# Conventional Selection of Mechanical Fasteners for Flat Belts
## 1 Introduction
It shows a general scheme of the structure of a belt conveyor with the use of a elastic belt and a mechanical fastener, in which the following can be distinguished: 1--mechanical fastener, 2--drive roller, 3--return idler, 4--elastic belt, and 5--tensioning roller (return roller). A condition of the operation mode of the conveyor without belt slippage is (1). The \(\mu\) is the coefficient of friction between the conveyor belt and the roller. \[\frac{S_{1}}{S_{2}}=e^{\mu\alpha} \tag{1}\]  
From the calculus of vectors, it follows that (2). \[P=S_{1}-S_{2} \tag{2}\]  
After transforming (2) and inserting it into Formula (1), the following result was obtained. \[S_{2}=\frac{P}{e^{\mu\alpha}-1} \tag{3}\]The force in the upper bend can be calculated from the dependence (2). The value of the resultant force \(Q\) loading the shaft was calculated using the famous cosine law (4). \[Q=\sqrt{S_{1}^{2}+S_{2}^{2}+2S_{1}S_{2}\cos\delta} \tag{4}\]  
Figure 2 shows detail B in Figure 1, thanks to which it is possible to calculate the inclination angle \(\theta\) of the resultant force vector \(Q\), which is needed to decompose this force into a vertical and a horizontal component \(Q_{x}\) and \(Q_{y}\). Two auxiliary parameters \(x\) and \(y\) were introduced, and the angle \(\theta\) was calculated using trigonometric functions (5) and (6). \[x=S_{2}\sin\delta;\ y=S_{2}\cos\delta \tag{5}\]  
\[\tan\theta=\frac{x}{S_{1}+y}=\frac{S_{2}\sin\delta}{S_{1}+S_{2}\cos\delta} \tag{6}\]  
\[\theta=\tan^{-1}\biggl{(}\frac{S_{2}\sin\delta}{S_{1}+S_{2}\cos\delta}\biggr{)}\]  
The calculation of the resultant force loading the main shaft is important in the context of this work, because, as it will turn out in a moment, the mechanical connection generates additional forces that also load the shaft. Figure 3 shows detail A from Figure 1. This detail is presented in two specific phases of the conveyor's work cycle--on the left: at the moment of contact of the contact edge, the mechanical connection with the cylindrical surface of the roller (during the run-up on the roller); on the right: when the face of the mechanical connection is tangent to the cylindrical surface of the roller (while circling the roller). This shows in a simplified way the distribution of stresses resulting from the application of a mechanical connection. Figure 1:0 Stress distribution in the roller-belt-mechanical fastener configuration during: **(left)** ramp-up, **(right)** circulating around the roll. Explanations: \(F_{R}\)—radial force during the run-up, \(F^{\prime}_{R}\)—radial force while going around the roller, \(F_{T}\)—tangential force during the run-up, \(F^{\prime}_{T}\)—tangential force while going around the roller (which is the same as the friction force on the bottom surface of the joint), \(F_{TR}\)—transverse force, \(F_{L}\)—longitudinal force. Figure 1:1 Scheme for calculating the inclination angle \(\theta\) of the resultant force vector \(Q\). When the joint is pushed onto the roller (at the moment of contact), the mechanical fastener acts with a force \(S_{1}\) on the cylindrical surface of the driving roller. By distributing this force into two components, a radial force \(F_{R}\) was obtained, which additionally (apart from the \(Q\) force) loads the main shaft, and the tangential force \(F_{T}\). This phase is a kind of impulse/shock impact of the joint against the roller, and it is during this phase that the radial load is the greatest. As the joint circulates around the drive roller, the distribution of forces is similar, except that now the tangential force \(F_{T}\) is at its greatest value. Some dependencies can be noticed (7). Figure 1:2  
Figure 4 also shows detail A from Figure 1, but this time in cross-section. It shows an example of a mechanical joint and the forces acting on it. The connecting element can be, for example, either a screw or a rivet. The previously derived forces that load the joint are: \(F_{TR}\)--transverse force, which generates tensile of the core of the link (bolt/rivet, etc.)