Ci a ion: H uzik, L.; S uz, J.; T och a,
M.; Zacal, J. Unsymme ically
Dis ibu ed Bol Axial Fo ces
be ween Symme ically Spaced Bol s
o Clamping Slee e. Symme y 2023,
15, 1893. h ps://doi.o g/10.3390/
sym15101893
Academic Edi o : Mikhail She eme
Recei ed: 30 Augus 2023
Re ised: 4 Oc obe 2023
Accep ed: 6 Oc obe 2023
Published: 9 Oc obe 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
symme y
S
S
A icle
Unsymme ically Dis ibu ed Bol Axial Fo ces be ween
Symme ically Spaced Bol s o Clamping Slee e
Lukas H uzik * , Ji i S uz , Mi osla T och a and Ji i Zacal
Depa men o Machine Pa s and Mechanisms, Facul y o Mechanical Enginee ing,
VSB-Technical Uni e si y o Os a a, 70800 Os a a, Czech Republic; [email p o ec ed] (J.S.);
mi osla [email p o ec ed] (M.T.); [email p o ec ed] (J.Z.)
*Co espondence: lukas.h [email p o ec ed]; Tel.: +420-596-993-218
Abs ac :
This a icle deals wi h he uni o mi y o he dis ibu ion o axial o ces be ween he
indi idual bol s o he clamping slee e. Clamping slee es a e machine componen s connec ing he
sha o he hub using con ac o ce. Fo slee e p e ension, bol s a e used and posi ioned symme ically
wi h espec o he axis o he clamping slee e. Though he bol s a e placed symme ically a ound
he axis, he bol axial o ces a e no uni o mly dis ibu ed when he bol s a e igh ened acco ding
o he manu ac u e ’s ca alog. Fi s , his a icle desc ibes he p ocedu e o igh ening he slee e
bol s acco ding o he manu ac u e ’s ca alog. In he nex pa o he a icle, an FEA simula ion
o slee e igh ening is pe o med. The FEM simula ion is hen compa ed wi h he alues ound
by measu emen .
Keywo ds: clamping slee e; bol s; s ain gauges; FEM; measu emen s
1. In oduc ion
Clamping slee es a e machine componen s connec ing he sha and hub. In oday’s
echnical p ac ice, hey a e e y o en used o hei ad an ages, bu hey also show ailu es,
o example in [
1
,
2
]. They wo k on he p inciple o a ic ional o ce. To de elop con ac
p essu e, which is impo an o he unc ion o he join , i is necessa y o p e ension
he clamping slee e bol s. These bol s a e o en dis ibu ed symme ically a ound he
axis o he clamping slee e. In his a icle i is he Rexno d Tollok TLK 400 clamping
slee e in dimensions 50
×
80 mm used [
3
]. Mo e de ailed in o ma ion abou he clamping
connec ions in gene al can be ound, o example, in [4].
This slee e is p e ensioned wi h eigh M8 bol s placed symme ically a ound axis o
he clamping slee e. A iew o he slee e and a desc ip ion o he bol s is shown in Figu e 1.
E en wi h he symme ical dis ibu ion o he clamp slee e bol s, a e igh ening he clamp
slee e bol s acco ding o he manu ac u e ’s ca alog, he bol s axial o ces be ween bol s
a e no symme ically dis ibu ed. In gene al, i can be said ha a ela i ely la ge e o is
in oduced in o he igh ening by he con e sion o he igh ening momen o he bol s,
which is gi en by he manu ac u e , in o he axial o ce, which is impo an o ensioning.
The p ecise de e mina ion o he axial o ce om he igh ening momen is p oblema ic due
o se e al complexly de e mined alues. The solu ion o his p oblem can be a specialized
de ice ha ensions he bol by axial o ce and hen igh ens he nu eely [
5
,
6
]. I is
no possible o use his me hod o igh ening he bol s o he clamping slee e due o he
sho e bol s.
