# Bending and Torsional Stress Factors in Hypotrochoidal H-Profiled Shafts Standardised According to DIN 3689-1

## Abstract

**:**

## 1. Introduction

## 2. Geometry of H-Profiles

#### 2.1. Geometric Properties

#### Area

#### 2.2. Radius of Curvature at Profile Corners and Flanks

#### 2.3. Bending Stresses

#### 2.4. Bending Deformations

#### 2.5. Moments of Inertia

#### 2.6. Example

#### 2.7. Stress Factor for Bending Loads

#### 2.8. Rotating Bending Stress

#### 2.9. Deflection

#### 2.10. Example

#### 2.11. H-Profiles According to DIN3689-1

#### 2.12. Stress Factor for Bending

#### 2.13. Stress Factor for Torsion

## 3. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Formula Symbols: | ||

$A$ | mm^{2} | Area of profile cross-section |

$e$ | mm | Profile eccentricity |

$\mathrm{e}$ | - | Euler’s number |

${e}_{lim}$ | mm | Profile overlap eccentricity limit |

$E$ | MPa | Young’s modulus |

$n$ | - | Profile periodicity (number of sides) |

${I}_{0}$ | mm^{4} | Corresponding reference moments of inertia for a round cross-section with radius r |

${I}_{t,ref}$ | mm^{4} | Torsional moment of inertia for reference shaft |

${I}_{x},{I}_{y},{I}_{xy}$ | mm^{4} | Surface moments of inertia in the Cartesian coordinate system |

${I}_{y,ref}$ | mm^{4} | Surface moment of inertia about y-axis for reference shaft |

${I}_{\xi},{I}_{\eta},{I}_{\xi \eta}$ | mm^{4} | Surface moments of inertia in the rotated coordinate system |

$l$ | mm | Length of profile shaft |

${l}_{b}$ | mm | Distance to strain gage in experimental test |

${M}_{b}$ | Nm | Bending moment |

$r$ | mm | Nominal or mean radius |

$t$ | - | Profile parameter angle |

${u}_{x}$ | mm | Displacement in x direction |

$x,y,z$ | mm | Cartesian coordinates |

Greek Formula Symbols: | ||

${\alpha}_{bh}$ | - | Bending stress factor for profile head |

${\alpha}_{bf}$ | - | Bending stress factor for profile foot |

${\alpha}_{t}$ | - | Torsional stress factor for profile foot |

${\delta}_{x}$ | mm | deflection |

$\epsilon =e/r$ | - | Relative eccentricity |

$\varphi $ | - | Rotation angle of the coordinate system |

$\lambda ={e}^{i\theta}$ | - | Physical plane unit circle |

$\theta $ | - | Polar angle |

${\sigma}_{b},{\sigma}_{z}$ | MPa | Bending stress (z-component of stress vector) |

${\tau}_{t}$ | MPa | Torsional stress |

$\omega \left(\zeta \right)$ | - | Completed mapping function |

${\omega}_{0}\left(\zeta \right)$ | - | Contour edge mapping function |

$\zeta $ | - | Complex variable in model plane |

$\xi ,\eta $ | - | Coordinates in rotated system |

## References

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**Figure 1.**Description of exemplary hypotrochoid (H-profile) with four concave sides. A detailed explanation of the parameters is given below in Section 2.

**Figure 2.**Some H-profiles manufactured by two-spindle process, Iprotec GmbH, © Guido Kochsiek, www.iprotec.de, Zwiesel, Germany [5].

**Figure 3.**Roller milling manufacturing for H-profile [7].

**Figure 7.**Circumferential distribution of the bending stress on the lateral surface of a standardised H3 profile.

**Figure 8.**Bending loads test bench (Machine Elements Laboratory at West Saxon University of Zwickau).

**Figure 10.**Stress factors for the bending stress at the profile head (Equation (18)) with varying relative eccentricity and number of sides.

**Figure 12.**Distributions of the bending stresses on the profile contour for different angles of rotation $\varphi $, with $r=18.18$ mm, $n=3$, $e=1.818$ mm, and ${M}_{b}$ = 500 Nm.

**Table 1.**Stress factors for bending and torsional loads for the H-profiles standardised according to DIN3689-1.

$\mathit{n}$ | $\mathit{\epsilon}$ | ${\mathit{\alpha}}_{\mathit{b}\mathit{h}}$ | ${\mathit{\alpha}}_{\mathit{b}\mathit{f}}$ | ${\mathit{I}}_{\mathit{y}}/{\mathit{I}}_{0}$ | ${\mathit{\alpha}}_{\mathit{t}}$ |
---|---|---|---|---|---|

3 | 0.100 | 1.12 | 0.92 | 0.98 | 1.23 |

4 | 0.056 | 1.07 | 0.96 | 0.99 | 1.17 |

4 | 0.111 | 1.17 | 0.94 | 0.95 | 1.37 |

5 | 0.031 | 1.04 | 0.97 | 0.99 | 1.12 |

5 | 0.062 | 1.09 | 0.96 | 0.98 | 1.24 |

5 | 0.094 | 1.16 | 0.96 | 0.95 | 1.38 |

6 | 0.020 | 1.02 | 0.98 | 1.00 | 1.10 |

6 | 0.040 | 1.05 | 0.97 | 0.99 | 1.18 |

6 | 0.062 | 1.10 | 0.97 | 0.97 | 1.37 |

7 | 0.028 | 1.04 | 0.98 | 0.99 | 1.15 |

7 | 0.056 | 1.09 | 0.97 | 0.97 | 1.29 |

7 | 0.083 | 1.16 | 0.99 | 0.93 | 1.43 |

9 | 0.023 | 1.03 | 0.98 | 0.99 | 1.17 |

9 | 0.047 | 1.08 | 0.98 | 0.97 | 1.31 |

9 | 0.062 | 1.12 | 0.99 | 0.95 | 1.39 |

12 | 0.017 | 1.02 | 0.99 | 0.99 | 1.16 |

12 | 0.033 | 1.06 | 0.99 | 0.98 | 1.28 |

12 | 0.050 | 1.10 | 1.00 | 0.95 | 1.38 |

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**MDPI and ACS Style**

Ziaei, M.
Bending and Torsional Stress Factors in Hypotrochoidal H-Profiled Shafts Standardised According to DIN 3689-1. *Eng* **2023**, *4*, 829-842.
https://doi.org/10.3390/eng4010050

**AMA Style**

Ziaei M.
Bending and Torsional Stress Factors in Hypotrochoidal H-Profiled Shafts Standardised According to DIN 3689-1. *Eng*. 2023; 4(1):829-842.
https://doi.org/10.3390/eng4010050

**Chicago/Turabian Style**

Ziaei, Masoud.
2023. "Bending and Torsional Stress Factors in Hypotrochoidal H-Profiled Shafts Standardised According to DIN 3689-1" *Eng* 4, no. 1: 829-842.
https://doi.org/10.3390/eng4010050