that you're comparing a lower bound vertical stiffness to a best-estimate lateral stiffness. The allowable bearing capacity is presumably the safe value not to exceed 25mm so you're effectively putting a FOS on the stiffness as well when you use allowable bearing in the stiffness calculation.

Another part of the difference would be using a completely different method for the lateral stiffness. To be consistent, wouldn't you divide a lateral allowable bearing pressure by 25mm also?

I've never seen lateral stiffness quoted as being higher than vertical, so would be interested if you find the reference for it.

 

[James B]

The 50 kPa is a factored (95 kPa unfactored) allowable for 25 mm settlement with the local understanding that the foundation is sitting on an overconsolidated crust with soft clay below (likely within 2-3 m). I provide an ultimate factored of 150 kPa for this case. It may be on the lower side but in this case it is highly preliminary with no investigation data at the site (to come and update parameters accordingly). As context, these parameters are used as a conservative desktop feasibility assessment and are to be refined later during future design phases if the green light is given.

From my understanding the horizontal modulus ends up being higher because it is not subjected to plastic settlement. It is truly an elastic stiffness only and does not need to incorporate plastic stiffness; whereas vertical does (or should). In this sense - I do not and have never provided a 'lateral bearing pressure' so could not adequately assess a Qa/S for lateral.

I was incorrect in saying ASCE paper - it is actually ASABE Paper 152190408. From this paper what I find interesting is that the Kh and Kv are different. Both are based on elastic theory from Pyke and Beikae (1984). Kh neglects v as the factor of 2 is a generic incorporation of v (would vary 1.8 to 2.3 based on the true v).

Kh = 2Es/B
Kv = q/S = Es / [CsB(1-v2)]

Since this is a rectangular footing of L/W = 3.5 I would select a Cs = 1.5 (NFEC 1986) and using my inputs for the three possible equations. Lateral ends up way higher than even the unfactored SLS Kv.

Kh = 2(10)/0.45 = 44.4 MN/m3
Kv = 95/0.025 = 3.8 MN/m3
Kv = (10) / [1.5*0.45*(1-0.4^2)] = 17.6 MN/m3

 

[geotechguy1]

You might find something useful in Chapter 6 of the below linked FEMA manual:

https://www.fema.gov/sites/default/files/documents/fema_soil-structure-interaction.pdf

 

[Settingsun]

A load applied to an elastic half-space would have a certain stiffness, call it X.

A load applied at a small distance into the half-space, acting away from the boundary, would have stiffness >X. This corresponds to the vertical load case.

A load applied within a whole-space has stiffness 2X.

A load applied within a half-space, parallel to the boundary, has stiffness <2X. This corresponds to the horizontal load case.

How does the difference get to a factor of 2.5? What have I missed?

 

[James B]

Steve, I don't think I'm following you there. But after reviewing the FEMA document also I have determined that the lateral modulus would be in the order of 12 MN/m3 and vertical in the order of 2 MN/m3. This "2x" approximation of lateral to vertical is also indicated in Bowles foundation manual CH.16 (relating to piles, but I am extending it to shallow foundations here). I find that things like these springs are endless rabbit holes that are best left to FEM analyses but in this case I have no option of having an experienced peer conduct such an analysis.

I have a Qu = 150 kPa (factored) and Qa = 50 kPa (factored). Note that my code mandates a 0.5 factor be applied to both (thus equating to an SF = 2). Still using my E = 10 MPa and v = 0.35 with B = 1.46 m. I will also add that the Qa is calculated following CFEM methods Qa = S*E / B*Is; where Is = 1.65 in my case and S is settlement.

Vertical:
Stress-strain theory (Vesic) - E/B*(1-v^2) = 7.8 MN/m3; Bowles 9-6a
Stress-strain theory approximation (Vesic & Bowles) - 40*SF*Qa = 4 MN/m3; Bowles 9-9
Elastic theory (Bowles) - Qa/S = 2 MN/m3 (Pg.501)

Now each theory has its problems and practices so I would stick to Bowles elastic theory formula (based on Winklers) as the most accepted practice Qa/S. This is my vertical subgrade modulus of 2 MN/m3 (very low indeed!).

