Accuracy of the spectral method in estimating fractal/spectral parameters for self-affine roughness profiles

Accurate quantification of roughness is important in modeling strength, deformability and fluid flow behaviors of rock joints. Self-affine fractals seem to have the potential to represent rock joint roughness profiles. Both stationary and non-stationary fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D, and input standard deviation, σ, were generated at different generation levels. A few smoothing techniques were used with the spectral method to calculate D, and two other spectral parameters K_s (a proportionality constant; see the text for the details) and CD (the cross-over dimension of the profile) for the fractional Brownian profiles. The effects of smoothing, generation level of the profile, seed value used in the generation, non-stationarity of the profile and σ on the accuracy of the calculated D were examined using the spectral method. The following conclusions were obtained: (a) To obtain accurate estimates for D, K_s and CD, it seems necessary to have at least 10 data points per unit length for a profile having a total length of 100 units (this is equivalent to a generation level of 10). (b) For accurate estimation of D, K_s and CD, the non-stationarity of profiles should be removed, if it exists. (c) The parameter combinations D and K_s, (which has the potential to capture scale effects), and D and CD are recommended for quantification of stationary roughness; in addition, extra parameters are required to quantify the non-stationarity. (d) Both the Parzen and Hanning smoothing techniques seem suitable to use with the spectral technique to obtain accurate estimates for D, K_s and CD. (e) To obtain accurate estimates for D, K_s, and CD, it is necessary to use a suitable bandwidth for the Parzen window and a suitable number of interactions for the Hanning window; this paper provides guidelines to choose these suitable values. (f) Seed value has negligible effect on the accuracy of estimated D, K_s and CD.     

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.

     

Quantitative three-dimensional description of a rough surface and parameter evolution with shearing

The choice of a general criterion to determine the shear strength of rough rock joints is a topic that has been investigated for many years. The major problem is how to measure and then to express the roughness with a number (e.g., joint roughness coefficient) or a mathematical expression in order to introduce the morphology of the joint into a shear strength criterion. In the present research a large number of surfaces have been digitised and reconstructed using a triangulation algorithm. This approach results in a discretisation of the joint surface into a finite number of triangles, whose geometric orientations have been calculated. Furthermore, during shear tests it was observed that the common characteristic among all the contact areas is that they are located in the steepest zones facing the shear direction. Based on this observations and using the triangulated surface data, it is possible to describe the variation of the potential contact area versus the apparent dip angle with the expression A_θ^*=A₀[(θ_max^*−θ^*)/θ_max^*]^C, where A₀ is the maximum possible contact area, θ_max^* is the maximum apparent dip angle in the shear direction, and C is a “roughness” parameter, calculated using a best-fit regression function, which characterises the distribution of the apparent dip angles over the surface. The close agreement between analytical curves and measured data therefore suggests the possibility of defining the influence of roughness on shear strength by the simple knowledge of A₀, C and θ_max^*. Based on the samples studied here, the values of these parameters capture the evolution of the surface during shearing. Moreover, they tend to be characteristic for specific rock types, indicating that it might be possible to determine ranges for each rock type based on laboratory measurements on representative samples.     

Assessment of rock fracture surface roughness using Riemannian statistics of linear profiles

The mechanical behaviour of rock masses is complex, due partly to the presence of discontinuities within them. Of the geometrical parameters of discontinuities, surface roughness, which encapsulates the topographical features of a rock surface, is known to play a significant role. Here, a new parameter for quantitative roughness determination based on the distribution of unit normal vectors to a rock profile is presented.The analysis of unit normal vectors in terms of directional statistics is customarily performed in Euclidean space using a Cartesian co-ordinate system. Here, the analysis is developed using Riemannian geometry, with Mahalanobis distances being proposed for discrimination between different rock profiles. Statistical parameters on the unit circle are extracted using Riemannian geometry, and from that a roughness parameter, D_R1, is obtained. This parameter corresponds to 1D Riemannian dispersion, and as such D_R1 increases as profile roughness increases.D_R1 is applied to the analysis of synthetic profiles and some real rock profiles. Conclusions are drawn that demonstrate the advantages of the new method in terms of investigating the scale effect in roughness determination as well as in comparing different profiles.A preliminary study into the correlation between D_R1 and the shear strength of a fracture, using analytical and numerical investigation of the strength of profiles comprising symmetric triangular asperities sheared at different normal stress levels, shows a clear relation between Riemannian roughness parameter and profile shear strength.     

