Chapter 5. Proposed Energy-Based Liquefaction Evaluation Procedure

5.1 Introduction

A critique of existing energy-based liquefaction evaluation procedures for possible use in

remedial ground densification design was presented in Chapter 2. Unfortunately, the

conclusion of the review was that none of the procedures were adequate for this purpose.

In this chapter, a new energy-based liquefaction evaluation procedure is proposed. As

will be shown, this new approach is a conceptual and mathematical unification of the

stress- and strain-based procedures, both of which were outlined in Chapter 2. The

Capacity curve for the proposed procedure was developed from the analyses of

earthquake case histories and compared with normalized data from numerous cyclic

triaxial tests. Because the Capacity curve provides the critical link between earthquake

liquefaction evaluation and remedial ground densification design, the same parameter

study used to critique the existing energy-based liquefaction evaluation procedures in

Chapter 2 (Section 2.3) is used to evaluate the proposed procedure.

As part of the proposed energy-based liquefaction evaluation procedure, a new

correlation was developed relating the number of equivalent cycles, earthquake

magnitude, and site-to-source distance. Aside from the focus of this thesis, in Appendix

5c, this new correlation is used as the basis for a new set of magnitude scaling factors

(MSF) for use in the stress-based liquefaction evaluation procedure. As opposed to the

current NCEER (1997) recommended MSF, presented in Chapter 2 (Figure 2-2, Section

2.2.1), the new set of MSF are functions of both earthquake magnitude and site-to-source

distance.

5.2 Mathematical Expression for Computing Dissipated Energy

An expression was presented in Chapter 3 defining the damping ratio (D) in terms of the

energy dissipated per unit volume of material in a single cycle of loading (Equation (3-2),

Section 3.3.2). This expression and related figure are repeated below as Equation (5-1)

and Figure 5-1.

129

γτ21

=W

τ

γ ∆W1

Gτ

γ

Figure 5-1. Quantities used in defining damping ratio (D).

WWD 1

41 ∆

⋅=π

(5-1)

where: D = Damping ratio.

∆W1 = Dissipated energy per unit volume of material in one cycle of loading.

W = Maximum energy stored in an elastic material having the same G as the visco-elastic material.

Simple rearrangement of Equation (5-1) and substitution of the definition of W given in

Figure 5-1 yields:

τγπDW 21 =∆ (5-2)

Using this expression in conjunction with the equivalent-number-of-cycles concept (e.g.,

Annaki and Lee 1977), the energy dissipated in the soil for the entire duration of the

earthquake motion can be estimated as:

eqveqv NDNWW ×=×∆=∆ )2(1 τγπ (5-3)

where: ∆W = Dissipated energy per unit volume for entire earthquake motion.

∆W1 = Dissipated energy per unit volume for one equivalent cycle of earthquake motion.

Neqv = Number of equivalent cycles in the earthquake motion.

D = Damping ratio.

τ = Shear stress.

130

γ = Shear strain.

By appropriately selecting τ and γ in Equation (5-3), the stress- and strain-based

liquefaction evaluation procedures can be unified. Expressing the shear strain (γ) in

terms of the shear stress (τ), i.e., γ = τ /G, Equation (5-3) can be re-written as:

eqvNGDW ×

=

22 τπ∆ (5-4)

The determination of τ, G, D, and Neqv will be discussed in order.

5.2.1 Determination of τ

As presented in Section 2.2.1, alternative to site response analyses, the average or

effective amplitude of the earthquake-induced shear stress (τave) at depth z in a soil profile

can be determined by the following expression:

dvoave rg

a⋅⋅⋅= στ max65.0 (5-5)

where: τave = Effective amplitude of the earthquake-induced shear stress.

amax = Maximum soil surface acceleration.

g = Acceleration due to gravity.

σvo = Total vertical stress at depth z.

rd = Dimensionless parameter that accounts for the stress reduction due to soil column deformability.

5.2.2 Determination of G and D

Because the damping ratio (D) and shear modulus (G) are functions of the induced shear-

strain, the iterative procedure used in the strain-based approach (Section 2.2.2) is

employed to determine the shear strain (γ) corresponding to τave. Starting with the

relation between stress and strain γ = τ /G, Dobry et al. (1982) derived the following

expression for determining the earthquake-induced shear strain at depth in a soil deposit:

131

γ

σγ

⋅

⋅⋅⋅=

maxmax

max65.0

GGG

rg

advo

(5-6)

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

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

This expression was previously given as Equation (2-11) in Chapter 2, and is solved

iteratively as illustrated in Figure 5-2. For 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 errorfinal iteration

iteration 2

iteration 1

10 % -4

1.0

(G/Gmax)γ

γ γ (log Scale)

G/G

max

1%

Figure 5-2. Iterative solution of Equation (5-6) to determ

(ine the effective shear-strain

γ) at a given depth in a soil profile.

Once γ is determined, the damping and shear modulus ratios are easily determined from

the respective degradation curves, as illustrated in Figure 5-3.

132

(G/Gmax)γ

Dγ

γ γ (log scale)

D

G/Gmax

1% 10-4%

Figure 5-3. The determination of shear modulus and damping ratios from the respective degradation curves.

Using the above expressions for τ, G, and D, the energy dissipated at depth in a soil

deposit for one equivalent cycle of loading is given as: 2

max

maxmax

1 65.02

⋅⋅⋅⋅

⋅

=∆ dvo rg

a

GGG

DW σ

π

γ

γ (5-7)

The various components of this expression are illustrated in Figure 5-4.

γ

⋅=

maxmax G

GGG

GD

W ave2

1

2 τπ γ=∆ Shear Strain

Shea

r Stre

ss

τave

γ

Figure 5-4. Graphical representation of the dissipated energy per unit volume for an equivalent cycle of loading.

133

5.2.3 Determination of Neqv

The final variable needed to determine the energy dissipated in the soil for the entire

duration of the earthquake is the number of equivalent cycles (Neqv). The equivalent-

number-of-cycles concept resulted from the inherent difficulties of imposing random

earthquake type motions to laboratory specimens. The concept is based on equating

transient earthquake motions, involving peaks of varying amplitudes, to a uniform cyclic

load having Neqv cycles.

Lee and Chan (1972) determined the equivalent number of cycles (Neqv) from earthquake

motions by computing the weighted sum of the number of spikes in an acceleration time

history, where the weighting factors were based on a normalized laboratory Capacity

curve. Using this same procedure, Seed et al. (1975) developed the correlation shown in

Figure 5-5. As may be seen in this figure, there is a large amount of scatter in the data,

which has brought criticism of the equivalent-number-of-cycles concept. However, an

alternative energy-based correlation relating Neqv, M, and site-to-source distance

developed as part of this thesis shows much less scatter. This new correlation is based on

equating the energy dissipated in the layers of a soil profile, as determined from site

response analyses, to the energy dissipated in an equivalent cycle of loading multiplied

by Neqv. This procedure is depicted graphically in Figure 5-6.

134

9 8 7 6 5 0

M6

M6-

3/4

M8.

5

Neqv ≈ 6

Neqv ≈ 10

Neqv ≈ 26

M7.

5

Neqv ≈ 15

S-I

A-1

Mean

Mean -1 Standard Deviation

Mean +1 Standard Deviation

40

30

20

10

Neq

v at τ

ave

Earthquake Magnitude

Figure 5-5. A correlation relating earthquake magnitude and equivalent number of cycles. The data points labeled S-I and A-1 are assumed to be from a different study than the rest of the data. Also shown in this figure are several M-Neqv pairs that are commonly presented in tabular form in published literature. (Adapted from Seed et al. 1975).

135

=

=

=

τ

γ

γ

Gγ

τ τave

γ

Gγ

τ τave

γ

τave

τ

Dissipated energy for an equivalent cycle

times Neqv

Gγ

Time (sec)

Soil Profile

Layer 3

Layer 2

Layer 1 γ

τ

Dissipated energy from site response

analysis

τ

γ

x Neqv

x Neqv

x Neqv

Acc

eler

atio

n

Figure 5-6. Illustration of the procedure used to develop the Neqv correlation. In this procedure, the dissipated in a layer of soil, as computed from a site response analysis, is equated to the energy dissipated in an equivalent cycle of loading multiplied by Neqv.

An option was added to the site response computer program SHAKE91 (Idriss and Sun

1992) to compute the energy dissipated in each layer of a soil deposit subjected to an

earthquake loading. This modified version of SHAKE91 is herein referred to as

SHAKEVT, with Figure 5-7 being an example of hysteresis loops output by the program.

A parametric study was performed by propagating 24 pairs of horizontal motions from

various magnitude earthquakes through 12 different soil profiles (i.e., 288 site response

analyses). The normalized energy demand (NED = ∆W/σ’mo) was computed for each

layer in the profiles. The acceleration time histories used in the parametric were recorded

on stiff soil and rock sites at varying site-to-source distances. Appendix 5a and Appendix

5b list the earthquake records and profiles, respectively, used in the parametric study.

136

The shear modulus and damping degradation curves proposed by Ishibashi and Zhang

(1993) were used in the site response analyses.

Shea

r Stre

ss (p

sf)

Shear Strain (%)

300

200

100

-100

-200

-300

0.05 0.025 -0.025 -0.05

Figure 5-7. Shear-stress shear-strain hysteresis loops, output by SHAKEVT, for a given depth in a soil profile subjected to an earthquake acceleration time history.

To compute the number of equivalent cycles corresponding to total motions experienced

at a site, the following procedure was used.

• Using Equation (5-7), the normalized energy dissipated in one equivalent cycle of

loading (i.e., NED1 = ∆W1/σ’mo) was computed for each layer in a profile. In

solving Equation (5-7), amax and rd values from the respective site response

analysis were used in conjunction with the Ishibashi and Zhang (1993) shear

modulus and damping degradation curves.

• The number of equivalent cycles for the motion in each layer is determined by:

i

iieqv NED

NEDN

1

= (5-8a)

where: Neqv i = Number of equivalent of cycles of the motion in layer i.

NEDi = Normalized energy dissipated in layer i, as computed by SHAKEVT.

NED1i = The normalized energy dissipated in one equivalent cycle of loading, as computed by Equation (5-7).

137

• Neqv for the entire profile subjected to a single component of horizontal motion

was computed as a weighted average of the Neqv i for layers lying between the

groundwater table and a depth of about 60ft, where the Neqv i values were weighted

according to the thickness of the corresponding layer:

gwt

i

iieqveqv dft

hNN

−⋅=∑ 60

(5-8b)

where: Neqv = Number of equivalent of cycles for entire profile.

Neqv i = Number of equivalent of cycles of the motion in layer i.

h i = Thickness of layer i.

d gwt = Depth to the groundwater table.

A comparison of the NED values computed by SHAKEVT and Equation (5-7)

multiplied by Neqv is shown in Figure 5-8.

• Finally, the Neqv values for each horizontal component in a pair were summed to

give the total Neqv of the motions experienced at the site.

0

Dep

th (f

t)

80

30

20

10

100

90

70

60

50

40

0

SHAKEVT × Neqv

∆W1

σ’mo

amax = 0.22gNeqv = 8.2

b)

0 15 20NED × 104

5 10

SHAKEVT × Neqv

∆W1

σ’mo

amax = 0.31g Neqv = 6.1

a) 10

20

30 40

Dep

th (f

t)

50

60 70 80

90

100 0

NED × 104 2 4 6

Figure 5-8. A comparison of the NED values computed by SHAKEVT and Equation (5-7) multiplied by Neqv. Figures a) and b) correspond to the two horizontal components of motion at the same site. The sum of the Neqv for each component represents the Neqv for total motions experienced at the site.

138

For each of the 12 profiles shown in Appendix 5b, the computed Neqv values for the 24

pairs of time histories were plotted as functions of epicentral distance (ED) and Richter

magnitude (M). Although epicentral distance and Richter magnitude are not the best

measures of site-to-source distance and earthquake magnitude, respectively, they were

used because these were the only measures available for the majority of the liquefaction

case histories. Figures 2-8a and 2-8b show the various definitions of site-to-source

distance and a comparison of various magnitude scales, respectively. Regression

analyses were performed to fit surfaces to the computed Neqv values for each profile. For

three dimensional regression analyses, an unlimited array of surfaces may be employed.

The simplest surface that provided a good fit of the data is shown in Figure 5-9.

Richter Magnitude Epicentral Distance (km)

Neq

v

Figure 5-9. A fit surface to the computed Neqv for one of the profiles shown in Appendix 5b. The black dots shown in this plot are the computed Neqv values.

All the surfaces fit to the data sets for the 12 profiles have the same general shapes but

vary slightly in the “steepness” of their inclines. An average of the 12 surfaces was

computed, a contour plot of which is shown in Figure 5-10. An approximation of the

average surface is given by Equation (5-9).

