Calculation of the reliability of brickwork according to the strength criterion

. In order to change the appearance of buildings or structures during major repairs and reconstruction, hinged ventilated facade systems are often used. To make a decision on the possibility of using a hinged ventilated facade, it is necessary to assess the technical condition of the bearing and enclosing structures of the building or structure in question, including determining their reliability. The article proposes a particular method for calculating the reliability of brickwork by the strength criterion, based on the combined method, for the case when the statistical information obtained as a result of the survey is complete in part of the controlled parameters, and in the other part of the parameters – not complete, since the use of a probabilistic method for calculating reliability will lead to the loss of some useful statistical information, since all parameters are represented as fuzzy variables, and the use of probabilistic statistical methods may lead to incorrect calculation results, since the assumption of the distribution law for the controlled parameters may be erroneous.


Introduction
The most common option for finishing the exterior walls of buildings are hinged ventilated facade systems (VF).Such facade systems have a certain set of advantages, such as: they protect the building from overheating in the warm season and keep warm in the cold season, i.e. they provide the necessary level of thermal protection of buildings corresponding to the current regulatory requirements, the level of humidity regime, allow you to create comfortable microclimate conditions inside the building, provide protection of the external enclosing walls of the building from aggressive atmospheric influences, allow you to realize the possibility of using vertical VF guides as a lightning protection system current collector, if necessary and the validity of such a decision, with the correct design of the structural facade design solutions have sound insulation properties, the VF cladding layer is durable and resistant to impacts, providing a certain degree of protection against vandalism, they have a large number of design solutions and a wide color palette of the cladding layer, allowing complex architectural tasks and various design solutions to be implemented, giving buildings architectural and artistic expressiveness, have high installation speed and maintainability, etc.Therefore, they are often chosen when carrying out reconstruction or major repairs of buildings and structures.The decision to use a hinged ventilated facade on already constructed buildings in operation must be carefully justified.One of the key points in making such a decision is to conduct a technical inspection of the bearing and enclosing structures of the building.There is a need to determine the reliability of the building foundation (the outer walls of the building) to which the VF is supposed to be attached.In the normative and technical document GOST 27751-2014, in force on the territory of the Russian Federation, it is proposed to use probabilistic and statistical methods when performing reliability calculations [1][2][3][4][5][6][7], using which it is possible to obtain an unambiguous assessment of reliability, however, the main condition under which these methods can be applied is the availability of complete statistical information on the parameters under consideration.By complete statistical information is meant that it is homogeneous, statistically independent and it is possible to establish the law and the parameters of its distribution.However, it is not always possible to obtain such information for all controlled parameters.In the theory of reliability of building structures, various approaches to reliability calculations have been formed, based on various theories, for example, the theory of possibilities, the Dempster-Schaefer theory, the theory of fuzzy sets, etc.The application of these theories in reliability calculations makes it possible to describe uncertainty in the absence of complete statistical information on the parameters under consideration.In practice, quite often there is a situation in which it is not possible to obtain complete statistical information only for a certain part of the parameters under consideration.In this case, when calculating reliability, the use of probabilistic statistical methods (based on probability theory) can lead to incorrect, so erroneous results, and the use of a probabilistic method (based on the theory of possibilities) [8][9][10] will lead to the loss of some useful statistical information on the parameter under consideration and, accordingly, the resulting reliability interval will be wider.That is, in order to obtain a more informative result, it becomes necessary to combine random variables (in terms of probability theory) and fuzzy variables (in terms of the theory of possibilities) when performing calculations on the reliability of building structures.This article presents the possibility of implementing such a combination of random variables and fuzzy variables on the basis of a modernized reliability calculation method [11,12] on the example of the proposed private method for calculating the reliability of brickwork of a building by the strength criterion.

