Stabfront: stability of the working face using limit analysis

The Stabfront calculation tool in Orbow allows for the assessment of the stability of tunnel faces in a multi-layered frictional-cohesive environment and in a purely cohesive single-layer environment (multi-layer is accepted, but in this case, only the lower layer is considered resistant, and the others are treated as simple overloads exerting pressure on the roof).

In a cohesive-frictional environment, the tool relies on the method of slices, combined with a wide range of empirical lateral pressure coefficients from the literature. An extensive finite element study was conducted in 2020 to automate the selection of the coefficient set based on the soil type (friction angle) and the choice of stress calculation method (Terzaghi's silo solution or earth pressure).

In a purely cohesive environment, the Davis solution (1980) is implemented and compared to the one obtained by the "cohesive-frictional" calculation method. The pressure required for stability corresponds to the minimum of the two approaches.

Consideration of water tables (drained calculation without flow) is also provided.

Tutorials for this module can be found ici.

Presentation of the tool

This tool has a calculation button in the tab of the same name: the results are determined/updated only when the calculation is initiated using this button.

Data

The inputs are divided into four categories:

General Data, which corresponds to the parameters defining the tunnel project, the calculation mode you want, and the parameters of the tunnel boring machine or bolts.
Soil Parameters, where stratigraphic data is entered.
Water, where groundwater data is provided.
Advanced Parameters, which include automatically chosen parameters, safety coefficients, and numerical settings.

In general, many buttons only appear when they are necessary (e.g., tunnel dimension buttons adapted to the chosen geometry).

General data

The first parameters concern the tunnel project, and are :

The excavation mode, either traditional or using a tunnel boring machine.
Fixed values for steering precision are associated with each tunnel boring machine type, but they can be manually adjusted by choosing the last option "Other." You can then select the precision value in kPa at the bottom of the page.
The choice of tunnel shape is only available in traditional excavation (circular if using a tunnel boring machine) and is between circular and horseshoe.
In the case of a circular tunnel, the diameter must be provided, along with the wedge geometry. Since the calculations are based on a rectangular section, the circle should be assimilated to a rectangle of the same area or to a circumscribed square.
In the case of a horseshoe, the width, height, and area of the section must be entered. The wedge is not to be entered; it is considered equivalent to a rectangle of the same dimensions. The equivalent diameter is used to determine the "cover over diameter" ratio $C/D," and in the case of a purely cohesive environment, it is also the diameter used for the Davis method (adapted to circular tunnels).
The keystone elevation helps locate the tunnel, and the base elevation is deduced from it and the tunnel dimensions.
The surface overburden completes the various options.

The following parameters concern the calculation methods. These depend on the type of soil (cohesive-frictional or purely cohesive).

They also depend, secondarily, on the selected calculation approach for the retained pressure. In all cases, calculations using approaches 2 and 3 will be performed; it's just a matter of choosing which pressure will be retained for post-processing (tunnel boring machine steering diagram, number of bolts).

In automatic mode, the retained pressure corresponds to the maximum of the two approaches. The user can also choose their preferred approach or manually enter the confinement density they want to be post-processed.

Finally, there are options specific to the excavation mode. If a tunnel boring machine is used, the shield weight (and possibly the steering precision) are required. If the traditional method is employed, the calculation of the number of bolts is possible:

Average useful bolt length. Note that this does not correspond to the total length of a bolt, but to its mobilised length, otherwise the resistance to sealing is overestimated.
Bolt section, elastic limit of the steel used and drill diameter.
The seal-to-soil friction resistance.

Soil parameters

For each layer, enter :

The name of the formation
the elevation of its roof
The saturated density $\gamma_{sat}$ (or wet in the absence of a water table)
Shear parameters $c$ and $varphi$.
The colour of the formation on the diagrams.

In case the lateral coefficients are manually specified in the advanced options, three additional columns are provided: $k_1$, $k_2$, and $k_c$, allowing to set, for each layer, the lateral coefficient in the silo and in the wedge, and the cohesion coefficient (silo + wedge) respectively.

Note

It is prohibited to enter layers out of order (upper layers must be entered at the top of the table), and no layer can have a roof elevation lower than the base elevation.

