Ground Response Curve Research Articles

Analytical solutions for cavity contraction in strain-softening materials with linear or exponential strength decay

This paper presents an analytical investigation into the contraction of spherical and cylindrical cavities excavated in strain-softening rock masses obeying the Mohr–Coulomb or Tresca yield criterion, with linear or exponential uniaxial compressive strength decay. The derivation of the ground response curves is based on the simplifying assumption that the strains inside the plastic zone are completely plastic. This significantly simplifies the mathematical formulation, enabling the derivation of closed-form solutions. An alternative simplifying approach which partially neglects the elastic strains inside the plastic zone and which is commonly adopted in the literature, is also examined. The accuracy of the simplified solutions is evaluated by comparing their predictions with rigorous solutions obtained by numerical finite-difference analyses. The investigation demonstrates that the proposed closed-form solutions represent a significant improvement on those based on the commonly-made simplifying assumption involving partial neglect of elastic strains.

Macro-micro analysis of ground response to circular tunnel excavation in granular media

Ground reaction in terms of stress and displacement analysis is one of the major challenges in tunnel design and is helpful for understanding the stability around the excavation zone. In this paper, the analysis of stress, strain responses and radial displacement around circular tunnel excavated in cohesive-frictional granular materials is discussed via an innovative multi-scale approach (FEM × DEM) that combined finite element (FEM) and discrete element (DEM) methods and compared to other solutions obtained with conventional constitutive laws. Thanks to simulations, (i) the Ground response curve is then established and provides a better understanding of the complex behaviour in the vicinity of the tunnel; (ii) the prediction by FEM × DEM approach differs considerably from conventional constitutive laws; and (iii) in a unique manner, macroscopic and microscopic analyses are simultaneously performed showing that at the macroscopic level (FE mesh), diffuse failure mode are observed, while at the microscopic level (DEM level), microscopic evolutions are observed to be strictly related to the macroscopic behaviour.

Elastoplastic coupling analysis of surrounding rock-prestressed yielding anchor bolts/cables based on unified strength theory

A whole-process analytical theory for the coupled deformation of deep circular tunnel surrounding rock and prestressed yielding anchor bolt (cable) system is derived based on the unified strength theory and non-associated flow rule. This theory considers the strain softening characteristic of rock mass, the yielding reinforced characteristics of anchor bolt (cable), and the support timing. An analytical method is proposed to calculate the difference in displacement of anchor bolts (cables). This study highlights the importance of monitoring anchor bolts (cables) elongation rate in high-stress tunnels by integrating theoretical analysis with engineering practice. The displacement difference that occurs during the coordinated deformation of the bolts (cables) and the surrounding rock provides the support resistance. Therefore, the safety margin standard of elongation rate can be used to assess the integrity and effectiveness of the support structure during the real-time bolts (cables) action. The present study aims to validate the rationality of the analytical theory by conducting numerical simulations using ABAQUS software. Specifically, we focus on analyzing the ground response curves (GRC) underthe bolt (cable) support and the elongation rate of the bolt (cable). Various factors such as prestress, yielding amount, support timing, intermediate principal stress coefficient, radial/longitudinal spacing, and length are considered tocompare and analyze their effects on the GRC. The results showed that anchor bolts (cables) can significantly reduce the surrounding rock convergence. With prestressed bolt (cable) support, the GRC exhibits a stepwise drop, significantly and timely reducing the support force required for the surrounding rock equilibrium. After setting the yielding structure, the increase in anchor cable stress was limited, thereby slightly increasing the surrounding rock displacement. However, the anchor cable elongation rate significantly decreased, thereby greatly enhancing the support structure safety. The support timing affects the starting point of bolts (cables) function. The displacement control effect is more pronounced when support is provided earlier. However, this also leads to an increase in the elongation rate of the bolt (cable). The intermediate principal stress coefficient reflects the self-bearing capacity of the surrounding rock, and increasing this value can improve the deformation control performance of the surrounding rock under anchor bolt (cable) support as a whole and reduce the elongation rate through the coupled function of the surrounding rock–anchor bolt (cable) system. The spacing between anchor cables significantly impacts the convergence control of surrounding rock. Smaller spacing leads to better control of the surrounding rock deformation and lower stress of the anchor cables. As the length of the anchor cables increases, the elongation rate and stress of the anchor cables continuously decrease. However, the surrounding rock deformation exhibits an initially decreasing and increasing pattern. This paper presents a theoretical and computational approach to analyze the effect of prestressed yielding anchor bolt (cable) support and the safety of support structures in tunnel engineering. The findings of this study can serve as a valuable reference for the convergence analysis of circular tunnels in underground engineering.

