Bayesian data analysis to quantify the uncertainty of intact rock strength

One of the main difficulties in the geotechnical design process lies in dealing with uncertainty. Uncertainty is associated with natural variation of properties, and the imprecision and unpredictability caused by insufficient information on parameters or models. Probabilistic methods are normally used to quantify uncertainty. However, the frequentist approach commonly used for this purpose has some drawbacks.First, it lacks a formal framework for incorporating knowledge not represented by data. Second, it has limitations in providing a proper measure of the confidence of parameters inferred from data. The Bayesian approach offers a better framework for treating uncertainty in geotechnical design. The advantages of the Bayesian approach for uncertainty quantification are highlighted in this paper with the Bayesian regression analysis of laboratory test data to infer the intact rock strength parameters σand mused in the Hoek-Brown strength criterion. Two case examples are used to illustrate different aspects of the Bayesian methodology and to contrast the approach with a frequentist approach represented by the nonlinear least squares(NLLS) method. The paper discusses the use of a Student’s t-distribution versus a normal distribution to handle outliers, the consideration of absolute versus relative residuals, and the comparison of quality of fitting results based on standard errors and Bayes factors. Uncertainty quantification with confidence and prediction intervals of the frequentist approach is compared with that based on scatter plots and bands of fitted envelopes of the Bayesian approach. Finally, the Bayesian method is extended to consider two improvements of the fitting analysis. The first is the case in which the Hoek-Brown parameter, a, is treated as a variable to improve the fitting in the triaxial region. The second is the incorporation of the uncertainty in the estimation of the direct tensile strength from Brazilian test results within the overall evaluation of the intact rock strength.

An analytical model for estimating rock strength parameters from small-scale drilling data

The small-scale drilling technique can be a fast and reliable method to estimate rock strength parameters. It needs to link the operational drilling parameters and strength properties of rock. The parameters such as bit geometry, bit movement, contact frictions and crushed zone affect the estimated parameters.An analytical model considering operational drilling data and effective parameters can be used for these purposes. In this research, an analytical model was developed based on limit equilibrium of forces in a Tshaped drag bit considering the effective parameters such as bit geometry, crushed zone and contact frictions in drilling process. Based on the model, a method was used to estimate rock strength parameters such as cohesion, internal friction angle and uniaxial compressive strength of different rock types from operational drilling data. Some drilling tests were conducted by a portable and powerful drilling machine which was developed for this work. The obtained results for strength properties of different rock types from the drilling experiments based on the proposed model are in good agreement with the results of standard tests. Experimental results show that the contact friction between the cutting face and rock is close to that between bit end wearing face and rock due to the same bit material. In this case,the strength parameters, especially internal friction angle and cohesion, are estimated only by using a blunt bit drilling data and the bit bluntness does not affect the estimated results.

Rock burst risk evaluation based on equivalent surrounding rock strength

On-site investigations consistently show that the rock burst inherent to coal seams varies greatly with coal seam thickness.In this study,impact factors related to coal seam thickness and surrounding rock strength were analyzed and a corresponding rock burst risk assessment method was constructed.The model reflects the influence of coal seam thickness on the stress distribution of surrounding rock at the roadway.Based on the roadway excavation range,a stress distribution model of surrounding roadway rock is established and the influence of coal seam thickness on rock burst risk is analyzed accordingly.The proposed rock burst risk assessment method is based on the equivalent surrounding rock strength and coal seam bursting liability.The proposed method was tested in a 3500 mining area to find that it yields rock burst risk assessment results as per coal seam thickness that are in accordance with real-world conditions.The results presented here suggest that coal seam thickness is a crucial factor in effective rock burst risk assessment.

Influence of heterogeneity on rock strength and stiffness using discrete element method and parallel bond model

The particulate discrete element method(DEM) can be employed to capture the response of rock,provided that appropriate bonding models are used to cement the particles to each other.Simulations of laboratory tests are important to establish the extent to which those models can capture realistic rock behaviors.Hitherto the focus in such comparison studies has either been on homogeneous specimens or use of two-dimensional(2D) models.In situ rock formations are often heterogeneous,thus exploring the ability of this type of models to capture heterogeneous material behavior is important to facilitate their use in design analysis.In situ stress states are basically three-dimensional(3D),and therefore it is important to develop 3D models for this purpose.This paper revisits an earlier experimental study on heterogeneous specimens,of which the relative proportions of weaker material(siltstone) and stronger,harder material(sandstone) were varied in a controlled manner.Using a 3D DEM model with the parallel bond model,virtual heterogeneous specimens were created.The overall responses in terms of variations in strength and stiffness with different percentages of weaker material(siltstone) were shown to agree with the experimental observations.There was also a good qualitative agreement in the failure patterns observed in the experiments and the simulations,suggesting that the DEM data enabled analysis of the initiation of localizations and micro fractures in the specimens.

