a layered beam theory for single lap joints
TRANSCRIPT
AD-A237 324
MTL TR 91-23 AD
A LAYERED BEAM THEORY FORSINGLE LAP JOINTS
DONALD W. OPLINGERMECHANICS AND STRUCTURES BRANCH
June 1991J
~4U~~J2 71991~
Approved for public release; distribution unlimited.
US ARMYLABORATORY COMMAND U.S. ARMY MATERIALS TECHNOLOGY LABORATORYMATEUNAS TECHNIXOGY LAORATOAY Watertown Ma~sarhusetts 02172-000 1
91-03134
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A LAYERED BEAM THEORY FOR SINGLE LAP JOINTS Final Report
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Donald W. Oplinger
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15 SUPPLEMENTARY NOTES
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Stress analysis AdhesionBonded joints
Structures
20. ABSTRACT (Continue an reyerse side if necesery end identfuy by block number)
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Block No. 20
ABSTRACT
The well known analysis of the single lap joint by Goland and Reissnerprovided important contributions to the literature on stress analysis of adhe-sive joints by clarifying not only the importance of adhesive peel stresses injoint failure, but also the role of bending deflections of the joint in control--ling the level of the stresses in the adhesive layer. Subsequent efforts havesuggested the need for corrections to the Goland and Reissner analysis becauseof what have been conceived as deficiencies in the model used to describe bend-ing deflections of the central part of the joint where a classical homogeneousbeam model without shear or thickness normal deflections were used. T -' nro-q"- raper addresses the issue through the use of a more realistic model inwhich adhesive layer deflections are allowed to decouple the two halves of thejoint in the overlap region in the bending deflection analysis, as well as inthe analysis of adhesive layer stresses where such a decoupling was allowed byGoland and Reissner. It is found that many of the predictions of the Golandand Reissner analysis are recovered in the limit of large adherend-to-adhesivelayer thickness ratios, although substantial differences from the Goland andReissner analysis can occur for relatively thin adherends.
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TABLE OF CONTENTS
INTRODUCTION ............................................... 1
THE ANALYSIS OF GOLAND AND REISSNER ........................ 5General Remarks ........................................... 5Detailed Relationships ....................................... 7
IMPROVED ANALYSIS ........................................ 11Deformations and Shear Stresses .............................. 11Peel Stresses in the Modified Analysis ......................... 19
COMMENTS ON THE HART SMITH ANALYSIS .................... 23
NUMERICAL RESULTS . ........................................ 24
CONCLUSIONS ............................................... 27
ACKNOWLEDGEMENT ......................................... 27
APPENDIX A EQUATIONS OF GOLAND REISSNER ANALYSIS ........ 29
APPENDIX B PEEL STRESS SOLUTION IN UPDATED ANALYSIS ....... 37
Acoesston For
NTIS ~&
_ Ditr.itution/. . .
AvaA' 1nl i t 7- Codes-Avail snn,' -
Dist Spclal
-i-i,
NOMENCLATURE
BU, BI. (-E't) -- Stretching stiffness, upper,lower adherendDu, DL k-= E't /12) -- Bending .... .. ..
E -- Adherend Young's modulusE' [- E/(1-v2)] -- " plane strain Young's modulusEb -- Bond Young's modulusGb -- " shear "HI,H2 [Eq(37] -- iV.{E'ktz/2}, factors in coefficients of solution for &rJIJ 2 [Eq(38)] H, divided by roots U2/8 (j=1) and 832 (j=2) of GR solutionsL (- ./2) half length of joint (see Fig. 2)L (subscript) -- reference to lower adherendMU,ML -- moments in individual adherendsM. -- moments in loaded adherend at ends of overlapR -- ratio, U40Rj (=1,2) -- ratios of roots pj to U/(8)1r2 (-I) or (8)lt2 5Tu,TL -- adherend resultants, tcaT (x--0,2L) -- resultants applied at ends of jointU [Eq(A-8)] -- t(T/Du) 1/
W (12,x) t
U (subscript) -- reference to upper adherendV -- lateral loads at erds of joint (see Fig. 2)V0 -- lateral loads at ends of overlapb (subscript) -- reference to bond layerc -- 1/2 in GR notationh -- height (combined thickness) of overlap region in thickness directionk [Eq(7)] -- ratio of M. to Tt/2 or of displacement at ends of overlap to t!2kis-- k expression obtained by Hart Smith [3]kn [Eq(41)] -- k-"new",i.e. k obtained from current analysisk21,k22 [Eq(35)] -- ratios of displacement solution coefficients to t/20 (Fig. 2) -- axial length of overlap region
.. -axial length of outer section of adherendst-- adherend thicknesstb -- bond layer thicknessu -- axial displacementw-- lateral displacementWU,WL -- adherend displacementsw [Eq(14.3)] -- (Wu+WL)/ 2
x -- axial coordinatez -- thickness-wise coordinateAh [Eq(22)] -- coefficients of solution of &r3 [Eq(A-6.3)] -- (PGPt) l' 2ar [Eq(14.1)] -- resultant difference TU-TL. -nominal applied strain, r.,/E'
Yb -- bond layer shear strainX-- dimensionless overlap length /t
.-- " length of outer adherend segment Q/tpl,p2 [Eq(22)] -- exponential coefficients of differential equations for w and &rPr -- Young's modulus ratio E'/E,
,-- modulus ratio E'/GbPt -- thickness ration t/t.
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INTRODUCTION
One of the most widely quoted papers in the literature on stresses in adhesive joints is thatof Goland and Reissner[1] (subsequently referred to as "GR") on single lap joints. This paperwas particula,!y significant in being the first effort to identify the effects of adherend bendingdeflections on the peel and shear stresses in the adhesive layer of a single lap join.
The portion of the GR analysis relating to bending deflections treated the actual joint, FigureI(A), as a stepped homogeneous beam, Figure 1(B), in which the height of the center sectionwas assumed to be twice the thicknesses of the adherends (thus ignoring the thickness of thebond layer). The forces due to tensile end-loading, assumed to lie along the line a-a' inFigure 1 (A), were found to produce varying moment about the neutral axis at any point alongthe representative beam which resulted in the bending deflections of interest. The deflectionanalysis for this system was treated essentially by "beam column" analysis [2], i.e. as theanalysis of deflections in a beam with combined column (i.e. axial) and lateral loading. (Notethat in the GR analysis the column loading is tensile rather than compressive. The latterrepresents the case used in [2] to introduce the concept of Euler column buckling.)
[I) GOLAND, M. and REISSNER, E.. Stresses in Cemented Joints, J. Applied Mechanics (ASME). v.11, (1944), p.A 17-A27[2] TIMOSHENKO, S. and GERE, J., Theory of Elastic Stability. 2nd ed., ChapL 1, McGraw Hill (1961)
(A) JOINT CONFIGLRATION M Vu, T a
T'MV, VV, TL= 0 t t b
TT h' 2t T-'
t(8) QtIIOENEOU5 BEAM MODEL
M u TT
tb
CC) FORCES CONTRIBUTING TO IIENT IN OVERLAP
Figure I Assumed Model
-I-
A question arises with the GR analysis in the manner in which the moment is represented inthe center section. The crux of the matter is the comparison of the classical moment-curvaturerelation for a homogeneous beam and the corresponding expression for a layered beam (i.e.Equation (1) vs. (2)):
MOMENT-CURVATURE RELATIONSHIPS
HomogeneousBeam: MN = (1)12 dx2
Layered Beam • M¢N - MU + ML + t(TL - TU) (2)2
In the above equations, MN represents the moment about the neutral axis, w is lateraldeflection and h is the height of the homogeneous beam -- ie. approximately twice theadherend thickness, t, for a thin bond layer. With z as the thickness-wise coordinate havingits origin at the mid plane of the bond layer, the remaining quantities in (1) and (2) are givenby
M = _-Ch/l2 za x dzMNN J-h12zad
Mu = J( h14-z)aAz , M =L (z + h/4)(xdz (3)
Tu= f c2Odz , TL= faXdz
Equation (1) is the classical moment-curvature relation for a homogeneous beam of heighth which ignores transverse shear and thickness-wise deformations. Equation (2), on the otherhand, is a statement of equilibrium of the components indicated at the right end of thesegment in Figure 1(C) which make up MN. Equation (1) happens to be equivalent to (2)under certain conditions, but these do not apply to the ends of the overlap where significantshear straining of the bond layer takes place. Because it provides for equilibrium, (2) is amandatory relationship which it is desirable to preserve in the formulation.
