Probabilistic analyses in Phase 2 8.0

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1 Probability desity Probabilistic aalyses i Phase 8.0 Beoît Valley & Damie Duff - CEMI - Ceter for Excellece i Miig Iovatio "Everythig must be made as simple as possible. But ot simpler." Albert Eistei Itroductio I order to develop a reliable desig approach, oe must use statistical methods to deal with the variability of the iput parameters. However tools usually used i geomechaics, like stress aalyses (e.g. Fiite Elemet Aalyses, FEM), are i essece determiistic (a sigle set of iput parameters leads to a sigle aswer). Also, these tools are ofte computig time itesive ad are ot well-suited for the multiple rus eeded for systematic sesitivity aalyses or statistical simulatios (e.g. Mote-Carlo). That's the reaso why the Ceter for Excellece i Miig Iovatio (CEMI) recetly cotracted RocSciece Ic. to itroduce a alterate method, the Roseblueth poit-estimate method (PEM, Roseblueth, 1975), a simple, computig efficiet probabilistic method, ito their FEM software Phase versio 8.0. This paper presets the approach ad discusses its applicability. Ucertaity, variability ad heterogeeity Whe cosiderig statistical distributios of iput parameters i geomechaics problems, three differet cocepts must be cosidered: ucertaity, variability ad heterogeeities. These three cocepts must be treated separately as they have various impacts o the rock mass behaviour ad, therefore, differet approaches must be used to tackle them. Ucertaities arise from the difficulty i measurig key geomechaical properties like rock stresses, rock modulus or rock stregth. Ay of these measuremets ivolves some error due to the samplig process, sample preparatio or sesitivity ad calibratio of the measurig devices. This ucertaity is usually evaluated ad reduced by acquirig repeated measuremets durig the developmet of a project (Fig. 1). Cosiderig a give desig criterio, the probability of failure is give by the gray areas o Fig. 1 which, whe combied with the cosequece of failure, allows the computatio of the risk of a give desig (takig the stadard defiitio of risk beig probability of occurrece times cosequece) ad therefore, a evaluatio of whether this risk is acceptable. Potetial for failure Demad Fial desig Detailed desig Prelimiary desig Capacity The PEM/FEM method preseted i this paper is particularly suited to hadle this kid of situatio, i.e., it allows oe to track how ucertaities i the iput parameters are propagated through the Desig parameter Fig. 1 Illustratio of the ucertaity reductio durig the developmet of a project util the potetial for failure is miimised to a acceptable level (after Hoek, 199)

2 aalyses ad produce ucertaity i the desig parameters. It allows the egieer to ot limit the desig to a sigle determiistic aalysis with the most probable parameters (the mode of the distributio of Fig. 1), but to evaluate the reliability of the desig by cosiderig the dispersio of the desig parameters. Variability is a iheret property of atural materials ad rocks or rock masses are o exceptio. It arises from the various formatio ad trasformatio processes of rock ad rock masses which have a local ifluece o their mechaical parameters ad characteristics. Due to this variability, rock mass properties will vary, for example, withi a rock uit alog the trace of a tuel. Thus, a failure mechaism will affect more or less severely various locatios alog this tuel. Here agai the PEM/FEM approach preseted i this paper is well-suited ad will, for example, allow the egieer to aticipate what percetage of a tuel sectio will be affected by a failure mechaism for a give severity level. It will also allow for a evaluatio of the rage of severity of a give failure mechaism that should be aticipated ad thereby permit the iclusio of flexibility i the desig to hadle the less probable but potetially more severe situatio. Havig a estimate of the distributio of the severity of a potetial failure mechaism will also permit the optimisatio of the support systems ad allow a better estimatio of the cost ad thus the ecoomical risk of a project. Heterogeeities eed to be treated separately as they will ifluece the severity of a failure mechaism ad, more importatly, chage the behaviour of the rocks or rock masses. For example, icreasig modulus heterogeeities i a rock will promote the developmet of local tesile stress eve i a overall compressive field, which will affect the failure mode, e.g. chage from a shear mechaism to a tesile domiated mechaism like spallig (e.g. Diederichs, 007). The PEM/FEM approach proposed i this paper does ot simulate the effect of heterogeeities. Heterogeeities must be hadled differetly, either by the use of a classificatio system, coupled with equivalet homogeeous properties (Hoek ad Brow, 1997) or by explicit modellig of the heterogeeities (e.g. Valley et al., 010a). The PEM approach I the simplest case, whe closed-form aalytical solutios are available for a aalysis ad whe the iput parameters are idepedet ad ucorrelated, the propagatio of errors ca be approximated by usig a first order Taylor series. However, such a approach requires that it is possible to extract a partial derivative for the solutio fuctio. This is ot always feasible ad is obviously impossible whe the solutio to a problem is foud by a umerical method like FEM. The poit estimate method proposed by Roseblueth (1975), allows oe to propagate error eve if o closed-form aalytical solutio is available. The priciple of PEM is to compute solutios at various estimatio poits ad to combie them with proper weightig i order to get a approximatio of the distributio of the solutio (see Fig. ). The PEM implemeted i Phase 8.0 is the two-poit estimate method for the first ad secod momet of ucorrelated variables. It eeds evaluatios of the solutio, where is the umber of radom variables. The distributio of the solutio for y f x, x,..., x ) is give by: ( 1 y i 1 wf i (1)

