Engineering Failure Analysis 112 (2020) 104530
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Engineering Failure Analysis
journal homepage: www.elsevier.com/locate/engfailanal
Fatigue failure analysis and multi-objective optimisation for the
hybrid (bolted/bonded) connection of magnesium–aluminium
alloy assembled wheel
T
Dengfeng Wang, Wenchao Xu
⁎
State Key Laboratory of Automotive Simulation and Control, Jilin University, No 5988, Renmin Road, Changchun, Jilin 130022, People’s Republic of
China
ARTICLE INFO
ABSTRACT
Keywords:
Bolted/bonded assembled wheel
Fatigue failure analysis
Multi-objective optimisation
Grey relational analysis
Hybrid weighting method
This study conducted fatigue failure analysis and multi-objective optimisation for bolted/bonded
magnesium–aluminium alloy assembled wheel. Firstly, the dynamic bending fatigue experiment
of the bolted assembled wheel was conducted, and then the validity of the simulation model was
verified. Secondly, the parameters of the constitutive model and the failure criterion of the
structural adhesive were respectively obtained through tensile and shear tests. Finally, the multiobjective optimisation based on grey relational analysis coupled with the hybrid weighting
method for bolted/bonded assembled wheel was conducted with optimisation objectives (the
fatigue life of spoke and connecting bolts and tensile and shear failure indices of structural adhesive) and design variables (the type of adhesive, thickness of the adhesive layer, pretension
force of bolts and the diameter of bolt holes). Optimisation results show that the fatigue life of
spoke and the connecting bolts reached 529,000 and 208,000 cycles, respectively. Meanwhile,
the adhesive had high reliability.
1. Introduction
Lightweight alloy wheels improve vehicle driving stability [1] and fuel economy. Researchers have designed an assembled wheel,
in which the aluminium alloy spoke and magnesium alloy rim are connected with bolts. The assembled wheel exhibited outstanding
radial fatigue and impact performance [2]. However, the connecting bolts fractured during the dynamic bending fatigue life test.
Therefore, the reliability of the bolt connection must be further improved.
Bolt joints are most commonly used in the connection of dissimilar materials. Many scholars have studied the factors affecting the
fatigue strength of bolt joints. The bolt clamping force improved the fatigue life of the bolt joint [3–5]. Further research by EsmaeiliGoldarag et al. [6] demonstrated that the clamping force of bolt joints changes on the basis of the load environment. Abazadeh et al.
[7–8] investigated the effects of bolt interference fit and cyclic load range on the fatigue life of aluminium alloy double-lap bolt joints.
Subsequently, the composite hybrid connection method of bolt and adhesive emerged. This method achieved continuous transmission
and distribution of loads and reduced the load transmitted by bolts, thereby decreasing the concentrated stress on the bolts and
improving their fatigue life. Hoang-Ngoc and Paroissien [9] established a simulation model of the composite hybrid connection joint
with bolt and flexible adhesive. Numerical analysis showed that the fatigue life of the composite joint is higher than that of the bolt
joint. Matsuzaki et al. [10] revealed that the fatigue strength of co-cured adhesive and bolt joint is higher than that of bolt joint or co-
⁎
Corresponding author.
E-mail address: xuwc17@mails.jlu.edu.cn (W. Xu).
https://doi.org/10.1016/j.engfailanal.2020.104530
Received 20 November 2019; Received in revised form 12 March 2020; Accepted 2 April 2020
Available online 10 April 2020
1350-6307/ © 2020 Elsevier Ltd. All rights reserved.
Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
cured adhesive joint separately. In the composite hybrid connection structure of bolt and adhesive, the thickness of the adhesive layer
[11–13], the type of adhesive [14–16] and the degree of adhesion between the adhesive layer and the connected sheet [17–18]
influence the fatigue strength of the hybrid (bolted/bonded) composite joint. In the hybrid joint, the load transfer ratio of the
adhesive and the bolts affects the stress distribution and the fatigue strength of the joint [19–23]. When the hybrid joint is in a
complex and variable load environment, the load transfer ratio of the adhesive and the bolts changes at any time. Thus, effectively
predicting the load form of the joint is difficult. Correlating the design of the structural and mechanical parameters of the bolt and
adhesive and improving the fatigue strength of the hybrid joint corresponding to different load environments are urgently needed for
the design of the composite connection of bolt and adhesive.
Grey relational analysis (GRA), which was proposed by Deng in the 1980s [24], has been recently used to solve multi-objective
and discrete data problems and mine valuable information from limited data [25,26]. Wang et al. [27] integrated the experiment
method design with GRA to solve multi-criteria decision-making problems and found that design of experiment–GRA is a simple,
robust and practical method with low sensitivity to weight changes. Xiong et al. [28] adopted the Taguchi and GRA methods to
optimise the structural parameters of novel foam-filled elliptical columns under multiple oblique impact loading. Renani and Mirsalehi [29] combined the Taguchi method with GRA to optimise the welding parameters based on the microstructural and mechanical
properties of the welding joint. Sameer and Birru [30] used the technique for ordering preferences by similarity to ideal solution and
GRA approaches to determine the optimum set of input parameters for dissimilar friction stir-welded joints.
The GRA method adopts an equal weighting strategy when dealing with multi-objective problems. Many scholars recently
combined GRA with multi-objective weight determination methods considering the weight difference in multi-objectives due to the
unsuitability of the equal weighting strategy. Cai and Wang [31] optimised the structural parameters of an S-beam using the GRA and
entropy weighting methods. This approach efficiently improved the crashworthiness and reduced the computational cost during the
design process of S-rail compared with the NSGA-II, MOPSO and ASA methods. Xiong et al. [32] optimised the body-in-white side
structure through the GRA and principal component analysis (PCA) methods and realised the light weight of the body structure whilst
ensuring the mechanical properties of the body. Xiong et al. [33] used the entropy weighting and GRA methods to optimise the
material and size parameters of the front end of the body-in-white structure. However, the single weight determination method often
exhibits the disadvantage of incomplete information during multi-index weight evaluation. The hybrid weight determination method
combined with multiple weight determination methods can produce effective weight results. Wu et al. [34] used the GRA and hybrid
entropy-based weighting methods to evaluate coal-fired power units comprehensively and verified the effectiveness of the method.
Wang et al. [35] proposed a hybrid interval analytic hierarchy process (AHP) entropy weighting method for electricity user evaluation and illustrated the effectiveness and advantages of the proposed method through numerical case studies.
However, few studies have addressed metal fatigue damage and tensile and shear failure analyses of adhesive and material–structure–process integration multi-objective optimisation problems using the GRA method. Firstly, the fatigue life of the wheel
structure and connecting bolts and the tensile and shear failures of the structural adhesive are determined as the research objectives
of this paper. The GRA coupled with the hybrid weighting method is then employed to select the optimal material, structure and
process parameters of the assembled wheel. Finally, the comparison results indicate that the hybrid weighting method is more
suitable than the PCA and AHP methods for the multi-objective optimisation design of the assembled wheel.
2. Fatigue failure analysis of bolted assembled wheel
2.1. Bending static analysis
The 16 in. assembled wheel comprises a 6061 aluminium alloy spoke and a ZK61M magnesium alloy rim. The spoke and rim are
connected by 20 M6 cylindrical hexagon socket head bolts distributed along the circumference of the wheel geometric rotation
centre. Bolt materials include SUS304 stainless steel or 45 medium carbon steel. The diameter of the bolt holes is 6.6 mm. The main
geometric dimensions of the spoke and rim are shown in Fig. S1. The material properties of the assembled wheel are shown in Table
S1.
