SSI Problem Definition, SSI Effects, Methods of Analysis, Interaction Analysis, FEM, BEM Analysis, Governing Equations, Staggered Solutions, Lumped Parameter Model, Travelling Wave effects, Dynamic SSI, Dynamic Soil-Struct Interaction,

1. Prof. Samirsinh P Parmar
Mail: samirddu@gmail.com
Asst. Prof. Dept. of Civil Engg.
Dharmsinh Desai University, Nadiad,
Gujarat , Bharatvarsh.
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
1

2. Content of the Presentation
• SSI – Problem Definition
• SSI Effects
• Methods of Analysis
• Interaction Analysis
• FEM, BEM Analysis
• Governing Equations
• Staggered Solutions
• Lumped Parameter Model
• Travelling Wave effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
2

3. SSI – Problem Definition
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
3
Earthquake Analysis
Structures supported by rigid foundations
Earthquakes=>Specified motion of base
Rigid
Base
Analysis
Tall Buildings
Acceptable
• Light & Flexible
• Firm Foundations
• Methods focus on
modeling of structure
• Displacements wrt fixed
base
• Finite Element Methods
Nuclear Power Plants
Wrong Assumption
• Massive & Stiff
• Soft Soils
• Interaction with supporting
soils becomes important

4. SSI – Problem Definition
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
4
Machine Foundation
Parameters
•Local Soil Conditions
•Peak Acceleration
•Frequency Content of
Motion
•Proximity to Fault
•Travel Path etc
Inertial Interaction
Inertial forces in structure are
transmitted to flexible soil
Kinematic Interaction
Stiffer foundation cannot conform to
the distortions of soil
TOTAL=INERTIAL + KINEMATIC
Seismic Excitation

5. SSI Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
5
0.00E+00
2.50E-05
5.00E-05
7.50E-05
1.00E-04
1.25E-04
1.50E-04
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
w/wn
Amplitude
in
ft.
SSI - Proposed BE-FE
SSI - Spring Dashpot
Model
Proposed BE-FE, Stiff
Soil
Fixed Base Analysis
Posin( w t)
Half Space
2b
H

6. SSI Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
6
-1.0E-05
-5.0E-06
0.0E+00
5.0E-06
1.0E-05
1.5E-05
2.0E-05
2.5E-05
0 25 50 75 100 125 150 175 200 225 250
Time x 1.08x10-4
(sec)
Horizontal
Amplitude
U1 SSI - Relative
U1 Fixed Base
U2 SSI - Relative
U2 Fixed Base
P(t)
Half Space
m
m

7. Cross Interaction Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
7
1. Moment is applied
2. Waves Propagate…
3. …Reach Receiver…
4. …and life goes on…

8. SSI Effects
• Alter the Natural Frequency of the Structure
• Add Damping
• Through the Soil Interaction Effects
• Traveling Wave Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
8

9. Methods of Analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
9
Objective:
Given the earthquake ground motions that would
occur on the surface of the ground in the absence of
the structure (control or design motions), find the
dynamic response of the structure.

10. Methods of Analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
10
Methods
Idealized
Complete
Direct MultiStep

11. Complete Interaction Analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
11
• Account for the variation of soil properties with depth.
• Consider the material nonlinear behavior of the soil
• Consider the 3-D nature of the problem
• Consider the nature of the wave propagation which produced
the ground motion
• Consider possible interaction with adjacent structures.
High Degree of Complexity

12. Idealized Interaction Analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
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Idealization
Horizontal Layers
Simplified Wave Mechanisms etc

13. Idealized Interaction Analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
13
Preliminary description of free field motion
before any structure has been built
The definition of the motion itself
the control motion in terms of response spectra, acceleration
records etc.
The location of the control motion
free surface, soil-rock interface
The generation mechanism at the control point vertically or
obliquely incident SH or SV waves, Rayleigh waves, etc.

14. Idealized Analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
14
Idealized Interaction Analysis
Tools: FEM, BEM, FDE, Analytical solutions
Direct Methods
Evaluation of Dynamic
Response in a Single
Step
MultiStep Methods
Evaluation of Dynamic Response in
Several Steps
SUPERPOSITION
• Two-Step
Kinematic+Inertia Interaction
• Three-Step
Rigid Foundations
Lumped Parameter Models
• Substructure
Division to Subsystems
Equilibrium & Compatibility
True Nonlinear
Solutions

15. Finite Element Method (FEM)
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
15
Governing Equation
• Modal Analysis
• Direct Integration
• Fourier Analysis - Complex Response
Solution Techniques

16. FEM Solution Techniques
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
16
Selection Criteria Cost and Feasibility
Paramount Consideration Accuracy
Differences
- Handling of Damping
- Ability to Handle High Frequency
Components of Motion

17. FEM - Modal Analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
17
 Damping is neglected during early stages
 Actual displacements are damped
 Damping is considered in arbitrary manner
 Structural Dynamics: First few modes need to be evaluated (<20)
 SSI: Acceleration response spectra over a large frequency range and
large number of modes need to be considered (>150)
 Not recommended for Direct SSI - Stiff Massive Structure Soft Soil
 OK for Substructure

