Vibration of a class of shallow shells bounded by edges described by polynomials, part I: theoretical approach and validation

Abstract The Ritz method is used to obtain an eigenvalue equation for the free vibration of thin shallow shells of curvilinear planform defined by polynomial expressions. The shell is discretized into four 90° sectorial elements allowing for up to four different outer curves and up to four different inner curves described by polynomials which define the planform of the shell; the elements are joined together through the use of artificial springs. Several complicating effects are included in the analysis, such as the presence of internal point or line supports, concentrated masses and stepped material thickness. In Part I of the paper, the theoretical approach is developed and its validity demonstrated through its application to several shells previously treated in the literature. In Part II of the paper, the proposed approach is applied to a number of shallow shells of various different planforms, with and without complicating effects, both demonstrating its versatility and giving results for problems as yet untreated in the open literature.