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Contact stresses between cylindrical shafts and sleeves
Znt.J. EngngSci. Vol. 5, pp. 541-554. Pergamon Press 1967. Printed in Great Britain
CONTACT STRESSES BETWEEN CYLINDRICAL SHAFTS AND SLEEVES H. D. CONWAY?and K. A. FARNHAM~ Department
of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York (Communicated
by I. N. SNEDDON)
Abstract-Numerical methods are used to tind the contact stresses and deformations between circular cylindrical shafts and sleeves. The first class of problems treated concerns the circular cylinder onto which a rigid sleeve is press-fitted or shrunk. Individual cases are worked out in which, (a) the contact is frictionless and (b) sufficient friction occurs to cause complete adherence of the surfaces in contact. The intermediate case is also discussed where Coulomb’s law of friction holds between normal and shearing contact stresses. In this case, certain portions of the contacting surfaces adhere, whereas others slide over one another. The problem of a cylinder loaded axially through rigid collars is then worked out. Finally, solutions are obtained for the case of a cylindrical shaft of finite length twisted by means of flexible concentric sleeves. In all cases, numerical values of the contact stresses and deformations are given for various cylinder and sleeve parameters. 1. INTRODUCTION
ONE of the most difficult class of contact stress problems to treat analytically and one which has received very scant attention in the literature, is that of finding the stresses between circular shafts and sleeves. At the same time, these problems have very considerable practical interest. They are the subject of the present article, which is the third in a series on the subject [l, 21. The first part of the article concerns the stresses and deformations produced when a sleeve is shrunk or press-fitted onto a cylinder. Two extreme cases are considered, in the first of which the effects of friction are ignored. The other extreme case is then treated where the friction induced at the contact surfaces is sufficient to cause complete adherence. Finally, the intermediate case is investigated where complete adherence occurs only over part of the contacting surfaces and relative slipping over the remainder. In the next case considered, a cylinder is elongated by forces applied through sleeves attached to the shaft and the shearing and normal stresses at the junctions of the cylinder and sleeves are worked out. In all the above cases, the rigidity of the sleeves is assumed to be very great compared with that of the cylinder. In addition, the cylinder is assumed long, so that end effects do not enter the problem. Thus, as a final contact stress problem, the case of a finite length cylinder twisted by couples applied through sleeves, is considered, the modulus of rigidity of the cylinder and sleeves being arbitrary. The method of solution is essentially adapted from that used in the previous articles [ 1,2]. The normal and shearing contact stress distributions are replaced by series of pressures and shears constant over large numbers of incremental lengths. Satisfaction of the boundary t Professor, Department of Theoretical and Applied Mechanics, Cornell University. IBM Corporation, Endicott, N.Y., U.S.A. $ Systems Development Division, IBM Corporation, Endicott, N.Y., U.S.A. 541
Consultant,
H. D.
542
CONWAY
and K. A. FARNHAM
conditions is then achieved at discrete points, leading to series of equations in the applied incremental pressures and shear, and hence, to approximations of the distributions. It has been shown [2] by comparison with known exact solutions that the techniques are capable of giving very accurate results. 2. FINITE
LENGTH
RIGID
SLEEVE SHRUNK CYLINDER
ON A CIRCULAR
Frictionless case
The cylindrical contact area was divided into 20 equal circular rings, but, because of it is only necessary to consider the radial displacements at the centers of 10 rings. The incremental pressures are denoted by pi, pz, . . . plo, and the resultant equal displacements at the centers of the 10 rings are found, making use of the solution for an infinite cylinder subjected to a single band of uniform pressure (Appendix I). Machine computations outlined in Appendix II were then used to sum the integrals involved and to solve the 10 simultaneous equations in the incremental pressures. Graphs of normal pressure distributions for various ratios of half sleeve length b to cylinder radius a are presented in Fig. 1. As might be imagined, the pressures for the larger b/a values are nearly uniform except near the ends. For b/a =&, the radius of the cylinder is very large compared with the length of the rigid sleeve, and hence, the pressure distributions shown closely approximate the exact solution of Sadowsky [3] for a rigid frictionless, axially-loaded punch applied to a half-space. The comparison made in Fig. 2 shows this to be the case and is added confirmation of the accuracy which can be expected from the present numerical method. symmetry,
2.6
2.6
1.6 P inab 1.2
0.8
I:
a
, ,
i---b+
,-.---J
FRICTIONLESS
-1
1.6
1.2
0.8
0.4
0.0
0.4
c
- 1.0
0.0 -0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
z 6
FIG. 1. Pressure distributions between a circular cylinder and a rigid sleeve for various values of b/a (frictionless contact).
and sleeves
Contact stresses between cylindrical shafts
543 4.0
3.5
3.0
3.0
2.5
2.5 P 4nab
X-EXACT SOLUTION b/a-O IFRICTIONLESSI 2.0
2.0
1.5
1.0
0.5
4 -1.0
-0.8
-0.6
-0'4
0.0
-0.2
0.2
0'4
0.6
0.8
0
1.0
2 b
FIG. 2. Comparison
of pressure distribution for b/u=1/32 (frictionless contact).
with the exact solution for b/u-+0
After the pressure distributions were computed, the radial displacement/force relationships were calculated. These radial displacements are presented in Table 1 as fractions of the corresponding radial displacement P(l -v)/4?rEb obtained for a cylinder of length 26 rather than infinite length. TABLE 1. RATIOOFRADULDISPLACEMENTBF FORAN MPINITECYLINDRRTO RADIAL DISPLACEMENT &=P (I--V)/4ZEb POR A CYLINDW OF LENGTH 2b POISSON'SRATIO V =0’3
Frictionless case l/32
b/a
0.185
WSO
l/4 O-635
1 0.882
0.9230
4 0946
Table 1 shows the supporting effect of the cylinder outside the sleeves. Clearly this effect is small where the length 2b of the sleeve is large compared with the diameter 2u of the cylinder. 3.
FINITE
LENGTH
RIGID
SLEEVE CYLINDER
SHRUNK
ON A CIRCULAR
Complete adherence case
The corresponding but far more complicated case where complete adherence takes place between the cylinder and sleeve was next investigated. As in the frictionless case, the contact area was divided into 20 equal incremental areas, and taking account of symmetry, these were assumed to be subjected to uniform normal stressespI,p2, . . . plo and uniform shearing stresses ql, q2, . . . qlo. Of course, the radial and axial displacements of the cylinder are both functions of pi, p2, . . . plo and ql, q2, . . . qlo.
H.D.
544
and
CONWAY
K. A. FARNHAM
These displacements were obtained by machine computation as indicated in Appendixes I and II. Setting the resultant radial displacements equal at the center of each incremental area and the resultant axial displacements zero, produced 20 simultaneous linear equations inpr,p,, . - *Ploy 419 q21 * * * qlo which were again solved by machine computation. Graphs of normal and shearing stress distributions found in the above manner for various b/a ratios are plotted in Figs. 3 and 4, respectively. When the results shown in Figs. 1 and 3 are compared, it appears that the normal stress distributions are similar, thus lending support to the frequently made assumption that friction does not markedly affect the contact normal stresses. However, it is noted from the complete adherence graphs in Fig. 3 that the positions of minimum normal stress for the larger values of b/a do not occur at the center, but move progressively away from the center. This effect is much more marked than in the frictionless graphs of Fig. 1, although it also occurs in these to a diminished extent.
