Solving a singular beam equation by the method of energy boundary functions

摘要

For a static singular beam equation and a static non-uniform beam equation under external static loads, we develop boundary functions method (BFM) and energy boundary functions method (EBFM) to find the deflection curves, which automatically satisfy the boundary conditions. Furthermore, the EBFM is also designed to preserve the energy. Both methods can quickly find accurate numerical solutions of static beam equations, and depict well the singular boundary layer behavior that appeared in the second-order differential term for the simply-supported and two-end fixed beams, and in the third-order differential term for the cantilever beam. Owing to the preservation of both the boundary conditions and energy, the EBFM is superior than the BFM, the shooting method, the weak-form method as well as the weak-form exponential trial functions method.