Influence of geometrical parameters on the strength of Hybrid CFRP-aluminium tubular adhesive joints

Tubular adhesive joints, used in truss structures to join pultruded carbon fibre-reinforced polymer members to aluminium nodes, are modelled with varying dimensions. The numerical model uses a Cohesive Zone Modelling formulation with a trapezoidal traction-separation law for the adhesive layer, and experimental tests are carried to validate it. The results showed that the joint strength increases significantly with the bonding area, with a limit on the overlap length above which it stops increasing. This upper limit is affected by the thickness and tapering angle of the adherends, due to their influence on the shear stress distribution along the overlap. On the other hand, the adhesive thickness has only a marginal influence on the joint strength.


Introduction
Composite materials are increasingly used in transportation industries, due to their low weight and high mechanical properties. Truss structures, which consist of straight members loaded axially and connected together by nodes, are a suitable application for the high axial strength of unidirectional composite materials, such as carbon fibrereinforced polymer (CFRP) pultrusions, and several applications of CFRP used in truss structures are being developed in the form of wound trusses and lattices [1][2][3]. A limitation to such structures is the load transfer between the members, therefore designing the nodes is an important step in designing the overall structure. The efficiency of a node is defined as the ratio of the load it can transfer to the load the members can carry. Mechanical fastening, such as riveting, provides a high efficiency but damages composite materials. In addition, it is difficult to apply to small-scale structures without significantly increasing their total weight. A solution is to adhesively bond the ends of the members to the nodes, as it can provide strengths comparable to or higher than mechanical fastening [4][5][6]. In a previous work, the authors simulated the strength of different node configurations and concluded that tubular joints, consisting of aluminium sockets overlapping the truss members, provided the best strength-to-weight ratio [7].
The dimensions of adhesive joints play a crucial role in their strength. Increasing the bonding area, i.e. the overlap length and width, greatly contributes to increasing the joint strength [8][9][10][11][12]. To a lower extent, an increase of the joint strength can be obtained by increasing the adherends thickness, which causes a higher bending stiffness [9,13]. Tapering the ends of the adherends also results in an increase of the joint strength by making the stress distribution in the adhesive more uniform [14], although this effect is marginal on tubular joints [15,16]. Additionally, tubular joints present a good robustness to manufacturing defects, such as bondline contamination or misalignment of the tubes relative to each other [17,18]. Increasing the adhesive bondline thickness can cause a reduction of the joint strength, although no generalized trend has been observed due to the numerous factors that can influence it [19,20]. The joint strength also depends on the type of adhesive used. Brittle adhesives generally have a higher strength than ductile adhesives, but their sensitivity to peak stresses limits this advantage to short overlaps, for which the stress distributions are more uniform [21,22]. Ductile adhesives plasticize when their peak strength is reached, and thereby redistribute the stresses in the less loaded parts of the overlap, which results in a higher performance than brittle adhesives for longer overlaps [23,24]. Knowing the influence of the geometrical parameters on the strength and weight of a joint is necessary to design and optimize it, since strength and weight objectives in this case are often of conflicting nature [15]. In particular, it is important to determine the design limits of each parameter, outside which no further strength or weight improvement is possible.
In order to determine these limits, a model able to predict the strength of the adhesive joint is required. Analytical models exist to analyse the stress distribution and predict the failure of adhesive joints, but they are not reliable for the modelling of adhesive displaying a highly non-linear behaviour, in which case numerical models are preferred [25,26]. Finite Element Modelling (FEM) has been used extensively to analyse adhesive bonds and predict their strength, using a variety of criteria to determine the failure of the joint. The criteria to be used depend on the geometry of the joint (lap, scarf, etc.), and on the type of adhesive used, with local stress-strain criteria being generally more suitable for brittle adhesives and global yielding criteria giving more accurate results for ductile adhesives [24,[27][28][29]. With the increase in computational power, more advanced modelling techniques are now being used, such as fracture mechanics and damage modelling, which comprise Cohesive Zone Modelling (CZM) and Extended Finite Element Method (XFEM) [30,31,32]. In the case of ductile adhesives, fracture mechanics are not suitable because the initial crack size and position is difficult to determine, and would require a large amount of computing time [29]. CZM combines the advantages of fracture and continuum mechanics to model the damage initiation and propagation through the adhesive layer until failure, and has been successfully used in the literature to predict the strength of adhesive joints. Even though XFEM shows satisfactory results compared to standard FEM, it has shown limited and less accurate results in the strength prediction of adhesive joints compared to CZM [20,21,33]. The choice of the traction-separation law to be used for the CZM depends on the adhesive. A triangular law, which models the stiffness degradation of the material immediately after damage initiation, shows more accurate results for brittle adhesives, while a trapezoidal law, by representing the plasticization of the material, is more suited to ductile adhesives [22].
The objective of the present work is to study the influence of several geometrical parameters on the strength of CFRP-aluminium tubular joints, in order to determine the limits of each parameter in improving the joint strength and weight and assess their relevance for a future optimization process. The parameters considered are the overlap length (L o ), adhesive thickness (t adh ), inner tube diameter (d in ), inner and outer tubes thickness (t in and t out , respectively), and inner and outer tubes taper lengths (l in and l out , respectively), as their influence on the joint strength and weight is expected to be significant. Considering the range of overlap length values, a ductile adhesive (Araldite 2015-1) is chosen, since a brittle adhesive is expected to result in lower strengths in most cases. A numerical model using CZM with a trapezoidal traction-separation law is built to predict the joint strength for varying parameter values. Tubular joints specimens are manufactured and tested in tension until failure for a selection of geometrical parameters, and validation of the numerical model is made by comparing the failure load predictions with those obtained experimentally. In addition, Computed Tomography (CT) scan images of a selection of specimens are made, in order to assess the importance of manufacturing defects on the joint strength and adapt the numerical model if necessary.

