A higher order theory for shells, plates and rods

• We develop a higher order theory for elastic shells, plates, roods and beams.
• We use for that Fourier series in terms of Legendre’s polynomials expansion.
• FEM implemented in the software MATLAB and Mathematica are used.
• The displacements, stresscylindrical shell and beamhave been studied.
• Comparison with elasticity and classical shell and beam solution has been done.

In this paper a new theory for shells, plates and rods has been developed. The proposed theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements, traction and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including Hooke׳s law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients has been obtained. The case of axially symmetric shell has been considered in more details. As a special case of the general approach the equations of the first and second approximations have been developed for circular plates, curvilinear rods and cylindrical shells. Numerical calculations have been done numerically using the finite element method (FEM) along with the Comsol Multiphysics, Matlab and Mathematica software.