H. Kim and K. Kedward1Stress Analysis of Adhesive Bonded Joints Under In-Plane Shear LoadingHyonny Kim* and Keith T. KedwardDepartment of Mechanical and Environmental EngineeringUniversity of California, Santa Barbara, CA 93106, USAA closed-form stress analysis of an adhesive bonded lap joint subjected to spatially varying in-plane shear loading is presented. The solution, while similar to Volkersen’s treatment of tensionloaded lap joints, is inherently two-dimensional, and in general predicts a multi-componentadhesive shear stress state. A finite difference numerical solution of the derived governingdifferential equation is used to verify the accuracy of the closed-form solution for a joint of semi-infinite geometry. The stress analysis of a finite sized doubler is also presented. This analysispredicts the adhesive stresses at the doubler boundaries, and can be performed independentlyfrom the complex stress state that would exist due to a patched crack or hole located within theinterior of the doubler. The analytical treatment of lap joints under combined tension and shearloading is now simplified since superposition principles allow the stress states predicted byseparate shear and tension cases to be added together. Applications and joint geometries arediscussed.Keywords: shear-load, bonded joint, doubler, composite adherend, crack patch,closed-form analysisNomenclaturex, y, zRectangular coordinatesr, θ, sCylindrical and shell coordinates2cOverlap length of adhesive jointaWidth of joint over which applied loading varies, or length of doubler inx-directionbLength of doubler in y-directionti, toThickness of inner, outer adherend * corresponding authorAccepted by Jo. Adhesion on 10/2000 for publication in 2001.H. Kim and K. Kedward2taThickness of adhesive layeriyE, oyEYoung’s modulus of inner, outer adherend in the y-directionixyG, oxyG Shear modulus of inner, outer adherendGaShear modulus of adhesive layerNx, NyApplied direct stress resultants (force per unit width)NxyApplied shear stress resultant (force per unit width)iyσ, oyσDirect stress in inner, outer adherend in the y-directionixyτ, oxyτShear stress in inner, outer adherendixyγ, oxyγShear strain in inner, outer adherendaxzτ, ayzτAdhesive shear stress components acting in x-z, y-z planeaxzγ, ayzγAdhesive shear strain components acting in x-z, y-z planeui, uoDisplacement of inner, outer adherend in x-directionvi, voDisplacement of inner, outer adherend in y-direction1. IntroductionAdhesive bonding has been applied successfully in many technologies. Foremost in applicationswhere primary loaded structures rely on adhesive bonding are aircraft and space structures.While bonding in large and small commercial aircraft has been practiced quite widely in Europe(sailplanes in Germany, SAAB 340 [1], and EXTRA EA-400 [2]), extensive adhesive bonding isbeing used in the United States for the assembly of newly emerging small all-composite aircraftstructures (Cirrus SR20 and Lancair Columbia 300) for reasons related to performance and cost.The analytical treatment of a bonded lap joint where the adherends are loaded in tension (seeFigure 1) has been considered extensively by many authors. Hart-Smith [3, 4] extended the shearlag theory that was presented by Volkersen [5] to include adhesive plasticity. Goland andReissner [6] and Oplinger [7] accounted for adherend bending deflections to predict the peelstress in the adhesive. Tsai, Oplinger, and Morton [8] provided a correction for adherend sheardeformation, resulting in a simple modification of the Volkersen’s theory based equations. All ofH. Kim and K. Kedward3these analytical treatments are formulated per unit width of the specimen which implies that thepredicted adhesive stress is independent of variations of loading through the width of the joint (x-direction in Figure 1). An extension of these solutions can be applied to the case of spatiallyvarying tensile loading, as shown in Figure 1, by performing the analysis using the value oftensile stress resultant at any particular x-axis location. The tension loaded lap joint analysis ispresented in Appendix A.Adhesively bonded lap geometries loaded by in-plane shear (see Figure 2) have been discussedby Hart-Smith [4], van Rijn [2], and the Engineering Sciences Data Unit [9]. The authors ofthese works indicate that shear loading can be analytically accounted for by simply replacing theadherend Young’s moduli in the tensile loaded lap joint solution with the respective adherendshear moduli. This approach is rigorously correct for only the case of spatially constant Nxy loadapplied to joints which are semi-infinite, as shown in Figure 2a.When the geometry is finite in size (see Figure 2b), an analytical treatment that is morecomprehensive than that suggested by Hart-Smith [4], van Rijn [2], and the EngineeringSciences Data Unit [9] is needed in order to account for adhesive stresses which would exist atthe bond terminations. Thus the objective of the theoretical work presented herein is to addressthe shear loaded lap joint problem in more general terms by allowing the applied in-plane shearload to vary in the spatial coordinates, and by accounting for a joint geometry of finite size.Closed-form analytical solutions are developed for single and double lap joint configurationssubject to general in-plane shear loading. It should be noted that the solution of this problemusing Finite Element Analysis (FEA) is difficult due to the inherent three-dimensional nature ofthe joint geometry and shear loading conditions. Since three-dimensional elements need to beH. Kim and K. Kedward4used in modeling shear transfer across a lap joint, creating a mesh having enough elementrefinement to capture the high stress gradients in the thin adhesive layer can easily result in aFEA model of formidable size. In comparison, a closed-form solution to this problem serves as acomputationally efficient tool that is useful for design and analysis.2. Structural ExamplesExamples of structures in which shear loading can challenge a bonded lap joint are shown inFigures 3 to 7. Figure 3 shows a generic section of an adhesive bonded fuselage assembly typicalof small aircraft. The fuselage halves are joined at the fuselage centerline, typically through ajoggled lap joint, as shown in the figure inset. Other joint configurations might include the use ofa splice strap. Shear loads are transmitted by this joint any time torsion is carried by the