Statistical representation of generalized distribution data for float-sink coal-cleaning devices: Sand cones

An earlier paper by this writer presented a method for combining distribution data for float-sink coal-cleaning devices into a single, generalized distribution curve which is independent of the specific gravity of separation. A nonlinear exponential-type equation was used to represent each of the curves and to determine the corresponding generalized probable error. Performance data for five common coal-cleaning devices were analyzed using this method.In this paper the method is applied to another common coal-cleaning device — the sand cone. Numerical values are presented for the constants in the exponential-type generalized distribution equation and for the corresponding generalized probable error. Several different feed size-fractions are considered.     

Analysis and optimization of mineral processing and coal-cleaning circuits — Circuit analysis

In mineral processing and coal cleaning operations, there are two classes of unit operations used in the separation processes. The first type — crushing, grinding, pelletizing and settling — modifies the size distribution of the feed. The second type — heavy media, sizing, flotation, magnetic, electrostatic and jigging — separates the particles. In these latter unit operations, the size distribution is not changed; rather, the feed stream is split into a product and waste stream based on subtle physical and physicochemical differences between the different types of feed particles. In separation unit operations, the mathematical functions, representing the behavior of the unit operations, are easy to manipulate. Thus, a number of theorems, useful to both the plant designer, and the plant engineer, can be proven.The work presented in this paper makes the fundamental assumption of linearity as do all current plant design analysis systems. The linearity assumption states that in a separation operation, there is no particle-particle interaction effecting the probability that a particle will be selected for the product or waste stream. In other words, if one doubles or triples the feed rate to a unit operation, the fraction of particles of a given characteristic selected to appear in the product stream remains unchanged. In actuality, this assumption is not true because higher feed rates affect the behavior of separation operations. However, when designing a plant, one can calculate the feed rate to a given unit operation and then select a piece of equipment large enough to handle the computed tonnage. Thus, the designed circuit behaves linearly.     

Fragility curves in operational dike reliability assessment

Operational flood prediction and flood risk assessment have become important components of flood management. One main aspect is the reliability assessment of the flood defence line during a flood event. This is generally performed by a comparison of the water level in the river to the crest height of the dikes whilst taking only hydraulic and geometric aspects into account. Additional information about material zones and material parameters are often available. However, these data are not in an appropriate shape when deriving the reliability of the flood defence line. This paper outlines how the fragility curve of a dike section is used to appropriately integrate geostatic and geohydraulic dike characteristics into operational flood management systems. Fragility curves are the result of a model-based reliability analysis and they summarise the dike performance depending on the water level. Failure modes such as piping or slope failure are included. In a case study, fragility curves for dike sections along the River Emscher (Germany) are determined. Their practical implementation in an operational flood management system shows an improvement in the operational reliability assessment due to the additional information taken into account. The use of fragility curves also supports the decision-making processes when emergency flood protection measures are required.     

Statistical representation of generalized distribution data for float-sink coal-cleaning devices: Baum jigs,Batac jigs,Dynawhirlpools

In an earlier paper (1978), Gottfried presented a method for combining distribution data for float-sink coal-cleaning devices into a single generalized distribution curve which, for a given device and feed size, is independent of specific gravity of separation. A non-linear, exponential-type equation was utilized to represent the generalized distribution curve, along with the corresponding generalized probable error. Distribution data for six common coal-cleaning devices have previously been treated by this method.This paper is an extension of two previous studies (Gottfried, 1978, 1980). The method described above is applied to three different float-sink coal-cleaning devices: Baum jig (replacing previously reported results), Batac jig and Dynawhirlpool separator. Results for the Baum jig and Batac jig reflect a two-stage separation process, with a set of generalized distribution curves obtained for each stage and another set for the overall separation. Several different feed size fractions are given for each vessel.     

Application of distribution functions to partition curves

The regular Tromp curve is a log-normal probability (K—Kolmogoroff) function; it is, however, also a modified RRS (Rosin-Rammler-Sperling) or GGS (Gates-Gaudin-Schuhmann) function. The exponent m (distribution modulus) is a measure of the slope of the functions, i.e. Tromp curves and can be used instead of the indices E_p or I to characterize the efficiency of the various separating (sorting or classifying or sizing) equipments. The relationship $\text{Im ?}\text{constant}$ is valid.     

