Chapter 2. Review of Liquefaction Evaluation Procedures · placed on energy-based procedures. The...

Chapter 2. Review of Liquefaction Evaluation Procedures

2.1 Introduction

Numerous methods have been proposed for evaluating the liquefaction potential of soil

deposits. The purpose of this chapter is to review selected approaches, with emphasis

placed on energy-based procedures. The intent of the review is to assess the procedures

for potential use in the design of remedial densification programs for liquefiable soils.

With the exception of Liang (1995), all the procedures discussed have simplified forms:

their implementation does not require site response analyses.

The procedures are presented in terms of Demand, Capacity, and Factor of Safety, where

Demand is the load imparted to the soil by the earthquake (both amplitude and duration),

Capacity is the Demand required to induce liquefaction, and Factor of Safety is defined

as the ratio of Capacity and Demand. The procedures are presented in this way for

consistency and to facilitate comparisons. To do this, several of the procedures are

presented in alternate forms from those published in the referenced literature.

Furthermore, to facilitate computations, curve-fitting techniques were employed to

develop equations corresponding to charts and graphs relied on by several of the

procedures. However, it is emphasized that the alternate forms of presentation and the

use of curve-fit equations in no way change the basic formulations and assumptions of

the procedures.

For the procedures that correlate soil Capacity to field tests, only correlations with SPT

N-values are presented. The reason for this is that the majority of the procedures only

have such correlations. Also, correlations with SPT N-values serve the intent of the

review, which is to assess the potential of the procedures to be used in the design of

remedial ground densification programs, not to assess the merits of the various field tests.

The appendix at the end of this chapter outlines the Youd et al. (2001) recommended

factors for normalizing measured SPT N-values for overburden pressure, hammer energy,

borehole diameter, rod length, and sampling method, with the result being designated as

N1,60. The liquefaction evaluation procedures proposed by Davis and Berrill (1982),

9

Berrill and Davis (1985), and Trifunac (1995) use correlations with SPT N-values

normalized only for overburden pressure. It is the opinion of the author that in

implementing these procedures, SPT N-values using all the normalizations given in the

appendix should be used. The normalizations standardize the SPT N-values and reduce

the variability in the measured blow counts by various drill rigs and operators.

Furthermore, several of the procedures require correction of the SPT N-values for fines

content. However, no two procedures use the same correction factors. Accordingly,

fines content correction factors are presented with the corresponding procedures.

Rather than delving into diversified theories of the various procedures, a “cookbook”

presentation is given for each. The procedures are presented in the following order: the

stress-based procedure, the strain-based procedure, and energy-based procedures. The

energy-based procedures are grouped into approaches developed using earthquake case

histories, approaches developed from laboratory data, and other approaches. This last

group includes studies that warrant mention but are either similar to other approaches

discussed more in depth or are not fully developed liquefaction evaluation procedures.

After the “cookbook” presentation, the results and observed trends of a parameter study

are presented. These results help to assess the procedures. Finally, general and specific

commentaries are given on the assumptions and formulations of several of the

procedures.

2.2 Overview of the Procedures

2.2.1 Stress-based procedure

The most widely used method for evaluating liquefaction is the stress-based procedure

first proposed by Seed and Idriss (1971) and Whitman (1971). This procedure is largely

based on empirical observations of laboratory and field data and has been continually

refined as a result of newer studies and the increase in the number of liquefaction case

histories (e.g., NRC 1985, NCEER 1997, Youd et al. 2001).

10

Demand:

The amplitude of the earthquake-induced Demand is quantified by the cyclic stress ratio

(CSR). CSR can be determined for any desired depth in a soil profile from site response

analyses or by using the “simplified” equation, Equation (2-1).

dvo

vo

vo

ave rg

aCSR

'65.0

'max

σσ

στ

== (2-1)

where: CSR = Cyclic stress ratio.

amax = Soil surface of acceleration.

g = Acceleration due to gravity.

σ’vo = Initial effective vertical stress at depth z.

σvo = Total vertical stress at depth z.

rd = Dimensionless parameter that accounts for the stress

reduction due to soil column deformability.

A consistent set of units should be used so that CSR is dimensionless. The range of rd, as

a function of depth, is shown in Figure 2-1. The average of this range can be computed

using Equation (2-2) (NCEER 1997).

0−

(2-2) ≤<−

mzfor 30. >

rd = mzmforz

mzmforzmzforz

503023008.0744.0

2315.90267.0174.115.900765.0.1

≤<−≤

where z is depth in m. Equation (2-3) yields essentially the same results as Equation (2-

2), but may be easier to program for use in spread sheet calculations (NCEER 1997):

25.15.0

5.15.0

001210.0006205.005729.04177.0)000.1()001753.004052.04113.0000.1(

zzzzzzzrd +−+−

++−= (2-3)

11

rd

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0

Simplified procedurenot verified with case history data in this region

Range for differentsoil profiles

Mean values of rd calculated from Equation (2-2)

Average values

3 6

9 12

Dep

th (m

)

15

18

21

24 27 30

Figure 2-1. Stress reduction factor (rd) to account for soil column deformability. (Adapted from NCEER 1997 and Seed and Idriss 1982).

To account for the duration of the earthquake motions, magnitude scaling factors (MSF)

are applied to the CSR:

vo

M

vo

aveM MSFMSF

CSRCSR''

5.75.7 σ

τσ

τ=

⋅== (2-4)

This expression defines the Demand imparted to the soil by the earthquake (i.e., Demand

= CSRM7.5).

Several different correlations for MSF have been proposed, as shown in Figure 2-2. The

bases for these relationships are given and discussed in NCEER (1997) and Youd et al.

(2001).

12

Earthquake Magnitude, M5.0

Average of NCEER

Recommended Range (MSFave)

Seed and Idriss, (1982)IdrissAmbraseys (1988) Arango (1996)Arango (1996)Andrus and StokoeYoud and Noble, PL<20%Youd and Noble, PL<32%Youd and Noble, PL<50%2.0

2.5 3.0 3.5 4.0 4.5

1.5

1.0

0.0 0.5

9.08.07.06.0

Mag

nitu

de S

calin

g Fa

ctor

, MSF

Figure 2-2. Magnitude scaling factors proposed by various investigators. (Adapted from Youd and Noble 1997).

The average values of the NCEER (1997) recommended range for the MSF can be

determined by Equation (2-5).

+

forMSF

MSFave = (2-5a)

MSF

5.7

5.72

>

≤−

Mfor

MMSF

Idriss

IdrissStokoeAndrus

where: 3.3

5.7

−

−

=

MMSF StokoeAndrus (2-5b)

56.2

24.210M

MSFIdriss = (2-5c)

13

Capacity:

The Capacity of the soil is quantified as a function of the cyclic resistance ratio (CRR),

which in turn is correlated to N1,60. Soils with fines have greater resistance to liquefaction

than clean sands having the same N1,60. To account for the fines content of the soil,

correction factors are applied to the N1,60, resulting in N1,60cs. N1,60cs can be determined

using Equation (2-6) (NCEER 1997, Youd et al. 2001).

N1,60cs = α + β⋅N1,60 (2-6a)

where:

(2-6b) 0 for FC ≤ 5%

α = exp[1.76 – (190/FC )] for 5% < FC ≤ 35%

5.0 for FC > 35%

2

(2-6c)

1.0 for FC ≤ 5%

β = [0.99 – (FC /1000)] for 5% < FC ≤ 35%

1.2 for FC > 35%

1.5

The CRR can be determined graphically from Figure 2-3. The curve shown in Figure 2-3

was developed by analyzing earthquake case histories. Sites containing liquefiable soils

and subjected to earthquake motions were categorized as liquefied and non-liquefied,

based largely on the presence or absence of surficial liquefaction features. For each of

the case histories, the Demand imparted to the soil was estimated using Equation (2-4)

and plotted as a function of the N1,60cs of the soil. The boundary giving a reasonable

separation of the liquefied and non-liquefied points defines the CRR. For spread sheet

calculations or other analytical techniques, CRR may be determined by Equation (2-7)

(NCEER 1997):

2001

)4510(50

135341

260,1

60,1

60,1

−+⋅

++−

=cs

cs

cs NN

NCRR (2-7)

14

No LiquefactionLiquefactionCRR

00.00

0.60

0.55 0.50 0.45 0.40

CSR

M7.

5, C

RR

0.35

0.30

0.25 0.20 0.15 0.10

0.05

5 10 15 20 25 30N 1,60cs

35 40 45 50

Figure 2-3. Cyclic resistance ratio (CRR) curve.

The Capacity of the soil is related to CRR as:

Capacity = CRR· Kσ·Kα (2-8)

where: Kσ = Correction factor for high overburden pressure.

Kα = Correction factor for initial static shear stresses.

Kσ is given in Figure 2-4. The history of the development of the Kσ correction factor and

data in support of the recommended values are given in Seed (1983), Seed and Harder

(1990) and Hynes and Olsen (1999).

15

Kσ = (σ’vo)f-1 Dr ≥ 80% (f = 0.6)

Dr ≈ 60% (f = 0.7)

Dr ≤ 40% (f = 0.8)

109 Vertical Effective Stress σ’vo (atm units, e.g., tsf)

8 6 751 2 3 40.0

0

0.2

0.4

0.6

0.8

1.0

1.2 Kσ

Figure 2-4. Recommended values for overburden pressure correction factor Kσ. (Adapted from Youd et al. 2001).

For the purposes of this research, only level ground conditions are considered, and

accordingly, Kα is set equal to 1.

