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Stress distribution in shrink fit with elastic-plastic hub exhibiting variable thickness
Int. J. Mech. Sei. Vol. 35, No. 1, pp. 39~16, 1993
0020-7403/93 $6.00 + .00 ~;) 1993 Pergamon Press Ltd
Printed in Great Britain.
STRESS DISTRIBUTION IN SHRINK FIT WITH ELASTIC-PLASTIC HUB EXHIBITING VARIABLE THICKNESS U . GI3VEN Department of Mechanical Engineering, Ylldlz Technical University, Besiktas, Istanbul, Turkey (Received 6 November 1991; and in revised form 13 July 1992) Abstract--The state of plane stress in shrink fit with an elastic-plastic hub exhibiting variable
thickness is studied. The thickness of the hub is assumed to vary exponentially with a general power of the radial distance from the inner boundary of the hub. The analysis is based on Tresca's yield condition, its associated flow rule and linear strain-hardening. Closed-form solutions are obtained in terms of confluent hypergeometric functions.
NOTATION O'r, (70 (70, O~y ~eq
de U E, v 2
a, b h(r) h o, n, k A,B,C,D,C 1 P
radial and circumferential stress components initial and subsequent yield stress, respectively equivalent plastic strain strain increment radial displacement Young's modulus and Poisson's ratio work-hardening parameter elastic-plastic interface radius inner and outer radii of the hub hub thickness as a function of radius r real constants in h(r) = hoe-"('/b)k constants of integration superscript denoting plastic component INTRODUCTION
Shrink fits are found frequently in mechanical engineering. The importance of shrink fits rests on the fact that they are capable of transmitting high moments at low production costs. To better utilize the hub material, plastic deformation is admitted in many cases. A useful approach to practical computations has been developed I-1] by using Tresca's yield condition. A generalization of Kollmann's work for linear strain-hardening materials has been given by Gamer and Lance [2]. The case of a shrink fit with a solid inclusion has been investigated [3, 4]. In these analyses, the Young's modulus, Poisson's ratio and yield stress of the hub and inclusion are different. Furthermore, it has been shown that hub material with an arbitrary nonlinear hardening law can be taken into account without excessive numerical calculations I-5, 6]. The papers mentioned above consider the thickness of the hub to be constant. The case of a hub with variable thickness has been investigated by the present author [7] by assuming a thickness function of the form h = ho r-n. In the present work, the thickness of the hub is assumed to vary exponentially with a general power of the radial distance from the inner boundary of the hub. The inclusion is a circular solid disk with uniform thickness (Fig. 1). The aim of this work is to develop a useful analytical solution for an elastic-plastic hub with thickness varying in an exponential form under the assumption of Tresca's yield condition, its associated flow rule and linear strain-hardening. BASIC EQUATIONS AND SOLUTION
We consider a state of plane stress and assume infinitesimal deformation. It is assumed that the variation of thickness is radial and symmetric with respect to the midplane. We shall confine ourselves to the problem of purely elastic behavior of the solid disk and concentrate on the elastic-plastic behavior of the hub with variable thickness. Therefore, for 145 3S:I-D
39
40
U. G0VEN
2(l+a) I
i'
I .
L
I
2'.
.
2b FIG. 1. Shrink-fit geometry prior to assemblage.
the investigation under consideration, it is assumed that the radial stress does not become positive inside the plastic region due to the interference. For a ~< r ~< z the hub material is plastic, while for z ~< r ~< b it is still in an elastic state. So long as the inequalities a, < 0 and ¢0 > 0 hold, the plastic deformation of the huh is governed by the yield condition ~0 -
~, = ~.
(1)
The work-hardening law for a linear isotropic hardening material is given by ay = %(1 + ~/e=q)
(2)
where ao is the initial tensile yield stress, ~/ is the hardening parameter and e=q is an equivalent plastic strain. According to the flow rule associated with Tresca's yield condition, de p + d e p = 0 ,
de p = 0 .
(3)
Consideration of the equivalence of the increment of plastic work yields. e=q = t;
(4)
The governing differential equation of equilibrium [8] is d
drr (hrtT,) - htTe = 0,
(5)
where h is the thickness function, and the geometric relations du t, = drr'
u t0 = -r
(6)
hold in the entire hub irrespective of material behavior. Total strains are decomposed into elastic and plastic components. The stress-strain relations are: 1
wr#) + e,p
(7)
e0 = ~ (~0 - ~<',) + e~.
