$1$D pre$-$tensioned string example $(5/10)$

Beam equilibrium equation under moderately large deflection:

$EI{w}_{,}+(N{w}_{,x}{)}_{,x}+q=0$

A $1$D string with negligible bending stiffness, i$.$e$.,EI$~~$0,$ is pre$-$stretched under tension $T.$

If the pre$-$tension dominates the behavior, i$.$e$.,T$ the axial load caused by string

deformation, please use the beam theory under moderately large deflection to answer the

following questions.

$2\mathrm{.}1.($Q$1$ in the above figure$)$ Given the known pre$-$tension $T,$ please write out the relation

between the distributed load $q$ and the string's curvature radius $.$

Hint: for a given curve $w=w(x),$ its curvature

$\frac{1}{(x)}=\frac{w_{,}}{[1+(w_{,x}{)}^2{]}^{\frac{3}{2}}}$~~$w_{,},(why?)$