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Shonan Institute of Technology

ShonanInstitute of Technology

MEMOIRS eF SHONAN INS7ITUTE OF TECHNOLeGY VoL 32,No. 1,1998

Stress

Analysis

of

Adhesive

Lap

Joints

of

Hollow

Shafts

Subjected

to

a

Tensile

Load

Yuichi

NAKANo*

and

Masataka

KAwAwAKi**

The stress and strain distributionsinadhesive lap

joints

of hollow shafts with dissimilardiameters subjected toan axial tensileloadare exarnined using an axisymmetric theory of elasticity. Inthe analysis, the jeintismodelled as an elastic three-body contact problem and the hollow shafts and the adhesive are respectively replaced byfinitehollowcylinders. Inthe numerical calculations, the effects ef the ratio of Young's rnodulus of the adhesive tothatof theshaft and ef thethickne$s of the adhesive on the stress distributions at the interfacesinthe

joint

are clarified. Itisshown that the stresses in radial and circumferential direction$ become singular at the ends of the interfacesand the stress increasesnear theends of theinterfaceswith a decreaseof Young'smodulus of the shaft and with an

increase ofthe thickness of the adhesive.

Key words: Elasticity;adhesive lap

joints;

hollowshafts; stress analysis; Yeung's modulus; axial

load;jointtensilestrength.

1.

Introduction

Transmitting shafts are usually connected together by using flanges which are attached by

means of keys and keyways and fastened

by

nuts and

bolts.

However, there are some problerns in

a flangetype shaft

joint;

forexample, a flange occupie$ a largervolume as compared with the size

of

the

shaft

itself

and

thus

the

weight of

the

joint

increasesand

the

stress concentrates at thecorner

of keyways during operation. On the other hand, ifadhesively bonded shaft

joints

are realized and

used forthetransmitting shafts, a power transmitting apparatus can be reduced insize, relaxation

of the stress concentration can be expected and

joints

composed of shafts of differentmaterials. such as metalicomposite combination shaft

joints,

can be obtained more easilyi}.

In

adhesive lap

joints

of shafts, the inner surface ef the outer hollow shaft isbonded to the outer surface of the inner solid or hollow shaft. Therefore. a wider bonded area than

in

adhesive

butt

joints

ofshafts can usually be achieved by increasing the overlap length of each shaft so that

the adhesive lap

joint

of shafts ismore suitable fortransmitting shaft

joint.

Lubkin and Reissner2)analyzed the stress distributioninadhesive lap

joints

of shafts of the

same material by thin-shelltheory for the shafts assuming the stress inthe adhesive as uniform

across itsthickness since the adhesive layer was very thin. Adams and Peppiatt3)analyzed the shear stress in adhesive

lap

joints

subjected to axial and torsional loads using a finiteelement method and compared with the results by Lubkin and Reissner2).Chon4),inhisanalysis, obtained

the stress

distribution

in

adhesive

lap

joints

intorsionwith adherends made of composite material assuming the stresses inthe adhesive as uniform across itsthickness. Hipo15) analyzed the stress

distribution

in

an adhesive

lap

joint

of metal and composite hollow shafts subjected to torsion using a finiteelement method, Lee et al,6) presented an approximate calculation method

for

the shear stress inadhesive lap

joints

subjected to torsion using a

finite

element analysis considering

*

uadiI\*l

Xnt

**

J<\eere

±

(thme)

ecE

uediIYgty

SIZut 9lii1O

H

11 HXN

(2)

-1-Shonan Institute of Technology

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ShonanInstitute of Technology

maMIN)S(\*eee eg 32 ts

eg

1

g

the nonlinear adhesive properties.

Reedy

and

Guess7)

performed the experimental study on the

tensilestrength and fatigue resistance of an

E-glass

composite-to-aluminum tubular

lap

joints

subjected totensile,compressive and

fatigue

loadings.

Itisclear from above thatmost of analytical

studies on the adhesive lap

joints

have been done assuming the uniform stress

distribution

across

the adhesive thickness. However, the stress and strain

distributions

in

theadhesive, especially near

the freeends of the adhesive where

high

stress and strain levelsare expected, usually have serious effects on the

joint

strength.

The purpose of thisstudy

is

toanalyze the stress and strain distributions

in

adhesive

lap

joint

of hollow shafts of differentmaterials subjected to an axial tensile

load

by

using an axisymmetric

theory of elasticity.

In

the analysis, the

joint

is

modelled as an elastic

three-body

contact problem

and the inner and outer hollow shafts and the adhesive are replaced with each finitehollow

cylinder.

In

the numerical calculations, theeffects of the ratio of Young's modulus of

the

adhesive

to thatof the hollow shafts, and of thethickness of the adhesive on the stress distributionsat the

interfaces

between

the shaft and the adhesive are clarified.

2.

Analysis

Figure 1shows theanalytical model of an adhesive

lap

joint

where an

inner

hollow shaft and an outer hollow shaft of

different

materials are bonded by an adhesive at an overlap

length

212

and an axial tensileloadP

is

applied at

both

ends of the shafts.

The

inner

and outer

diameters

and the

length of the

inner

hollow

shaft are

denoted

by

2ai,

2biand 21iand those of the outer hollow shaft

by 2a3,2b3 and 213,respectively.

Inthe analysis, the

inner

hollow

shaft, the adhesive and theouter hollow shaft are replaced by

finitehollow cylinders I,IIand

III,

respectively.

(The

inner

and outer

diameters

of the cylinder

II

are 2a2

(=2bi)

and

2b2

(==

2a3)).

Young's modulus, shear modulus and Poisson's ratio are

denoted

by

e

G and y,and the subscripts 1,2 and

3

designate

the

finite

cylinders

I,

II

and

III,

respectively. A coordinate system used

here

is

such that;r represents the radial direction,

e

the

hoop

direction

and g thelongitudinaldirectionof

the

joint.

