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Permanent Displacement Response Spectra

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Harmonic Motion Study
Four figures of seismic shear displacement.
Click on figure to view enlargement (70K).

Input motions and the natural resonance period of the displaced ground interact affecting the oscillation of the mass and shear stresses at the sliding base. When the shear wave velocity is high and the compliant mass is stiff, the compliant and rigid analyses compute approximately the same displacement. When the natural resonant frequency of the ground is a multiple of the input motion frequency displacements are amplified due to in-phase oscillations of the mass, as in plots (b) and (d). Very soft masses respond almost entirely out of phase with the input motion and dramatically reduce computed displacements relative to the rigid analysis (e).


Graph of permanent displacement response spectra. figure e

Click on figure '(e)' for enlargement (28K).

Slope spectra (Kramer, 1997) is a plot that presents the compliant maximum displacement normalized by the rigid displacement plotted against (a) the ground period, Tground or (b) the tuning ratio, Tground/Tmotion (same as ƒmotionground). Amplified deformation due to compliance is noted by normalized displacements >1.0. When Tground/Tmotion exceeds 1.2 (plot e) the harmonic stress-induced displacement falls to 0. Here, soft ground responds almost entirely out of phase (mü1 Nearly equal to symbol -müb) with the input motion, reducing the shear stress to 0.5 the rigid analysis stress. When the tuning ratio < 0.1 the stiff compliant upper-mass moves in unison with the base and the displacement is the same as the rigid analysis.


Earthquake Motion Study
Permanent displacement response spectra diagrams. figure 8figure 9
Click on figure to view larger image. (40K) (32K)
UNDERSTANDING A GROUND SHEAR DISPLACEMENT PLOT: Trajectories for displacement of rigid ground (blue) and compliant (red) ground are plotted above for a 2° slope subjected to the Newhall record. The maximum displacement occurs upslope to the north (~18cm). Downslope ratcheting of the ground ultimately drives the final displacements in the opposite direction (6 and 15 cm). The trajectories map the position of the ground during earthquake shaking. (Note: The modeled flexible displaced ground is 15m tall with Vs= 100m/s, Tground = 1.26 s.). Below is a similar ground displacement plot.

Seismic shear displacement Vs>>1000 m/s. Seismic shear displacement Vs=480 m/s
Seismic shear displacement Vs=278 m/s Seismic shear displacement Vs=250 m/s
Seismic shear displacement Vs=160 m/s Seismic shear displacement Vs=105 m/s
Seismic shear displacement Vs=700 m/s

Given a motion, Permanent Displacement Response Spectra (PDRS) plots the amplification and deamplification of permanent seismic displacements for slopes of varied resonant periods.


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