Principio de los trabajos virtuales para viga

Para un elemento con dos nodos:

\begin{equation} \iiint \delta \varepsilon \ \sigma \ dV = \int \delta w \ q \ dx + \sum_{i=1}^{2} \delta w_{i} \ Z_{i} + \sum_{i=1}^{2} \delta \theta_{i} \ M_{i} \end{equation}

Recordando expresiones conocidas:

\begin{eqnarray} \sigma &=& -z \ E \ \frac{\partial^{2} w}{\partial x^{2}} \\\ \delta \varepsilon &=& -z \ \delta \Big ( \frac{\partial^{2} w}{\partial x^{2}} \Big ) \end{eqnarray}

Reemplazando:

\begin{equation} \iiint z^{2} \ \delta \Big ( \frac{\partial^{2} w}{\partial x^{2}} \Big ) \ E \ \frac{\partial^{2} w}{\partial x^{2}} \ dx \ dy \ dz = \int \delta w \ q \ dx + \delta w_{1} \ Z_{1} + \delta w_{2} \ Z_{2} + \delta \theta_{1} \ M_{1} + \delta \theta_{2} \ M_{2} \end{equation}

Simplificando:

\begin{equation} \int \delta \Big ( \frac{\partial^{2} w}{\partial x^{2}} \Big ) \ EI \ \frac{\partial^{2} w}{\partial x^{2}} \ dx = \int \delta w \ q \ dx + \delta w_{1} \ Z_{1} + \delta w_{2} \ Z_{2} + \delta \theta_{1} \ M_{1} + \delta \theta_{2} \ M_{2} \end{equation}

Realizando un cambio de variable:

\begin{equation} \int \delta \chi \ EI \ \chi \ dx = \int \delta w \ q \ dx + \delta w_{1} \ Z_{1} + \delta w_{2} \ Z_{2} + \delta \theta_{1} \ M_{1} + \delta \theta_{2} \ M_{2} \end{equation}

Interpolando el campo $w$ (desplazamientos) y $\chi$ (curvatura):

\begin{eqnarray} w &=& N_{1} w_{1} + N_{2} \theta_{1} + N_{3} w_{2} + N_{4} \theta_{2} \\\ \chi &=& \frac{\partial^{2} w}{\partial x^{2}} = \frac{\partial^{2} N_{1}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \end{eqnarray}

Interpolando el campo $\delta w$ (deformaciones virtuales) y $\delta \chi$ (curvatura virtual):

\begin{eqnarray} \delta w &=& N_{1} \delta w_{1} + N_{2} \delta \theta_{1} + N_{3} \delta w_{2} + N_{4} \delta \theta_{2} \\\ \delta \chi &=& \frac{\partial^{2} \delta w}{\partial x^{2}} = \frac{\partial^{2} N_{1}}{\partial x^{2}} \delta w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \delta \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \delta w_{2} + \frac{\partial^{2} N_{4}}{\partial x^{2}} \delta \theta_{2} \end{eqnarray}

Reemplazando:

\begin{equation} \int \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \delta w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \delta \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \delta w_{2} + \frac{\partial^{2} N_{4}}{\partial x^{2}} \delta \theta_{2} \Big ) \ E I \ \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \ dx = \int ( N_{1} \delta w_{1} + N_{2} \delta \theta_{1} + N_{3} \delta w_{2} + N_{4} \delta \theta_{2} ) \ q \ dx + \delta w_{1} \ Z_{1} + \delta w_{2} \ Z_{2} + \delta \theta_{1} \ M_{1} + \delta \theta_{2} \ M_{2} \end{equation}

Expandiendo y agrupando términos:

\begin{equation} \int \Big [ \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{1}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \delta w_{1} + \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \delta \theta_{1} + \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \delta w_{2} + \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{4}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \delta \theta_{2} \Big ] \ E I \ dx = \int ( N_{1} \ q \ \delta w_{1} + N_{2} \ q \ \delta \theta_{1} + N_{3} \ q \ \delta w_{2} + N_{4} \ q \ \delta \theta_{2} ) \ dx + ( \delta w_{1} \ Z_{1} + \delta w_{2} \ Z_{2}) + (\delta \theta_{1} \ M_{1} + \delta \theta_{2} \ M_{2} ） \end{equation}

Las deformaciones virtuales y rotaciones virtuales son arbitrarias, para simplificar $\delta w_{1} = \delta \theta_{1} = \delta w_{2} = \delta \theta_{2} = 1$:

\begin{equation} \int \Big [ \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{1}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) + \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) + \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) + \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{4}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \Big ] \ E I \ dx = \int ( N_{1} \ q + N_{2} \ q + N_{3} \ q + N_{4} \ q ) \ dx + (Z_{1} + Z_{2}) + (M_{1} + M_{2} ） \end{equation}