Factor of Safety:

The factor of safety (FS) against failure is defined as the ratio of Capacity and Demand:

DemandCapacityFS = (2-9)

where failure is predicted when FS ≤ 1.0. Substituting the definitions for Capacity and

Demand in Equation (2-9), factor of safety against liquefaction can be computed by

Equation (2-10).

16

CSRMSFKKCRR

FS⋅⋅⋅

= ασ (2-10)

2.2.2 Strain-based procedure

As an alternative to the empirical stress-based procedure, Dobry et al. (1982) proposed a

strain-based procedure. This procedure was derived from the mechanics of two

interacting idealized sand grains and then generalized for natural soil deposits.

Demand:

The Demand is quantified by the amplitude of the earthquake-induced cyclic shear strain

(γ) at a given depth in the soil profile and is determined by Equation (2-11):

γ

σγ

⋅

⋅⋅⋅=

maxmax

max65.0

GGG

rg

advo

(2-11)

where: Gmax = shear modulus corresponding to γ = 10-4%.

(G /Gmax) = ratio of shear moduli corresponding to γ and γ = 10-4%.

The other variables in Equation (2-11) were defined previously. Although several of

these were proposed subsequent to Dobry et al. (1982), the following expression may be

used to relate N1,60 to Gmax:

( )5.0

21

3/160,1max

'440

=

PaPaNG moσ

(2-12)

σ’mo is the initial mean effective confining stress, and Pa1 and Pa2 are atmospheric

pressure having the same units as Gmax and σ’mo, respectively. Equation (2-12) is the

non-dimensional form an expression given in Seed et al. (1986). σ’mo is defined by

Equation (2-13).

voo

moK

'321

' σσ ⋅

+= (2-13)

17

where: )'sin(1 φ−=oK (2-14)

( ) 2020' 5.060,1 +⋅= Nφ (Hatanaka and Uchida 1996) (2-15)

Because the ratio (G/Gmax) is a function of shear strain (γ), Equation (2-11) has to be

solved iteratively using the shear modulus degradation curve associated with the soil

layer of interest. This process is shown graphically in Figure 2-5, where in the first

iteration a value of G/Gmax is assumed and γ computed. In the second iteration, the ratio

of G/Gmax corresponding to γ computed in the first iteration is used. The process is

repeated until the assumed and computed ratios are within a tolerable error.

Computed values of G/Gmax

Assumed values of G/Gmax

tolerable error final iteration

iteration 2

iteration 1

1% 10-4%

1.0

(G/Gmax)γ

γ γ (log Scale)

G/G

max

Figure 2-5. Iterative solution of Equation (2-11) to determ

strain (ine the effective shear-

γ) at a given depth in a soil profile.

Numerous shear modulus degradation curves have been proposed in literature. The

Ishibashi and Zhang (1993) shear modulus degradation curves were used throughout this

research. These curves were selected because they are presented in equation form and

are expressed as functions of both effective confining stress and plasticity index (Ip),

where other commonly used curves are expressed only as a function of Ip (e.g., Vucetic

and Dobry 1991). Figure 2-6 shows the Ishibashi and Zhang (1993) curves for different

initial mean effective confining stresses.

18

σ’mo ↓

0.001 10.1

G/G

max

Shear Strain (%)0.01

1.0

0.8

0.6

0.4

0.2

0.00.0001

σ’mo = 3533 psf σ’mo = 2947 psf σ’mo = 2413 psf σ’mo = 1933 psf

σ’mo = 1507 psfσ’mo = 1133 psfσ’mo = 813 psfσ’mo = 547 psf

σ’mo = 333 psf σ’mo = 173 psf σ’mo = 67 psf σ’mo = 13 psf

Figure 2-6. Plots of the Ishibashi and Zhang (1993) shear modulus degradation curves for various initial mean effective confining stresses.

The Ishibashi and Zhang (1993) shear modulus curves are given in equation form as:

( ) ),('

max

'),( pImmopIK

GG γσγ ⋅= (2-16a)

++⋅=

492.0)(000102.0lntanh15.0),(

γγ p

p

InIK (2-16b)

>⋅×≤<⋅×≤<⋅×

=

=

−

−

−

)(70107.2)(7015100.7

)(1501037.3)(00.0

)(

115.15

976.17

404.16

soilsplastichighIforIsoilsplasticmediumIforI

soilsplasticlowIforIsoilssandyIfor

In

pp

pp

pp

p

p (2-16c)

19

3.10145.04.0

000556.0lntanh1272.0),(' pIp eIm ⋅−⋅

+⋅=

γγ (2-16d)

where: γ = shear strain

Ip = plasticity index

The validity of the expressions for Gmax, (G/Gmax), and φ’ (Equations (2-12), (2-16), and

(2-15), respectively) may be questionable for low effective confining pressures. As will

be shown later in this chapter, the use of these expressions may under-predict γ in the

upper 3-5m of a soil profile, depending on the location of the ground water table.

Capacity:

The Capacity of the soil is quantified by the threshold shear strain (γth), which is defined

as the shear strain amplitude required to cause gross sliding across grain-to-grain contact

surfaces. Dobry et al. (1982) conducted a series of strain controlled cyclic tests on

saturated undrained specimens. In these tests, the threshold strain was determined as the

minimum amplitude shear strain inducing a non-zero excess pore pressure after the cyclic

loading stopped (i.e., residual excess pore pressure). Dobry et al. (1982) concluded that

γth is approximately 0.01% and is independent of the method of sample preparation,

relative density, and initial effective confining stress (at least for the range of initial mean

effective confining stresses used in the laboratory study). Figures 2-7a and 2-7b show

some of the results from Dobry et al. (1982).

20

γth

10-3 0.00

80 60 45

Dr (%) Symbol

Monterey No. 0 Sand σ’mo = 100kPa (2000 psf) n = 10 cycles

0.06

0.05 Ex

cess

Por

e Pr

essu

re R

atio

, ru

0.04

0.03

0.02

0.01

10-2 10-1 3 5 3 5 Shear Strain, γ, (%)

Figure 2-7a. Laboratory test results conducted on samples of varying relative densities. The shear strain at which excess pore pressures are measured for all the samples is slightly greater than 10-2 percent. (Adapted from Dobry et al. 1982).

21

1.2

n = 10 cycles Dr = 60%

γth

Monterey no. 0 sand, σ’mo = 2000psf, fresh samples, moist tamping Sand no. 1, σ’mo = 1400psf, staged testing, moist tamping Sand no. 1, σ’mo = 2800psf, staged testing, moist tamping Crystal silica sand, σ’mo = 2000psf, staged testing, dry vibration Crystal silica sand, σ’mo = 2000psf, fresh samples, wet rodding Crystal silica sand, σ’mo = 2000psf, fresh samples, dry vibration

1.0

Exce

ss P

ore

Pres

sure

Rat

io, r

u

0.8

0.6

0.4

0.2

0.0 10-3 10-2 10-1 1

Shear Strain, γ, (%)

F c

igure 2-7b. Laboratory test results conducted on samples of varying initial effective onfining stresses and prepared by various methods. The shear strain at which excess

pore pressures are measured for all the samples is slightly greater than 10-2 percent. (Adapted from Dobry et al. 1982).

Factor of Safety:

Using the definition for factor of safety (FS) given previously in Equation (2-9) (i.e.,

Capacity/Demand), yields:

γγ thFS = (2-17)

Although FS > 1.0 implies that liquefaction will not occur, FS ≤ 1.0 does not imply that

liquefaction will occur. Rather FS ≤ 1.0 predicts that there will be gross sliding of the

grain-to-grain contact surfaces, which is a requisite for excess pore pressure generation

and therefore a requisite for liquefaction. This is in contrast to the stress based procedure

for which FS > 1.0 implies that liquefaction will not occur, not necessarily that excess

pore pressures will not be generated. Accordingly, the ratio of γth and γ may be an overly

22

conservative method for computing the factor of safety against liquefaction. However,

FS ≤ 1.0 provides insight into the expected behavior of the soil in that the residual excess

pore pressures will be greater than zero.

2.2.3 Energy-based procedures

As opposed to using stress or strain as the base parameters to quantify Capacity and

Demand, the procedures presented in this section use various measures of energy. The

use of energy was a logical step in the evolution of liquefaction evaluations. The reason

for this is twofold. First, seismologists have long been quantifying the energy released

during earthquakes and have established simple correlations with common seismological

parameters (e.g., Gutenberg and Richter 1956). The second reason is the pioneering

study by Nemat-Nasser and Shokooh (1979) showing a functional relationship between

the dissipated energy in laboratory samples and generated pore pressures. Chapter 4

gives a brief presentation of Nemat-Nasser and Shokooh’s energy-based pore pressure

model, as well as other models.

In the presentation given below, the energy-based liquefaction evaluation procedures are

categorized into two main groups: 1) procedures developed from earthquake case

histories, and 2) procedures developed from laboratory data. A third section entitled

“Other Approaches” presents studies warranting mention but are either similar to other

procedures presented or are not fully developed liquefaction evaluation procedures. The

procedures developed from earthquake case histories are further subdivided into ones

quantifying energy using the Gutenberg-Richter energy relation and ones using Arias

intensity. The exception to this categorization is in the presentation of the procedures

proposed by Trifunac (1995). Trifunac (1995) proposes 5 different energy-based

liquefaction evaluation procedures, only one of which uses the Gutenberg-Richter

relation. However, for continuity of presentation, all 5 of them are presented in the

Gutenberg-Richter section.

