Abstract

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.

1. Introduction

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 [1–15]. In the past, Mohr–Coulomb failure criterion was widely used by many scholars for analysis of rock mass. However, experimental observations indicated that the strength envelope for most of rock materials is nonlinear. For the nonlinear failure criterion, the criterion by Hoek and Brown is generally accepted in rock mechanics community, as it can provide a reliable tool for predicting the strength of jointed rock mass. Subsequently, the Hoek–Brown criterion has been updated to the generalized form and has been widely used. Elastoplastic analysis of circular tunnels excavated in Hoek–Brow and Mohr–Coulomb rock mass was attempted by many scholars [13–16]. For the theoretical analysis, the expressions of stresses and displacements around the tunnel are mainly obtained based on the elastic-perfectly plastic model and elastic-brittle-plastic model under the nonassociated flow rule. Park and Kim [13] provided a procedure of strain-softening model for elastoplastic analysis of a circular opening considering elastoplastic coupling. Park et al. [14] presented the variation laws of deformations based on different softening indexes and dilatancy characteristics. Sharan [15] provided an analytical solution for stresses and displacements around a circular opening in a generalized Hoek–Brown rock mass. Theoretical formulation and solutions for poorly consolidated rocks surrounding a circular opening are presented by Wang [16]. Han et al. [17] analyzed variation regulations of the stresses and deformation around the tunnel for elastic-brittle-plastic behavior considering the load-bearing characteristics of the plastic zone. However, the variation laws of the stresses and deformation around the tunnel excavated in a strain-softening rock mass were studied by numerical analysis. Zheng et al. [18] pointed out that the convergent problem was existed in the finite-element analysis of strain-softening rock mass with a higher softening rate. Therefore, the plastic zone of a circular tunnel may be divided into many zones to be analyzed generally. For example, Brown et al. [2] thought that the elastic deformation of rock mass is a constant value; then, the stresses and displacement field can be calculated easily. Wang et al. [19, 20] presented an analytical approximation solution for a circular opening in an elasto-brittle-plastic rock. Lee and Pietruszczak [21] provided a new numerical procedure for elastoplastic analysis of a circular opening excavated in a strain-softening rock mass by using the difference method.

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:(1)The dimensionless method pointed out by Carranza-Torres and Fairhurst [22] is only applicable to the original Hoek–Brown rock mass. However, the dimensionless solutions in this study are based on the generalized Hoek–Brow and M-C rock mass. Thus, the dimensionless method of Carranza-Torres and Fairhurst [22] is a special case of that in this study, and the dimensionless method in this study is the expansion and extension of that of Carranza-Torres and Fairhurst [22].(2)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.(3)The dimensionless method pointed out by Carranza-Torres and Fairhurst [22] is only suitable for the elastoplastic rock mass. However, the dimensionless method in this study is used in strain-softening rock mass and can also be simplified for the elastic-brittle-plastic and elastoplastic rock mass, which means the dimensionless method in Carranza-Torres and Fairhurst [22] is a special case of that in this study, and the dimensionless method in this study is the expansion and extension of that in Carranza-Torres and Fairhurst [22].

2. Definition of Problem

Figure 1 illustrates that a circular tunnel with radius is imposed by a stress field throughout the domain before the tunnel is excavated. As the internal support pressure is less than the critical value , a plastic zone is formed around the circular tunnel. Plastic radius can be derived for the elastic-brittle-plastic or elastic-perfectly plastic behavior [23, 24]. Furthermore, when considering the strain-softening behavior, the plastic zone can be divided into softening and residual zones whose interface is expressed by in Figure 1. However, it is difficult to obtain a closed-form solution for strain-softening rock mass, and the distributions for stresses and displacement should be solved by numerical method.

Figure 1 

Plastic zone formed around circular opening.