The

origins of each

finite

cylinder are

denoted

by

Oi,

02

and

03 and the distance

between

Oi

and

02

is

denoted

as

Ci

and thatbetween 02 and 03 isas C3.

The boundary conditions of theadhesive lap

joint

are expressed as

follows

Eqs.

(IH5);

on the

inner hollow cylinder I,

213 1

Adhesive[ll]Hollow

--shaft[I] 8" .P.

8epapte

8-oJ '1 T- -y =pt-N

ft-

.. ・-2h

Hollowshaft[Ill]

2ii

Fig.1. Analyticalmodel of an adhesive lap jointof hollow shafts.

(3)

Shonan Institute of Technology

ShonanInstitute ofTechnology

StressAnatysis

of

Adhesive Lop foints

of

HbllowShaftsSubfectedtoa TlensiteLoad

zl=+ll zl=-ll

lzi1gti

: o!=Se==P

(l,

-cD Sz, Sl, :d= z(bi-al)

d.==O

:

(aDr--ai=

:

(oDr=bi=

,tl.=O

(Tlz)r=ai==O

(T

lrz) r=bl = O

(1)

on the

hollow

cylinder II

(adhesive),z2=

±l2: aLi=TIL=O

on the outer hollow cylinder III,

z3 :=+l3 : aLii=T-=o

a3==-l3 : aiii=S6=

n(bgP-ag) ,Tilj=O

-t3SZ3S(C3-l2):

(alll)r-a,=(Tilil)#-a,=O

lz31Kl3

:

<a;ii),-b,=(TIi.i).-b,=O

at the interfacebetween the cylinders Iand II.

-tiSZiS(l2-Ci),IZ21Sl2:

(Oi)r-b,=(OIJ)r-a,

(Tiz)T=bi=(TIIz)r=a2

(uf)r=bi=(ulJ)r=a2

(

0oWz,L

).-b,

=(

aaWz

3i

).=.,

and at

the

interfacebetween the cylinders

II

and

III,

lz2lSl2,(c3-l2)Sz3gl3:

<al[).==b2=(alii).-.,

(Tl:}r=b,=(TIizi)r;=a,

(ulrl)r;b2=(UIII)r=a3

(

0drW2i

)r-b2

=(

ObeWIiT

)r=a,

(2)

(3)

(4)

(5)

where u. and w. represent the displacements inr and z directions,respectively and superscripts I,

II

and

III

designate

the

corresponding finitecylinders I,IIand III,respectively. Particularly,since

the origins of each

finite

cylinder inthe longitudinal directionare fixed on the

different

points as shown

in

Figure

1,

the

derivative

of the axial displacement w. with respect toz should be equal to

each other at each interfacebetween the cylinder Ior IIIand the cylinder II

(adhesive>,

as expressed

in

Eqs.

(4)

and

(5).

From

the

differentiation,one of the unknown coeMcients inthe following stress

function iseliminated, and itcan be

determined

from the equilibrium condition of an axial

force

applied to the

joint,

Michell

stress function ¢

(n

z) isused inthisstudy in order to satisfy the above boundary conditions

Eqs,(1)-{5).

Taking

the

boundary

conditions intoconsideration,

Michell

stress function

¢ i

(L

zi> forthe hollow cylinder Iisexpressed as

Eq.

(6)

from the solution of variable separable of

biharrnonic

function8)'

(4)

-3-Shonan Institute of Technology

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Shonan  工nstitute  of  Teohnology

where ,        湘南工科大学 紀 要  第 32 巻 第 1 号        Φ1(r, z1)=Φ

5

十 Φ{十 Φ

L

十 Φ

k

十 Φと         Z3       Z72 Φ

Ab

 

6

−+跏 1・9・+C

 

 

 

・・1− ・・1)・4・

48

・1 ・・2−・… 1+

1

・〆}

… 1・9・− 1・ ・・一

[lb・・・・…

ll

・・

9

K

・・

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・…

Kl

・・ ・・…

es

・・1)

 

 

1…

1

ll… +K ・・ ・・・… 朏 ・・・・ ・… ,)

 

 

1

・・ ・… h ・劇 ・ ・・・ … h ・・

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[1・込・・

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… 翫 K 凶 ・・・ ・… ’ ・i         

 

 

・・…

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1

・・)・ … i… 鴫 )… h ・劇 ]C…

1

・r ・・一・

1

、、・

1

・・・・… ’ ・r

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’ ・…

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K

・瀞 ・・・・…

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・ ’ ・・〉

 

 

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・觚 ・・編 …

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・… 1)・・・・ … 鴫 … h ・・

1

・,)]C…

1

・・        コ ・ト

15

・丞

li

B

・・

ll

・丞

1

K

E

・ ・

… (

B

・Zi          ・

1

BL

 r  

llr

K5r

 

ts

 

b3

・麟 1

・・…

h

・劇 ・ ・…   h

lli

h

贓 △

k

= △

x

一 β監=

1

里,

 

β

X

− = (2n − 1)π n =

L2

,3,...)     Ila     Ilう        11α 一1〔瓰十       β。α1        一

lo

11δ 。十       β。

bl

  ll      211 2 (1一レ1lla+βnaiIOa 21一ン1)

llb

βnbiIOb − (1 − 2ン11(la一βnallla − 1 − 21IOb β n bi lib 4     一Kla     −

Klb

       

Kla

− Kαa一         β。ai        − KOb 一

Klb

    β。

bl

21 一ン1)Kla 一βnaiK 〔la 2(1 一ン1)Klb −

fin

 b1 KOb (1 − 2ン1)

Ko

α 一 βnalKla (1 − 2レ1)KOb 一βn b1 Klb (

6

) N工 工一Eleotronio  Library  

(5)

Shonan Institute of Technology

ShonanInstitute ofTechnology

StressAnalysis of Adhesive Lap

foints

of

HbllowShdis Suhiectedtoa TensileLoad

si:

=-

itL'-,

Ior=Io(BL

Ioa=Io

Cl3ka!),

I

Iob

:=

Io

(B:

tu1--

of

(vt,

li,r,)=- 2visinh(rslt)-r,l

ldi2=toLCvhl"r.)=-2vicosh(r.li)-r.t tok=ulk(yi,ti,r,)

to1=toh(yi,li,r,) C.(rsr)=

Jo,Ji,

Yo and Yi are Bessel functions and Ie,Ii,Ko and Ki are modified Bessel functions of orders

O

and 1 respectively, and rL

is

thes-th positiveroot ofequation

Ci

(r:ai)=O.

