In this paper we propose a refined finite element for the analysis of shell structures. A tensor-based finite element formulation that describes the mathematical shell model in a natural and simple way by using curvilinear coordinates is developed and implemented. A family of higher-order elements with Lagrangian interpolations is used to avoid membrane and shear locking, without resorting to mixed or assumed strain formulations. The formulation can be specialized to different shell theories by making appropriate kinematic approximations. For geometrically nonlinear analysis of shells, we use a first-order shell theory with seven parameters with exact nonlinear deformations and under the framework of the Lagrangian description. The theory takes into account thickness stretching and, therefore, 3D constitutive equations are utilized. Numerical simulations and comparisons of the present results with those found in the literature for benchmark problems involving isotropic and laminated composites, as well as functionally graded shells, are found to be excellent and show the accuracy and robustness of the developed shell element.