TOTAL RADIAI LOAD P
3.0
3.0
2.5
POISSON'S RATIO = 0.3
2.5
2.0 P G%
1.5
I.0
1.0
? 0.5
0.5
0 -1.0
-0.8
-0.6
-0.4
-0.2
0.0 Z ii
FIG.
0.2
0.4
0.6
0.8
1.0
3. Pressuredistributionsbetweena circular cylinder and a rigid sleevefor various values of b/a (completeadherence)Poisson’sratio =0.3.
The shearing stress distributions shown in Fig. 4 are very interesting. It will be seen that the magnitudes of the shearing stresses for certain b/a ratios (for example b/a=&) increase as one goes away from the center and then decrease and reverse sign near the edge. In addition, the direction of the shearing stress at a specific location depends upon the b/a ratio. It is also apparent that there is a value of b/a about &- (calculated but not plotted) for which there is very little shearing stress, except near the ends of the sleeve. Effects similar to these were also observed [I] in indented strips and slabs under adherence conditions.
54s
Contact stresses between cylindrical shafts and sleeves
2b TOTAL LOAD
0.6 Ir
2
i$+i POISSON’S
RADIAI P
RATIO
= 0.3
0-i P drrab
o-0
O’C
0.2
0.4
4
-1.0
-0.8
-0.6
-0.4
-0’2
0.0
0.2
0.4
0.6
0’8
1.0
z ii
Fro. 4. Shearing stress distributions between a circular cylinder and a rigid sleeve for various values of b/a (complete adherence), Poisson’s ratio=0.3.
The radial displacements 6, were next computed and these are presented in Table 2 as fractions of the corresponding radial displacements 8r for frictionless conditions. TABLE 2. RATIOOFRADIALDISPLACEMENT~A FORCOMPLETEADHERENCETO RADIALDISPLACEMENT~FFOR~'RICTIONLESSCONTACT POISSON'SRATIOV=0*3
% 6Al6P
l/32 130
l/4 1.00
1
2
4
0.93
0.86
0.82
As expected, the effects of friction on the radial displacement are only important for the higher values of b/a. 4. FINITE
LENGTH
RIGID
SLEEVE CYLINDER
SHRUNK
ON A CIRCULAR
Partial slipping case
After cases of frictionless contact and complete adherence were investigated, the intermediate case of partial slipping was considered. As in the previously investigated partial slipping cases of indented strips and slabs [2], the half-length of the contact area was divided into 10 equal increments and a specific ratio of central adherence length to total contact length 2b was selected. The normal pressures were designated pi-lo. If, for example, the central adherence length consists of eight increments (adherence length/ contact length=0*4) with four on either side of the center line, the shearing stresses for these are designated ql+. The average shearing stress intensities for the six increments in each slip length are then kp,_,,.
H. D. CONWAY and K. A. FARNHAM
546
Thus, there are 15 unknowns to be found-ten& four q’s and the coefficient of friction k. Ten of the equations required for the solution are found by assuming that the radial displacements at the centers of the increments are equal. Four more are obtained by making the axial displacements at the centers of the four adherence increments zero. The final equation is obtained by prescribing the axial displacement at the center of the increment immediately outside the contact length. The solution of these nonlinear equations can be avoided by noting that the radial pressure distribution changes little from the frictionless to the fully adhered case. From this fact, the set of 15 equations is reduced to five linear ones which can be readily solved. The results obtained using the frictionless and full adherence pressures agree well with one another. Figure 5 shows a typical graph of adherence length/sleeve length versus friction coefficient k for b/u =& and Poisson’s ratio =0*3. It is seen that a friction coefficient k =O-4 produces almost complete adherence. In view of the reversal of the shearing stresses for larger b/a values shown in Fig. 4, tracing the adherence lengths with varying coefficients of friction would be most complicated for these values. However, this would be of academic interest only, since it is apparent that a relatively small friction coefficient causes almost complete adherence.
0
0.2
0.4
0.6
0.8
1.0
FRICTION COEFFICIENT k FIG.
5.
Adherence
length/sleevelength vs. friction coefficientfor b/a= Poisson’s ratio =O-3.
5. AXIAL GOADING OF INFINITE CYLINDER THROUGH RIGID
l/32.
CIRCULAR SLEEVES
Complete aa%erence case The important practical problem of a circular cylindrical shaft loaded axially by forces applied through rigid sleeves was next investigated, using the above numerical techniques. Typical graphs of shearing and normal stresses at the junction of cylinder and sleeve are shown in Figs. 6 and 7, respectively. The centers af the two sleeves are assumed to be llb apart in all cases. As expected, the normal stresses are small compared with the corresponding shearing stresses.
Contact stresses between cylindrical shafts and sleeves
547
35
30
25
20 % na 15
10
5
0’0
0’1
0.2
0’3
0’4
0.5
0’6
0.7
0.8
0.9
1.0
2 6
FKL 6. Shearing stress distributions between an axially loaded circular cylinder and a tid sleeve for various values of b/u (complete adherence). Distance between sleeve centers = 1lb, Poisson’s ratio=0*3.
15
10
Ld4: 7 l-1 / / 2
5 Q 2
0.1
0
-5
0.2
__1..
0.3
4
-L____
0.4
0.5
t
0.6
0.8
0.9
1
:
-10
- 15
FIG. 7. Normal stress distributions between an axially loaded circular cylinder and a rigid sleeve for various values of b/u (complete adhuence). Distance between sleeve centers=llb, Poisson’s ratio=003.
548
H. D. CONWAYand K. A,
FARNHAM
It is very interesting to compute the relative axial movement these are presented in Table 3. TABLE 3. THROUGH
apart
of the sleeves,
RATI~~OF AXIALDISPLACEMENTBOFCLRCULARCYLINDERLOADED RIGID SLEEVES TO &=lOQb/nuW. DISTANCEBETWEENSLEEVE CENTERS=~~~
b/a
l/32
6160
4.202
l/4 1.319
1
2
1.063
1.031
As expected, the axial displacement 6 rapidly approaches 6, as the b/a ratio is increased. 6. TORSION
OF
FINITE CIRCULAR CYLINDER FLEXIBLE SLEEVES
THROUGH
In all the problems previously discussed, the circular cylinder has been assumed to be of infinite length and the sleeve completely rigid. The case of a finite length circular cylinder twisted through flexible sleeves will now be considered. In the numerical solution of the above problem the same technique is applied. The solutions for the finite length cylinder twisted by equal but opposite bands of uniform shearing stress and a finite thickness disc subjected to uniform shearing stress are given in Appendix III. In the present problem it is only necessary to match the tangential displacements at the junctions of the cylinder and sleeves. No normal stresses are induced either in the cylinder or the sleeves. This fact enables the latter to be of finite length in the investigation, since only one condition has to be satisfied at a junction rather than the customary two. Graphs of the shearing stress distributions at the junction for various values of the ratio G,/G, of the modulus of rigidity of the cylinder and sleeve are given in Fig. 8. The e/b and e/a values (see Fig. 8) are assumed to be equal and it was found that the stresses remain practically unchanged for e/b = e/a > 1.5.
1'2
-;-.-;
:I---;--
1'0
,, '-_--TORQUE T"-*__ 2 __._
O-8
0'6
_/
'____2e ; =f>
0.0
OS1
0.2
1.5
0.3
G,
__ -.
1
1 3
_,__ CYLINDER MODULUS
T--J ---~~
__
SLEFVE MODULUS
c.,
~~~_
POISSON'S RATIO = OS3
0'4
0.5
0.6
0.7
0.8
0.9
I.0
Z Ti FIG. 8.
