In this study a higher-order shell theory is proposed for arbitrary shell geometries which allows the cross-section to rotate with respect to the middle surface and to warp into a non-planar surface. This new kinematic assumption satisfies the shear-free surface boundary condition (BC) automatically. A new internal force expression is obtained based on this kinematic assumption. A new functional for arbitrary shell geometries is obtained employing Gateaux differential method. During this variational process the BC is constructed and introduced to the functional in a systematic way. Two different mixed elements PRSH52 and CRSH52 are derived for parabolic and circular cylindrical shells, respectively, using the new functional. The element does not suffer from shear locking. The excellent performance of the new elements is Verified by applying the method to some test problems. Copyright (C) 2000 John Wiley & Sons, Ltd.