Singh & Goel — N index¶

Singh and Göel (1999), for application in the field of tunnels, propose computing the Rock Mass Number \(N\) starting from Barton's Q classification, excluding the stress effect (i.e. by setting \(SRF = 1\)):

\[ N = \frac{RQD}{J_n} \cdot \frac{J_r}{J_a} \cdot J_w \]

In other words, \(N\) is the Q index without the \(SRF\) factor: see the Barton page for the full calculation of Q.

Input parameters¶

The indices \(J_n\), \(J_r\), \(J_a\), \(J_w\) are defined in the common tables:

For RQD, its nominal value is taken; if \(RQD < 10\), a value of 10 is assumed anyway (see RQD).

Coefficient A1¶

The parameter \(A_1\) is derived from the uniaxial compressive strength \(S_u\) of the intact rock, determined by a Point Load test, Schmidt hammer, or the standard ISRM procedure: see Uniaxial compressive strength Su.

The app derives \(A_1\) from the continuous equations that interpolate Bieniawski's charts, subdivided by intervals of \(S_u\):

Interval of \(S_u\) (MPa)
\(\leq 44{,}5\)
\(44{,}5 - 93{,}75\)
\(93{,}75 - 140\)
\(140 - 180\)
\(180 - 240\)
\(> 240\)

Above 240 MPa, \(A_1 = 15\) is assumed.

Continuous equations

If you have Point Load or Schmidt hammer test results, it is preferable to derive \(A_1\) from the equations of Bieniawski's charts rather than from the stepped tables: the result is continuous and less subjective.

Coefficient A6 (orientation of the discontinuities)¶

For the orientation of the discontinuities, the correction coefficient \(A_6\) is applied:

Very favourable	Favourable	Fair	Unfavourable	Very unfavourable
0	−2	−5	−10	−12

Results derived from the calculation of N¶

From the value of \(N\), the Rock Condition Rating (Singh-Goel) is obtained:

\[ RCR = 8\,\ln(N) + 30 \]

and from this the corrected and base values of RMR:

\[ RMR_{corretto} = RCR + (A_1 + A_6) \qquad RMR_{base} = RCR + A_1 \]

From Bieniawski's relation \(RMR = 9\,\ln(Q) + 44\), Barton's Q index is obtained:

\[ Q = e^{\frac{RMR - 44}{9}} \]

and therefore the Stress Reduction Factor and the normalised Q index:

\[ SRF = \frac{N}{Q} \qquad Q_c = Q \cdot \frac{\sigma_c}{100} \]

where \(\sigma_c\) is the uniaxial compressive strength of the rock.

Rock mass classes¶

As a function of \(RMR_c\), the rock mass is divided into five classes:

\(RMR_c\)	100–81	80–61	60–41	40–21	≤ 20
Class	I	II	III	IV	V
Description	Very good	Good	Fair	Poor	Very poor

The resulting Q index ranges from 0.001 to 1000 and is subdivided into 9 classes:

Q	Class	Description
0.001 – 0.01	IX	Exceptionally poor
0.01 – 0.1	VIII	Extremely poor
0.1 – 1	VII	Very poor
1 – 4	VI	Poor
4 – 10	V	Fair
10 – 40	IV	Good
40 – 100	III	Very good
100 – 400	II	Extremely good
400 – 1000	I	Exceptionally good

Characteristic parameters¶

From the value of \(RMR_b\) (Bieniawski), the peak characteristic parameters of the rock mass are derived:

\[ c_p\ [\text{kPa}] = 5\,RMR_b \qquad \varphi_p\ [°] = 0{,}5\,RMR_b + 5 \qquad E\ [\text{GPa}] = 1{,}5\,RMR_b - 100 \]

The residual values of cohesion and friction angle are obtained by introducing into the formulas a modified \(RMR_b\) according to \(RMR_b^{res} = RMR_b - 0{,}2\,RMR_b\) (Priest, 1983).

The formula for \(E\) is valid for \(RMR > 50\); for lower values, the expression of Serafim and Pereira (1983) is used:

\[ E\ [\text{GPa}] = 10^{\frac{RMR_b - 10}{40}} \]

Alternatively, as for Barton, the frictional and cohesive components of the rock mass can be extracted from N/Q:

\[ \varphi' = \arctan\!\left(\frac{J_r \cdot J_w}{J_a}\right) \qquad c' = \frac{RQD}{J_n} \cdot \frac{1}{SRF} \cdot \frac{\sigma_c}{100} \]

The bibliographic references are collected on the Bibliography page.

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