Bonded Joints in Composite Laminates

Peak adhesive shear and peel in a single lap joint, the safety factor as a load factor, and the adherend check at the overlap end.

A single lap joint bonds two plates over an overlap length L. The load path through the joint is offset by one plate thickness plus the bondline, so the joint rotates under load and bends the plates at the ends of the overlap². That bending puts a moment into each plate and raises two adhesive stresses at the overlap ends: the shear stress τ and the peel stress σ, a through-thickness tension. The middle of the overlap carries almost nothing.

The Bonded Joint tool gives the peak shear and peel in the adhesive, the bending moment at each overlap end, and a first-ply failure check of each laminate plate. Each check has a safety factor on the applied load. The overlap is two beams on adhesive springs, solved exactly for any pair of plates⁷. The end moments come from a closed form for dissimilar, coupled plates with long free arms⁸, which reduces to Goland-Reissner's bending moment factor when the two plates are identical^2,4. The results update after each edit.

Everything on this page is in SI-mm: stresses and moduli in MPa, lengths in mm, the load N in N/mm and moments in N·mm/mm. The tool shows whichever units you have selected.

When to use this tool

Use it to size a single lap bond between two plates. Each plate is a laminate or a metal, and the two can differ: a laminate bonded to aluminium, two different layups, or one layup turned over. A double lap, a scarf or a stepped lap is outside its scope. For a joint with fasteners, use Bolted Joints.

Inputs

A quasi-isotropic laminate bonded with a paste adhesive. Peel at the overlap ends governs.

• Plates: Plate 1 is on top and Plate 2 below it. Each is Composite or Metal. A composite plate takes the laminate card above it or a saved laminate, and its thickness comes from that laminate. A metal plate takes a metal preset and a thickness. With Plate 2 same as Plate 1 switched on, Plate 2 copies Plate 1. Switch it off to set Plate 2 on its own.
• Rotation and Flip: a laminate only. The load runs along x. Rotation turns every ply by the angle you enter, counter-clockwise, so 0° plies at a rotation of 90° run across the joint. Flip reverses the ply order, so the other face touches the adhesive. With neither, Plate 1 bonds through its last ply and Plate 2 through ply 1. The results name the bonded ply of each laminate.
• Temperature change ΔT: the service temperature minus the cure temperature, negative after cooldown. A cure at 180 °C with service at 25 °C gives ΔT = -155 °C. Enter 0 to leave out thermal residual stress. A laminate needs the ply CTEs when ΔT is not 0. A metal preset carries its own.
• Adhesive: a preset fills the shear modulus G_a, the tensile modulus E_a and Poisson's ratio ν_a (bulk resin, dry, room temperature). A film preset also fills the bondline thickness t_a. For a paste, enter t_a yourself. If you change a filled value, the preset changes to Custom. The adhesive moduli drop with temperature and moisture, so use the values for your design condition.
• Allowables: the shear allowable τ_allow and the peel allowable σ_allow. Each is optional. A blank allowable shows its check as N/A.
• Geometry and load: the overlap length L, the width b, and the load. Load input picks how the load reaches the overlap (see below). The width only sets the total load P = N·b. The stresses depend on N.

Load input: free arms or edge loads

Free arms is the classic single lap. You enter the tension N per unit width, and the tool assumes each plate runs on as a long free arm beyond the overlap. The load line through the arms is offset from the overlap, so the tool finds the moment at each overlap end from that offset, as the joint rotates under load. Free arms takes tension only: enter N above 0.

Edge loads is for a joint whose end loads you already know, for example from a finite element model of the structure around it. You enter three values at the overlap ends:

• N, the axial load per unit width. Tension is positive.
• M_a, the moment in Plate 1 at its loaded end (x = 0), about the Plate 1 mid-plane.
• M_b, the moment in Plate 2 at its loaded end (x = L), about the Plate 2 mid-plane.

The moments follow the sign convention of the laminate analysis pages, with z positive upward: a positive moment puts the plate's top face in tension. The transverse shear Q in the overlap is not an input. Moment balance of the overlap sets it from N, M_a and M_b, and the tool shows it next to the inputs.

A sketch in the Load section draws the overlap with the loads you entered: N and M_a on Plate 1 at the left end, N and M_b on Plate 2 at the right end, and Q dashed. Each arrow points the way its sign says, so a wrong sign shows before you read a result. A value of 0 or a blank field draws no arrow. The sketch is not to scale, and Q appears after the first result.