Material properties
The inner tubes of the joints are unidirectional pultruded tubes composed of TORAYCA® T700 fibres and epoxy resin. The outer tubes are made of aluminium 6086-T6. The properties of the CFRP and aluminium, to be implemented in the numerical model, were calculated from the manufacturers' data. They are disclosed in Tables 1 and 2, respectively.
In order for the numerical models to predict the behaviour of the adhesive joint accurately using CZM, the elastic moduli, failure strengths and fracture toughness of the adhesive, in tension and shear, are needed. The adhesive used in this work is the Araldite® 2015-1, a two-component (resin and hardener) ductile epoxy adhesive. It is a new version of the discontinued Araldite® 2015. As reported by the manufacturer (Huntsman International LLC), it shows slightly better performances than its predecessor did. The properties of the Araldite® 2015-1 to be used for the CZM are not expected to vary significantly from those of the Araldite® 2015, which has previously been characterized in the literature [34]. However, Campilho et al. [22] have shown that variations of the adhesive failure strength have a substantial impact on the failure load of the joint predicted by the CZM (up to 20% for a strength variation of 20%), while variations of the fracture toughness only have a marginal impact (up to 5% for a toughness variation of 20%). In order to increase the reliability of the model predictions, the elastic moduli and failure strengths of the Araldite® 2015-1, in tension and shear, are determined experimentally in this work, while its fracture toughness in tension and shear are considered identical to those previously determined for the Araldite® 2015.
Bulk adhesive specimens were tested to determine the tensile properties and Thick Adherend Shear Tests (TAST) were carried out to determine the shear properties. The bulk adhesive specimens were manufactured according to the ISO 15166-1 standard. The adhesive was cured at room temperature for five days between two plates coated with a release agent, with 2 mm thick spacers ensuring a constant adhesive thickness. The resulting adhesive plates were then machined to achieve the dogbone shape specified in the standard, as shown in Fig. 1a. Seven specimens were tested and the tensile properties of the adhesive were determined according to the ISO 527-2 standard. The TAST specimens were manufactured according to the ISO 11003-2:201 standard. Two steel plates were degreased with acetone, manually abraded using sandpaper and bonded together, with spacers guaranteeing an overlap thickness of 0.5 mm. After curing at room temperature for five days, the bonded assembly was machined to obtain specimens as specified in the standard, as shown in Fig. 1b. Instead of using an extensometer as suggested in the standard, Digital Image Correlation (DIC) was used to measure the displacement of the adherends relative to each other at the location of the overlap (Fig. 2). Seven specimens were tested with four yielding valid results, the remaining three having failed prematurely due to a misalignment in the testing apparatus. Fig. 3 shows the experimental stress-strain curves obtained from the bulk adhesive and TAST specimens. The properties of the Araldite® 2015-1 are summarized in Table 3, with the yield strengths in tension and shear calculated for a respective plastic strain of 0.2%.

