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Stress concentration and optimal design of pinned connections

Pedersen, Niels Leergaard

Published in: The Journal of Strain Analysis for Engineering Design Link to article, DOI: 10.1177/0309324719842766 Publication date: 2019 Document Version Peer reviewed version Link back to DTU Orbit

Citation (APA): Pedersen, N. L. (2019). Stress concentration and optimal design of pinned connections. The Journal of Strain Analysis for Engineering Design, 54(2), 95-104. https://doi.org/10.1177/0309324719842766

General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Stress concentration and optimal design of pinned connections

Niels Leergaard Pedersen Dept. of Mechanical Engineering, Solid Mechanics

Technical University of Denmark Nils Koppels Alle´, Building 404, DK-2800 Kgs. Lyngby, Denmark

email: [email protected]

Abstract

A pinned connection or lug joint is a common connection type used both in civil engineering and mechanical engineering. In civil engineering this connection is i.e. used for assembling truss members and in mechanical engineering this connection type is widely used in machine elements. The standard design is with a circular pin. The stress concentration factor size depends on the tolerances between pin and assembled parts and also by the 3D design. Relatively different maximum stress values are seen depending on the modelling being done in 2D (with assumptions) or in full 3D. The focus in the present paper is on the 2D design and minimizing the maximum stress. It is shown that not only the contact geometry is important for reducing the stress, the external design is equally important. By ﬁnite element analysis including contact modelling it is in the present paper shown that reductions in the stress concentration factor of up to 18% are possible.

Key words: Machine elements, pinned connection, lug joint, stress concentration, FEA, 2D.

1 Introduction

n many practical applications the assembly of different parts is done by pinned connections which are also termed lug joints. The simple assembly makes this type of connection preferable in cases where quick assembly and disassembly is needed, e.g. for the handling of shipping containers. In many civil engineering applications the lug joint is permanent, e.g. in truss assemblies ensuring that only axial load is transferred to the truss. Also in electrical connections this type of connection is used but the present paper focus is related to applications where it is the mechanical strength that is important. The loading of the connection is assumed to be cyclic so that the strength is related to fatigue and therefore to the maximum stress in the connection. In machine elements the pinned connections are typically used for shaft hub connections ensuring that the power can be transferred between them, in many cases the pin can also serve as a safety component if it is made by a signiﬁcantly weaker material than the hub and shaft. In all of the mentioned cases the pinned connection is a subset of positive connections. The pin design is for machine elements generally controlled by standards, see e.g. DS/EN ISO 2338 (1998).

The design of pinned connections is essentially 3 dimensional although most focus is on the cross sectional design that is typically extruded into a given thickness. The cross sectional

1

design is circular and stress concentration factor charts for tight ﬁts (no clearance) can be found in e.g. Pilkey (1997). The typical used modelling assumption is that it is sufﬁcient to use 2D with a further assumption of either plane stress (if the contact length in axial direction is relatively small) or plane strain (if the thickness is relatively large). For practical cases where the connection thickness is comparable to the pin diameter it is known that there is a signiﬁcant variation in the stress along the axial pin direction. The variation is also found in tight ﬁts and interference ﬁts as reported in e.g. Pedersen (2016). The 3D effect on the stress concentration is neglected in the present paper.

For the circular design a number of papers discuss the stress level in a pinned connection found both from experimental studies, see e.g. Frocht & Hill (1940), and using analytical work, see e.g. Theocaris (1956), and more recently using numerical methods, see e.g. Wang (1994), Strozzi, Baldini & Nascimbeni (2006), Pedersen (2007) and Strozzi, Baldini, Giacopini, Bertocchi & Bertocchi (2011). In many of these publications charts of stress concentration can be found also for different angles of attack for the load. The design is in all cases circular and no design optimization is performed, focus is on the stress level evaluation.

The present paper aim is to minimize the pinned connection stress concentration whereby an increase in the connection strength is achieved, i.e. we optimize with respect to fatigue due to a dynamic/oscillating load. The stress evaluation is performed using the ﬁnite element method (FEM) as the numerical tool. To validate that a design improvement has been achieved a signiﬁcant number of points need to be taken into consideration, these include:

• The stress concentration factor deﬁnition is important. The main point is which area the nominal stress is related to, i.e. the pin, the lug net area or the total lug width.

• The needed ﬁnite element mesh reﬁnement used in the non-linear contact analysis.

• Inﬂuence from head distance.

• Inﬂuence from value of Poisson’s ratio.

• Inﬂuence from modelling assumption, plane stress or plane strain.

• Inﬂuence from friction on the maximum stress concentration.

• Clearance and possible comparison to Hertzian pressure.

