Summary: A smoothed finite element method (SFEM) is presented to analyze linear and geometrically nonlinear problems of plates and shells using bilinear quadrilateral elements. The formulation is based on the first order shear deformation theory. In the present SFEM, the elements are further divided into smoothing cells to perform strain smoothing operation, and the strain energy in each smoothing cell is expressed as an explicit form of the smoothed strain. The effect of the number of divisions of smoothing cells in elements is investigated in detail. It is found that using three smoothing cells for bending strain energy integration and one smoothing cell for shear strain energy integration achieve most accurate results and hence these numbers recommended for plates and shells in this study. In the geometrically nonlinear analysis, the total Lagrangian approach is adopted. The arc-length technique in conjunction with the modified Newton-Raphson method is utilized to solve the nonlinear equations. The numerical examples demonstrate that the present SFEM provides very stable and most accurate results with the similar computational effort compared to the existing FEM techniques tested in this work.