Shearing stress distributions for a finite length circular cylinder twisted through flexible sleeves (complete adherence), e/b=e/a> 1.5. Poisson’s ratio =0.3.
Contact stressesbetween cylindricalshafts and sleeves
549
The torque T=KG,a3 per unit (radian) twist of the cylinder and sleeves was then computed for various values of G,/G, and e/b =e/a and is given in Table 4. TABLET.
VALUESOF
KIN TORQUE T=KGc a3 PER UNITTWIST FOR CYLINDER AND SLEEVESIN FIG. 8. POISSON’S RATIO=O-3
e/b =e/a
GclG~=j
Ge/Gs = 2
G,/G,=l
1.5
1.037
O-925
0 -777
2.5
0447
0.425
0.391
5
0.184
0.181
0.174
7.5
0.116
0.115
0.112
From simple strength of materials theory, the twist of the cylinder may be estimated as and that of each sleeve as T/4nGp2b. Thus, the torque To per unit radian twist of the cylinder and two sleeves together is, by strength of materials theory, 4Te/G,na4
To=
2nGsa2 b _ , I_\
.
Ratios of the torques T/T, are given in Table 5 for G,/G, = 1 and a = b. TABLE 5. RATIO OF TORQUE PER UNIT RADIAN TwISTTOCORRESpONDINGVALuEGlVENBYSIMPLE STRENGTH OF MATERIALS THEORY. G,/G,=l, POIsSGN'SRATIo=0*3
e/b =e/a
T/To
1.5
1.914
2.5
1.420
5
1.178
7.5
1.113
10
1.083
As expected, the T/T, values approach unity for the larger e/b = e/a values. REFERENCES M and S. So, Normal and shearing contact strew in [l] H. D. CONWAY,S. M. VOGEL,K. A. FARNHA indented strips and slabs. Znr..Z. Engng Sci 4,343 (1966). [2] H. D. CONWAYand K. A. FARNI-IAM, The contact stress problem for indented strips and slabs under conditions of partial slipping. Znf.J. Engng Sci. 5, 145 (1967). [3] S. TIMCWHENKO and J. N. GOODIER, Theory of Elasticity, p. 96. McGraw-Hill (1951). [4] A. I. LUR’E, Three Dimensional Problems of the i%eory of Elasticity. p. 394. Interscience (1964). (51 C. HASTINGS, Approximations for Digital Computers, p. 197. Princeton (1955).
550
H. D. CONWAYand K. A. FARNHAM
APPENDIX
I
To investigate the pressure and shearing stress distributions between a rigid sleeve and a flexible cylinder by the foregoing numerical methods, it is tirst necessary to find the surface displacements of the cylinder in Fig. 9.
FIG. 9. Cylinder subjected to band of uniform pressure. By expanding the normal surface loading into a Fourier integral, it is easy to show [4] that the radial displacements on the surface are given by
s
m 2( 1- v)1:(~)sin(/3c/a)cos(j?z/a) urEe= -- pa nG o B3CZ7d(8)-Z:(B)1-28(l-v)zt(~dB whereas the axial displacements
W,=,
_
Pa
61)
on the surface are
s
nG
*
o
v)zI(B)zO(B> + B[‘:(B) -‘~(B)]sinBesinSZdp 64.2) a a * S”CG
2(1-
In addition to the above, it is necessary to compute the radial and axial displacements due to the surface shear loading shown in Fig. 10. These are found in a similar manner [4] to those for the normal loading and are given by
u,=,_ _w “BCZ~(B)-Z:(B)1-2(1-v>Zo(B)ZliP> 71~~ -ml -au - vv:m s o B3CGW E w
=
s
=ga
’ =
aG
o&dj? a
,
(A.31
“(3-2v)B~~o-Bz:(8)-4(1-v>~o(B)z~(B>
o
B4CW)-ml
- v2u - vv:m
SHEARING
STRESS
q
q$--yf$(i
-7-
FIG. 10. Cylinder subjected to equal but opposite bands of axial shearing stress. It is neoessBTy to evaluate equations (A.lHA.4) in Appendix II.
numerically and the method for doing so is outlined
Contact stresses between cylindrical shafts and sleeves
551
APPENDIX II The following methods were used to obtain approximate numerical values for the infinite integrals Riven in Appendix I for various values of c/a, d/a and z/la. The given integrand was integrated numerically from p=O to B=A u&g a Simpson’s Rule integration. For values of jI>A, the integrand was replaced by asymptotic approximations for the Bessel functions and the integration was continued to j?=B. For values of fbB, the integrals were replaced by suitable sine or cosine integral values. Typical values of A and B were 40 and 200 respectively. The required asymptotic approximations for the Bessel functions include
From these expressions,
equation (A.]) becomes
A2(1- v)l&9)sin@c/a)cos(pz/a)dj?
-2/w- VW)
0 B3CG(B)-ma
+
s*
s
m2( 1- v)sin(/?c/a)cos(Bz/a)
The third of these. integrals upon integration 2(1
B2
I3
by parts produces
v)sin~c/a~20s@z~d,&
_
(l_
sin[(c; ‘jBal
v)
B
+ sin[(c - z)Ba] B
-~fjCi[(~)s]}
where
Ci[x]=
Ft.
s
In the same manner, equation (A.2) yields
s
+ ‘2 (‘-~)[‘-(3/‘8)1[‘+(‘/‘8)1-‘sin~sin~dp
A
--
/I’-2j3(1-v)[l-(3/4/I)]
(1-2~)
2
cos[(c+z)Ba] B
a
+[yjsi[r$)B:
1
+ cos[(c - z)Ba] B
+pbzjsi[(gq]}
where
si[x] = -
m s&tdt s x
t
and
si[x] = Si(x) - IL/~.
a
H. D. CONWAYand K. A. FARNHAM
552
The functions a(x) and S(x) are computed from excellent polynominal approximations given by Hastings [S]. The numerical evaluation of expressions (A.3) and (A.4) is accomplished in a similar manner. An appreciable saving in computation time can be realised by noting that equations (A.l) and (A.4) are symmetric and equations (A.2) and (A.3) have a transpose relationship.
APPENDIX
III
When analyzing the problem of a cylinder twisted by flexible sleeves (Fig. 8), it is necessary to obtain an expression for the tangential displacement at the boundary of the hole loaded by a band of shearing tractions q as shown in Fig. 11. This requires special treatment.
', /-
-;,
SHEARING
STRESS q
FIG. 11. Infinite plate subjected to band of uniform torsional boundary.
shearing stress on inner
Consider an elastic body in which the displacements u=w=O and the displacement u is independent of the coordinate 0. It follows that the three direct strains and one shearing strain are zero e,=es=e==O
9
Yrz=O.
(A.51
The stresses are given by
64.6) Of the three equations of equilibrium,
two are identically satisfied and the other becomes
a%
Ian rar
--$s----+-=o.
A particular solution of this equation is v=As/r.
D
r2
a20 a22
(A.71
For further solutions we write
u = V(r)sin kz, cos kz .
(A.@
v=o.
(A.9
It follows from equation (A.7) that
The solution is V=Al,(kr)+BK,(k~) where Zl and KI are moditied Bessel fractions. A=0 in the present problem.
As r+ ~0, I&b)-+ to and therefore, it is appropriate
(A.lO) to write
553
Contact stresses between cylindrical shafts and sleeves For the shearing stress distributions
%L. =+-2 We construct
shown in Fig. 11, we write
=!,
”
the displacement
k$.