18. FEM - Direct Integration
• Time Marching Schemes
Newmark’s Methods, Wilson Methods, Bathe and Wilson
Cubic Inertia Method
• Small Time Step for Accuracy
• Stability and Convergence
• Choice of Damping Matrix
 Frequency Dependent Damping Ratio - filters out high frequency components
 Proportional Damping
• Good Choice if True Dynamic Nonlinear Analysis is feasible
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
18

19. FEM - Complex Response
• Fourier Transformation - Transfer Functions
• Transfer Functions Independent of External Excitation
• Control of Accuracy
• Efficient
• Only Linear or Pseudo non-linear analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
19

20. FEM - Geometric Modeling
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
20

21. FEM Modeling
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
21










Matrix
Mass
Mixed
5
1
Matrix
Mass
Consistent
8
1
Matrix
Mass
Lumped
8
1
max
s
s
s
h



Max Element Size Governed by Highest frequency which must
be transmitted correctly within the element

22. FEM Modeling of Infinite Space
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
22

23. FEM Modeling of Infinite Space
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
23
Modeling Introduces Artificial Boundaries that Reflect Waves

24. FEM Modeling of Infinite Soil
• Absorbing Boundaries
 Viscous Boundary
 Variable Depth Method
 Damping proportional to Wave Velocities
• Radiating Boundaries (Hyper elements)
 Satisfy Boundary Conditions at Infinity
 Eigenvalue Analysis
 Frequency Domain Analysis
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
24

25. SSI – FEM Methods
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
25
FEM
Advantages
• Non-Linear Analysis
• Well Established
Shortcomings
• Finite Domains
• Volume
Discretization's

26. Boundary Element Methods
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
26
  j
j
ii
j
ij
i u
f
u
c
u
c
c 




 ,
2
2
,
2
2
2
1
Governing Equation
 Small Displacement
Field
 Homogeneous
 Isotropic
 Elastic

27. Boundary Element Method
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
27
GOVERNING EQUATION
BOUNDARY INTEGRAL EQUATION
Dynamic Reciprocal
Theorem
Indirect
DIRECT
Transform Domain TIME DOMAIN
Dirac- Step Impulse B-SPLINE
System of Algebraic Equations
Time Marching Scheme

28. Boundary Element Method
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
28
BOUNDARY INTEGRAL EQUATION
B-SPLINE FUNDAMENTAL SOLUTIONS
SPATIAL DISCRETIZATION
TEMPORAL DISCRETIZATION
BOUNDARY INTEGRAL EQUATION IN A DISCRETE FORM
TIME MARCHING SCHEME &
B-SPLINE IMPULSE RESPONSE
RESPONSE TO ARBITRARY EXCITATION
N
N
H
Ff 

29. BEM – Methods
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
29
BEM
Advantages
• Infinite Media
• Surface Discretization
Shortcomings
• Non-symmetric matrices
• Not Efficient for Nonlinear

30. SSI Methods
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
30
Combined BEM-FEM
eliminate disadvantages of each method
and retain advantages
Approach
• FEM Approach
• BEM Approach
• Staggered Solutions

31. Governing Equations
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
31
  j
j
ii
j
ij
i u
f
u
c
u
c
c 




 ,
2
2
,
2
2
2
1
   
   
t
t
t
t
f
Ku
u
C
u
M


 



32. FEM Method
Time Marching Scheme
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
32
       
t
t
t
t f
Ku
u
C
u
M 

 


N
N
f
Du 
Governing Equation
Discrete Form in Time

33. FEM-BEM Coupling Staggered Solutions
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
33
Can be Solved in a Staggered Approach...
N
N
BEM
N
BEM H
Ff
u 

N
FEM
N
FEM f
Du 
BEM
FEM

34. FEM-BEM Coupling Staggered Solutions
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
34
Compatibility of Displacements
at Interface
BEM
Solver
FEM
Solver
Equilibrium of Forces
at Interface
External
Excitation
External
Excitation
int
FEM
u
int
BEM
u
int
FEM
f
int
BEM
f
At Every Time Step...

35. FEM-BEM Coupling Advantages
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
35
 Independent Solutions for BEM and FEM
 Independent Time Step Selection
 Smaller Systems of Equations
 BEM System of Reduced Size
 In the Absence of Incidence Displacement Field
in Soil, BEM does not require Solution.