The finite element (FE) model of the bolted assembled wheel was established as shown in Fig. S2. The spoke was divided by
tetrahedron elements, whereas the remaining part was divided by hexahedron elements. The total number of elements and nodes in
the assembled wheel model is 1,294,034 and 849,379, respectively. A pretension force of 7900 N was set to each connecting bolt in
the FE model of the wheel based on the recommended value of the China automotive industry standard. Surface-to-surface contact
with a friction coefficient of 0.2 was established between the nut and rim and between the nut and spoke, whilst that with a friction
coefficient of 0.05 was established between the spoke and the rim.
Based on the test method for passenger car wheels (ISO 3006:2015), the simulation model of the bolted assembled wheel under
bending static condition was established, as shown in Fig. S3. The spoke was installed on the test axle divided by tetrahedron
elements with 5 RBE2 elements. The rim edge opposite to the spoke is constrained in all degrees of freedom. The centre of the end of
the test axle was applied with a loading force of 3305 N corresponding to actual experiment parameters.
2.2. Basic parameters of bending fatigue test
The nominal stress method (S–N method) is used for the fatigue life calculation of the assembled wheel. The S–N curve of the
wheel material describes the relationship between the constant amplitude cyclic stress and the corresponding fatigue life (stress cycle
times), as shown in Eq. (1) [36]. This curve can be obtained by the fatigue test of standard metal specimens.
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D. Wang and W. Xu
(a)
(b)
400
Expermental data for 6061
Expermental data for ZK61M
Fitting curve for 6061
Fitting curve for ZK61M
Stress Amplitude/MPa
350
300
250
200
150
100
50
1E+01 1E+02 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08
Reversals to Failure/
Fig. 1. Material fatigue test: (a) fatigue test machine; (b) S–N curve of the materials of the spoke and rim.
a
=
f
(2Nf ) b
(1)
where σa is the stress amplitude, σf‘ is the fatigue strength coefficient, b is the fatigue strength index and Nf is the number of load
cycles.
Standard metal samples of ZK61M magnesium alloy and 6061 aluminium alloy cut from the assembled wheel were prepared and
tested on an electro-hydraulic servo fatigue tester (EHF–EV series machine by Shimadzu Corporation, Japan, with a maximum range
of 100 kN and dynamic error of < 3%) based on the ISO 1099:2017 standard. Such testing was conducted to determine the life of the
specimens at different stress levels (Fig. 1(a)) and investigate the fatigue life of the assembled wheel accurately.
Statistical calculations were conducted on the data obtained from the experiment, and the relationship between the applied stress
level and the fatigue life of the samples was fitted to obtain the S–N curve of the materials of the spoke and rim, as shown in Fig. 1(b).
The S–N curves of SUS304 and 45 steel materials of the bolt were estimated on the basis of their tensile strength, yield strength,
elastic modulus and Poisson ratio using fatigue analysis software.
The dynamic bending fatigue test bench is shown in Fig. 2(a). The testing instrument was a passenger car wheel bending fatigue
test machine (CFT-3 type machine, Tianjin Jiurong Wheel Technology Co., Ltd, China). The bending torque accuracy was ± 1.5% full
scale, and the loading point offset accuracy was ± 1.0% full scale. The load curve of the bending fatigue test of the assembled wheel
is a sinusoidal function with a maximum value of 3305 N as calculated in Section 2.2, and the frequency value is 13.67 Hz as shown in
Fig. 2(b).
2.3. Fatigue analysis and verification of bolted wheel
In MSC Fatigue software, the S-N method was used for the fatigue life prediction of the bolted assembled wheel based on
Palmgren–Miner linear fatigue damage accumulation theory (described as Eq. (2)). This prediction also utilised the loading of the
bending fatigue analysis load curve and the S–N curve of the wheel materials and the bending static analysis result of the assembled
wheel under unit force. The Von Mises criterion was selected for the solution, and 50% was set as the survival rate corresponding to
Fig. 2. Bending fatigue test: (a) bending fatigue test bench; (b) loading curve of the bending fatigue test.
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Fig. 3. Simulation results and experimental comparison of fatigue life for rim: (a) simulation; (b) experiment.
the actual experimental parameters. Simulation results and experimental comparison of fatigue life for rim are shown in Fig. 3. The
simulation results and experimental comparison of fatigue damage for spoke are shown in Fig. 4, whereas those for the bolts of
SUS304 and 45 steel materials are presented in Fig. 5. The simulation results and experimental comparison of fracture location of the
SUS304 steel bolt are shown in Fig. 6. A scanning electron microscope (ZEISS EVO18 type machine, Carl Zeiss AG, Germany) was
used to observe the micro-morphology of the fracture surface of the bolt; the resolution can reach 3 nm at its maximum acceleration
voltage of 30 kV. The relative error between the simulation and experimental fatigue life value of the wheel and connecting bolts is
shown in Fig. 7.
m
D=
i=1
ni
=1
Ni
(2)
where m is the stress level series of the load amplitude, ni is the number of i-th load cycles, Ni is the fatigue life under the i-th load and
D is the total damage of the structure. The Miner criterion states that the structure undergoes fatigue damage when the sum of the
damage reaches 1.
Fig. 3 reveals that the test fatigue life of the rim was more than 800,000 cycles, which was consistent with the simulated value of
1020 cycles (close to infinite life). This finding indicates that the rim has a low probability of fatigue failure. Fig. 4 shows the initial
fatigue damage at the spoke in the wheel structure, and the damages of the spoke are located at its root based on the simulation and
experiment, as respectively shown in Fig. 4(a) and (b). Fig. 7 shows that the relative error of the fatigue life value of the spoke
obtained by simulation and experiment is 5.6%. Fig. 5 shows that the damage positions of the bolts of SUS304 and 45 steel materials
are both at the surface of the bolt rod near its head, as respectively obtained by the simulation (Fig. 5(a) and (b)) and the experiment
(Fig. 5(c) and (d)). The simulation and experimental fracture locations are comprehensively presented in Fig. 6 with a fractured
SUS304 steel bolt as an example. Fig. 6(a) reveals that the crack initiation was the circle edge of the bolt rod, which was consistent
with the simulation results in Fig. 6(b). The crack propagation type was model I (tensile open crack propagation), and the crack
propagation directions were marked on locations A1, A2, A3 and A4 of the fracture section, as shown in Fig. 6(a). Fig. 7 indicates that
the relative errors of the fatigue life values of the SUS304 and 45 steel bolts between the simulation and experiment are 5.4% and
Fig. 4. Simulation results and experimental comparison of fatigue damage for spoke: (a) simulation; (b) experiment.
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D. Wang and W. Xu
Fig. 5. Simulation results and experimental comparison of fatigue damage for connecting bolts: (a) simulation results for SUS304 steel bolts; (b)
simulation results for 45 steel bolts; (c) experimental results for SUS304 steel bolts; (c) experimental results for 45 steel bolts.
5.5%, respectively. Simulation and experimental results show that the simulated fatigue life of the wheel and the connecting bolts is
highly reliable.
Further analysis of Fig. 6 indicates that the stress of the bolt rod edge is concentrated, resulting in fatigue fracture of the bolt rod.
Reducing the normal and shear stresses in the vertical direction and in the plane, respectively, is necessary to improve the fatigue life
of the bolt. Thus, the hybrid bolted/bonded composite connection was proposed to connect the magnesium–aluminium alloy assembled wheel.
