Through-thickness shape optimisation of bonded repairs and lap-joints

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Abstract

For realistic applications, the design of bonded repairs and lap-joints, has often been undertaken through trial and error finite element analyses, or experiment. Recent experience indicates that for more complex practical applications, unacceptably high adhesive stresses can occur in the adhesive layer. In the present work, an automated sensitivity-based shape optimisation procedure has been developed for the optimal design of free-form bonded repairs and lap-joints, with the aim of achieving reduced adhesive stresses. The approach has been demonstrated through application to a number of single and double-sided configurations where both the shapes of the adhesive layer and the outer adherend are allowed to vary. Significant improvements over conventional designs are obtained, as assessed by the reduction in peak adhesive stresses. These results indicate that the numerical shape optimisation procedures presented can provide designs that offer substantial improvements over standard designs.

Introduction

Over approximately the last two decades, the use of high modulus bonded doublers has been demonstrated as an effective procedure for the life extension of cracked and uncracked metallic airframe components [1]. These repairs can be used either alone or in combination with shape reworking. The bonded repairs function by transferring some load from the component, through the adhesive layer into the repair, thereby reducing the magnitude of cyclic stresses in the repaired component. The thickness of the doubler (repair) determines how much load is drawn away from the damaged region and also the level of stress in the adhesive. Typically the doubler is a laminate consisting of unidirectional layers of boron fibre embedded in a matrix of epoxy resin. This allows for thinner doublers due to the high fibre modulus and it is better suited to inspection by NDI methods than laminates consisting of other fibres (i.e. carbon). Heat cured epoxy adhesives and refined methods of surface preparation have generally provided high strength and durable bonding. When applied to primary structural components, these repairs are typically used as a measure to prevent crack initiation or retard crack growth. It is generally required that the component has adequate static strength with or without the repair.

To date, most applications of bonded repairs have been to thin sections such as cracked airframe skins (i.e. a plate). Here, the stress analysis has been usually based on an analogy with a one-dimensional lap-joint analysis, where 100% of the load is carried by the doublers [2], [3]. A key quantity of interest is the adhesive stress concentrations at the extremities of load transfer regions [4], [5]. These occur at (i) the end of the bonded repair, and (ii) directly over the crack. Often yielding of the adhesive can occur at these locations, and for extreme cases, this can lead to premature adhesive failure depending on the specific service loading history. For more complex repair applications, involving curved thick sections, and/or involving significant bending, a particular feature is the existence of both high shear and peel stresses. This can potentially seriously compromise the integrity of the adhesive layer, as there will always be a trade off between achieving a suitable repair effectiveness (i.e. load transfer), and restricting adhesive stresses to be less than an acceptable limit [6], [7]. It should also be noted that practical applications to date have essentially used a constant adhesive thickness as well as a constant repair thickness (except for tapering at the ends of the repair). Typical designs have not been optimal, although various strategies have been used to reduce the magnitude of the stress concentration. These include linear tapering at the end of the joint, and/or increasing the local adhesive thickness at the end of the joint [8]. Generally, no closed form theoretical solutions are available for the design of more complex repairs, hence trial and error FE analyses or experiments have been used to obtain a suitable design.

Published work on the optimal design of bonded repairs/lap-joints to reduce adhesive stress is very limited. However some investigations of specific scope have been undertaken. For example an analytical treatment of the optimal tapering at the ends of an isotropic doubler for a uniaxialy loaded lap-joint is given by Ojalvo [9]. In other work the same problem is considered by using a 2D finite element gradientless optimisation method, (one thick section repair is also considered) in Refs. [10], [11]. In the work of Groth and Nordland [12] design sensitivity methods are used to optimise essentially the same configuration. In all four references above, the analyses are confined to the consideration of doubler tapering and do not consider variation in adhesive thickness. It should be noted that concurrent with the present investigation the authors have also recently completed a study for the through-thickness optimal design of a bonded repair, as a life extension option for the FS 470 bulkhead of the F/A-18 airframe [13]. Here sensitivity based finite element optimisation methods have been used and both the repair and the adhesive layer are allowed to vary in shape.

In the present work, an automated sensitivity-based shape optimisation procedure has been developed for the optimal design of free-form bonded repairs, with the aim of achieving reduced adhesive stresses. The approach has been demonstrated through application to a number of single and double-sided configurations. The analyses undertaken are completely relevant to the design of lap-joints as well. Particular features of the present approach include: (i) free form shapes, where the outer adherend and/or the adhesive thicknesses are allowed to be non-uniform, and are optimised, (ii) a ‘least squares’ objective function is used to undertake the optimisation, subject to specified constraints, and (iii) multiple shape-basis vectors from the analysis of an auxiliary model are used to specify allowable shape changes. Four general loading and geometric cases were considered as follows; where in each case the inner adherend is aluminium and fixed in geometry:

(i) In case 1 the geometry representative of a symmetric (i.e. double sided) crack repair is modelled. Here the crack opening is prevented from increasing during the optimisation process; hence the repair effectiveness (as compared to a nominal repair) is maintained as a constant while the adhesive stresses are minimised. In this model, aluminium doublers are bonded to an equivalent stiffness aluminium plate.

