Dynamic higher order equations for structural elements

Licentiate thesis, 2014

The subject of this thesis is to derive and evaluate governing equations and corresponding boundary conditions for solid isotropic cylinders and isotropic micropolar rectangular plates. This is achieved by a systematic power series expansion approach, by either adopting a generalized Hamilton's principle or a direct approach. For the solid cylinders a power series expansion in the radial coordinate is adopted. Equations of motion together with consistent sets of end boundary conditions are derived in a systematic fashion up to arbitrary order using a generalized Hamilton's principle. Governing equations are obtained for longitudinal, torsional, and flexural modes. In the case of the isotropic micropolar plate a power series expansion of the displacements and micro-rotations are adopted in the thickness coordinate. Governing equations of motion, for extensional and flexural case, together with consistent sets of edge boundary conditions are derived in a systematic fashion up to arbitrary order with use of the direct approach. Both the governing equations for the solid cylinder and the micropolar plate are asymptotically correct to all studied orders. Numerical examples are presented for different sorts of problems, using exact theory, the present series expansion theories of different order, various classical theories and other newly developed approximate theories. These results cover dispersion curves, eigenfrequencies, various curves of cross sectional quantities such as displacements, stresses and micro-rotations, as well as fixed frequency motion due to prescribed end displacement or lateral distributed forces. The results illustrate that the present approach may render benchmark solutions provided higher order truncations are used, and act as engineering equations using low order truncation.