Ground Reaction Curve — Convergence-Confinement Tunnel Support Analysis

The number that decides everything: when you install

A ground reaction curve is a two-dimensional object. It knows the tunnel cross-section and nothing about the face. But support does not meet a plane-strain tunnel — it is erected some metres behind an advancing face, and by then the wall has already moved. Where the support curve is anchored on the displacement axis is not a detail; it is the input that moves the answer more than any other, and two engineers with the same ground and the same lining will get different factors of safety if they disagree about it.

That conversion is the job of the longitudinal deformation profile. This tool defaults to Vlachopoulos & Diederichs (2009), which is the one to use, because it is the only common form that responds to the size of the plastic zone. That matters more than it sounds: a squeezing tunnel with a plastic radius five times its own has closed proportionally *less* at the face and continues closing far further back than an elastic profile admits. Using an elastic profile on squeezing ground puts your installation point in the wrong place and flatters the design. Panet (1995) is offered as well, because it is the form embedded in a great deal of published practice and you may need to reproduce a number somebody else calculated — it always concedes a quarter of the closure at the face, whatever the ground is doing.

You can also just type the pre-support closure in directly, which is the right move when you have monitoring data from the drive rather than an assumption.

How to read the figure below

The left panel is the main event: falling curves are ground reaction curves, rising straight lines are support. The small open circle marks the elastic/plastic transition — above it there is no yielded ground at all. The tick on the bottom axis is where the support was installed, and the filled dot with the callout is the equilibrium, spelled out so you do not have to read it off two axes.

The right panel shares the pressure axis and plots the plastic radius. Drop a horizontal line from the equilibrium point in the left panel and read the plastic zone off the right one. This is the number that sets bolt and cable length, and the rule is unchanged since Hoek wrote it down: anchor one to two metres beyond it, in ground that has not yielded.

The results table carries p_cr, the pre-support closure, the final closure, the strain, the mobilised pressure, the factor of safety and the plastic radius per scenario. Strain is the one to watch, and it is defined the way Hoek & Marinos define it — tunnel closure over tunnel diameter, as a percentage. Under 1 % you have few support problems. Between 1 and 2.5 % is minor squeezing. Past 5 % you are into very severe squeezing and face stability becomes its own problem. A comfortable factor of safety on a tunnel that has already closed 6 % is not a safe tunnel, and the strain column is what catches that.

Who this is for

Tunnel engineers doing preliminary support assessment — the stage where you are choosing between a shotcrete thickness and a bolt pattern, or working out whether a length of drive needs a fundamentally different support class. Engineering geologists converting a rock mass characterisation into a first estimate of what the ground will do. Anyone sanity-checking a numerical model, because a closed-form answer you can see the assumptions of is the best available check on a mesh you cannot.

It is also for the very common situation of being handed a GSI and asked what it means for support. You can drive this page straight off σci, GSI, mi and D — the same four numbers the Hoek-Brown envelope page takes — and it will fit the equivalent Mohr-Coulomb parameters internally and print them, so you can check the fit rather than take it on faith.

What this is NOT

It is not a finite-element analysis, and it is not a substitute for detailed support design. The closed-form solution behind it buys its elegance with five assumptions, all of which are printed on the figure itself and none of which is decorative: the opening is circular, the in-situ stress is hydrostatic, the rock mass is homogeneous and isotropic with failure not controlled by discontinuities, behaviour is elastic–perfectly-plastic in plane strain, and the support acts as a uniform pressure on a fully closed ring around the entire perimeter.

Real tunnels violate most of that most of the time. Stress fields are rarely hydrostatic, profiles are rarely circular, support rings are rarely closed — and Hoek, Kaiser & Bawden are explicit that an unclosed ring means "a drastic reduction in the capacity and stiffness" of a lining, so the numbers here are optimistic in exactly the way real construction is not. There is no face, no invert, no excavation sequence, no time dependence, no groundwater, and no bending moment anywhere in this analysis, because a perfectly symmetric problem induces none.

What it is genuinely good for is what parametric screening is always good for: understanding behaviour, comparing options, bracketing a problem, and knowing what to expect before you commission the model that will cost you three weeks. Hoek's own advice on this class of calculation is to use it "to explore behaviour patterns rather than to calculate support characteristics to three decimal places", and that is the right way to hold this page.

