Applying the generalized step-function to solve the bending problems of nonuniform beams with variable cross section

Abstract

The bending problems of nonuniform beams with variable cross-section can be approximated by that of a step beam under sectionally uniform load (including both concentrated forces and couples). In this paper, the concept of Heaviside function {x−a}⁰ will be generalized, and a new function {x−a}ⁿ,n=0, 1, 2..., will be defined, which may be named as a generalized step function. The rules of operation will also be given to {x−a}ⁿ{x−b}⁰. The reciprocal of the flexural of rigidity 1/EJ and the bending momentM(x) can all be expressed in terms of {x−a}ⁿ, and substituted into the differential equation of the elastic curve of the beam respectively. Thus we may establish a set of unified method to solve various types of bending problems of straight beams. The general solution of the deflection equation will be given.