Response spectra for differential motion of columns paper II: Out-of-plane response

It is shown that the common response spectrum method for synchronous ground motion can be extended to make it applicable for earthquake response analyses of extended structures experiencing differential out-of-plane ground motion. A relative displacement spectrum for design of first-story columns SDC (T, T_T, ζ, ζ_T, τ, δ) is defined. In addition to the natural period of the out-of-plane response, T, and the corresponding fraction of critical damping, ζ, this spectrum also depends on the fundamental period of torsional vibrations, T_T, and the corresponding fraction of critical damping, ζ_T, on the “travel time,” τ (of the waves in the soil over a distance of about one-half the length of the structure), and on a dimensionless factor δ, describing the relative response of the first floor. The new spectrum, SDC, can be estimated by using the empirical scaling equations for relative displacement spectra, SD, and for peak ground velocity, v_max. For recorded strong-motion acceleration, and for symmetric buildings, the new spectrum can be computed from Duhamel's integrals of two uncoupled equations for dynamics equilibrium describing translation and rotation of a two-degree-of-freedom system. This representation is accurate when the energy of the strong-motion is carried by waves in the ground the wavelengths of which are one order of magnitude or more longer than the characteristic length of the structure.