Settle3D Liquefaction Theory Manual
Liquefaction Analysis Theory Manual
© 2014 Rocscience Inc.
1
Table of Contents 1 Introduction ....................................................................................................................................... 2
2 Theory ................................................................................................................................................ 2
3 Standard Penetration Test (SPT) Based Calculations ................................................................... 3
3.1 Cyclic Stress Ratio (CSR) ........................................................................................................ 3
3.2 Stress Reduction Factor, rd ..................................................................................................... 4
3.3 SPT-N Value Correction Factors ............................................................................................ 7
3.3.1 Overburden Correction Factor, CN ..................................................................................... 7
3.3.2 Hammer Energy Efficiency Correction Factor, CE ............................................................ 9
3.3.3 Borehole Diameter Correction Factor, CB .............................................................................. 9
3.3.4 Rod Length Correction Factor, CR .................................................................................... 10
3.3.5 Sampler Correction Factor, CS .......................................................................................... 11
3.4 Cyclic Resistance Ratio (CRR) .............................................................................................. 11
3.5 Relative Density, DR .............................................................................................................. 16
3.6 Fines Content Correction ...................................................................................................... 17
3.7 Magnitude Scaling Factor, MSF ........................................................................................... 18
3.8 Overburden Correction Factor, Kσ ........................................................................................ 19
3.9 Shear Stress Correction Factor, Kα ...................................................................................... 22
4 Cone Penetration Test (CPT) Based Calculations ........................................................................ 23
5 Velocity (Vs) Measurement Based Calculations............................................................................ 30
6 Post-Liquefaction Lateral Displacement ....................................................................................... 32
7 Post-Liquefaction Settlement......................................................................................................... 38
References ............................................................................................................................................... 42
Table of Symbols .................................................................................................................................... 45
2
1 Introduction
Liquefaction of soils is a major cause of both damage and loss of life in earthquakes (e.g.; the 1964
Alaska and Niigata, 1983 Nihonakai-Chubu, 1987 Elmore Ranch and Superstition Hills, 1989 Loma
Prieta, 1993 Kushiro-Oki, 1994 Northridge, 1995 Hyogoken-Nambu (Kobe), 1999 Izmit earthquakes).
Researchers have tried to quantify seismic soil liquefaction initiation risk through the use of both
deterministic and probabilistic techniques based on laboratory test results and/or correlations of
insitu “index” tests with field case history performance data.
Settle3D offers different methods of calculating the factor of safety associated with liquefaction
resistance, probability of liquefaction, and the input parameters required for those calculations. This
manual also describes a simplified method for calculating the lateral spreading displacement as well
as the vertical settlement due to liquefaction.
2 Theory
The use of in situ “index” testing is the dominant approach for assessment of the likelihood of
“triggering” or initiation of liquefaction. There in-situ test methods have now reached a level of
sufficient maturity as to represent viable tools for this purpose. The following tests are often used:
Standard Penetration Test (SPT)
Cone Penetration Test (CPT)
Shear Wave Velocity (VST)
The potential for liquefaction can be evaluated by comparing the earthquake loading (CSR) with the
liquefaction resistance (CRR) - this is usually expressed as a factor of safety against
Liquefaction:
𝐹𝑆 =𝐶𝑅𝑅7.5𝑀𝑆𝐹
𝐶𝑆𝑅𝐾𝜎𝐾𝛼
where
𝐶𝑅𝑅7.5 = cyclic resistance ratio for an earthquake with magnitude 7.5
𝑀𝑆𝐹 = magnitude scaling factor
𝐾𝜎 = overburden stress correction factor
𝐾𝛼 = ground slope correction factor
3
3 Standard Penetration Test (SPT) Based Calculations
This section summarizes the methods available for calculating liquefaction resistance based on SPT
data. The following are presented:
Cyclic Stress Ratio (CSR)
Stress Reduction Factor (rd)
SPT N-Value Correction Factors
Cyclic Resistance Ratio (CRR)
Relative Density (DR)
Fines Content Correction
Magnitude Scaling Factor (MSF)
Overburden Correction Factor
Shear Stress Reduction Factor
3.1 Cyclic Stress Ratio (CSR)
The cyclic stress ratio, CSR, as proposed by Seed and Idriss (1971), is defined as the average cyclic
shear stress, 𝜏𝑎𝑣, developed on the horizontal surface of soil layers due to vertically propagating
shear waves normalized by the initial vertical effective stress, 𝜎𝑣′, to incorporate the increase in shear
strength due to increase in effective stress. By appropriately weighting the individual stress cycles
based on laboratory test data, it has been found that a reasonable amplitude to use for the “average”
or equivalent uniform stress, 𝜏𝑎𝑣, is about 65% of the maximum shear stress.
𝐶𝑆𝑅 =τ𝑎𝑣
𝜎𝑣′
= 0.65 (𝑎𝑚𝑎𝑥
𝑔) (
𝜎𝑣
𝜎𝑣′) 𝑟𝑑
where
𝑎𝑚𝑎𝑥 = maximum horizontal ground surface acceleration (g)
𝑔 = gravitational acceleration
𝜎𝑣 = total overburden pressure at depth z
𝜎𝑣′ = effective overburden pressure at depth z
𝑟𝑑 = stress reduction factor
4
3.2 Stress Reduction Factor, rd
The stress reduction factor, rd, is used to determine the maximum shear stress at different depths in
the soil. Values generally range from 1 at the ground surface to lower values at larger depths.
