Small deflection approximation

Figure 1: Infinitecimal segment of deflected beam
Figure 1: Infinitecimal segment of deflected beam

Curvature

The curvature of a line is given exactly by:

\[ \kappa = \frac{d \phi}{ds} \]

For small \(\phi\) we note that \(ds \approx dx\), therefore:

\[ \kappa \approx \frac{d \phi}{dx} \]

Deflection

\[ \tan \phi = \frac{dv}{dx} \]

For small deflections, \(\tan \phi \approx \phi\), therefore:

\[ \phi \approx \frac{dv}{dx} \]