Methods and materials
Let's consider the calculation of the reliability of a fragment of non-reinforced brickwork of the outer wall of a building by the criterion of strength under off-center compression (with a small eccentricity not exceeding the core of the section).Suppose that as a result of the examination of the external enclosing structure of the building in question, made of brickwork, no visible damage and defects were found.For the case under consideration, we use the mathematical model of the limit state given in the regulatory and technical document SP 15.13330.This model of the limit state, taking into account the variability of some parameters , will have the form: In eq. ( 1),  ̃ -the longitudinal force arising in brickwork from the action of a constant and temporary load. ̃=   +  ̃ , there   we assume a deterministic value, it can be determined by known methods of structural mechanics;  ̃ -a fuzzy variable, in view of the fact that the information obtained during the monitoring and inspection of the building about the time load may be heterogeneous and limited and, accordingly,  ̃ there will also be a -fuzzy variable.Coefficients:   (taking into account the impact of prolonged load),  1 (longitudinal bending),  (it is accepted depending on the type of masonry), and also   (the area of the compressed part of the section with a rectangular stress plot) -we take deterministic quantities. ̃ (brickwork compression resistance) -it can be determined in various ways, including non-destructive testing methods, for example, by using the elastic rebound method, methods based on the use of ultrasonic devices, etc., which allows you to obtain complete statistical information about the controlled parameter  ̃ and it can be considered as a random variable.
Calculation of the reliability of a fragment of brickwork of the exterior wall of a building according to criterion (1) for the case in which: - ̃ -a fuzzy variable characterized by an opportunity distribution function of the form: there -N ̃ult -the random variable will vary according to the normal law with the probability density of the distribution: A fuzzy variable  ̃ will be characterized by a set of distribution functions   () that will be located within the boundaries: from the lower bound   () to the upper bound   () of the probability of the distribution  ̃.If a fuzzy variable  ̃ is characterized by a function according to equation ( 2), then in accordance with the consistency condition (  () ≤   () ≤  ̅  ()) we get: -for the lower bound   () of the probability of the distribution ̃: -for the upper bound   () of the probability of the distribution  ̃: Many functions {P N (N)} are shown in Figure 1, they are located in the shaded area.The distribution function of the possibilities of the fuzzy variable  ̃ according to equation ( 2) and the probability distribution density function of the random variable N ̃ult according to equation ( 3) are shown in Figure 2. The reliability interval according to the criterion under consideration ( 1) is determined taking into account the provisions of the reliability calculation theory of A.R. Rzhanitsyn (to determine the values of the probability of an event:  ̃ ≤ N ̃ult ), according to which the probability of trouble-free operation for the option under consideration will be equal to: Using equation ( 6), we obtain formulas for determining the upper and lower values of the probability of trouble-free operation: Then taking into account the equations ( 2), ( 3), ( 4), ( 5), (7) we obtain formulas for determining the upper and lower values of the probability of trouble-free operation according to criterion (1) in the form: The reliability of the fragment of the considered exterior wall made of brickwork will be characterized by an interval of values [P; P], and the true value of reliability will be located within this interval.

Results and discussion
Let's consider a numerical example of determining the reliability (based on the use of the proposed private methodology) of a fragment of the external enclosing structure of a building made of brickwork, according to criterion (1).Let's assume that as a result of the survey and monitoring by parameter  ̃, the statistical information obtained was not complete, and accordingly, we consider this parameter as a fuzzy variable, and by parameter N ̃ult -managed to obtain complete statistical information and determine the parameters of its distribution.
For clarity, we will model a set of statistical data on the necessary controlled parameters:  ̃ and N ̃ult using a random number generator.After generating and analyzing the data, the following parameters were obtained: The reliability of a fragment of the external enclosing structure of a building made of brickwork, according to criterion (1) according to equations ( 8), ( 9 √ 57 2 +47 2 =1.37 , let 's determine the probability of failure -free work, which will be equal to P(  ≥ )=Ф(β)=Ф(1.37)=0.9147.This value (N ult ≥N)=0.9147falls within the reliability interval [P=0.835 ; P= 0.991], is unambiguous, which from the point of view of making a certain engineering decision is more attractive, however, one should take into account a rather significant moment in the calculation carried out (using the probabilisticstatistical method), namely: assumptions about the distribution law may turn out to be erroneous and, as a consequence, this will lead to incorrect calculation results.That is the interval reliability assessment obtained using the upgraded method makes it possible to make more correct decisions, since this method allows for more accurate consideration of the available statistical information about the controlled parameters by combining fuzzy variables and random variables.

Conclusion
The proposed private method for calculating reliability based on the upgraded method can be applied at the stage of operation of buildings and structures.Also, it is necessary to take into account that when carrying out calculations and assessing the reliability of building structures, including external enclosing structures, it is necessary to consider as a mechanical system, for example, as a sequential one.To make a final engineering decision, the reliability of the entire system should be determined.

Fig. 2 .
Fig. 2. The probability distribution function π N (N) , the probability distribution density function ρ N ult (N ult ) N ()=π N (N) P N ()=1-π N (N) E3S Web of Conferences 458, 07026 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345807026 590 kN; S  ult = 57 kN; a  = 489 kN; b  = 47 kN.Then, we use equations (8) to find the lower and upper values of the uptime probabilities: ) for the example under consideration, can be represented by a reliability interval in the form of: lower and upper values of the probability of trouble-free operation -[P=0.835; P= 0.991], inside of which is the true value.Let's consider the calculation of the reliability of this fragment of the external enclosing structure of the building according to criterion (1) using probabilistic and statistical methods.Suppose that the controlled parameters change according to the normal distribution law (Gauss-Laplace) with the distribution parameters: m  ult = 590 kN; S  ult = 57 kN; m  = 489 kN; S  = 47 kN, then after calculating the safety characteristic  = m  ult −m N E3S Web of Conferences 458, 07026 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345807026