Discussion on “A generalized three-dimensional failure criterion for rock masses”

There are many methods to construct true triaxial strength criteria for rocks.Jaiswal and Shrivastva(2012)proposed a strength criterion,named J–S criterion,in the deviatoric plane,which provides nearly the same misfts for true triaxial test data as the exponential criterion.It is diffcult to calculate the strength at given2and3using the J–S criterion,and the multiple solutions to the nonlinear equation may induce confusion and mistake.Strength envelopes in deviatoric planes are not geometric similar;therefore,true triaxial test data cannot be grouped in the mean stress to check strength criteria in the deviatoric plane.     

A simplified three-dimensional extension of Hoek-Brown strength criterion

The Hoek-Brown(HB) strength criterion has been applied widely in a large number of projects around the world.However,this criterion ignores the intermediate principal stress σ_2.Many evidences have demonstrated that the rock strength is dependent on σ_2. Thus it is necessary to extend the HB criterion into a three-dimensional(3D) form.In this study,the effect of σ_2 on the strength of rocks is identified by reviewing the true triaxial tests of various rock types reported in the literature.A simple 3D strength criterion is developed.The modified criterion is verified by the true triaxial tests of 13 rock types.The results indicate that the modified criterion can achieve a good fit to most of rock types.It can represent a series of criteria as b varies.For comparisons,several existing 3D versions of the HB criterion are selected to predict the strengths of these rock types.It is indicated that the proposed criterion works better than other criteria.A substantial relationship between parameter b and the unconfined compressive strength is established,which guarantees that the proposed criterion can still work well even in the absence of true triaxial test data.     

True-triaxial strength criteria for rock

This paper reviews some strength criteria which include the role of the intermediate principal stress, and proposes a new criterion. Strength criteria of the form σ_oct=f_N(σ_oct), such as Drucker–Prager, represent a rotation surface in the principal stress space, symmetric to the line σ₁=σ₂=σ₃ in the meridian plane. Because σ_oct=f_N(σ_oct) must fit the pseudo-triaxial compressive strength, it will have a non-physical outcome for triaxial extension. Mogi's criteria, σ_oct=g₁(σ_m,2) and σ_max=g₂(σ_β) are able to fit experimental data reasonably well, but the prediction of strength is not good and sometimes problematic. Strength criterion with the form λ(σ₁, σ₂, σ₃)=F[η(σ₁, σ₂, σ₃)], or a curve of two variables which can be decided by fitting pseudo-triaxial experimental data, is not expected to describe the strength under various stress states, no matter how high the correlation coefficient of λ and η is, or how low the misfit of the equation λ=F(η) is, as these seemingly good correlations usually result from the dominant influence of the maximum principal stress in the metrics of λ and η. The intermediate principal stress may improve the strength of rock specimen, but its influence will be restricted by σ₃. Also when σ₂ is high enough to cause failure in the σ₂–σ₃ direction, the strength will decrease with the increasing σ₂. The new strength criterion with exponent form has just such a character, and gives much lower misfits than do all seven criteria discussed by Colmenares and Zoback [Colmenares LB, Zoback MD. A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks. Int J Rock Mech Min Sci 2002;39:695–729].     

An analytical solution to wellbore stability using Mogi-Coulomb failure criterion

Deep wellbores/boreholes are generally drilled into rocks for oil and gas exploration, monitoring of tectonic stresses purposes. Wellbore and tunnel in depth are generally in true triaxial stress state, even if the ground is under axisymmetric loading condition. Stability of such wellbores is very critical and collapse of wellbore must be avoided. Mogi-Coulomb failure criterion is a better representation of rock strength under true triaxial condition. In this paper, an analytical solution is proposed using Mogi-Coulomb failure criterion. The solution is obtained for rock mass exhibiting elastic-perfectly plastic or elastic-brittle-plastic behaviour considering in-plane isotropic stresses. The proposed solution is then compared with exact analytical solution for incompressible material and experimental results of thick-wall cylinder. It is shown that the results obtained by the proposed analytical solution are in good agreement with the experimental results and exact analytical solution. A reduction of about 13%–20% in plastic zone from the proposed closed-form solution is observed, as compared to the results from the finite element method (FEM) based Mohr-Coulomb criterion. Next, the influences of various parameters such as Poisson's ratio, internal pressure (mud weight), dilation angle, and out-of-plane stress are studied in terms of stress and deformation responses of wellbore. The results of the parametric study reveal that variation in the out-of-plane stress has an inverse relation with the radius of plastic zone. Poisson's ratio does not have an appreciable influence on the tangential stress, radial stress and radial deformation. Dilation angle has a direct relation with the deformation. Internal pressure is found to have an inverse relation with the radial deformation and the radius of plastic zone.     

