The analytical calculation of circular rods of variable cross-section by the initial values method

In this work, the stress resultants and displacements (due to in-plane loading) of circular rods with variable cross-sections are calculated by using the initial values method. The principal matrix and its inverse required for the applicability of the method of initial values are obtained. The expression of the solution of a circular-rod loaded by both continuous and singular loads is given in a compact form. Irrespective of the number of singular loads and the degree of redundancy the system of algebraic equations to be solved contains no more than three variables as shown. Two examples exhibiting the advantages and priorities of the method are solved in full detail. The program Mathematica is used throughout.