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P-wave S-wave Particles oscillate back and forth Wave travels down rod, not particles Particle motion parallel to direction of wave propagation Particles oscillate back and forth Wave travels down rod, not particles Particle motion perpendicular to direction of wave propagation

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Boundary Effects

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At centerline, displacement is always zero Stress doubles momentarily as waves pass each other

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Boundary Effects (Fixed End)

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Response at boundary is exactly the same as for case of two waves of same polarity traveling toward each other At fixed end, displacement is zero and stress is momentarily doubled. Polarity of reflected wave is same as that of incident wave

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Boundary Effects (Fixed End) Response at boundary is exactly the same as for case of two waves of same polarity traveling toward each other At fixed end, displacement is zero and stress is momentarily doubled. Polarity of reflected stress wave is same as that of incident wave. Polarity of reflected displacement is reversed. Displacement

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Boundary Effects = 0

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Boundary Effects = 0

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Boundary Effects = 0

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Boundary Effects = 0

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Boundary Effects = 0 At centerline, stress is always zero Particle velocity doubles momentarily as waves pass each other

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Boundary Effects (Free End) = 0

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Boundary Effects (Free End) = 0

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Boundary Effects (Free End) = 0

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Boundary Effects (Free End) = 0

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Boundary Effects (Free End) = 0 Response at boundary is exactly the same as for case of two waves of opposite polarity traveling toward each other At free end, stress is zero and displacement is momentarily doubled. Polarity of reflected stress wave is opposite that of incident wave. Polarity of reflected displacement wave is unchanged. Displacement

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Boundary Effects (Material Boundaries) incident reflected transmitted

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incident reflected transmitted Boundary Effects (Material Boundaries) At material boundary, displacements must be continuous A i + A r = A t equilibrium must be satisfied i + r = t

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incident reflected transmitted z = Impedance ratio Boundary Effects (Material Boundaries) Using equilibrium and compatibility,

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incident reflected transmitted z = 0.5 Boundary Effects (Material Boundaries) Stiff Soft 2 = 1 v 2 = v 1 /2

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incident reflected transmitted Boundary Effects (Material Boundaries) Stiff Soft A r = A i / 3 A t = 4A i / 3 Displacement amplitude is reduced Displacement amplitude is increased

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incident reflected transmitted Boundary Effects (Material Boundaries) Stiff Soft r = - i / 3 t = 2 i / 3 Stress amplitude is reduced, reversed Displacement amplitude is reduced

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Boundary Effects (Material Boundaries) Stiff Soft Consider limiting condition: v 2 0 z = 0

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Boundary Effects (Material Boundaries) Stiff Soft Consider limiting condition: v 2 0 z = 0 A r = A i A t = 2A i Displacement amplitude is unchanged Displacement amplitude at end of rod is doubled - free surface effect

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Boundary Effects (Material Boundaries) Stiff Soft Consider limiting condition: v 2 0 z = 0 r = - i t = 0 Polarity of stress is reversed, amplitude unchanged Stress is zero - free surface effect

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Wave Propagation Example Dr Layer

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Three Dimensional Elastic Solids x y z xx yy zz xy yx zy xz zy zx Displacements on left Stresses on right Displacements on left Stresses on right

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Three Dimensional Elastic Solids or Using 3-dimensional stress-strain and strain-displacement relationships

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Three Dimensional Elastic Solids or Two types of waves can exist in an infinite body p-waves s-waves

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Waves in a Layered Body Incident P transmitted P reflected P Waves perpendicular to boundaries p-waves

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Incident SH transmitted SH reflected SH Waves perpendicular to boundaries SH-waves Waves in a Layered Body

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Incident P Refracted SV Refracted P reflected SV reflected P Inclined Waves Incident p-wave

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Incident SV Refracted SV Refracted P reflected SV reflected P Inclined Waves Incident SV-wave Waves in a Layered Body

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Incident SH Refracted SH Reflected SH Inclined Waves Incident SH-wave When wave passes from stiff to softer material, it is refracted to a path closer to being perpendicular to the layer boundary When wave passes from stiff to softer material, it is refracted to a path closer to being perpendicular to the layer boundary Waves in a Layered Body

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Vs=2,500 fps Vs=2,000 fps Vs=1,500 fps Vs=1,000 fps Vs=500 fps Waves in a Layered Body Waves are nearly vertical by the time they reach the ground surface

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Waves in a Semi-infinite Body The earth is obviously not an infinite body. For near-surface earthquake engineering problems the earth is idealized as a semi-infinite body with a planar free surface H1H1 H2H2 H3H3 incident reflected Surface wave Free surface

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Rayleigh-waves Love-waves Surface Waves

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Rayleigh-waves Comparison of Rayleigh wave and body wave velocities Rayleigh waves travel slightly more slowly than s-waves

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Horizontal and vertical motion of Rayleigh waves Rayleigh-waves Rayleigh wave amplitude decreases quickly with depth

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Attenuation of Stress Waves The amplitudes of stress waves in real materials decrease, or attenuate, with distance Material damping Radiation damping Two primary sources:

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Material damping A portion of the elastic energy of stress waves is lost due to heat generation Specific energy decreases as the waves travel through the material Consequently, the amplitude of the stress waves decreases with distance Attenuation of Stress Waves

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Radiation damping The specific energy can also decrease due to geometric spreading Consequently, the amplitude of the stress waves decreases with distance even though the total energy remains constant Attenuation of Stress Waves

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Both types of damping are important, though one may dominate the other in specific situations

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Transfer Function A Transfer function may be viewed as a filter that acts upon some input signal to produce an output signal. The transfer function determines how each frequency in the bedrock (input) motion is amplified, or deamplified by the soil deposit. Transfer Function (filter) input output

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Transfer Function Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) Ae i( t+kz) Be i( t-kz) At free surface (z = 0), u(z, t) = 2Acos kz e i t (0, t) = 0 (0, t) = 0 A = B Factor of 2 amplification

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Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) Amplification factor Transfer function relates input to output Transfer Function

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Zero damping Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) For undamped systems, infinite amplification can occur Extremely high amplification occurs over narrow frequency bands Amplification is sensitive to frequency Fundamental frequency Characteristic site period T s = 4H4H VsVs Transfer Function

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Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) Very high, but not infinite, amplification can occur Degree of amplification decreases with increasing frequency Amplification is still sensitive to frequency 1% damping Transfer Function

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Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) 2% damping Transfer Function

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Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) 5% damping Transfer Function

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Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) 10% damping Transfer Function

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Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) 20% damping Maximum level of amplification is low Amplification sensitive to fundamental frequency Transfer Function

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Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) All damping Amplification De-amplification Transfer Function

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Linear elastic layer on rigid base u z H u(0,t) u(H,t)u(H,t) 10% damping Stiffer, thinner Transfer Function

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Transfer Function example

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How is it used? Input motion convolved with transfer function – multiplication in freq domain Steps: 1.Express input motion as sum of series of sine waves (Fourier series) 2.Multiply each term in series by corresponding term of transfer function 3.Sum resulting terms to obtain output motion. Notes: 1.Some terms (frequencies) amplified, some de-amplified 2.Amplification/de-amp. behavior depends on position of transfer function Transfer Function

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