It is of interest to compare (1) and (2) for the case of a homogeneous beam. Noting thestandard expression for axial stress in the overlap region under combined bending andstretching (i.e. a linear function of distance from the neutral axis):
-t MNZ (where T =TL + TU ; I = h3/12) (4)
then making use of (1) to represent MN in (4) and introducing the latter into (3) leads to
-2-
M -M =E t3 w Hu 1 t t 3__ 3MU2 ML=E /w"IMN (5.1) TL = +- (5.2) tT - MN (5.3)
12 8 2 2 8 2 2 8
Since t-h/2, Eq(5) lead to
Mt+ML+ i(TLTtj) 2Et - JEh3d (6)2 3 dX2 12 dx 2
agreeing with (2) on inserting (1) into the latter. This implies that Eq(5) have to hold in orderfor (1) to be applicable. In the single lap joint, however, as noted in Fig I(A), ML and TL arezero at the left end of the overlap, while Mu and TU are zero at the right end. Furthermore, inthe GR analysis, Mu was represented at the left end of the overlap where (sec Figure 2) x=1oin terms of Tt/2 through a dimensionless parameter k:
MU(x =0)= kt i (7)
2
where k was a function of load in the range 0.26<k<1. Table 1 gives a comparison of requiredtraction conditions at x=Q with the values for these conditions which are implied by (5) when(7) is taken into account. It is apparent that if k is not equal to unity, the required end conditionson ML, Tu and TL cannot be realized if (5) apply. Moreover, the fact that MN is negative at x=1.while Mu is positive implies the presence of a curvature discontinuity in the upper adherend if(1) applies. Thus Equation (1) cannot, strictly speaking, be used to represent MN over thelength of the overlap. Hart Smith first pointed this out qualitatively in [3], concurring with theobservation that tne use of (1) requires the presence of nonzero tractions over the left end of thelower adherend. Rather than attempting to retain (1), it is desirable to use a formulation whichis consistent with (2) regardless of whether or not conditions apply for which (1) is valid.
['41 1ART-SMITH. L J.. Adhesive Bonded Single Lap Joints, NASA Conctor Repor 112236 (1973)
TABLE 1. COMPARISON OF CONDITIONS FROM EQ(5) WITH TRACTIONS ATx=Q.
REQUIRED END VALUES IMPLIED BY
MN -(1-k)Tt/2 -(1-k)Tt/2
MU kTt/2 kTt/2
ML 0 kTt/2
TU T (5+3k)T/8
TL 0 (11-3k)T/8
-3-
As suggested above, the lack of validity of Equation (1) is associated mainly with thepresence of shear strains in the bond layer. More specifically, the equivalent of the followingequation (see Appendix A, Eq(A-6) ) which GR used to obtain the gradient of shear strainin the bond layer:
dy 2 [(T -TL).+3(Mu+M,)]dx E't2 t 2
gives a value of zero when (5) are allowed for, so that a condition of constant or zero shearstrain in the bond is implied by (5). Lap joints typically have large shear stress and thereforestrain gradients along their length due to adherend deformations, contrary to the latter result.It should be understood that the impact of shear straining on the performance of the joint isassociated with the fact that organic bond materials are relatively soft compared with typicaladherends. The extreme version of this situation can be visualized in terms of a beam cutalong its neutral axis like a leaf spring. Since bending stiffness is a cubic function of beamdepth, the bending stiffnesses of each of the two halves would be one eighth of that of theoriginal beam. A simple analysis shows that the combined bending stiffness of the two half-beams would be the sum of the two components, amounting to 25% of that of the originalbeam. Thus, shear straining along the neutral axis can be expected to decrease the bendingstiffness of the composite beam making up the overlap region of the joint.In general, Equation (1) will overestimate the bending stiffness of the overlapping part of thejoint, preventing correct determination of bending deflections. Since the thrust of the GRanalysis was to demonstrate the influence of bending deflections on the stresses developedin the bond, an effort to correct the situation appears desirable. It should be expected thatdifferences between predictions of the original GR approach and those of the present analysiswhich provides for effects bond shear straining on decoupling of the two adherends will begreater in joints containing relatively thin adherends. This is due to the fact that in the caseof thicker adherends, the range of the decoupling effects is shorter and is confined to a regionclose to the ends of the overlap. It can be argued from St. Venant's principle that Eq(l)should be valid sufficiently far from the overlap ends, and the influence of the load diffusionprocess which makes it valid will be less when the effects of interest here are short in rangerelative to the adherend thicknesses.
In addition to the discussion of Hart Smith which was previously cited [31, problems with theGR treatment of the bending analysis have been noted by others[4-51. Benson[4], as discussedby Adarms[5], described a modification of the analysis which dealt with the situation. Inaddition, Hart-Smith[3] did an extensive amount of work on the single lap joint that providedan alternate method of introducing the appropriate bending deflection corrections which willbe discussed subsequently.
BENSON, N. K. Influence of Stress Distribution on Strength of Bonded Joints in Adhesion,Fundamentals and Practice , Gordon and Breach, NY (1969) p. 191-205151 ADAMS, R.C. and WAKE. W. C. Structural Adhesive Jointa in Engineering, Elsevier Applied Science Publishers, NY (1984)
-4-
V0T Z c---
w
F _ T
U
D I SPLACBDENTS
Figure 2 Joint GeometryIn the present case, the center section of the joint will be treated essentially as a pair ofcoupled beams throughout the analysis. Once again it is emphasized that GR did use the[4]coupled-beam approach to deal with of peel and shear stresses in the adhesive layer, but notin their deflection analysis.
THE ANALYSIS OF GOLAND AND REISSNER
General RemarksIt is useful here to repeat certain results which were obtained from the GR analysis. Figure
COORD I NATES DISPLACEMENTS
Z I
Ux
V2V
Figure 3 Sign Conventions
-5-
V
-- T/-'- -1 ---0X
xR -2 L - x L E L
2LL
2L
Figure 4 Antisymmetry Conditions
2 gives the basic parameters of interest. Here "U" and "L" denote the "upper" and "lower"adherends. The length of the adherends beyond the overlap is 9. while the overlap has alength I (-2c, in the GR notation). It is assumed that tension loads (i.e. resultants) T arepresent at the outer ends. In terms of T it is convenient to define nominal axial stress, iix, andstrain,'Tx: - (9.1) ; X (9.2) ; E 1=E(1 -v 2 ) (9.3)t E
M
- V0
Figure 5 End Conditions in Overlap Region
-6-
(E, v = Young's modulus, Poisson ratio for axial straining.) Along with T, there are assumedto be transverse shear forces V which produce moment free conditions at the end points,provided
V= aT ; =t/2L ; L=Qo + /2 a o + c (10)
As in Figure 2, the directions of the transverse forces are downward at the right end andupward at the left end for moment free conditions. Fig 3 recalls standard conventions for thesigns of moments, shear forces and tensions, as well as coordinates and displacements. Asusual, positive moments, shear forces and tensions correspond to those shown for the rightend of segment "1", while at the left end of a given segment the forces and moments areobviously the reactions to those applying to the right end of the neighboring segment (to theleft of the one under consideration). Thus in Figure 3, the reactions at the left end of Segment"1" are shown as the negatives of the quantities associated with the right end of Segment "2".
