Analytic solution of beams on Winkler foundation under complex conditions

Based on the differential equation for curvature of beams,the theory of finite element and the principle of minimum potential energy,a method for beams on Winkler foundation under complex conditions was deduced with compatibility conditions of displacement,angular rotation,moment and shear of adjacent beam elements.The method could be used to compute beams on Winkler foundation under complex conditions,such as variable Young’s modulus and section,complex foundation conditions,namely variable bedding modulus,and complex load consisting of concentrated force,moment and randomly distributed load.The displacement equations both for beam elements and for the whole beam,which had the same structure with the classic solution by Winkler,could be derived from the proposed method.Therefore,the equations for angular rotation,moment and shear of the beams could also be obtained from their differential relationship with displacement equation.It was shown by the computation of an example that the results obtaind by the proposed method were consistent with those by the classic finite element method when the elements were small enough.Furthermore,the element partition in the proposed method was thoroughly different from that in the classic finite element method.