A Closed-Form Solution for Circular Openings in an Elastic-Brittle-Plastic Extended Spatial Mobilized Plane Medium

Abstract

Based on the extended spatial mobilization plane (SMP) criterion, we present an elastic-brittle-plastic solution for an axisymmetric cylindrical tunnel. The influences of the intermediate principal compressive stress and material strain-softening behavior are considered. Closed-form formulas for the critical support force, radius of plastic zone, and distributions of stress and displacement in surrounding rock are proposed. The elastic-plastic solution based on SMP is compared with the Kastner solution to verify the credibility of the obtained elastic-plastic solution. The elastic-brittle-plastic solution following the SMP criterion and the current solution based on the Mohr-Coulomb criterion are also compared. The rock strain-softening rate and the intermediate principal stress affect the stability of the surrounding rock. The results provide guidance for optimizing the design of support systems for tunnels.

Introduction

Symmetrical circular tunnels are widely considered in geotechnical engineering, and many closed-form solutions and numerical methods have been developed for design of their support systems (Detournay, 1986; Wang, 1996; Carranza-Torres and Fairhurst, 1999; Egger, 2000; Alonso et al., 2003; Wang et al., 2010; Osman and Randolph, 2011; Zareifard and Fahimifar, 2014; Fahimifar et al., 2015; Zhao et al., 2016; Katebian and Molladavoodi, 2017). Many studies have focused on analytical solutions for a circular underground opening in brittle-plastic rock mass governed by the linear Mohr-Coulomb (M-C) and the nonlinear Hoek-Brown (H-B) failure criteria. Brown et al. (1983) reviewed previous work, and developed a closed-form solution for an elastic-brittle-plastic H-B medium and a numerical solution for a strain-softening H-B medium. Ogawa and Lo (1987) derived an analytical solution and analyzed the effects of dilatancy and yield criteria on the displacement and stress around tunnels. Sharan (2003, 2005) presented closed-form solutions for elastic-brittle-plastic H-B rock. Park and Kim (2006) derived a closed-form solution for a circular tunnel in a brittle-plastic rock mass governed by the H-B yield criterion, and compared the displacement in the plastic zone under different elastic strain conditions.

However, these previous solutions consider 2D failure criteria, which neglect the effect of the intermediate principal stress (σ₂) on the strength parameter and displacement of the surrounding rocks under excavation. To address these issues, this work considers the effect of the intermediate principal stress for an elastic-brittle-plastic medium. The extended spatially mobilized plane (SMP) is employed for frictional and cohesive materials in 3D stress space (termed ‘extended SMP’) (Nakai and Matsuoka, 1983; Matsuoka and Sun, 1995). This paper presents an analytical solution for circular tunnel excavation in brittle-plastic M-C and extended SMP media considering the effect of the intermediate principal stress. The solution was transformed from brittle-plastic solutions to perfect-plastic solutions when setting the residual strength parameters equal to the peak values. Comparison of the displacements and stresses of the solutions with the Kastner solution indicates that brittle-plastic behavior of rock mass greatly extends the plastic zone and the displacements around the cavity due to the effect of the intermediate principal stress. Ground response curves are produced for both M-C and extended SMP criteria, and the effects of the intermediate principal stress on cavity wall convergence and the plastic radius are discussed.

Model Description

Problem Definition

Fig. 1 shows a circular tunnel being excavated in an initially elastic rock mass subjected to a hydrostatic stress p₀. The tunnel surface is subjected to an internal pressure p. A plastic region develops around the opening when p reaches the yield stress. Upon yielding, the rock’s strength suddenly drops, and it then follows post-yield mechanical behavior as shown in Fig. 2. The H-B and M-C criteria are the most commonly used strength criteria for rock masses, both of which neglect the effect of the intermediate principal stress on the stress and displacement of the elastic and plastic regions. The extended SMP failure criterion, which considers the effect of the intermediate principal stress σ₂, is adopted here to solve the stresses and displacements in the plastic region.

Fig. 1. A circular opening in an infinite medium. r_i and r_y are the radii of the tunnel and plastic region, respectively, p is the internal pressure and p₀ is the hydrostatic stress of the rock mass.

