1. One-Dimensional Site Response Analysis
What do we mean?
One-dimensional = Waves propagate in one direction only

2. One-Dimensional Site Response Analysis
What do we mean?
One-dimensional = waves propagate in one direction only
Motion is identical on planes perpendicular to that motion
to infinity
to infinity

3. One-Dimensional Site Response Analysis
What do we mean?
One-dimensional = waves propagate in one direction only
Motion is identical on planes perpendicular to that motion
Can’t handle refraction so layer boundaries must be perpendicular to direction of wave propagation
Usual assumption is vertically-propagating shear (SH) waves
Horizontal surface motion
Horizontal input motion

4. One-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Stiffer
with
depth
Focus

5. Horizontal boundaries – waves tend to be refracted toward vertical
One-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Horizontal boundaries – waves tend to be refracted toward vertical
Stiffer
with
depth
Decreasing stiffness causes refraction of waves to increasingly vertical path
Focus

6. One-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Not appropriate here
Stiffer
with
depth

7. Not here! One-Dimensional Site Response Analysis Retaining structures
When are one-dimensional analyses appropriate?
Not here!
Retaining structures
Dams and
embankments
Inclined ground surface and/or non-horizontal boundaries can require use of two-dimensional analyses
Tunnels

8. Localized structures may require use of 3-D response analyses
One-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Not here!
Complex soil
conditions
Dams in
narrow
canyons
Multiple
structures
Localized structures may require use of 3-D response analyses

9. Rock outcropping motion
One-Dimensional Site Response Analysis
How should ground motions be applied?
Rock outcropping motion
2ui
Free surface motion us
Soil
Bedrock motion
ui + ur
Not the same!
Rock
Incoming motion ui

10. One-Dimensional Site Response Analysis
How should ground motions be applied?
Free surface motion us
Input (object) motion
If recorded at rock outcrop, apply as outcrop motion (program will remove free surface effect). Bedrock should be modeled as an elastic half-space.
If recorded in boring, apply as within-profile motion (recording does not include free surface effect). Bedrock should be modeled as rigid.
Object motion

11. Methods of One-Dimensional Site Response Analysis
Complex Response Method
Approach used in computer programs like SHAKE
Transfer function is used with input motion to compute surface motion (convolution)
For layered profiles, transfer function is “built” layer-by-layer to go from input motion to surface motion
Single elastic layer
Amplification
De-amplification

12. Amplitudes of upward- and downward-traveling waves in Layer j
Complex Response Method (Linear analysis)
Consider the soil deposit shown to the right. Within a given layer, say Layer j, the horizontal displacements will be given by
Layer j
Layer j+1
Amplitudes of upward- and downward-traveling waves in Layer j
At the boundary between layer j and layer j+1, compatibility of displacements requires that
No slip
Continuity of shear stresses requires that
Equilibrium satisfied

13. Complex Response Method (Linear analysis)
Defining a*j as the complex impedance ratio at the boundary between layers j and j+1, the wave amplitudes for layer j+1 can be obtained from the amplitudes of layer j by solving the previous two equations simultaneously
Propagation of wave energy from one layer to another is controlled by (complex) impedance ratio
Wave amplitudes in Layer j
Wave amplitudes in Layer j+1
So, if we can go from Layer j to Layer j+1, we can go from j+1 to j+2, etc.
This means we can apply this relationship recursively and express the amplitudes in any layer as functions of the amplitudes in any other layer. We can therefore “build” a transfer function by repeated application of the above equations.

14. Complex Response Method (Linear analysis)
Single layer on rigid base
H = 100 ft
Vs = 500 ft/sec
x = 10%

15. Complex Response Method (Linear analysis)
Single layer on rigid base
H = 50 ft
Vs = 1,500 ft/sec
x = 10%

16. Complex Response Method (Linear analysis)
Single layer on rigid base
H = 100 ft
Vs = 300 ft/sec
x = 5%

17. Complex Response Method (Linear analysis)

18. Complex Response Method (Linear analysis)

19. Complex Response Method (Linear analysis)
Different sequence of soil layers
Different transfer function
Different response

20. Complex Response Method (Linear analysis)
Another sequence of soil layers
Different transfer function
Different response

21. Complex Response Method (Linear analysis)
Complex response method operates in frequency domain
Input motion represented as sum of series of sine waves
Solution for each sine wave obtained
Solutions added together to get total response
Principle of superposition t g
Linear system
Can we capture important effects of nonlinearity with linear model?

