This is the "rst of two companion papers which collectively present a method for the analysis of built-up structures. One such structure is the machinery foundation of a ship which is constructed from a collection of large beams and #exible plates. The heavy vibration sources are supported by the large sti! beams. The power injected into and the power transmitted around the structure is controlled by long-wavelength waves generated in these beams by the vibration sources. As these long waves propagate along the sti! beams they generate short-wavelength #exural waves in the attached #exible plates. The long waves transmit some of their energy to the short-wavelength waves which therefore damp the long waves. The di!erence between the wavelengths of the long waves in the sti! beams and the short waves in the #exible plates is often very large. In this case, the short waves present a locally reacting impedance to the long waves at the structural joints. This paper argues that such a condition allows the vibration to predicted in three steps. First, the long-wave response of the sti! beams is analyzed in isolation of the short-wave response of the #exible plates; second, the short-wave response of the #exible plates is analyzed in isolation of the long-wave response; third, the two separate responses are combined to yield the response of the complete structure due to both the long and the short waves. The method is applied to a simple plate-sti!ened beam consisting of a directly excited sti! beam attached to a large #exible plate which is broadly representative of the machinery foundation. The method predicts the frequency response of the plate-sti!ened beam which compares well with laboratory measurements, thereby supporting the method. In this paper, all three steps are performed analytically which restricts the method to geometrically simple structures. The companion paper presents a hybrid numerical/analytical implementation which accommodates geometrically more diverse structures.