INTRODUCTION
During excavation in rock mass farfiled in situ stresses are redistributed
and cause a plastic deformation around the openings such as tunnels, shafts,
wellbores etc. Analytical solution in this case involves elastoplasticity which
have been used in previous texts (CarranzaTorres and Fairhurst,
1999; Chen et al., 1999; Hoek
and Brown, 1980; Li and Michel, 2009; Detournay,
1986; CarranzaTorres, 1998). Alani
(2002) and Alani and Nasser (2001) developed a new
cubic macroelement and quadratic analyses for plate with a hole under bending
and compared with closed form solution, respectively. Exact determination of
stress field in elastoplastic solution requires appropriate rock failure criterion.
The medium in elastoplastic solution for circular space presumed homogenous,
compressive and isotropic farfiled stress that subjected to internal pressure,
P_{i} that applied in plane strain condition (CarranzaTorres
and Fairhurst, 2000; Muhlhaus, 1985; Hoek,
1998; Malvern, 1969). Closed form solution of GRC
implementing in convergenceconfinement method by elastoplastic model are among
the most broadly used for general design evaluation, especially regarding excavations
and support design. Taha et al. (2009) applied
the MohrCoulomb material and simulated stress distribution around a pile in
cohesion less soil. It was found that dry soil condition gives more resistance
than others. Stress concentration analysis around a wellbore showed that in
addition to rockmud interaction drilling string vibration could cause many
problems (Ibrahim et al., 2004). Macro element
analysis and closed form solution of stress distribution around cavities in
plate bending were modeled and had excellent results with regards to conventional
finite element solutions (AlAni, 2010).
The MohrCoulomb criterion (MC) was the most common criterion that has been
used in elastoplastic solution of stress state around openings (Florence
and Schwer, 1978). However, The HoekBrown criterion (HB) could find wide
practical application as a method of describing the stress condition in rock
mass surrounded the opening. Hassani et al. (2008)
presented a 3D finite element analysis of Siah Bishe tunnels by using ABAQUS
software. It was observed maximum displacement occurs in the roof of the tunnels.
Stress concentration was intensified in transition zone of tunnel and shaft.
The rock mass condition under which HoekBrown criterion can be applied is only
intact rock or heavily jointed rock masses that can be considered homogenous
and isotropic (Hoek et al., 1998). The Classic
Tresca criterion related the difference between maximum and minimum principal
stresses to the cohesion without friction, like Von Mises (Hill,
1950). Stress analysis and hydro mechanical behavior of the Bisotun epigraph
showed that heterogeneity is one of the most significant factors on hydraulic
and mechanical properties of rock mass (Karimnia and Shahkarami,
2011). Fatigue behavior of a cylindrical hole in piston was studied by Rahman
et al. (2009) using the Tresca and Von Mises materials. It was observed
that more conservative prediction to use Signed Tresca parameter and Signed
von Mises stress gives the result that lie between the absolute maximum principal
stress and signed Tresca results.
Most of the cited failure criteria which applied in rock mechanics were extended
before the function of the intermediate principal stress was evident. According
to experimental data has shown that the intermediate principal stress has a
substantialalthough slightinfluence on the strength of several rock classes
(Colmenares and Zoback, 2002; Mogi,
1967). The MogiCoulomb Criterion (MGC) clearly showed the impact of intermediate
principal stress that was based on linear Mogi criterion in terms of first and
second stress invariants (AlAjmi and Zimmerman, 2006).
A mathematical model for couple coal/rock mass viscoelastic deformation was
presented by Sun (2006). It could properly show the
gas leak flow in these mediums.
