# Mechanics of Screw Joints Solved as Beams Placed in a Tangential Elastic Foundation
## 4 Experiment (Pull-Out Test for Screws) and Its Evaluation
In the next step, it is necessary to propose a suitable dependency from the relationship (5), i.e.,  
\[\mathrm{q}_{N}= \frac{N}{EA}\frac{dN}{du}. \tag{6}\]  
This is possible by evaluating the pull-out test experiments (see Figures 5 and 6 and Table 3). For the solution, finding a suitable regression equation is possible on the interval \(u_{\mathrm{B}}\in\left\langle 0;u_{\mathrm{E}}\right\rangle\) and \(\mathrm{F}\in\left\langle 0;\mathrm{F_{E}}\right\rangle\), which should meet three conditions:  
\[\left. \begin{array}{l}u_{\mathrm{B}}=0\,\ \mathrm{F}=0\dots i.e.,\ no\ load\ condition,\\ u_{\mathrm{B}}=u_{\mathrm{E}}\,\ \mathrm{F}=\mathrm{F}_{\mathrm{E}}\dots\ i.e.,\ the\ ultimate\ load\ (extreme),\\ u_{\mathrm{B}}=u_{\mathrm{E}}\,\frac{\mathrm{dF}}{du_{\mathrm{B}}}=0\ \dots\ i.e.,\ the\ ultimate\ load\ (extreme),\end{array} \right\}\, \tag{7}\]  
i.e., the dependency \(N_{\mathrm{B}}=\mathrm{F}=f(u_{\mathrm{B}})\) passes through the origin, passes through the extreme point and has the same extreme as the experiment. Mathematical regression models were applied for the interval \(u_{\mathrm{B}}\in\left\langle 0;u_{\mathrm{E}}\right\rangle\) (i.e., interval of applicability of a screw) (see Figures 5 and 6). Examples of mathematical regressions using sine series and square roots of power series (i.e., different regressions) are given in [18]. This paper only presents regressions using the \"elegant\" relationship \(N_{\mathrm{B}}=a[1-\cos(bu_{\mathrm{B}})]^{m}\), where \(a\) [N], \(b\) [1/m] and \(m\) [1] are constants. This dependency must meet the conditions (7). Hence, after the application of Equation (7), the aim is to find only one parameter \(m\) by regression (i.e., a simple and easy method of regression). Then, the following is applied  
\[N_{\mathrm{B}}= \frac{\mathrm{F_{E}}}{2^{m}}\bigg{[}1-\cos\bigg{(}\frac{\pi\ u_{ \mathrm{B}}}{u_{\mathrm{E}}}\bigg{)}\bigg{]}^{m}, \tag{8}\]  
where the constant \(m\) is found by regression, e.g., in sw Matlab (Toolbox Curve Fitting) (see Table 4 and Figure 8). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Measurement** \\ **Number** \\ \end{tabular} } & \multicolumn{2}{c}{**Approximation \(N_{\mathrm{B}}\)=\(\frac{\mathrm{F_{E}}}{2^{m}}\Big{[}1-\cos\bigg{(}\frac{\pi\ u_{\mathrm{B}}}{u_{ \mathrm{E}}}\bigg{)}\bigg{]}^{m}\)**} \\ \cline{2-4}  & **Regression** & **Measurement (Limit States)**, **See also Table 3 \\ \cline{2-4}  & \(m\) [1] & \(F_{E}\) [N] & \(u_{E}\) [m] \\ \hline
1 & 1.078 & 8634 & 0.001405 \\ \hline
2 & 0.5139 & 7475 & 0.001666 \\ \hline
3 & 0.7876 & 8541 & 0.001527 \\ \hline
4 & 0.5128 & 8649 & 0.001638 \\ \hline \end{tabular}
\end{table} 
Table 4: The pull-out test results (a simple one-parameter approximation of measured data). Similar consensus to the experiment, as shown in Figure 8 (i.e., measurement No. 1), is also shown by other measurements from Tables 3 and 4, which only confirms the correctness of the proposed model. Therefore, the introduction of a multi-parameter regression function does not seem to be necessary but is possible too. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Measurement** \\ **Number** \\ \end{tabular} } & \multicolumn{2}{c}{**Approximation \(N_{\text{B}}\)=\(\frac{F_{\text{E}}}{2^{m}}\left[1-cos\left(\frac{\pi\,u_{\text{B}}}{u_{\text{E}}}\right) \right]^{m}\)**} \\ \cline{2-4}  & **Regression** & **Measurement (Limit States), See also Table 3 \\ \cline{2-4}  & \(m\) [1] & \(F_{\text{E}}\) [N] & \(u_{\text{E}}\) [m] \\ \hline
5 & 0.6739 & 8770 & 0.001707 \\ \hline
6 & 0.7499 & 8942 & 0.001468 \\ \hline
7 & 0.621 & 8279 & 0.00114 \\ \hline
8 & 0.6648 & 7390 & 0.001385 \\ \hline
9 & 0.7506 & 7606 & 0.001544 \\ \hline
10 & 1.095 & 7322 & 0.001906 \\ \hline
11 & 0.5122 & 7381 & 0.001339 \\ \hline
12 & 0.6158 & 9105 & 0.001804 \\ \hline
13 & 0.5347 & 9481 & 0.002157 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 8: Pull-out test, measurement No.