# Beam Distortion

A beam distortion can be a displacement discontinuity or a rotation discontinuity. For a displacement discontinuity of $v$ at $x$ the rotation is then

$\theta_{0} = \theta_{1} = \frac{v}{l}$

For a rotational discontinuity of $\phi$ at $a$ the rotation is calculated by defining a displacement of $h$ at $a$ then the angle $\alpha$ and end 0 and $\beta$ at end 1 gives the set of equations

$b = l - a$
$\phi = \alpha + \beta$
$\tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{h}{a}$
$\tan\beta = \frac{\sin\beta}{\cos\beta} = \frac{h}{b}$

Substituting for $\beta$ in the last equation gives

$b\sin(\phi - \alpha) = h\cos(\phi - \alpha)$
$b\left( \sin\phi\cos\alpha - \cos\phi\sin\alpha \right) = h\left( \cos\phi\cos\alpha + \sin\phi\sin\alpha \right)$
$b\left( \sin\phi - \cos\phi\frac{\sin\alpha}{\cos\alpha} \right) = h\left( \cos\phi + \sin\phi\frac{\sin\alpha}{\cos\alpha} \right)$

Substituting for terms in $\alpha$

$b\left( a\sin\phi - h\cos\phi \right) = h\left( a\cos\phi + h\sin\phi \right)$
$\sin\phi h^{2} + (a + b)\cos\phi h - ab\sin\phi$

Then the rotation angles at end 0 and 1 are

\begin{aligned} \alpha = \tan^{- 1}\left( \frac{h}{a} \right) \\ \beta = \tan^{- 1}\left( \frac{h}{b} \right) \end{aligned}