3 INPUT Probabilistic iput variable 1 x 1 ± x1 y wfi wfi () i i OUPUT P - 1 P+ 1 Probabilistic iput variable x ± x P - P+ solutio (e.g. FEM solver) y=f( x,x ) 1 Compute solutios for combiatios of estimatio poits. - - y--=f( P 1, P ) + - y+ -=f( P 1, P ) - + y- + =f( P 1, P ) + + y ++ =f( P, P ) 1 Probabilistic output y ± y Combie the solutios with proper weightig to approximate the probabilistic output variable Fig. Illustratio of the computatio priciple of a approximatio of the output probabilistic variable usig the poit estimate method. I this example, the case with oly two probabilistic iput variables is assumed. where the weights w are give by 1/. f i are successive evaluatios of f at the possible combiatios of the radom variables at the poit estimate locatios, i.e. at x ad x x. I the solutio preseted here, all iput ad variable ad output variables are assumed to follow a ormal distributio give by their mea x ad stadard deviatio x. Example of applicatio I order to illustrate how to use the PEM with FEM let's look at the followig example: the stress distributio aroud a circular opeig has to be evaluated, but the estimatios of the far field stresses (S 1 ad S 3) are ucertai. Let's assume that this ucertaity ca be captured by a ormal distributio, i.e., a mea ad a stadard deviatio (see modellig properties give i Table 1). I order to evaluate the ucertaity associated with some desig parameter (let's assume, for example, the maximum pricipal stress at the excavatio boudary), four (, because there are two radom variables, S 1 ad S 3) models (Fig. 3a to d) must be ru, assumig the followig combiatios of iput for the far field stresses: [S 1=5 MPa; S 3=13 MPa], [S 1=5 MPa; S 3=17 MPa], [S 1=35 MPa; S 3=13 MPa] ad, [S 1=35 MPa; S 3=17 MPa]. These combiatios cosist of all possible combiatios of the mea ± oe stadard deviatio. The outputs of these models must the be combied usig Equatio (1) ad Equatio () i order to obtai the mea ad stadard deviatio of the output desig criteria (Fig. 3e ad f). It is iterestig to see that eve i this simple case of elastic stresses aroud a circular opeig, the patter of ucertaity (see Fig. 3f) is quite complex ad ot ituitive. The highest ucertaity is located where S 1 is maximum while a area of low ucertaity arises i the S 1 far field directio at about oe tuel radius (darker area o Fig. 3f). x

Desity of probability Table 1 Iput parameters for modellig Far field stress iput S =5 MPa =13 MPa S 1 3 S1=5 MPa S

far field pricipal stress S 3 (vertical) Mea Stadard deviatio 30 MPa 5 MPa 15 MPa MPa Out of plae stress S z 10

5 - a a These variables are ot cosidered as radom variables ad thus o stadard deviatio is defied for them.

1 ad ) S 1 cotour MPa MPa e) Mea of S 1 [MPa] a) 5.0 1.95 9.5 14.0 18.5 3.0 7.5 3.0 36.5 41.0 45.5 50.0.60 3.5 3.90 4.

0 90.5 95.0 10.40 11.05 11.70 1.35 13.00 13.65 14.30 14.95 S1=35 MPa S =13 MPa 3 c) g) 0.

05 ouput radom variable ( S 1 at the black dot above) S max 1 =73.6±14.

a), b), c) ad d) evaluatio of the maximum pricipal stress S1 (FEM elastic models) at the four combiatios of the

4 Desity of probability Table 1 Iput parameters for modellig Far field stress iput S =5 MPa =13 MPa S 1 3 S1=5 MPa S =17 MPa 3 Iput parameter Max. far field pricipal stress S 1 (horizotal) Mi. far field pricipal stress S 3 (vertical) Mea Stadard deviatio 30 MPa 5 MPa 15 MPa MPa Out of plae stress S z 10 MPa - a Youg modulus E 0 GPa - a Poisso ratio ν a a These variables are ot cosidered as radom variables ad thus o stadard deviatio is defied for them. Models ru at poit estimate locatios S 1 cotour Statistical output usig PEM (Eq. 1 ad ) S 1 cotour MPa MPa e) Mea of S 1 [MPa] a) b) f) Stadard deviatio of S 1 [MPa] S1=35 MPa S =13 MPa 3 c) g) S 3 =15± MPa iput radom variable (far field stress) d) 0.1 S 1 =30±5 MPa S =35 MPa =17 MPa S ouput radom variable ( S 1 at the black dot above) S max 1 =73.6±14.8 MPa Stress [MPa] Figure 3 Example of PEM/FEM computatio with parameters give i Table 1. a), b), c) ad d) evaluatio of the maximum pricipal stress S1 (FEM elastic models) at the four combiatios of the estimatio poits; e) ad f) probabilistic output (mea ad stadard deviatio) for S1 obtaied by combiig the FEM results i the left with the PEM (Equatios (1) ad ()); g) Probability desity fuctios of the iput (S1 ad S3, dashed lie ad dotted lie) ad the output of the aalyses where pricipal stress is maximum (S1 max ) (see black dot for locatio o e ad f).