For purposes of assessing the procedures for potential use in the design of remedial

densification programs, preference is automatically given to those developed from

23

earthquake case histories. This is because no study that the author has seen shows that

the Capacity of the soil, quantified by any measure of energy, is independent of soil

fabric, and none of the reviewed procedures developed from laboratory data accounts for

the influence of soil fabric in a way that can be used to evaluate the of Capacity actual

field deposits.

Finally, inherent to many of the energy-based procedures presented below are parameters

quantifying site-to-source distance and earthquake magnitude. However, not all the

procedures use the same definition of site-to-source distance or use the same magnitude

scale. To facilitate the presentation of the procedures, commonly used measures of site-

to-source distance and a comparison of various magnitude scales are shown in Figures 2-

8a and 2-8b, respectively.

D1: Hypocentral distance, where h is the focal depth D2: Epicentral distance D3: Distance to center of high-energy release (or high localized stress drop) D4: Closest distance to the slipped fault D5: Closest distance to the surface projection of fault rupture.

h

D1

D2 D3

D4 D5

Station

Surface

Hypocenter

Epicenter

Surface of fault slippage

High Stress zone

Figure 2-8a. Commonly used measures of site-to-source distance. (Adapted from Shakal and Bernreuter 1981).

24

MJMA: Magnitude scale used by the Japanese Meteorological Agency ML: Local Magnitude MS: Surface-wave Magnitude mb: Body-wave Magnitude, (short period body waves) mB: Body-wave Magnitude, (long period body waves) MW: Moment Magnitude

ML for M< 5.9 M: Richter Magnitude = MS for 5.9 < M < 8.0 MW for 8.0 < M < 8.3

Mag

nitu

de

2 2

3

8

9

4

5

7

6

Moment Magnitude MW 5 4 3 10 6 7 8 9

MSML

MW

MJMA

MS

mB

ML

mb

Figure 2-8b. Comparison of the central tendencies of various earthquake magnitude scales, with moment magnitude (MW) currently being the most accepted scale. Although in the past the term “Richter magnitude” was used synonymously with ML, it currently is used to refer to a conglomeration of magnitude scales (e.g., Krinitzski et al. 1993). (Adapted from Heaton et al. 1986).

25

2.2.3.1 Procedures developed from earthquake case histories

2.2.3.1.1 Gutenberg-Richter approaches

Most of the procedures presented in this section use the Gutenberg-Richter energy

correlation as the basis for computing the Demand imparted to the soil. The Gutenberg-

Richter energy relation is given as (Gutenberg and Richter 1956): 8.15.110 += M

oE (2-18)

where: Eo = Total radiated energy from the source (kJ).

M = Magnitude of earthquake (Richter scale).

2.2.3.1.1a Davis and Berrill (1982)

Demand:

In addition to the Gutenberg-Richter relation, Davis and Berrill (1982) make three other

assumptions in deriving their expression for Demand. First, it is assumed that the

magnitude of the energy decreases at a rate proportional to 1/r2, where r is distance from

site to the center of energy release. This attenuation model does not include energy lost

due to material damping, rather it only accounts for the geometrical spreading of a

spherically expanding wave front. Second, it is assumed that the increase in pore

pressure is a linear function of dissipated energy. Finally, it is assumed that energy

dissipation in the soil due to material damping is proportional to 1/(σ’vo)0.5, where σ’vo is

the initial effective overburden stress. Their derived expression for Demand is given as: 1

5.1

5.12

10'

−

= M

vorDemand σ (2-19)

where: r = Distance from site to the center of energy release (m).


σ’vo = Initial effective vertical stress at depth z ( kPa).

Capacity:

Similar to the procedure used to determine the CRR curve shown in Figure 2-3, Davis and

Berrill grouped earthquake case histories according to the occurrence or non-occurrence

of liquefaction. Using Equation (2-19), the Demand imparted to the soil was estimated

26

for each site. Because the distance from the site to the center of energy release was not

known for all the cases, hypocentral and epicentral distances were often used. In Davis

and Berrill’s presentation, they plotted the inverse of Demand (i.e., 1/Demand) as a

function of the N1, as shown in Figure 2-9.

Mvor5.1

5.12

10'σ

high Demand

Liq.No Liq.

1.0

10

100 low Demand

100 Corrected Standard Penetration Value, N1

1.0 0.1

10

Figure 2-9. Liquefaction chart proposed by Davis and Berrill (1982).

N1 is the measured SPT N-value adjusted to 1tsf; no hammer-energy correction factors or

fines correction factors were applied. Because Davis and Berrill (1982) plotted the

inverse of Demand, the boundary giving a reasonable separation of the liquefied and non-

liquefied data points defines the inverse of the Capacity of the soil. The Capacity is

given by Equation (2-20). 1

21

450−

=

NCapacity (2-20)

27

where: N1 = Measured SPT N-value normalized to 1tsf.

Factor of Safety Against Liquefaction:

Using the definition of factor of safety as the ratio of Capacity to Demand, substitution of

Equations (2-20) and (2-19) results in the following expression for the factor of safety

against liquefaction:

MvorN

FS 5.1

5.1221

10450)'(

⋅⋅⋅

=σ

(2-21)

2.2.3.1.1b Berrill and Davis (1985)

Demand:

Berrill and Davis (1985) proposed a revised model of their earlier work, Davis and Berrill

(1982). Two main revisions were included: 1) a revised pore pressure generation model

(i.e, pore pressure is proportional to the square root of dissipated energy), and 2) the

inclusion of an expression to account for the inelastic attenuation of seismic energy as it

travels from the source to the site. The resulting expression for Demand is given by

Equation (2-22). 5.0

5.1

5.12

10'

−

⋅= M

vo

Ar

Demandσ

(2-22)

where: A = Material attenuation factor, given by Equation (2-23a).

∫∞ −⋅=

0

22 )(4 dxexFxA ax

π (2-23a)

)1(1)( 2x

xF+

= (2-23b)

a = Dimensionless distance, given by Equation (2-23c).

dQrka = (2-23c)

where: r and d have same dimensions.

k = dimensionless constant that varies as a function of the source rupture model

28

(= 2.80 for the Brune model: Brune 1970, 1971).

d = Source dimension, radius of equivalent circular source, given by Equation (2-23d).

Q = Quality factor function of material attenuation (= 280, assumed).

( ) kmMd 35.1exp1014.1 3 ⋅×= − (2-23d)

The material attenuation factor, A, is plotted in Figure 2-10 as a function of dimensionless

distance, a. For typical cases, it can be expected that A ≥ 0.8, which implies that little

energy is lost due to anelastic attenuation as the seismic waves travel from the source to

the site. Accordingly, the main difference between the Demand expressions proposed in

Davis and Berrill (1982) and Berrill and Davis (1985) is that in the 1985 model the pore

pressure is assumed proportional to the square root of dissipated energy, as opposed to

the linear relationship assumed in the earlier model.

1.0

0.8

0.4

0.6

0.2

0.001 0.0

0.01 0.1 1.0 10

Mat

eria

l atte

nuat

ion

fact

or, A

Dimensionless distance, a = kr/Qd Figure 2-10. Material attenuation factor (A) as a function of the dimensionless

distance (a). (Adapted from Berrill and Davis 1985).

29

Capacity:

Again, categorizing sites according to the occurrence or non-occurrence of liquefaction,

Berrill and Davis (1985) plotted the inverse of energy demand (i.e., 1/Demand) as a

function of the N1, as shown in Figure 2-11.

Liquefaction No Liquefaction

5.0

5.1

5.12

10'

⋅ Mvo

Ar σ

0.5

1.0

5

10

50

1005010510.2

Corrected standard penetration value, N1

Figure 2-11. Liquefaction chart proposed by Berrill and Davis (1985).

The boundary giving a reasonable separation of the liquefied and non-liquefied data

points defines the inverse of the energy capacity of the soil. The Capacity is given as: 1

5.11

120−

=

NCapacity (2-24)

30

Factor of Safety:


Equations (2-24) and (2-22) yields:

Mvo

ArN

FS 75.05.0

75.05.11

10120)'(

⋅⋅⋅⋅

=σ

(2-25)

2.2.3.1.1c Law, Cao, and He (1990)

Demand:

Similar to Davis and Berrill (1982) and Berrill and Davis (1985), Law et al. (1990)

derived an expression for energy demand based on the Gutenberg-Richter relation.

However, Law et al. made several different assumptions regarding the functional

relationship between dissipated energy and excess pore pressure generation, anelastic

attenuation of seismic waves along the travel path, and material damping at the site. The

resulting expression for Demand, referred to as the seismic energy intensity function (T),

is given by Equation (2-26).

3.4

5.110r

TM

= (2-26)

where: T = The seismic energy intensity function (i.e., Demand).

r = Hypocentral distance from site to source (km).


Capacity:

Using the same procedure of grouping sites according to the observance and non-

observance of liquefaction, Law et al. (1990) computed the energy demand imparted to

the sites using Equation (2-26) and plotted the results versus N1,60, as shown in Figure 2-

12.

31

SANDSILTY SAND

LIQUEFIED SITENONLIQUEFIED SITE

100 1010.1101

107

SEIS

MIC

EN

ERG

Y IN

TEN

SITY

FU

NC

TIO

N, T

106

105

104

103

102

CORRECTED SPT RESISTANCE, N1,60

Figure 2-12. Liquefaction chart proposed by Law, Cao, and He (1990).

