The use of the direct optimized probabilistic calculation method in design of bolt reinforcement for underground and mining workings

Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 267593, 13 pages http://dx.doi.org/10.1155/2013/267593 Research Article The Use of the Direct Optimized Probabilistic Calculation Method in Design of Bolt Reinforcement for Underground and Mining Workings Martin Krejsa,1Petr Janas,1IGJkYilmaz, 2Marian Marschalko,3and Tomas Bouchal4 1Department of Structural Mechanics, Faculty of Civil Engineering, Vˇ SB-Technical University of Ostrava, 17 Listopadu 15, 70833Ostrava,CzechRepublic 2Department of Geological Engineering, Faculty of Engineering, Cumhuriyet University, 58140 Sivas, Turkey 3Faculty of Mining and Geology, Institute of Geological Engineering, Vˇ SB-Technical University of Ostrava, 17 Listopadu 15, 70833Ostrava,CzechRepublic 4Department of Environmental Engineering, Faculty of Mining and Geology, Vˇ SB-Technical University of Ostrava, 17 listopadu 15, 70833Ostrava,CzechRepublic Correspondence should be addressed to Is¸ık Yilmaz; [email protected] Received 9 April 2013; Accepted 10 June 2013 Academic Editors: K. Nemeth and U. Tinivella Copyright © 2013 Martin Krejsa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The load-carrying system of each construction should fulfill several conditions which represent reliable criteria in the assessment procedure. It is the theory of structural reliability which determines probability of keeping required properties of constructions. Using this theory, it is possible to apply probabilistic computations based on the probability theory and mathematic statistics. Development of those methods has become more and more popular; it is used, in particular, in designs of load-carrying structures with the required level or reliability when at least some input variables in the design are random. The objective of this paper is to indicate the current scope which might be covered by the new method—Direct Optimized Probabilistic Calculation (DOProC)in assessments of reliability of load-carrying structures. DOProC uses a purely numerical approach without any simulation techniques. This provides more accurate solutions to probabilistic tasks, and, in some cases, such approach results in considerably faster completion of computations. DOProC can be used to solve efficiently a number of probabilistic computations. A very good sphere of application for DOProC is the assessment of the bolt reinforcement in the underground and mining workings. For the purposes above, a special software application—“Anchor”—has been developed. 1. Introduction The designing process, assessment of the reliability, and individual stages of production, assembly, or operation of the underground structure are affected now by many uncertainties which influence reliability of such constructions by its random nature which cannot be neglected. This means that the design and implementation processes start being affected byvariabilityoffeaturesofthebuildingsandfacilities. It is possible to apply various calculation procedures basedontheprobabilitytheoryandmathematicstatistics in designs and assessment of the reliability, this approach being more and more popular now. The key feature of the probabilistic method is that it is possible to express variability of input quantities in a stochastic (probabilistic) form, for instance, by histograms. Unlike the applicable standards and procedures which are based on deterministic expression of input quantities (using a single value—a constant), the probabilistic methods provide more precise reliability assessment and improved safety for those who use the buildings and structures. 2. Interpreting Random Quantities in Probabilistic Calculations Histograms which are a part of the calculation in probability tasks should be regarded as approximation of the 2The Scientific World Journal (a) amin a1aiamax Δa/2 Δa Pai (b) bmin =b 1 Δb bibmax Pbi (c) amin amax Δa Pai (d) Figure 1: Approximation of the restricted probability distributions: (a) original approximation, (b) discrete approximation, (c) pure discrete approximation, and (d) piecewise uniform approximation. original distribution of probability of the random quantity (Figure 1(a)). If the distribution inside the histogram classes is even, such approximation is the approximation of the original distribution of random quantity probability by even parts (Figure 1(d)). If the histogram classes are represented by only one value, the original distribution of the random quantity probability is approximated by discrete distribution of the probabilities (Figures 1(b) and 1(c))whichareusedin DOProC calculations. 3. Function of Random Quantities In probabilistic calculations, the calculation model defines a function with generally 𝑛random quantities 𝑋1,𝑋2,...,𝑋𝑛. The resulting quantity—Zis expressed generally as follows: 𝑍=𝑓(𝑋1,𝑋2,...,𝑋𝑛). (1) It is also a random quantity which can be expressed by statistic moments, parametric distribution, or empirical distribution of probability using a nonparametrically defined histogram. 