Elastic perfect plastic rock mass analysis and equivalent GSI determination considering strain-softening behavior and three-dimensional strength

The elastic perfect plastic model (EPP) fails to accurately capture the strain-softening mechanical behaviors of rock masses, while the post-peak constitutive model of elastic strain softening (ESS) has great application limitations in practical engineering for complex numerical algorithms. In this paper, an EPP constitutive approximation of rock mass considering three-dimensional strength and strain-softening characteristics is proposed. The forward analysis method of the equivalent Geological Strength Index (GSIeq) is given considering the impact of in-situ stress, rock mass properties and engineering excavation. Extensive case studies demonstrate that the radial stress and displacement obtained by the proposed method agree well with those of the ESS model, with only small errors in the Ground Response Curve. The rock mass elastic modulus E and the initial stress level p0 significantly affect GSIeq, while the support pressure ps contributes little to that. As Rp,EPP/R0 > 3.0, usually GSI < 25, the residual strength parameter GSIr can be directly used as the calculation parameter of the EPP model, while for a highly qualified rock mass, such as GSI > 75, the peak strength parameter GSIp can be directly used as an approximate value of GSIeq. Furthermore, with the increase of the in-situ stress level, GSIeq gradually decreases to GSIr, and the difference of plastic stress distribution law based on ESS and EPP models gradually diminishes. Moreover, the proportion of plastic deformation to total deformation increases nonlinearly with buried depth. The proposed method offers a new calculation idea for the simplified analysis of strain-softening rock masses.

Numerical solution for circular tunnel excavated in strain-softening rock masses considering damaged zone

Despite the extensive investigation on the stress and displacement distributions in tunnels, few have considered the influences of the damaged zone around a tunnel on the strength and stiffness parameters of the surrounding rock, including the gradual variation in the damaged factor D and dimensionless damaged radius rho^{{text{d}}}, under the effect of excavation disturbance. In this paper, a numerical solution is presented for the stresses and displacement of a circular tunnel excavated in a Hoek–Brown rock mass considering the progressive destruction of the damaged zone. First, the results obtained using the proposed algorithm are compared with those obtained using other numerical solutions, demonstrating a high degree of accuracy through some examples. Second, the influences of the damaged factor D and dimensionless damaged radius rho^{d} on the stresses, radial displacement, plastic radii, and ground response curve are investigated. The results show that, as the damaged factor D increases, the radial displacement and plastic radius increase, whereas the tangential stress decreases. Both the plastic radius and displacement decrease with decreasing rho^{{text{d}}}. This shows that the damaged factor D has a significant effect on tunnel convergence.

A Proposed Algorithm to Compute the Stress-Strain Plastic Region and Displacement of a Deep-Lying Tunnel Considering Intermediate Stress and Strain-Softening Behavior

Past studies on deep-lying tunnels under the assumption of plane strain have generally neglected the influence of intermediate principal stress even though this affects the surrounding rocks in the plastic zone. This study proposes a finite difference method to compute the stress strain plastic region and displacement of a tunnel based on the Drucker–Prager (D–P) yield criterion and non-associated flow rule and considering the influences of intermediate principal stress and the strain-softening behavior of surrounding rock. The computed results were compared with those of other well-known solutions and the accuracy and validity of the method were confirmed through some examples. Parameter analysis was conducted to investigate the effects of intermediate principal stress on stress-strain, the plastic region, the ground response curve, and the dilatability of surrounding rock. The results showed that the plastic radius Rp, the residual radius Rd, and radial displacement of surrounding rock first decreased and then increased with increasing intermediate principal stress coefficient b from 0 to 1, with the minimums occurring at b = 0.75. On the contrary, the peak and rate of variation of the dilatancy coefficient first increased and then decreased with increasing b and the dilatancy coefficient Kψ gradually transitioned from nonlinear to linear variation. Meanwhile, the inhibition of the plastic radius and radial displacement gradually weakened with increasing support pressure, whereas the dilatancy coefficient of the tunnel opening gradually increased.

A Closed-Form Elastic Solution of Ground Response Curve for Noncircular Openings

Abstract Ground response curves (GRCs) can be used to characterize the evolution law of stress and displacement during tunnel excavation. In this study, an attempt is made to obtain the GRC and fin...

