Tunren : calculation of tunnels using the convergence-confinement method
The convergence-confinement method, popularized by Marc Panet starting from the 1970s, makes several strong assumptions:
- Tunnels are circular.
- The rock mass is homogeneous and isotropic.
- Initial stresses are isotropic.
- The depth is significant enough to neglect the stress gradient.
Assuming that these are verified, the method can be used to obtain an initial estimate of the performance of a type of primary lining structure in the context of a tunnel project. The Tunren tool implements this calculation approach.
Tutorials for this module can be found ici.
Presentation of the tool
This tool does not have a calculation button; calculations are made in real time, provided the data is compatible (non-zero Young's modulus, non-zero radius, etc.). Let's take a look at the different options, presented in the "Data" tab.
Basic assumptions
This first part reminds the user that the assumptions underlying the method are strong: any transgression potentially leads to a loss of the physical meaning of the method.
Calculation options
This second part groups together the general calculation and display options. Let's take a look at the various options available:
- The choice of deconfinement rate at laying \(\lambda_d\) can be dictated by one of the longitudinal profiles established in the literature, or be chosen directly by the user. In the last case, the implicit calculation option is no longer available.
- The user can choose to take into account the influence of the stiffness of the primary lining on the displacement at installation \(u_d\) and on the equilibrium displacement \(u_s\) using an implicit iterative method: the method of Nguyen-Minh & Guo (1993). This method, based on the results of axisymmetric simulations, tends however to overestimate the forces in the pavement for high stiffnesses. Initial results suggest that it can be used in conjunction with the classical method to control the displacement-solution when creep remains limited (\(N_s \leq 3.5\)).
- The long-term calculation option is used to evaluate the change in primary lining pressure during the transition to long-term parameters (creep), by solving a new equilibrium starting from the short-term equilibrium point.
The following options concern the display of the convergence-confinement graph, in particular the quantities corresponding to the different axes.
General data
This part corresponds to the definition of the physical problem: stress in place, radius of the tunnel, minimum distance between face and support \(d_1\) and advance step \(p\). The last two options are not proposed in the case of a deconfinement at arbitrary installation, since the displacement at installation is then entirely determined by this data and by the GRC.
In the opposite case, where a PDL is used, there are two ways of calculating \(d\) :
- From the average of the distances \(d_1\) and \(d_1+p\). This method, although based on an approximation, is the most commonly used.
- From the average displacement between \(d_1\) and \(d_2=d_1+p\). This method consists of calculating the average displacement between the two extreme values of unsupported distance, and deducting from this the corresponding distance to the front for the LDP.
Both options give relatively similar results in common cases.
Ground
The critical option for the ground is the choice of the behaviour law. It should also be noted that all the parameters are doubled if the long-term calculation option is activated.
- In elasticity, only Young's modulus \(E\) and Poisson's ratio \(\nu\) are proposed.
- In Mohr-Coulomb elastoplasticity, cohesion \(c\), friction \(\varphi\) and dilatancy \(\psi\) govern anelastic behaviour.
- In Hoek-Brown elastoplasticity, the GSI, the parameter \(m_i\), the intact strength in simple compression \(\sigma_i\) and the damage \(D\) allow the parameters \(m_b\) and \(\sigma_c\) to be calculated. Note that the exponent \(a\) is taken to be equal to 1/2 (a condition for obtaining a complete analytical formulation).
For Hoek-Brown in the short term, an assistant for choosing the modulus of the rock mass is proposed, based on the three methods proposed by Hoek & Diederichs (2006).
Reinforcement
The choice of reinforcements results in the determination of two parameters for the short and long term: the normal stiffness \(K_{sn}\) which reflects the increase in the force exerted by the support as a function of the orthoradial deformation, and the limit of primary lining structure pressure \(p_{s,max}\). These variables are global, in that they are taken to be equal to the sum of the contributions of each of the components. In particular, all stiffnesses contribute to the overall stiffness as long as the stress is less than the maximum overall pressure. Beyond this threshold, the stress is taken to be equal to \(p_{s,max}\). There is thus no differentiated creep, with weak lining creeps before the others.
