F6
_77,IT: ROOM
STRESSES IN A LAI D JOINT WITH
ELASTIC ADHESIVE
14o. 1864
September 1957
R A 4,,t.
FEB 171958
v.v
*PATE Co
This Report is One of a Series
Issued in Cooperation with the
ANC-23 PANEL ON SANDWICH.
CONSTRUCTION MR AIRCRAFT
of the Departments of the
AIR FORCE, NAVY, AND COMMERCE
H II111111111rIlm
in.1[111[,'
1111111111111l
FOREST PRODUCTS LABORATORY
MADISON 5, WISCONSIN
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
In Cooperation with the University of Wisconsin
t
TABLE OF CONTENTS
Page
I.
SUMMARY
1
II.
INTRODUCTION
1
III.
NOTATION
2
IV.
THEORETICAL ANALYSIS
4
V.
NUMERICAL COMPUTATIONS
24
VI.
CONCLUSIONS
27
VII.
REFERENCES
28
VIII.
ACKNOWLEDGEMENTS.
29
IX.
APPENDICES
30
X.
Appendix A -- Integration Formulas
30
Appendix B -- Euler Transformation for Divergent
Series
32
FIGURES
33
—
STRESSES IN A LAP JOINT WITH ELASTIC ADHESIVE !
By
R. E. SHERRER, Engineer
Forest Products Laboratory, ? Forest Service
U. S. Department of Agriculture
I. SUMMARY
A theoretical solution for the displacements and stresses in the adhesive of a lap joint loaded in tension is presented. The solution is applicable
for an anti-symmetrical jointwith arbitrary overlap, adhesive thickness and
plate thickness and for adhesive and plates of any arbitrary combination of
materials. Elementary theory of bending is used for the plates and theory
of elasticity is used for the adhesive. The usual limitations of these theories
apply to the solution. Numerical results and curves are included.
II. INTRODUCTION
The purpose of this report is to solve the following problem:
Given two rectangular sheets of any isotropic material and of equal
thickness t, of unit width, and of length (/ + 2c) (see figure 1), lap-jointed
!This report is one of a series (ANC-23, Item 51-4) prepared and distributed by the U. S. Forest Products Laboratory under U. S. Navy Bureau
of Aeronautics Order No. NAer 01628 and U. S. Air Force No. AF-18
(600)-102 and DO 33(616)-53-20. Results here reported are preliminary
and may be revised as additional data become available.
?Maintained at Madison, 'Wis., in cooperation with the University of Wisconsin.
Rept. No. 1864
2.
over a length 2c, bonded by an adhesive layer of any isotropic material
and of thickness 2b, and acted upon by tensile forces of magnitude T o per
unit sheet width, find the displacements and stresses of all points within
the adhesive.
For the solution in this report, a free body of the joint section (see
figure 2) acted upon by loads as computed in reference 1 will be used. Elementary theory of bending will be used on the rectangular sheets (henceforth
called plates) and theory of elasticity, using the Saint-Venant principle on
the free edges, will be used for the adhesive. Plane strain is assumed.
III. NOTATION
x, y
rectangular coordinates (figure 1)
2c
length of joint
2b
thickness of adhesive
t
thickness of each plate
G
shear modulus of elasticity for adhesive
Poisson' s ratio for adhesive
E
Young' s modulus for plates
D
flexural rigidity per unit width of plate
G'
2Gc non-dimensional rigidity factor
Et
D'
Dnon-dimensional rigidity factor
2
2Gtc
T
tension in plate per unit width
3.
M
moment in plate per unit width
V
transverse shear in plate per unit width
To
value of T at beginning of joint
Mo
estimated value of M at beginning of joint
T'
M'
Vt
To , non-dimensional tension factor
4Gc
Mc
12 D
, non-dimensional moment factor
Vo
• non-dimensional shear factor
4Gc2
Cro
normal unit stress of adhesive on plate
To
tangential unit stress of adhesive on plate
v odeflection of plate surface in contact with adhesive
uohorizontal displacement of plate surface in contact with
adhesive
vp
deflection of center plane of plate
up
horizontal displacement of center plane of plate
u, v
displacements of adhesive
o , o-y , , T xy plane stresses in adhesive
do
cosh an b
en
s inh ftn b
4.
fn
anb sinh anb - (1 - 2p.) cosh anb
gn
f3n b cosh Pn b - (1 - 2p.) sinh f3nb
h n
anb sinh an b - (3 - 4p.) cosh anb
in
Pn b cosh (an b - (3 - 4p.) sinh Pnb
in
anb cosh an b - 2(1 - p.) sinh anb
kn
sinh an b
fn
an b cosh anb
mn
Pn b sinh 13n b
nncosh On b
grib sinh Pnb - 2(1 - IA) cosh Pnb
on
IV. THEORETICAL ANALYSIS
To simplify the writing, the units of length will be taken such as
to make c equal to unity. This will have the effect of making b, t, u, v,
, vp
and the coordinates x and y non-dimensional, since b
u o , vo , up
will be written for b/c, t for t/c, etc.
