GeoStudio | PLAXIS - Weak rock tunnel excavation

Application	PLAXIS 2D PLAXIS 3D
Version	PLAXIS 2D PLAXIS 3D
Date created	10 June 2024
Date modified	10 June 2024
Original author	Richard Witasse - Bentley Principal Application Engineer

1. Introduction

Tunnelling in weak rock presents some special challenges to the geotechnical engineer since misjudgements in the design of support systems can lead to under-design and costly failures or over-design and high tunnelling costs. To understand the issues involved in the process of designing support for these tunnels, it is necessary to examine some basic concepts of how a rock mass surrounding a tunnel deforms and how the support system acts to control this deformation.

The stability of tunnels constructed in weak rock is primarily influenced by the ratio of the uniaxial compressive strength of the rock mass to the maximum in situ stress. This ratio is a critical factor in determining the initial support requirements necessary to control strain levels through empirical analysis. While preliminary estimates can be derived from idealized calculations for circular tunnels with isotropic initial stress states, a more reliable approach involves numerical analysis of the tunnel's response to sequential excavation and support installation.

The purpose of this article is to present how PLAXIS could be practically use for this purpose and optimally assess the tunnel behaviour and support associated with the underground excavation of weak rock mass.

2. Getting some order of magnitude: Empirical solutions

Hoek (1999) presented an analysis demonstrating that the ratio of the compressive strength σ_cm of a rock mass to the in-situ stress p₀ can serve as an indicator of potential tunnel squeezing issues. Building on Sakurai's (1983) recommendations, an analysis was conducted to establish the relationship between σ_cm / p₀and the tunnel's percentage "strain". The percentage strain ε is defined as 100 times the ratio of tunnel closure to tunnel diameter.

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Figure 1: Tunnel deformation versus ratio of rock mass strength to in-situ stress

Figure 1 displays the results of a study using closed-form analytical solutions for a circular tunnel in a hydrostatic stress field, as published by Fama (1993) and Carranza-Torres and Fairhurst (1999). The figure shows that the behaviour of these tunnels adheres to a well-defined pattern, accurately predicted by the equation included in the figure.

Figure 1 is valid for the condition of zero support pressure (p_i = 0). Similar analyses were run for a range of support pressures versus in situ stress ratios (p_i / p₀) over the results of which a statistical curve fitting process was used to determine the best fit curves for the generated data for each p_i / p₀value. This process finally led to the following equation:

Where u_wall is the tunnel sidewall displacement, r_i is the original tunnel radius, p_i is the internal support pressure, p₀ is the isotropic in-situ stress and σ_cm is the global rock mass strength. A similar analysis was carried out to determine the size of the plastic zone surrounding the tunnel:

with r_p the tunnel sidewall.

The idea of using strain ε as a basis for tunnel design can then be taken a step further by considering the amount of support (p_i) required to limit the strain to a specified level as suggested by Sakurai (1983). Typical support pressures for a variety of different systems for a range of tunnel sizes are available in literature especially those of Hoek and Brown (1980) and Brady and Brown (1985) to calculate the capacity of mechanically anchored rockbolts, shotcrete or concrete linings or steel sets for a circular tunnel.

3. Uniaxial compressive strength versus global strength

It is important to notice that the previous set of equations for evaluating the tunnel deformation and the extent of the plastic radius are defined as a function of σ_cm the global rock mass strength! The global rock mass strength is clearly different than the uniaxial compressive strength σ_c.

The uniaxial compressive strength σ_ccan be derived from the generalized Hoek-Brown criterion usually written as:

where σ′₁ and σ′₃ are the major and minor principal stresses respectively, σ_ci is the uniaxial compressive strength of the intact rock pieces. Finally, m_b, s and α are material constants determined from the GSI rating. Introducing σ′₃ = 0 into this equation gives the uniaxial compressive strength of the rock mass as:

The concept of using the global rock mass strength σ_cm rather than the uniaxial compressive strength σ_c originates from mining engineers dealing with pillar failures which resulted from the propagation of failure through the rock mass. In designing against such failures, mining engineers focus on the average strength of the pillar rather than the specifics of the failure mechanism development within the rock mass itself. In that sense, the uniaxial compressive strength of the rock mass, which applies to the pillar's boundaries, is of limited used for understanding the interior strength of the pillar.

