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Week6-WavePropagation_S11

# Week6-WavePropagation_S11 - 1 CE 316 Geotechnical...

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Unformatted text preview: 1 CE 316 Geotechnical Earthquake Engineering Week 6 Wave Propagation Infinitely Long Rod • Constrained infinite rod for 1-D wave propagation E A ρ ν dx …. …. ( 29 ( 29 E M ν ν ν 2 1 1 1- +- = Constrained Modulus… Constrained Modulus… 1-D Wave Propagation, Infinite Uniform Elastic Material, P Wave 2 2 m equilibriu Dynamic t u Adx A A dx x o o x x x ∂ ∂ =- ∂ ∂ + ρ σ σ σ x x M ε σ = 2 2 Equation of Motion x u x t σ ρ ∂ ∂ ⇒ = ∂ ∂ x u x ∂ ∂ = ε where… Inertial forces 1-D Wave Propagation, Infinite Uniform Elastic Material, P Wave 2 2 2 2 2 2 2 x u v x u M t u p ∂ ∂ = ∂ ∂ = ∂ ∂ ρ Particle Velocity: m x m m x m x m x x v v v v M t t v M t x t u u ρ σ ρ σ σ σ ε = = = ∂ ∂ = ∂ ∂ = ∂ ∂ = 2 ρ M v p = 1-D Wave Propagation, Infinite Uniform Elastic Material, S Wave 2 2 2 2 2 2 2 x v x G t s ∂ ∂ = ∂ ∂ = ∂ ∂ θ θ ρ θ ρ G v s = 1-D Harmonic Stress Wave, Infinite Uniform Elastic Material 2 2 2 2 2 x u v t u ∂ ∂ = ∂ ∂ ) cos( ) cos( ) , ( kx t B kx t A t x u- +- = ϖ ϖ wave number k v ϖ = t cos ) ( stress harmonic state- steady ϖ σ σ O t = 2 2 wavelength v vT v f k π π λ ϖ = = = = v = wave propagation velocity: time domain space domain General Three-Dimensional State of Stress on material element yx xy σ σ = zx xz σ σ = zy yz σ σ = General Three-Dimensional State of Strain on material element dx du xx = ε Strain-displacement components, ε dy dv yy = ε dz dw zz = ε dy du dx dv xy + = ε dz dv dy dw yz + = ε dx dw dz du zx + = ε Rotation-displacement components, Ω...
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Week6-WavePropagation_S11 - 1 CE 316 Geotechnical...

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