新的岩石节理粗糙度指标研究

首先,基于节理轮廓线提出一个剪切粗糙度指标(SRI),新指标由3个粗糙度参数构成,描述节理平均起伏度、起伏的方向性、起伏分布特点以及粗糙的分形特性,并建立新指标与JRC之间的拟合关系。其次,基于三角形面积单元离散的三维节理面,将剪切粗糙度指标推广到三维形式,从而使得新指标能够描述节理的三维形貌特征。最后,以张拉型花岗岩节理作为算例,计算结果表明新指标能较好地反映节理的各向异性粗糙特性。新指标物理意义明确,计算方便,能够表征节理的主要粗糙特性,为进一步研究岩石节理的剪切力学特性奠定基础,具有重要的理论意义。     

Flow-induced vibrations of two circular cylinders in tandem arrangement. Part 1: Characteristics of vibration

This paper presents the results of an investigation on the flow-induced vibration characteristics of two circular cylinders in tandem arrangement, with L/D=0.1–3.2 and reduced velocity U_r=1.5–26, where L is the gap spacing between the cylinders and D is the cylinder diameter. The cylinder vibration was restricted to a plane normal to the incident flow. Three different experimental conditions were examined: (i) both cylinders were allowed to vibrate; (ii) the downstream cylinder only was allowed to vibrate with the upstream cylinder fixed; and (iii) the upstream cylinder only was allowed to vibrate with the downstream cylinder fixed. Five Regimes I–V were identified, depending on L/D, fluctuating lift forces and vibration characteristics of the cylinders. In Regimes I (0.1≤L/D<0.2) and IV (2≤L/D<2.7), the cylinder vibration is absent. In Regime II (0.2≤L/D<0.6), both cylinders vibrate violently for U_r>6, including a divergent vibration of the upstream cylinder. In this regime, the vibration amplitude of the downstream cylinder is strongly dependent on whether the upstream cylinder is vibrating or fixed, whereas that of the upstream cylinder is weakly dependent on the downstream cylinder. In Regime III (0.6≤L/D<2), the convergent vibrations of the two cylinders occur at and around U_r≈6.7. In this regime, the upstream cylinder vibration is completely suppressed when the downstream cylinder is fixed, but the downstream cylinder vibration is almost independent on the upstream cylinder. Regime V corresponds to L/D≥2.7, where the two cylinders are separated sufficiently far, thus each vibrating like an isolated cylinder at and around U_r≈6. In this regime, the downstream cylinder vibration is strongly dependent on the upstream cylinder, but the upstream cylinder vibration is almost insensitive to the downstream cylinder condition.     

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.     

Flow-induced vibration of two circular cylinders in tandem arrangement. Part 2: Suppression of vibrations

This paper presents an experimental investigation on suppression of cross-flow vibrations of two circular cylinders in tandem arrangement, conducted at the fluid mechanics laboratory of Kitami Institute of Technology, Japan. To suppress the vibrations of the cylinders, tripping wires were deployed, attached symmetrically about the leading stagnation lines of the cylinders. Five spacing ratios were examined, i.e., L/D=0.1, 0.3, 0.8, 2.0 and 3.2 (L is the gap spacing between the two cylinders; D is the diameter of cylinder), which are representative for five Regimes I (0.1≤L/D<0.2), II (0.2≤L/D<0.6), III (0.6≤L/D<2), IV (2≤L/D<2.7) and V (L/D≥2.7), respectively, as classified in Part 1 [Kim et al., 2009. Flow-induced vibrations of two circular cylinders in tandem arrangement (part 1: characteristics of vibration). Journal of Wind Engineering and Industrial Aerodynamics, submitted together for publication]. Tripping wire position θ measured from the leading stagnation lines of the cylinders was changed from 20° to 60° to determine the optimum range of θ for suppressing structural vibrations. The shear layers separated from the two cylinders were investigated based on flow visualization. The main findings are: (i) the flow-induced vibration on the two cylinders depends strongly on θ, (ii) at θ=20–30° the vibrations on both cylinders are almost completely suppressed for all regimes except V, and (iii) for θ≥40° the vibration amplitudes of both cylinders are considerably larger than those of the plain cylinders, particularly at θ=40°, where the vibration of the upstream cylinder becomes divergent.     

