Elastic and Inelastic Response Spectra

Elastic and inelastic acceleration, velocity and displacement response spectra can be obtained in this module. 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 vs. amplitude graphs, commonly known as response spectra. In addition, and for the case of elastic spectra only, the pseudo-velocity and pseudo-acceleration response values, obtained through multiplication of the displacement response values by ω and ω2, respectively, are also given ("ω" stands for angular frequency). 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 defined below, 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 inelastic spectra for several ductility values has been requested. Upon completion, the elastic and inelastic spectra are automatically shown in the plotting window, where a number of different plotting combinations are made available to the user, including the possibility for plotting acceleration vs. displacement response spectrum. Moreover, the corresponding spectral values are provided in the 'Displacement', 'Velocity', 'Acceleration', 'Pseudo-Velocity' and 'Pseudo-Acceleration' tables, ready to be selected and copied to other Windows applications, if required. Look in here for further guidance on formatting and exporting results.

Elastic spectrum
The elastic spectrum is always computed in this module. Indeed, the minimum number of viscous damping values is 1, effectively meaning that even when constant-ductility inelastic spectra is computed, its elastic viscous damping counterpart is always calculated. The user, however, 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]. By defining relatively large values of equivalent viscous damping, overdamped elastic spectra can also be readily obtained.

Inelastic spectrum
Constant-ductility inelastic 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 modelled, with only a relatively small viscous damping quantity (usually not more than 5%) is added to the system, to somehow represent non-hysteretic energy dissipation mechanisms. Up to six levels of displacement ductility can be defined ranging from 1.5 to 10.0 (note that custom values can be typed-in), whilst the post-yield kinematic hardening ratio can be made to vary between 0.0 and 1.0 (the default is 0.0, i.e. no hardening, hence elasto-perfectly plastic system).

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

Spectral period range and step
The minimum and maximum periods of interest must also 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.

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 SeismoSignal, 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 analysed 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.

Constant-ductility spectra
For the computation of constant-ductility inelastic spectra, the post-yield hardening ratio needs to be defined (the default value is 2%). Further, when carrying out nonlinear dynamic analysis (needed in this case), it is required for a convergence criterion to be set, in order for the nonlinear analysis to progress from one step to the other. For this reason, a 'Ductility Tolerance' and an 'Out-of-balance Force Tolerance' are defined, with default values sufficiently small to warrant very good accuracy (users who wish to speed their analysis may obviously define larger tolerances).
Note also that, on some rare occasions, convergence difficulties may arise when computing constant-ductility inelastic response spectra, leading to a situation whereby the calculations continue on going endlessly. In such sporadic events, use the Cancel button to stop the calculations, adjust the input parameters (e.g. time-step, tolerance value, etc.), and restart the analysis. Please refer to available literature [e.g. Clough and Penzien, 1994; Cook et al., 1988; Crisfield, 1991] for further discussion and clarification on issues of step-by-step solution procedures, convergence criteria, and so on.

Notes

  1. On occasions, when comparing the acceleration response spectra for several levels of viscous damping, users may notice that higher damping values seem to inexplicably lead to higher response accelerations. This apparent paradox, which depends on the frequency content of the record, arises from the representation of structural damping as equivalent viscous damping and the fact that for higher damping levels, the ground vibrations on a genuinely viscously damped system are transmitted by the damper rather than by the stiffener. This behaviour can also be inferred from any standard transfer function for harmonic loading [e.g. Clough and Penzien, 1994].
  2. Note that the significantly diverse application framework of overdamped and constant-ductility inelastic spectra effectively means that when using the former to estimate the response of a structural system, users should enter the spectra with the effective period of vibration of the system, whilst when using the latter, users should consider the initial elastic period of vibration instead.
  3. It is re-emphasised that in SeismoSignal constant-ductility inelastic spectra is computed through nonlinear dynamic analyses of elasto-plastic hysteretic systems, rather than through the many simplified proposals available in the literature [e.g. Newmark and Hall, 1982; Krawinkler and Nassar, 1992; Miranda and Bertero, 1994].