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The adhesive scarf joint in pure bending
Int. Z mech. Sci., Vol. 18, pp. 223-228. Pergamon Press 1976. Printed in Great Britain
THE ADHESIVE SCARF JOINT IN PURE BENDING THEIN WAH Texas A & I University, Kingsville, Texas, U.S.A. ( R e c e i v e d 1 N o v e m b e r 1975, a n d in revised f o r m 21 A p r i l 1976)
Summary--The stress distribution in a scarf joint subjected to pure bending, and with arbitrary angle of scarf, is analyzed as a two dimensional elasticity problem in plane stress. Both the adherend and the adhesive are assumed to be elastic and isotropic. The two adherends may have differing moduli of elasticity. Numerical results are given. NOTATION
h o w e v e r , c o n f i n e d his a t t e n t i o n to t h e s c a r f j o i n t under pure tension and showed that the stresses w e r e c o n s t a n t a l o n g t h e j o i n t f o r all s c a r f angles p r o v i d e d t h e a d h e r e n d s h a d t h e s a m e elastic properties. In this p a p e r we h a v e a t t e m p t e d to give a s o l u t i o n of t h e s c a r f j o i n t u n d e r p u r e b e n d i n g f o r a d h e r e n d s with elastic c o n s t a n t s w h i c h m a y b e different. It is b e l i e v e d t h a t t h e s o l u t i o n is sufficiently a c c u r a t e f o r d e s i g n p u r p o s e s f o r s c a r f angles t h a t are n o t t o o small.
A , A ~ coefficients c length of adhesive E, E ' modulus of elasticity of adherends G, G ' shear rigidity of adherends Ec modulus of elasticity of cement Gc shear rigidity of cement Mo applied moment t adhesive thickness u., v,, radial and transverse displacements ux, vy x and y displacements r, 0 polar co-ordinates x, y rectangular co-ordinates a scarf angle 3' complex root of equation $ Airy's stress function v, v', vc Poisson's ratio for adherends and adhesive, respectively o'o = 6Molc2 sin2 a (r,, fro, 7,, stress components trx, try, ~'x, stress components ix K
=
FORMULATION OF THE PROBLEM The scarf joint is shown schematically in Fig. 1. The moments M0, applied at some distance from the scarf, are assumed to be produced by the linearly distributed stress shown. The cement is considered to have a very small thickness t compared to the length c of the joint. The problem is to determine the stress distribution in the cement and adjacent adherend material. The boundary conditions for the adherends are that the top and bottom surfaces be traction free, and that the stresses on the vertical sides be equivalent to a couple Mo. The use of eigenfunctions with undetermined constant coefficients enables the satisfaction of all these conditions. The constants are determined by imposing two sets of conditions; first, that the stresses at the adherendadhesive interfaces satisfy the equations of equilibrium of the adhesive treated as being very thin, and, second, that the displacements of the adhesive and adherends be compatible on the same interfaces.
GIG'
=-~/cot~/ Gt
)t 2
=
G~c Gt - E.c It
x, =~?vo A, A5
= A3/A2v~2 = A3/vo
Functions involved in the equations are defined in the Appendix. Primed symbols refer to the right hand adherend. INTRODUCTION WITH THE i n c r e a s i n g u s e o f c e m e n t e d j o i n t s in s t r u c t u r a l c o m p o n e n t s , the d i s t r i b u t i o n of s t r e s s e s at t h e a d h e r e n d - a d h e s i v e i n t e r f a c e of a s c a r f j o i n t h a s b e c o m e of i m p o r t a n c e in design. W h i l e c e m e n t e d j o i n t s o t h e r t h a n t h e scarf h a v e r e c e i v e d wide a t t e n t i o n f o r b o t h i s o t r o p i c a n d a n i s o t r o p i c m a t e r i a l s l-4 t h e o n l y paper, k n o w n to t h e a u t h o r , w h i c h specifically a i m e d at e l u c i d a t i n g t h e s c a r f j o i n t p r o b l e m is t h a t b y L u b k i n , 5 w h o ,
Y \
b-FIG. 1. Scarfjoint. 223
224
T. WArl
The scarf angle a is arbitrary and plane stress is assumed. The materials of both the adhesive and adherends are linear-elastic, but the two a d h e r e n d s m a y have different elastic constants. STRESS FUNCTION We use two co-ordinate s y s t e m s (r, O) and (x, y) as s h o w n in Fig. 2.
Equation (8) h a s complex roots, which occur in conjugate pairs. Choice of only those roots which have a positive real part e n s u r e s the satisfaction of the boundary conditions at large values of x > 0. T h e normal and shear stresses, o'0 and ~-~ along the c e m e n t line O = a, m a y be obtained by first calculating o-,, o-y and ~.~ from equations (5) and then transforming to the (r, O) system. T h e result is ~r~ = o-810=~ = 4Ai(3,2/sin 2 a ) X l ( z , 2 a ) + o-o sin s a(1 - 2z) (9)
Z = t'./C
\
~X I E ,G~u
~ ZI= f I/"C
I
yi
(10)
where the functions X I and X 2 are defined in the Appendix and the coordinate z = r l c is indicated in Fig. 2. Using H o o k e ' s L a w one m a y also derive expressions for the displacement c o m p o n e n t s in the x and y directions and from these obtain the radial and tangential components f r o m the formulas
>
X 4
~'~ = r~t0=~ = 4A~(3,2/sin2 a ) X 2 ( z , 2 a ) + cro sin a cos a ( 2 z - 1)
u~=u~cosO+v,
sinO,
v,=-u~sinO+v~cosO.
(11)
FIG. 2. Co-ordinates.
T h e displacements u, and v, along 0 = a written as follows
T h e stress function (1) where ¢,
=
(aod~16)(3y2ld~
2 y , / d ~)
-
m a y be
_ c 23" u~ = u,[e=~ - ~-~ sin a A , X 4 ( z , a, v)
(2)
(
')
c
23, sin a A i X 3 ( z , a, v)
+~-~o-0 sm a
(z~-z)+Vsina
and
+ H cos a ~b2=a'c~exp(-23,x]{Ksin3,(c
_
sma-l)
+Y(csina2Y _ l ) c o s 3 ' ( c s i n 2 Y a _ l ) }
v, = v0[e=~ - ~ (3)
o-o = 6Molc ~ sin 2 a.
