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Considering the effect of seepage force, a dimensionless approach was introduced to improve the stress and strain increment approach on the stresses and radial displacement around a circular tunnel excavated in a strain-softening generalized Hoek–Brown or Mohr–Coulomb rock mass. The circular tunnel can be simplified as axisymmetric problem, and the plastic zone was divided into a finite number of concentric rings which satisfy the equilibrium and compatibility equations. Increments of stresses and strains for each ring were obtained by solving the equilibrium and compatibility equations. Then, the stresses and displacements in softening zone can be calculated. The correctness and reliability of the proposed approach were performed by the existing solutions.

Analysis of stresses and displacements around circular tunnel excavated in isotropic rock mass is one of the fundamental problems in geotechnical engineering. A nonlinear method is needed to obtain a reliable solution since the deformation response depends on the stress path. Numerical method, elastoplastic methods, and limit analysis methods are popularly used for cavity expansion or contraction according to the existing literature surveys [

It can be seen from the researches worldwide that the theoretical analysis method and numerical simulation are two main techniques for analysis of strain-softening rock mass. The theoretical approaches can reflect the innate character of strain-softening rock mass, but the parameters for calculation which are usually not consistent with the actual values may cause computational errors as well. The numerical simulation shows the development of the softening rock mass. However, there exists some error in the solutions compared with the theoretical values. In the recent reports about the elastoplastic analysis of a circular opening excavated in a strain-softening rock mass, it is unusual to find a solution considering both the axial stress and seepage force in practical engineering. But the axial stress and seepage force have obvious influence on the surrounding rock’s stability, especially in strain-softening rock mass. And most diseases are related to the seepage force directly or indirectly. Only analyzing the influence of axial stress on the stresses and displacement of circular tunnel without considering seepage force does not accord with the practical engineering at abundant area. Therefore, the seepage fore is of great important for softening rock mass.

Therefore, on the basis of theoretical analysis, the dimensionless method is introduced in this study to eliminate the influence on the dimensions of variables and parameters based on the generalized Hoek–Brown and M-C failure criterion. Simultaneously, both the axial stress and seepage force are considered for reconstruction of the step-stress approach to analyze the strain-softening rock mass. Then, a new dimensionless method for the elastoplastic analysis of a circular opening excavated in a strain-softening rock mass considering seepage force is established in this study, which in hope of providing theoretical supports for the digging process and design of tunnel in a strain-softening rock mass.

The innovations of this study are listed as follows:

The dimensionless method pointed out by Carranza-Torres and Fairhurst [

Not only the generalized Hoek–Brown and M-C failure criteria but also the seepage force is considered in this study, and a new dimensionless method for analysis of the circular opening excavated in a strain-softening rock mass is presented, which is rarely studied from the existing literatures.

The dimensionless method pointed out by Carranza-Torres and Fairhurst [

Figure

Plastic zone formed around circular opening.

Assuming that the yielding of the rock mass is governed by the function,

Alonso et al. [

For the M-C rock mass,

For the generalized H-B rock mass,

The M-C criterion is regarded as the plastic potential function, and it may be written as follows:

The strength and deformation parameters presented in Equations (

Evolution of parameters in plastic regime.

When the internal support pressure

For H-B rock mass,

When

If

When the plastic zone is formed, radial stress

The plastic zone can be divided into

Normalized plastic zone with finite number of annuli.

A dimensionless method is proposed for calculating the increments of stresses and elastic strains. The circular tunnel can be simplified as axisymmetric problem, and the plastic zone is divided into a finite number of concentric rings which satisfy the equilibrium and compatibility equations in strain-softening rock mass. The increments of stresses and strains for each ring are obtained by solving the equilibrium and compatibility equations. Then, the stresses and displacements in the softening zone can be calculated.

