Response Spectrum: SDOF response calculator

Ground motion

Target PGA [g] Duration [s] Time step [s] Dominant freq ω_g [rad/s] Filter damping ζ_g Random seed

Kanai-Tajimi filter of Gaussian white noise multiplied by a Saragoni-Hart envelope and then scaled to the target PGA. Choose preset to mimic typical events:

Drop or choose a CSV / TXT One column = a_g (m/s²) at uniform dt · or two columns = t, a_g

Sample step [s] (single column) Units (input)

Input accelerogram

Ground acceleration a_g(t). PGA, Arias intensity, and significant duration (D_5-95) are reported in the record summary.

Spectrum settings

Damping ratio ζ [%] Period T_min [s] Period T_max [s] Periods (log-spaced)

Response spectra

S_D: peak relative displacement; PSV = ω_n S_D; PSA = ω_n² S_D. Each curve is computed independently for every period using the Nigam-Jennings closed-form recurrence (exact for piecewise-linear ground motion).

SDOF response — selected period

Period T [s] 1.00 s

SDOF relative displacement x(t) under the input ground motion at the selected natural period and global damping.

Method

The SDOF equation of motion is m·x″(t) + c·x′(t) + k·x(t) = -m·a_g(t), normalized as x″ + 2ζω_nx′ + ω_n²x = -a_g(t).

Between two consecutive samples of the accelerogram the ground acceleration is treated as linear in time. With this piecewise-linear assumption, the SDOF response advances by a 2×2 transition matrix plus a forcing contribution that depends only on the two end-point accelerations. The matrix entries are evaluated once per period from ζ, ω_n, Δt (Nigam & Jennings, 1969). The result is exact rather than an approximation and is the standard method used in modern response-spectrum codes.

The synthetic ground motion is filtered white noise through a Kanai-Tajimi filter H(ω) = (ω_g² + 2iζ_gω_gω)/((ω_g²-ω²) + 2iζ_gω_gω), modulated by the Saragoni-Hart envelope e(t) = (t/t₀)^α exp(-β(t-t₀)), and rescaled to the target peak ground acceleration.

References

Nigam, N.C.; Jennings, P.C. (1969). Calculation of Response Spectra from Strong-Motion Earthquake Records. Bulletin of the Seismological Society of America, 59(2), 909-922.
Chopra, A.K. (2017). Dynamics of Structures: Theory and Applications to Earthquake Engineering, 5th ed., Pearson. Sections 5.2 and 6.
Kanai, K. (1957). Semi-Empirical Formula for the Seismic Characteristics of the Ground. Bulletin of the Earthquake Research Institute, 35, 309-325. Tajimi, H. (1960). A Statistical Method of Determining the Maximum Response of a Building Structure During an Earthquake. Proc. 2nd WCEE.