In plotting their data, Law et al. distinguished between the soil types at the site (i.e., sand

and silty-sand). As shown in Figure 2-12, boundaries were drawn separating the points

representing liquefaction from those of no-liquefaction for each soil type. These

boundaries define the Capacities (i.e., ηLsand and ηLsilt) of the soils as a function of N1,60

and are expressed by Equation (2-27). 105.11

60,1 10)(28.2 −×= NsandLη (2-27a)

95.1160,1 10)(14.1 −×= NsiltLη (2-27b)

32

Factor of Safety:


Equations (2-27) and (2-26) results in the following expressions for the factor of safety

against liquefaction for sand and silty-sand:

( )Msand

rNFS 5.1

3.45.1160,1

10

101028.2 ⋅⋅×

=−

(2-28a)

( )Msilt

rNFS 5.1

3.45.1160,1

9

101014.1 ⋅⋅×

=−

(2-28b)

2.2.3.1.1d Trifunac (1995)

Trifunac (1995) proposed five separate pairs of expressions for energy Demand and

Capacity, only one of which is based on the Gutenberg-Richter energy relation.

However, all five are presented in this section and are referred to as Trifunac No.1,

Trifunac No.2, Trifunac No.3, etc. In deriving all of the energy demand expressions,

Trifunac (1995) retained two of the central assumptions made by Davis and Berrill

(1982), namely that the increase in pore pressure is a linear function of dissipated energy

and that energy dissipation in the soil due to material damping is proportional to

1/(σ’vo)0.5. Also, as with Davis and Berrill (1982) and Berrill and Davis (1985), all of the

capacity expressions proposed by Trifunac (1995) are expressed as functions of N1,

where N1 is the measured SPT N-value adjusted to 1tsf; no hammer-energy correction

factors or fines correction factors were applied.

More so than the other procedures presented above, the expressions for Demand and

Capacity are presented herein in alternate forms from those given in Trifunac (1995).

However, with the possible exception of differences resulting from the reduction in the

number of significant digits of various coefficients, the expressions presented below give

identical results to those given in Trifunac (1995).

33

Trifunac No.1

Demand:

The expression for Demand for Trifunac No.1 is based on the Gutenberg-Richter energy

relation and is identical to that derived by Davis and Berrill (1982). This expression was

presented previously as Equation (2-19) and is repeated below. 1

5.1

5.12

10'

−

= M

vorDemand

σ (2-19)

where: r = Epicentral distance from site to source (m).


σ’vo = Initial effective vertical stress at depth z (kPa).

Trifunac No.1 uses the epicentral distance from the site to the source as an approximation

for the distance from site to the center of energy release, the definition of r used by Davis

and Berrill (1982). This approximation is coupled with the imposed limitation on

Equation (2-19) for 50km ≤ r ≤ (100 to 150km), which is quite restricting. The basis of

this limitation is the validity of the assumed 1/r2 relation for geometrical spreading of the

seismic energy used in deriving Equation (2-19).

Capacity:

Although Trifunac No.1 uses the same expression for Demand as Davis and Berrill

(1982), a different expression is given for Capacity, presented as Equation (2-29). This

expression is based on a different regression of the same data used by Davis and Berrill

(1982). 8.3

1 15.392.9

−

−

=N

Capacity (2-29)

where: N1 = Measured SPT N-value normalized to 1tsf.

A comparison of the Capacity expressions proposed by Davis and Berrill (1982) and

Trifunac No.1 is shown in Figure 2-13.

34

M

vor5.1

5.12

10'log σ

LiquefactionNo Liquefaction

Trifunac No.1Davis & Berrill (1982)

0

1

2

3

0 -1

1Log(N1)

2

Figure 2-13. Comparison of the boundaries proposed by Davis and Berrill (1982) and Trifunac No.1 to separate the data points representing liquefaction and no-liquefaction. (Adapted from Trifunac 1995).

Factor of Safety:

Substituting Equations (2-29) and (2-19) into the definition of factor of safety as the ratio

of Capacity to Demand, yields:

( )M

vorNFS 5.1

5.128.31

106120)'(15.3

⋅⋅⋅−

=σ

(2-30)

Trifunac No.2

Demand:

Alternative to using the Gutenberg-Richter energy relation, Trifunac No.2 computes the

energy of the earthquake motions at the site from the Fourier amplitude spectrum of the

35

strong motion accelerations. Empirical regression models may be employed to estimate

the Fourier amplitude spectrum from the seismological parameters of the design

earthquake and local site conditions (e.g., Trifunac 1976b). Because the energy is

computed from the motions at the site (as represented by the Fourier amplitude

spectrum), the geometrical spreading and inelastic attenuation of the seismic waves

occurring along the travel path from the source to the site are inherently taken into

account. Accordingly, this approach alleviates the distance restriction placed on Trifunac

No.1. The expression for Demand for Trifunac No.2 is given as Equation (2-31). 15.1'

−

=

enDemand voσ

(2-31a)

∫=π

πω

ωω200

2

50

)( dFen (2-31b)

where: F(ω) = Fourier amplitude spectrum of strong motion acceleration at the site (units not specified: m/sec??).

ω = Frequency (rad/sec).

σ’vo = Initial effective vertical stress at depth z.

Capacity:

By performing a regression analysis of Demands computed using Equation (2-31a) to

obtain a reasonable separation of data corresponding to liquefaction and no liquefaction,

the following expression for Capacity was obtained for Trifunac No.2: 5.2

1 95.1572

−

+

=N

Capacity (2-32)

Factor of Safety:

From Equations (2-32) and (2-31a), the factor of safety against liquefaction for Trifunac

No.2 is given as:

( )en

NFS vo

⋅×⋅+

= 6

5.15.21

10825.7)'(95.1 σ

(2-33)

36

Trifunac No.3

Demand:

In Trifunac No.3, the energy of the earthquake motions at the site is computed from the

peak ground velocity (vmax) and the duration of the strong motions (dur). This expression

is given as Equation (2-34). 1

2max

5.1'−

=

durvDemand voσ

(2-34)

where: dur = Duration of strong ground motion at the site (sec).

vmax = Peak ground velocity at the site (m/sec).


The duration of the strong ground motion for alluvium deposits can be estimated using

Equation (2-35):

sec2

1

51

5

7

0

50 ≥

+

=

durdur

durdur (2-35a)

where: dur0 = 8.94 – 3.86M + 0.57M 2 + 0.07r (2-35b)

r = Epicentral distance (km).

dur7 = [email protected]

The median peak horizontal component of the ground velocity for alluvium sites can be

estimated using Equation (2-36).

log (2-36a) ( ) sec)/(;8135.9201.0059.3log max2

max minvMMAv o −−+=

where:

rforr 75≤

−

(2-36b) log Ao = ; r is in km kmrkmforr

km

35075200

525.2

504.1

≤<

+−

+

37

Capacity:

By performing a regression analysis of Demands computed using Equation (2-34) to


the energy Capacity for Trifunac No.3 is: 4.3

1 95.02.87

−

−

=N

Capacity (2-37)

Factor of Safety:

The factor of safety against liquefaction is given by Equation (2-38):

( )durv

NFS vo

⋅⋅×

⋅−= 2

max6

5.14.31

10961.3)'(95.0 σ

(2-38)

Trifunac No.4


Fourier amplitude of the velocity motions at a period of 0.39sec, which is near the plateau

of the Fourier spectrum amplitudes for a broad range of magnitudes and site conditions.

This expression is given as Equation (2-39). 1

2

5.1'−

=

FVDemand voσ

(2-39)

where: FV = Fourier amplitude of the horizontal velocity motions at a period of 0.39sec (m).


Trifunac (1995) gives the following empirical relation to compute the median value of

FV for alluvium sites,

( ) 657.7000709.01206.0053.2loglog 2 −+−+= rMMAFV o (2-40)

where log Ao was given previously in Equation (2-36b), FV is in m and r is in km. For the

M2 term, if M > 8.51, M = 8.51 should be used.

38

Capacity:



the following expression for Capacity was obtained for Trifunac No.4: 6.3

1 05.10.538

−

+

=N

Capacity (2-41)

Factor of Safety against Liquefaction:


( )29

5.16.31

10774.6)'(05.1

FVN

FS vo

⋅×⋅+

=σ

(2-42)

Trifunac No.5

Demand:


peak ground velocity at the site, the small strain shear modulus, and the duration of

strong ground motion at the site. This expression is given as Equation (2-43). 1

maxmax

5.1'−

⋅⋅=

durGvDemand voσ

(2-43)

where: Gmax = Small strain shear modulus (kPa), Equation (2-12).

vmax = Peak ground velocity at the site (m/sec), Equation (2-36a).

dur = Duration of strong ground motion at the site (sec), Equation (2-35a).


Capacity:



the following expression for Capacity was obtained for Trifunac No.5:

39

1.2

1 05.0313.0

−

−

=N

Capacity (2-44)

Factor of Safety:


( )durGv

NFS vo

⋅⋅⋅⋅−

=maxmax

5.11.21

08723.0)'(05.0 σ

(2-45)

2.2.3.1.2 Arias intensity approaches

In the following procedures, Arias intensity (Ih) is used as the quantitative measure of

earthquake shaking intensity at the site. As given by Kayen and Mitchell (1997), Ih is

“the sum of the two component energy per unit weight stored in a population of

undamped linear oscillators evenly distributed in frequency, at the end of earthquake

shaking.” Arias intensity is calculated by integrating the acceleration time histories, as

given by Equation (2-46) (Arias 1970).

+⋅= ∫ ∫

dur dur

yxh dttadttag

I0 0

22 )()(2π

(2-46)

where: Ih = Arias intensity of the earthquake motion at the top of the soil profile (m/sec).

ax(t) = Horizontal acceleration time history in the x-direction (m/sec2).

ay(t) = Horizontal acceleration time history in the y-direction (m/sec2).

g = Acceleration due to gravity (m/sec2).

dur = Duration of earthquake shaking (sec).

