INTRODUCTION
In practical structures, composite laminates are often fastened to other structural components and a mechanical joint by means of pin or bolt is widely used. High stress concentration near the contact area between plates and pin may cause failure. As a result, composite bolted joints could fail at much lower bearing stress than their counterparts made of isotropic materials such as metals. The bearing stress is defined as the compressive load applied onto the laminate divided by the projected cross sectional area of the bearing surface. The projected area is calculated as the product of the diameter of the hole and the thickness of the laminate. Therefore, exact stress field is first step to get true outputs.
Analytical method is done by Madenci and Ileri (1993) to predict contact stresses in mechanical joints with single fastener, with two fasteners and with a row of fasteners. Experimental results are a large part of literature published and effects of clearance (Kelly and Hallström, 2004) and geometry parameters (İçten and Sayman, 2003; Okutan, 2002; Karakuzu et al., 2006; Mevlüt Tercan and Aktas, 2005) are investigated. Also, numerical models are generated and results are investigated. Okutan and Karakuzu (2003) investigate the effects of geometric dimensions on the strength of pinned joints and show a definite dependence of bearing strength on stacking sequence. Two different symmetric carbonepoxy composite laminates are used to obtain the strength behavior and the optimum geometry of pinned joints using experimental and numerical models by Aktas and Dirikolu (2003, 2004). Three dimensional finite element analysis is used to examine the effects of bolthole clearance on behavior of composite bolted joints by McCarthy and McCarthy (2005).
In present study a three dimensional finite element model has been generated to solve for contact area in composite joints. As previously concluded, contact area is less than 180° (Chang, 1986) and its magnitude depends on geometry parameters, layer configurations, material properties and also friction and clearance between pin and plate. First, symmetric laminated composite models of quasiisotropic, crossply and angleply configurations are generated to study the separation angle, the magnitude and location of maximum radial stress in each ply. Then, nonsymmetric quasiisotropic laminate is used to evaluate the behavior of laminated composite plate at contact surface. No previous threedimensional studies on variation of separation angle through thickness and radial stress distribution in nonsymmetric laminated joints have been found in literature.
PROBLEM DESCRIPTION
Consider a composite rectangular plate of length L, width W and thickness t
with a hole of diameter D. The hole is at distance E from the free edge of the
plate. This geometry is used with single pin to form a single lap joint (Fig.
1) and geometry parameters are shown in Table 1.
Table 1: 
The geometry parameters of the model 

Table 2: 
Elastic properties for the unidirectional lamina (Truong et
al., 2002) 


Fig. 1: 
The geometry of the pinnedjoint problem 
The rigid pin is located at the center of the hole and a tensile load of 1.2
kN is applied to the plate. The laminated composite plate is consisted of eight
layers with properties as in Table 2. Plate’s layer configuration
is changed to investigate the effects of that on the contact area. Only the
case of equal pin and hole radius is investigated because it is sufficient for
present purposes.
FINITE ELEMENT MODEL
The finite element method is used to determine the stress field around a hole in a pinloaded laminate. The stress field around a hole in a laminate when subject to pin or bolt loading has been shown to be threedimensional (Camanho et al., 1998; Ireman, 1998) and thus a threedimensional finite element model is necessary to compute the multiaxial stress state which exists at the hole and the surrounding area. Finite element models of the pinloaded laminates are developed using the ABAQUS (2001) finite element code (Fig. 2).
Each ply is modeled using a layer of linear isoparametric threedimensional solid brick elements (C8D3), with the mesh being refined around the region adjacent to the hole in order to compute the contact stresses precisely (Fig. 3). Elements further afield from the hole where the aspect ratio of the elements was poorer are modeled using the reduced integration element (C8D3R) as reduced integration scheme alleviates the overstiffened behavior of such elements.

Fig. 3: 
Refined mesh around the hole at θ = ±90° 
The contact stress distribution and the full threedimensional contact problem
is solved in order to determine the actual contact pressure distribution and
contact area. The masterslave contact algorithm is used based on finite sliding
conditions (ABAQUS, 2001). The pin is modeled using an analytical rigid surface
based on the work of Eriksson (1986) and Hyer et al. (1987) who concluded
that the effect of pin modulus of elasticity is minor on the resulting contact
stress distribution in the laminate. The use of an analytical contact surface
is more computationally efficient and limits geometric discretisation error
in comparison to modeling the pin using solid finite elements.
A frictionless contact is assumed between the pin and the plate. Geometrically linear analysis is performed as both the pin and the laminate are stiff bodies which do not undergo large rotations.
To assess the accuracy of the present finite element calculation, the distributions
of the strain are verified by comparing results with those of previous work
as shown in Fig. 4 (Kelly and Hallström, 2004). Good
accuracy could be found between the results.

Fig. 4: 
Comparison of present results with previous results 
SEPARATION ANGLES AND STRESS DISTRIBUTIONS IN CONTACT AREA FOR DIFFERENT
CONFIGURATIONS
Tensile load, P and existence of interaction between the pin and the plate cause the deformation in the composite plate (Fig. 5). As a result of deformation, two separation locations occur at both sides of the pin and they are assumed to be constant through the thickness of the laminate in previous publications. In this study, analysis of the problem for different stacking sequences and various conditions is presented and in each section, reasons and explanations for the results have been given.
Symmetric laminated composite models of quasiisotropic, crossply and angleply configurations are generated first to study the separation angle, the magnitude and location of maximum radial stress in each ply. Then, nonsymmetric quasiisotropic laminate is used to study the behavior of laminated composite plate at contact area. The stresses at the center of each ply are plotted. For better presentation of the figures, stresses are normalized by the average bearing stress. The average bearing stress is defined as:
Where:
P 
= 
The load 
D 
= 
The hole diameter 
t 
= 
The laminate thickness 
Also θ_{max} and θ_{s} are the notations used to present maximum radial stress and separation locations.

Fig. 5: 
Separation locations around the hole 
Table 3: 
The
magnitude and location of maximum radial stress and separation angle in
each ply in symmetric quasiisotropic [0/45/90/45]_{s }laminate 

Symmetric quasiisotropic laminate [0/+45/90/45]_{s}: Radial stresses in each ply at the contact area are shown in Fig. 6. Because the stacking sequence is symmetric, results are only for four layers. The radial stresses are compressive regardless of ply orientation. Due to high stiffness in the loading direction, the 0° plies are under the highest stress. High radial stress levels are also evident in the ±45° plies with the peak load occurring close to the respective ply fiber directions where the stiffness is high. The radial stress distribution in these plies is nonsymmetric about the bearing plane and the stresses in +45° plies are slightly higher than the 45° plies. The maximum radial stress in the 90° plies is shown to occur at θ_{max} = 75°. The radial stress in 90° plies is significantly lower than the stresses in 0° and ±45° plies which is physically reasonable because the effective stiffness in the load direction is lower in 90° plies.
For better understanding, the maximum radial stresses, their locations (θ_{max})
and separation angle (θ_{s}) in each ply are shown in Table
3. The maximum radial stress is 313 Mpa and occurs in ply 1 (0°) at
θ_{max} = θ° which corresponds to the direction of applied
load. High radial stresses also occur in ±45° plies (296 Mpa +45°
plies and 273 Mpa in 45° plies). Due to the plies direction, maximum radial
stresses in plies 2 and 4 (+45° and 45°) are lower than the maximum
stress in ply 1 (0°). The maximum radial stress in ply 2 (+45°) is about
8% higher than the maximum stress in ply 4 (45° ply). The radial stress
in ply 3 (90°) is the lowest stress (151 Mpa) between plies which is physically
reasonable with respect to fibers orientation.

Fig. 6: 
Radial stress distribution around the hole in symmetric quasiisotropic
[0/45/90/45]_{s }laminate 

Fig. 7: 
Separation location in quasiisotropic [0/45/90/45]_{s
}laminate shown by () 
Table 4: 
The magnitude and location of maximum radial stress and separation
angle in each ply in cross ply [0_{2}/90_{2}]_{s}
laminate 

As it can be seen, in ply 1 (0°) and ply 3 (90°), the separation angles
(θ_{s}) are symmetric (±84° and ±87°) while
in ply 2 (45°) and ply 4 (45°), the separation angles (θ_{s})
are nonsymmetric (83°, 81 and 80°, 83°). Due to stiffness, for
ply 2 (45°) the magnitude of separation angle (θ_{s}) at clockwise
direction is lower than the magnitude of separation angle (θ_{s})
in counterclockwise direction with respect to bearing plane while for ply 4
(45°) is vise versa (Fig. 7).
Cross ply laminate [0_{2}/90_{2}]_{s}: The radial stress distribution for the cross ply laminate is shown in Fig. 8. Because the stacking sequence is symmetric, results are for only four layers. As can be seen, all the 0° plies have the maximum stresses at θ_{max} = 0°. The 90° plies experience their maximum stresses at an angle close to θ_{max} = 60°. The location of maximum radial stress θ_{max} is symmetric with respect to bearing plane as a result of symmetric stacking sequences. The magnitude and location of maximum radial stress θ_{max} and separation angle (θ_{s}) are shown in Table 4.
Due to crossply configuration, the separation angle (θ_{s}) is nearly constant through thickness. The separation angle (θ_{s}) is about 80° in all plies at both sides of contact area which is due to laminate stacking sequence (Fig. 9).
Angle ply laminate [+45/45]_{2s}: The radial stresses distributions
for the angle ply laminate are shown in Fig. 10. As can be
seen, peak stresses occur close to ±34°. As a result of plies orientations,
the location of maximum radial stress θ_{max} is nonsymmetric
with respect to bearing plane (Table 5, Fig.
11). The highest radial stress occurs in plies 2 (45°) and 3 (45°).

Fig. 8: 
Radial stress distribution around the hole in cross ply [0_{2}/90_{2}]_{s}
laminate 

Fig. 9: 
Separation location in cross ply [0_{2}/90_{2}]_{s}
laminate shown by ( ) 
Due to cross ply configuration, the separation angle (θ_{s}) is about 85° and nearly constant in all plies at both sides of contact area. The contact area through thickness is plotted in Fig. 11.
Nonsymmetric quasiisotropic laminate [0_{2}/+45_{2}/90_{2}/(
45)_{2}]: We shall now solve the problem for the nonsymmetric quasiisotropic
laminate. The deformed shape of the joint model is shown with actual deformation
in Fig. 12. More attention should be paid to this case for
studying the results, because bending moment occurs in the laminate. Thus, as
can be seen, results show a curvature in the laminated composite plate and the
separation angles and contact area change sharply through the thickness of laminate.
Table 5: 
The magnitude and location of maximum radial stress and separation
angle in each ply in angle ply [+45/45]_{2s} laminate 

This is caused by the nonsymmetry of the stacking sequence.
Radial stresses in each ply at the contact area are shown in Fig.
13. Because the stacking sequence is nonsymmetric, results are shown for
all layers. Due to high stiffness in the loading direction, the 0° plies
are under the highest stress. High radial stress levels are evident in the +45°
plies too with the maximum load occurring close to the respective ply fiber
directions. Stresses are also high in ply 7 (45°) but extremely decrease
in ply 8 (45°).

Fig. 10: 
Stress distribution around the hole in angle ply [+45/45]_{2s}
laminate 

Fig. 11: 
Separation location in angle ply [+45/45]_{2s} laminate
shown by () 

Fig. 12: 
Deformed shape of nonsymmetric quasiisotropic [0_{2}/+45_{2}/90_{2}/(45)_{2}]
laminate 
Table 6: 
The magnitude and location of maximum radial stress and separation
angle in each ply in nonsymmetric quasiisotropic [0_{2}/+45_{2}/90_{2}/(45)_{2}]
laminate 

This could be reasonable because of the bending moment and curvature produced
in laminated composite plate. The maximum radial stresses in 90° plies occur
at θ_{max} = ±76°.
For better understanding, the maximum radial stresses, their locations (θ_{max})
and separation angle (θ_{s}) in each ply are shown in Table
6. The maximum radial stress is 458 Mpa and occurs in ply 1 (0°) at
θ_{max} = 0° which corresponds to the direction of applied
load. High radial stresses also occur in +45° plies (336 Mpa in ply 3 and
305 Mpa in ply 4).

Fig. 13: 
Radial stress distribution around the hole in nonsymmetric
quasiisotropic [0_{2}/+45_{2}/90_{2}/(45)_{2}]
laminate 

Fig. 14: 
Separation location in nonsymmetric quasiisotropic ply [0_{2}/+45_{2}/90_{2}/(45)_{2}]
laminate shown by () 
The radial stress in ply 8 (45°) is the lowest stress (83.7 Mpa) between
plies which is physically reasonable with respect to fibers orientation and
laminated composite plate curvature. As it can be seen, the separation angles
(θ_{s}) are nonsymmetric with respect to bearing plane in all
plies. It should be noted that separation angle (θ_{s}) magnitude
is greater then 90° in ply 1 (0°), ply 2 (0°) and ply 8 (45°)
as a result of plate curvature (Fig. 14).
CONCLUSION
In this study, threedimensional finite element analysis was used to examine the effects of layer configuration on the contact area of singlelap laminated composite pinnedjoints. No previous threedimensional study on variation of the contact area in symmetric and nonsymmetric laminates through thickness has been found in the literature. The main findings from the study are summarized below.
In singlelap joints, the stress distribution in the laminate is nonuniform
through the thickness and stacking sequence causes threedimensional variation
in this stress distribution.
Stress field is different in composite laminates with different stacking sequences. The minimum magnitudes of stresses occur in the angleply laminate, because none of the layers is oriented in the loading direction. Also because all layers are oriented at the same angle with respect to the loading direction, the magnitude of maximum stress does not differ significantly in the layers. Magnitudes of stress are larger in the cross ply laminate than the angle ply laminate and in the quasiisotropic laminate are larger than the cross ply laminate. The cross ply laminate has four 0° plies which have more stiffness with respect to 90° plies in the loading direction and stresses are divided between these 0° plies. This causes that the maximum radial stress in symmetric laminates occurs in the quasiisotropic laminate which only has two 0° plies.
Results show the variation of the contact area through the thickness of the laminate even with symmetric stacking sequence. These variations are not investigated in previous works available in literature. So, this paper states why the contact problem of laminated composite joints should be solved in three dimensions.
In symmetric stacking sequences, the contact area is symmetric in 0° and 90° plies while it is nonsymmetric in plies with other fiber orientations. Also, the contact area is less than 180°. In nonsymmetric laminates, results show a curvature in the laminated composite plate and the separation angles and contact area change sharply through the thickness of laminate. Note that, the separation angle is greater than 90° in 0° and 45° plies but the contact area is less than 180° in all plies.