A 2D SEISMIC STABILITY AND DEFORMATION ANALYSIS.pdf

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A 2D Seismic Stability and Deformation Analysis

Yi Zhu1, Kuantsai Lee2, and Gary H. Collison3 1Golder Associates Inc., 3730 Chamblee Tucker Road, Atlanta, Georgia; phone (770) 496-1893; fax (770) 934-9476; email: [email protected] 2Golder Associates Inc., 39899 Balentine Drive, Suite 200, Newark, California; phone (510) 438-6865; fax (510) 438-6866; email: [email protected] 3Golder Associates Inc., 3730 Chamblee Tucker Road, Atlanta, Georgia Abstract Two-dimensional (2D) seismic stability and deformation calculations for a 16.5-meter high levee embankment are presented. The embankment is to be constructed on potentially liquefiable foundation soil. A liquefaction triggering analysis is conducted that includes stress calculations for the pre-earthquake condition under steady-state seepage using the finite element program PLAXIS, seismic response calculations using the finite element program TELDYN, and liquefaction resistance estimates. The results of the 2D analysis suggest that the potentially liquefiable soil is confined to approximately the uppermost 10 meters of the natural soil under a limited area of the downstream embankment toe. The 2D analysis clearly illustrates the increased liquefaction resistance and the safety factor against liquefaction in the foundation soil beneath the major portion of the embankment due to the increased confining stress of the embankment loading. The 2D liquefaction triggering analysis results are used to estimate the end-of-earthquake strength using the procedure proposed by Seed and Harder (1990). The two-dimensional distribution of end-of-earthquake strength is used for a limit equilibrium, post-earthquake, static stability analysis to calculate the factor of safety of the embankment-foundation system. The calculated factor of safety of 1.1 indicates that no excessive deformation is expected after the duration of earthquake shaking. The limit equilibrium stability analysis using the end-of-earthquake strength is repeated to locate the critical slip surface and the corresponding yield acceleration. The acceleration time history for the mass defined by this critical slip surface is calculated by TELDYN. The permanent deformation of the embankment during the design earthquake is estimated based on the acceleration time history and yield acceleration using the procedure developed by Newmark (1965).

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Text Box

2005 ASCE Geo-Frontiers, Austin, Texas, January 24-26, 2005

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Site Background and Design Earthquake A project located in a seismically active region involves construction of an embankment of moderate height with a chosen design earthquake having a 980-year return period (10 percent probability of exceedance in 100 years). In one area of the project site, the corresponding free surface peak ground acceleration (PGA) is 0.28g for an earthquake with a moment magnitude of 6.5. The response spectra of the free surface motion for the site-specific design acceleration time histories for the study area are shown on Figure 1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35

Frenquency (Hz)

Spe

ctra

l Acc

eler

atio

n (g

)

Damping Ratio = 5%Damping Ratio = 10%Damping Ratio = 20%

Figure 1: Response spectra of design free surface motion.

Design Profile and Soil Properties Figure 2 shows a typical design cross section of the embankment and foundation soil. The subsurface materials in the study area are predominantly loose to medium dense granular sediments consisting mostly of silt and fine to medium sand, occasionally with coarse sand and gravel. A major source of information used in the evaluation of the granular foundation soil is from the SPT blow counts (N). The field-recorded blow counts were corrected for an energy ratio of 60 percent to obtain a standard SPT value N60, and then corrected for depth by normalizing the N60 to an effective overburden pressure of 100 kPa, yielding corrected SPT values of (N1)60. The resulting (N1)60 values and soil descriptions were used to estimate other geotechnical properties, except for fines content which was determined from laboratory tests. Figure 3 shows the SPT (N1)60 profile of the granular foundation soil used in the

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analyses. Geotechnical properties used in the analyses are presented in Table 1 for foundation soil and the embankment material.

10 m

16.5

m

Foundation Soil

13

13

25 m

Original Ground and Groundwater Level

Water Level

bSediment

229 m

w

60 m 109 m 60 m

Bed Level

Water and bed levels for design: w - water level, 15.9 m for static related analyses, 13.4 m for dynamic related analyses; b - bed level, 13.4 m for both static and dynamic related analyses.