3. Constitutive model of structural adhesive
3.1. Constitutive and failure model of adhesive
The constitutive model of a structural adhesive includes an elasto-plastic model [37], cohesive zone model (CZM) [38] and
Gurson model [39]. Amongst these models, CZM and Gurson have additional parameters and complex calibration. The elasto-plastic
material model can accurately simulate the mechanical behaviour of adhesive at a certain strain rate, and the acquisition of the model
parameters is relatively simple. Thus, the elasto-plastic model was used as the constitutive model of the adhesive material as described in Eq. (3).
p
p
y ( eff , eff )
=
p
s
y ( eff )
+ SIGY × (
p
eff 1/ p
)
(3)
C
where
is the equivalent plastic strain,
is the strain rate, y (
is the dynamic yield stress,
is the static stress, SIGY is
the initial yield stress and C and P are Cowper–Symonds multipliers.
The classical secondary stress criterion [40] was used to determine the failure of the adhesive layer as shown in Eq. (4). The
tensile failure index λt and the shear failure index λs of the adhesive layer were defined as the evaluation indices of the adhesive layer
failure, as respectively shown in Eqs. (5) and (6). When the sum of λt and λs was more than 1, the adhesive layer failed; otherwise, the
adhesive layer was safe. Thus, the value of λt and λs was small, and the adhesive layer was less likely to fail. Moreover, the effective
p
eff
p
eff
p
p
eff , eff )
5
p
s
y ( eff )
Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
Fig. 6. Simulation results and experimental comparison of fracture location of SUS304 steel bolt: (a) experimental results; (b) simulation results.
Fatigue life/104 Cycles
24
Simulation
Experiment
25
20
Relative Error
20
16
15
12
10
8
5
4
0
Relative Error/%
30
28
Spoke
SUS304 bolt
45 steel bolt
0
Fig. 7. Relative error between the simulation and experimental fatigue life value of the wheel and connecting bolts.
service time of the adhesive layer without failure can be approximated as the fatigue life of the adhesive layer under the premise of
not considering the interface failure of the adhesive layer caused by factors, such as hot and humid environments [41]. Thus, λt and λs
can be approximately used as indicators to evaluate fatigue failure of the adhesive layer.
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(a)
(b)
50
Tensile stress-strain curve of Epoxy A
Tensile stress-strain curve of Acrylate B
40
Tensile stress-strain curve of Acrylate C
Stress/MPa
Tensile stress-strain curve of Epoxy D
30
115
80
33
25
1.25
20
25
6
10
0
0.0
0.2
0.4
0.6
0.8
1.0
Strain/
Fig. 8. Tensile test of the dumbbell adhesive specimens: (a) test machine; (b) true stress–strain curves of the four types of structural adhesives.
max( X , Y ,
Tmax
t
=
s
=
Z)
max( X , Y ,
Tmax
max(
2
+
Z)
max(
XY , YZ , ZX )
Smax
2
=1
(4)
2
XY , YZ , ZX )
(5)
2
(6)
Smax
where σX, σY and σZ represent the tensile stress of the adhesive in the X, Y and Z directions of space coordinate system, respectively.
Tmax represents the tensile strength of the adhesive. ΤXY, τYZ and τZX represent the shear stress of the adhesive in the X, Y and Z
directions of space coordinate system, respectively. Smax represents the shear strength of the adhesive.
3.2. Tensile and shear experiment of adhesive
Four types of structural adhesive, namely, epoxy A with high toughness, B and C for acrylate and epoxy D with small toughness
but high tensile strength, were prepared to bond the aluminium alloy spoke and magnesium alloy rim. Following the tensile property
measurement method of plastics (ISO 527-1:2019), the adhesive was sampled into specific layers of uniform thickness, cured at room
temperature for 72 h and then processed into standard dumbbell-shaped samples. The dumbbell specimens were stretched on a
microcomputer-controlled tensile testing machine (TCS-2000 type machine, GOTECH testing machines Co., Ltd, China, with load
accuracy of ± 0.5%) at the speed of 0.5 mm/min, as shown in Fig. 8(a). The true stress–strain curves of the four types of structural
adhesives were obtained as shown in Fig. 8(b).
Based on the tensile lap shear strength test standard of adhesive (ISO 4587:2003), samples of ZK61M magnesium alloy and 6061
aluminium alloy were designed and manufactured to constitute the single-lap joints shown in Fig. S4. These joins were used to obtain
the shear strength of the four types of structural adhesive. The bonding process of the single-lap joints includes the following. The
surface of adherends was firstly treated with mechanical polishing and acetone degreasing and cleaning. An adhesive with a thickness
of 0.2 mm was then evenly spread on the surface, and steel wires or balls with diameter values similar to those of the adhesive
thickness were placed symmetrically on the four sides of the surface. Finally, the adhesive joints were cured in a 23 °C incubator for
72 h. Shear tests of the adhesive joints were conducted on a microcomputer-controlled electronic universal testing machine (WDW100 series machine, Changchun Kexin Test Instrument Co., Ltd, China. Control error of force and velocity were 0.5% and 1%,
respectively) at a speed of 0.5 mm/min, as shown in Fig. 9(a). The experimental shear strength of the four types of structural adhesive
was obtained as shown in Fig. 9(b).
4. Multi-objective optimisation for bolted/bonded connection
The simulation model of the assembled wheel with a hybrid connection was established as shown in Fig. S5. The adhesive layer
was divided by hexahedral elements, the initial thickness of the adhesive layer was 0.2 mm and the constitutive model of the adhesive
was an elasto-plastic model. The adhesive layer was bonded with the spoke and rim using a binding contact. Each of the 20 connecting bolts was loaded with 7900 N pretension force.
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D. Wang and W. Xu
(a)
(b)
30
Upper
fixture
Experimental shear strength
Shear strength/Mpa
25
Shear
specimen
20
F
F
15
14.1
12.8
10
6
5
Lower
fixture
25
0
Epoxy A
Acrylate B
Acrylate C
Epoxy D
Adhesive type
Fig. 9. Shear test of the single-lap adhesive joints: (a) test machine; (b) experimental shear strength of four types of structural adhesives.
4.1. Effects of design variables on wheel performance
(1) Design variables
Current research indicates that the pretension force of bolt and the diameter of the bolt hole have a considerable influence on the
mechanical performance of bolted joint [3–5,42]. The material type and thickness of the adhesive layer have a considerable effect on
the tensile and shear properties of the adhesive joint [11–16]. Thus, the material type of adhesive, the thickness of the adhesive layer,
the pretension force of the bolts and the diameter of the bolt holes were selected as the design variables. The parametric model of the
assembled wheel with a hybrid connection was established on the basis of mesh deformation technology with DEP MeshWork
software. The single bolted/bonded joint extracted from the assembled wheel is shown in Fig. 10. The initial and range values for the
design variables are shown in Table 1. Adhesives 1, 2, 3 and 4 represent epoxy A, acrylate B, acrylate C and epoxy D, respectively.
(2) Optimisation objective
For the bolted/bonded assembled wheel, SUS304 steel bolts fractured after 61,400 cycles, fatigue cracks emerged on the spoke
after 250,000 cycles and the fatigue life of rim was infinite; the structural adhesive tended to fracture easily with the characteristic of
low strength, sensitivity to alternating load [43] and various failure modes [44]. Therefore, the safety of the bolted/bonded assembled wheel depends on the fatigue and failure behaviour of the spoke, connecting bolts and structural adhesive.
