Abstract: In this paper, a class of quadrilateral FEMOL (Finite Element Method of Lines) elements for arbitrarily curved shells are constructed based on degenerate shell theory. These elements are fully conforming elements with C⁰ continuity. The governing ordinary differential equations and the associated boundary conditions for static analysis using the above elements are derived, being in a consistent form as the FEMOL equation system derived from other C⁰ problems. The present paper addresses the basic theory, while a number of numerical examples will be given in a companion paper.