# Determination of Reliability of Selected Case
## 2 Materials and Methods  
The works on the creation of the mathematical apparatus and a web application enabling the determination of the reliability and duration of the warranty period were completed within the BaltSe@nioR project. The project aimed at equipping furniture manufacturers with new knowledge and tools facilitating the creation of furniture that is aging-friendly, thus not only safe and aesthetically pleasing, but also durable. As mentioned above, the reliability of the entire furniture can be calculated by taking into account the reliability of its individual furniture joints. Thus, the first stage of the investigation was dedicated to the reliability testing of selected cabinet furniture joints. Altogether 3600 samples of furniture joints were subjected to the reliability tests performed on a dedicated laboratory stand. The samples were made from laminated three-layer particleboard and MDF of 16, 18, and 22 mm in thickness. The furniture joints were manufactured by the professional furniture manufacturer Paged Meble S.A. (Poland). The quality grade of wood-based boards used to manufacture the furniture joints was in accordance with the standard [29]. Three types of mechanical connectors joining the elements of the samples were investigated; those included confirmat screws (\(\mathcal{O}5\)\(\times\) 50 mm), dowels (\(\mathcal{O}8\)\(\times\) 32 mm), and eccentric joints(Minifix type \(\mathcal{O}15\times\) 14,6 mm). Each of those configurations was represented by 200 testing samples (as presented in Table 1). The shape and dimensions of the tested samples are shown in Figure 1. Polyvinyl acetate (PVAc) adhesive was used for the joint with the dowel. The confirmat screws and eccentric joints were screwed under application of the same moment of 2.5 Nm with a torque wrench. The reliability testing of the joints was a necessary research step as the reliability characteristics of the furniture's individual joints are the main source of safety risk and durability of the use of the furniture. Usually, case furniture is treated as serial structures of connected constructional nodes [27]. It is assumed that an object (a piece of furniture) has a serial structure in terms of reliability when all its elements (construction nodes) are operational. In other words, the failure of one structural node leads to the failure of the entire system. In a mathematical notation, a serial structure is presented by the equation [30]:  
\[R(t)=\prod_{i=1}^{m}R_{i}(t). \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline  & & **Confirmat Screws** & **Dowels** & **Eccentric Joints** \\ \hline \multirow{3}{*}{MDF} & 16 mm & 200 & 200 & 200 \\ \cline{2-5}  & 18 mm & 200 & 200 & 200 \\ \cline{2-5}  & 22 mm & 200 & 200 & 200 \\ \hline \multirow{3}{*}{Particleboard} & 16 mm & 200 & 200 & 200 \\ \cline{2-5}  & 18 mm & 200 & 200 & 200 \\ \cline{1-1} \cline{2-5}  & 22 mm & 200 & 200 & 200 \\ \hline \end{tabular}
\end{table} 
Table 1: The number of testing samples made of various materials, connectors, and thickness of the elements of the furniture joints. Figure 1: The shape, dimensions, and the loading of a case-type furniture sample used in the reliability tests (dimensions are given in mm). The above mathematical notation means that damage to at least one structural node causes damage to the entire piece of furniture. A more advantageous solution from the point of view of reliability is the use of a parallel system, which remains reliable if at least one of its elements (structural nodes) remains operational [31]. In formal terms, a parallel system is presented as follows:  
\[R(t)=1-\prod_{i=1}^{m}[1-R_{i}(t)]. \tag{2}\]  
Thus, the use of several elements intended to perform the same function and connected in a parallel system in terms of reliability is an effective way of increasing (reserving) the reliability of the entire piece of furniture.