(S)
e, = ~ (o', -
1
Since we restrict ourselves to small strains, e, and t0 must satisfy the compatibility equation d dr (re0) = e,.
(9)
Substituting the strains e, and ee into the compatibility equation (9), and using (1)-(5), we obtain ~r2 +
3+r~-
dr +
r2 ~
h2
+
2 + 1+----~
r~
~'=I+H
Stress distribution in shrink fit
41
which is the differential equation for the radial stress, where a prime denotes differentiation with respect to r, and H = ~/aoE. The thickness of the hub is assumed to vary along the radius in the form
h = ho e-n('/b)u
(11)
where ho, n and k are real constants. Using (11), the differential equation (10) can be rewritten in the form
r -~-r2 +
3-kn
k
r--~r-
k+
a , = l ~ - H.
1-t-l+-~)Jkn~-~)
(12)
By using the transformation rl = n ~
(13)
we obtain the following differential equation of confluent hypergeometric type [9] d2tr, [ ( ~ ) rl-~-r~ + 1+
-Ida, -rlJ~rl-
1( ~ k+l+
1 +vH'~ 2fro 1 l+U /a,=k2(l+U)rl.
(14)
The general solution of this equation may be written as follows:
tr, = AP(rx) + BQ(rx) + R(rx)
(15)
where R(rl) is a particular integral of the equation, and P(r~) and Q(rl) are defined as follows:
P(rl)
ct ~(~+ 1) r 2 + . . . F(~;fl;r~)= 1 +~.flr~ + 2 ! p ( f l + 1)
(16)
Q(rl) = rl - p f ( c t - fl + 1 ; 2 - fl;rl)
l +vH'~ 1 ~ _ - ~ , J,
1( ~t=~ l+k+
(17)
fl=l+~.
2
(18)
F denotes the confluent hypergeometric function. P(rl) and Q(r:) are functions which are independent of each other as long as fl is not an integer. However, when fl is an integer, they become dependent. An independent solution may then be obtained by using the Frobenius method as follows: For fl/> 2
Q(rl) = F(fl ~ _ _ _- _ ) IF(or; fl;rl)lOg Irll + ( - 1)PG(ot;fl;rl) - ( - 1)PT(~t;fl; rl)] r(1 -- ct)
(19)
where ~rl
G(~t;fl;rl) = 1 + ~
[~,l(0t) - $1(1) - $#(1)]
ct(~t + 1)r 2 + 2!fl(fl + 1) [~b2(~) - ¢2(1) - ~bp+l(1)] + "'" r ( ~ ; B; r l ) = 1 + ¢ p - ~ ( 1 )
+
+ (~ -
1)!(B -
" "" + (1 - - ) ( 2 -
~)..-
2)!
'
(/~ -
(1 -
~)r-------~+ (1 -
1 - ~)r{ -1
1
~)(2 -
(20)
~)r~
(21)
in which, for s i> 1 1 #s(~) =
,.=l~+m--
1"
(22)
42
U. G0VEN
The particular solution can be found in series form, as follows:
R(r:)=Ao ~o+
-1
~-
20o
~r~
1 +
where -2 k[2 + (I + v)H]'
a 0
By using the equilibrium equation, the circumferential stress is found to be
cro= A (1-krl)P(rl)+r-~rP(rl)
+ B (1-krl)Q(rl)+
drQ(rt)
+ (1 - krl)R(rl) + r d R(rl).
(24)
Substituting for or, and a0 in (8), and using (1)-(4) and the circumferential strain-radial displacement relation, the radial displacement is obtained as follows: 1 d ] + B [ KoQ(r~)+~r~rQ(rl) 1 d 1 E--~U=Ar I K°P(r~)+'~rdrrP(rl) 1 d + KoR(rl)+-~r~R(r~)
ao
(25)
H
where
w2 =
H 1
__l- krl
(1-vH)
w2
H
+---H' Ko =
In the elastic region z ~< r ~< b, the stresses and radial displacement are obtained from (15), (24) and (25) for H ~ oo as follows: o- r ~
CPl(rl) + DQx(rl)
o- 0 ~
CI(1--krl)Pl(rl)+r~rPl(rl)
(26)
d
[
1
a
]
(27)
+ D (1 - v - krl)Ql(ri) + r _ Q l ( r l )
(28)
+ D (1 - krl)Ql(rl) + r ~ Q l ( r l ) Eu r
C[(1-
d v-krl)Pl(rl)+r~rPl(rl)]
where 0C1
Px(rx) = F(~;fl;r:)= I +l.-~r~ +
0~1(0~1 + 1) r~ + "'" 2~fl(fl+ 1)
Ql(rl) = r~-aF(~l - fl + 1; 2 - fl;rl) 1
~l=~(l+k+v)-
(29) (30) (31)
In the elastic inclusion 0 ~< r ~< a, the stresses and radial displacement are given by ar =tre = C1
Eu
--
r
= (1 -
v)C1.
