Elastic analysis of pin joints

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,=OandU=uocos@+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
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

~,)ll@d

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! cosm@ 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 (/.s-+@), 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 (a@to 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 n@ 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 I+@) 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\tIloCOS@,

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. REFERENCE3

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