ABSTRACT
In the following discussion, we treat rings and beams as special cases of arches. The arch equation is derived by reduction from Love’s equations for shells. Also by direct reduction from Love’s equation, we obtain the plate equation. In literature, reduction is usually not used for these relatively simple structures, and each special case is derived from basic principles (Timoshenko, 1955; Biezeno and Grammel, 1954; Thomson, 1972; Meirovitch, 1972)
The arch is a curved beam where all curvature is in one plane only, as shown in Fig. 1. Vibratory motion is assumed to occur only in that plane. Designating s as the coordinate along the neutral axis of the arch and y as the coordinate perpendicular to the neutral axis, the fundamental form becomes
ds′2=ds2+dy2 (4.1.1)
where ds′ is the fundamental form diagonal to avoid confusion with ds.
