Reliability analysis of the existing reinforced concrete beams with normal cracks by rebar strength criteria

. The paper describes the problem of the reliability analysis for individual reinforced concrete (RC) beams by the rebar strength criterion in cross section with a normal crack. It is proposed to evaluate the stress in the rebar by measurements of the crack width at the level of the rebar using the functional dependence of the crack width and the strain (deformation) in the reinforcement. It is also proposed to replace the modulus of elasticity of steel reinforcement on the secant modulus of elasticity, taking into account the increased reinforcement strain in cross section with crack. The work considers two options of the crack width in beam: less and greater than the ultimate crack width. The reliability analysis of RC beams by the rebar strength is based on the possibility theory and fuzzy set theory by the reason of small statistical data from measurements on existing individual RC beams. The use of offered reliability analysis methods will allow preventing the failures of reinforced concrete beams and in some cases to obtain economic benefit from the possibility of further operation of RC beams with cracks, even with a crack width more than ultimate value.


Introduction
Mechanical safety is the state of buildings and structures in which there is no unacceptable risk of a structural failure. According to the Eurocode 0 "Basis of structural design", the reliability is an ability of a structure or a structural part to meet the specified requirements, including the design working life, for which it has been designed. Reliability is usually expressed in probabilistic terms. General principles of structural reliability are regulated by Eurocode 0 and International Standard "General principles on reliability of structures". The Interstate Standard GOST 27751-2014 "Reliability of structures and foundation soils" regulates the general principles of structural reliability in the Russian Federation.
The failures of structures, including reinforced concrete beams and slabs, testify to the insufficient implementation of methods for the reliability analysis of structures during their inspection. It also shows the lack of methods and techniques for reliability (safety) analysis of structures on some limit state criteria. The measure of structural element reliability is probability of non-failure by all limit state criteria. The limit state criteria for reinforced concrete beams are: rebar and concrete strength, deflection, cracking, crack widths, etc. This article will consider the method for reliability analysis of existing RC beams with normal cracks on rebar strength.
Microcracks naturally present in the concrete of the existing reinforced concrete structures subject to the influence of various factors. The microcracks can become macrocracks. A series of normal cracks in beams are often formed in the tensile zone of the beam under the influence of a distributed load [1]. The ultimate crack width  The cracking leads to a decrease of load-bearing capacity and reliability (safety) of reinforced concrete beams, even in case when where m is a mean value; s is a standard deviation. However, the distance between the cracks s l is not taken into account in [1]. Moreover, it is difficult to get the full statistical data for individual reinforced concrete beam because of large variability of some statistical parameters. The work [2] considers experimental and theoretical calculation of the reliability index for reinforced concrete beams with different cross-sections. The paper [3] describes the calculation of the non-failure probability (reliability index) of reinforced concrete beams with carbon-fiber reinforced polymers (CFRP). A statistical analysis to control cracking in reinforced concrete structures based on the "68-95-99.7 rule" is presented in [4]. The paper [5] discusses the risk assessment of corrosion in the rebar of concrete structures and evaluation of their reliability and durability by probabilistic and statistical methods. The paper [6] discusses the measurement and monitoring of the crack width using piezo-ceramic sensors. The work [7] presents a study of the durability of reinforced concrete beams with cracks under the influence of corrosive environments and the influence of the crack width on corrosion of rebar. The reliability analysis of reinforced concrete beams using different probabilistic and statistical methods are also discussed in [8][9][10][11].
The considered literature shows the possibility of crack width measuring in reinforced concrete beams, and its influence on the load-bearing capacity and reliability (safety) of reinforced concrete elements. However, to the best of our knowledge, methods of reliability analysis according to the criterion of the rebar strength in the cross sections with crack are not thoroughly investigated in the scientific literature. In particular, they are not applied to reinforced concrete beams with various statistical information about controlled parameters in mathematical models of limit state. So, this consideration is the theme of the present work.

Methods
The mathematical model of the limit state for reliability analysis of RC beam on rebar strength can be written as: where s  is the stress in a rebar in the beam crosssection with a crack; it is a random variable (marked by a wavy line above the symbol); , (1) can be represented as: : In [12] academician I. N. Karpenko proposed to correct the equation (3'). Here where k is a constant value; X, Y, Z are fuzzy variables by the reason of small statistical data in the individual reinforced concrete beam.

Results
Consider the reliability analysis of reinforced concrete beam according to the mathematical model (4) by the theory of possibility approach [13] using the Zadeh's principle from the fuzzy sets theory [14]. The possibility distribution function ) (x X  can be accepted as: Further, the fuzzy function G can be created from fuzzy arguments X, Y, Z in accordance with the Zadeh's principle as: and with the left g g   and the right g g   branches. The reverse function g of G will be determined through the reverse functions x, y, z from X, Y, Z as described above.
From the left branch of ) (g G  , we have: and for the right branch: A minus sign is placed before the "b" term if from this value the  The results of calculation using equation (5)  . If s  is taken as a constant value, then reliability analysis will not differ from the above-considered algorithm. Further we consider a more complex version with a parameter s  as a random variable, which was determined by measurements of  The example shows that even when the crack width is more ultimate value, which equal to 0. 3

Discussion
In addition, the reliability of a reinforced concrete beam as the mechanical system should be estimated by the theorem of probabilities multiplication as where i P is the probability of non-failure by i-criterion of limit state. This paper presents the method of reliability analysis by only one of the limit state criteria (1).
Consider the case of beam reliability analysis, in which statistical information about the parameter .
The fuzzy variable T=X / Z and its values (the arguments) t can be introduced.    1 ) ( = t P . The reliability of reinforced concrete beam is characterized by interval [0.9999; 1], that is more informative than the interval [0.9996; 1] in the example 2.
Thus, precision of results of reliability analysis is increased with the growing of statistical data on the parameter Y l s =

.
The examples 1-4 show that normal cracks in reinforced concrete beams can be dangerous even if the width is less than 0.3 mm or the other ultimate value by any standard. Conversely, it is possible to allow the operation of reinforced concrete beams with normal cracks greater than the ultimate values, because their reliability index by the criterion of reinforcement bars strength can be quite high. In this case, special measures should be taken to protect the reinforcement bars from corrosion.

Conclusions
1. The article proposes new approaches of structural reliability analysis for the existing reinforced concrete beams by the rebar strength criterion in cross-section with a crack. The proposed approaches take into account different statistical data on the controlled parameters in the design mathematical models of a limit state; 2. The choice of reliability analysis method depends on the amount of statistical data on random variables. The larger is the volume of statistical information, the narrower (more informative) the reliability interval is. If the reliability interval is too wide for decision-making or risk analysis, additional tests of control samples should be performed. The most accurate reliability value can be obtained by identifying the exact probability distribution function and its parameters for each random variable in the mathematical model of limit states; 3. The use of the described reliability analysis methods allow one to prevent the failures of reinforced concrete beams with normal cracks, and to obtain economic benefit from the possibility of further operation of reinforced concrete beams with normal cracks in some cases.