Analysis of stress distribution around pin loaded holes in orthotropic plates

Composite Structures 86 (2008) 308–313

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Analysis of stress distribution around pin loaded holes in orthotropic plates O. Aluko a, H.A. Whitworth b,* a b

Department of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, United States 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

Article history: Available online 7 June 2008 Keywords: Composite materials Pinned joints Fiber reinforced laminates Contact stresses Frictional effects

a b s t r a c t An analysis was performed to evaluate the stress distribution in composite pin loaded joints. The analysis involves specification of displacement expressions in the form of a trigonometric series that satisfy the boundary conditions for the contact region in terms of a set of undetermined coefficients. Based on this assumed distribution, the Lekhnitskii complex variable approach is used to obtain the stress functions needed to evaluate the contact stresses within the joint. Unknown coefficients in the displacement expression were obtained by assuming coulomb friction within the contact region and evaluating the displacement at discrete points within this region. Material properties of carbon fiber reinforced plastic laminates were used for this study and the stress distribution for different values of coefficient of friction analyzed. The analysis revealed that friction affects the stress distribution around the hole boundary with, in general, the peak stresses varying with increasing values of frictional coefficient. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Due to their high specific strength and specific stiffness, composite materials and thus composite joints are finding increasing application in a variety of engineering structures and hence have received much attention by many investigators [1–18]. Mechanical joints are the only form of joints that permit disassembly without causing any damage to the structure. However, mechanical joining that requires rivets and/or bolt through holes result in stress concentrations, ultimately leading to possible failure. Accurate and proper design of mechanically fastened composite joints, require the determination of the stress distribution at pin-plate contact surface and within the plate followed by the use of an appropriate failure theory to determine the strength of the joint. There are two basic approaches used to analyze the problem of stress distribution in composites with stress concentration. The first and more mathematically rigorous approach is based on the anisotropic elasticity method of Lekhnitskii [4]. The second approach is using numerical techniques such as the finite element method. The elasticity solutions generally assume a pinned connection rather than a bolted connection due to the two-dimensional limitation of the elasticity solutions. Therefore, bolt clamping 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, two infinite plate solutions were superposed by de Jong [9] to approximate the finite geometry effects of orthotropic plates. de Jong * Corresponding author. Tel.: +1 202 806 6600; fax: +1 202 483 1396. E-mail address: [email protected] (H.A. Whitworth). 0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.06.001

[9,10] also showed the simultaneous influence of friction and load direction on the stresses in orthotropic plates with a single pinloaded hole. Hyer and Klang [12] modeled the pin and its interaction with the hole by including pin-elasticity. They showed that pin-elasticity is rather unimportant in stress prediction compared to clearance, friction and elastic properties of the plate material. Zhang and Ueng [11] presented a compact solution for a rigid, perfectly fitting-pin loading an infinite plate. They used a certain displacement expressions for the edge of the hole that satisfy the physical displacement requirements in conjunction with Lekhnitskii’s complex functions to evaluate the stress distribution in the contact region. Using the finite element analysis and a failure area index method, Ryu et al. [14] were able to predict the failure loads of carbon/epoxy composite laminates. In their analysis, the pin-plate interface was assumed to be frictionless and the results compared with experimental data. Lessard et al. [15] evaluated failure of mechanically fastened joints made from AS4-3501-6 graphite epoxy laminates. They tested laminates of varying geometric ratios in order to determine failure strengths using linear and non-linear finite element models. Whitworth et al. [16] also used finite element analysis and the Chang–Scott–Springer characteristic curve model [18] to evaluate the stress distribution around the fastener hole in composites. The Yamada–Sun failure criterion [19] was used to evaluate joint strength and good agreement observed between the theory and experimental data on bearing strength and failure modes for graphite/epoxy laminates. In this analysis, a method is proposed to evaluate the stress distribution around loaded holes in orthotropic plates using the method proposed by Zhang and Ueng [11]. The solution involves the

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determination of the complex stress functions used to calculate the contact stresses based on assumed displacement expressions that satisfy the displacement boundary conditions in the contact region. The plate is assumed to be infinite, the pin rigid and the coefficient of friction constant in the contact region. Additionally, it is assumed that under the action of the pin load, the circular hole deforms into an ellipse. Numerical solutions are obtained for different values of coefficient of friction. 2. Theoretical analysis of the joint Fig. 1 represents the geometry of the pin-loaded hole for the orthotropic plate. The plate is assumed to be infinite with a hole of radius r equal to radius of the pin and the pin is assumed to be acted on by a resultant force P causing displacement uo in the x-direction. For this case of zero clearance, contact between the pin 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

v¼0

ð1Þ

u ¼ uo

ð2Þ ð3Þ

srh < grrr Region II (slip region): 3p/2 6 h 6 k and k 6 h 6 p/2

v¼0

h ¼ p=2; 3p=2

ðuo  uÞ cos h ¼ v sin h

srh ¼ grrr

ð4Þ 3p=2 6 h 6 k and k 6 h 6 p=2 ð5Þ

3p=2 6 h 6 k and k 6 h 6 p=2

ð6Þ

ð7Þ

The displacements u and v that satisfy the boundary conditions in the contact region can be expressed by the following trigonometry series:

u¼

4 X

v¼

ð8Þ vi sin 2ih

ð9Þ

Similarly, the unknown coefficients vi can be obtained by satisfying Eq. (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  1017 uc þ 1:5uo v2 ¼ 0:833333ua  1:33333ub þ 0:5uc þ 0:75uo v4 ¼ 0:25ua  4:44089ub  1016 þ 0:5uc þ 0:25uo ð10Þ Lekhnitskii [4] has shown that if the known boundary displacement at the contour of the opening can be expressed in the form

u ¼ ao þ

ui cos 2ih

i¼1 4 X

uo 1 2  ua þ ub 3 2 6 uo 1 1 u2 ¼ þ ua  uc 2 4 4 1 2 u3 ¼  ua  ub 3 3 uo 1 1 u4 ¼ þ ua þ uc 2 4 4 u1 ¼

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

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

rrr ¼ srh ¼ 0 p=2 6 h 6 3p=2

where ui, vi (i = 1–4) are coefficients to be determined from the boundary conditions. To determine these coefficients, displacements are prescribed at a discrete number of points within the contact region. Since the displacement expression for u contains four unknown coefficients, the solution process requires displacement to be prescribed at four points within the contact region. Thus, in addition to the assumed displacement uo at h = 0, displacements ub, uc and ua were also prescribed at arbitrary points h = 30°, 45° and 90° within the contact region. These displacements can be determined from the laminate properties and the frictional condition between the pin and the plate. Substituting these prescribed displacements into Eq. (8), yields the following expressions for the unknown coefficients in terms of the prescribed displacements:

1 X  m rm g fam rm þ a m¼1



v ¼ bo þ

1 X

ð11Þ

m rm g fbm rm þ b

m¼1

i¼1

and the components of the resultant forces that cause the displacement are given, then the stress functions can be expressed by the following relations:

  1 p þ 1 xðibq þ ap Þ 1  1 q2  b /1 ðz1 Þ ¼ A ln f1 þ a 2 2 2 2 Df1 1 X 1 m m p Þf  m q2  b þ ða 2 1 D m¼2   1 p þ 1 xðibq þ ap Þ 1  1 q1  b /2 ðz2 Þ ¼ B ln f2  a 1 1 1 2 Df2 1 1X m p Þfm  m q1  b  ða 1 2 D m¼2

ð12Þ

In Eqs. (11) and (12), r = eih, bars represent conjugate values, am and bm are known coefficients that depend on the load distribution at the opening edge, ao, bo are arbitrary constants and D, p1, p2, q1 and q2 are constants that depend on the property of the plate and fk is the mapping function given by

fk ¼

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

zk 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2k  lk r 2  r2 r  ilk r

k ¼ 1; 2

ð13Þ

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

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

 

l1 l 1 þ l1 l2 þ l1 l 2  aa1222 l1 l2 l 1 l 2  1 Þðl1  l2 Þðl1  l  2Þ ðl1  l   a12    1l 2 P l2 l2 þ l2 l1 þ l2 l1  a22 l1 l2 l B¼  2 Þðl2  l1 Þðl2  l  1Þ pih ð l2  l

A¼

P pih

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

ð14Þ

rhh1 ¼

ð20Þ

1 2 4 ðða22 ð1 þ kÞnu1 þ kða12 þ a11 kÞv1 Þ cos 3h sin h gkr ah 2 4 þ cos h sin hða22 nu1 þ a22 knu1 þ a12 kv1 þ a11 k v1 3

þða22 n3 u1 þ a11 k v1 þ a12 kðknu1 þ kv1  n2 v1 ÞÞ cos 2h

where as previously indicated bars represent conjugate values, aij are the laminate elastic compliance and h is the thickness of the plate which is unity in this analysis. By expressing r in Eq. (11) in terms of trigonometric function defined by

2

2ða22 ðk  n2 Þu1 þ a12 kðku1  nv1 ÞÞ sin h þk cos5 hðkða12 ðnu1  v1 Þ þ a11 ðk þ n2 Þv1 Þ cos 2h 2

2

cos nh ¼

rn þ rn 2

;

sin nh ¼

rn  rn 2i

2

þkða11 kð1 þ 2k þ k  2n2 Þv1 2

þa12 ð2knu1 þ v1  2kv1  k v1 ÞÞÞ cos 2h 2

ð16Þ

rhh2 ¼

ð21Þ

1 4 2 ðk ða12 ðnu2  v2 Þ þ a11 ðk þ n2 Þv2 Þ cos5 h cos 4h gkr ah 2

þ 4kða22 ðu2 þ 2ku2 Þ þ a11 nð2k þ 2k  n2 Þv2 2

2

þ a12 ðku2 þ 2k u2 þ nðnu2 þ v2 ÞÞÞ cos5 h cos 2h sin h

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

2

2

2

 ða22 ð1 þ k Þnu2 þ kða11 kð1 þ 2k þ k  2n Þv2 2

2

þ a12 ð2knu2  v2 þ 2kv2 þ k v2 ÞÞÞ cos3 h cos 4h sin h

rrr ¼ 2Refðsin h  l1 cos hÞ2 /01 ðz1 Þ þ ðsin h  l2 cos hÞ2 /02 ðz2 Þg srh ¼ 2Refðsin h  l1 cos hÞðcos h þ l1 sin hÞ/01 ðz1 Þ þ ðsin h  l2 cos hÞðcos h þ l2 sin hÞ/02 ðz2 Þg rhh ¼ 2Refðl1 sin h þ cos hÞ2 /01 ðz1 Þ þ ðl2 sin h þ cos hÞ2 /02 ðz2 Þg

2

2

2

þ 4ða22 ð1  2k þ k þ 2n Þu2 þ kða11 ð1 þ k Þnv2 2

4

þ a12 ðð1  2k þ k Þu2 þ 2nv2 ÞÞÞ cos3 h cos 2h sin h 2

 ða22 nð2  2k þ n Þu2  kða11 kð2 þ kÞv2 4

þ a12 ðknu2 þ 2v2 þ kv2  n2 v2 ÞÞÞ cos h cos 4h sin h

ð17Þ

5

þ ða22 ðk þ n2 Þu2 þ a12 kðku2 þ nv2 ÞÞ sin h sin 4hÞ

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

srh ¼

2

þ2ða22 ð1  2k þ k þ 2n2 Þu1 þ kða11 ð1 þ k Þnv1

2 2 þa12 ðð1  2k þ k Þu1 þ 2nv1 ÞÞÞ sin hÞÞÞ

/1 ðz1 Þ ¼ A ln f1 þ

rrr ¼

2

2

þnðnu1 þ v1 ÞÞÞ sin hÞ þ cos3 h sin hðða22 ð1 þ k Þnu1 ð15Þ

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

1 4 ½ðu1 q2  iv1 p2 Þf2 1 þ ðu2 q2  iv2 p2 Þf1 2D 8 þ ðu3 q2  iv3 p2 Þf6 1 þ ðu4 q2  iv4 p2 Þf1  1 4 /2 ðz2 Þ ¼ B ln f2  ½ðu1 q1  iv1 p1 Þf2 2  ðu2 q1  iv2 p1 Þf1 2D 8  ðu3 q1  iv3 p1 Þf6 2  ðu4 q1  iv4 p1 Þf2 

2

þ2ða22 ðu1 þ 2ku1 Þ þ a11 nð2k þ 2k  n2 Þv1 þ a12 ðku1 þ 2k u1



 1 P ½a22 u1 ðn þ 1Þ  kða12 ðu1 þ v1 Þ  a11 v1 ðn þ kÞÞ  ka11 gr pr 1  cos h þ ½a22 ðu1 ðn  1Þ þ 2u2 ðn þ 1ÞÞ ka11 gr  kða12 ðu1  2u2  v1  2v2 Þ þ a11 ð2v2 ðn þ kÞ þ v1 ðn  kÞÞÞ 1  cos 3h þ ½a22 ð2u2 ðn  1Þ þ 3u3 ðn þ 1Þ ka11 gr  kða12 ð2u2  3u3  2v2  3v3 Þ þ a11 ð3v3 ðn þ kÞ 1 þ 2v2 ðn  kÞÞÞ cos 5h ½a22 ð3u3 ðn  1Þ þ 4u4 ðn þ 1ÞÞ ka11 gr  kða12 ð3u3  4u4  3v3  4v4 Þ þ a11 ð4v4 ðn þ kÞ 4 þ 3v3 ðn  kÞÞÞ cos 7h ½a22 u4 ðn  1Þ  kða12 ðu4  v4 Þ ka11 gr þ a11 v4 ðn  kÞÞ cos 9h ð18Þ  1 P ½a22 u1 ðn  1Þ  kða12 ðu1 þ v1 Þ  a11 v1 ðn þ kÞÞ þ ka11 gr pr 1 ½a22 ðu1 ðn þ 1Þ  2u2 ðn þ 1ÞÞ  sin h þ ka11 gr  kða12 ðu1 þ 2u2  v1 þ 2v2 Þ þ a11 ð2v2 ðn þ kÞ þ v1 ðn  kÞÞÞ 1 ½a22 ð2u2 ðn  1Þ  3u3 ðn þ 1Þ  sin 3h þ ka11 gr þ kða12 ð2u2 þ 3u3  2v2 þ 3v3 Þ þ a11 ð3v3 ðn þ kÞ 1 ½a22 ð3u3 ðn  1Þ  4u4 ðn þ 1ÞÞ þ 2v2 ðn  kÞÞÞ sin 5h ka11 gr þ kða12 ð3u3 þ 4u4  3v3 þ 4v4 Þ þ a11 ð4v4 ðn þ kÞ 4 þ 3v3 ðn  kÞÞÞ sin 7h ½a22 u4 ðn  1Þ  kða12 ðu4  v4 Þ ka11 gr ð19Þ þ a11 v4 ðn  kÞÞ sin 9h

rhh3 ¼

ð22Þ

1 6 2 ðk ða12 ðnu3  v3 Þ þ a11 ðk þ n2 Þv3 Þ cos5 h cos 6h gkr ah 2

þ 2kða22 ðu3 þ 2ku3 Þ þ a11 nð2k þ 2k  n2 Þv3 2

2

þ a12 ðku3 þ 2k u3 þ nðnu3 þ v3 ÞÞÞ cos5 hð1 þ 2 cos 4hÞ sin h 2

2

2

 ða22 ð1 þ k Þnu3 þ kða11 kð1 þ 2k þ k  2n Þv3 2

2

þ a12 ð2knu3  v3 þ 2kv3 þ k v3 ÞÞÞ cos3 h cos 6h sin h 2

2

2

þ 2ða22 ð1  2k þ k þ 2n Þu3 þ kða11 ð1 þ k Þnv3 2

4

þ a12 ðð1  2k þ k Þu3 þ 2nv3 ÞÞÞ cos3 hð1 þ 2 cos 4hÞ sin h 2

 ða22 nð2  2k þ n Þu3  kða11 kð2 þ kÞv3 4

þ a12 ðknu3 þ 2v3 þ kv3  n2 v3 ÞÞÞ cos h cos 6h sin h

5 þ ða22 ðk þ n2 Þu3 þ a12 kðku3 þ nv3 ÞÞ sin h sin 6hÞ



rhh4 ¼

ð23Þ

1 8 2 ðk ða12 ðnu4  v4 Þ þ a11 ðk þ n2 Þv4 Þ cos5 h cos 8h gkr ah 2

þ 4kða22 ðu4 þ 2ku4 Þ þ a11 nð2k þ 2k  n2 Þv4 2

þ a12 ðku4 þ 2k u4 þ nðnu4 þ v4 ÞÞÞ cos5 hðcos 2h þ cos 6hÞ 2

2

2

 sin h  ða22 ð1 þ k Þnu4 þ kða11 kð1 þ 2k þ k  2n2 Þv4 2

2

þ a12 ð2knu4  v4 þ 2kv4 þ k v4 ÞÞÞ cos3 h cos 8h sin h 2

2

þ 4ða22 ð1  2k þ k þ 2n2 Þu4 þ kða11 ð1 þ k Þnv4 2

4

þ a12 ðð1  2k þ k Þu4 þ 2nv4 ÞÞÞ cos3 hðcos 2h þ cos 6hÞ sin h  ða22 nð2  2k þ n2 Þu4  kða11 kð2 þ kÞv4 4

þ a12 ðknu4 þ 2v4 þ kv4  n2 v4 ÞÞÞ cos h cos 8h sin h

5 þ ða22 ðk þ n2 Þu4 þ a12 kðku4 þ nv4 ÞÞ sin h sin 8hÞ

ð24Þ

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

rhh5 ¼



a11 1 2 2 P cos hð7a22 þ 6a22 k þ 8a12 k  a22 k  3a22 n2 8a22 pr ah

2 4a22 ð2 þ 2k  n2 Þ cos 2h þ a22 ð1 þ 2k þ k  n2 Þ cos 4hÞ ð25Þ

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

12 a22 a11 

1 a12 a66 2 n ¼ iðl1 þ l2 Þ ¼ 2 k þ þ a11 a11

k ¼ l1 l2 ¼

Table 1 Laminate Properties [9] Laminate

[±45°]s

[04°/±45°]s

Ex (GPa) Ey (GPa) Gxy (GPa)

20.3 20.3 27.7 0.728 1.130 1

111.7 20.4 16.9 0.663 3.156 2.340

mxy n k

ð26Þ ð27Þ

0

As stated previously, in this analysis it is assumed that the hole deforms as an ellipse due to the application of the pin load. It was shown [11] that this assumption requires an additional term to be added to the hoop stress to account for this deformation. This term can be expresses as [13]

-0.2

rhh6

sffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 1 1 ¼ uo k  þ k ðk þ nÞ cos2 h ah a11 a11 a22 sffiffiffiffiffiffiffiffiffiffiffiffiffi ! !!, 1 1 2 2 k þ ð1 þ nÞðk þ nÞ sin h þ a11 a22 a11

1 nð1 þ k þ nÞr a11

θ 0

σ rr σb

20

80

η=0.4

-1.2

Present de Jong

η =0.2

-1

η=0

-1.4 Fig. 2. Radial stress for [±45°]s laminate.

ð29Þ

θ

0

3. Determination of constants ui and vi

-0.2 0

Assuming that the coefficient of friction g, is constant throughout the contact boundary, from Eq. (6), satisfaction of traction boundary condition at discrete points h = 25°, 50°, 75° and 90° results in the following relations:

-0.4

srh ð25 Þ þ grrr ð25 Þ ¼ 0 srh ð50 Þ þ grrr ð50 Þ ¼ 0 srh ð75 Þ þ grrr ð75 Þ ¼ 0 srh ð90 Þ ¼ 0

100

-0.6

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

60

-0.4

-0.8

ð28Þ

40

20

σ rr -0.6 σb -0.8

40

60

80

100

η=0.4

-1

ð30Þ

Using Eqs. (9), (10), (18) and (19), Eq. (30) can be solved to yield values of uo, ua, ub and uc in terms of material properties and coefficient of friction. Then, from Eqs. (9) and (10), the unknown displacement coefficients ui and vi can be determined and the results substituted into Eqs. (18), (19) and (29) to yield the values of the radial, tangential and hoop stresses, respectively. Finally, the angle k of Fig. 1 that describes the boundary between the slip and no-slip regions can be found by determining the value of k that satisfy the root of Eq. (6). 4. Results In this investigation, properties of [±45°]s and [04°/ ± 45°]s carbon fiber reinforced plastic laminates [9] presented in Table 1 were used to evaluate joint stress distribution for friction coefficient values of 0.0, 0.2 and 0.4, respectively. The resulting stress distribution are presented in terms of a dimensionless ratio obtained by normalizing these stresses by the bearing stress, rb = P/2r, for a plate of unit thickness. The results are displayed in Figs. 2–7. Also shown in these figures for comparison are the results generated in Ref. [9].

η=0.2

-1.2

Present de Jong

-1.4 -1.6

η=0

-1.8 Fig. 3. Radial stress for [04/±45°]s laminate.

In this analysis, a computer code was written using a Mathematica package that utilizes Newton’s method of iteration to determine the no slip region. From this analysis, the value of the angle k was found to be zero for all the values of coefficient of friction tested. 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 magnitude of the peak stresses decreases with increasing frictional value and the location also varies with increased friction. For g = 0, the maximum stress occurs at h = 0, beyond g = 0.2, the maximum occurs at larger values of h. The shear stress distribution, srh, at the pin-hole boundary for the [±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 the peak stress increases with increased value of the friction coefficient. Unlike de Jong [9], the present results show peaks occurring at two different locations within the contact region for the [±45°]s

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

1.6

0.45

Present de Jong

0.4

Present de Jong

1.2

0.35

τ rθ σb

1.4

η=0.4

σ θθ σb

0.3 0.25

1 0.8 0.6

η=0.2 0.2

η=0.4

0.4

0.15

0.2

0.1

0

η=0.2

0 -0.2

0.05

θ

0 0

20

40

60

80

50

100

θ

η =0

-0.4

100

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

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

0.45

η=0.4

0.4

Present de Jong

0.35

τ rθ σb

Figs. 6 and 7 show the hoop stress, rhh, for the laminates analyzed. As can be seen from these figures, the hoop stress initially increases with increasing values of h. However, for the [±45°]s laminate, this stress is compressive in the region h = 0 for all frictional values investigated with magnitude becoming increasingly tensile as h increases.

0.3

η=0.2

0.25 0.2

5. Conclusion

0.15 0.1 0.05

An analysis is presented to evaluate stress distributions in composite pin-loaded joints. The approach involves specification of displacement expressions in the form of a trigonometric series that satisfy the boundary conditions for the contact region in terms of a set of undetermined coefficients. Numerical results for [±45]s and [04/ ± 45°]s laminates indicate that friction has a significant influence on the radial, shear and hoop stresses for these laminates. In all cases, the maximum radial stress decreases with increasing value of the frictional coefficient. In addition, the location of this maximum also varies with increased friction coefficient. On the other hand, the shear stress distribution exhibits a peak stress that increases with increased value of the friction coefficient. For the hoop stress, the [±45°]s laminate experienced an initial compressive region followed by a tensile region and, in the tensile region, the peak stress and its location varied with increased friction coefficient.

θ

0 0

50

100

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

2

Present de Jong

1.5

σ θθ σb

1

0.5

References 0

-0.5

0 η=0.4

θ 20

40

60

80

100

η=0.2 η=0

-1 Fig. 6. Hoop stress for [±45°]s laminate.

laminate. However, while friction appears to influence the magnitude of the stress, it does not appear to influence the location at which the peak occurs.

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