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Unformatted text preview: CEG CEG5065C Soil Dynamics 5065C Soil Dynamics Lecture #16 Lecture #16 Elastic Stress Waves Luis A. PrietoPortar 2009 The propagation of stress in an elastic medium. When stress is applied to a body, that stress will propagate away from the point of application via stress waves . Different materials will propagate the stress at different speeds. For example, in sands the stress will propagate at about 1,000 feet/sec . On the other hand, in sandstones , the stress will propagate at 14,000 feet/sec . This lecture discusses the propagation of stress waves in elastic media in the form of rods, bars or beams. An example of this type of stress is that induced by a diesel pile driving hammer striking the head of a precast concrete pile. The propagation of stress is central to the understanding of how dynamic loads propagate in soils, whether the loads come from gravity, wind, explosions, machines or from earthquakes. A typical example of stress waves in a bar is the stress propagation along a concrete pile due to the impact of the diesel hammer shown below. The hammer is shown in its five stages of operation: (1) tripping, (2) fuel injection, (3) compression and impact, (4) “explosion” or rather, combustion, and (5) rebound . The diesel hammer strikes the pile with very high stresses, tolerated by steel but not concrete. Therefore, a reduction of the stresses is effected through the “bonnet” or pile head cushion. Stress and strain in elastic media. The notation for the normal and shear stresses in an idealized very small element of a much larger elastic body is shown below. x y z xy yx yz zy zx xz x Normal stresses, Shear stresses u Normal strains x v σ σ σ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ ε = = = ∂ = ∂ ∂ 1 2 1 2 1 2 y z xy x yz y zx z y w z v u w v Shear strains and x y y z w v u w y z z x u w v u z x x y ε ε γ ϖ γ ϖ γ ϖ γ ϖ γ ϖ γ ϖ γ ϖ γ ϖ γ ϖ γ ϖ γ ϖ γ ϖ = ∂ ∂ = ∂ & ¡ ∂ ∂ ∂ ∂ = + = ¢ £ ∂ ∂ ∂ ∂ ¤ ¥ ∂ ∂ ∂ ∂ & ¡ = + = ¢ £ ∂ ∂ ∂ ∂ ¤ ¥ & ∂ ∂ ∂ ∂ = + = ¢ ∂ ∂ ∂ ∂ ¤ ¡ £ ¥ where & is the components of rotation about the x, y and the z axes. Constitutive Relations (Hooke’s law). In elastic and isotropic media, the stresses and the strains are related to each other through relationships called constitutive relationships . These are: ( ) ( ) 1 2 1 2 1 & ¡ = + ¢ £ & ¡ = + ¢ £ ¡ + x x y z y y z x ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ ( ) 2 where is Young's elastic modulus and is Poisson's ratio. Shear stresses and shear strains are related via the shear modulus where 2 1 & ¡ = + ¢ £ = = + = = z z x y xy xy yz yz zx zx E G, E G G ( ) G G ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ ν τ γ τ γ τ γ τ γ ν τ γ τ γ τ γ τ γ τ γ τ γ τ γ τ γ ( )( ) Normal stresses can also be expressed in terms of strains,...
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 Fall '09
 STAFF
 Longitudinal wave, Wave mechanics, Normal mode

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