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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.