The lining that is stronger for being weaker

Almost every structure in this collection is loaded: something arrives, and the structure’s job is to carry it to the ground. A tunnel lining is in a different situation, and the difference is not a detail.

The ground is already stressed before the tunnel exists — vertically by everything above it, horizontally by $K_0$ times that. Boring a hole removes the material that was carrying those stresses across that space, and the ground closes in. The lining’s only function is to refuse, and the force it ends up carrying is whatever refusing costs.

That makes it an imposed-deformation problem, and imposed-deformation problems invert every intuition brought from load-carrying ones.

Which free body produced the number #

A slice of the ring, one metre long, with the ground’s pressure on the outside and the internal actions on two radial cuts.

The ground stress at the tunnel’s axis is $\sigma_v = \gamma H$ vertically and $\sigma_h = K_0\sigma_v$ horizontally. Split that into two parts, because they do completely different things.

The uniform part, $p_0 = (\sigma_v + \sigma_h)/2$, presses inward equally all round. A ring under uniform external pressure carries it as pure compression, $N = p_0R$, with no bending anywhere — the same free body as the pressure that needs no direction and the same answer with the sign reversed.

The second-harmonic part, $p_2\cos 2\theta$ with $p_2 = (\sigma_v - \sigma_h)/2$, presses in at the crown and invert and out at the springings. That is the part that squashes the circle into an ellipse, and it is the whole of the bending.

Two loadings, two mechanisms, and they do not interact. Which is why one of the two curves on the figure is flat.

The ring, and the ground pushing back #

For the second-harmonic part the ring is not alone. As it deforms it pushes into the ground, and the ground pushes back with a radial stiffness $k \approx E_s/R(1+\nu_s)$ per unit area.

An inextensible ring deforming in its $n = 2$ mode, $w = w_2\cos 2\theta$, has a curvature change of $-3w_2\cos 2\theta/R^2$, so its own restoring pressure is $9EIw_2/R^4$. Balance the applied second harmonic against that plus the ground springs:

$$w_2 = \frac{p_2R^4}{9EI + kR^4}, \qquad M = \frac{3EIw_2}{R^2} = \frac{p_2R^2/3}{1 + kR^4/9EI}$$

and the two limits are the argument.

Stiff lining, soft ground. $kR^4/9EI \to 0$ and $M \to p_2R^2/3$ — the free-ring answer, the whole of the ground’s distortion pressure, with no relief at all.

Flexible lining, stiff ground. The denominator runs away and $M \to 0$. The lining deforms into the shape the ground was going to take anyway, there is no curvature change left, and there is nothing to resist.

Between them the moment rises monotonically with the lining’s own bending stiffness, over the whole six orders of magnitude the figure sweeps. Thickening a lining moves it up that curve.

Thrust, which is the other story entirely #

The hoop thrust comes from the uniform part, and the uniform part does not care how stiff the ring is. So the thrust is $p_0R$ whatever the lining is made of, and every one of the moves that reduces the moment leaves it exactly where it was.

That is the reason the whole scheme works. A lining in ring compression is a very efficient structure: concrete is good in compression, the section is fully utilised, and there is no tension to reinforce for. A lining in bending is a poor one, because a thin ring has a small section modulus and the moment goes straight into tension on one face.

So the design objective is to be in ring compression and not in bending, and the way to get there is to be flexible. Which is exactly backwards from every other structure in this collection, and the reason is that the thrust is a load effect and the moment is an imposed-deformation effect. Built to the wrong length is the general statement of the trade: against a load, stiffness carries; against an imposed deformation, stiffness generates.

The joints, which are the design #

A segmental lining is not a ring. It is six or eight or twelve precast segments bolted together, and each joint is a place where the two faces bear on one another over a reduced contact width and rotate relative to each other under moment.

That reduces the ring’s effective bending stiffness, and by a lot. Muir Wood’s expression puts the effective second moment at $(4/n)^2$ times the segment’s own for $n$ joints, so eight joints leave a quarter of it and twelve leave an eighth. On the ring drawn, eight joints cut the moment to 34% of the solid ring’s and leave the thrust at exactly 100%.

Nobody puts joints in a tunnel lining for that reason — they are there because the segments have to fit through the machine and be erected inside the shield. But the structural consequence is the one that makes the scheme viable, and it has been understood since the 1970s that a jointed ring is a better ring.

The same logic runs the other way at the extreme. A sprayed concrete lining applied in a soft ground is thin and young and low-modulus for the first few hours — which is precisely when the ground is converging fastest — so the most flexible moment of its life coincides with the largest imposed deformation. That is not a defect to be engineered out; it is why the method works.

K₀, which decides more than the depth does #

Follow the two parts through and the design’s sensitivity to the ground turns out to be almost entirely a sensitivity to $K_0$ rather than to depth.

The thrust is $\frac{1}{2}(1+K_0)\gamma H R$ — proportional to depth, and to $(1+K_0)$, which for $K_0$ between 0.5 and 1.5 varies by a factor of 1.7. The moment is proportional to $\frac{1}{2}(1-K_0)\gamma HR^2$ — proportional to depth as well, but to $(1-K_0)$, which over the same range runs from $+0.5$ to $-0.5$ and passes through zero.

At $K_0 = 1$ there is no second harmonic at all, and a lining at any depth carries pure ring compression with no bending whatever. At $K_0 = 0.5$ and at $K_0 = 1.5$ the bending is the same magnitude and opposite in sign — the ring squashes vertically in one and horizontally in the other.

So a designer’s most valuable piece of ground information is not the depth or the unit weight, which are known to a few per cent, but $K_0$, which is known to a factor of two in most ground and which is the term the moment is proportional to. That is a general shape worth noticing: the sensitivity of a design is to the parameter multiplying the governing action, not to the parameter that is largest.

