2D Analysis - Simplified Methods

Homework Assignment #4

Assume the embankment is 8 m high and has a crest of 12 m. Both side

slopes are constructed on a 1H to 1V slope.

a.

Assume the embankment is constructed of granular material with an

effective stress friction angle of 25 degree, 10 kPa cohesion and a total unit weight of 2 Mg/m^3. The shear wave velocity is 175 m/s.

b.

The foundation soil below the embankment is a clayey soil with an

effective stress friction angle of 20 degrees, 30 kPa cohesion, dry unit weight of 1.2 Mg and a porosity of 0.3.

c.

Given the attached embankment properties and the attached shear modulus

reduction and damping curve and the attached acceleration response spectra, determine the maximum crest acceleration (g) of the embankment.

1.

From this information, calculate the crest acceleration using the design

spectrum attached with these notes.

2.

From this information, calculate the pseudostatic factor of safety against slope

failure using the average acceleration that develops within the critical circle. This may be done use the "Slide" software in conjunction with the Makdisi Seed

method.

3.

Using the Makidisi-Seed approach, make a plot of embankment displacement,

U in meters, as a function of yield acceleration, ky, for a M = 7.5 earthquake.

4.

Using the information given in problem 1 and the "Slide" software, calculate

the yield acceleration of the slope/foundation system.

5.

Use the yield acceleration determined in problem 5 to estimate the

displacement of the embankment/foundation system.

6.

© Steven F. Bartlett, 2014

Lecture Notes○

Pp. 423 - 449 Kramer○

Pp. 286-290 Kramer - Shear Beam Approach○

Makdisi-Seed Analysis (EERC).pdf○

Bray and Travasarou - 2007○

Reading Assignment

None○

Other Materials

http://www.rocscience.com/products/8/feature/87

2D Analysis - Simplified Methods Monday, February 03, 20142:32 PM

2D Analysis - Simplified Methods

http://www.rocscience.com/products/8/feature/87

Homework inputs


0

5

10

15

20

25

30

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.0001 0.001 0.01 0.1 1 10

Dam

pin

g (

%)

G/G

max

shear strain (%)

Sand (Seed and Idriss) Average

2D Analysis (cont.)Sunday, August 14, 20113:32 PM


Homework inputs


2D Analysis (cont.)Sunday, August 14, 20113:32 PM


Homework inputs


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5

damping 5 percent

damping 10 percent

damping 15 percent

damping 20 percent

2D Analysis (cont.)Monday, February 03, 20142:32 PM



Lower San Fernando Dam - 1971 San Fernando Valley Earthquake, Ca.

Main Issues in Seismic Assessment of Earthen Embankments and Dam:

• Stability: Is embankment stable during and after earthquake? • Deformation: How much deformation will occur in the embankment?

Two general types of analyses needed to answer these questions:

2D Dynamic Response Analysis○

2D Deformation Analysis○

In some approaches, these two analyses are coupled.

2-D Seismic Embankment and Slope Assessment and StabilityWednesday, August 17, 201112:45 PM



Pseudostatic Analysis (Stability)○

Makdisi and Seed (1978) used average accelerations computed by the

procedure of Chopra (1966) and sliding block analysis to compute earthquake-induced deformations of earth dams and embankments.

Newmark Sliding Block Analysis (Deformation)○

Quake/W□

Plaxis□

FEM

FLAC□

FDM

Numerically Based Analysis (Deformation)○

This course will focus on Pseudostatic and Newmark Sliding Block Analyses using

the Makdisi-Seed (1978) Method

General Types of 2D Seismic AnalysisSunday, August 14, 20113:32 PM



If the embankment and foundation materials are not susceptible to

liquefaction or strength reduction due to earthquake shaking, then the embankment will generally he stable and no catastrophic failure is expected

(Seed, 1979).

However, if the embankment or/and foundation comprise liquefiable

materials, it may experience flow failure depending on post-earthquake factor of safety against instability (FOSpe).

For high initial driving stress (steep geometry), the FOS will likely be much less

than unity, and flow failure may occur, as depicted by strain path A-B-C. Example of this is the failure of the Lower San Fernando Dam.

In this lecture we will not address the effects of liquefaction on embankment

stability. This will be discussed later in this course.

from:

Liquefaction EffectsWednesday, August 17, 201112:45 PM


Pseudostaic apply a static (non-varying) force the centroid of mass to

represent the dynamic earthquake force.

