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Introduction on One-dimensional Bars/Springs :

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.

Concepts and Formulas of One-dimensional Bars/Springs:

Governing differential equation (ODE):
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.

Types of boundary conditions
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):

Explicit equations for a two-node linear element:
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:

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Solved sample problems of One-dimensional Bars/Springs:

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