SlipSurface IQ Liquefaction — theory
What SlipSurface IQ Liquefaction computes, and where the method stops being valid. Depth z is measured from the ground surface; stresses are in kPa and Pa = 101.325 kPa (100 kPa in the TBDY 2018 procedure, as the code writes it).
1. Stresses
σv(z) = Σ γ·Δz above the water table and Σ γsat·Δz below it; u = γw·(z − z_w) below it; σ'v = σv − u. Two water tables are used:
- the one during the earthquake decides which soil is saturated, and gives σv and σ'v in the demand (CSR) and in the overburden factor Kσ;
- the one when the SPT was made gives σ'v in the overburden correction CN, since that is the stress the blow count was measured under.
A test at or above the design water table is dry and is not assessed.
2. SPT corrections
N60 = N · CE · CB · CR · CS
| factor | value |
|---|---|
| CE | ER / 60, ER the energy ratio of the hammer in per cent |
| CB | 1.00 (65 – 115 mm), 1.05 (150 mm), 1.15 (200 mm), interpolated |
| CR | 0.75 (rod < 3 m), 0.80 (3 – 4), 0.85 (4 – 6), 0.95 (6 – 10), 1.00 (> 10); rod length = depth + stick-up |
| CS | 1.0 with liners; 1 + (N1)60/100, between 1.1 and 1.3, without (Seed et al. 1984) |
The overburden correction is CN = (Pa/σ'v)^m ≤ 1.7:
- m = 0.5 (Liao & Whitman 1986) for Youd et al. (2001) and TBDY 2018 (Pa = 100 kPa there); Cetin et al. (2004) cap CN at 1.6;
- m = 0.784 − 0.0768·√(N1)60cs, (N1)60cs ≤ 46, for Idriss & Boulanger (2014). Since (N1)60cs depends on CN, the two are found by fixed-point iteration, which converges in a few steps.
Gravel. In a layer with D50 ≥ 2 mm the blow count is multiplied by 1 − 0.36·log10(D50/2) (Japan Road Association 1996), before the overburden normalisation: 0.64 at D50 = 20 mm. The SPT over-reads in gravel because a particle can block the spoon; the correction is an approximation, and a Vs or a large-penetration test (Becker, DPT) is the better tool. The program warns whenever a gravelly layer is present, corrected or not.
3. Fines content
The fines content FC is the fraction passing the No. 200 sieve (75 µm).
Idriss & Boulanger (2014): (N1)60cs = (N1)60 + Δ(N1)60,
Δ(N1)60 = exp(1.63 + 9.7/(FC + 0.01) − (15.7/(FC + 0.01))²)
which is nil for a clean sand and about 5.6 blows at FC = 35 %.
Youd et al. (2001), TBDY 2018: (N1)60cs = α + β·(N1)60,
| FC | α | β |
|---|---|---|
| ≤ 5 % | 0 | 1 |
| 5 – 35 % | exp(1.76 − 190/FC²) | 0.99 + FC^1.5/1000 |
| ≥ 35 % | 5 | 1.2 |
Cetin et al. (2004) carry FC inside the resistance (§ 5), with FC taken as 0 below 5 % and as 35 % above it.
4. Demand
CSR = 0.65 · (a_max/g) · (σv/σ'v) · rd
a_max is the peak acceleration at the ground surface, entered directly or taken as 0.4·S_DS (TBDY 2018). The stress reduction factor rd depends on the procedure:
Idriss (1999), used by Idriss & Boulanger: ln rd = α(z) + β(z)·M, α = −1.012 − 1.126·sin(z/11.73 + 5.133), β = 0.106 + 0.118·sin(z/11.28 + 5.142), for z ≤ 34 m; rd = 0.12·exp(0.22·M) below. (The fit exceeds 1 very slightly at the surface for large magnitudes; it is used as published.)
Blake's fit to Seed & Idriss, used for Youd et al. (2001), to 23 m: rd = (1 − 0.4113·z^0.5 + 0.04052·z + 0.001753·z^1.5) / (1 − 0.4177·z^0.5 + 0.05729·z − 0.006205·z^1.5 + 0.001210·z²); below 23 m the linear expressions.