○

Fh = ah W / g = kh W

Fv = av W/ g = kv W (often ignored)


Guidance on the Selection of Kh

Pseudostatic AnalysisSunday, August 14, 20113:32 PM


Recommendations for implementation of pseudostatic analysis (Bartlett)

General comment: The pseudostatic technique is dated and should only be

used for screening purposes. More elaborate techniques are generally warranted and are rather easy to do with modern computing software.


Representation of the complex, transient, dynamics of earthquake shaking by

a single, constant, unidirectional pseudostatic acceleration is quite crude.

○

Method has been shown to be unreliable for soils with significant pore

pressure buildup during cycling (i.e., not valid for liquefaction).

○

Some dams have failed with F.S. > 1 from the pseudostatic technique○

Cannot predict deformation.○

Is only a relative index of slope stability○

Limitations of Pseudostatic Technique

Pseudostatic Analysis (cont.)Sunday, August 14, 20113:32 PM



Layer (top to bottom)

(kN/m3)

γ (lb/ft3) E (kPa) v K (kPa) G (kPa) φ c (kPa) Ko Vs (m/s)

1 15.72 100 100000 0.37 128,205 36,496 24.37 0 0.5873 150.9

2 16.51 105 100000 0.37 128,205 36,496 24.37 0 0.5873 147.3

3 17.29 110 150000 0.35 166,667 55,556 27.49 0 0.5385 177.5

4 18.08 115 200000 0.3 166,667 76,923 34.85 0 0.4286 204.3

5 18.08 115 250000 0.3 208,333 96,154 34.85 0 0.4286 228.4

emban 21.22 135 300000 0.3 250,000 115,385 34.85 0 0.4286 230.9

Pasted from <file:///C:\Users\sfbartlett\Documents\GeoSlope\miscdynamic1.xls>

Example Geometry

Example Soil Properties

E = Young's Modulus

= Poisson's ratioK = Bulk modulusG = Shear Modulus

= drained friction anglec = cohesionKo = at-rest earth pressure coefficentVs = shear wave velocity

Pseudostatic Analysis - ExampleSunday, August 14, 20113:32 PM



Pseudostatic Results

FS = 1.252 (static with no seismic coefficient, Kh)

The analysis has been repeated by selecting only the critical circle. To do this,

only one radius point. This result can then be used with a Kh value to determine the factor of safety, FS.




Time [sec]

161514131211109876543210

Accele

ratio

n [g]

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

Acceleration time history

Damp. 5.0%

Period [sec]

3210

Response A

ccele

ratio

n [g]

1.4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

Response Spectrum for acceleration time history

pga = 0.6 gKh = 0.5 * pgaah = 0.3 g (This is applied in the software as a horizontal acceleration).




Reduce shear strength in stability model for all saturated soils to 80 percent of

peak strength as recommended by the Army Corp of Engineers. This is to account for pore pressure generation during cycling of non-liquefiable soils. (See table

below.) (If liquefaction is expected, this method is not appropriate.)

Layer (top to bottom)

(kN/m3)

γ (lb/ft3) E (kPa) v K (kPa) G (kPa) φ Tan φ

80 percent Tan φ

New phi angle for analysis

1 15.72 100 100000 0.37 128,205 36,496 24.37 0.4530 0.3624 19.92

2 16.51 105 100000 0.37 128,205 36,496 24.37 0.4530 0.3624 19.92

3 17.29 110 150000 0.35 166,667 55,556 27.49 0.5203 0.4162 22.60

4 18.08 115 200000 0.3 166,667 76,923 34.85 0.6963 0.5571 29.12

5 18.08 115 250000 0.3 208,333 96,154 34.85 0.6963 0.5571 29.12

embank 21.22 135 300000 0.3 250,000 115,385 34.85 0.6963 0.5571 29.12

Pasted from <file:///C:\Users\sfbartlett\Documents\GeoSlope\miscdynamic1.xls>

The analysis is redone with Kh = 0.3 and reduced shear strength (see below).

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1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106

107108 109 110 111 112

113114115116117118

119120 121 122 123 124

125 126 127 128129 130

131 132 133 134 135136

137 138 139 140 141142

143 144 145 146 147148

149 150 151 152 153154

0.651

1

2 3

4

5678910

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23

24 25 2627

2829

3031

3233

34

35

36

The resulting factor of safety is 0.651 (too low). Deformation is expected for this

system and should be calculated using deformation analysis (e.g., Newmark, Makdisi-Seed, FEM, FDM methods.)