The following formulations are provided in Settle3D:
NCEER (1997)
Idriss (1999)
Kayen (1992)
Cetin et al. (2004)
Liao and Whitman (1986a)
NCEER (1997)
𝑟𝑑 = 1.0 − 0.00765𝑧 𝑓𝑜𝑟 𝑧 ≤ 9.15𝑚
𝑟𝑑 = 1.174 − 0.0267𝑧 𝑓𝑜𝑟 9.15𝑚 < 𝑧 ≤ 23𝑚
𝑟𝑑 = 0.744 − 0.008𝑧 𝑓𝑜𝑟 23𝑚 < 𝑧 ≤ 30𝑚
𝑟𝑑 = 0.50 𝑓𝑜𝑟 𝑧 > 30𝑚
where
𝑧 = depth in meters
Idriss (1999)
ln(𝑟𝑑) = 𝛼(𝑧) + 𝛽(𝑧)𝑀𝑤
𝛼(𝑧) = −1.012 − 1.126 sin (𝑧
11.73+ 5.133)
𝛽(𝑧) = 0.106 + 0.118 sin (𝑧
11.28+ 5.142)
where
𝑧 = depth in meters ≤ 34𝑚
5
𝑀𝑤 = earthquake magnitude
Kayen (1992)
𝑟𝑑 = 1 − 0.012𝑧
where
𝑧 = depth in meters
Cetin et al. (2004)
𝑟𝑑(𝑧, 𝑀𝑤 , 𝑎𝑚𝑎𝑥 , 𝑉𝑠,12𝑚∗ ) =
[1 +−23.013 − 2.949𝑎𝑚𝑎𝑥 + 0.999𝑀𝑤 + 0.0525𝑉𝑠,12𝑚
∗
16.258 + 0.201𝑒0.341(−𝑑+0.0785𝑉𝑠,12𝑚∗ +7.586)
]
[1 +−23.013 − 2.949𝑎𝑚𝑎𝑥 + 0.999𝑀𝑤 + 0.0525𝑉𝑠,12𝑚
∗
16.258 + 0.201𝑒0.341(0.0785𝑉𝑠,12𝑚∗ +7.586)
]
± 𝜎𝜖𝑟𝑑
for z<20 m (65ft)
𝑟𝑑(𝑧, 𝑀𝑤 , 𝑎𝑚𝑎𝑥 , 𝑉𝑠,12𝑚∗ ) =
[1 +−23.013 − 2.949𝑎𝑚𝑎𝑥 + 0.999𝑀𝑤 + 0.0525𝑉𝑠,12𝑚
∗
16.258 + 0.201𝑒0.341(−20+0.0785𝑉𝑠,12𝑚∗ +7.586)
]
[1 +−23.013 − 2.949𝑎𝑚𝑎𝑥 + 0.999𝑀𝑤 + 0.0525𝑉𝑠,12𝑚
∗
16.258 + 0.201𝑒0.341(0.0785𝑉𝑠,12𝑚∗ +7.586)
]
− 0.0046(𝑧 − 20) ± 𝜎𝜖𝑟𝑑
for z≥20m (65ft)
𝜎𝜖𝑟𝑑(𝑧) = 𝑧0.8500 × 0.0198 for z<12m (40ft)
𝜎𝜖𝑟𝑑(𝑧) = 120.8500 × 0.0198 for z≥12m (40ft)
where
σϵrd = standard deviation (assumed to be zero)
𝑧 = depth in meters
𝑎𝑚𝑎𝑥 = gravitational acceleration
𝑉𝑠,12𝑚∗ = site shear wave velocity over the top 12m
6
Notes:
- If the site stiffness estimation is difficult, take 𝑉𝑠,12𝑚∗ 𝑎𝑠 150-200m/s.
- For very soft sites with 𝑉𝑠,12𝑚∗ less than 120m/s, use a limiting stiffness of 120m/s in
calculations.
- For very stiff sites, 𝑉𝑠,12𝑚∗ with stiffness greater than 250m/s, use 250m/s as the limiting value
in calculations.
Liao and Whitman (1986a)
𝑟𝑑 = 1.0 − 0.00765𝑧 for z≤9.15m
𝑟𝑑 = 1.174 − 0.0267𝑧 for 9.15 m<z≤ 23 m
where
𝑧 = depth below ground surface in meters
7
3.3 SPT-N Value Correction Factors
Before the CRR can be calculated, the N values obtained from the SPT must be corrected for the
following factors: overburden, rod length, non-standard sampler, borehole diameter, and hammer
energy efficiency, resulting in a(𝑁1)60 value. The equation below illustrates the correction.
𝑁60 = 𝑁𝐶𝑅𝐶𝑆𝐶𝐵𝐶𝐸
(𝑁1)60 = 𝑁60𝐶𝑁
The equations used to calculate the correction factors are summarized below.
Table 1.1: Summary of Correction Factors for Field SPT-N Values
Factor Equipment Variable Term Correction
Overburden Pressure CN Section 3.3.1
Energy Ratio
Donut hammer
CE
0.5-1.0
Safety hammer 0.7-1.2
Automatic hammer 0.8-1.3
Borehole Diameter
65 mm -115 mm
CB
1.0
150 mm 1.05
200 mm 1.12
Rod Length
<3 m
CR
0.75
3 m – 4 m 0.80
4 m - 6 m 0.85
6 m -10 m 0.95
10 m – 30 m 1.00
Sampling Method Standard Sampler
CS 1.0
Sampler without Liner 1.0-1.3
3.3.1 Overburden Correction Factor, CN
The overburden correction factor adjusts N values to the N1 value that would be measured at the
same depth if the effective overburden stress was 1 atm.
The following formulations are available:
Liao and Whitman (1986)
Bazaraa (1967)
Idriss and Boulanger (2004)
Peck (1974)
Kayen et al. (1992)
8
Liao and Whitman (1986)
𝐶𝑁 = (1
𝜎𝑣0′ )
0.5
≤ 1.7
𝐶𝑁 = (𝑃𝑎
𝜎𝑣𝑜′
)0.5
Bazaraa (1967)
𝐶𝑁 =4
1 + 2𝜎𝑣0′ 𝑓𝑜𝑟 𝜎𝑣0
′ ≤ 1.5
𝐶𝑁 =4
3.25 + 0.5𝜎𝑣0′ 𝑓𝑜𝑟 𝜎𝑣0
′ ′> 1.5
𝜎𝑣
′ 𝑖𝑠 𝑖𝑛 𝑘𝑠𝑓
𝐶𝑁 ≤ 2.0
Idriss and Boulanger (2004)
𝐶𝑁 = (𝑃𝑎
𝜎𝑣𝑜′
)0.784−0.0768√(𝑁1)60
≤ 1.7
(𝑁1)60 ≤ 46
Peck, Hansen, and Thorburn (1974)
𝐶𝑁 = 0.77 log (2000
𝜎𝑣0′ ) 𝜎𝑣0
′ 𝑖𝑠 𝑖𝑛 𝑘𝑃𝑎 ≤ 282 𝑘𝑝𝑎
Kayen et al. (1992)
𝐶𝑁 =2.2
1.2 +𝜎𝑣𝑜
′
𝑃𝑎
≤ 1.7
9
3.3.2 Hammer Energy Efficiency Correction Factor, CE
The energy efficiency correction factor is calculated using the measured energy ratio as follows.
𝐶𝐸 =𝐸𝑅𝑚
60
It varies from 0.5-1.3. The ranges are taken from Skempton (1986).
Hammer Type CE
Donut hammer 0.5-1.0
Safety hammer 0.7-1.2
Automatic hammer 0.8-1.3
More specifically,
Hammer Type CE
Automatic Trip 0.9-1.6
Europe Donut Free fall 1.0
China Donut Free Fall 1.0
China Donut Rope& Pulley 0.83
Japan Donut Free Fall 1.3
Japan Donut Rope& Pulley 1.12
United States Safety Rope& pulley 0.89
United States Donut Rope& pulley 0.72
United States Automatic Trip Rope& pulley 1.25
3.3.3 Borehole Diameter Correction Factor, CB
The following table, from Skempton (1986) summarizes the borehole diameter correction factors for
various borehole diameters.
Borehole Diameter (mm) CB
65-115 1.0
150 1.05
200 1.15
10
3.3.4 Rod Length Correction Factor, CR
The rod length correction factor accounts for how energy transferred to the sampling rods is affected
by the rod length.
Youd et al. (2001)
The following table from Youd et al (2001) summarizes the rod correction factor for various rod
lengths. The rod length above the ground must also be added to obtain the total rod length before
choosing the appropriate correction factor.
Rod Length (m) CR
<3 0.75
3-4 0.80
4-6 0.85
6-10 0.95
10-30 1.00
Cetin et al. (2004)
The figure below illustrates the recommended CR values (rod length from point of hammer impact to
tip of sampler).