NOT TO SCALE

Figure 2: Design cross section of embankment and foundation soil.

0

5

10

15

20

0 5 10 15 20

SPT (N 1)60

Dep

th (m

)

Figure 3: SPT (N1)60 profile of subsurface material used in the analyses.

Pre-Earthquake Condition Stress Calculations The 2D finite element program PLAXIS was used to perform stress calculations for the pre-earthquake condition under steady-state seepage. Effective stresses calculated by PLAXIS were used to calculate the effective confining stress, '

cσ , as

( )'3

'2

'1

'

31 σσσσ ++=c

4

where '1σ is the major effective principal stress, '

3σ is the minor effective principal stress, and '

2σ is the intermediate principal stress that is calculated from the major and minor principal stresses under the assumption of plane strain. Figures 4 and 5 show the calculated effective confining stress and in-plane shear stress, respectively.

Table 1: Assumed geotechnical properties of granular foundation soil and

embankment material.

Depth (m)

c' (kPa)

φ' (deg.)

E' (kPa) Gmax % Fines Ip (%) k (m/s)

0–5 0 30 15 5 3x10-5 Foundation Soil 5–25 0 30 Varies 5 5 2x10-4

Embankment 0 37 20,000 Varies

- 0 2x10-4

Notes: 5.0'

kPa 100' ⎟⎟

⎠

⎞⎜⎜⎝

⎛= c

refEE σ, where '

cσ is the effective confining stress and Eref =

11,000 kPa;

( ) 5.0'max,2max kPain 8.218(kPa) ckG σ= , where '

cσ is the effective confining stress and k2,max = 30 and 45 for foundation and embankment soils, respectively.

Horizontal Distance (m)

Ver

tical

Dis

tanc

e(m

)

0 50 100 150 200-30

-20

-10

0

10

20

0 50 100 150 200 250

Effective Confining Stress (kPa)

Figure 4: Effective confining stress distribution from PLAXIS.


Ver

tical

Dis

tanc

e(m

)

0 50 100 150 200-30

-20

-10

0

10

20

-82 -58 -33 -8 17

Static Shear Stress (kPa)

Figure 5: In-plane static shear stress distribution from PLAXIS.

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TELDYN Seismic Response Analysis The Finite Element program TELDYN (TAGAsoft 2000, version 1.39x) was used to perform 2D equivalent linear dynamic response analyses. The material properties for program input include:

1) Maximum (small strain) shear modulus, Gmax (Table 1) 2) Poisson’s ratio, ν, or constrained modulus, M. In the present analysis, the

value of the constrained modulus M, is assumed to equal 3.5 times the value of the maximum shear modulus ( max5.3 GM = ) for a corresponding Poisson’s ratio of 0.3. The constrained modulus thus varies with the confining stress but for a given confining stress is assumed to remain unchanged during an earthquake.

3) Variation of shear modulus ratio, G/Gmax, and the damping ratio, ξ, as functions of shear strain, γ. Equations proposed by Ishibashi and Zhang (1993) were used to estimate the variation of shear modulus and damping ratio as functions of shear strain (γ), effective confining stress ( '

cσ in kPa), and plasticity index (Ip in percent).

The average effective confining stress for the embankment was 58 kPa. This value and a plasticity index, Ip, of zero were used to estimate the variation of shear modulus ratio and damping ratio as functions of shear strain for the embankment material. The average effective confining stresses for the foundation soil under the embankment loading were 81, 146, and 193 kPa, respectively, for depths of zero to 10 meters, 10 to 20 meters, and 20 to 25 meters in the finite element model. These values of effective confining stress and Ip = 5 were used to obtain the variation of shear modulus ratio and the damping ratio as functions of shear strain for foundation soil. Figures 6 and 7 show the variation of shear modulus ratio and the damping ratio used for analyses, respectively. For comparison, curves without the embankment in place are also shown.

6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0001 0.001 0.01 0.1 1 10Shear Strain (%)

Mod

ulus

Rat

io G

/Gm

ax

Embankment Material

Foundation Soil (Depth = 0-10m)

Foundation Soil (Depth = 0-10m, with embankment)





Figure 6: Variation of modulus ratio as functions of shear strain.