(4)
The stress c o m p o n e n t s m a y be calculated f r o m the relations ,r. = a~4~lay ~,
o-y = a~4~lax ~,
~, = -a~laxay.
(5)
F r o m equations (2) and (4) it is easily seen that the stress function ~b~ satisfies the boundary conditions o',=r,,=0
at
y=0,
csina
(7)
T h e stress function
a + l-~v)z2-z
COS
a sina] (13)
In these equations, G is the shear m o d u l u s of the adherend and v is P o i s s o n ' s ratio. The c o n s t a n t s H, V and R define the displacements and rotation of the adherend as a rigid body. T h e functions X 4 and X 3 are defined in the Appendix. Equations (9), (10), (12) and (13) represent the stresses and displacements of the left h a n d adherend in Fig. 1. A similar set of equations m a y be written for the right hand adherend by replacing o-o by -o-o, A~ by A'~, z by z'=l-z,G byG',vby v ' , a n d V, H a n d R by V ' , H ' and R '. c and 3, remain the same for the two adherends.
(6)
and yields also ~r~ = o-o(1 - 2y/d).
2
- H sin a + V cos a + cRz.
in which A~ and K are constants, is biharmonic. In equation (2) d = c sin a,
+ ~ oC - o [ C O t a s i n
(12)
(8)
and K in equation (3) is taken to be equal to -3, cot 3,.
EQUATIONS OF EQUILIBRIUM FOR ADHESIVE T h e two equations of equilibrium for the adhesive are (Fig. 3) ao-=/On = -a.r=/cOz,
aerz/caz = - a r ~ / a n .
(14)
Since we a s s u m e the adhesive to be very thin we m a y replace the first of the preceding equations by (or" - o-o)It = - Or./caz and (o-~ - a ' ) / t = - ar" lcaz'.
(15)
The adhesive scarf joint in pure bending
225
E Q U A T I O N S O F COMPATIBILITY Approximately, the shear strains on the two surfaces of the adhesive are ~, = - ( u ~ + u ' ) l t + Ov,/cOz
and y" = - ( u , + u',)lt + o v " l c a z ' .
+ OZ dz
Using H o o k e ' s Law we get u, + u" = -(tl2Gc)(z~
+ ~')+(tl2c)(av,
lOz + Ov'/Oz')
(23) and also FIG. 3. Equilibrium of cement.
vo + v " = - ( t / 2 E c )[or. + o r " - vc (~z + ~r')]
Adding these two equations, we get Or.lOz + O'/.IOz' = 0.
(16)
From the second of (14) we may similarly derive O~r,lOz + oG'Joz' = 0.
1 as(y,.)
On s
c
c s Oz s
where Gc is the shear modulus of the adhesive. Substituting from (19) and (20) into (24), the latter becomes v . + v " = - ( t / 2 E ~ ) [ ( 1 - vc2)(~ + tr') -(voEclc)(au.lOz
(17)
+ au'laz')l.
OzOn
is automatically satisfied if we assume O~]On s = 0 since
Ouo I O z - Ou " l Oz' + ( t 12G~ )(Or./Oz - Or './ Oz') - ( t l 2 c ) ( O 2 v . IOz 2 - 0 2 v ' l O z '2) = 0
Oz s
(25)
Reference to equations (12) and (13) shows that equations (23) and (25) contain the rigid body terms H, V, R z and H ' , V' and R'(1 - z). These terms are eliminated by differentiating (23) once and (25) twice with respect to z. Equations (23) and (25) then take the form
The compatibility equation 0'E, ~ 10"~.
(24)
Oz s \ O n ]
(26)
OSv./Oz~ + OSv ' /Oz 's
+ (t(1 - v~S)12Ec )(OSo'./Oz s + OS~'lOz '2)
and
- (uotl2c)(O3u~ IOz ~ + O~u'/Oz '~) = 0.
as a s (o.+lO~=la so~ OzOn y" = ~ \o--n c ~ l c ozs On"
Now ~ = ,7,IE~ - J,c~.IEo
(18)
where Ec and z,~ are the elastic modulus and Poisson's ratio for the adhesive. Setting ~ = Ou.lcOz equation (18) gives
(27)
The stress and displacement components equations (9)-(13) and similar equations for the adherend are now substituted into equations (26) and (27). The resulting equations may be follows:
given by right hand (16), (22), written as
y, 2a) = 0
(28)
A , Q ~ ( z , y, 2 a ) + A ' l Q t ( z ' ,
A~Q2(z, y, a, v, 1) + A ~Qs(z', y, a, v ' , t~) =
-0.5A4~0 sin s a (s - / ~ s ' ) o-~ = E~Ou, IcOz + v ~ o .
(29)
(19) A , Q 3 ( z , y, a, v, 1) - A '~Q3(z', y, a, v ' , t~ )
Similarly,
= O,o{0.25(1 - 2z) sin a ( s - t~s') tr', = E~Ou'lcOz' + v~a'.
(20)
A~Q4(z, y, a, v, 1) + A~Q4(z', y, a, v', t~) = -0.125a0 sin 2 a ( s - t~s')
From (19) and (20) we obtain
oo-~ 4 oo-.' Oz
--
Oz'
E~'a2u a s "+~(ao-.+ao-;,~ - . . . . . 2c Oz s + 0 2 2 \Oz Oz']"
(21)
From equations (17) and (21) we get E~(o~.+a~'~
2c \ Oz"
vo/oo-= a,,'x
Oz'S ] + "~ [-~-z + "~'z') = 0.
+0.SA~cosa(s+t~s')-A~sin2acosa)
(22)
Equations (16) and (22) are the two equations of equilibrium for the adhesive.