Pasted from

<http://pubs.usgs.gov/of/1998/ofr-98-113/

ofr98-113.html>


Newmark’s method treats the mass as a rigid-plastic body; that is, the

mass does not deform internally, experiences no permanent displacement at accelerations below the critical or yield level, and

deforms plastically along a discrete basal shear surface when the critical acceleration is exceeded. Thus, for slope stability, Newmark’s method is

best applied to translational block slides and rotational slumps. Other limiting assumptions commonly are imposed for simplicity but are not

required by the analysis (Jibson, TRR 1411).

1. The static and dynamic shearing resistance of the soil are assumed to

be the same. (This is not strictly true due to strain rate effects 2. In some soils, the effects of dynamic pore pressure are neglected. This

assumption generally is valid for compacted or overconsolidated clays and very dense or dry sands. This is not valid for loose sands or normally

consolidated, or sensitive soils.3. The critical acceleration is not strain dependent and thus remains

constant throughout the analysis.4. The upslope resistance to sliding is taken to be infinitely large such that

upslope displacement is prohibited. (Jibson, TRR 1411)

Newmark Sliding Block AnalysisSunday, August 14, 20113:32 PM


http://pubs.usgs.gov/of/1998/ofr-98-113/ofr98-113.html

http://pubs.usgs.gov/of/1998/ofr-98-113/ofr98-113.html

Steps

Perform a slope stability analysis with a limit equilibrium method and find the

critical slip surface (i.e., surface with the lowest factor of safety) for the given soil conditions with no horizontal acceleration present in the model.

1.

Determine the yield acceleration for the critical slip circle found in step 1 by

applying a horizontal force in the outward direction on the failure mass until a factor of safety of 1 is reached for this surface. This is called the yield

acceleration.

2.

Develop a 2D ground response model and complete 2D response analysis for the

particular geometry. Use this 2D ground response analysis to calculate average horizontal acceleration in potential slide mass.

3.

Consider horizontal displacement is possible for each time interval where the

horizontal acceleration exceeds the yield acceleration (see previous page).

4.

Integrate the velocity and displacement time history for each interval where the

horizontal acceleration exceeds the yield acceleration (see previous page).

5.

The following approach is implemented using the QUAKE/WTM and SLOPE/WTM.


Acceleration vs. time at base of slope from 2D response analysis in Quake/W.

Newmark Sliding Block Analysis (cont.)Sunday, August 14, 20113:32 PM


Analysis perfromed using shear strength = 100 percent of peak value for all soils

(i.e., no shear strength loss during cycling).


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1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106

107108 109 110 111 112

113114115116117118

119120 121 122 123 124

125 126 127 128129 130

131 132 133 134 135136

137 138 139 140 141142

143 144 145 146 147148

149 150 151 152 153154

1.530

1

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Factor of Safety vs. Time

Fa

cto

r of

Safe

ty

Time

1.0

1.2

1.4

1.6

1.8

2.0

0 5 10 15 20

Note that critical

circle is obtained from the

pseudostatic analysis



Analysis repeated using shear strength = 80 percent of peak value for all soils to

account for some pore pressure generation during cycling.


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1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106

107108 109 110 111 112

113114115116117118

119120 121 122 123 124

125 126 127 128129 130

131 132 133 134 135136

137 138 139 140 141142143 144 145 146 147148

149 150 151 152 153154

1.365

1

2 3

4

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24 25 2627

2829

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34

35

36


Fa

cto

r of

Safe

ty

Time

1.0

1.2

1.4

1.6

1.8

0 5 10 15 20



Analysis repeated using shear strength in layer 1 equal to 5 kPa (100 psf) to

represent a very soft clay.


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21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106

107108 109 110 111 112

113114115116117118

119120 121 122 123 124

125 126 127 128129 130

131 132 133 134 135136

137 138 139 140 141142143 144 145 146 147148

149 150 151 152 153154

0.944

1

2 3

4

5678910

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24 25 2627

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35

36


Fa

cto

r of

Safe

ty

Time

0.8

0.9

1.0

1.1

1.2

0 5 10 15 20

Note FS < 1 for a

significant part of the time history.

Deformation vs. Time

De

form

atio

n

Time

0.0

0.5

1.0

1.5

2.0

2.5

0 5 10 15 20

Note that more than 2 m of

displacement have accumulated.




Makdisi - Seed Analysis - Crest AccelerationSunday, August 14, 20113:32 PM



Eq. 1

Eq. 2




Eq. 3

Eq. 3a

Eq. 4




Eq. 5

Eq. 6

Eq. 7a

Eq. 7b

Eq. 7c

See p. 533

Kramer




Eq. 8

Eq. 9




y / h




Eq. 10

Makdisi - Seed Analysis - Crest AcceleratiSunday, August 14, 20113:32 PM



Makdisi - Seed Analysis - DeformationsSunday, August 14, 20113:32 PM


Better chart for previous page


Interpolation on semi-log plot

If U/kh(max)gT is halfway between 0.01 and 0.1, then the exponent value for this

number is -1.5 (see red arrow on graph above). This can be converted back by 1 x 10-1.5 which is equal to 3.16 x 10-2.