(5-9a) )()()( MfEDfforEDf ≤

)()()( MfEDfforMf >

Neqv =

139

where:

7.5172.325453.110633.9 2 ≥+− MforMM

7.55.8 <Mfor

(5-9b)f(M) =

(5-9c)

kmEDforEDED 55.0487.8012.0011.0 2 ≥+−

kmEDfor 55.05.8 < f(ED) =

As a further examination of the proposed Neqv correlation, the near field - far field

boundary proposed by Krinitzsky et al. (1993) is superimposed on the contour surface

shown in Figure 5-10. The close coincidence of the “elbows” in the contours and the

near field - far field boundary suggests that the two are related. It is hypothesized that the

elbows result from setting the amplitude of τave proportional to a fixed percentage of amax

(i.e., 0.65 amax). In the far field, the acceleration time histories are more regular due to

the attenuation of the high frequencies motions, and therefore, a fixed fraction of amax

adequately represents the entire time history. However, in the near field the time

histories are much more chaotic, and often amax is associated with a high frequency spike,

which is not representative of the entire time history. The random nature of the

relationship between amax (or a fixed fraction of amax) and the rest of the acceleration time

history lead to the effective-peak-acceleration concept used by Newmark and Hall (1982)

in scaling design spectra.

140

Richter Magnitude (M)

Neqv = 10

5055

4045

3530

2520

15

20

40

60

80

100

50 6 7 8

NEAR FIELD

FAR FIELD

Epic

entra

l Dis

tanc

e (k

m)

Figure 5-10. Contour plot of the average fit surface for each of the 12 profiles used in the parameter study. The contours are of constant Neqv as a function of epicentral distance and Richter magnitude. The near field - far field boundary superimposed on the contour plot is that proposed by Krinitzsky et al. (1993); note this boundary is only given up to M7.5.

In comparing the Neqv correlation in Figures 5-10 to that developed by Seed et al. (1975)

shown in Figure 5-5, it needs to be realized that the correlation shown in Figure 5-5 is for

a single component of horizontal motion, while Figure 5-10 is for both components. The

vertical portion of the contours shown in Figure 5-10 indicates that Neqv is independent of

epicentral distance in the far field, similar to the correlation proposed by Seed et al.

(1975). However, the horizontal portion of the contours indicates that Neqv is independent

of earthquake magnitude in the near field. Again, this near field independence is

hypothesized to be a consequence of setting the amplitude of τave proportional to a fixed

percentage of the rather chaotic amax (i.e., 0.65 amax).

Aside from the focus of this thesis, in Appendix 5c, the correlation shown in Figure 5-10

is used in the derivation of a new set of magnitude scaling factors (MSF) for use in the

stress-based liquefaction evaluation procedure. As opposed to the current NCEER

141

(1997) recommended MSF, presented in Chapter 2 (Figure 2-2, Section 2.2.1), the new

set of MSF is a function of both earthquake magnitude and site-to-source distance.

5.3 Capacity Curve

Similar to the liquefaction evaluation procedures presented in Chapter 2, a Capacity

curve was developed by analyzing liquefaction case histories. The Demand imposed on

the soil by the earthquake can be estimated as:

eqvdvo

mo

eqvmo

Nrg

a

GGG

D

NWNED

⋅

⋅

⋅⋅

=

⋅∆

=

2max

maxmax

1

65.0'

2

'

σσ

π

σ

γ

γ

(5-10)

This expression was derived by substituting the values of τ, G, D, and Neqv, determined as

outlined above, into Equation (5-4) and normalizing the result by the initial mean

effective confining stress (σ’mo). Using this expression, the Demand (i.e., NED) imposed

on the soil was estimated for the 126 earthquake case histories listed in Table 5-1. The

data in this table came from liquefaction databases assembled by Liao and Whitman

(1986) and Fear and McRoberts (1995), where the Fear and McRoberts’ database is a

compendium and re-interpretation of the case histories listed in the databases of Seed et

al. (1984) and Ambraseys (1988). Furthermore, the Fear and McRoberts’ database

provides the available boring logs for each case history. Only the Liao and Whitman

database includes epicentral distances. The variables listed in Table 5-1 are those given

by Seed et al. (1984), with the epicentral distances coming from Liao and Whitman

(1986), and the N1,60cs values were computed using the latest NCEER (1997)

recommended procedure presented in Chapter 2. Because not all the case histories listed

in Seed et al. (1984) could be matched to corresponding case histories in Liao and

Whitman (1986), the data point numbers listed in Table 5-1, as assigned by Seed et al.

(1984), are not completely sequential. To facilitate the cross reference between the two

142

databases, the case numbers assigned by Liao and Whitman (1986) are listed in the last

column of the table.

Per Ambraseys (1988), the case histories were categorized as: Liq 0 – No Liquefaction;

Liq 1 – Marginal Widespread Liquefaction; Liq 2 – Sporadic Liquefaction; and Liq 3 –

Complete Liquefaction. A plot of the data is shown in Figure 5-11. For the purposes of

drawing a boundary separating the points corresponding to liquefaction and no

liquefaction, liquefaction was assumed to occur for all the case histories categorized as

Liq1, 2, and 3. Analogous to the procedures presented in Chapter 2, this boundary

quantifies the Capacity of the soil as a function of the penetration resistance and may be

approximated by Equation (5-11).

( )csNNEC 60,14 185.0exp10195.1 ⋅⋅= − for 3 ≤ N1,60cs ≤ 27 (5-11)

where: NEC = Normalized energy capacity (i.e., Capacity).

N1,60cs = SPT N-values corrected per NCEER (1997) recommendations.

The boundary shown in Figure 5-11 and defined by Equation (5-11) was drawn slightly

“less conservative” (i.e., to left and up) than may be desired for liquefaction evaluations.

However, the position of the boundary was selected because its eventual use is for

remedial ground densification design, and its placement is such as to ensure enough

energy is imparted in the ground to induce liquefaction. Accordingly, a curve that is

“unconservative” for liquefaction evaluations is “conservative” for remedial ground

densification design.

143

120 A3

73 122

113

28 23

26

99

96

92

33

3

54

107

1

83

84 108 53

85 114

109 0.020

0.018

0.000 5040 45 353025200 5 10 15

44

31 20

119 49 112 57 52

Marginal Widespread Liquefaction (Liq:1)

Capacity Curve

Sporadic Liquefaction (Liq:2)Complete Liquefaction (Liq:3)

See Figure 5-11b for enlargement of this area

A5

25

105 101

111 100

98

90

61 97

45 12

A7 27

43

4 82

102

A8 32

8 59 89

117 A1

106 104

46 88

103

35 94

7

55

87 86

36

2 A9

18

37

91

13

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002 48

6 62 63

A2

30

66

15

64

14 51

29

22

42

No Liquefaction (Liq:0)

Nor

mal

ized

Ene

rgy

Dem

and

(NED

)

N1,60cs

Figure 5-11a. Energy-based Capacity curve developed from 126 liquefaction field case histories.

144

Complete Liquefaction (Liq:3) Sporadic Liquefaction (Liq:2)Marginal Widespread Liquefaction (Liq:1)

Capacity Curve

No Liquefaction (Liq:0)

0.0035

0.0030

0.0025

0.0020

0.0005

0.0010

0.0015

121

66

64

47

6 118

A2 14

A6

62

22

15 95

9

21 30

110

24

5

44

58

119

31

0 0.0000

2

40,68

38 41

112

80

71 67

74

39 79

115

123 60 77

75 69

116

29

51

10

65

11

78

76

70 81 72 20

57 52

63

93

49

48

50

0.0040 N

orm

aliz

ed E

nerg

y D

eman

d (N

ED)

4 6 8 10 12 14 15

N1,60cs

Figure 5-11b. Energy-based Capacity curve developed from 126 liquefaction field case histories.

145

Table 5-1. Liquefaction case histories. Seed et al. Critical Depth of σvo σ’vo FC N1,60cs amax Richter ED L&W Data Pt # Depth (ft) gwt (ft) (psf) (psf) (%) (blws/ft) (g) M (km) NED Liq Case # 1 33 3 3960 2090 0 25.4 0.32 7.9 30 0.01328 3 0301 2 23 7 2760 1760 5 11.8 0.32 7.9 30 0.01012 3 0302 3 20 6 2400 1530 3 24.0 0.32 7.9 30 0.00772 3 0303 4 17 8 2040 1480 4 15.0 0.32 7.9 30 0.00583 3 0304 5 23 13 2630 2000 10 11.4 0.2 7.9 72 0.00345 2 0702 6 14 13 1550 1490 22 6.3 0.2 7.9 72 0.00207 2 0703 7 27 3 3240 1740 1 19.7 0.2 7.9 72 0.00841 3 0704 8 17 3 2040 1170 5 14.5 0.2 7.9 72 0.00727 3 0705 9 17 10 2040 1600 20 10.3 0.2 7.9 72 0.00323 1 0707 10 27 6 3095 1845 25 13.1 0.2 6.3 24 0.00151 0 1005 11 20 6 2270 1460 2 7.8 0.16 6.3 24 0.00069 0 1003 12 17 7 2040 1420 10 10.5 0.2 8.0 165 0.00494 3 1201 13 12 2 1440 820 27 6.1 0.2 8.0 165 0.01142 3 1202 14 10 8 1120 1000 30 7.8 0.2 8.0 165 0.00227 3 1203 15 23 7 2760 1760 5 11.8 0.16 8.0 200 0.00305 3 1204 18 13 4 1560 1000 0 10.6 0.4 7.3 6 0.01208 3 1304 20 10 8 1075 950 3 4.3 0.19 5.3 6 0.0004 3 1502 21 23 3 2760 1510 2 9.5 0.16 7.3 51 0.00336 3 1814 22 23 3 2760 1510 2 14.3 0.16 7.5 51 0.0028 1 1815 23 23 6 2760 1700 2 22.4 0.18 7.5 51 0.0025 0 1816 24 33 3 3960 2090 2 10.7 0.16 7.5 51 0.00337 3 1817 25 33 3 3960 2090 2 17.1 0.16 7.5 51 0.00283 1 1818 26 33 6 3960 2270 2 22.7 0.18 7.5 51 0.00297 0 1819 27 14 0 1680 810 10 6.8 0.16 7.5 51 0.00536 3 1820 28 20 4 2400 1400 0 30 0.18 7.5 51 0.00267 0 1821 29 20 8 2400 1650 0 13.7 0.18 7.5 51 0.00208 0 1822 30 15 2 1800 990 0 7.9 0.16 7.5 51 0.00333 3 1823

146

Seed et al. Critical Depth of σvo σ’vo FC N1,60cs amax Richter ED L&W Data Pt # Depth (ft) gwt (ft) (psf) (psf) (%) (blws/ft) (g) M (km) NED Liq Case # 31 3 3 330 330 0 2.8 0.13 6.3 56 0.00007 3 2001 32 13 3 1560 940 20 11.0 0.2 7.9 290 0.00655 3 2201 33 20 7 2400 1590 5 30 0.23 7.9 190 0.00614 0 2207 35 13 3 1560 940 5 22.6 0.23 7.9 190 0.00804 0 2208 36 20 10 2250 1800 20 10.9 0.45 6.6 14 0.00971 3 2603 37 20 15 2330 2000 >50 6.7 0.45 6.6 14 0.00992 3 2601 38 19.7 6.6 2325 1505 ? 10.4 0.1 7.3 90 0.00035 0 2803 39 42.7 6.6 5290 3038 ? 11.8 0.1 7.3 104 0.0004 0 2805 40 20.3 5 2505 1550 ? 6.5 0.1 7.3 95 0.00052 3 2806? 41 9.8 6.6 1180 980 ? 9.6 0.13 7.3 60 0.00034 3 2810 42 32.8 6.6 4055 2420 ? 7.8 0.2 7.3 55 0.00493 3 2811 43 33.8 6.6 4180 2480 ? 8.1 0.2 7.3 55 0.00479 3 2812 44 30 5 3675 2115 67 12.4 0.13 7.3 55 0.00118 3 2813 45 27 5 3300 1930 ? 10.6 0.2 7.3 53 0.00488 3 2809 46 27 5 3300 1930 48 20.1 0.2 7.3 50 0.00387 3 2814 47 27 5 3300 1930 ? 8.7 0.1 7.3 90 0.00055 0 2815 48 34 5 2910 1800 3 4.7 0.135 7.5 170 0.00167 3 2901 49 15 8 1150 705 3 8.6 0.135 7.5 170 0.00109 1 2902 50 38 8 3210 1860 3 11.7 0.135 7.5 170 0.00122 1 2902 51 35 12.5 2860 1490 0 13.9 0.135 7.5 170 0.00162 0 2903/4 52 7.5 4.5 875 685 ? 14 0.2 7.8 43 0.00110 3 3001 53 23 3.3 2740 1510 ? 4.4 0.35 7.8 25 0.04219 3 3002 54 17.5 10 2040 1570 10 30 0.5 7.6 1 0.00915 0 3006 55 17.4 3 2130 1230 20 23.5 0.35 7.6 20 0.01061 3 3007 57 6.7 5 760 655 12 14.7 0.2 7.6 80 0.00113 3 3009 58 20 4 2440 1440 12 13.1 0.13 7.6 90 0.00133 3 3010 59 20 3 2455 1395 ? 10.2 0.2 7.6 90 0.00705 3 3011 60 30 13 3555 2495 1 11.1 0.07 7.6 110 0.00013 0 3012 61 23 5 2800 1675 3 11.4 0.2 7.6 110 0.00574 3 3013

