In the present study, we propose a computational model for the linear and nonlinear analysis of shell structures. We consider a tensor-based finite element formulation which describes the mathematical shell model in a natural and simple way by using curvilinear coordinates. To avoid membrane and shear locking we develop a family of high-order elements with Lagrangian interpolations. The approach is first applied to linear deformations based on a novel and consistent third-order shear deformation shell theory for bending of composite shells. No simplification other than the assumption of linear elastic material is made in the computation of stress resultants and material stiffness coefficients. They are integrated numerically without any approximation in the shifter. Therefore, the formulation is valid for thin and thick shells. A conforming high-order element was derived with C0 continuity across the element boundaries. Next, we extend the formulation for the geometrically nonlinear analysis of multilayered composites and functionally graded shells. Again, Lagrangian elements with high-order interpolation polynomials are employed. The flexibility of these elements mitigates any locking problems. A first-order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the Lagrangian description. This approach takes into account thickness changes and, therefore, 3D constitutive equations are utilized. Finally, extensive numerical simulations and comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composites, as well as functionally graded shells, are found to be excellent and show the validity of the developed finite element model. Moreover, the simplicity of this approach makes it attractive for future applications in different topics of research, such as contact mechanics, damage propagation and viscoelastic behavior of shells.