E0-190-2008_(8)Chapter_6(Substructure_Method)

8 condition can be expressed by 23 in eq23 23 the

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Unformatted text preview: or Fig.8 condition: can be expressed by: (23) In Eq.(23): (23) The first equation becomes: (24) As shown in Fig.11: Because [KV] is the dynamic stiffness matrix of the soil part which will be excavated and {XSFq} is the displacement vector of the soil without the void, [KV] times {XSFq} coincides with the stress vector {σ} on the F. That is: (25) Substituting Eq.(25) into Eq.(24): (24) Then, (26) Joining Eq.(26): (26) with the second equation: (23-1) in Eq.(23): (23) leads to: (27) From Eq.(22): (22) the displacement vector {XSFq XSSq}T becomes: (28) where, {XSFq XSSq}T : earthquake response of the soil before the excavation {XFq XSq}T : earthquake response of the soil after the excavation {XsFq XsSq}T : response of the soil before the excavation due to the stress vector {σ} on the F Substituting Eq.(28): (28) into Eq.(27): (27) We obtain Eq.(29): (29) Note: The right hand side of Eq.(27) is decomposed. Eq.(29): (29) can be decomposed into: (30) (31) Of course, Eq.(30) is the same to Eq.(13...
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