147

Seed et al. Critical Depth of σvo σ’vo FC N1,60cs amax Richter ED L&W Data Pt # Depth (ft) gwt (ft) (psf) (psf) (%) (blws/ft) (g) M (km) NED Liq Case # 62 27 15 2970 2220 3 10.2 0.2 7.4 70 0.00251 3 3201 63 39 22 3190 2130 ? 8.7 0.2 7.4 70 0.00285 3 3203 64 12 4 1320 820 4 13.1 0.2 7.4 70 0.00337 0 3204 65 10 7 1100 910 3 12.5 0.2 7.4 70 0.00135 0 3205 66 17 6 1870 1180 50 11.0 0.2 7.4 70 0.00387 3 3206 67 21 3 2520 1400 0 12.4 0.1 6.7 140 0.00028 0 3401 68 11 2 1320 760 5 6.5 0.12 6.7 115 0.00052 2 3403 69 11 4 1320 880 4 8.4 0.12 6.7 115 0.00029 0 3404 70 18 6 2060 1310 60 7.5 0.12 6.7 115 0.00041 0 3405 71 14 3 1680 990 0 13.3 0.12 6.7 115 0.00037 0 3406 72 14 6 1680 1180 10 5.4 0.12 6.7 115 0.00035 0 3408 73 11 4 1320 880 7 17.8 0.12 6.7 115 0.00021 0 3409 74 14 1 1680 870 12 12.2 0.12 6.7 115 0.00056 0 3410 75 21 14 2220 1780 5 9.1 0.14 6.7 77 0.00032 0 3412 76 21 8 2390 1580 4 8.6 0.14 6.7 77 0.0006 0 3413 77 11 10 1300 1240 5 11.2 0.14 6.7 77 0.00015 0 3414 78 13 8 1560 1250 10 7.6 0.14 6.7 75 0.00033 0 3417 79 17 8 2040 1480 20 14.1 0.14 6.7 75 0.00036 0 3419 80 20 8 2400 1650 3 13.7 0.14 6.7 75 0.00044 0 3421 81 13 5 1560 1060 10 5.8 0.12 6.7 75 0.00036 0 3424 82 21 3 2520 1400 0 12.4 0.2 7.4 115 0.0058 3 3501 83 11 3 1320 820 4 26.6 0.32 7.4 112 0.01439 0 3502 84 11 2 1320 760 5 6.5 0.32 7.4 112 0.03956 3 3503 85 11 4 1320 880 4 8.4 0.32 7.4 112 0.01916 2 3504 86 18 6 2060 1310 60 7.5 0.24 7.4 115 0.00883 3 3505 87 14 3 1680 990 0 13.3 0.24 7.4 115 0.00825 2 3506 88 18 7 2160 1470 0 22.2 0.24 7.4 115 0.00456 0 3507 89 14 6 1680 1180 2 5.4 0.24 7.4 115 0.00753 2 3508 90 11 4 1320 880 7 17.8 0.24 7.4 115 0.0046 3 3509

148

Seed et al. Critical Depth of σvo σ’vo FC N1,60cs amax Richter ED L&W Data Pt # Depth (ft) gwt (ft) (psf) (psf) (%) (blws/ft) (g) M (km) NED Liq Case # 91 14 1 1680 870 12 12.2 0.24 7.4 115 0.01329 2 3510 92 24 4 2880 1630 17 24.3 0.24 7.4 115 0.0082 0 3511 93 21 14 2220 1780 5 9.1 0.24 7.4 92 0.0038 2 3512 94 21 8 2390 1580 4 8.6 0.24 7.4 92 0.00733 2 3513 95 11 10 1300 1240 5 11.2 0.28 7.4 95 0.00316 2 3514 96 20 10 2400 1780 0 28 0.28 7.4 95 0.00564 0 3515 97 20 8 2400 1680 10 12.4 0.24 7.4 85 0.00522 1 3516 98 13 8 1560 1250 10 7.6 0.24 7.4 85 0.00402 2 3517 99 23 8 2760 1820 5 25.3 0.24 7.4 85 0.00499 0 3518 100 17 8 2040 1480 20 14.1 0.24 7.4 85 0.00435 2 3519 101 15 8 1800 1360 26 17 0.24 7.4 85 0.00347 1 3520 102 20 8 2400 1650 3 13.7 0.24 7.4 85 0.0053 3 3521 103 20 8 2320 1570 11 21.2 0.24 7.4 85 0.00475 0 3522 104 23 12 2660 1970 12 21.9 0.24 7.4 85 0.00362 0 3523 105 13 5 1560 1060 10 5.8 0.2 7.4 80 0.00416 3 3524 106 20 5 2400 1460 10 21.5 0.2 7.4 80 0.00356 0 3525 107 12 6 1410 1035 25 30 0.78 6.6 10 0.09502 0 3901 108 12 6 1410 1035 29 6 0.78 6.6 10 0.21985 3 3902 109 14 6 1660 1160 37 23.3 0.78 6.6 10 0.15075 0 3903 110 6 1 735 425 80 9.8 0.24 6.6 42 0.00348 3 3904 111 14 1 1735 930 18 18.6 0.24 6.6 42 0.00442 3 3905 112 11 7 1270 1020 75 7.8 0.2 6.6 45 0.00089 3 3918 113 7.5 7 830 800 30 20.5 0.2 6.6 45 0.00028 0 3919 114 7 5 800 675 31 9.4 0.51 6.6 26 0.01939 3 3920/1? 115 20 3 2260 1200 13 8.8 0.13 6.1 15 0.0004 0 4101 116 47 3 5310 2560 27 8.8 0.13 6.1 15 0.00037 0 4103 117 16 3 1960 1150 40 17.7 0.26 5.6 10 0.00249 3 4206-11? 118 14 9 1615 1305 92 11.6 0.32 5.6 3 0.0021 3 4214-17? 119 6 1 735 425 80 9.8 0.21 5.6 15 0.00111 0 4204

149

Seed et al. Critical Depth of σvo σ’vo FC N1,60cs amax Richter ED L&W Data Pt # Depth (ft) gwt (ft) (psf) (psf) (%) (blws/ft) (g) M (km) NED Liq Case # 120 14 1 1735 930 18 18.6 0.21 5.6 15 0.00148 0 4205 121 11 7 1270 1020 75 7.8 0.2 5.6 14 0.00048 3 4218 122 7.5 7 830 800 30 20.5 0.2 5.6 14 0.00015 0 4219 123 7 5 800 675 31 9.4 0.09 5.6 28 0.00002 0 4221 Additional case histories listed in Fear and McRoberts (1995), but not listed in Seed et al. (1984). Critical Depth of σvo σ’vo FC N1,60cs amax Richter ED L&W Data Pt # Depth (m) gwt (m) (kg/cm2) (kg/cm2) (%) (blws/ft) (g) M (km) NED Liq Case # A1 9.7 0.5 1.8 0.92 ? 18.6 0.16 7.5 51 0.00307 0 - A2 9.5 2.0 1.8 1.07 ? 11.2 0.16 7.5 51 0.00238 3 - A3 16.5 16.5 3.22 3.22 50 16.9 0.45 6.6 14 0.00159 3 - A5 5.8 1.5 1.1 0.67 35 17.2 0.24 6.5 25 0.00257 0 3602 A6 4.3 1.2 0.8 0.49 1 9.2 0.2 6.9 50 0.00278 3 3801? A7 8 1 1.49 0.79 3 7 0.2 6.9 50 0.00549 3 3801? A8 4 2.1 0.72 0.53 10 9.7 0.35 6.7 23 0.00644 3 4001? A9 4.9 2.1 0.9 0.6 ? 8.7 0.35 6.7 23 0.01005 3 4001? Column headings: Seed et al. Data Pt #: Data point number assigned by Seed et al. (1984) Critical Depth (ft): Depth in the profile most susceptible to liquefaction Depth of gwt (ft): Depth of the ground water table σvo (psf): Total vertical stress at the critical depth σ’vo (psf): Initial vertical effective stress at the critical depth FC (%): Fines content of the soil at the critical depth N1,60cs (blws/ft): Corrected penetration resistance at the critical depth amax (g): Maximum horizontal acceleration at the soil surface Richter M: Richter magnitude ED (km): Epicentral distance from the site to the source NED: The normalized energy demand (i.e., Demand) imposed on the soil by the earthquake L&W Case #: Designation number for the case history assigned by Liao and Whitman (1986)

150

5.4 Liquefaction Evaluation

To perform a liquefaction evaluation, the Demand (i.e., NED) imposed on the soil by the

earthquake is determined from the iterative solution of Equation (5-10). The normalized

energy capacity (NEC) is then determined graphically using Figure 5-11 or numerically

using Equation (5-11). If NED < NEC, liquefaction is not predicted to occur, and if NED

≥ NEC liquefaction is predicted. The factor of safety (FS) against liquefaction is given

by Equation (5-12).

)105()115(

−−

=

=

EquationEquation

NEDNEC

DemandCapacityFS

(5-12)

5.5 Parameter Study

The proposed liquefaction procedure was evaluated in the same way as the procedures

presented in Chapter 2, where the factors of safety versus depth (FS-profile) are

computed for simple soil profiles. The profiles were assumed to be subjected to a M7.5

earthquake at an epicentral distance of 60km, resulting in amax = 0.13g. The following

profiles were examined:

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

of the ground water table (gwt) is equal to 0ft, 11ft, and 25ft.

• 100ft thick profile of clean sand with the gwt at a depth of approximately 11ft.

The profile is assumed to have constant N1,60 with depth equal to 5, 10, and

15blws/ft.

The resulting FS-profiles for the proposed procedure are shown in Figure 5-12. For

comparison purposes, the FS-profiles computed using both the stress- and strain-based

procedures (Figures 2-20 and 2-21 in Chapter 2) are presented below as Figures 5-13 and

5-14, respectively. Because the stress-based procedure has continually evolved with new

insights into soil behavior, the trends in the stress-based FS-profiles are assumed to be

151

correct, although the absolute values of the FS may not be. From examination of the

stress-based FS-profiles, the following observations are made:

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

• 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 critical depth is defined as the depth below the gwt corresponding to the lowest FS.

As may be recalled from Chapter 2, from FS-profiles for the strain-based procedure

excess pore pressures are expected for the entire depths of all the profiles. The high FS

towards the surface of the profiles shown in Figure 5-14 are attributed to the limitations

in the empirical correlation for Gmax, (G/Gmax), and φ’ vs. N1,60 at low effective confining

stresses and are not the expected behavior of the soil.

Because the proposed energy-based liquefaction evaluation procedure is a unification of

the stress- and strain-based approaches, the FS-profiles shown in Figure 5-12 have the

characteristics of those shown in Figures 5-13 and 5-14. Most notably:

• Except for the profile with the gwt at the surface, the critical depths are

essentially the same as those of the stress-based procedure.

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

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

lowered.

• The FS are over predicted at low effective confining stresses (i.e., near the

surface of the profiles).

Even with the last observation, the proposed liquefaction evaluation procedure is deemed

acceptable for remedial ground densification design. As a final comment, it may be

observed from comparisons of the FS-profiles that the absolute values of the predicted FS

152

are 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).

b)

12

3

Approximate Critical Depths

5 10 15

3 1 2

0

5

10

15

25

20

N1,60 = 15

3

2

1

30 0

a)

Approximate Critical Depth

155 10

5 10 N1,60 = 15

0

5

10

15

0

30

25

20

Dep

th (m

)

Factor of Safety Factor of Safety

Figure 5-12. FS-profiles computed using the proposed energy-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.

153

NCEER (1997)

1

2

3

Approximate Critical Depths

31 2

2 3 030

5 10 N1,60 = 15

b)0

5

10

15

25

20

N1,60 = 15

3

2

1

Approximate Critical Depth

0 a)

5

10

15

30 0 1

25

20

Dep

th (m

)

1 2 3 Factor of Safety Factor of Safety

Figure 5-13. 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.

154

Dobry et al. (1982)

Critical Depths difficult to determine

b)

Approximate Critical Depths

1

2

3

32

1

0

5

10

15

25

20

N1,60 = 15

3

2

1a)

Approximate Critical Depth

2 3 30 01

10 5

N1,60 = 15

0

5

10

15

30 0

25

20

Dep

th (m

)

1 2 3 Factor of Safety Factor of Safety

Figure 5-14. 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.