(30) (31)
in which ~ = G / G ' is the stiffness ratio of the adherends, and the Q~ axe given in the Appendix. In the preceding equations s = s i n 2 a + l / ( l + v ) , s'=sin 2a+l/(l+v') and the A's are defined under Numerical Results. It should be remarked that the set of equations (28)-(31) all become homogeneous when t~ = 1, v = v', and a = ~r/2. We then get the trivial solution A~ = A ' ~ = O. Thus only in the case of the 90 ° scarf with equal adherends is there a linear distribution of stress throughout, the solution being given by the second term on the
226
T. WAH
right of equations (9) and (10). For all other c a s e s the stress distribution is nonlinear. SOLUTION OF EQUATIONS The m e t h o d u s e d for solving the equations is point matching. W e have u s e d the first five roots of equation (8) given by
_.
(%'
3q.2.3.4.5= 3.7489 -+ i 1.3843, 6.9500 -- i 1.6761, 10.1193 -+ i 1.8584, 13.2772 --+i 1.9916, 16.4299 -+ i2.0966 (32)
%:
so that there are 20 u n k n o w n c o n s t a n t s in each of the equations. T h e equations are satisfied at 11 points z = 0, 0. ! . . . . . 1, resulting in a total of 44 equations in 20 u n k n o w n s . A linear least squares procedure is then used to obtain the 20 u n k n o w n constants. The stresses or,, %, o"~, r " and the average longitudinal stress Oz = (~r= + O=)/2 are then calculated. T h e equation for Oz is given in the Appendix. NUMERICAL RESULTS AND CONCLUSIONS Figs. 4-8 give the numerical results obtained for the normal and shear stress and the stress O, along the adhesive for various scarf angles a varying f r o m 90 ° to 60* and for several values of the stiffness ratio t~. W e have a s s u m e d that the P o i s s o n ' s ratio for both the a d h e r e n d s to be 0.3. Other parameters were arbitrarily c h o s e n taking care only to keep the t i c ratio small for the adhesive. T h e following numerical values were used in the calculations:
A, = G t / G c c = 0.1,
A2 = G t l E c c = Ad2.5,
A4=A3/A2/~c2, As=A3/v~, v~ =0.25,
o,-I~" }
I /'//))jo.,
-0.5 -0.4 - 0 3 - 0 2 - 0 I
<
/
~
02
03
0.4 0.5
r~l~ o
u
;0 08
60*
0204
/x = 1,0.5,0.1.
~75' 60.~ "~,,..,~~ -I
-08
-06
-04
-02
02
%1%
and ~ 1 %
04
06
08
-
FIG. 6. a = 75 °, 60 ° normal stresses.
08/ o.6o
~
0.5
z
o , ¢ ~
OI
.... ,;/%
FIG. 5. a = 90 ° shear stresses.
-
~J
0
~1%
A3 = vct/2c = 0.000125
We remark that the plane strain problem which more nearly approaches the practical case, m a y be obtained f r o m the present problem by substituting E / ( I - v 2) for E and H(1 v) for v w h e r e v e r t h e s e quantities occur. T h e pattern of normal and shear stresses distribution, as seen f r o m Figs. 4-7 is rather unexpected. Only in the
O5
p_ .o,
o.
~
p - 1.0,0.5,0.1
,/.L=O I
_..
Ga / G O
~ffiO. I
-04
-03
-02
-OJ
r J G o and r~lo-o - FI6.7.
0
, I oa/~O'J
02
I
08
04
I
06
06
I
04
08
I
02
FIG. 4. a = 90 ° normal stresses.
1.0 " ~ O ' a / ~ " 0
I
0
OI --
02
03
04-
% I G o and -i'ilO'o
a = 75 °, 60 ° shear stresses.
case of the 90 ° scarf is there f o u n d a strong nonlinearity in the stress distribution (Figs. 4 and 5). T h e nonlinearity b e c o m e s stronger the greater the deviation of ~ is f r o m unity. For the other two scarf angles 75 ° and 60 °, not only is the nonlinearity in the stress distribution negligible, but the distribution is almost identical for different values of/~
The adhesive scarf joint in pure bending
227
solution. The higher the root the larger the negative exponential involved (see equation (32)) and significant figures are lost in computation. This situation could become serious for small scarf angles and could necessitate using a smaller number of roots to attain reasonable results.