Watter

Two modes are available: the user can choose to enter either a pressure profile or a load profile. When the user switches modes, the data is automatically converted ($p \to h$ or $h \to p$).

Note

Entering data out of order is prohibited (upper elevations must be provided at the top of the table), and the last row must correspond to the base elevation. The number of rows must be zero (no water table) or greater than or equal to 2 (water table).

Advanced parameters

Warning

Some parameters in this tab are calculated automatically. To allow the user to manually adjust what they want, all automatic choices are disabled when the "Yes" button is checked. Conversely, even if changes have been made in the advanced parameters, switching to "No" reactivates automatic choices for all relevant buttons.

Among the automatic choices, there is the choice of lateral coefficients, and the four stress calculation options (inside and around both the silo and the wedge).

"Also present but not automated are the partial coefficients (approach 3) and global coefficients (approach 2). Finally, the choice of the dihedral angle increment."

Output

There are two types of outputs: visualization of input data and presentation of the results themselves.

Data Display

Four sub-tabs display the data entered:

3D: This window provides an overview of the stratigraphy and the tunnel's placement relative to it. The tunnel face is drawn. In the case of a horseshoe, the area is preserved whenever possible (drawing in a rectangle + 1/2 ellipse). Otherwise, an orange parabola-rectangle profile that does not meet the area criterion (but presents the correct height/width) is arbitrarily represented.
Stratigraphy: This cross-section helps identify the stratigraphy and the main characteristics of different layers (shear parameters). The tunnel is not represented.
Water Tables: This cross-section helps identify the tunnel in the hydrogeological context, with the representation of the pressure or load profile.
Data Table: This table gathers all the relevant parameters of the modeled problem.

Displaying results

Certainly! Here's the translation with British English:

The main window for reviewing the results is the "mechanisms" window. On this window, there are two curves for calculation approaches 2 and 3 of Eurocode 7, representing the required confinement pressure based on the angle of the considered mechanism. The maximum corresponds to the critical pressure.

In the case of tunnel excavation, the control diagram provides the pressure determination window (total).

The results table summarises the important results, both for tunnel excavation and traditional excavation.

"Report" tab

As with the other tools, essential components are integrated into the report generator.

Theoretical elements

Lateral coefficients

The sets of coefficients provided are as follows:

Jeu	$k_1$	$k_2$	$k_c$
Terzaghi & Jelinek (1954)	1	1	0
Melix (1987)	0.8	0.8	0
Anagnostou & Kovari (1994)	0.8	0.4	0
Jancsecz & Steiner (1994)	$k_a$	$(k_a+k_0)/2$	0
Mayer, Hartwig, Schwab (2003)	1	0	0
Kirsch & Kolymbas (2005)	$k_0$	$k_0$	0
Champagne (2017)	$\lambda$	$\lambda$	$k_c$

Where $k_0=1-\sin\varphi$, $k_a=\tan(\varphi/4-\varphi/2)^2$, $\lambda = \frac{\cos(\varphi)^2}{1+\sin(\varphi)^2}$ et $k_c=\frac{\sin(2\varphi)}{1+\sin(\varphi)^2}$.

Manual mode allows you to choose your own values for each layer.

The automatic choice is as follows:

If $\varphi < 30°$, the Terzaghi & Jelinek set is the most suitable.
If $30° \leq \varphi < 35°$, the Melix set is appropriate.
If $\varphi > 35°$, the Anagnostou & Kovari set is the best.

In the case of a multi-layer front, it is the minimum $\varphi$ that governs the choice of coefficients.

Stresses in and around the mechanism

Case of cohesive-frictional soil

The stress in the dihedral is always taken to be equal to the Terzaghi silo solution, and the stress around the silo is taken to be equal to the linear approximation without surcharge (zero stress at the key, and equal to the weight of the ground between the key and the slab at the level of the slab).

If the cover $C$ is less than $2D$ ($D$ is the diameter of the tunnel, or its equivalent via the horseshoe area formula), the shear resistance mobilised (weight of the earth in and around the dihedral) is considered negligible. In the case of sufficient coverage, the stress state of the Terzaghi silo is considered in and around the silo.

Case of purely cohesive soil

The weight-of-earth solution and the linear approximation without overload are applied both inside and outside the mechanism.