Connecting bolts and the adhesive layer jointly bear the shear and tensile stresses of the assembled wheel under bending fatigue
condition. A zero-sum relationship exists between the fatigue life of the bolts and the tensile and shear failure indices of the adhesive.
Meanwhile, the stress state of the adhesive and bolts will directly affect the fatigue life of the spoke. Thus, the fatigue life of the spoke
and connecting bolts and the tensile and shear failure indices of adhesive were determined as the optimisation objectives. The
established optimisation function can be expressed as follows:
Pretightening force
x3
Material type
x2
Adhesive
x4
Diameter
Spoke
x1
Thickness
Rim
Bolt
Fig. 10. Single bolted-bonded joint structure.
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Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
Table 1
Initial and range values of design variables.
Design variables
Description
Initial value
Range value
x1
x2
x3
x4
Adhesive layer thickness
Adhesive material type
Pretension force of bolt
Diameter of bolt hole
0.2 mm
4
7900 N
6.6 mm
0.1 mm ~ 1.0 mm
1,2,3,4
7500 N ~ 8300 N
6.2 mm ~ 7.0 mm
find DV = (DV 1, DV 2, DV 3, DV 4)T
min { Nb (x ), Ns (x ),
s. t . DVL DV DVU
t (x ),
s (x )}
(7)
where Nb(x) and Ns(x) are the bending fatigue life of the connecting bolts and the spoke (104 cycles), respectively; DVL and DVU are
the lower and upper limits of the design variables, respectively.
(3) Main effect and contribution analysis
The optimal Latin square sampling method was used to sample the design space of variables uniformly and randomly, and 40 sets
of samples were obtained as shown in Table S2. Then, the bending static and fatigue life simulations were conducted on the basis of
the parametric model, and the values of the design variables corresponding to 40 sets of samples were assigned. The values of the
target responses obtained by the simulation are shown in Table S2. The main effect and contribution analysis of the design variables
on the target responses were conducted, and the results are shown in Fig. S6.
Fig. S6 (a) indicates that amongst the design variables, the effect of the diameter of the bolt holes on the value of λs was the
highest, with contribution reaching 50.8%. The effect of the thickness of the adhesive layer is the second, with contributions reaching
23.9%. The value of λs increases with the thickness of the adhesive layer and the diameter of the bolt holes. Fig. S6 (b) shows that the
effect of the thickness of the adhesive layer on the value of λt is predominant, and its contribution reaches 90%, whilst the effect of
other factors is considerably small. The value of λt initially decreases and then increases with the thickness of the adhesive layer. Fig.
S6 (c) shows that the effect of the four design variables on the value of Nb is relatively balanced, and the contributions are all
approximately 25%. The value of Nb initially decreases and then increases with the thickness of the adhesive layer, the pretension
force of the bolt and the diameter of the bolt holes, and increases as the material type of the adhesive layer increases from 1 to 4. Fig.
S6 (d) shows that the effect of the adhesive layer material type on the value of Ns is predominant, the contribution reaches 71% and
the value of Ns initially decreases and then increases as the material type of adhesive layer increases from 1 to 4.
Fig. S6 indicates that the effects of the design variables on multiple target responses are inconsistent, and decision making is
difficult because of the lack of a unified judgement basis. Therefore, the optimisation method that combines the GRA with the hybrid
weight determination method is proposed. This method is used to convert multiple responses into a single response called grey
relational grade (GRG), which represents the relative closeness of a candidate to the ideal referential alternative and demonstrates the
‘larger-the-better’ characteristic, thereby facilitating multi-objective decision-making.
However, the multi-objective optimisation method implemented through multi-objective algorithms or surrogate models coupled
with multi-objective algorithms can also be used to resolve the inconsistent or contradictory relationship between multiple responses
in the connection of dissimilar materials [45,46]. Since adopting the multi-objective algorithm approach requires a large number of
computational resources to search for the optimal solution, and accurately establishing a surrogate model needs numerous samples to
address highly nonlinear responses of the fatigue life of the spoke and connecting bolts. The two methods are impractical for the
optimisation of wheel connection. The proposed GRA coupled with the hybrid weight determination method primarily mines the
global space information by dealing with discrete data, and the optimal solution can be obtained using limited experimental data.
Hence, the proposed method simultaneously considers computational accuracy, and computational efficiency can be used to solve the
multi-objective optimisation problem of wheel connection.
4.2. GRA coupled with a hybrid weighting method
Based on the previous optimal Latin hypercube orthogonal experiment results, the grey relational coefficient (GRC) of each
sample was obtained through GRA, combined with the hybrid weighting method comprising the PCA weighting and AHP weighting
methods. The GRG of each sample was calculated, and then the optimal combination of design parameters was selected by factor
analysis based on GRG. The specific steps are as follows.
Step 1: GRA
According to GRA, the orthogonal experiment calculation results must be normalised to dimensionless data between 0 and 1,
which is also called ‘grey relational generation’. GRA has three normalisation methods based on the different characteristics of the
target responses. If the target response has the characteristic ‘the higher, the better’, then the corresponding normalisation method
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Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
can be expressed as follows:
x i (k ) mink x i (k )
max k x i (k ) mink xi (k )
xi =
(8)
If a specific value is ideal for the target response, then the corresponding normalisation method can be expressed as follows:
| x i (k ) T |
max[max k x i (k ) T , T mink x i (k )]
x i (k ) = 1
(9)
If the target response has the characteristic ‘the lower, the better’, then the corresponding normalisation method can be expressed
as follows:
max k x i (k ) x i (k )
max k x i (k ) mink xi (k )
xi =
(10)
where xi*(k) denotes the kth response in the ith experiment after normalisation; xi(k) represents the initial sequence of the experiment results; minkxi(k) and maxkxi(k) respectively represent the minimum and maximum values of response parameters xi(k);
i = 1,2,…, m; k = 1,2,…, n; m denotes the number of experiments; n denotes the number of response parameters; and T is the specific
value. After grey relational generation, the corresponding GRC can be obtained as follows based on the normalised data.
min
(x 0 (k ), x i (k )) =
+
+
max
0i (k )
(11)
max
where xo*(k) denotes reference experiment sequence; xi*(k) denotes the initial experiment sequence; Δoi(k) = ∣xo*(k) − xi*(k)∣
denotes the absolute difference between xo*(k) and xi*(k); Δmax = maximaxkΔoi(k) and Δmin = miniminkΔoi(k) respectively denote the
maximum and minimum values of Δoi(k). ζ is the distinguishing coefficient, ζ ∈ [0, 1], which is generally defined as 0.5. The GRG can
be obtained as follows:
n
(x 0 , x i ) =
k=1
n
Wk (x 0 (k ), x i (k ))
k=1
Wk = 1
(12)
where Wk is the weight value of the kth response parameter. By calculation, the experiment with the highest GRG value represents the
optimal performance.
The normalised results were calculated according to Eqs. (8) and (10), as shown in Table S3. The GRC was calculated according to
Eq. (11), and Table S3 presents the results.
Step 2: PCA weighting method
The main steps for determining the weights of the response parameters of the wheel connection using the PCA weighting method
were as follows.
Firstly, the GRC in all experiments for each response parameter was normalised by the following:
i
(k ) =
i (k )
¯ (k )
(k )
(13)
where γi*(k) and γi(k) are the normalised and original GRC for the kth response parameter in the ith experiment, respectively;
1
m
m
¯ (k ) = m i = 1 i (k ) and (k ) = m 1 i = 1 ( i (k )
¯ (k ))2 are the mean and standard deviations of the original GRC for the kth
response parameter in all experiments, respectively.
