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Unformatted text preview: Soil Dynamics Lecture 09 The Behavior of Surface Soil Layers under Earthquake Loading The vibration of the surface layer due to an earthquake ensues from the upward propagation of the shear waves from below. The behavior of the surface stratum was studied and reported by Idriss and Seed in 1968. The discussion of this topic in this lecture, follows their analysis. Consider the following surface layer of soil, (rock) Swave input displacement of the soil layer due to the seismic input Case 1. A single top soil layer sitting upon a rock base. Let us consider the behavior of a column of soil of unit area, sitting upon a rock base. The equation of motion of the soil at any depth y and at any time t is, ( ) ( ) ( ) ( ) 2 2 2 2 where is the relative displacement at a depth and a time , is the shear modulus at any depth , is the damping coefficient at any depth , and is the density of the soil at any dep g u u u u y c y G y y t t y y t u y t G( y ) y c( y ) y ( y ) ρ ρ ρ ρ ρ ρ ρ ρ ρ ∂ ∂ ∂ ∂ ∂ + − = − ∂ ∂ ∂ ∂ ∂ th . The shear modulus increases with depth as, where A and B are soil properties. B y G( y ) Ay = ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 Replacing the shear modulus in the differential equation of motion, where B<0.5, but not zero. The solution is usually of the form, where g B n n n n n u u u u y c y Ay y t t y y t u( y ,t ) Y y X t Y ρ ρ ρ ρ ρ ρ ρ ρ =∞ = ∂ ∂ ∂ ∂ ∂ + − = − ∂ ∂ ∂ ∂ ∂ = ∑ ( ) ( ) 1 2 1 1 1 2 2 where is the Bessel function of the first order represents the roots of 0 where n =1,2,3... and b b n b n n n n n n n n g b n b n n n / y y y b J and H H X D X X R u J b J ( ) A / H θ θ θ θ θ θ θ θ β β β β β β β β ω ω ω ω ω ω ω ω β β β β β β β β β ρ β ρ β ρ β ρ ω θ − − − = Γ − + + = − − = = ɺɺ ɺ ɺɺ θ ( ) ( ) 1 1 1 The damping ratio in the th mode is, 2 and is the gamma function, such that 1 1 2 The values of b and are related as follows: B  +2 n n b n n b n n c / D R b J b ρ ω ρ ω ρ ω ρ ω β β β β β β β β θ θ θ θ θ θ θ θ θ − + − = Γ = Γ − = B  2 +2 A detailed analysis is given in Idriss and Seed (1967). θ θ θ θ θ θ θ θ = Procedure to obtain the relative displacement u at any depth y and at any time t : 1) Determine the shape function Y n (y) during the n th mode of vibration; 2) Determine X n (t) by direct numerical stepbystep procedure (see Wilson and Clough, 1962);...
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This note was uploaded on 08/29/2011 for the course CEG 5905 taught by Professor Staff during the Summer '10 term at FIU.
 Summer '10
 STAFF

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