From the free-body diagram shown in figure 3
0
a- o dV
-dx
.T.0 _ dT
0
dv dM
2
dx
dx
0
5.
where the moment M is given by
M=D
d 2 v,
(4)
Elimination of V and M gives
dv.
dr_
d4v
Dd
x4 P
+ Cr° = dx (T)
(5)
as the differential equation for the plates. It will be assumed that the
displacements of the central plane of the upper plate are related to the
displacements of the lower surface of the plate as follows:
v " = vo
t dvo
• u
— dx
p = uoZ
These assumptions are in accord with the elementary theory of
bending used for the plates. The unit strain at the central plane of the
plate is given by
dup T
dx Et
(8)
From equations (7) and (8)
T
Et
duo t d2vo
=0
dx
2 dx
(9)
Equations (5) and (9) are the two differential equations to be solved. It
6.
is noted that the term on the right side of equation (5) is non-linear
since both T and v pdepend on the applied load.
The differential equations for the adhesive based upon plane strain
are
I
8 au
av
1 - 2p. ax
+ -57 )
"2u = °
1
8 ,8u , Ov%
1
1 - 2p. 8y 8x
8y
(10)
v-v = 0
The boundary conditions for the upper plate are (see figure 2)
T = T o at x = 1
(12)
T = 0 at x = -1
(13)
M = Mo at x = 1
(14)
M = 0 at x = -1
(15)
V = -Vo at x = 1
(16)
V = 0 at x = -1
(17)
To simplify the analysis, the principle of Saint-Venant will be used
for the free boundaries (x = ±1) of the adhesive. This imposes the
following conditions:
crxdy =
x = ±1
0
(18)
7.
6x
y dy = 0
(19)
dy = 0
(20)
x = ±1
-b
T
xy
-b
x = *1
The conditions at the boundary between the adhesive and the upper plate
will be assumed to be
Oro
= Cry
To = T x
at y = b
(21)
at y = b
(22)
u 0 = u at y = b
(23)
vo = v at y = b
(24)
Similar conditions will apply for the lower plate, but these do not have
to be expressed if the necessary conditions of symmetry and anti-symmetry
are observed when writing the expressions for displacements. If any
term in the expressions for u and v is even in x, it must be odd in y
and vice versa.
Expressions for u and v satisfying equations (10) and (11) and
with sufficient arbitrary constants to satisfy all boundary conditions as
well as equations (5) and (9) will now be written. The details of how the
8.
various terms of these expressions were obtained is not important and
will not be given. That each term does satisfy the differential equations (10) and (11) may be verified by substitution. Equations (12) . to (20)
give nine conditions for which nine arbitrary constants (a 1 to a 9 ) have
been provided. The first seven of these are used in the expressions
for u and v, the eighth comes from the integration of equation (2) and
the ninth from integration of equation (1). The plate differential equations (5) and (9) together with conditions at the boundary between the
plate and adhesive make it necessary to add to the expressions for u
and_v terms which become Fourier series for this boundary. The expressions follow:
u = a y + a 2 (1 - µ)xx + a
1
5 5
a
7
+
n=1
x
5
[3
3
3xy2
a
x2y + 1
4 (3.3
1
- a6 —4x 4 y - —
+ (1 - 2p.)x 3y2 - (1 - p.)xy4
20
(1 - 2p.) x 4
y - (1 - p.)x2y 3 + (3 - 214 y5
4
Anorny cosh an y + A' Binh
20
a y
n
YE nf3ny sinh 13ny + Bn cosh 13n1
n=1
d
cos ax
an
sin Pnx
(25)
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13.
From equations (2), (22), and (29)
T = ZG 1 si x
-
w
a4 x 3 a 5 li bx 4 + (1 - p.)b3x2
2
3(1 - p,)
3a 3 bx 2 -
2
[2 - 11 ) 3,-5 _
(1
20
(2 -µ)O]
4
+a
µ)
n=1
n=1
d
a "Iµx 5
' 20
A n fn + A n' d
+ ZG
[3ngn + Bn e
- 2G
13 4x +
4
2
8
b 2x 3 _ - 1
CO5
(1 - /1) b2x3
2
sin anx
finx
n
(30)
Pn
From equations (4), (6) and (26)
a 3 b + a4 2(2 - /1) x +
M
( 1. -
-
1- a7x )
6(1 - ix) b 2 x
3
6a.