The normalised GRCs can be converted into a 40 × 4 matrix. The correlation coefficient array of the normalised matrix was then
calculated as follows:
Rkl =
1
Cov ( i (k ),
( i ) (k )
×
i
(l))
( i ) (l )
k = 1, 2,
, n;
l = 1, 2,
,n
(14)
where Cov ( i (k ), i (l)) is the covariance of sequences γi*(k) and γi*(l); ( i ) (k ) and ( i ) (l) are the standard deviations of sequences
γi*(k) and γi*(l), respectively.
The eigenvectors and eigenvalues of the correlation coefficient array were evaluated as follows:
(R
j Im ) Vij
(15)
=0
n
j=1
T
where λj is the eigenvalue, and
j = n , j = 1,2,…, n; Vij = [ak1 ak 2 akn ] is the eigenvector corresponding to the eigenvalue λj.
The eigenvalues, which were aligned in descending order considering the variance, are presented in Table S4 along with explained
variations. The eigenvector corresponding to each eigenvalue is listed in Table S5.
Based on the eigenvectors and eigenvalues, the uncorrelated principal components were formulated as follows:
Ymj =
n
(i )·Vij
i=1 m
(16)
where Ymj is the jth principal component. The principal components were arranged in descending order considering variance. Thus,
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Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
the first principal component Ym1 accounts for most of the variance in the data.
The variance contribution of the first principal component reached 45.02%, as shown in Table S4. Consequently, the squares of
the first principal eigenvectors were defined as the weights (WPCA) of the response parameters Nb, Ns, λs and λt, which were 0.3993,
0.0986, 0.2475 and 0.2546, respectively, as shown in Table S5.
Step 3: AHP weighting method
The main steps for determining the weights of the response parameters of the wheel connection by the AHP weighting method
were as follows.
(1) Construction of the hierarchical structure
The safety of the wheel structure is defined as the target hierarchy. The fatigue damage of spoke, the fatigue fracture of connecting bolts and the tensile and shear failures of structural adhesive were defined as the criterion hierarchies. The schematic of the
hierarchical structure with two hierarchies was obtained as shown in Fig. S7.
(2) Establishment of the judgement matrix
According to the expert scoring method, senior experts in automotive connection technology were invited to compare the relative
importance of the four criteria with respect to the safety of the wheel structure. The measurement scale of 1 to 9 was used to represent
such importance (Table S6), and the pairwise comparison matrix [B] was obtained as follows:
1 5 5/6 6/5
1/5 1 1/6 1/4
6/5 6 1 3
5/6 4 1/3 1
[B ] =
(17)
(3) Consistency check
The consistency check of the judgement matrix is required. The CI was introduced as an indicator to measure the deviation from
the consistency of the judgement matrix:
m
1
max
CI =
m
(18)
where λmax is the largest eigenvalue of the judgement matrix (the eigenvalue [λ] and eigenvector [V] of the judgement matrix satisfy
Eq. (19)), and m is the order of the judgement matrix.
(19)
BV = V
The average random consistency indicator RI was introduced for different orders of judgement matrices. The ratio CR of CI to RI is
defined as the average random consistency ratio:
CR =
CI
RI
(20)
The RI values of the 1–10 order judgement matrix are shown in Table S7. When CR < 0.1, the judgement matrix possesses
satisfactory consistency. Otherwise, the judgement matrix must be corrected.
According to Eqs. (18)–(20), the CR value of the judgement matrix [B] is 0.023, which is less than 0.1. Thus, the judgement matrix
[B] is feasible, consequently passing the consistency test. The eigenvector [V] of the judgement matrix [B] calculated by Eq. (19) was
normalised as shown in Eq. (21) to obtain the weight vector [WAHP] of the response parameters. The weights of responses Nb, Ns, λs
and λt are 0.06, 0.4361, 0.2066 and 0.2973, respectively.
[WAHP ] =
1
N
N
[Vi ]
(21)
i=1
Step 4: Hybrid weights determined by game theory
Game theory was used to reconcile the inconsistent relationships between the weights obtained by the PCA and AHP weighting
methods to improve the scientificality of weight determination. Assuming L types of methods can be used to weight response
parameters to construct a basic weight set w = {w1, w2, …, wL}, the arbitrary linear combination of L number of weight vectors wk is
as follows:
L
T
kw k
w=
(22)
k=1
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Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
W is a possible weight vector based on the basic weight set, and its entirety w
w=
L
T
k w k,
k=1
k
> 0 represents a possible set
of weight vectors. The assembly model of game theory is used to select the most satisfactory weight w* from the possible weights. The
basic idea of this method is to find consistency or compromise between different weights; that is, to minimise the respective deviations between the possible and the basic weights. Therefore, finding the most satisfactory weight vector can be attributed to
optimising the L number of linear combination coefficients αk in Eq. (23) under the optimisation goal of minimising the dispersion of
w from each wk. Thus, the following countermeasure model is derived.
L
T
j· w j
min
j=1
wi
i = 1,2, ,L
(23)
2
According to the differential characteristics of the matrix, the condition for obtaining the optimal first derivative of Eq. (23) is as
follows.
L
T
j·wi· w j
= wi·wT i i = 1,2, ,L
(24)
j=1
The linear combination coefficients (α1,α2,…,αL) are calculated by Eq. (24), and the linear combination coefficients are then
normalised as follows:
k
k
L
k=1
=
(25)
k
Finally, the hybrid weights can be expressed as follows:
L
T
k ·wk
w =
(26)
k=1
This study presents two weight vectors, namely, WPCA and WAHP, corresponding to two linear combination coefficients α1 and α2,
respectively. According to Eqs. (22)–(24), the values of α1 and α2 are 0.46 and 0.54, respectively. Finally, the hybrid weights of the
responses Nb, Ns, λs and λt are 0.2161, 0.2809, 0.2254 and 0.2777, respectively.
The weight values of the target responses under three weighting methods are compared in Fig. 11. Under the PCA weighting
method, the weight value of the fatigue life of connecting bolts occupies the leading role at 0.3993, which is 1.6 times that of the
tensile and shear failure indices of the adhesive. The weight value of the fatigue life of spoke is quite small at only 0.0986. Under the
AHP weighting method, the weight value of the fatigue life of spoke is predominant, reaching 0.4361. The weight value of the fatigue
life of connecting bolts is only 0.06, which is nearly a quarter of that of the tensile and shear failure indices of the adhesive. Owing to
the coupling and contradiction between target responses, the weight value of one of the target responses is substantially large or
small, resulting in deficits in the integral performance of target responses. Under the hybrid weighting method, extreme weight was
smoothed out, and the weight values of target responses achieved good uniformity with the standard deviation of 0.00087. Simultaneously, the main distinguishable characteristics inherited from the PCA and AHP weighting methods were retained. These
characteristics are beneficial for the comprehensive improvement of target responses.
Step 5: Factor effect analysis and optimal results
After acquiring the PCA, AHP and hybrid weights and GRCs, the GRG can be calculated according to Eq. (12). Thus, three series of
GRGs were calculated by multiplying GRCs with their corresponding weights of response parameters obtained by the PCA, AHP and
0.1
0.0
0.4361
0.0986
0.2
PCA
0.2254
0.2777
0.2161
0.2809
0.3
0.0600
0.4
0.2066
0.2973
Weight/
0.5
Shear failure index
Tensile failure index
Fatigue life of connecting bolts
Fatigue life of spoke
0.3993
0.6
0.2475
0.2546
0.7
AHP
Hybrid
Weighting methods/
Fig. 11. Weight values of target responses under three weighting methods.
