# Mechanics of Screw Joints Solved as Beams Placed in a Tangential Elastic Foundation

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## Abstract

**:**

## 1. Introduction

## 2. The Tangential (Tensile) Elastic Foundation

^{−1}] is directly proportional to the axial displacement of the center line $u=u\left(x\right)$ [m]. From equilibrium on the infinitesimal length section $dx$ [m], the equation of equilibrium in the axial direction can be derived (see Figure 3).

^{2}] is the cross-section area of the beam (screw) and $E$ [Pa] is the elastic modulus of the beam (screw). The relationship (2) is linear, however, the foundation and its deformation can generally be non-linear, which is a significant factor of the tangential elastic foundation.

## 3. Material and Methods

- To prepare test samples for woodwork screw joints in the laboratory.
- To determine experimentally the dependence of pulling force $\mathrm{F}=N=f\left(u,\dots \right)$ in the laboratory (i.e., the pull-out test).
- To determine the appropriate shapes of the reaction force ${\mathrm{q}}_{N}$ (i.e., to design a physical model of the tangential foundation) based on the evaluation of the experiment (laboratory measurements, regression methods, etc.).
- To solve differential Equation (3) based on the knowledge of the reaction force ${\mathrm{q}}_{N}$, i.e., to acquire the axial displacement $u$.
- To compare the obtained results with experiments and to evaluate proposed models of the tangential foundation, including the correctness of theory and use.
- To check dependencies on other materials such as bone (i.e., “similar” applications in biomechanics, traumatology and orthopaedics).
- To conduct a discussion, draw conclusions and suggest future possible applications and other possible solutions.

## 4. Experiment (Pull-Out Test for Screws) and Its Evaluation

## 5. Determination of the Screw Elongation

## 6. Discussion

## 7. Results

- 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.
- The screw (beam on the elastic foundation) is subjected to tension/compression loading in this article.
- Two types of screws (HB 6.5 and HB 7) were made of stainless steel 1.4441 (AISI 316L) and titanium alloy Ti6Al4V by machining technology and 3D printing + machining technology.
- 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 ${u}_{\mathrm{B}}$, $\mathrm{F}$ dependencies were obtained for 13 cases, and the limit failure states (${u}_{\mathrm{E}},{\mathrm{F}}_{\mathrm{E}}$) 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.
- A sufficient regression analysis of a sufficiently determined dependence of the normal force on the elongation (displacement) of the screw $N=$ $\frac{{\mathrm{F}}_{\mathrm{E}}}{{2}^{m}}$ ${\left[1-\mathrm{cos}\left(\frac{\pi u}{{u}_{\mathrm{E}}}\right)\right]}^{m}$ 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}_{\mathrm{B}}\in \langle 0;{u}_{\mathrm{E}}\rangle $).
- 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 $N=$ $\frac{{\mathrm{F}}_{\mathrm{E}}}{{2}^{m}}$ ${\left[1-\mathrm{cos}\left(\frac{\pi u}{{u}_{\mathrm{E}}}\right)\right]}^{m}$ for a given screw joint. However, this is only possible using the measurement evaluation.
- 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 $N=$ $\frac{{\mathrm{F}}_{\mathrm{E}}}{{2}^{m}}$ ${\left[1-\mathrm{cos}\left(\frac{\pi u}{{u}_{\mathrm{E}}}\right)\right]}^{m}$ or ${\mathrm{q}}_{N}=$ $\frac{N}{EA}\frac{dN}{du}$ 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]).
- Other possible approximations of the dependencies $N=$ $f\left(u\right)$, for interval ${u}_{\mathrm{B}}\in \langle 0;{u}_{\mathrm{E}}\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].
- It is clear from the first results and applications that the new tangential foundation model, which is also characterized by continuously distributed reaction force ${\mathrm{q}}_{N}=EA$ $\frac{{d}^{2}u}{d{x}^{2}}$, or ${\mathrm{q}}_{N}=$ $\frac{N}{EA}\frac{dN}{du}$, offers a new alternative as a sufficiently accurate and relatively simple solution to the real complex interaction of the screw and the fasteners. It is also advantageous that large deformations, non-linearities, anisotropy or inhomogeneities can occur in this model of tangential foundation.
- It is also appropriate to focus on the future. In other models of elastic foundations, the influence of self-weight and dynamics (see [21]) can also be taken into account (however, this would lead to solving partial differential equations in the most complex tasks). The application of large deformations (logarithmic strains ${\epsilon}_{log}=\mathrm{ln}(1+$ $\frac{du}{dx}$) applied in Equation (1)), etc. is significant, as well as the combined bending + tension/pressure loads (see [15]) or torsion + bending + tension/pressure loads and impact loads [21], which are common, and not only in mechanical, building or biomechanical structures. This will be the subject of further research to follow up this article. Other complex biomechanical/mechanical applications are possible too, for example, see [12,14,26].
- After further experiments, our model can be applied as an alternative method in design codes, for example, see [3], etc.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Elastic foundation due to the loads (points 1 and 2 are moved to new positions 1* and 2*). (

**a**) Bending according to Winkler (distributed reaction force in foundation ${q}_{R}=kv$ in linear cases); (

**b**) Tension/pressure—new tangential model according to Frydrýšek and Michenková (distributed reaction force ${q}_{N}={k}_{N}u$ in linear cases, where ${k}_{N}$ [Pa] is elastic foundation stiffness, $u$ [m] is axial displacement/elongation and ${F}_{i}$ [N] is a tensile/compression force).