(32) (33)
Stress distribution in shrink fit
43
THE ELASTIC HUB
For small values of interference, the stress field in the entire assembly is elastic. The integration constants C, D and C~ can be determined from the obvious boundary and interface conditions a,(a)li.¢. = a,(a)Ih,b = -- Pinterf,ce a,(b) = 0 and Uh.b -- Ui.¢. = 1. The integration constants are found as follows: D [ Ql(q,)
Pl(qX)p1 (n) Q(n)]=-Pi.t,m,¢,
C= C1 =
(34)
0 t ~..~.n.D Pl(n)
(35)
(36)
- - Pinterf,¢¢
where ql = n(a/b) k. The interface pressure can be found from the following equation
C[ Pl(ql)(1-v-kql)+
a ~rrPl(r,)l. d 1 +D [ (1-v-kql)Ql(ql)+a-~rrQ,(r,)l. d + (1 -
v)ei.te,f,c,
THE ELASTIC-PLASTIC
E1
= --.
]
(37)
d
HUB
In the elastic-plastic case, general expressions for the stresses and radial displacements contain the unknowns A, B, C, D and C1. Another unknown is the e!astic-plastic interface radius z. To determine these six unknowns there are six conditions available. The most convenient ones are: continuity of radial stress and displacement at r = z; continuity of radial stress at r = a; that the radial stress at the outer surface r = b vanishes; that at r = a, Uh,b -- Ui,c. = I; and, finally, the yield condition in Eqn (1) must be satisfied at r = z. When these conditions are enforced, the unknowns constants are found to be: D Z~rr Q~(rl)lz - kZlQl(Zl) - ~(n)[Z-drr P~(rx)lz - kzlP~(z~)
C-
= ao
0 ( ]~,.n.D
(38) (39)
P1 (n) B{ d zdrr Q(r~)lz-kzl Q(z~ ) - ~Q(Zl)[[_Z-~r d P(r~)lz - kzl P(zl ) ] }
=ao+
{R(z,) P(z,)
-kzlP(zl)
D
[
Q,(n)~,
,~][
d
P(z,) O,(z,)-p-~rltz~)j; z~P(r,)lz
] + kzlR(zx)- z~R(rl)l~ d
Q,(n)pl(zx)]_BQ(z,)_R(z,)} A = _p(z~){D[Q'(z~)-p---~ _1 C1 = AP(ql) + BQ(ql) + R(ql)
where
(4o) (41) (42)
44
U. GUVEN
The elastic-plastic interface radius z can be found from the following equation:
f
[
A KoP(ql) + ~ - ~ r r P ( r l ) l , + B KoQ(ql) + ~-~rr Q(r~)l, +
KoR(ql)
a ~rr d R(rl) l= - -~ ao -- (1 + ~-~
l
v)C1 = - E1 -. a
(43)
THE FULLY PLASTIC HUB
In the particular case, for z = b the hub becomes fully plastic. The integration constants and C~ can be determined from the boundary and interface conditions a,(a)[ine. = a , ( a ) [ h u b ---- - - P i n t e r f . . . . a,(b) = 0 and Uhub- U i n c . = I. The integration constants are found as follows:
A,B
B[Q(q~)-~Q(n)]
= -P(q') p - ~ R ( n ) - R ( q l ) - Pinterface 1
A = -- - - [BQ(n) + R(n)] P(n) C1 =
(44) (45) (46)
- - Pinterface-
The interface pressure can be found from the following equation:
A KoP(q~)+~-~rrP(rl)t.