36. Lumped Parameter Models for SSI
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
36
P(t) m
Half Space
P(t) m
Spring-Dashpot Model
Stick Model

37. Lumped Parameter Foundation Models
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
37
Reissner (1936) Analytic Solutions to Vertical Vibration of Circular Footing Due
to Harmonic Excitation
Assumptions:
Elastic ½-space
Material G,v,r
Uniform Vertical Pressure
Formed Basis of Almost All Analytical Studies

38. Lumped Parameter Foundation Models
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
38
Quinlan and Sung
Assumed Different Pressure Distributions
Richart & Whitman
Effects of Poisson’
Bycroft (1956)
Displacement Functions
Hsieh
K and C in terms of Soil and Foundation Parameters

39. Lumped Parameter Foundation Models
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
39
Lysmer Analog
Constant Lumped Parameters
Richart Hall & Wood(1970)
Gazetas (1983)
Wolf (1988)

40. Lumped Parameter Foundation Models
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
40
Representative Lumped Parameter Values - Square

41. Lumped Parameter Foundation Models
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
41
Mode K C B D
Vertical
(z)  
n

1
4 o
Gr
G
ro r
n
2
1
4
.
3
 3
4
1
o
r
m
r
n

z
B
425
.
0
Sliding
(x)  
n

2
8 o
Gr
G
ro r
n
2
2
6
.
4
 3
8
2
o
r
m
r
n

x
B
288
.
0
Rocking
()  
n

1
3
8 3
o
Gr
  

n
r
B
G
ro

 1
1
8
.
0 4
 
5
8
1
3
o
r
I
r
n 

  
 B
B

1
15
.
0
Torsional
() 3
6 3
o
Gr
 

r
B
G
B
2
1
4

5
o
r
I
r

 

B
2
1
5
.
0

Representative Lumped Parameter Values Circular

42. Lumped Parameter Foundation Models
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
42
Stehmeyer and Rizos (2003)
Properties k, and c are known to be frequency (w) dependent
The Real System Equivalent SDOF System
n
n
m
c
M
K
w

w
2



43. Lumped Parameter Foundation Models
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
43
Horizontal Displacement with Horizontal Impulse Applied
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
Time
Displacement
Discrete BEM Solution
Simplified Closed Form Solution
2b
B(t)
Half Space
y
z
x
B(t)
wn = 3.3
 = 0.975

44. SSI Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
44
0.00E+00
2.50E-05
5.00E-05
7.50E-05
1.00E-04
1.25E-04
1.50E-04
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
w/wn
Amplitude
in
ft.
SSI - Proposed BE-FE
SSI - Spring Dashpot
Model
Proposed BE-FE, Stiff
Soil
Fixed Base Analysis
Posin( w t)
Half Space
2b
H

45. SSI Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
45
-1.0E-05
-5.0E-06
0.0E+00
5.0E-06
1.0E-05
1.5E-05
2.0E-05
2.5E-05
0 25 50 75 100 125 150 175 200 225 250
Time x 1.08x10-4
(sec)
Horizontal
Amplitude
U1 SSI - Relative
U1 Fixed Base
U2 SSI - Relative
U2 Fixed Base
P(t)
Half Space
m
m

46. SSI Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
46
Based on the Simplified Lumped Parameter Models it can be
shown that

k
k
k
k
T
T h
h
2
1
~



P(t) m
Longer Period of Foundation-Structure System

47. SSI Effects – Cross Interaction
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
47
Receiver Foundation
Source Foundation

48. SSI Effects – Cross Interaction
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
48
0.0E+00
5.0E-11
1.0E-10
1.5E-10
2.0E-10
2.5E-10
0 0.5 1 1.5 2 2.5 3 3.5 4
Dimensionless Frequency ao
Horizontal
Amplitude

1
Source M=10
Receiver M=10
Source M=5
Receiver M=5
Source M=1
Receiver M=1
Receiver Foundation
Receiver Foundation
Source Foundation
Source Foundation
Receiver Foundation
Receiver Foundation
Source Foundation
Source Foundation

49. SSI Effects – Cross Interaction
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
49
0.0E+00
5.0E-11
1.0E-10
1.5E-10
2.0E-10
2.5E-10
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Dimensionless Frequency ao
Horizontal
Amplitude
d/a=0.25
Source Foundation
d/a=1.00
d/a=2.00
d/a=3.00
d/a=0.25
Receiver Foundation
d/a=1.00
d/a=2.00
d/a=3.00
Receiver Foundation
Receiver Foundation
Source Foundation
Source Foundation
Receiver Foundation
Receiver Foundation
Source Foundation
Source Foundation

50. Traveling Wave Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
50
After Betti et al.

51. Traveling Wave Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
51
After Betti et al.

52. Traveling Wave Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
52
After Betti et al.

53. Traveling Wave Effects
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
53
After Betti et al.

54. SH-Waves
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
54
After Betti et al.

55. P-Waves
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
55
After Betti et al.

56. SV-Waves
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
56
After Betti et al.

57. Rayleigh Waves
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
57
After Betti et al.

58. Traveling Wave Effects
• Inertia Effects were Not Important but yet SSI significantly
affects the response
• Asynchronous Motion Excite Antisymmetric Vibration Modes
• SSI effects cannot be ignored
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
58
After Betti et al.

59. Thank
You
Prof.
S.P.Parmar,M.Tech
Geotechnical
Engineering.
DDU-CL.
59