Terzaghi Rock Load Height

Karl Terzaghi's 1946 chapter in Proctor and White's Rock Tunneling with Steel Supports is the oldest tunnel support method still in daily use, and the reason it survives is that it asks a question a person can picture: how deep is the block of loosened rock that is going to come down onto the support? His answer was a load factor read from a table of rock conditions, multiplied by the sum of the tunnel's width and its height.

The first thing to understand is that this is a loosening load, not an in-situ stress, and confusing the two is the standard beginner's error. Terzaghi's picture is of a body of already-broken rock detaching above the crown, arching sideways onto the ground beside the tunnel, and settling onto whatever is underneath it. The support catches that body. It does not carry the whole weight of the overburden, and this is why the answer depends on the tunnel's size and not on its depth. Below a few tunnel diameters of cover, going deeper does not increase the rock load at all in this method — a result that surprises people and is entirely deliberate.

The second thing is what tunnel he was looking at. Drill and blast. Steel sets erected some distance behind the face. Timber blocking packed between the sets and the rock, with real gaps in it. Hours or days in which the roof could loosen before anything solid touched it. In that tunnel the loosening is real, the block is real, and Terzaghi's method is a fair description of what the sets are actually holding up.

Now consider a modern tunnel: a face sealed with fibre-reinforced shotcrete within the hour, bolted before the round is fully mucked out, with the support installed while the rock is still interlocked. The whole premise has changed. The job of the support is no longer to catch the loosened block; it is to prevent the block from ever forming. Applying Terzaghi's rock load to a lining designed on that philosophy will typically over-design it by a factor of two to three. That is not a reason to throw the method away — it is still taught, it is still the right tool for steel-set design in blocky ground, and it is still the sensible first estimate in squeezing or running ground where a loosening load genuinely does develop — but it is a reason to know which tunnel you are designing before you use it.

The load factor $k$ is a rating read from a table, and it is an input here. Terzaghi's own table ranges from zero for hard and intact rock, which carries no loosening load at all and only spalls occasionally, up through moderately blocky and seamy, very blocky and seamy, completely crushed, and on into squeezing and swelling ground where the factors are large and depth-dependent. Most of the entries give a range rather than a value, and several depend on whether the tunnel is above or below the water table. Use the range. A single value picked from the middle of a band that spans a factor of two is not more precise than the band; it is just less honest about it.

One thing the method genuinely cannot see. Width and height enter on completely equal terms, so a tall narrow tunnel and a wide flat one with the same $B + H_t$ carry the same rock load. Anyone who has stood under both knows that is wrong: the roof of a wide flat opening is a far harder thing to hold up. That is a consequence of Terzaghi's arched loosening geometry, it is one of the crudest parts of the method, and it is worth remembering before trusting the number on an unusually shaped section.