A finite element formulation via tensor notation for geometrically exact nonlinear laminated composite shells is presented. The present 5/6-DOF shell theory is based on total Lagrangian concept where the Green-Lagrange's strain tensor and and the work conjugate 2-nd Piola-Kirchoff stress tensor plays the central role. The two parametric incremental material rotation vector is considered to be orthogonal to the shell director. To improve the performance of the present 4/9-noded element, EAS scheme combined with ANS scheme is employed. The update procedure of the nodal rotation is kept additive within a load increment step. To avoid the singularity problem the load increment size is restricted to render the rotation magnitude less than $\pi$. The performance of the element is found to be exciting in linear and nonlinear regime for a wide class of structural problems. New results are presented for different structural problems under thermo-mechanical loads.