Response spectrum
Encyclopedia
A response spectrum is simply a plot of the peak or steady-state response (displacement, velocity or acceleration) of a series of oscillators of varying natural frequency, that are forced into motion by the same base vibration
or shock
. The resulting plot can then be used to pick off the response of any linear
system, given its natural frequency of oscillation. One such use is in assessing the peak response of buildings to earthquake
s. The science of strong ground motion
may use some values from the ground response spectrum (calculated from recordings of surface ground motion from seismographs) 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 of oscillation
(multi-degree of freedom systems), although they are only accurate for low levels of damping. Modal analysis
is performed to identify the modes, and the response in that mode can be picked from the response spectrum. This peak response is 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 for linear
systems. Response spectra can be generated for non-linear systems, 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.
for analyzing the performance of structures and equipment in earthquakes, since many behave principally as simple oscillators (also known as single degree of freedom systems). Thus, if you can find out the natural frequency of 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 Earthquake http://neic.usgs.gov/neis/eq_depot/world/1985_09_19.html where 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. Housner
began to publish calculations of response spectra from accelerograph
s. In the 1982 EERI
Monograph on "Earthquake Design and Spectra", 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 as non-linear static or dynamic analysis like in seismic performance analysis technique.
Vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...
or shock
Shock (mechanics)
A mechanical or physical shock is a sudden acceleration or deceleration caused, for example, by impact, drop, kick, earthquake, or explosion. Shock is a transient physical excitation....
. The resulting plot can then be used to pick off the response of any linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
system, given its natural frequency of oscillation. One such use is in assessing the peak response of buildings to earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...
s. The science of strong ground motion
Strong ground motion
"Peak ground velocity" redirects here.Seismologists usually define strong ground motion as the strong earthquake shaking that occurs close to a causative fault...
may use some values from the ground response spectrum (calculated from recordings of surface ground motion from seismographs) 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 of oscillation
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
(multi-degree of freedom systems), although they are only accurate for low levels of damping. Modal analysis
Modal analysis
Modal analysis is the study of the dynamic properties of structures under vibrational excitation.Modal analysis is the field of measuring and analysing the dynamic response of structures and or fluids when excited by an input...
is performed to identify the modes, and the response in that mode can be picked from the response spectrum. This peak response is 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 for linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
systems. Response spectra can be generated for non-linear systems, 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 of earthquake engineeringEarthquake engineering
Earthquake engineering is the scientific field concerned with protecting society, the natural and the man-made environment from earthquakes by limiting the seismic risk to socio-economically acceptable levels...
for analyzing the performance of structures and equipment in earthquakes, since many behave principally as simple oscillators (also known as single degree of freedom systems). Thus, if you can find out the natural frequency of 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
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
), which may be identified from the response spectrum. This was observed in the 1985 Mexico City Earthquake http://neic.usgs.gov/neis/eq_depot/world/1985_09_19.html where 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. Housner
George W. Housner
George W. Housner was an eminent authority on earthquake engineering and National Medal of Science laureate. Housner received his Bachelors degree in Civil Engineering from the University of Michigan where he was influenced by Stephen Timoshenko...
began to publish calculations of response spectra from accelerograph
Accelerograph
An accelerograph can be referred to as a strong motion seismograph, or simply as an earthquake accelerometer. They are usually constructed as a self-contained box, more commonly now being connected directly to the Internet....
s. In the 1982 EERI
Earthquake Engineering Research Institute
The Earthquake Engineering Research Institute is a leading technical society in dissemination of earthquake risk and earthquake engineering research both in the U.S. and globally. EERI members include researchers, geologists, geotechnical engineers, educators, government officials, and building...
Monograph on "Earthquake Design and Spectra", 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 as non-linear static or dynamic analysis like in seismic performance analysis technique.
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