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

Abstract

Hypotrochoidal profile contours have been produced in industrial applications in recent years using two-spindle processes, and they are considered effective high-quality solutions for form-fit shaft and hub connections. This study mainly concerns analytical approaches to determine the stresses and deformations in hypotrochoidal profile shafts due to pure bending loads. The formulation was developed according to bending principles using the mathematical theory of elasticity and conformal mappings. The loading was further used to investigate the rotating bending behaviour. The stress factors for the classical calculation of maximum bending stresses were also determined for all those profiles presented and compiled in the German standard DIN3689-1 for practical applications. The results were also compared with the corresponding numerical and experimental results, and very good agreement was observed. Additionally, based on previous work, the stress factor was determined for the case of torsional loading to calculate the maximum torsional stresses in the standardised profiles, and the results are listed in a table. This study contributes to the further refinement of the current DIN3689 standard.

Full text

Citation: 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
Academic Editor: Antonio Gil Bravo
Received: 14 December 2022
Revised: 8 February 2023
Accepted: 1 March 2023
Published: 6 March 2023
Copyright: © 2023 by the author.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
Bending and Torsional Stress Factors in Hypotrochoidal
H-Profiled Shafts Standardised According to DIN 3689-1
Masoud Ziaei
Department of Mechanical and Automotive Engineering, Institute for Machin Development,
Westsächsische Hochschule Zwickau, D-08056 Zwickau, Germany; masoud.ziaei@fh-zwickau.de
Abstract:
Hypotrochoidal profile contours have been produced in industrial applications in recent
years using two-spindle processes, and they are considered effective high-quality solutions for form-fit
shaft and hub connections. This study mainly concerns analytical approaches to determine the stresses
and deformations in hypotrochoidal profile shafts due to pure bending loads. The formulation was
developed according to bending principles using the mathematical theory of elasticity and conformal
mappings. The loading was further used to investigate the rotating bending behaviour. The stress
factors for the classical calculation of maximum bending stresses were also determined for all
those profiles presented and compiled in the German standard DIN3689-1 for practical applications.
The results were also compared with the corresponding numerical and experimental results, and
very good agreement was observed. Additionally, based on previous work, the stress factor was
determined for the case of torsional loading to calculate the maximum torsional stresses in the
standardised profiles, and the results are listed in a table. This study contributes to the further
refinement of the current DIN3689 standard.
Keywords:
hypotrochoidal profile shafts; DIN3689 H-profiles; bending stress; rotating bending loads
in profiled shafts; flexure; torsional stress in profiled shafts; noncircular shafts; bending stress factor;
torsional stress factor
1. Introduction
In the field of modern drive technology, there is an increasing demand for higher
power transmission in a smaller construction space. A necessary and important component
in drive trains is the form-fit shaft and hub connections. Thereby, a widely used standard
solution is the key-fit connection according to DIN 6885 [
1
]. However, this technique is
reaching its mechanical limitations, which is why industry focus has been increasingly on
form-fit connections with polygon profiles in the past few years. With the hypotrochoidal
polygonal connection (H-profiles in Figure 1), a polygonal contour has been the new
standard according to DIN 3689-1 [
2
] since November 2021. The great advantages of H-
profiles via key-fit connections were studied in [
3
]. These investigations display a significant
reduction of around 50% in the fatigue notch factor.
Additionally, a significant advantage of hypotrochoidal profiles (H-profiles) is their
manufacturability through two-spindle turning [
4
,
5
] (Figure 2) and oscillating–turning [
6
]
processes, as well as roller milling [7] (Figure 3). This allows time-efficient production.
Despite the excellent manufacturability described above and the great mechanical
advantages of H-profiles, there is currently no reliable and cost-effective calculation method
for the dimensioning of such profiles. The determination of the strength limit of H-profiles
is still performed by means of extensive numerical investigations.
DIN 3689-1 refers to geometric specifications for H-profiles. Design guidelines are
compiled in Part 2 of the standard. This paper represents an analytical solution for purely
bending-loaded H-profile shafts in general and specifically for all standardised H-profiles
for the first time. Furthermore, the author uses the analytical solution developed in another
Eng 2023,4, 829–842. https://doi.org/10.3390/eng4010050 https://www.mdpi.com/journal/eng

