Probability distribution of torsional response induced by lateral displacement and inertial force

In previous studies of torsional response, researchers considered torsional vibration of a building to be caused by an eccentric distribution of stiffness, damping, and the structure's mass, as well as spatially non-uniform ground motion input into a long or large base mat of a structure. However, we have discovered that the torque generated by horizontal displacement and the perpendicular inertial force, which we call the Q-? effect, can cause torsional vibration. To investigate this effect, we considered a single finite-size mass-linear elastic shear and torsion spring model and derived the resonance condition with respect to a torsional response from the equation of motion. In this study, we discuss the probability distribution of the induced torque and torsional-displacement response to the input ground acceleration of the uncorrelated Gaussian white-noise in two translational directions. In the frequency domain, the power spectral density function of the torque induced by the Q-? effect is proportional to a convolution of two frequency-response functions in two translational directions. In the time domain, the probability density function of the induced torque is a modified Bessel function of the second kind. Time-history response analysis was conducted to verify these theoretical results and to investigate the distribution of torsional response to Gaussian white-noise input ground acceleration.