0.5
04
02
0.~
0 2
04
0.6
0.8
I0
1.2
REFERENCES 1. M. GOLAND and E. REISSNER, J. appl. Mech. 11, A17 (1944). 2". F. SZEPE, Exp. Mech. 6, No. 5, 281 (1966). 3. O. VOLKERSEN, Lu[lfahrtforschung, 15, 41 (1938). 4. T. WAn, J. Materials Technol. pp. 174-181, July (1973). 5. J. L. LUBKIN, J. appl. Mech. 24, No. 2, 255 June (1957). 6. S. TIMOSrmNKO and J. N. GOODIES, Theory o / Elasticity. (3rd Edit.) pp. 61--62. McGraw-Hill, New York (1970). APPENDIX
FIG. 8. Axial stresses. (Figs. 6 and 7). There are slight differences in the curves for differing/~ but they could not be shown on the scale of the graph. Data for a 45 ° scarf (not shown) also exhibits little nonlinearity. The difference in stress magnitudes on the two sides of the adhesive were small and could not be shown without greatly expanding the scale of the figures. It was felt that the peculiar behavior of the 90 ° scarf needed further scrutiny. Stresses were therefore computed for an 85 ° scarf (not shown). This too exhibited little nonlinearity. One is thus forced to conclude that the 90 ° scarf is indeed a special case. A possible explanation may be found from an examination of equation (10) and Fig. 5. It is seen that for a 90* scarf (and for this scarf alone) there is a symmetrical distribution of shear stresses (identically zero f o r / ~ = 1) together with an antisymmetrical distribution of normal stresses. For other scarf angles both the shear stresses and the normal stresses are antisymmetrical. This happens to be true even for an 85* scarf. Thus any deviation from the right angle destroys symmetry. It is perhaps this freedom of the shear stresses to vary from strict symmetry that accounts for the smoother distribution of stresses when a # 90 °. Figure 8 shows the distribution of the average axial stress in the cement. For all cases this is antisymmetrical about the center line. Nevertheless the stress is much greaterwhentz # 1 for90°scarf.Yetthis s t r e s s d o e s n o t v a r y very much with change in V,. This is probably because 5, is an average stress and not the maximum stress which occurs at the interface with the stiffer adherend. The values of 5z shown in Fig. 8 are representative for all values of /~ investigated, although only the curves for ~ = 0.5 are shown. The only curve that deviates sharply from others is that f o r / z = 1 with a = 90 °. From the design standpoint it seems reasonable to conclude that when unequal adherends have to be used, it is better to use a scarf angle less than 90°. Not only does this permit a simple calculation of the stresses (assuming linearity), but obviates the possibility of sharp fluctuations in stress at the adherend-adhesive interface. A stress which could be of critical importance in design is the axial stress in the adherend. It could be large if the disparity in the stiffness of the adherends is very great. Finally, we remark that increasing the number of roots does not automatically increase the accuracy of the
H = exp (-23,z cot a )
Fl(Z, a ) = sin {y(2z - 1 ) - a}
(A1)
F2(z, a) = cos {~/(2z - 1) - a}
(A2)
F3(z, a) = y(2z - l) sin {y(2z - l) - a}
(A3)
F4(z, a ) = y(2z - 1) cos {y(2z - 1) - a}
(A4)
X I ( z , 2or)= [ ( r + 1)F,(z, 2 a ) + F,(z, 2 o r ) - F , ( z , 0)] H L J (A5) X2(z, 2 a ) = [(K + l)F2(z, 2a)-F3(z, 2 a ) ] H
(A6)
X 3 (z, a, v) = [ - ( ~ + 1)F2(z, a) + F,(z, a)
+~
F~(z,-~)+ F2(z,0) cos ~]n (A7)
X4(z, a, v) = [(K + 1)F~(z, a ) + F4(z, a)
+ ~
F,(z,-a)+ F2(z,O)sina]H.
(A8)
In the following equations the function F,, F2,/73 and F , on the right hand side have the same argument for a as shown on the left hand side, unless otherwise indicated. DXI(z, 2a) = [ -(K + 1) cot aF1 + (K + 2)F2 - F3 - cot aF4 - F2(z, 0) + cot aFt(z, 0)] H DX2(z, 2 a ) = [ - ( r + 2)F, - (K + 1) cot aF2 + cot aF3 - F4] H
(A10)
DX3(z, a, v) = [(K + 2)F~ + (K + 1) cot aF2 - cot aF3 + F4 (l-v)
{F,(z, - a ) + c o t aFt(z, - a ) }
- {F,(z, 0) + cot aF2(z, 0)} cos a ] H
(A11)
228
DX4(z, ~t, v)
T. WAn I [ - ( K + 1) cot aF~ + (K + 2 ) F 2 - F3 - cot aF4
+ (1 - 3 cot 2 a ) F ~ + cot a (3 - cot 2 a ) F , {cot a ( 3 - cot 2 a)F,(z, - a )
+~
+(1-v).~. / r 21.z, --a ) - cot aFt(z, - a )}
+ (3 cot ~ a - 1)F2(z, - a ) } {F,(z, 0) + cot aFt(z, 0)} sin a ] H
-
(A12)
+ sin a{(1
DDXI(z, 2 a ) = [{(K + 1) cot 2 a - K - 3}F,
-
3 cot 2 a)Fl(z, O)
+ cot a (3 - cot z a)F2(z, 0)}] H
- 2(K + 2) cot a f t + 2 cot aF3
Qj(z, y, 2 a ) = y3DX2(z, 2 a )
+ (cot 2 a - I)F4 - (cot 2 a - 1)F,(z, 0) + 2 cot aF2(z, 0)] H
(A 16) (AI7)
Q~(z, y, a, v, T) = 3, 3 [ s i n elA,TDDX4(z, a, v)
(A13)
+ DXI(z, 2 a ) ]
(A18)
DDX3(z, a, v) = [ - 2 ( r + 2) cot aF~ - {(r + 1) cot 2 a Q3(z, y, a, v, T ) = yz [ TDX4(z, a, v)+ 2yA1DX2(z, 2a )[ -
K - 3}F2 + (cot 2 a - 1)F3 - 2 cot aF4 sin
+ (1--~v) (1 - v)/,cot~ .( a - 1)F2(z,-a) + 2 cot aFt(z, - a ) } + cos a {(cot 2 a - 1)
x F2(z, 0) + 2 cot aFt(z, 0)}] H D D X 4( z, ol, i.,)
[{(r
+ 1) cot 2 a -
-
2TyA3 sin aDDDX4(z, a, v ) ]
(A20)
~z = C O F * 4 y 2 c o s e c a [ AtDX4(z, a, v)
+ A 'II.*DX4(z', a, v ' ) ] + 0.5vc(cr~ + ty'~)
2 cot oLF:(z, --a)} + sin a ( ( c o t ~ a - 1)
× F~(z, 0) + 2 cot aFl(z, 0)}l H
(A19)
+ 2yX2(1 - vc2)DDX l(z, 2a )
K - 3}F1 - 2(K + 2)
+ (1 - v) ltcot2 a - 1)F~(z, - c 0 (1 + v ) " -
2yAsTDDX3(z, a, v)]
-
Q4(z, y, a, v, T) = 3, 3 [ T sin aDDX3(z, a, v)
(A14)
x cot etF2 + 2 cot etF3 + (cot 2 a - 1)F,
a
(A15)
DDDX4(z, a, v) = [ c o t a{K(3 - cot 2 ~ ) + 9 - cot 2 a}F~ + {K (3 cot ~ a - 1)+ 6 cot 2 a - 4}F2
-
o'oCOF [ sin 2 a - 1/(I + v) + / ~ (sin 2 a - 1/ (1 + v'))] (1 - 2z) C O F = h3/2v.h2 r = - y cot y
(A21) (A22) (A23)
THE ADHESIVE SCARF JOINT IN PURE BENDING THEIN WAH Texas A & I University, Kingsville, Texas, U.S.A. ( R e c e i v e d 1 N o v e m b e r 1975, a n d in revised f o r m 21 A p r i l 1976)
Summary--The stress distribution in a scarf joint subjected to pure bending, and with arbitrary angle of scarf, is analyzed as a two dimensional elasticity problem in plane stress. Both the adherend and the adhesive are assumed to be elastic and isotropic. The two adherends may have differing moduli of elasticity. Numerical results are given. NOTATION
h o w e v e r , c o n f i n e d his a t t e n t i o n to t h e s c a r f j o i n t under pure tension and showed that the stresses w e r e c o n s t a n t a l o n g t h e j o i n t f o r all s c a r f angles p r o v i d e d t h e a d h e r e n d s h a d t h e s a m e elastic properties. In this p a p e r we h a v e a t t e m p t e d to give a s o l u t i o n of t h e s c a r f j o i n t u n d e r p u r e b e n d i n g f o r a d h e r e n d s with elastic c o n s t a n t s w h i c h m a y b e different. It is b e l i e v e d t h a t t h e s o l u t i o n is sufficiently a c c u r a t e f o r d e s i g n p u r p o s e s f o r s c a r f angles t h a t are n o t t o o small.