+B
KoQ(q~)+-~-~rrQ(rl)la
a d ao E1 + KoR(q~) + - ~ d r R(r~)[= - ~ + (1 - v)Pi,t=rface = --'a
(47)
It should be noted that for a sufficiently large interference, I, the pressure at the interface can reach the yield stress. For larger values of I, our assumption of purely elastic inclusion behavior fails. In this case, plastic flow of the inclusion must also be considered. NUMERICAL
RESULTS
Figure 2 shows the influence of thickness variation on the stress distribution of a fully elastic hub, for
l ao(a)- o,(a)I,=o = l cro(a) - err(a)I,=o.s = ao.
0"7~x,,
n=0
I "~-.
0
.
n = 0.5
3
5
~
,C
Z5 0
x
0.75 I
-0.2
-0.4 FIG. 2. Stress distribution in a fullv elastic h u b
1 I
Stress distribution in shrink fit
45
0.9
nffiO
---nffi 0.5
0.45
~=
o
0.50
x
0.75
1
-0.35
-0. FIG.3.
Stress distribution in an elastic-plastic hub.
0.5
--1.5
x
rlffiO n-O.5
0.75
1
-0.5 s
~
FIG. 4. Stress distribution in a fully plasticized hub.
2L
~',
nffiO
1
OI
0.5
i
0.75
i
1
X
FIG. 5. Radial displacement distribution in a fully plasticized hub.
46
U. G0VEN
E1
The corresponding interferences for n = 0 and n = 0.5 are I -
- 1 and 0.9830999,
o-oa
respectively. Figure 3 shows the influence of thickness variation on the stress distribution of an elastic-plastic hub, for H = 1/3 and ¢ = z/b = 0.75. The corresponding interferences for n = 0 and n = 0.5 are I -
E1
- 2.25 and 2.0359999, respectively.
o-0a
Figures 4 and 5 show the influence of thickness variation on the stress and radial displacement distributions of a fully plastic hub, for H = 1/3. The corresponding interferences for n = 0 and n = 0.5 are I -
E1
- 4 and 2.3111999, respectively.
O-oa
Numerical results are given for a hub of variable thickness for the case of exponential power, k = 2 and n = 0.5. Poisson's ratio v and the radius ratio q = a/b are taken to be 1/3 and 0.5, respectively, ai~ is defined as di~ = ao/ao, ~ is defined as ti = Eu/aob, and x = r/b. CONCLUSION
The stress and radial displacement distributions in an elastic-plastic shrink fit with hardening based on Tresca's yield condition are well k n o w n [2-6]. In the present work, the thickness variation was considered to be of a more general form, h = ho e-"('/b)k Closedform solutions for the stresses and radial displacement were obtained in terms of confluent hypergeometric functions. It can be seen from the present analysis that the elastic-plastic interference radius z is influenced by the hardening parameter t/; the stress and radial displacement distributions in the outer elastic region are also influenced by the occurrence of hardening. However, it was reported in [ 2 - 6 ] that the occurrence of hardening does not influence the elastic-plastic interface radius z and the outer elastic region of a hub with constant thickness. It is interesting to observe from the present analysis that the stress distribution in a fully plastic hub with variable thickness is influenced by Poisson's ratio v. However, it can be shown that the stress distribution in a fully plastic hub with constant thickness [2] is not influenced by Poisson's ratio v. REFERENCES 1. F. G. KOLLMANN, Die Auslegung elastisch-plastisch beanspruchter QuerpreBverb~nde. Forsch. Ing. -Wes. 44, 1 (1978). 2. U. GAMER and R. H. LANCE, Residual stress in shrink fits. Int. J. Mech. Sci. 25, 465 (1983). 3. U. GAMER and R. H. LANCE, Elastisch-plastische Spannungen im Schrupfsitz. Forsch. Ing. -Wes. 48, 192 (1982). 4. U. GAMER, Der elastisch-plastische QuerpreBverband bei verschiedenen FlieBgrenzen von Innen- und AuBenteil. Forsch. Ing. -Wes. $6, 166 (1990). 5. U. GAMER, The shrink fit with nonlinearly hardening elastic-plastic hub. J. Appl. Mech. 54, 474 (1987). 6. U. GAMER, Die Teilplastizierte Nabe eines PreBverbandes. Z A M M 67, 65 (1987). 7. U. Gi)VEN, The shrink fit with elastic-plastic hub exhibiting variable thickness. Acta Mech. 61, 548 (1991). 8. S. P. T~MOSHENKO and J. N. GOOD~ER, Theory of Elasticity. McGraw-Hill, New York (1970). 9. D. IT and L. J. SLAVER, Confluent Hypergeometric Functions. Cambridge University Press, Cambridge (1960).
