Introduction to Soil Dynamics Amit Prashant Indian Institute of Technology Gandhinagar

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

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

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

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

Internal forces Inertia force Damping force Stiffness force 1 1 1

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

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,

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

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

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 ??

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

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

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

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)

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

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

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

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

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

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

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

Seismic Ground Motion Response Time t Time t

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

Double Pendulum

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

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

Amit Prashant, IIT Gandhinagar Vibrations Amit Prashant, IIT Gandhinagar

Foundation Movement

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

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

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

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

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

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

Secant Modulus

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

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

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

Typical Values of Initial Shear Modulus (Source: FHWA-SA-97-076)

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

Correlations of initial shear Modulus (Source: FHWA-SA-97-076)

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

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

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

Modulus Reduction Curve Effect of Soil Type 28 　　　　σ’m0 (kPa)　　　 　Clay　　　100　　　　　 　Sand　　　50 ～300　　　 　Gravel　　50～830　　　 Clay Sand Gravel (Imazu & Fukutake, 1986)

Shear Modulus and Damping with Cyclic Strain τ γ 1.0 0.5 10－6 10－1

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

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

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

Typical values of Poisson’s Ratio

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

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

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 ?

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.

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

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