Sag cable

Sag is the vertical drop of a suspended cable from the support-to-support line to the lowest point of the cable.

As the conductor temperature increases, it thermally expands, and the sag increases for a fixed span between supports. The cable has an initial sag from installation due to its weight and typically on purpose shaped in order to accomodate the possible shrinkage when the temperature drops to values below those during the installation.

We provide three methods to calclate the sag:

Considering a catenary and use a function to numerically solve the catenary parameter $a_{cat}$ from inital length.
If the sag is modest, the cable can be modelled as a parabola. The cable length is approximately $L \approx S + \frac{8\,\mathrm{sag}_c^{2}}{3S}$
The small-slope geometric approximation considers the cable's extra thermal length accommodated by forming a half sine wave between the fixed points (a common first-mode deflected shape). It consider the arc length of a sine wave for small slopes to be $S\approx L_{0}+\frac{1}{2}\int_{0}^{L_{0}} \left(y'(x)\right)^{2}\,dx\qquad$ with $y(x)=h\cdot\sin\left(\frac{\pi\cdot x}{L_{0}}\right)$

Symbol

$sag_c$

Unit

Formulas

$a_{cat} cosh \left(\frac{L_{span}}{2a_{cat}}-1\right)$	catenary
$\sqrt{\frac{3L_{span} \delta{}L_c}{8}}$	small-sag parabolic approximation
$\frac{2L_{span}}{\pi} \sqrt{\chi_c \left(\theta_{max}-\theta_{pull}\right)}$	small-slope half-sine wave approximation

Related

$\chi_c$

Linear thermal expansion coefficient conductor material [1/K]

$\delta{}L_c$

Change length conductor [mm]

$L_{span}$

Span between clamp [mm]

$\pi$

Archimedes' constant $\pi$

$\theta_{max}$

Highest temperature of conductor over a load variation period [°C]

$\theta_{pull}$