**Figure 3.**Section of the beam (screw) resting on a tangential elastic foundation, where N [N] is a normal (axial, internal) force.

**Figure 5.**Pull-out test, measurement No. 1. (

**a**) Dependence of tensile force F on screw elongation ${u}_{B}$; (

**b**) the pulled out screw (end of measurement).

**Figure 6.**Pull-out test: A typical dependence of tensile force F on the screw elongation ${u}_{B}$ (determination of extreme values of the pull-out test for screws in the wood).

**Figure 8.**Pull-out test, measurement No. 1: Dependence of tensile force F on screw elongation ${u}_{B}$ (the experiment and its approximation by regression function $\mathrm{F}={N}_{\mathrm{B}}=$ $\frac{{\mathrm{F}}_{\mathrm{E}}}{{2}^{m}}$ ${\left[1-\mathrm{cos}\left(\frac{\pi {u}_{\mathrm{B}}}{{u}_{\mathrm{E}}}\right)\right]}^{m}$ for interval ${u}_{\mathrm{B}}\in \langle 0;{u}_{\mathrm{E}}\rangle $).

**Figure 9.**The course of the calculated continuous tangential reaction ${q}_{N}$ at the base at point B of the screw (measurement No. 1), see Figure 5.

**Figure 10.**The course of the screw elongation dependence $u=f\left({u}_{B},x\right)$ for measurement No. 1 (i.e., the solution of Equation (15)).

**Figure 11.**Biomechanics of human os femoris with factura collum femoris, osteosynthesis and application of a bone screw in connection with experiments.

Material | Modulus of Elasticity E [Pa] | Yield Limit [MPa] | Fracture Limit [MPa] |
---|---|---|---|

1.4441 | 2.1 × 10^{11} | 892 | 977 |

Ti6Al4V | 1.06 × 10^{11} | 919 | 1034 |

**Table 2.**Basic material information on the applied cancellous screws (see [20]).

Self-Tapping Cancellous Bone Screw HB6.5 | SSt | A | B |

129796911 | 45 mm | 32 mm | |

129796921 | 50 mm | ||

129796931 | 55 mm | ||

129796941 | 60 mm | ||

129796951 | 65 mm | ||

129796961 | 70 mm | ||

129796971 | 75 mm | ||

129796981 | 80 mm | ||

129796991 | 85 mm | ||

Thread diameter 6.5 mm | 129797001 | 90 mm | |

Shank diameter 4.5 mm | 129797011 | 95 mm | |

Core diameter 3 mm | 129797021 | 100 mm | |

Head diameter 8 mm | 129797031 | 105 mm | |

Drill bit for threaded hole 3.2/3.75 mm | 129797041 | 110 mm | |

Screwdriver Hex key 3.5 mm | 129797051 | 115 mm | |

Material: Stainless steel AISI 316 L (DIN 1.4441/316 L medical) or titanium alloy (Ti6Al4V). Dimension “B” = ${\mathrm{L}}_{2}$. | 129797061 | 120 mm | |

Cancellous Bone Screw HB7 | SSt | A | B |

129798210 | 40 mm | 32 mm | |

129798220 | 45 mm | ||

129798230 | 50 mm | ||

129798240 | 55 mm | ||

129798250 | 60 mm | ||

129798260 | 65 mm | ||

129798270 | 70 mm | ||

129798280 | 75 mm | ||

129798290 | 80 mm | ||

129798300 | 85 mm | ||

129798310 | 90 mm | ||

Thread diameter 7 mm | 129798320 | 95 mm | |

Shank diameter 5 mm | 129798330 | 100 mm | |

Core diameter 3.5 mm | 129798340 | 105 mm | |

Head diameter 8 mm | 129798350 | 110 mm | |

Drill bit for threaded hole 4 mm | 129798360 | 115 mm | |

Screwdriver Hex key 3.5 mm | 129798370 | 120 mm | |

Cannulation 1.8 mm | 129798380 | 125 mm | |

Material: Stainless steel AISI 316 L (DIN 1.4441/316 L medical) or titanium alloy (Ti6Al4V). Dimension “B” = ${\mathrm{L}}_{2}$. | 129798390 | 130 mm |