Eng 2023,4830
paper [
8
] for all standard profiles for torsional stresses and puts them together for practical
and industrial applications.
The results can be used for a reliable and cost-effective calculation method of H-profile
shafts with a simple pocket calculator for pure bending as well as torsional loads.
Eng
2023, 4, FOR PEER REVIEW 2
Figure 1. Description of exemplary hypotrochoid (H-profile) with four concave sides. A detailed
explanation of the parameters is given below in Section 2.
Additionally, a significant advantage of hypotrochoidal profiles (H-profiles) is their
manufacturability through two-spindle turning [4,5] (Figure 2) and oscillating–turning [6]
processes, as well as roller milling [7] (Figure 3). This allows time-efficient production.
Figure 2. Some H-profiles manufactured by two-spindle process, Iprotec GmbH, © Guido
Kochsiek, www.iprotec.de, Zwiesel, Germany [5].
e
x
y
t
ρ
h
P
ri
ra
r
rg
r
r
Figure 1.
Description of exemplary hypotrochoid (H-profile) with four concave sides. A detailed
explanation of the parameters is given below in Section 2.
Eng
2023, 4, FOR PEER REVIEW 2
Figure 1. Description of exemplary hypotrochoid (H-profile) with four concave sides. A detailed
explanation of the parameters is given below in Section 2.
Additionally, a significant advantage of hypotrochoidal profiles (H-profiles) is their
manufacturability through two-spindle turning [4,5] (Figure 2) and oscillating–turning [6]
processes, as well as roller milling [7] (Figure 3). This allows time-efficient production.
Figure 2. Some H-profiles manufactured by two-spindle process, Iprotec GmbH, © Guido
Kochsiek, www.iprotec.de, Zwiesel, Germany [5].
e
x
y
t
ρ
h
P
ri
ra
r
rg
r
r
Figure 2.
Some H-profiles manufactured by two-spindle process, Iprotec GmbH,
©
Guido Kochsiek,
www.iprotec.de, Zwiesel, Germany [5].
Eng 2023, 4, FOR PEER REVIEW 3
Figure 3. Roller milling manufacturing for H-profile [7].
Despite the excellent manufacturability described above and the great mechanical
advantages of H-profiles, there is currently no reliable and cost-effective calculation
method for the dimensioning of such profiles. The determination of the strength limit of
H-profiles is still performed by means of extensive numerical investigations.
DIN 3689-1 refers to geometric specifications for H-profiles. Design guidelines are
compiled in Part 2 of the standard. This paper represents an analytical solution for purely
bending-loaded H-profile shafts in general and specifically for all standardised H-profiles
for the first time. Furthermore, the author uses the analytical solution developed in an-
other paper [8] for all standard profiles for torsional stresses and puts them together for
practical and industrial applications.
The results can be used for a reliable and cost-effective calculation method of H-pro-
file shafts with a simple pocket calculator for pure bending as well as torsional loads.
2. Geometry of H-Profiles
A hypotrochoid (H-profile) is created by rolling a circle with radius 𝑟 (called a roll-
ing circle) on the inside of a guiding circle with radius 𝑟 with no slippage (see, for in-
stance, [9]). The distance between the centre point of the rolling circle and the generating
point P is defined as eccentricity (Figure 1). Depending on the diameter ratios of the two
circles and the location of the generating point P in the rolling circle, different H-profiles
may be formed.
The diameter ratio (𝑟/𝑟) defines the number of sides “n” and should be an integer
(𝑛>2) to obtain a closed curve without intersection. The coordinates of the generated
point P describe the parameter equations for the hypotrochoid (H-profile) as follows:
𝑥(𝑡)=𝑟⋅cos(𝑡)+𝑒⋅cos󰇟(𝑛−1)⋅𝑡󰇠
𝑦(𝑡)=𝑟⋅sin(𝑡)−𝑒⋅sin󰇟(𝑛−1)⋅𝑡󰇠 with 0°≤𝑡≤360
°. (1)
The overlapping of the profile contour starts from the limit eccentricities of 𝑒 =

 and, accordingly, the limit relative eccentricity of 𝜀 =
=
.
Figure 4 shows some examples of the H-profiles obtained for different numbers of
sides (n) and eccentricities.
ω
v
Rolling Circle
Rolling line
H-Profile
Workpi ece
Reference profile
Tool
Figure 3. Roller milling manufacturing for H-profile [7].

Eng 2023,4831
2. Geometry of H-Profiles
A hypotrochoid (H-profile) is created by rolling a circle with radius
rr
(called a rolling
circle) on the inside of a guiding circle with radius
rg
with no slippage (see, for instance, [
9
]).
The distance between the centre point of the rolling circle and the generating point P is
defined as eccentricity (Figure 1). Depending on the diameter ratios of the two circles
and the location of the generating point P in the rolling circle, different H-profiles may
be formed.
The diameter ratio (
rg/rr
) defines the number of sides “n” and should be an integer
(
n>
2) to obtain a closed curve without intersection. The coordinates of the generated
point P describe the parameter equations for the hypotrochoid (H-profile) as follows:
x(t)=r·cos(t)+e·cos[(n−1)·t]
y(t)=r·sin(t)−e·sin[(n−1)·t]with 0◦≤t≤360◦.(1)
The overlapping of the profile contour starts from the limit eccentricities of
elim =r
n−1
and, accordingly, the limit relative eccentricity of εl im =elim
r=1
n−1.
Figure 4shows some examples of the H-profiles obtained for different numbers of
sides (n) and eccentricities.
Eng 2023, 4, FOR PEER REVIEW 4
Figure 4. Examples of H-profiles with different numbers of sides (n) and eccentricities.
If a rolling circle rolls on the outside of a guiding circle, the profile generated is called
an epicycloid (E-profile).
2.1. Geometric Properties
Area
Starting from the parameter representation (1) for the hypotrochoidal contours, the
following complex mapping function is formulated as follows:
𝜔(𝜁)=𝑟⋅𝜁+ 𝑒
𝜁 (2)
This function conformally maps the perimeter of a unit circle to the contour of a H-
profile. However, when the area enclosed by the polygon was mapped, multiple poles
were formed at the corners of the contour. A complete conformal mapping is not essential
for the determination of bending stresses. However, for shear force bending, a complete
mapping of profile cross-section is necessary (analogue to torsion problem [8]).
By substituting mapping (2) into the equation for the area [10,11]:
𝐴
=1
2𝐼𝑚
󰇟𝜔(𝜁)⋅𝜔󰇗(𝜁)󰇠

