Predicting the deformation moduli of rock masses

Predictive empirical models for the mechanical properties of rock masses have been used in rock engineering because direct measurement of the properties is difficult due to the presence of discontinuities. Such empirical models are open to improvement because they are based on collected data. The purposes of the present study are to assess the existing empirical equations and to develop a new empirical approach. For this reason, in the first stage of the study, the prediction performance of the existing models proposed for predicting the deformation modulus of rock masses were evaluated statistically by using a database including 115 data values obtained from in situ plate loading and dilatometer tests. A new empirical approach with higher prediction capacity than the existing empirical models was developed in the subsequent stage of the study. The new empirical model considers the modulus ratio of intact rock (E_i/UCS), rock quality designation (RQD) and weathering degree (WD). Although, data obtained from very weak and weak rock masses were included in the development of the new empirical equation, the type of rocks employed in the study were limited. Therefore, a crosscheck between the new empirical equation and previous empirical approaches should be performed in the design stage.     

Genetic programming approach for estimating the deformation modulus of rock mass using sensitivity analysis by neural network

We use genetic programming (GP) to determine the deformation modulus of rock masses. A database of 150 data sets, including modulus of elasticity of intact rock (Ei), uniaxial compressive strength (UCS), rock mass quality designation (RQD), the number of joint per meter (J/m), porosity, and dry density for possible input parameters, and the modulus deformation of the rock mass determined by a plate loading test for output, was established. The values of geological strength index (GSI) system were also determined for all sites and considered as another input parameter. Sensitivity analyses are considered to find out the important parameters for predicting of the deformation modulus of rock mass. Two approaches of sensitivity analyses, based on “statistical analysis of RSE values” and “sensitivity analysis about the mean”, are performed. Evolution of the sensitivity analyses results establish the fact that variable of UCS, GSI, and RQD play more prominent roles for predicting modulus of the rock mass, and so those are considered as the predictors to design the GP model. Finally, two equations were achieved by GP. The statistical measures of root mean square error (RMSE) and variance account for (VAF) have been used to compare GP models with the well-known existing empirical equations proposed for predicting the deformation modulus. These performance criteria proved that the GP models give higher predictions over existing empirical models.     

Prediction of mode I fracture toughness of rock using linear multiple regression and gene expression programming

Prediction of mode I fracture toughness(KIC) of rock is of significant importance in rock engineering analyses. In this study, linear multiple regression(LMR) and gene expression programming(GEP)methods were used to provide a reliable relationship to determine mode I fracture toughness of rock. The presented model was developed based on 60 datasets taken from the previous literature. To predict fracture parameters, three mechanical parameters of rock mass including uniaxial compressive strength(...     

A geo-engineering classification for rocks and rock masses

In this article, an attempt is made to assess the reliability of predicting the uniaxial compressive strength and the corresponding modulus of a rock mass by current approaches. These two basic engineering properties, when estimated from rock mass rating (RMR), Q and geological strength index (GSI), indicate hardly any change in the modulus ratio with the change in the quality of the rock mass from very good to very poor. However, the modulus ratio obtained from the relations involving the joint factor, J_f, indicate a definite decrease in the modulus ratio with a decrease in the quality of the rock mass. The strength and modulus in the unconfined and confined states, the modulus ratio and failure strain in the unconfined case were linked to J_f in earlier publications based on a large experimental database. Some of these relations were adopted to verify the response of jointed test specimens, the response of the rock mass during excavations for mining and civil underground chambers, in establishing ground reaction curves including the extent of the broken zone, and the bearing capacity of shallow foundations.The joint factor is now linked to RMR, Q and GSI. The prediction of compressive strength and modulus of the rock mass appears to be more suitable. For classifying the rock, based on these properties, the Deere and Miller engineering classification, applicable to intact rocks, has been suitably modified and adopted. The results of different modes of failure of jointed specimens establish definite trends of changes in the modulus ratio originating from the intact rock value on the modified Deere and Miller plot. A geo-engineering classification is evolved by considering strength, modulus, quantifiable weathering index and lithological aspects of the rock.     

Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations

The rock mass failure process is characterized by several distinct deformation stages which include crack initiation, crack propagation and coalescence. It is important to know the stress levels associated with these deformation stages for engineering design and practice.Extensive theoretical, experimental and numerical studies on the failure process of intact rocks exist. It is generally understood that crack initiation starts at 0.3 to 0.5 times the peak uniaxial compressive stress. In confined conditions, the constant-deviatoric stress criterion was found to describe the crack initiation stress level.Here, generalized crack initiation and crack damage thresholds of rock masses are proposed. The crack initiation threshold is defined by σ₁−σ₃=A σ_cm and the crack damage threshold is defined by σ₁−σ₃=B σ_cm for jointed rock masses, where A and B are material constants and σ_cm is the uniaxial compressive strength of the rock masses. For a massive rock mass without joints, σ_cm is equal to σ_cd, the long-term uniaxial strength of intact rock. After examining data from intact rocks and jointed rock masses, it was found that for massive to moderately jointed rock masses, the material constants A and B are in the range of 0.4 to 0.5, 0.8 to 0.9, respectively, and for moderately to highly jointed rock masses, A and B are in the range of 0.5 to 0.6, 0.9 to 1.0, respectively. The generalized crack initiation and crack damage thresholds, when combined with simple linear elastic stress analysis, assist in assessing the rock mass integrity in low confinement conditions, greatly reducing the effort needed to obtain the required material constants for engineering design of underground excavations.     

Predicting uniaxial compressive strength,modulus of elasticity and index properties of rocks using the Schmidt hammer

The Schmidt hammer test is a non-destructive method which can be used in both laboratory and field to provide a quick and relatively inexpensive measure of rock hardness. The study investigated the relationship between the Schmidt hardness and modulus of elasticity, uniaxial compressive strength and index properties of nine types of rock including travertine, limestone, dolomitic limestone and schist. The empirical equations developed indicated the Schmidt hardness rebound values have a reliable relationship with the uniaxial compressive strength of rock (r = 0.92). Comparing the results with those reported by other researchers, it is concluded that no single relationship can be considered reliable for all rock types. Whilst the equations developed in this study may be useful at a preliminary stage of design, they should be used with caution and only for the specified rock types.      

Stress–strain–electrical resistance effects and associated state equations for uniaxial rock compression

This paper reports stress–strain–electric resistance experiments for diabase, limestone and marble containing NaCl solution during the whole process of uniaxial compression. We obtained the complete testing data for the stress–strain curve and the associated electrical resistance–strain curve. The change caused by internal cracking of the rock causes the corresponding variation of rock electrical resistance. There is a minimum value for all the electric resistance–strain curves, corresponding to the cracking stress σ_c or the initial cohesion c_i. Based on the experimental results and stochastic property analyses of the rock fracture variation, we put forward a group of state equations for rock in sections to express the characteristics of rock during the whole process of uniaxial compression. The three variables, stress, strain and electrical resistance, together with data-fitted parameters, α₁,α₂ and β, are contained in the equations. The equations are used to express the inelastic response which intensifies with the propagation of cracking.     

A modified Hoek–Brown failure criterion for anisotropic intact rock

The Hoek–Brown criterion parameters (σ_ci, m_i and s) are significantly influenced by the strength anisotropy of intact rock. In the present study, the criterion was modified by incorporating a new parameter (k_β) to account for the effect of strength anisotropy, thus being able to determine the strength of intact anisotropic rock under loading in different orientations of the plane of anisotropy. The range of the parameter (k_β) for the rocks tested has been analytically investigated by carrying out triaxial tests, in different orientations of the foliation plane. The proposed modification was studied for metamorphic rocks (gneiss, schist, marble), but could also be applied to other rock types exhibiting “inherent” anisotropy, e.g. sedimentary as well as igneous rocks. The proposed modified criterion is intended for use for prediction of strength of intact rock, but can also be extended to rock masses.     

