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Thermomechanical behaviour of pin joints subjected to sheet loads
Int. J. Mech. Sci. Vol. 35, No. 1, pp. 1 17, 1993
0020-7403/93 $6.00 + .00 © 1993 Pergamon Press Ltd
Printed in Great Britain.
THERMOMECHANICAL BEHAVIOUR OF PIN JOINTS SUBJECTED TO SHEET LOADS RIPUDAMAN SINGH a n d T. S. RAMAMURTHY Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India (Received 24 December 1991; and in revised form 16 June 1992)
Abstract--An important problem regarding pin joints in a thermal environment is addressed. The motivation emerges from structural safety requirements in nuclear and aerospace engineering. A two-dimensional model of a smooth, rigid misfit pin in a large isotropic sheet is considered as an abstraction. The sheet is subjected to a biaxial stress system and far-field unidirectional heat flow. The thermoelastic analysis is complex due to non-linear load-dependent contact and separation conditions at the pin-hole interface and the absence of existence and uniqueness theorems for the class of frictionless thermoelastic contact problems. Identification of relevant parameters and appropriate synthesis of thermal and mechanical variables enables the thermomechanical generalization of pin-joint behaviour. This paper then proceeds to explore the possibility of multiple solutions in such problems, especially interface contact configuration.
NOTATION a an
E g. H k Ki Sx, S~ T To T~ Uot
U Ol
6 Os Op (Osl, Os2 )°
2, V a
¢
radius of the hole free constants in elastic field (stress function) Young's modulus temperature field variables thermal inteffacial conductance coefficient of thermal conductivity sheet-to-pin thermal stiffness load component parameters, i = 1, 2 sheet loadings temperature uniform temperature change thermal loading (temperature gradient) rigid-body displacement of the pin radial displacement coefficient of linear thermal expansion Dirac delta semi-arc of separation (represents contact configuration) push-fit angle (represents contact configuration for push-fit pins) configuration with two semi-arcs of separation 0°1, 0~'2 geometric misfit parameter generalized misfit parameter Poisson's ratio stress Airy's stress function 1. I N T R O D U C T I O N
T h e pin j o i n t , achieving force transfer t h r o u g h rivets, studs a n d bolts, is extensively used to fasten s t r u c t u r a l c o m p o n e n t s r e q u i r i n g r e g u l a r a s s e m b l y a n d disassembly. E a c h p i n - h o l e c o m b i n a t i o n is a source of stress c o n c e n t r a t i o n a n d a p o t e n t i a l l o c a t i o n for failure. A m e c h a n i c a l joint, if p o o r l y designed, can be a cause of c a t a s t r o p h i c failure d u e to i n a d e q u a t e p h e n o m e n o l o g i c a l u n d e r s t a n d i n g of its b e h a v i o u r . Hence a n y a t t e m p t at the safe design of a structure w o u l d necessarily require a n u n d e r s t a n d i n g of the process of l o a d transfer t h r o u g h joints. Recent a d v a n c e s in the safety-conscious areas of n u c l e a r a n d a e r o s p a c e engineering call for a n i n - d e p t h s t u d y of the t h e r m a l effects on the p e r f o r m a n c e of structures. In p a r t i c u l a r , w e a k spots such as j o i n t s deserve extensive investigation to a t t a i n high levels of confidence.
2
R. SINGHand T. S. RAMAMURTHY
Blosser [1] recommends the shaping of the hole and the mating pin to alleviate the thermal stresses in joints. The concept is elegant but limited to uniform temperature changes only. A single pin in an isotropic sheet subjected to a thermal load of unidirectional heat flow, within the realm of small-deformation, linear, two-dimensional elasticity, appeared to be a good starting point to obtain some insight into the thermoelastic behaviour of the joint. The authors initiated the work with simpler studies of joint behaviour under pure thermal loading [2, 3], and went on to investigate the complex non-linear load-contact behaviour under combined thermal and pin-bearing load [4-6]. The study of partial contact behaviour of an unbonded circular misfit pin in a large isotropic and homogeneous elastic sheet is now carried out for another important case of a sheet under combined in-plane biaxial loading and unidirectional heat flow applied at the far field. We use the model of smooth interfacial surfaces and step interfacial conductance for the partial contact and separation conditions at the pin-hole interface. A unified, continuum method of analysis using Airy's stress function and the auxiliary displacement function approach [7, 8], together with inverse formulation (in which one seeks to estimate the load for a specified contact configuration [6,9, 10]), is employed to obtain the load-contact behaviour. By synthesising the thermal and mechanical parameters into generalized parameters, the possibility of presenting the load-contact relationship by a single curve is demonstrated. This Thermomechanical Generalization reduces the efforts of both analyst and designer by an order of magnitude at the preliminary design stage. Another intricate aspect of the present problem is that frictionless thermoelastic contact problems can exhibit multiple solutions as a result of the interaction between thermal expansion and contact conditions at the interface. Existence or uniqueness theorems cannot be established for such problems, particularly with classical boundary conditions. These aspects in relation to pin-joint models have already been observed by the authors for the cases of pure thermal loading [3] and combined pin-bearing and thermal loading [11]. They are now explored for the case of combined thermal load and applied sheet stresses. 2. CONFIGURATION Consider a large isotropic elastic plane sheet (E, v, ~s, ks) having a smooth hole of diameter 2a. A rigid pin (%, ko) of diameter 2a(1 + 2) is introduced into the hole, resulting in a misfit joint with geometrical proportional misfit 2 (Fig. 1). The pin-plate interface is presumed to be perfectly smooth so as to permit free tangential slip all over the interface. The thermal load is a far-field unidirectional heat flow along the x-axis, mathematically written as (1)
Ts = To + xT,,.
Uniform far-field applied stresses on the sheet are conveniently represented by a biaxial stress (Sx, St) described in terms of the parameters K1, Kz, So as Sx=(Kx+K2)~°
Sr=(K1-K2)~
Sxr=0
(2)
where K1 ~ and K= ~ represent respectively the axisymmetric and alternating (or pure shear) components of the general far-field stress. The effect of To is to alter the pin and hole diameters and modify the misfit parameter. The thermal gradient Tx pushes the pin along the x-axis through a distance of, say, Uo, [2]. Under mechanical loading, the pin-hole interface exhibits partial contact/separation [9, 10]. The thermal conductance at the pin-plate interface is modelled as a step function, viz. zero in the region where there is no mechanical contact and constant H in the contact region. Mathematically, 6H, 6 = 0 in separation and 6 = 1 in the contact zone. This is a reasonably good model as the heat transfer due to radiation in the region of separation can be neglected in comparison with the conductive heat transfer in the region of contact, and the normal interracial stress in most of the contact region is high enough that the contact conductance will be almost constant over the region of contact [6].
Pin joints subjected to sheet loads
(a)
3
SY
t __..
Q~
Sx
t
y
20
~
Sx
4"11" uot
SY
(b)
SY
.-_._L
t
t
D I1 in Sx
Q
Q
~ S
4"1~ Uot r
sY FIG. 1. Configuration; heat flow and sheet stresses: (a) Sx > Sy, (b) Sy > Sx.
The configuration (geometry, loading and boundary conditions) is symmetric about the x-axis and so only one half of the domain needs to be analysed. The stresses due to the interference 2 are axisymmetric and those due to the sheet loading are doubly symmetric [10]. In the case of an interference-fit pin for full contact situations, the normal interracial stress due to heat flow (loading Tx only) is zero [2]. Thus, with an increase in mechanical loading the separation will initiate symmetrically, leading to a doubly-symmetric separation configuration. Similarly for clearance-fit pins, as the net load on the pin must be zero, the initiation of spread of contact has to be doubly symmetric. Thus the contact/separation configuration obtained will be doubly symmetric. Depending upon whether Sx > Sy or Sy > S~, the configuration of Fig. l(a) or l(b) will be obtained, with the extent of separation as _ 0s about the horizontal or vertical axis through the centre of the pin, respectively. Working within the realm of small deformations the contact configuration can be treated as symmetric about the x- and y-axis. In some cases the interface may exhibit a contact configuration symmetric only about the x-axis. Such situations are dealt with using a slightly different approach [6], and are discussed in the latter half of this paper.
3. TEMPERATURE FIELD
For the partial contact configuration of Fig. l, the steady-state temperature fields should satisfy the governing differential equation, i.e. the Laplace equation V2T = 0 and the boundary conditions: (1) far-field temperature distribution Ts = To + xTx, (2) symmetry about the x-axis and antisymmetry about the y-axis, and (3) the interfacial condition, at r = a, - k p (3Tv -~r =
ks OTs dr "
4
R. SINGHand T. S. RAMAMURTHY
Such a temperature field can be expressed in series form as Ts = To + aTx
{(:) (a)} +
cosO - arx
(,.) (:)
gn r
~
cos nO
n=1,3,5...
Tp = To + aTx
--
g,
n=1,3,5...
cos nO.
(3)
The arbitrary constants gl, g3, g5 . . . . . gE,-i are to be determined by appropriately satisfying the interfacial boundary condition
-kpOTp~--
OTs
5H
ks ~rr - a (Tp-Ts)
(4)
6=00~ Sy
~ 6=1
5
Os <~ 0 <. rc - Os
0
rc-0s<0~
di-- 1 0 ~ < 0 ~ < ~ - 0 s and, for case B, when Sr > S x
0
rc - 0 s < 0 < ~ + 0rrs
1 ~+0s~<0~t 67.5 °. Points of transition between contact and separation were always on collocating points and treated as in the contact zone. Numerical data were obtained for 0s = 0°-90 ° in increments of 5.625 ° and k-2= --*0,0.5, 1.0, 2.0, --* oe ks
with
H ~ = 0.5
H k~ = -* 0, 0.5, 1.0, 2.0, --* oo
with
kp = ~ 0.5.
The variation of g-£ g with ~kp and H for both cases A and B is plotted in Figs 2 and 3, respectively. The variable g is given by [2] 2 kp g= ks
{ 1 + ks + H
-1 kp)
"
For full contact and point contact, the values ofga match the closed-form results of Ref. [2]. The variation is continuous between full contact and point contact and lies within reasonably narrow bands for the full range of conductivities. 4. E L A S T I C
FIELD
4.1. Full contact: interference-fit pin For the situation of positive interference (2 > 0), the interface exhibits full contact up to a certain load level (K~ So, K2So, To, Tx). So long as the pin and the sheet are in complete
Pin joints subjected to sheet loads
5
1
\\
-
\\
,'--.'..