Fig. 2. Relationship between stress and strain for an elastic-brittle-plastic medium.

The extension of SMP for cohesive and frictional materials by Matsuoka and Sun (1995) and Nakai and Matsuoka (1983) defined the extended SMP failure criterion as follows:

\(\begin{aligned}\frac{\hat{\tau}_{S M P}}{\hat{\sigma}_{S M P}}=\frac{2}{3} \sqrt{\frac{\left(\hat{\sigma}_{1}-\hat{\sigma}_{2}\right)^{2}}{4 \hat{\sigma}_{1} \hat{\sigma}_{2}}+\frac{\left(\hat{\sigma}_{2}-\hat{\sigma}_{3}\right)^{2}}{4 \hat{\sigma}_{2} \hat{\sigma}_{3}}+\frac{\left(\hat{\sigma}_{3}-\hat{\sigma}_{1}\right)^{2}}{4 \hat{\sigma}_{3} \hat{\sigma}_{1}}}=const,\end{aligned}\)       (1)

where, \(\begin{aligned}\hat{\sigma}_{S M P}\end{aligned}\) and \(\begin{aligned}\hat{\tau}_{S M P}\end{aligned}\) are normal stress and shear stress in extended SMP failure criterion, and \(\begin{aligned}\hat{\sigma}_1\end{aligned}\), \(\begin{aligned}\hat{\sigma}_2\end{aligned}\), and \(\begin{aligned}\hat{\sigma}_3\end{aligned}\) are the principal stresses for a cohesive material, and are given as

\(\begin{aligned}\begin{array}{l}\hat{\sigma}_{1}=\sigma_{1}+\sigma_{0} \\ \hat{\sigma}_{2}=\sigma_{2}+\sigma_{0} \\ \hat{\sigma}_{3}=\sigma_{3}+\sigma_{0}\end{array}\end{aligned}\),       (2)

where, σ₀ = c cot 𝜙, and c and 𝜙 are the cohesion and friction angle of the material, respectively.

Expressing the condition on the octahedral plane yields the SMP results as a smooth curve representing the Mohr-Coulomb criterion (solid line in Fig. 3) circumscribing a hexagon (dashed lines in Fig. 3). For a cohesive material, the extended SMP failure criterion for a planar strain state can be obtained based on the following equation:

\(\begin{aligned}\sigma_{2}+\sigma_{0}=\sqrt{\left(\sigma_{1}+\sigma_{0}\right)\left(\sigma_{3}+\sigma_{0}\right)}\end{aligned}\).       (3)

The axial symmetry of the problem leads to the radial stress σ_r and tangential stress σ_θ in the rock mass being the principal stresses, such that σ_{1 =} σ_r and σ_{3 =} σ_θ.

For a planar strain problem, Eq. (1) is rewritten as follows:

(i) in an elastic-plastic medium

\(\begin{aligned}\frac{\sigma_{\theta}+\sigma_{0}}{\sigma_{r}+\sigma_{0}}=\frac{1}{4}\left(A_{0}+\sqrt{B_{0}-2 A_{0}}-1\right)^{2}=R_{P S 0}\end{aligned}\),       (4)

(ii) in an elastic-brittle medium (Fig. 2)

\(\begin{aligned}\frac{\sigma_{\theta}+\sigma_{01}}{\sigma_{r}+\sigma_{01}}=\frac{1}{4}\left(A_{01}+\sqrt{B_{01}-2 A_{01}}-1\right)^{2}=R_{P S 1}\end{aligned}\),       (5)

where, \(\begin{aligned}A_{01}=\sqrt{8 \tan ^{2} \phi_{T C D}+9} ; A_{1}=\sqrt{8 \tan ^{2} \phi_{T C 1}+9} ; B_{0}=\sqrt{8 \tan ^{2} \phi_{T C D}+6} ; B_{01}=\sqrt{8 \tan ^{2} \phi_{T C 1}+6} ; \sigma_{0}=\end{aligned}\)= c_TC0/tan𝜙_TC0 ; and σ₀₁ = c_TC1/tan𝜙_TC1. R_PS0 and R_PS1 represent the ratio of major stress to minor stress at the peak strength and residual strength, respectively. 𝜙_TC0 and 𝜙_TC1 are the friction angle of the initial and residual states, respectively, under triaxial compressive stress (Fig. 3).