A , A ~ coefficients c length of adhesive E, E ' modulus of elasticity of adherends G, G ' shear rigidity of adherends Ec modulus of elasticity of cement Gc shear rigidity of cement Mo applied moment t adhesive thickness u., v,, radial and transverse displacements ux, vy x and y displacements r, 0 polar co-ordinates x, y rectangular co-ordinates a scarf angle 3' complex root of equation $ Airy's stress function v, v', vc Poisson's ratio for adherends and adhesive, respectively o'o = 6Molc2 sin2 a (r,, fro, 7,, stress components trx, try, ~'x, stress components ix K
=
FORMULATION OF THE PROBLEM The scarf joint is shown schematically in Fig. 1. The moments M0, applied at some distance from the scarf, are assumed to be produced by the linearly distributed stress shown. The cement is considered to have a very small thickness t compared to the length c of the joint. The problem is to determine the stress distribution in the cement and adjacent adherend material. The boundary conditions for the adherends are that the top and bottom surfaces be traction free, and that the stresses on the vertical sides be equivalent to a couple Mo. The use of eigenfunctions with undetermined constant coefficients enables the satisfaction of all these conditions. The constants are determined by imposing two sets of conditions; first, that the stresses at the adherendadhesive interfaces satisfy the equations of equilibrium of the adhesive treated as being very thin, and, second, that the displacements of the adhesive and adherends be compatible on the same interfaces.
GIG'
=-~/cot~/ Gt
)t 2
=
G~c Gt - E.c It
x, =~?vo A, A5
= A3/A2v~2 = A3/vo
Functions involved in the equations are defined in the Appendix. Primed symbols refer to the right hand adherend. INTRODUCTION WITH THE i n c r e a s i n g u s e o f c e m e n t e d j o i n t s in s t r u c t u r a l c o m p o n e n t s , the d i s t r i b u t i o n of s t r e s s e s at t h e a d h e r e n d - a d h e s i v e i n t e r f a c e of a s c a r f j o i n t h a s b e c o m e of i m p o r t a n c e in design. W h i l e c e m e n t e d j o i n t s o t h e r t h a n t h e scarf h a v e r e c e i v e d wide a t t e n t i o n f o r b o t h i s o t r o p i c a n d a n i s o t r o p i c m a t e r i a l s l-4 t h e o n l y paper, k n o w n to t h e a u t h o r , w h i c h specifically a i m e d at e l u c i d a t i n g t h e s c a r f j o i n t p r o b l e m is t h a t b y L u b k i n , 5 w h o ,
Y \
b-FIG. 1. Scarfjoint. 223
224
T. WArl
The scarf angle a is arbitrary and plane stress is assumed. The materials of both the adhesive and adherends are linear-elastic, but the two a d h e r e n d s m a y have different elastic constants. STRESS FUNCTION We use two co-ordinate s y s t e m s (r, O) and (x, y) as s h o w n in Fig. 2.
Equation (8) h a s complex roots, which occur in conjugate pairs. Choice of only those roots which have a positive real part e n s u r e s the satisfaction of the boundary conditions at large values of x > 0. T h e normal and shear stresses, o'0 and ~-~ along the c e m e n t line O = a, m a y be obtained by first calculating o-,, o-y and ~.~ from equations (5) and then transforming to the (r, O) system. T h e result is ~r~ = o-810=~ = 4Ai(3,2/sin 2 a ) X l ( z , 2 a ) + o-o sin s a(1 - 2z) (9)
Z = t'./C
\
~X I E ,G~u
~ ZI= f I/"C
I
yi
(10)
where the functions X I and X 2 are defined in the Appendix and the coordinate z = r l c is indicated in Fig. 2. Using H o o k e ' s L a w one m a y also derive expressions for the displacement c o m p o n e n t s in the x and y directions and from these obtain the radial and tangential components f r o m the formulas
>
X 4
~'~ = r~t0=~ = 4A~(3,2/sin2 a ) X 2 ( z , 2 a ) + cro sin a cos a ( 2 z - 1)
u~=u~cosO+v,
sinO,
v,=-u~sinO+v~cosO.
(11)
FIG. 2. Co-ordinates.
T h e displacements u, and v, along 0 = a written as follows
T h e stress function (1) where ¢,
=
(aod~16)(3y2ld~
2 y , / d ~)
-
m a y be
_ c 23" u~ = u,[e=~ - ~-~ sin a A , X 4 ( z , a, v)
(2)
(
')
c
23, sin a A i X 3 ( z , a, v)
+~-~o-0 sm a
(z~-z)+Vsina
and
+ H cos a ~b2=a'c~exp(-23,x]{Ksin3,(c
_
sma-l)
+Y(csina2Y _ l ) c o s 3 ' ( c s i n 2 Y a _ l ) }
v, = v0[e=~ - ~ (3)
o-o = 6Molc ~ sin 2 a.
