Hyperbolic Curve [Randolph, 1981]

The hyperbolic relationship between torque and the pile's angle of twist is illustrated in the formula below.

$$\tag{1} T = \frac{2Gt_u\theta}{t_u+2G\theta}$$

where $G$ = shear modulus of soil $t_u$ = ultimate shear stress $\theta$ = angle of twist

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[Randolph, 1981] Randolph, M. F. (1981). Piles subjected to torsion. Journal of the Geotechnical Engineering Division, 107(8):1095–1111.