Ah,

Bb,

C5,

D5,Eh,Ak,

BL,

CE,

...,

Ai.',

Ek'

and

ek'

(n,

s= 1,2,3,...)inEq.

(6)

are unknown coefficients. Similar stress functions Oii

(ny

z2)for

the

hollow cylinder II,i.e.forthe adhesive and ¢ M(r z3)

for

the

hollow

cylinder

III

are chosen by changing correspondingly the coefficients involved inEq.

(6).

The stresses and the

displacements

in

each

finite

hollow

cylinder can

be

obtained from

Michell

stress function and are written as Eq.

(7).

IIa

2(1-VDIia+BAailaa

-K;a

2<1-VDKia-BAaiK6a

If,

2(1-v,)Il,+BAb,I6, -Kl,

2(l-v,)K{b-BAb,K6,

-I6a+

ii3,

-(1-2vDI6arB:ail{a

-Kda-BII,[",

(1fi2vi)Ktu-BfiaiKi.

-I6,+

sl,SZ,

-(1-2v,)I6,-BAb,Il, -K6,-

B:,l(bb,

(1-2v,)K6,-BAb,K{,

--9,(rk)=sinh(rEti)cosh(r:ti)+rkti, 9L=9k'==9L'(rE)=sinh(rLli)cosh(rili)-r:l

r),IO=Io

CBI

r),Ii.=Ii

caLr),

I{.=Ii

avKr)

6a=Io

asL'

ai),Iia=Ii

osLai),

IIa=Ii

ueKai)

b,),

I6,=I,

CB:'

b,>,I,,=I,

C6L

b,),

I{,=I,

ueL'

b,)

i i ]icosh(rLID i i iisinh(r:ID T =(1-2pDcosh(r:li)-rLli sinh(r!ID i =:(1-2yi)sinh(rkli)-rklicosh(rLli)

J.(r.r)-Ji(T,bi)Y.(r,r}iYi(r.bi>

Cn=O,

1) q= ae= ole= Qr= Ur`= wi

2;}

(vv2o-

O,2,9)

£

(vvz

¢ - 'lill

l

¢ r

)

£

{(2-v)v2o-

Oa2z9}

g{(i-v)v2

¢ - 0,2.9} 1 02¢ 2G 0raz 21G

{(1-2")V2

¢ +

6o2.9

+1rooOr}

(7)

where,

V2v2

¢ =:o,

(v2=

oOr2,+

;

Infinite

simultaneous

linear

equations with th

oOr+

IS22).

e unknown coefficients are

derived

by

(6)

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ShonanInstitute ofTechnology

mamrptj(\seee

eg

32

g

eg

1

e

ing

the

stresses and displacements into the boundary conditions expressed in Eqs.

(1)-(5>.

(see

Appendix)

In

the numerical calculation, the number of terms

in

theseries

in

the

infinite

simulta-neous equations istruncated by N. Then the unknown coerncients are determined by solving the

(36N+15)

simultaneous equations and,

finally,

the stresses and the

displacernents

in

the

joint

can

be calculated from Eq.

(7).

3. Results and Discussion

In the following calculations, the number, N, of terms

in

the series of the simultaneous

A,C:leadedend B,D:un-leaded end 2.0 1.5 1.0iS o.ses o --O.5--1.0-1,O

-O.95 -O.9 -O.B5

z2/l2

near theun-loaded end D 4.8 2,O 1.5 1,Ome)B o.sbca n.s-1,OO,B

O,85 O.9 O.95 z2/l2

near theloadedend C

(a)

Innerinterface

(I-II)

1,O .e>8bop 1,5 1.0 O.5 o 4.5

-1.e.O,95 -O.9 .0.85 -O.8

Z2/t2

near the loadedend A

L5 1.0me)g o,sca o -O,5,

O.8 O,85 OS O,9S 1.0

z2tl2

near theun-loaded end B

Fig,2.

(b)

Outerinterface

(n-M)

Stress

distributions

near the ends of the interfaces

(Ei==E3,Vl=V3,

SO= Pn(b?-al))-

(7)

-6-Shonan Institute of Technology

ShonanInstitute of Technology

StressAnalysis

of

AdhesiveLop

foints

of

HOtlowShoftsSubjectedtoa 7lensileLoad

equations,

is

taken as

80

and the

length

2tiof the

inner

hollow shaft isthe same as the length 2t3

ef the outer Qne and all stress distributionsshown below are inthe adhesive.

Figure

2

(a)

shows the

distributions

of thestress components o,,ob, a. and T,,near theend of the

inner interface

(r=a2)

between the inner hollow shaft and

the

adhesive and

(b)

near theend of the

outer interface

<r=b2)

between the outer hollow shaft and the adhesive forthecase where the

joint

iscomposed of theshafts of thesame material and issubjected toan axial tensileload. The abscissa

denotes the axial positionat the interfacewhich isnormalized by halfthe length 212of the adhesive

and the ordinate denotes the stress component which isalso normalized by the mean tensilestress so applied at theends of the

joint,

Inthiscase, the

inner

and theouter radii of thehollow shafts are set so that the mean tensile stress so at one end of the

joint

isthe sarne as the mean tensilestress s6 at the other end.

