Radiation damping due to wave propagation in unbounded domains may cause a significant reduction of structural vibrations when excited near resonance. Here a novel matrix-valued algebraic Pad$acute{e}$-like stiffness formulation in the frequency-domain and a corresponding state equation in the time domain are elaborated for a soil-structure interaction problem with a layered soil excited in a transient manner by a flexible rotor during startup and shutdown. The contribution of radiation damping caused by a soil-layer upon a rigid bedrock is characterized by the corresponding amount of critical damping as it is used in structural dynamics.