𝑑𝑡 (3)
the following relationship can be derived for the area enclosed by an H-profile for any
number of flanks n and eccentricity 𝑒:
𝐴
=
𝐴
−𝜋⋅𝑒⋅󰇟𝑑+𝑒⋅(𝑛−2)󰇠 (4)
where 𝜔󰇗 =𝑑𝜔/𝑑𝑡 is the first derivative of the mapping function, t defines the parameter
angle, and 𝐴=
⋅𝑑
 is the area of the head circle (with 𝑑=2⋅𝑟
).
2.2. Radius of Curvature at Profile Corners and Flanks
From a manufacturing point of view, the radius of the curvature of the contour at
profile corners (on the head circle) plays an important role. Using the equation presented
in [11], the radius of curvature can be determined:
𝜌=2𝑖⋅ (𝜔󰇗 ⋅𝜔󰇗
󰆽)

𝜔󰇗
󰆽⋅𝜔
󰇘
−𝜔󰇗⋅𝜔
󰇘
󰆽=|𝜔󰇗|
𝐼𝑚(𝜔󰇗
󰆽⋅𝜔
󰇘
) (5)
r = 20 mm
e = 1.25 mm
n = 3
r = 20 mm
e= 0.8 mm
n= 4
r= 20 mm
e = 0.83 mm
n= 5
r = 20 mm
e = 0.47 mm
n = 7
r = 20 mm
e= 0.43 mm
n= 8
r= 20 mm
e= 0.24 mm
n= 11
Figure 4. Examples of H-profiles with different numbers of sides (n) and eccentricities.
If a rolling circle rolls on the outside of a guiding circle, the profile generated is called
an epicycloid (E-profile).
2.1. Geometric Properties
Area
Starting from the parameter representation (1) for the hypotrochoidal contours, the
following complex mapping function is formulated as follows:
ω(ζ)=r·ζ+e
ζn−1(2)
This function conformally maps the perimeter of a unit circle to the contour of a
H-profile. However, when the area enclosed by the polygon was mapped, multiple poles
were formed at the corners of the contour. A complete conformal mapping is not essential
for the determination of bending stresses. However, for shear force bending, a complete
mapping of profile cross-section is necessary (analogue to torsion problem [8]).

Eng 2023,4832
By substituting mapping (2) into the equation for the area [10,11]:
A=1
2Z2π
0Im¯
ω(ζ)·.
ω(ζ)dt (3)
the following relationship can be derived for the area enclosed by an H-profile for any
number of flanks nand eccentricity e:
A=Aa−π·e·[da+e·(n−2)] (4)
where
.
ω=dω/dt
is the first derivative of the mapping function, tdefines the parameter
angle, and Aa=π
4·d2
ais the area of the head circle (with da=2·ra).
2.2. Radius of Curvature at Profile Corners and Flanks
From a manufacturing point of view, the radius of the curvature of the contour at
profile corners (on the head circle) plays an important role. Using the equation presented
in [11], the radius of curvature can be determined:
ρ=2i·(.
ω·¯
.
ω)3
2
¯
.
ω·..
ω−.
ω·¯
..
ω=

.
ω

3
Im(¯
.
ω·..
ω)(5)
The second derivative of the mapping function in (5) is defined as ..
ω=d2ω
dt2.
The radius of curvature at profile corners (on the head circle in Figure 1) can be
determined by substituting mapping function (2) into Equation (5) for t= 0 as follows:
ρa=(da−2·e·n)2
2·[da+2·e·n·(n−2)] (6)
The radius of curvature at profile corners
ρa
is important in connection with the
minimum tool diameter regarding the manufacturability of the profile.
The radius of curvature of the profile in the profile flank
ρf
(Figure 1) can also be
determined using Equation (5) for t=π/n:
ρf=[da+2·e·(n−2)]2
2·[da−2·e·(n2−2·n+2)] (7)
The radius of curvature in the flank area
ρf
is a measure of the degree of the form
closure of profile contours.
2.3. Bending Stresses
In many practical applications, a failure may occur in the profiled shaft outside of the
connection due to the excessive stresses. For these cases, the following analytical approach
based on [12] is used to solve the bending problem.
It is assumed that the cross-sections remain flat (without warping) after bending. The
following relationships are valid for the stresses:
σx=σy=τxy =τyz =τxz =0
σz=−Mb
Iy·x,(8)
where
Iy
denotes the moment of inertia for profile cross-section relative to the y-axis
(Figure 5).