22. x Equivalent Linear Approach Equivalent shear modulus
Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions
Stiffness decreases and damping increases as cyclic strain amplitude increases
The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x
Equivalent shear modulus
Equivalent damping ratio

23. Assume some initial strain and use to estimate G and x
Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions
Stiffness decreases and damping increases as cyclic strain amplitude increases
The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x
g(1)
g(1)
Assume some initial strain and use to estimate G and x

24. Use these values to compute response
Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions
Stiffness decreases and damping increases as cyclic strain amplitude increases
The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties.
g(t) t x
g(1)
g(1)
Use these values to compute response

25. Determine peak strain and effective strain
Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions
Stiffness decreases and damping increases as cyclic strain amplitude increases
The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties.
g(t) t
gmax
geff x
g(1)
g(1)
Determine peak strain and effective strain
geff = Rg gmax

26. Select properties based on updated strain level
Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions
Stiffness decreases and damping increases as cyclic strain amplitude increases
The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x
g(1)
g(2)
g(1)
g(2)
Select properties based on updated strain level

27. x g(1) g(3) g(2) g(1) g(3) g(2) Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions
Stiffness decreases and damping increases as cyclic strain amplitude increases
The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x
g(1)
g(3)
g(2)
g(1)
g(3)
g(2)
Compute response with new properties and determine resulting effective shear strain

28. x geff geff Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions
Stiffness decreases and damping increases as cyclic strain amplitude increases
The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x
geff
geff
Repeat until computed effective strains are consistent with assumed effective strains

29. Equivalent Linear Approach
Advantages:
Can work in frequency domain
Compute transfer function at relatively small number of frequencies (compared to doing calculations at all time steps)
Increased speed not that significant for 1-D analyses
Increased speed can be significant for 2-D, 3-D analyses
Equivalent linear properties readily available for many soils – familiarity breeds comfort/confidence
Can make first-order approximation to effects of nonlinearity and inelasticity within framework of a linear model
The equivalent linear approach is an approximation. Nonlinear analyses are capable of representing the actual behavior of soils much more accurately.
… often, a very good one!

30. Nonlinear Analysis Divide time into series of time steps t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
Divide time into series of time steps t
Divide profile into series of layers z

31. Nonlinear Analysis Divide time into series of time steps tj t zi
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
Divide time into series of time steps tj t zi
Divide profile into series of layers
vij = v (z = zi, t = tj) z

32. Nonlinear Analysis tj t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium tj t
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi z

33. Nonlinear Analysis tj t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium tj t
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi
Can change material properties for use in next time step.
Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. z

34. Nonlinear Analysis tj t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium tj t
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi
Can change material properties for use in next time step.
Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. z

35. Nonlinear Analysis tj t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium tj t
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi
Can change material properties for use in next time step.
Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. z

36. Nonlinear Analysis tj t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium tj t
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi
Can change material properties for use in next time step.
Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. z

37. Nonlinear Analysis tj t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium tj t
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi
Step through time
Can change material properties for use in next time step.
Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response.
Procedure steps through time from beginning of earthquake to end. z

38. t t g g Nonlinear Behavior Actual Approximation Continuous
Linear segments
In a nonlinear analysis, we approximate the continuous actual stress-strain behavior with an incrementally-linear model. The finer our computational interval, the better the approximation.

39. Nonlinear Approach Advantages: Work in time domain
Can change properties after each time step to model nonlinearity
Can formulate model in terms of effective stresses
Can compute pore pressure generation
Can compute pore pressure redistribution, dissipation
Avoids spurious resonances (associated with linearity of EL approach)
Can compute permanent strain permanent deformations
Liquefaction
Nonlinear analyses can produce results that are consistent with equivalent linear analyses when strains are small to moderate, and more accurate results when strains are large.
They can also do important things that equivalent linear analyses can’t, such as compute pore pressures and permanent deformations.

