Beam theory

Beam

$M(z) = - EI v^{ii}(z)$

By integrating:

$\left\{ \begin{aligned} &v(z) = \frac{1}{EI} \iiint q(z) \,dz^3 + \frac{1}{EI} c_1 \frac{z^3}{6} + \frac{1}{EI} c_2 \frac{z^2}{2} + \frac{1}{EI} c_3 z + \frac{1}{EI} c_4 \\ &v^{I}(z) = \frac{1}{EI} \iint q(z) \,dz^2 + \frac{1}{EI} c_1 \frac{z^2}{2} + \frac{1}{EI} c_2 z + \frac{1}{EI} c_3 \\ &v^{II}(z) = \frac{1}{EI} \int q(z) \,dz + \frac{1}{EI} c_1 z + \frac{1}{EI} c_2 \\ &v^{III}(z) = \frac{1}{EI} q(z) + \frac{1}{EI} c_1 \\ &v^{IV}(z) = \frac{1}{EI} q(z) \\ &w(z) = -\frac{1}{EA} \iint p(z) \,dz^2 + \frac{1}{EA} c_5 z + \frac{1}{EA} c_6 \\ &w^{I}(z) = -\frac{1}{EA} \int p(z) \,dz + \frac{1}{EA} c_5 \\ &w^{II}(z) = -\frac{1}{EA} p(z) \end{aligned} \right. \quad + \text{B.C.}$

Beam theory

Examples

Cantilever beam

Simply supported beam

Load on the extremity

Movable load