2.2.3.1.2a Kayen and Mitchell (1997)

Kayen and Mitchell (1997) further developed the preliminary work of Egan and Rosidi

(1991) by establishing correlations for the occurrence and non-occurrence of liquefaction

40

as functions of the Arias intensity of the earthquake motion and penetration resistance of

the soil.

Demand:

To estimate the Arias intensity corresponding to the seismological parameters of the

design earthquake and the local site conditions, Kayen and Mitchell (1997) proposed the

following empirical relation:

(2-47)

soilsoftforM *2−−

Ih = (m/sec) r

alluviumforPrMlog4.3

63.0*log28.3 +−−

where: P = Probit (= ±1 for ± standard deviation; = 0 for mean).

r* = 22 ∆+r

∆ = Focal depth (km).

r = Distance from site to closest surface distance to the surface fault rupture at the focal depth (km).

r*, r, and ∆ are defined graphically in Figure 2-14.

Plan ViewFocus

site

Fault Rupture

r*

Elevation ViewFault Rupture

r*Focal Depth, ∆

site

Closest Surface Distance, r

Figure 2-14. Graphical illustration of the site-to-source distance r*. (Adapted from Kayen 1993).

41

Analogous to rd used in the stress-based procedure, Kayen (1993) and Kayen and

Mitchell (1998) developed a factor that relates the Arias intensity of surface motions to

the Arias intensity at depth in a profile. This relation is expressed by Equation (2-48).

bheqhb rII ⋅=, (2-48)

where: Ihb, eq = Arias intensity of the earthquake motions at a given depth in the profile (i.e., Demand).

rb = Arias intensity depth-of-burial reduction parameter.

Ih = Arias intensity at soil surface, Equation (2-47).

A statistical analysis of the variation of rb as a function of depth in a soil profile is shown

in Figure 2-15.

50

40

30

20

10

Mean(P=0)

SHAKE profile statistics

10.80.60.2 0.40 0

+1σ (P=+1)

-1σ (P=-1)

rb

Dep

th (f

t)

Figure 2-15. Results of a statistical analysis of the variation of rb as a function of depth in a soil profile. (Adapted from Kayen and Mitchell 1997).

42

Capacity:

Similar to the other procedures discussed previously, Kayen and Mitchell (1997) grouped

case histories according to the observance or non-observance of liquefaction. Using

Equation (2-48), the Demand imparted to soil was estimated for each. The computed

Demands were plotted as functions of the corresponding (N1,60cs)km, where (N1,60cs)km is

N1,60 corrected for fines content (FC). The correction factor fines content is given as

follows:

(2-49a) ( ) NNN kmcs ∆+= 60,160,1

; where FC is in % (2-49b) 3555( <<−=∆ FCforFC

%357

%%307)

%50

≥

≤

FCfor

N

FCfor

Equation (2-49) is an earlier version of the procedure used to correct for fines in the

stress-based approach. Either Equation (2-49) or Equation (2-6) should be acceptable for

use with the Kayen and Mitchell approach. The plot of Ihb,eq versus (N1,60cs)km is shown in

Figure 2-16, where the boundary separating the points representing liquefaction and no-

liquefaction defines the Capacity of the soil (Ihb,l).

43

Liquefaction ?? No Liquefaction Liquefaction

1

5

30250 0.1

(N1,60cs)km205 10 15

Ih < 0.10 m/s

LIQUEFACTION

NO LIQUEFACTION Clean-Sand Boundary

I hb (

m/s

ec)

Figure 2-16. Liquefaction curve proposed by Kayen and Mitchell (1997).

Kayen and Mitchell (1997) did not provide a mathematical expression for Capacity.

However, Equation (2-50) is proposed as a reasonable approximation of the boundary.

6227.0)(04162.0)(01132.0

)(001421.0)(10956.6)(10234.1)log(

60,160,1

60,160,160,1

2

34556,

−+−

+⋅−⋅= −−

kmkm

kmkmkmlhb

cscs

cscscs

NN

NNNI (2-50)

for 3 ≤ (N1,60cs)km ≤ 25

Factor of Safety:

The factor of safety against liquefaction is computed as follows:

eqhb

lhb

II

FS,

,= (2-51)

where Ihb,eq can be determined from Equation (2-48) and Ihb,l can be determined from

Equation (2-50) or Figure 2-16.

44

2.2.3.1.2b Running (1996)

Running (1996) proposed a hybrid liquefaction evaluation procedure combining facets of

the Kayen (1993) approach with critical state theory.

Demand:

Running uses the same definition for Demand as that proposed by Kayen and Mitchell

(1997) and given by Equation (2-48).

Capacity:

From critical state theory, Running derived an expression for the shear energy capacity of

soils at failure. This expression is given as Equation (2-52).

[ ][ ]

( ) ( ))/(

61'1000

'sinsincos3126.0

'cos'sin'3000 3

max

22

2

max

2

mJG

K

G

cU ovomo

sf−⋅

−−

−−⋅=

σ

φθθ

φφσ (2-52)

where: σ’mo = Initial mean effective confining stress (kPa), Equation (2-13).

c = Cohesive intercept of the soil strength envelope (kPa).

Gmax = Small strain shear modulus of the soil (kPa), Equation (2-12).

φ’ = Effective angle of internal friction of the soil, Equation (2-15).

θ = Lode angle, which is a function of the load path (for horizontal earthquake motions θ = 0.5210 rad).

Using the same case history data set as used by Kayen and Mitchell (1997), Running

plotted Ihb,eq as a function of sfU and drew a boundary giving a reasonable separation of

points representing liquefaction and no-liquefaction, as shown in Figure 2-17.

45

sfU

Liq.No Liq.Possible

No Liquefaction

Liquefaction

16014012080 100 (J/m3)

60402000

3

2

I hb

(m/s

)

1

Figure 2-17. Liquefaction curve proposed by Running (1996).

Running (1996) did not provide a mathematical expression for his proposed boundary,

but Equation (2-53) provides a reasonable approximation. Equation (2-53) allows the

shear energy capacity of the soil to be expressed in terms of Arias intensity required to

induce liquefaction (Ihb,l = Capacity).

28.0005.0, += sflhb UI (2-53)

where Ihb,l is in m/sec, and sfU is in J/m3.

Factor of Safety:

The factor of safety against liquefaction is computed as follows:

eqhb

lhb

II

FS,

,= (2-54)

where Ihb,eq can be determined from Equation (2-48) and Ihb,l can be determined from

Equation (2-53) or Figure 2-17.

46

2.2.3.2 Procedures developed from laboratory data

In all the energy-based procedures presented above, the Capacity of the soil was

determined from the analyzing earthquake case histories. Alternatively, the two

procedures discussed in this section rely on laboratory test data for determining the

Capacity of the soil.

2.2.3.2.1 Alkhatib (1994)

Demand:

Alkhatib (1994) quantified Demand by the dimensionless parameter Normalized

Maximum Energy (NME), which is computed by integrating the stress-strain time

histories at depth in a soil profile. From a series of site response analyses using scaled,

western United States acceleration time histories, Alkhatib proposed the relationship

shown in Figure 2-18 between the maximum acceleration (amax) of the acceleration time

history and NME.

Alkhatib (1994) did not provide a mathematical expression for the correlation shown in

Figure 2-18, but Equation (2-55) provides a reasonable approximation.

207.1)log(933.2)log( max −⋅= aNME (2-55)

where amax is in g, and NME is dimensionless.

Figure 2-18. Correlation of the dimensionless parameter Normalized Maximum Energy (NME) and maximum acceleration (amax). (Adapted from Alkhatib 1994).

0.010.000001 0.10.010.0010.00010.00001

Normalized Maximum Energy

1

Max

imum

Acc

eler

atio

n, g

0.1

47

Capacity:

From a series of cyclic triaxial tests conducted on Monterey sand subjected to earthquake

type load functions, Alkhatib proposed the correlation shown in Figure 2-19 between the

relative density (Dr) of the sand and energy ratio (ER). ER is the ratio of the energy

computed by integrating the stress-strain hysteresis loops to the initial effective confining

stress and was used to quantify the Capacity of the soil.

210

310

1008040 60Relative Density, %

200

Ener

gy R

atio

Figure 2-19. Correlation between relative density (Dr) and energy ratio (ER). (Adapted from Alkhatib 1994).

Alkhatib (1994) did not provide a mathematical expression for the correlation shown in

Figure 2-19, but Equation (2-56) provides a reasonable approximation.

6291.3)log(01747.0)log( −⋅= rDER (2-56)

where Dr is in %, and ER is dimensionless.

The following expressions can be used to relate penetration resistance to Dr.

( )vor

ND'107.1 σ+

= (Gibbs and Holtz 1957) (2-57)

where: Dr = Relative density in decimal.

N = Measured SPT N-value.

σ’vo = Initial vertical effective confining stress (psi).

48

( ) 5.060,115 NDr ⋅= ; 30% ≤ Dr ≤ 90% (Skempton 1986) (2-58)

where: Dr = Relative density (%).

N1,60 = Corrected SPT N-value.

Factor of Safety:

Using the definitions given above for Capacity and Demand, the author derived the

following expression for the factor of safety against liquefaction:

FS = alog{0.01747⋅ log(Dr) – 2.933⋅ log(amax) – 2.4221} (2-59)

2.2.3.2.2 Liang (1995)

Extensive research on energy-based liquefaction evaluations procedures has been

conducted at Case Western Reserve University under the direction of Professor J.L.