Consolidation behavior of dredged ultra-soft soil improved with prefabricated vertical drain at the Mae Moh mine,Thailand

The effectiveness of the prefabricated vertical drains (PVDs) in the consolidation of ultra-soft dredged soil with various soil water contents (W) in Mae Moh mine, Lampang, Thailand was researched via a series of large-scale model tests and numerical analysis. Large settlements with the delay of excess pore pressures is a distinct behavior of ultra-soft soil. The PVD dimensions were found to have a significant effect on the rate of consolidation and the delay of excess pore pressure at low total vertical stress (σ_v). The smaller PVD dimension resulted in the smaller rate of consolidation and longer delay of excess pore pressure. The undrained shear strength (S_u) of ultra-soft clay at various degrees of consolidation could be approximated by the vertical effective stress (σ′_v) based on the SHANSEP where the σ′_v was determined from the Asaoka's observational method. The finite element analysis with axisymmetric and plane strain models showed that the axisymmetric model produced an excellent settlement prediction. However, the excess pore pressures were not well predicted by the axisymmetric model, due to the delay of excess pore pressures at the early stages of consolidation. In practice, the plane strain models proposed by Chai et al. and Indraratna and Redana's methods are suggested to predict the consolidation settlement of the Mae Moh dredged soil improved with PVD. The outcome of this research will facilitate the geotechnical design of reclamation of ultra-soft dredged soil in Mae Moh mine and other similar soils.     

Stress redistribution and ground arch development during tunneling

Three-dimensional stress adjustment caused by tunneling is related with ground arching development. Investigating the development of ground arching by stress redistribution during tunneling is presented. A three-dimensional numerical simulation model was developed to gain understanding of adjustment of tunneling stress. Vertical stress adjustments σ_ZZ above the tunnel roof during tunneling are analyzed, using four different cover depths H and four lateral pressure coefficients K. Shift of the inflection points of σ_ZZ-depth curves above the tunnel roof during tunneling can be a reliable index to show the development of ground arching.     

Seismic bearing capacity of a strip footing on rock media

The bearing capacity factors for a rough strip footing placed on rock media, which is subjected to pseudo-static horizontal earthquake body forces, have been determined using the lower bound finite element limit analysis in conjunction with the power cone programming (PCP). The rock mass is assumed to follow the generalized Hoek-Brown (GHB) yield criterion. No assumption needs to be made to smoothen the GHB yield criterion and the convergence is found to achieve quite rapidly while performing the optimization with the usage of the PCP. While incorporating the variation in horizontal earthquake acceleration coefficient (k_h), the effect of changes in unit weight of rock mass (γ), ground surcharge pressure ($q_0$) and the associated GHB material shear strength parameters (geological strength index (GSI), yield parameter (m_i), uniaxial compressive strength (σ_ci)) on the bearing capacity factors has been thoroughly assessed. Non-dimensional charts have been developed for design purpose. The accuracy of the present analysis has been duly checked by comparing the obtained results with the different solutions reported in the literature. The failure patterns have also been examined in detail.     

Experimental verification of parameter m in Hoek–Brown failure criterion considering the effects of natural fractures

Hoek–Brown failure criterion is one of the widely used rock strength criteria in rock mechanics and mining engineering. Based on the theoretical expression of Hoek–Brown parameter m of an intact rock, the parameter m has been modified by crack parameters for fractured rocks. In this paper, the theoretical value range and theoretical expression form of the parameter m in Hoek–Brown failure criterion were discussed. A critical crack parameter B was defined to describe the influence of the critical crack when the stress was at the peak, while a parameter b was introduced to represent the distribution of the average initial fractures. The parameter m of a fractured rock contained the influences of critical crack (B), confining pressure (σ₃) and initial fractures (b). Then the triaxial test on naturally fractured limestones was conducted to verify the modification of the parameter m. From the ultrasonic test and loading test results of limestones, the parameter m can be obtained, which indicated that the confining pressure at a high level reduced the differences of m among all the specimens. The confining pressure σ₃ had an exponential impact on m, while the critical crack parameter B had a negative correlation with m. Then the expression of m for a naturally fractured limestone was also proposed.     

Risk of shear failure and extensional failure around over-stressed excavations in brittle rock

The authors investigate the failure modes surrounding over-stressed tunnels in rock.Three lines of investigation are employed:failure in over-stressed three-dimensional(3D) models of tunnels bored under 3D stress,failure modes in two-dimensional(2D) numerical simulations of 1000 m and 2000 m deep tunnels using FRACOD,both in intact rock and in rock masses with one or two joint sets,and finally,observations in TBM(tunnel boring machine) tunnels in hard and medium hard massive rocks.The reason for 'stress-induced' failure to initiate,when the assumed maximum tangential stress is approximately(0.4-0.5)σ_c(UCS,uniaxial compressive strength) in massive rock,is now known to be due to exceedance of a critical extensional strain which is generated by a Poisson's ratio effect.However,because similar 'stress/strength' failure limits are found in mining,nuclear waste research excavations,and deep road tunnels in Norway,one is easily misled into thinking of compressive stress induced failure.Because of this,the empirical SRF(stress reduction factor in the Q-system) is set to accelerate as the estimated ratio σ_(θmax)/σ_c 0.4.In mining,similar 'stress/strength' ratios are used to suggest depth of break-out.The reality behind the fracture initiation stress/strength ratio of '0.4' is actually because of combinations of familiar tensile and compressive strength ratios(such as 10) with Poisson's ratio(say0.25).We exceed the extensional strain limits and start to see acoustic emission(AE) when tangential stress σθ≈ 0.4σc,due to simple arithmetic.The combination of 2D theoretical FRACOD models and actual tunnelling suggests frequent initiation of failure by 'stable' extensional strain fracturing,but propagation in 'unstable' and therefore dynamic shearing.In the case of very deep tunnels(and 3D physical simulations),compressive stresses may be too high for extensional strain fracturing,and shearing will dominate,both ahead of the face and following the face.When shallower,the concept of 'extensional strain initiation but propagation' in shear is suggested.The various failure modes are richly illustrated,and the inability of conventional continuum modelling is emphasized,unless cohesion weakening and friction mobilization at different strain levels are used to reach a pseudo state of yield,but still considering a continuum.     

A modified empirical criterion for strength of transversely anisotropic rocks with metamorphic origin

A modified empirical criterion is proposed to determine the strength of transversely anisotropic rocks. In this regard, mechanical properties of intact anisotropic slate obtained from three different districts of Iran were taken into consideration. Afterward, triaxial rock strength criterion introduced by Rafiai was modified for transversely anisotropic rocks. The criterion was modified by adding a new parameter $ \alpha $ for taking the influence of strength anisotropy into consideration. The results obtained have shown that the parameter $ \alpha $ can be considered as the strength reduction parameter due to rock anisotropy. The modified criterion was compared to the modified Hoek–Brown (Saroglou and Tsiambaos) and Ramamurthy criteria for different anisotropic rocks. It was concluded that the criterion proposed in this paper is a more accurate and precise criterion in predicting the strength of anisotropic rocks.     

Effects of freestream turbulence on streamwise pressure measured on a square-section cylinder

Wind tunnel experiments were performed to study the effects of freestream turbulence on the flow around a square cylinder. The results indicate that the pressure fluctuations on streamwise surfaces are produced mainly by periodic vortex shedding when turbulence intensity, σ_u/U, is less than 12%. As σ_u/U approaches 20%, vortex shedding weakens significantly and pressure fluctuations occur over a broad range of frequency. Flow visualization experiments show that extreme negative pressure peaks in highly turbulent flow are associated with intermittent reattachment of separated shear layers. The results also show that an increase in turbulence scale causes peak pressure coefficients to increase.     

