SlipSurface IQ Correlation — theory

What SlipSurface IQ Correlation computes, and where the methods stop being valid. Depth z is measured from the ground surface; stresses are in kPa, pa = 101.325 kPa.

Every correlation below is empirical. Each was fitted to a particular set of soils, tests and equipment, and carries a scatter that is often a factor of two. The program's values are a first estimate, to be checked against the site's own laboratory and in-situ tests and against judgement — never a substitute for them.

1. In-situ stresses

σv0(z) = Σ γ·Δz above the water table and Σ γsat·Δz below it; u0 = γw·(z − zw) below the water table; σ'v0 = σv0 − u0. The last layer is taken as continuing below the profile for the stresses only. Each layer's representative stresses are those at its mid-depth.

A unit weight that is entered is used as it is. Otherwise it is taken from the phase relations and from the CPT (Robertson & Cabal 2010), combined as any other parameter; failing both, a typical value (18 / 20 kN/m³ granular, 18 / 18.5 kN/m³ cohesive) is used with a warning. The CPT's unit weight needs qt and Rf only, so it is found before the stresses.

2. Laboratory results

PI = LL − PL          LI = (w − PL)/PI          IC = (LL − w)/PI
e0 measured, or e0 = w·Gs for a saturated soil (a clay, or a layer below the water table)
γd = Gs·γw/(1 + e0)   γsat = (Gs + e0)·γw/(1 + e0)   γ = γd·(1 + w)   (S < 1)

The USCS group (ASTM D2487) follows from the gravel and fines contents and the plasticity of the fines on Casagrande's chart (A-line PI = 0.73·(LL − 20), CL-ML for 4 ≤ PI ≤ 7 above it). Without a grading curve Cu and Cc are unknown, so a clean coarse soil is reported as SW/SP (or GW/GP), and fines without Atterberg limits as M/C.

3. SPT

N60 = N·CE·CB·CS·CR                                 (Skempton 1986; Youd et al. 2001)
    CE = ER/60
    CB = 1.00 (65–115 mm), 1.05 (150 mm), 1.15 (200 mm)
    CS = 1.0 standard sampler, 1.2 without liners
    CR = 0.75 (< 4 m), 0.85 (4–6 m), 0.95 (6–10 m), 1.00 (≥ 10 m) of rod length
         (rod length = depth + rod above the ground)
(N1)60 = CN·N60,  CN = (pa/σ'v0)^0.5  (Liao & Whitman 1986)
                  CN = 2/(1 + σ'v0/pa) (Skempton 1986, normally consolidated fine sand)
                  CN ≤ 1.7

The correlations that were written for the field blow count of a 60 % hammer take N60. N ≥ 50 is flagged as refusal.

4. CPT(u)

Robertson (2009):

qt  = qc + u2·(1 − a)
Rf  = fs/qt,  Fr = fs/(qt − σv0),  Bq = (u2 − u0)/(qt − σv0),  Qt = (qt − σv0)/σ'v0
Qtn = [(qt − σv0)/pa]·(pa/σ'v0)^n
Ic  = √[(3.47 − log Qtn)² + (log Fr + 1.22)²]
n   = 0.381·Ic + 0.05·σ'v0/pa − 0.15 ≤ 1          (iterated to convergence)

Soil behaviour type zones from Ic: 7 (< 1.31), 6 (1.31–2.05), 5 (2.05–2.60), 4 (2.60–2.95), 3 (2.95–3.60), 2 (> 3.60). Zones 1, 8 and 9 need the chart position, not Ic, and are not assigned. A reading with qt ≤ σv0 or fs ≤ 0 cannot be normalised and is left out.