Design optimization using shape optimization in relation to reduced stress concentrations can be found in many papers. The special case of the present work is that the shape to be optimized also is the shape where the contact takes place between the two parts. Generally it is such that the shape is optimized if the stress level (the maximum stress) along major parts of the shape is constant. The evaluation of stress is here performed using FEM, and as stated in Ding (1986) it is important that the shape parameterization is not done by the FE nodal positions but by an independent parameterization that controls the nodal positions instead. In this way we will not optimize the FEM errors and at the same time a parameterization with a much lower number of parameters can be used, also leading to a design that can be easily communicated. With the shape ﬁxed, either numerically or analytically, a mesh reﬁnement can more simply be done to verify the stress level. The selected parameterization is here the super elliptical one which previously have been used to minimize stress in machine element related components, see e.g. Pedersen (2016) or Pedersen (2018). Shape optimization of a hole in an inﬁnite plate for external loading is well known and in the literature the reported optimal shape is the elliptical design where the ratio between the lengths of principal axes is equal to the ratio between the

2

two loads along the principal axes. To the authors knowledge no optimized design for the pin in the lug joint is reported in the literature.

The paper is organized as follows. In Section 2 the deﬁnition of stress concentration factor is discussed, speciﬁcally in relation to the subsequent evaluation of optimized designs. Section 3 presents the non-linear numerical modelling and important parameters in the modelling. The importance of the head distance is presented in Section 3.1, i.e. the exterior designs inﬂuence on the maximum stress level. Section 3.2 presents the inﬂuence from Poisson’s ratio on the stress level and also the modelling assumption of either plane stress or plane strain is discussed. The inclusion of friction in the connection is shortly discussed in Section 3.3 before Section 3.4 discuss the importance of including clearance in the modelling, a comparison to Hertzian pressure is also given. The shape optimization is presented in Section 4 before the conclusion in Section 5.

2 Deﬁnition of stress concentration factor

For a pinned connection or lug joint there is not one stress concentration factor deﬁnition that is valid for all cases, as also discussed in Pilkey (1997). In Figure 1 a lug joint is shown with dimension, the perpendicular thickness is assumed to be, T .

L

H

0001110001110001110001110001110001110100011101000111B010001110100011101000111000111 000111000111000111000111000111A000111000111000111000111000111000111000111000111000111000111C000111000111F W D 0001110001110001110001110001110001110100011101000111010001110100011101000111000111

Figure 1: Dimension deﬁnition of lug with circular pin, indicating tree points A, B and C along the hole boundary.

Traditionally the theoretical stress concentration factor (index t for theoretical) is deﬁned as

Kt = σmax

(1)

σnom

where σmax is the maximum normal stress, i.e. the numerical largest principal stress. The nominal stress deﬁnition, σnom, is in principal not important for evaluating the maximum stress from a given stress concentration factor as long as the deﬁnition is known. A stress concentration factor should however also preferably indicate how large the variation in the stress is. For the stress concentration value to give a direct physical interpretation it is important that the nominal stress deﬁnition do not make the stress concentration factor unreasonably low or high. A common choice for the nominal stress in a pinned connection is to relate it to the net area (of the lug)

σnnom = (W −F D)T (2)

3

i.e. relating the stress concentration to the lug strength with a pin hole. For small values of W/D the stress concentration will be low, although the variation in the stress level is high. For this reason one could also choose to use the bearing area instead

σnbom = DF· T (3)

i.e. relating the stress concentration indirectly to the pin strength. For large values of W/D the stress concentration will be low, although the maximum stress is not. Both relating the stress to the net area or to the pin area fails in giving a stress concentration measure that is reasonable for the whole variation range of W/D. Relating instead the stress concentration to the strength of the lug without a hole (for the pin) the nominal stress can be deﬁned as

σnl om = WF· T (4)

The latter choice is made in the present paper, as also done in Theocaris (1956). The reason being that this gives a direct measure for what the increase in stress is relative to a connection (with width, W , and thickness ,T ) without a pin. This stress concentration factor is also best suited for evaluating optimized designs, i.e. what is the optimal ratio of W/D. It is clear that there is a direct relation between the three nominal stress deﬁnitions and the stress concentrations

σnl om = 1 1 1 (5)

σn + σb

nom

nom

Leading to

Ktl = Ktn + Ktb

(6)

where Ktl = σmax/σnl om, Ktn = σmax/σnnom and Ktb = σmax/σnbom. In Figure 2 the experimental results from Frocht & Hill (1940) and the analytical results

from Theocaris (1956) are shown as they are presented in Pilkey (1997). The analytical results are based on an assumption of a strip, i.e. 2D, and inﬁnite size of H and L, see Figure 1. The experimental results are performed with a size of H and L sufﬁciently large so that they do not inﬂuence the stress level. In Figure 2 the stress concentration as deﬁned in (6) is also shown. From this line it can be seen that there is an optimum with lowest stress concentration for given width, D, around W/D = 2.5 as also described in Theocaris (1956).