Eng 2023,4833
Eng 2023, 4, FOR PEER REVIEW 5
The second derivative of the mapping function in (5) is defined as 𝜔󰇘 = 
.
The radius of curvature at profile corners (on the head circle in Figure 1) can be de-
termined by substituting mapping function (2) into Equation (5) for t = 0 as follows:
𝜌=(𝑑−2⋅𝑒⋅𝑛)
2⋅󰇟𝑑+2⋅𝑒⋅𝑛⋅(𝑛−2)󰇠 (6)
The radius of curvature at profile corners 𝜌 is important in connection with the
minimum tool diameter regarding the manufacturability of the profile.
The radius of curvature of the profile in the profile flank 𝜌 (Figure 1) can also be
determined using Equation (5) for 𝑡=𝜋/𝑛:
𝜌=󰇟𝑑+2⋅𝑒⋅(𝑛−2)󰇠
2⋅󰇟𝑑−2⋅𝑒⋅(𝑛−2⋅𝑛+2)󰇠 (7)
The radius of curvature in the flank area 𝜌 is a measure of the degree of the form
closure of profile contours.
2.3. Bending Stresses
In many practical applications, a failure may occur in the profiled shaft outside of the
connection due to the excessive stresses. For these cases, the following analytical approach
based on [12] is used to solve the bending problem.
It is assumed that the cross-sections remain flat (without warping) after bending. The
following relationships are valid for the stresses:
𝜎=𝜎
=𝜏
 =𝜏
 =𝜏
 =0
𝜎=−
⋅𝑥, (8)
where 𝐼 denotes the moment of inertia for profile cross-section relative to the y-axis (Fig-
ure 5).
Figure 5. The bending coordinate system for a loaded profile shaft.
2.4. Bending Deformations
Displacement is determined using Hooke’s law, and the corresponding correlation
between displacements and the strain is as follows (see [12,13]):
𝑢=𝑀
2⋅𝐸⋅𝐼
⋅󰇟𝑧+𝜈⋅(𝑦−𝑥)󰇠 (9)
2.5. Moments of Inertia
The moments of inertia involve a double integral over the profile’s cross-section, but
this can be reduced to a simple curvilinear integral over the profile contour using Green’s
theorem, as follows:
z
x
M
b
(+)
σ
bh
(−)
σ
bf
Figure 5. The bending coordinate system for a loaded profile shaft.
2.4. Bending Deformations
Displacement is determined using Hooke’s law, and the corresponding correlation
between displacements and the strain is as follows (see [12,13]):
ux=Mb
2·E·Iy
·hz2+ν·y2−x2i (9)
2.5. Moments of Inertia
The moments of inertia involve a double integral over the profile’s cross-section, but
this can be reduced to a simple curvilinear integral over the profile contour using Green’s
theorem, as follows:
Ix=−1
3Rγy3dx
Iy=1
3Rγx3dy
Ixy =1
2Rγx2ydy.
(10)
The contour description according to Equation (2) is also advantageous here. For the
contour of the profile’s cross-section, the following coordinates apply:
x=ω(λ)+ω(λ)
2
y=ω(λ)−ω(λ)
2·i.
(11)
By substituting Equation (11) in (10), Iy,Iy,Ixy can be determined as such:
Ix=−i
48 Rγω(λ)−ω(λ)3dω(λ)+ω(λ)
Iy=i
48 Rγω(λ)+ω(λ)3dω(λ)−ω(λ)
Ixy =−1
32 Rγω(λ)+ω(λ)2ω(λ)−ω(λ)dω(λ)−ω(λ),
(12)
where
λ=eit
. Function (12) facilitates the determination of moment of inertia with the
assistance of Equation (2).
The moment of inertia
Iy
is necessary for the calculation of the bending stress
σz
as
well as for the determination of bending deformation ux(Equations (8) and (9)).
Inserting the mapping function from (2) into Equation (12) for
Iy
, the following rela-
tionship is determined for the bending moment of inertia for an arbitrary number of flanks
nand eccentricity e:
Iy=π
4·r4−2e2(n−2)r2−e4(n−1)(13)

Eng 2023,4834
If one substitutes x(t) from (1) and
Iy
from (13) into Equation (8), the distribution of
the bending stress on the lateral surface of the profile can be determined as follows:
σb(t)=4Mb
π·rcos(t)+ecos((n−1)t)
r4−2e2(n−2)r2−e4(n−1)(14)
The maximum bending stress on the tension side occurs at
x=r+e
(on the profile
head, Figure 5), and therefore the following equation can be obtained:
σbh =4Mb
π·r+e
r4−2e2(n−2)r2−e4(n−1)(15)
The bending stress on the pressure side occurs at
x=r−e
in the middle of a profile
flank (on the profile foot, Figure 5) can also be determined as follows:
σbf =4Mb
π·r−e
r4−2e2(n−2)r2−e4(n−1)(16)
2.6. Example
An H-profile from DIN 3689-1 [
2
] with three sides, a head circle diameter of 40 mm
and eccentricity
e=
1.818 mm (
r=
18.18 mm; related eccentricity
ε=
0.1) was chosen as
the object of investigation. The bending load was chosen as Mb= 500 Nm.
In order to compare the analytical results, numerical investigations were carried out
using FE analyses, and the MSC-Marc programme system was used.
Figure 6shows the mesh structure and the corresponding boundary conditions. The
shaft is fixed on the right side. A bending moment is applied on the left side of the shaft
via a reference node using REB2s. Bending stresses were evaluated at an adequate distance
(
lb
) from the loading point. The FE mesh in Figure 6contains hexahedral elements with full
integration, type 7 according to the Marc Element Library [14].
Eng
2023, 4, FOR PEER REVIEW 7
In order to compare the analytical results, numerical investigations were carried out
using FE analyses, and the MSC-Marc programme system was used.
Figure 6 shows the mesh structure and the corresponding boundary conditions. The
shaft is fixed on the right side. A bending moment is applied on the left side of the shaft
via a reference node using REB2s. Bending stresses were evaluated at an adequate dis-
tance (𝑙) from the loading point. The FE mesh in Figure 6 contains hexahedral elements
with full integration, type 7 according to the Marc Element Library [14].
Figure 6. FE mesh and boundary conditions for the H-profile with n = 3 according to DIN 3689-1.
FE structures are generated by employing software written in Python language at the
Chair of Machine Elements at West Saxon University of Zwickau, Germany. The FE
meshes were then transferred to MSC-Marc program system and integrated into pre-pro-
cessing.
Figure 7 displays the distribution of bending stress on the circumference of the profile
according to Equation (14) and its comparison with the numerical result. A good agree-
ment between the results was observed.
Figure 7. Circumferential distribution of the bending stress on the lateral surface of a standardised
H3 profile.
-150
-100
-50
0
50
100
150
0 20406080100
Bending Stress N/mm
2
Circumferential Position mm
FEM
Analytic
0
Figure 6. FE mesh and boundary conditions for the H-profile with n= 3 according to DIN 3689-1.
FE structures are generated by employing software written in Python language at the
Chair of Machine Elements at West Saxon University of Zwickau, Germany. The FE meshes
were then transferred to MSC-Marc program system and integrated into pre-processing.
Figure 7displays the distribution of bending stress on the circumference of the profile
according to Equation (14) and its comparison with the numerical result. A good agreement
between the results was observed.

