Before the development of finite element and finite difference software, engineers relied on the use of closed form analytical methods. Subsequently many of these closed-form methods have been incorporated as modules into finite element programmes. In the author’s experience many of these modules are commonly used by the programmer without knowledge of the original context and underlying assumptions. The purpose of this article is to set out the historical development of some common closed-form analytical methods and discuss the limitations of their use.

**ELASTIC ANALYSIS METHODS**

As a preamble it is necessary to state that ‘elasticity’ refers to the property of reversibility of deformation in response to load. Many fresh hard rocks are elastic when tested as laboratory specimens, but on a field scale where the rock mass can contain fissures, fractures, bedding planes and zones of altered rock, most rocks do not exhibit perfect elasticity.

None the less the use of simple elastic analysis methods is often useful to gain an appreciation of areas of stress concentration and areas where the ground is likely to become overstressed. In this regard, it is perhaps useful to study the stress distribution in an unlined circular hole in an elastic medium.

Possibly the best reference is Poulos and Davies. Poulos quotes the elastic solutions developed by Terzaghi and Richart (1952) for the stresses around a tunnel in an infinite mass. For plain strain the formulae quoted are:

Due to uniform vertical Loading Pv:

If the ratio of horizontal to vertical pressure is defined by K, then for K=0 i.e. a uniaxial stress field, the stresses along the vertical and horizontal axis can be calculated as shown in Figure 2.

At the excavation boundary, for an unlined unpressurised tunnel, the radial stress σ_{r} is zero.

Along the horizontal axis, at the excavation boundary, the tangential stress σ_{t} is three times the vertical stress Pv and decreases rapidly to less than 4% greater than Pv at r=4a.

Along the vertical axis through the crown of the tunnel, the tangential stress σ_{t} is negative and comprises a tension equal to the vertical stress Pv. This tensile zone extends to approximately r=1.75a.

Elastic solutions to a homogeneous rock mass stressed within the elastic range offer a reasonably accurate solution. As predicted by elastic theory, tension cracks regularly developed in the crown of the Liverpool Loop and Link tunnels excavated through the Bunter Sandstone in the 1970s. Elastic theory can also be used to predict stress concentrations due to adjacent structures. As an example, for a cross passage excavated within the increased stress field of the main tunnel, by superposition, the material in the area of the collars may be subject to stress concentrations of up to nine times for K=0.

Similarly, in many parts of the world, due to tectonic forces, the horizontal stress may be considerably greater than the overburden pressure.

In this case, very high compressive stresses may be generated in the crown and invert of the excavation.

Figure 3 illustrates the stress concentrations that occur at the excavation boundary in the crown and axis of the tunnel as a function of K.

In situations where the stress concentrations at the excavation boundary do not exceed the unconfined compressive strength (UCS) of the rock mass, then the excavation is inherently stable and could, in theory, be left unlined.

However, in a jointed rock mass it is necessary to prevent mechanistic failure of unstable wedges and blocks of rock and traditionally this was achieved by timbering, steel supports or rock bolts.

**ELASTIC-PLASTIC ANALYTICAL METHODS**

The question therefore arises as to what happens if the stress concentration at the excavation boundary exceeds the unconfined compressive strength of the rock mass and how are the resultant conditions analysed? This was the problem exercising the minds of European tunnellers in the 1950s and 1960s, particularly our Austrian colleagues who were digging tunnels through the Alps under high overburden.

Goodman (1989) suggests that when the tangential stress around an opening is greater than about 50% of the UCS, cracks begin to form that tend to create slabs parallel to the periphery of the opening. There is usually also some rock breakage due to construction and this, in combination with the cracking, forms a ‘zone of relaxation’ around the excavation. Commonly the support experienced a radial pressure known as ‘squeeze’.

Many authors of the period hypothesised that this zone of relaxation could be modelled as the development of a plastic zone in the rock around the excavation. This hypothesis was perhaps more to do with the convenience of the mathematics rather than providing an accurate rheological model. In fact, few if any rocks behave as mild steel and exhibit the classic elasto-plastic yielding behaviour associated with steel. The UCS of most rocks is controlled by crushing and/or shearing. Plastic behaviour being limited to a few low-induration rock types with a significant clay content. In the extreme case of hard rocks at great depth, such as a South African gold mine, stress concentrations around the excavation can cause rock burst where brittle fracturing causes slabs of rock to violently spall into the excavation.

In order to further simplify the elasto-plastic analysis, it was common to assume K=1 i.e. that hydrostatic stress conditions existed with Pv=Ph. As can be seen from Figure 3, in an elastic medium a stress concentration of two times exists at all points around the excavation boundary.

It was found from the analysis that if the rock in the plastic zone is allowed to squeeze into the excavation in a controlled manner, then load is transferred into more remote areas, where with increasing σr confinement, it is able to sustain the increased stress. See Figure 4.

Studies were also made of the three-dimensional stress changes around an advancing tunnel face with a view to determining the convergence and virtual support pressures that occurred at various distances from the tunnel face. Studies by Panet and Guenot (1982) have yielded curves showing the relaxation percentage versus distance from the tunnel face for different conditions of plastification. The analysis assumed a purely cohesive material under hydrostatic pressure conditions i.e. K=1.

Parameter Ns shown on the curves is the Stability Number defined as:

in which Pv is the initial stress, pi is the internal pressure assumed zero in this analysis, and Cu is the undrained cohesion of the material. Note that for a uniaxial stress field, K=1 hence the tangential stress at the excavation boundary σ_{t}=2P v and for Ø=0 then UCS=2C_{u} A plastic zone will develop if σ_{t}>UCS i.e. when Ns>1 and conditions will remain elastic for Ns<1.

The residual radial movement and hence the virtual support pressure that will be applied to a lining installed at any point behind the tunnel face can be estimated as: p_{i}=(1-λ) P_{v}

It can be seen from Figure 5 that the relaxation percentage at the working face varies over a large range depending on the value of Ns. It ranges from 0.27 with Ns =1 to 0.6 with Ns=3. With increasing values of Ns a greater percentage of ground movement and consequent losses occur in advance of the face. Other work by Broms and Bennermark (1967) demonstrated that a vertical tunnel face became unstable at Ns≥6.

The ground-support interaction curve Figure 6 shows the rock/support interactions against radial deformations and provides a tool to idealise support stiffness and time of installation.

Initially, as the excavation advances, the deformation is elastic, but after point B the loads around the perimeter of the excavation exceed the UCS of the rock and a plastic zone starts to develop. After point B, ground behaviour becomes non-linear. When a stiff support (shown as ‘1’) is installed it will carry a larger load because the rock mass around the opening has not deformed enough to allow the plastic zone to fully develop.

If the support shown at ‘2’ is installed after a certain displacement has occurred, the system reaches equilibrium with the rock mass providing a greater percentage of the supporting structure and resulting in lower support loads with obvious economies.

The support required to limit the deformation of the crown drops to a minimum at point ‘C’ and then begins to rise again. This is because the downwards displacement of the zone of loosened rock in the roof causes additional rock to become loose, and the weight of this additional loose rock must be added to the required support pressure.

It can be seen from Figure 6 that the required strength, stiffness and time of installation can be sensitive to variations in the properties and behaviour of the rock mass.

In order to account for such variability, it became necessary to carefully instrument and monitor the convergence of the excavation to ensure stability.

In practice, load and support predictions were commonly used as an initial prediction in a predictor– corrector observational method. Shotcrete thickness and rock-bolt numbers being varied depending on the monitoring results, with additional support being provided if the monitoring showed slow convergence to a stable condition.

It can be seen from the preceding discussion how the mathematics related to the development of plastic zones around the excavation led to the essential elements of the so-called New Austrian Tunnelling Method (NATM) as set out in the following bullet points:

? Utilisation of the ground itself to form a major element of the supporting system.

? The use of a thin shotcrete shell to provide an internal radial support pressure ‘pi’ to stabilise the plastic zone.

? Application and closure of the shotcrete primary lining to reach equilibrium after the optimum deformation of the rock mass.

? The possible use of additional support elements such as rock bolts, lattice girders, forward spiling etc., to enhance shotcrete strength, control the excavation profile, prevent local mechanistic failures and protect the operatives.

? Careful instrumentation and monitoring of the radial convergence of the excavation and adjustment of the support requirements to ensure convergence to stable conditions.

? The flexibility of the method allowed the excavation portions to be dimensioned to suit the ground conditions and excavation plant. Dividing the excavation face into smaller areas also limited ground losses from in front of the excavation face.

? The use of dual-lining systems. The primary support, comprising predominantly a thin, semi-flexible shotcrete lining which allows the rock to deform in a controlled manner, thus reducing the loads on the secondary final lining.

In rock tunnels, the use of shotcrete and rock bolts offered considerable economic advantages over the then current American Arch Rib Method (AARM), hence NATM-type solutions were increasingly adopted. The brilliance of our Austrian colleagues was to self-brand the method and realise that the same mathematics developed for the study of plastic zones in rock under high overburden could also be applied to the construction of tunnels in soils at shallower depths. The application of rock mechanics to soils does however neglect the effect of pore water pressure changes in clays.

NATM enjoyed increasing usage throughout the 1970s and 1980s both in Europe, Asia and the UK but incurred a disproportionate number of collapses that generally occurred in and around the area of the heading face. On 21 October 1994, a major collapse occurred during the construction of the Heathrow Express (HEX) station tunnels. These tunnels comprised two parallel 9m-diameter platform tunnels constructed on either side of a 9m-diameter concourse tunnel. The London Clay soil pillars between the excavations in the area of the initial collapse were also approximately 9m wide.

Given the geometry of the adjacent tunnels, any plastic zone formed around the excavations would not be able to throw load into more remote regions, as illustrated previously in Figure 4. In fact, based upon an elastic analysis (K=1) illustrated in Figure 8, it is probable that the whole of the soil pillars between the tunnels were in a plastic condition. The stability of the soil pillars therefore relied entirely on the confining σ_{r} pressure provided by the thin shotcrete shells.

Unusually, the HEX collapse did not occur at the face but was initiated by the failure of the thin support shells in the down line platform where it connected to a cross adit connection to the main access shaft. On an elastic analysis, assuming K=1, the further stress concentration around the adit would have been up to 6.22 times the overburden pressure in the area of the adit collar.

The collapse was further exacerbated by ongoing repairs to the invert of the concourse tunnel which were being carried out to lower the profile. The ring was broken at this point for some distance and the capacity of the thin shotcrete shell to support the soil pillars between the excavations was severely compromised in this area. The subsequent HSE investigation also found that the flat invert profile, defective invert construction containing rebound and the simultaneous compensation grouting works carried out to protect surface structures were all contributory factors to the collapse.

Two main conclusions with regard to the application of NATM to soil tunnels can be drawn from the HEX collapse:

? Firstly, as reported by Clayton, instrumentation and monitoring arrays from late September onwards were showing continuing movement without any tendency to converge (see Figure 7). This warning of developing problems seems to have been largely ignored by management and reportedly, danger was only spotted two hours in advance of the collapse with insufficient time to effect remedial measures. Final failure in soils may develop with such rapidity that support systems that rely on observational measures are not appropriate.

? The thin-wall shotcrete primary support lacked robustness. A local failure of the shell in the region of the adit connection led to a loss of support to the soil pillars between the excavations. The failure of these soil pillars resulted in a progressive ‘unzipping’ collapse of the central concourse and both platform tunnels over a considerable length, causing massive ground loss.

In the aftermath of the HEX collapse, the HSE and ICE published special reports that required support systems using sprayed concrete linings (SCL) for soft ground applications to be fully engineered by the designer and then not usually varied. This generally resulted in much heavier shotcrete sections and the ring being closed as soon as possible. Instrumentation is used to monitor performance and thereby validate the design.

**OTHER PLAYERS**

Although the use of plastic design methods during the 1950s and 1960s is closely linked to the Austrians, in fact, at this time, our American colleagues were also expanding their transportation networks and were heavily engaged in digging nuclear bunkers and Minuteman silos. As a result, the United States Government commissioned the University of Illinois to advise on the problems. They produced a series of confidential reports co-authored variously by Peck, Deere and Schmidt. These reports were declassified in the early 1970s and various papers were published by these authors as a spin-off from these studies.

Deere (et al 1969) suggested that for an undrained clay with Ø=0 and K=1, that the radius of the plastic region Rp is given by:

where ‘a’ is the tunnel radius. The results of this equation are plotted in Figure 9 and show that for stability numbers greater than say Ns=3 the depth of the plastic zone around the tunnel begins to increase exponentially. For high values of stability number, the whole of the region around the tunnel effectively becomes plastic and rapid squeezing of the soil into the excavation will occur. At high values of stability number, shielded-type excavation methods should be considered.

Schmitt (1981) presented a stress path analysis for the undrained response of clays in the plastic zone. Again for simplicity, a value of K=1 was assumed. Schmidt’s approach was based upon Henkel’s pore pressure parameter. Figure 10 taken from Schmidt’s paper illustrates the total stresses and pore water pressure changes around a tunnel in clay for a Henkel pore pressure parameter α=0.12. This value is typical for a normal-to-slightly over consolidated clay. It can be seen that a zone of negative pore water pressure is created within the plastic zone and that positive pore water pressure is created around the plastic/elastic boundary and beyond.

With time these pore pressures dissipate, involving the suction of water into the zone of negative Δ_{u} from the zone of positive Δ_{u}. If the water demand of the negative zone is greater than the surplus of the positive zone, water must travel from greater distances and a potential volume increase near the tunnel results. If this volume increase cannot occur because of the incompressibility of the tunnel lining which is now in place, loading of the liner occurs and a new stress equilibrium state occurs.

Figure 10 assumes that the internal pressure pi at the tunnel boundary is zero and that a full plastic zone develops around the tunnel. However if the plastic radius is small, possibly because the inward movement of the soil is arrested by a shield, and little or no swelling occurs, Schmidt suggests that the build-up of lining pressure for a normally consolidated clay probably stops at or below the value of:

p_{l} = u_{o} + P_{v}’ (1 – sin?’) – c’ cos?’

where p_{l} is the pressure on the lining, u_{o} is the original pore water pressure and P_{v}’ is the effective overburden pressure.

However, for a relatively shallow tunnel in a clay that is normally consolidated, or nearly so, it is likely that an undrained shear zone will develop that very nearly reaches the ground surface. In these circumstances, disturbance and reconsolidation leads to a stress state that closely approximates to the original normally consolidated state. In other words, the vertical load on the tunnel liner approaches the full overburden pressure and the horizontal pressure on the sides of the tunnel approaches at least the normally consolidated at-rest pressure.

Schmidt suggests that for a normally consolidated clay, the average final lining pressure will lie in the range:

p_{l} = P_{v}’ (1 – 0.5 sin ?’) + u_{o} and p_{l} = P_{v}’ + u_{o}

Schmidt’s stress path analysis also explains the anomaly identified in the discussion of Muir-Wood and Curtis methods, i.e. why does a tunnel in an over consolidated clay, such as London Clay, with Ko>1 squat rather than ovalize? Consider the example of a tunnel constructed in an over consolidated clay with a Ko=2.5. In this instance, depending on the location of the water table, K in total stress terms might be around 1.8. Assume also that the stability number Ns=0.5. In this case, conditions should remain elastic if K is unity, however according to elastic theory, we can see from Figure 3 that the tangential stress at the axis is σ_{v} =P_{v}(3-K)=1.2P_{v} and at the crown the tangential stress σ_{h} =P_{v}(-1+3K)=4.4σ_{v}

For Ns=0.5, the plastic limit of the clay is approximately 4P_{v} hence conditions initially remain elastic at the axis, but plastic shear failure zones will form in the crown and invert of the tunnel. This overstress triggers the mechanism of swelling, creating increased load in these areas with accompanying squat of the tunnel lining.

The rate of lining pressure build-up has been shown by Peck (1969) to be linear with the logarithm of time both for soft and stiff clays. These mechanisms may continue for years.

**CONCLUSIONS**

The use of simple closed-form elastic and elasto-plastic design methods generally employ numerous approximations to simplify the analysis, and the validity of these assumptions – particularly with regard to K and the assumption of elastic and plastic behaviour – should be carefully considered in any application of the methods. In modern practice, these methods have been largely superseded by finite element models. However, as discussed in this article, closed-form models still provide from first principles, an insight into the behaviour of the rock and by extension soil around the tunnel during excavation.

Where thin shotcrete shell primary support linings are used to support multiple excavations in close proximity, the stress concentrations and interaction between the various structures should be considered at each stage to ensure the overall robustness of the multiple excavations.

Finite element programmes that employ a simple Mohr-Coulomb model to simulate elasto-plastic behaviour will model only the initial plastic squeezing of the soil onto the lining. In clays, they will not simulate the pore pressure changes around the excavation and the subsequent consolidation and swelling processes that will occur. As a consequence, the model will underestimate the final loads on the lining. In particular, as discussed by Schmidt, in over-consolidated clays considerable swelling pressures may develop in the invert, hence SCL profiles with overly flat invert profiles should be avoided.

Also, many 2D finite element programmes permit the simulation of 3D effects by introducing a relaxation factor during the excavation stage. Following Panet’s curves, the model automatically applies a percentage of the stresses released around the excavation boundary to the soil elements before the lining elements are introduced. In this way, the pressure applied to the lining is only (1-λ) times the released stress. Without a detailed study of the stability number and the excavation sequences, this procedure may also lead to an under-estimation of the loads on the lining. Again, when modelling tunnels in clay, the use of a λ-factor is unlikely to model the subsequent consolidation and swelling processes that will occur.