155

5.6 Comparison of Energy-Based Capacity Curve Derived from Field Data with

Laboratory Test Data

The Capacity curve shown in Figure 5-11, derived from the analysis of earthquake case

histories, cannot be directly compared to the cumulative energy dissipated in undrained

laboratory specimens subjected to cyclic loading. This is because the Capacity curve was

developed using numerical analyses that did not account for softening of the soil due to

increases in pore pressures (herein referred to as total stress analyses), while such

softening inherently occurs in the undrained cyclic loading of saturated sands. The

influence of pore pressure increases on energy dissipation can be understood by

comparing the respective hysteresis loops shown in Figure 5-15.

Axial Strain (%)

a) 60 40

20 0

-20

-40 Dev

iato

r Stre

ss (k

Pa)

0.2

b)

0 2 4 6 8Axial Strain (%)

0.1 0.0

Dev

iato

r Stre

ss (k

Pa)

-0.2

140

100

60

20 0

-0.3 -60

-20

-0.15

-60

0.3 0 0.15 Axial Strain (%)

-0.1

c)

-2 -4 6

80

Dev

iato

r Stre

ss (k

Pa) 60

40

20 0

-20 -40

--60

Figure 5-15. Stress-strain hysteresis loops for a) total stress numerical analysis, b) stress controlled cyclic triaxial test and c) strain controlled cyclic triaxial test.

156

The hysteresis loops shown in Figure 5-15a are those expected if an element of soil was

subjected to sinusoidal loading, and the boundary conditions were such that density of the

sample and the effective confining stress were not allowed to change (e.g., total stress

numerical analysis). Both the shear modulus and damping ratio remain constant for each

cycle of loading, and the resulting loops are the same for stress- and strain-controlled

loadings. On the contrary, for saturated samples subjected to undrained cyclic loading,

the shear modulus for each subsequent loop is less than the previous loop, and the

damping ratio for each subsequent loop is greater than the previous loop. As may be seen

in Figures 5-15b and 5-15c, the area bound by hysteresis loops increases as the loading

progresses for stress-controlled loading, while it decreases for strain-controlled loading.

Accordingly, the cumulative dissipated energy determined by a total stress numerical

analysis significantly differs from that which actually dissipates in undrained cyclic

laboratory specimens. As may be recalled from Chapter 2, several of the existing energy-

based procedures were criticized for the improperly comparing the energies from total

stress numerical analyses to undrained cyclic laboratory data.

A normalization technique is proposed for removing (or compensating for) the dissipated

energy resulting from the softening of the soil due to increases in pore pressures. Such a

procedure allows the dissipated energy computed from undrained cyclic laboratory tests

to be compared to the Capacity curve derived from total stress numerical analyses (i.e.,

Figure 5-11).

Two idealized hysteresis loops are shown in Figure 5-16 for a stress controlled cyclic

laboratory test, where loop 1 is measured earlier in the test than loop 2 (i.e., σ’m1 > σ’m2).

157

τ

σ’m1 > σ’m2∆W2

∆W1

γ2

τ

γ1 γ

Figure 5-16. Two idealized hysteresis loops for a stress-controlled, undrained cyclic test, where loop 1 occurs earlier in the test than loop 2.

Starting with Equation (5-2), the ratio of the areas (or dissipated energies) bound by loop

2 and loop 1 (ER) may be written:

21

12

11

22

1

2

22

GDGD

DD

WW

ER

=

=

∆∆

=

τγπτγπ

(5-13)

where: ER = Energy ratio.

∆W1 = Area bound by loop 1 (i.e., dissipated energy per unit volume associated with loop 1).

∆W2 = Area bound by loop 2 (i.e., dissipated energy per unit volume associated with loop 2).

τ = Amplitude of the shear stress for stress controlled loading.

γ1 = Amplitude of shear strain for loop 1.

γ2 = Amplitude of shear strain for loop 2.

D1 = Damping ratio for loop 1.

D2 = Damping ratio for loop 2.

158

G1 = Shear modulus for loop 1.

G2 = Shear modulus for loop 2.

Using the Ishibashi and Zhang (1993) shear modulus and damping degradation curves,

ER can be computed for different values of σ’m1 and σ’m2, different amplitudes of the

applied loading (CSR), and different relative densities of the soil (Dr). From examining

the ER values for several combinations of CSR and Dr, it was observed that these

parameters have relatively little influence on ER, so the following discussion is limited to

the influence of σ’m1 and σ’m2. Additionally, inherent to using the degradation curves in

computing ER is the assumption that changes in pore pressure only affects the effective

confining stress on the soil (i.e., undrained behavior can be approximated as drain, with

appropriate changes in effective confining pressure). This assumption is reasonable and

is one of the underlying assumptions of the commonly used strain-based pore pressure

generation model proposed by Martin et al. (1975).

Setting σ’m1 = σ’mo and σ’m2 = σ’m, ER is plotted as a function of the ratio σ’m /σ’mo in

Figure 5-17 for CSR = 0.1, σ’mo = 100kPa, and two different relative densities (Dr). For

isotropically consolidated specimens, the ratio σ’m/σ’mo is related to the excess pore

pressure ratio (ru) by: ru = 1-σ’m/σ’mo. Accordingly, when σ’m /σ’mo = 1.0, ru =0, and

when σ’m/σ’mo = 0, ru = 1.0 (i.e., liquefaction). From Figure 5-17, it can be observed that

ER increases as σ’m/σ’mo decreases. This implies that for a stress controlled cyclic test,

the area bound by a hysteresis loop increases as the pore pressures increase, which is

consistent with observed behavior in laboratory test data such as shown in Figure 5-15b.

In addition to ER curves, the function: (σ’m /σ’mo)-1.65 is also plotted in Figure 5-17. As

may be observed, the curve corresponding to this function reasonably approximates the

ER curves. Based on this approximation (i.e., ER ≈ (σ’m2/σ’m1)-1.65), the dissipated

energies associated with the hysteresis loops shown in Figure 5-16 can be related as:

159

65.1

1

22

21

''

⋅∆≈

∆=∆

m

mW

ERW

W

σσ

(5-14)

where: ER = Energy Ratio.

∆W1 = Dissipated energy per unit volume for loop 1.

∆W2 = Dissipated energy per unit volume for loop 2.

σ’m1 = Mean effective confining stress corresponding to loop 1.

σ’m2 = Mean effective confining stress corresponding to loop 2.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Dr = 70%

σ’m /σ’mo

25

ru = 0.8

0

5

10

15

20

CSR = 0.1, σ’mo = 100kPa

65.1

''

−

mo

m

σσ

Dr = 40%

ER = 1

Ener

gy R

atio

(ER)

ru = 0.0

Figure 5-17. The ratio of the energies dissipated during two cycles of loading in an undrained stress controlled cyclic test.

160

Equation (5-14) is quite significant and forms the basis of the proposed normalization

technique for removing (or compensating for) the dissipated energy resulting from the

softening of the soil due to increases in pore pressures. In short, if the energy dissipated

in a sample confined at σ’m2 is known, the energy that would have been dissipated in the

same sample had it been confined at another effective confining stress (σ’m1) can be

determined.

An expression for the cumulative energy dissipated in a cyclic triaxial specimen was

derived previously (i.e., Equation (3-18)) and is presented below as being divided by

initial mean effective confining stress.

∑ −+⋅=∆

++i

iaiaididmomo

W ))(('2

1' ,1,,1, εεσσ

σσ (5-15a)

where: ∆W = The cumulative energy dissipated in a cyclic triaxial sample.

σ’mo = Initial mean effective confining stress.

σd,i = The ith increment in deviatoric stress.

εa,i = The ith increment in axial strain.

By applying the approximation ER ≈ (σ’m /σ’mo)-1.65 (i.e., σ’m1 = σ’mo and σ’m2 = σ’m) to

remove the effects of strain softening due to increased pore pressures, the above

expression becomes: 65.1

,1,,1,,1,

'

'2''

))(('2

1'

+−+⋅=

∆ +++∑

mo

imim

iiaiaidid

momo

moW

σσσ

εεσσσσ

σ (5-15b)

where: ∆Wσ’mo = Energy dissipated in sample if effective confining stress remains σ’mo.

σ’m,i = The mean effective confining stress at the ith increment of loading.

Finally, if the above expression is used to integrate the stress-strain hysteresis loops up to

initial liquefaction, the normalized energy capacity (NEC) of the specimen, for which the

effects of strain softening due to increased pore pressures are removed, can be determined

from the following expression.

161

65.1,1,

1

1,1,,1, '2

''))((

'21

+−+⋅= +

−

=++∑

mo

imimn

iiaiaidid

mo

NECσ

σσεεσσ

σ (5-15c)

where: NEC = Normalized energy capacity (dimensionless).

n = Number of load increments up to failure (e.g., liquefaction).

The NEC determined from this expression can be compared to the Capacity curve shown

in Figure 5-11, which was developed from total stress numerical analyses of earthquake

case histories. The validity of Equation (5-15c) can be assessed from analyzing cyclic

triaxial test data. For a soil sample subjected to a stress controlled, sinusoidal loading

wherein strain softening occurs due to pore pressure increases, the rate of energy

dissipation will increase with time. This can be understood from examining Figure 5-

15b. The hysteresis loops shown in this figure were scribed at a constant rate and

increase in size as the loading progresses. Accordingly, if the cumulative energy

dissipated in the sample is plotted as a function of time, the curve would have a slope that

progressively increases with time (the slope represents the rate of energy dissipation).

Using Equation (5-15a), such a plot was computed and is shown in Figure 5-18.

Correspondingly, for a total stress numerical analysis of the same soil sample (e.g., using

SHAKE), the energy will dissipate at a constant rate. This can be understood from

examining Figure 5-15a, wherein the hysteresis loops remain unchanged from one cycle

to the next and are scribed at a constant rate. The corresponding plot of the cumulative

energy dissipated as a function of time will be a straight line with the slope representing

the rate of energy dissipation. Accordingly, the cumulative energy dissipated as a

function of time for cyclic laboratory data computed using Equation (5-15b) should also

plot as a straight line, if the proposed normalization technique is valid. The same cyclic

triaxial test data analyzed using Equation (5-15a) were reanalyzed using Equation (5-

15b), the results of which are shown in Figure 5-18. As may be seen in this figure, the

results of the reanalysis plot approximately as a straight line and is typical of the data

analyzed similarly for several hundred cyclic triaxial tests.

162

Nor

mal

ized

Dis

sipa

ted

Ener

gy

25

mo

liqW'@

σ∆

r u =

1.0

NEC

Normalized per Equation (5-15b) Normalized per Equation (5-15a)

302015Time (sec)

5 100

0.04

0.02

0.00

Figure 5-18. Normalized dissipated energies computed from cyclic triaxial test data using Equations (5-15a) and (5-15b). The data shown in this figure is that listed as m0e76c28 in Table 5-2a.

As stated previously, several researchers inappropriately compared the energies from

total stress numerical analyses to undrained cyclic laboratory data. The significance of

the error in this can be seen in Figure 5-18 by comparing the energies at initial

liquefaction as determined by Equations (5-15a) and (5-15b). As may be seen in this

figure, the Capacity of the soil determined from Equation (5-15a) is significantly higher

than that computed using Equation (5-15c). The appropriate procedure for computing

Capacity depends on how the Demand is determined, and errors are only introduced

when the two are determined inconsistently. For example, if total stress numerical

analyses are used to compute the Demand (e.g., using SHAKE), then the Equation (5-

15c) should be used to compute a comparable Capacity from undrained cyclic laboratory

test data. However, if effective stress numerical analyses (i.e., analyses that include

strain softening due to pore pressure increases) are used to compute the Demand, then

Equation (5-15a) should be used to compute the Capacity of the soil from undrained

cyclic laboratory data. This latter approach was used in the energy-based liquefaction

evaluation procedure proposed by Professor J.L. Figueroa and his associates at Case

Western Reserve University (e.g., Liang (1995)).

163

Using Equation (5-15c), NEC was computed from several hundred isotropically

consolidated, stress controlled cyclic triaxial data. The specimens were reconstituted,

moist tamped, non-plastic silt-sand mixtures having varying fines contents (FC) and

relative densities (Dr). The specimens were reconstituted using Monterey and Yatesville

sands and non-plastic silt, where the Monterey sand is medium grained and the Yatesville

sand is fine grained. The full details of the testing program and how Dr was determined

for silty samples are given in Polito (1999) and Polito and Martin (2001). As shown in

Polito (1999) and Polito and Martin (2001), the non-plastic silt has no influence on the

strength of the soil up to the limiting silt content of the base sand. The limiting silt

content is defined as that required to fill the voids of the sand skeleton. Above the

limiting silt content, the sand particles are suspended in the silt matrix and have no

influence on the soil behavior, while the silt particles control the behavior of the

specimen. For relative densities less than about 80%, the limiting silt content is between

25% and 35% for Monterey sand and between 37% and 50% for the Yatesville sand.

