Composite structures are extensively used in many industries, where they are subjected to a variety of loads and may undergo large deformations. Reliable utilisation of such structures requires prior knowledge of their failure response. In order to predict failure loads and modes, accurate, yet computationally efficient, evaluation of three-dimensional (3D) stress fields becomes important. In this paper, we present a modelling approach, based on the Unified Formulation, that accounts for geometric nonlinearity in laminated composites and predicts 3D stress fields for subsequent failure analysis. The approach builds upon the hierarchical Serendipity Lagrange finite elements and is able to capture high-order shear deformation, as well as local cross-sectional warping. A total Lagrangian approach is adopted and the classic Newton-Raphson method is employed to solve the nonlinear governing equations. A key novelty of the proposed formulation is its completeness and its applicability to fully anisotropic structures. In other words, using the Green-Lagrange strain components within the Unified Formulation framework, the explicit form of the tangent stiffness matrix is derived including general stiffness properties. This new model is benchmarked against 3D finite element solution, as well as other formulations available in the literature, by means of static analyses of highly nonlinear, laminated composite beam-like structures. Significant computational efficiency gains over 3D finite elements are observed for similar levels of accuracy. Furthermore, to show the enhanced capabilities of the present formulation, the postbuckling response of a composite stiffened panel is compared with experimental results from the literature. The 3D stress fields computed in the postbuckling regime are used to detect failure of the stiffened panel. The corresponding failure mode, as obtained by the new model, is shown to match with the experiment.