Substituting the real and the imaginary part of the

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Unformatted text preview: e is given by Eq.(3-21): * * λ = iωo exp(iφ) = ωo ( − sin φ + i cos φ) (3-21) where ω*o(=√k*/m) . Substituting the real and the imaginary part of the eigenvalue of Eq.(3-21) into Eq.(3-16), we obtain: 1 −1 K ′ ) = h (3-22) η = sin φ = sin( tan 2 K Eq.(3-22) equals damping factor of a single degree of freedom system with the complex spring constant. When the damping factor is small, the damping factor can be approximately expressed by: In a practical seismic design analysis, the rigidity is often given by: K′ h= 2K K + iK′ = K(1 + i2h) The figure shows the damping factors hHH of D.I.F. Concerning the damping factors hHH of the horizontal D.I.F, slight differences in the damping factors are only observed between the raft and the pile foundations. hHH are small and nearly constant in lower frequencies up to the cut-off frequency 1.25Hz, the values of hHH are around the damping factor h2=0.05 of the surface stratum. hHH becomes larger in higher frequencies over the cut-off frequency. The damping factors hRR of the rotational D.I.F for the raft foundation are larger than the pile foundation. Comparing hRR to hHH , hHH is larger than hRR in all frequencies for the both foundation types. 3.5 Foundation Input Motion A foundation input motion (F.I.M) is an effective input motion of earthquake disturbance, which is applied to D.I.F. The figure shows the amplitudes of F.I.M for the horizontal ∆ and the rotational Θ components. F.I.M are depicted by the ratio to the ground surface response amplitudes Us. The values of ∆/Us are less than...
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