5 The example preseted o Fig. 3 was selected for didactic purposes ad is very simple. The implemetatio of the PEM i Phase 8.0 permits iclusio of all the complexity that Phase typically allows, icludig complex geometry, excavatio stages, plasticity, etc. The Phase 8.0 iterface facilitates the iterpretatio of the probabilistic output by offerig the appropriate visualisatio tools, icludig stadard deviatio cotourig, coefficiet of variatio cotourig, ad lie plots with error bars. Whe icreasig the complexity of the model, oe must however be aware of the limitatios of the PEM. Particularly, i complex models, whe multiple behaviours occur cocomitatly, the actual output distributio ca sigificatly differ from a ormal distributio ad thus the PEM may have difficulty i capturig it accurately. This may happe, for example, whe lookig at a locatio close to the frige of a plasticity frot where both mechaisms affect the output distributio. These effects were studied i detail by comparig PEM ad Mote-Carlo output (Valley et al., 010b) ad the results are preseted schematically o Fig. 4. Whe all combiatios of estimatio poits, as well as the major part of the iput distributios, geerate the same mechaism (Fig. 4a), the PEM approximatio of the output distributio is accurate. However, whe mixed behaviour modes occur (Fig. 4b ad c), the PEM output ca be iaccurate ad caot capture the presece of tails i the output distributio (Fig. 4b) or the overall output distributio shape (Fig. 4c). Fig. 4: Illustratio of the effect of mixed behaviour o the accuracy of the PEM approximated output distributio compared to a actual output distributio evaluated usig Mote-Carlo computatios.

6 Coclusios The PEM/FEM approach implemeted i Phase 8.0 presets a attractive method for hadlig the ucertaity ad variability iheret i most geomechaical problems. The approximatio usig poit estimates makes it computatioally efficiet ad permits the performace of statistical aalyses for problems for which other methods like Mote-Carlo simulatio are ot practical. However, its simplicity brigs some limitatios. The PEM approach, as preseted here without correlatio, is based o ormal ad ucorrelated distributios. Whe a modelled case differs from these assumptios the results ca be iaccurate. Geerally, the cetral tedecy ad some variability aroud it is well captured, but i may cases the tails may ot be captured properly. Whe modellig ivolves behaviour discotiuities, as for example whe trasitioig from elastic to plastic domais, the poit estimate method shows further limitatios ad does ot accurately capture the distributio of the desig criteria. For this reaso, it is recommeded to test the effect of a limited umber of radom variables at a time. This will ot oly save computatio time ad allow deeper exploratio of the possible outcomes but will also permit a better uderstadig ad cotrol over the potetial bias itroduced by the PEM/FEM approach. I additio to the outputs obtaied usig the proposed PEM/FEM approach, it is recommeded to maually ru some extreme cases of the targeted distributios i order to determie if it captures the tails of the output distributio properly. Never forget that a model must be as simple as possible, but ot simpler. I summary, whe combied with a awareess of the assumptios ad potetial limitatios, the PEM/FEM approach offers a attractive ad very efficiet way of cosiderig ucertaity i FEM aalyses. It should lead to a broader use of the probabilistic approach i the miig idustry ad a better assessmet of the reliability level of the desig of udergroud opeigs. Refereces Hoek, E. (199) Whe is a desig i rock egieerig acceptable? Müller lecture, i Proceedigs 7th Cogress It. Soc. Rock Mech., Aache: A.A. Balkema, pp Hoek, E. ad Brow, E.T. (1997) Practical estimates of rock mass stregth, Iteratioal Joural of Rock Mechaics ad Miig Scieces, doi: /s (97)80069-x, Vol. 34 (8), pp Roseblueth, E. (1975) Poit estimates for probability momets, i Proceedigs of the Natioal Academy of Scieces, Vol. 7 (10), pp Valley, B., Suoriei, F.T. ad Kaiser, P.K. (010) Numerical aalyses of the effect of heterogeeities o rock failure process, i ARMA coferece, paper o , Salt Lake City. Valley, B., P. K. Kaiser, ad D. Duff (010). Cosideratio of ucertaity i modellig the behaviour of udergroud excavatios. I M. Va Sit Ja ad Y. Potvi (Eds.), 5th iteratioal semiar o deep ad high stress miig, pp Australia Cetre for Geomechaics.

Lecture 19: Convergence

Lecture 19: Convergence Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may