2.1. Yield Function

Assuming that the yielding of the rock mass is governed by the function,where and are the major principal stress and minor principal stress, respectively, and is the strain-softening parameter which reflects the evolution of the strength parameters in strain-softening rock mass and can be expressed as follows:

Alonso et al. [1] pointed out that no universal method was available to define the strain-softening parameter, but the definition in Equation (2) is widely accepted now.

For the M-C rock mass,where and are respectively strength parameters defined according to the friction angle and cohesion :

For the generalized H-B rock mass,where is the uniaxial compressive strength of rock and , , and are also the strength parameters for the H-B surrounding rock.

2.2. Plastic Potential Function

The M-C criterion is regarded as the plastic potential function, and it may be written as follows:where is the coefficient of dilation and may be written as follows:where is the angle of dilation, and when is equal to internal frictional angle , the plastic flow rule is related. When , no plastic volume change occurs. Therefore, the relationship between the radial and circumferential plastic strain increments can be obtained based on the plastic flow rule:

2.3. Evolution of Strength Parameters

The strength and deformation parameters presented in Equations (3), (5), and (6) are functions of . In plastic regime, these parameters can be described by bilinear functions of deviatoric plastic strain , which are shown in the following equation:where denotes one of the parameters , , , , , , and , and is the critical deviatoric plastic strain from which the residual behavior is firstly observed. The value of should be identified by experiments. This linear deterioration process of strength parameters is illustrated in Figure 2, where the subscripts and represent the peak and residual values of strength and deformation parameters, respectively.

Figure 2 

Evolution of parameters in plastic regime.

2.4. Critical Supporting Pressure

When the internal support pressure is lower than , the plastic zone develops. For M-C rock mass, can be calculated by the following equation:where and .

For H-B rock mass, can be obtained by solving the following nonlinear equation:

When , is expressed as follows:where .

If , can be calculated numerically based on suitable root-finding algorithm, such as Newton–Raphson method.

When the plastic zone is formed, radial stress is equal to on the elastic-plastic interface and is independent of radius :

3. Approximation of Strain-Softening Behavior

3.1. Preliminaries

The plastic zone can be divided into concentric annuli, and the annulus is bounded by two circles of normalized radii and in Figure 3. The thickness of each annulus is not uniform in general. On the outer boundary of plastic zone, where , stress and strain components under plane strain condition are obtained from Equations (14) and (15).

Figure 3 

Normalized plastic zone with finite number of annuli.

3.2. Increments of Stresses and Elastic Strains

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 , which maps the physical plane into a plane of coordinate based on the following transformation:

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 , and the wall of the cavity is defined 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 [22], the stress magnitude, , is used to normalize the stresses by

Based on the method pointed out by Brown et al. [2], radial stresses in the plastic zone may be divided into parts, and the increments of radial stress can be expressed as follows:

According to Equation (23), and Equation (26) can be written as follows:

And the stress components for the annulus can be given as follows:

In fact, if is sufficiently large, the circumferential stress can be written as follows:

3.3. Approximation of Displacements

When the number of annuli is sufficiently large and assuming that the strength parameters of rock mass is kept constant in each centric annulus, equilibrium equation can be transformed into

According to the transformation in Equation (25), the partial derivatives of the field functions with respect to the variables and are evaluated with the following operators:

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

Equation (34) can be approximated for the annulus aswhere . Then, the inner radius can be obtained as follows:

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 (38) may be reformulated as

Then,

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

So, Equation (43) may be reformulated as follows:

Combining Equations (36), (42), and (44), the following equation can be obtained:where and . The deviatoric plastic shear strain is updated as follows:

The total strain at the annulus can be given by

Recalling the relationship,then,

Then, the displacement normalized by plastic radius can be obtained as follows:where .

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 can be calculated from the following formula:

4. Verification Examples

4.1. Verifications for H-B and M-C Rock Mass

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 is equal to 0. However, the elastic-brittle-plastic case, , is the special case of strain-softening behavior. Comparing with the exact solutions for elastic-brittle-plastic rock mass, the dimensionless solutions can be verified.