(ii) Case 2 is the same as case 1 except that the doublers have been modelled using homogeneous orthotropic boron/epoxy properties.

(iii) Case 3 is similar to case 2 except, the boron/epoxy doubler is bonded to one side of the plate only. Hence due to the resulting offset of the neutral axis, the joint experiences significant local bending. This is a challenging case since very high adhesive stresses can be envisaged.

(iv) In case 4, the geometry is representative of a double-sided lap-joint rather than a crack repair where there is no restriction on joint opening. In this model aluminium doublers are bonded to an equivalent stiffness aluminium plate. Here while the adhesive thickness is allowed to vary in the optimisation process, the outer surfaces of the doublers are constrained to remain flat.

Detailed definition of the configurations follows in later sections with corresponding figures. In all cases, the objective has been to minimise the peak von Mises stress by making the adhesive stress distribution more uniform. This is achieved by defining an objective function as the sum of the squares of the stress deviations from the average value, over a selected region. While the work has been presented as a numerical study without supporting experimental data it has been shown experimentally in [8] that reducing adhesive stress concentrations leads to greater joint strength.

Section snippets

Lap-joint loading and initial configuration

For all analyses the initial configuration under study was a typical bonded double lap-joint as defined in Fig. 1. This configuration is perhaps more correctly termed a double strap joint but has been referred to as a double lap-joint in much of the literature. The geometry is also representative of the two dimensional idealisation of a bonded repair to a cracked plate [2], [3]. In this case the two outer adherends represent the doublers, while the two inner adherends represent the two sections

Symmetric crack repair with aluminium doublers: case 1

In case 1 the geometry representative of a double-sided crack repair was modelled, where both the doublers (outer adherends) and the plate (inner adherend) are aluminium alloy (Fig. 4(a)). The aim was to optimise the repair shape to minimise peak von Mises adhesive stresses, such that the repair effectiveness was unchanged as compared to a nominal geometry.

Symmetric crack repair with boron/epoxy doublers: case 2

This configuration is the same as for case 1 except for the modelling of boron/epoxy doublers instead of the aluminium alloy doublers (Fig. 6(a)). Again the aim is to optimise the repair shape to minimise peak adhesive stresses, such that the repair effectiveness is unchanged as compared to a nominal geometry. For the analysis of the initial geometry the finite element mesh was identical to that given for the aluminium doubler case, except the doubler thickness has been reduced everywhere by a

Non-symmetric crack repair with a boron/epoxy doubler: case 3

In this case we modelled the configuration where the doubler is bonded to one side of the plate only (Fig. 8(a)). Application of the remote stress will result in secondary bending, and hence a more severe adhesive stress distribution, (particularly for peel) will result. Again the aim is to optimise the repair shape to minimise peak adhesive stresses, such that the repair effectiveness is unchanged as compared to the nominal geometry.

Initial optimal solution

In this case it has been defined to represent a joint rather than a crack repair. The problem is the same as case 1, except for the following changes; (i) the outer doubler surface is kept flat by constraining design variables 10–16 to zero, (ii) the constraint on joint opening is removed (Fig. 10(a)). As before, basis shapes with zero centreline slope were used, and the initial geometry and material are the same as in case 1. The resultant optimal shape obtained after 5 iterations is given in

Conclusions

In the present work, an automated sensitivity-based shape optimisation procedure has been developed for the optimal design of free-form bonded repairs and lap-joints, with the aim of achieving reduced adhesive stresses. The approach has been demonstrated through application to a number of single and double-sided configurations where both the shapes of the adhesive layer and the outer adherand are allowed to vary. Significant improvements over conventional designs are obtained, as assessed by

References (17)

  • R.J. Chester et al.

    Adhesively bonded repairs to primary aircraft structure

    Int J Adhesion Adhesives

    (1999)
  • H.L. Groth et al.

    Shape optimisation of bonded joints

    Int J Adhesion Adhesives

    (1991)
  • Baker AA, Jones R, editors. Bonded repair of aircraft structures. Dordrecht: Martinus Nijhoff,...
  • L.F.R. Rose

    Theoretical analysis of crack patching

  • Hart-smith LJ. Adhesive bonded double lap joints. NASA CR-112235,...
  • Tran-Cong T, Heller M. Reduction in adhesive shear strains at the ends of bonded reinforcements. DSTO-RR-0115, Defence...
  • L.P. Shah et al.

    Reduction of plate stress concentration factors due to bonded reinforcements

    Int Aerosp Conf

    (1997)
  • Heller M, Kaye R, Whitehead S, Lubac J. Design and stress analysis. In: Chester R, editior. Life extension of F/A-18...
There are more references available in the full text version of this article.

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