Kirchhoff equation of thin elastic rod statics and dynamics must be written in piecewise at section with discontinuous or nonsmooth quantities such as external forces, mass geometry and physical parameters, which leads to inconvenience to the numerical calculation. According to singular function method in calculation of beam bending deformation, these discontinuous or nonsmooth quantities of the rod are expressed by singular function and become continues quantities alone centerline of the rod. Numerical simulations of equilibrium configuration of the rod acted by lateral concentrated load are made by means of Mathematics software. Results explain that introducing singular function to express discontinuous or nonsmooth quantities can avoid complicated calculation and improve the computational efficiency.

The Schrödinger analogy of thin elatic rod is studied. Compared with the Kirchhoff dynamics analogy, the Schrödinger analogy is proposed. By the new analogy, the Kirchhoff equation of elastic rod can be written as curvature equation which is similar to nonlinear Schrödinger equation. Thus, the Jacobi elliptic function solution of Schrödinger equation can be taken into elastic rod equation. The space configurations of the elastic rod described by the solution are given. Schrödinger analogy reveals the relations between the quantum state of wave function and the geometry configuration of elastic rod, and gives a new way to solve the Kirchhoff equation.