~sinkccoskdcoskz, 9*
(A.ll)
function
u=$‘+
i
k$
B,K,(kr)coskz,
n=l,
(A.12)
2,
whence
r,,=G,r;
=---2Wo
0 f
r2
i
B,kG,K,(kr)cos
kz
(A.13)
.
n=l,Z,
It follows from equating (A.11) and (A.13) that
&=-,
qca’
B,=-
G,b
4qb
sin(nnc/b)cos(md/b)
I127X2G S
(A.14)
K,(nxalb)
and finally
va
l&,=-+7 G,b
4qb ’ K1(ka)sin kc cos kd cos kz , n: G,,=,,2,~Kz(ka)
Ii
k=!!f.
(A.15)
It is interesting to note that the stresses associated with the summations in the above expressions produce no torque. The normal and shearing stresses on .z=O and b are, of course, zero. The other displacements required in the solution of the problem pertain to the circular cylinder loaded as shown in Fig. 12. These are obtained in the usual manner giving
u,=,=8qe f
(- 1)(“-‘)~211(n7ta/2e)
~2G,,=l,
3,
n2
I,(nxa/2e)
&!E&!E 2e
2e
,
(A.16)
The normal and shearing stresses on the ends z= +e of the cylinder are zero.
SHEARING
STRESS q
FIG. 12. Cylinder subjected to bands of uniform torsional shearing stress.
Machine methods were used to sum the series in equations (A.15) and (A.16). The results obtained using four- and then five-hundred terms were compared and found to agree very closely. (Received 17 October 1966)
554
H. D.
CONWAY
and K. A. FARNHAM
R&arm~On ttudie des methodes numbiques pour determiner les contraintes de contact et les deformations entre des arbres cylindriques circulaires et des manchons. Le premiet probleme trait& se rapporte au cas du cylindre circulaire sur iequel un manchon rigide a et6 monte a la presse ou fret& Des cas particuliers sont examines, dans lesquels (a) le contact se fait sans
frottement et (b) le frottement est suffisant pour assurer I’adhtsrence complete des surfaces en contact. On Ctudie aussi le cas intermkdiaire pour lequel la loi de frottement de Coulomb est applicable entre les contraintes normales et les contraintes de cisaillement. Dans ce cas, certaines parties des surfaces en contact sont adherentes, tandis que d’autres glissent les unes sur les autres. On Btudie egalement le probleme dun cylindre charge axidement par un collet rigide. On donne finalement des solutions pour le cas d’un arbre cylindrique, de longueur finie, soumis a la torsion par des manchons wncentriques flexibles. Dans tous les cas, it est don& des valeurs numeriques, pour les contraintes et les deformations de contact, en fonction de differents parametres du cylindre et des manchons. ZusammM-Zur Rerechnung der zwischen runden, zyhndrischen Wellen und Htilsen auftretenden KontaktkrMe und Deformationen werden numerische Methoden angewandt. Die erste Problemklasse, die behandeh wit-d, betrifft einen Rund~l~der, auf den die Hi&e aufgepresst bzw. aufg~chrumpft ist. Die verschiedenen behandelten Einzelfalle umfassen (a) den Fall der reibungslosen Beriihrung und (b) den Fall, wo die Reibung das vollkommene Zusammenhaften der Bert.ihrungsfl&chen bewirkt. Ein dazwischen liegender Fall, fur den das Coulombsche Reibungsgesetz zwischen normalen und scherenden Kontaktkriiften gilt, wird ebenfalls behandelt. In diesem Fall haften gewisse Zonen der Bertihrungsfl%chen aneinander, wahrend andere aufeinander gleiten. Es folgen Rerechnungen zu einem Problem, in dem ein Zyiinder durch steife Biinde achsial befastet wird. Zum Schluss wird ein weiterer Fat1 behandelt. in dem an einer zvlindrischen Welle bearenzter Lanae bei Reanspruchung durch biegsame konzentrische FRilsen Torsionsk&ten auftreten. Hie&r werden die Losungen erhalten. Ftir alle diese Ftille werden die numerischen Werte der Reriihrungsbeansprungen und Deformationen fur verschiedene Zylinder- und Htilsenparameter angegeben. SommarLei impiegano metodi numerici per scoprire Ie solle&azioni e le defor~oni di contatto fra alberi e maniwtti cihndrici cirwlari. La prima categoria di problemi trattati riguarda if ciliidro circolare sul quale B bloccato alla pressa o forzato a caldo un manicotto rigido. Si risolvono casi individuali in cui : (a) il wntatto b senza attriti, (b) si verifica un attrito sufficiente a provocare l’aderenza totale delle superfrci a contatto. I1 case intermedio i pure oggetto di discussione, in cui la legge di Coulomb sull’attrito vale nei riguardi delle sollecitazioni di contatto normaii e di tagho. In quest0 case, alcune parti delle super&i a contatto aderiscono, altre scorono una suti’aitra. Si risolve quindi ii problema di un cilindro caricato assialmente tram&e collari rigidi. Per ultimo, si ottengono soluzioni per il case dell’albero cilindrico di lunghezra 6nita sottoposto a torsione mediante manicotti wncentrici flessibili. In tutti i casi, si danno i valori numerici delle sollecitazioni e deformazioni di contatto per vari parametri di cihndri e di manicotti. Ai%rpa~3-kicno~r.B3yro~a rucJietiHbreM~TOKBI&KBOn~A~eH~~ KOHT~KTB~~X HaApB~e~ B Ae@OpMar@Br nptr conp~s~xBo~enmi KpyrnbrX nmnr~~p~recnri~ crepxBeJt B BT~~OK. DepBaK KaTeropailr pa3pa6aTbIBaeMbrX3Wax OTHOCHTCR K KpyrJtOMyI@rJtBB,Qpy,Ha KOTOpOMnOC%KeHaC WaTIITOMBBK HaC&KeHaB ropBBeM cocT0anmr XcecTKas BTyJrKa. PaspaliaTBmaroTca oco6eBBbre cnyuarr, Korm (a) BeT TpeBwa Mexw rro~epxrtoc~sr~rrCOII~XK~CHOB~HI~R mm (6) rr~ee~crr BocTaToluroe Tpemre &T.Bo6pa3oBaBBa nomioro cnemtemra no~epxsoc~e& wnp~~~o~~~. 06cywaeTcsr TaKxre npoMe~0~~~ cnyxa@, ~orjxa3a~orr ~periri5iKynoKr6a OcfaeTca B cmre BBa KorrTaKTBbrxrranpmKem&i, pacnonoxceBtibrX ~ie~oty copes B KaCaTeJI8BBrMB ~anpr~~~xh%~. B 3~0~ CnyKae HeKoTopbre y’aacr~ri conpAKacaroqBXca nOBepXBOCTett cnennerrbr Memny cotsoti, TOrm KaK Apymz CKOJIb3ffT OAm n0 Ap)TOMy. Pa3pa6aTbmewcff 3aTehf3a,qara,Kacamrqami lWlEkUXpa C oCeB0i-iHarpy3KOZinOCpeACTBoM?KeCTKBX @tamreB. HaKoBeu norryrarorca pemenmr ~rrri uryrarr ~P~KOrO cTepB%rsiorpaawemrott JrJmBbt, HCK$XiBJXHEO~O B3-3a J@CTBEa &ByXCoocBbrXBO,uaTBBBbrX BTyJrOK. RO BceX CJQ’%KXAarofcir BBCIreHBBte 3xaBeBBK KorrTaKTBbrXHa~~~e~~~ H A~pMa~ mra piufnirx rrapaMeTpoB BLOB R B~ynoK.