1.1. Uni o mi y wi h Respec o he Tigh ening Sequence
The uni o mi y o he dis ibu ion o axial o ces be ween he indi idual bol s can
a ec he con ac p essu e and he e o e he maximum ope a ing load ha he slee e will
ca y. The uni o mi y o axial o ces due o he igh ening p ocedu e is in es iga ed, o
example, in [
7
–
9
]. In none o he a icles men ioned abo e a e he uni o mi y o axial o ces
Symme y 2023,15, 1893. h ps://doi.o g/10.3390/sym15101893 h ps://www.mdpi.com/jou nal/symme y
Symme y 2023,15, 1893 2 o 17
and he e o e also he uni o mi y o s esses and p essu es o he clamping slee e add essed.
Ve y o en, he igh ening p ocedu e o langed sealing su aces is sol ed [
10
,
11
], whe e,
o cou se, he uni o mi y o he axial o ce in he bol s has an in luence on he uni o mi y
o he p essu e ac ing on he seal, which is needed o he co ec unc ioning o he seal.
I is a his poin ha he simila i y be ween his langed sealan join s and a clamping
join becomes appa en , whe e he p essu e ac ing on he ape ed su aces and cylind ical
su aces o ans e he load is essen ial o he co ec unc ion o he join . In he case o
hese complica ions, simul aneous igh ening o all bol s a he same ime seems o be o
g ea in e es ; equipmen o such igh ening is o e ed, o example, by SKF [12].
Figu e 1. Clamping slee e wi h numbe ed bol s.
1.2. Bol ed Connec ions
We can iden i y se e al ypes o clamping slee es, including he numbe o bol s and
hei loca ion. A clamping slee e wi h se e al bol s a ound he axis is p obably he mos
used. These bol ed connec ions in e ac wi h each o he , and he bol ed connec ion solu ion
we a e dealing wi h is a mul i-bol ed connec ion. The issue o mul iple bol ed connec ions
is add essed, o example, in VDI 2230 in i s second pa [
13
]. Acco ding o VDI 2230,
mul iple bol ed connec ions a e also ecommended o be sol ed using FEM. The FEM
simula ion e i ied by he expe imen is included, o example, in [14–16].
Due o se e al ypical cha ac e is ics o he clamping slee e, i is no possible o use
a common diag am o a p e ensioned bol ed join s, which can be ound in he common
li e a u e, o example, in [
17
,
18
]. Fundamen al o he impossibili y o using ypical analy i-
cal calcula ions a e he conical su aces, which ha e a e y small conici y, and ela i ely
la ge displacemen s o hese cones occu du ing igh ening. When conside ing he o al
de o ma ion (displacemen ), he s i ness o he langes is e y low. Due o he e y a ypical
p ope ies o he bol ed connec ions o he clamping slee e, i is no possible o apply he
knowledge o o ce dis ibu ion o con en ional bol ed connec ions well enough. Fo his
eason, which is also he poin o he pape , FEM and measu emen s a e used a he han
analy ical calcula ions using he p e ensioned bol ed join diag am.
Due o he impossibili y o using an analy ical calcula ion because o he e y a ypical
bol ed connec ion, wo al e na i e me hods a e used. The FEM simula ion o clamping
slee e bol ed connec ions is used. F om he li e a u e, in he case o he impossibili y
o analy ical calcula ion, i is qui e possible o use FEM [
19
]. FEM se es e y well in
applica ions whe e he bending momen o bol s can be expec ed [
20
], which is also he
case wi h he clamping slee e. In addi ion o he possibili y o using he FEM, ano he
op ion o de e mining he axial o ce in he bol is measu emen .
Symme y 2023,15, 1893 3 o 17
1.3. Measu ing Bol Axial Fo ce
Measu ing he axial o ce in bol s is qui e common in echnical p ac ice and is used
in cases whe e we do no know he axial o ce in he bol and we need o de e mine i
accu a ely. One a ian is he FEM simula ion, which de e mines he o ce in he bol .
The ad an age o his me hod is he abili y o accu a ely model, o example, he inpu
o ce. Howe e , we make simpli ica ions in se e al pa ame e s, such as su ace s uc u e,
ma e ial p ope ies o manu ac u ing inaccu acies, so we o en p oceed o measu e he
axial o ce di ec ly on he ac ual componen . The disad an age is he equen need o
adjus men and he e o e measu emen s di e om he ac ual s a e.