Polito (1999) and Polito and Martin (2001) do not specify the limiting silt contents as

functions of relative density. However, from reanalysis of their data, the author

hypothesizes that for samples less than about Dr = 80%, the soil particles can rearrange so

that the sand grains become in contact and control the behavior of the soil, even if in the

initial undisturbed state the silt lies between the sand grain contact surfaces. However, at

higher relative densities, the sand particles cannot easily rearrange, and the silt lying

between the sand grains prevents grain-to-grain contact of the sand, thus the silt controls

the soil behavior. More refined estimates of limiting silt contents as functions of relative

density and fines content cannot be determined from the available laboratory data.

The specimens were divided into two groups according to whether their silt contents were

above or below the limiting values of the sand. The computed NEC values are plotted as

functions of Dr in Figures 5-19 and 5-20 for samples having FC below and above the

limiting silt contents, respectively. Failure was defined as either initial liquefaction (ru =

1.0) or a double amplitude strain of 5% (εDA = 5%), whichever occurred first. Tables 5-

2a and 5-2b list the failure modes and the NEC values for the Monterey and Yatesville

silt-sand mixtures, respectively.

164

For comparison, the energy-based Capacity curve from Figure 5-11 is also shown in

Figures 5-19 and 5-20. To convert N1,60 to Dr, the following relation was used:

60,115 NDr ⋅= (%) (Skempton 1986) (5-16)

As can be seen in Figure 5-19, the field based curve gives an approximate lower bound of

the normalized energy capacities determined from laboratory data for samples below the

limiting silt content. This is consistent with the selected position of the boundary

separating the field data corresponding to liquefaction and no liquefaction.

From examining the NEC values listed in Table 5-2a, it may be observed that samples

subjected to higher amplitude CSR loadings have lower computed NEC values, similar to

the PEC discussed in Chapter 4. This trend could result from energy being dissipated in

the test equipment and the membrane surrounding the sample. At higher amplitude

loadings the energy lost due to these mechanisms may be small as compared to energy

dissipated due to friction between the sand grains. However, as the amplitude of the

loading decreases, these mechanisms may become important. Also, a viscous mechanism

due the relative movement of the pore water and the sand grains (i.e., viscous drag) may

become more prominent at lower amplitude loadings (Hall 1962). A detailed laboratory

study is required to examine the load dependence of NEC.

165

100 120806040200-20-40-600.00

0.02

Yatesville Lab Data (FC≤37%)Monterey Lab Data (FC≤25%)Field Based Liquefaction Curve

0.12

0.10

Nor

mal

ized

Ene

rgy

Cap

acity

(NEC

)

0.08

0.06

0.04

Relative Density (%)

Figure 5-19. NEC values computed from stress controlled cyclic triaxial tests for samples having silt contents below the limiting value. Also shown is the capacity curve derived from field case histories. The negative relative densities were generated by moist tamping the laboratory samples, thus creating specimens having larger void ratios than the maximum predicted by the index tests using dry soil.

From Figure 5-20, it can be observed that the field based curve is considerably greater

than the Capacities of samples having a silt content above the limiting values. A similar

trend was identified by Polito (1999) and Polito and Martin (2001) from the evaluation of

silt-sand mixtures using cyclic stress ratio (CSR) as the measure of Capacity and

Demand. As hypothesized by Polito (1999) and Polito and Martin (2001), above the

limiting silt content, the sand particles are suspended in the silt matrix and have no

influence on the soil behavior, while the silt particles control the behavior of the

specimen. The data presented in Figure 5-20 show that silt is less resistant to liquefaction

than sand having the same relative density (i.e., the zone representing liquefaction in

Figure 5-20 is larger than the corresponding zone for sands shown in Figure 5-19).

166

25%

FC=20%

0.06 25%

20%

20%25%

0.04

0.02

-600.00

Yatesville Lab Data (FC>37%)Monterey Lab Data (FC≤25%)Monterey Lab Data (FC>25%)Field Based Liquefaction Curve

0.12

0.10

Nor

mal

ized

Ene

rgy

Cap

acity

(NEC

)

0.08

-40 -20 0 20 40 60 80 100 120

Relative Density (%)

Figure 5-20. NEC values computed from stress controlled cyclic triaxial tests for samples having silt contents above the limiting value. Also shown is the capacity curve derived from field case histories.

In addition to the NEC values, Tables 5-2a and 5-2b also list the dissipated energy per

unit volume of soil (∆W), ∆W/σ’3o, and PEC. Liang (1995) used ∆W to define the soil

capacity, while Alkhatib (1994) used ∆W/σ’3o.

167

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168

Table 5-2a. Cyclic Triaxial Laboratory Test Data: Monterey Sand – Silt Mixtures. File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode m0e68c30 81.1 0.511 2 99.74 168.1 15.22 0.153 0.0469 22.46 0 ru = 1 m0e68c35 81.1 0.598 2 99.84 30.1 11.33 0.113 0.0289 12.66 0 ru = 1 m0e68c40 81.1 0.691 2 100.2 12.1 10.34 0.103 0.027 10.6 0 ru = 1 m0e72c27 65.3 0.439 2 100.4 73.1 7.68 0.076 0.021 0.07 0 ru = 1 m0e72c30 64.2 0.505 2 99.11 58.1 8.26 0.083 0.0273 11.29 0 ru = 1 m0e72c35 64.2 0.590 2 99.53 12.6 7.52 0.076 0.0184 8.51 0 ru = 1 m0e73c25 57.4 0.399 2 100 351.0 16.42 0.164 0.071 31.97 0 ru = 1 m0e73c31 58.4 0.525 2 99.11 15.1 5.02 0.051 0.0149 6.3 0 ru = 1 m0e73c37 58.4 0.597 2 99.74 5.1 4.18 0.042 0.01 6 0 ru = 1 m0e74c22 46.3 0.362 1 98.16 108.1 4.18 0.043 0.0136 5.99 0 ru = 1 m0e74c25 46.8 0.389 1 100.8 50.6 3.88 0.039 0.0102 4.6 0 ru = 1 m0e74c30 46.8 0.522 1 97.74 6.1 3.52 0.036 0.0086 4.03 0 ru = 1 m0e75c20 47.4 0.314 2 100 457.0 8.35 0.084 0.0318 14.5 0 ru = 1 m0e75c25 47.4 0.408 2 99.63 34.6 4.02 0.04 0.0117 5.13 0 ru = 1 m0e75c30 47.9 0.484 2 99.21 7.1 3.59 0.036 0.0079 4.55 0 ru = 1 m0e76c22 42.6 0.343 2 100.4 87.0 6.26 0.062 0.0187 8.46 0 ru = 1 m0e76c25 42.6 0.395 2 99.95 36.0 4.19 0.042 0.0112 4.86 0 ru = 1 m0e76c28 42.9 0.465 2 98.79 12.1 3.37 0.034 0.0087 4.52 0 ru = 1 m0e84c20 -4.7 0.179 1 99.53 3.9 1.04 0.0104 0.0011 0.54 0 ru = 1 m0e84c21 -3.7 0.18 1 99.95 5.0 0.48 0.0048 0.0008 0.38 0 ru = 1 m5e61c32 82.1 0.525 2 101.1 46.1 12.32 0.1218 0.0301 12.71 5 ru = 1 m5e61c35 79.9 0.591 2 100.8 28.1 12.16 0.1209 0.0297 11.46 5 ru = 1 m5e61c40 81.6 0.681 2 100.9 15.1 11.53 0.1143 0.0283 10.42 5 ru = 1 m5e65c27 64.7 0.426 2 101.4 83.1 10.17 0.1005 0.0314 13.61 5 ru = 1 m5e65c30 64.4 0.488 2 100.5 22.6 7.11 0.0708 0.016 6.88 5 ru = 1 m5e65c33 63.3 0.552 2 99.53 10.1 7.45 0.0748 0.0149 6.44 5 ru = 1

169

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode m5e67c22 53.4 0.349 2 99.63 444.3 18.94 0.1899 0.0677 34.55 5 ru = 1 m5e67c23 54 0.366 2 100.7 163.1 9.38 0.0933 0.025 13.33 5 ru = 1 m5e67c25 54.5 0.408 2 99.21 25.0 5.03 0.0507 0.0107 4.6 5 ru = 1 m5e67c38 55.6 0.478 2 97.42 9.0 4.89 0.0502 0.0104 4.49 5 ru = 1 m5e68c20 49.7 0.308 2 98.9 416.3 20.97 0.212 0.0667 36.64 5 ru = 1 m5e68c25 49.7 0.401 2 99.84 38.1 5.81 0.0582 0.0131 5.98 5 ru = 1 m5e68c30 53.4 0.516 2 93.74 10.1 6.17 0.0659 0.0127 4.64 5 ru = 1 m10e53c30 81.0 0.514 2 98.48 109.1 22.2 0.2258 0.0628 25.9 10 ru = 1 m10e53c35 79.7 0.601 2 99.84 44.0 17.74 0.1778 0.0455 18.23 10 ru = 1 m10e53c40 80.6 0.695 2 99.95 18.0 14.35 0.1434 0.0368 11.69 10 ru = 1 m10e57c25 62.9 0.403 2 100.3 112.1 11.08 0.1108 0.029 13.66 10 ru = 1 m10e57c35 62.9 0.595 2 98.69 9.1 8.37 0.0847 0.0187 6.74 10 ru = 1 m10e60c20 52.2 0.306 2 99.53 173.1 7.61 0.0766 0.0201 9.52 10 ru = 1 m10e60c25 51.7 0.398 2 99.21 35.0 5.71 0.0576 0.0141 5.88 10 ru = 1 m10e60c30 51.7 0.495 2 99.53 8.0 5.17 0.052 0.0103 3.73 10 ru = 1 m10e68c20 53.3 0.3 2 99.11 44.1 3.69 0.0373 0.0089 4.23 10 ru = 1 m10e68c22 54.1 0.340 2 100.2 21.1 3.40 0.0338 0.0073 3.2 10 ru = 1 m10e68c25 55.2 0.412 2 99.95 6.1 2.81 0.0281 0.0063 3.05 10 ru = 1 m15e48c30 81.7 0.501 2 100.9 49.0 12.38 0.1226 0.0307 11.69 15 ru = 1 m15e48c35 82.1 0.579 2 99.84 23.1 11.51 0.1151 0.0273 9.08 15 ru = 1 m15e48c38 81.7 0.673 2 99.53 20.1 15.34 0.1542 0.0427 13.59 15 ru = 1 m15e53c23 62.5 0.358 2 98.58 45.1 6.13 0.0622 0.0161 7.28 15 ru = 1 m15e53c28 62.9 0.469 2 98.58 10.1 4.49 0.0456 0.0106 4 15 ru = 1 m15e68c13 10.8 0.178 2 98.58 99.0 2.69 0.0273 0.0061 2.87 15 ru = 1 m15e68c15 11.2 0.215 2 99.32 38.0 2.61 0.0263 0.0059 2.79 15 ru = 1 m15e68c20 10.8 0.306 2 99.21 8.0 2.47 0.0249 0.0046 2 15 ru = 1 m20e36c25 96.8 0.417 2 99 338.2 27.59 0.2788 0.0705 33.77 20 ru = 1 m20e36c35 97.5 0.614 2 97.84 56.0 21.07 0.2149 0.0533 18.89 20 ru = 1 m20e36c40 96.8 0.701 2 99.42 42.7 34.97 0.3525 0.0782 20.11 20 εDA = 5%