Predicting the physico-mechanical properties of rocks from electrical impedance spectroscopy measurements

Electrical resistivity values of eight different samples cored from a fault breccia were measured using an impedance analyser. The uniaxial compressive strength, elastic modulus, point load strength, Schmidt hammer value, P-wave velocity, density and porosity values of the samples were determined in the laboratory. Electrical resistivity values were correlated with the corresponding physico-mechanical properties using the method of least-squares regression and the derived equations were checked by the t and F-test. A strong logarithmic relation between uniaxial compressive strength and resistivity was found. The relation between elastic modulus and resistivity is significant and follows a logarithmic function. Density was linearly, and porosity was exponentially correlated with resistivity.It may be concluded that electrical resistivity can be used as a representative measure of rock properties, particularly for characterizing rocks for which regularly shaped specimen are difficult to obtain. However, the effect of different rock types on the correlations must be further investigated. Additionally, the effect of certain minerals on the rock's resistivity must be taken into account, especially when testing dry or partially saturated rocks.     

Uncertainties in estimating the roughness coefficient of rock fracture surfaces

Joint roughness has a critical role in the deformation behavior of discontinuous rock masses. Several subjective (visual comparison) and quantitative (statistical and fractal) approaches have been proposed for estimating rock joint roughness coefficient (JRC). Using a large collection of 223 published joint profiles, this study investigates variability of the JRC estimates by these approaches. Among the profile parameters, maximum height (R _z), ultimate slope (λ), and fractal dimension (D _h–L, determined using the hypotenuse leg method) show a lower sensitivity to the sampling interval than the root mean square of the first deviation (Z ₂), profile elongation index (δ), fractal dimension (D _c, determined using the compass-walking method), and standard deviation of the angle i (σ _i ). Accordingly, this study proposes two separate sets of equations for quantitatively estimating JRC. The performances of these equations are examined by performing direct shear tests on 23 rock joint samples. The subjective approach is found to underestimate JRC by less than two units because it ignores (1) the main trend of the compared profile and (2) the limited scope of the standard profiles. Following these results, the visual comparison chart is updated by explicitly adding a scale bar for the y-axes of the standard profiles. Several basic rules for visual comparisons are also proposed.

     

Estimation of strength and deformation properties of Quaternary caliche deposits

The aim of this study was to develop and evaluate statistical models for predicting the uniaxial compressive strength (UCS) and average Young’s modulus (E _av) for caliches, using some index and physical properties. The caliche samples, from Adana, southern Turkey, were of low strength and difficult to sample. X-ray diffraction and microscopy were undertaken and the following physical parameters established: unit weight, apparent porosity, Schmidt rebound number, Shore hardness, P-wave velocity, slake durability, point load, uniaxial compressive strength and average Young’s modulus. Simple and linear regression variable selection analyses were performed. The best relationships were obtained for UCS with P-wave velocity and unit weight and for average Young’s modulus with P-wave velocity, porosity and slake durability. Empirical equations are proposed, although it is emphasised that these may only be applicable for caliche of a similar geological character.      

Characterizing rock masses by the RMi for use in practical rock engineering: Part 1: The development of the Rock Mass index (RMi)

The Rock Mass index, RMi, has been developed to satisfy a need for a strength characterization of rock masses for use in rock engineering and design. The method gives a measure of the reduction of intact rock strength caused by discontinuities given by RMi = σ · JP. Here, σ is the uniaxial compressive strength of the intact rock measured on 50 mm diameter samples, and JP is the jointing parameter which is a combined measure of block size (or intensity of jointing) and joint characteristics as measured by joint roughness, alteration and size. This paper describes the method of determining the RMi for a rock mass using various common field observations. The determination of a meaningful equivalent block size is a key issue which is discussed in detail. Several areas of application of the RMi are presented, among others for design of rock support. Discussion of these applications will be developed in Part 2 of this paper.     