5 bx 2
D
-9- a
6
(3 - 2µ) x3
[Anhn + An
an sin anx
n=1
+ DBin +B'n el
n gii cos Onx
n=1
[n
From equations (1), (21) and (28)
(31)
14.
V
= -
2G
a9 -
+
a (x 3 - 3b2x)
3
) x5
p.b 2x 3 -
a5
(1 -
a
p.) b 3x2
2
+
a
(1
b
4 (1 - p.)
2
x2
b4]
+
a bx4 4. (1 1-4 3x
2
7 4
a 6 ( 2 - /1) bx4
4
+
2G
n=1
+ An kd
c
13
anx
an
+ 2GZ
E3n o n + Bn'
n=1
n sin 13nx
(32)
Pn
It is advantageous to select an and On such that
siriai
cos
(33)
=0
Sri 0
(34)
These relations will be used wherever they apply.
The boundary conditions, equations (12) to (20), will now be used
to obtain nine equations between the constants a l to a9 inclusive. From
equations (12), (13) and (30)
- 3a
b -a5 E
2
3
b+ (1 - p)bl + a 8 = T'
(35)
15.
1a4
al 3(1 - p.)
2
+
a
3
+
a
k-. + ( 1aa) b 2 _(2 - ii) b 4
4
2
20
7
From equations
5
1 -11 4 2
2
- a6 ( 2 20
(14),
=
=
_
4
114
(36)
T'
(15) and (31)
- M'
(37)
b
a4 2(2 - P. ) + a6 (3 - 2p.) - 6(1 - lob ] - a7 = - 6M'
(1 - p.)
(38)
From equations (16), (17) and (32)
a 9 + a.4
b 3a7 Eb +
- (]L -
a6
4
(1 - p.)
4
2
[Awl n + A'. 1(11 cos an - V'
-
- 11) b3
2
(39)
an
n=1
(1- 1.0d
a 3 (1 - 3b 2 ) + a 5 (1 + /1)
2
10
Y
+
n=1
, n
B n o n + 13 n
n
sin gn
Pn
=-V
(40)
16.
From equations (18) and (27)
a2
+
a 3 (3 - b 2 )
+
a5
(1 - p.)b 2- IL:Lk) b4
10
+
=
0
(41)
From equations (19) and (27)
a
(1 - 11)
6
3
- 11 )
3
(2 5
b].
6
aE2
4
- (1 - 1.1.) b2
2
20 b
(1 - p) b2
2
( 2 P.)b]
20
µb
5
a
_
(42)
From equations (20) and (29)
al _a4
2
1 - 11)
a7 E
4
+
)-F An j n
n=i
cos a
+ A n' k ri a n b
n = 0
(43)
Equations (5) and (9) provide the additional relations that are required.
From equation (9)
17.
5
4
3
F0 +F1 x +F 2 x 2 + F3 x + F4 x + F5 x
Y
f
-
hat
A n + kn
n=1
n=1
En inP2
enG1
nt gn1
nn
Pn
8} cos O nx
n
dncrnt
drip'
– +
A ni } sin anx
2
enPnt
2
(44)
=0
From equation (5)
4
Ho + H 1 x+ H 2 x 2 + H 3x 3+ H4x
-
2 jn + a f
thnon3 D '
n n] A n + d n cr 3 D' + --k n
n
=1
+ an dd A ll sin
ni3n3 D + 2
– o n - fin gd Bn
a
x n=1
+ [e n i3 n D'
+ –2 n
t n
1 d dvn
e 13 n1 cos Pri x * _ --(T--'-.)
nn
(ix
Gt clx
+0
(45)
18.
Where
F0 = - a 2 (1 -
F1
1
-
al G
+ 3a 3b [b
+ a4
_
( 1211
+ E b4G
F =
3
(2 -
b
ti)]
+b
a 5 3(1 =- 2p.)b 2 + 3bt + (1 - p.)b 3G
G
a
+ a6 b
4 3(1 - p,)
+aE
±(3 - 2µ)t+ 1. (1
2
2
_
2p.) b + +
7
2
Ob2G]
aE + La be'
2
F5 =
Ho =
6 (2
20
[32
3 t
(46)
a6 3(1 - p.)b2t
[
2(1 - 1.1.)b 3 + (2 - FL) b1/41
4
4
F2 = -
+ a 5 (1 - p.)b4 + a G '
8
(47)
(48)
Ii)b 2 G '
(49)
(50)
G + a -L G'
7 20
+ a 5 libD'
(51)
_ (1
- 14 0 - 2(1 - p.)b1 (52)
t
19.
