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Dynamic deflection duct

The dynamic deflection of a duct can be calculated in principle in the same way as for a cable.

The mechanical forces acting on a duct installed in air can be modelled like for an uniformly loaded beam.
Depending on how the duct is fixed we consider different cases to calculate the maximal deflection at midpoint:

Symbol
$h_{dyn,d}$
Unit
mm
Formulas
$\frac{5\frac{F_{k,max}}{1000} {L_{span}}^4}{384JE_d}$simply supported beam
$\frac{1\frac{F_{k,max}}{1000} {L_{span}}^4}{384JE_d}$fixed-end beam
$\frac{\frac{F_{k,max}}{1000} {L_{span}}^4}{384JE_d} \frac{5+\frac{k_{\theta} L_{span}}{2JE_d}}{1+\frac{k_{\theta} L_{span}}{2JE_d}}$rotational spring
Related
$F_{k,max}$
Maximal mechanical force [kN/m]
$JE_d$
Flexural stiffness duct [kN.mm$^2$]
$L_{span}$
Span between clamp [mm]