0

5

10

15

20

25

30

35

0.0001 0.001 0.01 0.1 1 10Shear Strain (%)

Dam

ping

Rat

io (%

)

Embankment Material







Figure 7: Variation of damping ratio as functions of shear strain.

As in all dynamic finite element analyses, the sizes of the largest mesh elements are limited by the wavelength of the earthquake motion in the TELDYN model. Since

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the most significant motion in the analysis corresponds to vertical shear waves, the largest vertical element size was set to be

max81

fvy s=∆

where vs is the shear wave velocity and fmax is the maximum frequency of interest (cutoff frequency). The cutoff frequency was estimated from the natural frequencies using the average value of the maximum shear moduli of 70 MPa and a density of 2,100 kg/m3 for the embankment material. The corresponding shear wave velocity was estimated at 183 m/s. The first three natural frequencies of embankment are estimated as (Kramer 1996)

Hvf

Hvf

Hvf

s

s

s

)654.8(21

)52.5(21

)404.2(21

3

2

1

π

π

π

=

=

=

where H is embankment height = 16.5 meters. The corresponding frequencies for the embankment are 4.2, 9.7, and 15.3 Hz. Since the bulk of the energy in a seismic event is carried in the first three natural frequencies, a cutoff frequency of 20 Hz is used in the analyses. Using a cutoff frequency of 20 Hz and the estimated shear wave velocity of 183 m/s for the embankment material, the largest vertical dimension of the elements is about one meter. The finite element mesh used in TELDYN analysis has 9,457 nodes, 9,174 quadrilateral elements and satisfies the criteria that the vertical dimension of the element be less than one meter. The coordinates of the centroid of each element were calculated from those of the four corner nodes. The effective confining stress at the centroid in each element was then interpolated from the results of the PLAXIS stress calculations using an inverse-distance weighting scheme. Assuming that the effective confining stress within each element equals the value at the centroid, the interpolated effective confining stress was then used to estimate the maximum shear modulus and constrained modulus for that element. The program TELDYN requires that the input motion be provided at the base of the model as an “outcropping” motion. In the present analysis, the base is located at 25 meters below the ground surface. The motion at the base was obtained by de-convoluting the site-specific design acceleration time history at the free ground surface using the computer program SHAKE91 (Idriss and Sun 1992). Liquefaction Triggering Analysis Figure 8 shows the maximum earthquake-induced shear stresses (not including pre-earthquake static shear stresses shown on Figure 5), τmax, calculated in the first TELDYN run. The calculated maximum shear stress was used in the liquefaction triggering analysis.

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The calculated maximum shear stress allowed the embankment material to be specified and the embankment constructed to resist liquefaction under the design earthquake. For the foundation soil, the factor of safety against liquefaction was calculated by

max65.0 ττ L

LiqFS =

where τmax is the calculated maximum shear stress and τL is the foundation soil liquefaction resistance. The factor of 0.65 in the equation is an empirical value used to convert the peak shear stress (which only lasts a fraction of a second) to sustained uniform cycles of shear stress (Kramer 1996). The liquefaction resistance, τL, of a soil is defined as the maximum shear stress that can be sustained by the soil before the onset of liquefaction. For the saturated foundation soil, the liquefaction resistance was estimated from the design earthquake moment magnitude and soil parameters that include fines content, the corrected SPT resistance (N1)60, and effective vertical stress (Youd et al. 2001). The effect of initial static shear stress on liquefaction resistance was not considered in this study as per the recommendations by Youd et al. (2001).


Ver

tical

Dis

tanc

e(m

)

0 50 100 150 200-30

-20

-10

0

10

20

3.3653 11.5939 19.8224 28.0510 36.2796 44.5081 52.7367

Maximum Earthquake-Induced Shear Stress (kPa)

Figure 8: TELDYN calculated maximum shear stresses.

Figure 9 shows the variation in the calculated safety factor against liquefaction throughout the finite element model where regions susceptible to liquefaction (those with safety factors of one or less) are indicated by the warmer colors of red and orange. Zones of potential liquefaction exist in the foundation soil outside the embankment. Sediment that exists above the foundation soil on the upstream side (e.g., above the level of the bottom of the embankment, to the right of embankment shown on Figure 9) was not included in the analysis. The additional confining stress from the sediment is expected to decrease, or potentially eliminate, the liquefaction potential in the foundation soil on the upstream side.