Numerical model
The tubular joints are modelled for different values of overlap length (L o ), adhesive thickness (t adh ), inner tube diameter (d in ), inner and outer tubes thickness (t in and t out , respectively), and inner and outer tubes taper lengths (l in and l out , respectively). These parameters, shown in Fig. 4, are independent (the value of each parameter can be varied  without affecting the others). In order to represent the specimens used to validate the model as closely as possible, the centring hole and drill tips are also modelled (details about the purpose of the centring hole can be found in Section 2.4). The numerical model was built and analysed using the Finite Element software Abaqus (Dassault Systèmes). Given the tubular nature of the joints studied, an axisymmetric model was created, as it yields the same results as would a 3D model but with a lower computing time [12]. The adherends were modelled with 4-node axisymmetric quadrilateral elements (CAX4 in Abaqus), using the material properties presented in Tables 1 and 2. CZM can be applied to model the adhesive layer following two different approaches: the "continuum" or the "local" approach. In the continuum approach, the entire adhesive layer is modelled as a single row of cohesive elements acting as an interface of finite thickness between the two adherends. In the local approach, the adhesive is modelled using solid elements, while zero-thickness cohesive elements are inserted at the adhesive-adherends interfaces and within the adhesive layer to simulate different failure paths. For more details on the two approaches, the readers are referred to Da Silva & Campilho [29]. The continuum approach is used to predict the adhesive bond failure on a macroscopic level and requires a low computational cost, while the local approach allows for a more detailed analysis of the bond failure, for a higher computational cost. Since the objective of this work is limited to predicting the failure load of the tubular joints, the continuum approach was chosen. The adhesive layer was therefore modelled as a single row of 4-node axisymmetric cohesive elements (COHAX4 in Abaqus). A trapezoidal traction-separation law, detailed in Section 2.3, was implemented to model the damage initiation and propagation in the adhesive layer. A convergence study was carried out to determine suitable mesh sizes along the different dimensions, with bias effects applied to create smaller sized elements near the locations of stress concentrations, as shown in Fig. 5. The minimum size for the elements was set to 0.05 mm, and the maximum size to 1 mm. The boundary conditions shown in Fig. 4 were applied to the model to reproduce the testing conditions.

CZM implementation
Using the continuum approach, CZM simulates the elastic loading of an adhesive layer followed by local damage initiation and propagation until failure. It is formulated by a traction-separation law, which relates the stresses t and displacements δ between the paired nodes of a cohesive element, in tension and shear. The exact shape of a tractionseparation law can be determined experimentally, but requires an extensive testing campaign [35,36]. Approximate laws, such as the triangular and trapezoidal laws shown in Fig. 6, have been shown to give satisfactory results [22,34]. Therefore, the authors decided that using an approximate law was suitable for the purpose of this work. The choice of a shape to be used for the CZM depends on the adhesive type. Triangular and exponential laws, which model the stiffness degradation of the material immediately after damage initiation, show more accurate results for brittle adhesives, while a trapezoidal law, by representing the plasticization of the material, is more suited to ductile adhesives [22]. In this work, a trapezoidal law is used to represent the behaviour of the Araldite® 2015-1, using the material properties presented in Table 3 (with t 0 in tension and shear taken as equal to σ f and τ f , respectively).
The initial linear elastic behaviour of the adhesive is defined by Eq. (1), where n and s represent tension and shear, respectively. K n and K s are calculated as the ratios between the elastic stiffness of the adhesive in tension and shear, respectively, and the characteristic length of the adhesive layer, its thickness t adh [37]. The elastic loading of the adhesive is coupled between tension and shear, and damage initiation is considered when the quadratic nominal stress criterion, shown in Eq. (2), is fulfilled. The brackets 〈〉 indicate that the value is made equal to zero if negative, which prevents compressive stresses from initiating damage.
After damage initiation, the adhesive undergoes progressive degradation, characterized by the increase of a damage variable d defined Damage progresses until failure at δ f , whose value in tension and shear is calculated by considering the area under the respective curve to be equal to its respective fracture toughness G c . In the case of the trapezoidal law, a stress softening displacement δ s is introduced. After damage initiation, the adhesive maintains its peak strength despite the increase of d, and the softening up to failure starts at δ s . A simple predefined ratio of 0.8 was used between δ s and δ f , since the exact shape of the trapezoidal law has been shown to have a marginal impact on the joint strength [38,39]. It should be noted that Abaqus only provides the implementation of coupled degradation for the triangular and exponential laws. Consequently, when using a trapezoidal law, the tensile and shear behaviours of the cohesive elements are independent from damage initiation to failure. The values of the damage variable d with respect to the separation δ, in the constant stress and softening regions, were calculated from Eq. (4).
Since K n and K s depend on the value of t adh , new damage variable values must be calculated and implemented every time t adh is modified in the model, which increases the total computational cost. Due to the CFRP tubes having a relatively small diameter, it is difficult to grip them in the tensile machine without crushing them. Each specimen therefore consisted of one CFRP tube bonded at both ends to aluminium tubes enlarged at their extremities to be properly set in the tensile machine, as seen on Fig. 7. The specimens were fabricated via the following steps: (1) the adherends were degreased with acetone and manually abraded using sandpaper, (2) the CFRP tubes were inserted in the aluminium tubes and (3) the adhesive was injected in the overlap through an injection hole. A constant adhesive thickness and centring of the CFRP tube in the aluminium tube was ensured by a centring hole at the bottom of the overlap, drilled at a diameter equal to the CFRP tube's diameter. To ensure a proper curing of the adhesive, the joined assembly was left for one week at room temperature. Testing of the specimens was carried out until failure in a tensile testing machine equipped with a load cell of 20 kN, with a controlled displacement rate of 1 mm/min and at room temperature.