\
e
•..
.,,
,,
gl _
T
-
\\
_
x~-0.,---
,
Case
>,
"~'~\
;_ I
~
~~
__ I
I
,
I
I
','
A ~1
,
~t
0, FIG. 2. Effectof
I ~ . t " ~
--
= =.....
\
_
k__.Pv =
~.
ks
\
\
",,. \ \
--
.
05
~H%%% • %% • %% • \ ",7--.0-./',/, ~ 1 . o - - ' • /~ \/N/N,----,oo--Y• ~ I
\ \
-
on #1.
k,
•- . ~ . . .
gl g
"T
A
"~.
c.se~ _ _ _
~
, ,,,
~
~
~
~' l
,,,
0
2-
H FXG.3. Effectof~ on g~. contact, the stress and the displacement fields in the sheet can be expressed in closed form by a biharmonic Airy's stress function satisfying (i) (ii) (iii) (iv)
the far-field stress state [Eqn (2)] smooth interface; at r = a, a,o = 0 radial compatibility at the interface: at r = a, Us = Up + a2 + Uotcos 0, and zero load on the pin:
f~a, cos 0 r d0 =
0
at r = a .
The stress function in the presence of the temperature field for the full-contact situation (g~ = # and g. = 0 for n ~ 1) is
Ea22e (r)KlSoa2{1--v q~-
( l + v ) In
+ ~ 01
(r)
~ ' ~ v In
l(r) 2} +~
~, - ~ - {~ ( a ) + (at-) In ( r ) } cos 0
K2Soa 2 { ~ ( r ) z l : a ' ~ 2 2 +6\r] +
l+v( ~
1 (a)2)} 1--~ r cos 20.
(5)
From the above expression for the stress function, the rigid-body displacement of the pin with respect to the sheet was obtained as
not - ~,aTx - a
Yd':
(6)
6
R. SINGH and T. S. RAMAMURTHY
where 1
= ap_~s
ga--~(3+v)
1--gl~
and defined as the sheet-to-pin thermal stiffness, and agrees with the results of Ref. [2]. The load (So)c, for initiation of separation can be obtained from the condition a, = 0 at 0 = 0 7r
(for S~ > Sy) or 0 = ~ (for Sr > S~). This load can be derived as So
cr
=
Klat--~--v)
(7)
K2
and this agrees with the results of Rao [10]. Any further increase in load level (So) causes the separation to spread symmetrically about either the x- or y-axis. 4.2. Full separation: clearance-fit pin Initially with zero mechanical load (So = 0) an all-round uniform gap exists for the case of a clearance-fit pin (2 < 0). Under thermal loading with full separation at the interface, the problem of stress determination in the sheet reduces to the situation of a free hole, which has already been handled by Florence and Goodier [12]. Considering the pin to be a part of the system, the problem is ill-posed as the pin temperature is undefined and one more condition is required to evaluate the temperature field in the pin. To make the problem well-posed, consider the pin temperature to be To before the onset of contact. After the initiation of contact the pin temperature will be governed by the interfacial conditions. As the plate load So is applied and increased, initially the elastic field corresponds to that of an infinite plate with a circular hole loaded at infinity by sheet stresses and unidirectional heat flow [3, 6]. The uniform temperature To only alters the pin and hole diameters and the temperature gradient Tx causes a rigid geometrical displacement of the hole [2, 6]. The biaxial sheet loading distorts the circular hole to an elliptic one [6, 9, 10]. Thus, the initiation of contact largely depends upon the nature and the level of loading and is categorized in Table 1. After the initiation of contact, with increase in loading (mechanical or thermal) the pin may undergo a rigid-body movement and the point of contact may slip on the interface until the point-contact configuration becomes symmetric about both the x- and y-axis, i.e. either rc a point contact at 0 = ~ (for Sx > Sy) or point contacts at 0 = 0, rc (for Sy > Sx) are established. With further increase in loading, the contact will spread symmetrically about either the x- or y-axis. TABLE 1. INITIATIONAND SPREAD OF CONTACT IN CLEARANCE-FITPINS Mechanical loading
Location of initiation of contact
Thermal gradient
S~ > S r
Tx > 0
between 7~ ~ and n
T~ = 0
n ~
Tx < 0 Nature of spread of contact
between 7~ 0 and 2 symmetric about the x-axis
Sx = S r
Sx < Sy
71:
all-round simultaneous 0-r~
0 and n
0
all-round simultaneous
symmetric about the y-axis
Pin joints subjected to sheet loads
7
4.3. Partial contact: unified analysis of misfit pin For the analysis of the partial contact behaviour of a misfit pin, consider the case when the pin and sheet have separated over a part of the periphery. The boundary conditions are (i) biaxial sheet loading in the far field, Eqn (2) (ii) smooth interface, a,o = 0 at r = a, and (iii) no net load on the pin, i.e. f ~ a , cos 0 r dO = 0
(8)
at r = a.
(9)
Radial equilibrium and compatibility at the interface dictates that, at r = a, Us = Up + a2 + UotCOS0 in the contact zone
and
tr, = 0 in the separated zone.
(10)
Airy's stress function satisfying the boundary condition [Eqns (2), (8) and (9)], in the presence of a temperature field [Eqn (3)I, is obtained as
1 ~[K1Soa2(r) 2
+ n~2 { ( r ) 2
K2Soa2{(r) 2
-~
a
2
+l-n
-~-~n} (a)" a, cosnO.
(11)
The constants ao, a2, a3 a, are to be determined by appropriately satisfying the interfacial boundary condition [Eqn (10)]. Equidistant collocation is found to be satisfactory. . . . . .
4.4. Numerical evaluation For the unified analysis of a smooth pin-hole combination, the problem posed in inverse form [10] reads: Given the extent of separation, E, v, 2, what are the load levels To, Tx, K1 So, K2So? Obviously there cannot be a unique combination of all these loading values for a given contact configuration and hence the problem is treated by merging To with 2 and K2 assigning values to Tx and ~-~. Now the problem reduces to: Having specified the arcs of
separation~contact, what is the load level So for given values of E, v, 2, Ix, K2 ~v kp~
K I ' ~s' ks" A computer code was developed to determine the load-contact relationship. The conver-
gence of the load parameter (EK----~S~)was studied with respect to the order of truncation of the series. To limit the C P U time, all further computations were carried out with 65 points at the interface for 0s < 67.5 ° and 129 points for 0s t> 67.5 °. Numerical data were obtained for uniaxial tension along the x-axis (an example of case A) with ~- ~P-- 0 . 5 ;
~k p = 0.5;
~H= 0 . 5 ;
v=0.3;
oqaT~=O.O01, O.O, -0.001, - 0 . 0 0 2
for 0s = 00-90 ° in increments of 5.625 °. The load-contact relationship is plotted in Fig. 4. 4.5. Discussion For uniaxial tension along the x-axis (K1 = K2 = 1) for an interference-fit pin, the pin and sheet maintain full c°ntact uP t° a critical stress (SE-~,) cr = 0"376" As the sheet l°ad (S°, is increased further, the variation of the angle of separation (0,) with load (So) is non-linear and reaches an asymptotic value 0v as So ~ oo. This is shown in Fig. 4 by the dotted line. On the other hand, for clearance fits, contact starts spreading at So = - E2,. The contact angle
8
R. SINGH and T. S. RAMAMURTHY I I I I ' ~ I q
t
Ek e
KIS o
K1S 0
0s
Op=71
°
EZ.e
s ~ g
I !
-4
FIG. 4. L o a d - c o n t a c t behaviour: uniaxial tension
~x~aTx= 0.001, 0,
(
0c 0c = ~ - 0,
)
-4
I !
(~
k°
%
= 0.5, ~ = 0.5, --ct,= 0.5, v = 0.3, rigid pin,
- 0.001, - 0.002).
increases with an increase in load and tends asymptotically to ~ - 0p as
So ~ oo. This is more clearly understood by replotting the data with ~-o
as the ordinate
(represented by bold lines in Fig. 4). The intersection of this curve with the abscissa corresponds to So ~ oo for a given 2c or for any So with 2e = 0, i.e. the case of a push-fit pin, and the corresponding angle is 0 w The stability of the configurations was checked by slightly perturbing the contact configuration without changing the load and investigating the tendency of the contact to change. The configurations of the present cases were found to be stable. 5. T H E R M O M E C H A N I C A L
GENERALIZATION
5.1. Partial contact As long as full contact is maintained, thermal and mechanical effects are superposable. This was clearly brought out in Section 4.1, Eqn (5). For partial-contact situations with To merged into 2 the four load-contact curves for different values of T~ are found to be indistinguishable from each other. This implies that thermomechanical generalization has been achieved for the present case. The contact/separation configuration is independent of Tx and hence the curve of Fig. 4 holds good for all values of Tx ranging from - oo to + oo. A more general parameter for general biaxial sheet loading and unidirectional heat flow can / E 2 , - K1 So be \~ I(2So )" The data of Fig. 4 are replotted in Fig. 5 using this parameter along the /
ordinate. To confirm this generalization, numerical data were generated for the following combinations of parameters covering the full range of all variables involved: a---eP= ~ O, 0.5, 1.0 as
k-2 = ~ 0, 0.5, 1.0, 2.0, --* oo k, H
-- =
ks
~
0, 0.5,
1.0, 2.0,
~
v = 0.3 ct, aTx = 0.001, 0.0, -- 0.001, -- 0.002
K , = K2 = 1, uniaxial tension along the x-axis (Case A) K1 = 1, K2 = - 1, uniaxial tension along the y-axis (Case B).
Pin joints subjected to sheet loads 1.66 1
E~'e'K1S0
0
~5/7°
K2S 0
0 -1
-2
9
i
NNS \
-f
KI = - 1 K2
_
K~So (v = K2So
FIG. 5. Load--contact behaviour: use of generalized parameter E2c -
0.3, rigid
pro).