Figure 1: Recommended CR values
11
3.3.5 Sampler Correction Factor, CS
The sampler correction factor is applied in cases when the split spoon sampler has room for liner
rings, but those rings were not used.
Cetin et al. (2004)
CS Condition Reference
𝐶𝑆 = 1.0 Standard sampler (NCEER, 1997)
𝐶𝑆 = 1.0 − 1.3 Sampler without liners (NCEER, 1997)
𝐶𝑆 = 1.1 𝑁1,60 ≤ 10 (Cetin et al, 2004)
𝐶𝑆 = 1 +𝑁1,60
100 10 ≤ 𝑁1,60 ≤ 30 (Cetin et al, 2004)
𝐶𝑆 = 1.3 𝑁1,60 ≥ 30 (Cetin et al, 2004)
3.4 Cyclic Resistance Ratio (CRR)
The cyclic resistance ratio is the other term required to calculate the factor of safety against
liquefaction. The cyclic resistance ratio represents the maximum CSR at which a given soil can
resist liquefaction. The equation for CRR, corrected for overburden pressure and magnitude, is:
𝐶𝑅𝑅 = 𝐶𝑅𝑅7.5 𝑀𝑆𝐹
The following methods of calculating CRR are for CRR7.5 and still need to have the MSF correction
factors applied:
Seed et al. (1984)
Youd and Idriss (2001)
Idriss and Boulanger (2004)
Cetin et al. (2004) Deterministic and Probabilistic
Japanese Bridge Code (JRA 1990)
Liao et al. (1998) Probabilistic
Youd and Noble (1997) Probabilistic
12
Seed et al. (1984)
Figure 2: Liquefaction boundary curves (Seed et al. 1984)
Youd and Idriss (2001)
Figure 3: SPT clean-sand base curve for magnitude 7.5 earthquakes with data from liquefaction case
histories (modified from Seed et al. 1985)
13
The equation implemented in Settle3D is:
𝐶𝑅𝑅7.5 =1
34 − (𝑁1)60
+(𝑁1)60
135+
50
[10(𝑁1)60 + 45]2−
1
200
where (𝑁1)60 is the clean sand value (𝑁1)60𝑐𝑠.
Idriss and Boulanger (2004)
In this method, the N1,60 value is corrected for fines and adjusted to an equivalent clean sand value.
(𝑁1)60𝑐𝑠 = (𝑁1)60 + Δ(𝑁1)60
Δ(𝑁1)60 = exp (1.63 +9.7
𝐹𝐶 + 0.01− (
15.7
𝐹𝐶 + 0.01)
2
)
𝐶𝑅𝑅𝑀=7.5,𝜎=1 = exp ((𝑁1)60𝑐𝑠
14.1+ (
(𝑁1)60𝑐𝑠
126)
2
− ((𝑁1)60𝑐𝑠
23.6)
3
+ ((𝑁1)60𝑐𝑠
25.4)
4
− 2.8) for (𝑁1)60𝑐𝑠 < 37.5
𝐶𝑅𝑅𝑀=7.5,𝜎=1 = 2 𝑓𝑜𝑟 (𝑁1)60𝑐𝑠 > 37.5
Cetin et al. (2004) Deterministic and Probabilistic
The following formula calculates CRR for a given probability of liquefaction. In this equation, the
correction for fines content is built into the equations for PL and CRR.
𝐶𝑅𝑅((𝑁1)60, 𝑀𝑤 , 𝜎𝑣′ , 𝐹𝐶, 𝑃𝐿) = exp [
(𝑁1)60(1 + 0.004𝐹𝐶) − 29.53 𝑙𝑛(𝑀𝑤) − 3.70 𝑙𝑛 (𝜎𝑣
′
𝑃𝑎) + 0.05𝐹𝐶 + 16.85 + 2.70𝛷−1(𝑃𝐿)
13.32]
𝑃𝐿((𝑁1)60, 𝐶𝑆𝑅𝑒𝑞, 𝑀𝑤 , 𝜎𝑣′ , 𝐹𝐶) = Φ (−
(𝑁1)60(1 + 0.004𝐹𝐶) − 13.32 ln(𝐶𝑆𝑅𝑒𝑞) − 29.53 ln(𝑀𝑤) − 3.70 ln (𝜎𝑣
′
𝑃𝑎) + 0.05𝐹𝐶 + 16.85
2.70)
where
𝑃𝐿 = probability of liquefaction in decimals
𝐶𝑆𝑅𝑒𝑞 = CSR (not adjusted for magnitude or duration effects)
𝐹𝐶 = percent fines content expressed as an integer (5 ≤ 𝐹𝐶 ≤ 35)
𝑃𝑎 = atmospheric pressure (= 1 𝑎𝑡𝑚, ≈ 100 𝑘𝑃𝑎) (same units as a)
Φ = standard cumulative normal distribution
14
Φ−1(𝑃𝐿)= inverse of the standard cumulative normal distribution (ie. mean = 0, and standard
deviation = 1)
The deterministic analysis is done for a probability of liquefaction of 50% and a factor of safety of 1.
Japanese Bridge Code (JRA 1990)
This method is based on the equivalent clean sand value of N and the particle size distribution of
sand. The method of fines correction implemented is not necessarily the same as the one used in
Idriss and Boulanger (2004). The various methods of calculating fines correction factors will be
discussed in the next section.
𝐶𝑅𝑅𝑀=7.5,𝜎=1 = 0.0882√(𝑁1)60𝑐𝑠
𝜎𝑣′ + 0.7
+ 0.255 log (0.35
𝐷50
) + 𝑅3 𝑓𝑜𝑟 0.05𝑚𝑚 ≤ 𝐷50 < 0.6𝑚𝑚
𝐶𝑅𝑅𝑀=7.5,𝜎=1 = 0.0882√(𝑁1)60𝑐𝑠
𝜎𝑣′ + 0.7
− 0.05 𝑓𝑜𝑟 0.6𝑚𝑚 ≤ 𝐷50 < 2𝑚𝑚
𝑅3 = 0 𝑓𝑜𝑟 𝐹𝐶 < 40%
𝑅3 = 0.004𝐹𝐶 − 0.16 𝑓𝑜𝑟 𝐹𝐶 ≥ 40%
𝜎𝑣
′ 𝑖𝑠 𝑖𝑛 𝑘𝑔/𝑐𝑚2
Liao et al. (1998) Probabilistic
15
Youd and Noble (1997) Probabilistic
𝐿𝑜𝑔𝑖𝑡(𝑃𝐿) = ln (𝑃𝐿
1 − 𝑃𝐿
) = −7.0351 + 2.1738𝑀𝑤 − 0.2678(𝑁1)60𝑐𝑠 + 3.0265 ln(𝐶𝑅𝑅)
16
3.5 Relative Density, DR
The relative density of a soil is used in the calculation of the overburden correction factor, CN. The
following methods are provided in Settle3D:
Skempton (1986)
𝑁1,60 = 41 ∗ 𝐷𝑅2
Ishihara (1979)
𝐷𝑅 = 0.0676√𝑁1,60
Tatsuoka et al. (1980)
𝐷𝑅 = 0.0676√𝑁1,60 + 0.0035 ∗ 𝐹𝐶
Idriss and Boulanger (2003)
𝐷𝑅 = √𝑁1,60
46
Ishihara, Yasuda, and Yokota (1981)
𝐷𝑅 = 0.0676√𝑁1,60 + 0.085 𝑙𝑜𝑔10 (0.5
𝐷50
)
17
3.6 Fines Content Correction
The fines content has been shown to influence Δ𝑁1,60 and a number of equations have been proposed
to account for this.