Liao & Whitman (1986), linear, used by TBDY 2018: rd = 1 − 0.00765·z (z ≤ 9.15 m), 1.174 − 0.0267·z (≤ 23 m), 0.744 − 0.008·z (≤ 30 m), 0.5 below.
Cetin et al. (2004), from Mw, a_max and the average shear-wave velocity of the top 12 m, Vs,12 — entered, averaged from the layers' Vs (by travel time) where every layer of the top 12 m has one, or else estimated from N60 by Imai & Tonouchi (1982), Vs = 97·N^0.314:
rd = [1 + A/(16.258 + 0.201·e^{0.341(−d + 0.0785·Vs,12 + 7.586)})] / [1 + A/(16.258 + 0.201·e^{0.341(0.0785·Vs,12 + 7.586)})] A = −23.013 − 2.949·a_max + 0.999·Mw + 0.0525·Vs,12 d < 20 m rd = rd(20) − 0.0046·(d − 20) d ≥ 20 m
5. Resistance
The factor of safety is FS = CRR / CSR, with CRR = CRR7.5 · MSF · Kσ for the procedures that write the resistance at the reference conditions (M = 7.5, σ'v = 1 atm).
Idriss & Boulanger (2014)
CRR7.5 = exp(N/14.1 + (N/126)² − (N/23.6)³ + (N/25.4)⁴ − 2.8), N = (N1)60cs
MSF = 1 + (MSFmax − 1)·(8.64·exp(−M/4) − 1.325), MSFmax = 1.09 + (N/31.5)² ≤ 2.2
Kσ = 1 − Cσ·ln(σ'v/Pa) ≤ 1.1, Cσ = 1/(18.9 − 2.55·√N) ≤ 0.3, N ≤ 37
The curve reaches CRR = 2 at (N1)60cs ≈ 37.5; above it the soil is taken as too dense to liquefy.
Youd et al. (2001), NCEER
CRR7.5 = 1/(34 − N) + N/135 + 50/(10·N + 45)² − 1/200 N = (N1)60cs < 30
MSF = 10^2.24 / M^2.56 (Idriss; the lower bound of the NCEER range)
Kσ = (σ'v/Pa)^(f − 1) ≤ 1
with (N1)60cs ≥ 30 non-liquefiable. f is 0.8 at a relative density of 40 % and 0.6 at 80 % (Hynes & Olsen 1999), linearly between, with Dr = √((N1)60/46).
Cetin et al. (2004) — probabilistic; read at the probability of liquefaction PL (15 % is their deterministic recommendation):
CRR = exp[((N1)60·(1 + 0.004·FC) − 29.53·ln Mw − 3.70·ln(σ'v/Pa) + 0.05·FC
+ 16.85 + 2.70·Φ⁻¹(PL)) / 13.32]
The magnitude and the overburden are inside the expression; the MSF and Kσ the program reports for Cetin are the ratios that reproduce them against the M = 7.5, 1 atm value, and (N1)60·(1 + 0.004·FC) + 0.05·FC is reported as its clean-sand equivalent. This is the 2004 model; the 2018 revision of Cetin et al. is not implemented.
TBDY 2018 (§ 16.6) — the NCEER curve and fines correction, CN = (100/σ'v0)^0.5 ≤ 1.7, the linear rd, C_M = 10^2.24/M^2.56 and no Kσ:
τ_eq = 0.65 · σv0 · (a_max/g) · rd, τ_R = CRR7.5 · C_M · σ'v0, FL = τ_R / τ_eq
and FL ≥ 1.1 is required.
Andrus & Stokoe (2000), from the shear-wave velocity, where the layers carry one:
Vs1 = Vs · (Pa/σ'v)^0.25
Vs1* = 215 m/s (FC ≤ 5 %), 215 − 0.5·(FC − 5) (5 < FC < 35 %), 200 m/s (FC ≥ 35 %)
CRR7.5 = 0.022·(Vs1/100)² + 2.8·(1/(Vs1* − Vs1) − 1/Vs1*)
MSF = (M/7.5)^−2.56
with Vs1 ≥ Vs1* non-liquefiable. It is shown in the comparison, as an independent check; the chosen SPT procedure governs.