Additional relationships associated with the antisymmetry of the joint are illustrated in Figure4, where attention is focussed on points located at XL and XR - 2 L-XL , i.e. points located atthe same distance from each end. Equilibrium of vertical forces requires that the shear forcesat these points be equal, i.e.
V(xL) = V(2L -xL (11)
In addition, since the generic expression V=-dM/dx holds, the fact that the moment is afunction which is zero at x=L and whose derivative is symmetric about x=L requires that themoment itself be antisymmetric, i. e.
M(xL) = -M(2L -XL) (12)
so that as shown in Figure 5, the shear forces at the ends of the overlap are equal while themoments are equal but opposite in sign. In the specific case of the ends of the overlap, GRuses the notation Mo and Vo for the moment and shear force, as indicated in Figure 5. Inaddition, GR expresses Mo in terms of the parameter k introduced in (7) and Vo in terms ofan additional dimensionless parameter k':
M o =k t T • V = 2k'T/(32 (13)
where =t
For future reference, the end conditions on the individual adherends may be statedas in Table 2. The last four rows of Table 2 are used in forming end conditions fordifferential equations given in Appendix A.
Detailed RelationshipsThe equations encountered in the GR analysis are reviewed in Appendix A. Note thatthese exhibit the influence of lateral deflection on the moment distribution due to theshift in the lever arm, il!,istrated in Figure 6. Here the horizontal component of end load,
-7-
TABLE 2 CONDITIONS ON ADHERENDS AT ENDS OF OVERLAP
Vu -V0 = -2k'T/A 0
Mu Mo = ktT/2 0
TU T 0
VL 0 -V -2k'TIA.
ML 0 -M0 ktT/2
TL 0T
Mu + ML Mo -Mo
TL-TU -T T
MU-ML Mo MO
VU -VL -V0 V0
T, acts through a load line which shifts with respect to the neutral axis of the adherendsegments, as illustrated in Figure 6(A) for the outer adherend and Figure 6(B) and (C) forthe overlapping part of the joint.For subsequent reference it is useful to recall the most crucialresults given for the GR analysis in Appendix A at this point. These include the expressions forthe quantity k:Expressions for k
Thi (RPTA -7.6) ; k/= kXU (RPTA-7.7)Thi+ F(K8) Th2
4 Th)
where Thi=tanh(UXo) (RPTA-7.8) ; Th2 =tanh(U2/F) (RPTA-7.9)
For Thl -1
k 1 ; k' =lk7,U (RPTA-8)1 +/Th2 4
-8-
together with those for the shear and peel stress solutions:
Expression for Shear Stress
tb "~[F "[ F3 (1+3k) cosh[/ +(xL)/t] +.(l-k)] (RPTA-10.1)4 sinhvI, 4
Maximum Value ofxb(x=L±t/2 ; /83X/2•1 ; 0.262<k<1)
0.6313 < abs{L dma } p (I +3k) < 1.41410 (RPTA-10.2)(Ox 4
Maximum Value of Peel Stress (x = L±Q/2 ; y)12 • 1 )
-d i--(y+U) (RPTA -15.3)2
or, since U-cy in practical cases:
b)max k-?
Note that from the expressions given in (A-6.3) and (A-7.2) with (A-1.5), the following relationbetween the maximum peel and shear stress can be stated:
b)max "- k Eb
rb)max 1 +3k Gb (14)
If Eb/Gb is set equal to the expression for isotropic materials of 2 (1 +Vb) where Vb is the Poissonratio for the bond material, the max peel-to-shear stress ratio runs from about 2 for the upperlimit of k of unity, to a little less than I for the lower limit (k=0.26), for an assumed Poissonratio of around 0.5. The specific value of the assumed Poisson ratio is not important, but thisestimate emphasizes that two maximum stresses take on values which are proportional to eachother for various joint geometries and for which the proportionality constant is in the range of1 to2.
-9-
CA) MOMENT IN OUTER END T /cos o
S.. .M x +=0 Po dw/ dx
(B) WOVENT IN OVERLAP-- NO DEFLECTION
I- It t IM =vxN- -LLr 2 L NI -
CC) M ENT I N OVERLAP -- DEFLECTION PRESENT
2- MN =~ VX 2T + wT
t/2
Figure 6 Effect of Deflection on Moment Distribution
A certain amount of controversy has arisen over the results given in Appendix A for the peelstresses ( Eq(A-15) ) from which (A-15.3) is obtained. Kuenzi and Stevens[6] stated a versionof Eq(A- 15) (coefficients of the peel stress equation) with the signs of some terms reversed fromthose of the GR version (see discussion following Eq(A-15) ) without commenting on thejustification for differing from GR. Moreover, Refl3-4,7] claimed to have verified the versiongiven in [6], while Carpenter[81 found the original version as given above to be valid.
Considerable care has been taken here to identify correct boundary conditions at the ends of theoverlap in order to be sure that the correct expressions for peel stress coefficients were obtained.As far as can be determined, the original expressions given by GR are the correct ones. In anyevent, the terms in question which correspond to the term U in the second factor of (A-15.3)summarized above do not usually contribute strongly to the peel stress predictions.
161 KUENZI. E. and STEVENS. G., Determination of Mechanical Properties of Adhesives for Use in the Design of Bondaed Joint. ForestProducts Laboratory Note FPL-01 I(1963)(71 KUTSCIIA, D. and 110 FER, K.,FeasibiliyofJoiningAdvancedCompositeFligiu Vehicle Structures, Air Force Materials Laboratory ReportAFML-TR-68-391, (1968) p. 56,59181 CARPENTER, W. Goland and Reissner were Correct, J. Strain Analysis. v. 24, no. 3 (1989)
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IMPROVED ANALYSIS
Deformations and Shear Stresses
In the following discussion, it will be convenient to represent the individual adherenddisplacements and stress resultants in terms of differences and sums, since the bond layerstresses are more easily represented by differences in the adherend displacements and resultantsthan in terms of their individual values. Accordingly, the following notation is introduced:
I87<x) =Tu -TL (15.1) WW = --- ((wu+wr.) (15.2)
8. = wu -wL (15.3)
Note that the individual adherend resultants can be expressed in terms of T and &r(x),as follows: 1 - 1 -
Tu = -f [T+ T(X)I ; TL = 2[T-&T(x)] (16)
In addition, note that the quantity Mu+ML, which appears in Eq(2) as well as (A-6.1) for theassumed case of identical adherends, is given by
MU +ML = 2DU... (17)
For increased accuracy in the following, the quantity t/2 appearing in Eq(2) and (A-2.2) willbe replaced by (t+tb)/ 2 to provide for the effect of bond layer thickness. The replacement ofEq(1) by (2) modified in this way can then be accommodated by using Eq(A- 1.3, .4) to representMU and ML in terms of displacements (setting DU=D L) i.e.:
d2wu d 2 WL t+tbM N = Du(- + -- ) -.- (TU -TL)MN dU( 2 dx2 2
or, in terms of the notation just introduced,M =Du d 2; t+t b8T-()M N V-- - ' ) (18)
dx2 2
Then equating the modified form of (A-2.2) with (18) (letting the neutral axis displacement berepresented by W) leads to
D d 2 _t+tb 8 Tax-(.t+tb -)T (19)
Note that, as opposed to the original GR analysis, coupling occurs between w and the adherendresultants, so that w cannot be determined independently of 5T in the updated analysis. A second
-11-
equation in w and 5r is obtained by multiplying (A-6.1) by Gb to get dtbjdx on the left which,according to (A-3. 1), is equivalent to d2TU/dx 2, or equivalently (according to the left memberof (16) ) to (d25r/dx2)/2. Then again allowing for the assumption of identical adherends, makinguse of (15.1) and (17) leads to
d2 T 2 Gb 1 wd2 0dx2 t Bu dx2
We can combine Eq(19) and (20) with (A-5.1) (which there was no need to modify) and obtaina self-contained system of differential equations for w and 8T. After transposing and re-applyingpreviously developed notation, we get
Displacement in Outer Segment of Adherend (xQ)
d2wu U 2 U2
- ~WU = -(U (21.1)dx2 t2 t2
Displacement in Overlapping Segment of Joint (x<5,510 +9)
d2i U2 U2 t+tb t+tbdx2 8t2 8t2 2 "2 DU
ResultantDifference (x < :590 + 1
d 2 T - 23&5T = 2Gb±Id2 (21.3)
dx 2 t b dX2
Eq(21.2, .3) can be solved as a 2x2 system. Boundary conditions include the end conditions on&r. in terms of the values of Tu and TL given in Table 2, together with the joining conditions onthe displacements at x=Q0. The homogeneous part of this system can be written
-I U 2 - +( bWh " tWh -2 Th= 0 (22.1)
/2- 2132h-22Ew / =0 (22.2)8Th t2=
where U and P which are key parameters in the GR analysis (EQ(A-8, A-9) are defined by
U=t TrDu =- i ; =PG/pt where pg=Gb/E' ; pt= tbt
These represent exponential decay factors in the functions describing bending and bond shearstress distribution along the overlap region in the GR analysis. Solutions to (22) can beexpressed as exponential functions:When substituted into (22), these lead to a 2x2 system of algebraic equations with Ah,
-12-
Wh=Whe Th = AhePt (23)
WH and 1 as unknowns:(2u2 - U2)E'Wh - 6(1 +pt)Ah 0 (24.1)
-2p 2 p22ElWh + (122 -2p 2)Ah = 0 (24.2)
The following notation in terms of U and 0, will be useful:
R=U/15 (25.1) vj(j=1,2)=p1/ 2 (25.2)
R1 =F8p11U (25.3) R2= P12/1 ( 5} (25.4)
where pj are the roots of the polynomial
62 ( £ € R2_
4 - [(8 +6+R21.. p_]2p2 +R 2J2 4 =0 (26)P 2
obtained from setting the determinant of(24.1, .2) to zero.
