On the Probability of the Development of the Diagnostic Feature of Decline the Bearing Capacity of Structures under Random Dynamic Loads

The article is devoted to the problem of identifying the diagnostic feature of the structure bearing capacity decrease under random dynamic loads. The diagnostic features that characterize the decrease of the bearing capacity of structures and foundation are identified. The model of the dynamic system that describes the process of the diagnosed building structure functioning is presented. The article shows the difficulties encountered in the process of vibration diagnostics of reducing the bearing capacity of structures and foundations. It is pointed out that the output vibration signals can be recorded in the mode of displacements, velocities and accelerations. But at the same time, the probability spread of detection of diagnostic feature of the bearing capacity decrease of building elements in the energy spectra of the output vibration signals can be significant. To develop a method for determining this probability, the authors propose to use a regression analysis with the solution of the problem at three levels of “regression”. In the article the factors influencing the probability of detection the vibration diagnostic feature in bearing capacity decrease of building elements are analyzed. This probability characterizes the effectiveness of the vibration diagnostic method in monitoring systems of unique objects, including launch facilities. The authors have shown that the maximum efficiency is achieved by using systems for monitoring energy spectra of vibration accelerations at the output. At the same time, the authors conclude that the probability of effective functioning of vibration diagnostic systems is mainly influenced by such factors as the sensitivity of the vibration diagnostic method to the rigidity decrease of structures and bases. The equipment error of registration of the energy spectra of vibration accelerations is also to be taken into consideration


Introduction
In the functional vibration diagnostics of technical condition of long-span structures, the reduction of their bearing capacity is connected with their flexural rigidity C. Diagnostic feature of decrease in their bearing capacity is known to be the changes in dynamic parameters of structures [1][2][3][4][5]. These dynamic parameters are contained in their transfer functions [6] where ω is the circular frequency of the dynamic load impact Pt with the energy spectrum (spectral density function) P G [7,8].
In practice loads Ptare often random functions of space and time t [6][7][8]. Examples of such loads are: wind loads [9,10] for high-rise buildings and constructions; field of pressure pulsations of a gas  11,12] for launching facilities. In these cases informative diagnostic feature of reduction of the bearing capacity structures is a reduction of its frequency (for the first fundamental form) of the vibrations K by the same amount and the increase of the resonance extrema of the transfer function in some times [1-4, [11][12][13][14][15] (Figure 1). 0; decrease BC 1. Figure 2. Dynamical system. The ability to identify these F D is due to the functioning of the dynamic system (diagnosed building structure) by the algorithm (Figure 2) [4, 6,9]: (2) where the energy spectrum of the output vibrating signal G in the general case may be determined on the implementations of the vibration displacements Vt, velocities V and acceleration [16]. Diagnostic features (1) (Figure 1) can manifest itself in the spectrum of the output vibrating signal G in the form of the extremum on the natural (resonant) frequency of vibrations of the structure K in cases of sufficient energy input Pt with the spectrum P G for the functioning of the algorithm (2) with the graphical interpretation in Figure 3.  included in the algorithm (2). This circumstance is an obstacle to the detection of diagnostic feature (1) of reducing the bearing capacity of structures in the process of their long-term exploitation. This is especially true for buildings and constructions. When monitoring the technical condition by vibration method there is no possibility to register random input actions Pt and their energy spectra P G , for example, it is typical for high-rise buildings and constructions [9] and for launching facilities when launching the space-purpose missiles [12,13]. In this regard, when substantiating the expediency of using stationary monitoring systems [17][18][19][20][21], the probability estimation (6) of the fact that the informative extremum with the actual ordinate

Methods
To solve this problem it is advisable to use regressive proof (analysis), in which [22] "the course of reasoning goes from consequences to grounds". The event interesting for us: , is the result of functioning of dynamic system ( Figure 2) according to the algorithm (2). Thus, the probability П (3) of the event (4), according to the regression analysis [22], should be considered as a certain function, the basis of which is the product of functions .
i The functions i depend on the initial functions that are included in the algorithm (2): where sign is a logical symbol of "product" [22], and i are symbols of three functions, the product of which is written in the right part of equality (5).

Results
Recording (5) is the first level of regression analysis of the factors determining the probability П (3) of the event (4), i.e. the development of the resonance extremum in the spectrum of the output vibration signal G ( Figure 3).
The second level of "regression" [22] [that is, lowering the level of analysis to the basis of the function For launching facilities it was established [12,13] that the values of parameter  C and , M respectively, the rigidity and weight of the span reinforced concrete construction [17], and the parameter K -a coefficient depending on the conditions of fixing the ends of the span reinforced concrete construction.
In the third and the last function 3 G in (58) the symbol means the error [19] of recording the spectrum G of the output signals , It depends on the product of two functions: -is a function depending on the sensitivity µ [23] of the method of functional vibration diagnostics [24];

A
-function depending upon the registration equipment error A of spectra of the output vibration signals G [23].
The statistics of full-scale data on the launching facilities for the systems of Testing and long-term monitoring [13] allows to obtain the values of parameters  (Fig. 3) of identification of diagnostic feature of reducing the bearing capacity of reinforced concrete construction on launching facilities (1) (Figure 1) they give the following results [13]:

Conclusions
The result (8) (7) is interconnected with the parameters of the function 2 in (5) and that the sensitivity of the method vibration diagnostics increases sharply with the increase in the overall stiffness of the reinforced concrete structures [24,28] (see the function 2.1 C as a function 2 . This is especially true for the span concrete structures fixed rigidly, when the value of the function 2.3 K increases sharply in comparison with the hinged structures [28]. In this case, the error of the vibration diagnostics method itself is a fraction of a percent [13], and then the function values (7) are determined mainly only by the equipment errors of energy spectra registration of vibrational accelerations, which in modern technical means may not exceed 5%. In this case, the confidence probability of the method of functional vibration diagnostics [1] in comparison with the result (9) may increase to the desired values 0.95 in the construction practice for carriers of reinforced concrete construction [25]. Each figure should have a brief caption describing it and, if necessary, a key to interpret the various lines and symbols on the figure.