The e a e se e al ways o measu e he axial o ce in a bol . P obably he mos common
in p ac ice is he use o an axial o ce ansduce , which is mos o en a s ain gauge
de ice. In echnical p ac ice, i is o en no possible o use pu chased senso s due o hei
dimensions, so we usually use senso s made exac ly o he applica ion [
21
]. The senso
in he o m o a housing wi h a s ain gauge is discussed, o example, in he a icle [
22
].
Ano he me hod ha is less common is he op ical senso me hod [
23
]. These me hods need
a housing wi h s ain gauges, so hey change he s i ness o he langes. Ano he op ion is
o use he bol as a senso . In his case, i is necessa y o educe he p eload and conside
he change o he bol s i ness. Ve y o en he bol is i ed wi h wo s ain gauges [
24
],
which a e connec ed o one b idge.
1.4. FEM Simula ion
FEM simula ion in ol es con e ing a eal p oblem in o a 3D model and se ing
bounda y condi ions. I is he se ing o he bounda y condi ions ha is he bigges p oblem,
which is why he e i ica ion o he FEM model is usually achie ed using measu emen s,
such as in hese a icles [
25
–
27
]. The FEM model was se up conside ing in o ma ion om
he ollowing sou ces [
28
–
30
]. FEM is desc ibed in Sec ion 2.4. I used Ansys Wo kbench
2020 R2 om Ansys manu ac u e om Canonsbu g, PA, USA.
2. Ma e ials and Me hods
The basic equi emen is o use as many alues as possible om he manu ac u e ’s
ca alog o espec he eal applica ion si ua ion as much as possible.
2.1. Values om he Clamping Slee e Manu ac u e ’s Ca alogue
Fundamen al o his issue is he p ocedu e o igh ening he clamp slee e bol s and
lub ica ing he clamping slee e, which has a signi ican e ec on clamping slee e igh ening.
Fo he p e iously de ined clamping slee e, TLK 400 50
×
80, i is possible o ind he bol
igh ening o que alue in he manu ac u e ’s ca alog. The igh ening o que alue in he
manu ac u e ’s ca alog is o unmodi ied bol s and o ans e ing he wo king load o he
slee e. The TLK 400 50
×
80 slee e is p e- ensioned wi h 8 bol s wi h a igh ening o que
o 41 Nm. In addi ion o he igh ening o que, he igh ening p ocedu e is also de ined as
ollows:
1. Tigh en he bol s c osswise o hal o que;
2. Tigh en he bol s c osswise o ull o que;
3. Fi s ime a ound he ci cum e ence o ull igh ening o que;
4. Second ime a ound he ci cum e ence o ull igh ening o que.
So, i we di ide his p ocedu e in o indi idual s eps, we each he 32-consecu i e-s ep
inc emen s o bol p e ension o ce.
2.2. Inpu Da a o FEM
Fo he FEM model, i is i s necessa y o de e mine he inpu alues. These alues a e
no pa o he clamping slee e manu ac u e ’s ca alog. The calcula ion o inpu alues o
he FEM model was de e mined, o example, in [
31
]. The inpu alues we e he dimension
o he slee e, he igh ening oque o he Bol s and he p essu e o he sha o hub om
manu ac u e ’s ca alog. The ic ional p ope ies on he slee e, bol and he axial o ce in
Symme y 2023,15, 1893 4 o 17
he bol s a e unknown. Due o he necessi y o using a nume ical calcula ion me hod, he
solu ion was pe o med using Ma hCad om he PTC company based in Bos on, USA.
F om he con ac p essu e ac ing on he sha and he igh ening o que o he bol s, o he
pa ame e s needed o he FEM model we e de e mined.
In he manu ac u e ’s ca alog, he igh ening o que is de ined in he o m o a igh en-
ing o que. I is i s necessa y o de ine he equa ion o calcula ing he axial o ce in he
bol om he igh ening o que. Equa ions (1)–(4) a e om [32].
