Ecuación diferencial de una viga $$ \begin{equation*} EI \frac{d^{4} v(x)}{d x^{4}} - q(x) = 0 \end{equation*} $$
usando el método de Galerkin $$ \begin{equation*} \int_{0}^{L} R(x) W(x) \ dx = \int_{0}^{L} \bigg( EI \frac{d^{4} v(x)}{d x^{4}} - q(x) \bigg) W(x) \ dx = 0 \end{equation*} $$
puede escribirse como la suma de integrales $$ \begin{equation*} \int_{0}^{L} EI \frac{d^{4} v(x)}{d x^{4}} W(x) \ dx - \int_{0}^{L} q(x) W(x) \ dx = 0 \end{equation*} $$
integrando por partes para reducir el orden de la derivada $$ \begin{align*} \int u \ dv &= uv - \int v \ du \\ \int_{0}^{L} W(x) E I \frac{d^{4} v(x)}{d x^{4}} \ dx &= \bigg( W(x) E I \frac{d^{3} v(x)}{d x^{3}} \bigg) \bigg|_{0}^{L} - \int_{0}^{L} E I \frac{d^{3} v(x)}{d x^{3}} \frac{d W(x)}{d x} \ dx \end{align*} $$
integrando por partes para reducir el orden de la derivada $$ \begin{align*} \int u \ dv &= uv - \int v \ du \\ \int_{0}^{L} \frac{d W(x)}{d x} E I \frac{d^{3} v(x)}{d x^{3}} \ dx &= \bigg( \frac{d W(x)}{d x} E I \frac{d^{2} v(x)}{d x^{2}} \bigg) \bigg|_{0}^{L} - \int_{0}^{L} E I \frac{d^{2} v(x)}{d x^{2}} \frac{d^{2} W(x)}{d x^{2}} \ dx \end{align*} $$
reemplazando $$ \begin{equation*} \bigg( E I \frac{d^{3} v(x)}{d x^{3}} W(x) \bigg) \bigg|_{0}^{L} - \bigg( E I \frac{d^{2} v(x)}{d x^{2}} \frac{d W(x)}{d x} \bigg) \bigg|_{0}^{L} + \int_{0}^{L} E I \frac{d^{2} v(x)}{d x^{2}} \frac{d^{2} W(x)}{d x^{2}} \ dx - \int_{0}^{L} q(x) W(x) \ dx = 0 \end{equation*} $$
reordenando $$ \begin{equation*} \int_{0}^{L} E I \frac{d^{2} v(x)}{d x^{2}} \frac{d^{2} W(x)}{d x^{2}} \ dx = \int_{0}^{L} q(x) W(x) \ dx - \bigg( E I \frac{d^{3} v(x)}{d x^{3}} W(x) - E I \frac{d^{2} v(x)}{d x^{2}} \frac{d W(x)}{d x} \bigg) \bigg|_{0}^{L} \end{equation*} $$
para la solución se usará un elemento de dos nodos $$ \begin{align*} v(x) &= N_{1} v_{1} + N_{2} \theta_{1} + N_{3} v_{2} + N_{4} \theta_{2} \\ \frac{d^{2} v(x)}{d x^{2}} &= \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \\ W(x) &= N_{1} \delta v_{1} + N_{2} \delta \theta_{1} + N_{3} \delta v_{2} + N_{4} \delta \theta_{2} \\ \frac{d W(x)}{d x} &= \frac{d N_{1}}{d x} \delta v_{1} + \frac{d N_{2}}{d x} \delta \theta_{1} + \frac{d N_{3}}{d x} \delta v_{2} + \frac{d N_{4}}{d x} \delta \theta_{2} \\ \frac{d^{2} W(x)}{d x^{2}} &= \frac{d^{2} N_{1}}{d x^{2}} \delta v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \delta \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} \delta v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \delta \theta_{2} \end{align*} $$
reemplazando $$ \begin{equation*} \int_{0}^{L} E I \bigg( \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \bigg( \frac{d^{2} N_{1}}{d x^{2}} \delta v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \delta \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} \delta v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \delta \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) (N_{1} \delta v_{1} + N_{2} \delta \theta_{1} + N_{3} \delta v_{2} + N_{4} \delta \theta_{2}) \ dx - \bigg[ E I \frac{d^{3} v(x)}{d x^{3}} (N_{1} \delta v_{1} + N_{2} \delta \theta_{1} + N_{3} \delta v_{2} + N_{4} \delta \theta_{2}) - E I \frac{d^{2} v(x)}{d x^{2}} \bigg( \frac{d N_{1}}{d x} \delta v_{1} + \frac{d N_{2}}{d x} \delta \theta_{1} + \frac{d N_{3}}{d x} \delta v_{2} + \frac{d N_{4}}{d x} \delta \theta_{2} \bigg) \bigg] \bigg|_{0}^{L} \end{equation*} $$
