Toward Fuzzy block theory

(1) In contact with excavation (active block) (2) finite (3) movable (4) significant to other block movement. Base on this event “Goodman and Shi” proposed “block theory” (Goodma&shi, 1985). In this theory, analysis of key blocks in stability and identifications of key blocks are argued. Associated with this theory, different extensions, has been emerged such: probability analysis (Muldon, 1994), linear programming (Mauldon etal, 1997), key group method (Yarahmadi&Verdel, 2003). In this study, the blocks and key block method, from different view has been evaluated. Determination of blocks in fuzzy geometry and by possibility theory, can introduced direct and indirect combining between fuzzy theory and block theory. Background of this new combining can be induced from analyzing of following terms in fuzzy set theory: “approximation of blocks by linguistic variables”, “non-crisp boundary of blocks or vagueness in shape of blocks”, “modern uncertainty theories on analysis of key blocks”. In completing of static analysis on the fuzzy blocks, contact of blocks can be added. For example let minimum distance of two blocks is impression. Expression of distance in fuzzy numbers and using possibility theory can be lead to “possibility of blocks’ contact” (Owladeghaffari, unpublished). Parallelization of key block theory by Neuro Fuzzy Inference System (NFIS) may give a compressive view in possibility distribution of inputs and outputs. This procedure, in limit case, will be described in section2. In section 3, briefly, possibility theory and fuzzy geometry will be explained. Direct method, in section4, will be rendered.

Department of mining and metallurgical engineering, Amirkabir university of technology (Tehran polytechnic), Tehran, Iran ABSTRACT: This study, fundamentals of fuzzy block theory, and its application in assessment of stability in underground openings, has surveyed. Using fuzzy topics and inserting them in to key block theory, in two ways, fundamentals of fuzzy block theory has been presented. In indirect combining, by coupling of adaptive Neuro Fuzzy Inference System (NFIS) and classic block theory, we could extract possible damage parts around a tunnel. In direct solution, some principles of block theory, by means of different fuzzy facets theory, were rewritten.

Figure (1) summaries two branches of uncertainty .Modern uncertainty theory has been extended by Lotfi..A.Zadeh (Zadeh.1965):“fuzzy set theory”. Fuzzy logic (FL) is essentially coextension with fuzzy set theory and in narrow sense; fuzzy logic is logical system which is aimed at a formalization of modes of reasoning which are approximate rather than exact. FL in wide sense has four principal facets: The logical facet, FL/L; the set-theoretic facet (FL/S), the relational facet (FL/R) and the epistemic facet FL/E. (Dubois&Prade.2000) Figure1.schamatizatio of the uncertainty theory (Ayyub& Gupta, 1994-Zadeh, 2005)

Figure 3 shows a combining of KBT (key block theory) and TSK type inference system. One way to extension of this algorithm, can be carried out using multiple inputs/outputs systems, for example, CANFIS or MANFIS: coactive neuro-fuzzy inference systems; multiple ANFIS (Adaptive Neuro Fuzzy Inference System), respectively. (Jang etal.1997). In this study input parameters were dived in two facets :( 1) Fixed parameters (2) changeable parameters. Fixed parameters can be taken in such as shape of tunnel, unit weight of rock, some properties of joints….Changeable parameters must be inserted in different values, namely, in random data set, for example: joint properties, in situ stresses…After producing of KBT output, input data (changeable) and outputs of KBT must be rearranged. So these data sets must be normalized in defined range (for example in [-1, 1] -Step 1). Then normalized S.F, obtained from KBT, and mentioned data sets are gotten in ANFIS algorithm. In this step (2), the rules in if-then shape between input and output variables are obtained. Thus new predictions on S.F for new input can be performed. (Zadeh, 2005) Figure3. A combined algorithm on KBT, TSK Some results of the proposed algorithm can be highlighted as follows:

1-Detection of membership functions (MFs) for any input and output (figure 4) 2-The dominated rules in if-then format between inputs and output (safety factor for any block) 3-Possible damage parts around tunnel. In similar conditions; a compression between DDA (discontinuous deformation analysis)-MacLaughlin&Sitar.1995-and results of mentioned algorithm has been accomplished. See figure5. In this analysis, inputs were joint and tunnel pro