Hamiltonian structural analysis of curved beams with or without generalized two-parameter foundation

The solution of curved Timoshenko beams with or without generalized two-parameter elastic foundation is presented. Beam can be subjected to any kind of loads and imposed external actions, distributed or concentrated along the beam. It can have external and internal restraints and any kind of internal kinematical or mechanical discontinuity. Moreover, the beam may have any spatial curved geometry, by dividing the entire structure into segments of constant curvature and constant elastic properties, each segment resting or not on elastic foundation. The foundation has six parameters like a generalized Winkler soil with the addition of other two parameters involving the link between settlements in transverse direction, as occurs for Pasternak soil model. The solution is obtained by the Hamiltonian structural analysis method, based on an energetic approach, solving a mixed canonical Hamiltonian system of twelve differential equations, leading to the fundamental matrix. The solution, numerically expressed or in closed form, is fast and simple to be implemented on a software, being the solving matrix always of order 12 for any kind of geometry and loads, without increasing in computational complexity. Numerical examples are given, comparing the results of the proposed procedure with the literature data, for rectilinear and curved beams with different soil properties.