Plastic Radius Research Articles

A finite strain numerical procedure for a circular tunnel in strain-softening rock mass with large deformation

The reliable prediction of excavation responses for squeezing rock remains a challenging issue in rock engineering. This paper presents a numerical procedure for a strain-softening rock mass around highly deformed circular opening based on the finite strain theory, so the overall elasto-plastic analysis is a Lagrangian method which focuses on the motion of each material point and is well suited for the problems involving large deformation. The influence of axial in-situ stress is considered by introducing the initial axial stress into the elastic constitutive equations, and the axial plastic flow is realized by redefining the plastic internal variable and considering different stress distributions in the plastic zone. Thus, the numerical procedure enables prediction of the rock deformation and squeezing potential in a more rational and rigorous manner. The capability of the method for estimating the small strain and squeezing deformation is verified by being compared with the existing analytical results and the field measuring data. Negligence of the axial in-situ stress or the axial plastic flow underestimates the rock deformation, and the induced deviation essentially depends on the post-peak constitutive relation and the rock heterogeneity after rock yields. The higher level of the material homogeneity in the plastic zone, the lower effects of the axial in-situ stress on the rock responses. The stress distribution and strain-softening behavior of surrounding rock can be significantly affected by the axial in-situ stress and the critical plastic internal variable η*. Tunnel instability is independent of large deformation for the high η* but is closely associated with the squeezing problem when η* is small. As η* decreases, both of the plastic radius and wall convergence increase initially and remain constant afterwards. Lastly, the paper investigates the rock responses and ground-support interaction when the small strain theory is applied on the squeezing rock.

Difference solution for a circular tunnel excavated in strain-softening rock mass considering decayed confinement

Recent research regarding the strain-softening behavior of rock masses has demonstrated that the dilatancy angle ψ and the critical softening parameter η∗ are both strongly related to the confining stress. However, in many studies, ψ is assumed to be a constant or to vary linearly, and η∗ is also assumed to be a constant. The assumption of a constant ψ or η∗ is an approximation that does not correctly reflect the variable process in the plastic zone. In this paper, to describe the confinement-dependent characteristics of ψ and η∗ and their effects on the stress and displacement of tunnels in strain-softening rock masses, four dilatancy cases are defined as different combinations of ψ and η∗. First, two dilatancy models, the variable dilatancy model (VDM) and constant dilatancy model (CDM) are introduced. Then, a finite difference method for the strain-softening model is proposed to consider the variation of ψ and η∗ in analyzing the strain-softening behavior of rock masses. The accuracy of this method is verified by comparing the results with those calculated using the methods of Lee and Pietruszczak and Wang et al. Finally, using this proposed method, relevant comparisons are made among the four dilatancy cases for good-quality to poor-quality rock masses to reveal the confinement-dependent effects of ψ and η∗ on the evolution of several parameters, including the softening parameter, the critical softening parameter, the dilatancy coefficient, strength parameters, stress components, ground response curves, and plastic radii. It is concluded that in the plastic zone, ψ and η∗ varies nonlinearly with decreasing confining stress for underground tunnels excavated in different qualities of strain-softening Hoek-Brown rock masses. ψ and η∗ affect each other, and their relationship further influences the transition location from the plastic softening zone to the plastic residual zone. The effects of the confining dependency in ψ and η∗ for good-quality rock masses are less than those for poor quality rock masses. To guarantee simplicity and security in analyzing the stress and displacement of tunnel excavation problems, a constant ψ with a constant η∗ and a variable ψ with a variable η∗ are recommended as analysis models for good-quality and poor-quality rock masses, respectively.

Elasto-plastic solution for shallow tunnel in semi-infinite space

• Elasto-plastic solution for shallow tunnel in a semi-infinite space was proposed. • The gravitational effect on stress in the elastic surrounding rocks is notable. • The influence of gravity on the elasto-plastic model is limited. • Calculations of plastic zone shape are consistent with the results generated by Midas-GTS. This study focuses on a novel approach to develop an elasto-plastic solution for surrounding rocks around a shallow tunnel in an elasto-plastic semi-infinite space that incorporates the gravitational effect based on the bipolar coordinate system. A shallow circular tunnel was excavated in continuous, isotropic, and homogeneous rock masses. The solution for the surrounding rock in semi-infinite space was regarded as a semi-infinite plane problem, and the semi-infinite plane was partitioned along the curve trajectories of the bipolar coordinate. Based on the Mohr-Coulomb failure criterion and considering the gravitational effect, the closed-form solutions for the stress distribution of the surrounding rock were proposed. Furthermore, novel approaches for the calculation of the shape of the plastic zone around the tunnel and the critical internal support pressure were obtained. These presented approaches were validated by comparing the results of finite element method and the published literature. Using the numerical method, the sensitivity of gravity and the influences of the geological strength index, internal support pressure, surface loading, and buried depth of the tunnel on the plastic radius, shape of the plastic zone, and stress distribution of the surrounding rocks around the shallow circular tunnel were discussed.