5a. CPT
qt = qc + (1 − a)·u2
Q = (qt − σv)/Pa · (Pa/σ'v)^n, F = fs/(qt − σv)·100 %
Ic = √((3.47 − log Q)² + (log F + 1.22)²)
with σv, σ'v at the water table of the test. The stress exponent n follows Robertson & Wride (1998, as in Youd et al. 2001) for their procedure — n = 1 first; where Ic ≤ 2.6, n = 0.5 with Q = qt/Pa·(Pa/σ'v)^0.5; where that Ic exceeds 2.6, n = 0.75 — and Robertson (2009), n = 0.381·Ic + 0.05·σ'v/Pa − 0.15 ≤ 1 iterated, for Idriss & Boulanger. A reading with Ic above the limit (2.6) is clay-like.
Idriss & Boulanger (2014), CPT
FC = 80·(Ic + CFC) − 137, 0 – 100 %
qc1N = CN·qc/Pa, CN = (Pa/σ'v)^m ≤ 1.7, m = 1.338 − 0.249·qc1Ncs^0.264 (21 ≤ qc1Ncs ≤ 254)
Δqc1N = (11.9 + qc1N/14.6)·exp(1.63 − 9.7/(FC + 2) − (15.7/(FC + 2))²)
CRR7.5 = exp(q/113 + (q/1000)² − (q/140)³ + (q/137)⁴ − 2.8), q = qc1Ncs
MSFmax = 1.09 + (q/180)³ ≤ 2.2, Cσ = 1/(37.3 − 8.27·q^0.264) ≤ 0.3
with their rd and MSF form. The curve reaches CRR = 2 near qc1Ncs = 211.
Robertson & Wride (1998) — qc1N = CQ·qt/Pa with CQ = (Pa/σ'v)^n ≤ 1.7, qc1Ncs = Kc·qc1N, Kc = 1 for Ic ≤ 1.64 and −0.403·Ic⁴ + 5.581·Ic³ − 21.63·Ic² + 33.75·Ic − 17.88 above; CRR7.5 = 0.833·(q/1000) + 0.05 for q < 50, 93·(q/1000)³ + 0.08 for 50 ≤ q < 160, non-liquefiable from 160 on; the NCEER rd, MSF and Kσ.
For the strains, Dr = 0.478·qc1Ncs^0.264 − 1.063 (Idriss & Boulanger 2008).
6. Susceptibility of fine-grained soils
A sample is sand-like (the triggering curves apply) when its FC is below the threshold (35 % by default), or when it is non-plastic (PI = 0). A fine-grained, plastic sample is judged by the chosen criterion:
| criterion | susceptible | to be tested | not susceptible |
|---|---|---|---|
| Bray & Sancio (2006) | PI ≤ 12 and wn/LL ≥ 0.85 | 12 < PI ≤ 18 and wn/LL ≥ 0.8 | otherwise |
| Boulanger & Idriss (2006) | PI < 7 (sand-like) | — | PI ≥ 7 (clay-like) |
| Seed et al. (2003) | LL ≤ 37, PI ≤ 12, wn ≥ 0.8·LL (zone A) | LL ≤ 47, PI ≤ 20, wn ≥ 0.85·LL (zone B) | otherwise |
| Andrews & Martin (2000) | clay < 10 % and LL < 32 % | the other two combinations | clay ≥ 10 % and LL ≥ 32 % |
| Chinese criteria (Seed & Idriss 1982) | clay ≤ 15 %, LL ≤ 35 %, wn ≥ 0.9·LL | — | otherwise |
The clay fraction is entered as the fraction finer than 2 µm; the Chinese criteria were written for 5 µm, and the value entered is used as given. A plastic sample without a liquid limit cannot be judged and is put in the "to be tested" band. Samples in that band are taken as susceptible unless the option says otherwise.