Note that we can express R given in (25.1), in terms of U and j3, as
R- 12 rItb (27)Is Gb t
from which we observe that as the adherend thickness, t, grows, R approaches zero. In addition,Eq(25) give us, for R, and R2
R1P1l RPR[ (28.1) P2 1= F R,20 (28.2)
which, taking note of (25.2), leads to
R = 81 (28.3) , R 2 - - (28.4)
In terms of v (suppressing the j subscript), Eq(26) can be expressed as
-13-
3 2
v2 -[8(1+ ) +- . Jv +R2 =0 (29)4 P, 2
The roots of (29) are of the form
v1 ,V2 =a±b (30.1) where a =[4(l +pt) +---- (30.2) ; b-=_a2-R 2 (30.3)4 4
It is useful to observe here that the standard identity between the constant term of a quadraticexpression and the product of its roots, when applied to (29), implies that v2 R2/v1 . When thisrelationship is applied to (28.3, .4), the following relationship:
1 (31)
is obtained. For most practical situations, the value of R obtained from (27) will be on the orderof 1 or less. On the other hand, the quantity "a" given in (30.2) can not be less than 4, so that ingeneral we have R2<<a2. Accordingly, a binomial expansion of (30.3) gives the approximation
Rb =a - R (32)
2a
from which (30) lead to
R
v 1 =-R (33.1) ; v2 -=2a (33.2)2a
or, making use of (28.3, .4) followed by (28.1, .2):
2Ri = (34.1) " R2---F (34.2)
i.e.for large t,for which (p%, R) - 0 and a -4:
R1 =R 2 = 1 (34.3)
PI = R = U P2 = F 1 (35)
which are the roots of the differential equations in the original GR analysis for deflection of thecenter part of the joint and the bond shear stresses. In the updated analysis, the solution givenin (A-7. 1) for the outer part of the adherend is still applicable, while for the overlapping part ofthe joint, the complete solution corresponding to Eq(21.2, .3) is a combination of the exponentialfunctions appearing in (23) containing the coefficients +#1 and ±P2 (arranged terms ofhyperbolic functions to accommodate the anti-symmetry of the problem) plus a linear termrepresenting the particular solution of (21.2). In the following it is convenient to express thecoefficients of terms describing v as multiples of t/2, as was done earlier. Thus the coefficients
-14-
Whi (j=1,2), where j corresponds to which root of (26) is referenced, can be convenientlyexpressed in terms of t/2 by
Whj(J1,2) = kj- (36)
The complete solution for 7 and 8T is then given by
#o<X<I.o+I
- t sinh[pa1 (x-L)/t] t sinh[P2 (x -L)/t] t'f2' sinh(p1 A/2) + - 22" sinh(p 2 r2) 2 7
sinh [pI (x -L)/t] sinh [lp2(x -L)/t] (37.2)
sinh(plk/2) sinh(p 2X/2)
It is convenient to express the coefficients of Eq(37.2) as well as (37.1) in terms of k2j. For thispurpose note that (24.2) together with (36) imply that Ahj and k2j are related by
AHhE 1,2) = HjE/k 2j (38.1) where Hj = (38.2)
From Eq(9) together with the definition of U following (22), we can substitute
E't= 12U
2
into (38.1) to eliminate E't. In addition, it is useful to express the Hi's in terms of the rootsU/(8) 112 and (8)1/2 B of the GR deflection and bond sb-ar stress analysis, as follows:
H J U2 (Rp3) 2 (39.1) H H2 = j2 (8 32) (39.2)8 8
where the Jj's are constants to be determined. Substituting (39) into (38.1) together with theexpression for E't just given then leads to
Ah =3JIk21 T (40.1) ; Ah2=48J2 R-22 (40.2)4 R2
In addition, substitution of (38.2) into (39), representing the pj's in terms of (28.1, .2), andallowing for (31), leads to
2
J 2 R2 (41.1) ; J2 = 4R 2._ 1 (41.2)
R2 - 4R2 -16
-15-
In the following, the deflection equation, (A-7.1), for x< 0o, will be expressed in terms ofrevised notation in which "kn" (the subscript "n" designating "new") will be substituted for the"k" of the original GR analysis, ie. (A-7.1) is expressed as
X<1.