基于数字图像处理技术的岩石节理分形描述

岩石节理的粗糙度对结构面的剪切强度起着重要的作用.分形维数对节理的次级细微粗糙结构能够进行很好的描述,而对节理的一级起伏结构体现并不明显.考虑了节理形态的层次性,用分形维数代表节理的次级精细结构,即节理的粗糙度;用起伏度表示节理的一级波状形态,据此提出用分形维数D和起伏度W_d两个指标来描述JRC(节理粗糙度系数),并给出相应的计算公式.另外,针对工程实践中分形维数准确快速量测存在的困难,本文以摄影测量为基础,借助数字图像处理技术,在VC++平台下开发了节理轮廓线分形维数计算程序,为分维较准确方便地应用于工程实践做了有益的探索.     

Numerical simulation of the excavation damaged zone around an opening in brittle rock

The micromechanics-based damage model proposed by Golshani et al. [A micromechanical model for brittle failure of rock and its relation to crack growth observed in triaxial compression tests of granite. Mech Mater 2006;38:287–303] is extended so that time-dependent behavior of brittle material can be taken into account, with special attention to the numerical analysis of an excavation damaged zone (EDZ) around an opening, which is a major concern in assessing the safety of underground repositories. The present model is capable of reproducing the three characteristic stages of creep behavior (i.e., primary, secondary, and tertiary creep) commonly observed in the laboratory creep tests. The sub-critical microcrack growth parameters (i.e., n and A) can be determined for Inada granite by fitting the numerical results of elapse time to failure versus the creep stress ratio curve with the experimental data under both dry and wet conditions. It is found that moisture has a significant influence on the parameter A rather than the parameter n. Use of the extended model makes it possible to analyze not only the extension of microcrack length, but also the development of EDZ around an opening as a function of time. The damaged zones mainly develop in the sidewalls of the opening in the case that the vertical stress σ₂₂ is larger than the horizontal stress σ₁₁.     

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.     

A new parameter to assess hydromechanical effects in single-hole hydraulic testing and grouting

Grouting or filling of the open voids in fractured rock is done by introducing a fluid (grout) through boreholes under pressure. The grout may be either a Newtonian fluid or a Bingham fluid. The penetration of the grout and the resulting pressure profile may give rise to hydromechanical effects, which depend on factors such as the fracture aperture, the pressure at the borehole, and the rheological properties of the grout. In this paper, we postulate that a new parameter, Å, which is the areal integral of the fluid pressure change over the fracture plane, is an appropriate measure to describe the change in fracture aperture volume due to a change in effective stress. In many cases, analytic expressions are available to calculate pressure profiles and the Å parameter. The approach is verified using a fully coupled hydromechanical simulator for the case of a Newtonian fluid. Results of the verification exercise show that the new approach is reasonable and that the Å parameter is a good measure for the fracture volume change: i.e., the larger the Å parameter, the larger the fracture volume change, in an almost linear fashion. To demonstrate the application of the approach, short duration hydraulic tests and constant pressure grouting are studied. It is concluded that using analytic expressions for penetration lengths and pressure profiles to calculate the Å parameter provides a possibility to describe a complex situation and compare, discuss and weigh the impact of hydromechanical couplings for different alternative grouting strategies. Further, the analyses identify an effect of high-pressure grouting, that of uncontrolled grouting of larger fractures and insufficient (or less-than-expected) sealing of finer fractures under certain grouting conditions.     

隧道围岩断面轮廓分形维数与节理参数关系

隧道的超欠挖对衬砌结构稳定性影响比较大,研究超欠挖规律,对于掌握围岩受力,保证施工安全,具有较为深远的意义。根据小波分析的多尺度功能以及隧道断面轮廓超欠挖序列的统计自相似性特点,对白鹤隧道47个断面轮廓采用小波理论估算分形维数。结合现场调查的节理参数统计数据,分析隧道围岩断面轮廓分形维数与节理参数之间的关系。结果表明:分形维数越大,围岩稳定性越差;洞轴线与节理走向夹角越大,断面轮廓分形维数就越小;节理间距为0.25～0.40 m时,节理间距越大,分形维数越大;超挖百分比越大,分形维数越大。     

Tensile resistance of rock anchors

Using the Euler's variational method and assuming a rock mass failure criterion of Hoek and Brown type (Hoek E, Brown ET. Empirical strength criterion for rock masses. J Geotech Eng Division, American Society of Civil Engineers 1980;106(GT9)1013–35. Hoek E, Wood D, Shah S. A modified Hoek-Brown criterion for jointed rock masses. In: Hudson JA, editor. Proc., Rock Characterization Symp. of ISRM: Eurock 92. London: British Geotechnical Society, 1992. p. 209–14), the tensile resistance of rock anchors is obtained. The rupture surface shape through the rock mass is also obtained and checked against some published data. Depending on the slenderness ratio (L/D; L=anchor length, D=anchor diameter) two types of failure surfaces are obtained; short and long ones. This second type of surface is a complex one and it is composed by a cylinder and a surface of revolution with an hyperbolic type shape. The values of the ultimate pullout strength, depending on the rock type and its Bieniawski indexes, are obtained and compared with the values published in technical literature. A reasonable agreement is obtained.     