Idriss and Boulanger (2008)
(𝑁1)60𝑐𝑠 = (𝑁1)60 + Δ(𝑁1)60
Δ(𝑁1)60 = exp (1.63 +9.7
𝐹𝐶 + 0.01− (
15.7
𝐹𝐶 + 0.01)
2
)
Youd et al. (2001)
The following method can also be used with an N value corrected for fines content.
(𝑁1)60𝑐𝑠 = 𝛼 + 𝛽(𝑁1)60
𝛼 = 0 𝑓𝑜𝑟 𝐹𝐶 ≤ 5%
𝛼 = exp [1.76 − (190
𝐹𝐶)
2
] 𝑓𝑜𝑟 5% < 𝐹𝐶 < 35%
𝛼 = 5.0 𝑓𝑜𝑟 𝐹𝐶 ≥ 35%
𝛽 = 1.0 𝑓𝑜𝑟 𝐹𝐶 ≤ 5%
𝛽 = [0.99 + (𝐹𝐶1.5
1000)] 𝑓𝑜𝑟 5% < 𝐹𝐶 < 35%
𝛽 = 1.2 𝑓𝑜𝑟 𝐹𝐶 ≥ 35%
Cetin et al. (2004)
(𝑁1)60𝑐𝑠 = (𝑁1)60𝐶𝐹𝐼𝑁𝐸𝑆
𝐶𝐹𝐼𝑁𝐸𝑆 = (1 + 0.004𝐹𝐶) + 0.05 (𝐹𝐶
𝑁1,60
) 𝑓𝑜𝑟 5% ≤ 𝐹𝐶 ≤ 35%
18
3.7 Magnitude Scaling Factor, MSF
As mentioned previously, the CRR equations above need to be corrected for earthquake magnitude (if
the earthquake magnitude is not 7.5).
Tokimatsu and Seed (1987)
𝑀𝑆𝐹 = 2.5 − 0.2𝑀
Idriss (1999)
𝑀𝑆𝐹 = 6.9 exp (−𝑀
4) − 0.058 ≤ 1.8
This method can also be found in Idriss and Boulanger (2004).
Andrus and Stokoe (1997)
𝑀𝑆𝐹 = (𝑀𝑤
7.5)
−2.56
Idriss and Boulanger (2008)
𝑀𝑆𝐹 = 6.9 exp (−𝑀
4) − 0.058 ≤ 1.8
Youd and Noble (1997)
The summary of the 1996/1998 NCEER Workshop proceedings by Youd and Idriss (2001) outlines
various methods for calculating the MSF and provide recommendations for engineering practice.
The following MSF values are for calculated probabilities of liquefaction, the equation for which is
also shown.
𝐿𝑜𝑔𝑖𝑡(𝑃𝐿) = ln (𝑃𝐿
1 − 𝑃𝐿
) = −7.0351 + 2.1738𝑀𝑤 − 0.2678(𝑁1)60𝑐𝑠 + 3.0265 ln(𝐶𝑅𝑅)
𝑓𝑜𝑟 𝑃𝐿 < 20% 𝑀𝑆𝐹 =103.81
𝑀4.53 𝑓𝑜𝑟 𝑀𝑤 < 7
𝑓𝑜𝑟 𝑃𝐿 < 32% 𝑀𝑆𝐹 =103.74
𝑀4.33 𝑓𝑜𝑟 𝑀𝑤 < 7
𝑓𝑜𝑟 𝑃𝐿 < 50% 𝑀𝑆𝐹 =104.21
𝑀4.81 𝑓𝑜𝑟 𝑀𝑤 < 7.75
19
Cetin et al. (2012)
𝑀𝑆𝐹 = (𝑁
𝑁𝑀𝑤=7.5
)
𝑚
= (𝑁𝑀𝑤
7.5)
𝑐
Figure 6: Variation of “c” for different ru and 𝛾𝑚𝑎𝑥
3.8 Overburden Correction Factor, Kσ
In addition to magnitude, the CRR must be corrected for overburden. It is necessary to correct for
overburden because the CRR of sand depends on the effective overburden stress; liquefaction
resistance increases with increases confining stress.
Hynes and Olsen (1999) (NCEER)
𝐾𝜎 = (𝜎𝑣𝑜
′
𝑃𝑎
)
𝑓−1
𝜎𝑣𝑜′ = 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑜𝑣𝑒𝑟𝑏𝑢𝑟𝑑𝑒𝑛 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑃𝑎 = 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑓 = 0.7 − 0.8 𝑓𝑜𝑟 40% < 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 < 60% 𝑓 = 0.6 − 0.7 𝑓𝑜𝑟 60% < 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 < 80%
20
Figure 7: Recommended curves for estimating Kσ for engineering practice (from NCEER 1996
workshop)
The parameter “f” is a function of site conditions, and the estimates above are recommended
conservative values for clean and silty sands and gravels.
Idriss and Boulanger (2008)
This method is basically the same as Idriss and Boulanger (2004), except that the limit for K is
higher.
𝐾𝜎 = 1 − 𝐶𝜎 ln (𝜎𝑣𝑜
′
𝑃𝑎
) ≤ 1.1
𝐶𝜎 =1
18.9 − 17.3𝐷𝑅
≤ 0.3
𝐷𝑅 = √(𝑁1)60
46
𝐶𝜎 =1
(18.9 − 2.55√(𝑁1)60)
21
Cetin et al. (2004)
The following figure illustrates the recommended values.
Figure 8: Kσ values, shown with NCEER recommendations (for n=0.7 and DR<60%) for comparison
22
3.9 Shear Stress Correction Factor, Kα
Kα is the static shear stress correction factor, used to correct CRR values for the effects of static
shear stresses.
Idriss and Boulanger (2003)
𝐾𝛼 = 𝑎 + 𝑏 exp (𝜉𝑅
𝑐)
𝑎 = 1267 + 636𝛼2 − 634 exp(𝛼) − 632 exp(−𝛼)
𝑏 = exp[−1.11 + 12.3𝛼2 + 1.31 ln(𝛼 + 0.0001)]
𝑐 = 0.138 + 0.126𝛼 + 2.52𝛼3
𝜉𝑅 =1
𝑄 − ln (100𝑝′
𝑃𝑎)
− 𝐷𝑅
𝛼 ≤ 0.35
−0.6 ≤ 𝜉𝑅 ≤ 0
where
𝐷𝑅 = relative density
𝑝′ = mean effective normal stress
𝑄 = empirical constant which determines the value of p’ at which dilatancy is suppressed
and depends on the grain type (Q≈10 for quartz and feldspar, 8 for limestone, 7 for anthracite, and
5.5 for chalk)
𝑃𝑎 = atmospheric pressure
23
4 Cone Penetration Test (CPT) Based Calculations
The Magnitude Scaling Factor, MSF, and Stress Reduction Factor, Rd, equations are the same for
CPT as SPT. These equations can be found in sections 3.7 and 3.2, respectively.