Measurement | ||||||
---|---|---|---|---|---|---|

Measurement Number | Screw Type | Screw Material | Pre-Drilled Hole Diameter [m] | Limit State | Damage Limit State Description | |

${\mathit{F}}_{\mathit{E}}\left[\mathbf{N}\right]$ | ${\mathit{u}}_{\mathit{E}}\left[\mathbf{m}\right]$ | |||||

1 | HB 6.5 | 1.4441 (AISI 316L) | 0.0032 | 8634 | 0.001405 | Pulling the screw out of the wood |

2 | HB 6.5 | Ti6Al4V | 0.004 | 7475 | 0.001666 | |

3 | HB 7 | 1.4441 (AISI 316L) | 0.004 | 8541 | 0.001527 | Screw breakage |

4 | HB 7 | Ti6Al4V | 0.0045 | 8649 | 0.001638 | Pulling the screw out of the wood |

5 | HB 6.5 | 1.4441 (AISI 316L) | 0.004 | 8770 | 0.001707 | |

6 | HB 6.5 | 1.4441 (AISI 316L) | 0.004 | 8942 | 0.001468 | Screw breakage |

7 | HB 6.5 | 1.4441 (AISI 316L) | 0.004 | 8279 | 0.00114 | Pulling the screw out of the wood |

8 | HB 7 | 1.4441 (AISI 316L) | 0.005 | 7390 | 0.001385 | |

9 | HB 7 | 1.4441 (AISI 316L) | 0.005 | 7606 | 0.001544 | |

10 | HB 6.5 | Ti6Al4V | 0.005 | 7322 | 0.001906 | |

11 | HB 6.5 | Ti6Al4V | 0.005 | 7381 | 0.001339 | |

12 | HB 7 | Ti6Al4V | 0.0055 | 9105 | 0.001804 | |

13 | HB 7 | Ti6Al4V | 0.0055 | 9481 | 0.002157 | Screw breakage |

Measurement Number | $\mathbf{Approximation}{\mathit{N}}_{\mathit{B}}=$$\frac{{\mathit{F}}_{\mathit{E}}}{{2}^{\mathit{m}}}$${\left[1-\mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathit{\pi}{\mathit{u}}_{\mathit{B}}}{{\mathit{u}}_{\mathit{E}}}\right)\right]}^{\mathit{m}}$ | ||
---|---|---|---|

Regression | Measurement (Limit States), See also Table 3 | ||

$\mathit{m}$ [1] | ${\mathit{F}}_{\mathit{E}}\left[\mathbf{N}\right]$ | ${\mathit{u}}_{\mathit{E}}\left[\mathbf{m}\right]$ | |

1 | 1.078 | 8634 | 0.001405 |

2 | 0.5139 | 7475 | 0.001666 |

3 | 0.7876 | 8541 | 0.001527 |

4 | 0.5128 | 8649 | 0.001638 |

5 | 0.6739 | 8770 | 0.001707 |

6 | 0.7499 | 8942 | 0.001468 |

7 | 0.621 | 8279 | 0.00114 |

8 | 0.6648 | 7390 | 0.001385 |

9 | 0.7506 | 7606 | 0.001544 |

10 | 1.095 | 7322 | 0.001906 |

11 | 0.5122 | 7381 | 0.001339 |

12 | 0.6158 | 9105 | 0.001804 |

13 | 0.5347 | 9481 | 0.002157 |

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**MDPI and ACS Style**

Frydrýšek, K.; Michenková, Š.; Pleva, L.; Koutecký, J.; Fries, J.; Peterek Dědková, K.; Madeja, R.; Trefil, A.; Krpec, P.; Halo, T.;
et al. Mechanics of Screw Joints Solved as Beams Placed in a Tangential Elastic Foundation. *Appl. Sci.* **2021**, *11*, 5616.
https://doi.org/10.3390/app11125616

**AMA Style**

Frydrýšek K, Michenková Š, Pleva L, Koutecký J, Fries J, Peterek Dědková K, Madeja R, Trefil A, Krpec P, Halo T,
et al. Mechanics of Screw Joints Solved as Beams Placed in a Tangential Elastic Foundation. *Applied Sciences*. 2021; 11(12):5616.
https://doi.org/10.3390/app11125616

**Chicago/Turabian Style**

Frydrýšek, Karel, Šárka Michenková, Leopold Pleva, Jan Koutecký, Jiří Fries, Kateřina Peterek Dědková, Roman Madeja, Antonín Trefil, Pavel Krpec, Tomáš Halo,
and et al. 2021. "Mechanics of Screw Joints Solved as Beams Placed in a Tangential Elastic Foundation" *Applied Sciences* 11, no. 12: 5616.
https://doi.org/10.3390/app11125616