What the ground gives up, and what it takes back #

There is a second half to the imposed-deformation story and it is the part that makes tunnelling possible at all: the ground can carry a great deal of the hole itself.

Bore a circular hole in a stressed elastic medium and nothing needs to be put in it. The stress field reorganises round the opening — the tangential stress at the springing rises to $3\sigma_v - \sigma_h$, and at the crown falls to $3\sigma_h - \sigma_v$, which can be tension — and the ground stands as long as those stresses are inside its strength. Tunnels in rock are frequently unlined for exactly that reason, and a lining in good rock is a surface treatment rather than a structure.

What a lining is for is the cases where that does not work: ground with no tensile strength at the crown, ground whose strength is exceeded and which will converge indefinitely, ground that will ravel or flow, or a settlement limit at the surface that convergence would breach. In every one of those the lining is limiting a movement rather than carrying a weight.

Which gives a way to read the flexibility ratio that is more useful than its formula. It is the ratio of how much the ground would move on its own to how much the lining will let it move, and a large ratio means the lining is a passenger. Most modern linings are deliberately passengers.

Grout, and the day the ring is loaded #

A segmental lining is erected inside the tail of a machine at a diameter smaller than the hole the machine cut, and the annulus between the two is filled with grout. That gap and that grout are where most of the argument’s uncertainty lives.

Until the grout sets, the ring has no ground restraint at all — it is a free ring, at the stiff end of the curve, with only its own bending stiffness to resist whatever the ground has already started doing. It is also being pushed on by the machine’s rams, which apply a large axial force to a ring that is not yet round. The erection condition is the case that governs the segment’s reinforcement in a great many designs, and it has nothing to do with the ground load at all.

Once the grout sets, the ring is bedded, the flexibility ratio becomes what the calculation assumed, and the moment falls to the values on the figure. The design case and the analysis case are therefore days apart, which is the same structure of problem as every prop has its own worst day and has the same remedy: analyse the sequence.

And the grout’s own volume is the settlement control. If the annulus is not filled, the ground converges into it, and the surface settles by an amount proportional to the volume lost — which is measured, on every drive, as a percentage of the excavated volume and is the single number a tunnelling contract is judged on.

Buckling, which is the other way a ring fails #

A ring in compression can buckle, and a lining is a ring in compression by design, so the check has to be made.

It is not the check a column gets. A ring embedded in ground buckles into a lobed shape, and the ground has to move outward wherever the ring moves outward — so the ground’s stiffness enters the critical pressure directly and dominates it. The pressure that needs no direction works the mode numbers out: without restraint the critical pressure is $3EI/R^3$ in the two-lobe mode, and with a ground stiffness the mode number climbs and the critical pressure climbs faster.

The result is that an embedded lining is very hard to buckle. The same flexibility that sheds bending moment would make an unrestrained ring hopeless, and the ground that made the flexibility possible also supplies the restraint. The two arguments are the same argument: a lining is safe from buckling for the same reason it is free of bending, which is that it is not acting alone.

Where it does bite is before the ground is there — a segment ring inside the shield, a lining behind a machine that has over-excavated, a ring not yet grouted. The most dangerous day again — and the ring that has not yet been grouted is the same object as a structure that was never complete, analysed at the one moment it is least like the thing on the drawings.

Where the model stops #

The ring was inextensible. A real lining shortens under thrust as well as distorting, and the compressibility ratio — the analogue of the flexibility ratio for the uniform part — decides how much of the ground’s uniform pressure the lining actually receives rather than allowing the ground to relieve. For a thin lining in stiff ground it is a substantial reduction, and ignoring it is conservative.

The ground was elastic, and full slip was assumed. Whether shear can be transmitted between the lining and the ground changes the answer by tens of per cent and in a direction that depends on the flexibility ratio. Neither full slip nor no slip happens; the truth is a friction problem at an interface nobody can inspect.

The sequence was ignored. A great deal of the ground’s convergence happens before the lining is built, at the face and behind the shield, and the lining only receives what is left. The fraction that has already occurred — the “load reduction factor” — is between 30% and 70% and is the least defensible number in a tunnel calculation. It is also the one the answer is most nearly proportional to.

Water was ignored, and in most ground it is the largest term. Below the water table the lining carries the full hydrostatic pressure as a uniform component — which adds to the thrust, not to the moment, and which is the one part of the load that does not care how flexible the lining is. A drained lining sheds it and takes a seepage problem instead; an undrained one carries it and is thicker for it. That is the same choice a basement is a boat makes about a substructure, with the same arithmetic and the opposite sign.

And the loading was two-dimensional. A tunnel is a tube, and the effects nobody can draw in a ring — the face, a cross-passage, a junction, a shaft — are three-dimensional problems with no plane-strain answer at all.

The generalisation #

The idea to take away is the one the figure states in two curves: separate a load into the part that produces the mechanism being designed for and the part that does not, and check whether each is a load effect or a deformation effect.

Here the split is by harmonic: the mean stress gives thrust, which is a load effect, and the deviator gives bending, which is a deformation effect. Stiffness helps the first and hurts the second, so the design has to know which it is optimising before it can decide whether to add material.

The same decomposition works elsewhere. A frame under gravity plus wind has a load part and a sway part with opposite stiffness preferences. A restrained member under load plus temperature has one term that wants a stiff member and one that punishes it — the movement nobody applied. A bolt group under a direct shear and a torque has one term that rewards more bolts and one that rewards a wider group.

The habit is small and it saves the recurring mistake: before deciding that more material is the answer, ask what put the force there. If the answer is “something arrived”, more material helps. If the answer is “something moved and the structure refused”, more material makes it worse.