Plasticity solutions for ground deformation prediction of shallow tunnels in undrained clay

Elastic-plastic contraction solutions of a cavity within an infinite soil mass are widely used to estimate ground response curves of deep tunnels. The pressure-convergence response of soil around shallow tunnels, however, tends to vary with tunnel embedment ratio. To account for the embedment ratio effect, this paper presents a general solution procedure for undrained contraction analysis of a thick-wall cylinder or spherical shell of isotropically hardening soils, by which a set of analytical and semi-analytical large strain solutions for several Cam-Clay soil models is derived. Results obtained with the new solutions show that the cavity contraction response highly depends on the soil stress history (overconsolidation ratio) and the radial thickness of the surrounding soil. A limit ratio of the outer to the inner radii exists, beyond which the thickness ratio effect becomes negligible. The limit ratio for a spherical shell is smaller than that for a hollow cylinder, and it decreases with the overconsolidation ratio of soil. Based on the analogy between cavity contraction and tunnel convergence, theoretical methods for predicting ground response curves of shallow plane-strain tunnels in clays are proposed and validated by comparing with relevant experimental and numerical simulation results. The solutions can also provide valuable benchmark methods for verifying various numerical programmes.

Modified ground response curve (GRC) in strain-softening rock mass based on the generalized Zhang-Zhu strength criterion considering over-excavation

The ground response curve (GRC) depicts the relationship between support reaction force and ground displacement, which improves the understanding of ground-support interaction and provides important references to the tunnel design. However, it is difficult to anticipate the tunneling-induced large deformation with sufficient reliability in soft rock with high geostress since the small strain theory is not applicable. When large deformation occurs, the tunnel needs to be over-excavated. Thus, the GRC should be modified considering the enlarged excavation radius since the actual excavation radius is usually greater than the designed one. To overcome the shortcomings of small strain theory in recognizing ground-support interaction under large deformation circumstances, a new large strain numerical approach for modifying the GRC was proposed considering over-excavation in strain-softening rock masses based on the generalized Zhang-Zhu strength criterion. A case study was conducted based on the Lianchengshan tunnel in China. The modified GRC was employed to investigate the ground-support behavior for different support schemes and to explore the applicability of the stress release measures. Combined with field tests, the proposed approach was validated. By comparing with GRCs proposed by previous work, the present modified GRC was proved to be superior to others. Parametric studies were conducted and it is found that over-excavation, for example, reserving a very large clearance between the surrounding rock and the support, is necessary to reduce ground pressure to a large extent. The yielding supports which can provide high support pressure during the process of deformation are highly recommended when tunneling in high geostress environment. However, if the initial geostress is not very high, it is not necessary to pursue unwarranted over-excavation since the ground pressure applied on the support is mainly the loosening stress when the deformation is large. Ample support stiffness should be provided in the process of deformation to prevent uncontrolled large deformation of surrounding rock.

Numerical solutions for strain-softening surrounding rock under three-dimensional principal stress condition

Abstract In this study, elastoplastic model for the surrounding rock of axisymmetric circular tunnel is investigated under three-dimensional (3D) principal stress states. Novel numerical solutions for strain-softening surrounding rock were first proposed based on the modified 3D Hoek–Brown criterion and the associated flow rule. Under a 3D axisymmetric coordinate system, the distributions for stresses and displacement can be effectively determined on the basis of the redeveloped stress increment approach. The modified 3D Hoek–Brown strength criterion is also embedded into finite element software to characterize the yielding state of surrounding rock based on the modified yield surface and stress renewal algorithm. The Euler implicit constitutive integral algorithm and the consistent tangent stiffness matrix are reconstructed in terms of the 3D Hoek–Brown strength criterion. Therefore, the numerical solutions and finite element method (FEM) models for the deep buried tunnel under 3D principal stress condition are presented, so that the stability analysis of surrounding rock can be conducted in a direct and convenient way. The reliability of the proposed solutions was verified by comparison of the principal stresses obtained by the developed numerical approach and FEM model. From a practical point of view, the proposed approach can also be applied for the determination of ground response curve of the tunnel, which shows a satisfying accuracy compared with the measuring data.

A New Procedure for Calculation of Ground Response Curve of a Circular Tunnel Considering the Influence of Young’s Modulus Variation and the Plastic Weight Loading

In this paper, an analytical–numerical solution is presented for a circular tunnel in the hydrostatic condition using a generalized nonlinear Hoek–Brown failure criterion. To a deep insight into the problem, the influence of the plastic zone weight, and the Young's modulus on the ground response curve of a deep tunnel in a strain-softening media is highlighted. Experimental results show that the magnitude of the confined stress can have a direct effect on the critical softening parameter. Thus, a variable critical softening parameter is also assumed. That is, as the radial stress magnitude increases, the critical softening parameter is enhanced. The equations are derived for different directions (e.g. crown, floor, and sidewall) around the circular tunnel. The accuracy of the proposed method is validated by some examples. The results show that assuming a constant Young's modulus in the plastic zone imposes a significant error in the predicting of the tunnel convergence.

