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In this paper the smoothed strain based four-node flat element MISQ24 with driiling degrees of freedom is extended for geometrically nonlinear analysis of plate and shell structures. The von-Karman's large deflection theory and the Total Lagrangian (TL) approach are employed in the formulation of the elements to describe small strain geometric nonlinearity with large deformations using the first-order shear deformation theory (FSDT). The predictive capability of the present models is demonstrated by comparing the present results with analytical/experimental and other numerical solutions available in the literature. Numerical examples show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other quadrilateral elements with inplanes rotations.