Frorn

the results, the stresses q and ob

in

the radial and circumferential

directionsbecome tensile at the loaded end of the interface,i.e.at z2tl2= 1,Oin

(a)

and z2112= - 1.0in

<b)

and compressive at the opposite unloaded end of itand

increase

steeply at

both

ends of the

interface').

On

the other

hand,

both

thenorrnal stress a.

in

the axial

direction

and shear stress T,,

becorne largenear the loaded ends of the interfaces,and almost uniform along the rniddle section

of the

interface,

Figure 3shows theeffect of overlap

length

on the

distributions

of

the

stress components at

the

inner interface

(r=a2),

It

isseen that the stress distributionsare dependent on the overlap length and as the overlap length isdecreased, as shown

in

(b),

the stress components take

high

values especially near theloaded end of the interfaceand both the stresses a. and ae are compressive along the middle section of the interface,while they are almost zero

in

case of

long

overlap

length

as shown in

(a).

Inthe

following

numerical results, thestress

distributions

at the

interfaces

are shown

in

terms of normalized Mises equivalent stress aeqiso.

Figure

4

shows theeffects of the ratio of

Young's

modulus of the

hollow

shaft tothatof the

adhesive on the norrnalized Mises stress distributionat the

interfaces.

In thiscase, theinner and

the

outer radii of

the

hollow

shafts are also set so

that

themean tensilestresses so and s6 are the same,

(a)

shows thestresses near theloaded end of the

inner

interface

and

(b)

shows those near the

loaded

end of theouter interface.Frorn theresults, when the hollow shafts are made of the same material

(shown

by

thesolid

line

in

the

figures),

theequivalent stress at the

loaded

end of the

inner

interface

(z2it2=1)

is

largerthan thatat the

loaded

end of the outer

interface

(z21t2=

- 1);therefore,

.e)8uca

O.8O.6O,4e.2

o-O.2O.4rO,6

-1.o -o.s e o.s

z2/l2

(a)lill2=2

1,O oq$otsco O.8O.6O.4 O.2o.O.24.4 n.6 -1.0 ro.5 O O,5 z2/l2

(b)l,/t,=4

1.0

Fig.3. Effectof overlap length on the stress distributionsat the inner interface

(Ei=

E3, vi=:v3).

(8)

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ShonanInstitute of Technology

maMIptJ

±(#e-

eg

32

k

ee

1

e

3.5 3.0 2.5

qe

2.0 eg 1,s 1.0O.5 o 3.5 3.0 2.5.e 2.0 xbg 1.5 1.0O.5 o

o.gs o.96

O.97

O.98 O,99

1.0

-LO -O.99 -O.98 -097 -O.96 -O.95

z,/l, z,1t2

(a)

Innerinterface

(I-ll)

(b)

Outerinterface

(II-III)

Fig.4. Effectof Yeung's moduli ratio between the shaft and the adhesive on the Mises equivalent

stress distribution near theend of the interface.

2,5 2,O ..o 1,5x ye 1,o O.5 oO.95O,96O,97 O.98

zzfl2O,991,OFig,

5. Effectof thickness of the innerhollowshaft

on the Mises equivaLent stress distribution near

theend ef the innerinterface.

a fracturemay initiateat the

loaded

end of the inner interfaceas the tensile load

increases.

When the

hollow

shafts are made of differentmaterials

(the

dotted and

dashed-dotted

lines),the

equiva-lentstress islargeatthe loaded end of the

interface

between the hollow shaft of a smaller Young's

modulus and the adhesive,

i.e.

at z2tl2=1 shown by the dotted linein

(a)

and at z2tl2= - 1shown

by

the

dashed-dotted

linein

(b).

Frorn these results, the tensilestrength of a

joint

where an inner hollow shaft of a small

Young's

modulus

is

joined

toan outer one of a large

Young's

rnodulus

(Ei:

E2:E3=5 :2:1Oas shown

by the dotted line)issupposed to

be

smaller than that of the

joint

of shafts of the same Young's

modulus.

On

theother

hand,

the tensilestrength ofthe

joint

where an inner hollow shaft of a large

Young's modulus and an outer one of a small Young's modulus are

joined

(Ei:

E2:E3=1O :2 :5as

shown

by

the

dashed-dotted

line)

is

supposed to becorne larger,

if

the equivalent stresses at the

loaded ends of the inner and outer

interfaces

are equal toeach other. Moreover, frorna comparison

of the stresses near the

loaded

end of the

interface

z2ft2== 1 in

(a),

the dashed-dotted lineisslightly

smaller than the solid line. Therefore, itmay be possible that

by

choosing an appropriate combination of shafts of

different

Young's

rnoduli, the tensilestrength of the

joint

of shafts of

different

materials could exceed that of the

joint

of shafts of the same rnaterial of a

large

Young's

modulus.

(9)

Shonan Institute of Technology

ShonanInstitute of Technology

StressAnalysis

of

AdhesiveLmp

Jbints

of

llbtlowShdis Subjectedtoa TensileLoad 2.5 2.0 .e 1.5 -"s gb 1,O O.5 o euts.. gb 2.0 1.5 1.0 O.5 o

O.95 O,96 O.97 O.98 o.gg 1,o 1,O -O.99 -O.98 -O,97 -O,96 -o,9s

z,1l, z,1l,

(b)

Outerinterface

<ll-III)

(a)

lmer interface

(I-ll)

Fig.6. Effectof thickness ef the adhesive on the Mises equiva!ent stress distribution near the end of

the interface,

Figure

5

shows the effects of the thickness of the

inner

hollow

shaft hi

(=bi-ai)

on the equivalent stress distributionnear the loaded end of the inner interface

(r=a2).

