The study of soil's small strain dynamic behaviour was the main goal of this dissertation, for the case of a residual soil from Porto granite, a geotechnical material that possess a complex mechanic behaviour. To that end, a detailed experimental research was developed, in true triaxial conditions, considering the measurement of seismic wave velocities through bender elements. A three-dimensional numerical model was produced that characterized the behaviour of this particular residual soil using a finite difference program, FLAC3D. Its main advantages reside in its simplicity, versatility and the possibility of directly measuring seismic wave velocities, not only in the three principal directions, but also in inclined directions. This study requires the implementation of bender elements in the platens of the true triaxial apparatus, in order to assess the influence of its cubical geometry and boundary conditions (rigid or flexible platens, or even reflective and absorbent), as well as the validation of stiffness parameters attained from the measurement of seismic waves. For this purpose, a series of parametric and sensitivity studies were developed, considering the linear elastic constitutive model, with isotropic loading, in the previously mentioned software, to particularly evaluate the influence of each parameter in the numerical modelling of the true triaxial apparatus, and which values are better suited for its correct representation. These parameters are: time step, amplitude, frequency, damping, Poisson’s ratio and finally, boundary conditions and a cross-anisotropic constitutive model. With these studies completed, a comparison and validation between the numerical results attained and the laboratory results previously done by Ferreira (2009), regarding residual soil specimens from Porto granite was in order. Due to time limitations, only the dry specimens (w ≈ 0%), namely, R8D-TT and R4D-K0TT were used in this comparative study.

1. Numerical Modelling of the
Dynamic Behaviour of a Soil
in True Triaxial Tests with
Bender Elements
Ana Rita Silva - 201005317
Mestrado Integrado em Engenharia Civil – Especialização em Geotecnia
Porto, 17 de Julho de 2014

2. • Understanding of the characteristics and dynamic properties of soils
• Non-linear behaviour of soils Measurement of seismic waves
In situ and Laboratory methods
Bender Elements (BE)
True Triaxial Apparatus (TT)
• This numerical study through FLAC3D is a validation of the elastic and small strain stiffness
parameters from seismic wave measurements on a residual soil from Porto granite in a TT
- Ferreira (2009)
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions

3. Volumetric Seismic Waves
P-wave S-wave
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Seismic Waves

4. • Non-linearity
• Stress and strain
• Stiffness degradation curve
• “Small is beautiful” (Burland 1989)
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Dynamic Behaviour of Soils

5. Laboratory Methods
• Piezoelectric Transducers
• Piezoelectricity: ability of converting electrical energy into mechanical energy or vice-versa
• Bender Elements:
- Consists of two piezoceramic transdurcers composed of two piezoceramic plates rigidly attached
- Electrical connection ensures an accurate flexural movement, in order to propagate shear waves during its deformation
- Single transducer or T-shaped pair of transducers
Advantages:
- Simplicity (results interpretation and procedures)
- Versatility and portability
- Can be used in the small-strain domain
- Immediate registration of the results
- Quick, simple and low cost implementation of the
support equipment
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Seismic Wave Testing Methods

6. Laboratory Methods
• True Triaxial Apparatus
• Types of boundaries: Rigid, flexible, mixed
• Rigid boundary: six square platens assembled in a cubical frame of anodized aluminum
• Example: collaborative project between FEUP and University of Western Australia
Advantages:
- Versatility (uniform shear stresses and normal
stresses can be applied)
- Provdes much information
- Allow controlled gradual rotations of the principal
axes and strain
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Seismic Wave Testing Methods

7. • Three-dimensional finite difference program for geotechnical engineering calculations
• Possesses a lower processing capacity and models more complex behaviours than the finite
element method
• Explicit calculation: time domain – suited for the simulation of BE and the dynamic behaviour of
soil
• Possesses several constitutive models
• Suitable especification of boundary conditions
• Two types of hysteretic damping: local and Rayleigh
• Graphical output in a variety of formats
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Program FLAC3D

8. Brick shaped mesh Radially graded mesh around brick
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Mesh Generation

9. Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Mesh Generation

10. - Mesh Generation
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions

11. Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Mesh Generation

12. Elastic: - linear stress-strain behaviour
- deformation increases with applied forces
- homogeneous and continuous materials
Soil: residual soil from Porto granite
Soil Bender Element
Shear modulus (G)=80.00 MPa Shear Modulus (G)=1000 MPa
Poisson’s ratio (ν)=0.10 Poisson’s ratio (ν)=0.25
Young’s modulus (E)=176.00 MPa Young’s modulus (E)=2500 MPa
Bulk modulus (K)=73.33 MPa Bulk modulus (K)=1700 MPa
Density (ρ)=2000 kg/m3 Density (ρ)=3000 kg/m3
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Properties