Eng 2023,4835
Eng
2023, 4, FOR PEER REVIEW 7
In order to compare the analytical results, numerical investigations were carried out
using FE analyses, and the MSC-Marc programme system was used.
Figure 6 shows the mesh structure and the corresponding boundary conditions. The
shaft is fixed on the right side. A bending moment is applied on the left side of the shaft
via a reference node using REB2s. Bending stresses were evaluated at an adequate dis-
tance (𝑙) from the loading point. The FE mesh in Figure 6 contains hexahedral elements
with full integration, type 7 according to the Marc Element Library [14].
Figure 6. FE mesh and boundary conditions for the H-profile with n = 3 according to DIN 3689-1.
FE structures are generated by employing software written in Python language at the
Chair of Machine Elements at West Saxon University of Zwickau, Germany. The FE
meshes were then transferred to MSC-Marc program system and integrated into pre-pro-
cessing.
Figure 7 displays the distribution of bending stress on the circumference of the profile
according to Equation (14) and its comparison with the numerical result. A good agree-
ment between the results was observed.
Figure 7. Circumferential distribution of the bending stress on the lateral surface of a standardised
H3 profile.
-150
-100
-50
0
50
100
150
0 20406080100
Bending Stress N/mm
2
Circumferential Position mm
FEM
Analytic
0
Figure 7.
Circumferential distribution of the bending stress on the lateral surface of a standardised
H3 profile.
Additionally, bending stresses were experimentally determined for the profile head
and foot areas. Figure 8shows the test bench for bending load.
Eng
2023, 4, FOR PEER REVIEW 8
Additionally, bending stresses were experimentally determined for the profile head
and foot areas. Figure 8 shows the test bench for bending load.
Figure 8. Bending loads test bench (Machine Elements Laboratory at West Saxon University of
Zwickau).
Experimental results for head and foot areas are compared with Equations (15) and
(16) in Figure 9, where a good agreement of the results is evident.
Figure 9. Comparison of the experimental results with the analytical solutions.
2.7. Stress Factor for Bending Loads
The stress factor is defined as the ratio of bending stress in a profile shaft to a corre-
sponding reference stress for a round cross-section with radius 𝑟 (nomial radius of the
profile):
𝛼=𝜎
𝜎,
with: 𝜎, =

∙ 


,

and 𝐼
, =
∙𝑟. (17)
For the head of the profile, the stress factor is determined as follows:
-200
-100
0
100
200
0 150 300 450 600 750
Bending Stress N/mm
2
Bending Moment Nm
Head (Analytical)
Head (Strain Guage)
Foot (Analytical)
Foot (Strain Gauge)
Head
Foot
−100
−200
Figure 8.
Bending loads test bench (Machine Elements Laboratory at West Saxon University of Zwickau).
Experimental results for head and foot areas are compared with
Equations (15) and (16)
in Figure 9, where a good agreement of the results is evident.

Eng 2023,4836
Eng
2023, 4, FOR PEER REVIEW 8
Additionally, bending stresses were experimentally determined for the profile head
and foot areas. Figure 8 shows the test bench for bending load.
Figure 8. Bending loads test bench (Machine Elements Laboratory at West Saxon University of
Zwickau).
Experimental results for head and foot areas are compared with Equations (15) and
(16) in Figure 9, where a good agreement of the results is evident.
Figure 9. Comparison of the experimental results with the analytical solutions.
2.7. Stress Factor for Bending Loads
The stress factor is defined as the ratio of bending stress in a profile shaft to a corre-
sponding reference stress for a round cross-section with radius 𝑟 (nomial radius of the
profile):
𝛼=𝜎
𝜎,
with: 𝜎, =