In addition to the possibility of choosing manual parameters, four types of primary lining are available:
- Concrete ring
- Metal hangers
- Voussoirs
- Point-anchored bolts
It should be noted that one way of introducing the influence of distributed anchorage bolting would be to modify the GRC by means of finite difference calculations. However, it appears that the retaining action of the bolting is rather weak compared with that of other types of retaining: the primary mission of the bolting is to ensure stability, and consequently the elastoplastic behaviour modelled. This functionality is therefore not included.
Important note for the safety coefficient: In the current version, in accordance with the pre-existing Tunren spreadsheet, the safety coefficient for concrete or steel is less than 1 (contrary to the Eurocode formalism):
Data export
This option allows you to export a backup file, in JSON or XLSX format, which can be reimported from the "General" tab. **In particular, it stores the value of hidden/irrelevant buttons (e.g. a default value of cohesion even if the user chooses to work with Hoek-Brown material).
Users wishing to retrieve relevant inputs from a spreadsheet can select the desired rows in the 'Data Table' tab and download them. The same applies to outputs.
"Report" tab
As with the other tools, the integration of essential components is proposed within the report. There are three of them:
- Convergence-confinement curve
- Data table
- Results table
Technical notice
Assumptions and notations
Assumptions of the convergence-confinement method
Strictly speaking, the convergence-confinement method can only be applied to a specific class of problems, with the following characteristics the following:
-
Circular cylindrical tunnel
-
Stress isotropy (\(K_0 \approx 1\) & \(\Delta \sigma \ll \sigma\))
-
Homogeneous, isotropic mass
As viscoplasticity is not introduced at this stage, all deferred effects are also neglected. also neglected. The presence of the water table is not taken into account.
In addition, many empirical/numerical relationships require :
Notations and parameters used
The axes are configured as follows:
The adopted notations are summarised in the following table:
| Notation | Grandeur |
|---|---|
| \(R\) | Tunnel radius |
| \(R_p\) | unconfined plastic radius \(\lambda\) |
| \(R_{p,\text{max}}\) | Maximum plastic radius (\(\lambda=1\)) |
| \(\sigma_0\) | Initial isotropic stress |
| \(u_R\) | Parietal radial displacement |
| \(\sigma_R\) | Parietal radial stress |
| \(u_0\) | Displacement at the face (unsupported tunnel) |
| \(u_{s0}\) | Displacement at the face (supported tunnel) |
| \(u_d\) | Displacement on installation of the retaining structure (unsupported t.) |
| \(u_{sd}\) | Displacement during installation of the retaining wall (supported tunnel) |
| \(u_\infty\) | Displacement away from the tunnel face (unsupported tunnel) |
| \(u_{s\infty}\) | Displacement away from the tunnel face (supported tunnel) |
| \(u_\text{eq}\) | GRC-SRC equilibrium displacement |
| \(\lambda_e\) | Unconfined elastic limite |
| \(\lambda_d\) | Unconfined when the primary lining is installed |
| \(\lambda_\text{eq}\) | Equilibrium unconfined |
| \(E\), \(G\), \(\nu\) | Mechanical parameters of the mass |
| \(K_\text{sn},K_\text{sf}\) | Normal and flexural stiffness of the retaining structure |
For elastoplastic calculations, the following notations are also introduced are also introduced. In Mohr-Coulomb elastoplasticity :
In Hoek-Brown elastoplasticity :
Finally, \(\psi\) will designate the expansion angle, governing the plastic flow rule.
Principle of the method
Deconfinement concept
Tunnelling is fundamentally a three-dimensional problem, with the presence of the working face influencing the stresses and displacements measured behind it.
In the 1970s, Panet popularised a method in France that made it possible to account for this effect by introducing a fictitious pressure into a plane deformation problem: the deconfinement pressure. The deconfinement rate then corresponds to the proportion of the initial stress that has been deconfined.
Objective
Find the equivalent pressure \(p_d\) which, when applied to the tunnel walls in plane deformations, would restore the radial displacement measured when the primary lining was installed.
\(\rightarrow\) Using the behaviour of the retaining, it is then possible to calculate the equilibrium pressure of the tunnel.