This covers the full range of distortivity (6 = s/k) ratios of sheet to pin (6,/6p). The values corresponding to the above combination are: ~ 0, 0.25, 0.5, 1, 2 and ~ . E2e - K1 So The data obtained were plotted for generalized parameter and the curve of
K2So
Fig. 5 was obtained in each case. Each time the rigid-body displacement of the pin was found to obey the established relation [2]
Uot ~saTx a o,~Fzr Thus we see that the interpretation of the curve of Fig. 5 can bring out information regarding the load-separation behaviour for any combination of E, 2, K1, K2, So, To, Ix. The point T on the curve with ordinate
KISo_ the So level as E2eK2 So
-K-~2 represents the push-fit case irrespective of
K1 K2 leads to ~E2~ = 0. The portion of the curve above
T corresponds to interference and that below T corresponds to clearance. Also, if E2~ E2~ S---~-~> 1.66 then interfacial separation is not possible, while if -~o ~< - 2 then interracial contact cannot be achieved. The case of partial separation/contact arises only in the region E2,
1.66 > ~-o > - 2 .
K2 The position T for the push-fit case depends only on ~ and is independent of So. Thus, the angles of separation are invariant with respect to the load level so that the stresses due to push-fit in a loaded sheet increase linearly with the applied load system. 5.2. Interfacial stresses Consider the critical portion, i.e. the interface for stress-variation studies. The normal stress at the interface ( a ~ ) is invariant with respect to thermal loading. This is presented in Fig. 6. Thus the radial stresses change by (-~) with respect to those in the case of pure mechanical loading. No generalizing parameter could be identified for the tangential stress, but there is an interesting feature that can be seen logically. The thermal gradient Tx does not alter the contact configuration, and so the resulting change in hoop stress should vary linearly with Tx. This has been found to be so. The (a~-~)value for Tx = 0.0 is plotted in
(o0)
Fig. 7 and the change in ~
per unit Tx is plotted in Fig. 8. The hoop-stress parameter
R. SINGH and T. S. RAMAMURTHY
10 1 --
~
_
Or
-
~
-4 _
78"75°
0~t
\
\
22.50 ° 33.75 °
67.5 °
l s
FIG. 6. Normal interfacial stress for various contact configurations; uniaxial tension and heat flow: r, o q a T x (v = 0.3, rigid pin). for all values of ~H , ~k r,, o~ a-~'
10 0 s = 67.5 °
0
'
;
'
",.5
Fro. 7. |nteffacial hoop stress due to pure unia×ial tension (v = 0.3, rigid pin).
200
d(ajaTx)
0
0
-200
67.5 °
Flo. 8. Effect of heat flow on interracial hoop stress; sheet under uniaxial tension ~ = 0.5,
= 0.5,
c~2a,= 0.5, v = 0.3, rigid pin).
(~-~,) for any contact configuration can be estimated by summing up the contributions due to mechanical load (Fig. 7) and thermal loading (Tx times the value in Fig. 8). 6. P O S S I B I L I T Y
OF NON-UNIQUE
SOLUTIONS
Thermoelastic deformations, though small, are often sufficient to have a major effect on the mechanics of contact. Thermoelastic contact problems can exhibit multiple steady-state solutions as a result of the interaction between thermal expansion and contact conditions at the interface. Barber et al. [13-17] have extensively studied the thermoelastic behaviour of
Pin joints subjected to sheet loads
11
solid bodies in contact and the essence of their findings is consolidated in Chapter I of Ref. [13]. Very interesting features are related to aspects such as the non-existence and nonuniqueness of solutions to frictionless thermoelastic contact problems. Existence and uniqueness theorems have been proved for heat-conduction problems with any combination of boundary conditions. Furthermore, if the temperature is known at all points in the body, a particular thermoelastic solution can always be found which reduces the contact problem to an equivalent isothermal problem for which, again, existence and uniqueness theorems have been proved. However, no existence and uniqueness theorem can be established for frictionless thermoelastic contact problems with classical boundary conditions [13]. Barber et al. [14] and Comninou and Dundurs [17] show that the lack of existence is connected only to highly idealized boundary conditions, and existence can be achieved by appropriately modifying the conditions imposed at the contact interface. However, lack of uniqueness is not caused by idealized boundary conditions and is not purely mathematical. It has a physical basis and signals possible instabilities. In Section 4 it was seen that the spread of contact/separation takes place symmetrically about both the x- and y-axes and such a configuration is stable. Still there is a need to look for other solutions, in the absence of any uniqueness theorem, for this class of frictionless thermoelastic contact problems. Whatever the contact configuration may be, symmetry about the x-axis must exist because of the symmetry of geometry, properties, loading and initial and boundary conditions. There is a possibility that the configuration may not be symmetric about the y-axis. Typical possibilities are shown in Fig. 9. We analyse a typical configuration to gain more insight into the behaviour of the system. Consider the ease of uniaxial sheet tension along with on-axis heat flow. To understand the system all combinations with the following variables values were analysed: ~._eP__. ~ 0 , 0.5, 1.0~oo as
kp k,
--+0,0.5, 1.0, 2.0, ~ o o
H ks
-- =
~ 0, 0 . 5 , 1.0, 2 . 0 ,
~ oo.
(a)
SO
r- S O i.
- , - t l 4 - Uot
r
(b)
(c) e2
o 01
03D
D X
g
FIG. 9. Interracial contact configuration not symmetric about the y-axis.
12
R. SINGH a n d T. S. RAMAMURTHY
This closely represents the situation of a steel bolt in an aluminium sheet. The physical nature was found to be similar for all situations. In order not to clutter this paper with numerical data one typical set of results are presented. We choose the combination kp = 0.5 ~ H= 0 5 y,
-~
ap = 0.5
E~-~oo
6.1. Configuration Consider the configuration of Fig. l(a) extended to the present case of unequal arcs of separation under uniaxial tension along the x-axis as shown in Fig. 9(a). The two arcs of separation extend over + 0,1 and + 0sZ about the x-axis at 0 = zt and 0, respectively. For the sake of convenience, the concise notation (0,1, 0,z) ° will be used to represent such a configuration. In the present case of zero net load on the pin, both 0sl and 0,2 must be bounded between 0 and ~. Also, the configuration (0,1,0,2) ° with thermal gradient Tx is same as (0,2, 0,1)° with gradient - Tx. This will be confirmed from numerical results. 6.2. Temperature field The series solution for the temperature field in the sheet and the pin for unequal arcs of separation will now have all the terms rather than just the odd terms [of Eqn (3)] and can be written as Ts= To + aTx
cosO-aTx(kp~
+
cosn0
\ks,/.~1
T v = To + goaTx + aTx
g,
cos nO
(12)
n=l
with the unknown coefficients to be determined by numerically satisfying the interfacial conditions of conductance. Equidistant collocation with 65 points at the interface was used to obtain the solution. The variation of the important variables go and gl [2] is presented in r~ Fig. 10 for 0 ~ 0,2 ~< 0,1 ~<2" This is done to achieve computational economy since physically, gO,(O.2,0.t)o = --gO,(O.l,O.2)o
and gl,(O.2,o.t)o = gl,(O.l,o.:)o.
Hence the information regarding 0~2 > 0,~ need not be generated independently. There is another interesting feature regarding go. Since gO, (0, 0) ° ~
- - g0,(O, 0)o
Ie$1 = ~ $ 2 = O° L / ~75"75°
F/\
H
II-VVN l l h A Yx H[X X
0
IVV ,V, V
"/
67.50 ° 56.25 e 45.00 °
k
~:~:o.~
X",,
V"',.>~ ,V 33.75 °
~°°
,~f~'~ff 22.50°
°
11.25" ]
gl/g FIG. 10. V a r i a t i o n o f go a n d 0t for c o n f i g u r a t i o n s with 0sl ~ 0s2.
P i n j o i n t s s u b j e c t e d to sheet l o a d s
13
and go,(o,o)o = 0
we can write go,(o,o)o = go,(o,o)o + go,(o,o)o.
Physically, this can be extended to go,(o.~,o,2)o = g o , ( 0 , t , o ) o + g o , ( o , 0 , 2 ) o =
g O , (0,,1, O) ° - -
gO,(O,,2, O) °"
From the numerical data obtained this has been observed and is shown in Table 2 for a few typical values of 0sl and 0s2 chosen randomly. Thus, for go, the data for a single arc of separation (obtained earlier I-4,5]) can be used to approximately generate information regarding the double arc of separation configuration also. 6.3. Elastic field For the partial contact configuration (0~1, 0s2)°" there are two transition points placed asymmetrically and hence two load values can be uniquely determined using the inverse % kp H formulation. The problem posed reads: Given E, v, KI = K 2 = 1, ~ (including To) as' ks' ks' and the configuration (Osl, 0s2)°, what are the load levels Tx and So? The unknown coefficients ao, a~ . . . . . a. in the stress function q~ of Eqn (11) were evaluated numerically using the collocation technique and the boundary conditions of a, = 0 in separation, and gap = Us - Up - a2 - Uotcos 0 = 0 in the contact zone. Obviously, not all combinations of 0s~ and 0s2 will be feasible. So, for any configuration, having obtained the solution the feasibility was checked by looking at the interfacial normal stresses and displacements, i.e. by checking the inequality constraints of a, < 0 in the contact zone, and gap > 0 in the separation zone. A feasible configuration must have ar = 0, gap > 0
in the separation zone;
crr = 0, gap = 0
at the transition point; and
a, < 0, gap = 0
in the contact zone.
The feasibility envelope obtained for the configuration under study is presented in Fig. 11. Each point within the envelope AFGHBEDCA corresponds to a feasible combination of So and Tx. Points outside the envelope correspond to configurations which are non-existent. Although values of Tx and So can be obtained to satisfy the boundary conditions when using the inverse formulation, the solutions violate the above-mentioned inequality constraints and hence are unacceptable. The envelope is symmetric about the diagonal line APB and, in fact, the mirror-image points across this line represent the loadingcombinations So, Tx and So, - T~. Physically, this mirror-image symmetry confirms that one configuration is achievable by rotating the other through 180° about the y-axis. Thus, we need to discuss only one half of the envelope. The points on the diagonal line APB represent the equal-arcs-of-separation solution of Section 4, and each point refers to a unique value of So and any value of the thermal gradient Tx. The case of a push-fit pin is represented by the point P(0p, 0p)°. The variation of So along APB can be seen in Fig. 4. TABLE 2. INTERPRETATION OF THE VARIABLE go FOR TWO UNEQUAL ARCS OF SEPARATION FROM THE DATA FOR A SINGLE ARC OF SEPARATION S. n o
0sl
0s2
go,(o,=,o)
go,(e.2,o)
1 2 3 4 5
22.5 ° 45.0 ° 45.0 ° 67.5 ° 90.0 °
5.625 ° 16,875 ° 67.500 ° 28.125 ° 45.000 °
0.1663 0.4496 0.4496 0.7975 1.1635
0.0230 0.1106 0.7975 0.2286 0.4496
go,(e,t,o) -- go,(e,2,o) 0.1433 0.3390 -0.3479 0.5689 0.7139
go,(e,~.e.2) 0.1433 0.3400 -0.3455 0.5746 0.7354
%dev 0.00 0.29 0.69 1.00 3.01
14
R. SINGH and T. S. RAMAMURTHY
T E
H
I
i
0S2 C
I/"
/I F
Ao
I
I ~_
2
0sl
FIG. 11. The feasibility envelope: uniaxial loading [Ref. Table 3]
= 0.5,
= 0.5, - - = 0.5,
v = 0.3, rigid pin).