Slope hybrid reliability analysis considering the uncertainty of probability-interval using three-parameter Weibull distribution

A reliability model is proposed to solve the problem of hybrid uncertainty with both random and interval variables in slope engineering. A hybrid uncertainty model based on the dimension reduction method and Taylor expansion is constructed to approximate the limit state function. Using the polynomial theorem and variable transformation method, the origin and center moments’ interval of the limit state function are calculated. Moment information is applied to the expansion of a three-parameter Weibull distribution, and the cumulative distribution function and probability density function of limit state function are determined. As a result, the failure probability interval of the slope is calculated. The interval uncertainty problem is transformed into an interval certainty problem using Taylor expansion without solving for the statistical moment of limit state function using multiple integrals and iteratively searching for the most probable failure points. The numerical results from two slopes show that the proposed method is effective and feasible.

     

围岩强度和变形参数的分布特征及可靠性分析

从康家湾铅锌金矿Ⅲ-1号矿体上盘围岩取大量岩样,分别加工制作了50个压缩和拉伸试验的试样。利用RMT-150B伺服试验系统对试样进行单轴抗压、抗拉试验,各获得了50个试验结果。采用假设检验法,分别对50个单轴抗压强度和50个抗拉强度进行检验,结果表明,它们分别服从正态分布和对数正态分布;对50个 、 和50个C、 ,进行不放回抽样,组成50组E、 、C、 。利用FLAC计算软件,对硐室围岩中的应力进行了计算,分别获得了50个最大主应力和50个最小主应力;采用同样假设检验法,证明它们分别服从对数正态和正态分布;根据单轴抗压、抗拉强度及围岩中的最大主应力、最小主应力概率密度函数,计算了硐室围岩不发生拉伸破坏和压缩破坏的可靠度;并对硐室围岩抗剪强度的校核,得出了该地下硐室围岩稳定的结论。     

考虑能量-时间分布的边坡动力可靠性分析新方法

提出一种考虑能量-时间分布的边坡动力可靠性分析方法。该方法将动态最危险滑动面及其稳定系数以时间序列加以刻画,并根据边坡动力反应的能量分布特征,提取持续时间统计窗,用于对上述时间序列的统计分析,以获取边坡动力模糊失效概率、边坡动力可靠度指标和基于保证率的边坡动力稳定系数。以澳大利亚计算机应用协会边坡稳定考核题为例,应用上述新方法考察其在芦山7.0级主震波形条件下的稳定性,研究了在不同统计窗下的边坡动力可靠性。案例分析表明： （1）新方法能够抓住影响边坡动力稳定的主要时间段,使分析结果更为凝练、可信。（2）通过引入边坡失效状态的模糊判别,使得可靠性评价中能够考虑模糊性,解决了以往常规方法区分度不够的问题。（3）基于保证率的边坡动力稳定系数具有很好的应用前景,它在内涵上体现了可靠性分析,在形式上与静力稳定系数的定义兼容,在数值上反映了边坡瞬时动力稳定系数的保守估计值,在实践上与现行规范的拟静力法具有良好的可比性,因而具有多方面的优势。（4）就本案例而言,动力条件下最危险滑动面的发育位置趋向于静力条件下的最危险滑动面,体现了依据静力和拟静力理论框架所进行的防护工程设计,在动力条件下仍然具有积极的意义。（5）新方法对定量研究现行边坡规范的抗震设计冗余提供了一条途径。提出的新方法为边坡抗震研究提供了新的思路、方法和可供参考的实例。     

抗剪强度参数概率分布的最大熵估计及边坡可靠度分析

提出了基于最大熵原理的抗剪强度参数概率分布估计方法。首先简要介绍了最大熵原理估计抗剪强度参数概率分布的基本步骤。其次,基于模拟数据分别验证了最大熵原理估计抗剪强度参数概率分布和边坡失效概率的有效性,并比较了最大熵原理、选优识别和核密度估计方法的不确定性建模精度和稳健性。最后,以一组残积土抗剪强度参数试验数据并结合无限边坡稳定分析为例研究了最大熵原理在抗剪强度参数概率分布和边坡失效概率估计中的应用。结果表明：最大熵原理能够有效地估计抗剪强度参数的概率分布和边坡失效概率。与传统的选优识别和核密度估计相比,最大熵原理估计抗剪强度参数概率分布和边坡失效概率的精度和稳健性都更高。最大熵原理避免了核密度估计过分依赖有限样本数据的缺点,又克服了选优识别可能未将真实分布包括在备选概率分布集合中的缺陷。此外,基于有限数据估计的抗剪强度参数概率分布和边坡失效概率具有较大的离散性。     