The following methods are available in Settle3D for determining triggering of liquefaction.
Robertson and Wride (1997)
Modified Robertson and Wride (1998)
Boulanger and Idriss (2004)
Moss et al. (2006) Deterministic
Moss et al. (2006) Probabilistic
Robertson and Wride (1997)
𝑟𝑑 = 1.0 − 0.00765𝑧 𝑓𝑜𝑟 𝑧 ≤ 9.15𝑚 𝑟𝑑 = 1.174 − 0.0267𝑧 𝑓𝑜𝑟 9.15𝑚 < 𝑧 ≤ 23𝑚
𝑟𝑑 = 0.744 − 0.008𝑧 𝑓𝑜𝑟 23𝑚 < 𝑧 ≤ 30𝑚 𝑟𝑑 = 0.50 𝑓𝑜𝑟 𝑧 > 30𝑚
𝑧 = 𝑑𝑒𝑝𝑡ℎ 𝑖𝑛 𝑚𝑒𝑡𝑒𝑟𝑠
𝐼𝑐 = [(3.47 − log(𝑄))2 + (1.22 + log(𝐹))2]0.5
𝑄 = [𝑞𝑐 − 𝜎𝑣𝑜
𝑃𝑎2
] [(𝑃𝑎
𝜎𝑣𝑜′
)𝑛
]
𝐹 = [𝑓𝑠
𝑞𝑐 − 𝜎𝑣𝑜
] ∗ 100%
𝑛 = 1.0 𝑓𝑜𝑟 𝑐𝑙𝑎𝑦𝑒𝑦 𝑠𝑜𝑖𝑙𝑠 𝑛 = 0.5 𝑓𝑜𝑟 𝑐𝑙𝑒𝑎𝑛 𝑠𝑎𝑛𝑑𝑠
𝑛 = 1.0 𝑓𝑜𝑟 𝑠𝑖𝑙𝑡𝑠 𝑎𝑛𝑑 𝑠𝑖𝑙𝑡𝑦 𝑠𝑎𝑛𝑑𝑠
The recommended procedure for calculating the soil behaviour type index can be iterative. The
following procedure is outlined in the NCEER summary report (Robertson and Wride, 1997).
1) Assume n = 1.0 and calculate Q using the following equation and calculate Ic with the
equation above.
𝑄 = [𝑞𝑐 − 𝜎𝑣𝑜
𝑃𝑎
) [(𝑃𝑎
𝜎𝑣𝑜′
)1.0
] = [𝑞𝑐 − 𝜎𝑣𝑜
𝜎𝑣𝑜′
]
24
2) If Ic > 2.6, the soil is clayey and not susceptible to liquefaction.
3) If Ic < 2.6, recalculate Cq and Q using n = 0.5 and recalculate Ic.
4) If Ic <2.6, the soil is non-plastic and granular.
5) If Ic > 2.6, the soil is probably silty. Calculate qc1N using the equations below using n = 0.7
in the equation for CQ. Calculate Ic using the qc1N value.
𝑞𝑐1𝑁 = 𝐶𝑄 (𝑞𝑐
𝑃𝑎2
)
𝐶𝑄 = (𝑃𝑎
𝜎𝑣𝑜′
)𝑛
≤ 1.7
𝐾𝑐 = 1.0 𝑓𝑜𝑟 𝐼𝑐 ≤ 1.64
𝐾𝑐 = −0.403𝐼𝑐
4 + 5.581𝐼𝑐3 − 21.63𝐼𝑐
2 + 33.75𝐼𝑐 − 17.88 𝑓𝑜𝑟 𝐼𝑐 > 1.64
𝑞𝑐1𝑁𝑐𝑠 = 𝐾𝑐𝑞𝑐1𝑁
𝐶𝑅𝑅7.5 = 0.833 [𝑞𝑐1𝑁𝑐𝑠
1000] + 0.05 𝑖𝑓 𝑞𝑐1𝑁𝑐𝑠 < 50
𝐶𝑅𝑅7.5 = 93 [𝑞𝑐1𝑁𝑐𝑠
1000]
3
+ 0.08 𝑖𝑓 50 ≤ 𝑞𝑐1𝑁𝑐𝑠 < 160
Figure 9: Normalized CPT soil behaviour type chart, proposed by Robertson (1990)
OCR = overconsolidation ratio, φ’ = friction angle
25
The soil types from the chart above are listed below:
1 sensitive, fine grained
2 peats
3 silty clay to clay
4 clayey silt to silty clay
5 silty sand to sandy silt
6 clean sand to silty sand
7 gravelly sand to dense sand
8 very stiff sand to clayey sand (heavily overconsolidated or cemented)
9 very stiff, fine grained (heavily overconsolidated or cemented)
Modified Robertson and Wride (1998)
𝐼𝑐 = [(3.47 − log (𝑄))2 + (𝑙𝑜𝑔𝐹 + 1.22)2]0.5
𝑄 = (𝑞𝑐 − 𝜎𝑣𝑜
𝑃𝑎2
) (𝑃𝑎
𝜎𝑣𝑜′
)𝑛
𝐹 = [𝑓𝑠
𝑞𝑐 − 𝜎𝑣𝑜
] 100
𝐹 = 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 𝑖𝑛 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑓𝑠 = 𝐶𝑃𝑇 𝑠𝑙𝑒𝑒𝑣𝑒 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑠𝑡𝑟𝑒𝑠𝑠
𝑃𝑎 = 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑛 𝑠𝑎𝑚𝑒 𝑢𝑛𝑖𝑡𝑠 𝑎𝑠 𝜎𝑣𝑜′ (≈ 100𝑘𝑃𝑎)
𝑃𝑎2 = 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑝𝑟𝑒𝑠𝑢𝑟𝑒 𝑖𝑛 𝑠𝑎𝑚𝑒 𝑢𝑛𝑖𝑡𝑠 𝑎𝑠 𝑞𝑐𝑎𝑛𝑑 𝜎𝑣𝑜 (= 0.1𝑀𝑃𝑎 𝑖𝑓 𝑞𝑐𝑎𝑛𝑑 𝜎𝑣𝑜 𝑎𝑟𝑒 𝑖𝑛 𝑀𝑃𝑎)
This iterative procedure is:
1) Use n=1.0 to calculate Q.
2) If Ic≤2.6, the exponent for calculating Q changes to n=0.5, and Ic is calculated using qc1N and
F.
3) If Ic<2.6, the point can be plotted on the Robertson chart (shown below) using qc1N with
n=0.5.
4) If Ic iterates around 2.6 depending on n, then use n=0.75 to calculate qc1N as follows.
