GENERAL INFORMATION• 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 earthquakes.
GENERAL INFORMATION• 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.• 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.
GENERAL INFORMATION. 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.
Uses of Elastic Response Spectra• Elastic design response spectra are extremely useful to structural engineers. These spectra are the basis for:• Computing design displacements and forces in systems expected to remain elastic• Developing design forces and displacement systems that respond in elastically by: - Modifying elastic spectrum - Evaluating response of equivalent elastic structure• These elastic spectra can be obtained by several methods, which are:• Processing of site specific ground motion time histories• Statistical relationships• Empirical relationships• Code stipulations
Uses of Elastic Response Spectra• Elastic spectra can be presented in several formats, depending on the needs of the engineer and what information is being presented. Some of the most common formats are:• Spectral acceleration vs. period• Spectral velocity vs. period• Spectral displacement vs. period• Spectral acceleration vs. spectral displacement (capacity design spectrum)• Tripartite plots (Sa, Sv, and Sd vs. period)• Also, any of the above (except the capacity design spectrum) can be plotted versus frequency rather than period.• Factors which effect elastic spectra include the damping ratio, site conditions, and near fault ground motion effects such as rupture directivity.
SITE-SPECIFIC ELASTIC DESIGN SPECTRAResponse spectra for actual ground motions are quite irregular, asshown below. Do not use them for design — they can be used foranalysis to assess the response to a particular earthquake.Where site specific ground motions have been compiled, the response spectra for each record can be averaged. The resulting "mean“ spectrum will be smooth. The COE can be used to establish aspectrum with a desired probability of exceedance.
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• TYPES OF DESIGN RESPONSE SPECTRAa. Probability level. Design response spectra• are usually based statistically either on the mean, median (50th percentile probability level), or the median plus one standard deviation (84th percentile probability level), of the ground motion parameters or the records chosen.• Design response spectra used for design of new RCC dams or for evaluation of the safety and serviceability of existing dams shall be based on the mean level of the ground motion parameters
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• b. Type of spectrum required. Either a “site-specific”• or a “standard” design response spectra shall be used to describe the design earthquakes.• The type required shall be based on the seismic zone, the proximity of the seismic source, and the maximum height of the dam.• c. Site-specific design response spectra.• The site-specific design response spectra should be developed based on earthquake source conditions, propagation path properties, and local foundation characteristics associated with the specific site.• This type of design spectra may be established by anchoring a selected response spectral shape for the site to the estimated peak ground acceleration, or by estimating
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• HORIZONTAL AND VERTICAL DESIGN• a. Site-specific design response spectra.• When site-specific design response spectra are required in accordance with paragraph 5-5c, two independent design response spectra shall be developed, one to define the horizontal component of ground motion, and the second to define the vertical component.• The vertical component of ground motion usually contains much higher frequency content than the horizontal component, therefore the spectral shape is quite different than that of the horizontal component.
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• B. STANDARD DESIGN RESPONSE SPECTRA.• When it is acceptable to use standard design response spectra to define the design earthquakes, the horizontal component of ground motion shall be defined by anchoring the standard design response spectra for the appropriate damping factor with the scaling factor• The vertical component of ground motion shall utilize the same standard design response spectrum used for the horizontal component, but it shall be scaled using the appropriate ratio of the PGA for the vertical component to the PGA for the horizontal component
SITE-SPECIFIC ELASTIC DESIGN SPECTRAAMPLIFICATION CALCULATIONS• To compute the amplification functions, three different computer programs were used:• - ProShake  and CyberQuake  for 1D and• - Aki-Larner SH according to Bard and Gabriel  for 2D calculations. As input motion on the bedrock, time histories following as closely as possible the shape of the elastic• response spectra of the national application document of Euro code 8  (version ENV-1998-1-1) for• rock and zone 1 were taken. Zone 1 of the Swiss design code represents the largest part of the highly• populated area in Switzerland. The selected time histories had to fulfill the following criteria:
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• - Occurrence in similar tectonic conditions as Switzerland and• - Covering the target response spectra (split into the period ranges of 0.02-0.2s and 0.2-4s).• Table 1 contains the selected earthquakes. Figures 3 and 4 show the response spectra of the selected time• histories in comparison to the target spectra.
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• The multi-step code approach for calculating the Seismic Response Coefficient (Cs in NEHRP), is essentially a way of constructing a smoothed average response spectrum that accounts for the damping and ductility characteristics of the building, as well as the regional seismicity and underlying soil of the site.• Compare an elastic response spectrum for a Northridge 1994 earthquake motion, with a code design response spectrum developed with the NEHRP provisions.• The code spectrum is an approximation of an elastic response spectrum, scaled down by two factors:• It is reduced by the factor of safety used in allowable stress design to account for the fact to achieve the given yield strength, allowable stress design must aim at a lower strength. (for this case, Fs = 1.5)• It is reduced by the R factor to account for damping and ductility. This reduction creates an inelastic spectrum which accounts for the effect of ductility in limiting force levels. (for this case, R=6.5)
SITE-SPECIFIC ELASTIC DESIGN SPECTRAREDISTRIBUTION FOR HEIGHT• The response spectrum concept is based on the notion that the structure is a single degree of freedom system, but real structures are not.• In particular, the levels of acceleration are not constant throughout the structure.