New true-triaxial rock strength criteria considering intrinsic material characteristics

A reasonable strength criterion should reflect the hydrostatic pressure effect, minimum principal stress effect,and intermediate principal stress effect. The former two effects can be described by the meridian curves, and the last one mainly depends on the Lode angle dependence function. Among three conventional strength criteria, i.e.Mohr–Coulomb(MC), Hoek–Brown(HB), and Exponent(EP) criteria, the difference between generalized compression and extension strength of EP criterion experience a firstly increase then decrease process, and tends to be zero when hydrostatic pressure is big enough. This is in accordance with intrinsic rock strength characterization. Moreover, the critical hydrostatic pressure Icorresponding to the maximum difference of between generalized compression and extension strength can be easily adjusted by minimum principal stress influence parameter K. So, the exponent function is a more reasonable meridian curves, which well reflects the hydrostatic pressure effect and is employed to describe the generalized compression and extension strength.Meanwhile, three Lode angle dependence functions of L,L, and L, which unconditionally satisfy the convexity and differential requirements, are employed to represent the intermediate principal stress effect. Realizing the actual strength surface should be located between the generalized compression and extension surface, new true-triaxial criteria are proposed by combining the two states of EP criterion by Lode angle dependence function with a same lode angle. The proposed new true-triaxial criteria have the same strength parameters as EP criterion. Finally, 14 groups of triaxial test data are employed to validate the proposed criteria. The results show that the three new true-triaxial exponent criteria,especially the Exponent Willam-Warnke criterion(EPWW)criterion, give much lower misfits, which illustrates that the EP criterion and Lhave more reasonable meridian and deviatoric function form, respectively. The proposed new true-triaxial strength criteria can provide theoretical foundation for stability analysis and optimization of support design of rock engineering.

Influence of some rock strength properties on jaw crusher performance in granite quarry

The influence of rock strength properties on Jaw Crusher performance was carried out to determine the effect of rock strength on crushing time and grain size distribution of the rocks.Investigation was conducted on four different rock samples namely marble,dolomite,limestone and granite which were representatively selected from fragmented lumps in quarries.Unconfined compressive strength and Point load tests were carried out on each rock sample as well as crushing time and size analysis.The results of the strength parameters of each sample were correlated with the crushing time and the grain size distribution of the rock types.The results of the strength tests show that granite has the highest mean value of 101.67 MPa for Unconfined Compressive Strength(UCS) test,6.43 MPa for Point Load test while dolomite has the least mean value of 30.56 MPa for UCS test and 0.95 MPa for Point Load test.According to the International Society for Rock Mechanic(ISRM) standard,the granite rock sample may be classified as having very high strength and dolomite rock sample,low strength.Also,the granite rock has the highest crushing time(21.0 s) and dolomite rock has the least value(5.0 s).Based on the results of the investigation,it was found out that there is a great influence of strength properties on crushing time of rock types.

Rock strength interval analysis using theory of testing blind data and interval estimation

The concepts of rock strength intervals are presented in this work,furthermore,central values of intervals and their corresponding credibility are provided using two-case study based on blind data theory and fuzzy interval estimation.60 granite specimens are first tested,the compressive strength interval and tensile strength interval are[103.68,219.61]and[7.53,11.86]MPa,while the tested mean values of compressive strength and tensile strength are 152.86 and 10.14 MPa,the credibilities are less than58.4%and around 70.4%,respectively,the credibility of shear strength is between 40%and 60%.Then 70 other rock specimens are designed and tested,the similar conclusions can be reached.The results show that the conventional definite values are the particular values within the intervals,and the credibility of them often fails to reach the high-precision engineering requirement.The results demonstrate the feasibility and application potential of this proposed algorithm for the engineering practice.The references for engineering value selection of rock strength under different credibility or according to frequency distribution of central values are provided to increase the reliability and precision of calculation.

