1. Introduction to Soil Dynamics
Amit Prashant
Indian Institute of Technology Gandhinagar

2. Our Dear Pendulum – Revisited
Force Equilibrium:
Cord length,
Velocity,
Acceleration,
For small

3. Single Degree of Freedom Systems
Stiffness, k
Mass, m
Mass, m
Stiffness, k
Stiffness, k
Mass, m
Damping c
Stiffness, k
Mass, m
Damping, c

4. Single Degree of Freedom Systems
Structures which have
Most of their mass lumped at a single location
Only a single displacement as unknown
Elevated
Water
Tank
Bridges
Equivalent SDOF System

5. Dynamic Equilibrium Three independent properties Disturbance Mass, m
Building
Three independent properties
Mass, m
Stiffness, k
Damping, c
Disturbance
External force f(t)
Response
Displacement,
Velocity,
Acceleration,
u(t)
f(t)
Column
Roof

6. Internal forces
Inertia force
Damping force
Stiffness force 1 1 1

7. Force Equilibrium Dynamic equilibrium fI(t)+ fD(t) + fS(t) = f(t) f(t)
Inertia force
Stiffness force
Damping force
External force

8. Free Vibrations Initial disturbance No external force Divide by mass
Neutral position
Initial disturbance
Pull and release : Initial displacement
Impact : Initial velocity
No external force
Divide by mass
Extreme position
Natural frequency
Damping Ratio
Natural Period,

9. Free Vibration Response
Undamped system u0 d0 un
Displacement u(t)
Time t T v0 d0 u0 un Dt T
Displacement u(t)
Exponential decay
Time t
Damped system

10. Free Vibration Response of Damped Systemsu0
u(t) t
Overdamped d0 u0
u(t) t
Underdamped
In Civil Engineering Structures

11. Analogy of Swing Door with Dashpot Closing Mechanism
If the door oscillates through the closed position it is underdamped
If it creeps slowly to the closed position it is overdamped.
If it closes in the minimum possible time, with no overswing, it is critically damped.
Critical Damping: the smallest amount of damping for which no oscillation occurs
If it keeps on oscillating and does not stop, it is ??

12. Undamped System: Free vibrations
Equation of motion
Solution:
Initial velocity
Initial displacement
v0 =
Displacement u(t)
Time t u0

13. Undamped System: Free vibrations
m = 5000 kg
Example
m = 5000 kg
k = 8000 kN/m
k = 8000 kN/m

14. Damped System: Free vibrations
Equation of motion
Solution:
Initial velocity
Initial displacement
v0 =
Displacement u(t)
Time t u0

15. Damped System: Free vibrations
Example
m = 5000 kg
c = 20 kN/(m/s)
k = 8000 kN/m
m = 5000 kg
k = 8000 kN/m
c = 20 kN/(m/s)

16. Example: Damping From the given data, a0 = 5.5 m, a9 = 0.1 m
t9 - t0 = 4.5 s
Time t (s)
Damped natural period
TD = (tN - t0)/N
= 4.5/9 = 0.5 s
Damping ratio

17. Forced Vibrations Apply a sinusoidal loading with frequency,
The equilibrium equations becomes
Displacement u(t)
Time t

18. Forced Vibration Response
Sinusoidal Force
Constant Amplitude
Static w1 w2 w3 w4 w5 w6
ustatic u1 u2 u3 u4 u5 u6
Displacement
Frequency  n

19. Forced Vibration Response
Resonance at natural frequency of structure
Critically dependant on damping
Undamped
Magnification
Factor =
Normalised Displacement
umax/ustatic
Under-damped 1
Critically Damped
Frequency  n

20. Normalised Displacement
Evaluation of Damping
Half-Power Method
Normalised Displacement
umax/ustatic 1
A n B
Frequency 

21. From Earthquake Dynamics of Structures, Chopra (2005)
Seismic Ground Motion
From Earthquake Dynamics of Structures, Chopra (2005)

22. Seismic Ground Motion Response
Change of reference frame
Rigid body motion causes no stiffness & damping forces
Moving-base Structure
Fixed-base Structure
Mass m
Absolute
acceleration
Relative
Velocity/displacement

23. Seismic Ground Motion Response
Time t
Time t

24. From Earthquake Dynamics of Structures, Chopra (2005)
Deformation Response
From Earthquake Dynamics of Structures, Chopra (2005)

25. Double Pendulum

26. Multi Degree of Freedom (MDOF) Systems
u1(t)
MDOF?
Mass located at multiple locations
More than one displacement as unknowns
Equilibrium equation in matrix form
Solution is found by
Simultaneously solving the equation
Modal Analysis
u2(t)
Building

27. Seismic Waves Fault Line Earthquake Rock
Near the ground surface, most of the seismic waves arrive vertically
Rock

28. Amit Prashant, IIT Gandhinagar
Vibrations
Amit Prashant, IIT Gandhinagar

29. Foundation Movement

30. Machine Foundation Design Criteria
Static criteria
Safe against shear failure
Settlement within permissible limits
Dynamic criteria
Alignment – minimum eccentricity with foundation
No resonance – Displacement amplitudes within limit
Amit Prashant, IIT Gandhinagar

