Shonan Institute of Technology
ShonanInstitute of Technology
MEMOIRS eF SHONAN INS7ITUTE OF TECHNOLeGY VoL 32,No. 1,1998
Stress
Analysis
ofAdhesive
Lap
Joints
ofHollow
Shafts
Subjected
to
aTensile
Load
Yuichi
NAKANo*
andMasataka
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$s of the adhesive on the stress distributions at the interfacesinthejoint
are clarified. Itisshown that the stresses in radial and circumferential direction$ become singular at the ends of the interfacesand the stress increasesnear theends of theinterfaceswith a decreaseof Young'smodulus of the shaft and with anincrease ofthe thickness of the adhesive.
Key words: Elasticity;adhesive lap
joints;
hollowshafts; stress analysis; Yeung's modulus; axialload;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 andbolts.
However, there are some problerns ina flangetype shaft
joint;
forexample, a flange occupie$ a largervolume as compared with the sizeof
the
shaftitself
andthus
the
weight ofthe
joint
increasesandthe
stress concentrates at thecornerof keyways during operation. On the other hand, ifadhesively bonded shaft
joints
are realized andused 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 shaftjoints,
can be obtained more easilyi}.
In
adhesive lapjoints
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 thanin
adhesivebutt
joints
ofshafts can usually be achieved by increasing the overlap length of each shaft so thatthe adhesive lap
joint
of shafts ismore suitable fortransmitting shaftjoint.
Lubkin and Reissner2)analyzed the stress distributioninadhesive lap
joints
of shafts of thesame 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, obtainedthe stress
distribution
in
adhesivelap
joints
intorsionwith adherends made of composite material assuming the stresses inthe adhesive as uniform across itsthickness. Hipo15) analyzed the stressdistribution
in
an adhesivelap
joint
of metal and composite hollow shafts subjected to torsion using a finiteelement method, Lee et al,6) presented an approximate calculation methodfor
the shear stress inadhesive lapjoints
subjected to torsion using afinite
element analysis considering*
uadiI\*l
Xnt
**
J<\eere
±
(thme)
ecE
uediIYgty
SIZut 9lii1O
H
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maMIN)S(\*eee eg 32 ts
eg
1g
the nonlinear adhesive properties.
Reedy
andGuess7)
performed the experimental study on thetensilestrength and fatigue resistance of an
E-glass
composite-to-aluminum tubularlap
joints
subjected totensile,compressive and
fatigue
loadings.
Itisclear from above thatmost of analyticalstudies on the adhesive lap
joints
have been done assuming the uniform stressdistribution
acrossthe adhesive thickness. However, the stress and strain
distributions
in
theadhesive, especially nearthe freeends of the adhesive where
high
stress and strain levelsare expected, usually have serious effects on thejoint
strength.The purpose of thisstudy
is
toanalyze the stress and strain distributionsin
adhesivelap
joint
of hollow shafts of differentmaterials subjected to an axial tensile
load
by
using an axisymmetrictheory of elasticity.
In
the analysis, thejoint
is
modelled as an elasticthree-body
contact problemand 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 ofthe
adhesiveto 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 aninner
hollow shaft and an outer hollow shaft ofdifferent
materials are bonded by an adhesive at an overlaplength
212
and an axial tensileloadPis
applied atboth
ends of the shafts.The
inner
and outerdiameters
and thelength of the
inner
hollow
shaft aredenoted
by
2ai,
2biand 21iand those of the outer hollow shaftby 2a3,2b3 and 213,respectively.
Inthe analysis, the
inner
hollow
shaft, the adhesive and theouter hollow shaft are replaced byfinitehollow cylinders I,IIand
III,
respectively.(The
inner
and outerdiameters
of the cylinderII
are 2a2
(=2bi)
and2b2
(==
2a3)).
Young's modulus, shear modulus and Poisson's ratio aredenoted
by
e
G and y,and the subscripts 1,2 and3
designate
thefinite
cylindersI,
II
andIII,
respectively. A coordinate system usedhere
is
such that;r represents the radial direction,e
the
hoop
direction
and g thelongitudinaldirectionofthe
joint.