The stresses and displacement obtained by Sharan [15] are compared with the results in this study, and the equations given by Sharan [15] are obtained from the following equations:

According to the dimensionless method, compared with the results in this study in the same order of magnitude, Equations (53)–(55) can be normalized as follows:

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 [15]. The equations for the friction angle () and cohesive () are given by Yang and Pan [25]:where , , , is the rock mass strength, is the unit weight of the rock mass, and is the depth of the tunnel below the surface.

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 [15] were chosen as input parameters for calculation in this study: , , , , , , , , , , and . And two dilation angles, and , were used to evaluate the influence of plastic volume change.

Stresses and displacements in this study can be obtained by writing a program at the case of . Then, based on Equations (56)–(58), the elastic-brittle-plastic solutions and exact solutions can be compared. The results are shown in Figures 4 and 5.

Figure 4 

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

Figure 5 

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

Figure 4 shows the comparison between dimensionless solutions and exact solutions for Hoek–Brown rock mass. The ordinate stands for the dimensionless stresses.

From Figure 4, when the number of annuli , the distributions of radial and circumferential stresses match the exact solutions well, which validate the accuracy of the dimensionless solutions. Figure 5 gives the comparison between dimensionless displacement and exact displacement for Hoek–Brown rock mass. As can be seen in Figure 5, the distribution of radial displacement obtained by the proposed approach shows a good agreement with the exact solution. So, the dimensionless solutions prove to be accurate.

Figures 6 and 7 show the distribution of errors between dimensionless solutions and exact solutions. The abscissa denotes . The ordinate stands for the error percentage between dimensionless solutions and exact solutions. Moreover, in order to examine the influence of mesh density on dimensionless solutions, two values of are considered in this example, which are 10 and 500.

Figure 6 

Distribution of errors between dimensionless stresses and exact stresses.

Figure 7 

Distribution of errors between dimensionless displacement and exact displacement.

From Figures 6 and 7, radial stress has a little error to the exact solution with the maximum error 2.13% and the minimum error 0.06%. The circumferential stresses obtained in this study can finely match the exact solution with the maximum error 1.55% and minimum error . Furthermore, the value of has great influence on the dimensionless solutions, and the dimensionless solutions for are obviously more accurate than that for .

More different values of have been studied in this example, and we can find that the accuracy of dimensionless solutions will increase when the value of grows. However, when value of is larger than 500, the value of has little influence on the dimensionless solutions. Therefore, to increase computational efficiency, the results of stresses and displacement for can be selected.

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 [21] is compared. The parameters are presented as follows: , , , , , , , , , , and . The computed stresses and displacement using two numerical approaches are illustrated in Figures 8 and 9.

Figure 8 

Comparison of dimensionless stresses for M-C rock mass.

Figure 9 

Comparison of dimensionless displacement for M-C rock mass.

The finite difference approach is also adopted by Lee and Pietruszczak [21]. The stress increment approach for the mechanical states of strain-softening rock mass is developed in their study. It can be seen from Figures 8 and 9, when n = 10 and 100, the determined circumferential stresses are respectively 10.2% and 4.5% larger than the circumferential stress in Lee and Pietruszczak [21]; meanwhile, the calculated displacements are respectively 11.1% and 3.2% smaller than the result in Lee and Pietruszczak [21]. When n = 500 and 1000, the obtained stresses and displacements are almost the same as the counterparts in Lee and Pietruszczak [21]. Therefore, n = 500 is selected in this study in order to reduce the calculation load, which can also meet the safety requirements in practical engineering. The comparisons in Figures 8 and 9 also verify the reliability of the proposed dimensionless approach for the M-C surrounding rock.

4.2. Influence of the Deviatoric Plastic Shear Strain on the Dimensionless Solutions of the Generalized H-B and M-C Rock Mass

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 considered in this section are 0, 0.004, 0.008, 0.012, and 100. Where denotes the elastic-brittle-plastic behavior, and can be approximately regarded as elastic-perfectly plastic behavior. Furthermore, the data sets appearing in Carranza-Torres [26] was chosen as input parameters for the calculation in this section: , , , , , , , , , , , , and . The results are shown in Figures 10–13.

