… for torsion#
The differential equation for equilibrium relations for bending is based on the infinitesimal small loaded torsional bar:
Fig. 18 Free body diagram of an infinitesimal small loaded torsional bar#
Equilibrium of moments around the longitudinal axis results in the following differential equation:
\[\begin{split}
\begin{align*}
\sum T_{\rm{element}} &= 0 \\
- M_{\rm{t}} + q_{\rm{M}_{\rm{t}}} \cdot \Delta x + M_{\rm{t}} + \Delta M_{\rm{t}} &= 0 \\
q_{\rm{M}_{\rm{t}}} \cdot \Delta x + \Delta M_{\rm{t}} &= 0 \\
\mathop {\lim }\limits_{\Delta x \to 0 } \left( \cfrac{\Delta M_{\rm{t}}}{\Delta x} \right) &= \mathop {\lim }\limits_{\Delta x \to 0 } \left( - q_{\rm{M}_{\rm{t}}} \right) \\
\cfrac{dM_{\rm{t}}}{dx} &= - q_{\rm{M}_{\rm{t}}}
\end{align*}
\end{split}\]