Investigation on jointed rock strength based on fractal theory

Strength of discontinuities with complex structure is an important topic in rock engineering.A large number of studies have shown that fractal is applicable in the description of this discontinuity.Using fractal interpolation method for the generation of rock joints,numerical experiments of shear tests of the jointed rock mass model were carried out using FLAC^(3D).The test results show that the real rock joints can be simulated by fractal curves obtained by fractal interpolation.The fractal dimension is an important factor for the characterization of jointed rock mass;test results show that the fractal dimension of rock joints can be related to the equivalent cohesion strength and shear strength of the rock mass.When the fractal dimension of the joint surface is less than critical dimension Dc 1.404,the cohesion strength and shear strength of the rock mass increase as the fractal dimension increases;for larger fractal dimensions,all mechanical parameters decrease as the fractal dimension increases.Joint surfaces with different degrees of roughness were obtained by the fractal interpolation method.Three types of failure modes were observed in the tests:climbing slip failure,climbing gnawing fracture,and non-climbing gnawing fracture.

Dynamic strength of rocks and physical nature of rock strength

Time-dependence of rock deformation and fracturing is often ignored.However,the consideration of the time-dependence is essential to the study of the deformation and fracturing processes of materials,especially for those subject to strong dynamic loadings.In this paper,we investigate the deformation and fracturing of rocks,its physical origin at the microscopic scale,as well as the mechanisms of the time-dependence of rock strength.Using the thermo-activated and macro-viscous mechanisms,we explained the sensitivity of rock strength to strain rate.These mechanisms dominate the rock strength in different ranges of strain rates.It is also shown that a strain-rate dependent Mohr-Coulomb-type constitutive relationship can be used to describe the influence of strain rate on dynamic rock fragmentation.A relationship between the particle sizes of fractured rocks and the strain rate is also proposed.Several time-dependent fracture criteria are discussed,and their intrinsic relations are discussed.Finally,the application of dynamic strength theories is discussed.

Influence of grain size on strength of polymineralic crystalline rock: New insights from DEM grain-based modeling

Grain size effect on rock strength is a topic of great interest in geotechnical engineering.A consensus obtained from earlier laboratory tests is that rock strength generally decreases with the increase of grain size for both silicate and carbonate rocks;however,some recent numerical results conflict with such laboratory test results.To address this intriguing issue,the effect of grain size on strength of polymineralic crystalline rock with low porosity is investigated numerically using the grain-based modeling(GBM)approach in discrete element method(DEM)by interpreting micro-cracking process in response to loading.In agreement with some previous DEM simulation results,the simulated rock strength is found to increase with increasing grain size for both homogeneous and heterogeneous models,even when the number of assembled disks in one mineral grain changes.The mechanism of strength increase with increasing grain size is mainly associated with the number of assembled smooth-joint contacts along grain interfaces and the generation of grain boundary cracks in response to loading.The grain interfaces significantly weaken the integrity of the rock model,which is similar to effects of inherent defects in real rock.As the grain size increases,fewer grain interfaces are built in the model and the rock strength becomes much higher.Hence,by solely changing the mineral grain size in a model,the mechanism of grain size effect as observed in laboratory tests cannot be replicated.To address this issue,a method of degradation of grain boundary strength parameters is used to mimic the possible mechanism of grain size effect.The simulated strength using the method becomes comparable with those obtained from laboratory tests when the heterogeneity in the rock is considered.Degradation of grain boundary parameters with increasing grain size provides a plausible explanation for the grain size effect on rock strength.

Effects of pre-existing cracks and infillings on strength of natural rocks--Cases of sandstone, argillite and basalt

This study aims to examine the influence of pre-existing discontinuities on the strengths of four natural rocks of different origins.A series of unconfined compression tests was performed on specimens of two types of sandstones,argillite and basalt that contain open and filled cracks.It was found that the presence of cracks tends to decrease the overall strength for all studied rocks;however,the magnitude of strength reduction is related to the property of rock.The larger strength decrease was observed for the relatively harder argillite and basalt,compared to the softer sandstone.It was also found that the infill material could increase the strength of rock specimens,while the obtained strength depended on the characteristics of the fill material.

Statically accelerated experimental simulation on the deterioration of dynamic strength of rock

In order to reveal the strength deterioration law of rock mass under an acidified environment, a series of experiments on strength deterioration and the environmental effect on marble and diabase under the conditions of three kinds of acids （H2SO4, HNO3 and HC1） with three contents （1.0%, 5.0% and 10.0%） in a static fluid surrounding were carried out by single factor simulation. After the stones had been saturated in the above solutions for 90 days, an indoor weathering simulation had been observed for 300 days and its strength characteristics were measured under static/dynamic load and that results were compared with those under the natural condition. It is evident that the strength is emaciated greatly after the stones eroded by the liquids, whether under the static or dynamic load.