(4)
The stress c o m p o n e n t s m a y be calculated f r o m the relations ,r. = a~4~lay ~,
o-y = a~4~lax ~,
~, = -a~laxay.
(5)
F r o m equations (2) and (4) it is easily seen that the stress function ~b~ satisfies the boundary conditions o',=r,,=0
at
y=0,
csina
(7)
T h e stress function
a + l-~v)z2-z
COS
a sina] (13)
In these equations, G is the shear m o d u l u s of the adherend and v is P o i s s o n ' s ratio. The c o n s t a n t s H, V and R define the displacements and rotation of the adherend as a rigid body. T h e functions X 4 and X 3 are defined in the Appendix. Equations (9), (10), (12) and (13) represent the stresses and displacements of the left h a n d adherend in Fig. 1. A similar set of equations m a y be written for the right hand adherend by replacing o-o by -o-o, A~ by A'~, z by z'=l-z,G byG',vby v ' , a n d V, H a n d R by V ' , H ' and R '. c and 3, remain the same for the two adherends.
(6)
and yields also ~r~ = o-o(1 - 2y/d).
2
- H sin a + V cos a + cRz.
in which A~ and K are constants, is biharmonic. In equation (2) d = c sin a,
+ ~ oC - o [ C O t a s i n
(12)
(8)
and K in equation (3) is taken to be equal to -3, cot 3,.
EQUATIONS OF EQUILIBRIUM FOR ADHESIVE T h e two equations of equilibrium for the adhesive are (Fig. 3) ao-=/On = -a.r=/cOz,
aerz/caz = - a r ~ / a n .
(14)
Since we a s s u m e the adhesive to be very thin we m a y replace the first of the preceding equations by (or" - o-o)It = - Or./caz and (o-~ - a ' ) / t = - ar" lcaz'.
(15)
The adhesive scarf joint in pure bending
225
E Q U A T I O N S O F COMPATIBILITY Approximately, the shear strains on the two surfaces of the adhesive are ~, = - ( u ~ + u ' ) l t + Ov,/cOz
and y" = - ( u , + u',)lt + o v " l c a z ' .
+ OZ dz
Using H o o k e ' s Law we get u, + u" = -(tl2Gc)(z~
+ ~')+(tl2c)(av,
lOz + Ov'/Oz')
(23) and also FIG. 3. Equilibrium of cement.
vo + v " = - ( t / 2 E c )[or. + o r " - vc (~z + ~r')]
Adding these two equations, we get Or.lOz + O'/.IOz' = 0.
(16)
From the second of (14) we may similarly derive O~r,lOz + oG'Joz' = 0.
1 as(y,.)
On s
c
c s Oz s
where Gc is the shear modulus of the adhesive. Substituting from (19) and (20) into (24), the latter becomes v . + v " = - ( t / 2 E ~ ) [ ( 1 - vc2)(~ + tr') -(voEclc)(au.lOz
(17)
+ au'laz')l.
OzOn
is automatically satisfied if we assume O~]On s = 0 since
Ouo I O z - Ou " l Oz' + ( t 12G~ )(Or./Oz - Or './ Oz') - ( t l 2 c ) ( O 2 v . IOz 2 - 0 2 v ' l O z '2) = 0
Oz s
(25)
Reference to equations (12) and (13) shows that equations (23) and (25) contain the rigid body terms H, V, R z and H ' , V' and R'(1 - z). These terms are eliminated by differentiating (23) once and (25) twice with respect to z. Equations (23) and (25) then take the form
The compatibility equation 0'E, ~ 10"~.
(24)
Oz s \ O n ]
(26)
OSv./Oz~ + OSv ' /Oz 's
+ (t(1 - v~S)12Ec )(OSo'./Oz s + OS~'lOz '2)
and
- (uotl2c)(O3u~ IOz ~ + O~u'/Oz '~) = 0.
as a s (o.+lO~=la so~ OzOn y" = ~ \o--n c ~ l c ozs On"
Now ~ = ,7,IE~ - J,c~.IEo
(18)
where Ec and z,~ are the elastic modulus and Poisson's ratio for the adhesive. Setting ~ = Ou.lcOz equation (18) gives
(27)
The stress and displacement components equations (9)-(13) and similar equations for the adherend are now substituted into equations (26) and (27). The resulting equations may be follows:
given by right hand (16), (22), written as
y, 2a) = 0
(28)
A , Q ~ ( z , y, 2 a ) + A ' l Q t ( z ' ,
A~Q2(z, y, a, v, 1) + A ~Qs(z', y, a, v ' , t~) =
-0.5A4~0 sin s a (s - / ~ s ' ) o-~ = E~Ou, IcOz + v ~ o .
(29)
(19) A , Q 3 ( z , y, a, v, 1) - A '~Q3(z', y, a, v ' , t~ )
Similarly,
= O,o{0.25(1 - 2z) sin a ( s - t~s') tr', = E~Ou'lcOz' + v~a'.
(20)
A~Q4(z, y, a, v, 1) + A~Q4(z', y, a, v', t~) = -0.125a0 sin 2 a ( s - t~s')
From (19) and (20) we obtain
oo-~ 4 oo-.' Oz
--
Oz'
E~'a2u a s "+~(ao-.+ao-;,~ - . . . . . 2c Oz s + 0 2 2 \Oz Oz']"
(21)
From equations (17) and (21) we get E~(o~.+a~'~
2c \ Oz"
vo/oo-= a,,'x
Oz'S ] + "~ [-~-z + "~'z') = 0.
+0.SA~cosa(s+t~s')-A~sin2acosa)
(22)
Equations (16) and (22) are the two equations of equilibrium for the adhesive.