An improved numerical approach in surrounding rock incorporating rockbolt effectiveness and seepage force

The solutions of stress and displacement of a circular opening excavated in brittle and strain-softening rock mass incorporating rockbolts effectiveness and seepage force are presented in this study. The evolution equation is reconstructed for the strength parameters that incorporate these factors. Based on the evolution equation, an improved numerical method and stepwise procedure are presented which are compatible with the Mohr–Coulomb (M–C) and the generalized Hoek–Brown (H–B) failure criteria, respectively. Then given three interaction mechanisms between rockbolts and surrounding rock, solutions for stress and displacement are proposed in line with the improved numerical method and numerical stepwise procedure. The proposed approach can be reduced to Fahimifar and Soroush’s (Tunn Undergr Space Technol 20:333–343, 2005) solutions for special cases. The proposed method was validated by field monitoring data and FLAC results of Yanzidong tunnel. Examples under the M–C and generalized H–B failure criteria for rock mass are generated through MATLAB programming. Moreover, parametric studies are conducted to highlight the influence of rockbolts effectiveness in combination with seepage force on the stress and displacement of very good, average, and very poor surrounding rock. Results show that in this case, stress confinement is higher and tunnel convergences are lower than the corresponding stresses and displacements obtained in non-reinforced tunnels. Displacement and plastic radius are also higher than those without considering seepage force.

An improved radius-incremental-approach of stress and displacement for strain-softening surrounding rock considering hydraulic-mechanical coupling

This study focused on the mechanical and hydraulic characteristics of underwater tunnels based on Mohr-Coulomb (M-C), Hoek-Brown (H-B) and generalized H-B failure criteria. An improved approach for calculating stress, displacement and plastic radius of the circular tunnel considering hydraulic-mechanical coupling was developed. The innovation of this study was that the radius-incremental-approach was reconstructed (i.e., the whole plastic zone is divided into a finite number of concentric annuli by radius), stress and displacement of each annulus were determined in terms of numerical method and Terzaghi\'s effective stress principle. The validation of the proposed approach was conducted by comparing with the results in Brown and Bray (1982) and Park and Kim (2006). In addition, the Rp-pin curve (plastic radius-internal supporting pressure curve) was obtained using the numerical iterative method, and the plastic radius of the deep-buried tunnel could be obtained by interpolation method in terms of the known value of internal supporting pressure pin. Combining with the theories in Carranza and Fairhurst (2000), the improved technique for assessing the reliability of the tunnel support was proposed.

Convergence-confinement curve analysis of excavation stress and strain resulting from blast-induced damage

Using the convergence-confinement curves methodology, we analysed excavation behaviour for a range of rock masses of different geotechnical qualities, taking into account blast-induced damage. The novelty of the research is (i) we include blast-induced damage to the rock mass in ground reaction curve construction, and (ii) we analyse results for 54 rock mass and rock geotechnical quality scenarios. The research, an application of a previously developed methodology (González-Cao et al., 2013), provides practical guidelines for the preliminary design phase for an excavation resulting from blasting. Our main conclusions are (i) that rock mass quality has a greater bearing on the plastic radius and excavation maximum displacement than blast-induced damage, and (ii) that the plastic radius and maximum displacement around an excavation increase with the level of blast-induced damage, most especially for poor quality rock masses. This would justify the need to limit blast-induced damage in poor quality rock masses.

Influences of seepage force and out-of-plane stress on cavity contracting and tunnel opening

The effects of seepage force and out-of-plane stress on cavity contracting and tunnel opening was investigated in this study. The generalized Hoek-Brown (H-B) failure criterion and non-associated flow rule were adopted. Because of the complex solution of pore pressure in an arbitrary direction, only the pore pressure through the radial direction was assumed in this paper. In order to investigate the effect of out-of-plane stress and seepage force on the cavity contraction and circular tunnel opening, three cases of the out-of-plane stress being the minor, intermediate, or major principal stress are assumed separately. A method of plane strain problem is adopted to obtain the stress and strain for cavity contracting and circular tunnel opening for three cases, respectively, that incorporated the effects of seepage force. The proposed solutions were validated by the published results and the correction is verified. Several cases were analyzed, and parameter studies were conducted to highlight the effects of seepage force, H-B constants, and out-of-plane stress on stress, displacement, and plastic radius with the numerical method. The proposed method may be used to address the complex problems of cavity contraction and tunnel opening in rock mass.