A fast numerical method for evaluating the performance of float-sink coal-cleaning equipment

A method is presented which seeks to overcome a number of disadvantages that are inherent in the conventional practice of plotting the Tromp curve and analysing it for its various measures, e.g., specific gravity of separation, probable error and error area. A sufficient condition is established for extracting the appropriate criteria of the sharpness of classification, which are independent of coal-feed distribution in specific gravity. Partition area, a modified areal measure, similar to the conventional error area, of the equipment's deviation from ideal classification behavior is derived from the theoretical partition function. A purely numerical procedure is developed for locating the specific gravity of separation and the partition area. Unlike the Tromp-curve-based conventional procedure, the proposed histogram-based method is fast, more convenient and objective for analysing the coal-cleaning plant data.     

边坡模糊随机可靠性分析

为合理描述边坡稳定性分析中存在的多种不确定性,应同时考虑随机和模糊两种因素带来的影响.用L-R型模糊数描述随机变量的均值和方差,建立了模糊随机可靠性分析模型.为简化计算,将模糊数通过分解定理,取一系列λ截集,转换为一系列区间数进行计算.在一次二阶矩法基础上,推导了基于瑞典条分法及毕肖普法计算模糊随机可靠度的具体公式.算例表明,该法是合理实用的.     

土质边坡三维可靠度分析

基于空间随机场理论,利用局部平均法研究空间各向异性的土坡的可靠度问题,推导出相应计算公式,并通过计算边坡的最大破坏概率来进行滑坡的空间预测。     

Three-dimensional reliability analysis of earth slopes

Reliability of cohesive soil slopes is assessed using a three-dimensional (3D) probabilistic stability analysis algorithm. Spatial variability of soil properties is represented by an anisotropic random field. Parametric studies are performed for a typical earth structure. The influence of the model parameters, including expected value, variance and correlation distance of soil shear strength, on the reliability associated to particular failure mechanisms is evaluated. The effect on reliability of the dimensions and shape of potential slip surfaces for a given random field is also assessed. It is shown that the mechanisms that contribute most significantly to global probability of failure of the slope may be quite different from those identified as critical by standard deterministic evaluations assuming soil homogeneity. Some practical implications of this fact are discussed.     

Value of epeirogenic curves in interpreting spatial distribution of the geomagnetic field

Use of electronic computers has made possible a wider and at the same time more detailed interpretation of magnetic anomalies. It also permits the making of three dimensional computations and interpretations, which were practically prohibited with older methods. The present paper is a detailed magnetic and structural study of the east flank of the Urals from the extreme south beyond Chelkar, through Magnitogorsk, to a latitude just north of Sverdlovsk. A complex history of fault formation and development is worked out and presented on a map of the block structure (fig. 3) and two series of "epeirogenic curves" (fig. 4). It is pointed out that these curves can also be useful in a study of the origin of mineral deposits in the area.--E. Ingerson.     

Taguchi design-based seismic reliability analysis of geostructures

This paper presents a methodology to evaluate the seismic reliability of geostructures in an optimal way. Taguchi design of experiments are adopted to find the most efficient and cost-effective combination of material properties in the uncertainty domain. Twelve uniform and mixed design models are tested. A polynomial-based response surface meta-model is built for each one and the accuracy of perdition is examined using 10,000 Monte Carlo simulations. A two-dimensional gravity dam is used as a vehicle for probabilistic transient analyses. The ground motion record-to-record variability is added as well using over one hundred earthquake records selected based on probabilistic seismic hazard analysis. Dynamic sensitivity of epistemic random variables are evaluated for the first time. Finally, an efficient and practical procedure is proposed in order to determine the reliability index of the geostructures. This approach, in fact, can be generalised for any type of engineering structures dealing with multi-hazard problems.     

桩的模糊随机可靠性分析

利用一次二阶矩法,考虑可靠性分析模型中随机参数的模糊性,建立了模糊随机可靠性分析模型.利用分解定理,取一系列λ截集,将模糊数转换为一系列区间数进行运算,得到了一种模糊随机可靠度的计算方法.将该法应用到桩的可靠性分析中,实例表明,该方法是合理实用的.     