(30) (31)
in which ~ = G / G ' is the stiffness ratio of the adherends, and the Q~ axe given in the Appendix. In the preceding equations s = s i n 2 a + l / ( l + v ) , s'=sin 2a+l/(l+v') and the A's are defined under Numerical Results. It should be remarked that the set of equations (28)-(31) all become homogeneous when t~ = 1, v = v', and a = ~r/2. We then get the trivial solution A~ = A ' ~ = O. Thus only in the case of the 90 ° scarf with equal adherends is there a linear distribution of stress throughout, the solution being given by the second term on the
226
T. WAH
right of equations (9) and (10). For all other c a s e s the stress distribution is nonlinear. SOLUTION OF EQUATIONS The m e t h o d u s e d for solving the equations is point matching. W e have u s e d the first five roots of equation (8) given by
_.
(%'
3q.2.3.4.5= 3.7489 -+ i 1.3843, 6.9500 -- i 1.6761, 10.1193 -+ i 1.8584, 13.2772 --+i 1.9916, 16.4299 -+ i2.0966 (32)
%:
so that there are 20 u n k n o w n c o n s t a n t s in each of the equations. T h e equations are satisfied at 11 points z = 0, 0. ! . . . . . 1, resulting in a total of 44 equations in 20 u n k n o w n s . A linear least squares procedure is then used to obtain the 20 u n k n o w n constants. The stresses or,, %, o"~, r " and the average longitudinal stress Oz = (~r= + O=)/2 are then calculated. T h e equation for Oz is given in the Appendix. NUMERICAL RESULTS AND CONCLUSIONS Figs. 4-8 give the numerical results obtained for the normal and shear stress and the stress O, along the adhesive for various scarf angles a varying f r o m 90 ° to 60* and for several values of the stiffness ratio t~. W e have a s s u m e d that the P o i s s o n ' s ratio for both the a d h e r e n d s to be 0.3. Other parameters were arbitrarily c h o s e n taking care only to keep the t i c ratio small for the adhesive. T h e following numerical values were used in the calculations:
A, = G t / G c c = 0.1,
A2 = G t l E c c = Ad2.5,
A4=A3/A2/~c2, As=A3/v~, v~ =0.25,
o,-I~" }
I /'//))jo.,
-0.5 -0.4 - 0 3 - 0 2 - 0 I
<
/
~
02
03
0.4 0.5
r~l~ o
u
;0 08
60*
0204
/x = 1,0.5,0.1.
~75' 60.~ "~,,..,~~ -I
-08
-06
-04
-02
02
%1%
and ~ 1 %
04
06
08
-
FIG. 6. a = 75 °, 60 ° normal stresses.
08/ o.6o
~
0.5
z
o , ¢ ~
OI
.... ,;/%
FIG. 5. a = 90 ° shear stresses.
-
~J
0
~1%
A3 = vct/2c = 0.000125
We remark that the plane strain problem which more nearly approaches the practical case, m a y be obtained f r o m the present problem by substituting E / ( I - v 2) for E and H(1 v) for v w h e r e v e r t h e s e quantities occur. T h e pattern of normal and shear stresses distribution, as seen f r o m Figs. 4-7 is rather unexpected. Only in the
O5
p_ .o,
o.
~
p - 1.0,0.5,0.1
,/.L=O I
_..
Ga / G O
~ffiO. I
-04
-03
-02
-OJ
r J G o and r~lo-o - FI6.7.
0
, I oa/~O'J
02
I
08
04
I
06
06
I
04
08
I
02
FIG. 4. a = 90 ° normal stresses.
1.0 " ~ O ' a / ~ " 0
I
0
OI --
02
03
04-
% I G o and -i'ilO'o
a = 75 °, 60 ° shear stresses.
case of the 90 ° scarf is there f o u n d a strong nonlinearity in the stress distribution (Figs. 4 and 5). T h e nonlinearity b e c o m e s stronger the greater the deviation of ~ is f r o m unity. For the other two scarf angles 75 ° and 60 °, not only is the nonlinearity in the stress distribution negligible, but the distribution is almost identical for different values of/~
The adhesive scarf joint in pure bending
227
solution. The higher the root the larger the negative exponential involved (see equation (32)) and significant figures are lost in computation. This situation could become serious for small scarf angles and could necessitate using a smaller number of roots to attain reasonable results.
0.5
04
02
0.~
0 2
04
0.6
0.8
I0
1.2
REFERENCES 1. M. GOLAND and E. REISSNER, J. appl. Mech. 11, A17 (1944). 2". F. SZEPE, Exp. Mech. 6, No. 5, 281 (1966). 3. O. VOLKERSEN, Lu[lfahrtforschung, 15, 41 (1938). 4. T. WAn, J. Materials Technol. pp. 174-181, July (1973). 5. J. L. LUBKIN, J. appl. Mech. 24, No. 2, 255 June (1957). 6. S. TIMOSrmNKO and J. N. GOODIES, Theory o / Elasticity. (3rd Edit.) pp. 61--62. McGraw-Hill, New York (1970). APPENDIX
