General expressions for reactions, displacements, and stresses of nonuniform rods under arbitrary number of concentrated axial forces, torques, or bending moments

Closed-form expressions are derived for reactions, generalized displacements, and stresses in nonuniform linearly elastic rods with prescribed end conditions under an arbitrary number of applied concentrated axial forces, torques, or bending moments. The results are expressed in terms of the introduced nondimensional parameters α i and β i = 1 − α i , which represent the normalized effective compliances of the segments of the rod to the left and right of cross-section i . The Maxwell influence coefficients take on a simple form when expressed in terms of the parameters α i and β i , which greatly facilitates the analysis. The solution based on the transfer matrix method is presented, as well as the solution based on the reciprocal influence coefficients. The load-displacement relationships are also derived by the minimization of potential energy, as in the finite-element method. Examples are included to illustrate the incorporation of temperature and misfit effects. The developed systematic approach based on the parameters α i and β i is conceptually and pedagogically appealing, enabling stress and deformation analysis involving a large number of loads by simple analytical and computational means. As such, the presented analysis and the obtained results can be included in coverage of the topic of loaded nonuniform rods in engineering courses of strength of materials and structural mechanics, or conveniently used in design and optimization studies.