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The Journal of Strain Analysis for Engineering

http://sdj.sagepub.com/content/9/3/185The online version of this article can be found at:

DOI: 10.1243/03093247V093185

1974 9: 185The Journal of Strain Analysis for Engineering DesignR D Adams and N A Peppiatt

Stress analysis of adhesive-bonded lap joints

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Institution of Mechanical Engineers

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STRESS ANALYSIS OF ADHESIVE-BONDEDLAP JOINTS

R.D. mms Lecturer in Mechanical Engineering University of BristolN. A. Pl%PPIAm Research Assistant, Department of Mechanical Engineering, University ofBristolStresses in a standard metal-to-m etal adh esiv ebo nde d lap joi nt are analysed by a two-dimensional finit eele me ntmethod and comparisons are made with previous analyses. Particular attenti on is paid to the stresses at the endsof the adh esive layer. Unlike previous work, w hich assumes the adhesive to have a square edge, the adhesive spewis treated as a triangular fillet. The highest stresses exist at the adherend corner within the spew. Linear elasticbehaviour is assumed throughout.

A rubber model is reported which confirms these results physically.Go od agreement was also obtained between some practical results and the finiteelement predictions.

1 INTRODUCTIONIdeally there should be a uniform shear stress in theadhesive layer of an adhesive-bonded lap joint under tensileload in order to give maximum joint efficiency.Unfortunately, this ideal is rarely achieved in practicebecause of stress concentrations due to three separatefactors:

differential straining in the adherends- he shear-lageffect;bending induced by the non-axial loading;end effects caused by the free surfaces at the edges ofth e adhesive layer.Differential straining in the adherends gives rise t o anon-uniform shear-stress distribution in the adhesive, themaximum shear stress occurring at the ends of the overlap;the best-known analysis of this problem is that due toVolkersen (l)*. Demarkles (2) modified Volkersensanalysis to take into account the effects of sheardeformations in the adherends, and Adams and Peppia tt (3 )have shown tha t significant transverse shear stresses arise in

the adhesive layer because of Poissons ratio strainsassociated with the tensile loading of the adherends.The effect of adherend bending, which gives rise tonormal stresses across the adhesive layer, has beenconsidered for the single-lap case by Goland and Reissner(4). However, it should be n oted tha t their solution for thenormal stress in the adhesive is incorrectly written and hasunfortunately been used in this form by many authors. Thecorrected form is given in Appendix 1. Sneddon (5) andKuenzi and Stevens (6) have given the co rrect version b uthave not illustrated clearly the magnitude of the discrep-ancy. Fig. 1 shows the two solutions together with thepercentage error of Golands and Reissners values for arange of Youngs moduli for adhesives: it can be seen thatThe MS. of this paper w as received at the Institution o f MechanicalEngineers on 19th September 19 73 and accepted for publication on29th April 1974. 13*Referencesare given in Append ix 2.

their predic tion is considerably in e rror, especially for low-modulus adhesives. Volkersen (7) has more recently givenexpressions f or the shear and tensile stresses in th e adhesivewhen the effects of internal bending are considered in adouble-lap join t. All o f these solu tions allow for the effectsof non-axial loading bu t none takes in to acc ount the sheardeformations of the adherends.Coker (8), experimentally (using photoelasticity), andInghs (9) , analytically, have show n that in a plate subjectedto shear loading on tw o oppo site sides there are high tensileand compressive stresses at its comers, the magnitude ofthese being about four times the applied shear stress andthe direction being a t right-angles to th e sides on which the

60 OriginalsolutionI--I- -OO 1 2 3 4 5Youngs modulus of adhesiveFig 1. Original solution of Gohnd and Reissner and thecorrected solution for the maximum transverse tensilestresses in the adhesive hyer of a single-lap joint plottedagainst the Youngsmodulus of the adhesive

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R . D. ADAMS AN D N. A. PEPPIATTshear load is applied. These transverse direct stresses arisebecause the direct and shear stresses acting on the freesurface must be zero. Previous lap-joint analyses haveconsidered the adhesive layer as having a square edge (asshown in Fig. 2a) and because of the free surface on e wouldexpec t similar tensile and compressive stresses to o ccur inthe comers of this layer. Furthermore, the earlier analyses(1) and (2) must be in error in the important region at theend of the joint because they predict a maximum shearstress here, whereas the principle of complem entary shearsindicates that no such shear stress can exist on the freesurface, i.e. at right-angles to the predicted shears.Volkersen (7) has taken these effects into account by usingthe stress-equilibrium relation