CONTACT STRESSES BETWEEN CYLINDRICAL SHAFTS AND SLEEVES H. D. CONWAY?and K. A. FARNHAM~ Department
of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York (Communicated
by I. N. SNEDDON)
Abstract-Numerical methods are used to tind the contact stresses and deformations between circular cylindrical shafts and sleeves. The first class of problems treated concerns the circular cylinder onto which a rigid sleeve is press-fitted or shrunk. Individual cases are worked out in which, (a) the contact is frictionless and (b) sufficient friction occurs to cause complete adherence of the surfaces in contact. The intermediate case is also discussed where Coulomb’s law of friction holds between normal and shearing contact stresses. In this case, certain portions of the contacting surfaces adhere, whereas others slide over one another. The problem of a cylinder loaded axially through rigid collars is then worked out. Finally, solutions are obtained for the case of a cylindrical shaft of finite length twisted by means of flexible concentric sleeves. In all cases, numerical values of the contact stresses and deformations are given for various cylinder and sleeve parameters. 1. INTRODUCTION
ONE of the most difficult class of contact stress problems to treat analytically and one which has received very scant attention in the literature, is that of finding the stresses between circular shafts and sleeves. At the same time, these problems have very considerable practical interest. They are the subject of the present article, which is the third in a series on the subject [l, 21. The first part of the article concerns the stresses and deformations produced when a sleeve is shrunk or press-fitted onto a cylinder. Two extreme cases are considered, in the first of which the effects of friction are ignored. The other extreme case is then treated where the friction induced at the contact surfaces is sufficient to cause complete adherence. Finally, the intermediate case is investigated where complete adherence occurs only over part of the contacting surfaces and relative slipping over the remainder. In the next case considered, a cylinder is elongated by forces applied through sleeves attached to the shaft and the shearing and normal stresses at the junctions of the cylinder and sleeves are worked out. In all the above cases, the rigidity of the sleeves is assumed to be very great compared with that of the cylinder. In addition, the cylinder is assumed long, so that end effects do not enter the problem. Thus, as a final contact stress problem, the case of a finite length cylinder twisted by couples applied through sleeves, is considered, the modulus of rigidity of the cylinder and sleeves being arbitrary. The method of solution is essentially adapted from that used in the previous articles [ 1,2]. The normal and shearing contact stress distributions are replaced by series of pressures and shears constant over large numbers of incremental lengths. Satisfaction of the boundary t Professor, Department of Theoretical and Applied Mechanics, Cornell University. IBM Corporation, Endicott, N.Y., U.S.A. $ Systems Development Division, IBM Corporation, Endicott, N.Y., U.S.A. 541
Consultant,
H. D.
542
CONWAY
and K. A. FARNHAM
conditions is then achieved at discrete points, leading to series of equations in the applied incremental pressures and shear, and hence, to approximations of the distributions. It has been shown [2] by comparison with known exact solutions that the techniques are capable of giving very accurate results. 2. FINITE
LENGTH
RIGID
SLEEVE SHRUNK CYLINDER
ON A CIRCULAR
Frictionless case
The cylindrical contact area was divided into 20 equal circular rings, but, because of it is only necessary to consider the radial displacements at the centers of 10 rings. The incremental pressures are denoted by pi, pz, . . . plo, and the resultant equal displacements at the centers of the 10 rings are found, making use of the solution for an infinite cylinder subjected to a single band of uniform pressure (Appendix I). Machine computations outlined in Appendix II were then used to sum the integrals involved and to solve the 10 simultaneous equations in the incremental pressures. Graphs of normal pressure distributions for various ratios of half sleeve length b to cylinder radius a are presented in Fig. 1. As might be imagined, the pressures for the larger b/a values are nearly uniform except near the ends. For b/a =&, the radius of the cylinder is very large compared with the length of the rigid sleeve, and hence, the pressure distributions shown closely approximate the exact solution of Sadowsky [3] for a rigid frictionless, axially-loaded punch applied to a half-space. The comparison made in Fig. 2 shows this to be the case and is added confirmation of the accuracy which can be expected from the present numerical method. symmetry,
2.6
2.6
1.6 P inab 1.2
0.8
I:
a
, ,
i---b+
,-.---J
FRICTIONLESS
-1
1.6
1.2
0.8
0.4
0.0
0.4
c
- 1.0
0.0 -0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
z 6
FIG. 1. Pressure distributions between a circular cylinder and a rigid sleeve for various values of b/a (frictionless contact).
and sleeves
Contact stresses between cylindrical shafts
543 4.0
3.5
3.0
3.0
2.5
2.5 P 4nab
X-EXACT SOLUTION b/a-O IFRICTIONLESSI 2.0
2.0
1.5
1.0
0.5
4 -1.0
-0.8
-0.6
-0'4
0.0
-0.2
0.2
0'4
0.6
0.8
0
1.0
2 b
FIG. 2. Comparison
of pressure distribution for b/u=1/32 (frictionless contact).
with the exact solution for b/u-+0
After the pressure distributions were computed, the radial displacement/force relationships were calculated. These radial displacements are presented in Table 1 as fractions of the corresponding radial displacement P(l -v)/4?rEb obtained for a cylinder of length 26 rather than infinite length. TABLE 1. RATIOOFRADULDISPLACEMENTBF FORAN MPINITECYLINDRRTO RADIAL DISPLACEMENT &=P (I--V)/4ZEb POR A CYLINDW OF LENGTH 2b POISSON'SRATIO V =0’3
Frictionless case l/32
b/a
0.185
WSO
l/4 O-635
1 0.882
0.9230
4 0946
Table 1 shows the supporting effect of the cylinder outside the sleeves. Clearly this effect is small where the length 2b of the sleeve is large compared with the diameter 2u of the cylinder. 3.
FINITE
LENGTH
RIGID
SLEEVE CYLINDER
SHRUNK
ON A CIRCULAR
Complete adherence case
The corresponding but far more complicated case where complete adherence takes place between the cylinder and sleeve was next investigated. As in the frictionless case, the contact area was divided into 20 equal incremental areas, and taking account of symmetry, these were assumed to be subjected to uniform normal stressespI,p2, . . . plo and uniform shearing stresses ql, q2, . . . qlo. Of course, the radial and axial displacements of the cylinder are both functions of pi, p2, . . . plo and ql, q2, . . . qlo.
H.D.
544
and
CONWAY
K. A. FARNHAM
These displacements were obtained by machine computation as indicated in Appendixes I and II. Setting the resultant radial displacements equal at the center of each incremental area and the resultant axial displacements zero, produced 20 simultaneous linear equations inpr,p,, . - *Ploy 419 q21 * * * qlo which were again solved by machine computation. Graphs of normal and shearing stress distributions found in the above manner for various b/a ratios are plotted in Figs. 3 and 4, respectively. When the results shown in Figs. 1 and 3 are compared, it appears that the normal stress distributions are similar, thus lending support to the frequently made assumption that friction does not markedly affect the contact normal stresses. However, it is noted from the complete adherence graphs in Fig. 3 that the positions of minimum normal stress for the larger values of b/a do not occur at the center, but move progressively away from the center. This effect is much more marked than in the frictionless graphs of Fig. 1, although it also occurs in these to a diminished extent.
TOTAL RADIAI LOAD P
3.0
3.0
2.5
POISSON'S RATIO = 0.3
2.5
2.0 P G%
1.5
I.0
1.0
? 0.5
0.5
0 -1.0
-0.8
-0.6
-0.4
-0.2
0.0 Z ii
FIG.