Fo=Mu
·Ds
2+( an(ψ+ϕ))·d2
2(1)
Mu
is he igh ening momen o he bol s,
Fo
is he bol p e ension o ce, and
is he
ic ion coe icien unde bol head and on conical and cylind ical su aces.
Ds
2
is he middle
ic ion diame e , and
d2
2
is he majo diame e o h ead. The middle ic ion diame e is
he diame e be ween bo e diame e o he Bol and diame e o he Bol head.
Howe e , his equa ion is e y gene al and does no su icien ly de ine he exac
applica ion o his bol ed connec ion in he clamping slee e. A mo e accu a e applica ion
o his equa ion wi h he applica ion o a clamp slee e is possible by using he equa ion o
calcula e he con ac p essu e on he sha om igh ening he clamping join .
pH=
8·Mu
·(Ds
2)+( an(ψ+ϕ))·d2
2
π·dh·lh·( an(β)+ )(2)
The con ac p essu e
pH
ac ing on he sha is based on he clamping slee e as gi en
in he manu ac u e ’s ca alog. O he pa ame e s, such as he leng h o he slee e
lh
, sha
diame e dh, he angle o he cone β, e c. a e based on he 3D model o he clamp slee e.
To i he equa ion, i is necessa y o de ine wo mo e equa ions ep esen ing he
pa ame e s o he bol h ead:
Th ead pi ch angle:
ψ=a c anPh
π·d2(3)
whe e Phis a h ead pi ch.
Reduced ic ion angle:
ϕ=a c an z
cos α
2(4)
In Equa ion (4), a e adding he o he equa ions, we ha e h ee a iables: he axial
o ce in he bol
F0
, he coe icien o shea ic ion on he h ead
z
, and he coe icien o
shea ic ion on he conical and cylind ical su aces o he clamping slee e
. Due o he
numbe o a iables in one equa ion, i is necessa y o use he nume ical me hod o he
calcula ion. We used he nume ical calcula ion me hod in Ma hCad 15 om PTC based in
Bos on, MA, USA. The p ese ole ance in Ma hCAD was used; The ole ance = 0.001. Two
consecu i e i e a ions o he calcula ion mus no di e by mo e han 0.001. The solu ion
he e o e esul s in he ollowing alues and was applied o he igh ening momen om
ca alog 41 Nm: he coe icien o shea ic ion on he h ead
z=
0.1, and he coe icien o
shea ic ion on he conical and cylind ical su aces o he clamping slee e
=
0.13446.
The calcula ed o ce p e ension in he bol was F0=30,200 N.
These alues ound by he nume ical me hod we e e i ied by back subs i u ion in o
Equa ion (2), and he igh ening momen was calcula ed. The beha io o he FEM model
wi h he ollowing alues as pa ame e s was also e i ied: The co ec unc ion was checked
wi h espec o con ac p essu e. The con ac p essu e ac ing on he sha o he bo e in he
hub was compa ed o he p essu e speci ied by he clamping slee e manu ac u e in he
ca alog. FEM analysis se ings is he same as 2.4. E en bol p e ension is an o iginal o ce
om Equa ion (1).
Symme y 2023,15, 1893 5 o 17
The ca alog p essu e alue o he unmodi ied clamping slee e ac ing on he sha
is 147 MPa, and o he p essu e ac ing on he hub bo e, i is 95 MPa. These alues we e
he e o e compa ed o he alues o he a e age con ac p essu e ac ing on he su aces
o he sha o he hub bo e a e he ensioning o he slee e by axial o ces in he bol s.
The FEM mean con ac p essu e alues a e 144.4 MPa o he sha and 87.6 MPa o he
hub bo e. The con ac p essu e on he sha and hub bo e is shown on Figu es 2and 3.
The p essu e de e mined using FEM is lowe han he ca alog p essu e o bo h he sha
and he hub bo e, bu he di e ence is accep able. The inpu alues de e mined by he
nume ical calcula ion me hod a e co ec , he se ing o he FEM model wi h hese alues is
co ec . The se ings o his es model and he model used la e in his a icle a e he same.