Fig. 3. Failure surfaces in the π-plane. σ₁, σ₂ and σ₃ are the principal stresses.

Analytical Solution for Stresses and Displacement in the Plastic Region

Critical Internal Pressure p₀^cr

Lamé’s solutions for the elastic region are as follows (Fig. 1):

\(\begin{aligned}\begin{array}{l}\sigma_{r}=p_{0}\left(1-\frac{r_{i}^{2}}{r^{2}}\right)+p_{i}\left(\frac{r_{i}}{r}\right)^{2} \\ \sigma_{\theta}=p_{0}\left(1+\frac{r_{i}^{2}}{r^{2}}\right)-p_{i}\left(\frac{r_{i}}{r}\right)^{2,}\end{array}\end{aligned}\)       (6)

\(\begin{aligned}u=\frac{(1+\nu)\left(p_{0}-p_{i}\right) r_{i}^{2}}{E r}.\end{aligned}\)       (7)

An unsupported opening starts to fail when the in situ stress exceeds the critical pressure p₀^cr. By assuming that the plastic zone reaches the wall of the opening, the critical internal pressure p₀^cr is found by substituting Eq. (6) with r = r_i in Eq. (4):

\(\begin{aligned}p_{0}^{c r}=\frac{\left(R_{P S 0}-1\right) \sigma_{0}}{2}.\end{aligned}\)       (8)

For an opening with a supporting system (supporting force, p_i), when p₀ exceeds p₀^cr, the minimum value of p_i can be solved for the surroundings of the opening with no plastic zone:

\(\begin{aligned}p_{i}^{c r}=\frac{\sigma_{0}\left(1-R_{P S 0}\right)+2 p_{0}}{1+R_{P S 0}}.\end{aligned}\)       (9)

Stresses in the Plastic Region

Substituting the equilibrium equation in Eq. (5) obtains the stress in the plastic region for the extended SMP failure criterion:

\(\begin{aligned}\begin{array}{l}\sigma_{r}^{p}=-\sigma_{01}+\left(p_{i}+\sigma_{01}\right)\left(\frac{r_{i}}{r}\right)^{1-R_{P S 1}} \\ \sigma_{\theta}^{p}=-\sigma_{01}+R_{1}\left(p_{i}+\sigma_{01}\right)\left(\frac{r_{i}}{r}\right)^{1-R_{P S 1.}}\end{array}\end{aligned}\)       (10)

Based on equation (10), the tangential stress p_y on the elastic-plastic boundary is

\(\begin{aligned}p_{y}=\left.\sigma_{r}^{p}\right|_{r=r_{y}}=-\sigma_{01}+\left(p_{i}+\sigma_{01}\right)\left(\frac{r_{i}}{r_{y}}\right)^{1-R_{P S 1}}.\end{aligned}\)       (11)

Replacing r_i and p_i with r_y and p_y in Eq. (6) yields the stress in the plastic region:

\(\begin{aligned} \sigma_{r}^{e} & =p_{0}\left(1-\frac{r_{y}^{2}}{r^{2}}\right)+p_{y} \frac{r_{y}^{2}}{r^{2}} \\ \sigma_{\theta}^{e} & =p_{0}\left(1+\frac{r_{i}^{2}}{r^{2}}\right)-p_{y} \frac{r_{y}^{2.}}{r^{2}}\end{aligned}\)       (12)

Substituting Eq. (12) with r = r_y in Eq. (4), we obtain p_y. Considering the radial stress continuity at the elastic-plastic boundary, the plastic radius r_y is

\(\begin{aligned}r_{y}=r_{i}\left[\frac{\sigma_{0}\left(1-R_{P S 0}\right)+\sigma_{01}\left(1+R_{P S 0}\right)+2 p_{0}}{\left(R_{P S 0}+1\right)\left(p_{i}+\sigma_{01}\right)}\right]^{\frac{1}{R_{P S 1}-1}}.\end{aligned}\)       (13)

For no stress-drop or strain-softening, the radius of the plastic region can be found by replacing σ₀ and R_PS0 with σ₀₁ and R_PS1:

\(\begin{aligned}r_{y 0}=r_{i}\left[\frac{\sigma_{0}\left(1-R_{P S 0}\right)+\sigma_{0}\left(1+R_{P S 0}\right)+2 p_{0}}{\left(R_{P S 0}+1\right)\left(p_{i}+\sigma_{0}\right)}\right]^{\frac{1}{R_{P S 0}-1}}.\end{aligned}\)       (14)

Displacement in the Plastic Region

According to classical plasticity theory, the total strain consists of elastic and plastic parts. For planar strain, neglecting volumetric strain, the relationship between the radial and tangential strains is

\(\begin{aligned}\frac{d u}{d r}+\frac{u}{r}=0.\end{aligned}\)       (15)

From the condition of compatibility in the case of infinitesimal deformation, the displacement at the elastic-plastic boundary u^p is

\(\begin{aligned}u^{p}=\frac{r_{y}^{2}(1+\nu)\left(p_{0}-p_{y}\right)}{E r}.\end{aligned}\)       (16)

The displacement at the opening wall u_i is

\(\begin{aligned}u_{i}=\left.u^{p}\right|_{r=r_{i}}=\frac{r_{y}^{2}(1+\nu)\left(p_{0}-p_{y}\right)}{E_{i}}.\end{aligned}\)       (17)

The closed-form solution for the perfectly elastic-plastic problem can be obtained by replacing σ₀₁ and R_PS1 with σ₀ and R_PS0, respectively.

Comparison of Solutions

As rocks are strain-softening materials exhibiting peak and residual strength, it is reasonable to assume that they obey elastic-brittle-plastic behavior given that the mechanical parameters are accurate. Therefore, the effects of the strength parameters and internal pressure on the convergence of the opening wall and the plastic radius in the brittle-plastic extended SMP rock mass are assessed by comparing the presented solution with the Kastner solution (based on the M-C failure criterion) in both perfect-plastic and brittle-plastic media. Table 1 lists the geometric and mechanical parameters for three cases with different residual cohesions.

Table 1. Geometric and mechanical parameters

The failure stress p₀^SMP and critical support pressure p_i^SMP in M-C and extended SMP media can be obtained using Eqs. (8) and (9). The failure stress p₀^SMP (11.8 MPa) in the extended SMP medium is larger than that in the M-C medium (9.4 MPa). The critical support pressure p_i^SMP (9.5 MPa) in the extended SMP medium is smaller than that in the M-C medium (11.8 MPa). The result implies that σ₂ strengthens the opening wall in the extended SMP medium.

Perfect-Plastic Medium

Effect of Internal Cohesion and Friction Angle on the Plastic Radius

Fig. 4a shows the effect of internal cohesion on the dimensionless plastic radii in M-C and extended SMP media. The dimensionless plastic radius is the ratio of the plastic radius to the initial tunnel radius (r_y / r_i). For c > 3 MPa, a variation in c has a slight effect on the dimensionless plastic radius. In contrast, for c < 2 MPa, any variation in c has a strong effect on r_y / r_i. The ratio in the extended SMP medium is smaller than that in the M-C medium by 12.8% for c = 6 MPa and by 29.6% for c = 1 MPa.

Fig. 4. Effect of (a) internal friction angle and (b) cohesive force on the dimensionless plastic radii in perfect-plastic M-C and extended SMP media.

Fig. 4b shows the effect of friction angle on the dimensionless plastic radii in the M-C and extended SMP media. The trends are similar in both media. For 20° < 𝜙 < 30°, the variation in 𝜙 has a slight effect on the ratio. In contrast, when 𝜙 < 20°, r_y / r_i is strongly affected by any variation in 𝜙.

Effect of σ₂ on Displacement and Stress in the Plastic Region

Fig. 5a shows the stresses in the M-C and extended SMP media, in the case of p₀ (30 MPa) > p₀^SMP and p₀^MC. The hoop stress of the opening wall σ_θ^SMP in the M-C medium (23.5 MPa) is larger than that in the extended SMP medium (18.8 MPa). The maximum value of σ_θ^SMP (50.5 MPa) at the elastic-plastic boundary is larger than σ_θ^MC (48.2 MPa). The plastic radius r_y^SMP (4.1 m) is smaller than r_y^MC (4.7 m). The radial stress in the plastic region σ_r^SMP is larger than σ_r^MC, which implies that the rock is strengthened by σ₂.