Edge loads takes compression, and N = 0 with an end moment. Under compression the tool shows a warning, because it does not check the plates for buckling. Check the plates beyond the overlap for buckling yourself. In this mode the overlap responds in proportion to the loads, so the safety factor scales N, M_a and M_b together. With ΔT = 0 it equals the stress ratio.

Position x and the free arms

The Resultants and Stresses tabs show one position x, measured from the Plate 1 end of the overlap. In Free arms mode x also runs one overlap length into each arm: from -L to 0 on Plate 1's arm and from L to 2L on Plate 2's. Only one plate exists on an arm, so the Plate choice locks to it and there is no adhesive there. The arm values come from the same beam-column solution that gives the end moments, so the moment on an arm runs continuously into the overlap end. They are for display only: the safety factor checks stay on the overlap.

Edge loads mode models the overlap only. The arm positions are disabled, and the Shape tab draws the overlap pinned at x = 0. That plot shows how the overlap bends, not how the whole joint rotates.

Results

The header gives the safety factor, the governing location (an overlap end of the adhesive, or a plate at an overlap end) and the governing check. With every allowable blank there is no safety factor, and the header says so. The tabs below it:

• Margins: the ply table of one laminate plate at its loaded overlap end, under its N_x and M_x there with ΔT included. A Plate picker sets which plate. The SF column here is the strength ratio at those loads.
• Checks: one row each for adhesive shear, adhesive peel and each laminate plate. The columns are the peak at N, the peak from ΔT alone, the allowable, the stress ratio and the safety factor. A metal plate gets a row with no check, and a blank allowable skips its check.
• Resultants: one plate's N_x, M_x and Q at position x, in the joint axes, and a chart of N_x and M_x of both plates along the joint. Click the chart to set x. Use as analysis loads sends a laminate plate's loads at x to the laminate analysis pages (see The adherend check).
• Stresses: the adhesive shear τ and peel σ along the overlap with the allowables dashed, and the stress or strain through both plates and the adhesive at x. With ΔT not 0, dashed curves give the adhesive stress from ΔT alone.
• Shape: each plate drawn as a band that follows the deflection w of its mid-plane, over its unloaded outline. The Exaggeration slider sets the factor on w. In Free arms mode the joint's rigid rotation is removed, so w = 0 at both overlap ends. When ΔT is not 0, w includes cure warp.

The Resultants tab. The shaded bands on the chart are one overlap length of each free arm, where only that plate exists.

The Stresses tab. A click near the Plate 1 end sets x, and the plot below shows the stress through both plates and the adhesive there.

The Shape tab. The joint's rigid rotation is removed, so w = 0 at both overlap ends.

Which allowables to enter

The adhesive in this analysis is linear elastic, so the check compares a linear-elastic peak stress with the allowable. Enter a linear-limit or design allowable for the adhesive at your design condition.

Do not enter the lap-shear strength from a datasheet. That value is the failure load of a test coupon divided by its bonded area: an average over the test overlap, not a peak stress. The tool compares the peak at the overlap end with the allowable, and that peak is always higher than the average⁴. The two numbers do not measure the same thing. Datasheets usually give peel as a force per width (a peel test), not as a stress, so a peel allowable also has to come from your own data.

The safety factor is a load factor

The bending moment factor k falls as the load rises: the joint rotates further and the load path straightens. For identical plates the overlap-end moment is M₀ = k·N·t/2, so it does not grow in proportion to N, and neither do the peak stresses. Dissimilar plates behave the same way^2,8. The tool's safety factor is the factor n on the applied load N at which the peak reaches the allowable, with ΔT held fixed. It finds n by solving the joint again at n·N until the check reaches its limit.

The checks table also gives the stress ratio: the allowable divided by the peak at N. For a linear structure the two are the same number. Here the end moment and the fixed thermal part make them differ, and the safety factor is the one to size on. An SF of 2.00 means the joint carries twice N before that check fails. The header shows the lowest safety factor of the checks, and where it is.

The adherend check

Each laminate plate gets its own first-ply failure check. At every position along the overlap the plate carries its own N_x and M_x, plus its thermal residual stress at ΔT. The tool checks every ply there with the failure criterion you select and keeps the worst position, so a peak inside the overlap counts if it beats the end. The check names the plate, the overlap end and the ply that governs, and its safety factor is again a load factor. Each laminate enters the joint through its own membrane, coupling and bending stiffness^3,6, so a laminate with its 0° plies on the outside bends less than the same plies with 90° outside.