Define a dimensionless variable

Then, the plastic zone can be translated into a unit plane. In the unit plane, the position of the elastic-plastic interface is fixed by

In order to simplify the calculation process, the forms of the generalized Hoek–Brown and Mohr–Coulomb criteria need to be translated into the dimensionless forms which are in the same order of magnitude in this study. The yield function is expressed as follows:

For the Hoek–Brown criterion, the transformed radial and tangential stresses are expressed as follows:

The transformed internal pressure and far-field stress are

In order to be consistent with the definition of elastic strain rates, shear modulus can be scaled according to the following expression:

Then, the yield function for the generalized Hoek–Brown criteria may be rewritten by

For the M-C criterion, the transformed radial and tangential stresses are given by

And the transformed internal pressure and far-field stress are

Then, the yield function for the Mohr–Coulomb criteria can be rewritten as follows:

Based on the dimensionless method in Carranza-Torres and Fairhurst [

Based on the method pointed out by Brown et al. [

According to Equation (

And the stress components for the

In fact, if

When the number of annuli

According to the transformation in Equation (

In the unit plane, the equilibrium condition may be written as

Equation (

To solve the plastic strain increments, the displacement compatibility equation is considered:

In order to work with dimensionless field quantities, the strains may be normalized as follows:

So that the displacement compatibility equation can be written as follows:

In the plastic zone, the total strains can be decomposed into elastic and plastic parts as follows:

Equation (

Then,

According to Hooke’s law, under the plane strain condition,

So, Equation (

Combining Equations (

The total strain at the

Recalling the relationship,

Then, the displacement normalized by plastic radius

By using the dimensionless method, the stresses and displacements in the softening zone can be calculated. The actual value of displacement can be calculated by the following equation:

The plastic radius

In order to verify the correctness of the dimensionless solutions, the generalized Hoek–Brown criterion is firstly considered in this study. It is obvious that the strain-softening solutions converge to the brittle-plastic solutions when the critical deviatoric plastic strain

The stresses and displacement obtained by Sharan [

According to the dimensionless method, compared with the results in this study in the same order of magnitude, Equations (

In addition, compared with the results based on the generalized H-B and M-C failure criteria, the technique of equivalent M-C and generalized H-B strength parameters was adopted for the comparison between the developed method and the exact solutions in Sharan [

The above formulas can be used to transfer the parameters from the H-B yield criterion to M-C yield criterion. But that may create some errors in practical calculations.

The data sets appearing in Sharan [

Stresses and displacements in this study can be obtained by writing a program at the case of

Comparison between dimensionless stresses and exact stresses for Hoek–Brown rock mass.

Comparison between dimensionless displacement and exact displacement for Hoek–Brown rock mass.

Figure

From Figure

Figures

Distribution of errors between dimensionless stresses and exact stresses.

Distribution of errors between dimensionless displacement and exact displacement.

From Figures

More different values of

To analyze the reliability of solutions under the M-C criterion, the stresses and displacement for the M-C surrounding rock obtained by the proposed dimensionless approach, and the approach by Lee and Pietruszczak [

Comparison of dimensionless stresses for M-C rock mass.

Comparison of dimensionless displacement for M-C rock mass.

The finite difference approach is also adopted by Lee and Pietruszczak [

For the elastoplastic analysis in a strain-softening rock mass, the stresses and displacement are calculated in this study. The parameters which have great effects on the dimensionless solutions are chosen to be analyzed. Therefore, five values of the deviatoric plastic shear strain

Variation of the dimensionless radial and circumferential stresses with the change of

Variation of the dimensionless radial displacement with the change of

Variation of the dimensionless radial and circumferential stresses with the change of