Shear strength response of reinforced pond ash

Reinforced pond ash is a composite material, which can be used as an alternative construction material in the field of geotechnical engineering. To study the shear strength response of reinforced pond ash, a series of unconsolidated undrained (UU) triaxial test has been conducted on both unreinforced and reinforced pond ash. In the present investigation the effects of confining pressure (σ₃), number of geotextile layers (N), and types of geotextiles on shear strength response of pond ash are studied. The results demonstrate that normal stress at failure (σ_1f) increases with increase in confining pressure. The rate of increase of normal stress at failure (σ_1f) is maximum for three layers of reinforcement, while the corresponding percentage increase in σ_1f is around (103%), when the number of geotextile layers increases from two layers to three layers of reinforcement. With increase in confining pressure the increment in normal stress at failure, Δσ increases and attains a peak value at a certain confining pressure (threshold value) after that Δσ becomes more or less constant. The threshold value of confining pressure depends on N, dry unit weight (γ_d) of pond ash, type of geotextile, and also type of pond ash.     

Deformation and failure characteristics and fracture evolution of cryptocrystalline basalt

Cryptocrystalline basalt is one of the two major types of rocks exposed in the super large-scale underground powerhouse in Baihetan hydropower station in China.The rock of this type shows various sitespecific mechanical responses(e.g.fragmentation,fracturing,and relaxation) during excavation.Using conventional triaxial testing facility MTS 815.03,we obtained the stress-strain curves,macroscopic failure characteristics,and strength characteristics of cryptocrystalline basalt.On this basis,evolution of crack initiation and propagation was explored using the finite-discrete element method(FDEM) to understand the failure mechanism of cryptocrystalline basalt.The test results showed that:(1) under different confining stresses,almost all the pre-peak stress-strain curves of cryptocrystalline basalt were linear and the post-peak stresses decreased rapidly;(2) the cryptocrystalline basalt showed a failure mode in a form of fragmentation under low and medium confining stresses while fragmentation-shear coupling failure dominated at high confining stresses;and(3) the initial strength ratio(σ_(ci)/σ_f,where σ_(ci)and σ_f are the crack initiation strength and peak strength,respectively) ranged from 0.45 to 0.55 and the damage strength ratio(σ_(cd)/σ_f,where σ_(cd) is the crack damage strength) exceeded 0.9.The stress—strain curve characteristics and failure modes of cryptocrystalline basalt could be reflected numerically.For this,FDEM simulation was employed to reveal the characteristics of cryptocrystalline basalt,including high σ_(cd)/σ_f values and rapid failure after σ_(cd),with respect to the microscopic characteristics of mineral structures.The results showed that the fragmentation characteristics of cryptocrystalline basalt were closely related to the development of tensile cracks in rock samples prior to failure.Moreover,the decrease in degree of fragmentation with increasing confining stress was also correlated with the dominant effect of confining stress on the tensile cracks.     

Modelling tensile/compressive strength ratio of artificially cemented clean sand

The present work proposes a new theoretical model for predicting both the splitting tensile strength (q_t) and the compressive strength (q_u) of artificially cemented sand and assesses their ratio for a given material. The proposed model is based on the concept of the superposition of the failure strength contributions of the sand and cement phases. The sand matrix obeys the concept of critical state soil mechanics, while the strength of the cemented phase can be described using the Drucker-Prager failure criterion. The analytical solutions are compared against the results of tests on three different types of cemented clean sand cured for different time periods. While the analytical relation fits the experimental data well, it also provides a theoretical basis for the explanation of some features related to the experimentally derived strength relationships for cemented clean sand. The value of the power relationship between the strength and the porosity/cement ratio index seems to be governed by the soil matrix properties, while the interdependency of the strength and the curing time can also be captured. For a given cemented sand, the model equally confirms the existence of a unique tensile/compressive strength ratio (q_t/q_u), independent of the curing time and primarily governed by the compressive to tensile strength ratio (or the friction properties) of the cement. It is also confirmed that the q_t/q_u ratio changes within a narrow range for different frictional properties of the cementing phase.     

