Langrangian dynamic and static equation of the elastic plane deformation of beams

Basing on Hamilton´s principle, the second variation principle is deduced under the conditions of generalized coordinator, generalized force and the continuity of deformation. A Lagrangian basic elastic equation about plane deformation problem of beams structure is presented. By applying Hamilton´s function, the inverse problems in the continuum variation principle are avoided, and integration by parts is alternated into the calculation of partial difference. It not only simplifies the solving of direct problem, but also stylizes the solution process. In this paper, a set of dynamic and static partial difference equations and a set of primary and boundary equations are deduced in the cases of analyzing the geometric non-linear Timoshenko beam and Bernoulli-Euler beam under axial pressure.