W= t k sinh(Ux/t) -ax (42)2 sinh(U%,,)
Continuity conditions at x=#o can now be established to obtain equations for determining kntogether with the k2j's. Note here that for practical purposes, the displacements of the neutralaxis in the region of overlap may be taken as equivalent to that of either adherend, since theactual difference is on the order of the adhesive peel strain times the adhesive layer thickness,a much smaller quantity than wU, wL or w can be expected to amount to. Thus the displacementof the outer adherend can be considered equal to that of the neutral axis at x=10 and similarlywith the slopes. Continuity requirements then amount to equating the value and derivative of ivgiven by (37.1) with those of w given by (42) for x=Q0 [i.e. (x-L)/t =-X12] as well as setting &rgiven by (37.2) to T at that location. After canceling out common factors of T and t/2, threeequations relating kn, k21 and k22 are obtalned:
-3Jlk 21 + 48A2k 22 = -1 (43.1)
k2 1 +
22 + kn = 1 (43.2)
R 1Rk 2 1 + 8R2 Th2lk 22 - '8R.5-2L kn =0 (43.3)Th22 Thi
where Th2j = tanh(pjX/2) (43.4)
A straightforward elimination approach can be used, here in which (43.1) is first transposed toget k22 in terms of k21:
k2 2 = R 2 (C Ik 21 -C 2) (44.1)
where
Cl K ; C2 _ I K1 = 3Jl K2 = 48,2 (44.2)
-16-
which may be substituted into (43.2) to eliminate k22:
(1 +R2 C1)k 21 -R 2 C2 + k,, = 1
or, after transposing:
k 1 I-- 2 k (45)1 +R2 C1
Back-substituting this into (44.1) to eliminate k gives
k22 = R 2 (C1 1 +R 2 C2 -k C2) (46)1 +R2Cl
Finally, substituting (45) and (46) into (43.3) provides an expression for kn of theform
1 +R2C2-kn + Th21 R 2 (C 1 -C 2 ) -R 2 Ck, _ -Th2 =
1+R +C I +R Th22
which transposes to give:
RI(1 +R2 C2) + 8R 2 Th2lR(C1 -C2)
kn = Th22 (47)
+ 8R2 221RC1 + F (I + 21)2ZLTh22 T ,,-hl
As noted previously [ Eq(27) ], R tends toward zero for large t, so that all the termsin (47) containing R as a factor will disappear, while R1 and R2 approach unity. Wethen have:
-17-
t>>tb (R,pt--O)
k _ _ _ _ _ (48)1 + F8- T 1
i.e. k. approaches the value given in the GR analysis for their "k" given by (A-7.6). (It caneasily be shown that Th21 of the present analysis approaches Th2 of the GR analysis.) Inaddition, substituting (30.2) into (49.2) and the latter into (54) gives
t>>t b (R,Ptp-2)
Jl" 3+- p J2 --. 2- (49)
1 1P +(pt + 2
From (49) it is apparent that we can setJt--1, J2=1/3 (t>>tb), (50)
and making use of the notation of (44.2) as well as (45, 46) leads tot>>t b (R,Pt" O)
k~l I -. k 2 .13 1 -Q.- 11 -0 (51)
which is again consistent with GR. Applying (50, (51)) to (41) then leads tot>>t b (R,Pt- 0)
AM - _ 3 (1 -k3).t Ah [ 3(1 -kd - l]Zt (51)
4 4
At this point we focus attention on the bond layer shear stress distribution. Noting the relationbetween Tu and 5T, given in (15) and applying (A-3.1) to (36.2) with TU expressed in termsof &r leads to the following for 'rb:
P1 cosh[ p,(x-L)/tJ p2 cosh[p2(x-L)/t (_____AM___ + -A_ - (52)___t " sinh(p 1 X/2) 2t sinh(p 2 ./2)
For the case of t>>tb, substituting (51) into (52) and allowing for (49) results int>>t b (R,pt-)0)
Tb, ,{ F/80 (1 +3k.) cOsh[V/8(x-L)/tl + __ U(I -kQ coshLU(x -L)/Vl8t 1 (53)8 sinh(F8-3X/2) 8v'- sinh(uX/2V-)
from which the maximum value of shear stress in the bond layer is approximatelyt>>t b (R,pt-- 0)
_I + 3 _18 T 2 2 8- Th2 l
Note that as in the case of kn given by (46), Eq(54) is in essential agreement with Eq(A- 10)obtained by GR, except that the second term in the latter disappears for sufficiently large X
-18-
(i.e. dimensionless overlap length), whereas here the second term persists no matter how longthe joint is.
Peel Stresses in the Modified Analysis
Note that (see Eq(A-2.2), Appendix A) the same expression ( Eq(2) ) for MN, the momentabout the neutral axis, applies for the present analysis as that used by GR. However, for theindividual adherends, this needs to be examined further to insure a correct version of theformulation for the peel stresses. In particular, the influence of bond shear and peel stresseson the moments in the individual adherends has to be re-examined. In the following, it willturn out that the influence of peel stresses on the adherend moment distribution will beunchanged from what it was in the GR analysis. However, the contribution of the shearstresses on acherend moments is influenced by adherend deflections through the samemechanism as that which applied to the moments generated by the horizontal components ofthe end loads. A particular point which should be understood has to do with the way thatthese horizontal loads generate moments at various points in the adherends. These are notlarge deflection effects corresponding to large values of either lateral deflection or slope.Rather, they should be thought of as long range effects, since they correspond to relativelylarge horizontal offsets between the point where the load is generated and the point underconsideration where moments are being calculated. Figure 7(A) denotes the incremental loads,dF(R) and dF0 () generated on the upper adherend by the shear and peel stresses at a genericaxial position i, where
dF(,JR) = "rbd' ; dF0 (x) = ubdX
As stated previously, the theory developed here is a small strain, small deflection theory.Thus, for example, rotation will produce a vertical component of dF.r given approximately byw'dF t at i which will produce a moment contribution at x given by (i-x)w'dFT. The fact thatw' is assumed to be a negligible quantity does not rule out the possibility that (x-i)w' maynot be if x-i is large enough.
On the other hand, a similar horizontal component of dF o given approximately by w'dFoproduces a moment at x given by [w(x)-w(i)Jw'dFo; but since w(x)-w(i) is equivalent to(x-i)w')avc where w'),ve is the average slope in the interval between i and x, the contributionof the rotation of dFo is of the order of (x-i)w' 2dF0 , i.e. proportional to the square of theslope, and can be ignored in comparison with w'(x-i)dF.. Thus, while the rotation of dF. canbe ignored in a small-slope analysis, that of dF. cannot necessarily be similarly ignored.Accumulation of dF. and dF0 by integration along the adhesive-aaherend interface will thenproduce a moment at the point of observation, x, which is determined by the offset of theload lines of incremental loads dFl, and dF o , applied at i, from the point of observation, ie.by Q and Q shown in Figure 7(B). The adherend moments can thus be written as
-19-
(%,/ + (x )( w-(i) ]
(C)
2
Figure 7 Effects of Adherend Deflections on Moments in Overlap Region
Mu~~~~~~~x)~(b Ux /2 +W(X NUN £11(vud 1 i -; M(xi)(..~
0r -0 ~ X 0 0U(R
where Q0 -- x
for both adherends, while expressions for Q and are obtained as follows: for the upperadherend, as illustrated in Figure 7(C) and (D), the load produced by dF at i is offset from
L
the neutral axis at x by 0u given by
-20-
UpperAdherend
This is obtained from considering various contributions to 4, shown in Figure 7. It is clear,first of all, that a moment is generated by dF, from the difference in wu at f and x, as shownin Figure 7(C). In addition, as in Figure 7(D) (lower expression at right end of adherendsegment), the tilt of the tangent vector at i has the effect of counteracting the latterdisplacement difference by an amount (x-i)w', resulting in Eq(55)). As apparent in Figure7(B),however, the deflection of the lower adherend tends to move its neutral axis toward theload line of dF. rather than away from it as in the case of the upper adherend. As a result,the terms in Eq(55) which are in square brackets are subtracted for the lower adherendrather than added, resulting in
=-2(t+tb)- [wL(x)-wL(i)-(x-i)w(.)I (56)
inserting Eq(55) and (56) into the integral expressions just given for the moments in theadherends at a generic observation point x, ( xQ o) leads to:
MU- V + Twu(x ) + 'tb(1{(t+th + [wU(x)-w1(i)-(X-X)Wu(i)] )d' + (x-i)tb(i)di (57.1)
and
ML = 'l"t(){2(t +tb) - [W L(X) -WL (X -(x -)wL i). }di - (X -i)ab(i)di (57.2)
Derivatives of Eq(57) are needed for the subsequent development. These are obtained by useof the following identities which apply to arbitrary functions F(x), G(x) and H(x):
d 1.I1(1)di - H(x)
from which
F() () l= F(x) G(x) d :F() G(x) di = G'(x)VF(I)dI +G(x)F(x)
leading todg.. [G(x) -G(i)]F(.i)di = G(x)f'F(i)di (58)
Applying this to (57) leads to
- -... (xb) b -S+- V + Tw(x) 2 b(xX)-[ - W l',(1)di -o,(i)di (59.1)
d 1 -Ci+ rZrb(X z IW.X)L (59.2)- .- +)b(x)- L[w(x) -w()]'b(1)dl + .ab(1) di(5
while differentiating (59) to get second derivatives of Mu and ML results in
-21-
dx2 (+ Tg') b(x) + WU X) 1b(1)f- G(x) (60.1)
d2ML -= (t +tdrb(X) - wL (x) rb(1)dle + ab (x) (60.2)di 2
Making use of (16.2, .5) and (30) to eliminate the integral expressions in (60) thenleads to
d 2M u .d.2M = Tu(x)w(x) + (t + tt)s(x) -ab (61.1)
d2ML I (2(td '(x) 1(61.2)- = TL (x) WL (x) + c Gb
dx2 2
and subtracting (61.2) from (61.1) results ind2(MU _l Mr. ! 1d 2 - -(TUw v -TLw) -2b (62)
Note the identityTuWt TL wt =(r +T)wU-w. + .(Tu-Tt)(wl +wL.
or, making use of (16.7) together with the notation of (30),
T~wu - TLWL =, + 8, (63)
On allowing for (14.3, .4) and (17.2, .4), the second derivative of the difference betweenadherend moments can be expressed as :
d2(MU-M) d4s a4Eb - b d4 b(d =2 M- --DUlb d ='" fiDU _bd (64)
while from the same relationships, the right hand side of (63) can be expressed as
fb d 2ab +5 Ti (65)Eb dX2
Inserting (65) into (63) and the result of that substitution, together with (64), into (62) thenresults in
d4ab U2 d2ab 4y4 2- V
dx4 2P d2 4 - 4
The last equation is comparable to (20.1) obtained by GR for the peel stresses, with theaddition of the second order term on the left and the forcing function on the right. Thehomogeneous solution for (66) will be of a form similar to (26) for the GR peel stressanalysis, except for minor changes having to do with the fact that the coefficients for the
-22-
trigonometric factors in (24) will be slightly different from those for the factors involvinghyperbolic functions. In practice, the changes are so small as to be of no practicalsignificance, so that (26), with an appropriate modification of k, can be used for thehomogeneous solution for all intents and purposes.
The presence of a non-zero function on the right side of (66) requires the addition of aparticular solution to the homogeneous solution. Obviously, the factors from which the righthand side is formed are obtained from the solutions for 7V and aT given in (51). The detailsof this part of the solution will be provided in Appendix B. Again it will be found that inpractical situations, the differences introduced by these terms from those of the GR peel stressexpression (except for the replacement of k by kn) are essentially negligible.
COMMENTS ON THE HART-SMITH ANALYSISTo date, the only serious attempt to correct the deficiencies of the GR deflection analysis,other than the present one, appears to have been that of Hart-Smith[3] which was developedon a NASA contract in the early 70's. Those results have been frequently quoted in reportsand publications which have appeared since then, and a comparison with the present resultsis in order. The most crucial part of the Hart-Smith analysis is the expression given foradherend bending, which we may compare with Eq(21.2) of the present analysis. This isrepeated here for comparison:
--U2 a1 (RPT21.2)
dx 81 2 -2 2 DU
The comparable equation in [4] is equivalent to a twice differentiated version ofEq(35.2)
d4 _ U 2 d 2 = - 1 -B -t b1 .
dx 4 2t 2 dX2 d 2 U
but lacking the second order term which is needed to allow for the influence ofdeflection on moments, ie.
d 4,Z I Ia' T_.- = -f (1b +)---Ib
The absence of the second derivative term here completely changes the character of thesolution. As a result, the influence predicted by this approach gives much longer range effectsthan those of the present analysis. In effect, this prevents St. Venant's principle from beingsatisfied in the sense of not allowing Eq(l) to be applicable at any point in the system. ofeither GR or the present analysis is in the variation of k which is obtained. Table 3 comparesthe expression for k vs. overlap length obtained from the three approaches.
Considering k as a function of UXJ2 -(12Zx) 1 2t2/2t, i.e. dimensionless overlap length togetherwith square root of the loading strain, the GR analysis has the well known asymptotic limitk-0.262 for large UX/2. The present theory will likewise be found to produce an asymptoticlimit but one which may vary to some extent from 0.262, especially for thin adherends. In
-23-
TABLE 3 COMPARISON OF EXPRESSIONS FOR k vs. UV/2{ case of tanh(UXo)l 1}
1Goland & .XReissner[1] 1 +8tanh( )
(kGR)
Hart-Smith[3] 1(kHs) I + U- + 2
____2 2 2
Present Analysis R1 (1 +R2 C2) + 8R2 Th2tR (C1 -C 2)Th 2 2
(kn) R1 + 8R2 Th RCI + F"(1 +R2C1)Th21Yh2 2
where { Eq(27,28,30,41,44) }
R-a 12F;tb R = 18 R2 2
1b t R ' T
with vl,v 2 =a±b ; a=[4(+ .pt)+ ; b= a2 -R 2
4 4
04 2 2T2 ---R 4R2 - 1
contrast, the Hart-Smith expression has lower limit of zero for increasing UX. Note that helatter result corresponds to zero shear and peel stresses. This type of behavior is notconsistent with what is found in most problems of load diffusion which is what we aredealing with near the end of the overlap, and a lower limit on the stresses which persists evenfor indefinitely long overlaps and large loads seems to be more in line with practicalexpectations.
NUMERICAL RESULTS
The degree to which the results of the original Goland and Reissner analysis differ from thoseof the updated analysis of the present paper depends primarily on the R, the ratio of U to [3,
-24-
C
. ...............HART-SMITH 13]
0 2 4 6 1 10 1 2 14
Ux2
Figure 8 Comparison of Goland Reissner[1I and Hart-Smith Prediction of k vs. U
which is equivalent to (l2ixtbjGbt)1f2 according to Eq(27). Results which illustrate thesituation are given in Figure 8-10.
T
00
Q -------------------
1 6810 1 ,2IA2
Figure 9 Comparison of k vs. UX Prediction for Goland Reissner vs. Present Theories
-25-
0 0
0 0
0- 0 0 ] - n •U
"i 0 o
000 oh• G~n&R~nr1
D 00 @ 0 -. . Present Analys
80 - )-- Goland & Reissner (I
.. - - Present Analysis* - lki
E'- 2m, G, - S10kE Eb - SO.- 7 N at- .. o, .. O.O in
0.05 0.10 0.15Adherend Thickness (in)
Figure 10 Comparison of Stress Predictions -- Goland Reissner vs. Present Theories
Figure 8 gives a comparison of the GR and Hart-Smith predictions for k vs. UX, while Figure9 shows a similar comparison between GR and the results of the present analysis for k vs.Ux.The effect of R on the curve of k vs. U/2 is effectively brought out in these results. It isclear that the largest difference between GR and the present analysis occurs for smalladherend thicknesses.
Figure 10 shows the effect of adherend thickness on the shear and peel stress predictionsobtained from the present analysis for large enough UX so that effects of joint length havesettled out, i.e. with k having reached its lower limit. For the case considered here,
E'= 20x10 6 psi, Gb= 1 5 0 ksi, Eb= 5 0 0 ksi, tb = 0.008 in, s = 20 ksi.
It is interesting to note that the ratio of maximum peeling to shear stresses predicted inEq(15) for the GR analysis is close to unity, so that the GR curves lie nearly on top of eachother. Considerable deviation from the GR curves is noted in those of the present analysis.In particular, the shear stresses obtained from the present analysis are greater than thosepredicted by GR, while the peel stresses are somewhat less than those obtained by GR.
-26-
CONCLUSIONS
In the context of the use of beam theory for modelling adherend response, the approachemployed in this paper appears to provide the first completely consistent model for theanalysis of single lap joints. It is emphasized that the original Goland-Reissner model, inanalyzing the deflections of the joint, used a classical homogeneous beam model forcalculating deflections which was not consistent with the bi-layer model used by them topredict bond shear and peel stresses. Due to the long-range nature of the effect of bond shearstrains on the bending stiffness of the joint when the adherends are thin, the largestdifferences between the present approach and the GR approach occur the case of thinadherends. On the other hand, the numerical results which have been obtained demonstratethat the original GR approach produces numerical results which do not substantially differfrom those of the present analysis. It is clear that for large adherend thicknesses, the effectswhich are ignored in the original GR approach are so concentrated near the ends of theoverlap that their consequences are not significant. Thus for thicker adherends, the GRapproach can be considered to have been satisfactory from the stand-point of soundengineering. The insight that Goland and Reissner brought to bear on the effects of adherendbending deflections together with the role of peel stresses on the strength of adhesive jointsstill stand as a major advance in the understanding of factors important to successful adhesivejoint design.