170

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode m20e46c18 60.8 0.261 2 99.42 310.2 10.40 0.1048 0.0289 13.85 20 ru = 1 m20e46c20 61.1 0.317 2 99.32 55.0 5.98 0.0603 0.0138 6.11 20 ru = 1 m20e46c25 60.8 0.407 2 98.9 14.1 4.24 0.043 0.0094 3.72 20 ru = 1 m20e46c28 60.4 0.452 2 99.32 12.1 4.54 0.0458 0.0107 4.21 20 ru = 1 m20e68c13 -2.1 0.167 2 97.74 55.3 2.10 0.0215 0.0049 2.21 20 ru = 1 m20e68c17 -1.1 0.247 2 98.37 8.0 2.10 0.0214 0.004 1.66 20 ru = 1 m25e35c30 97.9 0.516 2 99.53 52.0 19.74 0.1979 0.0388 12.8 25 ru = 1 m25e35c40 97.9 0.709 2 99.21 25.9 27.65 0.2788 0.0654 15.06 25 εDA = 5% m25e35c50 98.3 0.903 2 99.11 11.8 21.42 0.2162 0.0621 20.84 25 εDA = 5% m25e42c20 75.9 0.305 2 99.95 124.1 8.80 0.0881 0.0266 12.43 25 ru = 1 m25e42c25 76.2 0.407 2 99.21 21.0 5.32 0.0537 0.0117 4.65 25 ru = 1 m25e42c28 75.5 0.47 2 99.63 14.1 5.16 0.0518 0.0115 4.28 25 ru = 1 m25e47c17 60.7 0.251 2 98.6 60.0 3.65 0.0372 0.0096 4.45 25 ru = 1 m25e47c20 60 0.311 2 99.21 29.1 4.23 0.0427 0.01 4.34 25 ru = 1 m25e47c30 61.4 0.450 2 99.42 6.1 3.62 0.0365 0.0073 2.8 25 ru = 1 m25e68c09 -2.1 0.114 2 99.11 155.3 2.20 0.0222 0.0052 2.45 25 ru = 1 m25e68c12 -0.3 0.149 2 95.95 55.0 2.29 0.0239 0.0043 1.99 25 ru = 1 m25e68c15 -1.7 0.226 2 97.42 8.0 2.26 0.0232 0.0034 1.47 25 εDA = 5% m35e61c11 55.8 0.081 1 100.2 42.4 0.56 0.0056 0.0011 0.45 35 εDA = 5% m35e61c13 69.2 0.110 2 99.42 11.5 1.00 0.01 0.001 0.31 35 ru = 1 m35e61c15 62.5 0.119 1 98.79 11.3 0.92 0.0093 0.0013 0.47 35 εDA = 5% m35e68c10 64.4 0.075 1 100.5 1198.0 0.03 0.00027 0.00017 0.171 35 ru = 1 m35e68c13 67.6 0.110 1 99.42 20.0 0.57 0.00571 0.00095 0.311 35 ru = 1 m35e68c15 72.8 0.119 1 98.79 11.3 0.92 0.0093 0.0013 0.473 35 εDA = 5% m50e68c14 73.9 0.112 1 97 466.0 1.25 0.0129 0.0014 0.388 50 εDA = 5% m50e68c16 76.9 0.125 1 98.9 9.4 1.89 0.01914 0.00161 0.442 50 ru = 1 m50e68c18 77.3 0.160 1 98.37 20.0 1.37 0.01396 0.00283 0.616 50 ru = 1 m50e68c20 79.6 0.148 1 98.68 20.0 1.85 0.01868 0.00197 0.51 50 ru = 1

171

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode m75e68c16 92 0.136 1 98.58 13.7 1.66 0.0169 0.0015 0.511 75 εDA = 5% m75e68c20 90.5 0.159 1 100.5 6.4 1.59 0.0158 0.0018 0.686 75 εDA = 5%

Table 5-2b. Cyclic Triaxial Laboratory Test Data: Yatesville Sand – Silt Mixtures.

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode y0e76c30 68 0.286 1 99.1 23.1 2.46 0.0248 0.0053 2.195 0 ru = 1 y0e76c33 70 0.323 1 99.63 11.1 2.59 0.026 0.00556 2.06 0 ru = 1 y0e76c35 68.3 0.342 1 98.58 11.1 2.54 0.0257 0.00554 2.179 0 ru = 1 y0e90c19 27.3 0.137 1 99.42 36.1 0.49 0.0049 0.00102 0.452 0 ru = 1 y0e90c22 24.8 0.168 1 101.3 14 0.62 0.0061 0.00094 0.429 0 ru = 1 y0e90c25 25.6 0.224 1 98.05 5 1.42 0.0144 0.0015 0.62 0 ru = 1 y0e11c10 -10.6 0.076 1 98.58 31.4 0.17 0.00172 0.000504 0.222 0 ru = 1 y0e11c12 -10.4 0.09 1 99.42 14.5 0.32 0.00322 0.00057 0.226 0 ru = 1 y0e12c09 -32.7 0.066 1 100.6 43.3 0.3 0.00302 0.000606 0.258 0 ru = 1 y0e12c10 -44.5 0.063 1 99.74 78.5 0.37 0.00372 0.000799 0.362 0 ru = 1 y0e12c11 -37.2 0.0668 1 98.47 60.4 0.37 0.00371 0.000716 0.306 0 ru = 1 y0e12c13 -32.3 0.087 1 99.84 11.3 0.53 0.00531 0.000866 0.313 0 ru = 1 y0e12c14 -32 0.110 1 98.26 2.3 0.49 0.00497 0.0000998 0.604 0 ru = 1 y4e86c17 27.5 0.136 1 99.53 8.6 0.38 0.00381 0.000784 0.334 4 ru = 1 y4e86c18 22.9 0.148 1 98.16 6.3 0.67 0.00683 0.000939 0.301 4 ru = 1 y4e86c20 28.2 0.163 1 98.16 3.2 0.74 0.00756 0.00102 0.408 4 ru = 1 y4e86c23 19.6 0.211 1 100.3 15 0.88 0.00881 0.001576 0.706 4 ru = 1 y4e76c27 52.1 0.249 1 99.84 33.1 2.56 0.0256 0.005406 2.449 4 ru = 1 y4e76c30 52.4 0.283 1 100.8 11.1 1.74 0.01724 0.003722 1.537 4 ru = 1

172

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode y4e76c35 52.1 0.335 1 99.21 7.1 2.01 0.02025 0.004107 1.384 4 ru = 1 y4e83c22 29.5 0.188 1 100.8 54 1.36 0.01349 0.002902 1.468 4 ru = 1 y4e83c25 30.8 0.229 1 99.31 19.1 1 0.01003 0.002185 0.99 4 ru = 1 y4e83c28 29.5 0.243 1 100.6 11.1 1.21 0.01196 0.002771 1.181 4 ru = 1 y4e97c12 -10.9 0.094 1 96.89 28.3 0.43 0.00447 0.000746 0.296 4 ru = 1 y4e97c14 -9 0.113 1 99.53 12 0.42 0.00418 0.000691 0.299 4 ru = 1 y4e97c15 -9.6 0.127 1 96.79 7.2 0.41 0.00419 0.000713 0.291 4 ru = 1 y4e11c10 -36.5 0.077 1 97.74 30.3 0.31 0.00322 0.000632 0.263 4 ru = 1 y4e11c12 -35.5 0.081 1 99.84 20.3 0.58 0.00576 0.000753 0.261 4 ru = 1 y4e11c14 -30.2 0.111 1 98.89 6.3 0.61 0.00616 0.000846 0.327 4 ru = 1 y7e76c26 34 0.241 1 97.21 20.1 1.95 0.02014 0.00398 1.77 7 ru = 1 y7e76c27 34.6 0.251 1 98.47 25.1 2.04 0.02074 0.004408 1.911 7 ru = 1 y7e76c30 33.7 0.283 1 100.6 9.1 1.33 0.01318 0.002986 1.247 7 ru = 1 y7e76c32 33.7 0.307 1 98.05 6 1.77 0.01803 0.003546 1.212 7 ru = 1 y7e78c25 27.3 0.225 1 100.5 58.1 2.44 0.0243 0.005938 2.896 7 ru = 1 y7e78c31 29.4 0.297 1 98.05 15.1 1.93 0.01963 0.004692 1.987 7 ru = 1 y7e78c34 30.7 0.323 1 100.9 7.1 1.81 0.01789 0.004277 1.611 7 ru = 1 y7e92c12 -9 0.0859 1 99.84 65 0.49 0.00494 0.000986 0.46 7 ru = 1 y7e92c15 -8.7 0.124 1 99.74 16.8 0.55 0.00548 0.000848 0.349 7 ru = 1 y7e92c18 -5.5 0.161 1 96.47 4.2 0.59 0.00616 0.000953 0.406 7 ru = 1 y7e10c11 -20.4 0.084 1 100.9 29.4 0.51 0.00502 0.000785 0.324 7 ru = 1 y7e10c12 -31.7 0.0884 1 102.5 29.6 0.33 0.00324 0.000745 0.328 7 ru = 1 y7e10c14 -25.3 0.113 1 97.63 8.3 0.51 0.00519 0.000797 0.321 7 ru = 1 y7e10c16 -25.3 0.132 1 98.05 4.3 0.61 0.00625 0.000934 0.352 7 ru = 1 y12e70c35 45.2 0.339 1 99.1 13.1 2.78 0.02801 0.006327 2.454 12 ru = 1 y12e70c38 42.4 0.37 1 100.4 15.1 3.48 0.03463 0.008331 3.131 12 ru = 1 y12e70c41 43.9 0.386 1 102.4 12.1 3.89 0.03805 0.009533 4.071 12 ru = 1 y12e67c28 54.4 0.255 1 99.63 11.7 2.81 0.0283 0.00423 1.578 12 εDA = 5% y12e67c31 55.1 0.284 1 98.89 11.1 1.59 0.01607 0.003844 1.637 12 ru = 1

173

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode y12e67c33 56.9 0.312 1 99 7.1 1.56 0.01579 0.004209 1.542 12 ru = 1 y12e76c23 25.7 0.201 1 100.3 47.1 2.12 0.0211 0.005565 2.688 12 ru = 1 y12e76c26 24.7 0.231 1 99.53 12.6 1.24 0.01245 0.003142 1.425 12 ru = 1 y12e76c28 27.5 0.267 1 98.89 7 1.79 0.01816 0.002898 0.998 12 ru = 1 y12e84c18 3.4 0.153 1 101.1 29 0.78 0.00772 0.002089 1 12 ru = 1 y12e84c22 2.8 0.188 1 100.6 10.9 1.65 0.0164 0.00152 0.642 12 εDA = 5% y12e84c25 4.6 0.224 1 101 4 2.14 0.02113 0.001704 0.706 12 ru = 1 y12e92c12 -11.1 0.09 1 98.68 28.3 0.41 0.00416 0.000825 0.339 12 ru = 1 y12e92c13 -11.8 0.097 1 100.9 31.4 0.43 0.00424 0.00095 0.403 12 ru = 1 y12e92c15 -9.6 0.116 1 99.31 9.3 0.61 0.00617 0.000834 0.332 12 ru = 1 y12e92c17 -9.9 0.144 1 98.89 3.4 1.05 0.01065 0.00128 0.66 12 ru = 1 y17e59c36 76.1 0.349 1 98.89 15.7 6.63 0.067 0.00939 1.968 17 εDA = 5% y17e59c39 77.8 0.376 1 100.6 16.7 7.17 0.0713 0.01215 3.358 17 εDA = 5% y17e59c42 76.3 0.406 1 100.3 15.1 7.08 0.07068 0.015383 3.645 17 ru = 1 y17e70c25 37.7 0.102 1 99.0 118.8 0.61 0.00614 0.000694 0.233 17 ru = 1 y17e70c30 41.0 0.156 1 98.37 16.7 0.98 0.00991 0.00219 0.75 17 ru = 1 y17e70c32 38.3 0.245 1 102.8 2.5 2.09 0.02029 0.002856 1.071 17 ru = 1 y17e73c22 25.3 0.201 1 98.68 46.6 1.5 0.01518 0.004733 1.972 17 ru = 1 y17e73c25 26.0 0.212 1 100.3 25 1.04 0.01036 0.002369 0.938 17 ru = 1 y17e73c27 27.7 0.248 1 100.2 7 1.2 0.01202 0.002149 0.836 17 ru = 1 y17e76c22 18.0 0.198 1 98.26 38 1.44 0.01469 0.003407 1.587 17 ru = 1 y17e76c25 18.0 0.222 1 100.1 14 1.31 0.01309 0.002409 1.027 17 ru = 1 y17e76c28 17.0 0.263 1 99.31 6 1.67 0.01682 0.002706 0.985 17 ru = 1 y17e84c12 -10.7 0.091 1 97.21 62.4 0.64 0.00662 0.001185 0.511 17 ru = 1 y17e84c15 -7.7 0.128 1 97.32 20.9 0.8 0.00818 0.001063 0.443 17 ru = 1 y17e84c18 -6.7 0.159 1 97.74 5.2 0.88 0.00899 0.001212 0.458 17 εDA = 5% y26e63c30 40.3 0.283 1 99.95 11.1 1.34 0.0134 0.002904 1.147 26 ru = 1 y26e63c33 42.8 0.301 1 99.74 14.1 2.1 0.021 0.004474 1.703 26 ru = 1 y26e63c36 43.2 0.33 1 99.42 14.1 2.68 0.02696 0.005865 2.132 26 ru = 1

