Theoretical Basis for Rotational Effects in Strong Motion and Some Results

 Vladimir Graizervgraizer@consrv.ca.gov

California Geological Survey

Menlo Park, February 16, 2006

Introduction

Most common instruments used in seismological measurements of ground motion are pendulum seismographs. Pendulums are sensitive to translational motion and rotations.

When seismology started measuring ground motion in the near-field of earthquakes and explosions the assumption that movement of the instrument’s base is purely translational was simply transferred from the far- to the near-source studies.

During the last half of century a number of attempts were made to measure or estimate rotational component of strong ground motion (Kharin & Simonov, 1969; Trifunac & Hudson, 1971; Lee & Trifunac, 1985; Niazi,1986; Lee & Trifunac, 1987; Graizer, 1987, 1989, 1991; Oliveira and Bolt, 1989; Nigbor, 1994; Takeo, 1998; Huang, 2003). But we still don’t have consistent measurements of rotations associated with strong-motion.

Schematic representation of an accelerograph

Equation of pendulum motion

Longitudinal: y1” + 21D1y1’ + 1

2y1 = -x1” + gψ2 - ψ3”l1 + x2”1

Vertical: y3” + 23D3y3’ + 3

2y3 = -x3” + gψ12/2 - ψ1”l3 +x2”3

Graizer, 1989

Golytsyn (1912) is not considering cross-axis sensitivity (item 4).Aki & Richards (1980) are not considering angular acceleration (item 3).

List of symbols

Where:

yi is recorded response of the instrument,

i is the angle of pendulum rotation,

li is the length of pendulum arm,

yi = i li ,i and Di are respectively the natural frequency

and fraction of critical damping of the ith transducer,g is acceleration due to gravity,xi” is ground acceleration in Ith direction,

ψi is a rotation of the ground surface about xi

axis.

Errors due to angular acceleration (a), tilt

(b) and cross axis sensitivity (c)

“Effective” equations of pendulums in strong-motion

Horizontal:y1” + 21D1y1’ + 1

2y1 = -x1” + gψ2

Vertical: y3” + 23D3y3’ + 3

2y3 = -x3”

What can be done in absence of rotations

y” + 2Dy’ + 2y = - Vx”

T1 T

W = [x’(t)]2dt + [x’(t)]2dt 0 T2

Graizer, 1979

Comparison of shake-table motion with displacement calculations

Comparison of the “true” displacement and displacement calculated using accelerogram contaminated by tilt

Response of Two-Pendulums System

Measurements of Displacement and Tilt in the near-field of 2 explosions using two-pendulums

instruments

Graizer et al., 1989

Method of estimating tilt using existing strong-motion records

Method of tilt evaluation using uncorrected strong-motion accelerograms based on the difference in the tilt sensitivity of the horizontal and vertical pendulums is suggested (Graizer, 1989).

Method was tested in a number of laboratory experiments with different strong-motion instruments (Moscow, 1988; Menlo Park, 1993).

“Effective” equations of pendulums in strong-motion

Horizontal:y1” + 21D1y1’ + 1

2y1 = -x1” + gψ2

Vertical: y3” + 23D3y3’ + 3

2y3 = -x3”

Response of Horizontal Pendulum to Acceleration and Tilt

Comparison of H to V (no tilt present)

Contamination with artificial tilt

Estimates of Tilts during Northridge Earthquake

Pacoima Dam

Fourier spectra and Spectral Ratios of H/V at Pacoima

Residual tilt extracted from the strong-motion record at the Pacoima Dam – Upper Left Abutment reached 3.10 (0.054 rad) in N450E direction. It was a result of local earthquake induced tilting due to high amplitude shaking (not source generated).

This value is in agreement with the residual tilt of 3.50 in N400E direction measured using electronic level few days after the earthquake by CSMIP staff (Shakal et al, 1994).

Tilting velocity is estimated to reach about 15 deg/sec (0.26 rad/sec).

Tilts at Pacoima Dam

Los Angeles – 6-story Office Bldg (24652)

Method was applied to the records of Northridge earthquake of 1994 at Los Angeles – 6-story Office Bldg.

According to the estimates, this building experienced residual tilting from 0.1 up to 3.5 degrees (0.002 to 0.06 rad) at the 1st and 3rd floors with no significant residual tilting at the basement and roof levels.

Tilts in a Building

Ventura – 12-story Hotel (25339)

Highlights Analysis of the equation of motion of horizontal and vertical pendulums has shown that horizontal sensors are sensitive not only to translational motion but also to tilts. In contrast to horizontal sensors, vertical sensors don’t have these limitations, since they are much less sensitive to tilts.

Ignoring tilt sensitivity may produce unreliable results, especially in residual displacement and long-period calculations.

Tilts at Pacoima Dam and in some CSMIP instrumented buildings reached more than 3 degrees (0.06 rad) during the Mw 6.7 Northridge earthquake. In general, only six-component systems measuring rotations and accelerations, or three-component systems similar to systems used in inertial navigation assuring purely translational motion of accelerometers can be used to calculate “true” displacements.

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References

Method Testing (Menlo Park, 1993)

Lee & Graizer, 1993