In thiscase, the

inner

and outer

hollow

shafts are made of the same material and only the

inner

radius ai of the

inner hollow shaft isvaried while the other dimensions of the shafts and the tensile loadP are kept constant.

The

equivalent stress near the

loaded

end of the

inner

interface

is

larger

than that near

the loaded end of the outer interfaceinthe case where the

joint

iscomposed of the hollow shafts

of the same material, as shown

in

Figure

4

(solid

line);

however,

by

increasing

the thickness of the

inner

hollow

shaft, the equivalent stress near the

loaded

end of the

inner

interfacedecreases and

the tensilestrength of the

joint

increases.

Meanwhile,

the effects of the thickness of the

inner

hollow

shaft on the equivalent stress distributionat the outer

interface

are negiigible,

Figure 6 shows theeffects of thethickness of theadhesive h2

(=b2-a2)

on theequivalent stress

distributions

near the

loaded

ends of the

inner

and outer

interfaces

when the

joint

is

composed of

the hollow shafts of thesame material and the inner and outer radii of each shaft are set so thatthe mean tensilestresses so and s6 at

both

ends of the

joint

are thesame.

In

the

figures,

the thickness of the adhesive h2isnormalized by thethickness of the inner hollow shaft hi.From the figures,the

equivalent stresses near

both

loaded ends of the interfacesdecrease as the thicknessof theadhesive

becomes smaller.

4. Conclusions

This

study

deals

with

the

stress analysis of adhesive

lap

joints

of

dissimilar

hollow

shafts

subjected toan axial load. The results obtained are as follows;

1> An exact analyzing method

for

thestress and strain distributionsinthe adhesive lap

joint

of

hollew

shafts, which

is

modelled as an elastic three-body contact problem, is

demonstrated

using an axisymmetric theory of elasticity,

2>

The

effects of theratio of

Young's

modulus of theadhesive tothatof the

hollow

shafts, and

of the thickness of the adhesive on the stress distributionsat interfacesbetween the shaft and the

adhesive are clarified

by

numerical calculations,

As

a result, the radial and

hoop

stresses are singular at the end of the interfaceand the Mises equivalent stress near the end of the interface

increases with a

decrease

of the ratio of

Young's

modulus of the shaft tothatof the adhesive and

with an

increase

of the thickness of the adhesive,

(10)

-9-Shonan Institute of Technology

NII-Electronic Library Service

ShonanInstitute ofTechnology

maMI*gd<\rest

eg

32

g

eg

1

e

3)

In

the case of

two

hollow

shafts made of

the

same rnaterial, the

Mises

equivalent stress at the

loaded

end of the

inner

interface

is

larger

than thatat the outer

interface

and as a result the

fractureinitiatesat theend of theinner interfaceof the

joint.

Inthecase of two hollow shafts made of

different

materials, the

joint

tensilestrength can

be

expected to

become

larger

when the

inner

hollow shaft of a relatively large Young's modulus

is

bonded

tothe outer one of a small

Young's

rnodulus.

References

1) A.Beevers,Mbtex Sct',and 7;2ehnoL,2.97-102

(1986).

2)

J.L

Lubkin and E.Reissner,7lans.,ASME 78,1213-1221

{1956).

3) R.D.Adams and N.A. Peppiatt,J Adhesioag9,1-18

(1977},

4) C.T,Chon,J CompositeMatex, 16,268-284

(1982),

5) P.J.HipoLJ Conrpositeimtex, 18,298-311

(1984).

6) D.G. Lee, K.S.Jeong and J.H.Choi,J Adhesion. 49, 37-56

(1995).

7) E,

D.

Reedy,Jr,and T.R,Guess,intl.

J

imcture,63,351-367

(1993).

8) T.Sawa, R,Sasakiand' M. Yoneno,ASMEJ PtessufeVesset71echnoL,117,298-304(1995).

Appendix

Stressesat the boundaries

(r=a,)

ai==Asv,+-

el-O

+cs(2p,-1)+,zO=e,cg'

C5r/i,[j,"')

/

ee oo

+.Z=,[Ak+,Z.-,CkPL,+,ZO=e,C:'P:g]cosuekz,)+.ze=O,[Xs+,zO.q,egPk,+,ze=O,e:'Pk]sintsk'z,)

Tl.== -Db(1 +vi)ai+E5

(1

-Yi)

+

:li

AX

sin

esIzi)+

:li]

Ak'

cos

esL

zi)

al n=1 n=1

(r=bi)

al=A5pi+ ,5,+c6(2vi-1)+.ZO=e,C:'

C5r/Ii,b')

+i,[Bk+S.,elQ},+tff,ek'QkZ]cosuekz,)+.xO=O,[Ek+,xO..D,eQL,+,ze=e,61'Qkk],i.esL'z,)

Tlr=-D5(1+p])bi+E5

(1

I;,Vi)

+.Z"!O,BXsinesXZi)+.ZO=O,Ek' COS

esLZi)

(r=a2)

o;i- -lt'

{Agv2+

:ll-i

+cg(2v,-o+,zOtO,eli' C5'r/f,ii,a2)}

+.ZOO]=,

Rikii

{A5i

v2+ -XIi+c5r

(2v2

- 1)+.Ze=O,C:i'

C5`

r/I,

ki,"2)

}

+

.zO.D.,ak-.ii

[A

-,

+ ,xD=O,e

r

P-,

.+

.E.:,

Cy'

Pk,

g]

+

.zD=O,ak-.ir

[A

h,

+ ,zO=O,6y p'iE, + .

£

co.