剪切过程中岩石节理粗糙度分形演化及力学特征

在系统测量和试验的基础上，研究了剪切过程中岩石节理粗糙面的分形特征和岩石节理的力学行为，阐述了岩石节理面分形维数Ｄ和截距Ａ与岩石节理在载荷作用下法向、切向变形以及抗剪强度之间的关系，得出岩石节理在剪切过程中由于表面损伤而引起的表面分维Ｄ和截距Ａ的演化规律。研究表明：分维Ｄ和截距Ａ是描述节理面粗糙性的两个重要的参数。前者反映节理粗糙面的不规则程度，后者则与节理面粗糙体（ａｓｐｅｒｉｔｉｅｓ）的坡度密切相关。仅依据分形维数Ｄ不足以确定岩石节理的粗糙性与岩石节理力学行为之间的关系。在许多情况下，岩石节理的力学性质对截距Ａ的依赖程度大于对分维Ｄ的依赖程度     

Variational design of open cross-section thin-walled beam under stability constraints

A thin-walled beam is in pure bending subjected to couples M₀. The open cross-section profile has two ribs, with cross-section A₀ and it is shaped symmetrically towards the plane perpendicular to the bending plane. The ribs are located at the profile ends. The shape of the profile line is searched for. Criterion is the minimal value of the cross-section area A₁ of the beam. The problem is described by means of variational calculus. Within the numerical calculations a Runge–Kutta method is used. The optimal shapes of beam profiles are shown graphically.     

THE FRACTAL CHARACTERISTICS OF ROCK FRACTURE IN DAM FOUNDATION OF THE THREE GORGES PROJECT UNDER UNIAXIAL COMPRESSION

The fractal structure of rock fracture in dam founda ti on of the Three Gorges Project under uniaxial compression is studied based on fr actal theory. The results show that the fracture system and the main fractures of rock under uniaxial compression tests are of highly self-similar characteris tic in statistics. The fractal dimension values of rock fracture raise with incr ease of pressure. The fractal dimension can be taken as a parameter to describe the process of rock fracture and there are good correlation among fractal dimens ion of rock fracture, uniaxial compressive strength and rock weathering deg ree. The fractal dimensions reflect the formational and evolutional features of rock fractures and the difference of rock weathering.     

Over-consolidation effect on shear behavior of rock joints

Although many researchers have studied the normal and shear behavior of fractures under stresses, the over-consolidation effect on the slip/shear behavior of discontinuities has not been considered. The over-consolidation behavior of non-planar rock fractures should be considered when deposition–consolidation–erosion (or excavation) sequences occur. Plaster replicas of representative natural rock joint surfaces were prepared for this study. In this case, the surface roughness and other geometrical properties remain constant during the laboratory direct shear tests. It was observed that the shear strength within a large range of roughness, joint wall strength and normal stress values significantly increases with increasing over-consolidation ratio. According to the test results, a new model is developed as an extended form of Barton's shear failure criterion for rock joints. This model considers the effect of various paths of normal loading/unloading before shearing and over-consolidation ratio in a fracture. A new joint over-closure (JOC) parameter is also introduced as the ratio of closure in over-closed to normally closed conditions.     

Correlations of steel column fire test data

Spray application is a method commonly used to protect structural steel columns from direct fire exposure. The endurance time for a column during a fire test is the time at which temperature at any level exceeds 1000°F.In this study a one-dimensional analysis of heat conduction was performed to define the pertinent parameters. For a fire resistive material this parameter was of the form (A/D)h, whereA is the cross-sectional area;D is the heated perimeter; andh is the thickness of fire resistive coating.Correlation of fire test data with the parameter (A/D)h showed that endurance point time for a range of wide flange columns could not be described with a single equation. When (A/D) ratios for the larger columns were determined on the basis of the web dimensions, a single linear relationship between (A/D)h and the endurance time for the complete size range of columns was obtainable. Data from tubular and pipe columns also correlated linearly with the (A/D)h parameter.