An approach to predicting the shear strength of soil-rock mixture based on rock block proportion

Soil-rock mixture (SRM) shows complicated mechanical behaviors due to their complex compositions and structures, leading to challenging instability problems during the construction process. Typical SRM are composed of rocks with high strength and fine grained soils, and the mechanical characteristic is largely controlled by the rock block proportion (RBP) and component properties. It is noted that the rock sizes of natural SRM make it difficult for laboratory or in situ tests. There are few studies on empirical formulas to predict the mechanical characteristics of SRM. In this study, the nonlinear relationship between SRM shear strength and RBP was investigated, and an empirical formula predicting the shear strength of mixtures consisted of strong rocks and a weak soil matrix was proposed. For this purpose, a database of shear strength and uniaxial compressive strength (UCS) of SRM with different RBPs was built firstly on the basis of the laboratory test results from previous literatures. In order to focus on the interactions of rock blocks and soil matrix in SRM, a RBP range of 30–90% was set as the applicable range of the empirical formula and both of the compositions are held to provide shear resistance in the applicable range. Subsequently, a nonlinear equation to calculate the shear strength of SRM with RBP range of 30–90% was proposed using regression analysis considering the strengths of components and soil-rock contact faces. Several representative properties of rocks and soil matrix, such as RBP, UCS of the matrix (UCS_m), and the friction angle of the blocks (φ_block), were chosen as the input parameters based on the mechanical properties of SRM. An additional parameter “A” was used to describe the connect strengths of the soil-rock contact faces. In addition, uniaxial compression tests and large-scale direct shear tests were performed on the Taoyuan SRM samples. The test results and other measured data from the database were used to compare with the corresponding estimated values. The results demonstrated that the empirical approach could predict the shear strength with R² = 0.75 and can be considered a practical tool in engineering designs when mechanical tests are not available.

     

An equivalent plasticity theory for jointed rock masses

A non-representative volume element (NRVE) approach to equivalent rock mass properties shows that the form of the elastic–plastic constitutive equations is the same for homogeneous material elements and multiple-material elements, subsequently homogenized. Thus the average stress and strain increments in an arbitrary jointed rock mass volume are related by {dσ}=([C*_ep]){dε} where σ is effective stress. The equivalent elastic-plastic properties matrix [C*_ep] is the sum of an equivalent elastic moduli matrix [C^*] and a plastic ‘correction’ matrix [C*_p, as usual. However, there are no equivalent plastic potentials Y^* or yield functions, failure criteria F^* or strengths. The equivalent elastic-plastic properties are constructed from the elastic moduli and strengths of the rock mass joints, the intact rock between and strain influence functions that relate local to overall average strains. Numerical examples that simulate laboratory-like tests on jointed rock cubes illustrate the approach.     

基于BQ的破碎岩体注浆加固强度增长理论

正确评估破碎岩体注浆加固后的强度对于岩石工程设计十分重要。基于莫尔库仑强度准则,建立了注浆前后破碎岩体强度增长理论,推导出单轴抗压强度增长率、单轴抗拉强度增长率、内摩擦系数增长率和内聚力增长率之间的关系方程。根据已有的基于BQ的岩体内聚力和内摩擦角的经验公式,推导出注浆后破碎岩体内摩擦系数增长率和内聚力增长率随岩体质量指标增长量（ΔBQ）变化的表达式。再根据已有的注浆后破碎岩体单轴抗压强度增长率的经验公式和各强度增长率之间的关系方程,得到隐含ΔBQ的非线性方程,可通过数值解法求出了ΔBQ的值,进而很容易求得各强度增长率的值。分析表明,随着岩体质量指标由小到大变化,注浆后强度增长率开始较大,并很快减小,而后趋于平缓;一般情况下,内聚力增长率约为摩擦系数增长率的2～5倍,单轴抗压强度增长率约为单轴抗拉强度增长率的2～3倍。     

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.     

Evaluation of empirical estimation of uniaxial compressive strength of rock using measurements from index and physical tests