The
origins of eachfinite
cylinder aredenoted
by
Oi,
02
and03 and the distance
between
Oi
and02
is
denoted
asCi
and thatbetween 02 and 03 isas C3.The boundary conditions of theadhesive lap
joint
are expressed asfollows
Eqs.
(IH5);
on theinner hollow cylinder I,
213 1
Adhesive[ll]Hollow
--shaft[I] 8" .P.8epapte
8-oJ '1 T- -y =pt-Nft-
.. ・-2hHollowshaft[Ill]
2iiFig.1. Analyticalmodel of an adhesive lap jointof hollow shafts.
Shonan Institute of Technology
ShonanInstitute ofTechnology
StressAnatysis
of
Adhesive Lop fointsof
HbllowShaftsSubfectedtoa TlensiteLoadzl=+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 cylindersII
andIII,
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
andIII
designate
the
corresponding finitecylinders I,IIand III,respectively. Particularly,sincethe origins of each
finite
cylinder inthe longitudinal directionare fixed on thedifferent
points as shownin
Figure
1,
thederivative
of the axial displacement w. with respect toz should be equal toeach other at each interfacebetween the cylinder Ior IIIand the cylinder II
(adhesive>,
as expressedin
Eqs.
(4)
and(5).
From
the
differentiation,one of the unknown coeMcients inthe following stressfunction iseliminated, and itcan be
determined
from the equilibrium condition of an axialforce
applied to the
joint,
Michell
stress function ¢(n
z) isused inthisstudy in order to satisfy the above boundary conditionsEqs,(1)-{5).
Taking
theboundary
conditions intoconsideration,Michell
stress function¢ i
(L
zi> forthe hollow cylinder Iisexpressed asEq.
(6)
from the solution of variable separable ofbiharrnonic
function8)'
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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
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・… 麟 ・麟 1) ・盞
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・・・…h
・劇 ・ ・… … h ・・lli
・…h
幽 ]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1)lla+βnaiIOa 2(1一ン1)llb
+βnbiIOb − (1 − 2ン1)1(la一βnallla − (1 − 2り1)IOb 一β n bi lib 4 一Kla −Klb
Kla
− Kαa一 β。ai − KOb 一Klb
β。bl
2(1 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 LibraryShonan 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,lldi2=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 ordersO
and 1 respectively, and rLis
thes-th positiveroot ofequationCi
(r:ai)=O.
Ah,
Bb,
C5,
D5,Eh,Ak,BL,
CE,
...,Ai.',
Ek'
andek'
(n,
s= 1,2,3,...)inEq.(6)
are unknown coefficients. Similar stress functions Oii(ny
z2)forthe
hollow cylinder II,i.e.forthe adhesive and ¢ M(r z3)for
thehollow
cylinderIII
are chosen by changing correspondingly the coefficients involved inEq.(6).
The stresses and the
displacements
in
eachfinite
hollow
cylinder canbe
obtained fromMichell
stress function and are written as Eq.
(7).
IIa
2(1-VDIia+BAailaa
-K;a2<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.=IicaLr),
I{.=Ii
avKr)
6a=Io
asL'
ai),Iia=IiosLai),
IIa=IiueKai)
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`= wi2;}
(vv2o-
O,2,9)£
(vvz
¢ - 'lilll
¢ 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
simultaneouslinear
equations with thoOr+
IS22).
e unknown coefficients are
derived
by
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ShonanInstitute ofTechnology
mamrptj(\seee
eg
32g
eg
1e
ing
the
stresses and displacements into the boundary conditions expressed in Eqs.(1)-(5>.
(see
Appendix)
In
the numerical calculation, the number of termsin
theseriesin
theinfinite
simulta-neous equations istruncated by N. Then the unknown coerncients are determined by solving the(36N+15)
simultaneous equations and,finally,
the stresses and thedisplacernents
in
thejoint
canbe calculated from Eq.
