# Funciones de forma para elementos viga

## Elementos de dos nodos

\begin{equation*} v = N_{1} v_{1} + N_{2} \theta_{1} + N_{3} v_{2} + N_{4} \theta_{2} = \sum_{i = 0}^{3} \alpha_{i} x^{i} = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} \end{equation*}

en forma matricial

\begin{equation*} v = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} = \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \end{equation*}

la deformación angular es

\begin{equation*} \theta = \frac{d v}{ d x} = \alpha_{1} + 2 \alpha_{2} x + 3 \alpha_{3} x^{2} \end{equation*}

reemplazando $x_{1} = 0$ y $x_{2} = L$

\begin{align*} \alpha_{0} + \alpha_{1} (0) + \alpha_{2} (0)^{2} + \alpha_{3} (0)^{3} &= v_{1} \\ \alpha_{1} + 2 \alpha_{2} (0) + 3 \alpha_{3} (0)^{2} &= \theta_{1} \\ \alpha_{0} + \alpha_{1} (L) + \alpha_{2} (L)^{2} + \alpha_{3} (L)^{3} &= v_{2} \\ \alpha_{1} + 2 \alpha_{2} (L) + 3 \alpha_{3} (L)^{2} &= \theta_{2} \end{align*}

simplificando

\begin{align*} \alpha_{0} &= v_{1} \\ \alpha_{1} &= \theta_{1} \\ \alpha_{0} + L \alpha_{1} + L^{2} \alpha_{2} + L^{3} \alpha_{3} &= v_{2} \\ \alpha_{1} + 2 L \alpha_{2} + 3 L^{2} \alpha_{3} &= \theta_{2} \end{align*}

en forma matricial

\begin{equation*} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & L & L^{2} & L^{3} \\ 0 & 1 & 2 L & 3 L^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*}

resolviendo el sistema

\begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{3}{L^{2}} & -\frac{2}{L} & \frac{3}{L^{2}} & -\frac{1}{L} \\ \frac{2}{L^{3}} & \frac{1}{L^{2}} & -\frac{2}{L^{3}} & \frac{1}{L^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*}

reemplazando las incógnitas

\begin{align*} v &= \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{3}{L^{2}} & -\frac{2}{L} & \frac{3}{L^{2}} & -\frac{1}{L} \\ \frac{2}{L^{3}} & \frac{1}{L^{2}} & -\frac{2}{L^{3}} & \frac{1}{L^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &=\begin{bmatrix} 1 - \frac{3}{L^{2}} x^{2} + \frac{2}{L^{3}} x^{3} & x - \frac{2}{L} x^{2} + \frac{1}{L^{2}} x^{3} & \frac{3}{L^{2}} x^{2} - \frac{2}{L^{3}} x^{3} & -\frac{1}{L} x^{2} + \frac{1}{L^{2}} x^{3} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &=\begin{bmatrix} N_{1} & N_{2} & N_{3} & N_{4} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{align*}

## Elemento de tres nodos

\begin{equation*} v = N_{1} v_{1} + N_{2} \theta_{1} + N_{3} v_{2} + N_{4} \theta_{2} + N_{5} v_{3} + N_{6} \theta_{3} = \sum_{i = 0}^{5} \alpha_{i} x^{i} = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} + \alpha_{4} x^{4} + \alpha_{5} x^{5} \end{equation*}

en forma matricial

\begin{equation*} v = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} + \alpha_{4} x^{4} + \alpha_{5} x^{5} = \begin{bmatrix} 1 & x & x^{2} & x^{3} & x^{4} & x^{5} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \end{bmatrix} \end{equation*}

la deformación angular es

\begin{equation*} \theta = \frac{d v}{ d x} = \alpha_{1} + 2 \alpha_{2} x + 3 \alpha_{3} x^{2} + 4 \alpha_{4} x^{3} + 5 \alpha_{5} x^{4} \end{equation*}

reemplazando $x_{1} = 0$, $x_{2} = \frac{L}{2}$ y $x_{3} = L$

\begin{align*} \alpha_{0} + \alpha_{1} (0) + \alpha_{2} (0)^{2} + \alpha_{3} (0)^{3} + \alpha_{4} (0)^{4} + \alpha_{5} (0)^{5} &= v_{1} \\ \alpha_{1} + 2 \alpha_{2} (0) + 3 \alpha_{3} (0)^{2} + 4 \alpha_{4} (0)^{3} + 5 \alpha_{5} (0)^{4} &= \theta_{1} \\ \alpha_{0} + \alpha_{1} \bigg( \frac{L}{2} \bigg) + \alpha_{2} \bigg( \frac{L}{2} \bigg)^{2} + \alpha_{3} \bigg( \frac{L}{2} \bigg)^{3} + \alpha_{4} \bigg( \frac{L}{2} \bigg)^{4} + \alpha_{5} \bigg( \frac{L}{2} \bigg)^{5} &= v_{2} \\ \alpha_{1} + 2 \alpha_{2} \bigg( \frac{L}{2} \bigg) + 3 \alpha_{3} \bigg( \frac{L}{2} \bigg)^{2} + 4 \alpha_{4} \bigg( \frac{L}{2} \bigg)^{3} + 5 \alpha_{5} \bigg( \frac{L}{2} \bigg)^{4} &= \theta_{2} \\ \alpha_{0} + \alpha_{1} (L) + \alpha_{2} (L)^{2} + \alpha_{3} (L)^{3} + \alpha_{4} (L)^{4} + \alpha_{5} (L)^{5} &= v_{3} \\ \alpha_{1} + 2 \alpha_{2} (L) + 3 \alpha_{3} (L)^{2} + 4 \alpha_{4} (L)^{3} + 5 \alpha_{5} (L)^{4} &= \theta_{3} \end{align*}