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Ver

tical

Dis

tanc

e(m

)

0 50 100 150 200-30

-20

-10

0

10

20 0.5565 0.8762 1.1958 1.5155 1.8352

Factor of Safety against Liquefaction Triggering

Figure 9: Foundation soil factor of safety against liquefaction triggering.

Figure 10 shows the safety factor against liquefaction triggering calculated at different depths from the present 2D analysis and from the 1D analysis following the established simplified method (Youd et al. 2001) at a location 50 meters downstream (left) of the downstream embankment toe. The distance of 50 meters was selected to minimize the potential influence of the embankment on the confining stresses and thus the liquefaction potential. The results of both analyses showed reasonable agreement at shallow depths, although at deeper depths the 2D analysis indicates higher factors of safety against liquefaction than 1D analysis.

0

5

10

15

20

25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Factor of Safety against Liquefaction Triggering

Dep

th (m

)

Level ground, no embankment, one-dimensional analysis using Youd et al. (2001)

With embankment, two-dimensional analysis using TELDYN

Foundation soil at a location of 50 meters left (downstream) of the downstream toe

Figure 10: Factor of safety against liquefaction at 50 meters left of downstream toe.

The safety factor against liquefaction triggering at the downstream toe, at the mid-point of the downstream slope, and at the downstream edge of the crest of the embankment as calculated in the 2D analyses are shown on Figure 11. The increase in the calculated safety factor from less than one at the toe to greater than one at the

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crest of the embankment illustrates the positive effect of increasing confining stress. Thus, the extent of the potentially liquefiable zone is limited to the immediate area of the downstream toe. End-of-Earthquake Strength During a seismic event, the compacted, granular embankment material is expected to experience little, if any, strength loss during and shortly after the design earthquake. However, it is recognized (Seed and Harder 1990; Marcuson et al. 1996; Finn 1998) that current knowledge on end-of-earthquake strength or residual strength is still incomplete. In recognition of the uncertainty associated with assigning end-of-earthquake shear strength, Seed and Harder (1990) recommended that the end-of-earthquake strength be taken at no more than 75 percent of the drained static strength. Accordingly, the end-of-earthquake shear strength for the embankment material was selected to be 75 percent of its drained static shear strength.

0

5

10

15

20

25

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

With Embankment, 2D Analysis Using TELDYNFactor of Safety against Liquefaction Triggering

Dep

th (m

)

At downstream toe

At mid-point of downstream slope

At downstream edge of embankmentcrest

Figure 11: Factor of safety against liquefaction triggering under the embankment.

For the foundation soil, the end-of-earthquake shear strength (EES) was estimated by the procedure proposed by Seed and Harder (1990) based on the calculated safety factor against liquefaction:

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⎪⎪

⎩

⎪⎪

⎨

⎧

=

>>

=≤

=≥

strength. residual andstrength static drained 75%between ion interpolat EES 1.1,1.4 If

strength; residual EES 1.1, If

strength; static drained 75% EES 1.4, If

Liq

Liq

Liq

FS

FS

FS

The residual strength of the foundation soil was estimated from SPT (N1)60 for clean sand, (N1)60cs (Youd et al. 2001), using two different procedures and adopting the smaller value of the two for subsequent calculations. The first procedure for estimating residual strength was based on the chart prepared by Seed and Harder (1990) and the average values between the upper and lower bound values were used. The second procedure was based on the equation proposed by Stark and Mesri (1992):

csv

r Ns601' )(0055.0=

σ

where sr = residual strength and 'vσ = effective vertical stress. Figure 12 shows the

estimated end-of-earthquake strength for the foundation soil.


