Pseudo-damping is a counter-intuitive phenomenon observed in a special class of linear structures that exhibit an impulse response characterized by a decaying amplitude, even in the absence of any dissipation mechanism. The conserved energy remains within but designated parts of the system. Pseudo-damping develops when the natural frequency distribution of the system includes condensation points. The recently formulated theoretical foundation of this phenomenon, based on mathematical properties of special trigonometric series, makes it possible to describe a class of mechanical systems capable of displaying pseudo-damping characteristics. They include systems with discrete oscillators and one-dimensional continuous beamlike structures already reported by the authors in recent studies. This paper examines development of pseudo-damping phenomenon in two-dimensional structures, using plates and shells as examples, and shows how a preloaded plate on an elastic foundation can lead to pseudo-damping. Moreover, in the case of curved shell elements examined here, pseudo-damping can result due to the curvature of the structure, which naturally introduces condensation points in the modal density.