Prototype One-Dimensional Boundary-Value Problem (BVP): Governing Differential Equation and Variational Formulations with Examples

In this chapter we introduce the prototype one-dimensional BVP and present the corresponding Principle of Minimum Potential Energy that will serve as the basis for the development of the finite element method for the solution of 1D problems. To fix ideas, we will consider as our first application the axial deformation of an inhomogeneous, linearly elastic bar with variable cross-section, fixed at the one end, with a concentrated load at the other end, and subjected to a distributed load along the length of the bar. We will initially derive the governing second-order differential equation (ODE) and associated boundary conditions (BCs) for the resulting BVP using the standard approach one encounters in an introductory undergraduate course in solid mechanics.