If the stress is multi-axial a different stress concentration factor deﬁnition might be more appropriate, i.e. in a Hertzian contact problem it is known that the maximum principal stress is at the contact surface but the largest equivalent stress measured by the von Mises stress, σvM is below the surface. So for multi-axial stress a stress concentration deﬁned relative to the von Mises stress might be a better choice.

KvM = σmvMax

(7)

σnvoMm

4

9 Kt

Ktl

7

Ktn

Ktb

5

3

1

W/D

1

3

5

7

Figure 2: The experimental stress concentration for lug with circular pin from Frocht & Hill (1940) and the analytical results from Theocaris (1956) as they are shown in Pilkey (1997) (Ktn and Ktb) together with the curve for Ktl derived from the two other curves and Eq. (6).

For the speciﬁc cases used in the present paper we will choose to use a stress concentration deﬁned as

Kl = σmvMax

(8)

vM σnl om

i.e. the nominal stress is given relative to the lug area as if there were no pin and in this case this will also correspond to the von Mises stress since it is the only stress component.

3 Numerical modelling and important parameters

The stress evaluation in pinned connections is in the present paper performed numerically using the ﬁnite element method (FEM). The numerical tool used is the COMSOL program COMSOL AB (1986 -). Due to symmetry only half the lug/pin connection is modelled. A high number of elements in the contact are used in order for the stress to converge, which have been checked by mesh reﬁnements. The elements used are triangles with a linear or quadratic displacement assumption. The modelling is done by an assumption of plane stress or strain as discussed later. An example of the used mesh is given in Figure 3.

Symmetry conditions are put on the lowest boundary and the pin is ﬁxed in the centre, a prescribed horizontal displacement is applied to the left boundary, see Figure 3. The maximum stress is evaluated at the contacting zone, i.e. not at the displaced left boundary. The size of the contact area between the two parts is determined as part of the non-linear ﬁnite element analysis (FEA).

3.1 Inﬂuence from head distance

It is initially assumed that there is no clearance between pin and lug i.e. a tight ﬁt. The stress concentration factor is not only controlled by the hole and pin design but also from the other geometric quantities as the ones seen in Figure 1. Keeping the ratio W/D constant and assuming L large enough for the stress to be fully developed towards the left boundary in Figure 1 there is still an inﬂuence on the maximum stress from the head distance H. This inﬂuence was also commented on in the experimental work presented in Frocht & Hill (1940). It is clear that when H approach D/2 there will be a large inﬂuence on the connection strength due to the possibility

5

a)

b)

c)

Figure 3: Mesh used for analysis. a) Full mesh, the left boundary is seen to have a ﬁne mesh in order to evaluate the force. The boundary on the bottom is the symmetry line. b) Zoom of pin, the small cut at the centre of the pin is where the constraint is applied. The contact zone is shown to have a ﬁne mesh. c) Zoom of part of contact zone close to point B in Figure 1.

of pulling out the pin. For large values of H where a fully developed stress ﬁeld is found in the head (in front of pin) there is no inﬂuence on the stress concentration by a further increase in H, however the inﬂuence from reducing the size of H to the somewhat intuitive choice

H = W (9) 2

is large. In Figure 4 the stress concentration is shown as a function of the relative head distance for

different width to diameter ratios. From the results it is clear that the head distance has a large inﬂuence for lower values of H/D. For larger values the stress concentration becomes constant as expected and we see that the transition point to constant maximum stress depends on the ratio W/D. The lowest stress concentration is found for the value W/D = 2.2, but for lower value of H/D the optimal value of W/D is changing.

In Figure 5 the stress concentration is shown as a function of width to diameter ratio for the case of head distance being large enough so that it has no inﬂuence on the stress level. This curve can be compared to the full line in Figure 2, the overall ﬁndings are that we ﬁnd a lower stress concentration level than that reported in Pilkey (1997). The reason for this can be multiple but primarily properly related to clearance in the case of experimentally found values or the 2D modelling. The graph is split in three (between point A and B, point B, point C) indicating where the maximum stress occurs, see Figure 1. If the maximum stress is between point A and B there is no contact at the point of maximum stress so the von Mises stress is identical to the maximum principal stress when an assumption of plane stress is used. Point B indicates the transition point from contact to no contact also here the von Mises stress is identical to

6

Kvl M 5.8 5.4 5.0

W/D = 2.2 W/D = 2.1 W/D = 2.0

W/D = 1.9

W/D = 1.8 W/D = 1.7 W/D = 1.6

W/D = 1.5

4.6 0.5 Kvl M 1.5

5.8 5.4 5.0

2.5

3.5

W/D = 2.2 W/D = 2.3 W/D = 2.4 W/D = 2.5 W/D = 3.0

H/D

4.6

H/D

0.5

1.5

2.5

3.5

Figure 4: Stress concentration factor as a function of head distance, under plane stress condition (FEA results).

the maximum principal stress, as seen in Figure 5 there is a continuous change between the two areas. At point C the von Mises stress is not identical to the maximum principal stress because of the contact pressure. In the ﬁgure we notice the discontinuity in the stress slope at the transition point (around W/D = 3.45).