Hi
= -
a
+ 3(1 -
H
H
2
(1 -
4b
(1 -
- a 7 6D' -
}L)b 2
2
a _6a
3 t
5
3
= - a 6 2(2 t
+
a6 6(3 - zon
2 (1
b 3 -
11 )
+ LL1.0.1,3
3(1 -
YID 2
(53)
(54)
61.1.1]
t
+
(2 - 4)
+
a 7 -21 1D
(55)
+
t
(56)
H 4 = a 5 (1 t
Relations for the constants,n
A A ''Bnand B , can be found by
———
applying integral transforms to equations (44) and (45). Using the sine
transform (see Appendix A) on equation (44) gives
2 cos an F
l
an
hnant
4
_
_a6 ) F3
InG '
An
a
2
n
+
[cn
_ 20 + 120 ) F5
an
dnant
2 +
cinG
a—
A n
[n
=0
(57)
20.
The cosine transform on equation (44) gives
2 sin /3
n
F0
, - 2
kl
) F2
On
+
Pn
12
- –2On
+
24% F4
Pn
–74/
en(3nt
2
n13nt gnGl
Bn
2
13n
-
e
mn
(58)
B' = 0
n
/3n
The sine transform on equation (45) gives
2 cos an
an
H1 +
(1
-
3
4
+
a3 D' +
nn
jn +
anf]
n
An
2 k + a d A' = P n
+ dna3
n 13' + t n
n n n
(59)
The cosine transform on equation (45) gives
2 sin I3n
+
H o
+
+
2 o
in
2
(1 - pz ) H
g 1 B n
nn
13
+
+
12
(1 - —
p2
3
enfin D I
+
+
24
—)
H4
p4
2 nn
i-
+
+
i
n13n3 D
nen B'n
=
Qn
(60 )
where
1
P=1
2Gt
v )
d (T dv
...1 dx
dx
sin anx dx
(61)
Qn
21.
1
1 f d (T dv ) cos Pnclx
2Gt
dx
clx
-1
(62)
Equations (35) to (43) and (57) to (60) form a matrix from which
the constants may be solved. A simplification of the matrix may be
made by multiplying equation (43) by b and adding the result to equation (39). The resulting equation,
a 9 + al
b a _Eb5 - a7
2
6 20
(2 - 114 5
20
= V
(63)
will replace equation (39). The nature of the matrix is such that it may
be broken into two independent matrices. By using the aforementioned
equations, together with equations (46) to (56), matrices (64) and (65)
are formed. Solving these matrices (with a finite number of equations)
for the constants and substituting the constants into equations (25) to
(29) will give the displacements and stresses for the adhesive.
24.
V. NUMERICAL COMPUTATIONS
Mo and Vo are computed from a given force T o by means of the
following equations:
Mo =k
2
= k (t
b) T
2
(66)
2b)
(67)
where the values of k and k' are taken from table 1 of reference 1.
In order to simplify the computations, the non-linear terms,
P n and Qn , of the matrices, were set equal to zero. Reference to
equations (61) and (62) together with equations (26) and (30) will show
that each term of P n and Qn involves the multiplication of two small
constants thereby making the contribution of P n and Qn to the matrix
significantly small.
The fifth equation of matrix (64) and the fourth equation of matrix
(65) each contain a divergent series. Both of these equations are
used to satisfy the shear stresses on the boundaries of the adhesive
and plates. Since they describe a physical condition of the problem,
they must sum to a finite value. The problem then is to determine the
values of the summations with a finite number of terms. There are
several methods of summing divergent series, but the one best suited
for this problem is the Euler transformation for divergent series (see
25.
Appendix B). The reasons are that in both series the terms in general
alternate in sign, remain fairly constant in magnitude (for large values
1
of c_) for matrix (64), and decrease nearly as — for matrix (65). These
are conditions for which the Euler transformation works very well.
Since one of the divergent series comes from equation (29) for
shear stress, it will be necessary to treat both equations the same way
in order to satisfy the boundary condition for shear. Hence, if the
Euler transform is used in the matrix, it must be used on equation (29).
For the other stresses and the displacements it is not so easy to determine whether to use the Euler transformation or not. If a series already converges iri a satisfactorily way, the application of the Euler
transformation may slow the convergence or even cause the series to
diverge.
Since it is difficult to determine the need of the Euler transformation analytical, the following three cases were studied numerically.
(a ) Transformation is not used on any equation
(b ) Euler transformation used on matrix equations only
(c Euler transformation used on matrices, displacement and
stress equations.
