Vibrations of singly curved thin shells: Analytical studies

The equations of motion for inhomogeneous thin shells of arbitrary derived by Pierce (1993) are specialized to singly curved shells, whose curvature is described by an arbitrary shape function Z (X) of only one Cartesian coordinate, X. It turns out that this specialization leads to considerable simplifications: The vector basis of the curvilinear coordinate system is orthonormal, the metric tensor is unity, the curvature tensor has only one non-zero component, and the Christoffel symbols all vanish, which means that the covariant derivatives become 'usual' derivatives. The resulting system of partial differential equations is further specialized by considering solutions which depend only on the Cartesian coordinate X or rather on the corresponding curvilinear coordinate. SH-like waves and solutions with coupled normal and in-shell motions are discussed. The second solution type can be described by an ordinary differential equation of sixth order consisting of 62 + 5 = 67 terms (left + right hand side). For shells with constant material properties and constant thickness this number reduces to 11 + 5, for constant curvature (circular cylinder) to 5 + 2. In case of shells with periodic shape functions Z (X) a Bloch ansatz leads to an infinite system of linear equations. The present approach opens up the possibility to asess the validity of the common equivalent-orthotropic-plate approximation for corrugated plates. Numerical results are presented in the companion paper by Aoki & Maysenhölder.