∙ 


,

and 𝐼
, =
∙𝑟. (17)
For the head of the profile, the stress factor is determined as follows:
-200
-100
0
100
200
0 150 300 450 600 750
Bending Stress N/mm
2
Bending Moment Nm
Head (Analytical)
Head (Strain Guage)
Foot (Analytical)
Foot (Strain Gauge)
Head
Foot
−100
−200
Figure 9. Comparison of the experimental results with the analytical solutions.
2.7. Stress Factor for Bending Loads
The stress factor is defined as the ratio of bending stress in a profile shaft to a correspond-
ing reference stress for a round cross-section with radius r(nomial radius of the profile):
αb=σb
σb,re f
with: σb,re f =Mb·r
Iy,re f
and Iy,re f =π
4·r4.
(17)
For the head of the profile, the stress factor is determined as follows:
αbh =1+e
1−2e2(n−2)−e4(n−1)(18)
Figure 10 shows the curves for the stress factor
αbh
as a function of the relative
eccentricity
ε
for different numbers of sides
n
. It can be recognised that the stress factor
rises with an increase in eccentricity and or the number of sides.
Eng 2023, 4, FOR PEER REVIEW 9
𝛼 =1+𝑒
1−2𝑒(𝑛−2)−𝑒(𝑛−1) (18)
Figure 10 shows the curves for the stress factor 𝛼 as a function of the relative ec-
centricity 𝜀 for different numbers of sides 𝑛. It can be recognised that the stress factor
rises with an increase in eccentricity and or the number of sides.
Figure 10. Stress factors for the bending stress at the profile head (Equation (18)) with varying rela-
tive eccentricity and number of sides.
For the profile base (foot), the following stress factor is analogously obtained:
𝛼 =1−𝑒
1−2𝑒(𝑛−2)−𝑒(𝑛−1) (19)
2.8. Rotating Bending Stress
During power transmission, the gear shaft always shows rotational movement.
Therefore, the rotating bending was also investigated.
Figure 11 schematically represents the rotated position of an H-profile with three
flanks according to the Cartesian coordinates.
Figure 11. Rotated coordinate system for determining the bending moment of inertia.
The moment of inertia remains invariant due to the periodic symmetry of the cross-
section of the H-profile presented based on Equation (2). Therefore, the following rela-
tionships are valid from Equation (12):
Relative Eccenticity
Stress Factor
α
bh
n=3
n=4
n=5
n=7
n=9
M
b,y
x
y
φ
Figure 10.
Stress factors for the bending stress at the profile head (Equation (18)) with varying relative
eccentricity and number of sides.

Eng 2023,4837
For the profile base (foot), the following stress factor is analogously obtained:
αb f =1−e
1−2e2(n−2)−e4(n−1)(19)
2.8. Rotating Bending Stress
During power transmission, the gear shaft always shows rotational movement. There-
fore, the rotating bending was also investigated.
Figure 11 schematically represents the rotated position of an H-profile with three
flanks according to the Cartesian coordinates.
Eng 2023, 4, FOR PEER REVIEW 9
𝛼 =1+𝑒
1−2𝑒(𝑛−2)−𝑒(𝑛−1) (18)
Figure 10 shows the curves for the stress factor 𝛼 as a function of the relative ec-
centricity 𝜀 for different numbers of sides 𝑛. It can be recognised that the stress factor
rises with an increase in eccentricity and or the number of sides.
Figure 10. Stress factors for the bending stress at the profile head (Equation (18)) with varying rela-
tive eccentricity and number of sides.
For the profile base (foot), the following stress factor is analogously obtained:
𝛼 =1−𝑒
1−2𝑒(𝑛−2)−𝑒(𝑛−1) (19)
2.8. Rotating Bending Stress
During power transmission, the gear shaft always shows rotational movement.
Therefore, the rotating bending was also investigated.
Figure 11 schematically represents the rotated position of an H-profile with three
flanks according to the Cartesian coordinates.
Figure 11. Rotated coordinate system for determining the bending moment of inertia.
The moment of inertia remains invariant due to the periodic symmetry of the cross-
section of the H-profile presented based on Equation (2). Therefore, the following rela-
tionships are valid from Equation (12):
Relative Eccenticity
Stress Factor
α
bh
n=3
n=4
n=5
n=7
n=9
M
b,y
x
y
φ
Figure 11. Rotated coordinate system for determining the bending moment of inertia.
The moment of inertia remains invariant due to the periodic symmetry of the cross-
section of the H-profile presented based on Equation (2). Therefore, the following relation-
ships are valid from Equation (12):
Ix=Iyand Ixy =0. (20)
From Equation (20) and the use of Mohr’s circle, it can be proven that the moment of
inertia is independent of the rotation angle φ(see also [10]):
Iξ=Iη=Ix=Iy
Iξη =Ixy =0. (21)
In order to obtain the general solution of the bending stress according to Equation (8)
for an arbitrary angle of rotation, the perpendicular distance
ξ
is to be calculated in the
rotated coordinate system:
ξ(φ)=ycos(φ)−xsin(φ)(22)

Eng 2023,4838
where
φ
denotes the angle of rotation. If the values for xand yfrom (1) are inserted into
the relationship (22), the following equation results for the perpendicular distance in the
rotated coordinate system (0 ≤t≤2π):
ξ(φ,t)=rsin(t−φ)−esin((n−1)t+φ)(23)
The distribution of bending stress on the profile contour may be determined by using
(23) in the relation of bending stress as follows:
σb(φ,t)=−Mb
Iη
·ξ(φ,t)=4Mb
π·rsin(t−φ)−esin((n−1)t+φ)
r4−2e2(n−2)r2−e4(n−1)(24)
Figure 12 shows the distributions of the bending stresses on the profile contour for
different angles of rotation, which were determined using Equation (24). As expected, the
maximum stress occurred at the profile head.
Eng 2023, 4, FOR PEER REVIEW 10
𝐼=𝐼
 and 𝐼 =0. (20)
From Equation (20) and the use of Mohr’s circle, it can be proven that the moment of
inertia is independent of the rotation angle 𝜙 (see also [10]):
𝐼=𝐼
= 𝐼=𝐼

