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... for bending

The differential equation for equilibrium relations for bending is based on the infinitesimal small loaded beam:

leading to :

qz=−dVdx{q_z} = - \frac{{{\rm{d}}V}}{{{\rm{d}}x}}
(1)
V=dMdx{V} = \frac{{{\rm{d}}M}}{{{\rm{d}}x}}
(2)

This is treated in chapter 11.1 of the book Engineering Mechanics Volume 1 Hartsuijker & Welleman, 2006.

References
  1. Hartsuijker, C., & Welleman, J. W. (2006). Engineering mechanics: Volume 1: Equilibrium. In Engineering Mechanics: Volume 1: Equilibrium. Springer Dordrecht. 10.1007/978-1-4020-5483-9
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