# Mechanics of Screw Joints Solved as Beams Placed in a Tangential Elastic Foundation
## 7 Results  
Since the abovementioned findings and analytical, experimental and numerical approaches have confirmed the theory and practice of the newly proposed tangential elastic foundation model, the following may be stated:  
1. We have been able to derive the new theoretical solution of the differential equation of the screw joint using the tangential elastic foundation theory, including the determination of the boundary conditions and derivation of the analytical solution procedure. 2. The screw (beam on the elastic foundation) is subjected to tension/compression loading in this article. 3. Two types of screws (HB 6.5 and HB 7) were made of stainless steel 1.4441 (AISI 316L) and titanium alloy Ti6 Al4V by machining technology and 3D printing + machining technology. 4. Experimental tests (pull-out tests) for joining dry spruce wood and bovine and human bones (i.e., three types of generally anisotropic and non-homogeneous materials) were carried out using pull-out tests up to limit states (screw breakage, pulling the screw out of the fasteners). The \(u_{\text{B}}\), F dependencies were obtained for 13 cases, and the limit failure states (\(u_{\text{E}}\), F\({}_{\text{E}}\)) were evaluated. Our experimental tests are in good agreement with different experiments with different materials (see [24; 26; 27; 28]). However, it is advisable to perform more experiments. 5. A sufficient regression analysis of a sufficiently determined dependence of the normal force on the elongation (displacement) of the screw \(N=\frac{\text{F}_{\text{E}}}{2^{m}}\left[1-\cos\left(\frac{\pi\,u}{u_{\text{E} }}\right)\right]^{m}\) was performed, i.e., the determination of the \(m\) factor from the evaluation of the pull-out tests. Regression approximates the course of the experiment accurately enough for desired applicability (i.e., for interval \(u_{\text{B}}\in\left\langle 0;u_{\text{E}}\right\rangle\)). 6. Thus, the application of the tangential elastic foundation model can be generalized to the joining of diverse materials by different types of screws made of different materials (forming, 3D printing). The condition of such an application of the tangential foundation is the knowledge of the relationship \(N=\frac{\text{F}_{\text{E}}}{2^{m}}\left[1-\cos\left(\frac{\pi\,u}{u_{\text{E} }}\right)\right]^{m}\) for a given screw joint. However, this is only possible using the measurement evaluation. 7. The obtained results confirm the correctness of the proposed tangential elastic foundation as an alternative and simpler solution of the complex 3D task of mechanical contact with friction, which is undoubtedly the case of the screw joint. Instead of the task of strongly non-linear mechanical contact, the simpler use of a suitable number of non-linear springs satisfying the relationships \(N=\frac{\text{F}_{\text{E}}}{2^{m}}\left[1-\cos\left(\frac{\pi\,u}{u_{\text{E} }}\right)\right]^{m}\) or \(\text{q}_{N}=\frac{N}{EA}\frac{dN}{du}\) can be applied. Such non-linear springs can be applied, for example, in nodes of the finite element mesh, etc. Thanks to the shortening of computational times, in addition to a simpler, more affordable and faster solution, it is also possible to apply the probabilistic approach very effectively in the design of the screw joint. Probabilistic approaches and probabilistic reliability assessments (based, e.g., on the Monte Carlo method and respecting the real variability of input and output variables) are in line with modern trends in science and technology (see [6; 29; 31; 32; 33]). 8. Other possible approximations of the dependencies \(N=f(u)\), for interval \(u_{\text{B}}\in\left\langle 0;u_{\text{E}}\right\rangle\), expressed by sine series, square roots of power series or other models, etc., are also other aspects of the research, but are not the subject of this article. For more details, see references [18; 42]. 9.