The three CV-CF curves
In its original form, the method combines three curves:
-
LDP (Longitudinal Displacement Profile), which associates the radial displacement distance \(x\) from the working face the radial displacement \(u_R\).
-
GRC (Ground Reaction Curve), which associates a radial displacement the active press exerted by the solid rock.
-
SCC (Support Confinement Curve) or SRC (Support Reaction Curve), which associates the radial displacement from the installation with the support reaction exerted by the retaining structure.
The equilibrium can then be read graphically. The displacement when the retaining structure is obtained from the unsupported distance (\(\varphi\) next paragraph) and the LDP. The SRC is then plotted from this point, and its intersection with the GRC gives the state of equilibrium.
Unsupported distance
When digging a tunnel, two distances characterise the advance mode. The first, which is fairly intuitive, is the excavation step \(p\), which represents the advance of the tunnel between the installation of two retaining elements.
The second is the retaining installation distance \(d_1\), which corresponds to the distance from the retaining wall to the quarry face when the latter is installed.
Panet [Panet, 1995] then defines the unsupported distance as the average of \(d_1\) and \(d_2 = p + d_1\), and it is at this distance that the deconfinement is evaluated when the retaining is laid.
Note
Strictly speaking, it would be better to calculate the average of the shape function between the distances \(d_1\) and \(d_2\). The previous relationship is only valid for a small digging step. The Tunren tool allows the user to adopt this approach.
The differents GRC
Nota
For detailed demonstrations, the reader may usefully refer to the bibliography. In the context of this technical note, we simply present the relationships used by the program.
For all elastoplastic GRCs, the underlying assumption is \(\sigma_{rr} < \sigma_{zz} < \sigma_{\theta\theta}\).
In the case of Plaxis calculations with weight taken into account, a factor \(K_0=1\) ensures this condition.
When the material has zero weight, the only way to impose the out-of-plane stress value is by using an initial field stress procedure.
Elastic material
In linear elasticity, the Lamé solution in displacements is written as [Panet, 1995] :
where, \(\sigma_R = (1-\lambda)\sigma_0\), therefore :
In linear elasticity, the law of convergence of the body (GRC) is therefore affine, and allows the stress associated with each displacement to be obtained directly.
Notion of plastic ring
When a plasticity criterion is associated with the material, beyond a certain deconfinement threshold, two zones appear: A ring encompassing the wall, in which the material has plasticised, surrounded by an elastic medium.
Beyond a threshold deconfinement \(\lambda_e\) obtained using the plasticity criterion, this plastic ring appears from the wall, and the law is no longer affine.
Mohr-Coulomb material
For Mohr-Coulomb material, the criterion can be stated as :
Using the notations already introduced, the yield stress deconfinement can be written as :
The plastic radius, once this deconfinement has been exceeded, is calculated using [Panet, 1995]:
By completely solving the elastoplastic problem, it is possible to obtain the expression for the parietal displacements as a function of \(R_p\). The GRC \(\sigma_R(u_R)\) is then an implicit function of \(\lambda\). [Panet, 1995] :
With :
Note
In the previous version of Tunren marketed by Terrasol, the GRC implemented for Mohr-Coulomb was not this one, but the one obtained by simplification considering zero elastic deformations in the plastic zone.
However, these are not entirely negligible. The current tool is therefore more accurate from this point of view.
For the record, the relationship was :
Hoek-Brown material
In the case where the GSI is greater than 25, it is possible to express the Hoek & Brown criterion via [Carranza-Torres, 1999] :
Attention
Compared with the formulation implemented in Plaxis, we note that the power \(a\) is taken to be independent of the GSI and equal to 1/2. This is the only way to obtain an analytical expression for the displacements [Carranza-Torres, 2004].
Carranza-Torres [Carranzza-Torres, 2000] proposes to normalise this criterion, which makes it possible to write :
Introducing deconfinement rates considerably complicates the expression. It is more direct to reason in terms of internal pressure \(p_i\) and elastic limit \(p_i^\text{cr}\).
