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Unformatted text preview: Soil Dynamics Lecture 05 Elastic Stress Waves in Bars Luis A. PrietoPortar, August 2006. 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. In sandstones, the stress will propagate at 14,000 feet/sec. This lecture deals with 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 pile driving hammer striking the head of a precast concrete pile (shown on the next three slides). The propagation of stress is central to the understanding of how dynamic loads propagate in soils, whether the loads come from gravity, wind, explosions, etc., or from earthquakes. 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. 1 2 1 2 1 2 x y z xy yx yz zy zx xz x y z xy x yz y zx z Normal stresses, Shear stresses u Normal strains x v 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 = = = = = = = + = = + = = + = where is the components of rotation about the x, y and the z axes. Constitutive Relations (Hookes 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 2 2 1 x x y z y y z x z z x y xy xy yz yz zx zx where E is Young' s elastic mod ulus and is Poisson' s ratio. Shear stresses and shear strains are related via the shear mod ulus G, E G where G ( ) G G = + = + = + = = + = = ( )( ) ( ) 2 2 2 1 1 2 2 x x y y z z x y z Normal stresses can also be exp ressed in t erms of strains, G G G where, E and called the volumetric strain Notice that, G = + = + = + = = + + + = + Longitudinal Elastic Stress Waves in a Bar.Longitudinal Elastic Stress Waves in a Bar....
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