Composite Structures 86 (2008) 308–313

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Analysis of stress distribution around pin loaded holes in orthotropic plates

O. Aluko a, H.A. Whitworth b,*

a Department of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, United Statesb Department of Mechanical Engineering, College of Engineering, Architecture and Computer Sciences, Howard University, Washington, DC 20059, United States

a r t i c l e i n f o a b s t r a c t

Article history:Available online 7 June 2008

Keywords:Composite materialsPinned jointsFiber reinforced laminatesContact stressesFrictional effects

0263-8223/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.compstruct.2008.06.001

* Corresponding author. Tel.: +1 202 806 6600; faxE-mail address: hwhitworth@howard.edu (H.A. W

An analysis was performed to evaluate the stress distribution in composite pin loaded joints. The analysisinvolves specification of displacement expressions in the form of a trigonometric series that satisfy theboundary conditions for the contact region in terms of a set of undetermined coefficients. Based on thisassumed distribution, the Lekhnitskii complex variable approach is used to obtain the stress functionsneeded to evaluate the contact stresses within the joint. Unknown coefficients in the displacementexpression were obtained by assuming coulomb friction within the contact region and evaluating the dis-placement at discrete points within this region. Material properties of carbon fiber reinforced plastic lam-inates were used for this study and the stress distribution for different values of coefficient of frictionanalyzed. The analysis revealed that friction affects the stress distribution around the hole boundarywith, in general, the peak stresses varying with increasing values of frictional coefficient.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Due to their high specific strength and specific stiffness, com-posite materials and thus composite joints are finding increasingapplication in a variety of engineering structures and hence havereceived much attention by many investigators [1–18]. Mechanicaljoints are the only form of joints that permit disassembly withoutcausing any damage to the structure. However, mechanical joiningthat requires rivets and/or bolt through holes result in stress con-centrations, ultimately leading to possible failure. Accurate andproper design of mechanically fastened composite joints, requirethe determination of the stress distribution at pin-plate contactsurface and within the plate followed by the use of an appropriatefailure theory to determine the strength of the joint.

There are two basic approaches used to analyze the problem ofstress distribution in composites with stress concentration. Thefirst and more mathematically rigorous approach is based on theanisotropic elasticity method of Lekhnitskii [4]. The second ap-proach is using numerical techniques such as the finite elementmethod. The elasticity solutions generally assume a pinned con-nection rather than a bolted connection due to the two-dimen-sional limitation of the elasticity solutions. Therefore, boltclamping force and interlaminar effects in composites, for example,are not accounted for with any of these elasticity solutions.

In evaluating the stress distribution in composite joints, twoinfinite plate solutions were superposed by de Jong [9] to approx-imate the finite geometry effects of orthotropic plates. de Jong

ll rights reserved.

: +1 202 483 1396.hitworth).

[9,10] also showed the simultaneous influence of friction and loaddirection on the stresses in orthotropic plates with a single pin-loaded hole. Hyer and Klang [12] modeled the pin and its interac-tion with the hole by including pin-elasticity. They showed thatpin-elasticity is rather unimportant in stress prediction comparedto clearance, friction and elastic properties of the plate material.Zhang and Ueng [11] presented a compact solution for a rigid, per-fectly fitting-pin loading an infinite plate. They used a certain dis-placement expressions for the edge of the hole that satisfy thephysical displacement requirements in conjunction with Lekhnit-skii’s complex functions to evaluate the stress distribution in thecontact region.

Using the finite element analysis and a failure area indexmethod, Ryu et al. [14] were able to predict the failure loads of car-bon/epoxy composite laminates. In their analysis, the pin-plateinterface was assumed to be frictionless and the results comparedwith experimental data. Lessard et al. [15] evaluated failure ofmechanically fastened joints made from AS4-3501-6 graphiteepoxy laminates. They tested laminates of varying geometric ratiosin order to determine failure strengths using linear and non-linearfinite element models. Whitworth et al. [16] also used finite ele-ment analysis and the Chang–Scott–Springer characteristic curvemodel [18] to evaluate the stress distribution around the fastenerhole in composites. The Yamada–Sun failure criterion [19] wasused to evaluate joint strength and good agreement observed be-tween the theory and experimental data on bearing strength andfailure modes for graphite/epoxy laminates.

In this analysis, a method is proposed to evaluate the stress dis-tribution around loaded holes in orthotropic plates using the meth-od proposed by Zhang and Ueng [11]. The solution involves the

O. Aluko, H.A. Whitworth / Composite Structures 86 (2008) 308–313 309

determination of the complex stress functions used to calculate thecontact stresses based on assumed displacement expressions thatsatisfy the displacement boundary conditions in the contact region.The plate is assumed to be infinite, the pin rigid and the coefficientof friction constant in the contact region. Additionally, it is as-sumed that under the action of the pin load, the circular hole de-forms into an ellipse. Numerical solutions are obtained fordifferent values of coefficient of friction.

2. Theoretical analysis of the joint

Fig. 1 represents the geometry of the pin-loaded hole for theorthotropic plate. The plate is assumed to be infinite with a holeof radius r equal to radius of the pin and the pin is assumed tobe acted on by a resultant force P causing displacement uo in thex-direction. For this case of zero clearance, contact between thepin and the plate spans through half of the hole’s circumference.The boundary conditions (Fig. 1) can be expressed as follows:

Region I (no-slip region): �k 6 h 6 k

v ¼ 0 ð1Þu ¼ uo ð2Þsrh < grrr ð3Þ

Region II (slip region): 3p/2 6 h 6 �k and k 6 h 6 p/2

v ¼ 0 h ¼ p=2; 3p=2 ð4Þðuo � uÞ cos h ¼ v sin h 3p=2 6 h 6 �k and k 6 h 6 p=2 ð5Þs�r�h ¼ �gr�r�r 3p=2 6 h 6 �k and k 6 h 6 p=2 ð6Þ

Region III (no contact region): p/2 6 h 6 3p/2

r�r�r ¼ s�r�h ¼ 0 p=2 6 h 6 3p=2 ð7Þ

The displacements u and v that satisfy the boundary conditions inthe contact region can be expressed by the following trigonometryseries:

u ¼X4

i¼1

ui cos 2ih

v ¼X4

i¼1

vi sin 2ih

ð8Þ

Fig. 1. Regions on the pin/plate boundary.

where ui, vi (i = 1–4) are coefficients to be determined from theboundary conditions.

To determine these coefficients, displacements are prescribed ata discrete number of points within the contact region. Since thedisplacement expression for u contains four unknown coefficients,the solution process requires displacement to be prescribed at fourpoints within the contact region. Thus, in addition to the assumeddisplacement uo at h = 0, displacements ub, uc and ua were also pre-scribed at arbitrary points h = 30�, 45� and 90� within the contactregion. These displacements can be determined from the laminateproperties and the frictional condition between the pin and theplate. Substituting these prescribed displacements into Eq. (8),yields the following expressions for the unknown coefficients interms of the prescribed displacements:

u1 ¼uo

2� 1

6ua þ

23

ub

u2 ¼uo

4þ 1

4ua �

12

uc

u3 ¼ �13

ua �23

ub

u4 ¼uo

4þ 1

4ua þ

12

uc

ð9Þ

Similarly, the unknown coefficients vi can be obtained by satisfyingEq. (5) at arbitrary points within the contact region. In this analysis,points h = 25�, 30�, 45� and 60� were selected to yield

v1 ¼ 0:166667ua � 0:666667ub þ 9:37293� 10�17uc þ 1:5uo

v2 ¼ 0:833333ua � 1:33333ub þ 0:5uc þ 0:75uo

v3 ¼ 0:166667ua � 0:666667ub þ 1:0uc þ 0:5uo

v4 ¼ 0:25ua � 4:44089ub � 10�16 þ 0:5uc þ 0:25uo

ð10Þ

Lekhnitskii [4] has shown that if the known boundary displacementat the contour of the opening can be expressed in the form

u� ¼ ao þX1m¼1

famrm þ �amr�mg

v� ¼ bo þX1m¼1

fbmrm þ �bmr�mgð11Þ

and the components of the resultant forces that cause the displace-ment are given, then the stress functions can be expressed by thefollowing relations:

/1ðz1Þ ¼ A ln f1 þ �a1q2 � �b1p2 þ12xðibq2 þ ap2Þ

� �1

Df1

þ 1D

X1m¼2

ð�amq2 � �bmp2Þf�m1

/2ðz2Þ ¼ B ln f2 � �a1q1 � �b1p1 þ12xðibq1 þ ap1Þ

� �1

Df2

� 1D

X1m¼2

ð�amq1 � �bmp1Þf�m2

ð12Þ

In Eqs. (11) and (12), r = eih, bars represent conjugate values, am andbm are known coefficients that depend on the load distribution atthe opening edge, ao, bo are arbitrary constants and D, p1, p2, q1

and q2 are constants that depend on the property of the plate andfk is the mapping function given by

fk ¼zk �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2

k � lkr2 � r2q

r � ilkrk ¼ 1;2 ð13Þ

where lk(k = 1, 2) are the roots of characteristics equation [4]. Addi-tionally, the constants A and B of Eq. (12) can be obtained from thefollowing relations [4]:

310 O. Aluko, H.A. Whitworth / Composite Structures 86 (2008) 308–313

A ¼ Ppih

l1 �l1 þ l1l2 þ l1 �l2 � a12a22

� �l1l2 �l1 �l2

ðl1 � �l1Þðl1 � l2Þðl1 � �l2Þ

B ¼ Ppih

l2 �l2 þ l2l1 þ l2 �l1 � a12a22

� �l1l2 �l1 �l2

ðl2 � �l2Þðl2 � l1Þðl2 � �l1Þ

ð14Þ

where as previously indicated bars represent conjugate values, aij

are the laminate elastic compliance and h is the thickness of theplate which is unity in this analysis.

By expressing r in Eq. (11) in terms of trigonometric functiondefined by

cos nh ¼ rn þ r�n

2; sin nh ¼ rn � r�n

2ið15Þ

and comparing Eqs. (8) and (11), the stress functions of Eq. (12) canbe expressed as [13]

/1ðz1Þ ¼ A ln f1 þ1

2D½ðu1q2 � iv1p2Þf�2

1 þ ðu2q2 � iv2p2Þf�41

þ ðu3q2 � iv3p2Þf�61 þ ðu4q2 � iv4p2Þf�8

1 �

/2ðz2Þ ¼ B ln f2 �1

2D½ðu1q1 � iv1p1Þf�2

2 � ðu2q1 � iv2p1Þf�41

� ðu3q1 � iv3p1Þf�62 � ðu4q1 � iv4p1Þf�8

2 �

ð16Þ

The radial, hoop and tangential stresses can be expressed in termsof the stress functions as [4]

rrr ¼ 2Refðsin h� l1 cos hÞ2/01ðz1Þ þ ðsin h� l2 cos hÞ2/02ðz2Þgsrh ¼ 2Refðsin h� l1 cos hÞðcos hþ l1 sin hÞ/01ðz1Þ

þ ðsin h� l2 cos hÞðcos hþ l2 sin hÞ/02ðz2Þgrhh ¼ 2Refðl1 sin hþ cos hÞ2/01ðz1Þ þ ðl2 sin hþ cos hÞ2/02ðz2Þg

ð17Þ

Aluko [13] has shown that the real parts of Eq. (17) can be ex-pressed as

rrr ¼�1

ka11gr½a22u1ðnþ 1Þ � kð�a12ðu1 þ v1Þ � a11v1ðnþ kÞÞ� � P

pr

� �

� cos hþ �1ka11gr

½a22ðu1ðn� 1Þ þ 2u2ðnþ 1ÞÞ

� kða12ðu1 � 2u2 � v1 � 2v2Þ þ a11ð�2v2ðnþ kÞ þ v1ðn� kÞÞÞ�

� cos 3hþ �1ka11gr

½a22ð2u2ðn� 1Þ þ 3u3ðnþ 1Þ

� kða12ð2u2 � 3u3 � 2v2 � 3v3Þ þ a11ð�3v3ðnþ kÞ

þ 2v2ðn� kÞÞÞ� cos 5h�1

ka11gr½a22ð3u3ðn� 1Þ þ 4u4ðnþ 1ÞÞ

� kða12ð3u3 � 4u4 � 3v3 � 4v4Þ þ a11ð�4v4ðnþ kÞ

þ 3v3ðn� kÞÞÞ� cos 7h�4

ka11gr½a22u4ðn� 1Þ � kða12ðu4 � v4Þ

þ a11v4ðn� kÞÞ� cos 9h ð18Þ

srh ¼�1

ka11gr½a22u1ðn� 1Þ � kð�a12ðu1 þ v1Þ � a11v1ðnþ kÞÞ� þ P

pr

� �

� sin hþ 1ka11gr

½a22ðu1ðnþ 1Þ � 2u2ðnþ 1ÞÞ

� kða12ðu1 þ 2u2 � v1 þ 2v2Þ þ a11ð2v2ðnþ kÞ þ v1ðn� kÞÞÞ�

� sin 3hþ �1ka11gr

½�a22ð2u2ðn� 1Þ � 3u3ðnþ 1Þ

þ kða12ð2u2 þ 3u3 � 2v2 þ 3v3Þ þ a11ð3v3ðnþ kÞ

þ 2v2ðn� kÞÞÞ� sin 5h�1

ka11gr½�a22ð3u3ðn� 1Þ � 4u4ðnþ 1ÞÞ

þ kða12ð3u3 þ 4u4 � 3v3 þ 4v4Þ þ a11ð4v4ðnþ kÞ

þ 3v3ðn� kÞÞÞ� sin 7h4

ka11gr½a22u4ðn� 1Þ � kða12ðu4 � v4Þ

þ a11v4ðn� kÞÞ� sin 9h ð19Þ

rhh ¼ rhh1 þ rhh2 þ rhh3 þ rhh4 þ rhh5 ð20Þwhere

rhh1 ¼1

gkr2ahðða22ð1þ kÞnu1 þ kða12 þ a11kÞv1Þ cos 3h sin4 h

þ cos h sin4 hða22nu1 þ a22knu1 þ a12kv1 þ a11k2v1

þð�a22n3u1 þ a11k3v1 þ a12kðknu1 þ kv1 � n2v1ÞÞ cos 2h

�2ða22ðk� n2Þu1 þ a12kðku1 � nv1ÞÞ sin2 h

þk cos5 hðkða12ðnu1 � v1Þ þ a11ð�kþ n2Þv1Þ cos 2h

þ2ða22ðu1 þ 2ku1Þ þ a11nð2kþ 2k2 � n2Þv1 þ a12ðku1 þ 2k2u1

þnð�nu1 þ v1ÞÞÞ sin2 hÞ þ cos3 h sin2 hðð�a22ð�1þ k2Þnu1

þkð�a11kð�1þ 2kþ k2 � 2n2Þv1

þa12ð2knu1 þ v1 � 2kv1 � k2v1ÞÞÞ cos 2h

þ2ða22ð�1� 2kþ k2 þ 2n2Þu1 þ kða11ð�1þ k2Þnv1

þa12ðð�1� 2kþ k2Þu1 þ 2nv1ÞÞÞ sin2 hÞÞÞ

ð21Þ

rhh2 ¼1

gkr4ahðk2ða12ðnu2 � v2Þ þ a11ð�kþ n2Þv2Þ cos5 h cos 4h

þ 4kða22ðu2 þ 2ku2Þ þ a11nð2kþ 2k2 � n2Þv2

þ a12ðku2 þ 2k2u2 þ nð�nu2 þ v2ÞÞÞ cos5 h cos 2h sin2 h

� ða22ð�1þ k2Þnu2 þ kða11kð�1þ 2kþ k2 � 2n2Þv2

þ a12ð�2knu2 � v2 þ 2kv2 þ k2v2ÞÞÞ cos3 h cos 4h sin2 h

þ 4ða22ð�1� 2kþ k2 þ 2n2Þu2 þ kða11ð�1þ k2Þnv2

þ a12ðð�1� 2kþ k2Þu2 þ 2nv2ÞÞÞ cos3 h cos 2h sin4 h

� ða22nð�2� 2kþ n2Þu2 � kða11kð2þ kÞv2

þ a12ðknu2 þ 2v2 þ kv2 � n2v2ÞÞÞ cos h cos 4h sin4 h

þ ða22ð�kþ n2Þu2 þ a12kð�ku2 þ nv2ÞÞ sin5 h sin 4hÞð22Þ

rhh3 ¼1

gkr6ahðk2ða12ðnu3 � v3Þ þ a11ð�kþ n2Þv3Þ cos5 h cos 6h

þ 2kða22ðu3 þ 2ku3Þ þ a11nð2kþ 2k2 � n2Þv3

þ a12ðku3 þ 2k2u3 þ nð�nu3 þ v3ÞÞÞ cos5 hð1þ 2 cos 4hÞ sin2 h

� ða22ð�1þ k2Þnu3 þ kða11kð�1þ 2kþ k2 � 2n2Þv3

þ a12ð�2knu3 � v3 þ 2kv3 þ k2v3ÞÞÞ cos3 h cos 6h sin2 h

þ 2ða22ð�1� 2kþ k2 þ 2n2Þu3 þ kða11ð�1þ k2Þnv3

þ a12ðð�1� 2kþ k2Þu3 þ 2nv3ÞÞÞ cos3 hð1þ 2 cos 4hÞ sin4 h

� ða22nð�2� 2kþ n2Þu3 � kða11kð2þ kÞv3

þ a12ðknu3 þ 2v3 þ kv3 � n2v3ÞÞÞ cos h cos 6h sin4 h

þ ða22ð�kþ n2Þu3 þ a12kð�ku3 þ nv3ÞÞ sin5 h sin 6hÞ

ð23Þ

rhh4 ¼1

gkr8ahðk2ða12ðnu4 � v4Þ þ a11ð�kþ n2Þv4Þcos5 hcos 8h

þ4kða22ðu4 þ2ku4Þ þ a11nð2kþ2k2 � n2Þv4

þ a12ðku4 þ 2k2u4 þ nð�nu4 þ v4ÞÞÞcos5 hðcos 2hþ cos 6hÞ� sin2 h� ða22ð�1þ k2Þnu4 þ kða11kð�1þ2kþ k2 � 2n2Þv4

þ a12ð�2knu4 � v4 þ 2kv4 þ k2v4ÞÞÞcos3 hcos 8hsin2 h

þ4ða22ð�1� 2kþ k2 þ 2n2Þu4 þ kða11ð�1þ k2Þnv4

þ a12ðð�1�2kþ k2Þu4 þ2nv4ÞÞÞcos3 hðcos 2hþ cos 6hÞsin4 h

� ða22nð�2�2kþ n2Þu4 � kða11kð2þ kÞv4

þ a12ðknu4 þ2v4 þ kv4 � n2v4ÞÞÞcoshcos 8hsin4 h

þ ða22ð�kþ n2Þu4 þ a12kð�ku4 þ nv4ÞÞsin5 hsin 8hÞ

ð24Þ

Table 1Laminate Properties [9]

Laminate [±45�]s [04�/±45�]s

Ex (GPa) 20.3 111.7Ey (GPa) 20.3 20.4Gxy (GPa) 27.7 16.9mxy 0.728 0.663n 1.130 3.156k 1 2.340

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

00 20 40 60 80 100

Present

de Jong

=0

=0.2

=0.4

rr

b

σσ

θ

ηη

η

Fig. 2. Radial stress for [±45�]s laminate.

θ

O. Aluko, H.A. Whitworth / Composite Structures 86 (2008) 308–313 311

rhh5 ¼a11

8a22pr1ah

P cos hð7a22 þ 6a22kþ 8a12k2 � a22k2 � 3a22n2

�4a22ð2þ 2k� n2Þ cos 2hþ a22ð1þ 2kþ k2 � n2Þ cos 4hÞð25Þ

In Eqs. (18)–(25), n and k can be obtained through the followingrelations [4]:

k ¼ �l1l2 ¼a22

a11

12

ð26Þ

n ¼ �iðl1 þ l2Þ ¼ 2 kþ a12

a11

þ a66

a11

� �12

ð27Þ

As stated previously, in this analysis it is assumed that the holedeforms as an ellipse due to the application of the pin load. It wasshown [11] that this assumption requires an additional term to beadded to the hoop stress to account for this deformation. This termcan be expresses as [13]

rhh6 ¼1ah

uo k � 1a11þ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1

a11a22

sk

!ðkþ nÞ cos2 h

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1

a11a22

sk2 þ 1

a11ð1þ nÞðkþ nÞ

!sin2 h

!!,

1a11

nð1þ kþ nÞr

ð28Þ

Thus the hoop stress can be obtained by adding this term to Eq. (20)and expressed as

rhh ¼ rhh1 þ rhh2 þ rhh3 þ rhh4 þ rhh5 þ rhh6 ð29Þ

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 20 40 60 80 100

Present

de Jong

η=0.4

η=0.2

η=0

rr

b

σσ

Fig. 3. Radial stress for [04/±45�]s laminate.

3. Determination of constants ui and vi

Assuming that the coefficient of friction g, is constant through-out the contact boundary, from Eq. (6), satisfaction of tractionboundary condition at discrete points h = 25�, 50�, 75� and 90� re-sults in the following relations:

srhð25�Þ þ grrrð25�Þ ¼ 0srhð50�Þ þ grrrð50�Þ ¼ 0srhð75�Þ þ grrrð75�Þ ¼ 0srhð90�Þ ¼ 0

ð30Þ

Using Eqs. (9), (10), (18) and (19), Eq. (30) can be solved to yieldvalues of uo, ua, ub and uc in terms of material properties and coef-ficient of friction. Then, from Eqs. (9) and (10), the unknown dis-placement coefficients ui and vi can be determined and theresults substituted into Eqs. (18), (19) and (29) to yield the valuesof the radial, tangential and hoop stresses, respectively. Finally, theangle k of Fig. 1 that describes the boundary between the slip andno-slip regions can be found by determining the value of k that sat-isfy the root of Eq. (6).

4. Results

In this investigation, properties of [±45�]s and [04�/ ± 45�]s car-bon fiber reinforced plastic laminates [9] presented in Table 1 wereused to evaluate joint stress distribution for friction coefficient val-ues of 0.0, 0.2 and 0.4, respectively. The resulting stress distribu-tion are presented in terms of a dimensionless ratio obtained bynormalizing these stresses by the bearing stress, rb = P/2r, for aplate of unit thickness. The results are displayed in Figs. 2–7. Alsoshown in these figures for comparison are the results generated inRef. [9].

In this analysis, a computer code was written using a Mathem-atica package that utilizes Newton’s method of iteration to deter-mine the no slip region. From this analysis, the value of the anglek was found to be zero for all the values of coefficient of frictiontested. Similar results were also obtained by de Jong [9].

Figs. 2 and 3 show the normal stress, rrr for the [±45�]s and[04� ± 45�]s laminates. As can be seen from these figures, the mag-nitude of the peak stresses decreases with increasing frictional va-lue and the location also varies with increased friction. For g = 0,the maximum stress occurs at h = 0, beyond g = 0.2, the maximumoccurs at larger values of h.

The shear stress distribution, srh, at the pin-hole boundary forthe [±45�]s and [04� ± 45�]s laminates are shown in Figs. 4 and 5.From these figures it can be observed that the magnitude of thepeak stress increases with increased value of the friction coeffi-cient. Unlike de Jong [9], the present results show peaks occurringat two different locations within the contact region for the [±45�]s

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80 100

Present

de Jong

η=0.4

η=0.2

η=0

b

θθσσ

θ

Fig. 6. Hoop stress for [±45�]s laminate.

-0. 4

-0. 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 0 100

Present

de Jong

=0.4

=0.2

=0

b

θθ σ σ

θ

η

η

η

Fig. 7. Hoop stress for [04/±45�]s laminate.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 20 40 60 80 100

Present

de Jong

η=0.4

η=0.2

r

b

θτσ

θ

Fig. 4. Shear stress for [±45�]s laminate.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 50 100

Present

de Jong

η=0.4

η=0.2r

b

θτσ

θ

Fig. 5. Shear stress for [04/±45�]s laminate.

312 O. Aluko, H.A. Whitworth / Composite Structures 86 (2008) 308–313

laminate. However, while friction appears to influence the magni-tude of the stress, it does not appear to influence the location atwhich the peak occurs.

Figs. 6 and 7 show the hoop stress, rhh, for the laminates ana-lyzed. As can be seen from these figures, the hoop stress initiallyincreases with increasing values of h. However, for the [±45�]s lam-inate, this stress is compressive in the region h = 0 for all frictionalvalues investigated with magnitude becoming increasingly tensileas h increases.

5. Conclusion

An analysis is presented to evaluate stress distributions in com-posite pin-loaded joints. The approach involves specification of dis-placement expressions in the form of a trigonometric series thatsatisfy the boundary conditions for the contact region in terms ofa set of undetermined coefficients. Numerical results for [±45]s

and [04/ ± 45�]s laminates indicate that friction has a significantinfluence on the radial, shear and hoop stresses for these lami-nates. In all cases, the maximum radial stress decreases withincreasing value of the frictional coefficient. In addition, the loca-tion of this maximum also varies with increased friction coeffi-cient. On the other hand, the shear stress distribution exhibits apeak stress that increases with increased value of the friction coef-ficient. For the hoop stress, the [±45�]s laminate experienced an ini-tial compressive region followed by a tensile region and, in thetensile region, the peak stress and its location varied with in-creased friction coefficient.

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