,esi

'

ih'

L]]cos

uak

z,)

+

.zO=O,

[bL-iT

{Ag

v,+ -:}1-Oi+

cdr2v,

- i)+ ,ze=:,

ey'

C5i;f,

Eii,"2}

}

eo ee

+

.Eca]=,bL-.ii

[Ag

+

tff

,eki

Ph,

,+ ,ZO=O,e:i '

P-g]

+ .Z= ,bL-.ii'

[A

h,

+

i-,6Si

Pi.,

,+ .S,

eki

' P.thg]]sin

(B

:'

z,)

(11)

-10-Shonan Institute of Technology

Shonan  工nstitute  of  Teohnology

(r=b2

(r=α3)

(r;b3)

(z=±‘)

   StreSS・Analysis of  Adhesive LaP 

Joints

 of  HollOW Sha丿「ts Subected  to a Tensile Load ・聾一

Dbt

1

+・・・・…

5i

( 1

21

貿

 

 

 

− D5i・

1

・・・… +・

5i

Llit2uev2

bk−・1 +

、 覦 厂+

1云蕾 ・

 

 

 

D5i

1

・・・… +・

911

レ2)

L

−II・

1入隙 +

1云

… (tllL・・〉 ・・一

・・…

C5i

・… − 1・・

9TilfXlg

) i (

f

ii

, b2)

 

 

 

1

が 1

… v ・+

・・

5i

・… − 1・・

C

21

    ca       co       co      co      oo

 

 

 

1

。臨 ・

Σ

6

1

且島 ・

1

δ撃

1

… (

Bkiiz

・)

 

 

 

・・i・・+

・ ・

6i

・… − 1・・

91

eVPI

lii

, b2

 

 

 

1 ・

B

盆・

    co      ua      DO      DO      Pt 昨

c

α’

・鼎

δ

Bkii

’・・) ・:・

Dbi・1・・・… + ・・ (1

レ2)

 

 

 

− D5i・

1

・… b・+ ・

8

(1

、 ン2

b

・i−・

1 ・噐珊 ’+ .

i

°  , 

Bts

,  ’

bkithiii

… 厳 ・

 

 

 

一・

5i

1

+・・… +・

bi

1

2)

rll

・器1「

1

… (

B

:ii… ・・II− A6ii・

・ ・

bll

(… − 1・・

・・厂

C8

g

 

 

 

1

CIII

1「

P

1

… (

BhL

’z・)・

1

i

’匸 +

こ煦 §

eSk

’” ・ ・〉 ,影 一D5ii1+ 。、)。,+畊 (1 一レ・) +

A

罸・si 。但

HI

・ 。 、)+

酔 c。 、 

eS

Lii

 z)        α3     n≡l      n−1 ・

II

』 Abii・

II

C6ii

− 1・・

C

3)

 

 

 

1

B

; i・

α・

Q

:・

1

α1’

Q

eSE

…)・

1

δ

IH

δ野 旦

uaLi

” ・・) 。!

Li

− − D 琴11+ 、)わ,+Egl (1 − v

BH

・si。 

aVki

・ ’ 。 、)+

罸・c。 、 

Uaki

・。 、)        ∂3   ・司       ・−1 σ。=

Ao

1

一 ン)十

2Co

2

− v)±

2Do

1

十レ)

1

 

 

 

    ca      @      oo      ロ       oo        ロ      ロ       co       ロC、 +ΣA 。R,。 +ΣB 。S、

n

±C。±ΣA 。R、

n

±ΣB 。S、 、    n三1           口‘l                n;1          n二

1

]・・r・・           − 1

(12)

Shonan Institute of Technology

NII-Electronic Library Service

Shonan  工nstitute  of  Teohnology

      湘南工科大学 紀 要 第 32 巻 第 1 号 T・…= 

D

・・

1

1

C1(r・・r) 一

。 }C・(… )

 

 

    co      ec       pm       co        ロ    ロ       @コ     コ±

Cl

±Σ

A

R

ζ。 ±Σ

B

S

‘。 +

C

‘ +Σ

A

R

ζ。+Σ

B

S

ζ 、       n ≡l           n =1              n=1         n耳

1

]C1(r・・

r

wher

C

   戸 。− 4− 1>

nr

・βksi ”h2(・・のC・(γ α)    孟, 一

Q

・・=  Ω、♂(γ 9 β監)・ Q − 2(−1)nγ・

C

(・・わ)

Qli

。 R 。

G

n

R

=       、、

1

, ,

,,

 ・氏一一1肇至

C

°

’  

a)    

2

( 纛 藩α){・・

1

・辮罸…

h

r

…h・r ・

臨 欝{ T1γ

1

・岬・

h

・r・ }.    4 ( 一1)

n

γ・β晝・i・

h2

(・ ・

C

γ ・わ)

Q

。 。一 一

4

1

)nr ・B:2c°

s

r

・の

C

・(・・う)

 

 

       

 ’               

l

γ詈+β益

2

2

t

、,,., ,,侮’づ

h

r

h・

r

・ ・

2(纛i牆δ

1

{ ・

2s

・・

hr ・

h

     

2

(−

1

n

§

A

     

                   

2

(−

1

) n 蠢B 五皿{ウ2C 叙γ、わ)一α

2

  C (γs α)}(γ欝+f ) ’              @     ー

2

(− 1 )nSA ;   S ・

S

・・;ム。{δ・C召( γ 、 )一。・ C ぎ(γ、α)}(,詈 + β孟・) 五

n

{b2 

C3

( s わ)一α2C 召(γ 、 〉}(γ詈+β鬢             ロ         ー

2

1

nSBn

b2C

(γ

sb

a2C召(γ α )}(γ +β

2

)     2 (−1)

n

S

ξ 。; ロ  Sl。= 2(−

1

)ns も   △’

n

b

@C

召 (

r

. 

