Teoría viga Euler-Bernoulli

Campo de desplazamientos

\begin{eqnarray} u(x, y, z) &=& -z \tan \theta (x) = -z \frac{\partial w (x)}{\partial 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^{2} w}{\partial x^{2}} \\\ \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} = 0 \\\ \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 y cortantes $\mathrm{v} = 0$ en $\lambda$ y $G$:

\begin{eqnarray} \sigma_{x} &=& -z \ E(x) \frac{\partial^{2} w}{\partial x^{2}} \\\ \sigma_{y} &=& 0 \\\ \sigma_{z} &=& 0 \\\ \tau_{xy} &=& 0 \\\ \tau_{xz} &=& 0 \\\ \tau_{yz} &=& 0 \end{eqnarray}

Momento flector

\begin{equation} M(x) = - \int_{A} z \ \sigma \ dA \end{equation}

Reemplazando:

\begin{equation} M(x) = \iint z^{2} \ E(x) \frac{\partial^{2} w}{\partial x^{2}} dy \ dz = E(x) \ I(x) \frac{\partial^{2} w}{\partial x^{2}} \end{equation}

Fuerza cortante

\begin{equation} V(x) = \frac{\partial M(x)}{\partial x} \end{equation}

Carga

\begin{eqnarray} q(x) = - \frac{\partial V(x)}{\partial x} \end{eqnarray}

Ecuación diferencial de la viga Euler-Bernoulli

\begin{equation} q(x) = - \frac{\partial^{2}}{\partial x^{2}} \Big [ E(x) \ I(x) \frac{\partial^{2} w}{\partial x^{2}} \Big ] \end{equation}