5. The correlations

parameter	correlation	formula
γ	Robertson & Cabal (2010)	γ/γw = 0.27·log Rf + 0.36·log(qt/pa) + 1.236
Dr	Skempton (1986)	Dr = √[(N1)60/60]
	Idriss & Boulanger (2008)	Dr = √[(N1)60/46]
	Meyerhof (1957)	Dr = 21·√[N60/(σ'v0/98.1 + 0.7)]
	Jamiolkowski et al. (1985)	Dr = −98 + 66·log[qc/√σ'v0] (t/m²)
	Baldi et al. (1986)	Dr = ln[qc/(157·σ'v0^0.55)]/2.41 (kPa)
	Kulhawy & Mayne (1990)	Dr = √(qt1/305), qt1 = (qt/pa)/√(σ'v0/pa)
φ' (sand)	Wolff (1989)	φ' = 27.1 + 0.3·N60 − 0.00054·N60²
	Hatanaka & Uchida (1996)	φ' = √[20·(N1)60] + 20
	Schmertmann (1975); Kulhawy & Mayne (1990)	φ' = arctan[N60/(12.2 + 20.3·σ'v0/pa)]^0.34
	Robertson & Campanella (1983)	φ' = arctan[0.1 + 0.38·log(qt/σ'v0)]
	Kulhawy & Mayne (1990)	φ' = 17.6 + 11·log qt1
φ' (clay)	Kenney (1959); Terzaghi, Peck & Mesri (1996)	sin φ' = 0.8 − 0.094·ln PI
	Sorensen & Okkels (2013)	φ' = 43 − 10·log PI
cu	Stroud (1974)	cu = f1·N60, f1 from 6.5 kPa (PI 15) to 4.2 kPa (PI ≥ 60)
	Hara et al. (1974)	cu = 29·N60^0.72
	Terzaghi & Peck (1967)	cu = qu/2 = 6.25·N60
	Sivrikaya & Toğrol (2006)	cu = 4.32·N60
	Lunne et al. (1997); Robertson (2009)	cu = (qt − σv0)/Nkt
	Lunne et al. (1997)	cu = (u2 − u0)/NΔu
	unconfined compression	cu = qu/2
	Mesri (1975)	cu = 0.22·σ'p
σ'p	Mayne et al. (2002)	σ'p = 0.47·N60^m·pa, m = 1 clays, 0.9 silts
	Kulhawy & Mayne (1990)	σ'p = 0.33·(qt − σv0)
	Stas & Kulhawy (1984)	σ'p = pa·10^(1.11 − 1.62·LI)
K0	Jaky (1944); Mayne & Kulhawy (1982)	K0 = (1 − sin φ')·OCR^sin φ'
E' (sand)	Kulhawy & Mayne (1990)	E = α·pa·N60, α = 10 (FC ≤ 12 %), 5 (FC > 12 %)
	Bowles (1996)	E = 500·(N60 + 15) kPa
	Schmertmann et al. (1978)	E = 2.5·qc
	Robertson (2009)	E = 0.015·10^(0.55·Ic + 1.68)·(qt − σv0)
Eu	Duncan & Buchignani (1976)	Eu = (Eu/cu)·cu, the ratio entered (default 300)
M	Robertson (2009)	M = αM·(qt − σv0); αM = Qt ≤ 14 for Ic > 2.2, else 0.0188·10^(0.55·Ic + 1.68)
Cc	Skempton (1944)	Cc = 0.009·(LL − 10)
	Azzouz et al. (1976)	Cc = 0.40·(e0 − 0.25)
	Nishida (1956)	Cc = 1.15·(e0 − 0.35)
	Koppula (1981)	Cc = 0.01·w
	Wroth & Wood (1978)	Cc = 0.5·Gs·PI/100
Cr	Nagaraj & Murty (1985)	Cr = 0.0463·(LL/100)·Gs
	Kulhawy & Mayne (1990)	Cr = 0.15·Cc (Cr/Cc ≈ 0.1–0.2)
Cα	Mesri & Godlewski (1977)	Cα = 0.04·Cc
	Mesri (1973)	Cα = 0.0001·w
cv	Terzaghi	cv = k·M/γw, k and M from the same CPT reading
Vs	Imai & Tonouchi (1982)	Vs = 97·N60^0.314
	Ohta & Goto (1978)	Vs = 85.35·N60^0.348
	İyisan (1996)	Vs = 51.5·N60^0.516
	Robertson (2009)	Vs = [αvs·(qt − σv0)/pa]^0.5, αvs = 10^(0.55·Ic + 1.68)
	Mayne (2006)	Vs = 118.8·log fs + 18.5
G0	elasticity	G0 = (γ/g)·Vs², γsat below the water table
k	Hazen (1911)	k = 0.01·D10² m/s, D10 in mm
	Robertson (2010)	k = 10^(0.952 − 3.04·Ic) (Ic ≤ 3.27), 10^(−4.52 − 1.37·Ic) (3.27 < Ic < 4)

Each correlation is applied only to the soil it was written for: the relative density, the friction angle from the tests and the drained modulus to granular layers; the undrained strength, σ'p, the compression indices and cv to cohesive layers; γ, M, Vs, G0 and k to both. In a granular layer OCR = 1 is assumed.

6. From the tests to a layer

Every correlation is evaluated at each SPT and CPT depth and the values are averaged over the tests in the layer — arithmetically, except for k and cv, which span orders of magnitude and are averaged geometrically. Laboratory correlations give one value per layer. The derived parameters follow in the order they need each other:

1. OCR = σ'p/σ'v0 at mid-layer, from the selected σ'p;
2. Mesri's cu = 0.22·σ'p joins the other undrained strengths;
3. K0 from the selected φ' and OCR;
4. Eu = (Eu/cu)·cu from the selected cu;
5. Cr = 0.15·Cc and Cα = 0.04·Cc join their correlations, from the selected Cc;
6. G0 from the selected Vs and the unit weight.

7. Selection

For each parameter, the value is the correlation the user chose for it — or, when that one has no data in the layer, all of them combined, with a warning. They are combined by

• the median (the default: one correlation far from the others moves it least),
• the mean, or
• a cautious estimate of the mean (Schneider 1997): Xm − 0.5·s for a strength or a stiffness (Dr, φ', cu, σ'p, OCR, E', Eu, M, Vs, G0, cv) and Xm + 0.5·s for a compressibility (Cc, Cr, Cα), kept within the range of the correlations. Other parameters take the median.

k and cv are always combined in logarithms. The report and the interface show every correlation's value beside the selection, with its source, the number of tests behind it and the range across the correlations.

8. Sending the parameters on

SlipSurface IQ Settle receives γ, γsat, E, ν, Cc, Cr, e0, OCR, cv and Cα of every layer and the water table. Settle's modulus of a cohesive layer is the undrained one, so Eu is sent with ν = 0.5. SlipSurface IQ Bearing receives γ, γsat, c' = 0, φ', cu, E and ν. Only the soil goes across; the foundation and the options keep the receiving program's defaults. A parameter that could not be derived is written as zero, and the receiving program points it out when it needs it.

References

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