Ae = e s 2 - 0 s 1 ~
Sl
~-22.500"
~ / 1 1 . 2 5 o
EX ..._&e So
"~ ~ ~ / 5 . 6 2 5 "
o -1
o
",,X\\\/0°
--
o..
_
° \.N\\ -- .x,~..
FIG. 12. L o a d - c o n t a c t behaviour for configurations with 0st ~ 0s2: mechanical load parameter
H = 0.5,~ = 0.5, -~, - = 0.5, v = 0.3, rigid pin .
The sub-envelopes A F G H P E D C A and BEPHB are regions of feasible configurations for interference- and clearance-fit pins, respectively. Line EPH represents the push-fit case. Each point I-configuration (0,1,0,2) °] within these sub-envelopes corresponds to a unique combination of loadings So and Tx. As we move from AC towards B the value of the load E2c parameter ~ decreases from a finite value to zero at EP, and falls further to - 1 at B. The E2c variation of -ff0-owhile moving parallel to APB (A0 = 0s2 - 0sl = constant) is plotted in Fig. 12. The thermal-load parameter
~
is positive for clearance fits (sub-region BEPB,
2e < 0), negative for interference fits (sub-region APDCA, 2e > 0) and zero for push fits (line oqaTx EP, 2c = 0). As we move from PB to EB, ~ increases from zero to + oo, and in moving
or,aT:, from AP to CDE ~ changes from zero to - o o . This indicates that high thermal
Pin joints subjected to sheet loads
15
gradients are required to sustain larger differences in extents of separation (0,2 - O~t). The
~saTx
line DP represents a special feature. The parameter ~
has a finite value on this line but
both T~ and 2. change sign across the line. In fact, the variables ~
1
and
tend to zero
along the line. In the six sub-envelopes of the feasibility envelope of Fig. 11, the various parameters have a specified nature and generally change sign across the boundaries. A comprehensive presentation of the nature of these various parameters is given in Table 3. E2~ 2, Figure 13 shows the relationship between the loading parameters ~ o and ~ for specified values of 0~ and A0 (A0 = 0~ - 0s2). Again, it is seen that as A0 increases the system requires high T~ values. Figure 14 shows the normal and hoop-stress variations along the interface for a situation in which one of the transition points is kept fixed (0,2 = constant) and the other is moved within the feasibility envelope. The hoop-stress
TABLE 3. QUALITATIVENATUREOF VARIABLESWITHINTHE FEASIBILITYENVELOPEOF FIG. 11
Line
Region Line Line
+
any
any
+
+
0
any
~
any
+ oo
AP
+
P
0
PB
So
-
-
any
any
+
+
+
-
-
+
+
CD
+
+
- ~
- oo
+
+ +
+
oo
- oo
-
~
DP(PEDP)
+
- oo
ov
-
- oo
PEDP
+
-
+
-
Line
So
APDCA DP(APDCA)
Region Line
~aTx 2~
2~
Point Line
T~
2
DE
-
-
+ +
+
-
+
- oo
-
+
EP(PEDP)
+
0
-
- oo
any
oo
EP(BEPB)
-
0
+
~
any
- oo
BEPB
-
-
-
+
+
-
EB
-
-
-
oo
+
-
Region Line
~O A0-*0 o
~saTx
i '6°
"~[~q~
i 0
1
~
14.0 °
~ 25.3•
2
Ek e So FIO. 13. L o a d - c o n t a c t
b e h a v i o u r f o r c o n f i g u r a t i o n s w i t h 0,1 # 0,2: t h e r m a l a n d m e c h a n i c a l l o a d
p a r a m e t e r s ( ~ , = 0 .5 , ~ kp = 0.5, --,t, ~p = 0.5, v = 0.3, r i g i d p i n ) . HS 35:1-B
16
R. SINGH a n d T. S. RAMAMURTHY
(a)
~r EX© -1
i o
0 (b) m
EX,c -4 0sl = 22.5 °
FIG. 14. Interfacial stresses for various contact configurations: (a) n o r m a l stress, (b) h o o p stress (configuration of Fig. 9a)
param0ter /
foun
= 0.5, ~ = 0,5, -as- = 0.5, v = 0.3, rigid pin .
to change drastica,,y whereas the norma, stress parameter
\
( a ~ _ ) changes very little. From the variation of load values (Tx, So) within the envelope, it can be seen that for any loading combination Tx and So there are two configurations, one corresponding to 0,1 = 0s2 (on the line APB) and the other corresponding to 0sl # 0,2 (within the envelope but not on the line APB). A typical example is: For the combination %aT~ -- -0.02634 and ESo = 0.003306, two solutions (Fig. 11) are realized as Sl ~ 0~1 = 0s2 = 48.08 °
and
$2--* 0~1 = 33.75 °, 0,2 = 42.1875 °.
In Section 4 it was seen that the contact/separation starts spreading symmetrically about the y-axis, and thus the true load-contact behaviour for misfits will, in general, be along line APB and configuration S1 will be realized in practice. However, by carefully steering the load magnitudes and directions, configuration $2 can be achieved, but any disturbance will make the configuration jump to solution S1 as $2 is an unstable configuration. The stability of $2 is checked by giving it a small perturbation, inspecting the inequality constraints mentioned earlier, and realizing that the configuration settles down to solution S1. It is due to this reason that the contact/separation regions cannot grow unsymmetrically. We have discovered that there are two configurations for all combinations of T~ and So. According to Barber and Comninou [131, there must be an odd number of solutions. Thus, there is at least one more solution, which probably corresponds to a configuration of
Pin joints subjected to sheet loads
17
the kind shown in Fig. 9(c) with three transition points. Possibly there can be more solutions with even more than three transition points. At this stage, we cannot analyse such complex configurations by the inverse technique, since we have only two free load variables, Tx and So. 7. C O N C L U S I O N
For a single misfit pin in a large isotropic plate, thermomechanical generalization of non-linear partial contact behaviour has been successfully achieved for combined in-plane biaxial sheet loading and on-axis unidirectional heat flow. The actual load--contact behaviour follows a stable doubly symmetric contact configuration. The stress concentration for stable configurations was found to be linearly dependent on the thermal gradient. Configurations analysed with single symmetry are unstable and non-realizable in the normal loading cycle. There is a possibility that some more solutions exist, which probably need tedious and complex procedures to obtain them. Acknowledgements--The authors acknowledge with pleasure the association of Prof. A. K. Rao, Emeritus Scientist, National Aeronautical Laboratory, and Prof. B. Dattaguru of the Department of Aerospace Engineering, Indian Institute of Science, Bangalore, with this work. The first author is grateful to the JRD Tata Trust for enhanced financial support. The general financial support of the Aeronautics R&D Board India, is also acknowledged. REFERENCES 1. M. L. BLOSSER,Thermal-stress-free fasteners for joining orthotropic materials. AIAA Jl, 27, 472 (1989). 2. R. SINGH, T. S. RAMAMURTHYand A. K. RAO, Thermomechanical generalisation in pin joints: A model. J. Aero. Soc. India 43, 35 (1991). 3. R. SINGHand T. S. RAMAMURTHY,Non-existence and non-uniqueness in thermoelastic contact problems: the circular inclusion model. J. Aero. Soc. India 43, 279 (1991). 4. R. SINGH, T. S. RAMAMURTHYand A. K. RAO, Thermoelastic analysis of misfit pin joints. Proc. Int. Conf. on Structural Testing Analysis and Design, Bangalore, India, pp. 736-741 (July 1990). 5. R. SINGH,T. S. RAMAMURTHYand A. K. RAO,Thermomechanical generalisation in joints: on-axis pin bearing load. Nucl. Engng. Design 135, 307 (1992). 6. R. SINGH, Thermomechanical generalisation of partial contact in pin joints. Ph.D. Thesis, Indian Institute of Science, Bangalore, India (1991). 7. K. S. RAO, M. N. BAPURAO and T. ARIMAN,Thermal stresses in plates with circular holes. Nucl. Engng Design 15, 97 (1971). 8. R. SINGH, Auxiliary displacement function approach to problems in 2-D elasticity with body forces and initial strains. Proc. 34th Congress of the Indian Society for Theoretical and Applied Mechanics, Coimbatore, India (1989). 9. V. A. ESHWAR, B. DATTAGURUand A. K. RAO, Partial contact and friction in pin joints. Report No. ARDB-STR-5010, Aeronautics R&D Board, Ministry of Defence, Directorate of Aeronautics, New Delhi, India (1977). 10. A. K. RAO, Elastic analysis of pin joints. Comput. Struct. 9, 125 (1978). 11. R. SINGH,T. S. RAMAMURTHYand A. K. RAO, Non-uniqueness and realisability of thermoelastic contact configurations in load bearing pin joints. Nucl. Engng. Design 135, 315 (1992). 12. A. L. FLORENCEand J. N. GOODIER,Thermal stress at spherical cavities and circular holes in uniform heat flow. J. Appl. Mech. 26, 293 (1959). 13. J.R. BARBERand M. COMNINOU,Thermoelastic contact problems, Chapter I in Thermal Stresses III (edited by R. B. HETNARSKI),pp. 1--106. Elsevier, Amsterdam (1989). 14. J. R. BARBER,J. DUNDURSand M. COMNINOU,Stability considerations in thermcclastic contact. J. Appl. Mech. 47, 871 (1980). 15. J. R. BARBER,Stability of thermoelastic contact for the Aldo model. J. Appl. Mech. 48, 555 (1981). 16. M. COMNINOUand J. DUNDURS,On the possibility of history dependence and instabilities in thermoelastic contact. J. Thermal Stresses 3, 427 (1980). 17. M. COMNINOUand J. DUNDURS,On lack of uniqueness in heat conduction through a solid to solid contact. J. Heat Transfer 102, 319 (1980).
0020-7403/93 $6.00 + .00 © 1993 Pergamon Press Ltd
Printed in Great Britain.
THERMOMECHANICAL BEHAVIOUR OF PIN JOINTS SUBJECTED TO SHEET LOADS RIPUDAMAN SINGH a n d T. S. RAMAMURTHY Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India (Received 24 December 1991; and in revised form 16 June 1992)
Abstract--An important problem regarding pin joints in a thermal environment is addressed. The motivation emerges from structural safety requirements in nuclear and aerospace engineering. A two-dimensional model of a smooth, rigid misfit pin in a large isotropic sheet is considered as an abstraction. The sheet is subjected to a biaxial stress system and far-field unidirectional heat flow. The thermoelastic analysis is complex due to non-linear load-dependent contact and separation conditions at the pin-hole interface and the absence of existence and uniqueness theorems for the class of frictionless thermoelastic contact problems. Identification of relevant parameters and appropriate synthesis of thermal and mechanical variables enables the thermomechanical generalization of pin-joint behaviour. This paper then proceeds to explore the possibility of multiple solutions in such problems, especially interface contact configuration.
NOTATION a an
E g. H k Ki Sx, S~ T To T~ Uot
U Ol
6 Os Op (Osl, Os2 )°
2, V a
¢
radius of the hole free constants in elastic field (stress function) Young's modulus temperature field variables thermal inteffacial conductance coefficient of thermal conductivity sheet-to-pin thermal stiffness load component parameters, i = 1, 2 sheet loadings temperature uniform temperature change thermal loading (temperature gradient) rigid-body displacement of the pin radial displacement coefficient of linear thermal expansion Dirac delta semi-arc of separation (represents contact configuration) push-fit angle (represents contact configuration for push-fit pins) configuration with two semi-arcs of separation 0°1, 0~'2 geometric misfit parameter generalized misfit parameter Poisson's ratio stress Airy's stress function 1. I N T R O D U C T I O N
T h e pin j o i n t , achieving force transfer t h r o u g h rivets, studs a n d bolts, is extensively used to fasten s t r u c t u r a l c o m p o n e n t s r e q u i r i n g r e g u l a r a s s e m b l y a n d disassembly. E a c h p i n - h o l e c o m b i n a t i o n is a source of stress c o n c e n t r a t i o n a n d a p o t e n t i a l l o c a t i o n for failure. A m e c h a n i c a l joint, if p o o r l y designed, can be a cause of c a t a s t r o p h i c failure d u e to i n a d e q u a t e p h e n o m e n o l o g i c a l u n d e r s t a n d i n g of its b e h a v i o u r . Hence a n y a t t e m p t at the safe design of a structure w o u l d necessarily require a n u n d e r s t a n d i n g of the process of l o a d transfer t h r o u g h joints. Recent a d v a n c e s in the safety-conscious areas of n u c l e a r a n d a e r o s p a c e engineering call for a n i n - d e p t h s t u d y of the t h e r m a l effects on the p e r f o r m a n c e of structures. In p a r t i c u l a r , w e a k spots such as j o i n t s deserve extensive investigation to a t t a i n high levels of confidence.
2
R. SINGHand T. S. RAMAMURTHY
Blosser [1] recommends the shaping of the hole and the mating pin to alleviate the thermal stresses in joints. The concept is elegant but limited to uniform temperature changes only. A single pin in an isotropic sheet subjected to a thermal load of unidirectional heat flow, within the realm of small-deformation, linear, two-dimensional elasticity, appeared to be a good starting point to obtain some insight into the thermoelastic behaviour of the joint. The authors initiated the work with simpler studies of joint behaviour under pure thermal loading [2, 3], and went on to investigate the complex non-linear load-contact behaviour under combined thermal and pin-bearing load [4-6]. The study of partial contact behaviour of an unbonded circular misfit pin in a large isotropic and homogeneous elastic sheet is now carried out for another important case of a sheet under combined in-plane biaxial loading and unidirectional heat flow applied at the far field. We use the model of smooth interfacial surfaces and step interfacial conductance for the partial contact and separation conditions at the pin-hole interface. A unified, continuum method of analysis using Airy's stress function and the auxiliary displacement function approach [7, 8], together with inverse formulation (in which one seeks to estimate the load for a specified contact configuration [6,9, 10]), is employed to obtain the load-contact behaviour. By synthesising the thermal and mechanical parameters into generalized parameters, the possibility of presenting the load-contact relationship by a single curve is demonstrated. This Thermomechanical Generalization reduces the efforts of both analyst and designer by an order of magnitude at the preliminary design stage. Another intricate aspect of the present problem is that frictionless thermoelastic contact problems can exhibit multiple solutions as a result of the interaction between thermal expansion and contact conditions at the interface. Existence or uniqueness theorems cannot be established for such problems, particularly with classical boundary conditions. These aspects in relation to pin-joint models have already been observed by the authors for the cases of pure thermal loading [3] and combined pin-bearing and thermal loading [11]. They are now explored for the case of combined thermal load and applied sheet stresses. 2. CONFIGURATION Consider a large isotropic elastic plane sheet (E, v, ~s, ks) having a smooth hole of diameter 2a. A rigid pin (%, ko) of diameter 2a(1 + 2) is introduced into the hole, resulting in a misfit joint with geometrical proportional misfit 2 (Fig. 1). The pin-plate interface is presumed to be perfectly smooth so as to permit free tangential slip all over the interface. The thermal load is a far-field unidirectional heat flow along the x-axis, mathematically written as (1)
Ts = To + xT,,.
Uniform far-field applied stresses on the sheet are conveniently represented by a biaxial stress (Sx, St) described in terms of the parameters K1, Kz, So as Sx=(Kx+K2)~°
Sr=(K1-K2)~
Sxr=0
(2)
where K1 ~ and K= ~ represent respectively the axisymmetric and alternating (or pure shear) components of the general far-field stress. The effect of To is to alter the pin and hole diameters and modify the misfit parameter. The thermal gradient Tx pushes the pin along the x-axis through a distance of, say, Uo, [2]. Under mechanical loading, the pin-hole interface exhibits partial contact/separation [9, 10]. The thermal conductance at the pin-plate interface is modelled as a step function, viz. zero in the region where there is no mechanical contact and constant H in the contact region. Mathematically, 6H, 6 = 0 in separation and 6 = 1 in the contact zone. This is a reasonably good model as the heat transfer due to radiation in the region of separation can be neglected in comparison with the conductive heat transfer in the region of contact, and the normal interracial stress in most of the contact region is high enough that the contact conductance will be almost constant over the region of contact [6].
Pin joints subjected to sheet loads
(a)
3
SY
t __..
Q~
Sx
t
y
20
~
Sx
4"11" uot
SY
(b)
SY
.-_._L
t
t
D I1 in Sx
Q
Q
~ S
4"1~ Uot r
sY FIG. 1. Configuration; heat flow and sheet stresses: (a) Sx > Sy, (b) Sy > Sx.
The configuration (geometry, loading and boundary conditions) is symmetric about the x-axis and so only one half of the domain needs to be analysed. The stresses due to the interference 2 are axisymmetric and those due to the sheet loading are doubly symmetric [10]. In the case of an interference-fit pin for full contact situations, the normal interracial stress due to heat flow (loading Tx only) is zero [2]. Thus, with an increase in mechanical loading the separation will initiate symmetrically, leading to a doubly-symmetric separation configuration. Similarly for clearance-fit pins, as the net load on the pin must be zero, the initiation of spread of contact has to be doubly symmetric. Thus the contact/separation configuration obtained will be doubly symmetric. Depending upon whether Sx > Sy or Sy > S~, the configuration of Fig. l(a) or l(b) will be obtained, with the extent of separation as _ 0s about the horizontal or vertical axis through the centre of the pin, respectively. Working within the realm of small deformations the contact configuration can be treated as symmetric about the x- and y-axis. In some cases the interface may exhibit a contact configuration symmetric only about the x-axis. Such situations are dealt with using a slightly different approach [6], and are discussed in the latter half of this paper.
3. TEMPERATURE FIELD
For the partial contact configuration of Fig. l, the steady-state temperature fields should satisfy the governing differential equation, i.e. the Laplace equation V2T = 0 and the boundary conditions: (1) far-field temperature distribution Ts = To + xTx, (2) symmetry about the x-axis and antisymmetry about the y-axis, and (3) the interfacial condition, at r = a, - k p (3Tv -~r =
ks OTs dr "
4
R. SINGHand T. S. RAMAMURTHY
Such a temperature field can be expressed in series form as Ts = To + aTx
{(:) (a)} +
cosO - arx
(,.) (:)
gn r
~
cos nO
n=1,3,5...
Tp = To + aTx
--
g,
n=1,3,5...
cos nO.
(3)
The arbitrary constants gl, g3, g5 . . . . . gE,-i are to be determined by appropriately satisfying the interfacial boundary condition
-kpOTp~--
OTs
5H
ks ~rr - a (Tp-Ts)
(4)
6=00~
~ 6=1
5
Os <~ 0 <. rc - Os
0
rc-0s<0~
di-- 1 0 ~ < 0 ~ < ~ - 0 s and, for case B, when Sr > S x
0
rc - 0 s < 0 < ~ + 0rrs
1 ~+0s~<0~
with
H ~ = 0.5
H k~ = -* 0, 0.5, 1.0, 2.0, --* oo
with
kp = ~ 0.5.
The variation of g-£ g with ~kp and H for both cases A and B is plotted in Figs 2 and 3, respectively. The variable g is given by [2] 2 kp g= ks
{ 1 + ks + H
-1 kp)
"
For full contact and point contact, the values ofga match the closed-form results of Ref. [2]. The variation is continuous between full contact and point contact and lies within reasonably narrow bands for the full range of conductivities. 4. E L A S T I C
FIELD
4.1. Full contact: interference-fit pin For the situation of positive interference (2 > 0), the interface exhibits full contact up to a certain load level (K~ So, K2So, To, Tx). So long as the pin and the sheet are in complete
Pin joints subjected to sheet loads
5
1
\\
-
\\
,'--.'..
\
e
•..
.,,
,,
gl _
T
-
\\
_
x~-0.,---
,
Case
>,
"~'~\
;_ I
~
~~
__ I
I
,
I
I
','
A ~1
,
~t
0, FIG. 2. Effectof
I ~ . t " ~
--
= =.....
\
_
k__.Pv =
~.
ks
\
\
",,. \ \
--
.
05
~H%%% • %% • %% • \ ",7--.0-./',/, ~ 1 . o - - ' • /~ \/N/N,----,oo--Y• ~ I
\ \
-
on #1.
k,
•- . ~ . . .
gl g
"T
A
"~.
c.se~ _ _ _
~
, ,,,
~
~
~
~' l
,,,
0
2-
H FXG.3. Effectof~ on g~. contact, the stress and the displacement fields in the sheet can be expressed in closed form by a biharmonic Airy's stress function satisfying (i) (ii) (iii) (iv)
the far-field stress state [Eqn (2)] smooth interface; at r = a, a,o = 0 radial compatibility at the interface: at r = a, Us = Up + a2 + Uotcos 0, and zero load on the pin:
f~a, cos 0 r d0 =
0
at r = a .
The stress function in the presence of the temperature field for the full-contact situation (g~ = # and g. = 0 for n ~ 1) is
Ea22e (r)KlSoa2{1--v q~-
( l + v ) In
+ ~ 01
(r)
~ ' ~ v In
l(r) 2} +~
~, - ~ - {~ ( a ) + (at-) In ( r ) } cos 0
K2Soa 2 { ~ ( r ) z l : a ' ~ 2 2 +6\r] +
l+v( ~
1 (a)2)} 1--~ r cos 20.
(5)
From the above expression for the stress function, the rigid-body displacement of the pin with respect to the sheet was obtained as
not - ~,aTx - a
Yd':
(6)
6
R. SINGH and T. S. RAMAMURTHY
where 1
= ap_~s
ga--~(3+v)
1--gl~
and defined as the sheet-to-pin thermal stiffness, and agrees with the results of Ref. [2]. The load (So)c, for initiation of separation can be obtained from the condition a, = 0 at 0 = 0 7r
(for S~ > Sy) or 0 = ~ (for Sr > S~). This load can be derived as So
cr
=
Klat--~--v)
(7)
K2
and this agrees with the results of Rao [10]. Any further increase in load level (So) causes the separation to spread symmetrically about either the x- or y-axis. 4.2. Full separation: clearance-fit pin Initially with zero mechanical load (So = 0) an all-round uniform gap exists for the case of a clearance-fit pin (2 < 0). Under thermal loading with full separation at the interface, the problem of stress determination in the sheet reduces to the situation of a free hole, which has already been handled by Florence and Goodier [12]. Considering the pin to be a part of the system, the problem is ill-posed as the pin temperature is undefined and one more condition is required to evaluate the temperature field in the pin. To make the problem well-posed, consider the pin temperature to be To before the onset of contact. After the initiation of contact the pin temperature will be governed by the interfacial conditions. As the plate load So is applied and increased, initially the elastic field corresponds to that of an infinite plate with a circular hole loaded at infinity by sheet stresses and unidirectional heat flow [3, 6]. The uniform temperature To only alters the pin and hole diameters and the temperature gradient Tx causes a rigid geometrical displacement of the hole [2, 6]. The biaxial sheet loading distorts the circular hole to an elliptic one [6, 9, 10]. Thus, the initiation of contact largely depends upon the nature and the level of loading and is categorized in Table 1. After the initiation of contact, with increase in loading (mechanical or thermal) the pin may undergo a rigid-body movement and the point of contact may slip on the interface until the point-contact configuration becomes symmetric about both the x- and y-axis, i.e. either rc a point contact at 0 = ~ (for Sx > Sy) or point contacts at 0 = 0, rc (for Sy > Sx) are established. With further increase in loading, the contact will spread symmetrically about either the x- or y-axis. TABLE 1. INITIATIONAND SPREAD OF CONTACT IN CLEARANCE-FITPINS Mechanical loading
Location of initiation of contact
Thermal gradient
S~ > S r
Tx > 0
between 7~ ~ and n
T~ = 0
n ~
Tx < 0 Nature of spread of contact
between 7~ 0 and 2 symmetric about the x-axis
Sx = S r
Sx < Sy
71:
all-round simultaneous 0-r~
0 and n
0
all-round simultaneous
symmetric about the y-axis
Pin joints subjected to sheet loads
7
4.3. Partial contact: unified analysis of misfit pin For the analysis of the partial contact behaviour of a misfit pin, consider the case when the pin and sheet have separated over a part of the periphery. The boundary conditions are (i) biaxial sheet loading in the far field, Eqn (2) (ii) smooth interface, a,o = 0 at r = a, and (iii) no net load on the pin, i.e. f ~ a , cos 0 r dO = 0
(8)
at r = a.
(9)
Radial equilibrium and compatibility at the interface dictates that, at r = a, Us = Up + a2 + UotCOS0 in the contact zone
and
tr, = 0 in the separated zone.
(10)
Airy's stress function satisfying the boundary condition [Eqns (2), (8) and (9)], in the presence of a temperature field [Eqn (3)I, is obtained as
1 ~[K1Soa2(r) 2
+ n~2 { ( r ) 2
K2Soa2{(r) 2
-~
a
2
+l-n
-~-~n} (a)" a, cosnO.
(11)
The constants ao, a2, a3 a, are to be determined by appropriately satisfying the interfacial boundary condition [Eqn (10)]. Equidistant collocation is found to be satisfactory. . . . . .
4.4. Numerical evaluation For the unified analysis of a smooth pin-hole combination, the problem posed in inverse form [10] reads: Given the extent of separation, E, v, 2, what are the load levels To, Tx, K1 So, K2So? Obviously there cannot be a unique combination of all these loading values for a given contact configuration and hence the problem is treated by merging To with 2 and K2 assigning values to Tx and ~-~. Now the problem reduces to: Having specified the arcs of
separation~contact, what is the load level So for given values of E, v, 2, Ix, K2 ~v kp~
K I ' ~s' ks" A computer code was developed to determine the load-contact relationship. The conver-
gence of the load parameter (EK----~S~)was studied with respect to the order of truncation of the series. To limit the C P U time, all further computations were carried out with 65 points at the interface for 0s < 67.5 ° and 129 points for 0s t> 67.5 °. Numerical data were obtained for uniaxial tension along the x-axis (an example of case A) with ~- ~P-- 0 . 5 ;
~k p = 0.5;
~H= 0 . 5 ;
v=0.3;
oqaT~=O.O01, O.O, -0.001, - 0 . 0 0 2
for 0s = 00-90 ° in increments of 5.625 °. The load-contact relationship is plotted in Fig. 4. 4.5. Discussion For uniaxial tension along the x-axis (K1 = K2 = 1) for an interference-fit pin, the pin and sheet maintain full c°ntact uP t° a critical stress (SE-~,) cr = 0"376" As the sheet l°ad (S°, is increased further, the variation of the angle of separation (0,) with load (So) is non-linear and reaches an asymptotic value 0v as So ~ oo. This is shown in Fig. 4 by the dotted line. On the other hand, for clearance fits, contact starts spreading at So = - E2,. The contact angle
8
R. SINGH and T. S. RAMAMURTHY I I I I ' ~ I q
t
Ek e
KIS o
K1S 0
0s
Op=71
°
EZ.e
s ~ g
I !
-4
FIG. 4. L o a d - c o n t a c t behaviour: uniaxial tension
~x~aTx= 0.001, 0,
(
0c 0c = ~ - 0,
)
-4
I !
(~
k°
%
= 0.5, ~ = 0.5, --ct,= 0.5, v = 0.3, rigid pin,
- 0.001, - 0.002).
increases with an increase in load and tends asymptotically to ~ - 0p as
So ~ oo. This is more clearly understood by replotting the data with ~-o
as the ordinate
(represented by bold lines in Fig. 4). The intersection of this curve with the abscissa corresponds to So ~ oo for a given 2c or for any So with 2e = 0, i.e. the case of a push-fit pin, and the corresponding angle is 0 w The stability of the configurations was checked by slightly perturbing the contact configuration without changing the load and investigating the tendency of the contact to change. The configurations of the present cases were found to be stable. 5. T H E R M O M E C H A N I C A L
GENERALIZATION
5.1. Partial contact As long as full contact is maintained, thermal and mechanical effects are superposable. This was clearly brought out in Section 4.1, Eqn (5). For partial-contact situations with To merged into 2 the four load-contact curves for different values of T~ are found to be indistinguishable from each other. This implies that thermomechanical generalization has been achieved for the present case. The contact/separation configuration is independent of Tx and hence the curve of Fig. 4 holds good for all values of Tx ranging from - oo to + oo. A more general parameter for general biaxial sheet loading and unidirectional heat flow can / E 2 , - K1 So be \~ I(2So )" The data of Fig. 4 are replotted in Fig. 5 using this parameter along the /
ordinate. To confirm this generalization, numerical data were generated for the following combinations of parameters covering the full range of all variables involved: a---eP= ~ O, 0.5, 1.0 as
k-2 = ~ 0, 0.5, 1.0, 2.0, --* oo k, H
-- =
ks
~
0, 0.5,
1.0, 2.0,
~
v = 0.3 ct, aTx = 0.001, 0.0, -- 0.001, -- 0.002
K , = K2 = 1, uniaxial tension along the x-axis (Case A) K1 = 1, K2 = - 1, uniaxial tension along the y-axis (Case B).
Pin joints subjected to sheet loads 1.66 1
E~'e'K1S0
0
~5/7°
K2S 0
0 -1
-2
9
i
NNS \
-f
KI = - 1 K2
_
K~So (v = K2So
FIG. 5. Load--contact behaviour: use of generalized parameter E2c -
0.3, rigid
pro).
This covers the full range of distortivity (6 = s/k) ratios of sheet to pin (6,/6p). The values corresponding to the above combination are: ~ 0, 0.25, 0.5, 1, 2 and ~ . E2e - K1 So The data obtained were plotted for generalized parameter and the curve of
K2So
Fig. 5 was obtained in each case. Each time the rigid-body displacement of the pin was found to obey the established relation [2]
Uot ~saTx a o,~Fzr Thus we see that the interpretation of the curve of Fig. 5 can bring out information regarding the load-separation behaviour for any combination of E, 2, K1, K2, So, To, Ix. The point T on the curve with ordinate
KISo_ the So level as E2eK2 So
-K-~2 represents the push-fit case irrespective of
K1 K2 leads to ~E2~ = 0. The portion of the curve above
T corresponds to interference and that below T corresponds to clearance. Also, if E2~ E2~ S---~-~> 1.66 then interfacial separation is not possible, while if -~o ~< - 2 then interracial contact cannot be achieved. The case of partial separation/contact arises only in the region E2,
1.66 > ~-o > - 2 .
K2 The position T for the push-fit case depends only on ~ and is independent of So. Thus, the angles of separation are invariant with respect to the load level so that the stresses due to push-fit in a loaded sheet increase linearly with the applied load system. 5.2. Interfacial stresses Consider the critical portion, i.e. the interface for stress-variation studies. The normal stress at the interface ( a ~ ) is invariant with respect to thermal loading. This is presented in Fig. 6. Thus the radial stresses change by (-~) with respect to those in the case of pure mechanical loading. No generalizing parameter could be identified for the tangential stress, but there is an interesting feature that can be seen logically. The thermal gradient Tx does not alter the contact configuration, and so the resulting change in hoop stress should vary linearly with Tx. This has been found to be so. The (a~-~)value for Tx = 0.0 is plotted in
(o0)
Fig. 7 and the change in ~
per unit Tx is plotted in Fig. 8. The hoop-stress parameter
R. SINGH and T. S. RAMAMURTHY
10 1 --
~
_
Or
-
~
-4 _
78"75°
0~t
\
\
22.50 ° 33.75 °
67.5 °
l s
FIG. 6. Normal interfacial stress for various contact configurations; uniaxial tension and heat flow: r, o q a T x (v = 0.3, rigid pin). for all values of ~H , ~k r,, o~ a-~'
10 0 s = 67.5 °
0
'
;
'
",.5
Fro. 7. |nteffacial hoop stress due to pure unia×ial tension (v = 0.3, rigid pin).
200
d(ajaTx)
0
0
-200
67.5 °
Flo. 8. Effect of heat flow on interracial hoop stress; sheet under uniaxial tension ~ = 0.5,
= 0.5,
c~2a,= 0.5, v = 0.3, rigid pin).
(~-~,) for any contact configuration can be estimated by summing up the contributions due to mechanical load (Fig. 7) and thermal loading (Tx times the value in Fig. 8). 6. P O S S I B I L I T Y
OF NON-UNIQUE
SOLUTIONS
Thermoelastic deformations, though small, are often sufficient to have a major effect on the mechanics of contact. Thermoelastic contact problems can exhibit multiple steady-state solutions as a result of the interaction between thermal expansion and contact conditions at the interface. Barber et al. [13-17] have extensively studied the thermoelastic behaviour of
Pin joints subjected to sheet loads
11
solid bodies in contact and the essence of their findings is consolidated in Chapter I of Ref. [13]. Very interesting features are related to aspects such as the non-existence and nonuniqueness of solutions to frictionless thermoelastic contact problems. Existence and uniqueness theorems have been proved for heat-conduction problems with any combination of boundary conditions. Furthermore, if the temperature is known at all points in the body, a particular thermoelastic solution can always be found which reduces the contact problem to an equivalent isothermal problem for which, again, existence and uniqueness theorems have been proved. However, no existence and uniqueness theorem can be established for frictionless thermoelastic contact problems with classical boundary conditions [13]. Barber et al. [14] and Comninou and Dundurs [17] show that the lack of existence is connected only to highly idealized boundary conditions, and existence can be achieved by appropriately modifying the conditions imposed at the contact interface. However, lack of uniqueness is not caused by idealized boundary conditions and is not purely mathematical. It has a physical basis and signals possible instabilities. In Section 4 it was seen that the spread of contact/separation takes place symmetrically about both the x- and y-axes and such a configuration is stable. Still there is a need to look for other solutions, in the absence of any uniqueness theorem, for this class of frictionless thermoelastic contact problems. Whatever the contact configuration may be, symmetry about the x-axis must exist because of the symmetry of geometry, properties, loading and initial and boundary conditions. There is a possibility that the configuration may not be symmetric about the y-axis. Typical possibilities are shown in Fig. 9. We analyse a typical configuration to gain more insight into the behaviour of the system. Consider the ease of uniaxial sheet tension along with on-axis heat flow. To understand the system all combinations with the following variables values were analysed: ~._eP__. ~ 0 , 0.5, 1.0~oo as
kp k,
--+0,0.5, 1.0, 2.0, ~ o o
H ks
-- =
~ 0, 0 . 5 , 1.0, 2 . 0 ,
~ oo.
(a)
SO
r- S O i.
- , - t l 4 - Uot
r
(b)
(c) e2
o 01
03D
D X
g
FIG. 9. Interracial contact configuration not symmetric about the y-axis.
12
R. SINGH a n d T. S. RAMAMURTHY
This closely represents the situation of a steel bolt in an aluminium sheet. The physical nature was found to be similar for all situations. In order not to clutter this paper with numerical data one typical set of results are presented. We choose the combination kp = 0.5 ~ H= 0 5 y,
-~
ap = 0.5
E~-~oo
6.1. Configuration Consider the configuration of Fig. l(a) extended to the present case of unequal arcs of separation under uniaxial tension along the x-axis as shown in Fig. 9(a). The two arcs of separation extend over + 0,1 and + 0sZ about the x-axis at 0 = zt and 0, respectively. For the sake of convenience, the concise notation (0,1, 0,z) ° will be used to represent such a configuration. In the present case of zero net load on the pin, both 0sl and 0,2 must be bounded between 0 and ~. Also, the configuration (0,1,0,2) ° with thermal gradient Tx is same as (0,2, 0,1)° with gradient - Tx. This will be confirmed from numerical results. 6.2. Temperature field The series solution for the temperature field in the sheet and the pin for unequal arcs of separation will now have all the terms rather than just the odd terms [of Eqn (3)] and can be written as Ts= To + aTx
cosO-aTx(kp~
+
cosn0
\ks,/.~1
T v = To + goaTx + aTx
g,
cos nO
(12)
n=l
with the unknown coefficients to be determined by numerically satisfying the interfacial conditions of conductance. Equidistant collocation with 65 points at the interface was used to obtain the solution. The variation of the important variables go and gl [2] is presented in r~ Fig. 10 for 0 ~ 0,2 ~< 0,1 ~<2" This is done to achieve computational economy since physically, gO,(O.2,0.t)o = --gO,(O.l,O.2)o
and gl,(O.2,o.t)o = gl,(O.l,o.:)o.
Hence the information regarding 0~2 > 0,~ need not be generated independently. There is another interesting feature regarding go. Since gO, (0, 0) ° ~
- - g0,(O, 0)o
Ie$1 = ~ $ 2 = O° L / ~75"75°
F/\
H
II-VVN l l h A Yx H[X X
0
IVV ,V, V
"/
67.50 ° 56.25 e 45.00 °
k
~:~:o.~
X",,
V"',.>~ ,V 33.75 °
~°°
,~f~'~ff 22.50°
°
11.25" ]
gl/g FIG. 10. V a r i a t i o n o f go a n d 0t for c o n f i g u r a t i o n s with 0sl ~ 0s2.
P i n j o i n t s s u b j e c t e d to sheet l o a d s
13
and go,(o,o)o = 0
we can write go,(o,o)o = go,(o,o)o + go,(o,o)o.
Physically, this can be extended to go,(o.~,o,2)o = g o , ( 0 , t , o ) o + g o , ( o , 0 , 2 ) o =
g O , (0,,1, O) ° - -
gO,(O,,2, O) °"
From the numerical data obtained this has been observed and is shown in Table 2 for a few typical values of 0sl and 0s2 chosen randomly. Thus, for go, the data for a single arc of separation (obtained earlier I-4,5]) can be used to approximately generate information regarding the double arc of separation configuration also. 6.3. Elastic field For the partial contact configuration (0~1, 0s2)°" there are two transition points placed asymmetrically and hence two load values can be uniquely determined using the inverse % kp H formulation. The problem posed reads: Given E, v, KI = K 2 = 1, ~ (including To) as' ks' ks' and the configuration (Osl, 0s2)°, what are the load levels Tx and So? The unknown coefficients ao, a~ . . . . . a. in the stress function q~ of Eqn (11) were evaluated numerically using the collocation technique and the boundary conditions of a, = 0 in separation, and gap = Us - Up - a2 - Uotcos 0 = 0 in the contact zone. Obviously, not all combinations of 0s~ and 0s2 will be feasible. So, for any configuration, having obtained the solution the feasibility was checked by looking at the interfacial normal stresses and displacements, i.e. by checking the inequality constraints of a, < 0 in the contact zone, and gap > 0 in the separation zone. A feasible configuration must have ar = 0, gap > 0
in the separation zone;
crr = 0, gap = 0
at the transition point; and
a, < 0, gap = 0
in the contact zone.
The feasibility envelope obtained for the configuration under study is presented in Fig. 11. Each point within the envelope AFGHBEDCA corresponds to a feasible combination of So and Tx. Points outside the envelope correspond to configurations which are non-existent. Although values of Tx and So can be obtained to satisfy the boundary conditions when using the inverse formulation, the solutions violate the above-mentioned inequality constraints and hence are unacceptable. The envelope is symmetric about the diagonal line APB and, in fact, the mirror-image points across this line represent the loadingcombinations So, Tx and So, - T~. Physically, this mirror-image symmetry confirms that one configuration is achievable by rotating the other through 180° about the y-axis. Thus, we need to discuss only one half of the envelope. The points on the diagonal line APB represent the equal-arcs-of-separation solution of Section 4, and each point refers to a unique value of So and any value of the thermal gradient Tx. The case of a push-fit pin is represented by the point P(0p, 0p)°. The variation of So along APB can be seen in Fig. 4. TABLE 2. INTERPRETATION OF THE VARIABLE go FOR TWO UNEQUAL ARCS OF SEPARATION FROM THE DATA FOR A SINGLE ARC OF SEPARATION S. n o
0sl
0s2
go,(o,=,o)
go,(e.2,o)
1 2 3 4 5
22.5 ° 45.0 ° 45.0 ° 67.5 ° 90.0 °
5.625 ° 16,875 ° 67.500 ° 28.125 ° 45.000 °
0.1663 0.4496 0.4496 0.7975 1.1635
0.0230 0.1106 0.7975 0.2286 0.4496
go,(e,t,o) -- go,(e,2,o) 0.1433 0.3390 -0.3479 0.5689 0.7139
go,(e,~.e.2) 0.1433 0.3400 -0.3455 0.5746 0.7354
%dev 0.00 0.29 0.69 1.00 3.01
14
R. SINGH and T. S. RAMAMURTHY
T E
H
I
i
0S2 C
I/"
/I F
Ao
I
I ~_
2
0sl
FIG. 11. The feasibility envelope: uniaxial loading [Ref. Table 3]
= 0.5,
= 0.5, - - = 0.5,
v = 0.3, rigid pin).
Ae = e s 2 - 0 s 1 ~
Sl
~-22.500"
~ / 1 1 . 2 5 o
EX ..._&e So
"~ ~ ~ / 5 . 6 2 5 "
o -1
o
",,X\\\/0°
--
o..
_
° \.N\\ -- .x,~..
FIG. 12. L o a d - c o n t a c t behaviour for configurations with 0st ~ 0s2: mechanical load parameter
H = 0.5,~ = 0.5, -~, - = 0.5, v = 0.3, rigid pin .
The sub-envelopes A F G H P E D C A and BEPHB are regions of feasible configurations for interference- and clearance-fit pins, respectively. Line EPH represents the push-fit case. Each point I-configuration (0,1,0,2) °] within these sub-envelopes corresponds to a unique combination of loadings So and Tx. As we move from AC towards B the value of the load E2c parameter ~ decreases from a finite value to zero at EP, and falls further to - 1 at B. The E2c variation of -ff0-owhile moving parallel to APB (A0 = 0s2 - 0sl = constant) is plotted in Fig. 12. The thermal-load parameter
~
is positive for clearance fits (sub-region BEPB,
2e < 0), negative for interference fits (sub-region APDCA, 2e > 0) and zero for push fits (line oqaTx EP, 2c = 0). As we move from PB to EB, ~ increases from zero to + oo, and in moving
or,aT:, from AP to CDE ~ changes from zero to - o o . This indicates that high thermal
Pin joints subjected to sheet loads
15
gradients are required to sustain larger differences in extents of separation (0,2 - O~t). The
~saTx
line DP represents a special feature. The parameter ~
has a finite value on this line but
both T~ and 2. change sign across the line. In fact, the variables ~
1
and
tend to zero
along the line. In the six sub-envelopes of the feasibility envelope of Fig. 11, the various parameters have a specified nature and generally change sign across the boundaries. A comprehensive presentation of the nature of these various parameters is given in Table 3. E2~ 2, Figure 13 shows the relationship between the loading parameters ~ o and ~ for specified values of 0~ and A0 (A0 = 0~ - 0s2). Again, it is seen that as A0 increases the system requires high T~ values. Figure 14 shows the normal and hoop-stress variations along the interface for a situation in which one of the transition points is kept fixed (0,2 = constant) and the other is moved within the feasibility envelope. The hoop-stress
TABLE 3. QUALITATIVENATUREOF VARIABLESWITHINTHE FEASIBILITYENVELOPEOF FIG. 11
Line
Region Line Line
+
any
any
+
+
0
any
~
any
+ oo
AP
+
P
0
PB
So
-
-
any
any
+
+
+
-
-
+
+
CD
+
+
- ~
- oo
+
+ +
+
oo
- oo
-
~
DP(PEDP)
+
- oo
ov
-
- oo
PEDP
+
-
+
-
Line
So
APDCA DP(APDCA)
Region Line
~aTx 2~
2~
Point Line
T~
2
DE
-
-
+ +
+
-
+
- oo
-
+
EP(PEDP)
+
0
-
- oo
any
oo
EP(BEPB)
-
0
+
~
any
- oo
BEPB
-
-
-
+
+
-
EB
-
-
-
oo
+
-
Region Line
~O A0-*0 o
~saTx
i '6°
"~[~q~
i 0
1
~
14.0 °
~ 25.3•
2
Ek e So FIO. 13. L o a d - c o n t a c t
b e h a v i o u r f o r c o n f i g u r a t i o n s w i t h 0,1 # 0,2: t h e r m a l a n d m e c h a n i c a l l o a d
p a r a m e t e r s ( ~ , = 0 .5 , ~ kp = 0.5, --,t, ~p = 0.5, v = 0.3, r i g i d p i n ) . HS 35:1-B
16
R. SINGH a n d T. S. RAMAMURTHY
(a)
~r EX© -1
i o
0 (b) m
EX,c -4 0sl = 22.5 °
FIG. 14. Interfacial stresses for various contact configurations: (a) n o r m a l stress, (b) h o o p stress (configuration of Fig. 9a)
param0ter /
foun
= 0.5, ~ = 0,5, -as- = 0.5, v = 0.3, rigid pin .
to change drastica,,y whereas the norma, stress parameter
\
( a ~ _ ) changes very little. From the variation of load values (Tx, So) within the envelope, it can be seen that for any loading combination Tx and So there are two configurations, one corresponding to 0,1 = 0s2 (on the line APB) and the other corresponding to 0sl # 0,2 (within the envelope but not on the line APB). A typical example is: For the combination %aT~ -- -0.02634 and ESo = 0.003306, two solutions (Fig. 11) are realized as Sl ~ 0~1 = 0s2 = 48.08 °
and
$2--* 0~1 = 33.75 °, 0,2 = 42.1875 °.
In Section 4 it was seen that the contact/separation starts spreading symmetrically about the y-axis, and thus the true load-contact behaviour for misfits will, in general, be along line APB and configuration S1 will be realized in practice. However, by carefully steering the load magnitudes and directions, configuration $2 can be achieved, but any disturbance will make the configuration jump to solution S1 as $2 is an unstable configuration. The stability of $2 is checked by giving it a small perturbation, inspecting the inequality constraints mentioned earlier, and realizing that the configuration settles down to solution S1. It is due to this reason that the contact/separation regions cannot grow unsymmetrically. We have discovered that there are two configurations for all combinations of T~ and So. According to Barber and Comninou [131, there must be an odd number of solutions. Thus, there is at least one more solution, which probably corresponds to a configuration of
Pin joints subjected to sheet loads
17
the kind shown in Fig. 9(c) with three transition points. Possibly there can be more solutions with even more than three transition points. At this stage, we cannot analyse such complex configurations by the inverse technique, since we have only two free load variables, Tx and So. 7. C O N C L U S I O N
For a single misfit pin in a large isotropic plate, thermomechanical generalization of non-linear partial contact behaviour has been successfully achieved for combined in-plane biaxial sheet loading and on-axis unidirectional heat flow. The actual load--contact behaviour follows a stable doubly symmetric contact configuration. The stress concentration for stable configurations was found to be linearly dependent on the thermal gradient. Configurations analysed with single symmetry are unstable and non-realizable in the normal loading cycle. There is a possibility that some more solutions exist, which probably need tedious and complex procedures to obtain them. Acknowledgements--The authors acknowledge with pleasure the association of Prof. A. K. Rao, Emeritus Scientist, National Aeronautical Laboratory, and Prof. B. Dattaguru of the Department of Aerospace Engineering, Indian Institute of Science, Bangalore, with this work. The first author is grateful to the JRD Tata Trust for enhanced financial support. The general financial support of the Aeronautics R&D Board India, is also acknowledged. REFERENCES 1. M. L. BLOSSER,Thermal-stress-free fasteners for joining orthotropic materials. AIAA Jl, 27, 472 (1989). 2. R. SINGH, T. S. RAMAMURTHYand A. K. RAO, Thermomechanical generalisation in pin joints: A model. J. Aero. Soc. India 43, 35 (1991). 3. R. SINGHand T. S. RAMAMURTHY,Non-existence and non-uniqueness in thermoelastic contact problems: the circular inclusion model. J. Aero. Soc. India 43, 279 (1991). 4. R. SINGH, T. S. RAMAMURTHYand A. K. RAO, Thermoelastic analysis of misfit pin joints. Proc. Int. Conf. on Structural Testing Analysis and Design, Bangalore, India, pp. 736-741 (July 1990). 5. R. SINGH,T. S. RAMAMURTHYand A. K. RAO,Thermomechanical generalisation in joints: on-axis pin bearing load. Nucl. Engng. Design 135, 307 (1992). 6. R. SINGH, Thermomechanical generalisation of partial contact in pin joints. Ph.D. Thesis, Indian Institute of Science, Bangalore, India (1991). 7. K. S. RAO, M. N. BAPURAO and T. ARIMAN,Thermal stresses in plates with circular holes. Nucl. Engng Design 15, 97 (1971). 8. R. SINGH, Auxiliary displacement function approach to problems in 2-D elasticity with body forces and initial strains. Proc. 34th Congress of the Indian Society for Theoretical and Applied Mechanics, Coimbatore, India (1989). 9. V. A. ESHWAR, B. DATTAGURUand A. K. RAO, Partial contact and friction in pin joints. Report No. ARDB-STR-5010, Aeronautics R&D Board, Ministry of Defence, Directorate of Aeronautics, New Delhi, India (1977). 10. A. K. RAO, Elastic analysis of pin joints. Comput. Struct. 9, 125 (1978). 11. R. SINGH,T. S. RAMAMURTHYand A. K. RAO, Non-uniqueness and realisability of thermoelastic contact configurations in load bearing pin joints. Nucl. Engng. Design 135, 315 (1992). 12. A. L. FLORENCEand J. N. GOODIER,Thermal stress at spherical cavities and circular holes in uniform heat flow. J. Appl. Mech. 26, 293 (1959). 13. J.R. BARBERand M. COMNINOU,Thermoelastic contact problems, Chapter I in Thermal Stresses III (edited by R. B. HETNARSKI),pp. 1--106. Elsevier, Amsterdam (1989). 14. J. R. BARBER,J. DUNDURSand M. COMNINOU,Stability considerations in thermcclastic contact. J. Appl. Mech. 47, 871 (1980). 15. J. R. BARBER,Stability of thermoelastic contact for the Aldo model. J. Appl. Mech. 48, 555 (1981). 16. M. COMNINOUand J. DUNDURS,On the possibility of history dependence and instabilities in thermoelastic contact. J. Thermal Stresses 3, 427 (1980). 17. M. COMNINOUand J. DUNDURS,On lack of uniqueness in heat conduction through a solid to solid contact. J. Heat Transfer 102, 319 (1980).