b

a2C

召(γ、

a

) 〔 γ尋+ β

fi2

) …i − ,、、誰論。},,,.,, , ・

忍{δ

2C

γ、わ)一 α 岩(γ, α)}(γぎ+β孟 2     −

2

(−1)

n

§ 1{

b2C

言(γ 、b)−a2C呂(γ、a)} ( γξ+β罸)

iSPIacen

ts

 

at

 

the

 

bo

u

ari θ

S

( r

b

u

お1( 旦 +

c

もわ

1

ゐ 1 ) 一 罍

12

謡 1

− 1

・ …

a − ・…

16

b

…+ KI ・・…

m

+・…臨 ・ttm ・ 一 盞

1

c1

1 ・識・一

i

・ ’・

bll6

・ ・

m +

K

・ …

K

幽    

、・

 

1

瓦 囎

即 駐呂入

r

1B 凶 瑚 ・ 茎 1 ・ ・ 【 ・ [ …1 碯 講 ・冗飜 麟 誉 臨一毒1且 ・▽・ ・ ]                        

(13)

Shonan Institute of Technology

ShonanInstitute ofTechnology

(r=a2)

(r=b2)

<r=a3)

StressAnatysis ofAdhesive LaP

Joints

of HotlouiShaftsSubl'ectedtoa TensiteLoad

+

.ZO=S,.i.O.,

[e

k--ii

[AhVV

h'

+

BkVi-'

+ ,ZO=O,A

l'

Ok'za

+ ,ZO=O,Bk'

VL'-'

]

+ein-thi'[D6vibi

i}1"I311.)tM,

+ALI.Wk'+EL,Vi.'-,Z"=ab,AL'(rV.b-,ZO=O,R'Vl'Rl]]cos(BLiz2)

+.Z".".,.Z";',[fk.ii[Ai.W-h,+BhVk,+,Zco..,AVOk'.,+,:e=C,BL'VL'k]

+fLTth'j[Di,p,b,

g[-iEit,1,)IM,

+?lklXrk'+EtsVla'-,zca=,Al'C

k'.,-,zD;:,Ei,'Vk'.,]]sincBg'g,)

2b2

(:ii

+cga,)

7 2}, .ZO=e,

AAi/V,SB-.,,・1,>l:

[rl{tnaA'lm-Bi."a2I6,..2S9.+Ki..A?.+Bk"a2K6..A't.]

-2b, .i.O.,

BAi

,5i,thJ12)t",

[rIimaZS'bmrBim"a2I6maA';m+K;maASm+BU'a2K6maA'Sm]

+

:;i

[AkiWkia+B:iVLla+

IiZ

Ai.''ek".a+

:ii

BU'Vh''.a]cos(Bkig2)

n=1 m=I m=1

+

.ZOtO,

[D

5i

v2a2

G2

i-,.,

1・

,};, +ALi17V

kla

+

ELi

VLla

-

iOO.].

,Ai."

ek"."

-.:O=b,Ei.i'V%r '.a] sin

ua

i."z,) u;i== -2},

(

71"

+cg b,)

L

tGl

.i=,

iAiLii.,'Si3i,it,・1,)t:

[-I(mbA'lm-Bi."b2I6.bAe.+KI.bhLT.+Bll'a2K6.bA't.]

-2t},.i.O., BA-E'.ISigi,i,・1,>l",

[-I(.,A'l.-Bi.,'b,I6.,Az.+K:.,tsz.+Bk,・b,K6.,A,s.]

+.Zna=,[AkiWLi'+BLiVllb+.ZO=O,Aim''Oh"n'+.i.Or,Bh''Vh''nb]cosuekig2)

+ .Ze=O,

[Db`

v2b2

G2i-.,,

l,)l",

+

71

ki

VV

kib+

Ek'

Vi.ib

- .Zb=e,All'

ek".'

-. £ OLe. ,Bi."

S7h'

'.b]sin

us

ll'z,)

u;ii--

2iG,

(Xgii

+cb"a,)

-.ZM=, 2GA,

E2'si

'

£

,il.,IM・2t,

[-I{ma

A'lm=BXi'a3

I6ma

2Sern

+K{..

Aff.+Ig-i

'a3

K6..

A't.]

'

.Ze=;,

2GB,Mill'l

t,il・iii.i>i:z

t,

[Ji{ma

Aqzm

-Bim'i'a3i6..

tsZ.

+K{..

AZ.+B:'i'a3

K6..

A's.]

+.]Ze"]=,ebHii[A.UiWXia+Bi."VX,la+,Z'=O,A.Lii'Oi,iKa+,Z'=C,BEi'VkiL'a]

+

.ZOO.].,ebHii '

[D5ii

.,a,

G2,(B-,thl.I:,

+

AX,i

W:,la

+B ,i

V

,la

-,Ze=',A.kll'

eki-'a-

,£ '.O.

,Ekii'

VkiKa]

(14)

-13-Shonan Institute of Technology

NII-Electronic Library Service

ShonanInstitute ofTechnology

(r=bi)

(r==a2)

(r=bz)

(r=ag)

vaptIrvJi(7*ept

ag

32

g

as

1

g

+

.ZO=O,.ZD=",

[e

llthiii

[A

-,i

VCX

X,la

+

B

,i

V:la

+ ,ZO=O,

ALii

'

O

kiL'a

+ ,=e.e.,

Bffi

'

VLih'a]

+effthiii'[Duiv,a3

G2i-..1.,)Ml,

+

A:i

S]Vula

+

E:i

Vi.,la

-

i.O.

,Age

(rllh'a

-,z"=ePffi'

i7LiL'a]]cos

ua:iz,)

+

.ZO=C,.Ze=e,

[f

:imiii

[A.

