Three-dimensional variational analysis of Paganoʹs problems for laminated composite plates

A brief survey of more than 50 previously published papers on numerical simulation of Paganoʹs 3-D (bidirectional bending) and 2-D (cylindrical bending) closed-form solutions for cross-ply and angle-ply composite plates is presented in the Introduction. Previous 3-D analysis approaches and numerical results obtained with the use of piece-wise polynomial splines and Bernstein polynomials are then reviewed. The major part of this work is devoted to the application of a novel 3-D displacement-assumed variational-analysis approach, which utilizes Bernstein basis functions in the three coordinate directions. Four classical transverse bending problems of laminated composite plates solved by Pagano are considered: (i) bidirectional transverse bending of simply supported cross-ply rectangular plates, (ii) cylindrical bending of simply supported cross-ply unsymmetric plates, (iii) cylindrical bending of single-layer monoclinic plates with different off-axis reinforcement angles, and (iv) cylindrical bending of simply supported angle-ply symmetric and unsymmetric plates. The problems (ii)–(iv) are 2-D (plane strain) in nature, and this feature has been utilized in Paganoʹs solutions. Nevertheless, the 3-D boundary value problem formulation used in the present work allows one to accurately simulate the cylindrical bending case and obtain numerical results that are nearly identical to the benchmark data for all of the examples solved. Overall, numerical results obtained in this work for all of the cases, (i)–(iv), are the closest to the benchmark data among up-to-date published results and can be used for future reference.