A {1,2}-order theory for elastodynamic analysis of thick orthotropic shells

Abstract

A higher-order shell theory is developed for elastodynamic analysis of orthotropic shells. The theory accounts for all basic deformations including transverse shear and transverse normal strains and stresses. The theory is developed in orthogonal curvilinear coordinates in which the reference surface components of the displacement vector vary linearly through the thickness while the transverse displacement is parabolic. Transverse shear and transverse normal strains are formulated to satisfy physical traction conditions at the top and bottom shell surfaces, and are also made least-squares compatible with the corresponding strains that are derived directly from the strain-displacement relations of three-dimensional elasticity. In these variational statements of strain compatibility, transverse shear and transverse normal correction factors are introduced, and are determined from dynamic considerations in the manner originally proposed by Mindlin. Equations of motion and associated engineering (Poisson) boundary conditions are derived from a three-dimensional variational principle. An important feature of the present theory is the requirement of only simple C0 and C−1 continuity for the shell kinematic variables. This aspect makes the theory particularly attractive for the development of efficient shell finite elements suitable for general purpose finite element analysis of thick shell structures. Analytical solutions for the free vibration of isotropic and orthotropic cylindrical shells are obtained for a wide range of thickness/radius and thickness/wavelength ratios and found to be in close agreement with the exact elasticity solutions.