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Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
0.9
0.8
Grey relational grade
0.8
Grey relational grade
Grey relational grade-PCA
Grey relational grade-AHP
Grey relational grade-Hybrid
0.7
0.6
0.5
Grey relational grade-PCA
Grey relational grade-AHP
Grey relational grade-Hybrid
0.7
0.6
0.5
0.4
0.4
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
1
2
(a)
Grey relational grade-PCA
Grey relational grade-AHP
Grey relational grade-Hybrid
Grey relational grade
Grey relational grade
Grey relational grade-PCA
Grey relational grade-AHP
Grey relational grade-Hybrid
0.8
0.7
0.6
0.5
0.7
0.6
0.5
0.4
0.4
7400
4
(b)
0.9
0.9
0.8
3
Type of adhesive/
Thickness of adhesive/mm
7600
7800
8000
8200
3.1
8400
3.2
3.3
3.4
3.5
Diameter of bolt hole/mm
Pretension force of bolts/N
(c)
(d)
Fig. 12. Effect of design variable levels of the wheel on multi-objective performance: (a) thickness of adhesive layer; (b) material type of adhesive;
(c) pretension force of bolts; (d) diameter of bolt hole.
Table 2
Values of design variables and responses before and after optimisation.
Initial design
GRA–PCA method
GRA–AHP method
GRA–Hybrid method
x1/mm
x2/
x3/N
x4/mm
Ns/×104cycles
λt/
λs/
λt + λs/
Nb/×104cycles
0.200
0.192
0.954
1.000
4
4
2
4
7900.000
7684.615
7643.590
7725.641
3.300
3.469
3.449
3.182
52.90
52.90
26.40
52.90
2.26
5.27
0.36
0.52
0.82
0.68
0.20
0.40
3.08
5.95
0.56
0.92
35.40
1000.00
3.40
20.80
hybrid weighting methods. The results are listed in Table S8.
The main effect analysis was performed on the basis of the three series of GRG. The influence of design variables for the wheel
structure can be observed from the main effect graph (Fig. 12). Under the PCA weighting method, the largest values of the GRGs for
the design variables x1, x2, x3 and x4 can be represented as 0.192 mm, 4, 7684.615 N and 3.469 mm, respectively, as shown in
Fig. 12(a)–(d). Therefore, the optimal combination of design variables for the assembled wheel was set to these values. Similarly, the
optimal combination of the design variables under the AHP and hybrid weighting methods can be obtained from Fig. 12(a)–(d). The
results are listed in Table 2.
4.3. Results and discussion
The values of responses corresponding to three series of optimal combinations of design variables are shown in Table 2. The
values of responses before optimisation are listed in Table 2 to compare the performance of the wheel before and after optimisation.
Table 2 shows that the fatigue life of spoke and connecting bolts respectively reached 529,000 and 354,000 cycles before optimisation, meeting the requirements of the ISO standards. The value of λt is 2.26, indicating the occurrence of tensile failure of the
adhesive layer. The value of λs is 0.82, suggesting that the possibility of shear failure of the adhesive layer is relatively high. Overall,
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Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
Fig. 13. Optimisation results with the hybrid weighting method: (a) fatigue cloud diagram of the spoke; (b) fatigue cloud diagram of the connecting
bolts; (c) tensile (in the Z direction) stress cloud diagram of the adhesive layer; (d) shear (in the YZ direction) stress cloud diagram of the adhesive
layer.
the fatigue life of the spoke and connecting bolts is guaranteed before optimisation, but the safety of the adhesive layer is low. Thus,
the safety of the entire wheel structure is unsatisfactory.
Under the PCA weighting method, the fatigue life of spoke reached 529,000 cycles, whilst that of connecting bolts was close to
infinity. This finding demonstrates the excellent fatigue resistance of the wheel structure. However, the tensile failure of the adhesive
layer occurred, and the value of λt reached 5.27, which is remarkably high. The value of λs is 0.68, and shear failure did not occur;
however, the value remained high. Thus, the safety of the wheel structure cannot be guaranteed due to the failure of the adhesive
layer under the PCA weighting method.
Under the AHP weighting method, the values of λs and λt of the adhesive layer are 0.36 and 0.20, respectively, and the sum is only
0.56. Hence, the probability of tensile and shear failures of the adhesive layer is low. The fatigue life of the spoke reached 264,000
cycles, but the connecting bolts fractured after 34,000 cycles. Thus, the safety of the wheel structure cannot be guaranteed because of
the fracture of the connecting bolts under the PCA weighting method.
Under the hybrid weighting method, the fatigue life of the spoke and connecting bolts reached 529,000 and 208,000 cycles,
respectively, far exceeding the requirements of the ISO standards. The fatigue cloud diagram of the spoke and connecting bolts are
respectively shown in Fig. 13(a) and (b). The values of λs and λt of the adhesive layer are 0.52 and 0.40, respectively, and the sum is
only 0.92. This finding indicates that the adhesive layer did not undergo shear and tensile failures. The tensile (in the Z direction) and
shear (in the YZ direction) stress cloud diagram of the adhesive layer are respectively shown in Fig. 13(c) and (d).
In GRA coupled with the weighting method, the weight value corresponding to the target response will directly affect the closeness degree of the target response to its ideal value after optimisation. The superiority of the hybrid weighting method is further
explained by the relationship between the surplus and deficit value of the target responses after optimisation, as shown in Fig. 14. The
fatigue life of connecting bolts shows a positive surplus with a rate of 429%, 164% and 429% under the PCA, AHP and hybrid
weighting methods, respectively. Under the PCA weighting method, the sum of tensile and shear failure indices of adhesive shows a
negative surplus with a rate of 495%, whereas that of connecting bolts shows a positive surplus with a rate of 9900%. Under the AHP
weighting method, the sum of tensile and shear failure indices of adhesive shows a positive deficit with a rate of 44%, whereas that of
connecting bolts shows a negative deficit with a rate of 66%. Under the hybrid weighting method, the sum of the tensile and shear
failure indices of adhesive shows a positive deficit with a rate of 8%, and the fatigue life of connecting bolts shows a positive surplus
with a rate of 108%. The contradiction between these responses has been balanced. Thus, the adhesive does not undergo failure, and
the fatigue life of connecting bolts far exceeds the standard requirements.
Overall, the safety of the entire wheel structure is guaranteed with the proposed GRA coupled with the hybrid weighting method.
The comprehensive performance of the wheel structure is considerably improved after optimisation.
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Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
Fig. 14. Surplus and deficit comparisons of target responses under three weighting methods.
5. Conclusion
(1) The bending fatigue simulation model of the bolted assembled wheel was established, and the simulated fatigue life of the wheel
structure and connecting bolts was obtained on the basis of the S–N method. The high accuracy of the simulation model was
verified by comparing the simulation and experimental results.
(2) An elasto-plastic constitutive model was used to describe the mechanical behaviour of the adhesive, and the secondary stress
criterion method was employed to determine the failure of the adhesive layer. Tensile and shear tests were conducted to obtain
the parameters of the constitutive and failure models of the adhesive. The tensile test of dumbbell specimens of the four types of
adhesive was performed to obtain their stress–strain curves and tensile strengths. The shear test of the single-lap adhesive joints
constructed by aluminium and magnesium alloy sheets was conducted to obtain the shear strengths of the adhesives.
