The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India

Determination of Ground Behavior using Fuzzy Logic

K. Oberste-Ufer, D. Hartmann Institute for Computational Engineering, Ruhr-University Bochum, Bochum, Germany N. Radoncic, W. Schubert Institute for Rock Mechanics and Tunnelling, TU Graz, Graz, Austria

Keywords: fuzzy logic, reasoning, expert system, natm, ground behavior types

The Institute for Computational Engineering at the Ruhr-University Bochum has been developed a fuzzy logic expert system for the determination of ground behavior in cooperation with the Institute for Rock Mechanics and Tunneling at the TU Graz. In the following paper the solution approach and the developed system is described. The approach extends the established calculation procedures by means of fuzzy logic components, expert rules and generates results that correlate to the distributed and uncertain nature of the problem. This paper elucidates the development of the fuzzy system starting with non-formal expert knowledge and its transformation into a fuzzy-based software system. Particular focus is placed on the modeling of the domain in terms of fuzzy sets and the fine tuning of the involved parameters.

1 Introduction Due to the high complexity, tunneling is a challenging problem that requires a tremendous amount of technology, knowledge and engineering experience. To obtain an appropriate pre-design of the underground structure, both the determination of the ground properties based on investigation data and the determination of the ground behavior with respect to associated failure modes are essential. Although geological investigation techniques have made great advantages over the recent years, information about ground conditions still contains considerable uncertainties. Geotechnical properties can vary in the observed area and are accurately determined only at the specific measurement points (e.g. drill hole, exposures, etc.). Nevertheless, these spatially scarce and inherently scattered data are used as the basis for the preliminary design of tunnels. Only experienced geotechnical/geological engineers are able to interpret these data properly and to draw conclusions leading to a safe and sustainable tunnel construction. Needless to say, the stated experience also leads to a certain amount of biasing, as with all decisions based on human thinking processes. Hence, the concept of “Ground Behavior Types” (GBT) has been introduced in order to minimize the amount of subjective reasoning in the course of tunnel design. Basic idea behind this concept is to classify the different types of ground behavior depending on the individual failure mechanisms. In the past, various groupings of ground behaviors have been proposed. Examples are the grouping of excavations in hard rock tunnels (Hoek et al., 1995) into “structurally controlled, gravity driven” and “stress induced, gravity assisted” or the classification of the Austrian guideline (Austrian Society for Geomechanics, 2001) as used in this work. In the NATM, the ground is considered as an integral load bearing element, not only as a loading element. The installed support is, in idealized case, merely a “help” to the surrounding rock mass to cope with the new stress distribution due to the creation of the cavity. In order to properly react and therefore, devise the most efficient design, the probable failure modes have to be correctly determined. The first step in such a design scheme is to determine the reaction of the ground to the excavation while ignoring the influences of the construction sequence, excavation method and support measures. Being a basis for the quantitative modeling the behavior of the rock mass for an infinite long and unsupported tunnel, without considering failures induced by dynamic loads, is classified into the following groups of basic behavior:

• gravity controlled failure of kinematically free blocks • ravelling of rock mass

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• stress induced failure of the rock mass • flowing of rock mass • swelling of rock mass

Due to its definition, each Behavior Type focuses mainly on one general kind of rock mass failure. Many failure modes combine into observed ground behaviour, influencing each other and standing in a clear causal relationship. On the other hand, certain failure modes cannot co-exist parallely due to a priori given mechanical reasons (for example: rock burst and flowing ground). Shortly put, the effort to force the reality (e.g. the true ground behavior) into rigidly defined classes (11 ground behavior types) is almost always connected to information loss. In order to circumvent this problem, the fuzzy-logic based expert system presented in this paper was developed.

2 General structure of the expert system The expert system uses the ground types (and the associated ground properties), the stress state, the ground water conditions and the orientiation of the geological structure relative to the underground structure as input for further evaluation. The schematic structure of the expert system is shown in Figure 1.

Figure 1: fuzzy-logic expert system with its main components

2.1 Classification variables A set of closed form solutions uses the ground properties, stress state and ground water conditions as input to calculate the consequences of excavation on the surrounding rock mass. The sweep through these relationships yields the following results:

• depth of the failure zone, • overbreak volume , • potential for daylighting failure, • potential for buckling failure, • potential for ravelling, • potential for flowing ground, • potential for rock burst, • potential for swelling.

The above quantities form the basis for the further classification process. A check against (partly theoretical, partly empirical) delimiting values (constraints) allows the determination of the likely failure mechanisms under given circumstances.

