Numerical Methods for Seismic Analysis of Damsby

Rajah AnandarajahJohns Hopkins University

•Introduction•Deformation Analysis

(a) Sliding Block Method(b) Elasto-Plastic Finite Element Method

•Some Observations

IntroductionMode of deformation

Importance of pore water pressure development

Methods of Deformation Analysis

• Newmark’s Sliding Block Method

• Elasto-Plastic Finite Element Method

Newmark’s Sliding Block MethodFurther developed by Goodman and Seed (1966)

)()()sin(cos)cossin(2

2

tbtaggdt

xd =++−= αµααµα

Integrate to get and x! x

t (sec)

a(g

)

0 2 4 6 8

-0.4

-0.2

0

0.2

0.4

t (sec)

Acc

(g)

0 2 4 6 8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

t (sec)ve

l(m

/sec

)0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

t (sec)

Dis

p(m

)

0 2 4 6 8-9.5455E-06

0.49999

0.99999

1.49999

1.99999

2.49999

Example 1:

Yield Acc: 0..25g

Slope angle: 25 deg

Coefficient of friction=0.5

t (sec)

Acc

(g)

0 2 4 6 8 10 12 14 16 18 20-0.4

-0.2

0

0.2

0.4

t (sec)

vel(

m/s

ec)

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

t (sec)

Disp

(m)

0 2 4 6 8 10 12 14 16 18 200

0.005

0.01

0.015

0.02

0.025

t (sec)

a(g

)

0 2 4 6 8 10 12 14 16 18 20

-0.2

0

0.2

0.4Example 2:

Yield Acc: 0..25g

Slope angle: 25 deg

Coefficient of friction=0.5

Elasto-Plastic Finite Element Method

1. Constitutive models for soil

2. Capability to perform drained, undrained or fully-coupled analyses

3. Capability to model interfaces (soil-to-soil, soil-to-structure)

Constitutive Models

Clays: e.g., The bounding surface modelIsotropic Clays: Dafalias and Herrmann (1986),J. Eng. Mech, ASCE.

Anisotropic Clays: Anandarajah and Dafalias (1986),J. Eng. Mech, ASCE.

Anisotropic Bounding Surface ModelAnandarajah and Dafalias, 1986, JEM, ASCE

Earthquake Response of a 2-Storey BuildingAnandarajah, A., Rashidi, H. and Arulanandan, K. (1995). “Elasto-Plastic Finite Element Analyses of Earthquake Pile-Soil-Structure Interaction Problems Tested in a Centrifuge.” Computers and Geotechnics, 17:301-325.

Sands:

Phenomenological Models:

e.g.,

Anandarajah (1992), J. Eng. Mech., ASCE

Sliding and Rolling Constitutive Theory for Granular Materials

Anandarajah, 2003, JEM, ASCE, In Press

)tan(1

2µφβ

σσ

−=

p

q

0 25 50 75 100 1250

10

20

30

40

50

60A

B

C

(a)

ε (%)q

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60A

B

C

q

(b)

An Example of Model Simulation

p (kPa)

q(k

Pa)

25 50 75 1000

2

4

6

8

10

Simulation of Sand Liquefaction

Deformation of Embankments by Elasto-Plastic Finite Element Method

Program: HOPDYNE (Anandarajah, 1990)

Soil Type: Normally consolidated Clay

Details of slip element

Normally Consolidated, No Slip Elements, Higher Resolution (Frame/5 steps)

V1

Disp

(m)

5 10 15

-5

-4

-3

-2

-1

0

t (s)

a(m

/sec

2 )

15 20 25 30-10

-5

0

5

t (s)

Acc

(m/s

ec2 )

5 10 15

-2

-1

0

1

(x=36m, y=30m)

t (s)V

el(m

/s)

5 10 15-1

-0.8

-0.6

-0.4

-0.2

0

0.2

No s lip e lements , OCR=1, Undrained

NC, No Slip Elements

Normally Consolidated, With Slip Elements, Frame/50 steps, Cohesion of interface = 100 kPa, Friction angle of interface = 0 deg

t (s)

a(m

/sec

2 )

15 20 25 30-10

-5

0

5

t (s)

Acc

(m/s

ec2 )

5 10 15-5

0

5

(x=36m, y=30m)

t (s)