A clay-like soil does not liquefy, but it may soften under cyclic loading. Below the water table it is checked by Boulanger & Idriss (2007), for level ground (Kα = 1):
CRR7.5 = C2D · 0.83 · su/σ'v = 0.8 · su/σ'v
MSF = 1.12·exp(−M/4) + 0.828 ≤ 1.13
FS = CRR7.5 · MSF / CSR
with su/σ'v measured (the layer's su), from the CPT as (qt − σv)/Nkt, or from SHANSEP as 0.22·OCR^0.8. A clay-like soil with FS < 1 is reported as "cyclic softening"; it adds nothing to the reconsolidation settlement, LPI or LSN, which belong to liquefaction.
7. Each test and the soil it stands for
A layer is shared among the tests that lie in it, at the mid-points between them: the first test of a layer stands for the soil from the top of the layer, the last to its bottom. A test never speaks for the layer next to it, so a sand test does not carry its factor of safety into the clay beneath. A layer without a test is not assessed, and the analysis warns about it.
8. Consequences
Volumetric strain — Ishihara & Yoshimine (1992), as fitted by Yoshimine et al. (2006) and written for the SPT by Idriss & Boulanger (2008):
Dr = √((N1)60cs / 46)
Fα = 0.032 + 4.7·Dr − 6.0·Dr² (Dr ≥ 0.4)
γlim = 1.859·(1.1 − Dr)³ ≥ 0
γmax = 0 FS ≥ 2
= min(γlim, 0.035·(2 − FS)·(1 − Fα)/(FS − Fα)) Fα ≤ FS < 2
= γlim FS < Fα
εv = 1.5 · exp(−2.5·Dr) · min(0.08, γmax)
Settlement — s = Σ εv·Δz over the saturated, assessed soil, down to the chosen depth (20 m by default). This is the one-dimensional reconsolidation settlement of the free field; the shear-induced settlement under a building and ejecta are not included.
Dry sand (Pradel 1998), above the water table, in sand-like soil:
τav = 0.65·(a_max/g)·σv·rd, p = (1 + 2K0)/3·σ'v, K0 = 0.5
Gmax = 447·Pa·(N1)60^(1/3)·(p/Pa)^0.5
a = 0.0389·(p/Pa) + 0.124, b = 6400·(p/Pa)^−0.6
γ = (1 + a·exp(b·τav/Gmax))/(1 + a) · τav/Gmax
ε15 = γ·((N1)60/20)^−1.2, εv = 2·ε15·(Neq/15)^0.45, Neq = (M − 4)^2.17
(capped at 5 %). For a CPT, N60 = (qt/Pa)/10^(1.1268 − 0.2817·Ic) (Robertson 2012). The total settlement is the saturated plus the dry part.
Lateral spreading (Zhang et al. 2004):
LDI = ∫ γmax dz over the saturated, liquefiable soil (to 2H for a free face)
LD = (S + 0.2)·LDI gently sloping ground, 0.2 < S < 3.5 %
LD = 6·(L/H)^−0.8·LDI level ground with a free face, 4 < L/H < 40
LD and LDI in metres, S in per cent. γmax is the maximum shear strain of the strain relationship above. Outside the ranges of the case histories the result is extrapolated, with a warning.
LPI (Iwasaki et al. 1978) — LPI = ∫₀²⁰ F(z)·w(z) dz, F = 1 − FS for FS < 1 (0 otherwise), w = 10 − 0.5·z. 0: very low; 0 – 5: low; 5 – 15: high; > 15: very high.
LSN (van Ballegooy et al. 2014) — LSN = 1000·∫ εv/z dz, εv as a fraction, over the saturated soil down to the settlement depth. Below 10: little to no expression; 10 – 20: minor; 20 – 30: moderate; 30 – 40: moderate to severe; 40 – 50: major; above 50: severe damage.
Both integrals are evaluated numerically in steps of 5 cm over each test's interval.
9. Limits
- The simplified procedure is a free-field, level-ground method. A sloping ground or a static shear stress (Kα) is not considered.
- Below about 20 – 30 m the procedures are poorly constrained by case histories; a site response analysis is the better tool there (the program warns for tests below 30 m with Idriss & Boulanger).
- The magnitude scaling factors were derived for 5.25 ≤ M ≤ 8.5 (warned outside it).
- Gravelly soils: the JRA correction is approximate; confirm with Vs or a large-penetration test.
- Cetin et al. (2018) is not implemented yet; the 2004 relationship is.