Representando como un sistema de ecuaciones:

\begin{eqnarray} \int \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{1}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \ E I \ dx &=& \int N_{1} \ q \ dx + Z_{1} + 0 \\\ \int \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \ E I \ dx &=& \int N_{2} \ q \ dx + 0 + M_{1} \\\ \int \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \ E I \ dx &=& \int N_{3} \ q \ dx + Z_{2} + 0 \\\ \int \Big ( \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} w_{1} + \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{1} + \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} w_{2} + \frac{\partial^{2} N_{4}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \theta_{2} \Big ) \ E I \ dx &=& \int N_{4} \ q \ dx + 0 + M_{2} \end{eqnarray}

Representando en forma matricial:

\begin{equation} \int \left [ \begin{matrix} \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{1}}{\partial x^{2}} & \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} & \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} & \frac{\partial^{2} N_{1}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \\\ \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{1}}{\partial x^{2}} & \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} & \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} & \frac{\partial^{2} N_{2}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \\\ \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{1}}{\partial x^{2}} & \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} & \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} & \frac{\partial^{2} N_{3}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \\\ \frac{\partial^{2} N_{4}}{\partial x^{2}} \frac{\partial^{2} N_{1}}{\partial x^{2}} & \frac{\partial^{2} N_{4}}{\partial x^{2}} \frac{\partial^{2} N_{2}}{\partial x^{2}} & \frac{\partial^{2} N_{4}}{\partial x^{2}} \frac{\partial^{2} N_{3}}{\partial x^{2}} & \frac{\partial^{2} N_{4}}{\partial x^{2}} \frac{\partial^{2} N_{4}}{\partial x^{2}} \end{matrix} \right ] E I \left [ \begin{matrix} w_{1} \\\ \theta_{1} \\\ w_{2} \\\ \theta_{2} \end{matrix} \right ] \ dx = \int \left [ \begin{matrix} N_{1} \\\ N_{2} \\\ N_{3} \\\ N_{4} \end{matrix} \right ] q \ dx + \left [ \begin{matrix} Z_{1} \\\ M_{1} \\\ Z_{2} \\\ M_{2} \end{matrix} \right ] \end{equation}

Factorizando:

\begin{equation} \int \left [ \begin{matrix} \frac{\partial^{2} N_{1}}{\partial x^{2}} \\\ \frac{\partial^{2} N_{2}}{\partial x^{2}} \\\ \frac{\partial^{2} N_{3}}{\partial x^{2}} \\\ \frac{\partial^{2} N_{4}}{\partial x^{2}} \end{matrix} \right ] E I \left [ \begin{matrix} \frac{\partial^{2} N_{1}}{\partial x^{2}} & \frac{\partial^{2} N_{2}}{\partial x^{2}} & \frac{\partial^{2} N_{3}}{\partial x^{2}} & \frac{\partial^{2} N_{4}}{\partial x^{2}} \end{matrix} \right ] \left [ \begin{matrix} w_{1} \\\ \theta_{1} \\\ w_{2} \\\ \theta_{2} \end{matrix} \right ] dx = \int \left [ \begin{matrix} N_{1} \\\ N_{2} \\\ N_{3} \\\ N_{4} \end{matrix} \right ] q \ dx + \left [ \begin{matrix} Z_{1} \\\ M_{1} \\\ Z_{2} \\\ M_{2} \end{matrix} \right ] \end{equation}

Representando en forma matricial reducida:

\begin{equation} \int \boldsymbol{B_{f}^{T}} \ E I \ \boldsymbol{B_{f}} \ dx \ \boldsymbol{u} = \int \boldsymbol{N^{T}} \ q \ dx + \boldsymbol{q} \end{equation}

Siendo la matriz constitutiva:

\begin{equation} \boldsymbol{D} = E I \end{equation}

Reemplazando:

\begin{equation} \int \boldsymbol{B_{f}^{T}} \ \boldsymbol{D} \ \boldsymbol{B_{f}} \ dx \ \boldsymbol{u} = \int \boldsymbol{N^{T}} \ q \ dx + \boldsymbol{q} \end{equation}

La anterior ecuación es una generalización para un elemento con cualquier número de nodos, teniendo en cuenta que está en coordenadas globales.

Reescribiendo la anterior expresión en coordenadas naturales:

\begin{equation} \int_{-1}^{+1} \color{Blue} {\boldsymbol{B_{f}^{T}}} \ \boldsymbol{D} \ \color{Blue} { \boldsymbol{B_{f}}} \ J \ d\xi \ \boldsymbol{u} = \int_{-1}^{+1} \color{Blue} {\boldsymbol{N^{T}}} \ q \ J \ d\xi + \boldsymbol{q} \end{equation}

También puede escribirse como:

\begin{equation} \boldsymbol{K} \ \boldsymbol{u} = \boldsymbol{f} + \boldsymbol{q} \end{equation}