An analytical solution for the analysis of symmetric
composite adhesively bonded joints
G.P.Zou,K.Shahin,F.Taheri
*
Department of Civil Engineering,Dalhousie University,1360Barrington Street,Halifax,NS,Canada B3J 1Z1
Available online 4February 2004
Abstract
Analytical solutions for adhesively bonded balanced composite and metallic joints are prented in this paper.The classical laminate plate theory and adhesive interface constitutive model are employed for this deduction.Both theoretical and numerical (finite element analysis)studies of the balanced joints are conducted to reveal the adhesive peel and shear stress.The methodology can be extended to the application of various joint configurations,such as single-lap and single-strap joints to name a few.The
缺火的女孩名字
methodology was ud to evaluate stress in veral balanced adhesively bonded metallic and composite joints subjected to the tensile,moment and transver shear loadings.The results showed good agreements with tho obtained through FEM.Ó2004Elvier Ltd.All rights rerved.
Keywords:Adhesives;Bonded joints;Composite;Analytical solution;Finite element analysis;Stress analysis;Lap joint;Strap joint;Panel to flange joint
1.Introduction
Over the recent decades,fiber reinforced composites have found veral applications as load-carrying compo-nents,particularly in the realms of aeronautical and marine industries.A typical aerospace structure may consist of thin composite panels or skins,adhesively bonded to stiffeners,webs,spars or frames.Stiffened panels are typically subjected to large in-plane and out-of-plane loads.As a result,considerable bending and shear stress are produced at the skin stiffeners interfaces.Many workers have studied the single-lap adhesively bonded analytically,numerically and experimentally with the primary purpo of predicting the extent of the load carried by the joint [1].Historically speaking,one would cite the work of Goland–Reissner [2]as one of the earliest investigations performed on the cylindrical bending plate analysis of a single-lap joint.Since then v
eral workers performed work on this subject;for instance,Hart-Smith [3]developed a layered beam model to solve the single-lap joint problem.Oplinger [4,5]also developed a beam method by considering the overlap bending moments and
introducing the individual tensile forces in the upper and lower adherents at the overlap ction.In order to ensure that the stress-free boundary conditions would be satisfied at the free ends,some rearchers employed two-dimen-sional elasticity theory in conjunction with the variational method,such as the minimum strain method [6,7]and the principle of complementary energy method [8].A 2D elasticity bad solution was also prented by Tsai and Morton [9].In a ries of papers,Bigwood and his coworkers [10–12]prented the details of their investi-gations proposing solutions and failure criteria for metal-lic adhesively bonded joints.In one of their papers,Bigwood and Crocombe [10]prented a good asssment of various approaches for analyzing adhesively bonded joints available at the time.They also ud the application of fracture mechanics to account for the singularities found at the joint of bimaterial interfaces,and propod failure criteria.They further delved into a single-lap joint analysis by accounting for the nonlinear and elasto-plastic respon of adherents,for a joint that was subjected to any combination of tensile,shear and moment loading.The adhesive was modeled by a ries of shear and tension springs.They prented the pro
blem by a t of six non-linear first-order ordinary differential equations,which had to be solved numerically using a finite-difference method.They did not however asss the integrity of composite joints.Recently,Li and Lee-Sullivan [13]also
*
Corresponding author.Tel.:+1-902-494-3935;fax:+1-902-484-6635.
E-mail address:farid.taheri@dal.ca (F.Taheri).0263-8223/$-e front matter Ó2004Elvier Ltd.All rights rerved.
doi:10.pstruct.2004.01.007
Composite Structures 65(2004)
499–510
investigated single-lap adhesively bonded balanced joints usingfinite element analysis and verified their results by comparing them to either experimentally or theoretically obtained results.They ud the geometrically nonlinear two-dimensionalfinite element method to investigate the effects of plane strain and plane stress conditions,the simply supported versus fullyfixed boundary conditions at the adherent ends,filleted and unfilleted overlap end geometries,as well as two different adhesive materials,on the bending moment factor,k,and the adhesive stress.