4. Reliability of the Supporting Construction During the construction design process, several computation operations are carried out with respect to the reliability assessment of specific structural part or the construction as a whole [1]. Various reliability criteria resulting from standards inforceshouldbefulfilled. The construction should be designed in such as way so that thestructuralresistance,𝑅,wouldbe higher than theload effects, 𝐸. Considering all random phenomena in the load, manufacturing and installation inaccuracies and inaccuracies where the construction is used, the structural resistance, 𝑅, and load effect, 𝐸, should be regarded as random quantities (Figure 2). The probabilistic reliability assessment is based on the reliability condition which can be expressed as follows: 𝑅−𝐸≥0, (2) where 𝑅is the structural resistance and 𝐸is the load effect. The left side of (2) is referred to as the reliability function, RF. Sometimes, it is also referred to as a failure function, 𝐺, The Scientific World Journal 3 Detail E Load effect E Detail Detail R Structural resistance R x emin emax rmin rmax Failure area Probability P(x) Figure 2: Probability density curves—load effect, 𝐸, structural resistance, 𝑅, and the area where a failure may occur. or reliability reserve, 𝑍. If the reliability condition (2)isnot fulfilled, such situation is undesirable in terms of reliability— it is a failure when the load effect, 𝐸, exceeds the magnitude of the structural reliability, 𝑅. The area where a failure may occur is shown in Figure 2. In the area where the histograms for the structural resistance, 𝑅,andloadeffect,𝐸,overlapinFigure 2,itis possible to determine the failure probability, 𝑃𝑓: 𝑃𝑓=𝑃(RF <0)=𝑃(𝑅−𝑆<0).(3) The magnitude of the failure probability is influenced bythenegativepartoftheRFhistogram.Thenonfailure probability, 𝑃𝑠,equals1−𝑃𝑓(see, e.g., Figure 3). The estimated failure probability, 𝑃𝑓,withrespecttothe reliability condition is defined by [2] 𝑃𝑓=𝑃(𝑅−𝑆<0) =∫𝐷𝑓𝑓(𝑋1,𝑋2,...,𝑋𝑛)d𝑋1,d𝑋2,...,d𝑋𝑛,(4) where 𝐷𝑓is the failure area and RF <0; a 𝑓(𝑋1,𝑋2,...,𝑋𝑛) is the function of combined probability density for random quantities 𝑋=𝑋1,𝑋2,...,𝑋𝑛. 5. Designed Failure Probability A degree of the structural reliability in the probabilistic calculation is the ultimate designed value of the failure probability, 𝑃𝑑, (the designed probability) or the reliability index, 𝛽.The structure is reliable only if the following reliability condition is fulfilled: 𝑃𝑓<𝑃 𝑑,(5) 𝛽𝑑<𝛽. (6) The designed failure probability, 𝑃𝑑,(orthereliability index, 𝛽) is determined on the basis of the required reliability level, type of the ultimate state, and estimated service life of the structure, 𝑇𝑑. Reference values for the designed probabilities, 𝑃𝑑, or reliability index, 𝛽, are specified in the European standards in force. In order to differentiate the reliability, the following classes of consequences were introduced in Eurocodes CC1, 𝛽·𝜎 G𝜑G(E, R) Failure 0𝜇GG(E, R) 𝜎G𝜎G P Pf 1−P f Figure 3: Determining the failure probability, 𝑃𝑓,andreliability index, 𝛽, by means of the failure reliability, RF (the failure function, 𝐺). CC2, and CC3 (where CC stands for consequences classes). Such consequence classes take into account consequences of failures or nonfunction incapacity of the construction. Reliability classes—RC1, RC2, and RC3—were defined on the basis of the reliability index, 𝛽.Thereliabilityclassesare related to the consequence classes CC1, CC2, and CC3. Figure 3 shows the curve based on the definition of the reliability structure (2) with a normal distribution of probabilities for the structural resistance, 𝑅, and load effect, 𝐸. In accordance with (3), the failure occurs also if the failure function 𝐺<0. The reliability index, 𝛽, is then the distance between the mean failure function, 𝐺, from the start defined in standard deviation units, 𝜎𝐺. For the reliability index, one obtains 𝛽=𝜇𝐺 𝜎𝐺,(7) where the mean value, 𝜇𝐺, is the difference: 𝜇𝐺=𝜇𝑅−𝜇𝑆(8) and the standard deviation, 𝜎𝐺, is expressed by 𝜎𝐺=√𝜎2 𝑅−𝜎2 𝐸,(9) where 𝜇𝑅,𝐸 arerespectivemeanvaluesofthestructural resistance, 𝑅,orloadeffects,𝐸,and𝜎𝑅,𝐸, are the standard deviations for the structural resistance and load effect. 