This study concerns analytical solution of stress distribution about an underground circular space via four rock failure criteria includes the generalized HoekBrown, the MohrCoulomb, the MogiCoulomb and the Tresca criterion which implemented in 3DEC by means of a FISH program. The aim of this paper was to compare elastoplastic solution of the rock failure criteria numerically in the 3DEC. Advantages and deficient of each criterion is presented. In addition, parametric study of the rock failure criteria in elastoplastic solution has been carried out.
FOUR ROCK FAILURE CRITERIA
The Tresca criterion: After a series of experiments, Tresca achieved
that the material will failed when a critical amount of shear stress is reached
(Tresca, 1868):
where, C is the cohesion and τ_{max} is the maximum shear stress of the material. Notice that the Tresca criterion can be considered as a particular type of the MC criterion, with φ = 0.
The generalized HoekBrown criterion: The generalized HB criterion concerns the maximum principal stress, σ_{1} to the minimum principal stress, σ_{3} via Eq. 2:
where, σ_{c} is the Uniaxial Compressive Strength (UCS) of intact rock, m, s and a are constants which depend on the rock mass properties:
where, m_{i} is the value of m for intact rock and can be obtained
from laboratory tests. While, D is the disturbance factor which varies from
0.0 for undisturbed rock masses to 1.0 for very disturbed rock mass e.g., by
stress release and drillblast. The Geological Strength Index (GSI) introduced
by Hoek indicates the characteristic of the rock mass (Hoek,
1994).
The MohrCoulomb criterion: A more general and frequently used criterion is the MohrCoulomb failure criterion. Failure will occur when in any (failure) plane the shear stress, τ reaches the failure shear stress, τ_{max} which is given by a functional relation of the form:
where, c is the cohesion of the rock mass, φ is the internal friction angle
of the rock mass and σ_{n} is the normal stress working on the
individual failure plane. The MC criterion can be written with regard to the
maximum and minimum principal stresses as follows (Benz
and Schwab, 2008):
The MogiCoulomb failure criterion: All three rock failure criteria
considered above, did not take into account the influence of intermediate principal
stress and determined from triaxial tests. Mogi’s experimental attempts
revealed that rock strength varied with the intermediate principal stress, σ_{2}
which was obtained from polyaxial (TrueTriaxial) tests (Mogi,
1971). He related the octahedral shear stress at failure to the sum of the
minimum and maximum principal stresses
where, f is a monotonically increasing function. The MGC criterion can be stated as:
According to AlAjmi and Zimmerman (2006) the linear
Mogi parameters a and b can be related to the Coulomb shear strength parameters
c and φ then can be extended as follows:
STRESSES AROUND CIRCULAR SPACE BY ELASTOPLASTIC SOLUTION
Elastoplastic solution around circular space using the Tresca failure criterion: The elastoplastic analytical solution is commonly carries out for simplified models. Consequently as shown in Fig. 1a circular space is utilized in plane strain condition which subjected to isotropic stresses, σ_{h} and σ_{v} at infinity and internal pressure P_{i}. Then, R_{i} is the primary radius of the underground space; R_{e} is the plastic zone radius; R and θ are the cylindrical coordinate of an assumed location, while σ_{R} and σ_{θ} are the related radial and tangential stresses, respectively.
According to equilibrium equation (Jaeger et al.,
2007):
And it is assumed that σ_{θ}, σ_{r} will be σ_{1} and σ_{3}, respectively. The Tresca failure criterion (Sec. 1.1) then requires:
in the plastic zone (R_{i}<R<R_{e}). Introducing Eq.
12 into Eq. 11 the stresses at boundary condition are
given by:
As the Tresca criterion does not comprise the intermediate principal stress
the hydrostatic ground pressure σ_{v} = σ_{h} is assumed
to solve the problem. Therefore, the induced stresses in the elastic region
can be found in Hiramatsu and Oka (1968) as follows:

Fig. 1: 
The circular space subjected to isotropic stresses, σ_{h}
and σ_{v}; internal pressure, P_{i} 
where, σ_{Re} is the radial stress at elasticplastic interface
i.e. (R = R_{e}). From Eq. 15 and 16
at this interface it can be obtained:
Substituting Eq. 17 into Eq. 12 the induced
radial stress can be determined as follows:
The plastic zone radius is determined from Eq. 18 and 13
as:
Assumed the circular space with R_{i} = 1 m and σ_{v} = σ_{h} = 30 MPa. Figure 2 illustrates the effects of internal pressure P_{i} on the plastic zone radius around the circular space by the Tresca criterion for three types of rocks. The plastic zone radius decreases by increasing of UCS, while internal pressure exceeds ground pressure, the results become vice versa. For an invariable UCS, increasing the internal pressure P_{i} decreases the plastic zone radius R_{e}.

Fig. 3: 
The effects of hydrostatic ground pressure on the plastic
zone radius at different compressive strengths 
Figure 3 displays the effects of ground pressure, σ_{v} = σ_{h} on the plastic zone radius by the Tresca criterion for three types of rocks where internal pressure P_{i} is 10 MPa. An increase in ground pressure leads to increase in the plastic zone radius in an invariable UCS. It can be seen that increasing UCS tends to decrease in the plastic zone radius.
Elastoplastic solution around circular space using the generalized HB failure criterion: In the plastic region, radial and tangential stresses can be found.
Thus, induced stresses by introducing Eq. 2 into Eq.
11 are as follows:
The radius of plastic zone can be determined from Eq. 17
and 20 that will be expressed as:
All the parameters in this paper were assumed, where the ground pressure P_{0}
is equal to hydrostatic stresses σ_{v} = σ_{h}. Figure
4 shows the effects of GSI and D on the plastic zone radius by the HB failure
criterion around circular space for different rock types where, m_{i}
= 4, P_{i} = 10 MPa, σv = σh = 30 MPa and σ_{c}
= 20 MPa. An increase in the D factor for the certain GSI leads to increasing
of the plastic zone radius. Whereas increasing the GSI decreases the plastic
zone radius.
Figure 5 shows the effects of internal pressure on the radius of plastic zone by HB failure criterion for different rock types where, σ_{v} = σ_{h} = 60 MPa, m_{i} = 4, σ_{c} = 20 MPa and D = 0.1. It can be understood with increasing of internal pressure the plastic zone radius will decrease in a certain GSI. Figure 6 displays the influence of the GSI on the radial stress around circular space for different rock types with σ_{v} = σ_{h} = 50 MPa, m_{i} = 4, σ_{c} = 20 MPa. It can be observed the radial stress around circular space will decrease when GSI increases.
The influence of ground pressure on tangential stress around circular space has been showed in Fig. 7 by the HB failure criterion with P_{i} = 0, σ_{c} = 20 MPa, D = 05 and GSI = 20. It can be noticed that for a certain ground pressure the increasing of m_{i} will decrease the tangential stress. On the other hand when the ground pressure increases for a fixed m_{i} the tangential stress raises.
Elastoplastic solution around circular space using the MogiCoulomb failure
criterion: As pointed out in Sec. 1.4 the strengthening effect of the intermediate
principal stress can be taking into account by utilizing the MogiCoulomb formula.
In terms of first and second stress invariants I_{1} and I_{2}
defined by AlAjmi and Zimmerman (2006):

Fig. 4: 
Variation of the plastic zone radius with disturbance factor
at different GSI using the HB failure criterion 

Fig. 5: 
Variation of the radius of plastic zone under increasing of
GSI at different internal pressures by the HB failure criterion 

Fig. 6: 
The influence of D on the radial stress at different GSI around
circular space using the HB failure criterion 

Fig. 7: 
The influence of ground pressure on the tangential stress
at different mi around circular space using the HB failure criterion 
Since the maximum and minimum principal stresses, σ_{1} and σ_{2} are corresponded to σ_{θ} and θ_{r} around the underground circular spaces then intermediate principal stress will be σ_{2} = σ_{z} along the circular space axis. In plane strain condition (ε_{z} = 0) the axial stress, σ_{z} can be determined as follows:
where, σ_{a} is the axial in situ stress, v is the Poisson’s ratio. In hydrostatic ground pressure around the circular space, σ_{v} = σ_{h} = σ_{a} the Eq. 15 can be rewritten as:
The radial stress in this case can be determined through Eq.
23 and 11 as follows:

Fig. 8: 
The influence of axial stress on the plastic zone radius at
different internal pressures around the circular space by the MGC failure
criterion 

Fig. 9: 
The influence of compressive strength on the plastic zone
radius at different Poisson’s ratios around circular space using MGC
criterion 
Substituting Eq. 26 and 25 into Eq.
23 the tangential stress around circular space using MGC criterion can
be found as:
As stated in previous sections the plastic zone radius is obtained using continues equations (σ_{Re} = σ_{R}) at elasticplastic interface. Figure 8 represents the influence of intermediate principal stress (axial stress) on the plastic zone radius around the circular space based on the MGC failure criterion. The features of the circular space and rock mass assumed as R_{i} = 1 and cohesion C = 3.45 MPa, v = 0.23 and φ = 30°. It can be understood that an increase in intermediate principal stress in a certain internal pressure, causes decreasing the plastic zone radius.

