Response spectrum

A series of mixed vertical oscillators

A plot of the peak acceleration for the mixed vertical oscillators

Aresponse spectrumis a plot of the peak or steady-state response (displacement, velocity or acceleration) of a series ofoscillatorsof varyingnatural frequency,that are forced into motion by the same basevibrationorshock.The resulting plot can then be used to pick off the response of anylinearsystem, given its natural frequency of oscillation. One such use is in assessing the peak response of buildings toearthquakes.The science ofstrong ground motionmay use some values from the ground response spectrum (calculated from recordings of surface ground motion fromseismographs) for correlation with seismic damage.

If the input used in calculating a response spectrum is steady-state periodic, then the steady-state result is recorded. Damping must be present, or else the response will be infinite. For transient input (such as seismic ground motion), the peak response is reported. Some level of damping is generally assumed, but a value will be obtained even with no damping.

Response spectra can also be used in assessing the response of linear systems with multiple modes ofoscillation(multi-degree of freedom systems), although they are only accurate for low levels of damping.Modal analysisis performed to identify the modes, and the response in that mode can be picked from the response spectrum. These peak responses are then combined to estimate a total response. A typical combination method is the square root of the sum of the squares (SRSS) if the modal frequencies are not close. The result is typically different from that which would be calculated directly from an input, since phase information is lost in the process of generating the response spectrum.

The main limitation of response spectra is that they are only universally applicable forlinearsystems. Response spectra can be generated fornon-linearsystems, but are only applicable to systems with the same non-linearity, although attempts have been made to develop non-linear seismic design spectra with wider structural application. The results of this cannot be directly combined for multi-mode response.

Seismic response spectra

Response spectra are very useful tools ofearthquake engineeringfor analyzing the performance ofstructuresand equipment in earthquakes, since many behave principally as simpleoscillators(also known assingle degree of freedomsystems). Thus, if you can find out thenatural frequencyof the structure, then the peak response of the building can be estimated by reading the value from the ground response spectrum for the appropriate frequency. In most building codes in seismic regions, this value forms the basis for calculating the forces that a structure must be designed to resist (seismic analysis).

As mentioned earlier, the ground response spectrum is the response plot done at the free surface of the earth. Significant seismic damage may occur if the building response is 'in tune' with components of the ground motion (resonance), which may be identified from the response spectrum. This was observed in the 1985 Mexico City Earthquakewhere the oscillation of the deep-soil lake bed was similar to the natural frequency of mid-rise concrete buildings, causing significant damage. Shorter (stiffer) and taller (more flexible) buildings suffered less damage.

In 1941 at Caltech,George W. Housnerbegan to publish calculations of response spectra fromaccelerographs.^[1]In the 1982EERIMonograph on "Earthquake Design and Spectra",^[2]Newmark and Hall describe how they developed an "idealized" seismic response spectrum based on a range of response spectra generated for available earthquake records. This was then further developed into a design response spectrum for use in structural design, and this basic form (with some modifications) is now the basis for structural design in seismic regions throughout the world (typically plotted against structural "period", the inverse of frequency). A nominal level of damping is assumed (5% of critical damping).

For "regular" low-rise buildings, the structural response to earthquakes is characterized by the fundamental mode (a "waving" back-and-forth), and most building codes permit design forces to be calculated from the design spectrum on the basis of that frequency, but for more complex structures, combination of the results for many modes (calculated through modal analysis) is often required. In extreme cases, where structures are either too irregular, too tall or of significance to a community in disaster response, the response spectrum approach is no longer appropriate, and more complex analysis is required, such asnon-linearstaticordynamic analysislike inseismic performance analysistechnique.

See also

• Peak ground acceleration
• Spectral acceleration

References

Other sources

• Report on 1985 Mexico City Earthquake] from "EQ Facts & Lists: Large Historical Earthquakes", USGS. (https://web.archive.org/web/20070206063939/http://neic.usgs.gov/neis/eq_depot/world/1985_09_19.html)
• ^"Historic Developments in the Evolution of Earthquake Engineering",illustrated essays by Robert Reitherman, CUREE, 1997, p10.
• ^Newmark, N. M., and Hall, W. J. 1982. “Earthquake Spectra and Design,” Engineering Monographs on Earthquake Criteria, Structural Design, and Strong Motion Records, Vol 3,Earthquake Engineering Research Institute,Oakland, CA.

External links

• "Illustration of Newmark-Hall Approach to Developing Design Response Spectra"– Appendix B of "Engineering and Design – Response Spectra and Seismic Analysis for Concrete Hydraulic Structures (EM 1110-2-6050)", US Army Corps of Engineers