Honest limits on the support side

Two support types ship, and both were checked against a published table before they were offered: a closed shotcrete or concrete ring, and ungrouted mechanically-anchored rockbolts. The lining formulae reproduce every entry of Table 9.1 of Hoek, Kaiser & Bawden (1995) — capacity and elastic range, three mixes across five tunnel diameters — to the two figures that table is printed to. The bolt capacity reproduces all four bolt sizes across the same five diameters.

Steel sets are deliberately not offered. The published stiffness needs the flange width, the section second moment of area, the block thickness and width, and the half-angle between blocking points. A tool that invented those five numbers would be making up its own answer and attributing it to Hoek & Brown. Grouted bolts and cables are not offered either, and that one is not for want of a formula: an equivalent-uniform-pressure idealisation genuinely cannot represent load transfer distributed along a grouted bar, which is a limitation RocSupport states about itself in as many words. For grouted reinforcement, take the plastic radius from this analysis and use it to set the anchorage length — that is the number the calculation legitimately gives you.

Everything runs in this browser tab. No ground model, no chainage and no support design is uploaded anywhere.

Common questions

The page says Pro — what can I actually do for free?

Everything except enter your own data. The figure below is the finished tool running live: change the lining thickness, the bolt pattern, the installation distance and the deformation profile, switch the plastic-radius panel and the callouts on and off, edit the caption, and export as SVG or PNG (watermarked, like every free export on this site). What Pro buys is the right to replace the sample scenarios with yours. There is no trial timer, no account, and the demo does not expire.

Which solution is this?

The Duncan Fama (1993) elastic–perfectly-plastic Mohr-Coulomb closed-form solution for a circular tunnel in a hydrostatic stress field — the one RocSupport documents, and the one Hoek presents in chapter 12 of Practical Rock Engineering and in chapter 9 of Hoek, Kaiser & Bawden (1995). Every equation is cited in the figure footer. The implementation is pinned in its test suite against two published worked examples, including all eleven tabulated points of the circular-shaft example in Figure 9.6 of that book and its support-interaction result.

I have a GSI, not a cohesion and friction angle. Can I use this?

Yes, and it is the expected case. Give σci, GSI, mi and D and the equivalent Mohr-Coulomb parameters are fitted internally using the closed-form relations of Hoek, Carranza-Torres & Corkum (2002) — equations 12 and 13, over the confinement range 0 to σ₃max, with σ₃max from that paper's deep-tunnel relation. The fitted c′ and φ′ are printed so you can check them, and the fit reproduces the 2002 paper's own worked example exactly. If you supply c and φ as well, the measured pair wins and the figure says so.

Why is my factor of safety so high when the tunnel has obviously closed too much?

Because they measure different things, and this is the classic trap in convergence-confinement. Install support very late and it picks up almost no load, so the ratio of its capacity to the pressure it carries is enormous — while the tunnel has quietly closed past anything you would accept. The factor of safety is a statement about the SUPPORT; the strain column is the statement about the TUNNEL. Read both, and read the strain first.

The curves do not cross below the support capacity. What does that mean?

That the support runs out of capacity before the ground stops moving. It is reported as a yielded support: the lining holds at its capacity while closure continues, the factor of safety is 1.0 by construction, and the design has no margin at all. You need more capacity, earlier installation, or a support type that can accept deformation on purpose — yielding elements exist precisely for this case, and they are outside what this analysis models.

Can I overlay several scenarios?

Yes, and it is the intended use: one row per ground scenario, one support configuration for the sheet. A drive through three rock mass qualities is three rows and one lining, and the question the figure answers is whether that one lining carries all three. The shipped sample is the simplest version of the same idea — the same ground with and without support, so the two ground curves are literally the same curve and everything that differs between them follows from one decision. Every scenario gets its own dash pattern as well as its own colour, so the figure survives a greyscale printer.

What units does it want?

SI throughout, and nothing else: radius and depth in metres, all stresses and moduli in MPa, friction angles in degrees, displacements reported in millimetres. Give a depth and p₀ is computed as γ·depth with the unit weight from the options panel, printed on the figure; give p₀ directly and that is used instead — which is the right move when you have a stress measurement, or when the horizontal stress governs. Supply both and any disagreement is flagged rather than silently resolved.

How long should the bolts be?

Longer than the plastic radius, which the figure reports at equilibrium in the results table and draws in the right-hand panel. The conventional rule is one to two metres of anchorage in ground outside the yielded zone. If you select rockbolts and the free length does not get past the plastic radius, the figure says so on its face — a bolt anchored inside yielded ground is not supporting anything.