Specimens manufacturing and testing
During the tensile tests, the machine measured the crosshead displacement, which comprises the deformation of the specimen as well as the displacement induced by the deformation of the testing apparatus. The latter was measured by testing, in the same conditions, a short specimen made of steel (for a high stiffness resulting in negligible specimen deformations). The deformation of each joint specimen was then calculated by subtracting the deformation of the apparatus from the crosshead displacement. Furthermore, the deformation of the aluminium between the grip and the section analysed by the FE model (see Fig. 5) is assumed negligible.

Joint strength prediction
All specimens were tested in tension failure, which was considered when the applied load reached a peak (P f ). In the sets of specimens with L o = 15 mm and 20 mm, a high standard deviation between the failure loads is observed (550 N and 380 N from the average failure load, respectively). CT scan images of the specimens with L o = 15 mm were taken after failure occurred (with the CFRP tubes still bonded to the aluminium tubes), in order to observe the adhesive layer and to try to   understand the cause for this high failure load deviation. Fig. 8 shows the three-dimensional images of each failed joint, reconstructed from the CT scan images, along with two-dimensional images of the corresponding midsections. The 3D images have been set to show only the material with the lowest density (air), so that air bubbles in the adhesive can be easily visualized. Thus, the grey column visible in each image represents the air inside the CFRP tube, and the clearer "masses" show the air trapped within the adhesive layer. The 2D images depict the adhesive layer in white, the adherends in shades of grey and the air in black. Fig. 9 shows the CFRP tubes after they were completely pulled out from the aluminium tubes. The surface of a non-bonded CFRP tube is provided as a reference, to be compared with the surfaces of the pulled-out tubes. Fig. 10 shows the respective failure loads of the specimens shown in Figs. 8 and 9.
The pulled-out CFRP tubes shown in Fig. 9 reveal that the specimens with the lowest failure loads (specimens 4 and 5) are still entirely covered by the adhesive layer, suggesting that the failure did not happen within the adhesive but at the interface between the adhesive and the aluminium. Conversely, the specimens with the highest failure loads (specimens 1 and 3) show adhesive only at the loaded end of the overlap, suggesting a cohesive failure originating from this location. It can be concluded that one cause for the high deviation in failure loads is an improper pre-treatment of the aluminium adherends. The bottom of the aluminium tubes was difficult to reach, which might have led to an insufficient abrasion in this area, with a higher risk of improper abrasion the higher the overlap length. It explains why the highest deviations were observed for the specimen sets with the highest overlap lengths. On specimen 2, an important amount of adhesive is still present, and the surface of the CFRP tube looks like it was not properly bonded, suggesting an improper pre-treatment of the CFRP tube at this location. For future similar tests, the use of a more homogeneous pretreatment technique on the aluminium adherends, such as chemical etching, is advised.
Another possible cause for the high deviation is a misalignment of the adherends. The 2D images in Fig. 8 show that, despite the specimens being assembled in a fixture designed to keep the tubes centred, the CFRP tubes are off-centre close to the open end of the overlap, resulting in an uneven adhesive thickness in this area. This misalignment being present in all of the five specimens scanned, it is difficult to gauge its impact on the failure load of the joints. Although previous studies have shown the robustness of tubular joints to manufacturing defects [17,18], the authors advise the use of a different centring method, such as inserting nylon threads as spacers between the adherends. Similarly, air bubbles of various sizes can be observed in the 3D images of the   specimens with no apparent relationship between the size of the bubbles and the failure loads of the specimens. The 2D images in Fig. 8 also reveal that, during the injection of the adhesive in the overlap, some of it went in the centring hole between the CFRP and the aluminium. It might have caused the joint strength obtained experimentally to be higher than the one predicted by the numerical models, due to the extra 2 mm of overlap it created. In order to verify this observation, a "corrected" numerical model was created by replacing the centring hole with an extra 2 mm length of adhesive. The comparison of the experimental failure loads with the predictions made by the original and corrected models are presented on Fig. 11. The original model under-predicts the failure loads by 19.9% on average, against 4.2% for the corrected model, which validates the use of the corrected model to predict the strength of tubular joints. In the following sections, the corrected values of L o (adding 2 mm of adhesive length) and the results from the corrected model are used. Fig. 12 reports the load-displacement curves obtained experimentally for the reference specimens, along with the FEM simulation from the corresponding model (displacement values are doubled for the numerical results, since the model represents one half of a specimen). The good agreement observed between the experimental and numerical curves validates the assumptions made in Section 2.4 about the measured displacements. The displacements applied in the simulation when the joint reaches its peak load and when it starts softening are labelled d p and d s , respectively. To explain the shape of the load-displacement curve given by the numerical simulation, the damage variable d and the shear stress τ xy (normalized by τ f ) along the adhesive overlap are plotted for different values of applied displacement around d p and d s (Fig. 13). It shows that damage initiates at the end of the overlap when the adhesive reaches its peak stress at this location (see the 42% d p curves on Fig. 13). As the applied displacement increases, the shear stresses are redistributed to the undamaged parts of the overlap and the load applied on the joint increases. The load reaches a peak when the  entire overlap has initiated damage (97% d p and 103% d p curves) and the stresses cannot be redistributed anymore. The applied load stays at this peak value when the applied displacement increases further, and starts dropping when the adhesive starts softening (102% d s curves), i.e. when d = 0.981 (calculated using the equations defined in Section 2.3).