Variation of the dimensionless radial displacement with the change of

Figures

Figures

Alonso et al. [

Distribution of radial and circumferential stresses for different values of

Distribution of radial displacement for different values of

Variation of radial displacement on the opening surface with different values of

Variation of plastic radius

Variation of radial displacement on the opening surface with the change of

Variation of radial displacement on the opening surface with the change of

Figure

It can be noted that the strength parameters

Figure

As can be seen in Figure

Figure

Figures

Overall, we can draw some conclusions: different values of

In practical engineering, the seepage force has significant influence on the rock mass surrounding tunnel, especially when considering the strain-softening behavior. Most diseases are related to the seepage force directly or indirectly. However, the elastoplastic analysis of a circular opening excavated in a strain-softening rock mass considering seepage force may be obtained by the dimensionless method, and the exact solutions are difficult to accomplish. Therefore, based on the generalized Hoek–Brown failure criterion, the dimensionless method is used for reconstruction of the step-stress approach to analyze the strain-softening rock mass considering seepage force. Then, a new dimensionless method for the elastoplastic analysis of a circular opening excavated in a strain-softening rock mass considering seepage force is established in this study.

For the circular tunnel with inner radius (

In Equation (

For the axisymmetric plane strain problem, seepage force is the volume force and is given by

Considering the influence of seepage force, stress equilibrium differential equation can be expressed as follows:

As is mentioned before, it is assumed that the yielding of the rock mass is governed by the yielding function:

In order to work with dimensionless field quantities, the stress magnitude,

Strains are normalized accordingly, considering the extra term 2G:

Displacement is normalized in terms of the radius

Similarly, the step-stress approach is used to divide the plastic zone into a finite number of concentric rings which satisfy the equilibrium and compatibility equations in strain-softening rock mass. The increments of stresses and strains for each annulus are obtained by solving the equilibrium and compatibility equations. Then, the stresses and displacements in the softening zone can be calculated.

When considering the effects of seepage force, a step-stress approach is used for solving the stresses and displacement in the plastic zone numerically. The whole plastic zone is divided into n annuli with a constant radial stress increment between the adjacent two annuli, which is defined as follows:

So that the stress components for the

Then, the circumferential stress can be written as follows:

According to Hooke’s law, the elastic strain increments are related to the stress increment. That is,

When the number of annuli

Equation (

Then, the inner radius can be obtained by

The strain components are given in Equation (

The total strain at the

According to Equations (

Then, the displacement normalized by plastic radius

Plastic radius

The dimensionless stresses and displacements in this study can be obtained by programming the dimensionless method into MATLAB codes. In order to examine the difference between the solutions with and without considering seepage force, the parameters are adopted as follows:

Comparison between dimensionless stresses with and without considering seepage force for Hoek–Brown rock mass.

Comparison between dimensionless displacement with and without considering seepage force for Hoek–Brown rock mass.

Figures

Figures

Variation of the dimensionless radial and circumferential stresses considering seepage force with the change of

Variation of the dimensionless radial displacement considering seepage force with the change of

Based on the generalized Hoek–Brown failure criterion, the parameter

Ground reaction curves for different

Distribution of radial and circumferential stresses for different values of

Evolution of plastic radii for different values of

Variation of radial displacements on the opening surface with the change of parameter

Figure

Figure

As can be seen from Figure

Figures

As are illustrated in Figure

From Figure

A dimensionless approach is developed for elastoplastic analysis of circular opening excavated in a strain-softening rock mass based on the generalized Hoek–Brown and M-C failure criterion. The plastic zone is divided into a finite number of concentric rings in this study. In order to solve the equilibrium and compatibility equations for each ring, the dimensionless method is used for calculating the stresses and displacement in the strain-softening zone. Through analysis of examples, some conclusions can be drawn:

As the number of annuli

Five values of the deviatoric plastic shear strain

The strength parameters

For the analysis of a circular opening excavated in a strain-softening rock mass considering seepage force, the stresses are smaller but the plastic radius and radial displacement are larger than the results without considering seepage force, which reflects that the seepage force can increase the effective stresses around a circular tunnel. The deviatoric plastic shear strain

The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.

The authors declare that they have no conflicts of interest.

This work was supported by National Key R&D Program of China (2017YFB1201204).