Thus, the normalized critical pressure \(P_i^\text{cr}\) can be written as [Carranza-Torres, 2000] :
The plastic radius is easily deduced from equilibrium (differential of a square root) [Carranza-Torres, 2000] :
The displacement expression is given by [Carranza-Torres, 2000] :
The different LDP
Corbetta's principle of similarity
Most of the old LDPs were defined on the basis of elastic simulations. To extend them to plastic situations, François Corbetta [Corbetta, 1990] observed that the longitudinal profiles of plastic convergence can be deduced from the elastic profiles by a simple homothety with centre \(O\).
If \(u_R^\text{el}\) designates the elastic profile, and \(u_R^\text{ep}\) the elastoplastic profile, he proposes to consider that :
Les LDP implémentées ici et concernées par ce principe sont celles de Panet et Corbetta. Les \(u_\infty\) sont déduits de la GRC pour \(\lambda=1\).
LDP statement
In elasticity, we have \(\xi = 1\)
-
Panet low [Panet, 1995] :
\[u_R(x) = \frac{1}{\xi} [\alpha_0 + (1-\alpha_0)a(x)] \frac{\sigma_0R}{2G}\]Where \(\displaystyle a(x) = 1-\left(\frac{mR}{mR+\xi x} \right)^2\)
Panet offers two sets of couples \((\alpha_0,m)\) : \((0.25,0.75)\) et \((0.27,0.84)\).
-
Corbetta law [Corbetta, 1990] :
\[u_R(x) = \frac{1}{\xi} \frac{\sigma_0R}{2G} \left[ 0.29 + 0.71 \times \left(1- e^{-1.5(\xi x/R)^{0.7}} \right) \right]\]Valid for Mohr-Coulomb elasticity and elastoplasticity.
-
Chern et al. law [Chern et al., 1998] :
\[u_R(x) = u_\infty \times \left(1+e^{\frac{-x}{1.1R}} \right)^{-1.7}\]Based on in situ measurements, for elastoplastic behaviour.
-
Unlu and Gercek's law (elastic) [Unlu & Gercek, 2003] :
\[u_R(x) = u_0 + \begin{cases} A_a\left(1-e^{B_ax/R}\right) u_\infty & \text{si $x<0$}\\[5pt] A_b\left(1-\left[\frac{B_b}{B_b+x/R} \right]^2 \right)u_\infty & \text{si $x>0$} \end{cases}\]\[A_a = -0.22\nu-0.19 \text{ ; } B_a = 0.73\nu + 0.81 \text{ ; }\]\[A_b = -0.22\nu+0.81 \text{ ; } B_b = 0.39\nu + 0.65 \text{ ; }\]\[u_0 = (0.22\nu+0.19)u_\infty\] -
Vlachopoulos & Diederichs law [Vlachopoulos & Diederichs, 2009] :
\[u_R(x) = \begin{cases} u_0 e^\frac{x}{R} & \text{, $x<0$} \\[5pt] u_\infty -(u_\infty - u_0) e^{- \frac{3x}{2R_{p,\text{max}}}} & \text{, $x>0$} \\ \end{cases}\]Where \(\displaystyle u_0 = \frac{1}{3} u_\infty e^{-0.15\frac{R_{p,\text{max}}}{R}}\).
The different types of retaining structure
The formulae developed in version 3 of Tunren are used in this section. They are taken from [Panet, 1995], except for the concrete ring, where the thin shell assumption is not necessary. The equations of the MMC easily give the result, we can refer for example to [Labriolle, 2017]. Also, the two quantities of interest are \(K_{sn}\) and \(p_{s,\text{max}}\), to be obtained for the sum of the supports.
In order to be able to use the Minh-Guo relationship, the tool does not is not interested in retaining structures successively installed at different distances from the face.
- Concrete ring
- Point-anchored bolting
- Metal hangers
- Voussoirs
Distributed anchorage bolting is only possible using a finite difference solution (not currently included in Tunren).
Concrete ring
The MMC equations for the hollow cylinder give \(K_b\), Tresca on the inside gives \(p_{s,b}\).
Note that in the tool, the external radius \(R_e\) is assimilated directly to the radius of the tunnel (and \(e = R_e - R_i\)).