0020-7403/93 $6.00 + .00 ~;) 1993 Pergamon Press Ltd
Printed in Great Britain.
STRESS DISTRIBUTION IN SHRINK FIT WITH ELASTIC-PLASTIC HUB EXHIBITING VARIABLE THICKNESS U . GI3VEN Department of Mechanical Engineering, Ylldlz Technical University, Besiktas, Istanbul, Turkey (Received 6 November 1991; and in revised form 13 July 1992) Abstract--The state of plane stress in shrink fit with an elastic-plastic hub exhibiting variable
thickness is studied. The thickness of the hub is assumed to vary exponentially with a general power of the radial distance from the inner boundary of the hub. The analysis is based on Tresca's yield condition, its associated flow rule and linear strain-hardening. Closed-form solutions are obtained in terms of confluent hypergeometric functions.
NOTATION O'r, (70 (70, O~y ~eq
de U E, v 2
a, b h(r) h o, n, k A,B,C,D,C 1 P
radial and circumferential stress components initial and subsequent yield stress, respectively equivalent plastic strain strain increment radial displacement Young's modulus and Poisson's ratio work-hardening parameter elastic-plastic interface radius inner and outer radii of the hub hub thickness as a function of radius r real constants in h(r) = hoe-"('/b)k constants of integration superscript denoting plastic component INTRODUCTION
Shrink fits are found frequently in mechanical engineering. The importance of shrink fits rests on the fact that they are capable of transmitting high moments at low production costs. To better utilize the hub material, plastic deformation is admitted in many cases. A useful approach to practical computations has been developed I-1] by using Tresca's yield condition. A generalization of Kollmann's work for linear strain-hardening materials has been given by Gamer and Lance [2]. The case of a shrink fit with a solid inclusion has been investigated [3, 4]. In these analyses, the Young's modulus, Poisson's ratio and yield stress of the hub and inclusion are different. Furthermore, it has been shown that hub material with an arbitrary nonlinear hardening law can be taken into account without excessive numerical calculations I-5, 6]. The papers mentioned above consider the thickness of the hub to be constant. The case of a hub with variable thickness has been investigated by the present author [7] by assuming a thickness function of the form h = ho r-n. In the present work, the thickness of the hub is assumed to vary exponentially with a general power of the radial distance from the inner boundary of the hub. The inclusion is a circular solid disk with uniform thickness (Fig. 1). The aim of this work is to develop a useful analytical solution for an elastic-plastic hub with thickness varying in an exponential form under the assumption of Tresca's yield condition, its associated flow rule and linear strain-hardening. BASIC EQUATIONS AND SOLUTION
We consider a state of plane stress and assume infinitesimal deformation. It is assumed that the variation of thickness is radial and symmetric with respect to the midplane. We shall confine ourselves to the problem of purely elastic behavior of the solid disk and concentrate on the elastic-plastic behavior of the hub with variable thickness. Therefore, for 145 3S:I-D
39
40
U. G0VEN
2(l+a) I
i'
I .
L
I
2'.
.
2b FIG. 1. Shrink-fit geometry prior to assemblage.
the investigation under consideration, it is assumed that the radial stress does not become positive inside the plastic region due to the interference. For a ~< r ~< z the hub material is plastic, while for z ~< r ~< b it is still in an elastic state. So long as the inequalities a, < 0 and ¢0 > 0 hold, the plastic deformation of the huh is governed by the yield condition ~0 -
~, = ~.
(1)
The work-hardening law for a linear isotropic hardening material is given by ay = %(1 + ~/e=q)
(2)
where ao is the initial tensile yield stress, ~/ is the hardening parameter and e=q is an equivalent plastic strain. According to the flow rule associated with Tresca's yield condition, de p + d e p = 0 ,
de p = 0 .
(3)
Consideration of the equivalence of the increment of plastic work yields. e=q = t;
(4)
The governing differential equation of equilibrium [8] is d
drr (hrtT,) - htTe = 0,
(5)
where h is the thickness function, and the geometric relations du t, = drr'
u t0 = -r
(6)
hold in the entire hub irrespective of material behavior. Total strains are decomposed into elastic and plastic components. The stress-strain relations are: 1
wr#) + e,p
(7)
e0 = ~ (~0 - ~<',) + e~.