Figure 10 

Variation of the dimensionless radial and circumferential stresses with the change of based on the generalized Hoek–Brown rock mass.

Figure 11 

Variation of the dimensionless radial displacement with the change of based on the generalized Hoek–Brown rock mass.

Figure 12 

Variation of the dimensionless radial and circumferential stresses with the change of based on the M-C rock mass.

Figure 13 

Variation of the dimensionless radial displacement with the change of based on the M-C rock mass.

Figures 10 and 11 show the distributions of stresses and radial displacement with the change of based on the generalized Hoek–Brown rock mass, respectively. It should be noted that the solutions finely match the brittle-plastic solutions for the case of . With the decrease of deviatoric plastic shear strain , the strain-softening solutions converge to the brittle-plastic solutions. The plastic radius will grow with the decrease of deviatoric plastic shear strain , which means the plastic zone becomes larger. For , the whole plastic zone is strain-softening zone.

Figures 12 and 13 show the distributions of stresses and radial displacement with the change of based on the generalized M-C rock mass, respectively. The results are similar to the generalized Hoek–Brown rock mass. With the decrease of deviatoric plastic shear strain , the strain-softening solutions converge to the brittle-plastic solutions. The plastic radius will reduce with the increase of deviatoric plastic shear strain , which means the plastic zone becomes smaller. For , the whole plastic zone is strain-softening zone. Generally, the results reflect that the strain-softening behavior causes a smaller plastic radius and the strain-softening behavior will become obvious with the decrease of plastic radius.

4.3. Influence of the Strength Parameters on the Dimensionless Solutions for the Generalized H-B Rock Mass

Alonso et al. [1] presented self-similar solutions in a strain-softening rock mass based on the generalized Hoek–Brown failure criterion for However, the general strain-softening solutions are still not available now. Many scholars have made deep studies of the strength parameter and find that it varies in accordance with the geological conditions. The strength parameter can be formulated bywhere GSI is the geological strength index which indicates the degree of fracturing and the condition of fracture surfaces of rock mass. Since the value of GSI is in the range from 10 to 100, can take a value in the range of 0.5–0.6. Therefore, it is necessary to find how the strength parameter affects the dimensionless solutions of the generalized H-B rock mass. In this study, different values of are analyzed, and the results are shown from Figures 14–19.

Figure 14 

Distribution of radial and circumferential stresses for different values of .

Figure 15 

Distribution of radial displacement for different values of .

Figure 16 

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

Figure 17 

Variation of plastic radius on the opening surface with the change of .

Figure 18 

Variation of radial displacement on the opening surface with the change of based on generalized Hoek–Brown failure criterion.

Figure 19 

Variation of radial displacement on the opening surface with the change of based on generalized Hoek–Brown failure criterion.

Figure 14 shows the distribution of the dimensionless stresses with different values of . When the softening zone is much thinner than the residual zone, the plastic behavior will be shifted quickly to residual regime.

It can be noted that the strength parameters and have an obvious influence on the dimensionless solutions. The plastic radius will grow when and become larger, which means the plastic zone becomes larger. When remains unchanged and becomes larger, the plastic radius will increase. In addition, the residual zone will increase with the increase of and .

Figure 15 shows the influence of different values of and on the radial displacement based on the generalized Hoek–brown failure criterion. Besides Figure 16 shows the variation of radial displacement on the opening surface with different values of .

As can be seen in Figure 15, the strength parameters and have an obvious influence on the dimensionless displacement. The radial displacement grows rapidly with the increase of and . From Figure 16, the dimensionless displacement for is about 1.4 times larger than that for . When translating the dimensionless displacement into actual displacement, we can find that the displacement for is about 3 times larger than that for .