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Composites Part A: …

Thermal and geometrically non-linear stress analyses of an adhesively bonded composite tee joint2003 •

Adhesive bonding technique is used successfully for joining the carbon fibre reinforced plastics to metals or composite structures. A good design of adhesive joint with either simple or more complex geometry requires its stress and deformation states to be known for different boundary conditions. In case the adhesive joint is subjected to thermal loads, the thermal and mechanical mismatches of the adhesive and adherends cause thermal stresses. The plate-end conditions may also result in the adhesive joint to undergo large displacements and rotations whereas the adhesive and adherends deform elastically (small strain). In this study, the thermal and geometrically non-linear stress analyses of an adhesively bonded composite tee joint with single support plus an angled reinforcement made of unidirectional CFRPs were carried out using the non-linear finite element method. In the stress analysis, the effects of the large displacements were considered using the small displacement–large displacement theory. The stress states in the plates and the adhesive layer of the tee joint configurations bonded to a rigid base and a composite plate were investigated. An initial uniform temperature distribution was attributed to the adhesive joint for a stress free state, and then variable thermal boundary conditions, i.e. air flows with different velocity and temperature were specified along the outer surfaces of the tee joints. The thermal analysis showed that a non-uniform temperature distribution occurred in the tee joints, and high heat fluxes took place along the free surfaces of the adhesive fillets at the adhesive free ends. Later, the geometrical non-linear thermal-stress analysis of the tee joint was carried out for the final temperature distribution and two edge conditions applied to the edges of the vertical and horizontal plates (HP). High stress concentrations occurred around the rounded adherend corners inside the adhesive fillets at the adhesive free ends, and along the adhesive–composite adherend interfaces due to their thermal–mechanical mismatches. The most critical joint regions were adhesive fillets subjected to high thermal gradients, the middle region of HP, the region of the vertical plate corresponding to the free end of the vertical adhesive layer–left support interface. In addition, the support length had a small effect of reducing the peak stresses at the critical adherend and adhesive locations.

46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference

3D Stress Analysis of Adhesively Bonded Composite Joints2005 •

International Journal of Adhesion and Adhesives

Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads2009 •

International Journal of Adhesion and Adhesives

Effect of the temperature on the strength of adhesively bonded single lap and T joints for the automotive industry2009 •

IAEME

ANALYSIS AND EXPERIMENTAL VALIDATION OF COMPOSITE LAP JOINT WITH CRACK FOR AEROSPACE APPLICATIONS2019 •

Composite materials have made way to several different fields, including aerospace structures, under water vehicles, automobiles and robot systems etc. Due to the high strength they are widely used in the low weight constructions and also used as a suitable alternative to metals. ANSYS FEA tool has been used for stress distribution characteristics of single lap joint with crack in-built in adhesive layer. In several different applications and also for joining various composite parts together, by using epoxy adhesives. Modeling and static analysis of 3D Models of lap joints were carried out and compared for two different composite materials. The present study deals with the analysis of single lap joint subjected to tensile load and the stress distribution in the joint members under various design conditions are to be found. The fracture characteristics of single-lap bonded joints in composites to be investigated by structural analysis and experimentally. The effects of bonding method, surface roughness, bond line thickness and the existence of fillet on the failure characteristics and strength of bonded single-lap joints were evaluated experimentally. The failure process, failure mode and the behavior of load-displacement curve was apparently different according to bonding method to be evaluated.

This study evaluates the effect of temperature upon adhesive properties and behavior of adhesively bonded T-joint. Finite element analysis is conducted to study the effect of this parameter on the durability joint and stress distribution within the adhesive layer. A series of temperatures and stress analyses has been performed for the T-joint configuration. The parametric studies on the FE model revealed that stress distributions are sensitive to the changes in adhesive properties due to changes in temperature. Conclusion, stresses were reduced with changes in the temperature which resulted in the ability of the adhesive layer to undergo plastic deformation.

Composite materials have made way to several different fields, including aerospace structures, under water vehicles, automobiles and robot systems etc. Due to the high strength they are widely used in the low weight constructions and also used as a suitable alternative to metals. ANSYS FEA tool has been used for stress distribution characteristics of single lap joint with crack in-built in adhesive layer. In several different applications and also for joining various composite parts together, by using epoxy adhesives. Modeling and static analysis of 3D Models of lap joints were carried out and compared for two different composite materials. The present study deals with the analysis of single lap joint subjected to tensile load and the stress distribution in the joint members under various design conditions are to be found. The fracture characteristics of single-lap bonded joints in composites to be investigated by structural analysis and experimentally. The effects of bonding method, surface roughness, bond line thickness and the existence of fillet on the failure characteristics and strength of bonded single-lap joints were evaluated experimentally. The failure process, failure mode and the behavior of load-displacement curve was apparently different according to bonding method to be evaluated.

International Journal of Adhesion and Adhesives

Design of adhesive joints based on peak elastic stresses2008 •

Composites: Part A 34 (2003) 135–150
www.elsevier.com/locate/compositesa
Thermal and geometrically non-linear stress analyses
of an adhesively bonded composite tee joint
M. Kemal Apalak*, Recep Gunes, M. Onder Turaman, A. Alper Cerit
Department of Mechanical Engineering, Erciyes University, Muhendislik Fakultesi, Kayseri 38039, Turkey
Received 3 January 2002; revised 24 October 2002; accepted 5 November 2002
Abstract
Adhesive bonding technique is used successfully for joining the carbon fibre reinforced plastics to metals or composite structures. A good
design of adhesive joint with either simple or more complex geometry requires its stress and deformation states to be known for different
boundary conditions. In case the adhesive joint is subjected to thermal loads, the thermal and mechanical mismatches of the adhesive and
adherends cause thermal stresses. The plate-end conditions may also result in the adhesive joint to undergo large displacements and rotations
whereas the adhesive and adherends deform elastically (small strain). In this study, the thermal and geometrically non-linear stress analyses
of an adhesively bonded composite tee joint with single support plus an angled reinforcement made of unidirectional CFRPs were carried out
using the non-linear finite element method. In the stress analysis, the effects of the large displacements were considered using the small
displacement – large displacement theory. The stress states in the plates and the adhesive layer of the tee joint configurations bonded to a rigid
base and a composite plate were investigated. An initial uniform temperature distribution was attributed to the adhesive joint for a stress free
state, and then variable thermal boundary conditions, i.e. air flows with different velocity and temperature were specified along the outer
surfaces of the tee joints. The thermal analysis showed that a non-uniform temperature distribution occurred in the tee joints, and high heat
fluxes took place along the free surfaces of the adhesive fillets at the adhesive free ends. Later, the geometrical non-linear thermal-stress
analysis of the tee joint was carried out for the final temperature distribution and two edge conditions applied to the edges of the vertical and
horizontal plates (HP). High stress concentrations occurred around the rounded adherend corners inside the adhesive fillets at the adhesive
free ends, and along the adhesive – composite adherend interfaces due to their thermal – mechanical mismatches. The most critical joint
regions were adhesive fillets subjected to high thermal gradients, the middle region of HP, the region of the vertical plate corresponding to the
free end of the vertical adhesive layer – left support interface. In addition, the support length had a small effect of reducing the peak stresses at
the critical adherend and adhesive locations.
q 2003 Elsevier Science Ltd. All rights reserved.
Keywords: B. Adhesion; E. Thermal analysis; Tee joint
1. Introduction
Adhesive bonding technique has been used successfully
to join structural components with different mechanical and
thermal properties, such as composites and metals. A good
design and a suitable adhesive type can provide a significant
strength for the adhesive joint. However, adhesion mechanism, design parameters, i.e. overlap length, adhesive
thickness, adherend edge geometry, adhesive fillet geometry
influence the strength of adhesive joints. In order to design a
suitable adhesive joint for specific purposes, the adhesion
* Corresponding author. Tel.: þ90-352-437-4901; fax: þ 90-352-4375784.
E-mail address: apalakmk@erciyes.edu.tr (M. K. Apalak).
mechanism and the deformation and stress states of the
adhesive and adherends based on joint geometry, loading
conditions and especially material properties should be
known in detail. Therefore, extensive analytical and
experimental studies have been carried out on the stress
and deformations of the adhesive joints, especially on the
single, double-lap joints and their modifications due to their
simple geometries and the ease of testing.
The elastic stress analyses of the different types of
adhesive joints in which the adhesive and adherends were
treated as elastic materials showed that the stress concentrations occurred around the adhesive free ends, thus at the
free ends of the adhesive –adherend interfaces, a large
region in the middle of the adhesive overlap region
experienced low stress distributions. Therefore, the studies
1359-835X/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.
PII: S 1 3 5 9 - 8 3 5 X ( 0 2 ) 0 0 2 3 3 - 6
136
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
have concentrated on the stress concentrations at the
adhesive free ends, called edge effect, and on reducing the
peak stresses [1 –3]. The effects of the geometrical factors,
such as adhesive thickness, overlap length, adherend
thickness and adhesive/adherend modulus ratio on the
peak stresses at the adhesive free ends were investigated in
detail. However, most studies assumed the adhesive free
ends to have square shape. Adams and Wake showed that in
case the adhesive layer have adhesive accumulations at its
free ends, called adhesive spew fillets, the peak stresses
could be reduced depending on the shape and size of the
adhesive fillets [2]. In addition, it is possible to reduce peak
stresses modifying the free edges of the adherends, i.e.
tapering partly or sharpening a knife-edge.
The adhesive joints may undergo thermal loads as well as
structural loads in practice. In this case, thermal strain
distributions may cause thermal stress distributions in the
adhesive joints as a result of thermal and mechanical
mismatches of the adhesive and adherends. Some of the
recent studies have accounted for the effect of the thermal
loads on the strength of the adhesive lap joints with some
limitations. Thus, a uniform temperature distribution is
attributed to the adhesive joints or a constant temperature
along their outer boundaries is specified, the effect of a nonuniform temperature distribution in the adhesive joint as a
consequence of variable thermal boundary conditions, i.e.
fluid-adherend or adhesive interaction was not considered.
Ioka et al. [4] used the boundary element method to
predict thermal residual stresses on the adhesive –adherend
interfaces and on the free surfaces of bonded dissimilar
materials, and showed that thermal stress singularity
disappeared for certain range of wedge angles of a pair of
materials. Reedy and Guess [5] studied thermal residual
stresses arising due to the cooling of the adhesive joint, and
their effect on the joint strength. They showed that the peak
adhesive stresses in the yield zone at the interface corner
could decay significantly when sufficient time is given. Kim
et al. [6] also investigated thermal stresses in adhesively
bonded tubular single lap joint considering non-linear
adhesive properties and presented a failure model. Kim
and Lee [7] presented a method for determining an optimum
design of an adhesive tubular lap joint based on failure
modes due to thermal stresses induced by fabrication.
Nakano et al. [8] investigated thermal stress distributions
around the circular holes and rigid fillers in the adhesive
layer of an adhesive butt joint subjected to a non-uniform
temperature distribution, and found that the size and
location of the circular holes and rigid fillers affected
evidently the thermal stresses occurring on the adhesive –
adherend interface and at the hole and filler peripheries.
Nagakawa et al. [9] showed that thermal stresses around
hole defects located near the centre of the adhesive layer
were larger than those around the hole defects located
near the free surface of the adhesive in an adhesive butt joint
with adhesive layer including adhesive defects
subjected to uniform temperature changes. Katsuo et al.
[10] investigated the effects of thermal expansion coefficient
and Young modulus ratios of the adhesive and adherends on
the transient thermal stress distribution in an adhesively
bonded butt joint whose upper and lower surfaces are
assumed to have different constant temperatures.
In case the adhesive joints experience large loads, two
types of non-linearities affect the stress and deformation
states of the adhesive joints. First, the loading and boundary
conditions may cause large displacements and rotations in
the adhesive joints with complex geometry, i.e. tee and
corner joints, whereas the strains are still small. Therefore,
stress concentration regions are affected considerably from
the large displacements in the adhesive joint. In case the
stress and strain variations are dependent on deformed
geometry, this non-linearity is termed as the geometrical
non-linearity. The small strain –small displacement (SSSD)
approach of the elasticity theory cannot predict the elastic
stress and deformation states in the adhesive joints.
Therefore, the small strain– large displacement (SSLD)
approach, which is an extension of the SSSD theory, should
be applied to the adhesive joint problem. Secondly, the
material properties of the adhesive and adherends may
introduce a non-linearity, called the material non-linearity.
In this case, the stress variations should obey constitutive
laws of adhesive and adherend materials.
Sawyer and Cooper [11] took into account the effect of
the geometrical non-linearity on the stress distribution in an
adhesive single lap joint whose adherends were preformed
in order to reduce the angle between the line of the applied
in-plane force and the bond line. The applied load has an
eccentricity since the adherends are deformed as the load is
applied. That the bending moment is dependent on the
applied load makes the problem geometrical non-linear.
Their analyses showed that pre-forming the adherends
reduced the peel and shear stresses at both free ends of the
overlap region and provided a more uniform shear stress
distribution in the adhesive layer.
Adams [12], Adams and Harris [13,14] investigated the
failure modes and loads of an adhesive single lap joint and
the effects of the local geometry of the adherend edges and
adhesive accumulations at the free ends on the peak stresses
using the non-linear finite element method. They considered
the non-linear behaviour of both adhesive and adherends
and the large displacement and rotation effects (geometrical
non-linearity). They showed that the non-linear constitutive
behaviour of the adhesive and adherends played important
role on the joint strength, i.e. failure modes and loads, and
that significant increases in the joint strength might be
achieved by filleting the adhesive accumulations at the free
edges of the of the overlap region. Adams et al. [15] also
investigated the shear and transverse tensile stresses in
carbon fibre reinforced plastic/steel double lap joints using
an elasto-plastic constitutive law for the adhesive material
(rubber-modified epoxy), and achieved considerable
increases in the joint strength modifying the local geometry
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
of the critical zones at the edges of the overlap region of
the composite-steel double lap joints.
Czarnocki and Pierkaski [16] carried out three dimensional stress analysis of an adhesive single lap joint treating
the adhesive as a non-linear material, and showed that the
joint width also played an important role on the peak
adhesive stresses. Reddy and Roy [17] investigated the
effect of large displacement gradients on the adhesive
stresses at the free ends of the adhesive layer in single lap
joint for different loading conditions. They applied the
geometrically non-linear finite element method based on the
large displacement-small strain approach to an adhesive
single lap joint for different plate end conditions, and
showed that the large displacement gradients had an evident
effect on the peak stresses at the adhesive free ends. Edlund
and Klarbring [18,19] developed a method for analysing the
three dimensional stress and deformation states of an
adhesive single lap joint considering both the geometrical
non-linearity and the non-linear material behaviour of the
adhesive and plates.
Due to their complex and unbalanced geometry the stress
analyses of the adhesive joints, i.e. corner or tee joints are
more complicated than those of the single and double-lap
joints. Shenoi and Violette [20] also carried out the
theoretical and experimental stress analyses of adhesively
bonded composite tee joints used in small boats and
investigated the influence of joint geometry on the ability
to transfer out-of-plane loads. Apalak et al. [21,22] carried
out the elastic stress analyses of two types of tee joints based
on the SSSD theory for different loading conditions.
Apalak and Engin [23] analysed the effect of the large
displacement gradients on the elastic adhesive and
adherend stresses of an adhesively bonded double
containment cantilever joint using the non-linear finite
element technique based on the SSLD theory and showed
that the SSSD theory might be misleading for predicting
the elastic stress and strain states of the adhesive joints.
Apalak [24] investigated the elastic stress and deformations in the various adhesively bonded corner joints
considering large displacement effects, and showed that
the geometrical non-linearity affected the elastic stress
levels occurring in the adherends and adhesive layer.
The previous thermal stress analyses of the adhesive
joints assumed that the heat transfer to occur by
conduction through the adhesive joint members. A
uniform temperature distribution was prescribed along
the boundaries of the adhesive joints, or a constant
temperature distribution was set through the adhesive
joint. However, the effects of variable thermal boundary
conditions and the heat transfer by convection were
neglected. The adhesive joint may interact with fluids
at a specific velocity, and temperature along its boundaries. In this case, the heat transfer takes place by
convection between the outer surfaces of the adhesive
joint and the fluid, and by conduction through the adhesive
joint. The flow direction, velocity and temperature of
137
the fluid, the free-surface temperatures of adherends and
adhesive play important role on the heat transfer by
convection between the fluid and the outer surfaces of the
adhesive joints since the convective heat transfer is
dependent on the film coefficient being function of these
factors. In this case, a non-uniform temperature distribution through the adhesive joint would be expected to
occur contrary to the constant temperature distribution
considered in the previous studies. Non-uniform thermal
strain and stress distributions would occur as a result of
non-uniform temperature distribution and the thermal and
mechanical mismatches of the adhesive and adherend
materials.
Apalak and Gunes [25] investigated elastic thermal stress
distributions in the adhesive layer and adherends of an
adhesively bonded single lap joint considering the effects of
variable thermal boundary conditions and the large
displacement gradients for the different plate end conditions. They showed that non-uniform temperature and
thermal strain distributions occurred through the adhesive
joint and that the stress concentrations occurred inside the
adhesive fillets around the adhesive free ends and along the
adhesive – adherend interfaces.
As a result the variable thermal conditions affect
considerably the stress and deformation states of the
adhesive joints their consideration in the analysis and
design of the adhesive joints would give more realistic
results. In this study, the steady state thermal analysis and
geometrically non-linear stress analysis of an adhesively
bonded tee joint were carried out using the non-linear finite
element method in which the variable thermal boundary
conditions were prescribed along the outer surfaces of the
adhesive joint, and for different plate end conditions.
2. Thermal model and finite element formulation
Application of the first law of thermodynamics to a
differential control volume yields
›T
þ vT 7T ¼ ﬄ
q þ 7ðD7TÞ
ð1Þ
rc
›t
with the Fourier’s law q ¼ 2D7T where v T is velocity
vector for mass transport of heat, q is heat flux vector, ﬄ
q is
heat generation rate per unit volume, and D is conductivity
matrix. The present problem assumes that the adhesive joint
is subjected to the specified convection surfaces (Newton’s
law of cooling) as
q·n ¼ 2hm ðT1 2 TS Þ
ð2Þ
T
n D7T ¼ hm ðT1 2 TÞ
where n is unit outward normal vector, hm is film coefficient,
T1 is bulk temperature of the adjacent fluid and TS is
temperature at the surface of the model. Pre-multiplying
Eq. (1) by a virtual change in temperature dT, integrating
138
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
over the volume of the element, and combining with Eq. (2)
yield
ð
›T
T
T
þ v LT þ L dTðDLTÞ dV
rc dT
›t
V
ð
ð
¼ dThm ðT1 2 TÞdS þ
dTﬄ
q dV
ð3Þ
S
V
where dX is an arbitrary infinitesimal line vector in the
initial position. The small strain tensor and the rotation
tensor are
!
›uj
1 ›ui
ð10aÞ
þ
1ij ¼
2 ›xj
›x i
and
where
LT ¼
›
›x
›
›y
1
Vij ¼
2
›
›z
is a vector operator.
The temperature is dependent on both space and time.
Therefore, the temperature can be written in terms of
element shape functions N and nodal temperature vector Te
as
T ¼ NT Te
ð4Þ
and the time derivatives of Eq. (4) may be written as
›T
›T
¼ NT e ¼ NT T_ e
›t
›t
ð5Þ
›uj
›u i
2
›x j
›xi
!
ð10bÞ
respectively. The large displacement-gradient components
make the strain to be characterized from the initial state
more difficult than in the small-strain case. Lagrangian
formulation allows the finite-strain to be defined based on
the material co-ordinates in the undeformed configuration.
The deformation equations for the movement of a particle
from its initial position X to the current position x are given
as
x ¼ xðX; tÞ or xi ¼ xi ðX1 ; X2 ; X3 Þ
ð11Þ
An arbitrary infinitesimal material vector dX at X can be
associated with a vector dx at x as follows
dT and LT also become
dT ¼ {dTe }T N and LT ¼ BTe
ð6Þ
dx ¼ F·dX ¼ dX·FT
ð12Þ
T
where B ¼ LN : The variational form of Eq. (3) can be
written as
ð
ð
ðrc{dTe }T NNT T_ e ÞdV þ ðrc{dTe }T NvT BTe ÞdV
V
V
þ
ð
T
V
¼
ð
T
ð{dTe } B DBTe ÞdV
T
S
T
{dTe } Nhm ðT1 2 N Te ÞdS þ
ð
¼
ð
V
{dTe } Nﬄ
q dV
ð7Þ
V
V
S
T1 hm N dS 2
ð
S
hm NNT Te dS þ
z
F ¼ x7
ð13Þ
In terms of the strain tensor the change in the squared length
of the material vector dX is given as follows
ðdsÞ2 2 ðdSÞ2 ¼ 2 dX·E·dX or ðdsÞ2 2 ðdSÞ2
T
If r is assumed to remain constant over the volume of the
element, and {dTe}T is dropped, Eq. (7) is reduced to the
final form as [26]
ð
ð
ð
r cNNT T_ e dV þ r cNvT BTe dV þ BT DBTe dV
V
using the deformation gradient vector
ð
Nﬄ
q dV
ð8Þ
V
¼ 2 dXI EIJ dXJ
ð14Þ
2
The new squared length (ds) of the element is written in
terms of the Green deformation tensor C referred to the
undeformed configuration as follows
ðdsÞ2 ¼ dX·C·dX or ðdsÞ2 ¼ dXI CIJ dXJ
Comparison of the Eqs. (14) and (15) gives the relationship
between the strain tensor E and the Green deformation
tensor C
2E ¼ C 2 1 or 2EIJ ¼ CIJ 2 dIJ
3. Small strain – large displacement theory
ð16Þ
2
The displacements and their gradients are assumed to be
infinitesimal in the SSSD theory, and the current configuration of a body is compared with its initial state. The
components of the unit relative displacement vector are
defined as
dui
›ui dXj
¼
›Xj dS
dS
ð15Þ
ð9Þ
The new squared length (ds) of the element is written using
the deformation gradient tensor as follows
ðdsÞ2 ¼ dx·dx ¼ ðdX·FT ÞðF·dXÞ ¼ dX·ðFT ·FÞ·dX
ð17Þ
Comparison of Eqs. (15) and (17) shows that the Green
deformation tensor
C ¼ FT ·F or CIJ ¼
›x k ›x k
› XI › XJ
ð18Þ
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
From Eq. (16) the strain tensor becomes
1 ›x k ›x k
T
1
2 dIJ
E ¼ 2 ðF ·F 2 1Þ or EIJ ¼
2 › XI › XJ
ð19Þ
Using the same reference axes for both xi and Xi, and using
lower-case subscripts for both, we have
x i ¼ X i þ ui
ð20Þ
with displacement components ui ¼ ui ðX1 ; X2 ; X3 ; tÞ:
In terms of the displacements, the general expression for
Eij in Eq. (19) takes the form
"
#
›uj
1 ›ui
›uk ›uk
þ
þ
ð21Þ
Eij ¼
2 › Xj
› Xi
› Xi › Xj
When Eqs. (10a) and (21) are compared it is evident that the
large displacement theory is an extension of the small
displacement theory since it includes the squares of the
displacement gradients. In case the displacement gradients
are not small compared to unity, the large displacement
theory should be used in order to consider their non-linear
effects on the stress and deformation states of the structures
[27,28].
4. Non-linear equilibrium equations
139
where N is a shape function array whose coefficients are
functions of the initial position x within the element
included in the expression of displacement within an
element
u ¼ Np
ð27Þ
where the nodal displacement vector
p ¼ ½u1 ;u2 ;…;un T
ð28Þ
and the strain matrix
B ¼ B0 þ BL
ð29Þ
where B0 is the linear strain matrix being a function of the
shape functions only, while BL is the non-linear strain
matrix being a function of the shape functions and
displacements [23]. The components of Green’s strain
vector E can be written as
h
i
E ¼ B0 þ 12 BL p
ð30Þ
in terms of the linear and non-linear Green’s strain vectors
E0 ¼ B0 p and EL ¼ 12 BL p
ð31Þ
The non-linear equilibrium Eq. (26) can be written as
ð
cðpÞ ¼ BT s dV 2 R ¼ w 2 R ¼ 0
ð32Þ
V
In order to establish the non-linear equilibrium equation
for the deformed body subjected to the external loads, the
virtual internal and external works done on the body are
equated. The non-linear equilibrium equation from the
virtual-work equation given in terms of the Lagrangian coordinate system is given as [26,29 –31]
ð
ð
ð
dET s dV ¼
rduT q dV þ duT q0 dA
ð22Þ
V
V
A
where the external work is due to the virtual displacements
du acting on the surface tractions, extending over the initial
undeformed surface A and, given by
q0 ¼ ½ q01
q03
q02
T
ð23Þ
and the body forces per unit mass, acting within the
undeformed volume V given by
q ¼ ½ q1
q2
q3 T
ð24Þ
r being the density of the undeformed body. The total
Lagrangian virtual work equation (22) can be approximated
by the finite element idealization as follows
ð
ð
ð
ð25Þ
dpT BT s dV ¼ dpT rNT q dV þ dpT NT q0 dA
V
V
A
Since the virtual nodal displacements dp are arbitrary, Eq.
(25) can be written as
ð
ð
ð
ð26Þ
BT s dV ¼ rNT q dV þ NT q0 dA
V
V
A
where R is the right hand side of Eq. (26) for convenience
and c(p) is termed the residual. The Newton – Raphson
method is used for the solution of the assembled non-linear
equations. The solution is achieved when c(p) is reduced to
zero or a given convergence criterion is satisfied [23,26,
29 – 31].
5. Joint configuration
Since the adhesive tee joint consists of a HP, a support,
a vertical plate (VP), and an angled reinforcement
member being an extension of VP and adhesive layer
bonding composite (CFRP) adherends, it is called an
adhesively bonded composite tee joint with single support
plus angled reinforcement as shown in Fig. 1. The VP and
HP lengths L ¼ 360 mm, plate thickness t ¼ 4 mm,
support length a ¼ 30 mm and adhesive thickness
(d ¼ 0.5 mm was taken as main joint dimensions (see
Fig. 1). In practice, since the adhesive layer is pressed
between the plates by applying pressure to the plates some
amount of adhesive are accumulated around the free ends
of the adhesive layer, called adhesive fillets. The shape
and size of the adhesive fillets play important role on
the magnitude and location of the peak stresses in the
adhesive layer [2]. Therefore, the adhesive fillets at the
adhesive free ends were considered as shown in Fig. 1,
and their shape was idealized to a triangle with a height
and width ft twice the adhesive thickness d.
140
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
Fig. 1. Dimensions of a tee joint with single support plus angled reinforcement bonded to a flexible base.
The previous studies on the stress analyses of the
adhesive lap joints showed that the stress concentrations
occurred around the adherend corners at the free ends of the
adhesive layer [2,3]. The geometry of the adherend corners
affects considerably the local stress distributions. In general,
the adherend corners are assumed to be sharp for the
simplicity of the analyses. However, the geometrical and
material discontinuities result in stress singularities at the
adherend corners. Adams and Harris [14] investigated the
effect of the local geometry on the joint strength of a single
lap joint using a model consisting adhesive layer around a
rigid rounded corner in order to simplify the analysis, and
found that the significant increases in the joint strength may
be achieved in single lap joints by filleting the adhesive
layer at the ends of the overlap region. Therefore the
adherend corners were rounded with a radius R3 of 0.1d.
The plates, support and angled reinforcement member of
the adhesive tee joint were made of carbon fibre reinforced
plastic, IM6/3501-6. An epoxy based adhesive was used to
bond plates, support and angled reinforcement member.
Each unidirectional ply consists of (IM6) carbon fibres and
(3501-6) epoxy with thickness of 0.1335 mm (Table 1). The
CFRP laminates (plates and supports) were made of thirty
unidirectional IM6/3501-6 plies. The engineering constants
of the unidirectional fibre-reinforced lamina IM6/3501-6
were computed by using the micro-mechanics approach
based on the engineering constants of the IM6 fiber and the
3501-6 epoxy in the material coordinates ðx1 ; x2 ; x3 Þ as [32]
E 1 ¼ Ef Vf þ Em Vm ;
E2 ¼
y 12 ¼ y f Vf þ y m Vm ;
Ef E m
;
E f Vm þ Em Vf
G12 ¼
Gf Gm
Gf Vm þ Gm Vf
ð33Þ
where E1 is the longitudinal modulus, E2 is the transverse
modulus, y 12 is the major Poisson’s ratio, G12 is the shear
modulus, and the shear modulus of the fiber and the matrix
Gf ¼
Ef
;
2ð1 þ y f Þ
Gm ¼
Em
2ð1 þ y m Þ
ð34Þ
respectively. The following reciprocal relations also exist
y 21
y
y 31
y
y 32
y
¼ 12 ;
¼ 13 ;
¼ 23
or
E2
E1
E3
E1
E3
E2
ð35Þ
y ij
y ji
¼
Ei
Ej
In case of a transversely isotropic material with the 2– 3
plane as the plane of isotropy
E 2 ¼ E3 ;
G12 ¼ G13
y 12 ¼ y 13
ð36Þ
Table 1
Base properties of graphite fibre (IM6), epoxy (3501-6) and lamina (IM6/3501-6)
Property
Unit
Graphite fibre (IM36)
Epoxy (3501-6)
Lamina (IM6/3501-6)
E11
E22 and E33
G12 and G13
G23
y 12 and y 13
y 23
a11
a22 and a33
Heat capacity (cp)
Heat conduction (k11)
GPa
GPa
GPa
GPa
259.3020
13.9380
50.9910
8.2800
0.2600
0.3300
0.8543
3.2374
0.2000
83.5090
4.347
4.347
1.598
1.598
0.360
0.360
40.470
40.470
0.250
0.179
166.100
9.618
6.316
3.584
0.297
0.342
20.460
19.530
m 1/8C
m 1/8C
cal/g 8C
watt/m 8C
82.726
141
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
The generalized Hooke’s law for an anisotropic material
under isothermal conditions is given by
or
sij ¼ Cijkl 1kl
In order to determine the strain – stress relations of a
laminate, it is necessary to convert strain components
referred to the material (lamina) coordinate system
ðx1 ;x2 ;x3 Þ to those referred to the problem (laminate)
coordinate system ðx;y;zÞ as
3 2 2
32 3
2
1xx
11
m
n2 0 0 0 2mn
7 6
76 7
6
7
61 7 6 2
12 7
6 yy 7 6 n
m2 0 0 0 mn 76
7
7 6
76
6
7
7 6
76
6
6
6 1zz 7 6 0
13 7
0 1 0 0
0 7
7
7 6
76
6
7
ð41aÞ
7 ¼6
76
6
7
6
6 21 7 6 0
7
1
0 0 m n
0 7
76
6 yz 7 6
4
7
7 6
76
6
7
7 6
76
6
7
76
6 21xz 7 6 0
1
0
0
2n
m
0
5
5 4
54 5
4
ð37Þ
or in contracted notation [32]
si ¼ Cij 1j
ð38Þ
If the coordinate planes are chosen parallel to the three
orthogonal planes of material symmetry, the Eq. (38) can be
written in matrix form in the material coordinates ðx1 ; x2 ; x3 Þ
as
32 3
2
2 3
11
C11 C12 C13 0
0
0
s1
76 7
6
6 7
76 7
6
6 7
6 C21 C22 C23 0
6 s2 7
0
0 7 6 12 7
76 7
6
6 7
76 7
6
6 7
76 7
6
6 7
0
0 7 6 13 7
6 C31 C32 C33 0
6 s3 7
76 7
6
6 7
ð39Þ
76 7
6 7 ¼6
76 7
6
6 7
0
0 C44 0
0 7 6 14 7
6 0
6 s4 7
76 7
6
6 7
76 7
6
6 7
6 7
6 0
6s 7
0
0
0 C55 0 7
7 6 15 7
6
6 57
54 5
4
4 5
s6
m
0
0
0
0
0
C66
16
m
The substitution of the engineering constants in to the
inverse relation of Eq. (39) gives
3
2
1
y 21 y 31
2
0
0
0
2
7
6 E1
E2
E3
7
6
7
6
72 3
2 3 6
7 s
6
y
1
y
12
32
11
7 1
62
0
0
0
2
76 7
6 7 6
E
E
E
1
3
2
76 7
6
6 7 6
7 6 s2 7
6 12 7 6
76 7
6 7 6 y
76 7
6 7 6 2 13 2 y 23 1
0 0 0 7
7
6 7 6
76
s 7
6 13 7 6 E1
E
E
76
3
2
6 7 6
6 37
7
7 ð40aÞ
6 7 ¼6
76
7
6 7 6
76
1
6 14 7 6
6 s4 7
7
0
0
0 0 76 7
6 7 6 0
7
6 7 6
G23
76
7
61 7 6
76
6 57 6
6 s5 7
7
5
4 5 6
4
7
1
7
6 0
0
0
0
0
7
6
16 m 6
G13
7 s6 m
7
6
7
6
4
1 5
0
0
0
0 0
G12
{1}m ¼½Sm {s}m
21xy
p
2mn 22mn 0 0 0 m2 2n2
ð40bÞ
16
m
where m¼cos u; n¼sin u; and similarly
{1}p ¼½RT {1}m
ð41bÞ
{s}p ¼½RT {s}m
ð42Þ
In order to relate compliance coefficients in the two
coordinate systems, lets substitute Eqs. (40b) and (42) into
Eq. (41b) as
{1}p ¼½RT {1}m ¼½RT ð½Sm {s}m Þ¼½RT ½Sm ð½R{s}p Þ
{1}p ¼½Sp {s}p
ð43Þ
and ½Sm ;½S
where ½Sp ;½S
T
½S¼½R
½S½R
The thermal expansion coefficients aij are also transformed
ða12 ¼ a13 ¼ a23 ¼0Þ as
axx ¼ a11 m2 þ a22 n2 ; ayy ¼ a11 n2 þ a22 m2 ;
ð44Þ
2axy ¼2ða11 2 a22 Þmn; 2axz ¼0; 2ayz ¼0; azz ¼0
In the thermal stress analysis of the adhesive tee joint, an
eight-noded isoparametric quadratic quadrilateral plane
element with four integration points was used to model
Fig. 2. Mesh details of a tee joint with single support plus angled reinforcement bonded to a flexible plate.
142
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
the horizontal and VPs, support, angled reinforcement
member and adhesive layers. The compliance matrix of the
adherends made of the unidirectional IM6/3501-6 laminate
was computed as mentioned before. Since the stress
concentrations occur around the adhesive free ends, mesh
areas around the adhesive free ends were refined in order to
obtain a reasonable accuracy in the computations as shown
in Fig. 2.
6. Thermal analysis
The adhesive joints consist of adherends and adhesive
layer with different mechanical and thermal properties. The
thermal loads prescribed along the boundaries of the
adhesive joints cause a non-uniform temperature distribution in the adhesive joint as a result of the heat transfer by
conduction throughout the adhesive joint. However, the
temperature distributions, consequently thermal strain
distributions are non-uniform due to mismatches in the
thermal conductivities of the adhesive and adherends. Since
the thermal strains are not compatible along the bi-material
interfaces, the thermal stresses arise along these interfaces
and in their neighbourhoods as a result of the thermal and
mechanical mismatches of the adhesive and adherends. The
final temperature distribution of the adhesive joint should be
known in order to compute the thermal strains, so this
requires the thermal analysis of the adhesive joint subjected
to thermal conditions.
In the previous thermal stress analyses of the bonded lap
joints considering the conductive heat transfer, a uniform
temperature difference all over the adhesive joint or a
constant temperature along its boundaries was prescribed
for the simplicity of the analysis. In fact, the adhesive joints
interact with an ambient having temperature and flow
conditions along their outer surfaces. However, these
variable thermal boundary conditions require a thermal
analysis considering the convective heat transfer between
the fluid and the outer surfaces of the adhesive joints and the
conductive heat transfer throughout the adhesive joint
members.
In this study, two configurations of the adhesive tee joint
are considered; one bonded to a rigid base and the other is
bonded to a flexible HP. The variable thermal boundary
conditions are prescribed along the outer surfaces of the
adhesive bonded tee joints as shown Fig. 3. It is assumed
that the left surfaces of the VP and the left support, and the
upper surface at the left side of the HP are subjected to an
air flow at a temperature T1 ¼ 80 8C and a velocity
U1 ¼ 1 m/s whereas the right surface of the VP and the
angled reinforcement member, and the upper surface at the
right side of the HP experience an air flow at a temperature
T1 ¼ 20 8C and a velocity U1 ¼ 1 m/s. In case of the rigid
base, the lower surface of the horizontal adhesive layer was
assumed to be an adiabatic surface whereas in case of the
flexible base (Fig. 3(b)), an air stream with a temperature
T1 ¼ 40 8C and a velocity U1 ¼ 1 m/s is prescribed along
the bottom surface of the HP. In both cases, all members of
the adhesive tee joint were assumed to have an initial
temperature distribution at Ti ¼ 20 8C. During the fluidadherend or adhesive interaction, the flow direction of the
air with respect to the adherend or adhesive surfaces play
important role on the convective heat transfer taking place
between the fluid (air) and the outer surfaces of the adhesive
tee joint since the film coefficients along the adherend
surfaces are dependent on the flow direction, the fluid
temperature and velocity, and the surface temperature. The
air flow was considered as normal to the adherend and
adhesive surfaces pointed by surface numbers 1 –2, 2– 3,
4 – 5, 5 –6, 7– 8, 8 – 9 and 10 – 11, 11 –12, 13– 14, 14 – 15 and
as horizontal along the other surfaces (Fig. 3(a) and (b)).
The averaged film coefficients can be computed using
the different empirical formula depending on the flow
direction of the air flows [33]. In the case of the vertical air
Fig. 3. Thermal boundary conditions of a tee joint with single support plus angled reinforcement bonded to (a) a rigid base, and (b) a flexible plate.
143
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
stream the film coefficient is given as
U1 Deqv 0:731 l0
hm ¼ 0:205
Deqv
v
Table 2
Thermal properties of air
ð45Þ
where U1 is air velocity (m/s), Deqv is equivalent diameter
(m), v is kinematic viscosity of the air (m2/s) and l0 is
thermal conductivity of the air (w/m 8C). In the case of the
horizontal air flow, the film coefficient can be computed as
follows
hm ¼
Nul
L
ð46Þ
(T1 þ TS)/2 (8C)
l0 (w/m 8C)
l (kcal/m 8C)
v (m2/s) £ 106
m (kg/ms) £ 105
cp (kcal/kg 8C)
20
0.0257
0.0221
15.11
1.82
0.240
30
0.02640
0.02270
16.04
1.865
0.2405
50
0.0278
0.0239
17.935
1.955
0.241
temperature T1 as follows:
T1 þ TS
2
where l is thermal conductivity of the air (kcal/m h 8C), and
Nusselt number
Tf ¼
Nu ¼ 0:836Re1=2 Pr1=3
The steady-state thermal analyses of the adhesive tee
joints were carried out for the thermal boundary
conditions (Fig. 3). The film coefficients along the
adherend surfaces were computed considering the thermal
properties of the air based on the averaged temperature
Tf (Table 2). The temperature distributions in the
adherends and in the adhesive fillets of the joint region
are plotted in Figs. 4 and 5 for the joints with rigid base
and flexible base, respectively. In case of the rigid base,
since a higher fluid temperature is specified at the left of
the adhesive joint the heat transfer takes place from the
left side of the adhesive tee joint to its right side;
therefore, the temperature levels decrease uniformly to its
right side (62.5 – 52.3 8C, Fig. 4(a)). Similar temperature
ð47Þ
where Reynolds and Prandtl numbers, respectively
Re ¼
U1 L
;
v
Pr ¼
cp m
l
ð48Þ
where cp is specific heat (kcal/kg 8C) and m is dynamic
viscosity (kg/m s). The substitution of Eqs. (47) and (48)
into Eq. (46) yields
U L 1=2 cp m 1=3 l
hm ¼ 0:836 1
ð49Þ
v
l
L
The thermal properties of the air were determined based on
the averaged value of the surface temperature TS and the air
ð50Þ
Fig. 4. The temperature distributions in critical regions: (a) the joint region, (b) the vertical adhesive fillet, (c) the left horizontal adhesive fillet, and (d) the right
horizontal adhesive fillet (rigid base, all temperatures in 8C).
144
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
Fig. 5. The temperature distributions in critical regions: (a) the joint region, (b) the vertical adhesive fillet, (c) the left horizontal adhesive fillet, and (d) the right
horizontal adhesive fillet (flexible base, all temperatures in 8C).
distributions are observed inside the vertical and
horizontal adhesive fillets. The higher temperature levels
are observed in the adhesive zones close to the left side
of the adhesive joint, and they decrease uniformly (Fig.
4(b) – (d)). In case of the flexible base, the temperature
levels decrease continuously from the leftmost edge of
the support to the rightmost edge of the angled
reinforcement member (57 – 47 8C, Fig. 5(a)). The
temperature levels in the vertical adhesive fillet (VAF)
decrease from the free surface of the adhesive fillet and
the left support – adhesive interface to the VP – adhesive
interface (Fig. 5(b)). In the left horizontal adhesive fillet
(LHAF), the temperature distribution reaches high levels
in the adhesive zones close to free surface of the
adhesive fillet, and lower levels in the adhesive regions
near the HP – adhesive layer (Fig. 5(c)). In the right
adhesive fillet, the temperature distribution varies uniformly from the high levels in the angled reinforcement
and HP – adhesive interfaces to lower levels along the
free surface of the adhesive fillet (Fig. 5(d)).
The thermal analyses of the adhesively bonded tee
joints with rigid base and flexible base subjected to
variable thermal boundary conditions show that the
temperature distributions in the members of the adhesive
tee joint are not uniform, and the thermal properties of
each joint member (adherends and adhesive layer) affect
considerably the heat transfer and its final temperature
distribution.
7. Geometrically non-linear stress analysis
This section concentrates on the geometrical non-linear
stress analysis of the adhesively bonded tee joint. The stress
analyses were carried out using the non-linear finite element
method based on the SSLD theory. The temperature
distributions determined based on the thermal analyses of
the tee joints with rigid base and flexible base were applied
to two adhesive tee joint configurations as thermal loads. In
Fig. 6. Structural bodyconditions of a tee joint with single support plus
angled reinforcement bonded to (a) a rigid base (BC-I), and (b) a flexible
plate (BC-II).
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
145
Fig. 7. The deformed geometries of the adhesively bonded tee joints bonded to (a) a rigid base (BC-I), and (b) a flexible plate (BC-II).
addition, the base of the adhesive tee joint and the upper
edge of the VP are fully fixed in the boundary condition-I
(BC-I, Fig. 6(a)) whereas the left and right bottom corners of
the flexible plate are fully fixed in the boundary condition-II
(BC-II, Fig. 6(b)). In the analyses, convergence criteria of
0.01 and 0.0001 were used for forces and displacements,
respectively.
The deformed and undeformed geometries of the
adhesive tee joints bonded to a rigid base and flexible
plate are compared in Fig. 7(a) and (b), respectively, based
on their geometrical non-linear stress analyses. In the rigid
base, (BC-I, Fig. 7(a)) the VP is forced to buckle since the
upper edge of the VP is fixed in all directions. The large
displacements are observed in its middle region and the free
end of the adhesive layer – VP interface is subjected to large
bending moment effects. Therefore, the large stresses and
deformations can be expected around the adhesive layer–
VP interface, and propagate throughout the vertical
adhesive layer to the joint base. In case of the tee joint
bonded to a flexible base (BC-II, Fig. 7(b)), both the VP and
HPs buckle significantly. The VP exhibits lower displacements and rotations than those in case of the rigid base (Fig.
7(a)). However, since the left and right regions of the HP
buckle considerably, the high bending moments causing
peeling stresses are expected around the free ends of the
horizontal adhesive layer. In addition, the free end of the
vertical adhesive layer is also subjected to high displacements and rotations, consequently, stress and strain
concentrations are expected around this region and
propagate throughout the vertical adhesive layer to the
joint base similar to the tee joint bonded to a rigid base.
The normal stress sxx, szz and shear stress sxz distributions
in the magnified joint region are plotted in Fig. 8(a) –(c),
respectively, in order to show the stress states of the tee joint
bonded to a rigid base (BC-I) in detail. The normal stresses
sxx concentrated around the free ends of horizontal adhesive
layer, and propagate through the horizontal adhesive layer to
Fig. 8. The normal stress (a) sxx, (b) szz, and (c) shear sxz stress
distributions in the joint region for the boundary condition BC-I (rigid base,
all stresses in MPa).
146
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
Fig. 9. The von Mises stress distributions in critical adhesive regions: (a) the left horizontal adhesive fillet, (b) the right horizontal adhesive fillet, (c) the vertical
adhesive fillet, (d) the bottom adhesive layer (rigid base, BC-I, all stresses in MPa).
the middle joint region (Fig. 8(a)). The horizontal regions of
the left support, the angled reinforcement member and
especially the elbow regions of the support and angled
reinforcement member also experience high normal stress
sxx distributions. The normal stress szz distributions reach
considerably high levels through the vertical region of the tee
joint in which the left support and the angled reinforcement
member being an extension of the VP are bonded by the
vertical adhesive layer (Fig. 8(b)). They concentrate around
the free end of the vertical adhesive layer, and in the elbow
regions of the support and the reinforcement member. In
addition, the shear stress sxz concentrations occur around the
free ends of the vertical and horizontal adhesive layers and in
the elbow region of the tee joint (Fig. 8(c)). However, when
the normal and shear stress components are compared, the
normal stresses appear as more critical stresses. Since the tee
joint bonded to a rigid base experiences considerable stress
levels in the joint region, the first adherend failure can be
expected from the outer layers of the laminate plates,
especially from the outer surfaces of elbows, and in the
horizontal sections of the support and the angled reinforcement member. Conversely the first adhesive failure can
initiate from the adhesive fillets at the adhesive free ends, and
then propagate along the adherend –adhesive interfaces or
through the adhesive layer. In this case, the critical adhesive
region would be the adhesive fillet at the free end of the
vertical adhesive layer.
The von Mises stress distributions are also plotted in the
left and right horizontal adhesive fillets (RHAF), in the VAF
and in the bottom adhesive layer (BAL) of the tee joint
bonded to a rigid base (BC-I) as shown in Fig. 9(a) – (d),
respectively. The stress concentrations in the LHAF occur
around the free end of the adhesive –rigid base interface,
and propagate towards the rounded support corner (Fig.
9(a)). The similar stress distributions are observed in the
RHAF, such that, the stress concentrations occur around the
free end of the adhesive –rigid base interface and spread
through the rounded support corner (Fig. 9(b)), but at lower
levels 50% in comparison with the LHAF (Fig. 9(a)). In the
VAF, the stress distribution varies from high levels at
the free end of the adhesive – VP interface to lower levels at
the rounded corner of the support (Fig. 9(c)). The adhesive
regions close to the support and angled reinforcement –
adhesive interfaces experience high stress levels in the BAL
(Fig. 9(d)). The LHAF and VAF are more critical adhesive
regions. However, since the stresses in the BAL and in the
RHAF are close to those in the VAF, the first adhesive
failures can be expected in the LHAF and in the VAF.
In addition, the joint region of the tee joint bonded to a
flexible base (BC-II) is magnified in order to show the stress
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
Fig. 10. The normal stress (a) sxx, (b) szz, and (c) shear sxz stress
distributions in the joint region for the boundary condition BC-II (flexible
base, all stresses in MPa).
distributions in detail, and the normal stress sxx, szz and
shear stress sxz distributions in the joint region are plotted in
Fig. 10(a) – (c), respectively. The normal stresses sxx
concentrate through the horizontal joint region along
which the HP, the left support and the angled reinforcement
member are bonded by the horizontal adhesive layer (Fig.
10(a)). The left and right free ends of the horizontal
adhesive layer and especially the middle region near the
horizontal adhesive layer of the HP experience considerably
high stress concentrations. Due to high normal stress sxx the
middle region of the HP, and its neighbourhood are more
critical regions, and the adhesive free ends. The normal
stresses sxx are 2– 3 times higher than the normal stress szz
(Fig. 10(b)) and the shear stress sxz (Fig. 10(c)),
respectively. On the contrary, the substantial normal stress
szz distributions occur the joint region through which the
support and the VP are bonded the vertical adhesive layer,
147
and in the elbow regions of the support and the angled
reinforcement member (Fig. 10(b)). In addition, the VAF
undergoes the normal stress szz concentrations propagating
through the vertical adhesive layer towards the base of the
tee joint. Therefore, the VAF and the elbow regions of the
support and the angled reinforcement member are critical
regions in terms of the normal stress szz (Fig. 10(b)). The
shear stress sxz concentrations are observed around the
horizontal and VAFs, and the elbow regions of the support
and the angled reinforcement member (Fig. 10(c)).
However, the shear stress sxz levels are 50 – 77% lower
than the normal stresses. When the normal and shear stress
distributions in the joint region are considered the horizontal
and vertical adhesive layers, the middle region of the HP
and the elbow regions are critical regions. However, the first
adhesive failure can be expected inside the adhesive fillets at
the adhesive free ends, and especially along the adhesivesupport, the adhesive – HP and the adhesive-angled
reinforcement member interfaces in the bottom of the
adhesive tee joint. The most critical adherend regions
appear as the elbows of the support and the VP, and
particularly the middle regions of the HP are bonded to
other joint members. It would be better to support the lower
surface of the HP by a thin metal sheet (by bonding) in order
to prevent the failure of the HP along its lower and upper
surfaces. When the normal and shear stresses (Fig. 10) are
compared with those in the tee joint bonded a rigid base
(Fig. 8) the normal stress sxx levels are eight times higher
than those in the rigid base configuration (Fig. 8) whereas
the normal stress szz and the shear stress szx distributions
remain at close levels.
The von Mises stress distributions are plotted inside the
LHAF and RHAFs, the VAF and in the BAL as shown in
Fig. 11(a) –(d), respectively, in order to show the critical
locations at which the first adhesive cracks may initiate. In
case of the left and right adhesive fillets, the stress
concentrations occur at the free ends of the adhesive – HP
interface, and the von Mises stresses propagate uniformly
through the adhesive caps to lower levels around the
rounded support or angled reinforcement corners (Fig. 11(a)
and (b)). Similarly, the free end of the adhesive – VP
interface in the VAF experiences high von Mises stress
concentrations (Fig. 11(c)), and stress levels decrease
uniformly towards the rounded corner of the support. In
the BAL, whereas a large adhesive region experiences
comparatively low von Mises stress distributions, the
adhesive regions close to the adhesive-support, adhesiveangled reinforcement, and adhesive – HP interfaces undergo
considerably high stresses (Fig. 11(d)). Although all these
adhesive regions exhibit close stress levels, the first
adhesive cracks may be expected to initiate inside the
LHAF and the VAF. When the adhesive stresses at the
critical locations in the rigid base and flexible base
configurations are compared, the von Mises stress levels
are higher by 32% than those in the LHAF of the rigid base
configuration (Fig. 9(a)) whereas the von Mises stresses in
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M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
Fig. 11. The von Mises stress distributions in critical adhesive regions: (a) the left horizontal adhesive fillet, (b) the right horizontal adhesive fillet, (c) the
vertical adhesive fillet, (d) the bottom adhesive layer (flexible base, BC-II, all stresses in MPa).
the other critical adhesive regions remain at similar levels
(Figs. 9 and 11).
The stress concentrations occur around the adhesive
free ends due to geometrical and material discontinuities
is known as the edge effects in the adhesive joints. Since
the first probable crack initiations can be expected from
these adhesive regions, the stress concentrations should be
reduced by modifying adherend edges or increasing
bonding area [1,2]. For this reason, the effect of the
support and angled reinforcement lengths on the peak
stresses at the critical locations inside the adhesive fillets,
and in the horizontal and VPs of the tee joint
configurations bonded to a rigid base and a flexible base
were investigated for the support and angled reinforcement lengths a ¼ 20, 30, 40, 50 and 60 mm based on both
the thermal analysis and the geometrically non-linear
elastic stress analysis.
In case of the tee joint bonded to a rigid base, the
variations of the normal stresses sxx and szz, and the shear
stress sxz at the critical locations inside LHAF, the RHAF,
the VAF, in BAL and at the VP are tabulated in Table 3 for
different support and angled reinforcement lengths. Increasing the support length resulted in decrease of 2.4, 17.6 and
7.4% in the peak normal sxx and szz, and shear sxz stresses
in the LHAF, respectively, while they are reduced by 5.9,
16.5 and 7.1% in the RHAF, respectively. However,
the support length has a negligible effect on the normal
and shear stress components in the VAF, BAL and VP,
respectively, except the shear stress in the BAL (a decrease
of 6.8%). In addition, after the support length a ¼ 30 mm
corresponding to a ratio of the VP length to the support
length L/a ¼ 12 its effect on the peak normal and shear
stresses become negligible.
In case of the flexible base, the variations of the normal
sxx, szz and shear sxz stress components at the critical
locations inside the LHAF, the RHAF, the VAF, in the
BAL, at the VP and the HP are tabulated in Table 4 for
different support and angled reinforcement lengths. In the
LHAF, the normal stress szz is reduced by 4.2% whereas
increasing the support length causes increase of 10 and 5.8%
in the normal sxx and shear sxz stresses, respectively. In the
RHAF, increasing the support length causes decrease of 2.6,
4.4 and 5.3% in the normal and shear stresses, respectively.
Its effect on the normal and shear stresses in the VAF, BAL
and at the VP is negligible. However, it results in increase of
14.5, 15.8 and 15% at the normal and shear stresses in the
HP. The LHAF, the VAF (Fig. 11(a) and (c)), and the HP
(Fig. 10(a) and (c)) are the most critical regions in the tee
joint with flexible base. The peak normal and shear stresses
in these critical regions cannot be reduced reasonably by
increasing the support and reinforcement lengths. Therefore, bonding a thin metal sheet along the lower surface of
149
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
Table 3
Effect of the support and angled reinforcement length a on the peak stress
components (in MPa) at the critical adhesive and plate locations of the
adhesively bonded tee joint with a rigid base (BC-I, rigid base; Fig. 6(a))
a (mm)
sxx
PVa
szz
PVa
sxz
PVa
Table 4
Effect of the support and angled reinforcement length a on the peak stress
components (in MPa) at the critical adhesive and plate locations of the
adhesively bonded tee joint with a flexible base (BC-II, flexible base; Fig.
6(b))
szz
PVa
sxz
Left horizontal adhesive fillet
20
214.00
0.00
30
214.40
2.86
40
214.90
6.43
50
215.90
13.57
60
215.40
10.00
212.00
211.50
211.50
210.50
211.70
0.00
24.17
24.17
212.50
22.50
28.85
28.81
29.04
29.50
29.36
0.00
20.45
2.15
7.34
5.76
0.00
24.25
26.52
24.82
27.08
Right horizontal adhesive fillet
20
11.60
0.00
30
11.40
21.72
40
11.40
21.72
50
12.20
5.17
60
11.30
22.59
9.50
9.10
9.08
8.09
9.15
0.00
24.21
24.42
214.84
23.68
6.93
6.64
6.62
6.87
6.56
0.00
24.18
24.47
20.87
25.34
29.14
29.20
29.21
29.45
29.15
0.00
0.66
0.77
3.39
0.11
Vertical adhesive fillet
20
11.20
0.00
30
11.20
0.00
40
11.20
0.00
50
10.10
29.82
60
11.10
20.89
13.20
13.30
13.30
14.00
13.20
0.00
0.76
0.76
6.06
0.00
8.19
8.22
8.21
8.45
8.17
0.00
0.37
0.24
3.17
20.24
0.00
0.66
0.66
0.66
0.00
24.41
24.24
24.17
24.13
24.11
0.00
23.85
25.44
26.35
26.80
Bottom adhesive layer
20
11.30
30
11.20
40
11.20
50
11.20
60
11.20
0.00
20.88
20.88
20.88
20.88
11.90
11.70
11.70
11.70
11.60
0.00
21.68
21.68
21.68
22.52
4.09
4.04
4.02
4.01
4.00
0.00
21.22
21.71
21.96
22.20
0.00
0.59
1.48
3.85
3.25
23.79
23.81
23.82
23.76
23.82
0.00
0.53
0.79
20.79
0.79
Horizontal plate
20
22.70
30
25.80
40
26.00
50
20.00
60
23.20
0.00
13.66
14.54
211.89
2.20
3.93
5.30
5.11
4.59
4.55
0.00
34.86
30.03
16.79
15.78
3.52
4.24
4.04
3.12
3.56
0.00
20.45
14.77
211.36
1.14
0.00
0.00
0.00
22.08
20.52
23.60
23.60
23.60
24.00
23.50
0.00
0.00
0.00
1.69
20.42
3.22
3.23
3.23
3.16
3.21
0.00
0.31
0.31
21.86
20.31
a (mm)
Left horizontal adhesive fillet
20
221.20
0.00
30
220.90 21.42
40
220.70 22.36
50
222.30
5.19
60
220.80 21.89
217.60
215.70
214.90
213.30
214.50
0.00
210.80
215.34
223.86
217.61
213.60
213.00
212.80
213.10
212.60
0.00
24.41
25.88
23.68
27.35
Right horizontal adhesive fillet
20
211.80
0.00
30
211.20 25.08
40
211.10 25.93
50
211.10 25.93
60
211.20 25.08
212.70
211.40
210.80
29.46
210.60
0.00
210.24
214.96
225.51
216.54
27.06
26.76
26.60
26.72
26.56
Vertical adhesive fillet
20
212.30
0.00
30
212.40
0.81
40
212.30
0.00
50
210.10 29.76
60
211.10 21.63
214.50
214.70
214.70
215.50
214.60
0.00
1.38
1.38
6.90
0.69
Bottom adhesive layer
20
215.30
30
215.40
40
215.50
50
215.50
60
215.50
0.00
0.65
1.31
1.31
1.31
215.10
215.20
215.20
215.10
215.10
0.00
0.45
0.45
21.57
20.22
233.80
234.00
234.30
235.10
234.90
Vertical plate
20
24.47
30
24.49
40
24.49
50
24.40
60
24.48
a
Percent variation ðPV; %Þ ¼
lStress value of any al 2 lStress value of a ¼ 20 mml
£ 100:
Stress value of a ¼ 20 mm
the HP, in which high normal stress concentrations occur,
would increase the stiffness of the HP.
Vertical plate
20
30
40
50
60
a
8. Conclusions
In this study, the thermal analysis and the geometrically
non-linear stress analysis of an adhesively bonded tee joint
were carried out using the non-linear finite element method.
The tee joint configurations bonded to a rigid base and a
flexible base were considered. The variable thermal
boundary conditions were specified along the outer
boundaries of the adhesive tee joints, i.e. the air flows
with different velocity, temperature and at the different flow
directions to the plate surfaces. Since the adhesive tee joints
consist of adhesive layer and adherends having different
mechanical and thermal properties, the non-uniform temperature distributions occurred in the adhesive tee joints,
PVa
sxx
3.85
3.85
3.85
3.77
3.83
PVa
Percent variation ðPV; %Þ ¼
lStress value of any al 2 lStress value of a ¼ 20 mml
£ 100:
Stress value of a ¼ 20 mm
consequently, non-uniform thermal strain distributions. In
addition, the edges of the HP and VPs were restrained partly
or completely, and then the thermal stress distributions of
two adhesive tee joints were determined based on their
non-uniform thermal strain distributions using the SSLD
theory. In case of the rigid base, the VP of the adhesive tee
joint was buckled and the upper free end of the vertical
adhesive layer experienced large displacements. The tee
joint with flexible base had similar deformations for the VP
and the free ends of the vertical adhesive layer; moreover,
its HP was deformed (buckled) considerably. Thus, the left
and right free ends of the horizontal adhesive layer and
150
M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150
the middle region of the HP had evident deformations. The
high normal stresses occurred in the horizontal and vertical
sections of the joint regions of both tee joints. Furthermore,
the horizontal and vertical adhesive fillets and the BAL were
subjected to high stress concentrations in both cases of the
rigid and flexible bases. In general, the stress levels reached
a maximum at the free ends of the HP or VP – adhesive
interfaces and distributed uniformly through adhesive fillets
towards the rounded corners of the support and the angled
reinforcement member. However, the most critical adhesive
regions were the LHAF, VAF and the BAL. The effect of
the support length on the peak stresses in the adhesive layer
and in the HP and VPs was also investigated for a certain
range of the support length. It was observed that in case of
the adhesive tee joint with a rigid base, the support length
has an effect of decreasing the peak stresses in the RHAFs
whereas its effect on the peak stresses in the VAF, in the
BAL and at the VP. The support length had an effect of
decreasing the peak stresses at the critical locations inside
the LHAF and RHAF, and in the HP whereas its effect on
the peak stresses in the VAF, in the BAL and in the VP is
insignificant. The thermal stress analyses for certain range
of the support and angled reinforcement length showed that
a ratio of the plate length to the support length L=a ¼ 12 is
reasonable for the design of the adhesive tee joints subjected
to similar thermal and structural boundary conditions.
Finally, the thermal loads result in non-uniform temperature distributions in the adhesive joints due to the different
thermal properties of the adhesive and composite plates,
consequently non-uniform thermal strain distributions arise.
When some additional structural constraints are attributed to
the adhesive joints, they may experience considerably high
stress and strain distributions. In case large displacements
and displacement gradients are observed, the SSLD theory
would predict reasonably accurate stress and deformation
states of the adhesive joints providing that the adhesive and
composite adherends have small strains.
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