(S)
e, = ~ (o', -
1
Since we restrict ourselves to small strains, e, and t0 must satisfy the compatibility equation d dr (re0) = e,.
(9)
Substituting the strains e, and ee into the compatibility equation (9), and using (1)-(5), we obtain ~r2 +
3+r~-
dr +
r2 ~
h2
+
2 + 1+----~
r~
~'=I+H
Stress distribution in shrink fit
41
which is the differential equation for the radial stress, where a prime denotes differentiation with respect to r, and H = ~/aoE. The thickness of the hub is assumed to vary along the radius in the form
h = ho e-n('/b)u
(11)
where ho, n and k are real constants. Using (11), the differential equation (10) can be rewritten in the form
r -~-r2 +
3-kn
k
r--~r-
k+
a , = l ~ - H.
1-t-l+-~)Jkn~-~)
(12)
By using the transformation rl = n ~
(13)
we obtain the following differential equation of confluent hypergeometric type [9] d2tr, [ ( ~ ) rl-~-r~ + 1+
-Ida, -rlJ~rl-
1( ~ k+l+
1 +vH'~ 2fro 1 l+U /a,=k2(l+U)rl.
(14)
The general solution of this equation may be written as follows:
tr, = AP(rx) + BQ(rx) + R(rx)
(15)
where R(rl) is a particular integral of the equation, and P(r~) and Q(rl) are defined as follows:
P(rl)
ct ~(~+ 1) r 2 + . . . F(~;fl;r~)= 1 +~.flr~ + 2 ! p ( f l + 1)
(16)
Q(rl) = rl - p f ( c t - fl + 1 ; 2 - fl;rl)
l +vH'~ 1 ~ _ - ~ , J,
1( ~t=~ l+k+
(17)
fl=l+~.
2
(18)
F denotes the confluent hypergeometric function. P(rl) and Q(r:) are functions which are independent of each other as long as fl is not an integer. However, when fl is an integer, they become dependent. An independent solution may then be obtained by using the Frobenius method as follows: For fl/> 2
Q(rl) = F(fl ~ _ _ _- _ ) IF(or; fl;rl)lOg Irll + ( - 1)PG(ot;fl;rl) - ( - 1)PT(~t;fl; rl)] r(1 -- ct)
(19)
where ~rl
G(~t;fl;rl) = 1 + ~
[~,l(0t) - $1(1) - $#(1)]
ct(~t + 1)r 2 + 2!fl(fl + 1) [~b2(~) - ¢2(1) - ~bp+l(1)] + "'" r ( ~ ; B; r l ) = 1 + ¢ p - ~ ( 1 )
+
+ (~ -
1)!(B -
" "" + (1 - - ) ( 2 -
~)..-
2)!
'
(/~ -
(1 -
~)r-------~+ (1 -
1 - ~)r{ -1
1
~)(2 -
(20)
~)r~
(21)
in which, for s i> 1 1 #s(~) =
,.=l~+m--
1"
(22)
42
U. G0VEN
The particular solution can be found in series form, as follows:
R(r:)=Ao ~o+
-1
~-
20o
~r~
1 +
where -2 k[2 + (I + v)H]'
a 0
By using the equilibrium equation, the circumferential stress is found to be
cro= A (1-krl)P(rl)+r-~rP(rl)
+ B (1-krl)Q(rl)+
drQ(rt)
+ (1 - krl)R(rl) + r d R(rl).
(24)
Substituting for or, and a0 in (8), and using (1)-(4) and the circumferential strain-radial displacement relation, the radial displacement is obtained as follows: 1 d ] + B [ KoQ(r~)+~r~rQ(rl) 1 d 1 E--~U=Ar I K°P(r~)+'~rdrrP(rl) 1 d + KoR(rl)+-~r~R(r~)
ao
(25)
H
where
w2 =
H 1
__l- krl
(1-vH)
w2
H
+---H' Ko =
In the elastic region z ~< r ~< b, the stresses and radial displacement are obtained from (15), (24) and (25) for H ~ oo as follows: o- r ~
CPl(rl) + DQx(rl)
o- 0 ~
CI(1--krl)Pl(rl)+r~rPl(rl)
(26)
d
[
1
a
]
(27)
+ D (1 - v - krl)Ql(ri) + r _ Q l ( r l )
(28)
+ D (1 - krl)Ql(rl) + r ~ Q l ( r l ) Eu r
C[(1-
d v-krl)Pl(rl)+r~rPl(rl)]
where 0C1
Px(rx) = F(~;fl;r:)= I +l.-~r~ +
0~1(0~1 + 1) r~ + "'" 2~fl(fl+ 1)
Ql(rl) = r~-aF(~l - fl + 1; 2 - fl;rl) 1
~l=~(l+k+v)-
(29) (30) (31)
In the elastic inclusion 0 ~< r ~< a, the stresses and radial displacement are given by ar =tre = C1
Eu
--
r
= (1 -
v)C1.