174

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode y26e67c25 26.9 0.19 1 100.2 69 1.12 0.01119 0.002688 1.005 26 ru = 1 y26e67c27 28.1 0.249 1 98.89 13 1.35 0.0136 0.002937 1.24 26 ru = 1 y26e67c29 28.5 0.273 1 100.9 5.1 1.06 0.01049 0.002224 0.883 26 ru = 1 y26e71c19 24.9 0.161 1 100.5 21 1.1 0.01095 0.001876 0.846 26 ru = 1 y26e71c22 31.0 0.192 1 99.0 14.1 1.1 0.0111 0.002609 1.149 26 ru = 1 y26e71c24 27.7 0.219 1 97.95 9 1.58 0.01608 0.002422 0.93 26 ru = 1 y26e76c15 -10.2 0.126 1 99.21 30.1 0.58 0.00585 0.001262 0.556 26 ru = 1 y26e76c18 -9 0.151 1 100.9 9.4 1.24 0.01229 0.001449 0.526 26 ru = 1 y26e76c20 -9 0.169 1 99.74 7.1 0.91 0.00911 0.001447 0.585 26 ru = 1 y37e62c25 25.1 0.224 1 102.6 12.1 0.75 0.00736 0.001423 0.592 37 ru = 1 y37e62c28 25.8 0.234 1 99.63 10 1.08 0.01082 0.002196 0.876 37 ru = 1 y37e62c31 24.4 0.285 1 100.6 4.1 2.48 0.02468 0.004575 1.004 37 ru = 1 y37e67c14 4.5 0.112 1 99.84 35.7 0.49 0.00492 0.001104 0.493 37 ru = 1 y37e67c15 8.4 0.119 1 99.42 17.9 0.74 0.00742 0.001011 0.434 37 ru = 1 y37e67c17 7.7 0.141 1 99.95 9.1 0.6 0.00598 0.00092 0.385 37 ru = 1 y37e69c19 1 0.116 1 98.79 29 0.4 0.00404 0.001062 0.39 37 ru = 1 y37e69c20 0 0.178 1 98.47 7 0.89 0.00906 0.00126 0.463 37 ru = 1 y37e69c22 3.8 0.194 1 99.63 3.4 1.83 0.0184 0.001872 0.436 37 εDA = 5% y37e76c10 -24.7 0.0779 1 95.32 46.3 0.32 0.00338 0.000658 0.278 37 ru = 1 y37e76c12 -11.1 0.0918 1 98.47 19.3 0.58 0.00592 0.000755 0.299 37 ru = 1 y37e76c14 -16.4 0.118 1 96.68 7.7 0.88 0.00912 0.001158 0.35 37 εDA = 5% y50e55c18 81.1 0.151 1 100.4 18.1 1.07 0.01068 0.000789 0.265 50 ru = 1 y50e55c20 80.0 0.168 1 100.1 9.7 2.18 0.0218 0.00193 0.59 50 εDA = 5% y50e55c23 91.2 0.205 1 100.2 3.8 1.87 0.0187 0.00395 2.386 50 εDA = 5% y50e67c14 72.2 0.111 1 99.31 12.5 0.74 0.00747 0.000583 0.227 50 ru = 1 y50e67c16 69.8 0.132 1 99.1 7.5 1.37 0.01384 0.001546 0.409 50 ru = 1 y50e67c18 71.6 0.154 1 98.79 4.2 1.14 0.01152 0.001417 0.573 50 ru = 1 y50e76c09 63.4 0.068 1 99.84 48.2 0.65 0.0065 0.00085 0.34 50 εDA = 5%

175

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode y50e76c13 62.2 0.107 1 98.68 11.2 0.73 0.0074 0.00078 0.21 50 εDA = 5% y50e76c15 58.6 0.125 1 96.37 5.7 1.13 0.0117 0.00187 0.555 50 εDA = 5% y50e78c10 62 0.076 1 99.95 46 1.1 0.01106 0.000952 0.394 50 ru = 1 y50e78c12 54.4 0.095 1 99.53 15.2 0.74 0.0074 0.00126 0.42 50 εDA = 5% y50e78c14 69.3 0.114 1 100.5 6.3 0.9 0.009 0.00156 0.499 50 εDA = 5% y75e76c13 85.5 0.105 1 102.5 26.2 0.53 0.00519 0.001037 0.499 75 ru = 1 y75e76c15 86.6 0.121 1 97.95 11.3 1.01 0.0103 0.00167 0.573 75 εDA = 5% y75e76c17 85.8 0.121 1 103.1 13.4 1.18 0.0115 0.00092 0.363 75 εDA = 5% y75e85c14 79.7 0.12 1 96.47 14.3 1.11 0.0115 0.00146 0.593 75 εDA = 5% y75e85c16 78 0.132 1 97.53 9.5 1.28 0.01317 0.001702 0.618 75 ru = 1 y75e85c18 78.2 0.152 1 96.89 5.4 1.57 0.01619 0.002326 0.865 75 ru = 1 y75e90c10 75.1 0.078 1 99.53 36.2 0.85 0.0086 0.00103 0.375 75 εDA = 5% y75e90c13 76.9 0.106 1 99.84 9.3 1.04 0.0104 0.00178 0.603 75 εDA = 5% y75e90c15 74.2 0.121 1 99.42 5.3 1.01 0.0101 0.00218 1.16 75 εDA = 5% se68c13 105.1 0.103 1 99.95 602.5 3.71 0.03712 0.00887 4.43 100 ru = 1 se68c40 104.3 0.364 1 101.1 15.2 3.07 0.03032 0.009069 10.218 100 ru = 1 se68c43 103.3 0.423 1 100.6 21.3 5.02 0.04991 0.018926 19.308 100 ru = 1 se76c25 97.5 0.231 1 98.79 180.2 8.32 0.08432 0.025773 14.232 100 ru = 1 se76c35 95.9 0.340 1 99.53 17.2 2.8 0.0281 0.008895 5.608 100 ru = 1 se76c38 98.5 0.365 1 99.31 3.2 2.33 0.02346 0.006764 4.076 100 ru = 1 se91c18 85.6 0.151 1 98.47 18.2 0.84 0.00853 0.00203 1.018 100 ru = 1 se91c20 86.2 0.171 1 100.9 9.8 1.84 0.0182 0.00244 0.986 100 εDA = 5% se91c22 87.2 0.194 1 98.47 3.7 1.8 0.0182 0.00289 1.325 100 εDA = 5% se95c10 87.6 0.079 1 100.3 52.2 0.98 0.0097 0.00122 0.516 100 εDA = 5% se95c13 88.4 0.098 1 99.95 20.8 1.07 0.0107 0.00148 0.574 100 εDA = 5% se95c15 87.2 0.119 1 100.1 8.8 1.38 0.0138 0.00176 0.757 100 εDA = 5% se99c10 84.3 0.078 1 99.63 46.1 0.88 0.00882 0.001121 0.473 100 ru = 1 se99c13 84.6 0.104 1 99.31 11.2 1 0.0101 0.00181 0.8 100 εDA = 5%

176

File Dr T σ’3o ∆W PEC FC Failure Name (%) CSR (sec) (kPa) Nliq (kPa) ∆W /σ’3o NEC (kPa) (%) Mode se99c15 84.7 0.114 1 100.7 7.8 2.3 0.0229 0.0025 0.973 100 εDA = 5%

177

Appendix 5a: Earthquake time histories used in the parameter study to develop the correlation relating Richter magnitude (M), epicentral distance (ED) from the site to the source, and number of equivalent cycles (Neqv).

All the time histories were recorded on either Site Class a or b profiles, where Site Class

a has an average shear wave velocity (vs) for the upper 30m of the profile greater than

1500m/s, and Site Class b: 760m/s ≤ vs ≤ 1500m/s.

San Francisco Earthquake (also called Daly City Earthquake), March 22,1957, M5.3 1) Golden Gate Park (ct 077): Lat.N. 37.770, Long W. 122.478, ED =11km Site Class a Azimuth 1: 10 Azimuth 2: 100 Sitka Earthquake (Alaska), July 30, 1972, M7.5 2) Sitka magnetic observatory: Lat. N. 57.060, Long W. 135.320, ED = 86km Site Class a Azimuth 1: 180 Azimuth 2: 90 Coyote Lake Earthquake, August 6, 1979, M5.7 3) Gilroy Array 1 (cdmg 47379): Lat. N. 36.973, Long W. 121.572, ED = 16.7km Site Class b Azimuth 1: 320 Azimuth 2: 230 Morgan Hill Earthquake, April 24, 1984, M6.1 4) Gilroy Array 1 (cdmg 47379): Lat. N. 36.973, Long W. 121.572, ED = 38.6km Site Class b Azimuth 1: 67 Azimuth 2: 337 Loma Prieta Earthquake, October 18, 1989, M7.0 5) Cherry Flat Reservoir (usgs 1696): Lat. N. 37.396, Long W. 121.756, ED = 41.1km Site Class a Azimuth 1: 360 Azimuth 2: 270 6) Hollister Sago Vault (usgs 1032): Lat. N. 36.765, Long W. 121.446, ED = 49.1km Site Class b Azimuth 1: 360 Azimuth 2: 270 7) Gilroy Array 1 (cdmg 47379): Lat. N. 36.973, Long W. 121.572, ED = 28.4km Site Class b Azimuth 1: 90 Azimuth 2: 0 8) Monterey City Hall (cdmg 47377): Lat. N. 36.597, Long W. 121.897, ED = 49.0km Site Class b Azimuth 1: 90 Azimuth 2: 0 9) San Fran Sierra Point (cdmg 58539): Lat. N. 37.674, Long W. 122.388, ED = 83.7km Site Class b Azimuth 1: 205 Azimuth 2: 115

178

Petrolia Earthquake (also called Cape Mendocino Earthquake), April 25, 1992, M7.0 10) Bunker Hill FAA (usgs 1584): Lat. N. 40.498, Long W. 124.294, ED = 15.0km Site Class b Azimuth 1: 360 Azimuth 2: 270 Landers Earthquake, June 28, 1992, M7.3 11) Twenty Nine Palms (cdmg 22161): Lat. N. 34.021, Long W. 116.009, ED = 44.2km Site Class b Azimuth 1: 90 Azimuth 2: 0 12) Silent Valley (cdmg 12206): Lat. N. 33.851, Long W. 116.852, ED = 54.7km Site Class b Azimuth 1: 90 Azimuth 2: 0 13) Amboy (cdmg 21081): Lat. N. 34.560, Long W. 115.743, ED = 75.2km Site Class b Azimuth 1: 90 Azimuth 2: 0 14) Whitewater Canyon (usgs 5072): Lat. N. 33.989, Long W. 116.655, ED = 31.0km Site Class b Azimuth 1: 270 Azimuth 2: 180 Northridge Earthquake, January 17, 1994, M6.7 15) Lake Hughes 9 (cdmg 24272): Lat. N. 34.608, Long W. 118.558, ED = 44.7km Site Class b Azimuth 1: 90 Azimuth 2: 360 16) Littlerock – Brainard Canyon (cdmg 23595): Lat. N. 34.486, Long W. 117.980, ED = 60.1km Site Class b Azimuth 1: 90 Azimuth 2: 180 17) Mt. Baldy – Elem. Sch (cdmg 23572): Lat. N. 34.233, Long W. 117.661, ED = 81.0km Site Class a Azimuth 1: 90 Azimuth 2: 180 18) Mt. Wilson (cdmg 24399): Lat. N. 34.224, Long W. 118.057, ED = 44.6km Site Class a Azimuth 1: 90 Azimuth 2: 360 19) Rancho Cucamnga – Deer Canyon (cdmg 23598): Lat. N. 34.169, Long W. 117.579, ED = 88.7km Site Class b Azimuth 1: 90 Azimuth 2: 180 20) Rancho Palos Verdes (cdmg 14404): Lat. N. 33.746, Long W. 118.396, ED = 53.2km Site Class b Azimuth 1: 90 Azimuth 2: 0 21) Wrightwood – Jackson Flat (cdmg 23590): Lat. N. 34.381, Long W. 117.737, ED = 76.4km Site Class b Azimuth 1: 90 Azimuth 2: 180

179

22) 8510 Wonderland Ave, LA (usc 17): Lat. N. 34.114, Long W. 118.380, ED = 18.2km Site Class b Azimuth 1: 185 Azimuth 2: 95 23) 1250 Howard Rd., Burbank (usc 59): Lat. N. 34.204, Long W. 118.302, ED = 22.0km Site Class b Azimuth 1: 330 Azimuth 2: 60 24) Griffith Observatory (usgs 141): Lat. N. 34.118, Long W. 118.299, ED = 24.3km Site Class b Azimuth 1: 360 Azimuth 2: 270

The station designations are as follows:

cdmg: California Division of Mines and Geology Strong Motion Instrumentation

Program

ct: California Institute of Technology Civil Engineering Department

usc: University of Southern California Civil Engineering Department

usgs: United States Geological Survey National Strong Motion Program

180

Appendix 5b: Profiles used in the parametric study to develop the correlation relating Richter magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv).

The profiles used in the parametric study were selected to be representative of actual

profiles. The profiles used had constant, decreasing, and increasing N1,60 with depth. The

water table was placed at the soil surface and at a depth of approximately 11ft. Both soft

and medium stiff profiles were examined. The following expressions were used to relate

N1,60 to Gmax.