Poroelastic contribution to the reservoir stress path

Pore pressure/stress coupling is the change in the smaller horizontal stress σ_h associated with changes in pore pressure P, and has been measured in numerous reservoirs worldwide. These measurements suggest that the change in minimum horizontal stress Δσ_h is on average ca. 64% of the change in the reservoir pore pressure ΔP, but can be as low as 34% and as high as 118%. Conventionally it is assumed that the total vertical stress σ_v, given by the overburden, is not affected by changes in pore pressure, in contrast to the horizontal stresses σ_H and σ_h. We investigate analytically and numerically the spatio-temporal pore pressure and stress evolution in poroelastic media for continuous fluid injection at a point source, and calculate from the numerical modelling results the ratio Δσ/ΔP. Analytically, we show that the measured average of Δσ_h/ΔP can mathematically be deduced from the long-term limit of the spatio-temporal evolution of pore pressure and horizontal stress caused by fluid injection at a point source. We compare our numerical results to the analytical solution for continuous point injection into homogeneous poroelastic media as well as to Δσ_h/ΔP values measured in the field, and show that all stress components change with a variation in P. We use the concept of poroelasticity to explain the observed coupling between pore pressure and stress in reservoirs, and we consider different measurement locations and measurement times as one possible reason for the measured variation in Δσ_h/ΔP in different oil fields worldwide.     

Linearization of the Hoek and Brown rock failure criterion for tunnelling in elasto-plastic rock masses

We present a novel methodology for estimation of equivalent Mohr–Coulomb strength parameters that can be used for design of supported tunnels in elasto-plastic rock masses satisfying the non-linear empirical Hoek–Brown failure criterion. We work with a general adimensional formulation of the Hoek–Brown failure criterion in the space of normalized Lambe's variables for plane stress, and we perform linearization considering the stress field in the plastic region around the tunnel. The procedure is validated using analytical solutions to a series of benchmark test cases. Numerical solutions are also employed to validate the procedure in cases for which analytical solutions are not available. Results indicate that the stress field in the plastic region around the tunnel, as well as the linearization method employed and the quality of the rock mass, has a significant impact on computed estimates of equivalent Mohr–Coulomb strength parameters. Results of numerical analyses also show that our proposed linearization method can be employed to estimate loads and moments on the tunnel support system. We recommend the equating model responses (EMR) method to compute equivalent Mohr–Coulomb strength parameters when the tunnel support pressure is accurately known, and we further show that our newly introduced linearization method can be employed as an alternative to the best fitting in the existing stress range (BFe) and best fitting in an artificial stress range (BFa) methods, providing performance estimates that are generally better than estimates of the BFe and BFa methods when differences with the response of the Hoek–Brown rock mass are of engineering significance (say more than 10%).     

Mechanical Properties of Lightweight Treated Soil Cured in Water Pressure

An artificial lightweight soil has been developed as a backfill to reduce the earth pressure behind port and harbor structures. To reduce the unit weight lightening ingredient such as air foam or EPS beads is mixed within slurry of dredged soft clay, while cement is used as stabilizer to warrant compressive strength. This experimental study aims to characterize the strength and deformation properties of lightweight treated soil cured in water pressure. Samples of two types of lightweight treated soil mixed with air foam or EPS were cured under various pressures, and subjected to undrained shearing tests on triaxial apparatus modified to detect volumetric change. Though high pressures inevitably compress lightener and consequently incur increment in unit weight, pressured curing did not reduce the compressive strength, q_max = (σ_a-σ_c)_max. It was also found that the deformation modulus E₅₀ greatly decreases with relative confining pressure σ_c/q_max. The lightweight soils maintained relatively large residual strengths, showing no significant sign of brittle failure as often confronted in unconfined compression test. It was observed that the critical state line exists when subjected to ultimate strains, and that the peak deviator stress envelop was identified in effective stress path plane for air foam mixed cases alone. K₀-consolidation tests were conducted on modified triaxial apparatus, showing that K₀ values from the quasi one dimensional tests decline to as small as 0.1 to 0.15 around axial strain of 0.5~1% at near yielding points. Poisson's ratios based on both undrained shearing and K₀-consolidation are compared in consistent tendency with minimal values of 0.1 to 0.2 near the identical yielding points. Yet it is revealed from the obtained compression curves that the compressibility increases drastically by some 100-fold when comparing before and after yielding for lightweight treated soil. This fact strikes the importance of not overloading lightweight treated soil by its compressive strength.     