Horizontal:
Bowles approximation - 40*Cm*SF*Qu = 40*1*2*150 = 12 MN/m3; Cm=1 for a square to limited rectangular footing
ASABE calculation - 2E/B = 13.7 MN/m3

From this I would gather that the horizontal should be in the order of 12-13 MN/m3 (6x that of vertical!).

To wrap my head around this I have to really understand the concept that vertical modulus is governed by the shearing of the soil below whereas the lateral modulus neglects shear in favor of the weight of soil above and confinement pressure. I follow it conceptually but in practice it just doesn't seem to make sense that I get a value 6x.

 

[geotechguy1]

Would be helpful to know what the loading circumstances / durations / load amounts are for the two different scenarios (horizontal and vertical), as that could make a difference.

For example, if the 'vertical modulus' is ' a conservative modulus relating long term settlement of the structure under the design loads applied for indefinite duration ' and the 'horizontal modulus' is a modulus to be used in calculations of transient loads applied for short durations with small strains, then maybe it isn't so unrealistic. The modulus depends on the stress, strain, duration of the loading. Probably even temperature if you want to get fiddly about it. You'd see if you try and do an advanced model of a pavement structure that modulus values used to assess what happens to a road pavement under small, short transient loads from traffic are significantly higher than those used to assess settlement of a building or a large pile of soil. Also, consider that soil is heterogeneous and there's a good chance that shallow soil in the vicinity of a footing is going to be overconsolidated and this the in-situ horizontal stress is actually much higher than the in-situ vertical stress.

Regarding steve's post, it's important to remember the difference between 'how we model soil' versus 'how soil actually behaves'. We have to know in what situations we have to step back from 'what we modelled the soil as' to 'how soil actually behaves' - differences in behaviour in the vertical direction under a static load vs horizontal direction under a transient load.

 

[geotechguy1]

Also, just adding on that usually what I've seen is giving a structural engineer a range of spring values and telling them to do a sensitivity analysis. Eg. you may give them 5 - 50 or something like that.

It's really an iterative problem because you need to know the load and deflection to know if the modulus is reasonable but the load and deflection probably depend on the modulus

 

[Settingsun]

Steve, I don't think I'm following you there.

I knew it needed sketches when I was typing it, but was typing on a phone. Hopefully the images below clarify why I expect a maximum of two as the ratio between vertical & horizontal stiffness in an elastic analysis. But researchers seem to go beyond that, so I'm wondering what the trick is.

For your vertical stiffness, it would help if you gave values for both sustained and transient loads. Peak live load doesn't usually occur for long enough for full consolidation to take place. This may make it seem more reasonable compared to the lateral stiffness (aside from being useful to the structural engineer).

Vertical: Stress-strain theory approximation (Vesic & Bowles) - 40*SF*Qa = 4 MN/m3; Bowles 9-9

Horizontal: Bowles approximation - 40*Cm*SF*Qu = 40*1*2*150 = 12 MN/m3; Cm=1 for a square to limited rectangular footing

These are fundamentally the same except that horizontal deflection is expected to be 25mm at the SF*Qu pressure, whilst vertical is at Qa pressure. Is that because the consolidation is built in, ie implicit assumption that the entire vertical load causes consolidation and none of the horizontal load does? For soft clays, consolidation can be 90% of the settlement (IIRC), so 10:1 ratio is quite believable.


(I'll trust that your terminology is common in your area. I wouldn't know what "Qa (factored)" means so would give you a call for clarification.)

 

[LRJ]

If this is a shallow foundation with vertical and lateral loads being applied simultaneously, then don't forget that the response will be 'coupled', i.e. you can't separate the vertical and lateral responses. If this is the case, then a stiffness matrix approach is likely to be more appropriate.