reemplazando \( E I \frac{d^{3} v(x)}{d x^{3}} = V(x) \) y \( E I \frac{d^{2} v(x)}{d x^{2}} = M(x) \) $$ \begin{equation*} \int_{0}^{L} E I \bigg( \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \bigg( \frac{d^{2} N_{1}}{d x^{2}} \delta v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \delta \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} \delta v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \delta \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) (N_{1} \delta v_{1} + N_{2} \delta \theta_{1} + N_{3} \delta v_{2} + N_{4} \delta \theta_{2}) \ dx - \bigg[ V(x) (N_{1} \delta v_{1} + N_{2} \delta \theta_{1} + N_{3} \delta v_{2} + N_{4} \delta \theta_{2}) - M(x) \bigg( \frac{d N_{1}}{d x} \delta v_{1} + \frac{d N_{2}}{d x} \delta \theta_{1} + \frac{d N_{3}}{d x} \delta v_{2} + \frac{d N_{4}}{d x} \delta \theta_{2} \bigg) \bigg] \bigg|_{0}^{L} \end{equation*} $$
multiplicando $$ \begin{equation*} \int_{0}^{L} E I \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} \delta v_{1} + E I \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} \delta v_{1} + E I \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} \delta v_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \delta v_{1} + E I \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} \delta \theta_{1} + E I \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} \delta \theta_{1} + E I \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} \delta \theta_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \delta \theta_{1} + E I \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} \delta v_{2} + E I \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} \delta v_{2} + E I \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} \delta v_{2} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \delta v_{2} + E I \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} \delta \theta_{2} + E I \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} \delta \theta_{2} + E I \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} \delta \theta_{2} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \delta \theta_{2} \ dx = \int_{0}^{L} q(x) N_{1} \delta v_{1} + q(x) N_{2} \delta \theta_{1} + q(x) N_{3} \delta v_{2} + q(x) N_{4} \delta \theta_{2} \ dx - \bigg( V(x) N_{1} \delta v_{1} + V(x) N_{2} \delta \theta_{1} + V(x) N_{3} \delta v_{2} + V(x) N_{4} \delta \theta_{2} - M(x) \frac{d N_{1}}{d x} \delta v_{1} - M(x) \frac{d N_{2}}{d x} \delta \theta_{1} - M(x) \frac{d N_{3}}{d x} \delta v_{2} - M(x) \frac{d N_{4}}{d x} \delta \theta_{2} \bigg) \bigg|_{0}^{L} \end{equation*} $$
reemplazando los límites de integración en el lado derecho $$ \begin{equation*} \int_{0}^{L} E I \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} \delta v_{1} + E I \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} \delta v_{1} + E I \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} \delta v_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \delta v_{1} + E I \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} \delta \theta_{1} + E I \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} \delta \theta_{1} + E I \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} \delta \theta_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \delta \theta_{1} + E I \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} \delta v_{2} + E I \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} \delta v_{2} + E I \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} \delta v_{2} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \delta v_{2} + E I \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} \delta \theta_{2} + E I \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} \delta \theta_{2} + E I \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} \delta \theta_{2} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \delta \theta_{2} \ dx = \int_{0}^{L} q(x) N_{1} \delta v_{1} + q(x) N_{2} \delta \theta_{1} + q(x) N_{3} \delta