0.2
0.4
0.6
0.8
1.0
3. Pressuredistributionsbetweena circular cylinder and a rigid sleevefor various values of b/a (completeadherence)Poisson’sratio =0.3.
The shearing stress distributions shown in Fig. 4 are very interesting. It will be seen that the magnitudes of the shearing stresses for certain b/a ratios (for example b/a=&) increase as one goes away from the center and then decrease and reverse sign near the edge. In addition, the direction of the shearing stress at a specific location depends upon the b/a ratio. It is also apparent that there is a value of b/a about &- (calculated but not plotted) for which there is very little shearing stress, except near the ends of the sleeve. Effects similar to these were also observed [I] in indented strips and slabs under adherence conditions.
54s
Contact stresses between cylindrical shafts and sleeves
2b TOTAL LOAD
0.6 Ir
2
i$+i POISSON’S
RADIAI P
RATIO
= 0.3
0-i P drrab
o-0
O’C
0.2
0.4
4
-1.0
-0.8
-0.6
-0.4
-0’2
0.0
0.2
0.4
0.6
0’8
1.0
z ii
Fro. 4. Shearing stress distributions between a circular cylinder and a rigid sleeve for various values of b/a (complete adherence), Poisson’s ratio=0.3.
The radial displacements 6, were next computed and these are presented in Table 2 as fractions of the corresponding radial displacements 8r for frictionless conditions. TABLE 2. RATIOOFRADIALDISPLACEMENT~A FORCOMPLETEADHERENCETO RADIALDISPLACEMENT~FFOR~'RICTIONLESSCONTACT POISSON'SRATIOV=0*3
% 6Al6P
l/32 130
l/4 1.00
1
2
4
0.93
0.86
0.82
As expected, the effects of friction on the radial displacement are only important for the higher values of b/a. 4. FINITE
LENGTH
RIGID
SLEEVE CYLINDER
SHRUNK
ON A CIRCULAR
Partial slipping case
After cases of frictionless contact and complete adherence were investigated, the intermediate case of partial slipping was considered. As in the previously investigated partial slipping cases of indented strips and slabs [2], the half-length of the contact area was divided into 10 equal increments and a specific ratio of central adherence length to total contact length 2b was selected. The normal pressures were designated pi-lo. If, for example, the central adherence length consists of eight increments (adherence length/ contact length=0*4) with four on either side of the center line, the shearing stresses for these are designated ql+. The average shearing stress intensities for the six increments in each slip length are then kp,_,,.
H. D. CONWAY and K. A. FARNHAM
546
Thus, there are 15 unknowns to be found-ten& four q’s and the coefficient of friction k. Ten of the equations required for the solution are found by assuming that the radial displacements at the centers of the increments are equal. Four more are obtained by making the axial displacements at the centers of the four adherence increments zero. The final equation is obtained by prescribing the axial displacement at the center of the increment immediately outside the contact length. The solution of these nonlinear equations can be avoided by noting that the radial pressure distribution changes little from the frictionless to the fully adhered case. From this fact, the set of 15 equations is reduced to five linear ones which can be readily solved. The results obtained using the frictionless and full adherence pressures agree well with one another. Figure 5 shows a typical graph of adherence length/sleeve length versus friction coefficient k for b/u =& and Poisson’s ratio =0*3. It is seen that a friction coefficient k =O-4 produces almost complete adherence. In view of the reversal of the shearing stresses for larger b/a values shown in Fig. 4, tracing the adherence lengths with varying coefficients of friction would be most complicated for these values. However, this would be of academic interest only, since it is apparent that a relatively small friction coefficient causes almost complete adherence.
0
0.2
0.4
0.6
0.8
1.0
FRICTION COEFFICIENT k FIG.
5.
Adherence
length/sleevelength vs. friction coefficientfor b/a= Poisson’s ratio =O-3.
5. AXIAL GOADING OF INFINITE CYLINDER THROUGH RIGID
l/32.
CIRCULAR SLEEVES
Complete aa%erence case The important practical problem of a circular cylindrical shaft loaded axially by forces applied through rigid sleeves was next investigated, using the above numerical techniques. Typical graphs of shearing and normal stresses at the junction of cylinder and sleeve are shown in Figs. 6 and 7, respectively. The centers af the two sleeves are assumed to be llb apart in all cases. As expected, the normal stresses are small compared with the corresponding shearing stresses.
Contact stresses between cylindrical shafts and sleeves
547
35
30
25
20 % na 15
10
5
0’0
0’1
0.2
0’3
0’4
0.5
0’6
0.7
0.8
0.9
1.0
2 6
FKL 6. Shearing stress distributions between an axially loaded circular cylinder and a tid sleeve for various values of b/u (complete adherence). Distance between sleeve centers = 1lb, Poisson’s ratio=0*3.
15
10
Ld4: 7 l-1 / / 2
5 Q 2
0.1
0
-5
0.2
__1..
0.3
4
-L____
0.4
0.5
t
0.6
0.8
0.9
1
:
-10
- 15
FIG. 7. Normal stress distributions between an axially loaded circular cylinder and a rigid sleeve for various values of b/u (complete adhuence). Distance between sleeve centers=llb, Poisson’s ratio=003.
548
H. D. CONWAYand K. A,
FARNHAM
It is very interesting to compute the relative axial movement these are presented in Table 3. TABLE 3. THROUGH
apart
of the sleeves,
RATI~~OF AXIALDISPLACEMENTBOFCLRCULARCYLINDERLOADED RIGID SLEEVES TO &=lOQb/nuW. DISTANCEBETWEENSLEEVE CENTERS=~~~
b/a
l/32
6160
4.202
l/4 1.319
1
2
1.063
1.031
As expected, the axial displacement 6 rapidly approaches 6, as the b/a ratio is increased. 6. TORSION
OF
FINITE CIRCULAR CYLINDER FLEXIBLE SLEEVES
THROUGH
In all the problems previously discussed, the circular cylinder has been assumed to be of infinite length and the sleeve completely rigid. The case of a finite length circular cylinder twisted through flexible sleeves will now be considered. In the numerical solution of the above problem the same technique is applied. The solutions for the finite length cylinder twisted by equal but opposite bands of uniform shearing stress and a finite thickness disc subjected to uniform shearing stress are given in Appendix III. In the present problem it is only necessary to match the tangential displacements at the junctions of the cylinder and sleeves. No normal stresses are induced either in the cylinder or the sleeves. This fact enables the latter to be of finite length in the investigation, since only one condition has to be satisfied at a junction rather than the customary two. Graphs of the shearing stress distributions at the junction for various values of the ratio G,/G, of the modulus of rigidity of the cylinder and sleeve are given in Fig. 8. The e/b and e/a values (see Fig. 8) are assumed to be equal and it was found that the stresses remain practically unchanged for e/b = e/a > 1.5.
1'2
-;-.-;
:I---;--
1'0
,, '-_--TORQUE T"-*__ 2 __._
O-8
0'6
_/
'____2e ; =f>
0.0
OS1
0.2
1.5
0.3
G,
__ -.
1
1 3
_,__ CYLINDER MODULUS
T--J ---~~
__
SLEFVE MODULUS
c.,
~~~_
POISSON'S RATIO = OS3
0'4
0.5
0.6
0.7
0.8
0.9
I.0
Z Ti FIG. 8.
Shearing stress distributions for a finite length circular cylinder twisted through flexible sleeves (complete adherence), e/b=e/a> 1.5. Poisson’s ratio =0.3.
