# One-dimensional Bars/Springs

**Courses > Finite Elements Method > Basics of Finite Elements > One-dimensional Bars/Springs**

### Introduction

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

#### 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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