Modeling uncertainty in cutter wear prediction for tunnel boring machines(2012)-C.Frenzel

Modeling uncertainty in cutter wear prediction for tunnel boring machinesC. Frenzel 11Department of Mining Engineering, Colorado School of Mines, 1600 Illinois St,Golden, CO, 80401; PH (303) 273-3714; FAX (303) 273-3719; email:cfrenzel@ABSTRACTWear of cutting tools and the excavation performance of tunnel boringmachines (TBM) is influenced by a number of geotechnical parameters. Theprediction model of the Colorado School of Mines uses Uniaxial CompressiveStrength, Brazilian Tensile Strength, and Cerchar Abrasivity Index. Due to thecomplex nature of rock each of these parameters shows variance and there arecorrelations between these parameters.The usual approach to performance and wear prediction is to use averageinput parameters to estimate the “average” result. It is shown that this is not alwayscorrect because of the statistical nature of the input parameters.The approach described in this paper establishes a statistical model for theperformance and wear prediction based on the previously developed model. Since ananalytical solution for variance prediction is not possible for this model a Monte-Carlo simulation is used.PREDICTION OF DISC CUTTER WEARA broadly used model for prediction tunnel boring machine performance and disccutter wear has been developed at the Colorado School of Mines. Its first version waspublished by Ozdemir et al. 1977 and the last version was published by Rostami1997.To predict the wear of disc cutters two steps need to be followed:1. Prediction of the penetration rate2. Prediction of disc cutter lifeThe reason is that cutter life has been shown to be proportional to the rolling distanceof a disc cutter (Frenzel 2010).Penetration Rate. The prediction of the penetration rate is an iterative process sincethere are three basic limitations:• Thrust limit: Depending on the size of the disc cutter the bearing capacity of the disc cutters varies and limits the maximum thrust force that a disc cuttercan withstand.• Torque limit: The main drive of a tunnel boring machine can overcome only a certain maximum torque at a given rotational speed.• Geometrical limit: The geometry of disc cutter rings does not support infinite penetration rates because the shoulders of the cutter ring will eventually touchthe rock surface.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T o n g j i U n i v e r s i t y o n 04/13/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r ve d .Based on geometrical considerations the opening angle ϕ of the contact area betweenthe disc cutter and the rock can be calculated bywhere R is the radius of the disc cutter and p is the penetration rate.The average stress P* in the contact area can be estimated bywhere σc is the uniaxial compressive strength, σt is the Brazilian Tensile Strength(BTS), S is the spacing between adjacent disc cutter positions on the cutterhead, andT is the width of the tip of the cutter ring.The resulting cutting force F on the disc cutter is given byThis resulting force can be separated into a normal and a rolling component. Therolling component is critical because the sum of all rolling forces times their distancefrom the center of the cutterhead yields the required torque for the main drive of theTBM:where nc is the number of disc cutters on the cutterhead and Dc is the excavationdiameter of the cutterhead.These calculations are repeated until one of the three limits as shown abovehas been reached.Disc Cutter Life. The maximum rolling distance until a disc cutter has to be changeddue to primary wear iswhere CAI is the Cerchar Abrasivity index and R the radius of the disc cutter in mm.The average rolling distance of a disc cutter during one revolution of the cutterheadcan be approximated byD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T o n g j i U n i v e r s i t y o n 04/13/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r ve d .where D c is the diameter of the cutterhead. Further analysis by Frenzel 2010revealed that actual cutter life was less than the expected value based on primarywear. Hence, a correctionis introduced where g i are the relative weights of the number of disc cutters in thecenter, face, and gage area. The corrections k k and k v can be determined fromTable 1.Table 1. Correction factors for cutter life at different cutterhead positionsCorrection k k Correction k vCentre cutters 0.561.0Face cutters 0.76 1.0Gage cutters 0.92 0.54The maximum number of cutterhead revolutions U cuntil a cutter changeoccurs can be calculated byand net cutter life V c is computed to bewhere S is the spacing of the disc cutters. Based on net cutter life the number of disccutter changes for a given tunnel can be predicted.INPUT PARAMETERSRock Strength. Due to inhomogeneity repeated tests for rock strengths yielddifferent results. If the number of tests is large enough (sufficient sample size) theresults can be used to estimate the (unknown) statistical distribution. That includesthe type of distribution and its parameters.Of course, measuring rock strengths already introduces a certain bias sinceany damage or imperfection of the sample will reduce its (apparent) strength. Bydefinition UCS should be the intact rock strength which means that there is a biastowards underestimating rock strengths using standard laboratory tests.Examples for histograms of UCS tests of rock samples from Hong Kong areshown in figures 1 and 2.