𝐼 =𝐼
 =0. (21)
In order to obtain the general solution of the bending stress according to Equation (8)
for an arbitrary angle of rotation, the perpendicular distance 𝜉 is to be calculated in the
rotated coordinate system:
𝜉(𝜙)=𝑦𝑐𝑜𝑠(𝜙)−𝑥𝑠𝑖𝑛(𝜙) (22)
where 𝜙 denotes the angle of rotation. If the values for x and y from (1) are inserted into
the relationship (22), the following equation results for the perpendicular distance in the
rotated coordinate system (0≤𝑡≤2𝜋):
𝜉(𝜙,𝑡)=𝑟𝑠𝑖𝑛(𝑡−𝜙)−𝑒𝑠𝑖𝑛(𝑛−1)𝑡+𝜙 (23)
The distribution of bending stress on the profile contour may be determined by using
(23) in the relation of bending stress as follows:
𝜎(𝜙,𝑡)=−𝑀𝑏
𝐼𝜂⋅𝜉(𝜙,𝑡)=4𝑀
𝜋⋅𝑟𝑠𝑖𝑛(𝑡−𝜙)−𝑒𝑠𝑖𝑛(𝑛−1)𝑡+𝜙
𝑟−2𝑒(𝑛−2)𝑟−𝑒(𝑛−1) (24)
Figure 12 shows the distributions of the bending stresses on the profile contour for
different angles of rotation, which were determined using Equation (24). As expected, the
maximum stress occurred at the profile head.
Figure 12. Distributions of the bending stresses on the profile contour for different angles of rotation
𝜙, with 𝑟 = 18.18 mm, 𝑛=3, 𝑒 = 1.818 mm, and 𝑀 = 500 Nm.
2.9. Deflection
The deflection of the profile shaft can also be determined with the help of the bending
moment of inertia 𝐼. As explained above, this is independent of the angular position of
the cross-section (Equation (21)).
φ
= 0°
t
t
φ
= 30°
t
φ
= 45°
t
φ
= 60°
t
Figure 12.
Distributions of the bending stresses on the profile contour for different angles of rotation
φ, with r=18.18 mm, n=3, e=1.818 mm, and Mb= 500 Nm.
2.9. Deflection
The deflection of the profile shaft can also be determined with the help of the bending
moment of inertia
Iy
. As explained above, this is independent of the angular position of
the cross-section (Equation (21)).
The deflection of the neutral axis is determined from Equation (9) for x=y= 0 as
follows:
δx=Mb
2·E·Iy
·z2(25)

Eng 2023,4839
Substituting (13) in (25), the deflection can be determined as
δx=2Mb
πE·z2
r4−2e2(n−2)r2−e4(n−1)(26)
2.10. Example
Figure 13 shows the deflection for an H-profile shaft with three flanks according to
DIN3689-1 with
da
= 40 mm (H3-40
×
32.73 with
ε
= 0.1) and a length of 160 mm made
of steel (E = 210,000
N
mm2
). The comparison with FE analysis shows very good agreement
with Equation (26), as can also be seen in Figure 13. The bending load was chosen as
Mb= 500 Nm.
Eng
2023, 4, FOR PEER REVIEW 11
The deflection of the neutral axis is determined from Equation (9) for x = y = 0 as
follows:
𝛿=𝑀
2⋅𝐸⋅𝐼
⋅𝑧 (25)
Substituting (13) in (25), the deflection can be determined as
𝛿=2𝑀
𝜋𝐸 ⋅𝑧
𝑟−2𝑒(𝑛−2)𝑟−𝑒(𝑛−1) (26)
2.10. Example
Figure 13 shows the deflection for an H-profile shaft with three flanks according to
DIN3689-1 with 𝑑 = 40 mm (H3-40 × 32.73 with
ε
= 0.1) and a length of 160 mm made of
steel (E = 210,000 


). The comparison with FE analysis shows very good agreement
with Equation (26), as can also be seen in Figure 13. The bending load was chosen as 𝑀
= 500 Nm.
Figure 13. Deflection in a DIN3689-H3-40 × 32.73 profile shaft.
2.11. H-Profiles According to DIN3689-1
DIN3689-1 is a new standard that was published for the first time in November 2021.
It describes the geometric properties of 18 specified H-profiles in two series. Series A is
based on the head diameter, and series B involves the foot diameter as the nominal size of
the profile. The respective corresponding profiles are geometrically similar. Each series
contains 48 nominal sizes, which remain geometrically similar amongst themselves. Con-
sequently, all standardised profiles are limited to 18 variants. This facilitates the pro-
cessing of a generally valid design concept.
2.12. Stress Factor for Bending
The maximum bending stresses at the head and foot of the profile are important from
a technical point of view for the design of a profile shaft subject to bending. Therefore, in
this section, the two stress factors 𝛼 and 𝛼 for all the 18 standard profile series were
determined using Equations (18) and (19).
2.13. Stress Factor for Torsion
Figure 13. Deflection in a DIN3689-H3-40 ×32.73 profile shaft.
2.11. H-Profiles According to DIN3689-1
DIN3689-1 is a new standard that was published for the first time in November 2021. It
describes the geometric properties of 18 specified H-profiles in two series. Series A is based
on the head diameter, and series B involves the foot diameter as the nominal size of the
profile. The respective corresponding profiles are geometrically similar. Each series contains
48 nominal sizes, which remain geometrically similar amongst themselves. Consequently,
all standardised profiles are limited to 18 variants. This facilitates the processing of a
generally valid design concept.
2.12. Stress Factor for Bending
The maximum bending stresses at the head and foot of the profile are important from
a technical point of view for the design of a profile shaft subject to bending. Therefore, in
this section, the two stress factors
αbh
and
αb f
for all the 18 standard profile series were
determined using Equations (18) and (19).
2.13. Stress Factor for Torsion
The stress concentration factor for torsion
αt
is defined as the ratio of the maximum
torsional stress τt,max (occurring in the middle of the profile flank) and the torsional stress
in a round reference shaft with radius r:
αt=τt,max
τt,ref
with: τt,ref =Mt·r
It,re f and It,re f =π
2·r4.
(27)