(7).
3. Results and Discussion
In the following calculations, the number, N, of terms
in
the series of the simultaneousA,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))--6-Shonan Institute of Technology
ShonanInstitute of Technology
StressAnalysis
of
AdhesiveLopfoints
of
HOtlowShoftsSubjectedtoa 7lensileLoadequations,
is
taken as80
and thelength
2tiof theinner
hollow shaft isthe same as the length 2t3ef the outer Qne and all stress distributionsshown below are inthe adhesive.
Figure
2
(a)
shows thedistributions
of thestress components o,,ob, a. and T,,near theend of theinner interface
(r=a2)
between the inner hollow shaft andthe
adhesive and(b)
near theend of theouter interface
<r=b2)
between the outer hollow shaft and the adhesive forthecase where thejoint
iscomposed of theshafts of thesame material and issubjected toan axial tensileload. The abscissadenotes 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, theinner
and theouter radii of thehollow shafts are set so that the mean tensile stress so at one end of thejoint
isthe sarne as the mean tensilestress s6 at the other end.Frorn
the results, the stresses q and obin
the radial and circumferentialdirectionsbecome 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 itandincrease
steeply atboth
ends of theinterface').
On
the otherhand,
both
thenorrnal stress a.in
the axialdirection
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 thedistributions
ofthe
stress components atthe
inner interface
(r=a2),
It
isseen that the stress distributionsare dependent on the overlap length and as the overlap length isdecreased, as shownin
(b),
the stress components takehigh
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 zeroin
case oflong
overlaplength
as shown in(a).
Inthe
following
numerical results, thestressdistributions
at theinterfaces
are shownin
terms of normalized Mises equivalent stress aeqiso.
Figure
4
shows theeffects of the ratio ofYoung's
modulus of thehollow
shaft tothatof theadhesive on the norrnalized Mises stress distributionat the
interfaces.
In thiscase, theinner andthe
outer radii ofthe
hollow
shafts are also set sothat
themean tensilestresses so and s6 are the same,(a)
shows thestresses near theloaded end of theinner
interface
and(b)
shows those near theloaded
end of theouter interface.Frorn theresults, when the hollow shafts are made of the same material(shown
by
thesolidline
in
thefigures),
theequivalent stress at theloaded
end of theinner
interface
(z2it2=1)
is
largerthan thatat theloaded
end of the outerinterface
(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
1.0Fig.3. Effectof overlap length on the stress distributionsat the inner interface
(Ei=
E3, vi=:v3).-7-Shonan Institute of Technology
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maMIptJ
±(#e-eg
32k
ee
1e
3.5 3.0 2.5qe
2.0 eg 1,s 1.0O.5 o 3.5 3.0 2.5.e 2.0 xbg 1.5 1.0O.5 oo.gs o.96
O.97
O.98 O,991.0
-LO -O.99 -O.98 -097 -O.96 -O.95z,/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 loadincreases.
When thehollow
shafts are made of differentmaterials(the
dotted anddashed-dotted
lines),theequiva-lentstress islargeatthe loaded end of the
interface
between the hollow shaft of a smaller Young'smodulus and the adhesive,
i.e.
at z2tl2=1 shown by the dotted linein(a)
and at z2tl2= - 1shownby
the
dashed-dotted
linein(b).
Frorn these results, the tensilestrength of a
joint
where an inner hollow shaft of a smallYoung's
modulusis
joined
toan outer one of a largeYoung's
rnodulus(Ei:
E2:E3=5 :2:1Oas shownby the dotted line)issupposed to
be
smaller than that of thejoint
of shafts of the same Young'smodulus.
