# Funciones de forma para elementos barra

## Elemento de dos nodos

\begin{equation*} u = N_{1} u_{1} + N_{2} u_{2} = \sum_{i = 0}^{1} \alpha_{i} x^{i} = \alpha_{0} + \alpha_{1} x \end{equation*}

en forma matricial

\begin{equation*} u = \alpha_{0} + \alpha_{1} x = \begin{bmatrix} 1 & x \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \end{bmatrix} \end{equation*}

reemplazando los puntos $x_{1}$ y $x_{2}$

\begin{align*} \alpha_{0} + \alpha_{1} x_{1} &= u_{1} \\ \alpha_{0} + \alpha_{1} x_{2} &= u_{2} \end{align*}

en forma matricial

\begin{equation*} \begin{bmatrix} 1 & x_{1} \\ 1 & x_{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \end{bmatrix} = \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} \end{equation*}

resolviendo el sistema

\begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \end{bmatrix} = \begin{bmatrix} \frac{x_{2}}{x_{2} - x_{1}} & -\frac{x_{1}}{x_{2} - x_{1}} \\ -\frac{1}{x_{2} - x_{1}} & \frac{1}{x_{2} - x_{1}} \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} \end{equation*}

reemplazando las incógnitas

\begin{align*} u &= \begin{bmatrix} 1 & x \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x \end{bmatrix} \begin{bmatrix} \frac{x_{2}}{x_{2} - x_{1}} & -\frac{x_{1}}{x_{2} - x_{1}} \\ -\frac{1}{x_{2} - x_{1}} & \frac{1}{x_{2} - x_{1}} \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} \\ &= \begin{bmatrix} \frac{x_{2}}{x_{2} - x_{1}} - \frac{1}{x_{2} - x_{1}} x & -\frac{x_{1}}{x_{2} - x_{1}} + \frac{1}{x_{2} - x_{1}} x \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} \\ &= \begin{bmatrix} N_{1} & N_{2} \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} \end{align*}

Reescribiendo $u$

\begin{equation*} u = \bigg( \frac{x_{2}}{x_{2} - x_{1}} - \frac{1}{x_{2} - x_{1}} x \bigg) u_{1} + \bigg( -\frac{x_{1}}{x_{2} - x_{1}} + \frac{1}{x_{2} - x_{1}} x \bigg) u_{2} \end{equation*}

## Elemento de tres nodos

\begin{equation*} u = N_{1} u_{1} + N_{2} u_{2} + N_{3} u_{3} = \sum_{i = 0}^{2} \alpha_{i} x^{i} = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} \end{equation*}

en forma matricial

\begin{equation*} u = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} = \begin{bmatrix} 1 & x & x^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \end{bmatrix} \end{equation*}

reemplazando los puntos $x_{1}$, $x_{2}$ y $x_{3}$

\begin{align*} \alpha_{0} + \alpha_{1} x_{1} + \alpha_{2} x_{1}^{2} &= u_{1} \\ \alpha_{0} + \alpha_{1} x_{2} + \alpha_{2} x_{2}^{2} &= u_{2} \\ \alpha_{0} + \alpha_{1} x_{3} + \alpha_{2} x_{3}^{2} &= u_{3} \end{align*}

en forma matricial

\begin{align*} \begin{bmatrix} 1 & x_{1} & x_{1}^{2} \\ 1 & x_{2} & x_{2}^{2} \\ 1 & x_{3} & x_{3}^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \end{bmatrix} = \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \end{align*}

resolviendo

\begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \end{bmatrix} = \begin{bmatrix} \frac{x_{2} x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & -\frac{x_{1}x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & \frac{x_{1} x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \\ -\frac{x_{2} + x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & \frac{x_{1} + x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & -\frac{x_{1} + x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \\ \frac{1}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & -\frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & \frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \end{equation*}

reemplazando las incógnitas

\begin{align*} u &= \begin{bmatrix} 1 & x & x^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x & x^{2} \end{bmatrix} \begin{bmatrix} \frac{x_{2} x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & -\frac{x_{1}x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & \frac{x_{1} x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \\ -\frac{x_{2} + x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & \frac{x_{1} + x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & -\frac{x_{1} + x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \\ \frac{1}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & -\frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & \frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \\ &= \begin{bmatrix} \frac{x_{2} x_{3}}{x_{1}^{2} - x_{1} x_{2} -x_{1} x_{3} + x_{2} x_{3}} - \frac{x_{2} + x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} x + \frac{1}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} x^{2} \\ - \frac{x_{1} x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} + \frac{x_{1} + x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} x - \frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} x^{2} \\ \frac{x_{1} x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} - \frac{x_{1} + x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} x + \frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} x^{2} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \\ &= \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \end{align*}


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