A typical comparison of u, v, cr x , o-„, and r xy is shown in figures (4)
to (8). A study of these curves and other similar comparison curves
led to the following conclusions:
2 6.
(1) use the Euler transformation on the matrices
(2) do not use the Euler transformation on u
(3) the same results are obtained with or without the Euler trans formation on v
(4) use the Euler transformation on
cr x
(5) the same results are obtained with or without the Euler transformation on
cr
(6) use the Euler transformation on T xy
It is possible that different choices of material and different ranges of
dimensions would lead to different conclusions. Also, some other transformation may work better than the Euler transformation.
Having established the numerical procedure, it remains to determine how many Fourier terms are needed to give satisfactory answers.
Figures (9) to (13) show the effects of adding these terms to the solution.
From a study of these curves and other similar curves, the following
conclusions may be drawn.
(1) displacement u converges in a satisfactory way, but will require several Fourier terms
(2) displacement v could not be made to converge by any method
tried
(3) stress Tx could not be made to converge by any method tried
27.
(4) the maximum value of stress
cr
is obtained with very few
Fourier terms, but the distribution will not be established
with these terms
(5) the shear stress T xy distribution could not be established with
the number of terms taken.
The most important stress in the design of lap joints is the normal stress cr y , as a study of the stress curves will verify. The maximum normal stress is well established with y = 4. Therefore, a study
of this normal stress is made under varying parameters. Figure (14)
shows the effect of varying the glue thickness while holding all other
variables constant. Likewise, figure (15) shows the effect of varying
the plate thickness and figure (16) shows the effect of varying
the
load.
VI. CONCLUSIONS
For the most important stress Ly , reliable computations can be
made from the solution. This is also true for the displacement u. The
shear stress .r xy requires further study and it appears that all that is
required is to take many more Fourier terms. For the displacement
v and the normal stress o- x
to be found.
some means of obtaining convergence has
28.
VII. REFERENCES
(1)
Goland, M. and Reissner, E.
1944. Stresses in Cemented Joints. Journal of Applied
Mechanics, Vol. II.
(2)
Kuenzi, E. W.
1956. Determination of Mechanical Properties of Adhesives
for Use in the Design of Bonded Joints. Forest Products
Laboratory Rept. No. 1851.
2 9.
VIII. ACKNOWLEDGEMENTS
The author is indebted to the Forest Products Laboratory and to
the Engineering Experimental Station of the University of Wisconsin for
providing financial support for this project. The author wishes to thank
Prof. Gerald Pickett of the Mechanics Department of the University of
Wisconsin and Mr. Charles Norris of the Forest Products Laboratory
for their advice, guidance and encouragement throughout the course of
this work.
30.
IX. APPENDICES
1. Appendix A -- Integration Formulas
Integration formulas used in this report include the following:
(48)
sin anx dx = 0
x sin a
x
S4
2
nx
2 cos a
n
dx =
sin a x dx =
n
n
4
(70)
0
3 .
x s in a x dx
x
(69)
2 cos a
n
an
(71)
(1 -)
(72)
sin anx dx = 0
5 s.
x m ax dx
n
2 cos a n
an
ki -
20
an
120
1
a4
n
(73)
31.
-1 cos
gn
2 sin
x dx =
13
n
(74)
n
x cos Pnx dx '= 0
(75)
2
x cos gnx dx = 2 Bin
(76)
gn
1311
1
x
3
(77)
cos gnx dx = 0
x4 cos gnx dx
2
x 5 cos Pax dx = 0
sin /3 n
12
24
on4
(78)
(79)
32.
2. Appendix B -- Euler Transformation for Divergent Series
The transformation is usually written as follows:
al
a + a
l
2
a a + 2a + a
2
1
2
3
a 2+ a a
+
3a + 3a + a
1
3
z
4
2
3
a
+ 2a + a
3a 2
3
4
a + a
3
4
a
4
(80)
1
where al , a2 , a 3 , etc. are terms of the original series. The new series
is formed as follows:
1
1
1 1
1 (a + 3a
—a + — (a 1 + aZ • ++ —
(a + 2a 2 + a 3 ) +
2
1
1
4
21
8
16
+ 3a
3
+ a ) + . .
4
(81)
In the series used in this report, the identity of the original terms has
to be maintained since the terms are used in amatrix. In order to do
this the series is rearranged in the following manner
Kl a l + K 2a2 + K 3a 3 + . . . . + Knan +
+ Kta.
k
(83)
where
Kn
(m - 1) !
ni=.n (rn - n) ! (n - 1)1 2m
and k is the total number of terms used in the series.
(83)