The uniaxial compressive strength(UCS) of rock is an important parameter required for design and analysis of rock structures,and rock mass classification.Uniaxial compression test is the direct method to obtain the UCS values.However,these tests are generally tedious,time-consuming,expensive,and sometimes impossible to perform due to difficult rock conditions.Therefore,several empirical equations have been developed to estimate the UCS from results of index and physical tests of rock.Nevertheless,numerous empirical models available in the literature often make it difficult for mining engineers to decide which empirical equation provides the most reliable estimate of UCS.This study evaluates estimation of UCS of rocks from several empirical equations.The study uses data of point load strength(Is(50)),Schmidt rebound hardness(SRH),block punch index(BPI),effective porosity(n) and density(ρ)as inputs to empirically estimate the UCS.The estimated UCS values from empirical equations are compared with experimentally obtained or measured UCS values,using statistical analyses.It shows that the reliability of UCS estimated from empirical equations depends on the quality of data used to develop the equations,type of input data used in the equations,and the quality of input data from index or physical tests.The results show that the point load strength(Is(50)) is the most reliable index for estimating UCS among the five types of tests evaluated.Because of type-specific nature of rock,restricting the use of empirical equations to the similar rock types for which they are developed is one of the measures to ensure satisfactory prediction performance of empirical equations.     

Correlating uniaxial compressive strength with schmidt hardness,point load index,Young's modulus,and mineralogy of gabbros and basalts (northern Greece)

An attempt has been made to correlate the uniaxial compressive strength and Young's modulus of gabbros and basalts with Schmidt hammer rebound number, the point load strength index, I_s(50) and the degree of weathering. Sixty three samples of gabbro and thirty of basalt from the ophiolitic comlex of Pindos zone (Northern Greece) have been collected by core drilling and tested accordingly. The results have been processed using techniques from the statistical software SPSS. Some of the equations produced show relatively high correlation coefficients, all significant at a significance level higher than 95%. The equations establish reliable prediction models for the uniaxial compressive strength and modulus of elasticity of the above rock types by means of simple tests which can be carried out in the field.     

Assessment of brittleness using rock strength and density with punch penetration test

Brittleness fracturing of rock is one of the most popular research areas in rock engineering, since some rocks show brittle fractures under loads. Direct standard testing method for measuring rock brittleness have not available yet. Therefore, rock brittleness is indirectly obtained as a function of rock strength. The aim of this study is not only to introduce direct method to measure rock brittleness as an index via punch penetration test, but also to investigate the relationship between intact rock properties (uniaxial compressive strength, Brazilian tensile strength, and density of rock) and the brittleness measured from the test. To obtain these objectives, rock cores were gathered from 48 tunnel projects throughout the world. Followings the sampling, the samples were prepared and relevant rock tests were carried out to establishment of dataset at the Earth Mechanics Institute of Colorado School of Mines in the USA. Consequently, using generated dataset, new brittleness index (BI_m) and rock brittleness classification was introduced base on type, strength and density of rock together with result of punch penetration test. Further, the rock brittleness index was predicted as a function of the uniaxial compressive strength, Brazilian tensile strength and density of rock with correlation coefficient of 0.94.     

Preferred orientations of open microcracks in granite and their relation with anisotropic elasticity

In order to study how to deal with open microcracks in rock, anisotropic behaviors of Oshima granite were investigated by carrying out wave velocity tests and uniaxial compression tests, together with observations of microcracks under an optical microscope equipped with a universal stage. Anisotropy in the longitudinal wave velocity V_L and secant deformation modulus E₁₀ at 10% strength is caused by pre-existing open microcracks, not by pre-existing healed microcracks. The structural anisotropy formed by open microcracks, which is quantitatively represented by a second-rank tensor (called crack tensor), is in good agreement with the directional changes of E₁₀ and V_L. The mechanical, as well as structural, anisotropy shows rhombic symmetry with orthogonal symmetry axes in the directions roughly normal to the rift, grain and hardway planes, which are parallel to the major joint sets in the field. Since longitudinal wave velocity changes drastically depending on the density and orientation of open microcracks in granitic rocks, it is suggested that the crack tensor can be determined from non-destructive wave velocity tests. The elastic modulus tensor theoretically formulated in terms of the second-rank crack tensor can be used, as a first-order approximation at least, to describe the anisotropic elasticity of Oshima granite induced by pre-existing open microcracks. It is of particular importance to point out that the micro-scale structure by open microcracks is geometrically similar to the macro-scale structure by joints and faults (scale independent). This finding strongly suggests that some of the conclusions related to open microcracks are applicable to deal with macro-scale cracks in rock masses.