The choice of \(E_b\) is left to the user, but it is generally set at 30 GPa for cast concrete and 10 GPa for shotcrete.
Metal hangers
The hangers have modulus \(E_a\), cross-section \(A\), and spacing \(\Delta x\). Let \(f_y\) be the limit of the steel:
Voussoirs
The formula for the stiffness of the voussoirs is adapted from that of the concrete ring, with an equivalent modulus.
\(\alpha\) and \(\beta\) are such that \(\alpha e\) corresponds to the thickness of a joint, and \(\frac{2\pi}{n}\beta\) is the angle corresponding to a joint.
Posing \(R_\pm = R_e + \frac{e}{2} (\pm \alpha - 1)\), we have:
Point-anchored bolting
Information
There is no method for dealing analytically with the case of a mass reinforced by continuously anchored bolts. In the literature, the use of finite difference schemes is proposed to establish a modified GRC (Ground Reaction Curve) in the presence of a bolt. The finite difference scheme has been internally implemented but not connected to Tunren (low influence on stiffness).
The only type of bolting implemented is point-anchored bolting. The stiffness of this retaining structure is calculated using :
- \(e_l\), \(e_t\) are the spacings in the longitudinal and circumferential directions respectively.
- \(Q\) is the load-deformation characteristic of the various parts of the bolt (\(Q = S_b/T_b\)).
We can also calculate the associated maximum support pressure, by noting \(d\) the diameter of the bolt, via :
Implicit methods
The classical confinement convergence method is based on the convergence curve of the unsupported tunnel, often obtained by the authors from numerical simulations. However, it should be noted that the retaining structure behind the front tends to modify the longitudinal convergence profile: the tunnel never actually experiences the maximum displacement associated with \(\lambda=1\).
Various solutions exist in the literature. proposed solutions:
- Nouvelle méthode implicite from Bernaud et Rousset [Bernaud & Rousset, 1992, 1995]. The two authors propose modifying the longitudinal displacement profile as a function of the stiffness of the retaining structure, and then solving the system to obtain the displacement at equilibrium and when the retaining structure is installed.
- La méthode implicite from Nguyen-Minh & Guo [Nguyen-Minh & Guo, 1993, 1998]. These two authors propose the use of a universal relationship between the relative displacements supported and unsupported during the installation of the retaining structure and at equilibrium.
- La nouvelle méthode convergence-confinement from Oke, Vlachopoulos & Diederichs [Vlachopoulos et al., 2018]. This iterative method proposes to determine a new supported LDP (Longitudinal Displacement Profile) based on the maximum supported displacement, approaching it at each step.
The latest method does not seem to produce convincing results, and the fits of certain parameters are very scattered, leaving us doubtful. The Bernaud & Rousset method sometimes gives larger equilibrium displacements than the classical method, probably because of the behaviour law used, which is quite old (1976).
Only the method of Nguyen-Minh & Guo is therefore implemented: in addition to giving results consistent with our numerical simulations, Panet in the re-edition of his book presents it as the implicit method giving the best results.
For information purposes, however, we present Bernaud and Rousset's approach, before focusing on Nguyen-Minh and Guo's method.
Bernaud and Rousset method
The authors retain the GRC and the SRC. Unsupported LDP is an old form proposed by Panet in the 1970s, and the method is based on the following three equations:
The shape function \(a^s\) is derived from the unsupported shape function \(a^0\) through a coefficient \(\alpha\) dependent on the adopted behavior law and the stiffness of the support.
To calculate \(\alpha\), the quantity \(\alpha^* = \alpha R/R_p\) is introduced. After calibrations, noting \(K'_s = K_\text{sn}/E\), the following relationship is proposed:
All that remains is to characterise \(u_0\), so that we can proceed with the calculations. In the elastic case, Bernaud and Rousset propose \(u_0 = 0.27 \times \sigma_0R/2G\).
In elastoplasticity (Mohr-Coulomb or Hoek-Brown), they propose :
The problem can then be reduced to the search for an unknown (\(u_R^\text{eq}\)), obtained for example by dichotomy. The other unknowns are then deduced from this equilibrium datum.