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T o n g j i U n i v e r s i t y o n 04/13/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r ve d .Figure 1: Histogram of Uniaxial Compressive Strength of Hong Kong granite (after: Frenzel 2010).Figure 2: Histogram of Uniaxial Compressive Strength of Hong Kong tuff (after: Frenzel 2010).Neither of these two examples represents a normal distribution, but both showa distinct asymmetry. That could be explained by geology: The Hong Kong granitehad its highest strength after being created and weathering started to degrade the freshrock which led to lower rock strength. That results in a skew towards the left (i.e.lower rock strengths).On the other hand the Hong Kong tuff is metamorphic which eventuallyincreased its strength, but not in a perfectly homogenous manner. That explains theskew towards the right (i.e. higher rock strengths).D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T o n g j i U n i v e r s i t y o n 04/13/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r ve d .The difficulty to estimate statistical distributions is to obtain a sufficientsample size. This could be substituted by a small number of tests if the type ofdistribution was linked to the geology of the rock. That should be subject of furtherresearch.Cerchar Abrasivity Index. Abrasivity of rock can be characterized by index tests orby its mineralogy. The most common index test is the Cerchar Abrasivity Index(CAI). However, since its results are influenced by a number of rock properties like:• Mineralogy • Strength of grain bounds• Grain size • Shape of grainsthere is an inherent variability to this parameter as well. In addition, several issues have been identified and discussed in regards to the test itself (Plinninger 2002,Rostami 2005, Käsling et al. 2007).Correlations. UCS and BTS are correlated for many specific rock types and this hasbeen shown as a ratio (Gunsallus & Kulhawy 1987, Heyne 1980). Since the ratiobetween UCS and BTS easily ranges from as low as 5 up to 20 there is the questionhow strong this correlation is.A correlation analysis for UCS, BTS, and CAI has been conducted to evaluatevariances and correlations between these parameters. The data were taken from 20different tunneling projects including a range of different rock types. The results aredepicted in table 2.Table 2: Empirical standard deviations and correlation coefficient for UCS,BTS, and CAI from 20 tunneling projects.UCS BTS CAIUCS s =63.3 MPa R =0.79 R =0.75BTS s = 5.6 MPa R =0.75CAI s = 1.5Although the dataset included many different rock types ranging fromsedimentary through metamorphic to igneous and volcanic there are clear positivecorrelations. This means that there is a bigger probability that a sample with a highUCS value also exhibits a high BTS and a high CAI value. The opposite also holdstrue. This has implications on the prediction of cutter life since using average valueswould underestimate the probability of extreme combinations of UCS, BTS, and CAI.STOCHASTIC EXTENSION OF THE PREDICTION MODELThe stochastic model of cutter life prediction can be expressed as the followingfunctionD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T o n g j i U n i v e r s i t y o n 04/13/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r ve d .where the penetration p and abrasivity CAI are the random variables. Randomvariables are printed in grey to distinguish them from non-random input parameters.In addition, the penetration p can be expressed as a function of the random variablesUCS and BTS:The stochastic model reflects the variances σ and correlations ρ of the random variables and assumes normal distributions as an approximation:For linear problems an analytical approach can be taken to predict thevariance of the results. However, in this specific case the prediction model is iterativeand non-linear which requires a different approach.Monte-Carlo-Simulation. A numerical solution to calculate the characteristics ofsuch a problem is the Monte-Carlo-Method. It is based