Elastic and Inelastic Response Spectra

Elastic and inelastic acceleration response spectra can be obtained in SeismoSpect. The spectra is computed by means of time-integration of the equation of motion of a series of single-degree-of-freedom systems (see below), from which the peak displacement, velocity and acceleration response quantities are then obtained and plotted in period, frequency or displacement vs. amplitude graphs, commonly known as response spectra. Users are referred to the literature [e.g. Clough and Penzien, 1994; Chopra, 1995] for further details on these procedures.

After defining the input parameters, users need only to click on the Refresh button to initiate the required calculations, noting that these might take some seconds if the computation of elastic or inelastic spectra for several accelerograms has been requested. Upon completion, the results are automatically shown on the Chart area on the right-hand-side, ready to be selected and copied to other Windows applications, if required. There is also a check box on the upper left corner that allows users to switch between having the results on a chart or in a table. Look in here for further guidance on formatting and exporting results.

Elastic spectrum
The user has the possibility of changing the level of viscous damping associated to such elastic spectrum, defined here as a percentage of the critical damping value, and usually featuring values ranging from 0 to 5% [e.g Chopra, 1995]. Damping values higher than this limit can also be employed to produce Overdamped spectra.

Inelastic spectrum
This type of spectra attempts to reproduce actual nonlinear structural response by means of an elasto-plastic representation of the system. In this way, energy dissipated through hysteresis comes explicitly modeled, with only a relatively small viscous damping quantity (usually not more than 5%) being added to the system, to somehow represent non-hysteretic energy dissipation mechanisms.

(Note: The following parameters can be defined/customised in the Program Settings menu)

Numerical integration parameters
Determination of elastic and inelastic response spectra requires the computation of peak response values of SDOF oscillators with varying periods of vibration that are subjected to the acceleration time-history under consideration. Therefore, linear and nonlinear dynamic analysis needs to be carried out, and a numerical direct integration scheme is employed in order to solve the system of equations of motion [e.g. Clough and Penzien, 1994; Chopra, 1995]. In SeismoSpect, such integration is carried out by means of the Newmark integration scheme [Newmark, 1959].
The Newmark integration scheme requires the definition of two parameters; beta (β) and gamma (γ). Unconditional stability, independent of time-step used, can be obtained for values of β ≥ 0.25 (γ + 0.5)². In addition, if γ = 0.5 is adopted, the integration scheme reduces to the well-known non-dissipative trapezoidal rule, whereby no amplitude numerical damping is introduced, a scenario that is clearly advantageous within the scope of the current application. The default values are therefore β = 0.25 and γ = 0.5.
It is noted, however, that the trapezoidal rule does require the use of relatively small time-steps in order to deliver sufficiently accurate solutions, the reason for which a maximum ratio between the integration time-step and the period of the oscillator being analyzed is imposed (max dt/T = 0.02, by default). As a starting point, the program uses the time-step of the loaded accelerogram as the time-step of the dynamic analysis, and then checks if this does not result to be larger than 2% (or any other threshold value adopted by the user) of the period of the system being analysed. If it is indeed greater, then the algorithm automatically changes the integration time-step so that the maximum dt/T ratio is respected, subdividing, through linear interpolation, the input accelerogram, as required for integration of the equation of motion. It is noted that the maximum dt/T default value (dt/T = 0.02) commonly leads to sufficiently accurate solutions, for which reason users are usually not required to change this value. In any case, if so interested, users need only to carry out a sensitive study in order to determine the largest dt/T value that provides fully precise solutions.

Spectral period range and step
The minimum and maximum periods of interest can be defined by the user, so as to characterise the spectral period range. Usually, for standard structural applications, these correspond to 0.02 and 4.0 seconds, respectively. In addition, the period step value, employed in the computation of the different values that make up the spectra, is also to be defined. The default value if 0.02 seconds. A ductility value and a hardening ratio needs to be defined in other to compute this kind of spectra.

Inelastic spectrum method
By default, SeismoSignal's inelastic response spectrum method, based on the dynamic analysis of nonlinear SDOF systems, is employed here to compute inelastic spectra. It consists of an iterative process to achieve the target ductility factor considering the hysteretic behaviour of the system; the displacement for each system is calculated and the computed ductility is compared with the target ductility, with the strength being then adjusted and the process continued until reaching a maximum number of iterations or until the computed ductility only differs 1% from the target ductility. Readers may refer to the literature for further info on this type of spectra computing approach [e.g. Carr, 2002].

Since in SeismoSpect the need to compute hundreds of spectral responses may arise, simplified and faster methods to compute the inelastic spectra, as opposed to the aforementioned default SeismoSignal method, have also been implemented:

Newmark and Hall [1982] - The force reduction factor given by this classic method is parameterized as a function of the yield level and three different formulas are provided depending of the natural period of the structure. The inelastic spectra computed using this method is similar to the elastic spectra for low periods and for higher periods.
Miranda and Bertero [1994] - The equation that gives the reduction factor in this method was obtained from a study of 124 ground motion records on a wide range of soil conditions and therefore, users need to select which type of soil should be considered (rock site, alluvium site or soft site). The yield level was taken into account to compute the parameters of each formula whilst a 5% critical damping was assumed.
Krawinkler and Nassar [1992] - This method uses a reduction factor that was obtained from a study of a set of ground motion records with magnitude varying between 5.7 and 7.7, recorded on alluvium and rock sites. The yield level and post-yield hardening coefficient were taken into account and a 5% damping value was assumed to create the formulas that compute the reduction factor.

These simplified methods provide estimates of acceleration response spectra ordinates (Sa), with (very approximate) displacement response spectra values being then computed by SeismoSpect through the product of the latter by the square of the angular frequency (ω²) and a ductility correction factor (μ / (μ + α∗ μ - α)), whereby "μ" is the target ductility and "α" the post-yield hardening ratio. Velocity response spectra ordinates, on the other hand, are estimated through the product of their acceleration counterparts by the angular frequency (ω) - it is noted, however, that such velocity spectral estimates are very unreliable, and users are strongly encouraged to use the SeismoSignal method instead, when velocity spectra (and, albeit to a more limited extent, also displacement spectra) are desired, even at the cost of longer analyses.

Note: In SeismoSpect, inelastic response spectrum computations using the SeismoSignal method appear faster than when the SeismoSignal program is employed; this is because in the latter, real-time display of iterations/convergence information is provided, amongst other reasons.