Principio de los trabajos virtuales para barras

Para un elemento con dos nodos:

\begin{equation} \iiint \delta \varepsilon \ \sigma \ dV = \int \delta u \ b \ dx + \sum_{i=1}^{2} \delta u_{i} \ X_{i} \end{equation}

Recordando expresiones conocidas:

\begin{eqnarray} \sigma &=& E \ \varepsilon \\\ dV &=& A \ dx \end{eqnarray}

Reemplazando:

\begin{equation} \int \delta \varepsilon \ E \ \varepsilon \ A \ dx = \int \delta u \ b \ dx + \delta u_{1} \ X_{1} + \delta u_{2} \ X_{2} \end{equation}

Interpolando el campo $u$ (deformaciones) y $\varepsilon$ (deformaciones unitarias):

\begin{eqnarray} u &=& N_{1} u_{1} + N_{2} u_{2} \\\ \varepsilon &=& \frac{\partial u}{\partial x} = \frac{\partial N_{1}}{\partial x} u_{1} + \frac{\partial N_{2}}{\partial x} u_{2} \end{eqnarray}

Interpolando el campo $\delta u$ (deformaciones virtuales) y $\delta \varepsilon$ (deformaciones unitarias virtuales):

\begin{eqnarray} \delta u &=& N_{1} \delta u_{1} + N_{2} \delta u_{2} \\\ \delta \varepsilon &=& \frac{\partial \delta u}{\partial x} = \frac{\partial N_{1}}{\partial x} \delta u_{1} + \frac{\partial N_{2}}{\partial x} \delta u_{2} \end{eqnarray}

Reemplazando:

\begin{equation} \int \Big ( \frac{\partial N_{1}}{\partial x} \delta u_{1} + \frac{\partial N_{2}}{\partial x} \delta u_{2} \Big ) \ A E \ \Big ( \frac{\partial N_{1}}{\partial x} u_{1} + \frac{\partial N_{2}}{\partial x} u_{2} \Big ) \ dx = \int (N_{1} \delta u_{1} + N_{2} \delta u_{2}) \ b \ dx + \delta u_{1} \ X_{1} + \delta u_{2} \ X_{2} \end{equation}

Expandiendo y agrupando términos:

\begin{equation} \int \Big [ \Big ( \frac{\partial N_{1}}{\partial x} \frac{\partial N_{1}}{\partial x} u_{1} + \frac{\partial N_{1}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{2} \Big ) \delta u_{1} + \Big ( \frac{\partial N_{1}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{1} + \frac{\partial N_{2}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{2} \Big ) \delta u_{2} \Big ] \ A E \ dx = \int (N_{1} \delta u_{1} b + N_{2} \delta u_{2} b) \ dx + \delta u_{1} \ X_{1} + \delta u_{2} \ X_{2} \end{equation}

Las deformaciones virtuales son arbitrarias, para simplificar $\delta u_{1} = \delta u_{2} = 1$:

\begin{equation} \int \Big [ \Big ( \frac{\partial N_{1}}{\partial x} \frac{\partial N_{1}}{\partial x} u_{1} + \frac{\partial N_{1}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{2} \Big ) + \Big ( \frac{\partial N_{1}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{1} + \frac{\partial N_{2}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{2} \Big ) \Big ] \ A E \ dx = \int (N_{1} b + N_{2} b) \ dx + (X_{1} + X_{2}) \end{equation}

Representando como un sistema de ecuaciones:

\begin{eqnarray} \int \Big ( \frac{\partial N_{1}}{\partial x} \frac{\partial N_{1}}{\partial x} u_{1} + \frac{\partial N_{1}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{2} \Big ) \ A E \ dx &=& \int N_{1} b \ dx + X_{1} \\\ \int \Big ( \frac{\partial N_{1}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{1} + \frac{\partial N_{2}}{\partial x} \frac{\partial N_{2}}{\partial x} u_{2} \Big ) \ A E \ dx &=& \int N_{2} b \ dx + X_{2} \end{eqnarray}

Representando en forma matricial:

\begin{equation} \int \left [ \begin{matrix} \frac{\partial N_{1}}{\partial x} \frac{\partial N_{1}}{\partial x} & \frac{\partial N_{1}}{\partial x} \frac{\partial N_{2}}{\partial x} \\\ \frac{\partial N_{1}}{\partial x} \frac{\partial N_{2}}{\partial x} & \frac{\partial N_{2}}{\partial x} \frac{\partial N_{2}}{\partial x} \end{matrix} \right ] A E \left [ \begin{matrix} u_{1} \\\ u_{2} \end{matrix} \right ] \ dx = \int \left [ \begin{matrix} N_{1} \\\ N_{2} \end{matrix} \right ] b \ dx + \left [ \begin{matrix} X_{1} \\\ X_{2} \end{matrix} \right ] \end{equation}

Factorizando:

\begin{equation} \int \left [ \begin{matrix} \frac{\partial N_{1}}{\partial x} \\\ \frac{\partial N_{2}}{\partial x} \end{matrix} \right ] A E \left [ \begin{matrix} \frac{\partial N_{1}}{\partial x} & \frac{\partial N_{2}}{\partial x} \end{matrix} \right ] \left [ \begin{matrix} u_{1} \\\ u_{2} \end{matrix} \right ] dx = \int \left [ \begin{matrix} N_{1} \\\ N_{2} \end{matrix} \right ] b \ dx + \left [ \begin{matrix} X_{1} \\\ X_{2} \end{matrix} \right ] \end{equation}

Representando en forma matricial reducida:

\begin{equation} \int \boldsymbol{B} \ A E \ \boldsymbol{B^{T}} \ dx \ \boldsymbol{u} = \int \boldsymbol{N^{T}} \ b \ dx + \boldsymbol{q} \end{equation}

Siendo la matriz constitutiva:

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

Reemplazando:

\begin{equation} \int \boldsymbol{B^{T}} \ \boldsymbol{D} \ \boldsymbol{B} \ dx \ \boldsymbol{u} = \int \boldsymbol{N^{T}} \ b \ 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^{T}}} \ \boldsymbol{D} \ \color{Blue} { \boldsymbol{B}} \ 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}