Ver

tical

Dis

tanc

e(m

)

0 50 100 150 200-30

-20

-10

0

10

2010 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

End-of-Earthquake Strength (kPa)

Figure 12: Foundation soil end-of-earthquake strength. Post-Earthquake Static Stability Post-earthquake static stability analyses using the end-of-earthquake strength were performed using the limit equilibrium slope stability program SLIDE, version 5 (Rocscience 2003). The generalized limit equilibrium method that satisfies both force and moment equilibrium (Abramson et al. 1996) was chosen for the analyses. A factor of safety of 1.1 was calculated as shown in Figure 13. This calculated factor of safety indicates that no excessive deformation is expected after the duration of earthquake shaking. The analysis was repeated to locate the critical slip surface and the corresponding yield acceleration, defined as the pseudo-static seismic coefficient resulting in a factor of safety of unity in a limit equilibrium slope stability analysis. Figure 14 illustrates the critical slip circle and the corresponding yield acceleration of 0.02g. The actual yield acceleration is expected to be larger than this value since the

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strength of the embankment and the foundation soil during the beginning of earthquake is also expected to be greater than the end-of-earthquake value. Earthquake Induced Deformation A second TELDYN model run was conducted to calculate the acceleration time history for the mass defined by the critical slip surface in Figure 14. The second model run was identical to the first run except that in the second run the program was used to calculate the acceleration time history of the critical sliding mass, which could only be identified using the results of the first TELDYN run. TELDYN determines the histories of average acceleration for the potential sliding mass by summing the horizontal forces acting on the boundaries of the mass at each solution time step and dividing the results by the mass to obtain accelerations. With the TELDYN-calculated acceleration time history for the critical slip mass and the yield acceleration of 0.02g, deformation analyses were performed using procedures developed by Newmark (1965). In the Newmark procedure, it is assumed that displacement would occur only during those (typically brief) periods when the acceleration of the mass exceeds its yield acceleration. Mathematically, the permanent deformation is calculated by double integration of the difference between the induced acceleration and yield acceleration. As shown in Figure 15, the estimated horizontal permanent displacement for the critical slip mass during the design earthquake is less than 0.3 meters. Assuming that the vertical settlement of embankment during the design earthquake would be of the same order of magnitude as the estimated horizontal displacement for the critical slip mass, vertical settlements would also be of the order of a fraction of a meter. In comparison, the difference between the elevations of the embankment crest and the water surface for the design seismic event is 3.1 meters, far greater than the estimated vertical settlement. The present analyses thus confirmed that the design crest level is expected to provide sufficient freeboard to protect against overtopping caused by vertical settlement that might occur during the design earthquake.

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1.11.11.11.1

-20

-10

0

10

40 50 60 70 80 90 100 110 120 130 140 1


Ver

tical

Dis

tanc

e (m

)

Global Minimum =Method: gle/morgenstern-priceFS: 1.1Center: 56.325, 30.305Radius: 38.730

EmbankmentStrength Type: Mohr-CoulombUnit Weight: 21 kN/m3Cohesion: 0 kPaFriction Angle: 29.5 degrees(75% drained shear strength)

SandEnd-of-Earthquake Strength Unit Weight: 19.2 kN/m3

Sediment

Figure 13: Critical slip surface using end-of-earthquake strength.

1.01.01.01.0

Seismic Load Coefficient (Horizontal): 0.02gGlobal Minimum =Method: gle/morgenstern-priceFS: 1.0Center: 56.039, 29.860Radius: 38.272

-20

-10

0

10

40 50 60 70 80 90 100 110 120 130 140 1


Ver

tical

Dis

tanc

e (m

)

EmbankmentStrength Type: Mohr-CoulombUnit Weight: 21 kN/m3Cohesion: 0 kPaFriction Angle: 29.5 degrees(75% drained shear strength)

SandEnd-of-Earthquake Strength Unit Weight: 19.2 kN/m3

Sediment

0.02

Figure 14: Critical slip surface and yield acceleration using end-of-earthquake

strength.

14

0 10 20 30 40Time (second)

-0.2

-0.1

0.0

0.1

0.2

Acc

eler

atio

n (g

)

Yield acceleration = 0.02 g

Yield acceleration = -0.02g

0 10 20 30 40Time (second)

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

Rel

ativ

e V

eloc

ity (m

/s)

0 10 20 30 40Time (second)

-0.30-0.25-0.20-0.15-0.10-0.050.000.050.100.150.20

Rel

ativ

e D

ispl

acem

ent (

m)

Figure 15: Estimate displacement by the Newmark procedure.