Figu e 2. Value o con ac p essu e ac ing on he sha .
Figu e 3. The alue o he con ac p essu e ac ing on he hole in he hub.
2.3. The P inciple o Measu emen
To e alua e he uni o mi y o he dis ibu ion o axial o ces be ween indi idual
bol s, he axial o ce in he bol s o he clamping slee e is e alua ed. The axial o ce was
de e mined using wo s ain gauges placed on each bol o he clamping slee e, A schema ic
ep esen a ion o he s ain gauges on he bol and he bol wi h he s ain gauges glued on
is shown in Figu es 4and 5. To que w ench has an unce ain y o 5%, and unce ain y om
he measu ing o he axial o ce wi h he s ain gauge is unde 2%. An HBM s ain gauge
1-LY41-1,5/120 was used [
33
]. R
1
was used wi h a s ain gauge wi h esis ance o 120 Ohm,
and R2, R3, and R4we e eplaced by esis o s wi h 120 Ohm esis ance.

Symme y 2023,15, 1893 6 o 17
Figu e 4. Schema ic ep esen a ion o a bol wi h s ain gauges.
Figu e 5. Bol wi h bonded s ain gauges.
Due o he modi ica ion o he bol s, he calib a ion o he bol s con inued. Calib a ion
was ca ied ou unde ension on a ea ing de ice. Bo h s ain gauges on he bol we e
always calib a ed simul aneously. The alues de e mined by he calib a ion we e plo ed
on a linea cu e, as shown in Figu e 6. The calib a ion cu es we e hen mo ed o he
o igin, and he cu e di ec i es we e added as a cons an o he measu emen p og am o
each s ain gauge sepa a ely. The ou pu o he p og am ha was s o ed is he axial o ce
unde each s ain gauge. The mean alue o he axial o ce in he bol can be de e mined as
he a e age be ween bo h s ain gauges. The sepa a e connec ion o each s ain gauge also
allows us o de ec he possible bending o he bol s.
Due o he educ ion in he bol diame e o s ain gauges and o he axial d illing o
he cable en y, he igh ening o que and hus he axial o ce in he bol need o be educed.
The calcula ion o he educed o que wi h espec o axial and adial bol pene a ion is
gi en in [
31
]. The igh ening o que was educed o 20 Nm. Fo his igh ening momen , he
axial o ce, which is he bounda y condi ion o he FEM, was calcula ed using
Equa ion (1)
.
The axial o ce in he bol and he igh ening o que a e gi en in Table 1.
Table 1. Tigh ening momen and axial o ce in bol s.
Tigh ening Momen Axial Fo ce
Tigh ening o hal o que a he c oss 10 Nm ±5% 7368.7 N
Tigh ening o ull o que a he c oss 20 Nm ±5% 14,707 N
Fi s igh ening a ound he pe ime e o ull o que 20 Nm ±5% 14,707 N
Second igh ening a ound he pe ime e o ull o que 20 Nm ±5% 14,707 N
Measu emen s we e aken a es . The hub was clamped solid and ixed. The only
inpu load ac ing on he cons uc ion is he load om he clamping slee e p eload. Machine
componen s eplacing he sha and hub we e manu ac u ed o measu emen pu poses.
The componen eplacing he sha is basically jus he cylind ical su ace o he sha on
which he clamping slee e is placed. The hub is essen ially a hollow cylinde . The ou e
diame e o he hub co esponds o he minimum equi ed hub diame e acco ding o he
clamping slee e manu ac u e ’s ca alog. The ixing o he clamping slee e is shown in
Figu e 7.
Symme y 2023,15, 1893 7 o 17
Figu e 6. S ain gauge calib a ion.
The measu emen unce ain y ep esen ed by he de ia ion ob ained om he calib a-
ion is up o 2%.
2.4. FEM Model Pa ame e s
Some pa ame e s o he FEM model ha e been e e ed o in he sec ion abo e. This
chap e will be ocused on he se up o he FEM model o he clamping slee e.
2.4.1. Geome y and Mesh
In e ms o geome y, he model was designed o be as close o eal as possible. This
model uses modi ied bol s ha ha e a educed ou e diame e o bond he s ain gauges.
Fo he easons o meshing and con e gence o he calcula ion, he axial and adial d illing
on he bol s we e igno ed. The bol is shown in Figu e 8.
Symme y 2023,15, 1893 8 o 17
Figu e 7. Hub and slee e wi hou sha .
Figu e 8. Bol om FEM analysis.
The mesh has a majo impac on he con e gence o he FEM model. The bes con-
e ging mesh was c ea ed au oma ically acco ding o he p ese pa ame e s o Ansys
Wo kbench 2020 R2 based on Canonsbu g, USA Figu e 9.
To achie e he inal quali y o he mesh, only he e inemen o he mesh on indi idual
elemen s was used Figu e 10. No o he modi ica ions o he mesh in e ms o he meshing
me hods used esul ed in he c ea ion o he mesh o con e gence o he calcula ion. The
complica ions wi h he c ea ion o he mesh a e mainly due o he e y small a eas c ea ed
by he modi ica ion o he bol s o he ela i ely hin s uc u es on he ings.
The modi ied FEM model mesh has a smalle elemen size. The global elemen size
is se o 5 mm. The cones and ings o he slee e a e esized o 3 mm. The mesh has a
22,643 elemen s
and 84,721 nodes. I uses 9269 e ahed ons (10 elemen s);
13,190
hexahe-
d ons; 20 elemen s; and 184 pen ahed ons (15 elemen s).
Symme y 2023,15, 1893 9 o 17
Figu e 9. Au oma ically gene a ed mesh om ANSYS.
Figu e 10. Modi ied mesh om ANSYS.
2.4.2. Con ac s
The con ac s shown in Figu e 11 be ween he indi idual componen s o he model a e
e y impo an om he FEM analysis se up and o con e gence. Unde he bol head,
ic ional con ac was used wi h a ic ion coe icien o 0.15. On he cone and on cylind ical
su aces be ween he sha and slee e as well as be ween he hub and slee e, ic ional
con ac was used wi h a ic ion coe icien o 0.13446. Fo he bol h ead, bonded con ac
was used.
2.4.3. Bounda y Condi ions
The bounda y condi ions o he FEM model slee e espec he eal measu emen s as
much as possible. The load is de ined by he bol p e ension unc ion, as shown in Figu e 12.
The cons uc ion is ixed in space by a “Fixed Suppo ” cons ain de ined on he hub head.
2.4.4. Ma e ial P ope ies
Only s uc u al s eel was used in his analysis. The ma e ial p ope ies o he s uc u al
s eel is desc ibed in Table 2. The linea ma e ial p ope ies a e om he Ansys da abase.
Symme y 2023,15, 1893 16 o 17
indi idual bol s a e as small as possible. I would be ad an ageous o use he FEM model
o de e mine a be e igh ening p ocedu e. I is as e and cheape han bonded s ain
gauges on bol s. In addi ion, we can simula e he FEM wi h he o iginal axial o ce in he
bol s. A complica ion a ises by no in oducing some o he p ope ies o he clamping
slee e. No in oducing he su ace s uc u e seems o be essen ial. The FEM model shows
a ela i ely uni o m dis ibu ion o axial o ces be ween he bol s, al hough he esponse
slee e shows e y di e en alues o axial o ces in he slee e bol s.
Au ho Con ibu ions:
Concep ualiza ion, L.H. and J.S.; da a cu a ion, L.H.; o mal analysis, L.H.;
in es iga ion, L.H.; me hodology, L.H. and J.S.; esou ces, L.H.; so wa e, M.T.; supe ision, J.Z.;
alida ion, L.H., J.S., M.T. and J.Z.; isualiza ion, L.H.; w i ing—o iginal d a , L.H.; w i ing— e iew
and edi ing, L.H. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This a icle was suppo ed speci ically by VSB Technical Uni e si y o Os a a, esea ch
numbe SP2023/077.
Da a A ailabili y S a emen :
The da a p esen ed in his s udy a e a ailable on eques om he
co esponding au ho .
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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