Damage analysis
The distinct separation values δ 0 and δ s of the trapezoidal law result in a load-displacement curve with similarly distinct displacements d p and d s when the peak load is reached and when the load starts dropping. In contrast, the real traction-separation law for an adhesive is nonlinear, i.e. it does not feature such distinct points delimitating linear zones, as shown in Fig. 6. It explains why the load-displacement curves obtained experimentally have a continuous shape instead of the trapezoidal one obtained numerically. This is a limitation inherent to the approximation of the traction-separation law, but given the good accuracy of the strength predictions obtained numerically, this approximation is judged acceptable for the purpose of this work.

Overlap length and inner tube diameter
As observed in Fig. 11, the overlap length (L o ) and inner tube diameter (d in ) are the parameters with the highest influence on the joint strength. The corrected model was used to predict the strength of joints with several combinations of L o and d in , and the results are presented in Fig. 14. The results show that a limit value of L o exists after which P f does not increase further, and that this limit slightly increases with d in .
On the other hand, no limit value of d in was observed. Fig. 15 shows the load-displacement curves obtained numerically for d in = 3 mm and different values of L o .
The part of the curve where the joint maintains its peak load (the flat part) becomes shorter as L o increases, and grows longer again after the limit value of L o is reached. In order to explain this behaviour, a damage analysis similar to the one made in Section 3.1.2 is made for the model with L o = 62 mm (Fig. 16). Similarly to the joint with L o = 12 mm, after damage has initiated at the end of the overlap (see the 8% d p curves on Fig. 16), the stresses are redistributed and the load applied to the joint keeps increasing. However, when the adhesive starts failing at the end of the overlap and the peak stress zone stops growing longer (89% d p and 101% d p curves), the load reaches P f , even though the rest of the overlap has not initiated damage yet. The load starts dropping when damage has been initiated in the entire overlap (99% d s and 101% d s curves) and the peak stress zone becomes shorter due to the adhesive gradually failing. In summary, before a certain limit value of L o (around 47 mm in this case) there is a proportional increase of P f with L o , due to the applied load reaching P f only when the entire overlap has initiated damage. Above this limit value of L o , the applied load reaches P f when the peak stress zone cannot get any longer and the adhesive starts failing at one end of the overlap. The length of the peak stress zone being independent from L o , further increasing L o does not cause P f to increase anymore. Fig. 11c shows that the adhesive thickness (t adh ) has no significant   Therefore considering the adhesive thickness as an optimization variable in the future is judged unnecessary, and the choice of a constant value is preferred.

