Advancing contact of a 2D elastic curved beam indented by a rigid pin with clearance

. A two-dimensional analytical solution is presented for stresses and displacements in an elastic curved beam forming an incomplete ring in frictionless and unbonded contact with a rigid pin loaded by a point force and in the presence of clearance. The circular beam is modelled as an incomplete elastic thick ring, constrained at both ends and in a plane stress state. The stress and displacement fields within the beam are derived from a biharmonic Airy stress function, according to the Michell solution in polar coordinates. The mixed boundary value problem is reduced to a set of dual series equations and then to a non-homogeneous linear system of infinite equations, which is then solved by truncation. The non-linear relations between the applied load and the contact angle or the pressure distribution are obtained by using an inverse method. The analytical results are compared with finite element predictions for a pin-lug connection and a reasonable agreement is observed for several typical geometries. The peaks of contact pressure and von Mises equivalent stress and their location within the curved beam are evidenced.


Introduction
Contact problems for a loaded pin connection with clearance are frequently met in mechanical and structural engineering, because a small amount of clearance is necessarily required for improving the rotational performance of the circular pin.On the other side, clearance is also responsible for vibration, noise and wear in mechanical connections, and thus its amount must be kept at the minimum allowable level.Due to their complexity arising from the progressive increase of the unknown contact region with the loading, which is responsible of the non-linear response though the material behavior is linear elastic, these problems are usually treated by advanced numerical techniques and only few analytical studies are available in the literature.Most of them concern the problem of the advancing contact of a circular pin in a hole in an infinite elastic medium [1][2][3][4][5][6], and interesting point is the location of the lug maximum equivalent stress.For low loads, the maximum stress occurs at the contact center.When the load is increased, the maximum stress moves abruptly laterally, at the sides of the bore.It is relevant to know for which load factor this shift of the maximum stress occurs.The above aspects describing the pin-lug mechanical response fully justify the attention of the researchers towards pinned connections and related contact problems.
Pinned connections provide a mechanical link between lugs without precluding relative rotations.
The determination of the contact angular width in terms of the applied load and of the initial clearance is a fundamental aspect in the understanding of the mechanical response of the pin-lug connections.The most employed model of this connection adopts a rigid, solid pin in contact with a lug described as an incomplete ring of constant radial width, clamped at both extremities.More precisely, two models of the incomplete ring are generally favored in the pertinent literature, namely a) a ring modelled as a curved beam according to the one-dimensional (1D) beam theory, and b) a ring mimicked as a two-dimensional (2D) annulus in terms of the 2D theory of elasticity.
To contain the mathematical difficulties, a purely flexural 1D beam model is often preferred, e.
g. [10].However, this model assumes transversal inextensibility of the beam and thus predicts underestimated contact lengths usually.Moreover, it demands that the contact reaction between the pin and the lug be described by a constant pressure profile along the pin-lug contact arc for the lug to exhibit a constant curvature, endowed with unrealistic radial concentrated forces directed outwards, see [14].These forces appear in order to avoid any unphysical compenetration between the pin contour and the inner border of the lug portion detached from the pin.The presence of

J o u r n a l P r e -p r o o f
Journal Pre-proof physically unrealistic concentrated forces still delivers a realistic mathematical connection between the pin-lug contact extent on one side, and the applied load and the initial clearance on the other side.Conversely, the presence of point forces precludes the possibility of predicting reliable stresses within the lug.
In the present work, a simplified geometry is adopted for the lug, modelled as an incomplete 2D elastic ring of constant radial width clamped at its extremities and indented by a rigid pin, in the presence of initial clearance.A Michell-type series representation is considered for the stress and displacement fields, according to the 2D theory of linear and isotropic elasticity, and a fully analytical approach based on the solution of a dual trigonometric series is adopted.This model is considerably more complex than its 1D curved beam counterpart, but it can consider the transversal deformability of the indented beam and it avoids the outcome of point forces at the pin-lug contact extremities.In fact, the 2D model produces realistic pressure bumps at the contact extremities, as confirmed by photoelastic studies [12].As a counterpart, the adopted 2D model does not simulate exactly the geometry of the lug, whose thickness actually increases at the transition between the lug curved shape and the lug shank.To simulate the presence of the lug shank, the angular extent of the 2D curved beam is extended beyond , based on a matching procedure with the predictions of a FEM analysis.
The analytical approach adopted here is suggested by the studies [15][16][17], which investigated the plane contact problems of a circular inclusion embedded in an infinite elastic medium or in a thickwalled tube, as well as by the work [18], which addressed the problem of the indentation of an incomplete elastic ring by a rigid flat surface.By using a modified Michell solution for a biharmonic stress function in polar coordinates, these authors reduced the mixed boundary value problems with circular boundaries arising in the study of contact problems in 2D elasticity to dual trigonometric series and then to integral equations of the Fredholm type.Here, we transform the dual series equations to a non-homogeneous linear system of infinite equations, which is then solved by truncation.Due to the progressive and nonlinear increase of the unknown contact length between pin and lug with the applied load, this procedure requires an inverse method of solution.
Namely, we impose a specified angular contact extent and then seek to determine the magnitude of the load causing that contact angle.By finding a number of these linear elastic solutions for increasing contact lengths, the non-linear relationship between the contact angle and a loading parameter proportional to the ratio between the load magnitude and the clearance is then obtained.
Finally, the analytical results in terms of the contact angular extent are compared with the FEM predictions for a pin-lug connection and a reasonable agreement is observed for a large range of aspect ratios, after proper calibration of the curved beam angular extent.

J o u r n a l P r e -p r o o f
Journal Pre-proof

Problem formulation
We consider the plane problem of an elastic circular thick beam in Fig. 1 with an angular extent of 2, whose inner and outer radii are denoted by ri and ro, respectively, indented by a rigid pin with radial clearance  = ri rp, where rp is the pin radius, under plane stress loading conditions.Due to the effect of clearance, the contact problem is progressive, and the contact length between the two components increases with the load P applied to the pin.Therefore, a special procedure based on the inverse method of solution is required to evaluate the contact extent.The problem aims to simulate the loading condition of a typical pin-lug connection.Reference is made to a polar coordinate system (0, r, ) whose origin lays at the center of the circular curved beam and the polar axis  = 0 coincides with the direction of the load P applied to the rigid pin (Fig. 1).The boundary conditions must require that the lower outer surface of the beam is stress free, namely 0, for 0 | | , .
whereas the upper inner surface is subject to mixed load and displacement boundary conditions due to the frictionless unbonded contact with the rigid pin, namely where  is the radial clearance and v0 denotes an unknown rigid body motion of the pin along the loading direction.The last contact condition on the radial displacement along the contact zone in the clearance case was obtained in [19,20] for the almost conforming contact problem between a Since the loading condition is symmetric with respect to the vertical axis ( = 0), then following [18] and [21] we take the following cylindrical components of the stress, which are obtained by means of a biharmonic Airy stress function containing only even terms in , according to the Michell solution (for analytical details see Appendix A) , 2 where m = n/,  is the elastic shear modulus of the curved beam, and a0, b0, an, bn, cn, and dn are arbitrary constants to be determined from the boundary conditions.The corresponding displacement fields are [21]: where  is the Kolosov's constant [21], namely 3 4 for plane strain, 3 for plane stress, 1

J o u r n a l P r e -p r o o f
Journal Pre-proof and  is the Poisson's ratio of the material.The boundary conditions (2.1) and (2.2)1 then provides the constants bn, cn, and dn in terms of an, for n = 0, 1, 2,.., namely where  = ri/ro < 1 denotes the aspect ratio of the curved beam.The remaining boundary conditions on the upper beam surface (2.2)2,3 then yield a set of dual trigonometric series and To remove the rigid body displacement v0, we apply the operator (D + D 1 ) on equation (2.9) as suggested in [16,17], where Then, we apply the procedure used in [18] for solving a set of dual series equations similar to eqns

J o u r n a l P r e -p r o o f
Journal Pre-proof (2.8) and (2.13).To this aim, we rearrange equations (2.8) and (2.13) in the form where t = /, = /, and ) The set of dual equations and (2.14) and (2.15) can now be solved by using the procedure proposed in [19] and adopted also in [18], based on the introduction of an auxiliary stress function  such that where H denotes the unit step function.Then, the coefficients An of the Fourier cosine series expansion (2.18), for n ≥ 0, are given in [18,22]: where Pn denotes the Legendre polynomials of order n [23,24].Now, using the following result ([25], eqn 2.6.31) The integral equation of the Abel type (2.22) Using the results provided in Appendix B of [15], the integrals in eqn (2.23) can be calculated in closed form, thus obtaining The introduction of eqn (2.24) in (2.19) and (2.20) then gives the following relations for the coefficients An, for n = 0, 1, 2,…: where  The results provided in Section 3 in [26] have been used in calculating the integrals (2.27) and (2.28) together with eqn (3.3) reported in [27].A detailed derivation of the result (2.28) is illustrated in Appendix B. Note that for n = m the coefficient Knn in (2.28) must be calculated as a limit.
Finally, the total load P acting on the pin must be equal to the resultant of the contact pressure distribution, namely (2.32)

Approximate solution
To solve the non-homogeneous linear system of infinite algebraic equations (2.25) and (2.26) for the unknown coefficients An, for n = 0, 1, 2,…, we choose a sufficiently large integer N and rewrite eqns (2.25) and (2.26) in the truncated form: thus obtaining a linear system of N+1 equations for the N+1 unknowns An, for n = 0, 1, 2,…, N.
Once the linear system (3.1) and (3.2) is solved, then the solution can be refined by considering an additional number M of terms, using the approximate relations which exploit the behavior (2.17) and (2.29) of hn and Jn for large n.
The relation between the applied load and the contact half-angle  in Fig. 1 is found by using an inverse method.Namely, for every assigned value of  the first N+1 coefficients of the series expansions of the stress and displacement fields (2.4) and (2.

Results
To define the beam angular extent  which better simulates the actual geometry of a pin-lug connection (Fig. 2), a matching procedure with the predictions of a FEM analysis is performed by comparing the analytical results concerning the contact half-angle  to the FEM predictions for the pin-lug connection, for four different values of the aspect ratio = ri/ro, also considered in previous works, namely 0.77, 0.667, 0.5 and 0.376, which generously cover the lug practically relevant geometries.Such a comparison showed that a reasonably good agreement between the analytical results and the FEM numerical predictions is attained for  = 0.56 .It can be observed that, for the considered value of , the analytical results (solid line) fit reasonably well with the FEM predictions (dashed line) for the considered four aspect ratios of the curved beam (lug), see Figs. 3a-6a.A small improvement would be achieved by making the angle  slightly depend on the aspect ratio , but this occurrence is not taken into consideration here for the sake of conciseness.
As the pin load increases, and so the loading factor  defined in (4.1), the contact angular extent between the pin and the beam also increases non-linearly.In particular, for a very thin lug, namely for  approaching 1, the contact half-angle  tends to /2 as the loading factor  becomes very large.For a thick lug instead, the contact half-angle  tends to an extent moderately smaller than Journal Pre-proof /2 as the loading factor  increases, in agreement with the result provided in [19], where a theoretical maximum contact half-angle around 84°, namely 0.467 , is found for a circular inclusion in an infinite elastic medium.As a general rule, for a very high pin-load that causes a contact angular extent close to the maximum asymptotic value, the size of the contact region increases very slowly with the load, until an increase in load produces an almost linear increase in the contact pressure, namely the response of the beam becomes almost linear within this range.
Note that the contact half-angle  depends on the aspect ratio , as well as on the ratio P/(E) appearing in the definition (4.1) of , where E = 2(1+) is the Young modulus of the beam material, and thus it does not depend on P, E and  separately.
The normalized distributions of the contact pressure along the contact zone p() =  rr(ri, ) are also plotted in Figs.3b-6b.There it can be observed that the distribution of the contact pressure is almost Hertzian when the loading level is low enough (black lines), as observed in [8] and more recently in [9].However, the pressure changes significantly as the loading magnitude increases, since two pressure peaks (bumps) take place just before the contact end points, where the pressure tends to vanish.In this case, the contact pressure is almost uniform in the central part of the contact region, in agreement with the predictions of the 1D beam model of the pin-lug connection proposed by the authors in the preliminary work [28].There, a uniform pressure distribution is found along the whole contact region, except at the lift off points, where a point force is introduced according to the 1D beam solution for an elastic open ring fitted on a rigid cylinder provided in [14].The contact angular extent predicted by the 1D beam model adopted in [28] turns out to be underestimated apparently due to the assumptions of transversal inextensibility of the classical beam theory as well as to the point force model applied at the contact end, which simulates the pressure bump observed in the present study but does not allow for pressure distribution.The 2D elastic model adopted here corrects these shortcomings at the cost of a greater analytical complexity.The distributions of the contact pressure for the limiting cases of a very thin beam ( = 0.95) and a very thick beam ( = 0.95) are plotted in Figs.7a and 7b, respectively.These results show that the location and the width of the contact pressure peak varies with the aspect ratio of the beam.Indeed, for a very thin beam, the pressure distribution displays a sharp peak just before the ends of the contact zone and is almost constant inside the contact zone (Fig. 7a), in agreement with the predictions of the 1D beam model proposed in [28], where the contact pressure peak is modelled by a concentrated load.As the beam thickness increases, the width of the contact pressure peak also increases and its location shifts toward the center of the contact zone (see also Figs. 3b-6b).It occurs exactly there as  tends to vanish, namely for a very thick beam (Fig. 7b), thus approaching the Hertzian pressure distribution.
In Fig. 7a, one can observe that the contact pressure exhibits numerical oscillations near a sharp peak due to the Gibbs phenomenon.These oscillations can be easily mitigated by using Lanczos sigma factors [29], e. g. as in [30,31].Both plots refer to the angular extent of the beam  = 0.56.
The normalized variations of the hoop stress along the contact region (ri, ) are plotted in Fig. 8 for the four aspect ratios considered in Figs 3-6.These results also show that the hoop stress is almost constant in the central part of the contact zone and it displays a large increase at the ends.
Moreover, as the load increases, and thus the size of the contact region also increases, the hoop stress changes its sign and the location of its maximum magnitude.Indeed, for low loading level the hoop stress is compressive (negative) along the contact zone and attains its largest magnitude at the center, namely at  = 0.As the load increases, the hoop stress becomes tensile starting from the ends of the contact region ( = ), where it attains the maximum values within the contact zone for sufficiently large load P. J o u r n a l P r e -p r o o f

Journal Pre-proof
The normalized variations of the von Mises equivalent stress, plotted in Fig. 9, also display an almost uniform stress level at the center of the contact zone, especially for thin lugs, and a remarkable increase at the end of the contact as the load P is increased.It is interesting to note that the maximum equivalent stress is attained at the center of the contact zone for low loading level, whereas it moves to the contact end points as the load is increased.The transition occurs for a loading level causing a contact half-angle of about 1 rad, namely   0.32 , and a bit less for thinner lugs (e. g. for the case ri/ro = 077).In terms of loading factor , the transition occurs for about 20, as it follows from Fig. 3a-7a, a bit more for thin lugs and a bit less for thick lugs.independent of the loading and geometry, in agreement with the findings in [32,33] for the hoop stress.
The results retrieved in this paper in terms of stresses are more realistic than those achieved by describing the lug in term of a curved, purely flexural 1D beam [28].In fact, the presence of an unrealistic concentrated contact force in the 1D beam-like model precludes the possibility of deriving reliable stresses.The 2D analytical model favoured in this paper corrects the above shortcome, since the contact pressure produces a bump of finite magnitude.Therefore, an interesting extension of this paper would be to examine whether the approach of [11] proposing an interpretation of the above concentrated force in terms of a Hertzian-type distribution describing the pressure bump, correctly modifies the unrealistic concentrated force of the 1D curved beam model into a more realistic pressure bump of finite maximum value, according to the results obtained here by using the 2D theory of elasticity.

Conclusions
An analytical 2D solution for the stress and displacement fields in an elastic circular beam indented by a rigid circular pin with clearance in frictionless contact with the upper inner surface of the beam is derived herein by using a modified Fourier-Michell solution and solving a set of dual series equations.The angular extent of the circular beam is chosen to best fit with the FEM predictions for a pin-lug connection.The progressive character of the contact in the presence of an initial clearance between the pin and the lug is discussed in detail.In particular, the contact region increases nonlinearly with the load factor and approaches an asymptotic value close to /2 as the load becomes very large.The pressure distributions along the contact zone are plotted for various aspect ratios and load factors.They show that the Hertzian distribution is valid only for very low load levels and small contact zone, while for high loading levels the central pressure remains reasonably constant, and two pressure bumps take place near the ends of the contact zone.The advantages of the proposed 2D modelling of the lug are clearly underlined with respect to the 1D counterpart.Finally,

Figure 1 .
Figure 1.2D Curved beam indented by a rigid pin.
r n a l P r e -p r o o f Journal Pre-proof cylindrical pin and a hole in a plate, namely for  << ri.Finally, the circumferential displacements and shear tractions are assumed to vanish at the beam ends, namely 0, for | | , .

5 )
is calculated by solving the linear system (3.1) and (3.2), and advantageously extended to a number M+1 of term according to(3.3).Their knowledge allows to evaluate the corresponding load P defined in (2.32) which produces the assigned contact angular amplitude  The results presented in the next Section are obtained by considering a number of terms N ≥ 30 in the truncated system (3.1)-(3.2) and a total number of terms M = 2N in the sums for stress and displacement.Preliminary tests proved that this number of terms is suitable for obtaining sufficiently accurate results.

Figure 2 .
Figure 2. Geometry and loading of the pin-lug connection considered in the FEM simulations.
r n a l P r e -p r o o f

Figure 3 .Figure 4 .Figure 5 .Figure 6 .
Figure 3. Normalized variations of the contact half-angle  with the loading factor  provided by the present analytical formulation and by FEM (a), and of the contact pressure p with the angular coordinate (b).Both plots refer to the angular extent of the beam  = 0.56 and to the aspect ratio ri/ro = 0.77.

Figure 7 .
Figure 7. Normalized variations of the contact pressure p with the angular coordinate for a very thin beam (a) and a very thick beam (b), for increasing values of the contact half-angle .

Figure 8 .
Figure 8. Normalized variations of the hoop stress  along the contact region with the angular coordinate  for the angular extent of the beam  = 0.56 and for four different aspect ratios.

Figure 9 .Figure 10 .Figure 11 . 1 and  = 1 . 5 .
Figure 9. Normalized variations of the von Mises equivalent stress eq along the contact region with the angular coordinate  for the angular extent of the beam  = 0.56 and for four different aspect ratios.
can be inverted to obtain the auxiliary stress function  