Nguyen-Minh and Guo method
In 1993, Nguyen-Minh and Guo discovered a universal relationship allowing the rigidity of retaining structures to be taken into account.
They thus relate the ratio of displacements at installation \(u_d^* = u_{sd}/u_d\) with the ratio of equilibrium displacements \(u_\infty^* = u_{s\infty}/u_\infty\) :
The solution is then based on this function, and we proceed by dichotomy. The problem can be reduced to the determination of a parameter, using the equation GRC = SRC at equilibrium.
The resolution diagram is shown below.
Wizard for the Hoek-Brown module
Tunren offers a wizard for calculating the modulus of a rock mass, using the formulae proposed by Hoek & Diederichs [Hoek & Diederichs, 2006].
They propose three methods:
- From the intact model :
- From uniaxial compressive strength. Hoek & Diederichs propose to obtain the modulus from the \(\sigma_\text{ci}\) from a proportionality factor MR, called Modulus ratio.
For various rocks (characterised by their nature, texture and mode of formation), the authors propose a median value MR and a relative deviation \(\Delta\)MR for this factor. In Tunren, the user is asked to choose a rock type and a barycentric factor \(t\). Noting \(\text{MR}_\pm=\text{MR}\pm\Delta \text{MR}\), the modulus ratio of the calculation is defined by the relation :
- By a *simplified formula :
Bibliography
Caquot, Kérisel, Traité de mécanique des sols, Gauthier-Villars (1956)
Corbetta, Nouvelles méthodes d'étude des tunnels profonds : Calculs analytiques et numériques, Thèse (1990)
Corbetta, Bernaud, Nguyen Minh, Contribution à la méthode convergence-confinement par le principe de la similitude, Rev. Franç. Géotech. n°54 (1991)
Bernaud, Rousset, La nouvelle méthode implicite pour l'étude du dimensionnement des tunnels, Rev. Franç. Géotech. n°60 (1992)
Panet, Le calcul des tunnels par la méthode convergence-confinement, Presses de l'École nationale des ponts et chaussées (1995)
Bernaud, Benamar, Rousset, La nouvelle méthode implicite pour le calcul des tunnels dans les milieux élastoplastiques et viscoplastiques, Rev. Franç. Géotech. n°68 (1994)
Nguyen Minh, Guo A new approach to convergence confinement method, (1998)
Chern, Shiao, Yu, An empirical safety criterion for tunnel construction, Proceedings of the Regional Symposium on Sedimentary Rock Engineering, Taipei (1998)
Carranza-Torres, Fairhurst, The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion, Int. J. Rock Mech. Min. Sci. (1999)
Carranza-Torres, Fairhurst, Analysis of tunnel support requirements using the Convergence-Confinement method and the Hoek-Brown rock failure criterion, Proceedings of GeoEng2000 (2000)
Unlu, Gercek, Effect of Poisson's ratio on the normalized radial displacements occuring around the face of a circular tunnel, Tunneling and Underground Space Technology 18 (2003)
Carranza-Torres, Elasto-plastic solution of tunnel problems using the generalized form of the Hoek-Brown failure criterion, (2004)
Hoek, Diederichs, Empirical estimation of rock mass modulus, Int. J. Rock Mech. Min. Sci. (2005)
Vlachopoulos, Diederichs, Improved Longitudinal Displacement Profiles for Convergence Confinement Analysis of Deep Tunnels, Rock Mech Rock Engng (2009)
Vlachopoulos, Diederichs, Appropriate Uses and Practical Limitations of 2D Numerical Analysis of Tunnels and Tunnel Support Response, Geotech Geol Eng (2014)
Champagne de Labriolle, Amélioration des méthodes analytiques basées sur des concepts simples pour le dimensionnement des tunnels superficiels et profonds en sol meuble, Rev. Franç. Géotech. (2017)
Oke, Vlachopoulos, Diederichs, Improvement to the Convergence-Confinement Method: Inclusion of Support Installation Proximity and Stiffness, Rock Mechanics and Rock Engineering (2018)
Panet, Sulem, Le calcul des tunnels par la méthode convergence-confinement, Presses des Ponts (2021)