6. Using Probabilistic Methods for Random Variable Models It is often very difficult to determine the failure probability, 𝑃𝑓, on the basis of the explicit calculation of the integral (4). A number of stochastic methods have been, and are being, developed [5]tosolve(4). The most frequently used and most numerous group of the computational method comprises the simulation methodswhicharebasedonthepopularsimulationtechnique— Monte Carlo (Direct Sampling, e.g., Bjerager [6]) or any 4The Scientific World Journal Figure 4: Desktop in the Anchor software [3]. advanced or stratified simulation methods (Latin Hypercube Sampling, (LHS), Stratified Sampling, Importance Sampling, Adaptive Sampling, Bucher [7]) which estimate the failure probability, 𝑃𝑓, using fewer simulations than the frequently used Monte Carlo. Eurocodes which are in force now mention the application of approximation methods—First/Second Order Reliability Method (abbreviated to FORM and SORM, der Kiureghian and Dakessian [8]) which are used mostly for calibration of partial coefficients. These computational methods employ for approximation of the final reliability function (the failure) a simple approximation—typically, a normal distribution of the probability. The integral (4)issolvedthen analytically. The response surface method [9,10]isoneofthe next approximation methods. Both the original method and the new method which are under development now—the Direct Optimized Probabilistic Calculation (DOProC)—use a purely numerical approach and basics of the probabilistic calculation without any simulation techniques to solve (4). This provides more accurate solutions to probabilistic tasks, and results, in some cases, in considerably faster completion of computations. 7. Direct Optimized Probabilistic Calculation (DOProC) The Direct Optimized Probabilistic Calculation (DOProC) has been developed since 2002. The original name of this method was the Direct Determined Fully Probabilistic Method (DDFPM). The word “Determined” in the name of themethodmeansthatthecalculationprocedureforacertain task is clearly determined by its algorithm, while Monte Carlo generates calculation data for simulation on a random basis. The name of the method was discussed and consulted with experts in the structural reliability, the conclusion being that the word “Determined” in the name of the method is somewhatmisleading.Consequently,thenameofthemethod was modified. The new term in the name of the method— “Optimized”—is based on the following facts. The number of variables that enter calculation of the failure probability, 𝑃𝑓,computationis,however,limitedbycapabilitiesofthe software to process the application numerically. If there are too many random variables, the application is extremely time demanding—even if high-performance computers are used. The computational complexity of DOProC is given, in particular, by (i) the number of random input quantities 𝑖 = 1,2,..., 𝑁; (ii) the number of histogram classes (intervals) for each random input quantity, 𝑛𝑖; (iii) complexity of the task (computational model), (iv) the probabilistic computation algorithm (the way used to define the computational model). Therefore,effortshavebeenmadetoreducethenumber of operations. The purpose of the DOProC optimizing techniques is to minimize the computing time since the algorithm is limited to a certain extent, in particular, for The Scientific World Journal 5 Figure 5: Histogram with empirical distribution of probabilities created from measured compression strength in carboniferous sandstone [MPa]. extensive applications where too many simulations exist. If the optimizing techniques are used in DOProC,thefailure probability, 𝑃𝑓, can be determined in a real time. On top of this, results are reliable and accurate enough even in relatively demanding probabilistic tasks. The optimizing techniques include the following. (a) Grouping of variable input quantities (such as load components) which may enter the calculation jointly andajointhistogramcanbepreparedinadvance. (b) Interval optimizing where the number of intervals of variable input quantities of individual histograms is decreased, while the whole range for each random input quantity is maintained. (c) Zone optimizing where only intervals affecting a certain value, for instance, the failure probability of astructure,𝑃𝑓are involved. (d) Trend optimizing which considers the correct direction (trend) in the algorithm of the probabilistic calculation. (e) Grouping of partial calculation results,forinstance,in creation of the resulting reliability function, RF. (f) Computation parallelization where the computation is carried out in several processors or cores at the same time. (g) Combinationoftheoptimizingproceduresabove. For instance, Janas et al. [11] include detailed theoretical background for the DOProC algorithm including the optimizing procedures which make it possible to determine in the reliability assessment the failure probability, 𝑃𝑓,fortwoor more random quantities. Currently, the DOProC along with the optimizing steps can address well several probabilistic tasks. It is possible to use ProbCalc in DOProC.ProbCalcis a software application which is still under development. It is rather easy and simple to implement quite a complicated analytical transformation model of a probabilistic task defined in a character form or as a dynamic DLL library similarly as in Tvedt [12], Thacker et al. [13], and Cervenka et al. [14]. A lite versionofthissoftwarecanbedownloadedfromthewebpage http://www.fast.vsb.cz/popv/ [15]. 