ACKNOWLEDGEMENTThe author is grateful to Dr. L. J. Hart-Smith for a number of discussions and observationsregarding the present effort.
-27-
APPENDIX A EQUATIONS OF GOLAND REISSNER ANALYSIS
BASIC RELATIONSHIPSAdherend Constitutive Relations
diw-uBu-d (A-1.1); TLff-B -L--.2);. Mu=fD d 2U(A-1.3); M-D d W(A-1.4)
DumDL =E t3 (A-1.5); BufSiBL =fiElt (A- 1.6)12
El =fE/(I -v 2) v - Poisson /sratio
Moment Distribution (Figure 6)
OuterEndoJVpperAdherend(O <x <t)
Mu -T(aX +W) (A-2.1)
Moment about Neutral Axis in O?,erlap Region(#o<x<Uf +E)
MN = Tcx-(t _w)T (A-2.2)
Equilibrium Relations
Upper Adherend Forces (FigA- 1(A))
dT --b (A-3.1) i.e. Tu-- + rdx' (A-3.2) d -- -- b (A-3.3) i.e. V =Vo0 dx1 (A -3.4)
d --(-3-Vu (A-3.5) "d- -
Global Equilibrium in Overlap Region (Fig.A-I(B))
TU+TL=T (A-3.7) ; VU+VL=-V. (A-3.8)
Bond Layer Constitutive and Kinematic Relations
Bond Layer Strains (FigA-2)
1 £ w - wL
yb [(UU -UL) +...(WUX +WLX)] (A-4.1) ; Ezb = W_-- (A-4.2)
BondLayerStresses
"b = GbYb (A-4.3) ; rb -EbZb (A-4.4)
Note that Fq(A-2) exhibit the influence of lateral deflection on the moment distribution dueto the shift tie lever arm through which the end load, T, acts with respect to the neutral axisof the adherend segnments under consideration (see Figure 6(A) for the outer adherendand Figure 6(B) and (C) for the overlapping part of the joint.
-29-
z
TU 'dx(D (A) Differential Relationships
-Tb dXUU~ =x
-v-i dx v + F~i+-~)ad=
E-Oj=dx +(V)adx0 d=0
M- t/ U Mu(ObOUt p)=M+ -M,7 +VUdx - b'd
Tb dx =T - U
t12 ~~ML(aboutp')=M -ME +VLdx-± d
TL (B) Global Relationships
vu FZ =TU +TL +(-T)=O0.- TU +TL =T
Figure A-i Equilibrium Relationships EIF 2 =VL + +io V .=0 V + VL=
-30-
w
DIS)I.AIENT
U
(A) EFFECT CF ADH" l U U LHORIZCNAL SLIDIG U b
-'tb - ULtb
(B) EFFECT OF ALHERENDROTATION
S, /2 >> t b6b= (t t tan 0 ;k t WU+2WL2 1 WU : W L
Ldw/dx w tan ey -- -- 5b, t (
tb 2 tb
e
Figure A-2 Bond Layer Shear Strain Components
-31-
DIFFERENTIAL EQUATIONSIn the following, the relationships on which the diferential equation under consideration isbased are given preceeding the DE's.
Adherend Deflections
Outer Segment of Adherend { Eq(A-1.3),(A-2.1))
d2wu U2 U2._ -Wu M m (A -5.1)dx2 t2 2
Overlapping Segment of Joint (Eq(1),(A-2.2))
d2w U2 u 2 t__-__ w -, - (aix -, (A -5.2)
dr2 8t2 8t 2
where Ut a F2/ F (A-5.3)
Bond Layer Shear Stress & Strain
Shear Strain fEq(A-4.1),(A-1.1) to (A-1.4)differentiated, (A-1.5)-(A-1.6)}
dyb 1 T TL I . M ML- - Ib ( . - .. )+ --( - + .- ) (A -6. 1)dx t b B TL 2D L
Shear Stress (Eq(A -4.3), (A - 1) & (A -6. 1) twice differentiated, (A-3.5,.6), (A-3.8) differentiated)
d3rb - 8 p2 d b - 0 (A-6.2)
dx t 2 dx
where ; PG- ; p, - (A -6.3)El
Bond Layer Peel Stress
(Eq(A -1.3, .4), (A -3.3), (A-3.5,.6), (A--4.1,.3))
d~4_ --0 (A-7.1)
Eb 114
where y (2....) (A-7.2)
-32-
SOLUTIONS
Adherend Deflections
In the following, the outer adherend deflection is represented as a multiple of kt/2,corresponding to the notation for M. given in Eq(A-1). The parameter k is determined bysatisfaction of joining conditions between the outer adherend and overlapping part of the jointat x=#,.
Generic Expressions
x<1o w,0 kt sinh(Ut) -cix (A-7.1) Eo<X<9 W + w -- W2 sinh[(U/8)(x-L)/t] + t -x (A-7.2)2 sinh(UX) sinh[UXIF8-1 2
Continuity Conditions at x-- Io
Displacement: 2 +t -oaI0 (A-7.3) k tU 1 U W2 (A-7.4)z---W2+ 2 t Th, F8" t Th2
Resulting Expressions
W2 -2(1-k) (A-7.5) ; k - T1, (A-7.6) • k'...JXU (A-7.7)2-Th,+F8)T2 4 4h
where TA, ,-tanh(UX,) (A-7.8) ; T2-tanh(U/2V-) (A-7.9)
Note that GR assume the outboard part of the adhere.d u L- 'zig cA,,u,,.6 tu ! ake UL>>I,leading to Thi =1, as a result of which they use the contracted expressions
k 1 k--kXU (A-8)I + /FTh 4
for k and k'.
Bond Layer Shear Stress
Generic Solution of Eq(A -6.2)
+c-cle +c2e + const. (A-9)
-33-
Bond Layer Shear Stress (con)
Boundary Conditions
(1) Using Eq(A -6.1) multiplied by Gb to generate dt/dr, end conditions determined by MU, ML, TU and TL,
inserted on RUlS of(A -6.1).
(2) Determine const. in (A -9) by integrating Tb,x then inserting into (A -3.2) to satisfy Tu=O condition at x=L +f
Final Form of Solution
'b r [2- (1 + 3k) cosh[C'8"P(x-L)t] + 3_(1 -k)] (A-10.1)4 sinhF8-O .
Maximum Valueofrb(x=L±t/2 ; V-X/2-1 ; 0.262<k<1)
0.631P3 < abs{F-V2- ( 1+b 3k ) < 1.414 (A-10.2)% 4
Peel Stresses
Functional Notation
Cc~) xL xL . x-L .x-LCc(x) -cosh(y.L) cos(y.....) (A-11.1) Ss(x) sinh(y..._) sin(y..__) (A -11.2)t t t t
Sc(x) -sinh(x-L )cos(x-L ) (A-11.3) Cs(x) cosh(y_..L)sin(y__.) (A-11.4)
t t t t
Derivative Relationships
(Cc)' =Y(Sc-Cs) (A-12.1) (Ss)l = Y(Sc+Cs) (A-12.2)t t
(Cc) = - 2 -. Ss (A-12.3) (Ss)" =2± Cc (A-12.4)12 t2
Generic Solution ( cc , a,, - -constants determined by boundary conditions)
ab(x) = (cCc + aSs (A-13)
-34-
Boundary Conditions
(1) Eq(A -4.2) combined with (A -4.1) differentiated twice (making use of (A -1.3,,4)):
d2arb =E b MuJ - ML (A-14.1)
dx2 tb Du
(2) Differemiation of (A - 14.1), making use of (A -3.5,.6):
d3ab Eb VU- VLdx_3 _ -t Du(A -14.2)
dx3 tb D U
(For x=Qo and Q+9o, differenced quantities appearing on the RHS of Eq(A-14) are taken fromthe last two rows of Table 2.)
ResultingExpressions
I (p 2 + 'P)Cc(x) TYPl ky 2crb(x) = Wt ( 2f- + 2j t4)C()+( lP)SX) (-51
where
PI = (Sc + Cs)(._LV*t .-X 2 P 2 (Sc -Cs)(-L), 2 P 3 = SS(-IyL,,2 p4 -CC(x-LYtL2
A = (PIP4 - P2P3)(x-L)lt-W2 (A- 15.2)
Maximum Value of PeelStress (x - L±1/2 ; y2 . 1)
Note that (Eq(A-8) 2k' k U
(2 +P2.'.Forx=L±t/2 ; ab--=FTY k Y ( P 2P 4 +P P3) + W 2
2 A 2- A2 2
P2 P4 +P1P 3 4~PAlso,foryX.12 .1 ; P2PA IP _P+4_A A
- y (Y + U) (A - 15.3)
2
-35-
APPENDIX BPEEL STRESS SOLUTION IN UPDATED ANALYSIS
Notation pertinent to this section are given in Table B. 1. Eq(66) which defines the peel stressesis repeated here for reference:
d'ab - U2 d2cab + Q(x) B-1.1)dX4 212 dX
2 tb4
where
Q(x) = T(x)Wl"(x) (B-1.2)
It is noted that the factors included in the function Q(x), on the right side of (B- 1. 1), involveonly the homogeneous parts of the solutions for &r and w(x), so that (B-1.2) can be written
Q() sinh[p, (x -L)/t] ' sinh[P2(X -L) / (IQ(x) --IAAI + Aja }sinh(JpWA2) sinh(p2 /2)
P, k t sinh[p 1(x-L)/t] p2 t sinh[p(x-L)/t)S{ -k2' 2 sinh(p, X/2 + kna sinh(P2 X/2) (B-2)
Defining M together with the qj's (ij = 1,2) as in Table B-1 and noting that from the definitionof M and U, we have that
7 -- a (B-3)
then using appropriate identities for the hyperbolic functions in (B-2) together with thefunctional notation in Table B-1 leads to
Q(x) = 3 - .{q" F "()- 1 Fn(4)- 1 F2 () -F 1 2(4) (7 F (X/2)-1 F22(./ 2) -1 + F2 (X/2) -F2(7)
Equation (B-1.1) can then be expressed as a sum of terms for each of which the RHS of (B-1.1) is an exponential function, giving a differential equation of the type
dab - U2 d2 ' + 4y( = Fe l' (B-5)
where (x and F are constants. Eq(B-5) has the particular solution1 I-Fea' (B-6)
a 4 -Ul 2 +4y'+ 2
2-?Decomposing the right side of (B- 1.1) into exponential terms and repeatedly applying (B-6)gives the particular solution that is needed here. The homogeneous solution of (B- 1. 1) consistsof terms of the form
ab-- a,e /1cos(yx/ t) + ae1"sin(yix/t)
+ Aje -' cos(yx/t) + oe "/'sin(yx/t) (B -7)
-37-
TABLE B-1 NOTATION FOR PEEL STRESS EXPRESSIONS
PARAMETRIC NOTATIONGeneral
M~--t :2 (x-L)
Definition of gi's2 2
8,2,A,, -- 21 URieEq, ., i t- 2 ' ,u q-k 22 e q-M-i U2 P22.
81f24 82 2t
2 2Pi t P&, 2 1
L IA1 R 2 . U2 P i 7 0APk2 A 2 -
3 2 R~2 32 3q1 s IN TERMS OF NOTATION OF EQ(39.3, .4), (52.2), (54)
_31 4R 3 2 M 2 - 3q = -; =2 , 4 2
Definition of Du2s
D -- +2 = D 2 22w +2 D 1 (u +p) - U2(p, +p) 2 +2 D -- +2272 2,e 2y' 2,(
FUNCTIONAL NOTATION
F11(t)=cosh2p,4 ; F2,( )=cosh2p 2 ; F1,(t)=cosh(p,-p) ; F,,( )=cosh(p,+p,)k
G11 -=sinh2p4 ; G.()=sinh2pt ; G( =)--sinh(p,+p k ; G21(,)=sinh(p 1 +p ,
-38-
in which ±(y.±iy1) [i=(-I)',] are complex roots of the quartic polynomialz" -U2 z 2 + 4- 0 (B-8)
associated with (B-I.1). The solution to (B-8) is
z2-[:(y, )12 [. ±_ i (4y'- U") I (B-9)
Because of the second derivative appearing in (B-1.1), it is found that 'y, yi, in contrast with theoriginal GR solution. However, the contribution of terms involving U in (B-9) compared withthose involving y is miniscule, and can be ignored with no risk of significant inaccuracy. Thusthe homogeneous part of the peel stress solution is essentially in agreement with that of the GRsolution will now be
c, = O,' + % (B-10)
where the homogeneous term, abh, is now assumed to have the same form as that of Eq(A-14)of the GR analysis of Appendix A
am =a Ss + aCc (B-11)
but with ac and o, appropriately modified:
P2?(M -M'+P 2f(V -V L - +' (ov-A t 2 (B-12)
in which 0 and V. are as before given in (A-13) which may be re-expressed here forconvenience, using notation which has been introduced in Appendis A (assuming Thl=l in (A-13.2) ) as
Mo-- k..T 2 V .a.t e (B-13)
while MP and Vp are obtained from derivatives of oy,, i.e.:-- 2_' 14 2 d3ah (8-14)
P 2('dX 2 P2y dX3
The in the notation of Table B-I, the particular solution to Eq(B-1) is
3 -( q 1 F ( ) - I + q 22 F 22(t) - I + D 1, - -15
D- 2 ' FDI1 (X/2)- D2 Fz2(2)-1 F 1( 12)-F 1 2( -/2) 1 (B-15)
Derivatives of abP required to evaluat (B-15) are then given by
At this point we wish to concentrate on evaluating quantities of interest at (x-L)/t = V2 wherethe boundary conditions are applied. We also assume the case of long overlap for which ptx, P2).
>>I. In such cases all functionaL ratios in (B-16) can be replaced by unity. The definingallows Kl and VP to be expressed as
M -- - 3 V- a Z (B-19)
-39-
dx3 2 D3-F, 2 (VI2) - 24 F .2(2 2)-
+12 (B 16)F,,(2) -F(12(X/2)
d 3 e, 3 328p13 q, F,,() ,q2 F 22(t)- i. . 8 I ' - + 81j-, .dx 3 2 D n F 1( -ID2)-1 F2 1
+ D21 D12 1(B -17)
3 3
H q P ,= , q12 .+42q. (PI+P) 1) (B-18)
' Dll 1 " yD 47( D21 Dz
y7 DI 7 -522 4y' D 2 D 12
and comparing (B-13) with (B-18) gives Mp and VP as the following ratios with respect to Moand V .
M 3
M0 8k. v. 2k.
The values of these ratios give a yardstick for judging how significant are the effects of theparticular solution on the peel stresses. Numerical evaluation indicates that G and H are on theorder of 1 to 2. Assuming the Goland Reissner limit for k (--0.262) applied to IC (considering thelong overlap case) and a value of I for G and H thus indicates that
M V--- O.I0.1 ; _-0.22(F- 1/
Mo V.
The adherend strain level will certainly not be greater than 0.01, so that the effect of theparticluar solution on the peel stresses is essentially negligible.
-40-
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