If the adherents of a joint have different geometries and/or mechanical properties,the joint is referred to as unsymmetric or unbalanced joints.The asssment of such joints have also been subjected to veral investi-gations.The review and treatment of this class of joints will be provided in a subquent paper.
1.1.Motivation
Most of the available solutions prently available for evaluating stress in adhesively bonded joints are tai-lored for a specific joint geometry or loading condition. It is therefore desirable to develop an explicit solution that can be applicable to most adhesively bonded geo-metrical and loading condition
s.
In this paper we prent the detail of an analytical solution that is applicable to most joint configuration and practical loading conditions.The solution is bad on the classical laminate plate theory in conjunction to an adhesive interface constitutive model for balanced composite joints subjected to in-plane and out-of-plane loads.Numerical results for stiffened joints and single-strap joints are produced and compared with thefinite element solutions.
The work was motivated by the need for a robust and fairly accurate analytical tool so that joint designers could gain greater confidence when designing adhesive bonded joints.
2.Constitutive and kinematics relations for composite laminates
The constitutive relations for a composite laminate considered as a beam are:
N i¼A i
11u0i;xÀB i
11
w i;xx
M i¼B i
11u0i;xÀD i
11
w i;xx
ð1Þ
where(i¼1,2)corresponds to the upper adherent and lower adherent.N i,M i are the adherents normal stress
and bending moment resultants.A i
11,B i
11
and D i
11
are the
adherents extensional,coupling and bending stiffness defined as:ðA i
11
;B i
11
;D i
11
Þ¼
Z h i=2
h i=2
Q i
11
ð1;z;z2Þd zð2Þ
where Q i
11
reprents the general stiffness in the adher-ents,h i is the adherent’s thickness.
The kinematics relations can be written bad on the ‘‘Euler–Bernoulli’’assumptions:
u i¼u0iþz b i;b i¼Àw i;xði¼1;2Þð3Þwhere u i is the in-plane displacement,u0i is in-plane displacement at the mid-plane,and w i is the out-of-plane displacement of the i th adherent(shown as Fig.1).
3.Adhesive stress
The coupling between the adherents is established through the constitutive relations for the interface
‘‘resin rich’’layer,which is assumed homogeneous,isotropic and linear elastic.The constitutive equations are sug-gested as follows:
r a¼
E a
h0
ðw2Àw1Þ
s a¼
G a
h0
u02
þ
h2
2
d w2
d x
Àu01þ
h1
2
d w1
d x
ð4Þ
where G a is the shear modulus,E a is the elastic modulus of the interface‘‘resin rich’’layer,and r a,s a are the interface‘‘resin rich’’layer transver normal and shear stress components.
4.Force equilibrium and overall system equations
Bad on the adhesive stress and the stress resul-tants in the bonded region as shown in Fig.2,the force equilibrium result is the following equations:
500G.P.Zou et al./Composite Structures65(2004)499–510
N1;x¼Às a;Q1;x¼Àr a;M1;x¼Q1Às a h1þh0
2
N2;x¼s a;Q2;x¼r a;M1;x¼Q2Às a h2þh0
2
ð5Þ
From the above equations,it is possible to form the complete t of system equations for the adhesive joint. ForÀ16x61,the displacement u i and w i are related to the resultant in-place forces and moments by:
u0i;x¼k i
11N iþk i
12
M i
w i;x¼Àb i
b i;x ¼k i
13
N iþk i
14
M i
ð6Þ
where for symmetric joint,in which the adherents are formed by symmetric laminates,the stiffness terms take the following form:
k i 11¼
1
A i
11
k i 12¼k i
13
¼0
k i 14¼
1
D i
11
ð7Þ
By differentiating the interface constitutive equation (2a),the adhesive shear stress can be expresd by:
d s a d x ¼
G a
h0
d u02
d x
À
d u01
d x
À
h2
2
d b2
d x
À
h1
2
d b1
d x
!