Simple solutions of an opening in elastic-brittle plastic rock mass by total strain and incremental approaches

This study deals with simple solutions for a spherical or circular opening excavated in elastic-brittle plastic rock mass compatible with a linear Mohr-Coulomb (M-C) or a nonlinear Hoek-Brown (H-B) yield criterion. Based on total strain approach, the closed-form solutions of stresses and displacement are derived simultaneously for circular and spherical openings using original H-B and M-C yield criteria. Two simple numerical procedures are proposed for the solution of generalized H-B and M-C yield criteria. Based on incremental approach, the similarity solution is derived for circular and spherical openings using generalized H-B and M-C yield criteria. The classical Runge-Kutta method is used to integrate the first-order ordinary differential equations. Using three data sets for M-C and H-B models, the results of the radial displacements, the spreading of the plastic radius with decreasing pressure, and the radial and circumferential stresses in the plastic region are compared. Excellent agreement among the solutions is obtained for all cases of spherical and circular openings. The importance of the use of proper initial values in the similarity solution is discussed.

A new procedure for ground response curve (GRC) in strain-softening surrounding rock

A new procedure for the ground response curve (GRC) is investigated in strain-softening surrounding rock for a circular opening. The procedure started each step with a radius increment and the analytical solutions of stress and strain in each annulus were presented. The plastic region is divided into a finite number of concentric annuli, whose thickness is uniformly determined by a small radius increment. Combining the equilibrium equation and failure criterion, stress for each annulus can be obtained analytically. The displacement for each step can be calculated analytically through solving the differential equation by invoking flow rule and Hooke’s law. The strains for each annulus can be obtained by the strain-displacement relationship. In the successive manner, the distributions of stress and displacement can be found. It should be noted that the finial stress and displacement at radial direction are the internal support pressure and deformation at the excavation surface, respectively. By assuming different plastic radii (using a plastic radius increment), GRC, the evolution curve of plastic radii and internal support pressure can be obtained analytically. Some numerical and engineering examples are performed to demonstrate the validity of the proposed procedure. It is shown that the results of the proposed procedure at the tunnel crown are basically consistent with field measuring data. The influence of the annulus number, plastic radius increment and dilation on the accuracy of the proposed approach is investigated. Results show that the solutions are more accurate and the calculation efficiency is higher.

The Improvement of Deformation Analysis for Circular Tunnel Based on Hoek-Brown Criterion in Strain-Softening Rock Mass

To overcome the defect that the support pressure is required to be measured or supposed in elasto-plastic analysis in strain-softening rock mass, by the connection of the support pressure imposed on surrounding rock mass, without assuming the support pressure is known a priori, an improved model of solving plastic radius and the distribution of radial displacement around the circular tunnel is proposed. Based on Hoek-Brown yield criterion, the results obtained by strain-softening model are compared with those of elastic perfectly brittle and elastic perfectly plastic models. The examples reveal that the difference of plastic radius obtained by the three models is comparatively smaller, but the difference of radial displacement is comparatively larger. The proposed model overcomes some defects of the previous studies in deformation analysis for tunnel.

Elastoplastic Analysis of Circular Openings in Elasto‐Brittle‐Plastic Rock Mass Based on Logarithmic Strain

Rock‐like materials, such as coal and soft rock, often manifest larger deformation features. The prediction values for displacement and failure region based on the commonly used small strain (SS) theory are generally larger than the field test results. Based on the Euler coordinate system, the logarithmic strain (LS) is employed to describe the actual deformation behavior. Both of the stresses and displacement of circular opening in elasto‐brittle‐plastic rock mass are formulated with differential equations. And a simple approach is proposed to solve the differential equations. The results show that the plastic radius depends on the elastic parameters, that is, Young’s modulus and Poisson’s ratio, which is different from SS theory. And the plastic radius and displacement of LS rock mass are smaller than those of SS rock mass, and the displacement of LS rock mass is absolutely smaller than the excavation radius. The proposed solutions can provide theoretical foundation for the optimization of supporting structure in underground engineering.

