Bars with Continuous Varying Cross Section

Let us now consider the case of a bar whose cross section is differing continuously as illustrated in Figure. The bar of (truncated) conical shape is subjected to an axial force of P. As in the matter of stepped bars, here also, we might apply the equation

δ = ∑δ_i = ∑ P_i L_i / A_i E_i, if we recognize that L_i is adequately small and A might be taken as uniform over the length of the small segment. i.e., the expression ∑ P_i L_i/ A_i E_i now becomes

Though, to simplify the expression for A, the origin might be conveniently is shifted to the apex of the generated cone as illustrated in given Figure. Hence,

The integral in above equation might be evaluated for solids with stepped as well as constant variation in cross section. Stepped bars needs no special treatment except that the equilibrium & deformations of each segment might be analysed separately and added wherever essential. Even a prismatic segment of the bar might have to be let as two or more members, if external loads are applied at interior points of the segment.