8. Probabilistic Calculation of Reliability of Bolt Reinforcements The probabilistic approach to the assessment and design of thestructureshasstartedappearinginpracticerecentlyonly. 6The Scientific World Journal Figure 6: Histogram with parametric distribution of probabilities for compression strength in carboniferous sandstone [MPa]. These computational procedures are used, in particular, in designs of load-carrying systems for ground structures—for instance, for steel structures [16–18], for reinforced concrete structures [19–21], or other engineering activities [22]. For underground and mining workings, this approach is used in rare cases only. The methods for the design of reinforcements in the underground workings were based, generally, on an assumption that the input values were clearly deterministic. This is the case not only of geological or technical conditions under whichtheboltswillbeappliedbutalsopropertiesofthe bolts that are influenced also by installation procedures. Most input data used in various design methods in connection with the bolts are random. When designing the underground workings, it is rather easy to use the deterministic approach. It, however, does not take into account the random nature of input quantities which, in turn, are almost neglected in the designing of the bolts. It is just this area where the probabilistic (stochastic) method appears to be very efficient for determination of the necessary load-carrying capacity of the bolt reinforcement. That method represents an entirely new approach to this field. Most successful applications of the DOProC include guidelines for probabilistic designs and reliability assessment of underground and mining workings [23,24] and creation of the software—Anchor (Janas et al. [3]; for the Anchor desktop see Figure 4). When designing the bolt support for certain conditions, the following parameters need to be defined: (i) the length of bolts; (ii) the number and location of the bolts near the mining working or underground working; (iii) parameters of the bolts (the type, diameters, material, anchoring method, etc.). Extensivemeasurementswerecarriedoutinthemining workings in the Ostrava-Karvin´ aColliery.Itfollowsfromthe measurements that the convergence, this means dislocation of rock into the mining working, can be calculated from the following formula: 𝑢=0,1𝐵⋅(1−𝑒−0,015𝑡)⋅(𝑒(1,2𝐻−𝑞)/45𝜎𝑟−1), (10) where 𝐻is the efficient depth under the surface (m), 𝐵 the dimension (typically, the width) of the mining working [m], 𝑡isthetimeindays,𝑞is the load-carrying capacity of The Scientific World Journal 7 Figure 7: Histogram of the width of the mining working 𝐵[m]. Figure8:Histogramofthereducedstrengthofhangingrock𝜎[MPa]. 8The Scientific World Journal Figure 9: Software desktop with a table for determination of rock mass rating (the geomechanical classification coefficient RMR), [4]. Figure 10: Histogram of rock mass rating (the geomechanical classification coefficient RMR) [4]. The Scientific World Journal 9 Figure11:Histogramofthelengthofthedesignedbolt𝑙[m]. the support [kNm−2], and 𝜎𝑟is the reduced strength of the hanging rock [MPa] which is determined as follows: 𝜎𝑟=𝛽∑𝑛 1𝜎𝑑𝑖𝑚𝑖 2𝐵 .(11) In relation (10)𝛽is the stratification coefficient pursuant (see Table 1), 𝜎𝑑𝑖 is the strength in one-axis compression of the 𝑖th strata, and 𝜇𝑖is the thickness of the 𝑖th strata. Nonelastic deformation range, 𝐵𝑛, which is the basis for specification of loading and length of the bolt can be described, using (10)andfor𝑡→∞, as follows: 𝐵𝑛=0,251189⋅𝐵⋅𝐾𝑛⋅(𝑒(1,2𝐻−𝑞)/45𝜎𝑟−1)0,6.(12) 𝐾𝑛characterizes the relation between the nonelastic deformation in the mining working or under working with the 𝐵dimension, 𝐵𝑛convergence, and 𝜎𝑟reduced strength. In past, a single one deterministic value was used in spite of thefactthatthisquantityisofarandomnature. The load to be transferred by the bolted support should be suitable for the nonelastic deformation range (𝐵𝑛), rock weight (𝛾) as well as for a certain level of self-bearing capacity of rock strata that does not exist in the nonelastic deformation range. Using the geomechanical classification parameter (RMR, rock mass rating) has proved to be a good solution [4].Then,theloadoftheboltedsupportwas determined by the following formula: 𝑄=𝐵𝑛⋅𝐵⋅𝛾⋅100−RMR 100 = 2,51189𝐵2𝛾100−RMR 100 𝐾𝑛(𝑒(1,2𝐻−𝑞)/45𝜎𝑟−1)0,6,(13) where 𝛾is the specific gravity of rock [103kg⋅m−3]and𝑄is the total load of the bolted reinforcement per running meter in the working [kN]. The assessment of reliability of bolted reinforcements in underground and mining workings is based on the reliability function (RF) analysis pursuant to (2) that is described using the following formula: RF =𝑄sv −𝑄, (14) where 𝑄sv is the load-carrying capacity of the bolts and 𝑄 is the bolt loading per running meter in the working. The load-carrying capacity of the bolts is based on the following formula: 𝑄sv =𝑛sv𝑞sv =𝑛⋅𝑞sv 𝑑𝑠=𝑛𝜋(𝑑1−𝑑2)2⋅𝜎sv 4𝑑𝑠,(15)