International Journal of Independent Research Studies

Volume 3, Number 1 (2016) pp 8-13 doi 10.20448/800.3.1.8.13 | Research Articles

 

Application of the Concept of Fractal for the Stress Assessment of the Condition of an Object

Barashkova T. 1
1 Associate Prof.[Tatjana], Tallinn University of Technology, Virumaa College, Jarvekula tee 75, 30322, Kohtla-Jarve, Estonia

Abstract

In the given paper application of fractal analysis to an estimation of the intense-deformed condition of a concrete slab with a hole-forming cores is observed. The new technique of an estimation of the strained condition of the specified concrete slab with use of the concept of a fractal is presented. The most widespread way of an estimation of a tension is connected with the use of a matrix of rigidity of system. In the paper the estimation of a course-of-value function of a measure which characterizes the loaded condition of a concrete slab is given. By means of a method of least squares and a method of the fastest descent, character of the intense-deformed condition of an object is assessed.

Keywords:  Matrix numbers, Fractal dimension, Course-of-value function, Fractal analysis.

DOI: 10.20448/800.3.1.8.13

Citation | Barasovahk T. (2016). Application of the Concept of Fractal for the Stress Assessment of the Condition of An Object. International Journal of Independent Research Studies, 3(1): 8-13.

Copyright: This work is licensed under a Creative Commons Attribution 3.0 License

Funding : The authors declare that they have no competing interests.

Competing Interests: The author declares that there are no conflict of interests regarding the publication of this paper.

History : Received: 25 May 2016/ Revised: 14 April 2016/ Accepted: 18 April 2016/Published: 22 April 2016

Publisher: Online Science Publishing

1. Problem Statement

The most widespread way of an estimation of a tension is connected with the use of a matrix of rigidity of system. We present short survey of the given technique. The armature is connected with concrete by means of special binding elements (BE), admitting mutual shifts of armature concerning the concrete, simulating the grip of armature with concrete. Consider that x is a relative shear of armature. The accepted mathematical model allows connecting a finite element of concrete with a finite element of armature at various length of a linear finite element (FE) of armature. In the capacity of a variable quantity the length of linear finite element of armature is accepted. Length of the finite element of the rectilinear, not unbent, stressed and not stressed armature are equal to a size of triangular finite element of concrete, and communication of finite element is carried out in each knot of a grid along the length of armature. The arrangement of binding elements is defined by means of mathematical model, the design scheme of a beam and an operating field of numbers. In such each nodal point in which binding element is absent, in the field of numbers it is marked as 0. In a point in which binding element occurs, the number distinct from 0 is established. And, if binding element unites finite element of concrete with the bottom rectilinear stressed armature in the corresponding knot of an operating field of numbers it is established 1, if with the bent stressed armature, number 2, and, if with upper not stressed number 3 is established. Such field of numbers can be changed depending on a slope and length of the bent part of armature. When formulating a matrix of rigidity of a system the following fact is considered: presence of binding element at any knot is taken into account by means of correction and change of corresponding sections of a matrix of rigidity of a system. Relationship between forces in finite element's bonding and mutual displacements of armature and concrete (where F-force in bonding) at level of a surface of contact is calculated by the following formula:

2. The Application of Fractal Analysis

Fiq-1. Matrix of numbers
Source: Allinson and Lawson (1990).

At the subsequent divisions we enlarge cages and again we make a matrix of numbers, for example on 4 cages, the square side doubles . Some boundary sections appear discarded. Then probabilities of realization of the loaded fragments, probabilities of that the binding element or certain quantity of binding elements got to corresponding cages, allow to define the function forming a fractal.

At the second division it was gained, for example:

Where

i-current parameter along all divisions, a-scaling parameter, q-dimension of a fractal by means of which the intense-deformed condition of an object is assessed. From all array of the fragments, any the most feasible, which characterizes the actual picture of a state of stress of a concrete slab.

Corresponding scaling parameter is realized for:

-density of a distribution of fragments  on parametere . The greatest contribution to an assessment of a course-of-value function  is given by those values of parameter , at which the parameter in sub integral expression (8) will aspire to a minimum, as the length  is small. The given condition means that:

The gained magnitude of a measure  allows defining a spectrum of dimensions of a quantity of a multifractal which assesses a character of a state of stress of an object.

3. Conclusion

In the given paper the probabilistic nature is used only and geometrical features of a multifractal are not used. Taking into account the fact that the length  of a finite element aspires to zero, it is possible to make fragment numbering different, considering, that the probability of a damage of a whole object equals to one. Dependence of a course-of-value on dimension of a multifractal allows estimating the character of intense-deformed condition

Research is supported by the program of ERASMUS-MUUNDUS

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About the Authors

Barashkova T.
Associate Prof.[Tatjana], Tallinn University of Technology, Virumaa College, Jarvekula tee 75, 30322, Kohtla-Jarve, Estonia

Corresponding Authors

Barashkova T.