simplificando

\begin{align*} \alpha_{0} &= v_{1} \\ \alpha_{1} &= \theta_{1} \\ \alpha_{0} + \frac{L}{2} \alpha_{1} + \frac{L^{2}}{4} \alpha_{2} + \frac{L^{3}}{8} \alpha_{3} + \frac{L^{4}}{16} \alpha_{4} + \frac{L^{5}}{32} \alpha_{5} &= v_{2} \\ \alpha_{1} + L \alpha_{2} + \frac{3 L^{2}}{4} \alpha_{3} + \frac{L^{3}}{2} \alpha_{4} + \frac{5 L^{4}}{16} \alpha_{5} &= \theta_{2} \\ \alpha_{0} + L \alpha_{1} + L^{2} \alpha_{2} + L^{3} \alpha_{3} + L^{4} \alpha_{4} + L^{5} \alpha_{5} &= v_{3} \\ \alpha_{1} + 2 L \alpha_{2} + 3 L^{2} \alpha_{3} + 4 L^{3} \alpha_{4} + 5 L^{4} \alpha_{5} &= \theta_{3} \end{align*}

en forma matricial

\begin{equation*} \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & \frac{L}{2} & \frac{L^{2}}{4} & \frac{L^{3}}{8} & \frac{L^{4}}{16} & \frac{L^{5}}{32} \\ 0 & 1 & L & \frac{3 L^{2}}{4} & \frac{L^{3}}{2} & \frac{5 L^{4}}{16} \\ 1 & L & L^{2} & L^{3} & L^{4} & L^{5} \\ 0 & 1 & 2 L & 3 L^{2} & 4 L^{3} & 5 L^{4} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \end{bmatrix} = \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \end{equation*}

resolviendo el sistema

\begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -\frac{23}{L^{2}} & -\frac{6}{L} & \frac{16}{L^{2}} & -\frac{8}{L} & \frac{7}{L^{2}} & -\frac{1}{L} \\ \frac{66}{L^{3}} & \frac{13}{L^{2}} & -\frac{32}{L^{3}} & \frac{32}{L^{2}} & -\frac{34}{L^{3}} & \frac{5}{L^{2}} \\ -\frac{68}{L^{4}} & -\frac{12}{L^{3}} & \frac{16}{L^{4}} & -\frac{40}{L^{3}} & \frac{52}{L^{4}} & -\frac{8}{L^{3}} \\ \frac{24}{L^{5}} & \frac{4}{L^{4}} & 0 & \frac{16}{L^{4}} & -\frac{24}{L^{5}} & \frac{4}{L^{4}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \end{equation*}

reemplazando las incógnitas

\begin{align*} v &= \begin{bmatrix} 1 & x & x^{2} & x^{3} & x^{4} & x^{5} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x & x^{2} & x^{3} & x^{4} & x^{5} \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -\frac{23}{L^{2}} & -\frac{6}{L} & \frac{16}{L^{2}} & -\frac{8}{L} & \frac{7}{L^{2}} & -\frac{1}{L} \\ \frac{66}{L^{3}} & \frac{13}{L^{2}} & -\frac{32}{L^{3}} & \frac{32}{L^{2}} & -\frac{34}{L^{3}} & \frac{5}{L^{2}} \\ -\frac{68}{L^{4}} & -\frac{12}{L^{3}} & \frac{16}{L^{4}} & -\frac{40}{L^{3}} & \frac{52}{L^{4}} & -\frac{8}{L^{3}} \\ \frac{24}{L^{5}} & \frac{4}{L^{4}} & 0 & \frac{16}{L^{4}} & -\frac{24}{L^{5}} & \frac{4}{L^{4}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \\ &= \begin{bmatrix} 1 - \frac{23}{L^{2}} {x}^{2} + \frac{66}{L^{3}} x^{3} - \frac {68}{L^{4}} x^{4} + \frac {24}{L^{5}} x^{5} \\ x - \frac{6}{L} x^{2} + \frac{13}{L^{2}} x^{3} - \frac{12}{L^{3}} x^{4} + \frac{4}{L^{4}} x^{5} \\ \frac{16}{L^{2}} x^{2} - \frac{32}{L^{3}} x^{3} + \frac{16}{L^{4}} x^{4} \\ -\frac{8}{L} x^{2} + \frac{32}{L^{2}} x^{3} - \frac{40}{L^{3}} x^{4} + \frac{16}{L^{4}} x^{5} \\ \frac{7}{L^{2}} x^{2} - \frac{34}{L^{3}} x^{3} + \frac{52}{L^{4}} x^{4} - \frac{24}{L^{5}} x^{5} \\ -\frac{1}{L} x^{2} + \frac{5}{L^{2}} x^{3} - \frac{8}{L^{3}} x^{4} + \frac{4}{L^{4}} x^{5} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \\ &= \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \\ N_{4} \\ N_{5} \\ N_{6} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \end{align*}