On
theotherhand,
the tensilestrength ofthejoint
where an inner hollow shaft of a largeYoung's modulus and an outer one of a small Young's modulus are
joined
(Ei:
E2:E3=1O :2 :5asshown
by
thedashed-dotted
line)is
supposed to becorne larger,if
the equivalent stresses at theloaded ends of the inner and outer
interfaces
are equal toeach other. Moreover, frorna comparisonof the stresses near the
loaded
end of theinterface
z2ft2== 1 in(a),
the dashed-dotted lineisslightlysmaller than the solid line. Therefore, itmay be possible that
by
choosing an appropriate combination of shafts ofdifferent
Young's
rnoduli, the tensilestrength of thejoint
of shafts ofdifferent
materials could exceed that of thejoint
of shafts of the same rnaterial of alarge
Young'smodulus.
Shonan Institute of Technology
ShonanInstitute of Technology
StressAnalysis
of
AdhesiveLmpJbints
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 oO.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 theinner
hollow
shaft hi(=bi-ai)
on the equivalent stress distributionnear the loaded end of the inner interface(r=a2).
In thiscase, theinner
and outerhollow
shafts are made of the same material and only theinner
radius ai of theinner hollow shaft isvaried while the other dimensions of the shafts and the tensile loadP are kept constant.
The
equivalent stress near theloaded
end of theinner
interface
is
larger
than that nearthe loaded end of the outer interfaceinthe case where the
joint
iscomposed of the hollow shaftsof the same material, as shown
in
Figure
4
(solid
line);
however,
by
increasing
the thickness of theinner
hollow
shaft, the equivalent stress near theloaded
end of theinner
interfacedecreases andthe tensilestrength of the
joint
increases.
Meanwhile,
the effects of the thickness of theinner
hollow
shaft on the equivalent stress distributionat the outerinterface
are negiigible,Figure 6 shows theeffects of thethickness of theadhesive h2
(=b2-a2)
on theequivalent stressdistributions
near theloaded
ends of theinner
and outerinterfaces
when thejoint
is
composed ofthe 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 thejoint
are thesame.In
thefigures,
the thickness of the adhesive h2isnormalized by thethickness of the inner hollow shaft hi.From the figures,theequivalent stresses near
both
loaded ends of the interfacesdecrease as the thicknessof theadhesivebecomes smaller.
4. Conclusions
This
studydeals
withthe
stress analysis of adhesivelap
joints
ofdissimilar
hollow
shaftssubjected toan axial load. The results obtained are as follows;
1> An exact analyzing method
for
thestress and strain distributionsinthe adhesive lapjoint
of
hollew
shafts, whichis
modelled as an elastic three-body contact problem, isdemonstrated
using an axisymmetric theory of elasticity,
2>
The
effects of theratio ofYoung's
modulus of theadhesive tothatof thehollow
shafts, andof 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 andhoop
stresses are singular at the end of the interfaceand the Mises equivalent stress near the end of the interfaceincreases with a
decrease
of the ratio ofYoung's
modulus of the shaft tothatof the adhesive andwith an
increase
of the thickness of the adhesive,-9-Shonan Institute of Technology
NII-Electronic Library Service
ShonanInstitute ofTechnology
maMI*gd<\rest
eg
32g
eg
1e
3)
In
the case oftwo
hollow
shafts made ofthe
same rnaterial, theMises
equivalent stress at theloaded
end of theinner
interface
is
larger
than thatat the outerinterface
and as a result thefractureinitiatesat theend of theinner interfaceof the
joint.
Inthecase of two hollow shafts made ofdifferent
materials, thejoint
tensilestrength canbe
expected tobecome
larger
when theinner
hollow shaft of a relatively large Young's modulus
is
bonded
tothe outer one of a smallYoung'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
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Tlr=-D5(1+p])bi+E5
(1
I;,Vi)
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Shonan 工nstitute of Teohnology
(r=b2>
(r=α3)
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湘南工科大学 紀 要 第 32 巻 第 1 号 T・…=
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ShonanInstitute ofTechnology
(r=a2)
(r=b2)
<r=a3)
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of HotlouiShaftsSubl'ectedtoa TensiteLoad+
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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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ShonanInstitute ofTechnology
StressAnalysis
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