# Esfuerzo plano

Ecuaciones de equilibrio interno del esfuerzo plano en función de los desplazamientos

\begin{align*} \frac{E}{1 - \nu^{2}} \bigg[ \frac{\partial^{2} u}{\partial x^{2}} + \nu \frac{\partial^{2} v}{\partial x \ \partial y} + \frac{1 - \nu}{2} \bigg( \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} v}{\partial y \ \partial x} \bigg) \bigg] + F_{x} &= 0 \\ \frac{E}{1 - \nu^{2}} \bigg[ \nu \frac{\partial^{2} u}{\partial y \ \partial x} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{1 - \nu}{2} \bigg( \frac{\partial^{2} u}{\partial x \ \partial y} + \frac{\partial^{2} v}{\partial x^{2}} \bigg) \bigg] + F_{y} &= 0 \end{align*}

Usando el método de Galerkin

\begin{align*} \int_{0}^{t} \int_{0}^{h} \int_{0}^{b} \bigg\{ \frac{E}{1 - \nu^{2}} \bigg[ \frac{\partial^{2} u}{\partial x^{2}} + \nu \frac{\partial^{2} v}{\partial x \ \partial y} + \frac{1 - \nu}{2} \bigg( \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} v}{\partial y \ \partial x} \bigg) \bigg] + F_{x} \bigg\} (N_{1} + N_{2} + N_{3} + N_{4}) \ dx \ dy \ dz &= 0 \\ \int_{0}^{t} \int_{0}^{h} \int_{0}^{b} \bigg\{ \frac{E}{1 - \nu^{2}} \bigg[ \nu \frac{\partial^{2} u}{\partial y \ \partial x} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{1 - \nu}{2} \bigg( \frac{\partial^{2} u}{\partial x \ \partial y} + \frac{\partial^{2} v}{\partial x^{2}} \bigg) \bigg] + F_{y} \bigg\} (N_{1} + N_{2} + N_{3} + N_{4}) \ dx \ dy \ dz &= 0 \end{align*}

El espesor es constante

\begin{align*} t \int_{0}^{h} \int_{0}^{b} \bigg\{ \frac{E}{1 - \nu^{2}} \bigg[ \frac{\partial^{2} u}{\partial x^{2}} + \nu \frac{\partial^{2} v}{\partial x \ \partial y} + \frac{1 - \nu}{2} \bigg( \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} v}{\partial y \ \partial x} \bigg) \bigg] + F_{x} \bigg\} (N_{1} + N_{2} + N_{3} + N_{4}) \ dx \ dy &= 0 \\ t \int_{0}^{h} \int_{0}^{b} \bigg\{ \frac{E}{1 - \nu^{2}} \bigg[ \nu \frac{\partial^{2} u}{\partial y \ \partial x} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{1 - \nu}{2} \bigg( \frac{\partial^{2} u}{\partial x \ \partial y} + \frac{\partial^{2} v}{\partial x^{2}} \bigg) \bigg] + F_{y} \bigg\} (N_{1} + N_{2} + N_{3} + N_{4}) \ dx \ dy &= 0 \end{align*}

Para reducir el tamaño de la ecuación se usara $N_{i}$

\begin{align*} t \int_{0}^{h} \int_{0}^{b} \bigg\{ \frac{E}{1 - \nu^{2}} \bigg[ \frac{\partial^{2} u}{\partial x^{2}} + \nu \frac{\partial^{2} v}{\partial x \ \partial y} + \frac{1 - \nu}{2} \bigg( \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} v}{\partial y \ \partial x} \bigg) \bigg] + F_{x} \bigg\} N_{i} \ dx \ dy &= 0 \\ t \int_{0}^{h} \int_{0}^{b} \bigg\{ \frac{E}{1 - \nu^{2}} \bigg[ \nu \frac{\partial^{2} u}{\partial y \ \partial x} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{1 - \nu}{2} \bigg( \frac{\partial^{2} u}{\partial x \ \partial y} + \frac{\partial^{2} v}{\partial x^{2}} \bigg) \bigg] + F_{y} \bigg\} N_{i} \ dx \ dy &= 0 \end{align*}

Expandiendo términos

\begin{align*} t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} u}{\partial x^{2}} + \nu N_{i} \frac{\partial^{2} v}{\partial x \ \partial y} + \frac{1 - \nu}{2} \bigg( N_{i} \frac{\partial^{2} u}{\partial y^{2}} + N_{i} \frac{\partial^{2} v}{\partial y \ \partial x} \bigg) \ dx \ dy + t \int_{0}^{h} \int_{0}^{b} N_{i} F_{x} \ dx \ dy &= 0 \\ t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} \nu N_{i} \frac{\partial^{2} u}{\partial y \ \partial x} + N_{i} \frac{\partial^{2} v}{\partial y^{2}} + \frac{1 - \nu}{2} \bigg( N_{i} \frac{\partial^{2} u}{\partial x \ \partial y} + N_{i} \frac{\partial^{2} v}{\partial x^{2}} \bigg) \ dx \ dy + t \int_{0}^{h} \int_{0}^{b} N_{i} F_{y} \ dx \ dy &= 0 \end{align*}

Se reducira el orden de las derivadas parciales mediante integración por partes

\begin{equation*} \int \alpha \ d\beta = \alpha \beta - \int \beta \ d\alpha \end{equation*}

Para $u_{xx}$

\begin{equation*} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} u}{\partial x^{2}} \ dx \ dy = \int_{0}^{h} \bigg( \int_{0}^{b} N_{i} \frac{\partial^{2} u}{\partial x^{2}} \ dx \bigg) \ dy \end{equation*}

identificando variables

\begin{matrix} \alpha = N_{i} & d\beta = \frac{\partial^{2} u}{\partial x^{2}} \ dx \\ d\alpha = \frac{\partial N_{i}}{\partial x} \ dx & \beta = \frac{\partial u}{\partial x} \end{matrix}

reemplazando

\begin{equation*} \int ( \alpha \beta - \int \beta \ d\alpha ) \ dy = \int_{0}^{h} \bigg( N_{i} \frac{\partial u}{\partial x} - \int_{0}^{b} \frac{\partial u}{\partial x} \frac{\partial N_{i}}{\partial x} \ dx \bigg) \ dy = \int_{0}^{h} N_{i} \frac{\partial u}{\partial x} \ dy - \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial x} \frac{\partial u}{\partial x} \ dx \ dy \end{equation*}

Para $v_{xy}$

\begin{equation*} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} v}{\partial x \ \partial y} \ dx \ dy = \int_{0}^{h} \bigg( \int_{0}^{b} N_{i} \frac{\partial^{2} v}{\partial x \ \partial y} \ dx \bigg) \ dy \end{equation*}

identificando variables

\begin{matrix} \alpha = N_{i} & d\beta = \frac{\partial^{2} v}{\partial x \ \partial y} \ dx \\ d\alpha = \frac{\partial N_{i}}{\partial x} \ dx & \beta = \frac{\partial v}{\partial y} \end{matrix}

reemplazando

\begin{equation*} \int ( \alpha \beta - \int \beta \ d\alpha ) \ dy = \int_{0}^{h} \bigg( N_{i} \frac{\partial v}{\partial y} - \int_{0}^{b} \frac{\partial v}{\partial y} \frac{\partial N_{i}}{\partial x} \ dx \bigg) \ dy = \int_{0}^{b} N_{i} \frac{\partial v}{\partial y} \ dy - \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial x} \frac{\partial v}{\partial y} \ dx \ dy \end{equation*}

Para $u_{yy}$

\begin{equation*} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} u}{\partial y^{2}} \ dx \ dy = \int_{0}^{b} \bigg( \int_{0}^{h} N_{i} \frac{\partial^{2} u}{\partial y^{2}} \ dy \bigg) \ dx \end{equation*}

identificando variables

\begin{matrix} \alpha = N_{i} & d\beta = \frac{\partial^{2} u}{\partial y^{2}} \ dy \\ d\alpha = \frac{\partial N_{i}}{\partial y} \ dy & \beta = \frac{\partial u}{\partial y} \end{matrix}

reemplazando

\begin{equation*} \int ( \alpha \beta - \int \beta \ d\alpha ) \ dx = \int_{0}^{b} \bigg( N_{i} \frac{\partial u}{\partial y} - \int_{0}^{h} \frac{\partial u}{\partial y} \frac{\partial N_{i}}{\partial y} \ dy \bigg) \ dx = \int_{0}^{b} N_{i} \frac{\partial u}{\partial y} \ dx - \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial y} \frac{\partial u}{\partial y} \ dx \ dy \end{equation*}

Para $v_{yx}$

\begin{equation*} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} v}{\partial y \ \partial x} \ dx \ dy = \int_{0}^{b} \bigg( \int_{0}^{h} N_{i} \frac{\partial^{2} v}{\partial y \ \partial x} \ dy \bigg) \ dx \end{equation*}

identificando variables

\begin{matrix} \alpha = N_{i} & d\beta = \frac{\partial^{2} v}{\partial y \ \partial x} \ dy \\ d\alpha = \frac{\partial N_{i}}{\partial y} \ dy & \beta = \frac{\partial v}{\partial x} \end{matrix}

reemplazando

\begin{equation*} \int ( \alpha \beta - \int \beta \ d\alpha ) \ dx = \int_{0}^{b} \bigg( N_{i} \frac{\partial v}{\partial x} - \int_{0}^{h} \frac{\partial v}{\partial x} \frac{\partial N_{i}}{\partial y} \ dy \bigg) \ dx = \int_{0}^{b} N_{i} \frac{\partial v}{\partial x} \ dx - \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial y} \frac{\partial v}{\partial x} \ dx \ dy \end{equation*}

Para $u_{yx}$

\begin{equation*} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} u}{\partial y \ \partial x} \ dx \ dy = \int_{0}^{b} \bigg( \int_{0}^{h} N_{i} \frac{\partial^{2} u}{\partial y \ \partial x} \ dy \bigg) \ dx \end{equation*}

identificando variables

\begin{matrix} \alpha = N_{i} & d\beta = \frac{\partial^{2} u}{\partial y \ \partial x} \ dy \\ d\alpha = \frac{\partial N_{i}}{\partial y} \ dy & \beta = \frac{\partial u}{\partial x} \end{matrix}

reemplazando

\begin{equation*} \int ( \alpha \beta - \int \beta \ d\alpha ) \ dx = \int_{0}^{b} \bigg( N_{i} \frac{\partial u}{\partial x} - \int_{0}^{h} \frac{\partial u}{\partial x} \frac{\partial N_{i}}{\partial y} \ dy \bigg) \ dx = \int_{0}^{b} N_{i} \frac{\partial u}{\partial x} \ dx - \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial y} \frac{\partial u}{\partial x} \ dx \ dy \end{equation*}

Para $v_{yy}$

\begin{equation*} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} v}{\partial y^{2}} \ dx \ dy = \int_{0}^{b} \bigg( \int_{0}^{h} N_{i} \frac{\partial^{2} v}{\partial y^{2}} \ dy \bigg) \ dx \end{equation*}

identificando variables

\begin{matrix} \alpha = N_{i} & d\beta = \frac{\partial^{2} v}{\partial y^{2}} \ dy \\ d\alpha = \frac{\partial N_{i}}{\partial y} \ dy & \beta = \frac{\partial v}{\partial y} \end{matrix}

reemplazando

\begin{equation*} \int ( \alpha \beta - \int \beta \ d\alpha ) \ dx = \int_{0}^{b} \bigg( N_{i} \frac{\partial v}{\partial y} - \int_{0}^{h} \frac{\partial v}{\partial y} \frac{\partial N_{i}}{\partial y} \ dy \bigg) \ dx = \int_{0}^{b} N_{i} \frac{\partial v}{\partial y} \ dx - \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial y} \frac{\partial v}{\partial y} \ dx \ dy \end{equation*}

Para $u_{xy}$

\begin{equation*} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} u}{\partial x \ \partial y} \ dx \ dy = \int_{0}^{h} \bigg( \int_{0}^{b} N_{i} \frac{\partial^{2} u}{\partial x \ \partial y} \ dx \bigg) \ dy \end{equation*}

identificando variables

\begin{matrix} \alpha = N_{i} & d\beta = \frac{\partial^{2} u}{\partial x \ \partial y} \ dx \\ d\alpha = \frac{\partial N_{i}}{\partial x} \ dx & \beta = \frac{\partial u}{\partial y} \end{matrix}

reemplazando

\begin{equation*} \int ( \alpha \beta - \int \beta \ d\alpha ) \ dy = \int_{0}^{h} \bigg( N_{i} \frac{\partial u}{\partial y} - \int_{0}^{b} \frac{\partial u}{\partial y} \frac{\partial N_{i}}{\partial x} \ dx \bigg) \ dy = \int_{0}^{b} N_{i} \frac{\partial u}{\partial y} \ dy - \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial x} \frac{\partial u}{\partial y} \ dx \ dy \end{equation*}

Para $v_{xx}$

\begin{equation*} \int_{0}^{h} \int_{0}^{b} N_{i} \frac{\partial^{2} v}{\partial x^{2}} \ dx \ dy = \int_{0}^{h} \bigg( \int_{0}^{b} N_{i} \frac{\partial^{2} v}{\partial x^{2}} \ dx \bigg) \ dy \end{equation*}

identificando variables

\begin{matrix} \alpha = N_{i} & d\beta = \frac{\partial^{2} v}{\partial x^{2}} \ dx \\ d\alpha = \frac{\partial N_{i}}{\partial x} \ dx & \beta = \frac{\partial v}{\partial x} \end{matrix}

reemplazando

\begin{equation*} \int ( \alpha \beta - \int \beta \ d\alpha ) \ dy = \int_{0}^{h} \bigg( N_{i} \frac{\partial v}{\partial x} - \int_{0}^{b} \frac{\partial v}{\partial x} \frac{\partial N_{i}}{\partial x} \ dx \bigg) \ dy = \int_{0}^{h} N_{i} \frac{\partial v}{\partial x} \ dy - \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial x} \frac{\partial v}{\partial x} \ dx \ dy \end{equation*}

Reemplazando y reordenando

\begin{align*} t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial x} \frac{\partial u}{\partial x} + \nu \frac{\partial N_{i}}{\partial x} \frac{\partial v}{\partial y} + \frac{1 - \nu}{2} \bigg( \frac{\partial N_{i}}{\partial y} \frac{\partial u}{\partial y} + \frac{\partial N_{i}}{\partial y} \frac{\partial v}{\partial x} \bigg) \ dx \ dy &= t \frac{E}{1 - \nu^{2}} \bigg[ \int_{0}^{h} N_{i} \frac{\partial u}{\partial x} + \nu N_{i} \frac{\partial v}{\partial y} \ dy + \frac{1 - \nu}{2} \int_{0}^{b} N_{i} \frac{\partial u}{\partial y} + N_{i} \frac{\partial v}{\partial x} \ dx \bigg] + t \int_{0}^{h} \int_{0}^{b} N_{i} F_{x} \ dx \ dy \\ t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} \nu \frac{\partial N_{i}}{\partial y} \frac{\partial u}{\partial x} + \frac{\partial N_{i}}{\partial y} \frac{\partial v}{\partial y} + \frac{1 - \nu}{2} \bigg( \frac{\partial N_{i}}{\partial x} \frac{\partial u}{\partial y} + \frac{\partial N_{i}}{\partial x} \frac{\partial v}{\partial x} \bigg) \ dx \ dy &= t \frac{E}{1 - \nu^{2}} \bigg[ \int_{0}^{b} \nu N_{i} \frac{\partial u}{\partial x} + N_{i} \frac{\partial v}{\partial y} \ dx + \frac{1 - \nu}{2} \int_{0}^{h} N_{i} \frac{\partial u}{\partial y} + N_{i} \frac{\partial v}{\partial x} \ dy \bigg] + t \int_{0}^{h} \int_{0}^{b} N_{i} F_{y} \ dx \ dy \end{align*}

El lado derecho es reemplazado por su equivalente nodal $F_{j}$

\begin{align*} t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial x} \frac{\partial u}{\partial x} + \nu \frac{\partial N_{i}}{\partial x} \frac{\partial v}{\partial y} + \frac{1 - \nu}{2} \bigg( \frac{\partial N_{i}}{\partial y} \frac{\partial u}{\partial y} + \frac{\partial N_{i}}{\partial y} \frac{\partial v}{\partial x} \bigg) \ dx \ dy &= F_{jx} \\ t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} \nu \frac{\partial N_{i}}{\partial y} \frac{\partial u}{\partial x} + \frac{\partial N_{i}}{\partial y} \frac{\partial v}{\partial y} + \frac{1 - \nu}{2} \bigg( \frac{\partial N_{i}}{\partial x} \frac{\partial u}{\partial y} + \frac{\partial N_{i}}{\partial x} \frac{\partial v}{\partial x} \bigg) \ dx \ dy &= F_{jy} \end{align*}

El desplazamiento horizontal es

\begin{equation*} u = \begin{bmatrix} N_{1} & N_{2} & N_{3} & N_{4} \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{bmatrix} \end{equation*}

El desplazamiento vertical es

\begin{equation*} v = \begin{bmatrix} N_{1} & N_{2} & N_{3} & N_{4} \end{bmatrix} \begin{bmatrix} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \end{bmatrix} \end{equation*}

Para reducir el tamaño de la ecuación $u = N_{j} u_{j}$ y $v = N_{j} v_{j}$; sus derivadas serán

\begin{equation*} \begin{matrix} \frac{\partial u}{\partial x} = \frac{\partial N_{j}}{\partial x} u_{j} & \frac{\partial u}{\partial y} = \frac{\partial N_{j}}{\partial y} u_{j} \\ \frac{\partial v}{\partial x} = \frac{\partial N_{j}}{\partial x} v_{j} & \frac{\partial v}{\partial y} = \frac{\partial N_{j}}{\partial y} v_{j} \end{matrix} \end{equation*}

Reemplazando

\begin{align*} t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} \frac{\partial N_{i}}{\partial x} \frac{\partial N_{j}}{\partial x} u_{j} + \nu \frac{\partial N_{i}}{\partial x} \frac{\partial N_{j}}{\partial y} v_{j} + \frac{1 - \nu}{2} \bigg( \frac{\partial N_{i}}{\partial y} \frac{\partial N_{j}}{\partial y} u_{j} + \frac{\partial N_{i}}{\partial y} \frac{\partial N_{j}}{\partial x} v_{j} \bigg) \ dx \ dy &= F_{jx} \\ t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} \nu \frac{\partial N_{i}}{\partial y} \frac{\partial N_{j}}{\partial x} u_{j} + \frac{\partial N_{i}}{\partial y} \frac{\partial N_{j}}{\partial y} v_{j} + \frac{1 - \nu}{2} \bigg( \frac{\partial N_{i}}{\partial x} \frac{\partial N_{j}}{\partial y} u_{j} + \frac{\partial N_{i}}{\partial x} \frac{\partial N_{j}}{\partial x} v_{j} \bigg) \ dx \ dy &= F_{jy} \end{align*}

En forma matricial

\begin{equation*} t \frac{E}{1 - \nu^{2}} \int_{0}^{h} \int_{0}^{b} \begin{bmatrix} \frac{\partial N_{i}}{\partial x} \frac{\partial N_{j}}{\partial x} u_{j} + \nu \frac{\partial N_{i}}{\partial x} \frac{\partial N_{j}}{\partial y} v_{j} + \frac{1 - \nu}{2} \bigg( \frac{\partial N_{i}}{\partial y} \frac{\partial N_{j}}{\partial y} u_{j} + \frac{\partial N_{i}}{\partial y} \frac{\partial N_{j}}{\partial x} v_{j} \bigg)\\ \nu \frac{\partial N_{i}}{\partial y} \frac{\partial N_{j}}{\partial x} u_{j} + \frac{\partial N_{i}}{\partial y} \frac{\partial N_{j}}{\partial y} v_{j} + \frac{1 - \nu}{2} \bigg( \frac{\partial N_{i}}{\partial x} \frac{\partial N_{j}}{\partial y} u_{j} + \frac{\partial N_{i}}{\partial x} \frac{\partial N_{j}}{\partial x} v_{j} \bigg) \end{bmatrix} \ dx \ dy = \begin{bmatrix} F_{jx} \\ F_{jy} \end{bmatrix} \end{equation*}

Factorizando y reordenando

\begin{equation*} \int_{0}^{h} \int_{0}^{b} \begin{bmatrix} \frac{\partial N_{i}}{\partial x} & 0 & \frac{\partial N_{i}}{\partial y} \\ 0 & \frac{\partial N_{i}}{\partial y} & \frac{\partial N_{i}}{\partial x} \\ \end{bmatrix} \frac{E}{1 - \nu^{2}} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1 - \nu}{2} \end{bmatrix} \begin{bmatrix} \frac{\partial N_{j}}{\partial x} & 0 \\ 0 & \frac{\partial N_{j}}{\partial y} \\ \frac{\partial N_{j}}{\partial y} & \frac{\partial N_{j}}{\partial x} \end{bmatrix} \ t \ dx \ dy \begin{bmatrix} u_{j} \\ v_{j} \end{bmatrix} = \begin{bmatrix} F_{jx} \\ F_{jy} \end{bmatrix} \end{equation*}

Reemplazando las cuatro funciones de forma

\begin{equation*} \int_{0}^{h} \int_{0}^{b} \begin{bmatrix} \frac{\partial N_{1}}{\partial x} & 0 & \frac{\partial N_{1}}{\partial y} \\ 0 & \frac{\partial N_{1}}{\partial y} & \frac{\partial N_{1}}{\partial x} \\ \frac{\partial N_{2}}{\partial x} & 0 & \frac{\partial N_{2}}{\partial y} \\ 0 & \frac{\partial N_{2}}{\partial y} & \frac{\partial N_{2}}{\partial x} \\ \frac{\partial N_{3}}{\partial x} & 0 & \frac{\partial N_{3}}{\partial y} \\ 0 & \frac{\partial N_{3}}{\partial y} & \frac{\partial N_{3}}{\partial x} \\ \frac{\partial N_{4}}{\partial x} & 0 & \frac{\partial N_{4}}{\partial y} \\ 0 & \frac{\partial N_{4}}{\partial y} & \frac{\partial N_{4}}{\partial x} \end{bmatrix} \frac{E}{1 - \nu^{2}} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1 - \nu}{2} \end{bmatrix} \begin{bmatrix} \frac{\partial N_{1}}{\partial x} & 0 & \frac{\partial N_{2}}{\partial x} & 0 & \frac{\partial N_{3}}{\partial x} & 0 & \frac{\partial N_{4}}{\partial x} & 0 \\ 0 & \frac{\partial N_{1}}{\partial y} & 0 & \frac{\partial N_{2}}{\partial y} & 0 & \frac{\partial N_{3}}{\partial y} & 0 & \frac{\partial N_{4}}{\partial y} \\ \frac{\partial N_{1}}{\partial y} & \frac{\partial N_{1}}{\partial x} & \frac{\partial N_{2}}{\partial y} & \frac{\partial N_{2}}{\partial x} & \frac{\partial N_{3}}{\partial y} & \frac{\partial N_{3}}{\partial x} & \frac{\partial N_{4}}{\partial y} & \frac{\partial N_{4}}{\partial x} \end{bmatrix} \ t \ dx \ dy \begin{bmatrix} u_{1} \\ v_{1} \\ u_{2} \\ v_{2} \\ u_{3} \\ v_{3} \\ u_{4} \\ v_{4} \end{bmatrix} = \begin{bmatrix} F_{1x} \\ F_{1y} \\ F_{2x} \\ F_{2y} \\ F_{3x} \\ F_{3y} \\ F_{4x} \\ F_{4y} \end{bmatrix} \end{equation*}

Puede reescribirse como

\begin{equation*} \int_{0}^{h} \int_{0}^{b} \begin{bmatrix} B_{1} \\ B_{2} \\ B_{3} \\ B_{4} \end{bmatrix} \frac{E}{1 - \nu^{2}} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1 - \nu}{2} \end{bmatrix} \begin{bmatrix} B_{1} & B_{2} & B_{3} & B_{4} \end{bmatrix} \ t \ dx \ dy \begin{bmatrix} d_{1} \\ d_{2} \\ d_{3} \\ d_{4} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} \\ F_{3} \\ F_{4} \end{bmatrix} \end{equation*}

y en forma compacta

\begin{equation*} \iint \limits_{A} \mathbf{B}^{\mathrm{T}} \ \mathbf{D} \ \mathbf{B} \ t \ dA \ \mathbf{d} = \mathbf{F} \end{equation*}

o en forma general

\begin{equation*} \mathbf{K} \ \mathbf{d} = \mathbf{F} \end{equation*}