Conclusion

• The results of the 2D liquefaction triggering analysis are consistent with 1D analysis results indicating that there is a potential for liquefaction of the granular foundation during the 10 percent 100-year design earthquake in areas where the presence of the embankment has little influence on the stress conditions.

• The results of the 2D liquefaction triggering analysis indicate that only limited portions of the granular foundation soil surrounding the downstream toe of the embankment are susceptible to liquefaction during the 10 percent 100-year design earthquake. Liquefaction is not expected during the design earthquake within most of the foundation soil beneath the embankment.

• The 2D analyses provided values of maximum shear stress in the embankment to aid the selection and the design of embankment material to resist liquefaction.

• The 2D analyses also indicate that the overall embankment-foundation system is expected to be stable during the design earthquake.

• Both the dynamic embankment displacement during the design earthquake and the permanent embankment displacement induced by such an earthquake are estimated to be less than 0.3 meters.

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• The available freeboard of 3.1 meters is expected to provide protection against overtopping due to potential embankment settlement that might occur during the design earthquake.

• The current analyses do not consider the in-plane static shear stress at pre-earthquake condition, which could be an important factor for liquefaction triggering and liquefaction induced deformation. However, there is no consensus in literature on how to take this factor into account.

References Abramson, L. W., Lee, T. S., Sharma, S., and Boyce, G. M. (1996). Slope Stability

and Stabilization Methods, John Wiley & Sons, Inc., New York. Brinkgreve, R. B. J., Al-Khoury, R., Bakker, K. J., Bonnier, P. G., Brand, P. J. W.,

Broere, W., Burd, H. J., Soltys, G., Vermeer, P. A., and Haag, DOC Den (2002). PLAXIS 2D – Version 8 Full Manual, A.A. Balkema, Netherlands.

Finn, W. D. L. (1998). "Seismic safety of embankment dams developments in research and practice 1988-1998." Geotechnical Earthquake Engineering and Soil Dynamics III, Seattle, WA, 812-852.

Idriss, I. M. and Sun, J. I. (1992). Users’ Manual for SHAKE91, University of California, Davis, California.

Ishibashi, I., and Zhang, X. (1993). "Unified dynamic shear moduli and damping ratios of sand and clay." Soils and Foundations, 33(1), 182-191.

Kramer, S. L. (1996). Geotechnical Earthquake Engineering, Prentice Hall Inc., Upper Saddle River, New Jersey.

Marcuson, W. F., Hadala, P. F., and Ledbetter, R. H. (1996). "Seismic rehabilitation of earth dams." Journal of Geotechnical Engineering, 122(1), 7-20.

Newmark, N. M. (1965). "Effects of earthquakes on dams and embankments." Geotechnique, 15(2), 139-160.

Rocscience Inc. (2003). Slide – 2D Limit Equilibrium Slope Stability for Soil and Rock Slopes, Version 5, Toronto, Canada.

Seed, R. B., and Harder, L. F. (1990). "SPT-based analysis of cyclic pore pressure generation and undrained residual strength." H. Bolton Seed Memorial Symposium Proceedings, 351-376.

Stark, T. D., and Mesri, G. (1992). "Undrained shear strength of liquefied sands for stability analysis." Journal of Geotechnical Engineering, 118(11), 1727-1747.

TAGAsoft Limited (1998). TELDYN – Users’ Manual, Lafayette, California. Youd, T. L., Idriss, I. M., Andrus, R. D., Arango, I., Castro, G., Christian, J. T.,

Dobry, R., Finn, W. D. L., Harder, L. F., Hynes, M. E., Ishihara, K., Koester, J. P., Liao, S. S. C., Marcuson, W. F., Martin, G. R., Mitchell, J. K., Moriwaki, Y., Power, M. S., Robertson, P. K., Seed, R. B., and Stokoe, K. H. (2001). "Liquefaction resistance of soils: summary report from the 1996 NCEER and 1998 NCEER/NSF workshops on evaluation of liquefaction resistance of soils." Journal of Geotechnical and Geoenvironmental Engineering, 127(10), 817-833.

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