Kvl M 8

7

between

point B

point A

6

and B

Point C

5

4 W/D

1

2

3

4

Figure 5: Stress concentration factor as a function of width to diameter ratio, under plane stress condition (FEA results).

3.2 Inﬂuence from Poisson’s ratio and plane stress/plane strain assumption

It is well known that Poisson’s ratio has an inﬂuence on the stress concentration factor value, see e.g. Dally & Riley (1991) or more recently Pedersen (2018). The previously shown results in this paper have all been found using the Poisson’s ration ν = 0.3. For the case of W/D = 2.2

7

and large head distance we show in Figure 6 the stress concentration factor as a function of Poisson’s ratio for the case of plane stress and plane strain.

Kvl M

4.4 Plane stress Plane strain

4.2

4.0

3.8 ν

0

0.1

0.2

0.3

0.4

0.5

Figure 6: Inﬂuence on stress concentration factor from Poisson’s ratio and assumption of either plane stress or plane strain (W/D = 2.2). (FEA results).

For the speciﬁc design with W/D = 2.2 the maximum stress is at the hole rim at a plane with no contact, i.e., we have a uniaxial stress state for the plane stress assumption and therefore no inﬂuence from Poisson’s ratio on the stress concentration factor. With the assumption of plane strain Figure 6 shows an inﬂuence from Poisson’s ratio due to the now double axial stress state at the point with maximum stress. We also see as expected that the results for plane strain and stress are identical for ν = 0. If a different design is selected, W/D = 8.0, we have that the maximum stress is found at the top of contact (point C), i.e., here we do not have unidirectional stress, independent of the 2D analysis type selected. Figure 7 shows the results.

Kvl M

6.0

5.6

Plane stress 5.2 Plane strain

4.8 ν

0

0.1

0.2

0.3

0.4

0.5

Figure 7: Inﬂuence on stress concentration factor from Poisson’s ratio and assumption of either plane stress or plane strain (W/D = 8.0). (FEA results).

For the case shown in Figure 7 we have with the assumption of plane stress a double axial stress state at the point with maximum stress and as seen the stress concentration factor depends on Poissons ratio to a large extend. For the plane strain case we have a general stress state at the point of maximum stress and as indicated a large inﬂuence on the maximum stress from Poisson’s ratio. In many cases it is therefore essential that the Poisson’s ratio is speciﬁed for evaluating the correct stress concentration factor. In the remaining part of the paper we will assume ν = 0.3.

8

3.3 Inﬂuence from friction on the maximum stress concentration

The inclusion of Coulomb friction in the contact modelling has shown, for the examples of the present paper, to have no inﬂuence on the result. The examples include both assumptions of either plane stress or plane strain and design cases where the point of maximum stress is a point with uni-axial, double axial or a general stress state. The inclusion of friction in the contact modelling is therefore omitted in the remaining examples.

3.4 Clearance and comparison to Hertzian pressure

Clearance between pin and lug can have a large inﬂuence on the maximum stress in the connection. Discussion on this inﬂuence can e.g. be found in Pedersen (2007) and more recent in Strozzi et al. (2011) where also the load direction relative to the lug is discussed.

The selected 2D modelling (with Poisson’s ration ν = 0.3) has a large inﬂuence on the maximum stress found. With plane stress assumption, as used in Strozzi et al. (2011), the maximum stress is found on the hole rim in the lug. However, with a plane strain assumption the maximum stress (von Mises) is for low values of the applied force found in front of the load direction but inside the lug, this is in agreement with the traditional Hertzian stress. The stress situation is in the pin/lug connection slightly different than the traditional Hertzian contact because the stress in the lug has to go around the hole into the support.

The Hertzian stress, see e.g. Gladwell (1980) or Norton (2006), is based on an assumption of plane strain, i.e. an inﬁnite half space. For parallel cylindrical contact, as for the pin/lug connection, the half-width of contact is

2

F

Lc = π (m1 + m2)φ T

(10)

where F is the contact force (load on pin), m1 and m2 are material constants which for the case of same material for lug and pin are given by

1 − ν2

m1 = m2 = E

(11)

with E being the modulus of elasticity. φ is a geometric constant expressing the radius difference for the two bodies. The expression for φ can also be found in the references Gladwell (1980) and Norton (2006). For the pinned connection we have a concave contact, and assuming the hole diameter in the lug to be given by D and the pin diameter by D − δ where δ is the diametral clearance the constant φ is deﬁned as

φ = D(D − δ) ≈ D2 (12)

δ

δ

where the last approximation holds for a relatively small clearance. T is the thickness (as deﬁne previously), i.e. the length of the contact zone in the axial direction of the pin. Perpendicular to this direction the distribution of the pressure is semi-elliptical and with the half-width known the maximum pressure at the centre is given by.

2F /T

pmax = πLc

(13)

9

Stress concentration and optimal design of pinned connections

Pedersen, Niels Leergaard

Published in: The Journal of Strain Analysis for Engineering Design Link to article, DOI: 10.1177/0309324719842766 Publication date: 2019 Document Version Peer reviewed version Link back to DTU Orbit

Citation (APA): Pedersen, N. L. (2019). Stress concentration and optimal design of pinned connections. The Journal of Strain Analysis for Engineering Design, 54(2), 95-104. https://doi.org/10.1177/0309324719842766

General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Stress concentration and optimal design of pinned connections

Niels Leergaard Pedersen Dept. of Mechanical Engineering, Solid Mechanics

Technical University of Denmark Nils Koppels Alle´, Building 404, DK-2800 Kgs. Lyngby, Denmark

email: [email protected]

Abstract

A pinned connection or lug joint is a common connection type used both in civil engineering and mechanical engineering. In civil engineering this connection is i.e. used for assembling truss members and in mechanical engineering this connection type is widely used in machine elements. The standard design is with a circular pin. The stress concentration factor size depends on the tolerances between pin and assembled parts and also by the 3D design. Relatively different maximum stress values are seen depending on the modelling being done in 2D (with assumptions) or in full 3D. The focus in the present paper is on the 2D design and minimizing the maximum stress. It is shown that not only the contact geometry is important for reducing the stress, the external design is equally important. By ﬁnite element analysis including contact modelling it is in the present paper shown that reductions in the stress concentration factor of up to 18% are possible.

Key words: Machine elements, pinned connection, lug joint, stress concentration, FEA, 2D.

1 Introduction

n many practical applications the assembly of different parts is done by pinned connections which are also termed lug joints. The simple assembly makes this type of connection preferable in cases where quick assembly and disassembly is needed, e.g. for the handling of shipping containers. In many civil engineering applications the lug joint is permanent, e.g. in truss assemblies ensuring that only axial load is transferred to the truss. Also in electrical connections this type of connection is used but the present paper focus is related to applications where it is the mechanical strength that is important. The loading of the connection is assumed to be cyclic so that the strength is related to fatigue and therefore to the maximum stress in the connection. In machine elements the pinned connections are typically used for shaft hub connections ensuring that the power can be transferred between them, in many cases the pin can also serve as a safety component if it is made by a signiﬁcantly weaker material than the hub and shaft. In all of the mentioned cases the pinned connection is a subset of positive connections. The pin design is for machine elements generally controlled by standards, see e.g. DS/EN ISO 2338 (1998).

The design of pinned connections is essentially 3 dimensional although most focus is on the cross sectional design that is typically extruded into a given thickness. The cross sectional

1

design is circular and stress concentration factor charts for tight ﬁts (no clearance) can be found in e.g. Pilkey (1997). The typical used modelling assumption is that it is sufﬁcient to use 2D with a further assumption of either plane stress (if the contact length in axial direction is relatively small) or plane strain (if the thickness is relatively large). For practical cases where the connection thickness is comparable to the pin diameter it is known that there is a signiﬁcant variation in the stress along the axial pin direction. The variation is also found in tight ﬁts and interference ﬁts as reported in e.g. Pedersen (2016). The 3D effect on the stress concentration is neglected in the present paper.

For the circular design a number of papers discuss the stress level in a pinned connection found both from experimental studies, see e.g. Frocht & Hill (1940), and using analytical work, see e.g. Theocaris (1956), and more recently using numerical methods, see e.g. Wang (1994), Strozzi, Baldini & Nascimbeni (2006), Pedersen (2007) and Strozzi, Baldini, Giacopini, Bertocchi & Bertocchi (2011). In many of these publications charts of stress concentration can be found also for different angles of attack for the load. The design is in all cases circular and no design optimization is performed, focus is on the stress level evaluation.

The present paper aim is to minimize the pinned connection stress concentration whereby an increase in the connection strength is achieved, i.e. we optimize with respect to fatigue due to a dynamic/oscillating load. The stress evaluation is performed using the ﬁnite element method (FEM) as the numerical tool. To validate that a design improvement has been achieved a signiﬁcant number of points need to be taken into consideration, these include:

• The stress concentration factor deﬁnition is important. The main point is which area the nominal stress is related to, i.e. the pin, the lug net area or the total lug width.

• The needed ﬁnite element mesh reﬁnement used in the non-linear contact analysis.

• Inﬂuence from head distance.

• Inﬂuence from value of Poisson’s ratio.

• Inﬂuence from modelling assumption, plane stress or plane strain.

• Inﬂuence from friction on the maximum stress concentration.

• Clearance and possible comparison to Hertzian pressure.

Design optimization using shape optimization in relation to reduced stress concentrations can be found in many papers. The special case of the present work is that the shape to be optimized also is the shape where the contact takes place between the two parts. Generally it is such that the shape is optimized if the stress level (the maximum stress) along major parts of the shape is constant. The evaluation of stress is here performed using FEM, and as stated in Ding (1986) it is important that the shape parameterization is not done by the FE nodal positions but by an independent parameterization that controls the nodal positions instead. In this way we will not optimize the FEM errors and at the same time a parameterization with a much lower number of parameters can be used, also leading to a design that can be easily communicated. With the shape ﬁxed, either numerically or analytically, a mesh reﬁnement can more simply be done to verify the stress level. The selected parameterization is here the super elliptical one which previously have been used to minimize stress in machine element related components, see e.g. Pedersen (2016) or Pedersen (2018). Shape optimization of a hole in an inﬁnite plate for external loading is well known and in the literature the reported optimal shape is the elliptical design where the ratio between the lengths of principal axes is equal to the ratio between the

2

two loads along the principal axes. To the authors knowledge no optimized design for the pin in the lug joint is reported in the literature.

The paper is organized as follows. In Section 2 the deﬁnition of stress concentration factor is discussed, speciﬁcally in relation to the subsequent evaluation of optimized designs. Section 3 presents the non-linear numerical modelling and important parameters in the modelling. The importance of the head distance is presented in Section 3.1, i.e. the exterior designs inﬂuence on the maximum stress level. Section 3.2 presents the inﬂuence from Poisson’s ratio on the stress level and also the modelling assumption of either plane stress or plane strain is discussed. The inclusion of friction in the connection is shortly discussed in Section 3.3 before Section 3.4 discuss the importance of including clearance in the modelling, a comparison to Hertzian pressure is also given. The shape optimization is presented in Section 4 before the conclusion in Section 5.

2 Deﬁnition of stress concentration factor

For a pinned connection or lug joint there is not one stress concentration factor deﬁnition that is valid for all cases, as also discussed in Pilkey (1997). In Figure 1 a lug joint is shown with dimension, the perpendicular thickness is assumed to be, T .

L

H

0001110001110001110001110001110001110100011101000111B010001110100011101000111000111 000111000111000111000111000111A000111000111000111000111000111000111000111000111000111000111C000111000111F W D 0001110001110001110001110001110001110100011101000111010001110100011101000111000111

Figure 1: Dimension deﬁnition of lug with circular pin, indicating tree points A, B and C along the hole boundary.

Traditionally the theoretical stress concentration factor (index t for theoretical) is deﬁned as

Kt = σmax

(1)

σnom

where σmax is the maximum normal stress, i.e. the numerical largest principal stress. The nominal stress deﬁnition, σnom, is in principal not important for evaluating the maximum stress from a given stress concentration factor as long as the deﬁnition is known. A stress concentration factor should however also preferably indicate how large the variation in the stress is. For the stress concentration value to give a direct physical interpretation it is important that the nominal stress deﬁnition do not make the stress concentration factor unreasonably low or high. A common choice for the nominal stress in a pinned connection is to relate it to the net area (of the lug)

σnnom = (W −F D)T (2)

3

i.e. relating the stress concentration to the lug strength with a pin hole. For small values of W/D the stress concentration will be low, although the variation in the stress level is high. For this reason one could also choose to use the bearing area instead

σnbom = DF· T (3)

i.e. relating the stress concentration indirectly to the pin strength. For large values of W/D the stress concentration will be low, although the maximum stress is not. Both relating the stress to the net area or to the pin area fails in giving a stress concentration measure that is reasonable for the whole variation range of W/D. Relating instead the stress concentration to the strength of the lug without a hole (for the pin) the nominal stress can be deﬁned as

σnl om = WF· T (4)

The latter choice is made in the present paper, as also done in Theocaris (1956). The reason being that this gives a direct measure for what the increase in stress is relative to a connection (with width, W , and thickness ,T ) without a pin. This stress concentration factor is also best suited for evaluating optimized designs, i.e. what is the optimal ratio of W/D. It is clear that there is a direct relation between the three nominal stress deﬁnitions and the stress concentrations

σnl om = 1 1 1 (5)

σn + σb

nom

nom

Leading to

Ktl = Ktn + Ktb

(6)

where Ktl = σmax/σnl om, Ktn = σmax/σnnom and Ktb = σmax/σnbom. In Figure 2 the experimental results from Frocht & Hill (1940) and the analytical results

from Theocaris (1956) are shown as they are presented in Pilkey (1997). The analytical results are based on an assumption of a strip, i.e. 2D, and inﬁnite size of H and L, see Figure 1. The experimental results are performed with a size of H and L sufﬁciently large so that they do not inﬂuence the stress level. In Figure 2 the stress concentration as deﬁned in (6) is also shown. From this line it can be seen that there is an optimum with lowest stress concentration for given width, D, around W/D = 2.5 as also described in Theocaris (1956).

If the stress is multi-axial a different stress concentration factor deﬁnition might be more appropriate, i.e. in a Hertzian contact problem it is known that the maximum principal stress is at the contact surface but the largest equivalent stress measured by the von Mises stress, σvM is below the surface. So for multi-axial stress a stress concentration deﬁned relative to the von Mises stress might be a better choice.

KvM = σmvMax

(7)

σnvoMm

4

9 Kt

Ktl

7

Ktn

Ktb

5

3

1

W/D

1

3

5

7

Figure 2: The experimental stress concentration for lug with circular pin from Frocht & Hill (1940) and the analytical results from Theocaris (1956) as they are shown in Pilkey (1997) (Ktn and Ktb) together with the curve for Ktl derived from the two other curves and Eq. (6).

For the speciﬁc cases used in the present paper we will choose to use a stress concentration deﬁned as

Kl = σmvMax

(8)

vM σnl om

i.e. the nominal stress is given relative to the lug area as if there were no pin and in this case this will also correspond to the von Mises stress since it is the only stress component.

3 Numerical modelling and important parameters

The stress evaluation in pinned connections is in the present paper performed numerically using the ﬁnite element method (FEM). The numerical tool used is the COMSOL program COMSOL AB (1986 -). Due to symmetry only half the lug/pin connection is modelled. A high number of elements in the contact are used in order for the stress to converge, which have been checked by mesh reﬁnements. The elements used are triangles with a linear or quadratic displacement assumption. The modelling is done by an assumption of plane stress or strain as discussed later. An example of the used mesh is given in Figure 3.

Symmetry conditions are put on the lowest boundary and the pin is ﬁxed in the centre, a prescribed horizontal displacement is applied to the left boundary, see Figure 3. The maximum stress is evaluated at the contacting zone, i.e. not at the displaced left boundary. The size of the contact area between the two parts is determined as part of the non-linear ﬁnite element analysis (FEA).

3.1 Inﬂuence from head distance

It is initially assumed that there is no clearance between pin and lug i.e. a tight ﬁt. The stress concentration factor is not only controlled by the hole and pin design but also from the other geometric quantities as the ones seen in Figure 1. Keeping the ratio W/D constant and assuming L large enough for the stress to be fully developed towards the left boundary in Figure 1 there is still an inﬂuence on the maximum stress from the head distance H. This inﬂuence was also commented on in the experimental work presented in Frocht & Hill (1940). It is clear that when H approach D/2 there will be a large inﬂuence on the connection strength due to the possibility

5

a)

b)

c)

Figure 3: Mesh used for analysis. a) Full mesh, the left boundary is seen to have a ﬁne mesh in order to evaluate the force. The boundary on the bottom is the symmetry line. b) Zoom of pin, the small cut at the centre of the pin is where the constraint is applied. The contact zone is shown to have a ﬁne mesh. c) Zoom of part of contact zone close to point B in Figure 1.

of pulling out the pin. For large values of H where a fully developed stress ﬁeld is found in the head (in front of pin) there is no inﬂuence on the stress concentration by a further increase in H, however the inﬂuence from reducing the size of H to the somewhat intuitive choice

H = W (9) 2

is large. In Figure 4 the stress concentration is shown as a function of the relative head distance for

different width to diameter ratios. From the results it is clear that the head distance has a large inﬂuence for lower values of H/D. For larger values the stress concentration becomes constant as expected and we see that the transition point to constant maximum stress depends on the ratio W/D. The lowest stress concentration is found for the value W/D = 2.2, but for lower value of H/D the optimal value of W/D is changing.

In Figure 5 the stress concentration is shown as a function of width to diameter ratio for the case of head distance being large enough so that it has no inﬂuence on the stress level. This curve can be compared to the full line in Figure 2, the overall ﬁndings are that we ﬁnd a lower stress concentration level than that reported in Pilkey (1997). The reason for this can be multiple but primarily properly related to clearance in the case of experimentally found values or the 2D modelling. The graph is split in three (between point A and B, point B, point C) indicating where the maximum stress occurs, see Figure 1. If the maximum stress is between point A and B there is no contact at the point of maximum stress so the von Mises stress is identical to the maximum principal stress when an assumption of plane stress is used. Point B indicates the transition point from contact to no contact also here the von Mises stress is identical to

6

Kvl M 5.8 5.4 5.0

W/D = 2.2 W/D = 2.1 W/D = 2.0

W/D = 1.9

W/D = 1.8 W/D = 1.7 W/D = 1.6

W/D = 1.5

4.6 0.5 Kvl M 1.5

5.8 5.4 5.0

2.5

3.5

W/D = 2.2 W/D = 2.3 W/D = 2.4 W/D = 2.5 W/D = 3.0

H/D

4.6

H/D

0.5

1.5

2.5

3.5

Figure 4: Stress concentration factor as a function of head distance, under plane stress condition (FEA results).

the maximum principal stress, as seen in Figure 5 there is a continuous change between the two areas. At point C the von Mises stress is not identical to the maximum principal stress because of the contact pressure. In the ﬁgure we notice the discontinuity in the stress slope at the transition point (around W/D = 3.45).

Kvl M 8

7

between

point B

point A

6

and B

Point C

5

4 W/D

1

2

3

4

Figure 5: Stress concentration factor as a function of width to diameter ratio, under plane stress condition (FEA results).

3.2 Inﬂuence from Poisson’s ratio and plane stress/plane strain assumption

It is well known that Poisson’s ratio has an inﬂuence on the stress concentration factor value, see e.g. Dally & Riley (1991) or more recently Pedersen (2018). The previously shown results in this paper have all been found using the Poisson’s ration ν = 0.3. For the case of W/D = 2.2

7

and large head distance we show in Figure 6 the stress concentration factor as a function of Poisson’s ratio for the case of plane stress and plane strain.

Kvl M

4.4 Plane stress Plane strain

4.2

4.0

3.8 ν

0

0.1

0.2

0.3

0.4

0.5

Figure 6: Inﬂuence on stress concentration factor from Poisson’s ratio and assumption of either plane stress or plane strain (W/D = 2.2). (FEA results).

For the speciﬁc design with W/D = 2.2 the maximum stress is at the hole rim at a plane with no contact, i.e., we have a uniaxial stress state for the plane stress assumption and therefore no inﬂuence from Poisson’s ratio on the stress concentration factor. With the assumption of plane strain Figure 6 shows an inﬂuence from Poisson’s ratio due to the now double axial stress state at the point with maximum stress. We also see as expected that the results for plane strain and stress are identical for ν = 0. If a different design is selected, W/D = 8.0, we have that the maximum stress is found at the top of contact (point C), i.e., here we do not have unidirectional stress, independent of the 2D analysis type selected. Figure 7 shows the results.

Kvl M

6.0

5.6

Plane stress 5.2 Plane strain

4.8 ν

0

0.1

0.2

0.3

0.4

0.5

Figure 7: Inﬂuence on stress concentration factor from Poisson’s ratio and assumption of either plane stress or plane strain (W/D = 8.0). (FEA results).

For the case shown in Figure 7 we have with the assumption of plane stress a double axial stress state at the point with maximum stress and as seen the stress concentration factor depends on Poissons ratio to a large extend. For the plane strain case we have a general stress state at the point of maximum stress and as indicated a large inﬂuence on the maximum stress from Poisson’s ratio. In many cases it is therefore essential that the Poisson’s ratio is speciﬁed for evaluating the correct stress concentration factor. In the remaining part of the paper we will assume ν = 0.3.

8

3.3 Inﬂuence from friction on the maximum stress concentration

The inclusion of Coulomb friction in the contact modelling has shown, for the examples of the present paper, to have no inﬂuence on the result. The examples include both assumptions of either plane stress or plane strain and design cases where the point of maximum stress is a point with uni-axial, double axial or a general stress state. The inclusion of friction in the contact modelling is therefore omitted in the remaining examples.

3.4 Clearance and comparison to Hertzian pressure

Clearance between pin and lug can have a large inﬂuence on the maximum stress in the connection. Discussion on this inﬂuence can e.g. be found in Pedersen (2007) and more recent in Strozzi et al. (2011) where also the load direction relative to the lug is discussed.

The selected 2D modelling (with Poisson’s ration ν = 0.3) has a large inﬂuence on the maximum stress found. With plane stress assumption, as used in Strozzi et al. (2011), the maximum stress is found on the hole rim in the lug. However, with a plane strain assumption the maximum stress (von Mises) is for low values of the applied force found in front of the load direction but inside the lug, this is in agreement with the traditional Hertzian stress. The stress situation is in the pin/lug connection slightly different than the traditional Hertzian contact because the stress in the lug has to go around the hole into the support.

The Hertzian stress, see e.g. Gladwell (1980) or Norton (2006), is based on an assumption of plane strain, i.e. an inﬁnite half space. For parallel cylindrical contact, as for the pin/lug connection, the half-width of contact is

2

F

Lc = π (m1 + m2)φ T

(10)

where F is the contact force (load on pin), m1 and m2 are material constants which for the case of same material for lug and pin are given by

1 − ν2

m1 = m2 = E

(11)

with E being the modulus of elasticity. φ is a geometric constant expressing the radius difference for the two bodies. The expression for φ can also be found in the references Gladwell (1980) and Norton (2006). For the pinned connection we have a concave contact, and assuming the hole diameter in the lug to be given by D and the pin diameter by D − δ where δ is the diametral clearance the constant φ is deﬁned as

φ = D(D − δ) ≈ D2 (12)

δ

δ

where the last approximation holds for a relatively small clearance. T is the thickness (as deﬁne previously), i.e. the length of the contact zone in the axial direction of the pin. Perpendicular to this direction the distribution of the pressure is semi-elliptical and with the half-width known the maximum pressure at the centre is given by.

2F /T

pmax = πLc

(13)

9

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