Soil Damping Ratio: Theoretical Aspect and Measurement

Transcription

1 Ratio: Theoretical Aspect and Measurement Sri Atmaja P. Rosyidi, Ph.D. Assistant Professor, Universitas Muhammadiyah Yogyakarta A Two Day Workshop on SASW for Practicing Engineer February 2011, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia

2 2 Outline of presentations Introduction: Definition of Ratio Dissipation Phenomenon and Mechanism of Seismic Waves in Soil Media Ratio: Material and Geometric

3 3 Chapter 1: Definition of damping ratio General definition in engineering term: damping ratio (D) is a dimensionless measure relating to decay mechanism of oscillations in a system after a disturbance. In physics, damping can be presented as any effect that tends to reduce the amplitude of oscillations in an oscillatory system.

4 4 Underdamped, critical damped, overdamped system from SDOF (Free Vibration) SDOF :

5 5 Significance of damping ratio in soil system (after Lai & Rix, 1998)

6 6 Hazards caused by earthquake (after Prof. Francesco Silvestri)

7 7 Chapter 2: Dissipation Phenomenon Dissipation Phenomenon: Seismic wave energy includes two parts, Kinetic and Potential or Strain energy. o Kinetic energy is the energy which be posed due to the motion of the material particles. o Strain energy is the product of force and displacement, in a macroscopic sense, is in a dimension of energy. The work done by stress is converted to elastic strain energy due amount and stored in the medium. When seismic waves are propagating through the soil column - a portion of their energy dissipates resulting in a reduction in the amplitude of the waves.

8 8 Energy dissipation phenomenon After Carlo G. Lai & Glenn J Rix (1998) After Jaime Horta-Rangel, Socorro Carmona, Victor M. Castaño (2008)

9 9 Energy dissipation example (after Prof. Jeremić, 2009)

10 10 Mechanism of energy dissipation Mechanism of energy dissipation in granular soil: o Cohesionless soils - frictional sliding at grain to grain contact surface (Whitman & Dobry, 1993) o Saturated soil - complex mechanism - frictional sliding and viscous drag of the pore fluid moving relative to the soil skeleton. o Soil plasticity mechanism. Mechanism of energy dissipation in fine grained soil: complex phenomenon controlled by electromagnetic interaction between water molecules and microscopic solid particles.

11 11 Relationship between energy imparted and dissipated by frictional and viscous Conceptualization of the cumulative energy imparted to the soil by an earthquake and the portions dissipated by frictional and viscous mechanisms. (after Hall & Mccabe, 1989)

12 12 Frictional sliding mechanism

13 13 Relative slippage of the spheres

14 14 Viscous dissipation mechanism Viscosity is the measure of a fluid's resistance to flow. Viscous drag is the force resisting the relative movement of a fluid and a solid. It is analogous to the frictional force between two solid. Theoretical evaluation of energy dissipation by viscous mechanism in soils: Biot s theory (1956) n is the porosity of soil, k is soil permeability coefficient, Ů is the velocity of soil particles and ů is the fluid particles velocity.

15 15 Effect of viscous drag Experimental study by Hall & Richart (1963) - influence of viscosity of the pore fluid on the total energy dissipated in granular materials

16 16 Soil plasticity in energy dissipation Rate of dissipation of energy (per unit volume) is computed from a plastic strain rate pattern because of soil plasticity (Drucker & Prager, 1951). The dissipation of the plastic energy is proportional to distortional strain increment and the mean effective stress acting on it, and can be calculated as: stress tensor plastic strain

17 17 Energy dissipation modeling: Hysteresis Loop Two spheres contact modeling - normal and tangential force resulting slippage. Load-unload tangential force, a plot of the resulting force-displacement relation scribes a hysteretic loop. Other approaches: T - hysteretic loop represented by bi- Linear (Idriss & Seed 1986), hyperbolic (Lee & Finn 1978), Ramberg-Osgood (Streeter et al. 1973)

18 18 Hysteresis Loop Stored energy and total dissipated energy From laboratory studies: shape of hysteresis loop is independent of the load rate (dry soil). quantity of energy dissipated by frictional mechanism is independent of the loading frequency (Hardin 1965). energy dissipated by viscous mechanism is proportional to the frequency of applied loading.

19 19 An example of hysteresis loop and its dissipated energy plot (Lenart, 2008)

20 20 Energy dissipation modeling: Equivalent Linearization Hysteretic loop represented by bi-linear (Idriss & Seed 1986), hyperbolic (Lee & Finn 1978), Ramberg-Osgood (Streeter et al. 1973) "A second order non linear partial differential equation" - energy dissipation phenomena by wave propagation. For simplification referred to as equivalent linearization is applied. Further reading: Dobry et al. 1971, Schnabel et al

21 21 Rheological model of non-linear hysteretic and linearized hysteretic system

22 22 coefficient in linearized hysteretic model

23 23 calculation from Hysteresis Loop Cyclic loading - damping Atriangle W 1 2 max max Aloop W D 4 W W where: D = damping ratio, W = dissipated energy per unit volume in one hysteretic loop, W = energy stored in an elastic material having the same G as the viscoelastic material.

24 24 Other energy dissipation or attenuation mechanism Geometric or radiation damping: spreading of energy over an expanding area as the wave front propagates away from the source. It causes the amplitude of waves to attenuate with increasing distance from the source (Abrahamson & Silva, 1995)

25 25 Soil attenuation In ground vibrations, internal attenuation is added to geometric attenuation. Theoretically, attenuation for P and S wave P and S wave: propagated by the free surface of a semi-infinite elastic body is proportional to 1/r 2 propagated by an infinite elastic body is 1/r. R wave, proportional to 1/ Ground (soil) attenuation constant. Depends on the type of ground, frequency and propagation velocity. It is larger the higher the frequency is and the slower the propagation velocity is (soft ground). Just for reference, the attenuation constant of clay is 0.02 ~ 0.01, while that for silt is 0.3 ~ r

26 26 Other energy dissipation or attenuation mechanism Apparent attenuation: Reflection and transmission of seismic waves at interfaces, mode conversions and scattering in non-homogeneous media. (Robertson Research International Ltd., 1998)

27 27 Dissipation factor Quality factor, Q is a description of material attenuation (geophysicists and seismologists). Inversed Q, Q -1, called as dissipation factor (specific attenuation factor), is defined as: For small value of material damping, the quality factor and damping ratio can be defined as the resonance frequency divided by the bandwidth (Stokoe et al., 1999):

28 28 Resonant column test for damping ratio (Stokoe et al., 1999)

29 29 Logarithmic decrement Attenuation can be obtained by the logarithmic decrement, of a harmonic wave. Attenuation is exponential function and its magnitude is describable by the exponential or logarithmic rates of decay.

30 30 Equation for logarithmic decrement (Stokoe et al., 1999)

31 31 Coefficient of attenuation The coefficient of attenuation is related to the logarithmic decrement, defined as:

32 32 Summary: ratio relationship D D 1 2Q V V 2 2 V for small D D for small D

33 33 ratio curve

34 34 Ishibashi and Zhang (1993)

35 35 Effect of soil plasticity in damping ratio curve (from Dobry and Vucetic, 1991)

36 Wavelength (m) Depth (m) Soil 36 Soil damping ratio profile 0 Attenuation Coefficient (1/m) Shear Ratio (%) 0.0% 0.5% 1.0% 1.5% 2.0% d theoretical Gm Experimental Theoretical

37 Thank you for your attention Material can be downloaded at: