This article is the first in a proposed series of articles by Stephen Doran discussing closed form tunnel analysis methods and the background behind them. These days when increasingly sophisticated software programmes are available, closed form methods can provide a sense check on the output and in many cases provide an insight into the underlying assumptions used in the software.

Both the Muir Wood and Curtis methods have their origins in the 1970’s design of the Channel Tunnel Project. John Curtis was senior structural engineer for Mott Hay and Anderson, the firm responsible for the design of the sub-sea section, and Sir Alan Muir Wood was the partner in Sir William Halcrow and Partners, which was responsible for the design of the landward tunnels.

Back in the 1970s, the standard design method used by UK engineers for the design of tunnel linings was based upon a 1961 paper by H. D. Morgan, a partner in Sir William Halcrow and Partners. Unfortunately in discussions with our French colleagues, they pointed out that there was a basic error in Morgan’s paper, in that the elastic solutions presented assumed plane strain conditions rather than plane stress. This realisation caused both design teams to rapidly redevelop their design methodologies, leading to the two publications in Geotechnique in 1975.

It is important to understand the underlying assumptions implicit in the elastic method:

? First, it is assumed that the tunnels are circular. Both methods integrate an Airy stress function in polar coordinates to an infinite boundary.

? Second, the material being excavated behaves elastically. In practice this assumption is only likely to be valid for an excavation in a massive rock at shallow depth where stresses do not exceed the elastic limit. For shielded excavations in clays, stresses are likely to create plastic conditions causing face ‘toothpasting’ and squeezing into the tail void, with consequent strain softening, squeezing and long-term swelling. None of these loading mechanisms can be modelled as elastic behaviour.

? Third, it is assumed that the lining is ‘wished in place’ i.e. that the tunnel lining has materialised instantaneously and that the lining has developed its full strength at the instant of excavation. This assumption is obviously conservative and is qualified in both papers.

**MUIR WOOD’S PAPER**

In discussing the Application of the Method, Muir Wood notes that “clearly if a tunnel were inserted in the ground without relaxation of the initial state of stress, then the at rest vertical overburden and horizontal soil pressures should be applied”. Ironically while open-faced shields typical of the 1970s did allow considerable “relaxation of the initial state of stress,” modern closed-face TBMs operating in pressurised face mode and equipped with tail-void grouting systems can come close to achieving this objective. Notwithstanding for tunnels in clay, it was commonly conservatively assumed that due to the combined effects of plastic squeezing, strain softening and swelling, at rest soil pressure conditions would eventually be re-established on the lining over the service life of the tunnel.

Instead, Muir Wood advocated that “The design method described [in his paper] should be applied to changes of loading in the ground and not necessarily directly to the overburden pressures at the level of the tunnel. Consideration of a circular tunnel seen as a hole being drilled through an elastic solid suggests that, at the face, the radial stresses to be applied to the tunnel lining are already only half the intact condition. The value of the loads on the tunnel lining can be derived by considering such an initial condition”.

Muir Wood’s comments acknowledge that even if installed directly at the tunnel face, it is practically impossible to install a tunnel lining without causing a degree of relaxation. In this regard, his comments are consistent with other contemporary European authors working to develop Convergence-Confinement methodology. It is noted that Panet’s elastic curve estimates a λ factor of 0.735 at the tunnel face, i.e. only 26.5% of the elastic movement had occurred at the face not 50% as estimated by Muir Wood.

**LOADING ASSUMPTIONS**

Muir Wood followed the methodology in Morgan’s original paper and studied the deformation of the lining due to the eccentricity of load by assuming “bending in the elliptical mode.”

The reactions from each load case are combined by elastic superposition.

Muir Wood gave two solutions for eliptical bending. First, following Morgan’s assumption of neglecting shear stresses between the lining extrados and the ground which, as acknowledged by Muir Wood, gives conservative results. Second, Muir Wood gives a further analysis considering shear. However, as noted by Curtis, there are problems with the loading conditions assumed for shear in the elliptical bending load case, hence for this reason, Muir Wood’s solution will not be discussed further in this article.

**EFFECT OF SEGMENTATION**

Muir Wood’s paper, equation 47, also gives an approximation for the reduction in bending stiffness of a tunnel lining due to the presence of joints:

Ie = Ij + (4/n)² I . . (47)

where:

Ie is the effective Moment of Inertia of the tunnel lining;

Ij is the Moment of Inertia of the joint:

I is the Moment of Inertia of the segment body; and

n is the number of segments. (n > 4)

It can be seen from Figure 2 that for an unjointed lining (shown in blue) points of contraflexure are formed at the shoulders and knees of the tunnel. These points of contraflexure act as virtual pin joints.

Hence, for a lining with four segments, if the joints are also located at the shoulders and knees Ie = I

However, if joints are located at the crown, axis and invert (shown in magenta) assuming that for an articulated lining where Ij « I , then Ie = I/4 i.e. equivalent to a lining with eight segments.

The foregoing illustrates the importance of joint location to the effective stiffness of a segmental lining. In the author’s opinion, Muir Wood’s equation 47 should at best be regarded as a first approximation. It is suggested that a method that accurately represents the location of the joints should be used in the final analysis of the lining behaviour.

**CURTIS FORMULA**

At this point I should declare an interest, in that during the period 1972-1975, I was a member of John Curtis’s design team. In this article, rather than the notation used by Curtis in his discussion of Muir Wood’s paper, I have used notation and formula used by the design team at this time. This notation is explained within the text and diagrams.

Curtis’s approach differed from the previous Morgan/ Muir Wood analysis of an elliptical bending load case. Instead he analysed uniform and skew-symmetric load cases as illustrated in Figure 3:

Where:

Pu = (Pv + Ph) /2 and Pd = (Pv – Ph )/2

Curtis further resolved the distortional load case into normal and tangential pressures such that:

Sn = – Pd cos 2θ and St = Pd sin 2θ.

The reactions from each load case are combined by elastic superposition.

Where the Elastic Modulus of the lining and soils have been obtained by uniaxial compression testing, for plane strain analysis it is convenient to write: Eg = Eg/(1 + vg) and where the lining is continuous El = El /(1 – vl² )

**UNIFORM LOAD CASE**

The uniform load case produces an axial compression Nu and radial movement Uu in the lining: Nu = Pu re [1/(1 + Eg Q1 )] and radial deflection Uu = –Nu Q1 where Q1 Is a compressibility factor: Q1 = rm/Elh and h is the lining thickness.

**WATER PRESSURE**

Water pressure acting on an impermeable lining is normally treated as a uniform load case applying the pressure at axis and neglecting any variation in pressure over the height of the tunnel. Water pressure is treated as a following load hence: Nw = Pwre and Uw = –Nw Q1

**DISTORTIONAL LOAD CASE**

To simplify the formula, Curtis defined further constants such that: Q2 = Eg rm 3 /12El I and Q3 = [12ElI (5 – 6vg ) + 4Egrm 3] where I is the moment of inertia of the lining per unit length.

The calculation of the lining’s reactions to the distortional load case is dependent upon the shear interaction between the lining and the soil.

It can be seen from Figure 4 that the maximum shear stress occurs at the shoulders and knees of the tunnel i.e. when θ = 45° and 135°. At these points Sn=0 hence the normal stress acting on the lining is Pu. The limiting shear strength of the soil at tunnel depth therfore is: τ = c’ + Pu tan O’ and the ratio to the distortional pressure Γg = τ /Pd

The shear strength required for full interaction between the lining and the ground can be calculated:

The lowest value of Γ is selected, i.e. if the shear strength required for full shear interaction is less than the shear capacity of the ground use Γl else use Γg in the following formula:

Then the axial load due to distortional ground loading can be calculated:

Similarly the radial deflection caused by the distortional ground loading:

And the bending moment caused by distortional ground loading at any point defined by θ:

By superposition: Nθ = Nw + Nu + Ndmax cos 2θ

and Uθ = Uw + Uu + Udmax cos 2θ

**EXTENSION OF THE METHOD – VISCO ELASTIC MODEL**

The subsea section of the Channel Tunnel was excavated largely through the chalk marl. This material was found to exhibit creep properties i.e. the strain in the ground increased under constant load. For the Channel Tunnel, the creep behaviour of various samples was measured in a modified Oedometer test rig.

Various rheological models are available to model creep behaviour and for the Channel Tunnel project a simple Kelvin visco-elastic cell comprising springs and a dashpot was adopted for both the ground and the lining.

In the Kelvin model (Figure 5) the uniaxial stress-strain and time are related by the equation: ε = σ–E [1 + ? (1 – e-γt )] where E is the immediate (elastic) modulus; σ is the applied stress; ? is the ratio of creep strain to immediate elastic strain ? = εc /εe and γ is a time constant.

Hence when t = 0 εe = σ–E and when t = ∞ εtotal = σ–E (1 + ?)

If we assume that the tunnel lining is installed some distance behind the face such that all the elastic ground movement has occurred but prior to any creep strain, then the load developed on the lining over time will only be due to creep in the ground. Assuming that Poisson’s ratio for creep is the same as the instantaneous elastic values, we can further modify the elastic properties of the ground and lining and apply the previously derived elastic formula:

When initially installed, the lining will not be loaded hence both elastic and creep strains will occur within the lining. This can simply be modelled by adjusting the elastic modulus of the lining in the previous equations. As t → ∞ then El → El/[(1 – νl 2) (1 + ?l )] Similarly for the ground, only creep strains will occur, hence as t → ∞ then Eg → Eg/[?g (1 + νg )]

Curtis showed that the creep loads acting upon the lining are similar to the uniform and distortional pressures used in the previous elastic interaction calculations, but multiplied by a ratio λ where:

Hence as time approaches infinity:

The lining reaction due to creep loading may therefore be expressed as:

By superposition:

Nθ = Nw + λNu + λNdmax cos 2θ

Uθ = Uw + λUu + λUdmax cos 2θ and

The visco elastic method therefore provided a simple method of assessing the ground loading on the tunnel linings installed in the chalk marl. If ?g = 1.3 a typical upper bound value, then the ratio ?g/(1 + ?g ) = 0.565 i.e. only 56.5% of the ground loading is applied to the lining.

**CONCLUSIONS**

It should be appreciated that in the early 1970s, finite element modelling was in its infancy and was generally limited to elastic behaviour of simple triangular elements. These early models were complicated and time consuming to set up and run. They offered few advantages over elastic closed form methods such as Muir Wood and Curtis that lent themselves to hand calculations and/or the use of early programmable calculators. Indeed, in some respects, closed form methods could be considered superior as they offered the integration of infinitely small elements to an infinite boundary; something which by definition finite element models could not achieve.

The Muir Wood and Curtis formulas assume elastic behaviour of both the ground and tunnel lining and, as discussed, this assumption is only likely to be the case in a few hard rock conditions. Both papers advocate that the design method is applied to “changes of loading in the ground”, Muir Wood suggesting a simplified 3D assessment of the stress at the tunnel face and Curtis proposing a visco-elastic model.

In soft soils, where the elastic limits are exceeded, to quote Muir Wood: “the three dimensional, time-dependent state of stress around an advancing tunnel face has a complexity beyond the scope of this paper.” It therefore falls to the engineer to form a judgement of the likely ground loadings. In soft soils, the relative stiffness of the lining to resist radial movement is likely to be considerably greater than the ground, hence the problem reduces to the analysis of an assessed ground loading applied to an elastic tunnel lining. Used in this manner, simple elastic models can give reasonably accurate results.

The importance of engineering judgement in assessing ground loading conditions cannot be overstated. As an example, the Ko value of London Clay is typically measured in the range 1.3 to 2.5. Given this observation, applying elastic modelling, it would be expected that the tunnel lining would ovalise. However, such is the complexity of loading that in practice, tunnel linings in London Clay invariably squat. Hence, when using elastic modelling of tunnels in London Clay it is common to adopt a K in the range 0.7 to 0.8 irrespective of the geotechnical data. This apparent conflict can be explained in terms of the load paths undergone by the clay during excavation and this will be discussed further in a future article.

In current practice, closed form solutions lend themselves to a simple spreadsheet type solution and enable rapid assessment of various boundary conditions, enabling early refinement of the design for more detailed analysis. However, closed form elastic methods do have limitations if:

? Soil conditions are not uniform. Layered soils with significantly different properties are difficult to model.

? Other structures, such as an adjacent tunnel, are in close proximity; and

? In rock tunnels where permanent arching effects can be relied upon.

Given these limitations, closed form elastic solutions remain a valuable tool in the tunnel designer’s locker.