Contact stressesbetween cylindricalshafts and sleeves
549
The torque T=KG,a3 per unit (radian) twist of the cylinder and sleeves was then computed for various values of G,/G, and e/b =e/a and is given in Table 4. TABLET.
VALUESOF
KIN TORQUE T=KGc a3 PER UNITTWIST FOR CYLINDER AND SLEEVESIN FIG. 8. POISSON’S RATIO=O-3
e/b =e/a
GclG~=j
Ge/Gs = 2
G,/G,=l
1.5
1.037
O-925
0 -777
2.5
0447
0.425
0.391
5
0.184
0.181
0.174
7.5
0.116
0.115
0.112
From simple strength of materials theory, the twist of the cylinder may be estimated as and that of each sleeve as T/4nGp2b. Thus, the torque To per unit radian twist of the cylinder and two sleeves together is, by strength of materials theory, 4Te/G,na4
To=
2nGsa2 b _ , I_\
.
Ratios of the torques T/T, are given in Table 5 for G,/G, = 1 and a = b. TABLE 5. RATIO OF TORQUE PER UNIT RADIAN TwISTTOCORRESpONDINGVALuEGlVENBYSIMPLE STRENGTH OF MATERIALS THEORY. G,/G,=l, POIsSGN'SRATIo=0*3
e/b =e/a
T/To
1.5
1.914
2.5
1.420
5
1.178
7.5
1.113
10
1.083
As expected, the T/T, values approach unity for the larger e/b = e/a values. REFERENCES M and S. So, Normal and shearing contact strew in [l] H. D. CONWAY,S. M. VOGEL,K. A. FARNHA indented strips and slabs. Znr..Z. Engng Sci 4,343 (1966). [2] H. D. CONWAYand K. A. FARNI-IAM, The contact stress problem for indented strips and slabs under conditions of partial slipping. Znf.J. Engng Sci. 5, 145 (1967). [3] S. TIMCWHENKO and J. N. GOODIER, Theory of Elasticity, p. 96. McGraw-Hill (1951). [4] A. I. LUR’E, Three Dimensional Problems of the i%eory of Elasticity. p. 394. Interscience (1964). (51 C. HASTINGS, Approximations for Digital Computers, p. 197. Princeton (1955).
550
H. D. CONWAYand K. A. FARNHAM
APPENDIX
I
To investigate the pressure and shearing stress distributions between a rigid sleeve and a flexible cylinder by the foregoing numerical methods, it is tirst necessary to find the surface displacements of the cylinder in Fig. 9.
FIG. 9. Cylinder subjected to band of uniform pressure. By expanding the normal surface loading into a Fourier integral, it is easy to show [4] that the radial displacements on the surface are given by
s
m 2( 1- v)1:(~)sin(/3c/a)cos(j?z/a) urEe= -- pa nG o B3CZ7d(8)-Z:(B)1-28(l-v)zt(~dB whereas the axial displacements
W,=,
_
Pa
61)
on the surface are
s
nG
*
o
v)zI(B)zO(B> + B[‘:(B) -‘~(B)]sinBesinSZdp 64.2) a a * S”CG
2(1-
In addition to the above, it is necessary to compute the radial and axial displacements due to the surface shear loading shown in Fig. 10. These are found in a similar manner [4] to those for the normal loading and are given by
u,=,_ _w “BCZ~(B)-Z:(B)1-2(1-v>Zo(B)ZliP> 71~~ -ml -au - vv:m s o B3CGW E w
=
s
=ga
’ =
aG
o&dj? a
,
(A.31
“(3-2v)B~~o-Bz:(8)-4(1-v>~o(B)z~(B>
o
B4CW)-ml
- v2u - vv:m
SHEARING
STRESS
q
q$--yf$(i
-7-
FIG. 10. Cylinder subjected to equal but opposite bands of axial shearing stress. It is neoessBTy to evaluate equations (A.lHA.4) in Appendix II.
numerically and the method for doing so is outlined
Contact stresses between cylindrical shafts and sleeves
551
APPENDIX II The following methods were used to obtain approximate numerical values for the infinite integrals Riven in Appendix I for various values of c/a, d/a and z/la. The given integrand was integrated numerically from p=O to B=A u&g a Simpson’s Rule integration. For values of jI>A, the integrand was replaced by asymptotic approximations for the Bessel functions and the integration was continued to j?=B. For values of fbB, the integrals were replaced by suitable sine or cosine integral values. Typical values of A and B were 40 and 200 respectively. The required asymptotic approximations for the Bessel functions include
From these expressions,
equation (A.]) becomes
A2(1- v)l&9)sin@c/a)cos(pz/a)dj?
-2/w- VW)
0 B3CG(B)-ma
+
s*
s
m2( 1- v)sin(/?c/a)cos(Bz/a)
The third of these. integrals upon integration 2(1
B2
I3
by parts produces
v)sin~c/a~20s@z~d,&
_
(l_
sin[(c; ‘jBal
v)
B
+ sin[(c - z)Ba] B
-~fjCi[(~)s]}
where
Ci[x]=
Ft.
s
In the same manner, equation (A.2) yields
s
+ ‘2 (‘-~)[‘-(3/‘8)1[‘+(‘/‘8)1-‘sin~sin~dp
A
--
/I’-2j3(1-v)[l-(3/4/I)]
(1-2~)
2
cos[(c+z)Ba] B
a
+[yjsi[r$)B:
1
+ cos[(c - z)Ba] B
+pbzjsi[(gq]}
where
si[x] = -
m s&tdt s x
t
and
si[x] = Si(x) - IL/~.
a
H. D. CONWAYand K. A. FARNHAM
552
The functions a(x) and S(x) are computed from excellent polynominal approximations given by Hastings [S]. The numerical evaluation of expressions (A.3) and (A.4) is accomplished in a similar manner. An appreciable saving in computation time can be realised by noting that equations (A.l) and (A.4) are symmetric and equations (A.2) and (A.3) have a transpose relationship.
APPENDIX
III
When analyzing the problem of a cylinder twisted by flexible sleeves (Fig. 8), it is necessary to obtain an expression for the tangential displacement at the boundary of the hole loaded by a band of shearing tractions q as shown in Fig. 11. This requires special treatment.
', /-
-;,
SHEARING
STRESS q
FIG. 11. Infinite plate subjected to band of uniform torsional boundary.
shearing stress on inner
Consider an elastic body in which the displacements u=w=O and the displacement u is independent of the coordinate 0. It follows that the three direct strains and one shearing strain are zero e,=es=e==O
9
Yrz=O.
(A.51
The stresses are given by
64.6) Of the three equations of equilibrium,
two are identically satisfied and the other becomes
a%
Ian rar
--$s----+-=o.
A particular solution of this equation is v=As/r.
D
r2
a20 a22
(A.71
For further solutions we write
u = V(r)sin kz, cos kz .
(A.@
v=o.
(A.9
It follows from equation (A.7) that
The solution is V=Al,(kr)+BK,(k~) where Zl and KI are moditied Bessel fractions. A=0 in the present problem.
As r+ ~0, I&b)-+ to and therefore, it is appropriate
(A.lO) to write
553
Contact stresses between cylindrical shafts and sleeves For the shearing stress distributions
%L. =+-2 We construct
shown in Fig. 11, we write
=!,
”
the displacement
k$.