Figueroa: Figueroa, 1990; Dahisaria, 1991; Figueroa and Dahisaria, 1991; Figueroa,

1993; Figueroa et al., 1994a; Figueroa et al., 1994b; Figueroa et al., 1995; Liang, 1995;

Liang et al., 1995a; Liang et al., 1995b; Kern, 1996; Kusky, 1996; Figueroa et al., 1997a;

Figueroa et al., 1997b; Rokoff, 1999; Figueroa et al., 1999. The most complete

presentation of this research is given in Liang (1995). The proposed liquefaction

evaluation procedure requires both laboratory testing of soil and site response analyses to

be performed. For the purposes of the present study, it is desired to develop an energy-

based procedure for designing remedial ground densification programs using simplified

procedures (i.e., that do not require site response analyses). Also, obtaining truly

undisturbed samples from the field is difficult. Accordingly, the author desires a

procedure that correlates energy Capacity to an in-situ field test (e.g., SPT), as opposed to

having to perform laboratory tests on undisturbed samples. Based on these conditions,

the work of Figueroa and his associates is only of general interest but not considered for

possible use in the design of remedial densification programs.

49

2.2.3.3 Other Approaches

2.2.3.3.1 Mostaghel and Habibaghi (1978, 1979)

Mostaghel and Habibaghi (1978, 1979) is the earliest energy-based liquefaction

evaluation procedure known to the author. In this procedure, an energy Capacity

relationship is assumed. As discussed in Chapter 4, Section 4.2.2 and 4.3, the assumed

Capacity relation is contrary to trends observed in the laboratory.

2.2.3.3.2 Moroto and Tanoue (1989)

The liquefaction evaluation procedure proposed by Moroto and Tanoue (1989) and

Moroto (1995) is similar to Davis and Berrill (1982) presented in Section 2.2.3.1.1a.

2.2.3.3.3 Ostadan, Deng, and Arango (1996, 1998)

Ostadan et al. (1996, 1998) analyzed 150 cyclic laboratory tests performed on high

quality undisturbed and reconstituted samples. They found that the strain-energy is a

function of the relative density for clean sands and a function of the fines content and

confining pressure for silty sands. Capacity curves analogous to that developed by

Alkhatib (1994) shown in Figure 2-19 are presented. However, as opposed to Alkhatib

(1994), Ostadan et al. do not normalize the strain-energy by the initial effective confining

stress.

2.3 Overview of the Parameter Study

2.3.1 Objective

A parameter study was performed using twelve of the procedures presented above:

Stress-based procedure, Strain-based procedure, Davis and Berrill (1982), Berrill and

Davis (1985), Law, Cao, and He (1990), Trifunac No.1, Trifunac No.3, Trifunac No.4,

Trifunac No.5, Kayen and Mitchell (1997), Running (1996), and Alkhatib (1994). The

two remaining procedures (i.e., Trifunac No.2 and Liang 1995) were not included

because Trifunac No.2 requires as an input the Fourier amplitude spectrum of the strong

motion acceleration and Liang (1995) requires both laboratory testing and a site response

analysis to be performed. For each of the procedures examined, the factors of safety

50

versus depth (FS-profile) were computed for simple soil profiles. The following profiles

were examined:

• 100ft thick profile of clean sand having N1,60 = 15blws/ft for all depths. The

depth to the groundwater table (gwt) is 0ft, 11ft, and 25ft.

• 100ft thick profile of clean sand with the groundwater table at a depth of

approximately 11ft. The profile is assumed to have constant N1,60 with depth

equal to 5, 10, and 15blws/ft.

In the first case, the profile is assumed to have a constant N1,60 with depth and the

elevation of the water table is varied. This scenario was selected because for a soil

having a given fabric, N1,60 correlates well with relative density (Dr) and the variation of

the water table give different effective confining stress profiles. Numerous laboratory

studies have examined the influence of effective confining stress on similar samples (i.e.,

same fabric and Dr). Accordingly, trends in the predicted FS-profiles can be evaluated

against the observed behavior from laboratory studies, which are incorporated in stress-

based procedure. In the second case, the elevation of the water table is held constant, but

the penetration resistance is varied. Changes in N1,60 values are often used to measure the

effectiveness of various remediation techniques, wherein increases in N1,60 reflects an

increases in the FS against liquefaction. Accordingly, the sensitivity of predicted FS-

profiles to changes in N1,60 is of practical interest.

The profiles were assumed to be subjected to a M7.5 earthquake at a distance of 60km,

where this distance is defined according to the requirements of each procedure (e.g.,

epicentral distance, center of energy release, etc.). Using the acceleration attenuation

relation proposed by Trifunac (1976a), given below as Equation (2-60), amax is 0.13g.

( ) )sec/();'log(loglog 2maxmax cminaaAMa o −+= (2-60a)

where:

log(a’) = 8.4466.1

5.78.4186.0789.1768.55.7186.0975.1163.7

2

2

≤<<+−

≥+−

MforMforMM

MforMM

(2-60b)

51

This expression gives the median peak horizontal acceleration for alluvium sites. An

expression for Ao was given previously as Equation (2-36b). All other seismological

parameters required by the various procedures (e.g., duration of strong motion, Arias

intensity) were determined using the empirical relations presented along with the

respective procedures.

2.3.2 Discussion of expected trends

The stress-based procedure was included in the parameter study to use as a benchmark

for assessing the other procedures. The stress-based procedure has been thoroughly

scrutinized, tested, and continually revised based on new laboratory and field

observations. Accordingly, the trends in the predicted FS are assumed to be correct. This

is not to say that the stress-based procedure is 100% accurate in predicting liquefaction,

but rather the variation in predicted factor of safeties tend in the correct direction as

parameters known to influence liquefaction are changed.

The strain-based procedure was also included in this study. Although this procedure only

provides the FS against gross slippage across particle contact surfaces, and not the FS

against liquefaction, it provides information as to whether excess pore pressures will be

generated. This information may be used as a lower bound for expected FS (i.e., if gross

slippage across the contact areas is not predicted, all the procedures should predict FS >

1).

Because the absolute values of the predicted FS will be unique to each procedure (e.g., a

FS = 1.2 for one procedure does not mean the same thing as it does for another

procedure), the focus of the parameter study is to examine the trends in the FS-profiles as

influenced by the variation of parameters. One of the things examined is the variation in

critical depth (i.e., the depth below the gwt corresponding to the lowest FS).

The FS-profiles for the stress- and strain-based procedures are shown in Figures 2-20 and

2-21. From examination of these figures, the following observations are made:

• The critical depth is unaffected by the changes in N1,60.

52

• The critical depth shifts downward as the elevation of the ground water table is

lowered.

• The FS increases as N1,60 increases and as the gwt is lowered. Both of these

trends correspond to the observed laboratory behavior that the Capacity of the

soil increases with increasing relative density and increasing effective confining

stress.

The strain-based procedure predicts a high FS towards the surface of the profile. This is

believed to be due to limitations in the empirical correlation for Gmax, (G/Gmax), and φ’ vs.

N1,60 at low effective confining stresses and is not the expected behavior of the soil. The

interpretation given to the results from the strain-based procedure is that excess pore

pressures will be generated for the entire depths of all the profiles.

53

NCEER (1997)

a)

3

b)

20

25

30

15

10

5

0

N1,60 = 15 10 5

0 1 2 3

Approximate Critical Depth

2 3

1

2

N1,60 = 15

20

25

30 0

3

1

1

2 3

2

1

Approximate Critical Depths

0

5

10

15

Dep

th (m

)

Factor of Safety Factor of Safety

Figure 2-20. FS-profiles for the stress-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

54

Dobry et al. (1982)

Critical Depths difficult to determine

b)


1

2

3

32

1

0

5

10

15

25

20

N1,60 = 15

3

2

1a)


2 3 30 01

5 10

N1,60 = 15

0

5

10

15

30 0

25

20

Dep

th (m

)

1 2 3 Factor of Safety Factor of Safety

Figure 2-21. FS-profiles for the strain-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

55

2.3.3 Results of parameter study and summary of observed trends

The results from Davis and Berrill (1982), Berrill and Davis (1985), Law, Cao, and He

(1990), Trifunac No.1, Trifunac No.3, Trifunac No.4, Trifunac No.5, Kayen and Mitchell

(1997), Running (1996), and Alkhatib (1994) are shown in Figures 2-22 to 2-31,

respectively. From examination of these figures, the following observations are made:

• Similar to the stress- and strain-based procedures, none of the approaches predict

a variation in the critical depth due to changes in N1,60, with several of the

procedures not showing a clearly identifiable critical depth.

• Unlike the stress- and strain-based procedures, none of the approaches predict a

variation in the critical depth due to changes in the water table (i.e., the critical

depth always corresponds to the depth of the water table).

• Davis and Berrill (1982), Berrill and Davis (1985), Trifunac No.1, Trifunac No.3,

Trifunac No.4, Trifunac No.5, and Running (1996) predict a continually

increasing FS with depth.

• Law, Cao, and He (1990) and Alkhatib (1994) show no variation of FS with

depth. Based on the formulations of these procedures, the predicted FS probably

correspond to an initial vertical effective confining stress of 100kPa. Depending

on the stratigraphy of the profile and elevation of the gwt, basing the FS on

100kPa may be under or overly conservative.

• The predicted FS-profiles by Davis and Berrill (1982), Trifunac No.1, Trifunac

No.3, and Trifunac No.4 are overly sensitive to variations in N1,60. Given the

variability of N1,60 values in actual “uniform” layers or profiles (e.g., Elton and

Hadj-Hamou 1990), the predicted FS using these procedures are unreliable.

• Running (1996) shows a decrease in FS in response to an increase in N1,60. This is

contrary to observed behavior in the laboratory that Capacity increases with

increasing Dr.