Overhanging rock slope by design:An integrated approach using rock mass strength characterisation,large-scale numerical modelling and limit equilibrium methods

Overhanging rock slopes(steeper than 90°) are typically avoided in rock engineering design, particularly where the scale of the slope exceeds the scale of fracturing present in the rock mass. This paper highlights an integrated approach of designing overhanging rock slopes where the relative dimensions of the slope exceed the scale of fracturing and the rock mass failure needs to be considered rather than kinematic release of individual blocks. The key to the method is a simplified limit equilibrium(LE) tool that was used for the support design and analysis of a multi-faceted overhanging rock slope. The overhanging slopes required complex geometries with constantly changing orientations. The overhanging rock varied in height from 30 m to 66 m. Geomechanical modelling combined with discrete fracture network(DFN)representation of the rock mass was used to validate the rock mass strength assumptions and the failure mechanism assumed in the LE model. The advantage of the simplified LE method is that buttress and support design iterations(along with sensitivity analysis of design parameters) can be completed for various cross-sections along the proposed overhanging rock sections in an efficient manner, compared to the more time-intensive, sophisticated methods that were used for the initial validation. The method described presents the development of this design tool and assumptions made for a specific overhanging rock slope design. Other locations will have different geological conditions that can control the potential behaviour of rock slopes, however, the approach presented can be applied as a general guiding design principle for overhanging rock cut slope.

Resist-decreasing effects of rock bolts on strength of rock mass around roadway——insight from numerical modeling

To investigate the resist-decreasing effects of rock bolts on the strength of the rock mass around a roadway,a comparative study has been carried out using the numerical analysis code FLAC3D.An unsupported and a rock bolt supported model have been built for comparison.Two types of rock mass strength,the uniaxial compressive strength(UCS) and the triaxial compressive strength(TCS) of rock mass have been obtained from each model,using a prepared Fish based on the Mohr-Coulomb criterion.The results indicate that when a roadway is excavated,both the UCS and TCS in a definite local rock mass around the roadway would inevitably decrease,no matter whether the roadway is supported or not.The major decreasing region did not settle in the middle of the roadway surface,but within a deeper horizon into the rock mass.The resist-decreasing effects of rock bolts both on the UCS and the TCS of rock mass around roadway are significant.

Shear strength criteria for rock,rock joints,rockfill and rock masses:Problems and some solutions

Although many intact rock types can be very strong,a critical confining pressure can eventually be reached in triaxial testing,such that the Mohr shear strength envelope becomes horizontal.This critical state has recently been better defined,and correct curvature or correct deviation from linear Mohr-Coulomb（MC） has finally been found.Standard shear testing procedures for rock joints,using multiple testing of the same sample,in case of insufficient samples,can be shown to exaggerate apparent cohesion.Even rough joints do not have any cohesion,but instead have very high friction angles at low stress,due to strong dilation.Rock masses,implying problems of large-scale interaction with engineering structures,may have both cohesive and frictional strength components.However,it is not correct to add these,following linear M-C or nonlinear Hoek-Brown（H-B） standard routines.Cohesion is broken at small strain,while friction is mobilized at larger strain and remains to the end of the shear deformation.The criterion ＇c then σn tan φ＇ should replace ＇c plus σn tan φ＇ for improved fit to reality.Transformation of principal stresses to a shear plane seems to ignore mobilized dilation,and caused great experimental difficulties until understood.There seems to be plenty of room for continued research,so that errors of judgement of the last 50 years can be corrected.