FIG. 8. Axial stresses. (Figs. 6 and 7). There are slight differences in the curves for differing/~ but they could not be shown on the scale of the graph. Data for a 45 ° scarf (not shown) also exhibits little nonlinearity. The difference in stress magnitudes on the two sides of the adhesive were small and could not be shown without greatly expanding the scale of the figures. It was felt that the peculiar behavior of the 90 ° scarf needed further scrutiny. Stresses were therefore computed for an 85 ° scarf (not shown). This too exhibited little nonlinearity. One is thus forced to conclude that the 90 ° scarf is indeed a special case. A possible explanation may be found from an examination of equation (10) and Fig. 5. It is seen that for a 90* scarf (and for this scarf alone) there is a symmetrical distribution of shear stresses (identically zero f o r / ~ = 1) together with an antisymmetrical distribution of normal stresses. For other scarf angles both the shear stresses and the normal stresses are antisymmetrical. This happens to be true even for an 85* scarf. Thus any deviation from the right angle destroys symmetry. It is perhaps this freedom of the shear stresses to vary from strict symmetry that accounts for the smoother distribution of stresses when a # 90 °. Figure 8 shows the distribution of the average axial stress in the cement. For all cases this is antisymmetrical about the center line. Nevertheless the stress is much greaterwhentz # 1 for90°scarf.Yetthis s t r e s s d o e s n o t v a r y very much with change in V,. This is probably because 5, is an average stress and not the maximum stress which occurs at the interface with the stiffer adherend. The values of 5z shown in Fig. 8 are representative for all values of /~ investigated, although only the curves for ~ = 0.5 are shown. The only curve that deviates sharply from others is that f o r / z = 1 with a = 90 °. From the design standpoint it seems reasonable to conclude that when unequal adherends have to be used, it is better to use a scarf angle less than 90°. Not only does this permit a simple calculation of the stresses (assuming linearity), but obviates the possibility of sharp fluctuations in stress at the adherend-adhesive interface. A stress which could be of critical importance in design is the axial stress in the adherend. It could be large if the disparity in the stiffness of the adherends is very great. Finally, we remark that increasing the number of roots does not automatically increase the accuracy of the
H = exp (-23,z cot a )
Fl(Z, a ) = sin {y(2z - 1 ) - a}
(A1)
F2(z, a) = cos {~/(2z - 1) - a}
(A2)
F3(z, a) = y(2z - l) sin {y(2z - l) - a}
(A3)
F4(z, a ) = y(2z - 1) cos {y(2z - 1) - a}
(A4)
X I ( z , 2or)= [ ( r + 1)F,(z, 2 a ) + F,(z, 2 o r ) - F , ( z , 0)] H L J (A5) X2(z, 2 a ) = [(K + l)F2(z, 2a)-F3(z, 2 a ) ] H
(A6)
X 3 (z, a, v) = [ - ( ~ + 1)F2(z, a) + F,(z, a)
+~
F~(z,-~)+ F2(z,0) cos ~]n (A7)
X4(z, a, v) = [(K + 1)F~(z, a ) + F4(z, a)
+ ~
F,(z,-a)+ F2(z,O)sina]H.
(A8)
In the following equations the function F,, F2,/73 and F , on the right hand side have the same argument for a as shown on the left hand side, unless otherwise indicated. DXI(z, 2a) = [ -(K + 1) cot aF1 + (K + 2)F2 - F3 - cot aF4 - F2(z, 0) + cot aFt(z, 0)] H DX2(z, 2 a ) = [ - ( r + 2)F, - (K + 1) cot aF2 + cot aF3 - F4] H
(A10)
DX3(z, a, v) = [(K + 2)F~ + (K + 1) cot aF2 - cot aF3 + F4 (l-v)
{F,(z, - a ) + c o t aFt(z, - a ) }
- {F,(z, 0) + cot aF2(z, 0)} cos a ] H
(A11)
228
DX4(z, ~t, v)
T. WAn I [ - ( K + 1) cot aF~ + (K + 2 ) F 2 - F3 - cot aF4
+ (1 - 3 cot 2 a ) F ~ + cot a (3 - cot 2 a ) F , {cot a ( 3 - cot 2 a)F,(z, - a )
+~
+(1-v).~. / r 21.z, --a ) - cot aFt(z, - a )}
+ (3 cot ~ a - 1)F2(z, - a ) } {F,(z, 0) + cot aFt(z, 0)} sin a ] H
-
(A12)
+ sin a{(1
DDXI(z, 2 a ) = [{(K + 1) cot 2 a - K - 3}F,
-
3 cot 2 a)Fl(z, O)
+ cot a (3 - cot z a)F2(z, 0)}] H
- 2(K + 2) cot a f t + 2 cot aF3
Qj(z, y, 2 a ) = y3DX2(z, 2 a )
+ (cot 2 a - I)F4 - (cot 2 a - 1)F,(z, 0) + 2 cot aF2(z, 0)] H
(A 16) (AI7)
Q~(z, y, a, v, T) = 3, 3 [ s i n elA,TDDX4(z, a, v)
(A13)
+ DXI(z, 2 a ) ]
(A18)
DDX3(z, a, v) = [ - 2 ( r + 2) cot aF~ - {(r + 1) cot 2 a Q3(z, y, a, v, T ) = yz [ TDX4(z, a, v)+ 2yA1DX2(z, 2a )[ -
K - 3}F2 + (cot 2 a - 1)F3 - 2 cot aF4 sin
+ (1--~v) (1 - v)/,cot~ .( a - 1)F2(z,-a) + 2 cot aFt(z, - a ) } + cos a {(cot 2 a - 1)
x F2(z, 0) + 2 cot aFt(z, 0)}] H D D X 4( z, ol, i.,)
[{(r
+ 1) cot 2 a -
-
2TyA3 sin aDDDX4(z, a, v ) ]
(A20)
~z = C O F * 4 y 2 c o s e c a [ AtDX4(z, a, v)
+ A 'II.*DX4(z', a, v ' ) ] + 0.5vc(cr~ + ty'~)
2 cot oLF:(z, --a)} + sin a ( ( c o t ~ a - 1)
× F~(z, 0) + 2 cot aFl(z, 0)}l H
(A19)
+ 2yX2(1 - vc2)DDX l(z, 2a )
K - 3}F1 - 2(K + 2)
+ (1 - v) ltcot2 a - 1)F~(z, - c 0 (1 + v ) " -
2yAsTDDX3(z, a, v)]
-
Q4(z, y, a, v, T) = 3, 3 [ T sin aDDX3(z, a, v)
(A14)
x cot etF2 + 2 cot etF3 + (cot 2 a - 1)F,
a
(A15)
DDDX4(z, a, v) = [ c o t a{K(3 - cot 2 ~ ) + 9 - cot 2 a}F~ + {K (3 cot ~ a - 1)+ 6 cot 2 a - 4}F2
-
o'oCOF [ sin 2 a - 1/(I + v) + / ~ (sin 2 a - 1/ (1 + v'))] (1 - 2z) C O F = h3/2v.h2 r = - y cot y
(A21) (A22) (A23)