Abstract: Plate structures made of advanced composite materials or functionally graded materials have been widely used in engineering practice recently, which is characterized by the variation of material properties along the plate thickness. Several plate elements have been presented utilizing the finite element formulation based on Reddy's higher-order shear deformation theory which yields more accurate transverse shear strain distributions of these structures. However, the C^1 continuous plate elements is very limited. Based on Reddy's higher-order shear deformation theory, a C^1 continuous quadrilateral plate element is established using the weak form quadrature element method in this work. The element presented here is then used for linear flexural and free vibrational analyses of the rectangular and skew plates made of homogenous or composite materials with constant thickness as well as the homogenous rectangular plates with variable thickness. The numerical results of quadrature element formulation are compared with those of other numerical method from the open literatures in order to validate the correctness and efficiency of the presented quadrature plate element. It is shown that only one quadrature element is fully competent for linear analysis of a quadrilateral plate regardless of its thickness variation and component materials. As for rectangular plates with constant or variable thickness, one quadrature element with only 9\times 9 numerical integration points is needed. And for skew plates, the number of numerical integration points required for acceptable accuracy gradually increases to 15\times 15 with the skew angle enlargement. As a completive numerical formulation, the quadrilateral quadrature plate element can be further applied in nonlinear and transient analyses of composite material plate structures.