𝑞𝑐1𝑁 = (𝑞𝑐
𝑃𝑎2
) 𝐶𝑄 =𝑞𝑐1
𝑃𝑎2
𝐶𝑄 = (𝑃𝑎
𝜎′𝑣𝑜
)𝑛
26
𝑖𝑓 𝐼𝑐 < 1.26 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑓𝑖𝑛𝑒𝑠 𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝐹𝐶(%) = 0
𝑖𝑓 1.26 ≤ 𝐼𝑐 ≤ 3.5 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑓𝑖𝑛𝑒𝑠 𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝐹𝐶(%) = 1.75𝐼𝑐3.25 − 3.7
𝑖𝑓 𝐼𝑐 > 3.5 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑓𝑖𝑛𝑒𝑠 𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝐹𝐶(%) = 100
𝐾𝑐 = 0 𝑓𝑜𝑟 𝐹𝐶 ≤ 5% 𝐾𝑐 = 0.0267(𝐹𝐶 − 5) 𝑓𝑜𝑟 5 < 𝐹𝐶 < 35%
𝐾𝑐 = 0.8 𝑓𝑜𝑟 𝐹𝐶 ≥ 35%
Δ𝑞𝑐1𝑁 =𝐾𝑐
1 − 𝐾𝑐
𝑞𝑐1𝑁
𝑞𝑐1𝑁𝑐𝑠 = 𝑞𝑐1𝑁 + Δ𝑞𝑐1𝑁
𝐶𝑅𝑅7.5 = 0.833 [𝑞𝑐1𝑁𝑐𝑠
1000] + 0.05 𝑖𝑓 𝑞𝑐1𝑁𝑐𝑠 < 50
𝐶𝑅𝑅7.5 = 93 [𝑞𝑐1𝑁𝑐𝑠
1000]
3
+ 0.08 𝑖𝑓 50 ≤ 𝑞𝑐1𝑁𝑐𝑠 < 160
Boulanger and Idriss (2004)
𝑟𝑑 = exp(𝛼(𝑧) + 𝛽(𝑧)𝑀)
𝛼(𝑧) = −1.012 − 1.126 sin (𝑧
11.73+ 5.133)
𝛽(𝑧) = 0.106 + 0.118 sin (𝑧
11.28+ 5.142)
𝑧 = 𝑑𝑒𝑝𝑡ℎ 𝑖𝑛 𝑚𝑒𝑡𝑒𝑟𝑠 𝑀 = 𝑚𝑜𝑚𝑒𝑛𝑡 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒
𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
𝐶𝑆𝑅 = 0.65(𝑃𝐺𝐴) (𝜎𝑣𝑜
𝜎𝑣𝑜′
) 𝑟𝑑
𝐶𝑁 = (𝑃𝑎
𝜎𝑣𝑐′
)1.338−0.249(𝑞𝑐1𝑁)0.264
≤ 1.7
𝑤ℎ𝑒𝑟𝑒 21 ≤ 𝑞𝑐1𝑁 ≤ 254
𝐼𝑐 = [(3.47 − log(𝑄))2 + (log(𝐹) + 1.22)2]0.5
27
𝑄 =𝑞𝑐𝑁 ∗ (101 − 𝜎𝑣𝑜)
𝜎𝑣𝑜′
𝐹 =𝑓𝑠𝑁
𝑞𝑐𝑁 −𝜎𝑣𝑜
101
∗ 100
𝑞𝑐𝑁 = 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑡𝑖𝑝 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑓𝑠𝑁 = 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝐶𝑃𝑇 𝑠𝑙𝑒𝑒𝑣𝑒 𝑓𝑟𝑖𝑐𝑖𝑡𝑜𝑛 𝑠𝑡𝑟𝑒𝑠𝑠 𝜎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑟𝑒 𝑖𝑛 𝑢𝑛𝑖𝑡𝑠 𝑜𝑓 𝑘𝑃𝑎
𝑞𝑐1𝑁𝑐𝑠 = 𝑞𝑐1𝑁 + Δ𝑞𝑐1𝑁
Δ𝑞𝑐1𝑁 = (5.4 +𝑞𝑐1𝑁
16) exp (1.63 +
9.7
𝐹𝐶 + 0.01− (
15.7
𝐹𝐶 + 0.01)
2
)
𝐶𝑅𝑅𝑀=7.5,𝜎𝑣𝑐′ =1 = exp (
𝑞𝑐1𝑁𝑐𝑠
540+ (
𝑞𝑐1𝑁𝑐𝑠
67)
2
− (𝑞𝑐1𝑁𝑐𝑠
80)
3
+ (𝑞𝑐1𝑁𝑐𝑠
114)
4
− 3)
𝑀𝑆𝐹 = 6.9 exp (−𝑀𝑤
4) − 0.058 ≤ 1.8
𝐾𝜎 = 1 − 𝐶𝜎 ln (𝜎𝑣𝑜
′
𝑃𝑎
) ≤ 1.0
𝐶𝜎 =1
37.3 − 8.27(𝑞𝑐1𝑁)0.264≤ 0.3
𝑞𝑐1𝑁 ≤ 211
𝐶𝑅𝑅 = 𝐶𝑅𝑅7.5𝐾𝜎𝑀𝑆𝐹
𝐹𝑆 =𝐶𝑅𝑅
𝐶𝑆𝑅
𝛾𝑙𝑖𝑚 = 1.859(1.1 − 𝐷𝑅)3 ≥ 0
𝛾𝑙𝑖𝑚 = 1.859(2.163 − 0.478(𝑞𝑐1𝑁𝑐𝑠)0.264)3 ≥ 0
𝐹𝛼 = 0.032 + 4.7𝐷𝑅 − 6.0(𝐷𝑅)2 𝑤ℎ𝑒𝑟𝑒 𝐷𝑅 ≥ 0.4
𝐹𝛼 = −11.74 + 8.34(𝑞𝑐1𝑁𝑐𝑠)0.264 − 1.371(𝑞𝑐1𝑁𝑐𝑠)0.528 𝑤ℎ𝑒𝑟𝑒 𝑞𝑐1𝑁𝑐𝑠 ≥ 69
𝛾𝑚𝑎𝑥 = 0 𝑖𝑓 𝐹𝑆 ≥ 2
28
𝛾𝑚𝑎𝑥 = min (𝛾𝑙𝑖𝑚 , 0.035(2 − 𝐹𝑆) (1 − 𝐹𝛼
𝐹𝑆 − 𝐹𝛼
)) 𝑖𝑓 𝐹𝛼 < 𝐹𝑆 < 2
𝛾𝑚𝑎𝑥 = 𝛾𝑙𝑖𝑚 𝑖𝑓 𝐹𝑆 ≤ 𝐹𝛼
𝜀𝑣 = 1.5 ∗ exp(−2.5𝐷𝑅) ∗ min(0.08, 𝛾𝑚𝑎𝑥)
𝜀𝑣 = 1.5 ∗ exp(2.551 − 1.147(𝑞𝑐1𝑁𝑐𝑠)0.264) ∗ min(0.08, 𝛾𝑚𝑎𝑥) 𝑤ℎ𝑒𝑟𝑒 𝑞𝑐1𝑁𝑐𝑠 ≥ 21
𝑆𝑣−1𝐷 = ∫ 𝜀𝑣 𝑑𝑧𝑧𝑚𝑎𝑥
0
Δ𝑞𝑐1𝑁−𝑆𝑟 = −0.00005(𝐹𝐶)3 − 0.0011(𝐹𝐶)2 + 1.0827(𝐹𝐶) − 0.6731
𝑆𝑟
𝜎𝑣𝑜′
= exp (𝑞𝑐1𝑁𝑐𝑠−𝑆𝑟
24.5− (
𝑞𝑐1𝑁𝑐𝑠−𝑆𝑟
61.7)
2
+ (𝑞𝑐1𝑁𝑐𝑠−𝑆𝑟
106)
3
− 4.42) ≤ 𝑡𝑎𝑛𝜙′
𝑆𝑟
𝜎𝑣𝑜′
= exp (𝑞𝑐1𝑁𝑐𝑠−𝑆𝑟
24.5− (
𝑞𝑐1𝑁𝑐𝑠−𝑆𝑟
61.7)
2
+ (𝑞𝑐1𝑁𝑐𝑠−𝑆𝑟
106)
3
− 4.42) ∗ (1 + exp (𝑞𝑐1𝑁𝑐𝑠−𝑆𝑟
11.1− 9.82)) ≤ 𝑡𝑎𝑛𝜙′
Δ𝑞𝑐1𝑁−𝑆𝑟 = −0.00005(𝐹𝐶)3 − 0.0011(𝐹𝐶)2 + 1.0827(𝐹𝐶) − 0.6731
𝑞𝑐1𝑁−𝑆𝑟 = 𝑞𝑐1𝑁 + Δ𝑞𝑐1𝑁−𝑆𝑟
Moss et al. (2006)
𝐶𝑅𝑅 = exp {𝑞𝑐1
1.045 + 𝑞𝑐1(0.110𝐹) + (0.001𝐹) + 𝑐(1 + 0.850𝐹) − 0.848 ln(𝑀𝑤) − 0.002 ln(𝜎𝑣′) − 20.923 + 1.632𝛷−1(𝑃𝐿)
7.177}