Eng 2023,4840
In [
15
], purely numerical investigations were carried out on the torsional stresses in
H-profile shafts to calculate the stress factor.
The analytical solution for torsion may be performed using the approach of Muskhel-
ishvili [
12
]. However, this requires a conformal mapping of the unit circle onto the polygon’s
cross-section. For H-profiles, the mapping function derived from the parametric equation,
Equation (1), cannot be directly used to solve the torsional stresses due to the multiple
poles. The authors of [
16
] employed an elaborate computational process to determine
the polynomials required for the description of the mappings of H-profiles. In [
8
,
17
–
19
],
successive methods according to Kantorovich [
20
] were used to develop a suitable mapping
function in the form of a series converging to the profile contour. The convergence quality
and limit were examined and presented depending on the number of terms in the series
developed in [
8
], calculating the torsional deformations for all standardised profiles. In
the presented work, this method, accompanied by FEA, was used for all the 18 standard-
ised profile geometries of DIN3689-1 to determine the maximum torsional stresses, which
occur in the middle of the profile flank at the profile foot. A stress concentration factor
for torsional loading
αt
was also determined analogously to that defined for the case of
bending load.
For practical applications, the results for the bending and torsional stress factors are
compiled in Table 1. Using the relative eccentricity, no dependence on the shaft diameter
appears. Table 1lists the results obtained for the bending and torsional stress factors for all
standardised profile geometries according to DIN3689-1 (rounded to two decimal places).
Table 1.
Stress factors for bending and torsional loads for the H-profiles standardised according to
DIN3689-1.
nε αbh αbf Iy/I0αt
30.100 1.12 0.92 0.98 1.23
40.056 1.07 0.96 0.99 1.17
40.111 1.17 0.94 0.95 1.37
50.031 1.04 0.97 0.99 1.12
50.062 1.09 0.96 0.98 1.24
50.094 1.16 0.96 0.95 1.38
60.020 1.02 0.98 1.00 1.10
60.040 1.05 0.97 0.99 1.18
60.062 1.10 0.97 0.97 1.37
70.028 1.04 0.98 0.99 1.15
70.056 1.09 0.97 0.97 1.29
70.083 1.16 0.99 0.93 1.43
90.023 1.03 0.98 0.99 1.17
90.047 1.08 0.98 0.97 1.31
90.062 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
The bending moment of the inertia of a circular cross-section with radius ris defined
as a reference moment of inertia and labelled
I0
. The ratio between
Iy
and
I0
is also listed
in Table 1for the standardised profiles. The H-profiles are normally slightly more flexible
than round profiles.

Eng 2023,4841
3. Conclusions
In this paper, an analytical approach was presented to determine the bending stresses
and deformations in the hypotrochoidal profile shafts. Valid calculation equations for the
area, radii of curvature of the profile contour, and the bending moment of inertia were
derived for such profiles. Furthermore, the solutions for bending stresses and deformations
were presented. For practical applications, a stress factor was defined for the critical
locations on the profile contour.
The analytical results demonstrated very good agreement with both numerical and
experimentally determined results.
The stress factors of the bending stresses were determined for all profile geometries
standardised according to DIN3689-1, and the values obtained were compiled in a table
for practical applications. Based on previous works of the author, the stress factors for
torsional stresses were also determined and added to the table. The data allow a reliable
and cost-effective calculation of H-profile shafts with a pocket calculator for pure bending
as well as torsional loads. This can be very advantageous for SMEs.
Funding:
This research was funded by DFG (Deutsche Forschungsgemeinschaft) grant number [DFG
ZI 1161/2].
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The author declares no conflict of interest.
Abbreviations
Formula Symbols:
Amm2Area of profile cross-section
emm Profile eccentricity
e - Euler’s number
elim mm Profile overlap eccentricity limit
EMPa Young’s modulus
n- Profile periodicity (number of sides)
I0mm4Corresponding reference moments of inertia for a round cross-section with radius r
It,re f mm4Torsional moment of inertia for reference shaft
Ix,Iy,Ixy mm4Surface moments of inertia in the Cartesian coordinate system
Iy,re f mm4Surface moment of inertia about y-axis for reference shaft
Iξ,Iη,Iξη mm4Surface moments of inertia in the rotated coordinate system
lmm Length of profile shaft
lbmm Distance to strain gage in experimental test
MbNm Bending moment
rmm Nominal or mean radius
t- Profile parameter angle
uxmm Displacement in xdirection
x,y,zmm Cartesian coordinates
Greek Formula Symbols:
αbh - Bending stress factor for profile head
αb f - Bending stress factor for profile foot
αt- Torsional stress factor for profile foot

Eng 2023,4842
δxmm deflection
ε=e/r- Relative eccentricity
φ- Rotation angle of the coordinate system
λ=eiθ- Physical plane unit circle
θ- Polar angle
σb,σzMPa Bending stress (z-component of stress vector)
τtMPa Torsional stress
ω(ζ)- Completed mapping function
ω0(ζ)- Contour edge mapping function
ζ- Complex variable in model plane
ξ,η- Coordinates in rotated system
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