(32) (33)
Stress distribution in shrink fit
43
THE ELASTIC HUB
For small values of interference, the stress field in the entire assembly is elastic. The integration constants C, D and C~ can be determined from the obvious boundary and interface conditions a,(a)li.¢. = a,(a)Ih,b = -- Pinterf,ce a,(b) = 0 and Uh.b -- Ui.¢. = 1. The integration constants are found as follows: D [ Ql(q,)
Pl(qX)p1 (n) Q(n)]=-Pi.t,m,¢,
C= C1 =
(34)
0 t ~..~.n.D Pl(n)
(35)
(36)
- - Pinterf,¢¢
where ql = n(a/b) k. The interface pressure can be found from the following equation
C[ Pl(ql)(1-v-kql)+
a ~rrPl(r,)l. d 1 +D [ (1-v-kql)Ql(ql)+a-~rrQ,(r,)l. d + (1 -
v)ei.te,f,c,
THE ELASTIC-PLASTIC
E1
= --.
]
(37)
d
HUB
In the elastic-plastic case, general expressions for the stresses and radial displacements contain the unknowns A, B, C, D and C1. Another unknown is the e!astic-plastic interface radius z. To determine these six unknowns there are six conditions available. The most convenient ones are: continuity of radial stress and displacement at r = z; continuity of radial stress at r = a; that the radial stress at the outer surface r = b vanishes; that at r = a, Uh,b -- Ui,c. = I; and, finally, the yield condition in Eqn (1) must be satisfied at r = z. When these conditions are enforced, the unknowns constants are found to be: D Z~rr Q~(rl)lz - kZlQl(Zl) - ~(n)[Z-drr P~(rx)lz - kzlP~(z~)
C-
= ao
0 ( ]~,.n.D
(38) (39)
P1 (n) B{ d zdrr Q(r~)lz-kzl Q(z~ ) - ~Q(Zl)[[_Z-~r d P(r~)lz - kzl P(zl ) ] }
=ao+
{R(z,) P(z,)
-kzlP(zl)
D
[
Q,(n)~,
,~][
d
P(z,) O,(z,)-p-~rltz~)j; z~P(r,)lz
] + kzlR(zx)- z~R(rl)l~ d
Q,(n)pl(zx)]_BQ(z,)_R(z,)} A = _p(z~){D[Q'(z~)-p---~ _1 C1 = AP(ql) + BQ(ql) + R(ql)
where
(4o) (41) (42)
44
U. GUVEN
The elastic-plastic interface radius z can be found from the following equation:
f
[
A KoP(ql) + ~ - ~ r r P ( r l ) l , + B KoQ(ql) + ~-~rr Q(r~)l, +
KoR(ql)
a ~rr d R(rl) l= - -~ ao -- (1 + ~-~
l
v)C1 = - E1 -. a
(43)
THE FULLY PLASTIC HUB
In the particular case, for z = b the hub becomes fully plastic. The integration constants and C~ can be determined from the boundary and interface conditions a,(a)[ine. = a , ( a ) [ h u b ---- - - P i n t e r f . . . . a,(b) = 0 and Uhub- U i n c . = I. The integration constants are found as follows:
A,B
B[Q(q~)-~Q(n)]
= -P(q') p - ~ R ( n ) - R ( q l ) - Pinterface 1
A = -- - - [BQ(n) + R(n)] P(n) C1 =
(44) (45) (46)
- - Pinterface-
The interface pressure can be found from the following equation:
A KoP(q~)+~-~rrP(rl)t.
+B
KoQ(q~)+-~-~rrQ(rl)la
a d ao E1 + KoR(q~) + - ~ d r R(r~)[= - ~ + (1 - v)Pi,t=rface = --'a
(47)
It should be noted that for a sufficiently large interference, I, the pressure at the interface can reach the yield stress. For larger values of I, our assumption of purely elastic inclusion behavior fails. In this case, plastic flow of the inclusion must also be considered. NUMERICAL
RESULTS
Figure 2 shows the influence of thickness variation on the stress distribution of a fully elastic hub, for
l ao(a)- o,(a)I,=o = l cro(a) - err(a)I,=o.s = ao.