Exponent

Makdisi - Seed Analysis - DeformationsSunday, August 14, 20113:32 PM


Example

Design Spectra


Values in red must be adjusted until convergenceIs obtained

Makdisi - Seed Analysis - ExampleSunday, August 14, 20113:32 PM



Shear modulus reduction and damping curves

Calculations




Calculations (cont.)

Charts for deformation analysis

Z = depth to

base of potential

failure plane (i.e., critical

circle from pseudostatic

analysis)

toe circle




(See regression equations on next page for M7.5 and M6.5 events



y = 1.7531e-8.401x

R² = 0.988

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0 0.2 0.4 0.6 0.8 1

U /

(kh

ma

x*g

*T1)

ky/khmax

Deformation versus ky/kymax curve for M = 7.5

y = 0.7469e-7.753x

R² = 0.9613

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0 0.2 0.4 0.6 0.8 1

U /

(kh

ma

x*g

*T1)

ky/khmax

Deformation versus ky/kymax curve for M = 6.5





Bray and TravasarouWednesday, February 05, 20142:32 PM



Bray and Travasarou (cont.)Wednesday, February 05, 20142:32 PM


In this application, probabilistic methodologies usually involve three steps:

(I) establishing a model for prediction of seismic slope displacements. where

seismic displacements are conditioned on a number of variables characterizing the important ground motion characteristics and slope properties:

(2) computing the joint hazard of the conditioning ground motion variables,

(3) integrating the above-mentioned two steps to compute the seismic

displacement hazard. Focusing on the first step.

Step 1 - Developing the Model

Compared to the rigid sliding block model, a nonlinear coupled stick-slip

deformable sliding block model offers a more realistic representation of the dynamic response of an earth/waste structure by accounting for (he

deformability of the sliding mass and by considering the simultaneousoccurrence of its nonlinear dynamic response and periodic sliding episodes.

In addition, its validation against shaking table experiments provides confidence in its use (Wartman et al. 2003).




Step 1 - Developing the Model (cont.)

The ground motion database used to generate the seismic displacement data

comprises available records from shallow crustal earthquakes (hat occurred in active Plate margins (PEER strong motion database

(http://peer.bcrkeley.edu/smcat/index.html)).

These records conform to the following criteria:

(1) 5.5 < Mw < 7.6

(2) R < 100 km

(3) Simplified Geotechnical Sites B C, or D (4) frequencies in the range of 0.25— 10 Hz have not been filtered out.

Earthquake records totaling 688 from 41 earthquakes comprise the ground

motion database for this study [see Travasarou (2003) for a list of records used]. The two horizontal components of each record were used to calculate

an average seismic displacement for each side of the records, and the maximum of these values was assigned to that record.

The seismic response of the sliding mass is captured by:

1. 1D equivalent-linear viscoelastic modal2. strain-dependent material properties to capture the nonlinear response3. single mode shape. but the effect of including three modes was shown to be

small.

The results from this model have been shown to compare favorably with those

from a fully nonlinear D-MOD-type stick-slip analysis (Rathje and Bray 2000), but this model can be utilized in a more straightforward and transparent manner.

The model used herein is one dimensional (i.e.. a relatively wide vertical column of deformable soil) to allow for the use of a large number ground motions with

wide range of properties of the potential sliding mass in this study. One-dimensional (1D) analysis has been found to provide a reasonably conservative

estimate of the dynamic stresses at the base of two-dimensional (2D) sliding systems




http://peer.bcrkeley.edu/smcat/index.html))


This nonlinear coupled stick-slip deformable sliding model can be

characterized by: (1) its strength as represented by its yield coefficient (ky.). and (2) its dynamic stiffness as represented by its initial fundamental period

(Ts). Seismic displacement values were generated by computing the response of the idealized sliding mass model with specified values of its yield

coefficient (i.e., ky=0.02. 0.05, 0.075. 0.1, 0.15. 0.2, 0.25. 0.3, 0.35, and 0.4) and its initial fundamental period (i.e., T=0. 0.2, 0.3. 0.5. 0.7, 1.0. 1.4. and 2.0

s) to the entire set of recorded earthquake motions described previously.



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2D Analysis - Simplified Methods

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