13. -3.00E-06
-2.00E-06
-1.00E-06
0.00E+00
1.00E-06
2.00E-06
3.00E-06
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03
Y-displacement(m)
Time (s)
Input 10.0 kHz 1.00E-04 1.78E-05 (automatic)
• Automatic time step: self-analysis
• Lack of data/points to correctly translate
the shape of the output signal
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Time Step

14. • Higher degrees of magnitude: more
continuous signal and an improvement in
quality and number of data
-3.00E-06
-2.00E-06
-1.00E-06
0.00E+00
1.00E-06
2.00E-06
3.00E-06
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03
Y-displacement(m)
Time (s)
Input 10.0 kHz 1.00E-04 1.78E-05 (automatic) 5.00E-06
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Time Step

15. -3.00E-06
-2.00E-06
-1.00E-06
0.00E+00
1.00E-06
2.00E-06
3.00E-06
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03
Y-displacement(m)
Time (s)
Input 10.0 kHz 1.00E-04 1.78E-05 (automatic) 5.00E-06 2.50E-07
• Chosen time step
• Perfect balance: data, quality and
processing capacity
Varies from model to model
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Time Step

16. Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Frequency: Time Domain (TD)
-1,6E-03
-1,4E-03
-1,2E-03
-1,0E-03
-8,0E-04
-6,0E-04
-4,0E-04
-2,0E-04
0,0E+00
2,0E-04
0,0E+00 1,0E-03 2,0E-03 3,0E-03 4,0E-03 5,0E-03 6,0E-03
Y-displacement(m)
Time (s)
Input 0.5 kHz 0.5 kHz
Input 1.0 kHz 1.0 kHz
Input 2.0 kHz 2.0 kHz
Input 4.0 kHz 4.0 kHz
Input 5.0 kHz 5.0 kHz
Input 10.0 kHz 10.0 kHz
Input 25.0 kHz 25.0 kHz

17. Fmax:
1.63 kHz
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 5 10 15
Output(V)
Input(V)
Frequency (kHz)
Input 0.5 kHz Output 0.5 kHz
Fmax:
1.75 kHz
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 5 10 15
Output(V)
Input(V)
Frequency (kHz)
Input 1.0 kHz Output 1.0 kHz
Fmax:
1.88 kHz
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 5 10 15
Output(V)
Input(V)
Frequency (kHz)
Input 2.0 kHz Output 2.0 kHz
Fmax:…
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 5 10 15
Output(V)
Input(V)
Frequency (kHz)
Input 4.0 kHz Output 4.0 kHz
Fmax:
1.63 kHz
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 50 100 150
Output(V)
Input(V)
Frequency (kHz)
Input 5.0 kHz Output 5.0 kHz
Fmax:
2.13…
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 50 100 150
Output(V)
Input(V)
Frequency (kHz)
Input 10.0 kHz Output 10.0 kHz
Fmax:
1.88…
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 50 100 150
Output(V)
Input(V)
Frequency (kHz)
Input 25.0 kHz Output 25.0 kHz
Fmax:
1.88 kHz
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 50 100 150
Output(V)
Input(V)
Frequency (kHz)
Input 50.0 kHz Output 50.0 kHz
a) b) c) d)
e) f) g) h)
Variation of frequency: a) 0.5 kHz; b) 1.0 kHz; c) 2.0 kHz; d) 4.0 kHz; e) 5.0 kHz; f) 10.0 kHz; g) 25.0 kHz; h)50.0 kHz
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Frequency: Frequency Domain (FD)

18. LOCAL Damping
-1.2E-04
-7.0E-05
-2.0E-05
3.0E-05
8.0E-05
1.3E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 8.0E-03
Y-displacement(m)
Time (s)
Input 10 kHz 0.01 0.05 0.10 0.50 0.90
-1.2E-04
-7.0E-05
-2.0E-05
3.0E-05
8.0E-05
1.3E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 8.0E-03
Y-displacement(m)
Time (s)
Input 10 kHz L: 0.05
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Damping

19. RAYLEIGH Damping
-1.2E-04
-7.0E-05
-2.0E-05
3.0E-05
8.0E-05
1.3E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 8.0E-03
Y-displacement(m)
Time (s)
Input 10 kHz R: 10 kHz
-1.2E-04
-7.0E-05
-2.0E-05
3.0E-05
8.0E-05
1.3E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 8.0E-03
Y-displacement(m)
Time (s)
Input 10.0 kHz 2.5 kHz 5.0 kHz 8.0 kHz 10.0 kHz 12.0 kHz
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Damping