Numerical evaluation of strength and deformability of fractured rocks

Knowledge of the strength and deformability of fractured rocks is important for design, construction and stability evaluation of slopes, foundations and underground excavations in civil and mining engineering. However, laboratory tests of intact rock samples cannot provide information about the strength and deformation behaviors of fractured rock masses that include many fractures of varying sizes, orientations and locations. On the other hand, large-scale in situ tests of fractured rock masses are economically costly and often not practical in reality at present. Therefore, numerical modeling becomes necessary. Numerical predicting using discrete element methods(DEM) is a suitable approach for such modeling because of their advantages of explicit representations of both fractures system geometry and their constitutive behaviors of fractures, besides that of intact rock matrix. In this study, to generically determine the compressive strength of fractured rock masses, a series of numerical experiments were performed on two-dimensional discrete fracture network models based on the realistic geometrical and mechanical data of fracture systems from feld mapping. We used the UDEC code and a numerical servo-controlled program for controlling the progressive compressive loading process to avoid sudden violent failure of the models. The two loading conditions applied are similar to the standard laboratory testing for intact rock samples in order to check possible differences caused by such loading conditions. Numerical results show that the strength of fractured rocks increases with the increasing confning pressure, and that deformation behavior of fractured rocks follows elasto-plastic model with a trend of strain hardening. The stresses and strains obtained from these numerical experiments were used to ft the well-known Mohr-Coulomb(MC) and Hoek-Brown(H-B) failure criteria, represented by equivalent material properties defning these two criteria. The results show that both criteria can provide fair estimates of the compressive strengths for all tested numerical models. Parameters of the elastic deformability of fractured models during elastic deformation stages were also evaluated, and represented as equivalent Young’s modulus and Poisson’s ratio as functions of lateral confning pressure. It is the frst time that such systematic numerical predicting for strength of fractured rocks was performed considering different loading conditions, with important fndings for different behaviors of fractured rock masses, compared with testing intact rock samples under similar loading conditions.

Strength evolution law of cracked rock based on localized progressive damage model

In the light of the localized progressive damage model,the evolution law of cohesive and frictional strength with irreversible strains was determined.Then,the location and the extent of the excavation disturbed zone in one deep rock engineering were predicted by using the strength evolution law.The theoretical result is close to the result of in-situ test.The strength evolution law excels the elastic-perfectly plastic model and elasto-brittle plastic model in which the cohesive and frictional strength are mobilized simultaneously.The results obtained indicate that the essential failure mechanism of the cracked rock can be described by the cohesion weakening and friction strengthening evolution law.

Numerical investigation on the sensitivity of jointed rock mass strength to various factors

The mechanical properties of jointed rock masses, such as strength, deformation and the failure mechanism, can be understood only by studying the sensitivity of jointed rock mass strength (both the peak and residual strengths) to the factors that affect it. An orthogonal design of uniaxial compression tests was simulated on eighteen groups of jointed rock specimens having different geometric and mechanical properties using RFPA2D (Rock Failure Process Analysis) code. The results show that the peak strength is controlled by the geometric parameters of the joints, but that the residual strength is controlled by the mechanical prop- erties of the joint interfaces. The failure mode of jointed rock specimens is mainly shear failure. Joint quantity, or density, is the most important index that affects jointed rock mass strength and engineering quality.

Effect of intermediate principal stress on strength of soft rock under complex stress states

A series of numerical simulations of conventional and true triaxial tests for soft rock materials using the three-dimensional finite difference code FLAC3D were presented. A hexahedral element and a strain hardening/softening constitutive model based on the unified strength theory(UST) were used to simulate both the consolidated-undrained(CU) triaxial and the consolidated-drained(CD) true triaxial tests. Based on the results of the true triaxial tests simulation, the effect of the intermediate principal stress on the strength of soft rock was investigated. Finally, an example of an axial compression test for a hard rock pillar with a soft rock interlayer was analyzed using the two-dimensional finite difference code FLAC. The CD true triaxial test simulations for diatomaceous soft rock suggest the peak and residual strengths increase by 30% when the effect of the intermediate principal stress is taken into account. The axial compression for a rock pillar indicated the peak and residual strengths increase six-fold when the soft rock interlayer approached the vertical and the effect of the intermediate principal stress is taken into account.

Modified Mohr-Coulomb strength criterion considering rock mass intrinsic material strength factorization

With the increase of mining depth of mineral resources,the rock mass stress state is being more and more complex.The rock mass show different features,namely,with the increase of hydrostatic pressure,rock mass failure mode turns from brittle tension failure to structure ductile failure and its limit strength also increases.The restriction of minimal principal stress on the initiation and development of microcrack and the change of micro-unit stress state by the intermediate principal stress play a decisive role in the increase of rock mass limit strength.Based on the rock mass failure behavior law under complex stress state and the σ2-dependence on the rock mass strength,we proposed a Modified Mohr-Coulomb(M-MC) strength criterion which is smooth and convex.Finally,the M-MC criterion is validated by multiaxial test data of eight kinds of rock mass.We also compared the fitting results with Mohr-Coulomb criterion(MC).It shows that the new criterion fits the test data better than the Mohr-Coulomb criterion.So the M-MC strength criterion well reveals the rock mass bearing behavior and can be widely used in the rock mass strength analysis.The results can provide theoretical foundations for stability analysis and reinforcement design of complex underground engineering.