¼
G a
h0
k2
11
À
h2
2
k2
13
N2Àk1
11
þ
h1
2
k1
13
N1
þk2
12
À
h2
2
k2
14
M2Àk1
12
þ
h1
2
k1
14
M1
!
ð8Þ
d2s a d x2¼
G a
h0
k2
11
&
þk1
11
À
h2
2
k2
13
þ
h1
2
k1
13
Àk1
12
À
h2
2
k2
14
h
2
þh0
2
þk1
12
þ
h1
2
k1
14
h
1
þh0
2
!
s a
þk2
12
À
h2
2
k2
14
Q2Àk1
12
þ
h1
2
k1
14
Q2
'
ð9Þ
and
d3s a
d x3
þa1
d s a
d x
¼0ð10Þ
where
a1¼À
G a
h0
k2
11
þk1
11
À
h2
2
k2
13
þ
h1
2
k1
13
Àk1
12
À
h2
2
k2
14
Â
h2þh0
2
þk1
12
þ
h1
2
k1
14
h
1
þh0
2
!
ð11Þ
a2¼À
G a
h0
k1
12
þk1
12
À
h2
2
k2
物流的七大功能
14
þ
h1
2
k1
14
!
Differentiating the adhesive interface equation,the
adhesive normal stress,r a,can be expresd as:
d2r a
d x2
¼
E a
h0
È
Àk2
13
N2Àk2
14
M2þk1
13
N1þk1
14
M1
É
ð12Þ
d3r a
d x3
¼
E a
h0
&
Àk1
13
þk2
13
þk1
14
h1þh0
2
Àk3
14
h2þh0
2
!
s a
þk1
14
Q1Àk2
14
Q2
'
ð13Þ
and
d4r a
d x4
þg1r aþg2
d s a
d x
¼0ð14Þ
where
g1¼
E a马字开头的成语
h0
k1
14
À
þk2
14
Á
g
2
¼
E a
h0
k1
13
þk2
13
þk1
14
h1þh0
2
Àk2
14
h2þh0
2
!ð15Þ
When the adherents are made of symmetric and equal
thickness laminates,then the coefficients a2and g2(in
Eqs.(11)and(15))become equal to zero.The adhesive
shear stress and normal stress are therefore uncoupled,
the system equations can be written as:
d3s a
d x3
À12
绨袍剑
d s a
d x
¼0
d4r a
d x4
þg1r a¼0
8
>><
>>:ð16a;b
ÞG.P.Zou et al./Composite Structures65(2004)499–510501
where 12¼a 1;
g 1¼4ðn Þ4
ð17ÞThe general solution for Eq.(16)can be stated as:s a ðx Þ¼C 0þC 1cosh ð1x ÞþC 2sinh ð1x Þð18a Þ
r a ðx Þ¼C 3cosh ðn x Þcos ðn x ÞþC 4sinh ðn x Þsin ðn x Þ
þC 5½cosh ðn x Þsin ðn x Þþsinh ðn x Þcos ðn x Þ
þC 6½cosh ðn x Þsin ðn x ÞÀsinh ðn x Þcosh ðn x Þ
ð18b Þ
the integration constants,C 0;...;C 6will be determined by the boundary conditions.
5.Boundary conditions
It can be shown that r a and s a along the boundary x ¼Æl can be expresd by the following relations:Z l
Àl s a ðx Þd x ¼N 1j x ¼Àl ÀN 1j x ¼l ð19a ÞZ
l
Àl r a ðx Þd x ¼Q 1j x ¼Àl ÀQ 1j x ¼l
ð19b Þ
Z
l
Àl
r a ðx ÞÁx d x ¼M 1j x ¼l &
ÀM 1j x ¼Àl Àl ÁQ 1j x ¼l Àl ÁQ 1j x ¼Àl
Àh 0þh 12N 1j x ¼l Â
ÀN 1j x ¼Àl Ã'ð19c Þd s a ðx Þd x j x ¼l ¼G a h 0k 2
11 &Àh 22
k 213
N 2j x ¼l Àk 111 Àh 12k 113 N 1j x ¼l þk 212 Àh 22
k 2
14
M 2j x ¼l
Àk 1
12 þh 12
k 114 M 1j x ¼l
'
ð19d Þd 2s a ðx Þd x 2j x ¼l þa 1s a ðx Þj x ¼l ¼G a h 0k 2
12 &Àh 22燃气灶开关
k 214
Q 2j x ¼l Àk 112 þh 12
k 1
14 Q 1j x ¼l
'
ð19e Þ
d 2r a ðx Þd x 2j x ¼l ¼E a h 0
ÈÀk 213N 2j x ¼l Àk 2
14M 2j x ¼l þk 113N 1j x ¼l þk 1
14M 1j x ¼l
Éð19f Þ
d 3r a ðx Þd x 3j x ¼l þg 2s a ðx Þj x ¼l ¼E a h 0k 114Q 1j x ¼l
ÈÀk 2
14Q 2j x ¼l
Éð19g Þwhere N i ðx ¼Çl Þ,Q i ðx ¼Çl Þand M i ðx ¼Çl Þði ¼1;2Þ,are known constants (note that the prescribed boundary conditions must be satisfied for the gross static equilib-rium conditions for the adhesive joint).The cas of
single-lap and stiffened adhesive joints will be considered in the following ction.
5.1.Ca (a):Stiffenedpanel und er axial tension A stiffened composite (panel-to-flange)joint is shown in Fig.1(a).The prescribed boundary conditions are stated as:
N 2j x ¼l ¼M 2j x ¼l ¼Q 2j x ¼l ¼M 1j x ¼Àl ¼M 1j x ¼l
¼Q 1j x ¼Àl ¼Q 1j x ¼l ¼0
N 1j x ¼l ¼N ;
N 1j x ¼Àl ¼N
ð20Þ
Applying the above boundary conditions to Eq.(19),
shear stress in the adhesive layer can be obtained through Eq.(18).For a balanced stiffened-panel joint under tension,for example,the shear stress distribution can be obtained by the following equation,noting that the equation predicts no normal stress in the adhesive layer:s ðx Þ¼
ÀG a k 11N
f h 0cosh ðf l Þ
sinh ðf x Þ
ð21Þ
5.2.Ca (b):Stiffened panel under bending moment Stiffened composite (panel-to-flange)joint is shown in Fig.1(b).The prescribed boundary conditions are stated as:
N 2j x ¼l ¼M 2j x ¼l ¼Q 2j x ¼l ¼N 1j x ¼Àl ¼N 1j x ¼l
¼Q 1j x ¼Àl ¼Q 1j x ¼l ¼0;M 1j x ¼l ¼M ;
M 1j x ¼Àl ¼M
ð22Þ
The boundary conditions described above result in adhesive shear and normal stress prented by:s ðx Þ¼
ÀG a k 14hM
2f h 0cosh ðf l Þ
sinh ðf x Þ
ð23Þ
r ðx Þ¼C 3cosh ðn x Þcos ðn x ÞþC 4sinh ðn x Þsin ðn x Þ
C 3¼À
E a k 14Me n l ðe 2n l sin ðn l ÞÀcos ðn l ÞðÞþsin ðn l Þþcos ðn l ÞÞ
n 2h 0ðe 4n l þ4e 2n l sin ðn l Þcos ðn l ÞÀ1ÞC 4¼
E a k 14Me n l ðe 2n l ðsin ðn l Þþcos ðn l ÞÞþsin ðn l Þþcos ðn l ÞÞ
实习期n h 0ðe 4n l þ4e 2n l sin ðn l Þcos ðn l ÞÀ1Þ
ð24Þ
The expressions obtained for constants C 3and C 4shown in Eq.(24)can be greatly simplified.For exam-ple,for relatively long adhesive joints,the first term in the numerator and the denominator becomes much larger than the other terms.In fact,for almost all practical joint dimensions,only the leading terms in the numerator and the denominator need to be considered,with little to no loss in accuracy.The simplified expressions for constants C 3and C 4take the form:
502G.P.Zou et al./Composite Structures 65(2004)499–510
C3¼ÀE a k14Mðsinðn lÞÀcosðn lÞÞ
n h0e n l
C4¼E a k14Mðsinðn lÞþcosðn lÞÞ
n2h0e n l
ð25Þ
5.3.Ca(c):Single-strap joint under tension
Single-strap joint under tensile load is shown in Fig. 3(a).Since the geometry is symmetric about the y-axis, only one-half of the geometry is modeled(thus the ori-gin coordinate is t at the center of the half length of the adhesive).The prescribed boundary conditions are stated as:
N2j
x¼l=2¼M2j
x¼l=2
¼Q2j
x¼l=2
¼M1j
x¼Àl=2
¼M1j
x¼l=2¼Q1j
x¼Àl=2
¼Q1j
x¼l=2
¼N1j
x¼Àl=2
¼0
N1j
x¼l=2
¼N
ð26ÞThe application of the above boundary condition to Eq.(19)enables us to obtain the values of the constants C0through C6,which subquently enable us to obtain the adhesive stress through Eq.(18)as:
sðxÞ¼Nðf2h0À2G a k11Þ
2f h0coshðf l=2Þ
sinhðf xÞþ
N f
2sinhðf l=2Þ
coshðf xÞ
rðxÞ¼ðcoshðn xÞþsinhðn xÞÞða cosðn xÞþb sinðn xÞÞ
ð27ÞIt is interesting to note the new arrangement of the normal stress expression,for which only two constants need be evaluated,which can be reprented by:
As with the ca of adhesive normal stress in stiff-ened joints subject to a bending moment,the constants in the single-strap adhesive joint expression can be greatly simplified.For most practical joint geometries (i.e.,with large n l),the constants found in the adhe-sive normal stress expression can be evaluated using Eq.(29).
a¼ðh0þhÞN n2ðcosðn l=2ÞÀsinðn l=2ÞÞ
e n l=2
b¼ðh0þhÞN n2ðsinðn l=2Þþcosðn l=2ÞÞ
e n l=2
ð29Þ
5.4.Ca(d):Single-lap joint under tension
A typical single-lap joint is shown as Fig.3(b).The
distance between the neutral plane of the top or bottom
adherents to the adhesive’s neutral plane is expresd by:
r1¼
h1þh0
2
ð30Þ
The prescribed boundary force conditions can be
expresd Cheng et al.[14]by:
N2j
x¼l
¼M2j
x¼l
¼Q2j x¼l¼N1j
x¼Àl
¼M1j
x¼Àl
¼Q1j
x¼Àl
M1j
x¼l
¼kÁ
N
2
h1;M2j
x¼Àl
¼kÁ
N
2
h1
Q1j
x¼l
¼
1
2l
N
2
ð2h1þh0ÞÀM1j
x¼l
ÀM2j
x¼Àl
!
ð31Þ
where
k¼
2
1
r1coshð2k lÞþk
k1
主要工作经历sinhð2k lÞ
h i
þh1
2
h i
2coshð2k lÞþk
1
þk1
sinhð2k lÞ
h ið32Þ
and
k1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
包子馅N
ðD11Þ
adherent
s
and k¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
D overlap
s
ð33Þ
in which D11,and D are the bending stiffness of either
adherents and the overlap ction,respectively.
a¼ðh0þhÞN n2e n l=2½e n lðcosðn l=2ÞÀsinðn l=2ÞÞþsinðn l=2Þþcosðn l=2Þ e2n lþe n lð4sinðn l=2ÞÀ4n l sinðn l=2Þcosðn l=2ÞÀ2Þþ1
b¼ðh0þhÞN n2e n l=2½e n lðsinðn l=2Þþcosðn l=2ÞÞþsinðn l=2ÞÀcosðn l=2Þ
e2n lþe n lð4sin2ðn l=2ÞÀ4n l sinðn l=2Þcosðn l=2ÞÀ2Þþ1
ð28Þ
G.P.Zou et al./Composite Structures65(2004)499–510503