20. • Face C is the zone most influenced by
ν, given the higher reflection of the
signal, due to the large dimensions of
the model
• P waves (vary – E ≠ constant)
• : Model’s first arrival (first inflection)
• : Theoretical first arrival
• S waves (constant – G = constant)
• : Theoretical first arrival
Poisson’s ratio chosen = 0,10
-1.2E-03
-1.0E-03
-8.0E-04
-6.0E-04
-4.0E-04
-2.0E-04
0.0E+00
2.0E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03
Y-displacement(m)
Time (s)
input 5.0 kHz ν = 0.00 tS tP 0.00
input ν = 0.10 tS tP 0.10
input ν = 0.20 tS tP 0.20
input ν = 0.30 tS tP 0.30
input ν = 0.40 tS tP 0.40
input ν = 0.48 tS tP 0.48
Variation of ν and the first arrival of both P and S waves in face C
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Poisson’s Ratio (ν)

21. ABSORBENT
-6.00E-04
-5.00E-04
-4.00E-04
-3.00E-04
-2.00E-04
-1.00E-04
0.00E+00
1.00E-04
0.00E+00 1.00E-03 2.00E-03 3.00E-03 4.00E-03 5.00E-03
Y-displacement(m)
Time (s)
A: Transmitter A: Midpoint A: Receiver
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Boundary Conditions

22. • Transmitter: No difference
• Midpoint and Receiver:
- reflection, energy, absorbency
(midpoint behaves as if it was in an infinite medium)
• The further from the transmitter, the
more difference between first arrival
of P and S waves
• P waves
• : Model’s first arrival
• : Theoretical first arrival
• S waves
• : Theoretical first arrivalREFLECTED VERSUS ABSORBENT
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Boundary Conditions
-6.00E-04
-5.00E-04
-4.00E-04
-3.00E-04
-2.00E-04
-1.00E-04
0.00E+00
1.00E-04
0.00E+00 1.00E-03 2.00E-03 3.00E-03 4.00E-03 5.00E-03
Y-displacement(m)
Time (s)
R: Transmitter A: Transmitter
R: Midpoint A: Midpoint
R: Receiver A: Receiver
tP tS
tS: Reflection tS

23. Displacements (example: EV = 0.7 EH)
VERTICAL DisplacementHORIZONTAL Displacement
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Anisotropy

24. Displacements (example: EV = 0.7 EH)
7.E-05
0.0E+00
3.0E-05
6.0E-05
9.0E-05
1.2E-04
1.5E-04
1.8E-04
Z
XY
Isotropic: EV = EH Anisotropic: EV = 0.7xEH
HORIZONTAL and Vertical
Displacements for isotropic and
anisotropic models
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Anisotropy
• Isotropic behaviour: great similarity
between the values of the
displacements
• Anisotropic behaviour: higher V
displacement, given the lower
stiffness (enough to verify the
presence of anisotropy)

25. Anisotropic and Isotropic behaviour from face B to D
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Anisotropy
Waves (for EV = 0.7 EH; EH = EH isotropic)
Horizontal Polarization
-1,5E-04
-1,0E-04
-5,0E-05
0,0E+00
5,0E-05
1,0E-04
1,5E-04
0,0E+00 1,0E-03 2,0E-03 3,0E-03 4,0E-03 5,0E-03 6,0E-03
Displacement(m)
Time (s)
Input 5.0 kHz B to D: Anisotropic B to D: Isotropic
Horizontal Displacement

26. Anisotropic and Isotropic behaviour from face B to D and A to C
Vertical
polarization
Horizontal
polarization
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Anisotropy
-3,5E-04
-3,0E-04
-2,5E-04
-2,0E-04
-1,5E-04
-1,0E-04
-5,0E-05
0,0E+00
5,0E-05
1,0E-04
1,5E-04
0,0E+00 1,0E-03 2,0E-03 3,0E-03 4,0E-03 5,0E-03 6,0E-03
Displacement(m)
Time (s)
Input 5.0 kHz A to C: Anisotropic A to C: Isotropic
B to D: Anisotropic B to D: Isotropic tP: EV = 0.70EH
tP: EV = EH tS: EV = 0.70EH tS: EV = EH
Vertical Displacement
Horizontal Displacement
Waves (for EV = 0.7 EH; EH = EH isotropic)

27. Specimen
γ
[kN/m3]
w0
[%]
e0
R2W-TT 19.1 28.4 0.770
R3W-K0TT 18.8 30.7 0.850
R4D-K0TT 14.1 1.2 0.883
R8D-TT 12.7 1.0 1.067
• Comparison already considering the validity of the simulation model
• The tests made in the TT by Ferreira (2009) consisted in dry (w ≈ 0%) and wet (w ≈ 30%)
reconstituted residual soil specimens from Porto granite
Physical properties of the reconstituted specimens tested in the TT
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Comparison with Laboratory Tests

