A large number of practical problems are governed by the one-dimensional boundary value problem (1D BVP) of the following form:
Since the differential equation is of second order, for a unique solution, there must be at least two specified boundary conditions.
The general governing differential equation for axial deformation of bars/springs is:
where A(x) is the area of cross section and can vary along the length; E is the elastic modulus; and q(x) is the applied distributed load in the axial direction.
The boundary conditions of the following forms may be specified at one or more points:
1- Essential boundary conditions (EBC):
2- Natural boundary conditions (NBC):
The interpolation functions are simple linear functions of x. Furthermore, if we assume that k,p, and q are constant over an element, then it is easy to carry out integrations and write explicit formulas for element equations:
the position of element nodes:
Interpolation functions (shape functions):
and consequently derivatives of shape functions:
Thus, the element stiffness matrix is:
and the explicit equations for a linear element: