Terzaghi Support Pressure

This is the second half of the 1946 method and it is pure bookkeeping: take the height of loosened rock, multiply by what rock weighs, and you have a pressure a steel set or a lining can be designed against. Nothing in the arithmetic is in doubt. Everything questionable about the answer lives upstream, in the rock load height and in the table its load factor came from.

Turning any support pressure back into a height of rock is the single most useful habit in this subject. A pressure of 164 kPa means nothing to anybody standing at a face. Six and a bit metres of rock means a great deal — it is a quantity a person can look at, compare with the joints they can see, and form a judgement about. It is also the only fair way to compare the 1946 method with the 1974 one, because both end up as some number of metres of rock hanging on the same support, and expressed that way the comparison is immediate. Run a Barton support pressure through this page in reverse and you will usually find the Q-system asking for a couple of metres where Terzaghi asks for six.

It is a uniform vertical pressure, and a tunnel is not uniform. The method produces one number for the crown and says almost nothing about the walls, where a separate horizontal rock load applies and is conventionally taken as some fraction of the vertical one. It also says nothing about where on the crown the load acts. A real roof fall is a wedge defined by particular joints in a particular place; it arrives as a concentrated load on two steel sets, not as a uniform pressure spread over all of them, and the sets that matter are the ones that happen to be under it.

And it is emphatically not the in-situ stress. At 500 m of cover the vertical stress in the ground is on the order of 13 MPa. This pressure is a few hundred kilopascals — two orders of magnitude smaller. They answer different questions. The rock load is what a loosened block weighs; the in-situ stress is what drives squeezing, spalling, rockburst and the convergence of the opening, and it is the number the convergence-confinement pages on this site are built on. Putting one where the other belongs produces answers that are wrong by a hundredfold and look entirely reasonable.

Sizing a lining from this pressure is a separate job, and this site keeps it separate. The thin-cylinder hoop stress relation and Barlow's formula both live on their own pages here, under strength of materials and piping respectively, and they are not repeated in the tunnelling shard because they are the same equations doing the same work. Bring the pressure to them. What none of those pages will do is choose the support: that is a decision made against the ground support standard the job is built to, by the engineer who signs it, with the face in front of them.

A last practical note on the unit weight. Most rock runs 25 to 28 kN/m³, or 160 to 180 lbf/ft³. Coal, weak shales and heavily weathered material can be nearer 20. It is not a number worth agonising over — a 10% error in the unit weight is invisible beside the uncertainty in the load factor — but it is worth entering rather than assuming, because it is the one number on this page that somebody can actually measure.