… for bending

The differential equation for kinematic relations for bending is based on a displaced the infinitesimal small beam:

leading to :

\[{\varphi} = - \frac{{{{d}}w}}{{{{d}}x}}\]

and

leading to :

\[{\kappa} = \frac{{{{d}}\varphi}}{{{{d}}x}}\]

This is treated in chapter 4.3.1 of the book Engineering Mechanics Volume 2 (Hartsuijker and Welleman, 2007).