Estimating the reliability of forestry machine elements with possibility theory application

The working conditions of forestry machines differ from those of agricultural machines. The presence of obstacles during the clearing of forest areas increases loading of machine components and assemblies and, consequently, leads to their failures. This paper deals with an improvement of probabilistic methods of forestry machine design by applying fracture mechanics and possibility theory. The main fracture mechanics expressions linking stress intensity factor with crack-like defect length are presented in the introduction. Fracture toughness and crack-like defect length are viewed as Gaussian random values, maximum applied stress is presented as a fuzzy variable with unknown distribution law in the second part of the paper. Analytical equations for reliability evaluation are obtained by estimation of upper and lower bounds of reliability function. The real value of reliability function is located within this interval. The proposed approach may be applied to give recommendations for engineering of forestry machine and equipment elements in the case of limited statistical information.


Introduction
Attaining high-quality performance of forest management works related to forest restoration, forest protection, fire protection and other measures is an important task. The conditions of forestry machine operation are characterized by forest-growing zone, category of forest-cultivated or other area (cuttings, young trees, plantings, etc.), terrain, size of slopes, tractor passability (number of stumps, swampiness, slope) [1].
Working in rough terrain requires significant tractor driving force, good stability, high cross-country ability and maneuverability. The presence of obstacles (stumps, stones, logging waste) causes the need to operate in various speed and load modes, therefore, increasing the load on machine components and parts leads to their failures [2].
Probabilistic methods of estimation of forest machine reliability are sufficiently developed now [3][4][5][6][7].When using such methods, statistical information on material characteristics, form and dimensions of structural units, actual loads and other parameters that determine the object's reliability is supposed to be complete. Thus, by applying the well-known random values distribution law, it is possible to determine quite accurately some reliability indices, e.g. the probability of failure-free operation [4]. In practice, the availability of appropriate equipment and experimental data enables establishing laws of mechanical properties distribution of materials, parts dimensions, and defect sizes. However, complete statistical information on the character and level of actual loads is difficult to obtain as the conditions of forest and agricultural machines performance vary a lot and depend on a multitude of both objective and subjective factors. Supposing that strength and size characteristics have been determined and their distribution laws are known these characteristics may be described by using the probability theory. Actual loads are characterized by incomplete information due to insufficient statistical data; therefore here the method of possibility theory [8][9][10], evidence theory by Dempster-Shafer [11], Bayesian approach [12] and the interval average method [13] may be applied. These methods have been applied to make conventional calculations based on material resistivity equations [10] or structural theory when deformation criteria are used to account for crack effect but not the principles of fracture mechanics.
According to Irwin's fundamental concept of a stress intensity factor [14], the condition of operable state is written as is a dimensionless geometry factor, depending on the machine part shape and crack's length (semilength); 1 σ is the maximum applied stress; l~ is the crack length (semilength); IC K is the critical plane-strain fracture toughness.

Materials and methods
We consider the case when: - , that applies in the case in which the crack length is far less than the machine part dimensions; -l~ and IC K are stochastic quantities with ( ) The evaluation of parameters max 1 σ and min 1 σ is based on the experimental loading process-related data analysis. Obviously, this statistical information is incomplete and, therefore, variable 1 σ can be viewed as a fuzzy variable.
The pair of distributions (upper and lower probability distribution functions) are known as a probability box (p-box) in the possibility theory [11]. The unknown "true" distribution ( ) is probability distribution function of the random variable IC K .

Results and discussion
where ( ) ( ) and the fracture toughness probability distribution function is

Conclusion
It is of importance to improve the probabilistic and statistical methods of fracture mechanics, which allow accounting for the influence of crack-like defects on the level of reliability of forestry and agricultural machine parts and structural elements. However, in some cases there is not enough statistical information, which determines the direction of further research.
The method of probability function estimation of forestry machine parts and structural elements under the influence of extreme load using force criteria of fracture mechanics with random and fuzzy parameters is developed. It is recommended to consider the maximum applied stress as a fuzzy variable described by the possibility distribution law. The crack length and fracture toughness were considered as random variables with known distribution laws. The suggested method may be applied to give recommendations for engineering of forestry machine and equipment elements in the case of limited statistical information.