on generating a large numberof input data sets and to calculate the result for each input data set. The results canthen be characterized using descriptive statistics.Generating input data sets requires some specific steps to ensure that the datasets exhibit the correct distribution functions and show the correct correlations.Typically, programming language only provide functions for pseudo-random numbergeneration that follow an equal distribution. Three of these random numbers areconverted into normal-distributed random numbers using the central limit theorem.Finally, correlations are introduced by using a Cholesky repartition of the variance-covariance matrix and transformation by the mean.Example. To analyze the consequences of the considerations above an example hasbeen studied. The input parameters are summarized in table 3.Table 3: Input parameters for cutter life predictionMean UCS 150 MPaExcavation diameter 9.0 mMean BTS 10 MPa Disc cutter diameter 432 mmMean CAI 3.8 Max cutter load 267 kNVariance of UCS 50 MPa Spacing 90 mmVariance of BTS 3 MPa Disc cutter tip width 19 mmVariance of CAI 0.5 Number of disc cutters 57Covariance UCS-BTS 0.78 Cutterhead speed 5.3 1/minCovariance UCS-CAI 0.7 Cutterhead torque 5.77 MNmCovariance BTS-CAI 0.7D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T o n g j i U n i v e r s i t y o n 04/13/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r ve d .The simulation was run using 10,000 iterations. The resulting distribution ofpenetration and cutter life is shown in figures 3 and 4.Figure 3: Histogram of penetration rate of Monte-Carlo-SimulationFigure 4: Histogram of cutter life of Monte-Carlo-SimulationBoth histograms show a right-skewed distribution which is a result of the non-linearity of the prediction model. The peak at a penetration rate of 10 mm in figure 3D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T o n g j i U n i v e r s i t y o n 04/13/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r ve d .can be explained by the transition between thrust and torque limitation as discussedearlier. For comparison purposes table 4 shows the comparison between a predictionusing average values versus the results obtained through Monte-Carlo simulation.Table 4: Comparison of Monte-Carlo simulation and static prediction usingmean input valuesPenetration Rate [mm] Cutter Life [m 3]Monte-Carlo simulation 9.1 486 Mean values without simulation 8.3 398DISCUSSIONThe differences between the Monte-Carlo simulation and the static prediction show adifference of 10% for the average penetration rate and a difference of 22% for cutterlife. The difference for penetration rate prediction is probably neglectable forpractical purposes. However, the difference of 22% for cutter life is significant andcannot be disregarded easily. However, it is important to run the simulation on bothvalues because the penetration rate influencing the cutter life.CONCLUSIONBased on the prediction model for penetration rate and cutter life prediction for tunnelboring machines developed at the Colorado School of Mines a methodologyaccounting for the inherent uncertainties of rock is introduced. The model usesUniaxial Compressive Strength, Brazilian Tensile Strength, and the CercharAbrasivity Index as geotechnical input parameters.Through extensive testing the statistical distribution of these parameters canbe determined. Typically, the type of distribution is a normal distribution, butdepending on geology this may be skewed significantly as shown on the example ofgranite and tuff from Hong Kong.In addition, there are high correlations between these three parameters whichmean that there is a higher probability of a high UCS value occurring along with ahigh BTS value and a high CAI value. This leads to an increased variance of theresulting penetration rate and cutter life compared to the variance of the inputparameters.The existing prediction model can be used as the basis for a Monte-Carlosimulation which incorporates the random nature of the input parameters and whichallows to study the characteristics of the results.Through a calculated example it becomes clear that although the differencesbetween a simulation and a static calculation using average input values may lead tosignificant differences for predicted cutter life.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T o n g j i U n i v e r s i t y o n 04/13/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r ve d .REFERENCESFrenzel, Christian (2010): Verschleisskostenprognose für Schneidrollen beimaschinellen Tunnelvortrieben in Festgesteinen , Münchner GeowissenschaftlicheAbhandlungen, Reihe B, Dr. Friedrich Pfeil, MünchenGunsallus, K.L.; Kulhawy, F. H. 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