• The FS-profiles predicted by Law, Cao, and He (1990), Kayen and Mitchell

(1997), and Alkhatib (1994) are insensitive to changes in the elevation of the

water table, assuming a constant N1,60. This is contrary to observed behavior in

the laboratory that Capacity varies as a function of effective confining stress.

56

Davis and Berrill (1982)

b)


1

2

3

0

5

10

15

25

20

3 1 2

N1,60 = 15

3

2

1a)


5 10 N1,60 = 15

0

5

10

15

25

20

40 60 10 20 30 Factor of Safety

50 70 30

00 30

Dep

th (m

)

40 60 50 70 10 20 30

Factor of Safety

Figure 2-22. FS-profiles for Davis and Berrill (1982): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

57

Berrill and Davis (1985)


b)

3

2

1

1 2 3

1

2

3

N1,60 = 15

20

25

30

15

10

5

0

0 4 61 2 3 5 7 8

a)

7 851 30

0

20

25

15

10

5

0

N1,60 = 15


5 10

Factor of Safety32 64

Dep

th (m

)

Factor of Safety

Figure 2-23. FS-profiles for Berrill and Davis (1985): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

58

Law, Cao, and He (1990)

0.0

3

1.0 1.50.5 Factor of Safety

2.0

b)

FS constant Critical Depths

can’t be determined

2.0

1 2

10

5

0

15

30 0.0 1.0 1.5 0.5

25

20

N1,60 = 15

3

2

1a)



N1,60 = 15 10 5

5

0

10

15

30

25

20

Dep

th (m

)

Factor of Safety

Figure 2-24. FS-profiles for Law, Cao, and He (1990): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

59

Trifunac No.1

0

5

10

15

25

20

1

2

3

N1,60 = 15

Factor of Safety

N1,60 = 15 10 5


Factor of Safety150 3000

30


20

25

30

15

10

5

0

0 150

1

2

3

300

1 2 3

Dep

th (m

)

Figure 2-25. FS-profiles for Trifunac No.1: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

60

Trifunac No.3

b)


1

2

3

40 50 30 2010

2 3

0

5

10

15

30 0

25

20

N1,60 = 15

3

2

1

1

a) Approximate Critical Depth

5040

5 10 N1,60 = 15

0

5

10

15

30 0 10 20 30

25

20

Dep

th (m

)



61

Trifunac No.4

b)


1

2

3

31 2

0

5

10

15

30 0

25

20

N1,60 = 15

3

2

1a)


504020 30

5 10 N1,60 = 15

0

5

10

15

30 0 10

25

20

Dep

th (m

)

10 20 30 40 50



62

Trifunac No.5

b)


1

2

3

1 2 3 4 5

3 1 2

0

5

10

15

25

20

N1,60 = 15

3

2

1a)


5 10 N1,60 = 15

0

5

10

15

30 0 1 2 3 4 5 30 0

25

20

Dep

th (m

)



63

Kayen and Mitchell (1997)

b)


1

2

3

1 2 3

0

5

10

15

25

20

N1,60 = 15

3

2

1a)


5 10

N1,60 = 15

0

5

10

15

25

20

Factor of Safety1 2 3

30 0

30 0

Dep

th (m

)

Factor of Safety 2 3 1

Figure 2-29. FS-profiles for Kayen and Mitchell (1997): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

64

Running (1996)

a) b)


1

2

3

2 3 1

0

5

10

15

30 0

25

20

Factor of Safety 8 12 16 204

N1,60 = 15

3

2

1


2016

510N1,60 = 15

0

5

10

15

30 0

25

20

Factor of Safety4 8 12

Dep

th (m

)

Figure 2-30. FS-profiles for Running (1996): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

65

Alkhatib (1994)

b)



20 25 30155 10

1 2 3

0

5

10

15

30 0

25

20

N1,60 = 15

3

2

1a)



302510 15 20

5 10 N1,60 = 15

0

5

10

15

30 0 5

25

20

Dep

th (m

)


Figure 2-31. FS-profiles for Alkhatib (1994): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

66

2.3.4 Commentary on procedures

A detailed commentary is not given for each procedure, rather the results of the

parameter study generally speak for themselves. However, general commentary is given

for several of the procedures, and the Arias intensity procedure proposed by Kayen and

Mitchell (1997) is examined in depth.

2.3.4.1 Gutenberg-Richter approaches

One of the assumptions central to Davis and Berrill (1982), Berrill and Davis (1985),

Trifunac No.1, Trifunac No.2, Trifunac No.3, Trifunac No.4, and Trifunac No.5 is that

energy dissipation in the soil due to material damping is proportional to 1/(σ’vo)0.5. The

basis for this assumption is the laboratory study by Hardin (1965). However, Hardin’s

study was conducted on dry sands (i.e., the effective confining pressure remains constant

for the duration of the test). During the process of liquefaction, the effective confining

stress is continually changing. Accordingly, it would seem more reasonable to interpret

Hardin’s results as energy dissipation due to material damping is proportional to

1/(σ’v)0.5, where σ’v is the effective overburden stress at a specific time, not the initial

effective overburden stress.

Davis and Berrill (1982), Trifunac No.1, Trifunac No.2, Trifunac No.3, Trifunac No.4,

and Trifunac No.5 all assume a linear relationship between dissipated energy and excess

pore pressure generation. As discussed in more detail in Chapter 4 and as revised in

Berrill and Davis (1985), excess pore pressure generation varies as a function of the

square root of dissipated energy.

It is unclear as to whether the amax value used in Alkhatib (1994) should be for the rock

outcrop or the soil surface. The correlation shown in Figure 2-18 between amax and NME

appears to have been developed from site response analyses using scaled acceleration

time histories. Based on this, it is assumed that amax corresponds to that for a rock

outcrop. However, from examining the case histories presented in Alkhatib (1994), it

appears that soil surface amax values were used. In computing the FS-profile shown in

Figure 2-31, it was assumed amax corresponds to the soil surface. Furthermore, the

67

correlation shown in Figure 2-18 between amax and NME was developed from total stress

site response analyses (i.e., SHAKE), and the Capacity correlation shown in Figure 2-19

was based on laboratory tests, which are inherently effective stress. It cannot be expected

that the dissipated energies computed from total stress analyses will correspond to those

from effective stress analyses. A more detailed discussion on this subject is presented in

Chapter 5, Section 5.6.

2.3.4.2 Arias intensity approaches

Because Kayen and Mitchell’s Arias intensity approach was considered the most

promising for remedial ground densification design, it is reviewed more in depth than the

others. From the parameter study, it may be observed that unlike the results from stress-

based procedure shown in Figure 2-20, the FS profiles for Kayen and Mitchell’s Arias

intensity approach do not vary in response to changes in the groundwater table (Figure 2-

29). As stated previously, N1,60 correlates to the relative density (Dr) of the soil (e.g.,

Equation (2-58)). Therefore, the scenario of a varying water table with constant N1,60 is

analogous to laboratory specimens prepared to the same Dr, but subjected to different

effective confining pressures. In order for N1,60 to remain constant for all elevations of

the gwt, it is inherently implied that N60 (i.e., the SPT penetration resistance normalized

to 60% of the hammer energy) will change, which may be unrealistic. Although this

parameter study is academic in nature, it highlights the Arias intensity approach’s failure

to properly account for changes in effective confining pressure, as well as several of the

other energy-based liquefaction evaluation procedures.

In addition to the observation from the parameter study discussed above, the following

simple examination provides further insight into Kayen and Mitchell’s Arias intensity

procedure. Consider a liquefiable sand layer lying below a stiff desiccated crust, such as

shown in Figure 2-32a. Due to the stiffness of the crust, it can be assumed that it will

respond as a rigid body during an earthquake (i.e., the accelerations experienced at the

surface of the crust will be the same as the accelerations experienced at its base). The

earthquake induced stresses acting on a soil element in the liquefiable layer can be

68

modeled in the laboratory using a cyclic simple shear device, as represented in Figure 2-

32b.

AW

vo =σ

τ(t)

a(t)

z

τ(t)

W Stiff Desiccated Crust

Liquefiable Layer

b)

a)

Figure 2-32. a) Soil profile having a liquefiable layer lying below a stiff desiccated crust. b) Duplication of in-situ stresses using simple shear device on a soil sample from the liquefiable layer.

From Newton’s second law, it can be shown that the acceleration time history of the rigid

desiccated crust, a(t), and the shear stress time history, τ(t), acting on the top of the

liquefiable layer are related according to Equation (2-61).

vogtat στ ⋅=)()( (2-61)

A simple rearrangement yields:

vo

gttaσ

τ ⋅=

)()( (2-62)

Substitution of Equation (2-62) into the expression for Arias intensity, given previously

as Equation (2-46) and repeated below for a single component of motion, allows the

Arias intensity to be expressed as a function of shear stress time history.

∫⋅=dur

h dttag

I0

2 )(2π (2-63)

∫⋅=dur

voh dttgI

0

22 )(

2τ

σπ (2-64)

69

Using Equation (2-64), the Arias intensity required to cause liquefaction (Ih,l) can be

computed for a laboratory specimen subjected to cyclic simple shear. Assume that the

laboratory specimen is subjected to a sinusoidal loading having amplitude τmax and

frequency f. If the sample liquefies in n cycles of loading, the Arias intensity required to

cause liquefaction is given by Equation (2-65).

f

gnIvo

lh

2max

2, 4τ

σπ

⋅= (2-65)

This expression is revealing in two ways. First, Ih,l is independent of effective confining

stress. This verifies the results of the parameter study, which showed the procedure is

insensitive to changes in the elevation of the groundwater table. Second, Ih,l (i.e.,

Capacity) is a function of the frequency of loading. However, laboratory studies have

shown that for a given amplitude load, the number of cycles required to induce

liquefaction is relatively independent of the frequency of the applied load (e.g., Arango

1994).