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2.2 Fuzzy delimiters The fuzzy definition of threshold values is dictated by the limitations in the knowledge of the ground properties and shortcomings of calculation models. Each obtained classification variable indicates rather a potential for a certain failure to occur than a strict and deterministic prediction. Hence, the incorporation of fuzzy logic delimiters is used for “softening” the sharp boundary between occurrence and non-occurrence. Needless to say, the definition of the membership functions has to be soundly derived from the mechanical background of the classification variable the membership functions are applied upon. The result of this evaluation by the means of membership functions results in values ranging between 0 (no potential detected) and 1 (very likely). Once the occurrence of every ground behaviour type has been quantified by the membership functions, the results of the quantification can be written down in vector form (1), every column representing one ground behaviour type.

1110987654321 GBTGBTGBTGBTGBTGBTGBTGBTGBTGBTGBT [ ]0,01,00,00,00,00,10,00,03,00,10,0=GBTvr (1)

2.3 Compatibility Matrix The imperative of mechanical soundness while combining detected ground behaviour types results in the relationship shown in Figure 2. Due to the differentiation between “primary” and “secondary” occurrence the presented relationship is not symmetrical.

Secondary ground behaviour type

1 2 3 4 5 6 7 8 9 10 11

1 ○ ○ ○ ○ ○ ○ ○ ○ + ○

2 ○ ○ ○ ○ ○ ○ ○ ○ + ○

3 ○ + ○ ○ + + ○ ○ + ○

4 ○ + ○ ○ + ○ ○ ○ + ○

5 ○ + ○ ○ ○ ○ ○ ○ + ○

6 ○ + ○ ○ ○ ○ ○ ○ + ○

7 ○ ○ ○ ○ ○ ○ + + + ○

8 ○ ○ + ○ ○ ○ ○ ○ ○ ○

9 ○ ○ + ○ ○ ○ ○ ○ ○ ○

10 ○ + ○ ○ ○ ○ ○ ○ ○ ○

Prim

ary

grou

nd b

ehav

iour

type

11 ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

Legend: + possible, ○ impossible

Figure 2: Overview of valid ground behaviour type combinations

This relationship is easily written down in matrix form, where “1” denotes a valid and “0” an invalid GBT combination (2).

=

1000000000001000000010001000001000001000010001111000000010001000100100001001001000101010010011001100100000001001000000001

Comp

(2)

The main matrix diagonal has to consist of “ones”, enabling rowwise multiplication with the GBT vector (Hadamard product) – hence resulting in a new matrix with the dimensions 11 x 11, where now every row represents one primary ground behavior type associated with its combinations (3).

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=•

0.0000000000001.000000000.10000.0000003.0000000.000003.00001.00.00.00.000000001.00000.10000.1001.000000.0000.1001.0000100.000.1001.00010.1003.00.1001.000000000.1001.0000000000.0

CompvGBTr

(3)

2.4 Disambiguation As already stated, the matrix obtained by Hadamard multiplication of the GBT vector and the compatibility matrix contains all detected ground behaviour types and their valid combinations. The final result is obtained by taking the row with the highest primary ground behaviour type rating and the highest sum of its components. This rule ensures that the most prominent failure mechanism is materialized, and that the information loss is as low as possible.

0.0000000000001.000000000.10000.0000003.0000000.000003.00001.00.00.00.000000001.00000.10000.1001.000000.0000.1001.0000100.000.1001.00010.1003.00.1001.000000000.1001.0000000000.0

0.0000000000001.000000000.10000.0000003.0000000.000003.00001.00.00.00.000000001.00000.10000.1001.000000.0000.1001.0000100.000.1001.00010.1003.00.1001.000000000.1001.0000000000.0

Figure 3: the selected ground behaviour type combination

Figure 3 shows the result of the application of this rule for the matrix above. The final ground behaviour can be described as primary type 6, “Buckling of rock mass with a closely spaced discontinuities, accompanied by structurally controlled, gravity driven block failure. A low potential for swelling of the rock mass caused by the mineral structure is present”.

3 Fuzzy logic and inference Many scientific and engineering problems have a uncertain nature that cannot be appropriately handled by classical analytical approaches in a reasonable way. The reason for the difficulties often lies in the blurred boundaries of the involved system properties, but also in the exact mathematical representation. To this end, fuzzy sets offer an alternative approach for system representation and conceptual modeling. The theory of fuzzy sets was developed in 1965 by Prof. Lofti Zadeh (Zadeh, 1965) at the University of Berkeley and has become an important solution concept for problems which are characterized by an uncertain nature. In contrast to classical boolean logic, fuzzy logic operates on fuzzy sets that represent states other than true or false. The fuzzy set theory allows the mapping of interim values between zero (false) and one (true) for a particular system state and therefore the mathematical modeling and the computational representation of systems states like “very much” or “small”. In particular, the transformation of human uncertain knowledge into a fuzzy model offers more realistic evaluation possibilities than classical approaches. Transformation of problems into a fuzzy logic representation and evaluation of the fuzzy system requires three steps. The first step is the fuzzyfication of the problem values, that is, the transformation of the problem specific values into a fuzzy set. Each characterizing value (linguistic variable) of the problem is represented as a fuzzy set that consists of different mathematical functions (membership functions) representing different numerical ranges of the modeled problem value (linguistic term). The individual functions define the degree of membership of a crisp value to the corresponding linguistic term. A fuzzy set is therefore defined as the set of all points (x, µ(x)) where x represents the crisp value and µ(x) the membership function.