Probability of liquefaction according to this method is also calculated by the following correlation;
𝑃𝐿 = Φ {−𝑞𝑐1
1.045 + 𝑞𝑐1(0.110𝐹) + (0.001𝐹) + 𝑐(1 + 0.850𝐹) − 7.177 ln(𝐶𝑆𝑅) − 0.848 ln(𝑀𝑤) − 0.002 ln(𝜎𝑣′) − 20.923
1.632}
29
where
𝑞𝑐1 = normalized tip resistance (MPa)
𝑅𝑓 = friction ratio (percent)
𝑐 = normalized exponent
𝐶𝑆𝑅 = equivalent uniform stress ratio
𝜎𝑣′ = effective overburden stress (kPa)
Φ(𝑃𝐿) = cumulative normal distribution
Φ−1(𝑃𝐿) = inverse cumulative normal distribution function
The deterministic analysis is done for a probability of liquefaction of 50% and a factor of safety of 1.
A revised estimate of the normalization exponent is found using the normalized tip resistance,
shown in the figure below.
Figure 10: Proposed CPT normalization exponent curves from Moss et al. (2006), labeled by
normalization exponent, c, values
30
5 Velocity (Vs) Measurement Based Calculations
The Magnitude Scaling Factor, MSF, and Stress Reduction Factor, Rd, equations are the same for
CPT as SPT. These equations can be found in sections 3.7 and 3.2, respectively.
Two methods are available for calculating triggering of liquefaction from velocity data.
NCEER (1997)
These investigators normalized VS1:
𝑉𝑠1 = 𝑉𝑠 (𝑃𝑎
𝜎𝑣0′ )
0.25
Pa is a reference stress of 100 kPa
𝐶𝑅𝑅 = 𝑎(𝑉𝑠1
100)2 + 𝑏/(𝑉𝑠1𝑐 − 𝑉𝑠1) − 𝑏/𝑉𝑠1𝑐
where Vs1c is the limiting upper value of Vs1 for liquefaction occurrence, and
Vs1c=220 m/s for sands and gravel with FC≤ 5%
Vs1c=210 m/s for sands and gravel with FC≈ 20%
Vs1c=200 m/s for sands and gravel with FC≥ 35%
Figure 11: Proposed cyclic stress ratio curves for different fines content (FC)
(Andrus and Stokoe 2000)
31
Juang et al. (2001) (Probabilistic)
𝑉𝑠1,𝑐𝑠 = 𝑉𝑠1𝐾𝑓𝑐
𝐾𝑓𝑐 = 1 𝑓𝑜𝑟 𝐹𝐶 ≤ 5%
𝐾𝑓𝑐 = 1 + 𝑇(𝐹𝐶 − 5) 𝑓𝑜𝑟 5% < 𝐹𝐶 < 35%
𝐾𝑓𝑐 = 1 + 30𝑇 𝑓𝑜𝑟 𝐹𝐶 ≥ 35%
where
𝑇 = 0.009 − 0.0109(𝑉𝑠1/100) + 0.0038(𝑉𝑠1/100)2
where Vs1 is overburden stress-corrected shear wave velocity
ln [𝑃𝐿
1 − 𝑃𝐿
] = 14.8967 − 0.0611𝑉𝑠1,𝑐𝑠 + 2.6418ln (𝐶𝑆𝑅7.5)
Figure 12: Vs-based probability cyclic stress ratio curves logistic regression (Juang et al. 2002)
32
6 Post-Liquefaction Lateral Displacement
The LDI is basically the lateral spreading that occurs due to liquefaction. It is calculated by
integrating the maximum shear strains over depth.
𝐿𝐷𝐼 = ∫ 𝛾𝑚𝑎𝑥𝑑𝑧𝑧𝑚𝑎𝑥
0
𝛾𝑙𝑖𝑚 = 1.859 (1.1 − √(𝑁1)60𝑐𝑠
46)
3
≥ 0
The spreadsheet example in Idriss and Boulanger (2008) also imposes the following limit.
𝛾𝑙𝑖𝑚: 1.859 (1.1 − √(𝑁1)60𝑐𝑠
46)
3
≤ 0.5
The limiting shear strain can also be calculated from the relative density, with the same limits as
above.
𝛾𝑙𝑖𝑚 = 1.859(1.1 − 𝐷𝑅)3
𝐹𝛼 = 0.032 + 4.7𝐷𝑅 − 6.0(𝐷𝑅)2 𝑤ℎ𝑒𝑟𝑒 𝐷𝑅 ≥ 0.4
𝐹𝛼 = 0.032 + 0.69√(𝑁1)60𝑐𝑠 − 0.13(𝑁1)60𝑐𝑠 𝑤ℎ𝑒𝑟𝑒 (𝑁1)60𝑐𝑠 ≥ 7
Using the limiting shear strain, 𝐹𝛼, and the factor of safety, the maximum shear strain can be
calculated.
