> For the complete documentation index, see [llms.txt](https://docs.openbrim.org/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.openbrim.org/technical-background/soil-structure-interaction/torsional-soil-models/hyperbolic-curve-randolph-1981.md).

# 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

***

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


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