In this context, the main challenge is defining the average or "global" rock mass strength as illustrated in Figure 2. The average strength of a rock mass in various structures depends on the degree of confinement, as even minimal confinement can significantly increase rock mass strength due to the Hoek-Brown failure envelope's curvilinear nature at low stress levels.

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Figure 2: Plot of the failure envelopes for the Generalized Hoek Brown failure criterion and the associated Mohr Coulomb criterion.

Calculating average strength is complex due to varying confining stresses, necessitating a compromise solution. One of the most common approaches is to fit an equivalent Mohr-Failure envelope to the failure. This can be done using a solution published by Balmer (1952) in which the normal and shear stresses σ_n and τ are expressed in terms of the corresponding principal stresses σ′₁ and σ′₃and then calculate the average cohesion c_eq and friction angle φ_eq values by linear regression analysis, in which the best fitting straight line is calculated for a relevant range of (σ_n, τ) pairs. The uniaxial compressive strength of a rock mass is finally defined by a cohesive strength c_eq and a friction angle φ_eq is given by:

The actual process of curve fitting has been discussed in detail in Hoek, Carranza-Torres and Corkum (2002) and the following relationship between the global rock mass strength σ_cm and the Hoek Brown parameters has finally been proposed:

for the confining stress range: 0 <σ_cm < σ_c_i / 4. As one can imagine the resulting global rock mass strength depends on the range of stresses over which the linear regression is being performed. The stress range 0 <σ_cm < σ_c_i / 4 was chosen based on experience and was found to work well for a wide variety of practical situations.

4. PLAXIS Finite Element Numerical Analysis for Preliminary Tunnel Deformation

Empirical methods are useful for initial estimates of support pressure needed to limit plastic zone size or tunnel closure. However, these methods often fall short for practical applications, requiring a detailed numerical analysis to assess more accurately tunnel closure and plastic zone but also to optimize the rock support interaction.

Rock Mass Properties and Constitutive Rock Model

Critical mechanical properties of the rock mass, such as compressive strength σcm, angle of friction φ, modulus of deformation E, and initial state of stresses, must be assessed in the first place. Classification systems like the Geological Strength Index (GSI), Q-system, and Rock Mass Rating (RMR) are commonly employed for these estimations. Hoek-Brown failure criterion is often used for the assessment of the rock strength.

PLAXIS software provides valuable constitutive models with that respect:

Hoek-Brown model: Based on the Hoek-Brown strength criterion, it is a linear elastic perfectly plastic model. It may optionally incorporate strain-softening for a even better description of failure propagation in a rock mass.
Mohr-Coulomb model: A simpler model that can be easily and quickly set-up to account for rock mass behaviour.

Geometry and Initial Stress Definition

Defining the geometry and initial stress conditions accurately is crucial. Complex tunnel shape can easily be defined through the PLAXIS Tunnel Designer. One should first make sure the model dimensions are sufficiently large compared to the tunnel dimensions to prevent any boundary effects (see Figure 3).

Initial vertical stress is calculated based on the weight of the overburden rock, while horizontal stresses are defined from stress ratios with the possible consideration of locked-in stress. Two relevant methods in PLAXIS for evaluating in-situ stresses are the K0-procedure and the field stress method.

Support Pressure Evaluation

Estimating support capacity can be guided by published equations from Hoek and Brown (1980) and Brady and Brown (1985). These equations calculate the capacity of various supports, including mechanically anchored rockbolts, shotcrete, concrete linings, and steel sets. Although care must be taken when applying these idealized calculations to real-world problems, they can still be considered to be representative of the tunnel support as a first approximation for the evaluation of tunnel closure and plastic zone.

Figure 3: Geometry Definition

Possible Convergence Issues

Non-associated plasticity in numerical simulations can lead to convergence issues which are characterized by non-uniqueness of the numerical solution, sensitivity to perturbations, and discontinuous responses.

This could be particularly pronounced when using the Hoek-Brown model with large plastic strain development (i.e. poor rock conditions and/or low compressive-strength-to-initial-stress ratio). Indeed, the very high level of non-associativity induced by the Hoek-Brown criterion at very low stress confinement level (difference between the friction angle that corresponds to the tangent of the HB criterion and the dilatancy angle which is 0 by default in PLAXIS HB model) will significantly affect the convergence rate and lead to lightly premature termination of the PLAXIS calculation.

