1 adhesive bonded joints - altair university2018/06/05 · v.1.2 version 4.4 theoretical background...
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1 ADHESIVE BONDED JOINTS
Flemming Mortensen and Ole Thybo Thomsen (Aalborg University, Institute of Mechanical Engineering, Denmark, 2000)
The method used in ESAComp for engineering analysis of adhesive bonded joints of various complexities is
presented. The joints considered are divided in two types: standard and advanced. The standard joints consist of
two or three adherends bonded together with a straight continuous adhesive layer parallel to the in-plane
direction of the adherends. The advanced joints consist of two adherends bonded together with either a single or double-sided scarfed adhesive interface. The adherends are modelled as beams or plates in cylindrical bending.
They are formed from laminates with arbitrary lay-ups using the classical lamination theory (CLT). The adhesive
layer is modelled by a two-parameter elastic foundation model, where the adhesive layer is assumed composed
of a continuous layer of linear tension/compression and shear springs. Since non-linear effects in the form of
adhesive plasticity play an important role in the load transfer, the analysis allows inclusion of non-linear
adhesive properties by an iterative method based upon the linear-elastic approach. The load and boundary
conditions can be chosen arbitrarily. Approaches for predicting the cohesive failure in the adhesive layers and
laminate failure in the joint area are also presented.
SYMBOLS
i
jkA Element of the adherend in-plane stiffness matrix
i
jkB Element of the adherend coupling stiffness matrix
cfi Constant used in system equations
CS Constant used in effective stress formulation
CV Constant used in effective stress formulation
i
jkD Element of the adherend bending stiffness matrix
Ea Adhesive elastic modulus
e Effective strain
eN Effective strain in the N’th iteration step
Ga Adhesive elastic shear modulus
hfi Constant used in system equations
I1 First invariant of the general strain tensor
I2D Second invariant of the deviatoric strain tensor
J1 First invariant of the general stress tensor
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J2D Second invariant of the deviatoric stress tensor
kgi Constant used in system equations
L Length of the overlap zone in the adhesive joint
L1, L2 Length of adherends outside the overlap zone
i
xxM , i
xyM , i
yyM Adherend moment resultants
mfi Constant used in system equations
i
xxN , i
xyN , i
yyN Adherend in-plane stress resultants
i
xxQ , i
xyQ , i
yyQ Adherend shear force resultants
RFadh Reserve factor for cohesive failure of adhesive (linear or non-linear
adhesive model)
RFadh,prop Reserve factor for proportional limit of adhesive (non-linear adhesive
model)
RFFPF Reserve factor for adherend (laminate) first ply failure in the vicinity
of the joint
s Effective stress
*
Ns Calculated stress in the N’th iteration step
sN Experimental stress in the N’th iteration step
sprop Stress proportional limit
DsN Difference between calculated and experimental stress
ti Adherend thickness
ti(x) Adherend thickness as a function of x
ta Adhesive layer thickness
x Adherend in-plane coordinate system in the longitudinal direction
iu0 Longitudinal displacement of the adherend mid-plane (x-direction)
ui Longitudinal displacement of the adherend (x-direction)
iv0 Displacement of the adherend mid-plane in the width direction (y-
direction)
vi Displacement of the adherend in the width direction (y-direction)
wi Transverse displacement of the adherend (z-direction)
a, a1, a2 Transition angles of scarfed adherend
i
xb , i
yb Rotation of mid-plane normal to the adherend
d Weight factor for the change in elastic modulus
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ef Principal strains (f = 1, 2, 3)
l Ratio between compressive and tensile yield stress
sa Adhesive layer out-of-plane normal stress
san Adhesive layer out-of-plane normal stress
sani Adhesive layer out-of-plane normal stress
sc Compressive yield stress
sf Principal stresses (f = 1, 2, 3)
st Tensile yield stress
tax, tay Adhesive layer shear stress
tan, Adhesive layer shear stress
taxi Adhesive layer shear stress
Subscripts
a Adhesive layer
i Adherend (i = 1, 1a, 1b, 2, 2a, 2b, 3)
N Iteration number for non-linear tangent modulus
,x Differentiation with respect to the x-coordinate
,y Differentiation with respect to the y-coordinate
ult Ultimate
Superscripts
i Adherend (i = 1, 1a, 1b, 2, 2a, 2b, 3)
end Adherend end thickness at the overlap zone
end,L Adherend thickness at the left end of the overlap zone
end,R Adherend thickness at the right end of the overlap zone
t Identifier used for non-linear tangent modulus
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1.1 INTRODUCTION
Joining of composite structures can be achieved through use of bolted, riveted or adhesive
bonded joints. The performances of the mentioned joint types are severely influenced by the
characteristics of the layered composite materials, but adhesive bonded joints provide a much
more efficient load transfer than mechanically fastened joints. Accurate analysis of adhesive
bonded joints, for instance by using the finite element method, is an elaborate and
computationally demanding task as described by Crocrombe et al. [3], Harris et al. [11] and
Frostig et al. [5]. Hence, there is an obvious need for analysis and design tools that can
provide accurate results for preliminary design purposes.
This chapter introduces the analysis approach used in ESAComp for determining the stress
and displacement fields in commonly used adhesive bonded joint configurations. The last
sections deal with the handling of plasticity effects in the adhesive layers and failure
prediction of bonded joints.
The bonded joint types considered in ESAComp are:
· Single lap joint (SL)
· Single strap joint (SS)
· Bonded doubler (BD)
· Double lap joint (DL)
· Double strap joint (DS)
· Single sided scarfed lap joint (SSC)
· Double sided scarfed lap joint (DSC)
These joint types are illustrated in Figure 1.1. All the joint configurations can be composed of
similar or dissimilar laminates with an arbitrary lay-up. The joints are subjected to a general
loading condition as shown in Figure 1.2.
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Figure 1.2 Schematic illustration of an adhesive single lap joint subjected to a general loading condition.
According to the complexity of the joints, the lap and strap joints and bonded doublers can be
referred to as standard joints. In these joints, the adhesive layer or layers are parallel to the in-
plane direction of the adherends. Correspondingly, the scarfed joints can be referred to as
advanced joints. The advanced joints are more efficient due to the reduced eccentricity of the
load path, but the advanced joints are also much more expensive to manufacture and they are
therefore only used for high-performance applications.
1.2 STRUCTURAL MODELLING
The structural modelling is carried out by adopting a set of basic restrictive assumptions for
the behaviour of bonded joints. Based on these restrictions, the constitutive and kinematic
relations for the adherends are derived, and the constitutive relations for the adhesive layers
are adopted. Finally, the equilibrium equations for the joints are derived and, by combining all
these equations and relations, the set of governing equations is obtained.
1.2.1 Model dimensions
The adhesive bonded joint configurations were introduced in Section 1.1. The adherend
thicknesses are given by t1 and t2 for all the joints outside the overlap zone. For the double lap
joint, the thickness of the third adherend (lower adherend) is t3. The adherend length outside
the overlap is L1 and L2, and the length of the adhesive layer is L as illustrated in Figure 1.3.
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Figure 1.3 Illustration of adherend lengths and adhesive layer length and thickness.
Inside the overlap zone (0 £ x £ L) the thicknesses are:
· Single lap joint, single strap joint, and bonded doubler:
( ) ( ) 2211 , txttxt == (1.2.1)
· Double lap and double strap joint:
( ) ( ) ( ) 332211 ,, txttxttxt === (1.2.2)
· Single sided scarfed lap joint:
( ) ( ) xL
tttxtx
L
tttxt
endend
end
2222
1111 ,
--=
--= (1.2.3)
Where the superscript end in endt1 and endt2 refers to the thicknesses of the adherends at the free
ends of the overlap, see Figure 1.4.
Figure 1.4 Thicknesses and scarf angle for single sided scarfed lap joint.
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· Double sided scarfed lap joint:
( ) ( )
( ) ( )Lx
xL
tttxtx
L
tttxt
xL
tttxtx
L
tttxt
Rend
b
Lend
bLend
bb
Rend
a
Lend
aLend
aa
Lend
b
Rend
bb
Lend
a
Rend
aa
££
ïïþ
ïïý
ü
--=
--=
--=
--=
0
,
,
,
2
,
2,
22
,
2
,
2,
22
,
2
,
212
11
,
2
,
212
11
(1.2.4)
Where the subscripts a and b and the superscript end in Lend
at ,
2 and Rend
bt ,
2 refer to the thickness
of adherend 2 at the left (L) and right (R) ends of the overlap above and below adherend 1
(Figure 1.5).
Figure 1.5 Thicknesses for double sided scarfed lap joint.
1.2.2 Basic assumptions for the structural modelling
The basic restrictive assumptions for the structural modelling are the following:
Adherends
· The adherends are modelled as beams or plates in cylindrical bending, using ordinary
“Kirchhoff” plate theory (“Love-Kirchhoff” assumptions).
· The constitutive behaviour of the adherends is obtained using the classical lamination
theory (CLT). No restrictions are set on the laminate lay-up, i.e. unsymmetric and
unbalanced laminates can be included in the analysis.
· The laminates are assumed to obey linear-elastic constitutive laws.
· The strains are small and the rotations are very small.
Adhesive layers
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· The adhesive layers are modelled as continuously distributed linear tension/compression
and shear springs.
· Non-linear adhesive properties are considered by using a secant modulus approach for the
non-linear tensile stress-strain relationship in conjunction with a modified Von Mises
yield criterion.
Loads and boundary conditions
· The structural model allows boundary conditions to be chosen arbitrarily as long as the
system is in equilibrium. Sets of prescribed external loads (in-plane and out-of-plane
forces and bending moments) and geometric boundary conditions are defined in
ESAComp to avoid selections of inconsistent boundary conditions, which can lead to
singularity problems in the system of equations.
The system of governing equations is set up for two different cases, i.e. the adherends are
modelled as plates in cylindrical bending or as wide beams. In the following, the case where
the adherends are modelled as plates in cylindrical bending is primarily considered since the
modelling of the adherends as beams is a reduced case of this.
1.2.3 Constitutive relations for adherends modelled as plates
For the purposes of the present investigation, and with references to Figures 1.6 and 1.2,
cylindrical bending can be defined as a wide plate (in the y-direction), where the displacement
field can be described as a function of the longitudinal coordinate only. As a consequence, the
displacement field in the width direction is uniform. Thus, the displacement field can be
described as
( ) ( ) ( )xwwxvvxuu iiiiii === ,, 0000 (1.2.5)
where u0 is the mid-plane displacement in the longitudinal direction (x-direction), v0 is the
mid-plane displacement in the width direction (y-direction), and w is the displacement in the
transverse direction (z-direction). The displacement components u0, v0 and w are all defined
relative to the mid-plane of the laminates, and i = 1, 2, 3 corresponds to the laminates 1, 2 and
3, respectively.
Based on the earlier assumptions, the following holds also true:
0,,,0,0 ==== i
yy
i
y
i
y
i
y wwvu (1.2.6)
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Figure 1.6 Schematic illustration of adhesive single lap joint “clamped” between two vertical laminates, which
prevent the adherends of the single lap joint from moving and rotating freely in the width direction. This
represents the conceptual interpretation of cylindrical bending as defined in the present formulation.
In the concept of “cylindrical bending”, the boundary conditions at the boundaries in the
width direction are not well defined. However, it is assumed that there are some restrictive
constraints on the boundaries, such that the boundaries are not capable of moving freely. It
should be noted that the concept of “cylindrical bending” is not unique, and that other
definitions than the one used in the present formulation can be adopted, see Whitney [20].
Substitution of the quantities in Eq. (1.2.5) into the constitutive relations for a laminated
composite material gives the constitutive relations for a laminate (i) in cylindrical bending
[20]:
i
xx
ii
x
ii
x
ii
xy
i
xx
ii
x
ii
x
ii
xy
i
xx
ii
x
ii
x
ii
yy
i
xx
ii
x
ii
x
ii
yy
i
xx
ii
x
ii
x
ii
xx
i
xx
ii
x
ii
x
ii
xx
w-DvBuBMw-BvAuAN
w-DvBuBMw-BvAuAN
w-DvBuBMw-BvAuAN
,16,066,016,16,066,016
,12,026,012,12,026,012
,11,016,011,11,016,011
,
,
,
+=+=
+=+=
+=+=
(1.2.7)
where i
jkA , i
jkB and i
jkD (j,k = 1,2,6) are the extensional, coupling and flexural rigidities based
on the classical lamination theory (see Part III, Chapter 2). i
xxN , i
yyN and i
xyN are the in-plane
stress resultants i
xxM , i
yyM and i
xyM are the moment resultants. For the joints with scarfed
adherends the rigidities i
jkA , i
jkB and i
jkD (j,k = 1,2,6) within the overlap zone are changed as a
function of the longitudinal coordinate in accordance with their definition, i.e. i
jkA is changed
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linearly, i
jkB is changed parabolically and i
jkD is changed cubically (j,k = 1,2,6) between their
values at the ends of the overlap zone. This is of course an approximation since the actual
stiffnesses of the laminates are changing by changes within the layers as a function of the
longitudinal direction.
1.2.4 Constitutive relations for adherends modelled as beams
Modelling of the adherends as wide beams can be considered as a special case of cylindrical
bending. When the adherends are modelled as beams, the width direction displacements are
not considered, and only the longitudinal and vertical displacements are included. Thus, the
displacement field in Eq. (1.2.5) is reduced to
( ) ( )xwwxuu iiii == ,00 (1.2.8)
For this case the constitutive relations for a composite beam are reduced to
i
xx
ii
x
ii
xx
i
xx
ii
x
ii
xx w-DuBMw-BuAN ,11,011,11,011 , == (1.2.9)
1.2.5 Kinematic relations
From the “Love-Kirchhoff” assumptions, the following kinematic relations for the laminates
in cylindrical bending are derived:
0,, ,0 =-=+= i
y
i
x
i
x
i
x
ii wzuu bbb (1.2.10)
here ui is the longitudinal displacement, iu0 is the longitudinal displacement of the mid-plane,
and wi is the vertical displacement of the i’th laminate.
The kinematic relations of Eq. (1.2.10) are the same for the beam case as for the cylindrical
bending case except that all the variables associated with the width direction are nil.
1.2.6 Constitutive relations for the adhesive layer
The coupling between the adherends is established through the constitutive relations for the
adhesive layer, which as a first approximation is assumed homogeneous, isotropic and linear
elastic. The constitutive relations for the adhesive layer are established by use of a two-
parameter elastic foundation approach, where the adhesive layer is assumed to be composed
of continuously distributed shear and tension/compression springs. The constitutive relations
of the adhesive layer are suggested in accordance with Thomsen [16–17], Thomsen et al. [18]
and Tong [19]:
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( ) ( ) ( )( )( ) ( )( )
( )( )ji
ji
ww
vvvv
xtuxtuuu
ji
t
E
a
ji
t
Gji
t
G
ay
j
xj
ji
xi
i
t
Gji
t
G
ax
a
a
a
a
a
a
a
a
a
a
¹
=
ïþ
ïý
ü
-=
-=-=
---=-=,3,2,1,
00
00
st
bbt (1.2.11)
where i and j are the numbers of the adherends, Ga is the shear modulus, and Ea is the elastic
modulus of the adhesive layer.
The consequence of using the simple spring model approach for the modelling of the adhesive
layers is that it is not possible to satisfy the equilibrium conditions at the (free) edges of the
adhesive. However, in real adhesive joints no free edges are present at the ends of the overlap,
since a fillet of surplus adhesive, a so-called spew-fillet, is formed at the ends of the overlap
zone. This spew fillet allows for the transfer of shear stresses at the overlap ends. Modelling
of the adhesive layer by spring models has been compared with other known analysis methods
such as finite element analysis (Crocrombe et al. [3] and Frostig et al. [5]) and a high-order
theory approach including spew fillets (Frostig et al. [5]). The results show that the overall
stress distribution and the predicted values are in very good agreement.
1.3 EQUILIBRIUM EQUATIONS
The equilibrium equations are derived based on equilibrium elements inside and outside the
overlap zone for each of the considered joint types.
1.3.1 Adherends outside the overlap zone
The equilibrium equations are derived for plates in cylindrical bending since the equilibrium
equations for the beam modelling can be considered as a reduced case of this. The equilibrium
equations outside the overlap zone for each of the adherends, i.e. in the regions -L1 £ x £ 0
and L £ x £ L + L2, are all the same (see Figure 1.2) and are derived based on Figure 1.7:
21
,
,
,
,
,
00
0
0
LLxLandxL
QM
QM
Q
N
N
i
y
i
xxy
i
x
i
xxx
i
xx
i
xxy
i
xxx
+££££-
ïïï
þ
ïïï
ý
ü
=
=
=
=
=
(1.3.1)
where i correspond to the adherends i = 1, 2, 3.
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Figure 1.7 Equilibrium elements of adherend outside the overlap zone; -L1 £ x £ 0 and L £ x £ L + L2.
1.3.2 Single lap and single strap joints
The equilibrium equations inside the overlap zone for the single lap joint and the single strap
are derived based on Figure 1.8. For the single lap joint the adherend thickness’ will remain
the same in the entire overlap zone as specified by Eqs. (1.2.1), thus giving the equations:
Lx
ttQM
ttQM
ttQM
ttQM
NN
NN
aayyxxy
aayyxxy
aaxxxxx
aaxxxxx
axxaxx
ayxxyayxxy
axxxxaxxxx
££
ïïïï
þ
ïïïï
ý
ü
++=
++=
++=
++=
-==
=-=
=-=
0
2,
2
2,
2
,
,
,
222
,
111
,
222
,
111
,
2
,
1
,
2
,
1
,
2
,
1
,
tt
tt
sstttt
(1.3.2)
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Figure 1.8 Equilibrium element of adherends inside the overlap zone for joints with one adhesive layer and
straight adherends; 0 £ x £ L.
1.3.3 Bonded doubler
Inside the overlap zone for bonded doubler joint the equilibrium equations are derived based
on Figure 1.8. and Eqs. (1.2.1) and yields exactly the same equations as for the single lap joint
(see the previous section).
1.3.4 Double lap and double strap joint
The equilibrium equations inside the overlap zone for the double lap joint and the double strap
are derived based on Figure 1.9:
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Figure 1.9 Equilibrium element of adherends inside the overlap zone for joints with two adhesive layers and
straight adherends; 0 £ x £ L.
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Lx
ttQM
ttQM
Q
N
N
ttQM
ttQM
Q
N
N
ttttQM
ttttQM
Q
N
N
aayyxxy
aaxxxxx
axx
ayxxy
axxxx
aayyxxy
aaxxxxx
axx
ayxxy
axxxx
aay
aayyxxy
aax
aaxxxxx
aaxx
ayayxxy
axaxxxx
££
ïïïïïïïïïïïïïï
þ
ïïïïïïïïïïïïïï
ý
ü
+-=
+-=
=
=
=
++=
++=
-=
=
=
+-
++=
+-
++=
-=
--=
--=
0
2
2
2
2
22
22
23
2
33
,
232
33
,
2
3
,
2
3
,
2
3
,
12
1
22
,
12
1
22
,
1
2
,
1
2
,
1
2
,
21
2
11
1
11
,
21
2
11
1
11
,
21
1
,
21
1
,
21
1
,
t
t
s
tt
t
t
st
t
tt
tt
sstt
tt
(1.3.3)
1.3.5 Single sided scarfed lap joint
The equilibrium equations inside the overlap zone for the single sided scarfed lap joint are
derived based on Figure 1.10. They are different from the earlier ones due to the linear change
of the adherend thicknesses and the sloping bond line:
( ) ( )
( ) ( )
Lx
L
ttN
txtQ
M
L
ttN
txtQ
M
L
ttN
txtQ
M
L
ttN
txtQ
M
NN
NN
end
xy
aayy
xxyend
xy
aayy
xxy
end
xx
aaxx
xxxend
xx
aaxx
xxx
axxaxx
ayxxyayxxy
axxxxaxxxx
££
ïïïïïïï
þ
ïïïïïïï
ý
ü
--
++
=-
-
++
=
--
++
=-
-
++
=
-==
=-=
=-=
0
2
2
,2
2
2
2
,2
2
,
,
,
221
22
2
,
111
11
1
,
221
22
2
,
111
11
1
,
2
,
1
,
2
,
1
,
2
,
1
,
tt
tt
ss
tttt
(1.3.4)
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where the relationship between tax, sa in Eq. (1.3.4) and tan, san shown in Figure 1.10 is
established through equilibrium:
asaatsaasatt 22 coscossin,cossincos aaxanaaxan +=+= (1.3.5)
where a is the scarf angle of the adherends in the overlap zone (see Figure 1.4).
Figure 1.10 Equilibrium elements in the overlap zone for a single sided scarfed lap joint (scarf angles a); 0 £ x £ L.
The adherend thicknesses t1(x), t2(x) vary linearly through the overlap length as specified by
Eq. (1.2.3).
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1.3.6 Double sided scarfed lap joint
Finally, the equilibrium equations inside the overlap zone for the double-sided scarfed lap
joint are derived based on Figure 1.11:
( ) ( )
( ) ( )
( )
( )
( )
( )
Lx
L
ttN
txtQM
L
ttN
txtQM
Q
N
N
L
ttN
txtQM
L
ttN
txtQM
Q
N
N
txt
txtQM
txt
txtQM
Q
N
N
Lend
b
Rend
bb
xyab
ax
b
y
b
xxy
Lend
b
Rend
bb
xxab
ax
b
x
b
xxx
a
b
xx
ay
b
xxy
ax
b
xxx
Lend
a
Rend
aa
xyaa
ay
a
y
a
xxy
Lend
a
Rend
aa
xxaa
ax
a
x
a
xxx
a
a
xx
ay
a
xxy
ax
a
xxx
abay
aaayyxxy
abax
aaaxxxxx
aaxx
ayayxxy
axaxxxx
££
ïïïïïïïïïïïïïï
þ
ïïïïïïïïïïïïïï
ý
ü
-+
+-=
-+
+-=
=
=
=
--
++=
--
++=
-=
=
=
÷ø
öçè
æ+-÷
ø
öçè
æ++=
÷ø
öçè
æ+-÷
ø
öçè
æ++=
-=
--=
--=
0
22
22
22
22
22
22
,
2
,
2222
2
22
,
,
2
,
2222
2
22
,
2
2
,
2
2
,
2
2
,
,
2
,
22121
22
,
,
2
,
2212
1
22
,
1
2
,
1
2
,
1
2
,
212
111
11
,
2
12
1
11
11
,
21
1
,
21
1
,
21
1
,
t
t
st
t
t
t
s
tt
tt
tt
sstttt
(1.3.6)
where t1(x), t2a(x) and t2b(x) are the adherend thicknesses, according to Eq. (1.2.4), and ta1 and
ta2 are the adhesive layer thicknesses.
V.1.18 Version 4.4
Theoretical Background of ESAComp Analyses 3.12.2012
V Joints
1 Adhesive Bonded Joints
Figure 1.11 Equilibrium element of adherends inside the overlap zone for double sided scarfed lap joints; 0 £ x
£ L.
The relationship between tax1, sa1 in Eq. (1.3.6) and tan1, san1 shown in Figure 1.11 as well as
the relationship between tax2, sa2 in Eq. (1.3.6) and tan2, san2 shown in Figure 1.11 is
established through equilibrium:
( )2,1,coscossin,cossincos 22 =+=-= iiaiiiaxianiiiaiiaxiani asaatsaasatt (1.3.7)
Version 4.4 V.1.19
3.12.2012 Theoretical Background of ESAComp Analyses
V Joints
1 Adhesive Bonded Joints
where ai (i=1,2) is the scarf angles of the adherends in the overlap zone (see Figure 1.5).
1.4 THE COMPLETE SET OF SYSTEM EQUATIONS
From the equations derived, it is possible to form the complete set of system equations for
each of the bonded joint configurations. Thus, combination of the constitutive and kinematic
relations, i.e. Eqs. (1.2.7) and (1.2.10), together with the constitutive relations for the adhesive
layers, i.e. Eqs. (1.2.11), and the equilibrium equations lead to a set of 8 linear coupled first-
order ordinary differential equations describing the system behaviour of each of the
adherends. The total number of coupled first-order ordinary differential equations within the
overlap zone is therefore 16 for joints with two adherends inside the overlap zone, and 24 for
the joints with three adherends inside the overlap zone. Outside the overlap zones the system
behaviour for all the joints is described by 8 linear coupled first-order ordinary differential
equations, except for the double lap joint which has two adherends in the region L £ x £ L +
L2 and therefore is described by a set of 16 linear coupled first-order ordinary differential
equations in this region.
The set of governing equations for all the considered adhesive bonded joint types are
presented in this section. The governing equations presented are those where the adherends
are modelled as plates in cylindrical bending. The case where the adherends are modelled as
wide beams can be considered as a special case of cylindrical bending, and results in a
reduced set of the governing equations. However, to demonstrate that this is true the
governing equations for the case of adherends modelled as beams are also shown for the
single lap joint.
1.4.1 Single lap and single strap joints
From the equations derived, it is possible to form the complete set of system equations for the
problem. Thus combination of Eqs. (1.2.7), (1.2.10) and (1.3.1) yields for the laminate 1 and 2
in the areas -L1 £ x £ 0 and L £ x £ L + L2 (outside of overlap):
2,1
0
0
0
,
,
,
,
,9,8,7,0
,6,5,4,
,
,3,2,1,0
=
ïïïïï
þ
ïïïïï
ý
ü
=
=
=
=
++=
---=
-=
++=
i
Q
QM
N
N
MkNkNkv
MkNkNk
w
MkNkNku
i
xx
i
x
i
xxx
i
xxy
i
xxx
i
xxxi
i
xxyi
i
xxxi
i
x
i
xxxi
i
xxyi
i
xxxi
i
xx
i
x
i
x
i
xxxi
i
xxyi
i
xxxi
i
x
bb
(1.4.1)
V.1.20 Version 4.4
Theoretical Background of ESAComp Analyses 3.12.2012
V Joints
1 Adhesive Bonded Joints
Eqs. (1.4.1) constitute a set of eight linear coupled first-order ordinary differential equations.
The coefficients k1I – k9i (i = 1, 2) contain laminate stiffness parameters and are a result of
isolating i
xu ,0 , i
xv ,0 and i
xxw from i
xxN , i
xyN and i
xxM in Eqs. (1.2.7):
i
i
i
iii
i
ii
i
i
i
iii
iiiiiiiiiii
i
ii
i
ii
i
i
c
c
c
kck
c
kc
ck
c
kck
hkhkhkhkkhk
m
mk
m
mk
mk
2
3
2
319
2
21
2
8
2
117
23163215114
1
23
1
32
1
1
,1
,
,
,,1
--=-=-=
+=+==
-=-==
(1.4.2)
where the coefficients cji, hji and mji (j=1,2,3) are:
i
i
i
i
ii
i
i
i
i
i
i
i
i
i
ii
i
i
i
i
ii
i
i
i
i
i
i
i
i
i
i
i
i
i
i
ii
iii
ii
iii
i
hBc
AmhB
c
cAm
c
cAhBAm
cD
Bh
DcD
cBh
cD
cB
D
Bh
D
Bc
D
BBAc
D
BBAc
311
2
163211
2
3162
2
116111111
211
163
11211
3162
211
116
11
111
11
163
11
1616662
11
1116161
1,
,,
-=--=--=
=-=-=
=-=-=
(1.4.3)
Within the overlap zone, i.e. for 0 £ x £ L, combination of Eqs. (1.2.7), (1.2.10), (1.2.11) and
(1.3.2) yields for the laminate 1 and 2:
Version 4.4 V.1.21
3.12.2012 Theoretical Background of ESAComp Analyses
V Joints
1 Adhesive Bonded Joints
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
212
,
2222
0
21211
0
212
,
2
0
1
0
2
,
222
0
111
0
2
,
2
,92
2
,82
2
,72
2
,0
2
,62
2
,52
2
,42
2
,
22
,
2
,32
2
,22
2
,12
2
,0
211
,
2122
0
11111
0
111
,
2
0
1
0
1
,
222
0
111
0
1
,
1
,91
1
,81
1
,71
1
,0
1
,61
1
,51
1
,41
1
,
11
,
1
,31
1
,21
1
,11
1
,0
4242
22
4242
22
wt
Ew
t
EQ
t
tttGu
t
ttG
t
tttGu
t
ttGQM
vt
Gv
t
GN
t
tGu
t
G
t
tGu
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
wt
Ew
t
EQ
t
tttGu
t
ttG
t
tttGu
t
ttGQM
vt
Gv
t
GN
t
tGu
t
G
t
tGu
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
a
a
a
axx
x
a
aa
a
aax
a
aa
a
aaxxxx
a
a
a
axxy
x
a
a
a
ax
a
a
a
axxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
a
a
a
axx
x
a
aa
a
aax
a
aa
a
aaxxxx
a
a
a
axxy
x
a
a
a
ax
a
a
a
axxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
+-=
++
++
++
+-=
+-=
+++-=
++=
---=
-=
++=
-=
++
++
++
+-=
-=
---=
++=
---=
-=
++=
bb
bb
bb
bb
bb
bb
(1.4.4)
Eqs. (1.4.4) constitute a set of 16 linear coupled first-order ordinary differential equations.
1.4.2 Single lap joint with adherends modelled as beam
For the laminates 1 and 2 in the areas -L1 £ x £ 0 and L £ x £ L + L2 (outside of overlap)
combining Eqs. (1.2.10), (1.2.9), (1.2.11) together with the equilibrium equations yields:
2,1
0
0
,
,
,
,6,5,4,
,
,3,2,1,0
=
ïïïï
þ
ïïïï
ý
ü
=
=
=
---=
-=
++=
i
Q
QM
N
MkNkNk
w
MkNkNku
i
xx
i
x
i
xxx
i
xxx
i
xxxi
i
xxyi
i
xxxi
i
xx
i
x
i
x
i
xxxi
i
xxyi
i
xxxi
i
x
bb
(1.4.5)
V.1.22 Version 4.4
Theoretical Background of ESAComp Analyses 3.12.2012
V Joints
1 Adhesive Bonded Joints
Eqs. (1.4.5) constitute a set of six linear coupled first-order ordinary differential equations.
The coefficients k1i – k9i (i = 1, 2) contain the laminate stiffness parameters and are
determined by isolation of i
xu ,0 and i
xxw from i
xxN and i
xxM in Eq. (1.2.9):
( )iiiii
ii
iiii
iiii
i
i
D
BBii
BBADD
BB
Dkkk
BBAD
Bk
Ak
i
ii
1111111111
1111
11
423
11111111
112
11
1
1,
,1
11
1111
-+==
--=
-=
(1.4.6)
By comparison with the coefficients for the cylindrical bending case it is seen that the
coefficients for the beam case are strongly reduced and only contain few of the laminate
stiffness parameters.
Within the overlap zone, i.e. for 0 £ x £ L, the governing for laminate 1 and 2 are:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
212
,
2222
0
21211
0
212
,
222
0
111
0
2
,
2
,42
2
,32
2
,
22
,
2
,22
2
,12
2
,0
211
,
2122
0
11111
0
111
,
222
0
111
0
1
,
1
,41
1
,31
1
,
11
,
1
,21
1
,11
1
,0
4242
22
4242
22
wt
Ew
t
EQ
t
tttGu
t
ttG
t
tttGu
t
ttGQM
t
tGu
t
G
t
tGu
t
GN
MkNk
w
MkNku
wt
Ew
t
EQ
t
tttGu
t
ttG
t
tttGu
t
ttGQM
t
tGu
t
G
t
tGu
t
GN
MkNk
w
MkNku
a
a
a
a
xx
x
a
aa
a
aa
x
a
aa
a
aa
xxxx
x
a
a
a
a
x
a
a
a
a
xxx
xxxxxxxx
xx
xxxxxxx
a
a
a
axx
x
a
aa
a
aax
a
aa
a
aaxxxx
x
a
a
a
ax
a
a
a
axxx
xxxxxxxx
xx
xxxxxxx
+-=
++
++
++
+-=
+++-=
--=
-=
+=
-=
++
++
++
+-=
---=
--=
-=
+=
bb
bb
bb
bb
bb
bb
(1.4.7)
Eqs. (1.4.7) constitute a set of 12 linear coupled first-order ordinary differential equations. By
comparison with the equations for the cylindrical bending case it is seen that the equations
Version 4.4 V.1.23
3.12.2012 Theoretical Background of ESAComp Analyses
V Joints
1 Adhesive Bonded Joints
display the same overall appearance except that all variables associated with the width
direction are nil in Eqs. (1.4.7).
1.4.3 Bonded doubler
The governing equations for the bonded doubler joint are exactly the same as for the single
lap joint in the overlap zone and outside the overlap zone in the region L £ x £ L + L2.
1.4.4 Double lap joint
The governing equations for the double lap joint are exactly the same as for the single lap
joint in the region L1 £ x £ 0. The governing equations for laminate 1, 2 and 3 within the
overlap zone, i.e. for 0 £ x £ L, are derived by combining Eqs. (1.2.7), (1.2.10), (1.2.11) and
(1.3.3):
V.1.24 Version 4.4
Theoretical Background of ESAComp Analyses 3.12.2012
V Joints
1 Adhesive Bonded Joints
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2
1
11
1
12
,
2
1
21112
0
1
1111
1
11211
0
1
12122
,
2
0
1
11
0
1
12
,
2
1
122
0
1
11
1
111
0
1
12
,
2
,92
2
,82
2
,72
2
,0
2
,62
2
,52
2
,42
2
,
22
,
2
,32
2
,22
2
,12
2
,0
3
2
22
1
11
2
2
1
11
,
2
2
32123
0
2
2122
1
21112
0
1
111
1
2
1212
1
11111
0
2
212
1
1111
1
,
3
0
2
22
0
1
11
0
2
2
1
11
,
3
2
233
0
2
32
1
122
0
1
11
2
21
1
111
0
2
2
1
11
,
1
,91
1
,81
1
,71
1
,0
1
,61
1
,51
1
,41
1
,
11
,
1
,31
1
,21
1
,11
1
,0
4242
22
4242
4422
2222
wt
Ew
t
EQ
t
tttGu
t
ttG
t
tttGu
t
ttGQM
vt
Gv
t
GN
t
Gtu
t
G
t
Gtu
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
wt
Ew
t
Ew
t
E
t
EQ
t
tttGu
t
ttG
t
tttGu
t
ttG
t
tttG
t
tttGu
t
ttG
t
ttGQ
M
vt
Gv
t
Gv
t
G
t
GN
t
Gtu
t
G
t
Gtu
t
G
t
Gt
t
Gtu
t
G
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
a
a
a
axx
x
a
aa
a
aax
a
aa
a
aaxxxx
a
a
a
axxy
x
a
a
a
ax
a
a
a
axxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
a
a
a
a
a
a
a
axx
x
a
aa
a
aax
a
aa
a
aa
x
a
aa
a
aa
a
aa
a
aax
xxx
a
a
a
a
a
a
a
axxy
x
a
a
a
ax
a
a
a
ax
a
a
a
a
a
a
a
axxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
+-=
++
++
++
+-=
+-=
+++-=
++=
---=
-=
++=
--÷÷ø
öççè
æ+=
++
+-
++
++
÷÷ø
öççè
æ +-
++÷÷
ø
öççè
æ +-
+-
=
--÷÷ø
öççè
æ+=
+---÷÷ø
öççè
æ--÷÷
ø
öççè
æ+=
++=
---=
-=
++=
bb
bb
bb
bb
b
bbb
bb
(1.4.8)
( ) ( ) ( ) ( )
3
2
21
2
23
,
3
2
32323
0
2
2321
2
12321
0
2
23233
,
3
0
2
21
0
2
23
,
3
2
233
0
2
21
2
211
0
2
23
,
2
,93
2
,83
2
,73
3
,0
2
,63
2
,53
2
,43
3
,
23
,
2
,33
2
,23
2
,13
3
,0
4242
22
wt
Ew
t
EQ
t
tttGu
t
ttG
t
tttGu
t
ttGQM
vt
Gv
t
GN
t
Gtu
t
G
t
Gtu
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
a
a
a
axx
x
a
aa
a
aax
a
aa
a
aaxxxx
a
a
a
axxy
x
a
a
a
ax
a
a
a
axxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
+-=
++
+-
++
++=
+-=
-+--=
++=
---=
-=
++=
bb
bb
bb
Version 4.4 V.1.25
3.12.2012 Theoretical Background of ESAComp Analyses
V Joints
1 Adhesive Bonded Joints
Eqs. (1.4.8) constitute a set of 24 linear coupled first-order ordinary differential equations
within the overlap zone.
The governing equation for the laminates 2 and 3 in the region L £ x £ L + L1 (outside of
overlap) are derived by combining Eqs. (1.2.7), (1.2.10) and (1.3.1):
0
0
0
0
0
0
3
,
33
,
3
,
3
,
3
,93
3
,83
3
,73
3
,0
3
,63
3
,53
3
,43
3
,
33
,
3
,33
3
,23
3
,13
3
,0
2
,
22
,
2
,
2
,
2
,92
2
,82
2
,72
2
,0
2
,62
2
,52
2
,42
2
,
22
,
2
,32
2
,22
2
,12
2
,0
=
=
=
=
++=
---=
-=
++=
=
=
=
=
++=
---=
-=
++=
xx
xxxx
xxy
xxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
xx
xxxx
xxy
xxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
Q
QM
N
N
MkNkNkv
MkNkNk
w
MkNkNku
Q
QM
N
N
MkNkNkv
MkNkNk
w
MkNkNku
b
b
b
b
(1.4.9)
Eqs. (1.4.9) constitute a set of 16 linear coupled first-order ordinary differential equations.
1.4.5 Single sided scarfed lap joint
The governing equations outside the overlap zone are the same as for the single lap joint, i.e.
Eqs. (1.4.1). Within the overlap zone, i.e. for 0 £ x £ L, the governing equations for the
laminates 1 and 2 are derived by combining of Eqs. (1.2.7), (1.2.10), (1.2.11) and (1.3.4):
V.1.26 Version 4.4
Theoretical Background of ESAComp Analyses 3.12.2012
V Joints
1 Adhesive Bonded Joints
( )( ) ( ) ( )( ) ( )( )
( ) ( )( )
( )( ) ( ) ( )( ) ( )( )
( ) ( )( )
212
,
222222
2
021211
022
2
,
2
0
1
0
2
,
222
0
111
0
2
,
2
,92
2
,82
2
,72
2
,0
2
,62
2
,52
2
,42
2
,
22
,
2
,32
2
,22
2
,12
2
,0
211
,
111212
2
011111
011
1
,
2
0
1
0
1
,
222
0
111
0
1
,
1
,91
1
,81
1
,71
1
,0
1
,61
1
,51
1
,41
1
,
11
,
1
,31
1
,21
1
,11
1
,0
24
242
22
24
242
22
wt
Ew
t
EQ
NL
tt
t
txtxtG
ut
txtG
t
txtxtGu
t
txtGQ
M
vt
Gv
t
GN
t
tGu
t
G
t
tGu
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
wt
Ew
t
EQ
NL
tt
t
txtxtG
ut
txtG
t
txtxtGu
t
txtGQ
M
vt
Gv
t
GN
t
tGu
t
G
t
tGu
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
a
a
a
axx
xx
end
x
a
aa
a
aax
a
aa
a
aax
xxx
a
a
a
axxy
x
a
a
a
ax
a
a
a
axxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
a
a
a
axx
xx
end
x
a
aa
a
aax
a
aa
a
aax
xxx
a
a
a
axxy
x
a
a
a
ax
a
a
a
axxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
+-=
--
++
++
++
+-
=
+-=
+++-=
++=
---=
-=
++=
-=
--
++
++
++
+-
=
-=
---=
++=
---=
-=
++=
b
b
bb
bb
b
b
bb
bb
(1.4.10)
Eqs. (1.4.10) constitute a set of 16 linear coupled first-order ordinary differential equations.
1.4.6 Double sided scarfed lap joint
The governing equations for the double-sided scarfed lap joint are the same as for the single
lap joint outside the overlap zone.
The governing equations for the laminates within the overlap zone, i.e. for 0 £ x £ L, are
derived by combining of Eqs. (1.2.7), (1.2.10), (1.2.11) and (1.3.6):
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( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
a
a
a
a
aa
xx
a
xx
end
aaa
x
a
aa
a
a
aax
a
aa
a
aaa
x
a
xxx
a
a
a
a
aa
xxy
a
x
a
aa
a
ax
a
a
a
aa
xxx
a
xxx
a
xxy
a
xxx
a
x
a
xxx
a
xxy
a
xxx
a
xx
a
x
a
x
a
xxx
a
xxy
a
xxx
a
x
a
a
a
a
a
a
a
axx
x
a
aa
a
aax
a
aa
a
aa
x
a
aa
a
aa
a
aa
a
aax
xxx
a
a
a
a
a
a
a
axxy
x
a
a
a
ax
a
a
a
ax
a
a
a
a
a
a
a
axxx
xxxxxyxxxx
xxxxxyxxxxx
xx
xxxxxyxxxx
wt
Ew
t
EQ
NL
tt
t
tttG
ut
ttG
t
tttGu
t
ttGQ
M
vt
Gv
t
GN
t
Gtu
t
G
t
Gtu
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
wt
Ew
t
Ew
t
E
t
EQ
t
tttGu
t
ttG
t
tttGu
t
ttG
t
tttG
t
tttGu
t
ttG
t
ttGQ
M
vt
Gv
t
Gv
t
G
t
GN
t
Gtu
t
G
t
Gtu
t
G
t
Gt
t
Gtu
t
G
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
2
1
11
1
12
,
2222
1
2111
2
0
1
1111
1
11211
0
1
1212
2
,
2
0
1
11
0
1
12
,
2
1
122
0
1
11
1
111
0
1
12
,
2
,92
2
,82
2
,72
2
,0
2
,62
2
,52
2
,42
2
,
22
,
2
,32
2
,22
2
,12
2
,0
3
2
22
1
11
2
2
1
11
,
3
2
32123
0
2
2122
1
21112
0
1
111
1
2
1212
1
11111
0
2
212
1
1111
1
,
3
0
2
22
0
1
11
0
2
2
1
11
,
3
2
233
0
2
32
1
122
0
1
11
2
21
1
111
0
2
2
1
11
,
1
,91
1
,81
1
,71
1
,0
1
,61
1
,51
1
,41
1
,
11
,
1
,31
1
,21
1
,11
1
,0
24
242
22
4242
4422
2222
+-=
--
++
++
++
+-
=
+-=
+++-=
++=
---=
-=
++=
--÷÷ø
öççè
æ+=
++
+-
++
++
÷÷ø
öççè
æ ++
++÷÷
ø
öççè
æ +-
+-
=
--÷÷ø
öççè
æ+=
+---÷÷ø
öççè
æ--÷÷
ø
öççè
æ+=
++=
---=
-=
++=
b
b
bb
bb
bb
b
bbb
bb
(1.4.11)
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( ) ( ) ( )
( )
b
a
a
a
ab
xx
b
xx
end
bbb
x
a
aa
b
a
aax
a
aa
a
aab
x
b
xxx
b
a
a
a
ab
xxy
b
x
a
ab
a
ax
a
a
a
ab
xxx
b
xxx
b
xxy
b
xxx
b
x
b
xxx
b
xxy
b
xxx
b
xx
b
x
b
x
b
xxx
b
xxy
b
xxx
b
x
wt
Ew
t
EQ
NL
tt
t
tttG
ut
ttG
t
tttGu
t
ttGQ
M
vt
Gv
t
GN
t
Gtu
t
G
t
Gtu
t
GN
MkNkNkv
MkNkNk
w
MkNkNku
2
1
11
1
12
,
2222
1
2111
2
0
1
1111
1
11211
0
1
1212
2
,
2
0
1
11
0
1
12
,
2
1
122
0
1
11
1
111
0
1
12
,
2
,92
2
,82
2
,72
2
,0
2
,62
2
,52
2
,42
2
,
22
,
2
,32
2
,22
2
,12
2
,0
24
242
22
+-=
-+
++
+-
++
++
=
+-=
-+--=
++=
---=
-=
++=
b
b
bb
bb
Eqs. (1.4.11) constitute a set of 24 linear coupled first-order ordinary differential equations
within the overlap zone.
1.5 BOUNDARY CONDITIONS
To solve the adhesive bonded joint problems the boundary conditions and continuity
conditions have to be stated. The continuity conditions must be stated at the ends of the
regions in which the joint is divided as shown in Figure 1.1. In the following the boundary
conditions and continuity conditions are stated for the different joint types.
1.5.1 Single lap joint
The boundary conditions for a single lap joint are
junctionacrossContinuity
QMNN
adherend
adherendLx
QMNN
junctionacrossContinuity
adherend
adherendx
iNorvMor
QorwNoruprescribedLLLx
xxxxyxx
xxxxyxx
i
xy
i
o
i
xx
i
i
x
ii
xx
i
0
:2
:1:
0:2
:1:0
2,1,
,,::,
1111
2222
0
021
=====
====
=
=ïþ
ïýü+-=
b
(1.5.1)
The boundary conditions for the adherend 2 at x = 0 and for adherend 1 at x = L are derived
from the assumption that the adherend edges are free, see Figure 1.1.
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1.5.2 Single strap joint
The boundary conditions for a single strap joint are
0:2
:1:
0
0
0
0
:)0(2
:)0(2
:)0(1
:)0(1
::0
,
,,::
2222
2222
0
22
0
22
0
1111
0
1111
2
1
2
1
0
2
11
0
2
11
0
1111
1111
02
====
=
====
====
====
====
+==
+==
+=<
+=>
=
ïþ
ïýü+=
xxxxyxx
xxyxy
xxy
xxxxy
xxxxyxx
xy
xx
xx
xyoxxx
xxx
QMNN
junctionacrossContinuity
adherend
adherendLx
QNu
Qvu
QMNu
QMNN
LLxatNifadherend
LLxatvifadherend
LLxatNoruifadherend
LLxatNoruifadherend
Symmetryx
NorvMor
QorwNoruprescribedLLx
b
b
b
(1.5.1)
The boundary conditions at x = 0 are derived from the assumptions that there is symmetry
around the centerline of the strap joint. For adherend 2 at x = L, it is assumed that the
adherend edge is free.
1.5.3 Bonded doubler
Similar assumptions as for single strap joints are used bonded doublers, which yields the
following boundary conditions:
ïþ
ïýü
====+=
=
====+==
====+==
====+==
====+==
=
1111
0
1111
0
2222
2
2222
02
1
22
0
22
02
1
0
1111
02
1
11
0
11
02
1
0
,
,,
:
0
:1
:2
:1
:
:
0:)0(2
0:)0(2
0:)0(1
0:)0(1
::0
xyoxx
xxx
xxxxyxx
xxyxyxy
xxy
xxyxyxy
xxy
NorvMor
QorwNoru
prescribed
QMNN
junctionacrossContinuity
adherend
adherend
adherend
LLx
Lx
QNuLLxatNifadherend
QvuLLxatvifadherend
QNuLLxatNifadherend
QvuLLxatvifadherend
Symmetryx
b
b
b
b
b
(1.5.2)
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1.5.4 Double lap joint
The boundary conditions for a double lap joint are
0
0
:3
:2
:1:
0
:3
:2
:1:0
3,2,1,
,,::,
3333
2222
1111
0
021
====
====
=
=====
=ïþ
ïýü+-=
xxxxyxx
xxxxyxx
xxxxyxx
i
xy
i
o
i
xx
i
i
x
ii
xx
i
QMNN
QMNN
junctionacrossContinuity
adherend
adherend
adherendLx
junctionacrossContinuity
junctionacrossContinuity
QMNN
adherend
adherend
adherendx
iNorvMor
QorwNoruprescribedLLLx
b
(1.5.3)
For the adherend 2 and 3 at x = 0 and for the adherend 1 at x = L, the boundary conditions are
derived from the assumption that the adherend edges are free.
1.5.5 Single sided scarfed lap joint
The boundary and continuity conditions at the ends of the joint and at the ends of the overlap
zone are the same as for the single lap joint in Eq. (1.5.1).
1.5.6 Double sided scarfed lap joint
The boundary and continuity conditions are the same as for the double sided stepped lap joint
in Eq. (1.5.3), except that no continuity conditions within the overlap zone is required for the
double sided scarfed lap joint.
1.6 MULTI-SEGMENT METHOD
Each set of governing equations, together with the appropriate boundary conditions for the
particular bonded joint problem considered, constitutes a multiple-point boundary value
problem to which no general closed-form solution is obtainable. Thus, a numerical solution
procedure must be used to solve the bonded joint problems. In general, this can be done by
using methods such as finite difference methods or direct integration methods. The use of a
normal direct integration approach will involve some disadvantages, from which the most
important is that a complete loss of accuracy invariably will occur if the length of the
integration interval is increased beyond a certain value. The loss of accuracy is caused by
subtraction of almost equal and very large numbers in the process of determination of the
unknown boundary values. However, the use of a modified direct integration method, called
the “multi-segment method of integration”, can overcome the loss of accuracy experienced
with the “normal” direct integration methods.
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The method is based on a transformation of the original “multiple-point” boundary value
problem into a series of initial value problems. The principle behind the method is to divide
the original problem into a finite number of segments where the solution within each segment
can be accomplished by means of direct integration. Fulfilment of the boundary conditions, as
well as fulfilment of continuity requirements across the segment junctions is assured by
formulation and solving a set of linear algebraic equations. As an example, the single lap joint
configuration shown in Figure 1.12 is divided into three regions.
Figure 1.12 Schematic illustration of a single lap joint divided into M1 + M2 + M3 segments.
According to Figure 1.12, the three regions are:
· The region to the left side of the overlap zone, i.e. -L1 £ x £ 0
· The overlap zone, i.e. 0 £ x £ L
· The region to the right side of the overlap zone, i.e. L £ x £ L + L2.
Each of the regions r (r = 1, 2, … nr) are then divided into a finite number of segments Mr, see
Figure 1.12. The segments within a region are denoted by r
jS (j = 1, 2, … Mr) and the j'th
segment extends from r
jx to r
jx 1+ .
The solution procedure adopted in the “multi-segment method of integration” includes four
steps:
· Solution of the governing equation within each segment r
jS in each region r.
· Specification of continuity conditions between each segment within each region r.
· Specification of boundary and continuity conditions at the ends of the regions.
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· Formulation and solution of a set of linear algebraic equations containing the unknown
variables.
It is beyond the scope of this document to go into further details of the solution method and
the implementation in the ESAComp software (see Mortensen [13] for details). It is essential,
though, to emphasise that the direct integration of the initial value problems is performed by
an embedded Runge-Kutta method with adaptive step size control based on a prescribed
accuracy which enables ESAComp to control the number of segments used.
1.7 PLASTICITY EFFECTS IN THE ADHESIVE LAYER
1.7.1 Introduction
The structural modelling described in Section 1.2 is based on the assumption that the adhesive
layers behave as a linear elastic material. This is a good approximation for most brittle
adhesives, especially at low load levels, and the approach is useful to predict the stress
distribution and the location of peak stress values. However, most polymeric structural
adhesives exhibit inelastic behaviour, in the sense that plastic residual strains are induced
even at low levels of external loading, and plastic yielding will appear in most adhesive
bonded joints as the load is increased to failure, see Hart-Smith [8–10], Pickett [14–15],
Adams [2], Gali [6–7] and Thomsen [16–17]. Thus, the assumption on linear elasticity of the
adhesive is clearly an approximation.
Based on the structural analysis described in Section 1.2, the bonded joint analysis has been
extended to include adhesive plasticity. However, non-linear time and temperature dependent
effects including visco-elasticity, creep and thermal straining are not considered.
1.7.2 Non-linear formulation and solution procedure
The concept of effective stress/strain is one way of approaching the non-linear problem. In
this approach it is assumed that for a ductile material the plastic residual strains are large
compared with the creep strains at normal loading rates. Therefore, a plastic yield hypothesis
can be applied, and the multidirectional state of stress can be related to a simple unidirectional
stress state through a function similar to that of Von Mises.
However, it is widely accepted that the yield behaviour of polymeric structural adhesives is
dependent on both deviatoric and hydrostatic stress components. A consequence of this
phenomenon is a difference between the yield stresses in uniaxial tension and compression,
see Adams et al. [2], [4], Gali et al. [6–7], Harris et al. [11] and Thomsen [16–17]. This
behaviour has been incorporated into the analysis by the application of a modified Von Mises
criterion suggested by Gali et al. [7]:
( )
t
cVSVDS CCJCJCs
ss
ll
lll
=-
=+
=+= ,2
1,
2
13,12 (1.7.1)
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where s is the effective stress, J2D is the second invariant of the deviatoric stress tensor, J1 is
the first invariant of the general stress tensor and l is the ratio between the compressive and
tensile yield stresses. J2D and J1 are defined by:
( ) ( ) ( )( )
3211
2
13
2
32
2
2161
2
sssssssss
++=
-+-+-=
J
J D (1.7.2)
For l = 1, Eqs. (1.7.1) are reduced to the ordinary Von Mises criterion. At the failure load
level, the first of Eqs. (1.7.1) is transformed into the expression:
( ) ( )ultultVultDultSult JCJCs 1,2,
2
1
+= (1.7.3)
where the subscript ult denotes “ultimate”. Eq. (1.7.1) describes the failure envelope for the
general case of a ductile material. In three-dimensional stress space Eq. (1.7.1) represents a
paraboloid with its axis coincident with the line s1 = s2 = s3.
The effective strain e is given by Gali et al. [7]:
1221
1
1
1ICICe VDS nn -
++
= (1.7.4)
where n is the Poisson's ratio, I2D is the second invariant of the deviatoric strain tensor and I1
is the first invariant of the general strain tensor. I2D and I1 are defined by:
( ) ( ) ( )( )
3211
2
13
2
32
2
2161
2
eeeeeeeee
++=
-+-+-=
I
I D (1.7.5)
The non-linear adhesive properties are included by implementing an effective stress-strain
relationship derived experimentally from tests on adhesive bulk specimens Thomsen [16–17]
and Tong [19]. Thus, it is assumed that the bulk and “in-situ” mechanical properties of the
structural adhesive are closely correlated as discusses by Gali et al. [7] and shown by
Lilleheden [12].
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Figure 1.13 (a) Effective stress-strain relationship obtained from tensile test on bulk specimen. (b) Illustration of
piece-wise linear approximation to the curve and the solution procedure for the stress analysis in the non-linear
range.
Based on a secant modulus approach for the non-linear effective stress-strain relationship for
the adhesive, as shown in Figure 1.13, the solution procedure for determining the stress
distribution in the adhesive layer can be described by the following steps:
(1) Calculate the effective strains e1 and stresses *
1s (Eqs. (1.7.1) and (1.7.4)) for each point of
the adhesive layer using the linear elastic solution procedure and assuming a uniform
elastic modulus E1 for the adhesive.
(2) If the calculated effective stresses *
1s are above the proportional limit denoted by sprop,
determine the effective stresses s1 for each point of the adhesive layer according to the
corresponding effective strains e1 (using the experimental relationship given by Figure
1.13) calculated in step (1).
(3) Calculate the difference Ds1 = *
1s -s1 between the “calculated” and the “experimental”
effective stress, and determine the specific secant-modulus tE2 defined by:
1
1
12 }1{ E
s
sE t
÷÷ø
öççè
æ D-= d (1.7.6)
where d is a weight-factor, which determines the change of the modulus in each iteration.
(4) Rerun the procedure (steps (1)-(2)) with the elastic modulus E1 for each adhesive point
modified as per step (3).
(5) Compare the “calculated” effective stresses s* for each adhesive point with the
“experimental” values s obtained from the effective stress-strain curve (Figure 1.12).
(6) Repeat steps (4)-(5) until the difference between the “calculated” and “experimental”
stresses (Ds) drops below a specified fraction (2%) of the “experimental” stress value.
Convergence is usually achieved within a few iterations. The non-linear stress-strain
relationship obtained from tensile test on bulk specimen as illustrated in Figure 1.12a is in
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ESAComp defined by a piece-wise linear approximation to the curve as illustrated in Figure
1.12b.
The procedure described above has previously been used for the analysis of non-linear
adhesive behaviour in tubular lap joints by Thomsen [16–17].
The maximum effective stress and strain criteria have been investigated by [2] and [3] by
incorporating the two criteria in a finite element analysis of double and single lap joints. Their
investigations showed that for brittle adhesives there was a very close correlation with
experimental results by using the maximum effective stress criterion. For toughened ductile
adhesives they found that the maximum effective strain criterion gave the best prediction of
the joint strength. From the finite element analyses it was also possible to predict the failure
mode fairly accurately.
1.8 VALIDATION OF THE MODELS
The validation of the adhesive layer model by comparison with a high-order theory approach
and finite element models has been presented by Mortensen in reference [13].
1.9 FAILURE ANALYSIS
Failure in adhesive bonded joints can be divided into the following four types:
1. The adhesive may fail due to high shear and transverse normal stresses (cohesive failure).
2. The adhesive/adherend interfaces may fail due to high shear and transverse normal
stresses (interface failure).
3. The adherends may fail due to the external loads coupled with the large bending moment
concentrations induced in the regions near to the ends of the overlap.
4. If the adherends are made of composite material they may fail due to ply-failure caused by
high interlaminar shear stresses.
The failure types 1 and 3 are considered in the ESAComp implementation as described in the
following subsections. The failure type 2 usually appears due to insufficient bonding or
surface preparation and is therefore primarily a question of proper manufacturing. The failure
type 4 is not predicted in the current ESAComp implementation, but the adhesive shear
stresses and adherent resultant shear forces obtained form the joint analysis can be used as the
basis for assessing the criticality of this mode.
1.9.1 Cohesive failure analysis
The cohesive failure analyses are divided into to types of analysis – linear and non-linear
adhesive failure analysis. The procedure for determine the to failure levels are described in
this section.
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Linear cohesive failure analysis
The linear cohesive failure level is reached when the effective adhesive stresses *
1s are equal
to the proportional limit (sprop) for the adhesive bulk data.
The procedure for determining the reserve factor (RFadh) is as follows:
1. Calculate the adhesive layer effective stresses corresponding to the applied load vector
{F} and determine the maximum value ( *
maxs ).
2. Calculate the reserve factor as RFadh = sprop /*
maxs .
3. If the reserve factor RFadh ³ 1.0, the results from load response analysis equal to the applied
load vector are shown.
4. If the reserve factor RFadh < 1.0, the load vector is multiplied with the reserve factor, i.e.
RFadh{F}, and the load response analysis is performed again with the reduced load vector.
The results from the analysis are shown, i.e. the results at failure load level.
5. If the joint poses two adhesive layers, step 3 and 4 are performed for each of the adhesive
layers. The lowest reserve factor of the two adhesive layers is displayed as the reserve factor
for the joint (RFadh).
Non-linear cohesive failure analysis
In the non-linear cohesive failure analysis two, reserve factors are displayed – the reserve
factor at which the proportional limit is reached, i.e. where plasticity starts (RFadh,prop) and the
reserve factor to failure level (RFadh). The reserve factor to the proportional limit (RFadh,prop) is
determined as for the linear cohesive failure analysis described above. The reserve factor to
failure level or the ultimate load-bearing capability of the bonded joints are determined by an
iterative use of the non-linear solution procedure described in Subsection 1.7.2, where the
external loads are modified between each iteration. The iteration scheme is repeated until the
calculated maximum effective strain reaches the ultimate value eult.
The procedure for determining the proportional adhesive reserve factor (RFadh,prop) and the
failure reserve factor (RFadh) is as follows:
1. Calculate the adhesive layer effective stresses corresponding to the applied load vector
{F} and determine the maximum value ( *
maxs ).
2. Calculate the reserve factor as RFadh,prop = sprop / *
maxs .
3. Call the non-linear solution procedure described in Subsection 1.7.2.
4. Increase or decrease the load vector by multiplying it with the fraction of the ultimate
effective adhesive strain and the maximum effective adhesive layer strain (eult / *
maxe ), i.e.
{F}i+1
= {F}i* eult /
*
maxe , where i is the iteration number starting from 1 to nfailure.
5. Call the non-linear solution procedure described in Subsection 1.7.2, with the modified load
vector input {F}i+1
.
6. Repeat step 5 and 6 until the calculated maximum effective strain reaches the ultimate
value eult, i.e. until emax = eult.
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7. Calculate the reserve factor to failure RFadh = {F}failure
/{F}applied
.
8. If the reserve factor RFadh ³ 1.0, call the non-linear solution procedure with the applied load
vector {F}applied
to display output from the load response equal to the applied load vector
together with the reserve factors.
Due to the simple way of modelling the adhesive layer (the adhesive is not modelled as a
continuum), it is not possible to predict the failure mode with this approach. However, it
should be possible to predict the joint strength with reasonable accuracy by applying the
maximum effective stress or strain criteria, since equally simple models of the adhesive layer
have been used successfully for the prediction of the joint strength by [8–10]. However, the
predictions should be used for comparative purposes only. For a realistic evaluation of the
predicted results they should be compared with experimental results.
1.9.2 Laminate failure
The failure of the adherends due to external loads combined with joint induced bending
moments is predicted using the laminate first ply failure (FPF) analysis of ESAComp (Part III,
Chapter 5). Potentially critical locations in the vicinity of the joint are considered as illustrated
in Figure X. The in-plane forces and bending moments acting at these locations are obtained
from the joint analysis. For comparison, the FPF reserve factors at the end supports are also
computed. As a result, the reserve factor for adherend (laminate) FPF are
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REFERENCES
1. Adams, Robert D. and Wake, William C., Structural Adhesive Joints in
Engineering, Elsevier Applied Science Publishers, 1984, 1. ed.
2. Adams, R. D. and Coppendale, J. and Peppiatt, N. A., Failure Analysis of
aluminium-aluminium bonded joints, Adhesion, 1978, vol. 2, pp. 105–119.
3. Crocombe, A. D. and Adams, R. D., An effective stress/strain concept in
mechanical characterization of structural adhesive bonding, Journal of Adhesion,
1981, vol. 13, 2, pp. 141–155.
4. Adams, R. D., Stress analysis: a finite element analysis approach, Developments in
Adhesives, Applied Science Publishers, London, 1981, 2 ed.
5. Frostig, Y. and Thomsen, O. T. and Mortensen, F., Analysis of Adhesive Bonded
Joints, Square-end and Spew-Fillet: Closed-Form Higher-Order Theory Approach,
Report No. 81, Institute of Mechanical Engineering, Aalborg University, Denmark,
1997, submitted.
6. Gali, S. and Ishai, O., Interlaminar stress distribution within an adhesive layer in
the nonlinear range, Journal of Adhesion, 1978, vol. 9, pp. 253–266.
7. Gali, S. and Dolev, G. and Ishai, O., An effective stress/strain concept in
mechanical characterization of structural adhesive bonding, International Journal of
Adhesion and Adhesives, 1981, vol. 1, pp. 135–140.
8. Hart-Smith, L. J., Adhesive bonded single lap joints, Technical report NASA CR
112236, Douglas Aircraft Company, McDonnell Douglas Corporation, USA, 1973.
9. Hart-Smith, L. J., Adhesive bonded double lap joints, Technical report NASA CR
112237, Douglas Aircraft Company, McDonnell Douglas Corporation, USA, 1973.
10. Hart-Smith, L. J., Adhesive bonded scarf and stepped-lap joints, Technical report
NASA CR 112235, Douglas Aircraft Company, McDonnell Douglas Corporation,
USA, 1973.
11. Harris, J. A. and Adams, R. D., Strength prediction of bonded single lap joints by
non-linear finite element methods, International Journal of Adhesion and
Adhesives, 1984, vol. 4, pp. 65–78.
12. Lilleheden, L., Properties of adhesive in situ and in bulk, International Journal of
Adhesion and Adhesive, 1994, vol. 14, 1, pp. 31–37.
13. Mortensen, F., Development of Tools for Engineering Analysis and Design of
High-Performance FRP-Composite Structural Elements, Ph.D.-thesis, Institute of
Mechanical Engineering, Aalborg University, Denmark, 1998, Special Report No.
37. (http://www.aub.auc.dk/phd/)
14. Pickett, A. K., Stress analysis of adhesive bonded lap joints, Ph.D. thesis,
University of Surrey, 1983.
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15. Pickett, A. K. and Hollaway, L., The analysis of elasto-plastic adhesive stress in
adhesive bonded lap joints, Composite Structures, 1985, vol. 4, pp. 135–160.
16. Thomsen, O. T., Analysis of adhesive bonded generally orthotropic circular
cylindrical shells, Ph.D.-thesis, Institute of Mechanical Engineering, Aalborg
University, Denmark, 1989, Special Report No. 4.
17. Thomsen, O. T., Elasto-static and elasto-plastic stress analysis of adhesive bonded
tubular lap joints, Composite Structures, 1992, vol. 21, pp. 249–259.
18. Thomsen, O. T. and Rits, W. and Eaton, D. C. G. and Brown, S., Ply Drop-off
effects in sandwich panels - theory, Composites Science and Technology, 1996,
vol. 56, pp. 407–422.
19. Tong, L., Bond strength for adhesive-bonded single-lap joints., Acta Mechanic,
Springer-Verlag, 1996, vol. 117, pp. 101–113.
20. Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates, Technomic
Publishing Company. Inc., Lancaster, 1987.
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2 MECHANICAL JOINTS
Timo Brander (HUT/LLS, 2002)
The procedure for analyzing uniaxial in-plane load induced stresses in mechanically fastened single lap and
double lap joints is presented. The external load can be either tensile or compressive uniaxial load in the joint
length direction. First, the fastener loads are calculated. The stresses of an infinite adherend on the fastener hole
are calculated from the fastener load and from the by-pass load. The joint failure load and the failure mode are
calculated. The procedure is primarily intended for analyzing bolted joints, but also riveted joints with solid
rivets can be analyzed using this procedure providing that proper values for certain parameters describing
fastener flexibility are defined.
SYMBOLS
Ab Fastener cross-sectional area
Aij In-plane stiffness matrix of a laminate
Ai1, Ai2 Cross-sectional area of adherend (= W h)
As Effective area of plate over which the fastener shear acts
[A] Adherend extension matrix
[B] Adherend coupling stiffness matrix
{B} Fastener/hole extension vector
C Fastener head rotational stiffness
D Fastener diameter
d0 Characteristic distance
E Young's modulus
f Fastener/hole flexibility
G Shear modulus
h Adherend thickness
I Stiffness moment of inertia
JF Joint flexibility
JS Joint stiffness
k Effective stiffness, per unit thickness, of adherend supporting
fastener
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L Overall joint length
el,, er Length of adherend before fastener 1 (left) and fastener N (right),
respectively
M Internal moment in a fastener
N Number of fasteners along joint
P Shear load in a fastener
{P} Fastener load vector
Q Shear load
q Reaction of plate supporting fastener per unit length
r Radius of a fastener hole or fastener
S Stiffness of fastener in a single adherend
W Width of a single line of fasteners
X Total in-plane longitudinal load of a single line of fasteners of
width W
b Rotation of fastener axis due to shear, b = Q/(lGbAb)
d In-plane longitudinal extension; Total deformation of a fastener and
adherend
e Strain
l Shape factor for circular beam, l = 6(1+nb)/(7+6nb )
m Coefficient of friction
n Poisson's ratio
q Direction angle from the x-axis
s Stress
t Shear stress
yb Rotation of a fastener axis due to bending
Subscripts
A,B Single lap joints composing a double-lap joint, bearing
b Fastener
bp By-pass
c Compression, characteristic
i Index, ith fastener or pitch along joint
f Failure
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p Adherend (plate), pitch
r Radial direction
t Tension
xy Orthotropic in-plane coordinate system
q Tangential direction on the hole boundary
1,2 Adherends in a single lap joint
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2.1 INTRODUCTION
Practically all real life structures consist of several sub-structures, or are connected to other
structures. Thus, structures almost inevitably contain joints. The two most commonly used
joining methods of composite structures are mechanical joining and bonding.
Highly loaded mechanical joints use rivet or bolt fasteners. The ESAComp analysis
procedures are basically similar in joints using either fastener. When the fastener loads are
calculated it is assumed that the fasteners are bolts. It is also assumed, by default, that the
bolts are tightened to torque that gives adequate rotational stiffness to the bolt head but does
not damage the adherends by through-the-thickness loads. The friction between adherends is
not considered. The fastener load induced stresses are calculated assuming a pin type fastener.
This means that no clamping in the laminate thickness direction is considered at this stage of
analysis. Thus, in this respect, the procedure gives conservative joint failure loads with
respect to bolted joints where at least some clamping or constraint is present.
2.2 ANALYSIS APPROACH
The analysis of mechanical joints is based on the following assumptions:
1. Adherend thicknesses are constant
2. Effects of adherend bending are neglected
3. Adherend strains are assumed constant through the thickness of the adherend
4. Load from a fastener to the adherend is transferred purely by bearing (pin joint).
The analysis procedure can be outlined as follows:
1. The fastener loads are solved according to the theory presented in ESDU 85035 [1] and
85034 [2] in which the flexibility of the components (adherends) and fasteners is included.
A new ESDU should replace the aforementioned ESDUs. A draft version of that, ESDU
S681D [3], is also used. For unsymmetric laminates zero-curvature moduli are used.
2. The stress field at the fastener hole is solved using the theory of anisotropic plates.
However the present solution is limited to orthotropic adherends where A16 = A26 = 0 and
[B] = 0. The solution includes the effects of the fastener load and by-pass load. The
applied theory applies for infinite adherends only. The procedure is as follows:
a) The stresses induced by the fastener are solved according to presentation of Zhang and
Ueng [4]. The solution is based on the theory of anisotropic plates by Lekhnitskii [5].
b) The by-pass load induced stresses are solved using the circular open hole solution for
infinite plate (see Part IV, Chapter 3).
c) In both cases the solution is based on linear elastic behavior of materials. Thus, the
stress fields can be summed.
3. The failure modes and margins of safety are evaluated at various points around the
fastener hole to assess the load carrying capability and the potential failure mode of the
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joint.
4. It is also possible to analyze the case of finite width adherends. However, this procedure
can be applied only to tangential stress at q = ± p/2, i.e. at locations deviating ±90° from
the fastener load direction.
2.3 MECHANICAL JOINT LOAD RESPONSE
2.3.1 Fastener and by-pass loads
The fastener loads are determined according to principles presented in references [1–3]. The
approach is valid only for single row, single lap joints subjected to in-plane tension loads.
However, the same analysis can also be applied to double lap joints and multiple row joints
provided the rows are identical. The analysis provides the loads carried by each fastener, by-
pass loads and the overall in-plane flexibility of the joint.
The approach is in principle valid only for isotropic materials. However, isotropy has a
significant effect only on the fastener/hole flexibility. Thus, it is believed that this approach
can be applied also to orthotropic materials with adequate confidence.
The analysis approach is based on small displacement elastic theory and it does not consider
the moment effects due to the eccentricity of the loading. The joint geometry and notation is
shown in Figure 2.1.
el er
L
p1 pi pN-1
X
X X-P1 X-åPk PN
P1 åPk X-PN
Faste
ner
1
Faste
ner
2
Faste
ner
i
Faste
ner
i+1
Faste
ner
N-1
Faste
ner
N
I 1 I 2
Figure 2.1 Single lap joint geometry and notation
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Each pitch between the fasteners is considered separately, and the effects are summed. For
compatibility, the pitch extensions for adherends 1 and 2 at fastener positions i are equal
dd 21 = ii (2.3.1)
The pitch extensions are made up of two components a) adherend extensions (dip1, dip2) and b)
fastener/hole deformations (di).
The adherend extensions for the adherend 1 and for the adherend 2 are
úû
ùêë
é-å P X
E A
p =
i
1 = k1x1
1p1 k
ii
iid (2.3.2)
úû
ùêë
éå P
E A
p =
i
1 = k2x2
2p2 k
ii
iid (2.3.3)
where Pk is the fastener load of kth fastener and X the total load of a single line of fasteners.
Adherend stiffnesses are zero curvature stiffnesses. The fastener flexibility, f, is related to
both adherend 1 and adherend 2. The pitch extension due to the fastener and fastener hole
flexibility is
i i i i id = P f P f+1 +1 - (2.3.4)
Combining Equations (2.3.1 - 4) the compatibility equation for each pitch becomes
0iiiik
ii
i
k
ii
i =-+úû
ùêë
é
÷÷÷
ø
ö
ççç
è
æ-ú
û
ùêë
é-÷÷
ø
öççè
æ åå fPfPP
EA
p
P XE A
p
1+1+
i
1 = k2x2
2i
1 = k1x1
1 (2.3.5)
where fi is the sum of the individual fastener/hole flexibility in each adherend as explained in
the next subsection.
Overall load compatibility yields
X = Pk=1
N
å k (2.3.6)
Equations (2.3.5) and (2.3.6) represent N simultaneous equations which may be written in the
matrix form
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[ ]{ } { }B = PA (2.3.7)
The fastener loads Pi can be solved from this equation.
The by-pass load for each pitch is
å=
-=i
k
kbpi PXP1
(2.3.8)
The joint extension is found by adding (a) the adherend extensions of the top adherend, (b) the
extensions due to the flexibility of the end fastener N, and (c) the free adherend extensions in
adherends 1 and 2 at each end, between fasteners 1 and N and points I1 and I2, respectively:
EA
eX +
EA
eX + fPP X
EA
p =
x22
r
x11
li
=1k1x1
1
1N
=1i
FF
NNk
ii
i +úû
ùêë
éúû
ùêë
é-åå
-
d (2.3.9)
The joint flexibility is defined as
X
JFd
= (2.3.10)
and the corresponding joint stiffness is
JSJF
=1
(2.3.11)
The flexibility of a double lap joint is obtained by analyzing two single lap joints, which are
partitioned from the double lap joint about the midplane of the center adherend as shown in
Figure 2.2. The flexibility of the double lap joint is
J F = J F J F
J F + J FBA
A B
A B
(2.3.12)
where A and B refer to single lap joints composing the double lap joint.
The corresponding stiffness is
JS JS JSAB A B= + (2.3.13)
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Center line of center plate
h1
h2 X
Single lap joint A
Single lap joint B
Adherend 1 Adherend 2
X/2
X/2
Figure 2.2 Double lap joint modeled as two single lap joints
The bearing stress for each fastener is
hD
PiB =s (2.3.14)
2.3.2 Fastener flexibility
In Equation (2.3.5) the only unknown parameter is the fastener/hole flexibility, f. It can be
solved by considering a single fastener in shear in a single flat plate (adherend) as shown in
Figure 2.3. The fastener is restrained against rotation. The local deformations of the plate are
included in the analysis. The fastener flexibility is defined as
2
,2,1,h
zjP
= f j -==d
(2.3.15)
If two unequal adherends are connected as a single lap joint, the fastener flexibility is
obtained by summing the flexibility of the fastener in the individual adherends
f + f = f21 (2.3.16)
and the stiffness is
f + f
1 = S
21
(2.3.17)
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z
x
Load P
Center line ofunloaded bolt
Center line ofloaded bolt
Unloaded bolt
d
h
Elastic support
Figure 2.3 Fastener/hole flexibility
The basic governing differential equation for the fastener acting as a beam on an isotropic
elastic foundation is
0 =y I E
k +
zd
xd
I E
A G +
A G
k
zd
xd
A G
A G + 1
bb2
2
bb
szx
bb4
4
bb
szx
úû
ùêë
é-ú
û
ùêë
é
ll (2.3.18)
where l is the shape factor for circular beam, As is the effective area of plate over which the
bolt shear acts, k is the effective stiffness of the plate supporting the bolt and Ib is the stiffness
moment of inertia of the bolt.
The reaction of the plate foundation is
2
2
dz
xdAGxkq szxpz +-= (2.3.19)
The linear small displacement assumption is made which gives
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dz
d I E = M b
bbbz
y (2.3.20)
The following four boundary conditions are applied:
2
,h
zdz
dxAGPQ szxbbz -=-= (2.3.21a)
2
,0h
zM zb -== (2.3.21b)
2
,h
zdz
dxAGQ szxzb =-= (2.3.21c)
2
,h
zCM bzb =Y-= (2.3.21d)
Equation (2.3.18) can be written as
02
''
1
'''' =++ xaxax (2.3.22)
where the primes refer to the differentiation with respect to z. The equation is solved using the
boundary conditions of (2.3.21a - d).
The equation for the rotation of fastener axis due to bending, y, required in Equation
(2.3.21d) is
bb
zb
b
b
AG
Q
dz
dx =
dz
dx =
+ = dz
dx
lby
yb
--
(2.3.23a,b)
From Equations (2.3.20) and (2.3.23) follows
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÷÷ø
öççè
æ
bb
zp
2
2
bbzbAG
q +
zd
xd IE = M
l (2.3.24)
to which the foundation reaction, Equation (2.3.19), is placed. The shear force, Q, is obtained
by derivation from Equation (2.3.24)
÷÷ø
öççè
æ--
dz
dq
AG
1 +
zd
xd IE =
dz
dM = Q
pz
bb
3
3
bbbz
bz l (2.3.25)
Equation (2.3.18) includes three quantities As, C, and k, which are typically not available. To
provide the highest possible accuracy the values for these parameters should be determined
through tests. The following estimates can mainly be used for metallic adherends. If the
fastener head is effectively restrained against rotation, the fastener head rotational stiffness, C,
may be considered as infinite. The effective area of adherend over which shear acts, As, may
be estimated to be As = 0.1 D2. The effective adherend stiffness, k, should be determined
experimentally according to the method described in ESDU 85034 Appendix B [2]. The value
k = 0.18 Ep given in [2] should only be used when titanium alloy fasteners are used in an
aluminum alloy adherend.
2.3.3 Fastener load induced stresses at the pin-loaded hole
The fastener load induced stresses at the fastener hole are determined according to the
presentation of Zhang and Ueng [4]. The analytical solution is based on the theory of
anisotropic plates, but in the present solution, it has been restricted to orthotropic plates to
obtain compact analytical solutions.
The expression of radial stress is
qnqs 3cos)1(2
1)33()1(5cos)1(
)1( 00 úû
ùêë
é --
+--++-+
= knc
nkkcrgc
uknu
rgc
cxyr
qnnp
cos)()1(
)22(2
)1( 00
úû
ùêë
é+-
++++-
-+- nk
rgc
ucnknk
rgc
uc
r
Pxyxy (2.3.26)
and for shear stress
qnqt q 3sin)1(2
1)2()1(5sin)1(
)1( 00 úû
ùêë
é --
+++-+--+
-= knc
nknkcrgc
uknu
rgc
cxyr
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qnnp
sin)()1(
)22(2
)1( 00
úû
ùêë
é+-
+-++-
--+ nk
rgc
ucnknk
rgc
uc
r
Pxyxy
(2.3.27)
When the effect of friction is considered, the following condition applies
qsmqtpp
q drdr rr òò -=2/
0
2/
0
(2.3.28)
In Equations (2.3.29) and (2.3.30) following parameters are used
2/1
÷÷ø
öççè
æ=
y
x
E
Ek (2.3.31)
2/1
)(2úúû
ù
êêë
é+-=
xy
xxy
G
Ekn n (2.3.32)
xyy
yxxy
G
k
Eg +
-=
nn1 (2.3.33)
1
11
A
ABc
-= (2.3.34)
( ) ( )knkBknkA
ABPgu
xyxy +----
-=
nnp 11
110
2 (2.3.35)
where
( ) ( )xyxy knknknknA nmn 1515611101011191 -+-+-++= (2.3.36)
( ) ( )nnkkknB xy ++-+-= 233101101 nm (2.3.37)
The tangential stress sq can be expressed as
54321 qqqqqq ssssss ++++= (2.3.38)
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where
[ qqnqnp
s qq
2224
1 sincos)21(cos xyxy
x
kr
P
E
E+---=
] qqn cossin)22( 42nk xy --++ (2.3.39)
[îíì
+-+úû
ùêë
é ---
= qnqqqs qq
22202 cos)()21()cossin(2cos
2
)1(nkkk
n
rgc
uc
E
Exy
x
] úû
ùêë
é +-+--+- qqqqqqqnn 2sin2
)cossin(2cos2sinsinsin)( 222222 nknkknk xyxy
[ ] }qqnqnn cossin)()2(cos)( 222
xyxyxy kknknkn -++++-+- (2.3.40)
[îíì
+úû
ùêë
é --+
= qqqqs qq
2220
3 cos)21()cossin(2cos2
)1(2knk
n
rgc
uc
E
E
x
] [ ( )qqqqqqn 222 cossin2cos2sinsinsin kn xy -++
][ ] }qqnqnq cossin)2()(cos)(2sin2
2222 kkknn
xyxy +-+-+-+ (2.3.41)
[îíì
+-+úû
ùêë
é --+
= qnqqqqs qq
2220
4 cos)()21()cossin(2cos24cos2
)1(2nkkk
n
rgc
uc
E
Exy
x
] úû
ùêë
é +-+--+- qqqqqqqqnn 4sin2sin2
)cossin(4cos2sinsinsin)( 22222 nknkknk xyxy
[ ] }qqnqnn cossin)()2(cos)( 222
xyxyxy kknknkn -++++-+- (2.3.42)
qs qq2
05 sin2
1uE
rc
c -= (2.3.43)
Eq in the previous equations is the adherend Young’s modulus in the q-direction.
2.3.4 By-pass load induced stresses
The by-pass load induced stresses is solved according to Part IV, Chapter 3 of this document.
The stresses are calculated either on the fastener hole or on a characteristic curve (Point Stress
Criterion), where the point of calculation is determined as
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22
,0
pq
p££-+= drrc (2.3.44)
where d0 is the characteristic distance.
The stresses calculated above apply to a hole in an infinite plate.
2.3.5 Failure of the joint
The failure of the joint is determined primarily on the fastener hole according to the following
procedure based on reference [6]:
1. The fastener load induced laminate stresses sr, sq and trq are calculated.
2. The by-pass load induced stresses are added to the fastener load induced stresses.
3. The FPF analysis approach (Part III, Chapter 5) with the selected failure criterion is
applied to predict the failure load in terms of reserve factor or margin of safety. In tension
loading the failure mode is determined as follows
-15° £ qf £ 15° bearing failure mode
30° < qf < 60° shear-out failure mode
75° < qf < 90° tension failure mode
qf is the angle of the point where the combined stress reaches the critical value. At
intermediate values of qf failure may be caused by a combination of the modes.
In compression loading the failure mode is
-15° £ qf £ 15° bearing failure mode
75° < qf < 90° compression failure mode
At intermediate angles, the failure mode is undefined.
Alternatively, if the characteristic distances are known, the failure can be calculated as
follows:
1. The fastener load induced laminate stresses sr, sq and trq are calculated along the
characteristic curve
rc r
c
c
c
r c r
c
= r
r, =
r
r, =
r
r s s s s t tq q q q (2.3.45)
2. The by-pass load induced stresses are added to the fastener load induced stresses along the
characteristic curve.
The final step is identical to the step 3 above.
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Some comments concerning the approach presented above are:
1. Characteristic distances should be determined experimentally for the laminate if relevant
values are not found in literature.
2. The approach does not take into account the stresses induced by the other closely situated
fasteners/holes.
3. The approach, where the stresses are calculated on the hole boundary, gives conservative
failure loads for the joint compared to the case, where the values are calculated on the
characteristic curve.
2.3.6 Finite width joint
The theory presented above for fastener induced stresses and by-pass load induced stresses is
valid for infinite plates only. However, in typical laminates with realistic end and side
distances and pitches the theory is applicable with reasonable accuracy.
The finite width correction is applied only to tangential normal stress at q = ± p/2. No other
stress components are corrected nor included in the failure analysis. The corrected stress for
the fastener load and for the by-pass load is
2
,p
qsss qqq ±=÷÷ø
öççè
æ+÷÷
ø
öççè
æ= ¥
¥¥
¥ pb
bpT
Tb
bT
T
K
K
K
K (2.3.46)
where sq¥ is the tangential stress in an infinite adherend and subscripts b and bp refer to
fastener and by-pass load, respectively. KT¥ and KT denote the stress concentration on the hole
boundary on the axis normal to the applied load for infinite plate and finite plate, respectively
[7]. The ration of KT and KT¥ is called Finite Width Correction (FWC) factor [7].
The FWC for the fastener load is [8]
32
167.29196.23820.82880.0 ÷ø
öçè
æ+÷ø
öçè
æ-÷ø
öçè
æ+=÷÷ø
öççè
æ¥ W
D
W
D
W
D
K
K
bT
T (2.3.47)
where W is the width of a plate containing a central opening or fastener. The formula is valid
for isotropic materials but it is used here due to the lack of a corresponding formula for
orthotropic materials. In addition, typical laminate structures for mechanical joints have close
to quasi-isotropic properties.
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2 Mechanical Joints
The FWC for the by-pass load is [7]
( )( )
( )úúû
ù
êêë
é÷ø
öçè
æ--÷ø
öçè
æ+-+
-=÷÷
ø
öççè
æ ¥¥
26
313
2
1
/12
/131 M
W
DKM
W
D
WD
WD
K
KT
bpT
T (2.3.48)
where
÷÷ø
öççè
æ -+-+=¥
66
2
122211122211
66 2
21
A
AAAAAA
AKT (2.3.49)
and
( )( )( )2
3
2
/2
11/12
/1381
WD
WD
WD
M
-úû
ùêë
é-
-+
--
= (2.3.50)
Equations 2.3.46–48 apply for orthotropic plates (laminates).
Version 4.4 V.2.17
3.12.2012 Theoretical Background of ESAComp Analyses
V Joints
2 Mechanical Joints
REFERENCES
1. "Computer program for the flexibility of single and double lap thin plate joints
loaded in tension". ESDU 85035. Engineering Sciences Data Unit, London, 1985.
2. "Flexibility of a single bolt shear joint". ESDU 85034. Engineering Sciences Data
Unit, London, 1985.
3. "Flexibility of, and load distribution in, multi bolt lap joints subjected to in-plane
axial loads". Draft data item S681D. ESDU International Plc., Fifth Draft,
September 1996.
4. Zhang, K. and Ueng, C.E.S. "Stresses Around a Pin-loaded Hole in Orthotropic
Plates". Journal of Composite Materials, Vol. 18, September 1984, pp. 432–446.
5. Lekhnitskii, S.G., Anisotropic Plates, English edition (Translated by S.W. Tsai and
T. Cheron), Gordon and Breach, London, 1968.
6. Ueng, C.E.S. and Zhang, K. "Strength Prediction of a Mechanically Fastened Joint
in Laminated Composites". AIAA Journal, Vol. 23, No. 11, November 1985, pp.
1832–1834.
7. Seng C. Tan, Stress Concentrations in Laminated Composites, Technomic
Publishing Company, Inc., USA, 1994
8. Walter D. Pilkey, Peterson’s Stress Concentration Factors, John Wiley & Sons,
Inc., USA, 1997