Theoretical solutions for displacement and stress of a circular opening reinforced by grouted rock bolt

This paper presented solutions of displacement and stress for a circular opening which is reinforced with grouted rock bolt. It satisfies the Mohr-Coulomb (M-C) or generalized Hoek-Brown (H-B) failure criterion, and exhibits elastic-brittle-plastic or strain-softening behavior. The numerical stepwise produce for strain-softening rock mass reinforced with grouted rock bolt was developed with non-associative flow rules and two segments piecewise linear functions related to a principle strain-dependent plastic parameter, to model the transition from peak to residual strength. Three models of the interaction mechanism between grouted rock bolt and surrounding rock proposed by Fahimifar and Soroush (2005) were adopted. Based on the axial symmetrical plane strain assumption, the theoretical solution of the displacement and stress were proposed for a circular tunnel excavated in elastic-brittle-plastic and strain-softening rock mass compatible with M-C or generalized H-B failure criterion, which is reinforced with grouted rock bolt. It showed that Fahimifar and Soroush's (2005) solution is a special case of the proposed solution for n = 0.5. Further, the proposed method is validated through example comparison calculated by MATLAB programming. Meanwhile, some particular examples for M-C or generalized H-B failure criterion have been conducted, and parametric studies were carried out to highlight the influence of different parameters (e.g., the very good, average and very poor rock mass). The results showed that, stress field in plastic region of surrounding rock with considering the supporting effectiveness of the grouted rock bolt is more than that without considering the effectiveness of the grouted rock bolt, and the convergence and plastic radius are reduced.

Estimation of Maximum Annular Pressure during HDD in Noncohesive Soils

AbstractThis study investigated the effectiveness of estimating the maximum allowable annular pressure during horizontal directional drilling (HDD) in noncohesive soils by using two theoretical cavity-expansion solutions (small strain and large strain); several laboratory and field tests were used for verification. The results indicated that the measurements were overestimated when using the plastic radius equal to two-thirds of the overburden depth and underestimated when using the hoop strain equal to 2%. Although the calculated limit pressures based on the two cavity-expansion solutions were higher than the measured failure pressures, a high correlation (R2 = 0.96) existed between the measurements and the calculations based on the large-strain cavity-expansion solution. This correlation may be used to estimate the maximum allowable annular pressure during HDD in noncohesive soils, provided that further validation is conducted with more field data for deep depths rather than shallow depths.

Numerical approach for strain-softening rock with axial stress

This study proposes a numerical approach that considers the effect of out-of-plane stress for circular tunnels excavated in strain-softening rock. The entire plastic region is divided into a finite number of concentric rings whose thickness is determined to satisfy equilibrium and compatibility equations internally. The increments of stress and strain for each ring are obtained by successively incorporating the effect of out-of-plane stress. The initial value of the outermost ring is determined using equilibrium and compatibility equations. Numerical solutions of the stress, displacement and plastic radius are derived by considering the effect of out-of-plane stress for circular tunnels excavated in strain-softening rock obeying Mohr–Coulomb or generalised Hoek–Brown failure criteria. The proposed approach is compared with previously proposed solutions. Several examples are presented to validate the accuracy of the proposed approach and the effect of different parameters is determined. The results show that the out-of-plane stress influences stress and displacement distributions in the plastic region. The results also show that the residual and softening plastic radii increase with increasing out-of-plane stress when out-of-plane stress is the major or intermediate principal stress, and the residual and softening radii increase with decreasing out-of-plane stress when out-of-plane stress is the minor principal stress.

A Combined Analytical and Numerical Approach for the Evaluation of Radial Loads on the Lining of Vertical Shafts

The evaluation of the load acting on a shaft support is of fundamental importance for the correct dimensioning of the structure. The load acting on the support can appear somewhat complex. One approach to define the load on the lining may be to use the convergence-confinement method (CCM) normally used in the tunneling design. This process involves intersecting the convergence-confinement (CC) curve with the support reaction line. However, in order to be able to adopt this technique, it is necessary to know the radial displacement of the shaft wall at the point in which the support is to be installed. Using the equations of Vlachopoulos and Diederichs (Rock Mech Rock Eng 42:131–146, 2009) the reaction line of the support can be calculated. Numerical models developed with Flac 2D v.6.0 considering the Mohr–Coulomb criterion and an ideal elasto-plastic behavior simulating stepwise excavation and support installation were developed. The relation between applied internal stress and radial displacement of the wall shaft, obtained by the numerical simulation was compared with the CC curve obtained by the CCM and it showed a good match between the two methods. However, an iterative procedure has also been used to insert the reaction line in the CC graph. The result shows lower initial displacements (and therefore greater radial stress) when compared with the values obtained by numerical calculation with the axisymmetric model. It is therefore recommended the combined use of the CCM (analytical method) and the axisymmetric numerical model (step by step simulation) to obtain the values of the final load on the lining and the final plastic radius, necessary for the correct design of supporting structures on the shaft wall.

