Comparison Between Experimental And Numerical Analysis Of A Double-lap .

Comparison between experimental and numerical analysis of a double-lap jointISAT - --&I -- I--same adherend stresses as does the second theory of Goland and Reissner, so it has the same bounds ofvalidity. The bending in the double-lapjoint does not cause rotations of the overlap region, and so theadhesive stress per unit load is not dependent on the load applied. Thus, the applied load is explicitlyfactorable from the solution functions for the shear and normal stresses in the adhesive layer.2.4. Recent theoriesIn more recent analysis of the single-lapjoint the end effects are taken into account [ 5 ] . Because theend face of the adhesive is a free surface, there is no shear stress on it. Another hypothesis is that thereare no transverse strains at the d o a d d ends. Tinis can be clarified by a simple explanation. if theadherends are in uniform tension up to the joint, there should be a uniform contraction both ofthickness and of width because of Poisson's ratio. However, there is no load in the other adherend atthis end section and so there can be no transverse contraction. This effect, called lateral straining, isshown in Figure 5.Figure 5: Deformation in single-lapjoint due to lateral strairThe analysis of Adams and Peppiatt (1973) 163 took the end effects and lateral straining into account.However, the effects of bending have been neglected and therefore, their results are more applicable todouble- rather than single-lapjoints. The tearing and peeling stresses and the longitudmal normalstresses in the adhesive were also ignored. But they did apply the adherend shears.They found a set of equations which has no closed analytical solution (Equations 4 ancl 5):- Kaal, - KbOl, caaxz- a2cT1zK a q , - K,q, C,az2From these equations an approximate analytical solution, which is exact at the boundaries, can beobtained.

Comparison between experimental and numerical analysis of a double-lap jointWith the following boundary conditions:atx O3o,,--P andUSAT -sw-*r)-w anlrmrpar CT,, Ob61atx Io,, -3Pand o,, Ob62b3 o,, O and o,, Oatz f2and assuming that olXis independent of z and Kbo,, and Kbolxare small compared withKaqXmcl &qZ,so riegligible, the sohtioti5 ofthese eqcztiens cm- be expressed as:(1- y(1- c0shaI))sinh COColX l-v(l-coshW)balSinhal'[p a (1 - y(1- cosh al))cosh OX-l/ISinhair7,, -[bSinhal73, 6,6,a(v,o,,E, - v,o,,E,) sinh (xzmere:a &Equations 6 and 7 are exact at the boundaries z & b/2 and Equations 8 and 9 are exact at theboundaries x Q, and x I.The Equations 4 and 5 were also solved by a fuite-difference method, and the results were comparedwith the approximate analytical solution. These two solutions for the adherend tensile stress and theadhesive shear stress are shown in Figure 6. A good correlation between these two methods can beremarked.14I2inXo---Approximate analytical solution (exact at z f6/2).Finite-difference solution at joint centre-line (z O).Approximate analytical solution neglecting adherendshears.Figure 6: Tensile stress in adherend and shear stress inadhesive against x for linxlin single-lap joint [5]6

Comparison between experimental and numerical analysis of a double-lapjointUSAT -sup(Nuw --oendnTrmupnn2.5. Double-lap jointIn a symmetrical double-lap joint, different of the single-lagjoint case, the centre adherend has nobending moment. However, the outer adherends bend, giving rise to tensile stresses across the adhesivelayer at the end of the overlap where they are not loaded, and compressive stresses at the end wherethey are loaded. Figure 7 gives a diagrammatically representation of this effect.Doublr lap, loadcd almost Io l u l u r eAdhesivr ïransvrrselensionnormal sirrisI Figure 7: Bending moments induced in the outer adherends of a double-lap joint 131In faei, as mentimed befoïe, the double-lapjoint cm be considered as i? back-to-back arrangement oftwo single-lapjoints. Therefore, one may conclude-that the stresses in the double-lap joint are of thesame shape as the ones in the single-lapjoint. In other words: the shear stresses, oxr, in the adhesiveare of the same shape as the graph of and the normal stresses, o,, in the adherends are of the shapeof 9,shown in Figure 6 . As seen above, the peel stresses, ow,in the adhesive should have the sameshape as given in the upper-left plot of Figure 7.So, the analytical formulations described above can be used to compare with the results calculated bythe finite element method, so as to test the validity of the finite element grogram.