31. Reduce Machine Vibrations
Minimize Eccentricity of moving parts
Counter Balancing
Amit Prashant, IIT Gandhinagar

32. Amit Prashant, IIT Gandhinagar
Placement of Machines
In basements
not at higher floors
Minimum structural response
Away from vibration sensitive areas
On improved ground conditions – more rigid
Vibration Isolation
Amit Prashant, IIT Gandhinagar

33. Idealization of Machine on Foundation
Fo.sin(wt)
Fo.sin(wt)
Fo.sin(wt)
Mass, m
Mass, m
Mass, m
Stiffness and Damping of Foundation block and Soil
Damping c
Stiffness, k
Forced Vibrations?
Amit Prashant, IIT Gandhinagar

34. Important Properties Propagation of Shear Waves
Density = Mass per unit volume
Shear Modulus
Damping Characteristics

35. Shear Modulus Secant Useful in Equivalent Linear Analysis1
Secant
Shear Modulus 1
Initial Shear Modulus 1
Tangent
Shear Modulus
Used in Nonlinear Analysis

36. Secant Modulus

37. Cyclic Loading – Secant Shear Modulus
Equivalent
Linear
Analysis
Branch curve
(Hysteresis loop)
Skeleton curve
Nonlinear Analysis
(step by step)

38. Hysteretic Damping
DW = Loss of Energy per cycle
W = Strain Energy

39. Modulus Reduction Curve
Plasticity index
Threshold Strain
After Vucetic, 1994
Threshold Strain
(Below this strain the behaviour is linear)

40. Typical Values of Initial Shear Modulus
(Source: FHWA-SA )

41. Initial Shear Modulus Increasing Factor Go Effective Stress Increases
Void Ratio
Decreases
Geologic age
Cementation
Overconsolidation
Plasticity Index
Negligible to small increase
Strain Rate
No effect on sand
Increases for clay
Number of loading cycles
Increases for sand
Decreases for clay

42. Correlations of initial shear Modulus
(Source: FHWA-SA )

43. Modulus Reduction Curve Effect of Confining Pressure
Non-plastic soil
(After Iwasaki et al., 1978)

44. Modulus Reduction Curve Effect of Confining Pressure
Non-plastic soil
Plastic soil
(After Ishibashi, 1992)

45. Modulus Ratio, G/Go Increasing Factor G/Go Cyclic Strain Decreases
Effective Stress
Increases
Void Ratio
Geologic age
May Increase
Cementation
Overconsolidation
No effect
Plasticity Index
Strain Rate
Number of loading cycles
Increases for drained sand
Decreases for undrained sand
Decreases for clay

46. Modulus Reduction Curve Effect of Soil Type28
　　　　σ’m0 (kPa)　　　
　Clay　　　100　　　　　
　Sand　　　50 ～300　　　
　Gravel　　50～830　　　
Clay
Sand
Gravel
(Imazu & Fukutake, 1986)

47. Shear Modulus and Damping with Cyclic Strainτ γ
1.0
0.5
10－6
10－1

48. Modulus Reduction Curve with Hysteresis and Damping along Depth
Liquefaction  Reduction on effective overburden? τ γ
Increasing Overburden
Deeper Strata τ γ τ γ τ γ τ γ

49. Shear Modulus and Damping Effect of Plasticity Index
For sand
(After Vucetic and
Dobry, 1991)

50. Damping Ratio, x Increasing Factor x Cyclic Strain Increases
Effective Stress
Decreases
Void Ratio
Geologic age
Cementation
May decrease
Overconsolidation
No effect
Plasticity Index
Strain Rate
May Increase
Number of loading cycles
No significant change

51. Typical values of Poisson’s Ratio

52. Volume Change or Evolution of Pore Water Pressure During Shearing
Drained
Shearing
Slow Loading
Undrained
Shearing
Fast Loading
Initially loose configuration
Increase in Pore Water Pressure
Contractive
Reduced effective stress
Settlement
Initially Dense configuration
Dilative
Decrease in Pore Water Pressure

53. Sand Behavior during Cyclic Loading
Drained/Slow Loading: τ γ γ
Compression
Undrained/Fast Loading:
Pore water pressure, p γ τ
Liquefaction
N (cycle)

54. Stress Strain Curve for soils
Peak Shear Strength
Zone of instability
Due to Initial
Overburden
Steady State Shear Strength OR
Residual Shear Strength
Cyclic loading
Strength reduced to 80% of Undrained shear strength is often considered to avoid large deformations and Instabilities ?

55. Residual Shear Strength
Difficult to measure in the laboratory
Triaxial test not reliable at very high strains
Ring Shear Test
Sampling issues
Time consuming
Field Test
Correlation with SPT, N60 corrected for fines content.

56. Residual Strength and SPT
(After Seed and Harder, 1990)

57. Thank You