𝛾𝑚𝑎𝑥 = 0 𝑓𝑜𝑟 𝐹𝑆 ≥ 2 𝛾𝑚𝑎𝑥 = 𝛾𝑙𝑖𝑚 𝑓𝑜𝑟 𝐹𝑆 ≤ 𝐹𝛼
𝛾𝑚𝑎𝑥 = min (𝛾𝑙𝑖𝑚 , 0.035(2 − 𝐹𝑆) (1 − 𝐹𝛼
𝐹𝑆 − 𝐹𝛼
)) 𝑓𝑜𝑟 𝐹𝛼 < 𝐹𝑆 < 2
The following methods are available for calculating 𝛾𝑚𝑎𝑥:
Zhang, Robertson and Brachman (2004)
Shamoto et al. (1998)
33
Wu et al. (2003)
Cetin et al. (2009)
Zhang, Robertson and Brachman (2004)
Figure 13: Relationship between maximum cyclic shear strain and factor of safety for different
relative densities, Dr, for clean sands
[based on data from Ishihara and Yoshimine (1992) and Seed (1979)]
34
Shamoto et al. (1998)
Figure 14: Relationship between normalized SPT N-value, dynamic shear stress ratio and residual
shear strain potential for clean sands
Figure 15: Relationship between normalized SPT N-value, dynamic shear stress ratio and residual
shear strain potential for FC=10%
35
Figure 16: Relationship between normalized SPT-N value, dynamic shear stress ratio, and residual
shear strain potential for FC=20%
Wu et al. (2003)
Figure 17: Estimation of cyclically induced deviatoric strains (Wu et al. 2003)
36
Cetin et al. (2009)
ln(𝛾𝑚𝑎𝑥) = ln [−0.025(𝑁1)60,𝐶𝑆 + 𝑙𝑛(𝐶𝑆𝑅𝑠𝑠,20,1𝐷,1𝑎𝑡𝑚) + 2.613
0.004(𝑁1)60,𝐶𝑆 + 0.001] ± 1.880
lim : 5 ≤ (𝑁1)60,𝐶𝑆 ≤ 40 0.05 ≤ 𝐶𝑆𝑅𝑠𝑠,20,1𝐷,1𝑎𝑡𝑚 ≤ 0.60 0% ≤ 𝛾𝑚𝑎𝑥 ≤ 100%
where 𝐶𝑆𝑅𝑠𝑠,20,1𝐷,1𝑎𝑡𝑚 are CSR values, corresponding to one-dimensional, 20 uniform loading cycles,
under a confining pressure of 100 kPa = 1 atm.
𝐶𝑆𝑅𝑠𝑠,20,1𝐷,1𝑎𝑡𝑚 = 𝐶𝑆𝑅𝑡𝑒𝑠𝑡,20𝐾𝜎𝐾𝑚𝑐𝐾𝑟
𝐾𝜎 = (𝜎𝑣𝑜
′
𝑃𝑎
)
𝑓−1
, 𝑓 = 1 − 0.005𝐷𝑅
𝐾𝑚𝑐 = −3𝑥10−5𝐷𝑅2 + 0.0048𝐷𝑅 + 0.7222
𝐾𝑟 = 1
where
𝐾𝑚𝑐 = cyclic simple shear tests correction factor
𝐾𝑟 = cyclic triaxial tests correction factor
The recommended maximum double amplitude shear strain boundary curves are also shown in the
figure below.
37
Figure 14: Recommended maximum double amplitude shear strain boundary curves
38
7 Post-Liquefaction Settlement
The liquefaction settlement calculations in Settle3D are completely separate from the settlements
from the stress calculations. They are not superimposed in any way.
Vertical displacements from liquefaction occur due to settlement from reconsolidation as well as
shear deformation from lateral spreading.
The liquefaction settlement calculated in Settle3D is caused by the reconsolidation of the liquefied
soil. Reconsolidation strains are calculated, based on the maximum shear strains that developed
during the cyclic loading.
𝑆𝜈−1𝐷 = ∫ 𝜀𝜈
𝑧𝑚𝑎𝑥
0
𝑑𝑧
The following formulations (largely graphical methods) for 𝜀𝑣 are provided in Settle3D:
Ishihara and Yoshimine (1992)
Tokimatsu and Seed (1987)
Shamoto et al. (1998)
Wu et al. (2003)
Cetin et al. (2009)
Ishihara and Yoshimine (1992)
Figure 15: Deviatoric and volumetric strain assessment charts
(adapted from Ishihara and Yoshimine 1992)
39
Tokimatsu and Seed (1987)
Figure 16: volumetric strain assessment chart (adapted from Tokimatsu and Seed 1984)
Shamoto et al. (1998)
Figure 17: Relationship between normalized SPT-N value, dynamic shear stress ratio and residual
volumetric strain potential for clean sands
40
Figure 18: Relationship between normalized SPT-N value, dynamic shear stress ratio, and residual
volumetric strain potential for FC=10%
Figure 19: Relationship between normalized SPT N-value, dynamic shear stress ratio, and residual
volumetric strain potential for FC=20%
41
Wu et al. (2003)
Figure 20: Cyclically induced volumetric strains (adapted from Wu and Seed 2004)
Cetin et al. (2009)
𝑙𝑛(𝜀𝑣) = ln [1.879 ln [780.416 𝑙𝑛(𝐶𝑆𝑅𝑠𝑠,20,1𝐷,1𝑎𝑡𝑚) − (𝑁1)60,𝐶𝑆 + +2,442.465
636.613(𝑁1)60,𝐶𝑆 + 306.732]] ± 0.689
𝑙𝑖𝑚: 5 ≤ (𝑁1)60,𝐶𝑆 ≤ 40 0.05 ≤ 𝐶𝑆𝑅𝑠𝑠,20,1𝐷,1𝑎𝑡𝑚 ≤ 0.60
Figure 21: Post-cyclic volumetric strain boundary curves (Cetin et al. 2009)
42
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45
Table of Symbols
amax maximum ground acceleration
Mw earthquake magnitude
N60 SPT value corrected for energy, rod length, etc.
%FC percentage fines content
γ unit weight
D50 particlediameter corresponding to 50% passing
ΔH influence depth of SPT reading
Cfines correction factor for fines content
ΔN1,60 correction factor for fines content (instead of Cfines)
N1,60cs equivalent clean sand N value, fully corrected
CN overburden normalization factor
N1,60 SPT normalized to 100 kPa overburden
CSR cyclic stress ratio
N1,60−Sr post-liquefaction N value
PL probability of liquefaction
Sr residual shear strength
Fα factor for calculating maximum shear strain
γlim limiting shear strain
CRRM=7.5,σ=1 cyclic resistance ratio for magnitude M=7.5 and overburden stress=100kPa
CRR cyclic resistance ratio, corrected for magnitude and overburden
FS factor of safety
γmax maximum shear strain
LDI lateral displacement index
εv vertical reconsolidation strain
S vertical reconsolidation settlement
SPT Standard Penetration Test
CPT Cone Penetration Test
Vs Shear Wave Velocity
Vs,12m∗ Site shear wave velocity over the top 12 m
FC Percent fines content passing sieve #200 (clay and silt)
Mw Earthquake magnitude
z Soil depth
τav Average cyclic shear stress
g Gravitational acceleration
σv Total overburden pressure at the depth z
σv′ Effective overburden pressure at the depth z
46
rd Stress reduction factor
MSF Magnitude scaling factor
Kσ Overburden stress correction factor
Kα Ground slope correction
Pa atmospheric pressure (=1 atm,≈100 kPa)
Φ standard cumulative normal distribution
Φ -1 (PL) inverse of the standard cumulative normal distribution
ru excess pore pressure ratio
γmax maximum double amplitude shear strain
CSRss,20,1D,1atm CSR values, corresponding to one-dimensional, 20 uniform loading cycles,
under a confining pressure of 100 kPa =1 atm