0"7~x,,
n=0
I "~-.
0
.
n = 0.5
3
5
~
,C
Z5 0
x
0.75 I
-0.2
-0.4 FIG. 2. Stress distribution in a fullv elastic h u b
1 I
Stress distribution in shrink fit
45
0.9
nffiO
---nffi 0.5
0.45
~=
o
0.50
x
0.75
1
-0.35
-0. FIG.3.
Stress distribution in an elastic-plastic hub.
0.5
--1.5
x
rlffiO n-O.5
0.75
1
-0.5 s
~
FIG. 4. Stress distribution in a fully plasticized hub.
2L
~',
nffiO
1
OI
0.5
i
0.75
i
1
X
FIG. 5. Radial displacement distribution in a fully plasticized hub.
46
U. G0VEN
E1
The corresponding interferences for n = 0 and n = 0.5 are I -
- 1 and 0.9830999,
o-oa
respectively. Figure 3 shows the influence of thickness variation on the stress distribution of an elastic-plastic hub, for H = 1/3 and ¢ = z/b = 0.75. The corresponding interferences for n = 0 and n = 0.5 are I -
E1
- 2.25 and 2.0359999, respectively.
o-0a
Figures 4 and 5 show the influence of thickness variation on the stress and radial displacement distributions of a fully plastic hub, for H = 1/3. The corresponding interferences for n = 0 and n = 0.5 are I -
E1
- 4 and 2.3111999, respectively.
O-oa
Numerical results are given for a hub of variable thickness for the case of exponential power, k = 2 and n = 0.5. Poisson's ratio v and the radius ratio q = a/b are taken to be 1/3 and 0.5, respectively, ai~ is defined as di~ = ao/ao, ~ is defined as ti = Eu/aob, and x = r/b. CONCLUSION
The stress and radial displacement distributions in an elastic-plastic shrink fit with hardening based on Tresca's yield condition are well k n o w n [2-6]. In the present work, the thickness variation was considered to be of a more general form, h = ho e-"('/b)k Closedform solutions for the stresses and radial displacement were obtained in terms of confluent hypergeometric functions. It can be seen from the present analysis that the elastic-plastic interference radius z is influenced by the hardening parameter t/; the stress and radial displacement distributions in the outer elastic region are also influenced by the occurrence of hardening. However, it was reported in [ 2 - 6 ] that the occurrence of hardening does not influence the elastic-plastic interface radius z and the outer elastic region of a hub with constant thickness. It is interesting to observe from the present analysis that the stress distribution in a fully plastic hub with variable thickness is influenced by Poisson's ratio v. However, it can be shown that the stress distribution in a fully plastic hub with constant thickness [2] is not influenced by Poisson's ratio v. REFERENCES 1. F. G. KOLLMANN, Die Auslegung elastisch-plastisch beanspruchter QuerpreBverb~nde. Forsch. Ing. -Wes. 44, 1 (1978). 2. U. GAMER and R. H. LANCE, Residual stress in shrink fits. Int. J. Mech. Sci. 25, 465 (1983). 3. U. GAMER and R. H. LANCE, Elastisch-plastische Spannungen im Schrupfsitz. Forsch. Ing. -Wes. 48, 192 (1982). 4. U. GAMER, Der elastisch-plastische QuerpreBverband bei verschiedenen FlieBgrenzen von Innen- und AuBenteil. Forsch. Ing. -Wes. $6, 166 (1990). 5. U. GAMER, The shrink fit with nonlinearly hardening elastic-plastic hub. J. Appl. Mech. 54, 474 (1987). 6. U. GAMER, Die Teilplastizierte Nabe eines PreBverbandes. Z A M M 67, 65 (1987). 7. U. Gi)VEN, The shrink fit with elastic-plastic hub exhibiting variable thickness. Acta Mech. 61, 548 (1991). 8. S. P. T~MOSHENKO and J. N. GOOD~ER, Theory of Elasticity. McGraw-Hill, New York (1970). 9. D. IT and L. J. SLAVER, Confluent Hypergeometric Functions. Cambridge University Press, Cambridge (1960).