Their main characteristics are the development of localized shear bands, around the excavated area along with repetitive sequences of unloading steps (as an attempt to overcome local numerical instabilities and therefore considerably lowering the convergence rate) although each numerical step manages to fulfill all convergence criteria (See Figure 4). Usually, the finer the mesh or the highest the element order used, the more pronounced the convergence issues will be.

Strategies to address these issues include:

Adjusting convergence settings.
Adding dilatancy to reduce non-associativity.
Challenge the use of extremely low uniaxial compressive strength values associated with GSI for poor rock conditions as presented in Figure 5.

Indeed, extremely low compressive strength values associated with GSI may not accurately represent the true strength and global mechanical behaviour of the rock mass in an average sense at the scale of the RVE (representative volume element) for the considered tunnel analysis. Using an equivalent Mohr-Coulomb model might provide a more representative depiction in poor conditions.


(a) Incremental shear strain	(b) Convergence log

Figure 4: Main characteristics of the non-convergence behaviour

Figure 5: Rock uniaxial compressive strength as a function of GSI

5. Detailed tunnel support modelling

Although the consideration of an equivalent support pressure is often useful for the evaluation of the tunnel closure and the extent of the plastic zone, detailed structural member analysis of tunnel support might be required either for a more accurate representation of the support action onto the excavated rock mass or because structural analysis for the support system design is required.

PLAXIS is a powerful tool for detailed tunnel support modelling. It allows for the simulation of different support systems and their interaction with the rock mass.

Tunnel excavation support

Tunnel excavation support, which may be steel sets, rockbolts or shotcrete or some combination of these, is installed some distance behind the tunnel advancing front. It acts like a spring and the support action that it provides to the tunnel increases with convergence of the tunnel.

Shotcrete can be modelled as a plate element or volume element, which are interacting with the rock mass through interface elements as shown in Figure 6.

Figure 6: Shotcrete modelling in PLAXIS

Steel sets are typically used in combination with shotcrete and can be modelled either as individual beam elements (option only possible in PLAXIS 3D) as shown in or using a homogenization approach consisting in defining a composite plate the properties of which are calculated by averaging the contribution of the shotcrete and spaced steel sets (equivalent section).

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Figure 7: Spaced steel set modelling (pink beam elements) in PLAXIS 3D

The equivalent stiffnesses of the composite plate is obtained by summing the rigidity of shotcrete and steel sets:

with

where:

is the Young's modulus of the steel
is the cross-sectional area of each steel set
is the moment of inertia of each steel set
is the Poisson's ratio of the steel
is the Young's modulus of the shotcrete
is the thickness of the shotcrete
is the Poisson's ratio of the shotcrete
s is the spacing between two consecutive steel sets

Once and have been computed, equivalent thickness and equivalent Young's modulus can be evaluated as:

and

Once structural forces are being calculated for the composite plate, it is possible to redistribute these thrusts and moments back onto the individual support elements. The equations for the redistribution of the moment Meq, axial thrust Neq and shear forces Qeq induced in any one of the beam elements representing the equivalent shell are (thin beam solution only):

For shotcrete:	For steel sets:

Rockbolts and cables are represented as cable elements in PLAXIS, suitable for various types of bolts and cables. Cable elements can interact with rock through bond element allowing for skin friction development (see Figure 8). The cable can support a normal force up to user-defined maximum values (in both compression and tension) at which it will behave plastically. The skin friction (at the cable/grout interface or at the grout/rock interface) can also be made dependent on the effective confining stress:

where is the grout cohesive strength, the grout friction angle, and p is the grout exposed perimeter.

Figure 8: Cable element formulation

Tunnel face stability

In tunnel construction, ensuring face stability is crucial to prevent collapses and provide safe working conditions. Various methods can be employed to address face instability, depending on the geological conditions, tunnel depth, and specific project requirements.

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Figure 9: Modelling fiberglass dowels in PLAXIS 3D

A very common method to protect the rock core ahead of the face, consists in using grouted fiberglass dowels. In PLAXIS, these can be modelled using cable elements (see Figure 9), although an equivalent stabilizing front pressure is often applied to the excavation face for practical purpose. Indeed, the detailed analysis of the grouted fiberglass dowels and their interaction with surrounding rock could be quite cumbersome. Considering that their purpose is to stabilize the tunnel front face and that they are progressively removed as the tunnel front advances, an equivalent stabilizing front pressure as suggested by Peila (1994) is often preferred from a numerical modelling perspective:

where:

N is the number of bolts,
A is the cross-sectional area of the bolt,
is the yielding strength of the bolt material,
S is the tunnel face surface,
s_l is the lateral surface of the bolt,
is the soil-bar limit skin friction.