Vel

(m/s

)

5 10 15

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

V1

Dis

p(m

)

5 10 15

-10

-8

-6

-4

-2

0

With no slip elementsWith slip elements

NC, Cslip=100 kPa, Fri. Ang=0 deg

NC, Comparison of results with and without slip elements

t (s)

X-R

elat

ive

Dis

p(m

)

5 10 15

-2

-1

0

1OCR=1, (x=55m, y=40m)With s lip e le ments

t (s)

Pore

pres

sure

(kPa

)

5 10 150

50

100

150

200

250

63 42m 2.5m

Elem X YNo.

t (sec)

Pore

pres

sure

(kPa

)

5 10 15

-5

0

5

10

1521 47m 47m1 5m 5m2 10m 10m

NC, With Slip Elements

Pore pressures and Relative Displacements

Normally Consolidated, With Slip Elements, Frame/10 steps, Cohesion of interface = 0, Friction angle of interface = 35 deg

t (s)

a(m

/sec

2 )

15 20 25 30-10

-5

0

5

t (s)

Acc

(m/s

ec2 )

5 10 15

-5

0

5

10

(x=36m, y=30m)

V1

Dis

p(m

)

5 10 15

-120

-100

-80

-60

-40

-20

0

Cslip=0, Fri. angle=35 degCslip=100, Fri. angle=0 deg

t (s)

Vel

(m/s

)

5 10 15

-15

-10

-5

0

With s lip e lements , NC

NC, Comparison of results from analysis with (c=100 kpa, phi=0 deg) and analysis with (c=0, phi=35 deg)

OCR=5, No Slip Elements, Frame/50 steps

t (s)

a(m

/sec

2 )

15 20 25 30-10

-5

0

5

t (s)

Acc

(m/s

ec2 )

5 10 15-5

0

5

(x=36m, y=30m)

t (s)V

el(m

/s)

5 10 15-1

-0.5

0

0.5

1

V1

Disp

(m)

5 10 15

-4

-2

0

OCR=1OCR=5

No s lip e lements

Comparison of results of OC and NC soils

OCR=5, With Slip Elements, Frame/50 steps, Cohesion of interface= 0,

Friction angle of interface = 35 deg

t (s)

a(m

/sec

2 )

15 20 25 30-10

-5

0

5

t (s)

Acc

(m/se

c2 )

5 10 15

-5

0

5

(x=36m, y=30m)

V1

Disp

(m)

5 10 15-6

-4

-2

0

2

4

6With s lip e lementsWith no s lip e lements

t (s)V

el(m

/s)

5 10 15-1

-0.5

0

0.5

1OCR=5

Comparison of results for OC soil with and without slip elements

t (s)

X-R

elat

ive

Dis

p(m

)

5 10 15

-0.05

-0.04

-0.03

-0.02

-0.01

0 OCR=5, With s lip e lements(x=40m, 30m)

t (s)

Pore

pres

sure

(kPa

)

5 10 150

50

100

150

200

250

OCR=1OCR=5

(X=42m, Y=2.5m)

t (sec)

Pore

pres

sure

(kPa

)

5 10 15

-30

-20

-10

0

10

(x=10m, y=10m)

Comparison of results for NC and OC soil

OCR=5, With Slip Elements, Frame/50 steps, Cohesion of interface= 0,

Friction angle of interface :

%1000 and ,

where,)(

0

minmaxmax

==

−−=

∑ 0|d| γγγ

γγ

φφφφ

-1.50E+01

-1.00E+01

-5.00E+00

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

0 5 10 15

Series1

OCR=5, With Slip Elements, Frame/50 steps, Cohesion of interface= 0,

Friction angle of interface :

%1000 and ,

where,)(

0

minmaxmax

==

−−=

∑ 0|d| γγγ

γγ

φφφφ

Observations

• Analysis of pore-pressure dominated problems require an accurate estimate of pore water pressure accumulation during a seismic loading.

• The deformation of dams where a significant pore water pressure is likely to buildup consists of slumping deformation and sliding deformation

• The behavior at the interface between a potential sliding mass and the slope depends heavily on the pore water pressure accumulation on the interface.

• Degradation of shear strength parameters needs to be modeled in simulating failure in stiff soils.