~sinkccoskdcoskz, 9*
(A.ll)
function
u=$‘+
i
k$
B,K,(kr)coskz,
n=l,
(A.12)
2,
whence
r,,=G,r;
=---2Wo
0 f
r2
i
B,kG,K,(kr)cos
kz
(A.13)
.
n=l,Z,
It follows from equating (A.11) and (A.13) that
&=-,
qca’
B,=-
G,b
4qb
sin(nnc/b)cos(md/b)
I127X2G S
(A.14)
K,(nxalb)
and finally
va
l&,=-+7 G,b
4qb ’ K1(ka)sin kc cos kd cos kz , n: G,,=,,2,~Kz(ka)
Ii
k=!!f.
(A.15)
It is interesting to note that the stresses associated with the summations in the above expressions produce no torque. The normal and shearing stresses on .z=O and b are, of course, zero. The other displacements required in the solution of the problem pertain to the circular cylinder loaded as shown in Fig. 12. These are obtained in the usual manner giving
u,=,=8qe f
(- 1)(“-‘)~211(n7ta/2e)
~2G,,=l,
3,
n2
I,(nxa/2e)
&!E&!E 2e
2e
,
(A.16)
The normal and shearing stresses on the ends z= +e of the cylinder are zero.
SHEARING
STRESS q
FIG. 12. Cylinder subjected to bands of uniform torsional shearing stress.
Machine methods were used to sum the series in equations (A.15) and (A.16). The results obtained using four- and then five-hundred terms were compared and found to agree very closely. (Received 17 October 1966)
554
H. D.
CONWAY
and K. A. FARNHAM
R&arm~On ttudie des methodes numbiques pour determiner les contraintes de contact et les deformations entre des arbres cylindriques circulaires et des manchons. Le premiet probleme trait& se rapporte au cas du cylindre circulaire sur iequel un manchon rigide a et6 monte a la presse ou fret& Des cas particuliers sont examines, dans lesquels (a) le contact se fait sans
frottement et (b) le frottement est suffisant pour assurer I’adhtsrence complete des surfaces en contact. On Ctudie aussi le cas intermkdiaire pour lequel la loi de frottement de Coulomb est applicable entre les contraintes normales et les contraintes de cisaillement. Dans ce cas, certaines parties des surfaces en contact sont adherentes, tandis que d’autres glissent les unes sur les autres. On Btudie egalement le probleme dun cylindre charge axidement par un collet rigide. On donne finalement des solutions pour le cas d’un arbre cylindrique, de longueur finie, soumis a la torsion par des manchons wncentriques flexibles. Dans tous les cas, it est don& des valeurs numeriques, pour les contraintes et les deformations de contact, en fonction de differents parametres du cylindre et des manchons. ZusammM-Zur Rerechnung der zwischen runden, zyhndrischen Wellen und Htilsen auftretenden KontaktkrMe und Deformationen werden numerische Methoden angewandt. Die erste Problemklasse, die behandeh wit-d, betrifft einen Rund~l~der, auf den die Hi&e aufgepresst bzw. aufg~chrumpft ist. Die verschiedenen behandelten Einzelfalle umfassen (a) den Fall der reibungslosen Beriihrung und (b) den Fall, wo die Reibung das vollkommene Zusammenhaften der Bert.ihrungsfl&chen bewirkt. Ein dazwischen liegender Fall, fur den das Coulombsche Reibungsgesetz zwischen normalen und scherenden Kontaktkriiften gilt, wird ebenfalls behandelt. In diesem Fall haften gewisse Zonen der Bertihrungsfl%chen aneinander, wahrend andere aufeinander gleiten. Es folgen Rerechnungen zu einem Problem, in dem ein Zyiinder durch steife Biinde achsial befastet wird. Zum Schluss wird ein weiterer Fat1 behandelt. in dem an einer zvlindrischen Welle bearenzter Lanae bei Reanspruchung durch biegsame konzentrische FRilsen Torsionsk&ten auftreten. Hie&r werden die Losungen erhalten. Ftir alle diese Ftille werden die numerischen Werte der Reriihrungsbeansprungen und Deformationen fur verschiedene Zylinder- und Htilsenparameter angegeben. SommarLei impiegano metodi numerici per scoprire Ie solle&azioni e le defor~oni di contatto fra alberi e maniwtti cihndrici cirwlari. La prima categoria di problemi trattati riguarda if ciliidro circolare sul quale B bloccato alla pressa o forzato a caldo un manicotto rigido. Si risolvono casi individuali in cui : (a) il wntatto b senza attriti, (b) si verifica un attrito sufficiente a provocare l’aderenza totale delle superfrci a contatto. I1 case intermedio i pure oggetto di discussione, in cui la legge di Coulomb sull’attrito vale nei riguardi delle sollecitazioni di contatto normaii e di tagho. In quest0 case, alcune parti delle super&i a contatto aderiscono, altre scorono una suti’aitra. Si risolve quindi ii problema di un cilindro caricato assialmente tram&e collari rigidi. Per ultimo, si ottengono soluzioni per il case dell’albero cilindrico di lunghezra 6nita sottoposto a torsione mediante manicotti wncentrici flessibili. In tutti i casi, si danno i valori numerici delle sollecitazioni e deformazioni di contatto per vari parametri di cihndri e di manicotti. Ai%rpa~3-kicno~r.B3yro~a rucJietiHbreM~TOKBI&KBOn~A~eH~~ KOHT~KTB~~X HaApB~e~ B Ae@OpMar@Br nptr conp~s~xBo~enmi KpyrnbrX nmnr~~p~recnri~ crepxBeJt B BT~~OK. DepBaK KaTeropailr pa3pa6aTbIBaeMbrX3Wax OTHOCHTCR K KpyrJtOMyI@rJtBB,Qpy,Ha KOTOpOMnOC%KeHaC WaTIITOMBBK HaC&KeHaB ropBBeM cocT0anmr XcecTKas BTyJrKa. PaspaliaTBmaroTca oco6eBBbre cnyuarr, Korm (a) BeT TpeBwa Mexw rro~epxrtoc~sr~rrCOII~XK~CHOB~HI~R mm (6) rr~ee~crr BocTaToluroe Tpemre &T.Bo6pa3oBaBBa nomioro cnemtemra no~epxsoc~e& wnp~~~o~~~. 06cywaeTcsr TaKxre npoMe~0~~~ cnyxa@, ~orjxa3a~orr ~periri5iKynoKr6a OcfaeTca B cmre BBa KorrTaKTBbrxrranpmKem&i, pacnonoxceBtibrX ~ie~oty copes B KaCaTeJI8BBrMB ~anpr~~~xh%~. B 3~0~ CnyKae HeKoTopbre y’aacr~ri conpAKacaroqBXca nOBepXBOCTett cnennerrbr Memny cotsoti, TOrm KaK Apymz CKOJIb3ffT OAm n0 Ap)TOMy. Pa3pa6aTbmewcff 3aTehf3a,qara,Kacamrqami lWlEkUXpa C oCeB0i-iHarpy3KOZinOCpeACTBoM?KeCTKBX @tamreB. HaKoBeu norryrarorca pemenmr ~rrri uryrarr ~P~KOrO cTepB%rsiorpaawemrott JrJmBbt, HCK$XiBJXHEO~O B3-3a J@CTBEa &ByXCoocBbrXBO,uaTBBBbrX BTyJrOK. RO BceX CJQ’%KXAarofcir BBCIreHBBte 3xaBeBBK KorrTaKTBbrXHa~~~e~~~ H A~pMa~ mra piufnirx rrapaMeTpoB BLOB R B~ynoK.