(3) The parametric simulation model of the bolted/bonded assembled wheel was established, and the multi-objective optimisation
was defined. GRA coupled with the hybrid weighting method was employed to convert multi-objective optimisation of the wheel
into single-objective optimisation. GRG was the ultimate goal of decision-making. Therefore, the optimal combination of design
variables was obtained through the design variable level–GRG curves. Furthermore, the superiority of the proposed hybrid
weighting method to the PCA and AHP methods was verified. The optimisation results showed that the fatigue life of the spoke
and connecting bolts reached 529 and 208 thousand cycles, respectively. The sum of the tensile and shear failure indices of the
adhesive is less than 1, indicating the high reliability of the adhesive.
Declaration of Competing Interest
None.
Acknowledgments
This work was supported by the Jilin Province and Jilin University Jointly Sponsor Special Foundation under Grant SXGJSF20172-1-5; National Natural Science Foundation of China under Grant 51475201. The authors would like to express their appreciations for
the fund supports.
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfailanal.2020.104530.
References
[1] A. Hirano, Study on Wheel Stiffness Considering Balance between Driving Stability and Weight, SAE Int. J. Commer. Veh. 8 (2015) 205–212, https://doi.org/10.
4271/2015-01-1755.
[2] D. Wang, S. Zhang, W. Xu, Multi-objective optimization design of wheel based on the performance of 13 degrees and 90 degrees impact tests, Int. J.
Crashworthiness. 24 (2019) 336–361, https://doi.org/10.1080/13588265.2018.1451229.
[3] F. Esmaeili, T.N. Chakherlou, M. Zehsaz, Prediction of fatigue life in aircraft double lap bolted joints using several multiaxial fatigue criteria, Mater. Des. 59
(2014) 430–438, https://doi.org/10.1016/j.matdes.2014.03.019.
[4] T.N. Chakherlou, M.J. Razavi, B. Abazadeh, Finite element investigations of bolt clamping force and friction coefficient effect on the fatigue behavior of
15
Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
aluminum alloy 2024–T3 in double shear lap joint, Eng. Fail. Anal. 29 (2013) 62–74, https://doi.org/10.1016/j.engfailanal.2012.11.004.
[5] T.N. Chakherlou, M.J. Razavi, A.B. Aghdam, B. Abazadeh, An experimental investigation of the bolt clamping force and friction effect on the fatigue behavior of
aluminum alloy 2024–T3 double shear lap joint, Mater. Des. 32 (2011) 4641–4649, https://doi.org/10.1016/j.matdes.2011.04.022.
[6] F. Esmaeili-Goldarag, A. Babaei, H. Jafarzadeh, An experimental and numerical investigation of clamping force variation in simple bolted and hybrid (boltedbonded) double lap joints due to applied longitudinal loads, Eng. Fail. Anal. 91 (2018) 327–340, https://doi.org/10.1016/j.engfailanal.2018.04.047.
[7] B. Abazadeh, T.N. Chakherlou, G.H. Farrahi, R.C. Alderliesten, Fatigue life estimation of bolt clamped and interference fitted-bolt clamped double shear lap
joints using multiaxial fatigue criteria, Mater. Des. 43 (2013) 327–336, https://doi.org/10.1016/j.matdes.2012.06.050.
[8] B. Abazadeh, T.N. Chakherlou, R.C. Alderliesten, Effect of interference fitting and/or bolt clamping on the fatigue behavior of Al alloy 2024–T3 double shear lap
joints in different cyclic load ranges, Int. J. Mech. Sci. 72 (2013) 2–12, https://doi.org/10.1016/j.ijmecsci.2013.03.001.
[9] C.-T. Hoang-Ngoc, E. Paroissien, Simulation of single-lap bonded and hybrid (bolted/bonded) joints with flexible adhesive, Int. J. Adhes. Adhes. 30 (2010)
117–129, https://doi.org/10.1016/j.ijadhadh.2009.12.002.
[10] R. Matsuzaki, M. Shibata, A. Todoroki, Improving performance of GFRP/aluminum single lap joints using bolted/co-cured hybrid method, Compos. Part aApplied Sci. Manuf. 39 (2008) 154–163, https://doi.org/10.1016/j.compositesa.2007.11.009.
[11] R.D.S.G. Campilho, D.C. Moura, M.D. Banea, L.F.M. da Silva, Adhesive thickness effects of a ductile adhesive by optical measurement techniques, Int. J. Adhes.
Adhes. 57 (2015) 125–132, https://doi.org/10.1016/j.ijadhadh.2014.12.004.
[12] K. Machalicka, M. Eliasova, Adhesive joints in glass structures: effects of various materials in the connection, thickness of the adhesive layer, and ageing, Int. J.
Adhes. Adhes. 72 (2017) 10–22, https://doi.org/10.1016/j.ijadhadh.2016.09.007.
[13] K. Naito, M. Onta, Y. Kogo, The effect of adhesive thickness on tensile and shear strength of polyimide adhesive, Int. J. Adhes. Adhes. 36 (2012) 77–85, https://
doi.org/10.1016/j.ijadhadh.2012.03.007.
[14] N.R.E. Domingues, R.D.S.G. Campilho, R.J.C. Carbas, L.F.M. da Silva, Experimental and numerical failure analysis of aluminium/composite single-L joints, Int. J.
Adhes. Adhes. 64 (2016) 86–96, https://doi.org/10.1016/j.ijadhadh.2015.10.011.
[15] E. Paroissien, F. Lachaud, S. Schwartz, A. Da Veiga, P. Barriere, Simplified stress analysis of hybrid (bolted/bonded) joints, Int. J. Adhes. Adhes. 77 (2017)
183–197, https://doi.org/10.1016/j.ijadhadh.2017.05.003.
[16] L. Liao, C. Huang, T. Sawa, Effect of adhesive thickness, adhesive type and scarf angle on the mechanical properties of scarf adhesive joints, Int. J. Solids Struct.
50 (2013) 4333–4340, https://doi.org/10.1016/j.ijsolstr.2013.09.005.
[17] Y. Boutar, S. Naimi, S. Mezlini, R.J.C. Carbas, L.F.M. da Silva, M.B.S. Ali, Fatigue resistance of an aluminium one-component polyurethane adhesive joint for the
automotive industry: Effect of surface roughness and adhesive thickness, Int. J. Adhes. Adhes. 83 (2018) 143–152, https://doi.org/10.1016/j.ijadhadh.2018.02.
012.
[18] S. Correia, V. Anes, L. Reis, Effect of surface treatment on adhesively bonded aluminium-aluminium joints regarding aeronautical structures, Eng. Fail. Anal. 84
(2018) 34–45, https://doi.org/10.1016/j.engfailanal.2017.10.010.
[19] S. Baha II, Numerical Investigation of the Rivet Installation in an Adhesively Bonded Joint and the Load Transfer in a Bolted/Bonded Hybrid Joint, Sae Int. J.
Mater. Manuf. 8 (2015) 45–55, https://doi.org/10.4271/2014-01-2242.
[20] G.-H. Lim, K. Bodjona, K.P. Raju, S. Fielding, V. Romanov, L. Lessard, Evolution of mechanical properties of flexible epoxy adhesives under cyclic loading and its
effects on composite hybrid bolted/bonded joint design, Compos. Struct. 189 (2018) 54–60, https://doi.org/10.1016/j.compstruct.2018.01.049.