Fig. 10: 
Variation of the plastic zone radius under increasing rock
cohesion at different friction using the MGC criterion 
Figure 9 indicates the effects of the Poisson’s ratio and uniaxial compressive strength on the plastic zone radius around the circular space by using the MGC failure criterion for different rock strengths. In this case, the internal pressure P_{i} is 10 MPa, ground pressure σ_{h} = σ_{v} is 30 MPa and the initial radius of underground space R_{i} is 1 m. It can be obtained that the plastic zone radius increases by increasing of the Poisson’s ratio in a particular UCS. Also in a fixed Poisson’s ratio an increase in UCS decreases the plastic zone radius. As illustrated in Fig. 10 when cohesion of the rock around the circular space increases, in a fixed friction angle the plastic zone radius decreases. On the other hand by increasing friction angle the plastic zone radius decreases subsequently.
NUMERICAL ANALYSIS
As a common criterion the elastoplastic solution for the circular space using
MC failure criterion has been given in Salencon (1969).
For this reason the solution has not mentioned in the previous section. Here,
the analytical solution of the Tresca, generalized HB, MC and MGC failure
criteria implemented in 3DEC using a FISH program (ITASCA,
2003). Because of axisymmetric and plain strain conditions only onefourth
of the sketch in the Fig. 1 has been modeled.
All the parameters in these solutions were assumed, where the initial radius of the underground circular space R_{i} is 1 m, the in situ stress σ_{h} = σ_{v} is 30 MPa and the internal pressure P_{i} is 5 MPa. The compressive strength of rock mass σ_{c} is 50 MPa, constant parameter m is 4.5, the Geological Strength Index GSI is 40, the Poisson’s ratio v is 0.22, cohesion C is 3.45 MPa and internal friction φ is 30°. Figure 11 compares the results of the generalized HB and Tresca failure criteria with the 3DEC.

Fig. 11: 
The comparison of the radial and tangential stresses using
the Tresca, the generalized HB criteria with the 3DEC around the circular
space 

Fig. 12: 
The comparison of the radial and tangential stresses using
the MGC, the generalized HB criteria with the 3DEC around the circular
space 

Fig. 13: 
The comparison of the radial and tangential stresses using
the MGC and MC criteria with the 3DEC around the circular space 

Fig. 14: 
The influence of intermediate principal stress using MGC
criterion (σ_{2} = 10 and σ_{2} = 15) on the stress
distribution around the circular space 
Table 1: 
The plastic zone radius around the circular space using four
rock failure criteria compared with 3DEC 

It can be seen that stress distribution around the circular space using the generalized HB criterion (the tangential stress in turquoise line and the radial stress in blue line) is close to the 3DEC (the tangential stress in red line and the radial stress in green line) than the Tresca criterion. Because the Tresca criterion does not concern with internal friction of the rock mass then it cannot be used in rock properties determination and the elastoplastic solution around the underground spaces.
The results of the generalized HB and MGC failure criteria, compared with the 3DEC have shown in Fig. 12. The stress distribution around the circular space using the MGC criterion (the tangential stress in turquoise line and the radial stress in brown line) is more similar to the 3DEC than the generalized HB criterion.
In Fig. 13, the comparison of stress distribution using the MGC and the MC criteria has represented. The both results of criteria are similar to the 3DEC, completely. The effect of intermediate principal stress (axial stress in this case) is illustrated in Fig. 14. It can be discerned that an increase in intermediate principal stress from σ_{2} = 10 MPa (the tangential stress in turquoise line and the radial stress in brown line) to σ_{2} = 15 MPa (the tangential stress in cyan line and the radial stress in red line) increases the stress distribution around the circular space using the MogiCoulomb (MGC) failure criterion.
The plastic zone radius around the underground circular space using four rock failure criteria are given in Table 1.
It can be found that the Tresca failure criterion overestimated the plastic zone radius more than other criteria and an increase in axial stress using MGC failure criterion tends to decrease of the plastic zone radius. The generalized HB failure criterion evaluated the least amount with regard to other criteria and numerical analysis in 3DEC.
CONCLUSIONS
Elastoplastic analytical solution of stress distribution around the underground circular space using four rock failure criteria has been given. The following consequences attained:
• 
The Tresca failure criterion overestimated the stress distribution
and radius of the plastic zone around the underground circular space. Because
of neglecting internal friction, this criterion does not answer the rock
mass problems 
• 
Increasing of uniaxial compressive strength, GSI, m_{i} and internal
pressure P_{i} decreases the plastic zone radius and the induced
stresses while an increase in ground pressure P_{0}, disturbance
factor D and Poisson’s ratio v increases the induced stresses and the
plastic zone radius around the underground circular space 
• 
Numerical analysis of elastoplastic solution using four rock failure
criteria has been carried out using 3DEC. The criterion that has closest
fitting to 3DEC was the MohrCoulomb, MogiCoulomb and generalized HoekBrown
failure criterion, respectively 
• 
The advantage of MogiCoulomb failure criterion to the others is that
comprising the axial stress (intermediate principal stress) in this criterion.
It was declared that an increase in axial stress increases the stress distribution
while decrease the plastic zone radius 
• 
The generalized HB failure criterion estimated the smallest amount of
the plastic zone radius visàvis other criteria and 3DEC 