1、NASA CR-112235ADHESIVE-BONDED DOUBLE-LAP JOINTSTECHNICAL REPORTbyL o J= HART-SMITHPrepared under Contract NAS1-11234Douglas Aircraft CompanyMcDonnell Douglas Corporation3855 kakewood BlvdLong Beach, California 90846January 1973forLangley Research CenterHampton, Virginia 23366NATIONAL AERONAUTICS AND
2、 SPACE ADMINISTRATIONProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASA CR I12235ADHESIVE-BONDED DOUBLE-LAP JOINTSDTECHNICAL REPORTbyL. J. HART-SMITHPrepared under Contract NASl-ll234Douglas Aircraft CompanyMcDonnell Douglas Corporation3855 Lakewoo
3、d Blvd.Long Beach, California 90846JANUARY 1973forLangley Research CenterHampton, Virginia 23366NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ABSTRACTExplicit analytical solutions are derived for the sta
4、tic load carrying capacityof double-lap adhesive-bonded joints. The analyses extend the elastic solutionof Volkersen and cover adhesive plasticity, adherend stiffness imbalance andthermal mismatch between the adherends. Both elastic-plastic and bi-elasticadhesive representations lead to the explicit
5、 result that the influence of theadhesive on the maximum potential bond strength is defined uniquely by thestrain energy in shear per unit area of bond. Failures induced by peelstresses at the ends of the joint are examined. This failure mode is partic-ularly important for composite adherends. The e
6、xplicit solutions are suffic-iently simple to be used for design purposes.KEYWORD DESCRIPTORSBonded JointsAdhesive Stresses and StrainsAdherend Thermal MismatchAdherend Stiffness ImbalancePeel StressesFatigue LoadsDouble-Lap JointsStatic StrengthShear Strain EnergyElastic-Plastic FormulationBi-Elast
7、ic FormulationAdvanced Composite JointsiiiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FOREWORDThis report was prepared by the Douglas Aircraft Company, McDonnell DouglasCorporation, Long Beach, California under the terms of Contract NASl-l1234.On
8、e summary report (NASA CR 2218) and four technical reports (NASA CR I12235,-6, -7, and -8) cover the work, which was performed between November 1971 andJanuary 1973. The program was sponsored by the National Aeronautics and SpaceAdministrations Langley Research Center, Hampton, Virginia. Dr. M. F. C
9、ardand Mr. H. G. Bush were the Contracting Agencys Technical Monitors.The basic concept of bonded joint shear analysis by classical mechanics ofcontinuous structures in terms of the elastic-plastic adhesive model was devel-oped initially under Douglas IRAD funding between 1968 and 1970. This contrac
10、thas permitted the work to be expanded greatly in both scope and detail. All ofthe peel-stress studies were performed under this contract.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Section123456789.CONTENTSPageSymbols xiSummary . 1Introduction 3
11、Balanced Double-Lap Joints (Elastic-Plastic Analysis) . 7Balanced Double-Lap Joints (Bi-Elastic Analysis) . 13Effect of Thermal Mismatch Between Adherends . 19Effect of Stiffness Imbalance Between Adherends 23Peel Stresses in Double-Lap Joints 27Adherend Induced Failures . 33Change in Joint Strength
12、 Between Tensile and Compressive Shear Loading 35In-Plane (Edgewise) Shear Loading . 37I0. Mixed-Modulus Adhesive Joints . 43If. Parametric Effects and Joint Efficiency Charts 4512. Design Method . 5113. Conclusions . 55References . 57Illustrations 59Appendices 85A.l General Analysis Including Adher
13、end Imbalances . 85A.l.l Fully-Elastic Analysis 86A.l.2 Analysis for Plastic Strains at Only One End of Joint 89A.l.3 Analysis for Plastic Strains at Both Ends of Joint . . 92A.l.4 Analysis for Fully-Plastic Adhesive-Bonded Joint . . . 95A.2 Computer Program A4EB for Shear Strength of Double-LapBond
14、ed Joints . 99viiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FigureI.2.3.4.ILLUSTRATIONSPageSchematic Explanation of Shearing in Adhesive . 59Co-ordinate System and Deformations in Bonded Joints . 60Shear Strength of Double-Lap Bonded Joints 61Av
15、erage Shear Stresses in Double-Lap Adhesive Bonds 625. Stresses and Strains in Bonded Joints . 636. Analytical Representations for Actual Adhesive Characteristics 647. Deformations and Adhesive Shear Strains in Thermally-MismatchedBonded Joints . 658. Co-ordinate System and Deformations in Bonded Jo
16、ints (DissimilarAdherends) 669. Strength Reduction Factor in Double-Lap Bonded Joints, due toAdherend Thermal Mismatch . 67I0. Deformations and Adhesive Shear Strains in Stiffness-UnbalancedBonded Joints . 68II. Strength Reduction Factor in Double-Lap Bonded Joints, due toAdherend Stiffness Imbalanc
17、e . 6912. Strength Reduction in Bonded Joints Due to Adherend Stiffness Imbalance 7013. Peel-Stress Failure of Thick Composite Bonded Joints . 7114. Co-ordinate System and Peel Deformations in Double-Lap Bonded Joints . 7215. Elastic Peel Stresses in Double-Lap Bonded Joints . 7316. Adhesive-Bonded