A metal plate gets no first-ply check, and the failure criterion does not apply to it. On the Resultants tab, Use as analysis loads copies a laminate's N_x, M_x and Q_x at position x, with ΔT, to the laminate analysis pages, so you can look at the plies there in more detail. It converts them to the laminate's own axes, undoing the rotation and the flip.

Thermal residual stress

Two plates with different thermal expansion, bonded at the cure temperature, would shrink by different amounts on cooldown. The adhesive stops them. That leaves shear and peel at the overlap ends with no load applied⁹. Aluminium bonded to carbon-epoxy is the common case: about 23 × 10^-6/°C against 2 to 3 × 10^-6/°C along the fibres. Two equal symmetric laminates, or two plates of one metal, carry no thermal stress.

The tool adds the thermal state to the mechanical one. The checks table gives each peak at the applied load and at ΔT alone. The safety factor scales the applied load only and holds ΔT fixed, because the cure stress does not grow with the load. If ΔT alone already exceeds an allowable, that check shows a safety factor of 0 and says so.

Unsymmetric laminates and twist

An unsymmetric laminate has bending-extension coupling (B ≠ 0): a membrane load bends it and a moment stretches it. The tool includes that coupling in the overlap and in the end moment⁸, so an unsymmetric plate needs no special treatment.

A laminate with A₁₆, B₁₆ or D₁₆ ≠ 0 also twists when it is pulled or bent. The tool treats each plate as a narrow strip that is free to twist. A wide joint, or one clamped against twisting, carries extra stress that the tool does not compute. The sketch shows a note on each plate this applies to. A quasi-isotropic laminate gets one, because a narrow strip of it twists under bending.

What the tool does not check

• Shear and peel together. Each adhesive stress is checked on its own, with no interaction criterion.
• Interlaminar tension in the adherend. The peel stress loads the first ply through its thickness, and laminate theory has no through-thickness stress.
• Adhesive plasticity. A ductile adhesive yields at the overlap ends and spreads the load. The tool does not model that.
• Cure curvature in the end moment. A joint that curls on cooldown moves its arms off the load line, and the tool leaves that out. On a quasi-isotropic laminate bonded to 2024-T3 after ΔT = -155 °C, two models with finite arms put the peak peel at three to six times the tool's value. Treat the peel of a cured laminate-to-metal joint as unconservative.
• Stresses across the width and at the side edges. Each plate is a narrow strip, so nothing varies across the joint¹⁰.
• Adhesive thermal expansion and moisture.
• Double lap, scarf and stepped lap joints.

A green margin here does not mean the joint is fine.

References

1. Volkersen, O. (1938). Die Nietkraftverteilung in zugbeanspruchten Nietverbindungen mit konstanten Laschenquerschnitten. Luftfahrtforschung, 15, 41-47.
2. Goland, M. & Reissner, E. (1944). The stresses in cemented joints. Journal of Applied Mechanics, 11, A17-A27.
3. Hart-Smith, L.J. (1973). Adhesive-bonded single-lap joints. NASA CR-112236, Douglas Aircraft Company. §2, Eq. (1).
4. Zhu, Y. & Kedward, K. (2005). Methods of analysis and failure predictions for adhesively bonded joints of uniform and variable bondline thickness. DOT/FAA/AR-05/12, Federal Aviation Administration. Eq. 3-5 and Tables 3-1, 3-2.
5. Tsai, M.Y., Oplinger, D.W. & Morton, J. (1998). Improved theoretical solutions for adhesive lap joints. International Journal of Solids and Structures, 35, 1163-1185.
6. Smeltzer, S.S. & Klang, E.C. (2003). Single-lap bonded joints with laminated composite adherends. Proceedings of the American Society for Composites 18th Technical Conference, paper 104. Eqs. 19-20.
7. Mortensen, F. & Thomsen, O.T. (2002). Analysis of adhesive bonded joints: a unified approach. Composites Science and Technology, 62, 1011-1031.
8. Talmon l'Armée, A., Stein, N. & Becker, W. (2016). Bending moment calculation for single lap joints with composite laminate adherends including bending-extensional coupling. International Journal of Adhesion and Adhesives, 66, 41-52.
9. Chen, W.T. & Nelson, C.W. (1979). Thermal stress in bonded joints. IBM Journal of Research and Development, 23(2), 179-188.
10. Adams, R.D. & Peppiatt, N.A. (1974). Stress analysis of adhesive-bonded lap joints. Journal of Strain Analysis, 9(3), 185-196.

Frequently Asked Questions