The overall bending of circular ring shells subjected to bending moments and lateral forces is discussed. The derivation of the equations was based upon the theory of flexible shells generalized by E. L. Axelrad and the assumption of the moderately slender ratio less than 1/3 ( i. e. , ratio between curvature radius of the meridian and distance from the meridional curvature center to the axis of revolution ). The present general solution is an analytical one convergent in the whole domain of the shell and with the necessary integral constants for the boundary value problems. It can be used to calculate the stresses and displacements of the related bellows. The whole work is arranged into four parts: ( Ⅰ )Governing equation and general solution; ( Ⅱ ) Calculation for Omega-shaped bellows;( Ⅲ ) Calculation for C-shaped bellows; ( Ⅳ ) Calculation for U-shaped bellows. This paper is the first part.