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

)'sin(1 φ−=oK

voo

moK

'321

' σσ ⋅

+=

( )5.0

21

3/160,1max

'440

=

PaPaNG moσ

(Seed et al. 1986)

where Pa1 and Pa2 are atmospheric pressure having the same units as Gmax and σ’mo,

respectively. The shear modulus and damping degradation curves proposed by Ishibashi

and Zhang (1993) were used in the site response analyses.

Figures 5b-1 through 5b-4 show the profiles used in the parametric study.

181

b)

γt = 120pcf

Profile 1 Profile 5 Profile 9

Dep

th (f

t)

0

10

20

30

40

50

60

70

80

90

0100

a)

Profile 1 Profile 5 Profile 9

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000Shear Wave Velocity (psf)

100

0

Dep

th (f

t)

6 12 18N1,60

24 30

γrock = 140pcf, Gmax =59521700psf (Site Class b) Gmax =108695700psf (Site Class a)

Figure 5b-1. Profiles 1, 5, and 9 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60.

182

b)

γt = 125pcf

Profile 2 Profile 6 Profile 10

Dep

th (f

t)

0

10

20

30

40

50

60

70

80

90

12 18 N1,60

60100

a)

Profile 2 Profile 6 Profile 10

0 200 400 600 800 1000Shear Wave Velocity (psf)

0 10 20 30

40

Dep

th (f

t)

50

60 70 80 90

100 24 30

γrock = 140pcf, Gmax =59521700psf (Site Class b)

Gmax =108695700psf (Site Class a)

Figure 5b-2. Profiles 2, 6, and 10 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60.

183

b)

γt = 120pcf

Profile 3 Profile 7 Profile 11

0

10

20

30

40

0 6 12 18 100

90

80

70

60

50

N1,60

Dep

th (f

t)

24

a)

Profile 3 Profile 7 Profile 11

0 200 400 600 800 1000Shear Wave Velocity (psf)

0 10 20

30

40

Dep

th (f

t)

50

60 70 80

90

100 30 γrock = 140pcf, Gmax =59521700psf (Site Class b)

Gmax =108695700psf (Site Class a)

Figure 5b-3. Profiles 3, 7, and 11 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60.

184

b)

γt = 125pcf

Profile 4 Profile 8 Profile 12

Dep

th (f

t)

0

20

10

30

40

50

60

70

80

90

0100

a)

Profile 4 Profile 8 Profile 12

0 200 400 600 800 1000Shear Wave Velocity (psf)

0 10

20

30

40

Dep

th (f

t)

50 60 70

80

90

100 6 12 18N1,60

24 30 γrock = 140pcf, Gmax =59521700psf (Site Class b)

Gmax =108695700psf (Site Class a) Figure 5b-4. Profiles 4, 8, and 12 used in the parameter study to develop the

correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60.

185

Appendix 5c: Energy-based magnitude scaling factors (MSF)

As noted in Chapter 2 (Section 2.2.1), magnitude scaling factors (MSF) are used in the

stress-based procedure to account for the duration of earthquake motions and are defined

as:

5.7MCSRCSRMSF = (5c-1)

where CSRM7.5 is the cyclic stress ratio associated with a M7.5 earthquake and CSR is the

cyclic stress ratio associated with an earthquake of magnitude M. The earliest MSF were

developed from laboratory liquefaction curves similar to the one shown in Figure 5c-1.

In this figure, the amplitude of the applied loading, as quantified by CSR, is plotted as a

function of the number of cycles required cause liquefaction (Nliq). Cyclic triaxial and

cyclic simple shear test apparatus were primarily used for developing these curves.

10.1 3 10 30Number of Cycles to Initial Liquefaction, Nliq

100

0.2

0.3

0.4

0.5 Dr = 60% σ’o =100 kPa Monterey No. 0 Sand Moist Tamped Samples

Cyc

lic S

tress

Rat

io (σ

d/2σ’

o)

Figure 5c-1. Capacity curve developed from cyclic triaxial laboratory tests. Each point on the plot represents a separate test. Each test is conducted on similar samples subjected to varying amplitude CSR. For isotropically consolidated cyclic triaxial tests, the CSR = σd/2⋅σ’o, where σd is the applied deviator stress, and σ’o is the initial effective confining stress.

As illustrated in Figure 5c-2, the MSF are the factors required to flatten the laboratory

liquefaction curve. Fifteen cycles was used as the base value (i.e., the Nliq value at which

186

MSF = 1.0) because early correlations equated 15 equivalent cycles to M7.5 (e.g., Figure

5-5). This was the earthquake magnitude for which the largest amount of liquefaction

case histories existed when the MSF were first proposed. Using laboratory liquefaction

curves in conjunction with correlations relating the number of equivalent cycles (Neqv) to

earthquake magnitude, the Demand imposed on the soil can be normalized to a M7.5

earthquake. As the liquefaction case history database increased in size, MSF were

developed directly from field data (e.g., Ambraseys 1988). However, laboratory

liquefaction curves can still play an important role in developing MSF by complementing

the field case histories.

10.1 3 10 30 100

0.2

0.3

0.4

0.5

Number of Cycles to Initial Liquefaction, Nliq

MSFCSR

Representative Laboratory Curve

15

M7.

5

Cyc

lic S

tress

Rat

io (σ

d/2σ’

o)

Figure 5c-2. Illustration of how MSF were developed from laboratory liquefaction curves.

As shown in Annaki and Lee (1977) and Idriss (1997), laboratory liquefaction curves

often plot as straight lines on log-log scales having a slope of -m. This is illustrated in

Figure 5c-3 and allows the MSF to be expressed in terms of Neqv and –m.

187

10.1 3 10 30 100

0.3

1.0

Number of Cycles to Initial Liquefaction, Nliq

MSFCSR

-m

Representative Laboratory Curve (log-log scale)

15

M7.

5

Cyc

lic S

tress

Rat

io (σ

d/2σ’

o)

Figure 5c-3. Illustration of liquefaction curve plotted on log-log scales.

The following is a derivation of an expression for the MSF in terms of the number of

equivalent cycles (Neqv) and the slope of the liquefaction curve (-m):

)log()log()log()log(

21

21

liqliq NNCSRCSRm

−−

=−

=

−−

=

2

1

1

2

21

12

log

log

)log()log()log()log(

liq

liq

liqliq

NNCSRCSR

NNCSRCSRm

=

⋅

1

2

2

1 loglogCSRCSR

NN

mliq

liq

188

m

liq

liq

NN

CSRCSR

=

2

1

1

2 (5c-2)

By setting CSR1 = CSRM7.5, Nliq 1 = Neqv M7.5, CSR2 = CSR, and Nliq 2 = Neqv,

Equation (5c-2) reduces to:

m

MCSRCSRMSF

=

5.7

m

eqv

Meqv

NN

MSF

= 5.7 (5c-3)

where: MSF = Magnitude scaling factor.

CSR M7.5 = Cyclic stress ratio of an earthquake of magnitude M7.5.

CSR = Cyclic stress ratio of an earthquake of magnitude M.

Neqv M7.5 = Number of equivalent cycles for magnitude M7.5 earthquake motions.

Neqv = Number of equivalent cycles for magnitude M earthquake motions.

m = Slope of the laboratory curve plotted on log-log scales.

As opposed to early correlations which expressed Neqv M7.5 as a function of magnitude

only, the correlation developed as part of this research shows Neqv M7.5 is dependent on the

site-to-source distance, as well as magnitude. This is illustrated in Figure 5c-4, which

shows the correlation presented previously in Figure 5-10.

189

Richter Magnitude (M)8

Neqv = 10

5055

4045

3530

2520

15

40

60

80

100

50 6 7

Epic

entra

l Dis

tanc

e (k

m)

20

Figure 5c-4. Contour plot of Neqv as a function of epicentral distance and Richter magnitude.

Because most of the liquefaction case histories are for far field, Equation (5c-4) is

proposed as a new expression for MSF. m

eqv

fieldfarMeqv

NN

MSF

= 5.7 (5c-4)

where: Neqv M7.5 far field = Number of equivalent cycles for magnitude M7.5 earthquake motions in the far field.

Neqv = Number of equivalent cycles for magnitude M earthquake motions in the near or far field.

m = Slope of the laboratory curve plotted on log-log scales.

In applying Equation (5c-4), the near field-far field boundary is defined by the elbows of

the contours shown in Figure 5c-4.

Attention is now focused on the slope of the laboratory liquefaction curve, m. A

literature review resulted in the following range of m values:

190

m = 0.23 Idriss (1997), data from DeAlba et al. (1976)

m = 0.27 Seed et al. (1975) (approximately)

m = 0.34 Idriss (1997), data from Yoshimi et al. (1984)

m = 0.5 Arango (1996) (from theory, not lab data)

As may be observed, there is a large variation is published values of m. However, as

opposed to using any of these values directly, for this study, m was selected such that the

resulting MSF curve, using Equation (5c-4) and the Neqv correlation shown in Figure 5c-4,

matched closest the MSF curve proposed by Ambraseys (1988) in the far field. This

avoids any reliance on laboratory data because Ambraseys’ MSF were developed directly

from field case histories, predominately far field case histories. Additionally, the

Ambraseys’ MSF are close to the average of the NCEER (1997) recommended range for

MSF, as shown in Figure 2-2 in Chapter 2. From the calibration of Equation (5c-4) to

Ambraseys’ MSF in the far field, m = 0.48, which is with in the limits of the values given

in the literature.

A comparison between the proposed MSF and Ambraseys’ MSF is shown in Figure 5c-5.

As may be seen in this figure, there are two significant differences between MSF curves.

The first is related to the proposed Neqv correlation for near and far field conditions. As

indicated by the vertical and horizontal portions of the contours shown in Figure 5c-4,

respectively, Neqv are only dependent on magnitude in the far field and only dependent on

epicentral distance in the near field. The proposed MSF are similarly dependent. The

near field dependency results in the series of horizontal lines corresponding to different

epicentral distances shown Figure 5c-5. The intersections of the horizontal lines with the

curved line defines the near field-far field boundary for the corresponding magnitudes.

For example, the near field-far field boundary for M6.75 earthquake is approximately

30km. For far field conditions, the proposed MSF and Ambraseys’ MSF are

approximately equal because m in Equation (5c-4) was selected accordingly. The

proposed MSF could be matched to any of the MSF shown in Figure 2-2 for far field

conditions by appropriately selecting m. The reasonableness of the resulting m can be

191

judged by comparing it to the range of values taken from published literature presented

above (i.e., 0.23 ≤ m ≤ 0.5).

The second significant difference between Ambraseys’ and the proposed MSF curves is

for magnitudes less than M6.0, the proposed MSF reach a low magnitude plateau. This

results from the interaction of setting the stress amplitude of the equivalent cycle

proportional to 0.65amax and the duration of the earthquake motions. It is interesting to

note that Idriss (1997) proposed MSF also having a low magnitude plateau. The basis for

his plateau is related to the minimum fraction of a loading cycle required to induce

dynamic liquefaction in a laboratory sample as opposed to failing the sample

monotonically.

far field

M6.

75

near field

Proposed MSF curve Ambraseys’ MSF curve

30km40km50km60km

20kmEpicentral Distance = 10km

4.0

3.5

3.0

2.5

2.0

1.5

1.0

5.0 0.5

Mag

nitu

de S

calin

g Fa

ctor

8.05.5 6.0 7.56.5 7.0

Magnitude

Figure 5c-5. A comparison of Ambraseys’ MSF curve and the set of curves developed in this study.

The implications of the differences between the currently used MSF and the proposed

MSF are that the former may under predict the Demand imposed on the soil at low

magnitudes (i.e., M<6) and over predict the Demand for large magnitude earthquakes in

the near field.

192

The earthquake case histories listed in Table 5-1 were analyzed using both the

Ambraseys’ MSF curve and the set developed in this study. The results are shown in

Figures 5c-6 and 5c-7, respectively. Because the vast majority of the liquefaction case

histories are for far field conditions, there is little difference between the two figures.

However, in performing liquefaction evaluations using the NCEER (1997) recommended

MSF, the trends stated above apply: the currently recommended MSF may under predict

the Demand imposed on the soil at low magnitudes (i.e., M<6) and over predict the

Demand at imposed by large magnitude earthquakes in the near field.

No LiquefactionLiquefactionNCEER proposed boundary

00.00

0.60

0.55 0.50 0.45

0.40

CSR

M7.

5 A

mbr

asey

s

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 5c-6. Liquefaction data computed using Ambraseys’ MSF curve.

193

00.00

N 1,60cs

3025

No LiquefactionLiquefactionNCEER proposed boundary

0.60

0.55

0.45

0.50

0.35

0.40

CSR

M7.

5 G

reen

0.30 0.25 0.20 0.15

0.10

0.05

5 10 15 20 35 40 45 50

Figure 5c-7. Liquefaction data computed using the set of MSF curves developed in this study.

194