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• Effects of local soil conditions on response spectra
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• TYPES OF SOIL
SITE-SPECIFIC ELASTIC DESIGN SPECTRA• At periods above about 0.5 s, spectral amplifications for soil sites are much higher for soil sites than for rock sites. Deep and soft soil deposits produce greater proportions of long period motion. Use of single response spectrum shape for all site conditions is not appropriate
STATISTICALLY DERIVED RESPONSE SPECTRA• Elastic design response spectra can be predicted in the same statistical manner as ground motion parameters such as peak ground acceleration or velocity. Numerous researchers have developed attenuation relationships for elastic spectra, which are listed in the references.• The general procedure for generating statistically derived spectra is as follows:• Classes of ground motions are selected (based on soil, magnitude, distance, etc.)• Response spectra for a large number of corresponding ground motions are generated and averaged• Curves are fit to match computed mean spectra• Resulting equations are used to develop a design response spectrum with desired probability of exceedance
STATISTICALLY DERIVED RESPONSE SPECTRA• Attenuation relationships are developed by statistical analyses performed on a large number of records which were obtained in compatible geomorphic regions. Most of these relationships are updated as new strong ground motion data becomes available and many now include additional parameters such as fault type and site soil conditions.• The relationships are grouped by region, with note to those which are applicable only to a particular area, such as the Cascadian seduction zone. It is important to realize that the relationships are only as good as the data they were generated from, and therefore the relations for Western North America are the most reliable because the database for that area is quite rich. The relationships for Eastern and Central North America should be used with caution because the relationships have been calibrated with only a few events. In any event, more than one relation should be used to predict the motions at a site.
STATISTICALLY DERIVED RESPONSE SPECTRA• where PHA = peak horizontal ground acceleration, PVA = peak vertical ground acceleration, PHV = peak horizontal ground velocity, PVV = peak vertical ground velocity, Sah = horizontal spectral acceleration, and Sav = vertical spectral acceleration Distance definitions: rrup = closest distance to the rupture surface, rjb = closest horizontal distance to the vertical projection of the rupture, rhypo = hypo central distance, rseis = closest distance to the seismogenic rupture zone
EMPIRICALLY DERIVED RESPONSE SPECTRAN. M. Newark and W. J. Halls procedure for developing elastic design spectra starts with the peak values of ground acceleration, velocity, and displacement. The values of peak ground acceleration and velocity should be obtained from a deterministic or probabilistic seismic hazard analysis.The value of peak ground displacement is a bit more difficult to obtain due to the lack of reliable attenuation relationships. A typical baseline curve plotted on tripartite axes is shown below.
EMPIRICALLY DERIVED RESPONSE SPECTRA• Use response amplification factors (listed in table on previous page to determine spectral values in the following period ranges:• Short period (Tn < 0.03 sec) Sa = ag• Transition• Constant amplified acceleration range (Tn > 0.13 sec) Sa = aaag• Intermediate period range Sv = avvg• Long period range Sd = addg• Very long period range Sd = dg (transition unclear)
EMPIRICALLY DERIVED RESPONSE SPECTRAConnect the lower bound response lines. If desired, plot the spectrum in a different format, such as the one shown here.STRUCTURAL RESPONSE AMPLIFICATION FACTORS• Structural response amplification factors are then applied to the different period-dependent regions of the baseline curve. These factors differ for acceleration, velocity, and displacement, especially at low values of damping. The factors decrease rapidly with increasing damping, especially at small damping values. These factors are shown in the table below.Newmark and Halls structural response amplification factors can also be used to change the damping value of other spectra, such as those generated using attenuation relationships. This modification technique is presented in the viscous damping section of the notes.
EMPIRICALLY DERIVED RESPONSE SPECTRATRIPARTITE PLOTS• Newmark and Halls spectra are plotted on a four-way log plot called a tripartite plot. This is made possible by the simple relation between spectral acceleration, velocity, and displacement:• Sa/w = Sv = Sdw• A tripartite plot begins as a log-log plot of spectral velocity versus period as shown.
EMPIRICALLY DERIVED RESPONSE SPECTRAThen spectral acceleration and spectral displacement axes aresuperimposed on the plot at 45 degree angles. All three types ofspectrum (Sa vs. T, Sv vs. T, and Sd vs. T) can be plotted as a singlegraph, and three spectral values for a particular period can easily bedetermined. The Sa, Sv, and Sd values for a period of 1 second areshown below.
EMPIRICALLY DERIVED RESPONSE SPECTRAEFFECT OF VARIOUS FACTORS ON SPECTRAL VALUES For soft soils, ag remains the same or decreases relative to firm soil, but vg and dg increase, generally. Layers of soft clay, such as the Young Bay Mud found in the San Francisco Bay area, can also act as a filter, and will amplify motion at the period close to the natural period of the soil deposit. Layers of deep, stiff clay can also have a large effect on site response. For more information on site effects, see Geotechnical Earthquake Engineering by Kramer.
EMPIRICALLY DERIVED RESPONSE SPECTRAREFERENCES:-Wikipedia (Response spectrum, general information)-Design of earthquake resistant building Using site-specific responsespectra”,(Site-specific elastic design spectra)http://peer.Berkeley.Edu/course_modules/eqrd/index.Htm?C227top.Htm&227cont.Htm&eqdef/eqdef4.Htm(eqrd,interactive), Site-specific elastic designspectra.http://peer.Berkeley.Edu/course_modules/eqrd/index.Htm?C227top.Htm&227cont.Htm&eqdef/eqdef4.Htm(eqrd,interactive), Empirically derivedresponse spectra. Bakir P. G., Roeck G. D., Degrade G. and Wong K. K. F., “Site dependentresponse spectra and analysis of the characteristics of the strong groundmotion,(Empirically derived response spectra)