X,i

Wli,la

+

B:,i

Vi.,la

+

,Ze!O,

AIii

'

Okiin'a

+

ieO]=

,

BJ,ii

'

WL'a]

+fLithiiT'[D6`iv3a3

G2,(B-..l,)Ml,

+

ii

lli

Srcr:la

+

EX'i

VX'la

- ,Zco=,Akii'

i!IL`kl'a

-,ZO=O,ELii'

S7ltiSi'a]]sin

cBll'z2)

Oat.VZ

=Pco=,[-. £ a=e,c:-.ii'

E;7Bl.,)Ml,

{A5(1-2v,)+4c6(1-.,)}-.ZO=O,ck-.iiD5

21;iBl.,)rn

+.Xe=e,ck-.ii'[,Ze=b,ALELb.+,ZO=",BkFLb.-,Ze=O,CIGkb.+Bi.'Fh'b+,Z"=O,el'Glt'.,]

+

.Ze=e,.L-.ii

[,Ze=O

,Alt

Ekb.

+ ,ZO=D,

BL

itb.

+ ,Zco=,6lt

di,b.

-

Ein'

Fh'b

+

,Zco=,

ct'

GL'.,

]]cos

(Bh,

'z,)

+.Ze=",[-.]EOO..],dS'th''

iili,.)I,

{Ab(1-2vi)+4ch(1-vi)}-.i.,d:-.iiD5

2iliii21.,)rn

co co

+.Z=,dk-.ii'[iL,AftELb.+,£e=O,BftFLb.-,Zco=,CLdLb.+E:FKb+,Ze=',CiL'Gk'.,]

OO DO

+

.£= ,dk-.Ti

[,Z=

,

71

ft

ELb.

+ ,ZO=O,

EL

FLb.

+ ,Zco=,elt

Glb.

-

EK

FL'b

+

,ZD=O,

ek・

av.,

]]

,.,

os

h,

.,)

{II:a

=-.Zco=,

tiilBl,l"l,

{A8i(1-2v2)+4cs(1-v,)}coseski'.,)

+

.Z{=O,

[II

,A-

Ek'".

+ .ZO.'.,B-'

Fhla.

-.Z'.O.,e#

Ghla.

+

All

'

Eki

'a +

BLi

'

Fki

'a + .ZO.e.,eh, '

G#

'.a] cos

cag

'z,)

+.=e=O,[-Dsi

2b;fi'.,2"

+.20=O,A-'

Eina.+.zO=O,Bh'

Finla.+.zD=e,ein'

Cinla.-Aki'Fll'a-Bg'F#'a

+ .Ze.e.,

6#

'

GH

ka]

,i.

ue

ki

.,)

(lllZ

=-.]!"O] ±,

6iX,li,

{A5i(1-2tr2)+4Cbi(1-v2)}cosvaki'z2)

+

.Z"=O,

[.ZO='

,A-'

E#bn

+

.ZO±

O

,BX'

Fll

bn

-.ZO=O,Cin'

Gts'bn

+

A:[

'

Ell

'b +

BLi

'

Fki

"+

i.O.

,eh"

G-'nb] cos

as

ki

'z2)

+.Z"=e,[-D6i211iiBl.,,)"+.Ze!O,Ak'

Fx'b.+.Zoo=,EL,

Fub.+.z"=e

,ein,

G:,b.-7tg'Ege-E:i'iLi'b

+ .i=

Sh"

C-'

k']

sin

ua

ki

g2)

OaW,Z

=.Ze=',[-.Ze=e,c:i-iii'

zi

IB

±

,l.Ml,

{Agea-2v,)+4cbii(1-v,)}-.zO=O,cffiiiiDsu

2iii;Bl.,IM -14-NII-Electronic

(15)

Shonan Institute of Technology ShonanInstitute ofTechnology where, 2GAnSn

V:--

?GAnS

Wa"--

2GA.B:

[-Ila

Ak -B:aIha Ag+KT. Ag +B:aKha Aa]

K7k--

2G

2g.BA

['I(a

A1

'B:aIda

Ak+K;a

A?

+B:aK6.

A2]

Uk"=

GA,.

I(Bz-Bs'

[rI{ma

Alm

-BAaI6ma AYm+Kfma Al.+

B-aK6..

A't.]

Vkn=

G

ZSt.

I

esl-B.a>

ek.-G

Vk"=

GA,.

I(Bk-B7)

[-Iima

A'lm

-SrnaIema AZm+Kima

AZm+

S.aKe.,.

A'S.]

Wfi= '2G

2XnB

2GEnBn

IN7:-

2GE.BA

[-IIbA5

-BAbl6,

Ag+Kl,

Ag+BfibK6,

A4,]

Va=

-l}(s-2g'I.-kz'B.

[-I{b

Ak

'B:

b

I6b

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(16)

Shonan Institute of Technology

NII-Electronic Library Service

Shonan  工nstitute  of  Teohnology

       湘南工科 大 学 紀要 第 32 巻 第 1 号

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(17)

Shonan Institute of Technology

ShonanInstitute ofTechnology

StressAnalysis

of

AdhesiveLap

loints

of HollowShaftsSub)'ectedtoa TensileLoad

E:n=

GB".(

-l

(Bl.,)"-'

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2K

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:,

O+(rif"eA,b))

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3rr:m2+-'

£

.,i2

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(r.I)}

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=

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-G gl

)."

i!:rC.,

etlirBM.,?

)

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I"

(

3rr:rn2 ++iglli

-2v)sinh

(r.

i)cos h

(r.

I)}

Grz.=

2(-i)"GB"g/i"i(hr2X+m);l,)Co(rmb)

{v-,zrl"x3z}

Fig. 1. Analytical model of an adhesive lap joint of hollow shafts.
Fig. 3. Effect of overlap length on the stress distributions at the inner interface (Ei= E3, vi =:v3).
Fig. 4. Effect of Yeung's moduli ratio between the shaft and the adhesive on the Mises equivalent       stress distribution near the end of the interface.
Fig. 6. Effect of thickness ef the adhesive on the Mises equiva!ent stress distribution near the end of       the interface,

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