The arching effect in rubber–sand mixtures

Arching effect in geomaterial refers to the stress redistribution as a result of induced displacements. It is most likely to occur in different situations such as retaining walls, piled foundations, tunnelling, and so on. The arching effects that occur in rubber–sand layers are examined in this paper. The digital image correlation technique and a series of pressure sensors were employed to capture the deformation characteristics and stress evolution developed in the sand and rubber–sand layers, which were subjected to active arching with and without a central surcharge in a trapdoor apparatus. A range of deformation measures, including horizontal and vertical displacements and strains, shear strains, and volume change variables were obtained and compared for different backfill materials. Ground response curves for sand and rubber–sand mixtures were developed using the stress measurement data. From the deformation data, an arch of equal displacement was introduced. It was observed that the use of rubber particles reduced the surface settlement of the sand layer and the arch heights, and higher stress reduction was obtained in the rubberised backfills.

Analytical study on pretensioned bolt-cable combined support of large cross-section tunnel

To study the mechanical responses of large cross-section tunnel reinforced by pretensioned rock bolts and anchor cables, an analytical model is proposed. Considering the interaction between rock mass and bolt-cable support, the strain softening characteristic of rock mass, the elastic-plastic characteristic of bolt-cable support, and the delay effect of installation are considered in the model. To solve the different mechanical cases of tunneling reinforced by bolt-cable support, an analytical approach has been put forward to get the solutions of stress and displacement associated with tunneling. The proposed analytical model is verified by numerical simulation. Moreover, parametric analysis is performed to study the effects of pretension force, cross-section area, length, and supporting density of bolt-cable support on tunnel reinforcement, which can provide references for determining these parameters in tunnel design. Based on the analytical model, a new Ground Response Curve (GRC) considering the reinforcement of bolt-cable support is obtained, which shows the pretension forces and the timely installation are important in bolt-cable support. In addition, the proposed model is applied to the analysis of the Great Wall Station Tunnel, a high-speed railway tunnel with a super large cross-section, which shows that the analytical model of bolt-cable support was a useful tool for preliminary design of large cross-section tunnel.

A novel numerical-iterative-approach for strain-softening surrounding rock incorporating rockbolts effectiveness and hydraulic-mechanical coupling based on Three-Dimensional Hoek-Brown strength criterion

The numerical solutions for strain-softening surrounding rock under 3D Hoek-Brown strength criterion are developed incorporating the influence of passive bolts and the seepage flow field. Based on 3D Hoek-Brown failure criterion, a numerical-increment-approach is reconstructed for the first time under the spatial axisymmetric conditions, in which the elastic and plastic zones are divided into a finite number of concentric annuli with constant increments of radial stress and radius between the adjacent annuli, respectively. Furthermore, the hydraulic-mechanical coupling model is improved by considering the effects of out-of-plane plastic strain and the spring-slider model is improved by considering the axial force at rockbolt head. On the basis of the reconstructed numerical-increment-approach and the improved spring-slider model, the bolt self-equilibrium conditions for five cases of passive bolts are also reconstructed (e.g., neutral point in elastic or plastic zones, neutral length in elastic, plastic or elastic-plastic zones, respectively). Ultimately, an improved numerical-iterative-approach is proposed in terms of the developed numerical-increment-approach and the reconstructed bolt self-equilibrium conditions, through which stresses and displacement of surrounding rock can be obtained by incorporating the effects of hydraulic-mechanical coupling and passive bolts. The corresponding calculation program is also presented. The reliability of the presented approach has been verified by the numerical and existed results. From a practical point of view, the proposed approach can be applied to estimate the reinforced ground response curves and optimize the length of passive bolts.

A new analytical-numerical solution to analyze a circular tunnel using 3D Hoek-Brown failure criterion

In this study, a new analytical-numerical procedure is developed to give the stresses and strains around a circular tunnel in rock masses exhibiting different stress-strain behavior. The calculation starts from the tunnel wall and continues toward the unknown elastic-plastic boundary by a finite difference method in the annular discretized plastic zone. From the known stresses in the tunnel boundary, the strains are calculated using the elastic-plastic stiffness matrix in which three dimensional Hoek-Brown failure criterion (Jiang and Zhao 2015) and Mohr-Coulomb potential function with proper dilation angle (i.e., non-associated flow rule) are employed in terms of stress invariants. The illustrative examples give ground response curve and show correctness of the proposed approach. Finally, from the results of a great number of analyses, a simple relationship is presented to find out the closure of circular tunnel in terms of rock mass strength and tunnel depth. It can be valuable for the preliminary decision of tunnel support and for prediction of tunnel problems.