Numerical modeling of time-dependent deformation and induced stresses in concrete pipes constructed in Queenston shale using micro-tunneling technique

Effects of time-dependent deformation (TDD) on a tunnel constructed using the micro-tunneling technique in Queenston shale (QS) are investigated employing the finite element method. The TDD and strength parameters of the QS were measured from tests conducted on QS specimens soaked in water and lubricant fluids (LFs) used in micro-tunneling such as bentonite and polymer solutions. The numerical model was verified using the results of TDD tests performed on QS samples, field measurements of some documented projects, and the closed-form solutions to circular tunnels in swelling rock. The verified model was then employed to conduct a parametric study considering important micro-tunneling design parameters, such as depth and diameter of the tunnel, in situ stress ratio (K_o), and the time lapse prior to replacing LFs with permanent cement grout around the tunnel. It was revealed that the time lapse plays a vital role in controlling deformations and associated stresses developed in the tunnel lining. The critical case of a pipe or tunnel in which the maximum tensile stress develops at its springline occurs when it is constructed at shallow depths in the QS layer. The results of the parametric study were used to suggest recommendations for the construction of tunnels in QS employing micro-tunneling.     

Effective stress law for the permeability of a limestone

The effective stress law for the permeability of a limestone is studied experimentally by performing constant-head permeability tests in a triaxial cell with different conditions of confining pressure σ and pore pressure p_f. Test results show that a pore pressure increase and a confining pressure decrease both result in an increase of the permeability, and that the effect of the pore pressure change on the variation of the permeability is more important than the effect of a change of the confining pressure. A power law is proposed for the variation of the permeability with the effective stress (σ′=σ–n_kp_f). The permeability-effective stress coefficient n_k increases linearly with the differential pressure and is greater than unity as soon as the differential pressure exceeds few bars. The test results are well reproduced using the proposed permeability-effective stress law. A conceptual pore-shell model based on a detailed observation of the microstructure of the studied limestone is proposed. This model is able to explain the experimental observations on the effect of the total stress and of the pore pressure on the permeability of the limestone. Effective stress coefficients for the stress-dependent permeability which are greater than one are obtained. It is shown that the controlling factor is the ratio of the different bulk moduli of the various constituents of the rock. This ratio is studied experimentally by performing microhardness tests.     

A generalized three-dimensional failure criterion for rock masses

The smooth convex generalized failure function, which represents 1/6 part of envelope in the deviatoric plane, is proposed. The proposed function relies on four shape parameters (Ls, a, b and c), in which two parameters (a and b) are dependent on the others. The parameter L s is called extension ratio. The proposed failure function could be incorporated with any two-dimensional (2D) failure criteria to make it a three-dimensional (3D) version. In this paper, a mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is presented. The Hoek-Brown failure criterion is the most suited 2D failure criterion for geomaterials. Two types of analyses for best-fitting solution of published true tri-axial test data were made by considering (1) constant extension ratio and (2) variable extension ratio. The shape and strength parameters for different types of rocks have been determined by best-fitting the published true tri-axial test data for both the analyses. It is observed from the best-fitting solution by considering uniform extension ratio (Ls) that shape constants have a correlation with Hoek-Brown strength parameters. Thus, only two parameters (σc and m) are needed for representing the 3D failure criterion for intact rock. The statistical expression between shape and Hoek-Brown strength parameters is given. In the second analysis, when considering varying extension ratio, another parameter fis introduced. The modified extension ratio is related to f and extension ratio. The results at minimum mean misfit for all the nine rocks indicate that the range of f varies from 0.7 to 1.0. It is found that mean misfit by considering varying extension ratio is lower than that in the first analysis. But it requires three parameters. A statistical expression between f and Hoek-Brown strength parameters has been established. Though coefficient of correlation is not reasonable, we may eliminate it as an extra parameter. At the end of the paper, a methodology has also been given for its application to isotropic jointed rock mass, so that it can be implemented in a numerical code for stability analysis of jointed rock mass structures.