18、Joint Loaded by In-Plane Shear 7417. Co-ordinate System and Shear Deformations in Bonded Joint underIn-Plane Shear 7518. Mixed-Modulus Adhesive Bonded Joints (Shear Stress Distributions) . . 7619. Influence of Governing Parameters on Shear Strength of Double-LapJoints 77ixProvided by IHSNot for Resa
19、leNo reproduction or networking permitted without license from IHS-,-,-20. Maximum Efficiency and Joint Strength for Metal Adherends . 7821. Maximum Efficiency and Joint Strength for Composite Adherends . 79Table II. Material Properties for Figures 20 and 21 . 8022. Stress-Strain Characteristics of
20、Adhesive Film in Shear, ShowingTemperature Dependence . 8123. Comparison of Ductile and Brittle Adhesive Strengths at VariousTemperatures . 8224. Double-Lap Bonded Joints under Partial Loads . 8325. Bonded Joint Analysis in Terms of Upper and Lower Bounds throughFully-Plastic Analysis 84XProvided by
21、 IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SYMBOLSA,B,CF,H,Ja,b,cCTHERMDdEEcE CETRGGe,G pGi,GOkl ,k2MPQSTATTt= Integration constants= Extents of plastic stress state in adhesive at ends of bondedjoint (in.)= Non-dimensionalized adherend thermal mismatch co
22、efficient= Flexural rigidity of adherends (Ib in.2)= Length of elastic zone in adhesive bond (in.)= Youngs modulus (longitudinal) for adherend (psi)= Adhesive peel (transverse tension) modulus (psi)= Adherend extensional stiffness ratio= Adhesive shear modulus for elastic-plastic representation (psi
23、)= Initial and final, respectively, adhesive shear moduli forbi-elastic representation (psi)= Adherend in-plane shear moduli (psi)= Factors see Equation(68)= Overlap (length of bond) (in.)= Bending moment in adherend (Ib in. / in.)= Applied direct load on entire joint (Ib / in.)= Applied running she
24、ar load on joint (Ib / in.)= Shear stress resultant (in-plane) in adherend (Ib / in.)= Temperature (F)= Temperature change (Toperating - Tcure) (F)= Direct stress resultants in adherends (Ib / in.)= Thickness of adherend (in.)= Transverse shear force on adherend (Ib / in.)xiProvided by IHSNot for Re
25、saleNo reproduction or networking permitted without license from IHS-,-,-W -_X =O the outer rowsof rivets are the most highly stressed. The analyses predict that practicallyall of the load transferred between these adherends passes through these effect-ive end zones which are found to have a charact
26、eristic extent independent of thetotal overlap. The analyses cited above, like those of Plantema (Reference 3)and Szepe (Reference 4) also, are based on the assumption of linearly elasticmaterials. Therefore their analyses lack the plastic end zones inherent in theanalysis presented herein, which es
27、tablishes that muchof the past discrepancybetween theory and experiment may be ascribed to this factor. The influence ofplasticity in the adhesive is revealed to be so great an increase in the potent-ial shear strength of the bond that in manycases failure must be initiated inthe adherends, as obser
28、ved experimentally.The technique of the present analysis is illustrated in the body of the reportby considering one factor at a time. The completely general analysis is pre-sented in Appendix Al. The report begins with the effect of adhesive plasticityon a balanced joint(Section 2). Next, (Section 3
29、) the arbitrary bi-elasticadhesive characterization is shownto lead to the sameresult, justifying theuse of the mathematically more convenient elastic-plastic formulation. Then,in turn, the strength reductions associated with adherend thermal and stiffnessimbalances, acting independently, are explai
30、ned in Sections 4 and 5 respective-ly. The problem of the peel stresses at the ends of the outer adherends foll-ows, (Section 6), and it is found to dominate the behavior of the thickercomposite adherends. Accounting for adhesive plasticity has raised the predict-Provided by IHSNot for ResaleNo repr
31、oduction or networking permitted without license from IHS-,-,-ed shear strength capacity of bonded joints to the level at which it becomesnecessary to examine also adherend-induced failures of, or at the edge of, thejoint (Section 7). The inclusion of thermal effects necessitates a distinctionbetwee
32、n compressive and tensile shear loading of a double-lap joint (Section8). In-plane (edgewise) shear loading is shown in Section 9 to be governed bydifferential equations of the same form and the equivalent joint parameters areidentified. Mixed-modulus adhesive-bonded joints are discussed in Section
33、lO.The effects of the various joint parameters are elucidated in Section II, whileSection 12 explains how to use the design methods, and prepare design charts,for double-lap bonded joints. Conclusions are reached in Section 13 and thecomputer program developed is listed in Appendix A2, along with sa
34、mple outputsand brief user instructions.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2. BALANCEDDOUBLE-LAPJOINTS(ELASTIC-PLASTICANALYSIS)The classical analysis by Volkersen (Reference l) allows for only the stressesarising from the differential st
35、raining in lap joints and is restricted toelastic adhesive behavior. It forms the basis of the development here account-ing for adhesive plasticity and thermal mismatch between the adherends, alongwith the adherend stiffness imbalance previously included. This section isrestricted to consideration o
36、f adhesive non-linear behavior, independently ofany adherend imbalance effects, to demonstrate the important conclusion thatthe maximumjoint strength betweengiven identical adherends is defined uniquelyby the adhesive strain energy in shear. This conclusion is independent of theprecise form of the s
37、tress-strain curve for the adhesive (as demonstrated by thebi-elastic solution in Section 3), and of any individual adhesive characteristicsuch as the initial elastic modulus.Figure 2 depicts the geometry and nomenclature for the analysis of a symmetricdouble-lap joint. The conditions of longitudina
38、l force-equilibrium for adifferential element cl_within the joint aredT dT.i = o (1)+ T = 0 , - ,where the subscripts o and i refer, respectively, to the outer and inner adher-ends, the inner adherend being acted on by the shear stresses in the adhesiveon each side. The stress-strain relations for t
39、he assumed elastic adherendyieldd_ T d6. T.0 0 1 1dx E t dx E.t.0 0 i 1As a first approximation, the adhesive shear strain is taken as(2)(a.-a)1 0 (3)Within the elastic region (of length a), the adhesive shear stress isProvided by IHSNot for ResaleNo reproduction or networking permitted without lice
40、nse from IHS-,-,-assumed * to be= Gy = ) = f(x)nwhile, throughout the remaining plastic region, the adhesive shear stress istaken asT = Tp = constantEliminating _i and ady 1 / T i T= _ Odx n Eit i E to oThe use of equations (I) to eliminate T. and Tl 0differential equationbetween equations (2) and (
41、3) produces)then yields the governingd2y I( I 2)dx 2 n E t E.t.0 0 i I(4)(s)(6)o (7)Within the elastic region around the center of the joint, this equationbecomes* (The relation (4) used both by Volkersen (Reference l) and by Goland andReissner (Reference 2) implies not only a linear stress-strain r
42、elation in shearbut also a uniform shear stress distribution across the thickness. It is thislatter assumption which leads the theory to predict a non-zero shear stress atthe load-free ends of the adhesive. However, the finite-element analyses ofbonded joints by Teodosiadis (Reference 5) have reveal
43、ed that this error affectsthe shear stress distribution only within a distance, from the ends, of a fewtimes the adhesive layer thickness. This factor is important for the purely-elastic solution, with its characteristically sharp spikes at the ends of theshear stress distribution, which significant
44、ly over-estimates the shear stressat the ends of the joints. However, the practical adhesives all exhibit somenon-linear behavior prior to failure and this proves to be a more powerfulfactor in softening the shear stress peaks at the ends of the joints than isinclusion of the variation of shear stre
45、ss across the thickness of the bondline. For an elastic-plastic adhesive, it is slightly conservative to neglectthis effect.)8Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-d2T G( 1 2) 2G 2Tp- X2_ = 0 X2 = +- = - =dx 2 n Eoto Eiti Etn Et(ny e)and ha
46、s the solution(8)T = A cosh(lx) + B sinh(lx) . (9)The constant B is found to be identically zero for balanced Joints (2E t =0 0E.t.) since the adhesive shear stress distribution is necessarily symmetric1 1about the x - origin adopted in Figure 2, being the mid-point of the joint.In the plastic regio
47、n, of length (6 - d) / 2 at each end of the joint, thesolution of equations (5) and (6) is(_2) 2y = _ Tp_ + C_ + Fin which _ _ d_, the origin for _ being at x = +a/2.F and the unknown d/ are found _ satis_ing the bounda_ conditionsY : Ye at x = + d / 2 , = 0 ,Y = Ye + YP at _ = + (6 - d) / 2 ,(lO)Th
48、e constants A, C, and(ll)(12)d_y_ d_y_ at x : + d / 2 = 0 , (13)dx : d_anddy P 2x X2_ = = av av at _ : (6 - d) / 2 , (14)d_ Eitin Eitin 2Gof which equation (13) ensures continuity in the adherend stresses. (Equation(14) could alternatively have been derived by consideration of gross horizontalequilibrium of the joint). HenceF = Ye = IP / G , (15)A : Xp I cosh(Xdl2) , (16)iXZp)C : ,_- tanh(Xd/2) , (17)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,