基于Copula函数的CFG桩复合地基载荷-变形曲线的概率分析

CFG桩(cement-fly ash-gravel pile)复合地基是一种重要的地基处理形式,在日益增加的大面积住宅和商业开发中作用越来越突出,然而该种桩型的加卸荷-沉降变形特性仍然需深入研究,尤其在概率评估方面。根据北京星光影视股份有限公司生产科研基地项目工地中的21根CFG桩单桩静载试验和32个复合地基静载试验的原位加卸载测试成果,采用两参数的双曲线或幂曲线回归拟合了每一条加荷-变形曲线。由于土体的内在各向异性和其强度的变异性,评估整个场地的加荷-变形曲线时,其回归参数表现出了较大的离散性。将一个场地的多组回归参数组成一个随机向量,其加载-位移曲线的不确定性可由简单的两变量随机向量体现,引入双变量联结函数（Copula）描述随机回归参数间的相依性。最后,考虑正常使用极限状态,采用基于Copula函数的模拟模型计算了CFG桩复合地基的可靠度指标。研究结果有助于改进CFG桩复合地基的概率设计与评估。     

Influence of rock property correlation on reliability analysis of rock slope stability: From property characterization to reliability analysis

Cohesion(c) and friction angle(φ) of rock are important parameters required for reliability analysis of rock slope stability. There is correlation between c and φ which affects results of reliability analysis of rock slope stability. However, the characterization of joint probability distribution of c and φ through which their correlation can be estimated requires a large amount of rock property data, which are often not available for most rock engineering projects. As a result, the correlation between c and φ is often ignored or simply assumed during reliability studies, which may lead to bias estimation of failure probability. In probabilistic rock slope stability analysis, the influence of ignoring or simply assuming the correlation of the rock strength parameters(i.e., c and φ) on the reliability of rock slopes has not been fully investigated. In this study, a Bayesian approach is developed to characterize the correlation between c and φ, and an expanded reliability-based design(RBD) approach is developed to assess the influence of correlation between c and φ on reliability of a rock slope. The Bayesian approach characterizes the sitespecific joint probability distribution of c and φ, and quantifies the correlation between c and φ using available limited data pairs of c and φ from a rock project. The expanded RBD approach uses the joint probability distribution of c and φ obtained through the Bayesian approach as inputs, to determine the reliability of a rock slope. The approach gives insight into the propagation of the correlation between c and φ through their joint probability into the reliability analysis, and their influence on the calculated reliability of the rock slope. The approaches may be applied in practice with little additional effort from a conventional analysis. The proposed approaches are illustrated using real c and φ data pairs obtained from laboratory tests of fractured rock at Forsmark, Sweden.     

土-水特征曲线的分形特性及其分析拟合

为了建立一种参数有明确物理意义的土-水特征曲线拟合方法,基于分形理论,给出了一种直接通过土-水特征曲线实测数据求解分维数的计算方法,建议了一种土-水特征曲线拟合分析的分形模型。为验证所建议拟合方法的合理性,采用液压千斤顶制作了7个不同干密度黏性土试样,利用压力板仪测量了土-水特征曲线。基于土-水特征曲线实测值计算了分维数,在此基础之上对土-水特征曲线实测数据进行分析拟合。结果表明：分维数计算及拟合分析时,7个试样相关系数都集中在0.97～0.99之间,从而证明了土-水特征曲线具有良好的分形特性,建议的拟合方法效果较好。此外,研究发现,分维数、进气值随干密度几乎呈线性增加,据此建议了一种不同压实度下的土-水特征曲线预测方法,其预测结果与实测值吻合较好。     

Finite element reliability analysis of reinforced retaining walls

This paper presents the reliability analysis of reinforced retaining wall using finite element method. Response surface approach is used to approximate the performance function and a first-order reliability method (FORM) is used to evaluate the reliability index. In the analysis, displacement response of the reinforced retaining wall is considered as performance function and the corresponding reliability index is evaluated with the aid of a spreadsheet. Uncertainties associated with the soil and reinforcement properties are explicitly taken into account in the analysis. A parametric sensitivity analysis has been performed to bring out the effect of important uncertain parameters by evaluating the sensitivity of the reliability index with respect to each of the uncertain parameters. Results of the response surface method coupled with finite element analysis show the ease and successful implementation of the reliability analysis procedure for the reinforced retaining walls.