E0-190-2008_(10)1D_Shear_Propagation(SHAKE)

16 16 into eq18 18 gives 20 and also putting

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Unformatted text preview: Eq.(18) : (18) gives: (20) And also, putting Eq.(17) : (17) into Eq.(19) : (19) gives: (21) The coefficient in the right hand side of Eq.(21) can be expressed by: (22) αj denotes the ratio of complex wave impedance between jth and (j+1)-th stratum. Using αj , Eq.(21) is transformed into Eq.(23): (23) Eq.(23) plus Eq.(20): (20) gives: (24) Eq.(20) minus Eq.(23) gives: (25) Eq.(24) and (25): (24) (25) are rearranged into the matrix-vector formulation as: (25) (26) where, (26) (27) For simplicity, putting: (28) then, Eq.(26) : (26) (25) can be expressed by: {C j +1 } = [ A j ]{C j } (29) Eq.(29): {C j +1 } = [ A j ]{C j } (29) represents the relation of the amplitude vector of the waves propagating upward and downward between in (j+1)-th stratum: and these in j-th stratum: { C j } = {E j F j } T (30) (j-1)-th stratum Ej-1 ( j) (31) j-th stratum Ej Fj-1 Fj ( j+1) Ej+1 (j+1)-th stratum Fj+1 The displacement amplitude at the (j+1)-th interface between j-th stratum and (j+1)-th stratum is given by: Uj+1=Uj+1(zj+1=0)=Ej+1+Fj+1 The shear strain amplitude Γj in the j-th stratum is evaluated at the middle depth (zj=dj/2) of the stratum. That is, Γj = [ (32) Uj =Uj(z...
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This note was uploaded on 03/14/2014 for the course CE 5680 taught by Professor Drgrd during the Summer '14 term at Indian Institute of Technology, Chennai.

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