Rock Mass Deformation Modulus (Hoek–Diederichs)

Strength decides whether the ground fails. Stiffness decides how much it moves before it does, and for most underground work the movement is what you actually design against — tunnel convergence, the load a lining picks up, the settlement under a dam abutment. That number is the rock mass deformation modulus, and measuring it properly means a plate load test, a dilatometer, or back-analysis of a monitored excavation, all of which are expensive.

Hoek and Diederichs' 2006 paper collected a large database of in-situ measurements from China and Taiwan and fitted a sigmoid through it. The simplified form takes only GSI and the disturbance factor: $E_{rm} = 100000\left[(1 - D/2)/(1 + e^{(75+25D-GSI)/11})\right]$ MPa. Put a jointed mass in beside its own intact core and the reduction is the message — the mass is a fraction as stiff, because most of the deformation happens by joints closing and blocks rotating rather than by the rock itself straining.

The 100 000 MPa in front is not a rock property. It is the top of the sigmoid, approached only at GSI 100 with $D = 0$, which means this expression quietly assumes every intact rock has a modulus near 100 GPa. Strong crystalline rock is in that neighbourhood; a weak sandstone is nowhere near it. That is why Hoek and Diederichs also give a FULL form that takes your measured intact modulus $E_i$ as an input and scales it. If you have $E_i$ from a laboratory test, the full form is the better tool, and this simplified one is the field estimate you use when you do not.

$D$ appears twice here — once shifting the curve sideways through the $25D$ in the exponent, and once as the $(1 - D/2)$ multiplier out front — so it bites harder in this equation than anywhere else in the system. At GSI 50, going from $D = 0$ to $D = 1$ takes the modulus from about 9.3 GPa to about 0.53 GPa. A factor of seventeen and a half, on a parameter chosen by describing how the ground was excavated. Nothing else in rock mechanics is that sensitive to a judgement call, and nothing else is as easy to choose optimistically.

The sigmoid is at its steepest right through the middle of the GSI range where most real rock masses sit, so the GSI sensitivity is worst exactly where it is least welcome. Five points either way is commonly a factor of about 1.5 on the modulus.

Two definitional points that get confused. A DEFORMATION modulus is not a Young's modulus of a material — it includes the irrecoverable part of the movement, so it is not what a plate test measures on the second, unload–reload cycle. And it is not the dynamic modulus a seismic survey returns either: dynamic moduli are measured at tiny strains and high frequencies and come out substantially higher than the static value that governs an excavation. Comparing the two without saying which is which is a common way to make a rock mass look stiffer than it will behave.