Analysis of the effects of blast-induced damage zone with attenuating disturbance factor on the ground support interaction

ABSTRACT The rock mass properties are typically influenced by the excavation technique and the changes in the state of stress due to the rock excavation. The amount and extent of damage introduced in the rock mass depends on the excavation technique and practice quality. The influence of blasting in the rock mass near the tunnel periphery is far more significant due to the energy of waves and redistribution of stresses and the severity of the damage diminishes as the radial distance from the tunnel opening increases. Therefore, it is important to consider the effect of the damaged zone when analysing the stresses and deformations around a tunnel. This study aimed at providing a new numerical solution for the determination of the ground response (reaction) curve with the consideration of the non-uniform damage zone around the tunnel periphery. A deep circular tunnel subjected to hydrostatic stress condition and excavated in rock materials obeying the Hoek-Brown failure criterion is considered. A solution for the determination of stresses, strains, and deformations around the circular deep tunnel is presented in order to correctly assess the attenuation of damaged rock as the distance from the tunnel perimeter increases considering the loads applied to the supporting structure.

Ground response curve of deep circular tunnel in rock mass exhibiting Hoek–Brown strain-softening behaviour considering the dead weight loading

In definition of the ground response curve for a tunnel, convergences of side wall, roof and floor may be different, because of the weight of the failed or plastic rock mass. In this paper, an analytical-numerical solution of the ground response curve for a tunnel excavated in an elastoplastic strain-softening rock mass considering the gravitational loading. In the proposed method, Hoek–Brown yield criterion and Mohr-Coulomb plastic potential function are used for the ground medium. The gravity loading is considered as a radial body force being applied to the ground medium. Two flexible and inflexible lining theories are introduced for the analysis of concrete lining. The derived differential equations cannot be integrated analytically; thus, they are solved by invoking finite difference approximations. Several illustrative examples are given to demonstrate the performance of the proposed method and to examine the effects of the gravity loading. The results obtained by the proposed method show that the gravity loads may affect the tunnel convergence, especially when the plastic region is wide.

Prediction of Top Coal Cavability Character of a Deep Coal Mine by Empirical and Numerical Methods

Longwall top coal caving mining provides high productivity where thick coal seams exist. The study aims to predict the cavability character of the top coal for deep and thick coal seam in Soma lignite basin located at Western Turkey. Active longwall top coal caving mines are at a depth of 100–400 m and they were used for comparison purposes. New coal mining operations will be initiated in deep sectors of the basin in the future. Future longwall top coal caving operations will be unique under a depth of 700–1200 m with a varied thickness. Several empirical and numerical methods are utilized in the study. Pre-existing empirical approaches lack of essential data and additional numerical modeling is necessary to be employed in order to assess cavability character of the projected new mines. Numerical modeling provides a practical platform for construction of ground response curves. Existing mines and future mining operations can be evaluated and compared by ground reaction curves and a final conclusion about cavability character is reached.

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.

Mechanical Analysis of the Circular Tunnel considering the Interaction between the Ground Response Curve and Support Response Curve

The viewpoint that the ground initial elastic displacement and the interaction between the ground response curve (GRC) and support response curve (SRC) in the surrounding rock should be considered at the same time in the mechanical analysis of the circular tunnel is proposed, and its solution method is also established. Meanwhile, in order to consider the effect of the intermediate principle stress, Drucker-Prager criterion is introduced to describe the yield property of the surrounding rock. The calculation example indicates that the final radial displacement of the tunnel circumference will increase when the ground initial elastic displacement in the surrounding rock is considered before the support structure is applied, which indicates that it is necessary to consider the ground initial elastic displacement in the surrounding rock before the support structure is applied. With increasing the support resistance force and the initial field stress, the plastic zone radius in the surrounding rock and the radial displacement of the tunnel circumference will decrease and increase, respectively, while with increasing the rock internal friction angle and cohesion, the plastic zone radius in the surrounding rock and the radial displacement of the tunnel circumference both decrease. Meanwhile, with the stress Lode parameter increasing from −1 to 1, the plastic zone radius in the surrounding rock and the radial displacement of the tunnel circumference both greatly decrease and then slightly increase. It indicates that the intermediate principle stress has some effect on the calculation results.