The obvious question becomes “why does the field data shown in Figure 2-16 show a

reasonable separation of points representing liquefaction and no-liquefaction?” The

answer to this can be understood by reformulating the expression for Arias intensity. By

observation, Equation (2-63) is similar in form to the equation for root mean square

acceleration (arms) given as Equation (2-66).

∫⋅=dur

rms dttadur

a0

2 )(1 (2-66)

where: arms = Root mean square acceleration.

a(t) = Acceleration time history.

dur = Duration of earthquake motion.

A simple rearrangement yields:

2

0

2 )( rms

duradurdtta ⋅=∫ (2-67)

70

Substituting Equation (2-67) into Equation (2-63) and multiplying by rb allows the Arias

intensity of the earthquake motions at depth in a soil profile (Ihb,eq) to be expressed as a

function of the duration of the earthquake shaking (dur) and arms:

brmseqhb radurg

I ⋅⋅⋅= 2, 2

π (2-68)

Seismological studies have shown (Hanks and McGuire 1981): 32 max ≤≤rmsa

a

For this illustration, it is assumed max31 aarms ⋅≈ . Substitution of this relation into

Equation (2-68) yields:

beqhb radurg

I ⋅⋅⋅= 2max,

056.0 π (2-69)

Dobry et al. (1978) proposed the following relationship between the duration of strong

motion (dur) and magnitude (M).

dur = alog(0.432M – 1.83) (2-70)

By empirical observation, the duration of strong motion may also be approximated as: 2

125

⋅≈

−StokoeAndrusMSFdur (2-71)

MSFAndrus-Stokoe is the magnitude scaling factor relation proposed by Andrus and Stokoe

(Youd and Noble 1997) for use in the stress-based liquefaction evaluation procedure.

This relation was presented previously as Equation (2-5b) and is repeated below. 3.3

5.7

−

−

=

MMSF StokoeAndrus (2-5b)

A comparison of Equations (2-70) and (2-71) is shown in Figure 2-33.

71

50

Equation (2-71)Equation (2-70)

40

Dur

atio

n (s

ec)

30

20

10

0 5 6 7 8

Magnitude

Figure 2-33. Comparison of expressions for duration of strong ground motion as a function of magnitude.

Substituting Equation (2-71) into Equation (2-69) yields: 22

max,

14.1

⋅⋅

⋅=

−StokoeAndrusbeqhb MSF

rg

agI π (2-72)

In reviewing the liquefaction case histories used by Kayen and Mitchell (1997) in

developing their liquefaction chart, a reasonable average of the of the total vertical

stresses at the critical depths may be assumed: σvo = 200kPa. Based on this assumption,

1.4πg ≈ 0.00255⋅(0.65⋅σvo)2, where g has units of (m/sec2). Furthermore, as shown in

Figure 2-34, rb ≈ rd2, where rb and rd are the "depth reduction factors" for the Arias

intensity and stress-based liquefaction evaluation procedures, respectively. The curves

shown in this figure represent the means, about which considerable scatter exists.

Although the curves in Figure 2-34 do not lie on top of each other, there is considerable

overlap in the scatter corresponding to each of the curves; scatter not shown in figure.

72

2

2(rd)rb

50

40

30

20

10

10.80.60.40.20 0

rb and (rd)

Dep

th (f

t)

Figure 2-34. Comparison of the “depth reduction factors” rb for the Arias Intensity approach and (rd) from the stress-based approach. 2

Using the above approximations, the Arias intensity of the earthquake motions at depth in

a soil profile (Ihb,eq) can be written: 2

max,

165.000255.0

⋅⋅⋅⋅⋅≈

−StokoeAndrusdvoeqhb MSF

rg

aI σ (2-73)

It may be observed that the expression inside the parentheses is the amplitude of the

earthquake-induced cyclic stress adjusted to a M7.5 event (refer to Equations (2-1) and

(2-4)).

( 25.7, 00255.0 MeqhbI τ⋅≈ )

)

(2-74)

For the case of liquefaction where FS =1.0 (i.e., τM7.5 = CRR⋅σ’vo), Equation (2-74)

becomes:

( 2, '00255.0 volhb CRRI σ⋅⋅≈ (2-75)

73

Again, in reviewing the liquefaction case histories used by Kayen and Mitchell (1997) in

developing their liquefaction chart, a reasonable average of the of the initial effective

vertical effective stresses at the critical depths may be assumed: σ’vo = 100kPa. Based on

this assumption, CRR and Ihb,l are related as:

5,lhbI

CRR ≈ (2-76)

A comparison of the CRR curve, given previously as Equation (2-7), and 0.2⋅(Ihb,l)0.5,

where Ihb,l is quantified by Equation (2-50), is shown in Figure 2-35.

5,lhbI

No LiquefactionLiquefaction

CRR

0.6

0.5

0.4

0.3

0.2

0.1

0 0.0

CRR

5 10 15 20 25 30

N1,60cs

Figure 2-35. Comparison of Arias intensity liquefaction curve and CRR.

Attention is now returned to the question “why does the field data shown in Figure 2-16

show a reasonable separation of points representing liquefaction and no-liquefaction?”

The Arias intensity approach proposed by Kayen and Mitchell (1997) is an alternate

formulation of the stress-based procedure. However, this alternate form introduces two

errors: a relative insensitivity of the soil Capacity to effective confining stress and an

74

erroneous sensitivity to frequency. The magnitudes of these errors are probably small for

the case histories used to develop the liquefaction chart. This is because for the cases

analyzed there is not a large variation in the initial vertical effective stresses at the critical

depth for liquefaction, and the frequency content of western US earthquakes probably

does not vary that much. However, it should be expected that the induced errors may be

significant if the procedure is used to perform liquefaction evaluations where conditions

differ significantly from those of the data base used in deriving the procedure (e.g., the

upstream toe of a dam located in the central or eastern US).

Finally, it needs to be emphasized that Equation (2-73) and all subsequently derived

expressions are only valid around the critical depth of liquefaction (i.e., σvo ≈ 200kPa)

and do not apply to other depths. This limitation can be understood by comparing the

Ihb,eq values computed using Equations (2-72) and (2-73) at the surface of a profile. The

Ihb,eq value computed using Equation (2-72) will be non-zero, while the value computed

using Equation (2-73) will equal zero. This dichotomy results from the surface of the soil

profile being a zero shear stress boundary (i.e., τM7.5 = 0) but the Arias intensity of the

surface motion being non-zero and typically greater than the Arias intensities of the

motions at depth. However, because the data points used by Kayen and Mitchell to

derive their Capacity curve correspond to the critical depths of liquefaction, the depth

limitation of Equation (2-73) does not detract from the hypothesis: the reasonable

separation of points representing liquefaction and no-liquefaction in Figure 2-16 is

because the Arias intensity procedure is an alternate form of the stress-based procedure.

2.4 Conclusions

A significant amount of work has been done in energy-based evaluations of liquefiable

soils, and several pioneering evaluation procedures have been developed (e.g., Davis and

Berrill 1982 and Kayen and Mitchell 1997). Although a lot can be learned from the

procedures reviewed, none are considered comprehensive enough in their present state of

development for the use in remedial ground densification design. This does not preclude

the use of the procedures for performing earthquake liquefaction evaluations, or upon

further development, possible use in remedial densification design.

75

Appendix 2a: Normalization of measured SPT N-values

This appendix outlines the NCEER(1997) and Youd et al. (2001) recommended

normalization factors for measured SPT N-values for overburden pressure, hammer

energy, borehole diameter, rod length, and sampling method. Measured SPT N-values

should be normalized as follows:

sRBEN CCCCCNN ⋅⋅⋅⋅⋅=60,1 (2a-1)

where: N = Measured SPT N-value.

CN = Normalization factor for overburden pressure.

CE = Normalization factor for hammer energy.

CB = Normalization factor for borehole diameter.

CR = Normalization factor for rod length.

CS = Normalization factor for sampler method.

Values for the different normalization factors in Equation (2a-1) are listed in the

following table.

76

Table 2a-1. SPT Normalization and Correction Factors.Factor Test Variable Term Correction

Overburden Pressure1 CN

5.0

'

vo

aPσ

1.15

1.05

1.0

200mm

150mm

65mm to 115mm CB Borehole Diameter

Automatic-Trip Donut-Type Hammer

Safety Hammer

Donut Hammer

0.8 to 1.3

0.7 to 1.2

0.5 to 1.0 CE Energy Ratio

3m to 4m

10m to 30m6m to 10m

4m to 6m

< 3m

1.0 0.95

0.85 0.8 0.75 CR Rod Length2

Standard Sampler

Sampler without Liners

1.0

0.1 to 1.3

CS Sampling Method

CN ≤ 1.7

(Modified from Skempton 1986 and Robertson and Wride 1998)

1The effective overburden pressure should be the value corresponding to that at the time of drilling and testing. A higher groundwater level might be assumed for conservatism in the liquefaction resistance calculations. 2Rod corrections were not applied for lengths greater than 3m in the formulation of the simplified procedure; therefore, corrections are not required in applying the procedure for lengths greater than 3m.

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Chapter 2. Review of Liquefaction Evaluation Procedures · placed on energy-based procedures. The...

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Transcript of Chapter 2. Review of Liquefaction Evaluation Procedures · placed on energy-based procedures. The...