40. Equivalent Linear vs. Nonlinear Approaches
What are people using in practice?
Equivalent linear analyses
One-dimensional –
2-D / 3-D –
SHAKE
QUAD4, FLUSH
Nonlinear analyses
One-dimensional –
2-D / 3-D –
DESRA, DMOD
TARA, FLAC, PLAXIS

41. Equivalent Linear vs. Nonlinear Approaches
What are people using in practice?
Equivalent linear analyses
One-dimensional –
2-D / 3-D –
SHAKE
QUAD4, FLUSH
Nonlinear analyses
One-dimensional –
2-D / 3-D –
DESRA
TARA

42. Since early 1970s, numerous computer programs developed for site
Available Codes
Since early 1970s, numerous computer programs developed for site
response analysis
Can be categorized according to computational procedure, number of
dimensions, and operating system
Dimensions OS
Equivalent Linear
Nonlinear
1-D
DOS
Dyneq, Shake91
AMPLE, DESRA, DMOD, FLIP, SUMDES, TESS
Windows
ShakeEdit, ProShake, Shake2000, EERA
CyberQuake, DeepSoil, NERA, FLAC, DMOD2000
2-D / 3-D
FLUSH, QUAD4/QUAD4M, TLUSH
DYNAFLOW, TARA-3, FLIP, VERSAT, DYSAC2, LIQCA, OpenSees
QUAKE/W, SASSI2000
FLAC, PLAXIS

43. Attendees at ICSDEE/ICEGE Berkeley conference (non-academic)
Current Practice
Informal survey developed to obtain input on site response modeling approaches actually used in practice
ed to 204 people
Attendees at ICSDEE/ICEGE Berkeley conference (non-academic)
Geotechnical EERI members – 2003 Roster (non-academic)
55 responses
Western North America (WNA)
Eastern North America (ENA)
Overseas
Private firms
Public agencies
Survey
Respondents
WNA
ENA
Overseas
Private
Public
Number of responses 35 3 6 1 5

44. Current Practice Method of Analysis
Of the total number of site response analyses you perform, indicate the
approximate percentages that fall within each of the following categories:
[ ] a. One-dimensional equivalent linear
[ ] b. One-dimensional nonlinear
[ ] c. Two- or three-dimensional equivalent linear
[ ] d. Two- or three-dimensional nonlinear
Method of
Analysis
WNA
ENA
Overseas
Private
(35)
Public
(3)
(6)
(1)
(5)
1-D Equivalent Linear 68 52 86 50 24 5
1-D Nonlinear 11 17 12 48
2-D/3-D Equiv. Linear 9 28 1 25 6
2-D/3-D Nonlinear 3 23 90
One-dimensional equivalent linear analyses dominate North American practice; nonlinear analyses are more frequently performed overseas

45. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
u(0,t)
1 m
15 m
29 m
Ts = 0.4 sec
30 m
Vs = 300 m/sec
u(H,t)
Topanga record (Northridge)
Vs = 762 m/sec
Topanga record (Northridge)

46. Low degree of nonlinearity
Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.05 g
Weak motion +
stiff soil
Low strains
Low degree of nonlinearity
Similar response

47. Low degree of nonlinearity
Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.05 g
Weak motion +
stiff soil
Low strains
Low degree of nonlinearity
Similar response

48. Low degree of nonlinearity
Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.05 g
Weak motion +
stiff soil
Low strains
Low degree of nonlinearity
Similar response

49. Low degree of nonlinearity
Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.05 g
Weak motion +
stiff soil
Low strains
Low degree of nonlinearity
Similar response

50. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.20 g
Moderate motion +
stiff soil
Relatively low strains
Relatively low degree of nonlinearity
Similar response

51. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Acceleration
Topanga motion scaled to 0.20 g
Moderate motion +
stiff soil
Velocity
Relatively low strains
Relatively low degree of nonlinearity
Similar response

52. Nonlinear Behavior
Equivalent linear overpredicts nonlinear response at certain frequencies – “spurious resonances”
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.20 g
Moderate motion +
stiff soil
Relatively low strains
Stress-strain response becoming more complicated – more variable stiffness and less “elliptical” shape
Relatively low degree of nonlinearity
Similar response

53. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.20 g
Moderate motion +
stiff soil
Relatively low strains
Stiffness starting to vary more significantly over course of ground motion
Relatively low degree of nonlinearity
Similar response

54. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Acceleration
Topanga motion scaled to 0.50 g
Strong motion +
stiff soil
Moderate strains
Low – moderate degree of nonlinearity
Noticeably different response

55. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.50 g
Strong motion +
stiff soil
Moderate strains
Low – moderate degree of nonlinearity
Noticeably different response

56. Softening by EL method causes underprediction
Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Acceleration
Topanga motion scaled to 1.0 g
Substantial softening by EL method causes underprediction of initial portion of record
Very strong motion +
stiff soil
Softening by EL method causes underprediction
Linearity inherent in EL method causes overprediction response in strongest portion of record
Moderate strains
Moderate degree of nonlinearity
Noticeably different response

57. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Topanga motion scaled to 0.50 g
Very strong motion +
stiff soil
Moderate strains
Moderate degree of nonlinearity
Noticeably different response

58. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
u(0,t)
1 m
15 m
29 m
16 m
Vs = 100 m/sec
Vs = 300 m/sec
14 m
u(H,t)
Vs = 762 m/sec

59. Nonlinear Behavior
Acceleration
Equivalent linear vs nonlinear analysis – how much difference does it make?
EL model predicts very soft behavior at beginning of earthquake, before any large strains have developed.
Large strain levels (~6%) near bottom of upper layer
EL model converges to low G and high x
High-frequency components cannot be transmitted through over-softened EL model
NL model: Stiffness stays relatively high except for a few large-amplitude cycles

60. Nonlinear Behavior
Acceleration
Equivalent linear vs nonlinear analysis – how much difference does it make?
Large strain levels (~6%) near bottom of upper layer
More consistency, but NL model can transmit high-frequency oscillations superimposed on low-frequency cycles – too much?
EL model converges to low G and high x
High-frequency components cannot be transmitted through over-softened EL model
NL model: Stiffness stays relatively high except for a few large-amplitude cycles

61. Nonlinear Behavior
Acceleration
Equivalent linear vs nonlinear analysis – how much difference does it make?
Large strain levels (~6%) near bottom of upper layer
NL model exhibits stiff behavior following strongest part of record; EL maintains low stiffness, high damping behavior throughout.
EL model converges to low G and high x
High-frequency components cannot be transmitted through over-softened EL model
NL model: Stiffness stays relatively high except for a few large-amplitude cycles

62. Nonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Large strain levels (~6%) near bottom of upper layer
EL model converges to low G and high x
High-frequency components cannot be transmitted through over-softened EL model
NL model: Stiffness stays relatively high except for a few large-amplitude cycles

63. Nonlinear Soil Behavior
Small cycle superimposed on large cycle (after Assimaki and Kausel, 2002)
Time
High stiffness
Equivalent linear model maintains constant stiffness and damping – higher stiffness excursions associated with higher frequency oscillations aren’t seen.
Low stiffness

64. Nonlinear Soil Behavior
Small cycle superimposed on large cycle (after Assimaki and Kausel, 2002)
Time
Low damping
Equivalent linear model maintains constant stiffness and damping – higher stiffness excursions associated with higher frequency oscillations aren’t seen.
High damping

65. Modified Equivalent Linear Approach
High frequencies are associated with smaller strains
High stiffness and low damping are associated with smaller strains
Make stiffness and damping frequency-dependent
Normalized strain spectra from five motions
Normalized strain spectrum from one motion
Three orders of magnitude
Frequency (Hz)
Frequency (Hz)

66. Frequency-dependent model
Modified Equivalent Linear Approach
Assimaki and Kausel
Frequency-dependent model
Conventional model
High frequencies oversoftened and overdamped
Excellent agreement with nonlinear model

67. Benchmarking of Nonlinear Analyses
Stewart and Kwok
PEER study to determine proper manner in which to use nonlinear analyses
Worked with five existing nonlinear codes; hired developers to run their codes and comment on results
Established advisory committee to oversee analyses and assist with interpretation
Met regularly with advisory committee and developers

68. Benchmarking of Nonlinear Analyses
Stewart and Kwok
Considered codes

69. Benchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic)
Enhanced version of D-MOD, which is enhanced version of DESRA
Lumped mass model
Rayleigh damping
Rayleigh
Damping ratio
Stiffness-proportional
Mass-proportional
Frequency

70. Benchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic)
Enhanced version of D-MOD, which is enhanced version of DESRA
Lumped mass model
Rayleigh damping
Newmark b method for time integration
Variable slice width – simulating response of dams, embankments on rock
Decreasing stiffness due to geometry

71. Benchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic)
Enhanced version of D-MOD, which is enhanced version of DESRA
Lumped mass model
Rayleigh damping
Newmark b method for time integration
Variable slice width – simulating response of dams, embankments on rock
Can simulate slip on weak interfaces
Uses MKZ soil model (modified hyperbola – needs Gmax, tmax, a and s)
Can soften backbone curve to model cyclic degradation

72. Benchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic)
Enhanced version of D-MOD, which is enhanced version of DESRA
Lumped mass model
Rayleigh damping
Newmark b method for time integration
Variable slice width – simulating response of dams, embankments on rock
Can simulate slip on weak interfaces
Uses MKZ soil model (modified hyperbola – needs Gmax, tmax, a and s)
Can soften backbone curve to model cyclic degradation
Uses Masing rules for unloading-reloading behavior
Need input parameters for:
MKZ backbone curve (4)
Cyclic degradation (3 for clay, 4 for sand)
Pore pressure generation (4 for clay, 4 for sand)
Pore pressure redistribution/dissipation (at least 2)
Rayleigh damping coefficients (2)
Basic layer properties (density, shear wave velocity, half-space properties)

73. Benchmarking of Nonlinear Analyses
DEEPSOIL (Hashash)
Similar to DMOD-2 (lumped mass, derives from DESRA-2)
More advanced Rayleigh damping scheme (lower frequency dependence)
TESS (Pyke)
Finite difference wave propagation analysis (not lumped mass)
Cundall-Pyke hypothesis for loading-unloading behavior
Similar backbone curve to DMOD-2 and DEEPSOIL
Inviscid (sort of) low-strain damping scheme
OpenSees (Yang, Elgamal)
Finite element model (1D, 2D, 3D capabilities)
Multi-surface plasticity model (von Mises yield surface, kinematic hardening, non-associative flow rule)
Full Rayleigh damping
SUMDES
Finite element model
Bounding surface plasticity model (Lade-like yield surface, kinematic hardening, non-associative flow rule)
Simplified Rayleigh damping

74. Benchmarking of Nonlinear Analyses
Recommendations
Specification of control motion
For outcropping motion, use recorded motion with elastic base
For motions recorded at depth, use recorded motion with rigid base
Specification of viscous damping
Use full or extended Rayleigh damping – iterate on selection of control frequencies to match equivalent linear response for low loading levels (linear response domain). If not possible, use full Rayleigh damping with targets at fo and 5fo.
Backbone curve parameters
Adjust, if possible, to produce correct shear strength at large strains
Bound nonlinear, inelastic behavior by running analyses with:
Backbone curve fit to match G/Gmax behavior
Backbone curve fit to minimize error in G/Gmax and damping curves

75. Benchmarking of Nonlinear Analyses
Performance
Based on validations against vertical array data
Models produce reasonable results
Some indication of overdamping at high frequencies, overamplification at site frequency
Variability of predictions due to backbone curves and damping models most pronounced at T<0.5 sec and is significant only for relatively thick profiles. Model-to-model variability most pronounced at low periods.
Nonlinearity modeled well up to levels for which adequate data is available (generally up to about 0.2g). Data for stronger shaking being sought (centrifuge tests, recent Nigaata earthquake).
DMOD-2, DEEPSOIL, and OpenSees generally produced similar amplification factors and spectral shapes; TESS produced different response at high frequencies (different damping formulation), SUMDES results were significantly different than all others for deep sites (probably due to simplified Rayleigh damping).

76. Nonlinear Behavior – Effective Stress Analyses
Wildlife – Superstition Hills recordings

77. Nonlinear Behavior – Effective Stress Analyses
Wildlife – Superstition Hills recordings

78. Nonlinear Behavior – Effective Stress Analyses
Wildlife – Elmore Ranch recordings

79. Nonlinear Behavior – Effective Stress Analyses
Wildlife – Superstition Hills recordings
Ground surface record
High frequency
???
Low frequency

80. Site Effects Elmore Ranch record – no liquefaction
Ratio of wavelet amplitudes – variation with frequency and time
Frequency (Hz)
Time (sec)

81. Site Effects Elmore Ranch record – no liquefaction
Ratio of wavelet amplitudes – variation with frequency and time
Frequency (Hz)
Time (sec)

82. Nonlinear Behavior – Effective Stress Analyses
Wildlife – Superstition Hills recordings

83. Nonlinear Behavior – Effective Stress Analyses
Wildlife – Superstition Hills recordings

84. One-Dimensional Site Response Analysis
Summary
Must be aware of assumptions
Uni-directional wave propagation (normal to layer boundaries)
Uni-directional particle motion (no surface waves)
Particularly useful for profiles with high impedance contrasts
Equivalent linear approach works very well for most cases
Material properties readily available
Computations performed rapidly
Nonlinear analyses match equivalent linear when strains are small
Nonlinear analyses are preferred when strains are high – soft soils and/or strong shaking
Can account for shear strength of soil
Can handle pore pressure generation – some well, some poorly
Can predict permanent deformations – for common for 2-D analyses

85. Thank you