## 2 Research Method
### Research Method
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline _dcfct No_ & _type of dcfct_ & \(\lambda(1)2\) & \(\lambda(1)3\) & \(\lambda(1)4\) & \(\lambda(1)5\) & \(\lambda(1)6\) & \(\lambda(1)7\) & \(\lambda(1)8\) & \(\lambda(1)9\) & \(\lambda(1)10\) & \(\lambda(1)11\) \\ \hline d1 & Mechanical defects & & & & & & & & & & \\ d2 & Leaks & & & & & & & & & & \\ d3 & Mechanical decrements of bricks & 0.13 & 0.28 & 0.08 & **0.19** & & 0.03 & & & & \\ d4 & Mechanical decrements of mortar & & 0.28 & & 0.30 & & & & & \\ d5 & Decrements caused by rotten bricks & 0.14 & 0.07 & 0.00 & 0.17 & & & & & \\ d6 & Decrements caused by rotten mortar & & 0.05 & & 0.09 & & & & & & \\ d7 & Pain coating’s peeling off & & & & & & & & & \\ d8 & Pain coating’s falling off & & & & & & & & & \\ d9 & Craks of bricks & 0.05 & 0.01 & 0.05 & 0.11 & & & & & \\ d10 & Craks of plaster & & 0.03 & & & & & & & \\ d11 & Scratching of walls & & & & & & & & & \\ d12 & Scratching of plaster & & & & 0.12 & 0.05 & & & & & \\ d13 & Loosening of plaster & & & & & & & & & & \\ d14 & Paster’s falling off & & & & & & & & & & \\ d15 & Signs of permanent damp & 0.70 & 0.74 & 0.58 & 0.56 & 0.07 & & 0.43 & 0.83 & 0.70 & 0.84 \\ d16 & Weeping & 0.64 & 0.52 & 0.67 & 0.46 & 0.27 & 0.59 & 0.50 & 0.74 & 0.61 & 0.79 \\ d17 & Biological corrosion of bricks & 0.35 & 0.31 & & 0.67 & & & & & \\ d18 & House fungus & & & & & & & & & \\ d19 & Mould \& decay & & & & & & & & \\ d20 & Localized corrosion of steel beams & & & & & & & & & \\ d21 & Surface corrosion of steel beams & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average values of the damage intensity function \(\lambda(i)\) for the ten most important elements of residential buildings. A typical distribution for objects subjected to damage in the first period of exploitation may be the gamma distribution in the form:  
\[f(t)=\frac{\lambda(\lambda(t))^{\beta-1}e^{-\lambda(t)}}{\int_{0}^{\infty}e^{-t }t^{\beta-1}dt} \tag{22}\]  
in which the denominator is the so-called Euler's integral. This distribution represents the processes of loss of serviceability well, which are based on the cumulative effects of external factors. Ultimately, as a result of the elimination of all the defects and faults covered by the warranty, it is assumed that gradual and sudden defects (the latter ones that result from the step operation of stimuli in the conditions of accumulation of wear) are completely eliminated. This characterizes the end of the adaptation period of a residential building and the beginning of its normal operation period. The shape of the intensity function becomes \"smooth\". Hence, value 1 for the shape parameter (\(\beta\)) can be assumed for both the Weibull distribution and gamma distribution. It is a very characteristic period of a residential building's service life, in which the risk function takes a constant value (\(\lambda(t)=\lambda\)). This results from the following transformations of the mentioned functions:  
* for the Weibull function: \[\lambda(t)=\frac{\beta}{\alpha}t^{\beta-1}=\frac{1}{\alpha}t^{1-1}=\frac{1}{ \alpha}=\lambda\] (23)
* for the gamma function (using a shortcut for long calculations): \[\lambda(t)=\frac{f(t)}{R(t)}\] (24) \[f(t)=\ \lambda e^{-\lambda t}\] (25) \[F(t)=1-e^{-\lambda t}\] (26) and using dependence (2): \[R(t)=e^{-\lambda t}\] (27)  
ultimately:  
\[\lambda(t)=\frac{\lambda e^{-\lambda t}}{e^{-\lambda t}}=\lambda(=const) \tag{28}\]  
If the risk function has a domain defined by segment ti of the exploitation time (\(0<t_{i}\leq t\)), then \(\lambda(t)=\) const (for \(\lambda>0\) and \(t>0\)) and the time of correct operation of a residential building has an exponential distribution.