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1)(0 ≤≤ xµ (4) In the first step the formalization of knowledge in the form of declarative rules that represent the system behavior and problem definition is established. The rules use the defined fuzzy sets and define the rule-base of the system. By concatenating and associating different rules from the rule-base, the fuzzy logic inference engine can draw conclusions that incorporate all relevant aspects of the modeled problem. In the second step the inference process is organized based on declarative rules defined in the rule-base and the modeled membership functions. Inference in a fuzzy-based system denotes the superposition of the individual fuzzy sets to a final result. The last step of the process encompasses the defuzzification of the obtained fuzzy set result. This step transforms the final fuzzy set to a sharp “crisp” value for further processing. Several methods like “center of gravity” or “maximum membership” can be used to perform the transformation. For further information about fuzzy logic and inference refer to (Ross 2004).

3.1 Application of fuzzy logic in the expert system The expert system for determination of ground behavior types uses fuzzy logic to classify a given ground into predefined types. Classification is done on the basis of rock mass properties measured in situ on the construction site or in advance by borehole measurements or lab tests and entered to the system. Output of the system consist of a list of possible predefined ground behavior types and their related degree of membership for the given ground condition. For the implementation of the prototype the Java programming language has been used. The software itself has been designed and developed as a rich client application using the Eclipse RCP development framework (Eclipse RCP). This framework offers advanced development features for rich client applications like predefined components for graphical user interfaces along with configuration and validation options. The developed fuzzy logic expert system is based on the open-source fuzzyJ-Toolkit (FuzzyJ) that provides predefined membership-functions and methods for fuzzy-set creation, editing and inference. The framework also provides integration for the rule-engine “Jess” (Jess) that delivers advanced reasoning methods for rule driven expert systems. In particular, the Jess rule-engine uses the Rete algorithm (Forgy CL, 1982) for rule processing and can perform forward and backward chaining. A further advantage of the Jess rule-engine is that fuzzy rules can be applied within the reasoning process. Thus, conclusions drawn from the reasoning process incorporate the uncertain nature of the input values and reflect this in the calculated “exact” output. To illustrate the process of fuzzyfication, knowledge representation and expert system reasoning, the implementation of GBT1/2 and GBT3/4 is depicted. GBT1/2 represents intact rock mass with a small chance of failure because of gravity induced falling or sliding blocks (GBT1) as well as deep reaching, discontinuity- controlled failure (GBT2). GBT3/4 subsumes the ground behavior types 3 and 4 and represents failure due to stress. According to the (Austrian Society for Geomechanics 2001), GBT3 represents shallow stress-induced failure and GBT4 deep seated stress-induced failure. The critical parameter for the determination of the above GBTs is the depth of the failure zone (DFZ). The empirical factor lim1 characterizes the boundary between GBT1/2 and GBT3/4 and is given by equation (5).

rC ∗= 11lim (5) In equation (5), r is the tunnel radius and the factor C1 is usually set to 1.0. Values for DFZ lower than lim1 other, more likely failure mechanisms and values higher than lim1 indicate GBT3/4. The boundary between GBT3 (shallow stress-induced failure) and GBT4 (deep seated stress-induced failure) is defined by

rC ∗= 22lim , (6)

where r is the radius of the excavation and the parameter C2 is usually set to 2.5. Values exceeding the limit lim2 indicate GBT4. Values for DFZ that lie between lim1 and lim2 represent GBT3. Subsequently, a fuzzy set is shown for the depth of the broken zone parameter. The term-set for the linguistic variable “DFZ” consists of three linguistic values that range from “LOW” to “HIGH” and represent the different states the parameter value can obtain. Triangular and trapezoid functions are used to map the membership functions whereas the overlapping edges of the functions indicate the blurred nature of the ranges. By using the fuzzy representation in terms of fuzzy sets and membership functions, the degree of membership of a given value x is not mapped to only one single range. Rather it is described by different values representing the degree of

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membership to different ranges of the fuzzy set. Figure 4 shows the fuzzy set representation of the DFZ indicator.