ISAT Comparison between experimental and numerical analysis of a double-lap joint sw-w*--bYcodnInnipail3. Finite element calculationsThe implementation in different Finite Element Programs has to be generalized for a part for one singleproblem. This generalization is necessary to be able to compare the results of the different programs.First the genera! implementationwill be discussed, followed by the explanation of the different FiniteElement Programs.3.í. Geometry and maieriallSo far, it is assumed that the adhesive layer ends in a square edge. However, instead of a square edge,real structural adhesivejoints are formed with a fillet of adhesive spew (see Figure 8), which issqueezed out under pressure while the joint is manufactured. Therefore, an adhesive layer with a squareedge Lnniikely to be realistic. . Figure 8: Diagrammatic lap joints to show adhesive layers with (a) square edge; (b) spew filletIn addition, under a shear load, the transverse tensile and compressive stresses become much higher inthe adhesive layer of a square edge, while the shear stress on the free surface must be zero. These hightransverse stresses penalize the joint's resistance. But in reality, the spew fillet improves well the stressstate.]In theirearly work, Adam and Peppiatt [7] have investigatedthe influence of a spew fillet on theaverage shear stresses in the adhesive using finite-element analyses. They calculated the average shearstress distributions along the adhesive layer obtained for joints with varying sizes of spew. This isshown in Figure 9. The highest stresses occur within the spew at the corner of the unloaded adherend.parently, the presence of the 90' corner introduces a stress-concentrationeffect. 60r \II aSOUARE EDGE-*.-o0 2 5 m m SPEW IEOUALS GLUE.LINE THICKNESS),-0- c O 50m SPEW-a-d-A-- I O m m SPEWFULL DEPTH SPEW'//',ootO0kJIIIIIO203040I Obi. Figure 9: Influence of spew size on shear-stress distribution at tension end of double-lapjoint [7]8

Comparison between experimental and numerical analysis of a double-lap jointLZATSFor above mentioned reasons, an implementation of a spew fillet with a 45' angle and a spew width thatequals two times the adhesive layer thickness has been chosen.For the linea& of the Finite Element Method, it is only necessary to calculate a single force in a singledirection, while other forces in that direction can be found by extrapolation.Also, results w i t h anelement can be found by interpolation, because of its linearity. But, the best results will be obtained bymesh-refinement at important places and at places with higher stresses. In case of the double-lap jointthat is at the overlap-ends, the adhesive layer and in our case at the places where the gauges areattached. The gauge positions and geometry are given InFigure 10.i'.'iII'3.7,I'4.5, ,3.6I'6.81.,A9I',l.3.85.2I'88411I',112.5I'Figure IO: Position (in mm) and name ofgauges on double-lapjoint [i]The caracteristics of the materials, used for the experiments, were found experimentally [i]. The innaerand outer adherends consist of the same steel with the following caracteristics:E, 207700 MPa and V, 0.29.The adhesive layer consists of a polymer with the name Eponoi 3 17 and has the caracteristics:Ej 5800 MPa and vJ 0.33.3.2. Finite Element Programs32.1. RDMThe program called "RDM Calcul des Structures par Elements Finis" is a finte element program for thecalculation of 2-dimensional, linear, elastic, mechanical andor thermal problems [SI. It is able tocalculate problems of thermo-elastic planes, axisymmetry or flexion of plates. Thus in our case, thelateral straining of the lap-joint is neglected. The program calculates the displacements ofthe nodes anduses these to determine the stresses at the Gauss-points. The stresses at the nodes me related to theseGauss-points via extrapolation and are different for every element to which the node is attached.Therefore, the stresses at the nodes are averages ofthe values of the connected elements.Rectmgulcis, q adratic,isoparmetric dements hiwe beeE chosen, became they give i? betterapproximation than triangular dements. Cdculations with higher dements take more time and do notgive a significantly better approximation.9