ao,+a7,,=ay axHigh tensile and compressive stresses are predicted in thecomers of the adhesive layer. These stresses are about 5.5times the average applied shear stress in a typical lap joint0.5 in (12.7 mm) long bonded with a high-modulusadhesive. These end-effect stresses must be added to thenormal stresses induced by bending. In fact, Volkersensanalysis (7) gives solutions for shear and transverse tensilestresses which take both bending and end effects intoaccount. Unfortunately, there are a number of errors in thepaper as published, mainly in the expressions given for theboundary conditions of the equations. For comparisonlater, simplified approaches based on his work have beenused.Real adhesive joints are, however, formed with a f d e t ofadhesive spew which is squeezed out under pressure whilethe joint is being manufactured. This fdet is showndiagramm atically in Fig. 2b. The assumption that theadhesive layer has a square end is thus unlikely to berealistic. Mylonas (lo), using photoelastic techniques, hasinvestigated the stresses induced at the end of an adhesivelayer for a number of adhesive edge shapes and has shownthat the position of the maximum stress is dependent onthe edge shape. The adherends in his model were rigid, andnone of the edge shapes studied is typical of the normalspew fillet, where the adhesive flows round, and bonds to,the end of the adherend.A full analysis of l ap jo in t stresses, which should includethe effects of bending, adh erend shears, and end effects, hasnot yet been reported but these factors can be consideredby using the finiteelement method of stress analysis.Previous authors have used the finite-element method, buthave not considered a l l three factors. Wooley and Carver(11) have written a general-purpose computer programmeto describe a single lap joint. They obtained goodagreement with the original analytical solution of Golandand Reissner (4), bu t they have not considered the stressescaused by end effects. The model of Harrison and Harrison(12) is that of a square-edged adhesive layer between rigidadherends (Fig. 2a) and so the effects of spew, bending, anddifferential straining are n ot considered.The purpose of this w ork was to investigate analyticallyby the finite-element method joints with either squareedges or spew fdets and to compare the results with thetheoretical solutions of previous authors. The models aremade up of constant-strain triangular elements and onlyHookean adhesives and adherends are considered. This is arealistic assumption for some high-strength adhesives at186

f ree surface

Fig.2. Diagrammatic lap joints to show adhesive layers with( a )square edge and ( b ) pew filletroom temperature. A silicone-rubber model, developedfrom the models reported by Adams et al. (13), is alsointroduced to show the effect of the spew filet physically.Finally, some practical consequences of the results aregiven.

2 FINITE-ELEMENT MODELS OF LAP JOINTSBecause of stress gradients across the adhesive layer andbecause of possible high stress gradients along the adhesivelayer, sufficient resolution is only likely to be obtained if aconsiderable number of elements is used. However,although Adams and Peppiatt (3) showed that the lap-jointproblem is three-dimensional, they also showed that theshear and tensile stresses in the direction of the applied loadare not significantly influenced by the transverse stressescaused by Poissons ratio strains in the adherends. Since ahigh matrix bandwidth is inherent in the use of three-dimensional elements, we decided to formulate th e problemusing simple constant-strain twodimensional triangularelements to give stresses at their centroids. This decisionwas further influenced by the fact that many adhesives arenon-linear, and furth er work would require a finiteelem entprogramme t o take into account plastic deformations. Sucha programme is available internally for non-linear analysisusing the simple triangular element.

As the joint is wide compared with its thickness, theproblem is considered as one of plane strain. Thisassumption should be satisfactory for the adhesive layer,but less so for the adherend. Unless otherwise stated, thelap joints studied are of standard lap-joint testpiece size(0.5 in, 12.7 mm, long by 1 in, 25 .4 mm, wide) and aremade of 16 gauge aluminium (1.62 mm thick) bonded witha high-strength modified epoxy-resin adhesive of Youngs

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STRESS ANALYSIS OF ADHESIVE-BONDED LAP JOINTSmodulus 7.0 X lo5 lbf in-' (4.8 2 GNm-'), shear mo dul us2.5 X lo5 lbf in-' (1.72 GNm-'), an d a glue-line thick nessof 0.010 in (0.25 mm). In the double-lap joints, the centreadherend is twice t he thickness of the o uter adherends (i.e.0.128 in, 3.24 mm). The mod ulus values were ob tainedfrom unpublished work by the authors on AF-130 adhesivewhich was found to have a substantially linear stress-strainrelation to failure.Bo th double-lap and single-lap join ts have been treated.The double-lap joi nt has been considered in tw o ways asshown in Figs 3a and 3b. The first model uses symmetryabo ut th e centre-line of the doub le lap and lateral restraintsare applied along this centre-line. The second model usesthe fact, revealed by simple analysis, that, if the adherendsare equally thick, th e loads in th e adh erends are equa l atthe centre of the overlap and the double lap can berepresented by a 'half-length' model, with conseque nteconomy of computing effort. Because of bending effects,restraints must t e applied in two ways to obtain th e stressesat the end where the adhesive is in tension and at the endwhere it is in compression. The load is applied uniformlyacross the adherends a t the mid-section of th e join t(Fig. 3b). These boun dary cond itions are exac t if theoverlap length is long enough for the shear stress at thispoint to be zero. For the case considered here, thiscondition is by no means fulfilled, and these boundaryconditions are therefore approximate. However, SaintVenant's principle suggests that th e stress dist ribu tion inthe outer quarters of the overlap length, as obtained fromthis model, was correct. Single-lap results were obtainedfrom the full-length lap model; the constraints used areshown in Fig. 3c.The spew was approxim ated to a 45" triangular fillet ofvarying size. To find the effect of the spew size on thestress distribution the half-length double-lap type modelwas used in order t o reduce com puting time. A summary ofthe models is given in Table 1.One disadvantage of the finite-element method is th etime required to prepare the element data for differentjoin t geometries. A programme was therefore written which

computed the geometrical data from the followingparameters:adherend length;overlap leng th;spew size (within the limits 0.015 in , 0.38 mm, to 0.9times the adhe rend thickness);adhesive-layer thickness;adhe rend thickness.Thus a joint of differing geometry only requires thechanging of a few data cards and it is equally easy to varythe elastic properties of the adhesive and adherends. Thismodel can be used to obtain results for both single- anddouble-lap joints by means of suitable nodal constraints.The mesh used for the full-length lap joint with spew isshown in Fig. 4 to three different scales to enable the m eshto be seen in perspective and w ith clarity.

3 RUBBER MODEL TO SHOW T H E E F F E C T O F SPEWPrevious rubber models fo r lap join ts as reported by Adamset aL (13) show the no n-uniform shear-stress distribution inan adhesive along a lap joint, bu t d o not have sufficientresolution to show deformations caused by the high, stressgradients at the ends of the adhesive layer. Accordingly, asimplified model has been designed for an experimentalinvestigation of end effects. The mo del is show n under loadin Fig. 5 and the undeformed outline is drawn in Fig. 6 .The part representing the adhesive layer was cast in siliconerubber and has both a 45" fillet to represent the adhesivespew and a square edge. The silicone rubber was cast in amould of which the steel adherends formed the sides(except on the two free surfaces). T he adherends weretreated with a suitable primer before they were cast toensure a good bo nd; the 'free' surfaces of the mou ld werecoated with a release agent. Because of the difficulty ofmarking t he silicone ru bber with a grid, V-grooves weremachined into the surface o f the base of th e mould so thata raised orthogonal grid of silicone rubber was createddirectly on th e side face of the 'adhesive'.The effects of differential straining and adherend

1 ---cP.F

I:-'

a Full-leng th double-lap join t. b Half-length double -lap join t. c Single-lap joint .Fig 3. Constraintsfor initeelement models of ap joints

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R. D. ADAMS AND N.A. PEPPIATT

Glue-linethicknesspew heightype of Type ofmodel joint Spew shapeTable 1. Table of initeelem ent models

Number ofnodes

0.010 in (0.25 m m)from adherend

Half length

I lo30.010 in (0.25 mm)Double-lap No spew.Square ende d

0.020 in (0.5 m m )from adherend

I 960.010 in (0.25 mm)0.010 in (0.25 nlm) I 135

I

Half length

Half lengthFull length

Double-lap Large spew 0.074 in (1.9 mm) 0.010 in (0.25 mm) 149Double-lap No spew. - 0.010 in (0.25 mm) 432Square ended

Half length

Full length

Half lengthSingle-lap No spew. - 0.010 in (0.25 mm) 432Square endedDouble-lap Large spew. 0.074 in (1.9 mm ) 0.010 in (0.25 mm ) 117Adherendcorners with0.016 in radius

General purpose Double-lap Spew fromfull length I or I adherendendsingle-lap Generally0.040 in (1.0 mm)but variable Variable ~I 338

The sec t ion be tw een the l ines i s s h o w n e n l a r g e d b e l o w

Number ofelements153

171

23 1

256754

754

20 2

562

188Fig. 4. Finite-element mesh fo r full-length lap joint with spew

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STRESS ANALYSIS OF ADHESIVE-BONDED LA P JOINTS

The black crosses are the finite-element predictions of the intersections of the grid lines of the model.Fig. 5. Comparison between calculated and experimental displacements of th esilicone-rubber model

bending are not represented on this model because theadherends are very much stiffer than the adhesive.Restraints could be appljed to prevent the rubber frombending owing to rotation of the adherend in the plane ofthe grid face when loading was applied.Comparison with the finite-element results was obtainedby comparing the grid deformations for a givendisplacement at the loading point of the model with thenodal displacements at corresponding points given by thefinite-element method for th e same displacement. A graphplotter drew the positions of the displaced nodes withvertical crosses and the plot was reproduced on a trans-parent film. The film was placed over the model so that adirect comparison could be made between the finite-element deformations and those of the model (see Fig. s).The experimentally determined Youngs modulus of thesilicone rubber used in the model was 2841bf in-2(1.96 MNm-) and the shear modulus was 9 7 lbf i n- (0.67MNm-). These values were obtained from the init ial regionof the stress-strain curve where the relation is approxi-mately linear. Sim ilar stress levels were used in the tests onth e model.

4 DISCUSSION OF RESULTS4.1 Results from rubber modelFig. 5 shows that close agreement was obtained betweenthe deformed grid of the silicone-rubber model and thenodal displacements predicted by the finite-elementanalysis. As surface displacements of the model are used forcomparison, the finite-element results were obtained withplane-stress stiffnesses. The errors were principally due tothe difficulty of applying a restraint in the correct positionwhile s t i l l allowing a photograph to be taken of the wholemodel.JOURNAL O F STRAIN ANALYSIS VOL 9 NO 3 1974 OlMechE 1974

Fig. 6 shows the principal-stress patt ern obtained by th efinite-element analysis. The length and direction o f the linesrepresent respectively the magnitude and direction of theprincipal stresses at the centroid of each finite element. Abar at the end of the line implies a negative principal stress,i.e. compressive. It is evident t ha t th e presence of the filletcauses the stress pattern to differ significantly from thepattern at the end with no fillet. At the points Al and A1the high tensile and com pressive stresses predic ted b y Inglis( 9 ) are shown, the absolute magnitude of the largestelemental principal stresses being at least 3.6 times theshear stress in the rubber between the plates. This is ofsimilar size to the value predicted by Inglis who says thatthe normal stress is more than four times as large as theapplied shear stress. It should be noted that the rubberaway from the ends of the steel plates is in pure shear, as isshown by the equal and opposite principal stresses in th eelements in this region. The stresses in the fillet arepredominantly tensile, the m aximum stress concentrationat this end being at the sharp corner B. The maximum stressis at least 3.5 times the shear stress in the rubber betweenthe plates.4.2 The square-en ded adhesive layerFig. 7 shows the shear-stress distributions in the adhesivelayer of a double-lap joi nt as predicted by the following sixanalyses.

The simple Volkersen analysis (1).A modification of the Volkersen analysis t otake in to account the fac t that the shear stress mustbe zero at the ends. This is based on Volkersensmore recent work (7).Demarkless modification of Volkersens originalanalysis to take into ac count adheren d shears (2).

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R. D. ADAMS AND N . A. PEPPIATT

1 -

I I I 1 I I I I I I I I I

I

Fig. 6. Principal-stress pat tern fo r rubber model, showing end effe cts

6 t h - 2 /

2 Volkersen's analysis (7), allowing for end effects.3 Analysis of D emarkles (2).4 Volkersen's analysis (7), allowing for bending.5 Full-length doublelap finiteelemen t modeL6 o Points from half-length finte -elem ent m odel.--- --- ---

4 'IFig. 7. Shear-stress distributions fro m double-lap theories and finite-element models

(4) An analysis allowing for the effects of bending ofthe adherends in a double-lap joint derived from theequa tions given by V olkersen (7).( 5 ) Finite-element results from the full-length lapmodel.(6) Finite-element results from the half-length lapmodel.

The finiteelement results should take into account endeffects, bending of the adherends, and shear straining of theadherendsComparing results (1) and (2), one can see that, if endeffects are neglected, the requirement that shear stress inthe adhesive should be zero at the joint ends only affectsthe shear-stress distribution significantly for ab out the end

4 per cent of the overlap length. Curves (3) and (4) showthat the maximum shear stress predicted in the adhesivelayer is reduced both by the shear deformation of theadherends and by the reduction of adherend strain at theadhesive-adherend interface caused by the bending of theadheren ds There is good agreement between the two curvesfrom the finite-element results ( 5 ) and (6) except at thecentre of the joint. The variation around the mid-point isdue partly to the free surface present at the centre of thehalf-length lap model where the shear stresses mu st be zero,and partly to the fact tha t' the loading was onlyapproximate. However, the good agreement at the endsshows that the half-length lap model is valid in ths regionfor the combination of adhesive and adherend propertiesconsidered. Problems do arise if the adhesive and adherend

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STRESS ANALYSIS OF A DHESIVE-BONDED LAP JOINTS

D--5:Ln1.

shear moduli are of the same order, but this is no t usual inpractice.The stress distributions from the finiteelement resultsare closest to the solutions taking into account the sheardeformations of the adherends and the bending of theadherends, which shows that these solutions give areasonable representation of the shear-stress distr ibutio nwithin the lap. However, each neglects a different butimportant aspect of the analysis, so the correlation may besomewhat fortuitous.The tensile stresses across the adhesive layers of adouble-lap oint from analyses (4), (5) and (6 ) ,are shown inFig. 8. There is reasonable agreement between the twofinite-elem ent solutions, half-length lap and fu ll-length lapin the outer quarters of the joint. However, there is amarked difference between the finite-element curves andthe curve obtained from the analytical solution (4),especially at the end of the joint where the adhesive is incompression: here the finite-element s o h ions predict acompressive-stress concen tration abo ut half th at given by

The shear-stress distribution in a single-lap oint is shownin Fig. 9 and th e transverse direct-stress distribution inFig. 10. Finite-element results from a full-length single lapwith no spew are compared with the solution of Golandand Reissner which is the only analytical solution whichreally ap plies to the single-lap case. There is good agreem entbetween the two for the shear stresses (Fig. 9). In Fig. 10the general shapes of the normal-stress distribution curvesare similar, although the tensile-stress concentration at theends of the overlap is predicted to be higher by the finite-element solution. Because of the high stress gradient, theactual stress concentration is difficult to determine. Thecurves of the finite-element solutions taking spew intoaccoun t will be discussed later.The good agreement between the theoretical and thefinite-element results for both the single- and double-lapjoints, and the agreement between the silicone-rubbermodel and its finite-element representation show the

(4).

I4 -3 -2 - 4

4 Volkersen's analysis (7), allowingfor bending.5 Full-length doublelap finiteelem ent model.6 o Points from half-length finite elem ent model

---(/l - / '.:F

- 4 4-J

suitability of this particular finiteelement method foranalysing the stresses in an adhesive-bonded lap joint.4.3 End effectsThe stress pattern at the end of a squareedged adhesivelayer is shown in Fig. 11. This plot was obtained from thetension end of a half-length double-lap joint model. This isagain a plot of principal stresses, the interpretation ofwhich is given in the discussion of the rubber model(section 4.1). The highest tensile stress exists at the come rof the adhesive adjacent to the loaded adherend andrepresents a stress concentration of at least 10 times theapplied shear stress on the joint. It should be noted th at,because constant-stress elements are used and the stressgradients are high, it is impossible t o determine the stressesjust below the surface very accurately without usinginfinitesimally small elements: the stresses acting on thesurface are, of course, zero. The effect of bending of theouter adherends modifies the stress distribution from thatobtained in the simple rubber model, the absolute value ofthe largest principal stress at this end being about fourtimes the absolute value of the largest principal stress at theother side of the adhesive layer (compare points A, and A2in Fig. 6).The effect of the spew on the stress pattern is shown inFig. 12, which is at the tension end of a double-lap joint.The spew is represented by a triangular fillet 0.020 in(0.5 mm ) high. It can be seen that, because of thepredominance of the major principal stress, the adhesive atthe ends of the adhesive layer and in the spew fillet isessentially subjected to a tensile load at about 45" to theaxis of loading. The highest stresses occur within the spewat the corner of the unloaded adherend, the presence of the90" comer introducing a stress-concentrating effect. As th emaximum stress occurs within the spew and no t at or nearthe adhesive surface, it is unlikely that the approximationto the spew shape by the triangular fdet has a significanteffect on the stress distribution.The stress pattern shown in Fig. 12 suggests tha t the area

cJI

Fig. 8. Transverse normal-stress distributions fr om double-lap th eory and finite-e lemen t modelsJOURNAL OF STRAIN ANALYSIS VOL 9 NO 3 1914 OlMechE 1914 191



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R. D. ADAMS AN D N. A. PEPPIATT

12r - i n i t e - e le m e n t m o d e l , s q u o r e e n d--- G o l a n d a n d R ei s s n er s o l u t i o n ( c o l c u l a t e d f o r 4. 5 kN l o a d-- -- F i n i t e - el e m e n t m o d e l w i t h s p e woA v e r o g e a p p l i e d s h e ar s t r e s s'I .

I I 1 I I 1 I I I I I I8 9 10 11 12 1 33 4 5 6 7D i s t a n c e o l o n g o v e r l a p

m m

Fig. 9. Shear-stress dism'butions from finire-elemenrmodels with and without spew, and fromGoland and Ressner (4)

-4L

Fig. 10. nanmerse normal-stressdistribution fio m finite-element models w ithand without spew, and fro m Goland and Reissner (4)

of transfer of load between the adherends is effectivelylengthened. Fig. 13 shows the average shea r-stress distribu -tions across the adhesive layer obtained for joints withvarying sizes of spew. The d istribution s are show n dot tedoutside the overlap length to show an area of load transferwithin the spew. As the loading in the spew is predomi-nantly tensile, a meaningful distribution of shear stress isdifficult to obtain. All curves, except the curve for the0.040 in (1 O mm) spew, were obtained from half-length lapsolutions. The stress distribution for the square-ended casesis shown for comparison. The values for the 0.040 in

(1.0 mm) spew were obtained from the full-length lap data-generation programme. The maxim um shear stress obtainedfrom the largest spew size, which extends com pletely acrossthe ends of the adherends, is 70 per cent of that obtainedfor the squareended adhesive layer. A spew of 0.040 in(1.Omm) height gives a shear-stress reduction of 15 percent. It can be seen that, when the effects of spew are takeninto account, the maximum shear stress obtained isconsiderably less than that predicted by the earlieranalytical theories.Figs 9 and 10compare the distributions of longitudinal192 JOURNAL O F STRAIN ANALYSIS VOL 9 NO 3 1974 OIMechE 1974



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STRESS ANALYSIS OF ADHESIVE-BONDED LAP JOINTS

/ /

Fig. 11. Finite-element prediction of the principal-stresspattern at the end o f a square-edged adhesive layer

n%/.//.I/ / /

Fig. 12. Finite-element prediction o f the principal-stresspattern at the end of an adhesive layer with 5 mm spewshear and of transverse direct stress (i.e. across the adhesivelayer, not principal) within the overlap length for asingle-lap oint having a 0.040 in (1.0 mm) spew height withthe distributions when the adhesive is square-ended. Themaximum shear stress predicted is reduced to 70 per centof that given by the Coland and Reissner theory. For thetensile stress, the analytical prediction lies between the twofinite-element curves at the ends of the lap, but themaximum transverse tensile stress when spew is present isreduced to 80 per cent of that given by Coland andReissner.4.4 Practical aspectsDirection o f cracks in failed jointsIt has been observed that in the spew of aluminium toaluminium joints bonded with low-ductility adhesives,cracks are formed approximately at right-angles to thedirections of the maximum principal stresses predicted bythe finite-element analysis. In general, these cracks runclose to the corners of the adherends. The region wherecracks are formed in the spew is indicated in Fig. 12. Theseobservations give weight to the view presented here thatfailure in a lap joint is initiated by the high tensile stresseswithin the spew. The cohesive failure of the adhesive

occurs in this manner in normal, well-bonded joints. (Inadhesives terminology , adhesive failure means the destru c-tion of the bond between adherend and adhesive; cohesivefailure means a fracture wholly within the adhesivematerial.) If, however, the boundary of the adherend andadhesive is weak, the spew is no t cracked but is pulled awayfrom the loaded adherend surface by the tensile stresses inthe spew.It should be noted th at co mplete removal of the spew bymachining would be difficult without machining either ofthe adherends, so that some spew, similar in size to theglue-line thickness, is likely to be left at the end of anyjoint. Moreover, machining may initiate cracks in theadhesive, especially if it is one of the more brittlehigh-tempers ure adhesives. Thus an adhesive layer with asquare edge is not only undesirable but is unlikely, and evendifficult to obtain in practice.Reduction o f stress concentrationOne of the results of the finite-element analysis is that thehighest principal stresses in the adhesive layer of a lap jointoccur in the region of the co rner of the unloaded adherend.This suggests that it may be possible to increase jointstrength by radiusing the corner of the adherend. A finite-element model with a 0.016 in (0.4 mm) radius wasanalysed and the maximum principal stress in the spew wasshown to be 40 per cent less than that obtained with aright-angle corner.Standard 1 in (25.4mm) by 0.5 n (12.7 mm) lap jointswere prepared with L73 aluminium alloy adh erends, half ofwhich had a 0.016 in (0.4 mm) radius hand-reamed on theadherend comer. The surfaces were prepared by a standardetching procedure and bonded with AF130 adhesive. Thejoints were then tested in tension in accordance withASTM D1002-64. he strengths obtained are given in Table2. No significant increase in strength was obtained, thesmall improvement (less than 3 per cent) with roundedcomers certainly not being of the size predicted by thefinite-element model. The reason for ths is that etching theadherends produces a radius on the adherend comer, evenon the nominally rectangular edges. The finiteelementmodel was therefore modified so that the adherend had asmall chamfer (0.001 in X 0.001 in , 0.025 mm X0.025mm) at its comer. The maximum adhesive stresspredicted was now 30 per cent less than th at obtained whenthe adherend corner was square. Thus, to obtain a morerepresentative picture of the maximum stresses likely to beobtained in a lap joint, the adherends are now alwaystreated as having a small chamfer o n the com er within thespew.Comparisonof doub le- and single-lap resultsThe effec t of greater bending in the single-lap jointappreciably affects the magnitude, but n ot the direction, ofthe principal stresses within the spew. There is an 80 pe rcent increase in the magnitude of the maximum principalstresses in t he single-lap join t over those in the do uble-lapjoint for the same load applied t o th e adhesive layer.Tests on a num ber of single- and double-lap joints madewith AF130 adhesive give the ratio of maximum jointloads, i.e. (failure load of a single-lap joint)/(half-failureload of doub le-lap joint), as 0.52.The finiteelement results for a 0.010 in (0.25mm)bond-line thickness indicate the stress-concentration ratio,

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R. D. ADAMS AND N. A. PEPPIATT

CornerAverage strengthMaximum strengthMinimum strength

u Square edge.. b 0.25mm spew (equals glueline thickness). c0.50 m m spew. d 1 m m spew (full-length solution) e Adherendthickness spew

Round Square(nine joints) (eight joints)0.556 tonf (5.53 kN)0.608 tonf (6.05 kN)0.486 tonf (4.83 kN)

0.540 tonf (5.37 kN)0.575 tonf (5.72 kN )0.490 tonf (4.87 kN)

Distance from centre o f overloemm

Fig. 13. Influence ojspew size on shear-stress distribution at tension end of double-lap joint

Table 2. Strength of round- and square-cornered joint s

i.e. (maximum principal stress in doub le-lapjoint)/(m aximum principal stress in single-lap oint), as 0.54.The stress-concentration values were obtained from theaverage of four adjacent elements in and near the chamferand are based on he increase over the applied average shearstress.The agreement between the results is good, indicatingthat the principal stresses in the spew are the cause of join tfailure.It is unrealistic to compare the earlier analytical theoriesin this way since these are known not to allow for theeffect of spew.

Effect ofglue line thickness on m xi m u m s tressesThe strengths of lap joints of epoxy adhesives do not varysignificantly over the range of adhesive-layer thicknessesused in practice. A plot of lap-joint strength againstglue-line thickness for BSL308 adhesive, as presented byBennett (14), is shown in Fig. 14.The stresses predicted by most stress analyses, however,are very dependent on glue-line thickness. Generally, if allother joint parameters are constant, the maximum stresspredicted is proportional to the reciprocal of the squareroot of the glue-line thickness, provided the thickness issufficiently small. Thus, it would be expected from thesetheories that joint strength increases as the glue-linethickness is increased. I t is for ths reason that Adams et a1(13) and Semerdjiev (15), using a shear-failure criterion,have proposed that joints with profded adherends should beused to give a varying glue-line thickness.Fig. 14 shows the strengths predicted by a number ofanalyses over a range of glue-line thicknesses, based on themaximum strength of the adhesive calculated for the failureload for a joint with an adhesive-layer thickness of 0.005 in

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STRESS ANALYSIS OF ADHESIVE-BONDED LAP JOINTS

4 I I I01 02 0 5 0 4 0 5Glue-lint l h i c k n mm m

1 Demarkles (2) (shear stres s).2 Volkersen (7 ) end-effect solution (tensile stress).3 Coland and Reissner (4) (tensile stress).4 Finitee lemen t single-lap (principal stress).5 BSL 308 adhesive.Fig 14. Influence of glueline thickness on the predictedand actual lap oi nt strengths(0.13 mm). The following analyses are considered.

(1) Demarkles's analysis (2) which gives the maximumshear stress only.(2) An analysis, based on Volkersen's work (7), for themaximum tensile stress caused by end effects.(3) The analysis of Goland and Reissner (4). Thecorrected maximum tensile stress is used here,(4) Finite-e lement analysis of a single-lap joint w ithspew, considering the maximum principal stresses inthe region of the chamfer.The analyses (1)-(3) all predict a significant increase in

strength by increasing the glue-line thicknesses over therange shown. The least increase is given by Demarkles'sshear-stress analysis. On the other hand, the fmiteelementanalysis is reasonably close to the curve shown for theBSL308 adhesive, althoug h the analysis still predicts a slightincrease in strength as the glue-line thickness is increased.The decrease in strength of the actual adhesive joint,however, as glue-line thickness is increased, may be causedby factors which cannot easily be taken into account byany predictive technique.Niranjan (16) suggests that there are three separatefactors, dependent on glue-line thickness, which influencethe strength of a lap join t:stress concentration;the porosity of the glue line;the strain rate.The results of the finite-element method suggest that thestrain-rate factor is not important because the maximumstress, and hence the maximum strain rate for a givencrosshead speed, does not vary significantly with glue-linethickness. It is thus concluded that th e decrease in strengthof actual joints, as the glueline thickness is increased, iscaused by the increase in porosity and the number ofmicrocracks in the adhesive.

5 CONCLUSIONSThe stresses in adhesivebonded lap joints of standardtestpiece size have been exam ined two-dimensionally. Planes t r a i n has been a p m e d and constant-strain triangular finiteelements have been used.

It has been shown that there is reasonable agreementbetween the stress distributions obtained for an adhesive-bonded lap joint by the finiteelement method and classicalanalytical techniques, the best agreement being obtainedwith those theories making the most realistic assumptions.The finite-element method has been used to give shear,transverse normal, and principal-stress distributions in b othsingle- and double-lap oints. Most of the classical analyticaltechniques do not allow for end effects and none canpredict the stresses in the adhesive spew.With the assum ption that the joint m aterials are linearlyelastic, good agreement with practical results has beenobtained. The maximum principal stresses at the ends ofthe adhesive layer are predicted to be at right-angles to thedirection of cracks formed in the spew of failed lap joints.The etching process in the preparation of aluminiumadherends leaves a small radius at the adherend comerswhich relieves part of the stress concentration. The finite-element method shows that if the adherend is assumed tohave a small chamfer on its comer due to the etchingprocess, good agreement between the strength ratios ahdthe predicted maxim umstress ratios of single- anddouble-lap joints is obtained. The influence of glue-linethickness on the maximum stress within the joint is morerealistically predicted than it was by earlier theories.The classical, closed-form analytical results for adhesive-bonded lap joints are limited by the assumptions made inorder that a solution may be obtained. Moreover, theeffects of spew or etching cannot be included because ofthe complexity of the geometry. However, it is relativelyeasy to consider these effects by using a fmiteelementtechnique and it has been shown that it is most importantto include them if realistic stress distributions are to beobtained, particularly in the highly stressed areas at theends of the joint.

APPENDIX 1The expression given by Goland and Reissner (4) for thetensile stresses across the adhesive layer isa. = @ - [ ( R 2'AI h 2 t + Xk'coshXcosX C

I

whereas the corrected expression, given in similar form bySneddon (S), ish x x-M' oshXcosh C1

Similarly, the expression for the maximum transversetensile stress must be corrected to giver

- p t 2 1X z 5sinh2h-sin2X)2 (sinh2X 4- sin2X)ao)max -7 L 11cosh2X +C O D A )-Ak' (sinh2h + in2X)JOURNAL O F STRAIN ANALYSIS VOL NO 3 1914 QIMechE 1974 195



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R. D. ADAMS AND N. A. PEPPIATTAPPENDIX 2REFERENCES

(1) VOLKERSEN, 0. Die N ietkraftve rteilu ng in Zugbeans-pruchten Nietverbindungen mit Konstanten Laschenquer-schnitten, Luftfarhtforschung 1938 15, 41-7.(2) DEMARKLES, L. R. Investigation of the use of a rubberanalog in the study of stress distribution in riveted andcemented joints, Tech Notes natn. advis Comm Aeronaut.,Wash 3413 1955.(3) ADAMS, R D. an d PEPPIAIT, N. A. Eff ect of Poissons ratiostrains in adherends on stresses of an idealized lap joint, J .Strain Analysis 1973 8,134-9.(4) GOLAND, M. and REISSNER , E. Stresses in cem ented joints,J appl Mech, nuns A m SOC mech Engrs 1944 66,(5) SNEDDON, 1. The distribu tion of stress in adhesive joints,Adhesives (ed. Eley, D.) 1961 chapter 9 (Oxford University

press).(6) KUENZI, E. W. and STEVENS, C. H. Determination of themechanical properties of adhesives for use in the design ofbonded joints, U.S. Forest Products Laborato ry Report

A17-A27.

FPL-011, 963.

(7) VOLKERSEN, 0. Recherches sur la thiorie des assemblagescolles, Construction mktalliaue 1965 (No. 4). 3-13.(8) COKER, E. C. A n optical determination of the v ariation ofstress in a thin rectangula r plate sub jected to shear, h c .Soc Series A 1912 86,291-319.(9) INGLIS , C. E. Stress distribution in a re ctangular plate havingtwo opposing edges sheared in opposite directions, h c .Soc Series A 1923 103,598-610.(10) MYLONAS. C. Experim ents on comp osite models withapplication to cemented joints, h c . SOCexpl Stress Analysis

(11) WOOLEY, G. R and CARVER, D. R St ress concentrationfactors for bonded lap joints, J . Aircr. 1971 8,817-20.(12) HARRISON, N. L. and HARRISON, W. J. The tresses in anadhesive layer, J. Adhesion 1972 3, 195-212.(13) ADAMS, R D., CHAMBERS, S. It, DEL STROTHER, P. J.A., and PEPPIATT, N. A. Rubber model for adhesive lapjoints,J. Strain Analysis 1973 8,52-7.(14) BENNETT, W. F. on-d estructive testing of adhesivebonding, P aper presented at the 9th Annual Confe rence onNon-Destructive Testing, Lo ughborou gh University, 1972.(15) SEMERDJIEV, S . Metal to metal adhesive bonding, 1970(Business Books, London).(16) NIRANJAN, V. Bonded joints - review for engineers, 1970UTIAS Review No. 28 (University of T oronto).

1954 12,129-42.

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