[21] K.P. Raju, K. Bodjona, G.-H. Lim, L. Lessard, Improving load sharing in hybrid bonded/bolted composite joints using an interference-fit bolt, Compos. Struct. 149
(2016) 329–338, https://doi.org/10.1016/j.compstruct.2016.04.025.
[22] G. Kelly, Quasi-static strength and fatigue life of hybrid (bonded/bolted) composite single-lap joints, Compos. Struct. 72 (2006) 119–129, https://doi.org/10.
1016/j.compstruct.2004.11.002.
[23] K. Bodjona, L. Lessard, Load sharing in single-lap bonded/bolted composite joints. Part II: Global sensitivity analysis, Compos. Struct. 129 (2015) 276–283,
https://doi.org/10.1016/j.compstruct.2015.03.069.
[24] J. Deng, Control problems of grey systems, Syst. Control. Lett. 1 (1982) 288–294, https://doi.org/10.1016/S0167-6911(82)80025-X.
[25] A.N. Siddiquee, Z.A. Khan, Z. Mallick, Grey relational analysis coupled with principal component analysis for optimisation design of the process parameters in infeed centreless cylindrical grinding, Int. J. Adv. Manuf. Technol. 46 (2010) 983–992, https://doi.org/10.1007/s00170-009-2159-8.
[26] A. Varun, N. Venkaiah, Simultaneous optimization of WEDM responses using grey relational analysis coupled with genetic algorithm while machining EN 353,
Int. J. Adv. Manuf. Technol. 76 (2015) 675–690, https://doi.org/10.1007/s00170-014-6198-4.
[27] P. Wang, P. Meng, J.-Y. Zhai, Z.-Q. Zhu, A hybrid method using experiment design and grey relational analysis for multiple criteria decision making problems,
Knowledge-Based Syst. 53 (2013) 100–107, https://doi.org/10.1016/j.knosys.2013.08.025.
[28] F. Xiong, D. Wang, S. Yin, Optimization analysis of novel foam-filled elliptical columns under multiple oblique impact loading, Mater. Des. 156 (2018) 198–214,
https://doi.org/10.1016/j.matdes.2018.06.057.
[29] H.K. Renani, S.E. Mirsalehi, Optimization of FSW lap joining of pure copper using Taguchi method and grey relational analysis, Mater. Res. Express. 6 (2019),
https://doi.org/10.1088/2053-1591/ab021d.
[30] M.D. Sameer, A.K. Birru, Optimization and characterization of dissimilar friction stir welded DP600 dual phase steel and AA6082-T6 aluminium alloy sheets
using TOPSIS and grey relational analysis, Mater. Res. Express. 6 (2019), https://doi.org/10.1088/2053-1591/aafba4.
[31] K. Cai, D. Wang, Optimizing the design of automotive S-rail using grey relational analysis coupled with grey entropy measurement to improve crashworthiness,
Struct. Multidiscip. Optim. 56 (2017) 1539–1553, https://doi.org/10.1007/s00158-017-1728-y.
[32] F. Xiong, D. Wang, Z. Ma, T. Lv, L. Ji, Lightweight optimization of the front end structure of an automobile body using entropy-based grey relational analysis,
Proc. Inst. Mech. Eng. Part D-Journal Automob. Eng. 233 (2019) 917–934, https://doi.org/10.1177/0954407018755844.
[33] F. Xiong, D. Wang, S. Zhang, K. Cai, S. Wang, F. Lu, Lightweight optimization of the side structure of automobile body using combined grey relational and
principal component analysis, Struct. Multidiscip. Optim. 57 (2018) 441–461, https://doi.org/10.1007/s00158-017-1749-6.
[34] D. Wu, N. Wang, Z. Yang, C. Li, Y. Yang, Comprehensive Evaluation of Coal-Fired Power Units Using Grey Relational Analysis and a Hybrid Entropy-Based
Weighting Method, Entropy. 20 (2018), https://doi.org/10.3390/e20040215.
[35] S. Wang, L. Ge, S. Cai, L. Wu, Hybrid interval AHP-entropy method for electricity user evaluation in smart electricity utilization, J. Mod. Power Syst. Clean
Energy. 6 (2018) 701–711, https://doi.org/10.1007/s40565-017-0355-3.
[36] C.L. Petracconi, S.E. Ferreira, E.S. Palma, Fatigue life simulation of a rear tow hook assembly of a passenger car, Eng. Fail. Anal. 17 (2010) 455–463, https://doi.
org/10.1016/j.engfailanal.2009.09.002.
[37] N. Stein, P.L. Rosendahl, W. Becker, Modelling load transfer and mixed-mode fracture of ductile adhesive composite joints, Int. J. Adhes. Adhes. 82 (2018)
299–310, https://doi.org/10.1016/j.ijadhadh.2018.01.013.
[38] D.L. Alves, R.D.S.G. Campilho, R.D.F. Moreira, F.J.G. Silva, L.F.M. da Silva, Experimental and numerical analysis of hybrid adhesively-bonded scarf joints, Int. J.
Adhes. Adhes. 83 (2018) 87–95, https://doi.org/10.1016/j.ijadhadh.2018.05.011.
[39] M. Imanaka, M. Takagishi, N. Nakamura, M. Kimoto, R-curve characteristics of adhesives modified with high rubber content under mode I loading, Int. J. Adhes.
Adhes. 30 (2010) 478–484, https://doi.org/10.1016/j.ijadhadh.2010.03.005.
[40] M. Lee, E. Yeo, M. Blacklock, M. Janardhana, S. Feih, C.H. Wang, Predicting the strength of adhesively bonded joints of variable thickness using a cohesive
element approach, Int. J. Adhes. Adhes. 58 (2015) 44–52, https://doi.org/10.1016/j.ijadhadh.2015.01.006.
[41] W. Mu, G. Qin, J. Na, W. Tan, H. Liu, J. Luan, Effect of alternating load on the residual strength of environmentally aged adhesively bonded CFRP-aluminum
alloy joints, Compos. Part B-Engineering. 168 (2019) 87–97, https://doi.org/10.1016/j.compositesb.2018.12.070.
16
Engineering Failure Analysis 112 (2020) 104530
D. Wang and W. Xu
[42] F. Liu, M. Shan, L. Zhao, J. Zhang, Probabilistic bolt load distribution analysis of composite single-lap multi-bolt joints considering random bolt-hole clearances
and tightening torques, Compos. Struct. 194 (2018) 12–20, https://doi.org/10.1016/j.compstruct.2018.03.100.
[43] V. Shenoy, I.A. Ashcroft, G.W. CritchloW, A.D. Crocombe, M.M.A. Wahab, Strength wearout of adhesively bonded joints under constant amplitude fatigue, Int. J.
Fatigue. 31 (2009) 820–830, https://doi.org/10.1016/j.ijfatigue.2008.11.007.
[44] N. Choupani, Interfacial mixed-mode fracture characterization of adhesively bonded joints, Int. J. Adhes. Adhes. 28 (2008) 267–282, https://doi.org/10.1016/j.
ijadhadh.2007.08.002.
[45] J.M. Arenas, C. Alia, J.J. Narbon, R. Ocana, C. Gonzalez, Considerations for the industrial application of structural adhesive joints in the aluminium-composite
material bonding, Compos. Part B-Engineering. 44 (2013) 417–423, https://doi.org/10.1016/j.compositesb.2012.04.026.
[46] V. Kradinov, E. Madenci, D.R. Ambur, Application of genetic algorithm for optimum design of bolted composite lap joints, Compos. Struct. 77 (2007) 148–159,
https://doi.org/10.1016/j.compstruct.2005.06.009.
17