# One-dimensional Shape Functions

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## Latest revision as of 08:59, 4 November 2009

**Shape Functions for 1D problems**

- If a polynomial is selected as basis function, then:

## Contents |

## Linear case

- Two nodes:

- with: the determinant of the matrix.

## Quadratic case

- Three nodes:

## Cubic case

- Four nodes:

- and so on... (note that the expressions for the N
_{2}, N_{3}and N_{4}can be easily obtained by swapping the x_{2}values for the x_{1}values in the first case, x_{3}for x_{1}in the second case and x_{4}for x_{1}in the last one.

*To check by yourself the functions, use this***Matlab code**

## Lagrangian elements

- To avoid solving this so complex system of equations, the well-known properties of the Lagragian polynomials can be used.

- All these shape functions are based in the polynomial Lagrange[1] and can be written as follows:

- This equation is easier to implement, as can be checked using this
**Matlab code**.

- An example of using nine nodes for each element, for which the expressions are unwriteable is shown.

## Normalised Shape Functions

- All these expressions can be normalised using the
**natural coordinate system**, based in , a variable defined in terms of , the central coordinate of the element, as follows:

- Note that in the left node, in the central point of the element, and in the right node of the element.

- In this way, all the shape functions can be expressed and, therefore obtained, independently of the real geometry, and then easier to implement.

- For the
**Linear case**, this transformation can be illustrated:

- The final specific expressions for the 1D Linear element are:

- For
**1D quadratic elements**are:

- Finally, for
**1D cubic elements**the normalised shape functions are: