# Teoría viga Timoshenko

## Campo de desplazamientos

\begin{eqnarray} u(x, y, z) &=& -z \tan \theta (x) \approx -z \ \theta (x) \\\ v(x, y, z) &=& 0 \\\ w(x, y, z) &=& w(x) \end{eqnarray}

## Campo de deformaciones

\begin{eqnarray} \varepsilon_{x}(x, y, z) &=& \frac{\partial u}{\partial x} = -z \frac{\partial \theta}{\partial x} \\\ \varepsilon_{y}(x, y, z) &=& \frac{\partial v}{\partial x} = 0 \\\ \varepsilon_{z}(x, y, z) &=& \frac{\partial w}{\partial x} = 0 \\\ \gamma_{xy}(x, y, z) &=& \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} = 0 \\\ \gamma_{xz}(x, y, z) &=& \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} = -\theta + \frac{\partial w}{\partial x} \\\ \gamma_{yz}(x, y, z) &=& \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} = 0 \end{eqnarray}

## Campo de esfuerzos

\begin{eqnarray} \sigma_{x} &=& \lambda e + 2 G \varepsilon_{x} \\\ \sigma_{y} &=& \lambda e + 2 G \varepsilon_{y} \\\ \sigma_{z} &=& \lambda e + 2 G \varepsilon_{z} \\\ \tau_{xy} &=& G \gamma_{xy} \\\ \tau_{xz} &=& G \gamma_{xz} \\\ \tau_{yz} &=& G \gamma_{yz} \\\ \lambda &=& \frac{\mathrm{v} E}{(1 + \mathrm{v})(1 - 2 \mathrm{v})} \\\ G &=& \frac{E}{2 (1 + \mathrm{v})} \end{eqnarray}

Para los esfuerzos normales $\mathrm{v} = 0$ en $\lambda$ y $G$, para el esfuerzo cortante $\mathrm{v} = 0$ en $\lambda$ y $\mathrm{v} \neq 0$ en $G$:

\begin{eqnarray} \sigma_{x} &=& -z \ E(x) \frac{\partial \theta}{\partial x} \\\ \sigma_{y} &=& 0 \\\ \sigma_{z} &=& 0 \\\ \tau_{xy} &=& 0 \\\ \tau_{xz} &=& G \Big ( \frac{\partial w}{\partial x} -\theta \Big ) \\\ \tau_{yz} &=& 0 \end{eqnarray}

## Momento flector

El momento flector en función de $x$:

$$M(x) = - \int_{A} z \ \sigma \ dA$$

Reemplazando:

$$M(x) = \iint z^{2} \ E(x) \frac{\partial \theta}{\partial x} dy \ dz = E(x) \ I(x) \frac{\partial \theta}{\partial x}$$

## Fuerza cortante

$$V(x) = - \int_{A} \tau_{xz} \ dA = - G A \Big ( \frac{\partial w}{\partial x} -\theta \Big )$$

## Carga

$$q(x) = - \frac{\partial V(x)}{\partial x}$$

## Ecuación diferencial de la viga Timoshenko

$$q(x) = ???$$


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