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Rao1978-ElasticAnalysis-PinJoints.pdf
ELASTIC ANALYSIS OF PIN JOINTS?
A. K. F&o
Department of Aeronautical Engineering, Indian Institute of Science, Bangalore 560012,India
(Received 29 November1977; receivedfor publication20 March 1978)
Ah&a&-The advent of huge and fast digital computers and development of numerical techniques suited to these
have made it possible to review the analysis of important fundamental and practical problems and phenomena of
engineering which have remained intractable for a long time. The unders~nding of the load transfer between pin
and plate is one such. Inspite of continuous attack on these problems for over half a century, classical solutions
have remained limited in their approach and value to the understanding of the phenomena and the generation of
design data. Cm the other hand, the finite element methods that have grown simultaneously with the recent
development of computers have been helpful in analysing specific problems and answering specific questions, but
are yet to be harnessed to assist in obtaining with economy a clearer understanding of the phenomena of partial
separation and contact, friction and slip, and fretting and fatigue in pin joints. Against this background. it is useful
to explore the application of the classical simple differential equation methods with the aid of computer power to
open up this very important area. In this pauer we describe some of the recent and current work at the Indian
Institute of Science in this last direction.
“Simpleis bear&l”-J.
NOMENCLAWRE
free constants in stress functions
diameter of hole
elastic constants, Young’s modulus, Poisson’s ratio and rigidity modulus
E&E,, pin to plate modular ratio
axisymme~ic and alternating (pure shear)
applied far field stress systems
KzlKi
number of equidistant points in coilocation
P pin load per unit thickness
P pin load parameter EaUP
r,@;X,Y polar and Cartesian coordinates
applied far field sheet stresses
applied stress level parameter
s plate load parameter linking EA. K,, K2,
iJ
I”, so
(km
,.& f,
~0s
ne
P
5
radial and tangential displacements
~splacements in Cartesian coordinates
rigid body component of pin displa~ment
along X-axis
arcs of slip
arcs of separation/contact with proper
subscript
proportional interference (takes negative
values for clearance)
polar variation of interference
friction coefficient
direct and shear stresses
t/co: + fQ*- a#@t 3&)
Aiiy stress function
displacement function auxiliary to cp
beginning, end of slip region
semi arcs of contact, separation and
separation for push tit
Cf critical
initiation of slip or separation
s,P sheet and pin
Y, r, e directions
tThis paper was prepared in honor of Professor John H.
Argyris on his 65th birthday. Unfortunately, it could not be
published in the Special Issue of Computers and Structures (Vol.
8, No. 3,4) dedicated to Professor Argyris.
H. Argyris
1. INTRODU~ION
The pin in a sheet is an impo~ant abstraction in
engineering practice. It represents the most common
method of, connecting together components of an
assembly, with or without the facility of articulation. The
essentials of the configuration are, a round pin introduced into a nominally round hole of a larger, equal or
smaller diameter, yielding respectively a clearance, push
(snug) or interference fit. The contact surfaces may be
ideally smooth permitting free relative slip or ideally
rough inhibiting any slip at all or with finite friction
permitting selective slip. The load environment may be
thermal, mechanical, or a combination. The force system
may be applied as a traction on the pIate alone or as a
force transfer between pin and plate or as a combination.
Clearly, each pin-hole combination is a potential source
of structural weakness, by overloading, or cumulative
fatigue damage or progressive creep. So the problem has,
for many years, attracted much analytical and experimental attention. Yet, we seem to have achieved relatively little systematic understanding and information for
practical purposes, as is evident from the literature and
the relevant data sheets of the Engineering Sciences
Data Unit (ESDU) [7]. This is true even with the simplifying abstraction of considering a plate of infinite extent.
In brief, the pin joint appears to be one of the deceptively simple problems where technological practice is
far ahead of analytical development.
Consider the simpie abstraction of a single pin in an
in~nitely large plate assuming two~imensional plane
stress or plane strain condition. The problem is relatively
straight forward, if full pin-hole interfacial contact
around the periphery is maintained at all stages of loading, and, the interfacial friction is ideally zero or ideally
large, so that uniform homogeneous boundary conditions
can be stipulated. In fact, for such cases, closed form
solutions have been obtained over and over again and by
alternative methods. On the other hand, when pin-plate
contact exists only around part of interface, this contact
area varies non-linearly with the applied tractions and we
have a problem of mixed boundary conditions with mov125
A. K. Ibo
126
ing boundaries. Analytically, following the KolosovMuskhelishvili formulation, this would lead to singular
integro-differential equations, which, by clever handling
of the formulation and manipulations, may at best
simplify to singular integral equations, which in turn may
yield closed form solutions in a limited sense, for only
one or two very special cases. The situation gets further
complicated when one introduces friction into the
formulation. In fact, we then have difficulty in satisfactory formulation, not to mention satisfactory analysis.
The foregoing situation led, in earlier years, to
experimental efforts, particularly with the photoelastic
technique. Here too, there have been serious problems of
specimen preparation and experimental observations of
slip and separation phenomena. In the last two decades
the appearance of large electronic digital computers and
the simultaneous development of the powerful finite
element methods by Argyris and others has led to a
different approach of anaiysing specific problems in great
detail. But the possibility of using the fast and large
computational capability of the electronic digital
computer to develop simple analytical tools for the
analysis, understanding and data gene~tion for the pin
joint, either as an abstraction or as part of a structural
system, appears to have remained relatively dormant. It
is the purpose of this paper to present work in this later
direction, carried out by the author et al., as a tribute to
Argyris whose emphasis has been on clarity of physical
concepts, simplicity of technique and expanse of scope.
2.OUTLINROFTREPRORLJiM
For the present purpose, we will consider a round pin
in a round hole in a Largesheet, Figs. 1 and 2. Nominally,
the hole and pin diameters are 2a and 2a(l+ A). A is the
proportions interference of the pin-hole combination.
When negative, it represents clearance, and when zero, it
represents a push fit. Fu~hermore, to account for nonuniformity of the interference around the periphery,
arising from design considerations, manufacturing errors
or service wear and tear, we may assign it a polar
variation. The plate and pin material properties are
to)
INTERFERENCE,
SEPARATION
POSITIVE
OVER
A,
characterised by their elastic constants (6, v,; E,, v,).
At this stage we presume isotropic, homogeneous
materials and small deformation linear elasticity. To
justify a two-dimensional treatment, the pin length is
identified with the plate thickness. The interface has, in
general, a finite coefficient of friction p. To obtain clarity
of the phenomena with relatively simple analysis, we first
examine extensively the two extreme cases of a smooth
zero-shear interface &+O) and a rough zero-slip interface (p +m). For a quantitative appreciation of the effect
of finite size of plate, one could study finite width
infinitely long strips, or square or rectangular or annular
plates. Without loss of generality in formulation, we will
consider two-dimensional states of plane stress arising
from application of an in-plane biaxial stress state to the
sheet alone (Fig. 1) or a transfer of load from pin to sheet
(Fig. 2).
The biaxial stress state on the plate at infinity is
conveniently represented by the combination,
S, = (Kr - K&%/Z and S,, = 0
(1)
s, = (K, + &&/2,
K&/2 and K&/2 respectively represent the axisymmetric and alternating (or pure shear) components of a
general far field loading. The pin loading is represented
by the orthogonal components Px, P, along the X and Y
directions. This loading is reacted at infinity by vanishingly small stress systems with resultants P,, P,.
The applied plate stress component K,&/2 being
axisymmetric has no preferential direction; it effectively
modifies the geometrical interference. The component
K&,/2 introduces a pair of orthogonal axes of symmetry
into the system and a preferential axis for the elongation
of the hole. Thus, if only K&/2 (tension) is applied to a
state of interference and is monotonic~ly increased,
interface separation is initiated simul~neously all around
the periphery. On the other hand, if only K&/2 (tension
in X-direction) is superposed on the interference, and
increased monotonically, separation is initiated first at
the two points A, I3 on the X-axis. The pin load P,
( b)
CLEARANCE,
CONTACT
2&i
NEGATIVE
OVER
X ,
2ec
Fig. 1. Configuration and coordinate system: Partial contact or separation due to plate load, KzSopositive,hole dia
20, pin dia 2a(l+ A).
f f~ 1
Fig.
2.
I~~RFERENCE
SEPARATION
, POSITIVE
OV?B 2&
h ,
f bf
CLEARANCE,
NEGATIVE
CONTACT
OVER 20,
X ,
ConEgurationand coordinatesystem: Par&l contact or separation due to pin load, hole dia 2n, pin dia
2a(l+ A).
Elastic
analysis of pin joints
introduces symmetry about the X-axis and a preferenti~
direction (+X) for the elongation of the hole. it initiates
contact at A(X = a) for clearance fits and separation at
S(X = -a) for interference fits. A similar statement can
be made for PY In fact, an axisymmetric stress KISo=
EA just suppresses the initial interference or clearance A,
leaving neither gap not contact stress at the interface,
while K& is the applied stress component which introduces a preferential direction for interface contact, so
that (EA -K&,)/K&
would primo facie appear to be
an important non-dimensional parameter for the
problem.
For an interference fit and small applied loads, full
interfacial contact is maintained around the periphery. In
all other cases, we have a situation of partial contact and
partial separation. As the load is appiied, the two surfaces where they are in contact, slip freely when p is
zero, are in rigid linkage (i.e. slip is completely inhibited)
when p -+ and exhibit partial or selective slip for finite
friction coefficients. That is, in the most general case,
one has distinct regions of separation, contact without
slip (rigid linkage), and contact with slip. The extent of
each region is, ab initio unknown, and generally va,ies
nonlinearly with the applied load.
So we have a formidable problem even when the
simplifications of infinite sheet, two-dimensional state of
stress, small deformation linear elasticity and isotropic
homogeneous materials are invoked. We wili therefore
start with a simple version of the problem with un~orm
A, rigid pin and perfectly smooth interface to develop a
method of analysis and proceed in steps to introduce the
complexities due to elasticity of pin, polarly variable
interference and interfacial friction. Once the method of
analysis is established and the phenomena appreciated in
relation to a pin in the infinite isotropic sheet, one can
introduce finite boundaries for the plate, combined
loading, interactions between pairs and clusters of pins,
and anisotropy of materials.
3. A BRIEF REVIEW OF THE ~~~
ON PINJOINTS
Over the years, the pin-joint has inspired many
outstanding
mathematical
inand ex~riment~
vestigators. The work on interference fits upto 1966 was
competently reviewed by Venkataraman [lo]. This work
almost exclusively presumed a state of full contact and
either full or zero slip on the pin-plate interface. Investigations of partial seaparation and partial slip have
been very few. Scarce too are investigations on any
aspects of clearance fit joints. Comparatively, there is
reasonably extensive literature on the attempts to
analyse push fit probIems. These papers mostly consider
infinite domains and apply the Kolosov-Muskhelishvili
complex variable formulation[2, 4, 51, with considerable
ingenuity and insight. They develop and go through
sequences of extensive and intricate mathematics
manipulations finaliy leading to inte~~ifferential
equations which occasionally can be simplified to
singular integral equations for a few special cases, such
as pin and plate material properties being identical or
bearing a simple relationship such as G,/(l - 2~~)=
G,/(l - ~vP). Invariably the final equation needs another
elaborate scheme of approximate numerical procedures
to obtain either phenomenological or quantitative information. The more recent trend, quite predictably, has
been to apply finite element formulation and iterative
schemes. One cannot escape a sense of frustration at so
CAS Vol. 9, No. 2-B
127
much ingenuity, competence and effort leading to such
limited i~ormation.
Consequently, efforts of scientific workers to
economically obtain an insight into specific aspects of
the problem has taken directions of ad hoc investigations. The analytical workers side step such questions as, how to determine the progress of separation of
slip with increasing load, and attempt to obtain the stress
state presuming knowledge of the areas of contact and
slip. The experimental workers have tended to plan
limited purpose experiments from which to draw out
specific data for ad hoc design purposes.
Phenomenologically, friction plays a major role in
determining the static or fatigue performance of a pinjoint. But there is extremely little analytical work reported on the effects of friction and the more promising
analytical papers, on closer examination, turn out to be
unacceptable in either their formulations or their generalised conclusions.
In view of the above situation, not much purpose can
be served here by an elaborate survey of the literature.
So, we will now attempt only a very brief review to bring
out the highlights. Table 1 presents a condensed picture
of the work heing reviewed and the new work on which
the present paper is based. More extensive bibliographical information is available in Refs. [2,4, IO, 341.
Bickley[lZ] considers the application of a load to a
rivet in an infinite ptate. He postulates, inter ah, the two
situations of push and clearance fits. In the push fit case,
he presumes contact over an arc of 180” and a cos 8
distribution for the contact pressure. Fourier analysing
this radial pressure in the full O-360”range, he sets up a
stress function in the form of the Michell solution in
polar trigonometric series [6]. The actual angle of contact
for a push fit is now known[l9, 351 to be only 165”.For
the clearance fit, Bickley proposes, by analogy with the
Hertzian solution, a pressure distribution p[l - @/a2]“*
and proceeds to set up a stress function as in the push fit
case. He does not concern himself with the determination of (Y,the semi angle of contact, which in fact
depends on the degree of clearance and the load
level.
Kni~t[31] follows Bickley’s procedure in formulation
and analysis but extends the solution to a strip of finite
width, and presents numerical results for a hole diameter
equal to half the strip width. His solutiod is useful to
ascertain the effect of boundaries and symmetrically
displaced neighbouring holes.
The analysis of a pin in an infinite sheet was given
tremendous impetus by the Kolosov-Muskhelishvili[4]
and Stevenson[43] method of complex variables and
complex potentials.
Tiffin and Sharfuddin[43] study the mixed boundary
value problem of a clearance fit insert with a smooth
interface in an infinite plate. Based on Stevenson’s
complex variable tech~que and certain manipulations,
they arrive at an integral equation for which, there is no
known method of solution except in the case when the
contact region can be taken to be equal to a semi-circle.
In this special case, they arrive at a singular integrodifferential equation which they point out is similar to
that given by Prandtl in connection with wing theory and
the solution of which can be obtained by Multhopp or
Muskhelishvili methods.
Sharfuddin[41] goes on to consider the problem with a
rough interface. Unfortunately his formulation used a
boundary condition V =0 in the rough contact zone
(B) PUSH FIT
(a) Partial separation
(smooth interface)
(d) Initiation of slip
(Finite a)
(e) Partial slip
(Finite p)
(c) Partial separation
(Rough interface)
(A) INTERFERENCEFIT
(a) Initiation of
separation (smooth
and rough surfaces)
(b) Partial separation
(Smooth interface)
(1)
Type of Fits
and
phenomenon
of past and current work
1. Bickley (1928)
2. Knight (1935)
3. VAE, BD and
AKR(1977)
4. SPG, BD and
AKR(1977)
WW+
1. VAE,BDand
AKR (1977)
1. VAE, BDand
AKR (1977)
2. SPG, BDand
AKR (1977)
1. VAE, BDand
AKR(1977)
2. SPG, BD and
AKR (1977)
1. NSV,AKR
(1966)+
1. NSV,AKR
(2)
(1973)
5. Kalandiya (1973)
(1969)
4. Keer et al.
1. Muskhelishvili (1949)
2. Stippes et al. (1%2)
3. Noble and Hussain
1. Margetson and
Morland (1970)
(3)
ANALYTICAL
Infinite Plates Except at ( + )
Continuum approach
Continuum approach
with complex Varia.
with stress
function
Sign. Int. eqn
Table I. Survey
I. Harris
et al. ( 1970)
(1973)
I. Ghadiali
et al. (1974)
2. Brombolich
I. Harris et al.
(1970)
(4)
FINITE
ELEMENTS
I. Coker(l925)
2. Jessop et al.
(1956)
3. Jessop et al.
(1958)
I. Lambert and
Brailey (1%2)
2. Jessop et ul.
(1956)
I. Lambert
and Brailey (1%2)
(5)
PhotoElasticity
(6)
Strain
Gauge
I. Jessop
et a/.
(1956)
EXPERIMENTAL
1. SPG,BDand
AKR (1977)
1. AKR, VAE and
BD (1977)
1. Bickley (1928)
2. VAE (1977)
3. SPG, BD and
AKR (1977)
1. VAE,BDand
AKR (1977)
2. SPG, BD and
AKR(1977)
1. V AE, BD and
AKR (1977)
2. SPG, BDand
AKR (1977)
1. Mus~elishv~i ( 1949)
2. Tin and Sharfuddin
(1964)
3. Kalandiya (1973)
1. Muskhelishv~i(l949)
2. Sh~udd~n (1966)
1. Hussain and Pu
(1971)
tlndicates finite plate
AKR-A. K. ho, BD-B. Dattaguru, SPG-S. P. Ghosh, VAE-V. A. Eshwar, NSV-N. S. Venka~aman.
(E) VARIABLE INTERFERENCE
AND CLE~NCE
FITS
(smooth interface)
(D) UNIFIED APPROACH
TO ALL TliREE FlTS
(smooth interface)
(b) Partial contact
(rough interface)
(0 CLEARANCE FIT
(a) Partial contact
(smooth interface)
(c) Partial slip
(b) Partial separation
(rough interface)
(1970)
I. Harris et al.
I. FrochtBnd
Hill (IWO)
2. Coxand
Brown(l964)
Hill (1940)
I. Frocht and
I. Frocht
and Hill
(1940)
130
A. K. RAO
which is untenable for such a problem of progressive
contact in the presence of friction.
Margetson and Morland[33] consider an oversized
circular inclusion with a smooth interface in an infinite
plate under uniaxial plate load. They study the separation
of the ,inclusion from the plate boundary using Muskhelishvili’s complex series formulation leading to a
singular integral equation for the unknown contact pressure over an unknown arc of contact. The integral has a
difference kernel which has a discontinuous first derivative (i.e. a cusp), at the origin. In order that the cusp may
be eliminated, the elastic constants of the two materials
should be such that the value of G( 1+ v)/( 1 - V) is identical. For the case of identical materials, they apply an
analytical-numerical procedure involving Chebyshev
series and determine the load levels for stipulated arcs of
contact. Two points are of special interest in this paper.
The authors do incidentally achieve results for clearance
fits, but by ascribing a negative sign to the load instead of
to the interference value, they fail to recognise their
solution for the clearance fit problem. Secondly, by
their procedure, they are also able to formulate and solve
the problem for viscoelastic materials.
Stippes et al. [42], Nobel and Hussain [351,and Keer et
al. [30] have systematically and with ingenuity developed
the application of the Muskhelishvili complex variable
technique to the push fit problem with a smooth interface
and an infinite domain.
Stippes et a/.[421 consider a uniaxial stress field at
infinity. They obtain a singular integro-differential equation which reduces to a singular integral equation when
the plate and disc materials are identical. This special
case is analysed and numerical results are obtained by
adopting Muskhelishvili’s solution of the Hilbert problem. The authors mention that for a general combination
of materials, this solution can be numerically evaluated
only by approximations.
Noble and Hussain[35] consider both plate and pin
loads and obtain a dual series representation, deriving it
from the two distinct boundary conditions on the arcs of
contact and separation. Again, presuming a special case
of G/(1 - 2~) identical for the two materials, they are
able to obtain a Fredholm integral equation for the
unknown contact stress and angle. With further substitutions and manipulations, they reduce this integral equation to an airfoil equation and obtain an equation for the
contact stress in terms of the contact angle and another
equation for the determination of the contact angle. They
state that, for arbitrary material constants, an approximate solution has to be sought based on variational
methods given by Noble.
Keer et al.[30] consider a biaxial load system at
infinity. Their work proceeds from methods for solutions
of crack problems and uses a whole series of steps of
physical intuition, mathematical manipulations and
numerical methods. Once again they present numerical
results only for identical pin and plate materials. They
also find difficulty in numerical evaluation as the angle of
separation approaches 90”.
Hussain and Pu[27] present the only successful analytical attempt to study the effect of friction at the interface. They consider a push fit pin with a rough interface
in an infinite plate under uniaxial tension at infinity. The
interface consists of a region of separation (u, = u* = 0),
a region of slip (a* = v,) and a region of zero slip.
Placing the constraint of a common value of G/( 1 - 2~)
for the pin and plate materials, these boundary condi-
tions yield triple sets of series equations. By mathematical manipulations, these are reduced to a pair of coupled
integral equations and an equality. the integral equations
being approximately solved by variational techniques.
From the numerical data, they make two important
observations, namely, the angle of separation is not
sensitive to friction while the slip region is quite sensitive
to the value of the friction coefficient. The first observation does not appear intuitively satisfactory. In fact,
our work[&lO] shows that the angle of contact is, in
general, sensitive to the friction coefficient. It is fortuitous that for Hussain and Pu’s case of a push fit joint
with uniaxial plate tension and identical pin and plate
materials, the sensitivity to interface friction is indeed
small. This is shown in Fig. 6.
The extensive work of the Russian School on push and
clearance fits with smooth interfaces is listed and summarised by Muskhelishvili and Kalandiya[2, 41. Their
formulations are also based on the complex variable
method, singular integro-differential equations, singular
integral equations, and the development of suitable
numerical techniques.
Harris ef al.[26], as part of an extensive study of
mechanical fastener joints, have made an iterative finite
element formulation for the problem of pin to plate load
transfer and studied the problem when the pin is rigid,
the plate is a finite rectangle, the interface is smooth and
the pin to plate fit is either interference or clearance. The
problem is posed as one where, on a specified number of
contact points around the periphery of the hole, the
redundant radial reactions must be determined for a
given load level and a particular initial fit condition.
Brombolich[l3] and Ghadiali et al. [22] introduce friction into finite element formulations. Ghadiali et al. claim
that their programme can undertake the evaluation of
frictional forces in interference fit joints. Brombolich
considers load sequence effects and effect of friction and
fretting on the stress distribution around the hole. He
makes an important observation that the values of the
maximum stresses are unaffected by the frictional
coefficient upto p = 1.0. Indications from some of our
recent work are that a change of p from 0 to 1 could lead
to changes of l&15% in the maximum stresses. Hopefully, in the above work, special elements or appropriate
procedures for the interfacial conditions, ensure a proper
formulation of the boundary conditions relevant to progressive contact and progressive separation in the
presence of friction.
Coker[14] in 1925, applied the photoelastic technique
to a push fit problem by loading a xylonite plate by a
steel pin and xylonite bush combination. Next, in 1940,
Frocht and Hi11(21]applied the photoelastic and strain
gage techniques to the study of clearance and push fit
joints with small Bakelite and large aluminium alloy
specimens respectively. They also considered the effect
of lubrication of the joint. They concluded, inter aiia,
that clearance increased stress concentration, the maximum stress does not always occur at the ends of the
horizontal diameter, increase of pin modulus increased
the maximum stress very slightly and that lubrication
slightly reduces the stress concentration. The last
conclusion is not in line either with physical considerations or with the findings of later analytical workill, 251.
It would appear that some of their conclusions are invalid because the errors in the experimental techniques
were comparable to the orders of variations being studied.
131
Elastic analysis of pin joints
Jessop et a!. [28, 291, carried out extensive photoelastic
investigations using brass and bakelite pins in araldite
plates. They applied pin load, plate load and pin-plate
load combinations and presented extensive data on
stresses and stress concentrations.
The only investigation of partial separation of rough
interference available is the excellent photoelastic study
of Lambert and Brailey in 1%1[32]. They used a brass
pin in a rectangular plate of araldite, with the interfacial
friction coefficient being determined as 0.3. An electric
wire embedded inside the araldite plate provided a make
and brake circuit which identified the initiation of
separation. They present some data on the influence of
the coefficient of friction on the elastic stress concentration factor for interference fit joints.
One of the problems faced in experimental work on
pin joints is the determination of interference or
clearance to the necessary degree of accuracy. In
general, workers have side stepped the issue by presenting data with suitable normalising parameters. One can,
however, develop relatively simple procedures for
determining the interference as an integral part of the
experimental procedure, as for example in Ref.[36].
Cox and Brown[lS] applied photoelasticity to study
the effect of varying the clearance and varying the plate
width. In their paper they also make a critical assessment
of the work of Frocht and Hill, Jessop et al. and Brown.
explore the possibilities of empirical relationships, and
finally review and coordinate all the then available information for introduction into data sheets[7] for the
purposes of design against fatigue.
In view of the complexity of the pin-joint problem,
irrespective of the degree of success with mathematical
analysis, it would be essential to have the backing of
reliable experimental investigations. With the recent
rapid strides in experimental techniques and instrumentation, a fresh approach to experimental investigation of
various phenomena in pin joints should be very fruitful.
4. UNIFIEDTREATMENT OF INTCE AND CLEARANCE
FIT RIGID PINS WITH SMOOTH INTERFACES
4.1 Plate load
4.1.1. Interference fit : Initiation of separation.
Consider the configuration of Fig. l(a) with an interference fit (A) rigid pin in an infinitely large elastic sheet
(E, v). Due to the oversize of the pin, initially, a state of
uniform compressive contact pressure, with zero friction
or shear, prevails around the interface. As loads K&,
KzSo are applied to the plate maintaining their ratio
(K = KJK,) constant, the compressive interfacial stress
is relieved around a part (or all) of the periphery, the
actual extent depending upon EA, K&, K& and i?
Without any loss of generality, we can align the X-axis to
pass through the points (A, B) where this relief is most
pronounced and consequently stipulate that K& is
positive (K& > 0). Thus for specified, A, at a specific
value of So = S,,, the interfacial radial compression just
disappears at A, B so that interfacial separation is initiated at these two locations. Further increase of load
causes the separation to spread out from each of these
points.
Before the onset of separation, the stress state in the
elastic _sheet can
~_ be obtained from a closed form Airy
stress function[6]
which identically satisfies the far field conditions of eqn
(1) and all the interface boundary conditions:
u&=0,
U=ahonr=a.
(3)
Following Coker and Filon[l], we find it convenient to
establish an auxiliary displacement function $ with the
following relationships for cp, IG;the stresses and the
displacements:
V4p=0, -$ r$
=V*q, V’+=O
(
>
EU=-(l+v)$+r$;
EV=_(l+v)aq
-z+rr
1 acp
u,=--+~-_T;
- r ar
Za+
5;
1 a%p
r ae
From the above stress function, eqn (2) we derive the
level of So for onset of separation as,
(K&,/EA),,
= [1+6K(l+
v)/(S- v)]-’
Pa)
or
se, = [(EA - K,S,,)/KzS,,],, = 6(1+ v)/(S - v).
(5b)
In the first relationship, we identify (K&/EA) the ratio
of the axisymmetric stress to the axisymmetric proportional interference required to initiate separation for a
given ratio K of the directional and axisymmetric stress
components. In the second relationship we derive an
effective interference parameter,
A. = (AlKd - W,IKd(So/E)
(6)
or a relative load parameter
s = (EA -K&)/K&
= (EAlKzSo- KJKz)
(7)
whose physical significance we touched upon earlier.
4.1.2. Clearance fit: Initiation of contact. Reconsider
the configuration of Fig. l(a) with a clearance fit pin as in
Fig. l(b). Initially, with zero plate load, there is no
contact pressure or shear around the periphery of the
hole. As the plate load is applied and raised, at some
value of So, pin-plate contact is established at twodiametrically opposite points. Again, without any loss of
generality, we can assign K&> 0 and assume the
contact to initiate at C and D on the Y-axis. Further
increase of load causes the contact to spread out symmetrically on both sides of C and D.
Before the onset of contact, the stress and displacement fields correspond to those in an infinitely large plate
punched with a circular hole, and loaded at infinity[6].
The threshold load for the onset of contact is the one at
which the radial displacements at C, D in the plate are
132
A. K. RAO
equal to (ah). This is readily determined as
(K&/W),,
= (1-2X)-’
for the electronic digital computer take the form
t8a)
or
t&
Scr= [(EA - K,So)/K*SLll,, = -2.
=c,,
* 0 m 1.2. .
(8b)
Ad,,,(a) cos 2mO= -cos 20
in 0 5 e I 8,
We note that A should be assigned positive values for
interference and negative values for clearance.
4.1.3. Partial separation and contact behaviour: A
combined analysis for all fits. In contrast to the preseparation-precontact
analyses the partial separationpartial contact analyses for interference and clearance
can be conveniently unified. Beyond the critical loads,
eqns (5) and (8) the interface is partly separated and
partly in contact. In both cases, the zone around points A
and B, on the X-axis are separated and those around C
and D on the Y-axis are in contact. The problem
naturally posed is: given EA, v, KISo and K& what are
the arcs of contact 20, at C, D? and the arcs of separation 28, at A, B? The problem being one of mixed
boundary conditions with a moving boundary, its direct
analysis, irrespective of the basic method, needs an
iterative procedure. On the other hand, the analysis can
be direct and simple if we invert the problem to read:
given arcs of separation 219, or arcs of contact 28,
(= rr - 2&), what are the parameters EA, v, K&, K&?
In fact, for a given &, there is no unique combination of
the quantities, EA, v, K&, K&; from physical considerations, EA, K&
K&
are linked together by the
single parameter s = (EA -K&)/K&.
So, we should
really seek the s for any v and 0, (or v and 0,).
Thus, taking account of the double symmetry of the
stress field and interpretation for interference and
clearance, we formulate the boundary conditions:
EaA
(ltv)K,s,-Ta
+&
(l-v)
(EA-K&)
&SO
SE ,, A,,,gm(a) cos 2mfI =T-41+4COS2e
2 0 m l.Z,
in 8, 5 e 5 r/2.
(12)
We now effect the transformations
s = (EA -K&)/K&
(13)
and rewrite
EU:
-y
as- AW + da t 2
A,&,,(a)
m=1.*....
= --c0s2e
a(lt4
3
ine,Ie~~/2.
cos 2me
(14)
In this pair of series of equations v, 8, are specified and
A& A;, . . ., AL and ‘s’ are determined. Introducing
ati=0
onr=a,
values for E, A, K,, Kz the values of So and AC,,
osesd2
(94
A ,, . . ., A,,, are determined, to complete the solution.
u = ah on r = a, 8,s e 5 d2
A computer programme is written applying the direct
(9b)
collocation procedure for the solution of the above
u,=O
onr=a,
05858,
(SC) equations. ‘n’ equidistant points are chosen in the interval 8 = 0 to ?r/2, giving rise to ‘n’ equations in general. If
and, at the far field (r-9,
as in eqn (1).
the transition .point (0 = 0,) is coincident with one of
An Airy stress function and its auxiliary displacement
these equidistant points, there are (n t 1) equations.
function identically satisfying the boundary conditions
When it is not, it is advisable to include the transition
(9a) and (1) can be written as
point as an additional collocation point for both the
equations, thus using a total of (n t 2) equations.
The convergence of the solution is studied by succp. = AoEa2A In i +i KISo?
cessively increasing the number of equidistant colloca0
[
tion points. Halving the collocation interval in successive
steps is particularly advantageous in estimating the true
-K2So(+i)c0s2+EA
m=z,,,M
values from the converging sequences[40]. The results
indicate
good convergence even with single precison.
2m-1
*Illr4flC+2
- a2m+2r-2m cos 2me (10) With only 13 point collocation the boundary conditions
2mtl
I
around the periphery are satisfied within 0.2% of Eah or
K&. For each combination of parameters, the total time
taken for solution and evaluation of loads, stresses and
(G;= K&e t 2EA
2
A,,, q
sin 2me.
displacements on the boundary for four successive apm=1,2....M
(11) proximations (with 12, 24, 36,48 intervals) is only 30 set
on an IBM 360/44 computer.
4.1.4. An example for discussion: Uniaxial tension
The arbitrary constants A,‘s and the load parameter ‘s’
are to be determined by an appropriate method of satis- S,. Let us consider an interference fit under uniaxial
fying the boundary conditions (9b, c). The simple equi- loading (K, = K1 = K = 1). By eqn (Sa) the pin and the
distant collocation technique is found to be satisfactory
plate maintain full contact upto a critical stress,
for this problem.
(SJEA),, = (5 - v)/(ll t 5~). As the load is further inThe boundary condition equations to be programmed creased, the variation of So required for achieving
133
Elastic analysis of pin joints
different an&es of separation 19.is non-linear as shown
by the interference curve in Fig. 3. With still further
increase of the applied load, 8, asymptotically approaches a limiting value 0,. Numerical evaluations
obtained by stipulating an angle of separation greater
than $ lead to negative values of So/W. These negative
values correspond to the case of contact 8, = (r/2- &)
applicable to a clearance fit (negative A and positive So).
Thus the progress of contact with increasing load in a
clearance fit is as shown by the clearance curve in Fig.
3. The onset of contact with clearance is at WEA = - 1.
Further increase of load causes the contact to spread on
both sides of C and D, upto an asymptotic limit which
coincides with 0, = 0,.
An understanding of the problem can be added to and
0, accurately fixed by replotting Fig. 3 as ENS, vs 0,.
This we will do for general biaxial loading in Fig. 4.
Fig. 3. Progress of separation/contact with uniaxial plate tension
for interference/clearance fit joints, smooth interface, rigid pin
Y = 0.3.
4.1.5. Biaxial loading. Consider now general biaxial
loading. Figure 4 shows s = (EA -K&,)/K&
vs es for
v = 0, 0.3, 0.5. Much useful information can be drawn
from these curves. For instance, take the solid line
(v = 0.3). It is clear that if End&> 1.66, interface
separation is impossible, while if EAJ&: - 2, interface
contact cannot be achieved. Further, in the region 1.66>
EAd&> -2, the point T on the curve, where &/So =
(EA -K&)/K&
= - KIIK2 corresponds to W/S0 = 0
or A = 0 for general So. Hence for a biaxial load ratio
K = K2/KI this point T on the curve represents the push
fit case. The portion of the curve above ‘T’ (& < &.)
corresponds to interference fits and that below ‘T’ (0, >
0,) corresponds to clearance fits.
In various physical interpretations that can be drawn
and the models that can be built by using this curve, it is
helpful to remember that So increases indefinitely as we
approach the point T from either side, and that A
changes sign as we cross 7’ from one side to the other.
Considering the push fit case, we notice that its position
on the curve depends only on KJK, and is independent
of So. Thus, for push fits, the angles of separation and
contact are invariant with the load, so that the stresses
due to a push fit pin in a loaded plate increase linearly
with the applied load system. Physically it also means
that any small load would instantaneously give rise to the
full possible extent of contact for the relevant K2/K1.
4.2 Pin load
Consider now the case of pin load on the pin-plate
configuration (Fig. 2). Without any loss of generality we
can align the X-axis along the direction of the pin load.
4.2.1. Initiation of separation or contact. For an interference fit pin, the application of pin load P,, causes
relief of interferential stresses at B(t9 = 180”) and an
increase in interfacial compression at A(0 G 00). With
reference to the origin located at the centre of the hole,
the pin has a rigid body movement u. along the direction
of the load P,. When contact is fully maintained the
stress state in the elastic sheet is described by the Airy
stress function
PX
cp,=-T;;tisinB+$(l-v)rln
0
5 c0se
+~(L-v~~c0se--~*n(~)
(1%
satisfying all the boundary conditions of the problem,
viz. the
u,=uo=ue+Oaasr+q
u,[email protected]+aAonr=a
I0
u (cr, cos b - are sin e) de = -5,
(16)
It is readily seen that the stiffness relation between pin
and plate is
Euo = P,(l + v)(3 + v)/87r.
(17)
The onset of separation at I3 is given by the condition
a, = 0 at r = a, 0 = T. This yields the critical load
parameter
Fig. 4. Progress of separation/contact with biaxial plate load for
interference/clearance fit joints, smooth interface, rigid pin.
r~rr= (EaA/P~L, = (1 + vh.
(18)
134
A. K. RAO
Any further increase in loading causes the region of
separation to spread on the two sides of the point B and
leads to non-linear stiffness and stress-load dependence.
With a clearance fit, the pin receives a rigid body
translation uo= -aA by the application of any insignificantly smalt load P,. At that instant, the pin comes
into contact with the plate at B = 0” and the plate is
virtually stress free. Increase of pin load to finite values
causes a region of contact to spread on both sides of
point ‘A’.
4.2.2. Par&t contact. Let us consider now the general
case of partial contact due to pin load P,.
It is sufficient to analyse one half of the symmetrical
field. The boundary conditions on the hoie can be conveniently state as,
ufv3=o,o<e4?r,
r=a
(19a)
U=ah+UoC0Se,0(:BS&,
u,=O,
I
&.SBSP,
r=a
(I9b)
r=u
(19c)
Oe’(cr, cos 0 - uti sin 0) d0 = -5.
(19d)
A stress function identically satisfying the conditions,
(19a, d) can be written as
,
I
/
I
/
t
/
Fig. 5. Progress of separation/contact
with pin load for interference/clearance fit joints. smooth interface, rigid pin.
4.2.3 Contact pressure. With the present method of
analysis, once the unknown coefftcients of the problem
are determined, the stresses are very easily computed.
From the data compiled, polar plots of the contact pressure are given in Fig. 6 for the loading cases of uniaxial
tension, uniaxial compression and pin load. The nonlinear rise in stress with load in each case can be clearly
seen.
(20a)
the
and
function
corresponding
4 = - (1 + v)
-EA
T$[In(r) sin 6 -
c
A,,,
m=2.3....
m
auxiliary
displacement
0 cos O]
($msin(m0).
CURVE
(20b)
The arbitrary constants A,‘s, load P, and pin rigid body
displacement u. are to be determined by a suitable
approximate method of satisfying the remaining boundary conditions (19b, c). The procedure and computer
programming for this case are parallel to those for the
plate load case with the parameter p = EaA/P, replacing
‘s’, and introducing (&a) as an additional unknown. In
fact the similarity of the two sets of equations shows that
a single program can be written to apply for either
problem as desired.
The changing pattern of contact and separation with
increasing pin load is evaluated and shown by a single
continuous curve (V= 0.3) for interference and clearance
in Fig. 5. In this case the positive values of EaA/Px apply
for interference and negative values for clearance.
Again, as one approaches the push fit point ‘T’ from
either side, EAA/P, approaches zero and, for given A, P,
rises indefinitely. Curves are drawn for v = 0, 0.3,0.5. A
curve is also given for v = 1.0, which corresponds to the
plane strain case of Y=OS, for comparison with the
results of Noble and Hussain[35].
First No
Second
DE&NAilON
Load
No.
Parameter
Corresponding
I
Sx/EX,Sy/EX
Semi-arc
or
P,/EoX
J tact
of CO
I/
Fig. 6. Variation of interfacial contact pressure with pin and
plate loads. smooth interface, rigid pin. v = 0.3.
5. ELASTIC PINS WITH SMOOTH INTERFACES
The unified treatment of inte~erence and clearance
achieved in the last section for rigid pins is easily extended for elastic pins. The relevant pin and plate (or
sheet) material properties are (E,, v,) and (& v,)
respectively and the pin to plate modular ratio e =
E&Es is obviousty significant. An extended study on
elastic pins is in progress[9,23-251 and here we touch
upon a few salient features of elastic pins with smooth
interfaces. The interfacial boundary conditions for this
case are the continuity and equ~Iib~um conditions in the
arc of contact and stress free conditions, for both pin
and plate, in the arc of separation.
5.1 Plate loud
The threshold values of the biaxial stress parameter
for the onset of separation in interference fit and the
135
Elastic analysis of pin joints
onset of contact
in clearance
(for interference)
&r =
5.2 Pin load
With interference fit, the onset of separation for pin
load occurs at
fit are respectively
E,A - K,So
= - 2 (for clearance)
IL%
1 tr
(
(21b)
The plate load parameter is seen to be identical for rigid
and elastic pins. In the clearance case, the threshold
value of sCris identical for all combinations of materials,
but not so for interference. The partial contact and
separation behaviour can be analysed with the same
stress function as in eqn (10) for the plate, together with
a stress function (ppfor the pin satisfying the zero shear
conditions around the periphery:
(pp
=
E&3,$
-(S)
+ E,A 2..
.
.
cos 2mB.
(oLL = to,),, US = U, + ah, along 8, S 0 5 ‘IT/2
‘1
INTERFERENCE ‘X3 CLEARANCE
FIT.SMOOTH INTERFACE (I* =O)
----
INTERFERENCE FIT,RUJG;1
INTERFACE (/++qll
, MONOTONIC
(24)
and the onset of contact for clearance occurs at an
infinitesimalIy smalf load, P, + 0. The partial separation
and contact behaviour is analysed by a stress function
identicat to eqn (20a) for the plate together with a stress
function rp, satisfying the load equihbrium condition and
the zero shear condition around the periphery:
pp = -(P,/2n)ti
-
sin Bt
r(l--+)ln
5 COSB
0
(1 - v,)r3 cos 19+ E,AAr’ - Epuor cos 0
(25)
The arbitrary constants in the stress functions for the pin
and the plate, the load parameter p, and the rigid body
movement of the pin uor become the unknowns of the
problem and the boundary conditions yet to be satisfied
are:
(0~)~=(u~)~p;U~=U~+ah;OieI6,
(23)
In Fig. 7 we have the load-separation-contact curves
for different pin-plate modular ratios (e = 3, 1, l/3). We
find that with increasing flexibility of the pin the
threshold value of Is’ for separation decreases. In brief,
the more flexible the pin the more it tends to cling to the
plate.
2,o
~,)[email protected]
CQf
The arbitrary constants in the stress function for the
sheet and the pin, and the load parameter ‘s’ are determined by suitably satisfying the boundary condtions,
along 0 S B I 8,.
= Ie(l + ~d+(l-
--m*2fr! [email protected]
I
Cqa-2”+2r2”
a-v-q
(a,), = 0, (G)p = 0,
pC,= (EaA/P&
(21a)
(u,)s = (ui)p = 0, 8, s e 5 %-.
(261
Figure 8 shows the progress of separation/contact with
the load parameter for pin-plate modular ratios, e = 3, 1,
l/3. In this case also, we find that the more flexible the
pin, the more it tends to adhere to the plate. In Fig. 9, we
plot the rigid body component of the pin movement,
rro/& with increasing pin load, P,lE#aA. There are’two
interesting points to note. The first, which is puzzling at
first sight is that with interference fit, when the pin
modulus is low the rigid body component of the pin
movement is opposite to the load direction. The second
is to confirm that changing from clearance to an equal
order of interference can increase the joint stiffness to
pin load by a large factor, say about 5 times.
REASING PIN RIGIDITY
Fig. 7. Progress of sep~ation~contact with biaxial plate load for
interference and clearance fit joints with smooth interface (fi = 0)
and interference fit joints with rough interface (/[email protected]), elastic
pin, v, = V, = 0.3.
Fig. 8. Progress of separation/contact with pin load for interferenceiciearance,
fit joints, smooth interface, elastic pin, v, =
VP = 0.3.
136
A. K.
RAO
given design h. There is a continuous and smooth progress of contact with increasing load in the clearance fit
joint, and the effect of pin elastic modulus is very small.
But in the interference fit joint the effect of pin elasticity
is highly significant and the behaviour changes from flat
response of ([email protected] PX)in the preseparation region to rapid
rise in the partial separation region. Considering a static
design criterion of maximum stress at design load, the
clearance fit is superior upto 85% of the interference
separation load and thereafter the interference fit takes
over. Considering fatigue design for which alternating
stress range is more important than the steady stress
component, the interference fit wins hands down for any
load, because the inte~erence visually stifles the alternating stress in the preseparation range and considerably
reduces the steady stress in the partial separation zone of
operation.
6. VARIABLEINTERFERENCE
AND CLEARANCEPINS:
sMm
INTERFACE
We will now consider an example of a smooth interface with polarly varying interference or clearance. It is
possible to introduce a general polar variation of the type
X
f,, cos [email protected] into the analysis, with the
“+X1.2...
stipulation that initially, under zero plate and pin loads,
the interference pin maintains contact all round and the
clearance pin leaves an all round clearance.
For example, consider a round pin in an oval hole in an
infinite plate under biaxial plate loading. Represent the
ovality of the hole by an interference variation A =
(A&!)VQ+ f~ cos 2f?), (with positive fO. fi and f. c fi = 2)
so that the variation of interference is similar to the
applied plate stress variation. The analysis can be
developed as a parallel to the analysis for uniform interA = (ha/k)
Fig. 9. Progressof
pin movement
with pin load,
face, elastic pin, V, = vP = 0.3.
smooth
inter-
A comparison of typical p~~ormance curves for interference and clearance fits over a range of elastic
moduli and friction coefficients can be highly illuminating. For instance, consider Fig. 10 in which we have for
the pin load, smooth interface case, for e = l/3, 1, 3, q
curves of u,JE,h vs P,/E,an drawn for important locations around the hole. Examine the stress at 0 = 90” for a
INTERFERENCE FIT
C_LEARANCE FIT
1.6
FRE SEW&RAT!0
1.2
;L
00
POSTSEPAWTION
I
I.0
I
3-o
I
20
40
Px/EsoA
1.6
AT 0=90”
r
12
4
g
0.8
bm
04
10
30
20
%/Es
4.0
I.0
ox
20
W%
30
4-o
ax
1
_ 0%
Fig.
10. Interracial
hoop stress concentrations
joints. smooth
due to pin load: Contrast between
interface, elastic pin, V, = V, = 0.3.
interference
and clearance
fit
Elastic analysis of pin joints
137
(29)
example of uniaxial tension for two alternative design
criteria, namely m~imum se (Fig. Ila), and the von
Mises ueff (Fig. lib).
In the interference fit joints, polar variation of interference results in a slightly lower stress ue/WO before
separation, a slightly higher stress after separation and
almost identical stress-load gradients in both regions. As
such, the variable interference has no significant advantage over uniform interference. On the other hand for
a clearance fit joint, variable clearance reduces the stress
bad gradient by about 30% and as such is beneficial. In
other words the ovaiity of a piugged hole does not have
any significant effect on the stresses if interference is
achieved; but when clearance is achieved, the ovaiity can
reduce the stress si~c~tIy
when the load is in the
direction of the major axis of the hole.
Next consider the situation where the performance
depends upon an effective stress by the von Mises criterion. We find that the stress level is substantially
reduced by the interference, but is almost uninfluenced
by varying the clearance. To sum up, the benefit or
otherwise, qualitatively or quantitatively, of varying interference or clearance around the interface has to be
carefully evaluated taking all the relevant factors into
account.
From the numerical results of par&J con~~t~sep~ation
it is co~med that the load~onta~t-sep~ation
relations
of Fig. 4, can be applied to the present case of nonuniform interference or clearance by using the redefined
load parameter s, eqn (28), for the ordinate.
A simple manipulation of the relationships shows that
given a load system (i.e. KdK, fixed), for a designated
angle of separation 0, and therefore, for equal ‘s’, the
applied load level ‘So’ for variable interference is cfis +
2f0)/4 times that for uniform interference. As, fO+fi = 2,
fo, f~ are positive and 2 > s > - 2, this ratio is never
greater than unity. Thus for the smooth interface-plate
load combination, variable interference or clearance
leads to smaller separation loads than uniform interference or clearance.
Let us consider with the aid of Fig. 11, the common
Interfacial friction plays a sign&ant role in determining the buoyance
of a pin joint. With an ideally
smooth interface the interfacial shear is zero, and load
transfer between pin and plate is through compressive
stresses normal to the interface. The surfaces in contact
have full freedom for unconstrained relative tangential
slips, but this slip has no deleterious effects. However,
the introduction of even a little friction changes the
picture. A tangential shear resistance is generated along
the unseparated part of the interface, so that the load
path is now bifurcated into a normal stress path and a
shear path; consequently the stress concentrations are
reduced. In fact, in permanent joints, if the interface can
be bonded or fused, the stress condensations are
~onsiderabIy dimi~shed and fretting completely elh-
ference, generally replacing the uniform i\ by the nonuniform (A~2)~* + ft cos 28). Thus, wherever contact
exists, the hole boundary conditions are to be modified to
u, = 0, U = (aAtJ2)Cfo+f2 cos 28).
(27)
Now, redefining the plate load parameter as,
s = - (K, So - EAof0/2NK&- EAoH4)
(28)
we find that the onset of separation for interference fits
is at s = 6(i + v)/(5 - V) and the onset of contact for
clearance pins is at s = -2. Thus the threshold values of
‘s’ are identid to those for uniform interference if we
use the redefined plate load parameter, eqn (28).
The analysis of partial contact and separation is carried out applying the stress function of eqn (Ii?), replacing A by AO.The load parameters, and the arbitrary
constants A,,,% are determined as before by a simple
equidistant collocation of the boundary conditions, written as:
a,=o,
OSBIB,
lJ = (a&J2)(fo + fi
cos
26%e, s e I 7~12.
PUSH FIT (o-e vs. SoI
( t 1 UNIX
MTEFtFERMCE
121 WWtYJNGINTEWEREWE
1-0
(3) UNJFORM CLEARANCE
(41 VARYING CLEARANCE
t
(a)
SQ/EX,
HOOP STRESS
f So/EXO
I bl
WN
MISES EFFECTIVE STRESS
Fig. It. Stress concentrations due to uniaxialplate tension due to variable interference and clearance fit joints, smooth
interface, rigid piti, v = 0.3.
138
A. K. RhO
increasing
fatigue
life
the
thereby
inated,
manyfold[lO]. The shear resistance puts a constraint on
the interfacial slip, inhibiting it completely over a part
and reducing it over the rest. However, such reduced
relative motion, acts against frictional shear and this
results in material and functional deterioration due to
wear, tear and fretting. Thus, a realistic analysis of a pin
joint should account for interfacial friction.
Unfortunately, the friction-slip combination makes the
system non-conservative and the stress state becomes a
function of the load sequence. This poses problems of
formulation and analysis in the mathematical approach
and of repeatability in experimental investigations.
However, with the numerical and logical capab~ities of
modern digital computers, we can proceed in discrete
steps to take the joint through any desired Ioad history
and obtain considerable insight into the phenomenological aspects of the problem and also to establish some
good approximations for the state of stress. Exploration
along these lines has already proved to be highly
profitable[& 181.
As we noticed earlier, the interface in general consists
of three types of segments: the separated, the slipping
and the non-slipping segments, Fig. 12. A6 initio, it is
generally not possible to demarcate these segments.
When such a demarcation is quantitatively possible, the
method of inverse formulation can be appiied. When the
load system is monotonic~y increased, the interface
experiences either receding contact (sep~a~~on increasing monotoni~~ly) or advancing contact (the arc of
contact increasing monotonically). In general, receding
contact is simpler to handle than advancing contact.
Arbitrary loading schedules complicate matters
considerably and need more elaborate and sophisticated
handling. One gets the feeling that in such situations
finite element methods (perhaps augmented by special
continuum elements [36, 381)may be better harnessed for
the problem than continuum methods.
7.1 Rigid interference fit pins
7.1.1. ~~~~i~e friction (p+m). An interface with
infinite friction (rough, p -+m) inhibits all relative slip
between pin and plate in regions where a firm contact is
already established. With an interference fit, in the load
regime when pin and plate continue to be in fulf contact,
the pin and the plate maintain continuity of deformations
across the interface and, the state of stress at any level
of loading corresponds to that due to a bonded interference pin. When the load is increased and interfacial
separation is initiated and progressed, the plate and the
pin displacements are independent in the regions of
separation, but maintain full continuity without slip in the
regions of contact.
Let us now analyse the stress and deformation state
due to a monotonically increasing load on a rigid pin in
an infinite plate, stipulating an interference h and a
non-slip (fi [email protected]) interface.
In the preseparation state we have the boundary
conditions
U=ahtuocostI;
AIJE:
I *
INITIATION
OF SLIP
NO SLIP
AF, GB +
cp= (-P,/2r)rli
sin B+ (I - v)(P,/47r)r In i
0
FIG
-w
SLIP
cos 8
-(l t ~)(~~n2~8~~)cos 8- (I + v)-‘EatA In 6
0
(31)
and the threshold value of the load parameter for
separation
pc, = Eahl P, = (1 + v)/2~.
(32)
The flexibility of the sheet-to-pin load is given by
E&P,
= (I + v)*/89r.
(33)
When separation has progressed over an arc (2~ - a) >
B> u, we have to apply the interfacial boundary conditions
r=a,(2n-(Y)>i?>cr:
r=a,a>8>-a:
u&=0,
(jyso
U=QA+U~COS~,
(34)
V = - u0 sin 19
to the stress function
rp=-$tisin0+$(1-v)rln
+A&%*hIn i
0
+ EA
2
d cos6
0
+A , %osff
r
[Amum+*rF’ + B,amr-mrz] cos m9.
m-2.3....
(35)
Obviously, for a given scheme of collocation (with n
equidistant points) the number of equations is twice that
for the smooth interface case. The progress of separation
with increasing pin load is evaluated from the solution
and shown as the interference curve in Fig. 13,
Cd)
NO SLIP
(30)
the stress function
(b)
POINT
V=-uosin60nr=a
AH--NO
MB-c
POINT
sup
SLIP
B~INITIATION
OF
SEPARATION
AK+NO
SLIP
KIL-,
SLIP
LB --SW-WATION
Fig. 12. Progress of slip and separation with pin load for interference fit joints, rigid pin.
139
Elastic analysis of pin joints
60
INTERFERENCE
8, DEGREES
Fig. 14. Polar variation of stress ratio (o&q) with pin load.
rough interface (c + m). rigid pin, Y= 0.3.
Fig. 13. Progress of separationlcontact with pin load for inter-
ferenceand clearancefit joints, roughinterface(p +m). rigidpin,
B= 0.3.
1.1.2. Finite friction and the phenomenon of
slip. Consider the preseparation regime for which we
established the stress function eqn (31). Under conditions of infinite friction there is a rigid linkage between
the adjoining points of pin and plate so that interfacial
slip is inhibited however high the appiied load. If now
we reduce the interfacial friction to a finite value (p), *the
rigid linkages can be broken and slip can occur over a
part of the interface. Slip is initiated when, and at the
location where, the maximum value of /o&r,/ on the
interface just exceeds the finite friction coefficient p. The
location at which slip is initiated (6 = ai) is therefore
obtained from d(u,lu,)/dt? = 0 which yields
cos ai = - (1 t v)P,lZrEaA = - (1 + u)/27rp
(36)
and the corresponding ratio a,&~, is
(c*/Ur)i = -Cot
(Yi.
(37)
aj = r - tan-‘(l/&).
(38)
In terms of the friction coefficient,
As the load is further increased, the slip region spreads
on either side of 0 = ai. For further discussion it is useful
to plot, from the stress function of eqn (31), the polar
variation of u&r, for different values of p, presuming
zero slip. This is done in Fig. 14.
Consider an interface with p = ~~ = 0.31 and increase
the load P monotonically from zero up, that is, reduce
the load parameter ‘p’ gradually until p is a little over 0.7.
Referring to Figs. 12 and 14, we see that u&u, is
everywhere lesser than 112and slip is totally inhibited.
When p = 0.7 is touched at I, o&r, equals p2 at B = ai2,
so that, at &hisIocation slip is initiated (Fig. 12a). With
further increase of load, that is decrease in p, say to a
level p =OS, slip spreads on either side of ai2 over the
region where urn exceeds w,. That is, to an arc, approximately F’G’, or a& > B> ab2. In this process of
slip, there is obviously a redistribution of the stress field
such that, in the slipping region a,0 = pur. As a result,
the actual curve of u&r, for p = 0.5, F = ~2, takes a
shape indicated by AFIGB and the actual arc of slip is
FG, ac2 > B > ab2 (Fig. 12b). Further monotonic increase
of Ioad will continue to extend the slip region until
separation is initiated at B(8 = P) as in Fig. 12(c).
Thereafter, with increasing load slip and separation
continue to spread together (uide Fig. 12d). The spread
of slip from its initiation upto the start of separation can
be further studied.
First consider the intersection of the curve p = pr =
0.5 with ~/a, = ~2 in Fig. 14. We find,
a&2,c2= ajt + [F -
COS-f(pcel
COS %2)]
ah+aL=2ai2.
(39)
(4)
Since the friction coefficient p2 limits 06 to pzo, in the
slip zone, the actual distribution of u,Ju, is as shown by
the readjusted shape AFGB. From the pattern of such
curves, we may assume ai - abt = ae2 - ai2, that Is, the
actual slip zone is equally spread on either side of (Y~z.
The error, if any, in this assumption will not introduce
any significant error in the solution, as from eqn (39),
a:2 - &2 = a& - at2 and the additive angles (a& - ab2)
and (a=2- ah) are relatively smalf.
The region of slip spreads upto the nearer axis of
symmetry, that is towards point B, and simultaneously
the radial pressure u, on the interface at B reduces to
zero. Thus separation between pin and plate is initiated
at this stage of loading. Further loading on the plate will
create three zones around the hole boundary as in Fig.
12(d). BL with no contact between pin and plate (~7,=
0, oe = O), LK with slip (limiting u,+,/u, = IL), and KA
with zero relative displacement (V = 0, a,.&, < p).
7.1.3. Analysis
of
post-slip,
preseparation
condition. The solution for post-slip preseparation
behaviour cannot be obtained in a simple closed form.
But it can be set up in the poi~-t~gonometric series of
eqn (35) by stipulating the region of slip FG and working
backwards for the applied Ioad P,.
Taking the symmetry about X-axis into account, the
boundary conditions for the case of load transfer through
the pin with partial slip are conveniently stated as:
U=aA+uocosBonr=a,arcAB
(i.e. contact is fully maintained)
(41a)
ati-pu,=Oonr=a,
arcFG
(ai-a)IBIfai+a)
Wb)
A. K. ho
140
V=-uosinBonr=a,arcAF,
OiBI(Ui-a)
(4,c)
UcGB,(ai+a)SBSr
and
I * (u, cos f?- o;e sin 8) de = - PJ2a.
(4fd)
An Ag stress function rp identic~ly satisfying the
boundary conditions (4la, d) can be written as,
- i esine+$C0 In0; case
[O
&f?
I
a
-~(~)cose]+~[~-*In(~)]
tEa2A
2
m=x3....
A,
amr-m _ m( 1 + u)am-2r-m+2
(m-2)(l+v)+4
(42)
I cosme’
The load parameter ‘p’. the displacement of the pin ‘rrd
and the constants A,,,% are to be determined by an
appropriate method of satisfying the boundary conditions (4lb, c). A simple collocation technique is again
found satisfactory for this purpose. The numerical solution for u = 0.3 for a range of friction coefficients (CL=
0.1-1.0) is presented in Fig. 15 as l/p vs iI. From this
figure we read off the arc of slip with a given p and for a
desired p.
158” and separation is initiated at B, 8 = 180”. Any
further increase in loading cusses three distinct regions,
i.e. AK with zero slip, KL with slip, and LB with partial
loss of contact between pin and plate.
Considering the limiting case, CL*m, PJEah =
Z?rf(l+ v), slip and separation are initiated simultaneouslv at B. B = 180”. In ohvsicd terms. seoaration
overtakes slip for very high f&ion c~~ci~nts~ On the
other hand, at c = 0, the initiation of slip occurs at
B = 90” and one can have the slip region spread over the
entire periphery for any insignificantly small load.
To summarise, we find, with increasing friction
coefficient (i) the slip initiation angle ai increases (ii) the
maximum arc of slip 2a(= a, - ab) decreases, and (iii)
the slip initiation load (PJEoh)i increases. Also, it is
clear that if the design interference, or the significant
load levels, or the friction coefficient, lead to the joint
operating around the slip initiation situation, the stress is
highly sensitive to the manufacturing tolerances on the
interference.
(b) Se~a~afi~~. The load at which separation initiates
depends on the friction c~~cient p. This dependence is
shown in Fig. I6 Two points are worth noting. Firstly,
P
6 o
$-j g SLIP REGION
Fii. 16. V~~tion of separation load with coefficient of friction
for interference frt joints with pin load, rigid pin, Y= 0.3.
50-
240;
fi
\
ay
30-
6
so-
l.O-
01
I
20
I
I
60
I/I
1
I
140
141
I
-
180
Fig. 15. Progress of slip region with pia load, for different
interface friction coefficients interference fk joints, rigid pin,
Y= 0.3.
(a) Slip. Consider a friction coefficient 1~=0.2 and
follow the corresponding curve in Fig. 15 along with Fig.
12. As the pin load is increased from zero, interfacial
friction inhibits slip until PJEah = 0.925, and the solution is identical with that for a perfectly rough interface.
At PJEah = 0.925, slip is initiated locally at I(0 = 101’).
The joint is now at a critical situation, when a small
decrease in load prevents slip everywhere while a small
increase in load spreads the slip over a large arc. In fact
an increase of 10% in the load spreads the slip over an
arc of 56” (i.e. 1W z BL 73”). Thereafter, increasing the
load further increases the slip region (e.g. to arc FG) but
at a slower rate, until the joint reaches another critical
situation, at PJEbEaA
= 2.864. Then the arc of slip BH is
the relationship of load parameter at separation to p is
virtually linear upto p = 1.0, which amply covers the
range of practical significance. Secondly, the load
parameter required to cover the increase of p from 1 to
m, is less than 5% of the limiting value at p *m. That is
to say, for highly rough surfaces (p > 1) the no-slip
solution may be practically adequate for at least some
purposes. Finally, referring to Figs. 12 and 14, we notice
that for values of friction coetkcients in the practical
range (say p = O-OS), the rate of reduction of the nonslip zone AK slows down as the initia~on of separation
is ~proach~, and there is a strong su~estion that as
separation progresses the arc of no-slip AK does not
contract significantly. In other words, for purposes of
practical analysis, the point ‘K’ can possibly be identified
with point Z-Z(Figs. 12c and Ed), without significant
errors in the estimation of the physical parameters of
interest. Once we confirm this assumption, the solution
of the three zone problem presents no difhculty with our
present method of analysis.
7.2 Clearance fits
The rough clearance fit, under monotonically increasiug pin loads, is an example of advancing contact which
needs rather careful handhng. An ~nvesti~tion of the
p j - case for a rigid pin can be ins~uctive. Consider
two discrete stages of loading PI, Pz when the arcs of
contact (in the semi circle 0-s) are al, o2 respective&.
At loading stage P,, the actual displacements V over the
141
Elastic analysis of pin joints
arc O-a,, are not known; but for all further loading
P > PI, they are frozen. At load P,, the tangential
deformations in the arc a~ > 0 > aI are non-zero and
unknown; they change continuously as the load progresses from P, to &. It is not possible, therefore, to make a
simple statement of the tangential displacement over the
arc of contact, as a boundary condition at load PZ, other
than to state that aV/dP = 0 in the arc aI > 0 > 0. This
does not provide a satisfactory statement of a boundary
condition. Nor is an alternative boundary condition apparent.
Our description of the process however, suggests a
possible method of solution, what one might describe as
a “marching solution” in which we build up the solution
in a series of small equal steps, of increasing contact. At
the end of the m th step, the V boundary condition on the
semi-arc of contact, m * Au, is obtained by superimposing the component of the rigid body translation during
this step (u,,.,,,- u,,.,,-r), on the V value estimated from
the stress function for the (m - 1)th stage. At the zeroth
stage, the boundary conditions are U = V = 0, and the
pin rigid body movement is uo.O= -ah.
The solution would be exact if the intervals Aa were
infinitesimally small. In practice, with discrete intervals
and finite number of steps, the accuracy would depend
upon the size of step. A simple computer programme is
written for this marching process and a convergence
study is incorporated by successive reductions of stepsize.
Using the polar trigonometric stress function as for the
earlier problems and applying the above procedure, we
have analysed the rough clearance fit problem for a rigid
pin for plate and pin load cases. Some results for pin
load case are incorporated in Fig. 13. For convenience
the curves of Fig. 13 are redrawn in Fig. 17 changing
from l/p to p for the ordinate. The curve labelled V = 0
presents the zeroth order approximation in which a
single step equal to the semi angle of contact (a) is used.
The tirst, second and third order approximations are
obtained using marching intervals of- 20, 10 and 5”. A
study of the convergence sequence permits us to extrapolate from these three approximations to a reason-
g-20
-2
JAI -3.0
-40
-50
Fig. 17. Progress of separation/contact with pin load for interference/clearance fit joints, rough interface (r+m), rigid pin.
Y= 0.3.
ably confident estimate of the true solution as shown by
dotted curves labelled “0” extrapolation” in Figs. 13 and
17.
As the load increases, the angle of contact should
approach an asymptotic limit. This limit is identified by
continuing to increase the angle of contact, until the
derived value of the load parameter changes sign. The
numerical data confirms that this asymptotic limit for 8,
for rough clearance fit pins coincides with the asymptotic 0, = 39” for rough interference fit pins.
7.3 Interaction of pin elasticity and friction
From our discussion in Section 3 of Hussain and Pu’s
paper[27] it is seen that there is a strong interaction
between pin elasticity and interface friction. These
effects are currently under study for different types of
fits and loads. Data compiled for an interference fit joint
under tensile plate load shows that for such a combination, increasing either pin modulus or interface friction,
reduces the stress concentration and the benefits are
non-linearly compounded.
06
04
02
I
0
5
;
-c
0
20
(b)
60
.9
000
140
160
6 1-o
3
b-
08
a
p 06
O6
i
=a
5
:
b
06
h
04
&
L
04
E
03
0
04
1.2
20
P/Ed
(c)
2-6
3.6
2
02
&
fJ
0
0.5
1.0
LOAD PARAh4ElER
15
VP=
20
P/Ed
Cd)
Fig. 18. Initiation of interfacial slip and separation in finite plates with pin load for interference fit joints, rough
interface (c +m), rigid pin, Y= 0.3.
142
A.
8.~~
K. KAO
OF FiNITE
PLATES
In the previous sections we considered at length the
analysis of pin joints in infinite plates. Our real interest,
however, is in the practical problems of tinite plates. The
analysis we have utilised for infinite plates is readily
extended to finite plates by including the relevant positive powers of r in the individual terms of the stress
function for the plate, i.e. by using the full Michell
series for the stress field in the plate(61 of which a
typical
term
’
(A,,$” + B,,J~+~+ A$,r-” +
BAr-“+*)cos me. Tiis function was applied by
Venkatraman[lO], for an extensive study of stresses in
finite lugs and plates due to interference fit pins. The two
additional constants in each term are to be determined by
satisf~ng the bound~y conditions on the external
Sunday of the finite plate along with those on the hoie
boundary. When the external boundary is fairly regular
and the pin size is small relative to the plate size, Fourier
expansion on a circle or direct collocation is adequate for
accounting for the external boundary conditions also.
Otherwise, a more sophisticated method, such as least
square collocation or successive integration or
polynomial expansion has to be utilised. Qualitative appreciation and reasonably accurate quantitative data for
preliminary estimates and design can be achieved with
the simple expedient of using a circular externai boundary with appropriate approximate boundary conditions.
To illustrate the procedure we will consider the initiation
of sep~ation and slip in the presence of friction in an
eye bar.
8.1 hifiation of separation ond slip in an eye bar with an
interference pin
Consider a simple eye bar, loaded through an interference fit rigid pin, Fig. 18(a). The conditions at the
interface, characterised by a friction coefficient CL,can be
reasonably well estimated by considering the simplified
problem of an annulus indicated in Fig. 18(d); the accuracy improves with increasing c = b/a. Consider the
situation of full contact, zero slip and monotonic increase of the pin load P,. The boundary conditions for
the elastic annular plate (I < I < b can be written down
as:
p = $ = c: o, = (P/4b)(l+ cos 26) 0 = 7rl2to 3?ri2
=o
CT+= -(P/4b) sin 20
=0
Application of ah bound~y conditions to the general stress
function of h&hell yields the stress function,
+B,[p’-e;]cosB
+z~,,B-C-m
..
t
pm+*+
I
cos me.
(45)
The constants A,,,, B, take the following values:
Be=
(l+ v)c
16[(1+ v)c2+(l - v)]
~,~_(~~~)~3(l-v)~4-2(l+3~~cz+3(5+~)J
487r[(I f v)c4- (3 - v)]
A2 = _ c’[( 1+ v)(3 - v)c” - 3(1 -I-v)*c*t 4(3 “t #‘I
16[(3- v)( 1 t v)c* + 4(1+ v)~c”- 6( 1 t v)‘c4
-4(3 t v2)c2+(3- v)(l+ v)]
(1 t v2)c3(c2- 1)
8[(3- v)(1 f v)c* t 4(1 t v)*c”- 6( 1t v)*c*
Bz=-
-4(3 + v2)c2t (3 - v)( 1t u)]
(46)
m=4,6,8,...;
A,=O,B,,,=O
m=3,5,7,...;
A,,, = (mg, -2~~)/m(m - l)Y,,
(47)
B, = (2h, - rnf~)/(rn + 1) Ym
Wf
where
e=-n12t0td2
y = gc GIL - gmhm)m(& _4)(_ 1)‘WWZ
m
3-v
0 = ~12 to 3~12
fm = c-9(3
e=-?r/2to7r/2
- v)c Zmt(m t2)(1t
-{mZ(1+v)+8(1-
U=Ql\[email protected],
V=-ucsinB.
Y
_,+2_m2(1+v)2t8(1-v)p_,
XP
m(1 t v)(3- v)
gm = c -2m-2[(m -2)(3p=;=l:
(m + l)(l t v)
3_
(43)
fi, = c-2m f(3 - v)?
v)c2-(m+l)(l
t v)]
v)~~“+~+(m +2)(m - l)(l
t v)c'
v)/(l+v)}]
-m(l+v)C*t(nl+
l)(l+ Y)]
k, = C-2”-2fm(3 - r$C2m+2- mfm - 1)(1+ v)c2
The external edge stresses can be Fourier analysed in the
full range B = 0 to 2r and written
V* = -(P/lb) sin2 e
(49)
Also the rigid body movement u0 of the pin is given by
ur = (P/Bb)(l t cos 20)
+ z
mz5,,. (- l)‘m-“‘2m-‘(m2-4)-’
(
)
+ m2(1+ v) + 8(1- v)l(l t v)].
cos mf3
uo=-2(1tv)
;
0
I?,.
(50)
An examination of the coefficient A,,,, B, and the
expressions for stresses and displacements gives clear
indication of very rapid convergence of the solution.
Elastic analysis of pin joints
particui~ly in the practical range of pin to plate size
parameter (a/b): for u/b = l/4,terms beyond A,, A3 may
be ignored.
It is interesting to study the initiation of separation and
slip in this problem. The solution is numerically evaiuated and relevant information is summarised in Fig.
18(b), (c). Separation is initiated when P/&IA exceeds
about 3.04. Figure 18(b), presents information that assists
in determining the effect of friction coefficient and the
load level on the initiation and extent of slip. Figure 18(c)
shows the growth of interfacial shear with increasing
load, while, Fig. 18(d) indicates the critical (or maximum
permissible) value of the friction coefficient for each load
level, to avoid interfacial slip. We notice that in this
example, Fig. 18(b), when ru.is very smai1, slip initiates at
about 112” from the load direction, instead of at 90” as
indicated by the infinite plate study. This is readity
explained. The infinite plate study implicity assumed the
stress field to be antisymmetric about the transverse (Y)
axis, while in the practical eye bar considered, such
antisymmetry is not possible. It can also be seen that this
lack of antisymmetry reduces the tendency ta slip, i.e.
increases the critical friction coefficient for any :oad
level, or a given friction coefficient requires larger load to
initiate slip.
9.CONCLU!SION
In the field of elastic analysis of pin joints we have
established that, although the success with sophisticated
mathematical methods has been very limited, application
of rather simple mathematics coupled with an inverse
formulation and the facility of modern computers, can
yield solutions, information and understanding on a wide
front. But what we have reported here is the proverbial
tip of the iceberg and all the work we have completed to
date, can only amount to a scratching of its surface. The
possibilities for probing deeper and exploring wider are
tremendous. We could study combined load systems,
simultaneous advance of separation and slip and a host
of other aspects. The writing of comprehensive computer
programmes and the development of special elements
and procedures to suit the finite elements method would
make the application of the technique to research, design
and data generation wider and simpler.
In conclusion, we find that analysis of pin joints is one
of those problems, where, simplicity works better than
sophistication, or, as Professor Argyris might quip
“Simple is beautiful”.
Acknowledgements-Theauthor has much pleasure in acknowledging the collaboration of his students and co-workers B.
Dattaguru, T. S. Ramamurthy, V. A. Eshwar, S. P. Ghosh, and
N. S. Venkataraman who have contributed so much to the
developments reported herein and to the preparation of this
paper. With their participation, work is a labour of love. Thanks
are due to the Indian Institute of Science, and to the Directorate
of Basic Sciences Research and the Aeronautics Research and
Development Board of the Government of India for support.
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