Three-dimensional plastic stress analysis of subsea tunnels: Nonlinear vs. linear - a comparison

The out-of-plane stress is very influential in the rock yield behavior; however, its effects on plastic stress field around subsea tunnels have not yet been analytically performed due to a lack of understanding of the relationship between the out-of-plane stress and the major/minor principal stress at failure. In this regard, a non-associated ow rule is utilized. With the established equilibrium equation, the nonlinear and linear solutions to the 3-Dimensional plastic stress field around subsea tunnels excavated in a Zhang 3D HB and Drucker-Prager rock mass are derived, and seepage body force is taken into account. After validation, a comparison of the changes in mechanical responses for various parameters in the two solutions is carried out. The analyses show that a significant difference exists in 3-Dimensional plastic stress as well as the plastic radius, which indicates that the analytical solution presented can be effectively used for the realistic design of subsea tunnels.

Solution of Strain-Softening Surrounding Rock in Deep Tunnel Incorporating 3D Hoek-Brown Failure Criterion and Flow Rule

In order to investigate the influence of the intermediate principal stress on the stress and displacement of surrounding rock, a novel approach based on 3D Hoek-Brown (H-B) failure criterion was proposed. Taking the strain-softening characteristic of rock mass into account, the potential plastic zone is subdivided into a finite number of concentric annulus and a numerical procedure for calculating the stress and displacement of each annulus was presented. Strains were obtained based on the nonassociated and associated flow rule and 3D plastic potential function. Stresses were achieved by the stress equilibrium equation and generalized Hoek-Brown failure criterion. Using the proposed approach, we can get the solutions of the stress and displacement of the surrounding rock considering the intermediate principal stress. Moreover, the proposed approach was validated with the published results. Compared with the results based on generalized Hoek-Brown failure criterion, it is shown that the plastic radius calculated by 3D Hoek-Brown failure criterion is smaller than those solved by generalized H-B failure criterion, and the influences of dilatancy effect on the results based on the generalized H-B failure criterion are greater than those based on 3D H-B failure criterion. The displacements considering the nonassociated flow rule are smaller than those considering associated flow rules.

Theoretical Solutions of a Circular Tunnel with the Influence of the Out-of-Plane Stress Based on the Generalized Hoek–Brown Failure Criterion

AbstractThe effect of out-of-plane stress on theoretical solutions of stress, displacement, and plastic radius for a circular tunnel were studied by using the generalized Hoek–Brown (HB) failure criterion and nonassociated flow rule in elastic–brittle–plastic rock masses. A technique of using equivalent Mohr–Coulomb and generalized HB strength parameters was adopted to compare results of the developed method and published results. The presented solution was validated by existing theory. The results of the developed method were found to be larger than those previously published, which were determined through programming.

The Evaluation of the Groundwater Influence on the Stress and Strain Behavior around a Tunnel by Analytical Methods

The presence of an aquifer in the soil causes a change in a relevant way of the stability conditions of a tunnel. The flow of water in the pores, in fact, modifies the stress and strain state of the soil and causes an increase in the thickness of the plastic zone. As a result the loads transmitted on the lining increase. In this study a calculation procedure by finite differences method, able to determine the stress and strain state of the soil in the presence of the water flow in the pores, is presented. More particularly, using the proposed procedure, it is possible to determine the convergence-confinement curve of the tunnel and the trend of the plastic radius varying the internal pressure. A calculation example is used to detect the great influence that the presence of the aquifer has on the stress and strain state of the soil and on the pressure-displacement relationship of the tunnel wall.

Theoretical solution for displacement and stress in strain-softening surrounding rock under hydraulic-mechanical coupling

This study focuses on the stress and displacement of a circular opening that is excavated in a strain-softening rock mass under hydraulic-mechanical coupling. It follows the generalized Hoek-Brown (H-B) failure criterion. Moreover, an improved numerical method and stepwise procedure are proposed. This method considers the deterioration of the strength, deformation, and dilation angle. It also incorporates the hydraulic-mechanical coupling and the variation of elastic strain in the plastic region. Several examples are conducted to demonstrate the validity and accuracy of the proposed solution through MATLAB programming and FLAC software. Parametric studies are also conducted to highlight the influence of hydraulic–mechanical coupling on stress and displacement. Results show that in this case, stress confinement is lower and tunnel convergences are higher than the corresponding stresses and displacements obtained when those factors are not considered. The displacement and plastic radius are also larger than those obtained when hydraulic-mechanical coupling is not considered.