Bending of laminated plates with mixed boundary conditions based on higher-order shear deformation theory

Abstract

The bending of laminated plates is considered using higher-order transverse shear deformation theory. The principle of virtual work is used to derive a new set of seven governing equations and corresponding boundary conditions. These equations, combined with eighteen relationships between the resultant stress and displacement components, compose a system of first-order partial differential equations that is solved by the generalized differential quadrature method. Numerical results for laminated plates with a variety of mixed boundary conditions are calculated using the proposed method, and good agreement is found with the corresponding solutions obtained using ANSYS. Fiber-reinforced laminated composite materials are widely used in a variety of engineering fields, such as aerospace, civil, marine, mechanical, nuclear, and petrochemical engineering. Such materials are popular for industrial applications due to high strength-to-weight ratios, long fatigue life, good stealth characteristics, and enhanced corrosion resistance. A number of theories describing laminated composite plates exist in the literature. Classical plate theory is based on the Kirchhoff kinematic hypothesis that straight lines normal to the undeformed midsurface remain straight and normal to the middle surface after deformation and undergo no thickness stretching. Neglecting transverse shear effects, this theory produces unacceptable approximations in the analysis of even thin laminated plates and shells. Surveys of various classic shell theories can be found in [Naghdi 1956; Bert and Francis 1974; Bert and Chen 1978]. The development of plate theories with transverse shear effects has improved the accuracy of results considerably. The refined theories are of different orders, based on discretization of the transverse shear effects and the number of terms included in the assumed displacement field. Reissner [1945] was the first to develop a plate theory that included transverse shear deformation for static analysis. Mindlin [1951] then expanded Reissner’s theory for dynamic analysis. Both approaches rely on first-order shear deformation theory (FSDT). Reissner’s theory is stress-based, whereas Mindlin’s is displacement-based. These theories do not satisfy the condition of zero transverse shear stress at the top and bottom surfaces of the plate, and consider a uniform transverse shear stress distribution across the thickness of the plate. Therefore they require the use of a shear correction factor to increase the precision of the results. Later a set of theories, generally known as higher-order shear deformation theories (HSDT), has been developed by a number of researchers. Basset [1890] appears to have been the first researcher to suggest that displacement fields can be expanded in a power series of the thickness coordinate. The higher-order