Barton Roof Support Pressure (with Jn)

In 1974 Nick Barton, Reidar Lien and Johan Lunde published a classification of rock masses at the Norwegian Geotechnical Institute, built from roughly two hundred tunnel case records. The famous part is the quality index $Q$. The part that gets used on site is what came after it: a set of curves that turn $Q$ into an estimate of how much support pressure a roof will need. This page is one of those curves.

It is worth being precise about what kind of statement that is. There is no free body diagram behind this equation. Nobody drew a wedge, resolved forces and arrived at $2\sqrt{J_n}/(3J_r)$. What happened is that a large number of tunnels were built, they mostly stood up, somebody recorded what had been installed in each of them alongside a quality index, and a line was drawn through the resulting cloud. The equation says tunnels at about this quality have historically needed about this much support. That is a genuinely valuable statement — a fit to two hundred real tunnels beats an elegant theory with no ground in it — and it is not the same statement as "the rock will push on your lining with this force".

Two published forms, and which one you are on. Barton and his colleagues gave the relation twice. Where the mass has fewer than three joint sets, the block size still matters and the expression keeps a $\sqrt{J_n}$ term; that is this page. Where it has three or more, $J_n$ is 9 or above, the block size has saturated and the simplified $2/J_r$ form takes over. The two are not rivals: at $J_n = 9$ exactly, $2\sqrt{9}/3 = 2$, and they give the identical answer. The pair meets cleanly at the boundary rather than jumping, which is a small piece of care in the original paper that is worth noticing.

The leading 2 is 2 kg/cm², not a pure number. The paper was written in the units the Institute worked in in 1974, and one kilogram-force per square centimetre is 98.0665 kPa. So the constant carries a pressure, it does not survive a change of units on its own, and this page converts it explicitly rather than quietly. You will meet the same equation elsewhere with 0.2 MPa in place of 2 kg/cm²; that is a rounding of 196.133 kPa and it runs about 2% high. Given everything else in the method, 2% is nothing — but it should be a known 2% rather than a mystery.

The mistake that costs money is one $Q$ for a whole drive. $Q$ spans six orders of magnitude and it can move two of them across a single fault crossing. It is also, in practice, logged where it is safe and convenient to stand and take the time, which tends to be the competent ground. Carrying a value logged in a good stretch across a kilometre of tunnel is how a support design ends up designed for the wrong rock. Support is designed by domain, mapped face by face, and revised when the ground says so — and the fact that the cube root makes $Q$ errors gentle in this equation is not a licence to be careless about which rock you are describing.

Finally, and this is the site's standing position rather than a comment on Barton: a support pressure is an input to a support decision, never the decision. Nothing here gives a bolt length, a bolt pattern, a shotcrete thickness or a steel set spacing. Those come from the ground support standard the job is built to, from the engineer of record, and from somebody standing in front of the face looking at it.