Comparison between experimental and numerical analysis of a double-lap jointUSAT -.w-*r-- a-.-m-In fact, the stress distribution in a double-lapjoint is a 3-dimensional problem. So, by assuming planestrain, E, O, or plane stress, 6, O, for the 2-dimensional case, an error will inevitably beintroduced. Calculations with both plane strain and plane stress have been executed for the distinctionbetween the two methods. The relative difference turned out to be less than 4% (see Figure 11). So, nosignificantly distinction between the two methods exists. In future, all calculations were performed withplane strain.Strain in v-direction for a double-lap joint: *.* exp; - RDMplane strain; -- RDM plane stress515Figure 11:Influence of plane strain or plane stress on results (WDM)The program only calculates nodal displacements, average stresses and reaction forces. However, weare interested in the strain in x and y direction, E, and that have been mesured in the experiments.The strains can easily be derived from the average stresses with the Equations 10 or 1 1 in case of planestrakE, -{(il vEEY -{(il v- v)crXx- v},.-v)aw -va,}EBecause of the use of average stresses, the strains calculated in this way are average values as well. Another problem that crossed our path, in using the RDM-program, was that the program did not haveenough memory to create a mesh-refinement at all eleven gauges at the same time. Therefore, for everygauge, a different mesh had to be created with refinements at their specific positions on the joint and attheir overlap ends. So, eleven calculationshad been made to obtain results of a s a t i s m g precision. Forevery calculation approximately 1150 elements and a total of 2350 degrees of freedom were used. Pergauge, the implementation, after knowing the program, took about an hour and the calculation time on aPersonal Computer (486 DX, 66 M p l z ) about 3 minutesTo know what kind of a mesh has to be used and to detect the influence of different meshes on theresinkts, calculations were made for one specific gmge wi?h different kind dmeshes. The chosen gaugewas gauge J15, because ofthe largest difference between experimental values and Calculations. Fourdifferent meshes were designed, from very small elements at the overlap ends and adhesive layer torelatively large ones. laii the four used designs had over 1000 elements. Figure 12 shows that there is nosignificant influence on the results for the chosen meshes.10

ISAT Comparison between experimental and numerical analysis of a double-lap joint-351100I150200I250I300I350I400,450I500550n-w- r -mcra -m600Force (daN)Figure 12: Infiuence of different meshes on results (RDM)The double-lap joint is symmetrical in a horizontal and a vertical line. Therefore, only a quarter ofthetotal joint was implemented.Thetypical mesh and the boundary conditions used for the calculations inthe RDM Finite Element Program are given in the Figures 13 and 14.IFigure 13: Finite-element mesh of a double-lap joint for RDMrI-1Figure 64: Lines of symmetry, boundary conditions and force of a double-lap joint in RDM1%

IISATaComparison between experimental and numerical analysis of a double-lap joint--&r)-- t-T wAs mentioned earlier in this report, the results have to be validated before comparing and concluding.Therefore, we look at the shape of the graphs containingthe normal stress in the inner adherend and theshear and peel stresses in the adhesive layer, plotted against the x-axis. These plots are sequentiallygiven in the Figures 15, 16 and 17.Normal Otms in inner adherend [RDMJShsar 8 1 m s in adhesive layer (RDM). . . . . .;.,. .,.i. i.;. .;. 1. :.- 5 .;. . . ;.;.;,.,.,.,.,,.*.;. .-. I. ;. ;.,.I:-4.0-1-2-3,.A.,I,I.,I,. . . .I,.I**,I,I0I.t .c . .*.c .I. . 2L.i. . :. .:.L :.,IO'. ,.;.:. 1.,IOII20II,.,. Li I*.II3040,,I,,50.,,,I.1I-2.L.L.II,, ;':II,.2 2IOa 304050,IIEU.,.2. . .:.:.,;.i . I. i.i.j,.i.i . .I.-. . . . 8 0 8 8O',.II60m. .,IB O MI coordinals along the meriap [rnrn]Figure 15: Normal stress in inner adherend O M )IFigure 16: Shear stress in adhesive layer (RDM)Peel slisss in sdhssivs layer RUM)- 4 ioa,3040I coordinale along the orsdspmmBO[mm]IFigure 17: Peel stress in adhesive layer (RDM)The shape of the graphs are as they ought to be. Figure 15 and Figure 16 have the same shapes as theplots given in Figure 6. The peel stress has tension at the overlap-end and compression at the loadedend, as also given in the upper-left plot of Figure 7. Therefore, it seems that the results obtained by theN I M program are useful.3.2.2. MEF-MosaicThe "MEF-Mosaic" Finite Element Program is also developped for linear, elastic problems 191, but itsbig advantage is the possibility to calculate in three dimensions. So, with this program the idhence ofthe lateral straining can be taken into account. Also, the validity of the assumption of plane strain in the2-dimensional case can be verified. Where the strains in the direction at right angles of the plane have tobe small in relation to the other two directions. Like the RDM Program, MEF-Mosaic calculatesthedisplacements at the nodes and the stresses at the Gauss-points, too. The stress at a node is extrapolatedfrom the Gauss-points of the concerning element and is different for every element to which the node isattached. To obtain a single value per node, the average of the results of the connected elements hasbeen taken.