28. 0.0E+00
1.0E-03
2.0E-03
3.0E-03
4.0E-03
5.0E-03
6.0E-03
7.0E-03
Z
XY
X, Y, Z axial displacements (m)
5 10 25 50
75 100 150 200
300 400
• Deformation of the model concomitant
with the applied stress
• Differences between experimental and
numerical results: compliance and
bedding errors (in the laboratory)
• Valid comparison, despite the
difference between the degrees of
magnitude of the displacement
Three-dimensional view of the strains measured for isotropic loading
1.3E-04
2.6E-04
6.7E-04
0.0E+00
3.0E-04
6.0E-04
9.0E-04
Z
XY
100 kPa
200 kPa
500 kPa
Experimental
results
Numerical
results
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Comparison with Laboratory Tests: R8D-TT
Displacements

29. • Shape and configuration of the waves
independent from the stresses applied –
characteristics of the constitutive
model (linear elastic with constant G)
• Experimental: increase of stiffness and
necessarily the seismic wave velocities
with increasing loads (soil hardening
due to particle rearrangement)
Simulations of seismic wave propagation for several stress values
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Comparison with Laboratory Tests: R8D-TT
Waves
-3,0E-04
-2,5E-04
-2,0E-04
-1,5E-04
-1,0E-04
-5,0E-05
0,0E+00
5,0E-05
1,0E-04
1,5E-04
2,0E-04
0,0E+00 1,0E-03 2,0E-03 3,0E-03 4,0E-03 5,0E-03 6,0E-03
Z-displacement(m)
Time (s)
Input 5.0 kHz 100 kPa 200 kPa 500 kPa tP tS

30. • The higher the stress applied, the
higher the displacement
• Isotropic material: V displacements are
higher than H, in agreement with the
loading conditions
• Anisotropic material: diferences
betwen H and V displacements in
agreement with loading conditions:
even though the soil is more rigid in the
V direction, it exhibits a higher strain
in the V direction due to the V stress
Horizontal and vertical displacements, correspondent to horizontal and
vertical stresses: a) 35 kPa and 100 kPa; b) 70 kPa and 200 kPa
Numerical results
2.E-05
0.0E+00
3.0E-05
6.0E-05
9.0E-05
1.2E-04
Z
XY Isotropic EV = EH
Anisotropic EV = 1.4xEH
Anisotropic EV = 0.7xEH
a)
4.E-05
0.0E+00
3.0E-05
6.0E-05
9.0E-05
1.2E-04
Z
XY Isotropic: EV = EH
Anisotropic: EV = 1.4xEH
Anisotropic: EV = 0.7xEH
b)
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Comparison with Laboratory Tests: R4D-K0TT
Displacements

31. • Different displacements, but similar
waves
• There is virtually no change between
both types of constitutive models: use
of another model
Stiffness (assumed constant) commands the
behaviour of the model
• Different from reality
-2.0E-04
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03
Z-displacement(m)
Time (s)
Input 5.0 kHz Isotropic: Z-disp 175/500 kPa
Anisotropic: Z-disp 175/500 kPa tP: EV = EH
tP: EV = 1.4xEH tS
Signal variation, considering a horizontal stress of 175 kPa and 500 kPa
Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
- Comparison with Laboratory Tests: R4D-K0TT
Waves

32. Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
• Even though there were no different properties that considered the presence of BE, a wave
signal could still be successfully created and evaluated and in considerably less amount of time
in the simpler model
Parametric and sensitivity studies proved the validity of the model by correctly characterizing
the soil
• Comparison between experimental and numerical results was successful and the
measurement of the stiffness parameters achieved in the laboratory verified, regardless of the
differences between the constitutive models
• Since this was the first numerical modelling approach to study the TT, a greater effort was made
in the implementation and improvement of the model (Parametric and Sensitivity Studies)
• Altogether, this was a pioneer work in a short amount of time and a learning experience for all
involved!

33. Contents
Scope and
objectives
State of
the Art
Numerical
Modelling
Parametric and
Sensitivity Studies
Results and
Discussion
Conclusions
Further Works
• Regarding the sensitivity studies:
• Study of the influence of the water level;
• Multi-parametric study regarding the relationship between the input frequency and the
Rayleigh damping;
• Include in the study a qualitative processing of the signal.
• Regarding the numerical modelling in FLAC3D:
• Investigation of all parametric and sensitivity studies in the frequency-domain;
• The simulation of a cubical cell to compare with more accuracy the influence of flexible or
absorbent boundaries;
• The use of a different constituive model that would consider the evaluation of stiffness;
• The evaluation of all the comparisons between numerical and experimental results with the
more complex model.