Figure 4: Fuzzy set of the depth of the broken zone parameter with respect to the classification of ground

behavior types 1/2, 3 and 4. The values lim1 and lim2 represent the sharp boundaries used in the conventional evaluation.

The rule-base consists of different rules representing the underlying expert knowledge on the Ground Type Behavior and its associations. The rules have been formulated by the Institute for Rock Mechanics and Tunneling and are based on experience and the evaluation of the available standards. The semantic structure of each rule is an IF-THEN construct with one or more premises on the left side and only one conclusion on the right hand side of the rule. The three rules that define the classification of GBT1/2, 3 and 4 are shown below. IF DFZ=LOW THEN assign GBT1/2 IF DFZ=MEDIUM THEN assign GBT3 IF DFZ=HIGH THEN assign GBT4 The expert system, in particular the Fuzzy-Jess engine, evaluates the rules provided. If a rule “fires”, that means the premises or antecedent part of the rule (if-part) matches a given condition, the right hand side assigns the correspondent GBT. The conclusion part of the rule represents a new fuzzy value that is used as input for a subsequent defuzzyfication process. For the determination of ground behavior types, no defuzzyfication process is necessary. It is sufficient to know the fraction of each possible ground behavior type.

Figure 5: Exemplary inference for given DFZ value. The input leads to a 66% assignment of GBT3 and a 33%

assignment of GBT1/2.

Using the above example, where a value is given for the DFZ indicator, the inference process leads to a 66% fraction of GBT3 and a 33% fraction for GBT1/2. However, GBT4 is not considered because the DFZ value is outside the value range of the membership function for “HIGH”. The described process of fuzzyfication and is repeated for all rules and all GBT using the inference engine which incorporates the rule base as a whole. Finally, the rule base evaluation leads to a set of possible ground behavior types and their associated fractions for the given rock mass. To determine the final solution, the Compatibility Matrix (see section 2.3) is applied to the preliminary result set. By that, impossible combinations of different ground behavior types are removed and the final system result in terms of a condensed and consistent classification is achieved. In contrast to the conventional method, which proposes Ground Behavior Types in an absolute way, the system considers fractions of predefined Ground Behavior Types. As a consequence, the system is able to represent the prevailing rock mass situation in a more realistic way.

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4 Results The current results obtained from the prototype version of the expert system draw a promising picture to the final system. A detailed comparison with existing determination approaches will be available after the completion of the expert system.

5 Summary The paper introduces a prototype of a fuzzy-based expert system developed at the Institute for Computational Engineering at the Ruhr-University Bochum in cooperation with the Institute for Rock Mechanics and Tunnelling at the TU Graz for determination of Ground Behavior Types used in NATM tunneling. Having given a survey on the current methods, procedures and standards for GBT classification, a short introduction into fuzzy based systems and reasoning has been given. Then, the expert system developed so far has been introduced. Also, an example was given describing the process of fuzzyfication, reasoning and evaluation within the expert system. The current results from the evaluation of exemplary rock mass properties demonstrate the ability of the system to represent mixed Ground Behavior Types. In contrast to the conventional approach, the combination of fuzzy sets, expert knowledge and sophisticated fuzzy reasoning methods lead to more realistic classification results.

6 Acknowledgements The authors express their thanks for the financial support obtained from the European Commission in the 6th integrated Framework Project TUNCONSTRUCT (Technology Innovation in Underground Construction).

7 References Austrian Society for Geomechanics. 2001: „Richtlinie für die geotechnische Planung von Untertagebauten mit zyklischem

Vortrieb”, Austrian Society for Geomechanics, Salzburg, Austria

Charles L. Forgy 1982. Rete: A Fast Algorithm for the Many Pattern/ Many Object Pattern Match Problem, Artificial Intelligence 19 (1982), 17-37.)

Eclipse RCP, Rich Client Platform, http://www.eclipse.org/home/categories/rcp.php

FuzzyJ ToolKit for the Java(tm) Platform, http://www.iit.nrc.ca/IR_public/fuzzy/fuzzyJToolkit2.html

Hoek, E., Kaiser, P.K., Bawden, W.F. 1995. Support of Underground Excavations in Hard Rock. Balkema, Rotterdam

Jess, the Rule Engine for the JavaTM Platform, http://herzberg.ca.sandia.gov/

Ross, T.J. 2004. Fuzzy Logic with Engineering Applications, John Wiley & Sons.

Wang, J.A. und Park, H.D (2001): Comprehensive prediction of rock burst based on analysis of strain energy in rocks, Tunneling and Underground Space Technology (49-57)

Zadeh LA. Fuzzy sets. Information and Control 1965; 8: 338-353.

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