v_{2} + q(x) N_{4} \delta \theta_{2} \ dx - \bigg( V(L) N_{1}(L) \delta v_{1} + V(L) N_{2}(L) \delta \theta_{1} + V(L) N_{3}(L) \delta v_{2} + V(L) N_{4}(L) \delta \theta_{2} - M(L) \frac{d N_{1}(L)}{d x} \delta v_{1} - M(L) \frac{d N_{2}(L)}{d x} \delta \theta_{1} - M(L) \frac{d N_{3}(L)}{d x} \delta v_{2} - M(L) \frac{d N_{4}(L)}{d x} \delta \theta_{2} \bigg) + \bigg( V(0) N_{1}(0) \delta v_{1} + V(0) N_{2}(0) \delta \theta_{1} + V(0) N_{3}(0) \delta v_{2} + V(0) N_{4}(0) \delta \theta_{2} - M(0) \frac{d N_{1}(0)}{d x} \delta v_{1} - M(0) \frac{d N_{2}(0)}{d x} \delta \theta_{1} - M(0) \frac{d N_{3}(0)}{d x} \delta v_{2} - M(0) \frac{d N_{4}(0)}{d x} \delta \theta_{2} \bigg) \end{equation*} $$
reordenando y agrupando $$ \begin{equation*} \int_{0}^{L} E I \bigg( \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \delta v_{1} + E I \bigg( \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \delta \theta_{1} + E I \bigg( \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \delta v_{2} + E I \bigg( \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \delta \theta_{2} \ dx = \int_{0}^{L} q(x) N_{1} \delta v_{1} + q(x) N_{2} \delta \theta_{1} + q(x) N_{3} \delta v_{2} + q(x) N_{4} \delta \theta_{2} \ dx + \bigg(- V(L) N_{1}(L) + M(L) \frac{d N_{1}(L)}{d x} + V(0) N_{1}(0) - M(0) \frac{d N_{1}(0)}{d x} \bigg) \delta v_{1} + \bigg(- V(L) N_{2}(L) + M(L) \frac{d N_{2}(L)}{d x} + V(0) N_{2}(0) - M(0) \frac{d N_{2}(0)}{d x} \bigg) \delta \theta_{1} + \bigg(- V(L) N_{3}(L) + M(L) \frac{d N_{3}(L)}{d x} + V(0) N_{3}(0) - M(0) \frac{d N_{3}(0)}{d x} \bigg) \delta v_{2} + \bigg(- V(L) N_{4}(L) + M(L) \frac{d N_{4}(L)}{d x} + V(0) N_{4}(0) - M(0) \frac{d N_{4}(0)}{d x} \bigg) \delta \theta_{2} \end{equation*} $$
formando un sistema $$ \begin{align*} \int_{0}^{L} E I \bigg( \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \delta v_{1} \ dx = \int_{0}^{L} q(x) N_{1} \delta v_{1} \ dx + \bigg(- V(L) N_{1}(L) + M(L) \frac{d N_{1}(L)}{d x} + V(0) N_{1}(0) - M(0) \frac{d N_{1}(0)}{d x} \bigg) \delta v_{1} \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \delta \theta_{1} \ dx = \int_{0}^{L} q(x) N_{2} \delta \theta_{1} \ dx + \bigg(- V(L) N_{2}(L) + M(L) \frac{d N_{2}(L)}{d x} + V(0) N_{2}(0) - M(0) \frac{d N_{2}(0)}{d x} \bigg) \delta \theta_{1} \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \delta v_{2} \ dx = \int_{0}^{L} q(x) N_{3} \delta v_{2} \ dx + \bigg(- V(L) N_{3}(L) + M(L) \frac{d N_{3}(L)}{d x} + V(0) N_{3}(0) - M(0) \frac{d N_{3}(0)}{d x} \bigg) \delta v_{2} \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \delta \theta_{2} \ dx = \int_{0}^{L} q(x) N_{4} \delta \theta_{2} \ dx + \bigg(- V(L) N_{4}(L) + M(L) \frac{d N_{4}(L)}{d x} + V(0) N_{4}(0) - M(0) \frac{d N_{4}(0)}{d x} \bigg) \delta \theta_{2} \end{align*} $$
simplificando constantes $$ \begin{align*} \int_{0}^{L} E I \bigg( \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) N_{1} \ dx + \bigg(- V(L) N_{1}(L) + M(L) \frac{d N_{1}(L)}{d x} + V(0) N_{1}(0) - M(0) \frac{d N_{1}(0)}{d x} \bigg) \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) N_{2} \ dx + \bigg(- V(L) N_{2}(L) + M(L) \frac{d N_{2}(L)}{d x} + V(0) N_{2}(0) - M(0) \frac{d N_{2}(0)}{d x} \bigg) \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) N_{3} \ dx + \bigg(- V(L) N_{3}(L) + M(L) \frac{d N_{3}(L)}{d x} + V(0) N_{3}(0) - M(0) \frac{d N_{3}(0)}{d x} \bigg) \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) N_{4} \ dx + \bigg(- V(L) N_{4}(L) + M(L) \frac{d N_{4}(L)}{d x} + V(0) N_{4}(0) - M(0) \frac{d N_{4}(0)}{d x} \bigg) \end{align*} $$
las funciones de forma tienen los siguientes valores