Adhesive and adherends thickness
Variations of the inner and outer tubes thickness (t in and t out , respectively) have an influence on P f for high overlap lengths. Fig. 17 shows the failure loads obtained numerically for different values of L o , t in and t out . Reducing t out results in an increase of P f for higher overlap lengths, while reducing t in results in a decrease of P f . This is due to the influence of the adherends geometry on the shear stress distribution along the adhesive overlap (see Fig. 18 for the case of L o = 62 mm). Reducing t out contributes to reducing the stiffness imbalance between the adherends, causing a better shear stress and damage distribution. A different distribution has no influence on P f for short overlap lengths, since in that case P f is reached when the entire overlap has initiated damage. However, for higher overlap lengths, P f is reached when one end of the overlap (the one with the highest peak stresses) starts failing, regardless of the amount of damage in the rest of the overlap. Lower shear stresses at this location cause the load required for the adhesive to fail to be higher, which results in a higher P f plateau (as seen in Fig. 17). Conversely, reducing t in contributes to increasing the stiffness imbalance, which results in a lower P f plateau. Additionally, there are lower limits on t out and t in , under which the applied load causes the adherend to fail before the adhesive layer.

Taper length
As with the adherends thickness, the inner and outer tubes taper lengths (l in and l out , respectively) have an influence on P f for high overlap lengths. Fig. 19 shows the failure loads obtained numerically for different values of l in and l out , for L o = 7 and 62 mm. To ensure that the tapering of an adherend does not reduce its thickness (given by t in or t out ), its length cannot exceed L o . An outer tube thickness of t out = 3 mm was used for the results shown in Fig. 19. The results for L o = 7 mm show no influence of the tapering on P f . For L o = 62 mm, high values of l out result in a higher P f , while high values of l in result in a lower P f . As explained in Section 3.2.2, the variation of the stiffness imbalance between the adherends, here caused by the tapering, results in a variation of the stress distribution (shown in Fig. 20 for L o = 62 mm). Strength improvements are obtained when the taper length of the stiffest adherend is increased, as shown with the increase in P f obtained with a high l out . On the other hand, tapering the least stiff adherend further increases the stiffness imbalance, resulting in a decrease of P f .

Conclusions
In this work, tubular joints consisting of CFRP tubes adhesively   bonded inside aluminium tubes were modelled for different values of overlap length (L o ), adhesive thickness (t adh ), inner tube diameter (d in ), inner and outer tubes thickness (t in and t out , respectively), and inner and outer tubes taper lengths (l in and l out , respectively), to study the influence of these parameters on the joint strength and weight. The numerical model used a CZM formulation with a trapezoidal traction-separation law for the adhesive layer, and experimental tests were carried to validate it. CT scan images of failed specimens showed that manufacturing defects, such as air bubbles in the adhesive layer or a misalignment of the adherends, do not have an apparent influence on the joint strength. However, it highlighted cases of improper pre-treatment of the adherends, leading to the premature failure of some specimens. This issue was attributed to the pre-treatment technique used (mechanical abrasion) having a higher risk of being non-uniform for longer overlaps, and the authors recommend the use of a more homogeneous technique such as chemical etching. Nevertheless, the failure loads predicted by the model were in good agreement with the ones obtained experimentally, validating the use of CZM as a design tool for this type of joint.
The parametric study showed that the joint strength increases significantly with L o and d in , with a limit on L o after which the strength stops increasing. A damage analysis showed that the initial increase of the failure load (P f ) with L o is due to the joint reaching P f when the entre adhesive overlap has initiated damage. Above a certain limit on L o , the adhesive layer starts failing before damage has initiated in the entire overlap, preventing P f from increasing further with L o . The adherends thickness (t in and t out ) and taper lengths (l in and l out ) have an influence on the uniformity of the shear stress distribution along the adhesive overlap, which in turn influences P f for high overlap lengths. Furthermore, there are lower limits on t in and t out , under which the applied load can cause the adherends to fail before the adhesive layer. In addition to their influence of the joint strength, these six parameter also have a significant influence on the total weight, which make them important variables to consider in a future optimization process. On the other hand, the adhesive thickness t adh has only a marginal influence on the joint strength and weight, and the extra computational time required when varying this parameter makes the choice of a fixed value preferable.

Funding
This research is funded by the Faculty of Aerospace Engineering of the Delft University of Technology.