During tunnel construction, sections with challenging ground conditions, such as high-water inflow or unstable fractured rock, can significantly impact face stability. In such cases, it may be necessary to stabilize and seal the section to be excavated using forepoling techniques like pipe roofing. This method involves driving poles, timber, or steel tubes into the ground before excavation. In PLAXIS, this technique can be modelled using embedded beam elements. The geometrical representation of these elements in PLAXIS 3D is simplified through the tunnel designer, allowing convenient definition from the theoretical excavation shape (corresponding to the tunnel crown), including their spacing, length, and angle (see Figure 9).

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Figure 9: Modelling forepoles pipes in PLAXIS 3D

6. Conclusion

Excavating tunnels in weak rock presents significant engineering challenges due to the inherent instability and potential for large deformations. These difficulties necessitate a comprehensive approach that combines empirical methods and advanced numerical modelling techniques to ensure safety and structural integrity.

The article highlights the importance of utilizing PLAXIS for detailed simulation of tunnel support systems. PLAXIS allows for precise modelling of various supports, including shotcrete, steel sets, rockbolts, and fiberglass dowels amongst others. These elements are crucial for stabilizing the tunnel face and controlling deformation during excavation. The use of plate, cable and embedded beam elements along with the tunnel designer feature in PLAXIS 3D facilitates accurate representation and efficient design of tunnel support systems.

Empirical methods, such as those developed by Hoek and Sakurai, provide valuable insights into the behaviour of tunnels in weak rock. These methods help predict ground response and support requirements, forming a basis for initial design considerations. However, the integration of numerical modelling is essential for refining these predictions and addressing complex geological conditions.

Numerical modelling is particularly valuable for designing tunnels in weak rock because it enables a detailed analysis of ground behaviour and support interaction. Through simulations, engineers can evaluate various support configurations and their effectiveness in real-time, allowing for optimization before actual construction begins. This predictive capability helps in anticipating potential issues and adapting the design accordingly, ensuring that the chosen support systems can handle the expected loads and deformations. By using PLAXIS, engineers can create a more robust and resilient tunnel design, minimizing risks and enhancing the overall safety and stability of the project.

In conclusion, the successful excavation of tunnels in weak rock relies on a multidisciplinary approach that integrates empirical analysis, numerical modelling, and practical support methods. This approach not only enhances the safety and stability of the excavation but also optimizes the efficiency and effectiveness of the support systems. By leveraging advanced tools like PLAXIS and implementing robust support strategies, engineers can achieve the structural integrity necessary for safe and successful tunnel construction in challenging geological environments.

References

Brady, B.H.G. and Brown, E.T. (1985). Rock mechanics for underground mining. Allen and Unwin, London.
Carranza-Torres, C. and Fairhurst, C. 1999. General formulation of the elasto-plastic response of openings in rock using the Hoek-Brown failure criterion. International Journal of Rock Mechanics and Mining Sciences. 36 (6), 777-809
Duncan Fama, M.E. 1993. Numerical modelling of yield zones in weak rocks. In Comprehensive rock engineering, (ed. J.A. Hudson) 2, 49-75. Pergamon, Oxford.
Hoek E., Caranza-Torres C.T. and Corkum B., 2002. Hoek-Brown failure criterion-2002 edition. In: Bawden, H.R.W., Curran, J., Telsenicki M., (Eds). Proc. North American Rock Mechanics Society (NARMS-TAC 2002). Mining Innovation and Technology, 267-273, Toronto, Canada.
Hoek, E., and Brown, E.T. (1980). Underground excavations in rock. Institution Min. Metall., London.
Hoek, E. 1999. Support for very weak rock associated with faults and shear zones. In Rock support and reinforcement practice in mining. (Villaescusa, E.., Windsor, C.R. and Thompson, A.G. eds.). Rotterdam: Balkema. 19-32.
Sakurai, S. 1983. Displacement measurements associated with the design of underground openings. Proc. Int. Symp. field measurements in geomechanics, Zurich 2, 1163-1178.

Weak rock tunnel excavation