Fig. 5. (a) Radial and hoop stresses and (b) radial displacement in perfect-plastic M-C and extended SMP media.

Fig. 5b shows the displacement in the plastic region. The plastic radius r_y^SMP in the extended SMP medium is 15.6% smaller than r_y^MC. The convergence displacement of the opening wall u_i^SMP in the extended SMP medium is 18.7% larger than that in M-C medium u_i^MC.

Fig. 6 shows the ground reaction curves in M-C and extended SMP perfect-plastic rock masses. The supporting force p_i/p₀ is decreased due to the presence of σ₂.

Fig. 6. Ground reaction curves for M-C and extended SMP perfect-plastic rock masses.

Elastic-Brittle-Plastic Medium

Effect of σ₂ on Stress, Plastic Radii, and Ground Reaction Curves

Table 1 lists the residual strength parameters for a brittle-plastic medium. Fig. 7a shows the stress and displacement in the plastic region. The plastic radius r_y^SMP (4.9 m) in the extended SMP medium is smaller than r_y^MC (6.1 m). The convergence displacement of the opening wall u_i^SMP (4.8 mm) in the extended SMP medium is 18.7% smaller than that in the M-C medium u_i^MC (5.7 mm). The radial stress in the plastic region σ_r^SMP is larger than σ_r^MC. The hoop stress of the opening wall σ_θ^SMP in the extended SMP medium (11.7 MPa) is larger than that in the M-C medium (9.4 MPa).

Fig. 7. (a) Radial and hoop stresses and (b) dimensionless radial displacement in M-C and extended SMP brittle-plastic media.

The maximum value of σ_θ^SMP (50.5 MPa) at the elastic-plastic boundary is larger than σ_θ^MC (48.2 MPa). These results suggest that the rock strength increases with increasing σ₂. Fig. 7b shows that the plastic radius r_y^SMP (4.9 m) is 24.6% smaller than r_y^MC (6.13 m) and that u_i^SMP (7.1 mm) in the extended SMP medium is 37.3% smaller than that in the M-C medium u_i^MC (9.7 mm). Fig. 8 shows that the supporting force (p_i/p₀) in the extended SMP medium is smaller than that in the M-C medium, and the dimensionless convergence displacement (u/r_i) in the extended SMP medium is also smaller than that in the M-C medium. These results indicate that the rock strength is significantly affected by σ₂.

Fig. 8. Ground reaction curves for M-C and extended SMP brittle-plastic rock masses.

Effect of Strength Reduction on Plastic Radius and Opening Wall Convergence

Fig. 9a shows the effects of residual cohesion on the dimensionless radial displacement of the opening (u/r_i). The plastic radius r_y^SMP increases by 24% as the residual cohesion decreases from 6 to 2 MPa. In addition, the critical supporting force (p_i/p₀) was not affected by the reduction in residual cohesion (Fig. 9b). The displacement of the plastic region increases by 100% as the residual cohesion is reduced from 6 to 2 MPa.

Fig. 9. (a) Dimensionless radial displacement and (b) ground reaction curves for an extended SMP brittle-plastic rock mass.

Conclusions

To estimate the effects of the intermediate principal stress on the plastic zone and the convergence of circular excavation in an elastic-brittle-plastic medium, closed-form solutions for both M-C and extended SMP media are presented for stresses and displacements in the plastic region.

The results for displacements and stresses are compared with the Kastner solution. The plastic radius and displacement obtained for an extended SMP medium increase significantly compared with those obtained for an M-C medium. The solutions can be used to investigate the effects of internal pressure on the convergence of opening and the extent of the plastic region in a brittle-plastic medium. The strength in the residual plastic region is higher in the extended SMP medium than in the M-C medium. For a decreasing residual strength of the rock mass, the opening wall convergence increases sharply when the residual cohesion decreases substantially from 6 to 2 MPa. This means that the stability of the underground opening is enhanced by the intermediate principal stress and the critical supporting pressure is decreased, which reduces the cost of the supporting system.