Comparison between experimental and numerical analysis of a double-lap jointLZAT&In this program, also rectangular, quadratic, isoparametric elements were used and but an eighth of thetotal joint was implemented The program has enough memory to calculate all gauges at the same time.The number of elements used for the calculation was 9960 with a total of 39375 degrees of freedom.After optimisation the rigidity matrix had a band yidth of 178. The implementationtime, after knowingthe program, concerned several days and the time to calculate, on a SPARC-10 machine, 2 hours and 28minutes (real, user and system time). The mesh and boundary conditions are given in Figure 18. A detailof the overlap end is shorn in Figure 19.N2TITRE:Baudaty Conditions and lolces0L.X)INTS1%- 1' B.0.DSOCIETEFRAMASOFT CSIMOSAIC - Fij ire 18: The boundary conditions and forces on the double-lap joint in MEF-MosaicP.V2 7 5 Because of its 3-dimensional implementation, the right side ofthe joint is oppressed in direction y iocomplete the boundary conditions. aso, for comparing, attention has to be paid to the fact that theorigin and direction of the axes in the MEF-Mosaic and RDM program are not the same. CompareFigure 18 to Figure 14. The z direction in MEF-Mosaic corresponds to the y direction in the RDMprogram.

ISAT Comparison between experimental and numerical analysis of a double-lapjoint-Sugkhm-* *sT-wTo validate the program, the normal stress in the inner adherend and the shear and peel stresses in theadhesive layer are plotted against the x-axis in the Figures 20,21 and 22. The distributions of thesestresses present the same tendency as the analytical ones (see Figure 6 and upper-left plot of Figure 7).So, the results obtained by using the MEF-Mosaic program seem valuable.Shear SWESS in odhss've layer (MEF-Mosaic)Norms1 slress in inner adherend (MEF-Mosaic)40IO2030405060I cooidinale slongthe wedap [mm]7080O'BBI2030.105060708 0 8 8I COOrdinaIs along ths oredap [mm]IFigure 20: Normal stress in inner adherend (MEFMosaic)Figure 21: Shear stress in adhesive layer (MEFMosaic)Figure 22: Peel stress in adhesive layer (MEFMosaic)3.2.3. Joint DesignJoint Design is a special purpose finite element program developed on a low cost microcomputer for theanalysis and design of adhesivesjoints [lO,l i]. It has been completely developed, planned and adaptedfor them with the following major features:i) the use of mixed (stress-displacement) interface finite elements (the mixed interface elementspermit the introduction of the component ofthe stress vector as a degree of freedom);i;) significant decrease ofthe computationaltime obtained by adequate simplifications andimprovements in numerical data storage and processing;iii) reanalysis under slightly modified conditions (loads, boundary conditions, materialproperties);iv) interactive user friendly package making possible the use ofthe program by designers withlittle or no howledge ofthe internal structure and methods.

USAT Comparison between experimental and numerical analysis of a double-lap joint- saphnrdrr mom dr.lrmpoinThe program uses a four-node linear triangular mixed interface finite element (TRL-l,[12]) developedfor plane elasticity problems, based on the Reissner's variational principle [13-151. This finite elementgives the opportunity to introduce the stress vector component as a degree of freedom in the elementformulation. Their use is limited to the interface and the zones near the interface. They are connected tothe linear triangular and rectangular displacement finite elements for the others zones of the joint.The formulation of the element is based on the uses of the relocalizationprocess to reduce a Reissner'sprinciple based finite element to an interface element [141. The excess variables are moved from thenodes at the corners and the middle of the interface.side to the inside ofthe element, so that they are notincluded

Comparison between experimental and numerical analysis of a double-lap joint ISAT rm.mn5uphmxd.l*u onioe&*I - Summary Experimental results on a double-lap joint have been compared with results of several numerical methods. A good correlation between the numerical and experimental values was found for positions not near to the overlap ends.