\begin{matrix} N_{1}(L) = 0 & \frac{N_{1}(L)}{d x} = 0 & N_{1}(0) = 1 & \frac{N_{1}(0)}{d x} = 0 \\ N_{2}(L) = 0 & \frac{N_{2}(L)}{d x} = 0 & N_{2}(0) = 0 & \frac{N_{2}(0)}{d x} = 1 \\ N_{3}(L) = 1 & \frac{N_{3}(L)}{d x} = 0 & N_{3}(0) = 0 & \frac{N_{3}(0)}{d x} = 0 \\ N_{4}(L) = 0 & \frac{N_{4}(L)}{d x} = 1 & N_{4}(0) = 0 & \frac{N_{4}(0)}{d x} = 0 \end{matrix}
reemplazando valores $$ \begin{align*} \int_{0}^{L} E I \bigg( \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) N_{1} \ dx + V(0) \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) N_{2} \ dx - M(0) \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) N_{3} \ dx - V(L) \\ \int_{0}^{L} E I \bigg( \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} v_{1} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} \theta_{1} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} v_{2} + \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \theta_{2} \bigg) \ dx = \int_{0}^{L} q(x) N_{4} \ dx + M(L) \end{align*} $$
en forma matricial $$ \begin{equation*} \int_{0}^{L} E I \begin{bmatrix} \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} & \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} & \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} & \frac{d^{2} N_{1}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \\ \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} & \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} & \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} & \frac{d^{2} N_{2}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \\ \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} & \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} & \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} & \frac{d^{2} N_{3}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \\ \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{1}}{d x^{2}} & \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{2}}{d x^{2}} & \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{3}}{d x^{2}} & \frac{d^{2} N_{4}}{d x^{2}} \frac{d^{2} N_{4}}{d x^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} dx = \int_{0}^{L} q(x) \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \\ N_{4} \end{bmatrix} dx + \begin{bmatrix} V(0) \\ -M(0) \\ -V(L) \\ M(L) \end{bmatrix} \end{equation*} $$
factorizando $$ \begin{equation*} \int_{0}^{L} \begin{bmatrix} \frac{d^{2} N_{1}}{d x^{2}} \\ \frac{d^{2} N_{2}}{d x^{2}} \\ \frac{d^{2} N_{3}}{d x^{2}} \\ \frac{d^{2} N_{4}}{d x^{2}} \end{bmatrix} E I \begin{bmatrix} \frac{d^{2} N_{1}}{d x^{2}} & \frac{d^{2} N_{2}}{d x^{2}} & \frac{d^{2} N_{3}}{d x^{2}} & \frac{d^{2} N_{4}}{d x^{2}} \end{bmatrix} dx \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} = \int_{0}^{L} q(x) \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \\ N_{4} \end{bmatrix} dx + \begin{bmatrix} V(0) \\ -M(0) \\ -V(L) \\ M(L) \end{bmatrix} \end{equation*} $$
reemplazando fuerzas y momentos en los nodos 1 y 2 $$ \begin{equation*} \int_{0}^{L} \begin{bmatrix} \frac{d^{2} N_{1}}{d x^{2}} \\ \frac{d^{2} N_{2}}{d x^{2}} \\ \frac{d^{2} N_{3}}{d x^{2}} \\ \frac{d^{2} N_{4}}{d x^{2}} \end{bmatrix} E I \begin{bmatrix} \frac{d^{2} N_{1}}{d x^{2}} & \frac{d^{2} N_{2}}{d x^{2}} & \frac{d^{2} N_{3}}{d x^{2}} & \frac{d^{2} N_{4}}{d x^{2}} \end{bmatrix} dx \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} = \int_{0}^{L} q(x) \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \\ N_{4} \end{bmatrix} dx + \begin{bmatrix} F_{1} \\ M_{1} \\ F_{2} \\ M_{2} \end{bmatrix} \end{equation*} $$
en forma compacta $$ \begin{equation*} \int_{0}^{L} \mathbf{B}^{\mathrm{T}} \mathbf{D} \ \mathbf{B} \ dx \ \mathbf{v} = \int_{0}^{L} q \ \mathbf{N}^{\mathrm{T}} dx + \mathbf{F} \end{equation*} $$